text stringlengths 4 2.78M |
|---|
---
author:
- 'Enea G. Bongiorno (enea.bongiorno@unimi.it)'
title: 'A Note on Fuzzy Set–Valued Brownian Motion'
---
Introduction
============
Stochastic (fuzzy) set–valued evolution is a relevant topic that was studied largely by different authors (e.g. [@li:gua07; @li:ogu:kre02; @mol05] and references therein). The following question was stated by Molchanov in [@mol05 Open Problem 1.24, p.316]:
> Define a set–valued analogue of the Wiener process and the corresponding stochastic integral.
In [@li:gua07], the authors tackle the proposed problem defining a fuzzy set–valued Brownian motion in ${\mathbb{F}_{kc}}$, the family of convex fuzzy subsets of ${{\mathbb{R}^d}}$ with compact support. In the sequel we shall prove that such a process is equivalent to consider simply a Wiener process in ${{\mathbb{R}^d}}$. This is based upon the fact that the Brownian motion is a zero–mean Gaussian (fuzzy set–valued) process.\
In fact, it is widely known (cf. [@li:ogu:kre02 Theorem 6.1.7]) that a Gaussian random fuzzy set decomposes according to $$\label{eq:gauss_decomposition}
X=\mathbb{E} X {\oplus}{\mathbb{I}_{\xi }},$$ where $\mathbb{E} X$ is in the Aumann sense, $\xi$ is a Gaussian random element in ${{\mathbb{R}^d}}$ with $\mathbb{E} \xi=0$ and ${\mathbb{I}_{A}}:{{\mathbb{R}^d}}\to\{0,1\}$ denotes the indicator function of any $A\subseteq {{\mathbb{R}^d}}$ $${\mathbb{I}_{A}}(x)=\left\{
\begin{array}{ll}
1, & \textrm{if } x\in A,\\
0, & \textrm{otherwise},
\end{array}
\right.$$ (for the sake of simplicity, whenever $A=\{a\}$ is a singleton we shall write ${\mathbb{I}_{ a }}$ instead of ${\mathbb{I}_{\{a\} }}$). Equation means that $X$ is just its expected value $\mathbb{E} X$ up to a random Gaussian translation $\xi$. In some sense, $\mathbb{E} X$ represents the “deterministic” part of $X$ whilst $\xi$ represents its random part. It is also known (cf. [@mol05 Proposition 1.30, p.161]) that a zero–mean random set is actually a random element in ${{\mathbb{R}^d}}$ with zero–mean. Such a result can be easily extended to the fuzzy case and, jointly to decomposition , implies $$X={\mathbb{I}_{0 }} {\oplus}{\mathbb{I}_{\xi }} = {\mathbb{I}_{\xi }}.$$ Roughly speaking, the definition of Brownian motion in [@li:gua07] for random fuzzy sets drives down the complexity of the chosen (fuzzy) framework. In fact, a Gaussian fuzzy random set with zero–mean is reduced to be a random Gaussian element in ${{\mathbb{R}^d}}$.\
In this paper we shall provide an alternative proof of the last fact using selections.
The paper is organized as follow. Section \[sec: preliminary results\] is devoted to preliminaries such as random (fuzzy) sets, embedding theorems and Brownian motion for fuzzy sets (according to [@li:gua07]). In Section \[sec:Brownian\_is\_singular\] we prove the main result of the paper, whilst in Section \[sec:proof\_Teo\_aumann\_integrall\_null\] we provide a proof to the statement zero–mean random set is a random element in ${{\mathbb{R}^d}}$ with zero–mean.
Preliminaries {#sec: preliminary results}
=============
Here we refer mainly to [@li:ogu:kre02]. Denote by ${\mathbb{K}_{kc}}$ the class of non–empty compact convex subsets of ${{\mathbb{R}^d}}$, endowed with the Hausdorff metric $${\delta_H}(A,B) = \max\{\sup_{a\in A} \inf_{b\in B}\|a-b\| ,
\sup_{b\in B} \inf_{a\in A}\|a-b\|\},$$ and the operations $$A{+}B = \{a+b:a\in A,\ b\in B\},\qquad \lambda\cdot A=\lambda A =\{\lambda a: a\in A\}.$$ A [*fuzzy set*]{} is a map $\nu: {{\mathbb{R}^d}}\to [0,1]$. Let ${\mathbb{F}_{kc}}$ denote the family of all fuzzy sets, which satisfy the following conditions.
1. Each $\nu$ is an upper semicontinuous function, i.e. for each $\alpha\in (0,1]$, the cut set $\nu_\alpha=\{x\in{{\mathbb{R}^d}}: \nu (x)\ge \alpha \}$ is a closed subset of ${{\mathbb{R}^d}}$.
2. The cut set $\nu_1=\{x\in{{\mathbb{R}^d}}: \nu (x)=1\}\neq\emptyset$.
3. The support set $\nu_{0+}=\overline{\{x\in{{\mathbb{R}^d}}: \nu(x)>0\}}$ of $\nu$ is compact; hence every $\nu_\alpha$ is compact for $\alpha\in(0,1]$.
4. For any $\alpha\in [0,1]$, $\nu_\alpha$ is a convex subset of ${{\mathbb{R}^d}}$.
Let us endow ${\mathbb{F}_{kc}}$ with the metric $${\delta_H}^{\infty}(\nu^1, \nu^2) = \sup \{\alpha\in [0,1] : {\delta_H}(\nu^1_\alpha, \nu^2_\alpha)\}.$$ and the operations $$(\nu^1{\oplus}\nu^2)_\alpha = \nu^1_\alpha {+}\nu^2_\alpha,\qquad (\lambda\odot\nu^1)_\alpha = \lambda \cdot\nu^1_\alpha.$$
Let $(\Omega,{\mathfrak{F}},{\mathbb{P}_{}})$ be a complete probability space. A [*fuzzy set–valued random variable*]{} ([[FRV]{}]{}) is a function $X:\Omega \to {\mathbb{F}_{kc}}$, such that $X_\alpha: \omega\mapsto X(\omega)_\alpha$ are random compact convex sets for every $\alpha\in (0,1]$ (i.e. $X_\alpha$ is a ${\mathbb{K}_{kc}}$–valued function measurable with respect to the ${\delta_H}$–Borel $\sigma$–algebra).
An [[FRV]{}]{} $X$ is *integrably bounded* and we shall write $X\in L^1[\Omega,{\mathfrak{F}},{\mu};{\mathbb{F}_{kc}}]=L^1[\Omega;{\mathbb{F}_{kc}}]$, if $\|X_{0+}\|_{H} :={\delta_H}(X_{0+},\{0\})\in L^1[\Omega;\mathbb{R}]$.\
The *expected value* of an [[FRV]{}]{} $X$, denoted by $\mathbb{E}[X]$, is a fuzzy set such that, for every $\alpha\in(0,1]$, $$(\mathbb{E}[X])_\alpha = \left(\int_\Omega X_\alpha {{\rm d}}{\mu}\right) = \{\mathbb{E}(f): f\in L^1[\Omega; {{\mathbb{R}^d}}], f \in X_\alpha \ \mu-\textrm{a.e.}\}.$$
#### Embedding Theorem.
Let ${S^{\, d-1}}$ be the unit sphere in ${{\mathbb{R}^d}}$. For any $\nu\in{\mathbb{F}_{kc}}$ define the *support function* of $\nu$ as follows: $${h_{\nu} (x,\alpha) } =\left\{
\begin{array}{ll}
{h_{\nu_\alpha} (x) } & {\rm if } \ \alpha >0, \\
{h_{\nu_{0^+}} (x) } & {\rm if } \ \alpha =0,
\end{array}
\right.$$ for $(x,\alpha)\in{S^{\, d-1}}\times [0,1]$ and where ${h_{K} (x) }=\sup\{ \langle x,a \rangle : a\in K \}$, for $x\in {S^{\, d-1}}$.\
It is known that support function satisfies the following properties:
1. for any $\nu^1, \nu^2 \in {\mathbb{F}_{kc}}$, ${h_{\nu^1{\oplus}\nu^2} (\cdot,\cdot) }={h_{\nu^1} (\cdot,\cdot) }+{h_{\nu^2} (\cdot,\cdot) }$,
2. for any $(x, \alpha)\in {{\mathbb{R}^d}}\times [0,1]$, ${h_{X(\cdot)} (x,\alpha) }\in L^1[\Omega;\mathbb{R}]$, $\mathbb{E}[{h_{X} (x,\alpha) }] = {h_{\mathbb{E}[X]} (x,\alpha) } $.
Let $C({S^{\, d-1}})$ denote the Banach space of all continuous functions $v$ on ${S^{\, d-1}}$ with respect to the norm $\|v\|_C = \sup_{x\in{S^{\, d-1}}} |v(x)|.$ Let $\overline{C}([0,1], C({S^{\, d-1}}) )$ be the set of all functions $f: [0,1]\to C({S^{\, d-1}})$ such that $f$ is bounded, left continuous with respect to $\alpha\in (0,1]$, right continuous at 0, and $f$ has right limit for any $\alpha\in (0,1)$. Then we have that $\overline{C}([0,1], C({S^{\, d-1}}) )$ is a Banach space with the norm $\|f\|_{\overline{C}} = \sup_{\alpha\in [0,1]} \|f(\alpha) \|_C$, and the following embedding theorem holds.
\[pro:embedding\] There exists a function $j:{\mathbb{F}_{kc}} \to \overline{C}([0,1], C({S^{\, d-1}}) )$ such that:
1. $j$ is an isometric mapping, i.e. $${\delta_H}^\infty (\nu^1, \nu^2) = \| j(\nu^1) - j(\nu^2) \|_{\overline{C}}, \quad \nu^1, \nu^2\in{\mathbb{F}_{kc}},$$
2. $j(r\nu^1 + t\nu^2) = rj(\nu^1) + tj(\nu^2)$, $\nu^1, \nu^2\in{\mathbb{F}_{kc}}$ and $r,t \ge 0$.
3. $j({\mathbb{F}_{kc}})$ is a closed subset in $\overline{C}( [0,1] , C({S^{\, d-1}}) )$.
As a matter of fact, we can define an injection $j:{\mathbb{F}_{kc}} \to \overline{C} ([0,1],C({S^{\, d-1}}))$ by $j(\nu)={h_{\nu} }$, i.e. $j(\nu)(x,\alpha) = {h_{\nu} (x,\alpha) }$ for every $(x,\alpha)\in {S^{\, d-1}}\times [0,1]$, and this mapping $j$ satisfies above theorem. For simplification, let $\overline{\mathbf{C}}:=\overline{C} ([0,1],C({S^{\, d-1}}))$.\
From Proposition \[pro:embedding\] it follows that every [[FRV]{}]{} $X$ can be regarded as a random element of $\overline{\mathbf{C}}$ by considering $j(X)={h_{X} } : \Omega \to \overline{\mathbf{C}}$, where ${h_{X} }(\omega)= {h_{X(\omega)} }$.
#### Fuzzy set–valued Brownian motion.
For the results in this subsection we refer to [@li:gua07] or we shall specify if otherwise.
A [[FRV]{}]{} $X:\Omega \to {\mathbb{F}_{kc}} $ is *Gaussian* if ${h_{X} }$ is a Gaussian random element of $\overline{\mathbf{C}}$.
A random element ${h_{X} }$ taking values in $\overline{\mathbf{C}}$ is Gaussian if and only if, for any $n\in\mathbb{N}$ and $f_1, f_2, \ldots, f_n\in \overline{\mathbf{C}}^*$, the real vector–valued random variable $( f_1({h_{X} }), f_2({h_{X} }), \ldots, f_n({h_{X} }) )$ is Gaussian, where $\overline{\mathbf{C}}^*$ is the conjugate space of $\overline{\mathbf{C}}$ (i.e. the set of all continuous linear functionals on $\overline{\mathbf{C}}$).\
It follows from the properties of ${h_{X} }$ and elements in $\overline{\mathbf{C}}^*$ that $X+Y$ is Gaussian if $X$ and $Y$ are Gaussian [[FRV]{}]{}. Also $\lambda X$ is Gaussian whenever $X$ is Gaussian and $\lambda\in\mathbb{R}$.
\[pro:gaussian\_process\] A [[FRV]{}]{} $X$ is Gaussian if and only if $X$ is representable in the form $$X=\mathbb{E}[X]{\oplus}{\mathbb{I}_{\xi}},$$ where $\xi$ is a Gaussian random element of ${{\mathbb{R}^d}}$ with zero mean.
Assume that $\{{\mathfrak{F}}_t: t\ge 0\}$ is a $\sigma$–filtration satisfying the usual condition (complete and right continuous). $\{X_t: t\ge 0\}$ is called an adaptive fuzzy set–valued stochastic process if for any $t\in\mathbb{R}_+$, $X_t$ is an ${\mathfrak{F}}_t$–measurable [[FRV]{}]{}. An adaptive fuzzy set–valued stochastic process $\{X_t : t\ge 0\}$ is called Gaussian if, for any $t\in\mathbb{R}_+$, $X_t$ is Gaussian.
An adaptive fuzzy set–valued stochastic process $X=\{X_t : t\ge 0\}$ is Gaussian if and only if $\{ ( f_1({h_{X_t} }), \ldots ,f_n ({h_{X_t} }) ) : t\ge 0 \}$ is a real vector–valued Gaussian process, for any $n\in\mathbb{N}$ and $f_1, f_2, \ldots, f_n \in \overline{\mathbf{C}}^*$. Further, the following theorem holds.
\[def:B\_t\] An adaptive fuzzy set–valued stochastic process $\{B_t: t\in\mathbb{R}_+\}$ is called a fuzzy set–valued Brownian motion if and only if $\{{h_{B_t} } : t\in\mathbb{R}_+\}$ is a Brownian motion in $\overline{\mathbf{C}}$.
\[pro:caratterizzazione\_Brownian\] Assume that a fuzzy set–valued stochastic process $\{B_t: t\ge 0\}$ satisfies $B_0={\mathbb{I}_{0}}$. Then $\{B_t : t\ge 0\}$ is a fuzzy set–valued Brownian motion if and only if it is a Gaussian process and
1. $\mathbb{E}[f_i({h_{B_t} })]=0$, for any $t\ge 0$, $f_i\in \overline{\mathbf{C}}^*$, $i=1,\ldots,n$,
2. $\mathbb{E}[f_i({h_{B_t} })f_i({h_{B_s} })]=t\wedge s$, for any $s,t\ge 0$, $f_i\in \overline{\mathbf{C}}^*$, $i=1,\ldots,n$,
3. $\mathbb{E}[f_i({h_{B_t} })f_j({h_{B_s} })]=0$, for any $s,t\ge 0$, $f_i,f_j\in \overline{\mathbf{C}}^*$, $i\neq j$, $i,j=1,\ldots,n$.
In [@li:gua07 Theorem 4.3 and Theorem 4.4] the authors provide also some properties of a fuzzy set–valued Brownian motion that are very similar to those of the real case.
\[pro:Brownian\_properties\] Let $\{B_t : t\ge 0\}$ be a fuzzy set–valued Brownian motion. The following hold.
1. $\{B_{t+t_0} \}_{t\ge 0}$ is a fuzzy set–valued Brownian motion for any $t_0\ge 0$.
2. \[teo:2.Brownian\_properties\] $\{\nu {\oplus}B_t \}_{t\ge 0}$ is a fuzzy set-valued Brownian motion for any fuzzy set $\nu\in{\mathbb{F}_{k}}$.
3. $\{\frac{1}{\sqrt{\lambda}} B_{\lambda t} \}_{t\ge 0}$ is a fuzzy set-valued Brownian motion for any $\lambda> 0$.
4. $\{t B_{\frac{1}{\sqrt{t}}} \}_{t\ge 0}$ is a fuzzy set-valued Brownian motion.
5. If ${\mathfrak{F}}_t=\sigma\{B_s : s\le t\}$, then $\{B_t, {\mathfrak{F}}_t\}_{t\ge 0}$ is a fuzzy set–valued martingale.
A [[FRV]{}]{} Brownian motion is a Wiener process in ${{\mathbb{R}^d}}$ {#sec:Brownian_is_singular}
=======================================================================
This section is devoted to prove Theorem \[teo:Brownian\_is\_singular\]: the main result of this paper.
\[teo:Brownian\_is\_singular\] A fuzzy set–valued process $\{B_t : t\ge 0\}$ is a Brownian motion, if and only if, $$B_t={\mathbb{I}_{b_t}}, \qquad {\mu}\textrm{--a.e.}$$ where $\{b_t : t\ge 0\}$ is a Wiener process in ${{\mathbb{R}^d}}$.
According to Definition \[def:B\_t\] a fuzzy set–valued Brownian motion $B_t$ is a process taking values in ${\mathbb{F}_{}}$ (that is a functional space over ${{\mathbb{R}^d}}$). On the other hand, the previous result provides a way to handle a fuzzy set–valued Brownian motion simply using a random vector of ${{\mathbb{R}^d}}$. In other words, we observe a complexity reduction, i.e. from ${\mathbb{F}_{}}$ to ${{\mathbb{R}^d}}$.\
Moreover, in view of Theorem \[teo:Brownian\_is\_singular\], Property \[teo:2.Brownian\_properties\] in Proposition \[pro:Brownian\_properties\] is true if and only if $\nu={\mathbb{I}_{0}}$, whilst the remain properties in Proposition \[pro:Brownian\_properties\] still hold due to the same properties of the driving Wiener process $b_t$ in $\mathbb{R}^d$.
Actually the complexity reduction stated in Theorem \[teo:Brownian\_is\_singular\] is strictly related to the characterization of Gaussian [[FRV]{}]{} (cf. Proposition \[pro:gaussian\_process\]), to Property 1 of Proposition \[pro:caratterizzazione\_Brownian\], and to the following result obtained for random closed sets.
\[pro:aumann integral null\] Let $X$ be in $L^1[\Omega;{\mathbb{K}_{}}]$ and let $a\in{{\mathbb{R}^d}}$. $\int_\Omega X d{\mu}=\{a\}$ if and only if there exists a $x\in L^1[\Omega;{{\mathbb{R}^d}}]$ such that $X=\{x\}$ ${\mu}$–a.e. and $\int_\Omega xd{\mu}=a$.
\[cor: aumann integral null\] Let $X$ be in $L^1[\Omega;{\mathbb{K}_{}}]$. $\int_\Omega X d{\mu}=\{0\}$ if and only if there exists a $x\in L^1[\Omega;{{\mathbb{R}^d}}]$ such that $X=\{x\}$ ${\mu}$-a.e. and $\int_\Omega xd{\mu}=0$.
Although Proposition \[pro:aumann integral null\] and Corollary \[cor: aumann integral null\] are proved by Molchanov in [@mol05 Proposition 1.30, p.161], we shall propose in Appendix \[sec:proof\_Teo\_aumann\_integrall\_null\] alternative proofs via selections avoiding the use of the support function as Molchanov did.
\[lem:phi\_in\_dual\] For each $(x,\alpha)\in{{\mathbb{R}^d}}\times [0,1]$, the following map belongs to $\overline{\mathbf{C}}^*$ $$\begin{array}{rccl}
\varphi_{x,\alpha}: & \overline{\mathbf{C}} & \to & \mathbb{R}\\
& s & \mapsto & \varphi_{x,\alpha}(s)=s(x,\alpha).
\end{array}$$
Map $\varphi_{x,\alpha}$ is linear since, for any $s_1$, $s_2$ in $\overline{\mathbf{C}}$ and $\lambda_1,\lambda_2\in\mathbb{R}$, the following chain of equalities hold. $$\begin{aligned}
\varphi_{x,\alpha}(\lambda_1 s_1+\lambda_2 s_2)= & [(\lambda_1 s_1+ \lambda_2 s_2)(\alpha)](x)= [\lambda_1 s_1(\alpha) + \lambda_2 s_2(\alpha)](x) \\
= & \lambda_1 s_1(\alpha,x) + \lambda_2 s_2(\alpha,x) = \lambda_1 \varphi_{x,\alpha}(s_1) + \lambda_2 \varphi_{x,\alpha}(s_2).\end{aligned}$$ For the continuity, let us consider any $s\in\overline{\mathbf{C}}$. For each $\varepsilon >0$ and $h\in\overline{\mathbf{C}}$ such that $\|h\|_{\overline{C}}<\varepsilon $, the following relations complete the proof. $$| \varphi_{x,\alpha}(s+h) - \varphi_{x,\alpha}(s) |= | \varphi_{x,\alpha}(h) | = | h(\alpha,x) | \le \|h\|_{\overline{C}} < \varepsilon.$$
The ifpart is trivial.
In order to prove the only ifpart let us consider the fuzzy set–valued Brownian motion $\{B_t : t\ge 0\}$.\
STEP 1. According to Proposition \[pro:caratterizzazione\_Brownian\] and Proposition \[pro:gaussian\_process\], for any $t\ge 0$ and $f\in \overline{\mathbf{C}}^*$, it satisfies $$\begin{aligned}
0=\mathbb{E}[f({h_{B_t} })]=\mathbb{E}[f({h_{\mathbb{E}[B_t]{\oplus}{\mathbb{I}_{\xi_t}}} })].\end{aligned}$$ where $\xi_t$ is an Gaussian random element of ${{\mathbb{R}^d}}$ with $\mathbb{E}\xi_t=0$. By the fact that, for any $\nu^1, \nu^2\in{\mathbb{F}_{c}}$, ${h_{\nu^1{\oplus}\nu^2} }= {h_{\nu^1} } + {h_{\nu^2} }$ (cf. Proposition \[pro:embedding\]), using the linearity of the expected value and of $f$, we get $$\begin{aligned}
0 &= \mathbb{E}[f({h_{\mathbb{E}[B_t]} })] + \mathbb{E}[f({h_{{\mathbb{I}_{\xi_t}}} })] = f({h_{\mathbb{E}[B_t]} }) + f (\mathbb{E}[{h_{{\mathbb{I}_{\xi_t}}} }]) \nonumber
\\
&= f({h_{\mathbb{E}[B_t]} }) + f ({h_{{\mathbb{I}_{\mathbb{E}[\xi_t]}}} }) = f({h_{\mathbb{E}[B_t]} }), \label{eq:f(s)=0}\end{aligned}$$ for any $t\ge 0$ and $f\in \overline{\mathbf{C}}^*$, where for the last two equalities we use ${h_{\mathbb{E}{X}} } = \mathbb{E} {h_{X} }$ and the fact that $\xi_t$ is zero mean.\
Clearly ${h_{\mathbb{E}[B_t]} }\equiv 0$. On the contrary, there will exists an $\alpha\in [0,1]$ such that ${h_{\mathbb{E}[B_t]} }(\alpha)\not\equiv 0 $; i.e. there exists an $\alpha\in [0,1]$ and $x\in{{\mathbb{R}^d}}$ such that ${h_{\mathbb{E}[B_t]} }(\alpha,x)\neq 0$. Let us consider the map defined by $\varphi_{x,\alpha}(s)=s(x,\alpha)$. It is an element of $\overline{\mathbf{C}}^*$ (cf. Lemma \[lem:phi\_in\_dual\]). Then $\varphi_{x,\alpha}({h_{\mathbb{E}[B_t]} })\neq 0$ contradicts Equation .\
As a consequence, $\mathbb{E}[B_t]={\mathbb{I}_{0}}$ for each $t\ge 0$; i.e. $$\label{eq:Bt_alpha_level_is_null}
\mathbb{E}[(B_t)_\alpha]=\{0\},$$ for each $t\ge 0$ and $\alpha\in (0,1]$.\
STEP 2. Combining Corollary \[cor: aumann integral null\] with Equation we obtain that, for each $t\ge 0$ and $\alpha\in (0,1]$, $(B_t)_\alpha$ is actually ${\mu}$–a.e. a random singleton with null mean value; i.e. $(B_t)_\alpha=\{b_t\}$ ${\mu}$–a.e. with $b_t$ being a random element of ${{\mathbb{R}^d}}$ such that $\mathbb{E}b_t=0$. By definition of $\alpha$–level sets for fuzzy set, $(B_t)_\alpha\supset (B_t)_\beta$ for any $0\le \alpha\le \beta\le 1$, and then $B_t={\mathbb{I}_{b_t}}$ ${\mu}$–a.e..\
Since $\{B_t\}_{t\ge 0}$ is a fuzzy set–valued Brownian motion, $\{b_t\}_{t\ge 0}$ is a Brownian motion in ${{\mathbb{R}^d}}$, and this fact concludes the proof.
Note that Proof of Theorem \[teo:Brownian\_is\_singular\] only uses the fact that $\{B_t\}$ is a Gaussian process for which any finite distribution, at any time $t$, has null expectation.
We want to point out that, although one can associate a fuzzy set–valued Brownian motion at any Brownian motion in $\overline{\mathbf{C}}$ (using the embedding in Proposition \[pro:embedding\]), in general, the contrary is not possible. This is due to the embedding properties. In fact, $j({\mathbb{F}_{kc}})$ is a proper subset of $\overline{C}([0,1], C({S^{\, d-1}}) )$.\
As a consequence, a Gaussian element in $\overline{C}([0,1], C({S^{\, d-1}}) )$ can assume different values (even negative), whilst this could not happen in ${\mathbb{F}_{kc}}$ since, the embedding $j$ could not carry back all the possible fluctuationsof gaussian element.
In this view, a definition of fuzzy set–valued Brownian motion, that take care completely the complexity of the (fuzzy) set–valued framework, has to take into account the above arguments and must pay attention to the possibly degeneracy.
Proof of Proposition \[pro:aumann integral null\] {#sec:proof_Teo_aumann_integrall_null}
=================================================
In [@mol05 Proposition 1.30, p.161] Molchanov proposed a proof of Proposition \[pro:aumann integral null\]. It involves the support function of a set. Here we propose a different approach, via random sets selections, that is interesting by itself, and that leads to the same result.
For the sake of generality, here we shall consider ${\mathfrak{X}}$ to be a separable Banach space with ${\mathcal{B}_{{\mathfrak{X}}}}$ its borel $\sigma$–algebra and $(\Omega, {\mathfrak{F}})$ to be a measurable space endowed with a positive finite measure ${\mu}$ (till now ${\mathfrak{X}}$ was $\mathbb{R}^d$ and ${\mu}$ a probability measure).\
In order to prove Proposition \[pro:aumann integral null\] we need the following two lemmas. Roughly speaking, the former says that any non–null vector in ${\mathfrak{X}}$ can be separated from zero using a suitable countable family of elements of ${{\mathfrak{X}^{*}}}$. The second lemma says that, for any couple of different (on some set of positive measure) integrable random elements in ${\mathfrak{X}}$, there exists an element of ${{\mathfrak{X}^{*}}}$ that separates (on a set of positive measure) these two random elements of ${\mathfrak{X}}$.
\[lem: exist separators in teo.aumann integral null\] There exists $\{\phi_n\}_{n\in\mathbb{N}}\subset{{\mathfrak{X}^{*}}}$ such that whenever $x\in{\mathfrak{X}}\setminus \{0\}$ there exists $n\in\mathbb{N}$ for which $\phi_n(x)\neq 0$.
Let $\{x_n\}_{n\in\mathbb{N}}$ be a dense subset of ${\mathfrak{X}}$. As a consequence of the Hahn-Banach Theorem (cf. [@dun:sch58 Corollary II.3.14, p. 65]) there exists $\{\phi_n\}_{n\in\mathbb{N}}\subset{{\mathfrak{X}^{*}}}$ such that $\phi_n(x_n)=\|x_n\|_{{\mathfrak{X}}}$ and $\|\phi_n\|_{{{\mathfrak{X}^{*}}}}= 1$ for all $n\in\mathbb{N}$. Then $$\label{eq: lem1 aumann integral null}
-\|{y}\|_{{\mathfrak{X}}} \le \phi_n(y) \le \|{y}\|_{{\mathfrak{X}}}, \qquad \forall y\in{\mathfrak{X}}\setminus\{0\}, \forall n\in\mathbb{N}.$$ Let $x\in{\mathfrak{X}}\setminus \{0\}$ and $n \in\mathbb{N}$ such that $\|{x-x_n}\|_{{\mathfrak{X}}}\le \frac{\|{x_n}\|_{{\mathfrak{X}}}}{2}$. By we have $$\phi_n(x)=\phi_n(x_n) + \phi_n(x-x_n) \ge\|{x_n}\|_{{\mathfrak{X}}} - \|{x-x_n}\|_{{\mathfrak{X}}} \ge
\frac{\|{x_n}\|_{{\mathfrak{X}}}}{2}
> 0$$ i.e. $\phi_n(x)>0$ that concludes the proof.
\[lem: varphi exists in teo.aumann integral null\] Let $x_1$, $x_2\in L^1[\Omega;{\mathfrak{X}}]$ and $A=\{\omega\in\Omega: x_1(\omega)\neq x_2(\omega)\}$ with ${\mu}(A) >0$. Then there exists $\varphi\in{{\mathfrak{X}^{*}}}$ such that $$A_\varphi = \{\omega\in \Omega: \varphi[x_1(\omega)] >
\varphi[x_2(\omega)]\}$$ has positive measure (i.e. ${\mu}(A_\varphi)>0$).
Let $x=(x_1-x_2)$ then $A=\{\omega\in\Omega: x(\omega)\neq 0\}$ and let $\{\phi_n\}_{n\in\mathbb{N}}\subset{{\mathfrak{X}^{*}}}$ as in Lemma \[lem: exist separators in teo.aumann integral null\]. We claim that there exists $n\in\mathbb{N}$ such that ${\mu}(A_{\phi_n})+ {\mu}(A_{-\phi_n}) >0$. By contradiction, if $A_n=A_{\phi_n}\cup A_{-\phi_n}$, we have $${\mu}(A_n)\le {\mu}(A_{\phi_n}) + {\mu}(A_{-\phi_n}) = 0, \qquad
\forall n\in\mathbb{N}.$$ Now we prove that $A\subseteq\bigcup_{n\in\mathbb{N}} A_n$: let $\omega\in A$ then $x(\omega)\neq 0$ and, by hypothesis, there exists $n\in\mathbb{N}$ such that $\phi_n(x(\omega))\neq 0$. Hence $\phi_n(x(\omega)) > 0$ or $\phi_n(x(\omega)) < 0$ i.e. $\omega\in A_n$ and thus $A\subseteq\bigcup_{n\in\mathbb{N}} A_n$.\
This means that ${\mu}(A)\le{\mu}(\bigcup_{n\in\mathbb{N}} A_n)= 0$ that contradicts hypothesis (${\mu}(A)>0$) and concludes the proof.
The “if” part is trivial. Vice versa, let us suppose that $\int_\Omega x d{\mu}=a$ holds for all $x\in S_X$, where integral is in the Bochner sense. Let us recall that a Bochner integrable map is also Pettis integrable and by definition (see [@tal84; @mus91]) we have $$\label{eq: teo.aumann integral null}
\int_\Omega \phi(x)d{\mu}=\phi(a),\qquad \forall\phi\in{{\mathfrak{X}^{*}}},\
\forall x\in S_X.$$ Now, by contradiction, let us suppose that $x_1,x_2$ are distinct elements of $S_X$ i.e. $A=\{\omega\in\Omega: x_1(\omega)\neq x_2(\omega)\}$ has positive measure. Then, by Lemma \[lem: varphi exists in teo.aumann integral null\], there exists $\varphi\in{{\mathfrak{X}^{*}}}$ such that $A_\varphi = \{\omega\in \Omega: \varphi[x_1(\omega)] > \varphi[x_2(\omega)]\}$ has positive measure. Let us consider $ x_{\varphi}= {\mathbb{I}_{A_\varphi}}x_1 + {\mathbb{I}_{A_\varphi^C}}x_2$. Clearly $x_\varphi$ is a selection of $X$ (i.e. $x_{\varphi}\in S_X$), and $$\begin{aligned}
\int_\Omega \varphi(x_{\varphi})d{\mu}= & \int_{A_\varphi}
\varphi(x_1)d{\mu}+ \int_{A_\varphi^C} \varphi(x_2)d{\mu}\\
> & \int_{A_\varphi} \varphi(x_2)d{\mu}+ \int_{A_\varphi^C}
\varphi(x_2)d{\mu}=\varphi(a)\end{aligned}$$ which contradicts Pettis integrability .
Conclusion
==========
We proved that a fuzzy set–valued Brownian motion is actually a degenerated process. In particular, it can actually be handle by a wiener process in the understanding space. This simplification is due mainly both to the well–known Gaussian degeneracy and to the nullexpectation.\
Moreover, we provided an alternative proof to Proposition \[pro:aumann integral null\]: an integrable set–valued map, which integral is a singleton, is almost everywhere an integrable singleton–valued map
We think that used hypothesis can be relaxed in different ways in order to get generalizations. For example, the space ${{\mathbb{R}^d}}$ can be replaced with a more general one. In this case, the difficulty lies in the fact that one have to redefine fuzzy set–valued Brownian motion in the new space as well as to use a different embedding theorem.
[10]{}
N. Dunford and J. T. Schwartz. . Wiley Classics Library. John Wiley & Sons Inc., New York, 1988.
S. Li and L. Guan. . , 177:3251–3259, 2007.
S. Li, Y. Ogura, and V. Kreinovich. . Kluwer Academic Publishers Group, Dordrecht, 2002.
I. Molchanov. . Springer. (2005)
K. Musial. , 23:177–262, 1991.
M. L. Puri and D. A. Ralescu. The concept of normality for fuzzy random variables. , 13:1373–1379, 1985.
M. Talagrand. , 307:224 p., 1984.
|
---
author:
- |
[Li Li, Alexandre Bartel, ]{}\
Jacques Klein, Yves Le Traon\
SnT\
University of Luxembourg\
[firstName.lastName@uni.lu]{}\
and Eric Bodden\
EC SPRIDE\
Technische Universität Darmstadt\
[firstName.lastName@ec-spride.de]{}\
Department of Computer Science and\
Engineering\
Pennsylvania State University\
{octeau,mcdaniel}@cse.psu.edu
bibliography:
- 'leaks.bib'
title: 'I know what leaked in your pocket: uncovering privacy leaks on Android Apps with Static Taint Analysis'
---
|
---
abstract: 'The fundamental collective degree of freedom of fractional quantum Hall states is identified as a unimodular two-dimensional spatial metric that characterizes the local shape of the correlations of the incompressible fluid. Its quantum fluctuations are controlled by a topologically-quantized “guiding-center spin”. Charge fluctuations are proportional to its Gaussian curvature.'
author:
- 'F. D. M. Haldane'
date: 'June 16, 2011'
title: Geometrical Description of the Fractional Quantum Hall Effect
---
In this Letter, I point out the apparently previously-unnoticed geometric degree of freedom of the fractional quantum Hall effect (FQHE), that fundamentally distinguishes it from the integer effect, and will provide the basis for a new description of its collective properties as a fluctuating quantum geometry.
The simplest model Hamiltonian for $N$ interacting electrons bound to a two-dimensional (2D) planar “Hall surface” traversed by a uniform magnetic flux density is $$H = \sum_{i=1}^N \frac{1}{2m}g^{ab}\pi_{ia}\pi_{ib} +
\frac{1}{A}\sum_{\bm q} V(\bm q) \sum_{i<j} e^{i\bm q\cdot (\bm r_i-\bm
r_j)}.$$ Here $\bm r_i - \bm r_j$ = $(r^a_i-r^a_j)\bm e_a$, $[r^a_i,r^b_j]$ = 0, are the relative displacements of the particles on the 2D surface with orthonormal tangent vectors $\bm e_a$, $a=1,2$, and $\pi_{ia}$ = $\bm e_a\cdot \bm \pi_i$ are the components of the gauge-invariant dynamical momenta, with commutation relations $$[r_i^a,\pi_{jb}]= i\delta_{ij}\hbar \delta^a_b, \quad
[\pi_{ia},\pi_{jb}] = i\delta_{ij}\epsilon_{ab}\hbar^2/\ell_B^2.$$ I use Einstein summation convention: $q_ar^a$ = $\bm q\cdot \bm r$ (index placement distinguishes real-space vectors $r^a$ from dual (reciprocal) vectors $q_a$); $\delta^a_b$ is the Kronecker symbol, and $\epsilon_{ab}$ = $\epsilon^{ab}$ is the 2D antisymmetric Levi-Civita symbol. A periodic boundary condition (pbc) can be imposed on a fundamental region of the plane with area $A$ = $2\pi\ell^2_BN_{\Phi}$, which restricts wavevectors $\bm q$ to the reciprocal lattice; $N_{\Phi}$ is an integer, and $2\pi \ell_B^2$ is the area through which a London magnetic flux quantum $h/e$ passes.
The parameters of the Hamiltonian are: (1) a Galileian effective mass tensor $mg_{ab}$, where $g_{ab}$ is a positive-definite “Galileian metric” with $\det g$ = 1 (*i.e.*, a *unimodular* metric) and inverse $g^{ab}$, and $m > 0$ is the effective mass that controls the cyclotron frequency $\omega_B$ = $\hbar/m\ell_B^2$; (2) $V(\bm q)$ which is the Fourier transform of an unretarded translationally-invariant two-body interaction potential.
In principle, the real function $V(\bm q)$ is the Fourier transform of the long-ranged unscreened Coulomb potential, with the small-$\bm q$ behavior $$\lim_{\lambda \rightarrow 0} \lambda V(\lambda \bm q) \rightarrow
\frac{e^2}{2\varepsilon}(\tilde g^{ab}q_aq_b)^{-1/2} ,$$ where $\tilde g^{ab}$ is the inverse of a unimodular *Coulomb metric* $\tilde g_{ab}$, controlled by the dielectric properties of the surrounding 3D insulating media, while the large-$\bm q$ behavior of $V(\bm q)$ is controlled by the quantum well that binds electrons to the surface. The singularity of $V(0)$ does not affect incompressibility, and can be screened by a metallic plane placed parallel to the surface.
There is no fundamental reason for the Coulomb and Galileian metrics to coincide, unless there is an atomic-scale discrete $(n>2)$-fold rotational symmetry of the surface, and no tangential magnetic flux. I will argue that the usual implicit assumption of rotational symmetry hides key geometric features of the FQHE.
In the presence of the magnetic field, the canonical degrees of freedom $\{\bm r_i, \bm p_i\}$ are reorganized into two independent sets, the dynamical momenta $\{\bm \pi_i\}$, which I will call “left-handed” degrees of freedom, and the “guiding centers” $\{ \bm
R_i\}$, the “right-handed” degrees of freedom, $$R_i^a = r_i^a -\hbar^{-1}\epsilon^{ab}\pi_{ib}\ell_B^2, \quad
[R^a_i,R^b_j] = -i\delta_{ij}\epsilon^{ab}\ell_B^2,
\label{qgeom}$$ with $[R^a_i,\pi_{jb}]$ = 0. The pbc further restricts the guiding-center variables to the set of unitary operators $\rho_{\bm q,i}$ = $\exp {i\bm q\cdot \bm R_i}$, which obey the Heisenberg algebra $$\rho_{\bm q,i}\rho_{\bm q',i} = e^{i\frac{1}{2}\bm q\times \bm
q'\ell_B^2}\rho_{\bm q + \bm q',i}, \quad \bm q \times \bm q' \equiv \epsilon^{ab}q_aq'_b;$$ reciprocal vectors $\bm q, \bm q'$ compatible with the pbc obey $( \exp {i\bm q\times \bm q'\ell_B^2} )^{N_{\Phi}}$ = 1. The pbc can be expressed as $$\left (\rho_{\bm q,i}\right )^{N_{\Phi}}|\Psi\rangle = \left
(\eta_{\bm q}\right )^{N_{\Phi}}|\Psi \rangle$$ for all states in the Hilbert space, where $\eta_{\bm q} = 1$ if $\frac{1}{2}\bm q$ is an allowed reciprocal vector, and $\eta_{\bm q}
= -1$ otherwise. This leads to the recurrence relation $$\rho_{\bm q + N_{\phi}\bm q',i} = \left (\eta_{\bm q'}e^{i\frac{1}{2}\bm q \times
\bm q'\ell_B^2}\right )^{N_{\Phi}}\rho_{\bm q,i} = \pm \rho_{\bm
q,i}.$$ For a given particle label $i$, the set of independent operators $\bm \rho_{\bm q,i}$ can be reduced to a set of $N_{\Phi}^2$ operators where $\bm q \in \text{BZ}$ takes one of a set of $N_{\Phi}^2$ distinct values that define an analog of a “Brillouin zone”. Let $$\delta^2_{\bm q,\bm q'} \equiv \frac{1}{N_{\Phi}}{\sum_{ \bm q''}}' e^{i\bm
q''\times (\bm q - \bm q')}.$$ (Primed sums are over the BZ.) Then $\delta^2_{\bm q,\bm q'} = 0$ if $\bm q $ and $\bm q'$ are distinct, and has the value $N_{\Phi}$ if they are equivalent; with this definition $\delta^2_{\bm q,\bm q'}$ becomes $2\pi \delta^2(\bm
q\ell_B-\bm q'\ell_B)$ in the limit $N_{\Phi}\rightarrow \infty$, where $\delta^2(\bm x)$ is the 2D Dirac delta-function. It is convenient to choose the BZ so it has inversion symmetry: $\bm q \in \text{BZ}$ $\rightarrow$ $-\bm q \in \text{BZ}$, and $\rho_{\bm q=0,i}$ is the identity. The set of $N_{\Phi}^2-1$ operators $\{\rho_{\bm q,i},\bm q \in \text {BZ}, \bm q \ne 0\}$ are the generators of the Lie algebra $SU(N_{\Phi})$. Both $\rho_{\bm q,i}$ and also (as noted by Girvin, MacDonald and Platzman[@GMP]) the “coproduct” $\rho_{\bm q}$ = $\sum_i \rho_{\bm q,i}$, obey $$[\rho_{\bm q},\rho_{\bm q'} ] = 2i \sin ( {\textstyle\frac{1}{2}}\bm q\times \bm
q'\ell_B^2)\rho_{\bm q+\bm q'}.$$ In this form of the Lie algebra, the quadratic Casimir is $$C_2 = \frac{1}{2N_{\Phi}}{\sum_{\bm q\ne 0}}'
\rho_{\bm q}\rho_{-\bm q} = \frac{N(N_{\Phi}^2 -N)}{2N_{\Phi}} + \sum_{i<j}P_{ij},$$ where $P_{ij}$ exchanges guiding centers of particles $i$ and $j$. For $N=1$, the $\rho_{\bm q,i}$ form the $N_{\Phi}$-dimensional fundamental (defining) $SU(N_{\Phi})$ representation of one-particle states of a Landau level, with $C_2$ = $(N_{\Phi}^2 -1)/2N_{\Phi}$.
The high-field condition is defined by $$\hbar \omega_B \gg \frac{1}{A}\sum_{\bm q} V(\bm q) f(\bm q)^2,
\quad f(\bm q) = e^{-\frac{1}{4}q_g^2\ell_B^2 },$$ where $f(\bm q)$ is the lowest-Landau-level form-factor, and $q_g^2$ $\equiv$ $ g^{ab}q_aq_b$. In this limit, the low-energy eigenstates of the model have all the particles in the lowest Landau level, and can be factorized into a simple *unentangled product* of states of right-handed and left-handed degrees of freedom: $$|\Psi_{0,\alpha}\rangle = |\Psi_0^L(g)\rangle \otimes |\Psi^R_{\alpha}\rangle,$$ where $|\Psi^L_0(g)\rangle$ is a trivial harmonic-oscillator coherent state, fully symmetric under interchange of the dynamical momenta of any pair of particles, and parametrized only by the Galileian metric $g_{ab}$; its defining property is $$a_i |\Psi^L_0( g)\rangle = 0,
\quad a_i \propto \omega^a(g)\pi_{ia},\quad i = 1,\ldots,N,
\label{lll}$$ where the complex unit vector $\omega^a(g)$ is obtained by solution of the generalized Hermitian eigenvector problem $$\omega_a(g) = g_{ab}\omega^b(g) = i\epsilon_{ab}\omega^b(g), \quad
\omega_a(g)^*\omega^a(g) = 1.$$ In contrast, the non-trivial states $|\Psi^R_{\alpha}\rangle$ are the eigenstates of the “right-handed” (guiding-center) Hamiltonian $$H_R = \frac{1}{2A}\sum_{\bm q}V(\bm q)f(\bm q)^2\rho_{\bm q}\rho_{-\bm q}.
\label{hamR}$$
The reduction of the problem by discarding “left-handed” degrees of freedom, “frozen out” by Landau quantization, makes numerical study of the problem by exact diagonalization of $H_R$ for finite $N, N_{\Phi}$ tractable. This may also be characterized as a “quantum geometry” description: once the “left-handed” degrees of freedom are removed, the notion of *locality*, fundamental to both classical geometry and Schrödinger’s formulation of quantum mechanics, is absent. The commutation relations (\[qgeom\]) imply a fundamental uncertainty in the “position” of the particles, now only described by their guiding centers. A Schrödinger wavefunction can only be constructed after “gluing” $|\Psi^R_{\alpha}\rangle$ together with some $|\Psi^L\rangle$, after which the composite state can be projected onto simultaneous eigenstates of the commuting set $\{\bm r_i\}$: *e.g.*, $$\Psi_{\alpha}(\{\bm r_i\},g) = \langle \{\bm r_i\}
|\Psi^L_0(g)\rangle \otimes |\Psi^R_{\alpha}\rangle .
\label{wf}$$ Note that the construction (\[wf\]) of a Schrödinger wavefunction involves an *extraneous quantity $(g_{ab})$ that is not directly determined by $|\Psi^R_{\alpha}\rangle$ itself*, and thus is a non-primitive construction that does not represent $|\Psi^R_{\alpha}\rangle$ in its purest form. This suggests a reconsideration of the meaning of the “Laughlin state”, usually presented in the form of the “Laughlin wavefunction”[@laughlin83], which is fundamental to current understanding of the FQHE.
The conventional presentation of FQHE states is as an $N$-particle Schrödinger wavefunction with the form $$\Psi(\{\bm r_i\}) =
F(\{z_i\})\prod_{i=1}^Ne^{-\frac{1}{2}z_i^*z_i},
\label{llwf}$$ where $z_i$ = $\omega_a(g)r^a_i/\surd 2 \ell_B$. Such wavefunctions, formulated in the “symmetric gauge”, obey (\[lll\]) with $a_i$ $\equiv$ ${\textstyle \frac{1}{2}}z_i +
{\partial}/{\partial z_i^*}$. The original Laughlin wavefunction[@laughlin83] was the polynomial $$F(\{z_i\}) = F_L^q(\{z_i\}) \equiv \prod_{i>j}(z_i-z_j)^q;
\label{polyl}$$ it was subsequently adapted[@halrez85] to a pbc with the form $$F^q_{L,\alpha}(\{z_i\}) = \prod_{i>j}w(z_i-z_j)^q \prod_{k=1}^q
w(({\textstyle \sum _i} z_i)-a_{k,\alpha}),
\label{lwfpbc}$$ where $w(z)$ is given in terms of an elliptic theta function: $w(z)$ = $\theta_1 (\pi z/L_1|L_2/L_1)\exp (z^2/2N_{\Phi} )$, with $L_1L_2^* - L_1^*L_2$ = $2\pi i N_{\Phi}$ (the wavefunction is (quasi) periodic under $z_i$ $\rightarrow$ $z_i + mL_1 + nL_2$). The additional $q$ parameters $a_{k,\alpha}$ of (\[lwfpbc\]), with $\sum_k a_{k,\alpha}$ = 0, characterize the $q$-fold topological degeneracy of the Laughlin state with a pbc.
The Laughlin wavefunction was originally presented as a “variational wavefunction”, albeit one with no continuously-tunable parameter, since $q$ is an integer fixed by statistics. Its initial success was that, as a “trial wavefunction”, it had a lower Coulomb energy than obtained in Hartree-Fock approximations, and explained the existence of a strong FQHE state at $\nu$ $\equiv$ $N/N_{\Phi}$ = 1/3, but not at $\nu$ = 1/2. In the wavefunction language, its defining characteristic is that, as a function of any particle coordinate $z_i$, there is an order-$q$ zero at the location of every other particle, which heuristically “keeps particles apart”, and lowers the Coulomb energy.
Subsequent to its introduction, the Laughlin state’s essential validity was further confirmed by this author’s observation[@haldane83] that, at $\nu = 1/q$, it is also uniquely characterized as the zero-energy eigenstate of a two-body “pseudopotential Hamiltonian” $$H_R = \sum_{m=0}^{q-1}V_mP_m(g) ,\quad V_m > 0,$$ where $$P_m(g) = \frac{1}{N_{\Phi}}\sum_{\bm q}
L_m(q_g^2\ell_B^2)e^{-{\textstyle\frac{1}{2}} q^2_g\ell_B^2}
\rho_{\bm q}\rho_{-\bm q},$$ where $L_m(x)$ is a Laguerre polynomial. Numerical finite-size diagonalization[@halrez85a] for $q$ = 3 showed that this $H_R$ had the gapped excitation spectrum of an incompressible FQHE state, and that this gap did not close along a path that adiabatically interpolated between it and the Hamiltonian of the Coulomb interaction with $\tilde g_{ab}$ = $g_{ab}$.
This raises the question that does not seem to have been previously considered: what if the “Coulomb metric” $\tilde g_{ab}$ and the “Galileian metric” $g_{ab}$ do *not* coincide? The “pseudopotential” definition of the Laughlin *state* (as opposed to the Laughlin *wavefunction*) defines a *continuously-parametrized family* of $\nu$ = $1/q$ Laughlin states $|\Psi^q_{L,\alpha} (\bar g)\rangle$ by $$P_m(\bar g)|\Psi^q_{L,\alpha}(\bar g)\rangle = 0 , m < q.$$ The continuously-variable parameter here is a unimodular *guiding-center metric* $\bar g_{ab}$ that is in principle distinct from the Galileian metric $g_{ab}$, and is *not* fixed by the one-body physics of the Landau orbits. Physically, it characterizes the *shape* of the correlation functions of the Laughlin state. If the shape of Landau orbits is used as the definition of “circular”, the correlation hole around the particles deforms to “elliptical” when $\bar g_{ab}$ $\ne $ $g_{ab}$.
If a wavefunction (\[lll\]) is constructed by “gluing together” $|\Psi^L_0( g)\rangle $ with the “Laughlin *state*” $|\Psi^R\rangle$ = $|\Psi^q_{L,\alpha}(\bar g)\rangle$, it does *not* correspond to the Laughlin *wavefunction* (\[lwfpbc\]) *unless* $\bar g_{ab}$ = $g_{ab}$, as there is no longer a $q$’th order zero of the wavefunction when $z_i$ = $z_j$. Despite this, I will not call $|\Psi^q_L(\bar g)\rangle$ with $\bar g_{ab}$ $\ne$ $g_{ab}$ a “generalization” of the Laughlin state, but propose it as a definition of the *family* of Laughlin states that exposes the geometrical degree of freedom $\bar g_{ab}$ hidden by the wavefunction-based formalism. I argue that FQHE states should be described completely within the framework of the “quantum geometry” of the guiding-center degrees of freedom alone, and no “preferred status” should be accorded to the metric choice $\bar g_{ab}$ = $g_{ab}$. If the states $|\Psi_{L,\alpha}^q(\bar g)\rangle$ are used as variational approximations to the ground state of a generic $H_R$ given by (\[hamR\]), $\bar g_{ab}$ must be chosen to minimize the correlation energy $E(\bar g)$ = $\langle \Psi_{L\alpha}^q(\bar g)|H_R
|\Psi_{L\alpha}^q(\bar g)\rangle$. If the Coulomb ($\tilde g_{ab}$) and Galileian ($g_{ab}$) metrics coincide, the energy will be minimized by the choice $\bar g_{ab}$ = $\tilde g_{ab}$= $g_{ab}$; otherwise, $\bar g_{ab}$ will be a compromise intermediate between the two physical metrics.
A more profound consequence of the identification of the variable geometric parameter $\bar g_{ab}$ follows from the observation that the correlation energy will be a quadratic function of local deformations $\bar g_{ab}(\bm r,t)$ around the minimizing value, whether or not this is equal to $g_{ab}$. This unimodular metric, or “shape of the circle” defined by the correlation function of the FQHE state, may be identified as the natural *local collective degree of freedom* of a FQHE state (defined on lengthscales large compared to $\ell_B$), and not merely a variational parameter.
In its finite-$N$ polynomial form (\[polyl\]), the Laughlin state $|\Psi^q_L(g)\rangle $ is an eigenstate of $L_R(g,0)$ where $L_R(g,\bm r)$ = $g_{ab}\Lambda^{ab}(\bm r)$ generates rotations of the guiding-centers about a point $\bm r$; here $\Lambda^{ab}(\bm r)$ = $\Lambda^{ba}(\bm r)$ are the three generators of area-preserving linear deformations[@fdmharxiv] of the guiding-centers around $\bm r$: $$\Lambda^{ab}(\bm r) = \frac{1}{4\ell_B^2}\sum_i \{\delta
R^a_i(\bm r),\delta R^b_i(\bm r)\},$$ with $\delta R^a_i(\bm r) \equiv R^a_i-\bm r$. Leaving $\bm r$ implicit, these obey the non-compact Lie algebra[@fdmharxiv] $$[\Lambda^{ab},\Lambda^{cd}] = -\frac{i}{2}\left (
\epsilon^{ac}\Lambda^{bd} +
\epsilon^{bd}\Lambda^{ac}
+ a \leftrightarrow b\right ),$$ which is isomorphic to $SO(2,1)$, $SL(2,R)$, and $SU(1,1)$, with a Casimir $C_2$ = $ -\frac{1}{2}\det \Lambda$ $\equiv$ $-\frac{1}{4}\epsilon_{ac}\epsilon_{bd}\Lambda^{ab}\Lambda^{cd}$.
FQHE states with $\nu$ = $ p/q$ can be simply understood as condensates of “composite bosons”[@gmpbos] which are “elementary droplets” of the incompressible fluid consisting of $p$ identical charge-$e$ particles “bound to $q$ London flux quanta” (*i.e.*, occupying $q$ one-particle orbitals of the Landau level), which behave as a boson under interchange. This requires that the Berry phase cancels any bare statistical phase: $(-1)^{pq}$ = $\xi^p$, where $\xi $ = $-1$ ($+1$) for fermions (bosons). For a condensate of charge-$pe$ objects, the elementary fractionally-charged vortex has charge $\pm e^*$ = $\pm (\nu e^2/h)
\times (h/pe)$ = $\pm e/q$. This work aims to extend the description of the “composite boson” by giving it (2D orbital) “spin” and geometry.
Polynomial FQHE wavefunctions like (\[polyl\]) that describe $\bar N$ = $N/p$ = $N_{\Phi}/q$ elementary droplets are generically eigenstates of $L_R(g,0)$ with eigenvalue $\frac{1}{2}pq\bar N^2 + \bar s\bar N$, where $\bar s$ is a variant of the so-called “shift” that I will identify as a fundamental FQHE parameter, the *guiding-center spin*, that characterizes the geometric degree of freedom of FQHE states. It can also be obtained as the limit $\bar N \rightarrow \infty$ of $$\bar s = \frac{1}{\bar N}\sum_{m= 0}^{q\bar N -1}
(m+{\textstyle\frac{1}{2}})(n_m(\bar g,\bm r)-\nu),$$ where $n_m(\bar g,\bm r)$, $m \ge 0 $ are the occupations of guiding-center orbitals defined as the eigenstates of $L_R(\bar g,\bm r)$.
Note that the “superextensive” ($\propto \bar N^2$) contribution to the eigenvalue derives from the uniform background density contribution $\nu \delta^2_{\bm q,0}$ to $\rho_{\bm q}$, and can be removed (regularized) by defining $\Lambda^{ab}(r)$ in the thermodynamic limit $N_{\Phi} = q\bar N\rightarrow \infty$ using the limit of the $\bm q\ne 0$ $SU(N_{\phi})$ generators alone, which become continuous functions $\rho(\bm q)$ of $\bm q$, with $\lim_
{\lambda \rightarrow 0} \rho(\lambda \bm q)$ = 0. Then $$\Lambda^{ab}(\bm r) =
\lim_{\lambda {\rightarrow 0}} \left (-\frac{1}{2}\frac{1}{(\lambda\ell_B)^2}\frac {\partial}{\partial q_a}\frac {\partial}{\partial q_b}
\rho(\lambda \bm q)e^{-i\lambda \bm q\cdot \bm r}\right ).$$ The Laughlin state $|\Psi^q_L(\bar g)\rangle$ is an eigenstate of $\bar g_{ab}\Lambda^{ab}(\bm r)$ with $\bar s$ = $\frac{1}{2}(1-q)$. Note that for fermionic particles ($\xi$ = $-1$), $\bar s$ is odd under particle-hole transformations, and vanishes when the Landau-level is completely filled (here $q$ = 1). A spin-statistics selection rule requires that $$(-1)^{2\bar s}(-1)^{2s} = (-1)^{pq} = \xi^p, \quad (-1)^{2s} =
(-1)^p,$$ where $s$ is the (orbital) “Landau-orbit spin” of the elementary droplet ($s$ = $-\frac{1}{2},-\frac{3}{2},\ldots$ for particles with Landau index $0,1,\ldots $). The expression for $\bar s$ may now be inverted to define the (local) unimodular guiding-center metric $\bar g_{ab}(\bm r)$ by the expectation value $$\lim_{\bar N \rightarrow \infty} \frac{1}{\bar N} \langle
\Psi^R|\Lambda^{ab}(\bm r)|\Psi^R\rangle =
{\textstyle\frac{1}{2}}\bar s \bar g^{ab}(\bm r), \quad \det \bar g = 1,$$ so if $\bar \rho(\bm r)$ is the local droplet density, $\frac{1}{2}\bar
s \bar \rho(\bm r) g^{ab}(\bm r)$ is the local density of the deformation generator.
The quantization of $2\bar s$ as an integer is a topological property deriving from the incompressibility of FQHE states. A simple picture that is reminiscent of Jain’s notion of “quasi-Landau-levels”[@jain] supports this: the “elementary droplet”, with a shape fixed by $\bar g_{ab}(\bm r)$, supports $q$ single-particle orbitals with guiding-center spins $\frac{1}{2},\frac{3}{2},\ldots
,\frac{q-1}{2}$. The way these are occupied by the $p$ particles of the droplet, determines the guiding-center spin of the droplet as the actual total guiding center spin of the configuration, minus that ($\frac{1}{2}pq$) given by putting $p/q$ particles in each orbital. The repulsive exchange and correlation fields of particles outside the droplet will give each of the internal levels a mean energy for orbiting around an effective potential minimum at its center. The droplet will be stable, with a quantized guiding center spin that is adiabatically conserved as the droplet changes shape, provided there is a finite positive energy gap between the highest occupied and lowest empty single-particle state in the droplet. Collapse of this gap implies that the system has become compressible with an unquantized or indeterminate value of $\bar s$.
The geometrical degree of freedom exposed here also suggests a new look at the problem of formulating a continuum description of incompressible FQHE states. Elsewhere, I will present a continuum description combining Chern-Simons fields with the geometry field $\bar \omega_a(\bm r,t)$, where $\bar
g_{ab}$ = $\bar \omega^*_a\bar \omega_b + \bar \omega^*_b\bar \omega_a$, but mention here some fundamental formulas that emerge. First, the electric charge density is given by $pe\bar \rho(\bm r)$, where $\bar \rho(\bm r)$ is the local elementary droplet (composite boson) density, and $$\bar \rho(\bm r,t) = \frac{1}{2\pi p q}\left ( \frac{ pe}{\hbar} B(\bm
r) + \bar s K(\bm r,t)\right ),$$ Here $B(\bm r)$ is the externally-imposed 2D (normal) magnetic flux density, (assumed to be time-independent, but not necessarily spatially uniform), and $K(\bm r, t)$ is the instantaneous Gaussian curvature of the unimodular guiding-center-metric field $\bar g_{ab}(\bm r,t)$, given by $K$ = $\epsilon^{ab}\partial_a\Omega^{\bar g}_b$, $\Omega^{\bar g}_a$ = $\epsilon^{bc}\bar \omega^*_b\nabla_a^{\bar
g}\bar \omega_c$, where $\Omega^{\bar g}_a$ is the spin connection gauge-field and $\nabla^{\bar g}_a$ is the covariant derivative (Levi-Civita connection) of $\bar g_{ab}$. This formula could perhaps have been anticipated from the work of Wen and Zee[@wenzee], who considered coupling Chern-Simons fields to curvature, but the curvature they apparently had in mind was not the collective dynamical internal degree of freedom described here, but that due to placing the FQHE system on a curved 2D surface embedded in 3D Euclidean space, as in formal calculations of the FQHE on a sphere surrounding a monopole[@haldane83; @halrez85]. The second formula is that the canonical conjugate of the geometry field $\bar \omega_a(\bm r)$ is $$\bar \pi^a_{\bar \omega}(\bm r) = \hbar \bar s \bar \rho(\bm r)
\epsilon^{ba}\bar \omega_b(\bm r)^*,$$ so the momentum density (translation generator density) is $\bar \pi^b_{\bar \omega}\nabla_a^{\bar g}\bar \omega_b $ = $\hbar s
\bar \rho \Omega^{\bar g}_a$. These formulas parallel those of quantum Hall ferromagnets, with guiding-center spin and Gaussian curvature replacing true electron spin and Berry curvature. On large lengthscales, the elementary charge $e^*$ = $\pm e/q$ quasiparticles appear as rational cone-singularities of the metric field $\bar g_{ab}(\bm
r,t)$ with localized Gaussian curvature $K$ = $\pm 4\pi/(2\bar s )$.
In summary, the prevalent assumption of rotational invariance of FQHE fluids conceals a fundamental geometric degree of freedom, the shape of their correlations, described by a unimodular spatial metric field that exhibits quantum dynamics.
This work was supported by DOE grant [DE]{}-[SC0002140]{}. The author thanks the Laboratoire Pierre Aigrain, École Normale Supérieure, Paris, for its hospitality during the final stages of this work.
[1]{} S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. Lett. [**54**]{}, 581 (1985)
R. B. Laughlin, Phys. Rev. Lett. [**50**]{}, 1395 (1983).
F. D. M. Haldane and E. H. Rezayi, Phys. Rev. B [**31**]{}, 2529 (1985).
F. D. M. Haldane, Phys. Rev. Lett. [**51**]{}, 605 (1983).
F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. [**54**]{}, 237 (1985).
S. M. Girvin and A. H. MacDonald, Phys. Rev [**58**]{}, 1252 (1987).
F. D. M. Haldane, arXiv:0906.1854 (unpublished).
X. G. Wu and J. K. Jain, Phys. Rev. B. [**51**]{}, 1752 (1995)
X. G. Wen and A. Zee, Phys. Rev. Lett. [**69)**]{}, 953 (1992).
|
---
abstract: 'The convergence to equilibrium of renormalized solutions to reaction-cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, By showing that any renormalized solution satisfies the conservation of masses and a weak entropy-entropy production inequality, it can be proved under the assumption of no boundary equilibria that [*all*]{} renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.'
address:
- 'Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria'
- 'Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria'
author:
- 'Esther S. Daus'
- Bao Quoc Tang
title: 'Trend to equilibrium of renormalized solutions to reaction-cross-diffusion systems'
---
[^1]
Introduction
============
Multi-species systems appear in many applications in biology, physics and chemistry, and can be modeled by reaction-cross-diffusion systems. We want to study the convergence to equilibrium of reaction-cross-diffusion systems with strongly growing reactions, where the system (without reactions) is of formal gradient-flow structure and thus admits an entropy estimate. But since the reactions do not obey any growth condition, this estimate is not enough to define weak solutions, which motivates the study of renormalized solutions *à la* J. Fischer [@Fi15]. Our goal is to show that any renormalized solution satisfies the conservation of masses and a weak entropy-entropy production inequality, and consequently, under the assumption of no boundary equilibria, all renormalized solutions converge to equilibrium with an exponential rate which is explicit up to a finite dimensional inequality.
The convergence to equilibrium for reaction-diffusion systems with linear diffusion has been studied extensively, see e.g. [@CDF14; @DeFe06; @DeFe14] and references therein, while much less is known for nonlinear diffusion or cross diffusion, see [@FLT17] for a porous-medium type diffusion and [@DJT18] for Maxwell-Stefan diffusion. In this work, we study the convergence to equilibrium for a cross-diffusion model originally introduced by Shigesada, Kawasaki and Teramoto [@SKT79] in population dynamics. The existence of global weak solutions for this class of cross-diffusion models with at most linearly growing reactions has been attracted a lot of attention recently by exploiting its formal gradient-flow structure, see *e.g* [@CDJ18; @ChJu04; @ChJu06; @DLM14; @DLMT15; @Jue15; @Jue16]. Unfortunately, for strongly growing reactions this does not provide enough regularity to define weak solutions. Hence, the notion of renormalized solutions was introduced in [@CJ17] for reaction-cross-diffusion systems in analogy to [@Fi15] for reaction-diffusion systems. The standard way for proving convergence to equilibrium via entropy method is to first prove the convergence for an approximate solution, and then by passing to the limit to obtain it also for the [*constructed*]{} weak solution (see *e.g.* [@DJT18]). But since uniqueness for cross diffusion is a very delicate topic (see *e.g.* [@CJ18]), it is desirable to prove convergence to equilibrium for [*all* ]{} solutions. This has been recently obtained in [@FT17a] for reaction-diffusion systems, and thus in this work, we extend these results to
More precisely, we consider $n$ chemical substances $S_1, \ldots, S_n$ reacting via $R$ reactions of the form [ $$\label{R}
y_{r,1}S_1 + \ldots + y_{r,n}S_n \xrightarrow{k_r} y_{r,1}'S_1 + \ldots + y_{r,n}'S_n \qquad \text{ or shortly } \qquad y_r\xrightarrow{k_r} y_r', \quad \qquad r=1,\ldots, R,$$]{} where $y_r = (y_{r,1},\ldots, y_{r,n}), y_r'= (y_{r,1}', \ldots, y_{r,n}') \in (\{0\}\cup [1,\infty))^n$ are the stoichiometric coefficients, and $k_r > 0$ are the reaction rate constants. The corresponding reaction-cross-diffusion system reads for each $i=1,\ldots, n$ as $$\label{S}\tag{S}
\begin{cases}
\partial_tu_i - \mathrm{div}\left(\sum_{j=1}^{n}A_{ij}(u){\nabla}u_j\right) = f_i(u), &\text{ for } (x,t)\in \Omega\times (0,T),\\
\left(\sum_{j=1}^{n}A_{ij}(u){\nabla}u_j\right)\cdot \nu = 0, &\text{ for } (x,t)\in {\partial}\Omega\times (0,T),\\
u_{i}(x,0) = u_{i,0}(x), &\text{ for } x\in \Omega,
\end{cases}$$ where $u = (u_1, \ldots, u_n)$ are the population densities and $\Omega$ is a bounded domain with smooth boundary $\partial\Omega$, . The reaction terms represent the reactions in , i.e. $$\label{reactions}
\color{black}{f_i(u) = \sum_{r=1}^{R}k_r(y_{r,i}' - y_{r,i})u^{y_r}} \quad \text{ with } \quad u^{y_r} = \prod_{i=1}^{n}u_i^{y_{r,i}},$$ while the diffusion matrix $A(u) = [A_{ij}(u)]_{i,j=1,\ldots, n}$ is given by $$\label{Aij}
A_{ij}(u) = \delta_{ij}\left(a_{i0} + \sum_{k=1}^{n}a_{ik}u_k \right) + a_{ij}u_i,$$ where $a_{i0}, a_{ij} \geq 0$ for all $i,j=1,\ldots, n$ and $\delta_{ij}$ denotes the Kronecker delta. They are assumed to satisfy (in analogy to [@CJ17]) either the weak cross-diffusion condition $$\label{weak-cross}
\alpha:= \min_{i=1,\ldots,n} \left(a_{ii} - \frac14 \sum_{i=1}^n \left(\sqrt{a_{ij}} - \sqrt{a_{ji}} \right)^2\right) > 0,$$ or the detailed-balance condition[^2] $$\label{detail-diffusion}
a_{ij} = a_{ji} \quad \mbox{for all} ~~1\leq i,j\leq n.$$ Let $m = \mathrm{codim}\{y_r'- y_r\}_{r=1,\ldots, R}^{\top}$, then if $m>0$ there exists a matrix $\mathbb Q \in \mathbb R^{m\times n}$ whose rows form a basis of $\mathrm{ker}\{y_r'- y_r\}_{r=1,\ldots, R} \in \mathbb R^{n\times R}$. From it follows that $\mathbb Q [f_1(u), \ldots, f_n(u)]^{\top} = 0$, and therefore formally possesses $m$ conservation laws $$\mathbb Q\overline{u}(t) = \mathbb Q \overline{u}_0 =: \mathbf M \quad \text{ for all } \quad t>0,$$ where $\overline{u} = (\overline{u}_1, \ldots, \overline{u}_n)$ and $\overline{u}_i = \frac{1}{|\Omega|}\int_{\Omega}u_idx$. The system is said to satisfy the *complex balanced condition* if there exists a positive [*complex balanced equilibrium*]{} $u_\infty = (u_{1,\infty}, \ldots, u_{n,\infty})\in (0,\infty)^n$, such that at $u_\infty$ the total out-flow and in-flow at each complex are balanced, i.e. $$\label{com-equi}
\sum_{\{r: y_r = y\}}k_ru_\infty^{y_r} = \sum_{\{s: y_s' = y\}}k_su_\infty^{y_s} \quad \text{ for all } \quad y\in \{y_r, y_r'\}_{r=1,\ldots, R}.$$ It was proved in [@Fei] that [if $m>0$ then]{} for each positive initial mass vector $\mathbf M$ there exists a unique positive complex balanced equilibrium $u_\infty \in (0,\infty)^n$, [while when $m=0$ the system has a unique positive complex balanced equilibrium for any positive initial data]{}. Note that there could possibly exist many [*boundary equilibria*]{}, i.e. $u^*\in \partial(0,\infty)^n$ and $u^*$ satisfies .
The main result of this paper reads as follows.
\[thm:main\] Let $\Omega$ be a bounded domain with smooth boundary $\partial\Omega$. Assume $a_{i0}$, $a_{ii}>0$, $a_{ij}\geq 0$, and let the diffusion matrix $A(u)$ satisfy either or . Assume that satisfies the complex balanced condition . Then, for any nonnegative measurable initial data $u_0\in L^1(\Omega)^n$ such that $u_{i,0}\log u_{i,0}\in L^1(\Omega)$ for all $i=1,\ldots, n$, there exists a global nonnegative renormalized solution $u = (u_1, \ldots, u_n)$ to , that is, for all $T>0$, $$u_i\log u_i \in L^\infty(0,T;L^1(\Omega)), \quad \mbox{and}~~\quad{\textcolor{black}}{ \|\sqrt{u_i}\|_{L^2(0,T;H^1(\Omega))}, \|u_i\|_{L^2(0,T;H^1(\Omega))} \leq C(T)}$$ and for any smooth function $\xi \in C^\infty([0,\infty)^n)$ with compactly supported $D\xi$, it holds for all test functions $\psi \in C^\infty_0(\overline{\Omega}\times [0,T))$ that $$\label{defi.renorm}
\begin{aligned}
-\int_{\Omega}\xi(u_0)\psi(\cdot, 0)dx - \int_0^T\int_{\Omega}\xi(u){\partial}_t\psi dxdt &= -\sum_{i,k=1}^n\int_0^T\int_{\Omega}{\partial}_i{\partial}_k\xi(u)\left(\sum_{j=1}^{n}A_{ij}(u){\nabla}u_j\right){\nabla}u_k\psi dxdt\\
&\quad - \sum_{i=1}^n\int_0^T\int_{\Omega}{\partial}_i\xi(u)\left(\sum_{j=1}^{n}A_{ij}(u){\nabla}u_j\right){\nabla}\psi dxdt + \sum_{i=1}^{n}\int_0^T\int_{\Omega}{\partial}_i\xi(u)f_i(u)\psi dxdt.
\end{aligned}$$ Assume additionally that does not have any boundary equilibria and fix an initial mass vector $\mathbf M$. Then, any renormalized solution to with positive initial mass $\mathbf M$, i.e. ${{\mathbb Q}}\overline{u}_0 = \mathbf M$, converges exponentially to the equilibrium, i.e. $$\sum_{i=1}^{n}\|u_i(t) - u_{i\infty}\|_{L^1(\Omega)} \leq Ce^{-\lambda t} \quad \text{ for all } \quad t>0,$$ where $C>0$ and $\lambda>0$ are constants which can be computed explicitly up to a finite dimensional inequality.
[The convergence result in Theorem \[thm:main\], [in case $m>0$]{}, depends only on the initial masses but not on the precise initial data. Thus, two solutions with different initial data but same initial masses converge exponentially to the same equilibrium. [When $m=0$, i.e. there are no conservation laws, then all renormalized solutions converge to the unique positive equilibirium for any positive initial data.]{}]{.nodecor}
The main tool in the proof of Theorem \[thm:main\] is to consider the relative entropy $$\label{re_entropy}
{\mathscr E}(u|u_\infty) = \int_{\Omega}E(u|u_\infty)dx, \quad \mbox{where} \quad E(u|u_\infty) = \sum_{i=1}^n \left(u_i\log(u_i/u_{i\infty}) - u_i + u_{i\infty}\right)\geq 0,$$ for which formally for any solution to the entropy production has the following form $$\label{eep}
\frac{d}{dt}{\mathscr E}(u|u_\infty) \leq -{\mathscr D}(u) \quad \text{ with } \quad {\mathscr D}(u) = \sum_{i=1}^{n}a_{i0}\int_{\Omega}\frac{|{\nabla}u_i|^2}{u_i}dx + \sum_{r=1}^{R}k_r u_{\infty}^{y_r}\int_{\Omega}\Psi\left(\frac{u^{y_r}}{u_\infty^{y_r}},\frac{u^{y_r'}}{u_\infty^{y_r'}} \right)dx,$$ where $\Psi(x,y) = x\log(x/y) - x + y$. For details we refer to [@CDJ18] for the cross-diffusion term and to [@FT17a] for the reaction term. Moreover, for all nonnegative measurable functions $u = (u_1, \ldots. u_n)$ satisfying the conservation laws $$\label{conservation}
\mathbb Q \overline{u} = \mathbf M,$$ it was proved (e.g. [@FT17a]) that $${\mathscr D}(u) \geq \lambda {\mathscr E}(u|u_\infty),$$ where $\lambda$ is an explicit constant up to a finite dimensional inequality. Then, still formally, one obtains the desired exponential decay $${\mathscr E}(u(t)|u_\infty) \leq e^{-\lambda t}{\mathscr E}(u_0|u_\infty).$$ Unfortunately, the notion of renormalized solutions is very weak, so that the entropy-entropy production inequality or even the conservation laws (which only concern the $L^1$-norm of the solution) are not easy to verify. As mentioned before, one can argue via approximate solutions, and thus obtain the convergence to equilibrium for [*one* ]{} renormalized solution, see e.g. [@DJT18]. However, it is not clear if all renormalized solutions () can be approximated in such a way. Our aim here is to prove that [*all* ]{} renormalized solutions with the same initial mass converge to the unique equilibrium. The main idea is to show that the conservation laws and a weaker version of the entropy-entropy production inequality (see Lemma ) hold for any renormalized solution. Our proof uses the techniques developed in [@Fi17].
Proof of the main result
========================
\[weak-eep\] For any renormalized solution $u$ of it holds that $${\mathscr E}(u(t)|u_\infty) + \int_s^t{\mathscr D}(u(\tau))d\tau \leq {\mathscr E}(u(s)|u_\infty) \quad \mbox{for a.e.} \quad t>s>0,$$ where ${\mathscr E}$ and ${\mathscr D}$ are defined in and respectively.
For $M>0$, let $\phi_M: [0,\infty) \to \mathbb{R}$ be a smooth function with $$\begin{aligned}
\phi_M(s) &= s, \quad \mbox{for}~~s \leq M, \qquad
\phi_M'(s) = 0, \quad \mbox{for}~~ s \geq M^C,\qquad
\phi_M'(s) \in [0,1],\nonumber \\
\left|\phi_M''(s)\right| &\leq \frac{C}{1 + s \log (1+s)} \quad \mbox{for all}~~ s\geq 0\label{phi}.
\end{aligned}$$ Moreover, we set $$\begin{aligned}
\xi(u) = \phi_M(E(u + \eta|u_\infty)),
\end{aligned}$$ where $u+\eta = (u_1 + \eta, \dots, u_n + \eta)$ for some $\eta >0$. [The regularization $\eta>0$ is needed to deal with the potential singularity of $\log u_i$ since a renormalized solution is non-negative but in general not strictly positive.]{} For simplicity, we will write $E(u)$ and $E(u+\eta)$ instead of $E(u|u_\infty)$ and $E(u+\eta|u_\infty)$ respectively inside this proof. Then we can compute $$\begin{aligned}
\partial_i \xi(u) &= \phi_M'(E(u + \eta))\log\left(\frac{u_i + \eta}{u_{i\infty}}\right), \\
\partial_i \partial_k \xi(u) &= \phi_M''(E(u + \eta))\log\left(\frac{u_k + \eta}{u_{k\infty}}\right)\log\left(\frac{u_i + \eta}{u_{i\infty}}\right)
+ \phi_M'(E(u + \eta))\frac{\delta_{ik}}{u_i + \eta}.
\end{aligned}$$ By choosing $\psi = 1$ [, or more precisely a smooth version of $1$ with compact support in $[0,T-\delta]$ then let $\delta \to 0$ (see [@CJ17 Lemma 11] for more details)]{} in the definition of the renormalized solutions, we get $$\begin{aligned}
\label{eq.star}
&I_1(\eta,M):=\int_{\Omega}\phi_M(E(u+\eta))\,dx\biggr|_s^t \notag \\
&= -\sum_{i,k=1}^n \int_s^t \int_{\Omega} \left(\phi_M''(E(u + \eta))\log\left(\frac{u_k + \eta}{u_{k\infty}}\right)\log\left(\frac{u_i + \eta}{u_{i\infty}}\right)
+ \phi_M'(E(u + \eta))\frac{\delta_{ik}}{u_i + \eta} \right)\left(\sum_{j=1}^n A_{ij}(u)\nabla u_j\right)\nabla u_k\,dxdt \\
& \quad + \sum_{i=1}^n \int_s^t \int_{\Omega} \phi'_M(E(u + \eta))\log\left(\frac{u_i + \eta}{u_{i\infty}}\right)f_i(u)\,dxdt \notag \\
& =: I_2(\eta,M) + I_3(\eta,M). \notag
\end{aligned}$$ Our first goal now is to pass to the limit $\eta \to 0$ in . Clearly, due to the dominated convergence theorem, we have for the left-hand side of that $$\begin{aligned}
\lim_{\eta \to 0} I_1(\eta,M) = \int_{\Omega}\phi_M(E(u))\,dx\, \biggr|_s^t.
\end{aligned}$$ Next, since $\phi'_M$ has compact support [the integrand of $I_3(\eta, M)$ vanishes when $|u|$ is large. Now for $|u| \leq C(M)$ we can use the property $f_i(u) \geq 0$ when $u_i = 0$ and the local Lipschitz continuity of $f_i(u)$ to estimate $f_i(u) \geq -C(M)u_i$. Hence by considering the signs of $f_i(u)$ and $\log(\frac{u_i+\eta}{u_{i\infty}})$ one obtains easily]{} $$\begin{aligned}
f_i(u)\log\left(\frac{u_i + \eta}{u_{i \infty}}\right) \leq C(M) u_i \left|\log \frac{u_i + \eta}{u_{i \infty}}\right|.
\end{aligned}$$ Thus, Fatou’s lemma yields $$\begin{aligned}
\limsup_{\eta \to 0} I_3(\eta,M)
\leq \sum_{i=1}^n \int_s^t \int_{\Omega} \phi'_M(E(u)) \log\left(\frac{u_i}{u_{i\infty}}\right)f_i(u)\,dxd\tau.
\end{aligned}$$ Next, we split $I_2(\eta,M)$ in into $$\begin{aligned}
I_2(\eta,M) &=
-\sum_{i,k=1}^n \int_s^t \int_{\Omega} \phi''_M(E(u + \eta))\log\left(\frac{u_k + \eta}{u_{k\infty}}\right) \log\left(\frac{u_i + \eta}{u_{i\infty}}\right) \left(\sum_{j=1}^n A_{ij}(u)\nabla u_j\right)\nabla u_k \,dxd\tau \\
&\quad - \sum_{i=1}^n \int_s^t \int_{\Omega} \phi'_M(E(u + \eta)) \frac{1}{u_i + \eta} \left(\sum_{j=1}^n A_{ij}(u)\nabla u_j\right)\nabla u_i \,dxd\tau \\
&=: I_4(\eta,M) + I_5(\eta,M).
\end{aligned}$$ In order to show the convergence of $I_4$, we use that $|A_{ij}(u)| \leq C\left(1 + \sum_{k=1}^n|u_k| \right)$ and [$\|\nabla u_j\|_{L^2(\Omega \times (0,T))} \leq C(T)$]{} thanks to the regularity of renormalized solutions. Then, recalling $\phi_M''$ has a compact support, we obtain by dominated convergence theorem that $$\begin{aligned}
\lim_{\eta \to 0} I_4(\eta,M) = - \sum_{i,k=1}^n \int_s^t \int_{\Omega} \phi''_M(E(u)) \log\left(\frac{u_k}{u_{k\infty}}\right)\log\left(\frac{u_i}{u_{i\infty}}\right)\left(\sum_{j=1}^n A_{ij}(u)\nabla u_j\right)\nabla u_k\,dxd\tau.
\end{aligned}$$ In a similar way, we obtain $$\begin{aligned}
\lim_{\eta \to 0} I_5(\eta,M) &= - \sum_{i=1}^n \int_s^t \int_{\Omega} \phi'_M(E(u))\frac{1}{u_i}\left(\sum_{j=1}^n A_{ij}(u)\nabla u_j\right)\nabla u_i \,dxd\tau.
\end{aligned}$$ From [@CJ18] we know that if $A(u)$ satisfies , then $$\sum_{i=1}^{n}\frac{1}{u_i}\left(\sum_{j=1}^n A_{ij}(u)\nabla u_j\right)\nabla u_i \geq 4\sum_{i=1}^{n}a_{i0}|{\nabla}\sqrt{u_i}|^2 + \alpha\sum_{i=1}^{n}|{\nabla}u_i|^2,$$ and if $A(u)$ satisfies , then $$\sum_{i=1}^{n}\frac{1}{u_i}\left(\sum_{j=1}^n A_{ij}(u)\nabla u_j\right)\nabla u_i \geq 4\sum_{i=1}^{n}a_{i0}|{\nabla}\sqrt{u_i}|^2 + 2\sum_{i=1}^{n}a_{ii}|{\nabla}u_i|^2 + 2\sum_{i\ne j} a_{ij}|{\nabla}\sqrt{u_iu_j}|^2.$$ From both cases we infer, by noticing that $a_{i0}>0$ and $4|{\nabla}\sqrt{u_i}|^2 = |{\nabla}u_i|^2/u_i$, $$\lim_{\eta\to 0}I_5(\eta,M) \leq - \sum_{i=1}^n \int_s^t \int_{\Omega} \phi'_M(E(u))a_{i0}\frac{|\nabla u_i|^2}{u_i}\,dxd\tau.$$ Putting everything together yields from $$\begin{aligned}
\label{eq.BigStar}
\int_{\Omega}\phi_M(E(u))\,dx\,\biggr|_s^t &\leq - \sum_{i,k=1}^n \int_s^t \int_{\Omega}\phi_M''(E(u))\log\left(\frac{u_k}{u_{k\infty}}\right)\log\left(\frac{u_i}{u_{i\infty}}\right)\left(\sum_{j=1}^n A_{ij}(u)\nabla u_j\right)\nabla u_k \,dxd\tau \\
&- \sum_{i=1}^n a_{i0}\int_s^t \int_{\Omega}\phi_M'(E(u))\frac{|\nabla u_i|^2}{u_i}\,dxd\tau
+ \sum_{i=1}^n \int_s^t \int_{\Omega}\phi'_M(E(u))\log\left(\frac{u_i}{u_{i\infty}}\right)f_i(u)\,dxd\tau \notag \\
&=: I_6(M) + I_7(M) + I_8(M). \notag
\end{aligned}$$ Our goal now is to pass to the limit $M\to \infty$ in . For the left-hand side of , the convergence is clear due the dominated convergence theorem. For $I_7$ we can use $\sqrt{u_i} \in L^2(0,T; H^1(\Omega))$ and the dominated convergence theorem to obtain $$\begin{aligned}
\color{black}{\lim_{M \to \infty} I_7(M) = -\sum_{i=1}^n a_{i0}\int_s^t \int_{\Omega} \frac{|\nabla u_i|^2}{u_i}\,dxd\tau.}
\end{aligned}$$ Since $\sum_{i=1}^n \log\left(\frac{u_i}{u_{i\infty}}\right)f_i(u)\leq 0$, we get by Fatou’s lemma that $$\begin{aligned}
\limsup_{M \to \infty} I_8(M) \leq \sum_{i=1}^n \int_s^t \int_{\Omega} \log\left(\frac{u_i}{u_{i\infty}}\right)f_i(u)\,dxd\tau.
\end{aligned}$$ [For $I_6(M)$ we first use the identity $\sum_{j=1}^nA_{ij}(u){\nabla}u_j = a_{i0}{\nabla}u_i + \sum_{j=1}^na_{ij}(u_j{\nabla}u_i + u_i{\nabla}u_j)$ to estimate $I_6(M) \leq I_{61}(M) + I_{62}(M) + I_{63}(M)$ where $$I_{61}(M) = C\sum_{i,k=1}^{n}\int_s^t\int_{\Omega}\left|\phi_M''(E(u))\log\left(\frac{u_k}{u_{k\infty}}\right)\log\left(\frac{u_i}{u_{i\infty}}\right){\nabla}u_i {\nabla}u_k\right|dxd\tau,$$ $$I_{62}(M) = C\sum_{i,j,k=1}^{n}\int_s^t\int_{\Omega}|\phi_M''(E(u))||u_j|\left|\log\left(\frac{u_k}{u_{k\infty}}\right){\nabla}u_k\right|\left|\log\left(\frac{u_i}{u_{i\infty}}\right){\nabla}u_i\right|dxd\tau,$$ $$I_{63}(M) = C\sum_{i,j,k=1}^{n}\int_s^t\int_{\Omega}\left|\phi_M''(E(u))\right|\left|\log\left(\frac{u_i}{u_{i\infty}}\right)u_i\right|\left|\log\left(\frac{u_k}{u_{k\infty}}\right){\nabla}u_k\right||{\nabla}u_j|dxd\tau.$$ For $I_{61}(M)$ we write ${\nabla}u_i{\nabla}u_k = 4\sqrt{u_i}\sqrt{u_k}({\nabla}\sqrt{u_i}{\nabla}\sqrt{u_k})$, then we use the property of $\phi_M''$ in to estimate $$\begin{aligned}
\left|\phi_M''(E(u))\log\left(\frac{u_k}{u_{k\infty}}\right)\log\left(\frac{u_i}{u_{i\infty}}\right)\right|\sqrt{u_i}\sqrt{u_k} \leq C \frac{\left|\log\left(\frac{u_k}{u_{k\infty}}\right)\right| \left|\log\left(\frac{u_i}{u_{i\infty}}\right)\right|\sqrt{u_k}\sqrt{u_i}}{1 + \sum_{j=1}^n u_j(\log(1 + u_j))^2} \leq C.
\end{aligned}$$ Hence, from the bound $\|{\nabla}\sqrt{u_i}\|_{L^2(\Omega\times(0,T))} \leq C(T)$ we obtain by dominated convergence that $\lim_{M\to+\infty}I_{61}(M) = 0$. To estimate $I_{62}(M)$ we have first $$\begin{aligned}
\left|\log\left(\frac{u_k}{u_{k\infty}} \right){\nabla}u_k \right| &\leq \chi_{\{u_k \geq 1\}}\left|\log\left(\frac{u_k}{u_{k\infty}} \right)\right|\left|{\nabla}u_k \right| + 2\chi_{\{0\leq u_k \leq 1\}}\left|\log\left(\frac{u_k}{u_{k\infty}} \right)\sqrt{u_k}\right|\left|{\nabla}\sqrt{u_k} \right|\\
&\leq \chi_{\{u_k \geq 1\}}\left|\log\left(\frac{u_k}{u_{k\infty}} \right)\right|\left|{\nabla}u_k \right| + C|{\nabla}\sqrt{u_k}|,
\end{aligned}$$ and similarly $\left|\log\left(\frac{u_i}{u_{i\infty}} \right){\nabla}u_i \right| \leq \chi_{\{u_i \geq 1\}}\left|\log\left(\frac{u_i}{u_{i\infty}} \right)\right|\left|{\nabla}u_i \right| + C|{\nabla}\sqrt{u_i}|.$ Therefore $$I_{62}(M) \leq C\sum_{i,j,k=1}^n\int_s^t\int_{\Omega}\biggl(J_1(M)|{\nabla}u_k||{\nabla}u_i| + J_2(M)|{\nabla}u_k||{\nabla}\sqrt{u_i}| + J_3(M)|{\nabla}\sqrt{u_k}||{\nabla}u_i| + J_4(M)|{\nabla}\sqrt{u_k}||{\nabla}\sqrt{u_i}|\biggr)dxd\tau$$ with $$J_1(M) = |\phi_M''(E(u))||u_j|\chi_{\{u_k \geq 1\}}\chi_{\{u_i \geq 1\}}\left|\log\left(\frac{u_i}{u_{i\infty}}\right)\right|\left|\log\left(\frac{u_k}{u_{k\infty}}\right)\right|, \quad J_4(M) = |\phi_M''(E(u))||u_j|,$$ $$J_2(M) = |\phi_M''(E(u))||u_j|\chi_{\{u_k \geq 1\}}\left|\log\left(\frac{u_k}{u_{k\infty}}\right)\right|, \quad J_3(M) = |\phi_M''(E(u))||u_j|\chi_{\{u_i \geq 1\}}\left|\log\left(\frac{u_i}{u_{i\infty}}\right)\right|.$$ Using we see that $|J_i(M)| \leq C$ for all $i=1,\ldots, 4$. Taking into account that $\|{\nabla}u_i\|_{L^2(\Omega\times(0,T))}, \|{\nabla}\sqrt{u_i}\|_{L^2(\Omega\times(0,T))}\leq C(T)$ we conclude by the dominated convergence theorem that $\lim_{M \to \infty}I_{62}(M) = 0$. The proof of $\lim_{M \to \infty} I_{63}(M) = 0$ is similar so we omit it.]{} Consequently, by collecting all results together and using the fact that $$\label{ineq.new}
\sum_{i=1}^nf_i(u)(\log u_i - \log u_{i\infty}) = -\sum_{r=1}^{R}k_ru_{\infty}^{y_r}\Psi\left(\frac{u^{y_r}}{u_\infty^{y_r}},\frac{u^{y_r'}}{u_\infty^{y_r'}} \right)\leq 0,$$ (see the computations in [@DFT17 Proposition 2.1]), we obtain the desired result.
\[cons\] When $m>0$, for any renormalized solution $u$ to it holds that $${{\mathbb Q}}\overline{u}(t) = {{\mathbb Q}}\overline{u}_0 \quad \text{ for all } \quad t>0.$$
We denote by $q = (q_1,\ldots,q_n)$ an arbitrary row of ${{\mathbb Q}}$. Thus, we have that $ \sum_{i=1}^n q_if_i(u) = 0. $ Let $\phi_M$ be chosen in the same way as in the proof Lemma \[weak-eep\]. By choosing $\xi$ as $\color{black}{\xi(u) = \phi_M\left(\beta \sum_{i=1}^n q_i u_i + E(u+\eta|u_\infty)\right) }$ where $\beta \in \mathbb{R}$ and $\psi=1$ in the definition of renormalized solutions, we can pass to the limits $\eta \to 0$ and $M \to +\infty$ like in the proof of Lemma \[weak-eep\] to obtain $$\begin{aligned}
\left(\beta \sum_{i=1}^n \int_{\Omega} q_i u_i \,dx + {\mathscr E}(u|u_\infty)\right)\,\biggr|_0^T \leq \int_0^T{\mathscr D}(u(\tau))\,d\tau.
\end{aligned}$$ By dividing both sides by $\beta >0$ and letting $\beta \to +\infty$, we get that $$\begin{aligned}
\sum_{i=1}^n \int_{\Omega} q_i u_i(T)\,dx \leq \sum_{i=1}^n \int_{\Omega}q_i u_{i0}(x)\,dx.
\end{aligned}$$ Repeating the arguments with $\beta<0$ and letting $\beta \to -\infty$, we obtain that $\sum_{i=1}^n \int_{\Omega} q_i u_i(T)\,dx \geq \sum_{i=1}^n \int_{\Omega}q_i u_{i0}(x)\,dx,$ which finishes the proof of the conservation laws.
We are now ready to give the proof of the main result.
The existence of a global renormalized solution follows from [@CJ17 Theorem 1] since under the complex balanced condition the reactions satisfy , which is (H4) in [@CJ17] with $\pi_i = 1$ and $\lambda_i = -\log u_{i\infty}$ for all $i=1,\ldots, n$.
We now turn to the convergence to equilibrium. Since the system possesses no boundary equilibria, it follows from [@FT17a Theorem 1.1] that $ {\mathscr D}(u) \geq \lambda {\mathscr E}(u|u_\infty)$ for all measurable nonnegative functions $u$ satisfying ${{\mathbb Q}}\overline{u} = {{\mathbb Q}}u_\infty$, where $\lambda>0$ is an explicit constant up to a finite dimensional inequality . Note that this inequality does not require any other higher regularity of $u$. Therefore, thanks to Lemma \[cons\], for any renormalized solution to it holds $${\mathscr D}(u(s)) \geq \lambda {\mathscr E}(u(s)|u_\infty) \quad \text{ for a.e. } \quad s>0.$$ Using this and Lemma \[weak-eep\] it follows that $${\mathscr E}(u(t)|u_\infty) + \lambda\int_s^t{\mathscr E}(u(\tau)|u_\infty)d\tau \leq {\mathscr E}(u(s)|u_\infty) \quad \text{ for a.e. } \quad t>s.$$ By Gronwall’s inequality we get $${\mathscr E}(u(t)|u_\infty) \leq e^{-\lambda t}{\mathscr E}(u_0|u_\infty),$$ and a Csiszár-Kullback-Pinsker type inequality (see e.g. [@FT17a Lemma 2.2]) completes the proof of Theorem \[thm:main\].
[**Acknowledgements:**]{} Both authors would like to thank Prof. Ansgar Jüngel for the fruitful discussions. The first author acknowledges partial support from the Austrian Science Fund (FWF), grants P27352 and P30000, while the second author is partially supported by the International Training Program IGDK 1754 and NAWI Graz.
[11]{} J. A. Cañizo, L. Desvillettes, and K. Fellner. Improved duality estimates and applications to reaction-diffusion equations. *Comm. Partial Differential Equations* 39 (2014) no.6, 1185–1204.
X. Chen, E. S. Daus, and A. Jüngel. Global existence analysis of cross-diffusion population systems for multiple species. *Arch. Ration. Mech. Anal.* 227 (2018), no. 2, 715–747.
L. Chen and A. Jüngel. Analysis of a multi-dimensional parabolic population model with strong cross-diffusion. [*SIAM J. Math. Anal.*]{} 36 (2004), 301–322.
L. Chen and A. Jüngel. Analysis of a parabolic cross-diffusion population model without self-diffusion. [*J. Diff. Eqs.*]{} 224 (2006), 39–59.
X. Chen and A. Jüngel. Global renormalized solutions to reaction-cross-diffusion systems. [*arXiv:1711.01463*]{}.
X. Chen and A. Jüngel. A note on the uniqueness of weak solutions to a class of cross-diffusion systems. To appear in [*J. Evol. Eqs.*]{}, 2018.
X. Chen and A. Jüngel. Weak-strong uniqueness of renormalized solutions to reaction-cross-diffusion systems. [*arXiv:1805.02950v1*]{}.
E. S. Daus, A. Jüngel, and B. Q. Tang. Exponential time decay of solutions to reaction-cross-diffusion systems of Maxwell-Stefan type. [*arXiv:1802.10274*]{}. L. Desvillettes and K. Fellner. Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations. [*J. Math. Anal. Appl.*]{} 319 (2006), 157–176. L. Desvillettes, K. Fellner, and B. Q. Tang. Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks. [*SIAM J. Math. Anal.*]{} 49 (2017), 2666–2709. L. Desvillettes and K. Fellner. Exponential convergence to equilibrium for nonlinear reaction-diffusion systems arising in reversible chemistry. In: C. Pötzsche, C. Heuberger, B. Kaltenbacher, and F. Rendl (eds.). [*System Modeling and Optimization*]{}. CSMO 2013, IFIP Advances in Information and Communication Technology, vol. 443, pp. 96–104. Springer, Berlin, 2014.
L. Desvillettes, T. Lepoutre, and A. Moussa. Entropy, duality, and cross diffusion. [*SIAM J. Math. Anal.*]{} 46 (2014), 820-853.
L. Desvillettes, T. Lepoutre, A. Moussa, and A. Trescases. On the entropic structure of reaction-cross diffusion systems. [*Commun. Partial Diff. Eqs.*]{} 40 (2015), 1705-1747.
M. Feinberg. The existence and uniqueness of steady states for a class of chemical reaction networks. [*Arch. Rational Mech. Anal.*]{} 132 (1995), 311–370.
J. Fischer. Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems. [*Arch. Rational Mech. Anal.*]{} 218 (2015), no.1, 553–587.
J. Fischer. Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations. *Nonlinear Anal.* 159 (2017), 181–207.
K. Fellner, E. Latos, and B. Q. Tang. Global regularity and convergence to equilibrium of reaction-diffusion systems with nonlinear diffusion. *arXiv:1711.02897*. K. Fellner and B. Q. Tang. Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems. [*Z. Angew. Math. Phys.*]{}, 69.3 (2018). A. Jüngel. The boundedness-by-entropy method for cross-diffusion systems. [*Nonlinearity*]{} 28 (2015), 1963–2001. A. Jüngel. [*Entropy Methods for Diffusive Partial Differential Equations*]{}. BCAM SpringerBriefs, 2016. N. Shigesada, K. Kawasaki, and E. Teramoto. Spatial segregation of interacting species. [*J. Theor. Biol.*]{} 79 (1979), 83-99.
[^1]:
[^2]: This should not be confused with the detailed balance condition occuring in reactions . Also note that in [@CJ17] the detailed balance diffusion condition was $\pi_i a_{ij} = \pi_j a_{ji}$ for some positive constants $\pi_i>0$. Here we choose $\pi_i=1$ for $i=1,\ldots, n$ for the compatibility with the reactions.
|
---
abstract: |
We use the graded eigenvalue method, a variant of the supersymmetry technique, to compute the universal spectral correlations of the QCD Dirac operator in the presence of massive dynamical quarks. The calculation is done for the chiral Gaussian unitary ensemble of random matrix theory with an arbitrary Hermitian matrix added to the Dirac matrix. This case is of interest for schematic models of QCD at finite temperature.\
[*PACS:*]{} 11.30.Rd; 12.38.Lg; 12.38.Aw\
[*Keywords:*]{} Spectrum of the QCD Dirac operator; Chiral random matrix models; Finite temperature models
address:
- 'Institut für Theoretische Physik, Technische Universität München, D-85747 Garching, Germany'
- 'Max-Planck-Institut für Kernphysik, Postfach 103980, D-69029 Heidelberg, Germany'
author:
- Burkhard Seif
- Tilo Wettig
- Thomas Guhr
date: 17 February 1999
title: Spectral correlations of the massive QCD Dirac operator at finite temperature
---
,
,
Introduction {#intro}
============
By now, it has been firmly established that the spectrum of the QCD Dirac operator possesses a number of universal features which can be described by chiral random matrix theory (RMT) [@Leut92; @Shur93]. In particular, the RMT predictions agree very well with data from lattice gauge simulations, both for the eigenvalue correlations in the bulk of the spectrum on the scale of the mean level spacing [@Hala95; @Mark98; @guhr98] and for the distribution and correlations of the low-lying eigenvalues [@Berb98a; @Ma98; @Berb98b; @Damg98]. It is the key assumption of chiral RMT that the matrix elements of the Dirac operator in an appropriate energy basis behave as random numbers. This concept has already been very successful in many other areas of physics, see the detailed review in Ref. [@review]. Thus, it is fair to say that RMT approaches can be viewed as thermodynamics for spectral fluctuations and related properties. In the context of QCD, deviations from pure RMT statistics have also been found in the microscopic [@osborn98; @bitsch98] and in the bulk region [@guhr98]. These findings provide evidence for the conjecture that lattice QCD may have much in common with disordered systems [@stern98; @janik98; @osborn98]. In the present work, however, we focus on some formal and theoretical aspects of chiral RMT. Hence, we shall not discuss in any detail the physical applications but refer to the existing literature [@Verb97; @Berb98c].
Our aim here is to compute the universal spectral correlations of the QCD Dirac operator in the presence of massive dynamical quarks for the chiral Gaussian unitary ensemble of RMT. Such a calculation has been done previously at zero temperature using orthogonal polynomials [@Damg97a; @Wilk98] and the finite volume partition function [@Akem98a; @Akem98b]. In Ref. [@Toub98], the connection between these two results was established in the framework of partially quenched chiral perturbation theory. In this paper, we employ the graded eigenvalue method [@Guhr91; @guhr96a; @guhr96b], which is a special variant of the supersymmetry method [@Efet83; @Verb85], for the following two reasons. First, it allows us to extent the calculation to the case where an arbitrary deterministic Hermitian matrix is added to the Dirac matrix. This is relevant for schematic random-matrix models of QCD at finite temperature [@Jack95; @Wett96]. The standard orthogonal-polynomial method cannot easily be applied in this case, because a certain rotation invariance in the space of the random matrices is lost. Second, the present problem leads to an interesting extension of the graded eigenvalue method in the context of chiral RMT. In ordinary RMT, this method was developed to calculate spectral correlations in crossover transitions from regularity to chaos [@guhr96a; @guhr96b]. This method was then extended to chiral RMT and used to compute the universal spectral correlations of the Dirac operator in the quenched approximation, i.e., without dynamical quarks [@Guhr97a; @Jack97a]. For an application of the supersymmetry method to the case of one flavor at zero temperature, see Ref. [@Jurk96].
In Refs. [@Damg97a; @Split98] it was shown that the RMT results are invariant under deformations of the distribution of the random matrix. While we suspect this statement to hold also for the results computed in the present paper, a rigorous proof would require an extension of the work of Ref. [@Zinn98]. We shall not address this issue here.
The outline of this paper is as follows. In Sec. \[setup\], we define the problem and outline the main idea for the solution. The graded eigenvalue method is applied in Sec. \[susy\]. The special case of energies and masses on the scale of the mean level spacing near zero, which is also the most interesting case for physical applications, is considered in Sec. \[micro\]. We conclude with a summary in Sec. \[summary\]. Technical details of the calculation are discussed in two appendixes.
Setup of the calculation {#setup}
========================
Since this paper is a natural extension of Ref. [@Guhr97a], we shall attempt to use a similar notation. The QCD Dirac operator in Euclidean space is defined by $D=\gamma_\mu\partial_\mu+ig\gamma_\mu
A_\mu$, where $g$ is the coupling constant and the $A_\mu$ are the gauge fields. Note that $D$ is anti-Hermitian. In a random matrix model in a chiral basis, the matrix $A$ representing the Dirac operator has the form [@Shur93] $$\label{eq1.1}
D\longrightarrow iA=
i\begin{bmatrix}0 & W+Y \\ W^\dagger+Y & 0\end{bmatrix}\:,$$ where $W$ is a square random matrix of dimension $N$ and $Y$ is an arbitrary Hermitian matrix. Expression is a [ *schematic*]{} model for the QCD Dirac operator at finite temperature. The matrix $Y$ represents the effects of the temperature on the Dirac spectrum. Its specific form depends on the choice of basis states [@Jack95; @Wett96]. Since we consider an arbitrary Hermitian matrix $Y$, we cover all possible choices of basis states.
One could also consider the more general problem of a rectangular matrix $W$, giving rise to exact zero modes of the Dirac operator whose number can be identified with the topological charge. At $Y=0$, this is not necessary since this problem is equivalent to introducing additional massless flavors [@Verb94a]. Although it is not obvious, we expect this equivalence to hold also for nonzero $Y$. We hope to address this problem in future work.
We will be interested in the correlations of the eigenvalues of the matrix $A$. In this paper, we study the chiral Gaussian unitary ensemble (chGUE) appropriate for QCD with three or more colors for which $W$ is a complex matrix without any symmetries [@Verb94a]. The probability distribution of $W$ is given by $$\begin{aligned}
\label{eq1.2}
P(W)&=\frac{1}{\mathcal{N}}\,P_0(W)\prod_{f=1}^{N_f} \det(im_f-A)\:,
\intertext{with}
\label{eq1.2a}
P_0(W)&=\exp\left(-N\Sigma^2\operatorname{tr}WW^\dagger\right)\:.\end{aligned}$$ Here, $N_f$ is the number of quark flavors with masses $m_f$, $\mathcal{N}$ is a normalization factor (see below), and $\Sigma$ is a real parameter which will turn out to be equal to the chiral condensate at zero temperature. Note that $\mathcal{N}$ depends on the quark masses. To be precise, the argument of the determinant in should have been $m_f+D=m_f+iA$. For convenience, we have pulled out a factor of $-i$ and absorbed it in $\mathcal{N}$.
We are interested in the $k$-point spectral correlation functions, defined as the probability of finding energies in infinitesimal intervals around the points $x_1,\dots,x_k$, regardless of labeling. Apart from some trivial contributions involving $\delta$-functions [@review] they are given by $$\label{eq1.3}
R_k(x_1,\dots,x_k)=\left(-\frac{1}{\pi}\right)^k \int d[W] P(W)
\prod_{p=1}^k \operatorname{Im}\operatorname{tr}\frac{1}{x_p^+-A} \:,$$ where $x_p^+=x_p+i\varepsilon$ with $\varepsilon$ positive infinitesimal. The measure $d[W]$ is simply the product of the differentials of the real and imaginary parts of the elements of $W$, i.e., of all independent variables. The integrations extend from $-\infty$ to $+\infty$. The normalization factor $\mathcal{N}$ is determined by the requirement $\int d[W]P(W)=1$. Since both the distribution (\[eq1.2\]) and the measure $d[W]$ are invariant under unitary transformations of $W$, only the relative unitary rotation between $W$ and $Y$ matters and we can, without loss of generality, write the Hermitian matrix $Y$ in diagonal form, $Y=\operatorname{diag}(y_1,\dots,y_N)$. Advantageously, the $k$-point functions can be obtained as $$\label{eq1.6}
R_k(x_1,\dots,x_k)=\left(-\frac{1}{\pi}\right)^k
\left.\frac{\partial^k}{\prod_{p=1}^k \partial J_p}Z_k(J)
\right|_{J_p=0}$$ with a generating function given by $$\label{eq1.5}
Z_k(J) = \int d[W] P(W) \prod_{p=1}^k\det(x_p-A)
\,\operatorname{Im}\frac{1}{\det(x_p^+-J_p-A)} \:,$$ where $J$ stands for $J_1,\dots,J_k$. The starting point of the graded eigenvalue method is to rewrite the determinants in as Gaussian integrals over commuting and anti-commuting variables. Note that the distribution $P(W)$ contains $N_f$ determinants in the numerator. To retain the determinant structure of the problem, it is highly desirable to have an equal number of determinants in numerator and denominator so that the bosonic and fermionic blocks in the supersymmetric representation of the generating function have the same size. Therefore, we introduce $N_f$ additional determinants in the denominator and write, in the large-$N$ limit, $$\label{eq1.7a}
Z_k(J)=\lim_{\{a_f\}\to\infty}\tilde Z_k(J)$$ with $$\begin{aligned}
\label{eq1.7}
\tilde Z_k(J)= \frac{1}{\mathcal{\tilde{N}}}\int d[W] \, P_0(W) &
\prod_{p=1}^k\det(x_p-A)\,\operatorname{Im}\frac{1}{\det(x_p^+-J_p-A)}\nonumber\\
\times&\prod_{f=1}^{N_f}\det(im_f-A)\,\operatorname{Im}\frac{1}{\det(a_f^+-A)}\:,\end{aligned}$$ where the $a_f$ ($f=1,\dots,N_f$) are dummy real variables. The modified normalization $\mathcal{\tilde{N}}$ depends on the $a_f$ and is given by $$\label{eq1.8}
\tilde{\mathcal{N}} = \int d[W] \, P_0(W) \prod_{f=1}^{N_f}
\det(im_f-A)\,\operatorname{Im}\frac{1}{\det(a_f^+-A)}\:.$$ The fact that Eq. holds in the limit $N\to\infty$ is proved in App. \[app:Zktilde\]. The introduction of the dummy determinants in is the main idea of the present calculation. As we shall see below, it allows us to use the results of Ref. [@Guhr97a] so that the generalization from the quenched approximation to the case with $N_f>0$ can be obtained with moderate effort.
We observe that is essentially a special case of with $k=0$. To simplify the notation, we will compute the generic quantity $$\label{generic}
G_\gamma(t)=\int d[W] \, P_0(W)\prod_{j=1}^\gamma\det(t_{j2}-A)
\,\operatorname{Im}\frac{1}{\det(t_{j1}^+-A)}\:,$$ where $\gamma$ is a nonnegative integer and $t=\operatorname{diag}(t_{11},\dots,t_{\gamma 1},t_{11},\dots,t_{\gamma 2})$ is a diagonal graded matrix of dimension $2\gamma$. Both and can be obtained from by choosing $\gamma$ and $t$ appropriately.
Supersymmetric representation and graded eigenvalue method {#susy}
==========================================================
Since we can employ the results of Ref. [@Guhr97a] to compute the function very efficiently, we only review the major steps in the derivation. First, the determinants in are rewritten as Gaussian integrals over commuting and anti-commuting variables which are arranged in a graded (or super) vector $\psi$. Then, the integration over $W$ can be performed, resulting in fourth-order terms in the $\psi$-variables. These terms can be removed by a Hubbard-Stratonovitch transformation at the expense of introducing additional integration variables which can be arranged in a complex graded (or super) matrix $\sigma$. The order of the integrations over $\sigma$ and $\psi$ can then be interchanged, provided that one is only interested in the imaginary parts in . This point has been discussed in Refs. [@Guhr97a; @Jack97a; @Jack96b]. The $\psi$-integration can then be performed trivially. The graded matrix $\sigma$ can be written in spherical coordinates as $us\bar v$ with graded (or super) unitary matrices $u$ and $\bar v$ and radial coordinates $s$. The integration over $u$ and $\bar v$ can be performed using the supersymmetric generalization of the Berezin-Karpelevich integral [@Guhr96]. It can be viewed as the extension of the supersymmetric Itzykson-Zuber integral [@Guhr91] to complex graded matrices. The final result for the function then becomes (see Eqs. (33) through (36) of Ref. [@Guhr97a]) $$\label{eq2.1}
G_\gamma(t)=\left(\frac{\pi}{N\Sigma^2}\right)^{N^2}
\frac{\exp(-N\Sigma^2\operatorname{trg}t^2)}{B_\gamma(t^2)}
\det[C_N(t_{i1},t_{j2})]_{i,j=1,\dots,\gamma}$$ with $$\begin{aligned}
\label{eq2.2}
C_N(x_1,x_2)=(2N\Sigma^2&)^2
\int\limits_0^\infty\int\limits_0^\infty ds_1ds_2
\frac{s_1s_2}{s_1^2+s_2^2}\exp\left(-N\Sigma^2(s_1^2+s_2^2)\right)
\nonumber\\
&\times I_0(2N\Sigma^2s_1x_1)J_0(2N\Sigma^2s_2x_2)\operatorname{Im}\prod_{n=1}^N
\frac{y_n^2+s_2^2}{y_n^2-(s_1^+)^2}\:,\end{aligned}$$ where $J$ and $I$ denote the Bessel and modified Bessel function, respectively. In Eq. , the symbol trg denotes the graded trace, and $$\label{berezinian}
B_\gamma(t^2)=\frac{\Delta_\gamma(t_1^2)\Delta_\gamma(t_2^2)}
{\prod_{ij}(t_{i1}^2-t_{j2}^2)}
\qquad\text{with}\qquad
\Delta_\gamma(x)=\prod_{i>j}^\gamma(x_i-x_j)\:.$$ Equations and can now be obtained by choosing $\gamma=k+N_f$ and $\gamma=N_f$, respectively, and substituting appropriate values for the entries of $t$. Performing the differentiations according to Eq. (with $Z_k(J)$ replaced by $\tilde Z_k(J)$, see Eq. ) then yields the $k$-point functions. We obtain after some algebra $$\label{eq2.5}
R_k(x_1, \dots, x_k) = \left(\frac2\pi\right)^k
\left(\prod_{p=1}^kx_p\right)
\lim_{\{a_f\}\to\infty}
\frac{\det[C_N(z_p,\zeta_q)]_{p,q=1,\dots,k+N_f}}
{\det[C_N(a_f,im_g)]_{f,g=1,\dots,N_f}}$$ with $$\begin{aligned}
\label{z_p}
z_p=&
\begin{cases}
x_p\phantom{ww}& \text{for $p=1, \dots, k$,}\\
a_{p-k}& \text{for $p=k+1, \dots, k+N_f$,}
\end{cases}
\\[3mm]
\label{zeta_q}
\zeta_q=&
\begin{cases}
x_q\phantom{ww}& \text{for $q=1, \dots, k$,}\\
im_{q-k}& \text{for $q=k+1, \dots, k+N_f$.}
\end{cases}\end{aligned}$$ Note that Eq. , just like Eq. , holds in the limit $N\to\infty$ (which is the interesting limit for physical applications) but not for finite $N$. Instead of introducing dummy determinants in Eq. , there is an alternative way to proceed which is exact for finite $N$. We briefly explain the idea for readers familiar with the graded eigenvalue method. Without the introduction of the dummy determinants in Eq. , the transformation from ordinary space to superspace leads to a graded (or super) matrix $\sigma$ whose boson-boson and fermion-fermion blocks have dimension $k$ and , respectively. The transformation of $\sigma$ to spherical coordinates involves a Berezinian which cannot be written as a determinant. The idea now is to enlarge the boson-boson block of $\sigma$ to dimension $k+N_f$ by introducing dummy integration variables in superspace. Then, the Berezinian resulting from the enlarged $\sigma$-matrix can be written as a determinant which is the prerequisite for expressing the $k$-point function in form of a determinant. However, we will not discuss this alternative way since we are only interested in the limit $N\to\infty$ for which the method of Eqs. through appears to be more economic.
Our conventions are such that the support of the spectral density is of order $\mathcal{O}(1)$ and the typical level spacing is of order $\mathcal{O}(1/N)$. While Eq. holds for all values of the $x_p$ and $m_f$, we are particularly interested in the microscopic region where the $x_p$ and $m_f$ are of order $\mathcal{O}(1/N)$. We now turn to this limit.
Microscopic limit {#micro}
=================
If $x_1$ and $x_2$ in Eq. are of order $\mathcal{O}(1/N)$, the integrals can be performed in saddle-point approximation in the large-$N$ limit. This was done in Ref. [@Guhr97a], and we obtain the Bessel kernel (see Eq. (63) of [@Guhr97a]) $$\label{eq3.1}
C_N(x_1,x_2)= \pi N\Xi\frac{x_1J_1(2N\Xi x_1)J_0(2N\Xi x_2)-
x_2J_0(2N\Xi x_1)J_1(2N\Xi x_2)}{x_1^2-x_2^2}\;,$$ where $\Xi$ is the only real and positive solution of $$1=\frac{1}{N}\sum_{n=1}^N\frac{1}{(\Sigma y_n)^2+(\Xi/\Sigma)^2}$$ or zero if no such solution exists [@Guhr97a]. As we shall show below, $\Xi=\Xi(Y)$ can be identified with the chiral condensate in the presence of the arbitrary offset $Y$. We now rescale the energies, the masses, and the dummy variables by $2N\Xi$ and define $u_p=2N\Xi x_p$, $\mu_f=2N\Xi m_f$, and $\alpha_f=2N\Xi a_f$. We thus obtain from $$\label{eq3.3}
R_k(x_1,\dots,x_k) = (2N\Xi)^k\left(\prod_{p=1}^ku_p\right)
\lim_{\{\alpha_f\}\to\infty}
\frac{\det[C(\tilde z_p,\tilde \zeta_q)]_{p,q=1,\dots,k+N_f}}
{\det[C(\alpha_f,i\mu_g)]_{f,g=1,\dots,N_f}}$$ with a kernel given by $$\label{kernel}
C(u_1,u_2)=\frac{u_1J_1(u_1)J_0(u_2)-u_2J_0(u_1)J_1(u_2)}
{u_1^2-u_2^2}\:,$$ where we have used the notation $\tilde z_p=2N\Xi z_p$ and $\tilde\zeta_p=2N\Xi\zeta_p$. We shall also need this kernel for the second argument purely imaginary, $$\label{ikernel}
C(u,i\mu)=\frac{uJ_1(u)I_0(\mu)+\mu J_0(u)I_1(\mu)}{u^2+\mu^2}\:,$$ and in the special case $u_1=u_2=u$, $$\label{ukernel}
C(u,u)=\frac12\left[J_0^2(u)+J_1^2(u)\right]\:.$$
The major difficulty now is to perform the limit $\{\alpha_f\}\to\infty$ in . For this purpose, it is convenient to divide both numerator and denominator of the last term in by $\Delta_{N_f}(\alpha)$ and to perform the limits separately in numerator and denominator. Since the derivation is somewhat technical it is presented in App. \[app:limit\]. The final result is $$\label{eq3.5}
\lim_{\{\alpha_f\}\to\infty}
\frac{\det[C(\tilde z_p,\tilde \zeta_q)]_{p,q=1,\dots,k+N_f}}
{\det[C(\alpha_f,i\mu_g)]_{f,g=1,\dots,N_f}}=
\frac{\det[\mathcal{A}_{pq}]_{p,q=1,\dots,k+N_f}}
{\det[\mathcal{B}_{fg}]_{f,g=1,\dots,N_f}}\:,$$ where $$\label{Bmatrix}
\mathcal{B}=
\begin{bmatrix}
I_0(\mu_1)&\cdots&I_0(\mu_{N_f})\\
-\mu_1I_1(\mu_1)&\cdots&-\mu_{N_f}I_1(\mu_{N_f})\\
\vdots&&\vdots\\
(-\mu_1)^{N_f-1}I_{N_f-1}(\mu_1)&\cdots&
(-\mu_{N_f})^{N_f-1}I_{N_f-1}(\mu_{N_f})
\end{bmatrix}$$ and $$\label{Amatrix}
\mathcal{A}=
\begin{bmatrix}
C(u_1,u_1)&\cdots&C(u_1,u_k)&
C(u_1,i\mu_1)&\!\cdots\!&C(u_1,i\mu_{N_f})\\
\vdots&&\vdots&\vdots&&\vdots&\\
C(u_k,u_1)&\cdots&C(u_k,u_k)&
C(u_k,i\mu_1)&\!\cdots\!&C(u_k,i\mu_{N_f})\\
J_0(u_1)&\cdots&J_0(u_k)&&&\\
u_1J_1(u_1)&\cdots&u_kJ_1(u_k)&&&\\
\vdots&&\vdots&&\mathcal{B}&\\
u_1^{N_f-1}J_{N_f-1}(u_1)&\cdots\!&u_k^{N_f-1}J_{N_f-1}(u_k)&&&
\end{bmatrix}$$ In compact notation, we have $$\label{A}
\mathcal{A}_{pq}=
\begin{cases}
C(u_p,u_q)&\text{for $1\le p,q\le k$};\\
C(u_p,i\mu_{q-k})&\text{for $1\le p\le k$; $k+1\le q\le k+N_f$};\\
u_q^{p-k-1}J_{p-k-1}(u_q)&
\text{for $k+1\le p\le k+N_f$; $1\le q\le k$};\\
(-\mu_{q-k})^{p-k-1}I_{p-k-1}(\mu_{q-k})&
\text{for $k+1\le p,q\le k+N_f$}
\end{cases}$$ and $$\label{B}
\mathcal{B}_{fg}=(-\mu_g)^{f-1} I_{f-1}(\mu_g) \qquad
\text{for $1\le f,g\le N_f$}\:.$$ Rescaling the $R_k$ by $(2N\Xi)^{-k}$, we arrive at the final result for the microscopic spectral correlations, $$\begin{aligned}
\label{final}
\rho_k(u_1,\dots,u_k)&
\equiv\frac{1}{(2N\Xi)^k}R_k(x_1,\dots,x_k)\nonumber\\
&=\left(\prod_{p=1}^ku_p\right)
\frac{\det[\mathcal{A}_{pq}]_{p,q=1,\dots,k+N_f}}
{\det[\mathcal{B}_{fg}]_{f,g=1,\dots,N_f}}\:.\end{aligned}$$ It remains to be shown that $\Xi$ can be identified with the absolute value of the chiral condensate $\langle\bar\psi\psi\rangle$. According to the Banks-Casher relation [@Bank80], we have $V|\langle\bar\psi\psi\rangle|=\pi R_1(0)$, where $R_1(0)$ is the spectral density at zero. The space-time volume $V$ can be identified with $2N$. Furthermore, we have $$R_1(0)=2N\Xi\lim_{u\to\infty}\rho_1(u)\:.$$ To compute this limit, we use the matrix $\mathcal{A}$ in with $k=1$ and $u_1=u$. We first observe that $C(u,u)\to1/(\pi u)$ as $u\to\infty$, see . In addition, the denominator of the entries $C(u,i\mu_f)$, see , can be written as a geometric series in $1/u^2$ so that $C(u,i\mu_f)$ is given as an expansion in $1/u^2$ with coefficients containing $I_0(\mu_f)$, $\mu_fI_1(\mu_f)$, etc. By subtracting appropriate multiples of rows 2 to $N_f+1$, the first $N_f$ of these terms can be eliminated, see also the discussion of Eq. in App. \[app:limit\]. The leading large-$u$ behavior of $C(u,u)$ is not modified by these subtractions. Higher-order terms in the expansion of $C(u,i\mu_f)$ are suppressed by powers of $1/u$, the leading term being of order $J_{(N_f+1)\,\text{mod}\,2}(u)/u^{N_f+1}$. Even when multiplied by the largest entry in the lower-left corner of $\mathcal{A}$, $u^{N_f-1}J_{N_f-1}(u)$, the result is suppressed compared to $1/(\pi
u)$. In the large-$u$ limit, the determinant of with $k=1$ thus becomes $1/(\pi u)\cdot\det\mathcal{B}$, and we obtain $$2N\Xi\lim_{u\to\infty}\rho_1(u)=2N\Xi u \frac{1}{\pi u}
=\frac1\pi 2N\Xi$$ and, hence, $2N\Xi=\pi R_1(0)$ as desired. Therefore, we have shown that the functional form of the microscopic spectral correlations in the presence of massive dynamical quarks does not change if a deterministic matrix $Y$ is added to the matrix of the Dirac operator, provided that $\Xi$, i.e., the chiral condensate, is nonzero. The only dependence of the final result on the matrix $Y$ appears in form of a rescaling of the energy scale, from $\Sigma$ at $Y=0$ to $\Xi$ at $Y\ne0$.
Our final result is given in terms of the determinant of a $(k+N_f)\times(k+N_f)$ matrix whose entries are simple functions. This structure arises naturally in the graded eigenvalue method. Two other forms for the microscopic spectral correlations have been obtain previously. In the orthogonal-polynomial method, the result is given as the determinant of a $k\times k$ matrix whose entries are $(N_f+2)\times(N_f+2)$ matrices [@Damg97a; @Wilk98]. From the finite volume partition function, the result is given as the determinant of a $(2k+N_f)\times(2k+N_f)$ matrix whose entries are simple functions [@Akem98b]. At the present time, we do not have a closed mathematical proof that these three results are identical. However, we have perfomed extensive checks for a large number of different values of $k$ and $N_f$, both numerically and using computer algebra. In all cases, the three results agree perfectly so that we do not have any doubt that they are identical. One of the virtues of the present method is that it allows us to include the deterministic matrix $Y$ as well. Moreover, it appears to lead to the most economical representation of the final result.
In Ref. [@Wilk98], the distribution of the smallest eigenvalue, $P(\lambda_{\rm min})$, was also computed. Its universality with respect to deformations of $P_0(W)$ was shown in Ref. [@Nish98]. Since $P(\lambda_{\rm min})$ follows directly from the microscopic spectral density $\rho_1(u)$ [@Wilk98], and since we have shown in this paper that the functional form of the latter quantity is not affected by the addition of $Y$, it follows that the functional form of $P(\lambda_{\rm min})$ also remains unchanged.
Summary
=======
In this paper, we have extended the graded eigenvalue method to the case where massive dynamical quarks enter the distribution of the random matrix. The virtue of this approach is twofold. First, we have obtained a novel representation of results computed previously with other methods. Our representation appears to be the most economical one. It is also very stable numerically, compared to the representations obtained from the orthogonal-polynomial method and the finite-volume partition function. Second, our method allows us to perform the calculation with a deterministic matrix added to the Dirac matrix. This is not easily possible in the standard orthogonal-polynomial method because this approach rests on the rotation invariance of the matrix ensemble which is fully broken due to the presence of the deterministic offset.
We point out again that the microscopic correlation functions computed in RMT are universal in the sense that they are expected to agree with the microscopic spectral correlations of the Dirac operator in full QCD. We hope to be able to compare the random matrix results with data from lattice gauge simulations with dynamical fermions in the near future.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank H.A. Weidenmüller for discussions. This work was supported in part by DFG project We 655/15-1 (TW) and by the Heisenberg foundation (TG).
Derivation of Eq. (\[eq1.7a\]) {#app:Zktilde}
==============================
To show that Eq. holds, we write the matrix $W$ in spherical coordinates, $W=U\Lambda\bar V$ with $U\in\mathrm{U}(N)$, $\bar V\in\mathrm{U}(N)/\mathrm{U}^N(1)$, and $\Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_N)$, where the $\lambda_n$ are real and nonnegative [@Guhr96]. The integration measure transforms according to $$d[W]=J(\Lambda)d[\Lambda]d\mu(U)d\mu(\bar V)$$ with $$d[\Lambda]=\prod_{n=1}^N d\lambda_n \qquad \text{and} \qquad
J(\Lambda)=\Delta_N^2(\Lambda^2)\prod_{n=1}^N \lambda_n\:.$$ The integrations over the $\lambda_n$ extend from 0 to $\infty$, $d\mu(U)$ and $d\mu(\bar V)$ are the invariant Haar measures, and the Vandermonde determinant is defined in Eq. . In the integration over $d[W]$ in Eq. , we shift $W$ and $W^\dagger$ by $-Y$ and obtain $$\begin{aligned}
\label{Zk}
Z_k(J)&=\frac{1}{\mathcal{N}}\int d[\Lambda]d\mu(U)d\mu(\bar V)
J(\Lambda)e^{-N\Sigma^2\operatorname{tr}(W-Y)(W^\dagger-Y)} \prod_{f=1}^{N_f}
\det\begin{bmatrix}im_f&-W\\-W^\dagger&im_f\end{bmatrix}\nonumber\\
&\qquad\qquad\times
\prod_{p=1}^k\det\begin{bmatrix}x_p&-W\\-W^\dagger&x_p\end{bmatrix}
\operatorname{Im}\frac{1}{\det\begin{bmatrix}x_p^+-J_p&-W\\
-W^\dagger&x_p^+-J_p\end{bmatrix}}\nonumber\\
&=\frac{e^{-N\Sigma^2\operatorname{tr}Y^2}}{\mathcal{N}}\int d[\Lambda]
F_N(\Lambda)G_N(\Lambda)H_N(\Lambda)\end{aligned}$$ with $$\begin{aligned}
F_N(\Lambda)&=\Delta_N^2(\Lambda^2)\prod_{n=1}^N
\lambda_n e^{-N\Sigma^2\lambda_n^2}
\prod_{f=1}^{N_f}(-m_f^2-\lambda_n^2)\:,\\
G_N(\Lambda)&=\prod_{p=1}^k \operatorname{Im}\prod_{n=1}^N
\frac{x_p^2-\lambda_n^2}{(x_p^+-J_p)^2-\lambda_n^2}\:,\\
\label{HNdef}
H_N(\Lambda)&=\int d\mu(U)d\mu(\bar V)
e^{N\Sigma^2\operatorname{tr}(W+W^\dagger)Y}\:.\end{aligned}$$ Analogously, we obtain for Eq. $$\label{Zktilde}
\tilde Z_k(J)=\frac{e^{-N\Sigma^2\operatorname{tr}Y^2}}{\tilde\mathcal{N}}
\int d[\Lambda]F_N(\Lambda)G_N(\Lambda)H_N(\Lambda)
\prod_{f=1}^{N_f}\operatorname{Im}\prod_{n=1}^N\frac{1}{(a_f^+)^2-\lambda_n^2}\:.$$ The normalization factors $\mathcal{N}$ and $\tilde\mathcal{N}$ follow by setting $G_N(\Lambda)$ to unity in Eqs. and , respectively. Throughout the remainder of this section, we assume that the dummy variables $a_f$ are pairwise different.
We proceed by converting the product over $n$ in to a sum, $$\label{sum}
\prod_{n=1}^N\frac{1}{(a_f^+)^2-\lambda_n^2}=
\frac{1}{\Delta_N(\Lambda^2)}\sum_{n=1}^N(-1)^{N-n}
\frac{\Delta_{N-1}(\Lambda_{(n)}^2)}{(a_f^+)^2-\lambda_n^2}\:,$$ where the subscript $(n)$ means that $\lambda_n$ is omitted in $\Lambda$. We now have $$\begin{aligned}
\label{delta}
\operatorname{Im}\frac{1}{(a_f^+)^2-\lambda_n^2}&=-\frac{\pi}{2a_f}\left[
\delta(\lambda_n-a_f)+\delta(\lambda_n+a_f)\right]\nonumber\\
&\longrightarrow -\frac{\pi}{2a_f}\,\delta(\lambda_n-a_f)\:,\end{aligned}$$ where in the last step we have used the fact that the integrations over the $\lambda_n$ extend from 0 to $\infty$ and that $a_f>0$ (since we consider the limit $a_f\to\infty$). Thus, out of the $N$ integrations over the $\lambda_n$ in Eq. , $N_f$ can be done using the $\delta$-functions. Using the symmetry of the integrand with respect to the labeling of the $\lambda_n$, we can choose to integrate over the last $N_f$ variables, $\lambda_{N-N_f+1},\dots,\lambda_N$, by relabeling the $\lambda_n$ appropriately and multiplying by a combinatorial factor. (The functions $F_N(\Lambda)$ and $G_N(\Lambda)$ are obviously symmetric under interchanges $\lambda_n \leftrightarrow \lambda_m$, for $H_N(\Lambda)$ this follows from Eq. below.)
We briefly pause to explain the essence of the proof of Eq. . After performing the $N_f$ integrations in Eq. and taking the limit $\{a_f\}\to\infty$, one obtains a result which is essentially equal to the expression for $Z_k(J)$ in Eq. , the only difference being that the integration is over $N-N_f$ variables $\lambda_n$ instead of over $N$ variables. In the limit $N\to\infty$, this difference can be neglected. (There are some additional prefactors which will be canceled by identical factors in the normalization $\tilde\mathcal{N}$.)
By performing the $N_f$ integrations in Eq. using the $\delta$-functions of Eq. the variables $\lambda_{N-N_f+1},\dots,\lambda_N$ are replaced by $a_1,\dots,a_{N_f}$, respectively. We define $\Lambda'=\operatorname{diag}(\lambda_1,\dots,\lambda_{N-N_f})$ and $\bar\Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_{N-N_f},a_1,\dots,a_{N_f})$. The various contributions in the integrand of become $$\begin{aligned}
d[\Lambda]&\longrightarrow d[\Lambda']\:,\\
F_N(\Lambda)&\longrightarrow F_{N-N_f}(\Lambda')\Delta_{N_f}^2(a^2)
\prod_{f=1}^{N_f}a_fe^{-N\Sigma^2a_f^2}\prod_{n=1}^{N-N_f}
(a_f^2-\lambda_n^2)^2\prod_{f'=1}^{N_f}(-m_{f'}^2-a_f^2)\nonumber\\
&\longrightarrow \mathcal{C}_1(a,m)F_{N-N_f}(\Lambda')
\quad\text{as $\{a_f\}\to\infty$}\:,\\
G_N(\Lambda)&\longrightarrow \prod_{p=1}^k \operatorname{Im}\prod_{n=1}^{N-N_f}
\frac{x_p^2-\lambda_n^2}{(x_p^+-J_p)^2-\lambda_n^2}
\prod_{f=1}^{N_f}\frac{x_p^2-a_f^2}{(x_p^+-J_p)^2-a_f^2}\nonumber\\
&\longrightarrow G_{N-N_f}(\Lambda') \quad\text{as
$\{a_f\}\to\infty$}\:. \end{aligned}$$ Here, $\mathcal{C}_1(a,m)$ is a function which no longer depends on the $\lambda_n$. From the product over the imaginary parts in Eq. we obtain with Eqs. and and after appropriate relabeling of the $\lambda_n$ $$\begin{aligned}
\int_0^\infty & d\lambda_{N-N_f+1}\cdots d\lambda_{N}
\prod_{f=1}^{N_f}\operatorname{Im}\prod_{n=1}^N\frac{1}{(a_f^+)^2-\lambda_n^2}
\nonumber\\
&\longrightarrow \frac{1}{\Delta_N^{N_f}(\bar\Lambda^2)}
\frac{(-\pi/2)^{N_f}}{a_1\cdots a_{N_f}}\frac{N!}{(N-N_f)!}
\prod_{f=1}^{N_f}(-1)^{N_f-f}
\Delta_{N-1}(\bar\Lambda_{(N-N_f+f)}^2)\nonumber\\
&\longrightarrow \mathcal{C}_2(a) \quad\text{as $\{a_f\}\to\infty$}\:,\end{aligned}$$ where $\mathcal{C}_2(a)$ is a function which depends only on the $a_f$.
Consider now the angular integrals in Eq. . Using Eq. (2.3) of Ref. [@Guhr96], we have $$\label{HN}
H_N(\Lambda)=
c\:\frac{\det[I_0(\lambda_n \tilde y_m)]_{n,m=1,\dots,N}}
{\Delta_N(\Lambda^2)\Delta_N(Y^2)}\:,$$ where we have defined $\tilde y_m=2N\Sigma^2y_m$. The constant $c$ depends neither on $\Lambda$ nor on $Y$. After the integration over $\lambda_{N-N_f+1},\dots,\lambda_N$, $\Lambda$ is replaced by $\bar\Lambda$, i.e., the last $N_f$ entries are $a_1,\dots,a_{N_f}$. We now expand the determinant in the numerator of Eq. with respect to the last row. Without loss of generality we can assume that $y_1<y_2<\ldots<y_N$. Using the asymptotic behavior of $I_0$, only the term proportional to $I_0(a_{N_f}\tilde y_N)$ remains in the limit $a_{N_f}\to\infty$. The other terms are suppressed by factors of $\exp(-a_{N_f}(\tilde y_N-\tilde y_n))$, where $n=1,\dots,N-1$. Using the same argument for the remaining $N_f-1$ bottom rows, we obtain for $\{a_f\}\to\infty$ $$\begin{aligned}
\det[I_0(\bar\lambda_n \tilde y_m)]_{n,m=1,\dots,N}\longrightarrow &
\det[I_0(\lambda_n \tilde y_m)]_{n,m=1,\dots,N-N_f}\nonumber\\
&\times\det[I_0(a_f\tilde y_{N-N_f+g})]_{f,g=1,\dots,N_f}\:.\end{aligned}$$ Denoting $Y'=\operatorname{diag}(y_1,\dots,y_{N-N_f})$ and $Y''=\operatorname{diag}(y_{N-N_f+1},\dots,y_N)$, we have $$\begin{aligned}
\Delta_N(\bar\Lambda^2)&=\Delta_{N-N_f}({\Lambda'}^2)\Delta_{N_f}(a^2)
\prod_{f=1}^{N_f}\prod_{n=1}^{N-N_f}(a_f^2-\lambda_n^2)\:,\\
\Delta_N(Y^2)&=\Delta_{N-N_f}({Y'}^2)\Delta_{N_f}({Y''}^2)
\prod_{f=1}^{N_f}\prod_{n=1}^{N-N_f}(y_{N-N_f+f}^2-y_n^2)\end{aligned}$$ and thus obtain $$H_N(\Lambda)\longrightarrow \mathcal{C}_3(a,Y) H_{N-N_f}(\Lambda')
\quad\text{as $\{a_f\}\to\infty$}\:.$$ The function $\mathcal{C}_3(a,Y)$ no longer depends on the $\lambda_n$. Note that the last $N_f$ entries of the diagonal matrix $Y$ have effectively disappeared from the problem. However, this effect is negligible in the limit $N\to\infty$.
Collecting the various terms, we finally obtain for $\{a_f\}\to\infty$ $$\begin{aligned}
\tilde Z_k(J)=&\frac{1}{\tilde\mathcal{N}}\,e^{-N\Sigma^2\operatorname{tr}Y^2}\,
\mathcal{C}_1(a,m)\mathcal{C}_2(a)\mathcal{C}_3(a,Y)\nonumber\\
&\times\int d[\Lambda']F_{N-N_f}(\Lambda')G_{N-N_f}(\Lambda')
H_{N-N_f}(\Lambda')\:.\end{aligned}$$ The normalization factor $\tilde\mathcal{N}$ is obtained by setting $G_{N-N_f}(\Lambda')$ to unity so that $$\lim_{\{a_f\}\to\infty}\tilde Z_k(J)=
\frac{\displaystyle{\int d[\Lambda']F_{N-N_f}(\Lambda')
G_{N-N_f}(\Lambda')H_{N-N_f}(\Lambda')}}
{\displaystyle{\int d[\Lambda']F_{N-N_f}(\Lambda')H_{N-N_f}(\Lambda')}}$$ which is equal to $Z_k(J)$ in Eq. with $N$ replaced by $N-N_f$. In the limit $N\to\infty$, this difference is negligible. This completes the proof.
Derivation of Eq. (\[eq3.5\]) {#app:limit}
=============================
We wish to compute the $\{\alpha_f\}\to\infty$ limit in Eq. . It is convenient to divide both numerator and denominator by the Vandermonde determinant $\Delta_{N_f}(\alpha)$, see Eq. , and to compute the $\{\alpha_f\}\to\infty$ limit of the quantity $$\mathcal{R}=
\frac{\det[C(\tilde z_p,\tilde \zeta_q)]_{p,q=1,\dots,k+N_f}}
{\Delta_{N_f}(\alpha)}$$ with a kernel $C$ given in Eq. , $\tilde z_p$ and $\tilde\zeta_q$ defined after Eq. , and $z_p$ and $\zeta_q$ given in Eqs. and , respectively. The denominator in Eq. then follows immediately by setting $k=0$. The $\alpha$-dependence in the numerator determinant of $\mathcal{R}$ is found in the last $N_f$ rows, $$\mathcal{R}=\frac{1}{\Delta_{N_f}(\alpha)}
\begin{vmatrix}
\hdotsfor[1.5]{6}\\
C(\alpha_1,u_1)&\cdots&C(\alpha_1,u_k)&
C(\alpha_1,i\mu_1)&\cdots&C(\alpha_1,i\mu_{N_f})\\
\vdots&&\vdots&\vdots&&\vdots\\
C(\alpha_{N_f},u_1)&\cdots&C(\alpha_{N_f},u_k)&
C(\alpha_{N_f},i\mu_1)&\cdots&C(\alpha_{N_f},i\mu_{N_f})
\nonumber
\end{vmatrix}\:,$$ or, indicating rows and their $\alpha_i$-dependence schematically by $r(\alpha_i)$, $$\mathcal{R}=\frac{1}{\Delta_{N_f}(\alpha)}
\begin{vmatrix}
\hdotsfor[1.5]{3}\\
\cdots&r(\alpha_1)&\cdots\\
&\vdots& \\
\cdots&r(\alpha_{N_f})&\cdots
\end{vmatrix}\:.$$ Before taking the $\{\alpha_f\}\to\infty$ limit, we need to take the limits $\alpha_f\to\alpha_g$ for all $f<g$. Let us start with $\alpha_1$. We subtract the row $r(\alpha_1)$ from all following rows, $r(\alpha_2)$ to $r(\alpha_{N_f})$, without changing the value of the determinant. The Vandermonde determinant supplies factors of the kind $1/(\alpha_i-\alpha_1)$ for $2\le i\le N_f$. Thus, for the $i$-th row $r(\alpha_i)$, we arrive at $$r(\alpha_i)\rightarrow r(\alpha_i)-r(\alpha_1)
\rightarrow \frac{r(\alpha_i)-r(\alpha_1)}{\alpha_i-\alpha_1}
\xrightarrow{\alpha_1 \rightarrow \alpha_i}
\partial_{\alpha_i} r(\alpha_i)\:.$$ Next, the resulting second row $\partial_{\alpha_2} r(\alpha_2)$ is subtracted from all following rows, $\partial_{\alpha_3}r(\alpha_3)$ to $\partial_{\alpha_{N_f}}r(\alpha_{N_f})$. Including factors supplied by the Vandermonde determinant, the $j$-th row $(3 \leq j
\leq N_f)$ thus becomes $$\partial_{\alpha_j}r(\alpha_j)\rightarrow
\partial_{\alpha_j}r(\alpha_j)-\partial_{\alpha_2}r(\alpha_2)
\rightarrow
\frac{\partial_{\alpha_j}r(\alpha_j)-\partial_{\alpha_2}r(\alpha_2)}
{\alpha_j-\alpha_2}
\xrightarrow{\alpha_2 \rightarrow \alpha_j}
\partial_{\alpha_j}^2 r(\alpha_j)$$ Analogous steps for the subsequent rows lead to higher derivatives, at the same time eating up all factors contained in the Vandermonde determinant. Finally, we obtain $$\begin{aligned}
\label{limR}
\lim_{\{\alpha_f\}\to\infty}\mathcal{R}=\lim_{\alpha\to\infty}
\begin{vmatrix}
\hdotsfor[2]{3}\\
\dots&r(\alpha)&\dots\\
\dots& \partial_{\alpha}r(\alpha)&\dots\\
&\vdots&\\
\dots& \partial_{\alpha}^{N_f-1}r(\alpha)&\dots
\end{vmatrix}\:,\end{aligned}$$ where $\alpha$ is now a simple number.
We now consider a generic derivative appearing in . Recall that $r(\alpha)$ stands for $C(\alpha,x)$, where $x$ can be one of the $u_p$ or one of the $i\mu_f$. The following manipulations will become more transparent by considering the first few special cases. We have $$\label{d0}
\partial^0_{\alpha} C(\alpha,x)=C(\alpha,x)
\xrightarrow{\alpha\to\infty}\frac1\alpha J_1(\alpha)J_0(x)\:.$$ The point is that this result is proportional to $J_0(x)$ in the large-$\alpha$ limit. The first derivative becomes $$\begin{aligned}
\label{d1}
\partial^1_{\alpha} C(\alpha,x)\xrightarrow{\alpha\to\infty}&
\left[\frac1\alpha J_0(\alpha)-\frac{2}{\alpha^2}J_1(\alpha)\right]
J_0(x)+\frac{1}{\alpha^2}J_0(\alpha)xJ_1(x)\nonumber\\
\xrightarrow{\phantom{\alpha\to\infty}}&
\:\frac{1}{\alpha^2}J_0(\alpha)xJ_1(x)\:,\end{aligned}$$ where the term proportional to $J_0(x)$ has been eliminated by subtracting an appropriate multiple of the row . Thus, the row containing the first derivative is proportional to $xJ_1(x)$ in the large-$\alpha$ limit. For the second derivative, we proceed analogously and obtain, in the large-$\alpha$ limit, terms proportional to $J_0(x)$, $xJ_1(x)$, and $x^2J_0(x)$. The first two of these terms can be eliminated by subtracting appropriate multiples of the rows and , respectively. Thus, the row containing the second derivative is proportional to $x^2J_0(x)$ in the large-$\alpha$ limit. We now proceed to a general $m-$fold derivative for which it is useful to expand the denominator of $C$ in a geometric series, leading to $$\begin{aligned}
\label{app1.2}
\partial^m_{\alpha} C(\alpha,x)=&
\sum_{l=0}^m \binom{m}{l}
\partial_{\alpha}^l\left(\frac{1}{\alpha^2-x^2} \right)
\partial_{\alpha}^{m-l}\left[
\alpha J_1(\alpha)J_0(x)-xJ_0(\alpha)J_1(x)\right]\nonumber\\
=&\sum_{l=0}^m \binom{m}{l}\partial_\alpha^l\left(
\sum_{n=1}^{\infty}\frac{x^{2n-2}}{\alpha^{2n}}\right)
\partial_{\alpha}^{m-l}\left[
\alpha J_1(\alpha)J_0(x)-xJ_0(\alpha)J_1(x)\right]\nonumber\\
=&\sum_{l=0}^m \binom{m}{l}\sum_{n=1}^{\infty}(-1)^l
\frac{(2n+l-1)!}{(2n-1)!}\frac{x^{2n-2}}{\alpha^{2n+l}}
\nonumber\\
&\!\!\times\left[\left(\alpha J_1^{(m-l)}(\alpha)
+(m-l)J_1^{(m-l-1)}(\alpha) \right)J_0(x)
- xJ_0^{(m-l)}(\alpha)J_1(x)\right] .\end{aligned}$$ It will not be necessary to perform the derivatives of $J_0(\alpha)$ and $J_1(\alpha)$. The expression contains terms proportional to $J_0(x)$, $xJ_1(x)$, $\dots$ up to $x^mJ_0(x)$ (if $m$ is even) or $x^mJ_1(x)$ (if $m$ is odd). All higher order terms in $x$ are suppressed by powers of $\alpha$ in the large-$\alpha$ limit, see the sum over $n$. Subtracting appropriate multiples of previous rows, only the term proportional to $x^mJ_{m\,\text{mod}\,2}(x)$ remains. By using Bessel function recursion relations and adding appropriate multiples of previous rows, $J_{m\,\text{mod}\,2}(x)$ can be replaced by $J_m(x)$. We thus obtain $$\partial^m_{\alpha} C(\alpha,x) \xrightarrow{\alpha\to\infty}
x^mJ_m(x) \times f_m(\alpha)$$ with unspecified functions $f_m(\alpha)$. Defining $$\label{F}
\mathcal{F}=\lim_{\alpha\to\infty}\prod_{m=0}^{N_f-1}f_m(\alpha)$$ and noting that $(i\mu)^mJ_m(i\mu)=(-\mu)^mI_m(\mu)$, we arrive at $$\label{numlimit}
\lim_{\{\alpha_f\}\to\infty}
\frac{\det[C(\tilde z_p,\tilde \zeta_q)]_{p,q=1,\dots,k+N_f}}
{\Delta_{N_f}(\alpha)} = \mathcal{F}\cdot\det\mathcal{A}$$ with $\mathcal{A}$ given in Eq. . Setting $k=0$, we obtain $$\label{denlimit}
\lim_{\{\alpha_f\}\to\infty}
\frac{\det[C(\alpha_f,i\mu_g)]_{f,g=1,\dots,N_f}}
{\Delta_{N_f}(\alpha)} = \mathcal{F}\cdot\det\mathcal{B}$$ with $\mathcal{B}$ given in Eq. . Taking the ratio of and , we finally arrive at . Note that it is not necessary to evaluate since $\mathcal{F}$ drops out of the final result.
[99]{} H. Leutwyler and A.V. Smilga, Phys. Rev. D 46 (1992) 5607. E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. A 560 (1993) 306. M.A. Halasz and J.J.M. Verbaarschot, Phys. Rev. Lett. 74 (1995) 3920; M.A. Halasz, T. Kalkreuter, and J.J.M. Verbaarschot, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 266. R. Pullirsch, K. Rabitsch, T. Wettig, and H. Markum, Phys. Lett. B 427 (1998) 119. T. Guhr, J.-Z. Ma, S. Meyer, and T. Wilke, hep-lat/9806003, Phys. Rev. D (in press). M.E. Berbenni-Bitsch, S. Meyer, A. Schäfer, J.J.M. Verbaarschot, and T. Wettig, Phys. Rev. Lett. 80 (1998) 1146. J.-Z. Ma, T. Guhr, and T. Wettig, Eur. Phys. J. A 2 (1998) 87. M.E. Berbenni-Bitsch, S. Meyer, and T. Wettig, Phys. Rev. D 58 (1998) 071502. P.H. Damgaard, U.M. Heller, and A. Krasnitz, hep-lat/9810060. T. Guhr, A. Müller–Groeling, and H.A. Weidenmüller, Phys. Rep. 299 (1998) 189. J.C. Osborn and J.J.M. Verbaarschot, Phys. Rev. Lett. 81 (1998) 268, Nucl. Phys. B 525 (1998) 738. M.E. Berbenni-Bitsch, M. Göckeler, T. Guhr, A.D. Jackson, J.-Z. Ma, S.Meyer, A. Schäfer, H.A. Weidenmüller, T. Wettig, and T. Wilke, Phys. Lett. B 438 (1998) 14. J. Stern, hep-ph/9801282. R.A. Janik, M.A. Nowak, G. Papp, and I. Zahed, Phys. Rev. Lett. 81 (1998) 264. J.J.M. Verbaarschot, hep-th/9710114. M.E. Berbenni-Bitsch, M. Göckeler, S. Meyer, A. Schäfer, and T. Wettig, hep-lat/9809058. P.H. Damgaard and S.M. Nishigaki, Nucl. Phys. B 518 (1998) 495. T. Wilke, T. Guhr, and T. Wettig, Phys. Rev. D 57 (1998) 6486. P.H. Damgaard, Phys. Lett. B 424 (98) 322; G. Akemann and P.H. Damgaard, Nucl. Phys. B 528 (98) 411. G. Akemann and P.H. Damgaard, Phys. Lett. B 432 (1998) 390. J.C. Osborn, D. Toublan, and J.J.M. Verbaarschot, hep-th/9806110. T. Guhr, J. Math. Phys. 32 (1991) 336. T. Guhr, Phys. Rev. Lett. 76 (1996) 2258. T. Guhr, Ann. Phys. (NY) 250 (1996) 145. K.B. Efetov, Adv. Phys. 32 (1983) 53. J.J.M. Verbaarschot, H.A. Weidenmüller, and M. Zirnbauer, Phys. Rep. 129 (1985) 367. A.D. Jackson and J.J.M. Verbaarschot, Phys. Rev. D 53 (1996) 7223. T. Wettig, A. Schäfer, and H.A. Weidenmüller, Phys. Lett. B 367 (1996) 28. T. Guhr and T. Wettig, Nucl. Phys. B 506 (1997) 589. A.D. Jackson, M.K. Şener, and J.J.M. Verbaarschot, Nucl. Phys. B 506 (1997) 612. J. Jurkiewicz, M.A. Nowak, and I. Zahed, Nucl. Phys. B 478 (1996) 605. K. Splittorff, hep-th/9810248. P. Zinn-Justin, Commun. Math. Phys. 194 (1998) 631. J.J.M. Verbaarschot, Phys. Rev. Lett. 72 (1994) 2531. A.D. Jackson, M.K. Şener, and J.J.M. Verbaarschot, Nucl. Phys. B 479 (1996) 707. T. Guhr and T. Wettig, J. Math. Phys. 37 (1996) 6395. T. Banks and A. Casher, Nucl. Phys. B 169 (1980) 103. S.M. Nishigaki, P.H. Damgaard, and T. Wettig, Phys. Rev. D 58 (1998) 087704.
|
---
abstract: 'The Wisconsin Plasma Astrophysics Laboratory (WiPAL) is a flexible user facility designed to study a range of astrophysically relevant plasma processes as well as novel geometries that mimic astrophysical systems. A multi-cusp magnetic bucket constructed from strong samarium cobalt permanent magnets now confines a 10 m$^3$, fully ionized, magnetic-field-free plasma in a spherical geometry. Plasma parameters of $ T_{e}\approx5$ to $20$ eV and $n_{e}\approx10^{11}$ to $5\times10^{12}$ cm$^{-3}$ provide an ideal testbed for a range of astrophysical experiments including self-exciting dynamos, collisionless magnetic reconnection, jet stability, stellar winds, and more. This article describes the capabilities of WiPAL, along with several experiments, in both operating and planning stages, that illustrate the range of possibilities for future users.'
author:
- 'C.B. Forest , K. Flanagan, M. Brookhart, M. Clark, C.M. Cooper, V. Désangles, J. Egedal, D. Endrizzi, I. V. Khalzov, H. Li, M. Miesch, J. Milhone, M. Nornberg, J. Olson, E. Peterson, F. Roesler, A. Schekochihin, O. Schmitz, R. Siller, A. Spitkovsky, A. Stemo, J. Wallace, D. Weisberg'
- 'E. Zweibel'
bibliography:
- 'WiPALJPP.bib'
title:
- The Wisconsin Plasma Astrophysics Laboratory
- The Wisconsin Plasma Astrophysics Laboratory
---
Introduction
============
During the past five years, a medium-scale multi-investigator experimental plasma facility—the Wisconsin Plasma Astrophysics Laboratory (WiPAL)—has been constructed and is now in operation at the University of Wisconsin-Madison. At the heart of the facility is a high bay that houses a 3 m diameter spherical multi-cusp confinement device (see figure \[fig:WiPAL1\]). Its flexibility and diagnostic access allow WiPAL to currently support two major experiments: the Madison Plasma Dynamo Experiment (MPDX) and the Terrestrial Reconnection Experiment (TREX), along with several new experiments being added in the near future. Both the flexible design of the facility and the shared hardware between experiments allow for a rapid turn-around between different configurations.
![The WiPAL facility (left) consists of a main laboratory space where the vessel is housed and several auxiliary spaces for high voltage power management, water pumping, computer control, and housing the TREX insert. Photo in upper right shows the 3 m diameter vessel covered with arrays of probes and clear viewports.[]{data-label="fig:WiPAL1"}](Figure1.pdf)
[r]{}[2.5in]{} 
\[fig:Cusp\]
The purpose of this paper is to describe the physical parameter regimes that can be achieved in WiPAL, to illustrate its flexibility, and to show how the plasma confinement scheme lends itself to many experiments of interest to the astrophysics and fundamental plasma physics communities. Already, this facility has transitioned from a single-purpose experiment focused on plasma dynamos (MPDX) to a multi-investigator facility accommodating a new reconnection experiment (TREX). In the future, WiPAL will transition into a collaborative user facility open to other laboratory astrophysics research groups. This paper illustrates the potential of the facility by describing (in addition to MPDX and TREX) several additional experiments that are now being pursued: acoustic and Alfvén wave propagation in connection with helioseismology; pulsar and stellar wind launching from a rotating dipolar magnetosphere; jet formation and propagation into background plasma; and small-scale high-power helicity injection. In addition to these experimental efforts, numerical and theoretical computations, often carried out with identical dimensionless parameters, are used to create predictive models and inform elements of design.
This article begins with an overview of WiPAL along with experimentally achieved plasma parameters. Then a description is given of the diagnostic suite and of the magnetic cusp confinement used in WiPAL. This is followed by brief summaries of experiments that are under way or in development.
Description of the Facility
===========================
The WiPAL facility (shown in figure \[fig:WiPAL1\]) consists of the plasma confinement vessel and the associated infrastructure (high bay, electrical power, water cooling, and associated laboratories). The vacuum vessel consists of two 1.5 m radius cast-aluminum hemispheres mounted on a track that can be separated during a vacuum opening. This provides the opportunity for quickly inserting different devices inside the WiPAL vessel. For example, a transition from full MPDX operation to the TREX configuration requires only a few days.
Nearly 3000 samarium cobalt (SmCo) permanent magnets (each with $|{\bf B}|>3$ kG at their surface) are held in axisymmetric rings on the inside of the vessel. These rings alternate in polarity to form a strong high-order multipole field that decreases in strength to the background earth field within 20 cm of the vessel wall (shown in figure \[fig:Cusp\]). This edge-localized cusp field provides sufficient plasma confinement to achieve the parameters listed in table \[tab:param\] while leaving the ions in the core unmagnetized. The rings were carefully produced to assure axisymmetry, thus reducing many aspects of WiPAL experiments to two dimensions. Since all the magnetic field variation is in the radial and polar directions, gradient and curvature drifts are azimuthal. This leads to an azimuthal symmetry of the generated plasmas, thereby reducing convective losses associated with magnetic ripple.
[r]{}[2.925in]{}
---------------------------- ---------- ---------- -----------
Parameter Achieved Achieved Projected
Gas He Ar He
$P_{in}(kW)$ 300 300 650
$T_{e}(eV)$ 20 10 40
$T_{i}(eV)$ 1.5 3 10
$n_{e}(10^{12}$ cm$^{-3})$ 2.5 4 10
$f_{\%}$ 75 90 99
$V$ (km s$^{-1}$) 10 5 20
$Rm=VL/\eta$ 900 350 5000
$Re=VL/\nu$ 600 1300 1000
$B_{equip}$ (G) 14 29 60
$\beta$ at 10 g 8 8 20
---------------------------- ---------- ---------- -----------
To date, argon and helium plasmas have been created and heated by an array of hot emissive lanthanum hexaboride (LaB$_6$) cathodes, each independently powered by separate 30 kW power supplies (currently 12, with 18 planned). See figure \[fig:Plasma1\] for an example discharge in a typical helium plasma. Five 20 kW magnetrons at 2.45 GHz are in the process of being added to independently heat electrons through electron cyclotron resonance (ECR) heating in the magnetized edge, bringing the maximum steady-state input power to $\sim650$ kW. These sources are quasi-stationary, meaning that plasmas can be sustained for tens of seconds limited only by vessel heating.
![Time traces of plasma parameters and input power from a typical WiPAL discharge in helium (left) and argon (right). Radiated power is measured with a bolometer, $n_{n}$ is measured with a pressure gauge at the edge of the vessel (assuming room temperature neutrals), $T_{e}$ is measured with a Langmuir probe inserted into the unmagnetized bulk, and $n_{e}$ is measured with the mm-wave interferometer system.[]{data-label="fig:Plasma1"}](Figure3.pdf)
A unique capability of the facility is the ability to control the plasma rotation at the boundary [@MPDXPOP]. Early experiments at the facility have been focused on studying highly conducting, flow-dominated plasmas and understanding the interface between magnetized and unmagnetized plasma conditions. Plasmas have been stirred at speeds up to 10 km s$^{-1}$ in the azimuthal direction by applying a ${\bf J}\times{\bf B}$ torque in the magnetized edge region. Each of the cathodes are mounted on a motorized stage so their insertion depth can be quickly and accurately set between shots. Cathodes pushed into the unmagnetized bulk of the plasma ionize and heat the plasma, while cathodes pulled back into the magnetized region draw current across magnetic field lines, injecting torque into the plasma. This torque is viscously coupled to the unmagnetized core. This boundary-driven flow scheme was designed for dynamo studies on MPDX, but will be used to study other features of fast-moving plasmas in the WiPAL facility.
Additionally, several capacitor banks with ignitron switching have been used to drive short pulse reconnection experiments in TREX (§\[sec:TREX\]) and to inject current for plasma jet experiments (§\[sec:jets\]). For example, initial reconnection experiments utilized an up to 10 kV, 100 $\mu$F capacitor bank to drive approximately 20 kA for 250 $\mu$s, generating a 10 G reconnection field from a two-turn internal coil. WiPAL is also equipped with a large Helmholtz coil pair which provides a uniform field throughout the plasma volume of up to 275 G.
Major Diagnostics
-----------------
The WiPAL vessel is covered with approximately 200 ports for diagnostic access of plasma discharges via robotic probe arrays and advanced optical measurements. Owing to the high temperature and steady-state nature of WiPAL discharges, [*in situ*]{} probes have been carefully designed to accommodate high heat flux. Additionally, the small loss area provided by the cusp confinement is increased significantly by inserting probes into the plasma. Motivated by these issues, a suite of non-invasive optical diagnostics has been developed and works in concert with inserted probes.
### Two-dimensional Probe Systems {#sec:Probes}
Most plasma astrophysics experiments require high-resolution, two-dimensional (2D) measurements obtained using internal probe arrays. To meet this need, WiPAL has several robotically controlled probe drive systems in place. Two types of scanning systems are used, as shown in figure \[fig:probecoverage\]. The first probe system is a set of sweeping probes (one currently installed, with six planned) that are inserted through a single ball-joint vacuum seal port, combining radial and angular motion to sweep out a 2D plane. Two stepper-motor-controlled transverse stages allow the regions shown to be scanned shot-to-shot with arbitrary resolution. The second probe system is an array of single-axis probes that are inserted every 10$^{\circ}$ in latitude. These probes have a single stepper-motor drive, allowing radial shot-to-shot scans.
[r]{}[2.5in]{}
\[fig:probecoverage\]
The main probe type to be used in WiPAL is a combination velocity-magnetic probe. Each probe tip consists of two Mach probes (four planar Langmuir faces collecting ion saturation current) to measure two orthogonal components of $ \mathbf{v}$. Development is under way to add three Hall sensors to measure the entire three-dimensional (3D) $\mathbf{B}$ field in each probe tip. These probes are designed to withstand long, warm plasma pulses, utilizing a thermally conductive copper shaft paired with an electrically and thermally insulating quartz shield. The lifetime of probes is ultimately limited by accumulated heat load over many plasma pulses, but thermocouple temperature monitoring facilitates optimization of plasma duty cycle versus probe temperature.
Increased probe coverage also leads to diminished plasma performance because plasma losses to the exposed probe area can be much larger than the edge losses through the permanent magnet cusp field. Motorized probes allow for flexibility in controlling the plasma loss area. For example, staggered radial scans of various probes can keep the loss area constant while still mapping out the full extent of probe coverage.
The 2D structure of the magnetic field on the surface of the plasma will be measured using an array of 64 external magnetic probes placed in the region between the plasma and the vessel wall. These probes will consist of three-axis Hall probes for measuring low-frequency magnetic fields as well as two orthogonal [*Mirnov*]{} coils for measuring higher-frequency fluctuations. The final implementation of this full array will resolve $\ell$=8 harmonics in the polar direction and $m$=8 harmonics in the toroidal direction.
In addition to this collection of internal probes, WiPAL is outfitted with an array of optical diagnostics. A millimeter-wave interferometer system measures the line-integrated electron density; a Fabry-Perot spectrometer provides measurements of ion temperature; and optical emission spectroscopy (OES) can be used to determine neutral and ion density profiles, electron temperature, and electron density. All of these systems provide precise, non-invasive measurements of key plasma parameters without increasing the plasma loss area.
### Absolute Density Measurements
WiPAL is equipped with a heterodyne millimeter-wave (320 GHz) interferometer system designed to measure chord-integrated absolute electron density via wave phase shifts. This system is very similar to interferometers used on the Helically Symmetric Experiment (HSX) and the Madison Symmetric Torus (MST) [@Deng2003; @Deng2006].Time resolution of density measurements is set by the intermediate frequency (IF) of the variable-frequency source. Typically, the IF is set to $f\sim1$ MHz, but is adjustable from 0.1-100 MHz.
At WiPAL densities of $n_{e}=10^{11}$ to $5\times10^{12}$ cm$^{-3}$, phase shifts of several fringes (one fringe corresponds to a $2\pi$ phase shift between the reference and plasma beams) must be measured. Additionally, owing to time resolution requirements, this measurement must be made at frequencies up to several megahertz. This constraint is met by using a high-speed field-programmable gate array (FPGA) programmed to compute the phase at each period of the reference beam. The ultimate density resolution is set by the speed of the FPGA clock relative to the IF frequency and yields $\delta n\sim10^{10}$ cm$^{-3}$ for an IF of 1 MHz.
This absolute electron density measurement is used to calibrate Langmuir probe data for point measurements as well as to constrain neutral emission models for spectroscopic measurements of the electron temperature (§\[sec:OES\]). Because of the good time and density resolution, this diagnostic will be used in TREX to measure density fluctuations associated with magnetic reconnection.
### Ion Temperature and Flow {#sec:FP}
Doppler-shifted line emission is measured using a Fabry-Perot spectrometer. Plasma light is passively collected from either the $\lambda=488$nm argon ion line or the $\lambda=468.6$nm helium ion line complex depending on the working gas. By imaging these ion lines at a high resolution, the Fabry-Perot captures the line-integrated ion velocity distribution function. Ion temperature is inferred from the thermal broadening of the lines, while velocity can be measured via the Doppler shift of the peak away from its stationary value.
The Fabry-Perot system has several distinct advantages over a grating-type spectrometer. For a comparable sized grating, the Fabry-Perot has an increase of nearly 100 times the resolving power. Additionally, because the output of the Fabry-Perot is a symmetric ring structure where $r^{2}\propto\lambda$, integration around the rings can greatly increase the signal-to-noise ratio. This procedure, called ring summing, results in a sensitivity gain of 10-30 compared to linear cuts of the pattern [@Coakley1996]. Additionally, it allows for relatively short integration times (roughly 1 s in this system) using a standard digital camera. WiPAL also has a high-performance intensified charge-coupled device (CCD) camera capable of taking even shorter exposures (approximately 0.1s) for multiple measurements during discharges. The typical resolution (i.e., error bars) of Fabry-Perot ion temperature measurements is $\delta T_{i}\approx0.05$ eV and the velocity resolution is of the order $\delta v\approx10$ m s$^{-1}$.
### Electron Temperature and Distribution {#sec:OES}
Several survey spectrometers are used to measure emission throughout the visible spectrum with lower spectral resolution than the Fabry-Perot. The spectrometers capture segments of the near-ultraviolet to the near-infrared range, either monitoring the entire spectrum at low resolution (300 nm $-$ 888 nm at 0.387 nm) or small segments at a higher resolution (381 nm $-$ 511 nm at 0.045 nm). Spectra are continuously captured through WiPAL discharges with exposure times of 2-80 ms and rates of 10-50 Hz which provide a time history relevant for long-pulse time scales.
The OES system, calibrated by the millimeter-wave absolute density measurement, can estimate the electron temperature and ionization fraction. Line ratios are compared to collisional radiative models calibrated to data from other experiments and used to infer $T_e$ while including additional corrections due to non-Maxwellian electron distribution functions in argon plasmas [@Boffard2010; @Wang2013]. In helium, line ratios can also be used to determine $T_e$ and $n_e$ with similar modeling [@Schmitz2008]. Good agreement has been found between the models and the data in the regimes calibrated for tokamak scrape-off layers (high $T_e$, $n_e$). Work is under way to benchmark these models for more relevant parameters. In addition to using the OES system for its own measurements, WiPAL offers an opportunity to extend these calibrations to regimes relevant for laboratory plasma astrophysics.
Major Experiments in WiPAL
==========================
Large Multi-Cusp Plasma Confinement {#sec:confinement}
-----------------------------------
WiPAL is the largest axisymmetric magnetic ring cusp ever constructed for confining plasmas. The permanent magnets effectively limit the loss area to $\sim1\%$ of the total vacuum vessel surface area, which in turn leads to high, steady-state confinement. The combination of WiPAL’s cusp confinement and diagnostic capabilities presents an opportunity for studying and characterizing the confinement of magnetic cusps. The cusp field is created by 36 rings of alternating-polarity magnets, as shown in figure \[fig:Cusp\]. This field is localized to the edge, dropping off to the background earth field within 20 cm of the vessel wall. Plasma losses in this configuration are limited to a small cusp width on the face of each magnet. To electrically isolate the plasma from the vessel wall, each magnet is covered by a thin insulating ceramic tile.
The long established empirical loss width for a magnetic cusp is the hybrid gyroradius $w\sim\sqrt{\rho_{i}\rho_{e}}$ [@Cusp], yet newer studies have found a scaling with neutral pressure which suggests more detailed physics than was previously employed [@hubble2014]. Since WiPAL is able to create and sustain plasmas at very low pressures, this neutral pressure scaling can be investigated over several orders of magnitude. Accurate measurements of the ion velocity distribution at the cusp can be made with the Fabry-Perot system. Additionally, owing to the insulating boundary condition, [*in situ*]{} probes at the ceramic tile can directly measure the loss width. Understanding the loss width scalings in a magnetic cusp is important not only for predicting plasma confinement in WiPAL but also to a number of other applications such as plasma processing [@malik1994], Hall-thrusters [@sengupta2009], and neutral beam injectors [@stirling1979].
Self-Exciting Dynamos {#sec:dynamo}
---------------------
Astrophysical plasmas are often characterized by high magnetic Reynolds numbers ($Rm = V L/ \eta$, where $\eta$ is the magnetic diffusivity, $V$ is a characteristic flow speed, and $L$ is the system size) wherein turbulent, flow-dominated plasmas continuously transform kinetic flow energy into magnetic energy. Understanding this energy transformation and predicting what form the magnetic field might take, be it small-scale turbulent magnetic fields or large-scale magnetic flux, is the dynamo problem. The theory of dynamos has shown that the scales at which the magnetic energy grows are largely determined by the relative values of the magnetic Reynolds number and the fluid Reynolds number ($Re=VL/\nu$, where $\nu$ is the dynamic viscosity). The ratio $Pm=Rm/Re$, called the magnetic Prandtl number, is thought to be a critical parameter that governs the nature of many astrophysical phenomena since it sets the relative collisional dissipation scales of the fields and flows. Astrophysical plasmas span a wide range of $Pm$; diffuse plasmas have $Pm\gg1$ whereas denser plasmas have $Pm\ll1$.
Dynamos can be classified as small-scale or large-scale. Small-scale dynamos tend to generate magnetic energy but little net magnetic flux, whereas large-scale dynamos generate both net flux and energy. While the process by which turbulence generates magnetic energy at small scales seems theoretically well understood [@schekochihin04; @iskakov2007_prl], understanding how a large-scale magnetic field self-organizes from small-scale magnetic fluctuations in astrophysical systems remains a grand challenge for plasma astrophysics.
![a) Hydrodynamic simulations of edge-driven, two-vortex axisymmetric flow in He for increasing $Re$. Axis of rotation and symmetry is vertical; left hemispheres show poloidal flow stream lines and poloidal flow magnitude, right hemispheres show toroidal flow contours. Poloidal flow scale is amplified 4x relative to toroidal flow scale. b) Kinematic dynamo growth rate of the flows shown in (a). Increasing $Re$ results in a larger amount of kinetic energy in poloidal flows, which in turn lowers the critical $Rm$ required for positive magnetic eigenmode growth. Flow is only axisymmetric for $Re < 300$; hydrodynamic instabilities produce non-axisymmetric modes at higher $Re$, requiring full 3-D solutions to the induction equation.[]{data-label="fig:MPDX"}](Figure5.pdf)
Studying plasma dynamos in the laboratory requires a previously unexplored regime of laboratory plasmas. Unmagnetized, fast-flowing, and highly conducting plasmas are required so that magnetic fields can be stretched and amplified by the plasma inertia. Cast in dimensionless terms, dynamo action requires plasmas with high $Rm$ and high Alfvén Mach number ($Ma=V/V_{A}\gg1$). The particular geometries being pursued in the MPDX configuration of WiPAL build upon similar geometries used by liquid-metal experiments. Mechanically stirred liquid-sodium experiments have observed spontaneous magnetic field generation [@monchaux2007; @monchaux2009; @gallet2012] and have added to our understanding of astrophysical and geophysical dynamos [@Lathrop2011]. A plasma experiment has the potential to extend these studies to parameters more relevant to astrophysics. Beyond the obvious fact that most naturally occurring dynamos are plasmas, the use of plasma rather than liquid metals corresponds to magnetic Reynolds numbers increased by a factor of 10 or larger. Additionally, viscosity can be varied independently of the conductivity, with $Pm$ ranging from $0.1$ to $10$, reflecting the wide range found in different astrophysical systems.
Extensive modeling and theoretical work have been carried out to find flow schemes for MPDX that excite a large-scale dynamo [@Spence.APJ.2009; @Khalzov2013; @Khalzov2012a; @Khalzov2012; @Katz2012RSI]. By biasing the cathodes in the magnetized edge, torque is injected at the boundary of the plasma and is viscously coupled to the core. Dynamo-relevant flows are found by solving the Navier-Stokes equation for a particular boundary condition imposed by the cathode drive. The resulting flow is then used to solve the kinematic dynamo equation. One such flow that results in dynamo action is shown in figure \[fig:MPDX\]. The imposed boundary condition drives two counter-rotating vortices in each hemisphere with a small flow direction reversal near the equator. For fairly modest $Rm$ and $Re$, positive dynamo growth is expected. Work on MPDX is currently directed at optimizing the flow drive and attempting to create this two-vortex flow.
![Numerical solutions for turbulence driven small scale dynamo and evidence for large scale field generation. To achieve $Rm=2000$ and $Re=400$ corresponds to a helium plasma with $V_{max}$=10 km s$^{-1}$, $T_e$=30 eV, $T_i$=1.2 eV, $n_e=$ 10$^{18}$ m$^{-3}$, and $Z_{\rm eff}=$1.2. The left plot shows where simulations find turbulent dynamos in $Re$ and $Rm$ space. []{data-label="fig:Re_Rm_small_scale"}](Figure6.pdf)
Fast small-scale dynamos can be excited by chaotic flows with the high values of $Rm$ and $Pm$ possible in WiPAL. Chaotic flows with positive Lyapunov exponents could be achieved with large $Re$ turbulence or by using time-dependent and highly viscous laminar flows [@Khalzov2013]. Three-dimensional numerical simulations using turbulent flows where $Re\sim500$ and $Rm$ is very high have confirmed that turbulent, small-scale magnetic fields naturally develop in high-$Pm$ plasmas. Moreover, the magnetic amplification of these flows is roughly independent of $Rm$ as $Rm$ continues to increase, a hallmark of fast dynamos. Figure \[fig:Re\_Rm\_small\_scale\] shows several example of flows where fast dynamo growth is possible. Simulations predict fast dynamo growth at plasma parameters that can be achieved with increased input power from ECR heating. Because $Rm$ is very large in most astrophysical systems, virtually any astrophysical dynamo must be fast. Understanding the conditions for fast dynamos and exploring whether they can generate magnetic fields on large scales is an open problem in plasma astrophysics.
Magnetic Reconnection {#sec:TREX}
---------------------
![Toroidal field coils and a cylindrical insert holding additional coils for driving reconnection (left) will be inserted into the WiPAL vessel. The large set of external Helmholtz coils seen in the figure (right) is a recent upgrade to the WiPAL facility with utility for many future experiments.[]{data-label="fig:TREX1"}](Figure7.pdf)
![Reconnection phase space diagrams for weak guide-field reconnection (left) and strong guide-field reconnection (right) identifying various regimes of reconnection. The star characterizes the first plasma obtained with TREX, demonstrating its ability to access the new collisionless reconnection regime.[]{data-label="fig:TREX2"}](Figure8.pdf)
One of the fundamental processes in nearly all magnetized plasmas is magnetic reconnection: a change in the magnetic topology of the plasma that converts stored magnetic energy into particle kinetic energy [@priest2000]. Given its importance, magnetic reconnection has been studied extensively through theory, numerical computation, spacecraft observations, and laboratory experiments. In recent years, new frontiers have emerged with an emphasis on particle energization, reconnection with multiple X-lines in large systems, the role of kinetic effects in collisionless reconnection, and extending the evolution of reconnection from two dimensions to three. For experiments to remain relevant and to advance this maturing field, new devices are needed that access these regimes. To address this need, we are now operating the Terrestrial Reconnection Experiment (TREX), which is the largest dedicated reconnection experiment to date. In addition to WiPAL’s existing vacuum vessel, plasma confinement, heating, and diagnostic suite, TREX consists of a cylindrical insert holding coils used for driving reconnection (shown in figure \[fig:TREX1\]). The parameter regimes expected for TREX are shown in table \[tab:TREX\]. The full operation of TREX within WiPAL renders the facility unique in its ability to address the expanding frontiers of reconnection research.
An important component to the implementation of TREX into the WiPAL user facility is the cylindrical insert housing the reconnection drive coils and the addition of a toroidal field coil. The insert and TF coil allow TREX to access multiple magnetic configurations (e.g. antiparallel reconnection, strong guide-field reconnection, and 3D reconnection), as well as a wider range of reconnection regimes. TREX also utilizes a suite of probe arrays specially developed to characterize reconnection dynamics. These include stationary magnetic and electric probes along with sweep probe arrays that allow for a 2D region to be mapped out over repeated shots (§\[sec:Probes\]). The necessary dynamics can be captured on time scales up to 20 MHz with the diagnostic suite. For a desired value of the reconnecting field, the configuration must be driven at the corresponding loop voltage $V_{loop}=2{\pi}R(0.1v_{A}B_{r})$. For example, a loop voltage of 5 kV for a 150 G reconnecting field is required to access fast reconnection in a hydrogen plasma. This consideration has necessitated the development of reliable, high-power pulsing circuits utilizing class D ignitron switches to drive reconnection. These drive circuits can be pulsed as fast as every 10 seconds.
So far, the term “collisionless reconnection" has referred to systems where the electron and ion distributions can remain near Maxwellian, but the collisionality is sufficiently low that their collective fluid behaviors decouple at the ion scale and Hall currents become important. However, in a truly collisionless plasma, pressure anisotropy develops which strongly impacts the properties of the reconnection process in ways not accounted for in traditional Hall reconnection. In fact, spacecraft observations [@hwang2013] and kinetic simulations at the full ion-to-electron mass ratio [@le2015] show that large-scale current layers are driven by electron pressure anisotropy that builds in the reconnection region due to kinetic electron trapping effects. To maintain pressure anisotropy, the time between electron collisions must be long compared with the full transit time of a fluid element through the reconnection layer [@egedal2013; @le2015].
$n_e (10^{18}\text{ m}^{-3})$ $T_{e}$ (eV) $B_{r}$ (T) $B_{g}$ (T) L (m)
------------------------------------- ------------------------------- -------------- ------------- ------------- -----------
Terrestrial Reconnection Experiment $0.1-10$ $8-40$ $0.04$ $0-0.3$ $0.8-1.8$
(TREX; UW Madison)
Magnetic Reconnection Experiment $2-100$ $5-10$ 0.03 $0-0.1$ 0.3-0.8
(MRX; Princeton)
Versatile Toroidal Facility $0.1-1$ $8-30$ 0.01 0.1 0.3
(VTF; MIT)
: Key parameters for various reconnection experiments in hydrogen plasmas where $B_r$ and $B_g$ are the reconnecting and guide fields, respectively.[]{data-label="tab:TREX"}
As a valuable tool for displaying the various regimes of reconnection and their transitions, @daughton2012 developed the reconnection phase diagram spanned by the Lundquist number $S$ and the normalized system size $\lambda$ with respect to the ion sound Larmor radius, $\rho_{s}=\sqrt{m_{i}(T_{e}+T_{i})}/eB$, or the ion skin depth, $d_{i}=c/\omega_{pi}$, for strong and weak guide-field reconnection, respectively [@ji2011]. A convenient way of representing the constraint for anisotropic pressure on a system is the condition $S>10(m_{i}/m_{e})(L/d_{i})$. The anisotropic pressure region of this phase space is shown in figure \[fig:TREX2\]. As indicated by the star, TREX has already demonstrated its ability to access this regime of collisionless reconnection. TREX is able to experimentally study the role that electron pressure anisotropy has on particle heating. In addition, the narrow current layers driven by the anisotropy may in 3D be unstable to reconnection at oblique angles. This effect of 3D reconnection may be important to the self-consistent evolution and generation of reconnection with multiple X-lines.
Acoustic Waves, Helioseismology, and Angular Momentum Transport
---------------------------------------------------------------
![A simulation of solar-like flow inside the WiPAL vessel with streamlines of poloidal flow (left). The internal rotation profile of the Sun inferred from helioseismic inversions [@solarpic]. Pink/red denote fast rotation and blue/green denote slower rotation as indicated by the color bar. Inversions near the rotation axis are unreliable and are thus omitted from the plot. The image of the angular velocity profile is reflected about the rotation axis in order to illustrate the full spherical geometry.[]{data-label="fig:solar"}](Figure9.pdf)
The WiPAL facility can study both basic solar plasma phenomena as well as longstanding mysteries explored by the heliophysics community, such as angular momentum transport in the Sun and other late-type stars. It is well established from observations and modeling that late-type stars with convective envelopes spin down as they age due to torques exerted by magnetized stellar winds [@Mestel1968; @barne07; @matt15]. It is less well understood how this angular momentum loss is transmitted through their convection zones and into their radiative cores. Helioseismic probing of our own Sun suggests that this coupling is efficient in the sense that the radiative core currently has a rotation rate that is comparable to the convective envelope [@thomp03]. However, these same helioseismic inversions reveal that the radiative core is rotating nearly uniformly while the convection zone rotates differentially, with a $\sim 30$% decrease in angular velocity from equator to pole [@thomp03]. The transition between these two distinct rotation regimes is called the solar tachocline, which overlaps with the base of the convection zone and the convective overshoot region.
Angular momentum transport within the solar convection zone is attributed to turbulent magnetized convection and occurs on convective time scales of weeks to months. By contrast, the transport of angular momentum across the solar tachocline, coupling the core and envelope, is thought to occur on much longer time scales of millions to billions of years [@spieg92]. Our current understanding of both short-term and long-term angular momentum transport in the Sun relies on incomplete and indirect observational data (e.g. helioseismic inversions) and numerical magnetohydrodynamic (MHD) models. WiPAL offers an opportunity to study some aspects of the relevant dynamics in the laboratory for the first time.
Using the drive system as MPDX, a solar-like flow can be programmed at the boundary of the WiPAL vessel. A calculation of the structure of this flow is shown on the left of figure \[fig:solar\]. This flow is aimed at mimicking the helioseismic inversion, shown on the right of figure \[fig:solar\], with strong toroidal rotation near the equator. Careful scrutiny of how such a boundary-forced differential rotation spreads into the interior through meridional flows, viscous diffusion, thermal gradients, and magnetic tension will provide insight into core-envelope coupling and the resulting spin-down of late-type stars. Magnetic tension in particular is thought to be responsible for maintaining the uniform rotation of the radiative interior by suppressing shear, as described by Ferraro’s theorem [@macgr99]. Furthermore, inserting a dipole electromagnet into the core of the WiPAL vessel will create a mock tachocline, testing the structure and stability of the boundary layer proposed by magnetic tachocline confinement models [e.g., @gough98].
By exciting the acoustic normal mode spectrum of a spherical plasma with external antennas, the stationary plasma response can be characterized. It is predicted by theory and observed in the Sun that these normal modes split when rotation is introduced [@CD2002]. Helioseismology relies on these mode splittings at different locations on the Sun to discern the flow profiles of the interior. A similar analysis is available at WiPAL with the addition of the external magnetic probe array described in §\[sec:Probes\]. When differential rotation is imposed at the boundary, the resulting spectra can be spatially decomposed and compared to the stationary mode structure. A mathematical inversion can then be used to propose likely candidates for internal global flow profiles, which can be benchmarked by Mach probe measurements under similar conditions. This would form the basis of a minimally invasive, global flow profile diagnostic at WiPAL akin to helioseismic inversions.
Using this helioseismic diagnostic and solar flow drive, WiPAL will study the short-term angular momentum transport in the solar convection zone. Experimental set-ups, such as that shown in figure \[fig:solar\], can be used to investigate the physics of turbulent transport. In the Sun, turbulent angular momentum transport is thought to be responsible not only for sustaining the solar differential rotation, but also for regulating the amplitude and structure of the meridional flow by means of gyroscopic pumping [@miesc11]. More generally, turbulent angular momentum transport, mean flows, and thermal gradients in the solar convection zone are all thought to be intimately linked through nonlinear feedbacks that can be explored with WiPAL.
Centrifugally Driven Stellar Winds
----------------------------------
Plasma wind launched from star surfaces carries stellar magnetic field into the local heliosphere and out to the interstellar medium. This system is complex and is governed by a range of processes, including magnetic reconnection, turbulence and particle heating, all topics of present-day heliospheric research. One aspect that has received little experimental attention is the magnetic topology of the advected magnetic field in the interface region between the magnetically dominated corona and the flow-dominated wind. This region is critical both in creating the Parker spiral and in determining the mass loss rates of stars. The Centrifugal Wind Experiment (CWE) at WiPAL will explore centrifugal breakout of wind from spinning dipolar magnetospheres.
The winds of rapidly rotating giant stars and pulsars are particularly relevant. For rapidly rotating giant stars, centrifugal breakout has been predicted to proceed in episodic bursts of plasmoids [@ud-Doula06]. Recent observations by the Microvariability and Oscillations of STars (MOST) telescope have called into question the validity of this breakout model, suggesting instead a smooth outflow [@Townsend2013]. Modeling centrifugal breakout in a laboratory setting can advance our understanding of this phenomenon. Pulsars have been extensively modeled using analytical MHD and 3D particle-in-cell (PIC) simulations of both aligned and obliquely rotating magnetospheres [@Spitkovsky06; @Philippov2014]. Both cases predict the creation of a magnetic Y-point where the closed magnetic topology breaks into an open configuration. This is expected to occur at the Alfvén radius, the location where the kinetic and magnetic energy densities are equal. WiPAL’€™s large volume, well-developed power supplies, and comprehensive diagnostics make it suitable for experimental analysis of these active areas of research.
![(a) WiPAL facility with centrifugal wind experiment installed and (b) proposed configuration showing the electromagnetic stirring scheme and Y-point.[]{data-label="fig:Wind"}](Figure10.pdf)
A laboratory analog of a rotating, stellar magnetosphere is possible using the electrostatic stirring techniques developed for MPDX with the addition of a dipole magnetic field at the center of the vessel. Through ECR heating, the dipole magnetosphere will be filled with confined plasma. Biased electrodes at the equatorial surface of the dipole will establish a cross-field potential gradient, leading to ${\bf E}\times{\bf B}$ stirring. As the plasma spins to critical velocity, a centrifugal wind will be launched. In this way, we can use a large cross-field potential to spin the magnetosphere and model the above astrophysical situations.
In preliminary experiments, an electrically insulated, spherical SmCo magnet ($r\simeq10$ cm, surface field $\sim4$ kG) will serve as the dipole source, shown in figure \[fig:Wind\]. ECR heating will produce plasma on the spherically symmetric 875 G surface. Using helium, expected temperatures and densities will be similar to prior MPDX experiments ($T_{e}\approx5-20 eV$ and $n_{e}\approx10^{12}$ cm$^{-ˆ’3}$). Equatorial cathodes, spaced $\approx10$ cm apart, will be biased up to 1 kV to establish a cross-field potential, producing axially aligned ${\bf E}\times{\bf B}$ rotation.
WiPAL’s existing diagnostics can be used for investigation of the Y-Point and current sheet regions. The standard suite of Langmuir and Mach probes will be used for basic plasma measurements at the edge and will supplement the non-invasive measurements. A 2D, motor-controlled, 20 MHz resolution magnetic probe array will serve as the primary diagnostic. The magnetic probe will look for the characteristic Y-point opening of the magnetic field lines at the Alfvén radius and will explore the equatorial current sheet at larger radii.
Planned upgrades will explore other regimes. For example, obliquely rotating pulsars generate larger spin-down power than aligned rotators [@Spitkovsky06] and provide a more realistic setting for studying current sheet reconnection. By supplementing the dipole magnet with phased coils, an obliquely rotating magnetosphere can be created. This will allow for experimental confirmation of the above prediction and characterization of the change in the undulating current sheets.
Stability of Astrophysical Jets in Background Plasma {#sec:jets}
----------------------------------------------------
Astrophysical jets are collimated magnetized plasma outflows from accreting bodies such as active galactic nuclei, binary systems, and young stellar objects. It is thought that jets could play a key role in angular momentum transport in accretion systems. Jets are launched and collimated in regions much smaller than the current resolution limits of observations, so the details of this process remain elusive. Certain models of the launching process predict that astrophysical jets arise due to magnetic fields in the accretion disk [@Lovelace1976; @1982MNRAS.199..883B; @Spruit2010]. In these models, a dipole-like magnetic field is sheared by Keplerian rotation in the disk. This creates a large electric field between the center and edge of the disk, driving current along the dipole field. Then ${\bf J}\times{\bf B}$ forces accelerate and collimate plasma, creating the jet structure. These jets should be susceptible to current-driven instabilities, but recently published theories suggest that the presence of external plasma pressure promotes collimation and enhances the stability of the jet structure out to large distances [@2003MNRAS.341.1360L; @Li2001; @Li2006].
![Experiment to study Jet expansion into background, finite pressure plasmas. Simulation of Jet is taken from [@Li2006] and superimposed into WiPAL plasma schematic, showing coaxial injection geometry. Right: photo of the assembled pulsed jet source preparing for first tests.[]{data-label="fig:PMPJS"}](Figure11.pdf)
The Permanent Magnet Pulsed Jet Source (PM-PJS), shown in figure \[fig:PMPJS\], creates an astrophysical jet-like plasma utilizing a design similar to that used by Hsu and Bellan [@Hsu2002]. A large voltage potential is created between two coplanar annuli (approximately 30 cm in diameter). This simulates the potential created via the aforementioned accretion disk shearing action. The background magnetic field in the astrophysical system is simulated using two antisymmetric rings of permanent magnets. The use of permanent magnets instead of electromagnets allows the source to maintain WiPAL’s cusp-confining boundary condition. Neutral gas is injected via internal channels in both annuli with ports placed at eight azimuthally symmetric pairs of magnetic field line foot points. This gas is ionized by the large voltage creating eight filamentary loop structures. These loops merge and collimate, driving a magnetic jet into the center of WiPAL.
This experiment complements Caltech experiments [@Hsu2002; @Hsu2003; @Moser2012] by studying the evolution of a magnetic jet evolving into a vacuum. Additionally, WiPAL allows the creation of a magnetic jet that propagates into a high-$\beta$ background plasma. This will help clarify the relationship between jet stability and external plasma conditions [@2003MNRAS.341.1360L; @Li2001; @Li2006]. Future work will focus on the shock and precursors that are formed between the magnetized jet and the unmagnetized background plasma as predicted in [@Li2006] and on how the development of the kink instability could depend on the strength of the background pressure. Experimentally it will be possible to vary the target plasma pressure by more than two orders of magnitude. The properties of the expanding magnetic plume (speed, density, magnetic field) as well as the shock precursors will be measured using scanning probes similar to those used in the reconnection and centrifugal wind experiments.
Helicity Injection and Decaying Turbulence
------------------------------------------
The generation of large-scale magnetic fields is a fundamental feature of many astrophysical systems. This dynamo action (as described in §\[sec:dynamo\]) is often attributed to a turbulent upscale transfer of small-scale magnetic helicity, $H=\int{\bf A}\cdot{\bf B}\,d\mathcal{V}$. The upscale transfer process is ideally described as a local, self-similar inverse cascade where helicity is conserved and transferred to larger scales [@frisch1975; @Ji1999; @Blackman2006]. In real systems however, the transfer process is much more complex and can involve non-local transfer directly from the small to large scales as described in the turbulent alpha effect. The upscale spectral transfer of magnetic helicity has been explored extensively via direct numerical simulations [e.g. @alexakis2006]. Understanding this upscale transfer of magnetic helicity is key to explaining the creation of the large-scale magnetic fields observed throughout the Universe.
In addition to upscale spectral transfer of helicity, transport in space is necessary for large-scale dynamo action. Large-scale dynamos can be sustained in systems with boundary conditions allowing the outflow of helicity. Roughly speaking, ejection of one sign of magnetic helicity allows helicity of the opposite sign to grow without violating helicity conservation. This idea was originally suggested on theoretical grounds by @blackman2001 and @vishniac2001. Later, it was shown that magnetic eruptions associated with solar activity transmit a net helicity flux, suggesting that the solar dynamo meets these boundary conditions [@rust2002; @kusano2002; @liu2014; @pevtsov2014]. In addition to the solar dynamo, this boundary condition effect is thought to occur in galactic dynamos driven by supernovae [@rafikov2000_mnras]. There is evidence from numerical simulations that the generation of large-scale fields is promoted by boundary conditions that permit escape of magnetic helicity [@brandenburg2004; @kapyla2010; @hubbard2012]. Yet none of these ideas have been tested in the laboratory.
In the plasma physics community, helicity injection is primarily explored as a startup mechanism for magnetically dominated tokamak plasmas [@raman2003; @raman2010; @battaglia2009]. Some laboratory astrophysical experiments that inject net helicity into plasma systems have also focused on magnetic reconnection and other instabilities that occur during Taylor relaxation of low-$\beta$ plasmas [@cothran2009; @gray2010; @Jarboe2005]. All of these experiments rely on helicity injection at large scales (comparable to the scale of the plasma volume) relaxing via magnetic instability to minimum-energy, helicity-conserving Taylor states in low-$\beta$ plasmas [@Taylor1986]. As a complement to these studies, we will inject helicity at small scales and use the large size of the WiPAL vessel to directly observe the upscale spectral transfer and transport of helicity in a high-$\beta$, dynamo-relevant plasma conditions.
[r]{}[2in]{} 
\[fig:helicity\]
Generalizing the PM-PJS experiment discussed above (§\[sec:jets\]), helicity can be injected using the edge-localized cusp field inside WiPAL. A large potential can be applied between alternating rings, which will drive a current along cusp field lines and lead to an injection of helicity into the system without a guide field present (figure \[fig:helicity\]) . By using all of the rings of permanent magnets inside the vessel, this helicity will be injected at a small ($\ell$=18, $m=0$) scale relative to the size of the vessel. The resulting rings of flux will then be blown off the wall into the bulk unmagnetized plasma with kinetic and magnetic energies near equipartition. In order to drive an upscale transfer instead of simple resistive diffusion, we require the resistive decay time to be much longer than the Alfvén time of this system. Cast in dimensionless terms, this means that the Lundquist number, $S\equiv \tau_{r}/\tau_{A}$, must be large. Using the high-power capacitor banks created for TREX, magnetic fields of $B\simeq100$ G can be induced from current driven along the cusp field lines. For WiPAL discharge parameters, this corresponds to $S\simeq1000$. Under these conditions, we expect helicity to undergo a turbulent transfer before resistive diffusion can dissipate the injected magnetic energy. Observing the transfer and transport of helicity in this set-up will complement dynamo studies conducted by MPDX and provide the astrophysics community with a laboratory environment to probe this fundamental process.
Conclusion
==========
The Wisconsin Plasma Astrophysics Laboratory (WiPAL) provides the plasma astrophysics and fundamental plasma physics communities with a unique opportunity to study plasma phenomena in a laboratory setting. Hot, dense, unmagnetized, and fully ionized plasmas are routinely created and confined in quiescent states for seconds as astrophysics experiments are performed. This user facility has been designed and operated with the goal of maximizing both plasma performance and flexibility of use.
To date, WiPAL has confined steady-state plasmas which are hotter and denser than any other large-scale non-magnetically dominated plasma device. These parameters have proven to be sufficient for both studying dynamo relevant regimes (MPDX) and providing high-Lundquist-number target plasmas for reconnection studies (TREX). In the longer term, the heating power will more than double as the cathode system is completed and the ECR system is installed.
Diagnosing the high-performance unmagnetized plasmas in WiPAL has been an area of intense focus. Arrays of robotically controlled magnetic and electrostatic probes are capable of mapping out large areas of WiPAL plasmas. Advanced optical diagnostics have been developed with the goal of reducing the additional loss area added by this suite of [*in situ*]{} probes. Millimeter-wave technology is used to power a compact heterodyne interferometer system for measuring electron density with high resolution. A Fabry-Perot interferometer is used to extract the ion velocity distribution. Finally, a complement of low-cost spectrometers coupled with collaborative modeling makes reliable estimates of the electron temperature and ionization fraction. All of these diagnostics provide a valuable set of data for every WiPAL discharge regardless of experiment.
Perhaps most significantly, the WiPAL facility has already demonstrated the ability to quickly change experimental configurations. This paper has outlined two major experiments already running on WiPAL (MPDX and TREX) as well as several new experiments in various stages of planning and implementation. We envision WiPAL transitioning in the near future to a user facility model in which investigators from outside the collaboration groups listed above could apply for and receive support to carry out experiments with technical support from the WiPAL staff.
The construction of the facility was supported by a National Science Foundation (NSF) Major Research Instrumentation grant. The MPDX and TREX research is now supported by NSF and the U.S. Department of Energy (DoE) and the NSF Center for Magnetic Self Organization in Laboratory and Astrophysical Plasmas (CMSO). The helioseismology studies are part of a collaboration with NCAR (M. Miesch) and are supported by a NASA graduate fellowship (E. Peterson). The stellar wind experiment is a collaboration with Princeton University (A. Spitkovsky) and is supported by an NSF graduate fellowship (D. Endrizzi).
|
---
abstract: 'This paper is concerned with theoretical analysis of a heat and moisture transfer model arising from textile industries, which is described by a degenerate and strongly coupled parabolic system. We prove the global (in time) existence of weak solution by constructing an approximate solution with some standard smoothing. The proof is based on the physcial nature of gas convection, in which the heat (energy) flux in convection is determined by the mass flux in convection.'
author:
- 'Buyang Li [^1] , Weiwei Sun , and Yi Wang [^2]'
title: '**Global existence of weak solution to the heat and moisture transport system in fibrous porous media**'
---
[**Key words:**]{} Heat and moisture transfer, Porous media, Global weak solution.
Introduction
============
Mathemaitcal modeling for heat and moisture transport with phase change in porous textile materials was studied by many authors, $e.g.$, see [@FLL; @FCWS; @HYS; @LZ]. A typical application of these models is a clothing assembly, consisting of a thick porous fibrous batting sandwiched by two thin fabrics. The outside cover of the assembly is exposed to a cold environment with fixed temperature and relative humidity while the inside cover is exposed to a mixture of air and vapor at higher temperature and relative humidity. In general, the physical process can be viewed as a multiphase and single (or multi) component flow. In this process, the water vapor moves through the clothing assembly by convection which is induced by the pressure gradient. The heat is transferred by conduction in all phases (liquid, fiber and gas) and convection in gas. Phase changes occur in the form of evaporation/condensation and/or sublimation. Based on the conservation of mass and energy and the neglect of the water influence, the model can be described by $$\begin{aligned}
& &\frac{\partial}{\partial t}(\epsilon
C_v)+\frac{\partial}{\partial x}(u \epsilon C_v) =-\Gamma_{ce},
\label{Cv-e}\\
&&\frac{\partial }{\partial t} \left ( \epsilon C_{vg}M C_vT +
(1-\epsilon) C_{vs} T \right ) + \frac{\partial}{\partial x} \left
( \epsilon C_{vg} M u C_vT \right ) = \frac{\partial}{\partial x}
\left ( \kappa \frac {\partial T}{\partial x} \right ) + \lambda
M\Gamma_{ce} \, . \label{T-e}\end{aligned}$$ Here $C_v$ is the vapor concentration ($\rm mol/m^3$), $T$ is the temperature ($K$), $\epsilon$ the porosity of the fiber, $M$ the molecular weight of water and $\lambda$ the latent heat of evaporation/condensation in the wet zone while in frozen zone, it represents the latent heat of sublimation. $C_{vg}$ and $C_{vs}$ are the heat capacities of the gas and mixture solid, respectively.
The evaporation/condensation (molar) rate of phase change per unit volume is defined by the Hertz-Knudsen equation [@Jon] $$\Gamma_{ce} = -\frac{E}{R_f}
\sqrt{\frac{(1-\epsilon)(1-\epsilon')}{2\pi RM }} \left(\frac{P_{\rm
sat}(T)- P}{\sqrt{T}}\right) \label{phase}$$ where $R$ is the universal gas constant, $R_f$ is the radius of fibre and $E$ is the nondimensional phase change coefficient. The vapor pressure is given by $P=RC_vT$ because of the ideal gases’ assumption. The saturation pressure $P_{\rm sat}$ is determined from experimental measurements, see Figure \[sat\].
The vapor velocity (volumetric discharge) is given by the Darcy’s law $$u=-\frac{k k_{rg}}{\mu_{g}}\frac{\partial P}{\partial x
}\label{darcy-v}$$ where $k$ is the permeability, $k_{rg}$ and $\mu_{g}$ are the relative permeability and the viscosity of the vapor, respectively.
Numerical methods and simulations for the heat and moisture transport in porous textile materials have been studied by many authors with various applications [@CW; @HSY; @OT; @ST; @WMR]. However, no theoretical analysis has been explored for the above system of nonlinear equations. A simple heat and moisture model was studied in [@Val], where the model was described by a pure diffusion process (without convection and condensation) with a non-symmetric parabolic part. There are several related porous media flow problems from other physical applications. A popular one is a compressible (or incompressible) flow in porous media with applications in oil and underground water industries, which is described by an elliptic pressure equation coupled with a parabolic concentration equation for incompressible case and a system of parabolic equations for compressible case. The existence of weak solution for incompressible and compressible flows has been studied in [@CE; @Feng] and [@AS], respectively. However, in most of these works, the temperature is ignored and the phase change (condensation/evaporation) does not occur due to the nature of these applications, while both temperature and phase change play important roles in the textile model.
For the textile model, the water content in the batting area usually is relative small and one often assumes that all these physical parameters involved in the system (\[Cv-e\])-(\[T-e\]) are positive constants. With nondimensionalization, the system (\[Cv-e\])-(\[T-e\]) reduces to $$\left\{
\begin{array}{l}
\r_t-((\rho \theta)_x \r)_x=-\G, \\
(\rho \theta)_t+\s\t_t-((\rho \theta)_x \rho \theta
)_x-(\k\t_x)_x=\l\G, ,
\end{array}
\right.\label{sys}$$ for $x \in (0,1)$, $t>0$, where $(\cdot)_{\mu} =
\frac{\partial}{\partial \mu}$ for $\mu=x, t$, $\r=\r(x,t)$ and $\t=\t(x,t)$ represent the density of vapor and the temperature, respectively, $$\Gamma = \rho \theta^{1/2} - p_s(\theta) \,$$ and $p_s(\theta) \sim P_{\rm sat}(\theta)/\theta^{1/2}$. $\s$ and $\l$ are given positive constants and $\k=\k_1+\k_2\r^2$ is the heat conductivity coefficient with $\k_i~(i=1,2)$ being positive constants. We consider a class of commonly used Robin type boundary conditions [@FLL; @FCWS; @HYS; @YHFS] defined by $$(\rho \theta)_x\r|_{x=1}=\a^1(\bar\r^1-\r(1,t)),\quad (\rho
\theta)_x\r|_{x=0}=\a^0(\r(0,t)-\bar\r^0), \label{sys-b1}$$ and $$\k\t_x|_{x=1}=\b^1(\bar\t^1-\t(1,t)),\quad
\k\t_x|_{x=0}=\b^0(\t(0,t)-\bar\t^0),\label{sys-b2}$$ and the initial condition is $$\r(x,0)=\r_0(x),\quad\t(x,0)=\t_0(x),\qquad x\in (0,1)
\label{sys-i}$$ where $\a^0,\a^1$ represent the mass transfer coefficients, $\bar\r^0,\bar\r^1$ are the density of the gas in the inner background and outer background, respectively, $\b^0,\b^1$ the heat transfer coefficients, and $\bar\t^0,\bar\t^1$ the inner and outer background temperatures. Physically, all the parameters above are positive constants and $\r_0(x)>\underline{\r},\t_0(x)>\underline{\t}$, where $\underline{\r}$ and $\underline{\t}$ are positive constants. Based on the experimental data in Figure \[sat\], we assume that $p_s$ is a smooth, increasing and nonnegative function defined on $\R^+$ which satisfies $$\lim_{\theta \rightarrow 0 }
\frac{p_s(\theta)}{\theta}=0,\;\;\;\quad \lim_{\theta \rightarrow
\infty } \frac{p_s(\theta)}{\theta^{1+\eta}}=\infty \label{sat-a}$$ for some $\eta>0$. For physical reasons, we set $p_s(\t)=0$ for $\t\leq 0$.
The objective of this paper is to establish the global existence of weak solution to the initial-boundary value problem (\[sys\])-(\[sys-i\]) under the general physical hypotheses (\[sat-a\]) for the saturation pressure function $\Gamma$. The difficulty lies on the strong nonlinearity and the coupling of equations. To the best of our knowledge, there are no theoretical results for the underlying model. More important is its significant applications in textile industries. Also analysis presented in this paper may provide a fundamental tool for theoretical analysis of existing numerical methods. Our proof is based on the equivalence of mass and heat transfer in convection.
The main result
===============
Before we present our main result, we introduce some notations. Let $T$ be a given positive number in the following sections. We define $$\Omega=(0,1),\quad I=(0,T],\quad Q_t = \Omega\times (0,t], \quad
Q_T= \Omega\times I,$$ $$V_1(Q_T)=L^2(I;H^1(\Omega)), \quad V_2(Q_T)=\Big\{f\in
L^2(Q_T)~\Big|~\|f\|_{V_2(Q_T)}<+\i\Big\},$$ $$\|f\|_{V_2(Q_T)}= {\rm ess}\sup_{\!\!\!\!t\in [0,T]}
\|f(\cdot,t)\|_{L^2(\Omega)}+\|f_x\|_{L^2(Q_T)},$$ $$W_2^{2,1}(Q_T)=\Big\{f\in L^2(Q_T)~\Big|~f_t,f_x,f_{xx} \in
L^2(Q_T)\Big\}.$$ Let $\mathcal{D}(\overline\Omega\times[0,T))$ be the subspace of $C^\infty(\R^2)$ consisting of functions which have compact support in $\R\times[-\infty,T)$, restricted to $\overline\Omega\times[0,T)$.
Now we give the definition of weak solution to the system (\[sys\])-(\[sys-i\]) and then, state our main result. 0.1in
\[defweaksol\]$\!\!$ [**(Weak solution)**]{}$\;$ [*We say that the measurable function pair $(\r,\t)$ defined on $\overline\Omega\times[0,T)$ is a global weak solution to (\[sys\])-(\[sys-i\]) if $(\r,\t)\in (V_1(Q_T))^2$ and the density $\r$ and the temperature $\t$ are nonnegative functions satisfying $$\begin{aligned}
\label{rdefeq}
&\int_0^T\alpha^0(\r(0,t)-\bar\r^0)\phi(0,t)dt
+\int_0^T\alpha^1(\r(1,t)-\bar\r^1)\phi(1,t)dt
{\nonumber}\\
&+\int_0^T\int_\Omega (-\r\phi_t+(\r\t)_x\r \phi_x+\G\phi) dxdt
=\int_\Omega\rho_0\phi_0dx\end{aligned}$$ and $$\begin{aligned}
\label{tdefeq}
&\int_0^T[\alpha^0(\r(0,t)-\bar\r^0)\t(0,t)+\beta^0(\t(0,t)-\bar\t^0)]
\psi(0,t)dt
{\nonumber}\\
&~~+\int_0^T[\alpha^1(\r(1,t)-\bar\r^1)\t(1,t)
+\beta^1(\t(1,t)-\bar\t^1)]\psi(1,t)dt
{\nonumber}\\
&~~+\int_0^\infty\int_\Omega \Big[-(\r\t+\s\t)\psi_t+(\r\t)_x\r\t
\psi_x+\k\t_x\psi_x-\l\G\psi\Big] dxdt
{\nonumber}\\
&=\int_\Omega (\r_0\t_0+\s\t_0)\psi_0dx\end{aligned}$$ for any test functions $\phi,\psi\in
\mathcal{D}(\overline\Omega\times[0,T))$.* ]{}
0.1in
\[thm1\] [*If the initial value $(\r_0,\t_0)$ satisfies $\r_0\in
L^{1+\gamma}(\Omega)$ $(\gamma>0)$, $\t_0\in L^\i(\Omega)$ and $\rho_0 \ge 0$, $\t_0\geq\underline{\t}$ for some positive constant $\underline{\t}$, then there exists a global weak solution $(\r,\t)$, in the sense of Definition \[defweaksol\], to the initial-boundary value problem (\[sys\])-(\[sys-i\]) such that $$\r\ln \r\in L^\i(0,T;L^1(\Omega)),\quad \r\in
L^4(Q_T),\quad\r_x\in L^2(Q_T);$$ $$\t,\t^{-1}\in L^\i(Q_T),\quad (1+\r)\t_x\in L^2(Q_T).$$* ]{}
In the following sections, we denote by $C_{p_1,p_2,\cdots,p_k}$ a generic positive constant, which depends solely upon $p_1,p_2,\cdots,p_k$, the physical parameters $\kappa_1, \kappa_2,
\sigma$ and $\lambda$ and the parameters involved in initial and boundary conditions. In addition, we denote by $C(p_1,p_2,\cdots,p_k)$ a generic positive function, dependent upon the physical parameters $\kappa_1, \kappa_2, \sigma$ and $\lambda$ and the parameters involved in boundary conditions, which is bounded when $p_1,p_2,\cdots,p_k$ are bounded.
Construction of approximate solutions {#apprxsol}
=====================================
Throughout this section, we let $\v$ be a fixed positive number which satisfies $$0<\v\leq\min\{\bar\r^0,\bar\r^1,\bar\t^0,\bar\t^1,1\},$$ and $0<\nu<\v$. To prove the existence of global weak solutions to the system (\[sys\])-(\[sys-i\]), we introduce a regularized approximate system as follows: $$\begin{aligned}
&\r_t-((\v+(\r\t)_\nu)\r_x)_x-(\r(\r_\v\t_x)_\v)_x
=-\r\chi^\v(\sqrt{\t})+\chi^\v(p_s(\t)),
{\nonumber}\\[3mm]
&(\r\t+\s\t)_{t}-(\kappa^\v\t_x)_x-((\v+(\r\t)_\nu))
\r_x\t)_x-(\r(\r_\v\t_x)_\v\t)_x \label{asys2}
\\
&=\l \r\chi^\v(\sqrt{\t})-\l\chi^\v(p_s(\t)) +(\l+\t) \left (
\chi^\v(p_s(\t))-p_s(\t)\right ) , ~~{\rm in}~~~ Q_T,{\nonumber}\end{aligned}$$ where $\chi^\v$ is a cut-off function defined by $$\chi^\v(h)=\left\{
\begin{array}{lll}
h&\mbox{\rm if}& |h|\leq \v^{-1},\\[3mm]
\v^{-1}&\mbox{\rm if}& |h|\geq \v^{-1},
\end{array}
\right. {\nonumber}$$ and $$\kappa^\v=\kappa_1+\kappa_2|\r_\v|^2,$$ and the subscriptions $\v, \nu$ define the smoothing operators in general by $f_{\mu} = {\rm Ext}(f)*\eta_{\mu}$ with $\mu=\nu, \v$. Here $\eta_{\mu}$ is the standard mollifier and ${\rm Ext}(\cdot)$ is the extension operator which extends any measurable functions defined on $\Omega_T$ to be zero on $\R^2\backslash\Omega_T$.
The system (\[asys2\]) can be rewritten as $$\left\{
\begin{array}{l}
\r_t-((\v+(\r\t)_\nu)\r_x)_x-(\r(\r_\v\t_x)_\v)_x
+\r\chi^\v(\sqrt{\t})=\chi^\v(p_s(\t)),\\[5pt]
(\r+\s)\t_t-(\k^\v\t_x)_x-\left[(\v+(\r\t)_\nu)\r_x+\r(\r_\v\t_x)_\v\right]
\t_x -\r\chi^\v(\sqrt{\t})\t+(\l+\t)
p_s(\t)\di=\l\r\chi^\v(\sqrt{\t}) \, .
\end{array}
\right. \label{asys}$$ The corresponding initial and boundary conditions are given by $$\begin{array}{l}
\di (\v+(\r\t)_\nu)\r_x+\r(\r_\v\t_x)_\v\big|_{x=1}
=\a^1(\bar\r^1-\r(1,t)),\\[3mm]
\di (\v+(\r\t)_\nu)\r_x+\r(\r_\v\t_x)_\v\big|_{x=0}
=\a^0(\r(0,t)-\bar\r^0),\\[3mm]
\di \r(x,0)=\r_{0\v}(x):=(\r_0)_\v(x)+\v,\\[3mm]
\di \k^\v\t_x|_{x=1}=\b^1(\bar\t^1-\t(1,t)),\\[3mm]
\di \k^\v\t_x|_{x=0}=\b^0(\t(0,t)-\bar\t^0),\\[3mm]
\di \t(x,0)=\t_{0\v}(x):=(\t_0)_\v(x).
\end{array}
\label{asys-bi}$$
We prove the existence of solutions to the system (\[asys\])-(\[asys-bi\]) by using the Leray–Schauder fixed point theorem. The following lemma (see [@Lions1], [@Lions]) is useful in our proof. 0.1in
$\!\!$[**(Aubin–Lions)**]{}$\;$ [*Let $B_1\hookrightarrow\hookrightarrow B_2\hookrightarrow B_3$ be reflective and separable Banach spaces. Then $$\{u\in L^p(I;B_1)|\;u_t\in
L^q(I;B_3)\}\hookrightarrow\hookrightarrow L^p(I;B_2), \quad
1<p,q<\infty;$$ $$\{u\in L^q(I;B_2)\cap L^1(I;B_1)|\;u_t\in L^1(I;B_3)\}
\hookrightarrow\hookrightarrow L^p(I;B_2),\quad 1\leq
p<q<\infty.$$* ]{}
Existence of approximate solutions
----------------------------------
We define $$\begin{aligned}
X=\{u\in L^2(I;H^1(\Omega))|\;u\geq0\},\quad Y=\{u\in
W^{2,1}_2(Q_T)|\;u\geq0\}.\end{aligned}$$ By Aubin–Lions lemma, $Y\hookrightarrow\hookrightarrow X$. Let $\v$ and $\nu$ be given positive constants and the parameter $s\in[0,1]$. For any given $(\r^0,\t^0)\in X^2$, we define $\r$ to be the solution of the following linear parabolic equation $$\r_t-((\v+(\r^0\t^0)_\nu)\r_x)_x-(\r(\r^0_\v\t^0_x)_\v)_x
+s\r\chi^\v(\sqrt{\t^0})=s\chi^\v(p_s(\t^0)), \label{rho-e}$$ with the initial and boundary conditions $$\left\{
\begin{array}{lr}
\di(\v+(\r^0\t^0)_\nu)\r_x+\r(\r^0_\v\t^0_x)_\v
=\a^1(s\bar\r^1-\r),&\mbox{\rm at}~~~ x=1,
\\[3mm]
\di(\v+(\r^0\t^0)_\nu)\r_x+\r(\r^0_\v\t^0_x)_\v
=\a^0(\r-s\bar\r^0),&\mbox{\rm at}~~~
x=0,\\[3mm]
\r(x,0)=s\r_{0\v}(x),&\mbox{\rm for}~~~x\in\Omega.
\end{array}
\right. \label{rho-b}$$ Now with $\r$ in hand, we define $\t$ to be the solution of the linear parabolic equation $$\begin{array}{r}
(\r+\s)\t_t-(\k^\v\t_x)_x-\big[(\v+(\r^0\t^0)_\nu)\r_x
+\r(\r^0_\v\t^0_x)_\v\big]\t_x\\[6pt]
-s\r\chi^\v(\sqrt{\t^0})\t+s(\lambda+\t)
p_s(\t)\di=s\lambda\r\chi^\v(\sqrt{\t^0}),
\end{array}
\label{theta-e}$$ with the initial and boundary conditions $$\left\{
\begin{array}{lr}
\di \k^\v\t_x=\b^1(s\bar\t^1-\t),&\mbox{\rm at}~~~
x=1,\\[3mm]
\di \k^\v\t_x=\b^0(\t-s\bar\t^0),&\mbox{\rm at}~~~
x=0,\\[3mm]
\t(x,0)=s\t_{0\v}(x),&\mbox{for}~~~x\in\Omega \, .
\end{array}
\right.\label{theta-b}$$ Let $M$ denote the mapping from $(\r^0,\t^0,s)$ to $(\r,\t)$. Then we have the following lemma.
\[mpcompconti\] [*The mapping $M:X^2\times[0,1]\rightarrow X^2$ is well defined, continuous and compact.* ]{}
[*Proof*]{}. By the $L^2$-theory of linear parabolic equations [@LSU], there exists a solution $\r\in W^{2,1}_2(Q_T)$ for the system (\[rho-e\])-(\[rho-b\]) such that $$\|\r\|_{W^{2,1}_2(Q_T)}\leq
C(\v^{-1},\|(\r^0\t^0)_\nu\|_{C^1(\overline
Q_T)},\|(\r^0_\v\t^0_x)_\v)_x\|_{C^1(\overline
Q_T)},\|\r_{0\v}\|_{H^1(\Omega)},T).$$ By noting the fact $$\|\r^0_\v\|_{H^1(\Omega)}\leq C_\v\|\r_0\|_{L^1(\Omega)}, \quad
\|(\r^0\t^0)_\nu\|_{C^1(\overline Q_T)}\leq
C_{\nu,T}\|\r^0\|_{L^2(Q_T)}\|\t^0\|_{L^2(Q_T)},$$ $$\|(\r^0_\v\t^0_x)_\v)_x\|_{C^1(\overline
Q_T)}\leq C_{\v,T}\|\r^0_\v\t^0_x\|_{L^1(Q_T)}\leq
C_{\v,T}\|\r^0\|_{L^2(Q_T)}\|\t^0_x\|_{L^2(Q_T)} \, ,$$ for the standard smoothing operator, we have $$\begin{aligned}
\label{rhomapping1}
\|\r\|_{W^{2,1}_2(Q_T)}\leq
C(\v^{-1},\nu^{-1},\|\r^0\|_X,\|\t^0\|_X,T)\end{aligned}$$ and therefore, $$\|\r\|_{L^\infty(Q_T)}\leq \|\r\|_{W^{2,1}_2(Q_T)}\leq
C(\v^{-1},\nu^{-1},\|\r^0\|_X,\|\t^0\|_X,T).$$
Let $\r^+=\max\{\r,0\}$, $\r^-=\max\{-\r,0\}$. Then $\r=\r^+-\r^-$. By multiplying $\r^-$ on both sides of the equation (\[rho-e\]) and integrating the resulting equation over $Q_t$, we have $$\begin{aligned}
&\int_0^1\f{|\r^-|^2}{2}dx+\int_0^t\int_0^1
\big(\v+(\r^0\t^0)_\nu\big)|\r^-_x|^2dxd\tau
+\int_0^t\int_0^1\big(s\chi^\v(\sqrt{\t})|\r^-|^2
+s\chi^\v(p_s(\t))\r^-\big)dxd\tau
{\nonumber}\\
&~~+\int_0^t\big(\alpha^0|\r^-(0,\tau)|^2
+\alpha^0s\bar\r^0\r^-(0,\tau)\big)d\tau
+\int_0^t\big(\alpha^1|\r^-(1,\tau)|^2
+\alpha^1s\bar\r^1\r^-(1,\tau)\big)d\tau
{\nonumber}\\
&=-\int_0^t\int_0^1\r^-\r^-_x(\r^0_\v\t^0_x)_\v dxd\tau
{\nonumber}\\
&\leq\int_0^t\int_0^1\biggl(\frac{\|(\r^0_\v\t^0_x)_\v
\|_{L^\infty(Q_T)}}{2\v} |\r^-|^2+\frac{\v}{2}|\r^-_x|^2\biggl)
dxd\tau. {\nonumber}\end{aligned}$$ Notice that $\r^-\geq 0$. Thus we have that $$\int_0^1|\r^-|^2dx \leq\frac{\|(\r^0_\v\t^0_x)_\v
\|_{L^\infty(Q_T)}}{2\v} \int_0^t\int_0^1|\r^-|^2 dxd\tau.$$
By Gronwall’s inequality, we can see that $\r^-\equiv 0$. Thus $\r=\r^+\geq0$. This and (3.8) imply that $\r\in Y\hookrightarrow\hookrightarrow X$.
Similarly, by the $L^2$-theory of quasi-linear parabolic equations [@LSU], there exists a solution $\t\in W^{2,1}_2(Q_T)$ for the system (\[theta-e\])-(\[theta-b\]) and $$\begin{aligned}
\label{rhomapping1}
\|\t\|_{W^{2,1}_2(Q_T)}\leq
C(\v^{-1},\nu^{-1},\|\r^0\|_X,\|\t^0\|_X,T).\end{aligned}$$ Let $\t^+=\max\{\t,0\}$, $\t^-=\max\{-\t,0\}$. Then $\t=\t^+-\t^-$. Multiplying $\t^-/(\r+\sigma)$ on both sides of the equation (\[theta-e\]) and integrating the resulting equation over $Q_t$, we can get $$\begin{array}{l}
\di\int_0^1\f{|\t^-|^2}{2}dx+\int_0^t\int_0^1\f{\k^\v}{\r+\s}|\t_x^-|^2dxd\tau+\int_0^t\int_0^1\f{s(\l+\t)p_s(\t)}{(\r+\s)}\t^-dxd\tau\\
\di
+\int_0^t\int_0^1s\l\r\chi^\v(\sqrt{\t^0})\f{\t^-}{\r+\s}dxd\tau+\int_0^t\f{\t^-(1,\tau)}{\r(1,\tau)+\s}\b^1(s\bar\t^1+\t^-(1,\tau))d\tau\\
\di+\int_0^t\f{\t^-(0,\tau)}{\r(0,\tau)+\s}\b^1(s\bar\t^1+\t^-(0,\tau))d\tau=\int_0^t\int_0^1s\r\chi^\v(\sqrt{\t^0})\f{|\t^-|^2}{\r+\s}dxd\tau\\
\di+\int_0^t\int_0^1\k^\v\t_x^-\f{\r_x\t^-}{(\r+\s)^2}dxd\tau+\int_0^t\int_0^1\big[(\v+(\r^0\t^0)_\nu)\r_x
+\r(\r^0_\v\t^0_x)_\v\big]\t^-_x\f{\t^-}{\r+\s}dxd\tau.
\end{array}$$ Since $p_s(\t)=0$ for $\t\leq0$, we observe that $(\l+\t)p_s(\t)\t^-=0$ a.e in $\Omega_T$. By Cauchy inequality and the estimations (3.8)-(3.9), we can estimate the terms in the right hand side of the above equality. Thus we obtain $$\int_0^1|\t^-|^2dx \leq C(\v^{-1},\nu^{-1},\|\r^0\|_X,\|\t^0\|_X,T)
\int_0^t\int_0^1|\t^-|^2 dxd\tau.$$ Gronwall’s inequality gives that $\t^-\equiv 0$. Thus $\t=\t^+\geq0$. This and (3.9) imply that $\t\in
Y\hookrightarrow\hookrightarrow X$.
We conclude that the mapping $M:X^2\times[0,1]\rightarrow X^2$ is a compact mapping.
Now we prove the continuity of the mapping $M$. For any $(\hat\r^0,\hat\t^0,\hat s)\in X^2\times[0,1]$, let $(\hat\r,\hat\t)=M(\hat\r^0,\hat\t^0,\hat s)$. Then $$\hat\r_t-[(\v+(\hat\r^0\hat\t^0)_\nu)\hat\r_x+\hat\r(\hat\r^0_\v\hat\t^0_x)_\v]_x
+\hat s\hat\r\chi^\v(\sqrt{\hat\t^0})=\hat s\chi^\v(p_s(\hat\t^0)),
\label{rho-e1}$$ $$\begin{array}{r}
(\hat\r+\s)\hat\t_t-(\hat\k^\v\hat\t_x)_x-\big[(\v+(\hat\r^0\hat\t^0)_\nu)\hat\r_x
+\hat\r(\hat\r^0_\v\hat\t^0_x)_\v\big]\hat\t_x\\[6pt]
-\hat s\hat\r\chi^\v(\sqrt{\hat\t^0})\hat\t+\hat s(\lambda+\hat\t)
p_s(\hat\t)\di=\hat s\lambda\hat\r\chi^\v(\sqrt{\hat\t^0}),
\end{array}
\label{theta-e1}$$ with the initial and boundary conditions $$\left\{
\begin{array}{lr}
\di(\v+(\hat\r^0\hat\t^0)_\nu)\hat\r_x+\hat\r(\hat\r^0_\v\hat\t^0_x)_\v
=\a^1(\hat s\bar\r^1-\hat\r),&\mbox{\rm at}~~~ x=1,
\\[3mm]
\di(\v+(\hat\r^0\hat\t^0)_\nu)\hat\r_x+\hat\r(\hat\r^0_\v\hat\t^0_x)_\v
=\a^0(\hat\r-\hat s\bar\r^0),&\mbox{\rm at}~~~
x=0,\\[3mm]
\r(x,0)=\hat s\r_{0\v}(x),&\mbox{\rm for}~~~x\in\Omega.
\end{array}
\right. \label{rho-b1}$$ and $$\left\{
\begin{array}{lr}
\di \hat\k^\v\hat\t_x=\b^1(\hat s\bar\t^1-\hat\t),&\mbox{\rm at}~~~
x=1,\\[3mm]
\di \hat\k^\v\hat\t_x=\b^0(\hat\t-\hat s\bar\t^0),&\mbox{\rm at}~~~
x=0,\\[3mm]
\t(x,0)=\hat s\t_{0\v}(x),&\mbox{for}~~~x\in\Omega \, .
\end{array}
\right.\label{theta-b1}$$
Denote $\tilde\r=\r-\hat\r$ and $\tilde\t=\t-\hat\t$. Then $\tilde\r$ satisfies the following equation, $$\begin{array}{l}
\di \tilde\r_t-(F-\hat F)_x+s\tilde\r\chi^\v(\sqrt{\t^0})+(s-\hat
s)\hat\r\chi^\v(\sqrt{\t^0})+\hat s\hat
\r[\chi^\v(\sqrt{\t^0})-\chi^\v(\sqrt{\hat\t^0})]\\
\di =(s-\hat s)\chi^\v(p_s(\t^0))+\hat
s[\chi^\v(p_s(\t^0))-\chi^\v(p_s(\hat\t^0))],
\end{array}
\label{density}$$ where $$F=(\v+(\r^0\t^0)_\nu)\r_x+\r(\r^0_\v\t^0_x)_\v,$$ $$\hat
F=(\v+(\hat\r^0\hat\t^0)_\nu)\hat\r_x+\hat\r(\hat\r^0_\v\hat\t^0_x)_\v.$$ Multiplying the equation (\[density\]) by $\tilde\r$ and integrating over $Q_t$ gives $$\int_0^1\tilde\r^2(x,t) dx+\int_0^t\int_0^1\tilde\r_x^2
dxd\tau\leq C\left[\int_0^t\int_0^1\tilde\r^2dxd\tau+(s-\hat
s)^2+\|\r^0-\hat\r^0\|_X^2+\|\t^0-\hat\t^0\|_X^2\right]$$ with $C=C(\v^{-1},\nu^{-1},\|\r_{0\v}\|_{L^2(\Omega)},\|\r^0\|_X,\|\t^0\|_X,\|\hat\r^0\|_X,\|\hat\t^0\|_X,T)$.
Thus Gronwall inequality implies that $$\|\tilde\r\|^2_X\leq C\left[(s-\hat
s)^2+\|\r^0-\hat\r^0\|_X^2+\|\t^0-\hat\t^0\|_X^2\right].$$ Similarly, we can derive the equation for $\tilde\t$ and get $$\|\tilde\t\|^2_X\leq C\left[(s-\hat
s)^2+\|\r^0-\hat\r^0\|_X^2+\|\t^0-\hat\t^0\|_X^2\right].$$ Thus, the mapping $M:X^2\times[0,1]\rightarrow X^2$ is continuous. The proof of Lemma 3.2 is complete. 0.1in
In addition, for $s=0$ we can see that $M(\r,\t,0)=0$ for any $(\r,\t)\in X^2$. Thus, by the Leray–Schauder fixed point theorem, there exists a fixed point for the mapping $M(\cdot,\cdot,1):X^2\rightarrow X^2$ if all the functions $(\r,\t)\in X^2$ satisfying $$\label{fixpointp}
(\r,\t)=M(\r,\t,s)$$ for some $s\in[0,1]$ are uniformly bounded in $X^2$. In fact, by the proof of Lemma \[mpcompconti\], $M$ maps $(\r,\t,s)\in
X^2\times[0,1]$ into $Y^2$. Therefore, if $(\r,\t)$ is a fixed point of $M(\cdot,\cdot,1)$, then $(\r,\t)\in W^{2,1}_2(Q_T)$.
\[exapps\][*Under the assumptions of Theorem \[thm1\], the system (\[asys\])-(\[asys-bi\]) has a (strong) solution $(\r,\t)\in
W^{2,1}_2(Q_T)$ which satisfies $$\begin{aligned}
&\displaystyle\rho\geq\underline{\r}_{\v,T}
\quad\mbox{and}\quad\underline{\t}_T\leq\t\leq
\overline{\t}_{T}\quad\mbox{for}\;\;\; (x,t) \in Q_T.
\label{positive}\\[5pt]
&\displaystyle\|\r\|_{L^\infty(I;L^4(\Omega))},\;\;
\|\r_x\|_{L^2(Q_T)},\;\;\|\r\r_x\|_{L^2(Q_T)}\;
\leq C_{\v,T},{\nonumber}\\[5pt]
&\displaystyle\|\t\|_{L^\infty(Q_T)},\;\;
\|\r\|_{L^\i(I;L^1(\Omega))},\;\; \| \t_x \|_{L^2(Q_T)}, \;\;\|
\rho_\v\t_x \|_{L^2(Q_T)}\;\leq C_{T} \, . \label{(3.17-2)}\end{aligned}$$ where $\underline{\r}_{\v,T}$ and $C_{\v,T}$ are positive constants which depend on $\v$ and $T$, independent of $\nu$; $\underline{\t}_T$, and $\overline{\t}_{T}$ and $C_{T}$ are positive constants, dependent upon $T$ and independent of $\v$ and $\nu$.* ]{}
By the Leray-Schauder fixed point theorem, it suffices to prove the uniform boundedness of functions $(\r,\t)\in X^2$ satisfying the equation (\[fixpointp\]) and (\[positive\]).
Uniform estimates
-----------------
We assume that $(\r,\t)\in X^2$ and therefore, $(\r,\t)=M(\r,\t,s)
\in Y^2$, for $s\in[0,1]$, $i.e.$, $(\r,\t)$ is a (strong) solution of the following system, $$\begin{aligned}
&\r_t-((\v+(\r\t)_\nu)\r_x)_x-(\r(\r_\v\t_x)_\v)_x
+s\r\chi^\v(\sqrt{\t})=s\chi^\v(p_s(\t)), \label{rho-ef}\\[5pt]
&(\r+\s)\t_t-(\k\t_x)_x-\left[(\v+(\r\t)_\nu)\r_x
+\r(\r_\v\t_x)_\v\right]\t_x{\nonumber}\\[2pt]
&~~~~~~~~~~-s\r\chi^\v(\sqrt{\t})\t+s(\l+\t)
p_s(\t)\di=s\lambda\r\chi^\v(\sqrt{\t}), \label{theta-ef}\end{aligned}$$ with the initial and boundary conditions $$\left\{
\begin{array}{lr}
\di(\v+(\r\t)_\nu)\r_x+\r(\r_\v\t_x)_\v
=\a^1(s\bar\r^1-\r),&\mbox{\rm at}~~~
x=1,\\[3mm]
\di(\v+(\r\t)_\nu)\r_x+\r(\r_\v\t_x)_\v
=\a^0(\r-s\bar\r^0),&\mbox{\rm at}~~~
x=0,\\[3mm]
\r(x,0)=s\r_{0\v}(x),&\mbox{\rm for}~~~x\in\Omega,
\end{array}
\right. \label{rho-bf}$$ and $$\left\{
\begin{array}{lr}
\di \k^\v\t_x=\b^1(s\bar\t^1-\t),&\mbox{\rm at}~~~
x=1,\\[3mm]
\di \k^\v\t_x=\b^0(\t-s\bar\t^0),&\mbox{\rm at}~~~
x=0,\\[3mm]
\t(x,0)=s\t_{0\v}(x),&\mbox{for}~~~x\in\Omega,
\end{array}
\right.\label{theta-bf}$$
In this subsection, we derive some uniform estimates for solutions to the above initial-boundary value problems.
Firstly we add the equation (\[rho-ef\]) multiplying by $(\lambda+\t)$ into (\[theta-ef\]) and then, integrate the resulting equation over $Q_t$. We arrive at $$\int_0^1(\l\r+\r\t+\s\t)(x,t)dx- \left. \int_0^tH_2(x,\tau) \right
|_{x=0}^{x=1}
d\tau\leq\int_0^1(\l\r_{0\v}+\r_{0\v}\t_{0\v}+\s\t_{0\v})(x)dx {\nonumber}$$ where $$H_2(x,\tau)=[\v\r_x+(\r\t)_\nu\r_x+\r(\r_\v\t_x)_\v](\l+\t)+\k^\v\t_x
\, . {\nonumber}$$ With boundary conditions in (\[rho-bf\])-(\[theta-bf\]), we have $$\begin{aligned}
-H_2(x,\tau) \Big |_{x=0}^{x=1} &=&
\a^1(\r(1,\tau)-s\bar\r^1)(\l+\t(1,\tau))
+\a^0(\r(0,\tau)-s\bar\r^0)(\l+\t(0,\tau)) {\nonumber}\\ [2mm] && +
\b^1(\t(1,\tau)-s\bar\t^1)+\b^0(\t(0,\tau)-s\bar\t^0) {\nonumber}\\ [3mm]
&\geq & -\a^1s\bar\r^1\t(1,\tau)-\a^0s\bar\r^0\t(0,\tau) -\l
s(\a^1\bar\r^1+\a^0\bar\r^0)-s(\b^1\bar\t^1+\b^0\bar\t^0) {\nonumber}\end{aligned}$$ and therefore, $$\int_0^1(\l\r+\r\t+\s\t)(x,t)dx\leq
C_T+C\int_0^t\|\t(\cdot,\tau)\|_{C(\bar\Omega)}d\tau,
\label{(3.14)}$$ where $$C_T=(\l+\|\t_{0\v}\|_{L^\i})\|\r_{0\v}\|_{L^1}+\s\|\t_{0\v}\|_{L^\i}
+\left[\l(\a^1\bar\r^1+\a^0\bar\r^0)+(\b^1\bar\t^1+\b^0\bar\t^0)\right]T.$$
Similarly, subtracting the equation (\[theta-ef\]) multiplied by $\t^l/l$ from the equation (\[rho-ef\]) multiplied by $\t^{l+1}/(l+1)$ and integrating the resulting equation over $Q_t$, we arrive at $$\begin{aligned}
\label{thetalp}
&& \int_0^1 (\r+\s)\t^{l+1}(x,t) dx- \left.
\int_0^tH_3(x,\tau)\right |_{x=0}^{x=1}d\tau +\int_0^t\int_0^1\k^\v
l(l+1)\t^{l-1}|\t_x|^2 dxd\tau
{\nonumber}\\
&& +s(l+1)\int_0^t\int_0^1(\lambda+\t)p_s(\t)\t^ldxd\tau =\int_0^1
(\r_{0\v}+\s)(\t_{0\v})^{l+1}(x) dx \label{rtk}
\\
&&+s\int_0^t\int_0^1\Big[l\t^{l+1} +\l(l+1)\t^l\Big]
\r\chi^\v(\sqrt{\t}) dxd\tau
+s\int_0^t\int_0^1\chi^\v(p_s(\t))\t^{l+1}dxd\tau {\nonumber},\end{aligned}$$ where $$\begin{array}{lll}
\di -H_3(x,\tau) \Big |_{x=0}^{x=1}&=&\di
\a^1(\r(1,\tau)-s\bar\r^1)[\t(1,\tau)]^{l+1}
+\a^0(\r(0,\tau)-s\bar\r^0)[\t(0,\tau)]^{l+1}
\\[3mm]
&&\di+ (l+1)\b^1(\t(1,\tau)-s\bar\t^1)[\t(1,\tau)]^{l}+(l+1)
\b^0(\t(0,\tau)-s\bar\t^0)[\t(0,\tau)]^{l}\\[3mm]
&=&\di [\a^1\r(1,\tau)+(l+1)\b^1-\a^1s\bar\r^1]
[\t(1,\tau)]^{l+1}-(l+1)\b^1s\bar\t^1[\t(1,\tau)]^{l}\\[3mm]
&&\di+[\a^0\r(0,\tau)+(l+1)\b^0-\a^0s\bar\r^0]
[\t(0,\tau)]^{l+1}-(l+1)\b^0s\bar\t^0[\t(0,\tau)]^{l},
\\[3mm]
&\geq&\di -2^l(l+1)[\b^1(s\bar\t^1)^{l+1}+\b^0(s\bar\t^0)^{l+1}] {\nonumber}\label{(3.43+)}
\end{array}$$ when $l$ is large enough. Since $\t^l\leq \t^{1/2}+\t^{l+1}$ for any $\t\geq0$ and $l\geq1$, by (\[(3.14)\])-(\[thetalp\]), $$\begin{aligned}
\label{prertl} && \int_0^1 (\r+\s)\t^{l+1}(x,t) dx
+l(l+1)\int_0^t\int_0^1\k^\v \t^{l-1}|\t_x|^2
dxd\tau+sl\int_0^t\int_0^1(\lambda+\t)p_s(\t)\t^ldxd\tau
{\nonumber}\\
&&\leq C_{l,T}+C_0s\int_0^t\int_0^1l\big(1+\t^{l+1/2}\big) \r\t
dxd\tau
{\nonumber}\\
&& \leq C'_{l,T}+C_0sl\int_0^t \|\t(\cdot,\tau)
\|_{L^\infty(\Omega)}^{l+3/2} d\tau
$$ where $$C_{l,T}=\int_0^1 (\r_0+\s)\t_0^{l+1}(x) dx +
2^l(l+1)[\b^1(s\bar\t^1)^{l+1}+\b^0(s\bar\t^0)^{l+1}]$$ and $C_{l,T}'=C_{l,T}+Cl$. Recall the Gagliardo–Nirenberg inequality $$\|f\|_{L^\infty(\Omega)}\leq C
\|f\|_{L^2(\Omega)}+C\|f\|_{L^2(\Omega)}^{1/2}
\|f_x\|_{L^2(\Omega)}^{1/2},\quad \forall\,f\in H^1(\Omega).$$ With $f=\t^{\frac{l+1}{2}}$ in the above inequality, we obtain $$\label{ttttt}
\|\t(\cdot,\tau)\|_{L^\infty(\Omega)}^{l+3/2}\leq
\frac{C_2}{2}\int_0^1\t^{l+3/2}(x,\tau)dx
+C_1\big\|\t^{\f{l+1}{2}}(\cdot,\tau)\big\|_{L^2(\Omega)}^{\f{2l+3}{2l+2}}
\big\|(\t^{\f{l+1}{2}})_x(\cdot,\tau)\big\|_{L^2(\Omega)}^{\f{2l+3}{2l+2}}
{\nonumber}$$ and by Hölder’s inequality, $$\begin{aligned}
&&\int_0^t\big\|\t^{\f{l+1}{2}}
(\cdot,\tau)\big\|_{L^2(\Omega)}^{\f{2l+3}{2l+2}}
\big\|(\t^{\f{l+1}{2}})_x(\cdot,\tau)
\big\|_{L^2(\Omega)}^{\f{2l+3}{2l+2}}d\tau{\nonumber}\\
&&\leq Cl\int_0^t\int_0^1(\t^\f{l+1}{2})^{\f{4l+6}{2l+1}}dxd\tau
+\frac{1}{(l+1)C_0C_1}\int_0^t\int_0^1\k|(\t^\f{l+1}{2})_x|^2dxd\tau
{\nonumber}\\
&&\leq Cl\int_0^t\int_0^1\t^\f{(l+1)(2l+3)}{2l+1}dxd\tau
+\frac{l+1}{4C_0C_1}\int_0^t\int_0^1\k\t^{l-1}|\t_x|^2dxd\tau. {\nonumber}\end{aligned}$$ It follows that $$\begin{aligned}
\int_0^t\|\t(\cdot,\tau)\|_{L^\infty(\Omega)}^{l+3/2}d\tau \leq
\frac{C_2}{2} \int_0^t\int_0^1\t^{l+3/2}dxd\tau
+Cl\int_0^t\int_0^1\t^\f{(l+1)(2l+3)}{2l+1}dxd\tau
{\nonumber}\\
+\frac{l+1}{4C_0}\int_0^t\int_0^1\k_1\t^{l-1}|\t_x|^2dxd\tau {\nonumber}\, .\end{aligned}$$ By the assumption (\[sat-a\]), we observe that $C_0 C_2
\t^{l+\f32}\leq p_s(\t)\t^{l+1}+C$ for all $\t\geq0$. Substituting the last inequality into (\[prertl\]) gives $$\begin{aligned}
\label{prertf}
&&\int_0^1 (\r+\s)\t^{l+1}(x,t) dx
+\frac{l(l+1)}{2}\int_0^t\int_0^1\k^\v
\t^{l-1}|\t_x|^2dxd\tau+\frac{sl}{2}\int_0^t\int_0^1p_s(\t)\t^{l+1}dxd\tau
{\nonumber}\\
&&\leq C_{l,T}'+C_3sl^2
\int_0^t\int_0^1\t^\f{(l+1)(2l+3)}{2l+1}dxd\tau,\end{aligned}$$ for $l$ being large enough. Let $l_0$ be a positive integer satisfying $$\frac{(l_0+1)(2l_0+3)}{2l_0+1} =l_0+1 + \frac{2l_0+2}{2l_0+1}<l_0
+ 1 +(1+\eta)$$ where $\eta$ is defined in (\[sat-a\]). By noting the fact $$C_3 \t^\f{(l_0+1)(2l_0+3)}{2l_0+1} \le \frac{1}{4l_0} p_s(\theta)
\theta^{l_0+1} + (C l_0)^{l_1}$$ with $l_1 = 2(l_0+2+\eta)/\eta$, we have $$\label{prertff}
\int_0^1 (\r+\s)\t^{l_0+1}(x,t) dx
+\frac{l_0(l_0+1)}{2}\int_0^t\int_0^1\k^\v \t^{l_0-1}|\t_x|^2dxd\tau
+\frac{sl}{4}\int_0^t\int_0^1p_s(\t)\t^{l_0+1}dxd\tau\leq
C_{l_0,T}'', {\nonumber}$$ where $C_{l_0,T}''=C_{l_0,T}'+C_T (C l_0)^{l_1}$ for some constant $C_T$ independent of $l_0$. Furthermore, $$\label{tff}
\sup_{0\leq t\leq
T}\int_0^1\t^{l_0+1}(x,t)dx+\int_0^T\int_0^1|(\t^{\frac{l_0+1}{2}})_x|^2
dxdt \leq C_{l_0,T}'' {\nonumber}$$ and by the Sobolev embedding inequality, $$\label{tff2}
\int_0^T\|\t\|_{L^\infty(\Omega)}^{l_0+1} dxdt \leq
C_T^{l_0+1}C_{l_0,T}''. {\nonumber}$$ Since $l_0$ is a fixed positive integer dependent solely upon $\eta$, we obtain the estimate $$\label{tff3}
\int_0^T\|\t\|_{L^\infty(\Omega)} dxdt \leq C_T.$$ From (\[(3.14)\]) and (\[prertl\]), we get $$\label{rrtl1}
\sup_{0\leq t\leq T}\int_0^1(\r+\r\t)dx\leq C_T$$ and $$\begin{aligned}
\label{prertl22}
&& \int_0^1 (\r+\s)\t^{l+1}(x,t) dx +l(l+1)\int_0^t\int_0^1\k^\v
\t^{l-1}|\t_x|^2
dxd\tau+sl\int_0^t\int_0^1(\lambda+\t)p_s(\t)\t^ldxd\tau
{\nonumber}\\
&&\leq C_{l,T}+Csl\int_0^t\int_0^1\r\t
dxd\tau+Csl\int_0^t\|\t\|_{L^\infty(\Omega)}^{1/2}\int_0^1
\r\t^{l+1} dxd\tau
{\nonumber}\\
&&\leq (C_{l,T}+C_Tl)+Csl\int_0^t\|\t\|_{L^\infty(\Omega)}^{1/2}
\int_0^1(\r+\sigma)\t^{l+1} dxd\tau. {\nonumber}\end{aligned}$$ Moreover, by using Gronwall’s inequality, $$\int_0^1 (\r+\s)\t^{l+1}(x,t) dx\leq (C_{l,T}+C_Tl)+(C_{l,T}+C_Tl)
e^{C_T l}$$ and $$\| \t \|_{L^{l+1}(Q_T)} \le [2(C_{l,T}+C_Tl)]^{\frac{1}{l+1}}
e^{C_T} \, .$$ On the other hand, by taking $l\rightarrow\infty$, we have $$\label{tlinfn} \|\t\|_{L^\infty(Q_T)}\leq C_T$$ where we have noted the fact $$C_{l,T}^{\frac{1}{l+1}} \leq C_T \, .$$ Moreover, by taking $l=1$ in the equation (\[prertl\]), we obtain $$\di \int_0^1 (\r+\s)\t^{2}(x,t) dx+\f{1}{2}\int_0^t\int_0^1
\big((\k_1+\k_2|\r_\v|^2)|\t_x|^2+s\t^2 p_s(\t)\big) dxd\tau\leq C_T
{\nonumber},
$$ which implies that $$\| \t_x \|_{L^2(Q_T)},\;\;\| \rho_\v\t_x \|_{L^2(Q_T)}\leq C_{T}
\, \, . \label{(3.17)}$$
Secondly we present some estimates for $\rho$. By multiplying $\r$ on both sides of the equation (\[rho-ef\]) and integrating the resulting equation over $Q_T$, with Gronwall’s inequality we get $$\begin{aligned}
\label{rho2pv}
\sup_{0\leq t\leq T}\int_0^1\r^2dx+\int_0^T\int_0^1|\r_x|^2dxdt
\leq C_{\v,T}+C(\v,\|(\r_\v\t_x)_\v\|_{L^\infty(Q_T)}) \le
C_{\v,T},\end{aligned}$$ which together with the Sobolev embedding inequality gives $$\int_0^T\int_0^1\r^6dxdt\leq C_{\v,T}.$$
Once again, multiplying $\r^3$ on both sides of the equation (\[rho-ef\]) and integrating the resulting equation over $Q_T$ lead to $$\begin{aligned}
\label{rho2pv2}
\sup_{0\leq t\leq
T}\int_0^1\r^4dx+\int_0^T\int_0^1\r^2|\r_x|^2dxdt \leq C_{\v,T}.\end{aligned}$$
From (\[tlinfn\]), (\[(3.17)\]) and (\[rho2pv\]), we conclude that $(\r,\t)$ is uniformly bounded in $X^2$. Thus, by the Leray–Schauder fixed point theorem, there exists a fixed point $(\r^{\v,\nu},\t^{\v,\nu})$ for the mapping $M(\cdot,\cdot,1):X^2\rightarrow X^2$ and $(\r^{\v,\nu},\t^{\v,\nu})$ is a solution of the system (\[asys\])-(\[asys-bi\]).
Positivity of the approximate solutions {#uniest}
---------------------------------------
Finally we prove the positivity of the approximate solutions $(\r^{\v,\nu},\t^{\v,\nu})$. Let $\tilde\t^\delta=\t e^t-\delta$. Then $\tilde\t^\delta$ is the solution of the following problem, $$\begin{aligned}
&(\r+\s)\tilde\t_t^\delta-(\k^\v\tilde\t_x^\delta)_x
-\left[(\v+(\r\t)_\nu)\r_x +\r(\r_\v\t_x)_\v\right]\tilde\t_x^\delta
-(\r+\sigma)\tilde\t^\delta-\r\chi^\v(\sqrt{\t})
\tilde\t^\delta{\nonumber}\\[6pt]
&~+\tilde q(\t e^t,\delta)\tilde\t^\delta\di =\r\chi^\v(\sqrt{\t})\t
e^t+\l\r\chi^\v(\sqrt{\t})e^t+(\r+\sigma)\delta -(\l+e^{-t}\delta)
p_s(e^{-t}\delta)e^t, \label{theta-e3}\end{aligned}$$ with the initial and boundary conditions $$\left\{
\begin{array}{lr}
\di
\k^\v\tilde\t^\delta_x+\b^1\tilde\t^\delta=\b^1(\bar\t^1e^t-\delta),&\mbox{\rm
at}~~~ x=1,\\[3mm]
\di -\k^\v\tilde\t^\delta_x+\b^0\tilde\t^\delta
=\b^1(\bar\t^0e^t-\delta),&\mbox{\rm at}~~~ x=0,\\[3mm]
\tilde\t^\delta(x,0)=\t_{0\v}(x)-\delta,&\mbox{for}~~~x\in\Omega,
\end{array}
\right.\label{theta-b3}$$ where $$\tilde q(\tilde\t,\delta)=\frac{(\l+e^{-t}\tilde\t)
p_s(e^{-t}\tilde\t)-(\l+e^{-t}\delta)
p_s(e^{-t}\delta)}{\tilde\t-\delta}e^t\geq0.$$ By the assumption (\[sat-a\]), the right hand side of the equations (\[theta-e3\])-(\[theta-b3\]) are nonnegative if $\delta$ is small enough (independent of $\v$ and $\nu$). Multiplying $(\tilde\t^\delta)^-/(\r+\sigma)$ on both sides of the equation (\[theta-e3\]) and integrating the resulting equation over $Q_t$, we derive $\tilde\t^\delta\geq 0$, i.e. $\t\geq
e^{-T}\delta$, which together with (\[tlinfn\]) implies that $$\label{unibdtheta}
\underline{\t}_{T}\leq\theta(x,t)\leq\overline{\t}_{T}
\;\;\;\mbox{for}\;\;\; (x,t) \in Q_T.$$ where $\underline{\t}_T$ and $\overline{\t}_{T}$ are positive constants independent of $\v$ and $\nu$.
For $\r$, we define $\r^\delta=\r-\delta$. Then $\r^\delta$ is the solution of the following equation $$\label{rho-e2}
\r^\delta_t-((\v+(\r\t)_\nu)\r^\delta_x)_x-(\r^\delta(\r_\v\t_x)_\v)_x
+\r^\delta\chi^\v(\sqrt{\t})=\chi^\v(p_s(\t))+\delta[(\r_\v\t_x)_\v]_x
-\delta\chi^\v(\sqrt{\t}),$$ with the initial and boundary conditions $$\left\{
\begin{array}{lr}
\di(\v+(\r\t)_\nu)\r^\delta_x+\r^\delta(\r_\v\t_x)_\v+\a^1\r^\delta
=\a^1(\bar\r^1-\delta)-\delta(\r_\v\t_x)_\v,&\mbox{\rm
at}~~~ x=1,\\[3mm]
\di-(\v+(\r\t)_\nu)\r^\delta_x-\r^\delta(\r_\v\t_x)_\v
+\a^0\r^\delta=\a^0(\bar\r^0-\delta)+\delta(\r_\v\t_x)_\v,&\mbox{\rm
at}~~~ x=0,\\[3mm]
\r^\delta(x,0)=\r_{0\v}(x)-\delta,&\mbox{\rm for}~~~x\in\Omega.
\end{array}
\right. \label{rho-b2}$$ Since $\chi^\v(p_s(\t))\geq\v$, the right hand side of the equations (\[rho-e2\])-(\[rho-b2\]) are nonnegative if $$\delta=\min\biggl\{\frac{\v}{2},\;\;
\frac{\v}{1+2\|(\r_\v\t_x)_\v\|_{C^1(\overline Q_T)}}\biggl\},$$ in which case $\r^\delta\geq0$, or eqivalently $\r\geq\delta$. On the other hand, from (\[(3.17)\]) we have $$\|(\r_\v\t_x)_\v\|_{C^1(\overline
Q_T)}\leq\frac{1}{\v^2}\|\r_\v\t_x\|_{L^1(Q_T)}\leq C_\v.$$ Thus, there exists a positive constant $\underline{\r}_{\v,T}$ such that $$\label{lowerbdrho}
\rho\geq\underline{\r}_{\v,T}\;\;\;\mbox{for}\;\;\; (x,t) \in Q_T
\,.$$
Global existence
================
We have constructed an approximate solution $(\rho^{\v, \nu},
\theta^{\v, \nu})$ to the system (\[asys2\]) and (\[asys-bi\]) (or equvilently (\[asys\])-(\[asys-bi\])) in the last section. In this section, we prove the global existence of weak solutions for the system (\[sys\])-(\[sys-i\]). Firstly we fix $\v>0$ and study the convergence as $\nu\rightarrow0$.
Since the system (\[rho-ef\])-(\[theta-ef\]) reduces to (\[asys\])-(\[asys-bi\]) when $s=1$, the uniform estimates (\[tlinfn\]), (\[(3.17)\]), (\[rho2pv\]) and (\[unibdtheta\]) given in the last section still hold for the approximate solution $(\rho^{\epsilon, \nu}, \theta^{\epsilon,
\nu})$. We rewrite the first equation in (\[asys\]) by $$\r_t=-f_x+g$$ with $g$ uniformly bounded in $L^2(Q_T)$ and $$f=(\v+(\r\t)_\nu)\r_x+\r(\r_\v\t_x)_\v.$$ Since $\r$ is uniformly bounded in $L^\infty(I;L^2(\Omega))\cap
L^2(I;H^1(\Omega))\hookrightarrow L^6(Q_T)$, we derive that $$\|\r(\r_\v\t_x)_\v\|_{L^2(Q_T)},\;\;\|(\r\t)_\nu\|_{L^6(Q_T)},\;\;
\|(\r\t)_\nu\r_x\|_{L^\frac{5}{4}(Q_T)}\; \leq C_{\v,T}$$ and $$\|\r_t\|_{L^{5/4}(I;W_0^{-1,{5/4}}(\Omega))}\leq C_{\v,T}.$$ From the first equation in (\[asys2\]) we derive that $$\|(\r\t+\sigma\t)_t\|_{L^{5/4}(I;W_0^{-1,{5/4}}(\Omega))}\leq
C_{\v,T}$$ where we have noted (\[tlinfn\]), and moreover, from (\[rho2pv\]), we observe that $\r^{\v,\nu}$ is uniformly bounded in $L^6(I;L^6(\Omega))\cap L^2(I;H^1(\Omega))$ and $\r^{\v,\nu}_t$ is uniformly bounded in $L^{5/4}(I;W_0^{-1,5/4}(\Omega))$. Using Aubin–Lions lemma, we conclude that there exists a sequence $\nu_j\rightarrow0$ such that $$\begin{array}{ll}
&\r^{\v,\nu_j}\rightarrow \r^\v~~ \mbox{\rm strongly~~
in}~~~L^p(Q_T)~(\forall 1\leq p<6), \\ [2mm]
&\r^{\v,\nu_j}\rightarrow \r^\v~~ \mbox{\rm strongly~~
in}~~~L^2(I;C(\overline\Omega)),\\ [2mm]
&\r^{\v,\nu_j}{\rightharpoonup} \r^\v ~~~ \mbox{\rm weakly ~~
in}~~~L^2(I,H^1(\Omega)),\\ [2mm]
&\r^{\v,\nu_j}_t{\rightharpoonup} \r^\v_t ~~~ \mbox{\rm weakly ~~
in}~~~L^{5/4}(I;W_0^{-1,{5/4}}(\Omega))
\end{array}
\label{convnurho}$$ and $$\r^{\v,\nu_j}(0,\cdot)\rightarrow
\r^\v(0,\cdot)\quad\mbox{and}\quad\r^{\v,\nu_j}(1,\cdot)\rightarrow
\r^\v(1,\cdot)\quad\mbox{strongly \;in}\;\;\;L^2(0,T).$$ Similarly, by noting the uniform estimates (\[tlinfn\]), (\[(3.17)\]) and (\[unibdtheta\]), we conclude that there exists a subsequence of $\t^{\v,\nu_j}$ (also denoted by $\t^{\v,\nu_j}$) such that $$\begin{array}{ll}
&\t^{\v,\nu_j}\rightarrow \r^\v~~ \mbox{\rm strongly~~
in}~~~L^p(Q_T)~(\forall 1\leq p<\infty), \\ [2mm]
&\t^{\v,\nu_j}\rightarrow \t^\v~~ \mbox{\rm strongly~~
in}~~~L^2(I;C(\overline\Omega)),\\ [2mm]
&\t^{\v,\nu_j}{\rightharpoonup} \t^\v ~~~ \mbox{\rm weakly ~~
in}~~~L^2(I,H^1(\Omega)),\\ [2mm]
&(\r^{\v,\nu_j}\t^{\v,\nu_j}+\sigma\t^{\v,\nu_j})_t{\rightharpoonup}
(\r^\v\t^\v+\sigma\t^\v)_t ~~~ \mbox{\rm weakly ~~
in}~~~L^{5/4}(I;W_0^{-1,{5/4}}(\Omega))
\end{array}
\label{convnutheta}$$ and $$\t^{\v,\nu_j}(0,\cdot)\rightarrow
\t^\v(0,\cdot)\quad\mbox{and}\quad\t^{\v,\nu_j}(1,\cdot)\rightarrow
\t^\v(1,\cdot)\quad\mbox{strongly \;in}\;\;\;L^p(0,T),\;\;1\leq
p<\infty.$$ Since $(\r^{\v,\nu_j},\t^{\v,\nu_j})$ is a strong solution of the system (\[asys2\]) and (\[asys-bi\]), it satisfies $$\begin{aligned}
&\int_0^T\alpha^0(\r^{\v,\nu_j}(0,t)-\bar\r^0)\phi(0,t)dt
+\int_0^T\alpha^1(\r^{\v,\nu_j}(1,t)-\bar\r^1)\phi(1,t)dt
{\nonumber}\\
&~~+\int_0^T\int_\Omega\rho_t^{\v,\nu_j}\phi dxdt
+\int_0^T\big[(\v+(\r^{\v,\nu_j}\t^{\v,\nu_j})_\nu)\r^{\v,\nu_j}_x
+\r^{\v,\nu_j}(\r^{\v,\nu_j}_\v\t^{\v,\nu_j}_x)_\v\big]\phi_xdxdt
\\
&=\int_0^T\int_\Omega\chi^\v(p_s(\t^{\v,\nu_j}))dxdt
-\int_0^T\int_\Omega\r^{\v,\nu_j}\chi^\v(\sqrt{\t^{\v,\nu_j}})\phi
dxdt\end{aligned}$$ and $$\begin{aligned}
&\int_0^T\int_0^1[(\r^{\v,\nu_j}+\s)\t^{\v,\nu_j}]_t\psi dxdt
+\int_0^T\beta^0(\t^{\v,\nu_j}(0,t)-\bar\t^0)\psi(0,t)dt
+\int_0^T\beta^1(\t^{\v,\nu_j}(1,t)-\bar\t^1)\psi(1,t)dt
\\[6pt]
&~~+\int_0^T\alpha^0(\r^{\v,\nu_j}(0,t)-\bar\r^0)\t^{\v,\nu_j}(0,t)\psi(0,t)dt
+\int_0^T\alpha^1(\r^{\v,\nu_j}(1,t)-\bar\r^1)\t^{\v,\nu_j}(1,t)\psi(1,t)dt
{\nonumber}\\
&~~+\int_0^T\int_0^1\k^\v\t^{\v,\nu_j}_x\psi_x dxdt
+\int_0^T\int_0^1\big[(\v+(\r^{\v,\nu_j}\t^{\v,\nu_j})_\nu)
\r^{\v,\nu_j}_x\t^{\v,\nu_j}+\r^{\v,\nu_j}(\r^{\v,\nu_j}_\v
\t^{\v,\nu_j}_x)_\v\t^{\v,\nu_j}\big]\psi_x dxdt
\\[6pt]
&~~+\int_0^T\int_0^1(\l+\t^{\v,\nu_j}) p_s(\t^{\v,\nu_j})\psi\big]
dxdt
\\[6pt]
&=\lambda\int_0^T\int_0^1\r^{\v,\nu_j}\chi^\v(\sqrt{\t^{\v,\nu_j}})\psi
dxdt
+\lambda\int_0^T\int_0^1\t^{\v,\nu_j}\chi^\v(p_s(\t^{\v,\nu_j}))\psi
dxdt,\end{aligned}$$ for any $\phi,\psi \in L^5(I;W^{1,5}(\Omega))$. By taking the limit $j\rightarrow\infty$, we obtain a global weak solution $(\r^\v,\t^\v)$ to the approximate system $$\begin{aligned}
&\r_t-((\v+\r\t)\r_x)_x-(\r(\r_\v\t_x)_\v)_x=-\G_\v, {\nonumber}\\[3mm]
&(\r\t+\s\t)_{t}-(\kappa^\v\t_x)_x-((\v+\r\t))
\r_x\t)_x-(\r(\r_\v\t_x)_\v\t)_x \label{asys3}
\\
&=\l\G_\v+(\l+\t) \left ( \chi^\v(p_s(\t))-p_s(\t)\right ) , {\nonumber}\end{aligned}$$ with the boundary and initial conditions $$\begin{array}{l}
\di (\v+\r\t)\r_x+\r(\r\t_x)_\v\big|_{x=1}=\a^1(\bar\r^1-\r(1,t)),\\[3mm]
\di (\v+\r\t)\r_x+\r(\r\t_x)_\v\big|_{x=0}=\a^0(\r(0,t)-\bar\r^0),\\[3mm]
\di \r(x,0)=\r_{0\v}(x):=\r_0\ast\eta_\v(x)+\v,\\[3mm]
\di \k^\v\t_x|_{x=1}=\b^1(\bar\t^1-\t(1,t)),\\[3mm]
\di \k^\v\t_x|_{x=0}=\b^0(\t(0,t)-\bar\t^0),\\[3mm]
\di \t(x,0)=\t_{0\v}(x):=\t_0\ast\eta_\v(x).
\end{array}
\label{asys-bi3}$$
Secondly, we study the convergence as $\v\rightarrow0$. To take the limit $\,\v\rightarrow0$, we need more uniform estimates for $\rho$ with respect to $\v$.
Clearly the system (\[asys\])-(\[asys-bi\]) reduces to the system (\[asys3\])-(\[asys-bi3\]) when $\nu=0$. Then the uniform estimates (\[positive\]) and (\[(3.17-2)\]) hold for the obtained solution $(\r^\v,\t^\v)$. From (\[rho2pv2\]) we see that $$\|\r\t\r_x\|_{L^2(Q_T)}\leq C_{\v,T}$$ and from the first equation of (\[asys3\]) we deduce that $\r_t\in
L^2(I;H^{-1}_0(\Omega))$. Note that $\ln\r\in L^2(I;H^1(\Omega))$. By multiplying the first equation of (\[asys3\]) by $\ln\r$ and integrating the equation over $Q_t$, we arrive at $$\begin{array}{l}
\di \int_0^1 \r\ln\r(x,t)dx-\int_0^1 \r(x,t)
dx+\int_0^t[\v\r_x+\r\t\r_x +\r(\r_\v\t_x)_\v]\ln\r\Big
|_{x=0}^{x=1}d\tau +\int_0^t\int_0^1\t\r_x^2dxd\tau
\\ [4mm]
\di\leq \int_0^1 \r_{0\v}\ln\r_{0\v}(x)dx-\int_0^1
\r_{0\v}dx-\int_0^t\int_0^1(\r_\v\t_x)_\v\r_xdxd\tau
-\int_0^t\int_0^1 (\r\sqrt{\t}- p_s(\t))\ln\r\, dxd\tau.
\end{array}$$ Since $$\begin{array}{ll}
\di \int_0^t\int_0^1|(\r_\v\t_x)_\v\r_x|dxd\tau&\di \leq
\f12\int_0^t\int_0^1\t\r_x^2dxd\tau
+\f12\int_0^t\int_0^1\f{|(\r_\v\t_x)_\v|^2}{\t}dxd\tau
\\ [4mm]
&\di \leq
\f12\int_0^t\int_0^1\t\r_x^2dxd\tau+C_T\|\r_\v\t_x\|_{L^2(Q_T)}^2
\\ [4mm]
&\di \leq \f12\int_0^t\int_0^1\t\r_x^2dxd\tau+C_T \, ,
\\ [4mm]
\end{array}
{\nonumber}\label{(3.56)}$$ we get $$\begin{array}{l}
\di \int_{[0,1]\cap\{\r\geq 1\}}
\r\ln\r(x,t)dx+\f12\int_0^t\int_0^1\t\r_x^2dxd\tau
+\int\int_{[0,1]\times[0,t]\cap\{\r\geq 1\}}\r\ln\r dxd\tau
\\ [4mm]
\di \leq\int_0^1 \r_{0\v}|\ln\r_{0\v}|(x)dx
+\int_{[0,1]\cap\{\r\leq 1\}} \r|\ln\r|(x,t)dx
\\ [4mm]
\di\quad +\int\int_{[0,1]\times[0,t]\cap\{\r\leq 1\}}\r|\ln\r|
dxd\tau+\int\int_{[0,1]\times[0,t]\cap\{\r\geq 1\}} p_s(\t) \ln\r
dxd\tau+C_T
\\ [6mm]
\leq \di C_T,
\end{array}
{\nonumber}\label{(3.55)}$$ which, together with (\[unibdtheta\]), leads to $$\|\r\ln\r\|_{L^\i(0,T;L^1(\Omega))},\quad \|\r_x\|_{L^2(Q_T)}\leq
C_T \, . \label{(3.24)}$$ From the inequalities (\[rrtl1\]) and (\[(3.24)\]) we derive $$\|\r\|_{L^2(0,T;H^1(\Omega))}\leq C_T\label{(3.25)}$$ and $$\begin{aligned}
\|\r\|_{L^\infty(\Omega)}^3 &\leq\|\r\|_{L^1(\Omega)}^3
+\|\r\|_{L^2(\Omega)}^{3/2}\|\r_x\|_{L^2(\Omega)}^{3/2}
{\nonumber}\\
&\leq
C_T+C\|\r\|_{L^1(\Omega)}^{3/4}\|\r\|_{L^\infty(\Omega)}^{3/4}\|\r_x\|_{L^2(\Omega)}^{3/2}
{\nonumber}\\
&\leq
C_T+\frac{1}{2}\|\r\|_{L^\infty(\Omega)}^3+C_T\|\r_x\|_{L^2(\Omega)}^2,
{\nonumber}\end{aligned}$$ which results in $$\int_0^T\|\r\|_{L^\infty(\Omega)}^3dt\leq
C_T+C_T\int_0^T\|\r_x\|_{L^2(\Omega)}^2dt\leq C_T.$$ Moreover, we have $$\label{ur4}
\int_0^T\int_0^1\r^4dxdt\leq
\biggl(\int_0^T\|\r\|_{L^\infty(\Omega)}^3dt\biggl)\biggl(\sup_{0\leq
t\leq T}\int_0^1\r dx\biggl)\leq C_T,$$ i.e. $\r$ is uniformly bounded in $L^4(Q_T)$.
Finally, we let $$B_1=H^1(\Omega),~B_2=L^4(\Omega),~B_3=W_0^{-1,6/5}(\Omega). {\nonumber}\label{(3.32)}$$ Then $B_1\hookrightarrow\hookrightarrow B_2\hookrightarrow B_3$ and $\{\r^\v\}$ is uniformly bounded in $L^4(I;B_2)\cap
L^2(I;B_1)$. From the first equation in (\[asys\]), i.e. $$\r_t=\Big[\v\r_{x}+\r\t\r_x +\r(\r\t_x)_\v\Big]_x
-\r\chi^\v(\sqrt{\t})+\chi^\v(p_s(\t)) \,,$$ we observe that $\{\r^\v_t\}$ is uniformly bounded in $L^{6/5}(I;B_3)$. By Aubin–Lions lemma, $\{\r^\v\}$ is relatively compact in $L^p(I;L^4(\Omega))$ for $(1\leq p<4)$. Thus, there exists a sequence $\r^{\v_j}$ such that $\lim_{j\rightarrow\infty}\v_j=0$ and $$\begin{array}{ll}
&\r^{\v_j}\rightarrow \r~~ \mbox{\rm strongly~~
in}~~~L^p(I,L^4(\Omega))~(\forall 1\leq p<4), \\ [2mm]
&\r^{\v_j}\rightarrow \r~~ \mbox{\rm strongly~~
in}~~~L^2(I,C(\overline\Omega)), \\ [2mm]
&\r^{\v_j}{\rightharpoonup} \r ~~~ \mbox{\rm weakly ~~
in}~~~L^2(I,H^1(\Omega)),\\[2mm]
&\r_t^{\v_j}{\rightharpoonup} \r_t ~~ \mbox{\rm weakly ~~
in}~~~L^{6/5}(I;W_0^{-1,6/5}(\Omega)).
\end{array}
\label{(3.27)}$$ Similarly, by (\[positive\]) and (\[(3.17-2)\]), there exists a subsequence of $\t^{\v_j}$ (also denoted by $\t^{\v_j}$) such that $$\begin{array}{ll}
&\t^{\v_j}\rightarrow \t~~ \mbox{\rm strongly~~
in}~~~L^p(Q_T)~(\forall 1\leq p<\i), \\ [2mm]
&\t^{\v_j}\rightarrow\t~~ \mbox{\rm strongly~~
in}~~~L^2(I,C(\overline\Omega)), \\ [2mm]
&\t^{\v_j}{\rightharpoonup} \t ~~~ \mbox{\rm weakly ~~
in}~~~L^2(I,H^1(\Omega)),\\[2mm]
&(\r^{\v_j}\t^{\v_j}+\sigma\t^{\v_j})_t{\rightharpoonup}
(\r\t+\sigma\t)_t ~~ \mbox{\rm weakly ~~
in}~~~L^{6/5}(I;W_0^{-1,6/5}(\Omega)).
\end{array}
\label{(3.28)}$$ Now we take the limit $j\rightarrow\infty$ and by (\[(3.27)\]) and (\[(3.28)\]), we obtain a weak solution $(\r,\t)$ which satisfies (\[rdefeq\]) and (\[tdefeq\]). 0.1in
[**Acknowledgements**]{} The authors wish to thank Professors P. Lei, T. Yang, G. Yuan and X. Xu for helpful discussions.
[99]{}
Y. Amirat and V. Shelukhin, [*Global weak solutions to equations of compressible miscible flow in porous media*]{}, SIAM J. Math. Anal., 38 (2007), 1825-1846.
Y. Z. Chen, *Parabolic Partial Differential Equations of Second Order*, Peking University Press, 2003, (in Chinese).
Z. Chen and R. Ewing, Mathematical analysis for reservoir models, [*SIAM J. Math. Anal.*]{}, 30 (1999), pp. 431–453.
A. Cheng and H. Wang, An error estimate on a Galerkin method for modeling heat and moisture transfer in fibrous insulation, [*Numer. Methods Partial Differential Equations*]{}, 24(2008), pp. 504–517.
J. Fan, Z. Luo and Y. Li, Heat and moisture transfer with sorption and condensation in porous clothing assemblies and numerical simulation, [*Int. J. Heat Mass Transfer*]{}, 43 (2000), pp. 2989-3000.
J. Fan, X. Cheng, X. Wen and W. Sun, An improved model of heat and moisture transfer with phase change and mobile condensates in fibrous insulation and comparison with experimental results, [*Int. J. Heat Mass Transfer*]{}, 47 (2004), pp. 2343-2352.
X. Feng, On existence and uniqueness results for a coupled system modeling miscible displacement in porous media, [*J. Math. Anal. Appl.*]{}, 194 (1995), pp. 883–910.
X. Hang, W. Sun and C. Ye, Finite volume solution of heat and moisture transfer in three-dimensional textile materials, submitted.
H. Huang, C. Ye and W. Sun, [*Moisture transport in fibrous clothing assemblies*]{}, J. Engrg. Math., 61 (2008), pp. 35–54.
F.E. Jones, *Evaporation of Water*, Lewis Publishers Inc., Michigan, 1992, pp. 25-43.
O. A. Ladyzenskaja, V. Solonnikov and N. N. Uralceva, [*Linear and nonlinear equations of parabolic type*]{}, Translations of Mathematical Monographs, 23, 1968.
Y. Li and Q. Zhu, Simultaneous heat and moisture transfer with moisture sorption, condensation, and capillary liquid diffusion in porous textiles, [*Textile Res. J.*]{}, 73 (2003), pp. 515-524.
J. L. Lions, [*Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires*]{}, Dunod, Paris (in French).
P. L. Lions, [*Mathematical topics in fluid mechanics*]{}, Vol. 2 of Oxford Lecture series in Mathematics and its its applications. Clarendon Press, Oxford 1998.
A. Novotny, I. Straskraba, *Introduction to the mathematical theory of compressible flow*, Oxford University Press 2004.
Y. Ogniewicz and C.L. Tien, Analysis of condensation in porous insulation, [*J. Heat Mass Transfer*]{}, 24 (1981), pp. 421-429.
P. Smith and E.T. Twizell, A transient model of thermoregulation in a clothed human, [*Applied Math. Modeling*]{}, 8 (1984), pp. 211-216.
J. Vala, On a system of equations of evolution with a non-symmetrical parabolic part occuring in the analysis of moisture and heat transfer in porous media, [*Applications of Math.*]{}, 47 (2002), pp.187–214.
J.A. Wehner, B. Miller, and L. Rebenfeld Moisture Dynamics of water vapor transmission through fabric barriers, [*Textile Research Journal*]{} 58 (1988) 581-592.
C. Ye, H. Huang, J. Fan and W. Sun, Numerical study of heat and moisture transfer in textile materials by a finite volume method, [*Communications in computational Physics*]{}, 4 (2008), pp. 929-948.
[^1]: Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong ([buyangli2@student.cityu.edu.hk, maweiw@math.cityu.edu.hk]{}). The work of the authors was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102005).
[^2]: Institute of Applied Mathematics, AMSS, CAS, Beijing 100190, China ([wangyi@amss.ac.cn]{}). The work of this author was supported in part by the NSFC grant (No. 10801128) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102005).
|
---
abstract: |
We consider the short time asymptotics of the heat content $E$ of a domain $D$ of $\R^d$. The novelty of this paper is that we consider the situation where $D$ is a domain whose boundary $\partial D$ is a random Koch type curve.
When $\partial D$ is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of $\partial D$ from the short time behaviour of $E(s)$. Furthermore, in some situations where the Minkowski dimension exists, finer geometric fluctuations can be recovered and the heat content is controlled by $s^\alpha e^{f(\log(1/s))}$ for small $s$, for some $\alpha \in (0, \infty)$ and some regularly varying function $f$. The function $f$ is not constant is general and carries some geometric information.
When $\partial D$ is statistically self-similar, then the Minkowski dimension and content of $\partial D$ typically exist and can be recovered from $E(s)$. Furthermore, the heat content has an almost sure expansion $E(s) = c s^{\alpha} N_\infty + o(s^\alpha)$ for small $s$, for some $c$ and $\alpha \in (0, \infty)$ and some positive random variable $N_\infty$ with unit expectation arising as the limit of some martingale.
address: 'Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Oxford OX1 6GG, United Kingdom'
author:
- 'Philippe H. A. Charmoy'
bibliography:
- 'references.bib'
date: '.'
title: Heat content asymptotics of some random Koch type snowflakes
---
Introduction
============
Let $D$ be a bounded domain in $\R^d$ and $\Delta$ be the Dirichlet Laplacian. Then the spectrum $\Lambda$ of $-\frac 12 \Delta$ is discrete and of the form $$0 < \lambda_1 \leq \lambda_2 \leq \cdots,$$ where the eigenvalues are repeated according to their multiplicity. Interest in the geometric information about $D$ contained in $\Lambda$ started more than 100 years ago and was crystallised by Kac in his paper [@Kac1966] entitled *Can one hear the shape of a drum?* In other words: Are isospectral domains always isometric? The answer is no in general, as shown, for example, in [@Buseretal1994; @Gordonetal1992; @Milnor1964]. But it is natural to enquire how much information about the geometry of $D$ is encoded by $\Lambda$, and, when $D$ is a random domain, how much of its distribution can be recovered.
As the spectral decomposition of the heat kernel with absorption on the boundary $p_D$ indicates, the heat content $$E_D(s) = \int_D \left(1- \int_D p_D(s,x,y) dy \right) dx$$ provides a natural proxy for the eigenvalues of $- \frac 12 \Delta$. Recall that $E$ may also be expressed more intuitively as $$\label{eq::defHeatContent}
E_D(s) = \int_D u_D(s,x) dx,$$ where $u_D$ is the solution to the heat equation with unit Dirichlet boundary condition and zero initial condition, i.e. is the solution to $$\label{eq::heatEquationSpecProblems}
\begin{aligned}
\partial_s u_D(s,x) &= \frac 12 \Delta u_D(s,x), &
(s, x) \in (0, \infty) \times D,\\
u_D(0, x) & = u_D(0+,x) = 0, &
x \in D,\\
u_D(-, x) & = 1,&
x \times \partial D.
\end{aligned}$$ We will omit the dependence on $D$ from the notation when there is no risk of confusion.
The heat content presents the advantage that it is amenable to probabilistic techniques. Furthermore, it is a convenient object to recover information about the geometry of the boundary of $D$. Interest in this naturally intensified when Berry studied the spectral properties of domains with a fractal boundary in [@Berry1979; @Berry1980] and conjectured that the Hausdorff dimension of $\partial D$ should be encoded by $\Lambda$. This was first disproved by [@BC1986] who showed that the Minkowski dimension was the relevant measure of roughness.
Since then, the short time asymptotics of the heat content have been studied extensively. Planar domains with polygonal boundary are discussed in [@vdBS1988; @vdBS1990]. Some domains with fractal boundary, including the triadic Koch snowflake, are discussed in [@FLV1994; @LV1996; @vdBdH1999]. In [@vdB1994], van den Berg proved that, under some regularity conditions, if the Minkowski dimension $\gamma$ of $\partial D$ exists, then $$E(s) \asymp s^{(d-\gamma)/2}$$ for small $s$, where $f(x) \asymp g(x)$ means that $c^{-1} f(x) \leq g(x) \leq c f(x)$ for some $c \in (0, \infty)$; this is the heat content analogue of the results in [@BC1986].
Here, we study the heat content asymptotics of two families of Koch type snowflakes, thereby addressing a situation closely related to [@FLV1994]. First, we discuss scale homogeneous snowflakes whose boundary is constructed from a sequence $(\xi_n, n \in \N)$ of natural numbers determining the size and number of linear pieces used at each iteration of the construction throughout the set. This is related to the Sierpinski gaskets discussed in [@Hambly1992; @BH1997]. Second, we discuss snowflakes whose boundary is statistically self-similar, and so we use the theory of general branching processes to study their geometry and heat content.
The paper is organised as follows. In Section \[sec::heatContentEstimates\], we discuss estimates for the heat content along the lines of [@vdB1994] and use this to show that one can recover the lower and upper Minkowski dimension of $\partial D$ from the heat content in the following way.
Let $D$ be a bounded domain in $\R^d$. Write $\alpha$, respectively $\beta$, for the lower, respectively upper, Minkowski dimension of $\partial D$. Then, under the regularity conditions given in Assumptions \[ass::regularityBrownianMotion\] and \[ass::capacitaryDensity\], $$\liminf_{s \to 0} \left(\frac d2 - \frac{\log E(s)}{\log s} \right) = \frac{\alpha}{2} \quad \text{and} \quad \limsup_{s \to 0} \left(\frac d2 - \frac{\log E(s)}{\log s} \right) = \frac{\beta}{2}.$$
Intuitively, this means that *one can hear the lower and upper Minkowski dimensions of $\partial D$*, and in particular determine whether they are equal.
In Section \[sec::scaleHomogeneous\], we detail the construction of the family of scale homogeneous snowflakes mentioned above. We then look at their Minkowski dimension and content and use that to study their heat content. In particular, we show that when the sequence $(\xi_n, n \in \N)$ used to build the snowflake is stationary and ergodic, the Minkowski dimension exists, and discuss how to construct examples where it does not. Furthermore, we show that the rate of convergence in the ergodic theorem dictates the short time asymptotics of the heat content. An important example is when $(\xi_n, n \in \N)$ is i.i.d., in which case we use the law of the iterated logarithm to prove the following theorem.
Let $D$ be a scale homogeneous snowflake constructed with an i.i.d. sequence. Then the Minkowski dimension $\gamma$ of the boundary of the snowflake exists almost surely. Furthermore, under the regularity conditions given in Assumptions \[ass::regularityBrownianMotion\] and \[ass::capacitaryDensity\], for some positive constants $c_1, \dots, c_6$, we have $$c_1 s^{1- \gamma/2} e^{-c_2 \psi(1/s)} \leq E(s) \leq c_3 s^{1- \gamma/2} e^{c_4 \psi(1/s)}$$ for small $s$, while $$\liminf_{s \to 0} \frac{E(s) e^{c_5 \psi(1/s)}}{s^{1-\gamma/2}} < \infty \quad \text{and} \quad \limsup_{s \to 0} \frac{E(s) e^{-c_6 \psi(1/s)}}{s^{1-\gamma/2}} > 0,$$ where $$\psi(x) = \sqrt{\log x \log \log \log x}.$$
The function $\psi$ is in some sense the best possible, and this result intuitively implies that *one can hear the law of the iterated logarithm*.
In Section \[sec::gbp\], we give a brief introduction to the theory of general branching processes and introduce the notation necessary to discuss statistically self-similar sets.
Finally, in Section \[sec::selfSimilar\], we detail the construction of our statistically self-similar snowflakes. Using the theory of general branching processes, we show that the Minkowski dimension and content of the boundary of these snowflakes typically exist and that this implies the following result for the heat content.
Under the regularity conditions given in Assumptions \[ass::regularityBrownianMotion\] and \[ass::capacitaryDensity\], the heat content of the statistically self-similar snowflakes satisfies $$s^{-(1-\gamma/2)} E(s) \to c_7 N_\infty, \text{ a.s.\ and in } L^1,$$ as $s \to 0$, for some positive constant $c_7$ and some positive random variable $N_\infty$ with unit expectation arising as the limit of some martingale.
Together with a similar result for the geometry of the boundary of the snowflake, this result has the intuitive interpretation that *one can hear the Minkowski dimension and content of the boundary*.
Notation {#notation .unnumbered}
--------
Throughout the document, the symbol $c_i$ with $i \in \N$ will mean *some positive constant* whose value is typically fixed for the length of a proof or a subsection.
Heat content estimates {#sec::heatContentEstimates}
======================
In this section, we start by recalling the definition of inner Minkowski dimension and content and then derive bounds on the heat content in a fashion inspired by [@BC1986; @vdB1994]. Finally, we look at an example.
Inner Minkowski dimension and content {#subsec::dimensionAndContentDefinition}
-------------------------------------
Let $K$ be a bounded subset of $\R^d$. The *$\epsilon$-neighbourhood* of $K$ is defined as $$K_\epsilon = \{x \in \R^d : d(x, K) \leq \epsilon\},$$ where $d(x, A)$ is the Euclidean distance between the point $x$ and the set $A$.
For a bounded domain $D$ of $\R^d$, we call $(\partial D)_\epsilon \cap D$ the *inner $\epsilon$-tubular neighbourhood* and will write $\mu_D(\epsilon)$ for the volume of that neighbourhood, i.e.$$\mu_D(\epsilon) = {\mathrm{vol}}_d((\partial D)_\epsilon \cap D),$$ where ${\mathrm{vol}}_d$ denotes the Lebesgue measure on $\R^d$; again, we will omit the dependence on $D$ when there can be no confusion. We call *inner lower, respectively upper, Minkowski dimension* of $\partial D$ the quantity $$\underline{\dim}_M \partial D = d - \limsup_{\epsilon \to 0} \frac{ \log \mu(\epsilon)}{\log \epsilon}, \quad \text{respectively} \quad \overline{\dim}_M \partial D = d - \liminf_{\epsilon \to 0} \frac{\log \mu(\epsilon)}{ \log \epsilon}.$$ When these two quantities are equal, we say that the inner Minkowski dimension exists and use the notation $\dim_M \partial D$ instead. In all cases of interest here, the inner Minkowski dimension is equal to the usual Minkowski dimension defined, for example, in [@Falconer1986a].
When the Minkowski dimension of $\partial D$ exists, we define the *inner lower, respectively upper, Minkowski content* as $$\cM_* = \liminf_{\epsilon \to 0} \epsilon^{\dim_M \partial D - d} \mu(\epsilon), \quad \text{respectively} \quad \cM^* = \limsup_{\epsilon \to 0} \epsilon^{\dim_M \partial D- d} \mu(\epsilon).$$ When these two quantities are equal, we say that the inner Minkowski content exists and use the notation $\cM$ instead.
The estimates
-------------
It will be convenient to be able to solve the heat equation under the conditions in using the probabilistic representation $$\label{eq::probabilisticSolution}
u(s,x) = \bP_x(T_{D^c} \leq s),$$ where $T_{D^c}$ is the hitting time of $D^c$ of Brownian motion and $\bP_x$ is the law of Brownian motion started at $x$. Therefore, we will always make the following assumption.
\[ass::regularityBrownianMotion\] All the points of $\partial D$ are regular for $D^c$.
Recall that this assumption is always satisfied for simply connected planar domains; see Proposition II.1.14 of [@Bass1995]. This will cover all the examples discussed here.
We now prove a first upper bound for the heat content following the argument of [@vdB1994].
\[thm::upperBoundHeatContent\] Let $D$ be a bounded domain in $\R^d$ and let $\omega: \R_+ \to \R_+$ be an increasing function with $\omega(0) = 0$. Then, for every $s \geq 0$, $$E(s) \leq \mu(\omega(s)) + 2^{(d+2)/2} {\mathrm{vol}}_d(D)e^{-\omega(s)^2/4s}.$$
Let $s \geq 0$ and put $$A_s = \{x \in D: d(x, \partial D) \leq \omega(s)\}.$$ By , we have $$\label{eq::firstStepBoundHeatContent}
\begin{aligned}
E(s) & = \int_{A_s} \bP_x(T_{D^c} \leq s) dx + \int_{D\setminus A_s} \bP_x(T_{D^c} \leq s) dx \\
& \leq \mu(\omega(s)) + \int_{D\setminus A_s} \bP_x(T_{D^c} \leq s) dx.
\end{aligned}$$
But now, by LŽvy’s maximal inequality, e.g. Theorem 3.6.5 of [@Simon2005], $$\begin{aligned}
\bP_x(T_{D^c} \leq s) & \leq \bP_x(T_{\cB(x, d(x, \partial D))^c} \leq s)\\
& = \bP_0 \left( \sup_{0 \leq u \leq s} |B(u)| \geq d(x, \partial D) \right)\\
& \leq 2 \bP_0 (|B(s)| \geq d(x, \partial D))\\
& = 2 (2 \pi s)^{-d/2} \int_{|y| \geq d(x, \partial D)} e^{-|y|^2/2s} dy,
\end{aligned}$$ where $\cB(x, \delta)$ denotes the open ball of radius $\delta$ centred at $x$. Changing variables and standard Gaussian estimates then yield $$\label{eq::gaussianEstimate}
\bP_x(T_{D^c} \leq s) \leq 2^{(d+2)/2} e^{-d(x, \partial D)^2/4s}.$$ Since $d(x, \partial D) > \omega(s)$ for $x \in D \setminus A_s$, using this estimate in completes the proof.
The other estimates that we will use are those of Theorems 1.2 to 1.4 of [@vdB1994] which we recall here for convenience.
\[thm::vdBUpperBound\] Let $D$ be a bounded domain in $\R^d$. Then, for every $s \geq 0$, $$E(s) \leq 2^{d/2} s^{-1} \int_0^\infty \epsilon e^{-\epsilon^2/4s} \mu(\epsilon) d\epsilon.$$
The lower bound for the heat content is proved under a capacitary condition that we state now; see [@BC1986; @vdB1994] for more background. We write $\operatorname*{Cap}(A)$ for the Newtonian capacity of the set $A$.
\[ass::capacitaryDensity\] For the bounded domain $D$ of $\R^d$ with $d \geq 2$ there exists a positive constant $c$ such that, for all $x \in D$ and $r \in (0, \operatorname*{diam}(D))$, $$\operatorname*{Cap}(B(x, r) \cap \partial D) \geq c \operatorname*{Cap}(B(x, r)).$$
This assumption is usual in these problems and we say that the *capacitary density of $\partial D$ is bounded below* when it is satisfied. This is always the case when $D$ is a simply connected planar domain; see [@vdB1994].
\[thm::vdBLowerBound\] Let $D$ be a bounded domain of $\R^d$ and assume that either $d= 1$ or $d \geq 2$ and the capacitary density of $\partial D$ is bounded below. Then, for every $s \geq 0$, $$E(s) \geq c_1 \mu(c_2 s^{1/2}).$$
These results illustrate the role of the Minkowski dimension in establishing a lower bound on $E(s)$ for small $s$. Furthermore, the change of variables $\eta = \epsilon^2/4$ yields $$\label{eq::changeVariables}
E(s) \leq 2^{(d+2)/2} s^{-1} \int_0^\infty e^{-\eta /s} \mu(2 \eta^{1/2}) d \eta.$$ If $\mu(\epsilon)$ does not oscillate too much for small $\epsilon$, then an Abelian theorem can be used to deduce the behaviour of $E(s)$ for small $s$ as we show now.
\[thm::heatContentAbelianArgument\] Let $D$ be a bounded domain of $\R^d$ and assume that either $d= 1$ or $d \geq 2$ and the capacitary density of $\partial D$ is bounded below. Assume further that there exist $\gamma \in (0, \infty)$ and a slowly varying function $L$ such that $$\mu(\epsilon) \asymp \epsilon^{d-\gamma} L(1/\epsilon)$$ for small $\epsilon$. Then, $$E(s) \asymp s^{(d- \gamma)/2} \tilde L(1/s)$$ for small $s$, where $\tilde L$ is the slowly varying function defined by $$\tilde L(x) = L(x^{1/2}/2).$$
Define $$\tilde \mu(\eta) = \mu(2 \eta^{1/2}) = 2^{d- \gamma} \eta^{(d - \gamma)/2} \tilde L(1/\eta).$$ Using , an integration by parts and applying Theorem XIII.5.3 in [@Feller1968], we have $$E(s) \leq c_3 s^{-1} \int_{0}^\infty e^{-\eta/s} \tilde \mu(\eta) d \eta = c_3 \int_0^\infty e^{-\eta/s} \tilde \mu(d \eta)\sim c_4 s^{(d- \gamma)/2} \tilde L(1/s),$$ as $s \to 0$, where the notation $f(x) \sim g(x)$ means that $f(x)/g(x) \to 1$. A similar lower bound follows immediately using Theorem \[thm::vdBLowerBound\].
When $\mu(\epsilon)$ oscillates too wildly, for example when the Minkowski dimension does not exist, the Abelian theorem is not applicable. But we can then rely on Theorem \[thm::upperBoundHeatContent\] to study the short time asymptotics of the heat content and get the following announced result.
\[thm::lowerAndUpperHeatDimension\] Let $D$ be a bounded open domain of $\R^d$ and assume that either $d= 1$ or $d \geq 2$ and the capacitary density of $\partial D$ is bounded below. Then, $$\liminf_{s \to 0} \left(\frac d2 - \frac{\log E(s)}{\log s} \right) = \frac 12 \underline{\dim}_M \partial D$$ and $$\limsup_{s \to 0} \left(\frac d2 - \frac{\log E(s)}{\log s} \right) = \frac 12 \overline{\dim}_M \partial D.$$
Put $\alpha = \underline{\dim}_M \partial D$ and $\beta = \overline{\dim}_M \partial D$ and let $\delta \in (0, \infty)$ be small.
By definition of the Minkowski dimension, we have, on the one hand, that there exists a sequence $(\epsilon_n, n \in \N)$ converging to 0 along which $$\mu(\epsilon_n) \leq \epsilon_n^{d- (\alpha+ \delta)},$$ and, on the other hand, that $$\mu(\epsilon) \geq \epsilon^{d-(\alpha - \delta)}$$ for $\epsilon$ small enough.
Setting $\omega(s) = \sqrt{2 d s \log(1/s)}$, which is increasing around 0 and putting $\omega(s_n) = \epsilon_n$ together with Theorems \[thm::upperBoundHeatContent\] and \[thm::vdBLowerBound\] shows that $$E(s_n) \leq s_n^{(d-(\alpha + \delta))/2} \log(1/s_n)^{(d-(\alpha + \delta))/2} + c_5 s_n^{d/2}.$$ and $$E(s) \geq c_6 s^{(d-(\alpha - \delta))/2}$$ for $s$ small enough. The result for the liminf follows.
The proof of the limsup is similar, and somewhat simpler, relying on Theorems \[thm::vdBUpperBound\] and \[thm::vdBLowerBound\].
Self-affine boundaries
----------------------
We conclude this section with a brief mention of a domain whose boundary’s Hausdorff and Minkowski dimensions disagree and where the results discussed here can be applied. This is an alternative example to that of [@BC1986] showing that the Minkowski dimension is the relevant measure of roughness for heat conduction problems; in the example presented here, however, the domain is connected.
Following [@PS2013], we define the boundary of the domain using the self-affine carpets of Bedford [@Bedford1984] and McMullen [@McMullen1984] whose construction we recall briefly now. Let $m < n$ be two integers. Divide the unit square $[0,1]^2$ into $m n$ rectangles of height $m^{-1}$ and width $n^{-1}$. Keep some rectangles according to a pattern $P$ and discard the others. This produces a compact set $K_1$. For each rectangle of the pattern, repeat the procedure. This produces a compact subset $K_2$ of $K_1$. Continue indefinitely to get a compact set $$K(P) = \bigcap_{j= 1}^\infty K_j.$$
A natural way to represent a pattern is to use an $m \times n$ matrix with entries in $\{0,1\}$ where each 1 indicates a rectangle that we choose to keep. The carpet corresponding to pattern $P$ is then $$K(P) = \left\{ \sum_{k = 1}^\infty (a_k n^{-k}, b_k m^{-k}) : (a_k, b_k) \in D \right\},$$ where $D = \{(i,j) : P(j,i) = 1\}$. Here, the rows and columns of $P$ are numbered starting from 0, and the rows are numbered from bottom to top. As an illustration, the first 3 iterations corresponding to the pattern $$A = \begin{pmatrix}
0 & 1 & 1 & 1\\
1 & 0 & 0 & 0
\end{pmatrix}$$ are shown in Figure \[fig:selfaffinecarpet\].
The Hausdorff dimension of the self-affine carpet built from pattern $P$ is $$\log_m \left(\sum_{j=0}^{m-1} r(j)^{\log_n m} \right),$$ where $r(j)$ is the number of chosen rectangles of the pattern in row $j$, while its Minkowski dimension (which exists) is given by $$1 + \log_n \left( \frac 1m \sum_{j=0}^{m-1} r(j) \right);$$ see [@Bedford1984; @McMullen1984; @P1994].
Of course, the set $K(P)$ is not a continuous curve in general. But, for appropriate patterns, we can alter the construction of the carpet using the reflected pattern $$P^r(i,j) = P(i, n-1-j)$$ to create a set that is the graph of a continuous function. This can be done for pattern $A$ using a procedure that we describe now.
As in the construction of $K(A)$, start with pattern $A$. Reproduce pattern $A^r$ in the chosen rectangle of column 2 and $A$ in the others. Repeat this procedure for chosen rectangles inside a pattern $A$; and, inside a pattern $A^r$, reproduce pattern $A$ for the chosen rectangle of column 1 and $A^r$ for the others. Continue indefinitely to get a compact set $K_c(A)$; this set is the graph of a continuous function $f_A: [0,1] \to [0,1]$. The first 3 iterations are shown in Figure \[fig:contselfaffinefun\]. Examining the calculations in [@P1994] shows that this alteration of the construction does not change the Hausdorff or the Minkowski dimension; see also [@PS2013].
Now define the continuous function $g: [0,2] \to [1,3]$ by $$g(t) = 1 + f(t) \bone_{[0,1]}(t) + (2 - f(2-t))\bone_{(1,2]}(t),$$ and let $D$ be the simply connected domain inside the Jordan curve defined by $$\{(t, g(t)) : t \in [0,2]\} \cup (\{0\}\times[0,1]) \cup ([0,2] \times \{0\}) \cup ( \{2\}\times[0,3]).$$ Furthermore, put $$\nu(\epsilon) = {\mathrm{vol}}_2(\{ x \in \R^2 : d(x, \{(t,g(t)) : t \in [0,2]\}) \leq \epsilon\}).$$ Because of the symmetry of $g$, we have $$\nu(\epsilon) = 2 {\mathrm{vol}}_2(\{x \in D :d(x, \{(t,g(t)) : t \in [0,2]\}) \leq \epsilon\}) + \pi \epsilon^2,$$ and therefore $$\nu(\epsilon) = 2 \, \mu (\epsilon) + O(\epsilon^2).$$ It follows that the inner Minkowski dimension of $\partial D$ is equal to the Minkowski dimension of $K_c(A)$, which is different from its Hausdorff dimension. But because $D$ is simply connected, Theorem \[thm::lowerAndUpperHeatDimension\] can be used to show that $$\lim_{s \to 0} \left(1 - \frac{\log E(s)}{\log s} \right) = \frac 12 + \frac 12 \log_n \left( \frac 1m \sum_{j=0}^{m-1} r(j) \right).$$
Scale homogeneous snowflakes {#sec::scaleHomogeneous}
============================
Construction
------------
We introduce a family of scale homogeneous random snowflakes by generalising the Koch curve. In the construction of the usual, triadic, Koch curve, the segment $[0,1]$ is replaced by a curve $K(1)$, say, made of 4 segments of length $1/3$ arranged to produce a spike in the middle. This procedure is then iterated and produces a limiting self-similar curve.
Here we proceed similarly, but using different building blocks with different numbers of spikes. More precisely, for $a \in A$, a bounded subset of $\N$, put $$m(a) = 3 a +1 \quad \text{and} \quad \ell(a) = 2a + 1,$$ and let $K(a)$ be the curve made of $m(a)$ segments of length $\ell(a)^{-1}$ arranged to produce $a$ spikes as depicted in Figure \[fig::buildingBlocksSnowflakes\]. Now, let $\xi = (\xi_n, n \in \N)$ be a sequence of elements of $A$. We will write $K(\xi)$ for the Koch curve where we used $K(\xi_n)$ as a building block at iteration $n$. For example, the curve formed from the first iterations for $\xi = (1, 3, 2, 1, \dots)$ is shown in Figure \[fig::exampleFlake\].
It will be convenient to use the notation $$\ell_n = \ell(\xi_n), \quad m_n = m(\xi_n), \quad \epsilon_n^{-1} = L_n = \prod_{i =1}^n \ell_i \quad \text{and}\quad M_n = \prod_{i = 1}^n m_i.$$ With these definitions, the $n$th iteration in the construction of $K(\xi)$ consists of $M_n$ segments of length $\epsilon_n = L_n^{-1}$.
The domain $D = D(\xi)$ in which we are interested is the one enclosed by the Jordan curve made of three copies of $K(\xi)$ arranged as in the construction of the usual Koch snowflake. In particular, $D$ is simply connected.
Fractal dimension and content
-----------------------------
Since $A$ is bounded, we have $$\label{eq::coverinNumber}
\epsilon_n \asymp \epsilon_{n+1} \quad \text{and} \quad M_n \asymp N(\epsilon_n, K(\xi)) \asymp N(\epsilon_{n+1}, K(\xi)),$$ where $N(\epsilon, K)$ is the covering number of $K$ by balls of radius $\epsilon$. This observation and the mass distribution principle enable us to calculate the dimension of $K(\xi)$.
\[thm::dimensionScaleHomogeneousFlake\] The Hausdorff and Minkowski dimensions of the set $K(\xi)$ are given by $$\dim K(\xi) = \underline{\dim}_M K(\xi) = \liminf_{n \to \infty} \frac{\log M_n}{\log L_n}$$ and $$\overline{\dim}_M K(\xi) = \limsup_{n \to \infty} \frac{\log M_n}{\log L_n}.$$
The calculation of the lower and upper Minkowski dimension follows directly from .
To simplify the notation for the calculation of the Hausdorff dimension, put $\alpha = \underline{\dim}_M K(\xi)$. It is standard that $\dim K(\xi) \leq \alpha$. To prove the other inequality, let $\delta \in (0, \infty)$. By , there exists $m$ large such that, for every $n \geq m$, we have $M_n^{-1} \leq \epsilon_n^{\alpha - \delta}$. Furthermore, the set $K(\xi)$ is made of $M_m$ (disjoint up to $m-1$ points) copies of the set $$\tilde K = L_m^{-1} K(\xi_{m+1}, \xi_{m+2}, \dots).$$
Now consider the Borel measure $\nu$ assigning mass $M_n^{-1}$ to sets of size $L_n^{-1}$ in the obvious way for $n \geq m$. Let $U$ be a subset of $\tilde K$ and let $n$ be such that $\epsilon_{n+1} \leq \operatorname*{diam}U < \epsilon_n$. Then, we have $$\nu(U) \leq M_n^{-1} \leq \epsilon_n^{\alpha- \delta} \leq c \epsilon_{n+1}^{\alpha- \delta} \leq c (\operatorname*{diam}U)^{\alpha- \delta},$$ for some positive constant $c$, thanks to .
By the mass distribution principle, it follows that $\tilde K$, and hence $K(\xi)$, has Hausdorff dimension at least $\alpha - \delta$. Since $\delta$ is arbitrary, this completes the proof.
To continue the geometric description of the scale homogeneous snowflakes, we now look at the volume of the inner tubular neighbourhoods of $D$. The following result immediately implies that the inner Minkowski dimensions agree with the usual definitions and calculations of Theorem \[thm::dimensionScaleHomogeneousFlake\].
\[lem::tubNeighbourhoodScaleHomo\] The volume of the inner tubular neighbourhoods of $D$ satisfies $$\mu(\epsilon_n) \asymp M_n L_n^{-2}.$$
On the one hand, the volume of the $\epsilon_n$ tubular neighbourhood in one third of the snowflake is bounded below by the volume of the level $n$ spike of a third of the boundary, i.e. a copy of $K(\xi)$. This means that $$\frac 13 \mu(\epsilon_n) \geq \frac{\sqrt{3}}{4} M_{n-1} \xi_n \epsilon_n^2 \asymp M_n \epsilon_n^2,$$ where we used that $A$ is bounded.
On the other hand, this same quantity is bounded above by the volume of the set $$\{x \in \R^d : d(x, K(\xi_1, \dots, \xi_n)) \leq \epsilon_n\},$$ where $K(\xi_1, \dots, \xi_n)$ is the curve obtained at the $n$th iteration in the construction of the $K(\xi)$. This means that $$\frac 13 \mu(\epsilon_n) \leq 4 M_n \epsilon_n^2.$$ The result follows.
Notice that, $$\log L_n = n \sum_{a \in A} p_a(n) \ell(a) \quad \text{and} \quad \log M_n = n \sum_{a \in A} p_a(n) m(a),$$ where $(p_a(n), a \in A)$ is the empirical distribution for the proportions of the elements of $A$, i.e.$$p_a(n) = \frac 1n \sum_{i = 1}^n \bone_{\xi_i = a}.$$ Therefore, the key to understanding the log asymptotics of the inner tubular neighbourhoods, and hence the fractal dimension of $\partial D$, is the convergence of the empirical distribution to some limiting probability measure $(p_a, a \in A)$. This observation can easily be used to produce examples such as the following one of snowflakes whose boundary does not have a Minkowski dimension.
\[ex::noMinkowskiDim\] Consider the sequence $(\xi_n, n \in \N)$ taking values in $\{1,2\}$ defined by $$\xi_n = \begin{cases} 1, & n \in S,\\ 2,& n \in \N \setminus S, \end{cases} \quad \text{where} \quad S = \bigcup_{n = 1}^\infty \{2^{2n} +1, \dots, 2^{2n+1}\}.$$ Then, for $a \in \{1, 2\}$, it is easily checked that $$\liminf_{n \to \infty} p_a(n) = \frac 13 \quad \text{and} \quad \limsup_{n \to \infty} p_a(n) = \frac 23.$$ Therefore, $$\liminf_{n \to \infty} \frac{\log M_n}{\log L_n} = \liminf_{n \to \infty} \frac{n^{-1} \sum_{i \leq n} \log m_i}{n^{-1} \sum_{i \leq n} \log \ell_i} = \frac{ \frac 13 \log 4 + \frac 23 \log 7}{\frac 13 \log 3 + \frac 23 \log 5} \simeq 1.2225,$$ while $$\limsup_{n \to \infty} \frac{\log M_n}{\log L_n} = \limsup_{n \to \infty} \frac{n^{-1} \sum_{i \leq n} \log m_i}{n^{-1} \sum_{i \leq n} \log \ell_i} = \frac{ \frac 23 \log 4 + \frac 13 \log 7}{\frac 23 \log 3 + \frac 13 \log 5} \simeq 1.2395.$$ So the Minkowski dimension of $K(\xi)$ does not exist.
On the other hand, when the sequence $(\xi_n, n \in \N)$ is stationary and ergodic, we have, for every $a \in A$, $$p_a(n) \to p_a,$$ as $n \to \infty$, for some probability measure $(p_a, a \in A)$. In this case, we have the following elementary result.
\[thm::dimensionScaleHomo\] Suppose that $(\xi_n, n \in \N)$ is stationary and ergodic. Then, the Minkowski dimension of $\partial D$ exists, is equal to its Hausdorff dimension, and is given by $$\label{eq::gammaHausdorffMinkowski}
\frac{\sum_{a \in A} p_a m(a)}{\sum_{a \in A} p_a \ell(a)}.$$
In a fashion inspired by [@BH1997], let us now focus on the stationary and ergodic case and study how the speed of convergence in the ergodic theorem affects the geometry of the snowflake. Assume that $p_a(n) \to p_a$ as $n \to \infty$ for some probability measure $(p_a, a \in A)$ and that $$\label{eq::definitionFunG}
\sup_{a \in A} |p_a(n) - p_a| \leq n^{-1} g(n),$$ where $g$ is a regularly varying function. If $p_a \in (0,1)$ for every $a \in A$, then $$\liminf_{n \to \infty} |n p_a(n) - n p_a| > 0.$$ Therefore, the best rate of convergence that one can get in general is $g(n) = O(1)$. As such, we will always assume that $g$ is non-decreasing.
We now show how the rate of convergence can be used to control the volume of the inner tubular neighbourhoods.
\[thm::boundsVolumeScaleHomo\] Suppose that $(\xi_n, n \in \N)$ is stationary and ergodic and satisfies for some regularly varying function $g$. Then, $$c_1 \epsilon^{2- \gamma} e^{-c_2 g(c_3 \log (1/\epsilon)) } \leq \mu(\epsilon)\leq c_4 \epsilon^{2- \gamma} e^{c_2 g(c_3 \log (1/\epsilon))}$$ for small $\epsilon$, where $\gamma = \dim_M \partial D$.
By Lemma \[lem::tubNeighbourhoodScaleHomo\], we know that $$\mu(\epsilon_n) \asymp \epsilon_n^{2- \gamma}M_n L_n^{-\gamma}.$$ But, using Theorem \[thm::dimensionScaleHomo\], note that $$\begin{aligned}
\log (M_n L_n^{-\gamma}) & = n \sum_{a \in A} [\log m(a) - \gamma \log \ell(a)]p_a(n)\\
& = n \sum_{a \in A} [\log m(a) - \gamma \log \ell(a)][p_a(n)- p_a] \\
& = O(g(n)).
\end{aligned}$$
Putting these observations together shows that $$c_1 \epsilon_n^{2- \gamma} e^{-c_2 g(n)} \leq \mu(\epsilon_n) \leq c_4 \epsilon_n^{2- \gamma} e^{c_2 g(n)}.$$ The result follows after using that $g$ is non-increasing, and that $\log(1/\epsilon_n) \asymp n$ and $\epsilon_n \asymp \epsilon_{n+1}$, because $A$ is bounded.
One of the consequences of this result is that we can only have $$0 < \cM_*(\partial D) \leq \cM^*(\partial D) < \infty,$$ if the rate of convergence is the best possible, i.e. $g(n) = O(1)$. When this is not the case, but the sharpest function $g$ is known, one can ask about the fluctuations of the volume of the inner tubular neighbourhoods between the bounds given above. A central example where this can be done is when the sequence $(\xi_n, n \in \N)$ is i.i.d., in which case the rate of convergence $$g(n) = \sqrt{n \log \log n}$$ is given by the law of the iterated logarithm; we discuss this case in the following theorem.
Suppose that $(\xi_n, n \in \N)$ is i.i.d. Then, $$c_1 \epsilon^{2- \gamma} e^{-c_2 \psi(1/\epsilon)} \leq \mu(\epsilon) \leq c_3 \epsilon^{2- \gamma } e^{c_2 \psi(1/\epsilon)}$$ for small $\epsilon$, where $\gamma = \dim_M \partial D$ and $$\psi(x) = \sqrt{\log x \log \log \log x}.$$ Furthermore, $$\liminf_{\epsilon \to 0} \frac{\mu(\epsilon) e^{c_4 \psi(1/\epsilon)}}{\epsilon^{2- \gamma}} < \infty \quad \text{and} \quad \limsup_{\epsilon \to 0} \frac{\mu(\epsilon) e^{-c_4 \psi(1/\epsilon)}}{\epsilon^{2- \gamma}} > 0.$$
The first part of the result follows from Theorem \[thm::boundsVolumeScaleHomo\] using the form of $g$ for the i.i.d. case.
For the second part, notice that, by the law of the iterated logarithm, $$\log (M_n L_n^{-\gamma}) \leq - \frac 12 g(n), \text{ i.o.} \quad \text{and} \quad \log (M_n L_n^{-\gamma}) \geq \frac 12 g(n), \text{ i.o.}$$ Proceeding as in the proof of Theorem \[thm::boundsVolumeScaleHomo\] along the appropriate subsequences yields the result.
Heat content asymptotics
------------------------
We now study the heat content asymptotics of the scale homogeneous snowflakes. We rely on the results of Section \[sec::heatContentEstimates\] which are readily applicable since the snowflakes are simply connected.
Let us start by finishing the discussion of Example \[ex::noMinkowskiDim\], a case where the heat content has non-trivial log asymptotics reflecting that the Minkowski dimension of the boundary does not exist.
For the snowflake constructed in Example \[ex::noMinkowskiDim\], we get, by Theorem \[thm::lowerAndUpperHeatDimension\], that $$\liminf_{s \to 0} \left(\frac d2 - \frac{\log E(s)}{\log s} \right) \simeq 0.6112,$$ while $$\limsup_{s \to 0} \left(\frac d2 - \frac{\log E(s)}{\log s} \right) \simeq 0.6198.$$ In other words, the log asymptotics of the heat content oscillate between those dictated by the lower and upper Minkowski dimensions.
As we did when we studied the volume of the inner tubular neighbourhood above, we now focus on the case where the sequence $(\xi_n, n \in \N)$ is stationary and ergodic with a rate of convergence given by the regularly varying function $g$ in .
Using Theorems \[thm::vdBLowerBound\], \[thm::upperBoundHeatContent\] or \[thm::heatContentAbelianArgument\], and \[thm::boundsVolumeScaleHomo\], it is straightforward to get lower and upper bounds for the heat content. Furthermore, when the volume of the inner tubular neighbourhoods oscillates between the lower and upper bounds given by $g$, then a similar reasoning along appropriate subsequences shows that the heat content oscillates in a similar fashion.
Notice, however, that dealing with the upper bound for the heat content is more delicate. This is because the function $\omega$ in Theorem \[thm::upperBoundHeatContent\] typically involves a logarithmic correction (see the proof of Theorem \[thm::lowerAndUpperHeatDimension\]). As we now show, this is never a problem when $g$ is not slowly varying; we postpone the discussion of the other situation to the next subsection.
\[thm::heatContentFluctuationRegularlyVarying\] Suppose that $(\xi_n, n \in \N)$ is stationary and ergodic and that $g$ is regularly varying, but not slowly varying, i.e. has the form $$g(x) = x^\theta L(x),$$ where $\theta \in(0, \infty)$ and $L$ is slowly varying. Then, $$\label{eq::boundHeatContent}
c_1 s^{1- \gamma/2} e^{-c_2 g(c_3 \log(1/s))} \leq E(s) \leq c_4 s^{1- \gamma/2} e^{c_2 g(c_3 \log(1/s))}$$ for small $s$, where $\gamma = \dim_M \partial D$. Furthermore, if $$\label{eq::fluctuationsVolumeTubNeighbourhood}
\liminf_{\epsilon \to 0} \frac{\mu(\epsilon) e^{c_5 g(c_6 \log(1/\epsilon))}}{\epsilon^{2- \gamma}} < \infty \quad \text{and} \quad \limsup_{\epsilon \to 0} \frac{\mu(\epsilon) e^{-c_5 g(c_6 \log(1/\epsilon))}}{\epsilon^{2- \gamma}}> 0,$$ then $$\label{eq::fluctuationsHeatContent}
\liminf_{s \to 0} \frac{E(s) e^{c_7 g(c_8 \log(1/s))}}{s^{1- \gamma/2}} < \infty \quad \text{and} \quad \limsup_{s \to 0} \frac{E(s) e^{-c_7 g(c_8 \log(1/s))}}{s^{1- \gamma/2}}> 0.$$
The lower bound in follows from Theorems \[thm::vdBLowerBound\] and \[thm::boundsVolumeScaleHomo\]. The limsup part of implies that $$\mu(\epsilon) \geq c_9 \epsilon^{2- \gamma} e^{c_{5} g(c_{6} \log (1/\epsilon))}, \text{ i.o.}$$ Together with Theorem \[thm::vdBLowerBound\], this shows that $$E(s) \geq c_{10} s^{1- \gamma/2} e^{c_7 g(c_8 \log(1/s))}, \text{ i.o.}$$ from which the limsup part of follows.
To get the upper bound in , we rely on Theorem \[thm::upperBoundHeatContent\] with the choice $$\label{eq::defnOmega}
\omega(s) = \sqrt{4 s \log (1/s)}$$ and Theorem \[thm::boundsVolumeScaleHomo\] to get that $$\label{eq::technicalEqOne}
\begin{aligned}
E(s) & \leq \mu(\omega(s)) + 2^{(d+2)/2} {\mathrm{vol}}_d (D) e^{-\omega(s)^2/4s}\\
& \leq c_{11} s^{1- \gamma/2} \left( \log \frac 1s \right)^{1- \gamma/2} e^{c_{12} g \left(c_{13} \log \frac{1}{s \log( 1/s)} \right)} + 2^{(d+2)/2} {\mathrm{vol}}_d (D) s\\
& \leq c_{11} s^{1- \gamma/2} \left( \log \frac 1s \right)^{1- \gamma/2} e^{c_{12} g (c_3 \log (1/s))},
\end{aligned}$$ for small $s$. Now using that that $g(x) = x^\theta L(x)$ (this is where we use the assumption that $g$ is not slowly varying), we get that $$\label{eq::technicalEqTwo}
E(s) \leq c_4 s^{1- \gamma/2} e^{c_2 g (c_3 \log (1/s))},$$ for small $s$, as required.
Finally, the liminf part of implies that $$\mu(\epsilon) \leq c_{13} \epsilon^{2- \gamma} e^{-c_{14} g(c_{15} \log (1/\epsilon))}, \text{ i.o.}$$ Reasoning as above then shows that $$E(s) \leq c_{19} s^{1- \gamma/2} e^{-c_7 g(c_8 \log(1/s))}, \text{ i.o.}$$ This establishes the liminf part of and completes the proof.
A particular instance of this is when $(\xi_n, n \in \N)$ is i.i.d. and $g(x) = \sqrt{x \log \log x}$; we state it in the following corollary.
Suppose that $(\xi_n, n \in \N)$ is i.i.d. Then, $$c_1 s^{1- \gamma/2} e^{-c_2 \psi(1/s)} \leq E(s) \leq c_3 s^{1- \gamma/2} e^{ -c_2 \psi(1/s)}$$ for small $s$, where $\gamma = \dim_M \partial D$ and $$\psi(x) = \sqrt{ \log x \log \log \log x}.$$ Furthermore, $$\liminf_{s \to 0} \frac{E(s) e^{c_4 \psi(1/s)}}{s^{1- \gamma/2}} < \infty \quad \text{and} \quad \limsup_{s \to 0} \frac{E(s) e^{-c_4 \psi(1/s)}}{s^{1- \gamma/2}}> 0.$$
Slowly varying rates of convergence in the ergodic theorem
----------------------------------------------------------
The key element in the proof of Theorem \[thm::heatContentFluctuationRegularlyVarying\] is that the logarithmic correction introduced in is *not* felt because $g$ is *not* slowly varying. This is what enables us to go from to , which is no longer possible if $g(x) = \log \log x$ or $g$ is constant, for example.
These difficulties as $g$ becomes ‘closer to a constant’ are expected. For example, for the triadic Koch snowflake studied intensively in [@FLV1994; @LP2006], it is known that the volume of the inner tubular neighbourhood behaves like $$\mu(\epsilon) = p(\log \epsilon) \epsilon^{2- \gamma} + o(\epsilon^{2- \gamma}),$$ as $\epsilon \to 0$, for some non-constant $\log 3$ periodic function $p$; this corresponds to $g$ constant. By a renewal argument, this is known to imply that $$E(s) = q(\log s) s^{1- \gamma/2} +o (s^{1- \gamma/2}),$$ as $s \to 0$, for some $\log 9$ periodic function $q$. But, to the best of my knowledge, it is not known whether or not $q$ is periodic. In other words, it is not known whether $E(s)$ fluctuates between the upper and lower bounds used in Theorem \[thm::heatContentFluctuationRegularlyVarying\] for this constant function $g$.
Similar problems have also been studied by Lapidus and coauthors for the eigenvalue counting function $$N(\lambda) = \#\{ \text{eigenvalues of }-\Delta/2 \leq \lambda\};$$ see [@LvF2000; @LP1993; @LP1996] and references therein. For a family of open sets called *fractal strings*, the eigenvalue counting function satisfies $$c_1\lambda^{\gamma/2} \leq N(\lambda) - \sqrt{2/\pi} \lambda^{1/2} \leq c_2 \lambda^{\gamma/2},$$ where $\gamma$ is the Minkowski dimension of the boundary of the fractal string, provided $$0 < \cM_* (\partial U) \leq \cM^*(\partial U) < \infty.$$ This again corresponds to a setup where $g$ is constant; see the remark after Theorem \[thm::boundsVolumeScaleHomo\]. As it turns out, the question of whether $N(\lambda) - \sqrt{2/\pi} \lambda^{d/2}$ fluctuates between its lower and upper bounds is equivalent to the Riemann hypothesis; see [@LvF2000] for further information.
Our aim in this subsection it to discuss one possible refinement of Theorem \[thm::upperBoundHeatContent\] which enables us to extend Theorem \[thm::heatContentFluctuationRegularlyVarying\] to some some cases where $g$ is slowly varying.
Let $D$ be a bounded domain of $\R^d$. Suppose that $$\mu(\epsilon) \leq c_1 \epsilon^{d- \gamma} e^{\pm c_2 g(c_3 \log(1/\epsilon))}$$ for small $\epsilon$, for some slowly varying function $g$. Suppose further that, for some $k \in \N$, $$\log^k(x) = O(g(x)),$$ as $x \to \infty$. Then, $$E(s) \leq c_4 s^{(d- \gamma)/2} e^{\pm c_5 g(c_6 \log(1/s))}$$ for small $s$.
This somewhat technical result acts as a substitute for the steps in and in the proof of Theorem \[thm::heatContentFluctuationRegularlyVarying\]. Therefore, it readily extends Theorem \[thm::heatContentFluctuationRegularlyVarying\] to situations where $g$ is slowly varying but satisfies the conditions of the lemma.
For $i \in \{1, \dots, k + 1\}$, define $$\omega_i(s) = \sqrt{2 d s \log^i (1/s)},$$ where $\log^i = \log \circ \cdots \circ \log$, with $i-1$ compositions, and put $$\begin{aligned}
B^{k+1}_s &= \{x \in D : d(x, \partial D) \leq \omega_{k+1}(s) \},\\
B_s^i &= \{ x \in D: d(x, \partial D) \leq \omega_i(s)\} \setminus B_s^{i+1}, \quad i \in\{1, \dots, k-1\},\\
B^0_s &= D \setminus \bigcup_{i = 1}^{k+1} B_s^i.
\end{aligned}$$
By and , we see that $$\begin{aligned}
E(s)&\leq \sum_{i = 0}^{k+1} \int_{B_s^i} \bP_x(T_{D^c} \leq s) dx\\
& \leq 2^{(d+2)/2} {\mathrm{vol}}_d(D) s^{d/2} + 2^{(d+2)/2} \sum_{i=1}^{k} \mu(\omega_{i}(s)) e^{- \omega_{i+1}(s)^2/4s} + \mu(\omega_{k+1}(s)).
\end{aligned}$$ By our assumption on $g$, we have $$\begin{aligned}
\mu(\omega_{k+1}(s)) & \leq c_1 s^{(d- \gamma)/2} \left( \log^{k+1} \frac 1s\right)^{(d- \gamma)/2} e^{\pm c_2 g(c_6 \log(1/s))}\\
& \leq c_1 s^{(d- \gamma)/2} e^{\pm c_7 g(c_6 \log(1/s))}
\end{aligned}$$ for $s$ small. Furthermore, for $i \in \{1, \dots, k\}$, $$\begin{aligned}
\mu(\omega_i(s)) e^{-\omega_{i+1}(s)^2/4s} & \leq c_1 s^{(d- \gamma)/2} \left(\log^i \frac 1s \right)^{(d- \gamma)/2} e^{\pm c_2 g(c_6 \log (1/s))} e^{- \frac d2 \log^{i+1}(1/s)}\\
& = c_1 s^{(d- \gamma)/2} \left( \log^i \frac 1s \right)^{- \gamma/2} e^{\pm c_2 g(c_6 \log(1/s))}\\
&\leq c_8 s^{(d- \gamma)/2} e^{\pm c_2 g(c_6 \log(1/s))}
\end{aligned}$$ for $s$ small.
Combining these estimates gives the desired bound on the heat content.
General branching processes {#sec::gbp}
===========================
In this section, we provide a brief introduction to general branching processes and introduce the relevant notation. The presentation is inspired by [@Hambly2000; @Jagers1975; @Nerman1981], where the reader is referred for further information.
Definitions and elementary properties
-------------------------------------
The typical individual $x$ in a general branching process has offspring whose birth times are modelled by a point process $\xi_x$ on $(0, \infty)$, a lifetime modelled as a random variable $L_x$, and a *characteristic* which is a (possibly random) càdlàg function $\phi$ on $\R$. The triples $(\xi_x, L_x, \phi_x)_x$ are sometimes assumed to be i.i.d. but we will allow $\phi_x$ to depend on the progeny of $x$; also, we do *not* make any assumptions about the joint distribution of $(\xi_x, L_x, \phi_x)$. When discussing a generic individual, it is convenient to drop the dependence on $x$ and write $(\xi, L, \phi)$. We shall use the notation $$\xi(t) = \xi((0, t]), \quad \nu(dt) = \bE \xi(dt), \quad \xi_\gamma(dt) = e^{-\gamma t} \xi(dt) \quad \text{and} \quad \nu_\gamma(dt) = \bE \xi_\gamma(dt).$$
We assume that the process has a *Malthusian parameter* $\gamma \in (0, \infty)$ for which $$\label{eq::MalthusianParameter}
\nu_\gamma(\infty) = 1.$$ In particular, the general branching process is super-critical, i.e. $\nu(\infty) > 1$. We also assume that $\nu_\gamma$ has a finite first moment.
It is natural to index the individuals of the population by their ancestry, which is the random subtree $\cT$ of the set of finite words $$\label{eq::addressSpace}
I = \bigcup_{k = 0}^\infty \N^{k}, \quad \text{with} \quad \N^0 = \emptyset,$$ generated by the underlying Galton-Watson process. The birth time of $x$ is written $\sigma_x$ and we have the relation $$\xi_x = \sum_{i = 1}^{\xi_x(\infty)} \delta_{\sigma_{x,i} - \sigma_x},$$ where $\delta$ is the Dirac measure and $x,i$ is the concatenation of the words $x$ and $i$.
The individuals of the population are counted using the characteristic $\phi$ through the *characteristic counting process* $Z^\phi$ defined by $$Z^\phi(t) = \sum_{x \in \cT} \phi_x (t - \sigma_x) = \phi_\emptyset(t) + \sum_{i = 1}^{\xi_\emptyset(\infty)} Z_i^\phi(t- \sigma_i),$$ where the $Z^\phi_i$ are i.i.d. copies of $Z^\phi$. Later, we will define characteristic functions whose corresponding counting process contains information about the inner Minkowski content and the heat content.
Results are often proved under the assumption that $\phi$ vanishes for negative times; e.g. [@Nerman1981]. When this assumption is not satisfied, we may work with $Z^\phi \bone_{[0, \infty)}$ instead of $Z^\phi$ to study the asymptotics of $Z^\phi$ as $t \to \infty$. Indeed, $Z^\phi \bone_{[0, \infty)}$ also appears as a counting process since $$\label{eq::oneSidedToTwoSided}
\begin{aligned}
Z^\chi(t) & = Z^\phi(t) \bone_{t \geq 0}\\
& = \phi_\emptyset(t) \bone_{t \geq 0} + \sum_{i = 1}^{\xi_\emptyset(\infty)} Z_i^\phi(t- \sigma_i) \bone_{0 \leq t < \sigma_i} + \sum_{i=1}^{\xi_\emptyset(\infty)} Z_i^\phi(t- \sigma_i) \bone_{t- \sigma_i \geq 0}\\
& = \chi_\emptyset(t) + \sum_{i=1}^{\xi_\emptyset(\infty)} Z_i^\chi(t- \sigma_i),
\end{aligned}$$ where the first, respectively last, equation defines $Z^\chi$, respectively $\chi$. It is clear that $\chi$ is a characteristic (that depends on the progeny in general), that $Z^\chi$ is its corresponding counting process, and that the asymptotics of $Z^\phi$ and $Z^\chi$ as $t \to \infty$ are the same.
The growth of the population of a general branching process is captured by the process $M$ defined by $$\label{eq::fundamentalMartingale}
M_t = \sum_{x \in \Lambda_t} e^{-\gamma \sigma_x}, \quad \text{where} \quad \Lambda_t = \{x = (y, i) : \sigma_{y} \leq t < \sigma_x\}$$ is the set of individuals born after time $t$ to parents born up to time $t$. The process $M$ is a non-negative càdlàg $\cF_t$-martingale with unit expectation, where $$\cF_t = \sigma((\xi_x, L_x) : \sigma_x \leq t).$$ By martingale convergence, $M_t \to M_\infty$, almost surely, as $t \to \infty$. Furthermore, if $$\label{eq::xlogxCondition}
\bE\left[\xi_\gamma(\infty) (\log \xi_\gamma(\infty))_+ \right] < \infty,$$ then $M$ is uniformly integrable and $M_\infty$ is positive on the event that there is no extinction. Proofs of these facts may be found in [@Jagers1975; @Nerman1981; @Doney1972; @Doney1976].
Strong law of large numbers
---------------------------
In this subsection, we state the strong law of large numbers proved by Nerman in [@Nerman1981]. The result assumes the characteristic is non-negative; in applications, it suffices to write the characteristic as the difference of its positive and negative parts.
We will need the following regularity condition.
\[cond::ghFunCondition\] There exist non-increasing bounded positive integrable càdlàg functions $g$ and $h$ on $[0, \infty)$ such that $$\bE \left[ \sup_{t \geq 0} \frac{ \xi_\gamma(\infty) - \xi_\gamma(t)}{g(t)} \right] < \infty \quad \text{and} \quad \bE \left[ \sup_{t \geq 0} \frac{e^{-\gamma t} \phi(t)}{h(t)} \right] < \infty.$$
The first part of the condition is satisfied if the expected number of offspring is finite because then, choosing $g(t) = 1 \wedge t^{-2}$, we have $$\frac{\xi_\gamma(\infty)- \xi_\gamma(t)}{g(t)} \leq \int_t^\infty \frac{1}{g(s)}\xi_\gamma(ds) \leq \int_0^\infty \frac{1}{g(s)} \xi_\gamma(ds) \leq \sup_{u \geq 0} \{(1 \wedge u^2) e^{-\gamma u}\} \xi(\infty),$$ which has finite expectation.
We can now state the strong law of large numbers.
\[thm::NermanSLLN\] Let $(\xi_x, L_x, \phi_x)_x$ be a general branching process with Malthusian parameter $\gamma$, where $\phi \geq 0$ and $\phi(t) = 0$ for $t < 0$. Assume that $\nu_\gamma$ is non-lattice. Assume further that Condition \[cond::ghFunCondition\] is satisfied.
Then, $$z^\phi(t) \to z^\phi(\infty) = \frac{\int_0^\infty e^{- \gamma s} \bE \phi(s) ds}{\int_0^\infty s \nu_\gamma(ds)},$$ some finite constant, and $$e^{-\gamma t} Z^\phi(t) \to z^\phi(\infty) M_\infty, \text{ a.s.},$$ as $t \to \infty$, where $M_\infty$ is the almost sure limit of the fundamental martingale of the general branching process. Furthermore, if $M$ is uniformly integrable, then the convergence also takes place in $L^1$.
A similar result holds when $\nu_\gamma$ is lattice; see [@Gatzouras2000]. But we will carefully avoid this case here.
Applications to statistically self-similar fractals {#subsec::appGBPtoFractals}
---------------------------------------------------
Intuitively, a random compact subset $K$ of $\R^d$ is statistically self-similar if there is a random number $N$ and random contracting similitudes $\Phi_1, \dots, \Phi_N$ such that $$K = \bigcup_{i=1}^N \Phi_i(K_i), \text { a.s.},$$ where $K_1, \dots, K_N$ are i.i.d. copies of $K$. The reader is referred to [@Falconer1986; @Graf1987; @MW1986] for more information.
To encode $K$ as general branching process, we use the address space $I$ defined in . To each $x \in I$, we associate a random collection $(N_x, \Phi_{x,1}, \dots \Phi_{x,N_x})_{x \in I}$, where $N_x$ is a natural number and $\Phi_{x,i}$ are contracting similitudes whose ratios we write $R_{x,i}$. We assume that the collection is i.i.d. in $x$.
Write $\cT$ for the path of the Galton-Watson process generated by the random numbers $(N_x, x \in I)$, i.e. $\emptyset \in \cT$ and $$\quad y = y_1 \dots y_n \in \cT \iff y_1 \dots y_{n-1} \in \cT \text{ and } y_n \leq N_{y_1\dots y_{n-1}}.$$ Starting with a compact set $K_\emptyset$, define, for $x = x_1 \dots x_n \in \cT$, $$K_x = \Phi_{x_1} \circ \dots \circ \Phi_{x_1 \dots x_n}(K_\emptyset) \quad \text{and} \quad K = \bigcap_{n = 1}^\infty \bigcup_{|x| = n} K_x,$$ where $|x|$ is the length of the word $x$. Then $K$ is the statistically self-similar set corresponding to $K_\emptyset$ and $(N, \Phi_1, \dots, \Phi_N)$.[^1]
The Hausdorff dimension of statistically self-similar sets is given by the following formula *à la* Moran and Hutchinson [@Moran1946; @Hutchinson1981]. The statement is adapted from [@Falconer1986; @Graf1987; @MW1986].
\[thm::dimHausStatSelfSimilar\] Let $K$ be a random statistically self-similar set as above. Then, on the event that the set $K$ is not empty, $$\dim K = \inf\left\{ s : \bE \left(\sum_{i=1}^N R_i^s \right) \leq 1 \right\}, \text{ a.s.}$$
To make the underlying Galton-Watson process structure of statistically self-similar sets into a general branching process, we specify birth times by setting $$\xi_x = \sum_{i=1}^{N_x} \delta_{-\log R_{x,i}},$$ and the lifetimes by setting $L_x = \sup_i (\sigma_{x,i} - \sigma_x)$.
For the first generation of offspring $e^{-\sigma_i} = R_i$. More generally, with this parametrisation, the offspring $x$ born around time $t$ correspond to compact sets $K_x$ of size roughly $e^{-t}$ in the construction.
Notice that $$\label{eq::calculationHausdorffDim}
\bE\int_{0}^\infty e^{-s x} \xi(dx) = \bE \left( \sum_{i=1}^N R_i^s \right),$$ so that the Malthusian parameter of the underlying general branching process is equal to the almost sure Hausdorff dimension of the set $K$; compare with .
Statistically self-similar snowflakes {#sec::selfSimilar}
=====================================
In this section, we study the geometry and heat content asymptotics of a family of Koch type snowflakes whose boundary is statistically self-similar. Our main tool will be Nerman’s strong law of large numbers.
Let us stress that the discussion here can be extended to any other snowflakes with statistically self-similar boundary so long as the corresponding general branching process satisfies the assumptions of the strong law of large numbers. However, for brevity, we will focus on a particular class of snowflakes that can easily be connected to those discussed in Section \[sec::scaleHomogeneous\]. Our aim is to emphasise that the behaviour of the heat content is qualitatively different when the boundary of the snowflake is statistically self-similar and not space homogeneous, because of additional spatial independence.
Construction
------------
Start with the segment $[0,1]$ and pick $a \in A$, a bounded subset of $\N$, randomly according to the probability distribution $(p_a, a \in A)$. Consider the building block $K(a)$ described in Section \[sec::scaleHomogeneous\]. Then replace each linear piece of $K(a)$ by a scaled i.i.d. copy of itself, i.e. again using $(p_a, a \in A)$. Iterating indefinitely, we obtain a sequence of curves converging to a statistically self-similar curve $K$. Indeed, in the notation of Subsection \[subsec::appGBPtoFractals\], it suffices to set $$(N(a), R_1(a), \dots, R_N(a)) = (m(a), \ell(a)^{-1}, \dots, \ell(a)^{-1});$$ the maps $(\Phi_1, \dots, \Phi_N)$ can easily be deduced from this.
The corresponding general branching process $(\xi,L)$ (no characteristic just yet) is obtained as detailed above. Figure \[fig::selfSimilarCurve\] contains an approximation of $K$ when $A = \{1, 2,3 \}$ and $(p_a, a \in A)$ is uniform.
Finally, the snowflake $D$ is defined as the simply connected interior of the Jordan curve created using three i.i.d. copies of $K$.
To ensure that $\nu_\gamma$ is non-lattice, we will make the following assumption.
\[ass::nonLattice\] The set $A$ contains two elements $a_1$ and $a_2$ such that $$p_{a_1} \text{ and } p_{a_2} \in (0,1) \quad \text{and} \quad \frac{\log \ell(a_1)}{\log \ell(a_2)} \notin \Q.$$
Fractal dimension and Minkowski content
---------------------------------------
Now that we have defined the statistically self-similar snowflakes, we will use the theory of general branching processes to study the volume of their inner tubular neighbourhoods.
\[thm::minkowskiContentSnowflakeSLLN\] For the statistically self-similar snowflake $D$, we have $$\epsilon^{\gamma- 2} \mu(\epsilon) \to \cM N_\infty, \text{ a.s.} \text{ and in } L^1,$$ as $\epsilon \to 0$, for some positive constant $\cM$ and positive random variable $N_\infty$ with unit expectation.
In particular, the Minkowski dimension of $\partial D$ exists and is equal to $\gamma$ almost surely, and the inner Minkowski content exists, is finite, and given by $\cM N_\infty$ almost surely.
Notice that the reasoning used in Lemma \[lem::tubNeighbourhoodScaleHomo\] shows that in this situation, again, the inner Minkowski dimension coincides with the standard definition.
The snowflake $D$ is built from three i.i.d. copies of $U$, the open set whose boundary is made of two linear pieces and $K$, as depicted in Figure \[fig::selfSimilarCurve\].
We will focus on how to deal with one such third $U$ for now. To do that, put $$\tilde \mu_U(\epsilon) = {\mathrm{vol}}_2(\{x \in U : d(x, K) \leq \epsilon\}),$$ i.e. $\tilde \mu$ measures the volume of the part of the inner tubular neighbourhood close to the fractal part $K$ of the boundary of $U$ only.
By construction of $K$, the set $U$ is made of a polygonal region $V$ (shaded in Figure in Figure \[fig::selfSimilarCurve\]), say, and $m(a)$ copies $U_i$ of itself scaled by a random factor $R_i$. Therefore, $$\label{eq::decompostitionVolumeTube}
\tilde \mu_U(\epsilon) = \theta(\epsilon) + \sum_{i=1}^{m(a)} \tilde \mu_{R_i U_i}(\epsilon),$$ where $\theta(\epsilon)$ is an error term bounded by $c_1 \epsilon^2$, as is clear from Figure \[fig::selfSimilarCurve\] (the existence of the constant $c_1$ also uses that the set $A$ and therefore $m(a)$ is bounded). Furthermore, by scaling, $$\label{eq::volumeScaling}
\tilde \mu_{R_i U_i}(\epsilon) = R_i^2 \tilde \mu_{U_i}(R_i^{-1} \epsilon).$$
In the language of the general branching process, each $R_i U_i$ corresponds to an offspring of the fractal $K$ born at time $\sigma_i = - \log R_i$. Therefore, putting, for $t \in \R$, $$Z^\phi(t) = e^{2t} \tilde \mu_U (e^{-t}) \quad \text{and} \quad \phi(t) = e^{2t} \theta(e^{-t}),$$ the relations and combine to produce $$Z^\phi(t) = \phi(t) + \sum_{i = 1}^{\xi(\infty)} Z_i^\phi(t- \sigma_i),$$ where the $Z_i^\phi$ are i.i.d. copies of $Z$ and $\phi$ is bounded. Therefore, $Z^\phi$ is the counting process of the characteristic $\phi$.
To apply the strong law of large numbers, we need to consider $$Z^\chi(t) = Z^\phi(t) \bone_{t \geq 0} = \chi(t) + \sum_{i = 1}^{\xi(\infty)} Z_i^\chi(t- \sigma_i),$$ defined using .
Since $\xi(\infty)$ is bounded (because $A$ is) and $Z^\phi(t)$ is bounded for negative times (by ${\mathrm{vol}}_2(U)$), the characteristic $\chi$ must be bounded as well. These observations imply that Condition \[cond::ghFunCondition\] is satisfied as well as the integrability condition of . Therefore, by Theorem \[thm::NermanSLLN\], $$e^{-\gamma t} Z^\chi(t) \to z^\chi(\infty) M_\infty \text{ a.s.} \text{ and in } L^1,$$ as $t \to \infty$, where $$z^\chi(\infty) = \frac{\int_0^\infty u^\chi(t) dt}{\int_0^\infty t \nu_\gamma(dt)} \in (0, \infty).$$ By definition of $Z^\chi$, this shows that $$\label{eq::muepsilonconvergencethird}
\epsilon^{\gamma - 2} \tilde \mu_U (\epsilon) \to z^\chi(\infty) M_\infty.$$
As $\mu$ is the sum of three i.i.d. copies of $\tilde \mu_U$, we get the desired result by putting $\cM = 3 z^\chi(\infty)$ and setting $N_\infty$ to be a third of the sum of the three i.i.d. copies of $M_\infty$ given by the general branching process.
Heat content asymptotics
------------------------
Following the analysis performed above for the volume of the inner tubular neighbourhoods of $D$, we now use the theory of general branching processes to study the heat content of $D$. Our aim is to prove the following result.
\[thm::heatContentAsymptotics\] For the statistically self-similar Koch snowflake $D$ $$s^{\gamma/2-1} E(s) \to \cE N_\infty, \text{a.s.} \text{ and in } L^1,$$ as $s \to 0$, for some positive constant $\cE$ and positive random variable $N_\infty$ with unit expectation, where $\gamma = \dim_M \partial D$.
The random variable appearing in this theorem is the same as that appearing in Theorem \[thm::minkowskiContentSnowflakeSLLN\]. So this theorem implies that, for statistically self-similar snowflakes, one can recover both the Minkowski dimension and the inner Minkowski content from short time asymptotics of the heat content.
In the proof, we will use the elementary fact that if $D$ is a domain in $\R^d$ and $r \in (0, \infty)$, then $$\label{eq::scalingHeatContent}
E_{rD}(s) = r^{d} E_D(r^{-2} s).$$
As in the proof for the volume of the inner tubular neighbourhood, we first only consider a third $U$ of the snowflake and let $w$ be the solution of the heat equation with the boundary condition $$w(s, x) = \begin{cases} 1, & x \text{ in the fractal part of } \partial U,\\
0, & x \text{ is in the linear part of } \partial U, \end{cases}$$ and the initial condition $$w(0, x) = 0,\quad x \in U.$$ Now write $F_U$ for the heat content of $U$ with this altered boundary condition, i.e. $$F_U(s) = \int_{U} w(s,x) dx.$$
Before we can use the theory of general branching processes, we need some notation to understand the effect of adding extra cooling inside $U$. More precisely, write $\tilde w$ for the solution of the heat equation with boundary condition $$\tilde w(s, x) = \begin{cases} 1, & x \text{ in the fractal part of } \partial U,\\
0, & x \text{ is in the linear part of } \partial U \text{ or in }\partial V,
\end{cases}$$ where $V$ is the polygonal region defined in the proof of Theorem \[thm::minkowskiContentSnowflakeSLLN\], and the initial condition $$\tilde w(0, x) = 0,\quad x \in U.$$ Let $\tilde F_U$ be the corresponding heat content $$\tilde F_U(s) = \int_{U} \tilde w(s,x) dx.$$
With these definitions, we have $$F_U(s) = F_U(s) - \tilde F_U(s) + \sum_{i = 1}^{m(a)} F_{U_i} (s) = \psi(s) + \sum_{i = 1}^{m(a)} F_{U_i} (s),$$ say. By scaling , $$F_{R_i U_i}(s) = R_i^2 F_{U_i}(R_i^{-2} s) = e^{-2\sigma_i} F_{U_i} (e^{2\sigma_i} s).$$ Putting, for $t \in \R$, $$Z^\phi(t) = e^{2t} F_U(e^{-2t}) \quad \text{and} \quad \phi(t) = e^{2t}\psi(e^{-2t}),$$ yields $$Z^\phi(t) = \phi(t) + \sum_{i = 1}^{\xi(\infty)} Z_i(t- \sigma_i),$$ where the $Z_i^\phi$ are i.i.d. copies of $Z^\phi$.
Let us now show that $\phi$ is bounded. This is done using the results of [@vdBG1997] about the impact on the heat content of imposing extra cooling. Put $$\lambda(\epsilon) = {\mathrm{vol}}_2 (\{ x \in \cup_{i} U_i : d(x,S) < \epsilon/\sqrt{2} \text{ and } d(x, V) < \epsilon/\sqrt{2} \}),$$ where $S$ is the fractal part of $\partial U$, i.e. $$S = \overline{\{x \in \partial U : w(x, - ) = 1\}}.$$ It is easy to check (by drawing a picture) that $$\lambda(\epsilon) \leq c_1 \epsilon^2.$$ Therefore, by Corollary 1.3 of [@vdBG1997] and an integration by parts, $$0 \leq \psi(s) \leq c_2 \int_0^\infty e^{-\epsilon^2/4s} \lambda(d \epsilon)
= c_2 2^{-1} s^{-1} \int_0^\infty \lambda(\epsilon) \epsilon e^{- \epsilon^2/4s} d \epsilon
\leq c_3 s \wedge c_4.$$ From this, it follows that $\phi$ is bounded.
Now reasoning as in the proof of Theorem \[thm::minkowskiContentSnowflakeSLLN\] shows that $$s^{\gamma/2 -1} F_U(s) \to \cF M_\infty, \text{ a.s.\ and in } L^1,$$ as $s \to 0$ for some positive constant $\cF$.
To conclude, recall that $D$ is the union of three i.i.d. copies of $U$, say $U_1, \dots, U_3$. Furthermore, by the estimate of [@vdBG1997] again, we have $$E_D(s) = \sum_{i= 1}^3 F_{U_i}(s) + O(s),$$ as $s \to 0$. It follows that $$s^{\gamma/2 -1} E_D(s) \to \cE N_\infty, \text{ a.s.\ and in } L^1,$$ as $s \to 0$, where $\cE = 3 \cF$ and $N_\infty$ is defined as in the proof of Theorem \[thm::minkowskiContentSnowflakeSLLN\], as required.
Open question
=============
For the statistically self-similar snowflakes, it is natural to ask about the fluctuations of the heat content around the almost sure short time asymptotics.
In the forthcoming paper [@CCH2014], we discuss a central limit theorem for general branching processes and apply it to study the fluctuations of the spectrum of some statistically self-similar fractals with Dirichlet weights.
More work on general branching processes is required before that central limit theorem can be applied to the snowflakes discussed here. However, it naturally leads to the following conjecture.
Let $D$ be a statistically self-similar snowflake of Section \[sec::selfSimilar\]. Then, $$s^{- \gamma/4}(s^{\gamma/2 - 1 }E_D(s) - \cE N_\infty) \to_d Y_\infty,$$ where $\cE$ and $N_\infty$ are defined in Section \[sec::selfSimilar\] and $Y_\infty$ is a random variable whose characteristic function has the form $$\bE\left[e^{i \theta Y_\infty}\right] = \bE \left[e^{-\frac 12 \theta^2 \sigma^2 N_\infty}\right],$$ for some $\sigma \in (0, \infty)$.
Acknowledgments
===============
I would like to thank Dmitry Belyaev, Ben Hambly, Sean Ledger and Sam Watson for related discussions, and Kolyan Ray for comments on an earlier version of this paper. The financial support of the Berrow Foundation and the Swiss National Science Foundation is gratefully acknowledged.
[^1]: To be completely rigorous, we assume that the sets $(\interior K_x, x \in \cT)$ *form a net*, i.e.$$x \leq y \implies \interior K_y \subset \interior K_x \quad \text{and} \quad \interior K_x \cap \interior K_y = \emptyset \text{ if neither $x \leq y$ nor $y \leq x$}.$$
We also assume that the construction described above is *proper* in the sense of [@Falconer1986a], i.e. that every cut of $\cC$ of $\cT$ satisfies the condition: For every $x \in \cC$, there exists a point in $K_x$ that does not lie in any other $K_y$ with $y \in \cC$.
|
---
author:
- |
Lucas Sourrouille$^a$\
[*$^a$Departamento de Física, FCEyN, Universidad de Buenos Aires*]{}\
[*Pab.1, Ciudad Universitaria, 1428, Ciudad de Buenos Aires, Argentina*]{}\
[sourrou@df.uba.ar]{}
title: '**Stability analysis for soliton solutions in a gauged CP(1) theory**'
---
[**Keywords**]{}:CP(1) nonlinear sigma model, Gauge theory, Topological solitons
[**PACS numbers**]{}: 11.10.Lm, 11.15.-q
Introduction
============
The two dimensional $CP(n)$ sigma model was introduced in the late seventies [@golo; @golo1; @golo2], in the search of understanding the strong coupling effects in $QCD$. This model captures several interesting properties, many of them present in four dimensional $QCD$[@witten; @witten1; @witten2; @witten3]. Whereas in four dimensional $QCD$ is difficult to demonstrate the existence of these properties, in two dimensional $CP(n)$ sigma model it becomes comparatively simple. An important issue related to this type of models concern to the existence of soliton type solutions. For the simplest $CP(1)$ model topological solutions have been shown to exist[@polyakov]. Nevertheless, the solutions are of arbitrary size due to scale invariance. As argued originally by Dzyaloshinsky, Polyakov and Wiegmann[@polyakov1] a Chern-Simons term can naturally arise in this type of models and the presence of a dimensional parameter could play some role stabilizing the soliton solutions. A first detailed consideration of this problem was done in Ref.[@voru] where a perturbative analysis around the scale invariant solutions (i.e no Chern-Simons coupling $\kappa=0$) showed that the solutions were pushed to infinite size. More recently, in Ref.[@my3], a nonperturbative analysis of the solutions was done, showing that the Chern-Simons-CP(1) system, without a potential term, admits only trivial solutions in $\rm R^2$. Nevertheless, in Ref.[@my5], it was shown that the Chern-Simons-CP(1) model in absence, has a non-trivial solution if the theory is defined in ${\rm R}^2\setminus D(0,\epsilon)$, where $D(0,\epsilon)$, is a disc centered at the origin and with an arbitrary radius $\epsilon$.
This paper pretends to be a continuation of the work [@my3] and a generalization of the results obtained there. We will show that a Chern-Simons-CP(1) model with a potential term, which was proposed in the reference [@Z], presents a stable soliton solution if there is a critical radius $S_c$ such that the equality $$\begin{aligned}
\int_{D_S} d^2 x \;\; B^2 = \int_{D_S} d^2 x \;\; V\end{aligned}$$ is satisfied for all radius $S \geq S_c$, where here the subindex $D_S$ indicates that the region of integration is a disc of radius $\hat{S} $ and the letters $V$ and $B$ represent the potential term and the magnetic field.
The model
=========
We begin by considering a $(2+1)$-dimensional Chern-Simons model coupled to a complex two component field $n(x)$ described by the action
$$\begin{aligned}
S&=& S_{cs}+\int_{D} d^3 x |D_\mu n|^2 + V
\label{S1}\end{aligned}$$
The subindex $D$ indicates that the region of integration is a disc $D$ of radius $R$[@my3]. Here $D_{\mu}= \partial_{\mu} - iA_{\mu}$ $(\mu =0,1,2)$ is the covariant derivative and $V$ is the potential term to be determined later. The term $S_{cs}$ is the Chern-Simons action given by
$$\begin{aligned}
S_{cs}= \kappa\int_{D} d^3 x \epsilon^{\mu \nu \rho} A_\mu \partial_\nu A_\rho\end{aligned}$$
where
$$\begin{aligned}
F_{\mu \nu}=\partial_{\mu}A_{\nu}-
\partial_{\nu}A_{\mu} \label{F}\end{aligned}$$
The metric signature is $(1,-1,-1)$ and the two component field $n(x)$ is subject to the constraint $n^\dagger n = 1$. The constraint can be introduced in the variational process with a Lagrange multiplier. Then, we extremise the following action
$$\begin{aligned}
S&=& S_{cs}+\int_{D} d^3 x \Big(|D_\mu n|^2 + V + \lambda (n^\dagger n -1)\Big)
\label{}\end{aligned}$$
The variation of this action yields the field equations
$$\begin{aligned}
D_\mu D^\mu n -\frac{\partial V}{\partial n^\dagger} +\lambda n =0
\label{motion1}\end{aligned}$$
$$\kappa\epsilon_{\mu \nu \rho} F^{ \nu \rho} = - J_\mu=i [n^\dagger D_\mu n - n(D_\mu n)^\dagger]
\label{motion}$$
From the first of these equations we get $\lambda= -n^\dagger (D_\mu D^\mu n - \frac{\partial V}{\partial n^\dagger})$, so that
$$\begin{aligned}
D_\mu D^\mu n -\frac{\partial V}{\partial n^\dagger} = \Big(n^\dagger (D_\mu D^\mu n -\frac{\partial V}{\partial n^\dagger})\Big)n
\label{motion2}\end{aligned}$$
The time component of Eq.(\[motion\]) $$\begin{aligned}
2\kappa F_{12} = -J_0 \label{gauss}\end{aligned}$$ is Gauss’s law of Chern-Simons dynamics. Integrating it over the entire plane one obtains the important consequence that any object with charge $Q =\int_{D} d^2 x J_0$ also carries magnetic flux $\Phi = \int_{D} B d^2 x$ [@Echarge; @E1; @E2]:
$$\begin{aligned}
\Phi = -\frac{1}{2\kappa} Q,
\label{Q}\end{aligned}$$
where in the expression of magnetic flux we renamed $F_{12}$ as $B$.
Defining the stress-tensor as[@Z] $$\begin{aligned}
T_{ \mu \nu}= (D_{\mu} n)^{\dagger} D_{\nu} n +(D_{\nu} n)^{\dagger} D_{\mu} n - g_{\mu \nu} \Big( (D_\eta n)^{\dagger} D^{\eta} n - V\Big)\;\;,\end{aligned}$$ we get
$$\begin{aligned}
E= \int_{D} d^2 x \Big(\kappa^2 B^2 + |D_i n|^2 + V \Big) \,,
\;\;\;\;\;\
i = 1,2 \;, \label{statich}\end{aligned}$$
which is the expression of the energy functional for the static field configuration.
Here, It is convenient to specified the potential term $V$. Following the reference [@Z] we define the potential as $$\begin{aligned}
V(n)= \eta \Big(1-n^{\dagger} ( \sigma_3 n)\Big)\end{aligned}$$ Where $$\begin{aligned}
\sigma_3 = \ \left( \begin{array}{cc}
1 & 0 \\
0 & -1 \end{array} \right)\;\;,\end{aligned}$$ is the third Pauli spin-matrix and $\eta$ is the coupling strength .
Let us consider, now, the following ansatz with cylindrical symmetry for the $N$ soliton solutions:
$$\begin{aligned}
n(\phi, r)= \left( \begin{array}{c}
\cos(\frac{\theta(r)}{2})e^{i N \phi}\\
\sin(\frac{\theta(r)}{2} )\end{array} \right)
\,,
\;\;\;\;\;\
A_\phi (r)= a(r)
\,,
\;\;\;\;\;\
A_r =0\,,
\label{ansatz}\end{aligned}$$
In terms of this ansatz the energy (\[statich\]) reads as $$\begin{aligned}
E= 2\pi \int_0^R r dr \Big(\kappa^2 \left( \frac{a(r)}{r}+ \partial_r a(r) \right)^2 +\frac{1}{4}(\partial_r \theta(r))^2 \nonumber \\
+ \left(\frac{N^2}{r^2} + \frac{2Na(r)}{r}\right)\cos^2(\frac{\theta(r)}{2})+ a^2(r) + \eta(1-\cos(\theta(r))) \Big) \;,
\label{statich1}\end{aligned}$$ whereas the field equations become
$$\begin{aligned}
\partial_r^2 a(r)+\frac{\partial_r a(r)}{r}- \frac{a(r)}{r^2} - \frac{a(r)}{\kappa^2}=\cos^2(\frac{\theta(r)}{2})\frac{N}{r\kappa^2}
\label{m1}\end{aligned}$$
$$\begin{aligned}
r\partial_r(r\partial_r \theta(r)) + \Big(N^2 + 2Nra(r)\Big)\sin(\theta(r)) = \eta 2 r^2 \sin(\theta(r)) \;,
\label{m2}\end{aligned}$$
In order to ensure the regularity of the field at the origin, we impose
$$\begin{aligned}
\lim_{r \to 0} \theta(r) = \pi
\,,
\;\;\;\;\;\
\lim_{r \to 0} a(r) = 0
\label{b1}\end{aligned}$$
On the other hand, the conditions at the boundary of the disk are in principle more general. This is because the length of the radius R is arbitrary. However, if the size of the disc becomes infinite, then, we must impose boundary conditions ensuring finite energy, that is
$$\begin{aligned}
\lim_{r \to \infty} \theta(r) = 0
\,,
\;\;\;\;\;\
\lim_{r \to \infty}a(r) =-\frac{ N}{r}
\label{b2}\end{aligned}$$
Therefore, it is convenient to use this boundary condition independently of the length of the radius. So, we fix the boundary conditions at $R$ to be
$$\begin{aligned}
\lim_{r \to R} \theta(r) = 0
\,,
\;\;\;\;\;\
\lim_{r \to R}a(r) =-\frac{ N}{r}
\label{b2}\end{aligned}$$
this boundary conditions imply the quantization of the magnetic flux
$$\begin{aligned}
\Phi = 2\pi\int_0^R r dr\,\, \frac{\partial_r(r\,\,a(r))}{r}=-2\pi N
\label{flux}\end{aligned}$$
If the solutions of (\[m1\]) and (\[m2\]) exist their scale must be set by the quantity $\kappa$. Following Ref.[@mehta], we introduce the dimensionless quantities
$$\begin{aligned}
A = \kappa a
\,,
\;\;\;\;\;\
s =\frac{r}{\kappa}\end{aligned}$$
in terms of which (\[m1\]) and (\[m2\]) become
$$\begin{aligned}
\partial_s^2 A +\frac{\partial_s A}{s}- \frac{A}{s^2} - A =\cos^2(\frac{\theta}{2})\frac{N}{s}
\label{m11}\end{aligned}$$
$$\begin{aligned}
s\partial_s(s\partial_s \theta) + \Big(N^2 + 2NsA\Big)\sin(\theta) = \eta 2 s^2 \sin(\theta)
\label{m22}\end{aligned}$$
The energy functional (\[statich1\]) in terms of these new variables reads as
$$\begin{aligned}
E(S)= 2\pi \int_0^S s ds \Big( \left( \frac{A}{s}+ \partial_s A \right)^2 +\frac{1}{4}(\partial_s \theta)^2 \nonumber \\
+ \left(\frac{N^2}{s^2} + \frac{2NA}{s}\right)\cos^2(\frac{\theta}{2})+ A^2 + \eta(1-\cos(\theta))\Big) \,
\;\;\;\;\;\
\label{H3}\end{aligned}$$
For the origin we choose the following boundary conditions, $$\begin{aligned}
\lim_{s \to 0} \theta = \pi
\,,
\;\;\;\;\;\
\lim_{s \to 0} A = 0
\label{b11}\end{aligned}$$ while for the boundary $S=R/\kappa$ we choose,
$$\begin{aligned}
& &\lim_{s \to S}A =-\frac{ N}{S} \nonumber \\
& & \lim_{s \to S} \theta =0
\;\;\;\;\;\
\label{b22}\end{aligned}$$
Stability analysis
==================
In this section we analyze the stability of soliton solutions corresponding to the equations (\[m11\]) and (\[m22\]).
Consider the following configuration defined in the interval of length $\lambda S$
$$\begin{aligned}
\tilde{A}_{\lambda S}(s) =\frac{A_{ S}(\frac{s}{\lambda})}{\lambda}
\,,
\;\;\;\;\;\
\tilde{\theta}_{\lambda S}(s) = \theta_{S}(\frac{s}{\lambda})
\label{config}\end{aligned}$$
Here $\lambda$ is a positive real number such that $\lambda >1$ and the configurations (\[config\]) satisfy the boundary conditions
$$\begin{aligned}
\lim_{s \to 0} \tilde{\theta}_{\lambda S}(s) = \pi
\,,
\;\;\;\;\;\
\lim_{s \to 0} \tilde{A}_{\lambda S}(s) = 0
\label{b111}\end{aligned}$$
$$\begin{aligned}
\lim_{s \to \lambda S}\tilde{\theta}_{\lambda S}(s) = 0
\,,
\;\;\;\;\;\
\lim_{s \to \lambda S}\tilde{A}_{\lambda S}(s) =\frac{-N}{\lambda S}
\label{b222}\end{aligned}$$
We can evaluate the energy functional (\[H3\]) for the configuration (\[config\]) in an interval of length $\lambda S$ $$\begin{aligned}
\tilde{E}(\lambda S)= 2\pi \int_0^{\lambda S} s ds \Big( \left( \frac{\tilde{A}_{\lambda S}(s)}{s}+ \partial_s \tilde{A}_{\lambda S}(s) \right)^2 +\frac{1}{4}(\partial_s \tilde{\theta}_{\lambda S})^2 \nonumber \\
+ \left(\frac{N^2}{s^2} + \frac{2N\tilde{A}_{\lambda S}(s)}{s}\right)\cos^2(\frac{\tilde{\theta}_{\lambda S}}{2})+ \tilde{A}^2_{\lambda S}(s) + \eta(1-\cos(\tilde{\theta}_{\lambda S})) \Big) \,
\;\;\;\;\;\
\label{H4}\end{aligned}$$ We denote the solution corresponding to the interval $\lambda S$ as $A_{\lambda S} (s)$ and $\theta_{\lambda S} (s)$ and its energy as $E(\lambda S)$. Since the configuration (\[config\]) satisfy the same boundary condition as $A_{\lambda S} (s)$ and $\theta_{\lambda S} (s)$, we have that $$\begin{aligned}
E(\lambda S) \leq \tilde{E}(\lambda S)
\label{inq11}\end{aligned}$$ Under the transformation $s= x\lambda$ the functional (\[H4\]) becomes $$\begin{aligned}
\tilde{E}(\lambda S)= 2\pi \int_0^{S} x dx \Big(\frac{1}{{\lambda}^2} \left( \frac{A_{S}(x)}{x}+ \partial_x A_{S}(x) \right)^2 +\frac{1}{4}(\partial_x \theta_{S})^2 \nonumber \\
+ \left(\frac{N^2}{x^2} + \frac{2NA_{S}(x)}{x}\right)\cos^2(\frac{\theta_{S}}{2})+ A^2_{S}(x) + \eta \lambda^2(1 -\cos(\theta_{S})) \Big) \,
\;\;\;\;\;\
\label{H5}\end{aligned}$$ Since there are evaluated in the same interval, we can compare this expression with the formula (\[H3\]). For this purpose we can look for the values of $\lambda$ for which the equality $\tilde{E}(\lambda S)= E(S)$ is held. Using the equations (\[H3\]) and (\[H5\]), we obtain $$\begin{aligned}
\int_0^{S} x dx \Big(\frac{1}{{\lambda}^2} \left( \frac{A_{S}(x)}{x}+ \partial_x A_{S}(x) \right)^2 + \eta \lambda^2(1 -\cos(\theta_{S})) \Big)
\nonumber \\
= \int_0^{S} x dx \Big( \left( \frac{A_{S}(x)}{x}+ \partial_x A_{S}(x) \right)^2 + \eta (1 -\cos(\theta_{S})) \Big)
\,
\;\;\;\;\;\
\label{H6}\end{aligned}$$ Renamed
$$\begin{aligned}
\lambda^2 &=& \omega
\nonumber \\[3mm]
\int_0^{S} x dx \eta (1 -\cos(\theta_{S})) &=& a
\nonumber \\[3mm]
\int_0^{S} x dx \left( \frac{A_{S}(x)}{x}+ \partial_x A_{S}(x) \right)^2 &=&b\;,\end{aligned}$$
the expression (\[H6\]) reduce to
$$\begin{aligned}
\omega^2 a -(a +b)\omega + b=0
\label{poly}\end{aligned}$$
The roots of this polynomial are
$$\begin{aligned}
\omega =1
\,,
\;\;\;\;\;\
\omega = \frac{b}{a}\end{aligned}$$
and then the possible values of $\lambda$ are
$$\begin{aligned}
\lambda =1
\,,
\;\;\;\;\;\
\lambda_{c} = \sqrt{\frac{b}{a}}
\label{val1}\end{aligned}$$
Now, suppose that $b>a$ and choose the values of $\lambda$ satisfying $1<\lambda < \sqrt{\frac{b}{a}}$. It is not difficult to see that for this values of $\lambda$ the following inequality is held
$$\begin{aligned}
\int_0^{S} x dx \Big(\frac{1}{{\lambda}^2} \left( \frac{A_{S}(x)}{x}+ \partial_x A_{S}(x) \right)^2 + \eta \lambda^2(1 -\cos(\theta_{S})) \Big)
\nonumber \\
< \int_0^{S} x dx \Big( \left( \frac{A_{S}(x)}{x}+ \partial_x A_{S}(x) \right)^2 + \eta (1 -\cos(\theta_{S})) \Big)
\,
\;\;\;\;\;\
\label{}\end{aligned}$$
and therefore
$$\begin{aligned}
\tilde{E}(\lambda S)< E(S)
\label{inq1a}\end{aligned}$$
Comparing (\[inq11\]) and (\[inq1a\]) we have that $$\begin{aligned}
E(\lambda S)< E(S)\end{aligned}$$ that is, the energy decreases when we enlarge the interval S. This process is valid only for $1<\lambda < \sqrt{\frac{b}{a}}$. We can repeat the proses by considering an interval of length $\lambda_c S$ instead of the interval $S$. Now, we have
$$\begin{aligned}
\int_0^{\lambda_c S} x dx \eta (1 -\cos(\theta_{\lambda_c S})) &=& a_1
\nonumber \\[3mm]
\int_0^{\lambda_c S} x dx \left( \frac{A_{\lambda_c S}(x)}{x}+ \partial_x A_{\lambda_c S}(x) \right)^2 &=&b_1\;,\end{aligned}$$
again, if $a_1 <b_1$, we can conclude that the energy decreases when the interval is enlarged. That is
$$\begin{aligned}
E(\lambda_{1} \lambda_c S)< E(\lambda_c S)\end{aligned}$$
where $1<\lambda_1 < \sqrt{\frac{b_1}{a_1}}$. The process can be repeated for successive intervals provided that $b>a$ in each of this intervals. Therefore if the relation
$$\begin{aligned}
\int_0^{S} x dx \eta (1 -\cos(\theta))
<
\int_0^{S} x dx \left( \frac{A_{S}(x)}{x}+ \partial_x A_{S}(x) \right)^2
\label{inq1}\end{aligned}$$
is held for all interval, the energy decreases indeterminably as $S\to \infty$, and thus there are no finite size soliton solution in ${\bf R}^2$.
Now, suppose $$\begin{aligned}
\int_0^{\rho S_1} x dx \eta (1 -\cos(\theta_{\rho S_1}))
>
\int_0^{\rho S_1} x dx \left( \frac{A_{\rho S_1}(x)}{x}+ \partial_x A_{\rho S_1}(x) \right)^2
\label{inq}\end{aligned}$$ where $S_1$ indicates the length of an arbitrary interval and $\rho$ is a real number such that $\rho >1$. Consider the following configuration defined in the interval of length $S_1$
$$\begin{aligned}
\tilde{A}_{S_1}(s) = \rho A_{\rho S_1}(\rho s)
\,,
\;\;\;\;\;\
\tilde{\theta}_{S_1}(s) = \theta_{\rho S_1}(\rho s)
\label{config1}\end{aligned}$$
where $A_{\rho S_1}(s)$ and $\theta_{\rho S_1}(s)$ are the solutions of the field equations in the interval $\rho S_1$. In virtue of equation (\[b11\]) and (\[b22\]), $A_{\rho S_1}(s)$ and $\theta_{\rho S_1}(s)$ must satisfied
$$\begin{aligned}
\lim_{s \to 0} \theta_{\rho S_1}(s) = \pi
\,,
\;\;\;\;\;\
\lim_{s \to 0} A_{\rho S_1}(s) = 0
\label{b1111}\end{aligned}$$
$$\begin{aligned}
\lim_{s \to \rho S_1} \theta_{\rho S_1}(s) = 0
\,,
\;\;\;\;\;\
\lim_{s \to \rho S_1} A_{\rho S_1}(s) =\frac{-N}{\rho S_1}
\label{b2222}\end{aligned}$$
Then, the configuration (\[config1\]) is subject to the following boundary conditions
$$\begin{aligned}
\lim_{s \to 0} \tilde{\theta}_{S_1}(s) = \pi
\,,
\;\;\;\;\;\
\lim_{s \to 0} \tilde{A}_{S_1}(s) = 0\;,
\label{b11111}\end{aligned}$$
$$\begin{aligned}
\lim_{s \to S_1}\tilde{\theta}_{S_1}(s) &=& 0
\nonumber \\
\lim_{s \to S_1}\tilde{A}_{S_1}(s)& =& -\rho \frac{N}{\rho S_1}= \frac{-N}{S_1}
\label{b22222}\end{aligned}$$
The solutions of the field equations in the interval $S_1$, which we denote by $A_{S_1}(s)$ and $\theta_{S_1}(s)$, also satisfied the boundary conditions (\[b11111\]) and (\[b22222\]). Therefore, we have $$\begin{aligned}
E(S_1) \leq \tilde{E}(S_1)
\label{inq3}\end{aligned}$$ where $E(S_1)$ is the energy corresponding to the solution $A_{S_1}(s)$ and $\theta_{S_1}(s)$, and
$$\begin{aligned}
\tilde{E}(S_1)&=& 2\pi \int_0^{S_1} s ds \Big( \left( \frac{\tilde{A}_{S_1}(s)}{s}+ \partial_s \tilde{A}_{S_1}(s) \right)^2 +\frac{1}{4}(\partial_s \tilde{\theta}_{S_1})^2 \nonumber \\
&+& \left(\frac{N^2}{s^2} + \frac{2N\tilde{A}_{S_1}(s)}{s}\right)\cos^2(\frac{\tilde{\theta}_{S_1}}{2})+ \tilde{A}^2_{S_1}(s) + \eta(1-\cos(\tilde{\theta}_{S_1})) \Big) \,
\;\;\;\;\;\
\label{H7}\end{aligned}$$
Using the configuration (\[config1\]) and under the transformation $x=\rho s$, the expression (\[H7\]) reads as
$$\begin{aligned}
\tilde{E}(S_1)&=& 2\pi \int_0^{\rho S_1} x dx \Big( \rho^2 \left( \frac{A_{\rho S_1}(x)}{x}+ \partial_x A_{\rho S_1}(x) \right)^2 +\frac{1}{4}(\partial_x \theta_{\rho S_1})^2 \nonumber \\
&+& \left(\frac{N^2}{x^2} + \frac{2N A_{\rho S_1}(x)}{x}\right)\cos^2(\frac{\theta_{\rho S_1}}{2})+ A^2_{\rho S_1}(x) + \frac{\eta}{\rho^2}(1-\cos(\theta_{\rho S_1})) \Big) \,
\;\;\;\;\;\
\label{H8}\end{aligned}$$
The energy for the solutions in the interval $\rho S_1$ is
$$\begin{aligned}
E(\rho S_1)&=& 2\pi \int_0^{\rho S_1} x dx \Big( \left( \frac{A_{\rho S_1}(x)}{x}+ \partial_x A_{\rho S_1}(x) \right)^2 +\frac{1}{4}(\partial_x \theta_{\rho S_1})^2 \nonumber \\
&+& \left(\frac{N^2}{x^2} + \frac{2N A_{\rho S_1}(x)}{x}\right)\cos^2(\frac{\theta_{\rho S_1}}{2})+ A^2_{\rho S_1}(x) + \eta (1-\cos(\theta_{\rho S_1})) \Big) \,
\;\;\;\;\;\
\label{H9}\end{aligned}$$
Since (\[H8\]) and (\[H9\]) are evaluated in the same interval we can compare this formulas. In this case we look for the values of $\rho$ for which the equality $\tilde{E}(S_1)= E(\rho S_1)$ is held. Following the same steps that we do previously we arrive to the polynomial
$$\begin{aligned}
\omega^2 a -(a +b)\omega + b=0\end{aligned}$$
However, in this case $a$ and $b$ are different from the constants present in the polynomial (\[poly\]). In fact we have
$$\begin{aligned}
\rho^2 &=& \omega
\nonumber \\[3mm]
\int_0^{\rho S_1} x dx \left( \frac{A_{\rho S_1}(x)}{x}+ \partial_x A_{\rho S_1}(x) \right)^2&=&a
\nonumber \\[3mm]
\int_0^{\rho S_1} x dx \eta (1 -\cos(\theta_{\rho S_1})) &=& b\end{aligned}$$
As in formula (\[val1\]) the roots of the polynomial produce the following values of $\rho$
$$\begin{aligned}
\rho =1
\,,
\;\;\;\;\;\
\rho_{c} = \sqrt{\frac{b}{a}}
\label{val2}\end{aligned}$$
Again, it is easy to show that choosing the values of $\rho$ satisfying the relation $1< \rho < \sqrt{\frac{b}{a}}$ we have
$$\begin{aligned}
\tilde{E}(S_1)< E(\rho S_1)\end{aligned}$$
and therefore in virtue of (\[inq3\])
$$\begin{aligned}
E(S_1)< E(\rho S_1)\end{aligned}$$
We can repeat the process by choosing $\rho_c S_1$ instead of $S_1$. In that case the formulas (\[inq\]) and (\[config1\]) read as
$$\begin{aligned}
\int_0^{\rho_1 \rho_c S_1} x dx \eta (1 -\cos(\theta_{\rho_1 \rho_c S_1}))
>
\int_0^{\rho_1 \rho_c S_1} x dx \left( \frac{A_{\rho_1 \rho_c S_1}(x)}{x}+ \partial_x A_{\rho_1 \rho_c S_1}(x) \right)^2
\label{}\end{aligned}$$
$$\begin{aligned}
\tilde{A}_{\rho_1 S_1}(s) = \rho_1 A_{\rho_1 \rho_c S_1}(\rho_1 s)
\,,
\;\;\;\;\;\
\tilde{\theta}_{\rho_1 S_1}(s) = \theta_{\rho_1 \rho_c S_1}(\rho_1 s)
\label{config3}\end{aligned}$$
and finally we arrive to
$$\begin{aligned}
E(\rho_c S_1)< E(\rho_1 \rho_c S_1)\end{aligned}$$
Of course the process can be repeated indefinitely provided that $b>a$ in all intervals. This implies that if the relation
$$\begin{aligned}
\int_0^{ S_1} x dx \eta (1 -\cos(\theta_{ S_1}))
>
\int_0^{ S_1} x dx \left( \frac{A_{ S_1}(x)}{x}+ \partial_x A_{ S_1}(x) \right)^2
\label{}\end{aligned}$$
is valid for all interval $S_1$, the energy increases. Certainly, we have
$$\begin{aligned}
0\leq E(\rho S_1)- \tilde{E}(S_1)\leq E(\rho S_1)- E(S_1)\;,\end{aligned}$$
where $$\begin{aligned}
E(\rho S_1)- \tilde{E}(S_1) = (1-\rho^2)a + (1-\frac{1}{\rho^2})b\end{aligned}$$ The roots of this equation are $$\begin{aligned}
\rho =1
\,,
\;\;\;\;\;\
\rho = \sqrt{\frac{b}{a}}
\label{val2}\end{aligned}$$ Since $1<\rho<\sqrt{\frac{b}{a}}$ for all interval, the energy increases indeterminably as $S\to\infty$. The stability only can take place if $|E(\rho S_1)- \tilde{E}(S_1)| \to 0$ and this implies that the interval $1<\rho<\sqrt{\frac{b}{a}}$ must be contracted to a point, that means $a=b$ and then $\rho=1$.
We conclude that if a soliton solution exist, then there is a critical radius $S_c$ such that the equality
$$\begin{aligned}
\int_0^{ S} x dx \eta (1 -\cos(\theta_{\hat{ S}}))
=
\int_0^{S} x dx \left( \frac{A_{\hat{ S}}(x)}{x}+ \partial_x A_{\hat{ S}}(x) \right)^2
\label{68}\end{aligned}$$
is verified for all radius $S \geq S_c$.
![[ The energy density, without a potential term, as a function of scaled radial coordinate $s$ for different disc sizes, from top to bottom, $S$=$30$, $50$, $60$, $70$, $100$, $150$. ]{}[]{data-label="E0"}](E0.eps){height="70mm"}
Numerical Results
=================
![[ The energy density, including a potential term, for different disc sizes, from top to bottom, $S$=$10$, $30$, $40$, $50$, $60$. The solution was calculated using a coupling constant $\eta =10^{-6}$. ]{}[]{data-label="E0001"}](E0001.eps){height="70mm"}
The field equations (\[m11\]) and (\[m22\]) can be resolved numerically. From the numerical point of view one solve the field equations in a finite disc and then the analysis of the solution in an infinite plane can be carried out by increasing successively the radius of disc. In order to solve numerically the field equations for arbitrary disc sizes, the relaxation method is suitable. In Fig.\[E0\] we show the behavior of the energy density, for $N=1$ and $S=30,50,60,70,100$, in the especial case in which there is no potential term and therefore the inequality (\[inq1\]) is held for arbitrary disc sizes. This fact is due to the constant value of the magnetic flux, which implies $B\not=0$ and therefore $\int_{D_S} d^2 x B^2 >0$ for all finite disc. This corresponds to problem analyzed in Ref.[@my3]. The figure clearly show that as the disc size increases the energy density tends to zero and then the solution is not stable in a finite disc. These results are in concordance with our previous analysis which point out that if the inequality (\[inq1\]) is held for arbitrary disc sizes, then the energy tends to zero.
In Fig.\[E0001\] show the energy density, for $N=1$ and for disc sizes $S= 10,30,40,50,60$, with the inclusion of a potential term. We can see that the energy density is highly instable for small disc sizes the size (i.e. $S=10$, $20$) and becomes more stable as S increases. In fact the energy density corresponding to $S=50$ and $S=60$ are practically indistinguishable in the graph and becomes completely stabilized for $S=60$. From theoretical point of view we have that the inequality (\[inq1\]) is held for $S<60$ and it becomes an equality for $S\geq 60$. So, $S_c=60$ is the critical radius from which the equality (\[68\]) is verified.
Conclusion
==========
In summary we have studied the classical solution of the Chern-Simons-CP(1) model with a potential term. Specifically we have shown that if the soliton solution exits, then there is a disc $D_{S_c}$ such that
$$\begin{aligned}
\int_{D_S} d^2 x \;\; B^2 = \int_{D_S} d^2 x\;\; V(n)\end{aligned}$$
is satisfied for all disc $D_S \geq D_{S_c}$. In addition we resolved numerically two situations. In the first situation we analyzed the model without a potential term, showing that the energy decreases as $S\to\infty$ , which implies the instability of the solutions. As a second case we analyzed the model with a potential term. We showed the energy decreases for $S<60$ and becomes stable for $S\geq 60$.
[99]{} H. Eichenherr, Nucl. Phys. B [**146**]{} (1978) 215 \[Erratum-ibid. B 155 (1979) 544\].\
V. L. Golo and A. M. Perelomov, Lett. Math. Phys. 2, 477 (1978); Phys. Lett. B [**79**]{}, 112 (1978).\
E. Cremmer and J. Scherk, Phys. Lett. B [**74**]{}, 341 (1978).\
A. D’Adda, M. Luscher and P. Di Vecchia, Nucl. Phys. B [**146**]{}, 63 (1978).\
A. D’Adda, P. Di Vecchia and M. Luscher, Nucl. Phys. B [**152**]{}, 125 (1979).\
E. Witten, Nucl. Phys. B [**149**]{}, 285 (1979).\
R. Rajaraman, Solitons and instantons, Elsevier Science, Amsterdam, (1987). ISBN 0-444-87047-4\
A.A. Belavin and A.M. Polyakov, JETP Lett. [**22**]{}, 245 (1975)\
I.E. Dzyaloshiskii, A.M. Polyakov and P.B. Wiegmann, Phys. Lett. A [**127**]{}, 112 (1988)\
P. Voruganti, Phys. Lett. B [**223**]{} (1989) 181.\
L. Sourrouille, A. Caso and G. S. Lozano, \[hep-th/1002.4847\], Mod. Phys. Lett. A [**26**]{}, 637 (2011)\
Lucas Sourrouille, \[arXiv:hep-th/1104.5045\], Mod. Phys. Lett. A, Vol. [**26**]{}, No. 33 (2011) pp. 2523-2531.\
B.M.A.G. Piette, D.H. Tchrakian, W.J. Zakrzewski, Phys. Lett. B [**339**]{} (1994) 95.\
J. Schonfeld, Nucl. Phys. B [**185**]{}, 157 (1981).\
S. Deser, R. Jackiw, and S. Templeton, Phys. Rev. Lett. [**48**]{}, 975 (1982).\
S. Deser, R. Jackiw, and S. Templeton, Ann. Phys.(N.Y.) [**140**]{}, 372 (1982).\
M.A. Mehta, J.A. Davis and I.J.R. Aitchison, Phys. Lett. B [**281**]{} (1992) 86.
|
[EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH ]{}
[CERN-EP-2002-045]{}\
[[1 July 2002]{}]{}
[**Search for Associated Production\
of Massive States\
Decaying into Two Photons\
in Annihilations at ${\sqrt{s}}=88-209$ GeV\
**]{}
[The OPAL Collaboration ]{}
[Abstract]{}
A search is performed for production of short-lived particles in ${{\mathrm e}^+ {\mathrm e}^-}{\rightarrow}\mrm {X Y}$, with $\mrm X {\rightarrow}{\gamma\gamma}$ and $\mrm Y {\rightarrow}{{\mathrm f} \bar{\mathrm f}}$, for scalar $\mrm X$ and scalar or vector $\mrm Y$. Model-independent limits in the range of 25-60 femtobarns are presented on $\sigma({{\mathrm e}^+ {\mathrm e}^-}{\rightarrow}\mrm {X Y})\times B({\mrm X} {\rightarrow}{\gamma\gamma})\times B(\mrm Y {\rightarrow}{{\mathrm f} \bar{\mathrm f}})$ for centre-of-mass energies in the range $205-207$ GeV. The data from all LEP centre-of-mass energies $88-209$ GeV are also interpreted in the context of fermiophobic Higgs boson models, for which a lower mass limit of 105.5 GeV is obtained for a “benchmark” fermiophobic Higgs boson.
[(Submitted to Physics Letters B)\
]{}
[The OPAL Collaboration ]{}
[ G.Abbiendi$^{ 2}$, C.Ainsley$^{ 5}$, P.F.kesson$^{ 3}$, G.Alexander$^{ 22}$, J.Allison$^{ 16}$, P.Amaral$^{ 9}$, G.Anagnostou$^{ 1}$, K.J.Anderson$^{ 9}$, S.Arcelli$^{ 2}$, S.Asai$^{ 23}$, D.Axen$^{ 27}$, G.Azuelos$^{ 18, a}$, I.Bailey$^{ 26}$, E.Barberio$^{ 8}$, R.J.Barlow$^{ 16}$, R.J.Batley$^{ 5}$, P.Bechtle$^{ 25}$, T.Behnke$^{ 25}$, K.W.Bell$^{ 20}$, P.J.Bell$^{ 1}$, G.Bella$^{ 22}$, A.Bellerive$^{ 6}$, G.Benelli$^{ 4}$, S.Bethke$^{ 32}$, O.Biebel$^{ 32}$, I.J.Bloodworth$^{ 1}$, O.Boeriu$^{ 10}$, P.Bock$^{ 11}$, D.Bonacorsi$^{ 2}$, M.Boutemeur$^{ 31}$, S.Braibant$^{ 8}$, L.Brigliadori$^{ 2}$, R.M.Brown$^{ 20}$, K.Buesser$^{ 25}$, H.J.Burckhart$^{ 8}$, J.Cammin$^{ 3}$, S.Campana$^{ 4}$, R.K.Carnegie$^{ 6}$, B.Caron$^{ 28}$, A.A.Carter$^{ 13}$, J.R.Carter$^{ 5}$, C.Y.Chang$^{ 17}$, D.G.Charlton$^{ 1, b}$, I.Cohen$^{ 22}$, A.Csilling$^{ 8, g}$, M.Cuffiani$^{ 2}$, S.Dado$^{ 21}$, G.M.Dallavalle$^{ 2}$, S.Dallison$^{ 16}$, A.De Roeck$^{ 8}$, E.A.De Wolf$^{ 8}$, K.Desch$^{ 25}$, M.Donkers$^{ 6}$, J.Dubbert$^{ 31}$, E.Duchovni$^{ 24}$, G.Duckeck$^{ 31}$, I.P.Duerdoth$^{ 16}$, E.Elfgren$^{ 18}$, E.Etzion$^{ 22}$, F.Fabbri$^{ 2}$, L.Feld$^{ 10}$, P.Ferrari$^{ 12}$, F.Fiedler$^{ 31}$, I.Fleck$^{ 10}$, M.Ford$^{ 5}$, A.Frey$^{ 8}$, A.Fürtjes$^{ 8}$, P.Gagnon$^{ 12}$, J.W.Gary$^{ 4}$, G.Gaycken$^{ 25}$, C.Geich-Gimbel$^{ 3}$, G.Giacomelli$^{ 2}$, P.Giacomelli$^{ 2}$, M.Giunta$^{ 4}$, J.Goldberg$^{ 21}$, E.Gross$^{ 24}$, J.Grunhaus$^{ 22}$, M.Gruwé$^{ 8}$, P.O.Günther$^{ 3}$, A.Gupta$^{ 9}$, C.Hajdu$^{ 29}$, M.Hamann$^{ 25}$, G.G.Hanson$^{ 4}$, K.Harder$^{ 25}$, A.Harel$^{ 21}$, M.Harin-Dirac$^{ 4}$, M.Hauschild$^{ 8}$, J.Hauschildt$^{ 25}$, C.M.Hawkes$^{ 1}$, R.Hawkings$^{ 8}$, R.J.Hemingway$^{ 6}$, C.Hensel$^{ 25}$, G.Herten$^{ 10}$, R.D.Heuer$^{ 25}$, J.C.Hill$^{ 5}$, K.Hoffman$^{ 9}$, R.J.Homer$^{ 1}$, D.Horváth$^{ 29, c}$, R.Howard$^{ 27}$, P.Hüntemeyer$^{ 25}$, P.Igo-Kemenes$^{ 11}$, K.Ishii$^{ 23}$, H.Jeremie$^{ 18}$, P.Jovanovic$^{ 1}$, T.R.Junk$^{ 6}$, N.Kanaya$^{ 26}$, J.Kanzaki$^{ 23}$, G.Karapetian$^{ 18}$, D.Karlen$^{ 6}$, V.Kartvelishvili$^{ 16}$, K.Kawagoe$^{ 23}$, T.Kawamoto$^{ 23}$, R.K.Keeler$^{ 26}$, R.G.Kellogg$^{ 17}$, B.W.Kennedy$^{ 20}$, D.H.Kim$^{ 19}$, K.Klein$^{ 11}$, A.Klier$^{ 24}$, S.Kluth$^{ 32}$, T.Kobayashi$^{ 23}$, M.Kobel$^{ 3}$, T.P.Kokott$^{ 3}$, S.Komamiya$^{ 23}$, L.Kormos$^{ 26}$, R.V.Kowalewski$^{ 26}$, T.Krämer$^{ 25}$, T.Kress$^{ 4}$, P.Krieger$^{ 6, l}$, J.von Krogh$^{ 11}$, D.Krop$^{ 12}$, M.Kupper$^{ 24}$, P.Kyberd$^{ 13}$, G.D.Lafferty$^{ 16}$, H.Landsman$^{ 21}$, D.Lanske$^{ 14}$, J.G.Layter$^{ 4}$, A.Leins$^{ 31}$, D.Lellouch$^{ 24}$, J.Letts$^{ 12}$, L.Levinson$^{ 24}$, J.Lillich$^{ 10}$, S.L.Lloyd$^{ 13}$, F.K.Loebinger$^{ 16}$, J.Lu$^{ 27}$, J.Ludwig$^{ 10}$, A.Macpherson$^{ 28, i}$, W.Mader$^{ 3}$, S.Marcellini$^{ 2}$, T.E.Marchant$^{ 16}$, A.J.Martin$^{ 13}$, J.P.Martin$^{ 18}$, G.Masetti$^{ 2}$, T.Mashimo$^{ 23}$, P.Mättig$^{ m}$, W.J.McDonald$^{ 28}$, J.McKenna$^{ 27}$, T.J.McMahon$^{ 1}$, R.A.McPherson$^{ 26}$, F.Meijers$^{ 8}$, P.Mendez-Lorenzo$^{ 31}$, W.Menges$^{ 25}$, F.S.Merritt$^{ 9}$, H.Mes$^{ 6, a}$, A.Michelini$^{ 2}$, S.Mihara$^{ 23}$, G.Mikenberg$^{ 24}$, D.J.Miller$^{ 15}$, S.Moed$^{ 21}$, W.Mohr$^{ 10}$, T.Mori$^{ 23}$, A.Mutter$^{ 10}$, K.Nagai$^{ 13}$, I.Nakamura$^{ 23}$, H.A.Neal$^{ 33}$, R.Nisius$^{ 8}$, S.W.O’Neale$^{ 1}$, A.Oh$^{ 8}$, A.Okpara$^{ 11}$, M.J.Oreglia$^{ 9}$, S.Orito$^{ 23}$, C.Pahl$^{ 32}$, G.Pásztor$^{ 8, g}$, J.R.Pater$^{ 16}$, G.N.Patrick$^{ 20}$, J.E.Pilcher$^{ 9}$, J.Pinfold$^{ 28}$, D.E.Plane$^{ 8}$, B.Poli$^{ 2}$, J.Polok$^{ 8}$, O.Pooth$^{ 14}$, M.Przybycień$^{ 8, n}$, A.Quadt$^{ 3}$, K.Rabbertz$^{ 8}$, C.Rembser$^{ 8}$, P.Renkel$^{ 24}$, H.Rick$^{ 4}$, J.M.Roney$^{ 26}$, S.Rosati$^{ 3}$, Y.Rozen$^{ 21}$, K.Runge$^{ 10}$, D.R.Rust$^{ 12}$, K.Sachs$^{ 6}$, T.Saeki$^{ 23}$, O.Sahr$^{ 31}$, E.K.G.Sarkisyan$^{ 8, j}$, A.D.Schaile$^{ 31}$, O.Schaile$^{ 31}$, P.Scharff-Hansen$^{ 8}$, J.Schieck$^{ 32}$, T.Schoerner-Sadenius$^{ 8}$, M.Schröder$^{ 8}$, M.Schumacher$^{ 3}$, C.Schwick$^{ 8}$, W.G.Scott$^{ 20}$, R.Seuster$^{ 14, f}$, T.G.Shears$^{ 8, h}$, B.C.Shen$^{ 4}$, C.H.Shepherd-Themistocleous$^{ 5}$, P.Sherwood$^{ 15}$, G.Siroli$^{ 2}$, A.Skuja$^{ 17}$, A.M.Smith$^{ 8}$, R.Sobie$^{ 26}$, S.Söldner-Rembold$^{ 10, d}$, S.Spagnolo$^{ 20}$, F.Spano$^{ 9}$, A.Stahl$^{ 3}$, K.Stephens$^{ 16}$, D.Strom$^{ 19}$, R.Ströhmer$^{ 31}$, S.Tarem$^{ 21}$, M.Tasevsky$^{ 8}$, R.J.Taylor$^{ 15}$, R.Teuscher$^{ 9}$, M.A.Thomson$^{ 5}$, E.Torrence$^{ 19}$, D.Toya$^{ 23}$, P.Tran$^{ 4}$, T.Trefzger$^{ 31}$, A.Tricoli$^{ 2}$, I.Trigger$^{ 8}$, Z.Trócsányi$^{ 30, e}$, E.Tsur$^{ 22}$, A.S.Turcot$^{ 9, o}$, M.F.Turner-Watson$^{ 1}$, I.Ueda$^{ 23}$, B.Ujvári$^{ 30, e}$, B.Vachon$^{ 26}$, C.F.Vollmer$^{ 31}$, P.Vannerem$^{ 10}$, M.Verzocchi$^{ 17}$, H.Voss$^{ 8}$, J.Vossebeld$^{ 8}$, D.Waller$^{ 6}$, C.P.Ward$^{ 5}$, D.R.Ward$^{ 5}$, P.M.Watkins$^{ 1}$, A.T.Watson$^{ 1}$, N.K.Watson$^{ 1}$, P.S.Wells$^{ 8}$, T.Wengler$^{ 8}$, N.Wermes$^{ 3}$, D.Wetterling$^{ 11}$ G.W.Wilson$^{ 16, k}$, J.A.Wilson$^{ 1}$, G.Wolf$^{ 24}$, T.R.Wyatt$^{ 16}$, S.Yamashita$^{ 23}$, V.Zacek$^{ 18}$, D.Zer-Zion$^{ 4}$, L.Zivkovic$^{ 24}$ ]{}
$^{ 1}$School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK $^{ 2}$Dipartimento di Fisica dell’ Università di Bologna and INFN, I-40126 Bologna, Italy $^{ 3}$Physikalisches Institut, Universität Bonn, D-53115 Bonn, Germany $^{ 4}$Department of Physics, University of California, Riverside CA 92521, USA $^{ 5}$Cavendish Laboratory, Cambridge CB3 0HE, UK $^{ 6}$Ottawa-Carleton Institute for Physics, Department of Physics, Carleton University, Ottawa, Ontario K1S 5B6, Canada $^{ 8}$CERN, European Organisation for Nuclear Research, CH-1211 Geneva 23, Switzerland $^{ 9}$Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago IL 60637, USA $^{ 10}$Fakultät für Physik, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany $^{ 11}$Physikalisches Institut, Universität Heidelberg, D-69120 Heidelberg, Germany $^{ 12}$Indiana University, Department of Physics, Swain Hall West 117, Bloomington IN 47405, USA $^{ 13}$Queen Mary and Westfield College, University of London, London E1 4NS, UK $^{ 14}$Technische Hochschule Aachen, III Physikalisches Institut, Sommerfeldstrasse 26-28, D-52056 Aachen, Germany $^{ 15}$University College London, London WC1E 6BT, UK $^{ 16}$Department of Physics, Schuster Laboratory, The University, Manchester M13 9PL, UK $^{ 17}$Department of Physics, University of Maryland, College Park, MD 20742, USA $^{ 18}$Laboratoire de Physique Nucléaire, Université de Montréal, Montréal, Quebec H3C 3J7, Canada $^{ 19}$University of Oregon, Department of Physics, Eugene OR 97403, USA $^{ 20}$CLRC Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK $^{ 21}$Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel $^{ 22}$Department of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel $^{ 23}$International Centre for Elementary Particle Physics and Department of Physics, University of Tokyo, Tokyo 113-0033, and Kobe University, Kobe 657-8501, Japan $^{ 24}$Particle Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel $^{ 25}$Universität Hamburg/DESY, Institut für Experimentalphysik, Notkestrasse 85, D-22607 Hamburg, Germany $^{ 26}$University of Victoria, Department of Physics, P O Box 3055, Victoria BC V8W 3P6, Canada $^{ 27}$University of British Columbia, Department of Physics, Vancouver BC V6T 1Z1, Canada $^{ 28}$University of Alberta, Department of Physics, Edmonton AB T6G 2J1, Canada $^{ 29}$Research Institute for Particle and Nuclear Physics, H-1525 Budapest, P O Box 49, Hungary $^{ 30}$Institute of Nuclear Research, H-4001 Debrecen, P O Box 51, Hungary $^{ 31}$Ludwig-Maximilians-Universität München, Sektion Physik, Am Coulombwall 1, D-85748 Garching, Germany $^{ 32}$Max-Planck-Institute für Physik, Föhringer Ring 6, D-80805 München, Germany $^{ 33}$Yale University, Department of Physics, New Haven, CT 06520, USA $^{ a}$ and at TRIUMF, Vancouver, Canada V6T 2A3 $^{ b}$ and Royal Society University Research Fellow $^{ c}$ and Institute of Nuclear Research, Debrecen, Hungary $^{ d}$ and Heisenberg Fellow $^{ e}$ and Department of Experimental Physics, Lajos Kossuth University, Debrecen, Hungary $^{ f}$ and MPI München $^{ g}$ and Research Institute for Particle and Nuclear Physics, Budapest, Hungary $^{ h}$ now at University of Liverpool, Dept of Physics, Liverpool L69 3BX, UK $^{ i}$ and CERN, EP Div, 1211 Geneva 23 $^{ j}$ and Universitaire Instelling Antwerpen, Physics Department, B-2610 Antwerpen, Belgium $^{ k}$ now at University of Kansas, Dept of Physics and Astronomy, Lawrence, KS 66045, USA $^{ l}$ now at University of Toronto, Dept of Physics, Toronto, Canada $^{ m}$ current address Bergische Universität, Wuppertal, Germany $^{ n}$ and University of Mining and Metallurgy, Cracow, Poland $^{ o}$ now at Brookhaven National Laboratory, Upton, NY 11973, USA
Introduction {#sec:intro}
============
This paper presents the results of two types of search for the production of a di-photon system recoiling from another massive scalar or vector object. The searches are sensitive to the processes ${{\mathrm e}^+ {\mathrm e}^-}{\rightarrow}\mrm {X Y}$, with $\mrm X {\rightarrow}{\gamma\gamma}$ and $\mrm Y {\rightarrow}{{\mathrm f} \bar{\mathrm f}}$, where ${{\mathrm f} \bar{\mathrm f}}$ is a hadronic system (jets), a pair of charged leptons, or neutrinos resulting in missing energy. In the [[*general*]{}]{} search mode $\mrm X$ must be a scalar, $\mrm Y$ can be any scalar or vector particle of any mass, and both particles must be short-lived so that they decay close to the interaction point. The other search mode is referred to as the search; it requires $\mrm Y$ to be a boson and is applied to data taken at all energies. The data used for these searches were recorded by the OPAL detector at centre-of-mass energies () $88-209$ GeV, the entire energy range achieved at LEP.
These searches are largely motivated by “fermiophobic” scenarios where one of the Higgs bosons decays primarily into a boson pair. In the fermiophobic interpretation, Y would be a and X a Higgs boson decaying into two photons. Indeed, the Higgs boson predicted in the Standard Model decays into two photons via a quark- or W-boson loop [@HBR], but with a rate too low for observation of the process at LEP luminosities. Processes ${{\mathrm e}^+ {\mathrm e}^-}{\rightarrow}{{\mathrm h}^{0}}{{\mathrm Z}^{0}}{\rightarrow}{\gamma\gamma}{{\mathrm f} \bar{\mathrm f}}$ at near-[[Standard Model]{}]{} production rate and having large di-photon branching ratios have been predicted in a number of alternative theories [@Hagiwara; @Bosonic; @Akeroyd; @Santos; @Gunion]; here refers to the lightest neutral boson where extended Higgs sector models are discussed. A particularly natural situation for fermiophobic Higgs bosons occurs in two Higgs doublet models (2HDM) [@TypeI] of “Type-I”, where one Higgs doublet couples only to bosons. Because there are different fermiophobic models, it is not possible to present search results for the entire parameter space of the various theories. In the present paper a benchmark fermiophobic model is defined as having [[Standard Model]{}]{}production strength and a Higgs boson di-photon branching fraction calculated by turning off the fermion couplings to the Higgs boson in the [[Standard Model]{}]{}. The OPAL Collaboration has presented searches similar to those reported here for LEP energies up to = 189 GeV [@189paper; @183paper; @172paper; @OPAL_ggjj_1]; this paper extends those searches with the addition of data taken at = $192-209$ GeV. Fermiophobic Higgs boson searches have also been presented by other LEP collaborations [@ALEPHpaper; @Delphipaper; @L3paper] and by hadron collider experiments [@D0paper; @CDFpaper]. To date, no evidence of a fermiophobic Higgs boson has been seen.
Data, Simulated Backgrounds and Signals
=======================================
The data used in this analysis were recorded using the OPAL detector [@detector] at LEP. The 1999 data consisted of $217.0\pm0.7$ collected at = $192-202$ GeV. The 2000 LEP data consisted of $211.1\pm0.8$ collected at = $200-209$ GeV, with the majority of the data taken at 205 and 207 GeV. The data sets are summarized in Table \[T:lumi\].
The backgrounds from Standard Model processes were modelled using Monte Carlo simulations at $\sqrt{s}=192$, 196, 202, and 206 GeV for the 1999 and 2000 data. Simulated events were processed using the full OPAL detector Monte Carlo [@GOPAL] and analysed in the same manner as the data. The full-detector simulations were reweighted for the $\sqrt{s}$ distribution of the data using the Monte Carlo generators.
The dominant background to this search arises from the emission of two energetic initial state radiation (ISR) photons. This process was simulated using the KK2f/CEEX [@CEEX] generator with hadronisation and fragmentation by PYTHIA 6.125 [@PYTHIA]. The CEEX modelling of ISR employs full second-order QED corrections to the matrix element, and applies coherent exponentiation of the QED corrections from interference between ISR and final-state radiation. Four-fermion processes were modelled using the grc4f [@grc4f] and KORALW [@KORALW] generators. Two-lepton final states were simulated using BHWIDE [@BHWIDE], TEEGG [@TEEGG] and KORALZ [@KORALZ]. The NUNUGPV [@nunugpv] program was used to generate events of the type ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\rightarrow}{\nu \bar{\nu}}\gamma\gamma(\gamma)$. The process ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\rightarrow}\gamma\gamma$ was simulated using the RADCOR generator [@RADCOR]. Tau lepton decays were modelled using Tauola 2.4 [@tauola].
The process ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\rightarrow}{\mbox{$\mathrm{h}^{0}$}}{\mbox{$\mathrm{Z}^{0}$}}$, ${\mbox{$\mathrm{h}^{0}$}}{\rightarrow}\gamma\gamma$ was simulated for each decay channel using the HZHA3 generator [@HZHA3]. For the [[*general*]{}]{} search, which is applied to the data taken in 2000 only, the role of the was replaced by scalar or vector particles having masses from $10-200$ GeV. Efficiencies for signals were estimated by generating Monte Carlo for both scalar and vector signals in mass steps of 5 GeV; the scalar and vector efficiencies agreed within systematic errors and therefore are not treated as separate cases.
Event Selection
===============
The analysis described in the following is identical to the one used in the paper for OPAL data taken at 189 GeV [@189paper]. Slightly different analysis cuts were used on the lower energy data sets, as described in the earlier publications [@183paper; @172paper; @OPAL_ggjj_1].
Events were selected if there were at least two photons recoiling from some other system decaying into one of the following three topologies:
- a ${{\mathrm q}\bar{\mathrm q}}$ pair (“Hadronic Channel”), or
- one or two charged leptons (“Leptonic Channel”), or
- a ${\nu \bar{\nu}}$ pair (“Missing Energy Channel”).
Photons were identified as clusters in the electromagnetic calorimeter (EC) which were not associated with tracks if the lateral spread of the clusters satisfied the criteria described in reference [@172paper]. The efficiencies were increased by approximately $10-20$% by including photon conversions into pairs using the methods described in reference [@183paper].
The dominant background to the searches arises from ISR producing mostly low-energy photons along the beam direction. Therefore, we required the two highest-energy photons in the event to satisfy the following:
- The two photon candidates were required to be in the fiducial region $|\cos(\theta_{\gamma})| < 0.875$, where the polar angle $\theta_{\gamma}$ is the angle of the photon with respect to the $\mrm e^-$ beam direction.
- The highest-energy photon was required to have ${\mbox{$E_{\gamma1}$}}/{E_{\mathrm{beam}}}> 0.10$ and the second-highest-energy photon was required to have ${\mbox{$E_{\gamma2}$}}/{E_{\mathrm{beam}}}> 0.05$.
- The sum of track momenta and extra electromagnetic cluster energies in a 15 degree cone about the photons had to be less than 2 GeV.
The remaining cuts depend on the particular recoil topologies. In order to assess the background modelling, the photon cuts are not applied until after preselection cuts for the three final state topologies. For all topologies, tracks and EC clusters that are not associated to tracks are required to satisfy the criteria defined in reference [@CTSEL]. The criteria for the definition of tracks and EC clusters in the OPAL detector are described in reference [@Zedometry].
Hadronic Channel {#s:qqgg}
----------------
The hadronic channel is characterised by two photons recoiling against a hadronic system. Candidate events were required to satisfy the following criteria:
- The standard OPAL hadronic event preselection in Ref. [@hadsel]; ${\mbox{$R_{\mathrm{vis}}$}}> 0.5$; $|\Sigma~p_{\mrm{z}}^{\mrm{vis}}| < 0.6 {E_{\mathrm{beam}}}$; and at least two electromagnetic clusters with $E/{E_{\mathrm{beam}}}> 0.05$. The quantities ${\mbox{$E_{\mathrm{vis}}$}}$ and $\vec{p}_{\mrm{vis}}$ are the scalar and vector sums of track momenta, unassociated EC and unassociated hadron calorimeter cluster energies, and ${\mbox{$R_{\mathrm{vis}}$}}\equiv \frac{\mbox{{\mbox{$E_{\mathrm{vis}}$}}}}{{\mbox{$E_{\mathrm{cm}}$}}}$. The visible momentum along the beam direction, obtained from the sum of all tracks and unassociated clusters, is denoted by $|\Sigma~p_{\mrm{z}}^{\mrm{vis}}|$.
- The photon pair criteria G1$-$G3.
- Photon isolation: both photon candidates were required to satisfy $p_{\mrm {T,~jet}-\gamma} > 5$ GeV, where $p_{\mrm {T,~jet}-\gamma}$ is the photon momentum transverse to the axis of the closest jet out of two jets formed with the Durham [@Durham] scheme (excluding the photon pair).
- Photon energy balance: $(E_{\gamma1}-E_{\gamma2})/E_{\mrm{o}} < 0.5$, where $E_{\mrm{o}} \equiv (s - {M_{\mathrm Z}}^{2})/(2\sqrt{s})$ would be the energy of a single photon recoiling from the . This cut discriminates against ISR photon pairs.
- The recoil mass from the di-photon system, ${M_{\mrm{recoil}}}$, is required to be consistent with the ${\mbox{$\mathrm{Z}^{0}$}}$: $|{M_{\mrm{recoil}}}- {M_{\mathrm Z}}| < 20$ GeV (not used in the [[*general*]{}]{} search mode).
Charged Lepton Channel {#s:llgg}
----------------------
This channel searches for events in the ${\gamma\gamma}{\ell^+ \ell^-}$ final state. Events having only one well-identified lepton are accepted to avoid efficiency loss for lepton tracks at low polar angles. The lepton tracks are treated as jets to include tau lepton final state topologies. Leptonic channel candidates were required to satisfy the following criteria:
- The standard OPAL low multiplicity preselection of Ref. [@lowmsel]; ${\mbox{$R_{\mathrm{vis}}$}}>0.2$; $|\Sigma~p_{\mrm{z}}^{\mrm{vis}}|<0.8 {E_{\mathrm{beam}}}$; number of EC clusters not associated with tracks $\leq 10$; number of tracks $N_{\rm T}$ satisfies $1 \leq N_{\rm T} \leq 7$; at least two electromagnetic clusters with $E/{E_{\mathrm{beam}}}> 0.05$.
- The photon pair criteria G1$-$G3.
- For events having only one track and a converted photon, the EC cluster associated with the track must not also be associated with the conversion.
- For events having two or more tracks, the event is forced to have two jets within the Durham scheme, excluding the identified di-photon candidate, and both jets are required to have energies above 3 GeV.
- $|{M_{\mrm{recoil}}}- {M_{\mathrm Z}}| < 20$ GeV (not used in the [[*general*]{}]{} search mode).
Missing Energy Channel {#s:nngg}
----------------------
The missing energy channel is characterised by two photons and no other significant detector activity. Candidates in the missing energy channel were required to satisfy the following criteria:
- The standard OPAL low multiplicity preselection of Ref. [@lowmsel]; the vetoes in Ref. [@photsel] against cosmic ray and beam-wall/beam-gas backgrounds; number of EC clusters not associated with tracks $\leq 4$; number of tracks $\leq 3$; $|\Sigma~p_{\mrm{z}}^{\mrm{vis}}|<0.8 {E_{\mathrm{beam}}}$; and at least two electromagnetic clusters with $E/{E_{\mathrm{beam}}}> 0.05$.
- The photon pair criteria G1$-$G3.
- $p_T ({\gamma\gamma})>0.05 {E_{\mathrm{beam}}}$ where $p_T ({\gamma\gamma})$ is the transverse momentum of the di-photon system; the angle between the two photons in the plane transverse to the beam axis: $|\phi_{{\gamma\gamma}}-180{^\circ}| > 2.5{^\circ}$; the polar angle of the momentum of the di-photon system: $|\cos\theta_{{\gamma\gamma}}| < 0.966$.
- No track candidates other than those associated with an identified photon conversion.
- Veto on unassociated calorimeter energy: the energy observed in the EC not associated with the two photons is required to be less than 3 GeV.
- $|{M_{\mrm{recoil}}}- {M_{\mathrm Z}}| < 20$ GeV (not used in the [[*general*]{}]{} search mode).
Results {#s:results}
=======
For the 1999 and 2000 data, the numbers of events passing the cuts are listed for the three recoil topologies in Table \[T:gg1999-2000\]. There are no events in which more than one photon pair satisfying the cuts was found. The numbers of candidates passing cuts are generally in good agreement with the expected numbers of Standard Model backgrounds; this was also the case in earlier OPAL publications for the lower [@189paper; @183paper; @172paper; @OPAL_ggjj_1]. The one noteworthy discrepancy is for cut C1 in the missing energy channel. In this channel there is a large background from Bhabha electrons lost in the beampipe. The ISR photons for this background have a steeply rising population in the forward direction. Cut C1 is made before the cut on polar angle, and therefore a discrepancy arises because of the steep angular distribution and the inadequate modelling of material in the very low polar angle regions.
Combining both the 1999 and 2000 data in the three topologies, 112 candidates pass the [[*general*]{}]{} cuts compared to 118.3$\pm$7.9 expected background, and 42 candidates pass the cuts compared to 51.9$\pm$2.9 expected background.
Systematic Errors
-----------------
The uncertainty on the modelling of ISR is the most important component of the systematic error because of the irreducible background arising from this process. This uncertainty is estimated from the comparison of data with the [[Standard Model]{}]{} background simulation for events passing cuts A2, B2, or C2. The shapes of the distributions for and are modelled well by the simulations. The simulations also reproduce well the number of events observed in the three channels combined. For the 1999 and 2000 runs the simulations predict 3.7% and 5.0% fewer events than observed, respectively. The statistical error on the 1999 data is 4%, and similarly for the 2000 data. Modelling of photon conversions has an uncertainty of approximately 1%. Uncertainties on the integrated luminosities of the data sets are negligible compared to the other uncertainties. Combined, these error sources result in a total background uncertainty estimate of 10%. The experimental results which follow are not very sensitive to this number.
The dominant systematic uncertainty for the signal acceptances arises from the photon detection efficiency, primarily due to the simulation of the photon isolation criterion G3 [@OPAL_ggjj_1], and is estimated to be 3%. The uncertainty from Monte Carlo statistics is typically better than 4%. A systematic error on the photon energy scale is estimated by comparing the fitted single-photon ISR energy peak to the expected value based on the precisely known beam energy and mass. For this study a sample of single-photon events was generated and compared to the data for photon energies above 5 GeV and polar angles greater than 25 degrees. This leads to a systematic uncertainty on the di-photon mass of 0.35 GeV at a mass of 100 GeV. The resolution on the di-photon mass ranges from approximately 0.5 GeV at [$m_{\gamma \gamma}$]{} = 10 GeV to 2.3 GeV at [$m_{\gamma \gamma}$]{} = 100 GeV.
General Search Results
----------------------
Figure \[COMGG2\] shows the di-photon mass versus the recoil mass for all candidate events passing the [[*general*]{}]{} search cuts for the year 2000 data only (where all the data were taken at near 206 GeV). The events at recoil masses near zero are expected from ${{\mathrm e}^+ {\mathrm e}^-}{\rightarrow}{\gamma\gamma}$ background. This plot also shows no unexpected structure for the lower data. In the absence of an indication for signal, limits are placed on the production at $\sim$206 GeV of a massive state decaying into photon pairs.
For the [[*general*]{}]{} search, the system recoiling from the di-photon system is not assumed to be a and hence the branching fractions ${\mrm X} {\rightarrow}{\gamma\gamma}$ and Y into topology A, B or C are not uniquely predicted. Here X is a scalar particle and $\mrm Y$ is a scalar or vector particle. Furthermore, X and Y must be a short-lived particles so that they decay near the interaction point. In order to be independent of models we do not combine data from different and therefore we restrict this part of the analysis to the highest energy data, in the range of $205-207$ GeV; this represents 200.0 of the 2000 data. We choose to present upper limits on $\sigma({{\mathrm e}^+ {\mathrm e}^-}{\rightarrow}\mrm {X Y})\times B({\mrm X} {\rightarrow}{\gamma\gamma})\times B(\mrm Y {\rightarrow}{{\mathrm f} \bar{\mathrm f}})$ as a function of ${M_{\mathrm{X}}}$. When presenting production upper limits as functions of ${M_{\mathrm{X}}}$, we show the limit obtained for the value of ${M_{\mathrm{Y}}}$ that gives the smallest efficiency in the region ${M_{\mathrm Z}}- {M_{\mathrm{X}}}< {M_{\mathrm{Y}}}< \mrm{E}_{\mrm{cm}} - {M_{\mathrm{X}}}$. The lower bounds on ${M_{\mathrm{Y}}}$ are used because searches for di-photon resonances at LEP1 [@OPAL_ggjj_1; @DelphiLEP1; @L3LEP1] have already set good limits on the lower-mass phase space. ${M_{\mathrm{X}}}$ and ${M_{\mathrm{Y}}}$ are also required to be above 10 GeV and below 200 GeV in order to allow the decay products to have sufficient energies and momenta to give reasonable search acceptances at = 206 GeV. For a scalar/vector hypothesis for X/Y, the efficiency is found to be the same to within 5% as that for a scalar/scalar hypothesis; the lower of these efficiencies is used in setting the limits. For the lepton search channel, the efficiency for Y ${\rightarrow}{\mbox{$\tau^+\tau^-$}}$ is used, as it turns out to have the lowest of the dilepton efficiencies.
The event candidates from = $205-207$ GeV in the [[*general*]{}]{} search are used to calculate 95% CL upper limits on the number of events in 1 GeV \[${M_{\mathrm{X}}},{M_{\mathrm{Y}}}$\] mass bins. The acceptances used at each 1 GeV mass bin are obtained by interpolation using a 4th-order polynomial fit to the acceptances simulated on a 5 GeV grid. For each \[${M_{\mathrm{X}}},{M_{\mathrm{Y}}}$\] bin, the 95% CL upper limit on the number of signal events is computed using the frequentist method of reference [@JUNK], which takes into account the predicted [[Standard Model]{}]{} background. This statistical procedure also incorporates the di-photon mass resolution (typically less than 2 GeV for [$m_{\gamma \gamma}$]{}$<$100 GeV); the limit procedure is valid for resonance states narrower than this resolution. The effect of the 10% systematic error for background modelling is incorporated in the statistical procedure, as is the 4% uncertainty on signal. For these [[*general*]{}]{} search limits, an additional systematic uncertainty of 5% is added to the signal uncertainty to account for interpolation error in the efficiency grid (especially near kinematic limits) and for the differences in the acceptance calculations for the scalar versus vector nature of particle Y.
Figure \[limxy\] shows the 95% CL upper limits on $\sigma({{\mathrm e}^+ {\mathrm e}^-}{\rightarrow}\mrm {X Y})\times B({\mrm X} {\rightarrow}{\gamma\gamma})\times
B(\mrm Y {\rightarrow}{{\mathrm f} \bar{\mathrm f}})$. These results are valid independent of the nature of Y, provided it decays to two jets, a lepton pair, or missing energy, and has a width less than or equal to the experimental resolution. Limits of $25-60$ fb are obtained over $10 < {M_{\mathrm{X}}}< 180$ GeV. The limits for the leptonic final state are stronger, except in the case Y couples exclusively to (the final state with lowest acceptance).
Limits on with ${\rightarrow}{\gamma\gamma}$
---------------------------------------------
The distribution of di-photon masses for the search candidates for the 1999 data ( = $192-202$ GeV) and the 2000 data ( = $200-209$ GeV) is shown in Figure \[COMGG\]a together with the simulation of Standard Model backgrounds. The observation of 42 events is in reasonable agreement with the expected background of 51.9$\pm$2.9 events. Because the process has a production rate and branching fractions described by theory, the data taken at all LEP energies can be combined in this analysis. Figure \[COMGG\]b shows the distribution of [$m_{\gamma \gamma}$]{} for = $88-209$ GeV. This plot is restricted to [$m_{\gamma \gamma}$]{} larger than 20 GeV because there is no background estimate for the low-[$m_{\gamma \gamma}$]{} LEP1 data. The figure has no indication of a resonance, and the total of 124 candidates agrees with the predicted background of 135.2$\pm$10.8 events.
Also indicated on Figure \[COMGG\] is the hypothetical signal of a 100 GeV Higgs boson produced at [[Standard Model]{}]{} strength and decaying into di-photons with a branching fraction of 18% – the fraction predicted in the “benchmark fermiophobic model” calculated by simply turning off the Higgs-fermion coupling. In reality, the fermiophobic Higgs photon branching ratio depends on parameters and details of fermiophobic 2HDM models [@Akeroyd; @Santos], so this benchmark is simply a guide to the broad interpretation of the data. Here we use the HDECAY [@HDECAY] package to calculate the modified photonic branching fractions.
The events passing all cuts are used to set an upper limit on the di-photon branching ratio for a particle produced in association with a and having the Standard Model Higgs boson production rate. As described in the previous section, the frequentist method of reference [@JUNK] is used to determine the 95% confidence level upper limit on possible signal events at each di-photon mass. Figure \[bgglim\] shows the 95% CL upper limit for the di-photon branching ratio obtained by combining the candidate events in 1999 and 2000 data described in this paper with those from OPAL searches at $\sqrt{s}=88-189$ GeV [@189paper; @OPAL_ggjj_1; @183paper], where the Standard Model production cross-section is assumed at each centre-of-mass energy. Higgs bosons produced at [[Standard Model]{}]{} rate and decaying exclusively to di-photons are ruled out at the 95% confidence level over the mass range $20-117$ GeV. Figure \[bgglim\] also shows the ${\mbox{$\mathrm{h}^{0}$}}{\rightarrow}{\gamma\gamma}$ branching ratio computed using HDECAY with the fermionic couplings switched off; the photonic branching fraction falls as the ${{\mathrm W}^+{\mathrm W}^-}$ and ${{\mathrm Z}^{0}{\mathrm Z}^{0}}$ channels become kinematically favourable. A 95% CL lower mass limit for the benchmark fermiophobic Higgs bosons is set at 105.5 GeV, where the predicted branching ratio crosses the upper-limit curve. The median limit one would expect to obtain in an ensemble of experiments in the absence of a signal is 106.4 GeV. The benchmark fermiophobic branching ratios can also be calculated using the HZHA3 [@HZHA3] generator. HZHA3 produces slightly higher di-photon branching fractions than does HDECAY. The lower mass limit on the benchmark fermiophobic Higgs boson calculated with HZHA3 is 106.3 GeV.
Conclusions
===========
A search for the production of Higgs bosons and other new particles of width no larger than the experimental resolution and decaying to photon pairs has been performed using annihilation data with = $192-209$ GeV combined with $88-189$ GeV data from previous OPAL searches. Model independent upper limits are obtained for $\sim$206 GeV on $\sigma({{\mathrm e}^+ {\mathrm e}^-}{\rightarrow}\mrm {X Y})\times B({\mrm X} {\rightarrow}{\gamma\gamma})\times B(\mrm
Y {\rightarrow}{{\mathrm f} \bar{\mathrm f}})$, where limits of $25-60$ fb are obtained over $10 < {M_{\mathrm{X}}}< 180$ GeV, for $10 < {M_{\mathrm{Y}}}< 200$ GeV and ${M_{\mathrm{X}}}+ {M_{\mathrm{Y}}}> {M_{\mathrm Z}}$. The limits are valid for Y either a scalar or vector particle, provided that the Y decays to a fermion pair (interpreted as two jets, a lepton pair, or missing energy).
Using OPAL data from all LEP centre-of-mass energies, model-specific limits are placed on $B$(${{\mathrm h}^{0}}{\rightarrow}{\gamma\gamma}$) up to a Higgs boson mass of 117 GeV, provided the Higgs particle is produced via ${{\mathrm e}^+ {\mathrm e}^-}{\rightarrow}{{\mathrm h}^{0}}{{\mathrm Z}^{0}}$ at the Standard Model rate. A lower mass bound of 105.5 GeV is set at the 95% confidence level for benchmark fermiophobic Higgs bosons. Similar lower mass limits on benchmark fermiophobic Higgs bosons have been obtained by the other LEP experiments [@ALEPHpaper; @Delphipaper; @L3paper].
Acknowledgements:
We thank A. G. Akeroyd, L. Brücher, and R. Santos for helpful discussions. We particularly wish to thank the SL Division for the efficient operation of the LEP accelerator at all energies and for their close cooperation with our experimental group. In addition to the support staff at our own institutions we are pleased to acknowledge the\
Department of Energy, USA,\
National Science Foundation, USA,\
Particle Physics and Astronomy Research Council, UK,\
Natural Sciences and Engineering Research Council, Canada,\
Israel Science Foundation, administered by the Israel Academy of Science and Humanities,\
Benoziyo Center for High Energy Physics,\
Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and a grant under the MEXT International Science Research Program,\
Japanese Society for the Promotion of Science (JSPS),\
German Israeli Bi-national Science Foundation (GIF),\
Bundesministerium für Bildung und Forschung, Germany,\
National Research Council of Canada,\
Hungarian Foundation for Scientific Research, OTKA T-029328, and T-038240,\
Fund for Scientific Research, Flanders, F.W.O.-Vlaanderen, Belgium.\
[99]{}
J. Ellis, M.K. Gaillard, and D.V. Nanopoulos, [Nucl. Phys.]{} [**B106**]{} (1976) 292.
K. Hagiwara and M.L. Stong, [Z. Phys. [**C62**]{}]{} (1994) 99.
A. Stange, W. Marciano and S. Willenbrock, [Phys. Rev.]{} [**D49**]{} (1994) 1354.
A.G. Akeroyd, [Phys. Lett. [**B368**]{}]{} (1996) 89.
L. Brücher, R. Santos, [Eur. Phys. J. [**C12**]{}]{} (2000) 87.
J.F. Gunion, R. Vega and J. Wudka, [Phys. Rev.]{} [**D42**]{} (1990) 1673.
H. Haber, G. Kane and T. Sterling, Nucl. Phys. [**B161**]{} (1979) 493.
[OPAL Collab.]{}, G. Abbiendi , [Phys. Lett. [**B464**]{}]{} (1999) 311.
[OPAL Collab.]{}, K. Ackerstaff , [Phys. Lett. [**B437**]{}]{} (1998) 218.
[OPAL Collab.]{}, K. Ackerstaff , [Eur. Phys. J. [**C1**]{}]{} (1998) 31.
[OPAL Collab.]{}, P. Acton , [Phys. Lett.]{} [**311**]{} (1993) 391.
ALEPH Collab., CERN-EP-2002-044, submitted to Phys. Lett. B.
DELPHI Collab., P. Abreu , [Phys. Lett. [**B507**]{}]{} (2001) 89.
L3 Collab., P. Achard , [Phys. Lett. [**B534**]{}]{} (2002) 28. D0 Collab., B. Abbott , [Phys. Rev. Lett.]{} [**82**]{} (1999) 2244.
CDF Collab., T. Affolder , [Phys. Rev.]{}[**D64**]{} (2001) 092002.
[OPAL Collab.]{}, K. Ahmet , [ [**A[305]{}**]{}]{} (1991) 275;\
O. Biebel , [ [**A[323]{}**]{}]{} (1992) 169;\
M. Hauschild , [ [**A[314]{}**]{}]{} (1992) 74;\
S. Anderson , [ [**A[403]{}**]{}]{} (1998) 326.
J. Allison , [ [**A[305]{}**]{}]{} (1992) 47.
S. Jadach, B.F. Ward and Z. Was, [Phys. Rev.]{} [**D63**]{} (2001) 113009.
T. Sjöstrand , [Comp. Phys. Comm. [**135**]{}]{} (2001) 238.
J. Fujimoto et al., [Comp. Phys. Comm. [**100**]{}]{} (1997) 128. M. Skrzypek , [Comp. Phys. Comm. [**94**]{}]{} (1996) 216;M. Skrzypek , [Phys. Lett. [**B372**]{}]{} (1996) 289.
S. Jadach, W. Placzek and B. F. L. Ward,[Phys. Lett. [**B390**]{}]{} (1997) 298. D. Karlen, [Nucl. Phys.]{} [**B289**]{} (1987) 23.
S. Jadach et al., [Comp. Phys. Comm. [**66**]{}]{} (1991) 276.
G. Montagna , [Nucl. Phys.]{} [**B541**]{} (1999) 31.
F.A. Berends and R. Kleiss, [Nucl. Phys.]{} [**B186**]{} (1981) 22.
S. Jadach, Z. Was, [Comp. Phys. Comm. [**76**]{}]{} (1993) 361. P. Janot, in [*Physics at LEP2*]{}, edited by G. Altarelli, T. Sjöstrand and F. Zwirner, CERN 96-01 Vol. 2 p.309.\
For version 3, see [*http://alephwww.cern.ch/$\sim$janot/Generators.html*]{}.
[OPAL Collab.]{}, G. Alexander , [Z. Phys. [**C72**]{}]{} (1996) 191.
[OPAL Collab.]{}, G. Abbiendi , [Eur. Phys. J. [**C19**]{}]{} (2001) 587.
[OPAL Collab.]{}, G. Alexander , [Z. Phys. [**C52**]{}]{} (1991) 175.
N. Brown and W.J. Stirling, Phys. Lett. [**B252**]{} (1990) 657;\
S. Bethke, Z. Kunszt, D. Soper and W.J. Stirling, Nucl. Phys. [**B370**]{} (1992) 310;\
S. Catani , Phys. Lett. [**B269**]{} (1991) 432;\
N. Brown and W.J. Stirling, Z. Phys. [**C53**]{} (1992) 629.
[OPAL Collab.]{}, R. Akers , [Z. Phys. [**C61**]{}]{} (1994) 19.
[OPAL Collab.]{}, R. Akers , [Z. Phys. [**C65**]{}]{} (1995) 47.
DELPHI Collab., P. Abreu , [Phys. Lett. [**B458**]{}]{} (1999) 431;\
DELPHI Collab., P. Abreu , [Z. Phys. [**C72**]{}]{} (1996) 179.
L3 Collab., M. Acciarri , [Phys. Lett.]{} [**388**]{} (1996) 409;\
L3 Collab., O. Adriani , [Phys. Lett.]{} [**295**]{} (1992) 337.
T. Junk, [ [**A[434]{}**]{}]{} (1999) 435.
A. Djouadi, J. Kalinowski and M. Spira, [Comp. Phys. Comm. [**108**]{}]{} (1998) 56.
Run (year) Integrated Luminosity () (GeV)
------------ -------------------------- -----------
1990-95 173.00 $88-94$
1995 5.41 $130-140$
1996 10.32 $172.3$
1996 10.04 $161.3$
1997 57.73 $182.6$
1998 182.61 188.6
1999 28.90 191.6
1999 74.79 195.6
1999 77.21 199.6
1999 36.08 201.6
2000 0.82 $200-202$
2000 2.62 $202-204$
2000 76.81 $204-206$
2000 123.26 $206-208$
2000 7.54 $208-210$
: Summary of all data sets used in the searches. Results using data from $1990-1998$ were reported in earlier publications [@189paper; @183paper; @172paper; @OPAL_ggjj_1].[]{data-label="T:lumi"}
[|l||r||r|r|r|r|r|r||r|]{} Cut & Data & & & & &\
\
(A1) & 10645 & 10695.4 & 7535.0 & 3160.2 &&70\
(A2) & 62 & 56.1 & 53.6 & 2.6 &&62\
(A3) & 48 & 44.8 & 42.5 & 2.3 &&61\
(A4) & 29 & $22.0\pm1.8$ & 19.8 & 2.2 &&61\
(A5) & 15 & $10.9\pm0.9$ & 10.9 & 0.0 &&60\
\
(A1) & 9371 & 9152.4 & 6096.0 & 3056.2 &&71\
(A2) & 52 & 46.5 & 43.5 & 3.0 &&60\
(A3) & 39 & 38.3 & 36.0 & 2.3 &&60\
(A4) & 18 & $17.8\pm1.6$ & 15.7 & 2.1 &&58\
(A5) & 7 & $7.9\pm0.7$ & 7.8 & 0.1 &&57\
\
Cut & Data & & & & & & &\
\
(B1) & 41947 & 39060.6 & 37245.9 & 715.6 & 50.5 & 314.6 & 734.0 &82\
(B2) & 167 & 188.6 & 72.4 & 11.3 & 8.0 & 95.5 & 1.5 &69\
(B3) & 155 & 178.8 & 67.3 & 10.4 & 7.4 & 92.4 & 1.2 &63\
(B4) & 23 & $31.4\pm5.8$& 18.8 & 5.0 & 6.9 & 0.4 & 0.3 &50\
(B5) & 5 &$9.2\pm1.7$ & 4.6 & 1.7 & 2.9 & 0.0 & 0.0 &48\
\
(B1) & 37432 & 33928.9 & 32329.1 & 607.8 & 43.7 & 273.9 & 674.4 &82\
(B2) & 138 & 146.6 & 57.6 & 10.2 & 6.8 & 70.9 & 1.0 &71\
(B3) & 123 & 141.7 & 55.4 & 9.2 & 6.4 & 69.7 & 1.0 &67\
(B4) & 28 & $23.7\pm4.8$ & 13.0 & 4.4 & 5.9 & 0.2 & 0.3 &61\
(B5) & 11 &$9.8\pm2.0$ & 5.1 & 1.9 & 2.6 & 0.0 & 0.0 &52\
\
Cut & Data & & & & & & &\
\
(C1) & 224989 & 129713.4 & 50.5 & 3337.7 & 124769.9 & 157.4 & 1397.9 &88\
(C2) & 377 & 336.3 & 13.0 & 276.4 & 45.1 & 1.1 & 0.7 &75\
(C3) & 71 & 73.1 & 12.2 & 31.5 & 27.8 & 1.0 & 0.5 &69\
(C4) & 33 & 42.4 & 12.1 & 29.2 & 0.9 & 0.0 & 0.2 &69\
(C5) & 8 & $12.9\pm0.5$ & 11.4 & 0.3 & 0.9 & 0.0 & 0.2 &67\
(C6) & 3 &$7.6\pm0.3$ & 7.5 & 0.0 & 0.0 & 0.0 & 0.1 &66\
\
(C1) & 202649 & 113268.8 & 43.9 & 2737.6 & 109070.9& 136.8 &1279.7 &87\
(C2) & 345 & 315.1 & 11.5 & 267.5 & 34.6 & 0.9 & 0.5 &75\
(C3) & 66 & 62.1 & 11.0 & 30.1 & 19.7 & 0.8 & 0.3 &70\
(C4) & 34 & 39.0 & 11.0 & 27.8 & 0.0 & 0.0 & 0.2 &69\
(C5) & 6 & $10.5\pm0.5$ & 10.2 & 0.1 & 0.0 & 0.0 & 0.1 &68\
(C6) & 1 &$6.5\pm0.3$ & 6.5 & 0.0 & 0.0 & 0.0 & 0.0 &65\
|
---
abstract: 'We report a demonstration of quantum key distribution (QKD) over a standard telecom fiber exceeding 50 dB in loss and 250 km in length. The differential phase shift QKD protocol was chosen and implemented with 2 GHz system clock rate. By careful optimization of the 1-bit delayed Faraday-Michelson interferometer and the use of the super-conducting single photon detector (SSPD), we achieved a quantum bit error rate below 2% when the fiber length was no more than 205 km, and of 3.45% for the 260 km length fiber with 52.9 dB loss. We also improved the quantum efficiency of SSPD to obtain high key rate for 50 km length.'
address: |
$^{1}$Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei 230026, China\
$^2$Anhui Asky Quantum Technology Co.,Ltd., Wuhu 241002, China
author:
- 'Shuang Wang,$^{1}$ Wei Chen,$^{1}$ Jun-Fu Guo,$^{2}$ Zhen-Qiang Yin,$^{1}$ Hong-Wei Li,$^{1}$ Zheng Zhou,$^{1}$ Guang-Can Guo,$^{1}$ and Zheng-Fu Han$^{1}$'
title: '2-GHz clock quantum key distribution over 260 km of standard telecom fiber'
---
Quantum key distribution (QKD) enables two remote participants to share unconditionally secure keys based on the principles of quantum physics [@bb84; @gisin2002; @scarani]. Combined with one-time pad encryption, QKD is hopeful to effectively end the cat and mouse game between the guardians of secrets and their enemies [@broad], and has become one of the most dynamic research fields. After the past two decades of developments, experimental QKD has achieved significant improvements, the transmission distance from 32 cm [@first] to 250 km [@cow], the speed (or system clock rate) from 200 Hz [@first] to 10 GHz [@oe06; @nphoton; @townsend].
With the dispersion-shifted fiber, Takesue et al. realized the first QKD experiment over 42.1 dB channel loss and 200 km of distance [@nphoton]. Then, with the ultra low loss fiber, Stucki et al. implemented the first QKD experiment over 250 km of distance, but the channel loss is still 42.6 dB [@cow]. In this letter, focused on the transmission over the widely used standard (ITU-T G.652) telecom fiber, of which loss coefficient is about 0.2 dB/km and dispersion is about 17 ps/(km $\cdot$ nm) at 1550 nm region, we report a QKD experiment over 260 km of this standard telecom fiber with 52.9 dB channel loss. This is the first QKD experiment exceeding 50 dB in channel loss and 250 km in length.
We chose the differential phase shift QKD (DPS-QKD) protocol [@dps] to be implemented with 2 GHz rate. The experimental setup is outlined in Fig. \[exp\], including the transmitter – Alice, quantum channel, and receiver – Bob. At Alice’s site, a continuous wave (CW) laser, whose central wavelength is 1560.2 nm, is first modulated into a pulse train by an intensity modulator (IM). The phase modulator (PM) randomly encodes $\{-\frac{\pi}{2},\frac{\pi}{2}\}$ on each pulse, and the followed variable attenuator (VA) attenuates the average photon number per pulse to the optimal value. Alice’s pattern generator (PG) has three outputs – narrow pulse with 2 GHz rate to IM, pseudo-random data to PM and 0.5 MHz sync signals to Bob. The quantum channel is the standard telecom fiber (STF). A 3-port optical circulator (CIR) at Bob’s site is put before his 2-GHz, 1-bit delayed Faraday-Michelson interferometer (FMI), which makes one pulse interfere with pulses before and after it. The two outputs of the FMI are connected with a double-channel super-conducting single photon detector (SSPD), of which D0 channel clicks if the phase difference between two adjacent pulses is $0$, D1 channel clicks if the phase difference is $\pi$. Both sync signals and electrical pulses from SSPD are sent to the time-to-digit convertor (TDC). Once TDC records a click event, Bob and Alice can share one sifted key bit. Note that although we transmitted the sync signal via a cable in the lab, the best way to transmit the sync signal is over the quantum channel.
![(Color online) Schematic of the DPS-QKD setup.[]{data-label="exp"}](exp.eps){width="8.3cm"}
Suppose the insertion loss (IL) of Bob’s detection setup and the effective detection efficiency (EDE) of SSPD are $\alpha_{_{IL}}$ and $\eta_{_D}$, the overall transmission and detection efficiency between Alice and Bob can be expressed as [@ma] $$\label{eta}
\eta=\eta{_{_D}} \cdot 10^{-\frac{\alpha \cdot l +
\alpha_{_{IL}}}{10}},$$ here $\alpha$ and $l$ are the loss coefficient and the length of the fiber respectively. Let $\mu$ denote the average photon number per pulse set by Alice, the probability that one click event happens is given by $$p_{click}\approx p_{signal}+p_{dark},$$ where the signal contribution is $p_{signal}=1-e^{-\mu\eta}$, the dark count contribution is $p_{dark}=2\cdot D\cdot t_{_{W}}$, here $D$ is the dark count rate (DCR) of Bob’s detector, and $t_{_{W}}$ is the measurement time window of the system [@eleni]. Considering the dead time of the detection system $t_d$, the sifted key rate could be expressed as [@td] $$\label{siftedkey}
R_{sifted}=f \cdot p_{click} \cdot e^{-f \cdot p_{click} \cdot t_d},$$ where $f$ is the repetition rate of transmission. If the probability that a signal hit the wrong detector is $e_{s}$, which is the baseline system error rate [@eleni], the quantum bit error rate (QBER) is given by $$\label{qber}
e=\frac{e_s \cdot p_{signal} + e_d \cdot p_{dark}}{p_{click}},$$ where $e_d = 0.5$ means the dark count contribution is random. Finally, the secure key rate under general individual attacks is given by [@secure] $$\label{securekey}
R_{secure}=R_{sifted} \cdot \{\tau - f(e)\cdot H_2(e)\},$$ where $\tau=-(1-2\mu)log_2[1-e^2-(1-6e)^2/2]$ is the compression factor in the privacy amplification process, $f(x)$ characterizes the efficiency of error correction algorithm, and $H_2(x)$ is the binary Shannon entropy. In order to get a tighter security threshold [@zhangq], $f(e)$ in this paper was chosen as 1.2. Using equations from to , we can maximize the secure key rate by setting optimal $\mu$ for the specific experimental setup.
![(Color online) Quantum bit error rate (QBER) without phase modulation and active compensation.[]{data-label="vis"}](vis.eps){width="8.3cm"}
Bob’s experimental setup includes three main parts – the Faraday-Michelson interferometer, the super-conducting single photon detector, and the time-to-digit convertor. **(1)** One 50/50 beam splitter (BS) and two Faraday mirrors (FM) constitute FMI, the Faraday mirror which is a combination of a 45 degree Faraday rotator and an ordinary mirror could automatically compensate for any birefringence effect in fiber [@mo], so the 2-GHz, 1-bit delayed FMI is almost insensitive to polarization.The interferometer was insulated from the environment and actively compensated by piezoelectric ceramics for better interference. Without phase modulation and active compensation, the mean QBER was about 0.65% over 2450 seconds (Fig. \[vis\]). However, when we added random phase modulation signal on the phase modulator, QBER increased to 1.80%, which was the value of $e_{s}$ in equation . The IL of Bob’s detection setup is about 1.5 dB, including the IL of CIR (from port 1 to port 2) and FMI. **(2)** The double-channel SSPD was made by Scontel Ltd. from Russia, and worked with a refrigeration system [@sspd] which could obtain a temperature of 1.7 K. The detector had a counting rate more than 70 MHz, and a jitter value better than 50 ps. By carefully increasing the bias current, we achieved 3.0% average quantum efficiency with 1 Hz DCR. **(3)** The TDC not only recorded the sync and SSPD signals, but also set the measurement time window $t_{_W}$ during the experiment. The value $t_{_W}$ was set to 200 ps, and this set reduced the quantum efficiency by 17%. The dead time of TDC is 15 ns, which is the value of dead time of the whole detection system.
Based on these specific experimental parameters, the secure key rate under individual attacks could be maximized by choosing optimal $\mu$ for each fiber length, and the attainable maximum distance is 281 km (with 0.2 dB/km loss coefficient) in principle (Blue dot line in Fig. \[IM\]). With standard telecom fiber, the sifted key rates and QBERs were measured at seven different fiber lengths – 10 km, 50 km, 75 km, 100 km, 150 km, 205 km, and 260 km. We set $\mu=0.19$ for 10 km and 50 km fiber length, and $\mu=0.20$ for other length values.
![(Color online) Experimental results of DPS-QKD.[]{data-label="IM"}](pt.eps){width="8.3cm"}
The dark count rate of detectors is a major limiting factor for long-distance QKD [@cow]. Therefore the ultra low-noise SSPD was used in our DPS-QKD system. We first set the EDE and DCR of SSPD at 2.5% and 1 Hz respectively, and the measurement results were shown as open shapes in Fig. \[IM\]. For each data, we ran the QKD process ten times, and each run lasted ten minutes. The measured QBERs and sifted key rates are the average values of the ten runs, but secure key rates are calculated results by equation . Through careful optimization of the 1-bit delayed interferometer, we achieved the values of QBER below 2% for the first six lengths, and of 3.45% for the 260 km length fiber with 52.9 dB loss (Red open circles in Fig. \[IM\]). At 10 km and 50 km fiber length, 1.16 Mbits/s and 185 kbits/s secure key rate under individual attacks were achieved. At 205 km with 41.6 dB transmission loss, 99.2 bits/s secure key rate was obtained, this rate value was more than eight times of that achieved in 10-GHz DPS-QKD experiment at 200 km with 42.1 dB loss [@nphoton]. Although the channel loss of 260 km was one order of magnitude larger than the loss 42.6 dB in previous 250 km QKD experiment [@cow], in which the ultra low loss fiber with 0.164 dB/km loss coefficient was used, secure keys with 1.85 bits/s rate could still be shared between Alice and Bob.
When the transmission fiber length was short, the signal contribution $p_{signal}$ was much larger than the dark count contribution $p_{dark}$. In order to get higher key rate, we improved the quantum efficiency of SSPD by increasing the bias current, though the dark count rate increased faster as the current increased. In the 50 km fiber length experiment, another $\eta_{_D}
= 11.2\%$ value was tested, QBER was 1.89%, and the corresponding secure key rate got up to 0.81 Mbits/s, which was close to Dixon’s BB84 experiment [@yuan].
For the QKD system over long distance, the nonzero accumulated chromatic dispersion of standard telecom fiber would severely limit the performance of QKD, especially for gigahertz systems [@yuannjp]. The optical pulses are broadened during propagation through the optical fiber, so photons would spread outside the measurement time window, which reduces the effective quantum efficiency, and even overlap with photons of neighbor pulses, which degrades the encoded signal. Take the 10-GHz QKD system for example, in which pulses are separated by 100 ps, while the dispersion is up to 153 ps after propagating in 25 km single mode fiber [@eleni], so the dispersion-shifted fiber was used in [@oe06] and [@nphoton]. In our experiment, the full width at half maximum of the 2-GHz pulse train was 170 ps, this wide width limited the effects of chromatic dispersion to some extent. Fig. \[IM\] shows that the measured sifted key rates (or counting rates) deviate from the simulation results, and the reduction increases as the fiber length increases. At 205 km transmission distance, the measured sifted key rate was 69.1% of the simulation one. The transmission loss 41.6 dB was higher than $0.2\times205$ dB, and the set average photon number 0.20 was a little bit larger than the optimum one (0.19776). After removing effects of these differences, there was still 20.8% reduction. The chromatic dispersion of the fiber was the main cause of this reduction.
In summary, we have experimentally demonstrated that quantum key distribution is possible over 260 km standard telecom fiber with 52.9 dB loss. Using the ultra low loss fiber with 0.164 dB/km loss coefficient [@cow], the quantum key exchange over 340 km distance is in sight.
This work was supported by the National Basic Research Program of China (Grants No. 2011CBA00200 and No. 2011CB921200), National Natural Science Foundation of China (Grant No. 60921091), and National High Technology Research and Development Program of China (863 program) (Grant No. 2009AA01A349).
[99]{}
C. H. Bennett and G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), p. 175.
N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. [**74**]{}, 145 (2002).
V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, Rev. Mod. Phys. [**81**]{}, 1301(2009).
D. J. Rogers, “Broadband Quantum Therory,” (Morgan & Claypool, 2010).
C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, J. Cryptology, [**5**]{}, 3 (1992)
D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, New J. Phys. [**11**]{}, 075003 (2009).
H. Takesue, E. Diamanti, C. Langrock, M. M. Fejer, and Y. Yamamoto, Opt. Express [**14**]{}, 9522 (2006).
H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, Nature Photonics [**1**]{}, 343 (2007).
I. Choi, R. J. Young, and P. D. Townsend, New J. Phys. [**13**]{}, 063039 (2011).
K. Inoue, E. Waks, and Y. Yamamoto, Phys. Rev. Lett. [**89**]{}, 037902 (2002).
X. F. Ma, B. Qi, Y. Zhao, and H. K. Lo, Phys. Rev. A [**72**]{}, 012326 (2005).
E. Diamanti, “Security and implementation of differential phase shift quantum key distribution systems,” Ph.D. thesis, Stanford University (2006).
E. Diamanti, H. Takesue, C. Langrock, M. M. Fejer, and Y. Yamamoto, Opt. Express [**14**]{}, 13073 (2006).
E. Waks, H. Takesue, and Y. Yamamoto, Phys. Rev. A [**73**]{}, 012344 (2006).
Q. Zhang, H. Takesue, T. Honjo, K. Wen, T. Hirohata, M. Suyama, Y. Takiguchi, H. Kamada, Y. Tokura, O. Tadanaga, Y. Nishida, M. Asobe, and Y. Yamamoto, New J. Phys. [**11**]{}, 045010 (2009).
A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, Appl. Phys. Lett. [**70**]{}, 793 (1997).
G. Goltsman, A. Korneev, A. Divochiy, O. Minaeva, M. Tarkhov, N. Kaurova, V. Seleznev, B. Voronov, O. Okunev, A. Antipov, K. Smirnov, Yu. Vachtomin, I. Milostnaya, and G. Chulkova, J. Mod. Opt. [**56**]{}, 1670 (2009).
A. R. Dixon, Z. L. Yuan, J. F. Dynes, A. W. Sharpe, and A. J. Shields, Appl. Phys. Lett. [**96**]{}, 161102 (2010).
Z. L. Yuan, A. R. Dixon, J. F. Dynes, A. W. Sharpe, and A. J. Shields, New J. Phys. [**11**]{}, 045019 (2009).
|
c ł u v Ł ¶ §
ø
Ø
\
\
[*Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy and INFN, Pisa, Italy*]{}\
Flavour physics, from now up to the operation of the next high energy collider, will be an important tool for BSM searches at the TeV scale. Although far from exhaustive, a particularly relevant case is represented by the possibility that the Higgs be a composite Pseudo-Nambu-Goldstone-Boson (PNGB) at a scale $l_H=1/m_*$. While a totally model-independent assessment of the potential of flavour physics in this case is impossible, here we illustrate what is likely to be a minimal sensitivity on $m_*$ by considering suitable examples.\
Introduction
============
Among the indirect tests of new physics, those that aim at seeing evidence for (or setting limits on) the scale of Higgs compositeness, $l_H = 1/m_*$, as usually called in the literature, are among the most important ones, if not the dominant at all. The purpose of this note is to describe the sensitivity to $m_*$ of flavour physics with an eye to the progress foreseen before the operation of the next collider, be it an $e^+e^-$ or a $pp$ or even a muon collider, here dubbed “near” future. A comprehensive detailed analysis of foreseen experimental and theoretical developments in flavour physics within this temporal range is definitely beyond the scope of this note. Rather we focus our attention on an indispensable set of hypotheses required to describe flavour in composite Higgs models, in order to see their impact on the sensitivity to $l_H = 1/m_*$ in minimal examples.
Setting the framework {#framework}
=====================
We assume that a new strong interaction with a confinement scale $m_*$ and a strong coupling $1< g_* <4\pi$ gives rise, after spontaneous symmetry breaking, to the Higgs, $H$, as a Pseudo-Nambu-Goldstone-Boson (PNGB). This is the scheme adopted also when discussing the ElectroWeak Precision Tests or other flavour-less precision tests in this context. Here we are not concerned with the issue of the relation between $m_*$ and $m_H$.
The standard fermions do not feel directly this new strong interaction, but, to get a mass, they have to be connected in some (unspecified) way with a composite operator of the strong sector, $\mathcal{O}_H$ with $<0|\mathcal{O}_H|H>\neq 0$. To describe flavour, after integrating out all states at or above a flavour scale $\Lambda_F$, the standard fermions, $f^a_i$, will enter the effective Lagrangian below $\Lambda_F$ always multiplied by a dimensionless coupling $\lambda^a_i$. There can be more than one flavour scale $\Lambda_F$, in which case we concentrate on the scale, that we shall keep calling $\Lambda_F$, at which the top quark acquires its mass [@Panico:2016ull]. (See also [@KerenZur:2012fr; @Barbieri:2012tu] and references therein.) Below this scale we take as relevant effective Lagrangian $$\mathcal{L}^{top}= \frac{\Lambda_F^4}{g_{\Lambda_F}^2}[\mathcal{L}^0(\frac{\lambda^t_Lq_{L3} }{\Lambda_F^{3/2}}, \frac{\lambda^t_R t_{R}}{\Lambda_F^{3/2}},\frac{D_\mu}{\Lambda_F}, \frac{\mathcal{O}_H}{\Lambda_F^{d_H}})
+\frac{g_{\Lambda_F}^2}{16\pi^2}\mathcal{L}^1(\frac{\lambda^t_L q_{L3}}{\Lambda_F^{3/2}}, \frac{\lambda^t_R t_{R}}{\Lambda_F^{3/2}},\frac{D_\mu}{\Lambda_F}, \frac{\mathcal{O}_H}{\Lambda_F^{d_H}})+\dots]
\label{topL}$$ as it arises from Naive Dimensional Analysis if a single new coupling $g_{\Lambda_F}$ is involved, other than $\lambda^t_{L,R}$, and the operator $\mathcal{O}_H$ has anomalous dimension $d_H$. This Lagrangian fixes in particular the effective top Yukawa interaction at $\Lambda_F$ $$\mathcal{L}^{top}_Y(\Lambda_F)=\frac{x_L x_R}{\Lambda_F^{d_H-1}} \bar{q}_{L3} \mathcal{O}_H t_R,\quad\quad
x_{L,R}=\frac{\lambda^t_{L,R}}{g_{\Lambda_F}}$$ or the top Yukawa coupling $y_t$ at the compositeness scale $m_*$, after $\mathcal{O}_H\rightarrow g_* m_*^{d_H-1} H$, $$y_t=g_* x_L x_R (\frac{m_*}{\Lambda_F})^{d_H-1}.
\label{topY}$$
As it will be used in the following, the Lagrangian (\[topL\]) fixes as well all the effective interactions among $q_{L3}, t_R$, but not the entire set of Yukawa couplings $Y_{U,D,E}$, which, as said, will involve all the other $\lambda_i^a$ and possibly other flavour scales. As such, the full Yukawa couplings are model dependent. On the other hand the mixings of the third generation quarks, $q_{L3}$ and $t_R$ entering in (\[topL\]) with the lighter generations, required to diagonalise these full Yukawa couplings, are crucial to compare the predictions of (\[topL\]) in flavour experiments. To overcome this problem, calling $U_{L,R}$ and $D_{L,R}$ the unitary matrices that diagonalise $Y_U$ and $Y_D$ respectively, we shall consider the following 3 cases:
- Case 1: $D_L = V, U_L =\bold{1}, D_R = U_R=\bold{1}$
- Case 2: $U_L = V^+, D_L =\bold{1}, D_R = U_R=\bold{1}, $
- Case 3: $U_L = V^+, D_L =\bold{1}, D_R =\bold{1}, U_R= V^+$
where $V= U_L^+D_L$ is the CKM matrix.
We think that the overall consideration of these examples illustrates the power of flavour physics. It is easy to consider motivated cases more constraining on $m_*$, whereas, on the contrary, it is hard to conceive a situation less constraining on $m_*$ than each of these examples, especially Case 1 and 2.
$\Delta F =2$
=============
From the Lagrangian (\[topL\]), upon use of eq. (\[topY\]), the 4-Fermi interactions at $m_*$ of $q_{L3},t_R$ among themselves are $$\mathcal{L}^{4F}=\frac{y_t^2}{m_*^2} (\frac{\Lambda_F}{m_*})^{2(d_H-2)}
[x_t^2 (\bar{q}_{L3}\gamma_\mu q_{L3})^2+ (\bar{q}_{L3}t_R)(\bar{t}_R q_{L3}) +\frac{1}{x_t^2} (\bar{t}_R\gamma_\mu t_R)^2],\quad x_t = \frac{x_L}{x_R}.
\label{L4F}$$ After going to the physical bases, this Lagrangian generates in the three cases defined above the following $\Delta F =2$ effective Lagrangians:
$$\mathcal{L}^{\Delta F =2}_{Case 1} = \frac{y_t^2}{m_*^2} C
[x_t^2 (\bar{d}_{Li} \xi^d_{ij}\gamma_\mu d_{Lj})^2], \quad\quad \xi^d_{ij} = V_{tj}V^*_{ti}$$
$$\mathcal{L}^{\Delta F =2}_{Case 2} = \frac{y_t^2}{m_*^2} C
[x_t^2 (\bar{u}_{Li} \xi^u_{ij}\gamma_\mu u_{Lj})^2], \quad\quad \xi^u_{ij} = V_{ib}V^*_{jb}$$
$$\mathcal{L}^{\Delta F =2}_{Case 3} = \frac{y_t^2}{m_*^2} C
[x_t^2 (\bar{u}_{Li} \xi^u_{ij}\gamma_\mu u_{Lj})^2 + (\bar{u}_{Li} \xi^u_{ij} u_{Rj})(\bar{u}_{Rk} (\xi^u_{lk})^* u_{Ll})+
\frac{1}{x_t^2} (\bar{u}_{Ri} \xi^u_{ij}\gamma_\mu u_{Rj})^2]$$
The overall coefficient $$C = (\frac{\Lambda_F}{m_*})^{2(d_H-2)}$$ is bigger than one for any scale $\Lambda_F> m_*$ since $d_H \geq 2$ [@Rattazzi:2008pe].
From these effective Lagrangians a full fit of current flavour data [@Silvestrini:2019sey] allows to set the bounds on the compositeness scale $m_*$ summarised in Table \[DeltaF=2\]. In the “near” future these bounds are expected to go to the values indicated in parenthesis.
$$\begin{array}{c|c|c|c|c}
&\Delta S=2&\Delta C=2&\Delta B_d=2&\Delta B_s=2\\ \hline
Case 1&7(13)x_t&-&8(20)x_t&9(20)x_t\\ \hline
Case 2&-&3(10)x_t&-&-\\ \hline
Case 3&-&10(30)&-&-\\ \hline
\end{array}$$
$\Delta F=1$
=============
By analogous considerations to the ones developed in the previous Section, the effective Lagrangian most relevant to $\Delta F=1$ transitions is $$\mathcal{L}^{\Delta F=1}=\frac{g_*y_t}{m_*^2}C^{1/2}i (H^\dag \overleftrightarrow{D_\mu} H) [x_t(\bar q_{L3} \gamma^\mu q_{L3})+\frac{1}{x_t}(\bar t_{R} \gamma^\mu t_{R})]$$
From this Lagrangian the most relevant bounds are obtained for Case 1, i.e, after going to the physical basis, from $$\mathcal{L}^{\Delta F=1}_{Case 1}=\frac{g_*y_t}{m_*^2}C^{1/2}i (H^\dag \overleftrightarrow{D_\mu} H) [x_t(\bar d_{Li} \xi_{ij}^d\gamma^\mu d_{Lj})].
\label{eq:DeltaF=1}$$ In Fig.s \[fig:DeltaF=1\], we show the current constraints on the overall coefficients $C^{(1)}_{\phi q}|_{ij}$ of the operator $i (H^\dag \overleftrightarrow{D_\mu} H) (\bar d_{Li} \gamma^\mu d_{Lj})$ in the $12=ds$ (upper figure) and $23=sb$ channels (lower figure). The uncertainties of the observables in the $12=ds$ case are dominated by theory. An important input would come from a determination of the branching ratio of $K^+\rightarrow \pi^+ \nu\bar \nu$ at the $10\%$ level. Achieving this sensitivity would be equivalent to $Re [C^{(1)}_{\phi q}]_{12}\lesssim 4\cdot 10^{-5} TeV^{-2}$ (and might perhaps support or dilute the putative evidence for BSM contribution in $\epsilon^\prime/\epsilon$, hinted in Fig. \[fig:DeltaF=1\], upper). Due to the large number of observables relevant to the $23=sb$ case, the lower figure shows the result of an overall fit. Particularly important are the branching ratio of $B_s\rightarrow\mu^+\mu^-$ and the angular variables in $B\rightarrow K^*\mu\mu$, both statistically dominated at present [@Aebischer:2018iyb].
![Current constraints on the coefficient $C^{(1)}_{\phi q}|_{12}$, upper figure, and on the coefficient $C^{(1)}_{\phi q}|_{23}$, lower figure. See text. Courtesy of David Straub. []{data-label="fig:DeltaF=1"}](Cphiq1_12_observables.pdf "fig:"){width=".48\textwidth"} ![Current constraints on the coefficient $C^{(1)}_{\phi q}|_{12}$, upper figure, and on the coefficient $C^{(1)}_{\phi q}|_{23}$, lower figure. See text. Courtesy of David Straub. []{data-label="fig:DeltaF=1"}](Cphiq1_20TeV.pdf "fig:"){width=".66\textwidth"}
Based on Fig.s \[fig:DeltaF=1\] we consider $$Im [C^{(1)}_{\phi q}]_{12}\lesssim 6\cdot 10^{-5} TeV^{-2},\quad\quad
Re [C^{(1)}_{\phi q}]_{23}, Im [C^{(1)}_{\phi q}]_{23}\lesssim 2\cdot 10^{-3} TeV^{-2}$$ which correspond to the bounds on $m_*$ shown in Table \[DeltaF=1\].
Note that eq. \[eq:DeltaF=1\] contains as well a correction to the $Zb\bar b$ coupling $$\delta g_{bL} = C^{(1)}_{\phi q}|_{33}\frac{v^2}{2}$$ so that, requiring $\delta g_{bL} < 1.5\cdot 10^{-3}$, one gets $m_*\gtrsim 4.5 \sqrt{g_*x_t}~TeV$.
$$\begin{array}{c|c|c}
&12&23\\ \hline
Case 1&1.7 \sqrt{g_*x_t}&4.5\sqrt{g_*x_t}\\ \hline
\end{array}$$
We do not consider the bounds/sensitivity from $\Delta C=1$ operators due to the difficulty of estimating the SM contributions to these processes.
In some models with a suitable custodial parity [@Agashe:2006at] the relation $$C^{(1)}_{\phi q}|_{33} = - C^{(3)}_{\phi q}|_{33}$$ with the coefficient $C^{(3)}_{\phi q}|_{33}$ of the operator $i (H^\dag \overleftrightarrow{D_\mu} \sigma_aH) (\bar d_{L3} \gamma^\mu \sigma^ad_{L3})$ may suppress all these $\Delta F=1$ effects.
Dipole operators
================
Neutron Electric Dipole Moment
------------------------------
From the Lagrangian (\[topL\]) at one loop one obtains the dipole operators (identifying $\Lambda_F$ with $m_*$ and $g_{\Lambda_F}$ with $g_*$ for ease of exposition, but without influence on the bounds on $m_*$) $$\mathcal{L}^{dip} = \frac{g_*^2}{16\pi^2} \frac{m_t}{m_*^2}
[C^{dip}(\bar{t}_L\sigma_{\mu\nu}t_R)e F^{\mu\nu} + \tilde{C}^{dip}(\bar{t}_L\sigma_{\mu\nu}T^at_R)g_S G^{\mu\nu}_a],
\label{dip_top}$$ where $C^{dip}, \tilde{C}^{dip}$ are coefficients of order unity, in general complex. In Case 3, after going in the physical basis, this gives $$\mathcal{L}^{dip}_{Case 3} = \frac{g_*^2}{16\pi^2} \frac{m_t}{m_*^2}
[C^{dip}(\bar{u}_{Li}\xi_{ij}^u\sigma_{\mu\nu}u_{Rj})e F^{\mu\nu} + \tilde{C}^{dip}(\bar{u}_{Li}\xi_{ij}^u\sigma_{\mu\nu}T^a u_{Rj})g_S G^{\mu\nu}_a]$$
The cromo-electric dipole moments of the quarks feed into the neutron electric dipole moment, currently bound by [@Baker:2006ts] $$d_n < 2.9\cdot 10^{-26}~e~cm,$$ via their contribution to the Weinberg operator. As a consequence, from the up, charm and top contributions one gets [@Sala:2013osa] $$Im(\tilde{C}^{dip})\xi^u_{11} < 1.3\cdot 10^{-8},\quad\quad Im(\tilde{C}^{dip})\xi^u_{22} <1.8\cdot 10^{-5},\quad\quad Im(\tilde{C}^{dip})\xi^u_{33} <3.3\cdot 10^{-2},$$ with a common factor $(g_*^2/16\pi^2)(TeV/m_*)^2$ left understood on the left side of each of these bounds. Given the values of $\xi^u_{ii}$, the bound in Case 3 is dominated by the contribution from the up quark. Taking $Im(\tilde{C}^{dip})=1$, these bounds translate into the bounds on $m_*$ shown in Table \[EDMs\]. The sensitivity to the neutron electric dipole moment is expected to be improved by one order of magnitude in the near future by a dedicated experiment at PSI. Consequently, in absence of a signal, the limits shown in Table \[EDMs\] are expected to improve by about a factor of 3.
$$\begin{array}{c|c}
&m_*/TeV\\ \hline
Case 1&5.5 \frac{g_*}{4\pi}\\ \hline
Case 2&5.5 \frac{g_*}{4\pi}\\ \hline
Case 3&32 \frac{g_*}{4\pi}\\ \hline
\end{array}$$
Electron Electric Dipole Moment
-------------------------------
Within the restricted framework considered so far and specified in Section \[framework\] the most significant effect on the electron EDM arises at two loops via the Barr-Zee-type diagrams. This proceeds through a one loop contribution to the $H^2F\tilde{F}$ operator, which is then transferred to the electron EDM by running to the low energy scale [@Panico:2018hal]. An estimate of the overall effect to the EDM $d_e$ is $$\frac{d_e}{e} \approx \frac{g_*^2}{16\pi^2}\frac{e^2}{16\pi^2}\frac{y_tx_t}{g_*}\frac{m_e}{m_*^2}$$ where we have assumed an order one phase appearing in the coefficient of the $H^2F\tilde{F}$ operator. Requiring that this estimate be less than the recently reported result by the ACME collaboration, $|d_e|< 1.1\cdot 10^{-29}~e\cdot cm$ [@ACME], leads to the bound $$m_*< 6\sqrt{g_* x_t} TeV.$$ The large effective electromagnetic fields $(> 10~ GV/cm)$ present in heavy polar molecules may allow in the coming years a greatly improved sensitivity on the electron EDM, potentially improving the ACME result by orders of magnitude.
Leptonic flavour {#Leptons}
================
In all considerations developed so far no assumption had to be made about the scales at which the elementary fermions interact with the composite Higgs sector to get their Yukawa couplings. To discuss flavour in the case of leptons we assume that the $\tau$ lepton gets its Yukawa coupling at the same scale $\Lambda_F$ at which the top quark gets its own. In practice this means that $\mathcal{L}^0$ and $\mathcal{L}^1$ in eq. (\[topL\]) acquire a further dependence on $
\lambda^\tau_L l_{L3}/\Lambda_F^{3/2}, \lambda^\tau_R \tau_R/\Lambda_F^{3/2}$. Furthermore we assume that the mixing matrices in the leptons are related to their masses $(i,j=e,\mu, \tau)$ by $$|V^{Ll}_{i>j}| = |V^{Rl}_{i>j}| = \frac{m_j}{m_i}$$
In full analogy with the case of the quarks, in particular eq. (\[dip\_top\]), the operator that leads to the strongest contraints in the lepton case is $$\mathcal{L}^{dip}_\tau = \frac{g_*^2}{16\pi^2} \frac{m_\tau}{m_*^2}
C^{dip}_\tau(\bar{\tau}_L\sigma_{\mu\nu}\tau_R)e F^{\mu\nu}
\label{Ctau}$$ i.e., after going to the physical basis, $$\mathcal{L}^{dip}_l = \frac{g_*^2}{16\pi^2} \frac{m_\tau}{m_*^2}
C^{dip}_\tau[(\bar{\tau}_L\sigma_{\mu\nu}\xi^l_{\tau\mu}\mu_R)
+ (\bar{\mu}_L\sigma_{\mu\nu}\xi^l_{\tau\mu}e_R)]e F^{\mu\nu},\quad\quad \xi^l_{ij}=V^l_{\tau j}V_{\tau i}^{l*}$$ plus similar terms with the role of L and R reversed. In view of the current bounds, this leads to the lower limits on $m_*$ shown in Table \[taumugamma\]. Since the sensitivity to the relevant branching ratios is expected to improve by one order of magnitude, these bounds are expected to improve by about a factor of three.
![Case 1: Current bounds now (full lines) and the sensitivity expected in the “near” future (dotted lines) from $\Delta B_s=2$ (blue) and the neutron EDM (red). Also shown is the current bound from the electron EDM (yellow) and from $\Delta B_s=1$ (green) in models without a custodial parity. Everywhere $x_t=1/2$, eq. \[L4F\]. []{data-label="fig:Case1"}](Case1.pdf){width=".70\textwidth"}
$$\begin{array}{c|c|c}
&\tau\rightarrow \mu \gamma&\mu\rightarrow e \gamma \\ \hline
m_*/TeV&8\frac{g_*}{4\pi}&16\frac{g_*}{4\pi}\\ \hline
\end{array}$$
If $C^{dip}$ has a phase, again after going to the physical basis, eq. (\[Ctau\]) leads to electric dipole moments for the leptons. For the electron, $$\mathcal{L}^{dip}_e = \frac{g_*^2}{16\pi^2} \frac{m_\tau}{m_*^2}
C^{dip}_\tau (\frac{m_e}{m_\tau})^2 (\bar{\tau}_L\sigma_{\mu\nu}\tau_R)e F^{\mu\nu},
\label{Ce}$$ so that, with maximal phase, the current limit on the $eEDM < 1.1\cdot 10^{-29}~e\cdot cm$, leads to the bound $$m_* < 20\frac{g_*}{4\pi} TeV.$$
Summary
=======
![Case 2: Current bounds now (full lines) and the sensitivity expected in the “near” future (dotted lines) from $\Delta C=2$ (blue) and the neutron EDM (red). Also shown is the current bound from the electron EDM (yellow). Everywhere $x_t=1/2$, eq. \[L4F\].[]{data-label="fig:Case2"}](Case2.pdf){width=".70\textwidth"}
![ Current bounds now (dotted lines) and the sensitivity expected at the end of HL-LHC (full lines) from flavour-less Precision Tests and direct searches. Adapted from a talk by A. Wulzer for the European Strategy, Granada,13-16 May, 2019[]{data-label="fig:EWPT"}](EWPT.pdf){width=".70\textwidth"}
From now up to the operation of the next high energy accelerator, flavour physics will be an important tool for BSM searches at the TeV scale. A particularly relevant case is represented by the possibility that the Higgs be a composite PNGB at a scale $l_H=1/m_*$. While a totally model-independent assessment of the potential of flavour physics is impossible, one can nevertheless consider two examples, Case 1 and 2 defined in Section \[framework\], that illustrate what is likely to be a minimal sensitivity on $m_*$. This is summarised in Fig.s \[fig:Case1\] and \[fig:Case2\] for the two cases respectively. Other cases considered in the text (Case 3 in Section \[framework\], and Section \[Leptons\]) give stronger and/or additional constraints. For comparison we show in Fig. \[fig:EWPT\] the sensitivity expected from flavour-less Precision Tests on the same basic composite Higgs model. Taking into account that in all the three figures the various bounds can be moved by different $\mathcal{O}(1)$ factors, the complementarity of the two approaches is manifest.
Although the considerations developed in this note are far from exhausting the potential impact of flavour physics in the “near” future, with the general aim of finding clues to attack the flavour puzzle, we think that they illustrate concretely such potential in a particularly relevant example[^1].
I am grateful to Gino Isidori, Luca Silvestrini and in particular David Straub for their helpful comments and informations.
[99]{}
G. Panico and A. Pomarol, JHEP [**1607**]{} (2016) 097 doi:10.1007/JHEP07(2016)097 \[arXiv:1603.06609 \[hep-ph\]\].
B. Keren-Zur, P. Lodone, M. Nardecchia, D. Pappadopulo, R. Rattazzi and L. Vecchi, Nucl. Phys. B [**867**]{} (2013) 394 doi:10.1016/j.nuclphysb.2012.10.012 \[arXiv:1205.5803 \[hep-ph\]\].
R. Barbieri, D. Buttazzo, F. Sala, D. M. Straub and A. Tesi, JHEP [**1305**]{} (2013) 069 doi:10.1007/JHEP05(2013)069 \[arXiv:1211.5085 \[hep-ph\]\]. R. Rattazzi, V. S. Rychkov, E. Tonni and A. Vichi, JHEP [**0812**]{} (2008) 031 doi:10.1088/1126-6708/2008/12/031 \[arXiv:0807.0004 \[hep-th\]\].
L. Silvestrini, arXiv:1905.00798 \[hep-ph\]. J. Aebischer, J. Kumar, P. Stangl and D. M. Straub, Eur. Phys. J. C [**79**]{} (2019) no.6, 509 doi:10.1140/epjc/s10052-019-6977-z \[arXiv:1810.07698 \[hep-ph\]\]. K. Agashe, R. Contino, L. Da Rold and A. Pomarol, Phys. Lett. B [**641**]{} (2006) 62 doi:10.1016/j.physletb.2006.08.005 \[hep-ph/0605341\]. C. A. Baker [*et al.*]{}, Phys. Rev. Lett. [**97**]{} (2006) 131801 doi:10.1103/PhysRevLett.97.131801 \[hep-ex/0602020\].
F. Sala, JHEP [**1403**]{} (2014) 061 doi:10.1007/JHEP03(2014)061 \[arXiv:1312.2589 \[hep-ph\]\].
G. Panico, A. Pomarol and M. Riembau, JHEP [**1904**]{} (2019) 090 doi:10.1007/JHEP04(2019)090 \[arXiv:1810.09413 \[hep-ph\]\].
ACME collaboration, V. Andreev et al., Nature 562 (2018) 355Ð360.
[^1]: One may wonder if and how the putative anomalies currently observed in B-decays can fit into a composite Higgs picture. They could potentially fit in Case 2 with $m_*\gtrsim 2$ TeV and $g_*=(1\div 2)m_*/TeV$, Fig.\[fig:Case2\] (as well as $x_\tau = \lambda^\tau_L/\lambda^\tau_R$ close to $ g_*/y_\tau$) with a suppressed CP-violating phase in the electron EDM.
|
---
author:
- |
on behalf of the collaboration[^1]\
DESY, Notkestr. 85, 22607 Hamburg, Germany\
E-mail:
title: Determination of the strong coupling at NNLO from jet production in DIS
---
Introduction
============
The strong coupling constant is one of the least known parameters of the Standard Model (SM) and a precise knowledge is of crucial importance for precision physics and searches for physics beyond the SM at the LHC. Cross sections for jet production in deep-inelastic electron-proton scattering (DIS) are directly sensitive to the strong coupling constant [$\alpha_s(m_Z)$]{} already in leading order in perturbative QCD (pQCD) as these measurements are performed in the Breit frame of reference. The cross section calculations are performed in next-to-next-to-leading order (NNLO) accuracy, where the cross section predictions are obtained with the program [NNLO]{}[@Currie:2016ytq; @Currie:2017tpe].
Methodology
===========
Cross sections for jet production in $ep$ collisions have been measured by the H1 experiment at HERA at different center-of-mass energies and for different kinematic regions. Here, jet and dijet cross sections taken during the years 1995 to 2007 [@Adloff:2000tq; @Aaron:2010ac; @Aktas:2007aa; @Andreev:2014wwa; @Andreev:2016tgi] are considered. Consistent to all data sets, jets are defined using the $k_t$ jet-algorithm with a parameter of $R=1$, and jets are required to be contained in the pseudorapidity range $-1<{\ensuremath{\eta^{\mathrm{jet}}_{\mathrm{lab}}}\xspace}<2.5$ defined in the laboratory frame. For the selected data, inclusive jet cross sections have been measured double-differentially as a function of the photon virtuality [$Q^2$]{} and jet transverse momentum [$P_{\rm T}^{\rm jet}$]{}, and dijet cross sections as a function of [$Q^2$]{} and the average transverse momentum of the two hardest jets, [$\langle P_{\rm T} \rangle$]{}. A brief summary of the employed measurements and the kinematic range of the observables is given in table \[tab:datasets\].
----------------------------------------------------------- ---------------------------------------- -------------------- ------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------
$\sqrt{s}$ int. $\mathcal{L}$ DIS kinematic Inclusive jets Dijets
$[{\ensuremath{\mathrm{GeV}}\xspace}]$ $[{\rm pb}^{-1}]$ range $n_{\rm jets}\ge2 $
$300\,{\ensuremath{\mathrm{GeV}}\xspace}$[@Adloff:2000tq] 300 33 $150<{\ensuremath{Q^2}\xspace}<5000\,{\ensuremath{\mathrm{GeV}^2}\xspace}$ $7<{\ensuremath{P_{\rm T}^{\rm jet}}\xspace}<50\,{\ensuremath{\mathrm{GeV}}\xspace}$ $8.5<{\ensuremath{\langle P_{\rm T} \rangle}}<35\,{\ensuremath{\mathrm{GeV}}\xspace}$
HERA-I[@Aaron:2010ac] 319 43.5 $5<{\ensuremath{Q^2}\xspace}<100\,{\ensuremath{\mathrm{GeV}^2}\xspace}$ $5<{\ensuremath{P_{\rm T}^{\rm jet}}\xspace}<80\,{\ensuremath{\mathrm{GeV}}\xspace}$ $7<{\ensuremath{\langle P_{\rm T} \rangle}}<80\,{\ensuremath{\mathrm{GeV}}\xspace}$
HERA-I[@Aktas:2007aa] 319 65.4 $150<{\ensuremath{Q^2}\xspace}<15\,000\,{\ensuremath{\mathrm{GeV}^2}\xspace}$ $5<{\ensuremath{P_{\rm T}^{\rm jet}}\xspace}<50\,{\ensuremath{\mathrm{GeV}}\xspace}$ $-$
HERA-II[@Andreev:2016tgi] 319 290 $5.5<{\ensuremath{Q^2}\xspace}<80\,{\ensuremath{\mathrm{GeV}^2}\xspace}$ $4.5<{\ensuremath{P_{\rm T}^{\rm jet}}\xspace}<50\,{\ensuremath{\mathrm{GeV}}\xspace}$ $5<{\ensuremath{\langle P_{\rm T} \rangle}}<50\,{\ensuremath{\mathrm{GeV}}\xspace}$
HERA-II[@Andreev:2016tgi; @Andreev:2014wwa] 319 351 $150<{\ensuremath{Q^2}\xspace}<15\,000\,{\ensuremath{\mathrm{GeV}^2}\xspace}$ $5<{\ensuremath{P_{\rm T}^{\rm jet}}\xspace}<50\,{\ensuremath{\mathrm{GeV}}\xspace}$ $7<{\ensuremath{\langle P_{\rm T} \rangle}}<50\,{\ensuremath{\mathrm{GeV}}\xspace}$
----------------------------------------------------------- ---------------------------------------- -------------------- ------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------
: Summary of the kinematic ranges of the inclusive jet and dijet data taken by the H1 experiment.[]{data-label="tab:datasets"}
The data sets are separated into different data taking periods and two [$Q^2$]{}-regions, where the scattered lepton is identified in different experimental devices. In case of the dijet cross sections, regions of the phase space exhibiting an infrared sensitivity due to ‘back-to-back’ topologies are avoided by imposing asymmetric cuts on the transverse momenta of the two leading jets.
The predictions are calculated as a convolution of parton density functions (PDFs) and a partonic cross section. Both these components exhibit a dependence on [$\alpha_s(m_Z)$]{} and their impact on the results are assessed below. The partonic cross section has its [$\alpha_s$]{}-dependence explicit as it is calculated in terms of a perturbative expansion in orders of $\alpha_s^{(n)}$. The [$\alpha_s$]{}-dependence of the PDFs is given by the factorisation theorem and where it originates from the QCD splitting kernels and the $\beta$-functions. Once a PDF is determined for a given value of [$\alpha_s(m_Z)$]{} it can be translated to any other value of [$\alpha_s(m_Z)$]{} by an integration step. This translation defines the [$\alpha_s$]{}-dependence of the PDF. An equivalent solution to this explicit integration is obtained by evaluating the PDFs at a suitable value of [$\mu_{F}$]{}, which depends on [$\alpha_s(m_Z)$]{}, thus taking full benefit of the factorisation theorem [@thisprelim]. The PDF parameterisation is based on the NNPDF3.0 PDF set [@Ball:2014uwa], which was determined for a value of ${\ensuremath{\alpha_s(m_Z)}\xspace}=0.118$. Multiplicative correction factors are applied in order to account for non-perturbative hadronisation effects. The renormalisation and factorisation scales are chosen to be ${\ensuremath{\mu_{R}}\xspace}^2={\ensuremath{\mu_{F}}\xspace}^2={\ensuremath{Q^2}\xspace}+{\ensuremath{P_{\rm T}}\xspace}^2$, where [$P_{\rm T}$]{} denotes [$P_{\rm T}^{\rm jet}$]{} in case of inclusive jet and [$\langle P_{\rm T} \rangle$]{} for dijet cross sections.
The value of the strong coupling constant is determined in a fit of these NNLO calculations to the H1 jet data, where the [$\alpha_s$]{}-dependencies in the predictions, both in the partonic cross sections and in the PDF, are taken into account. The NNLO coefficients are stored in the fastNLO format [@Britzger:2012bs] in order to allow for a repeated calculation with different values of [$\alpha_s(m_Z)$]{} and different PDF sets. The fit [$\chi^{2}$]{}-definition accounts for experimental, hadronisation and PDF uncertainties. Correlations of systematic uncertainties and statistical correlations of the data are considered. The uncertainties on the resulting value of [$\alpha_s(m_Z)$]{} due to experimental and theoretical sources are estimated. The PDF and hadronisation uncertainties are obtained by repeating the fit with these uncertainties excluded in the fit and comparing the resulting fit uncertainty. The scale uncertainty is estimated by repeating the fit with scale factors of 0.5 and 2. The ‘PDFSet’ uncertainty is obtained as half of the maximum difference of the results from fits using alternatively the ABMP, CT14, HERAPDF2.0, MMHT or NNPDF3.0 PDF set, and the ‘PDF[$\alpha_s$]{}’ uncertainty is estimated as half of the difference of the results obtained from fits using PDFs which were determined with [$\alpha_s(m_Z)$]{}-values differing by 0.004.
![ Values of [$\alpha_s(m_Z)$]{} obtained from fits to inclusive jet or dijet cross sections obtained for different definitions of the renormalisation and factorisation scales. The lower pads show the values of [$\chi^{2}$]{}/[$n_{\rm dof}$]{} of the fit. The open circles display results obtained using NLO matrix elements. The vertical error bars indicate the scale uncertainty. []{data-label="fig:plot1b"}]({H1prelim-17-031.fig3}.eps){width="99.00000%"}
0.03
![ Values of [$\alpha_s(m_Z)$]{} obtained from fits to inclusive jet or dijet cross sections obtained for different definitions of the renormalisation and factorisation scales. The lower pads show the values of [$\chi^{2}$]{}/[$n_{\rm dof}$]{} of the fit. The open circles display results obtained using NLO matrix elements. The vertical error bars indicate the scale uncertainty. []{data-label="fig:plot1b"}]({H1prelim-17-031.fig4}.eps){width="99.00000%"}
Results
=======
The sensitivity of the data to [$\alpha_s(m_Z)$]{} is studied in fits with two free parameters representing the two [$\alpha_s$]{} contributions to the calculation, assuming those can be chosen independently, i.e. one parameter for the PDFs, ${\ensuremath{\alpha_s}\xspace}^f(m_Z)$, and another parameter for the hard coefficients, ${\ensuremath{\alpha_s}\xspace}^{\hat\sigma}(m_Z)$. The fits are performed using inclusive jet or dijet cross section measurements, with data points below or above the renormalisation value of 15[$\mathrm{GeV}$]{}. The contours displaying the 68% confidence level of the fitted results are displayed in figure \[fig:plot1a\]. The two [$\alpha_s(m_Z)$]{} values determined in the fit are consistent, while the sensitivity to [$\alpha_s(m_Z)$]{} of the hard coefficients outperforms the one of the PDF. The two fit parameters are negatively correlated, resulting in an increased sensitivity for fits using a common [$\alpha_s(m_Z)$]{}.
Fits are also performed employing alternative definitions for the renormalisation and factorisation scales. The resulting [$\alpha_s$]{}-values and related values of [$\chi^{2}$]{}/[$n_{\rm dof}$]{} for the individual fits are displayed in figure \[fig:plot1b\] for fits to inclusive jet and to dijet cross sections. The results obtained with alternative scale choices are typically covered by the scale uncertainty. Choosing ${\ensuremath{\mu_{R}}\xspace}^2={\ensuremath{\mu_{F}}\xspace}^2={\ensuremath{Q^2}\xspace}$ is disfavored, presumably because this scale is not sufficiently related to the dynamics of jet production. For comparison the fits are also repeated with hard coefficients calculated in NLO accuracy only. These calculations typically yield higher values of [$\chi^{2}$]{}/[$n_{\rm dof}$]{} of the fits and the scale choice has a higher impact on the NLO results. These observations emphasize the improved perturbative convergence of the NNLO calculations as compared to NLO accuracy.
![ Values of [$\alpha_s(m_Z)$]{} obtained from fits to ‘H1 jets’ data points with similar values of [$\mu_{R}$]{} (full circles) in comparison to values from other experiments and processes, where all values are obtained at least in NNLO accuracy. The fitted values of [$\alpha_s(m_Z)$]{} are translated to [$\alpha_s({\ensuremath{\mu_{R}}\xspace})$]{} using the solution of the QCD renormalisation group equation as they also enter the calculations. The inner error bars display the experimental uncertainties and the outer error bars indicate the total uncertainties. []{data-label="fig:plot2b"}]({H1prelim-17-031.fig13}.eps){width="90.00000%"}
0.03
![ Values of [$\alpha_s(m_Z)$]{} obtained from fits to ‘H1 jets’ data points with similar values of [$\mu_{R}$]{} (full circles) in comparison to values from other experiments and processes, where all values are obtained at least in NNLO accuracy. The fitted values of [$\alpha_s(m_Z)$]{} are translated to [$\alpha_s({\ensuremath{\mu_{R}}\xspace})$]{} using the solution of the QCD renormalisation group equation as they also enter the calculations. The inner error bars display the experimental uncertainties and the outer error bars indicate the total uncertainties. []{data-label="fig:plot2b"}]({H1prelim-17-031.fig11}.eps){width="90.00000%"}
The values for [$\alpha_s(m_Z)$]{} obtained from fits to the individual data sets are displayed in figure \[fig:plot2a\] and compared to the world average value of ${\ensuremath{\alpha_s(m_Z)}\xspace}=0.1181\pm0.0011$ [@Olive:2016xmw]. The results obtained when using only inclusive jet data or only dijet data are also shown. An overall reasonable consistency between the results from the individual data sets is found.
A fit to all H1 jet cross section data (denoted ‘H1 jets’), where however the HERA-I dijet cross sections are excluded from the fit because their statistical correlations to the inclusive jets are not precisely known, yields a value of ${\ensuremath{\chi^{2}}}/{\ensuremath{n_{\rm dof}}}= 1.03$ for 203 data points and the value of the strong coupling constant [$\alpha_s(m_Z)$]{} is determined to $${\ensuremath{\alpha_s(m_Z)}\xspace}= 0.1157\,(6)_{\rm exp}\, (3)_{\rm had}\, (6)_{\rm PDF}\, (12)_{\rm PDF{\ensuremath{\alpha_s}\xspace}}\, (2)_{\rm PDFset}\, (^{+27}_{-21})_{\rm scale}~.
\nonumber$$ This is consistent with the world average and with fits of the individual data sets.
The running of the strong coupling constant as a function of the renormalisation scale [$\mu_{R}$]{}, is studied by repeating the fit for groups of data points with comparable values of [$\mu_{R}$]{}. The resulting values of [$\alpha_s(m_Z)$]{} and [$\alpha_s({\ensuremath{\mu_{R}}\xspace})$]{} are displayed at a representative value [$\mu_{R}$]{} for the given range in figure \[fig:plot2b\]. The results confirm the expectations from the QCD renormalisation group equation within the accessible range in [$\mu_{R}$]{} of approximately 7 to 90[$\mathrm{GeV}$]{}. The [$\alpha_s$]{}-values are also compared to [$\alpha_s$]{}-determinations at NNLO in other reactions at similar scales and consistency is found.
Summary and conclusion
======================
The strong coupling constant is determined in a fit of new next-to-next-to-leading order (NNLO) QCD predictions to inclusive jet and dijet cross section measurements by the H1 experiment as ${\ensuremath{\alpha_s(m_Z)}\xspace}=0.1157\,(6)_{\rm exp}\,(^{+31}_{-26})_{\rm theo}$ [@thisprelim], which is in consistency with the world average value. The running of the strong coupling constant is probed over one order of magnitude and consistency with the QCD expectation is found. The NNLO calculations reduce significantly the dominating theoretical uncertainty in comparison to previously available NLO calculations. The experimental uncertainty is reduced by considering the entire set of inclusive jet and dijet cross section measurements by the H1 experiment.
[99]{}
J. Currie, T. Gehrmann, and J. Niehues, [[*Phys. Rev. Lett.*]{} [**117**]{} (2016) 042001](http://dx.doi.org/10.1103/PhysRevLett.117.042001), [[arXiv:1606.03991]{}](http://arxiv.org/abs/1606.03991).
J. Currie, T. Gehrmann, A. Huss, and J. Niehues, [[arXiv:1703.05977]{}](http://arxiv.org/abs/1703.05977).
H1 Collaboration, C. Adloff [*et al.*]{}, [[*Eur. Phys. J.*]{} [ **C19**]{} (2001) 289](http://dx.doi.org/10.1007/s100520100621), [[arXiv:hep-ex/0010054]{}](http://arxiv.org/abs/hep-ex/0010054). H1 Collaboration, F. D. Aaron [*et al.*]{}, [[*Eur. Phys. J.*]{} [**C67**]{} (2010) 1](http://dx.doi.org/10.1140/epjc/s10052-010-1282-x), [[arXiv:0911.5678]{}](http://arxiv.org/abs/0911.5678). H1 Collaboration, A. Aktas [*et al.*]{}, [[*Phys. Lett.*]{} [**B653**]{} (2007) 134](http://dx.doi.org/10.1016/j.physletb.2007.07.050), [[arXiv:0706.3722]{}](http://arxiv.org/abs/0706.3722). H1 Collaboration, V. Andreev [*et al.*]{}, [[arXiv:1611.03421]{}](http://arxiv.org/abs/1611.03421). H1 Collaboration, V. Andreev [*et al.*]{}, [[*Eur. Phys. J.*]{} [**C75**]{} (2015) 65](http://dx.doi.org/10.1140/epjc/s10052-014-3223-6), [[arXiv:1406.4709]{}](http://arxiv.org/abs/1406.4709). H1 Collaboration and V. Bertone [*et al.*]{}, H1 preliminary report, [[H1prelim-17-031](https://www-h1.desy.de/h1/www/publications/htmlsplit/H1prelim-17-031.long.html)]{} (2017),\
[available at:](https://www-h1.desy.de/h1/www/publications/htmlsplit/H1prelim-17-031.long.html) <https://www-h1.desy.de/publications/H1preliminary.short_list.html>.
NNPDF Collaboration, R. D. Ball [*et al.*]{}, [[*JHEP*]{} [**04**]{} (2015) 040](http://dx.doi.org/10.1007/JHEP04(2015)040), [[arXiv:1410.8849]{}](http://arxiv.org/abs/1410.8849). D. Britzger [*et al.*]{}, [[*Conf. Proc.*]{} [**C12-03-26.1**]{} (2012) 217](http://dx.doi.org/10.3204/DESY-PROC-2012-02/165), [[arXiv:1208.3641]{}](http://arxiv.org/abs/1208.3641).
Particle Data Group Collaboration, C. Patrignani [*et al.*]{}, [[*Chin. Phys.*]{} [**C40**]{} (2016) 100001](http://dx.doi.org/10.1088/1674-1137/40/10/100001).
[^1]: Work performed by the H1 Collaboration together with V. Bertone, J. Currie, C. Gwenlan, T. Gehrmann, A. Huss, J. Niehues and M. Sutton
|
---
abstract: 'We present an outflow survey toward 20 Low Luminosity Objects (LLOs), namely protostars with an internal luminosity lower than 0.2 $L_\odot$. Although a number of studies have reported the properties of individual LLOs, the reasons for their low luminosity remain uncertain. To answer this question, we need to know the evolutionary status of LLOs. Protostellar outflows are found to widen as their parent cores evolve, and therefore, the outflow opening angle could be used as an evolutionary indicator. The infrared scattered light escapes out through the outflow cavity and highlights the cavity wall, giving us the opportunity to measure the outflow opening angle. Using the Canada-France-Hawaii Telescope, we detected outflows toward eight LLOs out of 20 at Ks band, and based on archival *Spitzer* IRAC1 images, we added four outflow-driving sources from the remaining 12 sources. By fitting these images with radiative transfer models, we derive the outflow opening angles and inclination angles. To study the widening of outflow cavities, we compare our sample with the young stellar objects from @ar06 and @ve14 in the plot of opening angle versus bolometric temperature taken as an evolutionary indicator. Our LLO targets match well the trend of increasing opening angle with bolometric temperature reported by Arce & Sargent and are broadly consistent with that reported by Velusamy et al., suggesting that the opening angle could be a good evolutionary indicator for LLOs. Accordingly, we conclude that at least 40% of the outflow-driving LLOs in our sample are young Class 0 objects.'
author:
- 'Tien-Hao Hsieh$^{1,2}$, Shih-Ping Lai$^{1,2}$, and Arnaud Belloche$^{3}$'
title: 'Widening of Protostellar Outflows: an Infrared Outflow Survey in Low Luminosity Objects'
---
INTRODUCTION
============
Protostellar outflows are commonly seen toward Young Stellar Objects (YSOs), especially at the embedded stage. Studying the outflow properties allows us to probe the nature of the central protostars indirectly. In a protostellar core, the outflow can carve out a biconical cavity that widens as the core evolves [@ar06; @of11; @ve14 hereafter AS06 and VLT14 for the first and last references, respectively]. The widening may initially originate from the precession of a collimated jet, but the small precession angles found in protostars cannot account for the large opening angles observed at a more evolved stage [@re00; @ar04]. @ar04, therefore, suggest that the widening of the outflow cavity is likely produced by the widening of the stellar wind from the central YSO.
The near-infrared (NIR) continuum emission traces the scattered light that escapes through the outflow cavity, and has thus offered us the opportunity to study the outflow structures over the past decades [@lu97; @pa99; @ei05; @te06; @st06; @to07; @se08; @ve14]. For a given outflow opening angle and a given inclination angle, the scattered light image can be modeled with radiative transfer codes [@wh03a; @wh03b; @ro07]. Therefore, comparing the synthetic images with observed NIR images allows us to derive the outflow opening angles, and thus study their evolution.
Very Low Luminosity Objects (VeLLOs), first discovered by the *Spitzer* Space Telescope [@yo04], are defined as embedded protostars with an internal luminosity $L_{\rm int}$ $<$ 0.1 $L_{\odot}$ [@di07]; the internal luminosity is the total luminosity of the central protostar and circumstellar disk and, at the early stage, is likely dominated by the photospheric and accretion luminosities. Using the data from the *Spitzer* Legacy Project “From Molecular Cores to Planet Forming Disks” (c2d; Evans et al. 2003, 2009), @du08 identified 15 VeLLOs in the Perseus, Ophiuchus, Serpens, Lupus, and Chamaeleon molecular clouds plus 82 regions that contain 95 small, dense cores. During the last decade, studies of individual VeLLOs found that their properties vary much from one object to the other (IRAM 04191: André et al. 1999; Belloche et al. 2002; Dunham et al. 2006, L1014: Bourke et al. 2005; Huard et al. 2006, L1521F: Bourke et al. 2006; Takahashi et al. 2013, Cha-MMS1: Belloche et al. 2006, Tsitali et al. 2013, L328: Lee et al. 2009, 2013, L673-7: Dunham et al. 2010a, CB130: Kim et al. 2011, L1148: Kauffmann et al. 2011, IC 348-SMM2E: Palau et al. 2014, IRAS16253: Hsieh et al. 2016). Based on these studies, @du14 summarize three interpretations of the low luminosity, suggesting that VeLLOs can be (1) very low mass protostars, (2) extremely young protostars, or (3) protostars in a quiescent phase of the episodic accretion process, in which a protostar is at a quiescent accretion phase for most of the time and accretion bursts occasionally occur to deliver material onto the central protostar [@ke95; @le07; @du10b; @du12; @jo15; @ki16]. @sc12 conducted an outflow survey of VeLLOs, and from the outflow forces, derived the time-averaged accretion luminosities; they found that these time-averaged accretion luminosities are higher than their current internal luminosities, suggesting that VeLLOs are in a quiescent phase of an episodic accretion process. On the other hand, using the N$_2$D$^+$/N$_2$H$^+$ abundance ratio as a chemical evolutionary indicator and the line width as a dynamical evolutionary indicator, @hs15 suggest that VeLLOs tend to be young Class 0 protostars.
In this paper, we aim at studying the evolution of the outflow opening angle in protostellar objects at early stage. We describe the sample and the observations in Section \[sec:obs\]. In Section \[sec:result\], we report the observational results and describe how we derive the outflow opening angles using the radiative models from @wh03a [@wh03b]. In Section \[sec:discussion\], we discuss the correlation of the derived outflow opening angle with bolometric temperature ($T_{\rm bol}$, in comparison with AS06 and VLT14) and with the ratio of bolometric to submillimeter luminosity ($L_{\rm bol}/L_{\rm smm}$), both taken as evolutionary indicators. Finally, we summarize these results in Section \[sec:summary\].
=0.08cm
[cccccccccccc]{} DCE 001 & IRAM 04191 & 04:21:56.88 & +15:29:46.0 & 0.05 & 28 (1) & 140 & U$^*$ & Y & 25$^{+7.5}_{-7.5}$ & 20$^{+2.5}_{-17.5}$ & 1, 2\
DCE 004 & L1521F & 04:28:38.90 & +26:51:35.6 & 0.03 & 20 (1) & 140 & U$^*$ & N & 60$^{+2.5}_{-7.5}$ & 15$^{+2.5}_{-2.5}$ & 4, 11\
DCE 024 & CB130-1-IRS1 & 18:16:16.39 & -02:32:37.7 & 0.07 & 56 (7) & 270 & O & N & 15$^{+2.5}_{-2.5}$ & 55$^{+2.5}_{-2.5}$ & 9\
DCE 025 & L328-IRS & 18:16:59.47 & -18:02:30.5 & 0.07 & 68 (7) & 270 & O & N & 25$^{+2.5}_{-2.5}$ & 50$^{+2.5}_{-2.5}$ & 6, 10\
DCE 031 & L673-7 & 19:21:34.82 & +11:21:23.4 & 0.04 & 24 (4) & 300 & U & N & - & - & 7\
DCE 032 & L1148-IRS & 20:40:56.66 & +67:23:04.9 & 0.09 & 145 (6) & 325 & E & N & - & - & 8\
DCE 038 & L1014-IRS & 21:24:07.60 & +49:59:08.9 & 0.09 & 67 (14) & 250 & O & N & 65$^{+2.5}_{-2.5}$ & 20$^{+2.5}_{-2.5}$ & 3, 5\
DCE 063 & & 03:27:38.26 & +30:13:58.8 & 0.2 & 199 (26) & 250 & E & N & - & - &\
DCE 064 & & 03:28:32.57 & +31:11:05.3 & 0.03 & 65 (7) & 250 & O & N & 55$^{+2.5}_{-2.5}$ & 20$^{+2.5}_{-2.5}$ &\
DCE 065 & & 03:28:39.10 & +31:06:01.8 & 0.02 & 29 (1) & 250 & U & N & - & - &\
DCE 078 & & 03:29:23.47 & +31:33:29.5 & 0.2 & 60 (5) & 250 & O & N & 40$^{+2.5}_{-2.5}$ & 20$^{+2.5}_{-2.5}$ &\
DCE 081 & & 03:30:32.69 & +30:26:26.5 & 0.06 & 33 (2) & 250 & U & N & - & - &\
DCE 090 & & 03:32:29.18 & +31:02:40.9 & 0.2 & 114 (13) & 250 & O & N & 45$^{+2.5}_{-2.5}$ & 25$^{+2.5}_{-2.5}$ &\
DCE 092 & & 03:33:14.38 & +31:07:10.9 & 0.14 & 47 (5) & 250 & U$^*$ & N & 25$^{+12.5}_{-2.5}$ & 15$^{+7.5}_{-12.5}$ &\
DCE 107 & & 03:44:02.40 & +32:02:04.9 & 0.15 & 77 (3) & 250 & O & Y & 55$^{+2.5}_{-27.5}$& 5$^{+2.5}_{-2.5}$ &\
DCE 109 & & 03:44:21.36 & +31:59:32.6 & 0.11 & 348 (14) & 250 & E & Y & - & - &\
DCE 145 & & 15:40:51.62 & -34:21:04.7 & 0.03 &\
DCE 181 & & 16:26:48.48 & -24:28:38.6 & 0.05 & 429 (23) & 125 & P & N & - & - &\
DCE 182 & LFAM 26 & 16:27:05.23 & -24:36:29.5 & 0.15 & 105 (6) & 125 & O & N & 70$^{+2.5}_{-2.5}$ & 5$^{+2.5}_{-2.5}$ &\
DCE 185 & IRAS 16253-2429 & 16:28:21.60 & -24:36:23.4 & 0.09& 31 (1) & 125 & U$^*$ & Y & 35$^{+2.5}_{-2.5}$ & 5$^{+2.5}_{-2.5}$ & \[tab:targets\]
OBSERVATIONS {#sec:obs}
============
Sample
------
We selected the faintest targets from the catalog in @du08 which includes 50 low-luminosity protostars with $L_{\rm int}\leq1.0\ L_\odot$ (15 have $L_{\rm int}\leq0.1\ L_\odot$) in five nearby molecular clouds ($d<400$pc) mapped by the c2d team [@ev03; @ev09]. Hereafter we define the source name as the initials of the first three authors followed by the source number in @du08, e.g., DCE 185. Our sample includes 13 VeLLOs out of 15 from @du08, in which DCE 018 and 161 are excluded due to their low elevation ($<$35) from the Canada-France-Hawaii Telescope (CFHT) on Mauna Kea, Hawaii. In addition to these 13 VeLLOs, we select 7 targets out of 9 in Dunham’s catalog with $0.1\ L_{\odot} \leqslant L_{\rm int} \leqslant 0.2\ L_{\odot}$, namely Low Luminosity Objects (hereafter LLOs) in order to enlarge our sample. Table \[tab:targets\] lists the internal luminosities and the bolometric temperatures ($T_{\rm bol}$) that separate the targets into 13 Class 0 objects ($T_{\rm bol}<$ 70 K) and 7 Class I objects (70 K $\leqslant T_{\rm bol}\leqslant$ 650 K). We note that this sample includes all 15 objects of our previous study of the envelope properties [@hs15].
Ks-band images {#sec:obsk}
--------------
The observations were carried out using the Wide-field InfraRed Camera (WIRCam) at the 3.6 m CFHT in the Ks broad-band filter in March and April 2010. The WIRCam has a field of view (FOV) of $20\arcmin\times20\arcmin$ with a sampling of 03 per pixel. The Ks-band filter has a bandwidth of 0.325 $\mu$m and a central wavelength of 2.146 $\mu$m. Every target was observed with a cycled dither pattern of five positions. The exposure time per image was 25 s, and the total exposure time was 300–900s per source depending on its distance. Saturated sources were later observed with several exposures of 5s in July and October 2011. The seeing during our observations was about 07 from 05 to 12. The data were processed with the CFHT WIRCam standard pipeline. The data reduction was done with the TERAPIX[^1] software, that corrects the astrometry and image distortion against the Two Micron All Sky Survey (2MASS) catalog. The image flux levels were done by comparing with the 2MASS catalog.
H$_2$ images {#sec:obsh}
------------
The H$_2$ observations were carried out with the WIRCam at the CFHT in July and October 2011. The H$_2$ narrow-band filter has a bandwidth of 0.032 $\mu$m and a central wavelength of 2.122 $\mu$m. The seeing was about 06 (from 05 to 10) during these observations. The targets were observed with five dithered exposures. An individual exposure of 200s was taken in each frame and was repeated 11–16 times for each object. Again, in order to avoid saturation issues, we took several shorter exposures of 13s for the bright objects at NIR wavelength. The data calibration and reduction are the same as for the Ks-band observations (see Section \[sec:obsk\]).
Archival *Spitzer* data
-----------------------
We use the archival *Spitzer* InfraRed Array Camera (IRAC) images obtained by the c2d[^2] team [@ev03; @ev09]. The IRAC channel 1 and channel 2 (hereafter IRAC1 and IRAC2, respectively) have central wavelength of 3.6 $\mu$m and 4.5 $\mu$m. The spatial resolutions of IRAC1 and IRAC2 are 166 and 172, and the pixel sizes are 122 and 121, respectively. These data were processed through the c2d standard pipeline, as described in the c2d data delivery document [@ev07].
RESULTS AND ANALYSIS {#sec:result}
====================
Observed images {#sec:obs_r}
---------------
### Ks-band and IRAC images {#sec:ksband}
Our Ks-band observations reveal a number of sources driving protostellar outflows. Figure \[fig:kband\] shows the Ks-band images of the 20 selected LLOs. Thirteen objects are detected (see Table \[tab:targets\]). Based on the morphology in the Ks-band image, we identify DCE 145 as a background galaxy misidentified as a VeLLO and exclude it from our sample. Out of the remaining 12 detected objects, eight sources show a conical or biconical structure (see Figure \[fig:kband\] and Table \[tab:targets\]). The other detected sources show extended emission except for DCE 181. Although the extended emission may come from a nearly pole-on outflow, we cannot rule out the possibility that it is an infrared nebula or a background galaxy. In addition, we cannot derive the inclination angle of a nearly pole-on outflow because the inclination angle is estimated based on the contrast between the red- and blue-shifted emissions. Thus, we exclude those extended objects in our analysis. As a result, we identify eight outflow-driving LLOs on the basis of the Ks-band data.
Because the Ks-band images have a higher spatial resolution than the IRAC1 images, they allow us to explore the outflow structures better. However, the scattered light at the short wavelength (2.146 $\mu$m) could be greatly attenuated by the circumstellar envelope for deeply embedded protostars. Since the *Spitzer* IRAC1 images at a longer wavelength are less affected by extinction, they reveal outflow cavities in four additional embedded objects in our sample (DCE 004/L1521F, DCE 001/IRAM 04191, DCE 185/IRAS 16253, and DCE 092). In addition to extinction, the non-detections in the Ks band can also be explained by nearly edge-on configurations, as indicated by the modeling of the infrared images (see Section \[sec:res\]). As a result, outflow cavities are found in 12 LLOs out of 19 on the basis of the Ks-band or IRAC1 images.
To study the outflows, we compare their morphologies at NIR wavelength with that in CO emission reported in the literature. In the plane of the sky, the outflow orientations in the NIR observations are approximately consistent with that from the CO observations in DCE 004 (L1521F: Takahashi et al. 2012), DCE 025 (L328-IRS: Lee et al. 2013), DCE 038 (L1014-IRS: Bourke et al. 2005), DCE 185 (IRAS 16253: Stanke et al. 2006; Hsieh et al. 2016), DCE 092, and DCE 078 (M. Hiramatsu, private communication). However, the NIR and CO observations could sometimes trace different components of an outflow. Toward DCE 038, the CO (2–1) map (see Figure 1 in Bourke et al. 2005) reveals a compact ($\sim$5) bipolar outflow, but our Ks-band image shows a more extended structure ($\sim$10) with a large opening angle which is consistent with the H-band and Ks-band images in @hu06. Although the CO observations trace large-scale outflows ($\gtrsim$100) in DCE 001 (IRAM 04191: André et al. 1999, Lee et al. 2002) and DCE 025 (L328-IRS: Lee et al. 2013), the infrared images reveal only the inner regions ($\lesssim$10) of the outflows. In addition, the position angle found in the IRAC1 image of DCE 001 differs by $\sim$30 degree from the position angle in the CO observations [@an99; @le02]. Besides, the opening angle derived from the CO observations (45$\arcdeg$) is much larger than the angle derived from our fitting (25$\arcdeg$) (see Section \[sec:res\]). The reason for this discrepancy remains unclear, and high-angular-resolution, high-sensitivity data at NIR wavelength would be required to understand it.
![Two-color image of DCE 185 with red scale for Spitzer IRAC1 (3.6 $\mu$m) continuum emission and green scale for CFHT H$_2$ emission (2.12 $\mu$m). []{data-label="fig:16253"}](IRAS16253.pdf)
Through NIR scattered light, the outflow cavities show quite different morphologies. The cavity size varies from few hundreds to few thousands au. Most of the bipolar cavities are asymmetric, being detected (or stronger) in one side only as commonly seen at NIR wavelength due to inclination effects; the undetected/weak lobes are believed to be the red-shifted lobes hidden behind the foreground envelopes. As a matter of fact, the asymmetries in the infrared images are consistent with the blue/red-shifted lobes detected by CO observations in DCE 001 [@an99] and DCE 038 [@bo05]. CO blue-shifted and red-shifted emissions appear in both sides of the outflow in DCE 004 [@ta13] and DCE 025 [@le13], suggesting a nearly edge-on inclination and a probably large opening angle; this result is compatible with the NIR observations. However, for DCE 185, the brighter NIR lobe in the south-west overlaps with the red-shifted lobe found by multitransition CO observations [@hs16]. We speculate that this inconsistency is due to an intrinsic property; instead of inclination effects, the south-west lobe may contain more scattered light because of heating by the UV-radiation from the stronger south-west H$_2$ jet (Figure \[fig:16253\]).
This discussion reveals that the NIR asymmetric bipolar cavities provide clues to estimate the inclination angle of the outflows. In most cases, the bipolar cavities would appear more symmetric in the systems with smaller inclination angles (with respect to the plane of the sky) and would be more asymmetric in those with larger inclination angles. Thus, we use this property to derive the inclination angle by modeling the observed images (see Section \[sec:model\])
### H$_2$ images {#h_2-images}
Figure \[fig:hband\] shows the H$_2$ images toward the 19 LLOs (DCE 145 has been excluded, see Section \[sec:ksband\]). The H$_2$ observations were performed with a narrow filter but with a long integration time, allowing them to trace the continuum emission as well as the Ks-band image. The H$_2$ images resembling the Ks-band ones are likely dominated by the continuum emission (Figure \[fig:hband\]). Hereafter, we call Figure \[fig:hband\] as “H$_2$ image(s)” when it actually consist of H$_2$ emission and continuum emission. We note that the Ks broad-band filter includes the H$_2$ line, but with a 10 times larger filter width corresponding to $\sim$45000 km s$^{-1}$. Thus, the H$_2$ emission is greatly diluted in the Ks-band image.
Comparing with the Ks-band images, our H$_2$ survey reveals a robust jet only toward DCE 185 (IRAS 16253, Figure \[fig:16253\]) as in the 2.12 $\mu$m H$_2$ image of @kh04 and in the mid-infrared InfraRed Spectrometer (IRS) H$_2$ image of @ba10. The H$_2$ jet of DCE 185 shows a prominent “S-shaped” symmetry around the central source (Figure \[fig:16253\]). It is believed to originate from the tidal interactions between the disk where the jet originates and a companion in a noncoplanar orbit [@hs16].
Besides, we identify three sources with marginal H$_2$-jet detections by eye. Because the H$_2$ image can be dominated by the continuum emission, we search for jet-like structures that are seen in the H$_2$ image (Figure \[fig:hband\]) but not in the Ks-band image (Figure \[fig:kband\]). As a result, we find three H$_2$-jet driving candidates: DCE 001, 109, and 107. These sources show very different morphologies of jets: (1) DCE 001 shows very weak H$_2$ emission roughly along the outflow direction seen in the CO (2–1) map from the IRAM 30 m telescope [@an99]. (2) In DCE 109, a collimated H$_2$ emission appears from the central star to the north and may extend to a clear H$_2$ jet knot. (3) We found two H$_2$ knots in the north-west direction that are likely driven by DCE 107 along a collimated jet. Because DCE 001, 109, and 107 have only marginal detections of H$_2$ jets, we call these three objects “jet-driving candidates”.
Statistically, our H$_2$ jet survey has a low detection rate: only one prominent H$_2$ jet-driving source and three jet-driving candidates out of 19 LLOs. These three candidates, if real at all, show very weak H$_2$ emission. Because the H$_2$ line emission usually traces high-velocity gas ($>$100 km s$^{-1}$) that produces high-energy photons exciting H$_2$ [@wo91], this result implies that most of the LLOs drive low-velocity outflows rather than high-velocity jets. Therefore, the low detection rate of H$_2$ jets is consistent with a scenario in which LLOs are at a quiescent accretion phase and thus drive weak outflows.
Infrared image modeling {#sec:model}
-----------------------
Measuring the outflow opening angle requires a good knowledge of the inclination angle because, due to inclination effects, the apparent opening angle seen in the image is in fact larger than the true opening angle. Inclination angles can be estimated based on the intensity ratio between the blue-shifted and red-shifted lobes. Considering a spherically-symmetric envelope, the scattered light from the red-shifted lobe is more attenuated, by the thick foreground envelope, than that from the blue-shifted lobe. Therefore, a large inclination angle yields a large difference in brightness between the blue-shifted and red-shifted outflows, which allows us to derive the inclination angle, and in turn the opening angle.
We fit the NIR images (Ks-band or IRAC1) with the radiative transfer model of @wh03a for the 12 sources with outflow detections (see Section \[sec:ksband\] and Table \[tab:targets\]). Note that although we tried to fit the three extended sources (Table \[tab:targets\]), the best-fit results were very different from the observations (see Section \[sec:ksband\]). Thus, we exclude these three sources in our analysis. We remind the reader that the IRAC1 images have a higher sensitivity but a much lower angular resolution (see Section \[sec:ksband\]). Thus, the IRAC1 images enable us to probe outflows from more embedded objects, and the Ks-band images provide better constraints on the opening angles.
### Setup of the models {#sec:setup}
@wh03a provide a radiative transfer code to model the scattered light images of an outflow cavity (hereafter Whitney’s code/model). This code is based on the Monte Carlo radiative equilibrium routine developed by @bj01, which calculates the radiative transfer in a three-dimensional spherical-polar grid. Whitney’s code models a spherically-symmetric envelope with biconical cavities carved out by a bipolar outflow viewed at different inclinations. The default cavity shape follows $z\propto\sqrt{x^2+y^2}^b$ where $x$, $y$, and $z$ are the Cartesian coordinates and $b$ is the power-law exponent of 1.5. Using this code, @wh03b present an evolutionary sequence of models for a low mass protostar at stages of Class 0, Late 0, I, Late I, II, and III. Because our targets have bolometric temperatures ranging from 20 K to 114 K (Table \[tab:targets\]), we take the Class Late 0 model as the template model and then vary the opening angle and inclination angle to fit the observed Ks-band and IRAC1 images.
Here, we discuss how the parameter set in Whitney’s code may affect the modeled images. Whitney’s code categorizes several tens of parameters into three groups: central star, disk, and envelope. We speculate that the central star and disk may only weakly affect the large-scale outflow cavities at hundreds to thousands au, when the disk of a Class Late 0 source [@wh03b] has an outer radius of 50au ($\sim$02–04 in our sample). In addition, to remove the influence of the star and disk emission, we mask the central source in a circular region in our fitting process (see Section \[sec:res\]). Furthermore, we find that the internal luminosity (associated to parameters of star and disk) likely affects only the image brightness scale of the cavity but not the distribution or structure. Therefore, we use a linear function to scale the intensity map (see Section \[sec:res\]), which makes our fitting results independent of the internal luminosity. On the other hand, we test whether or not the envelope properties could change the brightness distributions of the outflow cavities. We find that the density profiles in the cavity can significantly change the intensity scale but not the distribution. We also find that the mass infall rate ($\dot{M}_{\rm env}$) can alter the intensity distribution but does not significantly affect the opening angle, and for the Class Late 0 objects, it is set to 10$^{-5}$ $M_\odot\rm~yr^{-1}$ [@wh03b].
### Fitting the opening angles and inclination angles {#sec:res}
There are several steps in our fitting process. Since the luminosity of the central source can change the image brightness, we linearly scale the intensity of the modeled image to fit the observations. To remove the effects of field stars and the central object (star$+$disk), we mask these sources with adequate circles determined by eye. Using Whitney’s code, we construct a grid of models that vary opening angle ($\theta_{\rm open}$), inclination angle ($\theta_{\rm inc}$), and position angle (P.A.) with a cell size of 5$\arcdeg$ for all dimensions. Then, we derive the $\chi^2$ values between the observed image and each model, and as a result, we obtain a $\chi^2$ distribution in the three-dimensional grid. Then, we find the best-fit opening angle and inclination angle, and taking a 99.7% confidence level with five free parameters ($\theta_{\rm open}$, $\theta_{\rm inc}$, P.A., and two in the linear function) ($\Delta\chi^2>18.2$, Press et al. 1992), we obtain its upper and lower limits (Table \[tab:targets\]). Figure \[fig:mod\] shows the images of the best-fit models and the corresponding observed images.
We discuss here two special cases, DCE 185 and 092, with arbitrary decisions in our fitting process. For DCE 185, as the south-west lobe is brighter than the north-east lobe in the IRAC1 image (Figure \[fig:16253\]), this brightness asymmetry is in conflict with the blue- and red-shifted outflows identified by CO observations [@st06; @ma13; @hs16]. Thus, we speculate that this asymmetric bipolar cavity is due to an intrinsic property rather than inclination effects (see Section \[sec:ksband\]). In the fitting, we restrict the red-shifted lobe to be in the south-west as seen in the CO observations, and since the model assumes the asymmetry comes from inclination effects, we obtain an inclination angle of 5, the lower limit of our model grid. This underestimate of the inclination angle yields an overestimate of the opening angle. Therefore, we take the derived opening angle of DCE 185 as an upper limit for later analysis. Besides, the $\chi^2$ distribution of DCE 185 shows two local minima at opening angles of 125$\arcdeg$ and 35$\arcdeg$. We eliminate the larger opening angle, since it is most likely caused by contamination of diffuse cloud emission. For DCE 092, the $\chi^2$ distribution also has two local minima located at opening angles of 100$\arcdeg$ and 35$\arcdeg$. We eliminate the larger opening angle because it is likely affected by the nearby bright sources.
[ccccccc]{} =0.06cm DCE 004 & - & - & 60$^{+2.5}_{-7.5}$ & 15$^{+2.5}_{-2.5}$ & 70$^{+2.5}_{-2.5}$ & 10$^{+2.5}_{-2.5}$\
DCE 064 & 55$^{+2.5}_{-2.5}$ & 20$^{+2.5}_{-2.5}$ & 45$^{+2.5}_{-2.5}$ & 20$^{+2.5}_{-2.5}$ & - & -\
DCE 078 & 40$^{+2.5}_{-2.5}$ & 20$^{+2.5}_{-2.5}$ & 40$^{+2.5}_{-2.5}$ & 20$^{+2.5}_{-2.5}$ & - & -\
DCE 090 & 45$^{+2.5}_{-2.5}$ & 25$^{+2.5}_{-2.5}$ & 30$^{+2.5}_{-2.5}$ & 55$^{+2.5}_{-2.5}$ & - & -\
DCE 092 & - & - & 25$^{+12.5}_{-2.5}$ & 15$^{+7.5}_{-12.5}$ & 45$^{+2.5}_{-17.5}$ & 5$^{+7.5}_{-2.5}$\
DCE 107 & 55$^{+2.5}_{-27.5}$& 5$^{+2.5}_{-2.5}$ & 40$^{+2.5}_{-2.5}$ & 5$^{+2.5}_{-2.5}$ & - & - \[tab:irac\]
To check our fitting results, we fit the IRAC1 images in four sources (DCE 064, 107, 090, and 078) that have outflow detections at IRAC1 in addition to the Ks-band, and the IRAC2 images of the following two sources: (1) DCE 004, which is an outlier in the $T_{\rm bol}-\theta_{\rm open}$ plot (see Section \[sec:opeTbol\] and Figure \[fig:wid\]), and (2) DCE 092, for which the inclination angle derived from the fitting of its IRAC1 image has a large uncertainty. Comparing the fitting results at two wavelengths, we obtain consistent inclination angles and similar opening angles (difference $\leqslant$ 10 or within the errors), except for DCE 090 (Table \[tab:irac\] and Figure \[fig:mod4\]). In DCE 090, we suspect that the discrepancy between $\theta_{\rm inc, IRAC1}=55\arcdeg$ and $\theta_{\rm inc, Ks}=25\arcdeg$ comes from an unreliable fitting in IRAC1 due to the contamination from the bright central object. Because the Ks-band image has a better resolution, we believe that the result from the Ks-band is more reliable for DCE 090. Overall, we conclude that fitting the images at different wavelengths gives comparable results. As a result, we list the best-fit results from the relatively shorter wavelengths in Table \[tab:targets\] and use these values in the following analysis.
![ (a) Plots of outflow opening angle versus bolometric temperature. (b) Same as (a) but with inclination correction for opening angles in AS06’s and VLT14’s samples (assuming $\theta_{\rm inc}=32\arcdeg7$). The color circles (triangles) indicate the LLOs with the opening angle obtained from Ks-band (IRAC1 band). The vertical dashed lines show the boundaries of Class 0, I and II in bolometric temperatures. The black dots and plus signs show the sources of AS06 and VLT14, and the respective gray solid and gray dashed lines indicate the correlations derived by these two studies (Equations \[eq:arce2\] and \[eq:vela\]). The green line shows our best-fit power-law for LLOs. The parameters of the best fit are written in the bottom right corner](Tbol_vs_Op_asymErrA.pdf)
(Equation \[eq:llo\]). \[fig:wid\]
### Fitting with streamline cavity
In addition to the curved cavity mentioned above, Whitney’s code provides an alternative cavity shape namely streamline which is conical on large scales and might be carved out by precessing jets [@wh03a]. To study the origin of the outflow widening, we also fit the observed images with the model of streamline cavity and compare the results with those obtained assuming a curved cavity. We find that, except for DCE 001 and DCE 024 that are better fitted with the streamline cavity, all other objects are better fitted with the curved cavity above a confidence level of 99.7%. DCE 001 and DCE 024 have the smallest opening angles and relatively low bolometric temperatures among our targets, suggesting that they are younger than other sources. Although, based on this, one could be tempted to conclude that the outflow cavity is carved by precessing jets at the earliest stage, a sample of only two sources is not large enough to be statistically meaningful. In addition, the large-scale CO outflows in DCE 001 do not show the streamline shape [@an99]. Therefore, we lack evidence to support the fact that the outflow widening is caused by carving of precessing jets.
DISCUSSION {#sec:discussion}
==========
Outflow opening angle versus bolometric temperature {#sec:opeTbol}
---------------------------------------------------
Here, we compare our LLOs with YSOs from AS06 and VLT14 in the plot of bolometric temperature versus opening angle (Figure \[fig:wid\]a). To study the evolution of protostellar outflows, previous works have studied the correlation between outflow opening angle and bolometric temperature taken as an evolutionary indicator. Based on their survey of CO outflows toward YSOs, AS06 found that the outflow opening angle widens as the core evolves from Class 0 to Class III. They found a correlation between the outflow opening angle ($\theta_{\rm open}$) and the bolometric temperature ($T_{\rm bol}$) as $$\log(\frac{\theta_{\rm open}}{\rm deg}) = (0.47 \pm 0.20)+(0.60 \pm 0.10)\log(\frac{T_{\rm bol}}{\rm yr}).
\label{eq:arce2}$$ Lately, using [*Spitzer*]{} IRAC images, VLT14 measured the outflow opening angles toward 31 YSOs and obtained a $T_{\rm bol}-\theta_{\rm open}$ correlation of
\[eq:vela\] $$\begin{aligned}
\log(\frac{\theta_{\rm open}}{\rm deg}) & = 0.54+0.77\log(\frac{T_{\rm bol}}{\rm yr}),~{\rm for}~T_{\rm bol}<100K\\
&= 1.92+0.05\log(\frac{T_{\rm bol}}{\rm yr}),~{\rm for}~T_{\rm bol}>100K. \end{aligned}$$
Before we compare our sample with AS06’s and VLT14’s results, two caveats should be made. First, because both AS06 and VLT14 measured the opening angles without inclination correction, their measurements should be viewed as upper limits in comparison with ours. Second, the NIR scattered light could highlight a broader outflow cavity than CO [@oh97; @ta97; @le06; @la08; @ar13], because CO may not be able to trace the full extent of the NIR reflection nebula due to its high opacity especially at the velocities close to the ambient cloud velocity [@ar13].
We find that most LLOs fit well the $T_{\rm bol}-\theta_{\rm open}$ trend found by AS06, except for DCE 004 (L1521F) and DCE 024 (CB-130-1-IRS1). Two reasons may explain the deviation of DCE 004: (1) DCE 004 may host a binary system driving two outflows, as SMA CO (2–1) observations suggest [@ta13]. The possible two outflows have different axes as seen in many other cases [@of16; @le16], which can affect our opening angle estimate with NIR observation. Therefore, we consider the measured opening angle as an upper limit. (2) Our fitting result implies a nearly edge-on configuration, which may cause the low bolometric temperature (see Section \[sec:corT\]). Although DCE 024 deviates from the power-law relation of AS06, it fits well the new power-law index that better describes the sources with $T_{\rm bol}\lesssim 120$ K (see below).
We use a single power-law to fit our LLO sample (Figure \[fig:wid\]a). The best fit is $$(\frac{\theta_{\rm open}}{\rm deg}) = 10^{-1.42\pm0.39} \times (\frac{T_{\rm bol}}{K})^{1.64\pm0.21}.
\label{eq:llo}$$ We have excluded DCE 004 and DCE 185 from the fitting since their fitted opening angles are considered as upper limits. The best-fit power-law index, 1.64, is much larger than the index derived by AS06. If we include the data points with $T_{\rm bol} \leqslant120$ K from AS06, we obtain $(\frac{\theta_{\rm open}}{\rm deg}) = 10^{-1.02\pm0.26} \times (\frac{T_{\rm bol}}{K})^{1.46\pm0.14}$, which is consistent with Equation \[eq:llo\] within $1\sigma$. This suggests that we need a broken power law or a more complicated model to describe how the outflow cavities widen. In addition, based on this large index, we suggest that the outflow opening angle may better discriminate the evolutionary state of protostars than the bolometric temperature in a range of $T_{\rm bol} \leqslant 120~K$.
VLT14 used two power-law components to fit their data and found a break at $T_{\rm bol} \approxeq 100~K$. However, the steeper power law at the lower $T_{\rm bol}$ end (Equation \[eq:vela\]a) is quite different from our fitting result (Equation \[eq:llo\]). We speculate that this discrepancy partially comes from inclination effects that were not taken into account by VLT14. To correct for this, we assume that all sources of VLT14 (as well as AS06) have an inclination angle of 327 (with respect to the plane of the sky) and we estimate their inclination-corrected opening angles (Figure \[fig:wid\]b); the angle of 327 corresponds to a mean inclination angle assuming all orientations are equally favorable [@bo96; @du14 note that the angle 573 in the references is with respect to the line of sight]. After the correction, the VLT14 sample is broadly consistent with our LLOs. We find a new-best fit of $\log(\frac{\theta_{\rm open}}{\rm deg}) =-0.38+1.25\log(\frac{T_{\rm bol}}{\rm yr})$ for $T_{\rm bol}<100K$ for the inclination-corrected sample of VLT14, which is consistent with our LLOs (Equation \[eq:llo\]) within $\sim$3-4$\sigma$.
We now compare the opening angles of our LLO sample with the Class 0 samples studied by AS06, @se08, and VLT14. The AS06 sample includes 11 Class 0 objects, Seale & Looney’s sample has 21 Class 0 objects, and the VLT14 sample contains 20 Class 0 objects. Out of the 12 LLOs with NIR outflow detections in our sample, five ($\sim$40%) have opening angles smaller than $25\arcdeg$. Such small opening angles are rare in the Class 0 samples of AS06, @se08, and VLT14 (3/11, 2/21, and 0/20, respectively). Note that the ratios remain the same after the inclination correction. Assuming that the opening angle is an evolutionary indicator, we conclude that our sample of outflow-driving LLOs contains a higher fraction of young objects than the samples of AS06, @se08, and VLT14. In turn, this suggests that at least the five LLOs with $\theta_{\rm open}\leqslant25\arcdeg$ may be very young Class 0 objects.
Approximate correction for inclination effects on $T_{\rm bol}$ {#sec:corT}
---------------------------------------------------------------
Figure \[fig:wid\]a shows that the LLOs below the best-fit line have relatively large inclination angles while the sources above have relatively small inclination angles. Because the inclination angle can affect the Spectral Energy Distribution (SED), and in turn the bolometric temperature [@wh03a; @wh03b; @ro06; @cr08], this geometrical effect may cause the LLOs to deviate from the best-fit relation.
![ (a) Same as Figure \[fig:wid\] but using $T_{\rm bol}$ from the best-fit modeled SED in LLOs. (b) Same as (a) but using $T_{\rm bol}$ from the synthetic SED with $\theta_{\rm inc}=27\arcdeg$ (see Section \[sec:corT\]). The black line shows our best-fit power-law.[]{data-label="fig:wid2"}](Tbol_vs_Op_asymErrB.pdf)
To investigate whether the inclination effects cause the dispersion in Figure \[fig:wid\]a, we correct the bolometric temperatures for inclination effects using the SED fitting tool of @ro06 [@ro07]. This SED fitting tool, constructed with the radiative transfer code of @wh03a, consists of 20,000 YSO models with SEDs computed at 10 viewing angles for each model, resulting in 200,000 SEDs in total. Since the SEDs’ uncertainty grows rapidly at wavelengths $\gtrsim$100 $\mu$m and becomes very high at wavelengths $\gtrsim$600 $\mu$m [see Figure 1 in @ro06], we ignore the SED data points at wavelengths $>$850 $\mu$m, and for the sources lacking 850 $\mu$m data, interpolate a flux at 850 $\mu$m logarithmically. From the processed SEDs, we calculate the bolometric temperatures (Table \[tab:targets\]). The derived temperatures are consistent with the results of @du08 within the uncertainties except for DCE 031 (DCE 031 has no outflow detection such that it will not affect the following analysis.). We fit the processed SEDs with the YSO models of @ro06 [@ro07]. Then, we identify the best-fit model as the one with the lowest $\chi^2$ and with the consistent $\theta_{\rm open}$ and $\theta_{\rm inc}$ from our image fits (see Section \[sec:res\]) within a difference $<$10. To get rid of inclination effects on $T_{\rm bol}$, we define an inclination-independent bolometric temperature as the bolometric temperature of the synthetic SED of the best-fit model computed for an arbitrary inclination angle of 27. This inclination angle corresponds to about the averaged value in our sample (Table \[tab:targets\]). We remind the readers that our survey at NIR wavelength cannot identify outflows at a nearly pole-on configuration. Therefore, a small inclination angle is reasonable. We calculate the bolometric temperatures from the best-fit SED and from the synthetic SED, namely best-fit $T_{\rm bol}$ and synthetic $T_{\rm bol}$, respectively. With the same inclination, we can use the synthetic $T_{\rm bol}$ as an evolutionary indicator without inclination effects. Figures \[fig:wid2\]a and \[fig:wid2\]b show the $T_{\rm bol}-\theta_{\rm open}$ relations with the best-fit $T_{\rm bol}$ and synthetic $T_{\rm bol}$, respectively, in which, for comparison, we assign the same uncertainty as the observed $T_{\rm bol}$ in percentage. By fitting a single power-law as Equation (\[eq:llo\]), we find that the $T_{\rm bol}-\theta_{\rm open}$ correlation with the synthetic $T_{\rm bol}$ has a smaller $\chi^2$ (see Figure \[fig:wid2\]). This result supports our speculation that the inclination effects may cause the deviation from the best-fit $T_{\rm bol}-\theta_{\rm open}$ relation.
There are three caveats about the inclination correction. First, although we use the image fits to constrain $\theta_{\rm open}$ and $\theta_{\rm inc}$, the SED fit may still be degenerated, preventing us from determining the physical parameters. Thus, our inclination correction on $T_{\rm bol}$ should be considered as a rough approximation rather than a precise determination. Second, the SED models of @ro06 do not include brown dwarfs while they are used to interpret the low luminosity of VeLLOs in the literature [@bo05; @ka11; @le13; @pa14]. Third, the synthetic $T_{\rm bol}-\theta_{\rm open}$ correlation is probably a self-consistent result produced artificially by the radiative transfer code [@wh03a; @ro06; @ro07] because both the synthetic $T_{\rm bol}$ and $\theta_{\rm open}$ correspond to the model rather than the observation. Despite these caveats, the synthetic $T_{\rm bol}-\theta_{\rm open}$ and the observed $T_{\rm bol}-\theta_{\rm open}$ have very similar power-law fits (see equations in Figures \[fig:wid\] and \[fig:wid2\]b). This implies that the radiative transfer code reproduces well the observed $T_{\rm bol}-\theta_{\rm open}$ correlation at least for these LLOs. This discussion tentatively suggests that the tighter synthetic $T_{\rm bol}-\theta_{\rm open}$ correlation is reliable.
To further test the inclination correction on $T_{\rm bol}$, we compare the best-fit $T_{\rm bol}$ and the synthetic $T_{\rm bol}$ with the ratio of bolometric to submillimeter luminosity ($L_{\rm bol}/L_{\rm smm}$) taken as a better evolutionary indicator. We exclude DCE 038 because of its problematic measurement of $L_{\rm bol}/L_{\rm smm}$ (see Section \[sec:LL\]). We find that $L_{\rm bol}/L_{\rm smm}$ is slightly better correlated to the synthetic $T_{\rm bol}$ with $r=0.57$ and $\rho=0.69$ (p-values of 0.11 and 0.04, respectively) than to the best-fit $T_{\rm bol}$ with $r=0.09$ and $\rho=0.50$ (p-values of 0.81 and 0.17, respectively). Although the sample is small, this result suggests that the synthetic $T_{\rm bol}$ could better reflect the evolutionary stage than the best-fit $T_{\rm bol}$.
![ Plot of observed bolometric temperature ($T_{\rm bol}$) versus the ratio of bolometric to submillimeter luminosity ($L_{\rm bol}/L_{\rm smm}$). The black points indicate the sources that were fitted in the $T_{\rm bol}-\theta_{\rm open}$ plane ([*i.e.*]{} Figures \[fig:wid\] and \[fig:wid2\]) and the white points indicate those that were not. The Pearson ($r$) and Spearman ($\rho$) correlation coefficients and their significance are displayed in the top left corner and are calculated without the data point of DCE 038. The dashed lines show the boundary of the Class 0 and Class I for each evolutionary indicator ([$T_{\rm bol}=70K$: @ch95], [$L_{\rm bol}/L_{\rm smm}$=200: @an93]).[]{data-label="fig:TbolLL"}](Tbol_vs_LL.pdf)
Outflow opening angles versus $L_{\rm bol}/L_{\rm smm}$ and other parameters from SED {#sec:LL}
-------------------------------------------------------------------------------------
To study what physical conditions may affect the outflow opening angle, we compare the latter to four parameters from @du08: the internal luminosity ($L_{\rm int}$), the bolometric luminosity ($L_{\rm bol}$), the ratio of bolometric to submillimeter luminosity ($L_{\rm bol}/L_{\rm smm}$), and the submillimeter luminosity ($L_{\rm smm}$). We exclude DCE 185 and 004 because their opening angles are considered as upper limits (see Sections \[sec:res\] and \[sec:opeTbol\]). We evaluate the significance of correlation between the outflow opening angle and these parameters using Pearson’s $r$ correlation test and Spearman’s $\rho$ rank correlation test [@co99]. Based on the correlation tests, we find no significant correlations between $\theta_{\rm open}$ and these parameters, except for $L_{\rm bol}/L_{\rm smm}$. We find a correlation coefficient $r=0.55$ for $\theta_{\rm open}-L_{\rm bol}/L_{\rm smm}$, suggesting a probability (p-value) of 10% for it being an uncorrelated/random distribution. Since $L_{\rm bol}/L_{\rm smm}$ is considered as an evolutionary indicator [@an93; @yo05], we conclude that the opening angle is most likely sensitive to the evolutionary status but not to other physical conditions such as envelope mass traced by $L_{\rm smm}$.
![ Plot of outflow opening angles versus the ratio of bolometric to submillimeter luminosity ($L_{\rm bol}/L_{\rm smm}$). The Pearson ($r$) and Spearman ($\rho$) correlation coefficients and their significance are displayed in the top right corner.[]{data-label="fig:par"}](cor_others.pdf)
@yo05 suggest that $L_{\rm bol}/L_{\rm smm}$ can better reflect the evolutionary status than $T_{\rm bol}$ because, in their evolutionary model, $L_{\rm bol}/L_{\rm smm}$ is sensitive to the fraction of mass accreted onto the central object but is less affected by the initial core mass. Figure \[fig:TbolLL\] shows our LLO sample in a $L_{\rm bol}/L_{\rm smm}$ versus $T_{\rm bol}$ diagram, along with the boundaries traditionally used to separate the Class 0 and I phases, either in bolometric temperature [$T_{\rm bol}=70K$ @ch95] or in bolometric-to-submm luminosity ratio [$L_{\rm bol}/L_{\rm smm}$=200 @an93]. The classification of our LLO sample into Class 0 and I objects depends somewhat on the chosen axis. Three objects out of 19 fall into the Class 0 category according to their $L_{\rm bol}/L_{\rm smm}$ ratio but into the Class I category according to their bolometric temperature. The classification of the rest of the sample is the same with both criteria. Figure \[fig:TbolLL\] shows a correlation between $T_{\rm bol}$ and $L_{\rm bol}/L_{\rm smm}$ toward the 19 LLOs ($r = 0.74$ and $\rho=0.72$ with both p-values $=0$), though the correlation is less significant when considering only the ten targets used to derive the $T_{\rm bol}-\theta_{\rm open}$ relation. DCE 038 seems to be an outlier in both Figures \[fig:TbolLL\] and \[fig:par\] with an extremely low $L_{\rm bol}/L_{\rm smm}$ ratio (3 $\pm$ 1). This source has a puzzling jump in SED with 3.2 $\pm$ 0.5 Jy at 350 $\mu$m and 21.5 $\pm$ 16.1 Jy at 450 $\mu$m (1.8 $\pm$ 0.4 Jy at 850 $\mu$m), which results in the low $L_{\rm bol}/L_{\rm smm}$ [@du08] when the submillimeter luminosity is defined as the luminosity at $\lambda>350~\mu$m [@an93]. $L_{\rm bol}/L_{\rm smm}$ would become 10 $\pm$ 1 if we ignore the flux density at 450 $\mu$m or 6 $\pm$ 3 if we replace the observed flux density at 350 $\mu$m by a linear interpolation between 70 and 450 $\mu$m. This suggests that the observed $L_{\rm bol}/L_{\rm smm}$ is underestimated. Therefore, we remove DCE 038 and, in turn, find better correlation coefficients of $r = 0.74$ and $\rho=0.80$ between $T_{\rm bol}$ and $L_{\rm bol}/L_{\rm smm}$ (Figure \[fig:TbolLL\]). Furthermore, comparing to the outflow opening angle, we obtain correlation coefficients of $r = 0.71$ and $\rho=0.77$ (p-values of 0.03 and 0.02, respectively) for $L_{\rm bol}/L_{\rm smm}-\theta_{\rm open}$ (Figure \[fig:par\]) and $r = 0.67$ and $\rho=0.70$ (p-values of 0.05 and 0.04, respectively) for $T_{\rm bol}-\theta_{\rm open}$. Therefore, we suggest that the distribution of opening angles is better correlated with evolutionary status using $L_{\rm bol}/L_{\rm smm}$ as an age indicator than using $T_{\rm bol}$. This result implies that the outflow opening angle likely reflects the evolutionary status well and could be considered as a good evolutionary indicator.
SUMMARY {#sec:summary}
=======
We conducted an outflow/jet survey of 20 Low Luminosity Objects (LLOs) using CFHT in Ks-band continuum emission and H$_2$ line emission. From the Ks-band observations, we identify a background galaxy and eliminate it from our sample. We detect outflow cavities in eight sources out of 19 LLOs on the basis of the Ks-band continuum observations. Among the remaining 11 LLOs, the archival *Spitzer* IRAC1 data reveals four outflow-driving sources. We derive the outflow opening angles and the inclination angles of these 12 LLOs by fitting the observed images with the radiative transfer models of @wh03a. The H$_2$ observations reveal only one LLO associated with a prominent H$_2$ jet. Three other LLOs out of the remaining 18 have marginal detections of an H$_2$ jet. Our main results are the following:
1. As we detect only one to four H$_2$ jets out of 19 LLOs, we suggest that most LLOs do not generate strong jets or outflows. This indirectly supports the idea that these LLOs are likely at a quiescent accretion phase.
2. Our LLOs follow a trend similar to the one found by AS06 in the plot of bolometric temperature versus opening angle ($T_{\rm bol}-\theta_{\rm open}$), and are broadly consistent with the correlation reported by VLT14, after correction for the inclination. Instead of a single power-law describing the distribution, we find a larger index for the sources at an early evolutionary stage ($T_{\rm bol}$ $\lesssim$ 120 K), in agreement with VLT14. We conclude that the outflow opening angle may better trace the evolutionary stage than $T_{\rm bol}$ for $T_{\rm bol} < 120$ K.
3. The LLOs located above the best-fit power-law index have relatively small inclination angles and those located below have large inclination angles. This suggests that the dispersion in the $T_{\rm bol}-\theta_{\rm open}$ plot may be in part due to inclination effects on $T_{\rm bol}$.
4. Using the outflow opening angle as an evolutionary indicator, we suggest that at least 40% of the outflow-driving LLOs in our sample are young Class 0 objects.
5. Out of the 12 targets with infrared outflow detections, ten are better fitted with a curved cavity than a streamline cavity, suggesting that the outflow cavities are not carved by precessing jets.
The authors thank Dr. Masaaki Hiramatsu for fruitful discussions and a comparison to his SMA outflow survey of LLOs in Perseus. We are grateful to Dr. Chi-Hung Yan for his assistance with the CFHT data reduction. We would like to thank Prof. Héctor Arce, Dr. Ian Stephens, and Dr. Thomas Robitaille for providing valuable discussions. The authors acknowledge the staff at CFHT for assistance with operations. The authors thank the referee for the insightful comments that improved this paper. T.H.H. and S.P.L. are thankful for the support of the Ministry of Science and Technology (MoST) of Taiwan through Grants NSC 98-2112- M-007-007-MY3, NSC 101-2119-M-007-004, MoST 102-2119-M-007-004-MY3 and MoST 105-2119-M-007-024.
André, P., Ward-Thompson, D., & Barsony, M. 1993, , 406, 122 André, P., Motte, F., & Bacmann, A. 1999, , 513, L57 André, P., Ward-Thompson, D., & Greaves, J. 2012, Sci, 337, 69 Arce, H. G., & Sargent, A. I. 2004, , 612, 342 Arce, H. G., & Sargent, A. I. 2006, , 646, 1070 Arce, H. G., Mardones, D., Corder, S. A., et al. 2013, , 774, 39 Barsony, M., Wolf-Chase, G. A., Ciardi, D. R., & O’linger, J. 2010, , 720, 64 Belloche, A., André, P., Despois, D., & Blinder, S. 2002, , 393, 927 Belloche, A., Parise, B., van der Tak, F. F. S., et al. 2006, , 454, L51 Bjorkman, J. E., & Wood, K. 2001, , 554, 615 Bontemps, S, André, P., Terebey, S., & Cabrit, S. 1996, , 311, 858 Bourke, T. L., Crapsi, A., Myers, P. C., et al. 2005, , 633, L129 Bourke, T. L., Myers, P. C., Evans, N. J., II, et al. 2006, , 649, L37 Chen, H., Myers, P. C., Ladd, E. F., & Wood, D. O. S. 1995, , 445, 377 Conover, W. J. 1999, Practical Nonparametric Statistics (3rd ed.; New York: Wiley), 271, 456 Crapsi, A., van Dishoeck, E. F., Hogerheijde, M. R., Pontoppidan, K. M., & Dullemond, C. P. 2008, , 486, 245 Di Francesco, J., Evans, N. J., II, Caselli, P., et al. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil (Tucson: Univ. Arizona Press), 17 Dullemond, C. P., & Dominik, C. 2004, , 417, 159 Dunham, M. M., Evans, N. J., Bourke, T. L., et al. 2006, , 651, 945 Dunham, M. M., Crapsi, A., Evans, N. J., II, Bourke, T. L., Huard, T. L., Myers, P. C., & Kauffmann, J. 2008, , 179, 249 Dunham, M. M., Evans, N. J., Bourke, T. L., et al. 2010, , 721, 995 Dunham, M. M., Evans, N. J., II, Terebey, S., Dullemond, C. P., & Young, C. H. 2010, , 710, 470 Dunham, M. M., & Vorobyov, E. I. 2012, , 747, 52 Dunham, M. M., Arce, H. G., Mardones, D., et al. 2014, , 783, 29 Eisner, J. A., Hillenbrand, L. A., & Carpenter, J. M. 2005, , 635, 396 Evans, N. J., II, Allen, L. E., Blake, G. A., et al. 2003, , 115, 965 Evans, N. J., II, Harvey, P. M., Dunham, M. M., et al. 2007, Final Delivery of Data from the c2d Legacy Project: IRAC and MIPS (Pasadena, CA: SSC) Evans, N. J., II, Dunham, M. M., Jørgensen, J. K., et al. 2009, , 181, 321 Hsieh, T.-H., Lai, S.-P., Belloche, A., Wyrowski, F., & Hung, C.-L. 2015, , 802, 126 Hsieh, T.-H., Lai, S.-P., Belloche, A., & Wyrowski, F. 2016, , Huard, T. L., Myers, C. M., Murphy, D. C., et al. 2006, , 640, 391 Jørgensen, J. K., Visser, R., Williams, J. P., & Bergin, E. A. 2015, , Khanzadyan, T., Gredel, R., Smith, M. D., & Stanke, T. 2004, , 426, 171 Kauffmann, J., Bertoldi, F., Bourke, T. L., et al. 2011, , 416, 2341 Kenyon, S. J., & Hartmann, L. 1995, , 101, 117 Kim, H. J., Evans, N. J., II, Dunham, M. M., et al. 2011, , 729, 84 Kim, M.-R.,Lee, C. W., Dunham, M. M., et al. 2016, , 225, 26 Ladd, E. F., Fuller, G. A., & Deane, J. R. 1998, 495, 871 Launhart, T., Pavlyuchenkov, Ya., Gueth, F., et al. 2008, , 494, 147L Lee, J.-E. 2007, Journal of The Korean Astronomical Society, 40, 83 Lee, C.-F., Mundy, L. G., Stone, J. M., & Ostriker, E. C. 2002, , 576, 294 Lee, C.-F., Hirano, N., & Shang, H. 2006, , 639, 292 Lee, C. W., Bourke, T. L., Myers, P. C., et al. 2009, , 693, 1290 Lee, C. W., Kim, M.-R., Kim, G., Saito, M., Myers, P. C., & Kurono, Y. 2013 , 777, 50 Lee, C. W., Kim, M.-R., Kim, G., et al. 2013, , 777, 50 Lee, K. I., Dunham, M. M., Myers, P. C., et al. 2016, , 820, L2 Lucas, P. W., & Roche, P. F. 1997, , 286, 895 Offner, S. S. R., Lee, E. J., Goodman, A. A., & Arce, H. G. 2011, , 743, 91 Offner, S. S. R., Dunham, M. M., Lee, K. I., Arce, H. G., & Fielding, D. B. 2016, , 827, L11 Ohashi, N., Hayashi, M., Ho, P. T. P., & Momose, M. 1997, , 475, 211 Padgett, D. L., Brandner, W., Stapelfeldt, K. R., et al. 1999, , 117, 1490 Padoan, P., & Nordlund, Å. 2004, , 617, 559 Palau, A., Zapata, L. A., Rodríguez, L. F., et al. 2014, , 444, 833 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical Recipes in C: The Art of Scientific Computing (2nd ed.; Cambridge: Cambridge Univ. Press) Reipurth, B. 2000, , 120, 3177 Robitaille, T. P., Whitney, B. A., Indebetouw, R., Wood, K., & Denzmore, P. 2006, , 167, 256 Robitaille, T. P., Whitney, B. A., Indebetouw, R., & Wood, K. 2007, , 169, 328 Schwarz K. R., Shirley Y. L., & Dunham M. M., 2012, , 144, 115 Seale, J. P., & Looney, L. W. 2008, , 675, 427 Stark, D. P., Whitney, B. A., Stassun, K., & Wood, K. 2006, , 649, 900 Takahashi, S., Ohashi, Nagayoshi, & Bourke, T. L. 2013, , 774, 20 Terebey, S., Van Buren, D., Drundage, M., & Hancock, T. 2006, , 637, 811 Terebey, S., Fich, M., Noriega-Crespo, A., et al. 2009, , 696. 1918 Tobin, J. J., Looney, L. W., Mundy, L. G., Kwon, W., & Hamidouche, M. 2007, , 659, 1404 Tsitali, A. E., Belloche, A., Commerçon, B., & Menten, K. M. 2013, , 557, A98 van der Marel, N., Kristensen, L. E., Visser, R., et al. 2013, , 556, A76 Velusamy, T., Langer, W. D., & Thompson, T. 2014, , 783, 6 Whitney, B. A., Wood, K., Bjorkman, J. E., & Wolff, M. J. 2003a, , 591, 1049 Whitney, B. A., Wood, K., Bjorkman, J. E., & Cohen, M. 2003b, , 598, 1079 Wolfire, M. G., & Königl, A. 1991, , 383, 205 Young, C. H., Jørgensen, J. K., Shirley, Y. L., et al. 2004, , 154, 396 Young, C. H., & Evans, N. J., II 2005, , 627, 293
[^1]: http://terapix.iap.fr
[^2]: http://peggysue.as.utexas.edu/SIRTF/
|
---
abstract: 'We analyze the correlation structure of bipartite arbitrary-dimensional Bell inequalities via novel conditions of correlations in terms of differences of joint probabilities called *correlators*. The conditions of correlations are shown to be necessary for the multi-level Bell state. In particular, we find that the bipartite arbitrary-dimensional Bell-type inequalities introduced by Collins-Gisin-Linden-Massar-Popescu \[Phys. Rev. Lett. **88**, 040404 (2002)\] and Son-Lee-Kim \[Phys. Rev. Lett. **96**, 060406 (2006)\] are composed of correlators, and we reveal that the maximal violations by the Bell state just fulfill the conditions of quantum correlations. Correlators can be considered as essential elements of Bell inequalities.'
author:
- 'Che-Ming Li$^{1}$'
- 'Der-San Chuu$^{1}$'
- 'Yueh-Nan Chen$^{2}$'
title: 'Quantum correlations for arbitrarily high-dimensional Bell inequality'
---
Introduction
============
The remarkable properties of entanglement goes essentially beyond the classical correlation constrained by two plausible assumptions, namely *locality* and *realism* (local realism) [@bell]. Local realism is also the central view of Einstein, Podolsky, and Rosen (EPR) [@epr] on the quantum mechanics. The assumption of *realism* states that outcomes of measurements are predeterministic, and the one of locality says that a measurement performed by one party of a system does not influence the result of the measurement performed by another party. By the assumptions of local realism, the *Bell inequalities* [@bell; @chsh; @bi] for two-level systems have been proposed to experimentally invalidate the point of view of EPR and to show that quantum mechanics is *not* locally realistic. For the aspect of quantum information processing [@qip], the nonlocal features of quantum correlations enable people to perform high-security and novel quantum communication [@ek; @tel]. Moreover, it helps to solve the problems that have no solutions in classical information theory [@cab].
In addition to entanglement for quantum two-level systems (qubits), entangled quantum multi-level systems (qudits) attract much attention for their nonlocal characters [@collins; @son; @dghz] and advantages in quantum information processing [@qdexp]. Collins *et al.* [@collins] have reformulate Bell inequalities to construct a large family of multi-level inequalities in terms of a novel constraint for local-realistic theories called Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Recently, Son, Lee, and Kim (SLK) [@son] presented generic Bell inequalities and their variants for arbitrary high-dimensional systems through the generalized GHZ nonlocality [@gghz].
In this work, we adopt a different approach to multilevel Bell inequalities. We address the following question. What are the essential properties of quantum correlations of entangled qudits that can be defined concretely and be detected efficiently. We wonder whether these essential features can be detected by Bell inequalities, i.e., whether the kernels of Bell inequalities consist some correlation conditions that are necessary for the entangled qudits. In order to attain this aim, we use novel conditions of correlations in terms of pairs of difference of joint probabilities called *correlators* to investigate correlation structure of multi-level Bell state and Bell inequalities. These correlators can be measured locally by each party, and by which the dependent properties of qudits can be revealed in different directions of measurements. In particular, we show that the CGLMP and SLK inequalities are comprised of conditions of quantum correlations in terms of correlators. In the following, an introduction to the conditions of correlations will be given as a preliminary to further discussions and results.
Correlation condition
=====================
Before proceeding further, let us revisit the scenario of a two-party Bell-type experiment for identifying the correlations between outcomes of measurements. Therein, measurements on each spatially-separated particle are assumed to be performed with two distinct results from different observables. In each run of the experiment, the first observer chooses $V_{1}
$ and the second one chooses $V_{2}$ for their local measurements on their particles respectively. After measurements, a set of results $v_{1}$ and $
v_{2}$, which can be either $0$ or $1$, is acquired. If sufficient runs of such measurements have been made under the chosen local measurement setting, the correlation between experimental outcomes can be revealed through the analytical analysis of experimental records. In analogy, the multi-level Bell type experiments work in the same way as mentioned above. The key idea of this work is to utilize the correlation relations in terms of differences of joint probabilities, i.e. correlators. We define two correlators in terms of the differences of joint probabilities, which are given by $$C_{0}=P(0,0)-P(1,0)\;\text{and}\;C_{1}=P(1,1)-P(0,1),$$ where $P(v_{1},v_{2})$ denote the joint probabilities for obtaining the sets of results $(v_{1},v_{2})$ under a given local measurement setting. We can show that outcomes of measurements performed on a system composed of *two uncorrelated parts* must satisfy the following criteria: $$C_{0}\geq0\;\text{and}\;C_{1}\leq0\:\:\:\text{or}\:\:\:C_{0}\leq0\;\text{and}
\;C_{1}\geq0.$$ To prove it, note that, for two uncorrelated parts, $C_{0}$ and $C_{1}$ can be recast as: $$C_{0}=[P_{1}(0)-P_{1}(1)]P_{2}(0), C_{1}=[P_{1}(1)-P_{1}(0)]P_{2}(1),
\nonumber$$ where $P_{k}(v_{k})$ represent the probabilities to get a specific measurement result for party $k=1,2$. Since $P_{k}(v_{k})\geq0$, thus we always get $C_{0}C_{1}\leq0$, hence it ends the proof. In analogy, we can formulate another set of correlators which is dual to the former one by $
\bar{C_{0}}=P(0,0)-P(0,1)$ and $\bar{C_{1}}=P(1,1)-P(1,0)$. Using these equations, we also can set a condition for correlation between measurements on pairs.
It is clear that, if the value of the product, $C_{0}C_{1}$, is positive, one can assert that there must be correlations between the outcomes of the measurements in the composite system in some way. In the quantum regime, we consider $C_{0}$ and $C_{1}$ for a two-qubit pure entangled state: $$\left|
\psi \right\rangle =\sin (\xi )\left| 0_{1}0_{2}\right\rangle_{z} +\cos (\xi)\left| 1_{1}1_{2}\right\rangle_{z},\nonumber$$ where $\left|v_{1}v_{2}\right\rangle_{z}=\left|v_{1}\right\rangle_{1z}\otimes\left|v_{2}
\right\rangle_{2z}$ and $\left|v_{k}\right\rangle_{kz} $ is the eigenstate of Pauli-operator $\sigma _{z}$ with eigenvalue $(-1)^{v_{k}}$ for the $k^{
\text{th}}$ party. If the local measurement setting is chosen as $(\sigma
_{x}$,$\sigma _{x})$, we obtain a violation of the criterion (2) by $$C_{0}=C_{1}=\sin (2\xi )/2,\nonumber$$ hence it turns out that the sum of $C_{0}$ and $
C_{1}$, denoted by $C^{(x)}$, equals to $\sin (2\xi )$. It is apparent that the criteria can also be violated by $$C_{0}=\sin ^{2}(\xi ),C_{1}=\cos^{2}(\xi),\nonumber$$ and that $C^{(z)}=1$ under the setting $(\sigma _{z}$,$\sigma
_{z})$. Thus, we have, $C_{\psi }\equiv C^{(x)}+C^{(z)}=1+\sin (2\xi )$, for the state $\left|\psi\right\rangle$. On the other hand, for example, the corresponding $C^{(z)}$ for the *separable* state $\rho_{\psi}=\sin
^{2}(\xi)\left| 0_{1}0_{2}\right\rangle_{zz}\left\langle 0_{1}0_{2}\right|
+\cos ^{2}(\xi)\left|
1_{1}1_{2}\right\rangle_{zz}\left\langle1_{1}1_{2}\right| $ also exhibits the same correlation in the results under the setting $(\sigma _{z}$,$\sigma
_{z})$ and is also equal to one. However, since the corresponding $C_{0}$ and $C_{1}$ are both zero under the local setting $(\sigma _{x}$,$\sigma
_{x})$, the correlation is erased. Hence the summation of $C^{(x)}$ and $
C^{(z)}$, denoted by $C_{\rho }$, is equal to $1$. With the fact that $C_{\psi }>C_{\rho_{\psi} }$, we then can distinguish the pure entangled state $\psi $ from the mixed $\rho_{\psi}$.
The above scenario for telling entanglement involves the summation of criteria, $C_{0}$ and $C_{1}$ under two measurement settings. Since entanglement manifests itself via quantum correlation in different directions of measurements, it makes $C_{\psi }>C_{\rho_{\psi}}$. For general cases, as will be discussed, we can prepare more settings of local measurements and introduce more terms with the same meanings as $C_{0}$ and $C_{1}$ to investigate quantum correlations embedded in entangled states and perform identification of their nonlocal properties further.
Quantum correlation of bipartite arbitrary-dimensional Bell state
=================================================================
The necessary conditions constructed by Eq. (1) for a system composed of two-level independent pair can be extended to general ones. We give a set of correlators for a system composed of two $d$-level parts, $$\begin{aligned}
&&C_{m}^{(12)}=P(v_{1}^{(1)}\doteq -m,v_{2}^{(2)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(1)}\doteq 1-m,v_{2}^{(2)}=m),\end{aligned}$$ for $m=0,1,...,d-1$, where $\doteq$ denotes equality modulus $d$. The superscripts, $(12)$, $(1)$, and $(2)$, mean that the first measurement, $
V_{1}^{(1)}$, and the second measurement, $V_{2}^{(2)}$, have been selected from two choices of each party. For a system composed of two *independent* $d$-level parts, it must *not* satisfy the following condition: either $C_{m}>0$ or $C_{m}<0$ for all $m$ ’s. If either of the above conditions is satisfied by a bipartite system, there must be correlations between the measurement outcomes. Let us give a concrete example with $d=3$ for above statement. From Eq. (3), we represent the correlators explicitly by $$\begin{aligned}
&&C_{0}^{(12)}=P(0,0)-P(1,0), \nonumber \\
&&C_{1}^{(12)}=P(2,1)-P(0,1), \nonumber \\
&&C_{2}^{(12)}=P(1,2)-P(2,2), \nonumber\end{aligned}$$ and furthermore if the particles composed of the bipartite system are independent we have the following relations between probabilities $$\begin{aligned}
&&C_{0}^{(12)}=[P_{1}(0)-P_{1}(1)]P_{2}(0), \nonumber \\
&&C_{1}^{(12)}=[P_{1}(2)-P_{1}(0)]P_{2}(1), \nonumber \\
&&C_{2}^{(12)}=[P_{1}(1)-P_{1}(2)]P_{2}(2). \nonumber\end{aligned}$$ If we have the results: $P_{1}(1)>P_{1}(2)$ and $P_{1}(2)>P_{1}(0)$, it turns out that $P_{1}(0)<P_{1}(1)$ which means that it is impossible to have $C_{m}^{(12)}>0$ for all $m$’s. Thus we can prove the statement for arbitrary $d$ by the same way proposed above.
Other types of correlators similar to Eq. (3) can be readily formulated: $$\begin{aligned}
&&C_{m}^{(21)}=P(v_{1}^{(2)}\doteq d-m-1,v_{2}^{(1)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(2)}\doteq -m,v_{2}^{(1)}=m), \\
&&C_{m}^{(ii)}=P(v_{1}^{(i)}\doteq -m,v_{2}^{(i)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(i)}\doteq d-m-1,v_{2}^{(i)}=m),\end{aligned}$$ for $i=1,2$.
Now, through the derived correlators, let us progress towards analysis of the correlation structure of the bipartite arbitrary-dimensional Bell state: $$\left\vert \psi _{d}\right\rangle =\frac{1}{\sqrt{d}}\sum_{v=0}^{d-1}\left
\vert v\right\rangle _{1z}\otimes \left\vert v\right\rangle _{2z}.$$ We represent the state $\left\vert \psi _{d}\right\rangle $ in the following eigenbasis of some observable $V_{k}^{(q)}$, $$\left\vert l\right\rangle _{kq}=\frac{1}{d}\sum_{m=0}^{d-1}\text{exp}[i\frac{
2\pi m}{d}(l+n_{k}^{(q)})]\left\vert m\right\rangle _{kz},$$ and then the joint probabilities for obtaining the measured outcome $
(v_{1}^{(i)},v_{2}^{(j)})$ for the state $\left\vert \psi _{d}\right\rangle $ are given by [@collins] $$\begin{aligned}
&&P_{\psi _{d}}(v_{1}^{(i)},v_{2}^{(j)}) \nonumber \\
&=&\frac{1}{2d^{3}\sin ^{2}[\frac{\pi }{d}
(v_{1}^{(i)}+v_{2}^{(j)}+n_{1}^{(i)}+n_{2}^{(j)})]},\end{aligned}$$ where $(n_{1}^{(i)},n_{2}^{(j)})$ denote the local parameters of the local measurement settings $(V_{1}^{(i)},V_{2}^{(j)})$. For the set of local parameters given by $$n_{1}^{(1)}=0,n_{2}^{(1)}=1/4,n_{1}^{(2)}=1/2,n_{2}^{(2)}=-1/4,$$ $C_{m}^{(ij)}$ can be evaluated analytically, and we arrive at $$C_{m,\psi _{d}}^{(ij)}=\frac{1}{2d^{3}}[\csc ^{2}(\frac{\pi }{4d})-\csc ^{2}(
\frac{3\pi }{4d})],$$ for $i,j=1,2$. Since $C_{m}^{(ij)}>0$ for all $m$’s with any finite value of $d$, we ensure that outcomes of measurements performed on the particles of the state $\left\vert \psi _{d}\right\rangle $ are dependent under four different local measurement settings. Thus, we can consider each set of the condition $C_{m}^{(ij)}>0$ as a necessary one of the Bell state $\left\vert
\psi _{d}\right\rangle $, and hence the corresponding correlation structure of $\left\vert \psi _{d}\right\rangle $ could be specified concretely and analytically via the correlators.
Furthermore, to compare the correlation embedded in $\left|\psi
_{d}\right\rangle$ predicted by quantum mechanics with the one by local-realistic theories, we could utilize the necessary conditions proposed above to achieve this aim. First, we combine all of the correlators involved in the necessary conditions of $\left|\psi _{d}\right\rangle$ and evaluate the summation of all $C_{m}^{(ij)}$ ’s, $$C_{d}=C^{(11)}+C^{(12)}+C^{(21)}+C^{(22)},$$ where $C^{(ij)}=\sum_{m=0}^{d-1}C_{m}^{(ij)}$. Then we have $$C_{d,\psi _{d}}=\frac{2}{d^{2}}[\csc^{2}(\frac{\pi}{4d})-\csc^{2}(\frac{3\pi
}{4d})].$$ One can find that $C_{d,\psi _{d}}$ is an increasing function of $d$. For instance, if $d=3$, one has $C_{3,\psi _{3}}\simeq 2.87293$. In the limit of large $d$, we obtain, $\lim_{d\rightarrow \infty }C_{d,\psi _{d}}=(16/3\pi
)^{2}\simeq 2.88202$.
We proceed to consider the maximum value of $C_{d}$ by local-realistic theories. The following derivation is based on deterministic local models, since any probabilistic model can be converted into a deterministic one. We substitute a chosen set, $(v_{1}^{(1)},v_{2}^{(1)},v_{1}^{(2)},v_{2}^{(2)})$ , into $C^{(ij)}$, and $C_{d}$ turns into $$\begin{aligned}
&&C_{d,\text{LR}} \nonumber \\
&=&\delta \lbrack (v_{1}^{(1)}+v_{2}^{(1)})\text{mod }d,0]-\delta \lbrack
-(v_{1}^{(1)}+v_{2}^{(1)})\text{mod }d,1] \nonumber \\
&&+\delta \lbrack (v_{1}^{(1)}+v_{2}^{(2)})\text{mod }d,0]-\delta \lbrack
(v_{1}^{(1)}+v_{2}^{(2)})\text{mod }d,1] \nonumber \\
&&+\delta \lbrack (v_{1}^{(2)}+v_{2}^{(2)})\text{mod }d,0]-\delta \lbrack
-(v_{1}^{(2)}+v_{2}^{(2)})\text{mod }d,1] \nonumber \\
&&+\delta \lbrack -(v_{1}^{(2)}+v_{2}^{(1)})\text{mod }d,1]-\delta \lbrack
(v_{1}^{(2)}+v_{2}^{(1)})\text{mod }d,0], \nonumber \\
&&\end{aligned}$$ where $\delta \lbrack x,y]$ represents the Kronecker delta symbol. It is apparent that there are three non-vanishing terms at most among the four positive delta functions under some specific condition for $(v_{1}^{(1)},v_{2}^{(1)},v_{1}^{(2)},v_{2}^{(2)})$. We also know that there must exist one non-vanishing negative delta function in $C_{d,\text{LR}}$ under the same condition. Therefore, in the regime governed by local-realistic theories, the value of $C_{d,\text{LR}}$ is bounded by $2$, i.e., $C_{d,\text{LR}}\leq 2$.
From the above discussions, we realize that $C_{d,\psi _{d}}>C_{d,\text{LR} }
$. Therefore, the quantum correlations are stronger than the ones predicted by the local-realistic theories. With this fact, the derived equation $C_{d}$ can be utilized to tell quantum correlations from classical ones. For $d=2$, $C_{2,\psi _{2}}=2\sqrt{2}$ and the equation $C_{d}$ is the same as that in the CHSH [@chsh] inequality. Moreover, the $C^{(ij)}$ terms are just the expectation values of the outcome products which appear in the CHSH inequality. Then, we can reinterpret the correlation functions as a summation of all $C_{0}^{(ij)}$ and $C_{1}^{(ij)}$ which formulate correlation criteria for measurements on pairs. This idea can be applied to arbitrary high-dimensional systems and to construct new types of correlation functions, $C^{(ij)}$. Although the values of maximal quantum violation are slightly smaller than the ones derived by Collins *et al.* [@collins] and Fu [@fu], the total number of joint probabilities required by each of the presented correlation functions $C^{(ij)}$ is only $2d$, which is much smaller than that in Fu’s general correlation function, which is about $O( d^{2})$. It implies that the proposed *correlation* functions include the essential parts of quantum correlation of the state $
\left|\psi _{d}\right\rangle$.
Another feature of the sum of all correlators will be discussed here is its robustness to noise. If the state $\left\vert \psi _{d}\right\rangle $ suffered from white noise and turns into a mixed one in the form $$\rho =p_{\text{noise}}/d^{2}\openone+(1-p_{\text{noise}})\left\vert \psi
_{d}\rangle \langle \psi _{d}\right\vert ,$$ where $p$ describes the noise fraction, the value of $C_{d}$ for state $\rho
$ becomes $C_{d,\rho }=(1-p_{\text{noise}})C_{d,\psi _{d}}$. If the criterion, $C_{d,\rho }>2$, i.e., $p_{\text{noise}}<1-2/C_{d,\psi _{d}}$, is imposed on the system, one ensures that the mixed state still exhibits quantum correlations in outcomes of measurements. For instance, to maintain the quantum correlation for the limit of large $d$, the system must have $p_{
\text{noise}}<0.30604$.
Through the work by Masanes about *tightness* of Bell inequality from a geometric point of view [@tight], we have examined our Bell-type inequality. The result shows that the inequality is non-tight, i.e., it is not an optimal detector of non-local-realistic correlation. The detailed proof and discussions are given in the appendix.
Correlation structure of CGLMP inequality
=========================================
Let us introduce more correlators like $C_{m}^{(ij)}$ to describe the quantum correlations of $\left|\psi_{d}\right\rangle$. The first four sets of correlators could be the one with the following form: $$\begin{aligned}
&&C_{m0}^{(ii)}=P(v_{1}^{(i)}\doteq m,v_{2}^{(i)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(i)}\doteq m-1,v_{2}^{(i)}=m), \nonumber \\
&&C_{m0}^{(12)}=P(v_{1}^{(1)}\doteq m,v_{2}^{(2)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(1)}\doteq m+1,v_{2}^{(2)}=m), \nonumber \\
&&C_{m0}^{(21)}=P(v_{1}^{(2)}\doteq m-1,v_{2}^{(1)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(2)}\doteq m,v_{2}^{(1)}=m),\end{aligned}$$ for $m=0,1,...,d-1$ and $i=1,2$. For $\left|\psi_{d}\right\rangle$, the values of $C_{m0}^{(ij)}$ can be evaluated analytically under the same measurement settings as the previous ones, and we have the result $$C_{m0,\psi_{d}}^{(ij)}=\frac{1}{2d^{3}}[\csc^{2}(\frac{\pi}{4d})-\csc^{2}(\frac{3\pi}{4d})],$$ which is the same with $C_{m,\psi_{d}}^{(ij)}$ and $C_{m0,\psi_{d}}^{(ij)}>0$ for all $m$’s with any finite $d$. Thus we know that the particles of the pair are dependent on each other. The second four sets of correlators are introduced by $$\begin{aligned}
&&C_{m1}^{(ii)}=P(v_{1}^{(i)}\doteq m+1,v_{2}^{(i)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(i)}\doteq m-2,v_{2}^{(i)}=m), \nonumber \\
&&C_{m1}^{(12)}=P(v_{1}^{(1)}\doteq m-1,v_{2}^{(2)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(1)}\doteq m+2,v_{2}^{(2)}=m), \nonumber \\
&&C_{m1}^{(21)}=P(v_{1}^{(2)}\doteq m-2,v_{2}^{(1)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(2)}\doteq m+1,v_{2}^{(1)}=m),\end{aligned}$$ and the corresponding expectation values for the state $\left|\psi_{d}\right\rangle$ are $$C_{m1,\psi_{d}}^{(ij)}=\frac{1}{2d^{3}}[\csc^{2}(\frac{5\pi}{4d})-\csc^{2}(\frac{7\pi}{4d})],$$ and are strictly greater than zero for all $m$’s with any finite $d$. Then we obtain another four sets of correlators which could be utilized to describe the dependence of the entangled pair.
Furthermore, let us progress towards to general sets of correlators which are formulated by $$\begin{aligned}
&&C_{mk}^{(ii)}=P(v_{1}^{(i)}\doteq m+k,v_{2}^{(i)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(i)}\doteq m-k-1,v_{2}^{(i)}=m), \nonumber \\
&&C_{mk}^{(12)}=P(v_{1}^{(1)}\doteq m-k,v_{2}^{(2)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(1)}\doteq m+k+1,v_{2}^{(2)}=m), \nonumber \\
&&C_{mk}^{(21)}=P(v_{1}^{(2)}\doteq m-k-1,v_{2}^{(1)}=m) \nonumber \\
&&\quad\quad\quad-P(v_{1}^{(2)}\doteq m+k,v_{2}^{(1)}=m),\end{aligned}$$ for $k=0,...,\lfloor d/2\rfloor$. We deduce that the particles composed of the Bell state are indeed dependent from the positive expectation values of correlators with the following general forms: $$C_{mk,\psi_{d}}^{(ij)}=\frac{1}{2d^{3}}\{\csc^{2}[\frac{(1+4k)\pi}{4d}
]-\csc^{2}[\frac{(3+4k)\pi}{4d}]\}.$$ Thus we can feature the quantum correlations embedded in the bipartite $d$-level Bell state in the $4(\lfloor d/2\rfloor+1)$ sets of correlators. Thus we could take a linear combination of all of these sets of correlators as a mens of identification: $$\mathsf{C}_{d}=\sum_{k=0}^{\lfloor
d/2\rfloor}\sum_{i,j=1}^{2}\sum_{m=0}^{d-1}f(k)C_{mk}^{(ij)},$$ where $f(k)$ denotes the coefficient of combination which is function of $k$.
If we let $f(k)$ be $$f(k)=1-\frac{2k}{d-1},$$ the summation of all of the correlators $\mathsf{C}_{d}$ becomes the kernel of the CGLMP inequality [@collins]: $$\mathsf{C}_{CGLMP}=\sum_{k=0}^{\lfloor d/2\rfloor}\sum_{i,j=1}^{2}(1-\frac{2k}{d-1})C_{k}^{(ij)},$$ where $C_{k}^{(ij)}=\sum_{m=0}^{d-1}C_{mk}^{(ij)}$. The local realistic constraint proposed by Collins *et al.* [@collins] specifies that the correlations exhibited by local realistic theories have to satisfy the condition: $$\mathsf{C}_{CGLMP,\text{LR}}\leq2.$$ On the other hand, by Eq. (20), quantum correlations of the Bell state will give a violation of the CGLMP inequality for arbitrary high-dimensional systems. Thus, through Eqs. (20) and (21) and the related discussions, we realize that the CGLMP inequality is composed of correlators for correlations and know that the corresponding violations for the Bell state just fulfill the conditions of quantum correlations of the entangled pair.
Correlation structure of SLK inequality
=======================================
From the discussions in the previous sections, we could realize that the features of entanglement of the Bell state are described by sets of correlators with positive expectation values. Hence, we could generalize the formulations of correlators for describing quantum correlations by the following specification. The entanglement of the bipartite $d$-level Bell state is featured in the correlators under different measurement settings: $$\begin{aligned}
&&C_{m}^{(l)}(\alpha ,\beta ) \nonumber \\
&=&P(v_{1}\doteq m+\alpha ,v_{2}=m)-P(v_{1}\doteq m+\beta ,v_{2}=m),
\nonumber \\
&&\end{aligned}$$ for $m=0,...,d-1$, where $l=[i,j]$ stands for measurement setting and $\alpha $ and $\beta $ are real numbers, and, most importantly, the values of correlators for $\left\vert \psi _{d}\right\rangle $ strictly fulfill the criterion $$C_{m,\psi _{d}}^{(l)}(\alpha ,\beta )>0,\text{ for }m=0,...,d-1,$$ or $$C_{m,\psi _{d}}^{(l)}(\alpha ,\beta )<0,\text{ for }m=0,...,d-1.$$ To have a compact form, it should be noted that we have omitted the denotations of measurement setting $(i,j)$ from the measured outcomes $v_{k}$ ’s. A linear combination of these correlators is utilized to identify the state $\left\vert \psi _{d}\right\rangle $: $$\tilde{\mathsf{C}}_{d}=\sum_{l}\sum_{\alpha ,\beta }f_{l}(\alpha ,\beta
)C^{(l)}(\alpha ,\beta )$$ where $C^{(l)}(\alpha ,\beta )=\sum_{m=0}^{d-1}C_{m}^{(l)}(\alpha ,\beta )$ and the coefficient of combination, $f_{l}(\alpha ,\beta )$, depends on $
\alpha $, $\beta $, and $l$.
Let us give a concrete example to show above formulation by the following sum of correlators: $$\sum_{l}\sum_{\alpha =0}^{d-1}f_{l}(\alpha )\sum_{m=0}^{d-1}P(v_{1}\doteq
m+\alpha ,v_{2}=m),$$ where $$\begin{aligned}
&&f_{l}(\alpha )=\sin (2\alpha _{l}\pi )[\cot (\alpha _{l}\pi /d)-\cot
(\alpha _{l}\pi )]/4, \\
&&\alpha _{l}=\nu +\alpha +\nu _{l},\end{aligned}$$ $\nu $, and $\nu _{l}$ are constants. It is worth to note that $$\sum_{\alpha =0}^{d-1}f_{l}(\alpha )=0,$$ which indicates that the sum of positive $f_{l}$’s and negative ones is zero and implies that one can always have the following relation:
$$\sum_{\alpha=0}^{d-1}f_{l}(\alpha)\sum_{m=0}^{d-1}P(v_{1}\doteq m+\alpha,v_{2}=m)=\sum_{\alpha,\beta}f_{l}(\alpha,\beta)\sum_{m=0}^{d-1}[P(v_{1}\doteq m+\alpha,v_{2}=m)-P(v_{1}\doteq m+\beta,v_{2}=m)].$$
If we choose the same measurement settings as the previous ones, please refer to Eqs. (7) and (9), the values of $C_{m}^{(l)}(\alpha
,\beta )$ for $\left\vert \psi _{d}\right\rangle $ strictly satisfy the criteria $(26)$ or $(27)$ by the facts that $$C_{m,\psi _{d}}^{(ij)}(\alpha ,\beta )=P(v_{1}^{(i)}=\alpha
,v_{2}^{(j)}=0)-P(v_{1}^{(i)}=\beta ,v_{2}^{(j)}=0)$$ for all $m$’s. Thus we conclude that Eq. (28) is indeed composed of correlators for entanglement of the Bell state.
To investigate the meaning of the sum of correlators further, we assign values to the parameters by $\nu=1/4$, $\nu_{11}=0$, $\nu_{22}=0$, $
\nu_{21}=1/2$, and $\nu_{12}=1/2$. We have the kernel of the SKL inequality: $$\mathsf{C}_{SLK}=\sum_{i,j=1}^{2}\sum_{\alpha,\beta}\sum_{m=0}^{d-1}f_{ij}(
\alpha,\beta)C^{(ij)}_{m}(\alpha,\beta).$$ By a direct calculation, we have $$\sum_{\alpha=0}^{d-1}f_{ij}(\alpha)\sum_{m=0}^{d-1}P_{\psi_{d}}(v_{1}^{(i)}
\doteq m+\alpha,v_{2}^{(j)}=m)=(d-1)/4,$$ for the Bell state, the value of the SLK kernel is $$\mathsf{C}_{SLK,\psi_{d}}=d-1.$$ Son *et al.* [@son] have shown that local-realistic theories predict the value of the kernel by $$\mathsf{C}_{SLK,\text{LR}}\leq \frac{1}{4}[3\cot(\frac{\pi}{4d})-\cot(\frac{
\pi}{3d})]-1,$$ which is called the SLK inequality. Thus the SLK inequality can be violated by the Bell state by a factor: $$\lim_{d\rightarrow \infty}\frac{\mathsf{C}_{SLK,\text{LR}}}{\mathsf{C}
_{SLK,\psi_{d}}}=\frac{8}{3\pi},$$ for arbitrary-high dimension.
Summary
=======
In this work, we have analyzed the structures of Bell inequalities for bipartite multi-level systems by conditions of correlations in terms of correlators. We start with an investigation into the correlation properties of the multi-level Bell state, and then we give specifications of the correlation structure in terms of correlators. Through these correlators for the Bell state, we construct Bell inequalities with fewer analyses of measured outcomes. We also show that the CGLMP [@collins] and SLK [@son] inequalities are composed of correlation conditions in terms of correlators. From the quantum mechanical point of view, we reveal that correlators are the essential elements of the Bell inequalities for arbitrarily high-diemensional systems.
*Note added.—*During preparation of our manuscript, we were aware of one related structure of Bell inequalities for *d*-level bipartite systems [@lee].
Tightness of Bell inequalities
==============================
Every tight Bell inequality fulfills the following conditions [@tight]:
*Condition 1.* All the generators of the convex polytope must belong either to the half-space or to the hyperplane.
*Condition 2.* There must be $4d(d-1)$ linear independent generators among the ones that belong to the hyperplane.
On the other hand, non-tight Bell inequalities satisfy only Condition 1. Then, we will examine the proposed BI by these conditions for tightness.
Firstly, we discuss condition 1 for the inequality. Although the same result has been shown in our paper, i.e., the derivation of the bound of the propose Bell inequality, we follow the approach presented by Masanes [@tight] for completness. The summation of all correlators of quantum correlation can be written as: $$\begin{aligned}
&&C_{d} \nonumber \\
&&=P(v_{1}^{(1)}+v_{2}^{(1)}\doteq0)-P(v_{1}^{(1)}+v_{2}^{(1)}\doteq-1)
\nonumber \\
&&\ +P(v_{1}^{(1)}+v_{2}^{(2)}\doteq0)-P(v_{1}^{(1)}+v_{2}^{(2)}\doteq1)
\nonumber \\
&&\ +P(v_{1}^{(2)}+v_{2}^{(2)}\doteq0)-P(v_{1}^{(2)}+v_{2}^{(2)}\doteq-1)
\nonumber \\
&&\ +P(v_{1}^{(2)}+v_{2}^{(1)}\doteq-1)-P(v_{1}^{(2)}+v_{2}^{(1)}\doteq0),\end{aligned}$$ To have an explicit form of $C_{d}$ for further discussion, we define the following variables: $$\begin{aligned}
&&\chi_{11}=v_{1}^{(1)}+v_{2}^{(1)}+\dot{d}_{11}, \nonumber \\
&&\chi_{12}=-v_{1}^{(1)}-v_{2}^{(2)}+\dot{d}_{12}, \nonumber \\
&&\chi_{22}=v_{1}^{(2)}+v_{2}^{(2)}+\dot{d}_{22}, \nonumber \\
&&\chi_{21}=-v_{1}^{(2)}-v_{2}^{(1)}-1+\dot{d}_{21},\end{aligned}$$ where $\dot{d}_{ij}$ denotes a multiple of $d$ and $\chi_{ij}\in\{-1,0\}$ for $i,j=1,2$. In particular, the sum of the variables satisfies the constrain: $$\sum_{i,j=1}^{2}\chi_{ij}\doteq-1.$$ With the defined variables, $C_{d}$ is written as $$\begin{aligned}
&&C_{d}=\sum_{ij=1}^{2}P(\chi_{ij}=0)-P(\chi_{ij}=-1).\end{aligned}$$ Next, we proceed to consider the extreme values of $C_{d}$ under the local realistic theories. The all possible sets of $(\chi_{11},\chi_{12},
\chi_{22},\chi_{21})$ which fulfill the constraint of the sum of the variables are as the following:
\(i) three of the variables are $0$ and the rest is $-1$;
\(ii) all of the variables are $-1$.The first class can be applied to arbitrary $d$, and, however, the second one is only applicable for $d=3$. Thus, we have $C_{d,\text{LHV}}=2$ for the class (i) and $C_{3, \text{LHV}}=-4$ for (ii), which mean that for all the generators of the convex polytope for $C_{d,\text{LHV}}$ the value $C_{d,\text{LHV}}$ is equal or less than $2$. Thus the proposed Bell inequality fulfills the first condition.
Second, we consider the second condition for the Bell inequality. All the generators of the convex polytope are written as $$\mathbf{G}=\left|v_{1}^{(1)},v_{2}^{(1)}\right\rangle\oplus
\left|v_{1}^{(1)},v_{2}^{(2)}\right\rangle\oplus
\left|v_{1}^{(2)},v_{2}^{(1)}\right\rangle\oplus
\left|v_{1}^{(2)},v_{2}^{(2)}\right\rangle,$$ which, with the defined variables, can also be represented as the following: $$\begin{aligned}
&&\left|v_{1}^{(1)},\chi_{11}-v_{1}^{(1)}\right\rangle\oplus
\left|v_{1}^{(1)},-\chi_{12}-v_{1}^{(1)}\right\rangle \nonumber \\
&&\
\oplus\left|v_{1}^{(1)}-\chi_{11}-\chi_{21}-1,\chi_{11}-v_{1}^{(1)}\right
\rangle \nonumber \\
&&\ \
\oplus\left|v_{1}^{(1)}+\chi_{12}+\chi_{22},-\chi_{12}-v_{1}^{(1)}\right
\rangle,\end{aligned}$$ where $\left|v_{1}^{(i)},v_{2}^{(j)}\right\rangle=\left|v_{1}^{(i)}\text{mod}
\ d\right\rangle\otimes\left|v_{2}^{(j)}\text{mod}\ d\right\rangle$. Through a linear transformation with the preservation of orthogonality which is given by $$\begin{aligned}
&&\sum_{v,k=0}^{d-1}(\left|v,\chi_{11}\right\rangle\left\langle
v,\chi_{11}-v\right|\oplus\left|v,\chi_{12}\right\rangle\left\langle
v,-\chi_{12}-v\right| \nonumber \\
&&\ \ \ \oplus\left|v-\chi_{11},\chi_{21}\right\rangle\left\langle
v-\chi_{11}-\chi_{21}-1,\chi_{11}-v\right| \nonumber \\
&&\ \ \ \oplus\left|v+\chi_{12},\chi_{22}\right\rangle\left\langle
v+\chi_{12}+\chi_{22},-\chi_{12}-v\right|),\end{aligned}$$ $\mathbf{G}$ can be transformed into $$\begin{aligned}
&&\left|v_{1}^{(1)},\chi_{11}\right\rangle\oplus\left|v_{1}^{(1)},-\chi_{12}
\right\rangle \nonumber \\
&&\ \ \ \ \oplus\left|v_{1}^{(1)}-\chi_{11},\chi_{21}\right\rangle\oplus
\left|v_{1}^{(1)}+\chi_{12},\chi_{22}\right\rangle.\end{aligned}$$ The generators which satisfy $C_{d,\text{LHV}}=2$ are the ones with variables belonging to (i). Thus, the generators contained in the hyperplane are shown as: $$\begin{aligned}
&&\left|v,-1\right\rangle\oplus\left|v,0\right\rangle\oplus\left|v+1,0\right
\rangle\oplus\left|v,0\right\rangle, \\
&&\left|v,0\right\rangle\oplus\left|v,-1\right\rangle\oplus\left|v,0\right
\rangle\oplus\left|v-1,0\right\rangle, \\
&&\left|v,0\right\rangle\oplus\left|v,0\right\rangle\oplus\left|v,-1\right
\rangle\oplus\left|v,0\right\rangle, \\
&&\left|v,0\right\rangle\oplus\left|v,0\right\rangle\oplus\left|v,0\right
\rangle\oplus\left|v,-1\right\rangle,\end{aligned}$$ for $v\in\{0,1,...,d-1\}$. The total number of linear independent generators is $4d$ which is smaller than $4d(d-1)$ involved in the condition of tightness. Then the proposed Bell inequality is non-tight.
J. S. Bell, Physics (Long Island City, N.Y.) **1**, 195 (1964); J. S. Bell, *Speakable and unspeakable in quantum mechanics* (Cambridge University Press, July 29, 1988). A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. **47**, 777 (1935). J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. **23**, 880 (1969). N. D. Mermin, Phys. Rev. Lett. **65**, 1838 (1990); S. M. Roy and V. Singh, *ibid* **67**, 2761 (1991); M. Ardehali, Phys. Rev. A **46**, 5375 (1992); D. N. Klyshko, Phys. Lett. A **172**, 399 (1993); A. V. Belinskii and D. N. Klyshko, Phys. Usp. **36**, 653 (1993); S. Popescu and D. Rohrlich, Phys. Lett. A **166**, 293 (1992); N. Gisin and H. Bechmann-Pasquinucci, *ibid* **246**, 1 (1998); R. F. Werner and M. M. Wolf, Phys. Rev. A **64**, 032112 (2001); M. Seevinck and G. Svetlichny, Phys. Rev. Lett. **89**, 060401 (2002); M. Żukowski and Č. Brukner, *ibid* 88, 210401 (2002); Kai Chen, Sergio Albeverio, and Shao-Ming Fei, Phys. Rev. A **74**, 050101R (2006). D. Bouwmeester, A. Ekert, and A. Zeilinger, *The Physics of Quantum Information* (Springer, Berlin, 2000). A. K. Ekert, Phys. Rev. Lett. **67**, 661 (1991). C. H. Bennett *et al.*, Phys. Rev. Lett. **70**, 1895 (1993). A. Cabello, Phys. Rev. Lett. **89**, 100402 (2002). D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Phys. Rev. Lett. **88**, 040404 (2002). W. Son, J. Lee, and M. S. Kim, Phys. Rev. Lett. **96**, 060406 (2006). N. J. Cerf, S. Massar, and S. Pironio, Phys. Rev. Lett. **89**, 080402 (2003). A. Mair *et al.*, Nature **412**, 3123 (2001); R.T.Thew *et al.*, Quant. Inf. Proc. **4**, 093 (2004); N. K. Langford *et al.*, Phys. Rev. Lett. **93**, 053601 (2004); Leonardo Neves *et al.*, *ibid* **94**, 100501 (2005); M. N. O’Sullivan-Hale, *ibid* **94**, 220501 (2005); G. Lima *et al.*, Phys. Rev. A, **73**, 032340(2006). J. Lee, S.-W Lee, and M. S. Kim, Phys. Rev. A **73**, 032316 (2006). L. B. Fu, Phys. Rev. Lett. **92**, 130404 (2004). L. Masanes, Quant. Inf. Comp. **3**, 345 (2003). Seung-Woo Lee, Yong Wook Cheong, and Jinhyoung Lee, Phys. Rev. A **76**, 032108 (2007).
|
---
abstract: 'We consider a natural generalization of the Carlsson-Okounkov Ext operator on the $K$–theory groups of the moduli spaces of stable sheaves on a smooth projective surface. We compute the commutation relations between the Ext operator and the action of the deformed $W$–algebra on $K$–theory, which was developed in [@Hecke]. The conclusion is that the Ext operator is closely related with a vertex operator, thus giving a mathematical incarnation of the Alday-Gaiotto-Tachikawa correspondence for a general algebraic surface.'
address:
- 'MIT, Department of Mathematics, Cambridge, MA, USA'
- 'Simion Stoilow Institute of Mathematics, Bucharest, Romania'
author:
- Andrei Negut
---
Introduction {#sec:introduction}
============
The main purpose of the present paper is to understand the $\Ext$ operator of Carlsson-Okounkov in the setting of moduli spaces of stable sheaves on general algebraic surfaces. In more detail, we fix a smooth projective surface $S$ over an algebraically closed field (henceforth denoted by $\BC$), an ample divisor $H$, a rank $r \in \BN$ and a first Chern class $c_1 \in H^2(S,\BZ)$, and consider the moduli space: $$\label{eqn:moduli}
\CM = \bigsqcup^\infty_{c_2 = \left \lceil \frac {r-1}{2r} c_1^2 \right \rceil} \CM_{(r,c_1,c_2)}$$ of $H$–stable sheaves on $S$ with the invariants $(r,c_1,c_2)$. The reason why $c_2$ is bounded below is called Bogomolov’s inequality, which states that there are no $H$–stable sheaves if $c_2 < \frac {r-1}{2r} c_1^2$. We make the same assumptions as in [@Univ], [@W; @surf], [@Hecke]:\
- Assumption A: $\gcd(r,c_1 \cdot H) = 1$\
- Assumption S: either $\begin{cases} \omega_S \cong \CO_S \qquad \quad \text{ or} \\ c_1(\omega_S) \cdot H < 0 \end{cases}$\
Assumption A implies that the space $\CM$ is proper and there exists a universal sheaf: $$\label{eqn:universal}
\xymatrix{\CU \ar@{.>}[d] & \\
\CM \times S}$$ Assumption S implies that the moduli space $\CM$ is smooth. Philosophically, none of these assumptions is absolutely fundamental to the constructions in the present paper. Indeed, Assumption A can be sidestepped if one is prepared to work with twisted universal sheaves (see [@C]) instead of universal sheaves. Meanwhile, Assumption S is necessary to define the pull-back maps in the correspondences of Definition \[def:am\] below, but it could be dropped if one appealed to virtual pull-backs (replacing $\CM$ by the derived scheme of [@BCHR] would likely be the correct setup). We make no claims about the technical difficulties involved in either of these generalizations. Let us consider the following algebraic $K$–theory groups with $\BQ$ coefficients: $$\label{eqn:km}
\km = \bigoplus^\infty_{c_2 = \left \lceil \frac {r-1}{2r} c_1^2 \right \rceil} K_0(\CM_{(r,c_1,c_2)}) \underset{\BZ}{\otimes} \BQ$$ Let $m \in \pic(S)$, and consider two copies $\CM$ and $\CM'$ of the moduli space . These two copies may be defined with respect to a different $c_1$ and stability condition $H$, but we assume that the rank $r$ of the sheaves parametrized by $\CM$ and $\CM'$ is the same. In this paper, we will mostly be concerned with the virtual vector bundle: $$\label{eqn:e}
\xymatrix{ & \CE_m \ar@{-->}[d] \\
& \CM \times \CM' \ar[ld]_{\pi_1} \ar[rd]^{\pi_2} \\
\CM & & \CM'}$$ (which is a straightforward generalization of the construction of [@CO]) given by: $$\label{eqn:sheaf e}
\CE_m = \text{R}\Gamma(m) - \text{R}\pi_* \left(\operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}\left(\CU', \CU \otimes m \right)\right)$$ The $\operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}$ is computed on $\CM \times \CM' \times S$: the notation $\CU$, $\CU'$, $m$ stands for the pull-back of the universal sheaves from $\CM \times S$ and $\CM' \times S$, respectively, as well as the pull-back of the line bundle $m$ from $S$. Similarly, $\pi : \CM \times \CM' \times S \rightarrow \CM \times \CM'$ is the standard projection, so $\CE_m$ is a complex of coherent sheaves on $\CM \times \CM'$.\
\[def:am\]
Consider the so-called **Ext operator** $\kmm \xrightarrow{A_m} \km$ given by: $$\label{eqn:a correspondence}
A_m = \pi_{1*} \left(\wedge^\bullet (\CE_m ) \cdot \pi_2^{*} \right)$$ The push-forward and pull-back are defined due to the smoothness and properness of $\CM$ and $\CM'$. See Subsection \[sub:exterior\] for the appropriate definition of $\wedge^\bullet(\CE_m)$.\
The main purpose of [@Univ], [@W; @surf], [@Hecke] was to construct an action $\CA_r \curvearrowright \km$, where $\CA_r$ is an integral form of the deformed $W$–algebra of type $\fgl_r$. The construction of this action will be recalled in Section \[sec:mod\], but for details, the reader may consult the aforementioned references as follows: it was shown in [@Hecke] that the $r\rightarrow \infty$ limit algebra, denoted therein by $\CA_\infty$, acts on $\km$ under Assumptions A and S (this action conjecturally even holds in the absence of Assumption S). Then, it was shown in Section 6 of [@W; @surf] that this action factors through the quotient $\CA_\infty \twoheadrightarrow \CA_r$. Explicitly, the action $\CA_r \curvearrowright \km$ is defined by specifying operators: $$\label{eqn:def w}
\km \xrightarrow{W_{n,k}} \kms$$ for all $n \in \BZ$ and $k \in \BN$, which satisfy the quadratic commutation relations from [@AKOS] and [@FF] (see for the form of these relations in our language). Let us write: $$\label{eqn:def q}
q = [\omega_S] \in K_S := K_0(S) \otimes_{\BZ} \BQ$$ Given two copies $\CM$ and $\CM'$ of the moduli space of stable sheaves, each with its own universal sheaf $\CU$ and $\CU'$, respectively, we may write: $$\label{eqn:def u}
u = \det \CU \qquad \text{and} \qquad u' = \det \CU'$$ for the determinant line bundles on $\CM \times S$ and $\CM' \times S$, respectively. We set: $$\label{eqn:def gamma}
\gamma = \frac {m^r u}{q^r u'}$$ which is the class of a line bundle on $\CM \times \CM' \times S$ (it is implicit that $m$ and $q$ are pulled back from $S$). Our main result, which will be proved in Section \[sec:ext\], is:\
\[thm:main\]
We have the following interaction between the Ext operator and the generators of the $W$–algebra action: $$\label{eqn:comm a}
A_m W_k(x) (1-x) = m^k W_k(x\gamma) A_m \left(1- \frac x{q^k} \right)$$ where $W_k(x) = \sum_{n \in \BZ} \frac {W_{n,k}}{x^{n}}$. Both sides of are maps $\kmm \rightarrow \kms$, and thus represented by $K$-theory classes on $\CM \times \CM' \times S$.[^1] Thus, the factors $q$ and $\gamma$ in the right-hand side refer to multiplication by the line bundles and .\
Let us present the physical implications of Theorem \[thm:main\]. The Ext operator $A_m$ encodes the contribution of bifundamental matter in partition functions of 5d supersymmetric gauge theory on the algebraic surface $S$ times a circle ([@CNO]). The deformed $W$–algebra $\CA_r$ encodes symmetries of Toda conformal field theory. In this language, becomes a mathematical manifestation of the Alday-Gaiotto-Tachikawa correspondence between gauge theory and conformal field theory, by describing the Ext operator $A_m$ in terms of its commutation with $W$–algebra generators. However, this is not enough to completely determine $A_m$ for a general smooth projective surface $S$, and one should instead work with a deformed vertex operator algebra which properly contains several deformed $W$–algebras $\CA_r$. In the non-deformed case, a potential candidate for such an larger algebra was studied in [@FG], where the authors expect that it contains operators which modify sheaves on $S$ along entire curves, on top of our operators $W_{n,k}$ which modify sheaves at individual points. While we give a complete algebro-geometric description of the latter operators, we do not have such a description for the former operators. Once such a description will be available, we expect that one can extend Theorem \[thm:main\] to a bigger vertex operator algebra properly containing $\CA_r$.\
When $S = \BA^2$, the analogues of our results were proved in [@W], although the formulas in are significantly weaker than our Theorems \[thm:main\] and \[thm:main heis\], as we explain in the first paragraph of Subsection \[sub:weak\]. In this situation, $\km$ was shown in to be isomorphic to the universal Verma module for the deformed $W$–algebra of type $\fgl_r$, and we will show in Theorem \[thm:unique\] that the results of the present paper determine the Ext operator $A_m$ uniquely (whereas the results of [@W] were not strong enough for this purpose). We note that the AGT correspondence over toric surfaces like ALE spaces was studied by many authors from many different points of view (see e.g. [@BFN], [@BPSS], [@MO], [@SV] in mathematical language). To the author’s knowledge, the present paper is the first mathematical treatment of AGT over general algebraic surfaces in rank $r>1$ (the reference [@CNO] used different techniques from ours to describe the Ext operator in the rank $r=1$ case).\
Alongside the operators , we constructed ([@Hecke]) $K$–theory lifts of the operators studied by Baranovsky, Grojnowski, Nakajima in cohomology ([@Ba], [@G], [@Nak]): $$\label{eqn:def heis}
\km \xrightarrow{P_n} \kms$$ $\forall n \in \BZ \backslash 0$. These operators satisfy the Heisenberg commutation relation , and interact with the deformed $W$–algebra generators according to relation .\
\[thm:main heis\]
We have the following interaction between the Ext operator and the Heisenberg operators $P_{\pm n}$: $$\label{eqn:comm a heis 1}
A_m P_{-n} - P_{-n} A_m \gamma^n = A_m(1-\gamma^n)$$ $$\label{eqn:comm a heis 2}
\ \ A_m P_n - P_n A_m \gamma^{-n} = A_m(\gamma^{-n} - q^{rn})$$ $\forall n > 0$. Recall that $\gamma$ is the line bundle , and we refer the reader to the last sentence of Theorem \[thm:main\] for how to make sense of it in formulas –.\
For any $\alpha \in K_S$, we will write $P_n\{\alpha\}$ for the composition: $$P_n\{\alpha\} : \km \xrightarrow{P_n} \kms \xrightarrow{\text{multiplication by } \proj_2^*(\alpha)} \kms \xrightarrow{\proj_{1*}} \km$$ where $\proj_1$, $\proj_2$ are the projections from $\CM \times S$ to $\CM$ and $S$, respectively. Define: $$\label{eqn:phi}
\Phi_m = A_m \exp \left[\sum_{n=1}^\infty \frac {P_n}n \left\{ \frac {(q^n - 1)q^{-nr}}{[q_1]_n [q_2]_n} \right\} \right]$$ where $[x]_n = 1+x+...+x^{n-1}$. The expression in the right-hand side makes sense because $[q_1]_n [q_2]_n$ is a unit in the ring $K_S$. Indeed, since Chern character gives us an isomorphism $\ks \cong A^*(S,\BQ)$, then $q_1+q_2 = [\Omega^1_S] \in 2 + \mathcal{N}$ and $q = [\omega_S] \in 1 + \mathcal{N}$, where $\mathcal{N} \subset \ks$ denotes the nilradical. Therefore, $[q_1]_n [q_2]_n \in n^2 + \mathcal{N}$, and is thus invertible in the ring $K_S$.\
\[cor:main\]
Formulas , , imply the following: $$\label{eqn:comm phi}
\Big[ \Phi_m W_k(x) - m^k W_k(x\gamma) \Phi_m \Big] \left(1- \frac x{q^k} \right) = 0$$ $$\label{eqn:comm phi heis}
\ \Phi_m P_{\pm n} - P_{\pm n} \Phi_m \gamma^{\mp n} = \pm \Phi_m(\gamma^{\mp n} - q^{\pm rn})$$ $\forall k,n > 0$. An operator $\Phi_m$ satisfying , is called a **vertex operator**.\
I would like to thank Boris Feigin, Sergei Gukov, Hiraku Nakajima, Nikita Nekrasov, Andrei Okounkov, Francesco Sala and Alexander Tsymbaliuk for many interesting discussions on the subject of Ext operators and $W$–algebras. I gratefully acknowledge the support of NSF grant DMS–1600375.
The moduli space of sheaves {#sec:mod}
===========================
{#sub:action}
In the introduction, we referred to various operators $\km \rightarrow \kms$ as defining an action of a certain algebra on $\km$, and we will now explain the meaning of this notion. Given two arbitrary homomorphisms: $$\label{eqn:operators}
\km \xrightarrow{x,y} \kms$$ their “product" $x y |_\Delta$ is defined as the composition: $$x y \Big|_\Delta : \km \xrightarrow{y} \kms \xrightarrow{x \times \text{Id}_S} \kmss \xrightarrow{\text{Id}_{\CM} \times \Delta^*} \kms$$ where $S \xrightarrow{\Delta} S \times S$ is the diagonal. It is easy to check that $(xy|_\Delta) z|_\Delta = x (yz|_\Delta)|_\Delta$, hence the aforementioned notion of product is associative, and it makes sense to define $x_1...x_n|_\Delta$ for arbitrarily many operators $x_1,...,x_n : \km \rightarrow \kms$.\
Similarly, given two operators , we may define their commutator: $$\km \xrightarrow{[x,y]} \kmss$$ as the difference of the two compositions: $$\begin{aligned}
&\km \xrightarrow{y} \kms \xrightarrow{x \times \text{Id}_S} \kmss \\
&\km \xrightarrow{x} \kms \xrightarrow{y \times \text{Id}_S} \kmss \xrightarrow{\text{Id}_{\CM} \times \text{swap}^*} \kmss\end{aligned}$$ where $\text{swap} : S \times S \rightarrow S \times S$ is the permutation of the two factors. In all cases studied in this paper, we will have: $$[x,y] = \Delta_*(z)$$ for some $\km \xrightarrow{z} \kms$ which is uniquely determined (the diagonal embedding $\Delta_*$ is injective because it has a left inverse), and which will be denoted by $z = [x,y]_\red$. We leave it as an exercise to the interested reader to prove that the commutator satisfies the Leibniz rule in the form $[xy|_\Delta,z]_\red = x[y,z]_\red|_\Delta + [x,z]_\red y|_\Delta$, and the Jacobi identity in the form $[[x,y]_\red,z]_\red + [[y,z]_\red,x]_\red + [[z,x]_\red,y]_\red = 0$.\
Finally, we consider the ring homomorphism $\BK = \BZ[q_1^{\pm 1}, q_2^{\pm 1}]^{\sym} \rightarrow \ks$ given by sending $q_1$ and $q_2$ to the Chern roots of the cotangent bundle of $S$ (therefore, $q=q_1q_2$ goes to the class of the canonical line bundle). We will often abuse notation, and write $q_1,q_2,q$ for the images of the indeterminates in the ring $\ks$. For any $\lambda \in \BK$ and any operator , we may define their product as the composition: $$\lambda \cdot x : \km \xrightarrow{x} \kms \xrightarrow{\text{Id}_\CM \times (\text{multiplication by }\lambda)} \kms$$ where we identify $\lambda \in \BK$ with its image in $\ks$. With this in mind, the ring $\ks$ can be thought of as the “ring of constants" for the algebra of operators .\
{#sub:basic mod}
We refer the reader to [@Univ], [@W; @surf], [@Hecke] for details on the contents of this Section. Since our assumptions imply the existence of the universal sheaf , we may set: $$\label{eqn:z1}
\fZ_1 = \BP_{\CM \times S} (\CU) \longrightarrow \CM \times S$$ Since $\CU$ is isomorphic to a quotient $\CV/\CW$ of vector bundles on $\CM \times S$ (see [@Univ]), the object in the right-hand side of is defined as the derived zero locus of a section of a vector bundle on the projective bundle $\BP_{\CM \times S}(\CV)$. However, it was shown in [@Univ] that under Assumption S, the derived zero locus is actually a smooth scheme: $$\fZ_1 = \bigsqcup_{c = \left \lceil \frac {r-1}{2r} c_1^2 \right \rceil}^\infty \fZ_{c+1,c}$$ whose components are given by: $$\label{eqn:z1 closed}
\fZ_{c+1,c} = \Big \{ (\CF_{c+1}, \CF_c) \text{ s.t. } \CF_{c+1} \subset_x \CF_c \text{ for some } x \in S \Big\} \subset \CM_{c+1} \times \CM_c$$ and $\CF' \subset_x \CF$ means that $\CF' \subset \CF$ and the quotient $\CF/\CF'$ is isomorphic to the length 1 skyscraper sheaf at the point $x \in S$. This scheme comes with projection maps: $$\label{eqn:z1 projections}
\xymatrix{ & \fZ_{c+1,c} \ar[ld]_{p_+} \ar[d]^{p_S} \ar[rd]^{p_-} & \\
\CM_{c+1} & S & \CM_c}$$ More generally, we defined a derived scheme $\fZ_2^\bullet$ in [@W; @surf]. We will not be concerned with the precise definition, but we note that under Assumption S, it was shown in to be a smooth scheme: $$\fZ_2^\bullet = \bigsqcup_{c = \left \lceil \frac {r-1}{2r} c_1^2 \right \rceil}^\infty \fZ_{c+2,c}^\bullet$$ whose components are given by: $$\label{eqn:z2 closed}
\fZ_{c+2,c}^\bullet = \Big \{ (\CF_{c+2} \subset_x \CF_{c+1} \subset_x \CF_c) \text{ for some } x \in S )\Big\} \subset \CM_{c+2} \times \CM_{c+1} \times \CM_c$$ This scheme is equipped with projection maps as in below, but we observe that the rhombus is not Cartesian (and this is key to our construction): $$\label{eqn:z2 projections}
\xymatrix{ & \fZ_{c+2,c}^\bullet \ar[ld]_{\pi_+} \ar[rd]^{\pi_-} & \\
\fZ_{c+2,c+1} \ar[rd]_{p_- \times p_S} & & \fZ_{c+1,c} \ar[ld]^{p_+ \times p_S} \\
& \CM_{c+1} \times S&}$$ Using the maps in diagram , we define: $$\label{eqn:zn components}
\fZ_n^\bullet = \bigsqcup_{c = \left \lceil \frac {r-1}{2r} c_1^2 \right \rceil}^\infty \fZ_{c+n,c}^\bullet$$ whose components are given by derived fiber products: $$\label{eqn:zn}
\fZ_{c+n,c}^\bullet = \fZ_{c+n,c+n-2}^\bullet \underset{\fZ_{c+n-1,c+n-2}}\times \dots \underset{\fZ_{c+2,c+1}}\times \fZ_{c+2,c}^\bullet \rightarrow \CM_{c+n} \times ... \times \CM_{c}$$ While $\fZ_n^\bullet$ is a derived scheme, we note that its closed points are all of the form: $$\label{eqn:zn points}
\fZ_{c+n,c}^\bullet = \{(\CF_{c+n},...,\CF_c) \text{ sheaves s.t. } \CF_{c+n} \subset_x ... \subset_x \CF_c \text{ for some } x \in S\}$$ Therefore, we have the following projection maps, which only remember the smallest and the largest sheaf in a flag (the notation below generalizes ): $$\label{eqn:zn projections}
\xymatrix{ & \fZ_{c+n,c}^\bullet \ar[ld]_{p_+} \ar[d]^{p_S} \ar[rd]^{p_-} & \\
\CM_{c+n} & S & \CM_c}$$ Moreover, we consider the line bundles $\CL_1,...,\CL_n$ of $\fZ_n^\bullet$, whose fibers are given by: $$\label{eqn:line bundles}
\CL_i |_{(\CF_{c+n},...,\CF_c)} = \CF_{c+n-i,x}/\CF_{c+n-i+1,x}$$ on the component $\fZ_{c+n,c}^\bullet \subset \fZ_n^\bullet$.\
{#sub:w action}
Using the derived scheme and the maps , we construct operators: $$\begin{aligned}
&K_{\CM} \xrightarrow{L_{n,k}} K_{\CM \times S}, & &L_{n,k} = (-1)^{k-1} (p_+ \times p_S)_* \left(\CL_n^k \cdot p_-^* \right) \label{eqn:op l} \\
&K_{\CM} \xrightarrow{U_{n,k}} K_{\CM \times S}, & &U_{n,k} = \frac{ (-1)^{rn + k-1} u^n}{q^{(r-1)n}} (p_- \times p_S)_* \left( \frac {\CL_n^k}{\CQ^r} \cdot p_+^* \right) \label{eqn:op u}\end{aligned}$$ where $\CQ = \CL_1...\CL_n$, and $u$ is the determinant of the universal sheaf on $\CM \times S$, as in .[^2] Implicit in the definitions and is that we define the operators therein for all components $\CM_c$ of the moduli space $\CM$. Finally, consider: $$\begin{aligned}
&\km \xrightarrow{E_k} \kms, & &E_k = \text{ multiplication by} \wedge^k(\CU) \label{eqn:op e}\end{aligned}$$ Since $\CU \cong \CV/\CW$ is a coherent sheaf of projective dimension 1 on $\CM \times S$ (see [@Univ]), the class $\wedge^k(\CU)$ in the right-hand side of is defined by setting: $$\label{eqn:first ext}
\wedge^\bullet \left( \frac {\CU}z \right) = \frac {\wedge^\bullet \left( \frac {\CV}z \right)}{\wedge^\bullet \left( \frac {\CW}z \right)}$$ and picking out the coefficient of $z^{-k}$ when expanding in negative powers of $z$. The reason for our notation of the operators , , is that these three operators are respectively lower triangular, upper triangular, and diagonal with respect to the grading on $\km$ by the second Chern class (see ).\
\[def:w\]
([@W], [@W; @surf]) For any $n \in \BZ$ and $k \in \BN$, consider the operators: $$\label{eqn:op w}
W_{n,k} = \sum^{n_2 - n_1 = n}_{k_0+k_1+k_2 = k} q^{(k-1)n_2} \cdot L_{n_1,k_1} E_{k_0} U_{n_2,k_2} \Big|_\Delta$$ where $k_1,k_2$ go over $\BN$, and $k_0, n_1,n_2$ go over $\BN \sqcup 0$.\
Note that is an infinite sum, but its action on $\km$ is well-defined because the operators $L_{n,k}$ (respectively $U_{n,k}$) increase (respectively decrease) the $c_2$ of stable sheaves by $n$, and Bogomolov’s inequality ensures that the moduli space of stable sheaves is empty if $c_2$ is small enough.\
{#section}
Similarly with and , we have the following operators: $$\begin{aligned}
&K_{\CM} \xrightarrow{P_{-n}} K_{\CM \times S}, & &P_{-n} = (p_+ \times p_S)_* \left(\sum_{i=0}^{n-1} \frac {q^{i-1} \CL_n}{\CL_{n-i}} \cdot p_-^* \right) \label{eqn:op p 1} \\
&K_{\CM} \xrightarrow{P_n} K_{\CM \times S}, & &P_n = (-1)^{rn} u^{n} (p_- \times p_S)_* \left( \sum_{i=0}^{n-1} \frac {q^{i-1} \CL_n}{\CQ^r \CL_{n-i}} \cdot p_+^* \right) \label{eqn:op p 2} \\
&K_{\CM} \xrightarrow{H_{-n}} K_{\CM \times S}, & &H_{-n} = (p_+ \times p_S)_* \left( p_-^* \right) \label{eqn:op h 1} \\
&K_{\CM} \xrightarrow{H_n} K_{\CM \times S}, & &H_n = (-1)^{rn} u^{n} (p_- \times p_S)_* \left( \frac 1{\CQ^r} \cdot p_+^* \right) \label{eqn:op h 2}\end{aligned}$$ As a consequence of [@W; @surf], [@Hecke], the operators $H_{\pm n}$ are to the operators $P_{\pm n}$ as complete symmetric functions are to power sum functions: $$\label{eqn:h to p}
\sum_{n=0}^\infty \frac {H_{\pm n}}{z^{\pm n}} = \exp \left(\sum_{n=1}^\infty \frac {P_{\pm n}}{nz^{\pm n}} \right) \Big|_\Delta$$ or, explicitly: $$\begin{aligned}
&H_{\pm 1} = P_{\pm 1} \\
&H_{\pm 2} = \frac {P_{\pm 1} P_{\pm 1} |_\Delta + P_{\pm 2}}2 \\
&H_{\pm 3} = \frac {P_{\pm 1} P_{\pm 1} P_{\pm 1}|_\Delta + 3P_{\pm 1} P_{\pm 2} |_\Delta + 2 P_{\pm 3}}6 \\
&...\end{aligned}$$
\[thm:w\]
([@W; @surf], Section 6) The operators satisfy, for all $n \in \BZ$: $$\begin{aligned}
&W_{n,r} = u \sum_{n_1, n_2 \geq 0}^{n_2 - n_1 = n} H_{-n_1} H_{n_2} \Big|_{\Delta} \label{eqn:w=r} \\
&W_{n,k} = 0 \label{eqn:w>r}\end{aligned}$$ for all $k>r$.\
{#section-1}
The interaction between the operators , , are all presented by recalling the commutator construction in Subsection \[sub:action\].\
\[thm:acts\]
([@W; @surf]) We have the following formulas for all $n,n' \in \BZ$, $k,k' \in \BN$: $$\begin{aligned}
&[W_{n,k}, W_{n',k'}] = \Delta_* \left( \mathop{\sum_{k+k' = l+l'}^{\min(l,l') \leq \min(k,k')}}_{m+m' = n+n'}^{\frac ml \leq \frac {m'}{l'}} c_{n,n',k,k'}^{m,m',l,l'} (q_1,q_2) \cdot W_{m,l} W_{m',l'} \Big|_\Delta \right) \label{eqn:comm w} \\
&[P_{n'}, P_n] = \Delta_* \begin{cases} 0 & \text{if } \sgn(n) = \sgn(n') \\
\delta_{n+n'}^0 n [q_1]_{n} [q_2]_{n} [q^{n}]_r \cdot \emph{proj}_{\CM}^* & \text{if }n'<0<n \end{cases} \label{eqn:q heis} \\
&[W_{n',k'}, P_{\pm n}] = \Delta_* \Big( \pm [q_1]_{n} [q_2]_{n} [q^n]_{k'} q^{n(r-k') \delta_\pm^+} \cdot W_{\pm n + n',k'} \Big) \label{eqn:w and p}\end{aligned}$$ where the coefficients $c_{n,n',k,k'}^{m,m',l,l'} (q_1,q_2) \in \ks$ were computed algorithmically in [@W; @surf].
Indeed, we show in [@W; @surf] that is the defining relation in the deformed $W$–algebra $\CA_r$ (with $\Delta_*$ replaced by the constant $(1-q_1)(1-q_2)$). Similarly, relation is the defining relation in the deformed Heisenberg algebra. As we explained in [@W; @surf] and [@Hecke], the fact that the operators , , satisfy the relations in Theorem \[thm:acts\] is precisely what we mean when we say that the deformed $W$–algebra $\CA_r$ and the deformed Heisenberg algebra act on the groups $\km$.\
{#sub:series}
Let us consider the operators of Subsection \[sub:w action\] and form the generating series: $$\label{eqn:series}
L_n(y) = \sum_{k=1}^\infty (-1)^{k} \frac {L_{n,k}}{y^k}, \qquad U_n(y) = \sum_{k=1}^\infty (-1)^{k} \frac {U_{n,k}}{y^k}$$ In other words, these power series are considered as operators: $$\begin{aligned}
&K_{\CM} \xrightarrow{L_n(y)} K_{\CM \times S} \left\llbracket \frac 1y \right\rrbracket, & &L_n(y) = (p_+ \times p_S)_* \left( \frac 1{1 - \frac {y}{\CL_n}} \cdot p_-^* \right) \\
&K_{\CM} \xrightarrow{U_n(y)} K_{\CM \times S} \left\llbracket \frac 1y \right\rrbracket, & &U_n(y) = \frac {(-1)^{rn} u^n}{q^{(r-1)n}} (p_- \times p_S)_* \left( \frac {\CQ^{-r}}{1 - \frac {y}{\CL_n}} \cdot p_+^* \right)\end{aligned}$$ We will also consider the operators: $$\km \xrightarrow{E(y)} \kms \left\llbracket \frac 1y \right\rrbracket, \qquad E(y) = \text{multiplication by }\wedge^\bullet \left( \frac {\CU}y \right) \qquad \qquad$$ Furthermore, we will consider the generating series: $$\label{eqn:big series}
L(x,y) = 1 + \sum_{n=1}^\infty \frac {L_n(y)}{x^{-n}} \qquad \qquad U(x,y) = 1 + \sum_{n=1}^\infty \frac {U_n(y)}{x^n}$$ and also set: $$\begin{aligned}
&W_k(x) = \sum_{n = -\infty}^\infty \frac {W_{n,k}}{x^n} \label{eqn:series w} \\
&W(x,y) = 1 + \sum_{k=1}^\infty \frac {W_k(x)}{y^k} \label{eqn:big series w}\end{aligned}$$ The definition of the $W$–algebra generators in is equivalent to the following: $$\label{eqn:op w series}
W\left(x, yD_x \right) = L \left(x, yD_x \right) E\left( yD_x \right) U \left(xq, yD_x \right) \Big|_\Delta$$ where $D_x$ is the $q$-difference operator in the variable $x$, i.e. $f(x) \leadsto f(x q)$. For formula to be correct, the powers of $D_x$ should be placed to the right of the powers of $x$ in all terms except for $U(xq, yD_x)$, in which the powers of $D_x$ should be placed to the left of the powers of $x$. Similarly, formula reads: $$\label{eqn:w to p series}
[W_k(x), P_{\pm n}] = \Delta_* \Big(\pm [q_1]_n [q_2]_n [q^n]_k q^{n(r-k) \delta_{\pm}^+} \cdot x^{\pm n} W_k(x) \Big)$$
{#section-2}
Consider the rational function: $$\label{eqn:zeta}
\zeta(x) = \frac {(1-x q_1)(1-x q_2)}{(1-x)(1-x q)} \in \ks(x)$$ $$$$
\[prop:corr\]
([@W; @surf]) We have the following formulas for the maps : $$\begin{gathered}
(p_{+} \times p_S)_* r(\CL_1,...,\CL_n) = \label{eqn:push one} \\
= \int_{\infty - 0}^{z_1 \prec ... \prec z_n \prec \{0, \infty\}} \frac {r(z_1,...,z_n) \prod_{i=1}^n \wedge^\bullet \left(\frac {z_iq}{\CU} \right)}{\left(1 - \frac {z_2 q}{z_1} \right) ... \left(1 - \frac {z_n q}{z_{n-1}} \right) \prod_{1 \leq i < j \leq n} \zeta \left(\frac {z_j}{z_i} \right)}\end{gathered}$$ $$\begin{gathered}
(p_{-} \times p_S)_* r(\CL_1,...,\CL_n) = \label{eqn:push two} \\
= \int_{\infty - 0}^{\{0, \infty\} \prec z_1 \prec ... \prec z_n} \frac {r(z_1,...,z_n) \prod_{i=1}^n \wedge^\bullet \left(- \frac {\CU}{z_i} \right)}{\left(1 - \frac {z_2 q}{z_1} \right) ... \left(1 - \frac {z_n q}{z_{n-1}} \right) \prod_{1 \leq i < j \leq n} \zeta \left(\frac {z_j}{z_i} \right)} \end{gathered}$$ for any Laurent polynomial $r$.\
The notation $\int_{\infty - 0}$ stands for the difference between the residues at $\infty$ and $0$: $$\int_{\infty - 0} f(z) = \underset{z = \infty}{\text{Res}} \frac {f(z)}z - \underset{z = 0}{\text{Res}} \frac {f(z)}z$$ The notation $z_1 \prec ... \prec z_n \prec \{0,\infty\}$ means that we integrate the right-hand side of over a collection of contours nested inside each other, with $z_1$ being the farthest and $z_n$ being the closest to $0$ and $\infty$. The notation $\{0, \infty\} \prec z_1 \prec ... \prec z_n$ means that $z_1$ is the closest and $z_n$ is the farthest from 0 and $\infty$. Thus, the right-hand sides of and entail successively evaluating the residues at $0$ and $\infty$ of a rational function with values in $\kms$. The result of the computation will be a Laurent polynomial in $q_1^{\pm 1}$, $q_2^{\pm 1}$ and the exterior powers of $\CU$.\
There is also a version of Proposition \[prop:corr\] when $r(z_1,...,z_n)$ is allowed to have poles, and then the contours of the $z$ variables must take care to keep such poles on the same side as $0$ and $\infty$. We will only need this general result when $r$ has a single pole:\
\[prop:corr y\]
([@W; @surf]) We have the following formulas for the maps : $$\begin{gathered}
(p_{+} \times p_S)_* r(\CL_1,...,\CL_n) = \label{eqn:push one y} \\
= \int_{\infty - 0}^{z_1 \prec ... \prec z_n \prec \{0, y, \infty\}} \frac {r(z_1,...,z_n) \prod_{i=1}^n \wedge^\bullet \left(\frac {z_iq}{\CU} \right)}{\left( 1 - \frac y{z_n} \right) \left(1 - \frac {z_2 q}{z_1} \right) ... \left(1 - \frac {z_n q}{z_{n-1}} \right) \prod_{1 \leq i < j \leq n} \zeta \left(\frac {z_j}{z_i} \right)}\end{gathered}$$ $$\begin{gathered}
(p_{-} \times p_S)_* r(\CL_1,...,\CL_n) = \label{eqn:push two y} \\
= \int_{\infty - 0}^{\{0, y, \infty\} \prec z_1 \prec ... \prec z_n} \frac {r(z_1,...,z_n) \prod_{i=1}^n \wedge^\bullet \left(- \frac {\CU}{z_i} \right)}{\left( 1 - \frac y{z_n} \right) \left(1 - \frac {z_2 q}{z_1} \right) ... \left(1 - \frac {z_n q}{z_{n-1}} \right) \prod_{1 \leq i < j \leq n} \zeta \left(\frac {z_j}{z_i} \right)} \end{gathered}$$ for any Laurent polynomial $r$, with $y$ being an indeterminate.\
{#sub:correspondences}
Our main Theorems \[thm:main\] and \[thm:main heis\] will be proved in the next Section, but much of our computation will require going back and forth between the language of operators on $K$–theory and the language of correspondences. In other words, the operators , , , , , , as well as various compositions thereof, fit naturally in the following language.\
Given smooth projective varieties $X$ and $Y$, any class $\Gamma \in K_{X \times Y}$ (called a “correspondence" in this setup) defines an operator: $$\label{eqn:correspondence}
K_Y \xrightarrow{\Psi_\Gamma} K_X, \qquad \Psi_\Gamma = \emph{proj}_{X*} \left( \Gamma \otimes \emph{proj}_Y^* \right)$$ where $\emph{proj}_X, \emph{proj}_Y$ denote the projection maps from $X \times Y$ to $X$ and $Y$, respectively.\
For example, the operator $H_{-n}$ of equals $\Psi_\Gamma$, where $\Gamma$ is the direct image of 1 (namely the $K$–theory class of the structure sheaf) under the map: $$\bigsqcup_{c = \left \lceil \frac {r-1}{2r} c_1^2 \right \rceil}^{\infty} \fZ_{c+n,c}^\bullet \longrightarrow \bigsqcup_{c = \left \lceil \frac {r-1}{2r} c_1^2 \right \rceil}^{\infty} \CM_{c+n} \times \CM_c \times S$$ $$(\CF_{c+n} \subset_x \CF_{c+n-1} \subset_x \dots \subset_x \CF_{c+1} \subset_x \CF_c) \mapsto (\CF_{c+n}, \CF_c, x)$$ The composition of operators can also be described as a correspondence: $$\label{eqn:composition corr}
\Psi_{\Gamma} \circ \Psi_{\Gamma'} = \Psi_{\Gamma''} : K_Z \rightarrow K_X$$ where $\Gamma'' = \proj_{X \times Z*}(\proj_{X \times Y}^*(\Gamma) \otimes \proj_{Y \times Z}^*(\Gamma'))$, where $\proj_{X \times Y}$, $\proj_{Y \times Z}$, $\proj_{X \times Z}$ denote the standard projection $X \times Y \times Z \rightarrow X \times Y$, $Y \times Z$, $X \times Z$.\
Computing the Ext operator {#sec:ext}
==========================
{#sub:exterior}
To properly define the Ext operator , note that the complex $\CE_m$ of can be written as a difference $\CV_1 - \CV_2$ of vector bundles. Then we define: $$\label{eqn:second ext}
\wedge^\bullet \left( \frac {\CE_m}s \right) = \frac {\wedge^\bullet \left( \frac {\CV_1}s \right)}{\wedge^\bullet \left( \frac {\CV_2}s \right)}$$ as in . However, instead of thinking about as a power series in $s^{-1}$, we will think of it as a rational function in $s$ with coefficients in the $K$–theory of $\CM \times \CM'$. Our notation in the present paper subsumes the fact that this rational function can be specialized at $s=1$. This is merely an artifice: if the reader does not wish to make this specialization, then one can simply replace $m$ by $\frac ms$ in formulas , , and throughout the current Section. Once one does this, then our main Theorems \[thm:main\], \[thm:main heis\] and Corollary \[cor:main\] will be equalities of operator-valued rational functions in $s$. Moreover, we will often use the notation: $$\wedge^\bullet \left( \frac s{\CU} \right) \qquad \text{instead of} \qquad \wedge^\bullet \left(\CU^\vee s \right)$$ for any coherent sheaf $\CU$ (all our coherent sheaves have finite projective dimension).\
{#section-3}
The main goal of the present Section is to compute the commutation relations between the Ext operator $A_m : \kmm \rightarrow \km$ of and the operators: $$\label{eqn:generators}
W_{n,k}, P_{\pm n'} : \km \rightarrow \kms$$ of , for all $n \in \BZ$, $n',k \in \BN$. One must be careful what one means by “commutation relation". While the operator: $$\begin{aligned}
&P_{\pm n} A_m & &\text{unambiguously refers to} & &\kmm \xrightarrow{A_m} \km \xrightarrow{P_{\pm n}} \kms \\
& A_m P_{\pm n} & &\text{henceforth refers to} & &\kmm \xrightarrow{P_{\pm n}} \kmms \xrightarrow{A_m \times \text{Id}_S} \kms \end{aligned}$$ and analogously for $W_{n,k}$ instead of $P_{\pm n}$. As opposed from the operators , the operator $A_m$ acts non-trivially between all components of the moduli space: $$\label{eqn:a in components}
A_m |_c^{c'} : K_{\CM_{c'}} \longrightarrow K_{\CM_c}$$ In principle, the moduli spaces of sheaves in the domain and codomain can correspond to different choices of first Chern class and stability condition, but we always require them to have the same rank $r$. Therefore, there are two universal sheaves: $$\xymatrix{\CU \ar@{..>}[d] \\ \CM \times S} \qquad \qquad \xymatrix{\CU' \ar@{..>}[d] \\ \CM' \times S}$$ where $\CM$ (respectively $\CM'$) is the union of the moduli spaces that appear in the codomain (respectively domain) of . The determinants of these universal sheaves are denoted by $u$ and $u'$, respectively, as in .\
{#section-4}
We must explain how to make sense of the symbols $q,m,\gamma$ in , , . In the language of correspondences from Subsection \[sub:correspondences\], an operator: $$\kmm \xrightarrow{z} \kms$$ (such as the compositions $W_{n,k} A_m$ or $P_{\pm n} A_m$ that appear in , , ) can be interpreted as a $K$–theory class $\zeta$ on $\CM \times \CM' \times S$. Then the product $q z$ refers to the operator corresponding to the $K$–theory class $\text{proj}_S^*(q) \cdot \zeta$. Similarly, the product $\gamma z$ refers to the operator corresponding to the $K$–theory class: $$\text{proj}_{S}^*\left(\frac mq\right)^r \cdot \frac {\proj_{\CM \times S}^*(\det \CU)}{\proj_{\CM' \times S}^*(\det \CU')} \cdot \zeta$$ where $\CM\times \CM' \times S \xrightarrow{\proj_{\CM \times S}, \proj_{\CM' \times S}, \proj_S} \CM \times S, \CM' \times S, S$ are the projections.\
\[prop:doesn’t matter\]
We have the equality of correspondences $K_{\CM_{c \pm n}} \rightarrow K_{\CM_c \times S}$: $$\label{eqn:doesn't matter}
P_{\pm n} \cdot \left( \det \CU_{c \pm n} \right) = \left( \det \CU_c \right) \cdot P_{\pm n}$$ Formula also holds with $P_{\pm n}$ replaced with $W_{n,k}$ or $H_{\pm n}$.\
Equation is best restated in the language of correspondences, from Subsection \[sub:correspondences\]. In these terms, $P_{\pm n}$ is given by a $K$–theory class supported on the locus: $$\mathfrak{C} = \{\CF_{c + n} \subset_x \CF_c\} \subset \CM_{c + n} \times \CM_c \times S$$ Then merely states that the universal sheaves $\CU_{c + n}$ and $\CU_c$ have isomorphic determinants when restricted to $\mathfrak{C}$. This is just the version “in families" of the well-known statement that a codimension 2 modification of a torsion-free sheaf does not change its determinant. As a consequence of Proposition \[prop:doesn’t matter\], $\gamma$ of will behave just like a constant in all our computations, i.e. it will not matter where we insert $\gamma$ in any product of operators. Specifically, this means that for all $x,y \in \{P_{\pm n}, H_{\pm n}, W_{n,k}\}$, the following compositions will be equal: $$\kmm \xrightarrow{A_m} \km \xrightarrow{x} \kms \xrightarrow{\gamma} \kms \xrightarrow{y \times \text{Id}_S} \kmss \xrightarrow{\text{Id}_{\CM} \times \Delta^*} \kms$$ $$\kmm \xrightarrow{A_m} \km \xrightarrow{x} \kms \xrightarrow{y \times \text{Id}_S} \kmss \xrightarrow{\text{Id}_{\CM} \times \Delta^*} \kms \xrightarrow{\gamma} \kms$$
{#section-5}
Our main intersection-theoretic computation is the following:\
\[lem:commute heis\]
We have the following relations involving the *Ext* operator $A_m$: $$\label{eqn:one heis}
A_m (H_{-n} - H_{-n+1}) = \gamma^n (H_{-n} - H_{-n+1}) A_m$$ $$\label{eqn:two heis}
\qquad \qquad A_m \left(H_n - H_{n-1} \gamma^{-1} \right) = \left( H_n \gamma^{-n} - H_{n-1} q^r \gamma^{-n+1} \right) A_m$$
Consider the following diagrams of spaces and arrows, for all $c$ and $c'$: $$\label{eqn:big diagram 1}
\xymatrix{ & & \CM_c \times S \times \CM_{c'} \ar@/_3pc/[llddd]_{\pi_1 \times \Id_S} \ar@/^3pc/[rrddd]^{\pi_2} & & \\
& & \CM_c \times \fZ^\bullet_{c'+n,c'} \ar[ld]_{\Id \times p_+ \times p_S} \ar[rd] \ar[u]^{\Id \times p_S \times p_-} & & \\
& \CM_c \times \CM_{c'+n} \times S \ar[ld] \ar[rd] & & \fZ^\bullet_{c'+n,c'} \ar[ld]^{p_+ \times p_S} \ar[rd]_{p_-} & \\
\CM_c \times S & & \CM_{c'+n} \times S & & \CM_{c'}}$$ $$\label{eqn:big diagram 2}
\xymatrix{ & & \CM_c \times S \times \CM_{c'} \ar@/_3pc/[llddd]_{\pi_1' \times \text{Id}_S} \ar@/^3pc/[rrddd]^{\pi'_2} & & \\
& & \fZ^\bullet_{c,c-n} \times \CM_{c'} \ar[ld] \ar[rd]^{p_-' \times \Id} \ar[u]_{p_+' \times p_S' \times \Id} & & \\
& \fZ^\bullet_{c,c-n}\ar[ld]^{p'_+ \times p_S'} \ar[rd]_{p'_-} & & \CM_{c-n} \times \CM_{c'} \ar[ld] \ar[rd] & \\
\CM_c \times S & & \CM_{c-n} & & \CM_{c'}}$$ Recall that $H_{-n} = (p_+ \times p_S)_* p_-^*$, in the notation of . Then the rule for composition of correspondences in gives us the following formulas: $$\label{eqn:comp 1}
A_m H_{-n} = (\pi_1 \times \Id_S)_*(\Upsilon_n \cdot \pi_2^{*})$$ $$\label{eqn:comp 2}
H_{-n} A_m = (\pi'_1 \times \Id_S)_*(\Upsilon'_n \cdot \pi_2'^{*})$$ where, in the notation of and : $$\begin{aligned}
&\Upsilon_n = \left( \Id \times p_S \times p_- \right)_* \Big[ \wedge^\bullet \left( (\Id \times p_+)^* \CE_{m} \right) \Big] \label{eqn:gamma 1} \\
&\Upsilon'_n = \left( p_+' \times p_S' \times \Id \right)_* \Big[ \wedge^\bullet \left( (p'_- \times \Id)^* \CE_{m} \right) \Big] \label{eqn:gamma 2}\end{aligned}$$ are certain classes on $\CM_c \times S \times \CM_{c'}$, that we will now compute.\
\[claim:joe\]
We have the following equalities in $K$–theory: $$\label{eqn:geo 1}
(\emph{Id} \times p_+)^* \CE_{m} = (\emph{Id} \times p_-)^* \CE_{m} + \left(\frac 1{\CL_1} +...+ \frac 1{\CL_n} \right) (\emph{Id} \times p_S)^*\left( \frac {\CU m}q \right)$$ on $\CM_c \times \fZ_{c'+n,c'}^\bullet$, and: $$\label{eqn:geo 2}
(p_-' \times \emph{Id})^* \CE_{m} = (p_+' \times \emph{Id})^* \CE_{m} - (\CL_1+...+\CL_n) (p'_S \times \emph{Id})^*\left( {\CU'}^\vee m \right)$$ on $\fZ_{c,c-n}^\bullet \times \CM_{c'}$.\
Consider the following diagram: $$\label{eqn:grothendieck 1}
\xymatrix{ & \CM_c \times \fZ_{c'+n,c'}^\bullet \times S \ar[dd]^{\rho} \ar[ld]_-{\text{Id} \times p_+ \times \text{Id}_S} \ar[rd]^-{\text{Id} \times p_- \times \text{Id}_S} & \\
\CM_c \times \CM_{c'+n} \times S \ar[dd]^{\rho} & & \CM_c \times \CM_{c'} \times S \ar[dd]^{\rho} \\
& \CM_c \times \fZ_{c'+n,c'}^\bullet \ar[ld]_-{\text{Id} \times p_+} \ar[rd]^-{\text{Id} \times p_-} & \\
\CM_c \times \CM_{c'+n} & & \CM_c \times \CM_{c'}}$$ where the vertical maps are the natural projections (we use the notation $\rho$ for all of them). We have the following short exact sequence of $\fZ_{c'+n,c'}^{\bullet} \times S$: $$\label{eqn:joe}
0 \rightarrow \CU'_+ \rightarrow \CU'_- \rightarrow \Gamma_*(\CL_1 ``\oplus" ... ``\oplus" \CL_n) \rightarrow 0$$ where $\CU'_\pm = (p_\pm^* \times \text{Id}_S)(\text{universal sheaf})$, while $\CL_1,...,\CL_n$ denote the tautological line bundles on $\fZ_{c'+n,c'}^\bullet$ that were defined in , and: $$\label{eqn:def Gamma}
\Gamma : \fZ_{c'+n,c'}^\bullet \rightarrow \fZ_{c'+n,c'}^\bullet \times S$$ is the graph of the map $p_S$. The notation $``\oplus"$ in refers to a coherent sheaf which is filtered by the line bundles $\CL_1$,...,$\CL_n$; since we work in $K$–theory, we make no distinction between this coherent sheaf and its associated graded object. We may also pull-back the short exact sequence to $\CM_c \times \fZ_{c'+n,c'}^\bullet \times S$. Now apply the functor $\operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(-,\CU \otimes m)$ to the short exact sequence , where $\CU$ is the universal sheaf pulled back from $\CM_c \times S$: $$\operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU'_+,\CU \otimes m) = \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU'_-,\CU \otimes m) - \sum_{i=1}^n \frac 1{\CL_i} \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CO_\Gamma,\CU \otimes m)$$ Now recall that the line bundles $\CL_i$ come from $\fZ_{c'+n,c'}^\bullet$, and so they are unaffected by the derived push-forward map $\rho_*$: $$\rho_* \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU'_+,\CU \otimes m) = \rho_* \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU'_-,\CU \otimes m) - \sum_{i=1}^n \frac 1{\CL_i} \rho_* \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CO_\Gamma,\CU \otimes m) \quad \Rightarrow$$ $$\label{eqn:kay}
\Rightarrow \quad (\text{Id} \times p_+)^* \CE_{m} = (\text{Id} \times p_-)^* \CE_{m} + \sum_{i=1}^n \frac 1{\CL_i} \rho_* \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CO_\Gamma,\CU \otimes m)$$ where the implication $\Rightarrow$ stems from . Then follows from the fact that: $$\label{eqn:rick}
\rho_* \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CO_\Gamma,\CU \otimes m) = \underbrace{\rho_* \circ \Gamma_*}_{\text{Id}} \left( \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CO, \Gamma^! (\CU \otimes m)) \right) = \CU m \Big|_\Gamma \otimes \Gamma^! \CO$$ (the first equality is coherent duality, and the second equality holds for any closed embedding $\Gamma$). The right-hand side of matches $(\text{Id} \times p_S)^*(\CU m/q)$ because the map $\Gamma: \fZ_n^\bullet \rightarrow \fZ_n^\bullet \times S$ is obtained by base change from the diagonal map $S \rightarrow S \times S$, and the ratio of dualizing objects on $S$ and $S \times S$ is precisely $q = [\omega_S]$.\
As for , consider the diagram: $$\label{eqn:grothendieck 2}
\xymatrix{ & \fZ_{c,c-n}^\bullet \times \CM_{c'} \times S \ar[dd]^{\rho} \ar[ld]_-{p'_+ \times \text{Id} \times \text{Id}_S} \ar[rd]^-{ p'_- \times \text{Id} \times \text{Id}_S} & \\
\CM_c \times \CM_{c'} \times S \ar[dd]^{\rho} & & \CM_{c-n} \times \CM_{c'} \times S \ar[dd]^{\rho} \\
& \fZ_{c,c-n}^\bullet \times \CM_{c'} \ar[ld]_-{p'_+ \times \text{Id}} \ar[rd]^-{p'_- \times \text{Id}} & \\
\CM_c \times \CM_{c'} & & \CM_{c-n} \times \CM_{c'}}$$ and consider the following analogue of : $$0 \rightarrow \CU_+ \rightarrow \CU_- \rightarrow \Gamma'_*(\CL_1 ``\oplus" ... ``\oplus" \CL_n) \rightarrow 0$$ where $\CU_\pm = ({p'}_\pm^* \times \text{Id}_S)(\CU)$, and $\Gamma'$ denotes the graph of the map $p_S : \fZ_{c,c-n}^\bullet \rightarrow S$. Let us apply the functor $\operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU', - \otimes m)$ to the short exact sequence above: $$\operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU',\CU_- \otimes m) = \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU',\CU_+ \otimes m) + \sum_{i=1}^n \CL_i \otimes \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU',\CO_{\Gamma'} \otimes m)$$ Let us apply $\rho_*$ to the equality above, and recall the definition of $\CE_m$ in : $$(p_-' \times \text{Id})^*\CE_m = (p_+' \times \text{Id})^*\CE_m - \sum_{i=1}^n \CL_i \otimes \rho_* \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU',\CO_{\Gamma'} \otimes m)$$ By adjunction, we have: $$\rho_* \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU',\CO_{\Gamma'} \otimes m) = \underbrace{\rho_* \circ \Gamma'_*}_{\text{Id}} \operatorname{{R}\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(\CU'|_{\Gamma'}, {p'_S}^*m) = (p_S' \times \text{Id})^* \left({\CU'}^\vee m \right)$$
Armed with and , we may rewrite and as: $$\begin{aligned}
&\Upsilon_n = \left[ \wedge^\bullet\CE_m \right] \left( \Id \times p_S \times p_- \right)_* \left[ \bigotimes_{i=1}^n \wedge^\bullet \left( \frac {\CU m}{\CL_i q}\right) \right] \\
&\Upsilon_n' = \left[ \wedge^\bullet\CE_m \right] \left( p_+' \times p_S' \times \Id \right)_* \left[ \bigotimes_{i=1}^n \wedge^\bullet \left(- \frac {\CL_i m}{\CU'}\right) \right]\end{aligned}$$ Therefore, Proposition \[prop:corr\] implies: $$\begin{aligned}
&\Upsilon_n = \left[ \wedge^\bullet\CE_m \right] \int_{\{0,\infty\} \prec z_1 \prec ... \prec z_n} \frac 1{\prod_{i=1}^{n-1} \left(1 - \frac {qz_{i+1}}{z_i} \right) \prod_{i < j} \zeta \left( \frac {z_j}{z_i} \right)} \prod_{i=1}^n \frac {\wedge^\bullet \left(\frac {\CU m}{z_i q} \right)}{\wedge^\bullet \left(\frac {\CU'}{z_i} \right)} \label{eqn:gam 1} \\
&\Upsilon_n' = \left[ \wedge^\bullet\CE_m \right] \int_{z_1 \prec ... \prec z_n \prec \{0,\infty\}} \frac 1{\prod_{i=1}^{n-1} \left(1 - \frac {qz_{i+1}}{z_i} \right) \prod_{i < j} \zeta \left( \frac {z_j}{z_i} \right)} \prod_{i=1}^n \frac {\wedge^\bullet \left(\frac {z_i q}{\CU} \right)}{\wedge^\bullet \left(\frac {z_i m}{\CU'} \right)} \label{eqn:gam 2}\end{aligned}$$ Consider the following rational function with coefficients in $K_{\CM_c \times S \times \CM_{c'}}$: $$\label{eqn:definition i}
I_n(z_1,...,z_n) = \frac 1{\prod_{i=1}^{n-1} \left(1 - \frac {qz_{i+1}}{z_i} \right) \prod_{i < j} \zeta \left( \frac {z_j}{z_i} \right)} \prod_{i=1}^n \frac {\wedge^\bullet \left(\frac {\CU m}{z_i q} \right)}{\wedge^\bullet \left(\frac {\CU'}{z_i} \right)}$$ One may then rewrite and as: $$\begin{aligned}
&\Upsilon_n = \left[ \wedge^\bullet\CE_m \right] \int_{\{0,\infty\} \prec z_1 \prec ... \prec z_n} I_n(z_1,...,z_n) \\
&\Upsilon_n' = \left[ \wedge^\bullet\CE_m \right] \int_{z_1 \prec ... \prec z_n \prec \{0,\infty\}} I_n(z_1m,...,z_nm)\cdot \gamma^{-n} \end{aligned}$$ Changing the variables $z_i \mapsto \frac {z_i}m$ in the second formula, we conclude that: $$\label{eqn:integral identity 1}
\Upsilon_n - \Upsilon_n' \cdot \gamma^n =$$ $$= \left[ \wedge^\bullet \CE_m \right] \left[ \int_{\{0,\infty\} \prec z_1 \prec ... \prec z_n} I_n \prod_{i=1}^n \frac {dz_i}{2\pi i z_i} - \int_{z_1 \prec ... \prec z_n \prec \{0,\infty\}} I_n \prod_{i=1}^n \frac {dz_i}{2\pi i z_i} \right]$$ The only difference between the two integrals is the order of the contours, specifically where the poles at $\{0,\infty\}$ lie in respect to the variables $z_1,...,z_n$. Therefore, we conclude that the difference above picks up the poles at 0 and $\infty$ in the various variables. However, all such residues are 0, except for: $$\begin{aligned}
&\text{Res}_{z_1 = \infty} \frac {I_n(z_1,...,z_n)}{z_1} = I_{n-1}(z_2,...,z_n) \label{eqn:residue i 1} \\
&\text{Res}_{z_n = 0} \frac {I_n(z_1,...,z_n)}{z_n} = \gamma \cdot I_{n-1}(z_1,...,z_{n-1}) \label{eqn:residue i 2}\end{aligned}$$ Therefore, formula implies that: $$\label{eqn:nice}
\Upsilon_n - \Upsilon_n' \cdot \gamma^n = \Upsilon_{n-1} - \Upsilon_{n-1}' \cdot \gamma^n$$ which, as an equality of classes on $\CM_c \times S \times \CM_{c'}$, precisely encodes . Let us run the analogous computation for (we will recycle all of our notations):
$$\label{eqn:big diagram 1 bis}
\xymatrix{ & & \CM_c \times S \times \CM_{c'} \ar@/_3pc/[llddd]_{\pi_1 \times \Id_S} \ar@/^3pc/[rrddd]^{\pi_2} & & \\
& & \CM_c \times \fZ^\bullet_{c',c'-n} \ar[ld]_{\Id \times p_- \times p_S} \ar[rd] \ar[u]^{\Id \times p_S \times p_+} & & \\
& \CM_c \times \CM_{c'-n} \times S \ar[ld] \ar[rd] & & \fZ^\bullet_{c',c'-n} \ar[ld]^{p_- \times p_S} \ar[rd]_{p_+} & \\
\CM_c \times S & & \CM_{c'-n} \times S & & \CM_{c'}}$$
$$\label{eqn:big diagram 2 bis}
\xymatrix{ & & \CM_c \times S \times \CM_{c'} \ar@/_3pc/[llddd]_{\pi_1' \times \text{Id}_S} \ar@/^3pc/[rrddd]^{\pi'_2} & & \\
& & \fZ^\bullet_{c+n,c} \times \CM_{c'} \ar[ld] \ar[rd]^{p_+' \times \Id} \ar[u]_{p_-' \times p_S' \times \Id} & & \\
& \fZ^\bullet_{c+n,c}\ar[ld]^{p'_- \times p_S'} \ar[rd]_{p'_+} & & \CM_{c+n} \times \CM_{c'} \ar[ld] \ar[rd] & \\
\CM_c \times S & & \CM_{c+n} & & \CM_{c'}}$$
Recall that $H_{n} = (-1)^{rn} u^n (p_- \times p_S)_* (\CQ^{-r} \cdot p_+^*)$, in the notation of . Then the rule for composition of correspondences in gives us the following: $$\label{eqn:comp 1 bis}
A_m H_{n} = (\pi_1 \times \Id_S)_*(\Upsilon_n \cdot \pi_2^{*})$$ $$\label{eqn:comp 2 bis}
H_{n} A_m = (\pi'_1 \times \Id_S)_*(\Upsilon'_n \cdot \pi_2'^{*})$$ where: $$\begin{aligned}
&\Upsilon_n = (-1)^{rn} {u'}^{n} \left(\Id \times p_S \times p_+ \right)_* \Big[\CQ^{-r} \cdot \wedge^\bullet \left( (\Id \times p_-)^* \CE_{m} \right) \Big] \label{eqn:gamma 1 bis} \\
&\Upsilon'_n = (-1)^{rn} u^{n} \left(p_-' \times p_S' \times \Id \right)_* \Big[\CQ^{-r} \cdot \wedge^\bullet \left( (p'_+ \times \Id)^* \CE_{m} \right) \Big] \label{eqn:gamma 2 bis}\end{aligned}$$ are certain classes on $\CM_c \times S \times \CM_{c'}$. The following are equivalent to , : $$\label{eqn:geo 1 bis}
(\text{Id} \times p_-)^* \CE_{m} = (\text{Id} \times p_+)^* \CE_{m} - \left(\frac 1{\CL_1} +...+ \frac 1{\CL_n} \right) (\text{Id} \times p_S)^*\left( \frac {\CU m}q \right)$$ on $\CM_c \times \fZ_{c',c'-n}^\bullet$, and: $$\label{eqn:geo 2 bis}
(p_+' \times \text{Id})^* \CE_{m} = (p_-' \times \text{Id})^* \CE_{m} + (\CL_1+...+\CL_n) (p'_S \times \text{Id})^*\left( {\CU'}^\vee m \right)$$ on $\fZ_{c+n,c}^\bullet \times \CM_{c'}$. Armed with , , we may rewrite , as: $$\begin{aligned}
&\Upsilon_n = (-1)^{rn} {u'}^{n} \left[ \wedge^\bullet\CE_m \right] \left( \Id \times p_S \times p_+ \right)_* \left[\CQ^{-r} \bigotimes_{i=1}^n \wedge^\bullet \left(-\frac {\CU m}{\CL_i q}\right) \right] \\
&\Upsilon_n' = (-1)^{rn} u^{n} \left[ \wedge^\bullet\CE_m \right] \left( p_-' \times p_S' \times \Id \right)_* \left[\CQ^{-r} \bigotimes_{i=1}^n \wedge^\bullet \left(\frac {\CL_i m}{\CU'}\right) \right]\end{aligned}$$ Therefore, Proposition \[prop:corr\] implies: $$\begin{aligned}
&\Upsilon_n = \left[ \wedge^\bullet\CE_m \right] \int_{z_1 \prec ... \prec z_n \prec \{0,\infty\}} \frac {(-1)^{rn} {u'}^{n} \cdot z_1^{-r} ... z_n^{-r}}{\prod_{i=1}^{n-1} \left(1 - \frac {qz_{i+1}}{z_i} \right) \prod_{i < j} \zeta \left( \frac {z_j}{z_i} \right)} \prod_{i=1}^n \frac {\wedge^\bullet \left(\frac {z_iq}{\CU'} \right)}{\wedge^\bullet \left(\frac {\CU m}{z_i q} \right)} \label{eqn:gam 1 bis} \\
&\Upsilon_n' = \left[ \wedge^\bullet\CE_m \right] \int_{\{0,\infty\} \prec z_1 \prec ... \prec z_n} \frac {(-1)^{rn} u^{n} \cdot z_1^{-r} ... z_n^{-r}}{\prod_{i=1}^{n-1} \left(1 - \frac {qz_{i+1}}{z_i} \right) \prod_{i < j} \zeta \left( \frac {z_j}{z_i} \right)} \prod_{i=1}^n \frac {\wedge^\bullet \left(\frac {z_i m}{\CU'} \right)}{\wedge^\bullet \left(\frac {\CU}{z_i} \right)} \label{eqn:gam 2 bis}\end{aligned}$$ Consider the following rational function with coefficients in $K_{\CM_c \times S \times \CM_{c'}}$: $$\label{eqn:definition i bis}
I_n(z_1,...,z_n) = \frac {q^{rn}}{\prod_{i=1}^{n-1} \left(1 - \frac {qz_{i+1}}{z_i} \right) \prod_{i < j} \zeta \left( \frac {z_j}{z_i} \right)} \prod_{i=1}^n \frac {\wedge^\bullet \left(\frac {\CU'}{z_iq} \right)}{\wedge^\bullet \left(\frac {\CU m}{z_i q} \right)}$$ One may then rewrite and as: $$\begin{aligned}
&\Upsilon_n = \left[ \wedge^\bullet\CE_m \right] \int_{z_1 \prec ... \prec z_n \prec \{0,\infty\}} I_n(z_1,...,z_n) \\
&\Upsilon_n' = \left[ \wedge^\bullet\CE_m \right] \int_{\{0,\infty\} \prec z_1 \prec ... \prec z_n} I_n\left( \frac {z_1m}q ,..., \frac {z_nm}q \right) \cdot \gamma^n\end{aligned}$$ Changing the variables $z_i \mapsto \frac {z_i q}m$ in the second formula, we conclude that: $$\label{eqn:integral identity 1 bis}
\Upsilon_n - \Upsilon_n' \cdot \gamma^{-n} =$$ $$= \left[ \wedge^\bullet \CE_m \right] \left[ \int_{z_1 \prec ... \prec z_n \prec \{0,\infty\} } I_n \prod_{i=1}^n \frac {dz_i}{2\pi i z_i} - \int_{\{0,\infty\} \prec z_1 \prec ... \prec z_n} I_n \prod_{i=1}^n \frac {dz_i}{2\pi i z_i} \right]$$ The only difference between the two integrals is the order of the contours, specifically where the poles at $\{0,\infty\}$ lie in respect to the variables $z_1,...,z_n$. Therefore, we conclude that the difference above picks up the poles at 0 and $\infty$ in the various variables. However, all such residues are 0, except for: $$\begin{aligned}
&\text{Res}_{z_n = 0} \frac {I_n(z_1,...,z_n)}{z_n} = \gamma^{-1} \cdot I_{n-1}(z_1,...,z_{n-1}) \\
&\text{Res}_{z_1 = \infty} \frac {I_n(z_1,...,z_n)}{z_1} = q^r \cdot I_{n-1}(z_2,...,z_n) \end{aligned}$$ Therefore, formula implies that: $$\label{eqn:nice bis}
\Upsilon_n - \Upsilon_n' \cdot \gamma^{-n} = \Upsilon_{n-1} \cdot \gamma^{-1} - \Upsilon'_{n-1} \cdot q^r \gamma^{-n+1}$$ which, as an equality of classes on $\CM_c \times S \times \CM_{c'}$, precisely encodes .\
{#section-6}
We will now show how Lemma \[lem:commute heis\] allows us to prove Theorem \[thm:main heis\].\
*of Theorem \[thm:main heis\]:* We will only prove , since is analogous. We will use formulas , which say that the $H$ operators are to the $P$ operators as complete symmetric functions are to power sum functions. Then let us place into a generating series that goes over all $n \in \BN$: $$\label{eqn:kc}
\sum_{n=0}^\infty A_m H_{-n} \left( z^n - z^{n+1} \right) = \sum_{n=0}^\infty \left( \left( \gamma z \right)^n - \left( \gamma z \right)^{n+1} \right) H_{-n} A_m$$ If we write $H_-(z)$ for the power series (with sign $\pm = -$), then reads: $$\label{eqn:a and h series}
A_m H_-(z) \cdot (1 - z) = H_-\left(z \gamma\right) A_m \cdot (1-\gamma z)$$ If $P$ is an operator $\km \rightarrow \kms$ which commutes with two line bundles $c$ and $c'$ (in the sense of Proposition \[prop:doesn’t matter\], and the discussion after it), then: $$\label{eqn:equivalence}
A \exp(P) \exp(c') \Big|_\Delta = \exp(c) \exp(P) \Big|_\Delta A \qquad \Leftrightarrow \qquad AP + A c' = P A + c A$$ (this claim uses the associativity of the operation $x,y \leadsto xy|_\Delta$, as discussed in Subsection \[sub:action\]). With this in mind, formula implies: $$A_m P_-(z) - \sum_{n=1}^\infty \frac {A_m}{n z^{-n}} = P_-\left(z \gamma\right) A_m - \sum_{n=1}^\infty \gamma^n \frac {A_m}{n z^{-n}}$$ where $P_-(z) = \sum_{n=1}^\infty \frac {P_{-n}}{nz^{-n}}$. Extracting the coefficient of $z^n$ yields precisely .
{#section-7}
We will now perform the analogous computations for the commutator of $A_m$ with the operators of Subsection \[sub:w action\]:\
\[lem:comm\]
We have the following relations involving the *Ext* operator $A_m$: $$\begin{gathered}
A_m L_n(y) - A_m L_{n-1}(y) = \label{eqn:one} \\
= L_n \left( \frac ym \right) A_m \cdot \gamma^n - L_{n-1} \left( \frac {yq}m \right) E \left( \frac {yq}m \right) A_m E \left( y \right)^{-1} \Big|_\Delta \cdot \gamma^{n-1} \end{gathered}$$ and: $$\begin{gathered}
U_n \left(\frac {yq}m \right) A_m \cdot \gamma^{-n} - U_{n-1} \left(\frac {yq}m \right) A_m \cdot q \gamma^{-n+1} = \label{eqn:two} \\
= A_m U_n(y) - E\left(\frac {yq}m\right)^{-1}A_m E(yq) U_{n-1}(yq) \Big|_\Delta \cdot q \end{gathered}$$ The right-hand side of equations and maps $\km$ to $\rightarrow \kms \left\llbracket y^{-1} \right \rrbracket$. The symbol $|_\Delta$ applied to any term that involves three of the series $L,E,U$ means that we restrict a certain operator $\km \rightarrow \kmsss \left\llbracket y^{-1} \right \rrbracket$ to the small diagonal.\
We will closely follow the proof of Lemma \[lem:commute heis\]. With the notation therein, one needs to replace and by: $$\begin{aligned}
&\Upsilon_{n,y} = \left( \Id \times p_S \times p_- \right)_* \left[\frac 1{1- \frac y{\CL_n}} \wedge^\bullet \left( (\Id \times p_+)^* \CE_{m} \right) \right] \\
&\Upsilon'_{n,y} = \left( p_+' \times p_S' \times \Id \right)_* \left[ \frac 1{1- \frac y{\CL_n}} \wedge^\bullet \left( (p'_- \times \Id)^* \CE_{m} \right) \right]\end{aligned}$$ This is reflected by inserting: $$\left(1 - \frac y{z_n} \right)^{-1}$$ into the right-hand sides of formulas and . Therefore, the function $I_n(z_1,...,z_n)$ defined in should be replaced by: $$I_{n,y}(z_1,...,z_n) = \frac {I_n(z_1,...,z_n)}{1- \frac y{z_n}}$$ It is easy to see that the non-zero residues of $I_{n,y}$ are: $$\begin{aligned}
&\text{Res}_{z_1 = \infty} \frac {I_{n,y}(z_1,...,z_n)}{z_1} = I_{n-1,y}(z_2,...,z_n) \\
&\text{Res}_{z_n = y} \frac {I_{n,y}(z_1,...,z_n)}{z_n} = \frac {\wedge^\bullet \left(\frac {\CU m}{yq} \right)}{\wedge^\bullet \left( \frac {\CU'}y \right)} I_{n-1,yq}(z_1,...,z_{n-1})\end{aligned}$$ (indeed, the contours of the integrals should be taken as in Proposition \[prop:corr y\] instead of as in Proposition \[prop:corr\]). Therefore, the analogue of identity is: $$\Upsilon_{n,y} - \Upsilon_{n, \frac ym}' \cdot \gamma^n = \Upsilon_{n-1,y} - \Upsilon_{n-1, \frac {yq}m}' \cdot \gamma^{n-1} \frac {\wedge^\bullet \left(\frac {\CU m}{yq} \right)}{\wedge^\bullet \left( \frac {\CU'}y \right)}$$ This equality of classes on $\CM_c \times S \times \CM_{c'}$ precisely underlies equality .\
As for , we proceed analogously. One needs to replace and by: $$\begin{aligned}
&\Upsilon_n = \frac {(-1)^{rn} {u'}^n}{q^{(r-1)n}} \left(\Id \times p_S \times p_+ \right)_* \left[ \frac {\CQ^{-r}}{1- \frac y{\CL_n}} \cdot \wedge^\bullet \left( (\Id \times p_-)^* \CE_{m} \right) \right] \\
&\Upsilon'_n = \frac {(-1)^{rn} u^n}{q^{(r-1)n}} \left(p_-' \times p_S' \times \Id \right)_* \left[ \frac {\CQ^{-r}}{1- \frac y{\CL_n}} \cdot \wedge^\bullet \left( (p'_+ \times \Id)^* \CE_{m} \right) \right]\end{aligned}$$ This is reflected by inserting: $$q^{n(1-r)}\left(1 - \frac y{z_n} \right)^{-1}$$ into the right-hand sides of formulas and . Therefore, the function $I_n$ defined in should be replaced by: $$I_{n,y}(z_1,...,z_n) = \frac {I_n(z_1,...,z_n)}{ q^{(r-1)n} \left(1- \frac y{z_n}\right)}$$ It is easy to see that the non-zero residues of $I_{n,y}$ are: $$\begin{aligned}
&\text{Res}_{z_n = y} \frac {I_{n,y}(z_1,...,z_n)}{z_n} = q \frac {\wedge^\bullet \left( \frac {\CU'}{yq} \right)}{\wedge^\bullet \left(\frac {\CU m}{yq} \right)} I_{n-1,yq}(z_1,...,z_{n-1}) \\
&\text{Res}_{z_1 = \infty} \frac {I_{n,y}(z_1,...,z_n)}{z_1} = q I_{n-1,y}(z_2,...,z_n) \end{aligned}$$ Therefore, the analogue of identity is: $$\Upsilon_{n,y} - \Upsilon_{n, \frac {yq}m}' \cdot \gamma^{-n} = \Upsilon_{n-1,yq} \cdot q \frac {\wedge^\bullet \left(\frac {\CU'}{yq} \right)}{\wedge^\bullet \left( \frac {\CU m}{yq} \right)} - \Upsilon_{n-1, \frac {yq}m}' \cdot q \gamma^{-n+1}$$ This equality of classes on $\CM_c \times S \times \CM_{c'}$ precisely underlies equality .
{#section-8}
In all formulas below, whenever one encounters a product of several $L$, $E$, $U$ operators, one needs to place the symbol $|_\Delta$ next to it, e.g. $L(...)E(...)U(...)|_\Delta$ as in . From now on, we will suppress the notation $|_\Delta$ from our formulas for brevity.\
*of Theorem \[thm:main\]:* In terms of the generating series , formulas and take the following form: $$\begin{aligned}
&\left(1 - x \right) A_m L(x,y) = L \left( x \gamma , \frac ym \right) A_m - x L \left( x \gamma , \frac {yq}m \right) E \left(\frac {yq}m \right) A_m E(y)^{-1} \\
&U\left( x \gamma, \frac {yq}m \right) A_m \left(1 - \frac qx \right) = A_m U(x,y) - \frac qx E \left( \frac {yq}m \right)^{-1} A_m E(yq) U(x,yq)\end{aligned}$$ Change the variables $x \mapsto xq$, $y \mapsto y/q$ in the second equation, and multiply the first equation by $E(y)$ and the second equation by $E(y/m)$. Thus we obtain: $$\begin{aligned}
&\left(1 - x \right) A_m L(x,y) E(y) = \\
& \qquad \qquad = L \left( x \gamma , \frac ym \right) A_m E(y) - x L \left( x \gamma, \frac {yq}m \right) E \left(\frac {yq}m \right) A_m \\
&E \left( \frac ym \right) U\left( xq \gamma , \frac ym\right) A_m \left(1 - \frac 1x \right) = \\
& \qquad \qquad = E \left( \frac ym \right) A_m U\left(xq, \frac {y}q \right) - \frac 1x A_m E(y) U\left( xq,y \right)\end{aligned}$$ Now let us replace the variable $y$ by the symbol $yD_x$, where $D_x$ denotes the $q$-difference operator $f(x) \leadsto f(xq)$. However, we make the following prescription. In the first equation, the $D_x$’s come to the right of all $x$’s, while in the second equation, the $D_x$’s come before the $x$’s: $$\begin{aligned}
&\left(1 - x \right) A_m L(x,yD_x) E(yD_x) = \\
& \qquad \qquad = L \left( x \gamma , \frac {yD_x}m \right) A_m E(yD_x) - x L \left( x \gamma, \frac {yD_xq}m \right) E \left(\frac {yD_xq}m \right) A_m \\
&E \left( \frac {yD_x}m \right) U\left( xq \gamma , \frac {yD_x} m\right) A_m (1-x) = \\
& \qquad \qquad = A_m E(yD_x) U\left( xq,yD_x \right) - E \left( \frac {yD_x}m \right) A_m U\left(xq, \frac {yD_x}q \right) x\end{aligned}$$ Now let us multiply the first equation on the right by $U(qx,yD_x)$ (with the $D_x$’s on the left of the $x$’s) and the second equation on the left by $L(x \gamma,yD_x/m)$ (with the $D_x$’s on the right of the $x$’s): $$\begin{aligned}
&(1-x) A_m L(x,yD_x) E(yD_x) U(xq,yD_x) = \\
& \quad = L \left( x\gamma , \frac {yD_x}m \right) A_m E(yD_x) U(xq,yD_x) - x L \left( x \gamma, \frac {yD_xq}m \right) E \left(\frac {yD_xq}m \right) A_m U(xq,yD_x) \\
&L \left(x \gamma,\frac {yD_x}m \right) E \left( \frac {yD_x}m \right) U\left( xq \gamma , \frac {yD_x} m\right) A_m (1-x) = \\
& \quad = L \left(x \gamma,\frac {yD_x}m \right) A_m E(yD_x) U\left( xq,yD_x \right) - L \left(x \gamma,\frac {yD_x}m \right) E \left( \frac {yD_x}m \right) A_m U\left(xq, \frac {yD_x}q \right) x\end{aligned}$$ The two terms in the right-hand sides of the above equations are pairwise equal to each other (this is not manifestly obvious for the second term, because $y$ differs from $yq$, but this is a consequence of the action of $D_x$ on $x$). We conclude that: $$\begin{gathered}
(1-x)A_m L(x,yD_x) E(yD_x) U(xq,yD_x) = \\ = L \left(x \gamma,\frac {yD_x}m \right) E \left( \frac {yD_x}m \right) U\left( xq \gamma , \frac {yD_x} m\right) A_m (1-x)\end{gathered}$$ Recalling the definition , this implies $$(1-x) A_m W(x,yD_x) = W \left(x\gamma,\frac {yD_x}m \right) A_m (1-x)$$ Taking the coefficient of $(yD_x)^{-k}$ implies . In doing so, the right-most factor $1-x$ changes into $1 - \frac x{q^k}$ due to the fact that the operators $\frac 1{D_x^{k}}$ must pass over it.
{#section-9}
Finally, we recall the operator $\Phi_m : \kmm \rightarrow \km$ defined in : $$\Phi_m = A_m \exp \left[\sum_{n=1}^\infty \frac {P_n}n \left\{ \frac {(q^n-1)q^{-rn}}{[q_1]_n [q_2]_n} \right\} \right]$$ and let us translate , , into commutation relations involving $\Phi_m$.\
*of Corollary \[cor:main\]:* Because $P_n$ commutes with $P_{n'}$ for all $n,n'>0$, $\Rightarrow$ when the sign is $+$. Let us now prove when the sign is $-$. We write: $$\Phi_m = A_m \cdot \exp$$ where $\exp$ is shorthand for $\exp \left[\sum_{n=1}^\infty \frac {P_n}n \left\{ \frac {(q^n-1)q^{-rn}}{[q_1]_n [q_2]_n} \right\} \right]$. Then reads: $$\Phi_m \cdot \exp^{-1} \cdot P_{-n} - P_{-n} \cdot \Phi_m \cdot \exp^{-1} \gamma^n = \Phi_m \cdot \exp^{-1} (1 - \gamma^n)$$ The relation above will establish for $\pm = -$ once we prove that: $$\label{eqn:rus 0}
[\exp^{-1}, P_{-n}] = (1-q^{-rn}) \exp^{-1}$$ If we take the logarithm of , it boils down to: $$\label{eqn:rus}
\left[P_{-n}, \frac {P_n}n \left\{ \frac {(q^n-1)q^{-nr}}{[q_1]_n [q_2]_n} \right\} \right] = 1-q^{-rn}$$ Relation is an equality of operators $\km \rightarrow \kms$ (the operator in the right-hand side is just pull-back multiplied with $\proj_S^*(1-q^{-rn})$). Relation is proved as follows: take relation , which is an equality of operators $\km \rightarrow \kmss$, and multiply it with the class: $$\frac 1n \cdot\frac {(q^n-1)q^{-nr}}{[q_1]_n [q_2]_n}$$ in the second factor of $S \times S$. Then integrate over the second factor of $S \times S$.\
Now let us prove $\Rightarrow$ . For that, we must take formula (which is a priori an equality of operators $\km \rightarrow \kmss$) for $\pm = +$, multiply it with: $$\frac {(q^n-1)q^{-nr}}{[q_1]^n[q_2]^n}$$ coming from the second factor of $S \times S$, and then integrate over the second factor of $S \times S$. The resulting equality reads: $$\left[ W_k(x), P_n \left\{ \frac {(q^n-1)q^{-nr}}{[q_1]^n[q_2]^n} \right\} \right] = (1 - q^{-nk}) x^n W_k(x)$$ It is an easy exercise the show that $[W,P] = cW$ implies that $\exp(-P)W = \exp(c) \cdot W\exp(-P)$ as long as $c$ commutes with both $W$ and $P$. Therefore, we infer that: $$\begin{aligned}
&\exp^{-1} W_k(x) = \exp\left[ \sum_{n=1}^{\infty} \frac {(1 - q^{-nk})x^n}n \right] W_k(x) \exp^{-1} \Rightarrow \\
&\Rightarrow \ \exp^{-1} W_k(x) = \frac {1-\frac x{q^k}}{1-x} \cdot W_k(x) \exp^{-1} \Rightarrow \\
&\Rightarrow \ \Phi_m \exp^{-1} W_k(x) \cdot (1-x) = \Phi_m W_k(x) \exp^{-1} \cdot \left(1 - \frac x{q^k} \right)\end{aligned}$$ With this in mind, and the fact that $\Phi_m \exp^{-1} = A_m$ imply that: $$m^kW_k(x\gamma) \Phi_m \exp^{-1} \cdot \left(1- \frac x{q^k} \right) = \Phi_m W_k(x) \exp^{-1} \cdot \left(1- \frac x{q^k} \right)$$ Multiplying on the right with $\exp$ yields .
The Verma module {#sec:verma}
================
{#section-10}
Let us now specialize to $S = \BA^2$, and explain all the necessary modifications to the constructions in the present paper (we refer the reader to [@W] for details). Let $\CM$ denote the moduli space of rank $r$ torsion-free sheaves $\CF$ on $\BP^2$, together with a trivialization along a fixed line $\infty \subset \BP^2$: $$\CM = \Big\{\CF, \CF|_\infty \stackrel{\phi}\cong \CO_\infty^r \Big\}$$ The $c_1$ of such sheaves is forced to be 0, but $c_2$ is free to vary over the non-negative integers, and so the moduli space breaks up into connected components as before: $$\CM = \bigsqcup_{c = 0}^\infty \CM_c$$ The space $\CM$ is acted on by the torus $T = \BC^* \times \BC^* \times (\BC^*)^r$, where the first two factors act by scaling $\BA^2$, and the latter $r$ factors act on the framing $\phi$. Note that: $$K^T_0(\pt) = \BZ[q_1^{\pm 1}, q_2^{\pm 1}, u_1^{\pm 1},...,u_r^{\pm 1}]$$ where $q_1,q_2,u_1,...,u_r$ are elementary characters of the torus $T$. We note that $q_1$ and $q_2$ are the equivariant weights of the cotangent space to $\BA^2$, and the determinant of the universal sheaf $\CU$ is the equivariant constant $u = u_1...u_r$. Consider the group: $$\km = \bigoplus_{c=0}^\infty K^T_0(\CM_c) \underset {\BZ[q_1^{\pm 1}, q_2^{\pm 1},u_1^{\pm 1},...,u_r^{\pm 1}]}{\otimes} \BQ(q_1,q_2,u_1,...,u_r)$$ The main goal of was to define operators as in , , : $$\label{eqn:ops}
W_{n,k}, P_{\pm n'} : \km \rightarrow \km$$ for all $n \in \BZ$ and $k,n' \in \BN$, which are shown to satisfy the relations in the deformed $W$–algebra of type $\fgl_r$ (since $S = \BA^2$, $\km \cong \kms$ naturally).\
The universal Verma module $M_{u_1,...,u_r}$ with highest weight $(u_1,...,u_r)$ is the $\BQ(q_1,q_2,u_1,...,u_r)$-vector space with basis given by: $$\label{eqn:basis}
W_{n_1,k_1}...W_{n_s,k_s} \vac$$ as the pairs $(n_i,k_i)$ range over $-\BN \times \{1,...,r\}$ and are ordered in non-decreasing order of the slope $n_i/k_i$. We make $M_{u_1,...,u_r}$ into a deformed $W$–algebra module as follows. The action of an arbitrary generator $W_{n,k}$ on the basis vector is prescribed by the commutation relations , together with the relations: $$\begin{aligned}
&W_{n,k} \vac = 0 & &\text{if } n > 0 \text{ or } k>r \\
&W_{0,k} \vac = e_k(u_1,...,u_r)\vac & &\text{for all }k\end{aligned}$$ where $e_k$ denotes the $k$–th elementary symmetric polynomial.\
\[thm:fock\] ([@W]) With respect to the action of the operators , we have: $$\label{eqn:iso fock}
\km \cong M_{u_1,...,u_r}$$ The highest weight is given by the equivariant parameters of $(\BC^*)^r$, and is assumed generic. The isomorphism sends the structure sheaf of $\CM_0 \subset \CM$ to $|\emptyset\rangle$.\
{#sub:weak}
The Ext (respectively vertex) operator $A_m$ (respectively $\Phi_m$) for $S = \BA^2$ was studied in [@W], where we obtained an analogue of Theorem \[thm:main\] in the case $k=1$ (some coefficients in the formulas of differ from those of the present paper, because their operator $A_m$ differs from ours by an equivariant constant). However, having only proved the case $k=1$ in led to weaker formulas than . Thus, the present paper strengthens the results of (see Remark 4.8 therein). Specifically, Corollary \[cor:main\] completely determines the operator $\Phi_m$ (hence also $A_m$) in the case $S = \BA^2$, because of Theorems \[thm:fock\] and Theorem \[thm:unique\] below:\
\[thm:unique\]
Given two Verma modules $M_{u_1,...,u_r}$ and $M_{u'_1,...,u'_r}$, there is a unique (up to constant multiple in $\BQ(q_1,q_2,u_1,...,u_r,u_1',...,u_r')$) linear map: $$\Phi_m : M_{u_1',...,u_r'} \rightarrow M_{u_1,...,u_r}$$ satisfying for all $k \geq 1$.\
The existence of such a linear map follows from the very fact that the operator satisfies . To show uniqueness, it is enough to prove $\langle \emptyset | \Phi_m | \emptyset \rangle = 0$ implies $\Phi_m = 0$, for any operator that satisfies the following relations for all $n$, $k$: $$\label{eqn:comm phi foil}
\Phi_m W_{n,k} - \Phi_m W_{n+1,k} \cdot q^{-k} = W_{n,k} \Phi_m \cdot m^k \gamma^{-nk} - W_{n+1,k} \Phi_m \cdot \frac {m^k}{q^k} \gamma^{-(n+1)k}$$ where $m$ and $\gamma$ are certain constants. One may think of $\langle v|v' \rangle$ as a Shapovalov form on the Verma module, namely the unique (up to constant multiple) bilinear form for which $W_{n,k}$ is the adjoint of $W_{-n,k}$. Therefore, it remains to show that: $$\label{eqn:adam}
\langle \emptyset |W_{-n_s,k_s}... W_{-n_1,k_1} \Phi_m W_{n_1',k_1'}... W_{n_t',k_t'}| \emptyset \rangle = 0$$ for all collections of indices $(n_i,k_i), (n_i',k_i') \in -\BN \times \{1,...,r\}$, ordered by slope: $$\frac {n_1}{k_1} \leq ... \leq \frac {n_s}{k_s}, \qquad \frac {n_1'}{k_1'} \leq ... \leq \frac {n_t'}{k_t'}$$ The matrix coefficient is non-zero only if the $n_i$’s and $n'_j$’s are all negative, so we may prove formula by induction on the natural number $ - \sum n_i - \sum n_i'$ (the base case when this number is zero is precisely the assumption $\langle \emptyset | \Phi_m | \emptyset \rangle = 0$). One may iterate relation to obtain: $$\Phi_m W_{n,k} = \sum_{\bar{n} \geq n} W_{\bar{n},k} \Phi_m \cdot \text{constant}$$ Although the sum in the right-hand side is infinite, only finitely many terms act in a non-trivial way on any fixed vector, due to the fact that $W_{\bar{n},k}$ annihilates any vector of the Verma module for $\bar{n}$ large enough. Therefore, the LHS of equals: $$\label{eqn:final}
\langle \emptyset |... W_{-n_1,k_1} \Phi_m W_{n_1',k_1'}... | \emptyset \rangle = \sum_{\bar{n}_1' \geq n_1} \langle \emptyset |... W_{-n_1,k_1} W_{\bar{n}_1',k_1'} \Phi_m ... | \emptyset \rangle \cdot \text{constant}$$ It was proved in [@W; @surf] that the product $W_{-n_s,k_s}... W_{-n_1,k_1} W_{\bar{n}_1',k_1'}$ is equal to a linear combination of products of the form: $$W_{-n_r'',k_r''}... W_{-n_1'',k_1''} \qquad \text{with} \quad \frac {n_1''}{k_1''} \leq ... \leq \frac {n_r''}{k_r''}$$ and $\sum n_i'' = \sum n_i - \bar{n}_1'$. Therefore, we conclude that the right-hand side of is equal to a linear combination of expressions of the following form: $$\label{eqn:terms}
\langle \emptyset | W_{-n_r'',k_r''}... W_{-n_1'',k_1''}\Phi_m W_{n_2',k_2'}... W_{n_t',k_t'} |\emptyset \rangle$$ Since $- \sum n_i'' - \sum n_i' + n_1' = - \sum n_i - \sum n_i' + n_1' + \bar{n}_1' < - \sum n_i - \sum n_i'$, then all the terms are 0 by the induction hypothesis, and therefore so is .
We note that the identification of $A_m$ (in the case $S = \BA^2$) with a vertex operator was also achieved in [@BFMZZ], who computed relations and for $n=1$ in the basis of fixed points. This uniquely determines the operator $A_m$ due to the well-known features of the Ding-Iohara-Miki algebra, but does not directly establish the connection with the generating currents of the $W$–algebra for type $\fgl_r$. From a geometric point of view, this is because the usual Nakajima-type correspondences only describe the operators $L_{1,k}$ and $U_{1,k}$. As we have seen in Subsection \[sub:basic mod\], in order to define the operators $L_{n,k}$ and $U_{n,k}$ for all $n$ (with the ultimate goal of defining the $W$–algebra generators $W_{n,k}$ in ), one needs to introduce the more complicated correspondences .
[XXX]{}
Awata H., Kubo H., Odake S. and Shiraishi J., [*Quantum $W_N$ algebras and Macdonald polynomials*]{} **Comm. Math. Phys.** 179 (1996), no.2, 401–416
Baranovsky V., [*Moduli of sheaves on surfaces and action of the oscillator algebra*]{}, **J. Diff. Geom.** 55 (2000), no. 2
Behrend K., Ciocan-Fontanine I., Hwang J., Rose M., [*The derived moduli space of stable sheaves*]{}, **Algebra Number Theory**, Volume 8, Number 4 (2014), 781–812
Bourgine J.-E., Fukuda M., Matsuo Y., Zhang H., Zhu R.-D. [*Coherent states in quantum $W_{1+\infty}$-algebra and qq-character for 5d Super Yang-Mills*]{}, **PTEP**, 2016(12):123B05, 2016
Braverman A., Finkelberg M., Nakajima H., [*Instanton moduli spaces and $\mathscr{W}$-algebras*]{}, **Astérisque** No. 385 (2016), vii+128 pp
Bruzzo U., Pedrini M., Sala F., Szabo R.J. [*Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces*]{}, **Adv. Math.**, 288 (2016) 1175–1308
Caldararu A., [*Derived categories of twisted sheaves on Calabi-Yau manifolds*]{}, **Ph.D. Thesis**, Cornell University (2000)
Carlsson E., Nekrasov N., Okounkov A., [*Five dimensional gauge theories and vertex operators*]{}, ar$\chi$iv:1308.2465
Carlsson E., Okounkov A., [*Exts and vertex operators*]{}, **Duke Math. J.** 161 (2012), no. 9, 1797–1815
Feigin B., Frenkel E., [*Quantum $W$–algebras and elliptic algebras*]{}, **Comm. Math. Phys.**, 178 (1996), no. 3, 653–678
Feigin B., Gukov S., [*VOA\[M4\]*]{}, ar$\chi$iv:1806.02470
Grojnowski I., [*Instantons and affine algebras I. The Hilbert scheme and vertex operators*]{}, **Math. Res. Lett.** 3 (1996), no. 2
Maulik D., Okounkov A., [*Quantum Groups and Quantum Cohomology*]{}, ar$\chi$iv:1211.1287
Nakajima H., [*Heisenberg algebra and Hilbert schemes of points on projective surfaces*]{}, **Ann. Math.** 145, No. 2 (Mar 1997), 379–388
Negut A., [*The $q$-AGT-W relations via shuffle algebras*]{}, **Comm. Math. Phys.**, February 2018, Volume 358, Issue 1, 101–170
Negut A., [*Shuffle algebras associated to surfaces*]{}, ar$\chi$iv: 1703.02027
Negut A., [*$W$–algebras associated to surfaces*]{}, ar$\chi$iv: 1710.03217
Negut A., [*Hecke correspondences for smooth moduli spaces of sheaves*]{}, ar$\chi$iv: 1804.03645
Schiffmann O., Vasserot E., [*Cherednik algebras, $W$–algebras and the equivariant cohomology of the moduli space of instantons on $\BA^2$*]{}, **Publ. math. IHES**, Volume 118, Issue 1, 213–342
[^1]: See Subsection \[sub:correspondences\] for a review of correspondences as $K$–theoretic operators
[^2]: Note that $u$ parametrizes the determinant of any one of the sheaves $\CF_{c+n},...,\CF_c$ in a flag , since these sheaves have canonically isomorphic determinants, see Proposition \[prop:doesn’t matter\]
|
---
abstract: 'Here we show that a particular one-parameter generalization of the exponential function is suitable to unify most of the popular one-species discrete population dynamics models into a simple formula. A physical interpretation is given to this new introduced parameter in the context of the continuous Richards model, which remains valid for the discrete case. From the discretization of the continuous Richards’ model (generalization of the Gompertz and Verhuslt models), one obtains a generalized logistic map and we briefly study its properties. Notice, however that the physical interpretation for the introduced parameter persists valid for the discrete case. Next, we generalize the (scramble competition) $\theta$-Ricker discrete model and analytically calculate the fixed points as well as their stability. In contrast to previous generalizations, from the generalized $\theta$-Ricker model one is able to retrieve either scramble or contest models.'
address:
- |
Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto,\
Universidade de São Paulo, and\
National Institute of Science and Technology for Complex Systems\
Avenida Bandeirantes, 3900\
14040-901, Ribeirão Preto, São Paulo, Brazil.
- |
Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto,\
Universidade de São Paulo,\
Avenida Bandeirantes, 3900\
14040-901, Ribeirão Preto, São Paulo, Brazil.
- |
Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto,\
Universidade de São Paulo,\
Avenida Bandeirantes, 3900\
14040-901, Ribeirão Preto, São Paulo, Brazil.
author:
- Alexandre Souto Martinez
- Rodrigo Silva González
- Aquino Lauri Espíndola
title: Generalized exponential function and discrete growth models
---
Complex Systems,Population dynamics (ecology) ,Nonlinear dynamics 89.75.-k ,87.23.-n ,87.23.Cc ,05.45.-a
Introduction
============
Recently, the generalizations of the logarithmic and exponential functions have attracted the attention of researchers. One-parameter logarithmic and exponential functions have been proposed in the context of non-extensive statistical mechanics [@tsallis_1988; @tsallis_qm; @nivanen_2003; @borges_2004; @kalogeropoulos_2005], relativistic statistical mechanics [@kaniadakis_2001; @PhysRevE.66.056125] and quantum group theory [@abe_1997]. Two and three-parameter generalization of these functions have also been proposed [@kaniadakis:046128; @kaniadakis:036108; @veit:2007]. These generalizations are in current use in a wide range of disciplines since they permit the generalization of special functions: hyperbolic and trigonometric [@borges_1998], Gaussian/Cauchy probability distribution function [@tsallis_levy] etc. Also, they permit the description of several complex systems [@tsallis_stariolo:1996; @albuquerque:2000; @cajueiro_2006; @cajueiro_2007; @anteneodo:1:2002; @holanda:2004], for instance in generalizing the stretched exponential function [@martinez:2008c].
As mentioned above, the one-parameter generalizations of the logarithm and exponential functions are not univoquous. The $\tilde q$-logarithm function $\ln_{\tilde q}(x)$ is defined as the value of the area underneath the non-symmetric hyperbole, $f_{\tilde q}(t)=1/t^{1-\tilde q}$, in the interval $t \in [1,x]$ [@tiago]: $$\begin{aligned}
\ln_{\tilde q}(x) & = & \int_1^x \frac{dt}{t^{1-{\tilde q}}}=\lim_{{\tilde q}^\prime \to {\tilde q}}\frac{x^{\tilde q^\prime}-1}{{\tilde q}^\prime}
\; .
\label{eq:gen_log}\end{aligned}$$ This function is [*not*]{} the ordinary logarithmic function in the basis $\tilde q$, namely $[\log_{\tilde q}(x)]$, but a generalization of the natural logarithmic function definition, which is recovered for $\tilde q=0$. The area is negative for $0<x<1$, it vanishes for $x=1$ and it is positive for $x>1$, independently of the $\tilde q$ values.
Given the area $x$ underneath the curve $f_{\tilde q}(t)$, for $t\in [0,y]$, the upper limit $y$ is the generalized $\tilde q$-exponential function: $y=e_{\tilde q}(x)$. This is the inverse function of the $\tilde q$-logarithmic $e_{\tilde q}[\ln_{\tilde q}(x)]=x=\ln_{\tilde q}[e_{\tilde q}(x)]$ and it is given by: $$e_{\tilde q}(x) =
\left\{
\begin{array}{ll}
0 & \; \mbox{for} \; \tilde{q} x < -1 \\
\lim_{{\tilde q}^\prime \to {\tilde q}}(1+{\tilde q}^\prime x)^{1/{\tilde q}^\prime} & \; \mbox{for} \; \tilde{q} x \ge -1
\end{array}
\right. \; .
\label{eq:limite}$$ This is a non-negative function $e_{\tilde q}(x) \geq 0$, with $e_{\tilde q}(0)=1$, for any $\tilde q$. For $\tilde q \to \pm \infty$, one has that $e_{- \infty}(x)=1$, for $x \le 0$ and $e_{\infty}(x)=1$, for $x \ge 0$. Notice that letting $x=1$ one has generalized the Euler’s number: $$e_{\tilde q} (1) = (1+\tilde q)^{1/\tilde q}.
\label{eq:eqtilde}$$
Instead of using the standard entropic index $q$ in Eqs. (\[eq:gen\_log\]) and (\[eq:eqtilde\]), we have adopted the notation $\tilde{q} = 1 - q$. The latter notation permits us to write simple relations as: $\ln_{\tilde{q}}(x) = - \ln_{-\tilde{q}}(x)$ or $e_{-\tilde{q}}(-x) = 1/e_{\tilde{q}}(x)$, bringing the inversion point around $\tilde{q} = 0$. These relations lead to simpler expressions in population dynamics problems [@martinez:2008b] and the generalized stretched exponential function [@martinez:2008c] contexts. Also, they simplify the generalized sum and product operators [@tiago], where a link to the aritmethical and geometrical averages of the generalized functions is established.
This logarithm generalization, as shown in Ref. [@montroll_west p. 83], is the one of non-extensive statistical mechanics [@tsallis_qm]. It turns out to be precisely the form proposed by Montroll and Badger [@badger_1974] to unify the Verhulst ($\tilde q = 1$) and Gompertz ($\tilde q = 0$) one-species population dynamics model. The $\tilde{q}$-logarithm leads exactly to the Richards’ growth model [@richards_1959; @martinez:2008b]: $$\frac{d \ln p(t)}{dt} = -\kappa \ln_{\tilde q}p(t),
\label{eq:richard_model}$$ where $p(t)=N(t)/N_\infty$, $N(t)$ is the population size at time $t$, $N_\infty$ is the carrying capacity and $\kappa$ is the intrinsic growth rate. The solution of Eq. (\[eq:richard\_model\]) is the [*$\tilde q$-generalized logistic*]{} equation $p(t) = 1/{e_{\tilde q}[\ln_{\tilde q}(p_0^{-1})e^{-\kappa t}]} = e_{-{\tilde q}}[-\ln_{\tilde q}(p_0^{-1})e^{-\kappa t}] = e_{-{\tilde q}}[\ln_{-\tilde q}(p_0)e^{-\kappa t}]$.
The competition among cells drive to replicate and inhibitory interactions, that are modeled by long range interaction among these cells. These interactions furnish an interesting microscopic mechanism to obtain Richards’ model [@idiart_2002; @idiart_2002-2]. The long range interaction is dependent on the distance $r$ between two cells as a power law $r^{\gamma}$. These cells have a fractal structure characterized by a fractal dimension $D_f$.
Here we call the attention to Eq. (7) of Ref. [@idiart_2002], namely $\dot n(t)= n(t)\{\left<G\right> - JI[n(t)]\}$, where $I(n(t)) = \omega\left\{[D_f n(t)/\omega]^{1-\gamma/D_f}-1\right\}/[D_f(1-\gamma/D_f)]$. Here, $\omega$ is a constant related to geometry of the problem, $\left< G \right>$ is the mean intrinsic replication rate of the cells and $J$ is the interaction factor. Using Eq. (\[eq:gen\_log\]), one can rewrite it simply as: $\mbox{d}\ln n(t)/\mbox{d}t = \langle G \rangle / n(t)- J \omega \ln_{\tilde{q}}[D_f n(t)/\omega]/{D_f}$. Calling, $p = D_f n/\omega$, $\kappa = J \omega/D_f$ and $\tilde{q} = 1 - \gamma/D_f$, this equation is the Richard’s model \[Eq. (\[eq:richard\_model\])\] with an effort rate $\langle G \rangle / n(t)$. In this context the parameter $\tilde{q}$ acquires a physical meaning related to the interaction range $\gamma$ and fractal dimension of the cellular structure $D_f$. If the interaction does not depend on the distance, $\gamma=0$, and it implies that $\tilde q=1$. This physical interpretation of $\tilde{q}$ has only been possible due to Richards’ model underlying microscopic description.
Introduced by Nicholson in 1954 [@hassell_1975], scramble and contest are types of intraspecific competition models that differ between themselves in the way that limited resources are shared among individuals. In scramble competition, the resource is equally shared among the individuals of the population as long as it is available. In this case, there is a critical population size $N_c$, above which, the amount of resource is not enough to assure population survival. In the contest competition, stronger individuals get the amount of resources they need to survive. If there is enough resources to all individuals, population grows, otherwise, only the strongest individuals survive (strong hierarchy), and the population maintains itself stable with size $N_\infty$.
From experimental data, it is known that other than the important parameter $\kappa$ (and sometimes $N_\infty$), additional parameters in more complex models are needed to adjust the model to the given population. One of the most general discrete model is the $\theta$-Ricker model [@bellows_1981; @berryman_1999]. This model describes well scramble competition models but it is unable to put into a unique formulation the contest competition models such as Hassel model [@hassell_1975], Beverton-Holt model [@beverton_holt] and Maynard-Smith-Slatkin model [@maynard-smith_1973].
Our main purpose is to show that Eq. (\[eq:limite\]) is suitable to unify most of the known discrete growth models into a simple formula. This is done in the following way. In Sec. \[sec:loquistic\], we show that the Richards’ model \[Eq. (\[eq:richard\_model\])\], which has an underlying microscopic model, has a physical interpretation to the parameter $\tilde{q}$, and its discretization leads to a generalized logistic map. We briefly study the properties of this map and show that some features of it (fixed points, cycles etc.) are given in terms of the $\tilde q$-exponential function. Curiously, the map attractor can be suitably written in terms of $\tilde{q}$-exponentials, even in the logistic case. In Sec. \[sec:generalized\_theta\_ricker\], using the $\tilde q$-exponential function, we generalize the $\theta$-Ricker model and analytically calculate the model fixed points, as well as their stability. In Sec. \[sec:generalizedskellam\], we consider the generalized Skellam model. These generalizations allow us to recover most of the well-known scramble/contest competition models. Final remarks are presented in Sec \[sec:conclusion\].
Discretization of the Richards’ model {#sec:loquistic}
=====================================
To discretize Eq. (\[eq:richard\_model\]), call $(p_{i+1}-p_i)/\Delta t = -kp_i(p_{i}^{\tilde q}-1)/{\tilde q},~\rho_{\tilde q}^{\prime}= 1 + k\Delta t/{\tilde q}$ and $x_i= p_i[(\rho_{\tilde q}-1)/\rho_{\tilde q}]^{\tilde q}$, which leads to: $$x_{i+1}=\rho_{\tilde q}^{\prime}x_i(1-x_i^{\tilde q})=-\rho_{\tilde q}x_i\ln_{\tilde q}(x_i)\;,
\label{eq:loquistic}$$ where $\rho_{\tilde q}={\tilde q}\rho_{\tilde q}^{\prime}$. We notice that $\tilde q$ keeps its physical interpretation of the continuous model.
In Eq. (\[eq:loquistic\]), if $\tilde{q} = 1$ and $\rho_1 = \rho_1^{\prime} = 4 a$, with $a \in [0,1]$, one obtains the *logistic map*, $x_{i+1} = 4 a x_i (1 - x_i)$, which is the classical example of a [*dynamic system*]{} obtained from the discretization of the Verhulst model. Although simple, this map presents a extremely rich behavior, universal period duplication, chaos etc. [@may_1976].
Let us digress considering the Feigenbaum’s map [@feigenbaum_1979]: $y_{i+1} = 1 - \mu y_i^{\tilde{q} + 1}$, with $\tilde{q} > 0$, $0 < \mu \le 2$ and $-1 \le y_{i} \le 1$. Firstly, let us consider the particular case $\tilde{q} = 1$. If one writes $y_i = \tilde{y}_i - b$, with $b$ being a constant, then: $\tilde{y}_{i+1} = - \mu b^2 + b + 1 + 2 \mu b \tilde{y}_i[1 - \tilde{y}_i/(2b)]$. Imposing that $ - \mu b^2 + b + 1 = 0$ leads to $b_{\pm} = (1 \mp \sqrt{1 + 4 \mu})/(2 \mu)$ and calling $x_i = \tilde{y}_i/(2b)$, one obtains the logistic map with $\rho_1 = \rho'_1 = 4a = 1 + \sqrt{1 + 4 \mu}$, so that $0 < \rho_1 \le 4$. One can easily relate the control parameter of these two maps, making the maps equivalent. For arbitrary values of $\tilde{q}$, there is not a general closed analytical form to expand $|\tilde{y}_i - b|^{\tilde{q} + 1}$ and one cannot simply transform the control parameters of Eq. (\[eq:loquistic\]) to the Feigenbaum’s map. Here, in general, these two maps are not equivalent. It would be then interesting, to study the sensitivity of Eq. (\[eq:loquistic\]) with respect to initial conditions as it has been extensively studied in the Feigenbaum’s map [@lyra_1997; @lyra_1998; @lyra_1999; @lyra_2000].
Returning to Eq. (\[eq:loquistic\]), in the domain $0\leq x \leq 1$, $f(x_i)=-\rho_{\tilde q}x_i\ln_{\tilde q}(x_i)\geq 0$ (non-negative), for $\rho_{\tilde q} > 0$. Since $e_{\tilde{q}}(x)$ is real only for $\tilde{q} x > -1$, $\tilde{f}$ is real only for $\tilde{q} > -1$. The maximum value of the function is $$\tilde{f} = f(\tilde{x}) = \frac{\rho_{\tilde q}}{e_1({\tilde q})e_{\tilde q}(1)} \; ,$$ which occurs at $$\tilde x = \frac{1}{e_{\tilde q}(1)} \; ,$$ i. e., the inverse of the generalized Euler’s number $e_{\tilde q}(1)$ \[Eq. (\[eq:eqtilde\])\]. For the generalized logistic map, $0 \le x \le 1$, so that $0 \le \tilde{f} \le 1$, it leads to the following domain for the control parameter $0 \le \rho_{\tilde{q}} \le \rho_{max}$: $$\rho_{max} = e_{\tilde{q}}(1) e_1(\tilde{q}) = (1 + \tilde{q})^{1+1/\tilde{q}} \; .
\label{eq:rho_max}$$
The map fixed points $[x^*=f(x^*)]$ are $$\begin{aligned}
x_1^* & = & 0, \label{eq:fp1} \\
x_2^* & = & e_{\tilde q}(-1/{\rho_{\tilde q}}) \label{eq:fp2} \; .\end{aligned}$$ The fixed point $x_1^*$ is stable for $0 \leq \rho_{\tilde q} < {\tilde q}$ and $x_2^*$ is stable for ${\tilde q} \leq \rho_{\tilde q} < \rho_{pd}$, where $$\rho_{pd} = \tilde{q} + 2 \; .
\label{eq:rho_pd}$$ Notice the presence of the $\tilde{q}$-exponentials in the description of the attractors, even for the logistic map $\tilde{q} = 1$.
The generalized logistic map also presents the rich behavior of the logistic map as depicted by the bifurcation diagram of Fig. \[fig1\]. The inset of Fig. \[fig1\] displays the Lyapunov exponents as function of the central parameter $\rho_{\tilde q}$.
In Fig. \[fig2\] we have scaled the axis to $\rho_{\tilde{q}}[- (\tilde{q}+2)/\tilde{q}]/ (\rho_{max}\tilde{q})$, where $\rho_{max}$ is given by Eq. (\[eq:rho\_max\]) and we plotted the bifurcation diagram for $\tilde{q} = 1/10, 1$ and $10$. We see that the diagrams display the same structure but each one has its own scaling parameters. The role of increasing $\tilde{q}$ is to lift the bifurcation diagram to relatively anticipating the chaotic phase. The period doubling region start at $x_2^{*}(\tilde{q}) = e_{\tilde{q}}[-1/(\tilde{q} + 2) = [1 - 1/(1 + 2/\tilde{q})]^{1/\tilde{q}}$, so that for $x_2^{*}(1/10) = (20/21)^{10} \approx 0.61$, $x_2^{*}(1) = 2/3 \approx 0.67$ and $x_2^{*}(10) = (1/6)^{1/10} \approx 0.84$.
When $\rho_{\tilde{q}} = e_{\tilde{q}}(1) e_1(\tilde{q})$, then $x_i \in (0,1)$. In Fig. \[fig3\] we show the histograms of the distribution of the variable $x_i$. We see that as $\tilde{q}$ increases, the histograms have the same shape as the logistic histogram has, but it is crooked in the counter clock sense around $x=1/2$.
The generalized $\theta$-Ricker model {#sec:generalized_theta_ricker}
=====================================
The [*$\theta$-Ricker*]{} model [@bellows_1981; @berryman_1999] is given by: $$x_{i+1} = x_i e^{r[1-(x_i/\kappa)^\theta]},
\label{eq:theta_ricker}$$ where $\theta > 0$.
Notice that $\tilde x= r^{1/\theta} x/\kappa $ is the relevant variable, where $\kappa_1 = e^r > 0$. In this way Eq. (\[eq:theta\_ricker\]) can be simply written as $\tilde x_{i+1}=k_1\tilde x_ie^{-\tilde x_i^\theta}$. For $\theta=1$, one finds the standard [*Ricker*]{} model [@ricker_1954]. For arbitrary $\theta$, expanding the exponential to the first order one obtains the generalized logistic map \[Eq. (\[eq:loquistic\])\] which becomes the logistic map, for $\theta=1$. The $\theta$-Ricker, Ricker and quadratic models are all scramble competion models.
If one switches the exponential function for the ${\tilde q}$-generalized exponential in Eq. (\[eq:theta\_ricker\]), one gets the [*generalized $\theta$-Ricker model*]{}: $$x_{i+1} = \kappa_1 x_i \; e_{-{\tilde q}} \left[ -r \, \left( \frac{x_i}{\kappa} \right)^\theta \right]
= \frac{\kappa_1 x_i}{\left[1 + \tilde{q} r \left( \frac{x_i}{\kappa}\right)^{\theta} \right]^{1/\tilde{q}}} \; .
\label{eq:generalized_theta_ricker_model}$$ To obtain standard notation, write $c = 1/{\tilde q}$ and $k_2=r/(kc)$, so that $x_{i+1} = k_1x_i/(1+k_2x_i)^c$ [@brannstrom_2005].
The generalized model with $\theta=1$, leads to the [*Hassel*]{} model [@hassell_1975], which can be a scramble or contest competition model. One well-known contest competition model is the [*Beverton-Holt*]{} model [@beverton_holt], which is obtained taking ${\tilde q}=c=1$. For ${\tilde q}=0$, one recovers the Ricker model and for ${\tilde q}=-1$, one recovers the logistic model.
It is interesting to mention that the Beverton-Holt model [@beverton_holt] is one of the few models that have the time evolution explicitly written: $\tilde{x}_{i} = \kappa_1^{i} \tilde{x}_0/[1 + (1 - \kappa_1^{i})\tilde{x}_0/(1 - \kappa_1)]$. From this equation, one sees that $x_{i (\gg 1)} = 0$ ,for $\kappa_1 \le 1$ and $x_{i (\gg 1)} = \kappa_1 - 1$ for $\kappa_1 \ge 1$.
Using arbitrary values of $\theta$ in Eq. (\[eq:generalized\_theta\_ricker\_model\]), for ${\tilde q}=0$ one recovers the $\theta$-Ricker model, and for ${\tilde q}=1$, the [*Maynard-Smith-Slatkin*]{} model [@maynard-smith_1973] is recovered. The latter is a scramble/contest competition model. For ${\tilde q}=-1$, one recovers the generalized logistic map. The trivial linear model is retrieved for $\tilde q \to -\infty$.
In terms of the relevant variable $\tilde{x}$, Eq. (\[eq:generalized\_theta\_ricker\_model\]) is rewritten as: $$\tilde{x}_{i+1} = \kappa_1 \tilde{x}_i e_{-{\tilde q}}(- \tilde{x}_i^\theta) \; ,
\label{eq:final}$$ where $\tilde{x}_i \ge 0$ and we stress that the important parameters are $\kappa_1 > 0$, $\tilde{q}$ and $\theta > 0$. Eq. (\[eq:final\]) is suitable for data analysis and the most usual known discrete growth models are recovered with the judicious choice of the $\tilde q$ and $\theta$ parameters as it shown in Table \[tabela\]. Some typical bifurcation diagrams of Eq. (\[eq:final\]) are displayed in Fig. \[figbdtrm\].
----------------------- ------------ ---------- ---------------------------------------------------
Model $\tilde q$ $\theta$ competition
type
Linear $-\infty$ $>0~$
Logistic $-1$ $1$ s
Generalized Logistic $-1$ $>0$ s
Ricker $0$ $1$ s
$\theta$-Ricker $0$ $>0$ s
Hassel $*$ 1 s ($\tilde{q} < 1/2$) or c ($\tilde{q} \ge 1/2$)
Maynard-Smith-Slatkin $1$ $ > 0$ s ($\theta > 2$ ) or c ($\theta \le 2$)
Beverton-Holt $1$ $1$ c
----------------------- ------------ ---------- ---------------------------------------------------
: Summary of the parameters to obtain discrete growth models from Eq. (\[eq:final\]). In the competition type column, *s* and *c* stand for scramble and contest models, respectively. The symbol $*$ stands for arbitrary values.[]{data-label="tabela"}
Now, let us obtain some analytical results for the map of Eq. (\[eq:final\]), which we write as $\tilde{x}_{i+1} = f_{gtr}(\tilde{x}_i)$, with $$f_{gtr}(\tilde{x}) = \kappa_1 \tilde{x} e_{-{\tilde q}}(- \tilde{x}^\theta) = \frac{\kappa_1 \tilde{x}}{(1 + \tilde{q} \tilde{x}^{\theta})^{1/\tilde{q}}}\; .
\label{eq:mapa_generalizado}$$ The $\tilde{x}$ domain is unbounded ($\tilde{x} \ge 0$), for $\tilde{q} \ge 0$. However, for $\tilde{q} < 0$, $f_{gtr}(\tilde{x}) = \kappa_1 \tilde{x} (1 - |\tilde{q}| \tilde{x}^{\theta})^{1/|\tilde{q}|}$ and the $\tilde{x}$-domain is bounded to the interval: $0 \le \tilde{x} \le \tilde{x}_m$, with $$\tilde{x}_m = \frac{1}{(- \tilde{q})^{1/\theta}} \; ,
\label{eq:xm}$$ so that for $|\tilde{q}| < 1$, $\tilde{x}_m > 1$; for $|\tilde{q}| = 1$, with $\tilde{x}_m = 1$ (for $\tilde{q} = -1$, it is the generalized logistic case \[Eq. (\[eq:loquistic\]) in $\theta$ instead of $\tilde{q}$\] and for $\tilde{q} = 1$, the Maynard-Smith-Slatkin model) and for $|\tilde{q}| > 1$, $\tilde{x}_m < 1$.
The derivative of $f_{gtr}$ with respect to $\tilde{x}$ is: $f_{gtr}'(\tilde{x}) = \kappa_1 [1 + (\tilde{q} - \theta)\tilde{x}^{\theta}]/(1 + \tilde{q} \tilde{x}^{\theta})^{1+1/\tilde{q}}$. Imposing $f_{gtr}'(\tilde{x}_{max}) = 0$, one obtains the maximum of Eq. (\[eq:mapa\_generalizado\]), $$\tilde{f}_{gtr} = f_{gtr}(\tilde{x}_{max}) = \frac{\kappa_1 \tilde{x}_m e_{\tilde{q}}(-1/\theta)}{e_{\theta}(-1/\tilde{q})} \; .$$ at $$\tilde{x}_{max} = \frac{1}{(\theta - \tilde{q})^{1/\theta}} = \frac{\tilde{x}_m}{e_{\theta}(-1/\tilde{q})} \; ,
\label{eq:pontomaximo}$$ The control parameter $\kappa_1$ is unbounded ($\kappa_1 > 0$), for $\tilde{q} \ge 0$, but for $\tilde{q} < 0$, since $\tilde{f}_{gtr} \le \tilde{x}_m$, it belongs to the interval $0 < \kappa_1 \le \kappa_{m}$, where: $$\kappa_m = \frac{e_{\theta}(-1/\tilde{q})}{e_{\tilde{q}}(-1/\theta)} = e_{\theta}(-1/\tilde{q}) e_{ - \tilde{q}}(1/\theta) \; .
\label{eq:km}$$
From Eq. (\[eq:pontomaximo\]), one sees that for $\theta < \tilde{q}$, $f_{gtr}(\tilde{x})$ does not have a hump, it is simply a monotonically increasing function of $\tilde{x}$, which characterizes the contest models. At the critical $\theta = \tilde{q}$ value, the function $f_{gtr}(\tilde{x})$ starts to have a maximum value at infinity. For $\theta > \tilde{q}$, the function $f_{gtr}(\tilde{x})$ has a hump, with maximum value at $x_{max}$ \[Eq. (\[eq:pontomaximo\])\] such that as $\theta \rightarrow \infty$ then $x_{max} \rightarrow 1$.
The map fixed points \[$\tilde{x}^{*} = f_{gtr}(\tilde{x}^*)$\] are: $$\begin{aligned}
\tilde{x}_1^{*} & = & 0 \label{eq:x1}\\
\tilde{x}_2^{*} & = & [\ln_{\tilde{q}}(\kappa_1)]^{1/\theta} \ge 0 \; .
\label{eq:x2}\end{aligned}$$ These fixed points are show as function of $\kappa_1$ in Fig. \[fig:cp\].
The fixed point $\tilde{x}_1^{*}$ represents the species extinction and is stable, for $0 < \kappa_1 < 1$. Both fixed points $\tilde{x}_1^{*}$ and $\tilde{x}_2^{*}$ are marginal, for $\kappa_1 =1$. For $1 < \kappa_1 < e_{-\tilde{q}}(2/\theta)$, $\tilde{x}_1^{*}$ becomes unstable and $\tilde{x}_2^{*} > 0$ is stable and represents the species survival. For $\kappa_1 = e_{-\tilde{q}}(2/\theta)$, $\tilde{x}_2^{*}$ becomes unstable and as $\kappa_1$ increases, a stable cycle-2 appears. For $\tilde{q} < 0$, as $\kappa_1$ increases further, the cycle-2 becomes unstable at some value of $\kappa_1$ giving rise to a route to chaos as in the logistic map, via period doubling. Nevertheless, for $\tilde{q} \ge 0$, several scenarios may take place. Even though $\tilde{q} > 0$ and $\theta > \tilde{q}$, if $\tilde{q} < \theta < 2\tilde{q}$, the map $f_{gtr}$ has a hump, but it is not thin enough to produce periods greater than unity. In this case, $f_{gtr}$ produces only the two fixed points $\tilde{x}_1^{*}$ and $\tilde{x}_2^{*}$, which characterize the context models. Nevertheless, scramble models ($\theta > 2\tilde{q}$) have maps with a hump thin enough to produce stable cycles with period greater than unity. Thus, in scramble models one has more complex scenarios such as period doubling, as a route to chaos, as $\kappa_1 > e_{-\tilde{q}}(2/\theta)$. We have not being able to obtain analytically the behavior of the system $\tilde{q} > 0$ and $\theta > 2 \tilde{q}$. The $\theta \times \tilde{q}$ diagram is depicted in Fig. \[fig:diagram\].
For $\tilde{q} = \theta = 1$, one retrieves the Beverton-Holt model, with the fixed points $\tilde{x}_1^{*} = 0$ and $\tilde{x}_1^{*} = \ln_{1}(\kappa_1) = \kappa_1 - 1$. For $\tilde{q}$, one retrieves the generalized logistic map.
Generalized Skellam model {#sec:generalizedskellam}
=========================
All the contest competition models generalized by Eq. (\[eq:generalized\_theta\_ricker\_model\]) are power-law-like models for $\tilde q \neq 0$. However, the Skellam contest model cannot be obtained from this approach. It is the complement of an exponential decay $x_{i+1}=\kappa(1-e^{-rx_i})$ [@skellam]. Nevertheless, it is interesting to replace the exponential function to the $\tilde q$-exponential in this model: $x_{i+1} = k [1 - e_{-\tilde q}(-rx_i)]$ and write $\tilde{x} = r x$ and $\kappa = r k$, which leads to: $$\tilde{x}_{i+1} = \kappa \left[1 - e_{-\tilde q}(- \tilde{x}_i)\right]\;.\
\label{eq:genskellan}$$ For $\tilde q \rightarrow -\infty$, Eq. (\[eq:genskellan\]) leads to the constant model, for $\tilde q = -1$, the trivial linear growth is found. If $\tilde q=0$, one recovers the Skellam model and finally, $\tilde q = 1$ leads to the Beverton-Holt contest model (see Table \[tabela2\]).
Model $\tilde q$
--------------- ------------
constant $-\infty$
linear $-1$
Skellam $0$
Beverton-Holt $1$
: Summary of the parameters to obtain contest competition discrete growth models from Eq. (\[eq:genskellan\]).[]{data-label="tabela2"}
Conclusion {#sec:conclusion}
==========
We have shown that the $\tilde{q}$-generalization of the exponential function is suitable to describe discrete growth models. The $\tilde q$ parameter is related to the range of a repulsive potential and the dimensionality of the fractal underlying structure. From the discretization of the Richard’s model, we have obtained a generalization for the logistic map and briefly studied its properties. An interesting generalization is the one of $\theta$-Ricker model, which allows to have several scramble or contest competition discrete growth models as particular cases. Equation (\[eq:final\]) allows the use of softwares to fit data to find the most suitable known model throughout the optimum choice of $\tilde q$ and $\theta$. Furthermore, one can also generalize the Skellam contest model. Only a few specific models mentioned in Ref. [@brannstrom_2005] are not retrieved from our generalization. Actually, we propose a general procedure where we do not necessarily need to be tied to a specific model, since one can have arbitrary values of $\tilde q$ and $\theta$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank C. A. S. Terçariol for fruitful discussions. ASM acknowledges the Brazilian agency CNPq (303990/2007-4 and 476862/2007-8) or support. RSG also acknowledges CNPq (140420/2007-0) for support. ALE acknowledges CNPq for the fellowship and FAPESP and MCT/CNPq Fundo Setorial de Infra-Estrutura (2006/60333-0).
[00]{} C. Tsallis, J. Stat. Phys. **52** (1988) 479–487.
C. Tsallis, Química Nova **17** (1994) 468–471.
L. Nivanen, A. L. M. A, Q. A. Wang, Rep. Math. Phys. **52** (2003) 437–444.
E. P. Borges, Physica A **340** (2004) 95–101.
N. Kalogeropoulos, Physica A **356** (2005) 408–418.
G. Kaniadakis, Physica A **296** (2001) 405–425.
G. Kaniadakis, Phys. Rev. E **66** (2002) 056125.
S. Abe, Phys. Lett. A **224** (1997) 326–330.
G. Kaniadakis, M. Lissia, A. M. Scarfone, Phys. Rev. E **71** (2005) 046128.
G. Kaniadakis, Phys. Rev. E **72** (2005) 036108.
V. Schwämmle, C. Tsallis, J. Math. Phys. **48** (2007) 113301.
E. P. Borges, J. Phys. A: Math. Gen. **31** (1998) 5281–5288.
C. Tsallis, S. V. F. Levy, A. M. C. Souza, R. Maynard, Phys. Rev. Lett. **75** (1995) 3589–3592, [*ibid*]{} [**77**]{}, 5442(E) (1996).
C. Tsallis, D. A. Stariolo, Physica A **233** (1996) 395–406.
C. Tsallis, M. P. de Albuquerque, Eur. Phys. J. B **13** (2000) 777–780.
D. O. Cajueiro, Physica A **364** (2006) 385–388.
D. O. Cajueiro, B. M. Tabak, Physica A **373** (2007) 593–602.
C. Anteneodo, C. Tsallis, A. S. Martinez, Europhys. Lett. **59** (2002) 635–641.
A. de Jesus Holanda, I. T. Pizza, O. Kinouchi, A. S. Martinez, E. E. S. Ruiz, Physica A **344** (2004) 530–536.
A. S. Martinez, R. S. González and C. A. S. Terçariol, [*Generalized exponential function and some of its applications to complex systems*]{}, arXiv:0812.3071v1 \[physics.data-an\].
T. J. Arruda, R. S. González, C. A. S. Terçariol, A. S. Martinez, Phys. Lett. A **372** (2008) 2578-2582.
E. W. Montroll, B. J. West, Fluctuation Phenomena, Elsevier Science Publishers B. V., Amsterdam, 1979, Ch. On an enriched collection of stochastic processes, pp. 61–205.
E. W. Montroll, L. W. Badger, Introduction to quantitative aspects of social phenomena, Gordon and Breach, New York, 1974.
F. J. Richards, J. Exp. Bot. **10** (1959) 290–300.
A. S. Martinez, R. S. González and C. A. S. Terçariol, Physica A **387** (2008) 5679�-5687.
J. C. M. Mombach, N. Lemke, B. E. J. Bodmann and M. A. P. Idiart, A mean-field theory of cellular growth, Europhys. Lett. 59(6) (2002) 923-928.
J. C. M. Mombach, N. Lemke, B. E. J. Bodmann and M. A. P. Idiart, A mean-field theory of cellular growth, Europhys. Lett. 60(3) (2002) 489.
M. P. Hassell, J. Anim. Ecol. **45** (1975) 283–296.
T. S. Bellows, J. Anim. Ecol. **50** (1981) 139–156.
A. A. Berryman, Principles of population dynamics and their applications, Cheltenham: Thornes, 1999.
R. J. H. Beverton, H. S. J, On the dynamics of exploited fish populations. Series 2 Vol. 19 London: Her Majesty�s Stationary Office.
J. Maynard-Smith, M. Slatkin, Ecology **54** (1973) 384–391.
R. M. May, Nature **261** (1976) 459–467.
M. J. Feigenbaum, J. Stat. Phys. **21** (1979) 669-706.
U. M. S. Costa, M. L. Lyra, A. R. Plastino and C. Tsallis, Phys. Rev. E **56** (1997) 245–250.
M. L. Lyra and C. Tsallis, Phys. Rev. Lett. **80** (1998) 53–56.
C. R. da Silva, H. R. da Cruz and M. L. Lyra, Braz. J. Phys. **29** (1999) 144–152
F. A. B. F. de Moura, U. Tirnakli and M. L. Lyra, Phys. Rev. E **2000** 6361–6365.
K. Briggs, Mathematics of Computation **57** (1991) 435-439.
W. E. Ricker, J. Fisheries Res. Board Can. **11** (1954) 559–623.
A. Brännström, D. J. T. Sumpter, Proc. R. Soc. B **272** (1576) (2005) 2065–2072.
J. G. Skellam, Biometrika **38** (1951) 196–218.
|
---
abstract: 'We derive the magnitude of fluctuations in total synchrotron intensity in the Milky Way and M33, from both observations and theory under various assumption about the relation between cosmic rays and interstellar magnetic fields. Given the relative magnitude of the fluctuations in the Galactic magnetic field (the ratio of the rms fluctuations to the mean magnetic field strength) suggested by Faraday rotation and synchrotron polarization, the observations are inconsistent with local energy equipartition between cosmic rays and magnetic fields. Our analysis of relative synchrotron intensity fluctuations indicates that the distribution of cosmic rays is nearly uniform at the scales of the order of and exceeding $100{\,{\rm pc}}$, in contrast to strong fluctuations in the interstellar magnetic field at those scales. A conservative upper limit on the ratio of the the fluctuation magnitude in the cosmic ray number density to its mean value is 0.2–0.4 at scales of order 100pc. Our results are consistent with a mild anticorrelation between cosmic-ray and magnetic energy densities at these scales, in both the Milky Way and M33. Energy equipartition between cosmic rays and magnetic fields may still hold, but at scales exceeding 1kpc. Therefore, we suggest that equipartition estimates be applied to the observed synchrotron intensity smoothed to a linear scale of kiloparsec order (in spiral galaxies) to obtain the cosmic ray distribution and a large-scale magnetic field. Then the resulting cosmic ray distribution can be used to derive the fluctuating magnetic field strength from the data at the original resolution. The resulting random magnetic field is likely to be significantly stronger than existing estimates.'
author:
- 'Rodion Stepanov$^{1,2,3}$[^1], Anvar Shukurov$^1$, Andrew Fletcher$^1$, Rainer Beck$^{4}$,'
- |
Laura La Porta$^{4,6}$, Fatemeh Tabatabaei$^5$\
$^1$School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK\
$^2$Institute of Continuous Media Mechanics, Academy of Sciences, Korolyov str. 1, Perm 614013, Russia\
$^3$Department of Applied Mathematics and Mechanics, National Research Polytechnic University of Perm, Komsomolskii Av. 29, 614990, Perm, Russia\
$^4$Max-Planck Institut für Radioastronomie, Auf dem Hügel 69, Bonn D-53121, Germany\
$^5$Max Planck Institut für Astronomie, Königstuhl 17, Heidelberg D-69117, Germany\
$^6$Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115 Bonn, Germany
bibliography:
- 'biblio.bib'
date: 'Accepted .... Received ....; in original form ...'
title: An observational test for correlations between cosmic rays and magnetic fields
---
\[firstpage\]
cosmic rays – magnetic fields – galaxies: ISM – galaxies: magnetic fields – radio continuum: galaxies – radio continuum: general
Motivation and background
=========================
The concept of energy equipartition between cosmic rays and magnetic fields and similar assumptions such as pressure equality [@Longair:1994; @BK05; @AUAPV11] are often used in the analysis and interpretation of radio astronomical observations. This idea was originally suggested in order to estimate the magnetic field and cosmic ray energies of the source *as a whole* [@Burbidge:1956a; @Burbidge:1956b], from a measurement of the synchrotron brightness of a radio source. A physically attractive feature of the equipartition state is that it approximately minimizes the total energy of the radio source.
The energy density of cosmic rays is mainly determined by their proton component, whereas the synchrotron intensity depends on the number density of relativistic electrons. Therefore, in order to estimate the magnetic field energy, an assumption needs to be made about the ratio of the energy densities of the relativistic protons and electrons; the often adopted value for this ratio is 100, as suggested by Milky Way data [@BK05]. This ratio is adopted to be unity in applications to galaxy clusters, radio galaxies and active objects .
However, more recently this concept has been extended to large-scale trends in synchrotron intensity and to local energy densities at sub-kiloparsec scales in well-resolved radio sources, such as spiral galaxies . Another important application of the equipartition hypothesis, first suggested by @P66 [@P69; @P79], is to the hydrostatic equilibrium of the interstellar gas. Here magnetic and cosmic ray pressures are assumed to be in a constant ratio, in practice taken to be unity. This application appeals to equipartition (or, more precisely, pressure equality) at larger scales of the order of kiloparsec. The spatial relation between fluctuations in magnetic field and cosmic rays is crucial for a proposed method to measure magnetic helicity in the ISM [@Oppermann11; @2010JETPL..90..637V].
The physical basis of the equipartition assumption remains elusive. Since cosmic rays are confined within a radio source by magnetic fields, it seems natural to expect that the two energy densities are somehow related: if the magnetic field energy density $\epsilon_B$ is smaller than that of the cosmic rays, $\epsilon{_\mathrm{cr}}$, the cosmic rays would be able to ‘break through’ the magnetic field and escape; whereas a larger magnetic energy density would result in the accumulation of cosmic rays. Thus, the system is likely to be self-regulated to energy equipartition, $\epsilon_B\approx\epsilon{_\mathrm{cr}}$. A slightly different version of these arguments refers to the equality of the two pressures,[^2] giving $\epsilon_B\approx{{\textstyle{\frac{1}{3}}}}\epsilon{_\mathrm{cr}}$.
However plausible one finds these arguments, it is difficult to substantiate them. In particular, models of cosmic ray confinement suggest that the cosmic ray diffusion tensor depends on the *ratio* $(\delta
B/B_0)^2$, where $\delta B$ is the magnitude of magnetic field fluctuations at a scale equal to the proton gyroradius and $B_0$ is the mean magnetic field [e.g., @BBDGP90]. The magnetic field strength can determine the streaming velocity of cosmic rays via the Alfvén speed, but the theory of cosmic ray propagation and confinement relates $\epsilon{_\mathrm{cr}}$ to the intensity of cosmic ray sources rather than to the local magnetic field strength. Despite their uncertain basis, equipartition arguments remain popular as they provide ‘reasonable’ estimates of magnetic fields in radio sources, and also because they often offer the only practical way to obtain such estimates.
Equipartition between cosmic rays and magnetic fields can rarely be tested observationally. @CW93 used $\gamma$-ray observations of the Magellanic clouds to calculate the energy density of cosmic rays independently of the equipartition assumption. They further calculated magnetic energy density from radio continuum data at a wavelength of about $\lambda12{\,{\rm cm}}$. The resulting magnetic energy density is two orders of magnitude larger than that of cosmic rays, and @CW93 argue that the discrepancy cannot be removed by assuming a proton-to-electron ratio for cosmic rays different from the standard value of 100 [see, however, @P93]. [ More recently, however, @2012ApJ...759...25M analysed [*Fermi*]{} Large Area Telescope observations of the LMC and concluded that the equipartition assumption does not appear to be violated.]{}
An independent estimate of magnetic field strength can be obtained for synchrotron sources of high surface brightness (e.g., active galactic nuclei) where the relativistic plasma absorbs an observable lower-frequency part of the radio emission (synchrotron self-absorption). Then the magnetic field strength can be estimated from the frequency, the flux density and the angular size of the synchrotron source at the turnover frequency [@S63; @W63; @SW68]. @SR77 and @R94 concluded, from low-frequency observations of compact radio sources whose angular size can be determined from interplanetary scintillations, that there is no significant evidence of strong departures from equipartition. In the sources with strong synchrotron self-absorption in their sample, the total energy is within a factor of 10 above the minimum energy. @OD08 observed, using VLBI, five young, very compact radio sources to suggest that magnetic fields in them are quite close to the equipartition value. Physical conditions in spiral galaxies are quite different from those in compact, active radio sources, and departures from equipartition by a factor of several in terms of magnetic field strength would be quite significant in the context of spiral galaxies.
Here we test the equipartition hypothesis using another approach [see also @2009IAUS..259...93S]. We calculate the relative magnitude of fluctuations in synchrotron intensity using model random magnetic field and cosmic ray distributions with a prescribed degree of cross-correlations. When the results are compared with observations, it becomes clear that local energy equipartition is implausible as it would produce stronger fluctuations of the synchrotron emissivity than are observed. Instead, the observed synchrotron intensity fluctuations suggest weak variations in the cosmic ray number density or an anticorrelation between cosmic rays and magnetic fields, perhaps indicative of pressure equilibrium. We conclude that local energy equipartition is unlikely in spiral galaxies at the integral scale of the fluctuations, of order 100[[pc]{}]{}. We discuss the dynamics of cosmic rays to argue in favour of equipartition at larger scales of order 1kpc, comparable to the scale of the mean magnetic field and to the cosmic-ray diffusion scale.
[ The paper is organized as follows. In Section \[MFIMF\] we discuss the relative strengths of the mean and fluctuating magnetic field in the Milky Way and M33. In Section \[data\] we use observational data to estimate the magnitude of synchrotron intensity variations at high galactic latitudes in the Milky Way and in the outer parts of M33. Theoretical models for synchrotron intensity fluctuations, allowing for controlled levels of cross-correlation between the magnetic field and cosmic ray distributions, are developed analytically in Section \[SIF\] and numerically in Section \[MFCRM\]: readers who are interested only in our results may wish to skip these rather mathematical sections. Section \[SR\] presents an interpretation of the observational data in terms of the theoretical models; here we estimate the cross-correlation coefficient between magnetic and cosmic-ray fluctuations. In Section \[SDCR\] we briefly discuss cosmic ray propagation models from the viewpoint of relation between the cosmic ray and magnetic field distributions. Our results are discussed in Section \[Disc\] and Appendix \[app\] contains the details of some of our calculations.]{}
The magnitude of fluctuations in interstellar magnetic fields {#MFIMF}
=============================================================
The ratio of the fluctuating-to-mean synchrotron intensity in the interstellar medium (ISM) is sensitive to the relative distributions of cosmic ray electrons and magnetic fields and hence to the extent that energy equipartition may hold locally: the synchrotron emission will fluctuate strongly if equipartition holds pointwise, i.e., if the number density of cosmic ray electrons is increased where the local magnetic field is stronger. (We assume that cosmic ray electrons and heavier relativistic particles are similarly distributed – see Section \[Disc\] for the justification.)
Interstellar magnetic fields are turbulent, with the ratio of the random magnetic field to its mean component known from observations of Faraday rotation, independently of the equipartition assumption. Denoting the standard deviation of the turbulent magnetic field by $\sigma_b^2={\overline{B^2}}-B_0^2$ and the mean field strength as $B_0=|{\overline{{\bmath{B}}}}|$, where bar denotes appropriate averaging (usually volume or line-of-sight averaging), the relative fluctuations in magnetic field strength in the Solar vicinity of the Milky Way is estimated as [@RSS88; @OS93; @BBMSS96] $$\label{ratioB}
\delta_b^2=\left(\frac{\sigma_b}{B_0}\right)^2\simeq3\mbox{--}10.$$
Similar estimates result from radio observations of nearby spiral galaxies where the degree of polarization of the integrated emission at 4.8GHz is a few per cent, with a range $p\simeq0.01$–$0.18$ [@2009ApJ...693.1392S]. These data are affected by beam depolarization, so they only give upper limits for $\delta_b^2$. More typical values of the fractional polarisation in spiral galaxies are $p=0.01$–0.05 within spiral arms and 0.1 on average. The degree of polarization at short wavelengths, where Faraday rotation is negligible, can be estimated as [@1966MNRAS.133...67B; @1998MNRAS.299..189S] $$\label{burn1}
p=p_0 \frac{B_{0\perp}^2}{B_{0\perp}^2+\tfrac{2}{3}\sigma_b^2}$$ where $B_{0\perp}$ is the strength of the large-scale magnetic field in the sky plane, [the intrinsic degree of polarisation]{} $p_0\approx0.75$, and the random magnetic field is assumed to be isotropic, ${\overline{b_\perp^2}}=\tfrac23\sigma_b^2$. This yields $$\label{burn}
\delta_b^2\simeq\tfrac32\left(\frac{p_0}{p}-1\right)\ga4
\quad\mbox{for}\quad p<0.2\,$$ in a good agreement with [the estimate for]{} the Milky Way data obtained from Faraday rotation measures. For $p=0.05$–0.1, we obtain $\delta_b^2\simeq10$–20.
It is important to note that Eq. (\[burn\]) has been obtained assuming that the cosmic ray number density $n{_\mathrm{cr}}$ is uniform, so that all the beam depolarization is attributed solely to the fluctuations in magnetic field. Under local equipartition, $n{_\mathrm{cr}}\propto B^2$, @1998MNRAS.299..189S [their Eq. (28)] calculated the degree of polarization at short wavelengths to be $$p = p_0\frac{1+\tfrac73\delta_b^2}{1+3\delta_b^2+\tfrac{10}{9}\delta_b^4}.$$ As might be expected, this expression leads to a smaller $\delta_b$ for a given $p/p_0$ than Eq. (\[burn1\]): $$\label{delta_b_eq}
\delta_b^2\approx \frac{2 p_0}{p}
\quad
\mbox{for}\quad \frac{p}{p_0}\ll1,$$ so that $$\delta_b^2\simeq15\quad\mbox{for} \quad p=0.1.$$ Since local equipartition between cosmic rays and magnetic fields maximizes beam depolarization, this is clearly a lower estimate of $\delta_b$.
[Anisotropic fluctuations]{} {#AF}
----------------------------
The above estimates apply to statistically isotropic random magnetic fields. However, the random part of the interstellar magnetic field can be expected to be anisotropic at scales of order 100pc, e.g., due to shearing by the galactic differential rotation, streaming motions and large-scale compression. Synchrotron emission arising in an anisotropic random magnetic field is polarized [@L80; @1998MNRAS.299..189S] and the resulting net polarization, from the combined random and mean field, can be either stronger or weaker than in the case of an isotropic random field depending on the sense of anisotropy relative to the orientation of the mean magnetic field. Note that the anisotropy of MHD turbulence resulting from the nature of the spectral energy cascade [@GS95; @LGS07; @GNNP00 and references therein] is important only at much smaller scales.
The case of M33 provides a suitable illustration of the refinements required if the anisotropy of the random magnetic field is significant. obtained [integrated]{} fractional polarization of about 0.1 at $\lambda3.6{\,{\rm cm}}$. Using Eq. (\[burn\]), this yields $\delta_b^2\simeq10$, whereas Eq. (\[delta\_b\_eq\]) leads to $\delta_b^2\simeq4$, consistent with their equipartition estimates $\sigma_b\simeq6{\,\mu{\rm G}}$ and $B_0\simeq2.5{\,\mu{\rm G}}$. However, their analysis of Faraday rotation between $\lambda3.6,\ 6.2$ and $20{\,{\rm cm}}$ suggests a weaker regular magnetic field, $B_0\simeq1{\,\mu{\rm G}}$, leading to $\delta_b^2\simeq40$ if $\sigma_b\simeq6{\,\mu{\rm G}}$.
[The latter estimate for $B_0$ is more reliable since the degree of polarization leads to an underestimated $\delta_b$ if magnetic field is anisotropic.]{} @1998MNRAS.299..189S [their Eq. (19)] have shown that the degree of polarization at short wavelengths in a partially ordered, anisotropic magnetic field is given by $$p=p_0\frac{1+\delta_{by}^2(1-\alpha_b^2)}{1+\delta_{by}^2(1+\alpha_b^2)},$$ where [the $(x,y)$-plane is the plane of the sky with the $y$-axis aligned with the large-scale magnetic field, i.e., $B_y=B_0$ and $B_x=0$; we further defined]{} $\delta_{by}^2=\sigma_{by}^2/B_0^2$ and likewise for $\delta_{bx}$, and introduced $\alpha_b^2=\sigma_{bx}^2/\sigma_{by}^2$ ($<1$) as a measure of the anisotropy of ${\bmath{b}}_\perp$. This approximation is relevant to spiral galaxies where the mean magnetic field is predominantly azimuthal (nearly aligned with the $y$-axis of the local reference frame used here) and the anisotropy in the random magnetic field is produced by the rotational shear, $\sigma_{by}>\sigma_{bx}$. For $\delta_{by}^2\gg1$, this yields, for $p=0.1$, $$\label{alphab}
\alpha_b^2\approx\frac{p_0-p}{p_0+p}\approx0.8.$$ Thus, a rather weak anisotropy of the random magnetic field can produce $p\simeq0.1$ and this allows us to reconcile the different estimates of $\delta_b$ obtained from the degree of polarization and Faraday rotation in M33.
The required anisotropy can readily be produced by the galactic differential rotation. Shearing of an initially isotropic random magnetic field [by rotation (directed along the $y$-axis)]{} leads, within one eddy turnover time, to an increase of its azimuthal component to $$\sigma_{by}\simeq\sigma_{bx}\left(1-r\frac{{\mathrm{d}}\Omega}{{\mathrm{d}}r}\,
\frac{{{l}}}{v}\right),$$ where $\Omega(r)$ is the angular velocity of the galactic rotation (with the rotational velocity along the local $y$-direction and $r$ the galactocentric radius) and ${{l}}$ and $v$ are the correlation length[^3] and r.m.s. speed of the interstellar turbulence, so that ${{l}}/v$ is the lifetime of a turbulent eddy. This leads to $$\alpha_b\simeq
\left(1-r\frac{{\mathrm{d}}\Omega}{{\mathrm{d}}r}\,\frac{{{l}}}{v}\right)^{-1}
\simeq 1-\frac{{{l}}V_0}{R_0 v},$$ where the last equality is based on the estimate $r{\mathrm{d}}\Omega/{\mathrm{d}}r\simeq
-V_0/R_0$, with $V_0=107{{\,{\rm km}}{\,{\rm s}}^{-1}}$ and $R_0=8{\,{\rm kpc}}$ being the parameters of Brandt’s approximation to the rotation curve of M33 [@RWL76]. With ${{l}}=0.1{\,{\rm kpc}}$ and $v=10{{\,{\rm km}}{\,{\rm s}}^{-1}}$ [values typical of spiral galaxies – e.g., Sect. VI.3 in @RSS88], we obtain $\alpha_b^2\simeq0.8$ , in perfect agreement with the degree of anisotropy required by [Eq. ]{} to explain the observations of .
[Summary]{}
-----------
To conclude, a typical value of the relative strength of the random magnetic field in spiral galaxies is, at least, $$\label{delta_b_st}
\delta_b^2\simeq10.$$ This estimate refers to the correlation scale of interstellar turbulence, ${{l}}\simeq50$–$100{\,{\rm pc}}$. [The correlation scale will be introduced in Section \[SIF\], but here we stress that this estimate refers to the larger scales in the turbulent spectrum]{}. Higher values, $\delta_b^2\simeq40$ are perhaps more plausible, [especially within spiral arms,]{} but our results are not very sensitive to this difference (see Fig. \[I1b\] and Section \[SIF\]). [The estimate of $\delta_b$ in the Milky Way refers to the solar vicinity, i.e., to a region between major spiral arms where the degree of polarization is higher than within the arms and, correspondingly, $\delta_b$ is larger. Consistent with this, our analysis of the observed synchrotron fluctuations in Section \[data\] is for high Galactic latitudes and the outer parts of M33 where the influence of the spiral arms is not strong. Overall, $3\la\delta_b^2\la40$ appears to be a representative range for spiral galaxies, excluding their central parts.]{}
Synchrotron intensity fluctuations derived from observations of the Milky Way and M33 {#data}
=====================================================================================
In this section we estimate the relative level of synchrotron intensity fluctuations from observations of the Milky Way and the spiral galaxy M33. An ideal data set for this analysis should: (i) resolve the fluctuations at their largest scale, (ii) only include emission from the ISM and not from discrete sources such as AGN and stars, (iii) not be dominated by structures that are large and bright due to their proximity, such as the North Polar Spur, (iv) be free of systematic trends such as arm-interarm variations or vertical stratification. The data should allow the ratio $$\delta_I=\frac{\sigma_I}{I_0} \label{deltaI},$$ [where $\sigma_I$ and $I_0$ are the standard deviation and the mean value of synchrotron intensity in a given region,]{} to be calculated separately in arm and inter-arm regions or at low and high latitudes as $I_0$ differs between these regions. Regarding item (i) above, we note that a turbulent cell of $100{\,{\rm pc}}$ in size subtends the angle of about $6\degr$ at a distance of $1{\,{\rm kpc}}$. Furthermore, most useful for our purposes are long wavelengths where the contribution of thermal radio emission is minimal. Unfortunately, ideal data satisfying all these criteria do not exist; we therefore use several radio maps, where each map possesses a few of the desirable properties listed above and collectively they have them all.
[The Milky Way maps that we use contain isotropic emission from faint, unresolved extra-galactic sources and the cosmic microwave background. The contribution of the extragalactic sources to the brightness temperature of the total radio emission of the Milky Way is estimated by]{} @LMOP87 [as $T_\mathrm{e}\simeq50{\,{\rm K}}(\nu/150{\, {\rm MHz}})^{-2.75}$, which amounts to about $10^4{\,{\rm K}}$ and $3{\,{\rm K}}$ at the frequencies $\nu=22{\, {\rm MHz}}$ and $408{\, {\rm MHz}}$, respectively. The $3{\,{\rm K}}$ temperature of the cosmic microwave background should also be taken into account at $408{\, {\rm MHz}}$. For comparison, the respective total values of the radio brightness temperature near the north Galactic pole are about $3\times10^4{\,{\rm K}}$ and $20{\,{\rm K}}$ at $22{\, {\rm MHz}}$ and $408{\, {\rm MHz}}$, respectively. In our estimates of $\delta_I$ obtained below, we have not subtracted this contribution from $I_0$. Thus, our estimates of $\delta_I$ are conservative, and more realistic values might be about 40% larger at both $22{\, {\rm MHz}}$ and $408{\, {\rm MHz}}$. The observations of M33 use a zero level that is set at the edges of the observed area of the sky; since this zero level includes the CMB and unresolved extra-galactic sources $\delta_I$ is unaffected by these components.]{}
The data
--------
### The 408MHz all-sky survey {#408MHz}
The survey of @Haslam:1982 covers the entire sky at a resolution of $51\arcmin$ [(about $50{\,{\rm pc}}$ for a distance of $1{\,{\rm kpc}}$)]{} and with an estimated noise level of about $0.67$ K. Synchrotron radiation is the dominant contribution to emission at the survey’s wavelength of $\lambda$74cm. The brightest structures in the map shown in Fig. \[sky408all\]a are the Galactic plane and several arcs due to nearby objects, especially the North Polar Spur.
We expect that results useful for our purposes arise at the angular scale of about $6\degr$ in all three Milky Way maps (i.e., the angular size of a turbulent cell at a 1kpc distance), whereas larger scales [ reflect]{} regular spatial variations of the radio intensity.
### The 408MHz all-sky survey, without discrete sources
@LaPorta:2008 removed the strongest discrete sources from the data of @Haslam:1982 using a two-dimensional Gaussian filter. We compared the results obtained from the original $408{\, {\rm MHz}}$ survey with those from this map to show that the effect of point sources on our results is negligible.
### The 22MHz part-sky survey {#22MHz}
@Roger:1999 produced a map, shown in Fig. \[sky22\]a, of about $73\%$ of the sky at $\lambda13.6$m, between declinations $-28\degr$ and $+80\degr$ at a resolution of approximately $1\degr\times 2\degr$ and an estimated noise level of $5$ kK. The emission is all synchrotron radiation, but H[ii]{} regions in the Galactic plane absorb some background emission at this low frequency. However, we are most interested in regions away from the Galactic plane, so our conclusions are not affected by the absorption in the H[ii]{} regions. The brightest point sources were removed by @Roger:1999 as they produced strong sidelobe contamination in the maps: this accounts for the four empty rectangles in Fig. \[sky22\]a.
### The 1.4GHz map of M33
The nearby, moderately inclined, spiral galaxy M33 was observed at $\lambda$21cm by @TKB07, using the VLA and Effelsberg telescopes, at a resolution of $51\arcsec$, or about $200{\,{\rm pc}}$ at the distance to M33 of $840{\,{\rm kpc}}$. The noise level is estimated to be $0.07$ mJy/beam. The resolution is sufficient to resolve arm and inter-arm regions, but is at the top end of the expected scale of random fluctuations due to turbulence. The beam area includes a few (nominally, four) correlation cells of the synchrotron intensity fluctuations. The emission is a mixture of thermal and synchrotron radiation. The overall thermal fraction is estimated to be $18\%$ but it is strongly enhanced in large H[ii]{} regions and spiral arms [@TBKKBGM07] whereas the synchrotron emission comes from the whole disc. The radio map used here is shown in Fig. \[M33\]a. The spiral pattern in notably weak in total synchrotron intensity, so the map appears almost featureless. This makes this galaxy especially well suited for our analysis since we are interested in quasi-homogeneous random fluctuations of the radio intensity. Nevertheless, systematic trends are noticeable in this map and we discuss their removal in Section \[ssM33\].
Statistical parameters of the synchrotron intensity fluctuations
----------------------------------------------------------------
For the three Milky Way data sets of Sections \[408MHz\]–\[22MHz\], we calculated the mean $I_0$ and standard deviation $\sigma_I$ of the synchrotron intensity $I$ at each point in the map. In each case, the data were smoothed with a Gaussian of an angular width $a$, resulting in the local mean intensity at the scale $a$, which we denote $I_{0a}$: $$\label{aver}
I_{0a} =S_a^{-1}\int\!\!\!\int I(l',b')
\exp\left(-\theta^2/a^2\right) \cos{b}\,{\mathrm{d}}l'\,{\mathrm{d}}b',$$ where integration extends over the data area, ${\bmath{r}}=(1,l,b)$ is the position vector on the unit sky sphere, with $l$ and $b$ the Galactic longitude and latitude (confusion with the small-scale magnetic field, denoted here ${\bmath{b}}$, should be avoided), $\theta=\arccos({\bmath{r}}\cdot{\bmath{r}}')$ is the angular separation between ${\bmath{r}}$ and ${\bmath{r}}'$, $$S_a(l,b) =\int\!\!\!\int
\exp\left(-\theta^2/a^2\right) \cos{b}\,{\mathrm{d}}l'\,{\mathrm{d}}b',$$ is the averaging area, and the integration extends over the whole area of the sky available in a given survey. The standard deviation $\sigma_{Ia}$ of radio intensity at a given position $(l,b)$ at a given scale $a$ is calculated as $$\sigma^2_{Ia}(l,b)={\left\langle I^2\right\rangle}_a-{\left\langle I\right\rangle}_a^2,$$ where angular brackets denote [spatial]{} averaging as defined in Eq. (\[aver\]).
In the case of M33, we selected nine areas which avoid the brightest H[ii]{} regions and whose radio continuum emission is thus likely to be dominated by synchrotron radiation. Each area encompasses several beams and $\delta_I$ was calculated for each area, using the mean value and the standard deviation of $I$ among all the pixels in the field obtained after removing regular trends (see Section \[ssM33\]).
![\[sky408all\] [ (a):]{} The $408{\, {\rm MHz}}$ all-sky map of the total synchrotron intensity [@Haslam:1982], with the Galactic disc area ($I>52\,$K) blanked out. The lower panels show the magnitude of the relative fluctuations of the synchrotron intensity, $\delta_I={\,\sigma_{I}/I_0}$, at various scales, [with the colour bar shown between Panels (a) and (b)]{}: [ (b)]{} $a=30\degr$, [ (c)]{} $a=15\degr$ and [ (d)]{} $a=7\degr$. The latter scale is about the angular size of a turbulent cell ($2{{l}}_\varepsilon=100{\,{\rm pc}}$) seen at a distance $1{\,{\rm kpc}}$.](sky408all.eps "fig:"){width="46.00000%"}\
![\[sky408all\] [ (a):]{} The $408{\, {\rm MHz}}$ all-sky map of the total synchrotron intensity [@Haslam:1982], with the Galactic disc area ($I>52\,$K) blanked out. The lower panels show the magnitude of the relative fluctuations of the synchrotron intensity, $\delta_I={\,\sigma_{I}/I_0}$, at various scales, [with the colour bar shown between Panels (a) and (b)]{}: [ (b)]{} $a=30\degr$, [ (c)]{} $a=15\degr$ and [ (d)]{} $a=7\degr$. The latter scale is about the angular size of a turbulent cell ($2{{l}}_\varepsilon=100{\,{\rm pc}}$) seen at a distance $1{\,{\rm kpc}}$.](colbar408.eps "fig:"){width="35.00000%"}\
![\[sky408all\] [ (a):]{} The $408{\, {\rm MHz}}$ all-sky map of the total synchrotron intensity [@Haslam:1982], with the Galactic disc area ($I>52\,$K) blanked out. The lower panels show the magnitude of the relative fluctuations of the synchrotron intensity, $\delta_I={\,\sigma_{I}/I_0}$, at various scales, [with the colour bar shown between Panels (a) and (b)]{}: [ (b)]{} $a=30\degr$, [ (c)]{} $a=15\degr$ and [ (d)]{} $a=7\degr$. The latter scale is about the angular size of a turbulent cell ($2{{l}}_\varepsilon=100{\,{\rm pc}}$) seen at a distance $1{\,{\rm kpc}}$.](a4081.eps "fig:"){width="46.00000%"}\
![\[sky408all\] [ (a):]{} The $408{\, {\rm MHz}}$ all-sky map of the total synchrotron intensity [@Haslam:1982], with the Galactic disc area ($I>52\,$K) blanked out. The lower panels show the magnitude of the relative fluctuations of the synchrotron intensity, $\delta_I={\,\sigma_{I}/I_0}$, at various scales, [with the colour bar shown between Panels (a) and (b)]{}: [ (b)]{} $a=30\degr$, [ (c)]{} $a=15\degr$ and [ (d)]{} $a=7\degr$. The latter scale is about the angular size of a turbulent cell ($2{{l}}_\varepsilon=100{\,{\rm pc}}$) seen at a distance $1{\,{\rm kpc}}$.](a4084.eps "fig:"){width="46.00000%"}\
![\[sky408all\] [ (a):]{} The $408{\, {\rm MHz}}$ all-sky map of the total synchrotron intensity [@Haslam:1982], with the Galactic disc area ($I>52\,$K) blanked out. The lower panels show the magnitude of the relative fluctuations of the synchrotron intensity, $\delta_I={\,\sigma_{I}/I_0}$, at various scales, [with the colour bar shown between Panels (a) and (b)]{}: [ (b)]{} $a=30\degr$, [ (c)]{} $a=15\degr$ and [ (d)]{} $a=7\degr$. The latter scale is about the angular size of a turbulent cell ($2{{l}}_\varepsilon=100{\,{\rm pc}}$) seen at a distance $1{\,{\rm kpc}}$.](a4087.eps "fig:"){width="46.00000%"}
### The 408MHz survey {#subsubsec:408}
To reduce the influence of the Galactic disc, where the structure in the radio maps is mainly due to systematic arm-interarm variations and localized radio sources such as supernova remnants, the original intensity data were truncated at 52K (1% of the maximum and 167% of the r.m.s. value of $I$) [: this blanking only affects emission at Galactic latitudes $|b|<20\degr$]{}. The resulting sky distribution of the relative radio intensity fluctuations $\sigma_{Ia}/I_{0a}$ is shown in Fig. \[sky408all\] for a selection of averaging scales, $a=30\degr, 15\degr$ and $7\degr$ ($a$ corresponds to the radius rather than the diameter of the region).
While panels (b) and (c) in Fig. \[sky408all\] reflect mainly systematic trends in radio intensity, we expect that panel (d) is dominated by the turbulent fluctuations. In particular, the $a=7\degr$ map shows a much weaker variation with Galactic latitude than those at larger scales, for $|b|\ga30\degr$. We note that the correlation scale of the synchrotron intensity fluctuations obtained by @DS87 is $8\degr$.
[At $|b|\gtrsim20\degr$, the maximum pathlength through the synchrotron layer is about $h_{\epsilon}/\sin{b}\simeq 6{\,{\rm kpc}}$, where the synchrotron scale height $h_{\epsilon}\simeq 1.8{\,{\rm kpc}}$ [@Beuermann:1985]. With an angular resolution of about $1\degr$ the linear beamwidth is about $100{\,{\rm pc}}$ at most. Give that the correlation length is about $l_{\epsilon}\simeq 50{\,{\rm pc}}$ (see Section \[subsec:corr\]) the beam encompasses one synchrotron cell at most.]{}
Contours outside the Galactic disc in Fig. \[sky408all\]d, $|b|\ga30\degr$ give $$\label{fluct}
\delta_I=0.1\mbox{--}0.2.$$
Results obtained from the $408{\, {\rm MHz}}$ map with point sources subtracted differ insignificantly from those obtained using the original map.
![ \[sky22\] As in Fig. \[sky408all\] but for the 22MHz survey [@Roger:1999] with the Galactic disc area ($I>10^5\,$K) blanked out.](22mhz.eps "fig:"){width="47.00000%"} ![ \[sky22\] As in Fig. \[sky408all\] but for the 22MHz survey [@Roger:1999] with the Galactic disc area ($I>10^5\,$K) blanked out.](colbar22.eps "fig:"){width="35.00000%"} ![ \[sky22\] As in Fig. \[sky408all\] but for the 22MHz survey [@Roger:1999] with the Galactic disc area ($I>10^5\,$K) blanked out.](a221.eps "fig:"){width="47.00000%"} ![ \[sky22\] As in Fig. \[sky408all\] but for the 22MHz survey [@Roger:1999] with the Galactic disc area ($I>10^5\,$K) blanked out.](a224.eps "fig:"){width="47.00000%"} ![ \[sky22\] As in Fig. \[sky408all\] but for the 22MHz survey [@Roger:1999] with the Galactic disc area ($I>10^5\,$K) blanked out.](a227.eps "fig:"){width="47.00000%"}
### The 22MHz partial sky survey
Contours of the relative intensity fluctuations obtained from the $22{\, {\rm MHz}}$ map are shown in Fig. \[sky22\]. As in Fig. \[sky408all\], averaging over scales $a=30\degr$ and $15\degr$ reveals the large-scale structure clearly visible in the original data. However, results at $a=7\degr$ show much less of such structure, and the statistically homogeneous part of the sky in this panel includes the same contours of 0.1 and 0.2 as in Fig. \[sky408all\], with the value of 0.3 confined to the bright ridges seen in Fig. \[sky22\]a. Thus, the $22{\, {\rm MHz}}$ data are in a good agreement with the values for $\delta_I$ obtained from the 408 MHz data. This suggests a weak frequency dependence of the relative synchrotron intensity fluctuations $\delta_I$.
![\[M33\] The $1.4{\,{\rm GHz}}$ radio map of M33 [@TKB07], with the rectangular fields used in our analysis shown. For orientation, we note that the size of Field 3 is $7'\times 7'$ equivalent to $1.6\times 1.9{\,{\rm kpc}}^2$.](M33.eps){width="44.00000%"}
Field No. $\sigma^{(0)}_I/I_0$ $\sigma^{(1)}_I/I_0$ $\sigma^{(2)}_I/I_0$ $I_0$ \[$\mu$Jy/beam\]
----------- ---------------------- ---------------------- ---------------------- ------------------------
1 677
2 545
3 609
4 606
5 672
6 795
7 565
8 902
9 810
Mean 687
: \[tab\]Relative radio intensity fluctuations $\sigma_I/I_0$ in M33 with systematic trends of various orders subtracted. The first column gives the field number as specified in Fig. , and the next three columns show the relative fluctuations of radio intensity, with $\sigma^{(m)}_I$ the standard deviation of $I$ within a given field, obtained with a trend of order $m$ subtracted [(the mean value of the trend vanishes across each field)]{}: $m=0$ corresponds to the original data, $m=1$, to a linear trend, and $m=2$, to a quadratic trend in the angular coordinates. The last column shows the mean value of the radio intensity in each field.
### M33 {#ssM33}
Our analysis for the Milky Way has a potential difficulty that long lines of sight might make it impossible to separate the contributions to $\sigma_I$ from small-scale (random) and large-scale (systematic) variations of the synchrotron emissivity. Therefore, we consider also the nearby galaxy M33 seen nearly face-on (inclination $56\degr$). To avoid excessive contribution from large-scale variations due to the spiral pattern and the radial gradient in $I$, we selected areas free of strong star formation in the outer regions of the galaxy disc in Fig. \[M33\]. The areas of the rectangular fields chosen range from $1.9\times 1.2{\,{\rm kpc}}^2$ to $1.9\times 4.7{\,{\rm kpc}}^2$.
The fields are big enough to make gradients in the mean quantities significant; in particular, the non-thermal disc of M33 has a strong radial gradient in radio intensity [@TBKKBGM07], so we subtract regular trends from the values of $I$. We fitted first- or second-order polynomials to $I(l,b)$ in $l$ and $b$ in each field and calculated $\delta_I$ after subtracting the trends [with [vanishing]{} mean value]{} from the original data. Results are shown in Table \[tab\], with $\sigma_I^{(m)}$ denoting the standard deviation of the radio intensity obtained upon the subtraction of a polynomial of order $m$ in the angular coordinates. The mean value of synchrotron intensity $I_0$ was calculated for each field.
We note that $\sigma_I^{(0)}$ (the standard deviation of $I$ in the original data) is noticeably larger than $\sigma_I^{(1)}$ and $\sigma_I^{(2)}$. Thus, the large-scale trends contribute significantly to the intensity variations. On the other hand, $\sigma_I^{(1)}$ and $\sigma_I^{(2)}$ have rather similar magnitudes of [about]{} 0.15 (and $\sigma_I^{(1)}>\sigma_I^{(2)}$ as expected), so that they can be adopted as an estimate of $\sigma_I$ corrected for the large-scale trends. We use the value of $\sigma_I^{(2)}/I_0$ averaged over the nine fields explored as the best estimate for $\delta_I$.
![\[smallNW\] The relative fluctuations in synchrotron intensity $\delta_I$ as a function of $N_W$, the number of synchrotron correlation cells across the beam area, with $\delta_I$ normalized by $\delta_0$, the relative fluctuations obtained along a single line of sight, i.e., for $N_W\to0$. The asymptotic dependence $\delta_I=2\delta_0 N_W^{-1/2}$ [ (dashed line)]{} emerges only for $N_W\ga20$–$30$. The calculation assumed a Gaussian beam, with the full width at half-maximum (FWHM) taken for the beam diameter.](smallNW.eps){width="40.00000%"}
However, $W\approx100{\,{\rm pc}}$ (half-width at half maximum of the Gaussian beam) in the observations of M33 used here. [Assuming the synchrotron correlation length $l_{\epsilon}=50{\,{\rm pc}}$]{} the beam area encompasses $N_W\approx4$ correlation cells. [Therefore, to make the M33 results (especially $\delta_I$) comparable to those of the Milky Way. we have to reduce them to a common pathlength and number of synchrotron correlation cells within the beam. We recall that the beams at 408 MHz and 22 MHz only cover at most one synchrotron cell (see Section \[subsubsec:408\]), so we have $N_W=1$ in the Milky Way.]{} For the small value of $N_W$ in the high-resolution observations of M33, the dependence of $\delta_I$ differs significantly from its asymptotic form $\delta_I\propto N_W^{-1/2}$. Figure \[smallNW\] shows the dependence of $\delta_I$ on $N_W$ obtained for a model synchrotron-emitting system described in detail in Section \[MFCRM\]: $\delta_I$ only weakly depends on $N_W$ for small $N_W$. Reduced to a single line of sight, the synchrotron intensity fluctuations in M33 then correspond to $\delta_I=0.13/0.7=0.2$ if the synchrotron correlation length is ${{l}}_\varepsilon=50{\,{\rm pc}}$ (i.e., $N_W=4$). We also recall that the typical pathlength through [the disc of]{} M33 is [about]{} twice that through the Milky Way [(where the observer is located not far from the midplane), and we adopt $L\approx 1{\,{\rm kpc}}/\cos(i)\approx 2{\,{\rm kpc}}$]{} for this galaxy. Then the value of $\delta_I$ in M33, reduced to [a standard value of]{} $L=1{\,{\rm kpc}}$ for compatibility with the Milky Way data, is further $\sqrt2$ times larger: $$\delta_I\simeq0.3.$$
Comparison with earlier results and summary
-------------------------------------------
[To date, analysis of synchrotron observations of the Milky Way has focused]{} primarily on the spectrum of the fluctuations while their magnitude has attracted surprisingly little attention. @MS57 observed fluctuations of the Galactic radio background near the Galactic south pole at $\lambda=3.5{\,{\rm m}}$, with the resolution of $50\degr$, to obtain $\sigma_I=3.3\times10^{-26}\,\mathrm{W\,m^{-2}}{\,{\rm Hz}}^{-1}$ per beam, which corresponds to $\delta_I=0.12$ [see also @G59].
@DS87 used observations at $102.5{\, {\rm MHz}}$ ($\lambda2.92{\,{\rm m}}$) near the North Galactic Pole, where the Galactic radio emission is minimum, to determine the synchrotron autocorrelation function and its anisotropy arising from the large-scale magnetic field. They obtain $\delta_I\approx 0.07$ but note that this estimate should be doubled if the isotropic extragalactic background (half the total flux) is to be subtracted, to yield $\delta_I\simeq0.14$. @Betal91a argue that only 17% of the total flux is of extragalactic origin, and then $\delta_I\simeq0.08$ at $102.5{\, {\rm MHz}}$.
The autocorrelation function of the brightness temperature fluctuations of the Galactic radio background was determined also by @Betal91a [@Betal91b] who used observations at 408 and 1420MHz, smoothed to a resolution of about $\mathrm{FWHM}=5\degr$. They observed a ‘quiet’ region with reduced fluctuations, $30\degr<\mathrm{DEC}<50\degr$, $180\degr<\mathrm{RA}<250\degr$, identified by @B67 as an interarm region, since they were interested in the cosmic microwave background fluctuations. For the Galactic synchrotron radiation, which dominates at these frequencies, they obtain $\delta_I\approx0.05$ at 408MHz and 0.08 at 1420MHz.
These estimates are somewhat lower than those obtained above. The value of $\delta_I$ obtained by @Betal91a can be lower due to their selection of a region with weaker synchrotron intensity fluctuations.
The relative fluctuations in radio intensity are remarkably similar in all the Milky Way maps and in all fields in M33 considered, with $$\delta_I\approx0.1\mbox{--}0.3,$$ when reduced to the common pathlength $L=1{\,{\rm kpc}}$ (see Section \[scale\] for further discussion). The lower values are more plausible. We believe that these estimates are not significantly affected by either large-scale trends in the radio intensity or by discrete radio sources or by thermal emission. Even if these effects still contribute to our estimate, it provides an *upper* limit on the fluctuations in synchrotron intensity arising in the interstellar medium of the Milky Way and M33.
Statistical analysis of synchrotron intensity fluctuations {#SIF}
==========================================================
[In order to interpret the results of our analysis of observations in Section \[data\], we use a model of partially ordered, random distributions of magnetic fields and cosmic rays, assuming various degrees of correlation (or anticorrelation) between them. We calculate the relative magnitude of synchrotron intensity fluctuations $\delta_I$ analytically and numerically and compare the results with the observational constraints described above in order to establish to estimate the degree of correlation or anticorrelation compatible with observations.]{}
[In this Section, first we relate fluctuations in synchrotron itensity to the underlying synchrotron emissivity, then we describe a model for defining distributions of magnetic field and cosmic rays with controlled cross-correlation. These magnetic field and cosmic ray distributions allow us to calculate the model synchrotron emissivity, using results derived in Appendix \[app\], and hence the model synchrotron intensity.]{}
[Relative fluctuations of synchrotron intensity]{} {#RFSI}
--------------------------------------------------
[Here we]{} estimate the synchrotron intensity $I_0={\left\langle I\right\rangle}_S$ [averaged over an area $S$]{} in the plane of the sky and the [corresponding]{} standard deviation, $\sigma_I=\left({\left\langle I^2\right\rangle}_S-{\left\langle I\right\rangle}_S^2\right)^{1/2}$, as a function of the synchrotron emissivity. [Angular brackets are used to denote various spatial averages (over an area $S$, volume $V$ or path length $L$, as indicated by the corresponding subscript), whereas overbar is used for statistical (ensemble) averages. The former arise naturally from observations and numerical models, whilst analytical calculations usually provide the latter. We assume that the two types of averaging procedure lead to identical results]{} [unlike @GSFSM13], although we distinguish them formally for the sake of clarity.
Assuming that the synchrotron spectral index is equal to $-1$ [(this simplifies analytical calculations significantly, without noticeable effect on the results – see Section \[Disc\] below)]{}, the intensity of synchrotron emission at a given position in the sky is given by $$\label{I}
I=\int_L \varepsilon\,{\mathrm{d}}s \propto\int_L n{_\mathrm{cr}}B_\perp^2 \,{\mathrm{d}}s ,$$ where $\varepsilon$ is the synchrotron emissivity, $n{_\mathrm{cr}}$ is the number density of cosmic ray electrons, $B_\perp$ is magnetic field in the plane of the sky and integration is carried along the line of sight ${\bmath{s}}$ over the path length $L$.
[In a random magnetic field and cosmic-ray distribution, the synchrotron emissivity $I$ is also a random variable. We can rewrite Eq. (\[I\]) in terms of the path-length average as $$\label{Iapp}
I = L {\left\langle \varepsilon\right\rangle}_L\,,
$$ where ${\left\langle \varepsilon\right\rangle}_L=L^{-1}\int_L \varepsilon\,{\mathrm{d}}s$ is the average synchrotron emissivity along the path length. We neglect a dimensional factor in Eq. and other similar equations; it is unimportant as we always consider relative fluctuations where it cancels. The average synchrotron intensity from the area $S$ in the sky plane is then related to the synchrotron emissivity averaged over the volume of a depth $L$ (the extent of the radio source along the line of sight) and cross-section $S$: $$\label{I0e}
I_0={\left\langle I\right\rangle}_S=L {\left\langle \varepsilon\right\rangle}_V=2N l_\varepsilon {\overline{\varepsilon}}\,,$$ where $l_\varepsilon$ is the correlation length of the synchrotron emissivity, $N=L/(2l_\varepsilon)$ $(\gg1)$ is the number of correlation cells of $\varepsilon$ along the path length $L$, and the volume average has been identified with the statistical average to obtain the last equality, $$\label{Vst}
{\left\langle \varepsilon\right\rangle}_V={\overline{\varepsilon}}\,.$$ If $S$ is sufficiently large, such an identification applies to the area average as well, but the linear resolution of observations often approaches the size of a turbulent cell in the source; in this case, Eq. is more appropriate. ]{}
[Fluctuations in $I$ arise from variations of both $\varepsilon$ and $L$ between different lines of sight. Neglecting the latter, the standard deviation of $I$ follows as $$\label{sigmaIe}
\sigma_I=N^{-1/2}L\sigma_\varepsilon=\sigma_\varepsilon(2l_\varepsilon L)^{1/2}\,,
\qquad N\gg1\,;$$ this quantity characterizes the scatter in the synchrotron intensity at different positions across the radio source separated by more than $l_\varepsilon$ to make them statistically independent. ]{}
Cosmic ray distribution partially correlated with magnetic field {#Cray}
----------------------------------------------------------------
[In this section we introduce a distribution]{} of cosmic rays which has a prescribed cross-correlation coefficient with a given magnetic energy density. [Magnetic field is represented as the sum of the mean and random parts, ${\bmath{B}}_0$ and ${\bmath{b}}$, respectively:]{} $${\bmath{B}}={\bmath{B}}_0+{\bmath{b}},$$ [with ${\overline{{\bmath{b}}}}=0$, ${\overline{{\bmath{B}}}}={\bmath{B}}_0$ and ${\overline{B^2}}=B_{0\perp}^2+B_{0\parallel}^2+\sigma_b^2$, where $B_{0\perp}$ and $B_{0\parallel}$ are the mean field components in the plane of the sky and parallel to the line of sight, respectively.]{} Each Cartesian vectorial component of the random magnetic field ${\bmath{b}}$ is assumed to be a Gaussian random variable with zero mean value, ${\overline{b}}_i=0$, and [to avoid unnecessary complicated calculations,]{} the random magnetic field is assumed to be isotropic, $$\label{biso}
{\overline{b_i^2}}=\tfrac{1}{3}{\overline{b^2}}=\tfrac{1}{3}\sigma_b^2,$$ where $i=x,y,z$ and overbar denotes ensemble averaging.
[The number density of cosmic-ray electrons is similarly represented as the sum of a mean $n_0\equiv{\overline{n}}{_\mathrm{cr}}$ (slowly varying in space) and random $n'$ parts,]{} $$\label{ncra0}
n{_\mathrm{cr}}=n_0+n'.$$ The cross-correlation coefficient $c$ of $n{_\mathrm{cr}}$ and $B^2$ is defined as $$\label{corr}
c(B^2,n{_\mathrm{cr}})=
\frac{{\overline{ n{_\mathrm{cr}}B^2}} -{\overline{n}}{_\mathrm{cr}}{\overline{B^2}}}{\sigma_{n} \sigma_{B^2}}.$$
To implement local equipartition between cosmic rays and magnetic field, which corresponds to $c=1$, we use a distribution of cosmic rays identical to that of the random part of the magnetic energy density, $n'=\alpha(B^2-{\overline{B^2}})$, where $\alpha$ is a coefficient that allows us to control independently the magnitude of the fluctuations in $n{_\mathrm{cr}}$.
To obtain partially correlated distributions of $n{_\mathrm{cr}}$ and $B^2$, we introduce an auxiliary positive-definite, scalar random field $F$, uncorrelated with the magnetic energy density: $$\label{cB2F}
c(B^2,F)=0.$$ Our specific choice of ${\bmath{B}}$ and $F$ is discussed below.
Now, we represent the random part of the cosmic-ray number density as $$\label{ncra}
n'=\alpha (B^2-{\overline{B^2}}) + \beta(F-{\overline{F}}),$$ where the coefficients $\alpha$ and $\beta$ are chosen to obtain $$\label{cC}
c(B^2,n{_\mathrm{cr}})=C,$$ with $C$ the desired value of the cross-correlation coefficient. The first term in Eq. is responsible for the part of the cosmic-ray distribution fully correlated with magnetic field energy density, whereas the second term reduces the cross-correlation to the desired level. In particular, Eq. ensures that $c(B^2,n{_\mathrm{cr}})=1$ for $\alpha=1$, $\beta=0$ (perfect correlation), and $c(B^2,n{_\mathrm{cr}})=0$ for $\alpha=0$ (uncorrelated distributions).
Let us find $\alpha$ and $\beta$ from Eq. . It follows from Eqs (\[ncra0\]) and (\[ncra\]) that $${\overline{n{_\mathrm{cr}}B^2}}=
\alpha \sigma^2_{B^2}+n_0 {\overline{B^2}}$$ given that Eq. implies $${\overline{B^2 F}}= {\overline{B^2}}\, {\overline{F}}.$$ For any uncorrelated random variables $X$ and $Y$, the variance of their sum $Z=X+Y$ is given by $\sigma_Z^2=\sigma_X^2+\sigma_Y^2$. Hence, Eq. implies $$\label{sigman}
\sigma^2_{n}=\alpha^2 \sigma^2_{B^2} + \beta^2 \sigma^2_{F}.$$ Then Eq. yields $$\label{aC}
\frac{\alpha}{\sqrt{\alpha^2\sigma^2_{B^2}+\beta^2\sigma^2_{F}}}=C,$$ where $\sigma_{B^2}={\overline{B^4}}-({\overline{B^2}})^2$ and $\sigma_F^2={\overline{F^2}}-{\overline{F}}^2$. Using Eq. to eliminate $\beta$ in Eq. , we obtain $$\alpha=\frac{\sigma_n}{\sigma_{B^2}}C
\quad\mbox{and}
\quad
\beta=\frac{\sigma_n}{\sigma_{F}}\sqrt{1-C^2 }.$$ Equation then reduces to $$\label{ncrafull}
n{_\mathrm{cr}}=n_0+\sigma_n\left( C \frac{B^2-{\overline{B^2}}}{\sigma_{B^2}}+\sqrt{1-C^2}\frac{F-{\overline{F}}}{\sigma_{F}} \right),$$ where the standard deviation of the cosmic-ray number density $\sigma_n$ is an independent parameter that we are free to vary.
Our specific model of magnetic field is described in Section \[MFCRM\]. However, for the analytical calculations of $\delta_I$ presented in Section \[SIFPC\], it is sufficient to know $B_0$ and $\sigma_B$. There is no need to specify $F$ in any more detail as long as $F$ and $B^2$ (more precisely, $B_\perp^2$) are uncorrelated. The more detailed model of Section \[MFCRM\] is only required for numerical calculations presented below to verify and refine the analytical results.
Synchrotron intensity fluctuations with partially correlated $n{_\mathrm{cr}}$ and $B^2$ {#SIFPC}
----------------------------------------------------------------------------------------
Details of the calculations of ${\overline{\varepsilon}}$ and $\sigma_\varepsilon$, and then, of the mean value and standard deviation of the synchrotron intensity using Eqs and together with Eq. can be found in Appendix \[app\], with $I_0$ given by Eq. (\[I0\]) and $\sigma_I$ by Eq. (\[sigmaI\]). Here we consider the key special cases.
The model developed here allows us to express $\delta_I$ in terms of the following dimensionless parameters: the number of correlation cells along the line of sight $N$ (assuming perfect angular resolution; a finite beam size can be allowed for using additional averaging across the beam as in the end of Section \[ssM33\]), the cross-correlation coefficient between cosmic rays and magnetic field $C$, the relative magnitude of the magnetic field fluctuations $\delta_b$, and the relative magnitude of fluctuations in cosmic ray number density,
$$\delta_n=\frac{\sigma_n}{n_0}.$$
![\[I1b\] The relative fluctuations in synchrotron intensity $\delta_I$ versus the relative magnitude of magnetic field fluctuations $\delta_b$ [obtained analytically in four special cases. Solid line, Eq. , is for complete correlation of cosmic rays and magnetic fields, $C=1$; dashed line is from Eq. , i.e., $n{_\mathrm{cr}}=\mbox{const}$; dotted and dash-dotted lines were obtained from Eq. for uncorrelated fluctuations in cosmic rays and magnetic field, $C=0$, with $\delta_n=1$ and $\delta_n=0.5$, respectively.]{} We note that the curves rapidly approach the horizontal asymptote, [so that]{} the approximation $\delta_b\to\infty$ is reasonably accurate for $\delta_b>1.5$–2 [as typically found in spiral galaxies]{}.](fig0.eps){width="40.00000%"}
In the case of detailed (local, or pointwise) equipartition between cosmic rays and magnetic fields, $n{_\mathrm{cr}}\propto B^2$, [$C=1$ and $\delta_n=\sigma_{B^2}/B^2$ (Eq. (\[eq:App:sigB2\]))]{} so the relative fluctuations in the synchrotron intensity follow as $$\label{dIeq}
\delta_I=\frac{\sigma_I}{I_0}=
\frac{\delta_b(54+295\delta_b^2+404\delta_b^4+101\delta_b^6)^{1/2}}{N^{1/2}\left(3+10\delta_b^2+5\delta_b^4\right)}.$$ We recall that $\delta_b=\sigma_b/B_0$ [and note that all such analytical results are valid only for $N\gg1$.]{} As $\delta_b$ increases, $\delta_I$ rapidly approaches the asymptotic value independent of $\delta_b$ (see Fig. \[I1b\]): $$\delta_I\simeq 2N^{-1/2} \quad \mbox{for}\quad \delta_b^2\gg1.$$
It is useful to note similar expressions for $\delta_I$ obtained under different assumptions about the correlation between cosmic rays and magnetic field. If cosmic ray fluctuations are [uncorrelated with those]{} in magnetic field, Eqs. (\[I0\]) and (\[sigmaI\]) yield for $C=0$ $$\label{bcrauncorr}
\delta_I=
\frac{[\delta_n^2+\delta_b^2(2+\delta_b^2)(1+2\delta_n^2)]^{1/2}}{N^{1/2}\left(1+\delta_b^2\right)}.$$ In particular, for $\delta_n\to0$ we obtain an asymptotic form for a homogeneous distribution of cosmic rays: $$\label{homogncra}
\delta_I=
\frac{\delta_b(2+\delta_b^2)^{1/2}}{N^{1/2}\left(1+\delta_b^2\right)}.$$ Thus, $\delta_I\simeq N^{-1/2}$ for $\delta_b^2\gg1$ and $n{_\mathrm{cr}}=\mbox{const}$. Figure \[I1b\] shows the dependence of $\delta_I$ on $\delta_b$ from Eqs , (\[bcrauncorr\]) and (\[homogncra\]).
[It is convenient to summarize these results by providing the corresponding values of a quantity independent of the number of correlation cells along the path length, $N^{1/2}\delta_I$, as obtained from Eqs , (\[bcrauncorr\]) and (\[homogncra\]) for $\delta_b\gg1$, which is applicable to $\delta_b\simeq3$ (Fig. \[I1b\]).]{} The relative magnitude of synchrotron intensity fluctuations expected under detailed equipartition follows from Eq. (\[dIeq\]) as $$\label{complete}
N^{1/2}\delta_I\approx2.0
\quad\mbox{(local equipartition)}.$$ Equation (\[bcrauncorr\]) yields, for $\delta_n=0.5$, $$N^{1/2}\delta_I\approx1.2
\quad\mbox{(uncorrelated fluctuations)},$$ and Eq. (\[homogncra\]) leads to $$N^{1/2}\delta_I\approx1.0
\quad (n{_\mathrm{cr}}=\mbox{const}).$$ As might be expected, detailed equipartition between cosmic rays and magnetic fields leads to the strongest synchrotron intensity fluctuations for a given $\delta_b$ and $N$. [For illustration, with the correlation length of the synchrotron intensity fluctuations ${{l}}_\varepsilon=50{\,{\rm pc}}$ and the path length $L=1{\,{\rm kpc}}$, we obtain $N=L/2{{l}}_\varepsilon\simeq 10$ for a beam narrower than the size of the correlation cell.]{} We note that the dependence of $\sqrt{N}\delta_I$ on $\delta_b$ is quite weak as long as $\delta_b^2\ga3$ [which is usually the case for spiral galaxies (see Section \[MFIMF\]). The difference in the level of synchrotron fluctuations in these limiting cases is strong enough to be observable under certain conditions clarified in Section \[SR\].]{}
A model of a partially ordered magnetic field {#MFCRM}
=============================================
[ To verify, strengthen and refine the analytical calculations presented above, we implement numerically the model of magnetic field and cosmic rays described in Section \[Cray\]. For this purpose, we introduce in this section a multi-scale magnetic field with prescribed spectral properties and the corresponding cosmic-ray distribution using Eq. . The phases and directions of individual modes in the magnetic field spectrum can be chosen at random without affecting the magnetic field correlation scale, the value of $\delta_B$ and the energy spectrum. We use this freedom to generate a large number of statistically independent realizations of the magnetic field and cosmic-ray distributions to compute the resulting values of $\delta_I$ and compare them with the analytical results. ]{} To prescribe a quasi-random magnetic field ${\bmath{b}}$ with vanishing mean value in a periodic box, we use a Fourier expansion in modes with randomly chosen directions of wave vectors ${\bmath{k}}$, but with amplitudes adjusted to reproduce any desired energy spectrum: $$\label{ft}
{\bmath{b}}({\bmath{x}})=\frac{1}{(2\pi)^{3/2}}\int {\bmath{\hat{b}}}({\bmath{k}})
e^{\mathrm{i} {\bmath{k}}\cdot{\bmath{x}}}\, {\mathrm{d}}^3{\bmath{k}},$$ where $ {\bmath{\hat{b}}}$ is the Fourier transform of ${\bmath{b}}$; the physical field is represented by the real part of this complex vector. The corresponding magnetic energy spectrum is given by $$\label{spec}
M(k)=\int_{|{\bmath{k}}'|=k} |{\bmath{\hat{b}}}({\bmath{k}}')|^2\, {\mathrm{d}}^3{\bmath{k}}',$$ where the integral is taken over the spherical surface of a radius $k$ in $k$-space. In the isotropic case, $M(k)=4\pi k^2
|{\bmath{\hat{b}}}(k)|^2$. In order to ensure periodicity within a computational box of size $L$, as required for the discrete Fourier transformation, the components of the wave vectors are restricted to be integer multiples of $2\pi/L$.
A solenoidal vector field ${\bmath{b}}$, i.e., that having ${\bmath{k}}\cdot{\bmath{\hat{b}}}({\bmath{k}})=0$, is specified by $${\bmath{\hat{b}}}({\bmath{k}})=\frac{{\bmath{k}}\times{\bmath{X}}}{|{\bmath{k}}\times{\bmath{X}}|}k^{-1}\sqrt{M(k)},$$ where ${\bmath{X}}$ is a complex vector chosen at random, to ensure that the Fourier modes have random phases. We consider a magnetic energy spectrum represented by two power-law ranges, $$\label{Mss}
M(k)=M_0
\left\{
\begin{array}{ll}
(k/k_0)^{s_0} &\mbox{for } k<k_0,\\
(k/k_0)^{-s_1} &\mbox{for } k\geq k_0,
\end{array}
\right.$$ with $s_0>0$ and $s_1>0$, where $k_0$ is the energy-range wavenumber. We use $s_1=5/3$ as in Kolmogorov’s spectrum and $s_0=2$ [see @2001PhRvE..64e6405C]. The standard deviation of the magnetic field is given by $$\label{sigmabs}
\sigma_b^2=\int_0^\infty M(k)\,{\mathrm{d}}k=M_0 k_0\frac{s_0+s_1}{(s_0+1)(s_1-1)}$$ for $s_1>1$. The correlation length ${{l}}_b$ of the resulting magnetic field (analogous to the radius of a correlation cell) differs from its dominant half-wavelength $\tfrac12\lambda_0=\pi/k_0$ for any finite values of $s_0$ and $s_1$ [[@Monin:1975]]{}: $$\begin{aligned}
{{l}}_{b}&
=\frac{\pi}{2}\,\frac{\int_0^\infty k^{-1}M(k)\,dk}{\int_0^\infty M(k)\,{\mathrm{d}}k}
= \frac{\pi}{2k_0} \left(1+\frac{1}{s_0}\right)\left(1-\frac{1}{s_1}\right)
\nonumber\\
&=\frac{3\pi}{10k_0},\label{Bcorr}\end{aligned}$$ where the last equality follows for $s_0=2$ and $s_1=5/3$. [For the Milky Way, suitable values are $l_b=50{\,{\rm pc}}$ and $\sigma_b\simeq 5$–$10{\,\mu{\rm G}}$ (see Sections \[MFIMF\] and \[Disc\]).]{}
The resulting solenoidal vector field is then added to a uniform component ${\bmath{B}}_0$ to produce a partially ordered magnetic field with controlled fluctuation level $\delta_b$ and energy spectrum $M(k)$. This approach has been used to generate synthetic polarization maps of the turbulent ISM by . Similar constructions were used by @GJ99 and @CLP02 in their modelling of cosmic ray propagation in random magnetic fields, by @MV99 for modelling turbulent flows, and by @WBS07 to study dynamo action in chaotic flows.
![\[I1a\] The relative fluctuations in synchrotron intensity $\delta_I$ as obtained from analytical formulae Eqs. (\[I0\] and \[sigmaI\]) (thicker curves) and from numerical calculations with the condition $n{_\mathrm{cr}}>0$ enforced (the corresponding thinner curves), for: $\delta_n=0.2$ (solid), $\delta_n=0.4$ (dashed) and $\delta_n=0.8$ (dash-dotted). The analytic formulae become inapplicable for $C<0$ and $\delta_n\simeq1$ (see the text). The magnetic field is purely random, $B_0=0$ and $N=10$.](Ian.eps){width="40.00000%"}
We will now verify, by direct calculation, that $C(B^2,n{_\mathrm{cr}})\approx C$. The reason for the approximate equality is first explained.
A shortcoming of the [ analytical]{} cosmic ray model defined by Eqs. (\[ncra0\]) and (\[ncra\]) is that $n{_\mathrm{cr}}$ can be negative at some positions (especially when $C<0$ and hence $\alpha<0$) because, at some positions and in some realizations, $B^2$ can be arbitrarily large (as a Gaussian random variable squared). This deficiency could be corrected by selecting a more realistic probability distribution for ${\bmath{b}}$ (e.g., a truncated Gaussian) but we do not feel that this would lead to any additional insight. In the numerical calculations described below, we truncate the negative values of $n{_\mathrm{cr}}$ by replacing them with zero. This, however, makes it impossible to achieve exact anticorrelation between cosmic rays and magnetic fields, so that $C(B^2,n{_\mathrm{cr}})>-1$.
In analytical calculations, we restrict ourselves to the cases with $\delta_n<1$ to reduce the extent of the problem (even if not to resolve it completely). For example, Eqs. (\[I0\]) and (\[sigmaI\]) yield for $\delta_b\to\infty$ [(note that $\delta_I$ is constant with respect to $\delta_b$ for $\delta_b\gtrsim 3$ (Fig. \[I1b\]))]{} $$\label{delIB0}
\delta_I=
\frac{\left[9+6\delta_n\left(11C^2\delta_n+3\sqrt{6}C+3\delta_n\right)\right]^{1/2}}{N^{1/2}(\sqrt{6} C \delta_n +3)}.$$ This dependence of $\delta_I$ on the cross-correlation coefficient $C$ is shown with thicker curves in Fig. \[I1a\] for various values of the relative fluctuations in cosmic rays, $\delta_n$. Thinner curves show similar results obtained from a numerical calculation where $n{_\mathrm{cr}}>0$ is enforced. It is clear from Fig. \[I1a\] that these analytical results are accurate for $C>0$.
However, for $C<0$, [analytic results]{} are useful only if the fluctuations in the cosmic ray number density are relatively weak: $C\ga-0.1$ for $\delta_n<0.8$, $C\ga-0.5$ for $\delta_n<0.4$, and almost any value of $C$ for $\delta_n<0.2$. We only use these analytical results for illustrative purposes, whereas all our conclusions are based on numerical results where $n{_\mathrm{cr}}>0$ at all positions. [Nevertheless, the analytic results presented here and in Appendix \[app\], however unwieldly, are simpler to use than constructing a numerical model and are accurate for $\delta_n$ small enough (say, $\delta_n\lesssim0.4$) and $C$ large enough (say, $C\gtrsim-0.5$) — see Fig. \[I1a\].]{}
Results {#SR}
=======
Synthetic radio maps {#SyRaMa}
--------------------
Each component of the magnetic field described by Eq. (\[ft\]) is the sum of a large number of independent contributions from different wave numbers. By virtue of the central limit theorem, each component of the resulting magnetic field is well approximated by a Gaussian random variable. Then the mean synchrotron intensity and its standard deviation over $N$ correlation cells can be expressed, using Eq. (\[I\]), in terms of $B_0$, $\sigma_b$, $\delta_n$ and $C$. Explicit analytic expressions for $I_0$ and $\sigma_I$ can be found in Appendix \[app\], and we illustrate these results in Fig. \[I1a\]. As might be expected, the relative level of the synchrotron intensity fluctuations increases with the cross-correlation coefficient between $B^2$ and $n{_\mathrm{cr}}$.
Since analytical results are of limited relevance for $C<0$, we performed numerical calculations of the synchrotron intensity where the cosmic-ray number density is truncated to be non-negative (i.e. $n{_\mathrm{cr}}=0$ wherever the model defined by Eq. (\[ncra0\]) returns a negative value). The model has four free parameters:
1. the relative level of magnetic field fluctuations $\delta_b=\sigma_b/B_0$,
2. the relative level of cosmic-ray number density fluctuations $\delta_n=\sigma_n/n_0$,
3. the cross-correlation coefficient between magnetic field and cosmic rays $C$, and
4. the [dominant energy wave number]{} of the turbulent magnetic field $k_0$.
We do not vary the spectral index of magnetic field and cosmic rays as these parameters are of secondary importance [in this context]{}.
The value of $k_0$ controls the correlation lengths of magnetic field (Eq. (\[Bcorr\])), cosmic rays and synchrotron emissivity, and hence the number of the correlation cells of synchrotron intensity fluctuations in the telescope beam $N$, which in turn affects the magnitude of synchrotron intensity fluctuation as $\delta_I\propto N^{-1/2}$. Since $N$ can vary widely between different lines of sight in the Milky Way and between galaxies with different inclination angles, we present our results in terms of $N^{1/2}\delta_I$ for both the observations and the model.
![\[figN\] The isocontours of the numerical factor $G$ in Eq. (\[prefacG\]) for $\sigma_b=1$ (dashed lines and labels in squares) and $\sigma_b\to\infty$ (solid lines and labels in ovals) shown in the $(\delta_n,C)$-plane.](corl.eps){width="45.00000%"}
A relation between the correlation lengths of the synchrotron emissivity and magnetic field {#scale}
-------------------------------------------------------------------------------------------
In the case of an infinitely narrow beam, the number of synchrotron correlation cells traversed by the emission is just the ratio $N=L/(2 {{l}}_\varepsilon)$, where $L$ is the path length through the synchrotron source and ${{l}}_\varepsilon$ is the correlation length of the fluctuations in the synchrotron emissivity. For a finite beam width $W$, this is the number of correlation cells within the beam cylinder, $N\simeq(3/16)LW^2/l_\varepsilon^3$, assuming a circular beam and spherical correlation cells. Unlike the correlation lengths of physical parameters such as the magnetic field, velocity or density fluctuations, the correlation length of the intensity (or emissivity) variations cannot be deduced independently (e.g., from the nature of the turbulence driving), but has to be calculated from the statistical parameters of the physical variables or from observations. [Here, we shall derive an expression for ${{l}}_\varepsilon$ in terms of ${{l}}_b$, which will allow us also to estimate $N$.]{}
To illustrate the difficulties arising, consider the autocorrelation function of $b_\perp^2$ as an example. If $V(x)$ is a stationary Gaussian random function, with vanishing mean value and the autocorrelation function $K_v(r)={\overline{V(x)V(x+r)}}$, the [autocorrelation function of $V^2(x)$ is given by $K_{v^2}(x)=2[K_v(r)]^2$]{} [see e.g. §13 in @S66]. Assuming that each [Cartesian vector]{} component $b_i$ of the random magnetic field ${\bmath{b}}$ is a Gaussian random variable, with the autocorrelation function $K_{b_i}$, we have $K_{{b_i}^2}=2K_{b_i}^2$. Assuming statistical isotropy of ${\bmath{b}}$, $K_{b_x}(r)=K_{b_y}(r)$, and neglecting any cross-correlations between $b_x$ and $b_y$, we obtain the autocorrelation function of $b_\perp^2$: $$K_{b_\perp^2}(r)=4 K_{b_i}^2(r).$$ The relation between the correlation scales of $b_i$ and $b_\perp^2$, denoted ${{l}}_{b_i}$ and ${{l}}_{b_\perp^2}$, respectively, depends on the form of the autocorrelation function of magnetic field: for $K_{b_i}=\tfrac13\sigma_b^2\exp(-r/{{l}}_b)$, we have ${{l}}_{b_\perp^2}={{l}}_{b_i}/2$. However, for $K_{b_i}=\tfrac13\sigma_b^2\exp(-\pi r^2/{{l}}_b^2)$ we have ${{l}}_{b_\perp^2}={{l}}_{b_i}/\sqrt2$. There is no universal relation between the correlation scales of even these simply connected variables. Such a relation should be established in each specific case from the statistical properties of each physical component of the system.
[Since the power spectrum is a Fourier transform of the correlation function, these arguments also apply to the power spectra of $b_i$ and $\epsilon$.]{}
We calculate the correlation length ${{l}}_\varepsilon$ of the synchrotron emissivity $\varepsilon\propto n{_\mathrm{cr}}B_\perp^2$ in the synthetic radio maps from its autocorrelation function $K_\varepsilon(r)$, for various values of the cross-correlation coefficients $C$, $B_0$ and $n_0$: $$\label{lcorr}
{{l}}_\varepsilon
=\sigma_\varepsilon^{-2} \left(
\int_{0}^{L} K_\varepsilon(r)\, {\mathrm{d}}r
- {\left\langle \varepsilon\right\rangle}^2
\right),$$ where $L$ is the path length and $\sigma_\varepsilon$ is the standard deviation of the synchrotron emissivity (assuming $L\gg {{l}}_\varepsilon$ to minimize the impact of statistical fluctuations).
The resulting dependence of ${{l}}_\varepsilon$ on the [correlation length]{} of the magnetic field ${{l}}_b$, obtained in Eq. (\[Bcorr\]) [for the spectrum given by Eq. (\[Mss\])]{}, can be approximated as $${{l}}_\varepsilon/L\approx G^{-1}(4{{l}}_b/L)^{0.65},
\label{prefacG}$$ where the numerical factor $G$ depends on the model parameters and the cross-correlation coefficient $C$. The contours of $G$ in the $(\delta_n,C)$-plane are shown in Fig. \[figN\]; $G=9$–10 are representative values for $\delta_b>2$–3, [independent of the exact choice for $\delta_n$ provided $\delta_n\la1$]{}. The resulting values of $N=L/(2{{l}}_\varepsilon)$ are used below to compare the synchrotron intensity fluctuations obtained from observations in Section \[data\] with the model of Section \[MFCRM\].
![\[I1\] Relative fluctuations of the synchrotron intensity in synthetic radio maps of Section \[SyRaMa\] versus the cross-correlation coefficient between $B^2$ and $n{_\mathrm{cr}}$ for various choices of the model parameters. Top panel: a selection of $\delta_b$ values for fixed $\delta_n=0.5$ (solid: $\delta_b^2=10$; dashed: 1; dotted: 0.1). Grey horizontal lines correspond to uniformly distributed cosmic rays, $\delta_n=0$, for the same values of $\delta_b$ (with $\delta_I$ decreasing with $\delta_b$). Lower panel: various values of $\delta_n$ for $\delta_b^2=10$ ($\delta_n=1$, solid; 0.6, dashed; 0.4, dotted; 0.2, dash-dotted).](I2.eps "fig:"){width="40.00000%"} ![\[I1\] Relative fluctuations of the synchrotron intensity in synthetic radio maps of Section \[SyRaMa\] versus the cross-correlation coefficient between $B^2$ and $n{_\mathrm{cr}}$ for various choices of the model parameters. Top panel: a selection of $\delta_b$ values for fixed $\delta_n=0.5$ (solid: $\delta_b^2=10$; dashed: 1; dotted: 0.1). Grey horizontal lines correspond to uniformly distributed cosmic rays, $\delta_n=0$, for the same values of $\delta_b$ (with $\delta_I$ decreasing with $\delta_b$). Lower panel: various values of $\delta_n$ for $\delta_b^2=10$ ($\delta_n=1$, solid; 0.6, dashed; 0.4, dotted; 0.2, dash-dotted).](I1.eps "fig:"){width="40.00000%"}
Correlation between cosmic rays and magnetic fields {#subsec:corr}
---------------------------------------------------
The relative intensity of synchrotron intensity fluctuations is sensitive to the number $N$ of correlation cells of synchrotron emissivity within the beam (or along the line of sight in case of a pencil beam). When comparing the theoretical model with observations, we adopt $L=1{\,{\rm kpc}}$ for the pathlength in the Milky Way, ${{l}}_{b_i}=50{\,{\rm pc}}$ for the correlation length of magnetic field, $\delta_b=3$ (the asymptotic limit $\delta_b\gg1$ is quite accurate in this case), and explore the range $-1\leq C\leq1$ for the cross-correlation coefficient between cosmic-ray and magnetic fluctuations. For ${{l}}_b/L=0.05$ and $G\approx9$ (see Fig. \[figN\]), we have ${{l}}_\varepsilon\approx40{\,{\rm pc}}$ and $N=L/(2{{l}}_\varepsilon)\simeq10$; we also discuss the effect of larger values of $N$.
Figure \[I1\] shows the dependence of $\delta_I \sqrt{N}$ on the cross-correlation coefficient $C$ for various values of the parameters $\delta_b$ and $\delta_n$. The calculations are based on 100 realizations of ${\bmath{B}}$, so the statistical errors of the mean values shown are negligible.
As can be seen from the upper panel of Fig. \[I1\], the relative magnitude of synchrotron intensity fluctuations, $\delta_I\sqrt{N}>0.7$, obtained for $\delta_b\geq1$ and $\delta_n=0.5$, is stronger than what is observed in the Milky Way, $\delta_I\sqrt{N}=0.3$–0.6 assuming $N=10$. If $N=20$, the conservative observational estimate $\delta_I=0.1$–0.2 translates into $\delta_I\sqrt{N}=0.4$–0.9, implying $C\la-0.6$ [ for the highest $\delta_I \sqrt{N}$]{}. Thus, $\delta_n<0.5$ seems to be justified, unless $N$ is significantly larger than 10 or, otherwise, $\delta_b<1$ (which is highly implausible). Since the estimate $\delta_I=0.1$–0.2 has been obtained for high Galactic latitudes, the path length is unlikely to be much longer than 1[[kpc]{}]{}, and the correlation length of the synchrotron intensity fluctuations can hardly be much shorter than about 50pc. Thus, excluding the case of simultaneously large $L$ and small ${{l}}_\varepsilon$, we conclude that the distribution of cosmic ray electrons is unlikely to have any significant variations at scales of order 50–100pc.
The lower panel in Fig. \[I1\], where a range of values of $\delta_n$ are used with $\delta_b^2=10$, suggests that any positive correlation between cosmic rays and magnetic fields is only compatible with the observational estimate $\delta_I\sqrt{N}=0.3$–0.6 (for $N=10$) if $\delta_n<0.2$. In fact, [a upper limit $\delta_I\sqrt{N}=0.9$ (for $N=20$) might be achieved for $n{_\mathrm{cr}}=\mbox{const}$]{}. The values of $\delta_I\sqrt{N}$ in this case are shown with grey horizontal lines in the upper panel: for example, $\delta_I\sqrt{N}=1$ is compatible with $\delta_b^2=10$. However, the lower values of synchrotron intensity fluctuations in the Milky Way in the range $\delta_I\sqrt{N}=0.3$–0.9 for $N=10$–20 [can be compatible with]{} the presence of fluctuations in cosmic ray density mildly anticorrelated with those in magnetic field. It is difficult to be precise here, but $\delta_n<0.2$ and $C<-0.5$ seems to be an acceptable combination of parameters.
Propagation of cosmic rays and equipartition with magnetic fields {#SDCR}
=================================================================
To illustrate the relation between cosmic rays and magnetic fields, consider a simple model of cosmic ray propagation near a magnetic flux tube. The number density of cosmic rays $n{_\mathrm{cr}}$ (or their energy density $\epsilon{_\mathrm{cr}}$) is assumed to obey the diffusion equation with the source $Q$ and diffusivity $D$ terms depending on the magnetic field [@P69; @KP83; @SL85]. Consider the steady state of a one-dimensional system with $Q=\mbox{const}$. The magnetic field is assumed to have a statistically uniform fluctuating component, ${\overline{b^2}}=\sigma_b^2=\mbox{const}$, whereas the mean field is confined to a Gaussian slab of half-thickness $L$ symmetric with respect to $x=0$: $B_y=B_0\exp[-x^2/(2d^2)]$, $B_x=B_z=0$. The cosmic ray diffusivity is assumed to depend on the relative strength of magnetic fluctuations, $D=D_0\sigma_b^2/B_y^2$. The resulting steady-state diffusion equation $$\frac{{\mathrm{d}}}{{\mathrm{d}}x}D\frac{{\mathrm{d}}}{{\mathrm{d}}x}n{_\mathrm{cr}}+ Q=0,$$ can easily be solved with the boundary conditions $$\frac{{\mathrm{d}}n{_\mathrm{cr}}}{{\mathrm{d}}x}(0)=0,\quad n{_\mathrm{cr}}|_{|x|\to\infty}=0,$$ to yield $$\label{esol}
n{_\mathrm{cr}}= \frac{QB_0^2d^2}{2D_0\sigma_b^2}e^{-x^2/d^2}.
$$ The total number (and energy) of cosmic rays remains finite despite the uniform distribution of their sources, $Q=\mbox{const}$, because $D\to\infty$ as $|x|\to\infty$ in this illustrative model.
This simple solution shows that, near a magnetic flux tube in a statistically homogeneous random magnetic field, cosmic rays concentrate where the total magnetic field is stronger because their diffusivity is smaller there. In this example, the spatial distributions of cosmic rays and magnetic field are tightly correlated.
Another type of argument relating cosmic ray energy density to parameters of the interstellar medium was suggested by @PS05. If both the magnetic flux and the cosmic ray flux are conserved, $BS=\mathrm{const}$ and $n{_\mathrm{cr}}US=\mathrm{const}$ (where $B$ is the magnetic field strength and $S$ is the area within a fluid contour, and $U$ is the cosmic ray streaming velocity), one obtains $n{_\mathrm{cr}}U/B=\mathrm{const}$, which yields $n{_\mathrm{cr}}\propto n^{1/2}$, given that $U=V_\mathrm{A}\propto B n^{-1/2}$, with $n$ the gas number density and $V_\mathrm{A}$ the Alfvén speed. Thus, the cosmic ray density is independent of the magnetic field strength, and scales with the thermal gas density. This result relies on the fact that the streaming velocity of the cosmic rays is proportional to the Alfvén speed. If, instead, $U=V$, with $V$ the gas speed, we obtain $n{_\mathrm{cr}}\propto B$ from these arguments. No clear scaling of the cosmic rays energy density $\epsilon{_\mathrm{cr}}$ with the magnetic field was observed in the simulations of @SBMS06 who use the gas velocity for $U$. There is indication that the average propagation length of CREs depends on the degree of field ordering and hence varies between galaxies [@2013arXiv1307.6253T].
Assumption of a detailed, point-wise (local) equipartition between cosmic rays and magnetic fields is dubious also because these two quantities have vastly different diffusivities, and therefore cannot be similarly distributed in space. Magnetic filaments and sheets produced by the small-scale dynamo in the diffuse warm gas can have scales as small as a few parsecs [@S07], and the strength of this turbulent magnetic field can be about $5{\,\mu{\rm G}}$. The large-scale magnetic field varies over scales of order 1kpc, consistent with the turbulent diffusivity of $10^{26}{\,{\rm cm}}^2{\,{\rm s}}^{-1}$ and time scale $5\times10^8{\,{\rm yr}}$. The diffusive length scale of cosmic rays, based on the diffusivity of $D\simeq10^{28}{\,{\rm cm}}^2{\,{\rm s}}^{-1}$ and the confinement time in the disc, $\tau\simeq10^7{\,{\rm yr}}$, is about $(2D\tau)^{1/2}\simeq1{\,{\rm kpc}}$, similar to [ the length scale]{} of the large-scale magnetic field. On these grounds, it is not impossible that the energy densities of cosmic rays and the *large-scale* magnetic field vary at similar scales, but this would be very implausible for the total magnetic field. Then equipartition arguments may be better applicable to observations of external galaxies, where the linear resolution is not better than a few hundred parsecs, than to the case of the Milky Way.
Discussion {#Disc}
==========
The general picture emerging from our results is that cosmic rays and magnetic fields are slightly anticorrelated at the relatively small, sub-kiloparsec scales explored here ($n{_\mathrm{cr}}=\text{const}$ is also a viable possibility). Such an anticorrelation can result from statistical pressure equilibrium [(i.e. a statistically constant total pressure)]{} in the ISM, where cosmic rays and magnetic fields make similar contributions to the total pressure. An additional effect leading to an anticorrelation is the increase in the synchrotron losses of relativistic electrons in stronger magnetic field.
[Strictly speaking, this conclusion applies to regions for which we have analysed the data: high Galactic latitudes in the Milky Way and the outer parts of M33. However, it is likely that this result reflects general features of cosmic ray propagation.]{}
[Local energy equipartition (or pressure equality) between cosmic rays and magnetic field would produce stronger fluctuations of synchrotron intensity than those observed.]{} However, equipartition between cosmic rays and magnetic field cannot be excluded at larger scales of order 1kpc and greater. @HBX98 indirectly make a similar conclusion concerning loss of equipartition at small scales from their analysis of the radio–far-infrared correlation in M31.
Since magnetic fields and cosmic rays have vastly different diffusivities, and therefore, must vary at very different scales, any strong correlation between them can hardly be expected at scales smaller than 1kpc. Correlated (or rather anticorrelated) fluctuations can, however, arise from such secondary processes as the adjustment to pressure equilibrium, etc.
Our arguments and conclusions are based on observations and modelling of synchrotron emission, a tracer of the electron component of cosmic rays. Thus, our conclusions strictly apply only to the cosmic ray electrons. However, the only significant difference between the behaviour of electrons and protons in this context is that the former experience higher energy losses due to synchrotron emission and inverse Compton scattering off the relic microwave photons. The energy loss time scale $\simeq4\times10^8{\,{\rm yr}}(E/1{\,{\rm GeV}})^{-1}$ in a magnetic field of $5{\,\mu{\rm G}}$ in strength, for particles emitting at wavelengths larger than 1cm, is much longer than the confinement time $10^7{\,{\rm yr}}$, so the energy losses are negligible unless the local magnetic field is unusually strong. Therefore, we extend our conclusions derived from analysis of synchrotron intensity fluctuations to cosmic rays as a whole. Moreover, energy losses can only make the distribution of the electrons more inhomogeneous than that of the protons, so that our conclusions are robust with respect to this caveat.
[Our model, data and their analysis arguably match each other in the level of detail. We do not include any latitudinal variation of the path length $L$ and the variation of the angular size of the turbulent cells, $l_0/L$ with Galactic latitude in the Milky Way. Instead, we restrict our analysis to the range $|b|>30^\circ$ within which both $l_0/L$ and $L$ vary by a factor of two. The important parameter, the square root of the number of turbulent cells along the path length, $N^{1/2}\simeq(L/2l_0)^{1/2}$ then varies by a factor of about 1.5. Since there are other parameters varying with galactic latitude (e.g., the magnitudes of the random and regular magnetic fields, cosmic-ray intensity, etc.), including the dependence of the path length and the correlation scale into the model would make it significantly more complicated, if possible at all. Therefore, we prefer, instead, to present our results in the form of plausible ranges that allow for the numerous effects that remain beyond the framework of the model.]{}
[To simplify analytical calculations, we have adopted $s=-1$ for the synchrotron spectrum, so that the synchrotron emissivity $\varepsilon$ is proportional to $B_\perp^{1-s}=B_\perp^2$. We have verified that numerical results obtained with the more commonly used value $s=-0.7$, where $\varepsilon\propto B_\perp^{1.7}$, differ insignificantly from those with $s=-1$.]{}
[Our model includes magnetic field and cosmic ray distributions represented by a wide range of scales, with the magnetic energy spectrum given by Eq. . However, the spectral index of magnetic fluctuations only appears in the expressions for the r.m.s. magnetic field fluctuations, Eq. , and the magnetic correlation length, Eq. , through which it affects the number of correlation cells $N$. Otherwise, the standard deviation of the synchrotron intensity is not sensitive to the magnetic spectrum.]{}
[We have adopted $l_0=50$–$100{\,{\rm pc}}$ for the correlation scale of the random magnetic field. Estimates of the turbulent scales in the magnetoionic medium of the Milky Way are numerous and divergent.]{} @RS77 [discuss in detail techniques for estimating turbulent scales from pulsar ${\mathrm{RM}}$ data and obtain $l_0=100$–$150{\,{\rm pc}}$ without any restrictive assumptions regarding the correlation between the fluctuations in magnetic field and thermals electrons.]{} @RK89 [estimate the size of a turbulent cells as $2l_0=55{\,{\rm pc}}$ (using our notation) from the Faraday rotation measures of pulsars. In fact, their result refers to the size of the correlation cell of ${\mathrm{RM}}$ fluctuations and these authors do not discuss how it is related to the correlation scale of magnetic field; this relation depends on the degree of (anti) correlation between the fluctuation in magnetic field and thermal electron density]{} [@BSSW03]. @OS93 [estimate the scale of magnetic field fluctuations from the ${\mathrm{RM}}$ of close pairs of pulsars to obtain $2l_0=10$–$100{\,{\rm pc}}$. Their model includes fluctuations in thermal electron density but they are, presumably, considered to be uncorrelated with magnetic field fluctuations; this assumption can significantly affect the result.]{} @HBGM08 [obtain the integral (correlation) scale from ${\mathrm{RM}}$ and depolarization of extragalactic radio sources; their sample probes the inner Galaxy avoiding its central part. These authors obtain $l_0=1$–$5{\,{\rm pc}}$ from the Faraday rotation measures and $l_0=3.5$–$8.7{\,{\rm pc}}$ from depolarization. The authors attribute the difference from other estimates of the outer scale to a correspondingly smaller energy input scale of interstellar turbulence of a few parsecs. Perhaps more plausibly, the fluctuations in ${\mathrm{RM}}$, depolarization or any other parameter can have a hierarchy of characteristic scales due, say, to interstellar shocks, intermittent small-scale magnetic filaments, etc., and different methods can be sensitive to only some of them.]{} @FBSBH11 [deduce the correlation scale of ${\mathrm{RM}}$ fluctuations from high-resolution observations of M51 by comparing the scatter in the values of ${\mathrm{RM}}$ observed under various degrees of spatial smoothing and assuming that the standard deviation of ${\mathrm{RM}}$ scales as $l_0^{1/2}$ as predicted by theory]{} [e.g., @1998MNRAS.299..189S]. [The resulting scale of ${\mathrm{RM}}$ fluctuations is $l_\mathrm{RM}=50{\,{\rm pc}}$.]{} @Houde:2013 [analysed the dispersion of synchrotron polarisation angles in high-resolution observations of M51 to estimate $l_0=98\pm5{\,{\rm pc}}$ and $l_0=54\pm3{\,{\rm pc}}$ parallel and perpendicular to the local mean-field direction respectively.]{}
[We discuss the relation between the correlation lengths of synchrotron intensity and magnetic field in Section \[scale\]; this discussion and conclusions apply to other observables as well. It is important that there is no universal relation between the correlation lengths of, say, $B_\perp^2$ and $B$: to establish such a relation, one has to know the auto-correlation functions of $B_\perp$ and $B_\parallel$. In addition, such observables as Faraday rotation measure, total or polarized radio intensity involve not only magnetic field but also number densities of thermal or relativistic electrons. The cross-correlation between these variables and magnetic field are also required to deduce the statistical properties of magnetic field.]{} [ The comprehensive statistical analysis is recently suggested by @Lazarian2012ApJ. However the theoretical predictions discussed there is hardly possible to compare with available observational data. Only the simplest statistical characteristics give robust results.]{}
Our results can significantly change the interpretation of high-resolution radio observations of the Milky Way and spiral galaxies. Present interpretations, aimed at estimating the strength and geometry of interstellar magnetic fields, rely heavily on the assumption of local equipartition between cosmic rays and magnetic fields, at a scale corresponding to the resolution of the observations. This assumption is acceptable if the resolution is not finer than the diffusion length of the cosmic rays, about [say,]{} $1{\,{\rm kpc}}$. However, this assumption is questionable when applied to observations at higher resolution. We suggest a different procedure to interpret such observations. The original total intensity radio maps should first be smoothed to the scale of the cosmic ray distribution, 1kpc, where the equipartition assumption *may* apply, and the distribution of cosmic rays can be deduced from the smoothed data. [(The smoothing length may depend on the local environment e.g. star formation rate, magnetic field etc. — this requires further investigation using suitable cosmic ray propagation models.)]{} After that, this distribution of cosmic rays can be used to deduce the magnetic field distribution from the data at the original resolution. Since a larger part of the synchrotron intensity fluctuations observed will be attributed to magnetic fields, it is clear that this procedure will result in a more inhomogeneous magnetic field than that arising from the assumption of local equipartition so often used now.
Acknowledgments {#acknowledgments .unnumbered}
===============
[We thank an anonymous referee for a careful and insightful report on the submitted version of the paper, which resulted in many improvements. We are also grateful to Wolfgang Reich for helpful comments.]{} Financial support from the Royal Society and Newcastle University is gratefully acknowledged. At various stages, this work was supported by the Leverhulme Trust under Research Grant RPG-097, the Royal Society, and by the National Science Foundation under Grant NSF PHY05-51164. RS acknowledges support from the grant YD-520.2013.2 of the Council of the President of the Russian Federation. Numerical simulations were partially performed on the supercomputer “URAN” of Institute of Mathematics and Mechanics UrB RAS.
[The mean and standard deviation of the synchrotron intensity]{} {#app}
================================================================
Here we derive analytical expressions for magnitude of relative synchrotron intensity fluctuations $\delta_I=\sigma_I/I_0$, with $\sigma_I$ the standard deviation of the synchrotron intensity $I$ and $I_0$, its mean value. Relations between the statistical characteristics of synchrotron intensity and synchrotron emissivity $\varepsilon=n{_\mathrm{cr}}B_\perp^2$ are given by Eqs. (\[I0e\]) and (\[sigmaIe\]). Here we calculate ${\overline{\varepsilon}}$ and $\sigma_\varepsilon^2={\overline{\varepsilon^2}}-{\overline{\varepsilon}}^2$ using the cosmic ray model, partially correlated with magnetic fields, introduced in Section \[Cray\]. Here overbar denotes ensemble averaging. The calculations are quite straightforward although somewhat cumbersome.
We start with calculating ${\overline{\varepsilon}}$ using $n{_\mathrm{cr}}$ from Eq. : $$\begin{aligned}
{\overline{\varepsilon}}&=&{\overline{n{_\mathrm{cr}}B_\perp^2}}\\
&=&n_0{\overline{B_\perp^2}}+\frac{\sigma_n C}{\sigma_{B^2}}({\overline{B^2 B_\perp^2}} - {\overline{B^2}}\,{\overline{B_\perp^2}})\\
&&\mbox{}+\frac{\sigma_n \sqrt{1-C^2}}{\sigma_{F}}({\overline{F B_\perp^2}} - {\overline{{\overline{F}} B_\perp^2}}).\end{aligned}$$ The last term vanishes since $F$ and $B_\perp^2$ are uncorrelated by construction, and hence ${\overline{F B_\perp^2}}={\overline{F}}\,{\overline{B_\perp^2}}$ and ${\overline{{\overline{F}}\,B_\perp^2}}={\overline{F}} {\overline{B_\perp^2}}$. For the same reason, all terms in ${\overline{\varepsilon^2}}$ that contain $F$ also vanish. As each Cartesian vectorial component of the random magnetic field ${\bmath{b}}$ is assumed to be a Gaussian random variable with zero mean value, ${\overline{b}}_i=0$, we have, for the higher statistical moments, $$\label{shizo}
{\overline{b_i^{2k}}}=\frac{(2k)!}{2^k k!}\sigma_b^{2k},
\quad
{\overline{b_i^{2k+1}}}=0,
\quad
k=1,2,\ldots,$$ where $i=x,y,z$. This allows us to calculate the higher-order moments that contribute to ${\overline{\varepsilon}}$ and ${\overline{\varepsilon^2}}$ as follows: $$\begin{aligned}
{\overline{B_\perp^4}}&=&B_{0\perp}^4+\frac{8}{3} B_{0\perp}^2 \sigma_b ^2+\frac{8}{9} \sigma_b ^4,\\
{\overline{B^4}} &=&B_{0\perp}^4+\frac{10}{3} B_{0\perp}^2 \sigma_b ^2+\frac{5}{3} \sigma_b ^4,\\
&&\mbox{}+2 B_{0\perp}^2 B_{0\parallel}^2+\frac{10}{3}B_{0\parallel}^2 \sigma_b^2+B_{0\parallel}^4\\
{\overline{B^2 B_{\perp}^4}}&=&B_{0\perp}^4 B_{0\parallel}^2
+\frac{8}{3} B_{0\perp}^2 \sigma_b ^2 B_z^2
+\frac{8}{9} \sigma_b ^4 B_{0\parallel}^2 \\
&&\mbox{}+B_{0\perp}^6+\frac{19 }{3}B_{0\perp}^4 \sigma_b ^2
+\frac{80}{9}B_{0\perp}^2 \sigma_b^4+\frac{56 }{27}\sigma_b ^6,\\
{\overline{B^4B_{\perp}^4}}&=&2 B_{0\perp}^6 B_{0\parallel}^2+14 B_{0\perp}^4 \sigma_b^2 B_{0\parallel}^2
+B_{0\perp}^4 B_{0\parallel}^4\\
&&\mbox{}+ \frac{64}{3} B_{0\perp}^2 \sigma_b ^4 B_{0\parallel}^2
+\frac{8}{3} B_{0\perp}^2 \sigma_b^2 B_{0\parallel}^4\\
&&\mbox{}+\frac{16}{3} \sigma_b^6 B_{0\parallel}^2+\frac{8}{9} \sigma_b ^4 B_{0\parallel}^4
+B_{0\perp}^8+\frac{34 }{3}B_{0\perp}^6 \sigma_b ^2 \\
&&\mbox{}+\frac{109}{3} B_{0\perp}^4 \sigma_b^4 +\frac{104 }{3}B_{0\perp}^2\sigma_b^6
+\frac{56}{9} \sigma_b^8.\end{aligned}$$ [The algebra involved in deriving these relations is rather daunting; we used symbolic algebra software to derive these relations.]{}
We finally have from Eqs. (\[I0e\]), (\[sigmaIe\]) and (\[ncrafull\]): $$I_0
=\frac{n_0 N{{l}}_\varepsilon}{9}
\left[4 C \sigma^{-1}_{B^2}\delta_n\sigma_b^4+6\sigma_b^2
+3\left(4 C \sigma^{-1}_{B^2}\delta_n\sigma_b^2+3\right) B_{0\perp}^2\right],
\label{I0}$$ and $$\begin{aligned}
\sigma_I^2&=\frac{n_0^2 N{{l}}_\varepsilon^2}{81}
\Bigl\{224 C^2 \sigma^{-2}_{B^2} \delta_n^2 \sigma_b^8-81 \delta_n^2 \left(C^2-1\right)B_{0\perp}^4+2\sigma_b ^4
\nonumber\\
&\mbox{}+18\sigma_b^4 \left[51 C^2 \sigma^{-2}_{B^2} \delta_n^2 B_{0\perp}^4+36 C \sigma^{-1}_{B^2} \delta_n
B_{0\perp}^2 -4 \delta_n^2(C^2-1)\right] \nonumber \\
&\mbox{}+12 C^2 B_{0\parallel}^2\frac{\sigma_b^2}{\sigma^{2}_{B^2}} \delta_n^2 \left(9 B_{0\perp}^4
+24 \sigma_b^2 B_{0\perp}^2 +8\sigma_b^4\right)\nonumber\\
&\mbox{}+108B_{0\perp}^2 \sigma_b ^2\left[\left(C \sigma^{-1}_{B^2}\delta_n B_{0\perp}^2+1\right)^2-2\delta_n^2(C^2-1)
\right] \nonumber\\
&\mbox{}+ 144 C \sigma^{-1}_{B^2} \delta_n \left(9C \sigma^{-1}_{B^2} \delta_n B_{0\perp}^2+1\right)\sigma_b^6\Bigr\},
\label{sigmaI}\end{aligned}$$ where $$\sigma_{B^2}^2={\overline{B^4}}-{\overline{B^2}}^2
=B_0^4+\frac{10}{3}B_0^2\sigma_b^2+3\sigma_b^4.
\label{eq:App:sigB2}$$
\[lastpage\]
[^1]: E-mail: rodion@icmm.ru (RS); anvar.shukurov@ncl.ac.uk (AS); andrew.fletcher@ncl.ac.uk (AF); rbeck@mpifr-bonn.mpg.de (RB); laporta@mpifr-bonn.mpg.de (LLP), taba@mpia.de (FT)
[^2]: It is useful to carefully distinguish between what can be called ‘pressure equality’ and ‘pressure equilibrium’: the former refers to the case where magnetic fields and cosmic rays have equal pressures locally, whereas the latter describes the situation where the sum of the two (or more) pressure contributions does not vary in space.
[^3]: The correlation length $l$ is also known as the integral scale and is defined as the integral of the variance-normalized autocorrelation function of a random variable. The typical linear size, or diameter, of a turbulent cell is $2l$.
|
---
abstract: 'We investigate the spectral correlations between different species used to observe molecular clouds. We use hydrodynamic simulations and a full chemical network to study the abundances of over 150 species in typical Milky Way molecular clouds. We perform synthetic observations in order to produce emission maps of a subset of these tracers. We study the effects of different lines of sight and spatial resolution on the emission distribution and perform a robust quantitative comparison of the species to each other. We use the Spectral Correlation Function (SCF), which quantifies the root mean squared difference between spectra separated by some length scale, to characterize the structure of the simulated cloud in position-position-velocity (PPV) space. We predict the observed SCF for a broad range of observational tracers, and thus, identify homologous species. In particular, we show that the pairs C and CO, C$^{+}$ and CN, NH$_3$ and H$_2$CS have very similar SCFs. We measure the SCF slope variation as a function of beam size for all species and demonstrate that the beam size has a distinct effect on different species emission. However, for beams of up to 10”, placing the cloud at 1 kpc, the change is not large enough to move the SCF slopes into different regions of parameter space. The results from this study provide observational guidance for choosing the best tracer to probe various cloud length scales.'
author:
- 'Brandt A.L. Gaches'
- 'Stella S.R. Offner'
- 'Erik W. Rosolowsky'
- 'Thomas G. Bisbas'
bibliography:
- 'converted\_to\_latex.bib'
title: Astrochemical Correlations in Molecular Clouds
---
Introduction
============
In molecular clouds, the largest component of the mass is in the form of molecular hydrogen. However, the lowest transition of H$_2$ is excited at temperatures greater than 500 K, an order of magnitude hotter than typical molecular clouds (10-100 K). Therefore, observational studies of molecular cloud properties and dynamics must use low-abundance tracer species, the most important being CO [see @2006MNRAS.371.1865B; @2011MNRAS.412..337G; @2011MNRAS.412.1686S]. Despite its utility, CO is an imperfect gas tracer in a variety of regimes. It is photo-dissociated at cloud boundaries and near stellar sources. It also becomes optically thick quickly in high density regions. Furthermore, CO suffers greatly from depletion onto dust grains at gas densities greater than $\sim$ 10$^4$ cm$^{-3}$, further reducing its ability to probe high density environments . In contrast, high-density tracers such as N$_2$H$^+$ provide a better means to study the dense gas where CO depletion becomes severe.
Consequently, in order to construct a complete picture of molecular clouds, it is necessary to piece together information from a variety of different tracers and line transitions, each of which is sensitive to different densities, temperatures, and size scales [e.g. @1998ApJ...504..223G]. For diffuse gas, species such as OH$^{+}$ [@2014ApJ...781L...8P] and C$^{+}$ are used. Shock dominated regions are traced by molecules such as SiO [@2014arXiv1402.5066O]. Observers also use a wide array of species, including HCN, NH$_3$, H$_2$CO, H$_2$CS, CS, HCO$^{+}$ and N$_2$H$^{+}$ to study dense environments [@2008ApJS..175..509R; @2011MNRAS.415.1977R; @2012ApJ...756...60S; @2013arXiv1312.1905H; @2014ApJ...780...85V]. However, astrochemistry is highly nonlinear and is sensitive to the gas density, temperature and radiation field. Detailed modelling is necessary to understand the distribution of species and how they are correlated with each other.
Observationally, new instruments make studying many different chemical species much more feasible. In particlar, the wide spectral bandwidth of Atacama Large Millimeter Array (ALMA) allows many different molecular species to be be mapped simultaneously. For example, the ALMA Band 3 contains CN (N=1=0, J=$\frac{1}{2}$-$\frac{1}{2}$), $^{\rm 12}$CO (J=1-0), $^{\rm 13}$CO (J=1-0), HCN (J=1-0), HCO$^+$ (J=1-0), HNC (J=1-0), and N$_2$H$^+$ (J=1-0). These species alone span environments ranging from diffuse, large scale structure ($^{\rm 12}$CO) to dense cores (N$_2$H$^+$). Altogether, these lines provide a rich and detailed view of star forming gas.
However, interpreting this data is not always straight forward. Astrochemistry is a highly nonlinear function of the underlying gas density and temperature. Numerical modelling of the underlying structure and astrochemistry is essential to provide an interpretive framework. There have been numerous recent advancements in astrochemistry codes . Reducing the dimensionality can be helpful for modelling disk or outflow cavities , where the underlying density structure is symmetric is some manner. However, observations show that molecular clouds contain complex density and velocity structure. To account for this, some studies have adopted analytic prescriptions for clumpiness . Such approaches have been able to model the correlation between $^{13}$CO and C, CO emission (the “X-factor") and reproduce observed line profiles , however, they underestimate the influence of cloud morphology in the abundances of species such as C [@2004MNRAS.351..147P; @2014arXiv1403.3530G; @2014MNRAS.tmpL..37O]. A fully three dimensional method for modelling photo-dissociation regions (PDRs) is necessary to represent the complex geometries of molecular clouds. However, a three dimensional time dependent astrochemistry simulation coupled to the hydrodynamics with an extended network is still computationally infeasible.
Many astrochemical studies have focused only on the formation of H$_2$ and CO. These species represent most of the molecular cloud gas and can be modelled with a simple chemical network [see @2010ApJ...716.1191W]. This can reduce the problem of solving thousands of coupled equations to merely dozens. However, as more elements are added to the network, the number of species increases drastically, sometimes with significant effects [@1999RvMP...71..173H]. For instance, the inclusion of Sulfur chemistry leads to the addition of CS, which in turn impacts the abundance of CO. Since CO and CS have similar formation rates in diffuse environments from their atomic constituents, CS will reduce some of the atomic Carbon available for the formation of CO. Likewise, CH$_3$ is a reactant with both H$_2$CS and H$_2$CO. Without the inclusion of Sulfur chemistry, the abundance of H$_2$CO cannot be properly modelled. When studying many species simultaneously, reducing the number of reactions while maintaining accuracy is nearly [*impossible*]{}, requiring a very large network to properly account for the gas phase chemistry alone . Including grain surface chemistry, which requires also modelling the dust distribution and gas-dust surface reactions, further complicates astrochemical studies.
The goal of this paper is to study the distribution of a large variety of species and understand the correlations between them. We model a Milky Way-like molecular cloud using the magnetohydrodynamic code, [orion]{}. We then post processes these results with the 3D astrochemistry code, [3d-pdr]{} [@2012MNRAS.427.2100B], using an extended chemical network to obtain the abundances of over 200 different species. For a subset of this network, we calculate synthetic emission maps of their lowest transition with the radiative transfer code [radmc-3d]{} We compare the spectral structure of all the species to determine the similarity of different species. This is the first study to quantitatively compare the underlying structure of many species. In Section 2, we discuss the models and the methodology we use to generate the emission maps, as well as the way we quantify their spectral structure. In Section 3 we investigate how the following factors affect the spectral structure: viewing angle, species and spatial resolution. Finally, in Section 4 we discuss how the results are relevant to observational studies and general implications.
Methods
=======
Hydrodynamic Simulation {#sec:hydsim}
-----------------------
We use a turbulent hydrodynamic simulation to model a typical Milky Way cloud. The simulation was performed using the Adaptive Mesh Refinement (AMR) code [orion]{} [@truelove98; @klein99]. It was previously discussed in @2013ApJ...770...49O and @2014MNRAS.tmpL..37O as simulation [*Rm6*]{}, so we only briefly summarize its properties here.
The simulation represents a piece of a typical local molecular cloud. The domain is 2 pc on a side and contains 600 $M_{\sun}$. The gas temperature is 10 K and the 1D gas velocity dispersion is 0.72 km s$^{-1}$ such that the cloud satisfies the linewidth-size relation (e.g., ). The gas turbulence is initialized by adding random velocity perturbations with wave numbers $k=1..2$ for two box crossing times without self-gravity.
Once self-gravity is turned on collapse proceeds. The base grid is $256^3$ but four additional AMR levels are inserted to ensure that the gas obeys the Jeans criterion with a Jeans number of 0.125 [@truelove97]. On the highest level ($\Delta x_{\rm min}$=0.001 pc), sink particles (“stars") are added when the gas exceeds the maximum density [@krumholz04]. The output we analyze in this study is at 1 global free fall time at which point $\sim18\%$ of the gas resides in stars.
Astrochemistry {#sec:astrochem}
--------------
The chemical abundances were computed using [3d-pdr]{}, a three-dimensional photodissociation code coupled with a full chemical network [@2012MNRAS.427.2100B]. @2014MNRAS.tmpL..37O presented an analysis of the line emission based upon the abundances of molecular hydrogen, atomic carbon, and carbon monoxide (run Rm6\_1.0\_12\_1f\_a). In this study, we perform an additional analysis of the most common astrochemical species in the chemical network. Here, we describe the astrochemistry calculation procedure and refer the reader to @2012MNRAS.427.2100B and @2014MNRAS.tmpL..37O for additional details.
We irradiated the simulated cloud by an isotropic 1 Draine FUV field, where “1 Draine" is the standard interstellar radiation field [@draine78]. [3d-pdr]{} uses the hydrodynamic densities and assumed external field to compute the temperature and abundance distribution for points in the cloud with densities $200 \leq n \leq 10^5$ cm$^{-3}$. Below $n = 200$ cm$^{-3}$ we consider the gas as ionized, [ using limiting conditions on abundances and gas temperature]{}, whereas above $n = 10^5$ cm$^{-3}$ we assume it is fully molecular. We do not calculate the chemistry in those two regimes. [3d-pdr]{} computes the radiation field using a resolution of 12 [healpix]{} rays [@gorski05], emanating from each grid point. The input grid is the density field of the hydrodynamic calculation resampled to a resolution of $256^3/12$, i.e. a new grid comprised of every 12th data point. (See @2013ApJ...770...49O for a discussion of spatial resolution convergence.)
[3d-pdr]{} employs the UMIST2012 chemical database [@mcelroy13], which includes 215 species and approximately 3000 reactions. The calculation includes the formation of H$_2$ on dust grains following @cazaux02 [@cazaux04], photodissociation of H$_2$ and CO and self-shielding. The initial elemental abundances are \[He\] = $1.0 \times 10^{-1}$, \[C\] $= 1.41 \times 10^{-4}$, \[O\] $= 3.16 \times 10^{-4}$, \[Mg\] $= 5.1 \times10^{-6}$, \[S\] $= 1.4 \times 10^{-6}$, and \[Fe\] $= 3.6 \times 10^{-7}$ , which are similar to values estimated for local molecular clouds. We adopt a cosmic ionization rate of $5\times10^{-17}$ s$^{-1}$, which is the average value found in the Milky Way. [3d-pdr]{} solves the chemical network in equilibrium, with the final time parameter representing the time allowed for the chemistry to come to equilibrium.We analyze the calculation after advancing the chemistry to equilibrium at 10 Myr.
The [3d-pdr]{} calculation does not consider shock chemistry or dust-grain chemistry, which includes the freeze-out of species such as CO onto dust grains, surface reactions (with the exception of H$_2$ formation), or release of grain mantle species into the gas phase by means of evaporation, photodesorption or desorption [e.g. @viti04]. While these processes may impact abundances under certain conditions, we expect them to have a minimal impact on our results. For example, turbulent intermittency in the form of strong, thin shocks within the diffuse gas may enhance the emission and abundance of tracers such as CH$^+$, H$_2$ and CO [@falgarone05; @falgarone09]. However, excitation by FUV photons likely dominates the populations of the lowest energy states, which are what we study here. Dust grain freeze-out primarily affects gas with densities $\geq 10^4$ cm$^{-3}$; here, only 1% of the volume of our model cloud has $n_{\rm H_2} \geq$ 10$^4$ cm$^{-3}$ (see further discussion of the impact of molecular freeze-out in the Appendix). Other grain processes, such as photodesorption, may impact the species abundances over a larger range of densities. Namely, [@guzman13] find that photodesorption is needed to reproduce the measured enhanced abundance of H$_2$CO in the Horsehead PDR region. In contrast, they find that grain-chemistry is not required to model the H$_2$CO abundance within a UV-shielded dense core, which is instead well-fit by a pure gas-phase model. This suggests that the higher UV radiated outer regions of our cloud may require consideration of additional processes, at least with respect to H$_2$CO. However, we note that grain chemistry involves a high-degree of uncertainty and is sensitive to the local UV field, grain properties, and cosmic-ray flux, which make it difficult to apply conclusions from case-studies in the literature to our particular conditions. We quantitatively examine the impact of potential effects of dust-grain chemistry and shock chemistry in the Appendix.
In this study, we essentially assume a “one-way" coupling between the hydrodynamics and chemistry. Performing the chemistry by post-processing the simulations allows us to consider much larger chemical networks that would otherwise be computationally impossible when evolving the chemistry in parallel. However, the chemistry is not coupled to the hydrodynamics, so, although [3d-pdr]{} calculates the gas temperature due to UV heating and cooling, it does not influence the gas evolution, and consequently the hydrodynamics and chemistry are not fully consistent. However, even for the warmer ( $\approx$ 100 K) gas, motions are dominated by turbulence rather than thermal broadening, so we do not expect large differences. For further discussion of the chemistry modelling, see @2013ApJ...770...49O. Even though the chemistry considered here is not time-dependent, the approach exhibits good agreement with @2012MNRAS.421..116G. While their chemistry network is considerably simpler than ours, the equilibrium time scales they find are the same order of magnitude of the times at which we evolve the chemistry to in this study, and are the same order of magnitude of the free fall timescale of the modelled cloud.
Synthetic Emission Maps {#sec:sem}
-----------------------
We carry out synthetic observations for the 16 difference species in Table \[table:trans\] and compare our theoretical results to observations. We study these species because they are commonly used tracers. We use the Leiden Atomic and Molecular Database (LAMBDA) [^1] for the reaction rates and cross sections for the different molecules. When performing these calculations, we use the collisional partners defined in the LAMBDA database files, mainly H$_2$, H and He, assuming most of the H$_2$ is para-H$_2$.
To compute the emission, we use the radiative transfer code [radmc-3d]{} [^2] with the Large Velocity Gradient (LVG) approach [@2011MNRAS.412.1686S], a radiative transfer method that does not assume local thermodynamic equilibrium (LTE). This method computes the molecular level populations given the density, velocity and temperature fields. We use the velocity information from the [orion]{} calculation, while [3d-pdr]{} calculates the temperature and abundance information. Before performing the radiative transfer, we interpolate all data to a 256$^3$ resolution. We calculate the synthetic spectra for velocities within $\pm$ 3 km s$^{-1}$ of the line center. While [3d-pdr]{} computes the level populations for all levels in the LAMBDA data, we only analyze the emission from ground level transitions.
We use a constant microturbulence value of 0.1 km s$^{-1}$ to account for unresolved turbulence. We adopt a “Doppler catching" parameter $d_c = 0.025$, which forces an interpolation of the velocity field between cells if there is a jump greater than 0.025 times the local linewidth. We use collisional excitation and line data from the Leiden atomic database for all 16 species .
Since the main goal is to study the structure of the tracer species, we do not include the dust continuum in the emission calculation. We neglect heating and UV feedback due to embedded protostellar sources. However, since the cloud is forming low-mass stars, radiative feedback likely has a small impact on the emission. We convert the line emission into a brightness temperature $\rm{T_b}$ using the Rayleigh-Jeans approximation: $$T_b = \frac{c^2 I_{\nu}}{2\nu_i^2 k_b}$$ where $I_{\nu}$ is the specific intensity and $\nu_i$ is the frequency of the line transition.
Statistical Analysis: Spectral Correlation Function {#sec:sa}
---------------------------------------------------
In this study, we use the Spectral Correlation Function (hereafter denoted as SCF), first introduced by @1999ApJ...524..887R to study the spectral structure of the emission cubes. We define a position-position-velocity (PPV) cube as spectral line data consisting of two spatial dimensions and one velocity dimension. Likewise, a position-position-position (PPP) cube is data consisting of the density information in all 3 spatial dimensions. The SCF is sensitive to the temperature and sonic Mach numbers and weakly sensitive to the the magnetic field stength [@2003ApJ...588..881P; @2014ApJ...783...93Y]. Previous studies have only focused on the SCF of $^{13}$CO emission, in both simulated and observed molecular clouds. We calculate the SCF for all 16 species emission and density cubes to study how the SCF changes for the same cloud but different observational tracers.
There are several different functional forms for the SCF. We use the form given in @2003ApJ...588..881P, where the SCF is a function of length scale, $l$ and denoted by $S(l)$: $$S(l) = \left < 1 - \left < \sqrt{\frac{\sum_v | O(\textbf{r}, v) - O(\textbf{r + l}, v)|^2}{\sum_v |O(\textbf{r}, v)|^2 + \sum_v |O(\textbf{r+l}, v)|^2}} \right >_{\textbf{r}} \right >_{|\textbf{l}| = l}$$ Here $\textbf{r}$ is a two dimensional position on the image plane, $\textbf{l}$ is the offset vector with length $l$, and $O(\textbf{r}, v)$ is any PPV spectral data set. By definition $S(0) \equiv 1$. The SCF is defined to be bounded between 0 and 1, with 0 indicating no correlation. @2003ApJ...588..881P found that in driven turbulence simulations, the SCF can be analytically fit by a power law on small scales. They fit the SCF for $^{13}$CO (1-0) maps of several observed and simulated molecular clouds and demonstrated that the parameter correlations can be used as theoretical model tests. Likewise, we fit the SCF on small length scales using a power law: $$S(l) = S_0 l^{\alpha}$$ where $S_0$ is the value of the SCF at $l = 1$ pc. We fit the power laws in log space for length scales between 3$\times$ $l_{\rm min}$, where $l_{\rm min}$ corresponds to either the simulation resolution, or the beam size, and $l = $1 pc corresponding to half the maximum length scale. This was to remove beam size effects from the SCF power law fit. Figure \[fig:scfs\] shows the SCF for four different tracers. The tracers follow a tight power law behavior for small values of $l$, as illustrated by the black lines fits. However, Figure \[fig:scfs\] also shows that at some length scale, the SCF function appears to flatten and become noisy.
When comparing SCFs for different spectral maps, it is useful to define a quantitative scalar value that describes how similar two SCFs are to one another. We use the distance metric between two SCFs as defined by @2014ApJ...783...93Y: $$d_{SCF} = \sqrt{\sum_l [S_1(l) - S_2(l)]^2}$$ Using this distance metric provides a quantitative measure of how similar the spectral structures are between different species, resolutions, and sight lines. @2014ApJ...783...93Y showed that this metric is sensitive to global hydrodynamic parameters, although not to the magnetic field strength. Since the distance metric is a 1D statistic we can’t compute a chi-squared value, i.e. a probability measure of uncertainty. In §\[sec:va\] we will describe an example of using the distance metric to quantitatively comparing SCFs.
![\[fig:scfs\] Spectral correlation functions for four different species. The black lines indicate a power law fit.](scfs.pdf){width="\columnwidth"}
Results
=======
Power Law Range {#sec:pl}
---------------
Figure \[fig:scfs\] shows that the SCFs of species which trace dense gas transition away from power law behavior at larger scales and shows that the power law description holds only up to some scale $l \ll L$ where $L$ is the cloud size. At larger length scales, the SCFs start to flatten for all species. For example, the CO SCF is a continuous power-law up to nearly 2 pc, while the SCF for NH$_3$ starts to flatten around 0.4 pc. This flattening indicates where the gas is not dense enough to thermally emit. CO remains correlated at large scales throughout the cloud, with a high surface filling fraction of $f_s = 0.82$, and samples both diffuse and dense structures. CO has an low volume filling fraction, $f_v = 0.07$ since it is still contained mostly in dense environments with emission coming from diffuse regions because of its substantially smaller critical density. The emission from high density regions is constrained by its high optical depth. In our study, the surface filling fraction, $f_s$, is the fraction of the area containing the brightest 95% of the emission. The volume filling fraction, $f_v$, is defined from the 3D abundances, being the fraction of the volume containing the top 95% of the mass. However, NH$_3$ becomes quickly uncorrelated, because it traces only compact emission, i.e. dense cores, illustrated its much smaller volume filling fraction, $f_v \sim 0.005$ and small surface filling fraction $f_s \sim 0.25$.
Figure \[fig:so\] displays the SCF slope as a function of offset, $S_0$ for 16 synthetic emission cubes, where $S_0$ is the value of the SCF at 1 pc. While we define the SCF in terms of a slope and an offset, we show in Figure \[fig:so\] that these two parameters are tightly correlated. Therefore, henceforth, we only discuss the slope of the SCF. @2003ApJ...588..881P also found a strong correlation between the slope and the offset for simulated clouds.
Viewing Angle {#sec:va}
-------------
A statistic describing cloud structure is only meaningful if it does not vary strongly as a function of cloud viewing angle. Here we verify that this is the case.
We calculate the emission for 9 different lines of sights through the cloud in 45 degree increments about the x-axis for all 16 tracers. We find that the SCF slopes change by only $\approx$ 10% over all viewing angles. This variation, though small, is useful as a benchmark to define when two tracers are truly similar. To define an effective uncertainty, we calculate the maximum distance between any two SCFs for all lines of sight. Figure \[fig:dalph\] shows this maximum distance for 6 different species. We find that there is a power law relation between the average slope, $<|\alpha|>$, and the maximum distance, $d_{max}$. This relation now provides a quantitative way of identifying homologous tracers. We define two tracers as “complementary" if their SCF distance for a given line of sight is less than the maximum $d_{\rm{max}}$ of either tracer. Therefore, species $A$ and $B$ are complementary if $$d_{\rm SCF}(B \Leftrightarrow A) \leq {\rm max}(d_{\rm max}(A), d_{\rm max}(B))$$ Figure \[fig:dalph\] also suggests that diffuse species are more likely to have complements because they have shallower slopes. We can see this by comparing the SCFs of CO and NH$_3$. If we fix the CO offset and alter the slope by the maximum $\pm$ 10% error, the total rms difference is geometrically $d_{max}$. Since the SCF slope is shallower for CO, the magnitude of the SCF remains higher. In contrast, if the NH$_3$ SCF is altered in a similar manner, the magnitude of the SCF becomes smaller much faster as a result of the steeper power law. Therefore, the rms difference between the SCFs is much smaller than what it would be for CO. As such, there is an expectation that $d_{max}$ should decrease with increasing slope.
Figure \[fig:dalphC\] shows the SCF slopes for all the species in this study with their line of sight scatter. The species are seemingly separated into three distinct groups, though only two of the groups show a large jump in slope. Based on these groupings, we will refer to [*diffuse*]{} tracers as those with $\alpha \sim -0.3$, [*intermediate*]{} tracers as those with $\alpha \sim -0.5$, and [*dense*]{} tracers as those with $\alpha \sim -0.75$. CO and C both have the same SCF slope within 1 $\sigma$ of each other and within 2 $\sigma$ of all the other diffuse tracers except C$^+$, where $\sigma$ is the line of sight slope error. The larger errors for C$^+$ seem to correspond to different diffuse structures being superimposed at different sight lines giving somewhat different structures. Note that while the species whose SCFs are within the line of sight error limits are homologous (meaning they trace similar density regimes), they are only complementary if they also satisfy the distance metric criteria (Equation 5).
The simulated molecular cloud that we study has no magnetic fields. While @2014ApJ...783...93Y find that the SCF only weakly depends on the magnetic field, the presence of a strong magnetic field could create asymmetries in the gas distribution. Large asymmetries could in turn increase the SCF dependence on the viewing angle.
Chemical Species {#sec:chemspec}
----------------
[ccccccc]{} C & J=1-0 & 492.160651 & 820 & 0.87 & 0.45 &\
C$^{+}$ & J=$\frac{3}{2}-\frac{1}{2}$ & 1900.5369 & 7700 & 0.87 & 0.48 &\
CN & N=1-0, J=$\frac{1}{2}-\frac{1}{2}$ & 113.1686723 & 1.3$\times$10$^{6}$ & 0.87 & 0.14 &\
CO & J=1-0 & 115.2712018 & 2180 & 0.82 & 0.07 &\
CS & J=1-0 & 48.9909549 & 5$\times$10$^{4}$ & 0.73 & 0.05&\
HCN & J=1-0 & 88.6316023 & 1$\times$10$^{6}$ & 0.65 & 0.40 & No hfs\
HCO$^{+}$ & J=1-0 & 89.188523 & 1.6$\times$ 10$^{5}$ & 0.62 & 0.14 &\
HNC & J=1-0 & 90.663568 & 2.7$\times$10$^{5}$ & 0.21 & 0.01 &\
OH & J=$\frac{3}{2}$,F = 1,P = +/- & 1.66 & 2.60 & 0.22 & 0.15 & No hfs\
OH$^{+}$ & N=1-0 & 909.15880 & 4270 & 0.63 & 0.36 &\
H$_2$CO & J=1-0 & 72.837948 & 1.5$\times$10$^{5}$ & 0.41 & 0.03 & p-H$_2$CO, hfs\
H$_2$CS & J=1-0 & 34.3543 & 8400 & 0.24 & 0.006 & p-H$_2$CS\
NH$_3$ & (J,K)=(1,1) & 23.6944955 & 1990 & 0.25 & 0.005 & p-NH$_3$, hfs\
N$_2$H$^{+}$ & J=1-0 & 93.17370 & 1.4$\times$10$^{5}$ & 0.21 & 0.005 & No hfs\
SiO & J=1-0 & 43.42376 & 3.8$\times$10$^{4}$ & 0.51 & 0.005 &\
SO & J=1-0,N=0-1 & 30.00158 & 7.7$\times$10$^{4}$ & 0.28 & 0.005 &
We use 16 common astrophysical tracers to investigate how the SCF depends on species. We generate synthetic emission maps for the lowest energy state transitions shown in Table \[table:trans\]. Figures \[fig:xvsy\] and \[fig:xvsv\] show integrated emission maps in position-position (PP) and position-velocity (PV) space, respectively. The various tracers span different ranges of position and velocity space depending on their abundance and excitation. For example, astronomers commonly use N$_2$H$^{+}$ to trace dense gas, so it is unsurprising that N$^{2}$H$^{+}$ exhibits very compact emission in both figures. The velocity plots indicate structure that may be hidden by projection. Dense cores stand out in both maps.
We find that the filling fractions in PP and PV are similar: tracers with compact spatial emission are also compact in velocity space. This is essentially Larson’s relation [@1981MNRAS.194..809L], which states that small structures should have small velocity extents. CO has a high surface filling factor ($f_s \sim 0.82$) and traces both high and low density regions, with a relatively high abundance across the entire spatial region, exhibited by its spatial emission distribution. CO is chemically connected to the high density tracer H$_2$CO through a number of reactions. H$_2$CO is photodissociated to form CO when the density becomes low enough that it is no longer shielded from the external UV field. CO can get turned into H$_2$CO through intermediaries, such as HCO$^+$ in both gas phase reactions and dust grain chemistry. As expected then, the H$_2$CO emission has a much smaller surface filling fraction of $f_s \sim 0.41$ and is resides mostly in dense environments, requiring a relatively high density to be excited (n$_c \sim 10^5$ cm$^{-3}$).
Optical depth indicates the degree of transparency. Tracers that are optically thin, such as NH$_3$, have emission that reflects the underlying density structure more accurately. Figure \[fig:tau\] shows the optical depth of each line transition at the line center calculated by [radmc-3d]{}. The figure shows that C, CO, CN and C+ all have high optical depths. This is a result of having a lower critical density with a relatively high abundance. The intermediate density tracers, CS, SiO, HCN, HNC and HCO+ are similar, with the gas only being optically thick towards the highest density regions. High density tracers, NH$_3$, H$_2$CO, H$_2$CS and N$_2$H$^+$ remain optically thin throughout almost all of the entire volume. NH$_3$ is an exception because of its fairly low critical density, allowing it to be excited down to lower densities, though still relatively optically thin except in the densest regions of the filaments. OH$^+$ suffers from a very low abundance, and it is not easy to excite, so its emission appears very diffuse.
Figure \[fig:2plot\] displays two different integrated maps with spectra at two different points. It shows that the CO emission is mostly dominated by gas motions. The red star spectrum shows a single feature with a width of $\sim$ 0.7 km s$^{-1}$, which is consistent with the characteristic turbulence velocity. The CO white star spectrum is broad in part because it is a superposition of multiple density features along that particular line of sight. Similarly, the NH$_3$ white star spectrum shows a superposition of several density features at different velocities along the line of sight. However, the NH$_3$ red star spectrum, which is along a rather diffuse line of sight, shows a much narrower feature with a width of $~ 0.3$ km s$^{-1}$. The average sound speed of a cold molecular cloud is approximately 0.2 km s$^{-1}$, indicating that the NH$_3$ in the red star region is undergoing purely thermal motion rather than dominated by turbulence.
Resolution {#sec:res}
----------
Beam resolution is one of the most important factors in observations and it impacts the apparent gas structure and mean optical depth. A larger beam averages out the emission within the beam size, lowering the overall optical depth. We convolve the emission maps with a Gaussian beam to simulate a realistic resolution observation. We place the simulated cloud at a distance of 1 kpc to establish the angular size. Larger beams significantly blend the dense cores and diffuse gas structure. Interestingly, we find that some species’ spectral structures are artificially similar at one resolution and then deviate significantly at some lower resolution. To quantify this, at each resolution we calculate the SCF for each species and then compare their distance metrics. Figure \[fig:sn\] shows the evolution of the SCF slopes for all 16 species as a function of spatial resolution. The overlapping regions indicate where tracers at a particular spatial resolution have similar emission distributions. Note that this only impacts the emission. Therefore, overlapping regions do not indicate similar density distributions, unless the SCF slopes are similar at good (near simulation) resolutions. Thus, Figure \[fig:sn\] quantifies which tracers are statistically similar and useful for studying particular densities and size scales.
At the highest resolution (i.e. no smoothing), C, CO, CN and C$^{+}$ are all very similar tracers since they all trace diffuse gas. The positive correlation between C and $^{12}$CO has been observationally studied for years [e.g. @1999ApJ...512..768P; @1999ApJ...527L..59I; @2002ApJS..139..467I; @2005ApJ...625..194K; @2013ApJ...774L..20S], although historically C was theoretically predicted to only exist in a PDR surface layer [@1999RvMP...71..173H]. @2004MNRAS.351..147P predicted that C should be more prevalent in clouds than previously predicted by 1D models due to a combination of non-equilibrium processes and clumpy cloud morphology. @2014arXiv1403.3530G and @2014MNRAS.tmpL..37O both qualitatively demonstrated the similarity between the C and CO distributions in 3D PDR calculations. Our SCF comparison demonstrates that CO and C spectral cubes are [*quantitatively*]{} similar.
At higher densities, species such as SO and NH$_3$ also appear very similar. As expected, HCN and HNC structures appear nearly identical in spectral space. At lower spatial resolution, several species intersect: NH$_3$, N$_2$H$^{+}$, SO and H$_2$CO. This occurs as larger beams increasingly blend dense, compact emission.
We note that some species have a non-monotonic dependence on resolution where the SCF slope decreases until $\sim$ 15-20” and then increases again. The decrease in the slope is due to overlapping Gaussian structures creating artificial cores. Figure \[fig:deres\] shows the emission spectral structure evolution with beam size visually for NH$_3$ emission in both PP and PV space. Figures \[fig:dE\] and \[fig:dOE\] illustrate these trends as a function of beam size using the distance metric. The change in the slope occurs because species with significant compact emission (those which are generally optically thin) have their emission smoothed on larger scales, making their emission more extended and thus appear similar to other more optically thick species. Smoothing also somewhat affects the more diffuse tracers C, CO, CN and C+, which experience blending on larger scales. This can give the appearance of a more core-like structure where these smoothed regions overlap. At lower density, the emission smooths into am even more diffuse looking component.
Comparison of Density and Emission {#sec:comparisonDE}
----------------------------------
In §3.3 we demonstrate that many tracers produce similar SCFs. However, similarity between the SCFs of two emission maps does not guarantee that the underlying densities are also similar. In this section, we compare the SCFs of the emission and density data directly. We obtain a PPV cube based on the gas density (hereafter denoted by PPV$_{\rho}$) by taking the simulated density and velocity cubes and constructing a PPV cube where each spectral bin contains the number density (calculated as $N_i = N_{\rm H}\cdot n_i$) within a given velocity bin. Here, “V" is the velocity vector projected along a given line of sight. For the PPV$_\rho$ cube, the (i,j,k)$^{\rm th}$ voxel contains the total mass along some line of sight through the point (x$_i$, y$_i$) within a particular line of sight velocity range $\Delta$v$_k$. We compare the emission and density SCFs in Figure \[fig:dOE\]. This figure shows the distance between the emission SCFs and PPV$_{\rho}$ SCFs for each species. For C, CO, CN and C+, PPV$_{\rho}$ does not match the emission structure well. In fact, as the spatial resolution of the PPV cube becomes coarser, agreement between density and emission worsens. High opacity tracers exhibit poor correspondence between emission and density.
In contrast, species with optically thin emission have very similar density and emission SCFs, as indicated by the darker cells in Figure \[fig:dOE\]. For example, CO has a low critical density, $n_c \sim$ 2000 cm$^{-3}$, and a relatively high abundance. It becomes optically thick as the gas density approaches 10$^4$ cm$^{-3}$. Figure \[fig:tau\] demonstrates that CO is very optically thick throughout most of the simulation box. High optical depth effectively flattens the perceived distribution, i.e. as the gas becomes optically thick the emission no longer traces higher density regions. On the other hand, N$_2$H$^{+}$ is optically thin throughout the domain and has a small distance (d $<$ 0.1) between its emission and density SCF. N$_2$H$^+$ has a high critical density of 10$^5$ cm$^{-3}$ and has a much lower abundance than CO.
Differences between the true density and the emission arises from a combination of chemistry, which changes the abundance, and excitation, which impacts the line shape. For example, OH$^+$ has a low critical density of $\sim$ 4000 cm$^{-3}$ but its emission remains optically thin due to its very low abundance. However, HCN is optically thin because it is only excited at gas densities above 10$^6$ cm$^{-3}$ despite its modest abundance: \[HCN\]/\[H$_2$\] $\sim$ 10$^{-7}$.
![\[fig:so\] Slope, $\alpha$, versus offset, $S_0$, for the SCFs calculated for 16 PPV emission maps. The same trend holds for the density data, so we only show the emission parameters.](slope_offset.pdf){width="\columnwidth"}
![ \[fig:dalph\] Maximum SCF distance, $d_{max}$, as a function of the average of the SCF slope magnitude, $<|\alpha|>$, for different tracers (black dots). The blue line indicates a power law fit to the points with a slope of -3.2. The shaded region shows $\pm$ 1 $\sigma$ from the fit.](d_plot2.pdf){width="\columnwidth"}
![ \[fig:dalphC\] Plot of the SCF slopes for all the species used in the study. The blue and gray boxes are the 1 and 2 $\sigma$ errors on the slope, respectively, from line of sight variations. The horizontal grey lines indicate the measured slopes for three common tracers. ](slopesPlot.pdf){width="\columnwidth"}
{width="80.00000%"}
{width="80.00000%"}
{width="80.00000%"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:2plot\] Integrated emission maps (left) and line-of-sight velocity spectra (right) for CO (top) and NH$_3$ (bottom). The white and red star locations represent emission from compact regions and diffuse regions, respectively.](CO_plot.pdf "fig:"){width="\columnwidth"}
![\[fig:2plot\] Integrated emission maps (left) and line-of-sight velocity spectra (right) for CO (top) and NH$_3$ (bottom). The white and red star locations represent emission from compact regions and diffuse regions, respectively.](NH3_plot.pdf "fig:"){width="\columnwidth"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:sn\] SCF slope, $\alpha$, versus beam resolution. We calculate the angular beam size by placing the model cloud at a distance of 1 kpc. The clouds are then convolved with a Gaussian of the given beam size. ](slope_new_T.pdf){width="\columnwidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:deres\] NH$_3$ integrated emission maps for PP (top) and PV (bottom) at different spatial resolutions for a distance of 1 kpc. The beam size appears in the top right.](NH3.pdf "fig:"){width="\columnwidth"}
![\[fig:deres\] NH$_3$ integrated emission maps for PP (top) and PV (bottom) at different spatial resolutions for a distance of 1 kpc. The beam size appears in the top right.](NH3_0.pdf "fig:"){width="\columnwidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
------------------------------------ -------------------------------------
{width="55.00000%"} {width="55.00000%"}
{width="55.00000%"} {width="55.00000%"}
------------------------------------ -------------------------------------
-------------------------------------- ---------------------------------------
{width="55.00000%"} {width="55.00000%"}
{width="55.00000%"} {width="55.00000%"}
-------------------------------------- ---------------------------------------
Discussion
==========
Comparisons with Observations {#sec:compobs}
-----------------------------
[ccc]{} Model & -0.29 & 0.01\
Model (Oph Res) & -0.13 & 0.01\
Model (Per Res) & -0.19 & 0.006\
Ophiuchus & -0.22 & 0.01\
Perseus & -0.21 & 0.007
We now compare our results to observations of two local molecular clouds (MCs), Perseus and Ophiuchus. We use the $^{12}$CO data from the COMPLETE survey [@2006AJ....131.2921R], which were taken using the Five College Radio Astronomy Observatory (FCRAO). The $^{12}$CO observations have an angular resolution of 46$''$. Since the Perseus and Ophiuchus clouds are approximately 250 pc and 150 pc away [@2006AJ....131.2921R] corresponding to physical scales of 0.06 pc and 0.03 pc, respectively, we can compare at the to length scales represented in our simulation. We calculate the $^{12}$CO SCF for the full extent of both clouds. We also divide Perseus into two parts and compute the SCF of each half. Table \[table:obs\] displays the resulting SCF slopes and power law fit errors. We plot the full SCFs in Figure \[fig:comp\].
In order to properly compare the model to the observations, we perform several procedures on the observational data. First, we smooth the data with a Gaussian beam corresponding to the FCRAO 46$''$ beam resolution. Then, we regrid the data so that each pixel has the same spatial pixel scale of the two different clouds. Third, we apply a detection limit where we remove pixels with emission less than $0.01 \times <T_b>$. Finally, we add noise corresponding to Gaussian thermal noise with a standard deviation of $\sigma_T \sim $ 0.3 K to model the noise in the Perseus COMPLETE data. We did not perform biased sampling, which may affect the longest length scales of the SCF due to the irregular observational stencil.
Table \[table:obs\] gives the values of the slopes and the associated error of the power law fit. The model cloud SCF initially has a slope of -0.29. However, after matching to the observational resolution the SCF slope changes by quite a bit to -0.13 and -0.19 for the Ophiuchus and Perseus resolutions, respectively. Furthermore, the fit error indicated in the table shows that the power law fits are all well constrained. Figure \[fig:comp\] shows that the model cloud and the Perseus cloud have similar SCFs when taking into account observational biases, such as beam size and pixel resolution. This is indicated by the similar power law fits shown in the bottom of the figure. The Ophiuchus SCF is steeper than the mode SCF at the Ophiuchus pixel and beam resolution, though it is similar to both the Perseus SCF and the model SCF at Perseus resolution. The flattening of the SCF at small scales is due to the beam resolution. While the emission is still correlated, the turbulence has been resolved out within the beam area. There is still some difference between the simulation and observations which may be due to other biases. However the model and observations match within the 10% error mentioned in §\[sec:va\].
In Figure \[fig:scfs\], the SCFs for all of the tracers appear to flatten out at some large scale, with the exact scale seemingly changing depending on the tracer. @2003ApJ...588..881P also found that their SCFs from simulated clouds traced by $^{13}$CO flatten, but the SCFs for local clouds, also traced by $^{13}$CO, steepen. We were not able to replicate this discrepancy with our simulations, though we conclude that the steepening effect is most likely due to some observational bias at large length scales.
Understanding Chemistry from the SCF {#sec:understand_chem}
------------------------------------
A fundamental question of this work is: what does the SCF reveal about the chemistry of various tracers? Using Figures \[fig:sn\] and \[fig:dalphC\] we group the SCF slopes into three emission categories: diffuse, intermediate, and compact. C is an example diffuse tracer which is excited down to lower densities of $\sim$ 800 cm$^{-3}$ and is abundant throughout the entire volume. HCN is excited at fairly high densities and it becomes quite abundant once it is shielded from the UV field, so it traces up to intermediate scales. A common dense core tracer is N$_2$H$^+$ which also gets excited at higher densities, where it is also shielded from photodissociation. Commonly studied tracers such as CO, C$^{+}$, HCN, NH$_3$ and N$_2$H$^{+}$ confirm that the SCF slope reflects the expected optical depth, filling fraction, and critical density of the emission. While it is impossible to compute the true density distribution from the SCF alone, the slope does indicate the scale the species traces. For instance, shallower slopes, such as that of CO, show that the gas remains correlated on larger scales.
CO is a species of particular astrophysical importance. Over the past several decades, CO has become the most prominently utilized cloud mass tracer and has received significant theoretical attention. Our results show that the SCF of CO traces diffuse regions, as indicated by its shallow slope. Our results also show that CO traces the gas up to high densities, although proper treatment of dust grain depletion could change this. Since CO has a low critical density (see Table \[table:trans\]), the lowest level transition is easily excited throughout most of the cloud. This allows the cloud to produce emission in both the lower density environments and the dense cores (where the emission saturates due to optical depth). For this reason, CO also has a high volume and surface filling fraction. Figure \[fig:tau\] confirms that CO has the highest average optical depth, which is due to its low critical density and higher abundance.
However, the SCF analysis identifies several tracers that exhibit similar emission. We find that C and CN are homologous to CO. The correlations between these species makes chemical sense since all three depend upon the abundance of neutral Carbon. The UMIST chemical network shows that both CO and CN form rapidly in the original diffuse environment. Their behavior at higher densities though is quite different. CN is photodissociated much more rapidly towards lower densities than CO because it cannot self shield. CN is depleted through reactions forming more complex molecules faster than CO. This results in a factor of several orders of magnitude difference between the abundances of the two molecules. Therefore, it is reasonable to expect CO and CN to be very similar in low density environments. In regions of high mass star formation, CN could be used as a proxy for CO, which is optically thick, or of the surrounding diffuse gas.
Another common molecular gas tracer is HCN, which is frequently used to trace gas with densities above 10$^6$ cm$^{-3}$. The SCF slope implies it also traces intermediate density regions in filaments, i.e. $n \sim$ 10$^3$ to 10$^4$ cm$^{-3}$, although, the emission level may be too low to detect. Recent work by @2014arXiv1406.0540F confirms that HCN and HNC are good tracers of dense environments over a wide range of extinctions. The optical depths we calculate for HCN are off by a factor of three since we do not model the HCN fine structure for the ground state transition.
Finally, there are a variety of species that trace dense cores, including NH$_3$ and N$_2$H$^{+}$. Figure \[fig:xvsv\] confirms the trend that species tracing more diffuse regions have shallower slopes than species that typically trace high density regions. Here, the regions the species trace are clear in the velocity information. Diffuse gas tracers have significant emission in a broad range of velocities, as illustrated by the horizontal bands in species CO, C and CN in Figure \[fig:xvsv\]. Species which trace higher densities show no diffuse component. Instead, the emission is located in clumps or filaments, which exhibit a smaller range of velocities.
While the density is important, the gas must also have a high enough temperature to excite the transition. We define a characteristic line temperature T, where $h\nu_i = k T_{i}$, where $\nu_i$ is the line frequency. All the species have line temperatures below 10 K except C, C$^+$ and OH$^+$, which have line temperatures of 23 K, 91 K and 44 K, respectively. The average gas temperature in diffuse regions is around 100 K, so even these species are easily excited from their ground states. However, OH$^+$ is mostly observed in absorption. This is due to its low abundance and higher excitation making measurements from absorption in the dust continuum easier than trying to detect the very small emission signal (as noted by the multiplicative factor of $\sim$ 10$^6$ in Figures \[fig:xvsy\] and \[fig:xvsv\]).
Complex molecules, such as NH$_3$ require higher densities to form, and photodissociate rapidly at lower densities where the UV field is higher. Simple light diatomic molecules such as CN form in diffuse regions. An exception is CS, which appears only in intermediate density regions. It photodissociates faster than CO but slower then CN. However, due to the much lower initial abundance of sulfur, it forms only where the sulfur is concentrated. Most of the tracers in this study that have a shallow slope also tend to have a very high optical depth. The only exception is OH$^{+}$ which has a very low abundance.
Discussion of Observational Implications {#sec:implic}
----------------------------------------
The results from Figures \[fig:sn\] and \[fig:dE\], as well as our definition of complementary species, suggest sets of homologous species. These are groups of species whose spectral structure is very similar in PPV space, indicating that, especially for the optically thin tracers, they should trace similar density regimes.
Recent theoretical studies, such as @2014arXiv1403.3530G and @2014MNRAS.tmpL..37O, found that C is a good alternative tracer to CO. C has several advantages, including a ground state transition at 609 $\rm{\mu m}$ and a lower optical depth than CO. These recent studies challenge the idea that C traces only the surface of the PDR. Our study confirms this picture by showing that C and CO have very similar SCFs, with CO being complementary to C. Our study [*also*]{} predicts that CN is an alternative tracer to CO and C. CN (1-0) has a similar transition frequency with an optical depth around an order of magnitude less than CO, although its slope is slightly flatter indicating that it traces even lower density regions, and it has a similarly high filling fraction. The slightly flatter slope is expected; CN is destroyed faster than CO in higher density environments due to its role as a reactant in reactions forming more complex molecules. Since CN and CO form very quickly and depend on the relatively high-abundance of C, they both have very high filling fractions. Their surface filling fractions are both over 90%, and their volume filling fractions are greater than 0.3. In fact, Table \[table:trans\] shows that CN is both more surface filling and volume filling than CO. The cosmic-ray induced photodissociation rates (i.e. CX + CRPHOT $\rightarrow$ C + X) given by the UMIST2012 network are R$_{\rm{CN}} = 1.4 \times 10^{-13}$ [ s$^{-1}$]{} and R$_{\rm{CO}} = 7.5 \times 10^{-16}$ [ s$^{-1}$]{}. Cosmic rays penetrate further into the cloud than the external UV radiation, indicating the CN will be destroyed by cosmic rays inside the cloud faster than CO. Indeed, CO has an abundance several orders of magnitude greater. Singly ionized carbon traces more diffuse regions but is not as abundant in higher density regions where it combines with O to form CO. The C$^{+}$ (1-0) transition is in the infrared, which can be observed using space-based instruments, such as the Herschel Space Observatory and the GREAT Spectrometer on Stratospheric Observatory for Infrared Astronomy (SOFIA).
A commonly used high density tracer is N$_2$H$^{+}$, which has a ground state transition at 93 GHz. Our study predicts several homologous tracers to N$_2$H$^+$ including: H$_2$CO, H$_2$CS, NH$_3$, and SO. All of these species exist in similar environments and have surface filling fractions between 0.01 and 0.25. However, some of these tracers only form the high density cores which such as N$_2$H$^+$, with a volume filling fraction of 0.21. H$_2$CO has a higher volume filling fraction than N$_2$H$^{+}$ (f$_v$ = 0.37) which indicates that it traces a larger fraction of the core gas. We find that H$_2$CO and SO have higher brightness temperatures indicating that they should be easier to detect than species with fainter emission such as H$_2$CS and N$_2$H$^+$. NH$_3$ and SO show significant emission in filaments, but N$_2$H$^{+}$ is brightest in dense “cores". NH$_3$ has a relatively low critical density (n$_c = 1991$ cm$^{-3}$), so it is excited at lower densities than high densities tracers like N$_2$H$^+$. However, it is not as bright as other low critical density tracers like CO ($n_c \sim 2000$ cm$^{-3}$) because of its low abundance outside of dense regions. Some of these correlations could change with the inclusion of dust grain chemistry. This would lower the abundances of the higher density tracers, such as NH$_3$, H$_2$CO, H$_2$CS and SO, and reduce their emission. However, these molecules only begin to deplete for H$_2$ number densities $\geq$ 10$^7$, greater than the maximum density in this simulation. Detailed treatment of gas-grain chemistry is beyond the scope of this work.
![\[fig:comp\] SCF as a function of size for two clouds in the COMPLETE survey and the model cloud at different resolutions. The dashed-dot red line is the SCF for the Ophiuchus cloud, while the solid blue line is the Perseus cloud. We divide the Perseus cloud into two components at the cloud center (dashed blue lines). The different black line styles indicate different spatial resolutions, where the dotted line represents the simulation resolution, the solid line represents a 46” beam at 250 pc, and the dot-dashed line represents 46” at 125 pc to match with the line styles of the observed clouds. The vertical lines represent the minimum length scales used for the power law fitting, with $\theta$ representing the beam size. The power law fits are shown below, with the line styles matching their corresponding SCF.](scfobs.pdf){width="\columnwidth"}
Conclusions
===========
We use numerical simulations of a Milky Way-like molecular cloud to study how the SCF varies between different species. We post process the hydrodynamical simulation with the astrochemistry code [3d-pdr]{}, which uses a full chemical network to obtain abundances of over 200 different species. We produce synthetic line observations for a subset of these. We calculate the SCF for each of the species for both the density and emission distributions and define a “distance" metric to compare them. On the basis of this, we draw the following conclusions:
1. The SCF is sensitive to the chemistry of the tracer. Species tracing diffuse gas tend to have shallower SCF slopes, whereas species that trace dense regions (“cores”) have steeper slopes.
2. We confirm that the emission structure, as characterized by the SCF, poorly traces the density structure for species with optically thick emission. The decoupling is due to line saturation in the highest density regions.
3. Spatial resolution has a distinct effect on the SCF slope, but even with relatively low beam resolution the slopes remain in a similar region of parameter space. However, for poor resolution, some species have artificially similar SCFs.
4. Velocity resolution has no effect on the SCF slope, since the SCF only measures the rms velocity between spatial regions. This will only hold while the spectral features are resolved, namely that the velocity channels are smaller than the linewidth. Noise variation also has little effect on the slope, since variations cancel as long as the noise is Gaussian.
5. We find that C, C$^{+}$, CN, CO, and OH$^{+}$ are homologous diffuse gas tracers. OH, CS, HCN, HNC and SiO are homologous intermediate gas tracers. Finally, H$_2$CO, H$_2$CS, N$_2$H$^{+}$ and NH$_3$ are homologous dense core tracers when gas phase chemistry is dominant. The statistical similarity of the SCFs suggest that they trace similar cloud structure and likely provide complementary information.
This study takes the first steps in exploring the 3D astrochemical correlations in molecular clouds. In this study, we show that the SCF slope can be used as an indicator of the density of environments where specific species form. This provides insight into the gas chemistry of particular species. Future work can still expand on this in several ways. We do not investigate the SCF evolution as a function of time. Also, we do not include either dust grain chemistry or shock chemistry. Future studies can investigate higher level transitions, such as CS (2-1) and CO (3-2), and isotopologues such as $^{13}$CO.
Chemistry Considerations
========================
In this appendix, we discuss briefly how effects such as dust-grain chemistry and shock chemistry could affect our results.
CO and HCN
----------
Both CO and HCN, as well as other high-density tracers, may be affected by depletion at high-densities. CO depletion is extreme at high-densities with the mean abundances declining by several orders of magnitude . HCN also freezes out, but at higher densities. In order to assess the impact of depletion on our CO and HCN results, we adopt crude treatments for dust freeze-out and recalculate the optical depth and SCF. We set the abundance of CO to 0 where the H$_2$ number density exceeds 10$^4$ cm$^{-3}$. For HCN we set a limit of \[HCN\]/\[H$_2$\] $=$ 10$^{-8}$ in cells where $n_{\rm H_2} \geq 10^4$ cm$^{-3}$. We compare the SCF slopes of the tracers with and without depletion and find that the change in slope is less than 10%. Setting a maximum HCN abundance of 10$^{-8}$ relative to H$_2$ reduced the HCN optical depths to $\tau < 10$ across the entire cube, with the typical optical depth being $\sim$ 1-4. Furthermore, since in these high densities the HCN gas will remain optically thick, it should not drastically impact the SCF correlations. Overall, we conclude that the exclusion of dust depletion does not significantly affect our results due to the small percentage of the simulation volume at high-densities (1%).
N$_2$H$^+$
----------
When CO starts to freeze-out onto dust grains, several molecules, such as N$_2$H$^+$ will see an increase in their abundances. This occurs because H$_3^+$ reacts with CO to form HCO$^+$ which is the main destruction mechanism for H$_3^+$. As the amount of CO decreases there is a surplus of H$_3^+$. In high density environments, H$_3^+$ can be destroyed to form N$_2$H$^+$ [@2001ApJ...552..639A; @2002ApJ...570L.101B; @2004MNRAS.352..600R see] by the following mechanism: $$\textrm{H}_3^+ + \textrm{N}_2 \rightarrow \textrm{N}_2\textrm{H}^+ + \textrm{H}_2$$ In order to test whether this effect has an impact, we adopt a simple approximation in which the amount of N$_2$H$^+$ increases at high densities. Since CO starts to deplete at H$_2$ densities above n$(H_2) \ge 10^4$ cm$^{-3}$, we multiply the N$_2$H$^+$ abundance by a factor of 4 where the gas fits this criteria. Since the emission is mostly optically thin, this merely scales the emission by a constant factor. Similar to the case for CO and HCN depletion, we find that this mechanism has no statistical impact on our results. It is noteworthy that the enhancement of N$_2$H$^+$ is not universal, with studies such as showing either no N$_2$H$^+$ increase or showing N$_2$H$^+$ depletion.
Shock Chemistry: SiO
--------------------
Since our hydrodynamic simulation doesn’t resolve shock fronts to the necessary resolution to calculate post-shock densities and temperatures, well known shock tracers such as SiO will not be properly modelled. In shocks, changes in the density and temperature can lead to a large enhancement of the abundance of SiO. Furthermore, at higher temperatures, the abundance of SiO can be increased by sputtering from dust grains. This mechanism is the ejection of Si and SiO from a grain surface following a high enough energy impact of gaseous species. At lower temperatures, the sputtering rate is expected to be small [@1979ApJ...231...77D]. Even though sputtering from shock chemistry, and other shock effects, will affect the abundance, the inclusion of shock chemistry is beyond the scope of this work. Finally, the increased brightness of the SiO emission is an artifact of using a non-depleted Si abundance, which is over an order of magnitude larger that observed depleted abundances [@1975ApJ...197...85M].
The authors acknowledge support from NASA through Hubble Fellowship grant \# 51311.01 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, INC., for NASA, under contract NAS 5-26555 (SSRO). ER is supported by a Discovery Grant from NSERC of Canada. TGB acknowledges the NORDITA programme on Photo-Evaporation in Astrophysical Systems (2013 June). The ORION and 3D-PDR calculations were per formed on the Trestles XSEDE cluster and DiRAC-II COSMOS supercomputer (ST/J005673/1, ST/H008586/1, ST/K00333X/1), respectively. The authors thank R. Snell, N. Evans and K. Öberg and the anonymous referee for their valuable comments and suggestions that greatly improved this work.
[^1]: [http://home.strw.leidenuniv.nl/ moldata/](http://home.strw.leidenuniv.nl/~moldata/)
[^2]: <http://www.ita.uni-heidelberg.de/dullemond/software/radmc-3d/>
|
---
abstract: 'Recent trend of research indicates that not only massive but also massless (asymptotic Newtonian mass zero) wormholes can reproduce post-merger initial ring-down gravitational waves characteristic of black hole horizon. In the massless case, it is the non-zero charge of other fields, equivalent to what we call here the “Wheelerian mass”, that is responsible for mimicking ring-down quasi-normal modes. In this paper, we enquire whether the same Wheelerian mass can reproduce black hole observables also in an altogether different experiment, viz., the strong field lensing. We examine two classes of massless wormholes, one in the Einstein-Maxwell-Dilaton (EMD) theory and the other in the Einstein-Minimally-coupled-Scalar field (EMS) theory. The observables such as the radius of the shadow, image separation and magnification of the corresponding Wheelerian masses are compared with those of a black hole (idealized SgrA\* chosen for illustration) assuming that the three types of lenses share the same minimum impact parameter and distance from the observer. It turns out that, while the massless EMS wormholes can closely mimic the black hole in terms of strong field lensing observables, the EMD wormholes show considerable differences due to the presence of dilatonic charge. The conclusion is that masslessless alone is enough to closely mimic Schwarzschild black hole strong lensing observables in the EMS theory but not in the other, where extra parameters also influence those observables. The motion of timelike particles is briefly discussed for completeness.'
author:
- 'Ramil N. Izmailov'
- Amrita Bhattacharya
- 'Eduard R. Zhdanov'
- 'Alexander A. Potapov'
- 'K.K. Nandi'
title: 'Can massless wormholes mimic a Schwarzschild black hole in the strong field lensing?'
---
Introduction {#Intro}
============
Wormhole spacetimes predicted by gravitational theories have non-trivial topological structures and their observational aspects can be one of the interesting topics to understand our Universe. Although wormholes require exotic matter and the formation mechanism of wormholes is not well understood, these objects have not been ruled out by observations to date. In fact, recently some remarkable developments have taken place: It has been shown that the gravity waves from a particle near a phantom thin-shell wormhole surgically created by matching two copies of Schwarzschild black hole can mimic the time domain of quasinormal ringing of the Schwarzschild black hole at early times and differ from it only at sufficiently late times [@1; @2]. Such a possibility was previously discussed in [@3] in connection with an analytic wormhole model. For some recent works relating to this wormhole, see [@4; @5; @6]. Ellis-Bronnikov wormhole of the EMS theory[^1] (hereinafter EMS wormhole) have been considered in [@7], who showed that the massless wormhole, depending on the values of its parameters, either rings exclusively as a black hole at all times or rings differently at all times. (By the term massless wormhole, we shall always mean that its asymptotic Newtonian mass of the parent metric is zero). Ringing by *massive* EMS wormhole mimicking Schwarzschild black hole has been analyzed from a different approach in [@8].
An important factor concerning wormholes in general is its stability. While thin-shell wormholes are stable within certain parameter limits, the stability of massless EMS wormhole sourced by ghost field seems to be still an open question: It has been argued that the wormhole is stable under arbitrary space-time perturbations [@9]. It has also been shown that the same wormhole metric can also be generated by a special type of non-ghost equation of state and that the wormhole is stable against spherically symmetric and non-spherical axial perturbations [@10; @11]. However, arguments against the stability of massless EMS wormhole persist showing that the instability leads either to scalar field inflation or to collapse to black hole [@12]. Some authors also argue for instability of the massive EMS wormhole collapsing into a Schwarzschild black hole with the exotic scalar field radiated away [@13]. Mathematically, it means that the massive wormhole parameter $\gamma$ has to drastically cross over from a real value to a complex value ($\gamma =-i$) during collapse to a final black hole [@8]. While the collapse scenario under arbitrary perturbations is not contended, it is unclear what physical phenomenon would mark the cross-over from the real to complex value of $\gamma$ during the evolution of collapse. With all these, it is fair to say that the stability of EMS wormhole, massive or massless, is not yet completely understood for arbitrary perturbations even in the linear approximation, as argued in [@7].
Returning to the main topic of the paper, we note that the gravitational lensing phenomenon is an important tool for understanding the nature of the lens that could be a black hole, wormhole or even a naked singularity. However, there is a fundamental difference between weak and strong field lensing. Strong field light deflection by a Schwarzschild black hole or by a spherically symmetric wormhole displays a logarithmic divergence at the photon sphere [@14; @15], where light rays get captured (making infinite number of loops on it) demarcating the strong field limit for those lenses. There is no way that the strong field deflection angle could be expanded to yield the known weak field deflection terms supposed to be valid far away from the photon sphere. This is so because, as shown in [@16], even the zeroth order affine parametric expansion of the exact deflection angle already shows a logarithmic divergence. On the other hand, while the weak field light deflection is useful in itself, strong field gravitational lensing is expected to provide a completely different genre of observables for different types of lenses. Also, wormholes mimicking post-merger quasi-normal mode ringing [@1; @2; @3; @4; @5; @7; @8] provides a strong motivation for enquiring whether or not massless wormholes of different theories can mimic black hole lensing observables.
Despite having zero asymptotic Newtonian masses, the wormholes are sourced by other types of fields, such as scalar or dilaton having energy, hence an equivalent non-zero mass can be attributed as secondary source that can curve the spacetime causing deflection of light and consequent lensing effects. For instance, the massless EMS wormhole [@17; @18] is sourced entirely by an exotic (negative energy density) scalar field (a.k.a. ghost field), with the wormhole structure consisting of an exterior positive and an interior negative mass, both of equal magnitude symmetrically lying on either side of the throat. We call each of them a *Wheelerian mass* following “Wheeleresque mantra: Mass without mass [@20]” as humorously phrased in [@21]. The positive Wheelerian mass causing the light rays to bend toward the mass eventually leads to observable lensing effects. The existence of such non-zero Wheelerian mass then allows us to compare how the strong lensing properties of massless wormholes in the two theories, EMD and EMS, differ between themselves as well as with those of the Schwarzschild black hole. Wheelerian mass is a useful concept already used in astrophysical application accounting for dark halo objects in the exterior of our galaxy (see [@22]).
The purpose of the present paper is to investigate the strong field lensing observables caused by lightlike motion in the massless EMD and EMS wormholes using Bozza’s method [@14]. We shall enquire if these observables can mimic those of Schwarzschild black hole assuming that the three types of lenses share the same minimum impact parameter $u_{m}$ and distance $D_{\textmd{\scriptsize{OL}}}$ from the observer. The stability of circular orbits of timelike particles is briefly discussed for completeness.
The paper is organized as follows. A brief preview of the method for obtaining strong field lensing observables would be worthwhile and is presented in Sec.2. The method is applied to massless EMD wormhole in Sec.3 and to EMS wormhole in Sec.4. Numerical estimates are presented in Sec.5 and a brief analysis of the motion of timelike particles is added in Sec.6. Sec.7 concludes the paper. We take units such that $8\pi G=1$, $c=1$.
Bozza’s method: A brief preview {#Sec.2}
===============================
This method has by now gained considerable attention for its usefulness, so the purpose of this preview is to let the readers readily see what quantities have been calculated to get to the final lensing observables. The method starts with a generic spherically symmetric static spacetime $$ds^{2}=A(x)dt^{2}-B(x)dx^{2}-C(x)\left( d\theta ^{2}+\sin ^{2}\theta \phi^{2}\right).$$The equation $$\frac{C^{\prime }(x)}{C(x)}=\frac{A^{\prime }(x)}{A(x)}$$is assumed to admit at least one positive root and the largest root is called the radius of the photon sphere $x_{m}$. The strong field expansion will take the photon sphere radius as the starting point, which is required to exceed the horizon radius of a black hole or throat radius of a wormhole as the case may be. A light ray coming in from infinity will reach the closest approach distance $x_{0}$ from the centre of the gravitating source before emerging in another direction. By the conservation of angular momentum, $x_{0}$ is related to the impact parameter $u$ by $$u=\sqrt{\frac{C_{0}}{A_{0}}},$$where the subscript $0$ indicates that the function is evaluated at $x_{0}$. The minimum impact parameter is defined by$$u_{m}=\sqrt{\frac{C_{m}}{A_{m}}},$$where $C_{m}\equiv C(x_{m})$ etc. From the null geodesics, the deflection angle $\alpha (x_{0})$ can then be calculated as a function of the closest approach: $$\alpha (x_{0})=I(x_{0})-\pi,$$ $$I(x_{0})=\int\limits_{x_{0}}^{\infty }\frac{2\sqrt{B}dx}{\sqrt{C}\sqrt{\frac{C}{C_{0}}\frac{A_{0}}{A}-1}}.$$
In the weak field limit of deflection, the integrand is expanded to any order in the gravitational potential and integrated. When we decrease the impact parameter $u$ (and consequently $x_{0}$), the deflection angle increases. Decreasing $u$ further bringing the ray infinitesimally closer to the photon sphere will cause the ray to wind up a large number of times before emerging out. Finally, at $x_{0}=x_{m}$, corresponding to an impact parameter $u=u_{m}$, the deflection angle will diverge and the ray will be captured, i.e., it will wind around the photon sphere indefinitely.
It has been shown that this divergence is logarithmic for all spherically symmetric metrics, which yields an analytical expansion for the deflection angle close to the divergence in the form [@14] $$\alpha (x_{0})=-a\log \left( \frac{x_{0}}{x_{m}}-1\right) +b+O\left(x_{0}-x_{m}\right).$$The coefficients $a,b$ depend on the metric functions evaluated at $x_{m}$, and the Eq.(7) is defined as the *strong field limit* of the light deflection angle. The coefficients $a,b$ are respectively redefined to $\overline{a}$ and $\overline{b}$ that are obtained as follows. Define two new variables $$\begin{aligned}
&&y=A(x), \\
&&z=\frac{y-y_{0}}{1-y_{0}},\end{aligned}$$where $y_{0}=A_{0}$. The integral (6) then becomes $$I(x_{0})=\int\limits_{0}^{1}R(z,x_{0})f(z,x_{0})dz,$$ $$R(z,x_{0})=\frac{2\sqrt{By}}{CA^{\prime }}\left( 1-y_{0}\right) \sqrt{C_{0}},$$ $$f(z,x_{0})=\frac{1}{\sqrt{y_{0}-\left[ \left( 1-y_{0}\right) z+y_{0}\right]\frac{C_{0}}{C}}},$$where all functions without the subscript $0$ are evaluated at $x=A^{-1}\left[ \left( 1-y_{0}\right) z+y_{0}\right] $. The function $R(z,x_{0})$ is regular for all values of $z$ and $x_{0}$, while $f(z,x_{0})$ diverges for $z\rightarrow 0$, where $$f(z,x_{0})\sim f_{0}(z,x_{0})=\frac{1}{\sqrt{\alpha z+\beta z^{2}}},$$ $$\alpha =\frac{1-y_{0}}{C_{0}A_{0}^{\prime }}\left( C_{0}^{\prime}y_{0}-C_{0}A_{0}^{\prime }\right),$$ $$\beta =\frac{\left( 1-y_{0}\right) ^{2}}{2C_{0}^{2}{A_{0}^{\prime }}^{3}}%
\left[ 2C_{0}C_{0}^{\prime }{A_{0}^{\prime }}^{2}+ \left( C_{0}C_{0}^{\prime \prime }-2{C_{0}^{\prime }}^{2}\right)
y_{0}A_{0}^{\prime }-C_{0}C_{0}^{\prime }y_{0}A_{0}^{\prime \prime }\right] ,$$ where primes denote differentiation with respect to $x$.
For the calculation of lensing observables, note that the angular separation of the image from the lens is $\tan \theta = \frac{u}{D_{\textmd{\scriptsize{OL}}}}$, where $D_{\textmd{\scriptsize{OL}}}$ is the distance between the observer and the lens. Specializing to the photon sphere $x_{0}=x_{m}$, the deflection angle in Eq.(7) can be rewritten into a final form $$\begin{aligned}
\alpha (\theta ) &=&-\overline{a}\log \left( \frac{u}{u_{m}}-1\right) +\overline{b}, \\
u &\simeq &\theta D_{\mathrm{OL}}\textmd{ (assuming small }\theta)\end{aligned}$$ $$\overline{a}=\frac{a}{2}=\frac{R(0,x_{m})}{2\sqrt{\beta _{m}}}, \; \beta_{m}=\beta |_{x_{0}=x_{m}},$$ $$\begin{aligned}
\overline{b} &=&-\pi +b_{R}+\overline{a}\log {\frac{2\beta _{m}}{y_{m}}}, \; y_{m}=A(x_{m}), \\
b_{R} &=&\int_{0}^{1}g(z,x_{m})dz, \\
g(z,x_{m}) &=&R(z,x_{m})f(z,x_{m})-R(0,x_{m})f_{0}(z,x_{m}).\end{aligned}$$
The impact parameter $u$ is related to the angular separation $\theta $ of images by the relationship given in Eq.(17). Using this, Bozza proposed three strong lensing observables as [@14] $$\begin{aligned}
\theta _{\infty } &=&\frac{u_{m}}{D_{\textmd{\scriptsize{OL}}}}, \\
s &=&\theta _{1}-\theta _{\infty }=\theta _{\infty }\exp \left( \frac{\bar{b}}{\bar{a}}-\frac{2\pi }{\bar{a}}\right) , \\
r &=&2.5\log _{10}\left[ \ \exp \left( \frac{2\pi }{\bar{a}}\right) \ \right],\end{aligned}$$where $\theta _{1}$ is the angular position of the outermost image, $\theta_{\infty }$ is the asymptotic position approached by a set of images in the limit of a large number of loops the rays make around the photon sphere, $s$ is the angular separation between the outermost images resolved as a single image and the set of other asymptotic images, all packed together. $r$ is ratio between the flux of the first image and the flux coming from all the other images.
We shall calculate in what follows the strong field lensing coefficients $\left\{\overline{a},\overline{b}\right\}$ and observables $\left(\theta_{\infty}, s, r\right)$ applying respectively the formulas (18,19) and (22,23,24) to the massless EMD and EMS wormholes with the understanding, once again, that only the asymptotic Newtonian mass of the parent metric is zero, while the Wheelerian mass is not.
Massless wormhole in EMD theory {#Sec.3}
===============================
The wormhole solution in the EMD theory together with its massless limit has been derived in the literature [@23] and light deflection in it using the Gauss-Bonnet method has been obtained in [@24]. The weak field lensing observables have been recently computed and compared with those of EMS wormhole in [@25]. The massive parent metric (redefining $r=x$ in it [@23]) is:$$\begin{aligned}
A(x)&=& 1/B(x)= \frac{(x-r_{1})(x-r_{2})}{(x+d_{0})(x+d_{1})}, \quad C(x)=(x+d_{0})(x+d_{1}), \\
e^{2\phi} &=&\frac{P(x-\Sigma )}{Q(x+\Sigma )},\quad F_{xt}=\frac{P}{(x+\Sigma )^{2}},\quad F_{\theta \varphi }=P\sin \theta,\end{aligned}$$ where $P$ is the magnetic charge, $Q$ is electric charge, $\Sigma$ is the dilatonic charge and $r_{1},r_{2},d_{0},d_{1}$ are constants. The metric expands in the weak field as $$\begin{aligned}
A(x) &=&1-\frac{2M}{x}+O(1/x^{2}), \\
M &=&\frac{r_{1}+r_{2}+d_{0}+d_{1}}{2}\end{aligned}$$is the asymptotic Newtonian mass, which is assumed to be zero under the choice of constants $r_{1}=-r_{2},d_{0}=-d_{1}$. By further redefining $q^{2}=2PQ$ and $k^{2}=\Sigma ^{2}+q^{2}$, and using certain other conditions (see \[23,24\] for details), the massless EMD wormhole metric components read $$\begin{aligned}
A(x)&=&\left(1+\frac{q^{2}}{x^{2}}\right)^{-1} \\
B(x)&=&\left( 1+\frac{q^{2}}{x^{2}}\right) \left(1+\frac{k^{2}}{x^{2}}\right)^{-1} \\
C(x)&=&x^{2}+q^{2}.\end{aligned}$$ In the “standard” radial coordinate $R=\sqrt{x^{2}+q^{2}}$, the throat appears at $R_{\textmd{\scriptsize{th}}}=q$, which translates to $x_{\textmd{\scriptsize{th}}}=0$. The two functions $R(z,x_{0})$ and $f(z,x_{0})$ work out to $$\begin{aligned}
R(z,x_{m})&=&\left( \sqrt{\frac{2}{1-z^{2}}}\right) \left( \sqrt{\frac{q^{2}(1+z)}{q^{2}(1+z)+k^{2}(1-z)}}\right) , \\
f(z,x_{m})&\sim & f_{0}(z,x_{m})=\frac{1}{\sqrt{\alpha z+\beta z^{2}}},\end{aligned}$$ $$\alpha =1-\frac{2q^{2}}{q^{2}+x_{m}^{2}},$$ $$\beta =\frac{q^{2}}{q^{2}+x_{m}^{2}}.$$The radius $x_{m}$ of the photon sphere can be found from Eq.(2) as (it also follows from $\alpha =0$), $$x_{m}=q\Rightarrow \beta _{m}=\beta |_{q=x_{m}}=\frac{1}{2},\; y_{m}=A(x_{m})=\frac{1}{2},$$and the minimum impact parameter $u_{m}$ follows from Eq.(4) as $$u_{m}=2q.$$To compare the strong field lensing observables of this massless EMD wormhole with those of Schwarzschild black hole, we shall non-dimensionalize the parameters by the Schwarzschild radius $R_{\textmd{\scriptsize{s}}}$ as $q\rightarrow \frac{q}{R_{\textmd{\scriptsize{s}}}}$, $\Sigma \rightarrow \frac{\Sigma}{R_{\textmd{\scriptsize{s}}}}$, $R_{\textmd{\scriptsize{s}}}=2M_{\textmd{\scriptsize{s}}}$, where the subscript “s” stands for Schwarzschild. Then it follows from Eqs.(18,19) that the exact coefficients, with ${\frac{2\beta _{m}}{y_{m}}=2,}
$ are $$\begin{aligned}
\overline{a} &=&\frac{q}{\sqrt{k^{2}+q^{2}}}\Rightarrow \overline{a}=\left(
\frac{q}{R_{\textmd{\scriptsize{s}}}}\right) /\sqrt{\left( \frac{\Sigma }{R_{\textmd{\scriptsize{s}}}}%
\right) ^{2}+2\left( \frac{q}{R_{\textmd{\scriptsize{s}}}}\right) ^{2}} \\
\overline{b} &=&-\pi +b_{R}+\overline{a}\log {2}, \\
g(z,x_{m}) &=&-\frac{2q}{z\sqrt{1-z^{2}}}\left[ \sqrt{\frac{1-z^{2}}{%
k^{2}+q^{2}}}+\sqrt{\frac{1+z}{k^{2}\left( 1-z\right) +q^{2}\left(
1+z\right) }}\right] \\
b_{R} &=&\int_{0}^{1}g(z,x_{m})dz=-\frac{8q\log\left(q\right) }{\sqrt{%
k^{2}+q^{2}}} \\
&=&-\left[ 8\left( \frac{q}{R_{\textmd{\scriptsize{s}}}}\right) \ln \left( \frac{q}{R_{%
\textmd{\scriptsize{s}}}}\right) \right] /\sqrt{\left( \frac{\Sigma }{R_{\textmd{\scriptsize{s}}}}\right)
^{2}+2\left( \frac{q}{R_{\textmd{\scriptsize{s}}}}\right) ^{2}},\end{aligned}$$For a correct comparison, the minimum impact parameter $u_{m}$ of rays in the Schwarzschild black hole and EMD wormhole spacetime should be the same, which implies$$\begin{aligned}
u_{m}^{\textmd{\scriptsize{Sch}}} &=&\left( \frac{3\sqrt{3}}{2}\right) R_{\textmd{\scriptsize{s}}}=\left( 3%
\sqrt{3}\right) M_{\textmd{\scriptsize{s}}}=u_{m}^{\textmd{\scriptsize{EMD}}}=2q \\
&\Rightarrow &\frac{q}{R_{\textmd{\scriptsize{s}}}}=\frac{3\sqrt{3}}{4}.\end{aligned}$$ The last equation yields a formal identification of the Wheelerian mass $q$ with the BH mass $M_{\textmd{\scriptsize{s}}}$ as $$q=\frac{\left( 3\sqrt{3}\right) M_{\textmd{\scriptsize{s}}}}{2}.$$ The only variable in the Eqs.(38,39) now is the adimensionalized dilatonic charge $\frac{\Sigma}{R_{\textmd{\scriptsize{s}}}}$, and by varying it, we shall tabulate below the observables for massless EMD wormhole.
Massless wormhole in EMS theory {#Sec.4}
===============================
This massless wormhole is the zero Newtonian mass limit of the massive EMS wormhole [@17; @18] sourced by an exotic scalar field $\phi$. The parent massive EMS wormhole metric is given by
$$\begin{aligned}
d\tau_{\textmd{\scriptsize{EMS}}}^{2} &=&Adt^{2}-Bdx^{2}-C(d\theta ^{2}+\sin ^{2}\theta d\varphi^{2})], \\
A(x) &=&\exp\left[-\pi\gamma +2\gamma\tan^{-1}\left( \frac{x}{m}\right)\right], \\
B(x) &=&A^{-1}(x),\quad C(x)=B(x)(x^{2}+m^{2}), \\
\phi(x) &=&\pm \kappa\left[\frac{\pi}{2}-2\tan^{-1}\left(\frac{x}{m}\right)\right],\quad 2\kappa^{2}=1+\gamma^{2},\end{aligned}$$
where $x\in (-\infty ,\infty )$, $m$ and $\gamma $ are arbitrary constants. This solution represents a static, horizonless, traversable, everywhere regular massive wormhole that has manifestly two asymptotically flat regions, one with Newtonian positive mass $M$ ($=m\gamma $) and the other with negative mass $-Me^{\pi \gamma }$, on either side of a regular throat at $x_{\textmd{\scriptsize{th}}}=M$ [@17]. The photon sphere has a radius $x_{m}=2M$. The metric (46) is indistinguishable from the Schwarzschild black hole as far as weak field predictions are concerned and that it can mimic the black hole with regard to the ring-down gravitational waves [@8].
For the strong field deflection $\alpha $ in the EMS wormhole metric (46), we find that the integrand $g(z,x_{m})$ has a formidable expression that has to be integrated only numerically. Thus, we can in general write, using $x_{m}=2M=2m\gamma $, $$b_{R}=\int_{0}^{1}g(z,x_{m})dz=\int_{0}^{1}g(z,M,\gamma )dz.$$For the exact but complicated expression of $g(z,M,\gamma )$, see [@8]. It can also be verified for the metric (42) that $$\log \left( \frac{2\beta _{m}}{y_{m}}\right) =\log \left[ \frac{\exp \left[
-4\gamma \tan ^{-1}(2\gamma )\right] \left[ \exp (\pi \gamma )-\exp \left\{
2\gamma \tan ^{-1}(2\gamma )\right\} \right] ^{2}\left[ 1+4\gamma ^{2}\right]
}{2\gamma ^{2}}\right] ,$$The exact Schwarzschild black hole and the strong field coefficients can be deduced by choosing $\gamma =-i$ in Eqs.(18,19,50,51). Carrying out the numerical integration for $b_{R}$ for $\gamma =-i$, we obtain $$\begin{aligned}
\overline{a} &=&1, \\
b_{R} &=&0.9496, \\
\overline{b} &=&-0.4002,\end{aligned}$$which are just the known Schwarzschild values [@14].
The massless EMS wormhole is defined by $M=m\gamma =0$. For this, one option is to choose $\gamma =0$ but $m\neq 0$ so that the metric reduces to $$\begin{aligned}
d\tau ^{2} &=&dt^{2}-dx^{2}-(x^{2}+m^{2})\left( d\theta ^{2}+\sin ^{2}\theta
d\varphi ^{2}\right) , \\
\phi (x) &=&\pm \sqrt{\frac{1}{2}}\left[ \frac{\pi }{2}-2\tan ^{-1}\left(
\frac{x}{m}\right) \right] \simeq \pm \phi _{0}\pm \frac{\sqrt{2}m}{x}%
+O\left( \frac{m^{3}}{x^{3}}\right) ,\end{aligned}$$which has been considered in [@7] for quasi-normal mode ringing. The other option $\gamma \neq 0$, $m=0$ yields trivial flat space and is of no physical consequence in our context.
The weak field deflection of light by this wormhole has been verified by three independent methods in [@26]. The meaning of $\pm m$ as Wheelerian masses, up to an unimportant constant factor of $\sqrt{2}$, is evident from Eq.(56). These masses can also be called scalar charges, since the total integrated energy of the ghost scalar field $\phi$ is $\pm m$. The mass $+m$ is responsible for inward bending of light on the positive side of the massless EMS wormhole. We find that$$\log \left( \frac{2\beta _{m}}{y_{m}}\right) =\log \left( \frac{\pi ^{2}}{2}%
\right),$$and a much simplified expression$$g(z,0,0)=\frac{\pi z-\sqrt{2}\sqrt{1-\cos \left( \pi z\right) }}{z\sin
\left( \frac{\pi z}{2}\right) },$$leading to $$b_{R}=\int_{0}^{1}g(z,0,0)dz=\log (16)-2\log \pi .$$Collecting the values from Eqs.(52), (57) and (59), we find $$\begin{aligned}
\overline{b} &=&-\pi +b_{R}+\overline{a}\log \frac{2\beta _{m}}{y_{m}}=-\pi
+3\log (2), \\
\overline{a} &=&1.\end{aligned}$$Further, since $x_{m}=0$ in the coordinate of the metric (55), it follows from Eq.(4) that$$u_{m}^{\textmd{\scriptsize{EMS}}}=m.$$To compare the strong field lensing observables of the Ellis-Bronnikov wormhole with those of Schwarzschild black hole, we impose that the minimum impact parameters $u_{m}$ be the same for both so that
$$\begin{aligned}
u_{m}^{\textmd{\scriptsize{Sch}}} &=&\left(\frac{3\sqrt{3}}{2}\right) R_{\textmd{\scriptsize{s}}}=\left( 3%
\sqrt{3}\right) M_{\textmd{\scriptsize{s}}}=u_{m}^{\textmd{\scriptsize{EMS}}}=m \\
&\Rightarrow &\frac{m}{R_{\textmd{\scriptsize{s}}}}=\frac{3\sqrt{3}}{2},\end{aligned}$$
which allows us to connect the Wheelerian mass $m$ with the black hole mass $M_{\textmd{\scriptsize{s}}}$ as $$m=\left( 3\sqrt{3}\right) M_{\textmd{\scriptsize{s}}}.$$There is no free variable like dilaton in this case so that the coefficients are fixed. We shall estimate the observables in the ensuing Table.
Numerical estimates {#Sec.5}
===================
For numerical estimates of the strong lensing signatures, we choose as Schwarzschild black hole the SgrA\* residing in our galactic center[^2] but add that any other black hole would be good enough for comparison with wormhole signatures provided they share the same $u_{m}$. If any other black hole is chosen, then only $u_{m}^{\textmd{\scriptsize{Sch}}}/R_{\textmd{\scriptsize{s}}}$ would numerically change as would the corresponding values of $q/R_{\textmd{\scriptsize{s}}}$ for EMD wormhole and $m/R_{\textmd{\scriptsize{s}}}$ for EMS wormhole. We adopt the latest observed data pertaining to SgrA\* from \[28\]: Mass $M_{\textmd{\scriptsize{s}}}=4.2\times 10^{6}M_{\odot }$, $D_{\textmd{\scriptsize{OL}}}=7.6$ kpc, which imply $R_{\textmd{\scriptsize{s}}}=2M_{\textmd{\scriptsize{s}}}$, $u_{m}^{\textmd{\scriptsize{Sch}}}=\left( \frac{3\sqrt{3}}{2}\right) R_{\textmd{\scriptsize{s}}}$, $\theta _{\infty }=\left( \frac{u_{m}^{\textmd{\scriptsize{Sch}}}}{D_{\textmd{\scriptsize{OL}}}}\right) \times 206265\times 10^{6}$ $\mu $arcsec $=28.41$ $\mu $arcsec.
For Ellis-Bronnikov wormhole, $u_{m}^{\textmd{\scriptsize{Sch}}}/R_{\textmd{\scriptsize{s}}}\rightarrow m/R_{\textmd{\scriptsize{s}}}=\frac{3\sqrt{3}}{2}$ \[see Eqs.(64,65)\] so that the deflection angle is $$\alpha (\Delta )=-\overline{a}\log \left( \frac{\Delta }{m/R_{\textmd{\scriptsize{s}}}}\right) +\overline{b},$$where $\Delta =\left( u-u_{m}\right) /R_{\textmd{\scriptsize{s}}}$ represents the proximity of rays to the photon sphere (hence we call it the “proximity parameter”) and ($\bar{a}$, $\overline{b}$) given by Eqs.(60,61).
For EMD wormhole, $u_{m}^{\textmd{\scriptsize{Sch}}}/R_{\textmd{\scriptsize{s}}}\rightarrow q/R_{\textmd{\scriptsize{s}}}=\frac{%
3\sqrt{3}}{4}$ \[see Eq.(43,44)\] so that $$\alpha (\Delta ,\Sigma /R_{\textmd{\scriptsize{s}}})=-\overline{a}\log \left( \frac{\Delta
}{q/R_{\textmd{\scriptsize{s}}}}\right) +\overline{b}\left( q/R_{\textmd{\scriptsize{s}}},\Sigma /R_{\textmd{\scriptsize{s}}}\right) ,$$
The table below shows comparison of strong lensing coefficients ($\bar{a}$, $%
\overline{b}$) and observables ($\theta _{\infty },s,r$) of wormhole lenses composed of Wheelerian mass vis-à-vis SgrA\* black hole lens of the same minimum impact parameter $u_{m}$ as stipulated. To calculate the relevant quantities, the algorithm is as follows: First we consider EMS wormholes, both massive and massless. Putting the $x_{m}$ of wormholes in the master Eqs.(18,19), we tabulate ($\bar{a}$, $\overline{b}$) in the 3rd and 4th rows of the Table. Second, putting the adimensionalized Wheelerian mass $m/R_{\textmd{\scriptsize{s}}}=%
\frac{3\sqrt{3}}{2}$ in the expression for $\alpha $, we end up with the remaining unknown quantity, the proximity parameter $\Delta $, in Eq.(66). To fix it next, we note that the outermost image appears where $%
\alpha $ just equals $2\pi$,$$\alpha (\Delta )-2\pi =0.$$
This equation is solved to find $\Delta $, which tells us how close the rays should pass by the photon sphere in order to enable the wormholes to mimic the deflection angle $2\pi $ caused by a black hole. However, the observables ($\theta _{\infty },s,r$), determined solely by $\bar{a}$ and $%
\overline{b}$, do not show much difference from those of black hole except a minor one in $s$. This means that the EMS wormholes, massive or massless, can closely mimic a black hole in terms of strong field lensing observables.
The story with the EMD wormhole is quite different since there is now a freely specifiable dilatonic charge $\Sigma /R_{\textmd{\scriptsize{s}}}$. Specifying different values to it, we find that the equation, with the Wheelerian mass $q/R_{\textmd{\scriptsize{s}}}=\frac{3\sqrt{3}}{4}$, viz., $$\alpha (\Delta ,\Sigma /R_{\textmd{\scriptsize{s}}})-2\pi =0$$corresponds to different sets $\Delta $ and of ($\theta _{\infty },s,r$) for EMD wormholes, the difference signalling the presence of dilatonic charge in the lens. These sets of values are markedly different from those of EMS wormhole and black hole. The near zero values of $s$ for the EMD wormhole show that the set of secondary asymptotic images will merge with the outermost image producing a single image of the source, with the characteristics that the flux ratio $r$ gradually increases as the parameter $\Delta $ decreases or equivalently as the rays pass gradually closer to the photon sphere.
=0.1cm
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Lens $u_{m}/R_{s}$ - $\Delta$ $\bar{a}$ $\bar{b}$ $% $s$ ($\mu$as) $r$(mag)
\theta_{\infty}$ ($\mu$as)
----------- --------------------------------- --------------------------------------- ----------------------- ----------- ----------- ---------------------------- ---------------------- ----------
SgrA\* BH $\frac{3\sqrt{3}}{2}$ - $3.25\times 10^{-3}$ $1$ $-0.40$ $28.41$ $0.035$ $6.82$
Massive $\frac{3\sqrt{3}}{2}$ - $2.66\times 10^{-3}$ $1$ $-0.60$ $28.41$ $0.029$ $6.82$
EMS WH
EMS WH $\frac{3\sqrt{3}}{2}$ - $1.68\times 10^{-3}$ $1$ $-1.06$ $% $0.018$ $6.82$
28.41$
EMD WH $q/R_{\textmd{\scriptsize{s}}}$ $\Sigma /R_{\textmd{\scriptsize{s}}}$ - - - - - -
- $\frac{3\sqrt{3}}{4}$ $\pm 10$ $2.95\times 10^{-33}$ $0.12$ $% $14.20$ $3.2\times 10^{-32}$ $4.23$
-3.32$
- - $\pm 5$ $5.26\times 10^{-18}$ $0.24$ $-3.48$ $14.20$ $% $3.53$
5.7\times 10^{-17}$
- - $\pm 1$ $8.22\times 10^{-8}$ $0.62$ $-4.01$ $14.20$ $% $2.51$
9.0\times 10^{-7}$
- - $\pm 0.5$ $3.21\times 10^{-7}$ $0.68$ $-4.09$ $14.20$ $% $2.41$
3.5\times 10^{-6}$
- - $\pm 0.1$ $5.11\times 10^{-7}$ $0.70$ $-4.13$ $14.20$ $% $2.37$
5.6\times 10^{-6}$
- - $\pm 0.0$ $5.21\times 10^{-7}$ $0.71$ $-4.13$ $14.20$ $% $2.37$
5.7\times 10^{-6}$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: []{data-label="Tab1"}
Motion of timelike particles: stability of circular motion {#Sec.5}
==========================================================
Although the main focus in this article has been the strong field lensing observables caused by lightlike particles (rest mass zero), we shall for completeness briefly touch upon the motion of timelike particles (rest mass nonzero)[^3]. In particular, we shall examine here the location of two asymptotically flat regions as required of a wormhole geometry and the existence of stable circular orbits around the positive mass side.
*(a) EMD theory*
We start with the massless metric (29-31). To establish that it truly represents a wormhole, we have to show that it has two asymptotically flat regions on either side of the throat at $x_{\textmd{\scriptsize{th}}}=0.$ For convenience, we put the metric (29-31) in the isotropic form ($t,R,\theta ,\varphi $) by introducing the transformation $$x=\frac{R^{2}-k^{2}}{2R},$$which when inverted yields the “isotropic” throat radius $R_{\textmd{\scriptsize{th}}}=%
\frac{1}{2}x\left( 1\pm \sqrt{1+k^{2}/x^{2}}\right) \rightarrow k$ as $%
x\rightarrow x_{\textmd{\scriptsize{th}}}=0$. Discarding the negative sign, we find $\frac{R%
}{x}\rightarrow 1$ as $x\rightarrow \infty $, so at large distances $R$ and $%
x$ coincide. The metric (29-31) under the transformation (70) assumes an isotropic form$$\begin{aligned}
d\tau ^{2} &=&A(R)dt^{2}-B(R)\left( dR^{2}+R^{2}d\theta ^{2}+\sin ^{2}\theta
d\varphi ^{2}\right) \nonumber \\
&=&\left[ \frac{1}{1+\frac{4R^{2}q^{2}}{\left( R^{2}-k^{2}\right) ^{2}}}%
\right] dt^{2} \nonumber \\
&&-\left[ \frac{k^{4}+4q^{2}R^{2}-2k^{2}R^{2}+R^{4}}{4R^{4}}\right] \left(
dR^{2}+R^{2}d\theta ^{2}+R^{2}\sin ^{2}\theta d\varphi ^{2}\right) .\end{aligned}$$This metric is flat as $R\rightarrow \infty $ and is also invariant under inversion: $R=\frac{k^{2}}{y}$, which implies the presence of another asymptotically flat region at $y=0$. So, the metric form (71) represents a twice asymptotically flat regular wormhole as the spacetime on either side of the throat shows no curvature divergence at any point. Now redefine $R=2%
\overline{R}$ so that $$d\tau ^{2}=A(\overline{R})dt^{2}-B(\overline{R})\left( d\overline{R}^{2}+%
\overline{R}^{2}d\theta ^{2}+\overline{R}^{2}\sin ^{2}\theta d\varphi
^{2}\right) .$$Then the metric functions expand as$$A(\overline{R})=1-\frac{q^{2}}{\overline{R}^{2}}-\frac{q^{4}}{\overline{R}%
^{4}}\left( \frac{k^{2}}{2q^{2}}\right) +...$$
$$B(\overline{R})=1+\frac{q^{2}}{\overline{R}^{2}}\left( 1-\frac{k^{2}}{2q^{2}}%
\right) +\frac{q^{4}}{\overline{R}^{4}}\left( \frac{k^{4}}{16q^{4}}\right)
+...$$
Timelike test particles, with arbitrarily controllable parameters like energy and angular momentum, are not relevant at least for lensing observations. To understand what those particles would nevertheless see as the “effective gravitating mass” of the massless wormhole, it would be necessary to study the orbital precession of timelike test particles in both the spacetimes. But if *weak field* light deflection angle $\alpha $ is any indication, then note that the leading order deflection by the massless EMD wormhole obtained in [@24] using the Gauss-Bonnet method: $$\alpha (b)=\frac{\pi }{4b^{2}}\left( 3q^{2}-\Sigma ^{2}\right) ,$$which reveals, following Schwarzschild formula, that the effective gravitating mass is $M_{0}=\sqrt{3q^{2}-\Sigma ^{2}}$, and not merely $q$. On the other hand, it can be verified that the PPN method of Keeton & Petters [@31] for the metric functions (73) and (74) using the gravitational potential $\Phi =\frac{q}{\overline{R}}$ (and not $\Phi =\frac{M_{0}}{%
\overline{R}}$) also yield the same deflection (75)[^4] although the first order deflection term is absent anyway! So what is the mass seen, $q$ or $M_{0}$? When dilaton is switched off, $\Sigma =0$, the metric (29-31) reduces to the famous Einstein-Rosen bridge [@32] and in this case, the mass is proportional to just $q$. While the intriguing question raised above warrants a separate detailed investigation, it is expected that both the parameters $q$ and $\Sigma $ would influence the motion of timelike particles. A good example for demonstrating it is to investigate the stability of circular orbits of timelike particles.
To this end, it is convenient to define $x^{2}+q^{2}=\rho ^{2}$ in metric (29-31) so that the metric functions assume the “standard” coordinate Morris-Thorne form [@33] $$\begin{aligned}
d\tau ^{2} &=&A(\rho )dt^{2}-B(\rho )d\rho ^{2}-C(\rho )\left( d\theta
^{2}+\sin ^{2}\theta d\varphi ^{2}\right) , \\
A(\rho ) &=&1-\frac{q^{2}}{\rho ^{2}},B(\rho )=\frac{\rho ^{4}}{\left( \rho
^{2}-q^{2}\right) \left( \rho ^{2}+\Sigma ^{2}\right) },C(\rho )=\rho ^{2}.\end{aligned}$$Henceforth, define affine parameter $\lambda $, use the four-velocity $%
U^{\mu}=\frac{dx^{\mu}}{d\lambda}$, and normalize the constants of motion by $\epsilon $, viz., $E=U_{0}/\epsilon $, $L=U_{3}/\epsilon $. The geodesic motion on the equatorial plane ($\theta =\pi /2$) has the equations $$\begin{aligned}
A(\rho )\frac{dt}{d\lambda } &=&E,\rho ^{2}\frac{d\varphi }{d\lambda }=L, \\
g_{\mu \mu }U^{\mu }U^{\nu } &=&-\epsilon ^{2},\end{aligned}$$where $\epsilon $ is the conserved rest mass of the test particle, $\epsilon
=1$ for timelike and $0$ for lightlike particles. The geodesic Eq.(79) can be recast in the form$$\begin{aligned}
\frac{1}{2}\left( \frac{d\rho }{d\lambda }\right) ^{2}+V(\rho ) &=&\frac{%
E^{2}}{2}, \\
V(\rho ) &=&\frac{\epsilon ^{2}}{2}+\frac{L^{2}-q^{2}\epsilon ^{2}+\Sigma
^{2}\left( \epsilon ^{2}-E^{2}\right) }{2\rho ^{2}} \nonumber \\
&&+\frac{L^{2}(\Sigma ^{2}-q^{2})-q^{2}\epsilon ^{2}\Sigma ^{2}}{2\rho ^{4}}-%
\frac{L^{2}q^{2}\Sigma ^{2}}{2\rho ^{6}}.\end{aligned}$$
With the potential $V(\rho )$ at hand, the rest of the algorithm is quite well known. At the constant critical radius $\rho =\rho _{c}$, one has $%
\left. \frac{d\rho }{d\lambda }\right\vert _{\rho =\rho _{c}}=0$ and $\left.
\frac{dV}{d\rho }\right\vert _{\rho =\rho _{c}}=0$, which provide $\rho
=\rho _{c}(q,\Sigma ,E,L)$ for $\epsilon =1$. If there is a stable circular timelike orbit at $\rho =\rho _{c}$, then we should find $V^{\prime \prime
}(q,\Sigma ,E,L)\equiv \left. \frac{d^{2}V}{d\rho ^{2}}\right\vert _{\rho
=\rho _{c}}<0$. The expressions for $\rho _{c}$ and $V^{\prime \prime }$ are rather large, hence omitted but what we find is that, for given spacetime parameter values $\left( q_{0},\Sigma _{0}\right) $, and for *some choices* of orbital angular momentum $L_{0}$ (all in suitable units), circular orbits with radii $\rho =\rho _{c}$ show $V^{\prime \prime }(\rho
_{c})<0$ for all values of energy $E$. Two typical Figs.1 and 2 are exhibited for illustration.
*(b) EMS theory*
We start with the massless wormhole metric (55) that is covered on two sides by the coordinate ranges $\infty <t<\infty $, $-\infty <x<\infty $, $0\leq
\theta \leq \pi $, $0\leq \varphi \leq 2\pi $. The metric is twice asymptotically flat at $x\rightarrow \pm \infty $ since the curvatures vanish there. For example, in the orthonormal frame () of an observer, the Riemann curvature components are$$R_{\hat{\theta}\hat{\varphi}\hat{\theta}\hat{\varphi}} = -R_{\hat{x}\hat{\theta}\hat{x}\hat{\theta}} = -R_{\hat{x}\hat{\varphi}\hat{x}\hat{\varphi}} = \frac{m^{2}}{\left(x^{2}+m^{2}\right)^{2}},$$which $\rightarrow 0$ as $x\rightarrow \pm \infty $. For the present purpose, we cast the metric (55) in the Morris-Thorne “standard” form [@33] using $x^{2}+m^{2}=r^{2}$, so that the metric on one side of the wormhole manifold becomes ($-\infty <t<\infty $, $r>m$, $0\leq \theta \leq \pi $, $%
0\leq \varphi \leq 2\pi $):$$\begin{aligned}
d\tau ^{2} &=&A(r)dt^{2}-B(r)dr^{2}-C(r)\left( d\theta ^{2}+\sin ^{2}\theta
d\varphi ^{2}\right) , \\
A(r) &=&1,B(r)=\frac{1}{1-m^{2}/r^{2}},C(r)=r^{2}.\end{aligned}$$The weak field light deflection angle $\alpha $, to leading order in the massless EMS wormhole, is [@26]:$$\alpha (b)=\frac{\pi m^{2}}{4b^{2}},$$which reveals, following Schwarzschild formula, that the “effective gravitating mass” is just $m$. The orbital precession of timelike particles should also involve only $m$, as there are no other extra solution parameters.
Proceeding exactly as in (a), we find for timelike particles ( $\epsilon =1$) the potential $$V(r)=\frac{1}{2}+\frac{L^{2}+m^{2}\left( E^{2}-1\right) }{2r^{2}}-\frac{%
L^{2}m^{2}}{2r^{4}},$$which yields the critical circular radius $r_{c}$ and $V^{\prime \prime
}(r_{c})$: $$\begin{aligned}
r_{c} &=&\frac{\sqrt{2}Lm}{\sqrt{L^{2}+E^{2}m^{2}-m^{2}}} \\
V^{\prime \prime }(r_{c}) &=&-\frac{\left[ L^{2}-m^{2}\left( 1-E^{2}\right) %
\right] ^{3}}{2L^{4}m^{4}}.\end{aligned}$$$V^{\prime \prime }(r_{c})<0$ implies the stability condition$$L^{2}-m^{2}\left( 1-E^{2}\right) >0.$$Thus stability is not decided by the mass $m$ alone, but also by the orbital parameters $L,E$ of timelike particles that are to be so adjusted as to satisfy the above inequality.
Conclusion {#Sec.6}
==========
Throughout the paper, by the term massless, we mean that only the asymptotic Newtonian mass is zero but the Wheelerian mass is not, so it can cause observable lensing effects and stable circular orbits for timelike particles. Since the nature of the mass of a black hole is not yet known, we speculated that the same mass could as well be constituted by scalar/dilaton/eletromagnetic fields. Although massless EMS wormhole has been studied in the literature for its non-trivial lensing properties in astrophysical applications, it is scarcely explained why, despite being massless, it still bends light and exhibits observable lensing properties.
The present paper attempts to provide an explanation by invoking the Wheelerian mass inherent in the massless wormholes and shows how its* strong field lensing* properties compare with those of Schwarzschild black hole, when both masses are quantitatively the same. Thus, for the massless EMS wormhole, the asymptotic masses on either side, viz., $M$ ($=m\gamma )$ and $-Me^{\pi \gamma }$, could be zero for $\gamma =0$, but the Wheelerian masses $\pm m$ need not be zero. The mass $+m$ is responsible for the inward deflection of light and observable lensing effects as well as for the motion of timelike particles. Likewise, the Wheelerian mass $q$ of the massless EMD wormhole is provided by electric charge $P$ and magnetic charge $Q$ in a combined form $q=\sqrt{2PQ}$ that is perceived in the observation of angular radius of the shadow $\theta _{\infty }$, while the dilatonic charge $\Sigma
$ remains a free parameter. *Our conclusion is that, while the massless EMS wormhole can closely mimic a black hole (SgrA\* chosen for illustration) in terms of strong field lensing observables, the massless EMD wormhole shows considerable deviation due to the presence of dilatonic charge* $\Sigma $*. Additionally, we showed that stable circular orbits of timelike test particles exist in both massless spacetimes depending on the choice of the orbital parameters.*
In a little more detail, we devised a proximity parameter $\Delta $, which tells us how close the rays should pass by the photon sphere in order to enable the wormholes to mimic the deflection angle $2\pi $ by a black hole producing the outermost image, while the lensing observables ($\theta
_{\infty },s,r$) are determined exclusively by $\bar{a}$ and $\overline{b}$. These observables are entered in the 4th row of the table, which show that they are the same as those of black hole SgrA\* except for a slight difference in $s$.
The EMD wormhole strong field lensing observables differ quite significantly from those of SgrA\* black hole depending on the value of the freely specifiable dilatonic charge $\Sigma $. Its presence in the lens renders the values of image separation $s$ to be near zero as is evident from the table. It indicates that the set of secondary asymptotic images will merge with the outermost image producing a single image of the source, with the characteristics that the flux ratio $r$ gradually increases as the parameter $\Delta $ decreases or equivalently as the rays pass gradually closer to the photon sphere. This feature seems to be a fundamental characteristic of EMD wormhole that could constitute a potential test of string theory.
The simplest strong field observable is the angular radius of the shadow $%
\theta _{\infty }$ [@29], which alone can distinguish between massless EMS and EMD wormholes since $\theta _{\infty }^{\textmd{\scriptsize{EMS wormhole}}}=2\theta
_{\infty }^{\textmd{\scriptsize{EMD wormhole}}}$. Remarkably, our table (2nd and 3rd rows) shows that the measurement of $\theta _{\infty }$ cannot distinguish between the black hole and a *massive* EMS wormhole at the level of current technology. A $\mu $as resolution is reachable in the next years by Very Long Baseline Interferometry projects although there are lots of challenges that would make the identification of the relativistic images very difficult, as already enumerated in [@30]. Ongoing Event Horizon Telescope project aims to achieve an accuracy of $\sim 15$ $\mu $as [@29], obviously still falling far short of the accuracy needed.
For completeness, we devoted Sec.6 to the study of timelike particles and argued that stable circular orbits exist provided the orbital parameters are suitably adjusted so as to satisfy the stability condition $V^{\prime \prime
}<0$. In addition, we had shown the EMD metric (71) is twice asymptotically flat as required of a wormhole: One flatness is located at $R\rightarrow
\infty $ and the metric is also invariant under inversion: $R=\frac{k^{2}}{y}
$, which implies the presence of another asymptotically flat region at $y=0$. The asymptotic expansion of the metric is shown in Eqs.(73,74). Similar properties of the EMS metric (55) are discussed in Sec.6(b). To answer what the timelike test particles would see as the gravitating mass, it would be more approriate to study the orbital precession of timelike test particles in both the massless spacetimes in the potentials (81) and (86). Work is underway.
Acknowledgement {#acknowledgement .unnumbered}
===============
The reported study was funded by RFBR according to the research project No. 18-32-00377.
[99]{} V. Cardoso, E. Franzin and P. Pani, Phys. Rev. Lett. **116**, 171101 (2016); **117**, 089902(E) (2016). V. Cardoso and P. Pani, Nature Astronomy **1**, 586 (2017). T. Damour and S.N. Soludukhin, Phys. Rev. D **76**, 024016 (2007). S.H. Völkel and K.D. Kokkotas, Class. Quantum Grav. **35**, 105018 (2018). P. Bueno, P.A. Cano, F. Goelen, T. Hertog and B. Vercnocke, Phys. Rev. D **97**, 024040 (2018). K.K. Nandi, R.N. Izmailov, E.R. Zhdanov and Amrita Bhattacharya, JCAP 07 (2018) 027. H.G. Ellis, J. Math. Phys.** 14**, 104 (1973); **15**, 520E (1974). K.A. Bronnikov, Acta Phys. Polon. B **4**, 251 (1973). K.K. Nandi, I. Nigmatzyanov, R.N. Izmailov and N.G. Migranov, Class. Quantum Grav. **25**, 165020 (2008). R.A. Konoplya and A. Zhidenko, JCAP 12 (2016) 043. K.K. Nandi, R.N. Izmailov, A.A. Yanbekov and A.A. Shayakhmetov, Phys. Rev. D **95**, 104011 (2017). C. Armendáriz-Pícon, Phys. Rev. D **65**, 104010 (2002). I.D. Novikov and A.A. Shatskiy, JETP, 114, 801 (2012). K.A. Bronnikov, L.N. Lipatova, I.D. Novikov and A.A. Shatskiy, Grav. & Cosmol. **19**, 269 (2013). H.-a. Shinkai and S.A. Hayward, Phys. Rev. D **66**, 044005 (2002). J.A. Gonzalez, F.S. Guzman and O. Sarbach, Class. Quantum Grav. **26**, 015010 (2009). V. Bozza, Phys. Rev. D **66**, 103001 (2002). K.K. Nandi, Y.-Z. Zhang and A.V. Zakharov, Phys. Rev. D **74**, 024020 (2006). S.V. Iyer and E.C. Hansen, Phys. Rev. D **80**, 124023 (2009). J.A. Wheeler, Phys. Rev.** 48**, 73 (1957). M. Visser, *Lorentzian Wormholes-From Einstein To Hawking* (AIP, New York, 1995). F. Abe, Astrophys. J. **725**, 787 (2010). P. Goulart, arXiv:1611.03164. K. Jusufi, A. Övgün and A. Banerjee, Phys. Rev. D **96**, 084036 (2017). R.F. Lukmanova, G.Y. Tuleganova, R.N. Izmailov and K.K. Nandi, Phys. Rev. D **97**, 124027 (2018). A. Bhattacharya and A.A. Potapov, Mod. Phys. Lett. A **25**, 2399 (2010). V. Bozza and L. Mancini, Astrophys. J. **753**, 56 (2012). Y. Kato, M. Miyoshi, R. Takahashi, H. Negoro and R. Matsumoto, Mon. Not. R. Astron. Soc. **403**, L74 (2010). T. Lacroix and J. Silk, Astron. Astrophys. 554, A 36 (2013). K.S. Virbhadra and G.F.R. Ellis, Phys. Rev. D **62**, 084003 (2000). C.R. Keeton and A.O. Petters, Phys. Rev. D **72**, 104006 (2005). A. Einstein and N. Rosen, Phys. Rev. **48**, 73 (1935). M.S. Morris and K.S. Thorne, Am. J. Phys. **56**, 395 (1988).
[^1]: It should be mentioned here that Ellis [@17] derived two classes of asymptotically flat solutions, one showing naked singularity and the other a regular massive traversable wormhole, which we call here the massive Ellis-Bronnikov wormhole since Bronnikov [@18] independently derived it \[see the metric (42-45) below\]. Interestingly, the two Ellis classes are not really independent but can be connected by a complex Wick rotation, as shown in [@19].
[^2]: Strictly speaking, SgrA\* black hole has spin and for comparison of observables, the appropriate wormhole examples should also be the spinning ones. However, choosing SgrA\* is not mandatory for our analysis, any known Schwarzschild black hole would do. We point out that Sgr A\* has nonetheless been modeled purely as a Schwarzschild black hole in the literature [@27].
[^3]: We thank an anonymous referee for drawing our attention to the motion of timelike particles. This is a useful topic since the influence of secondary gravitating sources on such particles has not yet been addressed in the literature, to our knowledge.
[^4]: Details omitted to save space but the calculation is straightforward.
|
---
abstract: 'Despite being a relatively new communication technology, Low-Power Wide Area Networks (LPWANs) have shown their suitability to empower a major part of Internet of Things (IoT) applications due to their energy-aware designs, large coverage areas, low cost, and scalability. Nonetheless, most LPWAN solutions are based on star topology networks, making the end-devices or stations (STAs) to communicate directly to the gateway (GW) in single-hop manner, often causing lifetime shortening in STAs located far from the GW. In this work we study the cost and benefits of identifying uplink multi-hop routings through energy-aware machine learning algorithms based on the exploration versus exploitation problem. We show that depending on the topology, different approaches may apply, resulting in savings of 50% in some of the studied scenarios.'
author:
-
bibliography:
- 'bib.bib'
title: 'Learning energy-aware uplink routing algorithms for multi-hop LPWANs'
---
Introduction {#sec:introduction}
============
Low-Power Wide Area Networks (LPWANs) have arisen as a promising complementary communication technology for IoT. LPWANs are wireless wide area networks designed for achieving large coverage ranges, extending end devices battery lifetime and reducing the operational cost of traditional cellular networks. They are characterized by exploiting the sub-1GHz unlicensed, industrial, scientific and medical (ISM) frequency band, and by sporadically transmitting small packets at low data rates, which leads to achieving very low receptor sensitivities. Therefore, LPWANs are expected to be completely suitable for supporting IoT services, which commonly require low data throughput communications and large coverage ranges. However, most LPWANs solutions like LoRaWAN[@springer2000spread] or SIGFOX[@sigfox2016main] are built following a star topology, where end devices, or stations (STAs), are connected directly to the base station (BS) or gateway (GW), making STAs to rely deeply on their transceiver’s capabilities (i.e., transmission power, antenna gain, data rate, etc.) as they are intended to reach the GW directly in one hop. This strong requirement may lead to rapid energy consumption in STAs located far from the GW as they are required to transmit in high power levels, shortening their lifetime as a consequence. Even though such topology has clear benefits like protocol stack simplification, centralized control, or even infrastructure re-use of traditional cellular networks, it may not be efficient in terms of energy saving. Moreover, single-hop topologies hinder the inclusion of devices with transmission power limitations because of such range constraint.
Authors in [@barrachina2016multi] present DRESG, a framework for analyzing the impact on LPWANs energy consumption of enabling multi-hop routing connections in the uplink by identifying the optimal routing paths in terms of energy saving by means of balancing the consumption among all the STAs in the network. Results showed in that for LPWANs of up to several thousands of STAs, enabling such multi-hop connections in the uplink leads to higher network lifetimes than with single-hop transmissions since the consumption of STAs located far from the GW is significantly reduced. Based on such framework, we study... In [@adame2017hare] authors present a novel LPWAN protocol stack enabling multi-hop communication in the uplink when proving energetically more efficient. Multi-hop alternatives for the uplink in LPWANs technologies have not been profoundly explored yet in networks operating at sub-1GHz.
The remainder of this paper is organized as follows: Section \[sec:lpwans\] describes the main features
DRESG framework for LPWANs {#sec:dresg}
==========================
Explain what is DRESG. enabling children-parent routing connections in LPWANs may be a proper alternative against the widely implemented single-hop or star topology. How it works.
Every STA in DRESG generates its payload and sends it to its parent, which aggregates its own and all the payloads received from its direct children. Thus, the number of packets to be sent by an STA depends on the amount of payloads received. Similarly, the time a parent STA is in RX state depends on the number of children and the amount of packets they transmit. A parent node could be the GW or another STA, depending on the routing connections established.
Topology
--------
In DRESG, STAs are spread in distance rings composing a tree-based network structure that can be defined by the following 4 parameters:
- **Number of rings ($\boldsymbol{R}$):** the number of rings in a DRESG network structure is defined by $R$ and STAs belonging to the same ring are located exactly at the same distance to the GW, which is set depending on the selected distance spreading model.
- **Maximum distance ($\boldsymbol{D}$):** STAs at the furthest ring (i.e., last ring) are placed at distance $D$, which is given by the theoretical coverage range provided the GW’s transceiver at maximum $P_{\text{tx}}$ and minimum $s_{\text{tx}}$.
- **Tree children ratio ($\boldsymbol{c}$):** number of *tree children*[^1] of every STA which does not belong to the last ring. STAs belonging to the last ring have no *tree children*. In Figure \[fig:net\_topo\], two examples of DRESG network structures are shown. The tree children ratio refers only to the network structure and it is independent to the topology (or routing connections). As shown in Figure \[fig:routings\], different topologies may exist for the same DRESG network structure.
- **Number of branches ($\boldsymbol{B}$):** a branch is a set of nodes composed of an STA in the ring and its direct and indirect *tree children*. The node load of a branch, or *branch load* ($b$), is defined as the number of STAs in a branch. In DRESG, all branches have the same branch load.
Hence, the number of STAs ($N$) in an DRESG network can be defined as $N = B\sum_{r=1}^{R}c^{r-1}$, being $c^{r-1}$ the number of nodes per branch in ring $r$ for all branches.
![DRESG network structures examples. $\text{net}_1$ = {$D$, $R = 2$, $c = 1$, $B = 4$} and $\text{net}_2$ = {$D$, $R = 3$, $c = 2$, $B = 3$}. The *branch loads* of networks $\text{net}_A$ and $\text{net}_B$ are 2 and 7, respectively.[]{data-label="fig:net_topo"}](img/netA_netB_topo.png)
Energy consumption modeling
---------------------------
STAs consume different amounts of energy per time unit depending on the states they are, which are commonly determined by two sources of energy consumption: microprocessor’s ($e_{p}$) and transceiver’s ($e_{t}$). Microprocessor states are low power mode (LPM) and processing (CPU), while transceiver states are sleeping, idle, receiving (RX) and transmitting (TX). For the TX state, it is needed to differentiate among each possible transmission power level ($p\in\{1, ..., p_{\text{min}}\}$)[^2].
However, DRESG is focused exclusively on the network topology in order to perform energy consumption analysis regardless of the MAC layer specification. Therefore, a simple and ideal time division multiple access (TDMA) MAC layer with no packet collisions is considered, where STAs do not need to listen to the channel before transmitting or receiving. Specifically, there are reserved time slots assigned at the network creation phase for child-parent packet transmissions, allowing parent STAs to listen to the channel exactly the same period of time required by its child to transmit. Also, the impact of LPM and CPU states in LPWANs is expected to be very small due to its low current consumption compared to transceiver’s states (see Figure \[fig:states\_consumption\_detailed\]). In addition, the time spent in such states are expected to be similar in network STAs, as they perform similar processing operations (e.g., gathering data from sensors, buffering, etc.) and they are expected to be most of the time in microprocessor’s LPM state. Hence, for an STA in DRESG, the energy consumption can be simplified to the sum of the TX and RX energies when transmitting and receiving data packets, respectively, i.e., $e = e_{\text{tx}} + e_{\text{rx}}$, where $e_{\text{tx}} = \sum_{p=1}^{p_{\text{min}}}t_{\text{\text{tx}},p}I_{\text{tx}}(p)V_{\text{DD}}$, and $e_{\text{rx}} = t_{\text{rx}} I_{\text{rx}}V_{\text{DD}}$ where $t_{\text{rx}}$ is the time period the STA is in RX state, and $I_{\text{rx}}$ is the corresponding current consumption. The time and current consumption in TX state at power level $p$ are defined by $t_{\text{tx},p}$ and $I_{\text{tx}}(p)$, respectively. The nominal voltage is represented by $V_{\text{DD}}$.
Regarding the transmission power and data rate, we define a *transmission configuration* as the ordered pair $(P_{\text{tx}},s_{\text{tx}})$ corresponding to the uplink child-parent connection. Depending on the node’s transceiver, one or more transmission power and data rate levels may be available (or programmable), being the maximum data rate dependent on the sensitivity, as the communication range is determined by the link budget, i.e., the difference between the receiver’s sensitivity and the transmission power of the transceiver.
This complex relation among the variables impacting on the transceiver consumption hardens the task of identifying in advance which are the best *transmission configurations* for each of the established routing connections. For example, raising $P_{\text{tx}}$ would also increase $I_{\text{tx}}$, impacting negatively on the energy consumption. However, a higher $s_{\text{tx}}$ can be used when less demanding sensitivities are required, and therefore, $t_{\text{tx}}$ and $t_{\text{rx}}$ could be decreased (e.g., a radio transmitting at 100 kbps remains in TX state approximately twice the time a radio transmitting at 200 kbps), having a positive impact on the energy consumption. Also, the higher the data rate, the less the channel is occupied. Moreover, the transmission power level (and power output) is not usually linear with the corresponding power consumption, which hardens even more identifying the most suitable *transmission configuration* beforehand [@cc1200].
Learning the optimal routing {#sec:learning_optimal_routing}
============================
While in the DRESG framework for LPWANs, all possible hops combinations are tried, in this paper we aim to analyze the impact of such approach in front of other “more intelligent” ones. Also, as the ring topology is considered to be static for intrinsic DRESG simplicity, the probability density function of the reward (i.e. consumption) corresponding to every action is deterministic. We can exploit that somehow TALK WITH GERGO.
It is an exploration versus exploitation problem.
The $\epsilon$-greedy approach
------------------------------
Epsilon greedy is BLA. The general algorithm is:
1. Pick action randomly \[step1\]
2. Compute the energy consumed by every node (i.e. the cost)
3. Add action to history and corresponding accumulated energy
4. With probability $\epsilon$ go to step \[step1\] and with probability $1 - \epsilon$ pick the explored actions providing the minimum known cost. If decreasing, update epsilon accordingly.
Intelligent Neural Network?
---------------------------
To harness the gathered knowledge.
1. Pick action randomly \[step1\]
2. Compute the energy consumed by every node (i.e. the cost)
3. Add action to history and corresponding accumulated energy
4. Apply intelligence **HEHE**
Performance evaluation {#sec:results}
======================
Scenarios
---------
Three scenarios have been considered with different DRESG topologies (see Table \[table:scenarios\]). The rest of parameters are fixed and are the same in every scenario. Specifically, the results presented in this work have been computed considering data packet aggregation, equidistant rings, the CC1200 transceiver model, one branch per topology, and an 868 MHz carrier frequency and an outdoor path loss model for 802.11ah pico/hot zone deployments defined by [@hazmi2012feasibility]. Besides, all the nodes in the evaluated LPWANs (i.e., both GW and STAs) use the same transceiver model and antennas with transmission gain set to 0 dBi and reception gain set to 3 dBi. The nominal voltage ($V_{\text{DD}}$) is 3 V. Regarding data packets, the parameters implemented in ENTOMATIC EU-project[^3] are used, where the data payload and header sizes were considered to be $L_{p}=15$ and $L_h=2$ bytes, respectively. The fixed packet size was set to $L_{\text{DP}} = 65$ bytes, allowing to aggregate a maximum number of $n^{\max}_p=4$ payloads per packet. In addition, only one branch per network was considered for simplicity.
Evaluation
----------
The following metrics are considered:
- Cumulated energy consumed by historic bottleneck: how actions history affect to the long-term.
- Global transmission bottleneck energy: how actions affect in the short-term.
- CDF of the all-explored iteration.
- CDF of the optimal iteration.
Conclusions
===========
Results have shown that the optimal routing really depends on the topology. Also, the cost of identifying such optimal routing is really tied to the algorithm used for it. While in networks with few rings and small pool of actions it is usually better to first completely explore and then exploit the best one, for networks with larger number of rings, it is the opposite. That is, it is better to start exploiting from the very beginning after a few exploration turns.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was partially supported by the ENTOMATIC FP7-SME-2013 EC project (605073). It has also been funded by the Catalan government through the project SGR-2014-1173.
[^1]: We distinguish between *tree children* and *topology children*. On the one hand, *tree children* refers to all STAs of an adjacent higher ring from which an STA (i.e., *tree parent*) may receive packets. On the other hand, *topology children* (children from now on) refer to the STAs in lower adjacent or non-adjacent rings from which an STA actually receives packets. Similarly, *topology parent* (parent from now on) refers to that STA to which a child actually transmits its own packets (after aggregating the ones from its own children) in its way to the GW.
[^2]: We use the typical power level notation found in the transceiver datasheets: level 1 for maximum transmission power and level $p_{\text{min}}\geq 1$ for minimum transmission power level.
[^3]: ENTOMATIC is an agriculture plague-tracking system that intends to fight the olive fruit fly. It relies on LPWANs where STAs periodically reporting information on pest population density are spread over large olive orchards (1 STA per hectare approximately). Detailed information about the project can be found in the ENTOMATIC main website: <https://entomatic.upf.edu/>
|
---
abstract: 'We consider a countable system of interacting (possibly non-Markovian) stochastic differential equations driven by independent Brownian motions and indexed by the vertices of a locally finite graph $G = (V,E)$. The drift of the process at each vertex is influenced by the states of that vertex and its neighbors, and the diffusion coefficient depends on the state of only that vertex. Such processes arise in a variety of applications including statistical physics, neuroscience, engineering and math finance. Under general conditions on the coefficients, we show that if the initial conditions form a second-order Markov random field on $d$-dimensional Euclidean space, then at any positive time, the collection of histories of the processes at different vertices forms a second-order Markov random field on path space. We also establish a bijection between (second-order) Gibbs measures on $(\R^d)^V$ (with finite second moments) and a set of space-time (second-order) Gibbs measures on path space, corresponding respectively to the initial law and the law of the solution to the stochastic differential equation. As a corollary, we establish a Gibbs uniqueness property that shows that for infinite graphs the joint distribution of the paths is completely determined by the initial condition and the specifications, namely the family of conditional distributions on finite vertex sets given the configuration on the complement. Along the way, we establish various approximation and projection results for Markov random fields on locally finite graphs that may be of independent interest.'
address:
- 'Columbia University, New York, New York'
- 'Division of Applied Mathematics, Brown University'
- ' University of Michigan, Ann Arbor'
author:
- Daniel Lacker
- Kavita Ramanan
- Ruoyu Wu
bibliography:
- 'condind.bib'
date:
-
-
title: 'Locally interacting diffusions as space-time Markov random fields'
---
Introduction {#sec-intro}
============
Discussion of results
---------------------
Given a finite or locally finite infinite graph $G$ with vertex set $V$ and edge set $E$, and a positive integer $d$, consider interacting diffusions that satisfy the following stochastic differential equation (SDE) system: $$dX_v(t) = b_v (t, X_v(t), X_{N_v(G)}(t))\,dt + \sigma_v (t, X_v (t))\,dW_v(t), \quad v \in V, \ \ t \ge 0,$$ where the initial condition $X(0) = (X_v(0))_{v \in V}$ is distributed according to some given probability measure on $(\R^d)^V$. Here, $N_v (G) \subset V$ denotes the neighborhood of $v$ in the graph $G$, $(b_v, \sigma_v)_{v \in V}$ are given drift and diffusion coefficients, and $(W_v)_{v \in V}$ are independent standard $d$-dimensional Brownian motions. Such diffusions arise in a variety of applications, including statistical physics [@Der03; @RedRoeRus10], neuroscience [@LucStan14; @Med18], and systemic risk [@nadtochiy2018mean]. Under suitable conditions on the coefficients that guarantee the existence of a unique weak solution to the SDE, for any $t > 0$, we study the random field on $\C^V_t$ generated by the collection of trajectories $(X_v[t] := (X_v(s))_{s \le t})_{v \in V}$, where $\C_t$ (resp. $\C$) denotes the space of $\R^d$-valued continuous functions on $[0,t]$ (resp. $[0,\infty)$).
Our first set of results (Theorems \[pr:conditionalindependence-finitegraph\] and \[th:conditionalindependence-infinitegraph\]) show that, under modest conditions on the drift and diffusion coefficients that guarantee a unique weak solution to the SDE system on any locally finite graph, if $(X_v(0))_{v \in V}$ is a second-order Markov random field on $(\R^d)^V$ (as specified in Definition \[def-MRFs\]) then for each $t > 0$, $(X_v[t])_{v \in V}$ is a second-order Markov random field on $\C_t^V$. In fact, we establish this result for a more general class of SDEs with possibly non-Markovian dynamics (and potentially infinite memory) defined in Section \[sec-main\].
Our next set of results relate to an interpretation of the law of the SDE as a space-time Gibbs measure (see Section \[se:gibbs\] for precise definitions). Specifically, Theorem \[th:gibbsuniqueness\] establishes a bijection between (second-order) Gibbs measures on $(\R^d)^V$ (with finite second moments) and a set of space-time (second-order) Gibbs measures on $\C^V$, corresponding respectively to the initial law and the law of the solution to the SDE. As a consequence, we deduce a Gibbs uniqueness property, which shows that the law of the SDE system is completely determined by its initial condition and its specifications, namely the family of conditional distributions on finite sets given the configuration on the complement. In particular, together these show (see Corollary \[cor:gibbs\]) that when the initial distribution is the unique second-order Gibbs measure associated with some specifications on $(\R^d)^V$, then for each $t > 0$, the law of the SDE system is the unique second-order Gibbs measure associated with corresponding specifications on path space $\C_t^V$.
Such questions of characterizations of SDEs in terms of space-time Markov random fields are of broad interest. Prior work on similar questions has been mainly restricted to infinite-dimensional Markovian diffusions with drifts that have a local gradient structure [@Deu87; @RoeZes93; @CatRoeZes96; @DerRoe04; @RedRoeRus10; @RoeRus14]. The earliest work in this direction is that of Deuschel [@Deu87], which considers smooth gradient systems of finite range, that is, where the drift is assumed to be of the form $b_v = \nabla_v h_v$, $v \in V$, for suitable “Hamiltonian” functions $h_v : \R^{\Z^m} \mapsto \R$ defined in terms of smooth finite-range potentials with uniformly bounded second partial derivatives. For such systems, it is shown in [@Deu87 Corollary 2.21, Theorem 3.12 and Proposition 4.2] that when the initial condition is a Gibbs measure with respect to a Gibbsian specification with a finite range interaction potential (in the sense of [@georgii2011gibbs Chapter 2]), the law $P$ of the process $X$ on $C([0,1];\R^{\Z^m})$ is a Gibbs state, and is in fact the unique Gibbs state associated with the family of conditional laws ${\mathcal Law} (X_v|X_u, u \neq v)$. The approach in [@Deu87] relies crucially on the gradient structure of the drift (see [@Deu87 Remark (3.5)i)]) as it uses regularity of the explicit conditional densities ${\mathcal Law} (X_v|X_u, u \neq v)$, in order to apply Dobrushin’s uniqueness condition. To show that the Dobrushin contraction coefficient on path space is strictly bounded by $1$, a uniform boundedness assumption is imposed on the drifts. This result was later generalized to the case of unbounded drifts, still of locally gradient form, by Cattiaux, Roelly and Zessin [@CatRoeZes96]. In particular, in [@CatRoeZes96 Theorems 3.7, 3.22 and 4.9], they establish a bijection between initial Gibbs states on $\R^{\Z^m}$ and Gibbs measures on path space $C([0,1]; \R^d)^{\Z^m}$. They adopted a different approach from the one we use here, involving Malliavin calculus, a variational characterization and a certain integration-by-parts formula (see also [@RoeZes93; @MinRoeZes00]), which all exploit the gradient structure of the drift. In [@MinRoeZes00], a cluster expansion method is also used when the gradient system can be viewed as a small perturbation of a free field.
An additional motivation for our study stems from a recent result of [@LacRamWu19a], which shows how a second-order Markov property is useful for obtaining an autonomous description of the marginal (local) dynamics of a particle and its neighborhood when the underlying graph $G$ is a tree. For this purpose, a stronger *global* Markov property is derived in [@LacRamWu19a Proposition 7.3] in the setting of an infinite regular tree $G$ (or, most generally, a unimodular Galton-Watson tree) and homogeneous coefficients, $(b_v,\sigma_v)=(b,\sigma)$ for all $v \in V$. Notably, the characterization of the local dynamics in [@LacRamWu19a] relies on the precise order of the Markov random field (equivalently, range of interaction of the Gibbs state), and not merely the Gibbs property.
We now briefly describe the techniques used to prove our results. First, to establish the Markov random field property for SDEs on finite graphs (Theorem \[pr:conditionalindependence-finitegraph\]) we derive and apply a version of the Hammersley-Clifford characterization of Markov random fields on finite graphs. The result for the infinite graph (Theorem \[th:conditionalindependence-infinitegraph\]) is obtained by a delicate approximation by finite-dimensional systems in a way that preserves the Markov random field property. Along the way, we establish various approximation results for Markov random fields on locally finite graphs (see Section \[se:2MRFs\]) that may be of independent interest. Thus, our methods are quite different from those of the prior works described above. They do not require any gradient structure of the drift, are not restricted to small perturbations of a free field, and allow for non-Markovian dynamics involving path-dependent coefficients with possibly unbounded memory. More importantly, our contribution clarifies the generic nature of the conditional independence structure in these kinds of models.
Finally, we also provide examples (see Section \[subs-ceg\]) that demonstrate that the Markov random field property we establish cannot in general be significantly strengthened. Precisely, even on a finite graph with gradient drift, in general the collection of histories $(X_v[t])_{v \in V}$ do not form a first-order Markov random field, nor do the time-$t$ marginals $(X_v(t))_{v \in V}$ exhibit any non-trivial conditional independence structure. This highlights the natural problem of identifying special classes of systems for which simpler Markov random field properties are preserved, a problem which we do not address but which has attracted considerable attention in certain contexts. Specifically, in the context of diffusions, the papers [@DerRoe04; @DerRoe05; @RedRoeRus10; @RoeRus14; @van2009gibbsianness; @van2010gibbs] have studied the phenomenon of Gibbs-non-Gibbs transitions and the propagation (or lack thereof) of the Gibbs property at the level of the time-$t$ marginals, specifically whether the initial law of $X(0)$ being a Gibbs state on $(\R^d)^V$ implies that the marginal law of $X(t)$ is also a Gibbs state on $(\R^d)^V$. See Remark \[rem-gibbsprop\] for a more detailed description of these works.
The next section introduces some common notation and basic definitions used throughout the paper. The main results of the paper are stated in Section \[sec-main\], with their proofs relegated to Sections \[sec-finitegraph\]–\[se:gibbsuniqueness\].
Notation and basic definitions {#subs-not}
------------------------------
For any vectors $a, b \in \R^d$, we use $a \cdot b$ or $\langle a, b\rangle$ to denote the inner product. In this paper, unless explicitly stated otherwise, a *graph* $G=(V,E)$ always has a finite or countably infinite vertex set, is simple (no self-edges or multi-edges), and is locally finite (i.e., the degree of each vertex is finite). We abuse notation by writing $v \in G$ to mean $v \in V$. For a graph $G=(V,E)$ and a vertex $v \in V$, we write $N_v(G) = \{u \in V : (u,v) \in E\}$ for the set of neighbors of $v$ in $G$, noting that this set is empty if $v$ is an isolated vertex. A *rooted* graph $G=(V,E,\o)$ is a graph equipped with a distinguished vertex $\o \in V$, called the *root*. For two vertices $u,v \in V$, let $d(u,v)$ denote the graph distance, i.e., the length of the shortest path from $u$ to $v$ (with $d(u,u) := 0$). Also, let $\mathrm{diam}(A)$ denote the diameter of a set $A \subset V$; precisely, $\mathrm{diam}(A) = \sup\{d(u,v) : u,v \in A\}$. For a subset $A \subset V$, we define the first and second boundaries $$\label{bdaries}
\begin{array}{rcl}
\partial_G A &=& \{u \in V \backslash A : (u,v) \in E \text{ for some }v \in A\}, \\
\partial^2_G A &=& \partial_G A \cup \partial_G (A \cup \partial_G A).
\end{array}$$ We will often omit the subscript, writing simply $\partial^2A$ in place of $\partial^2_GA$, when the underlying graph $G$ is clear. A *clique* in a graph $G$ is a complete subgraph of $G$, i.e., a set $A \subset V$ such that $(u,v) \in E$ for every $u,v \in A$. Equivalently, a clique is a set $A \subset V$ of diameter at most $1$. Define $\mathrm{cl}_1(G)$ to be the set of all cliques of the graph $G$. Similarly, we will say that any subset $A \subset V$ with diameter at most $2$ is a [*$2$-clique*]{} of the graph $G$ and let $\mathrm{cl}_2(G)$ denote the set of $2$-cliques of $G$. Moreover, given a graph $G = (V,E)$, $H = (V_H, E_H)$ is said to be an induced subgraph of $G$ if $V_H \subset V$ and $E_H = E \cap \{(u,v): u, v \in V_H\}$.
For a set $\X$ and a graph $G= (V,E)$, we may write either $\X^V$ or $\X^G$ for the configuration space $\{(x_v)_{v \in V}: x_v \in \X \mbox{ for every } v \in V\}$. We make use of a standard notation for configurations on subsets of vertices: For $x=(x_v)_{v \in V} \in \X^V$ and $A \subset V$, we write $x_A$ for the element $x_A=(x_v)_{v \in A}$ of $\X^A$. When $\X$ is a Polish space, we write $\P(\X)$ for the set of Borel probability measures on $\X$, endowed always with the topology of weak convergence. Given any measurable space $\X$, $A \subset V$, and measure ${\nu}\in \P(\X^V)$, ${\nu}[A]$ represents the restriction of ${\nu}$ to the set $\X^A$, that is, the image measure under the restriction map $\X^V \ni (x_v)_{v\in V} \mapsto (x_v)_{v \in A} \in \X^A$.
Fixing $d \in \N$, we let $\C = C(\R_+;\R^d)$ denote the path space of $\R^d$-valued continuous functions on $\R_+=[0,\infty)$, endowed with the topology of uniform convergence on compacts. For $t > 0$, let $\C_t = C([0,t];\R^d)$ denote its restriction to the time interval $[0,t]$, endowed with the uniform topology. For $x \in \C$ and $t > 0$ we define $\|x\|_{*,t} := \sup_{s \in [0,t]}|x(s)|$, and let $x[t] = (x(s))_{s \le t} \in \C_t$ denote the restriction of the path $x$ to the time interval $[0,t]$. We assume that $\C$ and $\C_t$ are endowed with their respective Borel $\sigma$-algebras. Also, for any countable set $A$ and probability measure $Q$ on $\C^A$, we write $Q_t$ for the image under $Q$ of the map $\C^A \ni (x_v)_{v \in A} \mapsto (x_v[t])_{v \in A} \in \C_t^A$. The $\sigma$-algebra on a product space will always be the product $\sigma$-algebra, unless explicitly stated otherwise. Given $J, m \in \N$, a measurable function $f:[0,\infty) \times \C^J \mapsto \R^m$, is said to be [*progressively measurable*]{} if for each $t \ge 0$, $f(t,(\tilde{x}_u)_{u=1, \ldots, J}) = f(t,(y_u)_{u = 1, \ldots, J})$ whenever $x_u[t]=y_u[t]$ for all $u = 1, \ldots, J$.
We end this section by recalling the notion of a (first-order or second-order) Markov random field, which plays a central role in the paper.
\[def-MRFs\] Given a measurable space $\X$, and a (possibly infinite) locally finite graph $G = (V,E)$, let $({Y}_v)_{v \in V}$ be a random element of $\X^V$ with some distribution ${\nu}\in \P(\X^V)$. We say that $({Y}_v)_{v \in V}$, or equivalently its law ${\nu}$, is a *first-order Markov random field* (abbreviated as [1MRF]{}) on $\X^V$ if ${Y}_A$ is conditionally independent of ${Y}_{(A \cup \partial A)^c}$ given ${Y}_{\partial A}$, for every finite set $A \subset V$. On the other hand, we say that $({Y}_v)_{v \in V}$, or equivalently its law ${\nu}$, is a *second-order Markov random field* (abbreviated as [2MRF]{}) on $\X^V$ if ${Y}_A$ is conditionally independent of ${Y}_{(A \cup \partial^2 A)^c}$ given ${Y}_{\partial^2 A}$, for every finite set $A \subset V$. When the space $\X^V$ is clear from the context, we will simply say that $({Y}_v)_{v \in V}$, or equivalently its law ${\nu}$, is a [1MRF]{} or [2MRF]{}.
\[rem-MRFs\] In Definition \[def-MRFs\], it is important to stress that the sets A are required to be finite even when the graph $G$ is infinite. Allowing infinite sets $A$ results in the stronger *global Markov property*, which we do not study in this paper.
Main results {#sec-main}
============
Given a locally finite graph $G = (V,E)$ with a finite or countably infinite vertex set, we are interested in a system of (possibly non-Markovian) interacting stochastic processes, indexed by the vertices of the graph, that satisfy an SDE of the form $$\begin{aligned}
\label{fingraph-SDE}
dX_v(t) &= b_v(t,X_v,X_{N_v(G)}) \,dt + \sigma_v(t,X_v) \,dW_v(t), \quad v \in V,\end{aligned}$$ where $(W_v)_{v \in V}$ are independent Brownian motions, and the initial law $\mu_0 \in \P((\R^d)^V)$, of $(X_v(0))_{v \in V}$ and the coefficients $(b_v, \sigma_v)_{v \in V}$ satisfy the conditions stated in Assumption \[assumption:A\] or Assumption \[assumption:B\] below, depending on whether the graph is finite or infinite. As mentioned in the introduction, our main results concern the characterization of the law of the solution to the SDE as a [2MRF]{} on $\C^V$ (see Definition \[def-MRFs\]).
The finite graph case {#subs-mainfin}
---------------------
We first consider the case when $G$ is finite, and the conditions stated in Assumption \[assumption:A\] below are satisfied. Recall, from Section \[subs-not\], the definition of $2$-cliques, the notation for trajectories, $x[t] = (x(s))_{s \le t} \in \C_t$ for $x \in \C$, and the notion of a progressively measurable functional.
[**A**]{} \[assumption:A\] $\ $ We say that $(G,b, \sigma, \mu_0)$ satisfy Assumption \[assumption:A\] if $G = (V,E)$ is a finite graph and if $b=(b_v)_{v \in V}$, $\sigma = (\sigma_v)_{v \in V}$, and $\mu_0 \in \P((\R^d)^V)$ satisfy the following:
1. There exist $\lambda_v \in \P(\R^d), v \in V$, such that the probability measure $\mu_0$ is absolutely continuous with respect to the product measure $\mu_0^* = \prod_{v \in V}\lambda_v \in \P((\R^d)^V)$ and the density satisfies $$\begin{aligned}
\frac{d\mu_0}{d\mu_0^*}(x) = \prod_{K \in \mathrm{cl}_2(G)}f_K(x_K), \quad\quad x \in (\R^d)^V, \label{asmp:A1factorization}\end{aligned}$$ for some measurable functions $f_K : (\R^d)^K \rightarrow \R_+$, $K \in \mathrm{cl}_2(G)$, where $\mathrm{cl}_2(G)$ is the set of $2$-cliques of $G$. In addition, $\mu_0$ has a finite second moment.
2. For each $v \in V$, the drift $b_v: \R_+ \times \C \times \C^{N_v(G)} \mapsto \R^d$ is progressively measurable. Moreover, for each $T \in (0,\infty)$ there exists $C_T < \infty$ such that $$|b_v(t,x,y_{N_v(G)})| \le C_T\left(1 + \|x\|_{*,t} + \sum_{u \in N_v}\|y_u\|_{*,t}\right),$$ for all $v \in V$, $t \in [0,T]$, $x \in \C$, and $y_{N_v(G)} =(y_u)_{u \in N_v(G)} \in \C^{N_v(G)}$.
3. The diffusion matrices $\sigma_v : \R_+ \times \C \rightarrow \R^{d\times d}$, $v \in V$, satisfy the following:
1. For each $v \in V$, $\sigma_v$ is bounded, progressively measurable and invertible, with bounded inverse.
2. For each $v \in V$, the following driftless SDE system admits a unique in law weak solution starting from any initial position $x \in \R^d$: $$\begin{aligned}
dX_v(t) = \sigma_v(t,X_v)\,dW_v(t), \quad X_v(0)=x. \label{eq:canonical_law-individual}\end{aligned}$$
A necessary condition for Assumption (\[assumption:A\].1) is that $\mu_0$ is a [2MRF]{} and is absolutely continuous with respect to the product measure $\mu_0^*$; this follows from a form of the Hammersley-Clifford theorem stated in Proposition \[th:hammersleyclifford2\] below. If the density $d\mu_0/d\mu_0^*$ is strictly positive, then it factorizes as in if and only if $\mu_0$ is a [2MRF]{}.
\[rem-assA3\] If $\sigma_v(t,x)=\widetilde\sigma_v(t,x(t))$ depends only on the current state, not the history, and satisfies the additional continuity condition $\lim_{y \rightarrow x} \sup_{0 \leq s \leq T}|\widetilde\sigma_v(s,y) - \widetilde\sigma_v(s,x)| = 0$ for all $v \in V$, then Assumption (\[assumption:A\].3b) holds as a consequence of Assumption (\[assumption:A\].3a) and [@StroockVaradhan Chapter 7].
The following proposition shows that, as a simple consequence of Girsanov’s theorem, Assumption \[assumption:A\] guarantees weak existence and uniqueness in law of the SDE system . Its proof is given in Section \[ap-finite\], along the way to proving Theorem \[pr:conditionalindependence-finitegraph\] below.
\[prop-unique-finite\] When $(G,b, \sigma, \mu_0)$ satisfy Assumption \[assumption:A\], the SDE system has a weak solution that is unique in law.
We now state our main result for the SDE system on finite graphs.
\[pr:conditionalindependence-finitegraph\] Suppose $(G=(V,E),b, \sigma, \mu_0)$ satisfy Assumption \[assumption:A\], and let $(X_v)_{v \in V}$ be the unique in law solution of the SDE system with initial law $\mu_0$. Then, for each $t > 0$, $(X_v[t])_{v \in V}$ is a [2MRF]{} on $\C_t^V$. Moreover, $(X_v)_{v \in V}$ is a [2MRF]{} on $\C^V$.
The proof of Theorem \[pr:conditionalindependence-finitegraph\], given in Section \[ap-finite\], relies on a certain factorization property (stated in Proposition \[th:hammersleyclifford2\]) of the density of the law of the SDE on finite graphs with respect to a reference measure. Notice that in Assumption \[assumption:A\], and throughout the paper, we assume there is no interaction in the diffusion coefficients (i.e., no dependence of $\sigma_v$ on $X_{N_v(G)}$), a restriction made also in the prior works [@Deu87; @RoeZes93; @CatRoeZes96; @MinRoeZes00]; the general case seems out of reach of our approach, because the reference measure in the factorization property must crucially be a *product measure*. This factorization property is also used in Sections \[se:example-firstorder\] and \[se:example-marginals\] to show that, even when the initial states $(X_v(0))_{v \in V}$ are i.i.d., for $t > 0$, in general $(X_v[t])_{v \in V}$ fails to be a [1MRF]{}, and the time-$t$ marginals $(X_v(t))_{v \in V}$ can fail to be a Markov random field of either first or second order. In fact, the counterexamples show that this does not hold even on a finite graph when $\sigma$ is the identity covariance matrix, and the drift is of gradient type. This shows that, in a sense, Theorem \[pr:conditionalindependence-finitegraph\] cannot be strengthened.
The infinite graph case {#subs-maininfin}
-----------------------
We now consider the SDE system in the case when $G$ is an infinite, though still locally finite, graph. The well-posedness of the SDE system is no longer obvious and in particular does not follow from Girsanov’s theorem as it did when the graph was finite. Indeed, on an infinite graph, when $b_v \equiv 1$ and $\sigma_v \equiv I_d$, $v \in V$, for instance, it is straightforward to argue, using the law of large numbers, that the law of a weak solution of up to some time $t > 0$ and the law of the corresponding drift-free equation are mutually singular. This necessitates the following additional assumptions compared to Assumption \[assumption:A\]. Recall from Section \[subs-not\] that given a measure $\nu$ on ${\mathcal X}^V$ for some Polish space ${\mathcal X}$, and $A \subset V$, $\nu[A]$ denotes the restriction of $\nu$ to $A$. Also, we use the notation $\pi_1 \sim \pi_2$ to denote that the measures $\pi_1$ and $\pi_2$ are equivalent, that is, mutually absolutely continuous.
[**B**]{} \[assumption:B\] $\ $ We say that $(G,b, \sigma, \mu_0)$ satisfy Assumption \[assumption:B\] if $G=(V, E)$ is a countable locally finite connected graph and if $b=(b_v)_{v \in V}$, $\sigma = (\sigma_v)_{v \in V}$ and $\mu_0 \in \P((\R^d)^V)$ satisfy the following:
1. The initial law $\mu_0$ is a [2MRF]{} on $(\R^d)^V$. Moreover, there exists a product measure $\mu_0^* = \prod_{v \in V}\lambda_v \in \P((\R^d)^V)$ such that $\mu_0[A] \sim\mu_0^*[A]$ for each finite set $A \subset V$. Further, the initial law $\mu_0$ satisfies $$\begin{aligned}
\sup_{v \in V}\int_{(\R^d)^V}|x_v|^2\,\mu_0(dx_V) < \infty. \label{assumption:initialsecondmoment}
\end{aligned}$$
2. The drift coefficients $(b_v)_{v \in V}$ satisfy Assumption (\[assumption:A\].2), for some constants $(C_T)_{T > 0}$.
3. The diffusion matrices $(\sigma_v)_{v \in V}$ satisfy Assumption (\[assumption:A\].3).
4. The SDE system is unique in law, and this law is denoted by $P = P^{\mu_0} \in \P(\C^V)$.
\[rem-driftless\] Using Assumption (\[assumption:A\].3b) if the graph is finite or Assumption (\[assumption:B\].3) if the graph is infinite, we may define for any initial law $\nu \in \P((\R^d)^V)$ the measure $P^{*,\nu} \in \P(\C^V)$ to be the law of the unique weak solution of the SDE system $$\begin{aligned}
dX_v(t) = \sigma_v(t,X_v)\,dW_v(t), \quad v \in V, \ \ (X_v(0))_{v \in V} \sim \nu. \label{eq:canonical_law}
\end{aligned}$$ Note in particular that if we take $\nu=\mu_0^*$, where $\mu_0^*$ is a product measure as in Assumption (\[assumption:A\].1) or (\[assumption:B\].1), then $P^{*,\mu_0^*}$ too is a product measure.
We show in Lemma \[le:infinitegraphlimit\] that existence of a solution to follows automatically from Assumptions (\[assumption:B\].1–3). However, it is worth commenting on the uniqueness condition in Assumption (\[assumption:B\].4). The following proposition shows that a suitable Lipschitz condition is enough to guarantee uniqueness; its proof is standard and hence relegated to Appendix \[ap:uniqueness-infSDE\]. Recall in the following that $\|x\|_{*,t} = \sup_{s \in [0,t]}|x(s)|$ for $x \in \C$.
\[pr:uniqueness-infiniteSDE\] Suppose Assumptions (\[assumption:B\].1–3) hold, and $(b_v)_{v \in V}$ and $(\sigma_v)_{v\in V}$ are uniformly Lipschitz in the sense that for each $T > 0$ there exist ${K}_T, {\bar{K}}_T < \infty$ such that $$\begin{aligned}
|b_v(t,x,y_{N_v(G)}) - b_v(t,x',y'_{N_v(G)})| &\le K_T\left(\|x-x'\|_{*,t} + \frac{1}{|N_v(G)|}\sum_{u \in N_v(G)}\|y_u-y'_u\|_{*,t}\right), \\
|\sigma_v(t,x) - \sigma_v(t,x')| &\le {\bar{K}}_T\|x-x'\|_{*,t},
\end{aligned}$$ for all $v \in V$, $t \in [0,T]$, $x,x' \in \C$, and $y_{N_v(G)},y'_{N_v(G)} \in \C^{N_v(G)}$. Then pathwise uniqueness holds for the SDE system . In particular, Assumption (\[assumption:B\].4) holds.
We now state our second main result.
\[th:conditionalindependence-infinitegraph\] Suppose $(G=(V,E),b, \sigma, \mu_0)$ satisfy Assumption \[assumption:B\], and let $(X_v)_{v \in V}$ be the unique in law solution of the SDE system with initial law $\mu_0$. Then, for each $t > 0$, $(X_v[t])_{v \in V}$ is a [2MRF]{} on $\C_t^V$. Moreover, $(X_v)_{v \in V}$ is a [2MRF]{} on $\C^V$.
The proof of Theorem \[th:conditionalindependence-infinitegraph\] is given in Section \[subs-pf-infingraph\], with preparatory results established in Sections \[se:2MRFs\] and \[subs-tightness\]. The factorization result used in the finite graph case is no longer applicable in the infinite graph case, and thus the proof employs a completely different approach, involving a rather subtle approximation argument, which is outlined in Section \[subs-approxmeas\].
Gibbs measures on path space {#se:gibbs}
----------------------------
Our final results interpret our SDE system in the spirit of Gibbs measures, for which we introduce the following notation. Given a Polish space $\X$, a graph $G=(V,E)$, a random $\X^V$-valued element $({Y}_v)_{v \in V}$ with law ${\nu}\in \P(\X^V)$, and two disjoint sets $A,B \subset V$, we write ${\nu}[A \, | \, B]$ to denote a version of the regular conditional law of ${Y}_A$ given ${Y}_B$. Precisely, we view ${\nu}[A \, | \, B]$ as a measurable map (kernel) from $\X^B$ to $\P(\X^A)$. Note that ${\nu}$ is a [2MRF]{} if and only if ${\nu}[A \, | \, V \backslash A](x_{V \backslash A}) = {\nu}[A \, | \, \partial^2A](x_{\partial^2A})$ for ${\nu}$-almost every $x \in \X^V$ and every finite set $A$. We make use of the following terminology of Gibbs measures (see [@georgii2011gibbs] or [@rassoul2015course] for further discussion of this classical framework).
\[def-Gibbs\] Given a Polish space $\X$, graph $G=(V,E)$, and $\gamma \in \P(\X^V)$, define $\G_2(\gamma)$ as the set of [2MRF]{}s $\nu \in \P(\X^V)$ such that, for each finite set $A \subset V$, we have $\nu[A] \sim \gamma[A]$ and also $\nu [A \, | \, \partial^2 A] = \gamma[A \, | \, \partial^2 A]$, almost everywhere with respect to $\gamma[\partial^2 A]$.
In other words, ${\mathcal G}_2(\gamma)$ is the set of (second-order, infinite volume) Gibbs measures corresponding to the *specification* $\{\gamma[A \, | \, \partial^2A] : A \subset V \ \rm{finite}\}$. Note that if $\gamma$ is itself a [2MRF]{} then $\G_2(\gamma)$ is nonempty, as it contains $\gamma$ itself. Moreover, it is straightforward to check that, if $\gamma$ and $\nu$ are [2MRF]{}s, then $\nu \in \G_2(\gamma)$ if and only if $\gamma \in \G_2(\nu)$. Recall that, by Assumption (\[assumption:B\].4), the SDE system is well-posed starting from any initial distribution. The following bijection result is proved in Section \[se:gibbsuniqueness\].
\[th:gibbsuniqueness\] Suppose $(G, b, \sigma, \mu_0)$ satisfy Assumption \[assumption:B\]. Let $P^{\mu_0} \in \P(\C^V)$ be the law of the solution of the SDE system with initial law $\mu_0$, and let $P^{*,\mu_0}$ be the law of the driftless SDE system with initial law $\mu_0$, and define $$\MLset (\mu_0) := \Big\{ \nu_0 \in \G_2(\mu_0) : \sup_{v \in V}\int_{\R^d}|x_v|^2\,\nu_0(dx) < \infty\Big\},$$ and $$\MRset (\mu_0) := \left\{ Q \in \P(\C^V) : Q_t \in \G_2(P^{\mu_0}_t) \ \forall t \ge 0, \ \sup_{v \in V}\int_{\C^V}|x_v(0)|^2Q(dx) < \infty\right\}.$$ Then it holds that $$\begin{aligned}
\MLset(\mu_0) = \Big\{Q \circ (X_V(0))^{-1} : Q \in \MRset(\mu_0) \}.
\label{claim} \end{aligned}$$ Moreover, the map $Q \mapsto Q \circ (X_V(0))^{-1}$ defines a bijection between $\MRset(\mu_0)$ and $\MLset(\mu_0)$. In particular, if $Q \in \P(\C^V)$ satisfies $Q_t \in \G_2(P^{\mu_0}_t)$ for all $t \ge 0$ and also $Q \circ (X_V(0))^{-1} = \mu_0$, then $Q=P^{\mu_0}$.
In fact, we will show in the proof of Theorem \[th:gibbsuniqueness\] that the bijection $Q \mapsto Q \circ (X_V(0))^{-1}$ between the sets $\MRset(\mu_0)$ and $\MLset(\mu_0)$ has inverse given by $\nu_0 \mapsto P^{\nu_0}$, where $P^{\nu_0}$ denotes the law of the solution of the SDE with initial law $\nu_0$, and we note that this SDE is unique in law by Assumption (\[assumption:B\].4). Additionally, if $\mu_0(K^V) =1$ for some compact set $K \subset \R^d$, then (recalling that membership in $\G_2(\cdot)$ requires absolute continuity) can be rewritten as $$\begin{aligned}
\G_2(\mu_0) = \Big\{Q \circ (X_V(0))^{-1} : Q \in \P(\C^V), \, Q_t \in \G_2(P^{\mu_0}_t) \ \forall t \ge 0\Big\}.\end{aligned}$$
We conclude this section with the following simple corollary of Theorems \[th:conditionalindependence-infinitegraph\] and \[th:gibbsuniqueness\].
\[cor:gibbs\] Suppose $(G, b, \sigma, \mu_0)$ satisfy Assumption \[assumption:B\] and $P^{\mu_0}$ represents the unique law of the SDE . If $\G_2(\mu_0)$ is a singleton, then the set $\MRset(\mu_0)$ defined in Theorem \[th:gibbsuniqueness\] is equal to the singleton $\{P^{\mu_0}\}$, and hence, $P^{\mu_0}$ is completely characterized by its specifications $P_t^{\mu_0}[A| \partial^2 A], t \geq 0,$ for finite $A \subset V$.
Since Assumption (\[assumption:B\].1) ensures that $\mu_0$ is a [2MRF]{} with finite second moment, the set on the left-hand side of always contains $\mu_0$. Thus, if $\G_2(\mu_0)$ is a singleton, then by Theorem \[th:gibbsuniqueness\] the set $\MRset(\mu_0)$ is also a singleton. On the other hand, by Theorem \[th:conditionalindependence-infinitegraph\], for each $t \ge 0$ it holds that $P_t^{\mu_0}$ is a [2MRF]{} and thus $P_t^{\mu_0} \in \G_2(P^{\mu_0}_t)$. Hence, $P^{\mu_0} \in \MRset(\mu_0)$.
Interacting diffusions on a finite graph {#sec-finitegraph}
========================================
In Section \[subs-clique\] (specifically Proposition \[th:hammersleyclifford2\]) a useful characterization of a (positive) [2MRF]{} is derived in an abstract setting. This is then used in Section \[ap-finite\] to prove Theorem \[pr:conditionalindependence-finitegraph\]; along the way Proposition \[prop-unique-finite\] is also established. In Sections \[se:example-firstorder\] and \[se:example-marginals\] this characterization is applied to demonstrate via explicit examples that the space-time [2MRF]{} property established in Theorem \[pr:conditionalindependence-finitegraph\] (and hence, Theorem \[th:conditionalindependence-infinitegraph\]) cannot in general be substantially improved.
Clique factorizations {#subs-clique}
---------------------
We start by studying the relationship between random fields and factorization properties of their joint density with respect to a given reference measure. Throughout this section, we work with a fixed finite graph $G=(V,E)$, as well as a fixed Polish space $\X$, the state space. Recall the definition of the diameter $\mathrm{diam}(A)$ of a set $A \subset V$, $1$-cliques and $2$-cliques of a graph, and 1st-order and 2nd-order MRFs given in Section \[subs-not\].
First, we recall a well-known theorem often attributed to Hammersley-Clifford, which can be found in various forms in [@georgii2011gibbs Theorem 2.30] and [@lauritzen1996graphical Proposition 3.8 and Theorem 3.9], the latter covering our precise setting.
\[th:hammersleyclifford\] Assume the graph $G=(V,E)$ is finite. Assume ${\nu}\in \P(\X^V)$ is absolutely continuous with respect to a product measure ${\nu}^* = \prod_{v \in V} {\theta}_v \in \P(\X^V)$ for some ${\theta}_v \in \P(\X)$, $v \in V$. Consider the following statements:
(1) ${\nu}$ is a [1MRF]{}.
(2) The density of ${\nu}$ with respect to ${\nu}^*$ factorizes in the form $$\frac{d{\nu}}{d{\nu}^*}(x) = \prod_{K \in \rm{cl}_1(G)}f_K(x_K), \qquad x \in \X^V,$$ for some measurable functions $f_K : \X^K \rightarrow \R_+$, for $K \in \rm{cl}_1(G)$.
Then (2) implies (1). If also $d{\nu}/d{\nu}^*$ is strictly positive, then (1) implies (2).
We next formulate an analogue for a [2MRF]{}.
\[th:hammersleyclifford2\] Assume the graph $G=(V,E)$ is finite. Assume ${\nu}\in \P(\X^V)$ is absolutely continuous with respect to a product measure ${\nu}^* = \prod_{v \in V}{\theta}_v \in \P(\X^V)$ for some ${\theta}_v \in \P(\X)$, $v \in V$. Consider the following statements:
(1) ${\nu}$ is a [2MRF]{}.
(2) The density of $\mu$ with respect to ${\nu}^*$ factorizes in the form $$\label{eq-mufactor}
\frac{d{\nu}}{d{\nu}^*}(x) = \prod_{K \in \rm{cl}_2(G)}f_K(x_K), \qquad x \in \X^V,$$ for some measurable functions $f_K : \X^K \rightarrow \R_+$, for $K \in \rm{cl}_2(G)$.
Then (2) implies (1). If also $d{\nu}/d{\nu}^*$ is strictly positive, then (1) implies (2).
Define the *square graph* $G^2 = (V,E')$ by connecting any two vertices of distance $2$. That is, let $$E' := \{(u,v) \in V^2 : 1 \le d(u,v) \le 2\},$$ where $d$ is the graph distance on $G$. It is straightforward to check the following properties:
(i) The $1$-cliques of $G^2$ are precisely the $2$-cliques of $G$. That is, $\mathrm{cl}_2(G)=\mathrm{cl}_1(G^2)$.
(ii) We have $\partial_{G^2}A = \partial^2_G A$ for any set $A \subset V$.
It follows from (ii) that the statement (1) is equivalent to
1. $\nu$ is a [1MRF]{} relative to the graph $G^2$.
On the other hand, it follows from (i) that (2) is equivalent to
1. The density of ${\nu}$ with respect to ${\nu}^*$ factorizes in the form $$\frac{d{\nu}}{d{\nu}^*}(x) = \prod_{K \in \rm{cl}_1(G^2)}f_K(x_K), \qquad x \in \X^V,$$ for some measurable functions $f_K : \X^K \rightarrow \R_+$, $K \in \rm{cl}_1(G^2)$.
The equivalence of (1’) and (2’) follows from Proposition \[th:hammersleyclifford\].
The [2MRF]{} property is the more intuitive, but the second property of Proposition \[th:hammersleyclifford2\] will be quite useful in the analysis as well. Hence, we give it a name.
\[def:2cliquefactorization\] We say that ${\nu}\in \P(\X^V)$ *has a $2$-clique factorization with respect to ${\nu}^*$* if the density $d{\nu}/d{\nu}^*$ can be written in the form .
\[rem-cutset\] For a finite graph $G=(V,E)$ and Polish space $\X$, the following *cutset* characterization of [1MRF]{}’s on $\X^V$ is well known: An $\X^V$-valued random element $({Y}_v)_{v \in V}$ is a [1MRF]{} if and only if ${Y}_A$ is conditionally independent of ${Y}_B$ given ${Y}_S$ for any disjoint sets $A,B,S \subset V$ with the property that every path starting in $A$ and ending in $B$ contains at least one vertex of $S$. Given the correspondence between a [2MRF]{} on a graph and a [1MRF]{} on the square graph (established in the proof of Proposition \[th:hammersleyclifford2\]), this is easily seen to imply the following *cutset* characterization of [2MRF]{}s: An $\X^V$-valued random element $({Y}_v)_{v \in V}$ is a [2MRF]{} if and only if ${Y}_A$ is conditionally independent of ${Y}_B$ given ${Y}_S$ for any disjoint sets $A,B,S \subset V$ with the property that every path starting in $A$ and ending in $B$ contains at least two adjacent vertices of $S$.
Proof of the second-order Markov random field property for a finite graph {#ap-finite}
-------------------------------------------------------------------------
We now present the proofs of Proposition \[prop-unique-finite\] and Theorem \[pr:conditionalindependence-finitegraph\]. Throughout this section, we work with a fixed finite graph $G=(V,E)$ and consider the canonical measurable space $\C^V = (\C^V,\text{Borel})$, and let $(X_v)_{v \in V} : \C^V \rightarrow \C^V$ denote the canonical processes, that is, $X_v((x_u)_{u \in V}) = x_v$ for $x = (x_u)_{u \in V} \in \C^V$, for $v \in V$. Let $\mu_0, \mu_0^* \in \P((\R_d)^V)$ be as in Assumption (\[assumption:A\].1), and let $P^* = P^{*, \mu_0^*} \in \P(\C^V)$ denote the law of the unique solution of the driftless SDE system starting from initial law $\mu_0^*$ (the well-posedness of which is given by Assumption (\[assumption:A\].3b)). Recall that $\mu_0^*$ and thus $P^*$ are both product measures. Then, recalling that $dX_v(t) = \sigma_v(t,X_v)\,dW_v(t)$ for $v \in V$, define the following local martingale (under $P^*$): $$\label{martv}
M_v(t) := \int_0^t (\sigma_v\sigma_v^\top)^{-1}b_v(s,X_v,X_{N_v(G)}) \cdot dX_v(s), \qquad t \geq 0,$$ where we use the shorthand notation $(\sigma_v\sigma_v^\top)^{-1}b_v(s, x_v, x_{N_v(G)})$ to denote the map $$\label{sigmamap}
\R_+ \times \C^V \ni (s,x) \mapsto (\sigma_v(s,x_v)\sigma_v^\top(s,x_v))^{-1}b_v(s, x_v, x_{N_v(G)}) \in \R^d.$$ Also, given any continuous local martingale $M$, we let $\EE(M)$ denote the Doleans exponential: $$\begin{aligned}
\EE_t(M) = \exp\left( M (t) - \frac12 [M] (t) \right), \qquad t \geq 0, \label{def:doleans-exponential}\end{aligned}$$ where $[M]$ denotes the (optional) quadratic variation process of $M$.
Let $(f_K)_{K \in \mathrm{cl}_2 (G)}$ be as in Assumption (\[assumption:A\].1). For each $t > 0$, define the measure $P_t \in \P(\C_t^V)$ by $$\begin{aligned}
\frac{dP_t}{dP^*_t} & := \dfrac{d\mu_0}{d\mu_0^*} (X_V(0)) \EE_t\left(\sum_{v \in V}M_v\right), \nonumber \\
&= \prod_{K \in \mathrm{cl}_2(G)}f_K(X_K(0))\,\prod_{v \in V}\EE_t(M_v), \label{P-factorization}\end{aligned}$$ with $\EE(M)$ and $P^*$ as defined in the previous paragraph. Note that $W_v := \int_0^\cdot \sigma^{-1}_v(s,X_v)\,dX_v(s)$, $v \in V,$ are independent $d$-dimensional Brownian motions under $P^*$ by Remark \[rem-driftless\]. Therefore the stochastic exponentials appearing in are true $P^*$-martingales due to the form of $M_v$ in , the linear growth assumption (\[assumption:A\].2) on the drifts and the non-degeneracy of $\sigma_v$; see Lemma \[lem:Girsanov-justification\] with $\QQ= P^{*}$, $X=(X_v)_{v \in V}$ and $f_v(t,x) = \sigma_v^{-1}b_v(t,x_v,x_{N_v(G)})$, $v \in V$. Further, observe that $(M_v)_{v \in V}$ are orthogonal under $P^*$. So Girsanov’s theorem [@karatzas-shreve Corollary 3.5.2] implies that under $P_t$, $\tilde{W}_v := W_v - \int_0^\cdot \sigma_v^{-1}(s, X_v)b_v (s, X_v, X_{N_v(G)})\,ds$, $v \in V$, are independent $d$-dimensional standard Brownian motions on $[0,t]$. From this it follows that under $P_t$, $X$ solves the SDE on $[0,t]$, and the same argument also shows that the restriction to $[0,t]$ of any solution to must have law $P_t$ on $\C_t^V$. Thus, we have uniqueness in law. Weak existence follows from Kolmogorov’s extension theorem on observing that $\{P_t, t \geq 0\}$ form a consistent family in the sense that $P_s$ is the restriction of $P_t$ to $\C_s^V$ for each $t > s > 0$ (due to the martingale property of $\frac{dP_t}{dP^*_t}$). This completes the proof of Proposition \[prop-unique-finite\].
On the other hand, the fact that for each $t > 0$, $(X_v[t])_{v \in V}$ is a 2MRF on $C_t^V$ follows from on applying Proposition \[th:hammersleyclifford2\] with ${\mathcal X} = \C_t$ and $\mu^* = P^*_t$, noting that $P^*_t$ is a product measure on $\C_t^V$ and that, for each $v \in V$, $\{v\} \cup N_v(G)$ is a $2$-clique and $M_v$ of is $X_{\{v\} \cup N_v(G)}$-measurable. This proves the first assertion of Theorem \[pr:conditionalindependence-finitegraph\]. For the second assertion of Theorem \[pr:conditionalindependence-finitegraph\], denote by $P = P^{\mu_0} \in \P(\C^V)$ the law of the unique solution of the SDE system with initial law $\mu_0$. Fix a finite set $A \subset V$ and bounded continuous functions $f,g,h$ on $\C^A,\C^{\partial^2 A}, \C^{V \setminus (A \cup \partial^2 A)}$, respectively. Fix $t > 0$ and let ${{\mathcal{G}}}_t := \sigma\{ X_{\partial^2A}[t] \}$ and ${{\mathcal{G}}}_\infty := \sigma\{ X_{\partial^2A} \}$. Below, with some abuse of notation, for any $B \subset V$, we will also interpret elements $y \in \C_t^B$ as elements of $\C^B$ by simply setting $y(s) = y(t)$ for $s \geq t$. Note that with this identification, for any $x \in \C^V$ and $B \subset V$, $x_B[t] \rightarrow x_B$ in $\C^B$ as $t \rightarrow \infty$. Then, noting that $\sigma(\cup_{t > 0}\G_t) = \G_\infty$, invoking the martingale convergence theorem (in the third equality below), and using the fact that $P_t=P \circ (X_V[t])^{-1}$ is a [2MRF]{} on $\C_t^V$ for each $t$ (in the second equality below), we have $$\begin{aligned}
& {{\mathbb{E}}}^P \left[ f(X_A) g(X_{\partial^2A}) h(X_{V \setminus (A\cup\partial^2A)}) \right] \notag \\
& = \lim_{s \to \infty} \lim_{t \to \infty} {{\mathbb{E}}}^P \left[ f(X_A[s]) g(X_{\partial^2A}[t]) h(X_{V \setminus (A\cup\partial^2A)}[s]) \right] \notag \\
& = \lim_{s \to \infty} \lim_{t \to \infty} {{\mathbb{E}}}^P \left[ {{\mathbb{E}}}^{P} \left[ f(X_A[s]) \mid {{\mathcal{G}}}_t \right] g(X_{\partial^2A}[t]) {{\mathbb{E}}}^{P} \left[ h(X_{V \setminus (A\cup\partial^2A)}[s]) \mid {{\mathcal{G}}}_t \right] \right] \notag \\
& = \lim_{s \to \infty} {{\mathbb{E}}}^P \left[ {{\mathbb{E}}}^P \left[ f(X_A[s]) \mid {{\mathcal{G}}}_\infty \right] g(X_{\partial^2A}) {{\mathbb{E}}}^P \left[ h(X_{V \setminus (A\cup\partial^2A)}[s]) \mid {{\mathcal{G}}}_\infty \right] \right] \notag \\
& = {{\mathbb{E}}}^P \left[ {{\mathbb{E}}}^P \left[ f(X_A) \mid {{\mathcal{G}}}_\infty \right] g(X_{\partial^2A}) {{\mathbb{E}}}^P \left[ h(X_{V \setminus (A\cup\partial^2A)}) \mid {{\mathcal{G}}}_\infty \right] \right], \label{eq:infinitetime}\end{aligned}$$ where we have also made repeated use of the boundedness and continuity of $f, g, h$ and the bounded convergence theorem. This shows that $X_A$ and $X_{V \setminus (A \cup \partial^2A)}$ are conditionally independent given $X_{\partial^2A}$ under $P$, that is, $P$ is a [2MRF]{} on $\C^V$. This completes the proof.
Illustrative examples {#subs-ceg}
---------------------
We now provide examples to show that the [2MRF]{} property cannot in general be strengthened.
### The failure of the first-order MRF property for trajectories {#se:example-firstorder}
In general, $P_t$ fails to be a first-order Markov random field on $\C^V_t$ for any $t > 0$, even if the initial states are i.i.d. To see why, notice that the density $dP_t/dP^*_t$ given by does not in general admit a clique factorization. Indeed, for $v \in V$ and $t > 0$, we recall the definition of $M_v$ from and $\EE_t(M_v)$ from , which we write in full as $$\begin{aligned}
\EE_t(M_v) = \exp\Bigg(&\int_0^t(\sigma_v\sigma_v^\top)^{-1}b_v(s,X_v,X_{N_v(G)})\cdot dX_v(s) \\
&- \frac12\int_0^t\left\langle b_v(s,X_v,X_{N_v(G)}),\,(\sigma_v\sigma_v^\top)^{-1}b_v(s,X_v,X_{N_v(G)}) \right\rangle ds\Bigg).\end{aligned}$$ Noting that $\{v\} \cup N_v(G)$ is a $2$-clique but not a $1$-clique, this reveals why one cannot hope for a factorization over $1$-cliques. For example, consider the “nice" case where $\sigma_v \equiv I$ and $b_v(s,x_v,x_{N_v(G)}) = \sum_{u \in N_v(G)}(x_u(s)-x_v(s))$. (Equivalently, $b_v(s,x_v,x_{N_v(G)}) = \nabla_{x_v}h(x)$ is of gradient-type with potential $h(x) = -\frac12\sum_{(u,v) \in E}|x_u-x_v|^2$, where $E$ is the edge set of $G$.) Then the first term in the above exponential splits nicely into a sum of pairwise interactions $\sum_{u \in N_v(G)}\int_0^t(X_u(s)-X_v(s)) \cdot dX_v(s)$, but the second term becomes $$-\frac12\sum_{u,w \in N_v(G)}\int_0^t{\langle}X_u(s)-X_v(s), \, X_w(s)-X_v(s){\rangle}\,ds.$$ It is this term which fails to factorize further over $1$-cliques as opposed to $2$-cliques and thus precludes the first-order Markov property whenever $d\mu_0/d\mu_0^*$ is strictly positive due to Proposition \[th:hammersleyclifford\].
To informally provide a different (but arguably more intuitive) perspective on why the first-order Markov property for past histories fails, consider the case when $G$ is a line segment of length $\ell = 3$, labelling the vertices $-1, 0, 1$. Then, although the driving Brownian motions are all independent and the dynamics of each of the two extreme vertices only depend on its own state and the state of the center vertex, at any time $t$, conditioning on the past history of the states of the center vertex, does not make $X_{-1}(t)$ independent of $X_1(t)$ because the conditioning correlates the Brownian motions $W_{-1}$ and $W_1$ on the interval $[0,t]$. This happens because the past history of $X_0$ is influenced by both $W_{-1}$ and $W_1$ via $X_{-1}$ and $X_1$. On the other hand, to see why the [2MRF]{} property nevertheless does hold, note that if $G$ were a line segment of length $4$, labeling the vertices $\{-2,-1,1,2\}$, then conditioning on the history of the states of the two center vertices $-1$ and $1$ no longer correlates the Brownian motions $W_{-2}$ and $W_2$ since the dynamics of each of the conditioned vertices depends on a different driving Brownian motion. Thus, although the conditioning changes the distribution of $W_{-2}$ and $W_2$ (for instance, they need no longer be Brownian motions), they remain independent, and hence $X_{-2}(t)$ is conditionally independent of $X_2(t)$ in this case.
There are certain situations in which $P_t$ is, in fact, a [1MRF]{}for each $t > 0$ (even though we know from the above examples that this is not in general the case). For example, suppose that for every $v \in V$, there exists a clique $K_v$ of $G$ with $v \in K_v \subset N_v(G)$ such that $b_v(t,x_v,x_{N_v(G)})=\widetilde{b}_v(t,x_v,x_{K_v})$ depends on $x_{N_v(G)}$ only through $x_{K_v}$. Suppose also that $d\mu_0/d\mu_0^*$ admits a $1$-clique factorization. Then, recalling , note that for each $v$ the martingale $M_v$ is measurable with respect to $X_{K_v}$, and deduce from Proposition \[th:hammersleyclifford\] that $P_t$ is a first-order Markov random field. For a concrete example that has the above form, consider the case when $G$ is a triangular lattice with $V = \{\o, 0, 1, \ldots, m\}$, for some $m \in \N$, with the central vertex ${\o}$ having the neighborhood $N_{\o}(G) = \{0, \ldots, m\}$ and for each $v \in V \setminus \{\o\}$, $N_v(G) := \{{\o}, v+1, v-1\}$, where the vertices are to be interpreted mod $m+1$. Further, suppose the initial conditions are i.i.d. and that for some $c \in \R$, $b_v (t, x_v, x_{N_v(G)}) = c(x_{{\o}} + x_{v+1})$ for $v \in V \setminus \{\o\}$ and $b_{\o} (t, x_{\o}, x_{N_{\o}(G)}) = cx_{{\o}}$. Then this provides a specific example with $K_v = \{\o,v+1\} \subset N_v(G)$ for $v \in V \setminus \{\o\}$ and $K_{\o}=\emptyset$. In a similar spirit, the directed cycle graph model of [@DetFouIch18] provides another example.
### The failure of MRF properties for time-$t$ marginals {#se:example-marginals}
It is natural to wonder if and when the time-$t$ marginals $P_t \circ X(t)^{-1} \in \P((\R^d)^V)$ remain a first- or second-order Markov random field, given that this property is true at time $0$, or even given i.i.d. initial conditions. This question is related to propagation of Gibbsianness and Gibbs-non-Gibbs transitions that have been studied in the literature, which is discussed in greater detail in Remark \[rem-gibbsprop\].
Here, we provide a simple example where both the first-order and second-order Markov property fail for time-$t$ marginals. In fact, in this simple model we will see that there is no non-trivial conditional independence structure. Consider the segment with $5$ vertices: $G=(V,E)$ given by $V=\{1,2,3,4,5\}$ and $E = \{(i,i+1) : i=1,2,3,4\}$, and consider the SDE system $$\begin{aligned}
\begin{split}
dX_1(t) & = (X_2(t)-2X_1(t))\,dt+dW_1(t), \\
dX_i(t) & = (X_{i-1}(t)+X_{i+1}(t) - 2X_i(t))\,dt+dW_i(t), \qquad i=2,3,4, \\
dX_5(t) & = (X_4(t)-2X_5(t))\,dt+dW_5(t),
\end{split}
\label{def:ex:SDE}\end{aligned}$$ with $X_i(0)=0$ for each $i$. Once again, note that the drift here is of gradient type with potential $h(x) =\sum_{i=1}^4 x_i x_{i+1}-\sum_{i=1}^5 x_i^2$. Letting $\bm{X}(t)$ denote the column vector $(X_1(t),\ldots,X_5(t))$ and similarly for $\bm{W}(t)$, we may write this in vector form as $$\begin{aligned}
d\bm{X}(t) = L\bm{X}(t)\,dt + d\bm{W}(t), \label{def:multi-OU}\end{aligned}$$ where $L = A - 2I$ is the adjacency matrix $A$ of the graph minus twice the identity $I$: $$L = \begin{pmatrix}
-2 & 1 & 0 & 0 & 0 \\
1 & -2 & 1 & 0 & 0 \\
0 & 1 & -2 & 1 & 0 \\
0 & 0 & 1 & -2 & 1 \\
0 & 0 & 0 & 1 & -2
\end{pmatrix}.$$ The solution of the SDE is given by $$\begin{aligned}
\bm{X}(t) = e^{Lt} \int_0^t e^{-Ls} \,d\bm{W}(s), \qquad t > 0.\end{aligned}$$ Noting that $L$ is symmetric and invertible, we deduce that $\bm{X}(t)$ is jointly Gaussian with mean zero and covariance matrix $$\begin{aligned}
\E[\bm{X}(t)\bm{X}(t)^\top] = \int_0^t e^{2Ls} \,ds = \frac12 L^{-1}(e^{2Lt}-I). \label{def:multi-OU-cov}\end{aligned}$$ This covariance matrix can easily be computed explicitly by noting that the tridiagonal Toeplitz matrix $A$ is explicitly diagonalizable. To spare the reader any tedium, we provide only some pertinent snapshots. At time $t=2$ the covariance matrix is $$\begin{aligned}
\E[\bm{X}(2)\bm{X}(2)^\top] = \begin{pmatrix}
0.3611 & 0.2388 & 0.1435 & 0.0767 & 0.0324 \\
0.2388 & 0.5046 & 0.3156 & 0.1759 & 0.0767 \\
0.1435 & 0.3156 & 0.5370 & 0.3156 & 0.1435 \\
0.0767 & 0.1759 & 0.3156 & 0.5046 & 0.2388 \\
0.0324 & 0.0767 & 0.1435 & 0.2388 & 0.3611
\end{pmatrix}. \label{def:ex-covmatrix}\end{aligned}$$ Using the well known formula for conditional measures of joint Gaussians, we compute from this that $$\begin{aligned}
\mathrm{Cov}(X_1(t),X_3(t)|X_2(t))=
\begin{pmatrix}
0.2481 & -0.0058 \\
-0.0058 & 0.3397
\end{pmatrix},\end{aligned}$$ which reveals that $X_1(t)$ and $X_3(t)$ are not conditionally independent given $X_2(t)$. Hence, $(X_i(t))_{i \in G}$ is not a first-order Markov random field. Similarly, by computing $$\begin{aligned}
\mathrm{Cov}(X_1(t),X_4(t)|X_2(t),X_3(t))=
\begin{pmatrix}
0.2480 & -0.0030 \\
-0.0030 & 0.3189
\end{pmatrix},\end{aligned}$$ we see that $X_1(t)$ and $X_4(t)$ are not conditionally independent given $(X_2(t),X_3(t))$. Hence, $(X_i(t))_{i \in G}$ is not a second-order Markov random field.
In fact, in this example, there is no non-trivial conditional independence structure, in the sense that there are no two vertices $i,j$ such that $X_i(t)$ and $X_j(t)$ are conditionally independent given $\{X_k(t) : k \in G\setminus\{i,j\}\}$ for some $t > 0$. This can be read off from the the so-called *precision matrix*, which is simply the inverse of the covariance matrix, $Q(t) := (\E[\bm{X}(t)\bm{X}(t)^{\top}])^{-1}$. As is well known and can easily be seen from the form of the multivariate Gaussian density, the precision matrix reveals the conditional independence structure (see, e.g., [@lauritzen1996graphical Proposition 5.2]), in the following sense: For $t > 0$ define the graph $\widetilde{G}(t) = (V,\widetilde{E}(t))$ with the same vertex set $V$ but with $(i,j) \in \widetilde{E}(t)$ if and only if $Q_{i,j}(t) \neq 0$. Then $\bm{X}(t)$ is a (first-order) Markov random field with respect to the graph $\widetilde{G}(t)$. In our example, $\widetilde{G}(t)$ is the complete graph for each $t > 0$, and this Markov property is vacuous. (Note, however, that $Q(t) \to 2L$ as $t\to\infty$ because $L$ is negative definite, and the unique invariant measure of this diffusion is a first-order Markov random field with respect to the original graph $G$.)
A variation on this example gives rise to another interesting phenomenon. Suppose we modify the example by replacing the diagonal entries of $L$ with zeros, i.e., remove all the $-2X$ terms from the drifts in . Then the covariance matrix is again invertible, and now $Q_{1,4}(t)=Q_{2,5}(t)=0$ for all $t > 0$, where we continue with the notation of the previous paragraph. That is, $\widetilde{G}(t)$ is not the complete graph, but rather the complete graph with the edges $(1,4)$ and $(2,5)$ removed, for each $t > 0$. In particular, $X_1(t)$ and $X_4(t)$ are conditionally independent given $(X_2(t),X_3(t),X_5(t))$, for each $t > 0$.
\[rem-gibbsprop\] As mentioned in the introduction, one motivation for studying such conditional independence questions is that (a stronger version of) the MRF structure of interacting SDEs can lead to a more succinct autonomous “local characterization" of the dynamics at a vertex and its neighborhood, as developed in the quite different setting of unimodular Galton-Watson trees in [@LacRamWu19a]. From this perspective, it would be of interest to investigate if there are non-trivial special cases when the first-order or second-order MRF property for time-$t$ marginals propagates. A different but related question that has been studied in the literature is propagation of Gibbsianness for an infinite system of interacting real-valued diffusions indexed by $\Z^d$. Specifically, the work [@DerRoe05] considers a collection of interacting diffusions, indexed by $\Z^d$, with identity covariance and a drift that is the gradient of a Hamiltonian function associated with a certain interaction potential $\Phi$, and with an initial distribution that is also a Gibbs measure (as in Section \[se:gibbs\]) with respect to a Gibbsian specification (in the sense of [@georgii2011gibbs Chapter 2]) associated with another interaction potential $\Phi_0$, where both interaction potentials $\Phi$ and $\Phi_0$ are assumed to be of finite range and satisfy certain smoothness conditions. It is shown in [@DerRoe05] that when either $t$ or the interaction strength is sufficiently small, the time-$t$ marginals are strongly Gibbsian, that is, associated with Gibbsian specifications that have an absolutely summable, though not necessarily finite range, interaction potential. Extensions of these results to the case of interacting real-valued diffusions on $\Z^d$ with non-Markovian drifts with finite memory (again with finite range interactions and identity covariance) were later obtained in [@RedRoeRus10] and [@RoeRus14]. The restrictions on the time and interaction strength in these works arise from the fact that perturbative arguments are used. However, in general for moderate interaction strengths and moderate times, the time-$t$ marginals can fail to be Gibbsian (see, e.g., [@van2009gibbsianness], as well as the survey [@van2010gibbs], which also discusses related results for spin systems).
Finite-graph approximations for Markov random fields {#se:2MRFs}
====================================================
In this section we establish some important preparatory results that are used in the proof of Theorem \[th:conditionalindependence-infinitegraph\], which extends the finite graph results of Theorem \[pr:conditionalindependence-finitegraph\] to the infinite graph setting. Fix $(G, b, \sigma, \mu_0)$ that satisfy Assumption \[assumption:B\] and suppose $G = (V,E)$ is countably infinite. Recall that $P = P^{\mu_0} \in \P(\C^V)$ and $P^{*,\mu_0} \in \P(\C^V)$ denote the unique law of the SDE systems and , respectively, both with initial laws $\mu_0$, which are well-posed by Assumptions (\[assumption:B\].4) and (\[assumption:B\].3). To show that $P_t = P^{\mu_0}_t$ forms a [2MRF]{} on $\C_t^V$, we can no longer apply the clique factorization arguments used for finite graphs because the formula does not extend to infinite graphs. Even worse, the density $dP_t/dP^*_t$ therein does not exist, and it seems impossible to establish directly that by projecting to a finite set $A \subset V$ we have a density $dP_t[A]/dP^*_t[A]$ that admits a 2-clique factorization. Instead, we approximate the measure on the infinite graph by [2MRF]{}s on a growing sequence of finite graphs, arguing that the desired [2MRF]{} property passes to the limit. To highlight some of the subtleties that arise in such an approximation argument, and to better motivate the other results established in this section, we first desribe the approximating sequence of measures in Section \[subs-approxmeas\]. Then in the subsequent two sections we establish some general properties of finite-graph [2MRF]{}s to be used in the proof of Theorem \[th:conditionalindependence-infinitegraph\] in Section \[subs-pf-infingraph\], which are also of independent interest.
Construction of the approximating sequence of SDEs {#subs-approxmeas}
--------------------------------------------------
We fix $(G,b,\sigma, \mu_0)$ that satisfy Assumption \[assumption:B\]. As in Section \[ap-finite\], we will work with the canonical measure space $\C^V = (\C^V,\text{Borel}, P^{*,\mu_0})$, and let $(X_v)_{v \in V} : \C^V \rightarrow \C^V$ again denote the canonical processes. Also, recall from Section \[subs-not\] that given any measurable space $\X$, measure $\mu \in \P( \X^V)$ and subset $U \subset V$, $\mu[U] \in \P(\X^U)$ denotes the restriction of $\mu$ to the set $\X^U$.
Let $V_n \subset V, n \in \N,$ be such that $\bigcup_{n} V_n = V$, and let $G_n = (V_n, E_n)$, for some edge set $E_n$ to be specified later. Also, for each $n \in \N$ and $v \in V_n$, let $b^n_v: \R_+ \times \C \times \C^{N_v(G_n)} \mapsto \R^d$ be any progressively measurable map that satisfies the same conditions as $b_v$ in Assumption (\[assumption:B\].2). Fix $t > 0$ and for each $n$, define $P_t^n = P_t^{\mu_0,n} \in \P(\C_t^V)$ by $$\begin{aligned}
\label{dPdPstar}
\frac{dP^n_t}{dP^{*,\mu_0}_t} = \frac{dP_t^{\mu_0,n}}{dP^{*,\mu_0}_t}= \prod_{v \in V_n}\EE_t\left(\int_0^\cdot (\sigma_v\sigma_v^\top)^{-1}b^n_v(s,X_v,X_{N_v(G_n)}) \cdot dX_v(s)\right), \end{aligned}$$ where, as before, $(\sigma_v \sigma_v^\top)^{-1} b^n_v(s,x,x_{N_v(G_n)})$ denotes the map . We can apply Lemma \[lem:Girsanov-justification\] with $\QQ= P^{*,\mu_0}$, $X=(X_v)_{v \in V_n}$ and $f(t,x) = (\sigma_v^{-1}b^n_v(t,x_v,x_{N_v(G_n)}))_{v \in V_n}$, to conclude that the stochastic exponential in is a true $P^{*,\mu_0}$-martingale, due to the linear growth, non-degeneracy and boundedness properties of $b^n_v$ and $\sigma_v$ in Assumptions (\[assumption:B\].2) and (\[assumption:B\].3). Hence, the family $(P^n_t)_{t > 0}$ is consistent in the sense that the restriction of $P^n_t$ to $\C_s^V$ is precisely $P^n_s$ for each $t > s > 0$. Thus, by the Kolmogorov extension theorem, $(P^n_t)_{t > 0}$ uniquely determines a probability measure $P^n$ on $\C^V$. Now, from , and Girsanov’s theorem [@karatzas-shreve Corollary 3.5.2], it follows that under $P^n$ the canonical process solves the SDE system $$\label{fingraph-SDEn}
\begin{array}{rcl}
dX_v(t) &=& b^n_v(t,X_v,X_{N_v(G_n)})\,dt + \sigma_v(t,X_v)\,dW_v(t), \quad v \in V_n, \\
dX_v(t) &=& \sigma_v(t,X_v)\,dW_v(t), \quad v \in V \setminus V_n,
\end{array}$$ with $(X_v(0))_{v \in V} \sim \mu_0$, where $(W_v)_{v \in V}$ are independent Brownian motions under $P^n$. Note that for $v \in V_n$, the third argument of $b^n_v$ looks only at the states in $N_v(G_n)$, and thus $b^n_v$ depends only on the states of vertices in $G_n$. Thus, $P^n[V_n]$ is precisely the law of the finite-graph SDE system with inputs $(G,(b^n_v)_{v \in V},(\sigma_v)_{v \in V},\mu_0[V_n])$.
In order to implement our approximation argument we would like to choose $G_n$ and $(b^n_v)_{v \in V}$ such that both $P^n \to P$ and each $P^n_t[V_n]$ is a [2MRF]{}. In order to have $P^n \to P$ we should naturally choose $V_n$ increasing to $V$ and $b^n_v$ to behave like $b_v$ for most $v$. But the [2MRF]{} property is more delicate. It would follow from the finite-graph result of Theorem \[pr:conditionalindependence-finitegraph\] that $P^n_t[V_n]$ is a [2MRF]{} on $\C^{V_n}$ only if $\mu_0[V_n]$ were a [2MRF]{} on $(\R^d)^{V_n}$. But $\mu_0[V_n]$ is not necessarily a [2MRF]{} for arbitrary $V_n$ (e.g., with $G_n$ the induced subgraph), even though $\mu_0$ is a [2MRF]{} on the full graph $G$ by assumption; in other words, the [2MRF]{} property is not in general preserved under projections, as illustrated in Example \[eg-fingraph\] below. However, in Section \[se:MRFs-projections\] we show that for any Markov random field on an infinite graph $G = (V,E)$, it is possible to identify a suitable increasing sequence of vertices $(V_n)_{n \in \N}$ and associated graph $G_n = (V_n,E_n)$ for each $n \in \N$ that is a slight modification of the induced subgraph on $V_n$, such that the desired projection property holds. Then, in Section \[se:conditional distributions of 2MRFS\] we prove some results on preservation of a class of conditional distributions of [2MRF]{}s under restriction to induced subgraphs. The above results are combined with tightness and convergence estimates for the approximating sequence $\{P^n\}$ obtained in Section \[subs-tightness\] to complete the proof of Theorem \[th:conditionalindependence-infinitegraph\] in Section \[subs-pf-infingraph\].
Projections of Markov random fields {#se:MRFs-projections}
-----------------------------------
We first provide a simple example to illustrate that the restriction of an MRF to an induced subgraph need not remain an MRF.
\[eg-fingraph\] Suppose $G$ is a finite two-dimensional lattice, with vertex set $V$ identified with $\{-n,\ldots,n\}^2$ and the usual nearest-neighbor edge set, and let $({Y}_v)_{v \in G}$ be a [1MRF]{} on $\R^V$. Consider the line subgraph $H=\{(i,0) : i=-n,\ldots,n\}$ in $G$, and consider the restriction $({Y}_v)_{v \in H}$ of the [1MRF]{} to $H$. Note that every path in $H$ that starts in $A := \{(0,0)\}$ and ends in $B = \{ (2,0)\}$ must traverse through the vertex of $S:= \{(1,0)\}$. Thus, by the cutset characterization of [1MRF]{}’s given in Remark \[rem-cutset\], for $({Y}_v)_{v \in H}$ to be an [1MRF]{} on $H$, ${Y}_{(0,0)}$ must be conditionally independent of ${Y}_{(2,0)}$ given ${Y}_{(1,0)}$. However, by the same cutset characterization, it is clear that this conditional independence cannot be deduced from the [1MRF]{} property of $({Y}_v)_{v \in G}$ on $G$ since there are paths in $G$ that start in $A$ and end in $B$ that are disjoint from $S$. Similarly, if we assume $({Y}_v)_{v \in G}$ is a [2MRF]{}, the configuration $({Y}_v)_{v \in H}$ can fail to be a [2MRF]{}.
This example does suggest, however, that we can restore the MRF property by enlarging the edge set of the induced subgraph to reflect the lost connectivity. The following lemma gives one way to do this which is certainly not the only way, but it serves our purpose. For a random element $({Y}_v)_{v \in V}$ of $\X^V$ with law ${\nu}\in \P(\X^V)$, and for a set $A \subset V$, recall that we write ${\nu}[A]$ to denote the law of ${Y}_A$, the coordinates in $A$.
\[le:MRFprojections\] Fix a rooted graph $G=(V,E,\o)$ and $n \ge 4$. Define $V_n := \{v \in V : d(v,\o) \le n\}$ and $U_n := V_n \setminus V_{n-2}$, where $d$ denotes the graph distance. Define a graph $G_n=(V_n,E_n)$, where $$E_n := \{(u,v) \in V_n \times V_n : (u,v) \in E\} \cup \{(u,v) \in U_n \times U_n, u \neq v \}.$$
(i) For any $A \subset V_{n-3}$, it holds that $\partial^2_G A = \partial^2_{G_n} A$. Also, for any $A' \subset V_{n-2}$, $\partial^2_G A' \subset \partial^2_{G_n} A'$.
(ii) If $K \in \mathrm{cl}_2(G)$ satisfies $K \subset V_n$, then $K \in \mathrm{cl}_2(G_n)$.
(iii) If a $\X^V$-valued random variable $({Y}_v)_{v \in V}$ is a [2MRF]{} with respect to $G$, then $({Y}_v)_{v \in V_n}$ is a [2MRF]{} with respect to $G_n$.
(iv) Suppose $V$ is finite and the law ${\nu}$ of ${Y}^V$ admits the following $2$-clique factorization with respect to a product measure ${\nu}^* = \prod_{v \in V}{\theta}_v \in \P(\X^V)$ for some ${\theta}_v \in \P(\X)$, $$\frac{d{\nu}}{d{\nu}^*}(x_V) = \prod_{K \in \mathrm{cl}_2(G)}f_K(x_K),$$ for some measurable functions $f_K : \X^K \rightarrow \R_+$, for $K \in \mathrm{cl}_2(G)$. Then ${\nu}[V_n]$ admits a $2$-clique factorization $$\frac{d{\nu}[V_n]}{d{\nu}^*[V_n]}(x_{V_n}) = \prod_{K \in \mathrm{cl}_2(G_n)}f^0_K(x_K),$$ for some measurable functions $f^0_K : \X^K \rightarrow \R_+$, for $K \in \mathrm{cl}_2(G_n)$, which additionally satisfy the consistency condition $f^0_K \equiv f_K$ for $K \in \mathrm{cl}_2(G)$ such that $K \subset V_{n-3}$.
[ ]{}
(i) From the definition of $E_n$ it follows quickly that (a) for $A' \subset V_{n-2}$, $\partial_G A' = \partial_{G_n} A'$, and (b) for $A'' \subset V_{n-1}$, $\partial_G A'' \subset \partial_{G_n} A''$. Iterate these observations to prove the claims.
(ii) Let $d_G$ and $d_{G_n}$ denote the graph distance in $G$ and $G_n$, repsectively. From the definition of $E_n$, it is straightforward to argue that $d_{G_n} \le d_G$ on $V_n \times V_n$. Indeed, for any $u,v \in V_n$ and any path from $u$ to $v$ in $G$, there is a path from $u$ to $v$ in $G_n$ which is not longer. This implies for every $u, v \in V_n$, $d_G(u,v) \leq 2$ implies $d_{G_n}(u,v) \leq 2$, which proves property (ii).
(iii) Let $({Y}_v)_{v \in V}$ be a [2MRF]{} with respect to $G$. Let $A \subset V_n$, $B = V_n \setminus (A \cup \partial^2_{G_n} A)$, and $S := \partial^2_{G_n} A$. Assuming without loss of generality that $A$ and $B$ are nonempty, we must prove that ${Y}_A$ and ${Y}_B$ are conditionally independent given ${Y}_S$. First notice that one cannot have both $A \cap U_n \ne \emptyset$ and $B \cap U_n \ne \emptyset$, as this would imply $d_{G_n}(A,B) \le 1$, contradicting the definition of $B$. Therefore we must have either $A \cap U_n = \emptyset$, $B \cap U_n = \emptyset$, or both.
*Case 1:* Suppose $A \cap U_n = \emptyset$. This means $A \subset V_{n-2}$ and hence $\partial^2_G A \subset S$ by (i). Since ${Y}_A$ and ${Y}_{V \setminus (A \cup \partial^2_G A)}$ are conditionally independent given ${Y}_{\partial^2_G A}$, we then have conditional independence of ${Y}_A$ and ${Y}_B$ given ${Y}_S$. Indeed, this uses the elementary fact that if $(Z_1,Z_2,Z_3,Z_4)$ are random variables with $Z_1$ conditionally independent of $(Z_2,Z_3)$ given $Z_4$, then $Z_1$ is conditionally independent of $Z_2$ given $(Z_3,Z_4)$.
*Case 2:* Suppose $B \cap U_n = \emptyset$. This means $B \subset V_{n-2}$ and hence, again by (i), $\partial^2_G B \subset \partial^2_{G_n} B$. Also note that $\partial^2_{G_n} B \subset S$ (since otherwise $A \cap \partial^2_{G_n} B \ne \emptyset$, which contradicts the definition of $B$). Since the [2MRF]{} property with respect to $G$ implies ${Y}_B$ and ${Y}_{V \setminus (B \cup \partial^2_G B)}$ are independent conditioned on ${Y}_{\partial^2_G B}$, we then have conditional independence of ${Y}_B$ and ${Y}_A$ given ${Y}_S$. Since $A \subset V_n$ was arbitrary, this proves that $({Y}_v)_{v \in V_n}$ is a [2MRF]{} with respect to $G_n$.
(iv) Let $\K_n$ denote the set of $K \in \mathrm{cl}_2(G)$ such that $K \subset V_n$. Recalling that ${\nu}^*$ is a product measure, using the assumed clique factorization of ${\nu}$, we can then write $$\begin{aligned}
\frac{d{\nu}[V_n]}{d{\nu}^*[V_n]}(x_{V_n}) &= \int_{\X^{V \backslash V_n}} \prod_{K \in \mathrm{cl}_2(G)}f_K(x_K) \,{\nu}^*[V \backslash V_n](dx_{V \backslash V_n}) \\
&= \prod_{K \in \K_n}f_K(x_K)\int_{\X^{V \backslash V_n}} \prod_{K \in \mathrm{cl}_2(G) \backslash \K_n}f_K(x_K) \,{\nu}^*[V \backslash V_n](dx_{V \backslash V_n}).\end{aligned}$$ Now note that any $K \in \mathrm{cl}_2(G) \backslash \K_n$ is not contained in $V_n$, and as a $2$-clique it can have no neighbors in $V_{n-2}$. Recalling that $U_n = V_n \backslash V_{n-2}$, we see that the integral expression is $x_{U_n}$-measurable; that is, there is a measurable function $g_n : \X^{U_n} \rightarrow \R_+$ such that $$g_n(x_{U_n}) = \int_{\X^{V \backslash V_n}} \prod_{K \in \mathrm{cl}_2(G) \backslash \K_n}f_K(x_K) \,{\nu}^*[V \backslash V_n](dx_{V \backslash V_n}).$$ Note that $U_n \in \mathrm{cl}_2(G_n)$ by definition of $G_n$. Since clearly $\K_n \subset \mathrm{cl}_2(G_n)$, we find that the expression $$\begin{aligned}
\frac{d{\nu}[V_n]}{d{\nu}^*[V_n]}(x_{V_n}) &= \prod_{K \in \K_n}f_K(x_K)g_n(x_{U_n})\end{aligned}$$ exhibits a $2$-clique factorization of ${\nu}[V_n]$ over the graph $G_n$ satisfying the desired consistency condition.
Conditional distributions of second-order Markov random fields {#se:conditional distributions of 2MRFS}
--------------------------------------------------------------
First, in Lemma \[le:specification-abstract\], given a [2MRF]{} with respect to a graph, and another [2MRF]{} on a subgraph, or more generally given MRFs on two overlapping graphs, we identify conditions under which the conditional distributions of a subset in the intersection (given its complement) coincide for both [2MRF]{}s. This will be used to establish, for a suitable choice of $b^n$, a certain consistency condition for the sequence of approximating measures $\{P^n\}$ used in the proof of Theorem \[th:conditionalindependence-infinitegraph\]. Let us briefly recall a notation we introduced more carefully just before Theorem \[th:gibbsuniqueness\]: For $\nu \in \P(\X^V)$ and $A,B \subset V$ we write $\nu[A \, | \, B]$ for the conditional law of the $A$-coordinates given the $B$-coordinates.
\[le:specification-abstract\] Let $G=(V_G,E_G)$ and $H=(V_H,E_H)$ be finite graphs, and assume $V^* \subset V_G \cap V_H$ satisfies $$\begin{aligned}
E_G \cap (V^* \times V^*) = E_H \cap (V^* \times V^*). \label{asmp:edgesets}\end{aligned}$$ Moreover, let $A \subset V^*$ satisfy $\partial^2_G A \subset V^*$ and $\partial^2_H A \subset V^*$. Then $\partial^2_H A = \partial^2_G A =: \partial^2A$, and it holds that $$\begin{aligned}
\{K \in \mathrm{cl}_2(G) : K \cap A \neq \emptyset\} = \{K \in \mathrm{cl}_2(H) : K \cap A \neq \emptyset\} =: \mathcal{K}_A. \label{eq:cliques}\end{aligned}$$ Next, let ${\nu}^H \in \P(\X^{V_H})$ and ${\nu}^G \in \P(\X^{V_G})$, and suppose there exists a product measure ${\nu}^* = \prod_{v \in V_G \cup V_H}{\theta}_v \in \P(\X^{V_G \cup V_H})$ for some ${\theta}_v \in \P(\X)$, $v \in V_G \cup V_H$, such that the densities factorize as $$\begin{aligned}
\frac{d{\nu}^H}{d{\nu}^*[V_H]}(x_{V_H}) = \prod_{K \in \mathrm{cl}_2(H)}f^H_K(x_K), \quad\quad \frac{d{\nu}^G}{d{\nu}^*[V_G]}(x_{V_G}) = \prod_{K \in \mathrm{cl}_2(G)}f^G_K(x_K), \end{aligned}$$ for measurable functions $(f_K^H: \X^K \mapsto \R_+)_{K \in \mathrm{cl}_2(H)}$ and $(f_K^G: \X^K \mapsto \R_+)_{K \in \mathrm{cl}_2(G)}$ satisfying $f_K^H \equiv f_K^G$ for all $K \in \mathcal{K}_A$. Then ${\nu}^H[ A\, | \, \partial^2A] = {\nu}^G[ A\, | \, \partial^2A]$, almost surely with respect to ${\nu}^*[\partial^2 A]$.
Let $A \subset V^*$ satisfy $\partial^2_G A \subset V^*$ and $\partial^2_H A \subset V^*$. It is immediate from that $\partial^2_H A = \partial^2_G A$, and we write simply $\partial^2A$ for this set. To check , note that if $K \in \mathrm{cl}_2(G)$ intersects $A$, then $K \subset A \cup \partial^2A \subset V^*$. By the edge sets of $G$ and $H$ agree when restricted to $V^*$, and we deduce that $K \in \mathrm{cl}_2(H)$; this proves $\subset$ in , but the reverse inclusion follows by the same argument. Note that, with $\mathcal{K}_A$ as defined in , we have also shown that $$\label{prop-cliques}
K \in \mathcal{K}_A \Rightarrow K \subset V^*.$$
Let us work in the rest of the proof on the canonical probability space $(\X^{V_G \cup V_H},\mathrm{Borel}, \nu^*)$, with $\E$ denoting expectation on this space, and all equations are understood to hold $\nu^*$-almost surely. Let ${I}=({I}_v)_{v \in V_G\cup V_H}$ denote the identity map on $\X^{V_G\cup V_H}$. By Proposition \[th:hammersleyclifford2\], $\nu^H$ is a [2MRF]{}, and so ${\nu}^H[ A\, | \, \partial^2A] ({I}_{\partial^2A}) = {\nu}^H[ A \, | \, V_H \backslash A] ({I}_{V_H \backslash A})$ a.s. Hence, $$\begin{aligned}
\dfrac{ d{\nu}^H[ A\, | \, \partial^2A] ({I}_{\partial^2A})}{d{\nu}^*[A]} ({I}_A)
&=\dfrac{\frac{d{\nu}^H}{d{\nu}^*[V_H]}({I}_{V_H})}{\E\left[\frac{d{\nu}^H}{d{\nu}^*[V_H]} ({I}_{V_H})\,| \,{I}_{V_H \backslash A}\right]} \\
&= \dfrac{\prod_{K \in \mathrm{cl}_2(H)}f^H_K({I}_K)}{\E[\prod_{K \in \mathrm{cl}_2(H)}f^H_K({I}_K) \, | \, {I}_{V_H \backslash A}]},\end{aligned}$$ where we emphasize that the expectation in the denominator is with respect to independent random variables $({I}_v)_{v \in V_G \cup V_H}$.
The key observation is that if $K \in \mathrm{cl}_2(H)$ does not intersect $A$, then the term $f_K^{H}({I}_K)$ factors out of the conditional expectation and cancels. Hence, with $\mathcal{K}_A$ as in , we see that $$\begin{aligned}
\dfrac{ d{\nu}^H[ A\, | \, \partial^2A] ({I}_{\partial^2A})}{d{\nu}^*[A]} ({I}_A)
&= \frac{\prod_{K \in \mathcal{K}_A}f^H_K({I}_K)}{\E[\prod_{K \in \mathcal{K}_A}f^H_K({I}_K) \, | \, {I}_{V_H \backslash A}]}. \label{pf:specification-abstract-1}\end{aligned}$$ Since ${I}_{V_H \setminus V^*}$ is independent of ${I}_{V^*}$, in view of , we may equivalently condition on ${I}_{V^* \backslash A}$ in the denominator of the term on the right-hand side of to obtain $$\begin{aligned}
\dfrac{ d{\nu}^H[ A\, | \, \partial^2A] ({I}_{\partial^2A})}{d{\nu}^*[A]} ({I}_A)
&= \frac{\prod_{K \in \mathcal{K}_A}f_K^H({I}_K)}{\E[\prod_{K \in \mathcal{K}_A}f_K^H({I}_K) \, | \, {I}_{V^* \backslash A}]}.\end{aligned}$$ Repeating the same arguments that led us to this point, we also find that $$\begin{aligned}
\dfrac{ d{\nu}^G[ A\, | \, \partial^2A] ({I}_{\partial^2A})}{d {\nu}^*[A]} ({I}_A)
&= \frac{\prod_{K \in \mathcal{K}_A}f^G_K({I}_K)}{\E[\prod_{K \in \mathcal{K}_A}f^G_K({I}_K) \, | \, {I}_{V^* \backslash A}]}.\end{aligned}$$ Recalling that $f^H_K \equiv f^G_K$ for $K \in \mathcal{K}_A$ by assumption, the proof is complete.
The last lemma allows us to deduce the following insensitivity result that shows that given a finite graph $G = (V,E)$ and associated SDE , the conditional law of trajectories of particles in a set $A \subset V$ given the trajectories of particles at the double-boundary $\partial^2 A$ of the set does not depend on the graph structure outside of $A \cup \partial^2A$.
\[pr:specification-finitegraph\] Let $G=(V_G,E_G)$ and $H=(V_H,E_H)$ be finite graphs, and assume $V^* \subset V_G \cap V_H$ satisfies . Let $A \subset V^*$ satisfy $\partial^2_G A \subset V^*$ and $\partial^2_H A \subset V^*$, so that Lemma \[le:specification-abstract\] ensures that $\partial^2_H A = \partial^2_G A =: \partial^2A$ and that holds (defining $\mathcal{K}_A$ as therein). Suppose $(G, b^G, \sigma^G, \mu_0^G)$ and $(H, b^H, \sigma^H, \mu_0^H)$ both satisfy Assumption \[assumption:A\], and let $P^G \in \P(\C^{V_G})$ and $P^H \in \P(\C^{V_H})$ be the corresponding unique laws of the SDE described in . Further, suppose the following consistency conditions hold:
(i) We have $$\begin{aligned}
b^H_v \equiv b^G_v, \quad \text{ for } v \in A \cup \partial^2 A, \label{def:b-consistency} \\
\sigma^H_v \equiv \sigma^G_v, \quad \text{ for } v \in V_G \cap V_H. \label{def:sigma-consistency}\end{aligned}$$
(ii) There is a product measure $\mu_0^* = \prod_{v \in V_G \cup V_H}\lambda_v \in \P((\R^d)^{V_G \cup V_H})$ for some $\lambda_v \in \P(\R^d), v \in V_G \cup V_H,$ such that $\mu^G_0$ and $\mu^H_0$ admit $2$-clique factorizations: $$\begin{aligned}
\label{mu0-factorization}
\frac{d\mu_0^G}{d\mu_0^*[V_G]}(x_{V_G}) = \prod_{K \in \mathrm{cl}_2(G)}f^G_K(x_K), \quad\quad \quad \frac{d\mu_0^H}{d\mu_0^*[V_H]}(x_{V_H}) = \prod_{K \in \mathrm{cl}_2(H)}f^H_K(x_K),
\end{aligned}$$ for some measurable functions $(f^G_K: (\R^d)^K \mapsto \R_+)_{K \in \mathrm{cl}_2(G)}$ and $(f^H_K: (\R^d)^K \mapsto \R_+)_{K \in \mathrm{cl}_2(H)}$ that satisfy the consistency condition $f^G_K \equiv f^H_K$ for every $K \in \mathcal{K}_A$.
Then $P^G_t[A \, | \, \partial^2A] = P^H_t[A \, | \, \partial^2A]$ for each $t > 0$, both in the sense of $P^H_t[\partial^2A]$-almost sure and $P^G_t[\partial^2A]$-almost sure equality.
As in , let $P^* \in \P(\C^{V_G \cup V_H})$ be the unique law of the solution $X=(X_v)_{v \in V_G \cup V_H}$ of the driftless SDE $$\begin{aligned}
dX_v(t) = \sigma_v^G(t,X_v)\,dW_v(t), \ \ v \in V_G, \qquad dX_v(t) = \sigma_v^H(t,X_v)\,dW_v(t), \ \ v \in V_H \setminus V_G,\end{aligned}$$ initialized with $X(0) \sim \mu_0^*$. Again working on the canonical probability space $(\C^{V_G \cup V_H},\text{Borel},P^*)$, define the martingales $M^H = (M^H_v)_{v \in H}$: $$M^H_v(t) := \int_0^t (\sigma_v^H(\sigma_v^H)^\top)^{-1}b^H_v(s,X_v,X_{N_v(H)})\cdot dX_v(s), \quad v \in V_H,$$ with $M^G = (M^G_v)_{v \in V_G}$, defined analogously, as in . Using and we can write $$\begin{aligned}
\frac{dP^H_t}{dP^*_t[V_H]} &= \prod_{K \in \mathrm{cl}_2(H)}f^H_K(X_K(0))\,\prod_{v \in V_H}\EE_t(M^H_v), \\
\frac{dP_t^G}{dP^*_t[V_G]} &= \prod_{K \in \mathrm{cl}_2(G)}f^G_K(X_K(0))\,\prod_{v \in V_G}\EE_t(M^G_v).\end{aligned}$$ with $\EE_t$ defined as in . Note that if $v \in A \cup \partial A$, then we have $N_v(H)=N_v(G)$; indeed, this is due to and the inclusions $\partial_G A \subset V^*$ and $\partial_H A \subset V^*$. Thus, by the consistency conditions and along with the expressions above for $M^H$ and $M^G$, we have $\EE_t(M^H_v) = \EE_t(M^G_v)$ for $v \in A \cup \partial A$. Applying Lemma \[le:specification-abstract\] with ${\mathcal X} = {\mathcal C}_t$, $\nu^* = P^*$, $\nu^H = P^H_t$ and $\nu^G = P^G_t$, it follows from the consistency conditions (i) and (ii) that $P^G_t[A \, | \, \partial^2A] = P^H_t[A \, | \, \partial^2A]$ holds in the sense of $P^*_t[\partial^2A]$-almost sure equality. Since both $P^H_t[\partial^2A]$ and $P^G_t[\partial^2A]$ are absolutely continuous with respect to $P^*_t[\partial^2A]$, the claim follows.
Markov random field property for infinite-dimensional diffusions {#sec-pf-infinitegraph}
================================================================
Fix a countably infinite connected graph $G = (V,E)$, and let $(G,b,\sigma,\mu_0)$ be as in Assumption \[assumption:B\]. As usual, let $P = P^{\mu_0}$ and $P_t = P^{\mu_0}_t$ denote the unique law of the SDE and its projection, and let $P^{*,\mu_0}$ be the law of the canonical SDE system started from initial law $\mu_0$. In this section we will prove Theorem \[th:conditionalindependence-infinitegraph\], that is, the [2MRF]{} property for $P_t$ and $P$. We will also use the same canonical space $(\C^V, \text{Borel}, P^{*,\mu_0})$ and canonical processes $(X_v)_{v \in V}$ as in Section \[subs-approxmeas\].
Throughout, choose an arbitrary vertex $\o$ in $V$ to be the root, and let $G_n = (V_n, E_n)$ and $U_n = V_n\setminus V_{n-2}$ be as defined in Lemma \[le:MRFprojections\]. Also, set $$\label{driftn}
b_v^n = b_v, \quad v \in V_{n-2}, \qquad b_v^n = 0, \quad v \in U_n,$$ (The family $\{b_v^n: n \geq 3, v \in U_n\}$ is arbitrary and set to zero for convenience, but more generally must merely be measurable and uniformly bounded.) Let $\{P^n\}_{n \in \N}$ and $\{P^n_t\}_{n \in \N}$ be the corresponding approximating sequence of measures and its projections, as defined in Section \[subs-approxmeas\]. We first establish tightness and convergence results for $\{P^n_t\}_{n \in \N}$ in Section \[subs-tightness\] and finally present the proof of Theorem \[th:conditionalindependence-infinitegraph\] in Section \[subs-pf-infingraph\].
Tightness and convergence results {#subs-tightness}
---------------------------------
In the following, let $H(\cdot \, | \, \cdot)$ denote relative entropy, defined for $\nu \ll \mu$ by $$H(\nu | \mu) = \int \frac{d\nu}{d\mu}\log \frac{d\nu}{d\mu}\,d\mu,$$ and $H(\nu|\mu) = \infty$ for $\nu \not\ll \mu$. Recall also our notation $\|x\|_{*,t} := \sup_{0 \le s \le t}|x_s|$ for the truncated supremum norm.
\[le:tightness\] Suppose Assumption \[assumption:B\] holds. For each $t > 0$ and each finite set $A \subset V$, we have $$\begin{aligned}
\sup_n\sup_{v \in V_n}\E^{P^n}\left[\|X_v\|_{*,t}^2 \right] &< \infty, \label{def:secondmomentbound} \\
\sup_n H\left(\left. P^n_t[A] \, \right| \, P^{*,\mu_0}_t[A]\right) &< \infty, \label{def:entropybound} \\
\sup_n H\left(\left. P^{*,\mu_0}_t[A] \, \right| \, P^n_t[A]\right) &< \infty. \label{def:entropybound-2}\end{aligned}$$
Fix $t > 0$. We begin with a standard estimate. Recall the definition of $b^n_v$ from , apply Itô’s formula to the SDE , and use the linear growth of $b_v$ from Assumption (\[assumption:B\].2) along with the uniform boundedness of $\sigma_v$ from Assumption (\[assumption:B\].3) to conclude that, for each $n \in \N$ and $v \in V_n$, $$\begin{aligned}
\E^{P^n} \left[ \|X_v\|_{*,t}^2 \right] & \le C\E^{P^n}
\left[|X_v(0)|^2 + \int_0^t |b^n_v(s,X_v,X_{N_v(G)})|^2 \,ds + \int_0^t |\sigma_v(s,X_v(s))|^2 \,ds \right] \\
&\le C \E^{P^n} \left[1 + |X_v(0)|^2 + \int_0^t\left( \|X_v\|_{*,s}^2 + \frac{1}{|N_v(G)|}\sum_{u \in N_v(G)}\|X_u\|_{*,s}^2\right)ds\right],\end{aligned}$$ where $C < \infty$ is a constant that can change from line to line but does not depend on $n$ or $v$. This implies that $$\begin{aligned}
\sup_{v \in V_n}\E^{P^n}\left[ \|X_v\|_{*,t}^2 \right] &\le C\left( 1 + \sup_{v \in V} \E^{P^n}\left[|X_v(0)|^2\right] + \int_0^t\sup_{v \in V_n}\E^{P^n}\left[\|X_v\|_{*,s}^2 \right] ds\right), \end{aligned}$$ where we have used the inclusion $V_n \subset V$. Apply Gronwall’s inequality to find $$\begin{aligned}
\sup_{v \in V_n}\E^{P^n}\left[\|X_v\|_{*,t}^2 \right]\le C\left(1 + \sup_{v \in V}\int_{(\R^d)^V} |x_v|^2\,\mu_0(dx_V)\right). \label{pf:quadraticestimate}\end{aligned}$$ The right-hand side is finite by Assumption (\[assumption:B\].1), and so follows.
Given a finite subset $A \subset V$, define $Q^n_t \in \P(\C_t^V)$ by $$\begin{aligned}
\frac{dQ^n_t}{dP^{*,\mu_0}_t} = \prod_{v \in V_n \setminus A}\EE_t( M_v^n ),
\ \ \mbox{ where } \ \
M_v^n := \int_0^\cdot (\sigma_v\sigma_v^\top)^{-1}b^n_v(s,X_v,X_{N_v(G_n)}) \cdot dX_v(s), \ v \in V. \end{aligned}$$ Due to Assumptions (\[assumption:B\].2) and (\[assumption:B\].3), and the definition of $P^{*,\mu_0}$ from Remark \[rem-driftless\], we can apply Lemma \[lem:Girsanov-justification\] with $\QQ= P^{*,\mu_0}$, $X=(X_v)_{v \in V_n}$ and $f(t,x) = ({{\boldsymbol{1}}}_{\{v \in V_n \setminus A\}} \sigma_v^{-1}b^n_v(t,x_v,x_{N_v(G_n)}))_{v \in V_n}$, to conclude that $\frac{dQ^n_t}{dP^{*,\mu_0}_t}$ is a true $P^{*,\mu_0}$-martingale. It then follows from Girsanov’s theorem [@karatzas-shreve Corollary 3.5.2] and the uniqueness in law of the driftless SDE that $Q^n_t[A] = P^{*,\mu_0}_t[A]$. By a similar argument, $$\begin{aligned}
\frac{dP^n_t}{dP^{*,\mu_0}_t} = \prod_{v \in V_n} \EE_t( M_v^n ), \quad \ \text{and thus} \ \quad \frac{dP^n_t}{dQ^n_t} = \prod_{v \in A \cap V_n} \EE_t( M_v^n ), \end{aligned}$$ where, in case $A \cap V_n = \emptyset$, we interpret the empty product as $1$. Once again invoking the linear growth of $b$, the boundedness of $\sigma_v$, , the fact that $Q_t^n[A] = P^{*,\mu}[A]$ and Remark \[rem-driftless\], note that Girsanov’s theorem also shows that for every $v \in A \cap V_n$, under $P^n$, $M_v^n - [M_v^n]$ is a martingale and $[M_v^n](t) =
\int_0^t |\sigma_v^{-1} b_v^n (s, X_v, X_{N_v(G_n)})|^2 \, ds$. Then, use the data processing inequality of relative entropy to obtain $$\begin{aligned}
H(P^n_t[A] \, | \, P^{*,\mu_0}_t[A]) &= H(P^n_t[A] \, | \, Q^n_t[A]) \\
&\le H(P^n_t \, | \, Q^n_t) \\
& = \sum_{v \in A \cap V_n} \E^{P^n} \left[ M_v^n(t) - \frac{1}{2} [M_v^n](t) \right] \\
&= \frac12 \sum_{v \in A \cap V_n} \E^{P^n} \left[\int_0^t\, | \sigma_v^{-1}b^n_v(s,X_v,X_{N_v(G_n)})|^2\,ds \right]\\
&\le C\sum_{v \in A \cap V_n}\E^{P^n}\left[1 + \|X_v\|_{*,t}^2 + \frac{1}{|N_v(G_n)|} \sum_{u \in N_v(G_n)} \|X_u\|_{*,t}^2\right] \\
&\le C|A|\left(1 + 2 \sup_{v \in V_n}\E^{P^n} \left[\|X_v\|_{*,t}^2 \right]\right).\end{aligned}$$ Therefore follows from .
Noting that due to the identity $Q_t^n[A] = P^{*,\mu}[A]$ and Remark \[rem-driftless\], under $Q^n$, $(X_v)_{v \in A}$ is driftless and $M_v^n$ is a martingale. Therefore, noting that $$\begin{aligned}
\frac{dQ^n_t}{dP^n_t} = \prod_{v \in A \cap V_n}
\exp \left( -M_v^n (t)
+ {{\frac{1}{2}}}\int_0^t |\sigma_v^{-1}b^n_v(s,X_v,X_{N_v(G_n)})|^2\,ds \right), \end{aligned}$$ another application of the the data processing inequality of relative entropy yields $$\begin{aligned}
H(P^{*,\mu_0}_t[A] \, | \, P^n_t[A]) &= H(Q^n_t[A] \, | \, P^n_t[A]) \\
&\le H(Q^n_t \, | \, P^n_t) \\
&= \frac12 \sum_{v \in A \cap V_n} \E^{Q^n} \left[\int_0^t| \sigma_v^{-1}b^n_v(s,X_v,X_{N_v(G_n)})|^2\,ds \right]\\
&\le C\sum_{v \in A \cap V_n}\E^{Q^n}\left[1 + \|X_v\|_{*,t}^2 + \frac{1}{|N_v(G_n)|} \sum_{u \in N_v(G_n)} \|X_u\|_{*,t}^2\right] \\
&\le C|A|\left(1 + 2 \sup_{v \in V_n}\E^{Q^n} \left[\|X_v\|_{*,t}^2 \right]\right).\end{aligned}$$ The same argument that was used to obtain can also be used to show that holds with $P^n$ replaced by $Q^n$. Therefore $\sup_n\sup_{v \in V_n}\E^{Q^n}\left[\|X_v\|_{*,t}^2 \right] < \infty$ by Assumption (\[assumption:B\].1), and hence the last display implies .
The next lemma will be used to show both that the *existence* of a weak solution to the infinite SDE system holds automatically and also that it arises as the limit of finite-graph systems. Recall that $P \in \P(\C^V)$ denotes the law of the solution of .
\[le:infinitegraphlimit\] Suppose Assumption \[assumption:B\] holds. Then $P^n \rightarrow P$ weakly on $\C^V$. Moreover, for any finite set $A' \subset V$, any $t > 0$, and any bounded measurable function $\psi : \C_t^{A'} \rightarrow \R$, we have $$\begin{aligned}
\lim_{n\rightarrow\infty}\E^{P^n}[\psi(X_{A'}[t])] = \E^{P}[\psi(X_{A'}[t])].\end{aligned}$$
Fix $t > 0$. The entropy bound of shows that $(P^n_t[A'])_{n \in \N}$ are precompact in the weak$^*$ topology induced on $\P(\C^{A'}_t)$ by the bounded measurable functions on $\C_t^{A'}$ [@dembozeitouni Lemma 6.2.16]. In particular, this sequence is tight, and since this holds for every finite set $A'$ and every $t > 0$ we deduce that the entire sequence $(P^n)_{n \in \N}$ is tight in $\C^V$. Note also that for sufficiently large $n$ it holds under $P^n$ that the processes $$\begin{aligned}
\int_0^s \sigma_v^{-1}(r,X_v)\,dX_v(r) - \int_0^s\sigma_v^{-1}(r,X_v)b_v(r,X_v,X_{N_v(G)})\,dr , \ \ \ s \ge 0, \ \ \ v \in V_{n-2}, \label{pf:infinitegraphlimit1}\end{aligned}$$ are independent standard Wiener processes, due to the consistency condition for the $b^n_v$’s and the identity $N_v(G_n)=N_v(G)$ valid for $v \in V_{n-2}$.
Now let $Q \in \P(\C^V)$ be any weak (in the usual sense) subsequential limit of $(P^n)_{n \in \N}$, with $P^{n_k} \rightarrow Q$ weakly. The aforementioned precompactness in the weak$^*$ topology implies that $$\lim_{k\rightarrow\infty}\E^{P^{n_k}}[\psi(X_{A'}[t])] = \E^{Q}[\psi(X_{A'}[t])],$$ for any finite set $A' \subset V$, any $t > 0$, and any bounded measurable function $\psi$ on $\C_t^{A'}$. We conclude that, under $Q$, the processes in are independent Wiener processes, for $v \in V$. This shows that $Q$ is the law of a weak solution of the SDE system , which we know to be unique by assumption (\[assumption:B\].5). Hence, $Q=P$.
Proof of the second-order Markov random field property on the infinite graph {#subs-pf-infingraph}
----------------------------------------------------------------------------
Fix $(G = (V, E), b, \sigma, \mu_0)$ and $X = (X_v)_{v \in V}$ as in the statement of the theorem. For $n \geq 4$, consider the sequence of graphs $G_n = (V_n, E_n), n \in \N$ constructed from $G$ as in Lemma \[le:MRFprojections\]. We first note that due to the fact that $\mu_0$ is a [2MRF]{} by Assumption (\[assumption:B\].1), part (iii) of Lemma \[le:MRFprojections\], with $\X = \R^d, {\nu}= \mu_0, {\nu}^* = \mu_0^*$, ensures that $\mu_0[V_n]$ is a [2MRF]{} with respect to the graph $G_n$. Moreover, since $d\mu_0[V_n]/d\mu_0^*[V_n]$ is strictly positive by Assumption (\[assumption:B\].1), Proposition \[th:hammersleyclifford2\] shows that $\mu_0[V_n]$ admits a $2$-clique factorization with respect to the product measure $\mu_0^*[V_n]$ for each $n$. Hence, $\mu_0[V_n]$ satisfies Assumption (\[assumption:A\].1), which when combined with the definition of $b^n=(b^n_v)_{v \in V_n}$ in and the fact that $b, \sigma$ satisfy Assumptions (\[assumption:B\].2) and (\[assumption:B\].3), shows that $(G_n, b^n, (\sigma_v)_{v \in V_n}, \mu_0[V_n])$ satisfy Assumption \[assumption:A\]. Since $P^n[V_n]$ is the law of the SDE on the finite graph $G_n$, it is a [2MRF]{} by Theorem \[pr:conditionalindependence-finitegraph\].
Now, fix two finite sets $A, B \subset V$ with $B$ disjoint of $A \cup \partial^2A$, where throughout, we use $\partial^2$ to denote $\partial_G^2$. Let $n_0$ denote the smallest integer greater than or equal to $4$ for which $A \cup \partial^2A \cup B \subset V_{n_0-3}$, and let $n \geq n_0$. Then, part (iv) of Lemma \[le:MRFprojections\], again with ${\mathcal X} = \R^d$, $\nu = \mu_0$, and $\nu^* = \mu^*_0$, ensures that $\mu_0[V_n]$ and $\mu_0[V_{n_0}]$ admit $2$-clique factorizations which are consistent in the sense that the corresponding measurable functions $f_K^{G_n}$ and $f_K^{G_{n_0}}$ agree for every $K \in \mathrm{cl}_2(G_{n_0})$ that intersects $A$ (equivalently, for every $K \in \mathrm{cl}_2(G)$ that intersects $A$). Since $b_v^{n} = b_v^{n_0} = b_v$ for all $v \in A \cup \partial^2 A$ by , and since $A \cup \partial^2_G A \subset V_{n_0-3}$, we may apply Proposition \[pr:specification-finitegraph\], with $G = G_{n}$, $H = G_{n_0}$, $V^*=V_{n_0-3}$, $\mu_0^{G_k} = \mu_0[V_k]$, and $(b^{G_k}_v,\sigma^{G_k}_v) = (b^k_v,\sigma_v)$ for $v \in G_k$ and $k \in \{n_0, n\}$, to deduce that $P^n_t[A \, | \, \partial^2A] = P_t^{n_0}[A \, | \, \partial^2A]$ for all $n \geq n_0$. In other words, this implies that given a bounded continuous function $f$, there exists a measurable function $\varphi$ (that does not depend on $n$) such that $$\label{phi-meas}
\varphi(X_{\partial^2A}[t]) = \E^{P^n}[f(X_A[t]) \, | \, X_{\partial^2A}[t]], \ \ \ P^n-a.s., \text{ for } n \ge n_0.$$
Now, fix additional bounded continuous functions $g,h$. For $t > 0$, taking the conditional expectation with respect to $X_{V_n \setminus A}[t]$ inside the expectation on the left-hand side below and using the [2MRF]{} property of $P^n$ we have $$\begin{aligned}
\E^{P^n}[f(X_A[t])g(X_{\partial^2A}[t])h(X_B[t])] = \E^{P^n}[\E^{P^n}[f(X_A[t]) \, | \, X_{\partial^2A}]g(X_{\partial^2A}[t])h(X_B[t])].\end{aligned}$$ When combined with , this implies $$\begin{aligned}
\E^{P^n}[f(X_A[t])g(X_{\partial^2A}[t])h(X_B[t])] = \E^{P^n}[\varphi(X_{\partial^2A}[t])g(X_{\partial^2A}[t])h(X_B[t])].\end{aligned}$$ Using the second part of Lemma \[le:infinitegraphlimit\] for the finite set $A' = A \cup \partial^2 A \cup B$, and for both $\psi (y_{A'}) := f(y_A) g(y_{\partial^2A}) h(y_B)$, and $\psi( y_{A'}) = \varphi(y_{\partial^2A}) g(y_{\partial^2 A}) h(y_B)$ for $y_{A'} \in \C_t^{A'}$, we may pass to the limit $n\rightarrow\infty$ and denote $P = P^{\mu_0}$ to get $$\begin{aligned}
\E^{P}[f(X_A[t])g(X_{\partial^2A}[t])h(X_B[t])] = \E^{P}[\varphi(X_{\partial^2A}[t])g(X_{\partial^2A}[t])h(X_B[t])].\end{aligned}$$ This at once shows both that $$\E^P[f(X_A[t]) \, | \, X_{\partial^2A}[t]] = \varphi(X_{\partial^2A}[t]) = \E^{P^n}[f(X_A[t]) \, | \, X_{\partial^2A}[t]],$$ for all bounded continuous $f$ and $n \ge n_0$, which proves Proposition \[th:specification-infinitegraph\] below, and also that $X_A[t]$ and $X_B[t]$ are conditionally independent given $X_{\partial^2A}[t]$ under $P$. The latter proves the first statement in Theorem \[th:conditionalindependence-infinitegraph\], except for the fact that we have only proven this conditional independence when $A$ and $B$ are finite. Because $B \subset V \setminus (A \cup \partial^2A)$ was an arbitrary finite set and $\{X_B[t] : B \subset A \cup \partial^2A\}$ generates the same $\sigma$-field as $X_{A \cup \partial^2A}[t]$, we deduce that that $P_t$ is a [2MRF]{}. The second statement follows from the same argument as in using the [2MRF]{} property of $P_t = P_t^{\mu_0}$. This completes the proof.
We recapitulate two results that were established in the course of the proof, which may be of independent interest, and which are used in the proof of Theorem \[th:gibbsuniqueness\] in the next section.
\[rem-pnmrf\] Note that the first paragraph of the proof above shows that if $(G, b, \sigma, \mu_0)$ satisfy Assumption \[assumption:B\] and for $G_n = (V_n, E_n)$, $n \in \N$, is as in Lemma \[le:MRFprojections\], and $b^n$, $P^n$, $P^n_t$, $n \in \N$, $t > 0$, are as defined at the beginning of Section \[sec-pf-infinitegraph\], then $P^n_t[V_n]$ is a [2MRF]{} for each $n \in \N$ and $t > 0$.
\[th:specification-infinitegraph\] Suppose $(G, b, \sigma, \mu_0)$ satisfy Assumption \[assumption:B\], and let $G_n = (V_n, E_n), n \in \N,$ be the sequence of graphs constructed from $G$ as in Lemma \[le:MRFprojections\]. Fix $t > 0$, and let $P_t = P_t^{\mu_0}$ be the law of the unique weak solution to the SDE with initial law $\mu_0$. Then, for $n \ge 3$, and $A \subset V_{n-3}$, $\partial^2_GA \subset V_n$ and, $P_t$-almost surely, $$P_t[A \, | \, \partial^2_G A] = P^n_t[A \, | \, \partial^2_G A].$$
Proof of Gibbs measure properties {#se:gibbsuniqueness}
=================================
In this section we prove the Gibbs uniqueness property of Theorem \[th:gibbsuniqueness\]. Recall the definition of $P^{*,\mu_0}$ as the law of the solution of initialized at $\mu_0$.\
*Proof of Theorem \[th:gibbsuniqueness\].* Let $(G, b, \sigma, \mu_0)$ satisfy Assumption \[assumption:B\], and let $P^{\mu_0}$ be the unique solution of the SDE system with initial law $\mu_0$. We work again on the canonical space $(\C^V,\mathrm{Borel},P^{*,\mu_0})$, with $X_V=(X_v)_{v \in V}$ denoting the canonical process. Define the sets $\MLset = \MLset (\mu_0)$ and $\MRset = \MRset (\mu_0)$ as in the statement of the theorem. For any $\nu_0 \in \MLset$, the SDE system is well-posed starting from $\nu_0$, and we let $P^{\nu_0} \in \P(\C^V)$ denote the law of this solution. The proof of the theorem is broken down into five claims.\
[*Claim 1.* ]{} Suppose $(G,b,\sigma,\mu_0)$ satisfies Assumption \[assumption:B\]. If $\nu_0 \in \MLset$, then $(G,b,\sigma,\nu_0)$ also satisfies Assumption \[assumption:B\], and for every finite set $A \subset V$ and $t > 0$ we have $P^{\nu_0}_t [A] \sim P^{\mu_0}_t[A]$.
First, suppose $\nu_0 \in \MLset$. By the definition of $\MLset$, $\nu_0$ has a finite second moment. Moreover, for each finite set $A \subset V$, we have $\nu_0[A] \sim \mu_0^*[A]$ since $\nu_0 \in \G_2(\mu_0)$ implies (by Definition \[def-Gibbs\]) that $\nu_0[A] \sim \mu_0[A]$, and Assumption (\[assumption:B\].1) ensures that $\mu_0[A] \sim \mu_0^*[A]$. Thus $\nu_0$ satisfies Assumption (\[assumption:B\].1).
Now let $t > 0$, and let $A \subset V$ be finite. From Lemma \[le:infinitegraphlimit\] it follows that $P^{n,\mu_0} \to P^{\mu_0}$ weakly. It then follows from , and the lower semicontinuity of relative entropy that $P^{\mu_0}_t[A] \ll P^{*,\mu_0}_t[A]$ and $P^{*,\mu_0}_t[A] \ll P^{\mu_0}_t[A]$. Therefore $P^{\mu_0}_t[A] \sim P^{*,\mu_0}_t[A]$, and similarly $P^{\nu_0}_t[A] \sim P^{*,\nu_0}_t[A]$. Finally, note that $P^{*,\mu_0}[A]$ (resp. $P^{*,\nu_0}[A]$) is the law of the solution of the SDE system $$dX_v(t) = \sigma_v(t,X_v)\,dW_v(t), \qquad v \in A,$$ with initial law $\mu_0[A]$ (resp. $\nu_0[A]$), and it follows from $\nu_0[A] \sim \mu_0[A]$ that $P_t^{*,\mu_0}[A] \sim P_t^{*,\nu_0}[A]$. Putting it together, we have $P^{\nu_0}_t [A] \sim P^{*,\nu_0}_t[A] \sim P^{*,\mu_0}_t[A] \sim P^{\mu_0}_t[A]$.
[*Claim 2.* ]{} For any $Q \in \MRset$, we have $Q_0 := Q\circ(X_V(0))^{-1} \in \MLset$.
The proof of this claim is straightforward: fix $Q \in \MRset$, and set $Q_0 := Q \circ (X_V(0))^{-1}$. Then by the definition of $\MRset$, we have $Q_t \in \G_2(P^{\mu_0}_t)$ for all $t \ge 0$ and $\sup_{v \in V} \int_{\R^d} |x_v|^2\,Q_0(dx) < \infty$. Taking $t=0$ gives $Q_0 \in \G_2(\mu_0)$, where we have used the elementary fact that $P^{\mu_0}_0= P^{\mu_0} \circ (X_V(0))^{-1}=\mu_0$. Thus $Q_0$ belongs to $\MLset$.
[*Claim 3.*]{} If $\nu_0 \in \MLset$ then $P^{\nu_0} \in \MRset$.
Fix $\nu_0 \in \MLset$. Then by the first assertion of Claim 1, for every finite set $A \subset V$ and $t \geq 0$, $P^{\nu_0}_t[A] \sim P^{\mu_0}_t[A]$. So it only remains to show that for every $t > 0$, $$\begin{aligned}
P^{\nu_0}_t[A \,|\, \partial^2 A] = P^{\mu_0}_t[A \,|\, \partial^2 A],
\quad \text{for finite } A \subset V. \label{pf:gibbsclaim0}\end{aligned}$$
First, recall that Claim 1 also shows that $(G,b,\sigma,\nu_0)$ satisfies Assumption \[assumption:B\]. Next, let $G_n = (V_n, E_n)$ be the increasing sequence of finite graphs defined in Lemma \[le:MRFprojections\], and let $P^{\mu_0,n}, P^{\nu_0,n} \in \P(\C^V)$ denote the law of the solution of the corresponding SDE system with initial laws $\mu_0[V_n]$ and $\nu_0[V_n]$, respectively. Throughout this proof, the boundary operator $\partial$ is always with respect to the infinite graph $G$. Fix $A \subset V$ finite, and fix $n$ large enough that $A \subset V_{n-3}$, recalling that $V_n$ was defined in Section \[subs-approxmeas\]. By Proposition \[th:specification-infinitegraph\], we have both $$\begin{aligned}
\begin{split}
P^{\mu_0}_t[A | \partial^2 A] &= P^{\mu_0,n}_t[A | \partial^2 A], \\
P^{\nu_0}_t[A | \partial^2 A] &= P^{\nu_0,n}_t[A | \partial^2 A].
\end{split} \label{pf:gibbsclaim1}\end{aligned}$$ By Lemma \[le:MRFprojections\](iii), $\nu_0[V_n]$ is a [2MRF]{}. Also since $\nu_0[V_n] \sim \mu_0^*[V_n]$ implies $d\nu_0[V_n]/d\mu_0^*[V_n] > 0$, by Proposition \[th:hammersleyclifford2\] there is a 2-clique factorization $$\begin{aligned}
\frac{d\nu_0[V_n]}{d\mu_0^*[V_n]}(x_{V_n}) = \prod_{K \in \mathrm{cl}_2(G_n)}g^n_K(x_K), \label{pf:gibbs1.5}\end{aligned}$$ for some measurable functions $g^n_K : (\R^d)^K \to \R_+$. Similarly $\mu_0[V_n]$ admits a 2-clique factorization, $$\begin{aligned}
\frac{d\mu_0[V_n]}{d\mu_0^*[V_n]}(x_{V_n}) = \prod_{K \in \mathrm{cl}_2(G_n)}f^n_K(x_K), \label{pf:gibbs2}\end{aligned}$$ for some measurable functions $f^n_K : (\R^d)^K \to \R_+$. We claim (and justify below) that $f^n_K$ and $g^n_K$ can be chosen to be consistent, i.e., so that $$\begin{aligned}
f^n_K \equiv g^n_K, \quad \text{for all } K \in \mathrm{cl}_2(G_n) \text{ with } K \cap A \neq \emptyset. \label{pf:gibbsclaim3}\end{aligned}$$ To see this, let $I_{V_n}=(I_v)_{v \in V_n}$ denote the canonical random variable on the probability space $(\R^d)^{V_n}$. Define $$\widehat{f}^n(I_{U_n}) = \E^{\mu_0^*}\left[\left.\frac{d\mu_0[V_n]}{d\mu_0^*[V_n]}\,\right|\,I_{U_n}\right], \qquad \widehat{g}^n(I_{U_n}) = \E^{\mu_0^*}\left[\left.\frac{d\nu_0[V_n]}{d\mu_0^*[V_n]}\,\right|\,I_{U_n}\right].$$ Recalling that $U_n = V_n \setminus V_{n-2}$, and using , we have $$\begin{aligned}
\frac{d\nu_0[V_{n-2} \, | \, U_n]}{d\mu_0^*[V_{n-2} \, | \, U_n]} &= \frac{ \frac{d\nu_0[V_n]}{d\mu_0^*[V_n]} }{ \widehat{g}^n(I_{U_n})} = \frac{1}{\widehat{g}^n(I_{U_n})}\prod_{K \in \mathrm{cl}_2(G_n)}g^n_K(I_K).\end{aligned}$$ Applying the same argument to $\mu_0$ rather than $\nu_0$ and using , we also obtain $$\begin{aligned}
\frac{d\mu_0[V_{n-2} \, | \, U_n]}{d\mu_0^*[V_{n-2} \, | \, U_n]} &= \frac{ \frac{d\mu_0[V_n]}{d\mu_0^*[V_n]} }{ \widehat{f}^n(I_{U_n})} .\end{aligned}$$ Further, recognizing that $U_n = \partial^2 V_{n-2}$, since $\nu_0 \in {\mathcal G}_2(\mu_0)$ we see that $$\nu_0 [V_{n-2}| U_n] = \mu_0 [V_{n-2}| U_n].$$ Combining the last three displays, we find $$\begin{aligned}
\frac{d\mu_0[V_n]}{d\mu_0^*[V_n]}(I_{V_n}) &= \frac{d\nu_0[V_{n-2} \, | \, U_n]}{d\mu_0^*[V_{n-2} \, | \, U_n]}\widehat{f}^n(I_{U_n}) = \frac{\widehat{f}^n(I_{U_n})}{\widehat{g}^n(I_{U_n})}\prod_{K \in \mathrm{cl}_2(G_n)}g^n_K(I_K).\end{aligned}$$ Comparing this with and noting that $U_n \in \mathrm{cl}_2(G_n)$, we can thus take $f^n_K \equiv g^n_K$ in for $K \in \mathrm{cl}_2(G_n) \setminus \{U_n\}$ and $f^n_{U_n} \equiv g^n_{U_n}\widehat{f}^n/\widehat{g}^n$. This proves the above consistency claim; indeed, since $A \subset V_{n-3}$, we know that $U_n$ does not intersect $A \cup \partial A$.
Let $\mathcal{K} = \{ K \in \mathrm{cl}_2(G_n) : K \cap A \neq \emptyset\}$. Using the consistency property , we can finally conclude from Proposition \[pr:specification-finitegraph\] (with $V_G=V_H=V_n$, $b_v^H = b_v^G = b_v$, $\sigma_v^H = \sigma_v^H = \sigma_v$, $\mu_0^H = \mu_0$ and $\mu_0^G = \nu_0$) that $P^{\nu_0,n}_t[A \,|\, \partial^2 A] = P^{\mu_0,n}_t[A \,|\, \partial^2 A]$. Recalling , this completes the proof of .
Together, claims 2 and 3 prove . We now prove the last assertion of the theorem.\
[*Claim 4.* ]{} If $Q \in \P(\C^V)$ satisfies $Q_t \in \G_2(P^{\mu_0}_t)$ for all $t \ge 0$ and also $Q \circ (X_V(0))^{-1} = \mu_0$, then $Q=P^{\mu_0}$.
As in the proof of Claim 3, we let $G_n = (V_n, E_n)$ be the increasing sequence of finite graphs defined in Lemma \[le:MRFprojections\] and let the boundary operator $\partial$ always be with respect to the infinite graph $G$. Also, let $P^n = P^{\mu_0,n} \in \P(\C^V)$ denote the law of the solution of the corresponding SDE system with initial law $\mu_0[V_n]$. Now, fix a finite set $A \subset V$ and $T \in (0,\infty)$. Let $n_0$ denote the smallest integer such that $A \cup \partial^2A \subset V_{n_0-3}$. Define the martingales $$\begin{aligned}
M^n_v(t) = \int_0^t (\sigma_v\sigma_v^\top)^{-1}b^n_v(s,X_v,X_{N_v(G_n)}) \cdot dX_v(s), \qquad n \in \N, v \in V_n. \end{aligned}$$ Due to Assumptions (\[assumption:B\].2) and (\[assumption:B\].3), it follows from Lemma \[lem:Girsanov-justification\] (with $\QQ= P^{*,\mu_0}$, $X=(X_v)_{v \in V_n}$ and $f(t,x) = (\sigma_v^{-1}b^n_v(t,x_v,x_{N_v(G_n)}))_{v \in V_n}$) that $\EE(M^n_v)$ is a $P^{*,\mu_0}$-martingale. Thus, by Girsanov’s theorem [@karatzas-shreve Corollary 3.5.2], we may write $$\begin{aligned}
\label{RNVn}
\frac{dP^n_t[V_n]}{dP^{*,\mu_0}_t[V_n]} &= \prod_{v \in V_n}\EE_t(M^n_v). \end{aligned}$$ Now, by applying Remark \[rem-pnmrf\] to $(G,b,\sigma,\mu_0)$ and $(G,0,\sigma,\mu_0)$, respectively, it follows that the measures $P^n_t[V_n]$ and $P^{*,\mu_0}_t[V_n]$ are [2MRF]{}s with respect to $G_n$. Hence, for $n \ge n_0$, $$\begin{aligned}
\frac{dP^n_t[A \, | \, \partial^2A]}{dP^{*,\mu_0}_t[A \, | \, \partial^2A]} &= \frac{dP^n_t[A \, | \, V_n \backslash A]}{dP^{*,\mu_0}_t[A \, | \, V_n \backslash A]} \\
&= \left.\frac{dP^n_t[V_n]}{dP^{*,\mu_0}_t[V_n]} \right/ \E^{P^{*,\mu_0}}\left[\left. \frac{dP^n_t[V_n]}{dP^{*,\mu_0}_t[V_n]} \right| X_{V_n \backslash A}[t]\right] \\
&= \left.\prod_{v \in V_n}\EE_t(M^n_v)\right/ \E^{P^{*,\mu_0}}\left[\left.\prod_{v \in V_n}\EE_t(M^n_v)\right| X_{V_n \backslash A}[t]\right].\end{aligned}$$ For $v \in V_n \backslash (A \cup \partial A)$, $\EE_t(M^n_v)$ is measurable with respect to $X_{V_n \backslash A}[t]$ and thus factors out of the conditional expectation and cancels. Thus, $$\begin{aligned}
\label{RNcond}
\frac{dP^n_t[A \, | \, \partial^2A]}{dP^{*,\mu_0}_t[A \, | \, \partial^2A]} &= \left.\prod_{v \in A \cup \partial A}\EE_t(M^n_v)\right/ \E^{P^{*,\mu_0}}\left[\left.\prod_{v \in A \cup \partial A}\EE_t(M^n_v)\right| X_{V_n \backslash A}[t]\right].\end{aligned}$$ Because $Q_t \in \G_2(P^{\mu_0}_t)$ by assumption, we have $Q_t[A \, | \, \partial^2A] = P^{\mu_0}_t[A \, | \, \partial^2A]$. By Proposition Proposition \[th:specification-infinitegraph\], we have $P^{\mu_0}_t[A \, | \, \partial^2A] = P^{\mu_0,n}_t[A \, | \, \partial^2A]$, and it follows that the density $dQ_t[A \, | \, \partial^2A] / dP^{*,\mu_0}_t[A \, | \, \partial^2A]$ is given by the same expression .
Now take $A=V_{n-2}$, and note that $U_n := V_n \setminus V_{n-2} = \partial^2 V_{n-2}$. Because $Q_t[U_n] \sim P^{*,\mu_0}_t[U_n]$ by assumption, and because both $Q$ and $P^{*,\mu_0}$ start from the same initial state distribution $\mu_0$, we may use the martingale representation theorem (specifically, apply Remark \[re:mtgrep\] below with $\xi= dQ_T[U_n]/dP^{*,\mu_0}_T[U_n]$, which clearly satisfies $\E^{P^{*,\mu_0}}[\xi] = 1$) to find progressively measurable functions $r^n_v : [0,T] \times \C^{U_n} \to \R^d, v \in U_n,$ which are $dt\otimes dP^{*,\mu_0}$-square-integrable such that in terms of the associated $X_{U_n}$-adapted continuous martingales $R^n_v(t) = \int_0^t r^n_v(s,X_{U_n}) \cdot dX_v(s)$, $t \in [0,T]$, $v \in U_n$, we can write for $t \in [0,T]$, $$\begin{aligned}
\frac{dQ_t[U_n]}{dP^{*,\mu_0}_t[U_n]} = \prod_{v \in U_n}\EE_t(R^n_v). \end{aligned}$$ Note that these martingales are orthogonal, that is, the covariation process $[R^n_v,R^n_u]$ is identically zero for $v \neq u$. Thus, since $U_n \cap V_{n-2} = \emptyset$, $U_n \cup V_{n-2} = V_n$ and $V_{n-1} = V_{n-2} \cup \partial V_{n-2}$, applying with $A = V_{n-2}$ we have $$\begin{aligned}
\frac{dQ_t[V_n]}{dP^{*,\mu_0}_t[V_n]} &= \frac{dQ_t[V_{n-2} \, | \, U_n]}{dP^{*,\mu_0}_t[V_{n-2} \, | \, U_n]} \ \frac{dQ_t[U_n]}{dP^{*,\mu_0}_t[U_n]} \\
&= \prod_{v \in U_n}\EE_t(R^n_v)\left.\prod_{v \in V_{n-1}}\EE_t(M^n_v)\right/ \E^{P^{*,\mu_0}}\left[\left.\prod_{v \in V_{n-1}}\EE_t(M^n_v)\, \right| \, X_{U_n}[t]\right].\end{aligned}$$ The process in the denominator is a positive martingale (as the optional projection of a martingale) adapted to $X_{U_n}$ and thus, again using the martingale representation theorem (this time applying Remark \[re:mtgrep\] below with $\xi = \E^{P^{*,\mu_0}}\left[\left.\prod_{v \in V_{n-1}}\EE_T(M^n_v) \, \right| \, X_{U_n}[T]\right]$ and invoking to conclude that $\E^{P*,\mu_0}[\xi] = \E^{P*,\mu_0}[ dP_t^n[V_{n-1}]/dP_t^{*, \mu_0}[V_{n-1}]] = 1$), there exist square integrable, progressively measurable functions $\widetilde{r}^n_v$ (as above) and associated $X_{U_n}$-adapted continuous martingales $\widetilde{R}^n_v(t) = \int_0^t \widetilde{r}^n_v (s, X_{U_n})\cdot dX_v(s),$ $v \in U_n$, such that $$\begin{aligned}
\E^{P^{*,\mu_0}}\left[\left.\prod_{v \in V_{n-1}}\EE_t(M^n_v) \, \right| \, X_{U_n}[t]\right] = \prod_{v \in U_n}\EE_t(\widetilde{R}^n_v),\end{aligned}$$ Now note that, for any continuous martingales $Z$ and $(Z_i)_{i \in I}$, with $I$ a finite index set, we have the identities $1/\EE(Z) = \EE(-Z)e^{[Z]}$ and $$\begin{aligned}
\prod_{i \in I} \EE(Z_i) &= \exp\left(\sum_{i \in I}Z_i - \frac12 \sum_{i \in I}[Z_i]\right) \\
&= \exp\left(\sum_{i \in I}Z_i - \frac12 \Big[ \sum_{i \in I}Z_i \Big] + \frac12\sum_{i,j \in I, \, i\neq j}[Z_i,Z_j]\right) \\
&= \EE\left(\sum_{i \in I}Z_i\right)\exp\left(\frac12\sum_{i,j \in I, \, i\neq j}[Z_i,Z_j]\right),\end{aligned}$$ where $[Z_i,Z_j]$ denotes the covariation process. Hence, $$\begin{aligned}
\frac{dQ_t[V_n]}{dP^{*,\mu_0}_t[V_n]} &= \prod_{v \in U_n}\EE_t(R^n_v)\EE_t(-\widetilde{R}^n_v)\exp([\widetilde{R}^n_v](t)) \prod_{v \in V_{n-1}}\EE_t(M^n_v) \\
&= \prod_{v \in V_{n-2}}\EE_t(M^n_v) \prod_{v \in V_{n-1} \setminus V_{n-2}}\EE_t(M^n_v + R^n_v - \widetilde{R}^n_v)\exp\left([M^n_v-\widetilde{R}^n_v,R^n_v-\widetilde{R}^n_v](t) \right) \\
&\quad \cdot \prod_{v \in V_n \setminus V_{n-1}}\EE_t(R^n_v - \widetilde{R}^n_v)\exp\left([\widetilde{R}^n_v,\widetilde{R}^n_v - R^n_v](t) \right).\end{aligned}$$ Recalling the orthogonality properties of $R^n_v$ and $\widetilde{R}^n_v$ mentioned above, we see that we can write $Z_t := dQ_t[V_n]/dP^{*,\mu_0}_t[V_n]$ in the form $Z(t) = \EE_t(N)e^{A(t)}$, where $N$ is a continuous square-integrable martingale and $A(t)$ is square-integrable and a.s. absolutely continuous with $A(0)=0$. Since $Z$ is a martingale, we necessarily have $A \equiv 0$; indeed, Itô’s formula gives $dZ(t)=Z(t)(dN(t) + dA(t))$, and for $Z$ to be a martingale we must have $dA(t)=0$. It follows that $$\begin{aligned}
\frac{dQ_t[V_n]}{dP^{*,\mu_0}_t[V_n]} &= \prod_{v \in V_{n-2}}\EE_t(M^n_v) \prod_{v \in V_{n-1} \setminus V_{n-2}}\EE_t(M^n_v + R^n_v - \widetilde{R}^n_v) \prod_{v \in V_n \setminus V_{n-1}}\EE_t(R^n_v - \widetilde{R}^n_v).\end{aligned}$$ Since $Z$ is a $P^{*,\mu_0}$-martingale, Girsanov’s theorem [@karatzas-shreve Corollary 3.5.2] can be applied, using the definition of $M_v^n$, to deduce that $Q_t[V_n]$ is the law of a solution $(X^n_v[t])_{v \in V_n}$ of the SDE system (perhaps on an auxiliary the probability space) $$\begin{aligned}
{3}
dX^n_v(s) &= b_v(s,X^n_v,X^n_{N_v(G)})\,ds + \sigma(s,X^n_v)\,dB_v(s), &&\text{ for } v \in V_{n-2}, \\
dX^n_v(s) &= \left((r^n_v- \widetilde{r}^n_v)(s,X^n_{U_n}) + b^n_v(s,X^n_v,X^n_{N_v(G_n)})\right)ds + \sigma(s,X^n_v)\,dB_v(s), &&\text{ for } v \in V_{n-1} \backslash V_{n-2}, \\
dX^n_v(s) &= (r^n_v- \widetilde{r}^n_v)(s,X^n_{U_n})\,ds + \sigma(s,X^n_v)\,dB_v(s), &&\text{ for } v \in V_n \backslash V_{n-1},\end{aligned}$$ where $(B_v)_{v \in V_n}$ are independent Brownian motions.
Define $X^n_v \equiv 0$ for $v \notin V_n$. Since the sets $V_n$ increase to $V$, it is easily shown as in Lemma \[le:infinitegraphlimit\] that, as $n \rightarrow \infty$, $(X^n_v[T])_{v \in V}$ converges in law in $\C_T^V$ to a solution of the infinite SDE system with initial distribution $\mu_0$, restricted to the interval $[0,T]$. Recalling that $(X^n_v(0))_{v \in V} \sim \mu_0$ and that the solution to the infinite SDE system is unique in law by Assumption (\[assumption:B\].4), we conclude that $(X^n_v[T])_{v \in V}$ converges in law to $P_T^{\mu_0}$. But $X^n_{V_n}[T]$ has law $Q_T[V_n]$ by construction, which implies $X^n_V[T]$ converges in law to $Q_T$. Therefore $Q_T = P_T^{\mu_0}$. Since $T \in (0,\infty)$ was arbitrary, $Q= P^{\mu_0}$, which completes the proof of Claim 4.
To complete the proof of the theorem, it only remains to establish the bijection between the two sets in . However, we now show that this is a simple consequence of the last claim.\
[*Claim 5.* ]{} The map $Q \mapsto Q \circ (X_V(0))^{-1}$ defines a bijection between the sets $\MRset$ and $\MLset$.
Let $Q \in \MRset$, and set $\nu_0 := Q \circ (X_V(0))^{-1}$. By Claim 2, $\nu_0$ belongs to $\MLset$, and by Claim 3, $P^{\nu_0}$ lies in $\MRset$. Since trivially $P^{\nu_0} \circ (X_V(0))^{-1} = \nu_0$, to prove the claim it suffices to prove that $Q = P^{\nu_0}$. By Claim 1, $(G,b,\sigma,\nu_0)$ satisfies Assumption \[assumption:B\], and thus Claim 4 applies with $\nu_0$ in place of $\mu_0$. That is, by applying Claim 4 to $\nu_0$ instead of $\mu_0$, we deduce that if $Q \in \P(\C^V)$ satisfies $Q_t \in \G_2(P^{\nu_0}_t)$ for all $t \ge 0$ and also $Q \circ (X_V(0))^{-1} = \nu_0$, then $Q=P^{\nu_0}$. By definition of $\MRset$ we have $Q_t \in \G_2(P^{\mu_0}_t)$ for all $t \ge 0$, and it follows from , which was established in the proof of Claim 3, that $\G_2(P^{\nu_0}_t)=\G_2(P^{\mu_0}_t)$. This proves Claim 5, which completes the proof of Theorem \[th:gibbsuniqueness\].
\[re:mtgrep\] We sketch here the argument behind the use of the martingale representation theorem in the proof of Theorem \[th:gibbsuniqueness\] above. Recall that by Assumption (\[assumption:A\].3b) the SDE system $dX_v(t) = \sigma_v(t,X_v)\,dW_v(t), \ v \in U_n$, with initial law $\mu_0$ is unique in law, with the law of the solution $X=(X_v)_{v \in U_n}$ given by $P^{*,\mu_0}[U_n]$. This implies uniqueness of the associated martingale problem (cf. [@karatzas-shreve Corollary 5.4.9]), which is known to imply that the solution has the predictable representation property (cf. [@rogers-williams Theorem V.25.1] or [@stroock1980extremal Theorem 2.7]), in the following sense: For $T < \infty$ and an $\F_T^X$-measurable random variable $\xi > 0$ with $\E[\xi]=1$, the martingale $Z(t)=\E[\xi \, | \, \F^X_t] > 0, t \in [0,T]$, where $\F^X_t = \sigma(X(s):s \le t)$, can be represented as $Z(t) = 1 + \int_0^t\varphi(s,X) \cdot dX(s)$ for some predictable process $\varphi \colon [0,T] \times \C_T \to {{\mathbb{R}}}$ satisfying $\int_0^T|\varphi(t,X)|^2\,dt < \infty$ a.s., recalling that $\sigma_v$ is uniformly bounded and nondegenerate. Then, for $t \in [0,T]$, setting $\psi(t,X)=\varphi(t,X)/Z(t)$, by Itô’s formula, we have $d\log Z(t) = \psi(t,X) \cdot dX(t) - \frac12\psi(t,X)^\top d[X](t)\psi(t,X)$. Hence, $Z(t) = \EE_t(\int_0^\cdot \psi(s,X) \cdot dX(s))$, $t \in [0,T]$.
Proof of pathwise uniqueness under Lipschitz assumptions {#ap:uniqueness-infSDE}
========================================================
Let $(X_v)_{v \in V}$ and $(\widetilde{X}_v)_{v \in V}$ denote two solutions driven by the same Wiener processes and starting from the same initial states. Fix $T < \infty$. For each $v \in V$ and $t \in [0,T]$, by Itô’s formula, the boundedness of $\sigma$ (see Assumption (\[assumption:B\].3)), the assumed Lipschitz condition on the drift and diffusion coefficients we have $$\begin{aligned}
\E \left[ \|X_v - \widetilde{X}_v\|_{*,t}^2 \right]& \le 2t
\E \left[ \int_0^t \left| b_v(s,X_v(s),X_{N_v(G)}(s)) - b_v(s,\widetilde{X}_v(s),\widetilde{X}_{N_v(G)}(s))\right|^2 ds \right] \\
& \quad + 8\E \left[\int_0^t \left| \sigma_v(s,X_v(s)) - \sigma_v(s,\widetilde{X}_v(s))\right|^2 ds \right] \\
& \le 4tK_T^2 \E \left[\int_0^t \left(\|X_v - \widetilde{X}_v\|_{*,s}^2 + \frac{1}{|N_v(G)|}\sum_{u \in N_v(G)} \|X_u - \widetilde{X}_u\|_{*,s}^2 \right) ds \right] \\
& \quad + 8{\bar{K}}_T^2\E \left[ \int_0^t \|X_v - \widetilde{X}_v\|_{*,s}^2 \,ds\right].
\end{aligned}$$ Hence, $$\begin{aligned}
\sup_{v \in V} \E \left[ \|X_v - \widetilde{X}_v\|_{*,t}^2 \right] & \le 8(tK_T^2+{\bar{K}}_T^2) \int_0^t \sup_{v \in V} \E \left[ \|X_v - \widetilde{X}_v\|_{*,s}^2\right] ds.\end{aligned}$$ Complete the proof using Gronwall’s inequality.
Justification for applying Girsanov’s theorem {#sec-Girsanov}
=============================================
In this section we state a result that justifies our repeated application of Girsanov’s theorem under the condition that the drift is progressively measurable and has linear growth. Lemma \[lem:Girsanov-justification\] below is in fact a path-dependent multi-dimensional version of [@KlebanerLiptser2014when Theorems 5.1 and 8.1]. A simpler proof is provided here for completeness.
Let $(\Omega,\F,\FF,\QQ)$ be a filtered probability space supporting a $\FF$-Brownian motion $W$ of dimension $m$ as well as an $\FF$-adapted process $X$ of dimension $d$ such that $X$ satisfies the SDE $$\begin{aligned}
\label{eq:X-appendix}
dX(t) = \sigma(t,X)\,dW(t), \quad X(0) \sim \mu, \end{aligned}$$ where $\mu \in {{\mathcal{P}}}({{\mathbb{R}}}^d)$ and $\sigma : \R_+ \times \C \rightarrow \R^{d\times m}$ is bounded and progressively measurable. Also, let $\E$ denote expectation with respect to $\QQ$. Fix a progressively measurable $f: \R_+ \times \C \mapsto \R^m$, and define the stochastic integral $$M_t := \int_0^t f(s,X) \cdot dW_s, \qquad t \in [0, \infty),$$ which is well defined (and a local martingale) due to the linear growth condition imposed on $f$ in the lemma below. Recall in what follows that $\|x\|_{*,t} = \sup_{s \in [0,t]} |x(s)|$.
\[lem:Girsanov-justification\] Under the above setting, suppose for each $T \in (0,\infty)$ there exists $C_T < \infty$ such that $$\label{linear-growth}
|f(t,x)| \le C_T\left(1 + \|x(s)\|_{*,t} \right),$$ for all $t \in [0,T]$ and $x \in \C$. Then the Doleans exponential $\{\EE_t(M)\}_{t \geq 0}$ defined in is a true $\QQ$-martingale.
Since $\{\EE_t(M)\}_{t \ge 0}$ is always a $\QQ$-supermartingale, it suffices to show that ${{\mathbb{E}}}[\EE_T(M)|X(0)=x]=1$ for each $x \in {{\mathbb{R}}}^d$ and $T \in (0,\infty)$. So fix $T \in (0,\infty)$ and assume without loss of generality that $X(0)=x \in {{\mathbb{R}}}^d$ in . Since $\sigma$ is bounded, $X$ is a martingale and, from standard concentration inequalities for martingales (see, e.g., [@Vandegeer1995exponential Lemma 2.1]), we can find some $C > 0$ such that $\QQ(\|X-x\|_{*,T} \ge a) \le \exp(-Ca^2)$ for each $a > 0$. It then follows from the equivalence between sub-Gaussian tails and finite square-exponential moments (see, e.g., [@BoucheronLugosiMassart2013concentration Section 2.3]) that there exists $c > 0$ such that $\E[ \exp( c \|X\|_{*,T}^2 ) ] < \infty$. Now taking $0=t_0<t_1<\dotsb<t_{n(T)}=T$ with $t_n-t_{n-1} \le c/C_T^2$, and using the linear growth condition on $f$, we have $$\begin{aligned}
\E \left[ \exp \left( {{\frac{1}{2}}}\int_{t_{n-1}}^{t_n} |f(s,X)|^2 \,ds \right) \right] & \le \E \left[ \exp \left( (t_n-t_{n-1}) C_T^2(1+\|X\|_{*,T}^2) \right) \right] < \infty.
\end{aligned}$$ It then follows from [@karatzas-shreve Corollary 3.5.14] that $\{\EE_t(M)\}_{t \geq 0}$ is a true $\QQ$-martingale.
|
---
abstract: |
The average ground state energies for spin glasses on Bethe lattices of connectivities $r=3,\ldots,15$ are studied numerically for a Gaussian bond distribution. The Extremal Optimization heuristic is employed which provides high-quality approximations to ground states. The energies obtained from extrapolation to the thermodynamic limit smoothly approach the ground-state energy of the Sherrington-Kirkpatrick model for $r\to\infty$. Consistently for all values of $r$ in this study, finite-size corrections are found to decay approximately with $\sim N^{-4/5}$. The possibility of $\sim
N^{-2/3}$ corrections, found previously for Bethe lattices with a bimodal $\pm J$ bond distribution and also for the Sherrington-Kirkpatrick model, are constrained to the additional assumption of very specific higher-order terms. Instance-to-instance fluctuations in the ground state energy appear to be asymmetric up to the limit of the accuracy of our heuristic. The data analysis provides insights into the origin of trivial fluctuations when using continuous bonds and/or sparse networks.
author:
- 'Stefan Boettcher[^1]'
bibliography:
- '/Users/stb/Boettcher.bib'
title: 'Numerical Results for Spin Glass Ground States on Bethe Lattices: Gaussian Bonds'
---
Introduction\[sec:Introduction\]
================================
We study the ground state ($T=0$) properties of spin glasses on Bethe lattices with Gaussian bond distribution, which are also considered at low but finite temperatures in Ref. [@Rizzo09b]. In many ways, this study resembles that for the bimodal $\pm J$ bond distribution in Refs. [@Boettcher03a; @Boettcher03b]. Yet, surprisingly, the behavior for finite-size corrections differ significantly between both distributions.
Bethe lattices are $r$-regular graphs [@Bollobas], i. e. randomly connected graphs consisting of $N$ vertices, each having a fixed number, $r$, of neighbors [@Mezard03]. We explore the large-$N$ regime of low-connectivity graphs, $r=3,\ldots,15$, which are of great theoretical interest as finite-connected, mean-field models for low-dimensional lattice spin glasses [@MPV; @Viana85]. A great number of studies have focused on various aspects of this conceptually simple model to hone the complex mathematical techniques required to treat disordered systems [@Mezard87; @mezard:01; @Mezard03; @Tria02; @dedominicis:89; @Mottishaw87; @Lai90] or optimization problems [@Monasson99; @MPZ; @Franz01; @Wong87; @Banavar87b; @Zdeborova10].
As before in Refs. [@Boettcher03a; @Boettcher03b], we use the Extremal Optimization (EO) heuristic [@Boettcher00; @Boettcher01a; @Dagstuhl04] to find approximations to spin glass ground states. In previous papers, we have demonstrated the capabilities of EO in determining near-optimal solutions for spin glasses, the coloring problem [@Boettcher01a; @Boettcher04a] and the graph partitioning problem [@Boettcher99a; @Boettcher00; @Boettcher01b; @Percus08; @Zdeborova10]. It is generally harder to find good approximations in complex energy landscapes with a local search heuristic, such as EO, for a problem with continuous weights [@Bauke04], but some encouraging results exist [@Middleton04]. While our results appear to be sufficiently accurate for the prediction of energy averages for systems up to size $N=2048$, more detailed features, such as their fluctuations over the ensemble of instances (requiring higher moments of the energy), are less reliable for system sizes $N>256$ and larger degree $r$. Hence, for Ising spin glass simulations with EO, discrete $\pm J$ bond weights are usually preferable. Based on the experience with the Sherrington-Kirkpatrick (SK) model, scaling properties of thermodynamic observables are generally believed to be universal, independent of the details of the bond distribution used. There, for instance, finite-size corrections are found to scale approximately with $\sim
N^{-2/3}$ [@EOSK; @Bouchaud03; @Katzgraber05; @Palassini08; @Aspelmeier07] (just as for the Bethe lattices with bimodal bonds) and even the average ground-state energy density $\left\langle e_{SK}\right\rangle
\approx-0.76317\ldots$ is universal for any symmetric bond distribution [@MPV]. It would be therefore remarkable to find such distinct scaling behavior between distributions at the level of finite-size corrections, as this investigation suggest. Some previous investigations of Bethe lattices with Gaussian bonds [@Bouchaud03; @Liers03] have been consistent with $N^{-2/3}$-corrections but were based on smaller sizes and significantly less statistics as in our study here. As a possible resolution, we would have to appeal to ad-hoc assumptions about higher-order corrections. (The distinct effects of discrete versus continuous bonds on defect energies in finite-dimensional lattice spin glasses have been studied numerically in Ref. [@Hartmann01], and also with the renomalization group in Ref. [@amoruso:03].) In turn, the variation of corrections with the details of the bond distribution may provide important clues towards extending replica theory to include finite-size effects.
While the value of this work lies in exploring the range of finite-size scaling on sparse random graphs for spin glasses with differing distributions, it also provides a cautionary note about the analysis of data obtained from such systems. In a sparse system, such as a random graph or a randomly diluted lattice, (trivial) normal fluctuations in geometry [^2] or in the bond distribution can obscure the physical essence of the problem at hand. We will argue that in such a system we need to focus attention to the actual “cost” $C$ of disorder in terms of the frustration, instead of the energy $E$ itself. Just consider a simple ferromagnet with fixed bonds $J$ on an ordinary random graph $G_{N,p}$ [@Bollobas], or alternatively, with continuously distributed bonds $J>0$ on a fixed-degree Bethe lattice of size $N$. In either case, the ground state energy has all bonds satisfied, i. e. no cost in the number of frustrated bonds ($C=0$), and ground state cost fluctuations exhibit a $\delta$-peak, correspondingly. But by the central limit theorem, the absolute sum of all bonds $$B=\sum_{i=1}^{rN/2}\left|J_{i}\right|
\label{Beq}$$ has a normal distribution, inherited by the ground state energy fluctuations via $E=2C-B$. We would claim that the relevant fluctuations at finite $N$ are captured by $C$, not $E$, although the thermodynamic averages $\langle E\rangle$ and $\langle C\rangle$ are completely equivalent, of course. In any case, careful distinction is advisable in an environment of competing finite-size corrections, especially when averaging inherently finite samples in simulations.
The effect of these trivial fluctuations is most pronounced in the study of fluctuations in the ground states of the systems, which have been intensely studied in recent years [@Bouchaud03; @andreanov:04; @EOSK; @Boettcher05e; @Katzgraber05; @Aspelmeier07; @Palassini08; @Parisi08; @Parisi09; @Rizzo09; @Rizzo09b]. Our numerical data here shows that those fluctuations in the *energy* density would predict a seemingly interesting cross-over between non-trivial to trivial scaling between system size and degree. In contrast, the cost density fluctuations would predict a slow drift towards triviality at larger system sizes that is setting in earlier for larger system sizes and might be attributable to a decay in accuracy. As these results are somewhat inconclusive, in a related publication [@Bo_unpub] we will show that these fluctuations on Bethe lattices with a *bimodal* distribution of bonds behave, again, similar to those of the SK model.
In the following Sec. \[sec:Spin-Glasses-on\], we introduce first the Bethe lattices we used in the numerical calculations. Then, we address the relation between energies and costs in Sec. \[sec:Sampling-Ground-States\]. In Sec. \[sec:-EO-Algorithm\], we briefly describe the EO algorithm, which is amply discussed elsewhere [@Boettcher00; @Boettcher01a; @Dagstuhl04]. In Sec. \[sec:Numerical-Results\], we finally present our numerical results. Some conclusions are presented in Sec. \[sec:Conclusion\].
Spin Glasses on Bethe Lattices\[sec:Spin-Glasses-on\]
=====================================================
Disordered spin systems on random graphs have been investigated as mean-field models for low-dimensional spin glasses or optimization problems, since variables are long-range connected yet have a small number of neighbors. Particularly simple are Bethe lattices of fixed vertex degree $r=k+1$ [@Mezard87; @mezard:01; @Mezard03], which are locally tree-like with vertices imagined as possessing one up-direction and $k$ downward branches. Yet, all $r$ directions are fully equivalent in Bethe lattices, and there is no root vertex or any boundary. In comparison to the otherwise more familiar random graphs studied by Erdös and Rény [@Bollobas], Bethe lattices at a given $N$ and $r$ avoid fluctuations in the vertex degree and in the total number of bonds.
There are slight variations in the generation of Bethe lattices. For instance, to add a bond one could choose at random two vertices of connectivities $<r$ to link until all vertices are $r$-connected. Instead, we have used the method described in Ref. [@Bollobas] to generate these graphs. Here, all the terminals on the vertices form a list of $rN$ independent variables. For each added bond, two available terminals are chosen at random to be linked and removed from the list. Furthermore, for algorithmic convenience, we reject graphs which possess self-loops, i. e. bonds that connect two terminals of the same vertex. Multiple bonds between any pair of vertices are allowed; otherwise it is too hard to generate feasible graphs for small $N$, especially at larger $r$. Since $r$ remains finite for $N\to\infty$, the energy and entropy per spin would only be effected to $O(1/N)$ by differences between these choices.
Sampling Ground States on a sparse Graph\[sec:Sampling-Ground-States\]
======================================================================
Once a graphical instance is generated, be it a Bethe lattice or any other sparse graph, we assign bonds $J_{i,j}$, here randomly chosen from a Gaussian distribution of zero mean and unit variance, to existing links between neighboring vertices $i$ and $j$. Each vertex $i$ is occupied by an Ising spin variable $x_{i}\in\{-1,+1\}$. The Hamiltonian $$\begin{aligned}
H=-\sum_{\{bonds\}}J_{i,j}x_{i}x_{j}.
\label{Heq}\end{aligned}$$ provides the energy of the system. For each instance $I$, the energy $E^{(I)}$ is defined as the difference in the absolute weight of all violated bonds, $C^{(I)}$ (the “cost”), and satisfied bonds, $S^{(I)}$, i. e. $$\begin{aligned}
E^{(I)}=C^{(I)}-S^{(I)};
\label{Eeq}\end{aligned}$$ the larger the satisfied bond-weight $S^{(I)}$ in the instance, the lower its energy $E^{(I)}$. While $C^{(I)}$ and $S^{(I)}$ vary depending on the spin configuration, for each instance $$\begin{aligned}
B^{(I)} & = & C^{(I)}+S^{(I)}
\label{eq:BI}\end{aligned}$$ is a constant, with $B^{(I)}$ given by Eq. (\[Beq\]). Hence, $$\begin{aligned}
E^{(I)}=2C^{(I)}-B^{(I)},
\label{E2eq}\end{aligned}$$ provides a direct relation between cost and energy of each instance. Thus, after averaging over a sample $n_{I}$ of instances $I$ of size $N$ (denoted by $\langle\ldots\rangle_{N}$), we obtain by definition for the averages $$\begin{aligned}
\left\langle E\right\rangle _{N} & = & 2\left\langle C\right\rangle
_{N}-\left\langle B\right\rangle _{N}
\label{eq:averageE}\end{aligned}$$ and for the variances: $$\begin{aligned}
\sigma_{N}^{2}\left(E\right) & = &
4\sigma_{N}^{2}\left(C\right)+\sigma_{N}^{2}\left(B\right)-4cov_{N}\left(C,B\right).
\label{eq:varianceE}\end{aligned}$$ We observe that for the sparse-graph systems with continuous bond weights under consideration here [\[]{}similar to the ferromagnetic example in the Introduction, where $\sigma_{N}^{2}\left(B\right)=\sigma_{N}^{2}\left(E\right)\sim N$ and $\sigma_{N}^{2}\left(C\right)=cov_{N}\left(C,B\right)\equiv0$[\]]{}, $$\begin{aligned}
\sigma_{N}^{2}\left(C\right) & \lesssim &
cov_{N}\left(C,B\right)\ll\sigma_{N}^{2}\left(B\right),
\label{eq:smallCOV}\end{aligned}$$ which we demonstrate in Fig. \[fig:covar\_test\]. Note that this condition becomes less satisfied when graphs get denser, i. e. $r$ increases. That trend is indicated by arrows in Fig. \[fig:covar\_test\]. The situation described by Eq. (\[eq:smallCOV\]) does not impact the relation between the averages in Eq. (\[eq:averageE\]) in any way, in particular, not the thermodynamic quantities and their finite-size corrections discussed below. But it does significantly affect both, the error analysis for that data and the interpretation of the ground state energy fluctuations, each derived from the variance. Similarly, $\sigma_{N}^{2}\left(B\right)$ can not be neglected in a fluctuating geometry, such as a random graph or a random-diluted lattice [@Boettcher04c; @Boettcher04b], even if the individual bond-weights are sharp, $\left|J_{i,j}\right|\equiv J_{0}$ with constant $J_{0}$.
Eq. (\[eq:smallCOV\]) does not apply for a discrete bond distribution on a Bethe lattice [@Boettcher03a; @Boettcher03b], where $\sigma_{N}^{2}\left(B\right)\equiv0$; $E$ and $C$ are then equivalent stochastic variables in every respect. Furthermore, in a dense graph such as the SK model [@Sherrington75], and independent of any symmetric bond distribution, this situation is entirely distinct, as even for ground states almost exactly half of all bonds are violated, i. e. $2C^{(I)}\approx B^{(I)}$ for large $N$, and the energy appears merely as an extreme value within the normal fluctuations of $2C$ and $B$. In this case, using $C$ to study fluctuations would be futile. As $r$ is increasing, this trend is already visible in Fig. \[fig:covar\_test\]: the gap between the variance in $E$ and $C$ is closing, and is bound to cross over at some larger degree $r$.
Eqs. (\[eq:varianceE\]-\[eq:smallCOV\]) imply that the standard error of the average energy is dominated by the error in $B$, $$\begin{aligned}
\Delta E_{N} & = &
\frac{\sigma_{N}\left(E\right)}{\sqrt{n_{I}}}\sim\Delta
B\left(1-\frac{2cov_{N}\left(C,B\right)}{\sigma_{N}^{2}\left(B\right)}\right),
\label{eq:DeltaE}\end{aligned}$$ which by Eq. (\[eq:smallCOV\]) is larger than that of $C$ for the range of degrees $r$ used in this study. Therefore, our strategy for determining the average ground state energy in the thermodynamic limit will be based on an extrapolation for large $N$ of the values for $\left\langle C\right\rangle _{N}$ and the evaluation of Eq. (\[eq:averageE\]) at $N=\infty$ using the exact value of the bond-density $\left\langle b\right\rangle _{\infty}$, here, $$\begin{aligned}
\left\langle b\right\rangle _{\infty} &
=\lim_{N\to\infty}\frac{\left\langle B\right\rangle _{N}}{N}= &
\frac{r}{2}\left\langle \left|J\right|\right\rangle
_{\infty}=\frac{r}{\sqrt{2\pi}}\qquad
\label{eq:averageB}\end{aligned}$$ for Bethe lattices with a Gaussian bond distribution.
More significantly, when Eq. (\[eq:smallCOV\]) holds, the variance in the ground state energy fluctuations tracks the variance of $B$, making it apparently trivial (i. e. normal): $$\begin{aligned}
\sigma_{N}^{2}\left(E\right) & \sim & \sigma_{N}^{2}\left(B\right)\sim N.
\label{eq:EvarN}\end{aligned}$$ The scaling of $\sigma_{N}\left(E\right)\sim N^{1-\rho}$ has been the focus of keen interest for various mean-field models recently [@Bouchaud03; @andreanov:04; @EOSK; @Boettcher05e; @Katzgraber05; @Aspelmeier07; @Palassini08; @Parisi08; @Parisi09; @Rizzo09; @Rizzo09b]. There, non-trivial behavior is typically associated with an exponent that obeys $\rho>\frac{1}{2}$.) We claim that due to Eq. (\[eq:EvarN\]) any non-trivial deviation for sparse graphs would have to be found in the cost $C$, if it exists at all.
In the following, we will therefore focus on the cost per spin $c=C/N$. We will study the finite-size corrections of the form $$\begin{aligned}
\langle c\rangle_{N} & =\frac{\langle C\rangle_{N}}{N} & \sim a+\frac{b}{N^{\omega}}\left[1+\epsilon(N)\right]
\label{eq:FSS}\end{aligned}$$ with $a\approx\langle c\rangle_{\infty}$, even taking some higher-order corrections $\epsilon(N)\ll1$ into account. Additionally, we consider the fluctuations in the ground-state *cost* density, in particular, the scaling of its deviation with finite size, $$\begin{aligned}
\sigma_{N}\left(c\right) & \sim & N^{-\rho}.
\label{eq:rho}\end{aligned}$$
![\[fig:covar\_test\] Plot of the variances and co-variances, appropriately rescaled, in Eq. (\[eq:varianceE\]) as a function of $N$ for $r=3,4,\ldots,10,15$. (Arrows indicate increasing $r$-values for the adjacent data sets.) All curves are either constant in $N$ throughout, or appear to approach a constant. In this scaling, the variance for the bond weights $B$ (red $\triangle$) collapses for all $r$ to $\sigma^{2}(B)/(rN)=\sigma^{2}\left(\left|J\right|\right)/2=\frac{1}{2}-\frac{1}{\pi}$, and the variances for the energies (green $\square$) are of similar magnitude but declining for larger $r$ (top to bottom). At some fixed (larger) $N$, both, the covariance between $B$ and $C$ (blue $\diamond$) as well as the variance of $C$ (black $\circ$) alone, are quite small, but they are *increasing* with $r$ (bottom to top).](covar_test)
$\tau$-EO Algorithm for Bethe Lattices\[sec:-EO-Algorithm\]
===========================================================
To obtain the numerical results in this paper, we used exactly the same implementation of $\tau$-EO to find ground states as in Ref. [@Boettcher03a], except that we assign to each spin $x_{i}$ in Eq. (\[Heq\]) a “fitness” $$\begin{aligned}
\lambda_{i}=\left\lfloor10\frac{x_{i}\sum_{<\dot{,}j>}J_{i,j}x_{j}}{\sum_{<\dot{,}j>}\left|J_{i,j}\right|}\right\rfloor\end{aligned}$$ Instead of using the local field of each variable, which may vary in range somewhat between variables due to the fluctuations in the bond values, we re-scale each variable into the same interval $[-10,-9,-8,\ldots,+10]$, thereby “hashing” the otherwise continuous state-space into a discrete set of up to 21 bins. Note that in this case the sum of the fitnesses is *not* proportional to the total energy, but good fitness sufficiently correlates with good costs for local search with EO to succeed, as explained in Ref. [@Boettcher00].
$N$ $\left\langle c_{3}\right\rangle _{N}$ $\left\langle c_{4}\right\rangle _{N}$ $\left\langle c_{5}\right\rangle _{N}$ $\left\langle c_{6}\right\rangle _{N}$ $\left\langle c_{7}\right\rangle _{N}$ $\left\langle c_{8}\right\rangle _{N}$ $\left\langle c_{9}\right\rangle _{N}$ $\left\langle c_{10}\right\rangle _{N}$ $\left\langle c_{15}\right\rangle _{N}$
---------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ----------------------------------------- ----------------------------------------- -- --
16 0.09043(5) 0.20163(7) 0.3302(1) 0.4703(1) 0.6182(1) 0.7732(2) 0.9326(2) 1.0976(2) 1.9630(3)
32 0.06934(3) 0.16825(5) 0.28476(6) 0.41254(7) 0.54844(8) 0.6908(1) 0.8383(1) 0.9904(1) 1.7954(2)
64 0.05759(2) 0.14934(3) 0.25882(4) 0.37961(4) 0.50894(5) 0.64416(6) 0.78435(6) 0.92878(7) 1.6953(1)
128 0.05087(1) 0.13842(2) 0.24387(2) 0.36103(3) 0.48600(3) 0.61714(4) 0.75324(4) 0.89347(5) 1.63804(6)
256 0.04717(1) 0.13225(1) 0.23544(2) 0.35003(2) 0.47271(2) 0.60153(3) 0.73536(3) 0.87325(3) 1.60540(7)
512 0.04491(1) 0.12853(1) 0.23026(2) 0.34363(2) 0.46465(3) 0.59203(4) 0.72440(4) 0.86153(4) 1.58710(9)
1024 0.04338(4) 0.12616(3) 0.22736(7) 0.33967(8) 0.4601(1) 0.5863(2) 0.7181(2) 0.8545(2) 1.5772(4)
2048 0.04302(2) 0.12577(4) 0.22666(7) 0.3384(1)
$\infty$ 0.0420(1) 0.1236(1) 0.2236(1) 0.3352(3) 0.4542(5) 0.5798(8) 0.7103(9) 0.8459(9) 1.561(1)
$r$ $a$ $b$ $\omega$ ndf $\chi^2$/ndf $Q$
----- ---------- ------ ---------- ----- -------------- ------------------------
3 0.041995 0.45 0.81 6 92.5 0
4 0.123592 0.70 0.79 6 58.6 0
5 0.223676 0.97 0.80 5 63.6 0
6 0.335139 1.22 0.80 5 30.3 $6.3 \times 10^{-31} $
7 0.454128 1.45 0.79 4 38.2 $5.5 \times 10^{-32}$
8 0.579745 1.72 0.79 4 43.6 $1.3 \times 10^{-36}$
9 0.710283 1.97 0.79 4 40.4 $6.5 \times 10^{-34}$
10 0.845875 2.32 0.80 4 1.8 0.12
15 1.560930 3.65 0.79 4 59.3 0
: \[power\] Fit of the data in Tab. \[alldata\] to $\langle c_r\rangle_N=a+\frac{b}{N^{\omega}}$.
$r$ $a$ $b$ $\omega$ $c$ ndf $\chi^2$/ndf $Q$
----- ---------- ---------- ---------- ------- ----- -------------- ---------
3 0.041654 0.61 0.78 -0.13 5 37.9 5.5e-39
4 0.123219 0.85 0.77 -0.12 5 50.1 0
5 0.223430 1.06 0.79 -0.07 4 75.8 0
6 0.334402 1.46 0.77 -0.20 4 20.2 1.1e-16
7 0.452726 1.86 0.76 -0.35 3 7.1 8.6e-05
8 0.577866 2.24 0.76 -0.45 3 7.6 4.3e-05
9 0.708515 2.47 0.76 -0.42 3 17.6 2e-11
10 0.845673 2.38 0.80 -0.05 3 2.1 0.1
15 unstable
: \[allfit\_log\] Fit of the data in Tab. \[alldata\] to $\langle c_r\rangle_N=a+\frac{b}{N^{\omega}}+\frac{c\ln N}{N}$.
$r$ $a$ $b$ $c$ ndf $\chi^2$/ndf $Q$
----- ---------- ------- ------- ----- -------------- -----
3 0.042773 -0.21 -0.45 6 2579.1 0
4 0.125082 -0.37 -0.74 6 1792.5 0
5 0.225941 -0.57 -1.07 5 1935.1 0
6 0.337383 -0.60 -1.25 5 1995.7 0
7 0.456151 -0.53 -1.36 4 2392.7 0
8 0.581872 -0.61 -1.59 4 2653.7 0
9 0.713004 -0.76 -1.87 4 2772.5 0
10 0.851120 -1.40 -2.60 4 3212.2 0
11 1.571370 -2.70 -4.58 4 3063.2 0
: \[ln23\] Fit of the data in Tab. \[alldata\] to $\langle c_r\rangle_N=a+\frac{b}{N^{\frac{2}{3}}}+\frac{c\ln N}{N}$.
$r$ $a$ $b$ $c$ ndf $\chi^2$/ndf $Q$
----- ---------- ------ ------- ----- -------------- ---------
3 0.041941 0.50 -0.03 6 67.9 0
4 0.123668 0.70 0.01 6 62.9 0
5 0.223727 0.97 0.01 5 64.4 0
6 0.335286 1.21 0.02 5 33.7 1.4e-34
7 0.454604 1.37 0.08 4 57.3 0
8 0.580229 1.66 0.07 4 59.0 0
9 0.710829 1.90 0.09 4 54.6 0
10 0.845843 2.33 -0.01 4 1.7 0.14
11 1.561210 3.44 0.16 4 46.4 4.4e-39
: \[ln45\] Fit of the data in Tab. \[alldata\] to $\langle c_r\rangle_N=a+\frac{b}{N^{\frac{4}{5}}}+\frac{c\ln N}{N}$.
$r$ $a$ $b$ $\omega$ $c$ ndf $\chi^2$/ndf $Q$
----- ---------- ---------- ---------- ------ ----- -------------- ------------------------
3 0.041500 0.12 0.62 0.43 5 35.8 $9 \times 10^{-37}$
4 0.123099 0.32 0.68 0.49 5 50.5 0
5 0.223414 0.69 0.75 0.34 4 76.6 0
6 0.334213 0.57 0.69 0.81 4 21.4 $1.2 \times 10^{-17}$
7 0.452095 0.52 0.63 1.21 3 7.0 $ 9.9 \times 10^{-5}$
8 0.576847 0.54 0.61 1.54 3 5.6 0.00073
9 0.707617 0.73 0.64 1.61 3 15.0 $9.6 \times 10^{-10} $
10 0.845614 2.02 0.78 0.35 3 2.0 0.11
15 unstable
: \[allfit\] Fit of the data in Tab. \[alldata\] to $\langle c_r\rangle_N=a+\frac{b}{N^{\omega}}+\frac{c}{N}$.
$r$ $a$ $b$ $c$ ndf $\chi^2$/ndf $Q$
----- ---------- ------ ------ ----- -------------- -----------------------
3 0.041656 0.16 0.37 6 34.3 $1.2 \times 10^{-41}$
4 0.123011 0.29 0.54 6 42.4 0
5 0.222800 0.39 0.74 5 70.5 0
6 0.333995 0.50 0.91 5 17.6 $1.7 \times 10^{-17}$
7 0.452686 0.64 1.02 4 7.3 $7.4 \times 10^{-6}$
8 0.577985 0.75 1.22 4 8.6 $6.6 \times 10^{-7}$
9 0.708256 0.87 1.41 4 12.4 $4.1 \times 10^{-10}$
10 0.843511 0.93 1.74 4 17.5 $2.1 \times 10^{-14}$
15 1.557000 1.55 2.63 4 146.8 0
: \[power23\] Fit of the data in Tab. \[alldata\] to $\langle c_r\rangle_N=a+\frac{b}{N^{\frac{2}{3}}}+\frac{c}{N}$.
To evaluate the proposed $\tau$-EO algorithm, we have benchmarked over a number of exactly solved instances obtained with a branch-and-bound method. Such an approach is clearly limited in attainable system sizes, here $N\leq64$, and can only be executed for a small test-bed of instances due to the exponential computational cost of exact methods. For larger systems, we have also applied the $\tau$-EO algorithm to a small test-bed of 10 instances of size $N=2^{10}$ for each value of $r$ with 5-times more updates per run and compared results. In the worst case, about 30% of the instances for some $r$ showed a systematic error of about 0.1% or less. Since we sampled over $n_I\approx10^{3}-10^4$ instances at this $N$, this systematic error is well below the statistical error of $\sim1/\sqrt{n_I}\gtrsim1\%$.
Alternatively, we can show that averaged properties obtained with EO are in some sense self-consistent and/or consistent with certain theoretical predictions. For example, Fig. \[fig:LargeK\] shows the extrapolation for $r\to\infty$ of the already extrapolated thermodynamic ($N\to\infty$) limit of the average energies for each $r$ and $N$. Despite of its derivative nature, the EO data still reproduces the exactly-known ground state energy of SK very accurately.
Finally, it should be kept in mind that settings which provide sufficiently accurate averages of a quantity may be less proficient in determining its higher moments, let alone its entire PDF. Ever higher moments are dominated by rare events (or the lack thereof for finite $n_I$) ever deeper in the exponentially suppressed tails of the PDF. For instance, variances are based on (subtractions involving) higher moments and thus can be expected to have substantially boosted systematic errors compared to those quoted for averages.
Numerical Results\[sec:Numerical-Results\]
==========================================
We have simulated Bethe lattices with the algorithm discussed in Sec. \[sec:-EO-Algorithm\] for $r$ between 3 and 15, and graph sizes $n=2^{l}$ for $l=4,5,6,\ldots,11$ to obtain results for ground state energies. Statistical errors in our averages have been kept small by generating a large number of instances for each $N$ and $r$, typically $n_{I}\approx10^{6}$ for $N\leq256$ and $n_{I}\approx10^{3}-10^{5}$ for $N\geq512$.
Average Ground-State Properties\[sub:Average-Ground-State-Properties\]
----------------------------------------------------------------------
In Tab. \[alldata\], we list the values of average costs per spin, $\left\langle c\right\rangle _{N}$. When plotted as a function of $1/N$ in Fig. \[fig:Extrapolation\], the average costs per spin for each given $r$ clearly do not extrapolate linearly. Instead, we attempt a fit according to Eq. (\[eq:FSS\]) with variable finite-size correction exponent $\omega$, at first ignoring higher-order corrections ($\epsilon\equiv0$). Listed in Tab. \[power\], we find that for the whole range of connectivities $r$ studied here, the fitted scaling corrections appear to be consistent with $\omega=4/5$, and a plot of the data in Fig. \[fig:Extrapolation\] on a rescaled abscissa, $1/N^{4/5}$, produces linear scaling. In Fig. \[fig:Qtest\], we assess the quality of a fit, free of any assumptions about unknown higher-order corrections, by fixing the value of $\omega$ over a range and then fitting for the remaining two parameters, $a$ and $b$, in Eq. (\[eq:FSS\]). Each of the *independent* data sets for $r=3,\ldots,10,15$ exhibits a strong preference for $\omega\approx4/5$. In Fig. \[fig:omega\], we plot those fitted values for $\omega$, which within the range of degree values $r$ studied here suggest no trend toward the value of $\omega_{SK}=\frac{2}{3}$ expected for the SK-limit $r\to\infty$, irrespective of bond distribution [@Bouchaud03; @Boettcher05e; @Aspelmeier07; @Palassini08]. Plotting the data for $\omega=\frac{2}{3}$ in Fig. \[fig:Extrapolation\] clearly does not produce a linear extrapolation as in Refs. [@Boettcher03a; @Boettcher03b] for discrete bonds.
![\[fig:Qtest\] Plot of the $\chi^2$ per numbers of degrees of freedom (ndf) for a fit of the data in Tab. \[alldata\] to Eq. (\[eq:FSS\]) with only the parameters $a$ and $b$ over a range of fixed $\omega$, ignoring higher-order corrections ($\epsilon\equiv0$). Independently, for each $r$, the data shows a distinct minimum near $\omega\approx\frac{4}{5}$. ](Qtest)
![\[fig:omega\] Plot of the fitted values for the finite-size scaling exponent $\omega$ obtained in Tabs. \[power\] ($\square$) and \[allfit\] ($\circ$) for the available values of $r$. Based on an error of $0.01$ estimated from the fits, all values from Tab. \[power\] are consistent with $\omega=\frac{4}{5}$ (dashed line), without trend for increasing degree $r$ towards $\frac{2}{3}$ (dash-dotted line). The values from Tab. \[allfit\] behave less uniformly but seem more centered around $\frac{2}{3}$.](cost_omega)
In Tabs. \[allfit\_log\]-\[power23\], we have considered alternative fits to the data involving also higher-order corrections $\epsilon(N)$ to the finite-size corrections, using the full form of Eq. (\[eq:FSS\]). While there are some theoretical results [@parisi:93; @parisi:93b], obtained for the internal energy near $T_c$ in SK, possibly justifying $\omega=\frac{2}{3}$ also below $T_c$, little is know to higher order. The expansion for the free energy at $T_c$ in Refs. [@parisi:93; @parisi:93b] provided corrections of the form $c\ln(N)/N$, which could be argued to eventually affect also the ground state energy. We have therefore attempted to fit the data also with higher-order corrections of that form, see Tab. \[allfit\_log\], and simply $c/N$, another plausible form. \[Unfortunately, a five-parameter fit with a higher-order correction of $c/N^\alpha$ does not provide stable results for our data.\] Assuming either correction provides the best-quality fit to the remaining four parameters, with the lowest $\chi^2$ value relative to the remaining numbers of degree of freedom (ndf), shown in Tabs. \[allfit\_log\] and \[allfit\]. The former fit yields almost identical results to that without higher-order correction, Tab. \[power\], only that the values for $\omega$ are consistently shifted down by a small amount, see Fig. \[fig:omega\]. The fit with $1/N$ drastically changes the fitted values. The values for $\omega$ are now somewhat consistent with $\frac{2}{3}$, but vary widely with degree $r$ in Fig. \[fig:omega\]. Fixing $\omega=\frac{2}{3}$ improves the quality of fit (with one less parameter) for $1/N$ corrections to the best fit overall, see Tab. \[power23\]. But it becomes entirely inconsistent with $\ln(N)/N$ corrections, as shown in Tab. \[ln23\], unlike for fixed $\omega=\frac{4}{5}$ in Tab. \[ln45\].
![\[fig:LargeK\] Extrapolated values of the energy densities $\left\langle e_{r}\right\rangle _{\infty}$ obtained from the cost densities $\left\langle c_{r}\right\rangle _{\infty}$ listed in Tab. \[alldata\] (yellow $\square$). The data is plotted as a function of inverse degree, $1/r$, and rescaled by a root of the degree, such that the large-$r$ limit approaches the SK model. Similar to the corresponding plot in Ref. [@Boettcher03a; @Boettcher03b], the data appears to reach the SK limit virtually on a linear trajectory: a linear fit (dashed line) to all data projects the ground-state of the SK model, $\left\langle e_{SK}\right\rangle =-0.76317$ (marked by $\times$) as $-0.76332$, or to within an error of $<0.1\%$. An extrapolation of corresponding data from Tab. \[power23\] (blue $\diamond$) misses the SK value noticeably (dash-dotted line). ](energy_extra)
In should be remarked that in all cases, the confidence-of-fit $Q$ is essentially zero in light of the rather tight error bars obtained from the statistics for the average cost densities. We would argue that this is due to the inherent limitations of fitting this data down to small system size $N$ with an asymptotic form valid for large $N$, Eq. (\[eq:FSS\]), as a stand-in for an entirely unknown function of $N$. Hence, these fits should not be dismissed on the basis of $Q$ alone.
The extrapolated values for $a=\left\langle c_{r}\right\rangle
_{\infty}$ obtained in Tab. \[power\] are also listed in Tab. \[alldata\]. To demonstrate the quality of the extrapolation, we plot in Fig. \[fig:LargeK\] the derived values for the energy densities by way of Eq. (\[eq:averageE\]), $\left\langle
e_{r}\right\rangle _{\infty}=2\left\langle c_{r}\right\rangle
_{\infty}-\left\langle b_{r}\right\rangle _{\infty}$, using Eq. (\[eq:averageB\]). As in Refs. [@Boettcher03b; @Boettcher03a], already a linear fit to the extrapolated values reproduces the exactly known value of the SK model [@MPV; @crisanti:02; @Oppermann07] to within an error of $<0.1\%$, confirming to a high degree the numerical accuracy of the data. The obtained slope of the extrapolation, called $f_{1}$ in Ref. [@Tria02], which is not expected to be a universal quantity, evaluates to $f_{1}\approx0.36$, much larger than for the corresponding problem with $\pm J$ bonds [@Boettcher03a; @Boettcher03b]. The extrapolated costs in Tab. \[power23\], also plotted as energies in Fig. \[fig:Extrapolation\], markably miss the SK value.
![\[fig:Scaling-collapse-for GSEF\] Scaling collapse for the deviation in the ground state fluctuations for the energy densities, $\sigma_{N}(e)$. The raw data for the Bethe lattices is plotted for the available degrees $r$ on top. On the bottom, the same data is plotted, now rescaled by the indicated powers of the degree, with the deviation also multiplied by $\sqrt{N}$, such that trivial, normal fluctuations should plateau (as indicated by the continuous horizontal line). For fixed degree, systematic deviations appear only for sufficiently small $N\lesssim r^{2}$, consistent with a non-trivial scaling $\sigma(e)\sim N^{-\rho}$ at a value of $\rho=\frac{3}{4}$ (dashed line) or even $\rho=\frac{5}{6}$, as expected for the SK model. If taken at face-value, one would conclude that for any fixed degree at large $N$ merely trivial scaling is obtained and only the SK model (or any model with degree growing as $r\gg\sqrt{N}$) has anomalous scaling, $\rho>\frac{1}{2}$.](bethe_unscale "fig:") ![\[fig:Scaling-collapse-for GSEF\] Scaling collapse for the deviation in the ground state fluctuations for the energy densities, $\sigma_{N}(e)$. The raw data for the Bethe lattices is plotted for the available degrees $r$ on top. On the bottom, the same data is plotted, now rescaled by the indicated powers of the degree, with the deviation also multiplied by $\sqrt{N}$, such that trivial, normal fluctuations should plateau (as indicated by the continuous horizontal line). For fixed degree, systematic deviations appear only for sufficiently small $N\lesssim r^{2}$, consistent with a non-trivial scaling $\sigma(e)\sim N^{-\rho}$ at a value of $\rho=\frac{3}{4}$ (dashed line) or even $\rho=\frac{5}{6}$, as expected for the SK model. If taken at face-value, one would conclude that for any fixed degree at large $N$ merely trivial scaling is obtained and only the SK model (or any model with degree growing as $r\gg\sqrt{N}$) has anomalous scaling, $\rho>\frac{1}{2}$.](bethescale "fig:")
![\[fig:Scaling-collapse-for GSCF\] Plot for the deviation in the ground state fluctuations for the cost densities, $\sigma_{N}(c)$. The data for the Bethe lattices is plotted for the available degrees, $r=z$, with the deviation multiplied by $\sqrt{N/r}$, such that trivial, normal fluctuations should plateau. No collapsible scaling regimes emerge, such as a non-trivial scaling $\sigma(e)\sim N^{-\rho}$ with $\rho=\frac{3}{4}$ (continuous line) and much less $\rho=\frac{5}{6}$, as expected for the SK model. Instead, all data appears to steadily approach trivial scaling at much larger system sizes $N$.](bethe_scaled_cost)
Ground-State Energy and Cost Fluctuations\[sub:Ground-State-Energy-Fluctuations\]
----------------------------------------------------------------------------------
While the previous section has demonstrated the numerical accuracy of the data at the level of the average of observables, higher moments or a measure of the entire probability density function (PDF) of ground state energy fluctuations provide a more confusing picture. No clear conclusion can be reach on the basis of this data, even when heeding the implications of Sec. \[sec:Sampling-Ground-States\]. In light of that discussion, we first present the (as we believe, incorrect) extrapolation for the ground-state *energy* fluctuations, followed by the corresponding discussion for the cost fluctuations.
In Fig. \[fig:Scaling-collapse-for GSEF\], we show the raw data for the deviations of the energy densities $\sigma_{N}\left(e\right)$ and their apparent collapse. All the data seems to approach trivial, normal fluctuations for sufficiently large system sizes, with a cross-over at ever higher degree $r$. Similar normal fluctuations for this model have been claimed by Ref. [@Bouchaud03]. In fact, rescaling the data for a collapse indicates that the cross-over between system size and degree in the Bethe lattice occurs at $N\sim
r^{2}$. Taken at face value, this would be a remarkable result. While at any fixed, finite degree a trivial scaling is reached, *only* in the SK-limit, where system size and degree would scale direct proportionally, $N\sim r$, we would flow towards the left onto the non-trivial branch of the scaling in Fig. \[fig:Scaling-collapse-for GSEF\]. Hence, the Bethe lattice results for Gaussian bonds (unlike for $\pm J$ bonds, as we will show in Ref. [@Bo_unpub]) are disconnected from the SK-limit. Although this may also explain the unusual finite-size scaling corrections $\omega\approx0.8$, as well disconnected from the SK-limit, we believe that this data collapse does not probe the true disorder-induced frustration. As we have shown in Sec. \[sec:Sampling-Ground-States\], energy fluctuations are dominated by the variance in the Gaussian distribution of $N$ bonds (although ever less so for larger $r$).
Remarkably, when we plot the deviations $\sigma_{N}\left(c\right)$ for the cost density in Fig. \[fig:Scaling-collapse-for GSCF\], a far more difficult-to-interpret picture results. Unlike for the energy densities in Fig. \[fig:Scaling-collapse-for GSEF\], there is no apparent cross-over but the data instead veers steadily towards normal fluctuations for much larger system sizes. While the data does not show any noticeable statistical errors, it is impossible to exclude a systematic bias in the sampling of ground states with the heuristic, that could manifest itself in a smooth drift away from any potential non-trivial scaling. We can only eliminate the systematic error up to $N\leq64$ through comparisons with exact ground states obtained with a branch-and-bound algorithm. Above such sizes, we can only argue for small systematic errors based on the internal consistency of the average cost or energies, as is displayed, e. g., in Figs. \[fig:Extrapolation\] and \[fig:LargeK\]. Though, this may prove insufficient to guarantee similar fidelity for higher cumulants, like the deviations in the cost. But if we assume sufficient accuracy for this data, it would imply that even for the cost deviations, like for the energies before in Fig. \[fig:Scaling-collapse-for GSEF\], ultimately pure normal fluctuations may result either way when Gaussian bonds are considered. It would not be unusual to find extended transient behavior in spin glasses with Gaussian bonds [@BoCo].
Conclusion\[sec:Conclusion\]
============================
We have found surprising differences in the finite-size scaling behavior between a continuous, Gaussian bond distribution and previous results for a bimodal, $\pm J$ distribution [@Boettcher03a; @Boettcher03b] for spin glasses on Bethe lattices of degree $r$. While either distribution leads to equivalent results for thermodynamic averages that smoothly extrapolate to the exactly known SK results with identical scaling in $r$ for $r\to\infty$, the finite-size corrections term $b/N^{\omega}$ not only differ in the correction amplitude $b$ but possibly in the scaling exponent $\omega$ itself. Only when higher-order corrections are postulated, more consistency can be obtained with the value $\omega\approx\frac{2}{3}$ found for discrete bonds on Bethe lattices and for SK with either bond distribution. The value obtained here for Gaussian bonds, $\omega\approx\frac{4}{5}$, raises the question about an eventual cross-over to the SK value at higher $r$. No tendency toward such a cross-over is apparent in our study up to $r=15$. In light of that, the vague expectation of some uncooperative higher-order corrections to $\omega=\frac{2}{3}$ seems preferable, but one might be forgiven to be struck by the solid persistence of $\omega=\frac{4}{5}$ suggested by Figs. \[fig:Extrapolation\] and \[fig:omega\].
Our study of fluctuations in the ground-state properties is not successful in determining clearly the scaling behavior of the deviations. But it sends a cautionary note about the origin of fluctuations and the interpretation of data when simulating spin glasses on sparse graphs with continuous bonds or a randomly fluctuating geometry.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by the U. S. National Science Foundation through grant DMR-0812204.
[^1]: http://www.physics.emory.edu/faculty/boettcher/
[^2]: The fluctuations referred to are those in the total number of bonds in a graph over the ensemble of graphs. These are normal, even if the (independent) degrees at each vertex follows a separate distribution of non-zero width, such as a Poissonian.
|
---
abstract: 'The evolution of the curvature perturbation after multi-field inflation is studied in the light of the curvaton mechanism. Past numerical studies show that many-field inflation causes significant evolution of the curvature perturbation after inflation, which generates significant non-Gaussianity at the same time. We reveal the underlying mechanism of the evolution and show that the evolution is possible in a typical two-field inflation model.'
author:
- Seishi Enomoto
- Tomohiro Matsuda
title: 'Curvaton mechanism after multi-field inflation'
---
Introduction
============
The primordial curvature perturbation $\zeta(k)$ is strongly constrained by observation and provides a unique window on the very early universe [@Lyth-book]. It is known to have the spectrum ${\cal P}_\zeta(k)\simeq (5\times
10^{-5})^2$ with spectral tilt $n-1\equiv d\ln {\cal P}_\zeta/d\ln k \simeq -0.04$, and in future one could detect the running $dn/d\ln k$ as well as non-Gaussianity signaled by the bispectrum and trispectrum.
The process of generating $\zeta$ begins presumably during inflation, when the vacuum fluctuations of one or more bosonic fields are converted to classical perturbations. Within this general framework, there exist many proposals [@Lyth-book].
One proposal is to use two or more inflaton fields, which drive inflation in the multi-field model. That paradigm has been widely investigated, but it has usually been supposed that $\zeta(x,t)$ evaluated at an epoch $t_\mathrm{end}$ just before (or sometimes just after) the end of inflation is to be identified with the observed quantities in the spectrum. For this reason, a great deal of effort has gone into the calculation of the spectrum, bispectrum and trispectrum of $\zeta$ at the end of inflation [@Multi1; @Multi2; @Multi3; @Multi4; @Multi5; @Multi6; @Multi-matsuda; @Multi-NG].
The evolution after many-field inflation has been studied numerically in Ref. [@CGJ] using the statistical distribution of the parameters [@Easther:2005zr; @Battefeld:2008bu]. Later in Ref. [@afterCGJ] the evolution of the non-Gaussianity has been investigated. In these studies it has been found that there is a minimal number of the inflaton field $N_f\ge 10^3$, which is needed to realize the late-time creation and the domination of the curvature perturbation. Also, the number $N_f$ has been related to the creation of the non-Gaussianity. On the other hand, the calculation is not analytic and it is not clear if the evolution is possible in a two (or a few)-field model.
In this paper, we point out that the actual calculation of the curvature perturbation might well depend on the evolution after multi-field inflation, even if the number $N_f$ is [*not large*]{}. We show that the minimum number is $N_f=2$, simply because the mechanism requires isocurvature perturbation.
Just for simplicity, consider $N_f=2$ with the light scalar fields ($\phi, \sigma$) during inflation. The adiabatic and the entropy directions of multi-field inflation are defined using those fields. Basically, the “inflaton” (the adiabatic field) is not identical to $\phi$, even if $\sigma$ plays the role of the curvaton. The mixing is negligible when $\sigma$ is much lighter than $\phi$; that is the limit where the usual curvaton scenario applies.
Alternatively, it is possible to consider the opposite limit, where the fields have nearly equal mass[^1]. Can the curvaton mechanism work in that limit? A naive speculation is that the biased initial condition ($\sigma/\phi\ll 1$) might lead to the curvaton mechanism in that limit. Indeed the speculation is correct; however to reach the correct conclusion we need quantitative calculation of the curvaton mechanism in the equal-mass limit. The calculation details are shown in the Appendix. The usual curvaton mechanism is reviewed in Sec.\[sec:prepare\], and the non-linear formalism of the curvaton mechanism is reviewed in Sec.\[sec:3\]. The basic idea of the equal-mass curvaton model is shown in Sec.\[sec:4\] for two-field inflation. Deviation from the equal-mass limit and the applications are discussed in Sec.\[sec:5\].
Curvaton mechanism {#sec:prepare}
==================
In this section we review $\delta N$ formalism used to calculate $\zeta$. To define $\zeta$ one smooths the energy density $\rho$ on a super-horizon scale shorter than any scale of interest. Then it satisfies the local energy continuity equation, $$\frac{\partial \rho(x,t) }{\partial t} = - \frac{3}{a(x,t)}
\frac{\partial a(x,t)}
{\partial t} \left( \rho(x,t) + p(x,t) \right)
,$$ where $t$ is time along a comoving thread of spacetime and $a$ is the local scale factor. Choosing the slicing of uniform $\rho$, the curvature perturbation is $\zeta\equiv \delta (\ln a)$ and $$\frac{\partial \zeta (x,t) }{\partial t} = \delta \left( \frac{\dot\rho(t)}
{\rho(t) + p(x,t) } \right)
.$$ If $p$ is a function purely of $\rho$, one will find $\dot\zeta=0$. That is the case of single field inflation when no other field perturbation is relevant. The inflaton field $\phi(x,t)$ determines the future evolution of both $\rho$ and $p$. Similarly, the component perturbations $\zeta_i$ are conserved if they scale like matter ($\rho_m\propto a^{-3}$) or radiation ($\rho_r\propto a^{-4}$).
During nearly exponential inflation, the vacuum fluctuation of each light scalar field $\phi_i$ is converted at horizon exit to a nearly Gaussian classical perturbation with spectrum $(H/2\pi)^2$, where $H\equiv \dot a(t)/a(t)$ in the unperturbed universe. Writing $$\zeta = \delta [ \ln (a(x,t)/a(t_1)] \equiv \delta N
,$$ and taking $t_*$ to be an epoch during inflation after relevant scales leave the horizon, we define $N(\phi_1(x,t_*),\phi_2(x,t_*),\cdots,t,t_*)$ so that $$\zeta(x,t) = N_i \delta \phi_i(x,t_*)
+ \frac{1}{2} N_{ij} \delta \phi_i(x,t_*)\delta \phi_j(x,t_*) + \cdots
,$$ where a subscript $i$ denotes $\partial/\partial \phi_i$ evaluated on the unperturbed trajectory. We find $$\begin{aligned}
n-1 &=& \frac{2\sum_i N_i N_j \eta_{ij}}{\sum_m N_m^2}
-2\epsilon - \frac{2}{M_p^2 \sum_m N_m^2} \\
\eta_{ij} &\equiv& M_p^2V_{ij}/V,\qquad \epsilon \equiv M_p^2
\sum_m V_m^2/V^2, \end{aligned}$$ where $M_p$ is the reduced Planck mass.
The standard curvaton model [@lm; @curvaton-paper] assumes that these expressions are dominated by the single ‘curvaton’ field $\sigma$, which starts to oscillate during radiation domination at a time when the component perturbation $\zeta_\sigma$ has negligible contribution to the curvature perturbation. Then the non-Gaussianity parameter is given by [@Lyth-gfc; @Lyth-general] $$\begin{aligned}
f_{NL} &\simeq& \frac{5}{4r_\sigma} \left( 1 +\frac{g''g}{g^2}\right)
-\frac{5}{3} - \frac{5}{6}r_\sigma,\end{aligned}$$ where $g(\sigma)$ is the initial amplitude of the oscillation as a function of the curvaton field at horizon exit [@Lyth-gfc]. Here $r_\sigma$ is identical to $r_1$, which will be defined in this paper[^2].
Non-linear formalism and the evolution of the perturbation {#sec:3}
==========================================================
In this paper we consider a clear separation of the adiabatic and the entropy perturbations in a two-field inflation model. The non-linear formalism for the component curvature perturbation is defined in Ref. [@Lyth-general; @Langlois:2008vk] as $$\begin{aligned}
\label{def-compzeta}
\zeta_i&=&\delta N+\int^{\rho}_{\bar{\rho}_i}H
\frac{d\tilde{\rho}_i}{3(1+w_i)\tilde{\rho}_i}\nonumber\\
&=& \label{ln-rho}\delta N + \frac{1}{3(1+w_i)}\ln
\left(\frac{\rho_i}{\bar{\rho}_i}\right)\nonumber\\
&\simeq & \delta N+\frac{1}{3(1+w_i)}\frac{\delta \rho_i^\mathrm{iso}}{\bar{\rho}_i},\end{aligned}$$ where $w_i=1/3$ for the radiation fluid and $w_i=0$ for the matter fluid. Here a bar is for a homogeneous quantity, and the curvature perturbation of the total fluid should be discriminated from the component curvature perturbation $\zeta_i$. The quantity $\delta \rho_i^\mathrm{iso} =\rho_i-\bar{\rho}_i$ in Eq.(\[def-compzeta\]) is the isocurvature perturbation (the fraction perturbation that satisfies $\sum \delta
\rho_i^\mathrm{iso}\equiv 0$), which is defined on the uniform density hypersurfaces.
In order to formulate the evolution of the curvature perturbation, which is caused by the adiabatic-isocurvature mixings, we need to define first the “starting point” perturbations at an epoch.
The primordial perturbations
----------------------------
For the first step, we define the primordial quantities. In this paper the quantities at the end of inflation are denoted by the subscript “end”, while the corresponding scale exited horizon at $t_*$. The subscript “$*$” is used for the quantities at the horizon exit. For our purpose, we define the primordial curvature and isocurvature perturbations at the end of the primordial inflation.
We find from Eq.(\[ln-rho\]); $$\begin{aligned}
\rho_i&=&\bar{\rho}_ie^{3(1+w_i)(\zeta_i-\delta N)}\nonumber\\
&\simeq& \bar{\rho}_i+3(1+w_i)\zeta_i^\mathrm{iso}\bar{\rho}_i\nonumber\\
&\equiv& \bar{\rho}_i+\delta \rho_i^\mathrm{iso}.\end{aligned}$$ Then we find from $\rho^\mathrm{tot}\equiv\rho_1+\rho_2=\bar{\rho}_1+\bar{\rho}_2$: $$\label{trivial-iso}
f_1e^{3(1+w_1)(\zeta_1-\delta N)}
+\left(1-f_1 \right)e^{3(1+w_2)(\zeta_2-\delta N)}=1,$$ where the fraction of the energy density is defined by $$f_1 \equiv \frac{\bar{\rho}_1}{\bar{\rho}_1+\bar{\rho}_2}.$$ Expanding Eq.(\[trivial-iso\]) and solving the equation for $\delta
N$, we find at first order [@Lyth-general] $$\begin{aligned}
\label{deltaN-1}
\delta N&=&r_1 \zeta_1+(1-r_1)\zeta_2\nonumber\\
&\equiv& \left[r_1\zeta_1^\mathrm{iso}+(1-r_1)\zeta_2^\mathrm{iso}\right]
+\zeta^\mathrm{adi},\end{aligned}$$ where $\zeta^\mathrm{iso}_i$ denotes the second component in Eq.(\[def-compzeta\]). $r_1$ is defined by $$\label{r1-basic}
r_1\equiv
\frac{3(1+w_1)\bar{\rho}_1}{3(1+w_1)\bar{\rho}_1+3(1+w_2)\bar{\rho}_2}.$$
Defining the primordial adiabatic curvature perturbation ($\zeta^\mathrm{inf}$) just at the end of inflation, the component curvature perturbation ($\zeta_i$) can be split into $\zeta^\mathrm{inf}$ and $\zeta_i^\mathrm{iso}$.
The obvious identity is $$\label{iso0}
r_{1,\mathrm{end}}\zeta_{1,\mathrm{end}}^\mathrm{iso}+(1-r_{1,\mathrm{end}})\zeta_{2,\mathrm{end}}^\mathrm{iso}
\equiv 0,$$ which is valid at the end of inflation. Apart from that point the deviation due to the evolution of $r_1$ becomes significant.
The parameter of the fluid ($w_i$) is constant when $\rho_i$ behaves like matter ($w_i=0$) or radiation ($w_i=1/3$), and a jump (e.g, $w_i=0 \rightarrow w_i=1/3$) is possible when instant transition is assumed. In this paper we are using the sudden-decay approximation for the curvaton mechanism. [^3] We also assume that the inflatons start sinusoidal oscillations just at the end of slow-roll.
The curvature perturbation in the standard curvaton scenario is usually expressed as $$\label{lang-eq}
\delta N =r_1\zeta_1+(1-r_1)\zeta^\mathrm{inf}.$$ Assuming that $\zeta_1^\mathrm{iso} \gg \zeta^\mathrm{inf} \gg
\zeta_2^\mathrm{iso}$, one will find $\zeta_1 \simeq \zeta_1^\mathrm{iso}$ and $\zeta_2\simeq
\zeta^\mathrm{inf}$, which gives Eq.(\[lang-eq\]) from Eq.(\[deltaN-1\]). Usually the above approximation is justified when $m_1\ll m_2$ and the curvaton is negligible during inflation.
In this paper we are considering the equal-mass limit ($m_1\simeq m_2$), which is in the opposite limit of the conventional curvaton. In Appendix we show the validity of the above approximations and derive the quantitative bound on the ratio $\phi_1/\phi_2$.
A basic model {#sec:4}
=============
In this section we show why the curvaton mechanism can create the dominant part of the curvature perturbation after conventional chaotic multi-field inflation, neither by adding extra light field (curvaton) nor by introducing many inflatons. The calculation clearly explains why and how the curvaton mechanism works in the equal-mass limit ($m_1\simeq m_2$).
We assume (for simplicity) that after inflation the field $\phi_2$ decays immediately into radiation and $\phi_1$ starts sinusoidal oscillation at the same time. Then $\phi_1$ decays late at $H_{d1}\ll H_I$. There is no mixing between these components. Here $H_I$ denotes the Hubble parameter during primordial inflation.
In this scenario, we consider two phases $(A,B)$ characterized by $w_{1A}=0$ and $w_{1B}=1/3$. Here the subscripts $A$ and $B$ denotes the quantities in the phase A and B. They are separated by the uniform density hypersurface $H_{d1}\simeq \Gamma_1$:
(A) $\rho_1$; oscillation, $\rho_2$; radiation\
($w_1 = 0$, $w_2 =1/3$)
(B) Radiation\
($w_1 =w_2 =1/3$).
The important assumption of the model is that the transition occurs on the uniform density hypersurfaces so that we can neglect additional creation of $\delta N$ (modulation) at the transition.
We find in phase (A); $$\begin{aligned}
\label{deltanbc}
\delta N &\equiv& r_{1A} \zeta_{1A} + (1-r_{1A})\zeta_{2},\end{aligned}$$ where the subscript “$A$” (or “$B$”) is omitted for $\zeta_{2}$, since $\zeta_2$ is constant during the evolution. Here we used the definition $$\label{rminus}
r_{1A} =\frac{3\bar{\rho}_1}{3\bar{\rho}_1+4\bar{\rho}_2}.$$
Consider a simple double-quadratic chaotic inflation model in the equal-mass limit. The potential is given by $$\label{sym-pot}
V(\phi_1,\phi_2)=\frac{1}{2}m^2\left(\phi_1^2+\phi_2^2\right)\equiv
\frac{1}{2}m^2 \phi_r^2,$$ where $\phi_{1,2}$ are real scalar fields. Besides the potential, we need the interaction that causes difference in the decay rates. Fig.\[fig:figure1\] shows the evolution of the densities after inflation.
![$t_\mathrm{end}$, $t_{d2}$, $t_\mathrm{osc}$ and $t_{d1}$ denote the time at the end of inflation, $\phi_2$ decay, the beginning of $\phi_1$ oscillation and $\phi_1$ decay, respectively. Our scenario is shown in the left-hand side, which gives the time-ordering $t_{end}\simeq t_{osc}<t_{d2}<t_{d1}$. The usual curvaton scenario is shown in the right-hand side, which gives $t_{end}<t_{d2}<t_{osc}<t_{d1}$.[]{data-label="fig:figure1"}](figure1.eps){width="1.0\columnwidth"}
The end of chaotic inflation is given by $$\phi_{1,\mathrm{end}}^2+\phi_{2,\mathrm{end}}^2\equiv
\phi_{r,\mathrm{end}}^2\simeq M_p^2.$$
![The straight dotted line with an arrow is the inflaton trajectory, and the circle gives the uniform-density surface along which the entropy perturbation $\delta s$ appears.[]{data-label="fig:equalmass"}](equalmass.eps){width="0.5\columnwidth"}
Since the potential is quadratic during inflation, we find $$\zeta^\mathrm{inf}=\frac{1}{\eta}\frac{\delta \phi_{r*}}{\phi_{r*}}.$$ In this section we consider $\theta\ll 1$, which leads to the simplifications $\sin\theta\sim \theta$ and $\cos\theta\sim 1$. Our approximations are based on the exact calculation in Appendix A.
From Eq.(\[zeta12ndorder\]), we find the component perturbation of the late-decaying component ($\phi_1$) at the end of inflation: $$\zeta_{1A} \simeq \frac{1}{3}\frac{\delta
\rho_{1,\mathrm{end}}^\mathrm{iso}}{\bar{\rho}_{1,\mathrm{end}}} \simeq
\frac{2}{3}\frac{\delta\theta}{\bar{\theta}}+
\frac{1}{3}\left(\frac{\delta\theta}{\bar{\theta}}\right)^2.$$ The usual approximation of the curvaton mechanism is $\zeta_{1A}^\mathrm{iso} \gg \zeta^\mathrm{inf}$. The validity of this approximation is examined in the Appendix.
From Eq.(\[zeta-losecond\]), the final curvature perturbation is $$\zeta^\mathrm{fin}\simeq \frac{2r_{1-}}{3}
\left[\frac{\delta \theta}{\bar{\theta}}+\frac{1}{2}\left(\frac{\delta
\theta}{\bar{\theta}}\right)^2
\right].$$ Defining the ratio $y\equiv \sqrt{\Gamma_1/\Gamma_2}$, $r_{1-}$ ($r_{1}$ evaluated in the phase (A) just before the decay) is given by $$r_{1-}\simeq \frac{3\bar{\theta}^2}{3\bar{\theta}^2+4y}.$$
The non-Gaussianity parameter has been calculated in Ref. [@Lyth-general]. We find for $\theta\ll 1$: $$\begin{aligned}
f_{NL} &\simeq& \frac{5}{4r_1} \left( 1 +\frac{g''g}{g^2}\right)
-\frac{5}{3} - \frac{5}{6}r_1\nonumber\\
&\sim&\frac{5}{4r_{1-}}.\end{aligned}$$
Further simplification is possible when $\delta \theta=\delta
s_*/\phi_{r*}$ and ${\cal P}_{\delta s_*}={\cal P}_{\delta \phi_{r*}}$. For the quadratic potential we have $\phi_{r*}=2\sqrt{N_e}M_p \gg
\phi_{e,\mathrm{end}}$, where $N_e$ is the number of e-foldings during the primordial inflation spent after the corresponding scale exits horizon. The condition of the curvaton mechanism $\zeta^\mathrm{fin}>\zeta^\mathrm{inf}$ gives $$\label{barthup}
\bar{\theta}<\frac{2}{3}r_{1-}\eta\simeq \frac{5}{6}\frac{\eta}{f_{NL}}.$$ Here $\bar{\theta}$ should be less than 1 but does not require many orders of magnitude. From the CMB spectrum we find the normalization given by $${\cal P}^{1/2}_{\zeta^\mathrm{fin}}\simeq
\frac{r_{1-}}{6\pi\sqrt{N_e}\bar{\theta}}\frac{H_I}{M_p}\simeq 5\times
10^{-5}.$$ Using Eq.(\[barthup\]), we find $$\frac{H_I}{M_p}< 5\eta \times 10^{-3},$$ which does not always require significant suppression. The ratio $y\equiv \sqrt{\Gamma_1/\Gamma_2}$ is calculated in Eq.(\[appy\]) and is given by $$y\simeq \frac{3}{5}f_{NL}\theta^2.$$ We thus find that the difference between $\phi_1$ and $\phi_2$ decay rates is in the conceivable range.
The above conditions tell us how small $\theta$ and $y$ have to be to get a given CMB spectrum and $f_{NL}$. They have to be some orders of magnitude below 1 but not very many.
If the potential during inflation is both symmetric and quadratic, we find $\eta\equiv \eta_1=\eta_2$. We thus find the spectral index $$n-1=-2\epsilon+\eta= 0,$$ which shows that the above model requires deviation from the symmetric potential.
Looking back into the many-field inflation, the model in Ref. [@CGJ] assumed that the inflaton masses are not exactly the same but may have statistical distribution around the mean value. In that case, the cancellation in the spectral index is not realistic. Since the deviation from the symmetric potential is expected, we need to examine what deviation is needed for the model. Then we can understand why and how the curvaton mechanism works in the many-field inflation model.
Deviation from the symmetric potential {#sec:5}
======================================
The deviation from the symmetric quadratic potential can be classified as follows;
1. The spectral index does not vanish when the double quadratic potential has different (but not so much different as the usual curvaton) mass. The slow-roll parameters are $$\begin{aligned}
\epsilon_H &\equiv& \frac{\dot{H}}{H^2}= \sum \epsilon_i
=\sum\eta_i f_i
\nonumber\\
%2\frac{M_p^2}{\phi_r^2}\\
\eta_i&\equiv& \frac{m_i^2}{3H_I^2},
%\simeq \frac{2 m_1^2M_p^2}{m_2^2\phi_r^2}.\end{aligned}$$ where the fraction of the density is given by $f_i\equiv
\frac{\rho_{i*}}{\rho_\mathrm{tot*}}$. The spectral index is shifted from $n_s-1=0$ and is given by $$\begin{aligned}
\label{spect-1}
n_s-1&=&-2\epsilon_H+2\eta_1\nonumber\\
&\simeq& -2[\eta_1 f_1+\eta_2(1-f_1)]+2\eta_1\nonumber\\
&\simeq& -2(\eta_2-\eta_1)\nonumber\\
&\equiv& -2P\eta_2\nonumber\\
&=&-\frac{P}{N_e},\end{aligned}$$ where $P\equiv \frac{m_2^2-m_1^2}{m_2^2}<1$. The observation [@WMAP7] shows $n_s-1=0.037\pm 0.014$, which suggests $N_e \lesssim 40$ and requires secondary inflation [@thermal-Inf].
Besides the spectral index, $m_1<m_2$ suggests that the oscillation of the field $\phi_1$ is slightly delayed compared to $\phi_2$. The delay may enhance the density of $\phi_1$ at the beginning of the oscillation, while the initial $\rho_1$ density may be reduced since $m_1$ is smaller. Defining $y_\mathrm{eff}\equiv
\sqrt{\Gamma_1/m_1}$ and $\theta_\mathrm{eff} \equiv \phi_1/\phi_2$ at the end of inflation, we find $$\begin{aligned}
r_{1A-}&\simeq&\frac{3m_1^2 \bar{\phi}_1^2}{3m_1^2 \bar{\phi}_1^2+4m_2^2\bar{\phi}_2^2
\left(\frac{m_1^2}{m_2^2}\right)y_\mathrm{eff}}\nonumber\\
&\simeq&\frac{3\bar{\theta}_\mathrm{eff}^2 }{3\bar{\theta}_\mathrm{eff}^2 +4 y_\mathrm{eff}},\end{aligned}$$ which gives a similar bound for $\theta_\mathrm{eff}$ ($y_\mathrm{eff}$).
2. Usually the curvaton is assumed to be much lighter than the inflaton; however this assumption could be avoided. We consider the curvaton mechanism when the curvaton is slightly heavier than the inflaton.
We assume $\rho_2>\rho_1$. Once it is assumed at the beginning of inflation, it remains true during inflation.[^4] Then $\phi_1$-oscillation starts [*during inflation*]{}. It begins when $$m_1^2=H^2_\mathrm{osc}\simeq\frac{m_2^2\phi_2^2|_\mathrm{osc}}{6M_p^2},$$ where the subscript “osc” denotes the beginning of $\phi_1$-oscillation. From the above equation and $\phi_2|_\mathrm{osc}\simeq2\sqrt{N_2}M_p$, where $N_2$ is the remaining number of e-foldings after the beginning of $\phi_1$-oscillation, we find $$\label{n2-mm}
N_2=\frac{3 m_1^2}{2m_2^2}.$$ Defining $y_\mathrm{eff}\equiv e^{3N_2}\sqrt{\Gamma_1/\Gamma_2}$ and $\theta_\mathrm{osc}\equiv
[\phi_1/\phi_2]_\mathrm{osc}$, we can estimate $$\begin{aligned}
r_{1A-}
&\sim&\frac{3\bar{\theta}_\mathrm{osc}^2
}{3\bar{\theta}_\mathrm{osc}^2 +4 y_\mathrm{eff}}.\end{aligned}$$ Unfortunately, the spectral index is $$\begin{aligned}
n_s-1
&\simeq& -2\epsilon_H+2\eta_1\nonumber\\
&\simeq &-2\eta_2+2\eta_1\nonumber\\
&\simeq& 2\eta_1>0.\end{aligned}$$
3. The potential could be dominated by a polynomial $V(\phi_r)\propto\phi_r^p$ at the moment when the perturbation exits horizon, while it can be approximated by the quadratic potential during the oscillation. For the polynomial we find $\phi_{r*}\simeq \sqrt{2pN_e}M_p$ and the slow-roll parameters $$\begin{aligned}
\epsilon_H &\simeq& \frac{1}{2}M_p^2\frac{p^2}{\phi_r^2} \\
\eta_1&\simeq& M_p^2\frac{p(p-1)}{\phi_r^2}.\end{aligned}$$ The spectral index is shifted and is given by $$\begin{aligned}
n_s-1&\simeq& - \frac{M_p^2}{\phi_r^2}\left[p^2-2p(p-1)\right]\nonumber\\
&\simeq& \frac{p-2}{2N_e}.\end{aligned}$$ The result suggests that $p<2$ is needed for the scenario. In that case the mass and the coefficient of the polynomial must run in the trans-Planckian [@run-inf]. $p=1$ would correspond to monodromy in the string theory and it requires $N_e\lesssim 20$. $p<1$ is an interesting possibility if the effective action allows fractional power.
A model with a complex scalar
-----------------------------
An inderesting application of the idea is that a conventional 2-field multiplet contains both inflation and the curvaton at the same time. Consider a complex scalar field $\Phi\equiv \phi_2+i\phi_1$, which gives the symmetric potential $$V(\Phi)=\frac{1}{2} m^2 |\Phi^2|^2 =\frac{1}{2}m^2(\phi_1^2+\phi_2^2).$$
First, consider a small symmetry breaking caused by $$\Delta V\sim \frac{\Lambda^4}{M^2}\left(\frac{\Phi+\Phi^*}{2}\right)^2,$$ where $\Lambda \ll M$ is assumed. $\phi_2$-oscillation may cause significant particle production when there is the interaction given by $${\cal L}_\mathrm{int}=g(\Phi+\Phi^*)\bar{\psi}\psi,$$ which can lead to significant $\psi$-production at the enhanced symmetric point ($\phi_2\sim0$) [@PR]. The coefficient of the interaction could be small ($g\sim
\Lambda/M \ll 1$) when it is suppressed by a cut-off scale. $\psi$ may decay quickly into radiation since the amplitude of the oscillation after chaotic inflation is very large [@PR].
Define $\delta m^2 \equiv \frac{2\Lambda^4}{M^2}$. If $\delta m^2$ is much smaller than $m^2$, the cancellation in Eq.(\[spect-1\]) is still significant. On the other hand, it is possible to assume $\delta m^2/m^2\sim O(1)$ (which is still within the conventional set-up of multi-field inflation) one obtains $P\sim 1$ and $n_s-1\sim
-\frac{1}{N_e}$. Again, the scenario requires additional inflation stage [@thermal-Inf].
Second, consider the case in which the potential during inflation is dominated by a polynomial $V(\Phi)\propto \Phi^p$. The curvaton can dominate the spectrum, however the spectral index becomes $$\begin{aligned}
n_s-1&\simeq& - \frac{M_p^2}{\phi_r^2}\left[p^2-2p(p-1)\right]\nonumber\\
&\simeq& \frac{p-2}{2N_e}.\end{aligned}$$ The scenario requires $p<2$.
Sneutrino inflation {#sn-inf}
-------------------
It is possible to assume small inflation “before” the multi-field inflation. The observed spectrum of the curvaton perturbation exits horizon during the first inflation. In that case $\epsilon_H$ is determined by the first inflation and the cancellation in the spectral index is avoided. This scenario uses multi-field inflation for the curvaton inflation [@Infcurv].
The usual sneutrino inflation [@Ellis:2003sq] uses $m\sim 10^{13}$ GeV to satisfy the CMB normalization. When the condition is combined with the gravitino problem, Yukawa coupling of the first generation sneutrino (single-field inflaton) must satisfy $(Y_\nu Y_\nu)^{\dagger}_{11}< 10^{-12}$, whilst other Yukawa couplings will not be so small. Here $Y_\nu$ is the neutrino Yukawa matrix.
In this section we consider multi-stage inflation, in which three sneutrinos play crucial role. We assume that the first single-field inflation is caused by the third generation sneutrino, and the secondary two-field inflation is caused by the first and the second generation sneutrinos with the mass $M_{1}=M_{2}\equiv \hat{M}$. We assume $M_{3}>\hat{M}$ for the third generation.
The reheating after two-field inflation is due to the decay of the second generation sneutrino, which gives the reheating temperature $$T_R=\left(\frac{90}{\pi^2g_*}\right)^{1/4}\sqrt{\Gamma_2 M_p},$$ where the decay rate is $$\Gamma_i\simeq\frac{1}{4\pi}\left(Y_\nu Y_\nu^{\dagger}\right)_{ii}\hat{M}.$$ From Eq.(\[zeta-lo\]), the curvaton mechanism is significant when $\Gamma_2\gg \Gamma_1$. For the two-field sneutrino inflation, which is the secondary inflation of the above scenario, we find $${\cal P}_{\zeta_1}^{1/2} < \frac{1}{3\pi}
\left(\frac{(Y_\nu Y_\nu^\dagger)_{22}}{(Y_\nu Y_\nu^\dagger)_{11}}\right)^{1/4}
\frac{\hat{M}}{M_p}.$$ Here the mass of the first (second) neutrino is $$(m_\nu)_{ii}\simeq (Y_\nu Y_\nu^\dagger)_{ii}\frac{<H_u>^2}{\hat{M}}.$$ We thus find for the given neutrino mass $(m_\nu)_{11}$ and $(m_\nu)_{22}$; $${\cal P}_{\zeta_1}^{1/2}< \frac{1}{3\pi}
\left(\frac{(m_\nu)_{22}}{(m_\nu)_{11}}\right)^{1/4}
\frac{\hat{M}}{M_p}.$$ The reheating temperature after inflation is given by $$T_R=\left(\frac{45}{8\pi^4g_*}\right)^{1/4}
\frac{\hat{M}}{<H_u>}
\sqrt{(m_{\nu})_{22} M_p},$$ while the temperature just after the curvaton decay is $$T_R'=\left(\frac{45}{8\pi^4g_*}\right)^{1/4}
\frac{\hat{M}}{<H_u>}
\sqrt{(m_{\nu})_{11} M_p}.$$ We may write the spectrum ${\cal P}_{\zeta_1}$ using $T_R$ and $T_R'$; $${\cal P}_{\zeta_1}^{1/2} < \frac{1}{3\pi}
\left(\frac{T_R}{T_R'}\right)^{1/2}
\frac{\hat{M}}{M_p}.$$
When the primary inflation gives the number of e-foldings $N_1$, the spectral index is $$n_s-1\simeq -2\epsilon_H\simeq-\frac{1}{N_1}.$$ The observation gives $n_s-1= 0.037 \pm 0.014$, which suggests $20\lesssim N_1\lesssim 40$ for the first inflation.
N-flation {#N-fla}
---------
The two-field inflation model considered in this paper is a simplification of the N-flation model [@N-flation]. The N-flation has been studied using statistical argument [@CGJ], which helps us understand the results obtained above for the two-field model.
Assuming (for simplicity) the same potential for all $N_f$ fields, we find $$V(\phi_n)=\sum^{N_f}_{n=1}\frac{1}{2}m^2 \phi_n^2.$$ Using the adiabatic field defined by $$\phi_r^2 \equiv \sum^{N_f}_{n=1}\phi_n^2,$$ we find the potential $$V(\phi_r)=\frac{1}{2}m^2 \phi_r^2.$$ If we assume uniform initial condition $\phi_n\simeq \phi_0$, the model is identical to the two-field model with $\theta
\sim 1/\sqrt{N_f}\ll 1$.
For the number of e-foldings $N_e\sim 60$, the usual curvature perturbation created at the horizon exit is given by $$\zeta^\mathrm{inf} =-H_I\left.\frac{\delta \phi_r}{\dot{\phi}_r}\right|_*
=2N_e\left.\frac{\delta \phi_r}{\phi_r}\right|_* .$$ where $H_I^2 \equiv \frac{N_f m^2\phi_0^2}{6M_p^2}$ is the Hubble parameter during the primordial N-flation.
Suppose that the decay rate $\Gamma_{n}$ is uniform [*except for a field $\phi_1$*]{}, which has $\Gamma_1\ll \Gamma_n$. Here the density ratio becomes $r_1^*\simeq \frac{1}{N_f}$. Repeating the same calculation, we find $$\zeta_1\equiv \frac{\delta \rho_1}{3\rho_1}=
\frac{2}{3} \frac{\delta \phi_1}{\phi_0}\simeq
\frac{2}{3}\sqrt{N_f} \frac{\delta s}{\phi_r}.$$ ${\cal P}_{\zeta^\mathrm{inf}}^{1/2}\ll {\cal P}_{\zeta_1}^{1/2}$ is possible when $N_f \gg N_e^2$. This gives the minimum number of the fields that is needed for the curvaton mechanism and it explains the numerical calculation in Ref. [@CGJ].
In the above scenario, the curvaton is one of the inflaton fields that are equally participating $1/N_f$ of the inflaton dynamics.
At the end of inflation, the fraction of $\rho_1$ is $$r_1(t_{end})=\frac{1}{N_f}\ll 1,$$ while at the decay of $\phi_1$ it can grow; $$r_1(t_{decay})=r_1(t_{end})\times
\left(\frac{\Gamma_n}{\Gamma_1}\right)^{1/2}.$$ We need for the curvaton mechanism (i.e, $\zeta_1$-domination) $$\frac{2}{3}\sqrt{N_f}\frac{{\cal P}_{\delta \phi_1}}{\phi_r}\times \frac{1}{N_f}
\left(\frac{\Gamma_n}{\Gamma_1}\right)^{1/2}>2N_e \frac{{\cal P}_{\delta \phi_r}}{\phi_r},$$ which leads to $$\left(\frac{\Gamma_1}{\Gamma_n}\right)^{1/2}< \frac{1}{3N_e\sqrt{N_f}}.$$ Significant non-Gaussianity ($f_{NL}$) requires $r(t_{decay})\sim 0.1$, which gives $$\left(\frac{\Gamma_1}{\Gamma_n}\right)^{1/2}\sim \frac{10}{N_f}.$$ If the distribution is statistical for the decay rate, we need $N_f \gg 1$ for the strong suppression ($\Gamma_1/\Gamma_n\ll
1$).
In this section we found that the evolution after inflation may dominate the curvature perturbation when $N_f$ is large. Our result explains the numerical calculation in Ref. [@CGJ].
Conclusions
===========
The evolution after multi-field inflation can change the curvature perturbation. In this paper we considered a conventional two-field inflation model and showed that the curvaton mechanism after multi-field inflation could be significant when the decay rates are not identical [^5]. Interestingly, the mechanism works for a complex scalar field $\Phi\equiv \phi_2+i\phi_1$.
The previous numerical study [@CGJ] showed that $N_f\gg 1$ causes significant evolution of the curvature perturbation after inflation as well as the creation of significant non-Gaussianity. We showed that the same is true for two-field inflation, in which $\theta\ll 1$ is required instead of $N_f\gg 1$.
The source of the curvaton mechanism is the entropy perturbation generated during multi-field inflation. Since the uniform density surface of the multi-field potential is flat by definition, the perturbation on that surface is inevitable.
Our results suggest that many-field inflation must be considered with care. A large number ($N_f\ge 10^{3}$) can easily explain the required condition for the curvaton domination.
Acknowledgment
==============
We thank D. H. Lyth for collaboration in the early stage of the paper. T.M thanks J. McDonald for many valuable discussions. S.E. is supported by the Grant-in-Aid for Nagoya University Global COE Program, “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos”.
Calculation details
===================
Evolution of the curvature perturbation
---------------------------------------
In this Appendix we show the calculation details of the evolution after inflation.
We first assume that the potential is quadratic and symmetric during chaotic inflation. In our formalism $\zeta^\mathrm{inf}$ is defined at the end of inflation. The entropy perturbation is realized by $\delta \theta$, which is the perturbation of the angle $\theta$ in Fig.\[fig:equalmass\].
The spectrum of the entropy perturbation during inflation is ${\cal
P}_{\delta s_*} \simeq (H_{*}/2\pi)^2$. The entropy perturbation causes the fraction perturbation between densities. Using $\delta\theta$, the densities of the components and the isocurvature perturbations at the end of inflation are given by $$\begin{aligned}
\bar{\rho}_{1,\mathrm{end}}&=&\frac{1}{2}m^2|\phi_r^\mathrm{end}|^{2}\sin^2 \bar{\theta} \simeq
\frac{m^2M_p^2}{2}\sin^2 \bar{\theta}\\
&&\delta \rho_{1,\mathrm{end}}^\mathrm{iso}\simeq m^2 M_p^2(\sin\bar{\theta} \cos\bar{\theta} )
\delta\theta
,\\
\bar{\rho}_{2,\mathrm{end}}&=&\frac{1}{2}m^2|\phi_r^\mathrm{end}|^{2}\cos^2 \bar{\theta} \simeq
\frac{m^2M_p^2}{2}\cos^2 \bar{\theta}\\
&&\delta \rho_{1,\mathrm{end}}^\mathrm{iso}+\delta \rho_{2,\mathrm{end}}^\mathrm{iso}=0.\end{aligned}$$ We find at the end of inflation: $$\begin{aligned}
f_1&\equiv& \frac{\bar{\rho}_1}{\bar{\rho}_1+\bar{\rho}_2}= \sin^2\bar{\theta},\\
\delta f_1&\simeq& \frac{\partial
f_1}{\partial \theta} \delta \theta
=2[\sin\bar{\theta}\cos\bar{\theta}]\delta \theta\nonumber\\
&=&[\sin2\bar{\theta}] \delta \theta.\end{aligned}$$ The expansion with respect to $\delta \theta$ makes no sense when $\delta \theta/\sin\theta\ge 1$ or $\delta
\theta/\cos\theta \ge 1$ [@Lyth-ngaus]. We are excluding those regions.
Creation of the curvature perturbation after inflation requires the decay rate $\Gamma_1 \ll \Gamma_2$. In the phase (A) we find $$\begin{aligned}
\zeta^\mathrm{iso}_{1A}&\simeq&\frac{2}{3}
\frac{\cos\bar{\theta}}{\sin\bar{\theta}}\delta
\theta\\
\zeta^\mathrm{iso}_{2A}&\simeq&-\frac{1}{2}\frac{\sin\bar{\theta}}{\cos\bar{\theta}}\delta
\theta.\end{aligned}$$ Using Eq.(\[deltaN-1\]), we find $$\begin{aligned}
\label{zeta-lo}
\zeta^\mathrm{fin}
&=&
\left[\frac{2}{3}r_{1-}\frac{\cos\bar{\theta}}{\sin\bar{\theta}}-
\frac{1}{2}(1-r_{1-})
\frac{\sin\bar{\theta}}{\cos\bar{\theta}} \right]\delta \theta
+\zeta^\mathrm{inf}\nonumber\\
&=&\left[\frac{4r_{1-}\cos^2\bar{\theta}-3(1-r_{1-})\sin^2\bar{\theta}}
{6\sin\bar{\theta}\cos\bar{\theta}}
\right]\delta \theta\nonumber\\
&&+\zeta^\mathrm{inf},\end{aligned}$$ where $r_{1-}$ denotes the value of $r_{1A}$ evaluated just before the end of the phase (A).
The evolution is $$\begin{aligned}
\bar{\rho}_{1-}&=&\left[\frac{m^2 M_p^2}{2}\sin^2\bar{\theta}\right]\times
\left(\frac{a_\mathrm{d1}}{a_\mathrm{end}}\right)^{-3}
\nonumber\\
\bar{\rho}_{2-}&=&\left[\frac{m^2 M_p^2}{2}\cos^2\bar{\theta}\right]\times
\left(\frac{a_\mathrm{d2}}{a_\mathrm{end}}\right)^{-3}
\left(\frac{a_\mathrm{d1}}{a_\mathrm{d2}}\right)^{-4},\end{aligned}$$ which leads to the ratio $$\begin{aligned}
\frac{\bar{\rho}_{2-}}{\bar{\rho}_{1-}}&=&\frac{\cos^2\bar{\theta}}{\sin^2\bar{\theta}}
\left(\frac{a_{d2}}{a_{d1}}\right).\end{aligned}$$ Therefore, in the radiation dominated Universe we find $$\begin{aligned}
\label{r1-}
r_{1-}&=&\frac{3\rho_{1-}}{3\rho_{1-}+4\rho_{2-}}\nonumber\\
&=&\frac{3\sin^2\bar{\theta}}{3\sin^2\bar{\theta} + 4\cos^2\bar{\theta}
\sqrt{\Gamma_1/\Gamma_2}}.\end{aligned}$$ Domination by the curvaton density ($r_{1-}\sim1$) requires $\sqrt{\Gamma_1/\Gamma_2}\le \tan^2\bar{\theta}$.
The CMB spectrum requires ${\cal P}_{\zeta^\mathrm{fin}}\simeq (5\times 10^{-5})^2$ [@WMAP7]. The requirement is trivial when $\zeta^\mathrm{fin}\simeq\zeta^\mathrm{inf}$, [^6] while in the opposite case $\zeta^\mathrm{fin}>\zeta^\mathrm{inf}$, in which the curvaton mechanism dominates, we need the condition $$\label{adi-iso-con}
\left[\frac{2}{3}r_{1-}\frac{\cos\bar{\theta}}{\sin\bar{\theta}}-
\frac{1}{2}(1-r_{1-})
\frac{\sin\bar{\theta}}{\cos\bar{\theta}} \right]\delta \theta >
\frac{\delta \phi_{r*}}{\eta \phi_{r*}}.$$ Solving Eq.(\[adi-iso-con\]) for $r_{1-}$ and using Eq.(\[r1-\]), we find $$\label{gamma12}
\sqrt{\frac{\Gamma_1}{\Gamma_2}}<
%\frac{\sin\bar{\theta}}{2\cos\bar{\theta}}
%\frac{2\eta\cos\bar{\theta}-3\sin\bar{\theta}}{2\cos\bar{\theta}+\eta\sin\bar{\theta}}
\frac{2\eta\tan\bar{\theta}-3\tan^2\bar{\theta}}{4+2\eta\tan\bar{\theta}}<1.$$ This equation also shows that $2\eta-3\tan\bar{\theta}>0$, which gives $$\label{justif-theta}
\tan\bar{\theta}<\frac{2}{3}\eta.$$
The CMB observation gives the normalization $$\begin{aligned}
\label{spectrum0}
\left[\frac{2}{3}r_{1-}\frac{\cos\bar{\theta}}{\sin\bar{\theta}}-
\frac{1}{2}(1-r_{1-})
\frac{\sin\bar{\theta}}{\cos\bar{\theta}} \right]
{\cal P}^{1/2}_{\delta \theta}
%\frac{H_I}{2\pi\phi_{r*}}
\simeq 5\times 10^{-5}.\end{aligned}$$
Defining $k\equiv \frac{{\cal P}^{1/2}_{\delta\theta}}{5\times
10^{-5}}$ and $y\equiv
\sqrt{\Gamma_1/\Gamma_2}$, we can solve Eq.(\[spectrum0\]) for $y$ and find $$\begin{aligned}
\label{zeta-sim}
y
&=& \frac{2k -3\tan\bar{\theta}}
{2k+4\tan^{-1}\bar{\theta}}\nonumber\\
&\simeq & \frac{k\bar{\theta}}{2}.\end{aligned}$$ To avoid $y<0$, we need the condition $$\frac{3}{2}\tan\bar{\theta}<k.$$
The perturbations can be expanded up to second order. We find $$\begin{aligned}
\label{zeta12ndorder}
\zeta^\mathrm{iso}_1&\simeq&\frac{2}{3}
\left[\cos\bar{\theta} \left(\frac{\delta \theta}{\sin\bar{\theta}}\right)
+\frac{1}{2}\cos2\bar{\theta} \left(\frac{\delta \theta}{\sin\bar{\theta}}\right)^2
\right]
\\
\zeta^\mathrm{iso}_2&\simeq&-\frac{1}{2}
\left[\sin\bar{\theta} \left(\frac{\delta \theta}{\cos\bar{\theta}}\right)
+\frac{1}{2}\cos2\bar{\theta} \left(\frac{\delta \theta}{\cos\bar{\theta}}\right)^2
\right].\end{aligned}$$ Using Eq.(\[deltanbc\]), the final curvature perturbation after the decay is $$\begin{aligned}
\label{zeta-losecond}
\zeta^\mathrm{fin}
&=&
\left[\frac{2}{3}r_{1-}\frac{\cos\bar{\theta}}{\sin\bar{\theta}}-
\frac{1}{2}(1-r_{1-})
\frac{\sin\bar{\theta}}{\cos\bar{\theta}} \right]\delta \theta\nonumber\\
&&
+\left[\frac{1}{3}r_{1-}\frac{\cos2\bar{\theta}}{\sin^2\bar{\theta}}
-\frac{1}{4}(1-r_{1-})
\frac{\cos2\bar{\theta}}{\cos^2\bar{\theta}} \right](\delta \theta)^2\nonumber\\
&&+\zeta^\mathrm{inf}\nonumber\\
&=&\left[\frac{4r_{1-}\cos^2\bar{\theta}-3(1-r_{1-})\sin^2\bar{\theta}}
%{r_{1-}(4\cos^2\bar{\theta}+3\sin^2\bar{\theta})-3\sin^2\bar{\theta}}
{3\sin2\bar{\theta}}
\right]\delta \theta\nonumber\\
&&+\frac{\cos2\bar{\theta}}{3\sin^2 2\bar{\theta}}
\left[4r_{1-}\cos^2\bar{\theta}-3(1-r_{1-})\sin^2\bar{\theta}
%r_{1-}(4\cos^2\bar{\theta}+3\sin^2\bar{\theta})-3\sin^2\bar{\theta}
\right](\delta
\theta)^2\nonumber\\
&&+\zeta^\mathrm{inf}\nonumber\\
&=&
\frac{4r_{1-}\cos^2\bar{\theta}-3(1-r_{1-})\sin^2\bar{\theta}}
%{r_{1-}(4\cos^2\bar{\theta}+3\sin^2\bar{\theta})-3\sin^2\bar{\theta}}
{3\sin2\bar{\theta}}\nonumber\\
&&\times\left[\delta \theta + \frac{\cos2\bar{\theta}}{\sin2\bar{\theta}}(\delta \theta)^2
\right]
+\zeta^\mathrm{inf}.\end{aligned}$$
When the curvaton perturbation dominates ($\theta \ll 1$), the non-Gaussianity of the spectrum is measured by $$\begin{aligned}
\label{fnl-stdd}
f_{NL}&\simeq&\frac{5\cos2\bar{\theta}}
{4r_{1-}\cos^2\bar{\theta}-3(1-r_{1-})\sin^2\bar{\theta}}.
%{r_{1-}(4\cos^2\bar{\theta}+3\sin^2\bar{\theta})-3\sin^2\bar{\theta}}\end{aligned}$$
Using Eq.(\[r1-\]), we can substitute $r_{1-}$ in Eq.(\[fnl-stdd\]). Then solving the equation for $y$, we find $$\begin{aligned}
\label{gammma12a}
y&\simeq&\frac{3}{4}\tan^2\bar{\theta}\left[
\frac{4}{5}\frac{\cos^2\bar{\theta}}{\cos2\bar{\theta}}f_{NL}-1\right].\end{aligned}$$ Barring cancellation, the above equation gives a simplified formula $$\begin{aligned}
\label{appy}
y&\simeq&
\frac{3}{5}f_{NL}\bar{\theta}^2.\end{aligned}$$ Being combined with Eq.(\[zeta-sim\]), which has been obtained using the CMB normalization, we find $$\begin{aligned}
k& \simeq& \frac{6}{5}f_{NL} \bar{\theta}.\end{aligned}$$ We thus find (from $f_{NL}$ and CMB using the definition of $k$) $$\begin{aligned}
\frac{{\cal P}^{1/2}_{\delta\theta}}{\bar{\theta}}&\simeq&6\times 10^{-5}
\times f_{NL}\end{aligned}$$ or equivalently $$\begin{aligned}
\label{HIeq}
H_I
&\simeq&
6\times 10^{-3}\times
f_{NL}\bar{\theta}M_p.\end{aligned}$$ Solving the equation for $\bar{\theta}$, it gives $$\begin{aligned}
\bar{\theta}&\simeq & \frac{1}{6f_{NL}}
\left[\frac{H_I}{M_p}\times 10^3\right].\end{aligned}$$
Using $H_I$ in Eq.(\[HIeq\]) and calculating the tensor to scalar ratio $r_g$, we find [@Lyth-bound] $$r_g\simeq f_{NL}^2 \bar{\theta}^2\times 10^4.$$
Considering the natural bound $\Gamma_2<H_I$ and $\Gamma_1>H_\mathrm{nuc}$, where $H_\mathrm{nuc}$ is the Hubble parameter at the time of the nucleosynthesis, Eq.(\[gammma12a\]) gives the lower bound for $\bar{\theta}$; $$\bar{\theta}> \left(\frac{H_\mathrm{nuc}}{H_I}\right)^{1/4}.$$ Besides the above condition, we have another condition coming from $\bar{\theta}>\delta \theta$. Since we are assuming quadratic potential in the trans-Planckian, we have $\delta \theta =\delta s/\phi_{r*}$ and $\phi_{r*}=2\sqrt{N_e}M_p$. Then $\bar{\theta}>\delta \theta$ leads to $$\bar{\theta} >0.01 \frac{H_I}{M_p}.$$
[1]{} D. H. Lyth, A. R. Liddle, Cambridge, UK: Cambridge Univ. Pr. (2009) 497 p.
D. Wands, K. A. Malik, D. H. Lyth and A. R. Liddle, Phys. Rev. D [**62**]{}, 043527 (2000) \[astro-ph/0003278\]. A. Gangui, F. Lucchin, S. Matarrese and S. Mollerach, Astrophys. J. [**430**]{}, 447 (1994) \[astro-ph/9312033\]. M. Sasaki and E. D. Stewart, Prog. Theor. Phys. [**95**]{}, 71 (1996) \[astro-ph/9507001\]. J. Garcia-Bellido and D. Wands, Phys. Rev. D [**53**]{}, 5437 (1996) \[astro-ph/9511029\]. V. F. Mukhanov and P. J. Steinhardt, Phys. Lett. B [**422**]{}, 52 (1998) \[astro-ph/9710038\]. D. Polarski and A. A. Starobinsky, Phys. Rev. D [**50**]{}, 6123 (1994) \[astro-ph/9404061\]. T. Matsuda, Phys. Lett. B [**682**]{}, 163 (2009) \[arXiv:0906.2525 \[hep-th\]\]; T. Matsuda, JCAP [**0609**]{}, 003 (2006) \[hep-ph/0606137\]. F. Bernardeau and J. -P. Uzan, Phys. Rev. D [**66**]{}, 103506 (2002) \[hep-ph/0207295\]; L. E. Allen, S. Gupta and D. Wands, JCAP [**0601**]{}, 006 (2006) \[astro-ph/0509719\]. K. -Y. Choi, J. -O. Gong and D. Jeong, JCAP [**0902**]{}, 032 (2009) \[arXiv:0810.2299 \[hep-ph\]\].
R. Easther and L. McAllister, JCAP [**0605**]{}, 018 (2006) \[hep-th/0512102\]. D. Battefeld and S. Kawai, Phys. Rev. D [**77**]{}, 123507 (2008) \[arXiv:0803.0321 \[astro-ph\]\]. J. Elliston, D. J. Mulryne, D. Seery and R. Tavakol, JCAP [**1111**]{}, 005 (2011) \[arXiv:1106.2153 \[astro-ph.CO\]\]; J. Elliston, D. Mulryne, D. Seery and R. Tavakol, Int. J. Mod. Phys. A [**26**]{}, 3821 (2011) \[arXiv:1107.2270 \[astro-ph.CO\]\].
A. D. Linde and V. F. Mukhanov, Phys. Rev. D [**56**]{} (1997) 535 \[arXiv:astro-ph/9610219\]. T. Moroi, T. Takahashi, Phys. Lett. [**B522**]{}, 215-221 (2001). \[hep-ph/0110096\]. D. H. Lyth, D. Wands, Phys. Lett. [**B524**]{}, 5-14 (2002). \[hep-ph/0110002\].
D. H. Lyth, Phys. Lett. B [**579**]{} (2004) 239 \[hep-th/0308110\].
D. H. Lyth, K. A. Malik and M. Sasaki, JCAP [**0505**]{}, 004 (2005) \[astro-ph/0411220\]; M. Sasaki, J. Valiviita and D. Wands, Phys. Rev. D [**74**]{}, 103003 (2006) \[astro-ph/0607627\]. D. Langlois, F. Vernizzi and D. Wands, JCAP [**0812**]{}, 004 (2008) \[arXiv:0809.4646 \[astro-ph\]\]. K. Enqvist, R. N. Lerner and O. Taanila, JCAP [**1112**]{}, 016 (2011) \[arXiv:1105.0498 \[astro-ph.CO\]\]. E. Komatsu [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl. [**192**]{}, 18 (2011). D. H. Lyth and E. D. Stewart, Phys. Rev. D [**53**]{}, 1784 (1996) \[hep-ph/9510204\]; T. Matsuda, Phys. Rev. D [**65**]{}, 103501 (2002) \[hep-ph/0202209\]. K. Enqvist, S. Kasuya and A. Mazumdar, Phys. Rev. D [**66**]{}, 043505 (2002) \[hep-ph/0206272\]. G. N. Felder, L. Kofman and A. D. Linde, Phys. Rev. D [**59**]{}, 123523 (1999) \[hep-ph/9812289\]; T. Matsuda, JCAP [**0703**]{}, 003 (2007) \[hep-th/0610232\]. A. R. Liddle, A. Mazumdar, F. E. Schunck, Phys. Rev. [**D58**]{}, 061301 (1998). \[astro-ph/9804177\]; S. Dimopoulos, S. Kachru, J. McGreevy, J. G. Wacker, JCAP [**0808**]{}, 003 (2008). \[hep-th/0507205\].
D. Polarski and A. A. Starobinsky, Nucl. Phys. B [**385**]{}, 623 (1992). K. Dimopoulos, K. Kohri, D. H. Lyth and T. Matsuda, JCAP [**1203**]{}, 022 (2012) \[arXiv:1110.2951 \[astro-ph.CO\]\]; K. Dimopoulos, K. Kohri and T. Matsuda, Phys. Rev. D [**85**]{}, 123541 (2012) \[arXiv:1201.6037 \[hep-ph\]\]; S. Enomoto, K. Kohri and T. Matsuda, arXiv:1210.7118 \[hep-ph\]; K. Kohri, C. -M. Lin and T. Matsuda, arXiv:1211.2371 \[hep-ph\]. J. R. Ellis, M. Raidal, T. Yanagida, Phys. Lett. [**B581**]{}, 9-18 (2004). \[hep-ph/0303242\]. T. Matsuda, JCAP [**1204**]{}, 020 (2012) \[arXiv:1204.0303 \[hep-ph\]\]. D. H. Lyth, JCAP [**0606**]{}, 015 (2006) \[astro-ph/0602285\]. D. H. Lyth, Phys. Rev. Lett. [**78**]{}, 1861 (1997) \[hep-ph/9606387\]. L. Boubekeur and D. .H. Lyth, Phys. Rev. D [**73**]{} (2006) 021301 \[astro-ph/0504046\].
[^1]: In Ref. [@CGJ; @Easther:2005zr; @Battefeld:2008bu; @afterCGJ], statistical distribution of the inflaton mass has been considered for N-flation. The deviation $m_\mathrm{Max}/m_\mathrm{min}\lesssim O(10)$ will be considered in this paper.
[^2]: In this paper we use $(\phi_1, \phi_2)$ for two-field inflation, instead of using the conventional $(\sigma, \phi)$ in the curvaton scenario.
[^3]: Authors of Ref. [@Enqvist:2011jf] consistently accounted the curvaton decay when the curvaton decays pertubatively and showed that the correction to the potential can be significant.
[^4]: The opposite condition ($\rho_1>\rho_2$) requires $\phi_{1*}>M_p$, which suppresses the component perturbation of the curvaton and does not realize the curvaton mechanism.
[^5]: A similar but another story has been discussed in Ref.[@matsuda-hybrid].
[^6]: Note however the non-Gaussianity is not trivial because the curvaton perturbation may still dominate the second-order perturbation [@Lyth-adi0].
|
---
abstract: 'Dynamical scaling in ageing systems, notably in phase-ordering kinetics, is well-established. New evidence in favour of Galilei-invariance in phase-ordering kinetics is described.'
author:
- Malte Henkel
bibliography:
- 'granada8-henkel.bib'
title: 'Phase-ordering kinetics: ageing and local scale-invariance'
---
[ address=[Laboratoire de Physique des Matériaux (CNRS UMR 7556), Université Henri Poincaré Nancy I, B.P. 239, F - 54506 Vand[œ]{}uvre-lès-Nancy Cedex, France]{} ]{}
Dynamical scaling in ageing systems
===================================
The study of the long-time dynamics of statistical systems far from equilibrium has been a topic of intensive study. In many instances, the relaxation times towards equilibrium can become extremely long such that the system stays for all intents and purposes out of equilibrium. A paradigmatic example is provided by glassy systems which might be considered as an extremely viscous liquid. In principle, the presence of very long relaxation time-scales might suggest that quantitative properties of glassy dynamics should depend on a huge variety of microscopic ‘details’ and furthermore, as the behaviour of a glass may depend on its previous (thermal, mechanic,…) history, those system age. However, as first pointed out by Struik in 1978 [@Stru78], the ageing dynamics of many physically very different glass-forming systems can be described in terms of [*universal*]{} master curves. The challenge is to try to understand the origin of this dynamical scale-invariance and to compute the form of these master curves from the essential characteristics of the system.
Here we shall consider an analogous situation in supposedly more simple systems without intrinsic disorder or frustration. From some initial state (typically one chooses a fully disordered initial state) the system is rapidly ‘quenched’ either onto its critical point or else into a region of the phase diagram with at least two stable stationary states. The dynamics and an eventual ageing behaviour (that is, a breaking of time-translation invariance) is then observed. One distinguishes [*physical ageing*]{}, where the underlying microscopic processes are reversible, from [*chemico-biological ageing*]{}, where irreversible microscopic processes may occur. In terms of models, a simple example for physical ageing is provided by the phase-ordering kinetics of a simple Ising ferromagnet with purely relaxational dynamics quenched to below its critical temperature $T_c>0$. On the other hand, the ageing behaviour of the contact process (particles of a single species $A$ move diffusively on a lattice and react according to $A+A\to\emptyset$ and $A\to2A$) provides a paradigmatic case of ageing with underlying irreversible processes.
Physically, these two kinds of ageing phenomena are quite distinct. In relaxing ferromagnets, see e.g. [@Bray94; @Cugl02; @Henk04b] for reviews, there is for $T<T_c$ a non-vanishing surface tension between the ordered domains which leads to the formation and growth of ordered clusters of linear size $L=L(t)\sim t^{1/z}$, where $z$ is the dynamical exponent. For purely relaxational dynamics, it can be shown that $z=2$ for $T<T_c$ whereas for $T=T_c$, the non-trivial value of $z$ equals the one found for equilibrium critical dynamics. If $\phi(t,\vec{r})$ denotes the time- and space-dependent order parameter, it is convenient to characterize the ageing behaviour in terms of the two-time correlation and linear response functions $$C(t,s;\vec{r}) = \left\langle \phi(t,\vec{r})\phi(s,\vec{0})\right\rangle
\;\; , \;\;
R(t,s;\vec{r}) = \left.
\frac{\delta\langle\phi(t,\vec{r})\rangle}{\delta h(s,\vec{0})}
\right|_{h=0}$$ where $h$ is the magnetic field conjugate to $\phi$. The autocorrelation and linear autoresponse functions are given by $C(t,s)=C(t,s;\vec{0})$ and $R(t,s)=R(t,s;\vec{0})$. Here and later space-translation invariance will be assumed. In phase-ordering, dynamical scaling is found in the regime where $$\label{agereg}
t\gg \tau_{\rm micro} \;\; , \;\;
s\gg \tau_{\rm micro} \;\; \mbox{\rm and} \;\;
t-s \gg \tau_{\rm micro}$$ where $\tau_{\rm micro}$ is some microscopic reference time-scale. In the ageing regime (\[agereg\]) the only relevant lengths scales are describes in terms of $L(t)$ and one expects $$\label{skCR}
C(t,s) = s^{-b} f_C(t/s) \;\; , \;\;
R(t,s) = s^{-1-a} f_R(t/s)$$ with the asymptotics $f_{C,R}(y)\sim y^{-\lambda_{C,R}/z}$ for $y$ large. We stress the importance of the third condition in eq. (\[agereg\]) for the validity of the scaling forms (\[skCR\]). Throughout, the scaling limit $t,s\to\infty$ with $y=t/s>1$ fixed will be implied. While the autocorrelation (autoresponse) exponents $\lambda_{C,R}$ are new, independent exponents, the exponents $a,b$ can be explicitly given. At criticality $T=T_c$, one has $a=b=(d-2+\eta)/z=2\beta/\nu z$, where $\beta,\nu,\eta$ are standard equilibrium critical exponents. For $T<T_c$, one has $b=0$ always. In simple scalar systems with short-ranged equilibrium correlations, such as the Ising or Potts models (with $d>1$), one has $a=1/z=1/2$. In the case of long-ranged equilibrium correlations, $a$ may be different, i.e. $a=(d-2)/z$ in the $d$-dimensional spherical model.
On the other hand, in irreversible ageing systems such as the critical voter-model or the critical contact-process, there is no surface tension and the dynamics proceeds through cluster dissolution [@Dorn01]. Still, in the ageing regime (\[agereg\]) one observes again the same formal scaling behaviour eq. (\[skCR\]). However, the exponents $a$ and $b$ need no longer be the same even if one considers the ageing at a phase-transition in the non-equilibrium steady-state. For example, there is recent numerical evidence from the critical contact process in both $1D$ and $2D$ that $b=2\beta/\nu_{\perp}z$ which naturally generalizes the result of critical systems with detailed balance, but $a=b-1$ [@Enss04; @Rama04].
Galilei-invariance in phase-ordering kinetics
=============================================
Having reviewed current knowledge on the dynamical scaling of ageing systems, notably on the values of the ageing exponents $a$ and $b$, we now consider the scaling functions $f_{C,R}(y)$ themselves. In particular, we ask whether there exist any generic, model-independent argument which might inform us about the form of the functions $f_{C,R}(y)$. In this context, the following general statements about the dynamical symmetries of phase-ordering kinetics can be made.
1. Time-translation invariance is broken.
2. There is dynamical scaling [@Bray94; @Cugl02], i.e. formally the order parameter satisfies the covariance condition $$\phi(t,\vec{r}) = \alpha^{x_{\phi}} \phi(\alpha^{2} t,\alpha\vec{r})$$ where $\alpha$ is a constant rescaling factor and $x_{\phi}$ an exponent.
3. There is new evidence for Galilei-invariance which we now discuss.
Conventionally, one starts from a coarse-grained order parameter which is assumed to satisfy a Langevin equation, for example the one for model A dynamics [@Bray94] $$\label{lang}
2{\cal M}\frac{\partial\phi}{\partial t} = \Delta \phi -
\frac{{\rm d} V(\phi)}{{\rm d}\phi} + \eta$$ where $\Gamma=(2{\cal M})^{-1}$ is a kinetic coefficient, $\Delta$ is the spatial Laplacian, $V(\phi)$ is a typical double-well potential (e.g. $V(\phi)=(\phi^2-1)^2$) and $\eta$ is the thermal gaussian noise with covariance $\langle\eta(t)\eta(t')\rangle=2 T\delta(t-t')$. In addition, one assumes a gaussian uncorrelated initial state with covariance $\langle \phi(0,\vec{r})\phi(0,\vec{0})\rangle=a_0\delta(\vec{r})$. The associated field-theoretic action in the Martin-Siggia-Rose (MSR) formalism is $$\label{msr}
S[\phi,\widetilde{\phi}]=S_0[\phi,\widetilde{\phi}]+S_b[\widetilde{\phi}]$$ where $\widetilde{\phi}$ is the response field and $$\begin{aligned}
S_0[\phi,\widetilde{\phi}] &=& \int\!{\rm d}t{\rm d}\vec{r}\:
\left[ \widetilde{\phi}(2{\cal M}\partial_t -\Delta)\phi +
\widetilde{\phi}\frac{\delta V(\phi)}{\delta \phi} \right]
\nonumber \\
S_b[\widetilde{\phi}] &=& - T\int\!{\rm d}t{\rm d}\vec{r}\;
\widetilde{\phi}^2(t,\vec{r}) - \frac{a_0}{2} \int\!{\rm d}\vec{r}\;
\widetilde{\phi}^2(0,\vec{r})\end{aligned}$$ Then autocorrelators and autoresponses can be found as follows $$C(t,s) = \left\langle \phi(t)\phi(s)\right\rangle \;\; , \;\;
R(t,s) = \left\langle \phi(t)\widetilde{\phi}(s)\right\rangle$$
We shall consider as an extension of dynamical scaling the transformations of the so-called Schrödinger group, defined by [@Nied72] $$t\to \frac{\alpha t+\beta}{\gamma t +\delta} \;\; , \;\;
\vec{r} \to \frac{{\cal R}\vec{r}+\vec{v}t+\vec{a}}{\gamma t+\delta}$$ where $\alpha\delta-\beta\gamma=1$ and $\cal R$ is a $d$-dimensional rotation. Besides translations in time and space, rotations and scale transformations with $z=2$, the Schrödinger group also contains the so-called ‘special’ transformations parametrized by $\gamma$. The Schrödinger group is the maximal dynamical symmetry group of the free (and also of several non-linear) Schrödinger/diffusion equations and acts projectively (i.e. up to a phase factor) on the wave function [@Nied72; @Fush93].
The relationship of the Schrödinger group to phase-ordering kinetics can now be formulated in terms of the following three theorems.
We consider an arbitrary space-time infinitesimal coordinate transformation $\delta r_{\mu}=\epsilon_{\mu}$ with $\mu=0,1,\ldots,d$. Let $\eta$ stand for the phase picked up by the wave function $\phi$ under such a transformation. We call a MSR-theory [*local*]{} if the MSR-action (\[msr\]) transforms as $$\delta S= \int\!{\rm d}t{\rm d}\vec{r}\:
\left( T_{\mu\nu}\partial_{\mu}\epsilon_{\nu} + J_{\mu}\partial_{\mu}\eta
\right) + \int_{(t=0)}\!{\rm d}\vec{r}\:
\left( U_{\nu}\epsilon_{\nu} +V \eta\right)$$ Here $T_{\mu\nu}$ is the energy-momentum tensor, $J_{\mu}$ a conserved current and the second integral is only over the initial line $t=0$.
[**Theorem 1:**]{} [*[@Henk03] For a local MSR-theory, one has*]{} $$\left. \begin{array}{l}
\mbox{\it phase-shift invariance} \\
\mbox{\it space-translation invariance} \\
\mbox{\it scale-invariance with $z=2$}\\
\mbox{\it Galilei-invariance}
\end{array} \right\} \Longrightarrow
\mbox{\it special Schr\"odinger invariance}$$ This result is completely analogous to a well-known result in conformal field-theory. We point out that the requirement of time-translation invariance is not required in order to derive the invariance under special Schrödinger transformations for local theories.
To prove this, it suffices to write down the various infinitesimal transformations explicitly. From invariance under phase shifts it follows $V=0$, space-translation invariance implies $U_1=\ldots=U_d=0$, from scale-invariance it follows $2T_{00}+T_{11}+\ldots+T_{dd}=0$ and finally Galilei-invariance implies $T_{0i}+2{\cal M}J_i=0$ for $i=1,\ldots,d$. Consequently $\delta S=0$ under special Schrödinger transformations, as asserted. q.e.d.
One may write down the tensor $T_{\mu\nu}$ and the current $J_{\mu}$ explicitly, for free fields this has been done in [@Henk03].
We now consider the rôle of Galilei-invariance more closely. It is well-known that a system coupled to a heat bath with a uniform temperature $T>0$ cannot be Galilei-invariant. This can be easily seen from the MSR-action (\[msr\]), since the noise terms $S_b[\widetilde{\phi}]$ are not invariant under a phase-shift $\phi\to e^{\eta}\phi$, $\widetilde{\phi}\to
e^{-\eta}\widetilde{\phi}$. At most, the deterministic part $S_0$ of the action might be Galilei-invariant. If that is the case, one has the following Bargman superselection rule $$\label{Barg}
\langle~ \underbrace{\phi\cdots\phi}_n ~~
\underbrace{\widetilde{\phi}\cdots\widetilde{\phi}}_m ~\rangle_0
\sim \delta_{n,m}$$ The index $0$ refers to the average taken with respect to the deterministic part $S_0$ only.
[**Theorem 2:**]{} [*[@Pico04] If the deterministic part $S_0$ of the MSR-action (\[msr\]) is Galilei-invariant such that (\[Barg\]) holds true, then*]{} $$\begin{aligned}
R(t,s) &=& R_0^{(2)}(t,s) \\
C(t,s) &=& \frac{a_0}{2}\int\!{\rm d}\vec{r}\;
R_0^{(3)}(t,s,0;\vec{r}) + T \int\!{\rm d}u{\rm d}\vec{r}\;
R_0^{(3)}(t,s,u;\vec{r})\end{aligned}$$ [*where the two-point function $R_0^{(2)}(t,s)=\langle\phi(t,\vec{y})\widetilde{\phi}(s,\vec{y})\rangle_0$ and the three-point function $R_0^{(3)}(t,s,u;\vec{r})=\langle \phi(t,\vec{y})\phi(s,\vec{y})
\widetilde{\phi}^2(u,\vec{r}+\vec{y})\rangle_0$ are fixed by the deterministic part $S_0$.*]{}
To see this, one merely has to include the noise term $S_b[\widetilde{\phi}]$ into the average, viz. $R=\langle\phi\widetilde{\phi}\rangle=
\langle\phi\widetilde{\phi}e^{-S_b[\widetilde{\phi}]}\rangle_0
=\langle\phi\widetilde{\phi}\rangle_0$ because of the Bargman superselection rule (\[Barg\]). $C(t,s)$ is found similarly. q.e.d.
Consequently, given the Galilei-invariance of the deterministic part, one can find the contributions of both the thermal and the initial noise. Remarkably, the response function is noise-independent, whereas the correlator vanishes in the absence of noise.
Finally, we have to address the question whether the deterministic part of the Langevin equation (\[lang\]) can be Galilei-invariant. At first sight, the answer seems to be negative, since a well-known mathematical fact [@Fush93] states that the non-linear Schrödinger equation $$\left(2m{\rm i}\partial_t - \Delta \right)\phi = F(t,\vec{r},\phi,\phi^*)$$ is Schrödinger-invariant only for the special potential $F=c(\phi\phi^*)^{2/d}\phi$, where $c$ is a constant. Furthermore, the solutions $\phi$ are necessarily complex. These difficulties can be circumvented by considering the mass $m$ as a new dynamical variable [@Giul96; @Henk03]. We introduce a new wave function $\psi$ via $$\phi(t,r) = \int_{-\infty}^{\infty}\!{\rm d}\zeta\; e^{-{\rm i}m\zeta}
\psi(\zeta,t,r)$$ and look for the symmetries of the new non-linear ‘Schrödinger’-equation $$\label{schpsi}
\left(2\partial_{\zeta}\partial_t-\partial_r^2\right)\psi =
g F(\zeta,t,r,\psi,\psi^*)$$ where we wrote the coupling constant $g$ of the non-linear term explicitly. Since from dimensional counting, $g$ is in general dimensionful, it should transform under the action of the Schrödinger group. We have systematically constructed all such representations of the Schrödinger Lie algebra and also of its subalgebra when time-translations are left out [@Stoi05]. We then find all invariant non-linear equations of the type (\[schpsi\]). The full analysis will be presented elsewhere, here we merely quote one special result.
[**Theorem 3:**]{} [*[@Stoi05] If we write the solutions of (\[schpsi\]) $\psi=\psi_g(\zeta,t,r)=g^{(1-2x)/(4y)}\Psi(\zeta,t,r)$, where $x$ is the scaling dimension of $\psi$ and $y$ the scaling dimension of $g$, then the equation*]{} $$\label{stoi}
\left(2\partial_{\zeta}\partial_t-\partial_r^2\right)\Psi = g^{-5/4y}
f\left(g^{1/4y}\Psi\right)$$ [*is Schrödinger-invariant, where $f$ is an arbitrary function.*]{}
We have obtained in this way a formulation of Schrödinger-invariance which allows for real-valued functions and furthermore should be flexible enough to include the double-well potentials which enter into the Langevin equation (\[lang\]).
Tests of Galilei-invariance
===========================
The results quoted above in the theorems 1-3 lead to quantitative predictions which have been successfully tested in simulations. We first consider the response function, which according to theorem 2 can be found from the assumption of its covariant transformation under the Schrödinger group. This leads to [@Henk03a; @Pico04] $$\begin{aligned}
R(t,s;\vec{r}) &=& R(t,s)
\exp\left( -\frac{\cal M}{2}\frac{\vec{r}^2}{t-s}\right)
\;\; , \;\;
R(t,s) = s^{-1-a} f_R(t/s) \nonumber \\
f_R(y) &=& f_0 y^{1+a'-\lambda_R/z} (y-1)^{-1-a'}
\label{RR}\end{aligned}$$ where $\cal M$ and $f_0$ are non-universal constants. In most cases, one has $a=a'$, but exceptions are known to occur, e.g. in the $1D$ Glauber-Ising model. The gaussian space-dependence of $R(t,s;\vec{r})$ is characteristic of Galilei-invariance. Since response functions are very much affected by noise and hence difficult to measure directly, a convenient way of testing (\[RR\]) is to study the spatio-temporally integrated response $$\label{iR}
\int_0^s\!{\rm d}u \int_{0}^{\sqrt{\mu s}}\!{\rm d}r\: r^{d-1} R(t,u;\vec{r})
= r_0 s^{d/2-a} \rho^{(2)}(t/s,\mu) + \cdots$$ where $\mu$ is a control parameter, $r_0$ a constant related to $f_0$ and the function $\rho^{(2)}$ can be found explicitly from (\[RR\]) [@Henk03a].
![Scaling function $\rho^{(2)}(y,\mu)$ in the $2D$ Ising model at $T=0.66 T_c$ for two values of the control parameter $\mu$, as a function of $y=t/s$ (after [@Henk03a]). \[fig1\]](granada8-henkel_fig1){height=".3\textheight"}
In figure \[fig1\] we compare the prediction derived from (\[RR\]) with simulational data [@Henk03a] in the $2D$ kinetic Ising model, quenched into its ordered phase and with a purely relaxational heat-bath dynamics. We used the exponents $z=2$, $a=a'=1/z=1/2$ and $\lambda_R=1.26$. From the plots, one sees a nice collapse of the data obtained for several values of the waiting time $s$. Finally, the full curve agrees perfectly with the data. Similar results also hold true in the $3D$ Ising model [@Henk03a] and also for the $2D$ three-state Potts model [@Lore05]. Furthermore, the prediction (\[RR\]) can be reproduced in the exactly solvable $d$-dimensional spherical and $1D$ Glauber-Ising models.
We remark that $R(t,s)$ is well reproduced in the critical $1D$ contact-process [@Enss04].
As a second example, we consider the calculation of the autocorrelator $C(t,s)$. From theorem 2, this requires the calculation of a noiseless three-point response function. Since Schrödinger-invariance fixes the three-point function only up to an undetermined scaling function, a further argument is needed in order to determine $C(t,s)$ completely. For models which are described by an underlying free-field theory, it turns out that a simple heuristic idea based on the absence of singularities in $C(t,s)$ leads to the following simple expression for $T=0$ in the scaling limit [@Pico04] $$C(t,s) = f_C(t/s) \;\; , \;\;
f_C(y) \approx C_0 \left( {(y+1)^2}/{y}\right)^{-\lambda_C/2}$$ This agrees indeed with the exact solutions of the spherical model, the spin-wave approximation of the XY model, the critical voter model and the free random walk. It can also be checked [@Pico04] that the terms coming from thermal noise are irrelevant, as expected from renormalization group arguments [@Bray94].
Finally, for models not described by a free-field theory one can invoke an extension of Schrödinger invariance to a new form of conformal invariance [@Henk04]. Then $C(t,s)$ can be written in terms of hypergeometric functions and this prediction again agrees nicely with simulational data in both the $2D$ Ising [@Henk04] and three-states Potts models [@Lore05].
Summarizing, we have presented evidence, both conceptual and simulational, which strongly suggest that dynamical scaling in phase-ordering kinetics can be extended to the larger Schrödinger group (without time-translations) of local scale-transformations. In particular, it appears that equations of the form (\[stoi\]), rather than the Langevin equation (\[lang\]) used traditionally, allow for a simple symmetry characterization. However, the derivation of equations of the type (\[stoi\]) from a physical argument remains to be understood.
[**Acknowledgements:**]{} It is a pleasure to thank A. Picone, M. Pleimling, S. Stoimenov and J. Unterberger for fruitful collaborations.
|
---
abstract: 'We discuss the breakdown of spatial coherence in networks of coupled oscillators with nonlocal interaction. By systematically analyzing the dependence of the spatio-temporal dynamics on the range and strength of coupling, we uncover a dynamical bifurcation scenario for the coherence-incoherence transition which starts with the appearance of narrow layers of incoherence occupying eventually the whole space. Our findings for coupled chaotic and periodic maps as well as for time-continuous R[ö]{}ssler systems reveal that intermediate, partially coherent states represent characteristic spatio-temporal patterns at the transition from coherence to incoherence.'
author:
- Iryna Omelchenko
- Yuri Maistrenko
- 'Philipp H[ö]{}vel'
- 'Eckehard Sch[ö]{}ll'
title: 'Loss of coherence in dynamical networks: spatial chaos and chimera states'
---
Understanding the dynamics on networks is at the heart of modern nonlinear science and has a wide applicability to various fields [@WAT98; @SON10a]. Thus, network science is a vibrant, interdisciplinary research area with strong connections to physics. For example, concepts of theoretical physics like the Turing instability, which is a known paradigm of non-equilibrium self-organization in space-continuous systems, have recently been transferred to complex networks [@NAK10]. While spatially extended systems show pattern formation mediated by diffusion, i.e., local interactions, a network takes also into account long-range and global interactions yielding more realistic spatial geometries.
Network topologies like all-to-all coupling of, for instance, phase oscillators (Kuramoto model) or chaotic maps (Kaneko model) were intensively studied , and numerous characteristic regimes were found [@KUR84; @MOS02; @KAN96a]. In particular, for globally coupled chaotic maps they range –for decreasing coupling strength – from complete chaotic synchronization via clustering and chaotic itineracy to complete desynchronization. The opposite case, i.e., nearest-neighbor coupling, is known as lattice dynamical systems of time-continuous oscillators, or coupled map lattices if the oscillator dynamics is discrete in time. These kinds of networks arise naturally as discrete approximation of systems with diffusion and have also been thoroughly studied. They can demonstrate rich dynamics such as solitons, kinks, etc. up to fully developed spatio-temporal chaos [@JEN85; @CHO95a; @AFR05; @NIZ02; @KAN96a].
The case of networks with nonlocal coupling, however, has been much less studied in spite of numerous applications in different fields. Characteristic examples pertain to neuroscience [@BAT07; @VIC08], chemical oscillators [@BET04a; @MIK06], electrochemical systems [@Krischer], and Josephson junctions [@WIE96a]. A new impulse to study such networks was given, in particular, by the discovery of so-called chimera states [@Chimera; @OME10a]. The main peculiarity of these spatio-temporal patterns is that they have a hybrid spatial structure, partially coherent and partially incoherent, which can develop in networks of identical oscillators without any sign of inhomogeneity.
In this Letter we discuss the transition between coherent and incoherent dynamics in networks of nonlocally coupled oscillators. We start with coupled chaotic maps $$\label{eq:map}
z_i^{t+1} = f\left(z_i^t\right) + \dfrac{\sigma}{2P} \sum\limits_{j=i-P}^{i+P} \left[ f\left(z_j^t\right) -
f\left(z_i^t\right) \right],$$ where $z_i$ are real dynamic variables ($i=1,...,N$, $N \gg 1$ and the index $i$ is periodic mod $N$), $t$ denotes the discrete time, $\sigma$ is the coupling strength, $P$ specifies the number of neighbors in each direction coupled with the $i$-th element, and $f(z)$ is a local one-dimensional map. We choose $f$ as the logistic map $f(z)=az(1-z)$ and fix the bifurcation parameter $a$ at the value $a=3.8$. This choice yields chaotic behavior of the map $f$ with positive Lyapunov exponent $\lambda \approx 0.431$.
![(Color online) Regions of coherence for system (\[eq:map\]) in the $(r,\sigma)$ parameter plane with wave numbers $k=1, 2$, and $3$. Snapshots of typical coherent states $z_i$ are shown in the insets. Color code inside the regions distinguishes different time-periods of the states. The CIB bifurcation curve separates regions with coherent and incoherent dynamics. In the hatched and shaded (color) regions below CIB two-cluster incoherent states exist. Completely synchronized chaotic states exist in the light hatched region bounded by the blowout bifurcation curve BB. Parameters: $a=3.8$ and $N=100$.[]{data-label="Fig1"}](ome11_Figure1){width="0.9\linewidth"}
Results of direct numerical simulation of the model (\[eq:map\]) in the two-parameter plane of the coupling radius $r=P/N$ and coupling strength $\sigma$ are presented in Fig. \[Fig1\]. This figure reveals the appearance of regions of spatial coherence, shown in shading (color), at an intermediate radius of coupling. Alternatively, if the oscillators are uncoupled ($r=0$) or coupling is only local ($r = 1/N$) the network displays high-dimensional space-time chaos [@KAN96a]. In the opposite situation, when the coupling is all-to-all ($r=0.5$), chaotic synchronization occurs: the oscillators behave identically, but chaotically in time following the dynamics of $f$ [@KAN96a; @MOS02]. The chaotic synchronization (hatched region $k=0$) persists for smaller $r$ or $\sigma$ up to the blowout bifurcation [@OTT94] indicated by the curve BB, where the synchronized state loses transverse stability, i.e., the dynamics becomes desynchronized. The spatially homogeneous state represents the simplest example of coherent dynamics.
In general we call a network state $z_i^t,$ $i=1,...,N$, *coherent* on the ring $\mathcal{S}^1$ as $N
\rightarrow \infty$ if for any point $x\in\mathcal{S}^1$ the limiting value $$\lim\limits_{N \rightarrow \infty} \lim\limits_{t \rightarrow \infty} \sup\limits_{i,j \in U_{\delta}^N (x)} \left|
z_i^t - z_j^t \right|\rightarrow 0, \quad \text{for} \quad \delta\rightarrow 0,
\label{Eq:CoherenceDef}$$ where $U_{\delta}^N (x) = \left\{ j:~0 \leq j \leq N,~ \left| {j}/{N} -x
\right| < \delta \right\}$ denotes a network-neighborhood of the point $x.$ If the limit (\[Eq:CoherenceDef\]) does not vanish for $\delta\rightarrow 0$, at least for one point $x$, the network state is considered incoherent.
Geometrically, coherence means that in the thermodynamic limit $N\rightarrow \infty$ snapshots of the state $z_i^t$ approach a smooth profile $z(x,t)$ of the spatially continuous version of Eq. (\[eq:map\]) given by $$\label{eq:continuous}
z^{t+1}(x)= f(z^t(x))+ \dfrac{\sigma}{2r} \int_{x-r}^{x+r} \left[f\left(z^{t}(y)\right) -f\left(z^{t}(x)\right)\right] dy.$$ According to the definition given above, a transition from coherence to incoherence occurs when the respective solution profile $z^t(x)$ becomes discontinuous. Note that for networks of phase oscillators the property of coherence and incoherence can also be established with use of the notion of a local order parameter [@PIK08; @WOL11a].
Regions of coherence and typical shapes of the respective coherent states $z_i^t$ are shown in Fig. \[Fig1\] as shaded (color) tongues and insets, respectively. A coherent state has a smooth profile characterized by the number of maxima, i.e., the wave number $k$. Only regions for wave numbers $k = 1, 2,$ and $3$ are shown. Further decrease of $r$ yields additional thin higher-order regions following a period-adding cascade $k =4,5,..$. Inside the regions, the states are coherent in space and periodic in time and undergo a period-doubling cascade of bifurcations in time as $r$ or $\sigma$ decrease. In the parameter space between the coherence regions the network dynamics remain coherent but not periodic anymore. The states alternate chaotically between the adjacent $k$-states and thus exhibit chaotic itineracy [@KAN96a; @KAN03]. The combination of [*period-adding*]{} in space and [*period-doubling*]{} in time represents a remarkable feature of networks of coupled chaotic oscillators with nonlocal coupling.
![(Color online) Coherence-incoherence bifurcation for coupled chaotic logistic maps for fixed coupling radius $r=0.32$ (black triangles in Fig. \[Fig1\]). For each value of the coupling parameter $\sigma$ (decreasing from top to bottom, $\sigma= 0.43, 0.4, 0.32, 0.3, 0.2,$ and $0.1$, respectively) snapshots (left columns) and space-time plots (right columns) are shown. Other parameters as in Fig. \[Fig1\].[]{data-label="Fig2"}](ome11_Figure2){width="\linewidth"}
A typical scenario of the coherence-incoherence transition is illustrated in Fig. \[Fig2\](a)-(f), where we fix the coupling radius $r=0.32$ and decrease the coupling strength $\sigma$ along the vertical line with triangles in Fig. \[Fig1\]. First, in Fig. \[Fig2\](a), the solution profile $z_i^t$ is clearly smooth for $\sigma=0.43$. Thus, the network dynamics is spatially coherent. For smaller $\sigma$, the profile $z_i^t$ sharpens up and, at some value $\sigma \cong 0.40$, loses smoothness in two points $x_1$ and $x_2$ as shown in Fig. \[Fig2\](b). This is a bifurcation point for the coherence-incoherence transition: Beyond this parameter value, the wave-like profile $z_i^t$ splits up into upper and lower branches, and two narrow boundary layers of incoherence are born around the points $x_1$ and $x_2$ (shaded yellow stripes $\alpha_1$ and $\alpha_2$ in Fig. \[Fig2\](c)). The incoherence stripes become wider with further decrease of $\sigma$ (Fig. \[Fig2\](d)) and, eventually, the dynamics becomes completely incoherent (Figs. \[Fig2\](e) and (f)).
In our numerical simulations, no coherent states were found below the bifurcation parameter value $\sigma \cong
0.40$. In contrast, numerous hybrid states arise, which are coherent on some intervals of the ring $\mathcal{S}^1$ and incoherent on the complementary intervals. Typical examples of these partially coherent states are shown in Figs. \[Fig2\](c) and (d). In the figure, black diamonds mark a threshold within the incoherent regions: If the initial value for a chosen oscillator is located above/below this diamond, with all other oscillators unchanged, it will be attracted by the upper/lower branch. This implies that within the incoherent intervals $\alpha_1$ and $\alpha_2$ any combinations of the upper and lower states– so-called *mosaic* [@CHO95a] or *skeleton* pattern [@NIZ02] – are admissible and can be realized by appropriate choice of the initial conditions. The coherence-incoherence transition is illustrated by the local order parameter $R_i$ shown in Fig. \[Fig3\](a) for the snapshots depicted in Fig. \[Fig2\]. It is defined as (cf. Ref. [@WOL11a]) $$R_i=\lim_{N\rightarrow\infty}\frac{1}{2\delta(N)}\left|\sum_{|j-i|\leq\delta}e^{i\psi_j}\right|, ~~~ (i=1,\dots,N)$$ with the phase $\psi_j$ introduced by the mapping $\sin\psi_j=(2z_j-\max_jz_j-\min_jz_j)/(\max_jz_j-\min_jz_j)$ and $\delta/N\rightarrow0$ for $N\rightarrow\infty$, such that a spatial half-period oscillation is mapped onto the polar angular interval $[-\pi/2, \pi/2]$. $R_i$ is close to unity for the coherent state and decreases in regions of spatial incoherence. For the two cases of complete incoherence (Fig. \[Fig2\](e),(f)) the local order parameter is much smaller than unity, and fluctuating strongly as a signature of spatial chaos. In case of very small coupling (Fig. \[Fig2\](f)) the values of $z_i$ are more spread out, and hence $R_i$ varies less strongly between neighboring sites and is on average larger than in Fig. \[Fig2\](e).
As it is illustrated in the space-time plots of Fig. \[Fig2\], the temporal dynamics before and after the coherence-incoherence bifurcation remains periodic up to very small coupling, when it is chaotic (Fig. \[Fig2\](f)). The system’s complexity results from the fact that the bifurcation gives rise to a huge multistability of partially coherent states as $N\rightarrow\infty$. Indeed, the number $c_N$ of different partially coherent states born in the bifurcation is $c_N=2^{d N}$, where $d$ is the fraction of oscillators in the incoherent part ($d=\alpha_1 + \alpha_2$ in the case of two incoherence intervals as in Figs. \[Fig2\](c) and (d)). It follows that the number of different states grows exponentially fast with $N$, and the *spatial entropy* $h$, which is defined as $h = \lim_{N \rightarrow \infty} (1/N) \ln c_N$, is positive and equals $h=d
\ln 2$. Positive spatial entropy means that the system displays [*spatial chaos*]{} [@CHO95a; @AFR05; @NIZ02], i.e., sensitive dependence on space coordinates. Therefore, the coherence-incoherence bifurcation results instantly in the appearance of spatial chaos that develops first at narrow incoherence intervals and, with decreasing $\sigma$, spreads onto the whole ring. Thus a chimera-like state of coexisting coherent and incoherent regions arises as a transitional state in the coherence-incoherence bifurcation scenario. However, in contrast to previously reported chimera states in time-continuous systems [@Chimera; @OME10a], the temporal behavior is periodic rather than chaotic, and the complexity arises due to the huge variety of multistable incoherent states corresponding to permutations of the sequence of upper and lower local states. With further decrease of $\sigma$, the chimera states disappear giving rise to completely incoherent behavior.
To identify the parameter range for partially coherent states, we define a mosaic of $z_i^t$ as a symbolic sequence of “$-$” or “$+$” if the value $z_i$ belongs to the lower or upper branch of the solution profile, respectively. Hence, states as in Fig. \[Fig2\](b) are given by the mosaic of the form $(...--++...++--...)$. Therefore, they may be considered as two-cluster states with the ratio of $(n_1:n_2)$, where $n_1$ and $n_2$ are the numbers of “$-$” and “$+$” ($n_1 + n_2=N$), respectively. The ratio $(n_1:n_2)$ indicates the level of asymmetry of the solution $z_i^t$, which is important for its stability. Indeed, as it is illustrated in Fig. \[Fig3\](b) for $r=0.35$, the symmetric solution $(50:50)$ has the widest stability interval $\sigma \in (0.189,0.41)$. As the asymmetry grows, the stability interval shrinks, and the solution with the mosaic ratio $(35:65)$ has the shortest stability interval $\sigma \in (0.314,0.317)$. Two-cluster solutions with larger asymmetry cannot be stabilized anymore. The states with more complex mosaics, examples of which are presented in Fig. \[Fig2\](c)-(e), are characterized by more involved mechanisms of stability.
![(Color online) Snapshots with decreasing coupling strength $\sigma$: Coherence-incoherence transition for a ring of (a) 100 coupled periodic logistic maps ($a=3.2$) for a coupling radius $r=0.1$ and (b) 100 nonlocally coupled R[ö]{}ssler systems ($a=0.42$, $b=2$, $c=4$) with $r=0.3$.[]{data-label="Fig4"}](ome11_Figure4){width="0.95\linewidth"}
To test if the coherence-incoherence bifurcation is a universal scenario, we have also investigated nonlocally coupled networks with different local dynamics. Figure \[Fig4\] shows the coherence-incoherence transition for nonlocally coupled logistic maps (\[eq:map\]) in the periodic regime ($a=3.2$, period $2$, Fig. \[Fig4\](a)) and for nonlocally coupled chaotic R[ö]{}ssler systems $$\label{eq:roessler}
\begin{array}{l}
\dot{x}_i = -y_i-z_i + \dfrac{\sigma}{2P} \sum\limits_{j=i-P}^{i+P} \left( x_j - x_i \right), \\
\dot{y}_i = x_i + ay_i, \\
\dot{z}_i = b+z_i(x_i-c) \qquad \qquad \qquad (i =1,...,N)
\end{array}$$ (Fig. \[Fig4\](b)). As it can be seen from the snapshots, both models display a transition from spatial coherence to incoherence, as the coupling strength $\sigma$ decreases, according to the scenario described above. Since network (\[eq:roessler\]) is time-continuous, the oscillators within the incoherence intervals are not only located at an upper or lower branch of the solution profile, but vary continuously. This gives rise to chaotic temporal dynamics in the incoherent intervals, which resembles known chimera states [@Chimera; @OME10a]. We conclude that chaotic chimera states typically arise in the nonlocally coupled R[ö]{}ssler systems, similar to nonlocally coupled Kuramoto-Sakaguchi phase oscillators [@OME10a].
In conclusion, we have identified a novel mechanism for the coherence-incoherence transition in networks with nonlocal coupling of variable range. It consists in the appearance of multistable chimera-like states. We have found similar bifurcation scenarios for coupled maps with both chaotic and periodic local dynamics as well as for time-continuous systems. This indicates a common, probably universal phenomenon in networks of very different nature, due to nonlocal coupling.
We thank B. Fiedler, M. Hasler, and M. Wolfrum for illuminating discussions. I. O. acknowledges support from DAAD and DFG (SFB 910). Y. M. acknowledges support and hospitality from TU Berlin.
[10]{}
D. J. Watts and S. H. Strogatz, Nature [**393**]{}, 440 (1998).
C. Song, T. Koren, P. Wang, and A.-L. Barabási, Nature Physics [**6**]{}, 818 (2010).
H. Nakao and A. S. Mikhailov, Nature Physics [**6**]{}, 544 (2010).
Y. Kuramoto, [*Chemical Oscillations, Waves, and Turbulence*]{} (Springer, Berlin Heidelberg, 1984), J.A. Acebr[ó]{}n et al. Rev. Mod. Phys. [**77**]{}, 137 (2005).
E. Mosekilde, Y. Maistrenko, and D. Postnov, [*Chaotic Synchronization: Applications to Living Systems*]{} (World Scientific, Singapore, 2002).
K. Kaneko and I. Tsuda, [*Chaos and Beyond, A Constructive Approach with Applications in Life Sciences*]{} (Springer, Berlin, 1996)
M. H. Jensen, Physica Scripta [**T9**]{}, 64 (1985); P. Coullet, C. Elphick, and D. Repaux, Phys. Rev. Lett. [**58**]{}, 431 (1987); W. Shen, SIAM J. Appl. Math. [**56**]{}, 1379 (1996); B. Fernandez, B. Luna, and E. Ugalde, Phys. Rev. E [**80**]{}, 025203(R) (2009).
S. N. Chow and J. Mallet-Paret, IEEE Trans. Circ. Sys. [**42**]{}, 746 (1995).
V. Afraimovich, [*Some topological properties of lattice dynamical systems*]{} in Dynamics of coupled map lattices and of related spatially extended systems, Lecture Notes in Phys. [**671**]{}, 153 (Springer, Berlin, 2005).
L. P. Nizhnik, I. L. Nizhnik, and M. Hasler, Int. J. Bif. Chaos [**12**]{}, 261 (2002).
D. Battaglia, N. Brunel, and D. Hansel, Phys. Rev. Lett. [**99**]{}, 238106 (2007).
R. Vicente, L. L. Gollo, C. R. Mirasso, I. Fischer, and P. Gordon, Proc. Natl. Acad. Sci. [**105**]{}, 17157 (2008).
C. Beta, M. G. Moula, A. S. Mikhailov, H. H. Rotermund, and G. Ertl, Phys. Rev. Lett. [**93**]{}, 188302 (2004).
A. S. Mikhailov and K. Showalter, Phys. Rep. [**425**]{}, 79 (2006).
N. Mazouz, G. Fl[ä]{}tgen, and K. Krischer, Phys. Rev. E [**55**]{}, 2260 (1997); V. García-Morales and K. Krischer, Phys. Rev. Lett. [**100**]{}, 054101 (2008).
K. Wiesenfeld, P. Colet, and S. H. Strogatz, Phys. Rev. Lett. [**76**]{}, 404 (1996).
Y. Kuramoto and D. Battogtokh, Nonlin. Phen. in Complex Sys. [**5**]{}, 380 (2002); D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. [**93**]{}, 174102 (2004); G. C. Sethia, A. Sen, and F. M. Atay, Phys. Rev. Lett. [**100**]{}, 144102 (2008); C. R. Laing, Physica D [**238**]{}, 1569 (2009); E. A. Martens, C. R. Laing, and S. H. Strogatz, Phys. Rev. Lett. [**104**]{}, 044101 (2010).
O. E. Omel’chenko, M. Wolfrum, and Y. L. Maistrenko, Phys. Rev. E [**81**]{}, 065201 (2010).
E. Ott and J. C. Sommerer, Phys. Lett. A **188**, 39 (1994).
A. S. Pikovsky and M. G. Rosenblum, Phys. Rev. Lett. [**101**]{}, 264103 (2008). M. Wolfrum, O. E. Omel’chenko, S. Yanchuk, and Y. L. Maistrenko, Chaos **21**, 013112 (2011).
K. Kaneko and I. Tsuda, Chaos [**13**]{}, 926 (2003).
|
---
abstract: 'We present a library of Penn State Fiber Optic Echelle (FOE) observations of a sample of field stars with spectral types F to M and luminosity classes V to I. The spectral coverage is from 3800 Å$\ $ to 10000 Å with nominal a resolving power 12000. These spectra include many of the spectral lines most widely used as optical and near-infrared indicators of chromospheric activity such as the Balmer lines (H$\alpha$ to H$\epsilon$), Ca [ii]{} H & K, Mg [i]{} b triplet, Na [i]{} D$_{1}$, D$_{2}$, He [i]{} D$_{3}$, and Ca [ii]{} IRT lines. There are also a large number of photospheric lines, which can also be affected by chromospheric activity, and temperature sensitive photospheric features such as TiO bands. The spectra have been compiled with the goal of providing a set of standards observed at medium resolution. We have extensively used such data for the study of active chromosphere stars by applying a spectral subtraction technique. However, the data set presented here can also be utilized in a wide variety of ways ranging from radial velocity templates to study of variable stars and stellar population synthesis. This library can also be used for spectral classification purposes and determination of atmospheric parameters (T$_{\rm eff}$, $\log{g}$, \[Fe/H\]). A digital version of all the fully reduced spectra is available via ftp and the World Wide Web (WWW) in FITS format.'
author:
- 'David Montes, Lawrence W. Ramsey, Alan D. Welty'
title: 'Library of medium-resolution fiber optic echelle spectra of F, G, K, and M field dwarfs to giants stars '
---
Introduction
============
Spectral libraries of late-type stars with medium to high resolution and large spectral coverage are an essential tool for the study of the chromospheric activity in multiwavelength optical observations using the spectral subtraction technique (see Barden 1985; Huenemoerder & Ramsey 1987; Hall & Ramsey 1992; Montes et al. 1995a, b, c, 1996a, b, 1997b, 1998). Furthermore, these libraries are also very useful in many areas of astrophysics such as the stellar spectral classification, determination of atmospheric parameters (T$_{\rm eff}$, $\log{g}$, \[Fe/H\]), modeling stellar atmospheres, spectral synthesis applied to composite systems, and spectral synthesis of the stellar population of galaxies.
In previous work Montes et al. (1997a, hereafter Paper I) presented a library of high and mid-resolution (3 to 0.2 Å) spectra in the Ca [ii]{} H & K, H$\alpha$, H$\beta$, Na [i]{} D$_{1}$, D$_{2}$, and He [i]{} D$_{3}$ line regions of F, G, K, and M field stars. A library of echelle spectra of a sample of F, G, K, and M field dwarf stars is presented in Montes & Martín (1998, hereafter Paper II) which is an extension of Paper I to higher spectral resolution (0.19 to 0.09 Å) covering a large spectral range (4800 to 10600 Å).
The spectral library presented here expands upon the data set in Papers I and II. This library consists of echelle spectra of a sample of F, G, K, and M field stars, mainly dwarfs (V), subgiant (IV), and giants (III) but also some supergiants (II, I). The spectral resolving power is intermediate, nominally R = 12000 ($\approx$ 0.5 Å in H$\alpha$), but the spectra have a nearly complete optical region coverage (from 3900 to 9000Å). These regions includes most of the spectral lines widely used as optical and near-infrared indicators of chromospheric activity such as the Balmer lines (H$\alpha$ to H$\epsilon$), Ca [ii]{} H & K, Mg [i]{} b triplet, Na [i]{} D$_{1}$, D$_{2}$, He [i]{} D$_{3}$, and Ca [ii]{} IRT lines, as well as temperature sensitive photospheric features such as TiO bands.
Recently, Pickles (1998) has taken available published spectra and combined them into a uniform stellar spectral flux library. This library have a wide wavelength, spectral type, and luminosity class coverage, but a low spectral resolution (R = 500) and their main purpose is the synthesis and modeling of the integrated light from composite populations. However, for other purposes as detailed studies of chromospheric activity, stellar spectral classification, and determination of atmospheric parameters, libraries of higher resolution, as the presented in Paper I and II, Soubiran, Katz, & Cayrel (1998), and the library presented here are needed.
In Sect. 2 we report the details of our observations and data reduction. The library is presented in Sect. 3.
Observations and data reduction
===============================
The echelle spectra presented here were obtained during several observing runs with the Penn State Fiber Optic Echelle (FOE) at the 0.9-m and 2.1-m telescopes of the Kitt Peak National Observatory (KPNO). The FOE is a fiber fed prism cross-dispersed echelle medium resolution spectrograph and is described in more detail in Ramsey & Huenemoerder (1986). It was designed specifically to obtain in a single exposure a wide spectral range encompassing all the visible chromospheric activity sensitive features. Typical data and performance of the FOE for the different observing runs are discussed in Ramsey et al. (1987); Huenemoerder, Buzasi, & Ramsey (1989); Newmark et al. (1990); Hall et al. (1990); Buzasi, Huenemoerder, & Ramsey (1991); Hall & Ramsey (1992); Welty & Ramsey (1995); and Welty (1995).
In Table \[tab:obs\] we give a summary of observations. For each observing run we list the date, the CCD detector used, the number of echelle orders included, the wavelength range covered ($\lambda$$_{i}$-$\lambda$$_{f}$) and the range of reciprocal dispersion achieved (Å/pixel) from the first to the last echelle orders. The Å/pixel value for each order can be found in the header of the spectra. The spectral resolution, determined by the FWHM of the arc comparison lines, ranges from 2.0 to 2.2 pixels. The signal to noise ratio is larger than 100 in all cases. Tables \[tab:orders\] gives for each observing run the spectral lines of interest in each echelle order.
The spectra have been extracted using the standard reduction procedures in the IRAF[^1] package (bias subtraction, flat-field division, and optimal extraction of the spectra). The wavelength calibration was obtained from concurrent spectra of a Th-Ar hollow cathode lamp. Finally, the spectra have been normalized by a polynomial fit to the observed continuum.
The library
===========
As in Papers I and II, the stars included in the library have been selected as stars with low levels of chromospheric activity, that is to say, stars that do not present any evidence of emission in the core of Ca [ii]{} H & K lines in our spectra (Montes et al. 1995c, 1996a), stars with the lower Ca [ii]{} H & K spectrophometric index S (Duncan et al. 1991; Baliunas et al. 1995), or stars known to be inactive and slowly rotating stars from other sources (see Strassmeier et al. 1990; Strassmeier & Fekel 1990; Hall & Ramsey 1992).
Table \[tab:par\] presents information about the observed stars. In this table we give the HD, HR and GJ numbers, name, spectral type and luminosity class (T$_{\rm sp}$), from the Bright Star Catalogue (Hoffleit & Jaschek 1982; Hoffleit & Warren 1991), the Catalogue of Nearby Stars (Gliese & Jahreiss 1991), and Keenan & McNeil (1989). The exception is some of the M dwarfs for which we list the more recent spectral type determination given by Henry, Kirkpatrick, & Simons (1994). In column (6) MK indicates if the star is a Morgan and Keenan (MK) Standard Star from García (1989) and Keenan & McNeil (1989). MK\* indicates if the star is included in the list of Anchor Points for the MK System compiled by Garrison (1994). Column (7) give the metallicity \[Fe/H\] from Taylor (1994; 1995) or Cayrel de Strobel (1992; 1997) and column (8) rotational period (P$_{\rm rot}$) and [*v*]{} sin[*i*]{} from Donahue (1993), Baliunas et al. (1995), Fekel (1997), and Delfosse et al. (1998). We also give, in column (9), the Ca [ii]{} H & K spectrophometric index S from Baliunas et al. (1995) and Duncan et al. (1991). In column (10) we list information about the observing run in which each star have been observed, using a code given in the first column of Table \[tab:obs\], the number between brackets give the number of spectra available. The last two columns indicate if the star was also included in Papers I and II.
Representative spectra (from F to M, dwarfs and giants stars) in different spectral regions are plotted in figures (\[fig:hb\] to \[fig:cairt\]) in order to show the behaviour of the more remarkable spectroscopic features with the spectral type and luminosity class. In order of increasing wavelength we have plotted the following line regions: H$\beta$ (Fig. \[fig:hb\]), Na [i]{} D$_{1}$, D$_{2}$, and He [i]{} D$_{3}$) (Fig. \[fig:na\]), H$\alpha$ (Fig. \[fig:ha\]),and Ca [ii]{} IRT $\lambda$8498, 8542 (Fig. \[fig:cairt\]). In each figure we have plotted main sequence stars (luminosity class V) in the left panel, and giants stars (III) in the right panel.
A total of 130 stars are included in this library. Many of them have been observed in several observing runs, and in some cases several nights during the same observing run being the total number of spectra 345. Using these spectra as well as those of Papers I and II a study of possible short and long term spectroscopic variability of some of the multiply observed stars is possible.
A description of the spectral lines most widely used as optical and near-infrared indicators of chromospheric activity, as well as other interesting spectral lines and molecular bands present in the spectral range covered by the spectra can be found in Papers I and II and references therein.
As an illustration of the use of these spectra and those of Papers I and II we intend to analyze temperature sensitive lines in order to improve the actual line-depth ratio temperature calibrations (Gray & Johanson 1991, Gray 1994) and spectral-class/temperature classifications (Strassmeier & Fekel 1990), as well as the determination of fundamental atmospheric parameters T$_{\rm eff}$, $\log{g}$, \[Fe/H\] (Katz et al. 1998 and Soubiran et al. 1998). This will be the subject of forthcoming papers.
In order to enable other investigators to make use of the spectra in this library for their own purposes, all the final reduced (flattened and wavelength calibrated) multidimensional spectra containing all the echelle orders of the stars listed in Table \[tab:par\] are available at the CDS in Strasbourg, France, via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5). They are also available via the World Wide Web at: . The data are in FITS format with pertinent header information included for each image. In order to further facilitate the use of this library one dimensional normalized and wavelength calibrated spectra, for the orders containing the more remarkable spectroscopic features, are also available as separate FITS format files.
In addition this library as well as the libraries presented in Papers I and II will be included in the [*Virtual Observatory*]{} (see [http://herbie.ucolick.org/vo/]{}). This is a project to establish a new spectroscopic database which will be contained digitized spectra of spectroscopic plates as well as spectra observed digitally from different observatories. [*Virtual Observatory*]{} is an International Astronomical Union (IAU) initiative through its Working Group for Spectroscopic Data Archives.
This research has made use of the SIMBAD data base, operated at CDS, Strasbourg, France. This work has been supported by the Universidad Complutense de Madrid and the Spanish Dirección General de Enseñanza Superior e Investigacióna Científica (DGESIC) under grant PB97-0259 and by National Science Foundation (NSF) grant AST 92-18008. We also acknowledge, with gratitude, KPNO supporting the FOE presence from 1987 until 1996.
Baliunas, S. L., et al. 1995, ApJ, 438, 269
Barden, S. C. 1985, ApJ, 295, 162
Buzasi, D. L., Huenemoerder, D. P., & Ramsey, L. W. 1991, PASP, 103, 1077.
Cayrel de Strobel, G., et al. 1992, A&AS, 95, 273
Cayrel de Strobel, G., Soubiran, C., Friel, E. D., Ralite, N., & Francois, P. 1997, A&AS, 124, 299
Delfosse, X., Forveille, T., Perrier, C., & Mayor, M. 1998, A&A, 331, 581
Donahue, R. A. 1993, Ph.D. Thesis, New Mexico State University
Duncan, D. K., et al. 1991, ApJS 76, 383
Fekel, F. C. 1997, PASP, 109, 514
García, B. 1989, Bull. Inform. CDS, 36, 27
Garrison, R. F. 1994, in: The MK Process at 50 Years: A Powerful Tool for Astrophysical Insight, ed. C. J. Corbally, R. O. Gray, & R. F. Garrison, ASP Conf. S, 60, 3
Gliese, W., & Jahreiss, H. 1991, Preliminary Version of the Third Catalogue of Nearby Stars, (Astron. Rechen-Institut, Heidelberg)
Hall, J. C., Huenemoerder, D. P., Ramsey, L. W., & Buzasi, D. L. 1990, ApJ, 358, 61
Hall, J. C., & Ramsey, L. W. 1992, AJ, 104, 1942
Henry, T. J., Kirkpatrick, J. D., & Simons, D. A. 1994, AJ, 108, 1437
Hoffleit, D., & Jaschek, C. 1982, [The Bright Star Catalogue]{}, (4th ed.), New Haven: Yale University Press
Hoffleit, D., & Warren, W. 1991, The Bright Star Catalogue, (5th ed.), in Astronomical Data Center CD ROM, Selected Astronomical Catalogs Vol. 1, eds. L. E. Brotzman, S. E. Gessmer
Huenemoerder, D. P., & Ramsey, L. W. 1987, ApJ, 319, 392
Huenemoerder, D. P., Buzasi, D. L., & Ramsey, L. W. 1989, AJ, 98, 1398
Keenan, P. C., & McNeil, R. C. 1989, ApJS, 71, 245
Katz, D., Soubiran, C., Cayrel, R., Adda, M., & Cautain, R. 1998, A&A, 338, 151
Montes, D., Fernández-Figueroa, M. J., De Castro, E., & Cornide, M. 1995a, A&A, 294, 165
Montes, D., Fernández-Figueroa, M.J., De Castro, E., & Cornide, M. 1995b, A&AS, 109, 135
Montes, D., De Castro, E., Fernández-Figueroa, M. J., & Cornide, M. 1995c, A&AS, 114, 287
Montes, D., Fernández-Figueroa, M.J., Cornide, M., & De Castro, E. 1996a, A&A, 312, 221
Montes, D., Sanz-Forcada, J., Fernández-Figueroa, M. J., & Lorente, R. 1996b, A&A 310, L29
Montes, D., Martín, E. L., Fernández-Figueroa, M. J., Cornide, M., & De Castro, E. 1997a, A&AS, 123, 473 (Paper I), ([http://www.ucm.es/info/Astrof/fgkmsl/fgkmsl.html]{})
Montes, D., Fernández-Figueroa, M. J., De Castro, & E., Sanz-Forcada J. 1997b, A&AS, 125, 263
Montes, D., Sanz-Forcada, J., Fernández-Figueroa, M. J., De Castro, E., & Poncet, A. 1998, A&A, 330, 155
Montes, D., & Martín, E. L. 1998, A&AS, 128, 485 (Paper II),\
([http://www.ucm.es/info/Astrof/fgkmsl/UESfgkmsl.html]{})
Newmark, J. S., Buzasi, D. L., Huenemoerder, D. P., Ramsey, L. W., Barden, S. C., Nations, H. L., & Seeds, M. A. 1990, AJ, 100, 560.
Pickles, A. J. 1998, PASP, 110, 863
Ramsey, L. W., & Huenemoerder, D. P. 1986, Proc. SPIE, 621, 282
Ramsey, L. W., Huenemoerder, D. P., Buzasi, D. L., & Barden, S. C. 1987, in: Cool star stellar systems and the Sun, ed. J. F. Linsky & R. E. Stencel (Springer, Berlin), 515
Strassmeier, K. G., Fekel, F. C., Bopp, B. W., Dempsey, R. C., & Henry, G. W. 1990, ApJS, 72, 191
Strassmeier, K. G., & Fekel, F. C. 1990, A&A, 230, 389
Soubiran, C., Katz, D., & Cayrel R. 1998, A&AS, 133, 221
Taylor, B. J. 1994, PASP, 106, 704
Taylor, B. J. 1995, PASP, 107, 734
Welty, A. D. 1995, AJ, 110, 776
Welty, A. D., & Ramsey, L. W. 1995, AJ, 110, 336
[c l c c c c c c c c c c]{} O & Date & CCD Detector & N. Or. & $\lambda$$_{i}$-$\lambda$$_{f}$ & Å/pixel\
1 & 1994/12 & T1KA (1024x1024) & 34 & 3875-9400 & 0.123-0.296\
2 & 1994/05 & T1KA (1024x1024) & 33 & 3875-9000 & 0.124-0.284\
3 & 1993/12 & T1KA (1024x1024) & 34 & 3875-9400 & 0.121-0.288\
4 & 1992/11 & T2KB (2048x2048) & 36 & 3700-9050 & 0.113-0.276\
5 & 1991/09 & TI3 (800x800) & 34 & 3810-8950 & 0.077-0.180\
6 & 1991/05 & TEK2 (512x512) & 32 & 3950-8975 & 0.152-0.310\
7 & 1990/10 & RCA1 (512x512) & 40 & 3690-10700& 0.130-0.378\
8 & 1989/12 & TI2 (800x800) & 15 & 7250-9000 & 0.130-0.158\
9 & 1989/04 & RCA3 (512x512) & 34 & 3890-9350 & 0.151-0.359\
10 & 1988/09 & RCA3 (512x512) & 33 & 3880-8950 & 0.150-0.344\
11 & 1987/03 & RCA1 (512x512) & 33 & 3880-8950 & 0.151-0.346\
[l l l l l l l l l l l l l]{} Or. No. & 1, 2, 3, 10, 11 & 4, 7 & 5 & 6 & 9 & 8\
1 & Ca [ii]{} K & & & Ca [ii]{} H &\
2 & Ca [ii]{} H & & & &\
3 & & & Ca [ii]{} H & H$\delta$ & Ca [ii]{} IRT &\
4 & H$\delta$ & Ca [ii]{} K & & &\
5 & & Ca [ii]{} H & H$\delta$ & &\
6 & & & & H$\gamma$ &\
7 & H$\gamma$ & H$\delta$ & & &\
8 & & & H$\gamma$ & &\
9 & & & & &\
10 & & H$\gamma$ & & & Li [i]{}\
11 & & & & H$\beta$ & H$\alpha$\
12 & H$\beta$ & & & & & Ca [ii]{} IRT\
13 & & & H$\beta$ & & & Ca [ii]{} IRT\
14 & & & & Mg [i]{} b &\
15 & Mg [i]{} b & H$\beta$ & & & Na [i]{} D\
16 & & & Mg [i]{} b & &\
17 & & & & &\
18 & & Mg [i]{} b & & &\
19 & & & & Na [i]{} D &\
20 & Na [i]{} D & & & & Mg [i]{} b\
21 & & & Na [i]{} D & &\
22 & & & & &\
23 & & Na [i]{} D & & H$\alpha$ & H$\beta$\
24 & H$\alpha$ & & & Li [i]{} &\
25 & Li [i]{} & & H$\alpha$ & &\
26 & & & Li [i]{} & &\
27 & & H$\alpha$ & & &\
28 & & Li [i]{} & & & H$\gamma$\
29 & & & & &\
30 & & & & &\
31 & & & & Ca [ii]{} IRT & H$\delta$\
32 & Ca [ii]{} IRT & & & &\
33 & & & Ca [ii]{} IRT & & Ca [ii]{} H\
34 & & & & & Ca [ii]{} K\
35 & & Ca [ii]{} IRT & & &\
36 & & & & &\
[l l l l l l l l l l l l l l]{} HD & HR & GJ & Name & T$_{\rm sp}$ & MK & \[Fe/H\] & P$_{\rm rot}$ & [*v*]{} sin[*i*]{} & S & Obs. & Pap.\
& & & & & & (dex) & (days) & (km s$^{-1}$) & & & (I, II)\
\
58946 & 2852 & 274 A & $\rho$ Gem & F0 V (SB?) & MK& - & - & 68 & - & 8 &\
15257 & 717 & - & 12 Tri & F0 III & & - & - & 78 & - & 8 &\
1457 & - & - & SAO 11104 & F0 Iab & & - & - & - & - & 3, 8 &\
128167 & 5447 & 557 & $\sigma$ Boo & F2 V & MK & -0.387 & - & 7.8 & 0.190 & 11 &\
210027 & 8430 & 848 & $\iota$ Peg & F5 V (SB1) & MK & -0.079 & - & - & - & 5 &\
87141 & 3954 & - & BD+54 1348 & F5 V & & 0.047 & - & 10 & - & 8 &\
55052 & 2706 & - & 48 Gem & F5 III-IV & & - & - & 74 & - & 11 &\
20902 & 1017 & - & $\alpha$ Per & F5 Ib: & MK\* & - & - & 18 & - & 8 &\
76572 & 3563 & - & 61 Cnc & F6 V & & - & - & $<$10 & 0.148 & 11 &\
11443 & 544 & 78.1 & $\alpha$ Tri & F6 IV (SB) & & 0.000 & - & 93 & 0.275 & 5 &\
8992 & - & - & SAO 22328 & F6 Ib & & - & - & - & - & 3 &\
187013 & 7534 & 767.1 A & 17 Cyg & F7 V & & -0.109& - & 10.0 & 0.154 & 2, 11(2) & I\
222368 & 8969 & 904 & $\iota$ Psc & F7 V (SB?) & MK & -0.127 & - & 5.6 & 0.153 & 4 &\
187691 & 7560 & 768.1 A & o Aql & F8 V & & 0.059& - & 3.1 & 0.148 & 1, 2, 3, 6(2), 7, 11(2) & I\
142373 & 5914 & 602 & $\chi$ Her & F8 V & & -0.431& - & 2.4 & 0.147 & 6, 11 & I, II\
9826 & 458 & 61 & $\upsilon$ And & F8 V & & -0.14 & - & 8 & 0.154 & 5 & II\
45067 & 2313 & - & BD-00 1287 & F8 V & & -0.16 & - & $<$ 15& 0.141 & 11 & I\
107213 & 4688 & - & 9 Com & F8 V & & 0.154& - & 10.0 & 0.135 & 11 & I\
122563 & 5270 & - & BD+10 2617 & F8 IV & & -2.74 & - & - & - & 6 &\
102870 & 4540 & 449 & $\beta$ Vir & F9 V (SB?) & MK & 0.180 & - & 4.5 & - & 11 &\
22484 & 1101 & 147 & 10 Tau & F9 IV-V (SB?) & & -0.106 & - & 2.8 & 0.147 & 4, 8 &\
114710 & 4983 & 502 & $\beta$ Com & F9.5 V & MK& 0.135& 12.35 & 4.3 & 0.201 & 2, 11(2) & I, II\
[l l l l l l l l l l l l l]{} HD & HR & GJ & Name & T$_{\rm sp}$ & MK & \[Fe/H\] & P$_{\rm rot}$ & [*v*]{} sin[*i*]{} & S & Obs. & Pap.\
& & & & & & (dex) & (days) & (km s$^{-1}$) & & & (I, II)\
\
39587 & 2047 & 222 AB & $\chi^{1}$ Ori& G0- V (SB1) & MK & -0.084& 5.36 & 8.6 & 0.325 & 10(6) & I, II\
143761 & 5968 & 606.2 & $\rho$ CrB & G0+ Va & MK & -0.185& - & 5.0 & 0.150 & 11 & I\
13974 & 660 & 92 & $\delta$ Tri& G0.5 V (SB2)& MK & -0.444& -& 10.0 & 0.232 & 1(2), 3, 4, 5, 10 & I\
26630 & 1303 & - & $\mu$ Per & G0 Ib (SB)& MK & -0.32 & - & 14 & 0.362 & 8 &\
126053 & 5384 & 547 & BD+01 2920 & G1 V & & - & - & 1 & 0.165 & 6, 11 &\
95128 & 4277 & 407 & 47 UMa & G1- V & MK & 0.026 & - & $<$3 & 0.165 & 8 &\
67228 & 3176 & - & $\mu^{2}$ Cnc & G1 IVb & & 0.052 & - & 3.0 & 0.138 & 1(2), 6, 11 &\
84441 & 3873 & - & $\epsilon$ Leo & G1 II & & 0.17 & - & $<$17 & - & 9(6), 11(2) &\
185758 & 7479 & - & $\alpha$ Sge & G1 II & MK & -0.15 & - & 6.0 & - & 2(2) &\
- & - & - & Sun & G2 V & & 0.00 & 25.72 & $<$ 1.7& 0.179 & 1 & I\
1835 & 88 & 17.3 & 9 Cet & G2.5 V & MK & 0.050 & 7.7 & 6 & 0.349 & 1, 3, 4(2), 5 &\
221170 & - & - & BD+29 4940 & G2 IV & & -1.96 & - & - & 0.106 & 2, 3, 5 &\
196755 & 7896 & - & $\kappa$ Del & G2 IV & MK & -0.02 & - & 2.7 & 0.152 & 1, 2(2), 3, 4, 5 &\
218658 & 8819 & - & $\pi$ Cep & G2 III (SB) & & 0.01 & - & 22 & 0.237 & 1, 2 &\
161239 & 6608 & - & 84 Her & G2 IIIb & MK & - & - & 6.0 & 0.138 & 11(2) &\
11544 & - & - & SAO 22740 & G2 Ib & & - & - & - & - & 1, 3 & &\
223047 & 9003 & - & $\psi$ And & G3 Ib-II & & 0.10 & - & $<$ 19 & 0.385 & 8 &\
117176 & 5072 & 512.1 & 70 Vir & G4 V & MK & -0.035 & - & 1.2 & 0.142 & 6, 11(2) &\
123 & 5 & 4.1 A & V640 Cas & G5 V & & - & - & - & - & 3, 5, 11 &\
20630 & 996 & 137 &$\kappa^{1}$ Cet& G5 V (SB?) & MK\*& 0.133& 9.24 & 3.9 & 0.366 & 1(3), 4, 7 & I, II\
59058 & - & - & BD+38 1771 & G5 V & & - & - & - & - & 8 &\
86873 & - & - & BD+32 1970 & G5 & & - & - & - & - & 8 &\
161797 & 6623 & 695 A & $\mu$ Her A & G5 IV & MK\* & 0.242 & - & 1.2 & 0.136 & 5, 6 &\
71369 & 3323 & - & o UMa & G5 III & & -0.21 & - & 3.4 & 0.120 & 1 &\
- & - & - & $\kappa$ Her & G5 III & & - & - & - & - & 2 &\
20825 & - & - & 62 Ari & G5 III & & -0.14 & - & - & - & 4, 5 &\
190360 & 7670 & 777 A & BD+29 3872 & G6 IV+M6 V& & 0.308 & - & - & 0.146 & 5, 6 & I\
221115 & 8923 & - & 70 Peg & G7+ III & & -0.03 & - & $<$ 19 & 0.147 & 2 &\
101501 & 4496 & 434 & 61 UMa & G8 V & MK\* & -0.070& 16.68 & 2.3 & 0.311 & 6, 11(2) & I\
103095 & 4550 & 451 A & BD+38 2285 & G8 Vp & & -1.266 & - & 2.2 & 0.188 & 11 &\
188512 & 7602 & 771 A & $\beta$ Aql & G8 IV & MK\* & -0.30 & - & 1.4 & 0.136 & 1, 2, 4, 5, 7 & I\
73593 & 3422 & - & 34 Lyn & G8 IV & & - & - & - & 0.117 & 3, 11(3) &\
218935 & 8827 & - & 60 Peg & G8 III-IV & & - & - & - & 0.120 & 5 &\
113226 & 4932 & - & $\epsilon$ Vir & G8 IIIab & MK\* & 0.00 & - & 3.2 & - & 1(3), 2(6), 3, 9(6), 11(6) &\
16161 & - & - & $\nu$ Cet & G8 III & & -0.38 & - & $<$ 17 & 0.111 & 4, 5 &\
104979 & 4608 & - & o Vir & G8 IIIa & MK & -0.33 & - & 2.5 & - & 6, 11(2) &\
191026 & 7689 & - & 27 Cyg & G8.5 IVa & & -0.10 & - & - & - & 4 &\
108225 & 4728 & - & 6 Cvn & G9 III & MK & -0.11 & - & $<$ 19 & - & 6 &\
76294 & 3547 & - & $\zeta$ Hya & G9 IIIa & MK & -0.21 & - & - & - & 1, 3 &\
4128 & 188 & 31 & $\beta$ Cet & G9.5 III & & 0.13 & - & 4.0 & 0.187 & 10 &\
[l l l l l l l l l l l l l]{} HD & HR & GJ & Name & T$_{\rm sp}$ & MK &
\[Fe/H\] & P$_{\rm rot}$ & [*v*]{} sin[*i*]{} & S & Obs. & Pap.\
& & & & & & (dex) & (days) & (km s$^{-1}$) & & & (I, II)\
\
185144 & 7462 & 764 & $\sigma$ Dra & K0 V & MK\*& -0.045& - & 0.6 & 0.215 & 2 & I, II\
3651 & 166 & 27 & 54 Psc & K0+ V & MK & -9.000& 48.00 & 2.2 & 0.176 & 1, 3, 4, 5, 8 & I\
198149 & 7957 & 807 & $\eta$ Cep & K0 IV & MK & -0.32 & & 0.6 & - & 1, 2 &\
6734 & - & - & 29 Cet & K0 IV & & -0.25 & - & - & 0.131 & 3 &\
168723 & 6869 & 711 & $\eta$ Ser & K0 III-IV & MK & -0.42 & - & 2.6 & 0.122 & 6, 11(3) &\
45410 & 2331 & - & 6 Lyn & K0 III-IV & & - & - & - & 0.127 & 11 & I\
28 & 3 & - & 33 Psc & K0 III-IV (SB1) & MK & -0.31 & - & $<$ 17 & - & 5 &\
188947 & 7615 & - & $\eta$ Cyg & K0 III & MK & -0.09 & - & 1.8 & 0.103 & 2(2), 8 &\
197989 & 7949 & 806.1 A & $\epsilon$ Cyg & K0 III & MK\* & -0.18 & - & 2.0 & 0.104 & 4(2) &\
19476 & 941 & - & $\kappa$ Per & K0 III & MK & 0.04 & - & $<$ 17 & 0.110 & 5 &\
182272 & 7359 & - & BD+33 3434 & K0 III & & - & - & - & - & 6 &\
19787 & 951 & - & $\delta$ Ari & K0 III & MK & -0.03 & - & $<$ 17 & 0.110 & 5 &\
8512 & 402 & - & $\theta$ Cet & K0 IIIb & MK & -0.22 & - & $<$ 17 & 0.105 & 4, 5 &\
12014 & - & - & SAO 22820 & K0 Ib & & - & - & - & - & 3 &\
10476 & 493 & 68 & 107 Psc & K1 V & MK& -0.123& 35.2 & 0.6 & 0.198 & 4(2), 5, 7(2) & I, II\
155885 & 6401 & 663 B & 36 Oph B & K1 V & & -0.305 & 22.9 & - & 0.384 & 11 &\
142091 & 5901 & - & $\kappa$ CrB& K1 IVa & MK & -0.04 & - & 0.6 & - & 6, 11 & I\
138716 & 5777 & - & 37 Lib & K1 III-IV & MK & -0.12 & - & $<$ 19 & - & 11 &\
203504 & 8173 & - & 1 Peg & K1 III & & -0.14 & - & $<$ 17 & 0.103 & 2(2) &\
124897 & 5340 & 541 & $\alpha$ Boo & K1.5 III & MK & -0.47 & - & 3.3 & 0.144 & 6(5), 9(3), 11(6) &\
6805 & 334 & - & $\eta$ Cet & K2- III & MK & 0.04 & - & $<$ 17 & 0.112 & 4 &\
210745 & 8465 & - & $\zeta$ Cep & K1.5 Ib & MK & 0.75 & - & $<$ 17 & 0.293 & 8 &\
166620 & 6806 & 706 & BD+38 3095 & K2 V & & -0.114 & 42.4 & 0.6 & 0.190 & 1(3), 2, 3, 4(2), 6, 11(2) & I\
4628 & 222 & 33 & BD+04 123 & K2 V & & -0.235 & 38.5 & - & 0.230 & 4, 5, 10 & I\
22049 & 1084 & 144 & $\epsilon$ Eri& K2 V & MK\*& -0.165 & 11.68 & 2.0 & 0.496 & 5, 7 & I\
149661 & 6171 & 631 & 12 Oph & K2 V & & -0.004 & 21.3 & 0.6 & 0.339 & 6, 11(2) &\
201196 & 8088 & - & BD+15 4340 & K2 IV & & - & - & - & - & 1(2), 2, 3(3), 4(2) &\
153210 & 6299 & - & $\kappa$ Oph & K2 III & MK\*& -0.03 & - & $<$ 17 & 0.102 & 6 &\
161096 & 6603 & - & $\beta$ Oph & K2 III & MK & 0.00 & - & 2.5 & 0.103 & 9, 10(4) &\
194317 & 7806 & - & 39 Cyg & K2.5 III & MK & -0.17 & - & $<$ 19 & 0.148 & 2(2), 4, 8 &\
16160 A & 753 & 105 A & BD+06 398 & K3- V & MK & -0.297 & 48.0 & - & 0.226 & 1, 3, 4(3), 5, 7 & I, II\
160346 & - & 688 & BD+03 3465 & K3- V & & - & 33.5 & - & 0.300 & 11(3) &\
219134 & 8832 & 892 & BD+56 2966 & K3 V & MK& -0.017 & - & 2.1 & 0.230 & 8 & I, II\
3627 & 165 & - & $\delta$ And & K3 III (SB)& MK & 0.04 & - & $\leq$ 3 & - & 5 &\
136514 & 5710 & - & 6 Ser & K3 III & & -0.14 & - & $<$ 17 & - & 6 &\
186791 & 7525 & - & $\gamma$ Aql & K3 II & MK & -0.29 & - & $<$ 17 & - & 7(3) &\
131156 B&5544 B& 566 B & $\xi$ Boo B & K4 V & & 0.19 & 12.28 & 20 & 1.381 & 11 & I, II\
201091 & 8085 & 820 A & 61 Cyg A & K5 V & MK\*& -0.06 & 35.37 & 0.6 & 0.658 & 1(2), 2(2), 3, 4(6), 5, 8 & I, II\
156026 & - & 664 & 36 Oph C & K5 V & & -0.279 & 18.0 & 2.2 & 0.770 & 11(2) &\
29139 & 1457 & 171.1 A & $\alpha$ Tau & K5+ III & MK & -0.16 & - & $<$ 17 & - & 1(5), 3, 4(2), 5(4), 8 &\
11800 & - & - & BD+59 363 & K5 Ib & & - & - & - & - & 3 &\
216946 & 8726 & - & BD+48 3887 & K5 Ib & MK & -0.03 & - & - & - & 8 &\
201092 & 8086 & 820 B & 61 Cyg B & K7 V & MK & -0.10 & 37.84 & 1.4 & 0.986 & 1(2), 2(2), 3, 4(2), 5, 8 & I, II\
80493 & 3705 & - & $\alpha$ Lyn & K7 IIIab & MK & -0.26 & - & - & - & 8 &\
[l l l l l l l l l l l l l]{} HD & HR & GJ & Name & T$_{\rm sp}$ & MK & \[Fe/H\] & P$_{\rm rot}$ & [*v*]{} sin[*i*]{} & S & Obs. & Pap.\
& & & & & & (dex) & (days) & (km s$^{-1}$) & & & I, II\
\
- & - & 906 & V347 & M0 V (K5) & & - & - & - & - & 8 &\
89758 & 4069 & - & $\mu$ UMa & M0 III (SB) & MK & - & - & - & - & 1, 11(2) &\
6860 & 337 & 53.3 & $\beta$ And & M0+ IIIa & MK\*& -0.10 & - & - & 0.319 & 4(2), 5, 8 &\
- & - & 4 B & BD+45 4408 B & M0.5 V (K7) & & - & - & - & - & 8 &\
232979 & - & 172 & BD+52 857 & M0.5 V (K8) & MK& - & - & - & 1.909 & 1 & II\
1326 A & - & 15 A & GX And & M1.5 V (1) (M2 V)& MK& - & - & $<$ 2.9 & - & 1, 2(2), 3, 8 & II\
218329 & 8795 & - & 55 Peg & M1 IIIab & MK & - & - & - & 0.234 & 1 &\
206330 & 8284 & - & 75 Cyg & M1 IIIab & MK & - & - & - & - & 8 &\
39801 & 2061 & - & $\alpha$ Ori & M1-M2 Ia-Iab & MK\*& - & - & - & - & 11(2) &\
95735 & - & 411 & BD+36 2147 & M2+ Ve (1) & MK & -0.20 & - & $<$ 2.9 & 0.424 & 11 &\
206936 & 8316 & - & $\mu$ Cep & M2- Ia & MK\*& - & - & - & - & 8 &\
133216 & 5603 & 574.1 & $\sigma$ Lib & M2.5 III & MK & - & - & - & - & 11(2) &\
42995 & 2216 & - & $\eta$ Gem & M2.5 III & & - & - & - & - & 11(2) &\
2411 & 103 & - & TV Psc & M3 III & & - & - & - & 0.211 & 2(2) &\
44478 & 2286 & - & $\mu$ Gem & M3 IIIab & MK & 0.11 & - & - & - & 8 &\
14270 & - & - & AD Per & M3 Iab & & - & - & - & - & 3 &\
- & - & 273 & BD+05 1668 & M3.5 V (1) & & - & - & $<$ 2.4 & - & 1(2) & II\
55383 & 2717 & - & 51 Gem & M4 IIIab & & - & - & - & - & 1, 11(2) &\
214665 & 8621 & - & BD+56 2821 & M4+ III & MK & - & - & - & 0.259 & 8 &\
120323 & 5192 & - & 2 Cen & M4.5 III & MK & - & - & - & - & 11 &\
130144 & 5512 & - & BD+15 2758 & M5 III & & - & - & - & - & 11 &\
94705 & 4267 & - & VY Leo & M5.5 III & MK & - & - & - & - & 11 &\
33664 & 1693 & - & RX Lep & M6 III & & - & - & - & - & 11(2) &\
84748 & 3882 & - & R Leo & M8 IIIe & & - & - & - & - & 1(2) & I\
(1): Henry et al. (1994)
SB: Spectroscopic Binary (Duquennoy & Mayor 1991)
[^1]: IRAF is distributed by the National Optical Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.
|
---
abstract: '[Several platforms are currently being explored for simulating physical systems whose complexity increases faster than polynomially with the number of particles or degrees of freedom in the system. Defects and vacancies in semiconductors or dielectric materials [@pla; @hanson], magnetic impurities embedded in solid helium [@lemeshko13], atoms in optical lattices [@saffman; @simon11], photons [@northup], trapped ions [@kim10; @lanyon] and superconducting q-bits [@corcoles] are among the candidates for predicting the behaviour of spin glasses, spin-liquids, and classical magnetism among other phenomena with practical technological applications. Here we investigate the potential of polariton graphs as an efficient simulator for finding the global minimum of the $XY$ Hamiltonian. By imprinting polariton condensate lattices of bespoke geometries we show that we can simulate a large variety of systems undergoing the U(1) symmetry breaking transitions. We realise various magnetic phases, such as ferromagnetic, anti-ferromagnetic, and frustrated spin configurations on unit cells of various lattices: square, triangular, linear and a disordered graph. Our results provide a route to study unconventional superfluids, spin-liquids, Berezinskii-Kosterlitz-Thouless phase transition, classical magnetism among the many systems that are described by the $XY$ Hamiltonian.]{}'
author:
- 'Natalia G. Berloff$^{1,2}$, Kirill Kalinin$^1$, Matteo Silva$^{3}$, Wolfgang Langbein$^{4}$ and Pavlos G. Lagoudakis$^{1,3}$'
title: Realizing the $XY$ Hamiltonian in polariton simulators
---
Many properties of strongly correlated spin systems, such as spin liquids and unconventional superfluids are difficult to study as strong interactions between $n$ particles become intractable for $n$ as low as $30$ [@sandvik2010]. Feynman envisioned that a quantum simulator – a special-purpose analogue processor – could be used to solve such problems [@feymann]. It is expected that quantum simulators would lead to accurate modelling of the dynamics of chemical reactions, motion of electrons in materials, new chemical compounds and new materials that could not be obtained with classical computers using advanced numerical algorithms [@lloyd]. More generally, quantum simulators can be used to solve hard optimization problems that are at the heart of almost any multicomponent system: new materials for energy, pharmaceuticals, and photosynthesis, among others [@qubit]. Many hard optimisation problems do not necessitate a [*quantum*]{} simulator as only recently realised through a network of optical parametric oscillators (OPOs) that simulated the Ising Hamiltonian of thousands of spins [@yamamoto11; @yamamoto14]. The Ising model corresponds to the $n=1$ case of the $n$-vector model of classical unit vector spins ${\bf s}_i$ with the Hamiltonian ${\cal H}_I=-\sum_{ij} J_{ij} {\bf s}_i \cdot { \bf s}_j$, where $J_{ij}$ is the coupling between the sites labelled $i$ and $j$. For $n=2$ the $n$-vector Hamiltonian becomes the $XY$ Hamiltonian ${\cal H}_{XY}=-\sum_{ij} J_{ij} \cos (\theta_i-\theta_j)$, where we have parameterized unit planar vectors using the polar coordinates ${\bf s}_i=(\cos \theta_i, \sin\theta_i)$. Since ${\cal H}_{XY}$ is invariant under rotation of all spins by the same angle $\theta_i \rightarrow \theta_i+ \phi$ the XY model is the simplest model that undergoes the $U(1)$ symmetry-breaking transition. As such, it is used to emulate Berezinskii-Kosterlitz-Thouless phase transition and the emergence of a topological order [@bkt; @bkt2], topological quantum information processing and storage [@nayak08], and to study quantum phase transitions, unconventional superfluids, quantum spin models, spin-liquid phases and high-$T_c$ superconductivity. The $XY$ Hamiltonian has been simulated on a triangular lattice of atomic condensates investigating a variety of magnetic phases and frustrated spin configurations [@struck11]. Whereas optical lattices offer a scalable platform, they are likely to reach a local rather than global minimum of the Hamiltonian and are limited to sub-$\mu$K temperatures [@georgescu14].
In this Article, we propose and experimentally demonstrate the use of polariton graphs as a scheme for finding the global minimum of the $XY$ Hamiltonian. Polaritons are the mixed light-matter quasi-particles that are formed in the strong exciton-photon coupling regime in semiconductor microcavities [@weisbuch]. Under non-resonant optical excitation, rapid relaxation of carriers and bosonic stimulation result in the formation of a non-equilibrium polariton condensate characterized by a single many-body wave-function [@Kasprzak]. Polariton condensates can be imprinted into any two-dimensional graph by spatial modulation of the pumping source, offering the scalability matched only by optical lattices [@georgescu14]. Optically injected polariton condensates can potentially be imprinted in multi-site configurations with arbitrary polarisation and density profiles offering the possibility to control the separation distance between sites. Such flexibility allows for unprecedented control of the interaction between neighbouring sites. Due to the finite cavity lifetime, polaritons decay in the form of photons (through quasi-mode coupling [@amo2010; @ciutiPRB2000]) that carry all information of the corresponding polariton state (energy, momentum, spin and phase). The continuous coupling of polaritons to free photons allows for the in-situ characterisation of static polariton graphs, but more importantly it also allows for the dynamic control of an arbitrary set of sites, whilst measuring in real time the kinetics and phase configuration of the modulated polariton graph.
In a graph of two or more coupled polariton vertices, with increasing excitation density, polariton condensation occurs at the state with the phase-configuration that carries the highest polariton occupation [@ohadi14]. This is due to the bosonic character of the condensate formation: the probability of a particle to relax in a particular state grows with the population of that state. Just above condensation threshold a macroscopic coherent state is formed described by the wavefunction $\Psi_g$. $\Psi_g$ can be written as a superposition of the wavefunctions $\Psi_j$ at the sites ${\bf x}_j$ with phase $\theta_j$; that is $\Psi_g\approx\sum_j \Psi_j \exp[i \theta_j]$. Below we will show that the system of an arbitrary polariton graph condenses into the global minimum of the $XY$ Hamiltonian: ${\cal H}_{XY}=-\sum J_{ij} \cos\theta_{ij}$ where $\theta_{ij}$ is the phase difference between two sites, $\theta_{ij}=\theta_i-\theta_j$ and $J_{ij}$ is the corresponding coupling strength; the latter depends on the density of the sites $i$ and $j$, the distance between them, $d_{ij}=|{\bf x}_i-{\bf x}_j|$, and the outflow condensate wavenumber $k_{c}$, which under non-resonant optical excitation depends on the pumping intensity and profile. The bottom-up approach for the search of the global minimum of the $XY$ Hamiltonian is achievable within the linewidth of the corresponding state similarly to a network of time-multiplexed OPOs [@yamamoto14] that guarantees a phase-transition to the global minimum of the Ising Hamiltonian. This is an advantage over classical or quantum annealing techniques, where the global ground state is reached through transitions over metastable excited states (local minima), with an increase of the cost of the search with the size of the system.
![ (a) Schematic of the condensate density map for a five-vertex polariton graph. The sign of the coupling is annotated for some of the edges of the graph: depending on the separation distance between the sites and the outflow wavevector $k_c$ the interactions are either ferromagnetic (solid-blue lines) or anti-ferromagnetic (dashed-red lines). At each vertex ${\bf x}_i$ of the graph polaritons have a local phase $\theta_{i}$ that is mapped to a classical vector spin ${\bf s}_i=(\cos\theta_i,\sin\theta_i)$. (b) the vertices (blue solid-circles) and edges of the polariton density map depicted in (a), showing the sign of the coupling and the spin vector ${\bf s}_i$ of each vertex.[]{data-label="schema"}](Figure1,cropped,final){width="8.6cm"}
[*Modelling the phase coupling:*]{} we model the phase coupling in polariton graphs using the complex Ginzburg-Landau equation (cGLE) with a saturable nonlinearity and energy relaxation [@Wouters; @Berloff]: $$\begin{aligned}
i \hbar \frac{\partial \psi}{\partial t} &=& - \frac{\hbar^2}{2m} \left(1 - i \eta_d {\cal R} \right) \nabla^2\psi + U_0 |\psi|^2 \psi+
\hbar g_R {\cal R} \psi \nonumber \\
&+&\frac{i\hbar}{2} \biggl(R_R {\cal R} - \gamma_C \biggr) \psi, \label{Initial GL equation}\\
\frac{\partial \cal R}{\partial t} &=& - \left( \gamma_R + R_R |\psi|^2 \right) {\cal R} + P({\bf r}) ,
\label{Initial Reservoir equation}\end{aligned}$$ where $\psi$ is the condensate wavefunction, ${\cal R}$ is the density profile of the hot exciton reservoir, $m$ is the polariton effective mass, $U_0$ and $g_R$ are the strengths of effective polariton-polariton interaction and the blue-shift due to interactions with non-condensed particles, respectively, $R_R$ is the rate at which the exciton reservoir feeds the condensate, $\gamma_C$ is the decay rate of condensed polaritons, $\gamma_R$ is the rate of redistribution of reservoir excitons between the different energy levels, $\eta_d$ is the energy relaxation coefficient specifying the rate at which gain decreases with increasing energy, and $P$ is the pumping into the exciton reservoir. In Eq. (\[Initial Reservoir equation\]) we neglected the diffusion of the reservoir as well as density-density repulsion with the condensate in the view of the large mass of the hot exciton as compared to the mass of the polariton (five orders of magnitude). We non-dimensionalize these equations using $
\psi \rightarrow \sqrt{\hbar^2 / 2m U_0 l_0^2} \psi,
{\bf r} \rightarrow l_0 {\bf r}, t \rightarrow 2m t l_0^2/ \hbar$ and introducing the notations $g = 2 g_R/R_R,$ $\gamma = m \gamma_C l_0^2/ \hbar $, $ p=m l_0^2 R_R P({\bf r})/ \hbar \gamma_R,\eta = \eta_d \hbar / mR_R l_0^2,$ and $ b = R_R \hbar^2 / 2m l_0^2\gamma_R U_0. $ We choose $l_0=1\mu m$ and consider the stationary states.
By using the Madelung transformation $\Psi=\sqrt{\rho}\exp[i S]$ in the dimensionless Eqs. (\[Initial GL equation\],\[Initial Reservoir equation\]), where $\rho=|\psi|^2$, ${\bf u}=\nabla S$ is the velocity, $S$ is the phase and separating the real and imaginary parts we obtain the mass continuity and the integrated form of the Bernoulli equation which we write for a steady state, and, therefore, introduce the chemical potential $\mu$ $$\begin{aligned}
\mu = - \frac{ \nabla^2\sqrt{\rho}}{\sqrt{\rho}} + {\bf u}^2 + \rho &+& \frac{p({\bf r})}{1 + b \rho} \biggl( g - \eta \frac{\nabla \cdot (\rho {\bf u})}{\rho} \biggr) ,\label{systemM1} \\
\frac{\nabla\cdot ( \rho {\bf u})}{\rho} = \frac{p({\bf r})}{1+b\rho} \biggl( 1 &+& \eta \left( \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}} - {\bf u}^2 \right) \biggr) - \gamma.
\label{systemM2}\end{aligned}$$ First, we consider a single pumping spot with a radially symmetric pumping profile. Asymptotics at large distances from the center of the pump gives the velocity $|u|=k_c=const$ and $\rho\sim \exp[-\gamma r/k_c]r^{-1}.$ From Eq. (\[systemM1\]) at infinity, therefore, we obtain $\mu=k_c^2 - \gamma^2 / 4 k_c^2$. We can estimate the chemical potential for a wide pumping spot so that the quantum pressure term $\nabla^2 \sqrt{\rho}/\sqrt{\rho}$ and $u_r$ are insignificant at the pumping center. Under this assumption $\rho_{\max} \approx (p_{\max}-1)/b$ and $\mu \approx(p_{\max}-1)/b + g.$ In [@ohadi14] we established experimentally for the pulsed excitation that the coupling between two pumping spots (a“polariton dyad") can be either in-phase or with a $\pi$ phase difference depending on the outflow wavenumber $k_c$ and the distance between the spots. Below, in the steady state excitation regime, we obtain a general criterion for the switching between the relative phases. We start by considering the wavefunction of the condensate as the sum of the wavefunctions of individual condensates located at $\pm {\bf d}/2$, where $\pm{\bf d}=(\pm d_{ij},0)$ with the phase difference $\theta_{ij}$: $
\tilde{\Psi}(\mathbf{r})\approx
{\Psi}\left(\mathbf{r}+\frac{\mathbf{d}}{2}\right)
+e^{i\theta_{ij}}{\Psi}\left(\mathbf{r}-\frac{\mathbf{d}}{2}\right).
$ The total number of condensed polaritons can be found in Fourier space as $$\label{eq:lifetime-integral}
N = \int\frac{d{\bf k}}{2\pi^2}
|\widehat{\Psi}(k)|^{2}[1 +\cos({\bf k}\cdot {\bf d}-\theta_{ij})],$$ where $\widehat{\Psi}(k)=2\pi\int_0^\infty\sqrt{\rho(r)} \exp[i k_c r]J_0(kr)r\,dr $ is the Hankel transform of the wavefunction of an individual condensate. We conclude that $$\begin{aligned}
N &=& N_i + N_j + J_{ij}\cos\theta_{ij},\label{N1}\\
J_{ij}&=&\frac{1}{\pi}\int_0^\infty
|\widehat{\Psi}(k)|^{2}J_0(kd_{ij})k\, dk.
\label{J1}\end{aligned}$$ which in the case of $n$ condensates generalizes to $$N = \sum^n_i N_i + \sum^n_{i<j}J_{ij}\cos\theta_{ij}. \label{nnn}$$ The oscillating behaviour of the Bessel function, $J_0(k d_{ij})$, brings about the sign change in the coupling constants $J_{ij}$ depending on the distance $d_{ij}$. When $J_{ij}$ is positive the coupling is said to be ferromagnetic and when $J_{ij}$ is negative the coupling is said to be anti-ferromagnetic. The state with the phase configuration that carries the highest number of particles in Eq. (\[nnn\]) corresponds to the solution that minimises the $XY$ Hamiltonian, ${\cal H}_{XY}=-\sum^n_{i<j}J_{ij}\cos\theta_{ij}$. Between any two polariton nodes the polariton wavefunction forms a standing wave with the density $|\Psi_g|^2\approx\rho_+
+\rho_-+ 2\sqrt{\rho_+\rho_-}\cos[k_c|x-d_{ij}/2|-k_c |x+d_{ij}/2|-\theta_{ij}]$, where $x$ is the coordinate along the line that connects the two nodes separated by a distance $d_{ij}$ and $\rho_\pm=\rho(x\pm d_{ij}/2,y)$. Between two polariton nodes the density oscillates as $1+\cos(2 k_c x + \theta_{ij})$, from which the phase difference $\theta_{ij}$ of a single shot realization can be extracted directly. In Fig. 1(a) we plot the density of a polariton graph, where for simplicity we have annotated the sign of the coupling for some of the edges of the graph. Depending on the separation distance between the vertices and the outflow wavevector $k_c$ the interactions are either ferromagnetic (solid-blue lines) or anti-ferromagnetic (dashed-red lines). At each vertex ${\bf x}_i$ of the graph polaritons have a local phase $\theta_{i}$, which in the following we map to a classical vector spin ${\bf s}_i=(\cos\theta_i,\sin\theta_i)$. In Fig. 1(b) we show the vertices of the polariton graph, the edges of Fig.1(a) depicting the sign of the coupling and the spin vector ${\bf s}_i$ of each vertex as calculated from the minimisation of the $XY$ Hamiltonian.
![ (a) The maximum number of particles, $N$, of a polariton condensate dyad formed under incoherent pumping of two nodes as the function of the product $k_c d$ between the nodes obtained by numerical integration of the cGLE for a fixed $k_c$. The solid black line corresponds to the maximum number of particles in the in-phase ferromagnetic configuration and the dashed black line to the $\pi$-phase difference anti-ferromagnetic configuration. The switching occurs with the periodicity $2\pi/k_c$ as the superimposed graph of $\cos(k_cd + \phi)$ illustrates in red, where $\phi\approx 225^\circ$. (b,c) Experimental realization of an Ising chain of five equidistant polariton nodes with lattice constants of $\sim$ 6.7$\mu m$ and $\sim$9.6$\mu m$ respectively. The false-grey scale images show the normalised photoluminescence intensity of the real-space tomography at the energy of the condensate; (c) is saturated at 0.5 to increase the visibility of the low intensity fringes between the nodes. The corresponding $k_c d$ (11.9 and 16.04) are shown by two solid circles on (a). []{data-label="NvsD"}](Figure2,cropped,final "fig:"){width="8.6cm"}\
[*The Ising polariton chain:*]{} we theoretically describe and experimentally address the minimization of the $XY$ Hamiltonian for the simple case of a linear polariton chain with equal spacing $d=d_{ij}$ between neighbours. For a given $k_c$ with increasing separation distance the coupling between the neighbors, $J_{ij}$, oscillates between negative and positive values. We approximate the switching of the coupling sign with $\cos(k_c d+\phi)$, where $\phi$ is fixed by the system parameters (see Supp. Mat. for the derivation). In the steady state excitation regime, we can calculate the maximum particle number of a polariton dyad as a function of the separation distance $d$ by numerically integrating the cGLE to find the solutions of Eqs. (\[systemM1\]-\[systemM2\]) for a given pumping profile $p({\bf r})=p_0[\exp(-\alpha |{\bf r-d}/2|^2)+\exp(-\alpha |{\bf r+d}/2|^2)]$ of a characteristic width $\alpha$; the results are shown in Fig. 2(a). The relative phases that realise the maximum particle number switch periodically between $0$ and $\pi$ with the period $2\pi/k_c$ as shown by superimposing the function $\cos(k_c d + \phi)$ in Fig.2(a); we have used the experimental parameters for the pumping profile and $k_c$ as described in “Wavevector Tomography” in Supp. Mat. Where the coupling is ferromagnetic (anti-ferromagnetic) the graph of the maximum number of particles is plotted with a solid (dashed) line. We experimentally address the Ising chain by injecting a linear chain of five equidistant polariton nodes through non-resonant, continuous wave and spatially modulated optical excitation of a multiple InGaAs quantum well semiconductor microcavity that allows for detection of the polariton photoluminescence in the transmission geometry (for the sample description read the “Microcavity sample" and for the description of the excitation/detection scheme read the “Experimental setup" in Supp. Mat.). Figures 2(b,c) show the real-space tomography of the photoluminescence intensity at the energy of the condensate from the linear chain with lattice constants of $\sim$ 6.7$\mu m$ and $\sim$9.6$\mu m$ respectively at condensation threshold. The relative phase difference realised between neighbours in the chain is either $\pi$ or zero. The patterns are clearly distinguishable by the number of fringes (density maxima) between the sites: zero or even for anti-ferromagnetic and odd for ferromagnetic coupling. In Fig.2(a) we have annotated the abscissa with solid circles for each of the two separation distances from which the expected sign of coupling is depicted showing good agreement with the experiment. The observed phase configurations realise the ferromagnetic and anti-ferromagnetic Ising spin chain of the $XY$ model.
![ Spin configurations of square polariton lattices. The diagrams of the numerically calculated spins vectors at the pumping sites ${\bf s}_i=(\cos\theta_i,\sin\theta_i)$, the real-space energy tomography of the experimental realisations, and the averaged condensate densities of the numerically simulated condensate wavefunctions for several realizations are shown on the left, central and right columns respectively. Solid and dashed blue lines on the spin vector diagrams (left column) indicate ferromagnetic and anti-ferromagnetic coupling, respectively. The false-grey scale images of the middle column show the normalised photoluminescence intensity of the real-space tomography at the energy of the condensate; (c) is saturated at 0.5 to increase the visibility of the low intensity fringes between the vertices. The configurations shown are some elementary building blocks of square lattices such as (a,c) anti-ferromagnetic, (b) ferromagnetic, (d) $90^\circ$-compass. The centers of the pumping spots are shown by white dashed circles on the numerical density profiles (right column). The parameters of the numerical simulations of Eqs. (\[Initial GL equation\],\[Initial Reservoir equation\]) are listed in the Supp. Mat.[]{data-label="summary"}](Figure3,cropped,final){width="8.6cm"}
[*Equidistant vertices across a circle:*]{} we consider a geometry of $n$ incoherently pumped equidistant polariton vertices positioned on the circumference of a circle. For equal separation distances $d=d_{ij}$ between adjacent sites the $XY$ Hamiltonian to minimise becomes ${\cal H}_{XY}=-J\sum_{i=1}^{n}\cos (\theta_{i,i+1}),$ where $J=J_{ij}$, the summation is cyclic and we took into account only nearest neighbour interactions. If $J$ is positive, then all sites lock in phase ($\theta_{i,i+1}=0$). If $J$ is negative, the minimum of ${\cal H}_{XY}$ occurs for $\theta_{i,i+1}= \pm \pi$, when $n$ is even and for $\theta_{i,i+1}= \pm\pi (n\pm1)/n$ when $n$ is odd ($n>1$). We experimentally access these two regimes through incoherent injection of polaritons at the vertices of a square; Figure 3(a,b,c) show the spin configuration, experimental results of the real-space tomography of the photoluminescence intensity at the energy of the condensate at condensation threshold and numerical simulation for a square with lattice constants that lead to anti-ferromagnetic, ferromagnetic and the next anti-ferromagnetic coupling respectively. Similar to the Ising polariton chain the type of coupling is clearly distinguishable by the number and symmetry of fringes between the vertices: zero or even for anti-ferromagnetic (Fig.3(a,c)) and odd for ferromagnetic coupling (Fig.3(b)). These observations are in agreement with the $\pi$ phase difference reported in Ref.[@tosi13]. We can thus summarise in the case of the square lattice cell that for ferromagnetic coupling polaritons at the vertices lock with zero phase difference and for anti-ferromagnetic coupling polaritons at neighbouring vertices lock with a $\pi$ phase difference.
[*$90^\circ$ compass model:*]{} in the context of topological quantum computing apart from the trivial all ferromagnetic or all anti-ferromagnetic coupling configurations in a square geometry, more complex coupling configurations are of interest. Examples of such configurations are the compass models, where the coupling between the internal spin components is inherently directionally dependent. Such compass-type coupling appears in various physical systems, where the interactions are sensitive to the spatial orientation of the involved orbitals. In polariton graphs the compass models with direction dependent coupling or spin glassy models with random couplings can be realised by changing the pumping intensity and preserving the square geometry, or alternatively, tuning the separation distances so that each vertex has one ferromagnetic and one anti-ferromagnetic coupling with its nearest neighbours. In Fig. 3(c) we have realised the $90^\circ$ compass model, where each vertex has one ferromagnetic and one anti-ferromagnetic coupling with its neighbours as it is clearly distinguishable by the number of fringes between nearest vertices. The $90^\circ$ compass, where both ferro- and anti-ferromagnetic coupling appear across the two orthogonal diagonals here, has been proposed as a model to Mott insulators with orbital degrees of freedom and frustrated magnets [@Nussinov2015]. Other compass-type models accessible through polariton graphs include the plaquette orbital model, where the ferromagnetic and anti-ferromagnetic coupling alternate along each direction [@biskup2010] and the orbital compass model on a checkerboard lattice [@nasu2012]. Fully random couplings in the square lattice describes the thermodynamic behaviour of several disordered systems, such as magnetic systems with random Dzyaloshinskii- Moriya interactions [@Rubinstein1983], disordered Josephson junction arrays [@kosterlitz1989], disordered substrates [@cha], and vortex glasses in high-$T_c$ cuprate superconductors [@gingras1996]. It should be possible to address these systems by considering a polariton lattice composed of the individual compass elements analogous to the one we realised here.
![ Spin configurations of the diamond-shaped polariton lattices. The columns of images are as described in the caption to Fig.3. The configurations shown are some elementary building blocks of triangular lattices such as (a,c) anti-ferromagnetic and (b) ferromagnetic rhombuses. The false-grey scale images of the middle column show the normalised photoluminescence intensity of the real-space tomography at the energy of the condensate saturated at 0.5 to increase the visibility of the low intensity fringes between the vertices.[]{data-label="summary"}](Figure4,cropped,final){width="8.6cm"}
[*Triangular lattice:*]{} the $XY$ Hamiltonian has been simulated on a triangular lattice of atomic condensates discovering variety of magnetic phases and frustrated spin configurations [@struck11]. In the case of an anti-ferromagnetically coupled polariton triad, arranged at the vertices of an equidistant triangle, the energy flux that minimizes the $XY$ Hamiltonian corresponds to $\pm1$ winding ($2\pi/3$ phase difference between the condensates) [@ohadi14]. Here, we experimentally realise an equidistant triangular lattice of two lattice cells (rhombus configuration) under incoherent injection of polaritons in the bespoke lattice configurations. Figure 4(a,b,c) show the spin configuration, experimental results of the real-space tomography of the photoluminescence intensity at the energy of the condensate at condensation threshold and numerical simulation for a rhombus with lattice constants that lead to anti-ferromagnetic, ferromagnetic and the next anti-ferromagnetic coupling respectively. In the case of ferromagnetic coupling between nearest neighbours and neglecting opposite neighbours interaction across the long diagonal axis of the rhombus, the $XY$ Hamiltonian is minimised at ${\cal H}_{XY}\sim-5J $ when all polariton sites lock in phase, as shown in Fig. 4(b). Similarly, in the case of anti-ferromagnetic coupling between nearest neighbours the $XY$ Hamiltonian is minimised at ${\cal H}_{XY}\sim-3J$ when there is $\pm\pi$ phase difference between the outer edges of the rhombus. This configuration forces the rhombus in a frustrated state wherein opposite vertices have the same phase. This type of frustrated spin configuration is experimentally realised in Fig. 4(a,c). The corresponding states in Figs. 4(a,b,c) are shown in the order of the increasing distance between the sites, therefore, the anti-ferromagnetic states of Figs. 4(a) and 4(c) belong to two different bands of anti-ferromagnetic regions separated by a ferromagnetic band (the alternating anti-ferromagnetic/ferromagnetic couplings bands are shown in Fig. 2(a)). The measured density profiles show some clear differences: the local minimum at the center of the rhombus along the long diagonal in Fig. 4(a) is replaced by a local maximum in Fig. 4(c).
![ Spin configurations of a random polariton graph. The panels of images are as described in the caption to Fig.3. The false-grey scale image of the middle column show the normalised photoluminescence intensity of the real-space tomography at the energy of the condensate saturated at 0.5 to increase the visibility of the low intensity fringes between the vertices.[]{data-label="summary"}](Figure5,cropped,final){width="8.6cm"}
[*Random polariton graph:*]{} Beyond the minimization of the $XY$ Hamiltonian of polariton condensates on regular lattices we test our platform on a disordered polariton graph of five vertices. We took a graph initially consisting of three equidistant triangular unit cells for a lattice constant that leads to anti-ferromagnetic coupling, but with one spot breaking the symmetry. This is achieved experimentally by slightly displacing one spot on the graph. Figure 5 shows the spin configuration, experimental results of the real-space tomography of the photoluminescence intensity at the energy of the condensate at condensation threshold and numerical simulations that correspond to this graph. For the symmetric configuration of three equidistant triangular cells and considering only nearest neighbours interactions, the $XY$ Hamiltonian is minimised at ${\cal H}_{XY}\sim-3.86J $ with an alternating winding around each cell slightly deviating from $2\pi/3$ difference reported for a single equilateral triangle (see Supp. Mat. for details). Breaking the symmetry leads to a different phase distribution, while maintaining the winding around each cell. The analysis of the fringes on the experimental image (with the different rows of local maxima along the two long diagonals) shows that the symmetry is explicitly broken.
In conclusion, we propose polariton graphs as an analog platform for minimizing the $XY$ Hamiltonian and experimentally demonstrate its experimental implementation for simple building blocks of lattices. We demonstrated that the search for the global ground state of a polariton graph is equivalent to the minimisation of the $XY$ Hamiltonian ${\cal H}_{XY}=-\sum J_{ij} \cos\theta_{ij}$. Polariton graphs offer the scalability of optical lattices, together with the potential to study disordered systems, and to control both the sign and the strength of the coupling for each edge independently. Similar to networks of time-multiplexed OPOs, phase transitions in polariton graphs occur at the global ground state. Furthermore, polariton graphs offer the potential to quench either all or individual vertices on variable time scales and study the complex relaxation dynamics. With the recent advances in the field of polariton condensates, such as room temperature operation [@organic] and condensation under electrical pumping [@electrical], polariton graph based simulators offer unprecedented opportunities in addressing the $XY$ Hamiltonian and therefore topological quantum information processing and the study of exotic phase transitions.
[00]{}
Pla, J. J. *et al.* High-fidelity readout and control of a nuclear spin qubit in silicon. *Nature* **489**, 541-545 (2013).
Hanson, R. & Awschalom, D. D. Coherent manipulation of single spins in semiconductors. *Nature* **453**, 1043-1049 (2008).
Lemeshko, M. *et al.* Controllable quantum spin glasses with magnetic impurities embedded in quantum solids. *Phys. Rev. B* **88**, 014426 (2013).
Saffman, M., Walker, T. G. & Molmer, K. Quantum information with Rydberg atoms. *Rev. Mod. Phys.* **82**, 2313 (2010).
Simon, J. *et al.* Atomic physics: a route to quantum magnetism. *Nature* **472**, 307-312 (2011).
Northup, T. E. & Blatt, R. Quantum information transfer using photons. *Nature Photonics* **8**, 356 (2014).
Kim, K. *et al.* Quantum simulation of frustrated Ising spins with trapped ions. *Nature* **465**, 590-593 (2010).
Lanyon, B. P. *et al.* Universal digital quantum simulation with trapped ions. *Science* **334**, 57, (2011).
Corcoles, A. D. *et al.* Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. *Nature Commun.* **6**, 6979 (2015).
Sandvik, A. W. Ground states of a frustrated quantum spin chain with long-range interactions. *Phys. Rev. Lett.* **104(13)**, 137204 (2010).
Feynman, R. Simulating physics with computers. *Int. J. Theor. Phys.* **21**, 467–488 (1982).
Lloyd, S. Universal Quantum Simulators. *Science* **273**, 1073 (1996).
Boixo, S. *et al.* Evidence for quantum annealing with more than one hundred qubits. *Nature Physics*, **10**, 218 (2014).
Utsunomiya, S., Takata, K. & Yamamoto, Y. Mapping of Ising models onto injection-locked laser systems. *Opt. Express* **19**, 18091 (2011).
Marandi, A., Wang, Z., Takata, K., Byer, R. L. & Yamamoto, Y. Network of time-multiplexed optical parametric oscillators as a coherent Ising machine. *Nature Photonics* **8**, 937-942 (2014).
Berezinskii, V. L. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems. *Sov. Phys. JETP* **32**, 493 (1971).
Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. *J. Phys. C* **6**, 1181 (1973).
Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. D. Non-Abelian anyons and topological quantum computation. *Rev. Mod. Phys.* **80**, 1083 (2008).
Struck, J. Quantum simulation of frustrated classical magnetism in triangular optical lattices. *et al.* *Science* **333**, 996 (2011).
Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. *Rev. Mod. Phys.* **86**, 153 (2014).
Weisbuch, C., Nishioka, M., Ishikawa, A. & Arakawa, Y. Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity. *Phys. Rev. Lett.* **69**, 3314 (1992)
Kasprzak, J. *et al.* Bose-Einstein condensation of exciton polaritons. *Nature* **443**, 409 (2006).
Amo, A. *et al.* Light engineering of the polariton landscape in semiconductor microcavities. *Phys. Rev. B* **82**, 081301(R) (2010).
Ciuti, C., Schwendimann, P., Deveaud, B. & Quattropani, A. Theory of the angle-resonant polariton amplifier. *Phys. Rev. B* **62**, R4825(R) �(2000).
Ohadi, H. *et al.* Non-trivial phase coupling in polariton multiplets. In press by Phys. Rev. X.
Wouters, M. & Carusotto, I. Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons. *Phys. Rev. Lett.* **99**, 140402 (2007).
Keeling, J. & Berloff, N. G. Spontaneous rotating vortex lattices in a pumped decaying condensate. *Phys. Rev. Lett.* **100**, 250401 (2008).
Tosi, G. *et al.* Geometrically locked vortex lattices in semiconductor quantum fluids. *Nat. Comm.*, **3**, 1243 (2012).
Nussinov, Z. & van den Brink, J. Compass models: theory and physical motivations. *Rev. Mod. Phys.* **87**, 1 (2015).
Biskup, M. & Kotecky, R. True nature of longrange order in a plaquette orbital model. *J. Stat.. Mech.* **11**, 11001 (2010).
Nasu, J. and Ishihara, S. Orbital compass model as an itinerant electron system. *Europhysics Letters* **97**, 27002 (2012).
Rubinstein, M., Shrainam, B. & Nelson, D. R. Two-dimensional XY magnets with random Dzyaloshinskii-Moriya interactions. *Phys. Rev. B* **27**, 1800 (1983).
Granato, E. & Kosterlitz, J. M. Disorder in Josephson-junction arrays in a magnetic field. *Phys. Rev. Lett.* **62**, 823 (1989).
Cha, M.C. & Fertig, H. A. Orientational order and depinning of the disordered electron solid. *Phys. Rev. Lett.* **73**, 870 (1994).
Gingras, M. J. P. & Huse, D. A. Topological defects in the random-field XY model and the pinned vortex lattice to vortex glass transition in type-II superconductors. *Phys. Rev. B* **53**, 15193 (1996).
Lucas, A. Ising formulations of many NP problems. *Frontiers in Physics* **2**, 5 (2014).
Plumhof, J. D., Stöferle, T., Mai, L., Scherf, U. & Mahrt, R. F. Room-temperature Bose-Einstein condensation of cavity exciton-polaritons in a polymer. *Nat. Mat.* **13**, 247 (2014).
Schneider, C. *et al.* An electrically pumped polariton laser. *Nature* **497**, 348 (2013).
**Supplemental Material: Realizing the $XY$ Hamiltonian in polariton simulators**
Microcavity Sample {#microcavity-sample .unnumbered}
==================
The semiconductor microcavity structure studied here is a planar, strain compensated 2$\lambda$ GaAs microcavity with embedded InGaAs quantum wells (QWs). Strain compensation was achieved by AlAs$_{0.98}$P$_{0.02}$/GaAs DBR layers instead of the thin AlP inserts in the AlAs layers used in Ref.\[\] as their effective composition could be better controlled. The bottom DBR consists of 26 pairs of GaAs and AlAs$_{0.98}$P$_{0.02}$ while the top has 23 of these pairs, resulting in very high reflectance (>99.9$\%$) in the stop-band region of the spectrum. The average density of hatches along the \[$110$\] direction was estimated from transmission imaging to be about 6/mm, while no hatches along the \[$1\bar{1}0$\] direction were observed. Three pairs of 6nm In$_{0.08}$Ga$_{0.92}$As QWs are embedded in the GaAs cavity at the anti-nodes of the field as well as two additional QWs at the first and last node to serve as carrier collection wells. The large number of QWs was chosen to increase the Rabi splitting and keep the exciton density per QW below the Mott density [@saturation] also for sufficiently high polariton densities to achieve polariton condensation under non-resonant excitation. The strong coupling between the exciton resonance and the cavity mode is observed with a vacuum Rabi-splitting of $2\hbar\Omega\sim8$meV. A wedge in the cavity thickness allows access to a wide range of exciton-cavity detuning. All measurements reported here are taken at $\Delta\approx -5.5$meV. The measured Q-factor is $\sim 12000$, while the calculated bare cavity Q-factor, neglecting in-plane disorder and residual absorption, is $\sim25000$. As the emission energy of the InGaAs QWs is lower than the absorption of the GaAs substrate we can study the photoluminescence of the sample both in reflection and transmission geometry. The transmission geometry, which is not available for GaAs QWs, allows to filter the surface reflection of the excitation, and has been widely utilized to probe the features of polariton fluids [@all-optical_2011; @nardin_hydrodynamic_2011] under resonant excitation of polaritons. Using real and reciprocal space spectroscopic imaging under non-resonant optical excitation, polariton condensation, and a second threshold marking the onset of photon lasing, i.e. the transition from the strong to the weak-coupling regime has been studied in this microcavity [@InGaAs].
Experimental setup {#experimental-setup .unnumbered}
==================
In the experiments described here the sample was held in a cold finger cryostat at a temperature of $T\approx 6$K. Continuous wave excitation is provided by a Ti:Sapphire laser. We use non-resonant excitation from the epi side, and detect the emission from the substrate side, so that the excitation is filtered by the absorption of the GaAs substrate. The optical excitation, for all the measurements reported in this work, is at the first reflectivity minimum above the cavity stop band. The spatial profile of the excitation beam is modulated to a graph with Gaussian profiles at each vertex of approximatelly equal in diameter spots using a reflective spatial light modulator (SLM). We use a high numerical aperture microscope objective (NA = 0.65) to focus the spatially modulated beam to $\sim $1-2$\, \mathrm{\mu m}$ in diameter at full width at half maximum (FWHM) excitation spots. The photoluminescence from the sample is collected in transmission geometry with $\pm 25^{\circ}$ collection angle, by a 0.42 NA microscope objective. Fourier (dispersion) imaging is performed by projecting the Fourier-space at the slit of a $300$mm spectrophotometer coupled a cooled charge coupled (CCD) device and using a $1200$ grooves/mm with 50 $\mu$eV energy-resolution. The real-space tomography images are acquired with sub-micron optical resolution using a CCD camera imaging configuration through a tunable Fabry-Perot etalon with $\sim$ 20 $\mu$eV FWHM bandwidth.
Wavevector Tomography {#wavevector-tomography .unnumbered}
=====================
The condensate wavevector, $k_c$, of the polariton Ising chains is measured using two dimensional Fourier-space tomography utilising a tunable Fabry-Perot etalon with $\sim$ 20 $\mu$eV FWHM bandwidth and a CCD camera imaging configuration. Figure 1(a,b) in Supp. Mat. shows the false-colour normalised photoluminescence intensity of the two dimensional cross-section of the Fourier-space at the energy of the condensate from the Ising chain configuration of Fig.2(b) and Fig.2(c) respectively at condensation threshold. The outer ring in both images corresponds to $k_c$, whereas the inner fringes correspond to self-diffraction from the Ising chain. For the anti-ferromagnetic Ising chain of Fig.2(a), $k_c \approx$1.79$\mu m^{-1}$ and for the ferromagnetic Ising chain of Fig.2(c), $k_c \approx$1.67$\mu m^{-1}$.
![(a) False-colour normalised photoluminescence intensity of the Fourier-space tomography at the energy of the condensate from the Ising chain configuration of Fig.2(b). (b) same as (a) but for the the Ising chain configuration of Fig.2(c). []{data-label="KK_suppmat"}](KK_suppmat){width="16.cm"}
Finding the expression for the coupling coefficients {#finding-the-expression-for-the-coupling-coefficients .unnumbered}
====================================================
The expression for the coupling coefficients $J_{ij}$ given by Eq. (8) can be estimated based on the width of the Hankel transformation of the wavefunction of an individual condensate given by $$\widehat{\Psi}(k)=2\pi\int_0^\infty\sqrt{\rho(r)} \exp[i k_c r]J_0(kr)r\,dr.
\label{A1}$$ The density of the Hankel transformation, $|\widehat{\Psi}(k)|^2$, peaks at $k=k_c$ with the width, characterized by $\epsilon$, inversely proportional to the width of the condensate density $\rho(r)$, which is set by the width of the pumping profile $p(r)$. We, therefore, approximate $|\widehat{\Psi}(k)|^2$ by $$|\widehat{\Psi}(k)|^2\approx |\widehat{\Psi}(k_c)|^2 \frac{{\rm rect}(\frac{k-k_c}{\epsilon})}{\epsilon}.
\label{A2}$$ We integrate Eq. (8) using Eq. (\[A2\]) to get $$\begin{aligned}
J_{ij}&=&\frac{1}{\pi}|\widehat{\Psi}(k_c)|^2\biggr[\biggl(\frac{k_c}{d_{ij}\epsilon}-\frac{1}{2d_{ij}}\biggr) J_1\biggl( \frac{d_{ij}\epsilon}{2}- k_c d_{ij}\biggr) \nonumber \\
&&+ \biggl(\frac{k_c}{d_{ij}\epsilon}+\frac{1}{2d_{ij}}\biggr) J_1 \biggl(\frac{d_{ij}\epsilon}{2}+ k_c d_{ij}) \biggr)\biggl].
\label{jj}
\end{aligned}$$ In the limit of $\epsilon\rightarrow 0$ we recover our $\delta-$function approximation $J_{ij}= k_c \vert \widehat\Psi(k_c) \vert^2 J_0(k_c d_{ij})/\pi$. The finite width of the Hankel transformation of the condensate wavefunction, as seen from Eq. \[jj\], induces a phase shift, so the criterion for the phase switching can be approximated by the sign switching of $\cos(k_c d_{ij}+\phi)$, where $\phi$ is the system parameter dependent term.
Minimization of the XY Hamiltonian for sample configurations {#minimization-of-the-xy-hamiltonian-for-sample-configurations .unnumbered}
============================================================
We find the global minimum of the XY Hamiltonian directly for the sample configurations considered in our paper. For the lattice sites arranged in a square the phases relative to one fixed phase that we set equal to zero, $\theta_0=0$, minimize the XY Hamiltonian $${\cal H}_\square=-J(\cos \theta_{10} + \cos\theta_{12}+\cos\theta_{23}+ \cos\theta_{30}) - J\delta (\cos \theta_{20}+\cos\theta_{13}),$$ where we denoted $\delta$ to be the ratio of the coupling of the diagonal cites to the coupling of the neighboring sites. The coupling strength decays with the distance between sites, therefore, $|\delta|<1$. If all couplings are ferromagnetic, $J, \delta>0$, the minimum of ${\cal H}_\square$ is for $\theta_{i0}=0$. If $J<0$, there is a $\pi$ phase difference between the neighboring sites $\theta_{10} =\pi, \theta_{20}=0, \theta_{30}=\pi$ even for nonzero $\delta$ (as long as $|\delta|<1$ is satisfied).
For a rhombus, consisting of two equilaterial triangles, the XY Hamiltonian becomes $${\cal H}_{rh}=-J(\cos \theta_{10} + \cos\theta_{20}+\cos\theta_{30}+ \cos\theta_{12}+\cos\theta_{23}) - J\delta\cos\theta_{13},
\label{hrhombus}$$ where we associated $\theta_0=0$ with one of the sites along the shorter diagonal. $\delta$ in this case represents the ratio of the coupling along the long diagonal to that between the neighbours. While for an equilaterial triangle the XY Hamiltonian ${\cal H}_\triangle=-J (\cos\theta_{10}+\cos\theta_{20}+\cos\theta_{12})$, $J<0$ is minimized by $\theta_{i0}=\pm 2\pi/3$, the XY Hamiltonian (\[hrhombus\]) is minimized by $\theta_{10}=\theta_{30}=\pi$ , $\theta_{20}=0$ as in the case of the square.
For three equilaterial triangles non-trivial winding around sites is again possible, since the XY Hamiltonian $${\cal H}_5=-J(\cos \theta_{10} + \cos\theta_{20}+\cos\theta_{30}+ \cos\theta_{40}+\cos\theta_{12}+\cos\theta_{23}+\cos\theta_{34})
\label{h5}$$ for $J<0$ reaches its minimum at $\theta_{10}=-\theta_{40}=\pm 0.73\pi, \theta_{20}=-\theta_{30}=\mp0.54\pi$, therefore, creating an alternating winding around each of the equilaterial triangles. Here we associated $\theta_0=0$ with the site that is connected to all other sites and neglected the interactions along two long diagonals. If the distances are close to the switching points between ferro- and antiferro- couplings the small deviation in the position of the sites may lead to an even more complex configurations as is illustrated on Fig. 5 of the main text.
Parameters of the numerical simulations {#parameters-of-the-numerical-simulations .unnumbered}
=======================================
In our numerical simulation we used a Gaussian pumping profile that produces the same width of the condensate as in experiment (FWHM $2.6 \mu m$) and choose the pumping intensity to obtain the correct outflow wavenumber for a single condensate. The common integration parameters used for all numerical simulations are, therefore, $g = 0.1, b = 1, \gamma = 0.3, \ \eta = 0.4, \ p = 9.5 \exp(-0.4 r^2).$ The numerical simulations were performed for various geometries and distances as the main text shows. For Fig. 3 we varied the distances between the nearest neighbors, so that $k_c d = 12.1, 15.3, 18.4, 16.8$ for Figs. 3a,b,c,d respectively. For Fig. 4 we used $k_c d=19, 22.3, 24.9$ for Figs. 4a,b,c respectively. For Fig.5 we used $k_c d = 18.2\pm0.5$.
[00]{} Zajac, J. M., Clarke, E. & Langbein, W. Suppression of cross-hatched polariton disorder in GaAs/AlAs microcavities by strain compensation. [*Appl. Phys. Lett.*]{} [**101**]{}, 041114 (2012).
Houdr´e, R. [*et al.*]{} Saturation of the strong-coupling regime in a semiconductor microcavity: Free-carrier bleaching of cavity polaritons. [*Phys. Rev. B*]{} [**52**]{}, 7810 (1995).
Sanvitto, D. [*et al.*]{} All-optical control of the quantum flow of a polariton condensate. [*Nature Photonics*]{} [**5**]{}, 610 (2011).
Nardin, G. [*et al.*]{} Hydrodynamic nucleation of quantized vortex pairs in a polariton quantum fluid. [*Nature Phys.*]{} [**7**]{}, 635 (2011)
Cilibrizzi, P. [*et al.*]{} Polariton condensation in a strain-compensated planar microcavity with InGaAs quantum wells. [*Appl. Phys. Letts.*]{} [**105**]{}, 191118 (2014).
|
---
abstract: 'The present article is devoted to the generalized Salem functions, the generailed shift operator, and certain related problems. A description of further investigations of the author of this article is given. These investigations (in terms of various representations of real numbers) include the generalized Salem functions and generalizations of the Gauss-Kuzmin problem.'
address: |
45 Shchukina St.\
Vinnytsia\
21012\
Ukraine
author:
- Symon Serbenyuk
title: On certain generalizations of one function and related problems
---
Introduction
============
Nowadays it is well known that functional equations and systems of functional equations are using widely in mathematics and other sciences. Modeling of functions with complicated local structure by systems of functional equations is a shining example of their applications in function theory.
Note that a class of functions with complicated local structure consists of singular (for example, ), continuous nowhere monotonic [@Symon2017; @Symon2019] and nowhere differentiable functions (for example, , etc.).
Now researchers are trying to find simpler examples of singular functions. Interest in such functions is explained by their connection with modeling of real objects, processes, and phenomena (in physics, economics, technology, etc.) and with different areas of mathematics (for example, see [@BK2000; @ACFS2011; @Kruppel2009; @OSS1995; @Sumi2009; @Takayasu1984; @TAS1993]). A brief historical remark on singular functions is given in [@ACFS2017].
One of the simplest examples of singular functions was introduced by Salem. In [@Salem1943], Salem modeled the function $$s(x)=s\left(\Delta^2 _{\alpha_1\alpha_2...\alpha_n...}\right)=\beta_{\alpha_1}+ \sum^{\infty} _{n=2} {\left(\beta_{\alpha_n}\prod^{n-1} _{i=1}{q_i}\right)}=y=\Delta^{Q_2} _{\alpha_1\alpha_2...\alpha_n...},$$ where $q_0>0$, $q_1>0$, and $q_0+q_1=1$. This function is a singular function. However, generalizations of the Salem function can be non-differentiable functions or do not have the derivative on a certain set.
Note that many researches are devoted to the Salem function and its generalizations (for example, see [@ACFS2017; @Kawamura2010; @Symon2015; @Symon2017; @Symon2019] and references in these papers).
Describing the present investigations, a certain generalization of the $q$-ary representation is considered and certain properties of generalized shift operator defined in terms of some of these representations are studied. Also, several related further researhes of the author of this paper are noted. The main attention is given to modelling some generalization of the Salem function by certain systems of functional equations and by using the generalized shift operator.
Some generalizations of $q$-ary expansions of real numbers
===========================================================
Let us consider the following representation introduced by G. Cantor in [@C1869] in 1869.
Let $Q\equiv (q_k)$ be a fixed sequence of positive integers, $q_k>1$, $\Theta_k$ be a sequence of the sets $\Theta_k\equiv\{0,1,\dots ,q_k-1\}$, and $i_k\in\Theta_k$. Then $$\label{eq: series1}
[0,1]\ni x=\Delta^{Q} _{i_1i_2...i_n...}\equiv \frac{i_1}{q_1}+\frac{i_2}{q_1q_2}+\dots +\frac{i_n}{q_1q_2\dots q_n}+\dots,$$ It is easy to see that the last expansion is the $q$-ary expansion $$\label{eq: q-series}
\frac{\alpha_1}{q}+\frac{\alpha_2}{q^2}+\dots+\frac{\alpha_n}{q^n}+\dots \equiv \Delta^q _{\alpha_1\alpha_2...\alpha_n...}$$ of numbers from the closed interval $[0,1]$ whenever the condition $q_k=q$ holds for all positive integers $k$. Here $q$ is a fixed positive integer, $q>1$, and $\alpha_n\in\{0,1,\dots , q-1\}$.
Let us note that certain numbers from $[0,1]$ have two different representations by series , i.e., $$\Delta^Q _{i_1i_2\ldots i_{m-1}i_m000\ldots}=\Delta^Q _{i_1i_2\ldots i_{m-1}[i_m-1][q_{m+1}-1][q_{m+2}-1]\ldots}=\sum^{m} _{k=1}{\frac{i_k}{q_1q_2\dots q_k}}.$$ Such numbers are called *$Q$-rational*. The other numbers in $[0,1]$ are called *$Q$-irrational*.
Let $c_1,c_2,\dots, c_m$ be an ordered tuple of integers such that $c_j\in\{0,1,\dots, q_j-~1\}$ for $j=\overline{1,m}$.
*A cylinder $\Delta^Q _{c_1c_2...c_m}$ of rank $m$ with base $c_1c_2\ldots c_m$* is the following set $$\Delta^Q _{c_1c_2...c_m}\equiv\{x: x=\Delta^Q _{c_1c_2...c_m i_{m+1}i_{m+2}\ldots i_{m+k}\ldots}\}.$$ That is, any cylinder $\Delta^Q _{c_1c_2...c_m}$ is a closed interval of the form $$\left[\Delta^Q _{c_1c_2...c_m000}, \Delta^Q _{c_1c_2...c_m[q_{m+1}-1][q_{m+2}-1][q_{m+3}-1]...}\right].$$
By analogy, in the case of representation , we get $$\Delta^q _{\alpha_1\alpha_2\ldots\alpha_{m-1}\alpha_m000\ldots}=\Delta^Q _{\alpha_1\alpha_2\ldots \alpha_{m-1}[\alpha_m-1][q-1][q-1][q-1]...\ldots}=\sum^{m} _{k=1}{\frac{\alpha_k}{q_1q_2\dots q_k}}.$$ Also, an arbitrary cylinder $\Delta^Q _{c_1c_2...c_m}$ is a closed interval of the form $$\left[\Delta^q _{c_1c_2...c_m000}, \Delta^q _{c_1c_2...c_m[q-1][q-1][q-1]...}\right].$$
Shift operators
===============
In this section, the shift operator is described and the generalized shift operator is studied for the cases of $q$-ary expansions and of expansions of numbers in series .
*The shift operator $\sigma$ of expansion* is the following form $$\sigma(x)=\sigma\left(\Delta^Q _{i_1i_2\ldots i_k\ldots}\right)=\sum^{\infty} _{k=2}{\frac{i_k}{q_2q_3\dots q_k}}=q_1\Delta^{Q} _{0i_2\ldots i_k\ldots}.$$ It is easy to see that $$\begin{split}
\sigma^n(x) &=\sigma^n\left(\Delta^Q _{i_1i_2\ldots i_k\ldots}\right)\\
& =\sum^{\infty} _{k=n+1}{\frac{i_k}{q_{n+1}q_{n+2}\dots q_k}}=q_1\dots q_n\Delta^{Q} _{\underbrace{0\ldots 0}_{n}i_{n+1}i_{n+2}\ldots}.
\end{split}$$ Therefore, $$\label{eq: Cantor series 3}
x=\sum^{n} _{k=1}{\frac{i_k}{q_1q_2\dots q_k}}+\frac{1}{q_1q_2\dots q_n}\sigma^n(x).$$
Note that $$\sigma^n\left(\Delta^q _{\alpha_1\alpha_2\ldots \alpha_k\ldots}\right) =\sum^{\infty} _{k=n+1}{\frac{\alpha_k}{q^{k-n}}}=\Delta^{q} _{\alpha_{n+1}\alpha_{n+2}\ldots}.$$
In [@S.; @Serbenyuk; @alternating; @Cantor; @series; @2013], the notion of the generalized shift operator was introduced in terms of series $$\label{eq: alternating series1}
x=\Delta^{-Q} _{i_1i_2...i_n...}\equiv\frac{i_1}{-q_1}+\frac{i_2}{(-q_1)(-q_2)}+\dots +\frac{i_n}{(-q_1)(-q_2)\dots (-q_n)}+\dots .$$ That is, $$\sigma_m\left(\sum^{\infty} _{k=1}{\frac{(-1)^ki_k}{q_1q_2\cdots q_k}}\right)$$ $$=-\frac{i_1}{q_1}+\frac{i_2}{q_1q_2}-\frac{i_3}{q_1q_2q_3}+\dots + \frac{(-1)^{m-1}i_{m-1}}{q_1q_2\cdots q_{m-1}}+\frac{(-1)^{m}i_{m+1}}{q_1q_2\cdots q_{m-1}q_{m+1}}+\frac{(-1)^{m+1}i_{m+2}}{q_1q_2\cdots q_{m-1}q_{m+2}}+\dots .$$ The idea includes the following: any number from a certain interval can be represented by two fixed sequences $(q_n)$ and $(i_n)$. The generalized shift operator maps the preimage into a number represented by the following two sequences $(q_1,q_2,\dots , q_{m-1}, q_{m+1}, q_{m+2}, \dots )$ and $(i_1,i_2,\dots , i_{m-1}, i_{m+1}, i_{m+2}, \dots )$. In terms of certain encodings of real numbers, this number can belong to another interval.
Let us remark that, in this section, the main attention is given to generalized shift operator defined in terms of series because this series is a generalization of a $q$-ary expansion and models (in the general case) a numeral system with a variable alphabet. Let us note that some numeral system is a numeral system with a variable alphabet whenever there exist at least two numbers $k$ and $l$ such that the condition $A_k\ne A_l$ holds for the representation $\Delta_{i_1i_2...i_n...}$ of numbers in terms of this numeral system, where $i_k\in A_k$ and $i_l\in A_l$, as well as $k\ne l$.
Suppose a number $x\in [0,1]$ represented by series . Then $$\sigma_m(x)=\sum^{m-1} _{k=1}{\frac{i_k}{q_1q_2\cdots q_k}}+\sum^{\infty} _{l=m+1}{\frac{i_l}{q_1q_2\cdots q_{m-1}q_{m+1}\cdots q_l}}.$$ Denote by $\zeta_{m+1}$ the sum $\sum^{\infty} _{l=m+1}{\frac{i_l}{q_1q_2\cdots q_{m-1}q_{m+1}\cdots q_l}}$ and by $\vartheta_{m-1}$ the sum $\sum^{m-1} _{k=1}{\frac{i_k}{q_1q_2\cdots q_k}}$. Then $\zeta_{m+1}=q_m(x-\vartheta_m)$ and $$\label{eq: generalized shift 1}
\sigma_m(x)=q_mx-(q_m-1)\vartheta_{m-1}-\frac{i_m}{q_1q_2\cdots q_{m-1}}.$$
Let us remark that $$\sigma(x)=\sigma_1(x)=\sum^{\infty} _{n=2}{\frac{i_n}{q_2q_3\cdots q_n}}=q_1\Delta^Q _{0i_2i_3...i_n...}$$ and $$\sigma_m(x)=\Delta^Q _{i_1i_2...i_{m-1}000...}+q_m\Delta^Q _{\underbrace{0...0}_{m}i_{m+1}i_{m+2}...}=\Delta^Q _{i_1i_2...i_{m-1}0i_{m+1}i_{m+2}...}+(q_m-1)\Delta^Q _{\underbrace{0...0}_{m}i_{m+1}i_{m+2}...}.$$
In the case of expansion , the generalized shift operator has the following properties:
- $\sigma \circ \sigma^{m} _{2}(x)=\sigma^{m+1} (x)$.
- Suppose $(k_n)$ is a sequence of positive integers such that $k_n=k_{n-1}+1$, $n=2,3, \dots$. Then $$\sigma^{k_1+1}\circ \sigma_{k_n}\circ \sigma_{k_{n-1}}\circ \ldots \circ \sigma_{k_1}(x)=\sigma^{k_n}(x).$$
- Suppose $(k_n)$ is an arbitrary finite subsequence of positive integers. Then $$\sigma^{k_n-n}\circ \sigma_{k_n}\circ \sigma_{k_{n-1}}\circ \ldots \circ \sigma_{k_1}(x)=\sigma^{k_n}(x).$$
- The mapping $\sigma_m$ is continuous at each point of the interval $\left(\inf\Delta^Q _{c_1c_2...c_m}, \sup\Delta^Q _{c_1c_2...c_m}\right)$. The endpoints of $\Delta^Q _{c_1c_2...c_m}$ are points of discontinuity of the mapping.
- The mapping $\sigma_m$ has a derivative almost everywhere (with respect to the Lebesgue measure). If the mapping has a derivative at the point $x=\Delta^Q _{\varepsilon_1\varepsilon_2...\varepsilon_k...}$, then $\left(\sigma_m\right)^{'}=q_m$.
- $$x-\sigma_m(x)=\frac{i_m}{q_1q_2\cdots q_m}+\frac{\sigma^m(x)}{q_1q_2\cdots q_m}(1-q_m).$$
All properties follow from the definition of $\sigma_m$ and equality .
Let us consider a cylinder $\Delta^Q _{c_1c_2...c_n}$. It is easy to see that $\lim_{x\to x_0}{\sigma_m(x)}=x_0$ holds for any Q-irrational point from $\Delta^Q _{c_1c_2...c_n}$ and all Q-rational points whenever $m\ne n$. If $m=n$, then $$\lim_{x\to x_0+0}{\sigma_m(x)}=\sigma_m(x^{(1)} _0)=\sigma_m\left(\Delta^Q _{i_1i_2...i_{n-1}i_n000...}\right),$$ $$\lim_{x\to x_0-0}{\sigma_m(x)}=\sigma_m(x^{(2)} _0)=\sigma_m\left(\Delta^Q _{i_1i_2...i_{n-1}[i_n-1][q_{n+1}-1][q_{n+2}-1]...}\right),$$ and $$\sigma_m(x^{(1)} _0)-\sigma_m(x^{(2)} _0)=-\frac{1}{q_1q_2\cdots q_{m-1}}.$$ In addition, $$x-\sigma_m(x)=\vartheta_m+\frac{\sigma^m (x)}{q_1q_2\cdots q_m}-\vartheta_{m-1}-\zeta_{m+1}=\frac{i_m}{q_1q_2\cdots q_m}+\frac{\sigma^m(x)}{q_1q_2\cdots q_m}(1-q_m).$$
Let us consider expansion . In this case, $$\sigma_m(x)=\sigma_m\left(\sum^{\infty} _{n=1}{\frac{(-1)^ni_n}{q_1q_2\cdots q_n}}\right)=\sum^{m-1} _{k=1}{\frac{(-1)^ki_k}{q_1q_2\cdots q_k}}+\sum^{\infty} _{j=m+1}{\frac{(-1)^{j-1}i_j}{q_1q_2\cdots q_{m-1}q_{m+1}\cdots q_j}}$$ $$=-q_mx+(1+q_m)\sum^{m-1} _{k=1}{\frac{(-1)^ki_k}{q_1q_2\cdots q_k}}+\frac{(-1)^mi_m}{q_1q_2\cdots q_{m-1}}.$$ so, $\sigma_m$ is a piecewise linear function since $\sum^{m} _{k=1}{\frac{(-1)^kc_k}{q_1q_2\cdots q_k}}$ is constant for the set $\Delta^Q _{c_1c_2...c_m}$.
Let us note that, in the case of $q$-ary expansions of real numbers, properties of the generalized shift operator are similar with properties of the generalized shift operator for expansions . Really, $$\label{eq: generalized shift q-ary}
\sigma_m\left(\Delta^q _{\alpha_1\alpha_2...\alpha_n...}\right)=qx-\frac{\alpha_m}{q^{m-1}}-(q-1)\sum^{m-1} _{k=1}{\frac{\alpha_k}{q^k}}.$$ However, $\sigma_m\left(\Delta^q _{\alpha_1\alpha_2...\alpha_n...}\right)=\Delta^q _{\alpha_1\alpha_2...\alpha_{m-1}\alpha_{m+1}...}$.
In the paper [@preprint2019], the generalized shift operator is investigated more detail. In the next articles of the author of this paper, the notion of the generalized shift operator will be investigated in more detail and applyed by the author of the present article in terms of various representation of real numbers (e.g., positive and alternating Cantor series and their generalizations, as well as Luroth, Engel series, etc., various continued fractions).
Let us consider certain applications of the generalized shift operator. One can model generalizations of the Gauss-Kuzmin problem and generalizations of the Salem function.
Generalizations of the Gauss-Kuzmin problem
===========================================
This problem is one of the first and still one of the most important results in the metrical theory of continued fractions [@Lasku]. The problem was formulated by the Gauss and the first solution was received by Kuzmin [@Kuzmin]. The problem is investigated by a number of researchers for different types of continued fractions (for example, see and references in the papers).
The Gauss-Kuzmin problem is to calculate the limit $$\lim_{n\to\infty}{\lambda\left(E_n(x)\right)},$$ where $\lambda(\cdot)$ is the Lebesgue measure of a set and the set $E_n(x)$ is a set of the form $$E_n=\left\{z: \sigma^n (z)<x\right\}.$$ Here $z=\Delta _{i_1i_2...i_k...}$, i.e., $\Delta _{i_1i_2...i_k...}$ is a certain representation of real numbers, $\sigma$ is the shift operator.
Ganeralizations of the Gauss-Kuzmin problem are to calculate the limit $$\lim_{k\to\infty}{\lambda\left(\tilde E^{} _{n_k}(x)\right)},$$ for sets of the following forms:
- $$\tilde E^{} _{n_k}(x)=\left\{z: \sigma_{n_k}\circ \sigma_{n_{k-1}}\circ \ldots \circ \sigma_{n_1}(z)<x\right\}$$ including (here $(n_k)$ is a certain fixed sequence of positive integers) the cases when $(n_k)$ is a constant sequence.
- the set $\tilde E^{} _{n_k}(x)$ under the condition that $n_k=\psi(k)$, where $\psi$ is a certain function of the positive integer argument.
- $$\tilde E^{} _{n_k}(x)=\left\{z: \underbrace{\sigma_{n_k}\circ \sigma_{n_{k-1}}\circ \ldots \circ \sigma_{n_1}}_{\varphi(m,k,c)}(z)<x\right\},$$ where $\varphi$ is a certain function and $m,c$ are some parameters (if applicable). That is, for example, $$\tilde E^{} _{n_k}(x)=\left\{z: \underbrace{\sigma_{m}\circ \sigma_{m}\circ \ldots \circ \sigma_{m}}_{k}(z)<x\right\},$$ where $k>c$ and $c$ is a fixed positive integer, or $$\tilde E^{} _{n_k}(x)=\left\{z: \underbrace{\sigma_{m}\circ \sigma_{m}\circ \ldots \circ \sigma_{m}}_{k}(z)<x\right\},$$ where $k \equiv 1 (\mod c) $ and $c>1$ is a fixed positive integer.
- In the general case, $$\tilde E^{} _{n_k}(x)=\left\{z: \underbrace{\sigma_{\psi(\varphi(m,k,c))}\circ \ldots \circ \sigma_{\psi(1)}}_{\varphi(m,k,c)}(z)<x\right\},$$
In addition, one can formulate such problems in terms of the shift operator. For example, one can formulate the Gauss-Kuzmin problem for the following sets: $$\tilde E^{} _{n_k}(z)=\left\{z: \sigma^{n_k}(z)<\sigma^{k_0}(z)\right\},$$ where $k_0$, $(n_k)$ are a fixed number and a fixed sequence. $$\tilde E^{} _{n_k}(x)=\left\{z: \sigma^{n_k}(z)<\sigma^{k_0}(x)\right\}.$$ $$\tilde E^{} _{n}(x)=\left\{z: \sigma^{\psi(n)}(z)<x\right\},$$ where $\psi(n)$ is a certain function of the positive integer argument.
In addition, $$\tilde E^{} _{n}(z)=\left\{z: \sigma^{\psi(n)}(z)<\sigma^{\varphi(n)}(z)\right\},$$ $$\tilde E^{} _{n}(x)=\left\{z: \sigma^{\psi(n)}(z)<\sigma^{\varphi(n)}(x)\right\},$$ where $\psi, \varphi$ are certain functions of the positive integer arguments.
It is easy to see that similar problems can be formulated for the case of the generalized shift operator.
In next articles of the author of the present article, such problems will be considered by the author of this article in terms of various numeral systems (with a finite or infinite alphabet, with a constant or variable alphabet, positive, alternating, and sign-variable expansions, etc.).
Generalizations of the Salem function
=====================================
Let us consider certain functions whose argument represented by the $q$-ary expansion.
Suppose $(n_k)$ is a fixed sequence of positive integers such that $n_i\ne n_j$ for $i\ne j$ and such that for any $n\in\mathbb N$ there exists a number $k_0$ for which the condition $n_{k_0}=n$ holds.
Let $P_q=\{p_0,p_1,\dots , p_{q-1}\}$ be a fixed tuple of real numbers such that $p_i\in (-1,1)$, where $i=\overline{0,q-1}$, $\sum_i {p_i}=1$, and $0=\beta_0<\beta_i=\sum^{i-1} _{j=0}{p_j}<1$ for all $i\ne 0$. Then the finite system of functional equations $$\label{eq: system-q}
f\left(\sigma_{n_{k-1}}\circ \sigma_{n_{k-2}}\circ \ldots \circ \sigma_{n_1}(x)\right)=\beta_{\alpha_{n_k}}+p_{\alpha_{n_k},}f\left(\sigma_{n_{k}}\circ \sigma_{n_{k-1}}\circ \ldots \circ \sigma_{n_1}(x)\right),$$ where $x=\Delta^q _{\alpha_1\alpha_2...\alpha_k...}$, has the unique solution $$g(x)=\beta_{\alpha_{n_1}}+\sum^{\infty} _{k=2}{\left(\beta_{\alpha_{n_k}}\prod^{k-1} _{j=1}{p_{\alpha_{n_j}}}\right)}$$ in the class of determined and bounded on $[0, 1]$ functions.
Since the function $g$ is a determined on $[0,1]$ function, using system , we get $$g(x)=\beta_{\alpha_{n_1}}+p_{\alpha_{n_1}}g(\sigma_{n_1}(x))$$ $$=\beta_{\alpha_{n_1}}+p_{\alpha_{n_1}}(\beta_{\alpha_{n_2}}+p_{\alpha_{n_2}}g(\sigma_{n_2}\circ\sigma_{n_1}(x)))=\dots$$ $$\dots =\beta_{\alpha_{n_1}}+\beta_{\alpha_{n_2}}p_{\alpha_{n_1}}+\beta_{\alpha_{n_3}}p_{\alpha_{n_1}}p_{\alpha_{n_2}}+\dots +\beta_{\alpha_{n_k}}\prod^{k-1} _{j=1}{p_{\alpha_{n_j}}}+\left(\prod^{k} _{t=1}{p_{\alpha_{n_t}}}\right)g(\sigma_{n_k}\circ \dots \circ \sigma_{n_2}\circ \sigma_{n_1}(x)).$$
So, $$g(x)=\beta_{\alpha_{n_1}}+\sum^{\infty} _{k=2}{\left(\beta_{\alpha_{n_k}}\prod^{k-1} _{j=1}{p_{\alpha_{n_j}}}\right)}$$ since $g$ is a determined and bounded on $[0,1]$ function and $$\lim_{k\to\infty}{g(\sigma_{n_k}\circ \dots \circ \sigma_{n_2}\circ \sigma_{n_1}(x))\prod^{k} _{t=1}{p_{\alpha_{n_t}}}}=0,$$ where $$\prod^{k} _{t=1}{p_{\alpha_{n_t}}}\le \left( \max_{0\le i\le q-1}{p_i}\right)^k\to 0, ~~~ k\to \infty.$$
The following properties hold:
- The function $g$ is continuous at any $q$-irrational point of $[0,1]$.
- The function $g$ is continuous at $q$-rational point $$x_0=\Delta^{q} _{\alpha_1\alpha_2...\alpha_{m-1}\alpha_m 000...}=\Delta^{q} _{\alpha_1\alpha_2...\alpha_{m-1}[\alpha_m-1] [q-1][q-1][q-1]...}$$ whenever a sequence $(n_k)$ is such that the conditions $k_0=\max\{k: n_k \in \{1,2,\dots, m\}\}$ and $n_{k_0}=m$ hold. In the other case, a $q$-rational point $x_0$ is a point of discontinuity.
- The set of all points of discontinuities of the function $g$ is a countable, finite, or empty set. It depends on a sequence $(n_k)$.
Let us note that a certain fixed function $g$ is given by a fixed sequence $(n_k)$ described above. One can write our mapping by the following: $$g: x=\Delta^q _{\alpha_1\alpha_2...\alpha_k...}\to ~\beta_{\alpha_{n_1}}+\sum^{\infty} _{k=2}{\left(\beta_{\alpha_{n_k}}\prod^{k-1} _{l=1}{p_{\alpha_{n_l}}}\right)}=\Delta^{g(x)} _{\alpha_{n_1}\alpha_{n_2}...\alpha_{n_k}...}=g(x)=y.$$
Let $x_0=\Delta^q _{\alpha_1\alpha_2...\alpha_k...}$ be an arbitrary $q$-irrational number from $[0,1]$. Let $x=\Delta^q _{\gamma_1\gamma_2...\gamma_k...}$ be a $q$-irrational number such that the condition $\gamma_{n_j}=\alpha_{n_j}$ holds for all $j=\overline{1,k_0}$, where $k_0$ is a certain positive integer. That is, $$x=\Delta^q _{\gamma_1...\gamma_{n_1-1}\alpha_{n_1}\gamma_{n_1+1}...\gamma_{n_2-1}\alpha_{n_2}...\gamma_{(n_{(k_0-1)}+1)}...\gamma_{(n_{k_0}-1)}\alpha_{n_{k_0}}\gamma_{n_{k_0}+1}...\gamma_{n_{k_0}+k}...}, ~k=1,2,\dots .$$ Then $$g(x_0)=\Delta^{g(x)} _{\alpha_{n_1}\alpha_{n_2}...\alpha_{n_{k_0}}\alpha_{n_{k_0+1}}...},$$ $$g(x)=\Delta^{g(x)} _{\alpha_{n_1}\alpha_{n_2}...\alpha_{n_{k_0}}\gamma_{n_{k_0+1}}...\gamma_{n_{k_0}+k}...}.$$ Since $0\le g(x)\le 1$, we have $g(x)-g(x_0)=$ $$=\left(\prod^{k_0} _{j=1}{p_{\alpha_{n_j}}}\right) \left(\beta_{\gamma_{n_{k_0+1}}}+\sum^{\infty} _{t=2}{\left(\beta_{\gamma_{n_{k_0+t}}}\prod^{k_0+t-1} _{r=k_0+1}{p_{\gamma_{n_r}}}\right)}-\beta_{\alpha_{n_{k_0+1}}}-\sum^{\infty} _{t=2}{\left(\beta_{\alpha_{n_{k_0+t}}}\prod^{k_0+t-1} _{r=k_0+1}{p_{\alpha_{n_r}}}\right)}\right)$$ $$=\left(\prod^{k_0} _{j=1}{p_{\alpha_{n_j}}}\right)\left(g(\sigma_{n_{k_0}}\circ\ldots \sigma_{n_2} \circ \sigma_{n_1}(x))-g(\sigma_{n_{k_0}}\circ\ldots \sigma_{n_2} \circ \sigma_{n_1}(x_0))\right),$$ and $$|g(x)-g(x_0)|\le \prod^{k_0} _{j=1}{p_{\alpha_{n_j}}}\le \left(\max\{p_0,\dots , p_{q-1}\}\right)^{k_0}\to 0 ~~~~~~~(k_0\to\infty).$$
So, $\lim_{x\to x_0}{g(x)}=g(x_0)$, i.e., the function $g$ is continuous at any $q$-irrational point.
Let $x_0$ be a $q$-rational number, i.e., $$x_0=x^{(1)} _0=\Delta^{q} _{\alpha_1\alpha_2...\alpha_{m-1}\alpha_m 000...}=\Delta^{q} _{\alpha_1\alpha_2...\alpha_{m-1}[\alpha_m-1] [q-1][q-1][q-1]...}=x^{(2)} _0.$$ Then there exist positive integers $k^{*}$ and $k_0$ such that $$y_1=g\left(x^{(1)} _0\right)=\Delta^{g(x)} _{\alpha_{n_1}\alpha_{n_2}...\alpha_{n_{k^{*}}}...\alpha_{n_{k_0}}000...},$$ $$y_2=g\left(x^{(2)} _0\right)=\Delta^{g(x)} _{\alpha_{n_1}\alpha_{n_2}...\alpha_{n_{k^{*}-1}}[\alpha_{n_{k^{*}}}-1]\alpha_{n_{k^{*}+1}}...\alpha_{n_{k_0}}[q-1][q-1][q-1]...}.$$ Here $n_{k^{*}}=m$, $n_{k^{*}}\le n_{k_0}$, and $k_0$ is a number such that $\alpha_{n_{k_0}}\in\{\alpha_1, \dots, \alpha_{m-1}, \alpha_m\}$ and ${k_0}$ is the maximum position of any number from $\{1,2,\dots , m\}$ in the sequence $(n_k)$.
Let us consider a fact (Section 2 in [@Symon2019] and the paper [@Salem1943], since such expansion of numbers is an analytic representation the Salem function) that a representation $\Delta^{g(x)} _{\alpha_{1}\alpha_{2}...\alpha_{k}...}$ is the following whenever the conditions $(n_k)=(k)$ and $p_j>0$ for all $j=\overline{0,q-1}$, where $k=1,2, \dots $, hold: $$\label{eq: Pq}
[0,1]\ni x=\Delta^{P_q} _{\alpha_{1}\alpha_{2}...\alpha_{k}...}=\beta_{\alpha_1}+\sum^{\infty} _{k=2}{\left(\beta_{\alpha_1}\prod^{k-1} _{l=1}{p_{\alpha_l}}\right)}$$ This representation is the $q$-ary representation whenever the condition $$0< p_0=p_1=\dots=p_{q-1}=\frac{1}{q}$$ holds. Also, certain numbers have two different such representations, and the rest of the numbers have the unique such representation. That is, $$z_1=\Delta^{P_q} _{\alpha_1\alpha_2...\alpha_{m-1}\alpha_m 000...}=\Delta^{P_q} _{\alpha_1\alpha_2...\alpha_{m-1}[\alpha_m-1] [q-1][q-1][q-1]...}=z_2.$$ Really, $$z_1-z_2=\beta_{\alpha_1}+\sum^{m} _{k=2}{\left(\beta_{\alpha_1}\prod^{k-1} _{l=1}{p_{\alpha_l}}\right)}-\beta_{\alpha_1}-\sum^{m-1} _{k=2}{\left(\beta_{\alpha_1}\prod^{k-1} _{l=1}{p_{\alpha_l}}\right)}-\beta_{\alpha_m-1}\prod^{m-1} _{l=1}{p_{\alpha_l}}-p_{\alpha_m-1}\prod^{m-1} _{l=1}{p_{\alpha_l}}=0.$$ Since the Salem function is a strictly increasing function, conditions for holding $x_1<x_2$ or $x_1>x_2$ are identical in terms of the $q$-ary representation and of representation .
Using the case of a $q$-ary irrational number, let us consider the limits $$\lim_{x\to x_0+0}{g(x)}=\lim_{x\to x^{(1)} _0}{g(x)}=g(x^{(1)} _0)=y_1,~~~\lim_{x\to x_0-0}{g(x)}=\lim_{x\to x^{(2)} _0}{g(x)}=g(x^{(2)} _0)=y_2.$$
Whence $y_1=y_2$ whenever a sequence $(n_k)$ is such that the conditions $n_{k_0}=m$ and $k_0=\max\{k: n_k \in \{1,2,\dots, m\}\}$ hold.
So, the set of all points of discontinuities of the function $g$ is a countable, finite, or empty set. It depends on a sequence $(n_k)$.
Suppose $(n_k)$ is a fixed sequence and $c_{n_1}, c_{n_2}, \dots , c_{n_r}$ is a fixed tuple of numbers $c_{n_j}\in\{0,1,\dots , q-1\}$, where $j=\overline{1,r}$ and $r$ is a fixed positive integer.
Let us consider the following set $$\mathbb S_{q, (c_{n_r})}\equiv \left\{x: x=\Delta^q _{\alpha_1\alpha_2...\alpha_{n_1-1}\overline{c_{n_1}}\alpha_{n_1+1}...\alpha_{n_2-1}\overline{c_{n_2}}...\alpha_{n_{r}-1}\overline{c_{n_r}}\alpha_{n_r+1}...\alpha_{n_r+k}...}\right\},$$ where $k=1,2,\dots $, $\overline{c_{n_j}}\in\{c_{n_1}, c_{n_2}, \dots , c_{n_r}\}$ for all $j=\overline{1,r}$. This set has non-zero Lebesgue measure (for example, similar sets are investigated in terms of other representations of numbers in [@S.; @Serbenyuk; @alternating; @Cantor; @series; @2013]). It is easy to see that $\mathbb S_{q, (c_{n_r})}$ maps to $$g\left(\mathbb S_{q, (c_{n_r})}\right)\equiv\left\{y: y=\Delta^{g(x)} _{c_{n_1} c_{n_2}\dots c_{n_r}\alpha_{n_{r+1}}...\alpha_{n_{r+k}}...}\right\}$$ under $g$.
For a value $\mu_g \left(\mathbb S_{q, (c_{n_r})}\right)$ of the increment, the following is true. $$\mu_g \left(\mathbb S_{q, (c_{n_r})}\right)=g\left(\sup\mathbb S_{q, (c_{n_r})}\right)-g\left(\inf\mathbb S_{q, (c_{n_r})}\right)=\Delta^{g(x)} _{c_{n_1} c_{n_2}\dots c_{n_r}[q-1][q-1][q-1]...}-\Delta^{g(x)} _{c_{n_1} c_{n_2}\dots c_{n_r}000...}=\prod^{r} _{j=1}{p_{c_{n_j}}}.$$ Let us note that one can consider the intervals $\left[\inf\mathbb S_{q, (c_{n_r})}, \sup\mathbb S_{q, (c_{n_r})}\right]$. Then $\sup\mathbb S_{q, (c_{n_r})}-\inf\mathbb S_{q, (c_{n_r})}=$ $$=\Delta^q _{\underbrace{[q-1][q-1]...[q-1]}_{n_1-1}\overline{c_{n_1}}\underbrace{[q-1][q-1]...[q-1]}_{n_2-1}\overline{c_{n_2}}...\underbrace{[q-1][q-1]...[q-1]}_{n_r-1}\overline{c_{n_r}}(q-1)}$$ $$-\Delta^q _{\underbrace{00...0}_{n_1-1}\overline{c_{n_1}}\underbrace{[00...0}_{n_2-1}\overline{c_{n_2}}...\underbrace{00...0}_{n_r-1}\overline{c_{n_r}}(0)}=1-\sum^{r} _{j=1}{\frac{q-1}{q^{n_j}}}$$ and $$\label{eq: increment}
\mu_g \left(\mathbb S_{q, (c_{n_r})}\right)=\mu_g \left(\left[\inf\mathbb S_{q, (c_{n_r})}, \sup\mathbb S_{q, (c_{n_r})}\right]\right)=\prod^{r} _{j=1}{p_{c_{n_j}}}.$$ So, one can formulate the following statements.
The function $g$ has the following properties:
1. If $p_j\ge 0$ or $p_j>0$ for all $j=\overline{0,q-1}$, then:
- $g$ does not have intervals of monotonicity on $[0,1]$ whenever the condition $n_k=k$ holds for no more than a finite number of values of $k$;
- $g$ has at least one interval of monotonicity on $[0,1]$ whenever the condition $n_k\ne k$ holds for a finite number of values of $k$;
- $g$ is a monotonic non-decreasing function (in the case when $p_j\ge 0$ for all $j=\overline{0,q-1}$) or is a strictly increasing function (in the case when $p_j> 0$ for all $j=\overline{0,q-1}$) whenever the condition $n_k=k$ holds for $k\in\mathbb N$.
2. If there exists $p_j=0$, where $j=\overline{0,q-1}$, then $g$ is a constant almost everywhere on $[0,1]$.
3. If there exists $p_j<0$ (other $p_j$ are positive), where $j=\overline{0,q-1}$, and the condition $n_k=k$ holds for almost all $k\in\mathbb N$, then $g$ does not have intervals of monotonicity on $[0,1]$.
Let us note that the last statements follow from .
Let us consider a cylinder $\Delta^q _{c_1c_2...c_n}$. We obtain $\mu_g \left(\Delta^q _{c_1c_2...c_n}\right)=$ $$=\Delta^{g(x)} _{\underbrace{[q-1][q-1]...[q-1]}_{e_1}\overline{c_{1}}\underbrace{[q-1][q-1]...[q-1]}_{e_2}\overline{c_{2}}...\underbrace{[q-1][q-1]...[q-1]}_{e_n}\overline{c_{n}}(q-1)}$$ $$-\Delta^{g(x)} _{\underbrace{00...0}_{e_1}\overline{c_{1}}\underbrace{[00...0}_{e_2}\overline{c_{2}}...\underbrace{00...0}_{e_n}\overline{c_{n}}(0)},$$ where $\overline{c_{n}}\in\{c_1,c_2,\dots , c_n\}$ and $(e_n)$ is a certain sequence of numbers from $\mathbb N \cup\{0\}$.
So, differential properties of $g$ depend on a sequence $(n_k)$ and the set of numbers $P_q=\{p_0,p_1,\dots , p_{q-1}\}$.
The function $g$ can be a singular or non-differentiable function. It depends on a sequence $(n_k)$ and $P_q=\{p_0,p_1,\dots , p_{q-1}\}$.
Differential properties including special partial cases will be considered in the next articles of the author of this paper since the technique of proofs introduced by Salem in [@Salem1943] is not suitable for proving statements in our general case.
In addition, let us note the following.
Let $\eta$ be a random variable defined by the following form $$\eta= \Delta^{q} _{\xi_{n_1}\xi_{n_2}...\xi_{n_k}...},$$ where $k=1,2,3,\dots $, the digits $\xi_{n_k}$ are random and taking the values $0,1,\dots ,q-1$ with probabilities ${p}_{0}, {p}_{1}, \dots , {p}_{q-1}$. That is $\xi_n$ are independent and $P\{\xi_{n_k}=\alpha_{n_k}\}=p_{\alpha_{n_k}}$, $\alpha_{n_k}\in\{0,1,\dots q-1\}$. Here $(n_k)$ is a sequence of positive integers such that $n_i\ne n_j$ for $i\ne j$ and such that for any $n\in\mathbb N$ there exists a number $k_0$ for which the condition $n_{k_0}=n$ holds.
The distribution function $\tilde{F}_{\eta}$ of the random variable $\eta$ can be represented by $$\tilde{F}_{\eta}(x)=\begin{cases}
0,&\text{ $x< 0$}\\
\beta_{\alpha_{n_1}(x)}+\sum^{\infty} _{k=2} {\left(\tilde{\beta}_{\alpha_{n_k}(x)} \prod^{k-1} _{r=1} {\tilde{p}_{\alpha_{n_k}(x)}}\right)},&\text{ $0 \le x<1$}\\
1,&\text{ $x\ge 1$,}
\end{cases}$$ where $x=\Delta^{q} _{\alpha_{n_1}\alpha_{n_2}...\alpha_{n_k}...}$.
A method of the corresponding proof is described in [@Symon2017].
The Lebesgue integral of the function $g$ can be calculated by the formula $$\int^1 _0 {g(x)dx}=\frac{1}{q-1}\sum^{q-1} _{j=0}{\beta_j}.$$
By $A$ denote the sum $\sum^{q-1} _{j=0}{\beta_j}$ and by $B$ denote the sum $\sum^{q-1} _{j=0}{p_j}$. Since $$x=\frac{1}{q}\sigma_m(x)+(q-1)\sum^{m-1} _{k=1}{\frac{\alpha_k}{q^k}}+\frac{\alpha_m}{q^{m-1}}$$ and $$dx=\frac{1}{q}d(\sigma_m(x)),$$ we have $$\int^1 _0 {g(x)dx}=\sum^{q-1} _{j=0}{\int^{\frac{j+1}{q}} _{\frac{j}{q}} {g(x)dx}}=\sum^{q-1} _{j=0}{\int^{\frac{j+1}{q}} _{\frac{j}{q}} {\left(\beta_j+p_jg(\sigma_{n_1}(x))\right)dx}}$$ $$=\frac{1}{q}\sum^{q-1} _{j=0}{\beta_j}+\frac{1}{q}\left(\sum^{q-1} _{j=0}{p_j}\right)\int^1 _0 {g(\sigma_{n_1}(x))d(\sigma_{n_1}(x))}$$ $$=\frac{1}{q}\sum^{q-1} _{j=0}{\beta_j}+\frac{1}{q}\left(\sum^{q-1} _{j=0}{p_j}\right)\left(\sum^{q-1} _{j=0}{\int^{\frac{j+1}{q}} _{\frac{j}{q}} {\left(\beta_j+p_jg(\sigma_{n_2}\circ \sigma_{n_1}(x))\right)d(\sigma_{n_1}(x))}}\right)$$ $$=A+B\left(A+B\int^1 _0 {g(\sigma_{n_2}\circ \sigma_{n_1}(x)))d(\sigma_{n_2}\circ \sigma_{n_1}(x))}\right)$$ $$=A+AB+B^2\left(\sum^{q-1} _{j=0}{\int^{\frac{j+1}{q}} _{\frac{j}{q}} {\left(\beta_j+p_jg(\sigma_{n_3}\circ\sigma_{n_2}\circ \sigma_{n_1}(x))\right)d(\sigma_{n_2}\circ\sigma_{n_1}(x))}}\right)$$ $$=A+AB+B^2\left(A+B\int^1 _0 {g(\sigma_{n_3}\circ\sigma_{n_2}\circ \sigma_{n_1}(x)))d(\sigma_{n_3}\circ\sigma_{n_2}\circ \sigma_{n_1}(x))}\right)$$ $$=A+AB+AB^2+B^3\left(A+B\int^1 _0 {g(\sigma_{n_4}\circ\sigma_{n_3}\circ\sigma_{n_2}\circ \sigma_{n_1}(x)))d(\sigma_{n_4}\circ\sigma_{n_3}\circ\sigma_{n_2}\circ \sigma_{n_1}(x))}\right)=\dots$$ $$\dots = A+AB+\dots +AB^{k-1}+B^k\left(A+B\int^1 _0 {g(\sigma_{n_{k+1}}\circ\sigma_{n_k}\circ\ldots \circ \sigma_{n_1}(x)))d(\sigma_{n_{k+1}}\circ\sigma_{n_k}\circ \ldots \circ \sigma_{n_1}(x))}\right).$$
Since $$B^{k+1}=\left(\frac{1}{q}\sum^{q-1} _{j=0}{p_j}\right)^{k+1}=\left(\frac{1}{q}\right)^{k+1} \to 0 ~\text{as}~ k\to\infty,$$ we obtain $$\int^1 _0{g(x)dx}=\lim_{k\to\infty}{\left(\sum^{k} _{t=0}{AB^t}+B^{k+1}\int^1 _0 {g(\sigma_{n_{k+1}}\circ\sigma_{n_k}\circ\ldots \circ \sigma_{n_1}(x)))d(\sigma_{n_{k+1}}\circ\sigma_{n_k}\circ \ldots \circ \sigma_{n_1}(x))}\right)}$$ $$=\sum^{\infty} _{k=0}{AB^k}=\left(\sum^{q-1} _{j=0}{\beta_j}\right)\left(\sum^{\infty} _{k=0}{\frac{1}{q^{k+1}}}\right)=\frac{1}{q-1}\sum^{q-1} _{j=0}{\beta_j}.$$
In the next articles of the author of this paper, generalizations and properties of solutions of system of functional equations will be investigated for the cases of various numeral systems (with a finite or infinite alphabet, with a constant or variable alphabet, positive, alternating, and sign-variable expansions, etc.).
It can be interesting to consider the case when $(n_k)$ is an arbitrary fixed sequence (finite or infinite) of positive integers. Then the function $g$ can be a constant function, a linear function, or a function having pathological (complicated) structure, etc. It depends on $(n_k)$. Such problems will be investigated in the next papers of the author of this article.
[9]{}
E. de Amo, M.D. Carrillo and J. Fernández-Sánchez, On duality of aggregation operators and k-negations, *Fuzzy Sets and Systems*, **181** (2011), 14–27.
E. de Amo, M.D. Carrillo and J. Fernández-Sánchez, A Salem generalized function, *Acta Math. Hungar.* **151** (2017), no. 2, 361–378. https://doi.org/10.1007/s10474-017-0690-x
L. Berg and M. Kruppel, De Rham’s singular function and related functions, *Z. Anal. Anwendungen.*, **19**(2000), no. 1, 227–237.
K. A. Bush, Continuous functions without derivatives, *Amer. Math. Monthly* **59** (1952), 222–225.
, [Ueber die einfachen Zahlensysteme,]{} [*Z. Math. Phys.*]{} [**14**]{} (1869), 121–128. (German)
S. Ito and T. Sadahiro, Beta-expansions with negative bases *Integers* **9** (2009), 239-259.
Sofia Kalpazidou, On a problem of Gauss-Kuzmin type for continued fraction with odd partial quotients, *Pacific J. Math.* **123** (1986), no. 1, 103–114. https://projecteuclid.org/euclid.pjm/1102701402
Kiko Kawamura, The derivative of Lebesgue’s singular function, *Real Analysis Exchange* Summer Symposium 2010, pp. 83–85.
M. Kruppel, De Rham’s singular function, its partial derivatives with respect to the parameter and binary digital sums, *Rostock. Math. Kolloq.* **64** (2009), 57–74.
R.O. Kuzmin, On a problem of Gauss, *Dokl. Akad. Nauk SSSR Ser. A* (1928) 375-380. \[Russian; French version in Atti Congr. Internaz.Mat. (Bologna, 1928), Tomo VI (1932) 83-89. Zanichelli, Bologna\].
Dan Lascu, A Gauss–Kuzmin-type problem for a family of continued fraction expansions, *Journal of Number Theory* **133** (2013), no. 7, 2153–2181. https://doi.org/10.1016/j.jnt.2012.12.007
Dan Lascu, A Gauss-Kuzmin Theorem for Continued Fractions Associated with Nonpositive Integer Powers of an Integer $m\ge 2$, *The Scientific World Journal* **2014** (2014), Article ID 984650, 8 pages. http://dx.doi.org/10.1155/2014/984650
H. Minkowski, Zur Geometrie der Zahlen. In: Minkowski, H. (ed.) Gesammeine Abhandlungen, Band 2, pp. 50–51. Druck und Verlag von B. G. Teubner, Leipzig und Berlin (1911)
T. Okada, T. Sekiguchi, and Y. Shiota, An explicit formula of the exponential sums of digital sums, *Japan J. Indust. Appl. Math.* **12** (1995), 425–438.
A. Rényi, Representations for real numbers and their ergodic properties, *Acta. Math. Acad. Sci. Hungar.* **8** (1957), 477–493.
, [On some singular monotonic functions which are stricly increasing,]{} [*Trans. Amer. Math. Soc.*]{} [**53**]{} (1943), 423–439.
, [Functions, that defined by functional equations systems in terms of Cantor series representation of numbers,]{} [*Naukovi Zapysky NaUKMA*]{} [**165**]{} (2015), 34–40. (Ukrainian), available at https://www.researchgate.net/publication/292606546
, [Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers,]{} [*Journal of Mathematical Physics, Analysis, Geometry*]{} [**13**]{} (2017), No. 1, 57–81. https://doi.org/10.15407/mag13.01.057
, [Nega-$\tilde Q$-representation as a generalization of certain alternating representations of real numbers,]{} [*Bull. Taras Shevchenko Natl. Univ. Kyiv Math. Mech.*]{} [**1 (35)**]{} (2016), 32–39. (Ukrainian), available at https://www.researchgate.net/publication/308273000
S. Serbenyuk, On one class of functions with complicated local structure, *Šiauliai Mathematical Seminar* **11 (19)** (2016), 75–88.
, Representation of real numbers by the alternating Cantor series, [*Integers*]{} [**17**]{} (2017), Paper No. A15, 27 pp.
S. Serbenyuk, On one fractal property of the Minkowski function, *Revista de la Real Academia de Ciencias Exactas, F' isicas y Naturales. Serie A. Matem' aticas* **112** (2018), no. 2, 555–559, doi:10.1007/s13398-017-0396-5
*Non-Differentiable functions defined in terms of classical representations of real numbers*, Zh. Mat. Fiz. Anal. Geom. **14** (2018), no. 2, 197–213. https://doi.org/10.15407/mag14.02.197
Symon Serbenyuk, Representation of real numbers by the alternating Cantor series, slides of talk (2013) (Ukrainian). Available from: https://www.researchgate.net/publication/303720347
Symon Serbenyuk, Representation of real numbers by the alternating Cantor series, preprint (2013) (Ukrainian). Available from: https://www.researchgate.net/publication/316787375
*Serbenyuk S.* On some generalizations of real numbers representations, arXiv:1602.07929v1 (in Ukrainian)
Symon Serbenyuk, Generalizations of certain representations of real numbers, arXiv:1801.10540v3, 8 pp.
Symon Serbenyuk, On one application of infinite systems of functional equations in function theory, *Tatra Mountains Mathematical Publications* **74** (2019), 117-144. https://doi.org/10.2478/tmmp-2019-0024
Symon Serbenyuk, Generalized shift operator of certain encodings of real numbers, arXiv:1911.12140v1, 6 pp.
H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane, *Sugaku* **61** (2009), no. 2, 133–161.
H. Takayasu, Physical models of fractal functions, *Japan J. Appl. Math.* **1** (1984), 201–205.
S. Tasaki, I. Antoniou, and Z. Suchanecki, Deterministic diffusion, De Rham equation and fractal eigenvectors, *Physics Letter A* **179** (1993), no. 1, 97–102.
T. Zamfirescu, Most monotone functions are singular, *Amer. Math. Mon.* **88** (1981), 47–49.
|
---
abstract: 'Recent experimental progress in the fields of cold quantum gases and ultrafast optical spectroscopy of quantum materials allows to controllably induce and probe non-adiabatic dynamics of superconductors and superfluids. The time-evolution of the gap function before relaxation with the lattice is determined by the superposition of coherently evolving individual Cooper pairs within the manifold of the Bardeen-Cooper-Schrieffer (BCS) wavefunction. While dynamics following an abrupt quench of the pairing interaction strength in the single-band BCS model has been exactly solved due to the integrability of the model, the dynamics of post-quench multi-band superconductors remain under scrutiny. Here, we develop a generalization of the Volkov-Kogan Laplace-space perturbative method that allows us to determine the non-adiabatic gap dynamics of two-band fully gapped superconductors for a wide range of quench amplitudes. Our approach expands the long-time dynamics around the steady-state asymptotic value of the gap, which is self-consistently determined, rather than around the equilibrium value of the gap. We explicitly demonstrate that this method recovers the exact solution of the long-time gap dynamics in the single-band case and perfectly agrees with a numerical solution of the two-band model. We discover that dephasing of Cooper pairs from different bands leads to faster collisionless relaxation of the gap oscillation with a power-law of $t^{-3/2}$ instead of the well-known $t^{-1/2}$ behavior found in the single-band case. Furthermore, the gap oscillations display beating patterns arising from the existence of two different asymptotic gap values. Our results have important implications to a variety of two-band superconductors driven out of equilibrium, such as iron-based superconductors, MgB$_{2}$, and SrTiO$_{3}$.'
author:
- Tianbai Cui
- Michael Schütt
- 'Peter P. Orth'
- 'Rafael M. Fernandes'
bibliography:
- 'Biblio.bib'
title: 'Post-quench gap dynamics of two-band superconductors'
---
Introduction {#sec:introduction}
============
Superconductors that are perturbed into a state away from equilibrium display an extremely rich and interesting dynamical behavior. This originates from the interplay between the dynamics of its fermionic quasi-particle excitations and that of the superconducting order parameter, as expressed, for example, in the superconducting gap equation. Close to equilibrium, various collective modes emerge such as the Anderson-Bogoliubov phase mode [@andersonRandomPhaseApproximationTheory1958] and the longitudinal Schmid (or Higgs) amplitude mode [@schmidTimeDependentGinzburgLandau1966; @pekkerAmplitudeHiggsModes2015], which describe phase and amplitude fluctuations of the order parameter. The transverse Carlson-Goldman mode describes the coupled oscillations of normal currents and supercurrents [@carlsonPropagatingOrderParameterCollective1975; @Pals-PhysRep-1989], whereas in multi-gap superconductors additional Leggett phase modes appear [@leggettNumberPhaseFluctuationsTwoBand1966], corresponding to oscillations of the relative phases of the different gaps. Interesting dynamics also occurs farther away from equilibrium, where one observes, for example, intriguing non-linear behaviors such as dynamic instabilities towards slowly damped [@Volkov1974; @Galperin_Kozub_Spivak-JETP-1981] or even undamped order parameter oscillations [@Barankov-Rabi-Oscillations; @Yuzbashyan_Altshuler_Enolskii-PRB-2005; @Yuzbashyan-2006-PRL; @Barankov-Synchronization].
Generally, the dynamic response of a superconductor depends on the type of perturbation that is applied, for example, whether it is adiabatic or non-adiabatic, linear or non-linear, and whether it is charge neutral or charged. It also depends on the hierarchy of a number of important timescales such as the quasi-particle energy relaxation time $\tau_{\varepsilon}$, the dynamical scale of the superconducting order parameter $\tau_{\Delta}$, the timescale of the external perturbation $\tau_{\text{pert}}$ and the characteristic observation time $t$ [@aronovBoltzmannequationDescriptionTransport1981; @langenbergNonequilibriumSuperconductivity1986; @kopninTheoryNonequilibriumSuperconductivity2001]. Here, we are interested in the case of $\tau_{\Delta}\ll\tau_{\varepsilon}$ and in fast, non-adiabatic perturbation occurring on a timescale $\tau_{\text{pert}}\ll\tau_{\Delta}\approx t$. This non-adiabatic, collisionless regime has been explored in a linearized approach close to equilibrium in the seminal work by Volkov and Kogan [@Volkov1974], who studied the gap dynamics of a single-band superconductor following a small and instantaneous perturbation. They found coherent gap oscillations that are only algebraically damped $\propto t^{-1/2}$, analogous to Landau damping in a collisionless plasma [@landauVibrationsElectronicPlasma1946; @Kamenev-NonEqFieldTheory-Book]. More recently, experimental progress on two distinct fronts have brought renewed interest to this field: (1) Ultrafast optical studies in the Terahertz regime have unveiled non-adiabatic, coherent gap dynamics in thin superconducting films [@shimanoHiggsModeSuperconductors2019], for example, in NbN [@Shimano-PRL-2012; @Shimano-PRL-2013; @Shimano-Science-2014; @PhysRevLett.107.177007] and Nb$_{3}$Sn [@Yang-NatureMaterials-2018; @Cui2018; @Yang-NaturePhotonics-2019]; (2) Cold-atom realizations of superfluids and Bose-Einstein condensates have provided a fruitful avenue to induce non-adiabatic dynamics by performing rapid parameter changes such as quenching the pairing interaction strength [@langenUltracoldAtomsOut2015; @bloch:885].
The situation of a rapid parameter quench is theoretically particularly interesting as it is amenable to analytical approaches. Going beyond the linear analysis of Volkov and Kogan and exploiting the integrability of the Bardeen-Cooper-Schrieffer (BCS) Hamiltonian [@Richardson-NuclPhys-1964; @gaudinBetheWavefunction2014; @Dukelsky-RMP-2004], a number of works have explored post-quench non-adiabatic dynamics of single-band BCS superconductors far-away from equilibrium [@Barankov-Rabi-Oscillations; @Yuzbashyan_Altshuler_Enolskii-PRB-2005; @Yuzbashyan-2005-JPA; @Barankov-Synchronization; @Yuzbashyan_Dzero_Gurarie_Foster-PRA-2015; @Yuzbashyan_Altshuler-PRB-2005]. It was discovered that non-equilibrium dynamics at times $\tau_{\Delta}\ll t\ll\tau_{\varepsilon}$ fall into one of three distinct classes (or “phases”) [@Barankov-Synchronization; @YuzbashyanDzero-PRL-2006], which can be topologically distinguished by the number of complex roots of the spectral polynomial [@Yuzbashyan_Altshuler_Enolskii-PRB-2005; @Yuzbashyan-2006-PRL]: Phase I, where the gap decays exponentially to zero; Phase II, where the gap oscillates with frequency $2\Delta_{\infty}$ and decays algebraically $\propto t^{-1/2}$ to a finite value $\Delta_{\infty}$; and phase III, where persistent undamped gap oscillations occur. Phase II in the non-equilibrium quench phase diagram [@Barankov-Synchronization; @YuzbashyanDzero-PRL-2006] contains the linear regime around equilibrium studied by Volkov and Kogan [@Volkov1974]. Finally, we note that the topological classification explains why terahertz induced gap dynamics is qualitatively similar to the case of a parameter quench, as has been observed in various numerical studies [@Chou_Foster-PRB-2017; @Papenkort-PRB-2007; @Papenkort_JPhys-2009; @krullSignaturesNonadiabaticBCS2014a; @Eremin-2band; @Cui2018].
In this paper, we extend these previous studies by addressing numerically and analytically the gap dynamics of two-band superconductors following an interaction quench. Our motivation is on the fact that multi-band superconductivity is realized in a variety of materials with conventional and unconventional pairing mechanisms. Primary examples are MgB$_{2}$ [@budkoSuperconductivityMagnesiumDiboride2015], the iron-based superconductors [@Chubukov-PRB-2008], Sr$_{2}$RuO$_{4}$ [@mackenzieSuperconductivityMathrmSrMathrmRuO2003], heavy fermions [@wirthExploringHeavyFermions2016], strontium titanate [@Fernandes13; @Trevisan18], and oxide heterostructures such as LaAlO$_{3}$/SrTiO$_{3}$ [@Scheurer-NatComm-2015]. Unconventional multi-orbital superfluidity has also been reported in cold-atom setups on the honeycomb lattice [@soltan-panahiQuantumPhaseTransition2012a]. While different superconducting gap symmetries are possible in the presence of multiple Fermi surfaces, we will focus on the simplest case of $s$-wave superconductivity. As the quench dynamics is identical for $s^{+-}$ and $s^{++}$ pairing, corresponding to gaps with opposite or same signs on the two Fermi surfaces, respectively, our results apply to both cases. Quenches in two-band $s$-wave superconductors have so far only been studied numerically [@Eremin-2band; @Krull2016-Leggett-Higgs; @Dzero-PRB-2015], focusing on the coupling between the Higgs and the Leggett mode [@Krull2016-Leggett-Higgs] or the competition between superconductivity and spin-density wave order [@Dzero-PRB-2015]. Generalizations to quenches between other pairing symmetries such as time-reversal symmetry breaking $s+is$ or $s+id$ pairing are interesting avenues for further work. Indeed, a recent numerical study of Terahertz induced gap dynamics for $s+is$ pairing has revealed an unusual coupling between the Higgs amplitude and the Leggett (relative) phase mode [@Mueller-PRB-2018].
Exact solutions of the time-dependent two-band BCS model only exist for special fine tuned values of the intra- and inter-band interaction parameters, where the problem effectively reduces to the single-band case (see below and Ref. ). It is an open question whether the generic two-band BCS model is integrable. Here, we develop a generalization of the Volkov-Kogan Laplace-space analysis in order to investigate the non-adiabatic post-quench dynamics in generic two-band BCS models. Like Volkov and Kogan we solve linearized equations of motion in Laplace space, but an important distinction of our work is that we expand around the long-time steady-state of the system instead of the equilibrium state. This allows us to explore the gap dynamics away from the weak-quench limit in a larger region of the non-equilibrium phase diagram. We achieve this methodological advancement by self-consistently solving for the steady-state value of the superconducting gap $\Delta_{\infty}$. We show in detail that our method reproduces the exact solution in phase II of the single-band model. For the two-band model we carefully check our analytical results by comparing to the numerical solution of the dynamics. We find that the oscillatory gap dynamics exhibits pronounced beating behavior due to the presence of two asymptotic gap values $\Delta_{1,\infty}$ and $\Delta_{2,\infty}$, which has been previously reported in a numerical investigation of Terahertz driven gap oscillations in two-band superconductors [@Eremin-2band]. A central new result of our work is that the decay of the gap oscillations due to Landau damping in two-band superconductors is governed by a power-law $\propto t^{-3/2}$ that is different from the one found in the single band case, where it is $\propto t^{-1/2}$ (see Fig. \[fig:summary\]). Earlier numerical studies of multi-gap superconductors have reported power-law decays of $t^{-1/2}$, although in that case the dynamics was driven not a by an interaction quench, but by laser pulses [@Eremin-2band]. Interestingly, faster than $t^{-1/2}$ decay was also seen in the case of superconducting nanowires, where electronic subbands arise due to confinement [@Zachmann_NJPhys-2013]. Finally, a similar $t^{-3/2}$ decay of the pairing amplitude has been found in quenches into the strong pairing (Bose-Einstein condensation (BEC)) regime in three dimensions, but by a different microscopic mechanism [@Gurarie-BEC-t-three-halves; @Yuzbashyan_Dzero_Gurarie_Foster-PRA-2015].
The remainder of the paper is organized as follows: in Sec. \[sec:model\], we define the two-band BCS model and formulate it in terms of Anderson pseudospins. We then derive equations of motion of the pseudospins that govern the non-adiabatic dynamics of individual Cooper pairs and the gap following an instantaneous quench of the BCS coupling strength. In Sec. \[sec:numerical\_results\], we present numerical solutions of the gap dynamics in the regime of weak quenches, which show the main features of oscillatory beating and algebraic decay $\propto t^{-3/2}$. In Sec. \[sec:long\_time\_gap\_dynamics\], we present our main analytical calculation and flesh out the details of our method to find the long-time dynamics of the gap using a self-consistent Laplace analysis. In Sec. \[subsec:linearized\_eom\], we derive linearized equations of motion around the long-time steady-state. We present the solution of these equations in Laplace space in Sec. \[subsec:sol\_laplace\], which depends on the steady-state values of the gap $\Delta_{\alpha,\infty}$ and the pseudo-spins $S_{\alpha,\infty}^{i}$. These values are determined in Sec. \[subsec:gap\_from\_self\_consistency\] by solving self-consistent equations via an ansatz for the non-equilibrium distribution function in the steady-state. We first show that our method yields the exact solution in the single-band model, and then apply it to the two-band case, where only numerical solutions are available. Finally, in Sec. \[subsec:gap\_oscillations\], we discuss the long-time gap dynamics in real-time by performing an inverse Laplace transformation. We explicitly show how the new power-law decay exponent emerges from a distinct analytical structure of the gap in Laplace space and demonstrate how one re-obtains the single-band result. We conclude in Sec. \[sec:conclusions\], and present additional details of our analytical calculations in the Appendices.
![Summary of our main results for the gap dynamics of a quenched two-band superconductor. In these figures, only inter-band pairing is included. (panel A) When the densities of states of the two bands are the same, $\eta\equiv\frac{\mathcal{N}_{1}}{\mathcal{N}_{2}}=1$, the behavior is the same as that of a single-band model. (panel B) When $\eta\protect\neq1$, the behavior is different in that the damping of the gap oscillations changes from $t^{-1/2}$ to $t^{-3/2}$ and a beating pattern occurs due to the existence of two oscillation frequencies (inset). In this figure, $\Delta_{1}$ is the gap of band $1$ and $\Delta_{1f}$ is the quenched value of the gap. The parameters used here were $v_{i}=0.19$, $v_{f}=0.2$ for both panel A and B.\[fig:summary\]](eta_dependence_new){width="1\columnwidth"}
BCS model and quench protocol {#sec:model}
=============================
Pseudospin formalism for equilibrium two-band superconductors {#subsec:pseudospin_formalism}
-------------------------------------------------------------
We start from the reduced BCS Hamiltonian [@BCS] for two-band superconductors $$\begin{aligned}
H_{\text{BCS}} & = & \sum_{\mathbf{k},\sigma,\alpha}\varepsilon_{\mathbf{k},\alpha}c_{\mathbf{k},\sigma,\alpha}^{\dagger}c_{\mathbf{k},\sigma,\alpha}\nonumber \\
& & +\frac{1}{N}\sum_{\mathbf{k},\mathbf{p},\alpha,\beta}V_{\alpha\beta}c_{\mathbf{k},\uparrow,\alpha}^{\dagger}c_{-\mathbf{k},\downarrow,\alpha}^{\dagger}c_{-\mathbf{p},\downarrow,\beta}c_{\mathbf{p},\uparrow,\beta}\end{aligned}$$ where $\alpha,\,\beta\in\left\{ 1,\,2\right\} $ are the band indices, $\varepsilon_{\mathbf{k},\alpha}$ is the electronic dispersion near the Fermi level in band $\alpha$ (including the chemical potential), and $V_{\alpha\beta}$ is the effective pairing interaction between band $\alpha$ and band $\beta$. Although not important in the following, one may assume parabolic dispersions, $\varepsilon_{\mathbf{k},\alpha}=\mathbf{k}^{2}/2m_{\alpha}-\mu$. The interaction constants $V_{\alpha\beta}$ are positive (negative) if the interaction is repulsive (attractive). In multi-band systems, different bands develop different values of the superconducting gap, depending on the values of the intra-band interactions, $V_{11}$ and $V_{22}$, and the inter-band interactions, $V_{12}$ and $V_{21}$ (see Fig. \[Fig\_FS\_PS\] (A)) as well as the density of states of the two bands at the Fermi level, $\mathcal{N}_{\alpha}$. We assume that the two bands have the same intra-band electronic interactions such that $V_{11}=V_{22}\equiv U$; by definition, $V_{12}=V_{21}\equiv V$. Due to the different density of states $\mathcal{N}_{1}\neq\mathcal{N}_{2}$, electrons in different bands experience different effective interaction strengths. The BCS gap equation is therefore band-dependent: $$\Delta_{\alpha}=\Delta_{\alpha}'+i\Delta_{\alpha}''=-\frac{1}{N}\sum_{\mathbf{p},\beta}V_{\alpha\beta}\left\langle c_{-\mathbf{p},\downarrow,\beta}c_{\mathbf{p},\uparrow,\beta}\right\rangle$$ Going from summation over momenta to integrations over energy using the density of states, we write the equilibrium BCS gap equations explicitly in matrix form in the band-space. $$\left(\begin{array}{c}
\Delta_{1}\\
\Delta_{2}
\end{array}\right)=\hat{\gamma}v\left(\begin{array}{c}
\int_{-\Lambda}^{\Lambda}d\varepsilon\frac{\Delta_{1}}{2E_{1}}\tanh\left(\frac{E_{1}}{2T}\right)\\
\int_{-\Lambda}^{\Lambda}d\varepsilon\frac{\Delta_{2}}{2E_{2}}\tanh\left(\frac{E_{2}}{2T}\right)
\end{array}\right)$$ where $\Lambda$ is a high-energy cutoff and $$\hat{\gamma}=\left(\begin{array}{cc}
r & -\eta\\
-1 & r\eta
\end{array}\right)\label{eq:gamma_matrix}$$ with $\eta=\mathcal{N}_{2}/\mathcal{N}_{1}$ being the ratio of the density of states of the two bands, $E_{\alpha}=\sqrt{\varepsilon^{2}+\Delta_{\alpha}^{2}}$ is the Bogoliubov quasiparticle dispersion in band $\alpha$ and $T$ is the temperature of the system. In the following, we restrict our analysis to the $T=0$ ground state as the initial pre-quench state of the system. We have also defined the dimensionless inter-band interaction coupling constant $v=V\mathcal{N}_{1}$, and the dimensionless ratio $r=-U/V$ between intra-band and inter-band interactions. Here, we include the minus sign in the definition, as we will assume that $U<0$ is negative, corresponding to attractive intra-band interaction.
Note that the ratio of the density of states in the two bands, $\eta=\mathcal{N}_{2}/\mathcal{N}_{1}$, determines the relative sizes of the superconducting gaps of the two bands. If the two bands have the same density of states near the Fermi energy, i.e. $\eta=1$, the matrix $\hat{\gamma}$ becomes symmetric. Therefore, the gap equations are solved by $\Delta_{1}=-\Delta_{2}$ for repulsive inter-band interaction ($v>0$), corresponding to $s^{+-}$ pairing, and $\Delta_{1}=\Delta_{2}$ for attractive inter-band interaction ($v<0$), corresponding to $s^{++}$ pairing. In this paper, we will focus on the case with $\eta\neq1$, in which case the amplitude of the two gaps is different in equilibrium $|\Delta_{1}|\neq|\Delta_{2}|$ and the multi-band nature of the system has a pronounced imprint on the non-equilibrium dynamics of the superconducting gap.
![(A) Schematics of the two bands and the interactions between them. (B) Schematics of the mapping between the electronic operators and the pseudo-spin operators.[]{data-label="Fig_FS_PS"}](Fig_2_Bands+Mapping){width="1\columnwidth"}
It is convenient to use the pseudospin formalism [@Anderson-RPA-BCS] to study the non-equilibrium dynamics of the superconducting state. In the mean-field approach, which is exact in the BCS regime we consider here, the BCS Hamiltonian can be described by pseudospins exposed to an effective magnetic field: $$H_{\text{BCS}}=-\sum_{\mathbf{k},\alpha}\mathbf{B}_{\mathbf{k},\alpha}\cdot\hat{\mathbf{S}}_{\mathbf{k},\alpha}+\text{const.}\label{PS_BCS}$$ where $\mathbf{B}_{\mathbf{k},\alpha}=2\left(\Delta_{\alpha}^{'},-\Delta_{\alpha}^{''},-\varepsilon_{\mathbf{k},\alpha}\right)$and $$\begin{aligned}
\hat{S}_{\mathbf{k},\alpha}^{-} & =c_{-\mathbf{k},\downarrow,\alpha}c_{\mathbf{k},\uparrow,\alpha}\\
\hat{S}_{\mathbf{k},\alpha}^{+} & =c_{\mathbf{k},\uparrow,\alpha}^{\dagger}c_{-\mathbf{k},\downarrow,\alpha}^{\dagger}\\
\hat{S}_{\mathbf{k},\alpha}^{z} & =\frac{1}{2}\left(c_{\mathbf{k},\uparrow,\alpha}^{\dagger}c_{\mathbf{k},\uparrow,\alpha}+c_{-\mathbf{k},\downarrow,\alpha}^{\dagger}c_{-\mathbf{k},\downarrow,\alpha}-1\right)\end{aligned}$$ The constant term contributes to the condensation energy, which will be ignored because it is not relevant to the dynamics out of equilibrium. The mapping between pseudo-spins and electronic pair operators is summarized in Fig. \[Fig\_FS\_PS\](B). The anti-commutation relation between the electronic operators ensures the spin commutation relation between $\mathbf{\hat{S}}_{\mathbf{k},\alpha}$. Notice that despite the simple form of the pseudospin Hamiltonian, the effective magnetic field is self-consistently determined by the pseudospins collectively via the gap equation: $$\Delta_{\alpha}=-\frac{1}{N}\sum_{\mathbf{k},\beta}V_{\alpha\beta}S_{\mathbf{k},\beta}^{-}\label{gap_eq}$$ where $S_{\mathbf{k},\alpha}^{-}=\left\langle \hat{S}_{\mathbf{k},\alpha}^{-}\right\rangle =\left\langle c_{-\mathbf{k},\downarrow,\alpha}c_{\mathbf{k},\uparrow,\alpha}\right\rangle $.
In equilibrium, the pseudospins are parallel to the effective magnetic field. It is convenient to work in a gauge where both the gaps are real. Then the expectation values of the pseudo-spins at temperature $T$ are given by
$$\begin{aligned}
S_{\mathbf{k},\alpha}^{x} & =\frac{\Delta_{\alpha}}{2E_{\alpha}}\tanh\left(\frac{E_{\alpha}}{2T}\right)\\
S_{\mathbf{k},\alpha}^{y} & =0\\
S_{\mathbf{k},\alpha}^{z} & =\frac{-\varepsilon_{\mathbf{k}}}{2E_{\alpha}}\tanh\left(\frac{E_{\alpha}}{2T}\right)\,.\label{aux}\end{aligned}$$
Note that the length of the pseudospins in equilibrium is determined by the Fermi-Dirac distribution, $n_{\text{F}}$, of the Bogoliubov quasiparticles, i.e. $\left|\mathbf{S}_{\mathbf{k},\alpha}\right|=\frac{1}{2}-n_{\text{F}}$. As mentioned above, we will focus hereafter on initial pre-quench states at zero temperature ($T=0$).
Equations of motion for the pseudospins {#subsec:eom}
---------------------------------------
We consider the situation where the system is driven out of equilibrium by a sudden quench of the pairing interaction. Specifically, we focus on a sudden change of the inter-band coupling $v_{i}\rightarrow v_{f}$ while keeping the ratios between intra- and inter-band interactions, $r=U/V$, and between the densities of states, $\eta$, unchanged, i.e., $r_{i}=r_{f}$ and $\eta_{i}=\eta_{f}$. The subscript $i$ and $f$ denote the initial and final values of the respective dimensionless constants. Note that this requires quenching both intra- and inter-band interactions $U$ and $V$ in such a way to keep their ratio $r$ fixed. We focus on these quench protocols to constrain the parameter space. Generally, one can also consider quenches of $r$, however, this is expected to not lead to qualitative changes to the non-equilibrium dynamics, as it corresponds to a different way to prepare the initial conditions.
If the two bands have different densities of states, i.e. $\eta\neq1$, the quench dynamics is intrinsically different from single-band systems. In the pseudospin formalism, the superconducting gap determines the intrinsic frequency of the pseudospin precession. Therefore, once the two bands have different densities of states, they develop different values of the gap, leading to two distinct intrinsic frequencies. In addition, the gap also serves as the effective magnetic field that drives the precession. Through the inter-band interaction, each band experiences an oscillating magnetic field with the intrinsic frequency of the other band. Hence, the dephasing of the pseudospin oscillations in multi-band systems is fundamentally different from single-band systems. The dynamics is described by two sets of equations of motion for the two bands, which are derived from Eq. (\[PS\_BCS\]) in terms of expectation values of the pseudospins operators, $$\frac{d}{dt}\mathbf{S}_{\mathbf{k},\alpha}\left(t\right)=\mathbf{S}_{\mathbf{k},\alpha}\left(t\right)\times\mathbf{B}_{\mathbf{k},\alpha}\left(t\right)\label{EOM}$$ which are similar to the one-band case, but now with an extra band index $\alpha$. More importantly, the pseudospin dynamics in the two bands are coupled via the gap equations with a time-dependent inter-band coupling strength $v\left(t\right)=v_{i}\theta\left(-t\right)+v_{f}\theta\left(t\right)$:
{width="0.8\paperwidth"}
$$\Delta_{\alpha}\left(t\right)=v\left(t\right)\sum_{\beta}\gamma_{\alpha\beta}\int d\varepsilon S_{\beta}^{-}\left(\varepsilon,t\right)\label{gap_eq_energy_space}$$
The equations of motion for the pseudospins, Eq. (\[EOM\]) and the time-dependent gap equation, Eq. (\[gap\_eq\_energy\_space\]), determine the post-quench gap dynamics of two-band superconductors.
In the following, we first solve these equations numerically and describe our results. Then, we analytically find the long-term asymptotic behavior of the gap oscillations using Laplace transforms. We develop a generalization of the well-known procedure pioneered by Volkov and Kogan in Ref. (see also Refs. [@YuzbashyanDzero-PRL-2006; @Yuzbashyan_Dzero_Gurarie_Foster-PRA-2015]). By expanding around the long-time *non-equilibrium* pseudo-spin steady state, instead of the final equilibrium state, we are able to not only determine the power-law decay of the gap oscillations, but also the steady-state non-equilibrium gap values $\Delta_{\alpha,\infty}$. We also explicitly show how our solution approaches the known single-band result as $\eta\rightarrow1$.
Numerical results for the post-quench gap dynamics {#sec:numerical_results}
==================================================
We solve the equations of motion , together with the gap equation , numerically using the Runge-Kutta method. We focus on the weak-quench limit, to later compare with our analytical expansion. Results for two different ratios of initial and final inter-gap couplings $v_{i}/v_{f}=0.95$ and $0.9$ (with fixed $v_{f}=0.2$) are shown in Fig. \[Fig\_Gap\_Oscillations\]. The other parameters are kept fixed: $r_{i}=r_{f}=0$, $\eta=0.8$, $T_{i}=0$. In equilibrium, this corresponds to the following gap ratios $\Delta_{1,i}/\Delta_{2,i}=-0.8852$ for $v_{i}=0.19$ , $\Delta_{1,i}/\Delta_{2,i}=-0.8857$ for $v_{i}=0.18$ and $\Delta_{1,f}/\Delta_{2,f}=-0.8847$ for $v_{f}=0.2$. The figure contains both the time traces of the gap oscillations as well as their Fourier transforms.
There are two important qualitative features that emerge in the two-band case: first, the gap oscillations are characterized by two frequencies, corresponding to the steady-state values $\Delta_{1,\infty}$ and $\Delta_{2,\infty}$. This leads to pronounced beating when these two frequencies are sufficiently close to each other. This phenomenon has been described previously in numerical studies of two-band (multi-band) superconductors exposed to terahertz laser pulses [@Eremin-2band; @Krull2016-Leggett-Higgs; @Zachmann_NJPhys-2013] Second, the algebraic decay of the gap oscillations ($\propto t^{-\alpha}$) occurs more rapidly than in the single-band case. We numerically determine the exponent to be $\alpha_{\text{2-band}}=3/2$ as opposed to $\alpha_{\text{1-band}}=1/2$.
This behavior seems insensitive to the actual value of $r$. In Fig. \[Fig\_Gap\_Oscillations\_r\], we compare the behavior of $\Delta_{1}(t)$ for the cases in which $r=0.5$ and $r=0$. The other parameters used were $v_{i}=0.19$ and $v_{f}=0.2$ We note that an exponent of $\alpha=3/2$ also emerges if one considers deep quenches into the Bose-Einstein condensate (BEC) regime in a three-dimensional system [@Gurarie-BEC-t-three-halves; @Yuzbashyan_Dzero_Gurarie_Foster-PRA-2015].
![(A) Gap oscillations for the case of inter-band pairing only ($r=0$) and inter-band and intra-band pairing ($r=0.5$). Here, we set $\eta=0.8$. (B) The $t^{-3/2}$ damping of the gap oscillations in a log-log scale for the case $r=0.5$.[]{data-label="Fig_Gap_Oscillations_r"}](Figure4_new){width="1\columnwidth"}
Long-time asymptotic gap dynamics {#sec:long_time_gap_dynamics}
=================================
In order to gain more insights on the transient dynamics of the superconducting gap in two-band systems, it is instructive to have analytic solutions for the superconducting gap evolution. The gap dynamics in single-band conventional superconductors with isotropic gap structures can be solved exactly due to the integrability of the BCS model [@Richardson-NuclPhys-1964; @Dukelsky-RMP-2004; @Yuzbashyan-2005-JPA; @Yuzbashyan_Altshuler-PRB-2005; @Yuzbashyan_Altshuler_Enolskii-PRB-2005; @Yuzbashyan-2006-PRL; @Barankov-Synchronization; @Yuzbashyan_Dzero_Gurarie_Foster-PRA-2015]. The two-band BCS model doubles the number of degrees of freedom compared to the single-band model. Due to the coupling between the two distinct bands, the integrals of motion that were constructed previously for the single-band BCS model [@Yuzbashyan_Altshuler-PRB-2005; @Yuzbashyan_Dzero_Gurarie_Foster-PRA-2015] do not commute between the two bands, except in the symmetric case $\eta=1$. In the single-band case, it was determined that there are three different “phases” depending on the strength of the quench $\Delta_{i}/\Delta_{f}$: in phase I, corresponding to $\Delta_{i}/\Delta_{f}>\mathrm{e}^{\pi/2}$, the gap asymptotically approaches zero in an exponential fashion; in phase II, for $\mathrm{e}^{-\pi/2}<\Delta_{i}/\Delta_{f}<\mathrm{e}^{\pi/2}$, the gap shows damped $t^{-1/2}$ oscillations around one asymptotic value; and in phase III, which takes place for $\Delta_{i}/\Delta_{f}<\mathrm{e}^{-\pi/2}$, the gap shows persistent oscillations between two asymptotic values.
Whether the two-band BCS model is integrable or not is beyond the scope of this work. Given the difficulties in finding the integrals of motion of the two-band case, in this section we employ instead a perturbative method to extract the long-time asymptotic dynamics of the superconducting gap in phase II, where the gap shows damped oscillations. This is precisely the behavior found numerically for weak quenches, shown in Fig. \[Fig\_Gap\_Oscillations\]. In particular, the method we develop here is a modified version of the one pioneered by Volkov and Kogan in Ref. , which allows us to also analytically determine the steady-state gap values $\Delta_{\alpha,\infty}$.
For convenience, we briefly review our notation scheme: subscripts $i$ and $f$ denote the thermal equilibrium value before ($i$) and after ($f$) the quench. The subscript $\infty$ denotes the long-time asymptotic steady-state value of the gap. For example, $\Delta_{\alpha,i}$ ($\Delta_{\alpha,f}$) is the equilibrium value of gap $\alpha$ before (after) the quench, and $\Delta_{\alpha,\infty}$ is its long-time asymptotic steady-state value following the time evolution governed by the BCS Hamiltonian. We note that our analysis is restricted to weak quenches, resulting in the system being in phase II, where the gap experiences Volkov-Kogan-like behavior.
Linearized equations of motion {#subsec:linearized_eom}
------------------------------
To analytically describe the post-quench gap dynamics at long times, we generalize the method used first by Volkov and Kogan in Ref. . Instead of expanding around the final equilibrium state $S_{\alpha,f}^{i}$ and $\Delta_{\alpha,f}$, however, we expand around the long-time non-equilibrium steady-state values $S_{\alpha,\infty}^{i}$ and $\Delta_{\alpha,\infty}$. Importantly, these steady-state values will be determined self-consistently in our calculation using Laplace’s final value theorem. We thus assume that in the long-time limit the superconducting gaps reach their long-time asymptotic values $\Delta_{\alpha,\infty}$. Specifically, we expand the equations of motion and the gap equations around the asymptotic steady-state values
$$\begin{aligned}
S_{\alpha}^{z}\left(\varepsilon,t\right) & =S_{\alpha,\infty}^{z}\left(\varepsilon\right)+g_{\alpha}\left(\varepsilon,t\right)\label{Linearization_Sz}\\
S_{\alpha}^{-}\left(\varepsilon,t\right) & =S_{\alpha,\infty}^{-}\left(\varepsilon\right)+f_{\alpha}\left(\varepsilon,t\right)\\
\Delta_{\alpha}\left(t\right) & =\Delta_{\alpha,\infty}+\delta_{\alpha}\left(t\right)\end{aligned}$$
\[eq:linearization\_around\_ss\_values\]
where, from the stationary condition of the equations of motion, $S_{\alpha,\infty}^{\pm}=S_{\alpha,\infty}^{x}$, $S_{\alpha,\infty}^{y}=0$, $\varepsilon S_{\alpha,\infty}^{x}=-\Delta_{\alpha,\infty}S_{\alpha,\infty}^{z}$. Note that $f_{\alpha}$ describes pairing amplitude fluctuations and $g_{\alpha}$ describes density fluctuations. The deviation of the gap from its long-time asymptotic value is denoted by $\delta_{\alpha}$, which is determined by the pairing-amplitude fluctuations $f_{\alpha}$ via the gap equation:
$$\delta_{\alpha}\left(t\right)=v_{f}\sum_{\beta}\gamma_{\alpha\beta}\int_{-\Lambda}^{\Lambda}d\varepsilon f_{\beta}\left(\varepsilon,t\right)\label{gap_eq_redux}$$
where $\gamma_{\alpha\beta}$ is given in Eq. \[eq:gamma\_matrix\].As we will show below, because $f''_{\alpha}$ is an odd function of $\varepsilon$, $\delta_{\alpha}$ is real, as long as we choose the initial equilibrium gaps of the two bands $\Delta_{\alpha,i}$ to be real. With this in mind, we linearize the equations of motion by inserting Eqs. into Eq. to obtain
$$\begin{aligned}
\dot{f}_{\alpha}' & =2\varepsilon f_{\alpha}''\\
\dot{f}_{\alpha}'' & =-2\varepsilon f_{\alpha}'-2\Delta_{\alpha,\infty}g_{\alpha}-2S_{\alpha,\infty}^{z}\delta_{\alpha}\left(t\right)\\
\dot{g}_{\alpha} & =2\Delta_{\alpha,\infty}f_{\alpha}''\label{Linearization_EOM}\end{aligned}$$
where $f_{\alpha}=f_{\alpha}'+if_{\alpha}''$ and the notation $\dot{f}\equiv\frac{df}{dt}$ is used. Note that, as anticipated, $f_{\alpha}''$ remains an odd function of $\varepsilon$ for all times, since $S_{z,\infty}$ and $g_{\alpha}$ are odd while $f_{\alpha}'$ is even. As a result, the gap remains real for all times. The fact that the phases of the gaps are constants of motion follows directly from the particle-hole symmetry of the BCS Hamiltonian [@Barankov-Synchronization]. Therefore, the relative phase of the two gaps is also a constant of motion and the Leggett (relative phase) mode, which would in any case be overdamped in the regime we study here of inter-band pairing interaction only, is not excited in our quench protocol. In order to excite it, one must break the particle-hole symmetry of the BCS Hamiltonian, for example, by external perturbations as in the pump-probe setups [@Krull2016-Leggett-Higgs].
The linearized equations of motion faithfully describe the long-time dynamics since at the long-time limit, the deviations from the asymptotic values are small, i.e. $\left(g_{\alpha},f_{\alpha},\delta_{\alpha}\right)\ll\left(S_{\alpha,\infty}^{z},S_{\alpha,\infty}^{-},\Delta_{\alpha,\infty}\right)$. To have a better description of the gap dynamics over a wider time range, we focus on relatively weak quenches where $v_{f}/v_{i}$ is close to 1. In this case, the oscillations around $\Delta_{\alpha,\infty}$ are small already at earlier times, allowing for a better comparison between numerics and analytics. Such weak quench regime is also the most relevant to experiments, where excess heating is suppressed.
Since we are interested in $\delta_{\alpha}$, which is only related to $f_{\alpha}$, see Eq. \[gap\_eq\_redux\], we can further simplify the above equations by eliminating $g_{\alpha}$ to find
$$\begin{aligned}
\ddot{f}_{\alpha}'' & =-4E_{\alpha,\infty}^{2}f_{\alpha}''-2S_{\alpha,\infty}^{z}\dot{\delta}_{\alpha}\left(t\right)\label{eq:EOM_Im_f}\\
\dddot{f}_{\alpha}' & =-4E_{\alpha,\infty}^{2}\dot{f}_{\alpha}'-4\varepsilon S_{\alpha,\infty}^{z}\dot{\delta}_{\alpha}\left(t\right)\,,\label{eq:EOM_Re_f}\end{aligned}$$
where $E_{\alpha,\infty}^{2}=\varepsilon^{2}+\Delta_{\alpha,\infty}^{2}$. Eqs. \[eq:EOM\_Im\_f\] and \[eq:EOM\_Re\_f\] describe the dynamics of the imaginary and real parts of the pairing amplitude fluctuations, respectively, which determine the time evolution of the gap.
Solution in Laplace space {#subsec:sol_laplace}
-------------------------
To solve the differential equations and , it is useful to perform a Laplace transformation $y\left(s\right)=\int_{0}^{\infty}y\left(t\right)e^{-st}dt$. We find the the following algebraic equations:
$$\begin{aligned}
f_{\alpha}''\left(s\right)+\frac{2sS_{\alpha,\infty}^{z}}{s^{2}+4E_{\alpha,\infty}^{2}}\delta_{\alpha}\left(s\right) & =\frac{sf_{\alpha,0}''+\dot{f}_{\alpha,0}''}{s^{2}+4E_{\alpha,\infty}^{2}}+\frac{2S_{\alpha,\infty}^{z}}{s^{2}+4E_{\alpha,\infty}^{2}}\delta_{\alpha,0}\label{EOM_Laplace_1}\\
f_{\alpha}'\left(s\right)-\frac{-4\varepsilon S_{\alpha,\infty}^{z}}{s^{2}+4E_{\alpha,\infty}^{2}}\delta_{\alpha}\left(s\right) & =\frac{1}{s}\left[f_{\alpha,0}'-\frac{-4\varepsilon S_{\alpha,\infty}^{z}}{s^{2}+4E_{\alpha,\infty}^{2}}\delta_{\alpha,0}\right]-\frac{2\varepsilon}{s}\frac{sf_{\alpha,0}''+\dot{f}_{\alpha,0}''}{s^{2}+4E_{\alpha,\infty}^{2}}\,.\label{EOM_Laplace_2}\end{aligned}$$
Here, $s$ is the complex frequency in the Laplace domain and the subscript $0$ indicates an initial condition, i.e. $f_{\alpha,0}\equiv f_{\alpha}\left(\varepsilon,t=0^{+}\right)$, $\delta_{\alpha,0}\equiv\delta_{\alpha}\left(t=0^{+}\right)$, etc. Physically, Eqs. and describe the phase and amplitude dynamics of the gap, respectively.
Since $\delta_{\alpha}$ and $f_{\alpha}$ are related through the gap equation \[gap\_eq\_redux\], it is convenient to integrate both sides of the above equations over $\varepsilon$. Then, Eq. is trivially satisfied, since $S_{\alpha,\infty}^{z}$ is an odd function of $\varepsilon$, by virtue of Eq. \[aux\], $f_{\alpha,0}^{''}=0$ by construction, and $\dot{f}_{\alpha,0}''$ is an odd function of $\varepsilon$, by virtue of the second equation of \[Linearization\_EOM\]. Consequently, we are left with a single equation for $f_{\alpha}'$ and $\delta_{\alpha}$, which are related through the gap equation \[gap\_eq\_redux\].
Expressing $f$ in terms of $\delta$, and recasting Eq. in matrix form, the deviations of the superconducting gaps from their asymptotic values, $\delta_{\alpha}$, are given by: $$\left(\hat{\Phi}^{\infty}\left(s\right)+\hat{\mathcal{M}}\right)\vec{\delta}\left(s\right)=\frac{\vec{I}\left(s\right)}{s}\,,\label{gap_Laplace}$$ where the hat (arrow) denote a matrix (vector) in band space. Here, we defined: $$\begin{aligned}
\hat{\Phi}_{\alpha\beta}^{\infty}\left(s\right) & =\mathbb{I}_{\alpha\beta}\left(s^{2}+4\Delta_{\alpha,\infty}^{2}\right)\left\langle \frac{S_{\alpha,\infty}^{x}/\Delta_{\alpha,\infty}}{s^{2}+4E_{\alpha,\infty}^{2}}\right\rangle \\
\hat{\mathcal{M}}_{\alpha\beta} & =\left(\hat{\gamma}^{-1}\right)_{\alpha\beta}-\mathbb{I}_{\alpha\beta}\left\langle \frac{S_{\beta,\infty}^{x}}{\Delta_{\beta,\infty}}\right\rangle \,.\end{aligned}$$ where $\mathbb{I}$ is the identity matrix in band space and the following notation is used: $$\left\langle \ldots\right\rangle =v_{f}\int d\varepsilon\left(\ldots\right)\,.$$
For convenience, we write $\hat{\Phi}_{\alpha\beta}^{\infty}\left(s\right)\equiv\mathbb{I}_{\alpha\beta}\Phi_{\alpha}^{\infty}$ and define:
$$\Phi_{\alpha}^{\infty}(s)=\left(s^{2}+4\Delta_{\alpha,\infty}^{2}\right)\left\langle \frac{S_{\alpha,\infty}^{x}/\Delta_{\alpha,\infty}}{s^{2}+4E_{\alpha,\infty}^{2}}\right\rangle \label{eq_Phi_infty}$$
The function $\vec{I}\left(s\right)$ on the right-hand side is given by (detailed derivation in Appendix \[sec:app\_initial\_conditions\]) $$\begin{aligned}
I_{\alpha}\left(s\right) & =\sum_{\beta}\left(\hat{\gamma}^{-1}\right)_{\alpha\beta}\delta_{\beta,0}+\left(\Delta_{\alpha,i}-\Delta_{\alpha,\infty}\right)\nonumber \\
& \quad\times\left[\Phi_{\alpha}^{i}\left(s\right)-\frac{v_{f}}{v_{i}}\sum_{\beta}\left(\hat{\gamma}^{-1}\right)_{\alpha\beta}\frac{\Delta_{\beta,i}}{\Delta_{\alpha,i}}\right]\,,\label{Initial_Conditions}\end{aligned}$$ with: $$\Phi_{\alpha}^{i}\left(s\right)=\left(s^{2}+4\Delta_{\alpha,\infty}^{2}\right)\left\langle \frac{S_{\alpha,i}^{x}/\Delta_{\alpha,i}}{s^{2}+4E_{\alpha,\infty}^{2}}\right\rangle \label{eq:Phi_i}$$
The solution for $\vec{\delta}(s)$ in Laplace space is then simply given by $$\vec{\delta}\left(s\right)=\left(\hat{\Phi}^{\infty}\left(s\right)+\hat{\mathcal{M}}\right)^{-1}\frac{\vec{I}\left(s\right)}{s}$$
It is clear that without inter-band interaction, $V=0$, $\hat{\mathcal{M}}_{\alpha\beta}$ becomes a diagonal matrix, since $\hat{\gamma}_{\alpha\beta}$ in Eq. \[eq:gamma\_matrix\] is diagonal. As a result, Eq. \[gap\_Laplace\] becomes diagonal in band space as well, and the two-band model reduces to two independent one-band models.
In the following subsections, we will extract the dynamics of the gaps in the long-time limit from their analytic behaviors in Laplace space. These are determined by the functions $\hat{\Phi}^{\infty}\left(s\right)$ and $\vec{I}\left(s\right)$, as they are the only $s$-dependent functions in Eq. . Their $s$-dependence comes from the two functions $\Phi_{\alpha}^{\infty}\left(s\right)$ and $\Phi_{\alpha}^{i}\left(s\right)$ defined above.
The function $\Phi_{\alpha}^{i}\left(s\right)$ is straightforward to calculate since the initial pseudospin configuration is given by the equilibrium value of the gap at $T=0$, i.e. $S_{\alpha,i}^{x}/\Delta_{\alpha,i}=\frac{1}{2\sqrt{\varepsilon^{2}+\Delta_{\alpha,i}^{2}}}$. Inserting this initial pseudo-spin state into Eq. , this can be brought to the form $$\Phi_{\alpha}^{i}\left(s\right)=\Upsilon\left(\tilde{\Delta}_{\alpha,i},\frac{s}{2\Delta_{\alpha,\infty}}\right)\label{phi_initial}$$ where we defined the dimensionless ratio $\tilde{\Delta}_{\alpha,i}=\Delta_{\alpha,i}/\Delta_{\alpha,\infty}$ and the function $$\Upsilon\left(\Delta,x\right)=v_{f}\frac{\sqrt{\frac{x^{2}+1}{\Delta^{2}}}\arccos\left(\sqrt{\frac{x^{2}+1}{\Delta^{2}}}\right)}{\sqrt{1-\frac{1+x^{2}}{\Delta^{2}}}}\label{eq:definition_Upsilon}$$
To find an explicit expression for $\Phi_{\alpha}^{\infty}\left(s\right)$, given by Eq. \[eq\_Phi\_infty\], we first need to compute the function $S_{\alpha,\infty}^{x}/\Delta_{\alpha,\infty}$. The gap equation (see Eq. ), which is satisfied regardless of whether the system is in thermal equilibrium or not, restricts the expectation value of this quantity to: $$\left\langle \frac{S_{\alpha,\infty}^{x}}{\Delta_{\alpha,\infty}}\right\rangle =\sum_{\beta}\left(\hat{\gamma}^{-1}\right)_{\alpha\beta}\frac{\Delta_{\beta,\infty}}{\Delta_{\alpha,\infty}}$$
As we discussed above, the non-zero inter-band interactions render the matrix $\hat{\mathcal{M}}$ off-diagonal and make the two-band model fundamentally different than the single-band case. While a generic discussion of arbitrary inter- and intra-band interactions is possible, the analysis is simplified considerably by focusing on the case of inter-band repulsion only, i.e. $r=0$. Indeed, our numerical results discussed in Fig. \[Fig\_Gap\_Oscillations\_r\] show that the general behavior of the two-band problem is the same for $r=0$ and $r\neq0$. Setting $r=0$ in Eq. yields an off-diagonal matrix $\hat{\gamma}=\left(\begin{array}{cc}
0 & -\eta\\
-1 & 0
\end{array}\right)$. As result, the equation above becomes:
$$\left\langle \frac{S_{1,\infty}^{x}}{\Delta_{2,\infty}}\right\rangle =\eta\left\langle \frac{S_{2,\infty}^{x}}{\Delta_{1,\infty}}\right\rangle =-1\,.\label{eq:constraint_pure_interband}$$
Note that this ratio involves the pseudospin of band $\alpha$ and the gap of the other band $\bar{\alpha}$, where $\bar{\alpha}=1(2)$ for $\alpha=2(1)$. To proceed, we note that, in equilibrium, the same relationship holds between the ratios of the pseudospin and the gap:
$$\left\langle \frac{S_{1,f}^{x}}{\Delta_{2,f}}\right\rangle =\eta\left\langle \frac{S_{2,f}^{x}}{\Delta_{1,f}}\right\rangle =-1\,.\label{eq:constraint_aux}$$
The difference is that, in equilibrium, from Eq. \[aux\], we know precisely the expression for $S_{\alpha,f}^{x}$:
$$\begin{aligned}
\left\langle \frac{S_{1,f}^{x}}{\Delta_{2,f}}\right\rangle & =\left\langle \frac{\Delta_{1,f}/\Delta_{2,f}}{2\sqrt{\varepsilon^{2}+\Delta_{1,f}^{2}}}\right\rangle =-1\\
\eta\left\langle \frac{S_{2,f}^{x}}{\Delta_{1,f}}\right\rangle & =\eta\left\langle \frac{\Delta_{2,f}/\Delta_{1,f}}{2\sqrt{\varepsilon^{2}+\Delta_{2,f}^{2}}}\right\rangle =-1\,,\end{aligned}$$
Based on this similarity, we propose the following ansatz: $$\frac{S_{\alpha,\infty}^{x}}{\Delta_{\alpha,\infty}}=\frac{\tilde{\Delta}_{\alpha,f}}{\tilde{\Delta}_{\bar{\alpha},f}}\left(\frac{1}{2\sqrt{\varepsilon^{2}+\Delta_{\alpha,f}^{2}}}\right)\label{eq:ansatz}$$ where $\tilde{\Delta}_{\alpha,f}=\Delta_{\alpha,f}/\Delta_{\alpha,\infty}$ is defined analogously to $\tilde{\Delta}_{\alpha,i}$. Clearly, this ansatz satisfies the constraint \[eq:constraint\_pure\_interband\]. For $r\neq0$, the constraint will likely have a more complicated form; thus, for the sake of clarity, we focus on the case $r=0.$ We will verify the validity of this ansatz later by an explicit comparison to numerical calculations and by comparison with the exact solution of the single-band case. For now, we proceed with this ansatz and perform the energy integration in the expression of $\Phi_{\alpha}^{\infty}\left(s\right)$. We obtain: $$\Phi_{\alpha}^{\infty}\left(s\right)=\frac{\tilde{\Delta}_{\alpha,f}}{\tilde{\Delta}_{\bar{\alpha},f}}\Upsilon\left(\tilde{\Delta}_{\alpha,f},\frac{s}{2\Delta_{\alpha,\infty}}\right)\label{phi_infinite}$$
Asymptotic gap values {#subsec:gap_from_self_consistency}
---------------------
In this subsection, we show how to extract the long-time asymptotic steady-state gap values $\Delta_{\alpha,\infty}$ self-consistently. To set the stage, and validate the ansatz proposed in the previous subsection, we first present the calculation for the single-band case, comparing the perturbative solution with the exact one.
### Asymptotic gap for the single-band model {#subsubsec:asymptotic_gap_1_band}
In the single-band BCS model with attractive pairing interaction $u\equiv U\mathcal{N}$, a quench suddenly changes the pairing interaction $u_{i}\rightarrow u_{f}$. It is convenient to use $\Delta_{i}/\Delta_{f}$ as the quench parameter, where $\Delta_{i}$ ($\Delta_{f}$) is the equilibrium value of the gap with pairing interaction $u_{i}$ ($u_{f}$). We employ the same linearization scheme for the single-band model as above in Eqs. - for the two-band case, and expand around the long-time asymptotic values, $S_{\infty}^{\alpha}$ and $\Delta_{\infty}$. The equation for the gap deviation $\delta$ in Laplace space, Eq. \[gap\_Laplace\], becomes in the single-band case: $$\delta\left(s\right)=-\left(1-\frac{u_{f}}{u_{i}}\right)\frac{\Delta_{\infty}}{s\Phi_{\infty}\left(s\right)}+\left(\Delta_{i}-\Delta_{\infty}\right)\frac{\Phi_{i}\left(s\right)}{s\Phi_{\infty}\left(s\right)}\label{eq:delta_of_s_single_band}$$ where
$$\begin{aligned}
\Phi_{i}\left(s\right) & =\left\langle \frac{s^{2}+4\Delta_{\infty}^{2}}{\left(s^{2}+4E_{\infty}^{2}\right)}\frac{S_{i}^{x}}{\Delta_{i}}\right\rangle \\
\Phi_{\infty}\left(s\right) & =\left\langle \frac{s^{2}+4\Delta_{\infty}^{2}}{\left(s^{2}+4E_{\infty}^{2}\right)}\frac{S_{\infty}^{x}}{\Delta_{\infty}}\right\rangle \end{aligned}$$
Here, $S_{i}^{x}/\Delta_{i}=1/(2E_{i})$ with $E_{i}=\sqrt{\varepsilon^{2}+\Delta_{i}^{2}}$ is given by its value in the initial $T=0$ ground state state prior to the quench. The ratio $S_{\infty}^{x}/\Delta_{\infty}$, according to our ansatz (\[eq:ansatz\]), becomes in the single-band case:
$$\frac{S_{\infty}^{x}}{\Delta_{\infty}}=\frac{1}{2\sqrt{\varepsilon^{2}+\Delta_{f}^{2}}}\label{single_band_ansatz}$$
This ansatz can be recast in an alternative way as an ansatz for the non-equilibrium distribution function. From the definition of $S_{f}^{x}$, Eq. \[aux\], we have:
$$S_{f}^{x}=\frac{\Delta_{f}\,n_{0}(\varepsilon)}{2\sqrt{\varepsilon^{2}+\Delta_{f}^{2}}}\label{eq:single_band_Sx}$$
where we defined the equilibrium distribution function $n_{0}(\varepsilon)=\tanh(\sqrt{\varepsilon^{2}+\Delta_{f}^{2}}/\left(2T\right)).$ From the gap equation, it follows that $\left\langle \frac{S_{f}^{x}}{\Delta_{f}}\right\rangle =1$. Analogously, we can express $S_{\infty}^{x}$ in terms of the non-equilibrium quasiparticle distribution function $n_{\text{eff}}\left(\varepsilon\right)$:
$$S_{\infty}^{x}=\frac{\Delta_{\infty}n_{\text{eff}}\left(\varepsilon\right)}{2\sqrt{\varepsilon^{2}+\Delta_{\infty}^{2}}}$$
Because the gap equation has to be satisfied also in non-equilibrium, it follows that: $$\left\langle \frac{S_{\infty}^{x}}{\Delta_{\infty}}\right\rangle =\left\langle \frac{n_{\text{eff}}\left(\varepsilon\right)}{2\sqrt{\varepsilon^{2}+\Delta_{\infty}^{2}}}\right\rangle =1\,.\label{eq:gap_single_band}$$
The ansatz \[single\_band\_ansatz\] thus can be recast as an ansatz for the non-equilibrium distribution function: $$n_{\text{eff}}\left(\varepsilon\right)=n_{0}(\varepsilon)\sqrt{\frac{\varepsilon^{2}+\Delta_{\infty}^{2}}{\varepsilon^{2}+\Delta_{f}^{2}}}\,,\label{eq:n_non_eq}$$
Having obtained an explicit expression for $S_{\infty}^{x}/\Delta_{\infty}$, we can derive analytic expressions for $\Phi_{i}\left(s\right)$ and $\Phi_{\infty}\left(s\right)$: $$\Phi_{i/\infty}\left(s\right)=u_{f}\frac{\sqrt{s^{2}+4\Delta_{\infty}^{2}}\arccos\left(\frac{\sqrt{s^{2}+4\Delta_{\infty}^{2}}}{2\left|\Delta_{i/f}\right|}\right)}{\sqrt{4\left(\Delta_{i/f}^{2}-\Delta_{\infty}^{2}\right)-s^{2}}}\label{eq:Phi_single_band_explicit_expression}$$
To find the long-time asymptotic value of the gap, we use the self-consistency condition that $\lim_{t\rightarrow\infty}\Delta\left(t\right)=\Delta_{\infty}$, or equivalently, $\lim_{t\rightarrow\infty}\delta\left(t\right)=0$. Using the final value theorem in Laplace space, this condition becomes $$\lim_{s\rightarrow0}s\delta\left(s\right)=0\,.\label{eq:final_value}$$
Using Eq. and inserting the explicit expressions from Eq. , we find that the asymptotic value of the gap $\Delta_{\infty}$ must satisfy $$\frac{\sqrt{\Delta_{f}^{2}-\Delta_{\infty}^{2}}}{\arccos\left(\frac{\Delta_{\infty}}{\Delta_{f}}\right)}\left[\ln\frac{\Delta_{i}}{\Delta_{f}}-\left(1-\frac{\Delta_{\infty}}{\Delta_{i}}\right)\frac{\arccos\left(\frac{\Delta_{\infty}}{\Delta_{i}}\right)}{\sqrt{1-\frac{\Delta_{\infty}^{2}}{\Delta_{i}^{2}}}}\right]=0\,.\label{eq:asymptotic_gap_equation}$$
It is straightforward to show that this equation is identical to the one that emerges in the exact solution of the single-band BCS gap dynamics using the method of the Lax vector [@Barankov-Synchronization; @Yuzbashyan-2006-PRL]. In Fig. \[fig:Lax\_vs\_Laplace\], we compare the results from both methods, which match perfectly. Interestingly, in phase III (persistent oscillations), our method gives the average value of the gap. Of course, our method formally breaks down in this phase, because the Laplace final value theorem ceases to hold for an oscillatory long-time solution.
This comparison validates the ansatz \[eq:ansatz\] for the single-band case, giving us confidence to apply it to the two-band case as well. Note that the perfect agreement with the exact solution does not necessarily imply that the non-equilibrium distribution function \[eq:n\_non\_eq\] is also exact .
![Comparison of $\Delta_{\infty}$ as a function of the quench parameter $\Delta_{i}/\Delta_{f}$ from our self-consistent perturbative method (Eq. ) and from the exact solution using the Lax vector technique (see Refs. ). The vertical red dashed lines denote the extent of the phase II, as obtained from the roots of the Lax operator. The asymptotic gap vanishes in the phase I (larger values of $\Delta_{i}/\Delta_{f}$) and performs persistent oscillations in the phase III (smaller values of $\Delta_{i}/\Delta_{f}$). Our method correctly yields a vanishing gap in phase I, and provides the average value of the gap in phase III (see Ref. [@Barankov-Synchronization], for example). \[fig:Lax\_vs\_Laplace\]](Delta_inf_1band_Laplace){width="1\linewidth"}
### Asymptotic gap for the two-band model {#subsubsec:asymptotic_gap_2_band}
We now perform the same calculation for the two-band model with pure inter-band repulsion ($r=0$). Using Eqs. and , we obtain the following expression for $\delta_{\alpha}\left(s\right)$ from Eq. \[gap\_Laplace\]: $$\begin{aligned}
s\delta_{\alpha}\left(s\right) & =\left(\Phi_{\bar{\alpha}}^{\infty}\left(s\right)+\frac{1}{\eta_{\bar{\alpha}}}\frac{\Delta_{\alpha,\infty}}{\Delta_{\bar{\alpha},\infty}}\right)\frac{I_{\alpha}\left(s\right)}{D\left(s\right)}+\frac{1}{\eta_{\alpha}}\frac{I_{\bar{\alpha}}\left(s\right)}{D\left(s\right)}\,,\label{sol_Laplace_1}\end{aligned}$$ where, for convenience of notation, we introduced $\eta_{1}=1$ and $\eta_{2}\equiv\eta$, $I_{\alpha}\left(s\right)$ is given by Eq. , and: $$D\left(s\right)=\Phi_{1}^{\infty}\left(s\right)\Phi_{2}^{\infty}\left(s\right)+\frac{\Delta_{2,\infty}}{\Delta_{1,\infty}}\Phi_{2}^{\infty}\left(s\right)+\frac{1}{\eta}\frac{\Delta_{1,\infty}}{\Delta_{2,\infty}}\Phi_{1}^{\infty}\left(s\right)\,.$$
To find the asymptotic long-time value of the gaps $\Delta_{\alpha,\infty}$, we employ once again the final value theorem in Laplace space, Eq. \[eq:final\_value\]. We numerically solve for $\Delta_{1,\infty}$ and $\Delta_{2,\infty}$ for a given quench protocol, $v_{i}\rightarrow v_{f}$, or equivalently $\Delta_{1,i}\rightarrow\Delta_{1,f}$. As shown in Fig. \[fig:Del\_inf\], we find that, in the case of pure inter-band interactions ($r=0$), the ratios between the asymptotic and final equilibrium gaps $\Delta_{\alpha,\infty}/\Delta_{\alpha,f}$ are, to a very good approximation (i.e. with a numerical deviation of less than $0.01\%$), equal for both bands, i.e. $\tilde{\Delta}_{1,f}=\tilde{\Delta}_{2,f}$. They are also identical to the single-band ratio if we adjust the definition of the quench amplitude accordingly, such that the $x$-axis corresponds to $\Delta_{i}/\Delta_{f}$ in the single-band case and to $\Delta_{1,i}/\Delta_{1,f}$ in the two-band case.
Using the result obtained here that $\tilde{\Delta}_{1,f}=\tilde{\Delta}_{2,f}$, the pre-factor of Eq. \[phi\_infinite\] becomes 1. Thus, both $\Phi_{\alpha}^{\infty}\left(s\right)$ and $\Phi_{\alpha}^{i}\left(s\right)$ have the same functional dependence: $\Phi_{\alpha}^{\infty}\left(s\right)=\Upsilon\left(\tilde{\Delta}_{\alpha,f},\frac{s}{2\Delta_{\alpha,\infty}}\right)$, $\Phi_{\alpha}^{i}\left(s\right)=\Upsilon\left(\tilde{\Delta}_{\alpha,i},\frac{s}{2\Delta_{\alpha,\infty}}\right)$.
![Asymptotic values of the gaps in the two band case as a function of the interaction quench parameter $\Delta_{i}/\Delta_{f}$. The dashed gray line is the result for the single-band BCS model. For the two-band model, we use $\Delta_{1,i}/\Delta_{1,f}$ as the quench parameter, and we choose the ratio between the density of states to be $\eta=0.8$. \[fig:Del\_inf\]](Figure6_new){width="1\columnwidth"}
Damped gap oscillations in the long-time limit {#subsec:gap_oscillations}
----------------------------------------------
The long-time behavior of the gap in the time domain $\Delta(t)$ can be obtained by applying the inverse Laplace transformation to Eq. . In order to perform the inverse Laplace transformation, we first need to study the analytical behavior of the solution in Laplace space and find its poles and branch cuts. They are determined by the analytic properties of the function $\Upsilon(\Delta,x)$, defined in Eq. and repeated here for convenience:
$$\Upsilon\left(\Delta,x\right)=v_{f}\frac{\sqrt{\frac{x^{2}+1}{\Delta^{2}}}\arccos\left(\sqrt{\frac{x^{2}+1}{\Delta^{2}}}\right)}{\sqrt{1-\frac{1+x^{2}}{\Delta^{2}}}}$$
The reason why only the analytical properties of $\Upsilon(\Delta,x)$ matter is because we can express both $\Phi_{\alpha}^{i}(s)$ and $\Phi_{\alpha}^{\infty}(s)$ in terms of this function:
$$\begin{aligned}
\Phi_{1}^{i/\infty}\left(s\right) & =\Upsilon\left(\tilde{\Delta}_{1,i/f},z\right)\\
\Phi_{2}^{i/\infty}\left(s\right) & =\Upsilon\left(\tilde{\Delta}_{2,i/f},\kappa z\right)\,,\end{aligned}$$
where $z=\frac{s}{2\Delta_{1,\infty}}$, $\tilde{\Delta}_{\alpha,i/f}=\frac{\Delta_{\alpha,i/f}}{\Delta_{\alpha,\infty}}$, and $\kappa=\frac{\Delta_{1,\infty}}{\Delta_{2,\infty}}$. For concreteness, in this section we consider the gap with $\alpha=1$ to be the one that is asymptotically smaller, implying that $\left|\kappa\right|<1$. But note that our results can be straightforwardly applied also to the case $\left|\kappa\right|>1$.
The function $\Upsilon\left(\Delta,\,z\right)$ has two branch cuts, one between $\left(-i\infty,\ -i\right)$ and another one between $\left(i,\ i\infty\right)$. The function is analytic elsewhere. Applying the Cauchy’s residue theorem (see Appendix \[sec:appendix\_inverse\_laplace\_tf\] and Fig. \[Fig\_Bromwich\_Contour\] for details), we convert the Bromwich integral into four integrals along the sides of the two branch cuts. Note that we have already eliminated the pole at the origin by imposing the final value theorem in Section \[subsubsec:asymptotic\_gap\_1\_band\]. In addition, we also use the following properties of the function $\Upsilon$:
$$\begin{aligned}
\Upsilon\left(\Delta,\,z\right) & =\Upsilon\left(\Delta,\,-z\right)\\
\text{Re}\left[\Upsilon\left(\Delta,\,0^{+}\pm iy\right)\right] & =\text{Re}\left[\Upsilon\left(\Delta,\,0^{-}\pm iy\right)\right],\,\text{for \ensuremath{y>1}}\\
\text{Im}\left[\Upsilon\left(\Delta,\,0^{+}\pm iy\right)\right] & =-\text{Im}\left[\Upsilon\left(\Delta,\,0^{-}\pm iy\right)\right],\,\text{for \ensuremath{y>1}.}\end{aligned}$$
As a result, the inverse Laplace transformation is given by the following integral: $$\delta_{\alpha}\left(t\right)=\frac{2}{\pi}\int_{i}^{i\infty}\text{Im}\left[z\delta_{\alpha}\left(z\right)\right]\frac{\cosh\left(2\Delta_{1,\infty}zt\right)}{z}dz\label{Inverse_Laplace}$$ where $z\delta_{\alpha}\left(z\right)$ is given by
$$\begin{aligned}
\frac{z\delta_{\alpha}\left(z\right)}{2\Delta_{\alpha,\infty}} & =-\frac{1}{\eta_{2}}\left[\frac{1}{2}\frac{v_{f}}{v_{i}}\left(\frac{\tilde{\Delta}_{\alpha,i}}{\tilde{\Delta}_{\bar{\alpha},i}}+\frac{\tilde{\Delta}_{\bar{\alpha},i}}{\tilde{\Delta}_{\alpha,i}}\right)-1\right]\frac{1}{\tilde{D}\left(z\right)}+\frac{\left(\tilde{\Delta}_{\alpha,i}-1\right)}{2}\frac{\Upsilon\left(\tilde{\Delta}_{\alpha,i},\frac{\Delta_{1,\infty}}{\Delta_{\alpha,\infty}}z\right)\Upsilon\left(\tilde{\Delta}_{\bar{\alpha},f},\frac{\Delta_{1,\infty}}{\Delta_{\bar{\alpha},\infty}}z\right)}{\tilde{D}\left(z\right)}\nonumber \\
& \qquad-\frac{1}{2\eta_{\alpha}}\left(\frac{\Delta_{\bar{\alpha},\infty}}{\Delta_{\alpha,\infty}}\right)\left(\frac{v_{f}}{v_{i}}\frac{\tilde{\Delta}_{\bar{\alpha},i}}{\tilde{\Delta}_{\alpha,i}}-1\right)\frac{\Upsilon\left(\tilde{\Delta}_{\bar{\alpha},f},\frac{\Delta_{1,\infty}}{\Delta_{\bar{\alpha},\infty}}z\right)}{\tilde{D}\left(z\right)}\nonumber \\
& \qquad+\frac{\left(\tilde{\Delta}_{\bar{\alpha},i}-1\right)}{2\eta_{\alpha}}\left(\frac{\Delta_{\bar{\alpha},\infty}}{\Delta_{\alpha,\infty}}\right)\frac{\Upsilon\left(\tilde{\Delta}_{\bar{\alpha},i},\frac{\Delta_{1,\infty}}{\Delta_{\bar{\alpha},\infty}}z\right)}{\tilde{D}\left(z\right)}+\frac{\left(\tilde{\Delta}_{\alpha,i}-1\right)}{2\eta_{\bar{\alpha}}}\left(\frac{\Delta_{\alpha,\infty}}{\Delta_{\bar{\alpha},\infty}}\right)\frac{\Upsilon\left(\tilde{\Delta}_{\alpha,i},\frac{\Delta_{1,\infty}}{\Delta_{\alpha,\infty}}z\right)}{\tilde{D}\left(z\right)}\label{eq:delta_alpha_Laplace}\end{aligned}$$
with
$$D\left(z\right)=\Upsilon(\tilde{\Delta}_{1,f},z)\Upsilon(\tilde{\Delta}_{2,f},\kappa z)+\frac{1}{\kappa}\Upsilon(\tilde{\Delta}_{2,f},\kappa z)+\frac{\kappa}{\eta_{2}}\Upsilon(\tilde{\Delta}_{1,f},z)\,.$$
In the long-time limit, where $2\Delta_{1,\infty}t\gg1$, the integrand of Eq. is highly oscillatory. Only singular behaviors of $\text{Im}\left[z\delta_{\alpha}\left(z\right)\right]$ will therefore make a contribution to the long-time dynamics of the superconducting gap. Indeed, $\text{Im}\left[z\delta_{\alpha}\left(z\right)\right]$ has two branch points along $z\in\left[i,\,i\infty\right)$: one is located at $z=i$ and the other one is located at $z=i/\left|\kappa\right|$. We expand $\text{Im}\left[z\delta_{\alpha}\left(z\right)\right]$ near these two branch points, i.e. $z=i+i\epsilon$ and $z=i/\left|\kappa\right|\pm i\epsilon$, and find that both exhibit $\sqrt{\epsilon}$ behavior (details shown in Appendix ). This is sharply distinct from the single-band case, where only one branch point is present along $z\in\left[i,\,i\infty\right)$. More importantly, the asymptotic behavior in the vicinity of the branch point in the single-band case is $1/\sqrt{\epsilon}$ rather than $\sqrt{\epsilon}$. The two cases are plotted and compared in Fig. \[Fig\_Sol-Laplace\]. The $1/\sqrt{\epsilon}$ behavior leads to a $t^{-1/2}$ decay of the gap oscillation amplitude at long times in the single-band case [@Volkov1974]. In contrast, the $\sqrt{\epsilon}$ behavior in Laplace space leads to a faster $t^{-3/2}$ decay in the two-band model
![Non-analyticity of the gaps in Laplace-space along the imaginary axis, $s''$. In the single-band case (blue dashed line), the only non-analyticity is the inverse square root branch point at $s''=\pm2\Delta_{\infty}$ (only the positive axis is shown here). In two-band systems (red solid line), however, the branch point at $s''=\pm2\Delta_{1,\infty}$ becomes square root like. Moreover, additional square root branch points appear at $s''=\pm2\Delta_{2,\infty}$, which gives rise to the additional oscillation frequency of the gaps.[]{data-label="Fig_Sol-Laplace"}](Laplace_Solution_new){width="1\columnwidth"}
$$\int_{1}^{\infty}\frac{\sqrt{y-1}}{y}\cos\left[y\left(2\Delta t\right)\right]dy\simeq-\frac{\sqrt{\pi}\sin\left(2\Delta t+\frac{\pi}{4}\right)}{2\left(2\Delta t\right)^{3/2}}$$
for $2\Delta t\gg1$ (details are shown in Appendix \[sec:appendix\_inverse\_laplace\_tf\]). The damping of the gap oscillations thus occurs faster for two-band superconductivity.
To find the full long-time expressions of the gap, including prefactors and oscillatory factors, we perform a careful asymptotic analysis of $\text{Im}\left[z\delta_{\alpha}\left(z\right)\right]$. The final result for the long-time gap oscillations reads
$$\begin{aligned}
\Delta_{1}\left(t\right) & \simeq\Delta_{1,\infty}+\mathcal{A}_{1}\frac{\sin\left(2\Delta_{1,\infty}t+\frac{\pi}{4}\right)}{\left(\Delta_{1,\infty}t\right)^{3/2}}+\mathcal{B}_{1}\frac{\sin\left(2\left|\Delta_{2,\infty}\right|t-\frac{\pi}{4}\right)}{\left(\left|\Delta_{2,\infty}\right|t\right)^{3/2}}+\mathcal{C}_{1}\frac{\sin\left(2\left|\Delta_{2,\infty}\right|t+\frac{\pi}{4}\right)}{\left(\left|\Delta_{2,\infty}\right|t\right)^{3/2}}\label{analytics_delta_1}\\
\Delta_{2}\left(t\right) & \simeq\Delta_{2,\infty}+\mathcal{A}_{2}\frac{\sin\left(2\left|\Delta_{2,\infty}\right|t+\frac{\pi}{4}\right)}{\left(\left|\Delta_{2,\infty}\right|t\right)^{3/2}}+\mathcal{B}_{2}\frac{\sin\left(2\Delta_{1,\infty}t-\frac{\pi}{4}\right)}{\left(\Delta_{1,\infty}t\right)^{3/2}}+\mathcal{C}_{2}\frac{\sin\left(2\Delta_{1,\infty}t+\frac{\pi}{4}\right)}{\left(\Delta_{1,\infty}t\right)^{3/2}}\label{analytics_delta_2}\end{aligned}$$
where the pre-factors $\mathcal{A}_{\alpha}$, $\mathcal{B}_{\alpha}$ and $\mathcal{C}_{\alpha}$ are calculated from the asymptotic analysis and explicitly shown in Appendix \[sec:appendix\_asymptotic\_analysis\_gap\]. The gap oscillation frequencies are determined by the asymptotic values of the gaps in the two different bands $\Delta_{\alpha,\infty}$. As discussed in the previous sections, the asymptotic values of the gaps are determined by the quench amplitude $\Delta_{\alpha,i}/\Delta_{\alpha,f}$ and the ratio of the density of states $\eta$ between the two bands. In general, they will also depend on $r=-U/V$, which we have set to zero for simplicity here. The same holds for the prefactors of the sinusoidal oscillations.
{width="0.75\paperwidth"}
In Fig. \[fig:analytics\_vs\_numerics\], we compare our analytical results to the numerical solution of the equations of motion for two different weak quench amplitudes in phase II. We find an excellent quantitative agreement between the two, which also justifies our analytical ansatz *a posteriori*.
We finish this section by commenting on how our solution gives the known single-band result in the limit where the ratio between the two densities of states approaches one, $\eta\rightarrow1$. In this limit, the gaps have the same asymptotic magnitude, i.e. $\Delta_{1,\infty}=\left|\Delta_{2,\infty}\right|$. The equilibrium gaps also have the same magnitude, leading to $\Upsilon\left(\tilde{\Delta}_{\alpha,f},z\right)=\Upsilon\left(\tilde{\Delta}_{\bar{\alpha},f},z\right)=\Upsilon\left(\tilde{\Delta}_{1,f},z\right)$. As a result, Eq. \[eq:delta\_alpha\_Laplace\] becomes
$$\frac{z\delta_{\alpha}\left(z\right)}{2\Delta_{\alpha,\infty}}=\left[\frac{1}{2}\left(\frac{v_{f}}{v_{i}}-1\right)+\frac{\left(\tilde{\Delta}_{1,i}-1\right)}{2}\Upsilon\left(\tilde{\Delta}_{1,i},z\right)\right]\frac{\left[\Upsilon\left(\tilde{\Delta}_{1,f},z\right)-2\right]}{\tilde{D}\left(z\right)}$$
where $D\left(z\right)=\Upsilon^{2}(\tilde{\Delta}_{1,f},z)-2\Upsilon(\tilde{\Delta}_{1,f},z)$. Further simplification of the above equation gives: $$\frac{z\delta_{\alpha}\left(z\right)}{2\Delta_{\alpha,\infty}}=\frac{1}{2}\left(\frac{v_{f}}{v_{i}}-1\right)\frac{1}{\Upsilon(\tilde{\Delta}_{\alpha,f},z)}+\frac{\left(\tilde{\Delta}_{\alpha,i}-1\right)}{2}\frac{\Upsilon\left(\tilde{\Delta}_{\alpha,i},z\right)}{\Upsilon(\tilde{\Delta}_{\alpha,f},z)}$$
In writing this last equation, we used the fact that $\Upsilon\left(\tilde{\Delta}_{\alpha,f},z\right)=\Upsilon\left(\tilde{\Delta}_{1,f},z\right)$. This is the same expression as the solution of the single-band case in Laplace-space, Eq. \[eq:delta\_of\_s\_single\_band\]. Using the asymptotic behavior of $\Upsilon\left(\tilde{\Delta}_{\alpha},iy\right)$ near the branch point $y\rightarrow1$ (details shown in Appendix \[sec:appendix\_asymptotic\_analysis\_gap\], see Eq. \[eq:Asymptote\_Upsilon\]), we arrive at the following asymptotic behavior: $$\text{Im}\left[iy\delta_{\alpha}\left(y\right)\right]\simeq\frac{v_{f}^{-1}-v_{i}^{-1}}{\pi}\left|\Delta_{\alpha,f}\right|\sqrt{\frac{2}{y-1}}$$
By applying the inverse Laplace transformation, we find that the gap dynamics is characterized by oscillations with frequency $2\Delta_{\infty}$ and $t^{-1/2}$ damping: $$\Delta_{\alpha}\left(t\right)\simeq\Delta_{\alpha,\infty}+\left(\frac{2}{\pi}\right)^{3/2}\Delta_{\alpha,f}\ln\left(\frac{\Delta_{\alpha,i}}{\Delta_{\alpha,f}}\right)\frac{\cos\left(2\Delta_{\alpha,\infty}t+\frac{\pi}{4}\right)}{\sqrt{2\Delta_{\alpha,\infty}t}}$$
Conclusions {#sec:conclusions}
===========
In this paper, we developed a generalization of the Volkov-Kogan Laplace-space analysis for the post-quench dynamics of $s$-wave BCS superconductors in the collisionless regime [@Volkov1974], and applied it to interaction quenches of two-band BCS superconductors. We showed that the two-band case is fundamentally different from the single-band case. Not only do the gap oscillations display beating associated with the two different gap values on the two bands, but they also display a faster $t^{-3/2}$ power-law damping, as opposed to the $t^{-1/2}$ damping of the single-band case. For weak quenches, our analytical results agree very well with the numerical results in the long-time limit, demonstrating that the gap dynamics of multi-band systems cannot be simply decomposed into the sum of the gap dynamics of single-band systems. Formally, this new power-law decay can be understood as arising from the “splitting” of the relevant branch point in Laplace space in two, as shown in Fig. . As a result, one expects the same $t^{-3/2}$ behavior to take place even when the number of bands is larger than $2$. From a more physical perspective, the stronger damping in the two-band case arises because the Cooper-pairs dephasing involves states from both bands due to the inter-band coupling. Such a dephasing is thusintrinsic to multi-band systems and independent on the quench amplitude.
From a methodological viewpoint, our analysis is distinct from the one introduced by Volkov and Kogan [@Volkov1974] (see also the more recent works by Yuzbashyan and co-workers [@YuzbashyanDzero-PRL-2006; @Yuzbashyan_Dzero_Gurarie_Foster-PRA-2015]), because we linearize the equations of motion around the asymptotic long-time pseudo-spin states as opposed to the final equilibrium states. This allows us to self-consistently determine the asymptotic long-time steady-state values of the gaps over the full range of quench amplitudes in phase II (and phase I, where the steady-state gaps vanish). We explicitly showed that the self-consistent equation for the steady-state gap in the single-band case agrees with the exact expression derived within the Lax vector analysis [@Yuzbashyan_Altshuler_Enolskii-PRB-2005; @Barankov-Synchronization; @Yuzbashyan-2006-PRL]. Like in the two-band model we investigate here, our method can be very useful in cases where an exact solution is not (yet) available, for example, to investigate quenches towards more exotic fully gapped pairing states such as $s+is$ or $s+id$. Other interesting future directions are to include a finite intra-band pairing interaction $r\neq0$, competing electronic order parameters such as spin-density waves [@Schuett18], or generalize and apply our Laplace method to study quenches in superconductors with a nodal gap structure such as those with $d$-wave symmetry[@Peronaci-PRL-2015-d-wave].
This work was supported by the by U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award DE-SC0012336. The authors acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesota, where the numerical calculations were performed. P.P.O. acknowledges support from Iowa State University Startup Funds. T.C. also acknowledges the support from the Doctoral Dissertation Fellowship awarded by the University of Minnesota.
Initial conditions for the interaction quench {#sec:app_initial_conditions}
=============================================
The system is at equilibrium before the interaction quench. For systems with only inter-band repulsion, the superconducting gap is given by
$$\begin{aligned}
\Delta_{1,i} & =-v_{i}\eta\int d\varepsilon\frac{\Delta_{2,i}}{2\sqrt{\varepsilon^{2}+\Delta_{2,i}^{2}}}\\
\Delta_{2,i} & =-v_{i}\int d\varepsilon\frac{\Delta_{1,i}}{2\sqrt{\varepsilon^{2}+\Delta_{1,i}^{2}}}\end{aligned}$$
where $v_{i}=V_{i}\mathcal{N}_{1}$ is the dimensionless inter-band repulsion, and $\eta=\frac{\mathcal{N}_{2}}{\mathcal{N}_{1}}$ is the ratio between the density of states near the Fermi level of the two bands. The pseudospins are
$$\begin{aligned}
S_{\alpha,i}^{x} & =\frac{\Delta_{\alpha,i}}{2\sqrt{\varepsilon^{2}+\Delta_{\alpha,i}^{2}}}\label{initial_Sx}\\
S_{\alpha,i}^{y} & =0\label{initial_Sy}\\
S_{\alpha,i}^{z} & =\frac{-\varepsilon}{2\sqrt{\varepsilon^{2}+\Delta_{\alpha,i}^{2}}}\label{initial_Sz}\end{aligned}$$
After the interaction quench, the inter-band repulsion is suddenly changed to a different value, $v_{f}$. The initial conditions of the post-quench dynamics of the superconducting gaps are thus given by replacing the inter-band repulsion with its post-quench value $v_{f}$.
$$\begin{aligned}
\Delta_{1}\left(0^{+}\right) & =-v_{f}\eta\int d\varepsilon\frac{\Delta_{2,i}}{2\sqrt{\varepsilon^{2}+\Delta_{2,i}^{2}}}=\frac{v_{f}}{v_{i}}\Delta_{1,i}\\
\Delta_{2}\left(0^{+}\right) & =-v_{f}\int d\varepsilon\frac{\Delta_{1,i}}{2\sqrt{\varepsilon^{2}+\Delta_{1,i}^{2}}}=\frac{v_{f}}{v_{i}}\Delta_{2,i}\end{aligned}$$
Substituting in the linearized equations \[Linearization\_Sz\] and \[Linearization\_EOM\], the initial conditions on the pseudospin deviations $f_{\alpha}$ become
$$\begin{aligned}
f_{\alpha,0}'' & =0\\
\dot{f}_{\alpha,0}'' & =-\frac{\varepsilon\left(\Delta_{\alpha,i}-\Delta_{\alpha,\infty}\right)}{\sqrt{\varepsilon^{2}+\Delta_{\alpha,i}^{2}}}-2\delta_{\alpha,0}S_{\alpha,\infty}^{z}\end{aligned}$$
We recall that $f_{\alpha,0}''$ and $\dot{f}_{\alpha,0}''$ are related to the dynamics of the superconducting gap in Laplace space via $I_{\alpha}\left(s\right)=\left\langle \frac{2\varepsilon\left[sf_{\alpha,0}'+\dot{f}_{\alpha,0}''\right]}{s^{2}+4E_{\alpha,\infty}^{2}}\right\rangle $, which yields Eq. \[Initial\_Conditions\].
Asymptotic analysis of the superconducting gap in Laplace space
===============================================================
\[sec:appendix\_asymptotic\_analysis\_gap\]
In this appendix, we analyze the asymptotic behavior of the gap in Laplace space near the branch points. From Eq. \[eq:delta\_alpha\_Laplace\], there are 7 terms that determine the analytic behavior of the gap. The branch points all come from the function $\Upsilon\left(\Delta,z\right)$, which opens branch cuts at at $\left(-i\infty,\ -i\right)$ and $\left(i,\ i\infty\right)$, as shown in Fig. \[Fig\_Bromwich\_Contour\]. Let $z=iy$, then, around $y=1$, we have $$\Upsilon\left(\Delta,y\right)\simeq\begin{cases}
\frac{v_{f}\pi}{\left|\Delta\right|}\sqrt{\frac{1-y}{2}}+\mathcal{O}\left(1-y\right) & ,\ y\rightarrow1-\epsilon\\
i\frac{v_{f}\pi}{\left|\Delta\right|}\sqrt{\frac{y-1}{2}}+\mathcal{O}\left(y-1\right) & ,\ y\rightarrow1+\epsilon
\end{cases}\label{eq:Asymptote_Upsilon}$$ where $\epsilon$ is an infinitesimal positive number.
We use the asymptotic behavior of $\Upsilon\left(\Delta,y\right)$ to expand all the terms in Eq. \[eq:delta\_alpha\_Laplace\], and obtain the following results:
$$\begin{aligned}
\text{Im}\left[\frac{1}{D\left(y\right)}\right] & \simeq\begin{cases}
-\kappa^{2}\frac{\Upsilon\left(\tilde{\Delta}_{2,f},\kappa\right)+\frac{\kappa}{\eta}}{\Upsilon^{2}\left(\tilde{\Delta}_{2,f},\kappa\right)}\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{y-1}{2}} & ,\ y\rightarrow1+\epsilon\\
\frac{\eta}{\kappa}\text{Im}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]-\left(\frac{\eta}{\kappa}\right)^{2}\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)+\frac{1}{\kappa}}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{1-\left|\kappa\right|y}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}-\epsilon\\
\frac{\eta}{\kappa}\text{Im}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]-\left(\frac{\eta}{\kappa}\right)^{2}\text{Re}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)+\frac{1}{\kappa}}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{\left|\kappa\right|y-1}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}+\epsilon
\end{cases}\\
\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,f},y\right)}{D\left(y\right)}\right] & \simeq\begin{cases}
\kappa\frac{1}{\Upsilon\left(\tilde{\Delta}_{2,f},\kappa\right)}\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{y-1}{2}} & ,\ y\rightarrow1+\epsilon\\
-\left(\frac{\eta}{\kappa}\right)^{2}\frac{1}{\kappa}\text{Im}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{1-\left|\kappa\right|y}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}-\epsilon\\
-\left(\frac{\eta}{\kappa}\right)^{2}\left(1+\frac{1}{\kappa}\text{Re}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right)\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{\left|\kappa\right|y-1}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}+\epsilon
\end{cases}\\
\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},y\right)}{D\left(y\right)}\right] & \simeq\begin{cases}
\kappa\frac{1}{\Upsilon\left(\tilde{\Delta}_{2,f},\kappa\right)}\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,i}\right|}\sqrt{\frac{y-1}{2}} & ,\ y\rightarrow1+\epsilon\\
\frac{\eta}{\kappa}\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]-\left(\frac{\eta}{\kappa}\right)^{2}\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}+\frac{\frac{1}{\kappa}\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{1-\left|\kappa\right|y}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}-\epsilon\\
\frac{\eta}{\kappa}\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]-\left(\frac{\eta}{\kappa}\right)^{2}\text{Re}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}+\frac{\frac{1}{\kappa}\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{\left|\kappa\right|y-1}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}+\epsilon
\end{cases}\\
\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{2,f},\kappa y\right)}{D\left(y\right)}\right] & \simeq\begin{cases}
\Upsilon\left(\tilde{\Delta}_{2,f},\kappa\right)\text{Im}\left[\frac{1}{D\left(y\right)}\right] & ,\ y\rightarrow1+\epsilon\\
\frac{\eta}{\kappa}\text{Im}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{1-\left|\kappa\right|y}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}-\epsilon\\
\frac{\eta}{\kappa}\text{Re}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{\left|\kappa\right|y-1}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}+\epsilon
\end{cases}\\
\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{2,i},\kappa y\right)}{D\left(y\right)}\right] & \simeq\begin{cases}
\Upsilon\left(\tilde{\Delta}_{2,i},\kappa\right)\text{Im}\left[\frac{1}{D\left(y\right)}\right] & ,\ y\rightarrow1+\epsilon\\
\frac{\eta}{\kappa}\text{Im}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{2,i}\right|}\sqrt{\frac{1-\left|\kappa\right|y}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}-\epsilon\\
\frac{\eta}{\kappa}\text{Re}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{2,i}\right|}\sqrt{\frac{\left|\kappa\right|y-1}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}+\epsilon
\end{cases}\\
\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,f},y\right)\Upsilon\left(\tilde{\Delta}_{2,i},\kappa y\right)}{D\left(y\right)}\right] & \simeq\begin{cases}
\kappa\frac{\Upsilon\left(\tilde{\Delta}_{2,i},\kappa\right)}{\Upsilon\left(\tilde{\Delta}_{2,f},\kappa\right)}\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{y-1}{2}} & ,\ y\rightarrow1+\epsilon\\
\mathcal{O}\left(\epsilon\right) & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}-\epsilon\\
\frac{\eta}{\kappa}\frac{v_{f}\pi}{\left|\tilde{\Delta}_{2,i}\right|}\sqrt{\frac{\left|\kappa\right|y-1}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}+\epsilon
\end{cases}\\
\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{2,f},\kappa y\right)\Upsilon\left(\tilde{\Delta}_{1,i},y\right)}{D\left(y\right)}\right] & \simeq\begin{cases}
\kappa\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,i}\right|}\sqrt{\frac{y-1}{2}} & ,\ y\rightarrow1+\epsilon\\
\frac{\eta}{\kappa}\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{1-\left|\kappa\right|y}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}-\epsilon\\
\frac{\eta}{\kappa}\text{Re}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\frac{v_{f}\pi}{\left|\tilde{\Delta}_{1,f}\right|}\sqrt{\frac{\left|\kappa\right|y-1}{2}} & ,\ y\rightarrow\frac{1}{\left|\kappa\right|}+\epsilon
\end{cases}\end{aligned}$$
Inverse Laplace transformation and useful integrals {#sec:appendix_inverse_laplace_tf}
===================================================
The inverse Laplace transformation is given by the Bromwich integral: $$y\left(t\right)=\mathcal{L}^{-1}\left\{ Y\right\} \left(t\right)=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}Y\left(s\right)e^{st}ds$$ where $\sigma$ is a real number that is larger than the real parts of all the singularities of $Y\left(s\right)$.
![Integration contour in the complex Laplace space.[]{data-label="Fig_Bromwich_Contour"}](Bromwich_integral){width="0.6\columnwidth"}
All the asymptotic behaviors of the gap in Laplace space are square root like. Consequently, transforming back to real time domain leads to a $t^{-3/2}$ decay. $$\begin{aligned}
\int_{1}^{\infty}\frac{\sqrt{y-1}}{y}\cos\left(2\Delta_{1,\infty}yt\right)dy & =\sqrt{\frac{\pi}{4\Delta_{1,\infty}t}}\left[\cos\left(2\Delta_{1,\infty}t\right)-\sin\left(2\Delta_{1,\infty}t\right)\right]+\pi\left[C\left(\sqrt{\frac{4\Delta_{1,\infty}t}{\pi}}\right)+S\left(\sqrt{\frac{4\Delta_{1,\infty}t}{\pi}}\right)-1\right]\nonumber \\
& \simeq-\frac{\sqrt{\pi}\sin\left(2\Delta_{1,\infty}t+\frac{\pi}{4}\right)}{2\left(2\Delta_{1,\infty}t\right)^{3/2}}\end{aligned}$$ for $2\Delta_{1,\infty}t\gg1$. Similarly, $\int_{\frac{1}{\left|\kappa\right|}}^{\infty}\frac{\sqrt{\left|\kappa\right|y-1}}{y}\cos\left(2\Delta_{1,\infty}yt\right)dy\simeq-\frac{\sqrt{\pi}\sin\left(2\left|\Delta_{2,\infty}\right|t+\frac{\pi}{4}\right)}{2\left(2\left|\Delta_{2,\infty}\right|t\right)^{3/2}}$, and $$\begin{aligned}
\int_{-\infty}^{\frac{1}{\left|\kappa\right|}}\frac{\sqrt{1-\left|\kappa\right|y}}{y}\cos\left(2\Delta_{1,\infty}yt\right)dy & \simeq\lim_{\Lambda\rightarrow\infty}\int_{0}^{\Lambda}\sqrt{x}\cos\left(2\left|\Delta_{2,\infty}\right|t-2\left|\Delta_{2,\infty}\right|xt\right)\nonumber \\
& \simeq\frac{\sqrt{\pi}\sin\left(2\left|\Delta_{2,\infty}\right|t-\frac{\pi}{4}\right)}{2\left(2\left|\Delta_{2,\infty}\right|t\right)^{3/2}}\end{aligned}$$
Analytical expressions for the gap dynamics {#sec:appendix_analytic_expression_for_weak_quenches}
===========================================
We wrote the long-time asymptotic expressions of the gap oscillations in Eqs. \[analytics\_delta\_1\] and \[analytics\_delta\_2\]. In this appendix, we provide the explicit expressions for the prefactors $\mathcal{A}_{\alpha}$, $\mathcal{B}_{\alpha}$ and $\mathcal{C}_{\alpha}$ that appear in the two equations. $$\begin{aligned}
\frac{\mathcal{A}_{1}}{\Delta_{1,\infty}} & = & -\frac{\sqrt{\pi}}{4}\left\{ \left[\frac{\kappa}{\eta}\left[\frac{v_{f}}{v_{i}}\left(\frac{\tilde{\Delta}_{1,i}}{\tilde{\Delta}_{2,i}}+\frac{\tilde{\Delta}_{2,i}}{\tilde{\Delta}_{1,i}}\right)-2\right]+\left(\frac{v_{f}}{v_{i}}\frac{\tilde{\Delta}_{2,i}}{\tilde{\Delta}_{1,i}}-1\right)\Upsilon\left(\tilde{\Delta}_{2,f},\kappa\right)-\left(\tilde{\Delta}_{2,i}-1\right)\Upsilon\left(\tilde{\Delta}_{2,i},\kappa\right)\right]\times\right.\nonumber \\
& & \left.\times\frac{\Upsilon\left(\tilde{\Delta}_{2,f},\kappa\right)+\frac{\kappa}{\eta}}{\Upsilon^{2}\left(\tilde{\Delta}_{2,f},\kappa\right)}\frac{\kappa v_{f}}{\left|\tilde{\Delta}_{2,f}\right|}+\left(1+\frac{\kappa}{\eta}\frac{1}{\Upsilon\left(\tilde{\Delta}_{2,f},\kappa\right)}\right)\left(\tilde{\Delta}_{1,i}-1\right)\frac{\kappa v_{f}}{\left|\tilde{\Delta}_{1,i}\right|}\right\} \\
\frac{\mathcal{B}_{1}}{\Delta_{1,\infty}} & = & \frac{\sqrt{\pi}}{4}\left\{ \left(\frac{v_{f}}{v_{i}}\frac{\Delta_{1,i}}{\Delta_{2,i}}-1+\tilde{\Delta}_{2,f}-\frac{\tilde{\Delta}_{2,f}}{\tilde{\Delta}_{2,i}}\right)\text{Im}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]+\left[\frac{v_{f}}{v_{i}}\left(\frac{\tilde{\Delta}_{1,i}}{\tilde{\Delta}_{2,i}}+\frac{\tilde{\Delta}_{2,i}}{\tilde{\Delta}_{1,i}}\right)-2\right]\frac{1}{\kappa}\text{Im}\left[\frac{1}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right.\nonumber \\
& & \left.-\left(\tilde{\Delta}_{1,i}-1\right)\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right\} \frac{\eta}{\kappa^{2}}\frac{v_{f}}{\left|\tilde{\Delta}_{2,f}\right|}\\
\frac{\mathcal{C}_{1}}{\Delta_{1,\infty}} & = & -\frac{\sqrt{\pi}}{4}\left\{ \left(\frac{v_{f}}{v_{i}}\frac{\Delta_{1,i}}{\Delta_{2,i}}-1+\tilde{\Delta}_{1,f}-\frac{\tilde{\Delta}_{1,f}}{\tilde{\Delta}_{2,i}}\right)\text{Re}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]+\left[\frac{v_{f}}{v_{i}}\left(\frac{\tilde{\Delta}_{1,i}}{\tilde{\Delta}_{2,i}}+\frac{\tilde{\Delta}_{2,i}}{\tilde{\Delta}_{1,i}}\right)-2\right]\frac{1}{\kappa}\text{Re}\left[\frac{1}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right.\nonumber \\
& & \left.-\left(\tilde{\Delta}_{1,i}-1\right)\text{Re}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right\} \frac{\eta}{\kappa^{2}}\frac{v_{f}}{\left|\tilde{\Delta}_{2,f}\right|}\end{aligned}$$ $$\begin{aligned}
\frac{\mathcal{A}_{2}}{\Delta_{2,\infty}} & = & -\frac{\sqrt{\pi}}{4}\left\{ \left(\frac{v_{f}}{v_{i}}\frac{\Delta_{1,i}}{\Delta_{2,i}}-1+\tilde{\Delta}_{1,f}-\frac{\tilde{\Delta}_{1,f}}{\tilde{\Delta}_{1,i}}\right)\frac{1}{\Upsilon\left(\tilde{\Delta}_{2,f},\kappa\right)}+\frac{\kappa}{\eta}\left[\frac{v_{f}}{v_{i}}\left(\frac{\tilde{\Delta}_{1,i}}{\tilde{\Delta}_{2,i}}+\frac{\tilde{\Delta}_{2,i}}{\tilde{\Delta}_{1,i}}\right)-2\right]\frac{1}{\Upsilon^{2}\left(\tilde{\Delta}_{2,f},\kappa\right)}\right.\nonumber \\
& & \left.-\left(\tilde{\Delta}_{2,i}-1\right)\frac{\Upsilon\left(\tilde{\Delta}_{2,i},\kappa\right)}{\Upsilon^{2}\left(\tilde{\Delta}_{2,f},\kappa\right)}\right\} \frac{\kappa^{2}}{\eta}\frac{v_{f}}{\left|\tilde{\Delta}_{1,f}\right|}\\
\frac{\mathcal{B}_{2}}{\Delta_{2,\infty}} & = & \frac{\sqrt{\pi}}{4}\left\{ \left[\frac{v_{f}}{v_{i}}\left(2\frac{\tilde{\Delta}_{1,i}}{\tilde{\Delta}_{2,i}}+\frac{\tilde{\Delta}_{2,i}}{\tilde{\Delta}_{1,i}}\right)-3+\tilde{\Delta}_{2,f}-\frac{\tilde{\Delta}_{2,f}}{\tilde{\Delta}_{2,i}}\right]\text{Im}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right.\nonumber \\
& & +\left[\frac{v_{f}}{v_{i}}\left(\frac{\tilde{\Delta}_{1,i}}{\tilde{\Delta}_{2,i}}+\frac{\tilde{\Delta}_{2,i}}{\tilde{\Delta}_{1,i}}\right)-2\right]\frac{1}{\kappa}\text{Im}\left[\frac{1}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\nonumber \\
& & \left.-\left(\tilde{\Delta}_{1,i}-1\right)\left(\kappa\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]+\text{Im}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right)\right\} \frac{\eta}{\kappa^{2}}\frac{v_{f}}{\left|\tilde{\Delta}_{2,f}\right|}\\
\frac{\mathcal{C}_{2}}{\Delta_{2,\infty}} & = & -\frac{\sqrt{\pi}}{4}\left\{ \left[\frac{v_{f}}{v_{i}}\left(\frac{\tilde{\Delta}_{1,i}}{\tilde{\Delta}_{2,i}}+\frac{\tilde{\Delta}_{2,i}}{\tilde{\Delta}_{1,i}}\right)-2\right]\text{Re}\left[\frac{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)+\frac{1}{\kappa}}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right.\nonumber \\
& & +\left[\frac{v_{f}}{v_{i}}\left(\frac{\tilde{\Delta}_{1,i}}{\tilde{\Delta}_{2,i}}+\frac{\tilde{\Delta}_{2,i}}{\tilde{\Delta}_{1,i}}\right)-2+\tilde{\Delta}_{2,f}-\frac{\tilde{\Delta}_{2,f}}{\tilde{\Delta}_{2,i}}\right]\left(\kappa+\text{Re}\left[\frac{1}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right)\nonumber \\
& & \left.-\left(\tilde{\Delta}_{1,i}-1\right)\text{Re}\left[\frac{\kappa\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}+\frac{\Upsilon\left(\tilde{\Delta}_{1,i},\frac{1}{\kappa}\right)}{\Upsilon^{2}\left(\tilde{\Delta}_{1,f},\frac{1}{\kappa}\right)}\right]\right\} \frac{\eta}{\kappa^{2}}\frac{v_{f}}{\left|\tilde{\Delta}_{2,f}\right|}\end{aligned}$$
|
---
abstract: 'We present a mission concept for high resolution X-ray spectroscopy with a resolving power, $R \sim$6000, (c.f. $R\ls$1000 for [*Chandra*]{}, XMM-Newton). This resolution is physics-driven, since it allows the thermal widths of coronal X-ray lines to be measured, and astrophysics-driven, since 50 km s$^{-1}$ resolves internal galaxy motions, and galaxy motions within larger structures. Such a mission could be small and have a rapid response allowing us to ‘X-ray the Universe’ using the afterglows of Gamma-ray Bursts (GRBs) as strong background sources of X-rays, and so illuminate the ‘Cosmic Web’. The Cosmic Web is predicted to contain most of the normal matter (baryons) in the nearby Universe.'
author:
- |
Martin Elvis, Fabrizio Fiore (for the CWE Team) Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge MA01238, USA\
Osservatorio Astronomico di Roma, Monteporzio, Via di Frascati 33, Rome I-00040, Italy
title: |
A High Resolution Intergalactic Explorer\
for the Soft X-ray/FUV
---
INTRODUCTION {#sect:intro}
============
We present a mission concept for high resolution X-ray spectroscopy at E$<$1 keV with a resolving power, $R \sim$6000, (c.f. $R\ls$1000 for [*Chandra*]{}, XMM-Newton). This resolution is physics-driven, since it allows the thermal widths of coronal X-ray lines to be measured, and astrophysics-driven, since 50 km s$^{-1}$ resolves internal galaxy motions, and galaxy motions within larger structures. We can then do galaxy dynamics in X-rays.
As [*Chandra*]{} and XMM have made clear the region of the spectrum below 1 keV is where most of the X-ray atomic transitions lie, and is comparable with the optical and ultraviolet bands in this richness of features. However, this band lags behind optical-UV spectroscopy. High resolution spectroscopy of this soft X-ray band is the only part of ‘discovery space’ for which present or future X-ray missions are not pushing for order of magnitude improvements. Inoue (2001) showed how future missions compare to present and past ones in term of throughput, energy band, and angular resolution. We have now [*Chandra*]{} that beats ground-based optical telescopes in terms of angular resolution (though not HST), XMM-Newton and then Con-X and XEUS, will have large throughput. Con-X, and other smaller missions, will image above 10 keV for the first time. Calorimeters will give high resolution in the iron K band (the ASTRO-E calorimeter has a resolution of 6 eV, the Con-X calorimeter could reach 2-3 eV). What is really missing is high resolution (E/$\Delta E>$5000) at low energy.
Because low energy X-rays only require short focal lengths, such a mission could be compact and rapidly repointed. This would allow us to ‘X-ray the Universe’ using the afterglows of Gamma-ray Bursts (GRBs) as strong background sources of X-rays, and so illuminate the ‘Cosmic Web’. The Cosmic Web is predicted to contain most of the normal matter (baryons) in the nearby Universe. A recent flurry of papers has shown that this warm-hot intergalactic medium (WHIM) does exist, both locally (Nicastro et al. 2002a b, Sembach et al. 2002) and at moderate redshift (Zappacosta et al. 2002). The importance of this topic has led to the approval of SPIDR, a new MIDEX mission. SPIDR will map the Cosmic Web of warm intergalactic gas in the Far-UV OVI emission line at 1052Å, and so will provide strong morphological tests of cosmic structure formation.
We see a compelling need to go to the next level of the physics of this major, but elusive, component of our environment: how is the plasma moving? How is it ionized? Which type of supernova enriched it with heavy elements? What is the history of the formation of the Cosmic Web, and how does this tie in with the destruction of the ‘Lyman-alpha forest’ of cooler material? High resolution soft X-ray and far-UV spectra are the [*only*]{} means of studying the detailed physics of the warm gas of the Cosmic Web.
Gamma-ray Bursts are excellent, though fleeting, background beacons for these investigations: GRBs explode in galaxies spanning just the right redshift range, 1$<$z$<$2. Despite their location in the distant universe, the X-ray afterglows of GRBs shine as brightly as the brightest sources in our own local Milky Way for a short time. By rapidly slewing a high resolution spectrometer into position to record the soft X-ray and FUV spectrum of a GRB afterglow, we can gather 100 times more photons than missions that point at the only steady distant sources, high redshift quasars. This will let us study many absorption lines and resolve their profiles, so telling us about the physics of the gas in the Cosmic Web.
The GRB soft X-ray/FUV spectra will carry the signatures of [*all*]{} material along the line of sight to the GRB, separated out cleanly by redshift. This will include matter in the GRB host galaxies, telling us the composition of galaxies during the ‘Age of Star Formation’ (1$<z<$2, Madau et al. 1996) and so testing theories of the star formation history of the universe. The intimate environs of a gamma-ray burst will also imprint their signature on the spectra, testing models for these most powerful explosions in the Universe.
This mission is the complement of the MAP and Planck missions: they put constraints on one end of the process of cosmic structure formation; this mission on the other.
The Mission
===========
This mission concept is the result of several discussions with a large group of scientists during two meetings at Johns Hopkins University [^1] . The heart of the mission is a high resolution soft X-ray spectrometer, with 6 times the spectral resolution of [*Chandra*]{}. A slitless FUV spectrograph will add the OVI line to the X-ray OVII and OVII lines, allowing us to distinguish photoionization from collisional ionization. A GRB detector will trigger the spacecraft to slew the X-ray spectrometer onto the GRB afterglow within a few minutes. (10 minutes requirement, 1 minute goal.)
The soft X-ray spectrometer will consist of: an [**X-ray mirror**]{} (with a 5 arcsec HPD, image size), optimized for low energy performance ($<$1 keV), feeding photons to [**diffraction gratings**]{} deployed in an out-of-plane reflection configuration. The gratings disperse the photons onto an [**array of CCD detectors**]{}, which are also optimized for low energies. The spectral resolution achieved will be $R$=6000 (50 km s$^{-1}$) over the energy range 0.1–1 keV, with a collecting area of 1000 cm$^{-2}$ - 2000 cm$^{-2}$. Replicated X-ray optics developed for Con-X will be the basis of this design. The short focal length (2.5-3 meters) of the X-ray telescope mimimizes the moment-of-inertia of the satellite, allowing faster slews.
To achieve the rapid response necessary to achieve these mission goals, the satellite will include a compact 2-stage GRB detection and localization system based on the highly successful Beppo-SAX design: a [**CsI dodecahedron**]{} to localize GRB to $\sim$1 degree, and a small [**X-ray coded aperture telescope**]{} with a 10-20 degree field of view that can localize bursts to an arcminute. This is sufficient to put the GRB afterglow into the (slitless) spectrometer apertures. The X-ray CCDs of the coded aperture telescope (e.g. an XMM EPIC-pn wafer) will record good quality X-ray spectra of the afterglows, a bonus for the mission science.
SCIENCE DRIVERS
===============
The ‘Missing Baryons’ Problem
-----------------------------
It is well known that at high redshifts (z$>$2) most of the baryons in the Universe lie in ‘Lyman-alpha forest clouds’, at temperatures of $10^4$ K or less, and with densities only slightly above the average density of the Universe (overdensities of only 1-10). But, in the second half of cosmic time, (i.e. for the 7 Gyr at z$<$1) the number of baryons in Lyman-alpha clouds decreases rapidly, while the number of baryons in galaxies and clusters of galaxies does not increase by the same factor. Therefore one of the major problems of the modern cosmology is: [*“Where do most baryons go at low redshift?”*]{}
The leading solution to the ‘missing baryons’ problem is the concept of a ‘Warm-Hot Intergalactic Medium’ (WHIM). This concept has been developed in the last few years by Cen & Ostriker (1999), and by other groups, (see Davé et al. 2000 and references therein): hydrodynamical simulations of the evolution of structures in the Universe showed, surprisingly, that 30-40% of the baryons at z$<$1 are in a warm ($10^5-10^7$ K) phase at overdensities between 10 and 200. Most (70%) of these baryons should be at only a weak overdensity $<$60, and so are not virialized and are unbound. The resulting X-ray ‘forest’ of absorption lines will show us the structure of the Universe developing over the last 7 Gyr (Hellsten et al. 1998). Since this structure depends on what physics causes the primordial fluctuations seen in the Cosmic Microwave Background (CMB) to grow into the galaxies and clusters of galaxies we see today, the X-ray forest will constrain that fundamental physics.
A [*Chandra*]{} spectrum was recently used to discover X-ray absorption at zero redshift due to the WHIM (Nicastro et al., 2002a), following years of frustrating upper limits (Aldcroft et al. 1994). The WHIM thus does exist, at least locally and along on a single line of sight, and with the predicted overdensity. The bare detection of collisional OVI, OVII and OVIII lines is a prime scientific result, strongly corroborating the results of a decade of forefront cosmological modeling. Most likely the IGM component found by Nicastro et al. is a cut through the ‘local filament’ associated with the Local Group of glaxies (which is dominated by the Milky Way and Andromeda spirals.)
Examined in this light FUSE detections of zero redshift absorption toward many AGN implies that we are embedded in a WHIM filament of the Cosmic Web. Nicastro et al. (2002b, see also Sembach et al. 2002) demonstrate that these OVI absorbers are at rest in the frame of our Local Group of galaxies, but not in any Milky Way frame of reference. This clinches the extragalactic location of this highly ionized gas. Moreover, the mass of gas implied is enough to bind the Local Group gravitationally, which begins to put interesting limits on the amount of ‘dark matter’ in our vicinity.
![Chandra discovery of IGM absorption lines (Nicastro et al. 2002a). Residuals after subtraction of the continuum of the spectra, in velocity space, of the OVII$_{K\alpha}$, NeIX, OVIII, OVII$K\beta$, and OI$_{1s-2p}$ resonant lines, from the LETGS spectrum.[]{data-label="nicastro"}](nicastro.ps){height="7cm"}
To understand the structure of the warm IGM making up the Cosmic Web will require many more directions to be probed to distances well beyond our parochial group of galaxies. Observing out to z=2 along many lines of sight will let us see the scales on which the Web has formed and will show us how the Web has grown and been filled with X-ray hot gas, gradually replacing the cooler ‘Lyman-alpha forest’ gas. As we show later, quasars are a poor choice of background source; GRBs are far better.
The breakthrough science of the Cosmic Web will come from [*measuring the widths of the oxygen lines*]{}. This will allow a series of new tests of cosmological models For example, hydrodynamical simulations suggest that the IGM matter is heated to $10^5-10^7$ K by shocks during the collapse of density perturbations. If this is the case one would expect that the turbulent velocities in the shocked gas of the same order of magnitude as its sound speed, i.e. of the order of 100-200 km/s. This is a several times the thermal velocity of the heavier oxygen atoms (at $4\times10^6$ K, the temperature at which the baryon density peaks in hydrodynamical simulations, Davé et al. 2000). Detecting line widths of 100-200 km s$^{-1}$ would point toward shocks as main source of large scale heating of the gas. This would contrast with competing sources such as supernova heating, and so determine the heating mechanism. GRB soft X-ray/FUV afterglow spectroscopy will allow these tests to be carried out.
![[*left:*]{} the baryon density as a function of the temperature (Davé et al. 2000). [*right*]{}: the oxygen ion fraction as a function of the temperature, assuming collisional equilibrium. Adapted from Davè et al. (2001).[]{data-label="compare"}](dave_baryontemp.ps "fig:"){height="7cm"} ![[*left:*]{} the baryon density as a function of the temperature (Davé et al. 2000). [*right*]{}: the oxygen ion fraction as a function of the temperature, assuming collisional equilibrium. Adapted from Davè et al. (2001).[]{data-label="compare"}](ionic.ps "fig:"){height="7cm"}
Galaxies in the Age of Star Formation
-------------------------------------
Prompt observations of GRB afterglows offer a new and distinctive path for the study of the matter in the immediate surroundings of the GRB (r$\sim$100 pc) and in the GRB host galaxy (Fiore et al. 2000). This is the second main science driver for this mission. Around $z=2$ is the age of star formation (Madau et al. 1998), and many GRBs are found at such redshifts.
As a result GRBs can be a powerful tool to determine the history of the metal enrichment in galaxies in the Universe (Fiore 2000, 2001, Savaglio, Fall & Fiore 2002), which can then be compared with enrichment predictions of theoretical models (e.g. Cen & Ostriker 1999), to pin down their several, now unconstrained, assumptions.
GRB host galaxies appear typical of normal star-forming field galaxies at the same large redshifts (Bloom et al. 2001, Djorgovski et al. 2001). Moreover GRBs occur well within the main body of their host galaxies, not in the outer haloes. Best of all, since GRB host galaxies are $\gamma$-ray and X-ray selected they will be virtually unbiased against dusty environments, a serious limitation of all present studies of high z galaxies.
Optical spectra demonstrate clearly that GRB observations can be used to probe the ISM of the GRB host galaxy. Using a GRB afterglow spectrum, Castro et al. (2001) discovered two absorption systems in the GRB host galaxy separated by just $\sim150$ km s$^{-1}$. But optical observations are unable to investigate the abundant hot material expected. Neither can normal ultraviolet, or infrared, telescopes detect this material. Only the far-UV and soft X-ray bands contain the necessary spectral signatures.
Spectroscopy Goals
==================
The soft X-ray band will measure OVII and OVIII, while the FUV band will measure OVI. The ratios of these lines will determine the temperature of the WHIM for each line of sight (figure 2). Crucially, the two line ratios come from 3 ions of the same atom. If the ratios disagree on the temperature, then we will know that we are not dealing with a simple collisional plasma (Nicastro et al. 2002a).
The soft X-ray spectra will also contain NeIX, while the FUV contains iron, magnesium, silicon, silicium, carbon and zinc and hydrogen Lyman-$\alpha$ lines. With the oxygen ratios distinguishing unambiguously between photoionization and collisional ionization, these other lines will give us abundances and enrichment histories. This will tell us the history of supernovae, and of the recycling of matter from galaxies and quasars. Some elements condense onto dust grains more easily than others (Pettini et al. 1997, 1999), so the dust content of the universe will also be measured. Dust is a catalyst for further star formation, and so the WHIM dust content should link with the star formation history of the Universe. Dust dims light from more distant objects, and allowing for this (probably small) effect might change the cosmological parameters derived from SN1a light curves.
Why GRBs are the Best Path
==========================
The spectroscopic goals above are challenging. They require a large number of photons in each spectrum. We can use the example of the Chandra LETGS/HRC PKS 2155-305 spectrum of Nicastro et al. (2002a) to determine how many photons are needed to obtain sufficient signal-to-noise.
The Chandra LETGS/HRC PKS 2155-305 spectrum has between $\sim$600 and $\sim$1200 counts per resolution element. The most intense lines detected in PKS 2155-304 have an equivalent width, EW=10.4 mA. The resolution element with the line centroid contains $N(line)$= 511 counts, and the continuum around the line has instead $N(cont)$= 692 counts/res.element. The difference $D(centroid)$= \[$N(cont)$ - $N(line)$\] = 181 counts/res.element. To detect such a line at $M\sigma$ we need at least $N(line)$ such that $D(centroid) >= N(line)[1 + M/\surd{N(line)}]$. $D$ depends both on the EW of the line and on the resolution. For an instrument with a resolving power $R = 6000$, $N(cont)$ is reduces so $D$ is bigger for a given EW, and a detection at a given $M\sigma$ is easier to obtain (figure \[ctsigma\]).
\[ctsigma\] ![[*Left:*]{} Number of counts per resolution element (left vertical axis) vs. detection significance, $\sigma$. The right vertical axis shows the Area, $A$, fluence, $F$, resolution, $\Delta\lambda$, efficiency, $\eta$, product needed to obtain the corresponding number of counts per resolution element. [*Right:*]{} Minimum detectable absorption line equivalent width vs. resolving power, $R$.](counts_vs_sigma.eps "fig:"){height="7cm"} ![[*Left:*]{} Number of counts per resolution element (left vertical axis) vs. detection significance, $\sigma$. The right vertical axis shows the Area, $A$, fluence, $F$, resolution, $\Delta\lambda$, efficiency, $\eta$, product needed to obtain the corresponding number of counts per resolution element. [*Right:*]{} Minimum detectable absorption line equivalent width vs. resolving power, $R$.](ew.ps "fig:"){height="7cm"}
To resolve a line, the above computation has to be scaled for the line profile in the contiguous resolution elements, with 3-5 elements being minimal. For a gaussian with a FWHM of 100 km s$^{-1}$, at 21.6Å and an instrument with a resolution of 6000 at the same wavelength, the line would be resolved into roughly 3-5 resolution elements. For a gaussian the 2 resolution elements either side of the one containing the centroid would each contain roughly 15% of the total number of counts. This is about 20% of the number of counts contained in the centroid-resolution element. So, $D(wings) \sim 0.2 D(centroid)$.
BeppoSAX results showed that at least 7 out of 11 SAX GRB with fluence (i.e. integrated flux) $>$1$\times$10${-5}$ have bright X-ray and optical afterglows (Fiore 2001). Hence these bursts will provide the photons that will yield the high quality spectra needed to carry out the science. Because GRB X-ray afterglows can be so bright, prompt GRB observations (i.e. within one or a few minutes - a [*Swift*]{}-like response) provide a huge advantage compared to bright quasar observations, by providing large fluence of X-ray photons (Fiore et al. 2000). For example: a 40ks (1/2 day) observation of a GRB with the same peak X-ray flux as GRB990123 or GRB010222 (In’t Zand et al. 2001, Jha et al. 2001, Masetti et al. 2001) that begins 1 minute after the burst onset provides a fluence (10$^{-5}$ erg cm$^{-2}$, 0.5-2 keV band) equivalent to a one million second (about 2 weeks) long observation of a bright ($F_X\sim10^{-11}$ , or 0.5 mCrab) z=1 AGN. At z$>$0.2 there are only a dozen AGNs in the sky as bright as this, while there are several such GRB each year.
Figure \[lnls\] gives the number of GRB per year (at high galactic latitude) which give a 0.5-2 keV fluence (in a 40 ks observation) of $10^{-5}$ and $10^{-6}$ erg cm$^{-2}$, as a function of the delay time between the GRB and the start of the observations. The GRB keV logN-logF of figure \[lnls\] (using a slope of -1.3) can give the total X-ray fluence from bright bursts ($fluence(X)\gs 10^{-6}$ erg cm$^{-2}$) per year. If 40% of the high Galactic latitude GRB are included (since about half will be occulted, or caught too close before occultation, by the Earth for a LEO satellite) with a delay of one minute, this gives a total fluence of about $\sim4\times10^{-5}$ erg cm$^{-2}$, equivalent to a 4 million seconds observation of a $F_X\sim10^{-11}$ erg cm$^{-2}$ s$^{-1}$ AGN. A HEO satellite would record double this total fluence. If the delay time rises to 10 min, then the total fluence would be reduced by two thirds.
\[lnls\] {height="7cm"}
In the optical and UV bands observations of GRB afterglows are probably less efficient with respect to quasars for the study of the IGM, since quasars have a much bluer spectrum than GRB optical afterglows. There are about 100 quasars with UV flux $>10^{-14}$ Hz$^{-1}$, accessible to FUSE, while to obtain spectra of good quality (S/N$>$20 per resolution element) of a GRB afterglow needs an instrument with an effective area about 10 times that of FUSE. However these 100 quasars would need a [*much*]{} larger X-ray telescope to give the OVII and OVIII lines. Only GRBs give both easily.
There will be about 20 GRBs/year with $fluence(X)=1\times
10^{-7}$. These bursts provide the driving science of the mission - velocity resolved absorption lines of the WHIM. They will determine the state of the X-ray forest lines. In addition to the primary GRBs there will be 80/year that are faint ($fluence(X)=3\times 10^{-7}$. For these we will detect strong lines at redshifts inaccessible to SPIDR. This will give us a robust statistical view of the formation of the Cosmic Web with time. About once per year a bright ($fluence(X)=1\times10^{-5}$) GRB will be caught, allowing resolved velocity structure to be seen even in faint lines, which will give much tighter constraints on nucleosynthesis in the Age of Star Formation.
The X-ray coded aperture GRB location instrument will collect a $\sim$5000 count CCD spectrum for a GRB of medium fluence. This is sufficient to detect an Fe-K line at the GRB redshift, freeing us of the need for optical ground-based follow-up (although this would still be valuable). Changes in Fe-K strength with time seem to be strong (Reeves et al. 2002), and can be studied with the mission.
The combined telescopes make up a powerful instrument package that will be able to study the IGM on a whole range of scales. Since we will follow about 120 GRBs per year, and each one will be detectable for no more than one day, the satellite will spend a sizeable fraction of the time, 240 days/year, observing the IGM on more local objects:
[**z$<$0.2, Filaments between Nearby Clusters:**]{} Long observations of the 10-20 brightest AGN in the sky with z$>$0.2. Each of these observations will take about 1 month, (for 50% efficiency in LEO), and so will fill a 2-year secondary science mission. Triggers from X-ray all sky monitors may allow us to catch some of these AGN in outburst, and so with fluxes 5-10 times normal. Most AGN have ‘warm absorbers’ similar to the IGM, but denser. They form a fast wind emanating from the AGN, and their large outflow velocities can be confused with intervening matter. Fortunately during this month of monitoring most AGN vary significantly, which will cause changes in any absorbing material close to the AGN, distinguishing them clearly from the IGM. These variations will determine the density of AGN winds, a valuable byproduct of the mission.
[**Damped Lyman-$\alpha$ Absorbers (DLAs):**]{} DLAs may be protogalaxies (Prochaska & Wolfe 1999). Soft X-ray/FUV spectra will determine the abundances and enrichment processes of DLAs (Bechtold et al. 2001).
[**The Local Group Filament:**]{} Observations of X-ray sources in the Local Group (the Magellanic Clouds, M31, M81/M82, M33) and of the Milky Way halo via globular cluster X-ray binaries, will separate out the warm gas in our galaxy halo from gas in the IGM filament in which our galaxy and the Local Group of galaxies lie.
INSTRUMENTATION
===============
Energy/Wavelength Coverage
--------------------------
All the interesting X-ray spectral features are at E$<$2 keV, even in the rest frame, and for z$>$1 they all lie at E$<$1 keV (figure 5). Hence the primary science goals of the mission require that the X-ray spectra should reach to as low an X-ray energy as possible ($\sim$0.1 keV), but need not reach to higher energies than 1-2 keV.
The FUV allows the study of the OVI 1054Å and Ly$\alpha$1215Å lines. The OUV band (1500-4000Å) is needed if we are to study lines of several elements including: carbon, silicon, iron and zinc.
\[lines\] {height="7cm"}
Spectral Resolution
-------------------
The resolution of a spectrometer is normally measured in terms of [*resolving power*]{}, which is the wavelength divided by the FWHM of the instrumental line width: $R$=$\lambda/\Delta\lambda$.
A resolving power of $R\sim$400 was just sufficient to detect the strongest absorption lines from the WHIM (Nicastro et al. 2002a), but is not enough to pin down cosmological models from their detailed properties. A clear example of the detail lost at $R=1000$ is given in figure 6, which compares in velocity space the absorption lines detected by FUSE and Chandra LETGS-HRS for the BL Lacertae object PKS2155-301 (Nicastro et al. 2002a). At the FUSE resolution the OVI system is resolved in at least two components, one narrow and one broad. Although this complex system is detected with a very high significance in the soft X-ray spectrum the 10 times lower resolution does not resolve the components. A resolution of 6000 ($\Delta v$=50 km s$^{-1}$) is needed to resolve the thermal oxygen lines (for T$\gs4\times10^6$ K, see also Elvis 2001).
The FUV allows the study of the OVI 1054Å and Ly$\alpha$1215Å lines. Here a resolution of $R$=10,000, comparable to FUSE, would allow the detection and the characterization of faint oxygen lines. A resolution of $R$=3000 is barely enough for the detection of strong OVI lines, but would allow detection of Ly$\alpha$ absorption lines.
\[2155\] {height="7cm"}
Rapid GRB Trigger and Localization
----------------------------------
To obtain spectra of GRB afterglows, first one must find them, quickly. Moreover there are only about 100 bright GRBs at high Galactic latitude per year, so few can be missed. Hence the initial GRB trigger detector needs the largest possible field of view: 2$\pi$-4$\pi$ steradians; yet to use the X-ray/FUV spectrometers requires a position accurate to at least an arcminute. These are hard requirements to reconcile in a single instrument. By concentrating solely on X-ray bright GRBs we can divide the problem into two parts: (1) triggering on a GRB detection, and (2) localizing the GRB accurately enough for the soft X-ray/FUV spectrometers. Moreover, since this is a soft X-ray spectroscopy mission the GRB localization instrument only needs to operate in the typical 0.5-10 keV band of CCD detectors. The high energy response of BAT on [*Swift*]{} is not needed. In this way the GRB detection and localization system can be far more compact and light weight than that of [*Swift*]{}. ([*Swift*]{} will pursue [*all*]{} GRBs.) Since [*Swift*]{} can slew 1 radian in a minute, our much smaller and lower moment-of-inertia satellite should have no problem slewing 2 radians in a similar time.
\(1) [*Trigger:*]{} A GRB monitor (GRBM) similar to (but less sensitive and $\sim10$ times smaller than) BATSE, and capable of providing positions with accuracies of 1-10 degrees. The CsI GRB alert monitor on Beppo-SAX provides a good model. The spacecraft autonomously decides whether each burst detected satisfies the trigger criteria for a re-pointing of the spacecraft, and then slews to the rough GRB position.
\(2) [*Localization:*]{} An X-ray Wide Field Coded Mask (WFCM) with a CCD detector (e.g. an XMM EPIC-pn wafer) provides a fine position. A field of view 2-3 times the size of the GRBM positional error regions ($10\times10$ or $20\times 20$ degrees) is sufficient to safely cover the error box provided by the GRBM, which the spacecraft will have placed near the center of the WFCM field of view. This relatively small field of view (c.f. BAT on [*Swift*]{}) allows the use of pixels small enough to provide positions accurate to within 1 arcmin. The WFCM provides the improved GRB position to the spacecraft to perform a fine maneuver to put the high resolution spectrometers onto the target GRB. The WFCM needs sufficient signal to locate the burst in a few seconds, else valuable fluence is lost. The minimum area needed to obtain such a position in a few seconds is somewhat smaller than that of an EPIC-pn chip ($\sim36$ cm$^2$). The computation of the GRB position will be greatly speeded up since there will be only a single source producing mask shadows on the detector.
X-ray Mirror Area
-----------------
Our science goals show that we need to design an X-ray telescope providing an effective area of at least 1000 cm$^2$ below 1 keV. The mirror needs a relatively sharp PSF (HPD$\sim$5 arcsec) in order to produce high spectral resolution using gratings. There are reasons for optimism that 5 arcsec HPD can be obtained. Silicon carbide shells built by the Merate (Italy) group (Citterio, Pareschi and collaborators) have 11 arcsec HEW figure errors. These shells are 2mm thick, 60cm diameter, 3.5m focal length, and have a weight/effective area ratio of 0.06 kg cm$^{-2}$. Mandrel quality is expected to improve (Pareschi et al. 2001), and the HEW should roughly scale linearly with the thickness down to about a HEW of 5 arcsec, where other effects start to be more important than the mirror deformations.
For a small mission weight is a major constraint. As a rough estimate the mirror will need a support structure of similar weight. The detector and its associated electronics will likely have a similar mass, and the total payload will likely be 50% of the total mass to orbit, the spacecraft taking the remaining fraction. The mirror should then account for about 10% of the total to-orbit mass. For a MIDEX the mass to orbit is about 1000 kg), this implies a mirror mass of $<$100 kg. We have explored the mirror parameter space with the [mirror]{} raytrace code kindly provided by Leon Van Speybroeck. This code was used extensively for [*Chandra*]{} design and development. We show here the results of three representative design options: thin vs. thick shells, short vs. long focal length, and iridium vs. nickel coating. They give a feel for the effective areas which the X-ray mirror can achieve within the parameter space. The effective area results for simulations of mirror designs assuming a ‘minimal’ mass telescope, and a ‘maximal’ mass telescope (figure 7). In both designs various combination of mirror coatings have been used, with Ni being used on the outer shells and Ir on the inner shells. Nickel provides better reflectivity below 0.8 keV, and iridium above 0.8 keV, so the mixed coating designs use iridium for the inner shells, and nickel for the outer shells.
\[mirror\] ![[*left:*]{} Raytraced mirror effective areas for a ‘minimal’ mirror design based on 20 2mm thick shells, 1.5 m focal length, 60 cm outer mirror diameter, and 30 cm parabola length. The total mirror weight (including a 100% support structure margin) is 96 kg. [*right:*]{} ‘maximal’ mirror design: 20 3mm thick shells, 2 m focal length, 80 cm outer mirror diameter, and 40 cm parabola length. The total mirror weight (including a 100% support structure margin) is 260 kg.](aef_60_30_1500.ps "fig:"){height="7cm"} ![[*left:*]{} Raytraced mirror effective areas for a ‘minimal’ mirror design based on 20 2mm thick shells, 1.5 m focal length, 60 cm outer mirror diameter, and 30 cm parabola length. The total mirror weight (including a 100% support structure margin) is 96 kg. [*right:*]{} ‘maximal’ mirror design: 20 3mm thick shells, 2 m focal length, 80 cm outer mirror diameter, and 40 cm parabola length. The total mirror weight (including a 100% support structure margin) is 260 kg.](aef_80_40_2000.ps "fig:"){height="7cm"}
X-ray Gratings
--------------
The high spectral resolution required by our science drivers is technically feasible. If the XMM-Newton RGS reflection grating facets were used behind the [*Chandra*]{} mirror the result would be $R\sim 5000$ (subject to better facet alignment, Elvis 2001). But is $R=6000$ feasible in a small mission?
X-ray calorimeters have excellent quantum efficiency and are the natural spectrometer of choice at the 6.4 keV Fe-K complex. However they are not a good choice at 0.5 keV, since their resolution is fixed in eV, so $R$ degrades linearly with energy. The Con-X calorimater goal is to have $\Delta E$=1 eV, or $R$=500 at 0.5 keV. Gratings are a requirement for this mission.
A route to achieving $R=6000$ with 5 arcsec HPD mirrors is offered by out-of-plane reflection gratings (ORGs, Cash 1991). ORGs are being considered to boost the resolution of the Con-X low energy spectrometers to $R$=1000-2000. Spectral resolution improves linearly with angular resolution. So with a factor 3 better mirror, and an optimized alignment, $R=6000$ becomes feasible. ORGs also have favorable efficiency, peaking at $\sim$10-15%. (c.f. 4% of the XMM RGS, and the 10% of the [*Chandra*]{} LETGS), so requiring a smaller mirror to feed them photons.
Filters are needed to eliminate stray optical light. To avoid losses at and above the 0.28 keV Carbon edge, we will use aluminum filters, which have an edge at 1.5 keV, above our primary range of interest. Similar filters have been successfully flown on other missions.
Figure of Merit: Comparison with other Missions
===============================================
A figure of merit (FoM) for a system mirror+gratings can be given by the following formula: $$FoM=A_{eff}(cm^2) \times \epsilon_{peak} \times R(0.5 keV).$$ This FoM measures the ability of various systems to detect faint absorption lines, since the minimum detectable line equivalent width, $EW_{min}\propto\surd{R(eV)/A_{eff}}$. Table \[fom\] shows that the mission we propose represents a large gain over existing or planned missions, even for warm IGM [*detection*]{}. Higher resolution spectroscopy not only allows the line detection, but is essential to the study of the physics and the dynamics of the absorbing systems. [*No other mission*]{} can resolve the X-ray absorption lines, and so take the subject to the next level of physics and cosmology.
Mission A$_{eff}$(cm$^2$)$\times$ R(0.5 keV)$\times$$\epsilon_{peak}$ FoM
--------------------------- --------------------------------------------------------------- -----------
Current Missions
XMM 1 RGS 70$\times$ 300$\times$ 0.1 2100
Chandra LETGS 100$\times$ 500$\times$ 0.1 5000
Chandra MEG 75$\times$ 1200$\times$ 0.1 9000
Future Missions
[**‘minimal’ mission**]{} 1000$\times$ 6000$\times$ 0.1 600,000
[*‘maximal’ mission*]{} 2000$\times$ 6000$\times$ 0.15 1,500,000
Swift 100$\times$ 10$\times$ 1 1,000
Con-X 1 grating 1250$\times$ 500$\times$ 0.1 62,500
Con-X 4 gratings 5000$\times$ 500$\times$ 0.1 250,000
: Comparison of Absorption Line Detection Capability of Relevant Missions[]{data-label="fom"}
Note: includes response time estimate of 4 hours for missions other than this, and 1 minute for this mission, assuming a power-law decay index of $-$1.3.
Conclusions
===========
High resolution ($R\ge$6000) soft X-ray spectroscopy is feasible within the scope of a modest mission. This resolving power would open up new realms of physics to astronomy.
A rapid response mission to obtain high resolution soft X-ray and Far-UV spectra of gamma-ray burst afterglows is also possible. Such a mission would go well beyond current and planned missions in determining the formation and nature of the warm-hot intergalactic medium, the Cosmic Web.
references
==========
Aldcroft T.L., Elvis M., McDowell J.C., and Fiore F., 1994, ApJ, 437, 584
Bechtold, J., Siemiginowska, A., Aldcroft, T., Elvis, M. & Dobrzycki, A. 2001, ApJ, 562, 133
Bloom, J.S., Kulkarni, S.R. & Djorgovski, S.G. 2001, AJ, in press, astro-ph/0010176
Cash, W. 1991, Applied Optics, 30-13, 1749
Castro, S., et al. 2001 ApJ, submitted, astro-ph/0110566
Cen, R., Ostriker, J.P. 1999a, ApJ, 514, 1
Davè, R. et al. 2001, ApJ, 552, 473
Djorgovski, S.G. et al. 2001, 562, 654
Elvis, M. 2001, proceedings of ‘New Century of X-ray Astronomy’, Yokohama, Japan, astro-ph/0106053
Fiore, F., Nicastro, F., Savaglio, S., Stella, L. & Vietri, M. 2000 ApJL, 544, L7
Fiore, F. 2001, proceedings of ‘New Century of X-ray Astronomy’, Yokohama, Japan, astro-ph/0107276
In’t Zand, J.J.M. et al. 2001, ApJ, 545, 266
Jha, S. et al. 2001, ApJL, 554, L155
Hellsten, U., Gnedin, N.Y., Miralda-Escudè, J. 1998, ApJ, 509, 56
Madau P., Pozzetti L. & Dickinson M., 1998, ApJ, 498, 106
Masetti, N. et al., 2001, A&A, 374, 382, astro-ph/0103296
Nicastro, F. et al. 2002a, ApJ, in press, astro-ph/0201058
Nicastro, F. et al. 2002b, [*Nature*]{}, submitted, [astro-ph/0208012]{}
Pareschi G. et al. 2001, NCXA Conf. 526.
Perna, R, & Loeb, A. 1998, ApJL, 503, L135
Pettini, M., Smith, L.J., King, D.L. & Hunstead, W. 1997, ApJ, 486, 665
Pettini, M., Ellison, S.L., Steidel, C.C. & Bowen, D.V. 1999, ApJ, 510, 576
Prochaska, J.X. & Wolfe, A.M. 1999, ApJS, 121, 369
Reeves J. et al. 2002, [*Nature*]{}, 416, 512
Savaglio, S., Fall, M., & Fiore, F. 2002, ApJ, submitted
Sembach K. et al. 2002, ApJS, submitted, [astro-ph/0207562]{}
Zappacosta L. et al. 2002, A&A, in press [astro-ph/0208033]{}
[^1]: Contributions by: Andy Szentgyorgyi, Fabrizio Nicastro, Rob Cameron (SAO), Giovanni Pareschi, Oberto Citterio (OABrera), W. Cash (U.Colorado), L. Stella (OAR), C. Norman, J. Rhoads, J. Krolik, S. Savaglio, K. Weaver, T. Yaqoob, T. Heckman, K. Sembach, D. Bowen, J. Kruk, M.B. Kaiser, S. McCandless (JHU), A. Fruchter (STScI), S. Mathur (OSU), J. Greiner, G. Hasinger (MPE). Many thanks to L. van Speybroeck (SAO), L. Angelini (GSFC), and P. Miotto (Draper Lab.)
|
---
abstract: |
We examine the solvent-mediated interaction between two neutral colloidal particles due to preferential adsorption in a near-critical binary mixture. We take into account the renormalization effect due to the critical fluctuations using the recent local functional theory $[$J. Chem. Phys. [**136**]{}, 114704 (2012)$]$. We calculate the free energy and the force between two colloidal particles as functions of the temperature $T$, the composition far from the colloidal particles $c_\infty$, and the colloid separation $\ell$. The interaction is much enhanced when the component favored by the colloid surfaces is poor in the reservoir. For such off-critical compositions, we find a surface of a first-order bridging transition $\ell=
\ell_{\rm cx}(T,c_\infty)$ in the $T$-$c_\infty$-$\ell$ space in a universal, scaled form, across which a discontinuous change occurs between separated and bridged states. This surface starts from the bulk coexistence surface (CX) and ends at a bridging critical line $\ell=
\ell_{ c}(T)$. On approaching the critical line, the discontinuity vanishes and the derivatives of the force with respect to $T$ and $\ell$ both diverge. Furthermore, bridged states continuously change into separated states if $c_\infty$ (or $T$) is varied from a value on CX to value far from CX with $\ell$ kept smaller than $\ell_c(T)$.
address: |
$^1$Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan\
$^2$Department of Physics, Kyoto University, Kyoto 606-8502, Japan
author:
- Ryuichi Okamoto$^1$ and Akira Onuki$^2$
title: |
Attractive interaction and bridging transition between neutral colloidal particles\
due to preferential adsorption in a near-critical binary mixture
---
Introduction
============
Much attention has been paid to the physics of fluids in restricted geometries [@Evansreview; @Gelb]. The microscopic interactions between the fluid molecules and the solid surface can greatly influence the phase transition behavior of the confined fluid [@Is]. The liquid phase is usually favored by the walls in fluids undergoing gas-liquid phase separation, while one component is preferentially attracted to the walls in binary mixtures. In the film geometry, narrow regions may be filled with the phase favored by the confining walls or may hold some fraction of the disfavored phase. Between these two states, there can be a first-order phase transition, called capillary condensation [@Evansreview; @Gelb; @Butt], depending on the temperature $T$, and the reservoir chemical potential $\mu_\infty$ for each given wall separation $D$. This phenomenon occurs both in one-component fluids and binary mixtures.
As another aspect, adsorption-induced density or composition disturbances are known to produce an attractive interaction between solid objects [@Hansen; @Evans-Hop]. In binary mixtures, it is amplified when the solvent far from these objects is poor in the component favored by the surfaces [@Evans-Hop]. Such solvent-mediate interactions should play an important role in reversible aggregation of colloidal particles in near-critical binary mixtures at off-critical compositions [@Beysens; @Maher; @Bonn; @Guo]. In such situations, strong preferenial adsorption was observed by light scattering [@Beysens]. It is worth noting that the colloid-wall interaction in a near-critical fluid has been measured directly [@Nature2008; @Nellen]. We mention some theoretical papers, which treated the solvent-mediated colloid interaction in an early stage [@Slu; @Two; @Lowen; @Netz; @Kaler].
However, other interactions come into play in real systems. First, we should account for the van der Waals (dispersion) interaction, which sometimes gives rise to intriguing effects in wetting behavior [@Is; @Butt; @Bonnreview; @Russel]. In this paper, we examine importance of the van der Waals interaction as compared to the adsorption-induced interaction. Second, in aqueous fluids, the colloid surface can be ionized and the counterions and added ions form an electric double layer, resulting in the screened Coulomb interaction [@Is; @Butt; @Russel]. This repulsive interaction can be very strong close to the surface, but it decays exponentially with the Debye screening length $\kappa^{-1}$. Third, in near-critical fluids, the ion distributions and the critical fluctuations become highly heterogeneous around the colloid surfaces [@Okamoto]. As a result, the wetting layer formation and the surface ionization are strongly coupled, which much complicates the colloid interaction.
On approaching the solvent criticality, the adsorption-induced interaction becomes long-ranged and universal [@Okamoto; @Fisher; @Fisher-Yang; @Gamb; @Upton], where the wall-induced heterogeneities extend over mesoscopic length scales. In the film geometry, some universal scaling relations are well-known and considerable efforts have been made to calculate [@Gamb; @Upton] or measure [@Law; @Nature2008; @Gamb] the so-called Casimir amplitudes (coefficients in universal relations) [@Casimir], In these papers, near-critical fluids at the critical composition have mostly been treated along the critical path $\mu_\infty=0$. On the other hand, Maciołek [*et al*]{} [@Evans-Anna] found strong enhancement of one of the amplitudes in two-dimensional Ising films under applied magnetic field. In accord with their finding, we have recently found growing of the amplitudes at off-critical compositions [@OkamotoCasimir], which is particularly marked near a first-order capillary condensation line in the $T$-$\mu_\infty$ plane. We have also examined phase separation dynamics around the capillary condensation line [@Yabunaka].
In this paper, we aim to investigate the interaction between two neutral colloidal particles due to preferential adsorption in a near-critical binary mixture. We shall see that the solvent-mediated interaction is much enhanced when the component favored by the colloid surfaces is poor in the reservoir, as in the case of the Casimir amplitudes. We also aim to examine the bridging transition between two colloidal particles [@Butt; @Butt1], which is analogous to the capillary condensation transition in a film. That is, two large particles (or one large particle and a plate) are connected by the phase favored by the walls in bridged states, while they are disconnected by intrusion of the disfavored phase in separated states. Bridged states appear near the bulk coexistence curve as the separation distance is decreased. As previous papers on bridging, we mention numerical calculations of phenomenological models [@Yeomans; @Vino], density functional theories [@Bauer; @Evans-Hop], and a Monte Carlo study [@Higashi]. We also note that a bubble bridging can occur between hydrophobic surfaces in water [@bubble], which is related to predrying of hydrophobic surfaces [@Teshi]. Similarly, in the isotropic phase of liquid crystals, a nematic domain can appear between closely separated solid objects [@Zu; @Fukuda].
The organization of this paper is as follows. In Sec.II, we will summarize the results of the local functional theory of near-critical binary mixtures. In Sec.III, we will present a theory on the adsorption-induced interaction among colloidal particles together with some simulation results. In Sec.IV, we will numerically investigate the bridging transition near the bulk criticality.
Renormalized Ginzburg-Landau free energy
========================================
We consider near-critical binary mixtures using our local functional theory taking into account the renormalization effect near the bulk criticality, which is similar to the linear parametric model by Schofield [*et al.*]{} [@Sc69; @Onukibook] and the local functional model by Fisher [*et al.*]{} [@Upton; @Fisher-Yang]. These authors treated near-critical fluids outside CX, while we define our model within CX. Furthermore, our model satisfies the two-scale-factor universality[@Onukibook]. The critical amplitude ratios from our model are in fair agreement with reliable estimates for Ising systems.
We assume an upper critical solution temperature $T_c$ at a given average pressure. The order parameter $\psi$ is proportional to $c-c_c$, where $c$ is the composition and $c_c$ is its critical value. The physical quantities exhibit singular dependence on $\psi$ and the reduced temperature, = (T-T\_c)/T\_c. Hereafter, $\alpha=0.110$,$\beta=0.325$, $\gamma=1.240$, $\nu=0.630$, $ \eta=0.0317$, and $ \delta=4.815$ are the usual critical exponents for Ising-like systems [@Onukibook]. At the critical composition with $\tau>0$, the correlation length is written as $\xi = \xi_0 \tau^{-\nu}$, where $\xi_0$ is a microscopic length. The coexistence curve in the region $\tau<0$ is denoted by CX. We write $\psi$ in the coexisting two phases as $\pm \psi_{\rm cx}$ with \_[cx]{} = b\_[cx]{}||\^, where $ b_{\rm cx}$ is a constant.
We set up the singular bulk free energy $F_b $, where the critical fluctuations with wave numbers larger than the inverse correlation length $\xi^{-1}$ have been coarse-grained or renormalized. Including the square gradient term, $F_b$ is of the local functional form [@Fisher-Yang; @Upton; @OkamotoCasimir], F\_b = d[r]{}\[f + k\_BT\_c C||\^2\]. where the integral $\int d{{\mbox{\boldmath$r$}}}$ is within a cell. Outside CX ($|\psi|>\psi_{\rm cx}$), the singular free energy density $f= f(\psi,\tau)$ is written in the Ginzburg-Landau form, = k\_BT\_c(r\^2+ u\^4). We do not write a constant term ($\propto
|\tau|^{2-\alpha}$),which is a singular contribution for $\psi=0$. In this paper, $C$ is made dimensionless. Then, $\xi_0^{1/2}\psi$ is dimensionless and $ b_{\rm cx}$ in Eq.(2.2) is of order $ \xi_0^{-1/2}$. In the mean field theory, $C$, $r/\tau$, and $u$ in $F_b$ are constants independent of $\tau$ and $\psi$. In our renormalized functional theory, they depend on a nonnegative variable $w $ representing the distance from the criticality in the $\tau$-$\psi$ plane. Outside CX, fractional powers of $w$ appear as [@Casimir-comment] C &=& w\^[-]{},\
r/&=& \_0\^[-2]{}w\^[-1]{} ,\
u &=& u\^\* \_0\^[-1]{}w\^[(1-2)]{}, where $u^*$ is a universal number and is set equal to $2\pi^2/9$ in our numerical analysis.
From $\eta \ll 1$, we have $C \cong 1$. We determine $w$ as a function of $\tau$ and $\psi$ by w= + (3u\^\* \_0) w\^[1-2]{}\^2. For $\psi=0$, we simply have $w=\tau$. For $\tau=0$, we obtain $w^\beta \propto |\psi|$, leading to the Fisher-Yang results [@Fisher-Yang]: $\xi \propto |\psi|^{-\nu/\beta}$ and $f \propto \xi^{-d} \propto
|\psi|^{1+\delta}$. These authors introduced the local correlation length $\xi(\psi)$ for $\tau=0$.
In our scheme, $\xi $ and the susceptibility $\chi$ are related to the second derivative $f''=
\p^2 f/\p \psi^2 $ by k\_BT\_cC/\^2=k\_B T\_c/=f” . For $\tau>0$ and $ \psi=0$, we find $\chi(\tau,0)= \xi_0^2 \tau^{-\gamma}$. On approaching CX ($\psi \to \psi_{\rm cx}$), we require $f'= \p f/\p \psi \to 0$ to obtain $b_{\rm cx}=1.50 /(3u^*\xi_0)^{1/2}$ and $w=1.714|\tau|$. The susceptibility on CX is determined by $\tau$ and is written as \_[cx]{}= (,\_[cx]{}) = R\_\_0\^2 ||\^[-]{}, with $R_\chi = 8.82$. The correlation length on CX is written as $\xi_{\rm cx}=
0.334 \xi_0|\tau|^{-\nu}$.
We also need to determine $f$ inside CX ($|\psi|<\psi_{\rm cx}$ and $\tau<0$) to discuss phase separation. Its simplest form is =[f\_[cx]{}]{}+ k\_B[T\_c]{} (\^2-\_[cx]{}\^2)\^2/ (8[\_[cx]{}]{}[ \_[cx]{}\^2 ]{}) , where $f_{\rm cx}$ is the free energy density on CX. Then, $f$, $ f'$, and $ f''$ are continuous across CX. We also set $C=C_{\rm cx}= |\sigma_{\rm cx}\tau|^{-\eta\nu}
$ inside CX, which is the value of $C$ in Eq.(2.5) on CX. Here, we neglect the thermal fluctuations longer than $\xi_{\rm cx}$. In our applications, the space regions inside CX are not wider than $\xi_{\rm cx}$ and the $\psi^4$ form in Eq.(2.11) is well justified. As an example, we may calculate the interface profile from Eqs.(2.3) and (2.11), where the surface tension is of the form [@Onukibook], = 0.075 k\_BT\_c/\_[cx]{}\^2. We shall see another example inside CX in Fig.1.
Colloidal particles in a near-critical fluid
=============================================
We consider identical colloidal particles with common radius $a$ much larger than $ \xi_0$ in a near-critical binary mixture. We seek equilibrium profiles of $\psi({{\mbox{\boldmath$r$}}})$ around these large particles. We assume $\psi\to\psi_\infty$ far from them, where $\psi_\infty$ is proportional to the composition deviation $c_\infty-c_c$ far from the colloidal particles. In its calculation, we take the limit of strong preferential adsorption. This $\psi({{\mbox{\boldmath$r$}}})$ minimizes the grand potential $\Omega$, giving rise to attraction among the colloidal particles [@Slu; @Two; @Lowen; @Netz; @Okamoto]. Typical reduced temperatures in this paper are from $-1$ to $-10$ in units of $ (\xi_0/a)^{1/\nu}$ and are very small for large $a$. Then the prewetting transition [@Bonnreview] may be assumed to occur at lower temperatures. In fact, we realize thick adsorption layers in our numerical analysis.
Equilibrium relations
---------------------
On the cell surface we assume ${{\mbox{\boldmath$n$}}}\cdot \nabla\psi=0$ for simplicity, but on the colloid surfaces we assume = -h\_1 /C where ${\mbox{\boldmath$n$}}$ is the normal unit vector from the interior to the exterior and $h_1$ is a large positive surface field arising from the short-range, fluid-surface interaction. In equilibrium, we minimize the grand potential, consisting of the bulk term and the surface term as = d[r]{}\_[loc]{} -k\_BT\_c dS h\_1 . Hereafter, $\int d{{\mbox{\boldmath$r$}}}$ is the space integral outside the colloidal particles and in the cell, while $\int dS$ is the surface integral on the colloid surfaces. We define the grand potential density including the gradient contribution, \_[loc]{}= f() -f\_ -\_(-\_) + [k\_BT\_c]{} ||\^2, where $f_\infty=f(\psi_\infty)$ and $\mu_\infty$ is related to $\psi_0$ by \_= f’(\_). In particular, $\mu_\infty\cong
(\psi_\infty+ \psi_{\rm cx})/\chi_{\rm cx}$ close to the negative branch of CX. The $\omega_{\rm loc}$ is nonnegative in our case, tending to $0$ far from the colloidal particles. Minimization of $\Omega$ yields Eq.(3.1) as the boundary condition and = f’()-k\_BT\_cC \^2-[k\_BT\_c]{}||\^2 =\_, in the fluid region, where $C'(\psi)=dC/d\psi$.
In equilibrium, $\Omega$ is a function of the colloid centers ${{\mbox{\boldmath$R$}}}_\alpha=
(R_{\alpha x},R_{\alpha y}, R_{\alpha z})
$ ($\alpha=1, 2,\cdots$). In Appendix A, we will derive the following equilibrium relation, = \_dS \_[j]{} (\_[ij]{}-\_\_[ij]{}) n\_[j]{} , where $i, j=x,y,z$. The integral $\int_\alpha dS$ is on the surface of the $\alpha$-th colloidal particle and ${{\mbox{\boldmath$n$}}}_\alpha= (n_{\alpha x}, n_{\alpha y},
n_{\alpha z})$ is the normal unit vector from the colloid interior to the exterior. The $\Pi_{\psi ij} $ is the stress tensor due to the order parameter deviations given by [@Onukibook] && \_[ij]{} = (F\_b/- f- k\_BT\_c C ||\^2/2) \_[ij]{}\
&& +k\_BT\_cC (\_i) (\_j ). This tensor satisfies the relation, \_j \_j \_[ij]{} = \_i (F\_b/), which vanishes in equilibrium or under Eq.(3.5). Here, $\Pi_{\psi ij} \to \Pi_{\infty}\delta_{ij}$ far from the colloidal particles with \_= \_\_- f(\_) . If we further use Eq.(3.5), we obtain a simpler expression, \_[ij]{} = (\_ - \_[loc]{}) \_[ij]{} +k\_BT\_cC (\_i) (\_j ).
The expression (3.7) and the relation (3.8) are valid even in nonequilibrium and have in fact been used in dynamics[@Onukibook; @Yabunaka]. Note that the total stress tensor may be expressed as $p_0\delta_{ij}
+ \Pi_{\psi ij}$ in binary mixtures, where $p_0$ is a large background pressure nearly uniform in the cell (with small variations arising from sounds and gravity).
Scaling and strong adsorption limit
-----------------------------------
We make Eq.(3.5) dimensionless by scaling the position ${\mbox{\boldmath$r$}}$ by $a$ and $\psi$ by $\psi_a$, where $\psi_a$ is is a characteristic order parameter around the colloidal particles of the form, \_a = (\_0/a)\^[/]{}/(3u\^\* \_0)\^[1/2]{}, Use of $b_{\rm cx}$ in Eq.(2.2) gives $\psi_a= 1.47 b_{\rm cx}(\xi_0/a)^{\beta/\nu}$. By scaling $\tau$ and $\psi_\infty$, we introduce two parameters, && = (a/\_0)\^[1/]{},\
&& = \_/\_a. The scaled correlation length $\xi/a$ is given by $\hat{t}^{-\nu}$ for $\tau>0$ on the critical path, $0.13 |\hat{s}|^{-\nu/\beta}$ for $\tau=0$, and $0.3|{\hat{t}}|^{-\nu}$ on CX. The CX curve is expressed as $\hat{s}= \pm {\hat s}_{\rm cx}$ with ${\hat s}_{\rm cx}=\psi_{\rm cx}/\psi_a=
0.66|\hat{t}|^{\beta}$ from Eq.(2.2). In our calculations, we may use the scaled quantities only, where we need not specify the ratio $\xi_0/a \ll 1$. The scaling factors $\tau/\hat{t} =(\xi_0/a)^{1/\nu}$ and $\psi_\infty/ b_{\rm cx} \hat{s}= 1.47 (\xi_0/a)^{\beta/\nu}$. are needed when our theoretical results are compared with experimental data. For example, if $a/\xi_0=10^4$, they are $0.40\times 10^{-6}$ and $ 0.011$, respectively.
We write the value of $\psi$ on the colloid surfaces as $\psi_0$. For sufficiently large $\psi_0$, the near-wall behaviors of $\psi$ and $\omega_{\rm loc}$ are expressed as [@OkamotoCasimir; @Rudnick; @Fisher-Yang] && \~\_0\^[/-1/2]{} (+\_0)\^[-/]{},\
&& \_[loc]{} \~k\_BT\_c (+\_0)\^[-3]{}. where $\lambda$ is the distance from such a surface. We here assume that $\lambda$ is shorter than the correlation length $\xi= \xi(\tau,\psi_\infty)$ far from the surface. The length $\ell_0$ is of the order of the local correlation length near the surface ($\propto \psi_0^{-\nu/\beta}$) [@Fisher-Yang]. In terms of $b_{\rm cx}$ in Eq.(2.2), we have \_0= 0.544 \_0(b\_[cx]{}/\_0)\^[/]{}. where we assume $b_{\rm cx}^{-1}
\psi_0 \sim \xi_0^{1/2}\psi_0
\ll 1$ so $\ell_0\gg \xi_0$. In terms of $\psi_a$, we also have $\ell_0/a=
(\beta/2\nu) (\psi_a/\psi_0)^{\nu/\beta}$. For $\lambda\gg \ell_0$, $\psi$ and $\omega_{\rm loc}$ become independent of $\ell_0$ or $\psi_0$. From Eq.(3.1), we obtain the scaling relation, h\_1 \~C(\_0) \_0/\_0 \~\_0\^[-/]{}, where $\delta-\nu/\beta =(3-\eta)/(1+\eta)\cong 3$. The strong adsorption condition $\xi_0^{1/2}\psi_0 \gg |\tau|^\beta$ is realized with increasing $h_1$ or on approaching the bulk criticality. In our numerical analysis, we assume $h_1 /C(\psi_0) =170 \psi_a /a$ to obtain $\psi_0/\psi_a\sim 10$. See Fig.1 for the near-wall behaviors of $\psi$ and $\omega_{\rm loc}$ in the strong adsorption.
The integral of $\psi$ in the near-wall layers with $0<\lambda< \ell_0$ is proportional to $\psi_0\ell_0 \sim \psi_0^{1-\nu/\beta}$ and becomes negligible for large $\psi_0$ (since $\nu/\beta \sim 2$), while that in the region $\ell_0<\lambda<\xi(\tau,\psi_\infty)$ grows as $\xi^{1-\beta/\nu}$(critical adsorption) [@Bonnreview]. It follows a well-defined preferential adsorption, = d[r]{}\[([r]{})-\_\], which is independent of $h_1$ for large $h_1$. On the other hand, the integral of $\omega_{\rm loc}$ in the layers with $0<\lambda<\ell_0$ and the surface free energy in Eq.(3.2) ($\propto h_1)$ are both proportional to $\psi_0^{2\nu/\beta}$ and are large in magnitude. However, they are constants nearly independent of $\tau$ and $\psi_\infty$ and are irrelevant in the capillary condensation and the bridging transition (see discussions below Eq.(3.24)), which much simplifies our results.
In the strong adsorption regime, the profile of $\psi$ is highly nontrivial for negative $\psi_\infty $, since $\psi$ changes from a large positive value near the surface to $\psi_\infty<0$ far from it. To illustrate this aspect, we here consider the simplest case of a single spherical particle [@curved], where $\psi(r)$ is a function of the distance $r$ from the particle center. In this case, if $\psi_\infty $ approaches the CX value $-\psi_{\rm cx} $ under the condition $\xi \cong \xi_{\rm cx}\ll a$, the thickness of the adsorption layer increases logarithmically with increasing $a$ as [@curved] \_[ad]{}= (a/). It is also known that the contribution to $\Omega$ from the transition region ($r-a \sim \xi_{\rm ad}$) is of order $4\pi a^2 \sigma$, where $\sigma$ is the surface tension in Eq.(2.12).
In Fig.1, $\psi(r)/\psi_a$ and $\omega_{\rm loc}(r) a^3/k_BT_c $ are displayed around a single colloidal particle for $\hat{t}=-8$, where $\hat s=\pm 1.30$ on the positive and negative branches of CX.. For ${\hat s}=1.31$, the adsorption layer thickness is $\xi=0.09a$. For ${\hat s}=-1.31$, it is thicker than $\xi$ by a few times and is of order $\xi_{\rm ad}$ in Eq.(2.19). Furthermore, for ${\hat s}=-1.31$, $\omega_{\rm loc}(r) a^3/k_BT_c $ exhibits a peak around $r-a \sim \xi_{\rm ad}$ with its area being about $\sigma a^2/k_BT_c$. However, the peak recedes and diminishes for smaller $\hat s$ ($-1.38$ and $-1.45$).
Two colloidal particles
------------------------
As in Fig.2, we consider two colloidal particles with equal radius $a$. In our numerical analysis, they are placed in the middle of a cylindrical cell with radius $R_0=8a$ and height $H_0=16a$. The system is then in the region $0< (x^2+y^2)^{1/2}<R_0$ and $0<z<H_0$. The particle centers are at $(0,0, \pm (\ell/2+a))$ with $\ell$ being the surface-to-surface separation distance. Hereafter, we set =/a.
When the system lengths ($R_0$ and $L_0$) much exceed $a$, it is convenient to write $\Omega$ as = \_-k\_BT\_c[G]{}, where $\Omega_\infty$ is the value of $\Omega$ for $\ell\gg a$. That is, if $\Omega_1$ is the grand potential for one isolated colloidal particle, we have $\Omega_\infty=2\Omega_1$. The dimensionless quantity $\cal G$ is a universal function of $\hat{t}$, $\hat{s}$, and $\hat{\ell}$ decaying to 0 for large $\hat\ell$. Note that it is independent of $h_1$ in the strong adsorption limit, as discussed in Subsec.IIIB. The adsorption-induced force between the two colloidal particles is given by = . The dimensionless functions ${\cal F}(\hat{\ell})$ and ${\cal G}(\hat{\ell})$ are related by = - . We also have ${\cal G}(\hat{\ell})
= \int_{\hat{\ell}}^\infty d{\hat{\ell}'}
{\cal F}(\hat{\ell}')$. In the derivative and the integral with respect to $\hat\ell$, $\hat{t}$ and $\hat{s}$ are fixed.
From the calculations in Appendix A, the normalized force $\cal F$ is expressed in a convenient form, = dxdy \_[loc]{}(x,y,0) . where the integral is on the $xy$ plane with $z=0$ (the midplane between the two colloidal particles). From the geometrical symmetry, $\p \psi/\p z=0$ on this plane, we may set $\Pi_{\psi zz}= \Pi_\infty -\omega_{\rm loc}$ from Eq.(3.10). Also the integral $\int dxdy$ may be replaced by $2\pi \int dr r$, since $\omega_{\rm loc}(x,y,0)$ depends only on $r=(x^2+y^2)^{1/2}$. If $\ell\gg \ell_0$, the midplane is far from the transition layers with thickness $\ell_0$ and $\omega_{\rm loc}(x,y,0)$ becomes independent of $\psi_0$ or $h_1$. In this paper, we thus calculate $\cal F$ from Eq.(3.24).
Notice that we may use Eq.(3.15) on the midplane between the two colloidal particles for small $\ell$ [@mid], where we set $\lambda=\ell+r^2/a\gg \ell_0$ with $r= (x^2+y^2)^{1/2}$. In Eq.(3.24), the integral in the range $r \ls
(\ell a)^{1/2}$ then becomes \~a\_0\^ drr (+r\^2/a)\^[-3]{} \~ \^[-2]{}. To be precise, Eq.(3.24) yields $\lim_{\hat{\ell} \to 0}
{\hat{\ell}}^{2}{\cal F}= 0.205\pi$ as the coefficient in Eq.(3.25). In Appendix B, the Derjaguin approximation [@Is; @Butt; @Russel] for small $\hat\ell$ will yield \_[cri]{} \^[-1]{}, \_[cri]{} \^[-2]{}, with $ \Delta_{\rm cri} \cong
0.279 $. The coefficient $\pi \Delta_{\rm cri}$ is somewhat larger than that from Eq.(3.24). This small-$\hat\ell$ behavior stems from the de Gennes-Fisher theory for near-critical films [@Fisher; @Gamb; @Nature2008]. Furthermore, in Appendix B, we shall see that if $\ell $ exceeds the correlation length $\xi$ without bridging, $\cal F$ and $\cal G$ decay exponentially as \~(a/)\^2 e\^[-/]{}, \~(a/) e\^[-/]{}, where $\xi$ is determined by $\tau$ and $\psi_\infty$ from Eq.(2.9). These relations follow in separated states if the midpoint value of $\psi$ at $z=x=y=0$ is close to $\psi_\infty$ [@OkamotoCasimir].
Note that Eq. (3.26) holds for $\hat{\ell} \ls \xi/a$ and Eq.(3.27) for $\xi/a \ls \hat{\ell} \ll 1$. However, the exponential decays in Eq.(3.27) are observed even for $\hat{\ell}\sim 1$ in our numerical analysis (see Figs.5 and 6). The same exponential form of $\cal G$ was found in the previous papers [@Bonn; @Gamb; @Gamb1].
Numerical results without bridging transition
---------------------------------------------
In Figs.3-6, we present numerical results where there is no bridging transition. We aim to show that $\cal G$ and $\cal F$ are much more enhanced for $\hat{s}<0$ than for $\hat{s}>0$.
In the left panel of Fig.3, we show curves of ${\hat\ell }^2 {\cal F}$ vs $\hat\ell$ calculated from Eq.(3.24) and those from the Derjaguin approximation for $(\hat{\tau}, \hat{s})=(0,-1)$ and $(5,0)$. They tend to a constant as $\hat{\ell} \to 0$ as in Eqs.(3.25) and (3.26). Remarkably, for ${\hat s}<0$ and ${\hat t}=0$, ${\hat\ell }^2 {\cal F}$ increases up to of order 10 to exhibit a peak as a function of $\hat\ell$, where the peak position is at $\ell
\cong 6.14\xi$ from Eq.(B10). On the other hand, for ${\hat t}>0$ and ${\hat s}=0$, ${\hat\ell }^2 {\cal F}$ exhibits only a rounded maximum of order 1 at $\ell\sim 1.64\xi$ from Eq.(B12). We recognize that the force is much enhanced for negative $\hat{s}$ and the Derjaguin approximation nicely holds for $\hat{\ell} \ls 1$. In the right panel of Fig.3, we present $\psi(x,0,z)/\psi_a$ in gradation for $(\hat{\tau}, \hat{s},\hat{\ell})=(0,-1,0.4)$ in the $xz$ plane, where $\psi$ is large in the region between the two colloidal particles.
In Fig.4, for $(\hat{\tau},\hat{s})=(3,-1)$, we show $\psi(r,0)/\psi_a$ and $r \omega_{\rm loc}(r,0)\times 2\pi a^2/k_BT_c$ vs $r/a=(x^2+y^2)^{1/2}/a$ at $z=0$. We change $\hat{\ell}$ as $1$, 0.4, 0.2, and 0.008. For $\hat{\ell}=1$, the two collloidal particles are so separated such that $\psi(r,0)<0$ resulting in a small ${\cal F}=1.1$. On the other curves of smaller $\hat \ell$, $\psi(r,0)$ decreases from positive to negative with increasing $r$ and ${\cal F}$ increases dramatically up to 219. The behavior of the latter curves are consistent with the theoretical expressions: $
\psi \propto [\hat{\ell}+ (r/a)^2]^{-\beta/\nu}
$ and $
\omega_{\rm loc}\propto [\hat{\ell}+ (r/a)^2]^{-3},
$ at $z=0$, which follow from Eqs.(3.14) and (3.15) with $
\lambda =\ell+ r^2/a\gg \ell_0
$ as in Eq.(3.25).
Next, we plot ${\cal G}$ and $\cal F$ vs $\hat\ell$ for six values of $\hat{s}$ at $\hat{\tau}=0$ in Fig.5 and for five values of $(\hat{t}, \hat{s})$ with $\hat{\tau}>0$ in Fig.6 on semi-logarithmic scales. In these examples, there is no bridging transition for any $\hat\ell$. For small $\hat\ell$, we have the behaviors in Eq.(3.26). For relatively large $\hat\ell$ ($\xi\ll \ell \ls a$), both ${\cal G}$ and $\cal F$ decay exponentially as $\exp(-\ell/\xi)$. We confirm that the slopes of these curves are close to $a/\xi$ for $\hat{\ell}\sim 1$, where $\xi$ is calculated from Eq.(2.9). We can again see that $\cal F$ is well approximated by the Derjaguin approximation for $\hat{\ell}\ls 1$.
Van der Waals interaction
-------------------------
So far, we have not explicitly accounted for the pairwise van der Waals interaction [@Is] among constituent molecules, which was treated as one of the main elements causing colloid aggregation [@Beysens; @Petit]. The resultant potential $U_{\rm vdw}(r)$ between two colloidal particles with equal radius $a$ is written as [@Butt; @Russel] U\_[vdw]{} = - . where $r=2a+\ell$ is the center-to-center distance. The Hamaker constant $A_{\rm H}$ is in many cases of order $10^{-19}$J, but it can change its sign [@Is; @Bonnreview] and can be very small for some systems of colloids and binary mixtures [@Bonn]. Without charges, the total potential is of the form, U\_[tot]{}= -k\_BT\_c [G]{} + U\_[vdw]{}, consisting of the adsorption-induced part and the van der Waals part. The former is very sensitive to $\tau$ and $\psi_\infty$ in the critical ranges, while the latter is insensitive to them. If we further include the charge effects, we should add an appropriate chrage-induced interaction $U_C$ in Eq.(3.29) [@Nature2008; @Okamoto; @Bonn; @Beysens; @Gamb] (see item (3) in Sec.V for more discussions).
The force from the van der Waals interaction reads F\_[vdw]{} = U\_[vdw]{} = As $\hat{\ell}\to 0$, we find $F_{\rm vdw}
\cong {A_{\rm H}}/{12a}{\hat{\ell}^2}$ This behavior is the same as that of $\cal F$ in Eq.(3.26). So we compare the coefficients in front of the power ${\hat{\ell}}^{-2}$ of the two forces, ${A_{\rm H}}/{12a}$ and $\pi \Delta_{\rm cri}\times k_BT_c/a$, to obtain the ratio, R\_[vdw]{}= [A\_H]{}/([12k\_BT\_c\_c]{}), where the denominator is $0.4\times 10^{-19}$J for $T_c \cong 300$K. If $|A_{\rm H}|$ is smaller than $0.4\times 10^{-19}$J, we have $|R_{\rm vdw}| < 1$ and the van der Waals interaction is weaker than the adsorption-induced interaction at least for small ${\hat{\ell}}$.
However, $\hat{\ell}^2{\cal F}$ grows for $\hat{s}<0$ with increasing $\hat{\ell}$ as in Fig.3, so we need to examine the relative importance of the van der Waals interaction and the adsorption-induced interaction for larger $\hat{\ell}$. To this end, in Fig.7, we plot ${\hat\ell}^2{\cal F}/\pi\Delta_{\rm cri}$ for four typical cases together with W\_[vdw]{} F\_[vdw]{} = In Fig.7, while all the curves start from unity for $\hat{\ell}\to 0$, the normalized quantity ${\hat\ell}^2{\cal F}/\pi\Delta_{\rm cri}$ increases up to a maximum about 10 for $\hat{s}<0$ without bridging formation and can even be of order 100 close to a bridging transition with increasing $\hat{\ell}$. Thus, at an off-critical composition with $\hat{s}<0$, the adsorption-induced interaction can well dominate over the van der Waals interaction (even for $|A_{\rm H}|\sim 10^{-19}$J).
Bridging transition between two colloidal particles
===================================================
In this section, we study the bridging transition for $\hat{s}<0$ between two colloidal particles in a near-critical binary mixture. In our case, $\psi$ assumes the profiles of $\hat{s}<0$ in Fig.1 in separated states, where the adsorption layer has a thickness of order $\xi_{\rm av}$ in Eq.(3.19). A bridging transition can then occur in a wide range of $\ell$($< 2.6a$) under the condition $\xi/a \sim 0.3 |\hat{t}|^{-\nu}<1$. In the previous papers [@Vino; @Higashi; @Bauer; @Yeomans; @Evans-Hop], bridging between two spheres or between a sphere and a plate were studied numerically for small separation $\ell$ (say $\sim 0.2a$) far from the criticality.
Phase diagrams
--------------
In Fig.8, we first show a phase diagram of the bridging transition in the $\hat{t}$-$\hat{s}$-$\hat{\ell}$ space outside CX. We find a surface of a first-order bridging transition, bounded by CX and a bridging critical line. As functions of $(\hat{t},\hat{s})$, the normalized separation $\hat\ell$ may be written on the transition surface and on the critical line as = \_[tr]{}(,), = \_[c]{}(), respectively. Across this surface, discontinuities appear in $\cal F$ and the adsorption $\Gamma$ in Eq.(3.18), which tend to vanish on approaching the critical line. The critical line tangentially ends on CX at $(\hat{t},\hat{s},\hat{\ell})\cong (-1.0, -0.66, 2.6)$. The maximum of $\hat{\ell}$ at a transition is thus 2.6.
In Fig.9, phase diagrams in the $\hat{t}$-$\hat{s}$ and $\hat{t}$-$\mu_\infty/\mu_a$ planes are presented, where $\mu_\infty$ is related to $\psi_\infty$ by Eq.(3.4) and scaled by $\mu_a=k_BT_c /a^3 \psi_a$. In these phase diagrams, a first-order bridging transition occurs at some $\hat\ell$ in the region between CX and the bridging critical line, where the latter approaches CX tangentially. We also write cross-sectional bridging transition lines at fixed $\ell$ (equal to $1.4$ and $1.04$) on the bridging transition surface, each starting from CX and ending at a point on the critical line. In our case, these lines are nearly straight in the two phase diagrams in Fig.9. Previously, bridging transition lines at fixed separation $\ell$ were drawn [@Yeomans; @Bauer; @Gamb]. For near-critical films, on the other hand, the capillary condensation line is detached from CX. As a result, it is considerably curved in the $\tau$-$\psi_\infty$ plane [@OkamotoCasimir], but is nearly straight in the $\tau$-$\mu_\infty$ plane [@Yabunaka].
The phase behavior at fixed separation $\ell$ is particularly intriguing. In the left panel of Fig.10, we show a phase diagram in the $\hat{t}$-$\hat{\ell}$ plane, where we write the critical line $\hat{\ell}= \hat{\ell}_{\rm c}(\hat{t})$ and the transition line $\hat{\ell}= \hat{\ell}_{\rm cx}(\hat{t})$ on CX. The latter is defined by \_[cx]{}()= \_[tr]{}(,-\_[cx]{}()), where $ -{\hat{s}}_{\rm cx}(\hat{t})
= -0.68 |\hat{t}|^{\beta/\nu}$ is the vallue of $\hat{s}$ on the negative branch of CX. These two lines merge at $(\hat{t},\hat{\ell})=(-1, 2.6)$ on CX. Then, let us vary $\hat s$ at fixed $\hat \ell$ and $\hat t$. (i) If $\hat{\ell}>\hat{\ell}_{\rm cx}(\hat{t})$, separated states are realized without bridging for any $\hat s$. (ii) If $\hat{\ell}_{\rm cx}(\hat{t})>\hat{\ell} >\hat{\ell}_{\rm cx}(\hat{t})$,we encounter the transition surface at a certain $\hat s$ to find a discontinuous change. (iii) For $\hat{\ell} <\hat{\ell}_{\rm cx}(\hat{t})$, a bridging domain appears with a well-defined interface close to CX, but disconnection occurs continuously with incresaing the distance from CX. In this changeover, it is puzzling how the interface becomes ill-defined gradually (see Fig.17).
In the right panel of Fig.10, we plot the bridging radius $r_b$ vs $\hat \ell$ at $\hat{t}=-8$ for $\hat{s}=-1.31, -1.35$, and $-1.4$, which correspond to the three marked points in the top panel of Fig.9. We determine $r_b$ from the condition $\psi(r_b,0)=0$ at $z=0$, where $\psi(r,0)$ changes from positive to negative at $r=r_b$ with a bridging domain in the range $\hat{\ell}<
\hat{\ell}_{\rm tr}(\hat{t},\hat{s})$. As a function of $\ell$ at each $(\hat{t},\hat{s})$, $r_b$ is shortest at the transition and increases with decreasing $\hat\ell$. It is about $a$ for sufficiently small $\hat \ell$. Also it is smaller near the critical line. In fact, $r_b\cong 0.2a$ at the transition for $\hat{s}=-1.4$.
The transition surface is determined from minimization of $\Omega$ or maximization of $\cal G$ from Eq.(3.21). In the left panels of Fig.11, we plot ${\cal G}$ and $\hat\ell^2{\cal F}$ vs $\hat\ell$ for $(\hat{t},\hat{s})=(-8,-1.31)$. The curve of $\cal F$ from the Derjaguin approximation nicely agrees with that from Eq.(3.24) for $\hat{\ell} \ls 0.6$. For this ($\hat{t}, \hat{s})$, we find two stationary solutions satisfying Eqs.(3.1) and (3.5) in a window range ($0.96 <\hat{\ell}<1.08$ for this example). Outside this range, one solution becomes unstable and the other one remains as a stable solution. In the bistable range, $\cal G$ is larger on the equilibrium branch and smaller on the metastable one, so the transition is at the crosspoint of the two branches of $\cal G$.
In Fig.11, the slope of $\cal G$ is very steep with bridging. It is $ -37.8$ at the transition, where $\hat{\ell}=1.04 \gg \xi/a=0.09$. It is further amplified for smaller $\hat\ell$ and is $-76.6$ at $\hat{\ell}=0.08\cong \xi/a$. Here, for $\ell\gg \xi$, a well-defined bridging domain exists and ${\cal G}$ changes with a change of its surface area. In fact, use of the surface tension $\sigma$ in Eq.(2.12) gives $ 2\pi a^2 \sigma /k_BT_c=57.9$ at ${\hat t}=-8$. Thus, with a well-defined bridge, Eq.(3.24) yields the capillary force [@Butt; @Butt1], \~2a\^2 /k\_BT\_c\~(a/)\^2 . This relation is valid for ${\ell}\gs \xi$. For smaller $\ell \ls \xi$, the growth ($\sim {\hat\ell}^{-2}$) in Eq.(3.25) becomes dominant. These features will be further examined in Figs.12, 13, and 16.
In the original units, the force with a well-defined bridge is of order $k_BT_c a/\xi^2$, which increases as we move away from the bulk criticality. Also in Fig.14 below, we shall see that $\cal F$ increases with lowering $\tau$, where bridging occurs continuously. However, the exponential tail of the interaction in Eq.(3.27) in separated states ($\propto e^{-\ell/\xi}$) increases as the bulk criticality is approached, which was indeed observed experimentally [@Nature2008].
In the right panels of Fig.11, we display $\psi(r,z)/\psi_a$ in the plane of $r= (x^2+y^2)^{1/2}$ and $z$ in two bridged and one separated states at different $\hat\ell$. We can see that the bridging radius $r_b$ is larger in (a) (far below the bridging transition) than in (b) (close to it). The midplane between the two particles is filled with the phase outside the particles in (c) in a separated state.
In these phase diagrams the lowest value of $\hat t$ is $-10$. With further loweing $\hat t$, there is still a tendency of decreasing $\hat{\ell}_{\rm cx}$ and $\hat{\ell}_{\rm c}$. For example, for $\hat{t}=-20$, we find $\hat{\ell}_{\rm cx}=0.69$ on CX and $(\hat{\ell}, \hat{s}) = ({\hat{\ell}}_c , {\hat{s}}_c)
= (0.395, -2.0)$ at the corresponding bridging critical point.
Profiles at transition and critical points
------------------------------------------
In Figs.12 and 13, we compare the profiles of $\psi(r,z)$ and $\omega_{\rm loc}(r,z)$ in separated and bridged states at two typical transition points on the bridging transition surface in Fig.8. That is, $(\hat{t},\hat{s},\hat{\ell})$ is $(-8, -1.31, 1.04)$ in Fig.12 and is $(-2,-0.83,1.99)$ in Fig.13. These points correspond to points (A) and (B) in Fig.9. The former in Fig.12 is relatively far from the bulk criticality with $\xi/a \cong 0.09$ and the interface is well-defined. The latter in Fig.13 is closer to it with $\xi/a \cong 0.19$ and the interface is broadened and the separation is widened to $\hat{\ell}=1.99$.
More remarks on Figs.12 and 13 are as follows. (i) The profiles of $\psi/\psi_a$ are distinctly different in the separated and bridged states. Its midpoint value is 1.27 in (a) and $ -1.07$ in (a’) in Fig.12, while it is 0.54 in (a) and $ -0.31$ in (a’) in Fig.13. (ii) We can see layer regions with a peak in $\omega_{\rm loc} a^3/k_BT_c$, which enclose the colloid surfaces at a distance of order $\xi_{\rm ad}$ in Eq.(3.19) except for the bridged surface regions. See Fig.1 for the profile of ${\hat s}=-1.31$ around a single particle. These layers around the two spheres are separated in (b), while they are detached from the colloid surfaces in the bridged parts in (b’). (iii) We also display $\omega_{\rm loc} a^3/k_BT_c$ in bird’s eye views in (c) and (c’), Comparing them, we recognize how a discontinuous change occurs with the total grand potential unchanged. (iv) In the bottom plates, we plot one-dimensional profiles of $\omega_{\rm loc} a^3/k_BT_c$ in the two states. They are presented along the $z$ axis at $r=0$ in (d) and along the $r$ axis at $z=0$ and $-0.2a$ in (e). Note that the integral $ \int_0^\infty dr r \omega_{\rm loc} a/k_BT_c$ is equal to ${\cal F}/2\pi$ from Eq.(3.24) and is of order $r_b a\sigma$ with $\sigma$ being the surface tension.
Figsures 12 and 13 demonstrate that there should be a balance between the free energy cost of creating a bridge ($\sim \pi \sigma \ell r_b$) and the free energy decrease on the colloid surfaces ($\sim - \pi \sigma r_b^2$) at the transition (see (d) and (e)). The origin of the latter is evident from comparison of the two curves of ${\hat s}= \pm 1.31$ in Fig.1. Also in Fig.13, we have ${\cal F}=6.79$ with bridging and $2\pi a^2 \sigma /k_BT_c =10.09$, in agreement with Eq.(4.3). The corresponding values in Fig.12 have already been given above Eq.(4.3).
We also examine the behavior of $\cal F$ and the profile of $\psi$ near the critical line. In Fig.14, we plot $\cal F$ vs $\hat{t}$ for four sets of $(\hat{s},\hat{\ell})$. Here, the force $\cal F$ increases with lowering $\hat{t} <0$. The right two curves are in regions with $\ell<\ell_{\rm c}$ in Fig.8 and there is no discontinuous change (as in Fig.17 below). The third curve meets a critical point $(\hat{t}, \hat{s},\hat{\ell})=(-0.67, -1.35,0.76)$. On these curves, bridging is achieved continuously as $\hat t$ is lowered. The fourth curve passes through the bridging transition surface and exhibits a discontinuous change. The slope $\p{\cal F}/\p {\hat t}$ becomes steep near the critical line and diverges on it.
Overall behaviors
-----------------
In Fig.15, we plot ${\cal G}$, $\cal F$, and the normalized excess adsorption $(\Gamma-\Gamma_\infty)/a^3\psi_a$ vs $\hat\ell$ at $\hat{t}=-8$ for six values of $\hat{s}$. Here, $\Gamma$ is the adsorption in Eq.(3.18) and $\Gamma_\infty$ is its value for large separation. A discontinuous bridging transition occurs for $\hat{s}= -1.31$, $-1.35$, and $-1.4$, while there is no discontinuity for $\hat{s}= -1.43$, $-1.5$, and $-1.6$. In the latter, $\Gamma-\Gamma_0$ becomes negative for $\hat{\ell}<0.08$ (not shown). For (d), the curves nearly pass through a critical point, where we have very steep slopes: $\p{\cal F}/\p {\hat\ell}= -3413$ and $ (\p \Gamma/\p \hat{\ell})/a^3\psi_a
= -39.5$. This behavior indicates divergence of $\p {\cal F}/\p \hat{\ell}$ and $\p { \Gamma}/\p \hat{\ell}$ on the bridging critical line. In addition, the curves of $(\Gamma-\Gamma_\infty)/a^3\psi_a$ vs $\hat\ell$ exhibit rounded maxima at an intermediate $\hat \ell$. This behavior can be understood from the right panel of Fig.10, where the bridging radius $r_b$ increases with decreasing $\hat\ell$.
In Fig.15, the formula (4.3) for bridged states holds for $\hat{\ell} \gs \xi/a= 0.09$. For smaller $\hat \ell$, $\cal F$ diverges as in Eq.(3.25). In Fig.16, we thus plot $\psi(r,0)/{\psi_a}$ and $r
\omega_{\rm loc}(r,0)\times 2\pi a^2/k_BT_c$ as functions of $r/a$ at $z=0$ in the midplane at $(\hat{t}, \hat{s})=(-8, -1.31)$ for $\hat{\ell}=0.4, 0.2$, and 0.08. We recognize growing of $\psi$ and $\omega_{\rm loc}$ on the midplane with decreasing ${\hat\ell}\ls \xi/a$.
We also examine how a continuous changeover between bridged and separated states is achieved for small $\hat{\ell}<\hat{\ell}_c(\hat{t})$. This case has been mentioned in the explanation of Fig.10. In Fig.17, we display $\psi(r,0)/{\psi_a}$ and $r
\omega_{\rm loc}(r,0)\times 2\pi a^2/k_BT_c$ vs $r/a$ at $z=0$ for $\hat{t}=-8$ and $\hat{\ell}=0.5$. Here, $\hat s$ is decreased from a value close to CX, $-1.31$, to smaller values away from CX. The profile of $\psi(r,0)/{\psi_a}$ at $\hat{s}= -1.31$ indicates the presence of a well-defined interface with a thickness of order $\xi$. However, with decreasing $\hat s$ from -1.31 to -1.7, the profile of $\psi(r,0)/{\psi_a}$ is gradually broadened and $\cal F$ decreases from $67.7$ to 13.8.
Stability of separation distance
--------------------------------
In Figs.14 and 15, the derivatives $\p {\cal F}/\p {\hat\ell} (\propto
\p^2\Omega/\p\ell^2)$ and $\p {\cal F}/\p {\hat t} (\propto
\p^2\Omega/\p\ell\p{\tau})$ are negative and tend to diverge as the bridging critical line is approached. We mention one implication of this singular behavior.
In this paper, we have been fixing the colloid separation $\ell$ at a constant. To achieve this constraint, let us suppose the presence of an externally applied potential $U_{\rm ext}(\ell)$ between two colloidal particles. It is worth noting that optical tweezers have been used to trap colloidal particles at small separation [@Nature2008; @Kimura]. In equilibrium, we should minimize the sum $\Omega+ U_{\rm ext}$ with respec to $\ell$. Then, the equilibrium separation $\ell$ is determined from F\_[ext]{}= -U\_[ext]{}= k\_BT\_c [F]{} . In order to ensure the stability of this equilibrium separation $\ell$, we need to require K\_[ext]{}= U\_[ext]{} > k\_BT\_c (- ) where $K_{\rm ext}(\ell)
$ is the spring constant of the externally applied potential. The thermal fluctuation of the separation $\ell$ is increased with decreasing the effective spring constant $K_{\rm eff}=
K_{\rm ext}
+ (k_BT_c/a^2) \p {\cal F}/\p {\hat\ell}$ while $K_{\rm eff}>0$. However, there is a possibility of violation of the inequality (4.5) or negativity of $K_{\rm eff}$ sufficiently close to the bridging critical line, where the colloid configuration determined from Eq.(4.4) is unstable.
Summary and remarks
===================
We have investigated the adsorption-induced interaction between two neutral colloidal particles with common radius $a$ in a near-critical binary mixture. Use has been made of our local functional theory [@OkamotoCasimir]. In the strong adsorption limit, we have calculated the normalized free energy deviation $\cal G$ (with minus sign) and the normalized force $\cal F$ as universal functions of scaled reduced temperature $\hat{t}= \tau/\tau_a$ (where $\tau_a= (\xi_0/a)^{1/\nu})$, scaled reservoir order parameter $\hat{s}= \psi_\infty/\psi_a$ (where $\psi_a \sim \tau_a^\beta$), and scaled separation distance $\hat{\ell}=\ell/a$.
Main results are as follows.\
(i) We have expressed the forces for many colloidal particles in Eq.(3.6) and the force between two neutral colloidal particles in Eq.(3.24) using the stress tensor due to the order parameter deviation. Some general discussions on this aspect are given in Appendix A. Generalization including charges will be presented in another paper.\
(ii) The interaction is much enhanced for $\hat{s}<0$ as in Figs.3-7, where the component favored by the colloid surfaces is poor in the reservoir and the order parameter disturbances around the surfaces are large as in Fig.1. It is $10$-$100$ times larger than at the bulk criticality.\
(iii) The Derjaguin approximation [@Is; @Butt] can be made on the force $\cal F$ for $\hat{\ell} \ls 1$ on the basis of the results for films in our previous paper [@OkamotoCasimir], as discussed in Appendix B. It cannot describe the bridging transition, but it predicts the short separation growth in Eq.(3.26) and the exponential decay for large $\hat{\ell}$ in Eq.(3.27). These results agree with the calculations from Eq.(3.24) for $\hat{\ell} \ls 1$.\
(iv) We have compared the van der Waals interaction and the adsorption-induced interactions. The former may be neglected at off-critical compositions and particularly at a bridging transition even for typical values of the Hamaker constant $A_{\rm H}(\sim 10^{-19}$J), as shown in Fig.7.\
(v) We have found a surface of a first-order bridging transition $\hat{\ell}=
\hat{\ell}_{\rm cx}(\hat{t},\hat{s})$ in the $\hat{t}$-$\hat{s}$-$\hat\ell$ space in Fig.8, across which a discontinuous change occurs between separated and bridged states. This surface starts from the bulk coexistence surface and ends at a bridging critical line $\hat{\ell}=
\hat{\ell}_{\rm c}(\hat{t})$. The discontinuity vanishes and the derivatives of the force with respect to $T$ and $\ell$ diverge on the critical line as in Figs.14 and 15. The critical separation ${\hat{\ell}}_c$ decreases with decreasing $\hat t$, which assumes the maximum 2.6 at ${\hat t}=-1.0$ and is $0.395$ at ${\hat t}=-20$.\
(vi) We have calculated $\cal G$, $\cal F$, and the excess adsorption $\Gamma-\Gamma_\infty$ for various parameters in Fig.15. With a well-defined bridging domain with $\ell \gs \xi$, ${\cal F}$ is given by the capillary force proportional to the surface tension $\sigma$ in Eq.(4.3). For $\ell \ls \xi$, ${\cal F}$ grows as $\hat{\ell}^{-2}$ in accord with the de Gennes-Fisher theory.\
(vii) We have changed $\hat s$ (or $\hat t$) away from the bulk coexistence surface fixing $\hat\ell$ below $\hat{\ell}_{\rm c}$ in Subsec.IVC. There, we have found continuous changeover between bridged and separated states as in Figs.14 and 16.\
(viii) We have pointed out a possibility of an instability of the colloid separation distance near the bridging critical line where $\p {\cal F}/\p {\hat \ell}$ diverges.\
We give some remarks below.\
(1) To measure the force between colloidal particles, the geometry of a sphere and a plate has mostly been used [@Nature2008; @Higashi], while the geometry of two spheres was also used in a liquid crystal solvent [@Kimura]. In these two geometries, we expect essentially the same theoretical results for near-critical fluids. Systematic experiments on the force and the bridge formation at off-critical compositions near the bulk criticality should be promising.\
(2) There can arise repulsion between solid objects with asymmetric boundary conditions (with different signs of $h_1$) [@Law; @Gamb; @Nellen]. The adsorption-induced interaction in such asymmetric conditions should also be studied.\
(3) Real colloidal particles are usually charged and the charge effect can be crucial [@Beysens; @Maher; @Nature2008; @Bonn; @Guo]. For example, between a sphere and a plate, Hertlein [*et al.*]{}[@Nature2008] measured the adsorption-induced attractive interaction for $\ell \gs 0.1 \mu$m with $a=1.85\mu$m for various $\tau$ at the critical composition. In their experiment, the screened Coulomb interaction was dominant for smaller $\ell$ and decayed exponentially $(\propto e^{-\kappa \ell})$ with salt, where the screening length $\kappa^{-1} (= 12$nm) was shorter than $\ell$ measured.\
(4) The degree of ionization depends on the composition and the ion densities. In aqueous fluids, the colloid surface can be hydophobic for weak ionization and hydrophilic with progress of ionization [@Okamoto; @Maher; @Beysens]. Futhermore, added salts act as selective impurities to cause precipitation forming a wetting layer on the surfaces [@Onukireview; @Okamoto]. Aggregation of colloids depends on these elements.\
(5) The colloidal particles interact with the two components differently in a mixture solvent. They constitute a three component system, where the phase separation behavior is greatly altered by a small amount of the colloidal particles acting as selective impurities [@Onukireview; @Kaler; @Maher].\
(6) Dynamics of bridging and aggregation of colloidal particles should be of great interest, where the hydrodynamic flow is crucial[@Yabunaka; @Teshi]. Dynamical aspects have not yet been fully studied experimentally. Simulations on the dynamics of charged colloids, is complicated, where we need to integrate the dynamic equations for the composition, the ions, and the collodal particles [@Yabunaka; @Onukibook].\
(7) We should examine the nanobubble bridging in water [@bubble]. From our viewpoint, nanobubbles can appear with addition of a small amount of hydrophobic impurities in water [@Onukireview]. Particularly intriguing is dynamics of bubble formation and disruption upon a pressure change [@Teshi].\
We would like to thank Daniel Beysens for valuable discussions, This work was supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
[**Appendix A: Adsorption-induced force between colloidal particles in terms of stress tensor**]{}\
We consider two configurations of colloid particles in a near-critical fluid. That is, the colloid centers are at ${{\mbox{\boldmath$R$}}}_\alpha$ in one configuration and at ${{\mbox{\boldmath$R$}}}_\alpha +\delta
{{\mbox{\boldmath$R$}}}_\alpha$ in another slightly displaced one ($\alpha=1, 2,\cdots$). In these two states, we write the profiles of $\psi$ as $\psi({{\mbox{\boldmath$r$}}}) $ and $\psi' ({{\mbox{\boldmath$r$}}}')$ using different symbols. The space positions are written as ${\mbox{\boldmath$r$}}$ and ${{\mbox{\boldmath$r$}}}'$, respectively. We are interested in the difference between the grand potentials, $\Omega= \Omega( \{{{\mbox{\boldmath$R$}}}_\alpha\})$ and $\Omega' = \Omega( \{{{\mbox{\boldmath$R$}}}_\alpha+\delta{{\mbox{\boldmath$R$}}}_\alpha\})$, for these two states. From Eqs.(3.2) and (3.3), the grand potential $\Omega'$ for $\psi'({{\mbox{\boldmath$r$}}}')$ is written as = ’ d[r]{}’ \[ (’) + |’’|\^2 \] - dS’ h\_1 ’ , where $\psi'=\psi'({{\mbox{\boldmath$r$}}}')$, $\nabla'= \p/\p {{\mbox{\boldmath$r$}}}'$, and () = \[f()-f\_- \_(-\_)\]/k\_BT\_c. The $\int' d{{\mbox{\boldmath$r$}}}'$ is the integral outside the displaced colloidal particle, while $\int dS'$ is that on their spherical surfaces. We assume $\psi\to \psi_\infty$ far from the colloidal particles.
As a mathematical technique, we assume a mapping relation between the positions ${{\mbox{\boldmath$r$}}}'$ and ${{\mbox{\boldmath$r$}}}$ as ’= [r]{}+ [u]{}([r]{}), where ${{\mbox{\boldmath$u$}}}$ is a [*displacement*]{} vector vanishing far from the colloidal particles. Its surface value on the $\alpha$-th colloid particle is given by $\delta{{\mbox{\boldmath$R$}}}_\alpha$. We rewrite the right hand side of Eq.(A.1) by changing ${{\mbox{\boldmath$r$}}}'=(x', y',z')=(x'_1, x'_2,x'_3)$ to ${{\mbox{\boldmath$r$}}} =(x, y,z)=(x_1, x_2,x_3)$. To first order in ${\mbox{\boldmath$u$}}$, we may set $d{{\mbox{\boldmath$r$}}}'=d{{\mbox{\boldmath$r$}}}(1+ \nabla\cdot{{\mbox{\boldmath$u$}}})$ and $\p/\p x_i' = \p/\p x_i - \sum_j D_{ij} \p/\p x_j$, where ${D_{ij}} $ is the strain tensor, D\_[ij]{}= u\_j/x\_i . The deviation of the order parameter is written as ([r]{}) = ’([r]{}’)- ([r]{}). To first order in ${\mbox{\boldmath$u$}}$ and $\delta\psi$, we calculate $\delta\Omega=\Omega'-\Omega$ as && = d[r]{}- dS h\_1, where $C'= \p C/\p \phi$, $\nabla_i= \p/\p x_i$, and $\int d{{\mbox{\boldmath$r$}}}$ is the integral outside the colloidal particles at the original colloid positions, and $\int dS$ is that on their surfaces. Using $\Pi_{\psi ij} $ in Eq.(3.7), $\Pi_{\infty}$ in Eq.(3.9), and $\delta\Omega/\delta\psi=\delta F_b/\delta\psi-
\mu_\infty$ (see Eq.(3.5)), we simplify Eq.(A6) as && = d[r]{} \_[ij]{} (\_\_[ij]{}-\_[ij]{}) D\_[ij]{} + d[r]{}\
&& -k\_BT\_c dS\[ C [n]{}+h\_1\]. This relation is general and valid even in nonequilibrium.
In this paper, we assume that the original state is in equilibrium. That is, we assume the equilibrium relations (3.1) and (3.5) for $\psi({{\mbox{\boldmath$r$}}})$. Then, only the first term remains in Eq.(A7). Further using the equilibrium relation $\sum_j \nabla_j \Pi_{\psi ij} =0$ in the fluid and ${{\mbox{\boldmath$u$}}} = \delta{{\mbox{\boldmath$R$}}}_{\alpha }$ on the surface of the $\alpha$-th colloidal particle, we may rewrite Eq.(A7) as = \_ \_dS \_[ij]{} (\_[ij]{}-p\_\_[ij]{}) n\_[i]{} R\_[j]{}, where $\int_\alpha dS $ is the integral on the surface of the $\alpha$-th colloidal particle and ${{\mbox{\boldmath$n$}}}_\alpha = (n_{\alpha x}, n_{\alpha y},
n_{\alpha z})$ is the normal unit vector. This yields Eq.(3.6).
In the equilibrium case of two colloidal particles in Fig.2, we set $\delta {{\mbox{\boldmath$R$}}}_1=
\delta\ell {{\mbox{\boldmath$e$}}}_z$ and $\delta {{\mbox{\boldmath$R$}}}_1= 0$, where ${{\mbox{\boldmath$e$}}}_z$ is the unit vector along the $z$ axis. From Eq.(A8), $\p\Omega/\p\ell=\lim_{\delta\ell\to 0}
\delta\Omega/\delta\ell$ is obtained as = \_1 dS \_[i]{} (\_[iz]{}-\_\_[iz]{}) n\_[1 i]{} . where the surface integral is on the surface of the first colloidal particle. However, the above formula is not suitable for numerical calculations in the strong adsorption case. To devise a more convenient one, we integrate the equilibrium equation $\sum_j
\nabla_j{\Pi}_{zj} =0$ in the fluid region bounded by the upper colloid surface, a semisphere surface $S_{\rm semi}$, and a circular surface $S_{\rm mid}$ (see Fig.1), where the latter surfaces are represented by && S\_[semi]{} ={(x,y,z) | z>0, x\^2+y\^2+z\^2=L\^2 },\
&& S\_[mid]{} ={(x,y,z) | z=0, x\^2+y\^2=L\^2 }. Then $\p\Omega/\p\ell$ in Eq.(A9) is equal to the sum of the surface integrals on $S_{\rm mid}$ and $S_{\rm semi}$. In the limit of large $L$, the integral on $S_{\rm semi}$ vanishes and that on $S_{\rm mid}$ yields Eq.(3.24), where we use Eq.(3.10) and the relation $\p\psi/\p z=0$ on the midplane.
[**Appendix B: Derjaguin approximation** ]{}\
In non-bridging situations (with $ \xi \ll a$), we may use the Derjaguin approximation for $\ell \ls a$ [@Is; @Butt]. Though not exact, it provides a simple relation between the interaction free energy between two colloidal particles and that between two plates in the common boundary conditions. It is justified when the two spheres are closely separated without formation of a bridging domain.
We write the grand potential for a film per unit area as $ \Omega_{\rm f}(D)$, where $D$ is the film thickness. Then, we have () -\_ a\_\^dD \[ \_[f]{} (D)-\_[[f]{}]{}\], where $\Omega_{{\rm f}\infty}$ is the limit of $\Omega_{\rm f}$ for large $D$. Here, we have set $D= \ell+ r^2/a$ to change the integration on the $xy$ plane as $\int dxdy= 2\pi \int dr r=
\pi a \int dD$, as in Eq.(3.25). For Ising-like near-critical systems, $ \Omega_{\rm f} (D)$ is expressed in the de Gennes-Fisher scaling form [@Fisher; @Fisher-Yang] as \_[f]{} (D) = \_[[f]{} ]{} -k\_BT\_c D\^[-2]{}(t, s), in three dimensions. In our previous paper [@OkamotoCasimir], we calculated $\Delta (t, s)$ in the strong adsorption limit as a universal function of two scaling parameters $t$ and $s$ defined by && t=(D/\_0)\^[1/]{},\
&& s=\_/\_D, where $\psi_D =
1.47 b_{\rm cx}(\xi_0/D)^{\beta/\nu}$. If $D$ is replaced by $a$, we obtain $\hat t$ and $\hat s$ in Eqs.(3.12) and (3.13). From Eqs.(3.21) and (3.23) ${\cal G}$ and $\cal F$ are expressed as &&[G]{} \^[-1]{} \_[1]{} \^ ([t]{}\_v\^[1/]{}, [s]{}\_v\^[/]{}).\
&& [F]{} \^[-2]{} ([t]{}\_, [s]{}\_). We introduce two new scaling parameters, &&t\_= \^[1/]{}=(/\_0)\^[1/]{},\
&& s\_= \^[/]{}=\_/\_, with $\psi_\ell =
1.47 b_{\rm cx}(\xi_0/\ell)^{\beta/\nu}$. Here, $D$ in $t$ and $s$ is replaced by $\ell$ in $t_\ell$ and $s_\ell$. For small $\hat\ell$, the products $\hat{\ell}{\cal G} $ and ${\hat{\ell}}^2 {\cal F} $ are determined only by $t_\ell$ and $s_\ell$ from Eqs.(B4) and (B5).
We previously introduced another universal amplitude for near-critical films, written as ${\cal A}(t,s)$, where the osmotic pressure is expressed as $\Pi=-k_BT {\cal A}/D^3$ [@OkamotoCasimir]. It is related to $\Delta(t,s)$ in three dimensions by (t,s)= (t,s) . We notice that differentiation of ${\cal F}$ in Eq.(B5) with respect to $\hat\ell$ at fixed $\hat t$ and $\hat s$ just yields -\^[-3]{}[A]{}(t\_,s\_).
We remark the following. (i) First, for small $t$ and $s$, $\Delta(t,s)$ approaches its critical-point value $ \Delta_{\rm cri} =\Delta(0,0) \cong
0.279 $ [@OkamotoCasimir]. Thus, ${\cal G}$ and ${\cal F}$ grow as in Eq.(3.26) for $\hat{\ell}\ll 1$. (ii) Second, for $\tau=0$ (at $T=T_c$), we obtain \^2 [F]{}/ (0,s\_). See Fig.4 of Ref.[@OkamotoCasimir] for $\Delta(0,s)$. For ${\hat s} >0$, ${\hat{\ell}}^2 {\cal F}/\pi$ decays from $\Delta_{\rm cri}$ to zero monotonously with increasing $\hat\ell$. For $\hat{s}<0$, it takes a large maximum about $3.73=13.4\Delta_{\rm cri}$ at $s_\ell=-0.90$ or at = 0.82 ||\^[-/]{}= 6.14/a, where $\xi$ is defined by Eq.(2.9) (see the sentences below Eq.(3.13)). This explains the large maximum of ${\hat{\ell}}^2 {\cal F}$ for $(\hat{t}, \hat{s})=(0,-1)$ in Fig.3. (iii) Third, for $\psi_\infty=0$ and $\tau>0$ (on the critical path), we obtain \^2 [F]{}/ (t\_,0). See Fig.8 of Ref.[@OkamotoCasimir] for $\Delta(t,0)$. For ${\hat t}>0$, ${\hat{\ell}}^2 {\cal F}/\pi$ starts from $\Delta_{\rm cri}$, takes a mild maximum about $0.544=1.95\Delta_{\rm cri}$ at $t_\ell =2.30$ or at 1.64\^[-]{}= 1.64/a , and goes to zero for larger $\hat\ell$. This yields the mild minimum of ${\hat{\ell}}^2 {\cal F}$ for $(\hat{t}, \hat{s})=(5,0)$ in Fig.3.
Finally, we discuss how $\cal F$ and $\cal G$ behave away from the criticality or for $|\hat{t}_\ell|\gg 1$ or $|\hat{s}_\ell|\gg 1$. Our previous work [@OkamotoCasimir] indicates that if $|t| \gg 1$ or $|s|\gg 1$, ${\cal A}(t,s)$ decays as (t,s)\~(D/)\^3 (-D/), where $\xi$ is defined by Eq.(2.9). Let the midplane value of $\psi$ at $z=D/2$ be denoted by $\psi_m$ for a film. Then, Eq.(B13) follows for $\psi_m \cong \psi_\infty$, where we hve $-\ln(\psi_m/\psi_\infty-1)
\sim D/2\xi$ and ${\cal A}\cong
D^3f''(\psi_\infty) (\psi_m-\psi_\infty)^2/k_BT_c$ [@OkamotoCasimir]. We now need to replace $(t,s)$ by $(t_\ell,s_\ell) $ in Eq.(B13). To this end, we consider the combinations, q||\^/= |[t]{}|\^/[s]{}=|[t]{}\_|\^/[s]{}\_= 1.47b\_[cx]{}||\^/\_, which do not depend on $a$, $D$, and $\ell$. We may assume the scaling relation $\xi= \xi_0 |\tau|^{-\nu} M(q)$ in the critical region (for small $\tau$ and $\psi_\infty$), where $M(q)$ is a scaling function of $q$. Then, $D/\xi= |t|^{\nu} M(q)^{-1}$ in Eq.(B13). Replacement $(t,s)\to (t_\ell,s_\ell) $ simply yields $D/\xi \to \ell/\xi$, leading to (t\_,s\_)\~(/)\^3 (-/). Substitution of this relation into Eqs.(B8) and use of Eq.(3.23) give ${\cal F}$ and $\cal G$ in Eq.(3.27).
R. Evans, J. Phys.: Condens. Matter [**2**]{}, 8989 (1990).
L.D. Gelb, K.E. Gubbins, R. Radhakrishnan, and M. Sliwinska-Bartkowiak, Rep. Prog. Phys. [**62**]{}, 1573 (1999).
J. N. Israelachvili, [*Intermolecular and Surface Forces*]{} (Academic Press, London, 1991).
Hans-J$\ddot{\rm u}$rgen Butt and M. Kappl, [*Surface and Interfacial Forces*]{} (Wiley-VCH, Weinheim, 2010)).
J. Dzubiella and J.-P. Hansen, J. Chem. Phys.[**121**]{}, 5514 (2004).
P. Hopkins, A.J. Archer, and R. Evans, J. Chem. Phys. [**131**]{}, 124704 (2009).
D. Beysens and D. Est$\grave{\rm e}$ve, Phys. Rev. Lett. [**54**]{}, 2123 (1985); D. Beysens, J. -M. Petit, T. Narayan, A. Kumar, and M. L. Broide, Ber. Bunsenges. Phys. Chem. [**98**]{}, 382 (1994). B.M. Law, J.- M. Petit, and D. Beysens, Phys. Rev. E, [**57**]{}, 5782(1998); D. Beysens and T. Narayanan, J. Stat. Phys. [**95**]{}, 997 (1999).
P. D. Gallagher and J. V. Maher, Phys. Rev. A 46, 2012 (1992); P. D. Gallagher, M. L. Kurnaz, and J. V. Maher, Phys. Rev. A 46, 7750 (1992).
H. Guo, T. Narayanan, M. Sztucki, P. Schall and G. Wegdam, Phys. Rev. Lett. [**100**]{}, 188303 (2008)
D. Bonn, J. Otwinowski, S. Sacanna, H. Guo, G. Wegdam and P. Schall, Phys. Rev. Lett. [**103**]{}, 156101 (2009).
C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, and C. Bechinger, Nature [**451**]{}, 172 (2008).
U. Nellen, L. Helden, and C. Bechinger, Europhys. Lett. [**88**]{}, 26001 (2009); U. Nellen, J. Dietrich, L. Helden, S. Chodankar, K. Nyg${\rm a}$ard, J. F. van der Veen, and C. Bechinger, Soft Matter [**7**]{}, 5360 (2011).
T.J. Sluckin, Phys. Rev. A [**41**]{}, 960 (1990). H. L$\ddot{\rm o}$wen, Phys. Rev. Lett. [**74**]{}, 1028 (1995). R. R. Netz Phys. Rev. Lett. [**76**]{}, 3646 (1996).
A. Hanke,1 F. Schlesener,1 E. Eisenriegler,2 and S. Dietrich Phys. Rev. Lett. [**81**]{}, 1885 (1998). Y. Jayalakshmi and E. W. Kaler, Phys. Rev. Lett. [**78**]{}, 1379 (1997). W. B. Russel, D. A. Saville, and W. R. Schowalter, [*Colloidal Dispersions*]{} (Cambridge University Press, Cambridge, 1989).
D. Bonn and Ross, Rep. Prog. Phys. [**64**]{}, 1085 (2001).
R. Okamoto and A. Onuki, Phys. Rev. E. [**84**]{} 051401 (2011).
M.E. Fisher and P.G. de Gennes, C. R. Acad. Sci. Paris Ser. B 287 207 (1978). M. E. Fisher and H. Au-Yamg, Physica [**101**]{}A, 255 (1980); M. E. Fisher and P. J. Upton, Phys. Rev. Lett. [**65**]{}, 3405 (1990).
Z. Borjan and P. J. Upton, Phys. Rev. Lett. [**81**]{}, 4911 (1998); Z. Borjan and P. J. Upton, Phys. Rev. Lett. [**101**]{}, 125702 (2008).
A. Gambassi, A. Maciołek, C. Hertlein, U. Nellen, L. Helden, C. Bechinger, and S. Dietrich, Phys. Rev. E [**80**]{}, 061143 (2009); A. Gambassi and S. Dietrich, Phys. Rev. Lett. [**105**]{}, 059601 (2010).
A. Mukhopadhyay and B. M. Law, Phys. Rev. E [**63**]{}, 041605 (2001); M. Fukuto, Y. Yano, P. Pershan, Phys. Rev. Lett. [**94**]{}, 135702 (2005); S. Rafai, D. Bonn, and J. Meunier, Physica A [**386**]{}, 31 (2007).
The Casimir interaction was originally found to be induced by the ground-state fluctuations of the electromagnetic field between two mirrors. In near-critical fluids, the solvent-mediated interaction is caused by preferential adsorption, where the composition heterogeneity is stationary for fixed colloid positions. From our viewpoint, it is misleading to call it the Casimir interaction, Thus we call it the adsorption-induced interaction [@Beysens].
A. Maciołek, A. Drzewiński,and R. Evans, Phys. Rev. E [**64**]{}, 056137 (2001).
R. Okamoto and A. Onuki, J. Chem. Phys. [**136**]{}, 114704 (2012).
S. Yabunaka, R. Okamoto, and A. Onuki, Phys. Rev. E [**87**]{}, 032405 (2013).
Hans-J$\ddot{\rm u}$rgen Butt and M. Kappl, Adv. in Colloid and Interface Sci. [**146**]{}, 48 (2009).
H. T. Dobbs, G. A. Darbellay, and J. M. Yeomans, Europhys.Lett. [**18**]{}, 439 (1992); H. T. Dobbs and J. M. Yeomans, J. Phys.: Condens. Matter [**4**]{}, 10133 (1992).
D. Andrienko, P. Patricio, and O. I. Vinogradova, J. Chem. Phys. [**121**]{}, 4414 (2004).
C. Bauer, T. Bieker, and S. Dietrich, Phys. Rev. E [**62**]{}, 5324 (2000).
H. Shinto, K. Uranishi, H. Miyahara, and K. Higashitani, J. Chem. Phys. [**116**]{}, 9500 (2002).
A. Carambassis, L. C. Jonker, P. Attard, and M. W. Rutland, Phys. Rev.Lett. [**80**]{}, 5357 (1998); G. E. Yakubov, H. J. Butt, and O. I. Vinogradova, J. Phys. Chem. B [**104**]{}, 3407 (2000).
R. Teshigawara and A. Onuki, Phys.Rev.E [**84**]{}, 041602 (2011).
K. Kocevar, A. Borstnik, I. Musevic, and S. Zumer, Phys. Rev. Lett. [**86**]{},5914 (2001).
H. Stark, J. Fukuda, and H. Yokoyama, Phys. Rev. Lett. [**92**]{}, 205502 (2004).
P. Schofield, Phys. Rev. Lett. [**22**]{}, 606 (1969); P. Schofield, J.D. Lister, and J.T. Ho, ibid. [**23**]{}, 1098 (1969).
A. Onuki, [*Phase Transition Dynamics*]{} (Cambridge University Press, Cambridge, 2002).
The coefficient $C_1$ in Ref.[@OkamotoCasimir] is set equal to 1 in the present paper. Then $C$ is dimensionless.
J. Rudnick and D. Jasnow, Phys. Rev. Lett. [**48**]{}, 1059 (1982); ibid. [**49**]{}, 1595 (1982)
R. Holyst and A. Poniewierski Phys. Rev. B [**36**]{}, 5628 (1987); M.P. Gefand and r. Lipowsky, Phys. Rev. B [**36**]{}, 8725 (1987); P.J. Upton, J.O. Indekeu, and J.M. Yeomans, Phys. Rev. B [**40**]{}, 666 (1989).
For $\tau=0$, $\psi$ and $\omega_{\rm loc}$ are calculated as $\psi_m = 0.7107 \psi_D$ and $\omega_m= 0.205 k_BT_c D^{-3}$, respectively, at $z=D/2$ in the film geometry [@OkamotoCasimir]. See Eq.(B3) for $\psi_D $.
The authors of Ref.[@Gamb] obtained ${\cal G}= \pi Aa\xi^{-1}\exp(-\ell/\xi)$ with $A=1.2-1.4$ for $\ell>\xi$ on the critical path ($\tau>0$ and $\psi_\infty=0$). In our scheme, the curve of $\ln({\cal G})$ for $(\hat{t}, \hat{s})=(5,0)$ in Fig.6 gives $A\cong 0.9$.
J.-M. Petit, B. M. Law, and D. Beysens, J.Colloid Interface Sci, [**202**]{}, 441 (1998).
K. Takahashi, M. Ichikawa, and Y. Kimura, Phys. Rev. E [**77**]{}, 020703(R) (2008).
A. Onuki, R. Okamoto, and T. Araki, Bull. Chem. Soc. Jpn. [**84**]{}, 569 (2011).
|
---
abstract: 'In this paper, we derive the CR Reilly’s formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded $\mathrm{p}$-minimal hypersurfaces. In particular, we obtain the first Dirichlet eigenvalue estimate in a compact pseudohermitian $(2n+1)$-manifold with boundary and the first eigenvalue estimate of the tangential sublaplacian on closed oriented embedded $\mathrm{p}$-minimal hypersurfaces in a closed pseudohermitian $(2n+1)$-manifold of vanishing torsion.'
address:
- '$^{1}$Department of Mathematics and Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei 10617, Taiwan.'
- '$^{2}$Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan, R.O.C. '
- '$^{3}$Department of applied Mathematics, National Pingtung University, Pingtung 90003, Taiwan R.O.C.'
author:
- '$^{\ast }$Shu-Cheng Chang$^{1}$'
- '$^{\ast }$Chih-Wei Chen$^{2}$'
- '$^{\ast }$Chin-Tung Wu$^{3}$'
title: On the CR Analogue of Reilly Formula and Yau Eigenvalue Conjecture
---
[^1]
Introduction
============
In the paper of [@Re], by integral version of Bochner-type formula, R. Reilly proved so-called Reilly formula which has numerous applications. For example, Reilly himself applied it to prove a Lichnerowicz type sharp lower bound for the first eigenvalue of Laplacian on compact Riemannian manifolds with boundary. In this paper, we will derive the CR version of Reilly’s formula and give some applications as well.
Let $(M,J,\theta )$ be a pseudohermitian $(2n+1)$-manifold (see next section for basic notions in pseudohermitian geometry). The CR Reilly’s formula ([CR Reilly’s formula]{}) is involved terms which has no analogue in the Riemannian case. However, one can relate these extra terms to a third-order operator $P$ which characterizes CR-pluriharmonic functions ([@l1]) and the fourth-order CR Paneitz operator $P_{0}$ ([@gl]).
([@gl], [@p]) Let $(M,J,\theta )$ be a pseudohermitian $(2n+1)$-manifold. We define $$\begin{array}{c}
P\varphi =\sum_{\gamma ,\beta =1}^{n}(\varphi _{\overline{\gamma }\; \ \beta
}^{\, \overline{\text{ }\gamma }}+inA_{\beta \gamma }\varphi ^{\gamma
})\theta ^{\beta }=\sum_{\beta =1}^{n}(P_{\beta }\varphi )\theta ^{\beta },\end{array}$$which is an operator that characterizes CR-pluriharmonic functions. Here $$\begin{array}{c}
P_{\beta }\varphi =\sum_{\gamma =1}^{n}(\varphi _{\overline{\gamma }\; \
\beta }^{\, \text{\ }\overline{\gamma }}+inA_{\beta \gamma }\varphi ^{\gamma
}),\text{ \ }\beta =1,\cdots ,n,\end{array}$$and $\overline{P}\varphi =\sum_{\beta =1}^{n}\overline{P}_{\beta }\theta ^{\overline{\beta }}$, the conjugate of $P$. The CR Paneitz operator $P_{0}$ is defined by $$P_{0}\varphi =4\delta _{b}(P\varphi )+4\overline{\delta }_{b}(\overline{P}\varphi ), \label{Paneitz}$$where $\delta _{b}$ is the divergence operator that takes $(1,0)$-forms to functions by $\delta _{b}(\sigma _{\beta }\theta ^{\beta })=\sigma _{\beta
}^{\; \ \beta }$, and similarly, $\overline{\delta }_{b}(\sigma _{\overline{\beta }}\theta ^{\overline{\beta }})=\sigma _{\overline{\beta }}^{\; \
\overline{\beta }}$.
We observe that ([@gl]) $$\begin{array}{lll}
P_{0}\varphi & = & 2\square _{b}\overline{\square }_{b}\varphi -4in(A^{\beta
\gamma }\varphi _{\beta }),_{\gamma } \\
& = & 2\overline{\square }_{b}\square _{b}\varphi +4in(A^{\overline{\beta }\overline{\gamma }}\varphi _{\overline{\beta }}),_{\overline{\gamma }} \\
& = & 2(\Delta _{b}^{2}+n^{2}T^{2})\varphi -4n\func{Re}(iA^{\beta \gamma
}\varphi _{\beta }),_{\gamma }\end{array}
\label{111a}$$for $\square _{b}\varphi =(\overline{\partial }_{b}\overline{\partial ^{\ast
}}_{b}+\overline{\partial ^{\ast }}_{b}\overline{\partial }_{b})\varphi
=(-\Delta _{b}+inT)\varphi =-2\varphi _{\overline{\beta }}^{\, \ \overline{\beta }}$.
By using integrating by parts to the CR Bochner formula (\[Bochnerformula\]), we derive the following CR version of Reilly’s formula.
\[Reilly’sformula\] Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with boundary $\Sigma $. Then for any real smooth function $\varphi $, we have$$\begin{array}{ll}
& \frac{n+1}{n}\int_{M}[(\Delta _{b}\varphi )^{2}-\frac{2n}{n+1}\sum_{\beta
,\gamma }|\varphi _{\beta \gamma }|^{2}]d\mu \\
= & \frac{n+2}{4n}\int_{M}\varphi P_{0}\varphi d\mu
+\int_{M}[2Ric-(n+1)Tor]((\nabla _{b}\varphi )_{\mathbb{C}},(\nabla
_{b}\varphi )_{\mathbb{C}})d\mu \\
& -\frac{n+2}{2n}iC_{n}\int_{\Sigma }\varphi \left( P_{n}\varphi -P_{\overline{n}}\varphi \right) d\Sigma _{p}+\frac{i}{2}C_{n}\int_{\Sigma
}(\varphi ^{\overline{\beta }}B_{n\overline{\beta }}\varphi -\varphi ^{\beta
}B_{\overline{n}\beta }\varphi )d\Sigma _{p} \\
& -\frac{1}{4}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma _{p}+\frac{3}{4n}C_{n}\int_{\Sigma }\varphi _{e_{2n}}\Delta _{b}\varphi d\Sigma
_{p}+C_{n}\int_{\Sigma }\varphi _{e_{2n}}\Delta _{b}^{t}\varphi d\Sigma _{p}
\\
& +\frac{1}{4}C_{n}\int_{\Sigma }H_{p.h}\varphi _{e_{2n}}^{2}d\Sigma _{p}+\frac{1}{2}C_{n}\int_{\Sigma }\alpha \varphi _{e_{n}}\varphi
_{e_{2n}}d\Sigma _{p} \\
& +\frac{1}{4}C_{n}\int_{\Sigma }\tsum_{j,k=1}^{2n-1}\left \langle \nabla
_{e_{j}}e_{2n},e_{k}\right \rangle \varphi _{e_{j}}\varphi _{e_{k}}d\Sigma
_{p}.\end{array}
\label{CR Reilly's formula}$$Here $P_{0}$ is the CR Paneitz operator on $M.\ C_{n}:=2^{n}n!;$ $B_{\beta
\overline{\gamma }}\varphi :=\varphi _{\beta \overline{\gamma }}-\frac{1}{n}\varphi _{\sigma }{}^{\sigma }h_{\beta \overline{\gamma }}$. $\Delta
_{b}^{t}:=\frac{1}{2}\tsum_{j=1}^{2n-1}[e_{j}{}^{2}-(\nabla
_{e_{j}}e_{j})^{t}]$ is the tangential sublaplacian of $\Sigma $ and $H_{p.h} $ is the $p$-mean curvature of $\Sigma $ with respect to the Legendrian normal $e_{2n},$ $\alpha e_{2n}+T\in T\Sigma $ for some function $\alpha $ on $\Sigma \backslash S_{\Sigma },$ the singular set $S_{\Sigma }$ consists of those points where the contact bundle $\xi =\ker \theta $ coincides with the tangent bundle $T\Sigma $ of $\Sigma .$ $(\nabla
_{b}\varphi )_{\mathbb{C}}=\varphi ^{\beta }Z_{\beta }$ is the corresponding complex $(1,0)$-vector field of $\nabla _{b}\varphi $ and $d\Sigma
_{p}=\theta \wedge e^{1}\wedge e^{n+1}\wedge \cdots \wedge e^{n-1}\wedge
e^{2n-1}\wedge e^{n}$ is the $p$-area element on $\Sigma .$
If $(M,J,\theta )$ is a compact pseudohermitian $(2n+1)$-manifold without boundary, one can check easily that the fourth-order CR Paneitz $P_{0}$ is self-adjoint. That is $$\begin{array}{c}
\int_{M}gP_{0}fd\mu =\int_{M}fP_{0}gd\mu\end{array}
\label{1}$$for all smooth functions $f$ and $g$. However, if $(M,J,\theta )$ is a compact pseudohermitian $(2n+1)$-manifold with the smooth boundary $\Sigma ,$ it follows from (\[A1\]) and (\[A2\]) that (\[1\]) folds for all smooth functions with the Dirichlet condition or the Neumann condition as in (\[222\]) and (\[333\]) on $\Sigma $. In particular, it holds in the situation as in Theorem \[TB\] and Theorem \[TC\].
That is, one can have the following Dirichlet eigenvalue problem or Neumann eigenvalue problem, respectively : $$\left \{
\begin{array}{ccl}
P_{0}\varphi & = & \mu _{_{D}}\varphi \ \ \mathrm{on\ }M, \\
\varphi & = & 0\ \ \ \ \ \ \mathrm{on\ }\Sigma , \\
\Delta _{b}\varphi & = & 0\ \ \ \ \ \ \mathrm{on}\ \Sigma ,\end{array}\right. \label{222}$$and $$\left \{
\begin{array}{ccl}
P_{0}\phi & = & \mu _{_{N}}\phi \ \ \mathrm{on\ }M, \\
\Delta _{b}\phi & = & 0\ \ \ \ \ \ \mathrm{on\ }\Sigma , \\
(\Delta _{b}\phi )_{e_{2n}} & = & 0\ \ \ \ \ \ \mathrm{on}\ \Sigma .\end{array}\right. \label{333}$$Hence$$\begin{array}{c}
\int_{M}\varphi P_{0}\varphi d\mu \geq \mu _{_{D}}^{1}\int_{M}\varphi
^{2}d\mu\end{array}
\label{1a}$$for the first Dirichlet eigenvalue $\mu _{_{D}}^{1}$ and all smooth functions with $\varphi =0=\Delta _{b}\varphi $ on $\Sigma .$ Similarly $$\begin{array}{c}
\int_{M}\phi P_{0}\phi d\mu \geq \mu _{_{N}}^{1}\int_{M}\phi ^{2}d\mu\end{array}
\label{1aa}$$for the first Neumann eigenvalue $\mu _{_{N}}^{1}$ and all smooth functions with $\Delta _{b}\phi =0=(\Delta _{b}\phi )_{e_{2n}}$ on $\Sigma $. In general, $\mu _{_{D}}^{1}$ and $\mu _{_{N}}^{1}$ are not always nonnegative.
\[d1\] Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with the smooth boundary $\Sigma .$ We say that the CR Paneitz operator $P_{0}$ with respect to $(J,\theta )$ is nonnegative if $$\begin{array}{c}
\int_{M}\varphi P_{0}\varphi d\mu \geq 0\end{array}$$for all smooth functions with suitable boundary conditions as in Dirichlet eigenvalue problem or Neumann eigenvalue problem of $P_{0}$.
\[r1\] Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold of vanishing torsion with the smooth boundary $\Sigma $. It follows from (\[111a\]) that the Kohn Laplacian $\square _{b}$ and $\overline{\square }$ commute and they are diagonalized simultaneously with $$P_{0}\varphi =2\square _{b}\overline{\square }_{b}\varphi =2\overline{\square }_{b}\square _{b}\varphi .$$Then the corresponding CR Paneitz operator $P_{0}$ is nonnegative ([@ccc]). That is $$\mu _{_{D}}^{1}=0=\mu _{_{N}}^{1}.$$
For the first consequence of CR Reilly formula, we can consider the following Dirichlet eigenvalue problem: $$\left \{
\begin{array}{ccll}
\Delta _{b}\varphi & = & -\lambda _{1}\varphi & \mathrm{on\ }M, \\
\varphi & = & 0 & \mathrm{on\ }\Sigma .\end{array}\right. \label{1b}$$Then we have the following first Dirichlet eigenvalue estimate:
\[TB\] Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with the smooth boundary $\Sigma $. If the pseudohermitian mean curvature $H_{p.h}$ is nonnegative and $$\begin{array}{c}
\lbrack Ric-\frac{n+1}{2}Tor](Z,Z)\geq k\left \langle Z,Z\right \rangle\end{array}$$for all $Z\in T_{1,0}$ and a positive constant $k$, then
\(i) For $n\geq 2,$ $$\begin{array}{c}
\lambda _{1}\geq \frac{nk}{n+1};\end{array}$$
(ii) For $n=1,$ $$\begin{array}{c}
\lambda _{1}\geq \frac{k+\sqrt{k^{2}+6\mu _{_{D}}^{1}}}{4}\end{array}$$with $\mu _{_{D}}^{1}\geq -\frac{k^{2}}{6}$. In addition if $P_{0}$ is nonnegative, in particular if the torsion is vanishing, then$$\begin{array}{c}
\lambda _{1}\geq \frac{k}{2}.\end{array}$$
It is known that the sharp first eigenvalue estimate is obtained as in [gr]{}, [@ll], [@ch], [@cc2] and [@fk] in a closed pseudohermitian $(2n+1)$-manifold.
Next we can state the second consequence of the CR Reilly formula (\[CR Reilly’s formula\]) which served as a CR analogue of Yau conjecture [@Y] on the first eigenvalue estimate of embedded oriented minimal hypersurfaces. We refer to papers of Choi-Wang [@cw] and Tang-Yan [@ty] which are related to Yau conjecture.
As before, $\{e_{1},e_{2},\cdots ,e_{n},e_{n+1},\cdots ,e_{2n-1},\alpha
e_{2n}+T\}$ is the base of $T\Sigma $ for some function $\alpha $ on $\Sigma
\backslash S_{\Sigma }$. It follows from (\[2014d\]) that $\Delta
_{b}^{t}+\alpha e_{n}$ is a self-adjoint operator with respect to the $p$-area element $d\Sigma _{p}$ on $\Sigma $. Hence it is natural to consider the following CR analogue of eigenvalue problem on the embedded closed $p$-minimal ($H_{p.h}=0$) hypersurface $\Sigma $ in a closed pseudohermitian $(2n+1)$-manifold $(M,J,\theta )$:$$L_{\alpha }u=-\lambda _{1}u, \label{2015}$$here $$\begin{array}{c}
L_{\alpha }:=\Delta _{b}^{t}+\alpha e_{n}.\end{array}
\label{2015-1}$$
In this paper, we consider the particular case that $\{e_{1},e_{2},\cdots
,e_{n},e_{n+1},\cdots ,e_{2n-1},T\}$ are always tangent to $\Sigma $ ($\alpha =0$) as following:$$\begin{array}{c}
L_{0}:=\Delta _{b}^{t}.\end{array}
\label{2015-2}$$That is, we have the first eigenvalue estimate of $L_{0}$ on embedded oriented hypersurfaces of nonnegative pseudohermitian mean curvature:
\[TC\] Let $\Sigma $ be a compact embedded oriented $p$-minimal hypersurface with $\alpha =0$ in a closed pseudohermitian $(2n+1)$-manifold $(M,J,\theta )$ of vanishing torsion. Suppose that the pseudohermitian Ricci curvature of $M$ is bounded from below by a positive constant $k$. Then
\(i) The first non-zero eigenvalue $\lambda _{1}$ of $L_{0}$ on $\Sigma $ has a lower bound given by $$\begin{array}{c}
\lambda _{1}\geq \frac{k}{2}.\end{array}$$
\(ii) In case of $n=1$ if the equality holds, $(M,J,\theta )$ must be a closed spherical pseudohermitian $3$-manifold and $\Sigma $ be a compact embedded oriented $p$-minimal surface of genus $\leq 1.$ Moreover, $(M,J,\theta )$ is the the standard CR $3$-sphere $(\mathbf{S}^{3},\widehat{J},\widehat{\theta })$ if it is simply connected.
Let $(M,J,\theta )$ be a closed spherical pseudohermitian $3$-manifold. Recall ([@cc1]) that we call a CR structure $J$ spherical if Cartan curvature tensor $Q_{11}$ vanishes identically. Here $$\begin{array}{c}
Q_{11}=\frac{1}{6}W_{11}+\frac{i}{2}WA_{11}-A_{11,0}-\frac{2i}{3}A_{11,\overset{\_}{1}1}.\end{array}$$Note that $(M,J,\theta )$ is called a spherical pseudohermitian $3$-manifold if $J$ is a spherical structure. We observe that the spherical structure is CR invariant and a closed spherical pseudohermitian $3$-manifold $(M,J,\theta )$ is locally CR equivalent to the standard pseudohermitian $3$-sphere $(\mathbf{S}^{3},\widehat{J},\widehat{\theta }).$
Note that for an $p$-minimal Clifford torus $\Sigma _{0}=S^{1}(\frac{\sqrt{2}}{2})\times S^{1}(\frac{\sqrt{2}}{2})\subset \mathbb{R}^{2}\times \mathbb{R}^{2}$ in the standard CR $3$-sphere $\mathbf{S}^{3}$ (i.e. $k=2$ and $A_{11}=0)$, $T$ is always tangent to $\Sigma _{0}$ (i.e. $\alpha =0$). Furthermore, the coordinate function $x_{i}$ of $\Sigma _{0}$ is the eigenfunction of the tangential sublaplacian $\Delta _{b}^{t}$ with $$\Delta _{b}^{t}x_{i}=-x_{i},\ \ i=1,...4.$$
Then in view of Theorem \[TC\], we have the following CR analogue of Yau conjecture on the first eigenvalue estimate of embedded oriented $p$-minimal surfaces.
The first eigenvalue of $L_{0}$ on any closed embedded $p$-minimal surface of genus $\leq 1$ in the standard CR $3$-sphere $(\mathbf{S}^{3},\widehat{J},\widehat{\theta })$ is just $1$.
Finally, we propose a CR analogue of Lawson conjecture ([@la]):
Any closed embedded $p$-minimal torus (with $\alpha =0$) in the standard CR $3$-sphere $\mathbf{S}^{3}$ is the Clifford torus.
If the Yau conjecture is true for the $2$-torus, it was proved in [@mr] that the Lawson conjecture holds which is to say that the only minimally embedded torus in $\mathbf{S}^{3}$ is the Clifford torus. However, Lawson conjecture was solved by S. Brendle [@b] recently.
We briefly describe the methods used in our proofs. In section $3$, by using integrating by parts to the CR Bochner formula (\[Bochnerformula\]), we can derive the CR version of Reilly’s formula which involving a third order operator $P$ which characterizes CR-pluriharmonic functions and the CR Paneitz operator $P_{0}.$ By applying the CR Reilly’s formula, we are able to obtain the first Dirichlet eigenvalue estimate as in section $4$ and derive the first non-zero eigenvalue estimate of (\[2015\]) on compact oriented embedded $p$-minimal hypersurfaces in a closed pseudohermitian $(2n+1)$-manifold of vanishing torsion as in section $5$.
Basic Notions in Pseudohermitian Geometry
=========================================
We first introduce some basic materials in a pseudohermitian $(2n+1)$-manifold. Let $(M,J,\theta )$ be a $(2n+1)$-dimensional, orientable, contact manifold with contact structure $\xi =\ker \theta $. A CR structure compatible with $\xi $ is an endomorphism $J:\xi \rightarrow \xi $ such that $J^{2}=-1$. We also assume that $J$ satisfies the following integrability condition: If $X$ and $Y$ are in $\xi $, then so is $[JX,Y]+[X,JY]$ and $J([JX,Y]+[X,JY])=[JX,JY]-[X,Y]$. A CR structure $J$ can extend to $\mathbb{C}\mathbf{\otimes }\xi $ and decomposes $\mathbb{C}\mathbf{\otimes }\xi $ into the direct sum of $T_{1,0}$ and $T_{0,1}$ which are eigenspaces of $J$ with respect to eigenvalues $i$ and $-i$, respectively. A manifold $M$ with a CR structure is called a CR manifold. A pseudohermitian structure compatible with $\xi $ is a $CR$ structure $J$ compatible with $\xi $ together with a choice of contact form $\theta $. Such a choice determines a unique real vector field $T$ transverse to $\xi $, which is called the characteristic vector field of $\theta $, such that ${\theta }(T)=1$ and $\mathcal{L}_{T}{\theta }=0$ or $d{\theta }(T,{\cdot })=0$. Let $\left \{ T,Z_{\beta },Z_{\overline{\beta }}\right \} $ be a frame of $TM\otimes \mathbb{C}$, where $Z_{\beta }$ is any local frame of $T_{1,0},\ Z_{\overline{\beta }}=\overline{Z_{\beta }}\in T_{0,1}$ and $T$ is the characteristic vector field. Then $\left \{ \theta ,\theta ^{\beta },\theta ^{\overline{\beta }}\right \} $, which is the coframe dual to $\left \{ T,Z_{\beta },Z_{\overline{\beta }}\right \} $, satisfies $$d\theta =ih_{\beta \overline{\gamma }}\theta ^{\beta }\wedge \theta ^{\overline{\gamma }}, \label{dtheta}$$for some positive definite Hermitian matrix of functions $(h_{\beta
\overline{\gamma }})$. Actually we can always choose $Z_{\beta }$ such that $h_{\beta \overline{\gamma }}=\delta _{\beta \gamma }$; hence, throughout this note, we assume $h_{\beta \overline{\gamma }}=\delta _{\beta \gamma }$.
The Levi form $\left \langle \ ,\ \right \rangle $ is the Hermitian form on $T_{1,0}$ defined by$$\left \langle Z,W\right \rangle =-i\left \langle d\theta ,Z\wedge \overline{W}\right \rangle .$$We can extend $\left \langle \ ,\ \right \rangle $ to $T_{0,1}$ by defining $\left \langle \overline{Z},\overline{W}\right \rangle =\overline{\left
\langle Z,W\right \rangle }$ for all $Z,W\in T_{1,0}$. The Levi form induces naturally a Hermitian form on the dual bundle of $T_{1,0}$, also denoted by $\left \langle \ ,\ \right \rangle $, and hence on all the induced tensor bundles. Integrating the Hermitian form (when acting on sections) over $M$ with respect to the volume form $d\mu =\theta \wedge
(d\theta )^{n}$, we get an inner product on the space of sections of each tensor bundle.
The pseudohermitian connection of $(J,\theta )$ is the connection $\nabla $ on $TM\otimes \mathbb{C}$ (and extended to tensors) given in terms of a local frame $Z_{\beta }\in T_{1,0}$ by$$\nabla Z_{\beta }=\theta _{\beta }{}^{\gamma }\otimes Z_{\gamma },\quad
\nabla Z_{\overline{\beta }}=\theta _{\overline{\beta }}{}^{\overline{\gamma
}}\otimes Z_{\overline{\gamma }},\quad \nabla T=0,$$where $\theta _{\beta }{}^{\gamma }$ are the $1$-forms uniquely determined by the following equations:$$\begin{split}
d\theta ^{\beta }& =\theta ^{\gamma }\wedge \theta _{\gamma }{}^{\beta
}+\theta \wedge \tau ^{\beta }, \\
0& =\tau _{\beta }\wedge \theta ^{\beta }, \\
0& =\theta _{\beta }{}^{\gamma }+\theta _{\overline{\gamma }}{}^{\overline{\beta }},
\end{split}
\label{structure equs}$$We can write (by Cartan lemma) $\tau _{\beta }=A_{\beta \gamma }\theta
^{\gamma }$ with $A_{\beta \gamma }=A_{\gamma \beta }$. The curvature of the Tanaka-Webster connection, expressed in terms of the coframe $\{ \theta
=\theta ^{0},\theta ^{\beta },\theta ^{\overline{\beta }}\}$, is $$\begin{split}
\Pi _{\beta }{}^{\gamma }& =\overline{\Pi _{\bar{\beta}}{}^{\overline{\gamma
}}}=d\theta _{\beta }{}^{\gamma }-\theta _{\beta }{}^{\sigma }\wedge \theta
_{\sigma }{}^{\gamma }, \\
\Pi _{0}{}^{\beta }& =\Pi _{\beta }{}^{0}=\Pi _{0}{}^{\bar{\beta}}=\Pi _{\bar{\beta}}{}^{0}=\Pi _{0}{}^{0}=0.
\end{split}$$Webster showed that $\Pi _{\beta }{}^{\gamma }$ can be written $$\Pi _{\beta }{}^{\gamma }=R_{\beta }{}^{\gamma }{}_{\rho \bar{\sigma}}\theta
^{\rho }\wedge \theta ^{\bar{\sigma}}+W_{\beta }{}^{\gamma }{}_{\rho }\theta
^{\rho }\wedge \theta -W^{\gamma }{}_{\beta \bar{\rho}}\theta ^{\bar{\rho}}\wedge \theta +i\theta _{\beta }\wedge \tau ^{\gamma }-i\tau _{\beta
}\wedge \theta ^{\gamma }$$where the coefficients satisfy $$R_{\beta \overline{\gamma }\rho \bar{\sigma}}=\overline{R_{\gamma \bar{\beta}\sigma \bar{\rho}}}=R_{\overline{\gamma }\beta \bar{\sigma}\rho }=R_{\rho
\overline{\gamma }\beta \bar{\sigma}},\ \ W_{\beta \overline{\gamma }\rho
}=W_{\rho \overline{\gamma }\beta }.$$
We will denote components of covariant derivatives with indices preceded by comma; thus write $A_{\rho \beta ,\gamma }$. The indices $\{0,\beta ,\overline{\beta }\}$ indicate derivatives with respect to $\{T,Z_{\beta },Z_{\overline{\beta }}\}$. For derivatives of a scalar function, we will often omit the comma, for instance, $u_{\beta }=Z_{\beta }u,\ u_{\gamma \bar{\beta}}=Z_{\bar{\beta}}Z_{\gamma }u-\theta _{\gamma }{}^{\rho }(Z_{\bar{\beta}})Z_{\rho }u,\ u_{0}=Tu$ for a smooth function $u$ .
For a real function $u$, the subgradient $\nabla _{b}$ is defined by $\nabla
_{b}u\in \xi $ and $\left \langle Z,\nabla _{b}u\right \rangle =du(Z)$ for all vector fields $Z$ tangent to contact plane. Locally $\nabla
_{b}u=u^{\beta }Z_{\beta }+u^{\overline{\beta }}Z_{\overline{\beta }}$. We can use the connection to define the subhessian as the complex linear map $$(\nabla ^{H})^{2}u:T_{1,0}\oplus T_{0,1}\rightarrow T_{1,0}\oplus T_{0,1}\text{ \ by \ }(\nabla ^{H})^{2}u(Z)=\nabla _{Z}\nabla _{b}u.$$In particular, $$\begin{array}{c}
|\nabla _{b}u|^{2}=2\sum_{\beta }u_{\beta }u^{\beta },\quad |\nabla
_{b}^{2}u|^{2}=2\sum_{\beta ,\gamma }(u_{\beta \gamma }u^{\beta \gamma
}+u_{\beta \overline{\gamma }}u^{\beta \overline{\gamma }}).\end{array}$$Also the sublaplacian is defined by $$\begin{array}{c}
\Delta _{b}u=Tr\left( (\nabla ^{H})^{2}u\right) =\sum_{\beta }(u_{\beta
}{}^{\beta }+u_{\overline{\beta }}{}^{\overline{\beta }}).\end{array}$$The pseudohermitian Ricci tensor and the torsion tensor on $T_{1,0}$ are defined by $$\begin{array}{l}
Ric(X,Y)=R_{\gamma \bar{\beta}}X^{\gamma }Y^{\bar{\beta}} \\
Tor(X,Y)=i\tsum_{\gamma ,\beta }(A_{\overline{\gamma }\bar{\beta}}X^{\overline{\gamma }}Y^{\bar{\beta}}-A_{\gamma \beta }X^{\gamma }Y^{\beta }),\end{array}$$where $X=X^{\gamma }Z_{\gamma },\ Y=Y^{\beta }Z_{\beta }$.
The CR Reilly’s Formula
=======================
Let $M$ be a compact pseudohermitian $(2n+1)$-manifold with boundary $\Sigma
$. We write $\theta _{\gamma }^{\; \text{\ }\beta }=\omega _{\gamma }^{\;\text{\ }\beta }+i\tilde{\omega}_{\gamma }^{\; \text{\ }\beta }$ with $\omega _{\gamma }^{\; \text{\ }\beta }=\QTR{up}{\func{Re}}(\theta _{\gamma
}^{\; \text{\ }\beta })$, $\tilde{\omega}_{\gamma }^{\; \text{\ }\beta }=\QTR{up}{\func{Im}}(\theta _{\gamma }^{\; \text{\ }\beta })$ and $Z_{\beta }=\frac{1}{2}(e_{\beta }-ie_{n+\beta })$ for real vectors $e_{\beta }$, $e_{n+\beta }$, $\beta =1,\cdots ,n$. It follows that $e_{n+\beta }=Je_{\beta
}$. Let $e^{\beta }=\QTR{up}{\func{Re}}(\theta ^{\beta })$, $e^{n+\beta }=\QTR{up}{\func{Im}}(\theta ^{\beta })$, $\beta =1,\cdots ,n$. Then $\{
\theta ,e^{\beta },e^{n+\beta }\}$ is dual to $\{T,e_{\beta },e_{n+\beta }\}$. Now in view of (\[dtheta\]) and (\[structure equs\]), we have the following real version of structure equations: $$\left \{
\begin{split}
& d\theta =2\tsum_{\beta }e^{\beta }\wedge e^{n+\beta }, \\
& \nabla e_{\gamma }=\omega _{\gamma }^{\; \text{\ }\beta }\otimes e_{\beta
}+\tilde{\omega}_{\gamma }^{\; \text{\ }\beta }\otimes e_{n+\beta },\text{ \
}\nabla e_{n+\gamma }=\omega _{\gamma }^{\; \text{\ }\beta }\otimes
e_{n+\beta }-\tilde{\omega}_{\gamma }^{\; \text{\ }\beta }\otimes e_{\beta },
\\
& de^{\gamma }=e^{\beta }\wedge \omega _{\beta }^{\; \text{\ }\gamma
}-e^{n+\beta }\wedge \tilde{\omega}_{\beta }^{\; \text{\ }\gamma }\text{
\textrm{mod} }\theta ;\text{ }de^{n+\gamma }=e^{\beta }\wedge \tilde{\omega}_{\beta }^{\text{ \ }\gamma }+e^{n+\beta }\wedge \omega _{\beta }^{\text{ \ }\gamma }\text{ \textrm{mod} }\theta .
\end{split}\right.$$
Let $\Sigma $ be a surface contained in $M$. The singular set $S_{\Sigma }$ consists of those points where $\xi $ coincides with the tangent bundle $T\Sigma $ of $\Sigma $. It is easy to see that $S_{\Sigma }$ is a closed set. On $\xi ,$ we can associate a natural metric $\langle $ $,$ $\rangle
_{G}=\frac{1}{2}d\theta (\cdot ,J\cdot )$ call the Levi metric. For a vector $v\in \xi ,$ we define the length of $v$ by $\left \vert v\right \vert
_{G}^{2}=\langle v,v\rangle _{G}.$ With respect to the Levi metric, we can take unit vector fields $e_{1},\cdots ,e_{2n-1}\in \xi \cap T\Sigma $ on $\Sigma \backslash S_{\Sigma }$, called the characteristic fields and $e_{2n}=Je_{n}$, called the Legendrian normal. The $p$(pseudohermitian)-mean curvature $H_{p.h}$ on $\Sigma \backslash S_{\Sigma }$ is defined by $$\begin{array}{c}
H_{p.h}=-\sum_{j=1}^{2n-1}\left \langle \nabla _{e_{j}}e_{2n},e_{j}\right
\rangle .\end{array}$$For $e_{1},\cdots ,e_{2n-1}$ being characteristic fields, we have the $p$-area element $$d\Sigma _{p}=\theta \wedge e^{1}\wedge e^{n+1}\wedge \cdots \wedge
e^{n-1}\wedge e^{2n-1}\wedge e^{n}$$on $\Sigma $ and all surface integrals over $\Sigma $ are with respect to this $2n$-form $d\Sigma _{p}$. Note that $d\Sigma _{p}$ continuously extends over the singular set $S_{\Sigma }$ and vanishes on $S_{\Sigma }$.
We also write $\varphi _{e_{j}}=e_{j}\varphi $ and $\nabla _{b}\varphi =\frac{1}{2}(\varphi _{e_{\beta }}e_{\beta }+\varphi _{e_{n+\beta
}}e_{n+\beta })$. Moreover, $\varphi _{e_{j}e_{k}}=e_{k}e_{j}\varphi -\nabla
_{e_{k}}e_{j}\varphi $ and $\Delta _{b}\varphi =\frac{1}{2}\tsum_{\beta
}(\varphi _{e_{\beta }e_{\beta }}+\varphi _{e_{n+\beta }e_{n+\beta }})$. Next we define the subdivergence operator $div_{b}(\cdot )$ by $div_{b}(W)=W^{\beta },_{\beta }+W^{\overline{\beta }},_{\overline{\beta }}$ for all vector fields $W=W^{\beta }Z_{\beta }+W^{\overline{\beta }}Z_{\overline{\beta }}$ and its real version is $div_{b}(W)=\varphi _{\beta
,e_{\beta }}+\psi _{n+\beta ,e_{n+\beta }}$ for $W=\varphi _{\beta }e_{\beta
}+\psi _{n+\beta }e_{n+\beta }$. We define the tangential subgradient $\nabla _{b}^{t}$ of a function $\varphi $ by $\nabla _{b}^{t}\varphi =\nabla
_{b}\varphi -\langle \nabla _{b}\varphi ,e_{2n}\rangle _{G}e_{2n}$ and the tangent sublaplacian $\Delta _{b}^{t}$ of $\varphi $ by $\Delta
_{b}^{t}\varphi =\frac{1}{2}\sum_{j=1}^{2n-1}[(e_{j})^{2}\varphi -(\nabla
_{e_{j}}e_{j})^{t}\varphi ],$ where $(\nabla _{e_{j}}e_{j})^{t}$ is the tangential part of $\nabla _{e_{j}}e_{j}$.
We first recall the following CR Bochner formula.
Let $(M,J,\theta )$ be a pseudohermitian $(2n+1)$-manifold. For a real function $\varphi $, we have$$\begin{array}{lll}
\frac{1}{2}\Delta _{b}|\nabla _{b}\varphi |^{2} & = & |\nabla
_{b}^{2}\varphi |^{2}+\langle \nabla _{b}\varphi ,\nabla _{b}\Delta
_{b}\varphi \rangle \\
& & +[2Ric-(n-2)Tor]((\nabla _{b}\varphi )_{\mathbb{C}},(\nabla _{b}\varphi
)_{\mathbb{C}})+2\langle J\nabla _{b}\varphi ,\nabla _{b}\varphi _{0}\rangle
,\end{array}
\label{Bochnerformula}$$where $(\nabla _{b}\varphi )_{\mathbb{C}}=\varphi ^{\beta }Z_{\beta }$ is the corresponding complex $(1,0)$-vector field of $\nabla _{b}\varphi $.
The proof of the above formula follows from the Bochner formula (Lemma 3 in [@gr]) derived by A. Greenleaf and using the commutation relation (see Lemma 2.2 in [@cc1]) $$\begin{array}{c}
i\tsum_{\beta }(\varphi _{\beta }\varphi _{\overline{\beta }0}-\varphi _{\overline{\beta }}\varphi _{\beta 0})=i\tsum_{\beta }(\varphi _{\beta
}\varphi _{0\overline{\beta }}-\varphi _{\overline{\beta }}\varphi _{0\beta
})-Tor((\nabla _{b}\varphi )_{\mathbb{C}},(\nabla _{b}\varphi )_{\mathbb{C}}).\end{array}$$From [@cc1], we can relate $\langle J\nabla _{b}\varphi ,\nabla
_{b}\varphi _{0}\rangle $ with $\langle \nabla _{b}\varphi ,\nabla
_{b}\Delta _{b}\varphi \rangle $ by$$\begin{array}{c}
\langle J\nabla _{b}\varphi ,\nabla _{b}\varphi _{0}\rangle =\frac{1}{n}\langle \nabla _{b}\varphi ,\nabla _{b}\Delta _{b}\varphi \rangle
-2Tor((\nabla _{b}\varphi )_{\mathbb{C}},(\nabla _{b}\varphi )_{\mathbb{C}})-\frac{2}{n}\langle P\varphi +\overline{P}\varphi ,d_{b}\varphi \rangle .\end{array}
\label{relation formula}$$
For the proof of Reilly’s formula, we first need a series of formulae. In particular, one derives the following CR version of divergence theorem and Green’s identity for a compact pseudohermitian $(2n+1)$-manifold $M$ with boundary $\Sigma $. Note that $d\Sigma _{p}$ vanishes on $S_{\Sigma }$.
\[DivergenceTheorem\] (Divergence Theorem) Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with boundary $\Sigma .$ For a real function $\varphi $, we have $$\begin{array}{c}
\int_{M}\Delta _{b}\varphi d\mu =\int_{M}\QTR{up}{div}_{b}(\nabla
_{b}\varphi )d\mu =\frac{1}{2}C_{n}\int_{\Sigma }\varphi _{e_{2n}}d\Sigma
_{p}=C_{n}\int_{\Sigma }\langle \nabla _{b}\varphi ,e_{2n}\rangle
_{G}d\Sigma _{p},\end{array}
\label{C}$$$$\begin{array}{c}
\int_{M}\varphi \varphi _{00}d\mu +\int_{M}\varphi _{0}^{2}d\mu
=-C_{n}\int_{\Sigma }\alpha \varphi \varphi _{0}d\Sigma _{p}.\end{array}
\label{C1}$$Here $d\Sigma _{p}=\theta \wedge e^{1}\wedge e^{n+1}\wedge \cdots \wedge
e^{n-1}\wedge e^{2n-1}\wedge e^{n}$ is the $p$-area element of $\Sigma $ and $C_{n}=2^{n}n!$.
By the Stoke’s theorem, we have$$\begin{array}{lll}
\int_{M}\Delta _{b}\varphi d\mu & = & \frac{1}{2}\int_{M}\tsum_{\beta
}(\varphi _{e_{\beta }e_{\beta }}+\varphi _{e_{n+\beta }e_{n+\beta
}})2^{n}n!\theta \wedge e^{1}\wedge e^{n+1}\wedge \cdots \wedge e^{n}\wedge
e^{2n} \\
& = & 2^{n-1}n!\int_{M}\tsum_{\beta }d[-\varphi _{e_{\beta }}\theta \wedge
e^{1}\wedge e^{n+1}\wedge \cdots \wedge \widehat{e^{\beta }}\wedge
e^{n+\beta }\wedge \cdots \wedge e^{n}\wedge e^{2n} \\
& & \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }+\varphi _{e_{n+\beta
}}\theta \wedge e^{1}\wedge e^{n+1}\wedge \cdots \wedge e^{\beta }\wedge
\widehat{e^{n+\beta }}\wedge \cdots \wedge e^{n}\wedge e^{2n}] \\
& = & 2^{n-1}n!\int_{\Sigma }\varphi _{e_{2n}}\theta \wedge e^{1}\wedge
e^{n+1}\wedge \cdots \wedge e^{n-1}\wedge e^{2n-1}\wedge e^{n} \\
& = & C_{n}\int_{\Sigma }\langle \nabla _{b}\varphi ,e_{2n}\rangle
_{G}d\Sigma _{p}.\end{array}$$Here we used $d\mu =\theta \wedge (d\theta )^{n}=C_{n}\theta \wedge
e^{1}\wedge e^{n+1}\wedge \cdots \wedge e^{n}\wedge e^{2n}$ and the fact that the $2n$-forms $\theta \wedge e^{1}\wedge e^{n+1}\wedge \cdots \wedge
\widehat{e^{\beta }}\wedge e^{n+\beta }\wedge \cdots \wedge e^{n}\wedge
e^{2n}$ vanish on $S_{\Sigma }$ for $\beta =1,\cdots ,n$ and so are $\theta
\wedge e^{1}\wedge e^{n+1}\wedge \cdots \wedge e^{\beta }\wedge \widehat{e^{n+\beta }}\wedge \cdots \wedge e^{n}\wedge e^{2n}$ for $\beta =1,\cdots
,n-1$, since $e_{j}$ are tangent along $\Sigma $ for $j=1,\cdots ,2n-1$.
The second equation follows easily from Stoke’s theorem as above $$\begin{array}{lll}
\int_{M}\varphi \varphi _{00}d\mu +\int_{M}\varphi _{0}^{2}d\mu & = &
C_{n}\int_{M}d(\varphi \varphi _{0}e^{1}\wedge e^{n+1}\wedge \cdots \wedge
e^{n}\wedge e^{2n}) \\
& = & C_{n}\int_{\Sigma }\varphi \varphi _{0}e^{1}\wedge e^{n+1}\wedge
\cdots \wedge e^{n}\wedge e^{2n}\end{array}$$and the help of the identity $e^{2n}\wedge e^{n}=\alpha \theta \wedge e^{n}$ on $\Sigma \backslash S_{\Sigma }.$
\[Green’s identity\] (Green’s identity) Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with boundary $\Sigma .$ For real functions $\varphi $ and $\psi $, $$\begin{array}{c}
\int_{M}\psi \Delta _{b}\varphi d\mu +\int_{M}\langle \nabla _{b}\varphi
,\nabla _{b}\psi \rangle d\mu =\frac{1}{2}C_{n}\int_{\Sigma }\psi \varphi
_{e_{2n}}d\Sigma _{p}.\end{array}
\label{B}$$
It is easy to check that $\QTR{up}{div}_{b}(\psi \nabla _{b}\varphi )=\psi
\Delta _{b}\varphi +\langle \nabla _{b}\varphi ,\nabla _{b}\psi \rangle $ and then the result follows from the CR version of divergence theorem.
\[Jvarphi0\] Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with boundary $\Sigma .$ For any real smooth function $\varphi $, $$\begin{array}{c}
\int_{M}\langle J\nabla _{b}\varphi ,\nabla _{b}\varphi _{0}\rangle d\mu
+n\int_{M}\varphi _{0}^{2}d\mu =\frac{1}{2}C_{n}\int_{\Sigma }\varphi
_{0}\varphi _{e_{n}}d\Sigma _{p}.\end{array}
\label{eqJvarphi0}$$
Since $\QTR{up}{div}_{b}(\left( J\nabla _{b}\varphi \right) \varphi
_{0})=\langle J\nabla _{b}\varphi ,\nabla _{b}\varphi _{0}\rangle +n\varphi
_{0}^{2}$ and by the divergence theorem (\[C\]), we have$$\begin{array}{ll}
& \int_{M}\langle J\nabla _{b}\varphi ,\nabla _{b}\varphi _{0}\rangle d\mu
+n\int_{M}\varphi _{0}^{2}d\mu \\
= & \int_{M}\QTR{up}{div}_{b}(\left( J\nabla _{b}\varphi \right) \varphi
_{0})d\mu =C_{n}\int_{\Sigma }\langle \left( J\nabla _{b}\varphi \right)
\varphi _{0},e_{2n}\rangle _{G}d\Sigma _{p}=\frac{1}{2}C_{n}\int_{\Sigma
}\varphi _{0}\varphi _{e_{n}}d\Sigma _{p}.\end{array}$$
\[P&Paneitz\] Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with boundary $\Sigma .$ For any real smooth function $\varphi $, $$\begin{array}{c}
\int_{M}\langle P\varphi +\overline{P}\varphi ,d_{b}\varphi \rangle d\mu +\frac{1}{4}\int_{M}(P_{0}\varphi )\varphi d\mu =\frac{1}{2}iC_{n}\int_{\Sigma }\varphi \left( P_{n}\varphi -P_{\overline{n}}\varphi
\right) d\Sigma _{p}.\end{array}
\label{eqP&Paneitz}$$
It can be easily checked that $$\begin{array}{c}
\QTR{up}{div}_{b}\left( (\varphi P^{\beta }\varphi )Z_{\beta }+(\varphi P^{\overline{\beta }}\varphi )Z_{\overline{\beta }}\right) =\langle P\varphi +\overline{P}\varphi ,d_{b}\varphi \rangle +\frac{1}{4}\varphi P_{0}\varphi .\end{array}$$We then have by the divergence theorem (\[C\]) $$\begin{array}{ll}
& \int_{M}\langle P\varphi +\overline{P}\varphi ,d_{b}\varphi \rangle d\mu +\frac{1}{4}\int_{M}(P_{0}\varphi )\varphi d\mu \\
= & C_{n}\int_{\Sigma }\left \langle (\varphi P^{\beta }\varphi )Z_{\beta
}+(\varphi P^{\overline{\beta }}\varphi )Z_{\overline{\beta }},e_{2n}\right
\rangle _{G}d\Sigma _{p}=\frac{1}{2}iC_{n}\int_{\Sigma }\varphi \left(
P_{n}\varphi -P_{\overline{n}}\varphi \right) d\Sigma _{p}.\end{array}$$
Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with boundary $\Sigma $. For real-valued functions $\varphi $ on $\Sigma ,$$$\begin{array}{c}
\int_{\Sigma }\left( \varphi _{e_{n}}+2\alpha \varphi \right) d\Sigma _{p}=0;\end{array}
\label{2014a}$$$$\begin{array}{c}
\int_{\Sigma }[\varphi _{\overline{\beta }}+(\tsum_{\gamma \neq n}\theta _{\overline{\beta }}^{\; \text{\ }\overline{\gamma }}(Z_{\overline{\gamma }})+\frac{1}{2}\theta _{\overline{\beta }}^{\; \text{\ }\overline{n}}(e_{n}))\varphi ]d\Sigma _{p}=0\text{ for any }\beta \neq n;\end{array}
\label{2014b}$$$$\begin{array}{c}
\int_{\Sigma }[\varphi _{0}+\alpha \varphi _{e_{2n}}-(\alpha \tilde{\omega}_{n}^{\; \text{\ }n}(e_{n})-\func{Re}A_{\overline{n}\overline{n}})\varphi
]d\Sigma _{p}=0.\end{array}
\label{2014c}$$
By the Stoke’s theorem, we have $$\begin{array}{lll}
\frac{1}{2}C_{n}\int_{\Sigma }\varphi _{e_{n}}d\Sigma _{p} & = &
\int_{\Sigma }\varphi _{e_{n}}\theta \wedge \left( d\theta \right)
^{n-1}\wedge e^{n} \\
& = & -\int_{\Sigma }d\varphi \wedge \theta \wedge \left( d\theta \right)
^{n-1}+\int_{\Sigma }\varphi _{e_{2n}}e^{2n}\wedge \theta \wedge \left(
d\theta \right) ^{n-1} \\
& = & -\int_{\Sigma }d(\varphi \theta \wedge \left( d\theta \right)
^{n-1})+\int_{\Sigma }\varphi d\theta \wedge \left( d\theta \right) ^{n-1}
\\
& = & \int_{\Sigma }2\varphi e^{n}\wedge e^{2n}\wedge \left( d\theta \right)
^{n-1}=-\int_{\Sigma }2\alpha \varphi \theta \wedge e^{n}\wedge \left(
d\theta \right) ^{n-1} \\
& = & -C_{n}\int_{\Sigma }\alpha \varphi d\Sigma _{p},\end{array}$$where we used the identities $\theta \wedge \left( d\theta \right)
^{n-1}\wedge e^{2n}=0$ on $\Sigma $ since $e_{n}$ is tangent along $\Sigma ,$ $d\theta =2\tsum_{\beta =1}^{n}e^{\beta }\wedge e^{n+\beta }$ and $e^{2n}\wedge e^{n}=\alpha \theta \wedge e^{n}$ on $\Sigma \backslash
S_{\Sigma }.$
For the second equation, we compute $$\begin{array}{ll}
& \int_{\Sigma }\varphi _{\overline{\beta }}\theta \wedge \left( d\theta
\right) ^{n-1}\wedge e^{n}=\int_{\Sigma }\varphi _{\overline{\beta }}\theta
\wedge \theta ^{\beta }\wedge \theta ^{\overline{\beta }}\wedge
(\tsum_{j=1}^{n-1}\underset{j\neq \beta }{\wedge }\theta ^{j}\wedge \theta ^{\overline{j}})\wedge e^{n} \\
= & \int_{\Sigma }d\varphi \wedge \theta \wedge \theta ^{\beta }\wedge
(\tsum_{j=1}^{n-1}\underset{j\neq \beta }{\wedge }\theta ^{j}\wedge \theta ^{\overline{j}})\wedge e^{n}=-\int_{\Sigma }\varphi d[\theta \wedge \theta
^{\beta }\wedge (\left( d\theta \right) ^{n-2})\wedge e^{n}] \\
= & \int_{\Sigma }\varphi \lbrack \theta \wedge d\theta ^{\beta }\wedge
(\left( d\theta \right) ^{n-2})\wedge e^{n}]-\int_{\Sigma }\varphi \lbrack
\theta \wedge \theta ^{\beta }\wedge (\left( d\theta \right) ^{n-2})\wedge
de^{n}] \\
= & \int_{\Sigma }\varphi \lbrack \theta \wedge (\theta ^{\gamma }\wedge
\theta _{\gamma }{}^{\beta }+\theta \wedge \tau ^{\beta })\wedge
(\tsum_{j=1}^{n-1}\underset{j\neq \gamma }{\wedge }\theta ^{j}\wedge \theta
^{\overline{j}})\wedge e^{n}] \\
& -\int_{\Sigma }\frac{1}{2}\varphi \lbrack \theta \wedge \theta ^{\beta
}\wedge (\tsum_{j=1}^{n-1}\underset{j\neq \beta }{\wedge }\theta ^{j}\wedge
\theta ^{\overline{j}})\wedge (\tsum_{\gamma \neq n}\theta _{\overline{\gamma }}{}^{\overline{n}}(e_{n})\theta ^{\overline{\gamma }})\wedge e^{n}]
\\
= & \int_{\Sigma }\left( \tsum_{\gamma \neq n}\theta _{\gamma }{}^{\beta
}(Z_{\overline{\gamma }})-\frac{1}{2}\theta _{\overline{\beta }}^{\;\overline{n}}(e_{n})\right) \varphi \theta \wedge \theta ^{\beta }\wedge
\theta ^{\overline{\beta }}\wedge (\tsum_{j=1}^{n-1}\underset{j\neq \beta }{\wedge }\theta ^{j}\wedge \theta ^{\overline{j}})\wedge e^{n} \\
= & -\int_{\Sigma }\left( \tsum_{\gamma \neq n}\theta _{\overline{\beta }}^{\; \overline{\gamma }}(Z_{\overline{\gamma }})+\frac{1}{2}\theta _{\overline{\beta }}^{\; \overline{n}}(e_{n})\right) \varphi \theta \wedge
\left( d\theta \right) ^{n-1}\wedge e^{n},\end{array}$$where we used $de^{n}=\frac{1}{2}(\theta ^{\gamma }\wedge \theta _{\gamma
}{}^{n}+\theta ^{\overline{\gamma }}\wedge \theta _{\overline{\gamma }}{}^{\overline{n}})=\frac{1}{2}\tsum_{\gamma \neq n}\theta _{\overline{\gamma }}{}^{\overline{n}}(e_{n})\theta ^{\overline{\gamma }}\wedge e^{n}$ $\mathrm{mod}$ $\theta ,$ $e^{2n}$ on $\Sigma .$
The same compute for the third equation yields$$\begin{array}{ll}
& \int_{\Sigma }\varphi _{0}\theta \wedge \left( d\theta \right)
^{n-1}\wedge e^{n} \\
= & \int_{\Sigma }d\varphi \wedge \left( d\theta \right) ^{n-1}\wedge
e^{n}-\int_{\Sigma }\varphi _{e_{2n}}e^{2n}\wedge e^{n}\wedge \left( d\theta
\right) ^{n-1} \\
= & \int_{\Sigma }d(\varphi \left( d\theta \right) ^{n-1}\wedge
e^{n})-\int_{\Sigma }\varphi \left( d\theta \right) ^{n-1}\wedge
de^{n}-\int_{\Sigma }\alpha \varphi _{e_{2n}}\theta \wedge \left( d\theta
\right) ^{n-1}\wedge e^{n} \\
= & \int_{\Sigma }\varphi \left( d\theta \right) ^{n-1}\wedge \lbrack \tilde{\omega}_{n}^{\;n}(e_{n})e^{2n}\wedge e^{n}-\func{Re}A_{\overline{n}\overline{n}}\theta \wedge e^{n}]-\int_{\Sigma }\alpha \varphi _{e_{2n}}\theta \wedge
\left( d\theta \right) ^{n-1}\wedge e^{n} \\
= & \int_{\Sigma }[(\alpha \tilde{\omega}_{n}^{\;n}(e_{n})-\func{Re}A_{\overline{n}\overline{n}})\varphi -\alpha \varphi _{e_{2n}}]\theta \wedge
\left( d\theta \right) ^{n-1}\wedge e^{n}.\end{array}$$
Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with boundary $\Sigma $. For real-valued functions $\varphi $ and $\psi $ on $\Sigma ,$ we have$$\begin{array}{c}
\int_{\Sigma }\psi (\Delta _{b}^{t}+\alpha e_{n})\varphi d\Sigma
_{p}=\int_{\Sigma }\varphi (\Delta _{b}^{t}+\alpha e_{n})\psi d\Sigma _{p}.\end{array}
\label{2014d}$$
This Lemma implies that $\Delta _{b}^{t}+\alpha e_{n}$ is a self-adjoint operator with respect to the $p$-area element $d\Sigma _{p}$ on $\Sigma .$
**The Proof of Theorem**
By integrating the CR version of Bochner formula (\[Bochnerformula\]), we have$$\begin{array}{lll}
\frac{1}{2}\int_{M}\Delta _{b}|\nabla _{b}\varphi |^{2}d\mu & = &
\int_{M}|\nabla _{b}^{2}\varphi |^{2}d\mu +\int_{M}\langle \nabla
_{b}\varphi ,\nabla _{b}\Delta _{b}\varphi \rangle d\mu \\
& & +\int_{M}[2Ric-(n-2)Tor]((\nabla _{b}\varphi )_{\mathbb{C}},(\nabla
_{b}\varphi )_{\mathbb{C}})d\mu \\
& & +2\int_{M}\langle J\nabla _{b}\varphi ,\nabla _{b}\varphi _{0}\rangle
d\mu .\end{array}$$Note that $$\begin{array}{c}
\tsum_{\beta ,\gamma }|\varphi _{\beta \overline{\gamma }}|^{2}=\tsum_{\beta
,\gamma }|\varphi _{\beta \overline{\gamma }}-\frac{1}{n}\varphi _{\sigma
}{}^{\sigma }h_{\beta \overline{\gamma }}|^{2}+\frac{1}{4n}\left( \Delta
_{b}\varphi \right) ^{2}+\frac{n}{4}\varphi _{0}^{2}.\end{array}$$It follows from the CR Green’s identity (\[B\]) with $\psi =\Delta
_{b}\varphi $ and (\[eqJvarphi0\]), that $$\begin{array}{ll}
& \frac{1}{2}\int_{M}\Delta _{b}|\nabla _{b}\varphi |^{2}d\mu \\
= & 2\int_{M}\tsum_{\beta ,\gamma }|\varphi _{\beta \gamma }|^{2}d\mu
+2\int_{M}\tsum_{\gamma ,\beta }|\varphi _{\beta \overline{\gamma }}-\frac{1}{n}\varphi _{\sigma }{}^{\sigma }h_{\beta \overline{\gamma }}|^{2}d\mu \\
& -\frac{3n}{2}\int_{M}\varphi _{0}^{2}d\mu +C_{n}\int_{\Sigma }\varphi
_{0}\varphi _{e_{n}}d\Sigma _{p}+\frac{1}{2}C_{n}\int_{\Sigma }(\Delta
_{b}\varphi )\varphi _{e_{2n}}d\Sigma _{p} \\
& -\frac{2n-1}{2n}\int_{M}(\Delta _{b}\varphi )^{2}d\mu
+\int_{M}[2Ric-(n-2)Tor]((\nabla _{b}\varphi )_{\mathbb{C}},(\nabla
_{b}\varphi )_{\mathbb{C}}).\end{array}
\label{temporary05}$$By combining (\[eqJvarphi0\]), (\[relation formula\]), (\[B\]) and (\[eqP&Paneitz\]), we have$$\begin{array}{lll}
n\int_{M}\varphi _{0}^{2}d\mu & = & \frac{1}{n}\int_{M}(\Delta _{b}\varphi
)^{2}d\mu -\frac{1}{2n}C_{n}\int_{\Sigma }(\Delta _{b}\varphi )\varphi
_{e_{2n}}d\Sigma _{p} \\
& & -\frac{1}{2n}\int_{M}\varphi P_{0}\varphi d\mu +\frac{1}{n}iC_{n}\int_{\Sigma }\varphi \left( P_{n}\varphi -P_{\overline{n}}\varphi
\right) d\Sigma _{p} \\
& & +\frac{1}{2}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma
_{p}+2\int_{M}Tor\left( (\nabla _{b}\varphi )_{\mathbb{C}},(\nabla
_{b}\varphi )_{\mathbb{C}}\right) d\mu .\end{array}
\label{100}$$Also applying the divergence Theorem to the equation$$\begin{array}{c}
(B^{\beta \overline{\gamma }}\varphi )(B_{\beta \overline{\gamma }}\varphi
)=(\varphi ^{\beta }B_{\beta \overline{\gamma }}\varphi ),^{\overline{\gamma
}}-\frac{n-1}{n}(\varphi P_{\beta }\varphi ),^{\beta }+\frac{n-1}{8n}\varphi
P_{0}\varphi\end{array}$$with $B_{\beta \overline{\gamma }}\varphi =\varphi _{\beta \overline{\gamma }}-\frac{1}{n}\varphi _{\sigma }{}^{\sigma }h_{\beta \overline{\gamma }},$ we obtain$$\begin{array}{ll}
& \int_{M}\tsum_{\beta ,\gamma }|\varphi _{\beta \overline{\gamma }}-\frac{1}{n}\varphi _{\sigma }{}^{\sigma }h_{\beta \overline{\gamma }}|^{2}d\mu \\
= & \frac{n-1}{8n}\int_{M}\varphi P_{0}\varphi d\mu -\frac{n-1}{4n}iC_{n}\int_{\Sigma }\varphi \left( P_{n}\varphi -P_{\overline{n}}\varphi
\right) d\Sigma _{p} \\
& +\frac{1}{4}iC_{n}\int_{\Sigma }(\varphi ^{\overline{\beta }}B_{n\overline{\beta }}\varphi -\varphi ^{\beta }B_{\overline{n}\beta }\varphi )d\Sigma
_{p}.\end{array}
\label{101}$$Here$$\begin{array}{ll}
& i(\varphi ^{\overline{\beta }}B_{n\overline{\beta }}\varphi -\varphi
^{\beta }B_{\overline{n}\beta }\varphi ) \\
= & \frac{1}{4}\tsum_{\beta \neq n}[\varphi _{e_{n+\beta }}(\varphi
_{e_{\beta }e_{n}}+\varphi _{e_{n+\beta }e_{2n}})+\varphi _{e_{\beta
}}(\varphi _{e_{\beta }e_{2n}}-\varphi _{e_{n+\beta }e_{n}})] \\
& +\frac{1}{4}\varphi _{e_{2n}}[(\varphi _{e_{n}e_{n}}+\varphi
_{e_{2n}e_{2n}})-\frac{2}{n}\Delta _{b}\varphi ].\end{array}$$Substituting these into the right hand side of (\[temporary05\]), we get$$\begin{array}{ll}
& \frac{1}{2}\int_{M}\Delta _{b}|\nabla _{b}\varphi |^{2}d\mu \\
= & 2\int_{M}\tsum_{\beta ,\gamma }|\varphi _{\beta \gamma }|^{2}d\mu -\frac{n+1}{n}\int_{M}(\Delta _{b}\varphi )^{2}d\mu \\
& +\frac{n+2}{4n}\int_{M}\varphi P_{0}\varphi d\mu -\frac{n+2}{2n}iC_{n}\int_{\Sigma }\varphi \left( P_{n}\varphi -P_{\overline{n}}\varphi
\right) d\Sigma _{p} \\
& +\int_{M}[2Ric-(n+1)Tor]((\nabla _{b}\varphi )_{\mathbb{C}},(\nabla
_{b}\varphi )_{\mathbb{C}})d\mu +\frac{1}{4}C_{n}\int_{\Sigma }\varphi
_{0}\varphi _{e_{n}}d\Sigma _{p} \\
& +\frac{1}{2}iC_{n}\int_{\Sigma }(\varphi ^{\overline{\beta }}B_{n\overline{\beta }}\varphi -\varphi ^{\beta }B_{\overline{n}\beta }\varphi )d\Sigma
_{p}+\frac{2n+3}{4n}C_{n}\int_{\Sigma }(\Delta _{b}\varphi )\varphi
_{e_{2n}}d\Sigma _{p}.\end{array}
\label{temporary06}$$
On the other hand, the divergence theorem (\[C\]) implies that$$\begin{array}{lll}
\frac{1}{2}\int_{M}\Delta _{b}|\nabla _{b}\varphi |^{2}d\mu & = & \frac{1}{4}C_{n}\int_{\Sigma }\left( |\nabla _{b}\varphi |^{2}\right) _{e_{2n}}d\Sigma
_{p} \\
& = & \frac{1}{4}C_{n}\int_{\Sigma }\tsum_{\beta \neq n}\left( \varphi
_{e_{\beta }}\varphi _{e_{\beta }e_{2n}}+\varphi _{e_{n+\beta }}\varphi
_{e_{n+\beta }e_{2n}}\right) d\Sigma _{p} \\
& & +\frac{1}{4}C_{n}\int_{\Sigma }\left( \varphi _{e_{n}}\varphi
_{e_{n}e_{2n}}+\varphi _{e_{2n}}\varphi _{e_{2n}e_{2n}}\right) d\Sigma _{p}.\end{array}$$Substituting the commutation relations $$\begin{array}{l}
\varphi _{e_{\beta }e_{n+\gamma }}=\varphi _{e_{n+\gamma }e_{\beta }},\
\varphi _{e_{n+\beta }e_{n+\gamma }}=\varphi _{e_{n+\gamma }e_{n+\beta }},\
\mathrm{for}\text{ }\mathrm{all}\text{ }{\small \beta \neq \gamma ,} \\
\text{ }\varphi _{e_{n}e_{2n}}=\varphi _{e_{2n}e_{n}}+2\varphi _{0},\end{array}$$and $$\begin{array}{c}
\tsum_{\beta \neq n}2(\varphi _{\beta \overline{\beta }}+\varphi _{\overline{\beta }\beta })+\varphi _{e_{n}e_{n}}=\sum_{j=1}^{2n-1}\varphi
_{e_{j}e_{j}}=2\Delta _{b}^{t}\varphi +H_{p.h}\varphi _{e_{2n}} \\
\varphi _{e_{2n}e_{2n}}=2\Delta _{b}\varphi -\sum_{j=1}^{2n-1}\varphi
_{e_{j}e_{j}}\end{array}
\label{A}$$into the above equation, also integrating by parts from (\[2014a\]) and (\[2014b\]) yields $$\begin{array}{ll}
& \frac{1}{2}\int_{M}\Delta _{b}|\nabla _{b}\varphi |^{2}d\mu \\
= & \frac{1}{4}C_{n}\int_{\Sigma }\tsum_{\beta \neq n}(\varphi _{e_{\beta
}}\varphi _{e_{2n}e_{\beta }}+\varphi _{e_{n+\beta }}\varphi
_{_{e_{2n}e_{n+\beta }}})d\Sigma _{p} \\
& +\frac{1}{4}C_{n}\int_{\Sigma }\varphi _{e_{n}}(\varphi
_{e_{2n}e_{n}}+2\varphi _{0})d\Sigma _{p}+\frac{1}{4}C_{n}\int_{\Sigma
}\varphi _{e_{2n}}\varphi _{e_{2n}e_{2n}}d\Sigma _{p} \\
= & \frac{1}{4}C_{n}\int_{\Sigma }[\tsum_{\beta \neq n}2(\varphi _{\overline{\beta }}\varphi _{e_{2n}Z_{\beta }}+\varphi _{\beta }\varphi _{_{e_{2n}Z_{\overline{\beta }}}})+\varphi _{e_{n}}\varphi _{e_{2n}e_{n}}]d\Sigma _{p} \\
& +\frac{1}{2}C_{n}\int_{\Sigma }\varphi _{e_{n}}\varphi _{0}d\Sigma _{p}+\frac{1}{4}C_{n}\int_{\Sigma }\varphi _{e_{2n}}\varphi
_{e_{2n}e_{2n}}d\Sigma _{p} \\
= & -\frac{1}{4}C_{n}\int_{\Sigma }\varphi _{e_{2n}}[\tsum_{\beta \neq
n}2(\varphi _{\beta \overline{\beta }}+\varphi _{\overline{\beta }\beta
})+\varphi _{e_{n}e_{n}}]d\Sigma _{p} \\
& -\frac{1}{2}C_{n}\int_{\Sigma }\alpha \varphi _{e_{n}}\varphi
_{e_{2n}}d\Sigma _{p}-\frac{1}{4}C_{n}\int_{\Sigma }\varphi _{e_{n}}(\nabla
_{e_{n}}e_{2n})\varphi d\Sigma _{p} \\
& +\frac{1}{2}C_{n}\int_{\Sigma }\varphi _{e_{n}}\varphi _{0}d\Sigma _{p}+\frac{1}{4}C_{n}\int_{\Sigma }\varphi _{e_{2n}}\varphi
_{e_{2n}e_{2n}}d\Sigma _{p} \\
& +\frac{1}{4}C_{n}\int_{\Sigma }\varphi _{e_{2n}}[\tsum_{\beta \neq
n}(\theta _{n}{}^{\beta }(e_{n})\varphi _{\beta }+\theta _{\overline{n}}{}^{\overline{\beta }}(e_{n})\varphi _{\overline{\beta }})-(\nabla
_{e_{n}}e_{n})^{t}\varphi ]d\Sigma _{p} \\
& +\frac{1}{2}C_{n}\int_{\Sigma }\tsum_{\beta ,\gamma \neq n}i(\theta _{\overline{n}}{}^{\overline{\gamma }}(Z_{\beta })\varphi _{\overline{\gamma }}-\theta _{n}{}^{\gamma }(Z_{\beta })\varphi _{\gamma })\varphi _{\overline{\beta }}d\Sigma _{p} \\
& -\frac{1}{2}C_{n}\int_{\Sigma }\tsum_{\beta ,\gamma \neq n}i(\theta
_{n}{}^{\gamma }(Z_{\overline{\beta }})\varphi _{\gamma }-\theta _{\overline{n}}{}^{\overline{\gamma }}(Z_{\overline{\beta }})\varphi _{\gamma })\varphi
_{\beta }d\Sigma _{p} \\
= & \frac{1}{2}C_{n}\int_{\Sigma }\varphi _{e_{2n}}\left( \Delta _{b}\varphi
-2\Delta _{b}^{t}\varphi \right) d\Sigma _{p}-\frac{1}{4}C_{n}\int_{\Sigma
}H_{p.h}\varphi _{e_{2n}}^{2}d\Sigma _{p} \\
& -\frac{1}{2}C_{n}\int_{\Sigma }\alpha \varphi _{e_{n}}\varphi
_{e_{2n}}d\Sigma _{p}+\frac{1}{2}C_{n}\int_{\Sigma }\varphi _{0}\varphi
_{e_{n}}d\Sigma _{p} \\
& -\frac{1}{4}C_{n}\int_{\Sigma }\tsum_{j,k=1}^{2n-1}\left \langle \nabla
_{e_{j}}e_{2n},e_{k}\right \rangle \varphi _{e_{j}}\varphi _{e_{k}}d\Sigma
_{p}.\end{array}
\label{temporary09}$$Here we use $\varphi _{\beta \overline{\beta }}=Z_{\overline{\beta }}Z_{\beta }\varphi -\tsum_{\gamma \neq n}\theta _{\beta }{}^{\gamma }(Z_{\overline{\beta }})\varphi _{\gamma }$ for each $\beta \neq n$ and $\varphi
_{e_{n}e_{n}}=e_{n}^{2}\varphi -(\nabla _{e_{n}}e_{n})^{t}\varphi $ on $\Sigma ,$ the fact that (\[A\]) holds only on $\Sigma \backslash S_{\Sigma
}.$ However, $d\Sigma _{p}$ can be continuously extends over the singular set $S_{\Sigma }$ and vanishes on $S_{\Sigma }.$ Finally, by combining the equations (\[temporary06\]) and (\[temporary09\]), we can then obtain (\[CR Reilly’s formula\]). This completes the proof of Theorem.
The CR First Non-Zero Dirichlet Eigenvalue Estimate
===================================================
In this section, we derive the first Dirichlet eigenvalue estimate in a compact pseudohermitian $(2n+1)$-manifold $(M,J,\theta )$ with boundary $\Sigma $.
Let $(M,J,\theta )$ be a compact pseudohermitian $(2n+1)$-manifold with the smooth boundary $\Sigma $ of pseudohermitian mean curvature $H_{p.h}$ for $n\geq 2$. For the first eigenfunction $\varphi $ of Dirichlet eigenvalue problem (\[1b\]), we have$$\begin{array}{l}
\frac{n-1}{8n}\int_{M}\varphi P_{0}\varphi d\mu =\int_{M}\tsum_{\beta
,\gamma }|\varphi _{\beta \overline{\gamma }}-\frac{1}{n}\varphi _{\sigma
}{}^{\sigma }h_{\beta \overline{\gamma }}|^{2}d\mu +\frac{1}{8}C_{n}\int_{\Sigma }H_{p.h}\varphi _{e_{2n}}^{2}d\Sigma _{p}\end{array}
\label{2aa}$$which implies $$\begin{array}{c}
\int_{M}\varphi P_{0}\varphi d\mu \geq 0\end{array}
\label{2bb}$$if $H_{p.h}$ is nonnegative.
Since $\varphi =0$ on $\Sigma $ and $e_{j}$ is tangent along $\Sigma $ for $1\leq j\leq 2n-1$, then $\varphi _{e_{j}}=0$ for $1\leq j\leq 2n-1$ and $\Delta _{b}^{t}\varphi =\frac{1}{2}\sum_{j=1}^{2n-1}[e_{j}{}^{2}\varphi
-(\nabla _{e_{j}}e_{j})^{t}\varphi ]=0$ on $\Sigma .$ Furthermore, since $\Delta _{b}\varphi =\lambda _{1}\varphi \ $on$\mathrm{\ }M$ and $\varphi =0$ on $\Sigma ,$ then $\Delta _{b}\varphi =0$ on $\Sigma $.
It follows from (\[2014a\]), (\[2014b\]) and (\[A\]) that $$\begin{array}{ll}
& iC_{n}\int_{\Sigma }(\varphi ^{\overline{\beta }}B_{n\overline{\beta }}\varphi -\varphi ^{\beta }B_{\overline{n}\beta }\varphi )d\Sigma _{p} \\
= & iC_{n}\int_{\Sigma }\sum_{\beta \neq n}(\varphi _{\beta }\varphi _{n\overline{\beta }}-\varphi _{\overline{\beta }}\varphi _{\overline{n}\beta
})d\Sigma _{p}+C_{n}\int_{\Sigma }(\varphi _{n}B_{n\overline{n}}\varphi
-\varphi _{\overline{n}}B_{\overline{n}n}\varphi )d\Sigma _{p} \\
= & iC_{n}\int_{\Sigma }\varphi _{n}[B_{n\overline{n}}\varphi -\sum_{\beta
\neq n}(\varphi _{\beta \overline{\beta }}-\frac{1}{2}\theta _{n}{}^{\beta
}(e_{n})\varphi _{\beta })]d\Sigma _{p}+\text{ }\mathrm{conjugate} \\
= & iC_{n}\int_{\Sigma }\varphi _{n}[2\varphi _{n\overline{n}}-\frac{n+1}{n}\varphi _{\gamma }{}^{\gamma }+\frac{1}{2}\sum_{\beta \neq n}\theta
_{n}{}^{\beta }(e_{n})\varphi _{\beta }]d\Sigma _{p}+\text{ }\mathrm{conjugate} \\
= & \frac{1}{2}C_{n}\int_{\Sigma }\varphi _{e_{2n}}[\varphi
_{e_{n}e_{n}}+(\nabla _{e_{n}}{}^{e_{n}})^{t}\varphi +\varphi
_{e_{2n}e_{2n}}-\frac{n+1}{n}\Delta _{b}\varphi ]d\Sigma _{p} \\
& +\frac{n-1}{2}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma _{p}+\frac{1}{2}C_{n}\int_{\Sigma }\varphi _{e_{n}}[(\nabla
_{e_{n}}{}^{e_{2n}})^{t}\varphi +\tilde{\omega}_{n}^{\;n}(e_{n})\varphi
_{e_{n}}]d\Sigma _{p} \\
= & \frac{1}{2}C_{n}\int_{\Sigma }\varphi _{e_{2n}}(\frac{n-1}{n}\Delta
_{b}\varphi -2\Delta _{b}^{t}\varphi -H_{p.h}\varphi _{e_{2n}})d\Sigma _{p}+\frac{n-1}{2}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma _{p} \\
& -\frac{1}{2}C_{n}\int_{\Sigma }\varphi _{e_{n}}\varphi
_{e_{2n}e_{n}}d\Sigma _{p}+\frac{1}{2}C_{n}\int_{\Sigma }\tilde{\omega}_{n}^{\;n}(e_{n})\varphi _{e_{n}}^{2}d\Sigma _{p}-C_{n}\int_{\Sigma }\alpha
\varphi _{e_{n}}\varphi _{e_{2n}}d\Sigma _{p},\end{array}
\label{2ab}$$where we used $B_{n\overline{\beta }}\varphi =\varphi _{n\overline{\beta }}$ for $\beta \neq n,$ $B_{n\overline{n}}\varphi =\varphi _{n\overline{n}}-\frac{1}{n}\varphi _{\gamma }{}^{\gamma }$ and $$\begin{array}{cl}
& \int_{\Sigma }[\varphi _{e_{2n}}(\varphi _{e_{n}e_{n}}+(\nabla
_{e_{n}}{}^{e_{n}})^{t}\varphi )+\varphi _{e_{n}}(\nabla
_{e_{n}}{}^{e_{2n}})^{t}\varphi ]d\Sigma _{p} \\
= & \int_{\Sigma }[\varphi _{e_{2n}}(e_{n})^{2}\varphi +\varphi
_{e_{n}}(\nabla _{e_{n}}{}^{e_{2n}})^{t}\varphi ]d\Sigma _{p} \\
= & -\int_{\Sigma }[\varphi _{e_{n}}(e_{n}e_{2n}\varphi -(\nabla
_{e_{n}}{}^{e_{2n}})^{t}\varphi )+2\alpha \varphi _{e_{n}}\varphi
_{e_{2n}}]d\Sigma _{p} \\
= & -\int_{\Sigma }(\varphi _{e_{n}}\varphi _{e_{2n}e_{n}}+2\alpha \varphi
_{e_{n}}\varphi _{e_{2n}})d\Sigma _{p}.\end{array}$$Substituting (\[2ab\]) into (\[101\]), we get$$\begin{array}{lll}
\frac{n-1}{8n}\int_{M}\varphi P_{0}\varphi d\mu & = & \int_{M}\tsum_{\beta
,\gamma }|\varphi _{\beta \overline{\gamma }}-\frac{1}{n}\varphi _{\sigma
}{}^{\sigma }h_{\beta \overline{\gamma }}|^{2}d\mu +\frac{n-1}{4n}iC_{n}\int_{\Sigma }\varphi \left( P_{n}\varphi -P_{\overline{n}}\varphi
\right) d\Sigma _{p} \\
& & -\frac{1}{4}iC_{n}\int_{\Sigma }(\varphi ^{\overline{\beta }}B_{n\overline{\beta }}\varphi -\varphi ^{\beta }B_{\overline{n}\beta }\varphi
)d\Sigma _{p} \\
& = & \int_{M}\tsum_{\beta ,\gamma }|\varphi _{\beta \overline{\gamma }}-\frac{1}{n}\varphi _{\sigma }{}^{\sigma }h_{\beta \overline{\gamma }}|^{2}d\mu +\frac{n-1}{4n}iC_{n}\int_{\Sigma }\varphi \left( P_{n}\varphi
-P_{\overline{n}}\varphi \right) d\Sigma _{p} \\
& & -\frac{1}{8}C_{n}\int_{\Sigma }\varphi _{e_{2n}}(\frac{n-1}{n}\Delta
_{b}\varphi -2\Delta _{b}^{t}\varphi -H_{p.h}\varphi _{e_{2n}})d\Sigma _{p}-\frac{n-1}{8}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma _{p} \\
& & +\frac{1}{8}C_{n}\int_{\Sigma }\varphi _{e_{n}}\varphi
_{e_{2n}e_{n}}d\Sigma _{p}-\frac{1}{8}C_{n}\int_{\Sigma }\tilde{\omega}_{n}^{\;n}(e_{n})\varphi _{e_{n}}^{2}d\Sigma _{p}+\frac{1}{4}C_{n}\int_{\Sigma }\alpha \varphi _{e_{n}}\varphi _{e_{2n}}d\Sigma _{p}\end{array}$$which is the equation (\[2aa\]) under the assumptions.
Now we are ready to prove Theorem \[TB\].
**The Proof of Theorem \[TB\]**:
It follows the CR Reilly formula (\[CR Reilly’s formula\]) that$$\begin{array}{c}
\frac{n+1}{n}\int_{M}(\Delta _{b}\varphi )^{2}d\mu \geq \frac{n+2}{4n}\int_{M}\varphi P_{0}\varphi d\mu +\int_{M}[2Ric-(n+1)Tor]((\nabla
_{b}\varphi )_{\mathbb{C}},(\nabla _{b}\varphi )_{\mathbb{C}})d\mu .\end{array}
\label{2a}$$Since $$\varphi =0\ \mathrm{and}\ \Delta _{b}\varphi =0\ \mathrm{on}\ \Sigma ,$$(\[1a\]) and (\[2a\]) imply $$\begin{array}{l}
\frac{n+1}{n}\int_{M}(\Delta _{b}\varphi )^{2}d\mu \geq \frac{n+2}{4n}\mu
_{_{D}}^{1}\int_{M}\varphi ^{2}d\mu +\int_{M}[2Ric-(n+1)Tor]((\nabla
_{b}\varphi )_{\mathbb{C}},(\nabla _{b}\varphi )_{\mathbb{C}})d\mu .\end{array}$$Moreover, by using $$\begin{array}{c}
\lbrack 2Ric-(n+1)Tor]((\nabla _{b}\varphi )_{\mathbb{C}},(\nabla
_{b}\varphi )_{\mathbb{C}})\geq k|\nabla _{b}\varphi |^{2}\end{array}$$and $$\begin{array}{c}
\int_{M}|\nabla _{b}\varphi |^{2}d\mu =\lambda _{1}\int_{M}\varphi ^{2}d\mu ,\end{array}$$we obtain$$\begin{array}{l}
\frac{n+1}{n}\lambda _{1}^{2}\int_{M}\varphi ^{2}d\mu \geq (k\lambda _{1}+\frac{n+2}{4n}\mu _{_{D}}^{1})\int_{M}\varphi ^{2}d\mu .\end{array}$$Hence $$\begin{array}{c}
\frac{n+1}{n}\lambda _{1}^{2}-k\lambda _{1}-\frac{n+2}{4n}\mu
_{_{D}}^{1}\geq 0\end{array}$$and thus$$\begin{array}{c}
\lambda _{1}\geq \frac{nk+\sqrt{n^{2}k^{2}+(n+1)(n+2)\mu _{_{D}}^{1}}}{2(n+1)}.\end{array}$$
\(i) In case for $n=1,$ we have$$\begin{array}{c}
\lambda _{1}\geq \frac{k+\sqrt{k^{2}+6\mu _{_{D}}^{1}}}{4},\end{array}$$for $\mu _{_{D}}^{1}\geq -\frac{k^{2}}{6}$. In addition if $P_{0}$ is nonnegative, we have $$\begin{array}{c}
\lambda _{1}\geq \frac{k}{2}.\end{array}$$
\(i) In case for $n\geq 2,$ it follows from (\[2bb\]) and (\[2a\]) that $$\frac{n+1}{n}\lambda _{1}^{2}-k\lambda _{1}\geq 0$$and then $$\begin{array}{c}
\lambda _{1}\geq \frac{nk}{(n+1)}.\end{array}$$
The First Eigenvalue Estimate of Embedded $P$-minimal hypersurfaces
===================================================================
In this section, we study a CR analogue of Yau conjecture [@Y] on the first eigenvalue estimate of embedded $p$-minimal hypersurfaces.
**The Proof of Theorem**
Since $M$ has vanishing torsion and positive pseudohermitian Ricci curvature, it follows from [@cc1] that $M$ has positive Ricci curvature with respect to the Webster metric. Hence its first homology group $H^{1}(M,\mathbb{R})$ is trivial. By an exact sequence argument, we conclude that $\Sigma $ divides $M$ into two connected components $M_{1}$ and $M_{2}$ with $\partial M_{1}=\Sigma =\partial M_{2}$. Let us denote $D$ to be one of two components to be chosen later. If $u$ is the first nonconstant eigenfunction on $\Sigma $, satisfying $$L_{\alpha }u=-\lambda _{1}u.$$We first let $\varphi $ be the solution of $$\Delta _{b}\varphi =0\text{\ \textrm{on}\ }D$$with the boundary condition $$\varphi =u\text{\ \textrm{on}\ }\Sigma .$$
If $D$ is a compact pseudohermitian $(2n+1)$-manifold with the smooth boundary $\Sigma ,$ then $P_{0}$ is self-adjoint on the space of all smooth functions with $\Delta _{b}\varphi =0$ and $(\Delta _{b}\varphi )_{e_{2n}}=0$ on $\Sigma $. In fact, it suffices to check that $$\begin{array}{ccl}
\int_{D}g\Delta _{b}^{2}fd\mu & = & -\int_{D}\left \langle \nabla
_{b}g,\nabla _{b}\Delta _{b}f\right \rangle d\mu +C_{n}\int_{\Sigma
}g(\Delta _{b}f)_{e_{2n}}d\Sigma _{p} \\
& = & \int_{D}\Delta _{b}f\Delta _{b}gd\mu -C_{n}\int_{\Sigma
}g_{e_{2n}}\Delta _{b}fd\Sigma _{p}+C_{n}\int_{\Sigma }g(\Delta
_{b}f)_{e_{2n}}d\Sigma _{p} \\
& = & \int_{D}\Delta _{b}f\Delta _{b}gd\mu =\int_{D}f\Delta _{b}^{2}gd\mu\end{array}
\label{A1}$$and for $\alpha =0$$$\begin{array}{ccl}
\int_{D}gf_{00}d\mu & = & -\int_{D}g_{0}f_{0}d\mu +2C_{n}\int_{\Sigma
}\alpha gf_{0}d\Sigma _{p} \\
& = & \int_{D}fg_{00}d\mu -2C_{n}\int_{\Sigma }\alpha fg_{0}d\Sigma
_{p}+2C_{n}\int_{\Sigma }\alpha gf_{0}d\Sigma _{p} \\
& = & \int_{D}fg_{00}d\mu .\end{array}
\label{A2}$$It follows that if the torsion is vanishing $$\begin{array}{c}
\int_{D}\varphi P_{0}\varphi d\mu \geq 0.\end{array}
\label{aaa}$$By applying the CR Reilly formula (\[Reilly’sformula\]), we have $$\begin{array}{lll}
0 & \geq & k\int_{D}|\nabla _{b}\varphi |^{2}d\mu -\frac{1}{4}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma _{p}-\frac{n+2}{2n}iC_{n}\int_{\Sigma }\varphi \left( P_{n}\varphi -P_{\overline{n}}\varphi
\right) d\Sigma _{p} \\
& & +\frac{i}{2}C_{n}\int_{\Sigma }(\varphi ^{\overline{\beta }}B_{n\overline{\beta }}\varphi -\varphi ^{\beta }B_{\overline{n}\beta }\varphi
)d\Sigma _{p}+\frac{3}{4n}C_{n}\int_{\Sigma }\varphi _{e_{2n}}\Delta
_{b}\varphi d\Sigma _{p} \\
& & +C_{n}\int_{\Sigma }\varphi _{e_{2n}}\Delta _{b}^{t}\varphi d\Sigma
_{p}+\frac{1}{2}C_{n}\int_{\Sigma }\alpha \varphi _{e_{n}}\varphi
_{e_{2n}}d\Sigma _{p} \\
& & +\frac{1}{4}C_{n}\int_{\Sigma }\left \langle \nabla
_{e_{i}}e_{2n},e_{j}\right \rangle \varphi _{e_{i}}\varphi _{e_{j}}d\Sigma
_{p}.\end{array}
\label{00}$$Now we are going to estimate all terms in RHS of (\[00\]):
\(i) By the CR divergence theorem and $\Delta _{b}\varphi ^{2}=2\varphi
\Delta _{b}\varphi +2|\nabla _{b}\varphi |^{2}=2|\nabla _{b}\varphi |^{2},$ we have $$\begin{array}{cl}
& C_{n}\int_{\Sigma }\varphi _{e_{2n}}\Delta _{b}^{t}\varphi d\Sigma _{p} \\
= & -C_{n}\int_{\Sigma }\alpha \varphi _{e_{n}}\varphi _{e_{2n}}d\Sigma
_{p}-\lambda _{1}C_{n}\int_{\Sigma }\varphi \varphi _{e_{2n}}d\Sigma _{p} \\
= & -C_{n}\int_{\Sigma }\alpha \varphi _{e_{n}}\varphi _{e_{2n}}d\Sigma _{p}-\frac{1}{2}\lambda _{1}C_{n}\int_{\Sigma }(\varphi ^{2})_{e_{2n}}d\Sigma _{p}
\\
= & -C_{n}\int_{\Sigma }\alpha \varphi _{e_{n}}\varphi _{e_{2n}}d\Sigma
_{p}-\lambda _{1}\int_{D}(\Delta _{b}\varphi ^{2})d\mu \\
= & -C_{n}\int_{\Sigma }\alpha \varphi _{e_{n}}\varphi _{e_{2n}}d\Sigma
_{p}-2\lambda _{1}\int_{D}|\nabla _{b}\varphi |^{2}d\mu .\end{array}
\label{01}$$
\(ii) By the CR Green theorem $$\begin{array}{c}
\frac{3}{4n}C_{n}\int_{\Sigma }\varphi _{e_{2n}}\Delta _{b}\varphi d\Sigma
_{p}=\frac{3}{4n}\int_{D}(\Delta _{b}\varphi )^{2}d\mu +\frac{3}{4n}\int_{D}\left \langle \nabla _{b}\Delta _{b}\varphi ,\nabla _{b}\varphi
\right \rangle d\mu =0.\end{array}
\label{02}$$
\(iii) The same computation as (\[2ab\]) for $\alpha =0$ and from (\[A\]) $$\begin{array}{ll}
& \frac{i}{2}C_{n}\int_{\Sigma }(\varphi ^{\overline{\beta }}B_{n\overline{\beta }}\varphi -\varphi ^{\beta }B_{\overline{n}\beta }\varphi )d\Sigma _{p}
\\
= & \frac{1}{4}C_{n}\int_{\Sigma }\varphi _{e_{2n}}(\varphi _{e_{n}e_{n}}+\frac{n-1}{n}\Delta _{b}\varphi -2\Delta _{b}^{t}\varphi -H_{p.h}\varphi
_{e_{2n}})d\Sigma _{p}+\frac{n-1}{4}C_{n}\int_{\Sigma }\varphi _{0}\varphi
_{e_{n}}d\Sigma _{p} \\
= & \frac{1}{4}C_{n}\int_{\Sigma }\varphi _{e_{2n}}(\frac{n-1}{n}\Delta
_{b}\varphi -\sum_{j\neq n,2n}\varphi _{e_{j}e_{j}})d\Sigma _{p}+\frac{n-1}{4}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma _{p}.\end{array}
\label{03}$$
\(iv) By straightforward computation, since $A_{\beta \gamma }=0$ $$\begin{array}{c}
i\left( P_{n}\varphi -P_{\overline{n}}\varphi \right) =i\left( \varphi _{\overline{\beta }}{}^{\overline{\beta }}{}_{n}-\varphi _{\beta }{}^{\beta
}{}_{\overline{n}}\right) =\frac{1}{2}[n\varphi _{0e_{n}}+(\Delta
_{b}\varphi )_{e_{2n}}].\end{array}$$From (\[2014b\]), (\[03\]) and $\int_{\Sigma }\varphi (\Delta
_{b}\varphi )_{e_{2n}}d\Sigma _{p}=0$ that $$\begin{array}{ccl}
-\frac{n+2}{2n}iC_{n}\int_{\Sigma }\varphi (P_{n}\varphi -P_{\overline{n}}\varphi )d\Sigma _{p} & = & -\frac{n+2}{4n}C_{n}\int_{\Sigma }\varphi
\lbrack n\varphi _{0e_{n}}+(\Delta _{b}\varphi )_{e_{2n}}]d\Sigma _{p} \\
& = & \frac{n+2}{4}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma
_{p}+\frac{n+2}{2}C_{n}\int_{\Sigma }\alpha \varphi _{0}\varphi d\Sigma _{p}.\end{array}
\label{04}$$By combining (\[00\]), (\[01\]), (\[02\]), (\[03\]) and (\[04\]) for $\alpha =0$$$\begin{array}{lll}
0 & \geq & (k-2\lambda _{1})\int_{D}|\nabla _{b}\varphi |^{2}d\mu +\frac{n}{2}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma _{p} \\
& & -\frac{1}{4}C_{n}\int_{\Sigma }\sum_{j\neq n,2n}\varphi
_{e_{j}e_{j}}\varphi _{e_{2n}}d\Sigma _{p}+\frac{1}{4}C_{n}\int_{\Sigma
}\left \langle \nabla _{e_{i}}e_{2n},e_{j}\right \rangle \varphi
_{e_{i}}\varphi _{e_{j}}d\Sigma _{p}.\end{array}$$Moreover if $\alpha =0$, then the $p$-area element $d\Sigma _{p}$ is the area form $d\Sigma $ on $\Sigma $ and
$$\begin{array}{lll}
0 & \geq & (k-2\lambda _{1})\int_{D}|\nabla _{b}\varphi |^{2}d\mu +\frac{n}{2}C_{n}\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma \\
& & -\frac{1}{4}C_{n}\int_{\Sigma }\sum_{j\neq n,2n}\varphi
_{e_{j}e_{j}}\varphi _{e_{2n}}d\Sigma +\frac{1}{4}C_{n}\int_{\Sigma
}\left \langle \nabla _{e_{i}}e_{2n},e_{j}\right \rangle \varphi
_{e_{i}}\varphi _{e_{j}}d\Sigma .\end{array}
\label{05}$$
Next we observe that $T$ is always tangent to $\Sigma $ due to $\alpha =0$. Then $\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma $ is independent of the extended function $\varphi $. If we choose a different component of $M\backslash \Sigma $ to perform this computation, $u_{e_{n}}u_{0},$ $\sum_{j\neq n,2n}u_{e_{j}e_{j}}u_{e_{2n}}$ and $\left \langle \nabla
_{e_{i}}e_{2n},e_{j}\right \rangle u_{e_{i}}u_{e_{j}}$ will differ by a sign, hence we may choose a component, say $M_{1}$, so that $$\begin{array}{c}
2n\int_{\Sigma }\varphi _{0}\varphi _{e_{n}}d\Sigma -\int_{\Sigma
}\sum_{j\neq n,2n}\varphi _{e_{j}e_{j}}\varphi _{e_{2n}}d\Sigma
+\int_{\Sigma }\left \langle \nabla _{e_{i}}e_{2n},e_{j}\right \rangle \varphi
_{e_{i}}\varphi _{e_{j}}d\Sigma \geq 0.\end{array}
\label{06}$$By combining (\[05\]) and (\[06\]) that we have $$\begin{array}{c}
0\geq (k-2\lambda _{1})\int_{D}|\nabla _{b}\varphi |^{2}d\mu\end{array}$$with $D=M_{1}.$ This implies $$0\geq k-2\lambda _{1}$$and thus$$\begin{array}{c}
\lambda _{1}\geq \frac{k}{2}\end{array}$$because $\varphi $ has boundary value $u$ which is nonconstant.
Now if the equality holds for $n=1.$, then $$W=k.$$Since $A_{11}=0$, $$Q_{11}=0$$and then $(M,J,\theta )$ is a closed spherical pseudohermitian $3$-manifold. On the other hand, it follows from ([@chmy]) that any embedded $p$-minimal surface in a closed spherical pseudohermitian $3$-manifold must have genus less than two. In additional, if $M$ is simply connected, then $(M,J,\theta )$ is the standard pseudohermitian $3$-sphere. This completes the proof.
[CHMY]{} S. Brendle, Embedded Minimal Tori in $\mathbf{S}^{3}$ and the Lawson Conjecture, Acta Math. 11 (2013), 177-190.
S.-C. Chang, J.-H. Cheng and H.-L. Chiu, The fourth-order $Q$-curvature flow on a CR $3$-manifold, Indiana Univ. Math. J., Vol. 56, No. 4 (2007), 1793-1826.
S.-C. Chang and H.-L. Chiu, Nonnegativity of CR Paneitz operator and its application to the CR Obata’s Theorem in a pseudohermitian $(2n+1)$-manifold, Journal of Geometric Analysis, Vol. 19 (2009), 261-287.
S.-C. Chang and H.-L. Chiu, On the Estimate of the First Eigenvalue of a Sublaplacian on a Pseudohermitian 3-Manifold. Pacific J. Math., Vol.232 (2007), 269-282.
H.-L. Chiu, The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold, Ann. Global Anal. Geom. 30 (2006), 81-96.
J.-H. Cheng, J.-F. Hwang, A. Malchiodi, and P. Yang, Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005), 129-177.
H. I. Choi and A. N. Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Diff . Geom. 18 (1983), 559-562.
Y.-W. Fan and T.-J. Kuo, On the estimate of first positive eigenvalue of a sublaplacian in a pseudihermitian manifold. AJM Vol. 18, No. 5 (2014), 859–902.
C. R. Graham and J. M. Lee, Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains, Duke Math. J., Vol. 57 (1988), 697-720.
A. Greenleaf, The first eigenvalue of a Sublaplacian on a Pseudohermitian manifold, Comm. Part. Diff. Equ. 10(2) (1985), no.3 191–217.
K. Hirachi, Scalar Pseudo-hermitian invariants and the Szegö kernel on $3$-dimensional CR manifolds, Lecture Notes in Pure and Appl. Math. 143, pp. 67-76, Dekker, 1992.
S. Montiel and A. Ros, Minimal immersions of surfaces by the first eigenfunctions and conformal area, Invent. Math. 83 (1986), 153–166
H. B. Lawson, Jr., The unknottedness of minimal embeddings, Invent. Math. 11 (1970), 183–187.
J. M. Lee, Pseudo-Einstein structure on CR manifolds, Amer. J. Math. 110 (1988), 157-178.
J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. A.M.S. 296 (1986), 411-429.
P. Li, Geometric analysis, Cambridge Studies in Advanced Mathematics, 134, Cambridge University Press, Cambridge, 2012. x+406 pp. ISBN: 978-1-107-02064-11996.
A. Lichnerowicz, Géometrie des groups de transformations, Dunod, Paris, 1958.
S.-Y. Li and H.-S. Luk, The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold, Proc. Amer. Math. Soc. 132 (2004), 789–798.
P. Li and S. T. Yau, Estimates of eigenvalues of a compact Riemannian manifold, AMS Proc. Symp. in Pure Math., Vol. 36 (1980), 205-239.
P. Li and S. T. Yau, On the parabolic kernel of the Schrődinger operator, Acta Math. 156 (1985), 153-201.
S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, preprint, 1983.
R. Reilly, Applications of the hessian operator in a Riemannian manifold, Indiana U. Math. J. 26 (1977), 459-472.
N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya Book Store Co., Ltd, Kyoto, 1975.
Z-Z.Tang and W.-J. Yan, Isoparametric foliation and Yau conjecture on the first eigenvalue, J. Diff. Geom. 94 (2013), 521–540.
S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Diff. Geom. 13 (1978), 25-41.
S.-T. Yau, Seminar on differential geometry, edited, Annals of Math. Studies 102, Princeton, New Jersey, 1982.
[^1]: $^{\ast }$Research supported in part by the MOST of Taiwan
|
---
author:
- 'Xiaofei Sun, Jiang Guo, Xiao Ding'
- |
Ting Liu\
Center for Social Computing and Information Retrieval, Harbin Institute of Technology, China\
[{xfsun, jguo, xding, tliu}@ir.hit.edu.cn]{}\
bibliography:
- 'main.bib'
title: 'A General Framework for Content-enhanced Network Representation Learning'
---
|
---
abstract: 'A VLA survey of nearby LINER galaxies at 15 GHz has revealed the presence of a compact radio core in many sources. The cores seem to be correlated with the optical activity. Follow-up VLBA observations of the ten brightest sources confirm that these cores have brightness temperatures $>10^8$ Kelvin, thus confirming their AGN nature. The two brightest radio sources show extended jet-like structures and the flat spectral indices of all cores also suggest a jet nature rather than emission from an ADAF.'
author:
- Heino Falcke
- 'Luis C. Ho'
- 'James S. Ulvestad'
- 'Andrew S. Wilson & Neil M. Nagar'
title: VLA and VLBA observations of compact radio cores in LINER galaxies
---
21.5cm
[to appear in: Proceedings of the International Symposium on Astrophysics Research And Science Education at The Vatican Observatory, June 14-21 1998, Castel Gandolfo, Italy, Ed. C. Impey]{}
Introduction
============
Quite a few nearby galaxies seem to have compact radio cores in their nuclei, the most prominent case being the Milky Way (Sgr A\*). These radio cores resemble the cores of radio-loud quasars, showing a very high brightness temperature and a flat to inverted radio spectrum that extends up to submm wavelengths. The size of Sgr A\* is only a few Schwarzschild radii of the central black hole (see Falcke 1996a for a review). Several models have been developed to explain those cores in the context of black hole accretion: spherical accretion models (Melia 1992), advection dominated accretion flows (ADAFs, Narayan et al. 1998), or scaled-down AGN jet models (Falcke, Mannheim, & Biermann 1993; Falcke & Biermann 1996). Recent observations have shown that the radio core in M81—in analogy to Sgr A\* labelled as M81\*—is very similar to Sgr A\* and also well explained by a jet model (Falcke 1996b). This suggests that similar ultra-compact sources can be detected in other galaxies as well.
We have therefore conducted a high-frequency survey to search for compact, flat-spectrum radio cores in nearby galaxies and to obtain a statistically significant sample which can help to understand the energetic phenomena in low-luminosity active galactic nuclei (LLAGN) and the nature of radio cores similar to Sgr A\* or M81\*. Here we report first results from VLA and VLBA surveys aiming to detect such cores.
VLA survey of LINERS
====================
It has been known for quite a while that early-type (E and S0) galaxies often do indeed have compact radio cores in their nuclei (Wrobel & Heeschen 1984; Slee et al. 1994) and that the probability of detecting a radio core is much higher for galaxies with nuclear optical emission-lines (O’Connell & Dressel 1978).
Recently Ho, Filippenko & Sargent (1995) have presented an extensive and sensitive spectroscopical study of a magnitude-limited and statistically well defined sample of 486 nearby elliptical and spiral galaxies (Palomar sample). One third of the galaxies surveyed show evidence for LINER-like activity and 13% turned out to be Seyferts (Ho et al. 1997). A subsample of 48 of the LINERS was recently observed with the VLA at 5 and 8 GHz in A and B configuration (van Dyk & Ho 1997). Surprisingly, almost all galaxies showed compact nuclei at a level of at least 0.5–2 mJy; but it is not clear whether this activity is due to a compact starburst or is AGN related. This sample is ideally suited to search for flat-spectrum, Sgr A\*-like radio cores at high frequencies. First results of such a VLA survey were presented in (Falcke et al. 1997). Almost half of the galaxies were indeed detected above a $\sim5\sigma$ detection level of 1 mJy at 15 GHz and at least a quarter of all LINERS had flat-spectrum cores. In contrast to the steep spectrum 5 GHz emission of these galaxies (van Dyk & Ho 1997), the flat-spectrum 15 GHz emission is well correlated with the H$\alpha$ flux, supporting the AGN interpretation for LINERs (see Figure 1). Moreover, the detected radio cores fall exactly on the H$\alpha$ vs. radio luminosity correlation predicted by the scaled down AGN jet model (Falcke & Biermann 1996; Falcke et al. 1997).
VLBA observation of brightest LINER cores
=========================================
In order to further test our hypothesis that the radio cores in LINERs are indeed related to AGN activity, we have selected the eleven brightest cores from our sample which have a flat spectrum and flux densities larger than 3mJy. The sample was observed with the VLBA at 5 GHz in phase-referencing and snapshot mode, i.e. the telescopes switched every few minutes from the program source to a nearby phase calibrator source. This enabled us to detect compact (i.e. milli-arcsecond) structure at the level of a few mJy and at spatial scales of less than 0.1 pc. Observations of one source were lost because of problems with the phase-calibrator; however, all the remaining ten sources selected from our VLA sample were indeed detected at the flux density levels expected from extrapolating the 15 GHz VLA flux densities to 5 GHz with a flat spectrum. The two brightest sources showed jet-like extended structure (Fig. 2), while the remaining eight sources were basically point-like.
This result has a number of important consequences. First of all it demonstrates how effective our selection criterion was, giving us a 100% detection rate despite a technically challenging project. It seems that basically all the flux detected with the VLA is concentrated at the milli-arcsecond scales which validates our approach to use the VLA at 15 GHz to preferentially detect compact radio cores. Second, the brightness temperature of a radio source of size 1 milli-arcsec and flux density 5 mJy at 5 GHz is T$_{b}$ $\sim$ 3 $\times$ 10$^{8}$K, and indeed all our sources have lower limits for $T_{\rm b}$ around $10^8 K$. Hence, we can probably exclude a thermal origin of the emission and argue that the radio emission is most likely due to an AGN. The fact that we get similar flux densities at 15 GHz and 5 GHz also suggests a relatively flat to inverted radio spectrum—the average 15/5 GHz spectral index of our non-simultaneous VLBA/VLA data is $\alpha=+0.1$. This is predicted in AGN jet models but is in stark contrast to the expectations for ADAF models, even though we cannot exclude an ADAF component at even higher frequencies. The idea, that these compact cores are jets is also strengthened by the extended structures seen in the two brightest sources.
Conclusion
==========
Our observations have shown that galaxies with LINER-type nuclear spectra frequently contain very compact ($<0.1$ pc) radio cores—most likely from low power radio jets similar to those seen in Seyferts or radio galaxies. The monochromatic luminosities at 5 GHz of the cores are in the range $10^{36-38}$ erg/sec and hence are comparable to M81\* but are at least four orders of magnitude more luminous than Sgr A\*. The VLBA observations have clearly demonstrated that at least some LINER galaxies are powered by an AGN-like engine and suggests a continuity of AGN activity from the most luminous quasars down to weakly active galaxies in our neighbourhood.
|
---
abstract: 'Hyperbolic gradient flow, geometry of manifold, global existence, smooth solution, shock wave.'
author:
- 'De-Xing Kong[^1] and Kefeng Liu[^2]'
title: |
\
\
**Hyperbolic Gradient Flow: Evolution of Graphs in $\mathbb{R}^{n+1}$**
---
=1
(41,10)(0,0) (0,2)[(1,0)[41]{}]{} (0,16)[(1,0)[41]{}]{}(0,12.)[*©Higher Education Press*]{} (0,7.8)[*and International Press*]{}(0,3.6)[[*Beijing–Boston*]{} ]{}
[[*The title of\
This book\*\*\*\*\**]{}\
SMM?, pp.1–?]{}
Introduction
============
Classical differential geometry has been the study of curved spaces, shapes and structures of manifolds in which the time does not play a role. However, in the last several decades geometers have made great strides in understanding the shapes and structures of manifolds that evolve in time. There are many processes in the evolution of a manifold, among them the Ricci flow is arguably the most successful (see Hamilton [@h]), since it plays a fundamental role in the solution of the famous Poincaré conjecture (see [@p1]-[@p3]). The Ricci flow is described by a fully nonlinear system of parabolic partial differential equations of second order. Another famous geometric flow — mean curvature flow is also described by a fully nonlinear system of parabolic partial differential equations of second order. The (inverse) mean curvature flow has been used to prove the Riemannian-Penrose inequality in general relativity by Huisken and Ilmanen (see [@hi]) and also has been used to study many problems arising from applied fields, i.e., imaging processing (see [@ak]). In fact, the traditional geometric analysis has been successfully applied the theory of elliptic and parabolic partial differential equations to differential geometry and physics (see [@sy4]). There are three typical examples: the Hamilton’s Ricci flow, the (inverse) mean curvature flow and the Schoen-Yau’s solution of the positive mass conjecture (see [@sy2]-[@sy3]). On the other hand, since the hyperbolic equation or system is one of the most natural models in the nature, a natural and important question is if we can apply the theory of hyperbolic differential equations to solve some problems arising from differential geometry and theoretical physics (in particular, general relativity). Recently, we introduced the hyperbolic geometric flow which is an attempt to answer the above question. The hyperbolic geometric flow is a very natural tool to understand the wave character of the metrics, wave phenomenon of the curvatures, the evolution of manifolds and their structures (see [@k], [@kl], [@klx], [@dkl0], [@dkl], [@klw], [@he], [@kw], [@kw2]).
In this paper we introduce a new geometric flow — the hyperbolic gradient flow for graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$. The flow is described by hyperbolic evolution partial differential equations of first order for a family of vector fields $X_t$ defined on $\mathbb{R}^{n}$. Roughly speaking, the hyperbolic gradient flow evolves the tangent planes of the graph under consideration, this is different from the Ricci flow, the mean curvature flow or our hyperbolic geometric flow. This kind of flow is new and very natural to understand deformation phenomena of manifolds (in particular, graphs in $\mathbb{R}^{n+1}$) as well as the geometry of manifolds. It possesses many interesting properties from both mathematics and physics. In the present paper, we particularly investigate the global existence of the evolution of convex hypersurfaces in $\mathbb{R}^{n+1}$ and the evolution of plane curves, and prove that, under the hyperbolic gradient flow, they converge to the hyperplane and the straight line, respectively, when $t$ goes to the infinity. Our results show that the theory of shock waves of hyperbolic conservation laws can be naturally applied to do surgery on manifolds. Some fundamental but open problems are also given.
Hyperbolic gradient flow for graphs in $\mathbb{R}^{n+1}$
=========================================================
Let $\Sigma_t$ be a family of graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$ with coordinates $(x_1,\cdots,
x_{n+1})$. Without loss of generality, we may assume that the graphs $\Sigma_t$ are given by $$x_{n+1}=f(t,x_1,\cdots,x_n),$$ where $f$ is a smooth function defined on $\mathbb{R}\times\mathbb{R}^{n}$. Let $X_t$ be a family of tangent vector fields induced by $\Sigma_t$, or say, $$X_t=(X_1,\cdots X_n)=(\partial_{x_1}f,\cdots, \partial_{x_n}f),$$ where $\partial_{x_i}f\;(i=1,\cdots,n)$ stand for $\frac{\partial
f}{\partial x_i}$. The hyperbolic gradient flow under considered here is given by the following evolution equations $$\label{2.1}\frac{\partial X_t}{\partial t}+\nabla\left(\frac{\|X_t\|^2}{2}\right)=0,$$ where $\nabla =(\partial_{x_1},\cdots,
\partial_{x_n})$ and $$\|\cdot\|^2=\langle\cdot,\cdot\rangle,$$ in which $\langle\cdot,\cdot\rangle$ stands for the inner product in $\mathbb{R}^{n}$.
By the definition, the hyperbolic gradient flow introduced in this note is a geometric flow for the evolution of a family of tangent vector fields induced by a family of graphs, it is quite different from the Ricci flow and the mean curvature flow: the Ricci flow is described by evolution equations for a family of Riemannian metrics $g_{ij}(t)$ defined on the manifold under consideration, while the mean curvature flow is on the evolution of the manifold itself.
The evolution of convex hypersurfaces in $\mathbb{R}^{n+1}$
===========================================================
In this section, we shall investigate the evolution of convex hypersurfaces in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$.
As before, let $\mathbb{R}^{n+1}$ be the $(n+1)$-dimensional Euclidean space with coordinates $(x_1,\cdots, x_{n+1})$, and $x_{n+1}=\mathscr{S}(t,x_1,\cdots, x_n)$ be a family of hypersurfaces in $\mathbb{R}^{n+1}$. Introduce the vector field $$\label{3.1}
\vec{v}=\{\mathscr{S}_1,\cdots,\mathscr{S}_n\},$$ where $\mathscr{S}_i\;(i=1,\cdots,n)$ stand for $\frac{\partial
\mathscr{S}}{\partial x_i}$. In the present situation, the hyperbolic gradient flow (\[2.1\]) reads $$\label{3.2}
\vec{v}_t+\vec{v}\cdot\nabla\vec{v}=0.$$ In this case, (\[3.2\]) is nothing but the transport equation for $\vec{v}$.
[**Example 3.1.**]{} [*Consider the evolution of the hypersurface $x_n=\frac12\left(x_1^2+\cdots+x_n^2 \right)$ under the hyperbolic gradient flow. In the present situation, we need to consider the Cauchy problem for the equation (\[3.2\]) with the following initial data $$\label{3.3}
t=0:\;\; \vec{v}=\vec{v}^0\triangleq (x_1,\cdots,x_n).$$ It is easy to see that the solution of the Cauchy problem (\[3.2\]), (\[3.3\]) reads $$\label{3.4}
\vec{v}=\left(\frac{x_1}{t+1},\cdots,\frac{x_n}{t+1}\right),$$ moreover, the solution is unique. Obviously, the vector field defined by (\[3.4\]) gives a potential function $x_n=\frac{1}{2(t+1)}\left(x_1^2+\cdots+x_n^2\right)+C$, where $C$ is a constant independent of $x$. Noting that the initial hypersurface is $x_n=\frac12\left(x_1^2+\cdots+x_n^2 \right)$ leads to that the constant $C$ must be zero. Thus, the evolution of the hypersurface $x_n=\frac12\left(x_1^2+\cdots+x_n^2 \right)$ under the hyperbolic gradient flow is described by the family of hypersurfaces $x_n=\frac{1}{2(t+1)}\left(x_1^2+\cdots+x_n^2\right)$. Clearly, for any fixed $x$, the hypersurfaces tend to flat under the hyperbolic gradient flow when $t$ goes to the infinity.*]{} $\qquad\qquad\Box$
Consider the Cauchy problem for the equation (\[3.2\]) with the following initial data $$\label{3.5}
t=0:\;\; \vec{v}=\vec{v}^0(x_1,\cdots,x_n),$$ where $\vec{v}^0$ is a smooth vector field defined on $\mathbb{R}^{n}$. We now consider the global existence and decay property of smooth solutions of the the Cauchy problem (\[3.2\]) and (\[3.5\]).
In fact, we can obtain a sufficient and necessary condition on the global existence of smooth solutions of the following Cauchy problem for more general quasilinear systems of first order $$\label{1.3.1}
\frac{\partial u}{\partial t}+\sum^n_j \lambda_j(u)\frac{\partial
u}{\partial x_j}=0,\quad \forall\; (t, x)\in \mathbb{R}^+\times
\mathbb{R}^n$$ with the initial data $$\label{1.3.2}
u(0,x)=\phi(x), \quad \forall\; x\in \mathbb{R}^n,$$ where $x=(x_1,\cdots ,x_n)$ stands for the special variable, $u=(u_1(x, t),\cdots u_m(x,t))^T$ is the unknown vector-valued function of $(t, x)=(t, x_1,\cdots ,x_n)\in \mathbb{R}^+\times
\mathbb{R}$, $\lambda_i(u)\; (i=1,\cdots , n)$ are given $C^1$ functions and $\phi(x)=(\phi_1(x),\cdots , \phi_m(x))^T$ is a given $C^1$ vector-valued function with bounded $C^1$ norm. The following lemma comes from Conway [@conway], Li [@li], Dafermos [@dafermos] or Kong [@kong].
Under the assumptions mentioned above, the Cauchy problem (\[1.3.1\])-(\[1.3.2\]) has a unique global $C^1$ smooth solution on the domain $\mathbb{R}^+\times \mathbb{R}^n$ if only if, for any given $x\in \mathbb{R}^n$, it holds that $$\label{1.3.3}
d(S_p V_0(x), \mathbb{R}^- )\geq0,$$ i.e., all eigenvalues of the $n\times n$ matrix $$\label{1.3.4}
V_0(x)=\left(\sum^m_{k=1}\frac{\partial \lambda_i}{\partial
u_k}(\phi(x))\frac{\partial \phi_k}{\partial x_j}\right)^n_{i,j=1}$$ are non-negative, where $S_p V_0(x)$ stands for the spectrum of the matrix $V_0(x)$.
Under the assumptions of Lemma 3.2, suppose that $\phi$ is a $C^2$ vector-valued function with bounded $C^2$ norm and suppose furthermore that there exists a positive constant $\delta > 0$ such that $$\label{1.3.5}
d(S_p V_0(x), \mathbb{R}^- )\geq \delta, \quad \forall\; x\in
\mathbb{R}^n.$$ Then the Cauchy problem (\[1.3.1\])-(\[1.3.2\]) admits a unique global $C^2$ smooth solution $u=u(t,x)$ on the domain $\mathbb{R}^+\times \mathbb{R}^n$, moreover it holds that $$\label{1.3.6}
\parallel Du(t,x)\parallel_{L^{\infty}(\mathbb{R}^n)}=C_1(1+t)^{-1}$$ and $$\label{1.3.7}
\parallel D^2u(t,x)\parallel_{L^{\infty}(\mathbb{R}^n)}\leq
C_2(1+t)^{-2},$$ where $C_1$ is a positive constant independent of $t$ but depending on $\delta$ and the $C^1$ norm of $\phi$, while $C_2$ is a positive constant independent of $t$ but depending on $\delta$ and the $C^2$ norm of $\phi$.
The proof of Lemma 3.3 can be found in Grassin [@g] for the case of scalar equation and in Kong [@kong] for general case.
If $m=n$ and $\lambda_i(u)=u_i$ $(i=1,\cdots,n)$ (in this case, the system (\[1.3.1\]) goes back to the system (\[3.2\])), then in the present situation, $V_0(x)$ defined by (\[1.3.4\]) reads $$\label{1.3.38}
V_0(x)=\left(\frac{\partial \phi_i}{\partial x_j}\right)^n_{i,j=1}.$$ In particular, if there exists a potential function $\Phi(x)$ such that $$\label{1.3.39}
\frac{\partial \Phi}{\partial x_i}=\phi_i(x) \quad (i=1,\cdots, n),$$ then $$\label{1.3.40}
V_0(x)={\rm Hess}\,(\Phi(x)).$$
We now turn to consider the Cauchy problem for this special case, i.e., $$\label{1.3.41} \left\{\begin{array}{l}{\displaystyle
\frac{\partial u}{\partial t}+\sum^n_{j=1}u_j\frac{\partial
u}{\partial x_j}=0,\quad
\forall\; (t,x)\in \mathbb{R}^+\times \mathbb{R}^n,}\vspace{2mm}\\
{\displaystyle t=0: u=\phi(x)=\left(\frac{\partial \Phi}{\partial
x_1},\cdots , \frac{\partial \Phi}{\partial x_n}\right)^T, \quad
\forall\; x\in \mathbb{R}^n.}
\end{array}\right.$$ By Lemma 3.1-3.2, we have
Suppose that the potential function $\Phi=\Phi(x)$ is $C^2$ smooth and its derivates $\Phi_{k_i}$ $(i=1,..., n)$ has a bounded $C^1$ norm. Then the Cauchy problem (\[1.3.41\]) has a unique global $C^1$ smooth solution on the domain $\mathbb{R}^+\times \mathbb{R}^n$ if only if ${\rm Hess}\,(\Phi(x))$ is non-negative for all $x\in \mathbb{R}^n$.
Moreover, if the following assumptions are satisfied: (i) $\Phi$ is a $C^3$ smooth function, (ii) the derivative $D\Phi=(\Phi_{x_1},\cdots, \Phi_{x_n})^T$ is a vector-valued function with bounded $C^2$ norm, (iii) there exists a positive constant $\delta$ independent of $x$ such that $$\label{1.3.42}
d({\rm Hess}\,(\Phi(x), \mathbb{R}^-)\geq \delta, \quad \forall\;
x\in \mathbb{R}^n,$$ then the global smooth solution $u=u(t,x)$ to the Cauchy problem (\[1.3.41\]) satisfies the following properties:
\(I) there exists a $C^3$ potential function $U=U(t,x)$ such that $$\label{1.3.43}
u_i(t,x)=\frac{\partial U}{\partial x_i}(t,x) \quad (i=1, ..., n),
\quad \forall\; (t,x)\in \mathbb{R}^+\times \mathbb{R}^n,$$
\(II) there exist two positive constants $C_3$ and $C_4$ independent of $t$ but depending on $\delta$ and the $C^1$ norm (for $C_3$), the $C^2$ norm (for $C_4$) of $D\Phi(x)$, respectively, such that $$\label{1.3.44}
\| D^2 U(t, \cdot)\|_{L^{\infty}(\mathbb{R}^n)}\leq C_3(1+t)^{-1},$$ and $$\label{1.3.45}
\| D^3 U(t, \cdot)\|_{L^{\infty}(\mathbb{R}^n)}\leq C_4(1+t)^{-2},$$ where $D=(\partial_{x_1}, ..., \partial_{x_n})$.
[**Proof.**]{} By Lemmas 3.1-3.2, we only need to prove (I) in Lemma 3.3.
In order to prove (I), it suffices to show $$\label{1.3.46}
\frac{\partial u_i(t,x)}{\partial x_j}=\frac{\partial
u_j(t,x)}{\partial x_i}, \quad \forall\; i\neq j, \quad \forall\;
(t,x) \in \mathbb{R}^+\times \mathbb{R}^n.$$
In fact, introduce $$\label{1.3.47}
\omega^i_j=\frac{\partial u_i}{\partial x_j}-\frac{\partial
u_j}{\partial x_i} \quad (i,j=1,\cdots,n;\quad i \neq j) .$$ Obviously, when $t=0$, $$\label{1.3.48}
\omega^i_j(t,0)=\frac{\partial}{\partial
x_j}\left(\frac{\partial}{\partial
x_i}\Phi\right)-\frac{\partial}{\partial
x_i}\left(\frac{\partial}{\partial x_j}\Phi\right)=0 \quad
(i,j=1,\cdots,n).$$ On the one hand, differentiating the $i$-th equation in (\[1.3.41\]) with respect to $x_j$ gives $$\label{1.3.49}
\frac{\partial}{\partial t}\left(\frac{\partial u_i}{\partial
x_j}\right)+ \sum^n_{k=1} u_k \frac{\partial}{\partial
x_k}\left(\frac{\partial u_i}{\partial x_j}\right)= -\sum^n_{k=1}
\frac{\partial u_i}{\partial x_k}\frac{\partial u_k}{\partial x_j}.$$ On the other hand, differentiating the $j$-th equation in (\[1.3.41\]) with respective to $x_i$ yields $$\label{1.3.50}
\frac{\partial}{\partial t}\left(\frac{\partial u_j}{\partial
x_i}\right)+ \sum^n_{k=1} u_k \frac{\partial}{\partial
x_k}\left(\frac{\partial u_j}{\partial x_i}\right)=
-\sum^n_{k=1}\frac{\partial u_j}{\partial x_k}\frac{\partial
u_k}{\partial x_i}.$$ Combing (\[1.3.49\])-(\[1.3.50\]) leads to $$\label{1.3.51}
\frac{\partial \omega^i_j}{\partial t}+ \sum^n_{k=1}
u_k\frac{\partial \omega^i_j}{\partial x_k}=\sum_{p \neq q}
\Gamma^{ij}_{pq} \omega ^q_p, \quad \forall\; i \neq j,$$ where $\Gamma^{ij}_{pq}$ stands for the coefficients of $\omega
^q_p$ which are smooth functions of $\frac{\partial u_l}{\partial
x_h}~(l,h=1 ,\cdots, n)$. Clearly, $\omega^i_j=0 ~(i,j=1,\cdots,n;
~i\neq j)$ is a solution of the Cauchy problem (\[1.3.51\]), (\[1.3.48\]). By the uniqueness of the smooth solution of the Cauchy problem for hyperbolic partial differential equation, we have $$\label{1.3.52}
\omega ^i_j \equiv 0 \quad (i\neq j),\quad \forall\; (t,x) \in
\mathbb{R}^+\times \mathbb{R}^n.$$ This proves (\[1.3.46\]). Thus the proof of Lemma 3.3 is completed.$\qquad\qquad \blacksquare$
In Lemma 3.2, we need the $C^1$ norm of $\phi$ and the $C^2$ norm of $\phi$ is bounded for the estimates (\[1.3.6\]) and (\[1.3.7\]), respectively. For Lemma 3.3, the situation is similar.
However, in many cases (i.g., Example 3.1), the assumption that the $C^1$ norm or $C^2$ norm of the initial data is bounded is not satisfied. The following discussion is devoted to the case of unbounded initial data. For simplicity, we only consider the Cauchy problem (\[1.3.41\]).
Suppose that $\Phi=\Phi(x)$ is a $C^3$ convex function, i.e., $\Phi(x) \in C^3
(\mathbb{R}^n)$ and $$\label{1.3.53}
{\rm Hess}\,(\Phi)\geq 0.$$ Then the Cauchy problem (\[1.3.41\]) admits a unique $C^2$ solution $u=u(t,x)$ on the domain $\mathbb{R}^+\times \mathbb{R}^n$. Moreover, there exists a potential function $U=U(t,x)\in C^3(
\mathbb{R}^+\times \mathbb{R}^n)$ such that (\[1.3.43\]) is satisfied. In particular, if there exists a positive constant $\delta$ independent of $x$ such that (\[1.3.42\]) holds, then for any fixed $\alpha \in \mathbb{R}^n$ along the characteristic curve $x=x(t,\alpha)$ it holds that $$\label{1.3.54}
|D^2U(t, x(t, \alpha))|\leq \tilde{C_1}(1+t)^{-1}$$ and $$\label{1.3.55}
|D^3U(t, x(t, \alpha))|\leq \tilde{C_2}(1+t)^{-2},$$ where $\tilde{C_1}$ and $\tilde{C_2}$ are tow constants independent of $t$ but depending on $\delta$ and $\alpha$.
The following corollary comes from Lemma 3.4 directly.
Under the assumptions of Lemma 3.4, for any compact set $\Omega \subseteq \mathbb{R}^n$ it holds that $$\label{1.3.56}
\|D^2U(t, \cdot)\|_{L^{\infty}(\Omega(t))}\leq
\tilde{C_3}(1+t)^{-1},$$ and $$\label{1.3.57}
\|D^3U(t, \cdot)\|_{L^{\infty}(\Omega(t))}\leq
\tilde{C_4}(1+t)^{-2},$$ where $$\label{1.3.58}
\Omega(t)=\{(t,x)| x=x(t, \alpha),\quad \alpha \in \Omega \},$$ $\tilde{C_3}$ and $\tilde{C_4}$ are two constants independent of $t$ but depending on $\delta$ and the set $\Omega$. $\qquad \Box$
[**Proof of Lemma 3.4.**]{} Noting (\[1.3.53\]), we have $$\label{1.3.59}
\Phi_{x_ix_i}(x)\geq 0 \quad (i=1,\cdots, n), \quad \forall\; x \in
\mathbb{R}^n.$$ In the present situation, the characteristic curve pasting through any fixed point $(0,\alpha)$ in the initial hyperplane $t=0$ reads $$\label{1.3.60}
x_i=\alpha_i + \frac{\partial \Phi}{\partial \alpha_i}(\alpha)t
\quad (i=1,\cdots, n).$$ By (\[1.3.59\]), it is easy to check that the mapping $\Pi_t:
\mathbb{R}^n \rightarrow \mathbb{R}^n $ defined by (\[1.3.60\]) is [*proper*]{}. On the other hand, $$\label{1.3.61}
J(\Pi_t)=I+t{\rm Hess}\,(\Phi).$$ Using (\[1.3.53\]) again, we have $$\label{1.3.62}
\det J(\Pi_t) \geq 1, \quad \forall\; (t,\alpha) \in
\mathbb{R}^+\times \mathbb{R}^n.$$ This implies that for any fixed $x \in \mathbb{R}^+$, the mapping $\Pi_t$ is a global $C^1$ deffeomorphism. Solving $\alpha$ from (\[1.3.60\]) gives $$\label{1.3.63}
\alpha=\alpha(t,x) \in C^2( \mathbb{R}^+\times \mathbb{R}^n).$$ The rest of the proof is standard (See [@li] or [@kong]), here we omit the details. The proof of Lemma 3.4 is completed.$\qquad\qquad \blacksquare$
Lemma 3.3 guarantees that, if the initial vector field is induced by a graph, then so does the solution vector-field. That is, if there exists a fuction $\varphi_0(x_1,\cdots,x_n)$ such that $v^0_i=\frac{\partial\varphi_0}{
\partial x_i}\;(i=1,\cdots,n)$, then there is a family of functions $\varphi(t,x_1,\cdots,x_n)$ such that $$\label{3.16}
v_i(t,x_1\cdots,x_n)=\frac{\partial\varphi}{\partial
x_i}(t,x_1\cdots,x_n)\quad (i=1,\cdots,n)$$ and $$\label{3.17}
\varphi(0,x_1,\cdots,x_n)=\varphi_0(x_1,\cdots,x_n).$$ From the point of view of geometry, the hyperbolic gradient flow evolves a graph as a family of graphs in the Euclidean space $\mathbb{R}^{n+1}$.
Summarizing the above argument leads to
For any given initial vector field induced by a convex graph $x_{n+1}=\varphi_0(x_1,\cdots,x_n)$, the solution $\vec{v}=\vec{v}(t,x_1,\cdots,x_n)$ to the hyperbolic gradient flow (\[3.2\]) exists for all time, and there exists a unique family of graphs $x_{n+1}=\varphi(t,x_1,\cdots,x_n)$ such that the solution vector-field $\vec{v}=\vec{v}(t,x_1,\cdots,x_n)$ is induced by the family of graphs $x_{n+1}=\varphi(t,x_1,\cdots,x_n)$. Moreover, if the initial graph is strictly convex, then for any fixed point $(x_1,\cdots,x_n)\in \mathbb{R}^{n}$ the graphs $x_{n+1}=\varphi(t,x_1,\cdots,x_n)$ tends to be flat at an algebraic rate $(t+1)^{-1}$, when $t$ goes to the infinity.
The evolution of plane curves
=============================
In this section, we particularly investigate the evolution of plane curves under the hyperbolic gradient flow, here we still consider the graph case, however we do not assume that the graph is convex.
Let $y=f(x)$ be a smooth curve in the $(x,y)$-plane, and $$\label{4.1}
v_0(x)=f^{\prime}(x)$$ be the slope function of the curve. In the present situation, the hyperbolic gradient flow equation (\[3.2\]) becomes one-dimensional case, i.e., $$\label{4.2}
v_t+vv_x=0.$$ This equation can be rewritten as a conservative form $$\label{4.3}
v_t+(v^2/2)_x=0.$$
We next consider the Cauchy problem for the conservation law (\[4.3\]) with the initial data $$\label{4.4}
t=0:\;\; v=v_0(x).$$
As in Lax [@lax], we introduce
A function $\psi$ has mean value $M$, if $$\label{4.5}
\lim_{L\rightarrow\infty}\frac1L\int_{a}^{a+L}\psi(x)dx=M$$ uniformly in $a$.
If a function $\psi$ is periodic with $p$ period, then it has mean value $M$, and $M$ is given by $$\label{4.5-1}
M=\frac1p\int_{0}^{p}\psi(x)dx.$$ If $\psi\in
L^1(\mathbb{R})$, then it has mean value 0.
The following lemma comes from Lax [@lax].
Let $v(t,x)$ be a bounded weak solution of the Cauchy problem (\[4.3\]), (\[4.4\]). Suppose that the initial data $v_0(x)$ has a mean value, then $v(t,x)$ has the same mean value for all $t$.
The following important lemma comes from Kruzkov [@kr].
Suppose that the initial data $v_0$ is bounded measurable, then the Cauchy problem (\[4.3\]), (\[4.4\]) has a unique entropy solution $v=v(t,x)$ on the half plane $t\ge 0$.
Under the assumption of Lemma 4.2, if the initial data $v_0$ is periodic with $p$ period, then the entropy solution $v=v(t,x)$ of the Cauchy problem (\[4.3\]), (\[4.4\]) tends to $M$ uniformly in $x$ at an algebraic rate $(t+1)^{-1}$, when $t$ tends to infinity, where $M$ is given by $$M=\frac1p\int_{0}^{p}v_0(x)dx.$$
Suppose that the initial data $v_0$ is in the class of $L^{1}(\mathbb{R})$, then the Cauchy problem (\[4.3\]), (\[4.4\]) has a unique entropy solution $v=v(t,x)$ on the half plane $t\ge 0$. Moreover, $v(t,x)$ tends to $0$ uniformly in $x$ at an algebraic rate $(t+1)^{-1}$, when $t$ tends to infinity.
Lemmas 4.3 and 4.4 can be found in Serre [@serre] and Bressan [@bressan], respectively.
The entropy solution $v=v(t,x)$ mentioned in Lemmas 4.2, 4.3 and 4.4 means that (i) $v=v(t,x)$ is a weak solution of the Cauchy problem (\[4.3\]), (\[4.4\]); (ii) it satisfies the entropy condition. In fact, the entropy solution may includes shock waves, rarefaction waves, and other physical discontinuities.
We now consider the evolution of the initial curve $y=f(x)$ under the hyperbolic gradient flow.
Without loss of generality, we may assume that $$\label{4.6}
f(0)=0.$$ By Lemmas 4.2-4.4, we have
Suppose that $f^{\prime}(x)$ is bounded measurable, and suppose furthermore that $f^{\prime}(x)$ is periodic or is in the class of $L^{1}(\mathbb{R})$, then the family of curves $y=F(t,x)$ tends to the straight line $y=Mx$ uniformly in $x$ at an algebraic rate $(t+1)^{-1}$, when $t$ tends to infinity, where $M$ is the mean value of $f^{\prime}(x)$, and $y=F(t,x)$ is generated by the hyperbolic gradient flow, i.e., $F(t,x)$ satisfies $$\label{4.6}
\frac{\partial F}{\partial x}(t,x)=v(t,x),$$ in which $v=v(t,x)$ is the entropy solution of the Cauchy problem (\[4.3\]), (\[4.4\]) (in the present situation, $v_0(x)=f^{\prime}(x)$).
In geometry, one is, in general, interested in the case that the initial data $f(x)$, or say $v_0$, is smooth and bounded. However, in the evolution process under the hyperbolic gradient flow, discontinuities may appear. See Example 4.1 below for the details.
[**Example 4.1.**]{}
*Consider the evolution of the curve $y=-\cos x$ in the $(x,y)$-plane under the hyperbolic gradient flow. In the present situation, the initial data reads $$\label{4.7}
t=0:\;\; v=v_0(x)=\sin x.$$ By the method of characteristics, the solution of the Cauchy problem (\[4.3\]), (\[4.7\]) can be constructed and is given by Figure 4.1.*
\[0.55\] -2.0cm
Notice that Figure 4.1 only describes the solution on one space-periodic domain, i.e., $ \mathbb{R}^+\times [0,2\pi]$. Corresponding to the solution shown in Figure 4.1, the evolution of the curve $y=-\cos x$ under the hyperbolic gradient flow can be described by Figure 4.2.
\[0.55\] -2.0cm
We observe from Figure 4.2 that the singularity have appeared in the evolutionary process (see Figure 4.3 for the details).
\[0.55\] -2.0cm
Figure 4.3 shows that the singularity of cusp type of $v=v(t,x)$ appears at $x=\pi$ when $t=1$. It is easy to see that the entropy solution $v=v(t,x)$ to the Cauchy problem (\[4.3\]), (\[4.7\]) includes space-periodic shock waves.
$\qquad\qquad\Box$
Conclusions and open problems
=============================
It is well known that there have been many successes of elliptic and parabolic equations applied to mathematics and physics. On the other hand, hyperbolic partial differential equation is a very important kind of PDEs, it can be used to describe the wave phenomena in the nature and engineering. Recently, we introduced the hyperbolic geometric flow, showed that the hyperbolic geometric flow possesses very interesting geometric properties and dynamical behavior, and obtain some interesting results. However, the hyperbolic geometric flow is described by a fully nonlinear system of hyperbolic partial differential equations of second order, which is very difficult to solve. In this paper we introduce a new geometric flow — the hyperbolic gradient flow, which is described by a quasilinear system of hyperbolic partial differential equations of first order. Comparing the hyperbolic geometric flow, the hyperbolic gradient flow is easier to solve. The key point of the hyperbolic gradient flow is to evolve the tangent planes of the graphs under consideration, this is different with the famous Ricci flow, the mean curvature flow or our hyperbolic geometric flow. In this paper, we investigate the evolution of convex hypersurfaces in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$ and the evolution of plane curves, and prove that, under the hyperbolic gradient flow, they converge to the hyperplane and straight line, respectively, when $t$ goes to the infinity. Our results obtained in this paper show that the theory of shock waves of hyperbolic conservation laws can be naturally applied to differential geometry. We believe that the hyperbolic gradient flow is a new and powerful tool to study some problems arising from geometry and physics. However, there are many fundamental but still open problems. In particular, the following open problems seem to us more interesting and important:
[**1. The evolution of plane curves.**]{} [*In Theorem 4.1 if we do not assume that $f^{\prime}(x)$ has a mean value, what is the limit of the family of curves $F(t,x)$ as $t$ goes to infinity? Moreover, what happens if the initial curve is not a graph, e.g., a closed curve?*]{}
[**2. The evolution of surfaces in $\mathbb{R}^3$.**]{} [*In Theorem 3.1 if the initial surface is a graph but is not convex, what about the limit of the family of surfaces $\varphi(t,x_1,x_2)$ as $t$ goes to infinity? A more difficult but more natural and important question is: how to define the hyperbolic gradient flow for a family of close surfaces? If so, what is the asymptotic behaviour of a close surface under “the hyperbolic gradient flow"? This problem is related to the theory of multi-dimensional hyperbolic systems of partial differential equations of first order.* ]{}
[**3. The evolution of hypersurfaces in $\mathbb{R}^n\;(n\ge 4)$.**]{} [*Investigate the hyperbolic gradient flow in multi-dimensional Euclidean space $\mathbb{R}^{n+1}\;(n\ge 4)$. In particular, how can we define a suitable “ hyperbolic gradient flow" to evolve a closed sub-maifold? if we can, what is the large time behaviour of a close hypersurface in $\mathbb{R}^n\;(n\ge 4)$ under this kind of hyperbolic gradient flow. The convex case maybe is easier to study.*]{}
We may also consider variations of the above hyperbolic gradient flow which can be defined intrinsically on any manifold. For example we let $(\mathscr{M}, g)$ be a Riemannian manifold, and $X_t\in
\Gamma(\mathscr{M}, TM)$ be a family of tangent vector fileds, the hyperbolic gradient flow under considered here is given by the following evolution equation $$\label{5.1}\frac{\partial X_t}{\partial t}+\frac{1}{2}\nabla (\|X_t\|^2)=0,$$ where, if in local coordinates ${\displaystyle X_t =\sum_{i=1}^n
X^i_t\frac{\partial}{\partial x_i}}$, then $\|X_t\|^2$ is defined by $$\label{5.2}\|X_t\|^2=g_{ij}X_t^iX_t^j$$ and $\nabla h$ stands for the gradient vector field of a function $h$ on the manifold, and $g_{ij}= g(\frac{\partial}{\partial
x_i},\frac{\partial}{\partial x_j})$. By definition, for any given $h\in C^{\infty}(\mathscr{M}, \mathbb{R})$ and $X\in TM$, we have $$\label{5.3}
g(X, \nabla h)=X(h).$$ The study of this flow will be very useful to understand the topological and geometrical structure of the manifold.
Finally, we would like to point out that, perhaps the method in the present paper is more important than the results obtained here. Our method may provide a new approach to some conjectures in differential geometry (see Yau [@sy4]).
This work was completed while Kong was visiting the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) during the summer of 2010. Kong thanks L. Andersson for his invitation and hospitality. This work was supported in part by the NNSF of China (Grant No. 10971190) and the Qiu-Shi Chair Professor Fellowship from Zhejiang University, China.
[99]{}
A. Bressan, Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem, Oxford University Press, 2000.
G. Aubert & P. Kornprobst, Mathematical Problems in Image Processing, Springer, 2006.
E. Conway, The formation and decay of shocks for a conservation law in several dimensions, [*Arch. Rat. Mech. Anal.*]{} 64 (1977), 47-57.
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin Heidelberg, 2005.
W.-R. Dai, D.-X. Kong & K.-F. Liu, Hyperbolic geometric flow (I): short-time existence and nonlinear stability, [*Pure and Applied Mathematics Quarterly (Special Issue: In honor of Michael Atiyah and Isadore Singer)*]{} 6 (2010), 331-359.
W.-R. Dai, D.-X. Kong & K.-F. Liu, Dissipative hyperbolic geometric flow, [*Asian J. Math.*]{} 12 (2008), 345-364.
M. Grassin, Global smooth solutions to Euler equations for a perfect gas, [*Indiana Univ. Math. J.*]{} 47 (1998), 1397-1432.
R. Hamilton, Three-manifolds with positive Ricci curvature, [*J. Differential Geom.*]{} 17 (1982), 255-306.
C.-L. He, D.-X. Kong and K.-F. Liu, Hyperbolic mean curvature flow, [*Journal of Differential Equations*]{} 246 (2009), 373-390.
G. Huisken & T. Ilmanen, The inverse mean curvature flow and the Riemannian-Penrose inequality, [*J. Differential Geom.*]{} 59 (2001), 353-437.
D.-X. Kong, Lectures on Quasilinear Hyperbolic Systems and Applications, Zhejiang University, Hangzhou, China, 2009.
D.-X. Kong, Hyperbolic geometric flow, [*the Proceedings of ICCM 2007*]{}, Vol. II, Higher Educationial Press, Beijing, 2007, 95-110.
D.-X. Kong & K.-F. Liu, Wave character of metrics and hyperbolic geometric flow, [*J. Math. Phys.*]{} 48 (2007), 103508.
D.-X. Kong, K.-F. Liu & Y.-Z. Wang, Life-span of classical solutions to hyperbolic geometric flow in two space variables with slow decay initial data, to appear in [*Communications in Partial Differential Equations*]{}.
D.-X. Kong, K.-F. Liu and Z.-G. Wang, Hyperbolic mean curvature flow: Evolution of plane curves, [*Acta Mathematica Scientia (A special issue dedicated to Professor Wu Wenjun’s 90th birthday)*]{} 29 (2009), 493-514.
D.-X. Kong, K.-F. Liu & D.-L. Xu, The hyperbolic geometric flow on Riemann surfaces, [*Communications in Partial Differential Equations*]{} 34 (2009), 553-580.
D.-X. Kong and Z.-G. Wang, Formation of singularities in the motion of plane curves under hyperbolic mean curvature flow, [*Journal of Differential Equations*]{} 247 (2009), 1694-1719.
S. Kruzkov, First-order quasilinear equations with several space variables, [*Mathematics of the USSR-Sbornik*]{} 10 (1970), 217-273.
P. D. Lax, Hyperbolic systems of conservation laws II, [*Commun. Pure Appl. Math.*]{} 10 (1957), 537-556.
T.-T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems. RAM: Research in Applied Mathematics, 32. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, [*arXiv.org*]{}, November 11, 2002.
G. Perelman, Ricci flow with surgery on three-manifolds, [*arXiv.org*]{}, March 10, 2003.
G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, [*arXiv.org*]{}, July 17, 2003.
R. Schoen & S.-T. Yau, On the proof of the positive mass conjecture in general relativity, [*Comm. Math. Phys.*]{} 65 (1979), 45-76.
R. Schoen & S.-T. Yau, Proof of the positive mass theorem II, [*Comm. Math. Phys.*]{} 79 (1981), 231-260.
R. Schoen & S.-T. Yau, [*Lectures on Differential Geometry*]{}, International Press, Cambridge, MA, 1994.
D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Canmbridge University Press, Cambridge, 1999.
[^1]: Department of Mathematics, Zhejiang University, Hangzhou 310027, China
[^2]: Department of Mathematics, University of California at Los Angeles, CA 90095, USA
|
---
abstract: 'Recent [*Hipparcos*]{} results have lent support to the idea that RR Lyrae variables in the halo field and in globular clusters differ in luminosity by $\approx 0.2$ mag. In this [*Letter*]{}, we study the pulsation properties of RR Lyraes in clusters with distances determined [*via*]{} main-sequence fitting to [*Hipparcos*]{} parallaxes for field subdwarfs, and compare them with the properties of field variables also analyzed with [*Hipparcos*]{}. We show that the period–temperature distributions for field and cluster variables are essentially indistinguishable, thus suggesting that there is no significant difference in luminosity between them.'
author:
- 'M. Catelan'
title: |
Is There a Difference in Luminosity between Field and Cluster\
RR Lyrae Variables?
---
Introduction
============
Accurate knowledge of the Population II distance scale is one of the most important goals in astronomy. Upon it depends, for instance, the determination of the ages of globular clusters (GC’s), and thus of a firm lower limit to the age of the Universe.
RR Lyrae variables are the natural Pop. II “standard candle." Several methods have been devised to estimate their luminosities, but a consensus has not yet been reached. In particular, there appears to be a “dichotomy" between “faint" (i.e., short distance scale and old ages for the GC’s) and “bright" (long distance scale and younger GC ages) calibrations. Walker (1992) and Catelan (1996) provide useful references covering the literature as of 1995. But the noted “dichotomy" has become even more clear-cut recently. Ground-based investigations have continued to appear supporting either the “short" or the “long" scale. Examples of the former include the Baade-Wesselink (e.g., Clementini et al. 1995) and statistical parallaxes (Layden et al. 1996; Popowski & Gould 1998) analyses of field RR Lyraes. An example of the latter has been provided by the extensive analysis of the variables in M15 by Silbermann & Smith (1995). The “persistent" nature of such a “dichotomy" has led some authors (e.g., VandenBerg, Bolte, & Stetson 1996; Sweigart 1998) to speculate that [*there might exist a real difference in luminosity between field and cluster RR Lyrae variables*]{}. That was based in part on the (somewhat uncertain) Baade-Wesselink results of Storm, Carney, & Latham (1994) for a few RR Lyrae variables in the GC’s M5 and M92 and field counterparts of comparable metallicity.
These speculations notwithstanding, there has been widespread belief that, once the [*Hipparcos*]{} satellite parallax results became available, we would finally be able to decide between the “short" and “long" RR Lyrae distance scales. However, that turned out [*not*]{} to be the case. Based upon [*Hipparcos*]{} parallaxes of field subdwarfs and main-sequence fitting to GC’s with well-defined deep color-magnitude diagrams, Gratton et al. (1997) and Reid (1997, 1998) have strongly claimed that the majority of the GC’s in their samples are substantially farther away than previously estimated using ground-based parallaxes (but see Pont et al. 1998). Similarly, McNamara (1997b) has concluded that the [*Hipparcos*]{} parallaxes of field SX Phoenicis variables favor the “long" GC distance scale. These claims were supported by [*Hipparcos*]{} data for Cepheids (Feast & Catchpole 1997; Madore & Freedman 1997; see also the latest ground-based results by Laney 1998) and Miras (van Leeuwen et al. 1997), applied to the Large Magellanic Cloud (LMC). The “long" distance to the LMC is supported by the latest analysis of the SN1987a ring (Panagia, Gilmozzi, & Kirshner 1998; but see Gould & Uza 1998). On the other hand, Gratton (1998) has analyzed [*Hipparcos*]{} data for field horizontal-branch (HB) stars including three RR Lyrae variables, and found that the faint HB luminosity scale was preferred. Fernley et al. (1998) and Tsujimoto, Miyamoto, & Yoshii (1998) have also reported, based on [*Hipparcos*]{} data for field RR Lyraes, luminosities which are consistent with the corresponding ground-based analyses. As argued by Gratton, [*the Hipparcos results thus seem to favor the existence of an intrinsic difference in luminosity (by $\approx 0.2$ mag) between GC and field RR Lyraes*]{}.
However, no independent tests have thus far been applied to verify this. As is well known, [*RR Lyrae pulsation properties depend strongly on their luminosities*]{}. The purpose of this [*Letter*]{} is to employ such properties to constrain the difference in luminosity between field and GC variables. [*Only GC’s and field stars analyzed with Hipparcos will be covered.*]{} We begin in Sec. 2 by discussing the employed methods for deriving RR Lyrae temperatures. In Sec. 3, the selection criteria we have adopted are described. In Sec. 4, we demonstrate that GC variables do not show substantial period shifts with respect to field variables of similar metallicity, as opposed to what would be expected if there were an intrinsic luminosity difference between them. Finally, our results are critically discussed in Sec. 5.[^1]
Estimating RR Lyrae Temperatures
================================
The Carney, Storm, & Jones (1992a) Approach
-------------------------------------------
In their Baade-Wesselink analysis of field RR Lyraes, Carney et al. (1992a, hereafter CSJ92) compiled parameters for a number of variables, including temperatures derived from near-infrared colors. Analyzing possible correlations in their database, they concluded that a simple equation exists \[their eq. (16)\] relating the “equilibrium temperature" $T_{\rm eq}$, blue amplitudes $A_B$, pulsation periods $P$, and metallicities \[Fe/H\] for ab-type RR Lyraes. This relationship formed the basis for their discussion of the period-shift effect, and will be adopted here as a first means of estimating temperatures.
The Catelan, Sweigart, & Borissova (1998) Approach
--------------------------------------------------
Catelan et al. (1998, hereafter CSB98) have recently reanalyzed temperatures based on the CSJ92 data. They argued that a relationship involving only $T_{\rm eq}$, $A_B$, and \[Fe/H\] would be safer to adopt in period-shift analyses than CSJ92’s (Sec. 2.1), since period shifts caused by luminosity variations could easily be misinterpreted as being due to temperature variations. The idea of employing $A_B$ values to determine $T_{\rm eq}$ (cf. Sandage 1981a,b) is supported by Jones et al. (1992), who state that “$\ldots$ at a fixed metallicity, it is likely that [*relative*]{} $A_B$ values are reliable indicators of [*relative*]{} temperatures."
We have rederived the CSB98 relationship for $T_{\rm eq}$ for the same selection criteria and parameters used in our period-shift analysis of the [*Hipparcos*]{} sample (Sec. 3). Thus, the star SW And was also removed from the CSJ92 database, because it presents the Blazhko effect. Furthermore, the $A_B$ values from Blanco (1992) were adopted. (The differences are generally small, with the exception of DX Del, for which Blanco’s $A_B$ is larger by $0.28$ mag.)
Our new relationship for $\Theta_{\rm eq} = 5040/T_{\rm eq}$ thus reads: $$\Theta_{\rm eq} = (0.868\pm 0.014) - (0.084\pm 0.009)\, A_B -
(0.005\pm 0.003)\, {\rm [Fe/H]},$$ with a multiple correlation coefficient $r = 0.97$ and a rms deviation of $\simeq 40$ K. Fig. 1 shows that this relationship does provide a superb match to the CSJ92 equilibrium temperatures.
RR Lyrae Stars: Adopted Samples
===============================
In the present section, we lay out the selection criteria employed in our analysis.
Field RR Lyrae Stars
--------------------
We have retrieved the list of 125 variables employed by Tsujimoto et al. (1998) in their [*Hipparcos*]{}-based analysis of field RR Lyraes, as kindly supplied by Dr. T. Tsujimoto.
We have selected stars from this sample according to the following criteria. i) [*Reliable classification as ab-type RR Lyrae stars*]{}: Variables whose RRab Lyrae nature has been questioned by Schmidt, Chab, & Reiswig (1995) or Fernley & Barnes (1997) were dismissed; ii) [*Well-behaved light curves*]{}: Stars which Blanco (1992) or Schmidt et al. pinpointed as Blazhko variables were discarded; iii) [*Metallicity values available from Layden*]{}: Stars for which Layden et al. (1996) do not provide metallicity values were not considered. We have also removed from the list all variables for which metallicities are based on Hemenway’s (1975) measurements, since we consider the corresponding Layden et al. \[Fe/H\] values quite uncertain; iv) [*Blue amplitudes and pulsation periods available from Blanco (1992)*]{}.
Cluster RR Lyrae Stars
----------------------
Among the 12 GC’s which have had distances determined using [*Hipparcos*]{} parallaxes for field subdwarfs, only 5 contain a sufficiently large number of RR Lyrae variables to justify their inclusion in the present analysis: NGC 362, M5, M68, M15, and M92. Since the Layden et al. (1996) field RR Lyrae metallicities are tied in to the Zinn & West (1984) abundance scale, we decided to adopt the \[Fe/H\] entries of Harris’ (1996) catalogue for consistency. Although the Zinn & West scale has been seriously questioned by Gratton et al. (1997 and references therein), this is of minor relevance for the present purposes, since our goal is to perform a period-shift analysis at [*fixed metallicity*]{}. Likewise, the criticism of McNamara (1997a) of the near-infrared temperatures is of secondary relevance for us.
The adopted sources of information for the GC RR Lyrae variables are provided in Table 1, along with the cluster \[Fe/H\] and Lee-Zinn HB morphology parameter (both from Harris 1996). According to such HB types, the only GC for which evolution away from the blue zero-age HB may bias the period-shift analysis is M92. As with the field star sample, Blazhko variables were discarded—as were those suspected to be non-cluster members.
Period-Shift Analysis: Clusters versus Field
============================================
Table 1 shows that our GC sample divides into two metallicity bins, with ${\rm [Fe/H]} \approx -1.2$ and $\approx -2.2$. We have thus split the comparison between GC and field variables into two metallicity regimes. For the more metal-rich end, we employ all field RR Lyraes (25 stars) falling in the range $-1.50 \leq {\rm [Fe/H]} \leq -0.95$ which have passed our selection criteria (Sec. 3.1); at the metal-poor end, we restrict the sample to the variables with ${\rm [Fe/H]} \leq -1.85$ (10 stars). The resulting $\log\,P - \log\,T_{\rm eq}$ diagrams using temperatures derived as in Sec. 2.1 and Sec. 2.2 are shown in Figs. 2 and 3, respectively.
The scatter is substantially larger in Fig. 3 than in Fig. 2. This, however, should not be taken as evidence that eq. (1) is less satisfactory at estimating $T_{\rm eq}$ values than eq. (16) of CSJ92. As previously argued (Sec. 2.2), CSJ92’s relationship, by including a [*period*]{} term, can “mask" luminosity variations at a [*fixed*]{} $T_{\rm eq}$, misinterpreting them as temperature variations. Thus, eq. (16) of CSJ92 artificially [*drives*]{} a tight $P-T_{\rm eq}$ distribution for a sample of stars with intrinsic luminosity scatter. Our eq. (1) does not have this bias, being more suitable for detecting luminosity variations at a given $T_{\rm eq}$. Consider, for instance, V9 in 47 Tuc ($P = 0.737$ d), which is brighter than field RR Lyraes of similar metallicity by $\approx 0.6$ mag (cf. Fig. 9 in Storm et al. 1994). Eq. (16) of CSJ92 underestimates V9’s $T_{\rm eq}$ by $\simeq 600$ K, while the underestimate from eq. (1) is only $\simeq 180$ K. In fact, Marconi’s (1997, priv. comm.) pulsation models show temperatures to be quite insensitive to $L$ at fixed [*blue*]{} amplitude over a range in $M_{\rm bol}$ of 0.75 mag and for $0.2 \lesssim A_B \lesssim 2.0$. In addition, SS Leo, which has ill-determined physical parameters, was not discarded by CSJ92 when deriving their eq. (16). Excluding this star from the CSJ92 sample, we find a reduction in the $\log\,P$ coefficient of their eq. (16) by $\approx$ a factor of two, and an increase (in absolute value) in the corresponding $A_B$ coefficient by a similar factor. The $T_{\rm eq}$ value adopted by CSJ92 for this star, $\sim 6400$ K, differs from the one expected on the basis of eq. (1) by $\simeq 300$ K—a factor of $\approx 3$ larger than the one for the largest-deviating star in our Fig. 1. Figs. 2 and 3 show that [*there is no detectable difference in period-shift properties between the studied field and cluster RR Lyraes, either at the metal-poor or at the more metal-rich end*]{}—irrespective of the approach used to estimate $T_{\rm eq}$. If Gratton’s (1998) suggestion were correct and the GC variables were brighter by $\approx 0.2$ mag, we would expect to see a difference as large as $\Delta\log\,P \approx +0.067$ at fixed $T_{\rm eq}$ between GC and field stars (Catelan 1996), which is most decidedly [*not*]{} present in our diagrams.
Other interesting conclusions that may be drawn from Figs. 2 and 3, but which we shall not discuss in the present [*Letter*]{}, are: i) There seems to be no offset in the $P - T_{\rm eq}$ diagrams between GC’s with widely different HB types but similar \[Fe/H\] (cf. Catelan 1994); ii) Metal-poor RRab Lyraes may have a cooler $T_{\rm eq}$ cutoff than the metal-rich ones (Sandage 1993); iii) There may be an offset of $\Delta\log\,P \approx +0.05$ at constant $T_{\rm eq}$ ($\Rightarrow$ $\Delta M_{\rm bol}$ $\approx 0.15$ mag for fixed mass) between the metal-poor and the more metal-rich RR Lyraes.
Discussion
==========
[*The present analysis does not substantiate Gratton’s (1998) suggestion, based on Hipparcos results, that there is a difference in luminosity between GC and field RR Lyrae variables*]{}, showing instead that they have essentially the same distribution in the $P - T_{\rm eq}$ plane, both at the metal-poor and at the more metal-rich ends.
Does this imply that there is really [*no*]{} difference between GC and field HB stars? Not necessarily. In fact, at ${\rm [Fe/H]} > -1$, Sweigart & Catelan (1998) found (following the same approach as in the present [*Letter*]{}) substantial differences between field and (some) GC RR Lyraes (see also Storm et al. 1994 and Layden 1995). Moreover, it should be noted that: i) The $A_B - T_{\rm eq}$ diagram (and possibly even spectroscopically-derived metallicities) may be sensitive to the helium abundance $Y$, so that an additional, $Y$-dependent term may be needed to put eq. (1) on a firmer basis (CSB98). In any case, available models suggest that it would not be possible for differences in $Y$ between field and GC stars to be consistent with both a luminosity difference of $\approx 0.2$ mag and the remarkable overlapping in the $P - T_{\rm eq}$ plane found in Figs. 2 and 3; ii) As well known, field red giants do not seem to show signatures of non-canonical deep mixing, whereas some GC’s do (cf. Kraft et al. 1997). It might be worth examining whether RR Lyrae temperatures and amplitudes might be sensitive to their (inherited) abundance anomalies; iii) As pointed out by Sweigart (1998), the stars which are more likely to be affected by “helium mixing" during the red giant branch phase are the blue-HB and extreme-HB stars, not the cooler RR Lyraes.
The present results (see also Fernley 1993) provide motivation for searching for systematic errors in methods employed to estimate the distances of GC’s (esp. main-sequence fitting) and the luminosities of RR Lyrae stars (esp. the statistical parallaxes and Baade-Wesselink methods). Unless some “cosmic conspiracy" is leading to the remarkable agreement between field and cluster stars in Figs. 2 and 3, [*the “long" and “short" Pop. II distance scales cannot be reconciled in the way suggested by Gratton (1998)*]{}. We cannot tell whether the “long" or the “short" scale should be preferred on the basis of a comparison with evolutionary models, due to extant systematic uncertainties in the empirical RR Lyrae temperatures (McNamara 1997a). However, the LMC provides a means of estimating RR Lyrae luminosities (Walker 1992), and several methods seem to favor the “long" distance scale (implying brighter RR Lyraes and younger GC ages) over the “short" one.
The author would like to thank C. Cacciari, A. Sweigart, D. VandenBerg, W. Landsman and the referee for useful suggestions, and M. Marconi and T. Tsujimoto for providing relevant information. This work was performed while the author held a National Research Council–NASA/GSFC Research Associateship.
Bingham, E. A., Cacciari, C., Dickens, R. J., & Fusi Pecci, F. 1984, , 209, 765
Blanco, V. M. 1992, , 104, 734
Brocato, E., Buonanno, R., Malakhova, Y., & Piersimoni, A. M. 1996, , 311, 778
Carney, B. W., Storm, J., & Jones, R. V. 1992a, , 386, 663 (CSJ92)
Carney, B. W., Storm, J., Trammell, S. R., & Jones, R. V. 1992b, , 104, 44
Catelan, M. 1994, , 107, 2077
Catelan, M. 1996, , 307, L13
Catelan, M., Sweigart, A. V., & Borissova, J. 1998, in ASP Conf. Ser. 135, A Half Century of Stellar Pulsation Interpretations: A Tribute to Arthur N. Cox, ed. P. A. Bradley & J. A. Guzik (San Francisco: ASP), 41 (CSB98)
Clement, C. M. 1997, Helen Sawyer Hogg’s Fourth Catalogue of Variable Stars in Globular Clusters, AAS Newsletter, 84, 15
Clementini, G., Carretta, E., Gratton, R., Merighi, R., Mould, J. R., & McCarthy, J. K. 1995, , 110, 2319
Feast, M. W., & Catchpole, R. M. 1997, , 286, L1
Fernley, J. 1993, , 268, 591
Fernley, J., & Barnes, T. G. 1997, , 125, 313
Fernley, J., Barnes, T. G., Skillen, I., Hawley, S. L., Hanley, C. J., Evans, D. W., Solano, E., & Garrido, R. 1998, , 330, 515
Gould, A., & Uza, O. 1998, , in press (astro-ph/9705051)
Gratton, R. G. 1998, , in press (astro-ph/9710271)
Gratton, R. G., Fusi Pecci, F., Carretta, E., Clementini, G., Corsi, C. E., & Lattanzi, M. 1997, , 491, 749
Harris, W. E. 1996, , 112, 1487
Hemenway, M. K. 1975, , 80, 199
Jones, R. V., Carney, B. W., Storm, J., & Latham, D. W. 1992, , 386, 646
Kraft, R. P., Sneden, C., Smith, G. H., Shetrone, M. D., Langer, G. E., & Pilachowski, C. A. 1997, , 113, 279
Laney, C. D. 1998, in ASP Conf. Ser. 135, A Half Century of Stellar Pulsation Interpretations: A Tribute to Arthur N. Cox, ed. P. A. Bradley & J. A. Guzik (San Francisco: ASP), 180
Layden, A. C. 1995, , 110, 2312
Layden, A. C., Hanson, R. B., Hawley, S. L., Klemola, A. R., & Hanley, C. J. 1996, , 112, 2110
Madore, B. F., & Freedman, W. L. 1997, , 492, 110
McNamara, D. H. 1997a, , 109, 857
McNamara, D. H. 1997b, , 109, 1221
Panagia, N., Gilmozzi, R., Kirshner, R. P. 1998, in ASP Conf. Ser., SN1987a: Ten Years After, ed. M. Phillips & N. Suntzeff, in press
Pont, F., Mayor, M., Turon, C., & VandenBerg, D. A. 1998, , 329, 87
Popowski, P., & Gould, A. 1998, , in press (astro-ph/9703140)
Reid, I. N. 1996, , 278, 367
Reid, I. N. 1997, , 114, 161
Reid, I. N. 1998, , in press (astro-ph/9710311)
Sandage, A. 1981a, , 244, L23
Sandage, A. 1981b, , 248, 161
Sandage, A. 1993, , 106, 703
Schmidt, E. G., Chab, J. R., & Reiswig, D. E. 1995, , 109, 1239; erratum: 1995, , 110, 2439
Silbermann, N. A., & Smith, H. A., 1995, , 110, 704; errata: 1996, , 111, 567
Storm, J., Carney, B. W., & Beck, J. A. 1991, , 103, 1264
Storm, J., Carney, B. W., & Latham, D. W. 1994, , 290, 443
Storm, J., Nordström, B., Carney, B. W., & Andersen, J. 1994, , 291, 121
Sweigart, A. V. 1998, in The Third Conference on Faint Blue Stars, ed. A. G. D. Philip, J. Liebert, & R. A. Saffer (Cambridge: CUP), in press (astro-ph/9708164)
Sweigart, A. V., & Catelan, M. 1998, in ASP Conf. Ser. 135, A Half Century of Stellar Pulsation Interpretations: A Tribute to Arthur N. Cox, ed. P. A. Bradley & J. A. Guzik (San Francisco: ASP), 39
Tsujimoto, T., Miyamoto, M., & Yoshii, Y. 1998, , 492, L79
van Leeuwen, F., Feast, M. W., Whitelock, P. A., & Yudin, B. 1997, , 287, 955
VandenBerg, D. A., Bolte, M., & Stetson, P. B. 1996, , 34, 461
Walker, A. R. 1992, , 390, L81
Walker, A. R. 1994, , 108, 555
Zinn, R., & West, M. J. 1984, , 55, 45
[^1]: We emphasize that the purpose of the present work is to perform a period-shift analysis at fixed temperature [*and*]{} metallicity. Thus, a careful analysis of the Sandage (1993) period-shift effect lies outside the scope of this [*Letter*]{}.
|
---
abstract: 'Applying security as a lifecycle practice is becoming increasingly important to combat targeted attacks in safety-critical systems. Among others there are two significant challenges in this area: (1) the need for models that can characterize a realistic system in the absence of an implementation and (2) an automated way to associate attack vector information; that is, historical data, to such system models. We propose the cybersecurity body of knowledge (CYBOK), which takes in sufficiently characteristic models of systems and acts as a search engine for potential attack vectors. CYBOK is fundamentally an algorithmic approach to vulnerability exploration, which is a significant extension to the body of knowledge it builds upon. By using CYBOK, security analysts and system designers can work together to assess the overall security posture of systems early in their lifecycle, during major design decisions and before final product designs. Consequently, assisting in applying security earlier and throughout the systems lifecycle.'
author:
- 'Georgios Bakirtzis, Brandon J. Simon, Aidan G. Collins, Cody H. Fleming, and Carl R. Elks'
bibliography:
- 'manuscript.bib'
title: |
Data Driven Vulnerability Exploration\
for Design Phase System Analysis
---
Cyber-physical systems, security, safety, model-based engineering.
Introduction {#sec:org0b008b2}
============
It has been estimated that 70% of security flaws are introduced prior to coding, most of which are due to the traditional practice of application developers sharing and reusing third party, legacy software—that is assumed to be reasonably secure and trustworthy. These flaws usually end up in the application software and not, as might be expected, in network-based software [@fong2007web; @third; @safecode]. Most security flaws are introduced as early design or development decisions.
Both the academic and practicing cybersecurity community agree that security engineering and analysis as a full lifecycle practice, especially early in the design process allows better awareness and leverage at managing the challenges surrounding the unintentional introduction of security flaws into complex systems. This is especially important in the domain of cyber-physical systems (CPS), where the exploitation of software flaws and hardware weaknesses—introduced by either importing software of unknown pedigree, incomplete security specifications, or general unawareness of security characteristics of given software, firmware, and/or hardware—can lead to unforeseen physical behaviors that have consequences in terms of safety, loss of vital service, and other societal impacts.
As modern CPS evolve into tightly integrated, extensible, and networked entities, we significantly increase the attack surface of these systems. CPS now routinely employ a wide variety of networks, for example, cloud, mobile services, industrial, internet of things to realize a range of applications from real time data analytics to autonomous vehicles control. The use of extensible operating systems and software to update code through loadable device drivers enhances productivity, but it exposes the system to considerable risks from attack injections.
With these insights and observations, we posit that secure system design and deployment requires (1) planning for cybersecurity from the outset as a strategic lifecycle activity, and (2) taking the attackers perspectives to best understand how to defend a system from threats and exposing weaknesses before they become vulnerabilities. To achieve this goal methods and tools are needed to allow security assessment throughout the systems lifecycle and especially at the concept development phase, where decision effectiveness is highest [@frola_system_1984; @strafaci_what_2008].
In recent years, a promising and rapidly growing approach to enhancing awareness and managing challenges of cybersecurity flaws in evolving complex CPS is model-based engineering [@nguyen:2017]. Model-based analysis is firmly entrenched in safety, dependability, and reliability engineering world as evidenced by such standards as IEC 61508 and ISO 26262, however model-based engineering is a late comer to security [@nicol_model-based_2004].
Models are generally treated as *living documents* maintained to reflect design choices and system revisions. These models can be a valuable resource for the security specialist by providing what IT professionals consider the “what’s”; that is, the rationale behind design choices and not simply the resulting architecture of a system [@chapman_what_2001].
An additional benefit of model-based security is it tends to look at security from a strategic point of view, which means it attempts to secure a system based on its expected service. Rather than beginning with tactical questions of how to protect a system against attacks, a strategic approach begins with questions about what essential services and functions must be secured against disruptions and what represents unacceptable losses. This is critical for CPS where losses or disruptions to service can have dire societal or safety impacts [@alemzadeh:2013; @kshetri:2017].
{width=".8\linewidth"}
However, one of the major impediments to effectively transition security assessment into the model-based engineering realm has been associating system models to applicable attack vectors. These models reside in a higher-level of abstraction than what is typically present in cybersecurity analysis. Our aim is to use the model to drive the attack vector analysis in the design phase. There are two things necessary to achieve this congruence: (1) understand the data available to security researchers and decide on which of those can inform early on and (2) capture lower-level information in the model, such that it can be used to associate the available data with the model. Such an approach bridges the gap between existing curated attack vector information and models of systems. Indeed, this paper presents one answer to examining cybersecurity concerns in the model-based engineering setting.
Towards this goal we have previously [@bakirtzis2018model] presented a CPS model that includes a schema with extra design information to manually associate attack vector data describing attack patterns, weaknesses, and vulnerabilities. The previous work proposed just a model, not an algorithmic solution to the vulnerability exploration problem. To explore the large amount of data compiled by security professionals, it is helpful to associate attack vectors algorithmically. This is precisely the topic of this paper.
**Contributions.** The contributions of this work are:
- an algorithmic implementation, called cybersecurity body of knowledge (CYBOK), that accepts as input such models and produces:
- a component-wise attack vector analysis using attack vector data,
- a notion of attack surface, which only depends on the model, and
- all exploit chains applicable to subsystems of the system model and
- a demonstration of the method on an unmanned aerial system (UAS).
Model-Based Security Analysis {#sec:org103a848}
=============================
Model-based security analysis is a relatively new field that attempts—as the name implies—to understand system threats through the use of models. In this paradigm, models are used either as an augmentation to other security strategies during deployment or as evidence to support design decisions early in the systems lifecycle. However, most current models are probabilistic in nature and, therefore, require a ground truth. These models also heavily depend on the modelers expertise and experience. Any such model captures exactly that expertise such that it is communicated to other stakeholders. To our knowledge, models used at the design phase have not achieved fidelity with security data collected and otherwise used in already realized systems, which would consist of an important addition to defending against increasingly sophisticated threats.
But why is that? To understand the difficulty of finding vulnerabilities in system models—instead of a deployed product—it is important to first define the difference between bugs, vulnerabilities, and exploits. Successful exploits take advantage of flaws (either serious design flaws or unexpected system behavior that is implementation specific). These flaws in the system are called bugs. However, not all bugs are vulnerabilities. Only a subset of bugs that can lead to exploitation are vulnerabilities. This notion leads to the first problem that is addressed in this paper.
Vulnerabilities are explicitly found at the level of code or hardware. However, to address system security early in the design cycle, there need to be methods that can identify potential vulnerabilities before code development.
To bridge the gap between models and realized solutions requires constructing an initial design of the system. This design needs to include both the *what’s*, the components of the system, for example GPS, and the *how’s*; that is a particular hardware, firmware, and software solution that implements some desired function. Additionally, any such model needs to include the interaction between components as is defined by their communication and data transfer.
One way to fulfill those requirements is to model CPS as a graph of assets but with added information in the form of descriptive keywords. This is a reasonable and appropriate model as it pertains to security analysis. Attackers typically think in terms of graphs, through a series of increasing violations based on concepts of connectivity, reachability, and dependence, not in lists of assets as—most commonly—defenders do [@lambert_defenders_2015].
In addition, this model must include extra information in the form of keywords that augment the model, resulting in a system model that captures the choices a designer is considering about hardware, firmware, and software. These augmentations can be done without overly specific details about its final implementation. This is a key feature that reflects how designs evolve in the construction of a system, where choices of specific hardware and software are done early in the development cycle. Furthermore, the ease of changing those keywords to describe a functionally equivalent system allows for modeling flexibility that is not available after code has been written and designs are finalized.
Formally, an architectural model of a CPS can be captured in a graph, $\Sigma \triangleq \left(\mathcal{V}, \mathcal{E}, \mathcal{D}\right)$, where,
$$\mathcal{V} \triangleq \left\{v \mid v = \text{a system asset} \right\}\text{,}$$ $$\mathcal{E} \triangleq \left\{e \mid e = (v_i, v_j); v_i, v_j \in \mathcal{V} \text{ dependent assets} \right\}\text{, and}$$ $$\mathcal{D} \triangleq \left\{ d \mid d = \left(w_1, w_2, \dots, w_n\right) \text{ descriptive keywords}\right\}\text{.}$$
In order to extract and use the descriptive information; that is, the extra keywords, from a given vertex or edge, we define the descriptor function, $\text{\textit{desc}}: \mathcal{V} \cup \mathcal{E} \rightarrow \mathcal{D}$.
The above definitions lead to the first practical challenge, which is to determine if a model is sufficient for security assessment (Fig. \[fig:grid\]). It is important at these early design stages for a system model to sufficiently describe functionally complete system—by adding hardware and software information that, if put together, implements the desired functional behaviors expected from the system.
Any model used for security analysis contains two main challenges. The first challenge, is the amount of data associated with the model. When using a realized system it is possible to mine all possible configuration settings including software and firmware versions. In the absence of the implementation such information can still be reflected in a modeling setting but it requires significant modeling cost in terms of time and expertise. It can also be less informative at the design phase than a more general model because a slight change in versioning will hide a class of weaknesses and vulnerabilities. The second challenge, is the level of abstraction the model resides in. No model is a direct reflection of a realized system but any model needs to be specific enough to be informative. This is a difficult task and largely depends on the given abstraction set overall by the modeling process as well as the expertise of the modeller.
These are precisely the challenges that the added keywords address. By changing the specificity and amount of keywords, we change the overall fidelity of the system model, $\Sigma$. It is through that *extra* design information that our solution, CYBOK, is able to take the graph of a system model and map applicable attacks from security databases (Fig \[fig:process\]). While there may be a number of different criteria for selection, in previous work we have found that the extra design information can be categorized through the following practical schema: operating system, device name, communication, hardware, firmware, software, and entry points [@bakirtzis2018model]. Each of the categories is expected to contain a string of keywords, $d \in \mathcal{D}$ that collectively describe a given system solution. A given category can also duplicate the descriptive keywords present in another category or simply contain the null set, $\emptyset$.
The fidelity of the model is still based on the choices of the modeller. On the one side of the *model sufficiency spectrum* there are designs that are too general and do not contain information about the system that would aid in determining security posture. On the other side of the spectrum there are designs that are too specific, to the extent that the effort to create and potentially modify is equivalent to constructing an actual implementation of the system and its functionality. Such systems are complete but impractical. There is a spot in the middle of the spectrum, where the information contained in the model can provide a reasonable idea about the system’s threat space without being so detailed it is inflexible and costly to construct and maintain throughout its lifecycle.
A perhaps less obvious but equally important challenge refers to the information necessary to associate the model, $\Sigma$, to potential attack vectors.
How can we associate vulnerability, weakness, and attack pattern information that is intended to be used by security analysts to a model?
This problem is difficult because of the way security experts record vulnerabilities. While several attempts have been made to standardize the form of an attack vector entry, the current situation is such that the different databases are based on a different schema, because they rely on the deduction and inference capabilities of a human. This means there is no straightforward approach to feeding that data into a machine to automatically find those mapping.
![The fidelity of the attributes describing a CPS has to associate to the attack vector information (reproduced from Bakirtzis et al. [@bakirtzis2018model]).[]{data-label="fig:grid"}](attribute_grid){width="45.00000%"}
Our solution is based on the set of descriptors, $\mathcal{D}$, present in the model and using standard practices from natural language processing to deconstruct and associate the contents of an attack vector entry—in the form of text—to the models keywords, $d \in \mathcal{D}$. This requires a separate set of attack vector entries, $\mathcal{AV} \triangleq \left\{ av \mid av = \text{a set of stemmed words}\right\}$. Therefore, the fundamental problem that CYBOK attempts to solve is then formalized as the function, $\text{\textit{associate}}: \text{\textit{desc}} \rightarrow \mathcal{AV}$. By doing so, the problem is reduced to associating keywords describing the system (which exist within the model) to stemmed words describing attack vector information (which are constructed using the contents of the database entries and natural language processing).
Applying security consideration to model-based engineering is difficult for several reasons. Three of the most important difficulties are: the curation of information (both from the model and the attack vector databases), the intuition surrounding the fidelity of the system model, and the development of an algorithmic approach that allows for filtering through a large number of attack vectors produced at the design phase.
Finding attack vectors for individual vertices or edges overcomes these challenges. However, it can be daunting to see the big picture when confronted with such a larger amount of data. Security professionals frequently use complimentary metrics to understand the overall security posture of a given system.
Two such useful metrics for security analysis are:
1. The *attack surface* captures all the entry points into a given system [@manadhata_attack_2011].
2. The potential for further spread; that is, further violations, after an element of the attack surface has been compromised, known as an *exploit chain*.
**Proposed Solution**. In general, CYBOK is an algorithmic solution that takes as input a sufficient system model (of $\Sigma$ form) to associate to the body of knowledge of attack vectors (of $\mathcal{AV}$ form). By knowing the associated attack vectors it then produces security metrics only based on the model; that is, the attack surface of the system model and the exploit chains for a particular element of the model.\
**Intrinsic Limitations**. Model-based security analysis is grounded on early design information. This early design information is usually incomplete and abstracted with respect to the final design solution. This leads to result spaces; that is, associated attack vectors, that are significantly larger than when analyzing a realized system. Navigating through the results can be challenging for systems engineers that are not familiar with security practices. Our aim is to provide a framework in which security analysts and systems engineers work synergistically to understand both necessary design decisions (that might affect potential security mitigations) and security considerations (that might affect the design of the system).
An additional limitation is the fidelity of the model. The model can only be as good as the person who is modeling the system. Therefore, a poorly constructed model might mislead instead of providing insight into the security posture of the system.
System Model {#sec:org5ceca0b}
============
To address the challenges in the previous section and construct a sufficient model with respect to vulnerability analysis, first we must elicit information from the stakeholders. The stakeholders of the system include the owners of the eventual system, the system designers, the safety engineers, and the security analysts. While it is outside the scope of this paper to address the systematic process in which such information is elicited (see Carter et al. [@carter:2018] for further details on the topic), it is important to place CYBOK within its larger framework. Without this framework it would not be possible to have complete or correct information to apply vulnerability exploration this early in the system’s lifecycle. Based on this elicitation, an initial design solution is modeled in the systems modeling language (SysML).
SysML uses visual representations to capture the system design process through objects. The main benefit of SysML is that it presents the same information in different views, which allows the same system to be modeled based on its requirements (through the requirements diagram), through its behavior (through, for example, the activity and/or state machine diagrams), and/or through its architecture (through block definition diagram (BDD) and/or internal block diagram (IBD)).
To model CPS architectures in SysML the system structure is captured as a set of BDD and IBD. The BDD view of the system shows the composition of the system. The IBD view refines those compositions to interconnections within the system and how those interconnections compose the system behavior.
However, each element of the system model is described by a standardized schema as presented by Bakirtzis et al. [@bakirtzis2018model]. It is, therefore, not necessary to capture this model in SysML. This model is flexible to design changes and has supported vulnerability analysis in a manual setting. In this work we use the modeling methodology to support automated vulnerability analysis.
Security specialists usually construct the following information implicitly through expertise. To automate this task this implicit information needs to be captured explicitly in the model. This information will also assist in constructing a living document describing the *what’s* of those choices. To recap, the schema is composed by the following categories that describe each system element:
- operating system,
- device name,
- communication,
- hardware,
- firmware,
- software, and
- entry points.
It is through that *extra* design information—gathered by eliciting stakeholder information and inspecting design documentation—that CYBOK is able to take the graph of a system model and map potential attacks from databases (Section \[sec:org27e0311\]). This is done by using the key terms presented in the schema for each element and checking if they are present in the documents composing the databases.
Reasoning in terms of the diagrams has several benefits during the design process that hold for security analysis in general (see Oates et al. [@oates_security_2013]), which is the benefit of starting with a SysML model instead of its graph representation. This is less true for matching attack vectors to the model. The exporting of models in a standardized format is, therefore, beneficial. The translation of IBD diagrams into a graph is encoded into GraphML—a simple XML format that is widely used to import and export graph structures [@brandes2013graph].
The two models—i.e., the visual representation in SysML and the graph structure—-must be isomorphic. This means that the transformation between the SysML model and the GraphML representation must not change the model of the system. To achieve an isomorphic transformation we apply a model transformation on the IBD model. This transformation produces a sufficient graph model for security analysis (Fig. \[fig:topology\] in Section \[sec:org7815955\]).
An <span style="font-variant:small-caps;">internal block diagram</span> is the graph $I \triangleq \left(\mathcal{V}, \mathcal{P}, \mathcal{D}\right)$, where $\mathcal{V}$ is the set of vertices of $I$ and $\mathcal{P}$ is the set of ports of $I$. Further, $\mathcal{V}$ represents the assets of a cyber-physical system, $\mathcal{P}$ the dependence between assets, and $D$ the descriptors of each asset. Therefore, the graph $I$ is isomorphic to our system graph $\Sigma$; that is to say $I \cong \Sigma$.
This system model represented as a graph allows us to think similarly to attackers and to construct connections that would not be obvious if treated as simple decoupled components. The graph of the system model assists with not only finding vulnerable subsystems individually but also with finding the attack surface of the system and composing exploit chains. Exploit chains are a subset of attacks that could traverse though the system model graph and elevate the impact to deteriorate system behavior. All exploit chains start at elements of the attack surface. The attack surface, $\mathcal{AS}$ is composed by all vertices that an attack from the databases is found to be potentially applicable at the entry point description.
In particular the graph model allows us to define the attack surface and exploit chains over the system model $\Sigma$ in a straightforward manner.
We define the attack surface of a system model, $\Sigma$ as $\mathcal{AS} \subseteq \mathcal{V}$, which is composed by all vertices that can be entry points into the system and allow the attacker to cause further spread within the system structure.
To construct the exploit chain we define a function, $\textit{paths}: \mathcal{AS} \times t \times \Sigma \rightarrow \mathcal{P}$, where $\mathcal{AS}$ the sources of all paths and $t$ a used specified target and $\mathcal{P}$, the set of all simple paths from the source to the target over $\Sigma$. Then to construct a single exploit chain, $ec \in \mathcal{EC}$, the set $\mathcal{P}$ is filtered by checking if every vertex and edge within each individual path associates to some attack vector from $\mathcal{AV}$.
Attack Vector Dataset {#sec:org27e0311}
=====================
CYBOK is composed by several databases to address two main challenges. The first challenge is finding applicable attack vectors based on a system model. The second challenge is to present a reasonable amount of data to the security analyst, such that they can erect barriers or add resilience solutions to strengthen the design of CPS using an evidence-based approach.
To address these challenges, CYBOK incorporates three collections curated by the MITRE corporation: CVE [@CVE; @NVD], CWE [@CWE], and CAPEC [@CAPEC]. CVE is the lowest level of attack vector expression, defining tested and recorded vulnerabilities on specific systems. CWE presents a hierarchy of known system weaknesses at different levels of abstraction, from which exploits can be derived. Finally, CAPEC provides a high level view of attacks against systems at varying levels of abstraction, in the form of a hierarchy organized by the goal or mechanism of each attack. These three collections also include relationships to one another (Fig. \[fig:datasets\]). Formally CAPEC, CWE, and CVE construct the set of attack patterns, weaknesses, and vulnerabilities $A \times W \times V \cong \mathcal{AV} $.
Other known datasets include the MITRE Common Platform Enumeration (CPE) [@CPE] and the Exploit Database (exploit-db) [@exploit-db]. The former includes information that is platform specific, including particular versions of software that a particular exploit is associated with. The latter provides samples of exploits for given CVE entries. The reasons for not including these two datasets is because CPE requires knowing the specific versions of software that are going to be on the system—which are not known at design phase—and exploit-db requires a realized system to test exploits against.
![The collection of the most common open attack vector datasets and their interconnections in the sense of topical relationships with regard to attacker and defender perspectives and level of specificity they contain about platform information. Edges denote which datasets have explicit relationships with one another. CYBOK only uses a subset of these databases marked by a $\Diamond$. []{data-label="fig:datasets"}](databases){width="1\linewidth"}
Using CVE entries has the benefit of finding additional attack vectors because the specificity of their descriptions may more closely associate to the descriptions in the model than those of CAPEC and CWE. When CVEs are matched to the system model, CYBOK uses that information to abstract upwards towards the weaknesses and the attack patterns. This is especially useful in the case where multiple CVE entries are associated with a subsystem that has the same associated weakness or attack pattern.
In general the CWE and CAPEC abstractions are more useful to designers over CVE entries because CVEs are too specific to be useful at the design phase. For example, being aware of a vulnerability in a specific version of software is less illuminating than knowing that a specific class of software bugs might consist of a vulnerability in the implementation of the system—and therefore can construct more concrete requirements or define specific mitigations. On the other hand, a number of applicable attack vectors from the model are going to reside in CVE. At the same time, CVE contains a significantly larger number of entries than both CAPEC and CWE (${\sim}100,000$ vs. ${\sim}1500$), meaning the addition of CVE entries will explode the number of results for a given system $\Sigma$. Therefore, all three are needed: CVE entries to be more thorough and complete in analyzing the system model and CAPEC, CWE to abstract to useful information to system designers.
While the three selected collections are complete in relation to the sufficient model, CVE is certainly not exhaustive. There are certainly instances where companies or government agencies curate more exhaustive databases from their non-disclosed findings. In those cases, the design of CYBOK can be extended to use the information contained in these private databases.
Architecture {#sec:org51a89ab}
============
![\[fig:architecture\] The architecture of CYBOK is modular, meaning that each of the functions can be replaced without needing to interfere with other functions. ](architecture){width="\columnwidth"}
The overall architecture of CYBOK is designed to be modular and robust with respect to potential architectural changes (Fig. \[fig:architecture\]). For example, the specific implementation of searching can be changed without needing to change the rest of the core tool functionality.
Data Extraction, Preprocessing, & Indexing
------------------------------------------
The first step to associating models to attack vector databases is to extract and preprocess the information contained in several individual databases. An automated mechanism for downloading the latest set of data is built into CYBOK because of the CVE, CWE, and CAPEC update cycle.
Specifically, CAPEC is refactored and potentially updated every six months to a year, CWE is extended every one month to three months, and CVE adds new entries daily. By doing automatic updates, the analyst is sure to have the latest set of data that might inform about new system violations. All database files are encoded in a standard xml file format. The steps that follow after updating the data files include preprocessing and constructing the search index to be used to associate system models to attack vectors.
The preprocessing step extracts the name of the database, the identification number, name of attack vector, associated attack patterns (if any), associated weaknesses (if any), associated vulnerabilities (if any), and the contents; that is the description, for each entry. These consist of the full search index schema; that is, all the information necessary to construct $A \times W \times V$.
After preprocessing CYBOK keeps a persistent record of all attack vectors, $\mathcal{AV}$, by constructing a search index through the schema defined above. This allows for efficient information retrieval of all attack vector information and avoids having to rebuild $\mathcal{AV}$ for each new search query.
System Models as Graphs {#sec:org748e81a}
-----------------------
Graph structures provide an important view in a computing system, and can extend the notion of violation to more than just a singular view of components. Indeed, the violation of a single component by an attacker might not be detrimental to the systems expected service. However, this component might be connected to other critical infrastructure. Therefore, this singular component could be a point of lateral pivot for an attacker. This in turn can cause significant malfunction during operation with catastrophic consequences. This is how attackers operate and, therefore, reasoning in graphs of assets provides an attacker’s view to defenders. For this reason, CYBOK views the system topology; that is, the design artifact, as a graph.
Finding Applicable Attack Vectors from a System Model
-----------------------------------------------------
$\mathcal{R}$ $\gets$ \[\] $\mathcal{R}$.append({$v \vee e, d, av$}) $\mathcal{R}$
To find applicable attack vectors, CYBOK extracts the descriptive keywords that define each vertex and edge. Then, using the descriptive keywords of the system CYBOK looks at all attack vector entries from $\mathcal{AV}$ to associate the descriptive keywords. A list of results is returned for the full system model, $\Sigma$, including the vertex or edge the attack vector can exploit, the descriptive keyword, $w_i$, that produced the attack vector, and the attack vector itself (Algorithm \[lst:attackVectors\]).
The search functionality of CYBOK currently uses a compound word filter. Other candidates for applying the searching include n-grams and Shingle filter.
Finding Attack Surface Elements
-------------------------------
$\mathcal{AS}$ $\gets$ \[\] $\mathcal{AS}$.append({$v$, $d$}) $\mathcal{AS}$
CYBOK views the attack surface as any vertex that has an associated attack vector specifically at the entry point. It constructs this set by going through all vertices and checking if a descriptive keyword, $\text{entry\_points}(w_{i})$, associatess to an applicable attack vector (Algorithm \[lst:attackSurface\]).
Finding Exploit Chains
----------------------
$\mathcal{EC}$ $\gets$ $[]$ admissible\_path $\gets$ $\top$ admissible\_path $\gets$ $\bot$ break $\mathcal{EC}$.append({$p$}) $\mathcal{EC}$
Exploit chains are paths from a source to a target that contain violation for every vertex or edge in that path. CYBOK finds exploit chains from all elements of the attack surface, $\mathcal{AS}$ to a user input target, $t$ (Algorithm \[lst:exploitChains\]). These paths are not necessarily the most efficient or direct paths from the elements of the attack surface to the given target. This is because it is often the case that attackers move laterally from the attack surface to a specific target without having full observability of the system. This way the analyst can be aware of all paths that are valid based on the system model and be better informed about potential mitigations.
Furthermore, not all paths from $\mathcal{AS}$ to $t$ are admissible. Admissible exploit chains require each vertex and each edge in that path has produced at least one result from $\mathcal{AV}$. Otherwise the path is not fully exploitable based on evidence and, therefore, does not consist of an exploit chain under this definition.
Visualizations
--------------
![The system topology, $\Sigma$, shows a static view of the system model.[]{data-label="fig:topology"}](topology){width="1\linewidth"}
To facilitate the analysis of results, CYBOK includes three main visualizations: (1) the system topology, (2) the system attack surface, and (3) the system exploit chains. This is an important feature of CYBOK because it allows both security analysts and system designers to project how exploits propagate over the system model, $\Sigma$. This way, they can be better informed about potential mitigation strategies. For example, changing the definition of a single element to one that has no recorded attacks might significantly increase the security posture of the overall CPS.[^1]
Moreover, a full GUI is developed based on this methodology to implement further interactivity functions on top of CYBOK [@bakirtzis2018looking]. This is a natural progression of CYBOK since in-depth analysis requires the analyst to interact with the data through interactivity functions, for example, filtering, to facilitate effective exploration of the diverse types of data input and output to and by CYBOK [@jacobs:2014].
Evaluation {#sec:org111049f}
==========
To evaluate CYBOK we will discuss in some detail the vulnerability analysis of one potential design solution for a UAS that contains a full set of descriptors. There is ongoing work in applying CYBOK to several other systems in the military and nuclear power domain [@beling2019model].
System Model {#sec:orgc3625a5}
------------
While modular approaches to flight control systems (FCS) have been demonstrated to provide flexible choices in hardware [@ward_modular_2014], it is not currently possible to assess the security of one design over another before building the system. By using models of systems it is possible to assess several system designs and provide evidence over the use of one hardware solution over another. In this work one such hardware solution is evaluated—through its system model (Fig. \[fig:topology\])—and present the evidence that stems from assessing the model’s security posture using CYBOK.
The potential design solution present in this paper uses several XBee radio modules to communicate between components, Dell Latitude E6420 ground control station (GCS) laptop, an ARM STM32F4 primary application processor, a BeagleBone Black imagery application processor, an ARM STM32F0 safety switch processor, MPU9150 accelerometer, gyroscope, and magnetometer, MS4525DO differential and absolute pressure sensors, a GoPro Hero5 camera, and an Adafruit Ultimate GPS. This information is part of the descriptive keywords captured in the vertices of the model. Further information is given for both vertices and edges to drive this analysis per the schema above (Section \[sec:org103a848\]).[^2]
![The attack surface, $\mathcal{AS}$, extends the system topology, $\Sigma$, by adding in the descriptive keywords at the entry point that associate to attack vectors.[]{data-label="fig:attack_surface"}](attack_surface){width="1\linewidth"}
Example Analysis {#sec:org7815955}
----------------
SysML is used as the modeling language and tool because it is often familiar to Systems Engineers. However, CYBOK is not restricted to SysML models, it merely requires a graph representation of the system that includes extra design information (see Bakirtzis et al. [@bakirtzis2018model] and [@bakirtzis:2018b] on how this translation is achieved in practice). Inputting the UAS system topology to CYBOK first constructs the attack surface (Fig. \[fig:attack\_surface\]). The attack surface is extended to show all the descriptive keywords at the entry points that attack vectors are found. For example, by inspection the use of the XBee module with the ZigBee protocol for all three radio modules can be problematic because an attacker can exploit the system remotely. Other such entry points have a different degree of potential exploitation. It is unlikely that the GPS will be violated but attacks for GPS exist and, therefore, are reported by CYBOK. Additionally, the analyst might be aware of hardening techniques on the Wi-Fi network used by the GCS laptop. Consequently, the analyst might decide that they consist of no threat to the systems mission.
From this initial understanding of the systems security posture (through its composition and attack surface) an analyst can further interrogate the model by finding all the potential exploit chains from the attack surface elements to the primary application processor. This is because violation of the primary application processor will cause full degradation of system functions and, therefore, full mission degradation overall. Specifically, an analyst might want to examine a potential exploit chain stemming from the XBee element of the attack surface. By providing a target, $t$, CYBOK finds the admissible paths and, therefore, exploit chains from the imagery radio module to the primary application processor. This path is admissible if each vertex *and* each edge within that path has produced evidence; that is, attack vectors.
![Exploit chains, $\mathcal{EC}$, show a possible lateral paths an attacker might take over the system topology to reach a specific system element. This is but one example of such exploit chain from the attack surface, $\mathcal{AS}$, to some target $t$—in this case the primary application processor.[]{data-label="fig:exploit_chain"}](exploit_chain){width="1\linewidth"}
Examining the results produced by CYBOK we find the following three associated entries: `CAPEC-67` “String Format Overflow in syslog(),” `CWE-20` “Improper Input Validation,” and `CVE-2015-8732` a specific attack on the ZigBee protocol used by XBee that allows remote attackers to cause a denial of service (DOS) via a crafted packet. Further, for the edges from the radio module to the primary application processor there are the following attack vectors produced by CYBOK, `CVE-2013-7266`, which is a specific attack that takes advantage of not ensuring length values matching the size of the data structure, `CWE-20` “Improper Input Validation,” `CWE-789` “Uncontrolled Memory Allocation,” `CWE-770` “Allocation of Resources without Limits or Throttling,” and `CAPEC-130` “Excessive Allocation.” Finally, the primary application processor uses the I2C and RS-232 protocols to communicate with the rest of the hardware (these are descriptive keywords contained in the edges of the graph), which produce the following, `CAPEC-272` “Protocol Manipulation” and `CAPEC-220` “Client-Server Protocol Manipulation.” All this information is used as evidence for the feasibility of one exploit chain from the attack surface to the primary application processor (Fig. \[fig:exploit\_chain\]).
By projecting the attack over the system structure it is evident when the same attacks are applicable to several parts of the system.This is important because attackers contain a specific skill set and they do not usually deviate from it if not necessary.
Model Element Attack Vector Description
------------------------------- --------------- -----------------------------------------------------------------------------
Radio Modules CVE-2015-6244 Relies on length fields in packet data, allows attacks from crafted packets
CWE-20 Improper input validation
CAPEC-67 String format overflow in syslog()
NMEA GPS CAPEC-627 Counterfeit GPS signals
CAPEC-628 Carry-Off GPS attack
Primary Application Processor CVE-2013-7266 Does not ensure length values match size of data structure
CWE-20 Improper input validation
CWE-789 Uncontrolled memory allocation
CWE-770 Allocation of resources without limits or throttling
CAPEC-130 Excessive allocation
I2C & RS-232 Protocols CAPEC-272 Protocol manipulation
CAPEC-220 Client-server protocol manipulation
Imagery Application Processor CWE-805 Buffer access with incorrect length value
CAPEC-100 Overflow buffers
Safety Switch Processor CWE-1037 Processor optimization removal or modification of security-critical code
Laptop CAPEC-615 Evil twin Wi-Fi attack
CAPEC-604 Wi-Fi jamming
Camera CVE-2014-6434 Allows remote attackers to execute commands in a restart action
Additionally, CYBOK allows flexible “what-if” analysis by changing the descriptive keywords in the model. For example, by changing the radio module definition from XBee using the ZigBee protocol to some other radio module offered in the market might exclude it from the attack surface. Since a larger attack surface implies more access points and, therefore, a less secure system an analyst might decide to propose changing the design of the system.
A full analysis consists of first identifying the important elements of the system; that is, the assets that might require protections. This might be informed from the outputs produced by CYBOK or from expert input and information elicitation. Then, of filtering the large space of attack vectors that associate to the model to find the most relevant and strong evidence (Table \[tab:results\]). This evidence is what ultimately informs other stakeholders, such that they can devise mitigative actions—changing the system solution to conform to mission needs, erecting security barriers at strategic points, or applying resilience solutions during operation.
Related Work
============
Little research has been done for evidence-based security assessment in a model-based setting. Usually work in this area requires *transcribing* already known vulnerabilities to a modeling tool and assessing if it might apply to a system design. Instead, the aim of this work is to employ models that can—by their fidelity—immediately produce a large number of potential attack vectors. These attack vectors stem from the model itself and are not informed from some a priori security knowledge.
We acknowledge that some current attack vector search tools could be repurposed for model-based systems engineering. One such search tool is cve-search [@cve-search]. However, cve-search cannot input a system model. It only provisions security datasets in one search engine. It is also limited with respect to visualization techniques.
Noel et al. [@noel2016cygraph] propose CyGraph which also is based on a graph-based understanding of the system but this work fundamentally differs in scope (mainly targets traditional networked systems) and approach (uses a traditional notion of attack graphs).
Adams et al. [@adams2018selecting] propose topic modeling for finding applicable attack vectors given a system model. However, they only examine CAPEC as a potential source of attack vectors, which is necessary but insufficient.
Ford et al. [@ford2013implementing] propose using the ADVISE security methodology [@lemay2011model] on top of the M[ö]{}bius tool [@courtney2006data] to provide an attackers’s view. However, the quantitative analysis is based on profiling and modeling attacker actions. The framework is largely unaware of a specific system model that could be used to implement a realized system.
The analysis presented in this paper is qualitative. This is because quantitative information for cyber-physical attacks is limited and ultimately expert input is necessary to understand what it means for a metric to show that a system is more susceptible to attacks over another. For example, a number of quantitative approaches incorporate CVSS as a potential metric for risk [@frigault2008measuring; @houmb2010quantifying; @wang2015data]. But, CVSS only defines severity of a given vulnerability and not risk [@CVSS; @collier2014cybersecurity].
In general, to the best of the authors’ knowledge, there is no direct comparison between the work in this paper and existing work in the literature. It is challenging to do a direct comparison with any existing models because previous work is based on an already implemented system or does not apply attack vector information directly to the model.
Conclusion {#sec:org03d22dc}
==========
In this paper we propose a method and implement a tool to support this method, CYBOK, that is able to find associated attack vectors given a sufficient system model. CYBOK provides flexibility in modifying the system model to represent different design solutions that implement the same desired behaviors. Therefore, moving security analysis earlier in the systems lifecycle—particularly at the design phase—and, therefore, building systems with security by design. Two important metrics are used for assessing a systems security posture; the attack surface and exploit chains. The results of this method and toolkit is illustrated and evaluated using a UAS—an important area for secure system design because exploits can cause hazardous behavior.
As a final observation we note the experience of using a systematic, model-driven process to conduct attack vector analysis often yields more information than just quantifying the vulnerability aspects of the system. The process itself is an iterative learning experience, allowing circumspection into how a system behaves in response to potential exploits.
Acknowledgments
===============
This material is based upon work supported in part by the Center for Complex Systems and Enterprises at the Stevens Institute of Technology and in part by the United States Department of Defense through the Systems Engineering Research Center (SERC) under Contract HQ0034-13-D-0004. SERC is a federally funded University Affiliated Research Center managed by Stevens Institute of Technology. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Department of Defense.
[^1]: We assume that a component with no recorded attacks is less susceptible to exploitation over one that has a large number of recorded attacks.
[^2]: The model is publicly distributed [@cybok].
|
---
abstract: 'First-principles density-functional calculations have been performed for zinc monochalcogenides with zinc-blende- and wurtzite-type structures. It is shown that the local-density approximation underestimates the band gap, misplaces the energy levels of the Zn-3$d$ states, and overestimates the crystal-field splitting energy. Without spin-orbit coupling, the order of the states at the top of VB is found to be normal for all the Zn$X$ phases considered. Upon inclusion of the spin-orbit coupling in calculations, ZnO in zinc-blende- and wurtzite-type phases become anomalous. It is shown that the Zn-3$d$ electrons are responsible for the anomalous order. The effective masses of electrons and holes have been calculated and found that holes are much anisotropic and heavier than the electrons in agreement with experimental findings. The typical errors in calculated band gaps and related parameters originate from strong Coulomb correlations, which are found to be highly significant in ZnO. The LDA+$U$ approach is found to correct the strong correlation of the Zn-3$d$ electrons, and thus improves the agreement with the experimentally established location of the Zn-3$d$ levels. Consequently, it increases significantly the parameters underestimated in the pure LDA calculations.'
author:
- 'S. Zh. Karazhanov'
- 'P. Ravindran'
- 'U. Grossner'
- 'A. Kjekhus'
- 'H. Fjellv[å]{}g'
- 'B. G. Svensson'
title: 'Electronic structure and band parameters for Zn$X$ ($X$=O, S, Se, Te)'
---
\[intro\] Introduction
======================
Investigation of properties of zinc monochalcogenides \[Zn$X$ ($X$=O, S, Se, Te) with zinc-blende-(z-) and wurtzite-(w-)type structure\] has promoted much interest because of their numerous applications in optoelectronic devices such as visual displays, high-density optical memories, transparent conductors, solid-state laser devices, photodetectors, solar cells etc. Despite frequent studies over more than four decades many fundamental questions still remain open. One of them is the eigenvalue problem related to a severe underestimation of the band gaps ($E_g$), energy levels of the Zn-3$d$ electrons ($E_d$) (for ZnO \[\], ZnS \[\], and ZnSe \[\]), splitting energies of the states at the top of the valence band (VB), spin-orbit (SO) coupling and crystal-field (CF) splitting energies ($\Delta_{\rm{SO}}$ and $\Delta_{\rm{CF}}$, respectively) calculated according to the density-functional theory (DFT) within the local-density approximation (LDA). These problems have been the subject of numerous studies with different methods such as the LDA plus self-interaction correction (LDA+$SIC$) \[\], LDA plus the multiorbital mean-field Hubbard potential (LDA+$U$) \[\] (which includes the on-site Coulomb interaction in the LDA Hamiltonian), and different version of GW approximation \[\]. In the latter approximation “G” stands for one-particle Green’s function as derived from many-body perturbation theory and “W” for Coulomb screened interactions. This approach takes into account both non-locality and energy-dependent features of correlations in many-body system. However, none of the above approaches has been able to remedy the eigenvalue problem except the combination of exact-exchange DFT calculations in the optimized-effective-potential approach with GW \[\], which is found to give better agreement with the experimental band gaps and the location of the Zn-3$d$ energy levels.
The order of the states at the top of the VB in ZnO-w is one of the topics which is still frequently debated \[\]. Another debated aspect is the effective masses of the charge carriers although these are more indefinite parameters for the Zn$X$ phases. At present the effective masses from different *ab initio* packages and experiments scatter appreciably in publications on ZnO \[\] and ZnTe \[\]. In this work Zn$X$ ($X$=O, S, Se, Te) in z- and w-type structural arrangements is studied by first-principles calculations within the LDA, generalized gradient approximation (GGA), and LDA+$U$ approaches. See Ref. \[\] for a more detailed account of our findings.
Computational details
=====================
The electronic band structure of the Zn$X$ phases is studied using the VASP–PAW package \[\], which calculates the Kohn–Sham eigenvalues by the DFT \[\] within the LDA \[\], GGA \[\], and LDA+$U$ \[\] approximations. The exchange and correlation energy per electron have been described by the Perdew–Zunger parametrization \[\] of the quantum Monte Carlo procedure of Ceperley–Alder \[\]. The interaction between electrons and atomic cores is described by means of non-norm-conserving pseudopotentials implemented in the VASP package, which are generated in accordance to the projector-augmented-wave (PAW) method \[\]. Self-consistent calculations were performed using a $10\times
10\times 10$ mesh frame according to Monkhorst–Pack scheme for z-type phases and to $\Gamma$-centered grids for w-type phases. The completely filled semicore-Zn-3$d$ states have been considered as valence states. Values for $\Delta_{\rm CF}$, $\Delta_{\rm
SO}$, and the average band gap $E_0$ are calculated within the quasi-cubic model \[\]. For band-structure calculations we used the experimentally determined crystal-structure parameters for all Zn$X$ phases considered.
Without SO coupling the top of the VB for phases with w-type structure is split into a doublet $\Gamma_5$ and a singlet $\Gamma_1$ state by the crystal-field. Inclusion of SO coupling gives rise to three twofold degenerate valence bands, which are denoted as $A$, $B$, and $C$ states. The symmetry of two of these three bands are of $\Gamma_7$ character and one of $\Gamma_9$ character. Band gaps $E_g, E_A, E_B$, and $E_C$ are defined as the difference between the conduction band (CB) minimum and energy levels of the $A$, $B$, and $C$ states, respectively. Without SO coupling the VB spectrum near the $\Gamma$ point for the Zn$X$-z phases originates from the sixfold degenerate $\Gamma_{15}$ state. The SO interaction splits the $\Gamma_{15}$ level into a fourfold degenerate $\Gamma_8$ and doubly degenerate $\Gamma_7$ levels.
\[Results\] Results and discussion
==================================
\[LDA+U\] Value of the parameters $U$ and $J$
---------------------------------------------
The simplified rotationally invariant LDA$+U$ approach \[\] used in this work requires knowledge of the values of the parameters $U$ and $J$. Since these parameters do not explicitly take into account the final state, values of $U$ and $J$ were found empirically from LDA+$U$ band structure calculations as a function of $U$ and $J$ such that the value of $E_d$ obtained for particular $U$ and $J$ fit with the experimentally determined location. For comparison, the values of $U$ and $J$ have been calculated for some of the compounds within the constrain DFT \[\], showing that the calculated values to some extent agrees with those extracted semiempirically.
\[eigenvalue\] Eigenvalues
--------------------------
The results of the band structure calculations are listed in Table \[band-gap\]. It is found that the band gaps ($E_g, E_A,
E_B, E_C$) and the mean energy level $E_d$ severely underestimated in the LDA calculations. The error in the LDA calculated energies is quite pronounced for ZnO (see e.g. Fig. \[zZnO+disp\] for ZnO-z) compared to the other Zn$X$ phases and the discrepancy exceeds the usual error for LDA calculations. Furthermore, compared to experimental data the CF splitting energy ($\Delta_{\rm CF}$) is severely overestimated for ZnO-w by 2.4 times and underestimated for ZnS-w by around 1.2 times. $\Delta_{\rm SO}$ is overestimated for ZnO-w to about 12.2 times, underestimated for ZnS-w by about 3.2 times, but agrees well with experimental data for ZnSe-z and ZnTe-z. Furthermore, $\Delta_{\rm
SO}$ is found to be negative for ZnO-z and -w phases in agreement with results of Refs. , and , while it is positive for the other Zn$X$ considered.
[llddddddddd]{} & & & & & & & & & &\
ZnO-w & [LDA]{} & 0.7442 & 0.7238 & 0.7561 & 0.8385 & 0.773 & \~5.00 & 0.0951 & 0.0928 &-0.0429\
& [GGA]{} & 0.8044 & 0.7831 & 0.8165 & 0.9000 & 0.833 & \~5.00 & 0.0967 & 0.0944 &-0.0443\
& [LDA+$U$]{} & 1.9884 & 2.0080 & 2.0528 & 2.0530 & 2.038 & \~10.00 & & &\
& Expt. \[\] & & 3.4410 & 3.4434 & 3.4817 & 3.455 & & & 0.0394 &-0.0035\
ZnS-w & [LDA]{} & 1.9896 & 1.9681 & 1.9947 & 2.0734 & 2.012 & \~6.50 & 0.0688 & 0.0518 & 0.0269\
& [GGA]{} & 2.2322 & 2.2114 & 2.2361 & 2.3104 & 2.253 & \~6.00 & 0.0661 & 0.0494 & 0.0249\
& [LDA+$U$]{} & 2.2828 & 2.2595 & 2.2858 & 2.3662 & 2.304 & \~8.20 & 0.0588 & 0.0549 & 0.0256\
&Expt. \[\] & & 3.8643 & 3.8932 & 3.9808 & & & & 0.0580 & 0.0860\
&Expt. \[\] & & 3.8715 & 3.8998 & & & & & 0.0055 & 0.0920\
ZnSe-w & [LDA]{} & 1.0704 & 0.9389 & 1.0080 & 1.3789 & 1.109 & \~6.50 & 0.1138 & 0.3243 & 0.0467\
& [GGA]{} & 1.3271 & 1.2004 & 1.2678 & 1.6243 & 1.364 & \~6.50 & 0.1122 & 0.3105 & 0.0460\
& [LDA+$U$]{} & 1.4039 & 1.2708 & 1.3336 & 1.7209 & 1.442 & \~9.30 & 0.1010 & 0.3465 & 0.0408\
&Expt. \[\]& & 2.8600 & 2.8760 & 2.9260 & & \~9.20 & & &\
ZnTe-w & [LDA]{} & 1.0519 & 0.7600 & 0.8200 & 1.6912 & 1.091 & \~7.50 & 0.0857 & 0.8383 & 0.0332\
& [GGA]{} & 1.2577 & 0.9743 & 1.0319 & 1.8754 & 1.294 & \~7.20 & 0.0835 & 0.8115 & 0.0320\
& [LDA+$U$]{} & 1.2826 & 0.9897 & 1.0434 & 1.8818 & 1.305 & \~9.50 & 0.0754 & 0.8088 & 0.0296\
ZnO-z & [LDA]{} & 0.5732 & 0.5552 & & 0.5883 & 0.577 & \~4.60 & & &-0.0331\
& [GGA]{} & 0.6409 & 0.6152 & & 0.6494 & 0.638 & \~4.60 & & &-0.0342\
& [LDA+$U$]{} & 1.4859 & 1.4953 & & 1.4973 & 1.496 & \~7.90 & & & 0.0020\
ZnS-z & [LDA]{} & 1.8745 & 1.8516 & & 1.9158 & 1.873 & \~6.10 & & & 0.0642\
& [GGA]{} & 2.1134 & 2.0921 & & 2.1513 & 2.112 & \~6.00 & & & 0.0592\
& [LDA+$U$]{} & 2.3324 & 2.3097 & & 2.3886 & 2.336 & \~9.00 & & & 0.0789\
&Expt. \[\] & & 3.6800 & & 3.7400 & & 9.00 & & & 0.0670\
&Expt. \[\] & & 3.7800 & & 3.8500 & & & & &\
ZnSe-z & [LDA]{} & 1.0793 & 0.9484 & & 1.3409 & 1.079 & \~6.60 & & & 0.3925\
& [GGA]{} & 1.3349 & 1.2089 & & 1.5858 & 1.335 & \~6.50 & & & 0.3769\
& [LDA+$U$]{} & 1.4214 & 1.2908 & & 1.6995 & 1.427 & \~9.05 & & & 0.4087\
&Expt. \[\] & & 2.7000 & & & & \~9.20 & & & 0.4000\
&Expt. \[\] & & 2.8200 & & & & & & & 0.4003\
ZnTe-z & [LDA]{} & 1.0607 & 0.7715 & & 1.6681 & 1.070 & \~7.10 & & & 0.8966\
& [GGA]{} & 1.2671 & 0.9862 & & 1.8533 & 1.275 & \~7.05 & & & 0.8671\
& [LDA+$U$]{} & 1.3287 & 1.0456 & & 1.9561 & 1.349 & \~9.90 & & & 0.9105\
&Expt. \[\] & & 2.3941 & & & & \~9.84 & & & 0.9700\
&Expt. \[\] & & & & & & \~10.30 & & &\
\[band-gap\]
![Band dispersion for ZnO-z calculated within LDA (solid lines) and LDA+$U$ (dotted lines) approaches. The Fermi level is set at zero of energy.[]{data-label="zZnO+disp"}](zZnO+disp.eps)
The GGA approach corrected the above mentioned LDA deficiency only to a little extent. However, LDA+$U$ significantly increased the values of $E_g, E_A, E_B$, and $E_C$, and shifted the energy levels of the Zn-3$d$ electrons up to experimentally determined limits, but only slightly changed the values of $\Delta_{\rm SO}$ and $\Delta_{\rm CF}$ for the Zn$X$ phases except ZnO-z and -w. The dependence of $\Delta_{\rm CF}$, $\Delta_{\rm SO}$ on $U$ for ZnO-w is displayed in Fig. \[ZnO+CF+SO+ABC\](a) and of $\Delta_{\rm SO}$ for ZnO-z in Fig. \[ZnO+CF+SO+ABC\](b). Analysis of these illustrations shows that $\Delta_{\rm SO}$ remains negative for $U \leq 9.0$ eV for ZnO-w and for $U \leq
8.0$ eV for ZnO-z. For higher values of $U$, $\Delta_{\rm SO}$ becomes a complex number for ZnO-w with a non-zero imaginary part (in itself physically meaningless), and changes sign for ZnO-z from negative to positive.
The SO splitting energy is much smaller than 1.0 eV for all phases except ZnSe-z ($\Delta_{\rm SO} = 0.4$ eV) and ZnTe-z ($0.97$ eV). Regardless of the computational approach used, the numerical values of $\Delta_{\rm SO}$ for all Zn$X$ phases remained almost unchanged. The present values for $\Delta_{\rm SO}$ for the Zn$X$-z phases are in good agreement with theoretical calculations \[\] by the LAPW method and available experimental data.
![(a) CF splitting, SO coupling energy for ZnO-w, (b) SO coupling energy for ZnO-z, and (c) splitting of the states at the top of VB a function of $U$ for ZnO-w. In panels a and c the solid and dotted lines represent calculated and experimental data, respectively.[]{data-label="ZnO+CF+SO+ABC"}](Graph2.eps)
The variation of the energy splitting between the $A, B,$ and $C$ states, viz. $E_A-E_B, E_A-E_C,$ and $E_A-E_C,$ at the top of the VB of ZnO-w is studied as a function of $U$ as shown in Fig. \[ZnO+CF+SO+ABC\](c). For $U<9.0$ eV the energy splitting decreases with increasing $U$. At higher values of $U, E_A-E_B$ becomes negative and decreases, $E_B-E_C$ increases, while $E_A-E_C$ stays more or less constant.
Density of states
-----------------
The density of states corresponding to the Zn-3$d$ levels according to the LDA calculations are inappropriately close to the CB (see e.g. Fig. \[zZnO+dos\] for ZnO-z), which contradicts the findings from XPS, XES, and UPS experiments \[\]. Furthermore, these states and the top of the VB are hybridized. Distinct from the other Zn$X$ phases considered, ZnO in both z- and w-type structure shows artificially widened Zn-3$d$ states. These findings changed only slightly according to the GGA calculations.
![Total density of states for ZnO-z calculated from the LDA (solid line) and LDA+$U$ (dotted line) approaches. Fermi energy ($E_F$) is set at zero of energy.[]{data-label="zZnO+dos"}](zZnO+dos.eps)
The LDA$+U$ approach \[\] was used to adjust the energy levels of the Zn-3$d$ electrons derived bands to experimentally established positions using semiempirical values for the parameters $U$ and $J$. Consequently, the band gaps calculated by this approach become more reasonable than the pure LDA-derived band gaps. Further, the height of the peaks in the DOS corresponding to the Zn-3$d$ states calculated by the LDA+$U$ becomes much larger than those calculated by the pure LDA. This indicates that the semicore Zn-3$d$ electrons have become more localized than according to the pure LDA. For ZnO-z and -w the width of the Zn-3$d$ band calculated by LDA+$U$ becomes much narrower than that calculated by LDA. For the other Zn$X$ phases, however, LDA+$U$ only slightly changed the width of the Zn-3$d$ bands, which leads one to conclude that the Coulomb correlation effects for ZnO is more pronounced than that in other phases considered.
Order of states at the top of the VB
------------------------------------
Optical and transport properties for semiconductors strongly depend on structure of the topmost VB. The order of states at the top of VB for ZnO-w is one of the topics which is frequently debated \[\]. It is found that the normal order $\Gamma_5 > \Gamma_1$ is obtained by LDA without SO coupling for all Zn$X$ phases. Calculations within GGA did not changed the order. However, the order of the states at the top of VB for the ZnO-w is changed upon using the LDA+$U$ approach with the semiempirical values of the parameter $U$, while there appears no change for the other Zn$X$ phases. For ZnO with SO coupling, the order of states is $\Gamma_7 > \Gamma_9 > \Gamma_7$, which is referred to as *anomalous* order, resulting from a negative SO splitting \[\].
The variation of the structure at the top of the VB on $U$ with and without SO coupling is systematically studied for the ZnO-z- and -w-type phases. It is found that at around the above considered values of $U$ ($U \approx 9.0$ eV for ZnO-w and $U \approx 8.0$ eV for ZnO-z), the LDA+$U$ interchanges the sequence of the VB states from $\Gamma_7 > \Gamma_9 > \Gamma_7$ to $\Gamma_7 > \Gamma_7 > \Gamma_9$.
Since in theoretical calculations one can freeze interaction between the valence electrons and Zn-3$d$ electrons by including the latter into the core, we studied this particular case for ZnO-z (with SO coupling) and -w (without SO coupling). It is found that the the Zn-3$d$ electrons are responsible for the order of the states. On comparing the top of the VB structures calculated within the LDA and GGA approaches it is found that only small quantitative changes have occurred. Hence, inhomogeneities in the electron gas do not affect the order of the states at the top of VB and only slightly changes the band dispersion.
Effective masses
----------------
The effective masses are calculated along $\Gamma \rightarrow
A(\|)$, and $\Gamma \rightarrow M (\bot)$ with and without the SO couplings (Table \[em\]). According to the conventional notations, carrier masses for the Zn$X$-z phases are distinguished by the indices $e$ (electron), $hh$ (heavy-hole), $lh$ (light-hole), and $sh$ (split-off hole). The carrier masses for the Zn$X$-w phases are distinguished by the indices $e$, $A$, $B$, and $C$. The calculated CB masses $m_e$ for the Zn$X$-z phases are more isotropic than those for the Zn$X$-w phases. The numerical values of $m_e$ for ZnO-w, ZnS-w, ZnSe-z, and ZnTe-z obtained by the LDA is underestimated by about 50 $\%$ compared to experimental findings \[\], while those for the other Zn$X$ phases agree fairly well with experimental data. GGA and LDA+$U$ calculations only slightly improved the LDA-derived $m_e$ values for all Zn$X$ phases except ZnO, whereas for ZnO the $m_e$ values calculated from LDA+$U$ gives better agreement with experiment.
The heavy holes along all directions (see Table \[em\]) and light holes along the $\Gamma \rightarrow A(\|)$ direction are much heavier than other holes and electrons. Hence, the carrier transport in Zn$X$ is dominated by electrons, while that by holes can in practice be ruled out. This in turn can be the reason for the experimentally established large disparity between electron and hole mobilities \[\], and also for the large optical nonlinearity in ZnO \[\]. The hole effective masses are more anisotropic than those of electrons. On comparison of the values for $m_e$ in Table \[em\] one sees that the influence of SO coupling on $m_e$ is very important for ZnSe-z and ZnTe-z, while for the other phases its effect is small.
[lldddddddddd]{} & & & & & & & & & & &\
& & \
ZnO-z & [LDA]{} & 0.110 & 0.390 & 0.571 & 0.385 & 1.520 & 1.100 & 1.330 & 0.174 & 0.164 & 0.169\
& [GGA]{} & 0.120 & 0.409 & 0.579 & 0.492 & 1.505 & 1.252 & 1.281 & 0.188 & 0.186 & 0.181\
& [LDA]{}+$U$ & 0.193 & 1.782 & 2.920 & 1.972 & 0.968 & 1.392 & 1.669 & 0.250 & 0.240 & 0.230\
ZnS-z & [LDA]{} & 0.150 & 0.775 & 1.766 & 2.755 & 0.224 & 0.188 & 0.188 & 0.385 & 0.355 & 0.365\
& [GGA]{} & 0.172 & 0.783 & 1.251 & 3.143 & 0.233 & 0.216 & 0.202 & 0.378 & 0.373 & 0.383\
& [LDA]{}+$U$ & 0.176 & 1.023 & 1.227 & 1.687 & 0.268 & 0.252 & 0.218 & 0.512 & 0.445 & 0.447\
&Expt. \[\]& 0.184 & & & 1.760 & & 0.230 & & & &\
&Expt. \[\]& 0.340 & & & & & & & & &\
ZnSe-z& [LDA]{} & 0.077 & 0.564 & 1.310 & 1.924 & 0.104 & 0.100 & 0.094 & 0.250 & 0.246 & 0.254\
& [GGA]{} & 0.098 & 0.568 & 0.922 & 1.901 & 0.126 & 0.122 & 0.111 & 0.271 & 0.273 & 0.267\
& [LDA]{}+$U$ & 0.100 & 0.636 & 1.670 & 1.920 & 0.129 & 0.120 & 0.117 & 0.287 & 0.297 & 0.309\
&Expt. \[\]& 0.130 & 0.570 & 0.750 & & & & & & &\
&Expt. \[\]& 0.170 & & & & & & & & &\
ZnTe-z& [LDA]{} & 0.064 & 0.381 & 0.822 & 1.119 & 0.071 & 0.067 & 0.066 & 0.254 & 0.253 & 0.256\
& [GGA]{} & 0.078 & 0.418 & 0.638 & 1.194 & 0.093 & 0.086 & 0.081 & 0.261 & 0.255 & 0.274\
& [LDA]{}+$U$ & 0.081 & 0.483 & 0.929 & 1.318 & 0.096 & 0.088 & 0.085 & 0.288 & 0.292 & 0.290\
&Expt. \[\]& 0.130 & & 0.600 & & & & & & &\
\
& & & & & & & & & & &\
ZnO-w & [LDA]{} & 0.137 & 0.130 & 2.447 & 2.063 & 2.979 & 0.227 & 0.169 & 0.288 &&\
& [GGA]{} & 0.144 & 0.143 & 2.266 & 0.351 & 3.227 & 0.300 & 0.165 & 0.537 &&\
& [LDA]{}+$U$ & 0.189 & 0.209 & 0.207 &11.401 & 4.330 & 3.111 & 0.330 & 0.270 &&\
&FP-LMTO. \[\]& 0.230 & 0.210 & 2.740 & 0.540 & 3.030 & 0.550 & 0.270 & 1.120 &&\
&Expt. \[\] & 0.24 & & 0.590 & 0.590 & 0.590 & 0.590 & 0.310 & 0.550 &&\
&LCAO. \[\] & 0.280 & 0.320 & 1.980 & 4.310 & & & & &&\
ZnS-w & [LDA]{} & 0.144 & 0.153 & 1.746 & 3.838 & 0.756 & 0.180 & 0.183 & 0.337 &&\
& [GGA]{} & 0.142 & 0.199 & 2.176 & 1.713 & 0.402 & 0.198 & 0.440 & 0.443 &&\
& [LDA]{}+$U$ & 0.138 & 0.157 & 1.785 & 2.194 & 0.621 & 0.195 & 0.339 & 0.303 &&\
&Expt. \[\] & 0.280 & & 1.400 & 0.490 & & & & &&\
&LCAO. \[\] & 0.260 & 0.330 & 1.510 & 1.470 & & & & &&\
ZnSe-w& [LDA]{} & 0.148 & 0.139 & 1.404 & 0.158 & 0.114 & 0.124 & 0.171 & 0.197 &&\
& [GGA]{} & 0.184 & 0.149 & 1.395 & 0.184 & 0.135 & 0.173 & 0.190 & 0.306 &&\
& [LDA]{}+$U$ & 0.185 & 0.149 & 1.629 & 0.189 & 0.137 & 0.187 & & 0.344 &&\
ZnTe-w& [LDA]{} & 0.108 & 0.128 & 1.042 & 0.118 & 0.070 & 0.105 & 0.229 & 0.237 &&\
& [GGA]{} & 0.134 & 0.182 & 1.044 & 0.122 & 0.102 & 0.145 & 0.239 & 0.246 &&\
& [LDA]{}+$U$ & 0.131 & 0.184 & 1.116 & 0.131 & 0.128 & 0.166 & & &&\
&Ref. \[\] & 0.130 & & 0.600 & & & & & &&\
\[em\]
Conclusion
==========
Electronic structure and band parameters for Zn$X$-w and -z phases are studied by first-principles calculations within the LDA, GGA, and LDA+$U$ approaches. It is found that LDA underestimates the band gaps, the actual positions of the energy levels of the Zn-3$d$ states as well as splitting energies between the states at the top of VB, but overestimates the crystal-field splitting energy. The spin-orbit coupling energy is overestimated for ZnO-w, underestimated for ZnS-w, and comes more or less accurate for ZnS-z, ZnSe-z, and ZnTe-z.
The LDA+$U$ calculation as a function of $U$ and $J$ has been used to adjust the Zn-3$d$ band derived from LDA to the experimentally established location of the Zn-3$d$ levels determined from X-ray emission spectra. Using the $U$ values corresponding to the experimentally found $E_d$ location the calculated band gaps and band parameters are improved compared to pure LDA approach.
The order of the states at the top of VB is systematically examined for ZnO-z and -w phases. It is shown that without SO coupling the band order is normal, but it becomes anomalous and $\Delta_{\rm{SO}}$ goes negative upon inclusion of SO coupling. It is found that in the LDA+$U$ calculations the anomalous order is maintained until $U \approx\ 8.0$ eV for ZnO-z and $U \approx\
9.0$ eV for ZnO-w. Above these $U$ values, the band order is inverted and becomes normal and $\Delta_{\rm{CF}}$ for ZnO-w goes from positive to negative, whereas $\Delta_{\rm{SO}}$ converts to a complex quantity. It indicates that either the quasicubic model, within which $\Delta_{\rm{CF}}$ and $\Delta_{\rm{SO}}$ are calculated, do not work for this particular case or such order is unphysical.
Upon excluding the interaction between the Zn-3$d$ and other valence electrons by including the former in the core, the order becomes anomalous. Based on these analyses one can conclude that the Zn-3$d$ electrons are responsible for the anomalous order of the states at the top of VB in ZnO.
Effective masses of electrons at the conduction-band minimum and of holes at the valence-band maximum have been calculated. The heavy holes in the VB are found to be much heavier than the CB electrons. The calculations, moreover, indicate that effective masses of the holes are much more anisotropic than those of the electrons. CB electron masses for ZnO-w, ZnS-w, ZnSe-z, and ZnTe-z calculated within LDA are underestimated by about 50 $\%$ compared with experimental data, while those for the other Zn$X$ phases considered agree with experimental data.
The GGA approach does not remedy the LDA derived error in energy gaps and band parameters. We found that SO coupling is important for calculation of the parameters for the z-ZnSe and z-ZnTe phases, while it is not significant for other Zn$X$ phases. However, the calculated $\Delta_{\rm CF}$ values within the different approaches do not differ much except for ZnO emphasizing that Coulomb correlation effects are more pronounced for ZnO than ZnS, ZnSe, and ZnTe.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has received financial and supercomputing support from the Research Council of Norway. SZK thanks R. Vidya, P. Vajeeston and A. Klaveness (Department of Chemistry, University of Oslo) for discussions and assistance. We also thank Professor M.A. Korotin (Institute of Metal Physics, Ekaterinburg, Russia) for help with the computations of the values of the parameters U and J within the constrain DFT.
[37]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , , **** () .
, , , , **** () .
, , , , , **** () .
, , , **** () .
, , , , , , , , , **** () .
, **** () .
, , , , , **** () .
, , , , , , **** () .
, **** () .
, , , **** () .
, **** () .
, , , , , **** () .
, ed., ** (, ).
, , , , , , ().
, **** () .
, **** () .
, **** () .
, , , **** () .
, , , , , **** () .
, , , , , **** () .
, **** () .
, **** () .
, **** () .
, **** () .
, **** () .
, , , **** () .
** (, ).
, ed., **, vol. (, ).
, , , , , **** () .
, , , , , **** () .
, , , **** () .
, **** () .
, eds., **, vol. of ** (, ).
, , , , eds., ** (, ).
, , , **** () .
, , , **** () .
, , , , , **** () .
, **** () .
, , , **** () .
, , , **** () .
, **** () .
, , , **** () .
, eds., **, vol. (, ).
, **** () .
|
---
abstract: 'The fundamental string length, which is an essential part of string theory, explicitly breaks scale invariance. However, in field theory we demonstrated recently that the gravitational constant, which is directly related to the string length, can be promoted to a dynamical field if the standard model coupled to gravity (SM+GR) is lifted to a locally scale (Weyl) invariant theory. The higher gauge symmetry reveals previously unknown field patches whose inclusion turn the classically conformally invariant SM+GR into a geodesically complete theory with new cosmological and possibly further physical consequences. In this paper this concept is extended to string theory by showing how it can be Weyl lifted with a local scale symmetry acting on target space background fields. In this process the string tension (fundamental string length) is promoted to a dynamical field, in agreement with the parallel developments in field theory. We then propose a string theory in a geodesically complete cosmological stringy background which suggests previously unimagined directions in the stringy exploration of the very early universe.'
author:
- Itzhak Bars
- 'Paul J. Steinhardt'
- Neil Turok
title: |
Dynamical String Tension in String Theory\
with Spacetime Weyl Invariance\
---
Introduction
============
The well known theory of strings propagating in flat or non-trivial backgrounds is not invariant under local scale (Weyl) transformations in *target space* since this formulation of strings contains a dimensionful parameter, namely the string length (equivalently the string tension or slope parameter $\alpha^{\prime}$), which is closely related to the gravitational constant in Einstein’s general relativity. Our goal in this paper is to reformulate string theory with a local scale symmetry in target space without any fundamental lengths so that the string tension $T=\left(
2\pi\alpha^{\prime}\right) ^{-1}$ emerges from gauge fixing a field to a constant value.
The motivation for seeking such a formalism is our recent work [@BST-ConfCosm] on the Weyl invariant formulation of the Standard Model coupled to gravity which led to a classical cosmology [@BST-ConfCosm]-[@BST-sailing] that is geodesically complete across big bang and big crunch singularities [@BST-sailing]. The benefit of introducing the higher gauge symmetry is the inclusion of missing patches of field space and spacetime that are not evident in the conventional formalism. The geodesically complete field space was helpful in discovering new cosmological phenomena especially in the vicinity of the big bang or big crunch type singularities [@BCST1-antigravity]. To better understand the new phenomena further analysis is needed in the context of quantum gravity for which string theory is the leading candidate. Having learned some new lessons in field theory, we are strongly motivated to study the role of Weyl symmetry in *target space* in the context of string theory.
The Weyl invariant formulation of the Standard Model coupled to gravity required the replacement of the gravitational constant by a field with certain special properties. We should expect a parallel treatment of string theory where the constant string tension is replaced by a dynamical field. This is the route we will follow in this paper to introduce the formulation of Weyl invariance in target space in string theory.
One possible application of such a string theory is to investigate strings propagating in geodesically complete cosmological backgrounds that include big crunch and big bang singularities, thus extending our earlier work on cosmology in the context of field theory. We believe that learning about string theory and quantum gravity in such backgrounds is essential for understanding the physics of the very early universe.
Weyl Invariant Low Energy Effective String Theory
=================================================
Strings propagating in background fields such as the gravitational metric $G_{\mu\nu}\left( X\right) $, antisymmetric field $B_{\mu\nu}\left(
X\right) $, dilaton $\Phi\left( X\right) $, etc., are described conventionally with an action that is conformally invariant on the world sheet with the following form up to order $\alpha^{\prime}$ (see Eqs.(3.4.45-3.4.54) in [@GSW]) $$S_{string}=-\frac{1}{4\pi\alpha^{\prime}}\int d^{2}\sigma\left[
\begin{array}
[c]{c}\left( \sqrt{-h}h^{ab}G_{\mu\nu}\left( X\right) +\varepsilon^{ab}B_{\mu\nu
}\left( X\right) \right) \partial_{a}X^{\mu}\partial_{b}X^{\nu}\\
-\alpha^{\prime}\sqrt{-h}R^{\left( 2\right) }\left( h\right) \Phi\left(
X\right) +\cdots
\end{array}
\right] \label{Sstring}$$ where the string represented by $X^{\mu}\left( \sigma^{a}\right) $ is the map from the worldsheet to target space in $d$ dimensions, $h_{ab}\left(
\sigma^{a}\right) $ is the worldsheet metric and $R^{\left( 2\right)
}\left( h\right) $ is its Riemann curvature. The dots $\cdots$ correspond to higher order corrections in powers of the string slope $\alpha^{\prime}$ (to insure the worldsheet conformal symmetry at the quantum level) as well as to additional background fields beyond $\left( G_{\mu\nu},B_{\mu\nu},\Phi\right) $. The slope $\alpha
^{\prime},$ which is inversely proportional to the string tension $T=\left(
2\pi\alpha^{\prime}\right) ^{-1}$, has the dimensions of $\left(
length\right) ^{2}.$
The conformal symmetry on the worldsheet requires at the quantum level that the beta functions corresponding to the couplings $\left( G_{\mu\nu},B_{\mu\nu},\Phi,\cdots\right) $ must vanish. The vanishing of the beta functions correspond to equations of motion for $\left( G_{\mu\nu},B_{\mu\nu
},\Phi,\cdots\right) $ that can be derived from the following effective low energy field theory action in $d$ dimensions,[^1] $$S_{eff}=\frac{1}{2\kappa_{d}^{2}}\int d^{d}x\sqrt{-G}e^{-2\Phi}\left\{
R\left( G\right) +4\left( \partial\Phi\right) ^{2}-\frac{1}{12}H^{2}-\frac{d-26}{3\alpha^{\prime}}+\cdots\right\} ,\label{Seff}$$ where the completely antisymmetric tensor $H_{\mu\nu\lambda}$ is the curl of $B_{\mu\nu}$, $$H_{\mu\nu\lambda}=\partial_{\lbrack\lambda}B_{\mu\nu]}.\label{Hs}$$ In this action the dimensionful gravitational constant parameter $\kappa
_{d}^{2}$, which has dimension $\left( length\right) ^{d-2}$, is proportional to a power of the string slope parameter $\left( \alpha^{\prime
}\right) ^{\left( d-2\right) /2}.$ The proportionality constant can be absorbed into a redefinition of the dimensionless string coupling constant $g_{s}=e^{a}$ which emerges from a constant shift of the dimensionless dilaton field $\Phi\left( X\right) \rightarrow\Phi\left( X\right) +a$ as seen in (\[Seff\]). By adopting an appropriate definition of a shifted $\Phi$, we can take $$\kappa_{d}^{2}=\left( 2\alpha^{\prime}\right) ^{\left( d-2\right)
/2}.\label{kappa}$$ This choice is convenient since string computations are often done by using units in which $2\alpha^{\prime}=1$ which then sets units with $\kappa_{d}=1$ in every dimension.
We now lift the effective string theory action to a Weyl invariant version by replacing the dimensionful $\kappa_{d}$ (equivalently $\alpha^{\prime}$) by the expectation value of a field. This step for the effective action will provide us with the hints for how to lift the string action itself (\[Sstring\]) to a Weyl invariant new version of string theory, which is our ultimate goal in this paper. We are guided by our recent work in [@BST-ConfCosm] of lifting any field theory coupled to gravity to a Weyl invariant version by replacing the gravitational constant $\kappa_{d}$ with a field. This process involves an apparent ghost-like scalar field $\phi\left(
x\right) $ with the wrong sign kinetic energy. Since this field is compensated by the local scale symmetry (Weyl) there is really no ghost and no new degree of freedom associated with the extra field. The benefit of introducing the higher symmetry together with the extra field was discussed in our papers. Namely, we found that the previous field space is extended to a geodesically complete spacetime by the inclusion of additional field patches that were missing since those are not evident in the conventional geodesically incomplete formulation of gravity in the Einstein frame. Having learned this lesson in field theory, we extend the idea now first to the effective string theory $S_{eff}$ and next to the string action $S_{string}$ to formulate a Weyl invariant version of string theory including background gravitational and other fields.
We begin with a generalization of the Weyl lifting methods in [@BST-ConfCosm] to $d$ dimensions. Instead of the dilaton $\Phi$ we now have two scalar fields $\phi^{i}=\left( \phi,s\right) $ (one combination is a gauge degree of freedom) and in addition to the metric $g_{\mu\nu}$ we include the antisymmetric field $b_{\mu\nu}$ in a Weyl invariant action $S=\int d^{d}x\mathcal{L}\left( x\right) $ given by $$\mathcal{L}\left( x\right) =\sqrt{-g}\left\{
\begin{array}
[c]{c}\frac{d-2}{8\left( d-1\right) }U\left( \phi^{k}\right) \left[ R\left(
g\right) -\frac{1}{12}H^{2}\left( b,\phi^{k}\right) \right] \\
-\frac{1}{2}C_{ij}\left( \phi^{k}\right) g^{\mu\nu}\partial_{\mu}\phi
^{i}\partial_{\nu}\phi^{j}-V\left( \phi^{k}\right)
\end{array}
\right\} \label{actionWinv}$$ where $H_{\mu\nu\lambda}$ is$$H_{\mu\nu\lambda}\left( b,\phi^{k}\right) \equiv\partial_{\lbrack\lambda
}b_{\mu\nu]}+ T\left( \phi^{k}\right) ^{-1} b_{[\mu\nu}\partial_{\lambda
]}T\left( \phi^{k}\right) , \label{HT}$$ where, as we will see, $T\left( \phi^{k}\right) $ will play the role of the dynamical string tension, although at this stage there is no apparent connection to it. This *modified* $H_{\mu\nu\lambda}$ is constructed to be invariant, $\delta_{\Lambda}H_{\mu\nu\lambda}=0$ for any $T\left(
\phi\right) ,$ under the following modified gauge transformation of the antisymmetric tensor $$\delta_{\Lambda}b_{\mu\nu}=\partial_{\lbrack\mu}\Lambda_{\nu]}+T\left(
\phi^{k}\right) ^{-1} \Lambda_{\lbrack\nu}\partial_{\mu]}T\left( \phi
^{k}\right) ,$$ where the gauge parameter is the vector $\Lambda_{\mu}\left( x\right) .$
For *any number of scalars* $\phi^{i},$ requiring this action to be also invariant under the local scale (Weyl) transformations in $d$ dimensions,$$\left( g_{\mu\nu},b_{\mu\nu}\right) \rightarrow\Omega^{-\frac{4}{d-2}}\left( g_{\mu\nu},b_{\mu\nu}\right) ,\text{ and ~}\phi^{i}\rightarrow
\Omega\phi^{i}, \label{WeylTransf}$$ imposes the following homothety conditions on the metric $C_{ij}\left(
\phi\right) $ in field space [@IB-2TSugra][@BST-ConfCosm]$$\frac{\partial U}{\partial\phi^{i}}=-2C_{ij}\phi^{i},\;C_{ij}\phi^{i}\phi
^{j}=-U, \label{homothety}$$ and also demands that $\left( C_{ij},U,V,T\right) $ be homogeneous functions of $\phi^{i}$ with homogeneity degrees $\left( 0,2,\frac{2d}{d-2},\frac
{4}{d-2}\right) $ respectively, namely $$\begin{array}
[c]{c}C_{ij}\left( \Omega\phi^{k}\right) =C_{ij}\left( \phi^{k}\right)
,\;U\left( \Omega\phi^{k}\right) =\Omega^{2}U\left( \phi^{k}\right) ,\\
V\left( \Omega\phi^{k}\right) =\Omega^{\frac{2d}{d-2}}U\left( \phi
^{k}\right) ,\;T\left( \Omega\phi^{k}\right) =\Omega^{\frac{4}{d-2}}T\left( \phi^{k}\right) .
\end{array}
\label{homogeneous}$$ Then $H_{\mu\nu\lambda}$ is covariant under local scale transformations, $H_{\mu\nu\lambda}\left( \Omega^{-\frac{4}{d-2}}b,\Omega\phi^{k}\right)
=\Omega^{-\frac{4}{d-2}}H_{\mu\nu\lambda}\left( b,\phi^{k}\right) ,$ for any homogeneous $T$ as indicated above. When $\left( C_{ij},U,V,T\right) $ satisfy the conditions (\[homothety\],\[homogeneous\]) one can verify that the Lagrangian (\[actionWinv\]) transforms into a total derivative under the local scale transformations, hence the action is invariant.
The general solution of the homothety and homogeneity conditions (\[homothety\],\[homogeneous\]) for any number of scalars $\phi^{i}$, and in particular for two scalars $\left( \phi,s\right) $, is given in [@BST-ConfCosm]. The general solution shows that there remains a lot of freedom in the choice of the functions $\left( C_{ij},U,V,T\right) $ to construct various models that are invariant under the local scale symmetry. However, for our case, we will be able to fix all of these functions so that the string effective action $S_{eff}$ in (\[Seff\]) emerges when we fix a gauge for the Weyl gauge symmetry. The gauge of interest here was called the string gauge or s-gauge in [@BCT-cyclic], where other useful gauges that will also be useful here were discussed (E-gauge, c-gauge, $\gamma$-gauge).
We now turn to our case of only two scalars $\phi^{i}=\left( \phi,s\right)
$. By doing field reparametrizations of the $\phi^{i}$ it is always possible to transform to a basis in which the metric in scalar field space is conformally flat and of indefinite signature, $ds^{2}=C_{ij}d\phi^{i}d\phi
^{j}$ $\Rightarrow A^{2}\left( \phi,s\right) \left( -d\phi^{2}+ds^{2}\right) .$ In this basis, the homogeneity and homothety conditions on $U$ and $C_{ij}$ lead to a unique solution for both $U$ and $C_{ij},$ namely $A^{2}=1$ (constant fixed by normalization) and $$U=\left( \phi^{2}-s^{2}\right) .$$ The indefinite metric $C_{ij}$ in field space and the related relative minus sign in $U$ emerge from physical considerations. It is necessary to allow for a region or a patch of field space $\left( \phi,s\right) $ where the effective gravitational constant $U\left( \phi,s\right) $ is positive in order to recover ordinary gravity in the Einstein frame when a gauge is fixed. In the solution for $C_{ij}$ given by $ds^{2}=\left( -d\phi^{2}+ds^{2}\right) $, the field $\phi$ is a ghost because it has the wrong sign kinetic energy term in the Lagrangian. If $\phi$ had positive kinetic energy (like $s$) then $U$ would come out to be purely negative $\left( -\phi
^{2}-s^{2}\right) $ which would imply an unacceptable purely negative gravitational constant in the action (\[actionWinv\]). This is why there has to be a relative minus sign and therefore there has to be a ghost in the Weyl invariant formulation[^2]. However, the local scale symmetry in the action (\[actionWinv\]) is just sufficient to remove this ghost by a gauge choice, so this ghost is not dangerous for unitarity and therefore it is of no concern.
Hence, for two scalars, using the field basis described above, the Weyl invariant effective action (\[actionWinv\]) becomes$$\mathcal{L}\left( x\right) =\sqrt{-g}\left[
\begin{array}
[c]{c}\frac{d-2}{8\left( d-1\right) }\left( \phi^{2}-s^{2}\right) \left(
R\left( g\right) -\frac{1}{12}H^{2}\left( b,\phi,s\right) \right) \\
+\frac{1}{2}\partial\phi\cdot\partial\phi-\frac{1}{2}\partial s\cdot\partial
s-V\left( \phi,s\right)
\end{array}
\right] \label{actionWinv2}$$ The reader can recognize that in this basis $\left( \phi,s\right) $ are both conformally coupled scalars, thus making the Weyl symmetry more evident. There remains two unknown functions, $V\left( \phi,s\right) $, $T\left(
\phi,s\right) ,$ that are constrained by homogeneity as in (\[homogeneous\]).
We now want to show how to recover the low energy effective string action $S_{eff}$ in Eq.(\[Seff\]) by choosing the so called string gauge for the local gauge symmetry. In this gauge we label all fields with an extra label $s$ to indicate that we are in the string gauge (we will be interested also in other gauge choices labeled by different symbols, $E,c,\gamma,f$). Thus, in the field patches that satisfy $\left(
\phi^{2}-s^{2}\right) >0,$ which we call the $\pm$ gravity patches, we parametrize the fields $\phi,s$ in terms of a single scalar field $\Phi$ (one degree of freedom is lost in the fixed gauge) and identify the metric and antisymmetric tensor in the s-gauge with the string backgrounds $G_{\mu\nu
},B_{\mu\nu}$ that appear in $S_{eff}$ and $S_{string}$$$\begin{array}
[c]{c}g_{\mu\nu}^{s}=G_{\mu\nu},\;b_{\mu\nu}^{s}=B_{\mu\nu},\;\\
\phi_{s}=\pm\sqrt{\frac{4\left( d-1\right) }{\left( d-2\right) }}\frac{e^{-\Phi}}{\kappa_{d}}\cosh\frac{\Phi}{\sqrt{d-1}},\;~\\
s_{s}=\pm\sqrt{\frac{4\left( d-1\right) }{\left( d-2\right) }}\frac{e^{-\Phi}}{\kappa_{d}}\sinh\frac{\Phi}{\sqrt{d-1}}.
\end{array}
\label{s-gauge}$$ where the $\pm$ is needed to cover all regions of field space $\left(
\phi,s\right) $ in the $\pm$ gravity patches in the $\left( \phi,s\right) $ plane as seen in Fig. 1. It is also possible to replace $s_{s}$ by $-s_{s}$ in Eq.(\[s-gauge\]) for another parametrization of the gauge choice as we will discuss below. Although the $\pm$ signs cancel out in the Lagrangian, it does not cancel out in the equations of motion, so solutions need to be continuous as $\phi$ or $s$ vanish and change sign. The s-gauge in (\[s-gauge\]) corresponds to a pair of curves in the $\left( \phi,s\right) $ plane where points on the curves are parametrized by $\Phi$ according to $\left( \phi
_{s}\left( \Phi\right) ,s_{s}\left( \Phi\right) \right) $ as shown in Fig. 1.
\[tbh\]
[Fig1-WeylInvariantString.eps]{}
If $s_{s}$ is replaced by $-s_{s}$ in (\[s-gauge\]) we get a new set of curves in the $\pm$ gravity regions that are similar to the pair of curves in Fig. 1, but mirror-reflected through the horizontal axis ($s\rightarrow-s$).
Later we will discuss the possibility of including patches in which $\left(
\phi^{2}-s^{2}\right) $ is negative by simply interchanging $cosh$ and $sinh$ that modifies the gauge choice above leading to curves that look like 90$^{o}$ rotated in relation to those described above. These extend the domain of the string gauge into patches of antigravity (negative gravitational constant, as seen from (\[actionWinv2\])) which are required for geodesically complete cosmological spacetimes as discussed in a similar setting in [@BCT-cyclic]-[@BST-sailing]. The existence of such additional patches is not evident in the effective string action (\[Seff\]), but the study of particle geodesics shows that the spacetime described by (\[Seff\]) is geodesically incomplete, and this is a hint that invites a study of the complete space by including antigravity patches, as discussed below.
With this choice of s-gauge in the gravity patch we see that the coefficients of the curvature $R$ in (\[actionWinv2\]) and (\[Seff\]) match each other, and furthermore the kinetic energy terms of $\left( \phi_{s},s_{s}\right) $ in (\[actionWinv2\]) precisely reproduce the kinetic energy term of the dilaton $\Phi$ in (\[Seff\]), with its unusual normalization, $\sqrt
{-G}\frac{e^{-2\Phi}}{2\kappa_{d}^{2}}4\left( \partial\Phi\right) ^{2}$. In order to have also $H_{\mu\nu\lambda}^{s}$ match the expression in (\[Hs\]), it is necessary to determine the homogeneous function $T\left( \phi,s\right)
$ such that in the string gauge (\[s-gauge\]) it reduces to a constant as a function of $\Phi,$ namely $T\left( \phi_{s}\left( \Phi\right)
,s_{s}\left( \Phi\right) \right) $=constant with either set of $\pm$ signs in (\[s-gauge\]). We find that before gauge fixing, and in the gravity patch $\left( \phi^{2}-s^{2}\right) >0,$ the homogeneous $T\left( \phi,s\right)
$ of degree $4/\left( d-2\right) $ must be given by $$T\left( \phi,s\right) =\left( \frac{d-2}{4\left( d-1\right) }\right)
^{\frac{2}{d-2}}\left( \phi+s\right) ^{2\frac{1+\sqrt{d-1}}{d-2}}\left(
\phi-s\right) ^{2\frac{1-\sqrt{d-1}}{d-2}},\label{T}$$ or the same expression with $s$ replaced by $-s$ in Eq.(\[T\]) as well as Eq.(\[s-gauge\]). This $T\left( \phi,s\right) $ is normalized such that it reduces to the constant string tension in the gravity patches in the s-gauge given above (recall (\[kappa\])) $$\pi T\left( \phi_{s}\left( \Phi\right) ,s_{s}\left( \Phi\right) \right)
=\left( \kappa_{d}^{2}\right) ^{-\frac{2}{d-2}}=\frac{1}{2\alpha^{\prime}}~.$$ As already mentioned, in antigravity patches where $\left( \phi^{2}-s^{2}\right) <0$ we must exchange $cosh$ and $sinh$ in equation (\[s-gauge\]). We may then choose signs in (\[s-gauge\]) to ensure that the string tension $T$ as defined in (\[T\]) behaves smoothly across the boundary between gravity and antigravity regions. Notice that the overall constant or sign in front of the tension is irrelevant at this point since only $T^{-1}\partial_{\lambda}T$ (which is insensitive to the sign of $T$) appears in $H_{\mu\nu\lambda}$.
It is notable that in the critical dimension $d=10$ for superstrings or heterotic strings, we may choose signs so that the power of $\phi+s$ (or $\phi-s$) appearing in the dynamical string tension (\[T\]) is unity, for both the gravity (or antigravity) patches. This means that, in a suitable Weyl gauge, smooth cosmological transitions from gravity to antigravity, or vice versa, are associated with a smooth analytic sign change of the dynamical string tension $T$. Such transitions do in fact occur in generic homogeneous cosmologies [@BCST2-solutions], in a suitable Weyl gauge, termed $\gamma
$-gauge, which will shortly be explained. We emphasize that the signs of $\left( \phi+s\right) $ and $\left( \phi-s\right) $ are Weyl gauge invariants, so their signs in every gauge are the same as those displayed in the geodesically complete $\gamma$-gauge solution displayed below.
Finally, the potential energy that matches $V\left( \phi_{s},s_{s}\right)
=\frac{d-26}{3\alpha^{\prime}}\frac{e^{-2\Phi}}{2\kappa_{d}^{2}},$ is also determined such that before gauge fixing $V\left( \phi,s\right) $ is given by $$V\left( \phi,s\right) =\frac{\left( d-26\right) \left( d-2\right)
}{12\left( d-1\right) }\left( \phi^{2}-s^{2}\right) T\left(
\phi,s\right) . \label{V}$$ This $V\left( \phi,s\right) $ is homogeneous of degree $\frac{2d}{d-2}$ as required in (\[homogeneous\]), it vanishes for $d=26$ and is negative for $d<26$ in the gravity patch. It is expected that higher perturbative terms in $\alpha^{\prime}$, non-perturbative, and string-string interaction effects will alter the effective low energy potential. Furthermore in supersymmetric or heterotic versions of string theory $V$ is not the same as above (e.g. $\left( d-26\right) $ is replaced by $\left( d-10\right) $ in SUSY). In any case it is evident that the final phenomenologically significant result for $V\left( \phi,s\right) $ can always be rewritten in a Weyl covariant form as, $V=\phi^{\frac{2d}{d-2}}f\left( s/\phi\right) ,$ for some Weyl invariant $f\left( s/\phi\right) $ as in our previous work.
By determining the functions $T\left( \phi,s\right) $ and $V\left(
\phi,s\right) $ as in (\[T\],\[V\]), we have completely fixed the effective low energy string action as a Weyl invariant action in the gravity patch $\left( \phi^{2}-s^{2}\right) \geq0$ in the form given in Eqs.(\[actionWinv2\],\[HT\]). A significant property of the new Weyl invariant effective action is that it becomes geodesically complete by including all field patches in the infinite $\left( \phi,s\right) $ plane in the sense discussed in our previous papers and later in this paper.
To complete the definition of the geodesically complete string action in this sense we need to weigh carefully the possibilities of whether the smooth continuation of the functions $T\left( \phi,s\right) $ and $V\left(
\phi,s\right) $ to all patches will change sign when crossing the transition lines $\left\vert \phi\right\vert =\left\vert s\right\vert $ in the $\left(
\phi,s\right) $ plane which separate gravity/antigravity patches. This will be discussed below in the context of the string action with a dynamical tension $T\left( \phi,s\right) $ after a description of geodesic completeness across gravity/antigravity boundaries given below.
Gauge choices for local scale symmetry and geodesic completeness
----------------------------------------------------------------
The Weyl invariant action (\[actionWinv2\]) has no dimensionful constants of any kind. Besides the string gauge given in (\[s-gauge\]) where the gravitational constant (or the string tension) is introduced as the gauge fixed value of a combination of fields, the same action can be gauge fixed to other gauges that are useful for various physical applications especially in cosmology as discussed in [@BC-inflation]-[@BST-sailing]. Some of these gauges will be useful in string theory as well, so we review here some convenient gauge choices of the geodesically complete Weyl invariant action (\[actionWinv2\]), and then describe the nature of geodesic completeness in each gauge.
- **s-gauge**: This is the string gauge we used above in Eq.(\[s-gauge\]). A property of this gauge is that it satisfies $\left\vert \phi_{-s}\right\vert
=c_{d}\left\vert \phi_{+s}\right\vert ^{\lambda_{d}}$ where $c_{d},\lambda
_{d}$ are some constants, and $\phi_{\pm s}\equiv\phi_{s}\pm s_{s}$. More precisely, as seen from (\[s-gauge\]), with the choice of $$c_{d}=\left( \frac{\kappa_{d}^{2}\left( d-2\right) }{4\left( d-1\right)
}\right) ^{\frac{1}{\sqrt{d-1}-1}},\;\lambda_{d}=\frac{\sqrt{d-1}+1}{\sqrt{d-1}-1},$$ $\phi_{s}\left( \Phi\right) ,s_{s}\left( \Phi\right) $ are parametrized in the $\pm$ gravity patches in terms of a single field $\Phi$ as in (\[s-gauge\]) so that the Weyl invariant action (\[actionWinv2\]) is reduced to the gauge fixed action that matches the effective string action $S_{eff}$ in (\[Seff\]). Other choices of $\left( c_{d},\lambda_{d}\right)
$ would yield some similarly gauge fixed action, but a different one than (\[Seff\]). The expressions in (\[s-gauge\]) by themselves are insufficient to express the gauge choice in all regions of the $\left(
\phi,s\right) $ plane. These equations must be supplemented by similar expressions that include more branches of the curve in Fig. 1. The additional branches correspond to curves obtained from Fig. 1 by a reflection through the horizontal axis generated by $s\rightarrow-s$ (new curve still in gravity), plus those obtained by a 90$^{o}$ rotations of the curves already mentioned, leading to additional branches in the antigravity patches. Later we will introduce a flip symmetry in Eq.(\[flip\]) which relates to a symmetry that interchanges gravity and antigravity. To express the generic geodesically complete cosmological solution, given in Fig. 2 and Eqs.(\[metric\]-\[phi-s\]) below in the $\gamma$-gauge as a function of conformal time $x^{0}$, we need all of those additional branches in the string gauge basis $\left( \phi_{s}\left(
\Phi\left( x^{0}\right) \right) ,s_{s}\left( \Phi\left( x^{0}\right)
\right) \right) $. Accordingly, the geodesically complete form of Fig. 1 must include connections among all the branches of the curves described above, such that any given branch connects continuously to another branch across the gravity$/$antigravity intersection by passing through the origin in Fig. 1 tangentially to one of the dashed lines while being perpendicular to the other.
- **E-gauge:** This gauge is useful to rewrite the theory in the Einstein frame where the main physical intuition about gravitational phenomena is developed. Starting in the $\pm$ gravity patches, it is defined by gauge fixing $\left( \phi^{2}-s^{2}\right) $ to a positive constant, $\frac{\left( d-2\right) }{8\left( d-1\right) }\left( \phi_{E}^{2}-s_{E}^{2}\right) =\frac{1}{2\tilde{\kappa}_{d}^{2}}.$ For the kinetic term of the remaining scalar (called $\sigma$) to be normalized conventionally, $-\frac{1}{2}\left( \partial\sigma\right) ^{2},$ the fields $\left(
\phi,s\right) $ are parametrized in terms of the field $\sigma$ as follows $$\begin{array}
[c]{c}\phi_{E}\left( \sigma\right) =\pm\sqrt{\frac{4\left( d-1\right) }{\left(
d-2\right) }}\frac{1}{\tilde{\kappa}_{d}}\cosh\left( \frac{\left(
d-2\right) \tilde{\kappa}_{d}}{4\left( d-1\right) }\sigma\right) ,\;~\\
s_{E}\left( \sigma\right) =\left( \pm\text{ or }\mp\right) \sqrt
{\frac{4\left( d-1\right) }{\left( d-2\right) }}\frac{1}{\tilde{\kappa
}_{d}}\sinh\left( \frac{\left( d-2\right) \tilde{\kappa}_{d}}{4\left(
d-1\right) }\sigma\right) ,
\end{array}
\label{E-gauge}$$ The geometrical fields are also labeled with the letter $E:$ $\left(
g_{\mu\nu}^{E},b_{\mu\nu}^{E}\right) .$ In this gauge the tension $T\left(
\phi_{E},s_{E}\right) $ obtained from (\[T\]) is not a constant and therefore $H_{\mu\nu\lambda}^{E}$ as defined in (\[HT\]) has a non-trivial structure[^3]. The E-gauge in the gravity patches alone, as given in (\[E-gauge\]), is geodesically incomplete because one can find classical solutions of the fields in which the gauge invariant quantity $\left( 1-s^{2}/\phi^{2}\right) $ changes sign as a function of time. This happens when both $\left( \phi_{E},s_{E}\right) $ are infinitely large (or $\sigma\rightarrow$large) so that $\left( 1-s_{E}^{2}/\phi_{E}^{2}\right) $ can vanish even though $\left( \phi_{E}^{2}-s_{E}^{2}\right) $ is a constant all the way to the boundary of the gravity/antigravity patches. Hence the gravity patch by itself is incomplete. This is confirmed by noting that particle geodesics reach cosmological singularities in a finite amount of time. Furthermore, solutions of the field equations that extend to the gravity/antigravity boundary (see Fig. 2) show that when the gauge invariant $\left( 1-s_{E}^{2}\left( x\right) /\phi_{E}^{2}\left( x\right) \right)
\rightarrow0$ the spacetime metric $g_{\mu\nu}^{E}\left( x\right) $ has also a curvature singularity in the E-gauge. Hence the transition from gravity to antigravity patches occurs only at spacetime singularities in the Einstein frame. To obtain the gauge fixed expressions for $\left( \phi,s\right) $ in the antigravity patches in the E-gauge the *cosh* and *sinh* in (\[E-gauge\]) are interchanged; then the kinetic term of $\sigma$ also changes sign.
- **c-gauge:** This gauge is useful to relate the Weyl invariant theory to low energy degrees of freedom [@2Tgravity], and was used in the construction and analysis of the standard model coupled to gravity [@BST-ConfCosm][@BST-HiggsCosmo]. It is introduced by gauge fixing the field $\phi$ to a constant, $\phi_{c}\left( x\right) =c,$ and then $s_{c}\left( x\right) $ is the remaining dynamical scalar, while the other fields are labeled as $\left( g_{\mu\nu}^{c},b_{\mu\nu}^{c}\right) $. When $s_{c}$ is much smaller than the constant $\phi_{c},$ which is true at energies much smaller than the Planck scale, the factor $\left( \phi_{c}^{2}-s_{c}^{2}\left( x\right) \right) $ is practically a constant from the perspective of low energy physics. This gauge is geodesically complete because $\left( \phi_{c}^{2}-s_{c}^{2}\left( x\right) \right) $ can change sign dynamically (as seen in explicit solutions, including Fig. 2) when the spacetime $g_{\mu\nu}^{c}$ transits between gravity and antigravity at curvature singularities.
- $\gamma$**-gauge:** This gauge has been the most useful in simplifying equations and leading to analytic results. The determinant of the spacetime metric $g_{\mu\nu}^{\gamma}$ is fixed to be unimodular, $\det\left(
g^{\gamma}\right) =-1$, while the other fields are labeled with $\gamma$ as $\left( \phi_{\gamma},s_{\gamma},b_{\mu\nu}^{\gamma}\right) .$ For example for a cosmological spacetime of the form $ds_{\gamma}^{2}=a_{\gamma}^{2}\left( \tau\right) \left( -d\tau^{2}+ds_{d-1}^{2}\right) $ where $\tau$ is the conformal time, and $ds_{d-1}^{2}$ includes space curvature and anisotropy, the Weyl symmetry is used to gauge fix the scale factor to a constant for all conformal time, $a_{\gamma}\left( \tau\right) =1.$ The $\gamma$-gauge is geodesically complete. It was used in cosmological applications in [@BC-inflation]-[@IB-BibBang] and led to the discovery of the geodesically complete nature of spacetime across big bang and big crunch singularities where the gravity-antigravity transition occurs [@BCST1-antigravity][@BST-sailing].
- $f\left( R\right) $ **gravity gauge**: If we choose $\left(
\phi-s\right) $ to be a constant, $\phi_{f}\left( x\right) -s_{f}\left(
x\right) =c,$ then the kinetic term for the remaining field $\phi_{+}\left(
x\right) \equiv\phi_{f}\left( x\right) +s_{f}\left( x\right) $ drops out of the action (\[actionWinv2\]). Ignoring the $H^{2}$ term in (\[actionWinv2\]), the field $\phi_{+}\left( x\right) $ becomes a purely algebraic field that can be determined via the equations of motion (for any $V$) to be a function of the curvature $\phi_{+}=\phi_{+}\left( R\left(
g^{f}\right) \right) .$ Inserting this solution back into the action (\[actionWinv2\]) reduces it to a function of only $R.$ This is $f\left(
R\right) $ gravity, where the function $f\left( R\right) $ is determined by $V\left( \phi,s\right) $ (see also [@bamba]).
The transformations of fields from one fixed gauge to another is easily obtained by considering gauge invariants under the Weyl transformations. Some useful gauge invariants that may be used for this purpose are $$\frac{s}{\phi},\;\left( \sqrt{-g}\right) ^{\frac{d-2}{2d}}\phi,\;\left(
\sqrt{-g}\right) ^{\frac{d-2}{2d}}s,\;\left\vert T\left( \phi,s\right)
\right\vert ^{-\frac{d-2}{4}}\phi,\;\left\vert T\left( \phi,s\right)
\right\vert ^{-\frac{d-2}{4}}s,\;\text{etc.}$$ To illustrate this, consider the example of the s-gauge versus the E-gauge. By comparing the gauge invariant, $\frac{s_{s}\left( \Phi\right) }{\phi
_{s}\left( \Phi\right) }=\frac{s_{E}\left( \sigma\right) }{\phi_{E}\left(
\sigma\right) },$ we obtain the relation between the dilaton $\Phi\left(
x\right) $ in the s-gauge (\[s-gauge\]) and the scalar field $\sigma\left(
x\right) $ in the E-gauge (\[E-gauge\]). This provides the local scale transformation $\Omega_{sE}\left( x\right) $ that relates the s and E gauges according to Eq.(\[WeylTransf\]), and using it we find the relation between the remaining degrees of freedom, $\left( G_{\mu\nu},B_{\mu\nu}\right)
=\left( \Omega_{sE}\right) ^{-\frac{4}{d-2}}\left( g_{\mu\nu}^{E},b_{\mu
\nu}^{E}\right) $. Thus, solutions of equations of motion in one gauge can be used to obtain solutions in all other gauges via the Weyl transformations among fixed gauges.
Such field transformations are *duality transformations*: in each gauge the form of the action, the spacetime and the dynamics appear different, but the Weyl invariant physical information is identical. Indeed the stringy T-duality of the effective string action discussed in [@Veneziano] is an automatic outcome of the Weyl invariant formalism discussed here. This is because the dualities based on inversion of scale factors in different directions [@Veneziano] can be re-stated as a combination of general coordinate reparametrizations and local Weyl transformations, both of which are symmetries of our low energy string formalism in (\[actionWinv2\]). In our setting here we can construct a richer set of dualities based on Weyl transformations between different gauge fixed versions of the same Weyl invariant effective low energy string theory (equivalently, dual backgrounds for string theory on the worldsheet)[^4].
Having shown that the low energy effective string theory (\[Seff\]) is just a gauge choice of the Weyl-lifted low energy string theory (\[actionWinv2\]), we now are in a position to argue that by allowing the fields to take values in the full $\left( \phi,h\right) $ plane, including the antigravity regions shown in Fig. 1, we obtain a geodesically complete theory. To see this, we solve analytically the cosmological equations of motion that are predicted by (\[actionWinv2\]) as shown in the next paragraph, and find that the dynamics requires that solutions that begin initially anywhere in the gravity patches (such as those shown in Fig. 1) evolve inevitably into the antigravity patches, thus requiring the inclusion of all patches. Then a geodesically complete field space includes all of the $\left( \phi,s\right)
$ plane as well as an extension of the spacetime metric $g_{\mu\nu}\left(
x\right) $ as a function of spacetime $x^{\mu}$ to include the antigravity patches. In this complete field space geodesics (appropriately defined to be Weyl invariant as consistent solutions of the same theory [@BST-sailing]) are complete curves that do not artificially end at cosmological singularities [@BST-sailing]. Hence we have argued that the Weyl invariant low energy string theory (\[actionWinv2\]) which includes all field patches is geodesically complete, at least for spacetimes of interest in cosmology.
The behavior of the general solution in the vicinity of cosmological singularities is unique due to an attractor mechanism discovered in [@BCST1-antigravity]. It can be argued that the generic solution is kinetic energy dominated so that the potential energy $V$ is negligible. Taking also $b_{\mu\nu}=0$ for simplicity, but adding conformally invariant massless fields (treated as radiation $\rho_{r}/a^{4}\left( x^{0}\right) $ ), we display the solution obtained in [@BCST1-antigravity] in four dimensions, $d=4,$ as follows. For our purposes here it is most simply described in the $\gamma$-gauge. The metric $g_{\mu\nu}^{\gamma}$ is diagonal and generically anisotropic near singularities and is parametrized as $$\begin{aligned}
ds^{2} & =a_{\gamma}^{2}\left( -\left( dx^{0}\right) ^{2}+\sum_{i=1}^{3}e_{i}^{2}\left( dx^{i}\right) ^{2}\right) ,\;a_{\gamma}^{2}=1\text{
(}\gamma\text{-gauge}),\label{metric}\\
e_{1}^{2} & =e^{2\left( \alpha_{1}+\sqrt{3}\alpha_{2}\right) },e_{2}^{2}=e^{2\left( \alpha_{1}-\sqrt{3}\alpha_{2}\right) },e_{3}^{2}=e^{-4\alpha_{1}},\end{aligned}$$ where $\alpha_{1,2}\left( x^{0}\right) $ are two dimensionless geometrical fields that parametrize the anisotropy such that $e_{1}^{2}e_{2}^{2}e_{3}^{2}=1,$ so the metric $g_{\mu\nu}^{\gamma}$ is unimodular. The general generic solution for $\alpha_{1},\alpha_{2},\phi_{\gamma},s_{\gamma}$ is [@BCST1-antigravity] $$\begin{aligned}
\alpha_{1} & =\frac{p_{1}}{2p}\ln\left\vert \frac{x^{0}}{l_{1}^{4}\rho
_{r}(x^{0}-x_{c}^{0})}\right\vert ,\;\label{alpha1}\\
\alpha_{2} & =\frac{p_{2}}{2p}\ln\left\vert \frac{x^{0}}{l_{2}^{4}\rho
_{r}(x^{0}-x_{c}^{0})}\right\vert ,\;\label{alpha2}\\
\phi_{\gamma}+s_{\gamma} & =l_{3}^{2}\rho_{r}(x^{0}-x_{c}^{0})\left\vert
\frac{x^{0}}{l_{3}^{4}\rho_{r}(x^{0}-x_{c}^{0})}\right\vert ^{(p+p_{3})/2p},\;\label{phi+s}\\
\phi_{\gamma}-s_{\gamma} & =\frac{2x^{0}}{l_{3}^{2}}\left\vert \frac{x^{0}}{l_{3}^{4}\rho_{r}(x^{0}-x_{c}^{0})}\right\vert ^{-(p+p_{3})/2p}.
\label{phi-s}$$ Here $x^{0}$ is conformal time as seen from the definition of the metric in (\[metric\]). The constant parameters $\left( l_{1},l_{2},l_{3}\right) $ are constants of integration that have dimension of $\left( length\right) $; the parameters $\left( p_{1},p_{2},p_{3}\right) $ are constants of integration that have dimension of $\left( length\right) ^{-2},$ while $p\equiv\sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}};$ the constant $\rho_{r}$ that represents the radiation density has dimension of $\left( length\right)
^{-4};$ finally $x_{c}^{0}\equiv-\frac{\sqrt{6}p}{\kappa\rho_{r}}$ is the time of the crunch while the bang is set at zero time. This solution is plotted parametrically in Fig. 2. The arrows on the curve show the evolution of the fields $\left( \phi_{\gamma}\left( x^{0}\right) ,s_{\gamma}\left(
x^{0}\right) \right) $ as conformal time $x^{0}$ increases from negative values to positive values as the system goes through the crunch singularity at $x^{0}=x_{c}^{0}$ and the bang singularity at $x^{0}=0.$
\[ptb\]
[Fig2-AntigravityLoop.eps]{}
At the instants of crunch or bang the scale factor $a_{E}\left( x^{0}\right)
$ in the *Einstein gauge*, which in $d=4$ is given by [@BCST1-antigravity], $a_{E}^{2}=\frac{\kappa^{2}}{6}\left\vert
\phi_{\gamma}^{2}-s_{\gamma}^{2}\right\vert =\frac{\kappa^{2}}{6}\rho
_{r}\left\vert x^{0}(x^{0}-x_{c}^{0})\right\vert ,$ vanishes, and the corresponding scalar curvature in the Einstein frame, $R\left( g_{E}\right)
,$ blows up. However in the $\gamma$-gauge, since $a_{\gamma}=1,$ the behavior is much milder since $a_{\gamma}$ is a constant, which is how we were able to obtain our geodesically complete analytic solutions in the $\gamma$-gauge. If the anisotropy coefficients $p_{1,2}$ are non-zero, then the Weyl invariant Weyl curvature tensor, $C_{~\nu\lambda\sigma}^{\mu}\left( g\right) ,$ is singular in all gauges, including in the $\gamma$-gauge. However, this fact did not prevent us from obtaining [@BCST1-antigravity] our geodesically complete solutions as given above explicitly, nor did it stop the geodesics in this geometry from sailing through the cosmological singularities in the presence of anisotropy [@BST-sailing], thus showing that information does go through singularities in our cosmological geometry in Eqs.(\[metric\]-\[phi-s\]). The underlying reason and technique for our ability to complete the geometry and geodesics across these singularities is the identification of a sufficient number of conserved quantities in our equations that allows us to match continuously all observables, finite or infinite, across the singularities [@BCST1-antigravity][@BST-sailing]. The completion of all geometrical features on both sides of singularities is evident since all observables, including those gauge invariants that blow up such as $C_{~\nu\lambda\sigma}^{\mu}\left( g^{\gamma}\right) ,$ are constructed from our solutions for $\alpha_{1},\alpha_{2},\phi_{\gamma},s_{\gamma}$ which are continuous through the singularities.
The solution displayed in Fig. 2 is generic, exact, and includes all possible initial conditions (isotropic limit as well), hence there are no other solutions in our setting (\[actionWinv2\]) under the conditions stated above (in particular, $V=0$). However, since $V$ cannot be neglected away from the singularities this solution is useful mainly to study the neighborhood of the singularities analytically, which is in fact our focus, and of course precisely where string theory would play a role. So this geodesically complete geometry is of great interest to string theory.
When $V\neq0,$ the complete set of solutions are known analytically for certain potentials $V$ when anisotropy is neglected [@BCST2-solutions]. The analytic expressions and their plots [@BCST2-solutions] are sufficient to infer the solution for more general potentials $V.$ Combining the attractor behavior near the singularities which is induced by anisotropy (Fig. 2) together with the known behavior away from singularities when anisotropy is negligible, the general generic behavior is pretty well understood as follows: When $V$ is included, the generic solution is such that the trajectory shown in Fig. 2 continues to move to large values of $\phi_{\gamma}$ in the gravity region, while $s_{\gamma}$ oscillates at much smaller amplitudes (see illustrations in [@BCT-cyclic],[@IB-cyclic],[@IB-BibBang],[@BST-HiggsCosmo]). So, when the trajectory in the $\left( \phi_{\gamma
},s_{\gamma}\right) $ plane shown in Fig. 2 is continued to larger times beyond the singularities, the curve turns around at large values of $\phi_{\gamma}$ and comes back for another crunch at the origin of the $\left( \phi_{\gamma},s_{\gamma}\right) $ plane, passes through another antigravity loop and after another bang continues to the opposite gravity region, doing this again and again an infinite number of times as conformal time keeps progressing from minus infinity to plus infinity.
The Weyl invariant system in Eq.(\[actionWinv2\]), coupled to ordinary matter (here symbolized by the radiation parameter $\rho_{r}$), is then like a perpetual motion machine, which absorbs energy from the gravitational field and creates additional entropy during antigravity in each cycle [@BST-HiggsCosmo], thus yielding generically a cyclic universe, whose global scale in the Einstein frame, $a_{E}\left( x^{0}\right) ,$ grows progressively in each cycle due to entropy production, without having an initial cycle in the multicycles in the far past or an end of multicycles in the far future [@BST-HiggsCosmo].
There are of course quantum corrections to this picture. Unfortunately, these will remain obscure until quantum gravity is under better control. In the interim, some new directions of research emerge from interesting new questions that arise about the quantum physics during the antigravity period and close to singularities. There are signals of instability because the kinetic energy term for the gravitons is negative during antigravity. It seems gravitons would be emitted copiously to transit to a lower energy state in response to this instability. However, to maintain the energy constraints due to general coordinate invariance ordinary matter must also be emitted simultaneously. This is good for entropy production from one cycle to the next as discussed in [@BST-HiggsCosmo]. Note that a measure of the amount of time spent in the antigravity regime is $\left\vert x_{c}^{0}\right\vert =\frac{\sqrt{6}p}{\kappa\rho_{r}}$ as seen from our analytic solution (\[metric\]-\[phi-s\]) and Fig. 2 above. If radiation $\rho_{r}$ (any relativistic matter) increases by entropy production during antigravity as mentioned above, then the antigravity period $\left\vert x_{c}^{0}\right\vert $ gets shorter. This is a signal that there seems to exist a built-in dynamical recovery process to get from antigravity back to gravity. We hope that the string version of our formulation below can shed light on the microscopic stringy details of this mechanism.
The low energy theory has produced a wealth of signals for previously unknown interesting phenomena. The general generic features we have noted in [@BST-ConfCosm]-[@BST-sailing] are likely to survive in the context of quantum gravity when we understand it better. To counter comments such as those in [@Kallosh][@Linde], as we did in [@BST-sailing], it may be worth remembering that in string theory singularities tend to get smoother or resolved, so we should expect less singular behavior as compared to the low energy theory as stringy features are taken into account. It is not advisable to try to include in our analysis stringy features in the form of higher derivative corrections in the effective action. This is because those corrections to the action are computed under the assumption of low energy away from singularities, so they are the wrong tool to analyze phenomena near singularities.
Rather, we need to tackle string theory directly and ask again in that context the types of questions we have been able to analyze and partially resolve in the low energy theory. However, the starting point of string theory does not seem to be well suited for our type of analysis since it begins with a dimensionful constant, namely the string tension. For this reason we seek a more general starting point, one in which the string tension is replaced by a background field just like the gravitational constant which was replaced by a field in the low energy theory. The construction of such a string theory is the topic of the next section.
Weyl Invariance and Dynamical Tension in String Theory
======================================================
The goal in this section is to develop a Weyl invariant formalism for strings propagating in backgrounds, so that the string tension $\left( 2\pi
\alpha^{\prime}\right) ^{-1}$ - equivalently the gravitational constant - emerges from the gauge fixing of a background field in a new string theory that has a local scaling gauge symmetry acting on *target space* fields.
Taking the clues developed in the previous section we propose the following string action with a dynamical string tension and target-space Weyl symmetry. We simply insert in (\[Sstring\]) our expressions for the Weyl covariant substitutes for $2\alpha^{\prime}$ and $\Phi$ in terms of $\left(
\phi,s\right) $ as given in Eq.(\[T\]) and Eq.(\[s-gauge\]). The result is $$S=\int d^{2}\sigma\left[
\begin{array}
[c]{c}-\frac{1}{2}T\left( \phi\left( X\right) ,s\left( X\right) \right)
\left( \sqrt{-h}h^{ab}g_{\mu\nu}\left( X\right) +\varepsilon^{ab}b_{\mu\nu
}\left( X\right) \right) \partial_{a}X^{\mu}\partial_{b}X^{\nu}\\
+\frac{\sqrt{d-1}}{16\pi}\sqrt{-h}R^{\left( 2\right) }\left( h\right)
\ln\left( \frac{\phi\left( X\right) +s\left( X\right) }{\phi\left(
X\right) -s\left( X\right) }\right) ^{2}+O\left( \frac{1}{T}\right)
\end{array}
\right] \label{SstringWeyl}$$ where $X^{\mu}\left( \tau,\sigma\right) $ is the string coordinate. If we insert the s-gauge expressions for $\left( \phi_{s},s_{s},g_{\mu\nu}^{s},b_{\mu\nu}^{s}\right) $ given in Eq.(\[s-gauge\]) for the $\pm$ gravity patches, this action reduces to the standard string action in Eq.(\[Sstring\]). For this more general action to be consistent with the *worldsheet conformal symmetry* up to order $\left( 1/T\right) ,$ the target space Weyl covariant background fields $\left( \phi,s,g_{\mu\nu
},b_{\mu\nu}\right) $ must be classical solutions of the equations of motion that are derived from the target space Weyl invariant effective action (\[actionWinv2\]). The relevant solutions were described in the previous section.
The appearance of the logarithm in the action (\[SstringWeyl\]) might be a cause for concern, since the Lagrangian density appears non-analytic. However, since the Euler density $\sqrt{-h}R^{\left( 2\right) }$ is a total derivative, one can integrate by parts, so that the resulting Lagrangian density, involving only first order derivatives, has no branch cuts and is single-valued on field space.
This string action contains no dimensionful constants. It is Weyl invariant when the target-space background fields $\left( \phi,s,g_{\mu\nu},b_{\mu\nu
}\right) $ are rescaled locally according to the rules in Eq.(\[WeylTransf\]). This is easily seen by noting that the combinations of fields $$\hat{g}_{\mu\nu}\equiv T\left( \phi,s\right) g_{\mu\nu},\;\hat{b}_{\mu\nu
}\equiv T\left( \phi,s\right) b_{\mu\nu},\;\frac{s}{\phi}
\label{weylInvariantBackgr}$$ that appear in the basic string action (\[SstringWeyl\]) are Weyl invariant.
We will see below that they also obey an additional symmetry which we call *flip symmetry*, that allows us to extend $T\left( \phi,s\right) ,$ from the $\pm$ gravity patches where it was initially defined in (\[T\]), to the geodesically complete entire $\left( \phi,s\right) $ plane.
Therefore it is possible to Weyl gauge fix the action in various ways (e.g. $s,E,c,\gamma,f$ gauges of the previous section) to investigate the physics of string theory in the corresponding spacetimes that look like different backgrounds from one another. Any results obtained in one Weyl gauge are easily transformed to another gauge, just like duality transformations (which are now realized as Weyl transformations as explained at the end of the last section).
This expression for the string action is so far uniquely determined in the gravity patches $\left( \phi^{2}-s^{2}\right) \geq0$ where the dynamical tension $T\left( \phi,s\right) $ is positive. In this sense, with Eq.(\[SstringWeyl\]), we have already achieved a non-trivial stage for the *target-space Weyl invariant* definition of string theory. The remaining questions involve the sign of the dynamical tension $T\left(
\phi,s\right) $ in the antigravity patches so that the theory is sensible according to sacred principles, such as unitarity, since a negative sign of $T$ in (\[SstringWeyl\]) may imply negative norm states. We will show below that unitarity is not a problem, but there are physics questions to consider. We emphasize that the issue is subtle because the antigravity patches occur only for a finite amount of time $\left\vert x_{c}^{0}\right\vert $ in our cosmological solutions and during that period those space-time patches are separated by cosmological singularities from our own observable gravity patch, as seen in Fig. 2.
In considering the sign of $T\left( \phi,s\right) $ in the antigravity patches, there is an additional clue in the low energy Weyl invariant action (\[actionWinv2\]) to take into account. We observe that all the *kinetic* terms of the action in (\[actionWinv2\]) flip sign if $\phi$ and $s$ are interchanged with each other. Independently, if the metric $g_{\mu\nu}$ flips signature, $g_{\mu\nu}\rightarrow-g_{\mu\nu},$ then noting $R\left( -g\right) =-R\left( g\right) ,$ and that $\partial\phi
\cdot\partial\phi,$ $\partial s\cdot\partial s,$ $H^{2}$ all contain an odd power of $g^{\mu\nu},$ while $\left( -\det g\right) $ is invariant for even $d,$ we see that all kinetic terms in (\[actionWinv2\]) flip sign under the signature flip. Therefore, under the simultaneous flip of signature and interchange of $\left( \phi,s\right) $ $$\text{flip symmetry:\ }\left( \phi\leftrightarrow s\right) \text{ and
}g_{\mu\nu}\rightarrow-g_{\mu\nu}, \label{flip}$$ all kinetic terms of the low energy action (\[actionWinv2\]) are invariant for even $d$. It is possible to include a flip of sign of $b_{\mu\nu
}\rightarrow-b_{\mu\nu},$ as an additional part of the flip transformations in (\[flip\]), since the flip of $b_{\mu\nu}$ is also a symmetry all by itself in the low energy action. Observe also that $H_{\mu\nu\lambda}$ as given in (\[HT\]) is automatically flip symmetric for the $T$ given in (\[T\]). The remaining term in (\[actionWinv2\]) is the potential energy $V\left(
\phi,s\right) ,$ whose properties under the flip transformation is unclear at this stage but should be determined by the underlying string theory.
Since the low energy action has a flip symmetry (at least in its kinetic terms) we expect that the underlying string theory action (\[SstringWeyl\]), which should be compatible with all properties of (\[actionWinv2\]), should be symmetric under the flip transformation of all its *background fields*. Requiring this symmetry uniquely determines the properties of $T\left( \phi,s\right) $ in all gravity/antigravity patches because the combinations $T\left( \phi,s\right) g_{\mu\nu}$ and $T\left( \phi,s\right)
b_{\mu\nu}$ that appear in (\[SstringWeyl\]) must now be required to be flip symmetric. This determines uniquely that $T$ must be antisymmetric under the interchange of $\left( \phi,s\right) $ $$T\left( \phi,s\right) =-T\left( s,\phi\right) .$$ Therefore the continuation of the expression $T\left( \phi,s\right) $ in (\[T\]) from the original gravity patch to all other gravity/antigravity patches must obey this requirement. This can be done by writing $T_{string}\left( \phi,s\right) =\left( \phi^{2}-s^{2}\right) |\frac{T\left(
\phi,s\right) }{\phi^{2}-s^{2}}|$ thus extending the $T\left( \phi,s\right)
$ in (\[T\]) to the entire $\left( \phi,s\right) $ plane. In what follows we will assume that this has been done, but to avoid clutter, we will continue to use the symbol $T\left( \phi,s\right) $ to mean the properly continued $T_{string}\left( \phi,s\right) .$ This completes the definition of the Weyl symmetric string action in (\[SstringWeyl\]).
String in Geodesically Complete Cosmological Background
-------------------------------------------------------
We will investigate the proposed action (\[SstringWeyl\]) for the geodesically complete cosmological background in Eqs.(\[metric\]-\[phi-s\]) that includes both gravity and antigravity patches as shown in Fig. 2. This background is consistent with worldsheet conformal invariance since it is the generic solution of the equations of motion of the effective action (\[actionWinv2\]). This background, which is an exact solution when $V=0,$ and otherwise is the correct approximation near the singularity even when $V\neq0,$ is suitable for our purpose of analyzing what happens near the gravity/antigravity transitions. The worldsheet conformal symmetry has been insured for any dimension $d$ (to lowest order in $\alpha^{\prime}$). Since the background in Eqs.(\[metric\]-\[phi-s\]) was computed in four dimensions$,$ we will specialize to $d=4$ without losing the (perturbatively valid) worldsheet conformal symmetry.
We first compute the dynamical string tension (\[T\]) for $d=4$ for this background using Eqs.(\[T\],\[phi+s\],\[phi-s\]) $$T_{4}\left( X^{0}\right) =\frac{1}{6}\left( \phi_{\gamma}^{2}-s_{\gamma
}^{2}\right) \left\vert \frac{\phi_{\gamma}+s_{\gamma}}{\phi_{\gamma
}-s_{\gamma}}\right\vert ^{\sqrt{3}}=\frac{1}{3}\rho_{r}X^{0}(X^{0}-x_{c}^{0})\left( \frac{1}{2}\left\vert \frac{X^{0}}{l_{3}^{4}\rho_{r}(X^{0}-x_{c}^{0})}\right\vert ^{p_{3}/p}\right) ^{\sqrt{3}} \label{T4}$$ Here $X^{0}\left( \tau,\sigma\right) $ is the string coordinate which is the conformal time in the given background geometry. This $T_{4}\left(
X^{0}\right) $ is flip antisymmetric as discussed in the previous section.
The metric $g_{\mu\nu}^{\gamma}$ was given in Eq.(\[metric\]). But we emphasize that only the Weyl invariant combination $\hat{g}_{\mu\nu}=T\left(
\phi_{\gamma},s_{\gamma}\right) g_{\mu\nu}^{\gamma}$ enters in our action, with$$d\hat{s}^{2}=\hat{g}_{\mu\nu}dx^{\mu}dx^{\nu}=T\left( x^{0}\right) \left(
\begin{array}
[c]{c}-\left( dx^{0}\right) ^{2}+e^{2\left( \alpha_{1}+\sqrt{3}\alpha_{2}\right)
}\left( dx^{1}\right) ^{2}\\
+e^{2\left( \alpha_{1}-\sqrt{3}\alpha_{2}\right) }\left( dx^{2}\right)
^{2}+e^{-4\alpha_{1}}\left( dx^{3}\right) ^{2}\end{array}
\right) . \label{g-hat}$$ Explicitly, this purely time (string $X^{0}$) dependent cosmological background for a string theory with dynamical tension is $$\begin{array}
[c]{l}\hat{g}_{00}\left( X\right) =-\frac{\rho_{r}X^{0}(X^{0}-x_{c}^{0})}{2^{\sqrt{3}}3}\left\vert \frac{X^{0}}{X^{0}-x_{c}^{0}}\right\vert
^{\frac{\sqrt{3}p_{3}}{p}}\left( \rho_{r}l_{3}^{4}\right) ^{-\frac{\sqrt
{3}p_{3}}{p}},\;\\
\hat{g}_{11}\left( X\right) =\frac{\rho_{r}X^{0}(X^{0}-x_{c}^{0})}{2^{\sqrt{3}}3}\left\vert \frac{X^{0}}{X^{0}-x_{c}^{0}}\right\vert
^{\frac{p_{1}}{p}+\frac{\sqrt{3}p_{2}}{p}+\frac{\sqrt{3}p_{3}}{p}}\left(
\rho_{r}l_{1}^{4}\right) ^{-\frac{p_{1}}{p}}\left( \rho_{r}l_{2}^{4}\right)
^{-\frac{\sqrt{3}p_{2}}{p}}\left( \rho_{r}l_{3}^{4}\right) ^{-\frac{\sqrt
{3}p_{3}}{p}},\\
\hat{g}_{22}\left( X\right) =\frac{\rho_{r}X^{0}(X^{0}-x_{c}^{0})}{2^{\sqrt{3}}3}\left\vert \frac{X^{0}}{X^{0}-x_{c}^{0}}\right\vert
^{\frac{p_{1}}{p}-\frac{\sqrt{3}p_{2}}{p}+\frac{\sqrt{3}p_{3}}{p}}\left(
\rho_{r}l_{1}^{4}\right) ^{-\frac{p_{1}}{p}}\left( \rho_{r}l_{2}^{4}\right)
^{+\frac{\sqrt{3}p_{2}}{p}}\left( \rho_{r}l_{3}^{4}\right) ^{-\frac{\sqrt
{3}p_{3}}{p}},\\
\hat{g}_{33}\left( X\right) =\frac{\rho_{r}X^{0}(X^{0}-x_{c}^{0})}{2^{\sqrt{3}}3}\left\vert \frac{X^{0}}{X^{0}-x_{c}^{0}}\right\vert
^{-\frac{2p_{1}}{p}+\frac{\sqrt{3}p_{3}}{p}}\left( \rho_{r}l_{1}^{4}\right)
^{\frac{2p_{1}}{p}}\left( \rho_{r}l_{3}^{4}\right) ^{-\frac{\sqrt{3}p_{3}}{p}}.
\end{array}
\label{g-hat2}$$ where the parameters $\left( p_{1},p_{2,}p_{3},p,l_{1},l_{2},l_{3},\rho
,x_{c}^{0}\right) $ are defined after Eq.(\[phi-s\]).
Note that in the low energy effective theory the original metric $g_{\mu\nu}$ in (\[actionWinv2\]) does not change sign as the trajectory in Fig. 2 transits from a gravity region to an antigravity region at cosmological singularities. Instead, the dynamical gravitational constant $\left( \phi^{2}-s^{2}\right) ,$ that multiplies the curvature in the low energy action (\[actionWinv2\]), changes sign at those singularities. However, in view of the flip symmetry (\[flip\]) of the geodesically complete low energy theory (\[actionWinv2\]), the sign flip of the factor $\left( \phi^{2}-s^{2}\right) $ can also be viewed equivalently as being a signature flip of the effective metric $\hat{g}_{\mu\nu}$ defined above, since $R\left( -\hat{g}\right) =-R\left( \hat{g}\right) $. Indeed the effective metric $\hat{g}_{\mu\nu}\left( X^{0}\right) $ in (\[g-hat2\]) changes overall signature precisely at the cosmological singularities. The period of antigravity (or opposite signature of $\hat
{g}_{\mu\nu}$) and its temporary duration, $\left\vert x_{c}^{0}\right\vert
=\frac{\sqrt{6}p}{\kappa\rho_{r}},$ is determined by the physical parameters of the background above.
Just as we could study particle geodesics in this background in [@BST-sailing], we can also study string geodesics in this geodesically complete geometry by solving the equations for $X^{\mu}\left( \tau
,\sigma\right) $ that follow from our action (\[SstringWeyl\]). This topic is partially explored in the next section.
Unitarity with a Temporary Flip of the Dynamical String Tension
---------------------------------------------------------------
In this section we study a formalism for solving generally the classical solutions and the quantization of the string theory described by our action (\[SstringWeyl\]), and its specialization to a cosmological background given in Eq.(\[g-hat2\]). To simplify this investigation we will drop the quantum correction terms for worldsheet conformal symmetry in (\[SstringWeyl\]) as well as the $b_{\mu\nu}$ background field since these are not the crucial terms for our questions involving the temporary sign flip of the dynamical string tension. Hence we will simply concentrate on the purely classical string action which is modified only by the dynamical string tension$$S_{cl}=-\frac{1}{2}\int d^{2}\sigma\sqrt{-h}h^{ab}\hat{g}_{\mu\nu}\left(
X\right) \partial_{a}X^{\mu}\partial_{b}X^{\nu},\label{Sclass}$$ where the tension is absorbed into the definition of $\hat{g}_{\mu\nu}=Tg_{\mu\nu},$ allowing $\hat{g}_{\mu\nu}$ to have a temporary signature flip. We are interested in obtaining the solutions for strings in such backgrounds, and quantizing them to answer the questions on unitarity and begin to interpret the physics when there is a signature or tension flip. Since such questions are more general than the specific background in (\[g-hat2\]), we will consider a more general $\hat{g}_{\mu\nu}.$ Much of the necessary canonical formalism was developed in Ref. [@TPS]. It is useful to introduce the momentum density defined by $P_{\mu}^{a}\equiv\partial
S_{cl}/\partial\left( \partial_{a}X^{\mu}\right) ,$ $$P_{\mu}^{a}=-\sqrt{-h}h^{ab}\hat{g}_{\mu\nu}\partial_{b}X^{\nu},\;\Rightarrow
\partial_{b}X^{\nu}=-\frac{h_{ba}}{\sqrt{-h}}\hat{g}^{\nu\mu}P_{\mu}^{a},$$ where the canonical conjugate to $X^{\mu}$ corresponds to the momentum density $P_{\mu}^{\tau},$ i.e. when $a=\tau.$ The classical string equations of motion and constraints that follow from this action are $$\begin{array}
[c]{l}\delta X^{\mu}:0=\partial_{a}\left( \sqrt{-h}h^{ab}\hat{g}_{\mu\nu}\partial_{b}X^{\nu}\right) -\frac{\sqrt{-h}h^{ab}}{2}\partial_{a}X^{\lambda
}\partial_{b}X^{\rho}\partial_{X^{\mu}}\left( \hat{g}_{\lambda\rho}\right)
\\
\delta h_{ab}:0=\;\hat{g}^{\mu\nu}P_{\mu}^{a}P_{\nu}^{b}-\frac{1}{2}h^{\alpha
b}h_{cd}\hat{g}^{\mu\nu}P_{\mu}^{c}P_{\nu}^{d}~.
\end{array}
\label{eoms}$$ To make progress we consider the worldsheet gauge $h_{\tau\sigma}=0$ in which $h_{ab}$ is diagonal. There is remaining worldsheet gauge symmetry to choose also the gauge $X^{0}=\tau$, which we will do later. Although a diagonal $h_{ab}~$contains two functions, the combination $\sqrt{-h}h^{ab}$ contains only one function since its determinant is $-1.$ Hence we parametrize $h_{ab}$ as follows$$\sqrt{-h}h^{ab}=\left(
\genfrac{}{}{0pt}{}{-h}{0}\genfrac{}{}{0pt}{}{0}{1/h}\right) .\label{h and 1/h}$$ and insert it in all the equations listed in (\[eoms\]). In what follows we will assume that the metric is diagonal as is the case in (\[g-hat2\])$$\hat{g}_{\mu\nu}=T\times diag\left( -1,e_{1}^{2},e_{2}^{2},e_{3}^{2},\cdots\right) .$$ Then, $$\begin{array}
[c]{l}P_{\mu}^{\tau}=h\hat{g}_{\mu\nu}\partial_{\tau}X^{\nu},\;P_{\mu}^{\sigma
}=-h^{-1}\hat{g}_{\mu\nu}\partial_{\sigma}X^{\nu}\\
\mu=i:\;0=\partial_{\tau}\left( -h\hat{g}_{ii}\partial_{\tau}X^{i}\right)
+\partial_{\sigma}\left( h^{-1}\hat{g}_{ii}\partial_{\sigma}X^{i}\right)
,\text{ no sum on }i\\
\mu=0:\;0=\left(
\begin{array}
[c]{c}\partial_{\tau}\left( -h\hat{g}_{00}\partial_{\tau}X^{0}\right)
+\partial_{\sigma}\left( h^{-1}\hat{g}_{00}\partial_{\sigma}X^{0}\right) \\
-\frac{1}{2}\left( -h\partial_{\tau}X^{\lambda}\partial_{\tau}X^{\rho}+h^{-1}\partial_{\sigma}X^{\lambda}\partial_{\sigma}X^{\rho}\right)
\partial_{X^{0}}\hat{g}_{\lambda\rho}\end{array}
\right) \\
0=\;\left( P^{\tau}\pm hP^{\sigma}\right) ^{2}\text{ or }0=\left(
h\partial_{\tau}X^{\nu}\pm\partial_{\sigma}X^{\nu}\right) ^{2},\text{ }\end{array}
\label{eoms-X0gauge}$$ The last constraint equations are solved as$$h=\varepsilon\left( X\right) \sqrt{\frac{g_{\mu\nu}\partial_{\sigma}X^{\mu
}\partial_{\sigma}X^{\nu}}{-g_{\mu\nu}\partial_{\tau}X^{\mu}\partial_{\tau
}X^{\nu}}},\text{ and }g_{\mu\nu}\partial_{\tau}X^{\mu}\partial_{\sigma}X^{\nu}=0,\label{h-epsilon}$$ Note that in these expressions we wrote $g_{\mu\nu}$ rather than $\hat{g}_{\mu\nu}=Tg_{\mu\nu}$ because the tension factor of $T\left( X\right) $ drops out. Here $\varepsilon\left( X\right) $ is a sign function that emerges from taking the square root. Assuming $h_{ab}$ has a fixed signature we must take $\varepsilon\left( X\right) =1$ so that the signature of the metric in (\[h and 1/h\]) remains fixed forever[^5].
Now, if we choose also the timelike gauge, $X^{0}=\tau,$ we obtain the following form for the Hamiltonian density $\mathcal{H}\left( \tau
,\sigma\right) ,$ which is the negative of the canonical conjugate to $X^{0}$ as given in Eq.(\[eoms-X0gauge\]), $$X^{0}=\tau\;\Rightarrow\text{ }\mathcal{H}\left( \tau,\sigma\right)
=-P_{0}^{\tau}=h\left( \tau,\sigma\right) T\left( \tau\right) ,$$ where we insert the solution for $h$ in Eq.(\[h-epsilon\]) to obtain the Hamiltonian density (with $\varepsilon(X)$ set to 1$)$ $$\mathcal{H}\left( \tau,\sigma\right) =T\left( \tau\right) \left(
\frac{e_{j}^{2}\left( \tau\right) \left( \partial_{\sigma}X^{j}\right)
^{2}}{1-e_{j}^{2}\left( \tau\right) \left( \partial_{\tau}X^{j}\right)
^{2}}\right) ^{1/2},$$ Here $T\left( X^{0}\right) $ and $e_{i}^{2}\left( X^{0}\right) $ are backgrounds, such as those in our suggested cosmological background in (\[g-hat2\]), evaluated at $X^{0}=\tau$.
This Hamiltonian may also be expressed in terms of the canonical conjugate to $X^{i},$ which is $P_{i}^{\tau},$ instead of the velocity $\partial_{\tau
}X^{i}.$ Using $$P_{i}^{\tau}=h\hat{g}_{ii}\partial_{\tau}X^{i}=T\times\left( \frac{e_{j}^{2}\left( \partial_{\sigma}X^{j}\right) ^{2}}{1-e_{j}^{2}\left(
\partial_{\tau}X^{j}\right) ^{2}}\right) ^{1/2}e_{i}^{2}\partial_{\tau}X^{i},$$ the Hamiltonian density in terms of canonical conjugates is obtained as $$\mathcal{H}\left( \tau,\sigma\right) =T\left( \tau\right) \left(
e_{j}^{-2}\left( \tau\right) \left( P_{j}^{\tau}\right) ^{2}+\frac
{T^{2}\left( \tau\right) }{\pi^{2}}e_{j}^{2}\left( \tau\right) \left(
\partial_{\sigma}X^{j}\right) ^{2}\right) ^{1/2}.\label{H-density}$$ while the velocity is also given in terms of canonical variables $$\partial_{\tau}X^{i}\left( \tau,\sigma\right) =\frac{T\left( \tau\right)
P_{i}^{\tau}\left( \tau,\sigma\right) }{e_{i}^{2}\left( \tau\right)
\mathcal{H}\left( \tau,\sigma\right) }=\frac{P_{i}^{\tau}}{e_{i}^{2}}\left(
e_{j}^{-2}\left( \tau\right) \left( P_{j}^{\tau}\right) ^{2}+T^{2}\left(
\tau\right) e_{j}^{2}\left( \tau\right) \left( \partial_{\sigma}X^{j}\right) ^{2}\right) ^{-1/2}.$$
Note that in a time dependent curved cosmological background there is no translation symmetry in the time coordinate $X^{0}.$ Therefore we do not expect that the time translation generator $P_{0}^{\tau}$ would be conserved in the gauge $X^{0}=\tau.$ Hence the Hamiltonian density or the total energy of the system which is given by the Hamiltonian $H,$ $$H\left( \tau\right) =\int d\sigma\mathcal{H}\left( \tau,\sigma\right) ,
\label{energy}$$ will generally be a function of time $\tau$ in any cosmological background.
Having dealt with the gauge fixed $X^{0}$ and its conjugate Hamiltonian, the remaining independent dynamical degrees of freedom $X^{i}\left( \tau
,\sigma\right) $ satisfy the following equations of motion and constraints $$\begin{aligned}
0 & =-\partial_{\tau}\left( \mathcal{H}e_{i}^{2}\partial_{\tau}X^{i}\right)
+\partial_{\sigma}\left( \frac{e_{i}^{2}}{\mathcal{H}}\partial_{\sigma}X^{i}\right) ,\text{ no sum on }i\\
0 & =\partial_{\sigma}X^{j}P_{j}^{\tau}\text{ or }0=e_{j}^{2}\partial_{\tau
}X^{j}\partial_{\sigma}X^{j}\text{ },\text{ summed over }j\end{aligned}$$ The meaning of the constraint is the imposition of the $\sigma$-reparametrization gauge symmetry on physical states.
The quantization of this system is now evident. The equal time quantum rules are $$\left[ X^{i}\left( \tau,\sigma\right) ,P_{j}^{\tau}\left( \tau
,\sigma^{\prime}\right) \right] =i\delta\left( \sigma-\sigma^{\prime
}\right) .$$ On the Hilbert space created by these canonical operators we must impose the physical state constraints which must be normal ordered at a fixed time $\tau$$$\left( :\partial_{\sigma}X^{i}\left( \tau,\sigma\right) P_{i}^{\tau}\left(
\tau,\sigma\right) :\right) |phys\rangle=0.$$
This Hilbert space is clearly unitary, since we have shown that a unitary gauge exists in which only space-like canonical degrees of freedom are involved in the quantization of our system. Space-like degrees of freedom can never create negative norm states and the evolution of the system is generated by a Hermitian Hamiltonian. So, *there are no issues with unitarity*.
Outlook
=======
In this section we address some open problems and point out some tools and new directions to make further progress.
We argued that there are no problems with unitarity. However, there is the unfamiliar feature, that the energy carried by any string configuration as predicted by Eq.(\[energy\]) will become negative once the string (along with the rest of the universe) enters the antigravity patch. Taking our theory as a cosmological model, the negative energy is supposed to happen in between two cycles in an infinitely cyclic universe, so the question relates in particular to what happens just before our current big bang. This is precisely where we hope string theory could shed some light on quantum gravity.
Is the negative energy during the antigravity period a problem like an instability, or just a new physics feature to investigate? How can we tell from our perspective as observers in the gravity patches separated from antigravity by cosmological singularities? One way to investigate this is to consider the analog of a scattering process in which we imagine a string probe that travels to the antigravity patch and returns to the gravity patch as an altered string that brings information from the antigravity region. Based on a set of solutions of this nature, we have already verified that the behavior of string solutions as they pass through the crunch/bang singularities is regular and is physically sensible. This analysis will appear in a forthcoming paper.
From the classical and quantum discussion above it is clear that the single string system represented by the Weyl invariant action (\[SstringWeyl\]), with the cosmological configuration (\[g-hat2\]) in mind, could not be unstable during antigravity. This is because when the energy is negative, it is negative for all classical and all quantum configurations of the single string during the entire antigravity period for the whole universe. There are no positive energy states of the single string during antigravity. Based on an adiabatic approximation of energy conservation, it is not possible to make a transition to a lower energy state because it is not possible to conserve energy adiabatically with only negative energy states. Hence there can be no transitions among the states of a single string that would indicate an instability in the system described by (\[SstringWeyl\]).
What about a multi-string system that includes string-string interactions? A hint is provided by the flip symmetry of the low energy action. We have seen already that all the kinetic terms flip sign when either $g_{\mu\nu}$ flips signature or when $\left( \phi,s\right) $ are interchanged. This transformation can be used to infer the properties of the theory in the antigravity regime. If all terms of the effective action flip sign under the $\left( \phi,s\right) $ interchange, then we deduce that all equations of motion, and hence all physics, is the same during antigravity as compared to gravity, and therefore nothing strange should be expected during antigravity. If the potential $V\left( \phi,s\right) $ does not flip sign or has no definite flip symmetry, then after multiplying the overall action by an overall minus sign, the system would behave as if the potential is effectively time dependent when one compares the antigravity/gravity periods. Like in any time-dependent system, this system would undergo transitions. However this does not imply that any sacred principles of physics would be violated. Our considerations based on the low energy action and the flip symmetry of the kinetic terms are reassuring that the physics is still familiar, but we need better tools to understand the stringy details.
For a more powerful tool to investigate string-string interactions we may turn to string field theory (SFT). This relates to our new action (\[SstringWeyl\]) since the corresponding SFT is constructed by using a BRST operator derived from the stress tensor for the Weyl invariant action (\[SstringWeyl\]). Therefore SFT provides a conceptual framework in which the relevant issues may be considered in a complete formalism that includes all the interactions. Although it is hard to compute in SFT, recent improvements of its formulation [@SFT-BR] with a new Moyal-type star product that can be used for any curved space, makes it possible to use this as a tool for more progress to answer our questions in quantum gravity. For our purpose here the general structure of the open string field theory action [@WittenSFT] is already a guide, as follows. In the so called Siegel gauge the BRST operator boils down to the kinetic operator given by the Virasoro operator $L_{0},$ while the basic interaction is cubic as in Chern-Simons theory. After some manipulations, the open string field theory action can be gauge fixed to the following form of non-commutative field theory $$S=Tr\left[ \frac{1}{2}AL_{0}A+\frac{g}{3}A\star A\star A\right] .$$ where $L_{0}=\int d\sigma\left[ \text{Stress Tensor}\right] ,$ is the zeroth Virasoro operator. In the new formalism [@SFT-BR] the string field $A$ is a function of *half* of the string phase space $A\left( X_{+}^{\mu
}\left( \sigma\right) ,P_{-\mu}^{\tau}\left( \sigma\right) \right) ,$ the trace $Tr$ is integration in the half phase space, while the star product $\star$ is the new *background independent* Moyal star product (which represents string joining/splitting) in this *half* phase space[^6]. The point here is that $L_{0},$ as a function of the stress tensor of our theory in (\[SstringWeyl\]) written in terms of canonical variables $\left( X^{\mu
}\left( \sigma\right) ,P_{\mu}^{\tau}\left( \sigma\right) \right) $, is the only part of SFT that contains the new information about the flipping dynamical tension $T\left( X^{0}\left( \sigma\right) \right) $ as a function of $X^{0}\left( \sigma\right) .$ Hence, this formalism is an arena that can be used to further investigate our questions including string-string interactions both perturbatively in the coupling $g$, and non-perturbatively including the search for the ground state as in any interacting field theory in the presence of interactions. In this context a first impression is that, cosmologically the negative energies will occur not only for single strings but for all strings in the universe at the same time, and hence there should be no instabilities, although there should be new physical features.
It would be very useful to construct examples of worldsheet-conformally-exact string models whose string tension is Weyl-lifted to a background field as in Eq.(\[SstringWeyl\]). This would greatly improve the $\alpha^{\prime}$ expansion of Eq.(\[Sstring\]) or the corresponding $1/T$ expansion in (\[SstringWeyl\]), to include all orders, and would also be essential in practice for the SFT approach discussed above. A starting point to pursue this idea could be the (gauged) Wess-Zumino-Witten models like those discussed in [@IB-WZWs]-[@tseytlin]. Ideally we would like to find a conformally exact cosmological string background whose behavior near the cosmological singularities matches the behavior explicitly computed in Eqs.(\[alpha1\]-\[phi-s\]) and Fig. 2, and whose effective metric near the cosmological singularities matches the form of Eq.(\[g-hat2\]) after including the dynamical tension.
Pursuing duality concepts is bound to be helpful to clarify the questions mentioned above and to generalize our Weyl invariance approach to other corners of M-theory. We mentioned earlier the connection of our target space Weyl symmetry to dualities within the bosonic string theory in connection to scale inversions [@Veneziano] and the more general cases involving transformations among the $\left( s,E,c,\gamma,f\right) $-type gauge choices (see also footnote (\[foot2T\])). There is more to consider on the duality path by pursuing the flip symmetry (\[flip\]) further. A version of our flip symmetry was considered in [@Duff1][@Duff2] for a variety of supersymmetric and heterotic strings in the context of low energy string theory, but without our Weyl lifting ideas. We emphasize that without Weyl lifting such a symmetry is not really realizable in the standard formulation of string theory because, as stated in [@Duff1][@Duff2], it involves the sign flip of the effective string coupling $e^{\Phi}$ to $-e^{\Phi}$. This cannot be achieved by any reasonable transformation of the dilaton $\Phi$ as a field that takes values only on the real line. However, with our Weyl lifting, there is also a factor related to the dynamical tension $T$ that multiplies $e^{\Phi}$ instead of the constant tension. It is the extra factor, not $\Phi,$ that transforms under our flip symmetry, and this is realized in our case by the interchange of $\left( \phi,s\right) $. We may consider the work in [@Duff1][@Duff2] as a possible guide toward the generalization of our Weyl lifted string theory (\[SstringWeyl\]) to a variety of dual corners of M-theory, including supersymmetric and heterotic versions, and to pursue duality concepts.
The tools and ideas mentioned above offer concrete approaches to further exploration of string theory with target-space Weyl invariance and a dynamical string tension.
We acknowledge conversations with Ed Witten on this topic. This research was partially supported by the U.S. Department of Energy under grant number DE-FG03-84ER40168 (IB) and under grant number DE-FG02-91ER40671 (PJS). 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.
[99]{}
I. Bars, P.J. Steinhardt, N. Turok, Local conformal symmetry in physics and cosmology," Phys. Rev. **D89** (2014) 043515 \[arXiv:1307.1848\].
I. Bars and S-H. Chen, The Big Bang and Inflation United by an Analytic Solution , Pays. *Rev. D***83** 043522 (2011) \[arXiv:1004.0752\].
I. Bars, S-H. Chen and N. Turok, Geodesically Complete Analytic Solutions for a Cyclic Universe, *Phys. Rev.* **D84** 083513 (2011) \[arXiv:1105.3606\].
I. Bars, Geodesically Complete Universe, arXiv:1109.5872, in the online Proceedings of the DPF- 2011 Conference, Providence, RI, August 8-13, 2011, http://www.slac.stanford.edu/econf/C110809/.
I. Bars, S.-H. Chen, P. J. Steinhardt and N. Turok, Antigravity and the Big Crunch/Big Bang Transition, Phys. Lett. **B715** (2012) 278-281 \[arXiv:1112.2470\].
I. Bars, S.-H. Chen, P. J. Steinhardt and N. Turok, Complete Set of Homogeneous Isotropic Analytic Solutions in Scalar-Tensor Cosmology with Radiation and Curvature, Phys. Rev. **D86** (2012) 083542 \[ arXiv:1207.1940\].
I. Bars, Traversing Cosmological Singularities, Complete Journeys Through Spacetime Including Antigravity, arXiv:1209.1068, in *Beyond the Big Bang, Competing Scenarios for an Eternal Universe*, Ed. Rudy Vaas, Springer \[2013\], ISBN 978-3-540-71422-4.
I. Bars, P.J. Steinhardt, N. Turok, Cyclic Cosmology, Conformal Symmetry and the Metastability of the Higgs, Phys. Lett. **B726** (2013) 50 \[arXiv:1307.8106\].
I. Bars, P.J. Steinhardt, N. Turok, Sailing Through the Big Crunch-Big Bang Transition, Phys. Rev. **D89** (2014) 061302 \[arXiv:1312.0739\].
M. Green, J. Schwarz and E. Witten, *Superstring Theory*, Vol.1, Cambridge Univ. Press 1987.
R. C. Myers, Phys. Lett. B **199**, 371 (1987).
I. Bars, Constraints on Interacting Scalars in 2T Field Theory and No Scale Models in 1T Field Theory, Phys. Rev. **D82** (2010) 125025 \[arXiv:1008.1540\].
K. Bamba, S. Nojiri, S. Odintsov, D. Sáez-Gómez Possible antigravity regions in F(R) theory? Phys. Lett. **B730** (2014) 136-140 \[arXiv:1401.1328\].
I. Bars, Gravity in 2T-Physics, Phys. Rev. **D77** (2008) 125027 \[arXiv:0804.1585\]; I. Bars, S. H. Chen Geometry and Symmetry Structures in 2T Gravity, Phys. Rev. **D79** (2009) 085021 \[arXiv:0811.2510v2\].
I. Araya and I. Bars, Generalized Dualities in 1T-Physics as Holographic Predictions from 2T-Physics, Phys. Rev. **D89** (2014) 066011 \[arXiv:1311.4205\].
M. Gasperini and G. Veneziano The Pre - Big Bang Scenario in String Cosmology, Phys. Rept. **373** (2003) 1 \[hep-th/0207130\]
J. Carrasco, W. Chemissany, R. Kallosh, Journeys Through Antigravity?, JHEP **1401** (2014) 130 \[arXiv:1311.3671\].
A. Linde, Inflationary Cosmology after Planck 2013, arXiv:1402.0526.
N. Turok, M. Perry and P. J. Steinhardt, Phys. Rev. D **70**, 106004 (2004) \[Erratum-ibid. D **71**, 029901 (2005)\] \[hep-th/0408083\].
E. Witten, Nucl. Phys. **B268** (1986) 253.
I. Bars and D. Rychkov, String Field Theory with Background Independent Moyal-Star Product, in preparation.
I. Bars and K. Sfetsos, Mod. Phys. Lett. **A7** (1992) 1091; Phys. Rev. **D46** (1992) 4495; Phys. Lett. **B301** (1993) 183; Phys. Rev. D 48 (1993) 844.
C. Nappi and E. Witten, Phys.Rev.Lett. 71 (1993) 3751.
A. Tseytlin and K. Sfetsos, Nucl.Phys. **B415** (1994) 116; Phys. Rev. **D49** (1994) 2933; Nucl. Phys. **B427** (1994) 245.
M. Duff and J. Kalkkinen, Signature Reversal Invariance, Nucl. Phys. **B758** (2006) 161 \[hep-th/0605273\]
M. Duff and J. Kalkkinen, Metric and Coupling Reversal in String Theory, Nucl. Phys. **B760** (2007) 64 \[hep-th/0605274\].
[^1]: The expression given is for the bosonic string. For superstrings or heterotic strings, the final term is different, but still zero in the critical dimension [@myers].
[^2]: There is an additional piece of this argument. It seems possible to choose a field basis such that $U\left( \phi,s\right) $ is always positive, for example, $U\left( \phi,s\right) =\left\vert \phi
^{2}-s^{2}\right\vert .$ But then, according to the homothety conditions (\[homothety\]), the metric $C_{ij}$ must also have an extra sign, $ds^{2}=sign\left( \phi^{2}-s^{2}\right) \left( -d\phi^{2}+ds^{2}\right)
.$ More examples of positive $U$ but complicated $C_{ij}\left( \phi,s\right)
$ are also possible as shown in [@BST-ConfCosm]. However, the problem with such alternative field bases with positive $U$ is that they are all geodesically incomplete, as demonstrated in our work with explicit analytic solutions of the field equations. So $U$ must be allowed to change sign.
[^3]: A straightforward Weyl rescaling of (\[Seff\]) to the Einstein frame in which only $G_{\mu\nu}$ is rescaled is also possible. But only the transformation of the metric produces a different expression for the transformed action. This is because the untransformed $H_{\mu\nu\lambda}$ or $B_{\mu\nu}$ differ form our definitions of $H_{\mu\nu\lambda}^{E}$ and $b_{\mu\nu}^{E}$ by the extra terms involving $b_{[\mu\nu}^{E}\partial
_{\lambda]}\ln\left\vert T^{E}\right\vert .$ After writing $b_{\mu\nu}^{E}$ in terms of $B_{\mu\nu}$ and expressing $\sigma$ in terms of $\Phi$ (see below) we do recover the same results.
[^4]: Weyl symmetry follows from 2T-physics in 4+2 dimensions as a requirement for its conformal shadow in 3+1 dimensions. Indeed, Weyl rescalings in 3+1 are simply local reparametrizations of the extra 1+1 dimensions as a function of the 3+1 dimensions [@2Tgravity]. In this connection see [@AB1-dualities] where an even larger concept of dualities in phase space (beyond just position space), that follow from 2T-physics, is discussed. \[foot2T\]
[^5]: We make some remarks here on the possibility that the worldsheet may also be allowed to flip signature. Although we will not pursue this to the end in this paper, it could be important to keep it in mind. The general solution of the $h_{ab}$ equation in (\[eoms\]) is expressed in several forms $$h^{ab}\left( X\right) =\hat{g}^{\mu\nu}\left( X\right) P_{\mu}^{a}P_{\nu
}^{b}\times\omega^{-1}\left( X\right) ,\text{ or }h_{ab}\left( X\right)
=\hat{g}_{\mu\nu}\left( X\right) \partial_{a}X^{\mu}\partial_{b}X^{\nu
}\times\omega\left( X\right) .\label{h-factor}$$ The undetermined conformal factor $\omega\left( X\right) $ of the worldsheet metric in (\[h-factor\]) is immaterial except for its sign because its absolute value always cancels in the worldsheet-Weyl invariant combination $\sqrt{-h}h^{ab}$ that appears in all equations. However, in view of the fact that the tension factor $T$ in the action (\[Sclass\]) can flip sign temporarily, we may entertain the possibility that the hitherto undetermined conformal factor $\omega\left( X\right) $ in (\[h-factor\]) could also flip sign, thus inducing $h^{ab}$ to flip or not to flip signature, perhaps simultaneously with $T.$ Since only the combination $T\left( X\right)
\sqrt{-h}h^{ab}g_{\mu\nu}$ appears everywhere, the sign flip of $T$ could be associated with $\hat{g}_{\mu\nu}$ as already mentioned above, but it is also possible that in the solution of the theory as in (\[h-factor\]) the flipping signature of $\hat{g}^{\mu\nu}$ could be canceled by a flipping sign of the factor $\omega\left( X\right) .$ Considering such solutions of the theory amounts to associating the sign of $T\left( X\right) $ with $h_{ab}$ to combine them into an $\hat{h}_{ab}\equiv sign(T)h_{ab}$ that flips signature at the gravity/antigravity boundaries, while leaving the signature of $g_{\mu\nu}$ unchanged. But to be able to admit such solutions, the string theory in Eq.(\[Sclass\]) needs to be defined from the outset such that its worldsheet is permitted to have such properties. The signature flip of the effective $\hat{h}_{ab}$ amounts to the interchange of the role of $\tau,\sigma$ as timelike/spacelike coordinates on the worldsheet as soon as the string crosses the gravity/antigravity boundary. It is worth emphasizing that this should not create any ghost problems since the worldsheet still has only one time coordinate before or after the flip. If the string theory is defined to include such properties, then the quantization of the theory and the computation of amplitudes via the string path integral would need to admit the possibility of such worldsheets. Since this possibility has not, as far as we know, been considered before, we shall not pursue it further here.
[^6]: The string phase space is $X^{\mu}\left( \sigma\right)
,P_{\mu}^{\tau}\left( \sigma\right) $ as defined above, and taken at fixed $\tau$ (no gauge choice for $X^{0}$). *Half* of the string phase space is $\left( X_{+}^{\mu}\left( \sigma\right) ,P_{-\mu}^{\tau}\left(
\sigma\right) \right) $ where the $\pm$ indicate parts of those string degrees of freedom that are symmetric and antisymmetric relative to the midpoint of the open string at $\sigma=\pi/2$ [@SFT-BR].
|
---
abstract: 'We study $tt^*$-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the holomorphic sectional curvature of this hermitian metric, which is the analogue of a similar result for classifying spaces of pure polarized Hodge structures.'
author:
- Claus Hertling
- Christian Sevenheck
title: Curvature of classifying spaces for Brieskorn lattices
---
[^1]
Introduction
============
\[secIntroduction\]
In this paper, we study a generalization of variations of Hodge structures and the associated period maps. These generalizations are called TERP-structures; they first appeared under the name topological-antitopological fusion (also called $tt^*$-geometry) in [@CV1; @CV2; @Du] and were rigourously defined and studied in [@He4] and [@HS1].
An important situation where TERP-structures naturally occur, is the theory of ($\mu$-constant families) of isolated hypersurface singularities, and more specifically, the Fourier-Laplace transformation of their Brieskorn lattices. In this case the TERP-structures are regular singular. Irregular TERP-structures arise by a similar though more general construction where the initial object is a regular function on an affine variety. These functions appear as mirror partners of the quantum cohomology algebra of smooth projective varieties or more generally, orbifolds. It is a challenging problem to study the induced TERP-structures on the quantum cohomology side, although progress seems to have been made very recently in this direction ([@Ir]). Let us notice that TERP-structures are intimately related to the theory of harmonic bundles, via the so called twistor structures, i.e. (families of) holomorphic bundles on ${{\mathds P}}^1$. Any TERP-structure gives rise to a twistor which is called pure if it is a trivial bundle on ${{\mathds P}}^1$ and pure polarized if a naturally defined hermitian metric on its space of global sections is positive definite. This has to be seen as a generalization of the notion of variations of (pure polarized) Hodge structures. By a basic result of Simpson ([@Si5]), variations of pure polarized twistor structures are equivalent to harmonic bundles on the parameter space. Given a variation of TERP-structures on a complex manifold, one obtains a variation of pure polarized twistor structures resp. a harmonic bundle on an open subset of this manifold, which is a union of connected components of the complement of a real analytic subvariety. Notice also ([@Sa8]) that the TERP-structure of a tame function on an affine manifold is always pure polarized.
The main topic of this paper are the classifying spaces that appear as targets of period maps of variations of regular singular TERP-structures. In fact, these spaces were already investigated under a different name (as classifying spaces of Brieskorn lattices) in [@He2]. The main new point treated here is to show how $tt^*$-geometry arises on the classifying spaces and to prove the analogue of a crucial result in classical Hodge theory (see [@Sch], [@GSch1], [@GSch2] and [@De3]): the negativity of the sectional curvature of the Hodge metric in horizontal directions. Similarly to the situation in Hodge theory, we expect this result to be a cornerstone in the study of the above mentioned period maps. We prove a few quite direct consequences of our result at the end of this paper.
Let us give a short overview on the content of this article. In section \[secClassSpace\], we recall the basic definitions both of variations of TERP-structures and of the classifying spaces for Brieskorn lattices resp. regular singular TERP-structures. In order to do that, we also recall the construction of the polarized mixed Hodge structure and its cohomological invariants, the spectral numbers associated to a regular singular TERP-structure. In section \[secTangent\], we construct a Kodaira-Spencer map from the tangent bundle of the classifying space to some auxiliary bundle which gives a local trivialization of the tangent bundle needed later. In particular, this induces a positive definite hermitian metric on the pure polarized part of the classifying space. We also consider the subsheaf of the tangent bundle of the classifying space consisting of horizontal directions. Contrary to the case of Hodge structures, it is not locally free in general. Finally, in section \[secHolSectCurv\] the main result of the paper is stated and proved. The proof is considerably more complicated than in the case of Hodge structures as the classifying spaces of TERP-structures/Brieskorn lattices are not homogenous. We finish the paper by deducing from our main theorem a rigidity result for variations of TERP-structures on affine spaces.
#### Notations:
For a complex manifold $X$, we write ${\mathcal{E}}\in{\textit{VB}}_X$ for a locally free sheaf of ${\mathcal{O}}_X$-modules ${\mathcal{E}}$, the associated vector bundle is denoted by $E$. If $E$ comes equipped with a flat connection, we denote by $E^\nabla$ the corresponding local system.
Classifying spaces
==================
\[secClassSpace\]
In this section we introduce the classifying spaces of regular singular TERP-structures which were considered, under a different name, in [@He2]. We start by recalling very briefly the basic definition of a TERP-structure and some of its associated data. After this, we give the definition of the classifying spaces.
For the following basic definition we also refer to [@He4] and [@HS1].
Let $X$ be a complex manifold and $w$ an integer. A variation of TERP-structures on $X$ of weight $w$ consists of a holomorphic vector bundle $H$ on ${{\mathds C}}\times X$, an integrable connection $\nabla:{\mathcal{H}}\rightarrow {\mathcal{H}}\otimes\Omega^1_{{{\mathds C}}\times X}(*\{0\}\times X)$, a flat real subbundle $H'_{{\mathds R}}$ of maximal rank of the restriction $H':=H_{|{{\mathds C}}^*\times X}$ and a flat non-degenerate $(-1)^w$-symmetric pairing $P:{\mathcal{H}}'\otimes
j^*{\mathcal{H}}' \rightarrow {\mathcal{O}}_{{{\mathds C}}^*\times X}$, where $j(z,t):=(-z,t)$, subject to the following conditions:
1. $\nabla$ has a pole of type one (also called of Poincaré rank one) along $\{0\}\times X$, i.e., the sheaf ${\mathcal{H}}$ is stable under $z^2\nabla_z$ and $z\nabla_T$ for any $T\in p^{-1}{\mathcal{T}}_X$, where $p:{{\mathds C}}\times X\twoheadrightarrow X$.
2. $P$ takes values in $i^w{{\mathds R}}$ on $H'_{{\mathds R}}$
3. $P$ extends as a non-degenerate pairing $P:{\mathcal{H}}\otimes j^*{\mathcal{H}}\rightarrow z^w{\mathcal{O}}_{{{\mathds C}}\times X}$, in particular, it induces a non-degenerate symmetric pairing $[z^{-w}P]:{\mathcal{H}}/z{\mathcal{H}}\otimes{\mathcal{H}}/z{\mathcal{H}}\rightarrow{\mathcal{O}}_X$.
${(H,H'_{{\mathds R}},\nabla,P,w)}$ is called regular singular, if $({\mathcal{H}},\nabla)$ is regular singular along $\{0\}\times X$, i.e, if sections of ${\mathcal{H}}$ have moderate growth along $\{0\}\times X$ compared to flat sections of ${\mathcal{H}}'$.
The case $X=\{pt\}$ is referred to as a single TERP-structure. There is a canonically associated set of data, which we call “topological”.
\[defTopDataTERP\] Let ${(H,H'_{{\mathds R}},\nabla,P,w)}$ be a TERP-structure, then we put $$H^\infty:=\{\textup{flat multivalued sections of }{\mathcal{H}}'\}.$$ We let $H^\infty_{{\mathds R}}$ be the subspace of real flat multivalued sections, then $H^\infty_{{\mathds R}}$ comes equipped with the monodromy endomorphism $M\in\mathit{Aut}(H^\infty_{{\mathds R}})$, which decomposes as $M=M_s\cdot M_u$ into semi-simple and unipotent part. Let $H^\infty:=\oplus H^\infty_\lambda$ be the decomposition into generalized eigenspaces with respect to $M$.
We restrict here to the case where all eigenvalues have absolute value $1$, as this is automatically the case for *mixed* TERP-structures, as defined in definition \[defMixedTERP\]. We put $H^\infty_{\neq 1}:=\oplus_{\lambda \neq 1} H^\infty_\lambda$, so that $H^\infty=H^\infty_1 \oplus H^\infty_{\neq 1}$, and let $N:=\log(M_u)$ be the nilpotent part of $M$.
$P$ induces a polarizing form $S$ on $H^\infty$ defined as follows: First note that $P$ corresponds (after a counter-clockwise shift in the second argument) to a pairing $L$ on the local system $(H')^\nabla$, then given $A,B\in H^\infty$, we put $S(A,B):= (-1)(2\pi i)^w L(A, t(B))$ where $$t(B)=
\left\{
\begin{array}{c}
(M-\mathit{Id})^{-1}(B) \;\;\;\;\;\; \forall B\in H^\infty_{\neq 1}\\ \\
-(\sum\limits_{k\geq1}\frac{1}{k!}N^{k-1})^{-1}(B)\;\;\;\;\;\; \forall B\in H^\infty_1.
\end{array}
\right.$$ $S$ is nondegenerate, monodromy invariant, $(-1)^w$-symmetric on $H^\infty_1$, $(-1)^{w-1}$-symmetric on $H^\infty_{\neq 1}$, and it takes real values on $H^\infty_{{\mathds R}}$ [@He4 Lemma 7.6]. We call the tuple $(H^\infty, H^\infty_{{\mathds R}}, M, S, w)$ the topological data of ${(H,H'_{{\mathds R}},\nabla,P,w)}$.
Note that by [@HS1 lemma 5.1] the topological data are equivalent to the data $(H',H'_{{\mathds R}}, \nabla, P, w)$.
Let us now suppose that ${(H,H'_{{\mathds R}},\nabla,P,w)}$ is regular singular. Then the following classical objects will play a key role in the sequel of this paper.
\[defHodgeFiltSpectrum\] Let ${(H,H'_{{\mathds R}},\nabla,P,w)}$ be a regular singular TERP-structure.
1. Define for any $\alpha\in{{\mathds C}}$, $C^\alpha:=z^{\alpha\mathit{Id}-\frac{N}{2\pi i}}H^\infty_{e^{-2\pi i \alpha}}
\subset i_*({\mathcal{H}}')_0$ to be the space of elementary sections of $H'$ of order $\alpha$. Let $V^{\alpha}$ (resp. $V^{>\alpha}$) the free ${\mathcal{O}}_{{\mathds C}}$-module generated by elementary sections of order at least (resp. strictly greater than) $\alpha$, i.e. $V^\alpha:=\sum_{\beta\geq\alpha} {\mathcal{O}}_{{\mathds C}}C^\beta$ and $V^{>\alpha}:=\sum_{\beta>\alpha} {\mathcal{O}}_{{\mathds C}}C^\beta$. The meromorphic bundle (i.e., locally free ${\mathcal{O}}_{{\mathds C}}[z^{-1}]$-module) $V^{>-\infty}$ is defined as $V^{>-\infty}:=\bigcup_\alpha V^\alpha$. Any $V^\alpha$ and $V^{>\alpha}$ is a lattice inside $V^{>-\infty}$ and the decreasing filtration $V^\bullet$ is called Kashiwara-Malgrange-filtration (or V-filtration) of $V^{>-\infty}$.
Notice that the objects $C^\alpha, V^\alpha, V^{>\alpha}$ and $V^{>-\infty}$ only depend on the topological data of the TERP-structure, i.e. on $(H',H'_{{\mathds R}}, \nabla, P, w)$, but not on the extension $H$ of the vector bundle $H'$ on ${{\mathds C}}^*$ to a vector bundle on ${{\mathds C}}$.
2. The regularity assumption on $(H,\nabla)$ can be rephrased by saying that ${\mathcal{H}}\subset V^{>-\infty}$. The V–filtration induces a filtration on ${\mathcal{H}}$, which is used to define a decreasing filtration on the space $H^\infty$ in the following way. Define for any $\alpha\in(0,1]+i{{\mathds R}}$ $$F^pH^\infty_{e^{-2\pi i \alpha}}:=z^{p+1-w-\alpha+\frac{N}{2\pi i}}Gr_V^{\alpha+w-1-p}{\mathcal{H}},$$ then $F^\bullet$ is a decreasing exhaustive filtration on $H^\infty$. We will use a twisted version of this filtration, which is obtained as $\widetilde{F}^\bullet:=G^{-1}F^\bullet$, where $G:=\sum_{\alpha \in(0,1]+i{{\mathds R}}} G^{(\alpha)} \in
\operatorname{\textup{Aut}}\left(H^\infty=\oplus_{\alpha}
H^\infty_{e^{-2\pi i\alpha}}\right) $ is defined as follows (see [@He4 (7.47)]): $$G^{(\alpha)} := \sum_{k\geq 0}\frac{1}{k!}\Gamma^{(k)}(\alpha)
\left( \frac{-N}{2\pi i}\right)^k
=: \Gamma \left(\alpha\cdot id - \frac{N}{2\pi i}\right) .$$ Here $\Gamma^{(k)}$ is the $k$-th derivative of the gamma function. In particular, $G$ depends only on $H'$ and induces the identity on $\operatorname{Gr}^W_\bullet$ where $W_\bullet$ is the weight filtration of the nilpotent endomorphism $N$. Note that the restriction of $W_\bullet(N)$ to $H^\infty_1$ is by definition centered around $w$, and the restriction to $H^\infty_{\neq 1}$ is centered around $w-1$.
$\widetilde{F}^\bullet$ is the Hodge filtration of Steenbrink if the TERP-structure is defined by an isolated hypersurface singularity.
As a matter of notation, we also write $\widetilde{F}^\bullet_{\mathcal{H}}$ for the filtration $\widetilde{F}^\bullet$ on $H^\infty$ defined by ${\mathcal{H}}$.
3. The $V$-filtration is also used to define the spectrum of a regular singular TERP-structure ${(H,H'_{{\mathds R}},\nabla,P,w)}$. Namely, let $\operatorname{\textup{Sp}}(H,\nabla) = \sum_{\alpha \in{{\mathds C}}}d(\alpha)\cdot\alpha\in{{\mathds Z}}[{{\mathds C}}]$ where $$\label{eqSpectrum}
d(\alpha):=\dim_{{\mathds C}}\left(\frac{Gr^\alpha_V {\mathcal{H}}}{Gr^\alpha_V z{\mathcal{H}}}\right)
=\dim_{{\mathds C}}\operatorname{Gr}_F^{\lfloor w-\alpha\rfloor} H^\infty_{e^{-2\pi i \alpha}}.$$ It is a tuple of $\mu$ complex numbers $\alpha_1 \leq \ldots \leq \alpha_\mu$. By definition, $d(\alpha)\neq 0$ only if $e^{-2\pi i \alpha}$ is an eigenvalue of $M$. We have the symmetry property $\alpha_1 + \alpha_\mu = w$. In most applications the eigenvalues of $M$ are roots of unity so that the spectrum actually lies in ${{\mathds Z}}[{{\mathds Q}}]$.
The following notion is quoted from [@HS1], where it is shown to correspond to “nilpotent orbits” of TERP-structures.
\[defMixedTERP\] A regular singular TERP-structure ${(H,H'_{{\mathds R}},\nabla,P,w)}$ of weight $w$ is called mixed if the tuple $$(H^\infty_{\neq 1}, (H^\infty_{\neq 1})_{{\mathds R}}, -N, S, \widetilde{F}^\bullet)
\mbox{ resp. }
(H^\infty_1, (H^\infty_1)_{{\mathds R}}, -N, S, \widetilde{F}^\bullet)$$ is a polarized mixed Hodge structure of weight $w-1$ resp. of weight $w$. We refer to [@He4] or [@HS1] for the notion of a polarized mixed Hodge structure (PMHS for short) used here. The data here are $M_s$-invariant. In [@HS1 lemma 5.9] it is shown that any semi-simple automorphism of a PMHS has eigenvalues in $S^1$. This is compatible with our assumptions in definition 2.2. that $M_s$ has all its eigenvalues in $S^1$ and justifies this assumption.
Next we reformulate the definitions of the classifying spaces $D_\mathit{BL}$ resp. $\operatorname{\mathit{D}_{\mathit{PMHS}}}$ for Brieskorn lattices resp. PMHS from [@He2] in terms of regular singular TERP-structures. We start with a PMHS of one single weight $w$ with a semisimple automorphism. As we have seen, a mixed TERP-structure defines a sum of PMHS’s of different weights on $H^\infty_1\oplus H^\infty_{\neq 1}$, so that later we need a slight adjustment of this situation (this is done in definition \[defClassSpaces\]).
The next lemma gives an equivalence of conditions for a filtration to induce a PMHS.
\[lemCondInDcPMHS\] Let $(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0)$ be a PMHS of weight $w$ and let $M_s$ be a semisimple automorphism of it. Then the eigenvalues of $M_s$ are elements of $S^1$. Let $W_\bullet$ be the weight filtration centered at weight $w$ which is induced by $N$. Let $P_{l}$ be the primitive subspace $P_l:=\ker (N^{l-w+1}:\operatorname{Gr}^W_{l}\to \operatorname{Gr}^W_{2w-l-2})$ of $\operatorname{Gr}^W_l$ (for $l\geq w$) and let $G_{{\mathds C}}$ be the group $G_{{\mathds C}}:=\operatorname{\textup{Aut}}(H^\infty,N,S,M_s)$. The primitive subspace $P_l$ decomposes into the eigenspaces of $M_s$, $P_l=\bigoplus_\lambda P_{l,\lambda}$. Then for any $M_s$-invariant filtration $F^\bullet$ on $H^\infty$, the following conditions are equivalent.
1. $\dim F^pP_{l,\lambda}=\dim F^p_0P_{l,\lambda},
\ N(F^p)\subset F^{p-1}$, $F^pN^jP_l=N^jF^{p+j}P_l$,\
$F^p\operatorname{Gr}^W_l=\bigoplus_{j\geq 0} F^pN^jP_{l+2j}$, $S(F^p,F^{w+1-p})=0$.
2. There exists an $M_s$-invariant common splitting $\widetilde{I}^{p,q}$ of $F^\bullet$ and $W_\bullet$ with the properties in lemma 2.3 (a)–(d) in [@He2].
3. $F^\bullet$ is the image of $F^\bullet_0$ by an element of $G_{{\mathds C}}$.
4. $\dim F^pP_{l,\lambda}=\dim F^p_0P_{l,\lambda},\ S(F^p,F^{w+1-p})=0$, and all powers of $N$ are strict with respect to $F^\bullet$.
As to the proof, let us just remark that obviously 2. implies 1., 3., and 4. The equivalence of 1., 2. and 3. is proved in [@He2 Ch. 2]. The only remaining point is that 4. implies 1.-3., which is rather technical. As we will not use the characterization 4., it is skipped here.
Also the following is proved in [@He2 chapter 2].
\[propClassSpacesOneWeight\] In the situation of lemma \[lemCondInDcPMHS\], consider the set $$\begin{aligned}
\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w) :=
\big\{\textup{filtrations }F^\bullet H^\infty\ |\ F^\bullet H^\infty
\textup{ is }M_s\textup{-invariant}\\
\textup{and satisfies the equivalent conditions in lemma \ref{lemCondInDcPMHS}}\big\}.\end{aligned}$$ It is a complex homogeneous space on which $G_{{\mathds C}}$ acts transitively. The set $$\begin{aligned}
\operatorname{\mathit{D}_{\mathit{PMHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w) :=
\big\{\textup{filtrations }F^\bullet H^\infty\ |\ F^\bullet H^\infty
\textup{ is }M_s\textup{-invariant,}\hspace*{2cm}\\
\dim F^pP_{l,\lambda}=\dim F^p_0P_{l,\lambda},
\ (H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet,M_s)\textup{ is a PMHS of weight }w\big\}\end{aligned}$$ is an open submanifold of $\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$ and a real homogeneous space with transitive action by a certain real group lying in between $G_{{\mathds C}}$ and $G_{{\mathds R}}=\operatorname{\textup{Aut}}(H^\infty_{{\mathds R}},N,S,M_s)$. It is a classifying space for the $M_s$-invariant PMHS with the same discrete data as the reference PMHS defined by $F^\bullet_0$.
Consider for $l\geq w$ the pairing $S_{l-w}:=S(-,N^l-)$ on $\operatorname{Gr}^W_l$, the primitive subspaces $P_l:=\mathit{Ker}(N^{l-w+1})\subset \operatorname{Gr}^W_l$. They decompose as $P_l=\oplus_\lambda P_{l,\lambda}$ into eigenspaces of $M_s$. Then we define for any $l\geq w$: $$\begin{array}{rcl}
\check{D}_l(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w)
& := & \big\{\textup{filtrations }F^\bullet P_l \,|\,F^p P_l \textup{ is }M_s
\textup{-invariant}, \\
& & \dim F^p P_{l,\lambda}=
\dim F_0^p P_{l,\lambda},\, S_{l-w}(F^pP_l ,F^{l-p+1}P_l)=0\big\}, \\ \\
D_l(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w)
& := & \left\{F^\bullet P_l \in \check{D}_l \,|\, F^\bullet P_l \textup{ gives
a PHS of weight }l\textup{ on }P_l \right\} ,\\ \\
\operatorname{\check{\mathit{D}}_{\mathit{PHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w) & := & \prod\limits_{l\geq w} \check{D}_l(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w) , \\ \\
\operatorname{\mathit{D}_{\mathit{PHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w) & := & \prod\limits_{l\geq w} D_l(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w).
\end{array}$$ $\operatorname{\check{\mathit{D}}_{\mathit{PHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w)$ is a projective manifold and a complex homogenous space, $\operatorname{\mathit{D}_{\mathit{PHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w)$ is an open submanifold and a real homogenous space. The projection $$\begin{array}{rcl}
\pi_\mathit{PMHS}:\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w) & \longrightarrow & \operatorname{\check{\mathit{D}}_{\mathit{PHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w) \\ \\
F^\bullet & \longmapsto & \prod_{l\geq w} F^\bullet P_l
\end{array}$$ is an affine fiber bundle with fibre isomorphic to ${{\mathds C}}^{N_\mathit{PMHS}}$ for some $N_\mathit{PMHS}\in{{\mathds N}}\cup\{0\}$. The classifying space $\operatorname{\mathit{D}_{\mathit{PMHS}}}$ is the restriction of the total space of this bundle to $\operatorname{\mathit{D}_{\mathit{PHS}}}$, in other words, we have the following diagram of projections and inclusions $$\begin{aligned}
\label{diagClassSpaces1}
\begin{matrix}
\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w) & \stackrel{\pi_\mathit{PMHS}}{\longrightarrow} & \operatorname{\check{\mathit{D}}_{\mathit{PHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w)\\
\cup & & \cup \\
\operatorname{\mathit{D}_{\mathit{PMHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w) & \longrightarrow & \operatorname{\mathit{D}_{\mathit{PHS}}}(H^\infty,H^\infty_{{\mathds R}},N,S,F^\bullet_0,M_s,w).
\end{matrix}\end{aligned}$$
The next definition introduces the main objects of this paper, namely, the classifying spaces of regular singular TERP-structures. The most direct way to fix the data needed to define these spaces is to consider a reference TERP-structure $(H^{(0)},
H'_{{\mathds R}}, \nabla, P, w)$, which is supposed to be mixed. This defines the set of discrete data needed, among them are the topological data of $(H^{(0)},
H'_{{\mathds R}}, \nabla, P, w)$ as well as its spectral numbers.
\[defClassSpaces\] Let $(H^{(0)},H'_{{\mathds R}},\nabla,P,w)$ be a regular singular mixed TERP-structure. Consider its topological data $(H^\infty, H_{{\mathds R}}^\infty, M, S, w)$ as defined in definition \[defTopDataTERP\]. We also have the filtration $\widetilde{F}^\bullet_0:=\widetilde{F}^\bullet_{{\mathcal{H}}^{(0)}}$ and the spectral numbers of ${\mathcal{H}}^{(0)}$ as in definition \[defHodgeFiltSpectrum\]. Then, using proposition \[propClassSpacesOneWeight\] we define $$\begin{array}{l}
\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}:= \operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}(H^\infty_1,(H^\infty_{{\mathds R}})_1,-N,S,\widetilde{F}^\bullet_0,\operatorname{id},w)
\\ \\ \quad \quad \quad \quad \times \operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}(H^\infty_{\neq 1},(H^\infty_{{\mathds R}})_{\neq 1},-N,S,\widetilde{F}^\bullet_0,M_s,w-1)
\end{array}$$ and similarly $\operatorname{\mathit{D}_{\mathit{PMHS}}}$, $\operatorname{\check{\mathit{D}}_{\mathit{PHS}}}$, $\operatorname{\mathit{D}_{\mathit{PHS}}}$.
Write, as before, $H':=H^{(0)}_{|{{\mathds C}}^*}$. Then we are interested in all possible extensions of $H'$ to a vector bundle $H$ on ${{\mathds C}}$ making ${(H,H'_{{\mathds R}},\nabla,P,w)}$ into a TERP-structure and such that the associated filtration is an element in the space $\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$ just defined. This leads to the following definition. $$\begin{aligned}
{{\check D_{BL}}}& := & \big\{\textup{free }{\mathcal{O}}_{{{\mathds C}},0}\textup{-submodules }{\mathcal{H}}_0
\textup{ of }V^{>-\infty}_0\,|\, (z^2\nabla_z){\mathcal{H}}_0\subset{\mathcal{H}}_0, \\
\nonumber & & \;P({\mathcal{H}}_0,{\mathcal{H}}_0)=z^w{\mathcal{O}}_{{\mathds C}},\widetilde{F}^\bullet_{\mathcal{H}}\in\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}\big\} \\ \nonumber \\
D_\mathit{BL} & := & \left\{{\mathcal{H}}_0\in{{\check D_{BL}}}\,|\,\widetilde{F}^\bullet_{\mathcal{H}}\in\operatorname{\mathit{D}_{\mathit{PMHS}}}\right\}\end{aligned}$$
Notice first that as any element ${\mathcal{H}}_0$ in ${{\check D_{BL}}}$ defines an extension of the fixed (flat) bundle $H'$ on ${{\mathds C}}^*$, one may rephrase the definition of ${{\check D_{BL}}}$ by saying that its elements are the bundles $H$ on ${{\mathds C}}$ extending $H'$ such that ${(H,H'_{{\mathds R}},\nabla,P,w)}$ is a regular singular TERP-structure and such that the associated filtration $\widetilde{F}^\bullet_{\mathcal{H}}$ lies in the same classifying space as $F^\bullet_{{\mathcal{H}}^{(0)}}$. Similarly, $D_\mathit{BL}$ consists of bundles $H$ such that ${(H,H'_{{\mathds R}},\nabla,P,w)}$ is a regular singular mixed TERP-structure with $\widetilde{F}^\bullet_{{\mathcal{H}}^{(0)}}\in\operatorname{\mathit{D}_{\mathit{PMHS}}}$.
We already remarked that $\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}, \operatorname{\mathit{D}_{\mathit{PMHS}}}, \operatorname{\check{\mathit{D}}_{\mathit{PHS}}}, \operatorname{\mathit{D}_{\mathit{PHS}}}$ are complex manifolds and complex resp. real homogenous spaces. A priori, the above definition describes ${{\check D_{BL}}}$ resp. $D_\mathit{BL}$ only as a set with no obvious topological or analytical structure. However, one of the main results of [@He2] (see also [@He3 proof of theorem 12.8]) is that ${{\check D_{BL}}}$ has a natural structure of a complex manifold and that the projection $\pi_\mathit{BL}:{{\check D_{BL}}}\rightarrow \operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$, ${\mathcal{H}}\mapsto\widetilde{F}^\bullet_{{\mathcal{H}}}$ is a affine fibre bundle over $\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$ with fibres isomorphic to ${{\mathds C}}^{N_\mathit{BL}}$ for some $N_\mathit{BL}\in{{\mathds N}}\cup\{0\}$. Moreover $D_\mathit{BL}$ is the restriction of the map $\pi_\mathit{BL}$ to $\operatorname{\mathit{D}_{\mathit{PMHS}}}$. Notice that contrary to $\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$, $\operatorname{\mathit{D}_{\mathit{PMHS}}}$, $\operatorname{\check{\mathit{D}}_{\mathit{PHS}}}$ and $\operatorname{\mathit{D}_{\mathit{PHS}}}$, the spaces ${{\check D_{BL}}}$ and $D_\mathit{BL}$ are not homogenous. However, there is a good ${{\mathds C}}^*$-action on the fibers of $\pi_\mathit{BL}$, the corresponding zero section $\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}\hookrightarrow {{\check D_{BL}}}$ consists of the regular singular TERP-structures in ${{\check D_{BL}}}$ which are generated by elementary sections, see [@He2 theorem 5.6].
Notice also that it follows from the definition of the space ${{\check D_{BL}}}$ that all elements ${\mathcal{H}}\in {{\check D_{BL}}}$ have the same spectral numbers, namely those of ${\mathcal{H}}^{(0)}$. The reason for this is that the spectrum is determined by the topological data (more precisely, by the eigenvalues of $M$) and the filtration $\widetilde{F}^\bullet_{\mathcal{H}}$ (namely, by its Hodge numbers, i.e., the dimensions $\dim\operatorname{Gr}^{\lfloor w-\alpha \rfloor}_F H^\infty_{e^{-2\pi i \alpha}}$, see formula ). Moreover, the definition of the space $\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$ (condition 1. in lemma \[lemCondInDcPMHS\]) shows that these numbers are constant for all $F^\bullet\in\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$, namely they are equal to those of $\widetilde{F}^\bullet_{{\mathcal{H}}^{(0)}}$. We will denote all along this article these spectral numbers by $\alpha_1, \ldots, \alpha_\mu$, where $\mu:=\dim_{{\mathds C}}(H^\infty)$.
The following diagram completes diagram and visualizes how all of the above defined manifolds are interrelated. $$\begin{aligned}
\label{diagClassSpaces2}
\begin{matrix}{{\check D_{BL}}}& \stackrel{\pi_\mathit{BL}}{\longrightarrow} & \operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}&
\stackrel{\pi_\mathit{PMHS}}{\longrightarrow} & \operatorname{\check{\mathit{D}}_{\mathit{PHS}}}\\
\cup & & \cup & & \cup \\
D_\mathit{BL} & \longrightarrow & \operatorname{\mathit{D}_{\mathit{PMHS}}}&
\longrightarrow & \operatorname{\mathit{D}_{\mathit{PHS}}}.
\end{matrix}\end{aligned}$$
The tangent bundle and horizontal directions
============================================
\[secTangent\]
This section gives a concrete description of the tangent bundle of the manifold ${{\check D_{BL}}}$ using a Kodaira-Spencer map. This description is used in an essential way in the curvature calculation in the next section.
Consider the flat bundle $H'\in {\textit{VB}}_{{{\mathds C}}^*}$ (with real structure) and the pairing $P:{\mathcal{H}}'\otimes j^*{\mathcal{H}}'\rightarrow{\mathcal{O}}_{{{\mathds C}}^*}$ which correspond to $(H^\infty, H_{{\mathds R}}^\infty, S, M, w)$ by [@HS1 lemma 5.1]. By abuse of notation, we will denote by $H'$ (resp. ${\mathcal{H}}'$ for its sheaf of sections) the pullback of this bundle by the projection $\pi':{{\mathds C}}^*\times {{\check D_{BL}}}\rightarrow {{\mathds C}}^*$. This bundle comes equipped with an integrable connection, which is the pullback of the original connection of $H'\in{\textit{VB}}_{{{\mathds C}}^*}$. Similarly, $P$ pulls back to a pairing $(\pi')^*P:{\mathcal{H}}'\otimes j^*{\mathcal{H}}' \rightarrow {\mathcal{O}}_{{{\mathds C}}^*\times {{\check D_{BL}}}}$, which we denote, by abuse of notation, again by $P$. We also consider the pullbacks of the Deligne lattices $V^\alpha$ and $V^{>\alpha}$ and of the meromorphic bundle $V^{>-\infty}$ under the projection $\pi:{{\mathds C}}\times{{\check D_{BL}}}\rightarrow{{\check D_{BL}}}$. We write ${\mathcal{V}}^\alpha$, ${\mathcal{V}}^{>\alpha}$ and ${\mathcal{V}}^{>-\infty}$ for the pull-backs $\pi^*V^\alpha$, $\pi^*{\mathcal{V}}^{>\alpha}$ and $\pi^*{\mathcal{V}}^{>-\infty}$, respectively. All of them are extensions of ${\mathcal{H}}'$ to ${{\mathds C}}\times{{\check D_{BL}}}$, i.e., we have ${\mathcal{V}}^\alpha,{\mathcal{V}}^{>\alpha},{\mathcal{V}}^{>-\infty}\subset i_*{\mathcal{H}}'$, where $i:{{\mathds C}}^*\times{{\check D_{BL}}}\hookrightarrow {{\mathds C}}\times{{\check D_{BL}}}$ is the inclusion. Notice however that ${\mathcal{V}}^\alpha$ and ${\mathcal{V}}^{>\alpha}$ are ${\mathcal{O}}_{{{\mathds C}}\times{{\check D_{BL}}}}$-locally free, whereas ${\mathcal{V}}^{>-\infty}$ is only ${\mathcal{O}}_{{{\mathds C}}\times{{\check D_{BL}}}}[z^{-1}]$-locally free. By definition, we have a connection operator $$\nabla:{\mathcal{V}}^\alpha\longrightarrow{\mathcal{V}}^\alpha\otimes\Omega^1_{{{\mathds C}}\times{{\check D_{BL}}}}(\log\,\{0\}\times{{\check D_{BL}}}),$$ and similarly for ${\mathcal{V}}^{>\alpha}$.
From the very definition of the space ${{\check D_{BL}}}$, we see that there is another naturally defined extension of ${\mathcal{H}}'$ to ${{\mathds C}}\times{{\check D_{BL}}}$, which we call ${\mathcal{L}}$. It is the universal family of Brieskorn lattices, i.e., ${\mathcal{L}}_{|{{\mathds C}}\times\{t\}} = {\mathcal{H}}(t)$ for any $t\in{{\check D_{BL}}}$. It can equally be described by gluing the locally defined bundles over ${{\mathds C}}\times U$, $U\subset {{\check D_{BL}}}$, given by the bases constructed in [@He2 lemma 5.2 to theorem 5.6]. The pairing $P$ has the property that $P({\mathcal{L}},{\mathcal{L}})\subset z^w{\mathcal{O}}_{{{\mathds C}}\times{{\check D_{BL}}}}$ and that $z^{-w}P$ defines a non-degenerate symmetric pairing on ${\mathcal{L}}/z{\mathcal{L}}$. In particular, the original $P$ on ${\mathcal{H}}'$ takes values in ${\mathcal{O}}_{{{\mathds C}}\times{{\check D_{BL}}}}[z^{-1}]$ when restricted to ${\mathcal{V}}^{>-\infty}$.
By definition, ${\mathcal{L}}$ comes equipped with a connection $$\nabla:{\mathcal{L}}\rightarrow{\mathcal{L}}\otimes
\left(z^{-2}\Omega^1_{{{\mathds C}}\times{{\check D_{BL}}}/{{\check D_{BL}}}}
\oplus
\Omega^1_{{{\mathds C}}\times{{\check D_{BL}}}/{{\mathds C}}}(*\{0\}\times{{\check D_{BL}}})\right).$$ Using the Deligne extensions ${\mathcal{V}}^\alpha$, we can give a precise statement on the pole order of $\nabla$ on ${\mathcal{L}}$. Define $n:=[\alpha_\mu-\alpha_1]$, note that if $n=0$ then the classifying space ${{\check D_{BL}}}$ consists of a single element only, namely, the lattice $V^{\alpha_1}$.
\[lemPoleOrder-n\] Suppose that $n>0$. Then ${\mathcal{L}}$ is stable under $z^n\nabla_X$ for any $X\in p^{-1}{\mathcal{T}}_{{\check D_{BL}}}$, i.e., we have a connection operator $$\nabla:{\mathcal{L}}\longrightarrow{\mathcal{L}}\otimes
\left(
z^{-2}\Omega^1_{{{\mathds C}}\times{{\check D_{BL}}}/{{\check D_{BL}}}}
\oplus
z^{-n}\Omega^1_{{{\mathds C}}\times{{\check D_{BL}}}/{{\mathds C}}}
\right)$$
As explained before, any ${\mathcal{H}}\in {{\check D_{BL}}}$ has the spectral numbers $\alpha_1,\ldots,\alpha_\mu$. It follows in particular hat $V^{\alpha_1} \supset {\mathcal{H}}\supset V^{>\alpha_\mu-1}$. The first inclusion is obvious, for the second, if we had any $s\in V^{>\alpha_\mu-1}$ which is not a section of ${\mathcal{H}}$, then there would be a $k\in{{\mathds N}}_{>0}$ with $z^k s \in {\mathcal{H}}$ which implies (if we take a minimal such $k$) that the principal part of $z^k s$ does not vanish in $Gr^V_\bullet({\mathcal{H}}/z{\mathcal{H}})$. In other words, we would get a spectral number larger then $\alpha_\mu$, which is impossible. As these inclusions of lattices hold true at any point of ${{\check D_{BL}}}$, we have $${\mathcal{V}}^{\alpha_1}\supset{\mathcal{L}}\supset{\mathcal{V}}^{>\alpha_\mu-1}.$$ Now let $s$ be any local section of ${\mathcal{L}}$, then $s\in {\mathcal{V}}^{\alpha_1}$, so that $$\nabla_X(s)\subset {\mathcal{V}}^{\alpha_1}
\subset {\mathcal{V}}^{>-n+\alpha_\mu-1}=z^{-n}{\mathcal{V}}^{>\alpha_\mu-1}\subset z^{-n}{\mathcal{L}}.$$
The following lemma will be useful for the proof of lemma \[lemKS\], but it will also be used in the computations in the next section.
\[lemCanExtBase\] Consider a module ${\mathcal{H}}\in {{\check D_{BL}}}$ and a local ${\mathcal{O}}_{{\mathds C}}$-basis $\underline{v}^{(0)}:=(v^{(0)}_1,\ldots,v^{(0)}_\mu)$ of ${\mathcal{H}}$.
1. There exists a small neighborhood $U_1\times U_2$ of $(0,{\mathcal{H}})\in {{\mathds C}}\times{{\check D_{BL}}}$ and a unique basis $\underline{v}:=(v_1,\dots,v_\mu)$ of ${\mathcal{L}}_{|U_1\times U_2}$ which is an extension of $\underline{v}^{(0)}$ and which satisfies $$\begin{aligned}
\label{v=v_0}
\underline{v}=\underline{v}^{(0)}\cdot
\left({{\mathds 1}}_\mu+\sum_{k=1}^n z^{-k} C_k\right)\end{aligned}$$ where $C_k\in M(\mu\times \mu,{\mathcal{O}}_{{\check D_{BL}}}(U_2))$. Here $\underline{v}^{(0)}$ is extended to a section in $\pi^{-1}V^{>-\infty}\subset{\mathcal{V}}^{>-\infty}$, and this equation holds in ${\mathcal{V}}^{>-\infty}$, i.e., meromorphically along $z=0$.
2. If $z^{-w}P((\underline{v}^{(0)})^{tr},\underline{v}^{(0)})$ is a constant $\mu\times\mu$-matrix then so is $z^{-w}P((v)^{tr},v)$.
3. If $\underline{v}^{(0)}$ is a good basis for ${\mathcal{H}}$ in the sense of [@SM] (i.e., if it projects to a basis of the vector space $\operatorname{Gr}^V_\bullet({\mathcal{H}}/z{\mathcal{H}})$) then $\underline{v}$ is a good basis for ${\mathcal{L}}_{|U_1\times U_2}$. Here being a good basis for ${\mathcal{L}}_{|U_1\times U_2}$ can be expressed in two equivalent ways: Either we require that for any $t\in U_2$, the restriction of $\underline{v}$ to $U_1\times \{t\}$ is a good basis of the restriction ${\mathcal{L}}_{|U_1\times\{t\}}$ or we ask that $\underline{v}$ projects to a ${\mathcal{O}}_{U_2}$-basis of $\operatorname{Gr}^{\mathcal{V}}_\bullet({\mathcal{L}}/z{\mathcal{L}})$. These requirements are equivalent as the latter module is locally free due to the fact that the spectral numbers of ${\mathcal{L}}_{|{{\mathds C}}\times\{t\}}$ are the same for each $t\in U_2$.
<!-- -->
1. Consider a holomorphic extension $\underline{v}'=(v'_1,...,v'_\mu)$ of $\underline{v}^{(0)}$ in a suitable neighborhood $\Delta_\varepsilon\times U'_2$ of $(0,{\mathcal{H}})$. The matrix $\Psi$ with $\underline{v}'=\underline{v}^{(0)}\cdot \Psi$ is holomorphic and invertible on $\Delta^*_\varepsilon \times U'_2$ and defines a cocycle in $H^1({{\mathds P}}^1\times U'_2,
\mathit{GL}(\mu,{\mathcal{O}}^*_{{{\mathds P}}^1\times U'_2}))$ and thus a vector bundle on ${{\mathds P}}^1\times U'_2$. Because the restriction to ${{\mathds P}}^1\times \{{\mathcal{H}}\}$ is trivial, the restriction to ${{\mathds P}}^1\times U_2$ for some $U_2\subset U_2'$ is a family of trivial vector bundles on ${{\mathds P}}^1$. The Birkhoff factorization (see, e.g., [@Mal2 proposition 4.1]) yields unique matrices $\Psi_0\in
\Gamma(\Delta_\varepsilon\times U_2,{\mathcal{O}}^*_{\Delta_\varepsilon\times U_2})$ and $\Psi_\infty\in \Gamma(({{\mathds P}}^1\backslash\{0\})\times U_2,
{\mathcal{O}}^*_{({{\mathds P}}^1-\{0\})\times U_2})$ with $\Psi_\infty|_{\{\infty\}\times U_2}={{\mathds 1}}_\mu$ and $\Psi=\Psi_\infty\cdot \Psi_0^{-1}$. Consider $$\underline{v}:=\underline{v}^{(0)}\cdot \Psi_\infty=\underline{v}'\cdot \Psi_0.$$ As in the proof of lemma \[lemPoleOrder-n\], we conclude from $${\mathcal{L}}\subset{\mathcal{V}}^{\alpha_1}\subset{\mathcal{V}}^{>\alpha_\mu-1-n}=z^{-n}{\mathcal{V}}^{>\alpha_\mu-1} \subset z^{-n}{\mathcal{O}}_{{{\mathds C}}\times {{\check D_{BL}}}}\otimes_{{\mathcal{O}}_{{\mathds C}}}{\mathcal{H}}$$ that ${\mathcal{L}}\subset z^{-n}{\mathcal{O}}_{{{\mathds C}}\times {{\check D_{BL}}}}\otimes_{{\mathcal{O}}_{{\mathds C}}}{\mathcal{H}}$. It follows that the matrix $\Psi_\infty$ satisfies $\Psi_\infty={{\mathds 1}}_\mu+\sum_{k=1}^{n} z^{-k} C_{k}.$ Uniqueness is now also clear.
2. This follows from $z^{-w}P({\mathcal{L}},{\mathcal{L}}) = {\mathcal{R}}$ and $z^{-w}P(v_i,v_j)-z^{-w}P(v_i^{(0)},v_j^{(0)})\in z^{-1} {\mathcal{O}}_{U_2}[z^{-1}]$.
3. First we introduce two notations: within ${\mathcal{V}}^{>-\infty}$ and ${\mathcal{V}}^\alpha$ we consider the $\pi^{-1}{\mathcal{O}}_{{\check D_{BL}}}$-module ${\mathcal{C}}^\alpha$ consisting of elementary sections of order $\alpha$ on ${{\mathds C}}^*\times {{\check D_{BL}}}$. Then any $v_i\in {\mathcal{L}}_{|U_1\times U_2}$ can be written as a sum $v_i = \sum_{\beta\geq \alpha_1} s(v_i,\beta)$ where $s(v_i,\beta)\in {\mathcal{C}}^\beta({{\mathds C}}\times U_2)$.
That $\underline{v}^{(0)}$ is a good basis means that $v_i^{(0)}=\sum_{\beta\geq\alpha_i} s(v_i^{(0)},\beta)$ and that $$\operatorname{Gr}^V_\alpha{\mathcal{H}}=\bigoplus_{i,k:k\geq 0,\alpha_i+k=\alpha}
{\mathcal{O}}_{{\mathds C}}\cdot z^k\cdot s(v_i^{(0)},\alpha_i).$$ Then for any $\beta$ all sections $z^k\cdot s(v^{(0)}_i,\alpha_i)$ where $k\in {{\mathds Z}},\alpha_i+k=\beta$ are linearly independent. In a small neighborhood $U_3\subset U_2$ of ${\mathcal{H}}$ the sections $(z^k\cdot s(v_i,\alpha_i))_{k\in{{\mathds Z}},\alpha_i+k=\beta}$ inherit this property of being linearly independent.
For any $v_j$, define $\beta_j$ to be the unique complex number such that $s(v_j,\beta_j)\neq 0$ and $s(v_j,\beta)=0$ for $\beta<\beta_j$. We have $\beta_j\leq \alpha_j$. Formula and the linear independence of the $z^k\cdot s(v_i^{(0)},\alpha_i)$ show $$\begin{aligned}
\label{v=v_02}
s(v_j,\beta_j)= \delta_{\beta_j,\alpha_j}\cdot s(v_j^{(0)},\alpha_j)
+ \sum_{k,i,j:\alpha_i-k=\beta_j} (C_k)_{ij}\cdot z^{-k}\cdot s(v_i^{(0)},\alpha_i)\end{aligned}$$ (remember that this is an equation in ${\mathcal{V}}^{>-\infty}$, where the sections $v_j$ of ${\mathcal{H}}\subset V^{>-\infty}$ has been extended to sections in $\pi^{-1}V^{>-\infty} \subset {\mathcal{V}}^{>-\infty}$).
The main point is to show $$\begin{aligned}
\label{v=v_03}
(C_k)_{ij}=0 \qquad \textup{ for }\alpha_i-k<\alpha_j.\end{aligned}$$ Then $\beta_j=\alpha_j$ and $s(v_j,\alpha_j)\in \operatorname{Gr}^V_{\alpha_j} {\mathcal{L}}(U_2)$. From this and the linear independence of the $z^k\cdot s(v_i,\alpha_i)$ it follows that $\underline{v}$ is a good basis, first on a small $U_3\subset U_2$, then on all of $U_2$.
In order to show we argue indirectly. Suppose $(C_k)_{ij}\neq 0$ for some $\alpha_i-k<\alpha_j$ and suppose that $\alpha_i-k$ is minimal with this property. Then $\beta_j=\alpha_i-k$ for this $j$, and $\beta_l=\alpha_l$ for all $l$ with $\alpha_l\leq \alpha_i-k$. Then in a neighborhood $U_4\subset U_2$ of ${\mathcal{H}}$ for any $\gamma\leq \beta_j=\alpha_i-k$ $$\bigoplus_{l,m:m\geq 0,\alpha_l+m=\gamma} {\mathcal{O}}_{{{\mathds C}}\times U_4}\cdot z^m \cdot s(v_l,\alpha_l)$$ is a submodule of $\operatorname{Gr}^V_\gamma {\mathcal{L}}_{|U_4}$ of the same rank and thus coincides with $\operatorname{Gr}^V_\gamma {\mathcal{L}}_{|U_4}$. But in the case $\gamma=\beta_j=\alpha_i-k$ we have additionally $s(v_j,\beta_j)\in \operatorname{Gr}^V_\gamma {\mathcal{L}}_{|U_4}$, and by and by the linear independence, it is not a linear combination of the sections above. Thus $(C_k)_{ij}=0$ if $\alpha_i-k<\alpha_j$.
Next we give a concrete description of the tangent bundle of ${{\check D_{BL}}}$ using the universal bundle ${\mathcal{L}}$. For this purpose, we will introduce some auxiliary holomorphic bundles on ${{\check D_{BL}}}$. We will describe local bases of these bundles, these will be written as row vectors. We use the convention that given a (sheaf of) ${\mathcal{A}}$-module(s) ${\mathcal{N}}$, one can multiply matrices with entries of ${\mathcal{N}}$ with matrices with entries in ${\mathcal{A}}$ by scalar multiplication. Moreover, we make use of the tensor product of matrices, and in particular of the following rules, where the matrices involved are supposed to have the appropriate size. $$\begin{aligned}
(A\otimes B)\cdot(C\otimes D)=(A\cdot C)\otimes (B \cdot D), \\
(A\otimes B)^{-1}=A^{-1}\otimes B^{-1}, \\
(A\otimes B)^{tr}=A^{tr}\otimes B^{tr}, \\
\label{eqMatVecTens} (X\otimes Y) \cdot A^{vec} = (Y \cdot A \cdot X^{tr})^{vec}.\end{aligned}$$ In the last formula, we denote for any matrix $A\in M(m\times n, {\mathcal{A}})$ by $A^{vec}\in M(nm\times 1,{\mathcal{A}})$ the column vector obtained by stacking the columns of $A$ in a single one. Finally, we denote the sheaf of rings ${\mathcal{O}}_{{{\mathds C}}\times{{\check D_{BL}}}}$ by ${\mathcal{R}}$ and its localization along $\{0\}\times{{\check D_{BL}}}$ by ${\mathcal{R}}[z^{-1}]$.
Define ${\mathcal{M}}:={{\mathcal{E}}\!}nd_{{\mathcal{R}}[z^{-1}]}({\mathcal{V}}^{>-\infty})$, then ${\mathcal{M}}$ is a meromorphic bundle with connection induced from ${\mathcal{V}}^{>-\infty}$. This connection is obviously regular singular so that ${\mathcal{M}}$ carries its own $V$-filtration, characterized by ${\mathcal{V}}^0{\mathcal{M}}=\left\{\phi\in{\mathcal{M}}\,|\,\phi({\mathcal{V}}^\alpha)\subset{\mathcal{V}}^\alpha\,;\;\forall\alpha\right\}$. Consider the ${\mathcal{R}}$-submodule $\operatorname{\widetilde{{\mathcal{G}}}^\mathit{I}}:={{\mathcal{H}}\!}om_{\mathcal{R}}({\mathcal{L}},z^{-n}{\mathcal{L}})$ of ${\mathcal{M}}$, and the quotient $$\operatorname{{{\mathcal{G}}}^\mathit{I}}:=\frac{\operatorname{\widetilde{{\mathcal{G}}}^\mathit{I}}}{{{\mathcal{E}}\!}nd_{\mathcal{R}}({\mathcal{L}})}$$ $\operatorname{{{\mathcal{G}}}^\mathit{I}}$ is a $z$-torsion sheaf and can be identified with ${{\mathcal{H}}}\!om_{\mathcal{R}}({\mathcal{L}},z^{-n}{\mathcal{L}}/{\mathcal{L}})$. As an ${\mathcal{O}}_{{\check D_{BL}}}$-module, it is locally free of rank $n\mu^2$. Any section $v$ of the projection ${\mathcal{L}}\twoheadrightarrow k_*({\mathcal{L}}/z{\mathcal{L}})$ (here $k:{{\check D_{BL}}}\hookrightarrow {{\mathds C}}\times{{\check D_{BL}}}, t\mapsto(0,t)$) yields an isomorphism $$\operatorname{{{\mathcal{G}}}^\mathit{I}}\cong\left[{{\mathcal{E}}}\!nd_{{\mathcal{O}}_{{\check D_{BL}}}}({\mathcal{L}}^{sp,v})\right]^n$$ where ${\mathcal{L}}^{sp,v}:=\mathit{Im}(v)$. Any local basis $\underline{v}=(v_1,\ldots,v_\mu)$ of ${\mathcal{L}}$ in a neighborhood of a point $(0,t)\in{{\mathds C}}\times{{\check D_{BL}}}$ (in particular, this gives a section $v:k_*({\mathcal{L}}/z{\mathcal{L}})\rightarrow {\mathcal{L}}$ locally) yields a local basis of $\operatorname{{{\mathcal{G}}}^\mathit{I}}$, namely $(\underline{z}\otimes\underline{v}^*\otimes\underline{v})\in M(1\times n\mu^2,\operatorname{{{\mathcal{G}}}^\mathit{I}})$, where the symbol $\underline{z}$ denotes the vector $(z^{-1},\ldots,z^{-n})$. A local section $\phi\in\operatorname{{{\mathcal{G}}}^\mathit{I}}$ is written in this basis as $$\label{eqIndBasis}
\phi=\sum_{
\begin{array}{c}
\scriptstyle k=1,\ldots,n\\
\scriptstyle i,j=1,\ldots,\mu
\end{array}
}(\Delta_k)_{ji} z^{-k}\otimes v^*_i\otimes v_j
=(\underline{z}\otimes \underline{v}^* \otimes \underline{v})(\sum_{k=1}^n e_k \otimes \Delta^{vec}_k)$$ where $\Delta_k\in M(\mu\times\mu, {\mathcal{O}}_{{\check D_{BL}}})$ and $e_k\in M(n\times 1, {{\mathds C}})$ is the $k$-th unit vector.
We will define a chain $\operatorname{{\mathcal{G}}^\mathit{IV}}\subset\operatorname{{\mathcal{G}}^\mathit{III}}\subset\operatorname{{\mathcal{G}}^\mathit{II}}\subset\operatorname{{{\mathcal{G}}}^\mathit{I}}$ of subbundles of $\operatorname{{{\mathcal{G}}}^\mathit{I}}$, and an injective morphism ${\mathcal{T}}_{{\check D_{BL}}}\hookrightarrow\operatorname{{{\mathcal{G}}}^\mathit{I}}$ with image equal to $\operatorname{{\mathcal{G}}^\mathit{IV}}$. This will give the description of the tangent bundle alluded to above.
Put $$\begin{array}{rcl}
\operatorname{\widetilde{{\mathcal{G}}}^\mathit{II}}& := & \left\{\phi\in\operatorname{\widetilde{{\mathcal{G}}}^\mathit{I}}\,|\,P(\phi(a),b)+P(a,\phi(b))\in z^w{\mathcal{R}}\quad\forall\,a,b\in{\mathcal{L}}\right\},\\ \\
\operatorname{\widetilde{{\mathcal{G}}}^\mathit{III}}& := & \operatorname{\widetilde{{\mathcal{G}}}^\mathit{II}}\cap{\mathcal{V}}^0{\mathcal{M}}, \\ \\
\operatorname{\widetilde{{\mathcal{G}}}^\mathit{IV}}& := & \left\{\phi\in\operatorname{\widetilde{{\mathcal{G}}}^\mathit{III}}\,|\,\mathit{ad}(z^2\nabla_z)(\phi)=[z^2\nabla_z,\phi]\in{{\mathcal{E}}}\!nd_{\mathcal{R}}({\mathcal{L}})\right\},
\end{array}$$ and define $\operatorname{{\mathcal{G}}^\mathit{II}}, \operatorname{{\mathcal{G}}^\mathit{III}}$ resp. $\operatorname{{\mathcal{G}}^\mathit{IV}}$ to be the images of $\operatorname{\widetilde{{\mathcal{G}}}^\mathit{II}}, \operatorname{\widetilde{{\mathcal{G}}}^\mathit{III}}$ resp. $\operatorname{\widetilde{{\mathcal{G}}}^\mathit{IV}}$ in $\operatorname{{{\mathcal{G}}}^\mathit{I}}$.
From the definition it is clear that $\operatorname{{\mathcal{G}}^\mathit{II}}, \operatorname{{\mathcal{G}}^\mathit{III}}$ and $\operatorname{{\mathcal{G}}^\mathit{IV}}$ are ${\mathcal{O}}_{{\check D_{BL}}}$-coherent. The following result yields local bases for $\operatorname{{\mathcal{G}}^\mathit{II}}$ and $\operatorname{{\mathcal{G}}^\mathit{III}}$ showing that they are in fact locally free. The same is true for $\operatorname{{\mathcal{G}}^\mathit{IV}}$, but it is more complicated to give an explicit local base for that bundle. Instead, we give a characterization of the elements of $\operatorname{{\mathcal{G}}^\mathit{IV}}$. Its local freeness will be shown in lemma \[lemKS\].
Let $\underline{v}$ be a local basis of ${\mathcal{L}}$ as above and suppose moreover that $P^{mat}:=(P(v_i,v_j))=(\delta_{i+j,\mu+1})$. Then we have $$\operatorname{{\mathcal{G}}^\mathit{II}}\cong \bigoplus_{(k,i,j)\in N}
{\mathcal{O}}_{{\check D_{BL}}}z^{-k}\otimes \left(v_i^*\otimes v_j+(-1)^{k+1} v_{\mu+1-j}^*\otimes v_{\mu+1-i}\right),$$ where $$N:=\left\{(k,i,j)\in\{1,\ldots,n\}\times\{1,\ldots,\mu\}^2\,|\,i+j<\mu+1 \textup{ if } k \textup{ is even, }
i+j\leq\mu+1\textup{ if }k\textup{ is odd } \right\}.$$ Suppose moreover that $\underline{v}$ is a good basis in the sense of [@SM], i.e., that $\underline{v}$ induces a basis of $\operatorname{Gr}^\bullet_{\mathcal{V}}({\mathcal{L}}/z{\mathcal{L}})$. Order the elements of $\underline{v}$ in such a way that $v_i\in\operatorname{Gr}^{\alpha_i}_{\mathcal{V}}({\mathcal{L}}/z{\mathcal{L}})$. Then $$\operatorname{{\mathcal{G}}^\mathit{III}}\cong \bigoplus_{
\begin{array}{c}
\scriptscriptstyle (k,i,j)\in N \\
\scriptscriptstyle \alpha_i-k\geq\alpha_j
\end{array}
} {\mathcal{O}}_{{\check D_{BL}}}z^{-k} \left(v_i^*\otimes v_j+(-1)^{k+1} v_{\mu+1-j}^*\otimes v_{\mu+1-i}\right).$$ Although there is no simple choice for a basis of $\operatorname{{\mathcal{G}}^\mathit{IV}}$, its elements can be characterized as follows: An endomorphism $\phi \in \operatorname{\widetilde{{\mathcal{G}}}^\mathit{III}}$ lies in $\operatorname{\widetilde{{\mathcal{G}}}^\mathit{IV}}$ iff the ${\mathcal{R}}[\epsilon]/(\epsilon^2)$-module $$\widetilde{{\mathcal{L}}}:=\bigoplus_{i=1}^\mu{\mathcal{R}}[\varepsilon]/(\varepsilon^2)
\left(v_i+\varepsilon \phi(v_i)\right),$$ is stable under $z^2\nabla_z$.
The first point follows from the simple computation $$P(\phi(\underline{v})^{tr},\underline{v})+P(\underline{v}^{tr},\phi(\underline{v}))=
\underline{v}\cdot\sum_{k=1}^n z^{-k}\left(\Delta_k^{tr}\cdot P^{mat} + (-1)^k P^{mat} \cdot \Delta_k\right)
=0\quad\quad\textup{mod }{\mathcal{L}}$$ From the condition $\Delta_k^{tr}\cdot P^{mat} + (-1)^k P^{mat} \cdot \Delta_k=0$ one easily deduces the above bases of $\operatorname{{\mathcal{G}}^\mathit{II}}$ and $\operatorname{{\mathcal{G}}^\mathit{III}}$. For the description of $\operatorname{{\mathcal{G}}^\mathit{IV}}$, note that $\widetilde{{\mathcal{L}}}$ is stable under $z^2\nabla_z$ iff there is $B' \in M(\mu\times\mu,{\mathcal{O}}_{{\check D_{BL}}})$ such that $$(z^2\nabla_z)(\underline{v}+\varepsilon\phi(\underline{v}))=\underline{v}\cdot(B+\varepsilon B'),$$ where $(z^2\nabla_z)(\underline{v})=\underline{v}\cdot B$. In the ring ${\mathcal{R}}[\varepsilon]/(\varepsilon^2)$, this is equivalent to $$[z^2\nabla_z,\phi](\underline{v})=\underline{v}\cdot B',$$ i.e., to $\phi\in\operatorname{{\mathcal{G}}^\mathit{IV}}$.
We are now in the position to compare the bundle $\operatorname{{{\mathcal{G}}}^\mathit{I}}$ with the tangent bundle of ${{\check D_{BL}}}$. Define the following morphism: $$\begin{array}{rcl}
\operatorname{\mathit{KS}}:{\mathcal{T}}_{{\check D_{BL}}}& \longrightarrow & \operatorname{{{\mathcal{G}}}^\mathit{I}}\\ \\
X & \longmapsto & [v\in{\mathcal{L}}\mapsto \nabla_X v]
\end{array}$$ where the brackets on the right-hand side denote the class in the quotient $\operatorname{{{\mathcal{G}}}^\mathit{I}}$.
\[lemKS\] $\operatorname{{\mathcal{G}}^\mathit{IV}}$ is a bundle, and $\operatorname{\mathit{KS}}$ is a bundle isomorphism from ${\mathcal{T}}_{{\check D_{BL}}}$ to $\operatorname{{\mathcal{G}}^\mathit{IV}}$.
First we will prove that given $X\in{\mathcal{T}}_{{\check D_{BL}}}$, the covariant derivative $\nabla_X$ really defines an element in $\operatorname{{{\mathcal{G}}}^\mathit{I}}$. It was already shown that for any $v\in{\mathcal{L}}$, $\nabla_X v$ lies in $z^{-n}{\mathcal{L}}$. On the other hand, if $f\in{\mathcal{O}}_{{\check D_{BL}}}$, then $\nabla_X(f\cdot v)=f\nabla_X(v)+X(f)\cdot v$, but $X(f)\cdot v\in{\mathcal{L}}$, so modulo ${{\mathcal{E}}}\!nd({\mathcal{L}})$ we have $\nabla_X(f\cdot v)=f\nabla_X(v)$. Moreover, the flatness of $P$ implies that $P(\nabla_X(-),-)+P(-,\nabla_X(-))=XP(-,-)$, so $\mathit{Im}(KS)\subset\operatorname{{\mathcal{G}}^\mathit{II}}$ and it is clear that the ${\mathcal{V}}$-filtration is respected, i.e., that we get an element in ${\mathcal{V}}^0{\mathcal{M}}$, as we derive only in parameter direction. This proves $\mathit{Im}(\operatorname{\mathit{KS}})\subset\operatorname{{\mathcal{G}}^\mathit{III}}$. From $[z^2\nabla_z,\nabla_X]=0$ it follows immediately that $\mathit{Im}(\operatorname{\mathit{KS}})\subset\operatorname{{\mathcal{G}}^\mathit{IV}}$.
The last step of the proof is to show that $\operatorname{\mathit{KS}}$ maps ${\mathcal{T}}_{{\check D_{BL}}}$ isomorphically onto $\operatorname{{\mathcal{G}}^\mathit{IV}}$. Then it follows that $\operatorname{{\mathcal{G}}^\mathit{IV}}$ is locally free. We will use the construction of coordinates on the fibers resp. on the base of the projection ${{\check D_{BL}}}\to\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$ in [@He2 Ch. 5] resp. in [@He3 proof of theorem 12.8]. We will rephrase the outcome of these constructions, without going into details. From that it will follow that $\operatorname{\mathit{KS}}$ is a bundle isomorphism onto $\operatorname{{\mathcal{G}}^\mathit{IV}}$.
Having fixed a reference TERP-structure, we first consider the larger spaces $$\begin{aligned}
D_{Sp}&=& \{{\mathcal{H}}\subset V^{\alpha_1}\ |\ {\mathcal{H}}\textup{ is a free }
{\mathcal{O}}_{{\mathds C}}-\textup{module} \\
&& \hspace*{2.2cm} \textup{with spectral numbers } \alpha_1,...,\alpha_\mu\},\\
D_{Fl}&=& \{\widetilde{F}^\bullet_{\mathcal{H}}|\ {\mathcal{H}}\in D_{Sp}\},\end{aligned}$$ where $\widetilde{F}^\bullet_{\mathcal{H}}$ denotes the filtration defined by ${\mathcal{H}}$ on the space $H^\infty$ (see definition \[defHodgeFiltSpectrum\]). We have an obvious projection $D_{Sp}\to D_{Fl}$. Here $D_{Fl}$ is a flag manifold, and $D_{Sp}$ is also a manifold, with fibers isomorphic to some $N_{Sp}\in{{\mathds N}}$. Local coordinates on $D_{Sp}$ and $D_{Fl}$ can be chosen as follows: For some ${\mathcal{H}}\in D_{Sp}$ one fixes a good basis $\underline{v}^0$. The analogue of lemma \[lemCanExtBase\] holds and provides a unique good basis $\underline{v}$ in a neighborhood $U$ of ${\mathcal{H}}\in D_{Sp}$ where $$\begin{aligned}
\underline{v}=\underline{v}^0\cdot
\left({{\mathds 1}}_\mu+\sum_{k=1}^n z^{-k} C_k\right)\end{aligned}$$ with $(C_k)_{ij}=0$ if $\alpha_i-k<\alpha_j$. Now the $(C_k)_{ij}$ with $\alpha_i-k\geq\alpha_j$ are local coordinates on $D_{Sp}$ and those with $\alpha_i-k=\alpha_j$ are local coordinates on $D_{Fl}$.
The construction of coordinates on ${{\check D_{BL}}}$ in [@He2][@He3] amounts to the following: For ${\mathcal{H}}\in {{\check D_{BL}}}\subset D_{Sp}$ a very special good basis was chosen. It gives local coordinates $(C_k)_{ij}$ with $\alpha_i-k\geq\alpha_j$ on $D_{Sp}$. The conditions from the pairing $P$ and the pole of order $2$, which determine ${{\check D_{BL}}}$ in $D_{Sp}$ locally, were shown to give a set of equations in $(C_k)_{ij}$ whose linear parts are linearly independent. This proved the smoothness of ${{\check D_{BL}}}$.
Now the definition of $\operatorname{{\mathcal{G}}^\mathit{IV}}$ uses exactly these linear parts. This shows the injectivity of $\operatorname{\mathit{KS}}$ and that $\mathit{Im}(\operatorname{\mathit{KS}})=\operatorname{{\mathcal{G}}^\mathit{IV}}$.
By abuse of notation, we call $\operatorname{\mathit{KS}}$ the Kodaira-Spencer morphism, although this is the correct name only if we consider $\operatorname{\mathit{KS}}$ as an isomorphism between ${\mathcal{T}}_{{\check D_{BL}}}$ and $\operatorname{{\mathcal{G}}^\mathit{IV}}$.
It will be useful to have a local characterization of $\operatorname{{\mathcal{G}}^\mathit{IV}}$ in the basis of $\operatorname{{\mathcal{G}}^\mathit{III}}$ given above. Let $\phi=
(\underline{z}\otimes \underline{v}^* \otimes \underline{v})(\sum_{k=1}^n e_k \otimes \Delta^{vec}_k)$ be a local section of $\operatorname{{\mathcal{G}}^\mathit{III}}$, i.e., $(\Delta_k)_{ij}+(-1)^k(\Delta_k)_{\mu+1-j,\mu+1-i}=0$ and $(\Delta)_{ij}=0$ for all $\alpha_i-k<\alpha_j$. Then $\Delta\in\operatorname{{\mathcal{G}}^\mathit{IV}}$ iff there is $B'\in M(\mu\times\mu, {\mathcal{R}})$ such that $$(z^2\nabla_z)(\underline{v}+\varepsilon \phi(\underline{v})) = \underline{v}\cdot(B+\varepsilon B')$$ where $B\in M(\mu\times\mu, {\mathcal{R}})$ is defined by $(z^2\nabla_z)(\underline{v})=\underline{v}\cdot B$. This is equivalent to $$B'=[B,\sum_{k=1}^n z^{-k}\cdot \Delta_k]
+\sum_{k=1}^n (-k)z^{-k+1}\cdot \Delta_k.$$ $B'$ is required to be holomorphic, so that the coefficients of all strictly negative powers of $z$ in this equation must vanish. Writing $B=\sum_{k=0}^\infty z^k\cdot B_{k-1}$, there are a priori conditions for any $l=1,...,n$ (with $\Delta_{n+1}:=0$): $$\begin{aligned}
\label{eqCondForGIV}
0=(\textup{coefficient of }z^{-l})
=(-1-l)\Delta_{l+1}+ \sum_{k=l}^{n} [B_{k-1-l},\Delta_{k}].\end{aligned}$$ However, as we are working with a good basis $v$ of ${\mathcal{L}}$, it follows that $v_i\in{\mathcal{V}}^{\alpha_i}$ and $(z\nabla_z)v_i\in{\mathcal{V}}^{\alpha_i}$. This gives $(B_k)_{ij}=0$ for all $\alpha_i+k<\alpha_j$. Remember also that $(\Delta_k)_{ij}=0$ for $\alpha_i-k<\alpha_j$. The equation $[\Delta_n, B_{-1}]=0$ is thus trivially satisfied, so that the conditions are non-empty only for $l\in\{1,\ldots,n-1\}$.
We will now define the analogue of the subbundle of horizontal tangent directions on the classifying spaces $\operatorname{\check{\mathit{D}}_{\mathit{PHS}}}$ in the sense of [@Sch] for the space ${{\check D_{BL}}}$.
\[defHorBundles\] Define the following subsheaf of $\operatorname{{{\mathcal{G}}}^\mathit{I}}$: $$\operatorname{{\mathcal{G}}^{\mathit{I},hor}}:=\textup{Image of }{{\mathcal{H}}}\!om_{\mathcal{R}}({\mathcal{L}},z^{-1}{\mathcal{L}})\textup{ in }\operatorname{{{\mathcal{G}}}^\mathit{I}},$$ and put $\operatorname{{\mathcal{G}}^{\mathit{II},hor}}:=\operatorname{{\mathcal{G}}^{\mathit{I},hor}}\cap \operatorname{{\mathcal{G}}^\mathit{II}}$, $\operatorname{{\mathcal{G}}^{\mathit{III},hor}}:=\operatorname{{\mathcal{G}}^{\mathit{I},hor}}\cap \operatorname{{\mathcal{G}}^\mathit{III}}$ and $\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}:=\operatorname{{\mathcal{G}}^{\mathit{I},hor}}\cap \operatorname{{\mathcal{G}}^\mathit{IV}}$. Then $\operatorname{{\mathcal{G}}^{\mathit{I},hor}}, \operatorname{{\mathcal{G}}^{\mathit{II},hor}}$ and $\operatorname{{\mathcal{G}}^{\mathit{III},hor}}$ are ${\mathcal{O}}_{{\check D_{BL}}}$-locally free. $\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}$ is an ${\mathcal{O}}_{{\check D_{BL}}}$-coherent subsheaf of $\operatorname{{\mathcal{G}}^\mathit{IV}}$. It is equal to $\operatorname{{\mathcal{G}}^\mathit{IV}}$ and thus locally free if $n=\lfloor\alpha_\mu-\alpha_1\rfloor=1$. We call ${\mathcal{T}}^{hor}_{{\check D_{BL}}}:=\operatorname{\mathit{KS}}^{-1}(\operatorname{{\mathcal{G}}^{\mathit{IV},hor}})$ the subsheaf of horizontal tangent directions or horizontal tangent sheaf for short.
The ${\mathcal{O}}_{{\check D_{BL}}}$-coherence of all of the sheaves in question is obvious from their definition, and one obtains local bases of $\operatorname{{\mathcal{G}}^{\mathit{I},hor}}$, $\operatorname{{\mathcal{G}}^{\mathit{II},hor}}$ resp. $\operatorname{{\mathcal{G}}^{\mathit{III},hor}}$ by restricting the bases of $\operatorname{{{\mathcal{G}}}^\mathit{I}}$, $\operatorname{{\mathcal{G}}^\mathit{II}}$ resp. $\operatorname{{\mathcal{G}}^\mathit{III}}$ to $k=1$. A section $\phi=
(\underline{z}\otimes \underline{v}^* \otimes \underline{v})(\sum_{k=1}^n e_k \otimes \Delta^{vec}_k)$ is contained in one of these subbundles iff $\Delta_k=0$ for $k>1$. The equality $\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}=\operatorname{{\mathcal{G}}^\mathit{IV}}$ for $n=1$ is obvious from the definition as we have $\operatorname{{\mathcal{G}}^{\mathit{I},hor}}=\operatorname{{{\mathcal{G}}}^\mathit{I}}$ in this case.
In the remainder of this section, we show that $\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}$ is not locally free in general. For that purpose, consider the local basis of $\operatorname{{\mathcal{G}}^{\mathit{III},hor}}$ given above. In this basis, the conditions for a section $\phi\in \operatorname{{\mathcal{G}}^{\mathit{III},hor}}$ to be an element in $\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}$ are simply obtained from formula by putting all $\Delta_k=0$ if $k>1$. This yields the unique equation $$[B_{-1},\Delta_1]=0.$$ Note that $B_{-1}$ is the matrix of the pole part of $\nabla_z$ with respect to $\underline{v}$, i.e., of the endomorphism ${\mathcal{U}}:=[z^2\nabla_z]\in{{\mathcal{E}}}\!nd_{{\mathcal{O}}_{{\check D_{BL}}}}({\mathcal{L}}/z{\mathcal{L}})$. Similarly, $\Delta_1$ is by definition the matrix of the class $[z\phi]\in {{\mathcal{E}}}\!nd_{{\mathcal{O}}_{{\check D_{BL}}}}({\mathcal{L}}/z{\mathcal{L}})$. This shows that we have the following simple characterization $$\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}=\left\{\phi\in\operatorname{{\mathcal{G}}^{\mathit{III},hor}}\,|\,[{\mathcal{U}},[z\phi]]=0\right\}.$$ We have ${\mathcal{U}},[z\phi]:{\mathcal{V}}^\bullet({\mathcal{L}}/z{\mathcal{L}})\subset{\mathcal{V}}^{\bullet+1}({\mathcal{L}}/z{\mathcal{L}})$, so that $[{\mathcal{U}},[z\phi]]=0$ if $n=1$. This implies $\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}=\operatorname{{\mathcal{G}}^{\mathit{III},hor}}$, in particular, $\operatorname{{\mathcal{G}}^\mathit{IV}}=\operatorname{{\mathcal{G}}^{\mathit{III},hor}}$ in this case.
If $n\geq 2$ then in general $\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}$ will not be locally free. The reason is that in general the rank of the condition $[{\mathcal{U}},[z\phi]]=0$ varies within ${{\check D_{BL}}}$. Note however that the base $\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$ carries a horizontal subbundle, as it is a homogeneous space, so that the obstruction for $\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}$ to be locally free lies in the fibers of ${{\check D_{BL}}}\to\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}$. We will describe a situation where this actually occurs.
For simplicity we restrict to a situation in which $\dim C^\alpha=1$ for all $\alpha$. Then $N=0$ and $\operatorname{\mathit{D}_{\mathit{PHS}}}=\operatorname{\check{\mathit{D}}_{\mathit{PHS}}}=\operatorname{\mathit{D}_{\mathit{PMHS}}}=\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}=\{pt\}$, so that $D_\mathit{BL}={{\check D_{BL}}}={{\mathds C}}^{N_\mathit{BL}}$ for some $N_\mathit{BL}$. We choose $v_i^0\in C^{\alpha_i}$ with $P(v_i^0,v_{\mu+1-j}^0)=\delta_{ij}$. By [@He2 Ch. 5] formula holds on all of ${{\check D_{BL}}}$, with $$\begin{aligned}
(C_k)_{ij}&=&0 \qquad \textup{ for }\alpha_i-k\leq \alpha_j,\\
(\alpha_i-k-\alpha_j)\cdot(C_k)_{ij}&=& \sum_{l}(\alpha_l-1-\alpha_j)\cdot
(C_{k-1})_{il}(C_1)_{lj},\\
(C_1)_{ij}&=&(C_1)_{\mu+1-j,\mu+1-i},\\
{\mathcal{U}}([\underline{v}]) &=& [\underline{v}]\cdot
((\alpha_i-1-\alpha_j)\cdot (C_1)_{ij}),\end{aligned}$$ and global coordinates on ${{\check D_{BL}}}$ are given by those $(C_1)_{ij}$ where $i+j\leq\mu+1$. The zero point of these coordinates is the TERP-structure ${\mathcal{H}}^0=\bigoplus{\mathcal{O}}_{{\mathds C}}\cdot v_i^0$.
In the basis of $\operatorname{{\mathcal{G}}^{\mathit{III},hor}}$ in definition-lemma \[defHorBundles\] $$[z\phi]([\underline{v}])=[\underline{v}]\cdot \Delta_1$$ with $(\Delta_1)_{ij}=0$ if $\alpha_i-1\leq \alpha_j$ (equality is impossible due to $\dim C^\alpha=1$) and $(\Delta_1)_{ij}=(\Delta_1)_{\mu+1-j,\mu+1-i}$. We have $\operatorname{rank}\operatorname{{\mathcal{G}}^{\mathit{III},hor}}=N_{BL}$ and $${\mathcal{T}}^{hor}_{|0}{{\check D_{BL}}}\cong {\mathcal{G}}^\mathit{IV,h}_{|0} = {\mathcal{G}}^\mathit{III,h}_{|0} \cong {\mathcal{T}}_{|0}{{\check D_{BL}}}$$ as ${\mathcal{U}}$ and $[z\phi]$ commute at the point $0$. To prove that $\operatorname{{\mathcal{G}}^{\mathit{IV},hor}}$ is not locally free it is sufficient to show that the condition $[{\mathcal{U}},[z\phi]]=0$ is non-empty at some point $t\in {{\check D_{BL}}}$ .
If there are $\alpha_i,\alpha_l,\alpha_m$ with $\alpha_i-2>\alpha_l-1>\alpha_m$ and $m\neq \mu+1-i$ then the $(i,m)$ entry of the commutator of the matrices corresponding to ${\mathcal{U}}$ and $[z\phi]$ is $$\sum_j \left((\alpha_i-1-\alpha_j)(C_1)_{ij}\cdot (\Delta_1)_{jm}
- (\Delta_1)_{ij}\cdot (\alpha_j-1-\alpha_m)(C_1)_{jm}\right).$$ This sum is non-empty because $j=l$ gives a term, and all present $(\Delta_1)$-coefficients are different. So if $(C_1)_{il}(t) \neq 0$ this gives a non-empty condition, and $\operatorname{rank}{\mathcal{T}}^{hor}_{|t}{{\check D_{BL}}}< \operatorname{rank}{\mathcal{T}}^{hor}_{|0}{{\check D_{BL}}}$.
An example of this type can be constructed starting with a suitable semiquasihomogeneous deformation of a Brieskorn-Pham singularity $f=x_0^{a_0}+x_1^{a_1}+x_2^{a_2}$ where $\textup{gcd}(a_i,a_j)=1$ for $i\neq j$ and such that $\frac{1}{a_0}+\frac{1}{a_1}+\frac{1}{a_2}$ is sufficiently small.
Holomorphic sectional curvature {#secHolSectCurv}
===============================
One of the most interesting features of a TERP-structure is the construction of a canonical extension to a twistor, i.e., a ${{\mathds P}}^1$-bundle. Starting with a family of TERP-structures, this yields a ${\mathcal{C}}^\infty$-family (which is actually real analytic) of twistors. Let us briefly recall how this is done (see [@He4] and [@HS1] for more details).
Given a TERP-structure ${(H,H'_{{\mathds R}},\nabla,P,w)}$, define for any $z\in{{\mathds C}}^*$ the anti-linear involution $$\begin{array}{rcl}
\tau: H_z & \longrightarrow & H_{1/\overline{z}}\\
s & \longmapsto & \nabla\textup{-parallel transport of }\overline{z^{-w}s}
\end{array}$$ The image $\tau({\mathcal{H}}_0)$ of the germ of sections of ${\mathcal{H}}$ at zero is contained in the germ $(\widetilde{i}({\mathcal{H}}'))_\infty$, where $\widetilde{i}:{{\mathds C}}^*\rightarrow {{\mathds P}}^1\backslash\{0\}$. This defines an extension of $H$ to infinity, which is a holomorphic ${{\mathds P}}^1$-bundle, i.e., a twistor. We will denote it by $\widehat{H}$. If $\widehat{H}$ is trivial, then we call the original TERP-structure pure. Moreover, in this case we can define a hermitian pairing $h$ on $H^0({{\mathds P}}^1,\widehat{{\mathcal{H}}})$ by the formula $h(a,b):=z^{-w}P(a,\tau b)$. If this form is positive definite, then ${(H,H'_{{\mathds R}},\nabla,P,w)}$ is called pure polarized. We only remark (this is discussed in detail in [@He4] and [@HS1]) that if we start with a family of TERP-structures, then this extension procedure yields a real analytic family of twistors. Define $${{\check D_{BL}^{pp}}}:= \left\{{\mathcal{H}}\in {{\check D_{BL}}}\,|\, \widehat{{\mathcal{H}}}\mbox{ is pure polarized }\right\}.$$ One of the main results in [@HS1], namely, theorem 6.6 says that a regular singular TERP-structure is an element of $D_\mathit{BL}$ iff it induces a nilpotent orbit, i.e. iff the pullback $$\pi_r^*(H,H'_{{\mathds R}},\nabla,P),$$ where $\pi_r:{{\mathds C}}\to{{\mathds C}},\;z\mapsto r\cdot z$, is a pure polarized TERP-structure for any $r\in {{\mathds C}}^*$ with $|r|$ sufficiently small.
Such a pullback is then also an element of $D_\mathit{BL}$. Therefore the set ${{\check D_{BL}^{pp}}}$ of all pure polarized TERP-structures in ${{\check D_{BL}}}$ is non-empty, and it intersects $D_\mathit{BL}$ nontrivially. The condition to be pure and polarized is open, so ${{\check D_{BL}^{pp}}}$ is an open submanifold of ${{\check D_{BL}}}$.
There is no reason to expect ${{\check D_{BL}^{pp}}}\subset D_\mathit{BL}$, but the intersection ${{\check D_{BL}^{pp}}}\cap D_\mathit{BL}$ seems to be most interesting for applications. If $N=0$ then $\operatorname{\mathit{D}_{\mathit{PHS}}}=\operatorname{\mathit{D}_{\mathit{PMHS}}}$ and moreover ${{\check D_{BL}^{pp}}}\cap D_\mathit{BL}$ contains a neighborhood of the zero section $\operatorname{\mathit{D}_{\mathit{PMHS}}}\hookrightarrow D_\mathit{BL}$. The reason is that if $N=0$, then the action by pullback $\pi_r^*$ coincides with the good ${{\mathds C}}^*$-action considered in [@He2 Theorem 5.6].
Performing the above construction on the whole classifying space ${{\check D_{BL}}}$ yields an extension of the universal bundle ${\mathcal{L}}$ to a real analytic family of twistors $\widehat{{\mathcal{L}}}$, that is, a locally free ${\mathcal{O}}_{{{\mathds P}}^1}{\mathcal{C}}^{an}_{{\check D_{BL}}}$-module. Moreover, on the subspace ${{\check D_{BL}^{pp}}}$ the sheaf of fibrewise global sections $p_*\widehat{{\mathcal{L}}}_{|{{\check D_{BL}^{pp}}}}$ is by definition locally free over ${\mathcal{C}}^{an}_{{\check D_{BL}^{pp}}}$ and comes equipped with a positive definite hermitian metric $h$. We will show that this induces positive definite hermitian metrics on the bundles $\operatorname{{{\mathcal{G}}}^\mathit{I}},\ldots,\operatorname{{\mathcal{G}}^\mathit{IV}}$, restricted to ${{\check D_{BL}^{pp}}}$.
Denote by ${\mathcal{K}}$ the sheaf $\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}\otimes{\mathcal{L}}_{|{{\check D_{BL}^{pp}}}}$ and put $\operatorname{{\mathcal{K}^\mathit{sp}}}:=p_*\widehat{{\mathcal{L}}}_{|{{\check D_{BL}^{pp}}}}$, then we have a hermitian metric $h:=z^{-w}P(-,\tau-)$ on the $\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}$-module $\operatorname{{\mathcal{K}^\mathit{sp}}}$. We obtain a splitting $$k^{-1}{\mathcal{K}}= {\mathcal{K}}^{sp} \oplus k^{-1}(z{\mathcal{K}})$$ where $k:{{\check D_{BL}^{pp}}}\hookrightarrow{{\mathds C}}\times{{\check D_{BL}^{pp}}}, t\mapsto (0,t)$. This yields $$\begin{array}{c}
\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}\otimes k^{-1}\operatorname{{{\mathcal{G}}}^\mathit{I}}=
{{\mathcal{H}}}\!om_{k^{-1}{\mathcal{O}}_{{\mathds C}}\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}}\left({\mathcal{K}}^{sp} \oplus k^{-1}(z{\mathcal{K}}), k^{-1}\left(\frac{z^{-n}{\mathcal{K}}}{{\mathcal{K}}}\right)\right) \cong \\ \\
{{\mathcal{H}}}\!om_{\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}}\left(\operatorname{{\mathcal{K}^\mathit{sp}}}, \oplus_{k=1}^n z^{-k}\operatorname{{\mathcal{K}^\mathit{sp}}}\right) = \oplus_{k=1}^n {{\mathcal{H}}}\!om_{\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}}\left(\operatorname{{\mathcal{K}^\mathit{sp}}}, z^{-k}\operatorname{{\mathcal{K}^\mathit{sp}}}\right)
\cong \left[{{\mathcal{E}}}\!nd_{\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}}(\operatorname{{\mathcal{K}^\mathit{sp}}})\right]^n
\end{array}$$ We obtain a hermitian metric on ${{\mathcal{E}}}\!nd_{\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}}(\operatorname{{\mathcal{K}^\mathit{sp}}})$ (and its powers) by $h(\phi,\phi')=Tr(\phi\cdot(\phi')^*)$, where $(-)^*$ denotes the hermitian adjoint. This induces a metric on $\operatorname{{{\mathcal{G}}}^\mathit{I}}$ and by restriction on the subbundles $\operatorname{{\mathcal{G}}^\mathit{II}}$, $\operatorname{{\mathcal{G}}^\mathit{III}}$ and $\operatorname{{\mathcal{G}}^\mathit{IV}}$. We denote these metrics by $h^{I}, \ldots, h^{IV}$. We remark that choosing any local basis $\underline{u}\in
M(1\times n, {\mathcal{K}})$ of ${\mathcal{K}}$ in a neighborhood of a point $(0,t)\in\{0\}\times{{\check D_{BL}^{pp}}}$ yields a similar splitting $$k^{-1}{\mathcal{K}}={\mathcal{K}}^{\mathit{sp},\underline{u}}\oplus k^{-1}(z {\mathcal{K}})\mbox{ and }
\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}\otimes \operatorname{{{\mathcal{G}}}^\mathit{I}}\cong \left[{{\mathcal{E}}}\!nd_{\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}}({\mathcal{K}}^{\mathit{sp},\underline{u}})\right]^n,$$ where ${\mathcal{K}}^{\mathit{sp},\underline{u}}:=\oplus_{i=1}^\mu \operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}u_i$. If $\underline{u}$ is a global basis of $\widehat{{\mathcal{L}}}_{|{{\check D_{BL}^{pp}}}}$, then ${\mathcal{K}}^{\mathit{sp},\underline{u}}=\operatorname{{\mathcal{K}^\mathit{sp}}}$. If $\underline{u}$ happens to be holomorphic, i.e., $\underline{u}\in M(1\times \mu, {\mathcal{L}})$, then $\operatorname{{{\mathcal{G}}}^\mathit{I}}\cong
\left[{{\mathcal{E}}}\!nd_{{\mathcal{O}}_{{\check D_{BL}^{pp}}}}({\mathcal{L}}^{\mathit{sp},\underline{u}})\right]^n$, this isomorphism was already considered in section \[secTangent\]. Similarly to the holomorphic basis from formula , we obtain a basis of $\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}\otimes \operatorname{{{\mathcal{G}}}^\mathit{I}}\cong \left[{{\mathcal{E}}}\!nd_{\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}}({\mathcal{K}}^{\mathit{sp},\underline{u}})\right]^n$, namely, $\underline{z}\otimes\underline{u}^*\otimes
\underline{u}\in M(1\times n\mu^2,\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}\otimes \operatorname{{{\mathcal{G}}}^\mathit{I}})$ and any section $\phi$ of $\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}\otimes \operatorname{{{\mathcal{G}}}^\mathit{I}}$ is written in the basis $\underline{z}\otimes\underline{u}^*\otimes
\underline{u}$ as $$\phi=\sum_{
\begin{array}{c}
\scriptstyle k=1,\ldots,n\\
\scriptstyle i,j=1,\ldots,\mu
\end{array}
}(\Gamma_k)_{ji} z^{-k}\otimes u^*_i \otimes u_j
=(\underline{z}\otimes \underline{u}^* \otimes \underline{u})(\sum_{k=1}^n e_k \otimes \Gamma^{vec}_k)$$ (remember that $\underline{z}:=(z^{-1},\ldots,z^{-n})$, that $e_k\in M(n\times 1, {{\mathds C}})$ is the $k$-th unit vector and that $A^{vec}$ denotes the column vector of a matrix $A$ as explained after formula ).
Recall the Kodaira-Spencer map from lemma \[lemKS\] $$\begin{array}{rcl}
\operatorname{\mathit{KS}}:{\mathcal{T}}_{{\check D_{BL}^{pp}}}& \hookrightarrow & \operatorname{{{\mathcal{G}}}^\mathit{I}}\\ \\
X & \longmapsto & [v\mapsto \nabla_X v]),
\end{array}$$ which endows ${\mathcal{T}}_{{\check D_{BL}^{pp}}}$ with a positive definite hermitian metric which we simply denote by $h$. Recall also that we denoted by ${\mathcal{T}}^{hor}_{{\check D_{BL}^{pp}}}$ the coherent subsheaf of ${\mathcal{T}}_{{\check D_{BL}^{pp}}}$ defined by ${\mathcal{T}}^{hor}_{{\check D_{BL}^{pp}}}:=\operatorname{\mathit{KS}}^{-1}({{\mathcal{H}}}\!om_{{\mathcal{O}}_{{{\mathds C}}\times{{\check D_{BL}^{pp}}}}}({\mathcal{H}},\frac{z^{-1}{\mathcal{H}}}{{\mathcal{H}}}))$, and that it is not locally free in general. However, it contains the zero section of $T_{{\check D_{BL}^{pp}}}\rightarrow {{\check D_{BL}^{pp}}}$, and we may consider the holomorphic sectional curvature of the metric $h$ on vectors of $T_{{\check D_{BL}^{pp}}}^{hor}\backslash\{\textup{zero section}\}$. Let us briefly recall its the definition: Given any holomorphic bundle $E$ on a complex manifold $M$ and a positive definite hermitian metric $h$ on $E$, there is a unique connection $D:{\mathcal{E}}\rightarrow {\mathcal{E}}\otimes{\mathcal{A}}^1_M$ such that $D(h)=0$ and such that the $(0,1)$-part of $D$ coincides with the operator defining the holomorphic structure of ${\mathcal{E}}$. $D$ is called the Chern connection of $(E,h)$. Its curvature is by definition the section $R$ of ${{\mathcal{E}}}\!nd_{\operatorname{{\mathcal{C}}^{an}_\mathit{M}}}(\operatorname{{\mathcal{C}}^{an}_\mathit{M}}\otimes{\mathcal{E}})\otimes {\mathcal{A}}^{1,1}_M$ given by $e\stackrel{R}{\mapsto} D^{(2)}(D(e))$, here $D^{(2)}:{\mathcal{E}}\otimes{\mathcal{A}}^1_M\rightarrow{\mathcal{E}}\otimes{\mathcal{A}}^2_M$, $D^{(2)}(e\otimes\alpha)
=D(e)\wedge\alpha+s\otimes d\alpha$. If $E$ is the holomorphic tangent bundle of $M$, then the function $$\begin{array}{rcl}
\kappa: T_M \backslash \{\mbox{zero section}\} & \longrightarrow & {{\mathds R}}\\ \\
\xi&\longmapsto& h(R(\xi,\overline{\xi})\xi,\xi)/h(\xi,\xi)^2
\end{array}$$ is called the holomorphic sectional curvature of $M$.
We are now able to state the main result of this section.
The restriction of the holomorphic sectional curvature $\kappa:T_{{\check D_{BL}^{pp}}}\backslash\{\mbox{zero-section}\} \rightarrow {{\mathds R}}$ to the (linear space associated to the) coherent subsheaf ${\mathcal{T}}_{{\check D_{BL}^{pp}}}^{hor}$ is bounded from above by a negative real number.
First recall a formula for the curvature tensor of the Chern connection on an arbitrary bundle. Let, as before, $E$ be a holomorphic vector bundle of rank $\mu$ on a complex manifold $M$ and $h:{\mathcal{E}}\otimes\overline{{\mathcal{E}}}
\rightarrow\operatorname{{\mathcal{C}}^{an}_\mathit{M}}$ a positive definite hermitian metric. For a local holomorphic basis $\underline{e}\in M(1\times\mu,{\mathcal{E}})$, put $H:=
\left(h(\underline{e}^{tr},\underline{e})\right)^{tr}\in M(\mu\times\mu,\operatorname{{\mathcal{C}}^{an}_\mathit{M}})$. The curvature $R$ is linear, thus $R(\underline{e})=\underline{e}M_R$, where $M_R\in M(\mu\times\mu,{\mathcal{A}}^{1,1}_M)$. It is well known (see, e.g., [@CarlStachPeters lemma 11.4]) that $$M_R=H^{-1}\overline{\partial}\partial H-H^{-1}\overline{\partial}(H) \wedge H^{-1}\partial H.$$ In particular, for any holomorphic vector field $X\in{\mathcal{T}}_M$, we have $$M_R(X,\overline{X})=
-H^{-1}(\overline{X}X)(H)+H^{-1}\overline{X}(H) H^{-1}X(H) \in M(\mu\times\mu,\operatorname{{\mathcal{C}}^{an}_\mathit{M}}).$$ If at a point $x\in M$, $H(x)=Id$, then $M_R(X,\overline{X})(x)=\overline{X}(H)(x)X(H)(x)-(\overline{X}X)(H)(x)$, or, if we write $\xi:=X(x)$, then $$\label{eqCurv}
M_R(X,\overline{X})(x)=\overline{\xi}(H)\xi(H)-(\overline{X}X)(H)(x).$$ This formula will allow a very significant simplification of the calculations. Let $t\in{{\check D_{BL}^{pp}}}$, and let $\xi\in T^{hor}_t({{\check D_{BL}^{pp}}})$ be any vector with $\xi\neq 0$. Choose local holomorphic coordinates $(t_1,\ldots,t_{\dim({{\check D_{BL}^{pp}}})})$ centered at $t$ such that $(\partial_{t_1})_{|t}=\xi$. Although we are interested in the curvature tensor $R^{IV}$ of the bundle ${\mathcal{G}}^{IV}_{|{{\check D_{BL}^{pp}}}}$ (which is isomorphic to ${\mathcal{T}}_{{\check D_{BL}^{pp}}}$), our first aim is to give an expression for the matrix $M_R(\partial_{t_1},\overline{\partial}_{t_1})(t)$ which represents $R^{I}(\partial_{t_1},\overline{\partial}_{t_1})(x) \in\mathit{{{\mathcal{E}}}\!nd}_{{\mathds C}}({\mathcal{G}}^{I}_{|t})$ with respect to a holomorphic basis in a neighborhood of $t$. This basis is induced from a holomorphic basis of ${\mathcal{L}}$ near $(0,t)$, which is obtained as follows: choose a basis $\underline{v}^0\in M(1 \times \mu, {\mathcal{L}}_{|{{\mathds C}}\times\{t\}})$ of ${\mathcal{L}}_{|{{\mathds C}}\times\{t\}}$ such that $P((\underline{v}^0)^{tr},\underline{v}^0)={{\mathds 1}}_\mu$ and $\tau(\underline{v}^0)=\underline{v}^0$. Then define $$\underline{v}:=\underline{v}^0\left({{\mathds 1}}_\mu+\sum_{k=1}^nz^{-k}C_k\right) \in M(1\times\mu,{\mathcal{L}})$$ to be the extension provided by lemma \[lemCanExtBase\]. It still satisfies $P(\underline{v}^{tr},\underline{v})={{\mathds 1}}_\mu$, but not necessarily $\tau(\underline{v})=\underline{v}$. Write $\operatorname{\mathit{KS}}(\xi)=\sum_{k,i,j}(\Delta_k)_{ji} \, (z^{-k}\otimes (v^{(0)}_i)^*\otimes v^{(0)}_j)$ (i.e. $\Delta_k\in M(\mu\times\mu,{{\mathds C}})$), then it follows from $\xi\in T_t^{hor}({{\check D_{BL}^{pp}}})$ that $(\Delta_k)=0$ for all $k>1$. Moreover, as $\kappa(\partial_{t_1})=(\underline{v}\mapsto\underline{v}(\sum_{k=1}^n z^{-k} \partial_{t_1}C_k))$, we conclude that $\xi(C_1)=\Delta_1$ and $\xi(C_k)=\Delta_k = 0$ for $k>1$. The matrices $H:=[h^I((\underline{z}\otimes\underline{v}^*\otimes\underline{v})^{tr},
\underline{z}\otimes\underline{v}^*\otimes\underline{v})]^{tr}$ and $M(\partial_{t_1},\overline{\partial}_{t_1})$ are elements of $M(\mu\times\mu,\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}})$, so that we conclude from formula that $M(\partial_{t_1},\overline{\partial}_{t_1})(x)$ can obtained from the image of $H$ under the reduction map $M(\mu\times\mu,\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}})\twoheadrightarrow M(\mu\times\mu,{\mathcal{Q}})$, where ${\mathcal{Q}}:=\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}/(t_1^2,\overline{t}_1^2,t_j,\overline{t}_j)_{j>1}$. In order to keep the notation simple, we still call this reduction $H$. Moreover, it is clear that this reduced matrix $H$ may be calculated from the image of the basis $\underline{v}$ under the map ${\mathcal{K}}\twoheadrightarrow {\mathcal{K}}\otimes\widetilde{{\mathcal{Q}}}$, where $\widetilde{{\mathcal{Q}}}:={\mathcal{O}}_{{\mathds C}}\operatorname{{\mathcal{C}}^\mathit{an}_\mathit{\check{D}^{pp}_{BL}}}/(t_1^2,\overline{t}_1^2,t_j,\overline{t}_j)_{j>1}$. Again we denote this image by $\underline{v}$. All subsequent calculations take place in either $\widetilde{{\mathcal{Q}}}$ or ${\mathcal{Q}}$. In particular, we have $C_1=t_1\Delta_1$ and $C_k=0$ for $k>1$ in ${\mathcal{Q}}$. This implies $\underline{v}=\underline{v}^{(0)}({{\mathds 1}}_\mu+z^{-1}C_1)$ and ${{\mathds 1}}_\mu=P(\underline{v}^{tr},\underline{v})=
({{\mathds 1}}_\mu+z^{-1}C_1)^{tr}P((\underline{v}^0)^{tr},\underline{v}^0)({{\mathds 1}}_\mu-z^{-1}C_1)=
({{\mathds 1}}_\mu+z^{-1}(C_1^{tr}-C_1))$ so that $C^{tr}_1=C_1$.
Consider the base change given by $\underline{w}:=\underline{v}({{\mathds 1}}_\mu+\frac12[\overline{C}_1,C_1]+z\overline{C}_1)$. It follows from $P(\underline{v}^{tr},\underline{v})={{\mathds 1}}_\mu$, $\overline{C}^{tr}_1=\overline{C}_1$ and $[\overline{C}_1,C_1]^{tr}=-[\overline{C}_1,C_1]$ that $P(\underline{w}^{tr},\underline{w})={{\mathds 1}}_\mu$. Moreover, as $$\underline{w}=\underline{v}^{(0)}\left({{\mathds 1}}_\mu+z\overline{C}_1+z^{-1}C_1+\frac12(\overline{C}_1C_1+C_1\overline{C}_1)\right),$$ we also have $\tau(\underline{w})=\underline{w}$. It is a simple calculation to show that the inverse base change is given by $$\underline{v}:=\underline{w}\cdot({{\mathds 1}}_\mu-\frac12[\overline{C}_1,C_1]-z\overline{C}_1)$$ We obtain an induced base change on $\operatorname{{{\mathcal{G}}}^\mathit{I}}$, given by $\left({{\mathds 1}}_\mu-\frac12[\overline{C}_1,C_1]-z\overline{C}_1)^{tr}\right)^{-1}
\otimes({{\mathds 1}}_\mu-\frac12[\overline{C}_1,C_1]-z\overline{C}_1)$, i.e.: $$\begin{array}{c}
\underline{z}\otimes \underline{v}^* \otimes \underline{v} =
(\underline{z}\otimes \underline{w}^* \otimes \underline{w})\cdot
\left[{{\mathds 1}}_n\otimes
\left(({{\mathds 1}}_\mu-\frac12[\overline{C_1},C_1]-z\overline{C_1})^{-1}\right)^{tr}\otimes
({{\mathds 1}}_\mu-\frac12[\overline{C_1},C_1]-z\overline{C_1})\right] = \\ \\
\underline{z}\otimes \underline{w}^* \otimes \underline{w} \cdot
\left(\underbrace{{{\mathds 1}}_n\otimes ({{\mathds 1}}_\mu\otimes{{\mathds 1}}_\mu-\frac12({{\mathds 1}}_\mu\otimes[\overline{C_1},C_1]+[\overline{C_1},C_1]\otimes{{\mathds 1}}_\mu))
+N_z\otimes(\overline{C_1}\otimes{{\mathds 1}}_\mu-{{\mathds 1}}_\mu\otimes \overline{C_1})}_{=:X}\right),
\end{array}$$ here $$N_z=\begin{pmatrix}0&1&\ldots &&0\\0 & 0 & 1 &\ldots&0\\&&\vdots\\0 &0 & \ldots & 0&1\\0&0& & \ldots& 0\end{pmatrix}.$$ Now we have $$\begin{array}{c}
H=[h^I((\underline{z}\otimes\underline{v}^*\otimes\underline{v})^{tr},
\underline{z}\otimes\underline{v}^*\otimes\underline{v})]^{tr}=
[X^{tr}h^I((\underline{z}\otimes\underline{w}^*\otimes\underline{w})^{tr},\underline{z}\otimes\underline{w}^*\otimes\underline{w})\overline{X}]^{tr}
\\ \\=\overline{X}^{tr}X=
{{\mathds 1}}_n\otimes({{\mathds 1}}_\mu\otimes{{\mathds 1}}_\mu-{{\mathds 1}}_\mu\otimes[\overline{C_1},C_1]-[\overline{C_1},C_1]\otimes{{\mathds 1}}_\mu)
+(N_z)^{tr}\otimes(C_1\otimes{{\mathds 1}}_\mu-{{\mathds 1}}_\mu\otimes C_1)\\ \\
+N_z\otimes(\overline{C_1}\otimes{{\mathds 1}}_\mu-{{\mathds 1}}_\mu\otimes \overline{C_1})
+{{\mathds 1}}_{n-1}\otimes(C_1\overline{C_1}\otimes{{\mathds 1}}_\mu+{{\mathds 1}}_\mu\otimes C_1\overline{C_1}-\overline{C_1}\otimes C_1 -C_1\otimes\overline{C_1})
\end{array}$$ where ${{\mathds 1}}_{n-1}=N_z^{tr}\cdot N_z=\textup{diag}(0,1,\ldots,1)$. The next step is to invoke formula to obtain the matrix $M_R(\partial_{t_1},\overline{\partial}_{t_1})(x)$. Using $C_1=t\Delta_1$, we get $$\begin{array}{rcl}
\overline{\xi}(H) & = & N_z\otimes(\overline{\Delta}_1\otimes{{\mathds 1}}_\mu-{{\mathds 1}}_\mu\otimes\overline{\Delta}_1), \\ \\
\xi(H) & = & N_z^{tr}\otimes(\Delta_1\otimes{{\mathds 1}}_\mu-{{\mathds 1}}_\mu\otimes\Delta_1), \\ \\
(\overline{\partial}_{t_1}\partial_{t_1}(H))(x)
& = & -{{\mathds 1}}_n\otimes({{\mathds 1}}_\mu\otimes[\overline{\Delta}_1,\Delta_1]+
[\overline{\Delta}_1,\Delta_1]\otimes{{\mathds 1}}_\mu)+\\ \\
& &
{{\mathds 1}}_{n-1}\otimes(\Delta_1\overline{\Delta_1}\otimes{{\mathds 1}}_\mu+{{\mathds 1}}_\mu\otimes \Delta_1\overline{\Delta_1}-\overline{\Delta_1}\otimes \Delta_1 -\Delta_1\otimes\overline{\Delta_1}),
\\ \\
M_R(\xi,\overline{\xi}) & = &
{{\mathds 1}}_n\otimes(\underbrace{{{\mathds 1}}_\mu\otimes[\overline{\Delta}_1,\Delta_1]+[\overline{\Delta}_1,\Delta_1]\otimes{{\mathds 1}}_\mu}_{=:S})\\\\
& & -{{\mathds 1}}_{n-1}\otimes(\Delta_1\overline{\Delta}_1\otimes{{\mathds 1}}_\mu+{{\mathds 1}}_\mu\otimes\Delta_1\overline{\Delta}_1-
\overline{\Delta_1}\otimes \Delta_1 - \Delta_1\otimes\overline{\Delta_1}) \\ \\
& & +{{\mathds 1}}'_{n-1}\otimes(\underbrace{\overline{\Delta}_1\Delta_1\otimes{{\mathds 1}}_\mu+{{\mathds 1}}_\mu\otimes\overline{\Delta}_1\Delta_1
-\overline{\Delta_1}\otimes \Delta_1 - \Delta_1\otimes\overline{\Delta_1}}_{=:R}),
\end{array}$$ where ${{\mathds 1}}'_{n-1}=N_z\cdot N_z^{tr}=\operatorname{\textup{diag}}(1,\ldots,1,0)$. What we are really interested in is to give a closed formula for the expression $h^I(R(\xi,\overline{\xi})\operatorname{\mathit{KS}}(\xi),\operatorname{\mathit{KS}}(\xi))$. As $\operatorname{\mathit{KS}}(\xi)=\sum_{i,j}(\Delta_1)_{j,i}(z^{-1}\otimes
(v_i^{(0)})^*\otimes v^{(0)}_j)$ and $h^I((\underline{z}\otimes \underline{v}^* \otimes \underline{v})^{tr},
(\underline{z}\otimes \underline{v}^* \otimes \underline{v}))=
{{\mathds 1}}_n\otimes h((\underline{v}^*)^{tr},\underline{v}^*)\otimes h(\underline{v}^{tr},\underline{v})$, we obtain that $$\begin{array}{c}
h^I(R(\xi,\overline{\xi})\operatorname{\mathit{KS}}(\xi),\operatorname{\mathit{KS}}(\xi)) = \\ \\
h^I\left(((z^{-1}\otimes(\underline{v}^{(0)})^*\otimes\underline{v}^{(0)})(R+S)(\Delta_1)^{vec})^{tr},(z^{-1}\otimes(\underline{v}^{(0)})^*\otimes\underline{v}^{(0)})(\Delta_1)^{vec}\right)=\\ \\
\left(S\cdot(\Delta_1)^{vec}\right)^{tr}({{\mathds 1}}_\mu\otimes{{\mathds 1}}_\mu)(\overline{\Delta}_1)^{vec}
=
\left(([[\overline{\Delta}_1,\Delta_1],\Delta_1])^{vec}\right)^{tr}(\overline{\Delta}_1)^{vec}
=\textup{Tr}([[\overline{\Delta}_1,\Delta_1],\Delta_1]\cdot \overline{\Delta}_1^{tr}) \\ \\
=-\textup{Tr}([\Delta_1,[\overline{\Delta}_1,\Delta_1]]\cdot \overline{\Delta}_1^{tr})
=-\textup{Tr}([\Delta_1,\overline{\Delta}_1],\overline{[\Delta_1,\overline{\Delta}_1]}^{tr})
\end{array}$$ The last computation uses formula and the fact that $R\cdot(\Delta_1)^{vec}=[[\Delta_1,\Delta_1],\overline{\Delta}_1]=0$.
It is well known that the curvature decreases on subbundles, see, e.g., [@Sch lemma (7.14)], thus we obtain the following estimate: $$h^{IV}(R^{IV}(\xi,\overline{\xi})\operatorname{\mathit{KS}}(\xi),\operatorname{\mathit{KS}}(\xi))
\leq h^I(R^I(\xi,\overline{\xi})\operatorname{\mathit{KS}}(\xi),\operatorname{\mathit{KS}}(\xi)).$$ This implies that the holomorphic sectional curvature $\frac{h^{IV}(R^{IV}(\xi,\overline{\xi})\xi,\xi)}{h^2(\xi,\xi)}$ is smaller than or equal to $$-\frac{\textup{Tr}([\Delta_1,\overline{\Delta}_1],\overline{[\Delta_1,\overline{\Delta}_1]}^{tr})}{\textup{Tr}(\Delta_1\cdot\overline{\Delta}^{tr}_1)^2}$$ As $\xi\in T^{hor}_t({{\check D_{BL}^{pp}}})$, i.e., $\operatorname{\mathit{KS}}(\partial_{t_1})\in\operatorname{{\mathcal{G}}^\mathit{IV}}\subset\operatorname{{\mathcal{G}}^\mathit{III}}$, the morphism $\operatorname{\mathit{KS}}(\partial_{t_1}):{\mathcal{L}}\rightarrow z^{-1}{\mathcal{L}}/{\mathcal{L}}$ respects the $V$-filtration, and $[z\operatorname{\mathit{KS}}(\partial_{t_1})]\in{{\mathcal{E}}}\!nd_{{\mathcal{O}}_{{\check D_{BL}^{pp}}}}({\mathcal{L}}/z{\mathcal{L}})$ shifts the (induced) $V$-filtration by one. By definition, $\Delta_1$ is the matrix representing $[z\operatorname{\mathit{KS}}(\xi)]\in{{\mathcal{E}}}\!nd_{{\mathds C}}(({\mathcal{L}}/z{\mathcal{L}})_{|(0,t)})$ with respect to the basis $\underline{v}$, which shows that it is nilpotent. Lemma \[lemMatrices\] then proves that the value of $-\frac{\textup{Tr}([\Delta_1,\overline{\Delta}_1],\overline{[\Delta_1,\overline{\Delta}_1]}^{tr})}{\textup{Tr}(\Delta_1\cdot\overline{\Delta}^{tr}_1)^2}$ and thus of the holomorphic sectional curvature on $T^{hor}_{{{\check D_{BL}^{pp}}}}\backslash \{\textup{zero section}\}$ is bounded from above by a negative real number.
\[lemMatrices\] Fix $\mu\in {{\mathds N}}$.
1. Consider a matrix $A\in M(\mu\times \mu,{{\mathds C}})$ which is symmetric and nilpotent. Then $$[A,\overline{A}]=0\iff A=0.$$
2. The map $$\begin{aligned}
\varphi:\big\{A\in M(\mu\times\mu,{{\mathds C}})\backslash\{0\} &|& A\textup{ is symmetric and nilpotent}\big\}
\longrightarrow{{\mathds R}},\\ \\
A &\mapsto& \frac{-\operatorname{\textup{Tr}}\left([A,\overline{A}]\cdot \overline{[A,\overline{A}]}^{tr}\right)}
{\operatorname{\textup{Tr}}\left(A\cdot\overline{A}^{tr}\right)^2}\end{aligned}$$ is bounded from above by a negative number.
<!-- -->
1. $\Re(A)$ and $\Im(A)$ are real symmetric matrices and thus diagonalizable. $[A,\overline{A}]=0$ is equivalent to $[\Re(A),\Im(A)]=0$ and to the simultaneous diagonalizability of $\Re(A)$ and $\Im(A)$. In that case also $A$ is diagonalizable. As $A$ is nilpotent, it vanishes.
2. The image of the map $\varphi$ does not change when we restrict $\varphi$ to the subset $$\{A\in M(\mu\times\mu,{{\mathds C}})\ |\ A\textup{ is symmetric and nilpotent and }
\operatorname{\textup{Tr}}{(A\cdot \overline{A}}^{tr})=1\}.$$ This set is compact, its image is contained in ${{\mathds R}}^-$, $\varphi$ is continuous so that the image is compact and therefore $\textup{Im}(\varphi)$ has a strictly negative upper bound.
In the remaining part of this section, we outline some rather direct consequences of the above curvature calculations. They are close in spirit to the work of Griffiths and Schmid on the classifying spaces of Hodge structures ([@GSch1; @GSch2]). The key tool is the following result. Let us call a holomorphic map $\phi:M\rightarrow {{\check D_{BL}^{pp}}}$ *horizontal* if $d\phi({\mathcal{T}}_M)\subset \phi^*{\mathcal{T}}^{hor}_{{\check D_{BL}^{pp}}}$, where $d\phi:{\mathcal{T}}_M\rightarrow \phi^*{\mathcal{T}}_{{\check D_{BL}^{pp}}}$ is the derivative of $\phi$.
\[propAhlfors\]
1. Write $\Delta$ for the open unit disc in ${{\mathds C}}$ and let $\phi:\Delta\rightarrow{{\check D_{BL}^{pp}}}$ be a horizontal map. Denote by $$\omega_\Delta:=\frac{1}{(1-|r|^2)^2}dr\wedge d\overline{r}$$ the (metric) $(1,1)$-form associated to the Poincaré metric on $\Delta$ and similarly by $\omega_h$ the form associated to the metric $h$ on $T_{{\check D_{BL}^{pp}}}$ defined above. Then the following inequality holds $$c\phi^*\omega_h \leq \omega_\Delta$$ for some $c\in{{\mathds R}}_{>0}$ meaning that $\omega_\Delta-c\phi^*\omega_h$ is a positive semi-definite form.
2. Let now $M$ be any complex manifold and $\phi:M\rightarrow {{\check D_{BL}^{pp}}}$ a horizontal map. Then $\phi$ is distance-decreasing with respect to the (suitably normalized) distance $d_h$ on ${{\check D_{BL}^{pp}}}$ induced by $h$ and the Kobayashi pseudo-distance on $M$.
The proof of the first part is well-known and uses Ahlfors’ lemma (see, e.g., [@CarlStachPeters 13.4]). The second part is an immediate consequence.
The following rather obvious lemma shows how to apply the above computations to the study of period mappings.
Let $H$ underly a variation of pure polarized, regular singular TERP-structures on $M$ with constant spectral numbers. Let $\pi:\widetilde{M}\rightarrow M$ be the universal cover. We obtain a period mapping $$\phi:\widetilde{M}\longrightarrow {{\check D_{BL}^{pp}}}$$ by associating to $\widetilde{x}\in\widetilde{M}$ the TERP-structure $H_{|{{\mathds C}}\times\{\pi(x)\}}\in{{\check D_{BL}^{pp}}}$. Then we have $d\phi({\mathcal{T}}_{\widetilde{M}})\subset\phi^*{\mathcal{T}}^{hor}_{{\check D_{BL}^{pp}}}$, i.e., $\phi$ is horizontal.
The pullback of the universal bundle $({\mathcal{L}},\nabla)$ under the map $id\times\phi:{{\mathds C}}\times \widetilde{M}\rightarrow {{\mathds C}}\times{{\check D_{BL}}}$ is isomorphic to $({\mathcal{H}},\nabla)$. By definition, for a *variation* of TERP-structures, the sheaf ${\mathcal{H}}$ is stable under $z\nabla_X$ for any $X\in (p')^{-1}{\mathcal{T}}_{\widetilde{M}}$ (where $p':{{\mathds C}}\times\widetilde{M}
\rightarrow \widetilde{M}$), and not only under $z^n\nabla_X$ as it is the case for ${\mathcal{L}}$. Therefore, $$(id\times\phi)^*(z\nabla_{d\phi(X)}){\mathcal{L}}= (z\nabla_X)(id\times\phi)^*{\mathcal{L}}\subset(id\times\phi)^*{\mathcal{L}}.$$ This implies that $\operatorname{\mathit{KS}}(d\phi({\mathcal{T}}_{\widetilde{M}}))$ is contained in ${{\mathcal{H}}}\!om_{\mathcal{R}}({\mathcal{L}},z^{-1}{\mathcal{L}}/{\mathcal{L}})$, so that by definition $\mathit{Im}(d\phi)\subset\phi^*{\mathcal{T}}^{hor}_{{\check D_{BL}^{pp}}}$.
As an example of possible applications we give the following rigidity result similar to the one for variations of Hodge structures.
\[corRigidTERP\] Let ${(H,H'_{{\mathds R}},\nabla,P,w)}$ be a variation of pure polarized regular singular TERP-structures on ${{\mathds C}}^m$ with constant spectral numbers. Then the variation is trivial, i.e., the corresponding map $\phi:{{\mathds C}}^m\rightarrow {{\check D_{BL}^{pp}}}$ is constant or, in other words, ${\mathcal{H}}$ is stable under $\nabla$.
The last lemma and the second point of proposition \[propAhlfors\] show that the period map $\phi:{{\mathds C}}^m\rightarrow {{\check D_{BL}^{pp}}}$ satisfies $d_{{{\mathds C}}^m}(x,y)\geq
d_h(\phi(x),\phi(y))$, where $d_{{{\mathds C}}^m}$ is the Kobayashi pseudo-distance on ${{\mathds C}}^m$ and $x,y\in{{\mathds C}}^m$. It is known that $d_{{{\mathds C}}^m}=0$, on the other hand, $d_h$ is a true distance, so that $\phi$ is necessarily constant.
Let us finish this paper by pointing out that the above construction has an a priori unpleasant feature: the metric space ${{\check D_{BL}^{pp}}}$ is not complete in general. We will give a concrete example showing this phenomenon. We will not carry out all details of the computations which are rather lengthy.
Consider the following topological data: Let $H^\infty_{{\mathds R}}$ be a three-dimensional real vector space, $H^\infty:=H^\infty_{{\mathds R}}\otimes {{\mathds C}}$ its complexification and choose a basis $H^\infty=\oplus_{i=1}^3{{\mathds C}}A_i$ such that $\overline{A}_1=A_3$ and $A_2\in H^\infty_{{\mathds R}}$. Moreover, choose a real number $\alpha_1\in(-3/2,-1)$, put $\alpha_2:=0$, $\alpha_3:=-\alpha_1$ and let $M\in\mathit{Aut}(H^\infty_{{\mathds C}})$ be given by $M(\underline{A})=\underline{A}\cdot\operatorname{\textup{diag}}(\lambda_1,\lambda_2,\lambda_3)$ where $\underline{A}:=(A_1,A_2,A_3)$ and $\lambda_i:=e^{-2\pi i \alpha_i}$. Putting $$\{0\}=F_0^2 \subsetneq
F_0^1:={{\mathds C}}A_1 \subsetneq
F_0^0 := {{\mathds C}}A_1 \oplus {{\mathds C}}A_2 =
F_0^{-1} \subsetneq
F_0^{-2}:= H^\infty$$ defines a sum of pure Hodge structures of weights $0$ and $-1$ on $H^\infty_1$ and $H^\infty_{\neq 1}$. A polarizing form is defined by $$S(\underline{A}^{tr},\underline{A}):=
\begin{pmatrix}
0 & 0 & \gamma \\
0 & 1 & 0 \\
-\gamma & 0 &0
\end{pmatrix},$$ where $\gamma:=\frac{-1}{2\pi i}\Gamma(\alpha_1+2)\Gamma(\alpha_3-1)$. In particular, we have for $p=1$ $$i^{p-(-1-p)}S(A_1,A_3)= (-1)iS(A_1,A_3)= \frac{\Gamma(\alpha_1+2)\Gamma(\alpha_3-1)}{2\pi} >0$$ and for $p=0$ $$i^{p-(-p)}S(A_2,A_2)=S(A_2,A_2)>0$$ so that $F_0^\bullet$ indeed induces a pure polarized Hodge structure of weight $-1$ on $H^\infty_{\neq 1}={{\mathds C}}A_1\oplus{{\mathds C}}A_2$ and a pure polarized Hodge structure of weight $0$ on $H^\infty_1={{\mathds C}}A_2$. As $M$ is semi-simple and its eigenspaces are one-dimensional, we have $\operatorname{\mathit{D}_{\mathit{PMHS}}}=\operatorname{\check{\mathit{D}}_{\mathit{PMHS}}}=\operatorname{\mathit{D}_{\mathit{PHS}}}=\operatorname{\check{\mathit{D}}_{\mathit{PHS}}}=\{F_0^\bullet\}$ and $F^\bullet_0=\widetilde{F}^\bullet_0$.
Let $(H',H_{{\mathds R}}',\nabla)$ be the flat holomorphic bundle on ${{\mathds C}}^*\times{{\mathds C}}$ with real flat subbundle corresponding to $(H^\infty,H^\infty_{{\mathds R}},M)$, and put $s_i:=z^{\alpha_i}A_i\in{\mathcal{H}}'$. According to [@HS1 formula (5.3), (5.4)], the pairing $P:{\mathcal{H}}'\otimes j^*{\mathcal{H}}'\rightarrow {\mathcal{O}}_{{{\mathds C}}^*\times {{\mathds C}}}$ is determined by the above chosen $S$, namely, $P(\underline{s}^{tr}, \underline{s}):=(\delta_{i+j,4})_{i,j\in\{1,\ldots,3\}}$.
It follows from the construction in [@He2 section 5] that the classifying space $D_\mathit{BL}={{\check D_{BL}}}$ associated with the given topological data and the spectrum $\alpha_1,\alpha_2,\alpha_3$ is ${{\check D_{BL}}}\cong{{\mathds C}}^2=\operatorname{\textup{Spec}\,}{{\mathds C}}[r,t]$, with the universal family of Brieskorn lattices given by ${\mathcal{H}}=\oplus_{i=1}^3{\mathcal{O}}_{{{\mathds C}}^3}v_i$, where $$\begin{array}{rcl}
v_1 & := & s_1 + r z^{-1} s_2 + \frac{r^2}{2} z^{-2} s_3 + t z^{-1}s_3,\\
v_2 & := & s_2 + r z^{-1} s_3,\\
v_3 & := & s_3.
\end{array}$$ $\widehat{{\mathcal{H}}}$ is pure outside of the real-analytic hypersurface $(1-\rho)^4-\theta=0$, where $\rho=\frac12r\overline{r}$ and $\theta=t\overline{t}$.
The complement has three components. $\widehat{{\mathcal{H}}}$ is polarized on two of them, those which contain $\{(r,0)\ |\ |r|<\sqrt{2}\}$ and $\{(r,0)\ |\ |r|>\sqrt{2}\}$, respectively. On the third component the metric on $p_*\widehat {\mathcal{H}}$ has signature $(+,-,-)$. So in this example ${{\check D_{BL}^{pp}}}$ has two connected components, one of them is bounded while the other is not.
If we restrict the metric $h$ on ${\mathcal{T}}_{{{\check D_{BL}^{pp}}}}$ to the tangent space of $\{(r,0)\ |\ |r|\neq \sqrt{2}\}$, then it is given by $$h(\partial_r,\partial_r) = 2\frac{1+\rho^2}{(1-\rho)^4}.$$
From this it is directly evident that the distance defined by $h$ on the unbounded component of ${{\check D_{BL}^{pp}}}$ cannot be complete, as we have $$h(\partial_{r^{-1}},\partial_{r^{-1}})=h(-r^2\partial_r,-r^2\partial_r)
=8\rho^2\frac{1+\rho^2}{(1-\rho)^4}
\stackrel{r\rightarrow \infty}{\longrightarrow} 8\neq\infty.$$ Comparing the situation to the one for classifying spaces of Hodge structure (where the distance induced by the Hodge metric on $\operatorname{\mathit{D}_{\mathit{PHS}}}$ is known to be complete due to the homogeneity of $\operatorname{\mathit{D}_{\mathit{PHS}}}$), it is clear that one needs to have a complete metric space as a possible target for period maps for variations of regular singular TERP-structures. We are able to construct such a space, it is in fact a partial compactification of ${{\check D_{BL}^{pp}}}$, on which the metric can be extended and becomes complete. However, this construction presents a number of technical difficulties and is somewhat beyond the scope of the present work. We will treat this and related questions in a subsequent paper.
[CMSP03]{}
James Carlson, Stefan M[ü]{}ller-Stach, and Chris Peters, *Period mappings and period domains*, Cambridge Studies in Advanced Mathematics, vol. 85, Cambridge University Press, Cambridge, 2003.
Sergio Cecotti and Cumrun Vafa, *Topological–anti-topological fusion*, Nuclear Phys. B **367** (1991), no. 2, 359–461.
[to3em]{}, *On classification of [$N=2$]{} supersymmetric theories*, Comm. Math. Phys. **158** (1993), no. 3, 569–644.
Pierre Deligne, *[Travaux de Griffiths.]{}*, [Séminaire Bourbaki. Vol. 1969/70: Exposés 364–381]{} (Berlin), Lecture Notes in Mathematics, Vol. 180, no. 376, Springer-Verlag, 1971, pp. 213–237.
Boris Dubrovin, *Geometry and integrability of topological-antitopological fusion*, Comm. Math. Phys. **152** (1993), no. 3, 539–564.
Phillip Griffiths and Wilfried Schmid, *Locally homogeneous complex manifolds*, Acta Math. **123** (1969), 253–302.
[to3em]{}, *Recent developments in [H]{}odge theory: a discussion of techniques and results*, Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, 1975, pp. 31–127.
Claus Hertling, *Classifying spaces for polarized mixed [H]{}odge structures and for [B]{}rieskorn lattices*, Compositio Math. **116** (1999), no. 1, 1–37.
[to3em]{}, *Frobenius manifolds and moduli spaces for singularities*, Cambridge Tracts in Mathematics, vol. 151, Cambridge University Press, Cambridge, 2002.
[to3em]{}, *[$tt\sp *$]{} geometry, [F]{}robenius manifolds, their connections, and the construction for singularities*, J. Reine Angew. Math. **555** (2003), 77–161.
Claus Hertling and Christian Sevenheck, *Nilpotent orbits of a generalization of [H]{}odge structures*, J. Reine Angew. Math. **609** (2007), 23–80.
Hiroshi Iritani, *[R]{}eal and integral structures in quantum cohomology [I]{}: toric orbifolds*, Preprint math.AG/0712.2204, 2007.
B. Malgrange, *Sur les déformations isomonodromiques. [I]{}. [S]{}ingularités régulières*, Mathematics and Physics (Paris, 1979/1982), Progr. Math., vol. 37, Birkhäuser Boston, Boston, MA, 1983, pp. 401–426.
Claude Sabbah, *[F]{}ourier-[L]{}aplace transform of a variation of polarized complex [H]{}odge structure.*, J. Reine Angew. Math. **621** (2008), 123–158.
Morihiko Saito, *On the structure of [B]{}rieskorn lattice*, Ann. Inst. Fourier (Grenoble) **39** (1989), no. 1, 27–72.
Wilfried Schmid, *Variation of [H]{}odge structure: the singularities of the period mapping*, Invent. Math. **22** (1973), 211–319.
Carlos T. Simpson, *Mixed twistor structures*, Preprint math.AG/9705006, 1997.
Lehrstuhl für Mathematik VI\
Institut für Mathematik\
Universität Mannheim, A 5, 6\
68131 Mannheim\
Germany
hertling@math.uni-mannheim.de\
sevenheck@math.uni-mannheim.de
[^1]: 2000 *Mathematics Subject Classification.* 14D07, 32S30, 32S40, 53C07, 32G20.\
Keywords: TERP-structures, twistor structures, classifying spaces, $tt^*$ geometry, mixed Hodge structures, curvature, hyperbolicity.\
C.H. acknowledges partial support by the ESF research grant MISGAM.
|
---
abstract: 'Shape memory materials have gained considerable attention thanks to their ability to change physical properties when subjected to external stimuli such as temperature, pH, humidity, electromagnetic fields, etc. These materials are increasingly used for a large number of biomedical applications. For applications inside the human body, contactless control can be achieved by the addition of electric and/or magnetic particles that can react to electromagnetic fields, thus leading to a composite biomaterial. The difficulty of developing accurate numerical models for smart materials results from their multiscale nature and from the multiphysics coupling of involved phenomena. This coupling involves electromagnetic, thermal and mechanical problems. This paper contributes to the multiphysics modeling of a shape memory polymer material used as a medical stent. The stent is excited by electromagnetic fields produced by a coil which can be wrapped around a failing organ. In this paper we develop large deformation formulations for the coupled electro-thermo-mechanical problem using the electric potential to solve the electric problem. The formulations are then discretized and solved using the finite element method. Results are validated by comparison with results in the literature.'
address: 'Columbia University, Department of Civil Engineering and Engineering Mechanics, New York, 10027 NY, USA.'
author:
- Innocent Niyonzima
- Yang Jiao
- Jacob Fish
bibliography:
- 'cmame\_smp.bib'
title: 'Modeling and simulation of nonlinear electro-thermo-mechanical continua with application to shape memory polymeric medical devices'
---
Multiphysics modeling, electro-thermo-mechanical coupling, shape memory polymers stents, large deformations. 34A34,34A36,34A37,65L20
Introduction {#sec:motivation}
============
The increase of life expectancy creates a need to maintain the functions of aging organs to allow greater independence for the elderly. Biomaterial implants have the potential to fulfill some of these functions. The total number of implants in the world exceeds four hundred million per year and grows every year [@uweb-biometarials-04]. Biomaterials are also increasingly used for a large number of biomedical applications such as the prevention and cure of coronary heart disease and stroke, as well as ophthalmological applications, biosensors and drug delivery systems [@bhatia-biomaterials-10; @rezaie-biomaterials-15]. They have the potential to contribute to the reduction of the cost of health and the improvment of the life conditions.
Among biomaterials, shape memory materials have gained considerable attention in the biomedical community thanks to their ability to change physical properties (morphing, structural rigidity, refractive index, etc.) when subjected to external stimuli such as temperature, pH, humidity, electromagnetic fields, etc. This special behavior results from the the shape memory effect observed in shape memory materials [@huang-smp-11]. They are used in minimally invasive surgery as embolic devices to treat aneurysm [@small-biomaterials-07; @maitland-biomaterials-07] and as vascular stents [@yakacki-stents-07; @baer-stents-09; @ajili-stents_09].
![Diagram illustrating the deployment of a stent in a blood vessel [@yahia-smp-15].[]{data-label="fig:sme-stents"}](SMP_SelfDeployingStent){width="100.00000%"}
Figure \[fig:sme-stents\] illustrate the deployment of a stent in a blood vessel. They can also be used as portable sensors to monitor heart and respiratory rates and in controlled drug delivery systems thus allowing to reduce the side effects of drugs [@wischke-drugdelivery-10; @nagahama-biomaterial-09]. For these different uses, biomaterials must possess a number of properties. They must be biocompatible to avoid toxicity in contact with biological tissues. Biodegradability is a desirable property for temporary implants, and for minimally invasive in vivo surgery applications, devices must be controllable without contact and self-expanding. All of these properties make polymers the best candidates for a wide range of biomedical applications. Contactless control can be achieved by the addition of electric/magnetic (nano)particles inclusions to produce a smart composite that can react to electromagnetic fields.
Accurate numerical models for smart composites must account for the multiphysics coupling which involve different domains of physics (electromagnetism, thermal, mechanics) and the multiscale nature of the materials. The present manuscript is concerned with multiphysics modeling of shape memory polymer materials used for biomedical devices for homogeneous materials. The difficulty of developing multiphysics models arises from a number of factors. (a) The *geometric non-linearity* resulting from large mechanical deformations leads to the modification of the equations that govern the electromagnetic and thermal problems to account for motion. Additional complexity for the electromagnetic problem results from the presence of electromagnetic fields in the air and vacuum. (b) The *material non-linearities* resulting from the presence of materials with nonlinear thermal constitutive laws, plastic and viscous laws for mechanics and nonlinear anhysteretic/hysteretic constitutive laws for electromagnetism.
Models and numerical simulations have already been developed for multiphysics problems in piezoelectric, magnetostrictive, and piezomagnetic materials in the case of small deformations [@fish-piezo-03; @elhadrouz-piezo-06; @anderson-coupling-07; @khalaquzzaman-piezoelectric-multiscale-12; @kuznetsov-hmm-12; @perevertov-coupling-15; @bishay-piezo-electro-magnetic-15]. Theoretical models have also been developed for thermomechanical and electro-magneto-thermomechanical problems in the case of large mechanical deformations [@eringen-electrodynamics-12; @pao-electrodynamics-78; @ogden-coupling-09; @saxena-coupledlargedefo-13]. Popular numerical implementations combine the Lagrangian approach for the mechanical problem and the Eulerian or arbitrary Lagrangian Eulerian approach for the electromagnetic problem [@stiemer-ale-09; @abali-largedefo-18-a]. In this paper we consider low frequency electromagnetic problems and solve for a scalar potential formulation only defined in the mechanical domain. Thus, a Lagrangian mesh can also be used for the electromagnetic problem.
The development of multiphysics models for smart composites controlled by electromagnetic fields is still in its infancy. Electro-magneto-mechanical models using electrostatic and magnostatic formulations have been developed for magneto-sensitive composites and magneto-electro-elastic composites (e.g., electro active polymers ) in large deformations [@ethiraj-coupling-16; @miehe-coupling-16; @bayat-coupling-18]. In [@homsi-DG-17], the discontinuous Galerkin method was used to solve the electro-thermo-mechanical problem in a SMP by solving an electrokinetic problem excited by surface currents. For a more effective contactless control, the multiphysics problem should include eddy currents and hysteretic losses as a means of controlling the temperature.
In this paper, we develop a simple multiphysics model for an electromagnetically controlled vascular stent excited by a coil. The paper extends the thermomechanical model developed in [@boatti-smp-16] by proposing a contactless electromagnetic control of the temperature, especially during the recovery step that takes place inside the human body. For the sake of clarity and in order to have a self-sufficient paper which is easily accessible by the mechanics and electromagnetic communities, we derive the general fully coupled problem from Maxwell equations and conservation laws using the Lagrangian and Eulerian formalisms. Then, a simplified, quasistatic electro-thermo-mechanical problem is derived and discretized using the finite element (FE) method.
The paper is organized as follows: in Section \[section:governing-equations\] we recall Maxwell’s equations and conservation laws using the Lagrangian and the Eulerian formalisms. In Section \[section:formulations\], we derive the simplified coupled problem and its strong and weak forms using potential formulations. The weak formulations are then semi-discretized in space using the FE method and in time using the backward Euler time stepping method. The resulting system of nonlinear algebraic equations is linearized and solved using the Newton–Raphson method. Section \[section:numerical\_tests\] deals with numerical examples. At first, we validate the thermo-mechanical formulation for SMP materials with simple geometries along the lines of [@boatti-smp-16]. We then study the behavior of the electromagnetically responsive SMP stent excited by a coil. In Section \[section:conclusions\] we close the paper with conclusions and perspectives.
Governing equations of the general multiphysics problem {#section:governing-equations}
=======================================================
In this section, the general electro-magneto-thermo-mechanical coupled problem is derived from Maxwell’s equations and conservation laws. Throughout the paper, we use the indices $E$ and $L$ to denote the Eulerian and Lagrangian quantities. Thus, ${\mbox{\boldmath$f$}}_E$ and ${\mbox{\boldmath$f$}}_L$ denote the forces in the Eulerian and Lagrangian framework, respectively. The open domains $\Omega_{\mathrm{0}}^{\mathrm{Mec}}$, $\Omega_{\mathrm{0}}^{\mathrm{The}}$ and $\Omega_{\mathrm{0}}^{\mathrm{Ele}}$ denote the undeformed computational domains for the mechanical, thermal and electromagnetic problems, respectively. Likewise, $\Omega_{\mathrm{t}}^{\mathrm{Mec}}$, $\Omega_{\mathrm{t}}^{\mathrm{The}}$ and $\Omega_{\mathrm{t}}^{\mathrm{Ele}}$ denote the deformed computational domains for the mechanical, thermal and electromagnetic problems at a time $t \in \mathcal{I}_t := ]t_0, t_{\mathrm{end}}[$. The domains of the mechanical and thermal problems are generally subdomains of the electromagnetic domain, i.e., $\Omega_i^{\mathrm{Mec}} \subseteq \Omega_i^{\mathrm{Ele}}$ and $\Omega_i^{\mathrm{The}} \subseteq \Omega_i^{\mathrm{Ele}}$ with $i = \{0, t\}$ as the electromagnetic fields can be defined in the entire domain including the surrounding air. The domains $\Omega_{c, i} \subseteq \Omega_{i}^{\mathrm{Ele}}$, $\Omega_{c, i}^C \subseteq \Omega_{i}^{\mathrm{Ele}}$ and $\Omega_{s, i} \subsetneq \Omega_{c, i}^C \subset \Omega_{i}^{\mathrm{Ele}}$ are the conductors, non-conductors and inductors where the electric currents source is imposed. The domains $\Gamma_i^{\mathrm{Ele}}$, $\Gamma_i^{\mathrm{Ele}}$ and $\Gamma_i^{\mathrm{Ele}}$ denote the boundaries of the electromagnetic, thermal and mechanical domains, respectively. The differential operators ${\text{{\text{\bf Grad}}}{}\,{}}$, ${\text{{\text{\bf Curl}}}{}\,{}}$ and ${\text{{\text{Div}}}{}\,{}}$ denote the gradient, rotational and divergence operators defined on the undeformed configurations while ${\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{}}$, ${\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{}}$ and ${\mathrm{{\mathrm{div}}}{}\,{}}$ denote the same operators defined on the deformed configurations.
Kinematics
----------
The motion is described by the mappings $\boldsymbol{{\mbox{\boldmath$\varphi$} }}_t$ and $\boldsymbol{{\mbox{\boldmath$\varphi$} }}$ assumed to be smooth enough (we do not consider fracture). The mapping $\boldsymbol{{\mbox{\boldmath$\varphi$} }}_t$ is also assumed to be bijective and defined by: $$\begin{aligned}
&{\mbox{\boldmath$\varphi$} }_t: \,\, && \Omega_0^{\mathrm{Mec}} \rightarrow \mathbb{E}^3, \\
& &&\boldsymbol{X} \mapsto {\mbox{\boldmath$x$}}= {\mbox{\boldmath$\varphi$} }_t(\boldsymbol{X}) = {\mbox{\boldmath$\varphi$} }(\boldsymbol{X}, t) = \boldsymbol{X} + {\mbox{\boldmath$u$}}(\boldsymbol{X}, t)
\end{aligned}
\label{eq:Large_Transformation_Mapping}$$ where $\boldsymbol{X}$ is the position of a particle point P in the undeformed configuration, ${\mbox{\boldmath$x$}}$ is the position of P in the deformed configuration, ${\mbox{\boldmath$u$}}$ is the vector of displacements and $\mathbb{E}^3$ is the three dimensional Euclidean space [@wriggers-fem-08]. The positions in the undeformed and deformed configurations are related by $\boldsymbol{X} = {\mbox{\boldmath$\varphi$} }_t^{-1}({\mbox{\boldmath$x$}})$ which is valid thanks to the bijection of ${\mbox{\boldmath$\varphi$} }_t$. For any time $t \in \mathcal{I}_t$, the deformed configurations are also defined as: $$\Omega_{\mathrm{t}}^{\mathrm{Mec}} := \varphi_t\left(\Omega_{\mathrm{0}}^{\mathrm{Mec}}\right), \quad
\Omega_{\mathrm{t}}^{\mathrm{The}} := \varphi_t\left(\Omega_{\mathrm{0}}^{\mathrm{The}}\right), \quad
\Omega_{\mathrm{t}}^{\mathrm{Ele}} := \varphi_t\left(\Omega_{\mathrm{0}}^{\mathrm{Ele}}\right).$$
The deformation gradient tensor and its determinant are given by: $${\boldsymbol{F}}:= \frac{\partial {\mbox{\boldmath$x$}}}{\partial {\boldsymbol{X}}} = {\text{{\text{\bf Grad}}}{}\,{\varphi}} = {\boldsymbol{\mathbbm{1}}}+ {\text{{\text{\bf Grad}}}{}\,{{\mbox{\boldmath$u$}}}} \quad,
\quad J = \mathrm{det}{\boldsymbol{F}}\label{eq:Deformation_Gradient_Tensor}$$ where ${\boldsymbol{\mathbbm{1}}}$ is the identity matrix. The velocity ${\mbox{\boldmath$v$}}$ and acceleration ${\mbox{\boldmath$a$}}$ are given by: $$\begin{gathered}
{\mbox{\boldmath$v$}}(\boldsymbol{X}, t) = \frac{\partial {\mbox{\boldmath$\varphi$} }}{\partial t}(\boldsymbol{X}, t), \quad
{\mbox{\boldmath$a$}}(\boldsymbol{X}, t) = \frac{\partial {\mbox{\boldmath$v$}}}{\partial t}({\boldsymbol{X}}, t) = \frac{\partial^2 {\mbox{\boldmath$\varphi$} }}{\partial t^2}(\boldsymbol{X}, t).
\label{eq:Velocity_Acceleration_Eulerian}\end{gathered}$$
Assuming the existence of a mapping ${\mbox{\boldmath$\Theta$} }$: $$\begin{aligned}
&{\mbox{\boldmath$\Theta$} }: \,\, && \mathbb{E}^3 \times \mathcal{I}_t \rightarrow \Omega_{0}^{\mathrm{Mec}} \subsetneq \mathbb{E}^3, \\
& &&({\mbox{\boldmath$x$}}, t) \mapsto {\boldsymbol{X}}= {\mbox{\boldmath$\Theta$} }({\mbox{\boldmath$x$}}, t) = {\mbox{\boldmath$\Theta$} }({\mbox{\boldmath$\varphi$} }({\boldsymbol{X}}, t), t),
\end{aligned}
\label{eq:Large_Transformation_Inverse_Mapping}$$ it is possible to derive the following relationship: $$\begin{gathered}
\frac{D {\boldsymbol{X}}}{D t}
= \frac{\partial \boldsymbol{\Theta}}{\partial {\mbox{\boldmath$\varphi$} }} \frac{\partial {\mbox{\boldmath$\varphi$} }}{\partial t}
+ \frac{\partial \boldsymbol{\Theta}}{\partial t}
= {\boldsymbol{F}}^{-1} {\mbox{\boldmath$v$}}+ {\boldsymbol{V}}= \boldsymbol{0},
\label{eq:Velocity_Lagrangian}\end{gathered}$$ which relates the *matter flow field* ${\boldsymbol{V}}$ to the velocity ${\mbox{\boldmath$v$}}$ as: $$\begin{gathered}
{\boldsymbol{V}}= -{\boldsymbol{F}}^{-1} {\mbox{\boldmath$v$}}.
\label{eq:Velocity_Lagrangian_V_v}\end{gathered}$$ The matter flow field is important for the definition of the electromagnetic problem on the undeformed configuration. The independence of the initial position ${\boldsymbol{X}}$ on the time was used to derive .
Maxwell’s equations {#sub:section_maxwell_equations}
-------------------
We recall non-relativistic Maxwell’s equations on moving domains under large deformations, using Eulerian and Lagrangian formalisms [@penfield-electrodynamics-63; @fano-electromagnetism-68; @pao-electrodynamics-78; @eringen-electrodynamics-12].
### Full Maxwell’s equations in Eulerian formalism
In Eulerian setting, the electromagnetic fields are governed by the following Maxwell’s equations [@jackson-electrodynamics-98; @eringen-electrodynamics-12]:
$${\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$h$}}}} = {\mbox{\boldmath$j$}}+ \partial_t {\mbox{\boldmath$d$}}, \quad
{\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$e$}}}} = -\partial_t {\mbox{\boldmath$b$}}, \quad
{\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$b$}}}} = 0, \quad
{\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$d$}}}} = \rho_{E},
\tag{\theequation\,a-d}$$
\[eq:Full\_Maxwell\_GoverningEquations\_Eulerian\]
and constitutive laws:
$$\begin{gathered}
{\mbox{\boldmath$h$}}= \mu_0^{-1} {\mbox{\boldmath$b$}}- \underset{{\mbox{\boldmath$m$}}_{\mathrm{eff}}}{\underbrace{({\mbox{\boldmath$m$}}- {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$p$}})}} = \boldsymbol{\nu}_E({\mbox{\boldmath$b$}}) \, {\mbox{\boldmath$b$}}+ {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$p$}}= {\boldsymbol{H}}({\mbox{\boldmath$e$}}, {\mbox{\boldmath$b$}}, {\mbox{\boldmath$v$}}), \\
{\mbox{\boldmath$d$}}= \epsilon_0 {\mbox{\boldmath$e$}}+ {\mbox{\boldmath$p$}}= \epsilon_0 \boldsymbol{\epsilon}_{r E}({\mbox{\boldmath$e$}}) \, {\mbox{\boldmath$e$}}= \boldsymbol{\epsilon}_E({\mbox{\boldmath$e$}}) \, {\mbox{\boldmath$e$}}= {\mbox{\boldmath$D$}}({\mbox{\boldmath$e$}}), \textcolor{white}{Inno Inno Inno Inno}\\
{\mbox{\boldmath$j$}}= {\mbox{\boldmath$\sigma$} }_E (\underset{{\mbox{\boldmath$e$}}_{\mathrm{eff}}}{\underbrace{{\mbox{\boldmath$e$}}+ {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$b$}}} } ) + {\mbox{\boldmath$j$}}_s + \rho_{E} {\mbox{\boldmath$v$}}= {\mbox{\boldmath$j$}}_{\mathrm{eff}} + \rho_{E} {\mbox{\boldmath$v$}}= {\mbox{\boldmath$J$}}({\mbox{\boldmath$e$}}, {\mbox{\boldmath$b$}}, {\mbox{\boldmath$v$}}).
\tag{\theequation\,a-c}
\end{gathered}$$
\[eq:Full\_Maxwell\_MaterialLaw\_Eulerian\]
In (\[eq:Full\_Maxwell\_GoverningEquations\_Eulerian\]a-d)–(\[eq:Full\_Maxwell\_MaterialLaw\_Eulerian\]a-c), ${\mbox{\boldmath$h$}}$ is the magnetic field (A/m), ${\mbox{\boldmath$b$}}$ the magnetic flux density (T), ${\mbox{\boldmath$j$}}$ the electric current density (A/m$^2$), ${\mbox{\boldmath$d$}}$ the electric flux density (C/m$^2$), ${\mbox{\boldmath$e$}}$ the electric field (V/m) and $\rho_{E}$ the electric charge density (C/m$^3$). The fields ${\mbox{\boldmath$e$}}_{\mathrm{eff}}$ and ${\mbox{\boldmath$j$}}_{\mathrm{eff}}$ are the effective electric field and effective electric current density, whereas ${\mbox{\boldmath$j$}}_s$ (A/m$^{2}$) is the electric current source defined in the inductors $\Omega_{s, t}$ and ${\mbox{\boldmath$\sigma$} }_E$ is the electric conductivity tensor ($\Omega$/m). The magnetization ${\mbox{\boldmath$m$}}$ (A/m) and polarization ${\mbox{\boldmath$p$}}$ (C/m$^2$) are defined by:
$${\mbox{\boldmath$m$}}= \mu_0^{-1} \, \boldsymbol{\chi}_{b E}({\mbox{\boldmath$b$}})\, {\mbox{\boldmath$b$}}, \quad
{\mbox{\boldmath$p$}}= \epsilon_0 \, \boldsymbol{\chi}_{e E}({\mbox{\boldmath$e$}}) \, {\mbox{\boldmath$e$}}= \frac{\boldsymbol{\chi}_{e E}}{{\boldsymbol{\mathbbm{1}}}+ \boldsymbol{\chi}_{e E}} \, {\mbox{\boldmath$d$}},
\tag{\theequation\,a-b}$$
\[eq:Full\_Maxwell\_MaterialLaw\_mb\_pe\_Eulerian\]
where $\boldsymbol{\chi}_{b E}$ and $\boldsymbol{\chi}_{e E}$ are the magnetic and electric susceptibility tensors, $\mu_0 = 4 \pi 10^{-7}$ is the magnetic permeability of the free space (H/m) and $\epsilon_0 \simeq 10^{-9}/36 \pi$ is the electric permittivity of the free space (C$^2$/Nm$^2$). Another definition of magnetic susceptibility $\boldsymbol{\chi}_{m E}$ with ${\mbox{\boldmath$m$}}= \boldsymbol{\chi}_{m E} {\mbox{\boldmath$h$}}$ is often used in the constitutive law dual to (\[eq:Full\_Maxwell\_MaterialLaw\_mb\_pe\_Eulerian\]a). Additionally, $\boldsymbol{\nu}_E = \mu_0^{-1} ({\boldsymbol{\mathbbm{1}}}- \boldsymbol{\chi}_{b E})$ is the magnetic reluctivity tensor, $\boldsymbol{\mu}_E = \boldsymbol{\nu}_E^{-1}$ is the magnetic permeability tensor, ${\mbox{\boldmath$\epsilon$} }_E$ is the electric susceptibility tensor, ${\mbox{\boldmath$\epsilon$} }_{r E}({\mbox{\boldmath$e$}})$ is the relative electric permittivity tensor with ${\mbox{\boldmath$\epsilon$} }_{r E}({\mbox{\boldmath$e$}}) = {\boldsymbol{\mathbbm{1}}}+ \boldsymbol{\chi}_{e E}({\mbox{\boldmath$e$}})$ and ${\mbox{\boldmath$v$}}$ is the velocity (m/s). The dependency of the mapping ${\boldsymbol{H}}$ on (\[eq:Full\_Maxwell\_MaterialLaw\_Eulerian\]a) on the electric field ${\mbox{\boldmath$e$}}$ results from (\[eq:Full\_Maxwell\_MaterialLaw\_mb\_pe\_Eulerian\]b). The motion is accounted for by the velocity terms in the constitutive laws (\[eq:Full\_Maxwell\_MaterialLaw\_Eulerian\]a-c).
### Full Maxwell’s equations in Lagrangian formalism
In Lagrangian setting, the electromagnetic fields are governed by the following Maxwell’s equations (see [@penfield-electrodynamics-63; @fano-electromagnetism-68; @pao-electrodynamics-78; @eringen-electrodynamics-12; @saxena-coupledlargedefo-13] and [@castro-electromagnetism-14 Appendix F]):
$${\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{H}}}_{\mathrm{eff}}}} = {\boldsymbol{\mathcal{J}}}+ \partial_t {\boldsymbol{\mathcal{D}}}, \,\,
{\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{E}}}_{\mathrm{eff}}}} = -\partial_t {\boldsymbol{\mathcal{B}}}, \,\,
{\text{{\text{Div}}}{}\,{{\boldsymbol{\mathcal{B}}}}} = 0, \,\,
{\text{{\text{Div}}}{}\,{{\boldsymbol{\mathcal{D}}}}} = \rho_{L},
\tag{\theequation\,a-d}$$
\[eq:Full\_Maxwell\_GoverningEquations\_Lagrangian\]
and constitutive laws: $$\begin{aligned}
{\boldsymbol{\mathcal{H}}}_{\mathrm{eff}} &= \underset{\boldsymbol{\nu}_L}{\underbrace{J^{-1} {\boldsymbol{F}}^{T} (\boldsymbol{\nu}_E \circ \varphi_t^{-1}) {\boldsymbol{F}}} } \, {\boldsymbol{\mathcal{B}}}+ ({\mbox{\boldmath$\epsilon$} }_{r E} \circ \varphi_t^{-1})^{-1} ({\boldsymbol{V}}\times {\boldsymbol{\mathcal{D}}})
\label{eq:Full_Maxwell_MaterialLaw_Lagrangian_H} \\
{\boldsymbol{\mathcal{D}}}&= \underset{{\mbox{\boldmath$\epsilon$} }_L}{\underbrace{J \, {\boldsymbol{F}}^{-1} ({\mbox{\boldmath$\epsilon$} }_E \circ \varphi_t^{-1}) {\boldsymbol{F}}^{-T}}} (\underset{{\boldsymbol{\mathcal{E}}}}{\underbrace{{\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} + {\boldsymbol{V}}\times {\boldsymbol{\mathcal{B}}}}}),
\label{eq:Full_Maxwell_MaterialLaw_Lagrangian_D}\\
{\boldsymbol{\mathcal{J}}}&= \underset{\boldsymbol{{\mbox{\boldmath$\sigma$} }}_L}{\underbrace{J \, {\boldsymbol{F}}^{-1} \, ({\mbox{\boldmath$\sigma$} }_E \circ \varphi_t^{-1}) \, {\boldsymbol{F}}^{-T} }} \, {\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} + {\boldsymbol{\mathcal{J}}}_s
\label{eq:Full_Maxwell_MaterialLaw_Lagrangian_J}\end{aligned}$$ defined on the undeformed configuration $\Omega_{0}^{\mathrm{Ele}}$. In (\[eq:Full\_Maxwell\_GoverningEquations\_Lagrangian\]a-d) and –, ${\boldsymbol{\mathcal{H}}}$ is the magnetic field (A/m), ${\boldsymbol{\mathcal{B}}}$ the magnetic flux density (T), ${\boldsymbol{\mathcal{J}}}$ the electric current density (A/m$^2$), ${\boldsymbol{\mathcal{D}}}$ the electric flux density (C/m$^2$), ${\boldsymbol{\mathcal{E}}}_{\mathrm{eff}}$ the effective electric field, ${\boldsymbol{\mathcal{E}}}$ the electric field (V/m), ${\boldsymbol{\mathcal{J}}}_s$ (A/m$^{2}$) the current source defined in the inductors $\Omega_{s, 0}$ and ${\boldsymbol{V}}$ is the matter flow field (m/s) defined in . The tensor ${\mbox{\boldmath$\sigma$} }_L$ is the electric conductivity tensor ($\Omega$/m) and $\rho_{L}$ is the electric charge density (C/m$^3$). The notation $({\mbox{\boldmath$f$}}\circ {{\mbox{\boldmath$\varphi$} }_t}^{-1})({\mbox{\boldmath$x$}}) := {\mbox{\boldmath$f$}}({{\mbox{\boldmath$\varphi$} }_t}^{-1}({\mbox{\boldmath$x$}}))$ is used.
The one differential forms are transformed as:
$$\begin{gathered}
{\boldsymbol{\mathcal{H}}}_{\mathrm{eff}} = {\boldsymbol{F}}^{T} \left({\mbox{\boldmath$h$}}- {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$d$}}\right), \, \,
{\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} = {\boldsymbol{F}}^{T} \left({\mbox{\boldmath$e$}}+ {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$b$}}\right), \\
{\boldsymbol{\mathcal{E}}}= {\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} + {\boldsymbol{V}}\times {\boldsymbol{\mathcal{B}}}= {\boldsymbol{F}}^{T} {\mbox{\boldmath$e$}}.
\tag{\theequation\,a-c}
\end{gathered}$$
\[eq:Simplified\_Maxwell\_Pullback\_H\_E\]
and the two differential forms are transformed as: $${\boldsymbol{\mathcal{B}}}= J \, {\boldsymbol{F}}^{-1} \, {\mbox{\boldmath$b$}}, \quad {\boldsymbol{\mathcal{D}}}= J \, {\boldsymbol{F}}^{-1} \, {\mbox{\boldmath$d$}}, \quad {\boldsymbol{\mathcal{J}}}= J \, {\boldsymbol{F}}^{-1} \, {\mbox{\boldmath$j$}}, \quad {\boldsymbol{\mathcal{J}}}_s = J \, {\boldsymbol{F}}^{-1} \, {\mbox{\boldmath$j$}}_s.\label{eq:Simplified_Maxwell_Pullback_B_D_J}$$ In (\[eq:Simplified\_Maxwell\_Pullback\_H\_E\]c), ${\boldsymbol{V}}\times {\boldsymbol{\mathcal{B}}}= -{\boldsymbol{F}}^{T} \left({\mbox{\boldmath$v$}}\times {\mbox{\boldmath$b$}}\right)$ results from the identity: $${\boldsymbol{F}}^{T} {\mbox{\boldmath$v$}}\times {\boldsymbol{F}}^{T} {\mbox{\boldmath$b$}}= J {\boldsymbol{F}}^{-1}({\mbox{\boldmath$v$}}\times {\mbox{\boldmath$b$}}),$$ which is valid for any matrix ${\boldsymbol{F}}\in GL_3(\mathbb{R})$ [@castro-electromagnetism-14 Formula B.11].
Additionally, the magnetization ${\boldsymbol{\mathcal{M}}}$ is related to the magnetic induction by: $$\begin{gathered}
{\boldsymbol{\mathcal{M}}}= {\boldsymbol{F}}^{T} {\mbox{\boldmath$m$}}= {\boldsymbol{F}}^{T} (\mu_0^{-1} \boldsymbol{\chi_{b E}} \, {\mbox{\boldmath$b$}})
= {\boldsymbol{F}}^{T} (\mu_0^{-1} \boldsymbol{\chi}_{b E} \, J^{-1} \, {\boldsymbol{F}}\, {\boldsymbol{\mathcal{B}}}) \\
= \mu_0^{-1} J^{-1} {\boldsymbol{F}}^T \boldsymbol{\chi_{b E}} {\boldsymbol{F}}\, {\boldsymbol{\mathcal{B}}}= \mu_0^{-1} \boldsymbol{\chi}_{b L} \, {\boldsymbol{\mathcal{B}}}= ( \mu_0^{-1} {\boldsymbol{\mathbbm{1}}}- \boldsymbol{\nu}_{L}) \, {\boldsymbol{\mathcal{B}}},
\label{eq:Simplified_MaterialLaw_M} \end{gathered}$$ while the effective magnetization ${\boldsymbol{\mathcal{M}}}_{\mathrm{eff}}$ is given by: $${\boldsymbol{\mathcal{M}}}_{\mathrm{eff}}
= {\boldsymbol{F}}^{T} ({\mbox{\boldmath$m$}}- {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$p$}})
= {\boldsymbol{\mathcal{M}}}+ {\boldsymbol{V}}\times {\boldsymbol{\mathcal{P}}},
= ( \mu_0^{-1} {\boldsymbol{\mathbbm{1}}}- \boldsymbol{\nu}_{L}) \, {\boldsymbol{\mathcal{B}}}+ {\boldsymbol{V}}\times {\boldsymbol{\mathcal{P}}}\label{eq:Simplified_MterialLaw_M_eff}$$ where the polarization is related to the electric flux density by [@castro-electromagnetism-14]: $${\boldsymbol{\mathcal{P}}}= \frac{{\mbox{\boldmath$\epsilon$} }_{r L} - {\boldsymbol{\mathbbm{1}}}}{{\mbox{\boldmath$\epsilon$} }_{r L}} \, {\boldsymbol{\mathcal{D}}}= \frac{\boldsymbol{\chi_{e E}}}{{\boldsymbol{\mathbbm{1}}}+ \boldsymbol{\chi_{e E}}} \, {\boldsymbol{\mathcal{D}}}.
\label{eq:Simplified_MterialLaw_M}$$ Combining all these results, the following transformations for second order tensors used in constitutive laws can be derived: $$\begin{gathered}
\boldsymbol{\nu}_{L} = J^{-1} {\boldsymbol{F}}^{T} \boldsymbol{\nu}_{E} {\boldsymbol{F}},
\boldsymbol{\mu}_{L} = J {\boldsymbol{F}}^{-1} \boldsymbol{\mu}_{E} {\boldsymbol{F}}^{-T}, \\
{\mbox{\boldmath$\epsilon$} }_{L} = J {\boldsymbol{F}}^{-1} {\mbox{\boldmath$\epsilon$} }_{E} {\boldsymbol{F}}^{-T},
{\mbox{\boldmath$\sigma$} }_{L} = J {\boldsymbol{F}}^{-1} {\mbox{\boldmath$\sigma$} }_{E} {\boldsymbol{F}}^{-T}.
\label{eq:Simplified_Maxwell_Pullback_2ndOrderTensor} \end{gathered}$$
Conservation equations
----------------------
We recall conservation equations using the Eulerian and Lagrangian formalisms.
### Conservation equations using the Eulerian description
In Eulerian setting, the conservation equations read [@simo-fem-06; @wriggers-fem-08; @belytschko-fem-13]:
$$\frac{D \rho}{D t} + \rho \, {\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$v$}}}} = 0, \quad
\rho \frac{D {\mbox{\boldmath$v$}}}{D t} - {\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$\sigma$} }}} = {\mbox{\boldmath$f$}}_E, \quad
{\mbox{\boldmath$\epsilon$} }_{P} \colon {\mbox{\boldmath$\sigma$} }+ {\mbox{\boldmath$L$}}_E = 0.
\tag{\theequation\,a-c}$$
\[eq:Full\_Conservation\_Equations\_Eulerian\]
$$\rho \, \frac{D U_E}{D t} + {\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$q$}}_E}} = {\mbox{\boldmath$\sigma$} }: {\boldsymbol{L}}+ w_E.
\label{eq:Full_Conservation_Energy_Eulerian}$$
Equations (\[eq:Full\_Conservation\_Equations\_Eulerian\]a-c) are balance equations of the mass, linear and angular momentum and is the balance equation of the internal energy. The quantity $\rho$ is the mass density (kgm$^{-3}$), ${\mbox{\boldmath$\sigma$} }$ is the Cauchy stress (N/m$^2$), ${\mbox{\boldmath$f$}}_E$ is the volume force (N/m$^3$), ${\mbox{\boldmath$\epsilon$} }_{P}$ is the third order permutation tensor also known as the Levi Civita tensor such that ${({\mbox{\boldmath$\epsilon$} }_p {\mbox{\boldmath$\sigma$} })}_i = ({\mbox{\boldmath$\epsilon$} }_p)_{ijk} {\mbox{\boldmath$\sigma$} }_{jk}$, ${\mbox{\boldmath$L$}}_E$ is the torque, $U_E$ is the density of internal energy, ${\mbox{\boldmath$q$}}_E$ is the heat flux density, ${\boldsymbol{L}}:= {\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{{\mbox{\boldmath$v$}}}}$ is the gradient deformation and $w_E$ is the electromagnetic source term for the heat problem.
The electromagnetic force, torque, internal energy and source term are given by: $$\begin{gathered}
\textcolor{white}{I\,\,} {\mbox{\boldmath$f$}}_E = \rho_{E} {\mbox{\boldmath$e$}}+ {\mbox{\boldmath$j$}}\times {\mbox{\boldmath$b$}}+ \left({\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{{\mbox{\boldmath$e$}}}} \right)^{T} {\mbox{\boldmath$p$}}+ \left({\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{{\mbox{\boldmath$b$}}}} \right)^{T} {\mbox{\boldmath$m$}}\\+ \partial_t ({\mbox{\boldmath$p$}}\times {\mbox{\boldmath$b$}}) + {\mathrm{{\mathrm{div}}}{}\,{[{\mbox{\boldmath$v$}}\otimes ({\mbox{\boldmath$p$}}\times {\mbox{\boldmath$b$}})]}},
\label{eq:Full_Electromagnetic_Force_Eulerian}\end{gathered}$$ $$\begin{aligned}
{\mbox{\boldmath$L$}}_E &= {\mbox{\boldmath$p$}}\times {\mbox{\boldmath$e$}}+ ({\mbox{\boldmath$m$}}- {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$p$}}) \times {\mbox{\boldmath$b$}},
\textcolor{white}{{\mbox{\boldmath$L$}}_E = {\mbox{\boldmath$p$}}\times {\mbox{\boldmath$e$}}+ ({\mbox{\boldmath$m$}}- {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$p$}}) \times {\mbox{\boldmath$b$}}. }
\label{eq:Full_Electromagnetic_Torque_Eulerian} \\
\rho \, \frac{D U_E}{D t} &= \rho \, c_p \frac{\partial \vartheta_E}{\partial t},
\textcolor{white}{{\mbox{\boldmath$L$}}_E = {\mbox{\boldmath$p$}}\times {\mbox{\boldmath$e$}}+ ({\mbox{\boldmath$m$}}+ {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$p$}}) \times {\mbox{\boldmath$b$}}, }
\label{eq:Full_Electromagnetic_Energy_Eulerian} \\
w_E &= {\mbox{\boldmath$j$}}_{\mathrm{eff}} \cdot {\mbox{\boldmath$e$}}_{\mathrm{eff}}
- {\mbox{\boldmath$m$}}_{\mathrm{eff}} \cdot \frac{\partial {\mbox{\boldmath$b$}}}{\partial t}
+ \rho \frac{\partial}{\partial t} \left( \frac{{\mbox{\boldmath$p$}}}{\rho} \right) \cdot {\mbox{\boldmath$e$}}_{\mathrm{eff}}
\label{eq:Full_Electromagnetic_Losses_Eulerian}\end{aligned}$$ where $c_p$ and $\vartheta_E$ are the heat capacity and the temperature, respectively. Equations (\[eq:Full\_Conservation\_Equations\_Eulerian\] c) and suggest that Cauchy stress ${\mbox{\boldmath$\sigma$} }$ may not be symmetric in presence of electromagnetic fields. However, symmetry may be kept for isotropic materials.
Equations – must be completed by constitutive laws derived from the *Clausius-Duhem inequality*. In this paper, we assume the following nonlinear constitutive law for the thermal problem [@belytschko-fem-13]: $${\mbox{\boldmath$q$}}_E = \boldsymbol{Q}(\vartheta_E, {\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{\vartheta_E}}) = -{\mbox{\boldmath$\kappa$} }_E(\vartheta_E) \, {\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{\vartheta_E}}
\label{eq:Full_Thermal_Mechanical_ConstitutiveLaws_Eulerian}$$ where ${\mbox{\boldmath$\kappa$} }_E$ is the thermal conductivity tensor (W/mK). The thermo-mechanical constitutive law involves the definition of an appropriate objective rate ${\mbox{\boldmath$\sigma$} }^{\nabla}$ (e.g., Jaumann rate, Truesdell rate or Green-Naghdi rate) which is related to the material derivative of the Cauchy stress $\dot{{\mbox{\boldmath$\sigma$} }}$ [@belytschko-fem-13] as:
$${\mbox{\boldmath$\sigma$} }^{\nabla} = \dot{{\mbox{\boldmath$\sigma$} }} - \boldsymbol{\psi}(\boldsymbol{L}, {\boldsymbol{F}})
= \boldsymbol{\Sigma}({\mbox{\boldmath$\sigma$} }, \boldsymbol{L}, {\boldsymbol{F}}, \vartheta_E, {\boldsymbol{Z}}_E(\tau \leq t)), \quad
\boldsymbol{L} = \dot{{\boldsymbol{F}}} \, {\boldsymbol{F}}^{-1}
\tag{\theequation\,a-b}$$
\[eq:Full\_ThermoMechanical\_ConstitutiveLaws\_Eulerian\]
where the velocity gradient ${\boldsymbol{L}}$ is related to the rate of the deformation gradient $\dot{{\boldsymbol{F}}}$ through (\[eq:Full\_ThermoMechanical\_ConstitutiveLaws\_Eulerian\]b) and ${\boldsymbol{Z}}_E(\tau \leq t)$ is the set of internal variables that account for the history of the loadings.
### Conservation equations using the Lagrangian description
In Lagrangian setting, the conservation equations read [@simo-fem-06; @wriggers-fem-08; @belytschko-fem-13]:
$$\rho_0 - \rho J = 0, \quad
\rho_0 \frac{D^2 {\mbox{\boldmath$u$}}}{D t^2} - {\text{{\text{Div}}}{}\,{{\boldsymbol{F}}{\boldsymbol{S}}}} = {\mbox{\boldmath$f$}}_L, \quad
{\mbox{\boldmath$\epsilon$} }_{P} \colon ({\boldsymbol{F}}{\boldsymbol{S}}{\boldsymbol{F}}^{T}) + {\mbox{\boldmath$L$}}_L = 0.
\tag{\theequation\,a-c}$$
\[eq:Full\_Conservation\_Equations\_Lagrangian\]
$$\rho_0 \, \frac{D U_L}{D t} + {\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$q$}}_L}} = -{\boldsymbol{P}}^{T} \colon {\text{{\text{\bf Grad}}}{}\,{({\boldsymbol{F}}{\boldsymbol{V}})}} + w_L.
\label{eq:Full_Conservation_Energy_Lagrangian}$$
Equations (\[eq:Full\_Conservation\_Equations\_Lagrangian\]a-c) are balance equations of the mass, linear and angular momentum and is the balance equation of the internal energy, expressed on the undeformed configuration $\Omega_0^{\mathrm{Mec}}$. The quantity $\rho_0$ is the mass density (kg/m$^{3}$), ${\boldsymbol{S}}$ is the second Piola–Kirchhoff stress (N/m$^2$) with ${\boldsymbol{S}}= {\boldsymbol{F}}^{-1} {\boldsymbol{P}}$ where ${\boldsymbol{P}}$ is the first Piola–Kirchhoff stress or nominal stress tensor with ${\boldsymbol{P}}= J \, {\mbox{\boldmath$\sigma$} }\, {\boldsymbol{F}}^{-T}$. The quantity ${\mbox{\boldmath$f$}}_L$ is the volume force (N/m$^3$), ${\mbox{\boldmath$L$}}_L$ is the torque, $U_L$ is the density of internal energy, ${\mbox{\boldmath$q$}}_L$ is the heat flux density and $w_L$ is the source term for the heat equation. The electromagnetic force which is a three differential form is transformed as: $$\begin{aligned}
\!\!{\mbox{\boldmath$f$}}_L \!=\! J \, {\mbox{\boldmath$f$}}_E
&= \rho_L {\boldsymbol{F}}^{-T} {\boldsymbol{\mathcal{E}}}+ J^{-1} \left( ({\boldsymbol{F}}\, {\boldsymbol{\mathcal{J}}}) \times ({\boldsymbol{F}}\, {\boldsymbol{\mathcal{B}}}) \right) \\
&+ J \left({\boldsymbol{F}}^{-T} {\text{{\text{\bf Grad}}}{}\,{({\boldsymbol{F}}^{-T} \, {\boldsymbol{\mathcal{E}}})}} \right)^{T} (J^{-1} \, {\boldsymbol{F}}\, {\boldsymbol{\mathcal{P}}}) \\
&+ \left({\boldsymbol{F}}^{-T} {\text{{\text{\bf Grad}}}{}\,{(J^{-1} \, {\boldsymbol{F}}\, {\boldsymbol{\mathcal{B}}})}} \right)^{T} ({\boldsymbol{F}}^{-T} \, {\boldsymbol{\mathcal{M}}}) \\
&+ J \frac{\partial}{\partial t}\left[ J^{-2} ({\boldsymbol{F}}{\boldsymbol{\mathcal{P}}}) \times ({\boldsymbol{F}}{\boldsymbol{\mathcal{B}}}) \right] +{\text{{\text{Div}}}{}\,{\left[J^{-1} {\boldsymbol{V}}\otimes \left[ ({\boldsymbol{F}}{\boldsymbol{\mathcal{P}}}) \times ({\boldsymbol{F}}{\boldsymbol{\mathcal{B}}}) \right] \right]}}.
\end{aligned}
\label{eq:Full_Electromagnetic_Force_Lagrangian}$$ The torque and the internal energy are transformed as:
$$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!\!\!{\mbox{\boldmath$L$}}_L
&= J \, {\mbox{\boldmath$L$}}_E = \left({\boldsymbol{F}}{\boldsymbol{\mathcal{P}}}\right) \times \left({\boldsymbol{F}}^{-T} {\boldsymbol{\mathcal{E}}}\right)
+ \left({\boldsymbol{F}}^{-T} {\boldsymbol{\mathcal{M}}}_{\mathrm{eff}} \right) \times \left({\boldsymbol{F}}{\boldsymbol{\mathcal{B}}}\right), \\
\!\!\!\!\!\!\!\!\!\!\!\!\rho_0 \frac{D U_L}{D t}
&= \rho_0 c_p \frac{\partial \vartheta_L}{\partial t},
\textcolor{white}{{\mbox{\boldmath$L$}}_E = {\mbox{\boldmath$p$}}\times {\mbox{\boldmath$e$}}+ ({\mbox{\boldmath$m$}}+ {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$p$}}) \times {\mbox{\boldmath$b$}}, }
\end{aligned}$$
\[eq:Full\_Electromagnetic\_Torque\_Losses\_Lagrangian\]
where $\vartheta_L$ is the temperature expressed on the undeformed configuration and the source term is obtained using the transformation:
$$\begin{aligned}
w_L = J \, w_E
&= ({\boldsymbol{F}}{\boldsymbol{\mathcal{J}}}) \cdot ({\boldsymbol{F}}^{-T} {\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} ) \\
&-J \left({\boldsymbol{F}}^{-T} {\boldsymbol{\mathcal{M}}}_{\mathrm{eff}} \right) \times \left( \partial_t (J^{-1} {\boldsymbol{F}}{\boldsymbol{\mathcal{B}}}) +
{\text{{\text{\bf Grad}}}{}\,{ \left( J^{-1} {\boldsymbol{F}}{\boldsymbol{\mathcal{B}}}\right) }} {\boldsymbol{V}}\right) \\
&- \rho_0 \left( \frac{\partial}{\partial t} \left(\frac{{\boldsymbol{F}}{\boldsymbol{\mathcal{P}}}}{\rho_0} \right)
+ {\text{{\text{\bf Grad}}}{}\,{\left( \frac{{\boldsymbol{F}}{\boldsymbol{\mathcal{P}}}}{\rho_0}\right) {\boldsymbol{V}}}}\right) \cdot \left( {\boldsymbol{F}}^{-T} {\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} \right).
\end{aligned}$$
\[eq:Full\_Electromagnetic\_Losses\_Lagrangian\]
Equations – must be completed by constitutive laws which relate the stress tensor to its associated conjugate strain tensor. In this paper, we use the second Piola–Kirchhoff stress and the Green–Lagrange strain tensors. We assume the following nonlinear constitutive laws for the thermal and thermo-mechanical problems [@belytschko-fem-13]: $$\begin{gathered}
{\mbox{\boldmath$q$}}_L
= J \, {\boldsymbol{F}}^{-1} \, \boldsymbol{Q}({\text{{\text{\bf Grad}}}{}\,{(\vartheta_E \circ {{\mbox{\boldmath$\varphi$} }_t}^{-1})}}) \\
= \underset{{\mbox{\boldmath$\kappa$} }_L}{\underbrace{J \, {\boldsymbol{F}}^{-1} \, {\mbox{\boldmath$\kappa$} }_E \, {\boldsymbol{F}}^{-T} } } {\text{{\text{\bf Grad}}}{}\,{\vartheta_L}}
= -{\mbox{\boldmath$\kappa$} }_L(\vartheta_L) \, {\text{{\text{\bf Grad}}}{}\,{\vartheta_L}},
\label{eq:Full_Thermal_ConstitutiveLaws_Lagrangian}\end{gathered}$$
$${\boldsymbol{S}}= {\boldsymbol{\mathcal{S}}}_{\mathrm{VEP}}({\mbox{\boldmath$E$}}, \dot{{\mbox{\boldmath$E$}}}, \vartheta_L, {\boldsymbol{Z}}_L(\tau \leq t)), \quad
{\mbox{\boldmath$E$}}= \frac{1}{2} ({\boldsymbol{F}}^T {\boldsymbol{F}}- {\boldsymbol{\mathbbm{1}}}),
\tag{\theequation\,a-b}$$
\[eq:Full\_ThermoMechanical\_ConstitutiveLaws\_Lagrangian\]
where ${\mbox{\boldmath$\kappa$} }_L$ is the thermal conductivity tensor (W/mK) and ${\boldsymbol{\mathcal{S}}}_{\mathrm{VEP}}$ is a mapping that represents the visco-elastoplastic constitutive law. Viscosity is reflected through the dependence of the stress on the rate of the Green–Lagrange tensor $\dot{{\mbox{\boldmath$E$}}}$, plasticity is accounted for using a set of internal variables ${\boldsymbol{Z}}_L(\tau \leq t)$ and the thermo-mechanical aspect is accounted for by the dependency on the temperature $\vartheta_L$.
Formulations, discretization and linearization {#section:formulations}
==============================================
In this section, we derive the simplified multiphysics problem from the fully coupled problem defined in section \[section:governing-equations\]. Using this simplified problem, strong and weak formulations of the multiphysics problems are derived using potentials. The weak formulations are then discretized in space using the continuous Galerkin approximation and in time using the backward Euler integrator. Finally, the resulting nonlinear system of algebraic equations is linearized.
Simplified governing equations of the multiphysics problem {#section:simplified_governing-equations}
----------------------------------------------------------
We make the following *magnetoquasistatic (MQS) assumptions*
$$\delta_i \simeq L_{\mathrm{sys, i}}, \quad \lambda \gg L_{\mathrm{sys, i}}.
\label{eq:Simplified_Maxwell_GoverningEquations_MQSAssumption}$$
In , $\delta_i := 1/\sqrt{\pi f \sigma_{E, i} \mu_{E, i}}$ is the skin depth in the spatial direction $i = x, y$ and $z$, $f$ is the frequency of the source term, $\sigma_{E, i}$ and $\mu_{E, i}$ are eigenvalues of ${\mbox{\boldmath$\sigma$} }_E$ and $\boldsymbol{\mu}_E$, $\lambda$ is the wavelength corresponding to the frequency $f$ and $L_{\mathrm{sys, i}}$ is the characteristic length of the structure in a spatial direction $i$ [@hiptmair-mqs-05]. The first term of explains the presence of eddy currents while the second explains the neglect of electromagnetic waves. Further in this section, we will relax the first condition in to $\delta_i = \alpha \, L_{\mathrm{sys, i}}$ with $\alpha$ which is big thus allowing to neglect the reaction field.
### The magnetoquasistatic problem
Using the MQS assumption, the following MQS problem in the deformed configuration can be defined:
$${\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$h$}}}} = {\mbox{\boldmath$j$}}, \quad
{\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$e$}}}} = -\partial_t {\mbox{\boldmath$b$}}, \quad
{\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$b$}}}} = 0,
\tag{\theequation\,a-c}$$
\[eq:Simplified\_Maxwell\_GoverningEquations\_Eulerian\]
together with the constitutive laws:
$${\mbox{\boldmath$h$}}= \boldsymbol{\nu}_E({\mbox{\boldmath$b$}}) \, {\mbox{\boldmath$b$}}= {\boldsymbol{H}}({\mbox{\boldmath$b$}}), \quad
{\mbox{\boldmath$j$}}= {\mbox{\boldmath$j$}}_{\mathrm{eff}} = {\mbox{\boldmath$\sigma$} }_E {\mbox{\boldmath$e$}}_{\mathrm{eff}} + {\mbox{\boldmath$j$}}_s = {\mbox{\boldmath$\sigma$} }_E ({\mbox{\boldmath$e$}}+ {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$b$}}) + {\mbox{\boldmath$j$}}_s.
\tag{\theequation\,a-b}$$
\[eq:Simplified\_Maxwell\_MaterialLaw\_Eulerian\]
In Lagrangian setting, the MQS problem is governed by Maxwell’s equations [@eringen-electrodynamics-12; @castro-electromagnetism-14]:
$${\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{H}}}_{\mathrm{eff}}}} = {\boldsymbol{\mathcal{J}}}, \quad
{\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{E}}}_{\mathrm{eff}}}} = -\partial_t {\boldsymbol{\mathcal{B}}}, \quad
{\text{{\text{Div}}}{}\,{{\boldsymbol{\mathcal{B}}}}} = 0
\tag{\theequation\,a-c}$$
\[eq:Simplified\_Maxwell\_GoverningEquations\_Lagrangian\]
completed by the following constitutive laws:
$${\boldsymbol{\mathcal{H}}}_{\mathrm{eff}} = \nu_{L} \, {\boldsymbol{\mathcal{B}}}, \quad
{\boldsymbol{\mathcal{J}}}= {\mbox{\boldmath$\sigma$} }_{L} \, {\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} + {\boldsymbol{\mathcal{J}}}_s.
\tag{\theequation\,a-b}$$
\[eq:Simplified\_Maxwell\_MaterialLaw\_Lagrangian\_H\_J\]
### Conservation equations
In addition to the MQS assumption, we assume *quasistatic mechanical problem* thus neglecting the inertia term in the balance of linear momentum and *isotropic magnetic materials* therefore restoring the symmetry of the Cauchy and the second Piola–Kirchhoff stress as ${\boldsymbol{L}}_E = {\boldsymbol{L}}_L = \boldsymbol{0}$ in (\[eq:Full\_Conservation\_Equations\_Eulerian\] c) and (\[eq:Full\_Conservation\_Equations\_Lagrangian\] c). Additionally, we neglect mechanical losses in the heat equation. Under these assumptions, balance equations in the deformed configuration become:
$$\frac{D \rho}{D t} + \rho {\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$v$}}}} = 0, \quad
{\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$\sigma$} }}} + {\mbox{\boldmath$f$}}_E = 0, \quad
{\mbox{\boldmath$\sigma$} }= {\mbox{\boldmath$\sigma$} }^T.
\tag{\theequation\,a-c}$$
\[eq:Simplified\_Conservation\_Equations\_Eulerian\]
$$\rho \, c_p \frac{\partial \vartheta_E}{\partial t} + {\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$q$}}_E}} = w_E
\label{eq:Simplified_Conservation_Energy_Eulerian}$$
and the electromagnetic force, torque and electromagnetic losses in – become:
$${\mbox{\boldmath$f$}}_E = {\mbox{\boldmath$j$}}\times {\mbox{\boldmath$b$}}+ \left({\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{{\mbox{\boldmath$b$}}}} \right)^{T} {\mbox{\boldmath$m$}}, \quad
{\mbox{\boldmath$L$}}_E = {\mbox{\boldmath$m$}}\times {\mbox{\boldmath$b$}}, \quad
w_E = {\mbox{\boldmath$j$}}_{\mathrm{eff}} \cdot {\mbox{\boldmath$e$}}_{\mathrm{eff}} - {\mbox{\boldmath$m$}}_{\mathrm{eff}} \cdot \frac{\partial {\mbox{\boldmath$b$}}}{\partial t}.
\tag{\theequation\,a-c}$$
\[eq:Simplified\_Electromagnetic\_Torque\_Losses\_Eulerian\]
In Lagrangian setting, the conservation equations become:
$$\rho_0 - \rho J = 0, \quad
{\text{{\text{Div}}}{}\,{{\boldsymbol{F}}{\boldsymbol{S}}}} + {\mbox{\boldmath$f$}}_L = \boldsymbol{0}, \quad
{\boldsymbol{S}}= {\boldsymbol{S}}^T,
\tag{\theequation\,a-c}$$
\[eq:Simplified\_Conservation\_Equations\_Lagrangian\]
$$\rho_0 \, c_p \, \frac{\partial \vartheta_L}{\partial t} + {\text{{\text{Div}}}{}\,{{\mbox{\boldmath$q$}}_L}} = w_L.
\label{eq:Simplified_Conservation_Energy_Lagrangian}$$
where the electromagnetic force and torque are given by: $$\begin{aligned}
{\mbox{\boldmath$f$}}_L &= J^{-1} \left( ({\boldsymbol{F}}\, {\boldsymbol{\mathcal{J}}}) \times ({\boldsymbol{F}}\, {\boldsymbol{\mathcal{B}}}) \right) +
J\left({\boldsymbol{F}}^{-T} {\text{{\text{\bf Grad}}}{}\,{(J^{-1} \, {\boldsymbol{F}}\, {\boldsymbol{\mathcal{B}}}}}) \right)^{T} ({\boldsymbol{F}}^{-T} \, {\boldsymbol{\mathcal{M}}}),
\label{eq:Simplified_Electromagnetic_Force_Lagrangian} \\
{\mbox{\boldmath$L$}}_L &= \left({\boldsymbol{F}}^{-T} {\boldsymbol{\mathcal{M}}}_{\mathrm{eff}} \right)
\times \left({\boldsymbol{F}}{\boldsymbol{\mathcal{B}}}\right),
\label{eq:Simplified_Electromagnetic_Torque_Lagrangian} \end{aligned}$$ and the source term is given by: $$\begin{gathered}
w_L = ({\boldsymbol{F}}{\boldsymbol{\mathcal{J}}}) \cdot ({\boldsymbol{F}}^{-T} {\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} ) - \\
J ({\boldsymbol{F}}^{-T} {\boldsymbol{\mathcal{M}}}_{\mathrm{eff}}) \times \left[ \partial_t (J^{-1} {\boldsymbol{F}}{\boldsymbol{\mathcal{B}}}) + {\text{{\text{\bf Grad}}}{}\,{ \left( J^{-1} {\boldsymbol{F}}{\boldsymbol{\mathcal{B}}}\right) }} {\boldsymbol{V}}\right].
\label{eq:Simplified_Electromagnetic_Losses_Lagrangian}\end{gathered}$$ Further in the paper, we consider elasto-plastic materials governed by the following constitutive law:
$${\boldsymbol{S}}= {\boldsymbol{\mathcal{S}}}_{EP}({\mbox{\boldmath$E$}}, \vartheta_L, {\boldsymbol{Z}}_L(\tau \leq t)), \quad
{\mbox{\boldmath$E$}}= \frac{1}{2} ({\boldsymbol{F}}^T {\boldsymbol{F}}- {\boldsymbol{\mathbbm{1}}}).
\tag{\theequation\,a-b}$$
\[eq:Simplified\_ThermoMechanical\_ConstitutiveLaws\_Lagrangian\]
In this case, the second Piola–Kirchhoff stress does not depend on the rate of Green–Lagrange strain as viscosity is not considered.
Strong forms
------------
### The magnetoquasistatic problem
The strong formulation of the MQS problem is derived using the so-called *magnetic induction conforming formulations* [@bossavit-cem-98]. In the deformed configuration, the derivation is achieved by verifying equations (\[eq:Simplified\_Maxwell\_GoverningEquations\_Eulerian\]b-c) in the strong sense: $${\mbox{\boldmath$b$}}= {\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$a$}}}} \simeq {\mbox{\boldmath$b$}}_s = {\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$a$}}_s}}, \, \,
{\mbox{\boldmath$e$}}= -\partial_t {\mbox{\boldmath$a$}}- {\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{\phi}} \simeq -\partial_t {\mbox{\boldmath$a$}}_s - {\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{\phi}}
\label{eq:Strong_Derivation_Electromagnetic_Potentials_Eulerian}$$ where ${\mbox{\boldmath$a$}}$ is the vector potential unknown field used for the eddy currents problem, ${\mbox{\boldmath$a$}}_s$ is the source vector potential which can be pre-computed based on the electric current source ${\mbox{\boldmath$j$}}_s$ imposed in inductors such as in coils and $\phi$ is the unknown scalar potential. The approximation ${\mbox{\boldmath$a$}}\simeq {\mbox{\boldmath$a$}}_s$ implies the neglect of the reaction field and is valid under the assumption $\delta_i = \alpha \, L_{\mathrm{sys, i}}$ with $\alpha$ which is big [@scorretti-formulations-12].
The electric potential $\phi$ is governed by the following problem where is obtained by applying the divergence ${\mathrm{{\mathrm{div}}}{}\,{}}$ operator to (\[eq:Simplified\_Maxwell\_GoverningEquations\_Eulerian\] a) and is derived from (\[eq:Simplified\_Maxwell\_MaterialLaw\_Eulerian\] b): $$\begin{aligned}
{\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$j$}}}} &= 0 && \text{ in } \Omega_{\mathrm{t}}^{\mathrm{Ele}},
\label{eq:Strong_Electrokinetic_Governing_Equation_Eulerian} \\
{\mbox{\boldmath$j$}}&= {\mbox{\boldmath$\sigma$} }_{E} ({\mbox{\boldmath$e$}}+ {\mbox{\boldmath$v$}}\times {\mbox{\boldmath$b$}}) + {\mbox{\boldmath$j$}}_s &&\\
&= {\mbox{\boldmath$\sigma$} }_{E} (-\partial_t {\mbox{\boldmath$a$}}_s - {\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{\phi}} + {\mbox{\boldmath$v$}}\times {\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$a$}}_s}}) + {\mbox{\boldmath$j$}}_s
&& \text{ in } \Omega_{\mathrm{t}}^{\mathrm{Ele}},
\label{eq:Strong_Electrokinetic_Constitutive_Law_Eulerian} \\
\phi({\mbox{\boldmath$x$}}, t) &= \phi_{D}({\mbox{\boldmath$x$}}, t)
&& \text{ on } \Gamma_t^{\mathrm{Diri, Ele}},
\label{eq:Strong_Electrokinetic_Dirichlet_BC_Eulerian} \\
{\mbox{\boldmath$n$}}_E \cdot {\mbox{\boldmath$j$}}&= 0
&& \text{ on } \Gamma_t^{\mathrm{Neu, Ele}}.
\label{eq:Strong_Electrokinetic_Neumann_BC_Eulerian}
$$ and are the Dirichlet and the Neumann boundary conditions.
In the undeformed configuration, the derivation is carried out by verifying equations (\[eq:Simplified\_Maxwell\_GoverningEquations\_Lagrangian\]b-c) in the strong sense: $$\begin{gathered}
{\boldsymbol{\mathcal{B}}}= {\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{A}}}}} \simeq {\boldsymbol{\mathcal{B}}}_s = {\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{A}}}_s}} = {\text{{\text{\bf Curl}}}{}\,{\left({\boldsymbol{F}}^{T} {\mbox{\boldmath$a$}}_s\right)}}, \\
{\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} = -\partial_t {\boldsymbol{\mathcal{A}}}- {\text{{\text{\bf Grad}}}{}\,{\Phi}} \simeq -\partial_t {\boldsymbol{\mathcal{A}}}_s - {\text{{\text{\bf Grad}}}{}\,{\Phi}}
= -{\boldsymbol{F}}^{T} \partial_t {\mbox{\boldmath$a$}}_s - {\text{{\text{\bf Grad}}}{}\,{\Phi}}.
\label{eq:Strong_Derivation_Electromagnetic_Potentials_Eulerian}\end{gathered}$$ In , the 1 differential form ${\boldsymbol{\mathcal{A}}}_s$ is transformed as ${\boldsymbol{\mathcal{A}}}_s = {\boldsymbol{F}}^{T} {\mbox{\boldmath$a$}}_s$. The electric potential $\Phi$ is therefore governed by the following problem where is obtained by applying the divergence operator ${\text{{\text{Div}}}{}\,{}}$ to (\[eq:Simplified\_Maxwell\_GoverningEquations\_Lagrangian\]a) and is derived from (\[eq:Simplified\_Maxwell\_MaterialLaw\_Lagrangian\_H\_J\]b): $$\begin{aligned}
{\text{{\text{Div}}}{}\,{{\boldsymbol{\mathcal{J}}}}} &= 0, && \text{ in } \Omega_0^{\mathrm{Ele}}
\label{eq:Strong_Electrokinetic_Governing_Equation_Lagrangian}, \\
{\boldsymbol{\mathcal{J}}}&= {\mbox{\boldmath$\sigma$} }_{L} \, {\boldsymbol{\mathcal{E}}}_{\mathrm{eff}} + {\boldsymbol{\mathcal{J}}}_s \\
&= -{\mbox{\boldmath$\sigma$} }_{L} ({\boldsymbol{F}}^{T} \partial_t {\mbox{\boldmath$a$}}_s + {\text{{\text{\bf Grad}}}{}\,{\Phi}}) + {\boldsymbol{\mathcal{J}}}_s , && \text{ in } \Omega_0^{\mathrm{Ele}}
\label{eq:Strong_Electrokinetic_Constitutive_Law_Lagrangian} \\
\Phi({\mbox{\boldmath$x$}}, t) &= \Phi_{D}({\mbox{\boldmath$x$}}, t)
&& \text{ on } \Gamma_0^{\mathrm{Diri, Ele}}
\label{eq:Strong_Electrokinetic_Dirichlet_BC_Lagrangian }, \\
{\mbox{\boldmath$n$}}_L \cdot {\boldsymbol{\mathcal{J}}}&= 0
&& \text{ on } \Gamma_0^{\mathrm{Neu, Ele}}
\label{eq:Strong_Electrokinetic_Neumann_BC_Lagrangian}.
$$ where the conductivity tensor is transformed as ${\mbox{\boldmath$\sigma$} }_{L} = J {\boldsymbol{F}}^{-1} {\mbox{\boldmath$\sigma$} }_{E}(\vartheta_E \circ \varphi_t^{-1}) {\boldsymbol{F}}^{-T}$ thanks to .
### The heat equation
In the deformed configuration, the evolution of the temperature is governed by the following problem derived from and (\[eq:Simplified\_Electromagnetic\_Torque\_Losses\_Eulerian\] c) and : $$\begin{aligned}
\rho c_{p} \frac{\partial \vartheta_E}{\partial t} + {\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$q$}}_E}} &= w_E
&&\text{ in } \Omega_{\mathrm{t}}^{\mathrm{The}},
\label{eq:Strong_Heat_Governing_Equation_Eulerian} \\
{\mbox{\boldmath$q$}}_E &= -{\mbox{\boldmath$\kappa$} }_E(\vartheta_E) \, {\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{\vartheta_E}}
&& \text{ in } \Omega_{\mathrm{t}}^{\mathrm{The}},
\label{eq:Strong_Heat_Constitutive_Law_Eulerian} \\
\vartheta_E({\mbox{\boldmath$x$}}, 0) &= \vartheta_{E,0}({\mbox{\boldmath$x$}})
&& \text{ in } \Omega_0^{\mathrm{The}}
\label{eq:Strong_Heat_IC_Eulerian}, \\
\vartheta_E({\mbox{\boldmath$x$}}, t) &= \vartheta_{E, D}({\mbox{\boldmath$x$}}, t)
&& \text{ on } \Gamma_t^{\mathrm{Diri, The}}
\label{eq:Strong_Heat_Dirichlet_BC_Eulerian}, \\
{\mbox{\boldmath$n$}}_E \cdot {\mbox{\boldmath$q$}}_E &= h_E(t)(\vartheta_E - \vartheta_{E, B})
&& \text{ on } \Gamma_t^{\mathrm{conv, The}}
\label{eq:Strong_Heat_Convective_BC_Eulerian}, \\
{\mbox{\boldmath$n$}}_E \cdot {\mbox{\boldmath$q$}}_E &= \epsilon_E^R \sigma_E^R (\vartheta_E^4 - \vartheta_{E, R}^4)
&& \text{ on } \Gamma_t^{\mathrm{rad, The}}
\label{eq:Strong_Heat_Radiative_BC_Eulerian}.
$$ The source term in can be expressed in terms of the potentials as: $$\begin{gathered}
w_E = \underset{w_E^{\mathrm{eddy}}}{\underbrace{\left({\mbox{\boldmath$\sigma$} }_E \left(\partial_t {\mbox{\boldmath$a$}}_s + {\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{\phi}} - {\mbox{\boldmath$v$}}\times {\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$a$}}_s}} \right) \right)^2}} \\
\underset{w_E^{\mathrm{eddy}}}{\underbrace{- {\mbox{\boldmath$j$}}_s \cdot ({\mbox{\boldmath$\sigma$} }_E \, (\partial_t {\mbox{\boldmath$a$}}_s + {\mathrm{{\mathrm{\boldsymbol{grad}}}}{}\,{\phi}} - {\mbox{\boldmath$v$}}\times {\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$a$}}_s}}))}}
- \underset{w_E^{\mathrm{hyst}}}{\underbrace{{\mbox{\boldmath$m$}}_{\mathrm{eff}} \cdot {\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{ \left(\frac{\partial {\mbox{\boldmath$a$}}_s}{\partial t} \right) }} } }\end{gathered}$$ where $w_E^{\mathrm{eddy}}$ and $w_E^{\mathrm{hyst}}$ represent eddy current and hysteretic losses. Equations , , and represent the initial condition, Dirichlet, convective and radiative boundary conditions, respectively.
The Lagrangian strong form of the heat problem is given by: $$\begin{aligned}
\rho_0 c_{p} \frac{\partial \vartheta_L}{\partial t} + {\text{{\text{Div}}}{}\,{{\mbox{\boldmath$q$}}_L}} &= w_L
&&\text{ in } \Omega_{\mathrm{0}}^{\mathrm{The}}
\label{eq:Strong_Heat_Governing_Equation_Lagrangian}, \\
{\mbox{\boldmath$q$}}_L &= -{\mbox{\boldmath$\kappa$} }_L(\vartheta_L) \, {\text{{\text{\bf Grad}}}{}\,{\vartheta_L}}
&& \text{ in } \Omega_{\mathrm{0}}^{\mathrm{The}}
\label{eq:Strong_Heat_Constitutive_Law_Eulerian}, \\
\vartheta_L({\mbox{\boldmath$x$}}, 0) &= \vartheta_{L,0}({\mbox{\boldmath$x$}})
&& \text{ in } \Omega_0^{\mathrm{The}}
\label{eq:Strong_Heat_IC_Lagrangian}, \\
\vartheta_L({\mbox{\boldmath$x$}}, t) &= \vartheta_{L, D}({\mbox{\boldmath$x$}}, t)
&& \text{ on } \Gamma_0^{\mathrm{Diri, The}}
\label{eq:Strong_Heat_Dirichlet_BC_Lagrangian}, \\
{\mbox{\boldmath$n$}}_L \cdot {\mbox{\boldmath$q$}}_L &= h_L(t)(\vartheta_L - \vartheta_{L, B})
&& \text{ on } \Gamma_0^{\mathrm{conv, The}}
\label{eq:Strong_Heat_Convective_BC_Lagrangian},\\
{\mbox{\boldmath$n$}}_L \cdot {\mbox{\boldmath$q$}}_L &= \epsilon_L^R \sigma_L^R (\vartheta_L^4 - \vartheta_{L, R}^4)
&& \text{ on } \Gamma_t^{\mathrm{rad, The}}
\label{eq:Strong_Heat_Radiative_BC_Lagrangian}
$$ where the thermal conductivity is transformed according to as ${\mbox{\boldmath$\kappa$} }_L = J\, {\boldsymbol{F}}^{-1} \, {\mbox{\boldmath$\kappa$} }_E {\boldsymbol{F}}^{-T}$ and the convective heat coefficient $h_L(t)$ is transformed using Nanson’s formula as $h_L(t) = J \, | {\boldsymbol{F}}^{-T} {\mbox{\boldmath$n$}}_L| \, h_E(t)$.
The source term in can be expressed in terms of the potentials as: $$\begin{gathered}
w_L = ({\boldsymbol{F}}^{\textcolor{white}{{-T}}} \!\!\!\!\!\!\!\! (\underset{-{\boldsymbol{\mathcal{J}}}}{\underbrace{{\mbox{\boldmath$\sigma$} }_L \left(\partial_t {\boldsymbol{\mathcal{A}}}_s
+ {\text{{\text{\bf Grad}}}{}\,{\Phi}} \right) - {\boldsymbol{\mathcal{J}}}_s } } ) )
\cdot
(\underset{-{\boldsymbol{\mathcal{E}}}_{\mathrm{eff}}}{\underbrace{{\boldsymbol{F}}^{-T} \left(\partial_t {\boldsymbol{\mathcal{A}}}_s + {\text{{\text{\bf Grad}}}{}\,{\Phi}} \right) } } ) +
\\
J {\boldsymbol{F}}^{-T} \underset{{\boldsymbol{\mathcal{M}}}_{\mathrm{eff}}}{\underbrace{\mu_0^{-1}({\boldsymbol{\mathbbm{1}}}- {\mbox{\boldmath$\nu$} }_L) {\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{A}}}_s}} } } \bigg[ \partial_t (J^{-1} {\boldsymbol{F}}{\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{A}}}_s}} ) + {\text{{\text{\bf Grad}}}{}\,{\left(J^{-1} {\boldsymbol{F}}{\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{A}}}_s}} \right)}} \bigg].\end{gathered}$$ Equations , , and represent the initial condition, Dirichlet, convective and radiative boundary conditions, respectively.
### The mechanical problem
Mechanical fields in the undeformed configuration are governed by the following problem: $$\begin{aligned}
{\text{{\text{Div}}}{}\,{({\boldsymbol{F}}\, {\boldsymbol{S}})}} + {\mbox{\boldmath$f$}}_L &= 0 &&\text{ in } \Omega_{\mathrm{0}}^{\mathrm{Mec}},
\label{eq:Strong_Mechanics_Governing_Equation_Lagrangian} \\
{\boldsymbol{S}}&= {\boldsymbol{\mathcal{S}}}_{EP}\left({\mbox{\boldmath$E$}}, \vartheta_L, {\boldsymbol{Z}}_{L}(\tau \leq t) \right)
&& \text{ in } \Omega_{0}^{\mathrm{Mec}},
\label{eq:Strong_Mechanics_Constitutive_Law_Lagrangian}\\
{\mbox{\boldmath$E$}}&= \frac{{\boldsymbol{F}}^T {\boldsymbol{F}}- {\boldsymbol{\mathbbm{1}}}}{2} && \text{ in } \Omega_{0}^{\mathrm{Mec}},
\label{eq:Strong_GreenLagrange_Deformation}\\
{\mbox{\boldmath$u$}}({\mbox{\boldmath$x$}}, t) &= {\mbox{\boldmath$u$}}_D({\mbox{\boldmath$x$}}, t) && \text{ on } \Gamma_{0, D}^{\mathrm{Mec}}.
\label{eq:Strong_Mechanics_Dirichlet_BC_Lagrangian}\\
{\mbox{\boldmath$n$}}_L \cdot ({\boldsymbol{F}}{\boldsymbol{S}}) &= {\mbox{\boldmath$t$}}_L && \text{ on } \Gamma_{0, N}^{\mathrm{Mec}}.
\label{eq:Strong_Mechanics_Neumann_BC_Lagrangian}
$$ In terms of the potential, the force ${\mbox{\boldmath$f$}}_L$ in is given by: $$\begin{gathered}
{\mbox{\boldmath$f$}}_L = {\boldsymbol{F}}_L({\mbox{\boldmath$u$}}, \Phi)
= J^{-1} \, {\boldsymbol{F}}\, \underset{{\boldsymbol{\mathcal{J}}}}{\underbrace{({\mbox{\boldmath$\sigma$} }_L (\partial_t {\boldsymbol{\mathcal{A}}}_s + {\text{{\text{\bf Grad}}}{}\,{\Phi}}) + {\boldsymbol{\mathcal{J}}}_s)}}
\times
{\boldsymbol{F}}\, \underset{{\boldsymbol{\mathcal{B}}}}{\underbrace{{\text{{\text{\bf Curl}}}{}\,{\left({\boldsymbol{\mathcal{A}}}_s\right)}}}} +
\\
J ({\boldsymbol{F}}^{-T} \, {\text{{\text{\bf Grad}}}{}\,{( J^{-1} \, {\boldsymbol{F}}\, \underset{{\boldsymbol{\mathcal{B}}}}{\underbrace{{\text{{\text{\bf Curl}}}{}\,{\left({\boldsymbol{\mathcal{A}}}_s\right)}}}} ) }} )^T
({\boldsymbol{F}}^{-T} \underset{{\boldsymbol{\mathcal{M}}}}{\underbrace{\mu_0^{-1} \boldsymbol{\chi_{B L} } J^{-1} \, {\boldsymbol{F}}^T \, {\boldsymbol{F}}\, {\text{{\text{\bf Curl}}}{}\,{{\boldsymbol{\mathcal{A}}}_s}} } } ).
\label{eq:Strong_Mechanics_Force_Lagrangian}\end{gathered}$$ The term ${\mbox{\boldmath$t$}}_L$ in represents the surface traction applied on part of the boundary $\Gamma_{0, N}^{\mathrm{Mec}}$. The thermo-mechanical constitutive law – is derived from (\[eq:Simplified\_ThermoMechanical\_ConstitutiveLaws\_Lagrangian\]a). In this paper, we use the constitutive law described in [@boatti-smp-16]. As a reminder, the total deformation gradient ${\boldsymbol{F}}$ and the total deformation gradients in the glassy and rubbery states were given by: $${\boldsymbol{F}}= {\boldsymbol{F}}^{tg} = {\boldsymbol{F}}^{tr} = {\boldsymbol{\mathbbm{1}}}+ {\text{{\text{\bf Grad}}}{}\,{{\mbox{\boldmath$u$}}}},
\label{eq:Strong_Mechanics_Constitutive_Law_Lagrangian_Kinematics_1_new}$$ where the superscript $t$ denotes the total deformation gradient and the superscripts $r$ and $g$ were used for the rubbery and the glassy states. The total deformation gradients of both phases are decomposed as: $${\boldsymbol{F}}^{tg} = {\boldsymbol{F}}^{g}{\boldsymbol{F}}^{f} = {\boldsymbol{F}}^{eg}{\boldsymbol{F}}^{pg}{\boldsymbol{F}}^{f}, {\boldsymbol{F}}^{tr} = {\boldsymbol{F}}^{r}{\boldsymbol{F}}^{p} = {\boldsymbol{F}}^{er}{\boldsymbol{F}}^{p},
\label{eq:Strong_Mechanics_Constitutive_Law_Lagrangian_Kinematics_2_new}$$ where ${\boldsymbol{F}}^{eg}$ and ${\boldsymbol{F}}^{pg}$ are the deformation gradients for the elastic and plastic phases in the glassy state, ${\boldsymbol{F}}^{f}$ is the frozen deformation gradient that represents the temporary deformation which is stored during high temperature shape fixing and ${\boldsymbol{F}}^{r}$ and ${\boldsymbol{F}}^{p}$ are the elastic and plastic deformation gradients for the rubbery state.
The total Cauchy stress was also given by: $${\mbox{\boldmath$\sigma$} }= z^g {\mbox{\boldmath$\sigma$} }^g + (1 - z^g) {\mbox{\boldmath$\sigma$} }^r,
\label{eq:Strong_Mechanics_Constitutive_Law_sigma_new}$$ where the temperature-dependent parameter $z^g$ is the ratio of the glassy state. Using the second Piola–Kirchhoff stress and Green-Lagrange strain tensors, the following expression of the stress tensor can be derived: $$\begin{aligned}
{\boldsymbol{S}}&= J {\boldsymbol{F}}^{-1} {\mbox{\boldmath$\sigma$} }{\boldsymbol{F}}^{-T} = z^g J {\boldsymbol{F}}^{-1} {\mbox{\boldmath$\sigma$} }^g {\boldsymbol{F}}^{-T} + (1 - z^g) J {\boldsymbol{F}}^{-1} {\mbox{\boldmath$\sigma$} }^r {\boldsymbol{F}}^{-T}\\
&= z^g J^f {{\boldsymbol{F}}^{f}}^{-1} \underset{{\boldsymbol{S}}^{g}}{\underbrace{\left(J^g {{\boldsymbol{F}}^{g}}^{-1} {\mbox{\boldmath$\sigma$} }^g {{\boldsymbol{F}}^{g}}^{-T} \right)}} {{\boldsymbol{F}}^{f}}^{-T} \\
& \textcolor{white}{Inno Inno Inno}+ (1 - z^g) J^p {{\boldsymbol{F}}^{p}}^{-1} \underset{{\boldsymbol{S}}^{r}}{\underbrace{\left(J^r {{\boldsymbol{F}}^{r}}^{-1} {\mbox{\boldmath$\sigma$} }^r {{\boldsymbol{F}}^{r}}^{-T} \right)}} {{\boldsymbol{F}}^{p}}^{-T}. \end{aligned}
\label{eq:Strong_Mechanics_Constitutive_Law_sigma_PK_new}$$ In , ${\boldsymbol{S}}^{g}$ and ${\boldsymbol{S}}^{r}$ are the second Piola–Kirchhoff stress tensors defined on the intermediate configurations [@boatti-smp-16] by: $$\begin{aligned}
&{\boldsymbol{S}}^g = \left({\boldsymbol{F}}^{pg}\right)^{-1} \left(\lambda^g \mathrm{tr}({\mbox{\boldmath$E$}}^{eg}) {\boldsymbol{\mathbbm{1}}}+ 2 \mu^g {\mbox{\boldmath$E$}}^{eg} \right)
\left({\boldsymbol{F}}^{pg}\right)^{-T},
\label{eq:Strong_Mechanics_Constitutive_Law_Lagrangian_1_g_new} \\
&{\boldsymbol{S}}^r = \lambda^r \mathrm{tr}({\mbox{\boldmath$E$}}^{er}) {\boldsymbol{\mathbbm{1}}}+ 2 \mu^r {\mbox{\boldmath$E$}}^{er}.
\label{eq:Strong_Mechanics_Constitutive_Law_Lagrangian_1_r_new}\end{aligned}$$ The parameters $\lambda^g$, $\lambda^r$, $\mu^g$ and $\mu^r$ are Lamé parameters for the glassy and the rubbery states. Determinants of deformation gradients of intermediate configurations are defined as $J^{i} = \mathrm{det}{\boldsymbol{F}}^{i}$ where the superscript $^\emph{i}$ refer to configurations of the glassy state ($^\emph{g}$, $^\emph{eg}$ and $^\emph{pg}$) or of the rubbery state ($^\emph{er}$ and $^\emph{p}$). The contribution to the stress due to thermal expansion have been neglected in –.
Details on the equations that govern the evolution of internal variables ${\boldsymbol{Z}}_{L}(\tau \leq t) := (z^g, {\boldsymbol{F}}^{f}, {\boldsymbol{F}}^{p},{\boldsymbol{F}}^{pg})$ and the numerical update of the internal variables can be found in [@boatti-smp-16].
Weak forms
----------
The weak forms of the electromagnetic problem –, the heat problem – and the mechanical problem – read [@bossavit-cem-98; @bachinger-cem-05; @wriggers-fem-08; @hughes-fem-12]:\
for each $t \in \mathcal{I}_t$, find $\left(\Phi \times \vartheta_L \times {\mbox{\boldmath$u$}}\right) \in U \times V \times {\boldsymbol{W}}$ such that $$\int_{\Omega_0^{\mathrm{Ele}}} {\mbox{\boldmath$\sigma$} }_L {\text{{\text{\bf Grad}}}{}\,{\Phi}} \cdot {\text{{\text{\bf Grad}}}{}\,{\Phi^{'}}} \, \mathrm{d} \Omega_0 +
\int_{\Omega_0^{\mathrm{Ele}}} {\mbox{\boldmath$\sigma$} }_L \partial_t {\boldsymbol{\mathcal{A}}}_s \cdot {\text{{\text{\bf Grad}}}{}\,{\Phi^{'}}} \, \mathrm{d} \Omega_0 = 0,
\label{eq:Weak_Electrokinetic_Equation_Lagrangian}$$ $$\begin{gathered}
\int_{\Omega_0^{\mathrm{The}}} \rho_0 c_p \frac{\partial \vartheta_L}{\partial t} \cdot \vartheta_{L}^{'} \, \mathrm{d} \Omega_0 +
\int_{\Omega_0^{\mathrm{The}}} \underset{-{\mbox{\boldmath$q$}}_L}{\underbrace{{\mbox{\boldmath$\kappa$} }_L \, {\text{{\text{\bf Grad}}}{}\,{\vartheta_{L}}}}} \cdot {\text{{\text{\bf Grad}}}{}\,{\vartheta_{L}^{'}}} \, \mathrm{d} \Omega_0
- \textcolor{white}{Inno Inno Inno Inno Inno Inno Inno Inno}
\\
\int_{\Omega_0^{\mathrm{The}}} \underset{w_L}{\underbrace{ ({\boldsymbol{F}}\left({\mbox{\boldmath$\sigma$} }_L \left(\partial_t {\boldsymbol{\mathcal{A}}}_s + {\text{{\text{\bf Grad}}}{}\,{\Phi}} \right) - {\boldsymbol{\mathcal{J}}}_s \right) )
\cdot
({\boldsymbol{F}}^{-T} \left(\partial_t {\boldsymbol{\mathcal{A}}}_s + {\text{{\text{\bf Grad}}}{}\,{\Phi}} \right) ) } } \cdot \vartheta_{L}^{'} \, \mathrm{d} \Omega_0
\\
+
\int_{\Gamma_0^{\mathrm{conv, The}}} \underset{{\mbox{\boldmath$n$}}_L \cdot {\mbox{\boldmath$q$}}_L}{\underbrace{h_L(t)(\vartheta_{L} - \vartheta_{L, B})}} \cdot \vartheta_L^{'} \, \mathrm{d} \Gamma_0
+
\\
\int_{\Gamma_0^{\mathrm{rad, The}}} \underset{{\mbox{\boldmath$n$}}_L \cdot {\mbox{\boldmath$q$}}_L}{\underbrace{ \epsilon_L^R \sigma_L^R (\vartheta_{L}^4 - \vartheta_{L, R}^4)}} \cdot \vartheta_L^{'} \, \mathrm{d} \Gamma_0
= 0
\label{eq:Weak_Thermal_Equation_Lagrangian}\end{gathered}$$ $$\begin{gathered}
\int_{\Omega_0^{\mathrm{Mec}}} {\boldsymbol{\mathcal{S}}}_{EP}({\mbox{\boldmath$u$}}, \vartheta_L, {\boldsymbol{Z}}_{L}) \colon \delta {\mbox{\boldmath$E$}}\, \mathrm{d} \Omega_0 -
\\
\int_{\Omega_0^{\mathrm{Mec}}} {\boldsymbol{F}}_L({\mbox{\boldmath$u$}}, \Phi) \cdot {\mbox{\boldmath$u$}}^{'} \, \mathrm{d} \Omega_0
=
\int_{\Gamma_0^{\mathrm{N}}} {\mbox{\boldmath$t$}}_L \cdot {\mbox{\boldmath$u$}}^{'} \, \mathrm{d} \Gamma_0
\label{eq:Weak_Mechanics_Equation_Lagrangian}\end{gathered}$$ holds for all test functions $\left(\Phi^{'} \times \vartheta_{L}^{'} \times {\mbox{\boldmath$u$}}^{'}\right) \in \left(U_0 \times V_0 \times {\boldsymbol{W}}_0 \right)$. The force ${\mbox{\boldmath$f$}}_L = {\boldsymbol{F}}_L({\mbox{\boldmath$u$}}, \Phi)$ is given by and the dependence on the displacement is achieved through $J$ and ${\boldsymbol{F}}$. The function spaces are defined such that $U \subseteq {H^1(\Omega_0^{\mathrm{Ele}})}$, $V \subseteq {H^1(\Omega_0^{\mathrm{The}})}$, ${\boldsymbol{W}}\subseteq {\boldsymbol{H}^1(\Omega^{\mathrm{Mec}})} \equiv \left({H^1(\Omega^{\mathrm{Mec}})}\right)^3$ and the source term of the electromagnetic problem ${\boldsymbol{\mathcal{A}}}_s$ belongs to a subspace of ${\boldsymbol{H}({\text{\bf Curl}};\Omega_0^{\mathrm{Ele}})}$. The virtual Green–Lagrange strain $\delta {\mbox{\boldmath$E$}}$ is related to the virtual displacement ${\mbox{\boldmath$u$}}^{'}$ through $\delta {\mbox{\boldmath$E$}}= {\boldsymbol{F}}^{T} {\text{{\text{\bf Grad}}}{}\,{{\mbox{\boldmath$u$}}^{'}}}$.
Spatial and temporal discretization
-----------------------------------
The unknown fields $\Phi$, $\vartheta_L$ and ${\mbox{\boldmath$u$}}$ in – belong to infinite dimensional functional spaces. For numerical simulation, these fields need to be approximated by finite dimensional spaces $$\Phi({\mbox{\boldmath$x$}}, t) \approx \bar{\Phi}({\mbox{\boldmath$x$}}, t), \quad
\vartheta_L({\mbox{\boldmath$x$}}, t) \approx \bar{\vartheta}_L({\mbox{\boldmath$x$}}, t), \quad
{\mbox{\boldmath$u$}}({\mbox{\boldmath$x$}}, t) \approx \bar{{\mbox{\boldmath$u$}}}({\mbox{\boldmath$x$}}, t)
\label{eq:Spatial_Discretization_Lagrangian_Approximation_Spaces}$$ defined by: $$\bar{\Phi}({\mbox{\boldmath$x$}}, t) = \sum_{i = 1}^{N^{\mathrm{Ele} } } \bar{\Phi}_i(t) N_i^{\mathrm{Ele} }({\mbox{\boldmath$x$}}), \, \,
{\text{{\text{\bf Grad}}}{}\,{\bar{\Phi}}}({\mbox{\boldmath$x$}}) = \sum_{i = 1}^{N^{\mathrm{Ele} } } \bar{\Phi}_i(t) {\text{{\text{\bf Grad}}}{}\,{N_i}}^{\mathrm{Ele} }({\mbox{\boldmath$x$}}),
\label{eq:Spatial_Discretization_Lagrangian_Ele_Phi}$$ $$\bar{\vartheta}_L({\mbox{\boldmath$x$}}, t) = \sum_{i = 1}^{N^{\mathrm{The} } } \bar{\vartheta}_i(t) N_i^{\mathrm{The} }({\mbox{\boldmath$x$}}), \,
{\text{{\text{\bf Grad}}}{}\,{\bar{\vartheta}_L}}({\mbox{\boldmath$x$}}) = \sum_{i = 1}^{N^{\mathrm{The}} } \bar{\vartheta}_i(t) {\text{{\text{\bf Grad}}}{}\,{N_i^{\mathrm{The} }}}({\mbox{\boldmath$x$}}),
\label{eq:Spatial_Discretization_Lagrangian_The_VarTheta}$$ $$\bar{{\mbox{\boldmath$u$}}}({\mbox{\boldmath$x$}}, t) = \sum_{i = 1}^{N^{\mathrm{Mec} } } \bar{{\mbox{\boldmath$u$}}}_i(t) N_i^{\mathrm{Mec} }({\mbox{\boldmath$x$}}), \, \,
{\text{{\text{\bf Grad}}}{}\,{\bar{{\mbox{\boldmath$u$}}}}}({\mbox{\boldmath$x$}}) = \sum_{i = 1}^{N^{\mathrm{Mec}} } \bar{{\mbox{\boldmath$u$}}}_i(t) {\text{{\text{\bf Grad}}}{}\,{N_i^{\mathrm{Mec} }}}({\mbox{\boldmath$x$}}),
\label{eq:Spatial_Discretization_Lagrangian_Mec_U}$$ where $N^{\mathrm{Ele}}$, $N^{\mathrm{The}}$ and $N^{\mathrm{Mec}}$ are the number of nodes of the electromagnetic, thermal and mechanical domains, $\bar{\Phi}_i$, $\bar{\vartheta}_i$ and $\bar{{\mbox{\boldmath$u$}}}_i = (\bar{u}_{i, x}, \bar{u}_{i, y}, \bar{u}_{i, z})$ are degrees of freedom and $N_i^{\mathrm{Ele} }$, $N_i^{\mathrm{The} }$ and $N_i^{\mathrm{Mec} }$ are shape functions for the electromagnetic, thermal and mechanical problems, respectively.
Inserting – into – leads to the following discrete system of equations: $$\begin{aligned}
\displaystyle \boldsymbol{K}^{\mathrm{Ele}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}) \bar{{\mbox{\boldmath$\Phi$} }}
+ {\boldsymbol{F}}^{\mathrm{Ele}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}) = \boldsymbol{0},
\label{eq:Spatial_Discretized_Lagrangian_a} \\
\displaystyle \boldsymbol{M}^{\mathrm{The}} \frac{D \bar{{\mbox{\boldmath$\vartheta$} }}}{D t}
+ \boldsymbol{K}^{\mathrm{The}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}) \bar{{\mbox{\boldmath$\vartheta$} }}
+ {\boldsymbol{F}}^{\mathrm{The}}(\bar{{\mbox{\boldmath$\Phi$} }}, \bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}) = \boldsymbol{0},
\label{eq:Spatial_Discretized_Lagrangian_b} \\
\displaystyle \boldsymbol{K}^{\mathrm{Mec}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}, {\boldsymbol{Z}}_{L})
+ \bar{{\boldsymbol{F}}}^{\mathrm{Mec}}(\bar{{\mbox{\boldmath$\Phi$} }}, \bar{{\mbox{\boldmath$u$}}})
= \boldsymbol{0}.
\label{eq:Spatial_Discretized_Lagrangian_c}\end{aligned}$$ where $\bar{{\mbox{\boldmath$\Phi$} }}$, $\bar{{\mbox{\boldmath$\vartheta$} }}_L$ and $\bar{{\mbox{\boldmath$u$}}}$ are vectors of degrees of freedom and the matrices in – are given by: $$\begin{aligned}
\boldsymbol{K}^{\mathrm{Ele} } &= \sum_{e = 1}^{N^{\mathrm{Ele} } } {{\boldsymbol{L}}^{e}}^T
\left[ \int_{\Omega_e} ({\mbox{\boldmath$B$}}^{e})^T \, {\mbox{\boldmath$\sigma$} }_L(\bar{{\mbox{\boldmath$u$}}}, \bar{{\mbox{\boldmath$\vartheta$} }}) \, {\mbox{\boldmath$B$}}^{e} \mathrm{d} \Omega_e \right] {{\boldsymbol{L}}^{e}}^T,
\label{eq:Spatial_Lagrangian_Terms_K_Ele} \\
{\boldsymbol{F}}^{\mathrm{Ele} } &= \sum_{e = 1}^{N^{\mathrm{Ele} } } {{\boldsymbol{L}}^{e}}^T
\left[ \int_{\Omega_e} ({\mbox{\boldmath$B$}}^{e})^T \, {\mbox{\boldmath$\sigma$} }_L(\bar{{\mbox{\boldmath$u$}}}, \bar{{\mbox{\boldmath$\vartheta$} }}) \, {\boldsymbol{\mathcal{A}}}_s^{e} \mathrm{d} \Omega_e \right],
\label{eq:Spatial_Lagrangian_Terms_F_Ele}
\\
\boldsymbol{M}^{\mathrm{The} } &= \sum_{e = 1}^{N^{\mathrm{The} } } {{\boldsymbol{L}}^{e}}^T
\left[ \int_{\Omega_e} ({\boldsymbol{N}}^{e})^T \, \rho_0 c_p \, {\boldsymbol{N}}^{e} \mathrm{d} \Omega_e \right] {{\boldsymbol{L}}^{e}}^T,
\label{eq:Spatial_Lagrangian_Terms_M_The} \\
\boldsymbol{K}^{\mathrm{The}} &= \sum_{e = 1}^{N^{\mathrm{The} } } {{\boldsymbol{L}}^{e}}^T
\left[ \int_{\Omega_e} ({\mbox{\boldmath$B$}}^{e})^T \, {\mbox{\boldmath$\kappa$} }_L(\bar{{\mbox{\boldmath$u$}}}, \bar{{\mbox{\boldmath$\vartheta$} }}) \, {\mbox{\boldmath$B$}}^{e} \mathrm{d} \Omega_e \right] {{\boldsymbol{L}}^{e}}^T,
\label{eq:Spatial_Lagrangian_Terms_K_The}
\\
{\boldsymbol{F}}^{\mathrm{The} } &= \sum_{e = 1}^{N^{\mathrm{The} } } {{\boldsymbol{L}}^{e}}^T
\left[ \int_{\Omega_e} ({\mbox{\boldmath$B$}}^{e})^T \, \bar{w}_L(\bar{{\mbox{\boldmath$u$}}}, \bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$\Phi$} }}) \, \mathrm{d} \Omega_e \right],
\label{eq:Spatial_Lagrangian_Terms_F_The} \\
\boldsymbol{K}^{\mathrm{Mec}} &= \sum_{e = 1}^{N^{\mathrm{Mec} } } {{\boldsymbol{L}}^{e}}^T
\left[ \int_{\Omega_e} ({\mbox{\boldmath$B$}}^{e})^T \, {\boldsymbol{\mathcal{S}}}_{EP}(\bar{{\mbox{\boldmath$u$}}}, \bar{{\mbox{\boldmath$\vartheta$} }}, {\boldsymbol{Z}}_{L}) \, \mathrm{d} \Omega_e \right],
\label{eq:Spatial_Lagrangian_Terms_K_Mec} \\
{\boldsymbol{F}}^{\mathrm{Mec} } &= \sum_{e = 1}^{N^{\mathrm{Mec}}} {{\boldsymbol{L}}^{e}}^T
\left[ \int_{\Omega_e} ({\boldsymbol{N}}^{e})^T \, {\boldsymbol{F}}_L(\bar{{\mbox{\boldmath$u$}}}, \bar{{\mbox{\boldmath$\Phi$} }}) \mathrm{d} \Omega_e \right]
\label{eq:Spatial_Lagrangian_Terms_F_Mec} \end{aligned}$$ where ${\boldsymbol{L}}^{e}$ is the gather matrix, ${\boldsymbol{N}}^{e}$ is the element shape function matrix, ${\mbox{\boldmath$B$}}^{e}$ is composed of the elements of the gradient of ${\boldsymbol{N}}^{e}$ [@fish-fem-07; @wriggers-fem-08] and where we neglected the boundary terms in –. The recurrent dependence on the displacement field in – and – results from the transformations between the deformed and undeformed configurations which involve the deformation gradient ${\boldsymbol{F}}$ and its determinant $J$, and the electric conductivity ${\mbox{\boldmath$\sigma$} }_L(\bar{{\mbox{\boldmath$u$}}}, \bar{{\mbox{\boldmath$\vartheta$} }})$ is considered to be temperature-dependent.
Equations – can be written as a system of differential algebraic equations: $$\underset{\displaystyle \boldsymbol{M}}{\underbrace{
\begin{pmatrix}
\boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} \\
\boldsymbol{0} & \boldsymbol{M}^{\mathrm{The}} & \boldsymbol{0} \\
\boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} \end{pmatrix}
}}
\displaystyle \frac{D }{D t}
\underset{\displaystyle \bar{{\mbox{\boldmath$v$}}}}{\underbrace{
\begin{pmatrix}
\bar{\boldsymbol{{\mbox{\boldmath$\Phi$} }}} \\
\bar{\boldsymbol{{\mbox{\boldmath$\vartheta$} }}} \\
\bar{{\mbox{\boldmath$u$}}}
\end{pmatrix}
}}
\!+\!
\underset{\displaystyle {\mbox{\boldmath$f$}}(\bar{{\mbox{\boldmath$v$}}}, {\boldsymbol{Z}}_{L})}{\underbrace{
\displaystyle \begin{pmatrix}
\boldsymbol{K}^{\mathrm{Ele}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}) \bar{{\mbox{\boldmath$\Phi$} }} + {\boldsymbol{F}}^{\mathrm{Ele}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}})
\\
\displaystyle \boldsymbol{K}^{\mathrm{The}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}) \bar{{\mbox{\boldmath$\vartheta$} }} + {\boldsymbol{F}}^{\mathrm{The}}(\bar{{\mbox{\boldmath$\Phi$} }}, \bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}})
\\
\displaystyle \boldsymbol{K}^{\mathrm{Mec}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}, {\boldsymbol{Z}}_{L}) + \bar{{\boldsymbol{F}}}^{\mathrm{Mec}}(\bar{{\mbox{\boldmath$\Phi$} }}, \bar{{\mbox{\boldmath$u$}}})
\end{pmatrix}
}}
\!=\!
\begin{pmatrix}
\boldsymbol{0} \\
\boldsymbol{0} \\
\boldsymbol{0}
\end{pmatrix}
\label{eq:Spatial_Discretized_Lagrangian_b}$$ or $$\displaystyle \boldsymbol{M} \frac{D \bar{{\mbox{\boldmath$v$}}} }{D t} + {\mbox{\boldmath$f$}}(\bar{{\mbox{\boldmath$v$}}}, {\boldsymbol{Z}}_{L}) = \boldsymbol{0},
\label{eq:DAE_Continuous_Lagrangian}$$ where $\boldsymbol{M}$ is a singular matrix and $$\begin{gathered}
{\mbox{\boldmath$f$}}_1 = \boldsymbol{K}^{\mathrm{Ele}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}) \bar{{\mbox{\boldmath$\Phi$} }}
+ {\boldsymbol{F}}^{\mathrm{Ele}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}),
{\mbox{\boldmath$f$}}_2 = \displaystyle \boldsymbol{K}^{\mathrm{The}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}) \bar{{\mbox{\boldmath$\vartheta$} }}
+ {\boldsymbol{F}}^{\mathrm{The}}(\bar{{\mbox{\boldmath$\Phi$} }}, \bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}), \\ {\mbox{\boldmath$f$}}_3 = \displaystyle \boldsymbol{K}^{\mathrm{Mec}}(\bar{{\mbox{\boldmath$\vartheta$} }}, \bar{{\mbox{\boldmath$u$}}}, {\boldsymbol{Z}}_{L})
+ \bar{{\boldsymbol{F}}}^{\mathrm{Mec}}(\bar{{\mbox{\boldmath$\Phi$} }}, \bar{{\mbox{\boldmath$u$}}}).
\label{eq:Spatial_Discretized_Lagrangian_Terms}\end{gathered}$$ Equation can be discretized in time using the backward Euler integrator: $$\displaystyle \boldsymbol{M} \frac{\bar{{\mbox{\boldmath$v$}}}^{n+1} - \bar{{\mbox{\boldmath$v$}}}^{n}}{\Delta t} + {\mbox{\boldmath$f$}}(\bar{{\mbox{\boldmath$v$}}}^{n+1}, {\boldsymbol{Z}}_{L})
= \boldsymbol{0},
\label{eq:DAE_Discretized_Lagrangian}$$ where $\bar{{\mbox{\boldmath$v$}}}^{n+1} = \bar{{\mbox{\boldmath$v$}}}(t_{n+1})$ with $t_{n+1} = t_0 + (n+1) \Delta t$ and $\Delta t$ which is the time step. After reordering the terms of , the following nonlinear equations can be derived: $$\begin{gathered}
\displaystyle \boldsymbol{M} \bar{{\mbox{\boldmath$v$}}}^{n+1} + {\Delta t} {\mbox{\boldmath$f$}}(\bar{{\mbox{\boldmath$v$}}}^{n+1}, {\boldsymbol{Z}}_{L}) - \boldsymbol{M} \bar{{\mbox{\boldmath$v$}}}^{n}
= {\boldsymbol{G}}(\bar{{\mbox{\boldmath$v$}}}^{n+1}, {\boldsymbol{Z}}_{L}) - \boldsymbol{M} \bar{{\mbox{\boldmath$v$}}}^{n} \\
= {\boldsymbol{H}}(\boldsymbol{\bar{\Phi}}^{n+1}, \boldsymbol{\bar{\vartheta}}^{n+1}, \bar{{\mbox{\boldmath$u$}}}^{n+1}, {\boldsymbol{Z}}_{L}) - \boldsymbol{M} \bar{{\mbox{\boldmath$v$}}}^{n}
= \boldsymbol{0}
\label{eq:Nonlinear_Lagrangian_1}\end{gathered}$$ with the vector function ${\boldsymbol{H}}= ({\boldsymbol{H}}_1, {\boldsymbol{H}}_2, {\boldsymbol{H}}_3)$ defined such that ${\boldsymbol{H}}_i(\boldsymbol{\bar{\Phi}}^{n+1}, \boldsymbol{\bar{\vartheta}}^{n+1}, \bar{{\mbox{\boldmath$u$}}}^{n+1}, {\boldsymbol{Z}}_{L}):= {\boldsymbol{G}}_i(\bar{{\mbox{\boldmath$v$}}}^{n+1}, {\boldsymbol{Z}}_{L})$ for i = 1, 2 or 3.
Linearization
-------------
Equation can be solved using the Newton–Raphson method. To do this, an iterative schema is used with the following linearization: $$\begin{gathered}
{\boldsymbol{G}}(\bar{{\mbox{\boldmath$v$}}}^{n+1}, {\boldsymbol{Z}}_{L}) \simeq \displaystyle {\boldsymbol{G}}(\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1}, {\boldsymbol{Z}}_{L})
+ \displaystyle \left(\frac{\partial {\boldsymbol{G}}}{\partial \bar{{\mbox{\boldmath$v$}}}^{n+1}} \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
\left(\bar{{\mbox{\boldmath$v$}}}_{m+1}^{n+1} - \bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} \right)
\\
= {\mbox{\boldmath$b$}}_{\mathrm{RHS}}({\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1}}) - \boldsymbol{A}({\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1}}) \, \Delta \bar{{\mbox{\boldmath$v$}}}_{m+1}^{n+1} = \boldsymbol{0},
\label{eq:Linearization_Lagrangian}\end{gathered}$$ where the index $m$ is used to denote the Newton–Raphson iteration. The terms in are given by: $$\begin{gathered}
{\mbox{\boldmath$b$}}_{\mathrm{RHS}} = {\boldsymbol{G}}(\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1}, {\boldsymbol{Z}}_{L})
= {\boldsymbol{H}}(\boldsymbol{\bar{\Phi}}_{m}^{n+1}, \boldsymbol{\bar{\vartheta}}_m^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}, {\boldsymbol{Z}}_{L})
\\
=
\begin{pmatrix}
{\boldsymbol{H}}_1(\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1})
\\
{\boldsymbol{H}}_2(\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1})
\\
{\boldsymbol{H}}_3(\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}, {\boldsymbol{Z}}_{L})
\end{pmatrix}.
\label{eq:Linearization_Lagrangian_NL_Term}\end{gathered}$$ The term ${\boldsymbol{H}}_3$ depends on the internal variables ${\boldsymbol{Z}}_L(\tau \leq t)$ through the second Piola–Kirchhoff stress ${\boldsymbol{S}}$. For each quadrature point, the stress is updated using the return mapping described in [@boatti-smp-16].
The stiffness matrix is given by: $$\!\!\left(\frac{\partial {\boldsymbol{G}}}{\partial \bar{{\mbox{\boldmath$v$}}}^{n+1}} \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
\!\!= \boldsymbol{A} =
\begin{pmatrix}
\displaystyle \frac{\partial {\boldsymbol{H}}_1}{\partial \bar{{\mbox{\boldmath$\Phi$} }}^{n+1} } &
\displaystyle \frac{\partial {\boldsymbol{H}}_1}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1} } &
\displaystyle \frac{\partial {\boldsymbol{H}}_1}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1} }
\\
\displaystyle \frac{\partial {\boldsymbol{H}}_2}{\partial \bar{{\mbox{\boldmath$\Phi$} }}^{n+1} } &
\displaystyle \frac{\partial {\boldsymbol{H}}_2}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1} } &
\displaystyle \frac{\partial {\boldsymbol{H}}_2}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1} }
\\
\displaystyle \frac{\partial {\boldsymbol{H}}_3}{\partial \bar{{\mbox{\boldmath$\Phi$} }}^{n+1} } &
\displaystyle \frac{\partial {\boldsymbol{H}}_3}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1} } &
\displaystyle \frac{\partial {\boldsymbol{H}}_3}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1} }
\end{pmatrix}.
\label{eq:Exact_Linearization_Lagrangian_NL_Term}$$ The terms of the tangent stiffness matrix in are given by: $$\begin{aligned}
\displaystyle \left( \frac{\partial {\boldsymbol{H}}_1}{\partial \bar{{\mbox{\boldmath$\Phi$} }}^{n+1} } \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
&= \left(\boldsymbol{K}^{\mathrm{Ele}} \right)_{\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}},
\label{eq:Inexact_Linearization_Lagrangian_NL_Term_a_1} \\
\displaystyle \left( \frac{\partial {\boldsymbol{H}}_1}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1} } \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
&= \left( \frac{\partial \boldsymbol{K}^{\mathrm{Ele}}}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1}} \, \bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}
+ \frac{\partial \boldsymbol{F}^{\mathrm{Ele}}}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1}} \right)_{\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}},
\label{eq:Inexact_Linearization_Lagrangian_NL_Term_a_2} \\
\displaystyle \left( \frac{\partial {\boldsymbol{H}}_1}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1} } \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
&= \left( \frac{\partial \boldsymbol{K}^{\mathrm{Ele}}}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1}} \, \bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1} + \frac{\partial \boldsymbol{F}^{\mathrm{Ele}}}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1}} \right)_{\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}},
\label{eq:Inexact_Linearization_Lagrangian_NL_Term_a_3} \\
\displaystyle \left( \frac{\partial {\boldsymbol{H}}_2}{\partial \bar{{\mbox{\boldmath$\Phi$} }}^{n+1} } \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
&= {\Delta t} \, \left( \frac{\partial \boldsymbol{F}^{\mathrm{The}}}{\partial \bar{{\mbox{\boldmath$\Phi$} }}^{n+1}} \right)_{\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}},
\label{eq:Inexact_Linearization_Lagrangian_NL_Term_b_1} \\
\displaystyle \left(\frac{\partial {\boldsymbol{H}}_2}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1} } \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
&= \left( \boldsymbol{M}^{\mathrm{The}}
+ {\Delta t} \left( \boldsymbol{K}^{\mathrm{The}}
+ \frac{\partial \boldsymbol{K}^{\mathrm{The}}}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1}} \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}
+ \frac{\partial \boldsymbol{F}^{\mathrm{The}}}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1}}
\right) \right), \label{eq:Inexact_Linearization_Lagrangian_NL_Term_b_2} \\
\displaystyle \left( \frac{\partial {\boldsymbol{H}}_2}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1} } \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
&= {\Delta t} \left( \frac{\partial \boldsymbol{K}^{\mathrm{The}}}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1}} \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}
+ \frac{\partial \boldsymbol{F}^{\mathrm{The}}}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1}} \right)_{\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}},
\label{eq:Inexact_Linearization_Lagrangian_NL_Term_b_3} \\
\displaystyle \left( \frac{\partial {\boldsymbol{H}}_3}{\partial \bar{{\mbox{\boldmath$\Phi$} }}^{n+1} } \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
&= \left( \frac{\partial \boldsymbol{F}^{\mathrm{Mec}}}{\partial \bar{{\mbox{\boldmath$\Phi$} }}^{n+1}} \right)_{\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}},
\label{eq:Inexact_Linearization_Lagrangian_NL_Term_c_1} \\
\displaystyle \left(\frac{\partial {\boldsymbol{H}}_3}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1} } \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
&= \left( \frac{\partial \boldsymbol{K}^{\mathrm{Mec}}}{\partial \bar{{\mbox{\boldmath$\vartheta$} }}^{n+1}} \right)_{\bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}},
\label{eq:Inexact_Linearization_Lagrangian_NL_Term_c_2} \\
\displaystyle \left(\frac{\partial {\boldsymbol{H}}_3}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1} } \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }
&= \left( \frac{\partial \boldsymbol{K}^{\mathrm{Mec}}}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1}}
+ \frac{\partial \boldsymbol{F}^{\mathrm{Mec}}}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1}} \right)_{\bar{{\mbox{\boldmath$\Phi$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$\vartheta$} }}_{m}^{n+1}, \bar{{\mbox{\boldmath$u$}}}_{m}^{n+1}}.
\label{eq:Inexact_Linearization_Lagrangian_NL_Term_c_3}\end{aligned}$$ In , the first term is given by $$\left( \frac{\partial \boldsymbol{K}^{\mathrm{Mec}}}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1}} \right)_{\bar{{\mbox{\boldmath$v$}}}_{m}^{n+1} }= \sum_{e = 1}^{N^{\mathrm{Mec} } } {{\boldsymbol{L}}^{e}}^T
\left[ \int_{\Omega_e} ({\mbox{\boldmath$B$}}^{e})^T \, \left(\frac{\partial {\boldsymbol{\mathcal{S}}}_{EP}}{\partial {\mbox{\boldmath$E$}}} \colon \frac{\partial {\mbox{\boldmath$E$}}}{\partial \bar{{\mbox{\boldmath$u$}}}^{n+1}} \right) \, \mathrm{d} \Omega_e \right].$$ The Jacobian of the mechanical problem ${\partial {\boldsymbol{\mathcal{S}}}_{EP}}/{\partial {\mbox{\boldmath$E$}}}$ in is also updated using the return mapping algorithm described in [@boatti-smp-16]. The pseudocode in Algorithm \[alg:smp\_multiphysics\] illustrates the flow of the numerical code used to solve the multiphysics problem.
Numerical Tests {#section:numerical_tests}
===============
This section is devoted to the numerical testing of the electro-thermo-mechanical problem. A set of numerical tests similar to the ones developed in [@boatti-smp-16] are herein proposed. Whereas the authors in [@boatti-smp-16] considered ideal and non-ideal shape memory polymer materials, the main focus of this paper is on the fully coupled problem. Therefore, in Section \[sec:appli\_mecha\] we consider an ideal and a non-ideal shape memory polymer single element similar to the one in [@boatti-smp-16] for the validation of the thermomechanical problem. In Section \[sec:appli\_fully\_coupled\] we only consider an ideal shape memory polymer stent for the fully coupled electro-thermo-mechanical problem.
The two considered tests are:
- the uniaxial tests on a $1 \times 1 \times 1$ mm$^3$ single-element cube (SEC),
- the simulation of a cylindrical vascular stent (CVS) similar to the one described in [@boatti-smp-16]. The stent has the same dimensions and material properties but without the small holes.
Symbol Value Unit
----------------- ------------------------- ------
$E^r$ 0.9 MPa
$E^g$ 771 MPa
$\nu^r$ 0.49 –
$\nu^g$ 0.29 –
$R^{pg}$ 10 MPa
$h$ 0 MPa
$\Delta \theta$ 30 (SEC) – 5 (CVS) K
$\theta_t$ 350 (SEC) – 344 (CVS) K
$w$ 0.2 (SEC) – 0.375 (CVS) 1/K
$c$ 1 –
$c^p$ 0 –
: Model parameters of the mechanical problem[]{data-label="table:mechanics"}
Material properties listed in Table \[table:mechanics\] are used for both cases. The software GetDP [@dular-getdp-98] was used to solve the fully coupled problem based on a total Lagrangian formulation.
Validation of the mechanical problem {#sec:appli_mecha}
------------------------------------
Results of the thermo-mechanical model developed in [@boatti-smp-16] are reproduced. This model consisted of a temperature-dependent elasto-plastic model with a constant temperature field imposed for all Gauss points at any given time instant $t$.
Results of the single element are reported in Figure \[fig:smp-mechanics-single-element\] for a high-temperature fixing similar to the one used in [@boatti-smp-16]. The material is progressively deformed at 400K, then cooled down to 200K while keeping constant the deformation. The material is then unloaded at 200K before re-heating it up to 400K to trigger the shape-recovery (see Test 1 in Fig. 6 of [@boatti-smp-16]). The results reported in Figure \[fig:smp-mechanics-single-element\] conform to those obtained in [@boatti-smp-16].
New results of the coupled problem {#sec:appli_fully_coupled}
----------------------------------
The results of the coupled problem are presented below. A description of the mechanical, thermal and electromagnetic problems is followed by the presentation of numerical results of the shape memory polymer. A best design can be obtained by choosing material properties for the thermal problem (thermal conductivity, mass density and heat capacity) that maintain a homogeneous temperature field in the stent, in order to avoid the appearance of regions with different phases during the recovery step. A non dimensionalization analysis of the thermal problem carried out in Section \[subsubsec:resultscoupledproblem\] facilitates this design. However, the control of the temperature is complicated by the dependence of the mechanical stress on the temperature-dependent ratio of the glassy state $z^g(\theta_L)$.
(0, 0) node\[inner sep=0\] [![Dirichlet boundary conditions for the mechanical problem. Zero displacement ${\mbox{\boldmath$u$}}_D^{\mathrm{Bottom}}(t) = \boldmath{0}$ is imposed on the bottom section and a time-dependent displacement ${\mbox{\boldmath$u$}}_D^{\mathrm{Top}}(t)$ similar to the one used for Test 1 of Fig. 6 in [@boatti-smp-16] is imposed on the top line.[]{data-label="fig:smp-mechanics-stent"}](smp_stent_geo.png "fig:"){width="10cm"}]{}; (A) at (0.0, 1.0) [${\mbox{\boldmath$u$}}_D^{\mathrm{Top}}$]{}; (B) at (0.0, 3.75) ; (C) at (-0.0, -1.0) [${\mbox{\boldmath$u$}}_D^{\mathrm{Bottom}}$]{}; (D) at (-0.0, -3.55) ; (A) edge (B); (C) edge (D);
The geometry of Figure \[fig:smp-mechanics-stent\] is used for the mechanical problem of the cylindrical vascular stent.
### Electromagnetic and thermal problems
We simulate the insertion of a vascular shape memory polymer stent in a vein of the arm. The stent contains electric particles that can react to electromagnetic source fields produced by a coil wrapped around the arm by producing heat by the Joule effect. For the sake of simplicity, we consider the resulting shape memory polymer composite to be homogeneous with homogenized macroscopic material properties, thus ignoring the multiscale nature of the composite.
The mechanical problem is similar to the one in [@boatti-smp-16], with the temperature field obtained by solving the thermal problem with the source generated by the eddy current losses. In the following, we define the electromagnetic and the thermal problems.
The temperature can be controlled by an electromagnetic field generated by a coil crossed by a current denoted $I_s(t)$. For all problems studied herein, we consider a single frequency source $$I_s(t) = I_0(t) (a + b \, \sin(\omega \, t)) = I_0(t) (a + b \, \sin(2 \, \pi \, f \, t)),$$ where $I_0(t)$ (A) is piecewise, linear, time-dependent amplitude of the electric current, $\omega$ is the angular velocity and $f$ the frequency of the signal. The design parameters for the electromagnetic and thermal problems are the amplitude of the current, the frequency and the material properties: the electric conductivity $\sigma$ (S/m), the magnetic permeability $\mu$ (H/m), the mass density $\rho$, the heat capacity $c_p$ and the thermal conductivity ${\mbox{\boldmath$\kappa$} }$. In all our applications, we consider frequencies small than 1000Hz, $\sigma = 10^4$S/m and $\mu = \mu_0 \mu_{\mathrm{rel}}$ with $\mu_{\mathrm{rel}} = 20$, which corresponds to the wavelength $\lambda$ and skin depth $\delta$: $$\lambda = \frac{c}{f} = \frac{1}{\sqrt{\mu \epsilon} f } \approx 300\text{km} \quad , \quad \delta = \sqrt{\frac{2}{\mu \sigma \omega}} \approx 40\text{mm}.$$
![Geometry and mesh used for the coupled problem. Top: The cylindrical cardiovascular stent surrounded by an exciting coil. Middle: Mesh of the stent and the coil. Bottom: Mesh of the stent, the coil and the surrounding air. The enclosing box is used to bound the computational domain for the electromagnetic problem assumed to be unbounded.[]{data-label="fig:smp_physical_chemical"}](smp_stent_geometry_stent_coil_geo "fig:"){width="85.00000%"} ![Geometry and mesh used for the coupled problem. Top: The cylindrical cardiovascular stent surrounded by an exciting coil. Middle: Mesh of the stent and the coil. Bottom: Mesh of the stent, the coil and the surrounding air. The enclosing box is used to bound the computational domain for the electromagnetic problem assumed to be unbounded.[]{data-label="fig:smp_physical_chemical"}](smp_stent_geometry_stent_coil_mesh "fig:"){width="85.00000%"} ![Geometry and mesh used for the coupled problem. Top: The cylindrical cardiovascular stent surrounded by an exciting coil. Middle: Mesh of the stent and the coil. Bottom: Mesh of the stent, the coil and the surrounding air. The enclosing box is used to bound the computational domain for the electromagnetic problem assumed to be unbounded.[]{data-label="fig:smp_physical_chemical"}](smp_stent_geometry_all_mesh "fig:"){width="75.00000%"}
The wavelength is very large compared to the dimensions of the structure (typically 20mm for the length and 1mm for the thickness) that the *quasistatic assumption* can be made [@rodriguez-10-eddycurrents]. Likewise, the skin depth is large compared to the dimensions of the stent that the eddy currents resulting from the reaction field can be neglected. Figure \[fig:smp\_physical\_chemical\] illustrates the geometry used for the coupled problem.
To determine the magnetic induction source ${\mbox{\boldmath$b$}}_s(t)$ for the electromagnetic problem, we consider a coil with a very large number of turns. The value of the magnetic field ${\mbox{\boldmath$h$}}_s(t)$ and the magnetic induction ${\mbox{\boldmath$b$}}_s(t)$ in the coil are homogeneous and given by [@field-coil] $${\mbox{\boldmath$h$}}_s(t) = h_s(t) \, {\mbox{\boldmath$e$}}_z = \frac{N}{L}I_s(t) \, {\mbox{\boldmath$e$}}_z \, \, , \, \, {\mbox{\boldmath$b$}}_s(t) = b_s(t) \, {\mbox{\boldmath$e$}}_z = \mu {\mbox{\boldmath$h$}}_s(t) = \mu \frac{N}{L}I_s(t) \, {\mbox{\boldmath$e$}}_z
\label{eq:Applications_Sources_h_b}$$ where $N$ is the number of turns, $L$ the length of the coil, $\mu$ the magnetic permeability of the material and ${\mbox{\boldmath$e$}}_z$ the direction oriented along the axis of the coil. From the Gauss magnetic law ${\mathrm{{\mathrm{div}}}{}\,{{\mbox{\boldmath$b$}}_s}} = 0$, a source vector potential ${\mbox{\boldmath$a$}}_s(t)$ can be derived from the magnetic induction ${\mbox{\boldmath$b$}}_s(t)$ as ${\mbox{\boldmath$b$}}_s(t) = {\mathrm{{\mathrm{\boldsymbol{curl}}}}{}\,{{\mbox{\boldmath$a$}}_s(t)}}$. In the computational domain of the stent, a possible vector potential that satisfies this equality and is symmetric with respect to the undeformed geometry of the stent is $${\mbox{\boldmath$a$}}_s(x, y, t) = 0.5 \, b_s(t) (-y, x, 0) = 0.5 \, \mu \frac{N}{L} I_s(t) (-y, x, 0)
\label{eq:Applications_Sources_a}$$ with $x = X + u_x$ and $y = Y + u_y$ where $X$ and $Y$ are the coordinates expressed in the undeformed configuration and $u_x$ and $u_y$ are components of the displacement ${\mbox{\boldmath$u$}}= (u_x, u_y, u_z)$. From and , it can be noted that the magnetic field ${\mbox{\boldmath$h$}}_s(t)$ and the magnetic induction ${\mbox{\boldmath$b$}}_s(t)$ in the coil do not depend on spatial coordinates whereas the vector potential ${\mbox{\boldmath$a$}}_s(t)$ depends on spatial coordinates $x$ and $y$. This vector potential is a one differential form that can be transformed as ${\boldsymbol{\mathcal{A}}}_s = {\boldsymbol{F}}^T {\mbox{\boldmath$a$}}_s$ thus leading to the source in .
Table \[table:elemag\_thermal\] contains model parameters of the electromagnetic and thermal problems.
Symbol Value Unit
---------- --------------------------- ----------------------------
$I_0(t)$ electric current waveform A
$f$ 1000 Hz
$\sigma$ 10$^4$ S/m
$\mu_r$ 20 –
$\rho$ 270 kg m$^{-3}$
$c_p$ 10 kg m$^2$ K$^{-1}$ s$^{-2}$
$k$ 237 W m$^{-1}$ K$^{-1}$
$h$ 500 W m$^{-2}$ K$^{-1}$
$N$ 1000 –
$L$ 1 m
: Model parameters of the electromagnetic and thermal problems[]{data-label="table:elemag_thermal"}
Defining the thermal problem resulting from the deployment of the actual stent is challenging. Though it is easy to control the temperature of the device during the first three stages (*loading* at high temperature, *cooling* followed by *unloading/insertion* of the stent) most of which are done outside the human body, the last step, the *recovery*, necessitates controlling the temperature using electromagnetic fields. In this paper, we simulate the control of the entire deployment process using the electromagnetic fields.
During the last step, different modes of heat exchange can be considered: (1) heat conduction in the stent, at the interface of the stent and the surrounding tissue and in the tissue itself and (2) forced convection at part of the boundary of surface of the stent in contact with the blood flowing in the vein. The surface of the stent in contact with the tissue/blood varies during the process of recovery and its detection would necessitate consideration of contact mechanics. For the sake of simplicity, we only consider forced convection.
Finally, thanks to the assumptions made of the electromagnetic and thermal problems, all three problems can only be solved on the computational domain of the stent thus neglecting the surrounding environment.
### Results of the coupled problems {#subsubsec:resultscoupledproblem}
Results of the coupled problem are herein reported. As mentioned earlier, the main difference between this section and section \[sec:appli\_mecha\] lies in the use of a temperature field obtained by solving the heat equation on a moving domain with the source obtained by solving the electromagnetic problem instead of a priori imposing a temperature field at each time instant $t$.
Figures \[fig:Applications\_Fields\_bu\_bj\] and \[fig:Applications\_Fields\_JL\_T\] show the displacement ${\mbox{\boldmath$u$}}$, the current density ${\mbox{\boldmath$j$}}$, Joule losses and the temperature $T$ at the instances $t = 4.78125 \times 10^{-3}$s and $t = 4.84375 \times 10^{-3}$s.
This can play an important role in the design of the stent, especially for the computation of the temperature. Indeed, the high dependency of the ratio of the glassy states on the temperature $z^g(\theta_L)$ necessitates selecting electromagnetic and thermal loadings as well as thermal material properties that allow for a quick diffusion of the heat sources throughout the stent, to avoid inhomogeneities of temperature that would cause different regions of the stent to be in different phases (rubbery/glassy) during the recovery step. Another issue concerns the use of the Newton–Raphson method to solve the nonlinear coupled problem. Considerably large and inhomogeneous increments of temperature computed especially during the first nonlinear iterations of the Newton–Raphson scheme may lead to inhomogeneities of temperature and cause slow convergence in the recovery process.
To avoid inhomogeneities of temperature in the recovery step, we developed the following normalization process, which makes the problem well conditioned.
The process starts with the linearized version of the *heat equation* : $$\displaystyle \rho_L c_p \frac{\partial \theta_L}{\partial t} + {\text{{\text{Div}}}{_X}\,{ \left[ {\mbox{\boldmath$\kappa$} }_L
{\text{{\text{\bf Grad}}}{_X}\,{\theta_L}} \right] }} = -w_L(\phi, \theta_L, {\mbox{\boldmath$u$}}).
\label{eq:Weak_Thermal_Equation_Lagrangian_Dimensionless_Eq_1}$$ A new coordinate system ($\tau$, $\boldsymbol{\eta}$) is introduced as: $$\begin{gathered}
t = T_{c} \, \tau \, \, , \, \, dt = T_{c} \, d \tau \, \, , \, \,
\frac{\partial (\cdot)}{\partial t} = \frac{1}{T_{c}} \frac{\partial (\cdot)}{\partial \tau}, \\
X_i = L_{c} \, \eta_i \, \, , \, \, dX_i = L_{c} \, d \eta_i \, \, , \, \,
\frac{\partial (\cdot)}{\partial X_i} = \frac{1}{L_{c}} \frac{\partial (\cdot)}{\partial \eta_i}, \quad \quad
\theta_L = \theta_{c} \, \bar{\theta}_L
\label{eq:Weak_Thermal_Equation_Lagrangian_Dimensionless_Eq_2}\end{gathered}$$ where $\bar{\theta}$, $T_{c}$, $L_{c}$, $\tau$, and $\eta_i$ are the characteristic temperature, the characteristic time, the characteristic length, the dimensionless temporal and spatial coordinates, respectively. The derivatives in are transformed as: $$\begin{gathered}
\displaystyle \frac{\partial \theta_L}{\partial t} = \frac{1}{T_{c}} \frac{\partial \left(\theta_{c} \,
\bar{\theta}_L \right) }{\partial \tau} = \frac{\theta_{c}}{T_{c}} \frac{\partial \bar{\theta}_L}{\partial \tau} \quad , \quad
\frac{\partial \theta_L}{\partial X_i} = \frac{1}{L_{c}}
\frac{\partial \left(\theta_{c} \, \bar{\theta}_L \right) }{\partial \eta_i} = \frac{\theta_{c}}{L_{c}}
\frac{\partial \bar{\theta}_L}{\partial \eta_i}, \\
\displaystyle \frac{\partial^2 \theta_L}{\partial X_i^2} = \frac{1}{L_{c}}
\frac{\partial \left( \displaystyle \frac{\partial \theta_L}{\partial x_i} \right) }{\partial \eta_i} =
\frac{\theta_{c}}{L_{c}^2} \frac{\partial^2 \bar{\theta}_L}{\partial \eta_i^2}
\label{eq:Weak_Thermal_Equation_Lagrangian_Dimensionless_Eq_3}\end{gathered}$$ which leads to the *dimensionless heat equation*: $$\displaystyle \frac{\partial \bar{\theta}_L}{\partial \tau} +
\frac{T_{c} \bar{{\mbox{\boldmath$\kappa$} }}_L}{\rho_L c_p L_{c}^2} {\text{{\text{Div}}}{_{\eta}}\,{ \left[
{\text{{\text{\bf Grad}}}{_{\eta}}\,{\bar{\theta}_L}} \right] }} = - \frac{T_{c}}{\rho_L c_p L_{c}^2} \bar{w}_L(\bar{\phi}, \bar{\theta}_L, \bar{{\mbox{\boldmath$u$}}})
\label{eq:Weak_Thermal_Equation_Lagrangian_Dimensionless_Eq_4}$$ where the thermal conductivity was assumed constant and barred quantities are defined in the new coordinate system. For the first two terms to be of the same order of magnitude, i.e., for the temperature to have enough time to diffuse in the stent, the material properties must be chosen such that $$T_{c} = \displaystyle \frac{\rho_L c_p L_{c}^2}{\bar{{\mbox{\boldmath$\kappa$} }}_L} .
\label{eq:Weak_Thermal_Equation_Lagrangian_Dimensionless_Eq_5}$$
Results of the coupled problem are reported in Figures \[fig:smp-coupled-stent\_1\]-\[fig:smp-coupled-stent\_2\] for a stent with the high-temperature fixing and slightly different material properties as those reported in [@boatti-smp-16]. In the case of the imposed temperature, the material is progressively deformed at 350K, then cooled down to 320K while the deformation is maintained constant, then unloaded at 320K, and finally re-heated up to 350K to trigger shape-recovery. We mimick the same trajectory of the temperature by changing material properties, the frequency and the amplitude of the excitation source. Results of the coupled problem are different from the ones obtained with the imposed temperature. An optimal control of the temperature using the source current $I_s(t)$ and the geometry of the coil as control parameters can allow to prescribe a temperature trajectory convenient for surgical purposes.
Conclusions {#section:conclusions}
===========
In this paper, the deployment of a vascular shape memory polymer stent in a vein of an arm is simulated. The temperature field used in the thermo-mechanical model of the stent is controlled by solving for electromagnetic fields generated by a coil wrapped around the arm. The controllability of the temperature depends on the choice of the electromagnetic source field determined by the amplitude and the excitation frequency of the current flowing through the coil, and on the material properties used for the thermal problem. An initial design of the stent which allows for the diffusion of heat and leads to a homogeneous distribution of temperature during the recovery step is proposed. The optimal control of the temperature of the devices can further be carried out, thus allowing the device to follow a prescribed temperature trajectory that might be convenient for surgical purposes.
Acknowledgments {#sec:motivation .unnumbered}
===============
The authors would like to thank Dr. Elisa Boatti at Georgia Institute of Technology, Prof. Ludovic Noels and Miguel Pareja Mu$\tilde{\text{n}}$oz at the University of Liège for fruitful discussions on mechanical models of shape memory polymers. They would also like to thank Prof. Christophe Geuzaine at the University of Liège for discussions about the implementation in GetDP. The first author is particularly indebted to Dr. François Henrotte at the University of Liège for the discussions on electromagnetic formulations under large deformations. During the time the research was carried out, Innocent Niyonzima was a postdoctoral Fellow with the Belgian American Educational Foundation (BAEF). He is also partially supported by an excellence grant from Wallonie-Bruxelles International (WBI).
|
---
abstract: 'In the framework of the Littlest Higgs Model with T-parity (LHT), we study the contributions of the new particles to $Zb\bar{b}$ couplings at one-loop level. Based on these results, we further study the branching ratio $R_{b}$ and the unpolarized forward-backward asymmetry ${A_{FB}^{b}}$. We find that the correction of the new particles to $Zb\bar{b}$ couplings is mainly on the left-handed coupling and has small part of the parameter space to alleviate the deviation between theoretical predictions and experimental values. The precision measurement value of $R_{b}$ can give severe constraints on the relevant parameters. The constraints from the precision measurement value of ${A_{FB}^{b}}$ are very weak.'
author:
- 'Bingfang Yang$^1$$^,$$^2$'
- Xuelei Wang$^1$
- Jinzhong Han$^1$
title: 'The Study of the contribution of the LHT model to $Zb\bar{b}$ coupling '
---
Introduction
=============
The Standard Model (SM) has been very successful, however, it is still believed to be a theory effective at the electroweak scale and some new physics (NP) must exist at higher energy regimes. So far there have been many speculations on the possible forms of the NP beyond the SM, one of the interesting possibilities is the Little Higgs model. The little Higgs theory was proposed [@1] as a possible solution to the hierarchy problem and remains a popular candidate for the NP. The Littlest Higgs (LH) model [@2] is a cute economical implementation of the little Higgs, but suffered from severe constraints from electroweak precision tests [@3], which would require raising the mass scale of the new particles to far above TeV scale and thus reintroduce the fine-tuning in the Higgs potential [@4]. The most serious constraints resulted from the tree-level corrections to precision electroweak observables due to the exchanges of the additional heavy gauge bosons present in the theories, as well as from the small but non-vanishing vacuum expectation value (VEV) of an additional weak-triplet scalar field. In order to solve this problem, a discrete symmetry called T-parity is proposed [@5], which explicitly forbids any tree-level contributions from the heavy gauge bosons to the observables involving only the SM particles as external states. The interactions that induce triplet VEV contributions is also forbidden. This model is called the Littlest Higgs Model with T-parity (LHT). In the LHT model, corrections to the precision electroweak observables are generated exclusively at loop level.
The branching ratio $R_{b}$ is very sensitive to the NP beyond the SM, the precision experimental value of $R_{b}$ may give a severe constraint on the NP [@6]. Experimentally, the electroweak observables have been precisely measured at the SLC and LEP, in the most recent analysis of the electroweak data, $R_{b}=0.21629\pm0.00066$ differs from the SM fit by $0.7\sigma$, $A_{FB}^{b}=0.0992\pm0.0016$ disagrees with the SM fit by $-2.9\sigma$ [@7]. Furthermore, the experimental value of $Zb\bar{b}$ couplings disagrees with the SM fit by about $3\sigma$, especially the deviation of the right-handed coupling is so large that it is very difficult to explain. These significant deviations from the ${A_{FB}^{b}}$ and the $Zb\bar{b}$ couplings might be the first window into the NP. With the running of the LHC, they will be further researched. In the LHT model, there are new fermions and new gauge bosons, which can contribute to the $Zb\bar{b}$ couplings and give modifications to the $R_{b}$ and $A_{FB}^{b}$. Therefore, it is possible to give some constraints on the relevant parameters via their radiative corrections to the $R_{b}$ and $A_{FB}^{b}$. In this paper, we calculate the contributions of the LHT model to the $Zb\bar{b}$ couplings. On this basis, we further study the $R_{b}$ and ${A_{FB}^{b}}$, then we give the constraints on the relevant parameters according to the precision measurements.
This paper is organized as follows. In Sec.II we recapitulate the LHT model and discuss the new flavor interactions which will contribute to the $Zb\bar{b}$ vertex. In Sec.III we calculate the one-loop contributions of the LHT model to the $Zb\bar{b}$ vertex, $R_{b}$ and $A_{FB}^{b}$, then the relevant numerical results are shown. Finally, we give our conclusions in Sec.IV.
A brief review of the LHT model
================================
The LHT [@5] is based on a non-linear sigma model describing the spontaneous breaking of a global $SU(5)$ down to a global $SO(5)$. This symmetry breaking takes place at the scale $f\sim\mathcal{O}(TeV)$ and originates from the VEV of an $SU(5)$ symmetric tensor $\Sigma $, given by $$\Sigma_{0}\equiv<\Sigma>=
\begin{pmatrix}
0_{2\times2}&0&1_{2\times2}\\
0&1&0\\
1_{2\times2}&0&0_{2\times2}
\end{pmatrix}$$ From the $SU(5)/SO(5)$ breaking, there arise 14 Goldstone bosons which are described by the “pion" matrix $\Pi$, given explicitly by $$\Pi=
\begin{pmatrix}
-\frac{\omega^0}{2}-\frac{\eta}{\sqrt{20}}&-\frac{\omega^+}{\sqrt{2}}
&-i\frac{\pi^+}{\sqrt{2}}&-i\phi^{++}&-i\frac{\phi^+}{\sqrt{2}}\\
-\frac{\omega^-}{\sqrt{2}}&\frac{\omega^0}{2}-\frac{\eta}{\sqrt{20}}
&\frac{v+h+i\pi^0}{2}&-i\frac{\phi^+}{\sqrt{2}}&\frac{-i\phi^0+\phi^P}{\sqrt{2}}\\
i\frac{\pi^-}{\sqrt{2}}&\frac{v+h-i\pi^0}{2}&\sqrt{4/5}\eta&-i\frac{\pi^+}{\sqrt{2}}&
\frac{v+h+i\pi^0}{2}\\
i\phi^{--}&i\frac{\phi^-}{\sqrt{2}}&i\frac{\pi^-}{\sqrt{2}}&
-\frac{\omega^0}{2}-\frac{\eta}{\sqrt{20}}&-\frac{\omega^-}{\sqrt{2}}\\
i\frac{\phi^-}{\sqrt{2}}&\frac{i\phi^0+\phi^P}{\sqrt{2}}&\frac{v+h-i\pi^0}{2}&-\frac{\omega^+}{\sqrt{2}}&
\frac{\omega^0}{2}-\frac{\eta}{\sqrt{20}}
\end{pmatrix}$$ Under T-parity the SM Higgs doublet, $H=(-i\pi^+\sqrt{2},(v+h+i\pi^0)/2)^T$ is T-even while other fields are T-odd.
The Goldstone bosons $\omega^{\pm},\omega^{0},\eta$ are respectively eaten by the new T-odd gauge bosons $W_{H}^{\pm},Z_{H},A_{H}$, which obtain masses at $\mathcal O(\upsilon^{2}/f^{2})$ $$M_{W_{H}}=M_{Z_{H}}=gf(1-\frac{\upsilon^{2}}{8f^{2}}),M_{A_{H}}=\frac{g'f}{\sqrt{5}}
(1-\frac{5\upsilon^{2}}{8f^{2}})$$ with $g$ and $g'$ being the SM $SU(2)$ and $U(1)$ gauge couplings, respectively.
The Goldstone bosons $\pi^{\pm},\pi^{0}$ are eaten by the T-even $W_{L}^{\pm}$and $Z_{L}$ bosons of the SM, which obtain masses at $\mathcal O(\upsilon^{2}/f^{2})$ $$M_{W_{L}}=\frac{g\upsilon}{2}(1-\frac{\upsilon^{2}}{12f^{2}}),M_{Z_{L}}=\frac{g\upsilon}
{2\cos\theta_{W}}(1-\frac{\upsilon^{2}}{12f^{2}})$$ The photon $A_{L}$ is also T-even and remains massless.
For each SM fermion, a copy of mirror fermion with T-odd quantum number is added in order to preserve the T-parity. For the mirror quarks, we denote them by $u_{H}^{i},d_{H}^{i}$, where i= 1, 2, 3 are the generation index. At the order of $\mathcal
O(\upsilon^{2}/f^{2})$ their masses are given by $$m_{d_{H}^{i}}=\sqrt{2}\kappa_if, m_{u_{H}^{i}}=
m_{d_{H}^{i}}(1-\frac{\upsilon^2}{8f^2})$$ where $\kappa_i$ are the diagonalized Yukawa couplings of the mirror quarks.
In order to cancel the quadratic divergence of the Higgs mass induced by top loops, an additional heavy quark $T^{+}$ is introduced, which is even under T-parity. The implementation of T-parity then requires also a T-odd partner $T^{-}$. Their masses are given by $$\begin{aligned}
m_{T^{+}}&=&\frac{f}{v}\frac{m_{t}}{\sqrt{x_{L}(1-x_{L})}}[1+\frac{v^{2}}{f^{2}}(\frac{1}{3}-x_{L}(1-x_{L}))]\\
m_{T^{-}}&=&\frac{f}{v}\frac{m_{t}}{\sqrt{x_{L}}}[1+\frac{v^{2}}{f^{2}}(\frac{1}{3}-\frac{1}{2}x_{L}(1-x_{L}))]\end{aligned}$$ where $x_{L}$ is the mixing parameter between the SM top-quark $t$ and the new top-quark $T^{+}$.
Just like the SM, the mirror sector in the LHT model also has weak mixing, parameterised by unitary mixing matrices: two for mirror quarks and two for mirror leptons: $$V_{Hu},V_{Hd},V_{Hl},V_{H\nu}$$ $V_{Hu}$ and $V_{Hd}$ are for the mirror quarks which are present in our analysis. $V_{Hu}$ and $V_{Hd}$ satisfy the physical constraints $V_{Hu}^{\dag}V_{Hd}=V_{CKM}$. We follow [@8] to parameterize $V_{Hd}$ with three angles $\theta^d_{12},\theta^d_{23},\theta^d_{13}$ and three phases $\delta^d_{12},\delta^d_{23},\delta^d_{13}$ $$\begin{aligned}
V_{Hd}=
\begin{pmatrix}
c^d_{12}c^d_{13}&s^d_{12}c^d_{13}e^{-i\delta^d_{12}}&s^d_{13}e^{-i\delta^d_{13}}\\
-s^d_{12}c^d_{23}e^{i\delta^d_{12}}-c^d_{12}s^d_{23}s^d_{13}e^{i(\delta^d_{13}-\delta^d_{23})}&
c^d_{12}c^d_{23}-s^d_{12}s^d_{23}s^d_{13}e^{i(\delta^d_{13}-\delta^d_{12}-\delta^d_{23})}&
s^d_{23}c^d_{13}e^{-i\delta^d_{23}}\\
s^d_{12}s^d_{23}e^{i(\delta^d_{12}+\delta^d_{23})}-c^d_{12}c^d_{23}s^d_{13}e^{i\delta^d_{13}}&
-c^d_{12}s^d_{23}e^{i\delta^d_{23}}-s^d_{12}c^d_{23}s^d_{13}e^{i(\delta^d_{13}-\delta^d_{12})}&
c^d_{23}c^d_{13}
\end{pmatrix}\end{aligned}$$
The one-loop corrections to $Zb\bar{b}$ couplings in the LHT model
==================================================================
We employ the following notation for the effective $Zb\bar{b}$ interaction: $$\begin{aligned}
L_{Zb\bar{b}}&=&\frac{e}{S_{W}C_{W}}(g_{L}^{b}\bar{b}\gamma^{\mu}bP_{L}+g_{R}^{b}\bar{b}\gamma^{\mu}bP_{R})Z_{\mu} \nonumber\\
&=&\frac{e}{2S_{W}C_{W}}\bar{b}\gamma^{\mu}(g_{V}^{b}-g_{A}^{b}\gamma_{5}bZ_{\mu})\end{aligned}$$ where $\theta_{W}$ is the Weinberg angle, $S_{W}=\sin\theta_{W}$, $C_{W}=\cos\theta_{W}$, $P_{L}=\frac{1-\gamma_{5}}{2}$ and $P_{R}=\frac{1+\gamma_{5}}{2}$. The effective couplings are then written as $$\begin{aligned}
\bar{g}_{L,R}^{b}&=&g_{L,R}^{b}+\delta g_{L,R}^{SM}+\delta
g_{L,R}^{NP}\\ \bar{g}_{V,A}^{b}&=&g_{V,A}^{b}+\delta
g_{V,A}^{SM}+\delta g_{V,A}^{NP}\end{aligned}$$ where $\bar{g}_{L,R}^{b}, \bar{g}_{V,A}^{b}$ are respectively the radiatively-corrected effective couplings, $g_{L,R}^{b}$ are respectively the left-handed and right-handed $Zb\bar{b}$ couplings at tree level, $\delta g_{L,R}^{SM} $ and $\delta g_{L,R}^{NP}$ are their corresponding one-loop corrections of the SM and the NP, $g_{V,A}^{b}$ are respectively the vector and axial vector coupling coefficients of $Zb\bar{b}$ interaction at tree level, $\delta
g_{V,A}^{SM}$ and $\delta g_{V,A}^{NP}$ are their corresponding one-loop corrections of the SM and the NP. The tree-level couplings are given by $$\begin{aligned}
g_{L}^{b}&=&-\frac{1}{2}+\frac{1}{3}S_{W}^{2}~~~,~~~g_{R}^{b}=\frac{1}{3}S_{W}^{2}\\
g_{V}^{b}=g_{L}^{b}+g_{R}^{b}&=&-\frac{1}{2}+\frac{2}{3}S_{W}^{2}
~~~,~~~g_{A}^{b}=g_{L}^{b}-g_{R}^{b}=-\frac{1}{2}\end{aligned}$$
The branching ratio is defined as $$R_{b}=\frac{\Gamma(Z\rightarrow b\bar{b})}{\Gamma(Z\rightarrow
hadrons)}$$
The full hadron width is the sum of widths of five quark channels: $$\Gamma(Z\rightarrow hadrons)=\Gamma(Z\rightarrow
u\bar{u})+\Gamma(Z\rightarrow d\bar{d})+\Gamma(Z\rightarrow
s\bar{s})+\Gamma(Z\rightarrow c\bar{c})+\Gamma(Z\rightarrow
b\bar{b})$$
For the decays to any of the five pairs of quarks $q\bar{q}$ we have[@9] $$\Gamma_{q}\equiv\Gamma(Z\rightarrow
q\bar{q})=12\Gamma_{0}(g_{Aq}^{2}R_{Aq}+g_{Vq}^{2}R_{Vq})$$ with $\Gamma_{0}=\frac{G_{F}M_{Z_{L}}^{3}}{24\sqrt{2}\pi}$, here $g_{Aq}$ and $g_{Vq}$ are the axial-vector and effective vector couplings. The radiators $R_{Aq}$ and $R_{Vq}$ contain contributions from the final state gluons and photons. In the crudest approximation $$R_{Vq}=R_{Aq}=1+\frac{\hat{\alpha_{s}}}{\pi}$$ where $\alpha_{s}(q^{2})$ is the QCD running coupling constant: $$\hat{\alpha_{s}}\equiv\alpha_{s}(q^{2}=M_{Z_{L}}^{2})$$
The expression of the radiative correction to $R_{b}$ can be expressed as [@10] $$\delta
R_{b}\simeq\frac{2R_{b}^{SM}(1-R_{b}^{SM})}{g_{Vb}^{2}(3-\beta^{2})+2g_{Ab}^{2}\beta^{2}}[g_{Vb}(3-\beta^{2})\delta
g_{Vb}+2g_{Ab}\beta^{2}\delta g_{Ab}]$$ with $\beta=\sqrt{1-\frac{4\hat{m}_{b}^{2}}{M_{Z_{L}}^{2}}}$ being the velocity of b-quark in $Z$ decay, here $\hat{m}_{b}$ is the value of the running mass of the b-quark at scale $M_{Z_{L}}$ calculated in $\overline{MS}$ scheme [@11].
The unpolarized forward-backward asymmetry in the decay to $b\bar{b}$ equals: $$A_{FB}^{b}=\frac{N_{F}-N_{B}}{N_{F}+N_{B}}$$ where $N_{F}$ is the cross section for finding the scattered fermion in the hemisphere defined by the incident electron direction and $N_{B}$ is the cross section for finding it in the positron hemisphere. It can be expressed as $$A_{FB}^{b}=\frac{3}{4}(1-\frac{k_{A}}{\pi})A_{e}A_{b}$$ where the factor $(1-\frac{k_{A}}{\pi})$ represents a QCD radiative correction, as in Ref. [@12], for which we use the numerical value 0.95, $A_{e}$ refers to the creation of $Z$ boson in $e^{+}e^{-}$ -annihilation, while $A_{b}$ is the left-right coupling constant asymmetry£¬refers to its decay in $b\bar{b}$ [@9] $$A_{b}=\frac{2g_{Ab}g_{Vb}}{\beta^{2}g_{Ab}^{2}+(3-\beta^{2})g_{Vb}^{2}/2}$$
The relevant Feynman diagrams for the LHT contributions are shown in Fig.1. We use the ’t Hooft-Feynman gauge, so the contributions of Goldstone bosons should be involved. In our calculation, $g_{Ab}$ and $g_{Vb}$ should be replaced by $\bar{g}_{Ab}$ and $\bar{g}_{Vb}$, $g_{V,A}^{b}+\delta g_{V,A}^{SM}$ can be found in Ref. [@13]. The calculations of the loop diagrams are straightforward. Each loop diagram is composed of some scalar loop functions [@14], which are calculated by using LOOPTOOLS [@15]. The relevant Feynman rules can be found in Ref. [@16]. We applied the on-shell renormalization scheme and have checked that the divergences are canceled.
In the numerical calculations we take the input parameters [@17] as Fermi constant $G_{F}=1.16637\times 10^{-5}GeV^{-2}$, the fine-structure constant $\alpha=1/128$, $Z$-boson mass $M_{Z_{L}}=91.2GeV$, fermion masses $m_{f}$, the electroweak mixing angle $S_{W}^{2}=0.231$ and the final-state asymmetry parameter $A_{e}=0.1515$. In our calculation, the relevant LHT parameters are the scale $f$, the mixing parameter $x_{L}$, the mirror quark masses and parameters in the matrices $V_{Hu}$ and $V_{Hd}$.
For the mirror quark masses, from Eq.(5) we get $m_{u_{H}^{i}}=m_{d_{H}^{i}}$ at $\mathcal O(\upsilon/f)$ and further assume $$m_{u_{H}^{1}}=m_{u_{H}^{2}}=m_{d_{H}^{1}}=m_{d_{H}^{2}}=M_{12},m_{u_{H}^{3}}=m_{d_{H}^{3}}=M_{3}$$
For the matrices $V_{Hu}$ and $V_{Hd}$, considering the constraints in Ref.[@18], we study the completely generic scenario, i.e.the six parameters of $V_{Hd}$ are arbitrary. After that, we follow Ref.[@19] to consider the following two scenarios for comparison:
Scenario I$:V_{Hd}=1,V_{Hu}=V_{CKM}^{\dag}$
Scenario II$
:S_{13}^{d}=0.5,\delta_{12}^{d}=\delta_{23}^{d}=0,\delta_{13}^{d}=\delta_{13}^{SM},S_{ij}^{d}=S_{ij}^{SM}$otherwise
Firstly, we discuss the $R_{b}$ changes with the LHT parameters, the numerical results are summarized in Fig.(2-3). To see the influence of the mixing parameter $x_{L}$ on the $R_{b}$, considering the existing constraints, we let the parameters vary randomly in the range: $M_{12}=300\sim3000GeV$, $M_{3}=300\sim3000GeV$, $f=400\sim3000GeV$. In these three scenarios, we can see the plots of $R_{b}$ decline with the $x_{L}$ increasing, which shows that the contribution of the mixing diagrams between $t$ and $T^{+}$ is negative and becomes larger with the $x_{L}$ increasing. When $x_{L}>0.7$, part of the plots are beyond the $2\sigma$ regions of its experimental value. This feature is similar in three different scenarios.
To see the influence of the first two generation mirror quarks mass $M_{12}$ on the $R_{b}$, considering the constraint from $R_{b}$ on the $x_{L}$, we let the parameters vary randomly in the range: $M_{3}=300\sim3000GeV$, $f=400\sim3000GeV$, $x_{L}=0.1\sim0.7$. In these three scenarios, we can see the plots of $R_{b}$ are almost in the $2\sigma$ regions of its experimental value. The noticeable feature is that the $R_{b}$ isn’t sensitive to $M_{12}$ so that the constraint from $R_{b}$ on $M_{12}$ is very loose.
Secondly, we discuss the $A_{FB}^{b}$ changes with the LHT parameters, the numerical results are summarized in Fig.(4-6). Same as the $R_{b}$, the $A_{FB}^{b}$ isn’t sensitive to $M_{12}$, so we don’t give the figures of the $A_{FB}^{b}$ as the function of $M_{12}$. To see the influence of the mixing parameter $x_{L}$ on the $A_{FB}^{b}$, we let the parameters vary randomly in the range: $M_{12}=300\sim3000GeV$, $M_{3}=300\sim3000GeV$, $f=400\sim3000GeV$. For the same reason, we can see the plots of $A_{FB}^{b}$ decline and become closer to the experimental central value with the $x_{L}$ increasing. However, the contribution of the new particles is not large enough so that the plots of the $A_{FB}^{b}$ are still entirely scattered between the $2\sigma$ and $3\sigma$ region of its experimental value.
To see the influence of the scale $f$ on the $A_{FB}^{b}$, we let the parameters vary randomly in the range: $M_{12}=300\sim3000GeV$, $M_{3}=300\sim3000GeV$, $x_{L}=0.1\sim0.7$. We can see the plots of $A_{FB}^{b}$ are entirely between the $2\sigma$ and $3\sigma$ region of its experimental value. The plots of $A_{FB}^{b}$ become closer to the SM with the $f$ increasing, which shows that the contribution of the heavy particles decouples with the $f$ increasing.
To see the influence of the third generation mirror quarks mass $M_{3}$ on the $A_{FB}^{b}$, we let the parameters vary randomly in the range: $M_{12}=300\sim3000GeV$, $f=400\sim3000GeV$, $x_{L}=0.1\sim0.7$. We can see the plots of $A_{FB}^{b}$ are entirely between the $2\sigma$ and $3\sigma$ region of its experimental value.
Finally, we discuss the $Zbb$ couplings in the LHT model. In our calculation, we still consider the above three scenarios and let the parameters vary randomly in the range: $M_{12}=300\sim3000GeV$, $M_{3}=300\sim3000GeV$, $f=400\sim3000GeV$, $x_{L}=0.1\sim0.7$, the numerical results are summarized in Figs.(7-8). We confirm the result of Ref.[@18], in which the correction from the mixing diagrams between $t$ and $T^{+}$ to $Zb\bar{b}$ couplings is mainly on the $g_{L}^{b}$ and doesn’t have the correct sign to alleviate the large deviation between theoretical predictions and experimental values. The plots scatter beyond the $3\sigma$ region their experimental values are mainly caused by these couplings. Furthermore, the correction on the $g_{R}^{b}$ is very small. However, there is a little difference when we consider the contributions involve other new particles. At this time, we can see part of the plots scatter in the $3\sigma$ internal region of their experimental values, where the deviation of $g_{L}^{b}$ can be alleviated. Unfortunately, the correction on the $g_{R}^{b}$ is still very small and the plots still scatter near the $3\sigma$ region of their experimental values so that the large deviation between theoretical predictions and experimental values can’t be explained. The similar results are found on the $g_{A}^{b}$ and $g_{V}^{b}$.
Conclusions
===========
In this paper,we studied the one-loop contributions of the new particles to the $R_{b}$ and $A_{FB}^{b}$ for three different scenarios in the framework of the LHT model. From the scatter plots of $R_{b}$ versus $x_{L}$, the precision measurement data of $R_{b}$ can give strong constraint on the $x_{L}$. Considering this constraint, we can see $R_{b}$ isn’t sensitive to the mass of the first two generation mirror quarks. The relevant parameters are weakly constrained by the precision measurement data of $A_{FB}^{b}$. In the given parameters space, the large deviation of $A_{FB}^{b}$ can’t be explained reasonably. From our study, the LHT model can provide the correction to the $g_{L}^{b}$ and have small part of the parameter space to alleviate the deviation between theoretical predictions and experimental values. But the LHT model can’t provide the large correction to the $g_{R}^{b}$ so that the large deviation between the SM prediction predictions and experimental values of the $Zbb$ couplings can’t be alleviated substantially.\
**Acknowledgments**\
We would thank Junjie Cao and Lei Wu for useful discussions and providing the calculation programs. This work is supported by the National Natural Science Foundation of China under Grant Nos.10775039, 11075045, by Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20094104110001 and by HASTIT under Grant No.2009HASTIT004.
**Appendix A: The expression of the renormalization vertex $\hat{\Gamma}^{\mu}_{Zb\bar{b}}$** [@20]
$$\begin{aligned}
\hat{\Gamma}^{\mu}_{Zb\bar{b}}&=&\Gamma^{\mu}_{Zb\bar{b}}-ie\gamma^{\mu}(v_{b}-a_{b}\gamma_{5})\frac{C_{W}}{2S_{W}}
\delta Z_{ZA}-ieQ_{b}\gamma^{\mu}\frac{1}{2} \delta
Z_{ZA}\nonumber\\
&+&ie\gamma^{\mu}(v_{b}-a_{b}\gamma_{5})\delta
Z_{V}^{b}-ie\gamma^{\mu}\gamma_{5}(v_{b}-a_{b}\gamma_{5})\delta
Z_{A}^{b}\nonumber\end{aligned}$$
where $$\begin{aligned}
v_{b}\equiv\frac{I_{b}^{3}-2Q_{b}S_{W}^{2}}{2C_{W}S_{W}},\quad
a_{b}\equiv\frac{I_{b}^{3}}{2C_{W}S_{W}},
\quad I_{b}^{3}=-\frac{1}{2},\quad Q_{b}=-\frac{1}{3}~~~~~~~~~~~\\
\quad \delta Z_{ZA}=2\frac{\Sigma_{T}^{AZ}(0)}{M_{Z_{L}}^{2}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\delta
Z_{L}^{b}=Re\Sigma_{L}^{b}(m_{b}^{2})+m_{b}^{2}\frac{\partial}{\partial
P_{b}^{2}}Re[\Sigma_{L}^{b}(P_{b}^{2})+\Sigma_{R}^{b}(P_{b}^{2})+2\Sigma_{S}^{b}(P_{b}^{2})]|_{P_{b}^{2}=m_{b}^{2}}\\
\delta
Z_{R}^{b}=Re\Sigma_{R}^{b}(m_{b}^{2})+m_{b}^{2}\frac{\partial}{\partial
P_{b}^{2}}Re[\Sigma_{L}^{b}(P_{b}^{2})+\Sigma_{R}^{b}(P_{b}^{2})+2\Sigma_{S}^{b}(P_{b}^{2})]|_{P_{b}^{2}=m_{b}^{2}}\\
\delta Z_{V}^{b}=\frac{1}{2}(\delta Z_{L}^{b}+\delta
Z_{R}^{b}),\delta Z_{A}^{b}=\frac{1}{2}(\delta Z_{L}^{b}-\delta
Z_{R}^{b})~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{aligned}$$ $$\begin{aligned}
\hat{\Gamma}^{LHT,\mu}_{Zb\bar{b}}&=&\Gamma^{\mu}_{Zb\bar{b}}(\pi^{\pm})+\Gamma^{\mu}_{Zb\bar{b}}(\eta)+
\Gamma^{\mu}_{Zb\bar{b}}(\omega^{0})+\Gamma^{\mu}_{Zb\bar{b}}(\omega^{\pm})+
\Gamma^{\mu}_{Zb\bar{b}}(W_{L}^{\pm})+\Gamma^{\mu}_{Zb\bar{b}}(A_{H})+\Gamma^{\mu}_{Zb\bar{b}}(Z_{H})\\
&+&\Gamma^{\mu}_{Zb\bar{b}}(W_{H}^{\pm})+\Gamma^{\mu}_{Zb\bar{b}}(\pi^{\pm},W_{L}^{\pm})
+\Gamma^{\mu}_{Zb\bar{b}}(\omega^{\pm},W_{H}^{\pm})
+\delta\Gamma^{\mu}_{Zb\bar{b}}(\pi^{\pm})+\delta\Gamma^{\mu}_{Zb\bar{b}}(\eta)+
\delta\Gamma^{\mu}_{Zb\bar{b}}(\omega^{0})\\
&+&\delta\Gamma^{\mu}_{Zb\bar{b}}(\omega^{\pm})+\delta\Gamma^{\mu}_{Zb\bar{b}}(W_{L}^{\pm})
+\delta\Gamma^{\mu}_{Zb\bar{b}}(A_{H})+\delta\Gamma^{\mu}_{Zb\bar{b}}(Z_{H})+
\delta\Gamma^{\mu}_{Zb\bar{b}}(W_{H}^{\pm})\end{aligned}$$
**Appendix B: The explicit expressions of the $\delta
g_{L,R}^{LHT}$**
They can be represented in form of 1-point, 2-point and 3-point standard functions $A,B_{0},B_{1},C_{ij}$. Here $P_{b}$ and $\bar{P_{b}}$ are outgoing. In all expressions, the mass of b-quark is ignored. $$\begin{aligned}
\delta
g_{L}&&=\frac{1}{16\pi^{2}}g^{2}C_{W}^{2}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}m_{u_{H}^{i}}^{2}
C_{0}^{a}\\
&&-\frac{1}{16\pi^{2}}\frac{g'^{2}}{100M_{A_{H}}^{2}}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}
\{(-\frac{1}{2}+\frac{1}{3}S_{W}^{2})[m_{d_{H}^{i}}^{4} C_{0}^{b}
-m_{d_{H}^{i}}^{2}M_{Z_{L}}^{2}C_{12}^{b}
-m_{d_{H}^{i}}^{2}M_{Z_{L}}^{2}C_{23}^{b}\\
&&-2m_{d_{H}^{i}}^{2}C_{24}^{b}+\frac{1}{2}m_{d_{H}^{i}}^{2}]-\frac{1}{2}[\frac{1}{2}m_{d_{H}^{i}}^{2}B_{0}
(-P_{b},m_{d_{H}^{i}},M_{A_{H}})\\
&&+\frac{1}{2}m_{d_{H}^{i}}^{2}(m_{d_{H}^{i}}^{2}-M_{A_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}
(-P_{b},m_{d_{H}^{i}},M_{A_{H}})]\\&&-\frac{1}{3}S_{W}^{2}[-\frac{1}{2}m_{d_{H}^{i}}^{2}B_{0}
(-P_{b},m_{d_{H}^{i}},M_{A_{H}})-\frac{1}{2}m_{d_{H}^{i}}^{2}(m_{d_{H}^{i}}^{2}-M_{A_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0} (-P_{b},m_{d_{H}^{i}},M_{A_{H}})]\}\\
&&-\frac{1}{16\pi^{2}}\frac{g^{2}}{4M_{Z_{H}}^{2}}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}\{(-\frac{1}{2}+\frac{1}{3}S_{W}^{2})[m_{d_{H}^{i}}^{4}
C_{0}^{c}-m_{d_{H}^{i}}^{2}M_{Z_{L}}^{2}C_{12}^{c}-m_{d_{H}^{i}}^{2}M_{Z_{L}}^{2}C_{23}^{c}\\
&&-2m_{d_{H}^{i}}^{2}C_{24}^{c}+\frac{1}{2}m_{d_{H}^{i}}^{2}]-\frac{1}{2}[\frac{1}{2}m_{d_{H}^{i}}^{2}B_{0}
(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})\\&&+\frac{1}{2}m_{d_{H}^{i}}^{2}(m_{d_{H}^{i}}^{2}-M_{Z_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}
(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})]-\frac{1}{3}S_{W}^{2}[-\frac{1}{2}m_{d_{H}^{i}}^{2}B_{0}
(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})\\&&-\frac{1}{2}m_{d_{H}^{i}}^{2}(m_{d_{H}^{i}}^{2}-M_{Z_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0} (-P_{b},m_{d_{H}^{i}},M_{Z_{H}})]\}\\
&&-\frac{1}{16\pi^{2}}\frac{g^{2}}{2M_{W_{H}}^{2}}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}\{(\frac{1}{2}-\frac{2}{3}S_{W}^{2})[m_{u_{H}^{i}}^{4}
C_{0}^{d}-m_{u_{H}^{i}}^{2}M_{Z_{L}}^{2}C_{12}^{d}
-m_{u_{H}^{i}}^{2}M_{Z_{L}}^{2}C_{23}^{d}\\
&&-2m_{u_{H}^{i}}^{2}C_{24}^{d}+\frac{1}{2}m_{u_{H}^{i}}^{2}]-\frac{1}{2}[\frac{1}{2}m_{u_{H}^{i}}^{2}B_{0}
(-P_{b},m_{u_{H}^{i}},M_{W_{H}})\\&&+\frac{1}{2}m_{u_{H}^{i}}^{2}(m_{u_{H}^{i}}^{2}-M_{W_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}
(-P_{b},m_{u_{H}^{i}},M_{W_{H}})]-\frac{1}{3}S_{W}^{2}[-\frac{1}{2}m_{u_{H}^{i}}^{2}B_{0}
(-P_{b},m_{u_{H}^{i}},M_{W_{H}})\\&&-\frac{1}{2}m_{u_{H}^{i}}^{2}(m_{u_{H}^{i}}^{2}-M_{W_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}
(-P_{b},m_{u_{H}^{i}},M_{W_{H}})]+2C_{W}^{2}m_{u_{H}^{i}}^{2}C_{24}^{e}\}\\
&&-\frac{1}{16\pi^{2}}\frac{g'^{2}}{100}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}
\{(-\frac{1}{2}+\frac{1}{3}S_{W}^{2})[-2m_{d_{H}^{i}}^{2}C_{0}^{f}
+2M_{Z_{L}}^{2}C_{11}^{f} +2M_{Z_{L}}^{2}C_{23}^{f}\\
&&+4C_{24}^{f}-2]+[\frac{1}{2}B_{0}(-P_{b},m_{d_{H}^{i}},M_{A_{H}})+\frac{1}{2}(m_{d_{H}^{i}}^{2}-M_{A_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}(-P_{b},m_{d_{H}^{i}},M_{A_{H}})\\&&-\frac{1}{3}S_{W}^{2}B_{0}(-P_{b},m_{d_{H}^{i}},M_{A_{H}})-\frac{1}{3}
S_{W}^{2}(m_{d_{H}^{i}}^{2}-M_{A_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}
(-P_{b},m_{d_{H}^{i}},M_{A_{H}})]-\frac{1}{2}+\frac{1}{3}S_{W}^{2}\}\end{aligned}$$ $$\begin{aligned}
&&-\frac{1}{16\pi^{2}}\frac{g^{2}}{4}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}
\{(-\frac{1}{2}+\frac{1}{3}S_{W}^{2})[-2m_{d_{H}^{i}}^{2}C_{0}^{g}
+2M_{Z_{L}}^{2}C_{11}^{g}
+2M_{Z_{L}}^{2}C_{23}^{g}\\
&&+4C_{24}^{g}-2]+[\frac{1}{2}B_{0}(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})+\frac{1}{2}(m_{d_{H}^{i}}^{2}-M_{Z_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})\\&&-\frac{1}{3}S_{W}^{2}B_{0}(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})-\frac{1}{3}
S_{W}^{2}(m_{d_{H}^{i}}^{2}-M_{Z_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0} (-P_{b},m_{d_{H}^{i}},M_{Z_{H}})]-\frac{1}{2}+\frac{1}{3}S_{W}^{2}\}\\
&&+\frac{1}{16\pi^{2}}\frac{g^{2}}{2}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}
(\frac{1}{2}-\frac{2}{3}S_{W}^{2})[2m_{u_{H}^{i}}^{2}C_{0}^{h}
+2M_{Z_{L}}^{2}C_{11}^{h}+2M_{Z_{L}}^{2}C_{23}^{h}+4C_{24}^{h}-2]\\
&&+\frac{1}{16\pi^{2}}\frac{g^{2}}{2}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}
C_{W}^{2}[-2M_{Z_{L}}^{2}C_{0}^{i}
-2M_{Z_{L}}^{2}C_{11}^{i}-2M_{Z_{L}}^{2}C_{23}^{i}-12C_{24}^{i}+2]\\
&&+\frac{1}{16\pi^{2}}\frac{g^{2}}{2}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}[\frac{1}{2}B_{0}(-P_{b},m_{u_{H}^{i}},M_{W_{H}})
+\frac{1}{2}(m_{u_{H}^{i}}^{2}-m_{W_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}(-P_{b},m_{u_{H}^{i}},M_{W_{H}})\\&&-\frac{1}{3}S_{W}^{2}B_{0}(-P_{b},m_{u_{H}^{i}},M_{W_{H}})-\frac{1}{3}
S_{W}^{2}(m_{u_{H}^{i}}^{2}-M_{W_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0} (-P_{b},m_{u_{H}^{i}},M_{W_{H}})-\frac{1}{2}+\frac{1}{3}S_{W}^{2}]\\
&&+\frac{1}{16\pi^{2}}\frac{g^{2}}{M_{Z_{L}}^{2}}C_{W}^{2}[-2A(M_{W_{H}})+2M_{W_{H}}^{2}B_{0}(0,M_{W_{H}},M_{W_{H}})+2M_{W_{H}}^{2}+M_{Z_{L}}^{2}B_{0}(0,M_{W_{H}},M_{W_{H}})]\\
&&-\frac{1}{16\pi^{2}}\frac{2g^{2}}{M_{Z_{L}}^{2}}\{\frac{2}{3}(\frac{1}{2}-\frac{2}{3}S_{W}^{2})[-\frac{2}{3}A(m_{u_{H}^{i}})
+\frac{2}{3}m_{u_{H}^{i}}^{2}B_{0}(0,m_{u_{H}^{i}},m_{u_{H}^{i}})+\frac{2}{3}m_{u_{H}^{i}}^{2}]\\
&&-\frac{1}{3}(-\frac{1}{2}+\frac{1}{3}S_{W}^{2})[-\frac{2}{3}A(m_{d_{H}^{i}})+\frac{2}{3}m_{d_{H}^{i}}^{2}B_{0}(0,m_{d_{H}^{i}},m_{d_{H}^{i}})
+\frac{2}{3}m_{d_{H}^{i}}^{2}]\\
&&-(-\frac{1}{2}+S_{W}^{2})[-\frac{2}{3}A(m_{l_{H}^{i}})+\frac{2}{3}m_{l_{H}^{i}}^{2}B_{0}(0,m_{l_{H}^{i}},m_{l_{H}^{i}})
+\frac{2}{3}m_{l_{H}^{i}}^{2}]\}\\
&&+\frac{g^2x^2_{L}}{4M^2_{W_{L}}}(1-2S_{W}^2)\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[-2m^2_{T^{+}}C_{24}^{j}]\\
&&+\frac{g^2x^2_{L}}{2M^2_{W_{L}}}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[\frac{2}{3}S_{W}^2 m^4_{T^{+}}C_{0}^{k}-m^2_{T^{+}}M_{Z_{L}}^{2}C_{12}^{k}-\frac{2}{3}S_{W}^2 m^2_{T^{+}}C_{23}^{k}\\
&&-\frac{4}{3}S_{W}^2 C_{24}^{k}+\frac{1}{3}S_{W}^2 m^2_{T^{+}}]\\
&&+\frac{g^2x^2_{L}}{4M^2_{W_{L}}}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}\{\frac{1}{2}m^2_TB_0(-P_b,m_{T^{+}},M_{W_{L}})\\
&&+\frac{1}{2}m^2_{T^{+}}(m^2_{T^{+}} -M^2_{W_L})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{T^{+}},M_{W_{L}})\\
&&-\frac{1}{3}S_{W}^2 [m^2_{T^{+}}B_0(-P_b,m_{T^{+}},M_{W_{L}})+ m^2_{T^{+}}(m^2_{T^{+}}-M^2_{W_L})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{T^{+}},M_{W_{L}})]\}\\
&&+\frac{g^2x^2_{L}}{4M^2_{W_{L}}}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[-m^2_{T^{+}}m^2_tC_{0}^{l}]\\
&&+\frac{g^2x^2_{L}}{4M^2_{W_{L}}}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[-m^2_{T^{+}}m^2_tC_{0}^{m}]\\
&&+\frac{g^2x^2_{L}}{2}C_{W}^2
\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[-2M_{Z_{L}}^{2}C_{0}^{j}-2M_{Z_{L}}^{2}C_{11}^{j}-2M_{Z_{L}}^{2}C_{23}^{j}-12C_{24}^{j}+2]\\
&&+\frac{g^2x^2_{L}}{2}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}\frac{2}{3}S_{W}^2 [m^2_{T^{+}}C_{0}^{k}-2M_{Z_{L}}^{2}C_{11}^{k}-2M_{Z_{L}}^{2}C_{23}^{k}-\frac{4}{3}C_{24}^{k}+2]\\\end{aligned}$$
$$\begin{aligned}
&&+\frac{g^2x^2_{L}}{4}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[2M_{Z_{L}}^{2}C_{11}^{l}+2M_{Z_{L}}^{2}C_{23}^{l}+4C_{24}^{l}-2]\\
&&+\frac{g^2x^2_{L}}{4}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[2M_{Z_{L}}^{2}C_{11}^{m}+2M_{Z_{L}}^{2}C_{23}^{m}+4C_{24}^{m}-2]\\
&&+\frac{g^2x^2_{L}}{2}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[\frac{1}{2}B_0(-P_b,m_{T^{+}},M_{W_L})\\
&&+(m^2_{T^{+}} -M^2_{W_L})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{T^{+}},M_{W_L})-\frac{1}{3}S_{W}^2 B_0(-P_b,m_{T^{+}},M_{W_L})\\
&&-\frac{1}{3}S_{W}^2 (m^2_{T^{+}}-M^2_{W_L})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{T^{+}},M_{W_L})-\frac{1}{2}+\frac{1}{3}S_{W}^2]\\
&&-\frac{g^2x^2_{L}}{2}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[\frac{2}{3}S_{W}^2 m^2_{T^{+}}C_{0}^{n}+2(1-\frac{2}{3}S_{W}^2) M_{Z_{L}}^{2}C_{11}^{n}\\
&&+2(1-\frac{2}{3}S_{W}^2) M_{Z_{L}}^{2}C_{23}^{n}+4(1-\frac{2}{3}S_{W}^2)C_{24}^{n}-2(1-\frac{2}{3}S_{W}^2)]\\
&&-\frac{g^2x^2_{L}}{2}C_{W}^2 \frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[-2M_{Z_{L}}^{2}C_{0}^{o}-2M_{Z_{L}}^{2}C_{11}^{o}-2M_{Z_{L}}^{2}C_{23}^{o}-12C_{24}^{o}+2]\\
&&-\frac{g^2x^2_{L}}{2}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[\frac{1}{2}B_0(-P_b,m_{t},M_{W_L})+\frac{1}{2}(m^2_t
-M^2_{W_L})\frac{\partial}{\partial
P_{b}^{2}}B_0(-P_b,m_{t},M_{W_L})\\
&&-\frac{1}{3}S_{W}^2 B_0(-P_b,m_{t},M_{W_L})
-\frac{1}{3}S_{W}^2 (m^2_t-M^2_{W_L})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{t},M_{W_L})-\frac{1}{2}+\frac{1}{3}S_{W}^2]\\
&&-\frac{g^2x^2_{L}}{2M^2_{W_{L}}}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[-m^4_t(1-\frac{2}{3}S_{W}^2)C_{0}^{o}-m^2_tM_{Z_{L}}^{2}C_{12}^{o}\\
&&-\frac{2}{3}S_{W}^2 m^2_tM_{Z_{L}}^{2}C_{23}^{o}-\frac{4}{3}m^2_tS_{W}^2 C_{24}^{o}(\Bar{P_b},P_b,m_{t},M_{W_{L}},m_{t})+\frac{1}{3}S_{W}^2 m^2_t]\\
&&-\frac{g^2x^2_{L}}{4M^2_{W_L}}(1-2S_{W}^2)\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[-2m^2_tC_{24}^{o}]\\
&&-\frac{g^2x^2_{L}}{4M^2_{W_L}}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}m^2_t\{\frac{1}{2}B_0(-P_b,m_{t},M_{W_{L}})\\
&&+\frac{1}{2}(m^2_t -M^2_{W_{L}})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{t},M_{W_{L}})\\
&&-\frac{1}{3}S_{W}^2[ B_0(-P_b,m_{t},M_{W_{L}})+(m^2_t-M^2_{W_{L}})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{t},M_{W_{L}})]\}\\
&&-\frac{g^2x^2_{L}}{2} \frac{v^2}{f^2}S_{W}^2(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[m^2_{T^{+}}C_{0}^{j}(\Bar{P_b},P_b,M_{W_{L}},m_{T^{+}},M_{W_L})]\\
&&-\frac{g^2x^2_{L}}{2} \frac{v^2}{f^2}S_{W}^2(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[m^2_{T^{+}}C_{0}^{j}(\Bar{P_b},P_b,M_{W_L},m_{T^{+}},M_{W_{L}})]\\
&&+\frac{g^2x^2_{L}}{2} \frac{v^2}{f^2}S_{W}^2(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[m^2_tC_{0}^{o}(\Bar{P_b},P_b,M_{W_{L}},m_{t},M_{W_L})]\\
&&+\frac{g^2x^2_{L}}{2} \frac{v^2}{f^2}S_{W}^2(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}[m^2_tC_{0}^{o}(\Bar{P_b},P_b,M_{W_L},m_{t},M_{W_{L}})]\\\end{aligned}$$
$$\begin{aligned}
C_{ij}^{a}&=&C_{ij}^{a}(\bar{P_{b}},P_{b},M_{W_{H}},m_{u_{H}^{i}},M_{W_{H}})\\
C_{ij}^{b}&=&C_{ij}^{b}(\bar{P_{b}},P_{b},m_{d_{H}^{i}},M_{A_{H}},m_{d_{H}^{i}})\\
C_{ij}^{c}&=&C_{ij}^{c}(\bar{P_{b}},P_{b},m_{d_{H}^{i}},M_{Z_{H}},m_{d_{H}^{i}})\\
C_{ij}^{d}&=&C_{ij}^{d}(\bar{P_{b}},P_{b},m_{u_{H}^{i}},M_{W_{H}},m_{u_{H}^{i}})\\
C_{ij}^{e}&=&C_{ij}^{e}(\bar{P_{b}},P_{b},M_{W_{H}},m_{u_{H}^{i}},M_{W_{H}})\\
C_{ij}^{f}&=&C_{ij}^{f}(\bar{P_{b}},P_{b},m_{d_{H}^{i}},M_{A_{H}},m_{d_{H}^{i}})\\
C_{ij}^{g}&=&C_{ij}^{g}(\bar{P_{b}},P_{b},m_{d_{H}^{i}},M_{Z_{H}},m_{d_{H}^{i}})\\
C_{ij}^{h}&=&C_{ij}^{h}(\bar{P_{b}},P_{b},m_{u_{H}^{i}},M_{W_{H}},m_{u_{H}^{i}})\\
C_{ij}^{i}&=&C_{ij}^{i}(\bar{P_{b}},P_{b},M_{W_{H}},m_{u_{H}^{i}},M_{W_{H}})\\
C_{ij}^{j}&=&C_{ij}^{j}(\bar{P_b},P_b,M_{W_{L}},m_{T^{+}},M_{W_{L}})\\
C_{ij}^{k}&=&C_{ij}^{k}(\Bar{P_b},P_b,m_{T^{+}},M_{W_{L}},m_{T^{+}})\\
C_{ij}^{l}&=&C_{ij}^{l}(\Bar{P_b},P_b,m_{t},M_{W_{L}},m_{T^{+}})\\
C_{ij}^{m}&=&C_{ij}^{m}(\Bar{P_b},P_b,m_{T^{+}},M_{W_L},m_{t})\\
C_{ij}^{n}&=&C_{ij}^{n}(\Bar{P_b},P_b,m_{t},M_{W_L},m_{t})\\
C_{ij}^{o}&=&C_{ij}^{o}(\Bar{P_b},P_b,M_{W_L},m_{t},M_{W_L})~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{aligned}$$
$$\begin{aligned}
\delta
g_{R}&=&-\frac{1}{16\pi^{2}}\frac{g'^{2}}{100M_{A_{H}}^{2}}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}\{\frac{1}{3}S_{W}^{2}m_{d_{H}^{i}}^{2}B_{1}
(-P_{b},m_{d_{H}^{i}},M_{A_{H}})\\&&-\frac{1}{3}S_{W}^{2}[-\frac{1}{2}m_{d_{H}^{i}}^{2}B_{0}
(-P_{b},m_{d_{H}^{i}},M_{A_{H}})
-\frac{1}{2}m_{d_{H}^{i}}^{2}(m_{d_{H}^{i}}^{2}-M_{A_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}(-P_{b},m_{d_{H}^{i}},M_{A_{H}})]\}\\
&&-\frac{1}{16\pi^{2}}\frac{g^{2}}{4M_{Z_{H}}^{2}}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}\{\frac{1}{3}S_{W}^{2}m_{d_{H}^{i}}B_{1}
(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})\\&&-\frac{1}{3}S_{W}^{2}[-\frac{1}{2}m_{d_{H}^{i}}^{2}B_{0}
(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})
-\frac{1}{2}m_{d_{H}^{i}}^{2}(m_{d_{H}^{i}}^{2}-M_{Z_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})]\}\\
&&+\frac{1}{16\pi^{2}}\frac{g^{2}}{2M_{W_{H}}^{2}}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}\{\frac{1}{3}S_{W}^{2}m_{u_{H}^{i}}B_{1}
(-P_{b},m_{u_{H}^{i}},M_{W_{H}})\\&&-\frac{1}{3}S_{W}^{2}[-\frac{1}{2}m_{u_{H}^{i}}^{2}B_{0}
(-P_{b},m_{u_{H}^{i}},M_{W_{H}})
-\frac{1}{2}m_{u_{H}^{i}}^{2}(m_{u_{H}^{i}}^{2}-M_{W_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}(-P_{b},m_{u_{H}^{i}},M_{W_{H}})]\}\\
&&-\frac{1}{16\pi^{2}}\frac{g'^{2}}{100}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}\{\frac{2}{3}S_{W}^{2}B_{1}
(-P_{b},m_{d_{H}^{i}},M_{A_{H}})+\frac{1}{3}S_{W}^{2}B_{0}(-P_{b},m_{d_{H}^{i}},M_{A_{H}})\\
&&+\frac{1}{3}S_{W}^{2}(m_{d_{H}^{i}}^{2}-M_{A_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}(-P_{b},m_{d_{H}^{i}},M_{A_{H}})\}\\
&&-\frac{1}{16\pi^{2}}\frac{g^{2}}{4}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}\{\frac{2}{3}S_{W}^{2}B_{1}
(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})+\frac{1}{3}S_{W}^{2}B_{0}(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})\\
&&+\frac{1}{3}S_{W}^{2}(m_{d_{H}^{i}}^{2}-M_{Z_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}(-P_{b},m_{d_{H}^{i}},M_{Z_{H}})\}\\
&&-\frac{1}{16\pi^{2}}\frac{g^{2}}{2}(V_{Hd})^{\ast}_{i3}(V_{Hd})_{i3}\{\frac{2}{3}S_{W}^{2}B_{1}
(-P_{b},m_{u_{H}^{i}},M_{W_{H}})+\frac{1}{3}S_{W}^{2}B_{0}(-P_{b},m_{u_{H}^{i}},M_{W_{H}})\\
&&+\frac{1}{3}S_{W}^{2}(m_{u_{H}^{i}}^{2}-M_{W_{H}}^{2})\frac{\partial}{\partial
P_{b}^{2}}B_{0}(-P_{b},m_{u_{H}^{i}},M_{W_{H}})\}\\
&&+\frac{g^2x^2_{L}}{4M^2_{W_{L}}}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}\{-\frac{2}{3}S_{W}^2 m^2_{T^{+}}B_{1}(-P_b,m_{T^{+}},M_{W_{L}})\\
&&-\frac{2}{3}S_{W}^2[\frac{1}{2}m^2_{T^{+}}B_{0}(-P_b,m_{T^{+}},M_{W_{L}})+\frac{1}{2}m^2_{T^{+}}(m^2_{T^{+}}-M^2_{W_L})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{T^{+}},M_{W_{L}})]\}\\
&&+\frac{g^2x^2_{L}}{2}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}\{-\frac{2}{3}S_{W}^2 B_{1}(-P_b,m_{T^{+}},M_{W_{L}})\\
&&-\frac{1}{3}S_{W}^2[B_{0}(-P_b,m_{T^{+}},M_{W_{L}})+(m^2_{T^{+}}-M^2_{W_L})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{T^{+}},M_{W_{L}})]\}\\
&&-\frac{g^2x^2_{L}}{4M^2_{W_{L}}}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}\{-\frac{2}{3}S_{W}^2 m^2_tB_{1}(-P_b,m_{t},M_{W_{L}})\\
&&-\frac{1}{3}S_{W}^2[m^2_tB_{0}(-P_b,m_{t},M_{W_{L}})+m^2_t(m^2_t-M^2_{W_L})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{t},M_{W_{L}})]\}\\
&&-\frac{g^2x^2_{L}}{2}\frac{v^2}{f^2}(V_{CKM})^{\ast}_{tb}(V_{CKM})_{tb}\frac{1}{16\pi^{2}}\{-\frac{2}{3}S_{W}^2 B_{1}(-P_b,m_{t},M_{W_{L}})\\
&&-\frac{1}{3}S_{W}^2[B_{0}(-P_b,m_{t},M_{W_{L}})+(m^2_t-M^2_{W_L})\frac{\partial}{\partial P_{b}^{2}}B_0(-P_b,m_{t},M_{W_{L}})]\}\\\end{aligned}$$
\
N. Arkani-Hamed, A. G. Cohen, and H. Georgi, Phys. Lett. B 513, 232 (2001); N. Arkani-Hamed, et al., JHEP 0208, 020 (2002); JHEP 0208, 021 (2002); I. Low, W. Skiba, and D.Smith, Phys. Rev. D 66, 072001 (2002); D. E. Kaplan and M. Schmaltz, JHEP 0310, 039(2003). N. Arkani-Hamed, A. G. Cohen, E. Katz, and A. E. Nelson, JHEP 0207, 034 (2002); S.Chang, JHEP 0312, 057 (2003); T. Han, H. E. Logan, B. McElrath, and L. T. Wang, Phys.Rev. D 67, 095004 (2003); M. Schmaltz and D. Tucker-smith, Ann. Rev. Nucl. Part. Sci. 55,229 (2005). C.Csaki,J.Hubisz,G.D.Kribs,P.Meade,J.Terning , Phys. Rev. D 67, 115002 (2003); Phys. Rev. D 68, 035009 (2003); J. L. Hewett,F. J. Petriello, and T. G. Rizzo, JHEP 0310, 062 (2003); M. C. Chen and S. Dawson, Phys.Rev. D 70, 015003 (2004); M. C. Chen, et al., Mod. Phys. Lett. A 21, 621 (2006); W. Kilian and J. Reuter, Phys. Rev. D 70, 015004 (2004). G. Marandella, C. Schappacher and A. Strumia, Phys. Rev. D 72, 035014 (2005). H. C. Cheng and I. Low, JHEP 0309, 051 (2003); JHEP 0408, 061 (2004); I. Low, JHEP0410, 067 (2004); J. Hubisz and P. Meade, Phys. Rev. D 71, 035016 (2005). D. Comelli and J. P. Silva, Phys. Rev. D54(1996)1176; V. D. Barger, K. M. Cheung,P. Langaclar, Phys. Lett. B381(1996)226; P. Bamert, C. P. Burgess, J. M. Cline,D. London, and E. Nardi, Phys. Rev. D54(1996)4275; D. Atwood, L. Reina, and A.Soni, Phys. Rev. D54(1996)3296. The ALEPH, CDF, D0, DELPHI, L3, OPAL, SLD Collaborations,the LEP Electroweak Working Group,the Tevatron Electroweak Working Group,and the SLD electroweak and heavy flavour groups ,arXiv:0811.4682v1 \[hep-ex\] 28 Nov 2008. M.Blanke, et al., Phys. Lett. B 646, 253 (2007). V.A.Novikov, L.B.Okun,A.N.Rozanov,M.I.Vysotsky, Rept.Prog.Phys. 62.1275-1332(1999); Morris L. Swartz, Int.J.Mod.Phys. A15S1 307-332(2000). M. Boulware and D. Finnell, Phys. Rev D44, 2054 (1991);Junjie Cao, Zhaohua Xiong and Jin Min Yang, Phys. Rev. Lett. 88, 111802 (2002). N.Gray,David J. Broadhurst, W. Grafe, K. Schilcher., Z.Phys. C48:673-680 (1990) 673;L.R. Surguladze,Phys.Lett. B341.60-72(1994). M.E. Peskin and T. Takeuchi, Phys. Rev. D46,381(1992). The ALEPH,DELPHI, L3, OPAL, SLD Collaborations,the LEP Electroweak Working Group,the SLD Electroweak and Heavy Flavour Groups,Phys.Rept.427:257(2006). G. ¡¯t Hooft and M. J. G. Veltman,Nucl. Phys. B 153, 365 (1979). T. Hahn and M. Perez-Victoria, Computl. Phys. Commun. 118, 153 (1999); T. Hahn, Nucl. Phys. Proc. Suppl. 135, 333 (2004). M.Blanke, et al., JHEP 0701:066 (2007). C. Amsler, etal., Phys. Lett. B 667,1 (2008). J.Hubisz, P.Meade, A.Noble, M.Perelstein,JHEP 0601:135(2006). J.Hubisz, S. J. Lee, and G. Paz, JHEP 0606:041 (2006). W.F.L.Hollik,Preprint,DESY 88-188(1988); A.Denner, Fortschr.Phys.41: 307-420 (1993).
|
---
abstract: 'A classical problem that arises in numerous signal processing applications asks for the reconstruction of an unknown, $k$-sparse signal $\x_0\in \R^n$ from underdetermined, noisy, linear measurements $\y=\A\x_0+\z\in \R^m$. One standard approach is to solve the following convex program $\hat\x=\arg\min_\x \|\y-\A\x\|_2 + \la \|\x\|_1$, which is known as the $\ell_2$-LASSO. We assume that the entries of the sensing matrix $\A$ and of the noise vector $\z$ are i.i.d Gaussian with variances $1/m$ and $\sigma^2$. In the large system limit when the problem dimensions grow to infinity, but in constant rates, we *precisely* characterize the limiting behavior of the normalized squared error $\|\hat\x-\x_0\|^2_2/\sigma^2$. Our numerical illustrations validate our theoretical predictions.'
address: |
\* Department of Electrical Engeeniring, Caltech, Pasadena, USA\
$\dagger$ Signal Processing Group, Chalmers University of Technology, Gothenburg, Sweden
bibliography:
- 'compbib.bib'
title: 'Precise Error Analysis of the $\ell_2$-LASSO'
---
|
---
author:
- Bernardino Spisso
---
[99]{}
H. Weyl, *The theory of Groups and Quantum Mechanics*, translation of *Gruppentheorie und Quantemmechanik*, . E.P. Wigner, *On the Quantum Correction For Thermodynamic Equilibrium*, Phys. Rev. 40 (1932) 749. P.Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, *Quantum Fields and Strings: A Course for Mathematicians Vol.2*, American Mathematical Society (1999). J. Zinn-Justin, *Quantum Field Theory and Critical Phenomena*, Oxford University Press, Second edition (1993). N. Seiberg and E. Witten, *String Theory and Noncommutative Geometry*, JHEP 9909 (1999) 032, \[hep-th/9908142\]. G. S. Agarwal and E. Wolf, *Calculus For Functions Of Noncommuting Operators And General Phase-Space Methods In Quantum Mechanics. Mapping Theorems And Ordering Of Functions Of Non-commuting Operators*, Phys. Rev. D **2** (1970) 2161. J.C. Varilly, J.M. Gracia-Bondia, *On the ultraviolet behaviour of quantum fields over noncommutative manifolds*, Int.J.Mod.Phys. A14 (1999) 1305, \[hep-th/9804001\]. M. Chaichian, A. Demichev and P. Presnajder, *Quantum Field Theory on Noncommutative Space-Times and the Persistence of Ultraviolet Divergences*, \[hep-th/9812180\]. S. Doplicher, K. Fredenhagen, J.E. Roberts, *The Quantum Structure of Space-Time at the Planck Scale and Quantum Fields*, Phys.Lett.B331:39-44,(1994). M.R. Douglas, *D-Geometry and Noncommutative Geometry*, \[hep-th/9901146\]. C.P. Martin and D. Sanchez-Ruiz, *The One-loop UV Divergent Structure of U(1) Yang-Mills Theory on Noncommutative $ \mathbb{R}^4$*, Phys.Rev.Lett. 83 (1999) 476-479, \[hep-th/9903077\]. T. Filk, *Divergencies in a field theory on quantum space*, Phys.Lett. B376 (1996) 53-58. T. Krajewski, R. Wulkenhaar,*Perturbative quantum gauge fields on the noncommutative torus*, \[hep-th/9903187\]. S. Minwalla, M. Van Raamsdonk, N. Seiberg, *Noncommutative perturbative dynamics*, JHEP 0002 (2000) 020 \[arXiv:hep-th/9912072\]. I. Chepelev, R. Roiban,*Convergence theorem for non-commutative Feynman graphs and renormalization*, JHEP 0103 (2001) 001 \[arXiv:hep-th/0008090\]. A. Matusis, L. Susskind and N. Toumbas, *The IR/UV connection in the non-commutative gauge theories*, JHEP 0012 (2000) 002 \[arXiv:hep-th/0002075\]. E. Langmann and R. J. Szabo, *Duality in scalar field theory on noncommutative phase spaces*, Phys. Lett. B533 (2002) 168–177, \[hep-th/0202039\]. V. Rivasseau, F. Vignes-Tourneret, R. Wulkenhaar, *Renormalization of non-commutative $\varphi^4$-theory by multi-scale analysis*, Phys. Lett. B533 (2002) 168–177, \[hep-th/0202039\]. V. Rivasseau, *Constructive Matrix Theory*, arXiv:0706.1224 \[hep-th\]. V. Rivasseau, F. Vignes-Tourneret and R. Wulkenhaar, *Renormalization of noncommutative $\varphi^4$-theory by multi-scale analysis*, Commun. Math. Phys. 262 (2006) 565 \[arXiv:hep-th/0501036\]. R. Gurau and V. Rivasseau, *Parametric representation of noncommutative field theory*,Commun. Math. Phys. 272 (2007) 811 \[arXiv:math-ph/0606030\]. J. Magnen and V. Rivasseau, *Constructive $\varphi^4$ field theory without tears*, arXiv:0706.2457\[math-ph\]. J. Madore, S. Schraml, P. Schupp, J. Wess, *Gauge theory on noncommutative spaces*, Eur. Phys. J. C 16 (2000) 161 \[arXiv:hep-th/0001203\]. J. Madore, *An Introduction to Noncommutative Differential Geometry and its Physical Applications*, Cambridge University Press (1995). J. C. T. Pool, *Mathematical Aspects of the Weyl Correspondence*, Phys. **7** (1), 66-76 (1996). H. Grönewold, *On principles of quantum mechanics*, Physica 12, (1946) 405. J.M. Gracia-Bondia, *Generalized Moyal quantization on Homogeneous Aymplectic Spaces*, Contemporary Mathematics, **134** (1992) 93. J.M. Gracia-Bondia, J.C. Varilly, H. Figueroa, *Elements of noncommutative geometry*,Birkhäuser Advanced Texts, Birkhäuser, Boston, (2001). J.M. Gracia-Bondia, J.C. Varilly, H. Figueroa, *Algebras of distributions suitable for phase-space quantum mechanics*, I, J. Math. Phys. 29, (1988). K. Gottfried, *Quantum mechanics Vol. I: Fundamentals*Benjamin, New York (1966). A. Connes, *Noncommutative Geometry*, Academic Press, Inc. (1994). A. Connes, *Geometry from the spectral point of view*, Lett. Math. Phys. 34 (1995), no. 3, 203–238. A. Connes, *On the spectral characterization of manifolds*, (2008) arXiv:0810.2088v1\[math.OA\]. A. Connes, *Essay on Physics and Non-commutative Geometry*, The Interface of Mathematics and Particle Physics, ed. D.G.Quillen et al. Clarendon press, Oxford, (1990). A. Connes, *Noncommutative geometry and reality*, J. Math. Phys. 36 (1995) 6194. A. Connes, *Gravity coupled with matter and the foundation of noncommutative geometry*, Comm. Math. Phys. 155 (1996) 109. A. Connes and J. Lott, *Particle models and noncommutative geometry (expanded version)*, Nucl. Phys. Proc. Suppl. 18B (1991) 29. A. Connes and J. Lott, *Particle Models and Noncommutative Geometry*, Nucl.Phys.B (Proc.Suppl.)18B (1990) 29-47. A. Connes, H. Moscovici, *The local index formula in noncommutative geometry*, GAFA 5 (1995) 174–243. H. Chamseddine and A. Connes, *The Spectral Action Principle*, Math.Phys.186:731-750 (1997). H. Chamseddine, Alain Connes, Matilde Marcolli, *Gravity and the standard model with neutrino mixing*, Adv.Theor.Math.Phys.11:991-1089, (2007) \[arXiv:hep-th/0610241v1\]. F. Lizzi, G. Mangano, G. Miele, and G. Sparano, *Fermion Hilbert space and fermion doubling in the noncommutative geometry approach to gauge theories*, Phys. Rev. D55:6357–6366, (1997). H. Grosse and R. Wulkenhaar, *Power-counting theorem for non-local matrix models and renormalisation*, \[arXiv:hep-th/0305066\]. H. Grosse and R. Wulkenhaar, *Renormalisation of $\phi^4$-theory on noncommutative $ \mathbb{R}^4$ in the matrix base*, Commun. Math. Phys. 256 (2005) 305 \[arXiv:hep-th/0401128\]. H. Grosse and R. Wulkenhaar, *$8D$-spectral triple on $4D$-Moyal space and the vacuum of noncommutative gauge theory*, arXiv:0709.0095\[hep-ht\]. R. Wulkenhaar, *Renormalisation of noncommutative $\varphi_4^4$-theory to all orders*, habilitation thesis, Vienna University of Technology (2005). W. Rudin, *Real and Complex Analysis*, McGraw-Hill, (1987). J. Dixmier, *Existence de traces non normals*, C.R. Acad. Sci. Paris, Ser. A-B, **262** (1966). G. Landi, *An Introduction to Noncommutative Spaces and Their Geometries*, Springer, Berlin Eidelberg(1997). T. Schücker, *Forces from Connes’ geometry*, Lect. Notes Phys. 659, 285–350 (2005). T. Schücker, *Spin group and almost commutative geometry*, hep-th/0007047. T. Schücker, *Forces from noncommutative geometry*, hep-th/0110068. V. Gayral, B. Iochum, *The spectral action for Moyal planes*, J. Math. Phys. 46 (2005). 043503 \[arXiv:hep-th/0402147\]. V. Gayral, J. H. Jureit, T. Krajewski and R. Wulkenhaar, *Quantum field theory on projectivemodules*, CPT-P67-2006 \[hep-th/0612048\]. V. Gayral, J. M. Gracia-Bondia. Iochum, T. Schücker and J. C. Varilly, *Moyal planes are spectral triples*, Commun. Math. Phys. 246 (2004) 569 \[arXiv:hep-th/0307241\]. V. Gayral, J. M. Gracia-Bondia and F. Ruiz, *Position-dependent noncommutative products: Classical construction and field theory*, Nucl. Phys. B 727 (2005) 513 \[arXiv:hep-th/0504022\]. V. Gayral and R. Wulkenhaar *In preparation*. B. De Witt, *Dynamical Theory of Groups and Fields*, New York, Gordon and Breach (1965). I. Gradshteyn and I. M. Ryzhik, *Table of integrals, series, and products*, Academic Press, San Diego, 6th ed., (2000). P. B. Gilkey, *Invariance theory, the heat equation, and the Atiyah-Singer index theorem*, volume 11 of Mathematics Lecture Series. Publish or Perish Inc., Wilmington, DE, (1984). R. P. Feynman and A. R. Hibbs, *Quantum mechanics and path integrals*, McGraw-Hill (1965). J. Zinn-Justin, *Quantum Field Theory and Critical Phenomena*, Oxford University Press, Second edition (1993). J. Glimm and A. Jaffe, *Quantum physics: A functional integral point of view*, Springer-Verlag (1981). K. Osterwalder and R. Schrader, *Axioms for Euclidean Green’ Functions*, Commun. Math. Phys. **31** (1973) 83; Commun. Math. Phys. **42** (1975) 281. G. Roepstorff, *Path Integral Approach to Quantum Physics*, Springer (1994). B. Simon, *Functional integration and quantum physics*, Academic Press, New York (1979). A. Rennie Joseph and C. Varilly *Reconstruction of manifolds in noncommutative geometry* , (2006) arXiv:math/0610418v4\[math.OA\]. A. Rennie *Commutative geometries are spin manifolds*, Rev. Math. Phys. 13 (2001). A. Rennie, *Smoothness and locality for nonunital spectral triples* , K-theory 28 (2003) 127. A. Carey, V. Gayral, A. Rennie, F. Sukochev, *Integration on locally compact noncommutative spaces*, \[arXiv:0912.2817\]. A. de Goursac, J.-C. Wallet, and R. Wulkenhaar, *On the vacuum states for noncommutative gauge theory*, Eur. Phys. J. C56 (2008) 293–304,arXiv:0803.3035 \[hep-th\]. I. S. Gradshteyn and I. M. Ryzhik, *Tables of Series, Produces, and Integrals. Sixth Edition*, Academic Press, San Diego (2000). N. Higson and J. Roe *Analytic K-Homology*, Oxford University Press, Oxford, (2000). P. N. Saeta, *The Metropolis Algorithm*, Physics 170 course notes. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth. A. H. Teller and E. Teller, *Equation of State Calculations by Fast Computing Machines*, J. Chem. Phys. 21 (1953) 1087-1092. I. Montvay and G. Münster, *Quantum Field Theory on a Lattice*, Cambridge University Press, (1997). M. E. Newman and G. T. Barkema, *Monte Carlo Methods in Statistical Physics* Oxford University Press (2002). Eric C. Anderson, *Monte Carlo Methods and Importance Sampling* Lecture Notes for Stat 578C Statistical Genetics. W. Janke, *Statistical Analysis of Simulations: Data Correlations and Error Estimation*, published in *Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms, Lecture Notes*, John von Neumann Institute for Computing, Jülich, NIC Series, Vol. 10, ISBN 3-00-009057-6 (2002) 423-445. D. P. Landau and K. Binde, *A guide to Monte Carlo Simulations in statistical Physics*Cambridge University Press, (2000). X. Martin, *A matrix phase for the $\phi^4$ scalar field on the fuzzy sphere*, JHEP 0404 (2004) 077 \[hep-th/0402230\]. J. Medina, *Fuzzy Scalar Field Theories: Numerical and Analytical Investigations*, arXiv:0801.1284v1 \[hep-th\]. N. Madras and A. D. Sokal, *The pivot algorithm: a highly efficient Monte Carlo method for the self-avoiding walk*, J. Stat. Phys. 50 (1988) 109. B. Efron, *The Jackknife, the Bootstrap and Other Resampling Plans*, Society for Industrial and Applied Mathematics \[SIAM\], Philadelphia, (1982). A.D. Sokal and L.E. Thomas, *Exponential convergence to equilibrium for a class of random-walk models* J. Stat. Phys. 54, 797 (1989). J. Ambjorn, B. Durhuus, T. Jonsson, *Quantum geometry: a statistical field theory approach*, Cambridge University Press (1997).
|
---
abstract: 'We study an oriented first passage percolation model for the evolution of a river delta. This model is exactly solvable and occurs as the low temperature limit of the beta random walk in random environment. We analyze the asymptotics of an exact formula from [@RWRE] to show that, at any fixed positive time, the width of a river delta of length $L$ approaches a constant times $L^{2/3}$ with Tracy-Widom GUE fluctuations of order $L^{4/9}$. This result can be rephrased in terms of particle systems. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations.'
address:
- 'G. Barraquand, Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA.'
- 'M. Rychnovsky, Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA.'
author:
- Guillaume Barraquand
- Mark Rychnovsky
title: 'Tracy-Widom asymptotics for a river delta model'
---
Model and results
=================
Introduction
------------
First passage percolation was introduced in 1965 to study a fluid spreading through a random environment [@hammersley1965first]. This model has motivated many tools in modern probability, most notably Kingman’s sub-additive ergodic theorem (see the review [@fppbook] and references therein); it has attracted attention from mathematicians and physicists alike due to the simplicity of its definition, and the ease with which fascinating conjectures can be stated.
The Kardar-Parisi-Zhang (KPZ) universality class has also become a central object of study in recent years [@universality]. Originally proposed to explain the behavior of growing interfaces in 1986 [@originalKPZ], it has grown to include many types of models including random matrices, directed polymers, interacting particle systems, percolation models, and traffic models. Much of the success in studying these has come from the detailed analysis of a few exactly solvable models of each type.
We study an exactly solvable model at the intersection of percolation theory and KPZ universality: Bernoulli-exponential first passage percolation (FPP). Here is a brief description (see Definition \[model\] for a more precise definition). Bernoulli-exponential FPP models the growth of a river delta beginning at the origin in ${\mathbb{Z}}_{\geq 0}^2$ and growing depending on two parameters $a,b>0$. At time $0$, the river is a single up-right path beginning from the origin chosen by the rule that whenever the river reaches a new vertex it travels north with probability $a/(a+b)$ and travels east with probability $b/(a+b)$ (thick black line in Figure \[fig:littlepicture\]). The line with slope $a/b$ can be thought of as giving the direction in which the expected elevation of our random terrain decreases fastest.
(0,0) node\[below left\][[$(0,0)$]{}]{}; (-.2,-.2) rectangle(14.4, 14.4); (0,0) grid(16,16); (4,3) – (5,3) – (6,3) – (7,3) – (7,4) – (8,4) – (8,5) – (9,5) – (9,6) – (10,6) – (11,6) – (11,7) – (11,8) – (11,9); (3,7) – (3,8) – (3,9) – (4,9) – (5,9) – (6,9) – (6,10) – (7,10) – (7,11) – (8,11) – (9,11) – (10,11) – (10,12) – (11,12) – (12,12) – (12,13) – (12,14) – (13,14) – (13,15) – (14,15); (2, 3) – (3,3) – (4,3) – (4,4) – (4,5) – (5,5) – (5,6) – (6,6) – (6,7); (0,0) – (0,1) – (0,2) – (1,2) – (1,3) – (2,3) – (2,4) – (2,5) – (2,6) – (3,6) – (3,7) – (4,7) –(5,7) –(6,7) –(7,7) – (8,7) – (8,8) – (9,8) – (9,9) – (10,9) – (11,9) – (12,9) – (12,10) – (13,10) – (14,10) – (14,11) – (14,12) – (14,13) – (15,13);
As time passes the river erodes its banks creating forks. At each vertex which the river leaves in the rightward (respectively upward) direction, it takes an amount of time distributed as an exponential random variable with rate $a$ (resp. $b$) for the river to erode through its upward (resp. rightward) bank. Once the river erodes one of its banks at a vertex, the flow at this vertex branches to create a tributary (see gray paths in Figure \[fig:littlepicture\]). The path of the tributary is selected by the same rule as the path of the time $0$ river, except that when the tributary meets an existing river it joins the river and follows the existing path. The full path of the tributary is added instantly when the river erodes its bank.
In this model the river is infinite, and the main object of study is the set of vertices included in the river at time $t$, i.e. the percolation cluster. We will also refer to the shape enclosed by the outermost tributaries at time $t$ as the river delta (see Figure \[largescaleBEFPP\] for a large scale illustration of the river delta).
The model defined above can also be seen as the low temperature limit of the beta random walk in random environment (RWRE) model [@RWRE], an exactly solvable model in the KPZ universality class. Bernoulli-exponential FPP is particularly amenable to study because an exact formula for the distribution of the percolation cluster’s upper border (Theorem \[exact\] below) can be extracted from an exact formula for the beta RWRE [@RWRE]. We perform an asymptotic analysis on this formula to prove that at any fixed time, the width of the river delta satisfies a law of large numbers type result with fluctuations converging weakly to the Tracy-Widom GUE distribution (see Theorem \[main theorem\]). Our law of large numbers result was predicted in [@RWRE] by taking a heuristic limit of [@RWRE Theorem 1.19]; we present this non-rigorous computation in Section \[main result\]. We also give other interpretations of this result. In Section \[interpretations\] we introduce an exactly solvable particle system and show that the position of a particle at finite time has Tracy-Widom fluctuations.
![The percolation cluster for $400 \times 400$ Bernoulli-exponential FPP at time $1$ with $a=b=1$. Paths occurring earlier are shaded darker, so the darkest paths occur near $t=0$ and the lightest paths occur near $t=1$.[]{data-label="largescaleBEFPP"}](percolation)
Definition of the model
-----------------------
We now define the model more precisely in terms of first passage percolation following [@RWRE].
\[model\] Let $E_e$ be a family of independent exponential random variables indexed by the edges $e$ of the lattice ${\mathbb{Z}}_{\geq0}^2$. Each $E_e$ is distributed as an exponential random variable with parameter $a$ if $e$ is a vertical edge, and with parameter $b$ if $e$ is a horizontal edge. Let $(\zeta_{i,j})$ be a family of independent Bernoulli random variables with parameter $b/(a+b)$. We define the passage time $t_e$ of each edge $e$ in the lattice ${\mathbb{Z}}_{\geq 0}^2$ by $$t_e=\begin{cases}
\zeta_{i,j} E_e \qquad \text{if $e$ is the vertical edge $(i,j) \to (i,j+1)$},\\
(1-\zeta_{i,j}) E_e \qquad \text{if $e$ is the horizontal edge $(i,j) \to (i+1,j)$}.\\
\end{cases}$$
We define the point to point passage time $T^{\mathrm{PP}}(n,m)$ by
$$T^{\mathrm{PP}}(n,m)=\min_{\pi:(0,0) \to (n,m)} \sum_{e \in \pi} t_e.$$ where the minimum is taken over all up-right paths from $(0,0)$ to $(n,m)$. We define the percolation cluster $C(t)$, at time $t$, by
$$C(t)=\left\{ (n,m): T^{\mathrm{PP}}(n,m) \leq t \right\}.$$ At each time $t$, the percolation cluster $C(t)$ is the set of points visited by a collection of up-right random walks in the quadrant ${\mathbb{Z}}_{\geq 0}^2$. $C(t)$ evolves in time as follows:
- At time $0$, the percolation cluster contains all points in the path of a directed random walk starting from $(0,0)$, because at any vertex $(i,j)$ we have passage time $0$ to either $(i,j+1)$ or $(i+1,j)$ according to the independent Bernoulli random variables $\zeta_{i,j}$.
- At each vertex $(i,j)$ in the percolation cluster $C(t)$, with an upward (resp. rightward) neighbor outside the cluster, we add a random walk starting from $(i,j)$ with an upward (resp. rightward) step to the percolation cluster with exponential rate $(a)$ (resp. $b$). This random walk will almost surely hit the percolation cluster after finitely many steps, and we add to the percolation cluster only those points that are in the path of the walk before the first hitting point (see Figure \[fig:littlepicture\]).
Define the height function $H_t(n)$ by $$H_t(n)=\sup \{ m \in {\mathbb{Z}}_{\geq 0}| T^{\mathrm{PP}}(n,m) \leq t)\},
\label{eq:defheighfunction}$$ so that $(n,H_t(n))$ is the upper border of $C(t)$.
History of the model and related results
----------------------------------------
Bernoulli-exponential FPP was first introduced in [@RWRE], which introduced an exactly solvable model called the beta random walk in random environment (RWRE) and studied Bernoulli-exponential FPP as a low temperature limit of this model (see also the physics works [@thiery2015integrable; @thiery2016exact] further studying the Beta RWRE and some variants). The beta RWRE was shown to be exactly solvable in [@RWRE] by viewing it as a limit of $q$-Hahn TASEP, a Bethe ansatz solvable particle system introduced in [@pov13]. The $q$-Hahn TASEP was further analyzed in [@borodin2015spectral; @cor14; @qhahnboson], and was recently realized as a degeneration of the higher spin stochastic six vertex model [@aggarwal2017dynamical; @borodin2017family; @borodin2018higher; @corwin2016stochastic], so that Bernoulli-exponential FPP fits as well in the framework of stochastic spin models.
Tracy-Widom GUE fluctuations were shown in [@RWRE] for Bernoulli-exponential FPP (see Theorem \[heuristic theorem\]) and for Beta RWRE. In the Beta RWRE these fluctuations occur in the quenched large deviation principle satisfied by the random walk and for the maximum of many random walkers in the same environment.
The connection to KPZ universality was strengthened in subsequent works. In [@corwin-gu] it was shown that the heat kernel for the time reversed Beta RWRE converges to the stochastic heat equation with multiplicative noise. In [@balazs-rassoul] it was shown using a stationary version of the model that a Beta RWRE conditioned to have atypical velocity has wandering exponent $2/3$ (see also [@chaumont2017fluctuation]), as expected in general for directed polymers in $1+1$ dimensions. The stationary structure of Bernoulli-exponential FPP was computed in [@thieryzerotemp] (In [@thieryzerotemp] Bernoulli-exponential FPP is referred to as the Bernoulli-exponential polymer).
The first occurrence of the Tracy-Widom distribution in the KPZ universality class dates back to the work of Baik, Deift and Johansson on longest increasing subsequences of random permutations [@baik1999distribution] (the connection to KPZ class was explained in e.g. [@prahofer2000universal]) and the work of Johansson on TASEP [@johansson2000shape]. In the past ten years, following Tracy and Widom’s work on ASEP [@tracy2008integral; @tracy2008fredholm; @asep] and Borodin and Corwin’s Macdonald processes [@borodin2014macdonald], a number of exactly solvable $1+1$ dimensional models in the KPZ universality class have been analyzed asymptotically. Most of them can be realized as more or less direct degenerations of the higher-spin stochastic six-vertex model. This includes particle systems such as exclusion processes (q-TASEP [@borodin2014duality; @barraquand2014phase; @qtasep; @orr2017stochastic] and other models [@barraquand2016q; @baik2017facilitated; @ghosalpush; @qhahnboson]), directed polymers ([@FreeEnergyCorwin; @borodin2013log; @borodin2015height; @corwin2014strict; @krishnan2016tracy; @o2014tracy]), and the stochastic six-vertex model [@aggarwal2016phase; @aggarwal2016current; @barraquand2017stochastic; @borodin2016stochastic; @borodin2016asep].
Main result {#main result}
-----------
The study of the large scale behavior of passage times $T^{\mathrm{PP}}(n,m)$ was initiated in [@RWRE]. At large times, the fluctuations of the upper border of the percolation cluster (described by the height function $H_t(n)$) has GUE Tracy-Widom fluctuations on the scale $n^{1/3}$.
\[[[@RWRE Theorem 1.19]]{}\]\[heuristic theorem\] Fix parameters $a,b>0$. For any $\theta>0$ and $x\in {\mathbb{R}}$, $$\lim_{n \to \infty} {\mathbb{P}}\left(\frac{H_{\tau(\theta)n} - \kappa(\theta)n}{\tilde\rho(\theta) n^{1/3}}\leq x \right)=F_{\mathrm{GUE}}(x),\label{eq:limittheoremlargetime}$$ where $F_{\textrm{GUE}}$ is the GUE Tracy-Widom distribution (see Definition \[tracywidomdist\]) and $\kappa(\theta)$, $\tau(\theta)$, $\tilde\rho(\theta)= \frac{\kappa'(\theta)}{\tau'(\theta)} \rho(\theta)$ are functions defined in [@RWRE] by $$\begin{aligned}
\kappa(\theta)&:=\frac{ \frac{1}{\theta^2}-\frac{1}{(a+\theta)^2}}{\frac{1}{(a+\theta)^2}-\frac{1}{(a+b+\theta)^2}},\\
\tau(\theta)&:=\frac{1}{a+\theta}-\frac{1}{\theta}+\kappa(\theta) \left( \frac{1}{a+\theta}-\frac{1}{a+b+\theta} \right)=\frac{a(a+b)}{\theta^2(2a+b+2\theta)},\\
\rho(\theta)&:=\left[ \frac{1}{\theta^3}-\frac{1}{(a+\theta)^3}+\kappa(\theta) \left(\frac{1}{(a+b+\theta)^3}-\frac{1}{(a+\theta)^3} \right) \right]^{1/3}.
\end{aligned}$$
Note that as $\theta$ ranges from $0$ to $\infty$, $\kappa(\theta)$ ranges from $+\infty$ to $a/b$ and $\tau(\theta)$ ranges from $+\infty$ to $0$.
In [@RWRE] the limit theorem is incorrectly stated as $$\lim_{n \to \infty} {\mathbb{P}}\left(\frac{\min_{i\leq n}T^{\textrm{PP}}(i, \kappa(\theta) n)-\tau(\theta) n}{\rho(\theta) n^{1/3}} \leq x \right)=F_{\mathrm{GUE}}(x),$$ but following the proof in [@RWRE Section 6.1], we can see that the inequality and the sign of $x$ should be reversed. Further, we have reinterpreted the limit theorem in terms of height function $H_t(n)$ instead of passage times $T^{\textrm{PP}}(n,m)$ using the relation .
In this paper, we are interested in the fluctuations of $H_t(n)$ for large $n$ but fixed time $t$. Let us scale $\theta$ in above as $$\theta = \left( \frac{na(a+b)}{2t} \right)^{1/3},$$ so that $$\tau(\theta)n = t +O(n^{-1/3}).$$ Let us introduce constants $$\lambda= \left( \frac{a(a+b)}{2t}\right)^{1/3}, \quad d=\frac{3a(a+b)}{2b \lambda}, \quad \sigma=\left(\frac{3a(a+b)\lambda}{2b^3}\right)^{1/3}. \label{constants}$$ Then, we have the approximations $$\begin{aligned}
\kappa(\theta)n &= \frac{a}{b}n + dn^{2/3} + o(n^{4/9}),\\
\tilde\rho(\theta)n^{1/3} &= \sigma n^{4/9} + o(n^{4/9}). \end{aligned}$$ Thus, formally letting $\theta$ and $n$ go to infinity in suggests that for a fixed time $t$, it is natural to scale the height function as $$H_{t}(n) = \frac{a}{b}n + dn^{2/3} + \sigma n^{4/9} \chi_n,$$ and study the asymptotics of the sequence of random variables $\chi_n$.
Our main result is the following.
\[main theorem\] Fix parameters $a,b>0$. For any $t>0$ and $x \in \mathbb{R}$, $$\lim_{n \to \infty} {\mathbb{P}}\left(\frac{H_t(n)-\frac{a}{b} n-d n^{2/3}}{\sigma n^{4/9}} \leq x\right) = F_{\textrm{GUE}}(x),$$ where $F_{\textrm{GUE}}$ is the GUE Tracy-Widom distribution.
Note that the heuristic argument presented above to guess the scaling exponents and the expression of constants $d$ and $\sigma$ is not rigorous, since Theorem \[heuristic theorem\] holds for fixed $\theta$. Theorem \[heuristic theorem\] could be extended without much effort to a weak convergence uniform in $\theta$ for $\theta$ varying in a fixed compact subset of $(0,+\infty)$. However the case of $\theta$ and $n$ simultaneously going to infinity requires more careful analysis. Indeed, for $\theta$ going to infinity very fast compared to $n$, Tracy-Widom fluctuations would certainly disappear as this would correspond to considering the height function at time $\tau(\theta)n\approx 0$, that is a simple random walk having Gaussian fluctuations on the $n^{1/2}$ scale. We explain in the next section how we shall prove Theorem \[main theorem\].
The scaling exponents in Theorem 2 might seem unusual, although the preceding heuristic computation explains how they result from rescaling a model which has the usual KPZ scaling exponents. A similar situation occurs for scaling exponents of the height function of directed last passage percolation in thin rectangles [@baik2005gue; @UniversalPropertyCloseToAxis] and for the free energy of directed polymers [@UniversalityRectanglesCorwin] under the same limit.
Outline of the Proof
--------------------
Recall that given an integral kernel ${\mathsf}{K}: \mathbb{C}^2 \to \mathbb{C}$, its Fredholm determinant is defined as $$\det(1+{\mathsf}{K})_{L^2(\mathcal{C})}:=\frac{1}{2 \pi {\mathbf{i}}}\sum_{n=0}^{\infty} \frac{1}{n!} \int_{{\mathcal}{C}^n} \det[{\mathsf}{K}(x_i,x_j)]_{i,j=1}^n dx_1...dx_n.$$ To prove Theorem \[main theorem\] we begin with the following Fredholm determinant formula for $\mathbb{P}(H_t(n)<m)$, and perform a saddle point analysis.
\[exact\] $${\mathbb{P}}(H_t(n) < m) =\det(I-\mathsf{K}_{n})_{{\mathbb{L}}^2(\mathcal{C}_0)},$$ where $\mathcal{C}_0$ is a small positively oriented circle containing $0$ but not $-a-b$, and $\mathsf{K}_{n}: {\mathbb{L}}^2(\mathcal{C}_0) \to {\mathbb{L}}^2(\mathcal{C}_0)$ is defined by its integral kernel $$\begin{aligned}
\mathsf{K}_{n}(u,u') &=\frac{1}{2\pi {\mathbf{i}}} \int_{1/2 -i \infty}^{1/2+i \infty} \frac{e^{ts}}{s} \frac{g(u)}{g(s+u)} \frac{ds}{s+u-u'}, \quad \textrm{where} \label{eq:defK}\\
g(u) &=\left(\frac{a+u}{u}\right)^n \left(\frac{a+u}{a+b+u}\right)^m \frac{1}{u}.
\end{aligned}$$
Note that [@RWRE Theorem 1.18] actually states ${\mathbb{P}}(H_t(n)<m) = \det(I+\mathsf{K}_{n})_{{\mathbb{L}}^2(\mathcal{C}_0)}$, instead of $\det(I-\mathsf{K}_{t,n})_{{\mathbb{L}}^2(\mathcal{C}_0)}$ due to a sign mistake.
This result was proved in [@RWRE] by taking a zero-temperature limit of a similar formula for the Beta RWRE obtained using the Bethe ansatz solvability of $q$-Hahn TASEP and techniques from [@borodin2014macdonald; @borodin2014duality]. The integral above is oscillatory and does not converge absolutely, but we may deform the contour so that it does. We will justify this deformation in Section $2.2$.
Theorem \[main theorem\] is proven in Section 2 by applying steep descent analysis to $\det(1-\mathsf{K}_n)$, however the proofs of several key lemmas are deferred to later sections. The main challenge in proving Theorem \[main theorem\] comes from the fact that, after a necessary change of variables ${\mathbb{\omega}}=n^{-1/3}u$, the contours of the Fredholm determinant are being pinched between poles of the kernel $\mathsf{K}_n$ at ${\mathbb{\omega}}=0$ and ${\mathbb{\omega}}=\frac{-a-b}{n^{1/3}}$ as $n \to \infty$. In order to show that the integral over the contour near $0$ does not affect the asymptotics, we prove bounds for $\mathsf{K}_n$ near $0$, and carefully choose a family of contours ${\mathcal}{C}_n$ on which we can control the kernel. This quite technical step is the main goal of Section 3. Section 4 is devoted to bounding the Fredholm determinant expansion of $\det(1-\mathsf{K}_n)_{L^2(\mathcal{C}_n)},$ in order to justify the use of dominated convergence in Section 2.
Other interpretations of the model {#interpretations}
----------------------------------
There are several equivalent interpretations of Bernoulli-exponential first passage percolation. We will present the most interesting here.
### A particle system on the integer line
The height function of the percolation cluster $H_t(n)$ is equivalent to the height function of an interacting particle system we call geometric jump pushTASEP, which generalizes pushTASEP (the $R=0$ limit of PushASEP introduced in [@BorodinFerrari]) by allowing jumps of length greater than 1. This model is similar to Hall-Littlewood pushTASEP introduced in [@ghosalpush], but has a slightly different particle interaction rule.
Let $\mathrm{Geom}(q)$ denote a geometric random variable with ${\mathbb{P}}(\mathrm{Geom}(q)=k)=q^k(1-q)$. Let $1 \leq p_1(t) < p_2(t) < ...<p_i(t)<...$ be the positions of ordered particles in ${\mathbb{Z}}_{\geq 1}$. At time $t=0$ the position $n \in {\mathbb{Z}}_{\geq 0}$ is occupied with probability $b/(a+b)$. Each particle has an independent exponential clock with parameter $a$, and when the clock corresponding to the particle at position $p_i$ rings, we update each particle position $p_j$ in increasing order of $j$ with the following procedure. ($p_i(t-)$ denotes the position of particle $i$ infinitesimally before time $t$.)
- If $j<i$, then $p_j$ does not change.
- $p_i$ jumps to the right so that the difference $p_i(t)-p_i(t-)$ is distributed as $1+\mathrm{Geom}(a/(a+b))$
- If $j>i$, then
- If the update for $p_{j-1}(t)$ causes $p_{j-1}(t) \geq p_{j}(t-)$, then $p_{j}(t)$ jumps right so that $p_{j}(t)-p_{j-1}(t)$ is distributed as $1+\mathrm{Geom}(a/(a+b))$.
- Otherwise $p_j$ does not change.
- All the geometric random variables in the update procedure are independent.
![This figure illustrates a single update for geometric jump pushTASEP. The clock corresponding to the leftmost particle rings, activating the particle. The first particle jumps 2 steps pushing the next particle and activating it. This particle jumps 1 step pushing the rightmost particle and activating it. The rightmost particle jumps 3 steps, and all particles are now in their original order, so the update is complete.[]{data-label="particlesystem"}](pushTASEP_particles)
Another way to state the update rule is that each particle jumps with exponential rate a, and the jump distance is distributed as $1 + \mathrm{Geom}(a/(a+b))$. When a jumping particle passes another particle, the passed particle is pushed a distance $1+\mathrm{Geom}(a/(a+b))$ past the jumping particle’s ending location (see Figure \[particlesystem\]).
The height function $\overline{H}_t(n)$ at position $n$ and time $t$ is the number of unoccupied sites weakly to the left of $n$. If we begin with the distribution of $(n,H_t(n))$ in our percolation model, and rotate the first quadrant clockwise $45$ degrees, the resulting distribution is that of $(n,\overline{H}_t(n))$. The horizontal segments in the upper border of the percolation cluster correspond to the particle positions, thus $$H_t(n)=p_t(n)-n=\sup\{k: \overline{H}_t(n+k) \geq k\}.$$ A direct translation of Theorem \[main theorem\] gives:
\[cor:particle\] Fix parameters $a,b>0$. For any $t>0$ and $x \in \mathbb{R}$, $$\lim_{n \to \infty} \mathbb{P}\left(\frac{p_t(n)-\left(\frac{a+b}{b}\right) n-d n^{2/3}}{\sigma n^{4/9}} \leq x\right)=F_{\mathrm{GUE}}(x),$$ where $F_{\mathrm{GUE}}(x)$ is the Tracy-Widom GUE distribution.
To the authors knowledge Corollary \[cor:particle\] is the first result in interacting particle systems showing Tracy-Widom fluctuations for the position of a particle at finite time.
### Degenerations
If we set $b=1, t'= t/a,$ and $a \to 0$, then in the new time variable $t'$ each particle performs a jump with rate 1 and with probability going to 1, each jump is distance 1, and each push is distance 1. This limit is pushTASEP on ${\mathbb{Z}}_{\geq 0}$ where every site is occupied by a particle at time $0$. Recall that in pushTASEP, the dynamics of a particle are only affected by the (finitely many) particles to its left, so this initial data makes sense.
We can also take a continuous space degeneration. Let $x$ be the spatial coordinate of geometric jump pushTASEP, and let $\exp(\lambda)$ denote an exponential random variable with rate $\lambda$. Choose a rate $\lambda>0$, and set $b=\frac{\lambda}{n}, x'=x/n, a=\frac{n-\lambda}{n}$, and let $n \to \infty$. Then our particles have jump rate $\frac{n-\lambda}{n} \to 1$, jump distance $\frac{\mathrm{Geom}(1-\lambda/n)}{n} \to \exp(\lambda)$, and push distance $\frac{\mathrm{Geom}(1-\lambda/n)}{n} \to \exp(\lambda)$. This is a continuous space version of pushTASEP on ${\mathbb{R}}_{\geq 0}$ with random initial conditions such that the distance between each particle position $p_i$ and its rightward neighbor $p_{i+1}$ is an independent exponential random variable of rate $\lambda$. Each particle has an exponential clock, and when the clock corresponding to the particle at position $p_i$ rings, an update occurs which is identical to the update for geometric jump pushTASEP except that each occurrence of the random variable $1 +\mathrm{Geom}(a/(a+b))$ is replaced by the random variable $\exp(\lambda).$
### A benchmark model for travel times in a square grid city
The first passage times of Bernoulli-exponential FPP can also be interpreted as the minimum amount of time a walker must wait at streetlights while navigating a city [@stoplight]. Consider a city, whose streets form a grid, and whose stoplights have i.i.d exponential clocks. The first passage time of a point $(n,m)$ in our model has the same distribution as the minimum amount of time a walker in the city has to wait at stoplights while walking $n$ streets east and $m$ streets north. Indeed at each intersection the walker encounters one green stoplight with zero passage time and one red stoplight at which they must wait for an exponential time. Note that while the first passage time is equal to the waiting time at stoplights along the best path, the joint distribution of waiting times of walkers along several paths is different from the joint passage times along several paths in Bernoulli-exponential FPP.
Further directions {#questions}
------------------
Bernoulli-exponential FPP has several features that merit further investigation. From the perspective of percolation theory, it would be interesting to study how long it takes for the percolation cluster to contain all vertices in a given region, or how geodesics from the origin coalesce as two points move together.
From the perspective of KPZ universality, it is natural to ask: what is the correlation length of the upper border of the percolation kernel, and what is the joint law of the topmost few paths.
Under diffusive scaling limit, the set of coalescing simple directed random walks originating from every point of ${\mathbb{Z}}^2$ converges to the Brownian web [@fontes2002brownian; @fontes2004brownian]. Hence the set of all possible tributaries in our model converges to the Brownian web. One may define a more involved set of coalescing and branching random walks which converges to a continuous object called the Brownian net ([@newman2010marking], [@sun2008brownian], see also the review [@schertzer2015brownian]). Thus, it is plausible that there exist a continuous limit of Bernoulli-Exponential FPP where tributaries follow Brownian web paths and branch at a certain rate at special points of the Brownian web used in the construction of the Brownian net.
After seeing Tracy-Widom fluctuations for the edge statistics it is natural to ask whether the density of vertices inside the river along a cross section is also connected to random matrix eigenvalues and whether a statistic of this model converges to the positions of the second, third, etc. eigenvalues of the Airy point process.
Notation and conventions
------------------------
We will use the following notation and conventions.
- $B_{{\varepsilon}}(x)$ will denote the open ball of radius ${\varepsilon}>0$ around the point $x$.
- ${\mathfrak{Re}}[x]$ will denote the real part of a complex number $x$, and ${\mathfrak{Im}}[x]$ denotes the imaginary part.
- $\mathcal{C}$ and $\mathcal{\gamma}$ with any upper or lower indices will always denote an integration contour in the complex plane. $\mathsf{K}$ with any upper or lower indices will always represent an integral kernel. A lower index like $\mathcal{\gamma}_r$, $\mathcal{C}_n$, or $\mathsf{K}_n$ will usually index a family of contours or kernels. An upper index such as $\mathcal{\gamma}^{{\varepsilon}}$, $\mathcal{C}^{{\varepsilon}}$, or $\mathsf{K}^{{\varepsilon}}$ will indicate that we are intersecting our contour with a ball of radius ${\varepsilon}$, or that the integral defining the kernel is being restricted to a ball of radius ${\varepsilon}$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank Ivan Corwin for many helpful discussions and for useful comments on an earlier draft of the paper. The authors thank an anonymous reviewer for detailed and helpful comments on the manuscript. G. B. was partially supported by the NSF grant DMS:1664650. M. R. was partially supported by the Fernholz Foundation’s “Summer Minerva Fellow” program, and also received summer support from Ivan Corwin’s NSF grant DMS:1811143.
Asymptotics
===========
Setup
-----
The steep descent method is a method for finding the asymptotics of an integral of the form $$I_M=\int_{\mathcal{C}} e^{Mf(z)}dz,$$ as $M \to \infty$, where $f$ is a holomorphic function and $\mathcal{C}$ is an integration contour in the complex plane. The technique is to find a critical point $z_0$ of $f$, deform the contour $\mathcal{C}$ so that it passes through $z_0$ and ${\mathfrak{Re}}[f(z)]$ decays quickly as $z$ moves along the contour $\mathcal{C}$ away from $z_0$. In this situation $e^{Mf(z_0)}/e^{Mf(z)}$ has exponential decay in $M$. We use this along with specific information about our $f$ and $\mathcal{C}$, to argue that the integral can be localized at $z_0$, i.e. the asymptotics of $\int_{\mathcal{C} \cap B_{{\varepsilon}}(z_0)} e^{M f(z)} dz$ are the same as those of $I_M$. Then we Taylor expand $f$ near $z_0$ and show that sufficiently high order terms do not contribute to the asymptotics. This converts the first term of the asymptotics of $I_M$ into a simpler integral that we can often evaluate.
In Section $2.1$ we will manipulate our formula for ${\mathbb{P}}(h(n) < m)$, and find a function $f_1$ so that the kernel ${\mathsf}{K}_{n}$ can be approximated by an integral of the form $\int_{\lambda+{\mathbf{i}}\mathbb{R}} e^{n^{1/3}[f_1(z)-f_1({\mathbb{\omega}})]} dz$. Approximating ${\mathsf}{K}_{n}$ in this way will allow us to apply the steep descent method to both the integral defining ${\mathsf}{K}_n$ and the integrals over $\mathcal{C}_0$ in the Fredholm determinant expansion.
For the remainder of the paper we fix a time $t>0$, and parameters $a,b>0$. All constants arising in the analysis below depend on those parameters $t,a,b$, though we will not recall this dependency explicitly for simplicity of notation.
We also fix henceforth $$m=\left\lfloor \frac{a}{b} n +d n^{2/3}+n^{4/9} \sigma x \right\rfloor. \label{m}$$ We consider ${\mathsf}{K}_n$ and change variables setting $\tilde{z}=s+u$, $d\tilde{z}=ds$ to obtain $$\tilde{\mathsf{K}}_n(u,u')=\frac{1}{2 \pi {\mathbf{i}}} \int_{1/2+u-{\mathbf{i}}\infty}^{1/2+u+{\mathbf{i}}\infty} \frac{e^{t(\tilde{z}-u)}}{(\tilde{z}-u)({\tilde}{z}-u')} \frac{g(u)}{g({\tilde}{z})} d{\tilde}{z}.$$ In the following lemma, we change our contour of integration in the ${\tilde}{z}$ variable so that it does not depend on $u$.
For every fixed $n$, $${\tilde}{\mathsf{K}}_n(u,u')=\frac{1}{2 \pi {\mathbf{i}}} \int_{n^{1/3} \lambda+{\mathbf{i}}\mathbb{R}} \frac{e^{t({\tilde}{z}-u)}}{({\tilde}{z}-u)({\tilde}{z}-u')} \frac{g(u)}{g({\tilde}{z})} d{\tilde}{z}.$$
Choose the contour $\mathcal{C}_0$ to have radius $0<r<\min[1/4,\lambda]$. This choice of $r$ means that we do not cross ${\mathcal}{C}_0$ when deforming the contour $1/2+u+{\mathbf{i}}\mathbb{R}$ to $\lambda+{\mathbf{i}}\mathbb{R}$. In this region $K$ is a holomorphic function, so this deformation does not change the integral provided that for $M$ real,
$$\frac{1}{2 \pi {\mathbf{i}}}\int_{1/2+u+{\mathbf{i}}M}^{n^{1/3}\lambda+{\mathbf{i}}M} \frac{e^{t(\tilde{z}-u)}}{(\tilde{z}-u)(\tilde{z}-u')} \frac{g(u)}{g(\tilde{z})} d\tilde{z} \xrightarrow[M \to \pm \infty]{} 0.$$ This integral converges to $0$ because for all $\tilde{z} \in [n^{1/3} \lambda-{\mathbf{i}}M,1/2+u-{\mathbf{i}}M] \cup [n^{1/3} \lambda+{\mathbf{i}}M,1/2+u+{\mathbf{i}}M]$ we have $$\left|\frac{1}{(\tilde{z}-u)(\tilde{z}-u')g(\tilde{z})}\right| \sim \frac{1}{M},$$ as $M \to \infty$.
Set $$\tilde{h}_n(z)=-n \log \left( \frac{a+z}{z}\right )-m \log \left(\frac{a+z}{a+b+z} \right), \qquad \text{so that} \qquad e^{\tilde{h}_n(z)}=\frac{z}{g(z)}.$$ Then $$\mathsf{K}_n(u,u')=\frac{1}{2 \pi {\mathbf{i}}} \int_{n^{1/3} \lambda+{\mathbf{i}}{\mathbb{R}}} \frac{e^{t{\tilde}{z}+\tilde{h}_n({\tilde}{z})}}{e^{tu+{\tilde}{h}_n(u)}} \frac{{\tilde}{z}}{u} \frac{d{\tilde}{z}}{({\tilde}{z}-u)({\tilde}{z}-u')}.$$ Now perform the change of variables $$z=n^{-1/3}{\tilde}{z}, {\mathbb{\omega}}=n^{-1/3} u, {\mathbb{\omega}}'=n^{-1/3} u'.$$ If we view our change of variables as occuring in the Fredholm determinant expansion, then due to the $d{\mathbb{\omega}}_i$s, we see that scaling all variables by the same constant does not change the Fredholm determinant $\det(1-{\ensuremath{\mathsf{K}}}_n)_{L^2(\mathcal{C})}$. Thus our change of variables gives $$\mathsf{K}_n({\mathbb{\omega}},{\mathbb{\omega}}')=\frac{1}{2 \pi {\mathbf{i}}} \int_{\lambda+{\mathbf{i}}{\mathbb{R}}} \frac{e^{n^{1/3} t(z-{\mathbb{\omega}})}}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} e^{h_n(z)-h_n({\mathbb{\omega}})} \frac{z}{{\mathbb{\omega}}} dz$$ where $$h_n(z)={\tilde}{h}_n(n^{1/3}z)=-n \log \left(\frac{a+n^{1/3}z}{n^{1/3} z} \right) -m \log \left( \frac{a+n^{1/3}z}{a+b+n^{1/3}z} \right).$$
The contour for ${\mathbb{\omega}}$, ${\mathbb{\omega}}'$ becomes $n^{-1/3} {\mathcal}{C}_0$ after the change of variables, but ${\ensuremath{\mathsf{K}}}_n({\mathbb{\omega}},{\mathbb{\omega}}')$ is holomorphic in most of the complex plane. Examining of the poles of the integrand for ${\ensuremath{\mathsf{K}}}_n({\mathbb{\omega}},{\mathbb{\omega}}')$, we see that we can deform the contour for ${\mathbb{\omega}},{\mathbb{\omega}}'$ in any way that does not cross the line $\lambda+ {\mathbf{i}}{\mathbb{R}}$, the pole at $-(a+b)/n^{1/3}$, or the pole at $0$, without changing the Fredholm determinant $\det(I-{\ensuremath{\mathsf{K}}}_n)_{L^2(n^{-1/3}\mathcal{C}_0)}$.
Taylor expanding the logarithm in the variable $n$ gives $$h_n(z)=-n^{1/3} \left(\frac{a(a+b)}{2z^2}-\frac{bd}{z}\right)-n^{1/9} \left( \frac{-b \sigma x}{z} \right)+r_n(z).$$ Here $r_n(z) = \mathcal O(1)$ in a sense that we make precise in Lemma \[chaos control\]. The kernel can be rewritten as $$\begin{gathered}
\mathsf{K}_n({\mathbb{\omega}},{\mathbb{\omega}}')= \\\frac{1}{ 2 \pi {\mathbf{i}}} \int_{\lambda+{\mathbf{i}}{\mathbb{R}}} \frac{\exp(n^{1/3} (f_1(z)-f_1(w))+n^{1/9}(f_2(z)-f_2({\mathbb{\omega}}))+(r_n(z)-r_n({\mathbb{\omega}})))}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} \frac{z}{{\mathbb{\omega}}} dz\end{gathered}$$ where $$f_1(z)=tz-\frac{a(a+b)}{2z^2}+\frac{bd}{z}, \qquad
f_2(z)=\frac{b \sigma x}{z}. \label{f def}$$ We have approximated the kernel as an integral of the form $\int e^{n^{1/3} [f_1(z)-f_1({\mathbb{\omega}})]}dz$. To apply the steep-descent method, we want to understand the critical points of the function $f_1$. We have $$f_1'(z)=t+\frac{a(a+b)}{z^3}-\frac{db}{z^2},\qquad
f_1''(z)=-\frac{3a(a+b)}{z^4}+\frac{2bd}{z^3}, \qquad
f_1'''(z)=\frac{12a(a+b)}{z^5}-\frac{6bd}{z^4}.$$ Where $a,b$ are the parameters associated to the model. Let the constant $\lambda$ be as defined in (\[constants\]), then $0=f_1'(\lambda)=f_1''(\lambda)=0$, and
$$f_1'''(\lambda)=\frac{3a(a+b)}{\lambda^5}=2\left(\frac{ b \sigma}{\lambda^2}\right)^3=2\left(\frac{-f_2'(\lambda)}{x}\right)^3 ,$$ is a positive real number. $\sigma$ is defined in equation (\[constants\]).
Recall the definition of the Tracy-Widom GUE distribution, which governs the largest eigenvalue of a gaussian hermitian random matrix.
\[tracywidomdist\] The Tracy-Widom distribution’s distribution function is defined as $F_{\textrm{GUE}}(x)=\det(1-\mathsf{K}_{\mathrm{Ai}})_{L^2(x,\infty)}$, where $K_{Ai}$ is the Airy kernel, $$\begin{aligned}
\mathsf{K}_{\mathrm{Ai}}(s,s')=\frac{1}{2\pi {\mathbf{i}}} \int_{e^{-2 \pi i/3} \infty}^{e^{2 \pi {\mathbf{i}}/3} \infty} d{\mathbb{\omega}}\frac{1}{2\pi {\mathbf{i}}} \int_{e^{-\pi {\mathbf{i}}/3} \infty}^{e^{\pi {\mathbf{i}}/3} \infty}dz \frac{e^{z^3/3-zs}}{e^{{\mathbb{\omega}}^3/3-{\mathbb{\omega}}s'}} \frac{1}{(z-{\mathbb{\omega}})}.\nonumber \end{aligned}$$
In the above integral the two contours do not intersect. We can think of the inner integral following the contour $(e^{-\pi {\mathbf{i}}/3} \infty, 1] \cup (1, e^{\pi {\mathbf{i}}/3} \infty)$, and the outer integral following the contour $(e^{-2\pi {\mathbf{i}}/3} \infty, 0] \cup (0, e^{2\pi {\mathbf{i}}/3} \infty)$. Our goal through the rest of the paper is to show that the Fredholm determinant $\det(I-{\mathsf}{K}_n)$ converges to the Tracy-Widom distribution as $n \to \infty$.
Steep descent contours
----------------------
We say that a path $\gamma: [a,b] \to \mathbb{C}$ is steep descent with respect to the function $f$ at the point $x= \gamma(0)$ if $\frac{d}{dt} {\mathfrak{Re}}[f(\gamma(t))]>0$ when $t>0$, and $\frac{d}{dt} {\mathfrak{Re}}[f(\gamma(t))]<0$ when $t<0$.
We say that a contour $\mathcal{C}$ is steep descent with respect to a function $f$ at a point $x$, if the contour can be parametrized as a path satisfy the above definition. Intuitively this statement means that as we move along the contour $\mathcal{C}$ away from the point $x$, the function $f$ is strictly decreasing.
In this section we will find a family of contours ${\mathcal}{\gamma}_r$ for the variable $z$ and so that ${\mathcal}{\gamma}_r$ is steep descent with respect to ${\mathfrak{Re}}[f_1(z)]$ at the point $\lambda,$ and study the behavior of ${\mathfrak{Re}}[f_1]$. The contours $\mathcal{C}_n$ for ${\mathbb{\omega}}$ are constructed in Section \[Cn\].
\[steep descent\]
The contour $\lambda+{\mathbf{i}}\mathbb{R}$ is steep descent with respect to the function ${\mathfrak{Re}}[f_1]$ at the point $\lambda$.
We have that
$$\frac{d}{dy} {\mathfrak{Re}}[f_1(\lambda+{\mathbf{i}}y)]=-{\mathfrak{Im}}[f_1'(\lambda+{\mathbf{i}}y)]=-{\mathfrak{Im}}\left[t+\frac{a(a+b)}{(\lambda+{\mathbf{i}}y)^3}-\frac{bd}{\lambda+{\mathbf{i}}y}\right].$$ Now using the relation $2 b d \lambda= 3a(a+b)$ and computing gives
$$\frac{d}{dy} {\mathfrak{Re}}[f_1(\lambda+{\mathbf{i}}y)]=\frac{-4a(a+b)y^3}{(\lambda^2+y^2)^3}.$$ This derivative is negative when $y>0$ and positive when $y<0$.
![The level lines of the function ${\mathfrak{Re}}[f_1(z)]$ at value ${\mathfrak{Re}}[f_1(\lambda)]$. In this image we take $a=b=t=1$.[]{data-label="contour"}](Contourf_1new)
Now we describe the contour lines of ${\mathfrak{Re}}[f_1(z)]$ seen in Figure \[contour\]. ${\mathfrak{Re}}[f_1]$ is the real part of a holomorphic function, so its level lines are constrained by its singularities, and because the singularities are not too complicated, we can describe its level lines. The contour lines of the real part of a holomorphic function intersect only at critical points and poles and the number of contour lines that intersect will be equal to the degree of the critical point or pole. We can see from the Taylor expansion of $f_1$ at $\lambda$, that there will be $3$ level lines intersecting at $\lambda$ with angles $\pi/6,\pi/2$, and $5\pi/6$. From the form of $f_1$, we see that there will be $2$ level lines intersecting at $0$ at angles $\pi/4$ and $3\pi/4$, and that a pair of contour lines will approach ${\mathbf{i}}\infty$ and $-{\mathbf{i}}\infty$ respectively with ${\mathfrak{Re}}[z]$ approaching $f_1(\lambda)/t$. This shows that, up to a noncrossing continuous deformation of paths, the lines in Figure $\ref{contour}$ are the contour lines ${\mathfrak{Re}}[f_1(z)]=f_1(\lambda)$. We can also see that on the right side of the figure, $tz$ will be the largest term of ${\mathfrak{Re}}[f_1(z)]$, so our function will be positive. This determines the sign of ${\mathfrak{Re}}[f_1(z)]$ in the other regions.
Our contour $\lambda+ {\mathbf{i}}\mathbb{R}$ is already steep descent, but we will deform the tails, so that we can use dominated convergence in the next section.
For any $r>0$, define the contour $\mathcal{\gamma}_r=(e^{-2 \pi \mathbf{i}/3} \infty, \lambda-r\mathbf{i}) \cup [\lambda-r\mathbf{i},\lambda+r\mathbf{i}] \cup (\lambda+r\mathbf{i},e^{2\pi \mathbf{i}/3}\infty)$ and $\mathcal{\gamma}_r^{{\varepsilon}}=\mathcal{\gamma}_r \cap B_{{\varepsilon}}(\lambda).$ These contours appear in Figure $\ref{contours}$.
(0,-1)–(0,1); (0,1)–(0,2); (0,1)–(0,-2); (0,-2) – +(-120:2) node\[below\][$e^{-2 \pi \mathbf{i}/3} \infty$]{}; (0,2)– +(120:2) node\[above\][$e^{2 \pi \mathbf{i}/3} \infty$]{};
at (-2,0) [$0$]{}; at (0,2) [$\lambda+{\mathbf{i}}r$]{}; at (0,-2) [$\lambda-{\mathbf{i}}r$]{}; at (0,0) [$\lambda$]{}; at (0,1) [$\lambda+{\mathbf{i}}{\varepsilon}$]{}; at (0,-1) [$\lambda- {\mathbf{i}}{\varepsilon}$]{};
(0,0) circle \[radius=.06\]; (0,2) circle \[radius=.06\]; (0,-2) circle \[radius=.06\]; (0,1) circle \[radius=.06\]; (0,-1) circle \[radius=.06\]; (-2,0) circle \[radius=.06\];
Because for any fixed $n$, we have $e^{h_n(z)} \to 1$ as $|z| \to \infty$, $\frac{z}{{\mathbb{\omega}}(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')}$ has linear decay in $z$, and $e^{n^{1/3}t(z-{\mathbb{\omega}})}$ has exponential decay in $z$, we can deform the vertical contour $\lambda+ {\mathbf{i}}\mathbb{R}$ to the contour $\mathcal{\gamma}_r$. Thus $$\mathsf{K}_n({\mathbb{\omega}},{\mathbb{\omega}}')=\int_{\mathcal{\gamma}_r} \frac{e^{n^{1/3}t(z-{\mathbb{\omega}})}}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} e^{h_n(z)-h_n({\mathbb{\omega}})} \frac{z}{{\mathbb{\omega}}} dz.$$
The function ${\mathfrak{Re}}[f_1]$ is still steep descent on the contour ${\mathcal}{\gamma}_r$ with respect to the point $\lambda$. Lemma \[steep descent\] shows that ${\mathfrak{Re}}[f_1]$ is steep descent on the segment $[\lambda-r\mathbf{i},\lambda+r\mathbf{i}]$, and on $(e^{-2 \pi \mathbf{i}/3} \infty, \lambda-r\mathbf{i}) \cup (\lambda+r\mathbf{i},e^{2\pi \mathbf{i}/3} \infty)$ we inspect $f'_1(z)$ and note that for $z$ sufficiently large, the constant term $t$ dominates the other terms. Because our paths are moving in a direction with negative real component the contour ${\mathcal}{\gamma}_r$ is steep descent.
Up to this point we have been concerned with contours being steep descent with respect to ${\mathfrak{Re}}[f_1]$, but the true function in our kernel is $\exp(n^{1/3}t(z-{\mathbb{\omega}})+h_n(z)-h_n({\mathbb{\omega}}))$. To show that ${\mathcal}{\gamma}_r$ is steep descent with respect to this function, we will need to control the error term $n^{1/3}tz+h_n(z)-n^{1/3}f_1(z)=n^{1/9}f_2(z)+r_n(z)$. The following lemma gives bounds on this error term away from $z=0$.
\[chaos control\] For any $N,{\varepsilon}>0$ there is a constant $C$ depending only on ${\varepsilon}, N$ such that $$|f_2({\mathbb{\omega}})| \leq C \text{ and } |r_n({\mathbb{\omega}})| \leq C, \label{control}$$ for all $n \geq N,$ and ${\mathbb{\omega}}\geq \frac{|a+b|+{\varepsilon}}{N^{1/3}}$.\
Similarly for any $\delta>0$, there exists $N_{\delta}$ and $C'$ depending only on $\delta$, such that $$|f'_2({\mathbb{\omega}})| \leq C' \text{ and } |r_n'({\mathbb{\omega}})| \leq C', \label{derivative control}$$ for all $n \geq N_{\delta}$, and ${\mathbb{\omega}}$ satisfying $|{\mathbb{\omega}}| \geq \delta.$
Lemma \[chaos control\] is proved in Section \[Cn\].
At this point we have a contour ${\mathcal}{\gamma}_r$ for the variable $z$, which is steep descent with respect to ${\mathfrak{Re}}[f_1]$. We want to find a suitable contour for ${\mathbb{\omega}}$. The following lemma shows the existence of such a contour $\mathcal{C}_n$, where property $(c)$ below takes the place of being steep descent. This lemma is fairly technical and its proof is the main goal of Section \[Cn\]. To see why observe that the function $n^{1/3}f_1({\mathbb{\omega}})$ does not approximate $n^{1/3}t {\mathbb{\omega}}-h_n({\mathbb{\omega}})$ well when ${\mathbb{\omega}}$ is near $0$. The fact that the contribution near $0$ is negligible is nontrivial because the function $n^{1/3}t {\mathbb{\omega}}-h_n({\mathbb{\omega}})$ has poles at $0$ and $\frac{-a-b}{n^{1/3}}$, and our contour ${\mathcal}{C}_n$ is being pinched between them; we will use Lemma \[C bound\] to show that the asymptotics of $\det(1-{\ensuremath{\mathsf{K}}}_n)_{L^2(\mathcal{C}_n)}$ are not affected by these poles
\[C bound\] There exists a sequence of contours $\{{\mathcal}{C}_n\}_{n \geq N}$ such that:
- For all $n$, the contour $\mathcal{C}_n$ encircles $0$ counterclockwise, but does not encircle $(-a-b)n^{-1/3}$.
- ${\mathcal}{C}_n$ intersects the point $\lambda$ at angles $-\pi/3$ and $-2\pi/3$.
- For all ${\varepsilon}>0$, there exists $\eta, N_{{\varepsilon}}>0$ such that for all $n>N_{{\varepsilon}}$, ${\mathbb{\omega}}\in \mathcal{C}_n \setminus \mathcal{C}_n^{{\varepsilon}}$ and $z \in {\mathcal}{\gamma}_r$, we have
$${\mathfrak{Re}}[n^{1/3}t(z-{\mathbb{\omega}})+h_n(z)-h_n({\mathbb{\omega}})] \leq -n^{1/3}\eta,$$ where ${\mathcal}{C}_n^{{\varepsilon}}={\mathcal}{C}_n \cap B_{{\varepsilon}}(\lambda).$
- There is a constant $C$ such that for all ${\mathbb{\omega}}\in \mathcal{C}_n$,
$${\mathfrak{Re}}[n^{1/3}t(\lambda-{\mathbb{\omega}})+h_n(\lambda)-h_n({\mathbb{\omega}})] \leq n^{1/9}C.$$
The next lemma allows us to control ${\mathfrak{Re}}[n^{1/3}tz+h_n(z)]$ on the contour ${\mathcal}{\gamma}_r$.
\[gamma bound\] For all ${\varepsilon}>0$, and for sufficiently large $r$, there exists $C,N_{{\varepsilon}}>0$, such that for all ${\mathbb{\omega}}\in {\mathcal}{C}_n$, and $z \in \mathcal{\gamma}_r \setminus \mathcal{\gamma}_r^{{\varepsilon}}$, then $${\mathfrak{Re}}[h_n(z)-h_n({\mathbb{\omega}})+n^{1/3}t(z-{\mathbb{\omega}})] \leq -n^{-1/3} C.$$
We have already shown that ${\mathcal}{\gamma}_r$ is steep descent with respect to $f_1(z)$.
By Lemma \[chaos control\], $|r_n| \leq C, |f_2| \leq Cn^{1/9}$ away from $0$. We have $$\begin{aligned}
h_n(z)-h_n({\mathbb{\omega}})+n^{1/3}t(z-{\mathbb{\omega}})=&n^{1/3}(f_1(z)-f_1({\mathbb{\omega}}))+n^{1/9}(f_2(z)-f_2({\mathbb{\omega}}))+(r_n(z)-r_n({\mathbb{\omega}}))\nonumber\\
\leq n^{1/3}(f_1(z)-&f_1({\mathbb{\omega}}))+n^{1/9}C+C \leq n^{1/3}(f_1(z)-f_1({\mathbb{\omega}})+\delta), \nonumber \end{aligned}$$ for any sufficiently small $\delta>0$. Because $f_1(z)$ is decreasing as we move away from $\lambda$, we have $$n^{1/3}tz+h_n(z)<n^{1/3}t \lambda+h_n(\lambda)+Cn^{1/9}.$$ Thus by $\ref{chaos control}$, we have that for all ${\varepsilon}>0$ there exists $C$ such that for $z \in \mathcal{\gamma}_r \setminus \mathcal{\gamma}_r^{{\varepsilon}}$, $${\mathfrak{Re}}[h_n(z)-h_n(\lambda)+n^{1/3}t(z-\lambda)] \leq -n^{1/3} C.$$ By Lemma \[C bound\] (d), we have $${\mathfrak{Re}}[h_n(\lambda)-h_n({\mathbb{\omega}})+n^{1/3}t(\lambda-{\mathbb{\omega}})] \leq n^{1/9}C,$$ for ${\mathbb{\omega}}\in \mathcal{C}_n.$ This completes the proof
Localizing the integral {#localize}
-----------------------
In this section we will use Lemma \[C bound\] and Lemma \[gamma bound\] to show that the asymptotics of $\det(1-\mathsf{K}_n)_{L^2(\mathcal{C}_n)}$ do not change if we replace $\mathcal{C}_n$ with $\mathcal{C}_n^{{\varepsilon}}=\mathcal{C}_n \cap B_{{\varepsilon}}(\lambda)$, and replace the contour $\mathcal{\gamma}_r$ defining $\mathsf{K}_n$ with the contour $\mathcal{\gamma}_r^{{\varepsilon}}=\mathcal{\gamma}_r \cap B_{{\varepsilon}}(0).$
First we change variables setting $z=\lambda+n^{-1/9} \overline{z},{\mathbb{\omega}}=\lambda+n^{-1/9} \overline{{\mathbb{\omega}}}$, and ${\mathbb{\omega}}'=\lambda+n^{-1/9} \overline{z}$.
Define the contours $\mathcal{D}_0=[-{\mathbf{i}}\infty, {\mathbf{i}}\infty]$, and $\mathcal{D}_0^{ \delta}={\mathcal}{D}_0 \cap B_{ \delta}(0).$ (We will often use $\delta=n^{1/9} {\varepsilon}$.)
Our change of variables applied to the kernel ${\mathsf}{K}_n^{{\varepsilon}}$ gives $$\begin{gathered}
\overline{\mathsf{K}}^{{\varepsilon}}_n(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}')=\frac{1}{2 \pi {\mathbf{i}}}\int_{\mathcal{D}_0^{n^{1/9} {\varepsilon}}} \frac{1}{(\overline{z}-\overline{{\mathbb{\omega}}})(\overline{z}-\overline{{\mathbb{\omega}}}')} \frac{(\lambda+n^{-1/9} \overline{z})}{(\lambda+n^{-1/9}\overline{{\mathbb{\omega}}})} e^{n^{1/3} f_1(\lambda+n^{-1/9} \overline{z})-f_1(\lambda+n^{-1/9} \overline{{\mathbb{\omega}}})} \\ \times e^{n^{1/9} f_2(\lambda+n^{-1/9} \overline{z})-f_2(\lambda+n^{-1/9} \overline{{\mathbb{\omega}}})} e^{r_n(\lambda+n^{-1/9} \overline{z})-r_n(\lambda+n^{-1/9} \overline{{\mathbb{\omega}}})} d \overline{z}. \label{K def}\end{gathered}$$
The contours $\mathcal{C}_{-1}$ and $\mathcal{C}_{-1}^{{\varepsilon}}$ are defined as $\mathcal{C}_{-1}=(e^{-2\pi {\mathbf{i}}/3} \infty, -1) \cup [-1,e^{2 \pi {\mathbf{i}}/3} \infty)$ and $\mathcal{C}_{-1}^{{\varepsilon}}=\mathcal{C}_{-1} \cap B_{n^{1/9} {\varepsilon}}(-1).$
By changing variables, for each $m$ we have
$$\int_{({\mathcal}{C}_{n}^{{\varepsilon}})^m} \det(\mathsf{K}^{{\varepsilon}}_n({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m d {\mathbb{\omega}}_1...d{\mathbb{\omega}}_m=\int_{(\mathcal{C}_{-1}^{n^{1/9}{\varepsilon}})^m} \det(\overline{\mathsf{K}}_{n}^{{\varepsilon}}(\overline{{\mathbb{\omega}}}_i,\overline{{\mathbb{\omega}}}_j))_{i,j=1}^m d \overline{{\mathbb{\omega}}}_1...d \overline{{\mathbb{\omega}}}_m.$$ This equality follows, because after rescaling the contour $C_{n}^{{\varepsilon}}$, we can deform it to the contour $\mathcal{C}_{-1}^{n^{1/9} {\varepsilon}}$ without changing its endpoints. The previous equality implies $$\det(1-\mathsf{K}^{{\varepsilon}}_n)_{L^2(\mathcal{C}_{{\varepsilon}}^{{\varepsilon}})}=\det(1-\overline{\mathsf{K}}_{n}^{ {\varepsilon}})_{L^2(\mathcal{C}_{-1}^{n^{1/9} {\varepsilon}})}.$$
We will make this change of variables often in the following arguments. Given a contour such as ${\mathcal}{C}_n$ or ${\mathcal}{\gamma}_r$, we denote the contour after the change of variables by ${\mathcal}{\overline{C}}_n$ or ${\mathcal}{\overline{\mathcal{\gamma}}}_r$. Now we are ready to localize our integrals.
\[cut\] For any sufficiently small ${\varepsilon}>0$,
$$\lim_{n \to \infty} \det(1-\mathsf{K}_n({\mathbb{\omega}},{\mathbb{\omega}}'))_{L^2(\mathcal{C})}=\lim_{n \to \infty} \det(1-\mathsf{K}_{n}^{{\varepsilon}}({\mathbb{\omega}},{\mathbb{\omega}}'))_{L^2(\mathcal{C}_{n}^{{\varepsilon}})},$$ where
$$\mathsf{K}_{n}^{{\varepsilon}}=\frac{1}{2 \pi {\mathbf{i}}} \int_{\mathcal{\gamma}_r^{{\varepsilon}}} \frac{e^{n^{1/3}t(z-{\mathbb{\omega}})+h_n(z)-h_n({\mathbb{\omega}})}}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} \frac{z}{w} dz.$$
The proof will have two steps, and will use several lemmas that are proved in Section 4. In the first step we localize the integral in the $z$ variable and show that $\lim_{n \to \infty}\det(1-\mathsf{K}_n)_{L^2(\mathcal{C}^{{\varepsilon}})}=\lim_{n \to \infty}\det(1-\mathsf{K}_n^{{\varepsilon}})_{L^2(\mathcal{C}^{{\varepsilon}})}$ using dominated convergence. In order to prove this, we appeal to Lemmas \[dom\] and \[dom 2\] to show that the Fredholm series expansions are indeed dominated. In the second step we localize the integral in the ${\mathbb{\omega}},{\mathbb{\omega}}'$ variables by using Lemma \[max bound lemma\] to find an upper bound for $\det(1+K_n)_{L^2({\mathcal}{C}_n)}-\det(1+K_n)_{L^2({\mathcal}{C}_n^{{\varepsilon}})}.$ Then we appeal to Lemma \[sum\] to show that this upper bound converges to $0$ as $n \to \infty$.
By Lemma \[gamma bound\], for any ${\varepsilon}>0$, there exists a $C',N>0$ such that if ${\mathbb{\omega}}\in C_n$ and $z \in \mathcal{\gamma}_r \setminus \mathcal{\gamma}_r^{{\varepsilon}}$, then for all $n>N$,
$${\mathfrak{Re}}[h_n(z)-h_n({\mathbb{\omega}})+n^{1/3}t(z-{\mathbb{\omega}})] \leq -n^{1/3} C'.$$
We bound our integrand on $\mathcal{\gamma}_r \setminus \mathcal{\gamma}_r^{{\varepsilon}}$, $ {\mathbb{\omega}},{\mathbb{\omega}}' \in {\mathcal}{C}_n^{{\varepsilon}},$ $$\left|\frac{e^{h_n(z)-h_n({\mathbb{\omega}})+n^{1/3}t(z-{\mathbb{\omega}})}}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} \frac{z}{{\mathbb{\omega}}}\right| \leq \frac{C}{\delta^2} z e^{-n^{1/3 C'}} \xrightarrow[n \to \infty]{pointwise} 0.$$ (the $\delta^2$ comes from the fact that $|z-{\mathbb{\omega}}| \geq \delta$). By Lemma \[chaos control\], there exists a $\eta>0$ such that for sufficiently large $n$, $$\left|\frac{e^{h_n(z)-h_n({\mathbb{\omega}})+n^{1/3}t(z-{\mathbb{\omega}})}}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} \frac{z}{{\mathbb{\omega}}}\right| <\left|\frac{e^{n^{1/3}(f_1(z)-f_1({\mathbb{\omega}})+\eta)}}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} \frac{z}{{\mathbb{\omega}}}\right|.$$ The linear term of $f_1(z)$ in (\[f def\]) implies $$\frac{1}{2 \pi {\mathbf{i}}}\int_{{\mathcal}{\gamma}_r} \left|\frac{e^{n^{1/3}(f_1(z)-f_1({\mathbb{\omega}})+\eta)}}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} \frac{z}{{\mathbb{\omega}}}\right| dz<\infty.$$
In the previous inequality we should write $|dz|$ instead of $dz$. We will often omit the absolute value in the $d{\mathbb{\omega}}$ portion of the complex integral when the integrand is a positive real valued function.
So for each ${\mathbb{\omega}},{\mathbb{\omega}}'$, by dominated convergence $$\frac{1}{2 \pi {\mathbf{i}}}\int_{\mathcal{\gamma}_r \setminus \mathcal{\gamma}_r^{{\varepsilon}}}\frac{e^{h_n(z)-h_n({\mathbb{\omega}})+n^{1/3}t(z-{\mathbb{\omega}})}}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} \frac{z}{{\mathbb{\omega}}} dz \to 0 \quad \text{as} \quad n \to \infty,$$ So $\lim_{n \to \infty}\mathsf{K}_n^{{\varepsilon}}({\mathbb{\omega}},{\mathbb{\omega}}') = \lim_{n \to \infty} \mathsf{K}_n({\mathbb{\omega}},{\mathbb{\omega}}').$
Now by Lemma \[dom\], and \[dom 2\], both Fredholm determinant expansions $\det(1-\mathsf{K}_n)_{L^2(\mathcal{C}^{{\varepsilon}})}$ and $\det(1-\mathsf{K}^{{\varepsilon}}_n)_{L^2(\mathcal{C}^{{\varepsilon}})}$, are absolutely bounded uniformly in $n$. Thus we can apply dominated convergence to get
$$\lim_{n \to \infty}\det(1-\mathsf{K}_n)_{L^2(\mathcal{C}^{{\varepsilon}})}=\lim_{n \to \infty}\det(1-\mathsf{K}_n^{{\varepsilon}})_{L^2(\mathcal{C}^{{\varepsilon}})}. \label{fred equal}$$
In the expansion $$\det(1-\mathsf{K}_n)_{L^2(\mathcal{C}_n)}=\sum_{m=0}^{\infty} \frac{1}{m!} \int_{({\mathcal}{C}_n)^m} \det(\mathsf{K}_n({\mathbb{\omega}}_i,{\mathbb{\omega}}'_j))_{i,j=1}^n d{\mathbb{\omega}}_1,...,d{\mathbb{\omega}}_m.$$ The $m$th term can be decomposed as the sum $$\int_{({\mathcal}{C}_n^{{\varepsilon}})^m} \det( \mathsf{K}_n({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^n d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m+\int_{{\mathcal}{C}_n^m \setminus ({\mathcal}{C}_n^{{\varepsilon}})^m} \det(\mathsf{K}_n({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^n d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m.$$ Lemma \[max bound lemma\] along with Hadamard’s bound on the determinant of a matrix in terms of it’s row norms, implies that when ${\mathbb{\omega}}_1\in {\mathcal}{C}_n \setminus {\mathcal}{C}_n^{{\varepsilon}}$ and ${\mathbb{\omega}}_2,...,{\mathbb{\omega}}_m \in {\mathcal}{C}^n$, $$|\det(\overline{\mathsf{K}}_{n}({\mathbb{\omega}}_i, {\mathbb{\omega}}_j))_{i,j=1}^m| \leq m^{m/2} M^{m-1/2} L_4 n^{4/9} e^{-n^{1/3} \eta} \to 0 \text{ as } n \to \infty. \label{determinant bound}$$
Now let $R$ be the maximum length of the paths ${\mathcal}{C}_n$. The rescaled paths $\overline{{\mathcal}{C}_n}$ will always have length less than $n^{1/9}R$. We have
$$\begin{aligned}
&\int_{{\mathcal}{C}_n^{m} \setminus ({\mathcal}{C}_n^{\varepsilon})^m} |\det(\mathsf{K}_n({\mathbb{\omega}}_i, {\mathbb{\omega}}_j))_{i,j=1}^m| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m\nonumber\\
&\hspace{2cm}\leq
m\int_{{\mathcal}{C}_n \setminus {\mathcal}{C}_n^{{\varepsilon}}} d{\mathbb{\omega}}_1 \int_{{\mathcal}{C}_n^{m-1}} |\det(\mathsf{K}_n({\mathbb{\omega}}_i, {\mathbb{\omega}}_j))_{i,j=1}^m| d{\mathbb{\omega}}_2...d{\mathbb{\omega}}_m \nonumber\\
&\hspace{2cm}\leq m\int_{\overline{{\mathcal}{C}}_n \setminus \overline{{\mathcal}{C}}_n^{{\varepsilon}}} d\overline{{\mathbb{\omega}}}_1 \int_{\overline{{\mathcal}{C}}_n^{m-1}} |\det(\overline{\mathsf{K}}_{n}(\overline{{\mathbb{\omega}}}_i, \overline{{\mathbb{\omega}}}_j))_{i,j=1}^m| d\overline{{\mathbb{\omega}}}_2...d\overline{{\mathbb{\omega}}}_m \nonumber\\
&\hspace{2cm}\leq \int_{\overline{{\mathcal}{C}}_n \setminus \overline{{\mathcal}{C}}_n^{{\varepsilon}}} d\overline{{\mathbb{\omega}}}_1 \int_{\overline{{\mathcal}{C}}_n^{m-1}} m^{m/2} M^{(m-1)/2} L_4 n^{4/9} e^{-n^{1/3} \eta} d\overline{{\mathbb{\omega}}}_2...d\overline{{\mathbb{\omega}}}_m \nonumber \\
&\hspace{2cm}\leq m (n^{1/9}R)^{m} m^{m/2} M^{(m-1)/2} L_4 n^{4/9} e^{-n^{1/3} \eta}\nonumber\\
&\hspace{2cm}\leq e^{-n^{1/3} \eta} (n^{1/9})^mm^{1+m/2} (MR)^m n^{4/9}.
\end{aligned}$$
The first inequality follows from symmetry of the integrand in the ${\mathbb{\omega}}_i$. In the second inequality, we change variables from ${\mathbb{\omega}}_i$ to $\overline{{\mathbb{\omega}}}_i$. In the third inequality we use the first inequality of (\[determinant bound\]). In the fourth inequality, we use the fact that the total volume of our multiple integral is less than $(n^{1/9}R)^m$. In the fifth inequality we rewrite and use $M^m>M^{(m-1)/2}$.
So we have
$$\begin{gathered}
\sum_{m=1}^{\infty} \frac{1}{m!} \int_{{\mathcal}{C}_n^m \setminus ({\mathcal}{C}_n^{{\varepsilon}})^m} |\det( \mathsf{K}_n({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m \\ \leq
\sum_{m=1}^{\infty} \frac{1}{m!}e^{-n^{1/3} \eta} (n^{1/9})^mm^{1+m/2} (MR)^m n^{4/9} \\
\\ = n^{4/9} e^{-n^{1/3} \eta} \sum_{m=1}^{\infty} \frac{1}{m!} (MRn^{1/9})^m m^{1+m/2} \label{sum bound}
\end{gathered}$$
Applying Lemma \[sum\] with $C=MRn^{1/9}$ gives.
$$n^{4/9} e^{-n^{1/3} \eta} \sum_{m=1}^{\infty} \frac{1}{m!} (MRn^{1/9})^m m^{1+m/2} \leq n^{4/9}e^{-n^{1/3}} 16 (MRn^{1/9})^4 e^{2 (MR)^2 n^{2/9}} \xrightarrow[n \to \infty]{} 0.$$ Thus $$\lim_{n \to \infty}\det(1-\mathsf{K}_n)_{L^2({\mathcal}{C}_n)}=\lim_{n \to \infty} \det(1-\mathsf{K}_n)_{L^2({\mathcal}{C}_n^{{\varepsilon}})}. \label{fred equal 2}$$ Combining (\[fred equal\]) and (\[fred equal 2\]) concludes the proof of Proposition 2.11.
Convergence of the kernel
-------------------------
In this section we approximate $h_n(z)-h_n({\mathbb{\omega}})+n^{1/3}t(z-{\mathbb{\omega}})$ by its Taylor expansion near $\lambda$, and show that this does not change the asymptotics of our Fredholm determinant.
\[taylor approximation\] For sufficiently small ${\varepsilon}>0$,
$$\lim_{n \to \infty} \det(1-\mathsf{K}_n^{{\varepsilon}})_{L^2(\mathcal{C}_{{\varepsilon}}^{{\varepsilon}})}=\lim_{n \to \infty}\det(1-\mathsf{K}_{(x)})_{L^2(\mathcal{C}_{-1})},$$ where $$\mathsf{K}_{(x)}(\overline{u},\overline{u}')=\frac{1}{2\pi {\mathbf{i}}}\int_{D'} \frac{e^{s^3/3-xs}}{e^{u^3-xu}} \frac{dz}{(z-u)(z-u')},$$ and $$D'=(e^{-\pi {\mathbf{i}}/3} \infty, 0) \cup [0,e^{\pi {\mathbf{i}}/3} \infty).$$
Let $$\mathsf{K}(\overline{\omega},\overline{\omega}')=\frac{1}{2 \pi {\mathbf{i}}} \int_{D'} \frac{d\overline{z}}{(\overline{z}-\overline{{\mathbb{\omega}}})(\overline{z}-\overline{{\mathbb{\omega}}}')} e^{ f_1'''(\lambda) (\overline{z}^3-\overline{{\mathbb{\omega}}}^3)/6+f_2'(\lambda)(\overline{z}-\overline{{\mathbb{\omega}}})}, \label{def:K}$$ We have seen in Section \[localize\] that
$$\det(1-\mathsf{K}_n^{{\varepsilon}}({\mathbb{\omega}},{\mathbb{\omega}}'))_{L^{2}(\mathcal{C}_{{\varepsilon}}^{{\varepsilon}})}=\det(1-\overline{\mathsf{K}}_{n}^{{\varepsilon}}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}'))_{L^{2}(\mathcal{C}_{-1}^{n^{1/9}{\varepsilon}})}.$$
The proof will have two main steps. In the first step we use dominated convergence to show that $$\lim_{n \to \infty} \det(1-\overline{\mathsf{K}}_{n}^{{\varepsilon}}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}'))_{L^{2}(\mathcal{C}_{-1}^{n^{1/9}{\varepsilon}})} = \lim_{n \to \infty} \det(1-\overline{\mathsf{K}}_{(x)}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}'))_{L^{2}(\mathcal{C}_{-1}^{n^{1/9}{\varepsilon}})}.$$ In the second step we control the tail of the Fredholm determinant expansion to show that $$\lim_{n \to \infty} \det(1-\overline{\mathsf{K}}_{(x)}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}'))_{L^{2}(\mathcal{C}_{-1}^{n^{1/9}{\varepsilon}})}=\det(1-\overline{\mathsf{K}}_{(x)}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}'))_{L^{2}(\mathcal{C}_{-1})}.$$ In step $1$ we will use Lemma \[dom\] to establish dominated convergence.
We have the following pointwise convengences
$$\frac{\lambda+n^{-1/9} \overline{z}}{\lambda + n^{-1/9} \overline{{\mathbb{\omega}}}} \to 1,$$ and for $z=\lambda+n^{-1/9} \bar{z}, {\mathbb{\omega}}=\lambda+n^{-1/9}\overline{{\mathbb{\omega}}}$, $$n^{1/3}( f_1(z)-f_1({\mathbb{\omega}}))+n^{1/9} (f_2(z)-f_2({\mathbb{\omega}}))+r_n(z)-r_n({\mathbb{\omega}}) \rightarrow \frac{1}{6}f_1'''(\lambda)(\overline{z}^3-\overline{{\mathbb{\omega}}}^3)+f'_2(\lambda)(\overline{z}-\overline{{\mathbb{\omega}}}). \label{exponent}$$ Because $z$ is purely imaginary, for each $\overline{{\mathbb{\omega}}}, \overline{{\mathbb{\omega}}}'$, the exponentiating the right hand side of (\[exponent\]) gives a bounded function of $\overline{z}$ and $z/{\mathbb{\omega}}\leq \frac{|\lambda+{\varepsilon}|}{|\lambda-{\varepsilon}|}$. The left hand side of (\[exponent\]) can be chosen to be within $\delta/n^{1/9}$ of the right hand side by choosing ${\varepsilon}$ small by Taylor’s theorem, because all the functions on the left hand side are holomorphic in $B_{{\varepsilon}}(\lambda)$. Thanks to the quadratic denominator $\frac{1}{(\overline{z}-\overline{{\mathbb{\omega}}})(\overline{z}-\overline{{\mathbb{\omega}}}')}$, we can apply dominated convergence to get
$$\overline{\mathsf{K}}_{n}^{{\varepsilon}}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}') \xrightarrow[n \to \infty]{pointwise} \frac{1}{2\pi {\mathbf{i}}} \int_{{\mathbf{i}}{\mathbb{R}}} \frac{d\overline{z}}{(\overline{z}-\overline{{\mathbb{\omega}}})(\overline{z}-\overline{{\mathbb{\omega}}}')} e^{ f_1'''(\lambda) (\overline{z}^3-\overline{{\mathbb{\omega}}}^3)/6+f_2'(\lambda)(\overline{z}-\overline{{\mathbb{\omega}}})}. \label{conv}$$
Because the integrand on the right hand side of (\[conv\]) has quadratic decay in $\overline{z}$, we can deform the contour from $\mathcal{\gamma}_0$ to $D'$ without changing the integral, so the right hand side is equal to $\mathsf{K}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}')$ from \[def:K\]. Now by Lemma \[dom\] we can apply dominated convergence to the expansion of the Fredholm determinant $\det(1-\overline{\mathsf{K}}_{n}^{{\varepsilon}})_{L^2(\mathcal{C}_{-1}^{n^{1/9}{\varepsilon}})}$, to get
$$\lim_{n \to \infty}\det(1-\overline{\mathsf{K}}_{n}^{{\varepsilon}})_{L^2(\mathcal{C}_{-1}^{n^{1/9}{\varepsilon}})}=\lim_{n \to \infty}\det(1-\mathsf{K})_{L^2(\mathcal{C}_{-1}^{n^{1/9} {\varepsilon}})}.$$
Now we make the change of variables $s=-(f_2'(\lambda)/x) \overline{z}$, $u=-(f_2'(\lambda)/x) \overline{{\mathbb{\omega}}}$, and $u'=-(f_2'(\lambda)/x) \overline{{\mathbb{\omega}}}'$. Keeping in mind that $-2(f_2'(\lambda)/x)^3=f_1'''(\lambda)$, we get
$$\mathsf{K}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}')=\mathsf{K}_{(x)}(u,u')=\frac{1}{2\pi {\mathbf{i}}}\int_{D'} \frac{e^{s^3/3-xs}}{e^{u^3/3-xu}} \frac{ds}{(s-u)(s-u')}.$$ Recall the expansion: $$\det(1-\mathsf{K}_{(x)})_{L^2(\mathcal{C}_{-1}^{{\varepsilon}})}=\sum_{m=0}^{\infty} \frac{(-1)^m}{m!} \int_{\mathcal{C}_{-1}^m} \det(\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m,$$
where $\mathcal{C}_{-1}=(e^{-2\pi {\mathbf{i}}/3} \infty, 1] \cup (1, e^{2\pi {\mathbf{i}}/3} \infty)$, and $\mathcal{C}_{-1}^m$ is a product of $m$ copies of $\mathcal{C}_{-1}.$
$$\begin{gathered}
| \det(1-\mathsf{K}_{(x)})_{L^2(\mathcal{C}_{-1})}-\det(1-\mathsf{K}_{(x)})_{L^2(\mathcal{C}_{-1}^{{\varepsilon}})} | \leq \\ \sum_{m=0}^{\infty} \frac{(-1)^m}{m!} \int_{\mathcal{C}_{-1}^m \setminus (\mathcal{C}_{-1}^{n^{1/9} {\varepsilon}})^m} |\det (\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m,
\end{gathered}$$
so to conclude the proof of the proposition, we are left with showing that
$$\sum_{m=0}^{\infty} \frac{1}{m!} \int_{\mathcal{C}_{-1}^m \setminus (\mathcal{C}_{-1}^{n^{1/9} {\varepsilon}})^m} |\det (\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m \xrightarrow[n \to \infty]{} 0 \label{step 2}$$
Note that $$\begin{gathered}
\int_{\mathcal{C}_{-1}^m \setminus (\mathcal{C}_{-1}^{ n^{1/9}{\varepsilon}})^m} |\det (\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m \leq \\ m \int_{\mathcal{C}_{-1} \setminus \mathcal{C}_{-1}^{n^{1/9}{\varepsilon}}} \int_{\mathcal{C}_{-1}^{m-1}} |\det (\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m.
\end{gathered}$$ Set $$M_1=\int_{D'}|\overline{z}e^{ f_1'''(\lambda) \overline{z}^3/6+f_2'(\lambda)\overline{z}}|d\overline{z}<\infty.$$
Then $\mathsf{K}_{(x)}({\mathbb{\omega}},{\mathbb{\omega}}') \leq M_1 e^{-|{\mathbb{\omega}}|^3-x |\omega|}$, and Hadamard’s bound gives $$|\det (\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m| \leq m^{m/2}M_1^m \prod_{i=1}^m |e^{-{\mathbb{\omega}}_i^3/3+x \omega_i}|.$$ We have
$$\begin{aligned}
&\int_{\mathcal{C}_{-1} \setminus \mathcal{C}_{-1}^{ n^{1/9}{\varepsilon}}} \int_{{\mathcal}{C}_{-1}^{m-1}} |\det (\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m\nonumber\\
&\hspace{2cm}\leq M_1\int_{\mathcal{C}_{-1} \setminus \mathcal{C}_{-1}^{ n^{1/9}{\varepsilon}}} \int_{{\mathcal}{C}_{-1}^{m-1}} \prod_{i=1}^m |e^{-{\mathbb{\omega}}_i^3/3+x \omega_i}| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m\nonumber \\
&\hspace{2cm}\leq m^{1+m/2} M_1^m M_2^{m-1} \int_{\mathcal{C}_{-1} \setminus \mathcal{C}_{-1}^{n^{1/9}{\varepsilon}}} | e^{-{\mathbb{\omega}}_1^3+x \omega_1}| d\omega_1, \label{to 0}
\end{aligned}$$
where $M_2=\int_{\mathcal{C}_{-1}} |e^{-{\mathbb{\omega}}^3-x \omega}| d \omega <\infty$ because $-{\mathbb{\omega}}^3$ lies on the negative real axis. (\[to 0\]) goes to zero because $n^{1/9} {\varepsilon}\to \infty$. So $$\int_{\mathcal{C}_{-1} \setminus \mathcal{C}_{-1}^{ n^{1/9}{\varepsilon}}} \int_{\mathcal{C}_{-1}^{m-1}} \left|\det (\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m\right| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m \xrightarrow[n \to \infty]{} 0.$$ Note also that
$$\begin{aligned}
\int_{\mathcal{C}_{-1}^{m}\setminus (\mathcal{C}_{-1}^{ n^{1/9}{\varepsilon}})^m} \left|\det (\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m \right| d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m &\leq
\int_{\mathcal{C}_{-1}^{m}} |\det (\mathsf{K}_{(x)}({\mathbb{\omega}}_i,{\mathbb{\omega}}_j))_{i,j=1}^m | d{\mathbb{\omega}}_1...d{\mathbb{\omega}}_m \\ &\leq m^{1+m/2} M_1 M_2^m.
\end{aligned}$$
By Stirling’s approximation $$\sum_{m=0}^{\infty} \frac{1}{m!} m^{1+m/2} M_1^m M_2^m< \infty.$$ So by dominated convergence (\[step 2\]) holds which concludes the proof of Proposition \[taylor approximation\].
Reformulation of the kernel
---------------------------
Now we use the standard $\det(1+AB)=\det(1+BA)$ trick [@FreeEnergyCorwin Lemma 8.6] to identify $\det(1-\mathsf{K}_{(x)})_{L^2(\mathcal{C}_{-1})}$ with the Tracy-Widom cumulative distribution function.
For $x \in \mathbb{R}$, $$\det(1-\mathsf{K}_{(x)})_{L^2(\mathcal{C}_{-1})} =\det(1-\mathsf{K}_{\mathrm{Ai}})_{L^2(x,\infty)}.$$
First note that because ${\mathfrak{Re}}[z-{\mathbb{\omega}}] > 0$ along the contours we have chosen, we can write $$\frac{1}{z-{\mathbb{\omega}}}=\int_{{\mathbb{R}}_+} e^{-\lambda(z-{\mathbb{\omega}})} d\lambda.$$ Now let $A:L^2(\mathcal{C}_{-1}) \to L^2({\mathbb{R}}_+)$, and $B:L^2({\mathbb{R}}_+) \to L^2(\mathcal{C}_{-1})$ be defined by the kernels $$\begin{aligned}
A(\omega,\lambda)&=e^{-{\mathbb{\omega}}^3/3+{\mathbb{\omega}}(x+\lambda)},\\
B(\lambda,\omega')&=\int_{e^{-\pi {\mathbf{i}}/3} \infty}^{e^{\pi {\mathbf{i}}/3} \infty} \frac{dz}{2\pi {\mathbf{i}}} \frac{e^{z^3/3-z(x+\lambda)}}{z-{\mathbb{\omega}}'}.
\end{aligned}$$ We compute $$\begin{aligned}
AB({\mathbb{\omega}},{\mathbb{\omega}}') &=\int_{{\mathbb{R}}_+} e^{-{\mathbb{\omega}}^3/3+{\mathbb{\omega}}(x+\lambda)} \int_{e^{-\pi {\mathbf{i}}/3}\infty}^{e^{\pi {\mathbf{i}}/3} \infty} \frac{dz}{2 \pi {\mathbf{i}}} \frac{e^{z^3/3-z(x+\lambda)}}{z-{\mathbb{\omega}}'}\\
&= \frac{1}{2\pi {\mathbf{i}}} \int_{e^{-\pi {\mathbf{i}}/3} \infty}^{e^{\pi {\mathbf{i}}/3} \infty} \frac{e^{z^3/3-zx}}{e^{{\mathbb{\omega}}^3/3-{\mathbb{\omega}}x}} \frac{dz}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')}\\
&=\mathsf{K}_{(x)}({\mathbb{\omega}},{\mathbb{\omega}}').
\end{aligned}$$ Similarly, $$BA(s,s')=\frac{1}{2\pi {\mathbf{i}}} \int_{e^{-2 \pi {\mathbf{i}}/3} \infty}^{e^{2 \pi {\mathbf{i}}/3} \infty} d{\mathbb{\omega}}\frac{1}{2\pi {\mathbf{i}}} \int_{e^{-\pi {\mathbf{i}}/3} \infty}^{e^{\pi {\mathbf{i}}/3} \infty} dz \frac{e^{z^3/3-z(x+s)}}{e^{{\mathbb{\omega}}^3/3-{\mathbb{\omega}}(x+s')}} \frac{1}{(z-{\mathbb{\omega}})}
=\mathsf{K}_{{\mathrm}{Ai}}(x+s,x+s').$$ Because both $A$ and $B$ are Hilbert-Schmidt operators, we have $$\begin{gathered}
\det(1-\mathsf{K}_{(x)})_{L^2(\mathcal{C})}=\det(1-AB)_{L^2({\mathbb{R}}_+)} = \det(1-BA)_{L^2({\mathbb{R}}_+)}\\ =\det(1-\mathsf{K}_{{\mathrm}{Ai}})_{L^2(x,\infty)}=F_{\textrm{GUE}}(x).\end{gathered}$$
Constructing the contour $\mathcal{C}_n$ {#Cn}
========================================
This section is devoted to constructing the contours $\mathcal{C}_n$ and proving Lemma \[C bound\]. We will prove several estimates for $n^{1/3}{\mathbb{\omega}}+h_n({\mathbb{\omega}})$; then we will construct the contour ${\mathcal}{C}_n$, and prove it satisfies the properties of Lemma \[C bound\]. We begin by proving that we can approximate $n^{1/3}{\mathbb{\omega}}+h_n({\mathbb{\omega}})$ by $n^{1/3}f_1({\mathbb{\omega}})$ away from $0$.
Estimates away from 0: proof of Lemma \[chaos control\]
-------------------------------------------------------
Both inequalities for $|f_2|=\frac{b \sigma x}{{\mathbb{\omega}}}$ follow from the fact that $f_2$ and $f_2'$ are bounded on ${\mathbb{C}}\setminus B_{{\varepsilon}}(0)$. Let $y=1/{\mathbb{\omega}}$, and let $m=n^{-1/9}$. Define the function $g(y,m)=r_n({\mathbb{\omega}}).$ First we prove (\[control\]). Note that $h_n({\mathbb{\omega}})$ is holomorphic in $y$ and $m$ except when $n =\infty$, $n^{1/3} \omega=0,-a-b$. By Taylor expanding $h_n({\mathbb{\omega}})$, we see that $r_n({\mathbb{\omega}})=g(y,m)$ is holomorphic in $y$ and $m$, except at points $(y,m)$ such that $n^{1/3} \omega=0,-a-b$, in particular there is no longer a pole when $n=\infty$. Thus for any $N$, $g(y,m)$ is holomorphic with variables $y$ and $m$, in the region $U=\{(y,m):n>N, {\mathbb{\omega}}> |a+b|/N^{1/3}\}$, because in this region $n^{1/3} \omega>|a+b|$. The region $U_{{\varepsilon}}=\{(y,m): n>N, {\mathbb{\omega}}\geq \frac{|a+b|+{\varepsilon}}{N^{1/3}} \}$ is compact in the variables $y$ and $m$, and because $U_{{\varepsilon}} \subset U$, the function $g(y,m)$ is holomorphic in the region $U_{{\varepsilon}}$. Thus $g(y,m)=r_n({\mathbb{\omega}})$ is bounded by a constant $C$ in the region $U_{{\varepsilon}}$.
Now we prove (\[derivative control\]). For any $\delta$, pick an arbitrary ${\varepsilon}$ and an $N_{\delta}$ large enough that $\frac{|a+b|+{\varepsilon}}{N_{\delta}^{1/3}} \leq \delta$. Because $g(y,m)=r_n({\mathbb{\omega}})$ is holomorphic in the variables $y$ and $m$ in the compact set $U_{{\varepsilon}}$, the function $\frac{\partial}{\partial y} g(y,m)=-{\mathbb{\omega}}^2 r_n'({\mathbb{\omega}})$, is also holomorphic in $y,m$. So $|{\mathbb{\omega}}^2 r_n'({\mathbb{\omega}})| \leq C$ on $U_{{\varepsilon}}$. We rewrite as $|r_n'({\mathbb{\omega}})| \leq C/|{\mathbb{\omega}}|^2$, and this gives $|r_n'({\mathbb{\omega}})| \leq \frac{C}{|\delta|^2} \leq C',$ on the set $U_{{\varepsilon}} \cap (\mathbb{N} \times B_{\delta}(0)^c)$. But by our choice of $N_{\delta}$, we have $U_{{\varepsilon}} \cap (\mathbb{N} \times B_{\delta}(0)^c)$ is just the set $\{(y,m): n \geq N_{\delta}, |{\mathbb{\omega}}| \geq \delta \}$.
Estimates near 0
----------------
The function $n^{1/3}f_1({\mathbb{\omega}})$ only approximates $-n^{1/3}t {\mathbb{\omega}}-h_n({\mathbb{\omega}})$ well away from $0$. In this section we give two estimates for $-n^{1/3}t {\mathbb{\omega}}-h_n({\mathbb{\omega}})$: one in Lemma \[one third bound\] when ${\mathbb{\omega}}$ is of order $n^{-1/3}$ and one in Lemma \[delta bound\] when ${\mathbb{\omega}}$ is of order $n^{\delta-1/3}$ for $\delta \in (0,1/3).$ Together with Lemma \[chaos control\] which gives an estimate when ${\mathbb{\omega}}$ is of order $1$, this will give us the tools we need to control $-n^{1/3}t {\mathbb{\omega}}-h_n({\mathbb{\omega}})$ along $\mathcal{C}_n$. First to prove the bound in Lemma \[one third bound\], we choose a path which crosses the real axis at $-a$, between the poles at $0$ and $-a-b$ before rescaling $\tilde{h}_n$ to $h_n$. We show that after the rescaling, we can bound ${\mathfrak{Re}}[-n^{-1/3} {\mathbb{\omega}}-h_n({\mathbb{\omega}})]$ on this path for small ${\mathbb{\omega}}$.
\[one third bound\] Fix any $c_0>1$ and let $s=c_0(a+b)$. For $C=\log\left(\sqrt{s^2+a^2}\right)-\log(s)>0$, we have $$\limsup_{n \to \infty} \frac{1}{n} \sup_{y \in [-s,s]} {\mathfrak{Re}}[h_n(\lambda)-h_n({\mathbf}{i}n^{-1/3}y-n^{-1/3}a)]< -C.$$
Let $y \in [-s,s]$ and expand $e^{{\mathfrak{Re}}[h_n(\lambda)-h_n(iy-a n^{-1/3})]}$ to get
$$\left(\frac{y}{\sqrt{y^2+a^2}}\right)^n \left(\frac{y}{\sqrt{y^2+b^2}} \right)^m \left(\frac{n^{1/3} \lambda}{n^{1/3} \lambda+a} \right)^n \left(\frac{a+b+n^{1/3} \lambda}{n^{1/3} \lambda+a} \right)^m.$$ The third factor is always less than $1$. For sufficiently large $n$, the second factor times the fourth factor is less than $1$, because $|y| \leq |s|$ while $n^{1/3} \lambda \to \infty$. We can bound the first factor by $$\left|\frac{y}{\sqrt{y^2+a^2}}\right|^n \leq \left(\frac{s}{\sqrt{s^2+a^2}}\right)^n = e^{-nC},$$ with $C=\log\left(\sqrt{(s^2+a^2)}\right)-\log(s)$.
Next we will prove the estimate for ${\mathbb{\omega}}$ of order $n^{\delta-1/3}$. In this proof we will consider ${\mathbb{\omega}}$ of the form ${\mathbb{\omega}}=-n^{-1/3}a+{\mathbf{i}}n^{\delta-1/3}c(a+b)$, choose $c$ sufficiently large, then let $n \to \infty$. The largest term in the expansion of $-n^{-1/3}{\mathbb{\omega}}-h_n({\mathbb{\omega}})$ will be of order $\frac{n^{1-2\delta}}{c^2}$. We introduce the following definition to let us ignore the terms which are negligible compared to $\frac{n^{1-2\delta}}{c^2}$ uniformly in $\delta.$
Let $A$ and $B$ be functions depending on $n$ and $c$, we say $A \sim_{\delta} B$ or $A$ is $\delta$-equivalent to $B$, if for sufficiently large $c$ and $n$, $$|A-B| \leq \frac{n^{2/3-2\delta}}{c^2}M_1+\frac{n^{1-3\delta}}{c^3}M_2+\frac{n^{4/9-\delta}}{c}M_3.$$ for some constants $M_1,M_2,M_3$ independent of $c$ and $n$.
Now we prove the estimate.
For all $\delta \in (0,1/3)$, setting ${\mathbb{\omega}}=-n^{-1/3}a+{\mathbf{i}}n^{\delta-1/3}c(a+b)$, gives $${\mathfrak{Re}}[n^{1/3}t {\mathbb{\omega}}+h_n({\mathbb{\omega}})] \sim_{\delta} {\mathfrak{Re}}[n^{1/3}f_1({\mathbb{\omega}})] \sim_{\delta} M\frac{n^{1-2\delta}}{c^2},$$ where $\sim_{\delta}$ is defined in Definition 8. \[delta bound\]
The proof of this Lemma \[delta bound\] comes from Taylor expanding $h_n$ and keeping track of the order of different terms with respect to $n$ and $c$.
Recall that $$h_n({\mathbb{\omega}})=-n \log \left(1+\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)+m \log\left(1+\frac{b}{a+n^{1/3}{\mathbb{\omega}}}\right). \label{hn}$$
For $|n^{1/3} {\mathbb{\omega}}|>a$ and $|a+n^{1/3}{\mathbb{\omega}}|>b$, we can Taylor expand in $n^{1/3} {\mathbb{\omega}}$ to get
$$h_n({\mathbb{\omega}})=-n \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)^k+m \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{b}{a+n^{1/3}{\mathbb{\omega}}}\right)^k.$$
Let ${\mathbb{\omega}}=-n^{-1/3}a+{\mathbf{i}}n^{\delta-1/3}c(a+b)$ for $\delta \in (0,1/3)$, so $ |n^{1/3} {\mathbb{\omega}}|, |a+n^{1/3}{\mathbb{\omega}}| > n^{\delta}c(a+b)>c(a+b)$, for a constant $c$ to be determined later. If $c>2$, we have
$$\sum_{k=1}^{\infty} \left|\left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)\right|^k \leq \sum_{k=1}^{\infty} \left(\frac{b}{n^{\delta}c(a+b)}\right)^k \leq \frac{a}{n^{\delta}c(a+b)}\sum_{k=0}^{\infty}\left(\frac{1}{2}\right)^k \leq \frac{2a}{n^{\delta}c(a+b)}=\frac{n^{-\delta}}{c}M, \label{sum 1}$$
and
$$\sum_{k=1}^{\infty} \left|\left(\frac{b}{a+n^{1/3}{\mathbb{\omega}}}\right)\right|^k \leq \sum_{k=1}^{\infty} \left(\frac{a}{n^{\delta}c(a+b)}\right)^k \leq \frac{a}{n^{\delta}c(a+b)}\sum_{k=0}^{\infty} \left(\frac{1}{2}\right)^k=\frac{2a}{n^{\delta}c(a+b)}=\frac{n^{-\delta}}{c} M. \label{sum 2}$$
In what follows, we will use (\[sum 1\]) or (\[sum 2\]) when we say that an infinite sum is $\delta$-equivalent to its first term.
We examine the first term in (\[hn\]). $$\begin{aligned}
-n\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)^k &= -\left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)+\frac{1}{2}\left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)^2-n\sum_{k=3}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)^k,\\
&\sim_{\delta} -\left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)+\frac{1}{2} \left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)^2.
\end{aligned}$$ where the $\delta-$equivalence follows because $\left|n\sum_{k=3}^{\infty} \frac{(-1)^{k+1}}{k} \left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)^k\right| \leq \frac{n^{1-3\delta}}{c^3}M$ for some $M$ by (\[sum 1\]).
Recall that $$m \sum_{k=1}^{\infty} \left(\frac{b}{a+n^{1/3}{\mathbb{\omega}}}\right)^k=\left[\left(\frac{a}{b}\right)n+dn^{2/3}+\sigma x n^{4/9}\right]\sum_{k=1}^{\infty}\left(\frac{b}{a+n^{1/3}{\mathbb{\omega}}}\right)^k.$$ We decompose this series as three sums. First the $\left(\frac{a}{b}\right)n$ term gives $$\begin{gathered}
\frac{a}{b} n \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{b}{a +n^{1/3}{\mathbb{\omega}}}\right)^k= \\ n\left(\frac{a}{b}\right) \left(\frac{b}{a +n^{1/3}{\mathbb{\omega}}}\right)-\frac{n}{2}\left(\frac{a}{b}\right)\left(\frac{b}{a +n^{1/3}{\mathbb{\omega}}}\right)^2 +\frac{a}{b} n \sum_{k=3}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{b}{a +n^{1/3}{\mathbb{\omega}}}\right)^k
\\ \sim_{\delta} n\left(\frac{a}{b}\right)\left(\frac{b}{a +n^{1/3}{\mathbb{\omega}}}\right)-\frac{n}{2}\left(\frac{b}{a +n^{1/3}{\mathbb{\omega}}}\right)^2,
\end{gathered}$$ because $\left|-\frac{a}{b} n \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{b}{a +n^{1/3}{\mathbb{\omega}}}\right)^k\right| \leq Mn^{1-3\delta}/c^3$ for some $M$. The second term is $$\begin{aligned}
dn^{2/3} \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right)^k &= dn^{2/3}\left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right)-dn^{2/3} \sum_{k=2}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right)^k\\
&\sim_{\delta} dn^{2/3}\left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right)
\end{aligned}$$ because $\left|dn^{2/3} \sum_{k=2}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right)^k\right| \leq Mn^{2/3-2\delta}/c^2$ for some $M$. The third term is $$n^{4/9} \sigma x \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right)^k \sim_{\delta} 0,$$ because the full sum $\left|n^{4/9} \sigma x \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right)^k\right| \leq \frac{Mn^{4/9-\delta}}{c}$ for some $M$. Now we have shown $$-n \log\left(1+\frac{a}{n^{1/3} {\mathbb{\omega}}}\right) \sim_{\delta} -n^{2/3} \frac{a}{{\mathbb{\omega}}}+n^{1/3} \frac{a^2}{2{\mathbb{\omega}}^2}, \label{term 1}$$ $$\begin{gathered}
m \log\left(1+\frac{b}{a+n^{1/3}{\mathbb{\omega}}}\right) \sim_{\delta} \\ n \left(\frac{a}{b}\right) \left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right)-n \left(\frac{a}{2b}\right) \left(\frac{b}{a+n^{1/3}{\mathbb{\omega}}}\right)^2+dn^{2/3} \left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right). \label{term 3}
\end{gathered}$$ Adding (\[term 1\]) and (\[term 3\]) together yields $$\begin{gathered}
h_n({\mathbb{\omega}}) \sim_{\delta} -n^{2/3} \frac{a}{{\mathbb{\omega}}}+n^{1/3} \frac{a^2}{2{\mathbb{\omega}}^2}+n \left(\frac{a}{b}\right) \left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right) \\
-n \left(\frac{a}{2b}\right) \left(\frac{b}{a+n^{1/3}{\mathbb{\omega}}}\right)^2+dn^{2/3} \left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right). \label{all terms} \end{gathered}$$ Adding the first and third terms from (\[all terms\]) gives the following cancellation. $$\begin{gathered}
-n^{2/3} \frac{a}{{\mathbb{\omega}}}+n \left(\frac{a}{b}\right) \left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right)= \\
-n^{2/3} \frac{a}{{\mathbb{\omega}}}+n^{2/3} \frac{a}{{\mathbb{\omega}}}\left[1-\frac{a}{n^{1/3} {\mathbb{\omega}}}+\sum_{k=2}^{\infty}(-1)^k\left(\frac{a}{n^{1/3} {\mathbb{\omega}}}\right)^k\right] \sim_{\delta} -n^{1/3}\frac{ a^2}{{\mathbb{\omega}}^2},
\end{gathered}$$ thus $$h_n({\mathbb{\omega}}) \sim_{\delta} -n^{1/3}\left(\frac{ a^2}{2{\mathbb{\omega}}^2} \right)-n \left(\frac{a}{2b}\right) \left(\frac{b}{a+n^{1/3}{\mathbb{\omega}}}\right)^2+dn^{2/3} \left(\frac{b}{a+n^{1/3} {\mathbb{\omega}}}\right).$$ When we expand $\frac{b}{a+n^{1/3} {\mathbb{\omega}}}=\frac{b}{n^{1/3} {\mathbb{\omega}}}+\left(\frac{b}{n^{1/3} {\mathbb{\omega}}}\right)\sum_{k=1}^{\infty}\left(\frac{-a}{n^{1/3} {\mathbb{\omega}}}\right)^k,$ we see that because $n^{1/3} {\mathbb{\omega}}\sim_{\delta} n^{\delta}{\mathbf{i}}c(a+b)$, the sum is of order $1/c$ times the first term. So we can take only the first terms in our expansion, just as when we Taylor expand. This approximation leads the $n^{2/3}$ terms to cancel giving $$h_n({\mathbb{\omega}}) \sim_{\delta} -n^{1/3} \left(\frac{a^2+ab}{2{\mathbb{\omega}}^2}\right) +dn^{1/3} \left(\frac{b}{ {\mathbb{\omega}}}\right) \sim_{\delta} n^{1/3}\left(f_1({\mathbb{\omega}})-t{\mathbb{\omega}}\right).$$
This implies that ${\mathfrak{Re}}[n^{1/3}t {\mathbb{\omega}}+h_n({\mathbb{\omega}})] \sim_{\delta} {\mathfrak{Re}}[n^{1/3}f_1({\mathbb{\omega}})].$ Completing the first $\delta$-equivalence in the statement of Lemma \[delta bound\].
Now observe that in $${\mathfrak{Re}}[n^{1/3}f_1({\mathbb{\omega}})]={\mathfrak{Re}}\left[n^{1/3}\left(t{\mathbb{\omega}}-\frac{a(a+b)}{2{\mathbb{\omega}}^2}+\frac{bd}{{\mathbb{\omega}}}\right)\right],$$ we can bound the first term $|{\mathfrak{Re}}[n^{1/3}t{\mathbb{\omega}}]| \leq n^{\delta}M$. We can bound the third term by ${\mathfrak{Re}}\left[n^{1/3}\frac{bd}{{\mathbb{\omega}}}\right] \leq M\frac{n^{2/3-\delta}}{c}$. For the second term, we have $\left|\frac{a(a+b)}{2{\mathbb{\omega}}^2}\right|\sim_{\delta} \left(\frac{a(a+b)}{2}\right)\left(\frac{n^{1-2\delta}}{c}\right).$ Thus $${\mathfrak{Re}}[n^{1/3}f_1({\mathbb{\omega}})] \sim_{\delta} \left(\frac{a(a+b)}{2}\right)\left(\frac{n^{1-2\delta}}{c}\right).$$ This gives the second $\delta$-equivalence in the statement of Lemma \[delta bound\], and completes the proof.
Construction of the contour $\mathcal{C}_n$
-------------------------------------------
To construct the contour $\mathcal{C}_n$ we will start with lines departing from $\lambda$ at angles $e^{\pm 2 \pi {\mathbf{i}}/3}$, and with a vertical line $-n^{1/3}a+{\mathbf{i}}\mathbb{R}$. We will cut both these infinite contours off at specific values $q$ and $p$ respectively which allow us to use our estimates from the previous section on these contours. We will then connect these contours using the level set $\{z: Re[-f_1(z)]=-f_1(\lambda)-{\varepsilon}\}$. The rest of this section is devoted to finding the values $p$ and $q$, showing that our explanation above actually produces a contour, and controlling the derivative of $f_1$ on the vertical segment near $0$.
We note $$f_1(\lambda)=3t^{2/3} \left(\frac{a(a+b)}{2}\right)^{1/3}>0, \label{positive}$$ and let $$p=\sqrt{\frac{1}{3}\left(\frac{a(a+b)}{2t}\right)^{2/3}}>0. \label{p definition}$$
By simple algebra, we see that ${\mathfrak{Re}}[-f_1(\pm {\mathbf{i}}y)]< {\mathfrak{Re}}[-f_1(\lambda)]<0$, when $y < p$, with equality at $y=p$.
\[derivative\] $\frac{d}{dy} {\mathfrak{Re}}[-f_1(n^{-1/3}a+{\mathbf{i}}y)]$ is positive for $y \in [n^{-1/3}|a+b|, p]$, and negative for $y \in [-n^{-1/3}|a+b|, -p].$
We compute $$\begin{aligned}
\frac{d}{dy} {\mathfrak{Re}}[f_1(n^{-1/3}a+{\mathbf{i}}y)]=&-{\mathfrak{Im}}({\mathfrak{Re}}[f_1(n^{-1/3}a+{\mathbf{i}}y)]) \\
=-\frac{y^3a(a+b)}{|n^{-1/3}a+{\mathbf{i}}y|^6}&+ \frac{a^2(a+b)n^{-2/3}y}{|n^{-1/3}a+{\mathbf{i}}y|^6}+\frac{ 3a^2(a+b)b n^{-1/3} y}{2b \lambda |n^{-1/3} a +{\mathbf{i}}y|^4}. \label{derivative near 0} \end{aligned}$$ Note that for $y \in [n^{-1/3}|a+b|, p] \cup [-n^{-1/3}|a+b|, -p]$, we have $|n^{-1/3} a +{\mathbf{i}}y| \sim |y|$, so the first term of (\[derivative near 0\]) is of order $y^{-3}$ and the third term of (\[derivative near 0\]) is of order $y^{-3}n^{-1/3}$. So for large enough $n$, the third term of (\[derivative near 0\]) is very small compared to the first term. For $y=\pm n^{-1/3}|a+b|,$ we have $| n^{-1}a(a+b)^4|=|y^3a(a+b)|>|a(a+b)n^{-2/3}ay|=|a^2(a+b)^2n^{-1/3}|$, and the derivative of $y^3a(a+b)$ is larger than the derivative of $a(a+b)n^{-2/3}ay$ for $y \in [n^{-1/3}|a+b|, p] \cup [-n^{-1/3}|a+b|, -p]$, so the first term of (\[derivative near 0\]) has larger norm than the second term for $y \in [n^{-1/3}|a+b|, p] \cup [-n^{-1/3}|a+b|, -p]$. Thus the sign $\frac{d}{dy} {\mathfrak{Re}}[-f_1(n^{-1/3}a+{\mathbf{i}}y)]$ is determined by the first term of (\[derivative near 0\]) in these intervals.
Now we can define the contour ${\mathcal}{C}_n$. We will give the definition, and then justify that it gives a well defined contour.
Let $q>0$ be a fixed real number such that for $0<y\leq q$, $\frac{d}{dy}{\mathfrak{Re}}[-f_1(\lambda\pm y e^{\pm 2 \pi {\mathbf{i}}/3})]<0$. Let $$\begin{gathered}
s=\max\left\lbrace {\mathfrak{Re}}[-f_1(\lambda+ q e^{-2 \pi {\mathbf{i}}/3})], {\mathfrak{Re}}[-f_1(\lambda+ q e^{ 2 \pi {\mathbf{i}}/3})], \right. \\ \left . {\mathfrak{Re}}[-f_1(n^{-1/3}(a-{\mathbf{i}}|a+b|))], {\mathfrak{Re}}[-f_1(n^{-1/3}(a+{\mathbf{i}}|a+b|))]\right\rbrace.\end{gathered}$$ Let $\alpha$ be the contourline $\alpha=\{{\mathbb{\omega}}:{\mathfrak{Re}}[-f_1({\mathbb{\omega}})]=s\}$, and define the set $$S_n=\{\lambda+ y e^{\pm 2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\} \cup \alpha \cup [-an^{-1/3}-{\mathbf{i}}p,-an^{-1/3}+{\mathbf{i}}p].$$ For sufficiently large $n$, define the path ${\mathcal}{C}_n$ to begin where $\alpha$ intersects $\{\lambda+ y e^{ -2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\}$, follow the path $\{\lambda+ y e^{ -2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\}$ toward $y=0$, then follow the path $\{\lambda+ y e^{ 2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\}$ until it intersects $\alpha.$ ${\mathcal}{C}_n$ then follows $\alpha$ in either direction (pick one arbitrarily) until it intersects $[-an^{-1/3}-{\mathbf{i}}p,-an^{-1/3}+{\mathbf{i}}p]$ in the upper half plane. ${\mathcal}{C}_n$ then follows the path $[-an^{-1/3}-{\mathbf{i}}p,-an^{-1/3}+{\mathbf{i}}p]$ toward $-an^{-1/3}- {\mathbf{i}}p$ until it intersects $\alpha$ in the negative half plane. Then ${\mathcal}{C}_n$ follows $\alpha$ in either direction (pick one arbitrarily) until it reaches its starting point where it intersects $\{\lambda+ y e^{ -2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\}$. See Figure \[Cn picture\]\[Cn def\]
We see that the $q$ in Definition \[Cn def\] exists by applying Taylor’s theorem along with the fact that $f_1'''(\lambda)>0$, and the $f_1'(\lambda)=f_1''(\lambda)=0$.
\[N’\] The sets $\{\lambda+ y e^{ 2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\}$ and $\{\lambda+ y e^{ -2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\}$ both intersect $\alpha$ at exactly one point. Lemma \[N\] and Lemma \[N’\] will show that $\mathcal{C}_n$ is a well defined contour.
This follows from the definition of $q$ and $s$.
\[N\] There exists $N>0$ such that for all $n>N$, the sets $[n^{-1/3}+{\mathbf{i}}n^{-1/3}|a+b|, n^{-1/3}a+p]$ and $[-an^{-1/3}-n^{-1/3}|a+b|, -an^{-1/3}-p]$ both intersect $\alpha$ exactly once.
This is true because $${\mathfrak{Re}}[-f_1(-n^{-1/3}(a\pm{\mathbf{i}}|a+b|))]<{\mathfrak{Re}}[-f_1(\lambda)]. \label{small a+b}$$ by the contour lines in Figure \[contour\]. This in addition to Lemma \[derivative\], and (\[positive\]) implies the lemma.
![${\mathcal}{C}_n$ is the thick, colored piecewise smooth curve, the contour lines $\{z:{\mathfrak{Re}}[-f_1(z)]=f_1(\lambda)\}$ are the thin black curves. On the right side of the image we see ${\mathcal}{C}_n$ as a thick blue curve sandwiched between the contour lines. On the left we zoom in near 0 and see ${\mathcal}{C}_n$ pass the real axis as a dotted line to the left of zero. The contour lines meet at the point $0$ on the left and $\lambda$ on the right. We will now describe what section of the proof of Theorem \[C bound\] bounds $h_n(z)-h_n({\mathbb{\omega}})+nt^{1/3}(z-{\mathbb{\omega}})$ on different portions of ${\mathcal}{C}_n$. The diagonal segments of ${\mathcal}{C}_n$ near $\lambda$ are bounded in (ii). The curved segments in the right image, and the solid dark blue vertical segments at the top and bottom of the left image are bounded in (i). The dark red dashed segment that crosses the real axis in the left image is distance $O(n^{-1/3})$ from $0$ and is bounded in (iii). The green dotted segments in the left image are distance $O(n^{\delta-1/3})$ from $0$ for $\delta \in (0,1)$ and are bounded in (iv).[]{data-label="Cn picture"}](contourlinescolorswap.pdf){width="11cm"}
Properties of the contour $\mathcal{C}_n$: proof of Lemma \[C bound\]
---------------------------------------------------------------------
Most of the work is used to prove part (c). The idea of this proof is to patch together the different estimates from the beginning of Section \[Cn\]. Away from $0$ we use Lemma \[chaos control\] and the fact that the contour is steep descent near $\lambda$. Very near $0$ on the scale $n^{-1/3}$ we use Lemma \[one third bound\]. Moderately near $0$ we use Lemma \[delta bound\], and our control of the derivative of $f_1$ on the vertical strip of $\mathcal{C}_n$ near $0$. This last argument allows us to get bounds uniform in $\delta \in (0,1/3)$ when ${\mathbb{\omega}}$ is on the scale $n^{1/3-\delta}$.
\(a) and (b) follow from the definition of $\mathcal{C}_n$. By a slight modification of the proof of Lemma $2.8$, we see that for $z \in {\mathcal}{\gamma}_r$, $${\mathfrak{Re}}[h_n(z)-h_n(\lambda)+n^{1/3}t(z -\lambda) \leq n^{1/9}C, \label{gamma lambda}$$ so to show (c) it suffices to show that for ${\mathbb{\omega}}\in \mathcal{C}_n\setminus \mathcal{C}_n^{{\varepsilon}}$, we have $${\mathfrak{Re}}[h_n(\lambda)-h_n({\mathbb{\omega}})+n^{1/3}t(\lambda-{\mathbb{\omega}})] \leq -n^{-1/3} \eta. \label{C lambda}$$ Below we split the contour into $4$ pieces and bound each separately. See Figure \[Cn picture\].
- By Lemma \[derivative\] and the construction of $\mathcal{C}_n$, we have ${\mathfrak{Re}}[-f_1({\mathbb{\omega}})] \leq s< {\mathfrak{Re}}[-f_1(\lambda)]$ for ${\mathbb{\omega}}\in {\mathcal}{C}_n \setminus (\{\lambda+ y e^{\pm 2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\} \cup [n^{-1/3}(-a-{\mathbf{i}}|a+b|),n^{-1/3}(-a+{\mathbf{i}}|a+b|)])$. So we can apply Lemma \[chaos control\] and the fact that $f_2$ is bounded outside a neighborhood of $0$ to show that for any $c_1<0$, we have ${\mathfrak{Re}}[h_n(z)-h_n(\lambda)+n^{1/3}t(z-\lambda)] \leq -n^{-1/3} \eta$ for $ {\mathbb{\omega}}\in {\mathcal}{C}_n \setminus (\{\lambda+ y e^{\pm 2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\} \cup [-n^{-1/3}a-{\mathbf{i}}c_1 |a+b|,-n^{-1/3}a+{\mathbf{i}}c_1 |a+b|]).$
- By the definition of $q$, The contour $\{\lambda+ y e^{\pm 2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\}$ is steep descent with respect to the function $f_1$ at the point $\lambda$, so we can apply Lemma \[chaos control\] and the fact that $f_2$ is bounded outside a neighborhood of $0$ to show ${\mathfrak{Re}}[h_n(z)-h_n(\lambda)+n^{1/3}t(z-\lambda)] \leq -n^{-1/3} \eta$ for $ {\mathbb{\omega}}\in \{\lambda+ y e^{\pm 2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\} \setminus B_{{\varepsilon}}(\lambda).$
- By Lemma \[one third bound\], for any $c_0$, we have ${\mathfrak{Re}}[h_n(z)-h_n(\lambda)+n^{1/3}t(z-\lambda)] \leq -n^{-1/3} \eta$ for all ${\mathbb{\omega}}\in [n^{-1/3}(-a-{\mathbf{i}}c_0|a+b|),n^{-1/3}(-a-{\mathbf{i}}c_0|a+b|)]$.
- Now we bound the ${\mathfrak{Re}}[h_n(z)-h_n(\lambda)+n^{1/3}t(z-\lambda)]$ on the last piece of our contour $[n^{-1/3}(-a-{\mathbf{i}}c_0|a+b|),-n^{-1/3}a+{\mathbf{i}}c_1 |a+b|] \cup [-n^{-1/3}a-{\mathbf{i}}c_1 |a+b|,n^{-1/3}(-a-{\mathbf{i}}c_0|a+b|)].$ We will do this by fixing a constant $c>c_1$, and bounding the function on ${\mathbb{\omega}}=n^{-1/3} a + {\mathbf{i}}n^{\delta-1/3} c (a+b)$ for all pairs $n>N,\delta \in (0,1/3)$ such that $n^{1/3} \leq c_1/c$.
By Lemma \[delta bound\], we have that when ${\mathbb{\omega}}= n^{-1/3} a + {\mathbf{i}}n^{\delta-1/3} c (a+b)$, there exist constants $M_1,M_2,M_3$, such that $${\mathfrak{Re}}[n^{1/3}t {\mathbb{\omega}}+h_n({\mathbb{\omega}})-n^{1/3}f_1({\mathbb{\omega}})] \leq \frac{n^{2/3-2\delta}}{c^2}M_1+\frac{n^{1-3\delta}}{c^3}M_2+\frac{n^{4/9-\delta}}{c}M_3,$$ and $$f_1({\mathbb{\omega}}) \sim_{\delta} M \frac{n^{1-2\delta}}{c^2}.$$ First we consider the case when $\delta \in (0,1/3-{\varepsilon})$. In this case, for any $r>0$ we can choose $c$ and $N_{r}$ large enough that for all $n>N_{r}$, $$\frac{\frac{n^{2/3-2\delta}}{c^2}M_1+\frac{n^{1-3\delta}}{c^3}M_2+\frac{n^{4/9-\delta}}{c}M_3}{{\mathfrak{Re}}[n^{1/3}f_1({\mathbb{\omega}})]}<r/2,$$ uniformly for all $\delta \in (0,1/3-{\varepsilon}).$ In this case we also have that, by Lemma \[chaos control\], $$|{\mathfrak{Re}}[n^{1/3}t z+h_n(z)]| \leq n^{1/3}f_1(\lambda)+n^{1/9}f_2(\lambda)+C.$$ By potentially increasing $N_r$, we have that for all $n>N_r$ $$\frac{|{\mathfrak{Re}}[n^{1/3}t z+h_n(z)]|}{{\mathfrak{Re}}[n^{1/3}f_1({\mathbb{\omega}})]} \leq r/2.$$ By Lemma \[derivative\] and (\[small a+b\]), for all pairs $n,\delta$ such that $n^{\delta-1/3}<c/c_1$, there is an $\eta>0$ such that $${\mathfrak{Re}}[-f_1({\mathbb{\omega}})] \leq {\mathfrak{Re}}[-f_1(\lambda)]-2\eta <-2\eta.$$ setting $r=1/2$ gives $${\mathfrak{Re}}[n^{1/3}t (z-{\mathbb{\omega}})+h_n(z)-h_n({\mathbb{\omega}})] \leq {\mathfrak{Re}}[-n^{1/3} f_1({\mathbb{\omega}})]+\frac{1}{2} {\mathfrak{Re}}[n^{-1/3}f_1({\mathbb{\omega}})]<-\eta n^{1/3}.$$
Now we prove the case $\delta \in (1/3-{\varepsilon}, 1/3)$. Note that in the expression $${\mathfrak{Re}}[n^{1/3}t {\mathbb{\omega}}+h_n({\mathbb{\omega}})-n^{1/3}f_1({\mathbb{\omega}})] \leq \frac{n^{2/3-2\delta}}{c^2}M_1+\frac{n^{1-3\delta}}{c^3}M_2+\frac{n^{4/9-\delta}}{c}M_3,$$ when $n$ is sufficiently large, we can bound the right hand side by $(M_1+M_2)n^{3{\varepsilon}} \leq (r/2) n^{1/3}$ for any $r>0$. We also have $$|{\mathfrak{Re}}[n^{1/3}t \lambda -h_n(\lambda)-n^{1/3}f_1(\lambda)]| \leq n^{1/9} f_1(\lambda)+C \leq (r/2)n^{1/3}.$$ The first inequality comes from Lemma \[chaos control\], and the second holds for large enough $n$. By Lemma \[derivative\] and (\[small a+b\]), for all pairs $n,\delta$ such that $n^{\delta-1/3}<c/c_1$, there is an $\eta>0$ such that $${\mathfrak{Re}}[-f_1({\mathbb{\omega}})] \leq {\mathfrak{Re}}[-f_1(\lambda)]-2\eta <-2\eta.$$ Setting $r=\eta$ gives $${\mathfrak{Re}}[n^{1/3}t(\lambda-{\mathbb{\omega}})+h_n(\lambda)-h_n({\mathbb{\omega}})] \leq n^{1/3}{\mathfrak{Re}}[f_1(\lambda)-f_1({\mathbb{\omega}})] +n^{1/3} \eta \leq -\eta n^{1/3}.$$
The $c_1$ in part $(i)$ can be chosen as small as desired, the $c$ in part $(iv)$ has already been chosen, and the $c_0$ in part $(iv)$ can be chosen as large as desired. Choose $c_1<c<c_0$ to complete the proof of (c).
Given inequalities (\[gamma lambda\]) and (\[C lambda\]), part (d) follows if we can show $${\mathfrak{Re}}[n^{1/3}t(\lambda-{\mathbb{\omega}})+h_n(\lambda)-h_n({\mathbb{\omega}})],$$ for ${\mathbb{\omega}}\in \mathcal{C}_n^{{\varepsilon}}.$ Indeed this follows from Lemma \[chaos control\] and the fact that the contour $\{\lambda+ y e^{\pm 2 \pi {\mathbf{i}}/3}: 0\leq y \leq q\}$ is steep descent with respect to the function ${\mathfrak{Re}}[-f_1]$ at the point $\lambda$.
Dominated convergence {#Dominated Convergence}
=====================
In this section we carefully prove that the series expansion for $\det(1-{\ensuremath{\mathsf{K}}}_n)_{L^2(\mathcal{C}_n^{{\varepsilon}})}$ gives an absolutely convergent series of integrals bounded uniformly in $n$. This allows us to use dominated convergence when we localize the integral in Proposition \[cut\], and again when we approximate the kernel by its Taylor expansion in Proposition \[taylor approximation\]. First we zoom in on a ball of radius epsilon and show that we can absolutely bound $\det(1-{\ensuremath{\mathsf{K}}}_n^{{\varepsilon}})_{L^2(\mathcal{C}_n^{{\varepsilon}})}$ uniformly in $n$.
\[dom\] For any sufficiently small ${\varepsilon}>0$, and sufficiently large $r$, there exists a function $\overline{F}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}')$, such that for all $\overline{{\mathbb{\omega}}}, \overline{{\mathbb{\omega}}}' \in \mathcal{C}_{-1}^{n^{1/9} {\varepsilon}}$, $z \in \mathcal{D}_0^{n^{1/9} {\varepsilon}}$, $n>N$ the integrand of $\overline{\mathsf{K}}_{n}^{{\varepsilon}}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}')$ in equation (\[K def\]) is absolutely bounded by $\overline{F}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}',\overline{z})$, and
$$\sum_{m=0}^{\infty}\int_{(\mathcal{C}_{-1}^{n^{1/9} {\varepsilon}})^m} \left|\det \left(\int_{\mathcal{D}_0^{n^{1/9} {\varepsilon}}}\overline{F}(\overline{{\mathbb{\omega}}}_i,\overline{{\mathbb{\omega}}}_j,\overline{z}) d\overline{z}\right)_{i,j=1}^m \right| d\overline{{\mathbb{\omega}}}_1...d\overline{{\mathbb{\omega}}}_m<\infty. \label{bar dominated}$$
For $\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}' \in \mathcal{C}_{-1}^{{\varepsilon}}$, and $\overline{z} \in \mathcal{D}_0^{{\varepsilon}}$, we have
$$\left|\frac{\lambda+n^{-1/9} \overline{z}}{\lambda+n^{-1/9} \overline{{\mathbb{\omega}}}}\right| \leq \left|\frac{\lambda+{\varepsilon}}{\lambda-{\varepsilon}}\right|,$$ and by Taylor approximation, we have the additional bounds
$$\begin{aligned}
n^{1/3} (f_1(\lambda+n^{-1/9}\overline{z})-f_1(\lambda+n^{-1/9} \overline{{\mathbb{\omega}}})) &\leq (f_1'''(\lambda)+\delta_1)(\overline{z}^3-\overline{{\mathbb{\omega}}}^3),
\label{f_1} \\
n^{1/9}(f_2(\lambda+n^{-1/9} \overline{z})-f_2(\lambda+n^{-1/9} (\overline{{\mathbb{\omega}}}))) &\leq (f_2'(\lambda)+\delta_2)(\overline{z}-\overline{{\mathbb{\omega}}}),
\label{f_2}\\
r_n(\lambda+n^{-1/9} \overline{z})-r_n(\lambda+n^{-1/9} \overline{{\mathbb{\omega}}}) &\leq C n^{-1/9} (\overline{z}-\overline{{\mathbb{\omega}}}) \leq C{\varepsilon}\leq \delta_3.
\label{f_3}\end{aligned}$$
Note that in these bounds we can make $\delta_1,\delta_2,\delta_3$ as small as desired by choosing ${\varepsilon}$ small. Equations (\[f\_1\]) and (\[f\_2\]) follow from the fact that $f_1$, and $f_2$ are holomorphic in the compact set $\overline{B}_{{\varepsilon}}(\lambda)$. And equation (\[f\_3\]) follows from Lemma \[chaos control\]. Note that along $\mathcal{D}_0$, $z$ is purely imaginary, so (\[f\_1\]),(\[f\_2\]), and (\[f\_3\]) show that the full exponential in the integrand in (\[K def\]) is bounded above by
$$e^{2 \delta_3} e^{-(f_1'''(\lambda)-\delta_1)\overline{{\mathbb{\omega}}}^3-(f'_2(\lambda)-\delta_2)\overline{{\mathbb{\omega}}}}. \label{omega bound}$$
We choose ${\varepsilon}$ small enough that $\delta_1 < f'''_1(\lambda)$, so that (\[omega bound\]) has exponential decay as ${\mathbb{\omega}}$ goes to $\infty$ in directions $e^{\pm 2 \pi {\mathbf{i}}/3}$. Set $$\overline{F}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}',\overline{z})=\left|\left(\frac{\lambda+{\varepsilon}}{\lambda-{\varepsilon}}\right) e^{2\delta_3}e^{-(f_1'''(\lambda)-\delta_1)\overline{{\mathbb{\omega}}}^3-(f'_2(\lambda)-\delta_2)}\frac{1}{(\overline{z}+1)(\overline{z}+1)}\right|.$$ By the sentence preceeding (\[omega bound\]) $\overline{F}$ absolutely bounds the integrand of $\overline{\mathsf{K}}_{n}^{{\varepsilon}}$. Now set $L_1= \frac{|\lambda+{\varepsilon}|}{|\lambda-{\varepsilon}|} e^{2\delta_3} \int_{\mathcal{D}_0} \frac{1}{(\overline{z}+1)(\overline{z}+1)} d \overline{z}$ so that $2 e^{2\delta_3}\int_{\mathcal{D}_0} \frac{1}{(\overline{z}-\overline{{\mathbb{\omega}}})(\overline{z}-\overline{{\mathbb{\omega}}}')} d \overline{z} \leq L_1.$ Then
$$\int_{\mathcal{D}_0^{{\varepsilon}}}\overline{F}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}',\overline{z}) \leq L_1 \left| e^{-(f_1'''(\lambda)-\delta_1)\overline{{\mathbb{\omega}}}^3-(f'_2(\lambda)-\delta_2)}\right| , \label{L}$$
By Hadamard’s bound
$$\left|\det\left(\int_{\mathcal{D}_0^{n^{1/9}} {\varepsilon}}\overline{F}(\overline{{\mathbb{\omega}}}_i,\overline{{\mathbb{\omega}}}_j',\overline{z}) d \overline{z}\right)_{i,j=1}^m\right| \leq m^{m/2} L_1^m \prod_{i=1}^m \left| e^{-(f'''_1(\lambda)-\delta)\overline{{\mathbb{\omega}}}^3-(f_2'(\lambda)-\delta)\overline{{\mathbb{\omega}}}} \right|.$$
Now because $\delta_1<f'''_1(\lambda)$, we can set
$$S=\int_{\mathcal{C}_{-1}^{n^{1/9} {\varepsilon}}} \left| e^{-(f'''_1(\lambda)-\delta)\overline{{\mathbb{\omega}}}^3-(f_2'(\lambda)-\delta)\overline{{\mathbb{\omega}}}} \right| d \overline{{\mathbb{\omega}}}<\infty.$$
Then we have the bound,
$$\int_{(\mathcal{C}_{-1}^{n^{1/9}{\varepsilon}})^m} \left|\det\left(\int_{\mathcal{D}_0^{n^{1/9}{\varepsilon}}} \overline{F}(\overline{{\mathbb{\omega}}}_i,\overline{{\mathbb{\omega}}}_j',\overline{z}) d\overline{z}\right)_{i,j=1}^m\right| d\overline{{\mathbb{\omega}}}_1...d\overline{{\mathbb{\omega}}}_m \leq m^{m/2} (SL_1)^m.$$
So by Stirling’s approximation
$$\sum_{m=0}^{\infty}\int_{(\mathcal{C}_{-1}^{n^{1/9} {\varepsilon}})^m} \left|\det\left(\int_{\mathcal{D}_0^{n^{1/9} {\varepsilon}}} \overline{F}(\overline{{\mathbb{\omega}}}_i,\overline{{\mathbb{\omega}}}_j,\overline{z}) d \overline{z}\right)_{i,j=1}^m\right| d\overline{{\mathbb{\omega}}}_1...d\overline{{\mathbb{\omega}}}_m< \infty.$$
The next lemma completes our dominated convergence argument, by controlling the contribution to $\det(I-K_n)_{L^2({\mathcal}{C}_n^{{\varepsilon}})}$ of $z \in \gamma_r \setminus \gamma_r^{{\varepsilon}}$.
\[dom 2\]
For any sufficiently small ${\varepsilon}>0$, and sufficiently large $r$, there is a function $\overline{G}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}',\overline{z})$, and a natural number $N$, such that for all $\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}' \in \overline{\mathcal{C}}_{n}^{{\varepsilon}}$ and $\overline{z} \in \overline{\mathcal{\gamma}}_r$, $n >N$, the integrand of $\overline{\mathsf{K}}_{n}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}')$ is absolutely bounded by $\overline{G}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}',\overline{z})$, and
$$\sum_{m=0}^{\infty} \frac{1}{m!} \int_{(\overline{C}^{{\varepsilon}})^m} \left| \det \left(\int_{\overline{\mathcal{\gamma}}_r}\overline{G}(\overline{{\mathbb{\omega}}}_i,\overline{{\mathbb{\omega}}}_j,\overline{z})dz \right)_{i,j=1}^m\right| d\overline{{\mathbb{\omega}}}_i...d\overline{{\mathbb{\omega}}}_j<\infty, \label{dominated}$$
where $\overline{\mathcal{\gamma}}_r$ and $\overline{C}_n^{{\varepsilon}}$ are the rescaled contours of ${\mathcal}{\gamma}_r$ and $C_n^{{\varepsilon}}$ respectively.
Let $\overline{G}=\overline{F}$ for $z \in \mathcal{\gamma}_r^{{\varepsilon}}$. We decompose the integral along $\mathcal{\gamma}_r$ in three parts: the integral along $\mathcal{\gamma}_r^{{\varepsilon}}$, the integral along $(e^{-2\pi {\mathbf{i}}/3} \infty,-r) \cup (r,e^{2\pi {\mathbf{i}}/3} \infty)$ and the integral along $[-r,-{\varepsilon}] \cup [{\varepsilon},r]$. For $z \in \mathcal{\gamma}_r \setminus \mathcal{\gamma}_r^{{\varepsilon}}$ we have the following bounds
$$\begin{aligned}
|e^{n^{1/3}t(z-{\mathbb{\omega}}) +h_n(z)-h_n({\mathbb{\omega}})}| &\leq |e^{n^{1/3}(f_1(z)-f_1({\mathbb{\omega}}))+n^{1/9}C_2+C_3}|\nonumber \\
&\leq |e^{n^{1/3}(f_1(z)-f_1({\mathbb{\omega}})+\delta)}| \nonumber \\
&\leq |e^{n^{1/3}(f_1(z)-f_1(\lambda)+\delta)}||e^{n^{1/3}(f_1(\lambda)-f_1({\mathbb{\omega}}))}| \label{exp}.
\end{aligned}$$
Where the first inequality follows from Lemma \[chaos control\]. If we choose $\delta< \eta/2$, and recall that if $z \in \mathcal{\gamma}_r \setminus \mathcal{\gamma}_r^{{\varepsilon}}$, then $f_1(z)-f_1(\lambda)< -\eta,$ so $f_1(z)-f_1(\lambda)+\delta< -\eta/2<0$. So if we wish we can bound (\[exp\]) by either of the following expressions
$$|e^{n^{1/3}(f_1(\lambda)-f_1({\mathbb{\omega}}))}| \label{exp 2}$$
$$|e^{n^{1/9}(-tz+t\lambda)}||e^{n^{1/3}(f_1(\lambda)-f_1({\mathbb{\omega}}))}| \label{exp 3}$$
The bound (\[exp 3\]) follows from the fact that we can choose $r$ large enough so that $|f_1(z)+tz| \leq \delta$ outside $B_r(0)$. Then because the exponent in the first factor of (\[exp\]) is negative, for large enough $n$ we can remove the constant $\delta$ in return for reducing $n^{1/3}$ to $n^{1/9}$.
Now for $z \in [-r,-{\varepsilon}] \cup [{\varepsilon},r]$, we have
$$\left|\frac{z}{{\mathbb{\omega}}}\right| \leq \left| \frac{r+\lambda}{\lambda-{\varepsilon}}\right|, \qquad \left|\frac{1}{(\overline{z}-\overline{{\mathbb{\omega}}})(\overline{z}-\overline{{\mathbb{\omega}}}')}\right| \leq 1.$$
So for $z \in [-r,-{\varepsilon}] \cup [{\varepsilon},r]$, we set
$$\overline{G}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}',\overline{z})=\left| \frac{r+\lambda}{\lambda-{\varepsilon}} \right| \left| \frac{1}{(\overline{z}-\overline{{\mathbb{\omega}}})(\overline{z}-\overline{{\mathbb{\omega}}}')} \right| \left| e^{n^{1/3}(f_1(\lambda)-f_1({\mathbb{\omega}}))}\right|.$$
Using the above bounds and (\[exp 2\]) we see that the integrand of $\overline{\mathsf{K}}_n$ is absolutely bounded by $\overline{G}$ in this region. Set $L_2=\int_{i\mathbb{R}}\frac{r+\lambda}{\lambda-{\varepsilon}}\frac{1}{(\overline{z}+1)(\overline{z}+1)} d \overline{z}$ so that the integral of $\overline{G}$ on the rescaled contour of $[-r,-{\varepsilon}] \cup [{\varepsilon},r]$ is bounded by $L_2|e^{n^{1/3}(f_1(\lambda)-f_1({\mathbb{\omega}}))}|$.
For $z \in (e^{-2\pi {\mathbf{i}}/3} \infty,-r) \cup (r,e^{2\pi {\mathbf{i}}/3} \infty)$, we have
$$\left|\frac{1}{(\overline{z}-\overline{{\mathbb{\omega}}})(\overline{z}-\overline{{\mathbb{\omega}}}')}\right| \leq 1.$$
So for $z \in (e^{-2\pi {\mathbf{i}}/3} \infty,-r) \cup (r,e^{2\pi {\mathbf{i}}/3} \infty)$, we set
$$\overline{G}(\overline{{\mathbb{\omega}}},\overline{{\mathbb{\omega}}}',\overline{z})= \left|\frac{z}{{\mathbb{\omega}}} \right| \left| e^{t(\lambda-\overline{z})} \right| \left| e^{(-f_1'''(\lambda)+\delta)\overline{{\mathbb{\omega}}}}\right|.$$
Thus by (\[exp 3\]), we can see that the integrand of $\overline{\mathsf{K}_n}$ is absolutely bounded by $\overline{G}$ in this region. Now let $L_3=\int_{(e^{-2 \pi {\mathbf{i}}/3} \infty,-r] \cup [r,e^{2\pi {\mathbf{i}}/3} \infty)} \left|\frac{\lambda+\overline{z}}{\lambda-{\varepsilon}}\right| |e^{t(\lambda-\overline{z})}| d \overline{z}$. For all $n$, the integral of $\overline{G}$ over the rescaled contour $(e^{-2 \pi {\mathbf{i}}/3} \infty,-r] \cup [r,e^{2\pi {\mathbf{i}}/3} \infty)$ is bounded above by $L_3 |e^{(-f_1'''(\lambda)+\delta)\overline{{\mathbb{\omega}}}^{3}}|$.
Let $\overline{\mathcal{\gamma}}_r$ be the rescaled contour $\mathcal{\gamma}_r$ in the variable $\overline{z}$
$$\int_{\overline{\mathcal{\gamma}}_r} \overline{G} d\overline{z} \leq (L_1+L_2+L_3) e^{(-f_1'''(\lambda)+\delta)\overline{{\mathbb{\omega}}}^{3}} \leq L e^{(-f_1'''(\lambda)+\delta)\overline{{\mathbb{\omega}}}^{3}}, \label{G bound}$$
where the constant $L$ comes from (\[L\]). Thus we have bounded $\int_{\overline{\mathcal{\gamma}}_r} \overline{G} d\overline{z}$ by a constant times a term which has exponential decay as $\overline{{\mathbb{\omega}}} \to e^{\pm 2\pi {\mathbf{i}}/3} \infty$. The same argument as in Lemma \[dom\] shows that
$$\sum_{m=0}^{\infty} \frac{1}{m!} \int_{(C^{{\varepsilon}})^m} \left|\det\left(\int_{\mathcal{\gamma}_r^{{\varepsilon}}}G({\mathbb{\omega}}_i,{\mathbb{\omega}}_j,z)dz\right)_{i,j=1}^m\right| d{\mathbb{\omega}}_i...d{\mathbb{\omega}}_j<\infty.$$
\[max bound lemma\] Let ${\mathbb{\omega}}_1 \in {\mathcal}{C}_n \setminus {\mathcal}{C}_n^{{\varepsilon}}$ and ${\mathbb{\omega}}_2,..,{\mathbb{\omega}}_m \in {\mathcal}{C}^n$. There exist positive constants $M,L_4, \eta>0$ so that for sufficiently large $n$, we have $$|\overline{\mathsf{K}}_{n}(\overline{{\mathbb{\omega}}}_i,\overline{{\mathbb{\omega}}}_j)| \leq M \qquad \nonumber$$ and $$|\overline{\mathsf{K}}_{n}(\overline{{\mathbb{\omega}}}_1, \overline{{\mathbb{\omega}}}_i)| \leq L_4 n^{4/9} e^{-n^{1/3} \eta}, \nonumber$$ for all $i,j$.
By Lemma \[C bound\], for any ${\varepsilon}>0$, there exists a $N,C>0$, such that if $v \in {\mathcal}{C}_{n}\setminus {\mathcal}{C}_n^{{\varepsilon}}$, and $z \in \mathcal{\gamma}_r$, then for all sufficiently large $n$, we have $${\mathfrak{Re}}[h_n(z)-h_n({\mathbb{\omega}})+n^{1/3}t(z-{\mathbb{\omega}})] \leq -n^{1/3} \eta.$$ For $z \in \mathcal{\gamma}_r$ and ${\mathbb{\omega}}, {\mathbb{\omega}}' \in {\mathcal}{C}_n \setminus {\mathcal}{C}_n^{{\varepsilon}}$, $n>N$ we have the following bounds: $$\frac{1}{(z-{\mathbb{\omega}})(z-{\mathbb{\omega}}')} \leq \left(\frac{2}{{\varepsilon}}\right)^2, \qquad \frac{1}{{\mathbb{\omega}}} \leq \frac{n^{1/3}}{a},$$ and $$\begin{aligned}
|e^{n^{1/3}t(z-{\mathbb{\omega}})+h_n(z)-h_n({\mathbb{\omega}})}| &\leq |e^{n^{1/3}(f_1(z)-f_1({\mathbb{\omega}})+\delta)}| \label{exponent 1} \\
&\leq |e^{n^{1/3}(f_1(z)-f_1(\lambda)}||e^{n^{1/3}(f_1(\lambda)-f_1({\mathbb{\omega}})+\delta)}| \label{exponent 2}
\end{aligned}$$ where (\[exponent 1\]) follows from (\[chaos control\]) and the fact that $f_2$ is bounded away from $0$. Note that for $z \in \mathcal{\gamma}_r$, $|f_1(z)-f_1(\lambda)| \leq 0$, and for ${\mathbb{\omega}}, {\mathbb{\omega}}' \in {\mathcal}{C}_n \setminus {\mathcal}{C}_n^{{\varepsilon}}$, $f_1(\lambda)-f_1({\mathbb{\omega}}) + \delta< -\eta$, so (\[exponent 2\]) is bounded above by $$|e^{(f_1(z)-f_1(\lambda)}||e^{-n^{1/3}\eta}|.$$ Thus if we set $L_4=\frac{2^2}{a {\varepsilon}^2} \int_{\mathcal{\gamma}_r} |z| |e^{f_1(z)-f_1(\lambda)}| dz< \infty$, we get $$|\mathsf{K}_n({\mathbb{\omega}},{\mathbb{\omega}}')| \leq L_4 n^{1/3} e^{-n^{1/3} \eta}.$$ So if we change the variable of integration to $d\overline{z}=n^{1/9} d z$ gives. $$|\overline{\mathsf{K}}_{n}(\overline{{\mathbb{\omega}}}, \overline{{\mathbb{\omega}}}')| \leq L_4 n^{4/9} e^{-n^{1/3} \eta} \qquad \text{for ${\mathbb{\omega}},{\mathbb{\omega}}' \in {\mathcal}{C}_n \setminus {\mathcal}{C}_n^{{\varepsilon}}$} \label{outer bound}$$ Let ${\mathbb{\omega}}_1 \in {\mathcal}{C}_n \setminus {\mathcal}{C}_n^{{\varepsilon}}$ and ${\mathbb{\omega}}_2,..,{\mathbb{\omega}}_m \in {\mathcal}{C}^n$, then for $i \neq 1$, $$\begin{aligned}
|\overline{\mathsf{K}}_{n}(\overline{{\mathbb{\omega}}}_1, \overline{{\mathbb{\omega}}}_i)| &\leq L_4 n^{4/9} e^{-n^{1/3} \eta},\nonumber \\
|\overline{\mathsf{K}}_{n}(\overline{{\mathbb{\omega}}}_i,\overline{{\mathbb{\omega}}}_j)| &\leq \max[Le^{(-f_1'''(\lambda)+\delta) \overline{{\mathbb{\omega}}}^{3}}, L_4 n^{4/9} e^{-n^{1/3} \eta}] \leq M.\end{aligned}$$ The first equality follows from (\[G bound\]) and the second inequality holds for large $n$, when we set $M=\max[L_4,L]$ because $-f_1'''(\lambda)+\delta<0$.
The last thing we need to complete the proof of Theorem \[main theorem\] is to bound (\[sum bound\]) from Proposition (\[localize\]). We do so in the following lemma.
\[sum\] For any $C>1$, we have $$\sum_{m=1}^{\infty} \frac{1}{m!} C^m m^{1+m/2} \leq 16 C^4 e^{2C^2}.$$
We have $$\frac{m^{1+m/2}}{m!} \leq \frac{m 2^{m/2}}{(\lfloor m/2 \rfloor)!},$$ so that $$\begin{aligned}
\sum_{m=1}^{\infty} \frac{1}{m!} C^m m^{1+m/2} &\leq \sum_{m=1}^{\infty} \frac{m}{(\lfloor m/2 \rfloor)!} (2C^2)^{m/2}\\
&\leq \sum_{k=1}^{\infty} \frac{2k (2C^2)^k}{k!}+\sum_{k=1}^{\infty} \frac{(2k+1)(2C^2)^{k+1}}{k!} \\
&\leq 16 C^4 e^{2 C^2}.
\end{aligned}$$
[10]{} Amol Aggarwal, *Current fluctuations of the stationary [ASEP]{} and six-vertex model*, Duke Math J., arXiv:1608.04726 (2016).
[to3em]{}, *Dynamical stochastic higher spin vertex models*, Selecta Math. (2017), 1–77.
Amol Aggarwal and Alexei Borodin, *Phase transitions in the [ASEP]{} and stochastic six-vertex model*, arXiv preprint arXiv:1607.08684 (2016).
Antonio Auffinger, Jinho Baik, and Ivan Corwin, *[U]{}niversality for directed polymers in thin rectangles*, arXiv preprint arXiv:1204.4445 (2012).
Antonio Auffinger, Michael Damron, and Jack Hanson, *50 years of first-passage percolation*, vol. 68, American Mathematical Soc., 2017.
Jinho Baik, Guillaume Barraquand, Ivan Corwin, and Toufic Suidan, *Facilitated exclusion process*, to appear in proceedings of Abel 2016 symposium, arXiv:1707.01923 (2017).
Jinho Baik, Percy Deift, and Kurt Johansson, *On the distribution of the length of the longest increasing subsequence of random permutations*, J. Amer. Math. Soc. **12** (1999), no. 4, 1119–1178.
Jinho Baik and Toufic M. Suidan, *A [GUE]{} central limit theorem and universality of directed first and last passage site percolation*, Int. Math. Res. Not. **2005** (2005), no. 6, 325–337.
M[á]{}rton Bal[á]{}zs, Firas Rassoul-Agha, and Timo Sepp[ä]{}l[ä]{}inen, *Large deviations and wandering exponent for random walk in a dynamic beta environment*, arXiv preprint arXiv:1801.08070 (2018).
Guillaume Barraquand, *A phase transition for q-[TASEP]{} with a few slower particles*, Stochastic Process. Appl. **125** (2015), no. 7, 2674 – 2699.
Guillaume Barraquand, Alexei Borodin, Ivan Corwin, and Michael Wheeler, *[Stochastic six-vertex model in a half-quadrant and half-line open ASEP]{}*, Duke Math. J., arXiv:1704.04309 (2017).
Guillaume Barraquand and Ivan Corwin, *The [$q$]{}-[H]{}ahn asymmetric exclusion process*, Ann. Appl. Probab. **26** (2016), no. 4, 2304–2356.
[to3em]{}, *[R]{}andom-walk in [B]{}eta-distributed random environment*, Prob. Theory Related Fields **167** (2017), no. 3-4, 1057–1116.
Thierry Bodineau and James Martin, *[A]{} universality property for last-passage percolation paths close to the axis*, Electron. Commun. Probab. **10** (2005), 105–112.
Alexei Borodin, *On a family of symmetric rational functions*, Adv. Math. **306** (2017), 973–1018.
Alexei Borodin and Ivan Corwin, *Macdonald processes*, Probab. Theory Related Fields **158** (2014), no. 1-2, 225–400.
Alexei Borodin, Ivan Corwin, and Patrik Ferrari, *[F]{}ree energy fluctuations for directed polymers in random media in 1+ 1 dimension*, Comm. Pure Appl. Math **67** (2014), no. 7, 1129–1214.
Alexei Borodin, Ivan Corwin, Patrik Ferrari, and Bálint Vet[ő]{}, *Height fluctuations for the stationary [KPZ]{} equation*, Math. Phys. Anal. Geom. **18** (2015), no. 1, Art. 20, 95.
Alexei Borodin, Ivan Corwin, and Vadim Gorin, *Stochastic six-vertex model*, Duke Math. J. **165** (2016), no. 3, 563–624.
Alexei Borodin, Ivan Corwin, Leonid Petrov, and Tomohiro Sasamoto, *Spectral theory for interacting particle systems solvable by coordinate [B]{}ethe ansatz*, Comm. Math. Phys. **339** (2015), no. 3, 1167–1245.
Alexei Borodin, Ivan Corwin, and Daniel Remenik, *Log-gamma polymer free energy fluctuations via a fredholm determinant identity*, Comm. Math. Phys. **324** (2013), no. 1, 215–232.
Alexei Borodin, Ivan Corwin, and Tomohiro Sasamoto, *From duality to determinants for [q-TASEP]{} and [ASEP]{}*, Ann. Probab. **42** (2014), no. 6, 2314–2382.
Alexei Borodin and Patrik Ferrari, *[L]{}arge time asymptotics of growth models on space-like paths [I]{}: Push[ASEP]{}*, Electron. J. Probab. **13** (2008), 1380–1418.
Alexei Borodin and Grigori Olshanski, *The [ASEP]{} and determinantal point processes*, Commun. Math, Phys. **353** (2017), no. 2, 853–903.
Alexei Borodin and Leonid Petrov, *Higher spin six vertex model and symmetric rational functions*, Selecta Math. **24** (2018), no. 2, 751–874.
Hans Chaumont and Christian Noack, *Fluctuation exponents for stationary exactly solvable lattice polymer models via a [M]{}ellin transform framework*, arXiv preprint arXiv:1711.08432 (2017).
Ivan Corwin, *[T]{}he [K]{}ardar-[P]{}arisi-[Z]{}hang equation and universality class*, Random Matrices Theory Appl. **1** (2012), no. 01, 1130001.
Ivan Corwin, *[T]{}he q-[H]{}ahn [B]{}oson process and q-[H]{}ahn [TASEP]{}*, Int. Math. Res. **2015** (2014), no. 14, 5577–5603.
Ivan Corwin, *[K]{}ardar-[P]{}arisi-[Z]{}hang universality*, Not. [A]{}[M]{}[S]{} (2016).
Ivan Corwin and Yu Gu, *[Kardar–Parisi–Zhang]{} equation and large deviations for random walks in weak random environments*, J. Stat. Phys. **166** (2017), no. 1, 150–168.
Ivan Corwin and Leonid Petrov, *Stochastic higher spin vertex models on the line*, Comm. Math. Phys. **343** (2016), no. 2, 651–700.
Ivan Corwin, Timo Sepp[ä]{}l[ä]{}inen, and Hao Shen, *The strict-weak lattice polymer*, J. Stat. Phys. (2015), 1–27.
Patrick Ferrari and Bálint Vet[ő]{}, *[T]{}racy [W]{}idom asymptotics for q-[T]{}[A]{}[S]{}[E]{}[P]{}*, **51** (2015), no. 4, 1465–1485.
Luiz Fontes, Marco Isopi, Charles Newman, and Krishnamurthi Ravishankar, *The [B]{}rownian web*, Proc. Nat. Acad. Sci. **99** (2002), no. 25, 15888–15893.
[to3em]{}, *The [B]{}rownian web: characterization and convergence*, Ann. Probab. **32** (2004), no. 4, 2857–2883.
Promit Ghosal, *[H]{}all-[L]{}ittlewood-push[T]{}[A]{}[S]{}[E]{}[P]{} and its [K]{}[P]{}[Z]{} limit*, arXiv preprint arXiv:1701.07308 (2017).
John Hammersley and Dominic Welsh, *First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory*, Bernoulli 1713, Bayes 1763, Laplace 1813, Springer, 1965, pp. 61–110.
Kurt Johansson, *Shape fluctuations and random matrices*, Comm. Math. Phys. **209** (2000), no. 2, 437–476.
Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang, *Dynamic scaling of growing interfaces*, Phys. Rev. Lett. **56** (1986), 889–892.
Arjun Krishnan and Jeremy Quastel, *[T]{}racy-[W]{}idom fluctuations for perturbations of the log-gamma polymer in intermediate disorder*, arXiv preprint arXiv:1610.06975 (2016).
Charles Newman, Krishnamurthi Ravishankar, and Emmanuel Schertzer, *Marking (1, 2) points of the [B]{}rownian web and applications*, **46** (2010), no. 2, 537–574.
Neil O’Connell and Janosch Ortmann, *[T]{}racy-[W]{}idom asymptotics for a random polymer model with gamma-distributed weights*, Electron. J. Probab. **20** (2015), no. 25, 1–18.
Daniel Orr and Leonid Petrov, *Stochastic higher spin six vertex model and q-[TASEP]{}s*, Adv. Math. **317** (2017), 473–525.
Alexander Povolotsky, *[O]{}n the integrability of zero-range chipping models with factorized steady states*, J. Phys. A **46** (2013), no. 46, 465205.
Michael Pr[ä]{}hofer and Herbert Spohn, *Universal distributions for growth processes in [1+1]{} dimensions and random matrices*, Phys. rev. lett. **84** (2000), no. 21, 4882.
Emmanuel Schertzer, Rongfeng Sun, and Jan Swart, *The [B]{}rownian web, the [B]{}rownian net, and their universality*, Advances in Disordered Systems, Random Processes and Some Applications (2015), 270–368.
Rongfeng Sun and Jan Swart, *The [B]{}rownian net*, Ann. Probab. (2008), 1153–1208.
Thimoth[é]{}e Thiery, *[S]{}tationary measures for two dual families of finite and zero temperature models of directed polymers on the square lattice*, J. Stat. Phys. **165** (2016), no. 1, 44–85.
Thimoth[é]{}e Thiery and Pierre Le Doussal, *On integrable directed polymer models on the square lattice*, J. Phys. A **48** (2015), no. 46, 465001.
[to3em]{}, *Exact solution for a random walk in a time-dependent [1D]{} random environment: the point-to-point [B]{}eta polymer*, J. Phys. A **50** (2016), no. 4, 045001.
Craig A. Tracy and Harold Widom, *A [F]{}redholm determinant representation in [ASEP]{}*, J. Stat. Phys. **132** (2008), no. 2, 291–300.
[to3em]{}, *Integral formulas for the asymmetric simple exclusion process*, Comm. Math. Phys. **279** (2008), no. 3, 815–844.
[to3em]{}, *Asymptotics in [A]{}[S]{}[E]{}[P]{} with step initial condition*, Comm. Math. Phys. **290** (2009), no. 1, 129–154.
B[á]{}lint Vet[ő]{}, *[T]{}racy-[W]{}idom limit of [$q$-Hahn]{} [T]{}[A]{}[S]{}[E]{}[P]{}*, Electron. J. Probab. **20** (2015).
|
---
abstract: |
We establish strict upper limits for the Casimir interaction between multilayered structures of arbitrary dielectric or diamagnetic materials. We discuss the appearance of different power laws due to frequency-dependent material constants. Simple analytical expressions are in good agreement with numerical calculations based on Lifshitz theory. We discuss the improvements required for current (meta) materials to achieve a repulsive Casimir force.\
Dated: 20 Oct 2005, *Europhysics Letters*, in press.\
PACS. 42.50.Pq – Cavity quantum electrodynamics; 42.50.Lc – Quantum fluctuations, quantum noise; [78.67.-n]{} – [Optical properties of low-dimensional, mesoscopic, and nanoscale materials and structures]{}
author:
- |
C. Henkel$^1$ and K. Joulain$^2$\
$^1$ Institut für Physik, Universität Potsdam, Germany\
$^2$ Laboratoire d’Etudes Thermiques, Ecole Nationale Supérieure\
de Mécanique Aéronautique, Poitiers, France
date: 20 October 2005
title: |
Casimir force between designed materials:\
what is possible and what not
---
Introduction
============
The optical properties of materials that show both a dielectric and magnetic response, have recently attracted much attention (see [@Ramakrishna05] for a review). A number of striking phenomena like perfect lensing and a reversed Doppler effect have been predicted, and experimenters have begun to explore the large parameter space of structural units that can be assembled into artificial materials. Breakthroughs have been reported on the way towards designed susceptibitilies in the near-infrared and visible spectral range [@Pendry04b; @Wegener04]. Quantum electrodynamics in meta materials has recently been explored with particular emphasis on left-handed or negative-index materials [@Klimov02d; @Fleischhauer05a]. We discuss here to what extent the Casimir interaction between two meta material plates can be manipulated by engineering their magneto-dielectric response. Strict limits for the Casimir interaction are proven that apply to all causal and linear materials, including both bulk and multilayer structures. We illustrate these results by computations of the Casimir pressure, considering materials with frequency-dispersive response functions like those encountered in effective medium theories. We derive power law exponents and prefactors and find that a strongly modified Casimir interaction is possible in a range of distances around the resonance wavelength of the response functions. We give estimates for the required temperature range and structure size: it is not unreasonable to expect that improvements in fabrication and detection will allow for experimental observations.
One of the most striking changes to the Casimir interaction is a cross over to repulsion. This has been predicted previously for idealized magnetodielectric materials [@Boyer74; @Kenneth02; @Boyer03] and objects suspended in a liquid [@Hartmann91; @Israelachvili]. In the latter case, repulsion has been observed experimentally with colloidal particles [@Sigmund01] and is also used in a recent proposal for measuring Casimir torques [@Capasso05]. Casimir repulsion between mirrors separated by vacuum requires a strong magnetic response [@Kenneth02; @Tomas05] that hardly occurs in conventional ferromagnets [@Camley98; @Kenneth02c-2]. Indeed, to manipulate the Casimir force in the micrometer range and below, where it can be conveniently measured, the key challenge is to achieve a magnetic susceptibility at high frequencies, approaching the visible range. Now, there is a well-known argument due to Landau, Lifshitz, and Pitaevskii that $\mu( \omega ) = 1$ in the visible [@Landau10]. This objection, however, only applies to materials whose magnetization is of atomic origin, where the magnetic susceptibility is $\chi_{\rm m} \sim (v/c)^2 \ll 1$. An array of split ring resonators with sub-wavelength size typically gives, on the contrary, $\chi_{\rm m} \sim (\omega/\vartheta)^2 f \sim 1$, where $\vartheta$ is the resonance frequency and $f$ the filling factor [@Pendry99d; @Smith04a]. As we illustrate below, artificial materials that are structured on the sub-micron scale are promising candidates for a strongly modified Casimir interaction.
Lifshitz theory
===============
For two perfectly conducting plates held at zero temperature and separated by a distance $d$, Casimir derived a force per unit area given by $
F_{C} =
\pi^2 \hbar c /( 240 \, d^4 )
$ [@Casimir48b]. We use the convention that $F_{C} > 0$ corresponds to attraction. For linear media with complex, frequency-dependent material parameters, the force can be computed from Lifshitz theory [@Lifshitz56]. This expression has been re-derived, for plates of arbitrary material and for multilayer mirrors, using different methods . At finite temperature, it can be written in the form $$\begin{aligned}
F_{L} &=& 2 k_{B} T
{\kern 4.5ex\raisebox{0.25ex}{$'$}\kern -4.5ex}\sum\limits_{n=0}^{\infty}
\int\limits_{\xi_{n}/c}^{\infty} \! \frac{ {\rm d}\kappa }{ 2 \pi } \,
\kappa^2
\sum_{\lambda}
\left(
\frac{ {\rm e}^{2 \kappa d } }{
r_{\lambda 1}
r_{\lambda 2}
} - 1
\right)^{-1}\!\!
,
\label{eq:Lifshitz-formula}\end{aligned}$$ where the sum is over the imaginary Matsubara frequencies $\omega_{n} = {\rm i} \xi_{n} \equiv 2\pi {\rm i} n k_{B} T / \hbar$ (the $n=0$ term being multiplied by $1/2$), and $\kappa$ is related to the wave vector component perpendicular to the mirrors, $k_{z} = (\omega^2_{n}/c^2 - k_{x}^2 - k_{y}^2)^{1/2}
\equiv {\rm i}\, \kappa$. The $r_{\lambda \alpha}$ ($\lambda = {\rm TE}, \, {\rm TM}$, $\alpha = 1, \, 2$) are the reflection coefficients at mirror $\alpha$ for electromagnetic waves with polarization $\lambda$ [@Parsegian70b; @Ninham71]. For homogeneous, thick plates, they are given by $$\begin{aligned}
r_{\rm TM}
&=&
\frac{ \varepsilon ( {\rm i} \xi_{n} ) c \kappa -
\sqrt{ \xi_{n}^2 (\varepsilon ( {\rm i} \xi_{n} ) \mu (
{\rm i} \xi_{n} ) - 1) + \kappa^2 c^2
}
}{
\varepsilon ( {\rm i} \xi_{n} ) c \kappa +
\sqrt{ \xi_{n}^2 (\varepsilon ( {\rm i} \xi_{n} ) \mu (
{\rm i} \xi_{n} ) - 1) + \kappa^2 c^2 }
}
\label{eq:Fresnel-r}\end{aligned}$$ (exchange $\varepsilon$ and $\mu$ for $r_{\rm TE}$). The zeros of $D_\lambda \equiv {\rm e}^{2 \kappa d } / (
r_{\lambda 1}
r_{\lambda 2}
) - 1$ at real frequencies define the eigenmodes of the cavity formed by the two mirrors.
Strict limits
=============
To derive upper and lower limits for $F_{L}$, we use that the Kramers-Kronig relations [@Landau10] imply real and positive material functions at imaginary frequencies, $\varepsilon( {\rm i}\xi ) \ge 1$, provided the material is passive (non-negative absorption ${\rm Im}\,\varepsilon( \omega ) \ge 0$). As a consequence, the Fresnel formulas (\[eq:Fresnel-r\]) imply $-1 \le r_{\lambda \alpha} \le 1$, and we find $$- \frac{ 1 }{ {\rm e}^{2\kappa d} + 1 } \le
\frac{ 1 }{ D_\lambda }
\le
\frac{ 1 }{ {\rm e}^{2\kappa d} - 1 }
\label{eq:D-limits}$$ with the stronger inequalities $0 \le 1/D_{\lambda} \le
1/({\rm e}^{2\kappa d} - 1)$ holding for identical plates. In the latter case, the Casimir force is hence necessarily attractive. The inequalities (\[eq:D-limits\]) saturate for a perfectly conducting mirror facing a perfectly permeable one ($\varepsilon_{1} = \infty$, $\mu_{2} = \infty$, say), and for identical, perfectly reflecting mirrors, respectively. The resulting forces at zero temperature are [@Lifshitz56; @Boyer74] $$T = 0: \quad
- \frac{7}{8} F_C \le
F_L
\le
F_C.
\label{eq:F-limits-zero-T}$$ In the high-temperature limit, we get similarly [@Tort99] $
-\frac{3}{4} F_T \le
F_L
\le
F_T \equiv \zeta(3) k_B T / (8\pi d^3)$ by keeping in Eq.(\[eq:Lifshitz-formula\]) only the $n=0$ term in the sum.
Consider now a mirror made from layers of arbitrary passive materials. Reflection coefficients for such a system can be obtained recursively. For a layer ‘$b$’ separating a medium ‘$a$’ from a substrate ‘$c$’, for example, $$r_{abc} = \frac{ r_{ab} + r_{bc} \,{\rm e}^{ 2 {\rm i} k_{b} w }
}{
1 + r_{ab} r_{bc} \,{\rm e}^{ 2 {\rm i} k_{b} w } }
\label{eq:multiple-layer-r}$$ where $r_{ab}$ ($r_{bc}$) describes the reflection from the interface $ab$ ($bc$), respectively, and $w$ is the layer thickness [@BornWolf; @Yeh]. If the substrate $c$ is a multilayer system itself, $r_{bc}$ is the corresponding reflection coefficient. For the imaginary frequencies occurring in the Lifshitz expression (\[eq:Lifshitz-formula\]), the wavevector in the layer is purely imaginary, $k_{b} = {\rm i} \, \kappa_b$, and single-interface coefficients are real \[Eq.(\[eq:Fresnel-r\])\]. From Eq.(\[eq:multiple-layer-r\]), they remain real for multilayer mirrors. In addition, the mapping $r_{ab}
\mapsto r_{abc}$ is a conformal one, and if $r_{bc}
\,{\rm e}^{ - 2 \kappa_b w }$ is real and $\in[-1,1]$, the interval $[-1,1]$ is mapped onto itself. For multilayer mirrors, we thus obtain again the inequalities $-1 \le r_{\lambda} \le 1$. This generalizes the limits of Refs. [@Lambrecht97; @Genet03c] that are obtained only for layered dielectric mirrors, using transfer matrices.
Casimir interaction between metamaterials
=========================================
To illustrate these generally valid results, we focus on meta materials described by effective medium theory [@Ramakrishna05; @Pendry99d; @Smith04a]. We adopt Lorentz-Drude formulas for $\varepsilon$ and $\mu$ $$\varepsilon_{\alpha}( {\rm i}\,\xi ) = 1 +
\frac{ \Omega_{\alpha}^2 }{ \omega_{\alpha}^2 + \xi^2},
\qquad
\mu_{\alpha}( {\rm i}\,\xi ) = 1 +
\frac{ \Theta_{\alpha}^2 }{ \vartheta_{\alpha}^2 + \xi^2 }
.
\label{eq:Lorentz-Drude-1}$$ Regarding the permeability, we have taken the limit of weak absorption and computed $\mu( {\rm i}\,\xi )$ in terms of ${\rm Im}\,\mu( \omega )$ using the Kramers-Kronig relations. This is necessary to ensure high-frequency transparency of the medium. We denote in the following by $\Omega$ a typical resonance or plasma frequency occurring in Eqs.(\[eq:Lorentz-Drude-1\]). The corresponding wavelength, $\Lambda = 2\pi c / \Omega$, provides a convenient distance scale. Note that a (magnetic) resonance wavelength as short as $\sim 3\,\mu{\rm m}$ has already been achieved with material nanofabrication [@Wegener04]. The key advantage of meta materials is that their electric and magnetic ‘plasma frequencies’ $\Omega_\alpha$ and $\Theta_\alpha$ are fairly large as well: a value of $\Theta_\alpha
\approx \vartheta_\alpha \sqrt{f} \le (c/a) \sqrt{f}$ is typical for a split-ring resonator array with period $a$ and filling factor $f$ [@Pendry99d]. This property is also necessary, of course, to achieve a left-handed medium ($\varepsilon_\alpha( \omega )$, $\mu_\alpha( \omega ) < 0$ for some real frequencies). The magnetic plasma frequencies occurring in conventional ferromagnets are much smaller [@Camley98], and the impact on the Casimir interaction is weak, as reported recently [@Tomas05].
In the plots shown below, the Casimir pressure is normalized to $\hbar \Omega / d^3$ \[see after Eq.(\[eq:c3-attraction\])\]. In order of magnitude, this corresponds to $10^4\, {\rm pN}\, {\rm mm}^{-2}/(\Lambda / \mu{\rm m})^4$ at a distance $d = \Lambda/2$. This can be measured with sensitive torsion balances [@Lamoreaux97a; @Chan01] or cantilevers [@Mohideen98; @Onofrio02]. We plot in Fig.\[fig:attr-rep\] the result of a numerical integration of Eq.(\[eq:Lifshitz-formula\]), the curves corresponding to different material pairings. One sees that in all cases, the force satisfies the limits (\[eq:F-limits-zero-T\]) that exclude the shaded areas. We observe that materials with negative index of refraction around $\Omega$ show a strongly reduced attraction (Fig.\[fig:attr-rep\](b)). This can be attributed to the reduced mirror reflectivity due to impedance matching. Casimir repulsion is achieved for some distances between mirrors made from different materials (Fig.\[fig:attr-rep\](c,d)). At short distance, i.e. $d \ll \Lambda / 2\pi$, even these pairings show attraction with a power law $1/d^3$. Coating one mirror with a magnetic layer (Fig.\[fig:attr-rep\](c)), there is a sign change around the layer thickness $w$: for $\Lambda/2\pi \ll d \ll w$, the layer behaves like a thick plate, and its material parameters lead to repulsion. The layer can be ignored for $w \ll d$, and one recovers the attraction between the (identical) substrates. This is consistent with asymptotic analysis based on the reflection coefficient (\[eq:multiple-layer-r\]), as we outline below. Detailed calculations show that a large resonance frequency is not sufficient to achieve repulsion, the oscillator strength of the resonances (proportional to $\Omega_{1}$ and $\Theta_{2}$) must be large enough so that $\hbar \Omega_{1}, \hbar \Theta_{2} \gg
\max( k_{B} T, \hbar c / d )$. As the temperature is raised, the distance range where repulsion is observed disappears, see Fig.\[fig:finite-T\]. One then finds a $1/d^3$ power law at large distance as well.
The different regimes of Fig.\[fig:attr-rep\] can be understood from an asymptotic analysis of Eq.(\[eq:Lifshitz-formula\]). At short distance ($d \ll \Lambda/2\pi$), the integral is dominated by a region in the $\kappa$-$\xi$-plane where the Fresnel coefficients (\[eq:Fresnel-r\]) take the nonretarded forms $r_{{\rm TM}}
\to R( \varepsilon) \equiv
(\varepsilon - 1)/( \varepsilon + 1 ) > 0$ assuming that $\varepsilon > 1$ and similarly $r_{{\rm TE}} \to R( \mu )
> 0$ unless $\mu = 1$. Proceeding like Lifshitz [@Lifshitz56; @Henkel04a], yields to leading order a power law $F_{L} = c_{3} / d^3$ with a *positive* Hamaker constant given by $$c_{3} = \frac{ k_{B} T }{ 4\pi }
{\kern 4.5ex\raisebox{0.25ex}{$'$}\kern -4.5ex}\sum\limits_{n=0}^{\infty}
\left\{
{\rm Li}_{3}[
R( \varepsilon_1(\xi_n) )
R( \varepsilon_2(\xi_n) )
]
+
{\rm Li}_{3}[
R( \mu_1(\xi_n) )
R( \mu_2(\xi_n) )
]
\right\}
,
\label{eq:c3-attraction}$$ where ${\rm Li}_{n}( z ) \equiv \sum_{k=1}^\infty z^k / k^n$. It must be noted that for the special case of homogeneous plates, this asymptotic expression actually provides another, much stricter, upper limit to the Casimir force, since $r_{{\rm TM}\alpha} \le R( \varepsilon_{\alpha} )$, $r_{{\rm
TE}\alpha} \le R( \mu_{\alpha} )$ and ${\rm Li}_{3}( z )$ is a monotonous function (see Fig.\[fig:attr-rep\]). In order of magnitude, $c_{3} \sim \hbar\Omega$ at low temperatures ($k_{B} T \ll \hbar \Omega$). Compared to ideal mirrors, dispersive plates thus show a much weaker Casimir interaction that is in general attractive (Fig.\[fig:attr-rep\] and Ref.[@Lambrecht97]). At larger distances, $\Lambda/2\pi \ll d \ll \Lambda_{T} \equiv \hbar c / k_{B} T$, the Casimir force follows a $1/d^4$ power law, and repulsion is found provided one of the materials is dominantly magnetic. Here, the non-dispersive results of Ref.[@Kenneth02] are recovered. Finally, for $d \gg \Lambda_{T}$, the leading order force is the term $n=0$ in the sum (\[eq:Lifshitz-formula\]), again an attractive $1/d^3$ law, with a coefficient given by an expression similar to (\[eq:c3-attraction\]), but involving the static material constants, see [@Kenneth02].
The impact of temperature is illustrated in Fig.\[fig:finite-T\]: at high temperature, $k_{B}T \gg \hbar\Omega$, the second Matsubara frequency $\xi_{1}$ falls already into the mirrors’ transparency zone, and the $1/d^3$ power law is valid at all distances. As $T \to 0$, the intermediate repulsive zone appears in the range $\Lambda/2\pi \ll d \ll \Lambda_T/2\pi$. A good agreement with the analytical $1/d^3$ asymptotics is found outside this zone, as shown by the dashed lines. For the resonance wavelength $\Lambda = 3\,\mu{\rm m}$ mentioned above, cooling to a temperature $T \approx 0.1 \,\hbar\Omega/ k_B
\sim 50\,{\rm K}$ is required to ‘open up’ the repulsive window. This temperature increases, of course, with materials whose response extends to higher frequencies.
Finally, we would like to illustrate the kind of peculiar asymptotics that becomes possible with carefully matched material parameters. This follows Ref.[@Parsegian69] that computes the Van der Waals force on a water film coated on both sides by lipid membranes, finding a weak dependence on the ultraviolet frequency range because both materials have a similar electron density. Consider thus a liquid-filled gap with a similar electron density as medium 2 so that $\varepsilon_{0} = \varepsilon_{2}$, and a permeability $\mu_{0} = \mu_{1} \equiv 1$ matched to medium 1. For simplicity, we assume that these equalities hold at all frequencies. In this case, we can show that the force is repulsive at all distances, even at finite temperature. Indeed, both contributions in Eq.(\[eq:c3-attraction\]) vanish, and the leading order term for high temperatures also vanishes. The high-temperature limit is given by the $n=1$ term in (\[eq:Lifshitz-formula\]). This gives a distance dependence proportional to $\exp( - 4 \pi d / \Lambda_{T})$ similar to what has been observed in some experiments with colloids (mentioned in [@Ackler96]). As the temperature is lowered, this exponential regime still applies for $d \gg \Lambda_{T}$. If $k_{B} T \ll \hbar \Omega$, the $1/d^4$ regime of Ref. [@Kenneth02] exists at intermediate distances $\Lambda/2\pi \ll d \ll
\Lambda_{T}/2\pi$. The short distance regime sets in for $d \ll \Lambda$, and an analysis similar to the one leading to Eq.(\[eq:c3-attraction\]) gives ($T = 0$) $$F_{L} = \frac{ \hbar }{ \pi }
\sum\limits_{n=1}^{\infty}
\left( 2 n d \right)^{2n-3}
\int\limits_0^\infty\!\frac{ {\rm d}\xi }{ 2\pi }
\Gamma( 3 - 2n, 2 n \xi \, d / c )
\left(
-
\frac{ \varepsilon_{1} - \varepsilon_{0} }{
\varepsilon_{1} + \varepsilon_{0} }
\frac{ (\mu_{2} - \mu_{1}) \xi^2 }{ 4 c^2 }
\right)^n
\label{eq:c1-repulsion}$$ with $\Gamma( k, z ) \equiv \int_{z}^{\infty}\!{\rm d}t \,t^{k-1}
{\rm e}^{-t}$. At short distance, the sum is dominated by the first term, so that to leading order, we get a repulsive power law $F_{L} = - c_{1} / d$ (Fig.\[fig:mismatch\](a)). In order of magnitude, $c_{1} \sim \hbar \Omega^3 / c^2$ and therefore again $-F_{L} \ll F_{C}$, with a cross over occurring around $\Lambda/2\pi$ (see Fig.\[fig:mismatch\](a)). Due to our assumption of a perfect matching $\varepsilon_2 =
\varepsilon_0$, this kind of behaviour seems quite remote from experimental reality. As shown in Fig.\[fig:mismatch\](b–d), a slight mismatch between the dielectric functions of liquid and plate leads back to an attractive force, first at short distances, then suppressing the repulsive window altogether.
Conclusion
==========
We have generalized strict upper and lower limits for the Casimir force. We have shown that a strongly modified Casimir force can occur between dispersive and absorbing mirrors with a sufficiently large magnetic susceptibility, extending results restricted to non-dispersive materials [@Kenneth02]. The most promising way to achieve this repulsion seems the use of meta materials engineered at scales between the nanometer and the micron because they provide a fairly large magnetic oscillator strength. Our results are intrinsically limited to distances $d \gg a$ by our use of effective medium theory. Sufficiently small structures and sufficiently low temperatures then ensure that in the range $a \ll d \ll \Lambda_T/2\pi$, the Casimir interaction can be strongly altered: even if repulsion cannot be achieved in a first step, we expect a significant reduction of the Casimir attraction at distances of a few microns (Fig.\[fig:attr-rep\]).
Our thanks for comments and discussion goes to J.-J. Greffet, J.-P. Mulet, M. Wilkens, M. Tomaš, C. Genet, A. Lambrecht, S. Reynaud, and L. Pitaevskii. We thank anonymous referees for constructive criticism. We acknowledge financial support from the bilateral French-German programme “Procope” under project numbers 03199RH and D/0205739.
[10]{}
S. A. Ramakrishna, Rep. Prog. Phys. [**68**]{}, 449 (2005).
T. J. Yen [*et al.*]{}, Science [**303**]{}, 1494 (2004).
S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, Science [**306**]{}, 1351 (2004).
V. V. Klimov, Opt. Commun. [**211**]{}, 183 (2002).
J. Kästel and M. Fleischhauer, Phys. Rev. A [**71**]{}, 011804 (2005).
T. H. Boyer, Phys. Rev. A [**9**]{}, 2078 (1974).
O. Kenneth, I. Klich, A. Mann, and M. Revzen, Phys. Rev. Lett. [**89**]{}, 033001 (2002).
T. H. Boyer, American Journal of Physics [**71**]{}, 990 (2003).
U. Hartmann, Phys. Rev. B [**43**]{}, 2404 (1991).
J. N. Israelachvili, [*Intermolecular and surface forces*]{}, 2nd ed. (Academic Press, San Diego, 2000).
S.-W. Lee and W. M. Sigmund, J. Coll. Int. Sci. [**243**]{}, 365 (2001).
J. N. Munday, D. Iannuzzi, Y. Barash, and F. Capasso, Phys. Rev. A [**71**]{}, 042102 (2005).
M. S. Tomaš, Phys. Lett. A [**342**]{}, 381 (2005).
R. E. Camley, M. R. F. Jensen, S. A. Feiven, and T. J. Parker, J. Appl. Phys. [**83**]{}, 6280 (1998).
D. Iannuzzi and F. Capasso, Phys. Rev. Lett. [**91**]{}, 029101 (2003); O. Kenneth, I. Klich, A. Mann, and M. Revzen, *ibid.* [**91**]{}, 029102 (2003).
L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, [*Electrodynamics of continuous media*]{}, 2nd ed. (Pergamon, Oxford, 1984).
J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. [**47**]{}, 2075 (1999).
D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, IEICE Trans. Electron. [**E87-C**]{}, 359 (2004).
H. B. G. Casimir, Proc. Kon. Ned. Akad. Wet. [**51**]{}, 793 (1948).
E. M. Lifshitz, Soviet Phys. JETP [**2**]{}, 73 (1956), \[[*J. Exper. Theoret. Phys. USSR*]{} [**29**]{}, 94 (1955)\].
V. A. Parsegian and B. W. Ninham, Nature [**224**]{}, 1197 (1969).
B. W. Ninham and V. A. Parsegian, J. Chem. Phys. [**53**]{}, 3398 (1970).
P. Richmond and B. W. Ninham, J. Phys. C: Solid State Phys. [ **4**]{}, 1988 (1971).
F. C. Santos, A. Tenório, and A. C. Tort, Phys. Rev. D [**60**]{}, 105022 (1999).
K. Schram, Phys. Lett. [**43A**]{}, 282 (1973).
J. Schwinger, L. L. DeRaad, Jr., and K. A. Milton, Ann. Phys. (N.Y.) [**115**]{}, 1 (1978).
D. Kupiszewska, Phys. Rev. A [**46**]{}, 2286 (1992).
F. Zhou and L. Spruch, Phys. Rev. A [**52**]{}, 297 (1995).
G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Phys. Rev. A [ **61**]{}, 062107 (2000).
R. Esquivel-Sirvent, C. Villarreal, and G. H. Cocoletzi, Phys. Rev. A [**64**]{}, 052108 (2001).
M. S. Tomaš, Phys. Rev. A [**66**]{}, 052103 (2002).
C. Genet, A. Lambrecht, and S. Reynaud, Phys. Rev. A [**67**]{}, 043811 (2003).
M. Born and E. Wolf, [*Principles of Optics*]{}, 6th ed. (Pergamon Press, Oxford, 1959).
P. Yeh, [*Optical Waves in Layered Media*]{} (John Wiley & Sons, New York, 1988).
A. Lambrecht, M.-T. Jaekel, and S. Reynaud, Phys. Lett. A [**225**]{}, 188 (1997).
S. K. Lamoreaux, Phys. Rev. Lett. [**78**]{}, 5 (1997), erratum: [**81**]{} (1998) 5475.
H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, Science [**291**]{}, 1941 (2001).
U. Mohideen and A. Roy, Phys. Rev. Lett. [**81**]{}, 4549 (1998).
G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, Phys. Rev. Lett. [**88**]{}, 041804 (2002).
C. Henkel, K. Joulain, J.-P. Mulet, and J.-J. Greffet, Phys. Rev. A [**69**]{}, 023808 (2004).
H. D. Ackler, R. H. French, and Y.-M. Chiang, J. Coll. Int. Sci. [**179**]{}, 460 (1996).
|
---
abstract: 'Many applications use sequences of $n$ consecutive symbols ($n$-grams). Hashing these $n$-grams can be a performance bottleneck. For more speed, recursive hash families compute hash values by updating previous values. We prove that recursive hash families cannot be more than pairwise independent. While hashing by irreducible polynomials is pairwise independent, our implementations either run in time $O(n)$ or use an exponential amount of memory. As a more scalable alternative, we make hashing by cyclic polynomials pairwise independent by ignoring $n-1$ bits. Experimentally, we show that hashing by cyclic polynomials is twice as fast as hashing by irreducible polynomials. We also show that randomized Karp-Rabin hash families are not pairwise independent.'
address:
- 'LICEF, Université du Québec à Montréal (UQAM), 100 Sherbrooke West, Montreal, QC, H2X 3P2 Canada '
- 'Dept. of CSAS, University of New Brunswick, 100 Tucker Park Road, Saint John, NB, Canada'
author:
- Daniel Lemire
- Owen Kaser
bibliography:
- 'lemur.bib'
title: 'Recursive n-gram hashing is pairwise independent, at best'
---
Rolling Hashing ,Rabin-Karp Hashing ,Hashing Strings
Introduction
============
An $n$-gram is a consecutive sequence of $n$ symbols from an alphabet $\Sigma$. An $n$-gram hash function $h$ maps $n$-grams to numbers in $[0,2^L)$. These functions have several applications from full-text matching [@cohen1998haa; @cohen1999mqr; @306482], pattern matching [@tan2006fpm], or language models [@cardenal2002fast; @zhang2002minimum; @schwenk2007csl; @li2007fam; @talbot2007smoothed; @talbot2007randomised; @talbot2008randomized] to plagiarism detection [@857699].
To prove that a hashing algorithm must work well, we typically need hash values to satisfy some statistical property. Indeed, a hash function that maps all $n$-grams to a single integer would not be useful. Yet, a single hash function is deterministic: it maps an $n$-gram to a single hash value. Thus, we may be able to choose the input data so that the hash values are biased. Therefore, we randomly pick a function from a family $\mathcal{H}$ of functions [@carter1979uch].
Such a family $\mathcal{H}$ is *uniform* (over $L$-bits) if all hash values are equiprobable. That is, considering $h$ selected uniformly at random from $\mathcal{H}$, we have $P(h(x)=y)=1/2^L$ for all $n$-grams $x$ and all hash values $y$. This condition is weak; the family of constant functions ($h(x)=c$) is uniform[^1].
Intuitively, we would want that if an adversary knows the hash value of one $n$-gram, it cannot deduce anything about the hash value of another $n$-gram. For example, with the family of constant functions, once we know one hash value, we know them all. The family $\mathcal{H}$ is *pairwise independent* if the hash value of $n$-gram $x_1$ is independent from the hash value of any other $n$-gram $x_2$. That is, we have $P(h(x_1)=y \land h(x_2)=z)= P(h(x_1)=y) P(h(x_2)=z)=1/4^L$ for all distinct $n$-grams $x_1$, $x_2$, and all hash values $y$, $z$ with $x_1 \not = x_2$. Pairwise independence implies uniformity. We refer to a particular hash function $h \in \mathcal{H}$ as “uniform” or “a pairwise independent hash function” when the family in question can be inferred from the context.
Moreover, the idea of pairwise independence can be generalized: a family of hash functions $\mathcal{H}$ is *$k$-wise independent* if given distinct $x_1,\ldots,x_k$ and given $h$ selected uniformly at random from $\mathcal{H}$, then $P(h(x_1)=y_1 \land \cdots \land h(x_k)=y_k)=1/2^{kL}$. Note that $k$-wise independence implies $k-1$-wise independence and uniformity. (Fully) independent families are $k$-wise independent for arbitrarily large $k$. For applications, non-independent families may fare as well as fully independent families if the entropy of the data source is sufficiently high [@1347164].
A hash function $h$ is *recursive* [@cohenhash]—or rolling[@winnowing]—if there is a function $F$ computing the hash value of the $n$-gram $x_2\ldots x_{n+1}$ from the hash value of the preceding $n$-gram ($x_1 \ldots x_n$) and the values of $x_1$ and $x_{n+1}$. That is, we have $$h(x_2,\ldots,x_{n+1})=F(h(x_1,\ldots,x_n),x_1,x_{n+1}).$$ Ideally, we could compute function $F$ in time $O(L)$ and not, for example, in time $O(Ln)$. The main contributions of this paper are:
- a proof that recursive hashing is no more than pairwise independent (§ \[section:not3wise\]);
- a proof that randomized Karp-Rabin can be uniform but never pairwise independent (§ \[sec:id37\]);
- a proof that hashing by irreducible polynomials is pairwise independent (§ \[sec:general\]);
- a proof that hashing by cyclic polynomials is not even uniform (§ \[sec:cyclic\]);
- a proof that hashing by cyclic polynomials is pairwise independent—after ignoring $n-1$ consecutive bits (§ \[cyclicalmost\]).
We conclude with an experimental section where we show that hashing by cyclic polynomials is faster than hashing by irreducible polynomials. Table \[sec:summaryofpaper\] summarizes the algorithms presented.
name cost per $n$-gram independence memory use
-------------------------------------------------------------------------------------------- ------------------------------ ------------------------------- -------------------------
non-recursive 3-wise (§ \[sec:RecursiveHashing\]) $O(Ln)$ 3-wise $O(n L |\Sigma|)$
Randomized Karp-Rabin (§ \[sec:id37\]) $O(L \log L 2^{O(\log^*L)})$ uniform $O(L |\Sigma|)$
<span style="font-variant:small-caps;">General</span> (§ \[sec:general\]) $O(Ln)$ pairwise $O(L |\Sigma|)$
RAM-Buffered <span style="font-variant:small-caps;">General</span> (§ \[sec:rambuffered\]) $O(L)$ pairwise $O(L |\Sigma| + L 2^n)$
<span style="font-variant:small-caps;">Cyclic</span> (§ \[sec:cyclic\]) $O(L+n)$ pairwise (§ \[cyclicalmost\]) $O( (L+n) |\Sigma|)$
: \[sec:summaryofpaper\]A summary of the hashing function presented and their properties. For <span style="font-variant:small-caps;">General</span> and <span style="font-variant:small-caps;">Cyclic</span>, we require $L\geq n$. To make <span style="font-variant:small-caps;">Cyclic</span> pairwise independent, we need to discard some bits—the resulting scheme is not formally recursive. Randomized Karp-Rabin is uniform under some conditions.
Trailing-zero independence
==========================
Some randomized algorithms [@flajolet1985pca; @Gibbons2001] merely require that the number of trailing zeroes be independent. For example, to estimate the number of distinct $n$-grams in a large document without enumerating them, we merely have to compute maximal numbers of leading zeroes $k$ among hash values [@durand2003lcl]. Naïvely, we may estimate that if a hash value with $k$ leading zeroes is found, we have $\approx 2^k$ distinct $n$-grams. Such estimates might be useful because the number of distinct $n$-grams grows large with $n$: Shakespeare’s First Folio [@Gutenberg] has over 3 million **distinct** 15-grams.
Formally, let $\textrm{zeros}(x)$ return the number of trailing zeros (0,1,…,$L$) of $x$, where $\textrm{zeros}(0) = L$. We say $h$ is *$k$-wise trailing-zero independent* if $P(
\textrm{zeros}(h(x_1)) \geq j_1 \wedge \textrm{zeros}(h(x_2)) \geq j_2
\wedge \ldots \wedge \textrm{zeros}(h(x_k)) \geq j_k) =
2^{-j_1-j_2-\cdots-j_k}$, for $j_i = 0, 1, \ldots, L$.
If $h$ is $k$-wise independent, it is $k$-wise trailing-zero independent. The converse is not true. If $h$ is a $k$-wise independent function, consider $g \circ h$ where $g$ makes zero all bits before the rightmost 1 (e.g., $g(0101100) = 0000100$). Hash $g \circ h$ is $k$-wise trailing-zero independent but not even uniform (consider that $P(g=0001)=8 P(g=1000)$).
Recursive hash functions are no more than pairwise independent {#section:not3wise}
==============================================================
Not only are recursive hash functions limited to pairwise independence: they cannot be 3-wise *trailing-zero* independent.
\[threewiseprop\] There is no 3-wise trailing-zero independent hashing function that is recursive.
Consider the ($n+2$)-gram $\texttt{a}^n\texttt{bb}$. Suppose $h$ is recursive and $3$-wise trailing-zero independent, then $$\begin{aligned}
\lefteqn{P \left (\textrm{zeros}(h({\texttt{a}},\ldots,{\texttt{a}}))\geq L\bigwedge \right.}\\
& &\left.
\textrm{zeros}(h({\texttt{a}},\ldots, {\texttt{a}},{\texttt{b}})) \geq L \bigwedge
\textrm{zeros}(h({\texttt{a}},\ldots, {\texttt{a}},{\texttt{b}},{\texttt{b}})) \geq L \right )\\
&= & P\left ( h({\texttt{a}},\ldots,{\texttt{a}})=0 \bigwedge F(0,{\texttt{a}},{\texttt{b}})=0 \bigwedge F(0,{\texttt{a}},{\texttt{b}})=0 \right)\\
&= & P\left (h({\texttt{a}},\ldots,{\texttt{a}})=0 \bigwedge F(0,{\texttt{a}},{\texttt{b}})=0\right )\\
& =& P\left (\textrm{zeros}(h({\texttt{a}},\ldots,{\texttt{a}}))\geq L \bigwedge
\textrm{zeros}(h({\texttt{a}},\ldots, {\texttt{a}},{\texttt{b}})) \geq L\right )\\
& = &2^{-2L} \mbox{\ by trailing-zero pairwise independence}\\
& \not =& 2^{-3L} \mbox{\ as required by trailing-zero 3-wise independence.}\end{aligned}$$ Hence, we have a contradiction and no such $h$ exists.
A non-recursive 3-wise independent hash function {#sec:RecursiveHashing}
================================================
A trivial way to generate an independent hash is to assign a random integer in $[0,2^L)$ to each new value $x$. Unfortunately, this requires as much processing and storage as a complete indexing of all values.
However, in a multidimensional setting this approach can be put to good use. Suppose that we have tuples in $K_1 \times K_2
\times \cdots \times K_n$ such that $\vert K_i \vert$ is small for all $i$. We can construct independent hash functions $h_i : K_i \rightarrow [0,2^L)$ for all $i$ and combine them. The hash function $h(x_1,x_2,\ldots,x_n)=h_1(x_1){\oplus}h_2(x_2) {\oplus}\cdots {\oplus}h_n(x_n)$ is then 3-wise independent (${\oplus}$ is the “exclusive or” function, XOR).
In time $O(\sum_{i=1}^n \vert K_i \vert)$, we can construct the hash function by generating $\sum_{i=1}^n \vert K_i \vert$ random numbers and storing them in a look-up table. With constant-time look-up, hashing an $n$-gram thus takes $O(L n)$ time. Algorithm \[algo:nwise\] is an application of this idea to $n$-grams.
$n$ $L$-bit hash functions $h_1, h_1, \ldots, h_n$ over $\Sigma$ from an independent hash family $s\leftarrow$ empty FIFO structure append $c$ to $s$ **yield** $h_1(s_1) {\oplus}h_2(s_2) {\oplus}\ldots {\oplus}h_n(s_n)$\
remove oldest character from $s$
This new family is not 4-wise independent for $n>1$. Consider the $n$-grams `ac`,`ad`, `bc`, `bd`. The [XOR]{} of their four hash values is zero. However, the family is 3-wise independent.
The family of hash functions $h(x)= h_1(x_1){\oplus}h_2(x_2) {\oplus}\ldots {\oplus}h_n(x_n)$, where the $L$-bit hash functions $h_1,\ldots, h_n$ are taken from an independent hash family, is 3-wise independent.
Consider any 3 distinct $n$-grams: $x^{(1)}=x_1^{(1)}\ldots x_n^{(1)}$, $x^{(2)}=x_1^{(2)}\ldots x_n^{(2)}$, and $x^{(3)}=x_1^{(3)}\ldots x_n^{(3)}$. Because the $n$-grams are distinct, at least one of two possibilities holds:
Case A
: For some $i\in \{1,\ldots,n\}$, the three values $x_i^{(1)}, x_i^{(2)}, x_i^{(3)}$ are distinct. Write $\chi_j=h_i(x_i^{(j)})$ for $j=1,2,3$. For example, consider the three 1-grams: `a`,`b`,`c`.
Case B
: (Up to a reordering of the three $n$-grams.) There are two values $i,j\in \{1,\ldots,n\}$ such that $x_i^{(1)}$ is distinct from the two identical values $x_i^{(2)}, x_i^{(3)}$, and such that $x_j^{(2)}$ is distinct from the two identical values $x_i^{(1)}, x_i^{(3)}$. Write $\chi_1=h_i(x_i^{(1)})$, $\chi_2=h_j(x_j^{(2)})$, and $\chi_3=h_i(x_i^{(3)})$. For example, consider the three 2-grams: `ad`,`bc`,`bd`.
Recall that the [XOR]{} operation is invertible: $a {\oplus}b = c$ if and only if $a = b {\oplus}c$.
We prove 3-wise independence for cases A and B.
#### Case A
Write $f^{(i)} = h(x^{(i)}){\oplus}\chi_i$ for $i=1,2,3$. We have that the values $\chi_1, \chi_2, \chi_3$ are mutually independent, and they are independent from the values $f^{(1)}, f^{(2)},f^{(3)}$[^2]: $$\begin{aligned}
P\left (\bigwedge_{i=1}^3 \chi_i=y_i \land \bigwedge_{i=1}^3 f^{(i)}=y'_i \right )=\prod_{i=1}^3 P(\chi_i=y_i)P\left (\bigwedge_{i=1}^3 f^{(i)}=y'_i\right )\end{aligned}$$ for all values $y_i, y'_i$. Hence, we have $$\begin{aligned}
\lefteqn{P\left (h(x^{(1)})= z^{(1)} \bigwedge
h(x^{(2)})= z^{(2)} \bigwedge
h(x^{(3)})= z^{(3)}\right )} \\
&=& P\left (\chi_1 = z^{(1)}{\oplus}f^{(1)}) \bigwedge \chi_2 = z^{(2)} {\oplus}f^{(2)}
\bigwedge \chi_3 = z^{(3)} {\oplus}f^{(3)}\right )\\
&=& \sum_{\eta,\eta',\eta''} P\left (\chi_1 = z^{(1)}{\oplus}\eta \bigwedge \chi_2 = z^{(2)} {\oplus}\eta'
\bigwedge \chi_3 = z^{(3)} {\oplus}\eta'' \right ) \times \\
& & P(f^{(1)}=\eta \land f^{(2)}=\eta' \land f^{(3)}=\eta'' ) \\
& = & \sum_{\eta,\eta',\eta''} \frac{1}{2^{3L}} P(f^{(1)}=\eta \land f^{(2)}=\eta' \land f^{(3)}=\eta'') \\ & = & \frac{1}{2^{3L}}.\end{aligned}$$ Thus, in this case, the hash values are 3-wise independent.
#### Case B
Write $f^{(1)} = h(x^{(1)}){\oplus}\chi_1$, $f^{(2)} = h(x^{(2)}){\oplus}\chi_2{\oplus}\chi_3$, $f^{(3)} = h(x^{(3)}){\oplus}\chi_3$. Again, the values $\chi_1, \chi_2, \chi_3$ are mutually independent, and independent from the values $f^{(1)}, f^{(2)},f^{(3)}$. We have $$\begin{aligned}
\lefteqn{P\left (h(x^{(1)})= z^{(1)} \bigwedge
h(x^{(2)})= z^{(2)} \bigwedge
h(x^{(3)})= z^{(3)}\right )} \\
&=& P\left (\chi_1 = z^{(1)}{\oplus}f^{(1)}) \bigwedge \chi_2 {\oplus}\chi_3 = z^{(2)} {\oplus}f^{(2)}
\bigwedge \chi_3 = z^{(3)} {\oplus}f^{(3)}\right )\\
&=& P\left (\chi_1 = z^{(1)}{\oplus}f^{(1)}) \bigwedge \chi_2 = z^{(2)} {\oplus}f^{(2)}{\oplus}z^{(3)} {\oplus}f^{(3)}
\bigwedge \chi_3 = z^{(3)} {\oplus}f^{(3)}\right )\\
&=& \sum_{\eta,\eta',\eta''} P\left (\chi_1 = z^{(1)}{\oplus}\eta \bigwedge \chi_2 = z^{(2)}{\oplus}z^{(3)} {\oplus}\eta' {\oplus}\eta''
\bigwedge \chi_3 = z^{(3)} {\oplus}\eta'' \right ) \times \\
& & P(f^{(1)}=\eta \land f^{(2)}=\eta' \land f^{(3)}=\eta'' ) \\
& = & \sum_{\eta,\eta',\eta''} \frac{1}{2^{3L}} P(f^{(1)}=\eta \land f^{(2)}=\eta' \land f^{(3)}=\eta'') \\ & = & \frac{1}{2^{3L}}.\end{aligned}$$ This concludes the proof.
Randomized Karp-Rabin is not independent {#sec:id37}
========================================
One of the most common recursive hash functions is commonly associated with the Karp-Rabin string-matching algorithm [@karp1987erp]. Given an integer $B$, the hash value over the sequence of integers $x_1, x_2,\ldots, x_n$ is $\sum_{i=1}^n x_i B^{n-i}$. A variation of the Karp-Rabin hash method is “Hashing by Power-of-2 Integer Division” [@cohenhash], where $h(x_1,\ldots,x_n) = \sum_{i=1}^n x_i B^{n-i} \bmod{2^L}$. In particular, the `hashcode` method of the Java String class uses this approach, with $L=32$ and $B=31$ [@j15doc:String]. A widely used textbook [@weis:dsaaj p. 157] recommends a similar Integer-Division hash function for strings with $B=37$.
Since such Integer-Division hash functions are recursive, quickly computed, and widely used, it is interesting to seek a randomized version of them. Assume that $h_1$ is a random hash function over symbols uniform in $[0,2^L)$, then define $h(x_1,\ldots,x_n)=B^{n-1} h_1(x_1)+B^{n-2} h_1(x_2)+ \cdots + h_1(x_n)\bmod{2^L}$ for some fixed integer $B$. We choose $B=37$ (calling the resulting randomized hash “ID37;” see Algorithm \[algo:id37\]). Our algorithm computes each hash value in time O($M(L)$), where $M(L)$ is the cost of multiplying two $L$-bit integers. (We precompute the value $B^n \bmod{2^L}$.) In many practical cases, $L$ bits can fit into a single machine word and the cost of multiplication can be considered constant. In general, $M(L)$ is in $O(L \log L 2^{O(\log^*L)})$ [@1250800].
an $L$-bit hash function $h_1$ over $\Sigma$ from an independent hash family $B\leftarrow 37$ $s\leftarrow$ empty FIFO structure $x\leftarrow 0$ ($L$-bit integer) $z\leftarrow 0$ ($L$-bit integer) append $c$ to $s$ $x\leftarrow B x - B^n z + h_1(c) \bmod{2^L}$ **yield** $x$ remove oldest character $y$ from $s$ $z \leftarrow h_1(y)$
The randomized Integer-Division functions mapping $n$-grams to $[0,2^L)$ are not pairwise independent. However, for some values of $B$ and $n$, they are uniform.
Randomized Integer-Division hashing is not uniform for $n$-grams, if $n$ is even and $B$ is odd. Otherwise, it is uniform for $B$ even and any $n$, or $B$ odd and $n$ odd. However, there is no value of $B$ for which it is pairwise independent when $n\geq 2$.
For $B$ odd, we see that $P(h(\texttt{a}^{2k}) = 0) > 2^{-L}$ since $h(\texttt{a}^{2k}) = h_1(\texttt{a}) ( B^0(1+B) + B^2(1+B) + \cdots + B^{2k-2}(1+B)) \bmod 2^L$ and since $(1+B)$ is even, we have $P(h(\texttt{a}^{2k}) = 0) \geq P(h_1(x_1)=2^{L-1} \lor h_1(x_1)=0)= 1/2^{L-1}$. Hence, for $B$ odd and $n$ even, we do not have uniformity.
Suppose that $B$ and $n$ are both odd. Consider any string $x_1, x_2, \ldots, x_n$. We can find a character value $x_j$ which is repeated an odd number of times in the string. Let $I$ be the set of indexes $i$ such that $x_i=x_j$. We have that the equation $h(x_1, x_2, \ldots, x_n)=y$ is equivalent to $(\sum_{i=1}^n B^{n-i} h_1(x_i))=y$. We can rewrite it as $(\sum_{i\in I} B^{n-i}) h_1(x_j)=y - (\sum_{i\not \in I} B^{n-i} h_1(x_i))$. There is a unique solution $h_1(x_j)$ to this equation because $(\sum_{i\in I} B^{n-i})$ is odd: the sum of an odd number of odd integers is an odd integer. Hence, we have uniformity when $B$ and $n$ are odd.
Consider $B$ even. Consider any string $x_1, x_2, \ldots, x_n$. We are interested in the last character $x_n$. It might be repeated several times in the string. Let $I$ be the set of indexes $i$ such that $x_i=x_n$. We have that $h(x_1, x_2, \ldots, x_n)=y$ is equivalent to $(\sum_{i=1}^n B^{n-i} h_1(x_i))=y$ or $(\sum_{i\in I} B^{n-i}) h_1(x_n)=y - (\sum_{i\not \in I} B^{n-i} h_1(x_i))$. We want to show that there is a unique solution $h_1(x_n)$ to this equation. This follows because we have that $(\sum_{i\in I} B^{n-i})$ is an odd number because $B$ is even and $n\in I$. Hence, we have uniformity when $B$ is even.
To show it is not pairwise independent, first suppose that $B$ is odd. For any string $\beta$ of length $n-2$, consider $n$-grams $w_1 = \beta \texttt{a} \texttt{a}$ and $w_2 = \beta \texttt{b} \texttt{b}$ for distinct $\texttt{a}, \texttt{b} \in \Sigma$. Then $P(h(w_1) = h(w_2)) = P(B^2 h(\beta) + B h_1(\texttt{a})+h_1(\texttt{a})
= B^2 h(\beta) + B h_1(\texttt{b})+h_1(\texttt{b}) \bmod 2^L)
=P( (1+B) (h_1(\texttt{a})-h_1(\texttt{b})) \bmod 2^L = 0)\geq
P(h_1(\texttt{a})-h_1(\texttt{b}) = 0)+P(h_1(\texttt{a})-h_1(\texttt{b}) = 2^{L-1})$. Because $h_1$ is independent, $P(h_1(\texttt{a})-h_1(\texttt{b})=0)=\sum_{c\in [0,2^L)} P(h_1(\texttt{a})=c)P(h_1(\texttt{b})=c)=\sum_{c\in [0,2^L)} 1/4^L=1/2^L$. Moreover, $P(h_1(\texttt{a})-h_1(\texttt{b})= 2^{L-1})>0$. Thus, we have that $P(h(w_1) = h(w_2))>1/2^L$ which contradicts pairwise independence. Second, if $B$ is even, a similar argument shows $P(h(w_3) = h(w_4)) > 1/2^L$, where $w_3 = \beta \texttt{a} \texttt{a}$ and $w_4 = \beta \texttt{b} \texttt{a}$. $P(h(\texttt{a},\texttt{a})=h(\texttt{b},\texttt{a}))=
P(B h_1(\texttt{a})+h_1(\texttt{a})=B h_1(\texttt{b})+h_1(\texttt{a}) \bmod 2^L)
=P(B(h_1(\texttt{a})-h_1(\texttt{b})) \bmod 2^L = 0)\geq
P(h_1(\texttt{a})-h_1(\texttt{b}) = 0)+P(h_1(\texttt{a})-h_1(\texttt{b}) = 2^{L-1})>1/2^L$. Hence, as long as we consider strings of length $n>1$ and an alphabet $\Sigma$ containing at least two distinct characters, we can find two strings with a collision probability greater than $1/2^L$ whether $B$ is even or odd.
A weaker condition than pairwise independence is 2-universality: a family is 2-universal if $P(h(x_1)=h(x_2))\leq 1/2^L$ [@1347164]. As a consequence of this proof, Randomized Integer-Division is not even 2-universal.
These results also hold for any Integer-Division hash where the modulo is by an even number, not necessarily a power of 2.
Generating hash families from polynomials over Galois fields {#sec:RecursiveHashingbyPolynomials}
============================================================
A practical form of hashing using the binary Galois field GF(2) is called “Recursive Hashing by Polynomials” and has been attributed to Kubina by Cohen [@cohenhash]. GF(2) contains only two values (1 and 0) with the addition (and hence subtraction) defined by [XOR]{}, $a + b = a {\oplus}b$ and the multiplication by AND, $a \times b = a \wedge b$. $\textrm{GF}(2)[x]$ is the vector space of all polynomials with coefficients from GF(2). Any integer in binary form (e.g., $c=1101$) can thus be interpreted as an element of $\textrm{GF}(2)[x]$ (e.g., $c=x^3+x^2+1$). If $p(x)\in \textrm{GF}(2)[x]$, then $\textrm{GF}(2)[x]/p(x)$ can be thought of as $\textrm{GF}(2)[x]$ modulo $p(x)$. As an example, if $p(x)=x^2$, then $\textrm{GF}(2)[x]/p(x)$ is the set of all linear polynomials. For instance, $x^3+ x^2+ x+1= x+1 \bmod{x^2}$ since, in $\textrm{GF}(2)[x]$, $(x+1) + x^2 (x+1) = x^3 + x^2 + x + 1$.
As a summary, we compute operations over $\textrm{GF}(2)[x]/p(x)$—where $p(x)$ is of degree $L$—as follows:
- the polynomial $\sum_{i=0}^{L-1} q_i x^i$ is represented as the $L$-bit integer $\sum_{i=0}^{L-1} q_i 2^i$;
- subtraction or addition of two polynomials is the [XOR]{} of their $L$-bit integers;
- multiplication of a polynomial $\sum_{i=0}^L q_i x^i$ by the monomial $x$ is represented either as $\sum_{i=0}^{L-1} q_i x^{i+1}$ if $q_{L-1}=0$ or as $p(x)+\sum_{i=0}^{L-1} q_i x^{i+1}$ otherwise. In other words, if the value of the last bit is 1, we merely apply a binary left shift, otherwise, we apply a binary left shift immediately followed by an [XOR]{} with the integer representing $p(x)$. In either case, we get an $L$-bit integer.
Hence, merely with the [XOR]{} operation, the binary left shift, and a way to evaluate the value of the last bit, we can compute all necessary operations over $\textrm{GF}(2)[x]/p(x)$ using integers.
Consider a hash function $h_1$ over characters taken from some independent family. Interpreting $h_1$ hash values as polynomials in $\textrm{GF}(2)[x]/p(x)$, and with the condition that $\textrm{degree}(p(x))\geq n$, we define a hash function as $h(a_1,a_2,\cdots,a_n)=h_1(a_1) x^{n-1} + h_1(a_2) x^{n-2}+
\cdots + h_1(a_n)$. It *is* recursive over the sequence $h_1(a_i)$. The combined hash can be computed by reusing previous hash values: $$h(a_2,a_3,\ldots,a_{n+1})= x h(a_1,a_2,\ldots,a_n) - h_1(a_1) x^{n} + h_1(a_{n+1}).$$ Depending on the choice of the polynomial $p(x)$ we get different hashing schemes, including <span style="font-variant:small-caps;">General</span> and <span style="font-variant:small-caps;">Cyclic</span>, which are presented in the next two sections.
Recursive hashing by irreducible polynomials is pairwise independent {#sec:general}
====================================================================
an $L$-bit hash function $h_1$ over $\Sigma$ from an independent hash family; an irreducible polynomial $p$ of degree $L$ in $\textrm{GF}(2)[x]$
$s\leftarrow$ empty FIFO structure $x\leftarrow 0$ ($L$-bit integer) $z\leftarrow 0$ ($L$-bit integer) append $c$ to $s$ $x \leftarrow \textrm{shift}(x)$ $z \leftarrow \textrm{shift}^n(z)$ $x \leftarrow x {\oplus}z {\oplus}h_1(c)$ **yield** $x$ remove oldest character $y$ from $s$ $z \leftarrow h_1(y)$
------------------------------------------------------------------------
**function** shift **input** $L$-bit integer $x$ shift $x$ left by 1 bit, storing result in an $L+1$-bit integer $x'$ $x' \leftarrow x' {\oplus}p$ **return** rightmost $L$ bits of $x'$
degree polynomial
-------- ----------------------
10 $1+x^3+x^{10}$
15 $1+x+x^{15}$
20 $1+x^3+x^{20}$
25 $1+x^3+x^{25}$
30 $1+x+x^4+x^6+x^{30}$
: \[table:irred\]Some irreducible polynomials over $\textrm{GF}(2)[x]$
We can choose $p(x)$ to be an irreducible polynomial of degree $L$ in $\textrm{GF}(2)[x]$: an irreducible polynomial cannot be factored into nontrivial polynomials (see Table \[table:irred\]). The resulting hash is called <span style="font-variant:small-caps;">General</span> (see Algorithm \[algo:general\]). The main benefit of setting $p(x)$ to be an irreducible polynomial is that $\textrm{GF}(2)[x]/p(x)$ is a field; in particular, it is impossible that $p_1(x) p_2(x) = 0 \bmod
{p(x)}$ unless either $p_1(x)=0$ or $p_2(x)=0$. The field property allows us to prove that the hash function is pairwise independent.
\[uniformlemma\] <span style="font-variant:small-caps;">General</span> is pairwise independent.
If $p(x)$ is irreducible, then any non-zero $q(x)\in \textrm{GF}(2)[x]/p(x)$ has an inverse, noted $q^{-1}(x)$ since $\textrm{GF}(2)[x]/p(x)$ is a field. Interpret hash values as polynomials in $\textrm{GF}(2)[x]/p(x)$.
Firstly, we prove that <span style="font-variant:small-caps;">General</span> is uniform. In fact, we show a stronger result: $P(q_1(x) h_1(a_1) + q_2(x) h_1(a_2)+\cdots+q_n(x)h_1(a_n)=y)= 1/2^L$ for any polynomials $q_i$ where at least one is different from zero. The result follows by induction on the number of non-zero polynomials: it is clearly true where there is a single non-zero polynomial $q_i(x)$, since $q_i(x) h_1(a_i)=y \iff q_i^{-1}(x) q_i(x) h_1(a_i) = q_i^{-1}(x)y$. Suppose it is true up to $k-1$ non-zero polynomials and consider a case where we have $k$ non-zero polynomials. Assume without loss of generality that $q_1(x)\neq 0$, we have $P(q_1(x) h_1(a_1) + q_2(x) h_1(a_2)+\cdots+q_n(x)h_1(a_n)=y)=
P( h_1(a_1) = q_1^{-1}(x)(y - q_2(x) h_1(a_2)-\cdots-q_n(x)h_1(a_n)))
=\sum_{y'} P( h_1(a_1) = q_1^{-1}(x)(y - y'))P(q_2(x) h_1(a_2)+\cdots+q_n(x)h_1(a_n)=y')
= \sum_{y'} \frac{1}{2^L}\frac{1}{2^L}=\frac{1}{2^L}$ by the induction argument. Hence the uniformity result is shown.
Consider two distinct sequences $a_1,a_2,\ldots,a_n$ and $a'_1,a'_2,\ldots,a'_n$. Write $H_a= h(a_1,a_2,\ldots,a_n)$ and $H_{a'}=h(a'_1,a'_2,\ldots,a'_n)$. We have that $P(H_a=y \land H_{a'}=y')
=P(H_a=y | H_{a'}=y') P(H_{a'}=y')$. Hence, to prove pairwise independence, it suffices to show that $P(H_a=y|H_{a'}=y')=1/2^L$.
Suppose that $a_i= a'_j$ for some $i,j$; if not, the result follows since by the (full) independence of the hashing function $h_1$, the values $H_a$ and $H_{a'}$ are independent. Write $q(x)= -(\sum_{k | a_k= a_i} x^{n-k} ) (\sum_{k | a'_k= a'_j} x^{n-k} )^{-1}$, then $H_a+q(x)H_{a'}$ is independent from $a_i= a'_j$ (and $h_1(a_i)=h_1(a'_j)$).
In $H_a+q(x)H_{a'}$, only hashed values $h_1(a_k)$ for $a_k \neq a_i$ and $h_1(a'_k)$ for $a'_k \neq a'_j$ remain: label them $h_1(b_1),\ldots, h_1(b_m)$. The result of the substitution can be written $H_a+q(x)H_{a'} = \sum_k q_k(x) h_1(b_k) $ where $q_k(x)$ are polynomials in $\textrm{GF}(2)[x]/p(x)$. All $q_k(x)$ are zero if and only if $H_a+q(x)H_{a'}=0$ for all values of $h_1(a_1),\ldots, h_1(a_n)$ and $h_1(a'_1),\ldots, h_1(a'_n)$ (but notice that the value $h_1(a_i)=h_1(a'_j)$ is irrelevant); in particular, it must be true when $h_1(a_k)=1$ and $h_1(a'_k)=1$ for all $k$, hence $(x^n+\cdots+x+1)+q(x)(x^n\ldots+x+1)=0\Rightarrow q(x)=-1$. Thus, all $q_k(x)$ are zero if and only if $H_a=H_{a'}$ for all values of $h_1(a_1),\ldots, h_1(a_n)$ and $h_1(a'_1),\ldots, h_1(a'_n)$ which only happens if the sequences $a$ and $a'$ are identical. Hence, not all $q_k(x)$ are zero.
Write $H_{y',a'}=(\sum_{k | a'_k= a'_j} x^{n-k} )^{-1} (y'- \sum_{k | a'_k\neq a'_j} x^{n-k} h_1(a'_k))$. On the one hand, the condition $H_{a'}=y'$ can be rewritten as $h_1(a'_j) = H_{y',a'}$. On the other hand, $H_a+q(x)H_{a'}=y+q(x)y'$ is independent from $h_1(a'_j)=h_1(a_i)$. Because $P(h_1(a'_j) = H_{y',a'})=1/2^L$ irrespective of $y'$ and $h_1(a'_k)$ for $k\in \{k | a'_k\neq a'_j\}$, then $P(h_1(a'_j) = H_{y',a'} | H_a+q(x)H_{a'}=y+q(x)y' )= P(h_1(a'_j) = H_{y',a'})$ which implies that $h_1(a'_j) = H_{y',a'}$ and $H_a+q(x)H_{a'}=y+q(x)y'$ are independent. Hence, we have $$\begin{aligned}
\lefteqn{P( H_a=y | H_{a'}=y' )}&&\\
& = &P( H_a+q(x)H_{a'}=y+q(x)y' | h_1(a'_j) = H_{y',a'})\\
& = &P( H_a+q(x)H_{a'}=y+q(x)y')\\
& = &P(\sum_k q_k(x) h_1(b_k) = y + q(x) y' )\end{aligned}$$ and by the earlier uniformity result, this last probability is equal to $1/2^L$. This concludes the proof.
Trading memory for speed: RAM-Buffered <span style="font-variant:small-caps;">General</span> {#sec:rambuffered}
============================================================================================
Unfortunately, <span style="font-variant:small-caps;">General</span>—as computed by Algorithm \[algo:general\]—requires $O(nL)$ time per $n$-gram. Indeed, shifting a value $n$ times in $\textrm{GF}(2)[x]/p(x)$ requires $O(nL)$ time. However, if we are willing to trade memory usage for speed, we can precompute these shifts. We call the resulting scheme RAM-Buffered <span style="font-variant:small-caps;">General</span>.
\[lemma:firsttradinglemma\] Pick any $p(x)$ in $\textrm{GF}(2)[x]$. The degree of $p(x)$ is $L$. Represent elements of $\textrm{GF}(2)[x]/p(x)$ as polynomials of degree at most $L-1$. Given any $h$ in $\textrm{GF}(2)[x]/p(x)$. we can compute $x^n h$ in O($L$) time given an $O(L 2^n)$-bit memory buffer.
Write $h$ as $\sum_{i=0}^{L-1} q_i x^i$. Divide $h$ into two parts, $h^{(1)}=\sum_{i=0}^{L-n-1} q_i x^i$ and $h^{(2)}=\sum_{i=L-n}^{L-1} q_i x^{i}$, so that $h=h^{(1)}+ h^{(2)}$. Then $x^n h =
x^n h^{(1)} + x^n h^{(2)}$. The first part, $x^n h^{(1)}$ is a polynomial of degree at most $L-1$ since the degree of $h^{(1)}$ is at most $L-1-n$. Hence, $x^n h^{(1)}$ as an $L$-bit value is just $q_{L-n-1}q_{L-n-2}\ldots q_{0} 0 \ldots 0$. which can be computed in time $O(L)$. So, only the computation of $x^n h^{(2)}$ is possibly more expensive than $O(L)$ time, but $h^{(2)}$ has only $n$ terms as a polynomial (since the first $L-n$ terms are always zero). Hence, if we precompute $x^n h^{(2)}$ for all $2^n$ possible values of $h^{(2)}$, and store them in an array with $O(L)$ time look-ups, we can compute $x^n h$ as an $L$-bit value in $O(L)$ time.
When $n$ is large, this precomputation requires excessive space and precomputation time. Fortunately, we can trade back some speed for memory. Consider the proof of Lemma \[lemma:firsttradinglemma\]. Instead of precomputing the shifts of all $2^n$ possible values of $h^{(2)}$ using an array of $2^n$ entries, we can further divide $h^{(2)}$ into $K$ parts. For simplicity, assume that the integer $K$ divides $n$. The $K$ parts $h^{(2,1)}, \ldots, h^{(2,K)}$ are made of the first $n/K$ bits, the next $n/K$ bits and so on. Because $x^n h^{(2)}= \sum_{i=1}^K x^n h^{(2,i)}$, we can shift $h^{(2)}$ by $n$ in $O(KL)$ operations using $K$ arrays of $2^{n/K}$ entries. To summarize, we have a time complexity of $O(K L)$ per $n$-gram using $O(L |\Sigma| + L K 2^{n/K})$ bits. We implemented the case $K=2$.
Recursive hashing by cyclic polynomials is not even uniform {#sec:cyclic}
===========================================================
Choosing $p(x)=x^{L}+1$ for $L\geq n$, for any polynomial $q(x) = \sum_{i=0}^{L-1} q_i x^i$, we have $$x^i q(x) = x^i ( q_{L-1} x^{L-1}+ \cdots +q_1 x + q_0) = q_{L-i-1} x^{L-i-2}+ \cdots + q_{L-i+1} x + q_{L-i}.$$ Thus, we have that multiplication by $x^i$ is a bitwise rotation, a cyclic left shift—which can be computed in $O(L)$ time. The resulting hash (see Algorithm \[algo:cyclic\]) is called <span style="font-variant:small-caps;">Cyclic</span>. It requires only $O(L)$ time per hash value. *Empirically*, Cohen showed that <span style="font-variant:small-caps;">Cyclic</span> is uniform [@cohenhash]. In contrast, we show that it is not formally uniform:
an $L$-bit hash function $h_1$ over $\Sigma$ from an independent hash family $s\leftarrow$ empty FIFO structure $x\leftarrow 0$ ($L$-bit integer) $z\leftarrow 0$ ($L$-bit integer) append $c$ to $s$ rotate $x$ left by 1 bit rotate $z$ left by n bits $x \leftarrow x {\oplus}z {\oplus}h_1(c)$ **yield** $x$ remove oldest character $y$ from $s$ $z \leftarrow h_1(y)$
\[not-uniformlemma\] <span style="font-variant:small-caps;">Cyclic</span> is not uniform for $n$ even and never 2-universal, and thus never pairwise independent.
If $n$ is even, use the fact that $x^{n-1}+\cdots+x+1$ is divisible by $x+1$ to write $x^{n-1}+\cdots+x+1=(x+1)r(x)$ for some polynomial $r(x)$. Clearly, $r(x) (x+1)(x^{L-1}+x^{L-2}+\cdots+x+1)=0 \bmod{x^L+1}$ for any $r(x)$ and so $P(h(a_1,a_1,\ldots, a_1)=0)=P((x^{n-1}+\cdots+x+1)h_1(a_1)=0)=
P((x+1) r(x) h_1(a_1)=0)
\geq P(h_1(a_1)=0 \lor h_1(a_1)=x^{L-1}+x^{L-2}+\cdots+x+1)= 1/2^{L-1}$. Therefore, <span style="font-variant:small-caps;">Cyclic</span> is not uniform for $n$ even.
To show <span style="font-variant:small-caps;">Cyclic</span> is never pairwise independent, consider $n=3$ (for simplicity), then $P(h(a_1,a_1,a_2)= h(a_1,a_2,a_1))=P((x+1)(h_1(a_1)+h_1(a_2))=0)\geq
P(h_1(a_1)+h_1(a_2)=0 \lor h_1(a_1)+h_1(a_2)=x^{L-1}+x^{L-2}+\cdots+x+1)=1/2^{L-1}$, but 2-universal hash values are equal with probability $1/2^L$. The result is shown.
Of the four recursive hashing functions investigated by Cohen [@cohenhash], <span style="font-variant:small-caps;">General</span> and <span style="font-variant:small-caps;">Cyclic</span> were superior both in terms of speed and uniformity, though <span style="font-variant:small-caps;">Cyclic</span> had a small edge over <span style="font-variant:small-caps;">General</span>. For $n$ large, the benefits of these recursive hash functions compared to the 3-wise independent hash function presented earlier can be substantial: $n$ table look-ups is much more expensive than a single look-up followed by binary shifts.
<span style="font-variant:small-caps;">Cyclic</span> is pairwise independent if you remove $n-1$ consecutive bits {#cyclicalmost}
=================================================================================================================
Because Cohen found empirically that <span style="font-variant:small-caps;">Cyclic</span> had good uniformity [@cohenhash], it is reasonable to expect <span style="font-variant:small-caps;">Cyclic</span> to be *almost* uniform and maybe even *almost* pairwise independent. To illustrate this intuition, consider Table \[tab:aa\] which shows that while $h(\texttt{a},\texttt{a})$ is not uniform ($h(\texttt{a},\texttt{a})=001$ is impossible), $h(\texttt{a},\texttt{a})$ minus any bit is indeed uniformly distributed. We will prove that this result holds in general.
The next lemma and the next theorem show that <span style="font-variant:small-caps;">Cyclic</span> is quasi-pairwise independent in the sense that $L-n+1$ consecutive bits (e.g., the first or last $L-n+1$ bits) are pairwise independent. In other words, <span style="font-variant:small-caps;">Cyclic</span> is pairwise independent if we are willing to sacrifice $n-1$ bits. (We say that $n$ bits are “consecutive modulo $L$” if the bits are located at indexes $i\bmod{L}$ for $n$ consecutive values of $i$ such as $i=k,k+1,\ldots,k+n-1$.)
\[lemma:magic\] If $q(x) \in \textrm{GF}(2)[x]/(x^L+1)$ (with $q(x) \neq 0$) has degree $n<L$, then
- the equation $q(x)w = y \bmod{x^L+1}$ modulo the first $n$ bits[^3] has exactly $2^n$ solutions for all $y$;
- more generally, the equation $q(x)w = y \bmod{x^L+1}$ modulo any consecutive $n$ bits (modulo $L$) has exactly $2^n$ solutions for all $y$.
Let $P$ be the set of polynomials of degree at most $L-n-1$. Take any $p(x)\in P$, then $q(x)p(x)$ has degree at most $L-n-1+n=L-1$ and thus if $q(x) \neq 0$ and $p(x)\neq 0$, then $q(x)p(x)\neq 0 \bmod{x^L+1}$. Hence, for any distinct $p_1,p_2\in P$ we have $q(x) p_1 \neq q(x) p_2 \bmod{x^L+1}$.
To prove the first item, we begin by showing that there is always exactly one solution in $P$. Consider that there are $2^{L-n}$ polynomials $p(x)$ in $P$, and that all values $q(x)p(x)$ are distinct. Suppose there are $p_1,p_2 \in P$ such that $q(x)p_1 = q(x) p_2 \bmod{x^L+1}$ modulo the first $n$ bits, then $q(x)(p_1 - p_2 )$ is a polynomial of degree at most $n-1$ while $p_1-p_2$ is a polynomial of degree at most $L-n-1$ and $q(x)$ is a polynomial of degree $n$, thus $p_1-p_2=0$. (If $p1-p2 \neq 0$ then $\textrm{degree}(q(x)(p1-p2) \bmod{x^L+1}) \geq \textrm{degree}(q(x))=n$, a contradiction.) Hence, all $p(x)$ in $P$ are mapped to distinct values modulo the first $n$ bits, and since there are $2^{L-n}$ such distinct values, the result is shown.
Any polynomial of degree $L-1$ can be decomposed into the form $p(x)+x^{L-n}z(x)$ where $z(x)$ is a polynomial of degree at most $n-1$ and $p(x) \in P$. By the preceding result, for distinct $p_1,p_2 \in P$, $q(x)( x^{L-n}z(x)+p_1 )$ and $q(x)( x^{L-n}z(x)+p_2 )$ must be distinct modulo the first $n$ bits. In other words, the equation $q(x)( x^{L-n} z(x) + p ) = y$ modulo the first $n$ bits has exactly one solution $p\in P$ for any $z(x)$ and since there are $2^n$ polynomials $z(x)$ of degree at most $n-1$, then $q(x)w=y$ (modulo the first $n$ bits) must have $2^n$ solutions.
To prove the second item, choose $j$ and use the first item to find any $w$ solving $q(x)w =yx^j \bmod{x^L+1}$ modulo the first $n$ bits. $j$. Then $w x^{L-j}$ is a solution to $q(x)w =y \bmod{x^L+1}$ modulo the bits in positions $j,j+1,\ldots,j+n-1 \bmod{L}$.
We have the following corollary to Lemma \[lemma:magic\].
\[corollary:magic\]If $w$ is chosen uniformly at random in $\textrm{GF}(2)[x]/(x^L+1)$, then $P(q(x) w = y\bmod{n-1 \textrm{\ bits}}) = 1/2^{L-n+1}$ where the $n-1$ bits are consecutive (modulo $L$).
\[thm:cyclicalmostind\] Consider the $L$-bit <span style="font-variant:small-caps;">Cyclic</span> $n$-gram hash family. Pick any $n-1$ consecutive bit locations, then remove these bits from all hash values. The resulting $L-n+1$-bit hash family is pairwise independent.
We show $P(q_1(x) h_1(a_1) + q_2(x) h_1(a_2)+\cdots+q_n(x)h_1(a_n)=y \bmod{n-1 \textrm{\ bits}})= 1/2^{L-n+1}$ for any polynomials $q_i$ where at least one is different from zero. It is true when there is a single non-zero polynomial $q_i(x)$ by Corollary \[corollary:magic\]. Suppose it is true up to $k-1$ non-zero polynomials and consider a case where we have $k$ non-zero polynomials. Assume without loss of generality that $q_1(x)\neq 0$, we have $P(q_1(x) h_1(a_1) + q_2(x) h_1(a_2)+\cdots+q_n(x)h_1(a_n)=y\bmod{n-1 \textrm{\ bits}})=
P( q_1(x) h_1(a_1) = y - q_2(x) h_1(a_2)-\cdots-q_n(x)h_1(a_n)\bmod{n-1 \textrm{\ bits}})
=\sum_{y'} P( q_1(x) h_1(a_1) = y - y'\bmod{n-1 \textrm{\ bits}})
P(q_2(x) h_1(a_2)+\cdots+q_n(x)h_1(a_n)=y'\bmod{n-1 \textrm{\ bits}})
= \sum_{y'} \frac{1}{2^{L-n+1}} \frac{1}{2^{L-n+1}}=1/2^{L-n+1}$ by the induction argument, where the sum is over $2^{L-n+1}$ values of $y'$. Hence the uniformity result is shown.
Consider two distinct sequences $a_1,a_2,\ldots,a_n$ and $a'_1,a'_2,\ldots,a'_n$. Write $H_a= h(a_1,a_2,\ldots,a_n)$ and $H_{a'}=h(a'_1,a'_2,\ldots,a'_n)$. To prove pairwise independence, it suffices to show that $P(H_a=y \bmod{n-1 \textrm{\ bits}}|H_{a'}=y'\bmod{n-1 \textrm{\ bits}})=1/2^{L-n+1}$. Suppose that $a_i= a'_j$ for some $i,j$; if not, the result follows by the (full) independence of the hashing function $h_1$. Using Lemma \[lemma:magic\], find $q(x)$ such that $q(x) \sum_{k | a'_k= a'_j} x^{n-k} = - \sum_{k | a_k= a_i} x^{n-k} \bmod{n-1 \textrm{\ bits}}$, then $H_a+q(x)H_{a'} \bmod{n-1 \textrm{\ bits}}$ is independent from $a_i= a'_j$ (and $h_1(a_i)=h_1(a'_j)$).
The hashed values $h_1(a_k)$ for $a_k \neq a_i$ and $h_1(a'_k)$ for $a'_k \neq a'_j$ are now relabelled as $h_1(b_1),\ldots, h_1(b_m)$. Write $H_a+q(x)H_{a'} = \sum_k q_k(x) h_1(b_k) \bmod{n-1 \textrm{\ bits}} $ where $q_k(x)$ are polynomials in $\textrm{GF}(2)[x]/(x^L+1)$ (not all $q_k(x)$ are zero). As in the proof of Lemma \[uniformlemma\], we have that $H_{a'}=y' \bmod{n-1 \textrm{\ bits}}$ and $H_a+q(x)H_{a'}=y+q(x)y' \bmod{n-1 \textrm{\ bits}}$ are independent[^4]: $P(H_{a'}=y' \bmod{n-1 \textrm{\ bits}} | y', b_1, b_2, \ldots, b_m)=1/2^{L-n+1}$ by Corollary \[corollary:magic\] since $H_{a'}=y$ can be written as $r(x) h_1(a'_j)=y- \sum_k r_k(x) h_1(b_k)$ for some polynomials $r(x), r_1(x), \ldots, r_m(x)$. Hence, we have $$\begin{aligned}
\lefteqn{P( H_a=y \bmod{n-1 \textrm{\ bits}}| H_{a'}=y' \bmod{n-1 \textrm{\ bits}})}&&\\
& = &P( H_a+q(x)H_{a'}=y+q(x)y' \bmod{n-1 \textrm{\ bits}}| H_{a'}=y' \bmod{n-1 \textrm{\ bits}})\\
& = &P( H_a+q(x)H_{a'}=y+q(x)y' \bmod{n-1 \textrm{\ bits}})\\
& = &P(\sum_k q_k(x) h_1(b_k) = y + q(x) y' \bmod{n-1 \textrm{\ bits}} )\end{aligned}$$ and by the earlier uniformity result, this last probability is equal to $1/2^{L-n+1}$.
Experimental comparison
=======================
Irrespective of $p(x)$, computing hash values has complexity $\Omega(L)$. For <span style="font-variant:small-caps;">General</span> and <span style="font-variant:small-caps;">Cyclic</span>, we require $L\geq n$. Hence, the computation of their hash values is in $\Omega(n)$. For moderate values of $L$ and $n$, this analysis is pessimistic because CPUs can process 32- or 64-bit words in one operation.
To assess their real-world performance, the various hashing algorithms[^5] were written in C++. We compiled them with the GNU GCC 4.0.1 compiler on an Apple MacBook with two Intel Core 2 Duo processors (2.4GHz) and 4GiB of RAM. The -O3 compiler flag was used since it provided slightly better performance for all algorithms. All hash values are stored using 32-bit integers, irrespective of the number of bits used.
All hashing functions generate 19-bit hash values, except for <span style="font-variant:small-caps;">Cyclic</span> which generates 19+$n$-bit hash values. We had <span style="font-variant:small-caps;">Cyclic</span> generate more bits to compensate for the fact that it is only pairwise independent after removal of $n-1$ consecutive bits. For <span style="font-variant:small-caps;">General</span>, we used the polynomial $p(x)=x^{19}+x^{5}+x^{2}+x+1$ [@COS]. For Randomized Karp-Rabin, we used the ID37 family. The character hash-values are stored in an array for fast look-up.
We report wall-clock time in Fig. \[fig:timings\] for hashing the $n$-grams of the King James Bible [@Gutenberg] which contains 4.3 million ASCII characters. <span style="font-variant:small-caps;">Cyclic</span> is twice as fast as <span style="font-variant:small-caps;">General</span>. As expected, the running time of the non-recursive hash function (3-wise) grows linearly with $n$: for $n=5$, 3-wise is already seven times slower than <span style="font-variant:small-caps;">Cyclic</span>. Speed-wise, Randomized Karp-Rabin (ID37) is the clear winner, being nearly twice as fast as <span style="font-variant:small-caps;">Cyclic</span>. The performance of <span style="font-variant:small-caps;">Cyclic</span> and ID37 is oblivious to $n$ in this test.
The RAM-Buffered <span style="font-variant:small-caps;">General</span> timings are—as expected—independent of $n$, but they are twice as large as the <span style="font-variant:small-caps;">Cyclic</span> timings. We do not show the modified version of RAM-Buffered <span style="font-variant:small-caps;">General</span> that uses two precomputed arrays instead of a single one. It was approximately 30% slower than ordinary RAM-Buffered <span style="font-variant:small-caps;">General</span>, even up to $n=25$. However, its RAM usage was 3 orders of magnitude smaller: from 135MB down to 25kB. Overall, we cannot recommend RAM-Buffered <span style="font-variant:small-caps;">General</span> or its modification considering that (1) its memory usage grows as $2^n$ and (2) it is slower than <span style="font-variant:small-caps;">Cyclic</span>.
![\[fig:timings\]Wall-clock running time to hash all $n$-grams in the King James Bible](realdataspeedtesting){width=".6\textwidth"}
Conclusion
==========
Considering speed and pairwise independence, we recommend <span style="font-variant:small-caps;">Cyclic</span>—after discarding $n-1$ consecutive bits. If we require only uniformity, Randomized Integer-Division is twice as fast.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by NSERC grants 155967, 261437 and by FQRNT grant 112381. The authors are grateful to the anonymous reviewers for their significant contributions.
[^1]: We omit families uniform over an arbitrary interval $[0,b)$—not of the form $[0,2^L)$. Indeed, several applications [@flajolet1985pca; @Gibbons2001] require uniformity over $L$-bits.
[^2]: The values $f^{(1)}, f^{(2)},f^{(3)}$ are not necessarily mutually independent.
[^3]: By “equality modulo $\langle$*some specified set of bit positions*$\rangle$”, we mean that the two quantities are bitwise identical, with exceptions permitted only at the specified positions. For our polynomials, “equality modulo the first $n$ bit positions” implies the difference of the two polynomials has degree at most $n-1$.
[^4]: We use the shorthand notation $P(f(x,y)=c | x, y)= b$ to mean $P(f(x,y)=c | x= z_1, y=z_2)=b$ for all values of $z_1, z_2$.
[^5]: <http://code.google.com/p/ngramhashing/>.
|
---
abstract: |
Let $M$ be a surface sum of 3-manifolds $M_1$ and $M_2$ along a bounded connected surface $F$ and $\partial_{i}$ be the component of $\partial M_{i}$ containing $F$. If $M_i$ has a high distance Heegaard splitting, then any minimal Heegaard splitting of $M$ is the amalgamation of those of $M^{1}, M^{2}$ and $M^{*}$, where $M^{i}=M_{i}\setminus
\partial_{i}\times I$, and $M^{*}=\partial_{1}\times I\cup_{F}
\partial_{2}\times I$. Furthermore, once both $\partial_i\setminus F$ are connected, then $g(M)=Min\bigl\{g(M_{1})+g(M_{2}), \alpha\bigr\}$, where $\alpha=
g(M_{1})+g(M_{2})+1/2(2\chi(F)+2-\chi(\partial _{1})-\chi(\partial
_{2}))-Max\bigl\{g(\partial _{1}), g(\partial _{2})\bigl\}$; in particular $g(M)=g(M_{1})+g(M_{2})$ if and only if $\chi(F)\geq
1/2Max\bigl\{\chi(\partial_{1}), \chi(\partial_{2})\bigr\}.$
The proofs rely on Scharlemann-Tomova’s theorem.
author:
- 'Ruifeng Qiu[^1], Shicheng Wang[^2], Mingxing Zhang'
title: Additivity of Heegaard genera of bounded surface sums
---
[**Keywords**]{}: Heegaard Distance and Genus, Surface Sum, Weakly incompressible.
AMS Classification: 57M25
Introduction
============
All surfaces and 3-manifolds in this paper are assumed to be compact and orientable.
Let $F$ be either a properly embedded surface in a 3-manifold $M$ or a sub-surface of $\partial M$. If there is an essential simple closed curve on $F$ which bounds a disk in $M$ or $F$ is a 2-sphere which bounds a 3-ball in $M$, then we say $F$ is compressible; otherwise, $F$ is said to be incompressible. If $F$ is an incompressible surface not parallel to $\partial M$, then $F$ is said to be essential. If $M$ contains an essential 2-sphere, then $M$ is said to be reducible. If $\partial M$ is compressible, then $M$ is said to be $\partial$-reducible.
Let $M$ be a 3-manifold. If there is a closed surface $S$ which cuts $M$ into two compression bodies $V$ and $W$ with $S=\partial_{+} W=\partial_{+} V$, then we say $M$ has a Heegaard splitting, denoted by $M=V\cup_{S} W$; and $S$ is called a Heegaard surface of $M$. Moreover, if the genus $g(S)$ of $S$ is minimal among all the Heegaard surfaces of $M$, then $g(S)$ is called the genus of $M$, denoted by $g(M)$.
Now let $M$ be a 3-manifold, and $F$ be a compact surface in $M$ which cuts $M$ into two 3-manifolds $M_{1}$ and $M_{2}$. Then $M$ is called a surface sum of $M_{1}$ and $M_{2}$ along $F$, denoted by $M=M_{1}\cup_{F} M_{2}$. Note that $F\subset\partial M_{i}$ for $i=1, 2$. A central topic in Heegaard splitting is to address relations between $g(M_1)$, $g(M_2)$ and $g(M)$.
Suppose first that $F$ is a closed surface. Let $M_{i}=V_{i}\cup_{S_{i}} W_{i}$ be a Heegaard splitting for $i= 1,
2$. Then $M$ has a natural Heegaard splitting called the amalgamation of $V_{1}\cup_{S_{1}} W_{1}$ and $V_{2}\cup_{S_{2}}
W_{2}$. From this view, $g(M)\leq g(M_{1})+g(M_{2})-g(F)$. If $F$ is a 2-sphere, the so-called Haken’s lemma claimed $g(M)=g(M_{1})+g(M_{2})$. For $g(F)>0$, there are some examples to show that it is possible that $g(M)\leq g(M_{1})+g(M_{2})-g(F)-n$ for any given $n>0$, see \[7\] and \[18\]. Philosophically, in such examples neither the gluing between $M_1$ and $M_2$ along $F$ nor the Heegaard splitting of $M_i$ are complicated.
Under various different conditions describing the complicated gluing maps, the equality $g(M)=g(M_{1})+g(M_{2})-g(F)$ was proved, see \[1\], \[9\], \[10\] and \[19\]. By invoke results of Hartshorn \[3\], Scharlemann \[13\] and Scharlemann and Tomova \[16\], it is just proved in \[6\] that $g(M)=g(M_{1})+g(M_{2})-g(F)$ if $M_1$ and $M_2$ have high distance Heegaard splittings, where the distance of a Heegaard splitting was introduced by Hempel \[4\].
Suppose that $F$ is a bounded surface. Then it is easy to see $g(M)\leq g(M_{1})+g(M_{2})$ (see Lemma 2.1). By the disk version of Haken’s lemma, $g(M)=g(M_{1})+g(M_{2})$ if $F$ is a disk. If $F$ is an annulus, various results about if $g(M)=g(M_{1})+g(M_{2})$ hold or not have been given, see \[5\], \[8\], \[11\] and \[12\].
In this paper we will address the additivity of Heegaard genus of surface sum of 3-manifolds along a bounded surface $F$ with $\chi(F)< 0$, which seems not touched before.
We first fix some notions. Suppose $P$ (resp. $H$) is a properly embedded surface (resp. an embedded 3-manifold) in a 3-manifold $M$. We use $M\setminus P$ (resp. $M\setminus H$) to denote the resulting manifold obtained by splitting $M$ along $P$ (resp. removing $\text{int} H$, the interior of $H$).
Let $M=M_{1}\cup_{F} M_{2}$, $\partial_{i}$ be the component of $\partial M_{i}$ containing $F$, and $\partial_{i}\times [0,1]$ be a regular neighborhood of $\partial_{i}$ in $M_{i}$ with $\partial
_{i}=\partial _{i}\times\bigl\{0\bigr\}$. We denote by $P^{i}$ the surface $\partial_{i}\times\bigl\{1\bigr\}$. Let $M^{i}=M_{i}\setminus \partial_{i}\times [0,1]$ for $i=1, 2$, and $M^{*}=\partial_{1}\times [0,1]\cup_{F}
\partial_{2}\times [0,1]$. Then $M=M^{1}\cup_{P^{1}}M^{*}\cup_{P^{2}}M^{2}$.
[**Theorem 1.**]{} Let $M$ be a surface sum of 3-manifolds $M_1$ and $M_2$ along a bounded surface $F$, and $\partial_{i}$ be the component of $\partial M_{i}$ containing $F$. If $M_{i}$ has a Heegaard splitting $V_{i}\cup_{S_{i}} W_{i}$ with $d(S_{i})>
2(g(S_{1})+g(S_{2}))$, $i=1,2$, then any minimal Heegaard splitting of $M$ is the amalgamation of Heegaard splittings of $M^{1}$, $M^{2}$ and $M^{*}$ along $\partial_{1}$ and $\partial_{2}$.
The proof of Theorem 1 invokes full energy of Scharlemann-Tomova’s deep result (Lemma 2.5).
[**Theorem 2.**]{} Under the assumptions of Theorem 1, if $\partial_i \setminus F$ is connected for $i=1,2$, then $g(M)=Min\bigl\{g(M_{1})+g(M_{2}), \alpha\bigr\}$, where $$\alpha=
g(M_{1})+g(M_{2})+1/2(2\chi(F)+2-\chi(\partial _{1})-\chi(\partial
_{2}))-Max\bigl\{g(\partial _{1}), g(\partial _{2})\bigl\}\qquad
(1.1).$$ Furthermore $g(M)=g(M_{1})+g(M_{2})$ if and only if $\chi(F)\geq 1/2Max\bigl\{\chi(\partial_{1}),
\chi(\partial_{2})\bigr\}.$
[**Corollary 3.**]{} Under the assumptions of Theorem 1, if $F$ is annulus, then $g(M)=g(M_{1})+g(M_{2})$.
[**Remark 4.**]{} It is remarkable to compare Theorem 2 with the main result in \[6\] for closed surface case. In Theorem 2
\(1) once $\chi(F)\geq 1/2Max\bigl\{\chi(\partial_{1}),
\chi(\partial_{2})\bigr\}$, $\chi(F)$, therefore $g(F)$, itself plays no role in the result.
\(2) once $\chi(F)< 1/2Min\bigl\{\chi(\partial_{1}),
\chi(\partial_{2})\bigr\}$, the contribution of $\chi(F)$ to $g(M)$ is non-trivial linear function. A particular case when $g(\partial_{1})=g(\partial_{2})$ and $\chi(F)=\chi(\partial_{1})$, then $g(M)=g(M_{1})+g(M_{2})-g(\partial_{1})+1$.
[**Acknowledgements.**]{} The authors would like to thank Tsuyoshi Kobayashi and Qing Zhou for helpful discussions on this paper.
Distance of Heegaard splitting
==============================
Weakly incompressible surface in 3-manifolds was introduced in \[16\]: Let $P$ be a separating connected closed surface in 3-manifold $M$ which cuts $M$ into two 3-manifolds $M_{1}$ and $M_{2}$. Then $P$ is said to be bicompressible if $P$ is compressible in both $M_{1}$ and $M_{2}$. $P$ is strongly compressible if there are compressing disks for $P$ in $M_{1}$ and $M_{2}$ which have disjoint boundaries in $P$; otherwise $P$ is weakly incompressible.
Let $M$ be a compact orientable 3-manifold, and $M=V\cup_{S} W$ be a Heegaard splitting. If there are essential disks $B\subset V$ and $D\subset W$ such that $\partial B=\partial D$ ($\partial
B\cap\partial D=\emptyset$), then $V\cup_{S} W$ is said to be reducible (weakly reducible). Otherwise, it is said to be irreducible (strongly irreducible), see \[2\]. It is easy to see that a strongly irreducible Heegaard surface is weakly incompressible.
Now let $P$ be a bicompressible surface in an irreducible 3-manifold $M$. By maximally compressing $P$ in both sides of $P$ and deleting the possible 2-sphere components, we get a surface sum structure of $M$ as follow: $$M=N_1\cup_{F^{P}_{1}}H^{P}_{1}\cup_{P}H^{P}_{2}\cup_{F^{P}_{2}}N_2,$$ where $H^{P}_{i}$ is a compression body with $\partial_{+}
H^{P}_{i}=P$, and $\partial_{-} H^{P}_{i}=F^{P}_{i}$ is a collection (may be empty) of incompressible closed surfaces of genus at least one in $N_{i}$, $i=1,2$. Note that, if $F^{P}_{i}$ is empty, then $H^{P}_{i}$ is a handlebody and $N_{i}$ is empty. It is easy to see that if $M$ has boundary, then $F^{P}_{1}$ and $F^{P}_{2}$ can not be both empty. Moreover if $P$ is weakly incompressible, then the Heegaard splitting $H^{P}_{1}\cup_{P}H^{P}_{2}$ is strongly irreducible.
Two weakly incompressible surfaces $P$ and $Q$ are said to be well-separated in $M$ if $H^{P}_{1}\cup_{P}H^{P}_{2}$ is disjoint from $H^{Q}_{1}\cup_{P}H^{Q}_{2}$ by isotopy.
[**Lemma 2.1.**]{} Suppose $P$ is a weakly incompressible surface in $M$. Then each component of $F^{P}_{i}$ is incompressible in $M$ for $i=1, 2$.
[**Proof.**]{} By the definition, each component of $F^{P}_{i}$ is incompressible in $N_{i}$ for $i=1,2$. Since $P$ is a bicompressible but weakly incompressible, $H^{P}_{1}\cup H^{P}_{2}$ is a non-trivial strongly Heegaard splitting. By the disc version of Haken’s Lemma, each component of $F^{P}_{i}$ is incompressible in $H^{P}_{1}\cup H^{P}_{2}$. Q.E.D.
[**Lemma 2.2.**]{} Let $S$ be a strongly irreducible Heegaard surface of a 3-manifold $M$, then $M\setminus (H_{1}^{S}\cup_{S}
H_{2}^{S})$ is homeomorphic to $\partial M\times I$. Fruthermore, if $P$ is a weakly incompressible surface in $M$, then either $H^{P}_{1}\cup_{P} H^{P}_{2}\subset\partial \times I$ is homeomorphic to $\partial\times I$ for one component $\partial$ of $\partial M$, or, $S$ and $P$ are not well-separated.
[**Proof.**]{} Now $M=V\cup_{S} W$, where $V$ and $W$ are compression bodies. We may assume that $H_{1}^{S}\subset V$ and $H_{2}^{P}\subset W$. Now there are essential disks $D_{1}, \ldots,
D_{n}$ in $V$ such that each component of $V \setminus
\cup_{i=1}^{n} D\times [0,1]$ is an I-bundle of a closed surface. Since $\partial_{-} H_{1}^{S}$ is incompressible in $M$, hence in $V$, each $D_{i}$ can be isotoped to be disjoint from $\partial_{-}
H_{1}^{S}$. Hence $\partial_{-} H_{1}^{S}\subset V \setminus
\cup_{i=1}^{n} D\times [0,1]$. This means that $M\setminus
(H_{1}^{S}\cup_{S} H_{2}^{S})$ is homeomorphic to $\partial M\times
I$.
Suppose $P$ and $S$ are well-separated, then $H^{P}_{1}\cup_{P}H^{P}_{2}\subset \text{closed surface $\times
I$}$. Since $F^{P}_{1}\cup F^{P}_{2}$ is incompressible in $M$, each component of $F^{P}_{i}$ is isotopic to $F\times\bigl\{0\bigr\}$ in $F\times I$. Hence the lemma holds. Q.E.D.
The distance between two essential simple closed curves $\alpha$ and $\beta$ on a compact surface $P$, denoted by $d(\alpha,\beta)$, is the smallest integer $n\geq 0$ so that there is a sequence of essential simple closed curves $\alpha_{0}=\alpha,\ldots,\alpha_{n}=\beta$ on $P$ such that $\alpha_{i-1}$ is disjoint from $\alpha_{i}$ for $1\leq i\leq n$. When $P$ is a bicompressible surface in a 3-manifold $M$, the distance of $P$ is $d(P)=Min\bigl\{d(\alpha,\beta)\bigr\}$, where $\alpha$ bounds a disk in $H^{P}_{1}$ and $\beta$ bounds a disk in $H^{P}_{2}$. See \[4\] and \[16\].
Lemma 2.3 follows from the definitions and the main result in \[15\]:
[**Lemma 2.3.**]{} (1) If $M=V\cup_{S}W$ is a reducible Heegaard splitting, then $d(S)=0$.
\(2) If $M=V\cup_{S} W$ is a weakly reducible Heegaard splitting, then $d(S)\leq 1$.
\(3) If $M=V\cup_{S} W$ is a non-trivial and $\partial$-reducible Heegaard splitting, then $d(S)\leq 1$.
\(4) If $C$ is a closed surface, and $V\cup_{S} W$ is a non-trivial Heegaard splitting of $C\times I$, then $d(S)\leq 2$ \[15\].
[**Lemma 2.4 (\[3\], \[13\]).**]{} Let $M=V\cup_{S} W$ be a Heegaard splitting, and $P$ be an incompressible surface in $M$. Then either $P$ can be isotoped to be disjoint from $S$ or $d(S)\leq 2-\chi(P)$.
[**Lemma 2.5 (\[16\]).**]{} Let $P$ and $Q$ be bicompressible but weakly incompressible connected closed separating surfaces in a 3-manifold $M$. Then either
\(1) $P$ and $Q$ are well-separated, or
\(2) $P$ and $Q$ are isotopic, or
\(3) $d(P)\leq 2g(Q)$.
[**Lemma 2.6 (\[16\]).**]{} Let $M=V\cup_{S} W$ be a Heegaard splitting of a 3-manifold $M$. If $d(S)>2g(S)$, then $M$ has the unique minimal Heegaard spliting $V\cup_{S}W$ up to isotopy.
The proof of Theorem 1
======================
Let $M=M_{1}\cup_{F} M_{2}$, and $F$ be a bounded surface. Then $M=M^{1}\cup_{P^{1}}M^{*}\cup_{P^{2}} M^{2}$, where $M^{1}, M^{2},
P^{1}, P^{2},
\partial_{1},
\partial_{2}$ are defined in Section 1.
[**Lemma 3.1.**]{} $g(M)\leq g(M_{1})+g(M_{2})$.
[**Proof.**]{} Let $M_{i}=V_{i}\cup_{S_{i}} W_{i}$ be a Heegaard splitting of $M_{i}$ such that $F\subset
\partial_{i}\subset \partial_{-} W_{i}$. Now let $\gamma_{i}$ be a unknotted arc in $W_{i}$ such that $\partial_{1} \gamma_{i}
\subset\partial_{+} W_{i}$, $\partial_{2}\gamma_{1}=\partial_{2}\gamma_{2}\subset intF$. Let $N(\gamma_{1}\cup\gamma_{2})$ be a regular neighborhood of $\gamma_{1}\cup\gamma_{2}$ in $W_{1}\cup_{F} W_{2}$. Let $V=V_{1}\cup N(\gamma_{1}\cup\gamma_{2})\cup V_{2}$, and $W$ be the closure of $(W_{1}\cup_{F} W_{2})\setminus
N(\gamma_{1}\cup\gamma_{2})$. Then $V\cup_{S} W$ is a Heegaard splitting of $M$, where $S=\partial_{+} V=\partial_{+} W$. Note that $g(S)=g(S_{1})+g(S_{2})$. Hence the lemma holds. See also \[17\]. Q.E.D.
[**Lemma 3.2.**]{} If $M_{i}$ has a Heegaard splitting $V_{i}\cup_{S_{i}} W_{i}$ with $d(S_{i})> 2(g(S_{1})+g(S_{2}))$, $i=1,2$, then any minimal Heegaard splitting of $M$ is irreducible and weakly reducible.
[**Proof.**]{} Since $d(S_{i})>2(g(S_{1})+g(S_{2}))$ for $i=1,2$, $M_i$ is irreducible and $F$ is an essential surface in $M_i$ by Lemma 2.3 (1) and (2). Then it follows that $M=M_1\cup _F M_2$ is irreducible and $F$ is an essential surface in $M$. Furthermore, $M_{i}$ is not homeomorphic to an I-bundle of a closed surface by Lemma 2.3 (4).
Let $M=V\cup_{S} W$ be a minimal Heegaard splitting of $M$. Since $M$ is irreducible and $V\cup_{S} W$ is minimal, $V\cup_{S} W$ is irreducible. By Lemma 2.6, $g(M_{i})=g(S_{i})$. By Lemma 3.1, $g(S)\leq g(S_{1})+g(S_{2})$.
Now suppose $V\cup_{S} W$ is strongly irreducible. Since $S_{i}$ is separating in $M$, $S_{i}$ is bicompressible but weakly incompressible in $M$. By Lemma 2.5, either $S$ and $S_{1}$ are well-separated, or $S$ and $S_{1}$ are isotopic, or $d(S_{1})\leq
2g(S)$. Since $S_1$ is a Heegaard surface of $M_1$, we have that $H^{S_{1}}_{1}\cup_{S_{1}}H^{S_{1}}_{2}$ is homeomorphic to $M_{1}$, which not a product as we just proved. By Lemma 2.2, $S$ and $S_{1}$ are not well-separated. Since $S$ is a Heegaard surface of $M$ and $S_{1}$ is a Heegaard surface of $M_{1}$, $S$ is not isotopic to $S_{1}$. Since $d(S_{1})>2(g(S_{1})+g(S_{2}))$ and $g(S)\leq
g(S_{1})+g(S_{2})$, $d(S_{1})>2g(S)$, a contradiction. Q.E.D.
[**The proof of Theorem 1.**]{} Under the assumptions of Theorem 1, by Lemma 3.2, any minimal Heegaard splitting of $M$ is irreducible and weakly reducible. Let $M=V\cup_{S} W$ be a minimal Heegaard splitting, then $V\cup _{S} W$ has a thin position as $$V\cup_{S}W=(V_{1}^{'}\cup_{S_{1}^{'}}
W_{1}^{'})\cup_{F_{1}}\ldots\cup_{F_{n-1}}(V_{n}^{'}\cup_{S_{n}^{'}}
W_{n}^{'}) \ \ \ (*)$$ where $n\geq 2$, and each component of $F_{1},\ldots, F_{n-1}$ is an incompressible closed surface in $M$, and each $V_{i}^{'}\cup_{S_{i}^{'}} W_{i}^{'}$ consists of a active component which is a non-trivial strong irreducible Heegaard splitting and possible some product components each of which is a trivial Heegaard splitting of an I-bundle of a closed surface. See \[14\].
Since $d(S_{i})>2(g(S_{1})+g(S_{2}))$ for $i=1,2$, $M$ is irreducible and $F$ is an essential surface in $M$ by Lemma 2.3. Furthermore, $M_{i}$ is not homeomorphic to an I-bundle of a closed surface.
[**Claim 1.**]{} $F_{i}$, $1\leq i\leq n-1$, can be isotoped so that $F_{i}\subset M^{*}$.
[**Proof.**]{} Suppose that $F_{i}\cap (M^{1}\cup
M^{2})\neq\emptyset$ for some $1\leq i\leq n-1$. We may assume that $F_{i}\cap M^{1}\neq\emptyset$, and $F_{i}\cap M^{1}$ is incompressible moreover. Note that $g(F_{i})\leq g(S)\leq
g(S_{1})+g(S_{2})$. Now $\chi(F_{i}\cap M^{1})\geq \chi(F_{i}\cap M_{1})\geq \chi(S)= 2-g(S) \ge
2-2(g(S_{1})+g(S_{2}))$, we have $d(S_1) > 2- \chi(F_i\cap M^{1})$. By Lemma 2.4, $F_{i}$ can be isotoped to be disjoint from $S_{1}$. Hence $F_{i}\cap M^{1}$ lies in one of $V_{1}$ and $W_{1}$ which contains $F$. It follows $F_i$ can be further isotoped to be disjoint from $M^{1}$. Q.E.D. (Claim 1)
[**Claim 2.**]{} There exists a component of $\bigcup_{1\leq i\leq n-1} F_{i}$ isotopic to $P^{1}$ (resp. $P^{2}$).
[**Proof.**]{} Suppose that each component of $\bigcup_{1\leq i\leq
n-1} F_{i}$ is not isotopic to $P^{1}$. By Claim 1, $M^{1}\subset
M_{i}^{'}=V_{i}^{'}\cup_{S_{i}^{'}} W_{i}^{'}$ for some $1\leq i\leq
n$.
If $S_1$ is contained in the product component of $M_{i}^{'}=V_{i}^{'}\cup_{S_{i}^{'}} W_{i}^{'}$, by the same argument in the proof of Lemma 2.2, $H^{S_{1}}_{1}\cup_{S_{1}}H^{S_{1}}_{2}$ is a Heegaard splitting of an I-bundle of a closed surface. That is $M_1$ is an I-bundle of a closed surface, it is a contradiction.
Now suppose $S_1$ is contained in the active component of $M_{i}^{'}=V_{i}^{'}\cup_{S_{i}^{'}} W_{i}^{'}$. We also use $M_{i}^{'}=V_{i}^{'}\cup_{S_{i}^{'}} W_{i}^{'}$ to denote the active component. Now $S_{i}^{'}$ and $S_{1}$ are both weakly incompressible in $M_{i}^{'}$ and $H^{S_{1}}_{1}\cup_{S_{1}}H^{S_{1}}_{2}$ is not product. By Lemma 2.2, $S_{i}^{'}$ and $S_{1}$ are not well separated. By Lemma 3.1, $g(S_{i}^{'})\leq g(S)\leq g(M_{1})+g(M_{2})$. By Lemma 2.6, $g(M_{1})+g(M_{2})=g(S_{1})+g(S_{2})$. Hence $2g(S_{i}^{'})<d(S_{1})$.
By Lemma 2.5, $S_{i}^{'}$ and $S_{1}$ are isotopic in $M_{i}^{'}$.
It follows $S_1$ is a strongly irreducible Heegaard surface of $M_i'$. Then by Lemma 2.2 $M_i'\setminus
(H^{S_{1}}_{1}\cup_{S_{1}}H^{S_{1}}_{2})$ is $\partial M_i'\times
I$. By Lemma 2.2 we may assume that $H^{S_{1}}_{1}\cup_{S_{1}}H^{S_{1}}_{2}=M^{1}$. Hence one component of $\partial M_{i}^{'}$ is isotopic to $P^{1}$.
Similarly, there exists a component of $\bigcup_{1\leq i\leq n-1}
F_{i}$ isotopic $P^{2}$. Q.E.D. (Claim 2)
By Claim 2, Theorem 1 holds. Q.E.D.
The proof of Theorem 2
======================
[**Lemma 4.1.**]{} Let $N=P_{1}\times I\cup_{F} P_{2}\times I$ be the surface sum of $P_{1}\times I$ and $P_{2}\times I$ along $F$, where $P_{1}$ and $P_{2}$ are orientable closed surfaces, and $F$ is a connected bounded surface in both $P_{1}\times\bigl\{0\bigr\}$ and $P_{2}\times\bigl\{0\bigr\}$.
\(1) If both $P_{1}\times\bigl\{0\bigr\}\setminus F$ and $P_{2}\times\bigl\{0\bigr\}\setminus F$ are connected, then $g(N)=Min\bigl\{g(P_{1})+g(P_{2}), \alpha\bigr\}$, where $$\alpha=1/2(2\chi(F)+2-\chi(P_{1})-\chi(P_{2}))+Min\bigl\{g(P_{1}),
g(P_{2})\bigr\}.$$
\(2) If $F$ is an annulus, then $g(N)=g(P_{1})+g(P_{2})$.
[**Proof.**]{} We first prove (1).
Since both $P_{1}\times\bigl\{0\bigr\} \setminus F$ and $P_{2}\times\bigl\{0\bigr\}\setminus F$ are connected, $N$ contains three boundary components $P_{1}\times\bigl\{1\bigr\}$, $P_{2}\times\bigl\{1\bigr\}$, and $P^{*}=(P_{1}\times\bigl\{0\bigr\}\setminus F)\cup
(P_{2}\times\bigl\{0\bigr\}\setminus F)$. See Figure 4.1. Hence
$$g(N)\geq
Min\bigl\{g(P_{1})+g(P_{2}), g(P_{1})+g(P^{*}),
g(P_{2})+g(P^{*})\bigr\}.$$
{height="2.5in"}
Figure 4.1
It is easy to see that $N$ is homeomorphic to both $P_{1}\times
I\cup_{P_{1}\times\bigl\{0\bigr\}\setminus F} P^{*}\times I$ and $P^{*}\times I\cup_{P_{2}\times\bigl\{0\bigr\} \setminus F}
P_{2}\times I$. See Figure 4.2.
{height="2in"}
Figure 4.2
By Lemma 3.1, $$g(N)\leq Min\bigl\{g(P_{1})+g(P_{2}),
g(P_{1})+g(P^{*}), g(P_{2})+g(P^{*})\bigr\}.$$ Now $$g(N)=Min\bigl\{g(P_{1})+g(P_{2}),
g(P_{1})+g(P^{*}), g(P_{2})+g(P^{*})\bigr\}.$$
Since $P^{*}=(P_{1}\times\bigl\{0\bigr\}\setminus F)\cup
(P_{2}\times\bigl\{0\bigr\}\setminus F)$, $$2-2g(P^{*})=\chi(P^{*})=\chi(P_{1})+\chi(P_{2})-2\chi(F),$$ and $$g(P^{*})=1/2(2\chi(F)+2-\chi(P_{1})-\chi(P_{2})).$$ Hence (1) holds.
Now we prove (2).
Suppose now that $F$ is an annulus. Now there are three cases:
Case 1. Both $P_{1}\times\bigl\{0\bigr\}\setminus F$ and $P_{2}\times\bigl\{0\bigr\}\setminus F$ are connected.
Now $N$ contains three boundary components $P_{1}\times\bigl\{1\bigr\}$, $P_{2}\times\bigl\{1\bigr\}$ and $P^{*}=(P_{1}\times\bigl\{0\bigr\}\setminus F)\cup
(P_{2}\times\bigl\{0\bigr\}\setminus F)$. Since $F$ is an annulus, $g(P^{*})\geq g(P_{1}), g(P_{2})$. By the argument in (1), (2) holds.
Case 2. One of $P_{1}\times\bigl\{0\bigr\}\setminus F$ and $P_{2}\times\bigl\{0\bigr\}\setminus F$ is connected while the other is non-connected.
The argument is the same with the one in Case 1.
Case 3. Both $P_{1}\times\bigl\{0\bigr\}\setminus F$ and $P_{2}\times\bigl\{0\bigr\}\setminus F$ are non-connected.
Now we denote by $F_{i}^{1}$ and $F_{i}^{2}$ the two components of $P_{i}\times\bigl\{0\bigr\}\setminus F$. We may assume that $\partial F_{1}^{j}=\partial F_{2}^{j}$. Then $N$ contains four boundary components $P_{1}\times\bigl\{1\bigr\}$, $P_{2}\times\bigl\{1\bigr\}$, $F^{1}=F_{1}^{1}\cup F_{2}^{1}$ and $F^{2}=F_{1}^{2}\cup F_{2}^{2}$. In this case, $g(F^{1})+g(F^{2})=g(P_{1})+g(P_{2})$. Hence (2) holds. Q.E.D.
[**The proof of Theorem 2.**]{} Recalling the definitions of $M^{i},
M^{*}$ defined in Section 1. Since $\partial_{i}$ is separating in $M$ for $i=1, 2$, and $M^{i}$ is homeomorphic to $M_{i}$ for $i=1,2$, by Theorem 1, $$g(M)=g(M_{1})+g(M_{2})+g(M^{*})-g(\partial_{1})-g(\partial_{2}).$$ By Lemma 4.1, $g(M^{*})=Min\bigl\{g(\partial_{1})+g(\partial_{2}),
\alpha\bigr\}$, where $$\alpha=1/2(2\chi(F)+2-\chi(\partial_{1})-\chi(\partial_{2}))+Min\bigl\{g(\partial_{1}),
g(\partial_{2})\bigr\}.$$ Hence $g(M)=Min\bigl\{g(M_{1})+g(M_{2}),
\alpha\bigr\}$, where $$\alpha=
g(M_{1})+g(M_{2})+1/2(2\chi(F)+2-\chi(\partial _{1})-\chi(\partial
_{2}))-Max\bigl\{g(\partial _{1}), g(\partial _{2})\bigl\}.$$
It is easy to see that $g(M)=g(M_{1})+g(M_{2})$ if and only if $\chi(F)\geq 1/2Max\bigl\{\chi(\partial_{1}),
\chi(\partial_{2})\bigr\}.$ Q.E.D.
[**Corollary 3.**]{} The proof follows immediately from Theorem 1 and Lemma 4.1 (2). Q.E.D.
[**Reference**]{}
\[1\] D. Bachman, S. Schleimer and E. Sedgwick, Sweepouts of amalgamated 3-manifolds, Algebr. Geom. Topol. 6(2006) 171-194.
\[2\] A. Casson and C. McA Gordon, Reducing Heegaard splittings, Topology Appl. 27(1987) 275-283.
\[3\] K. Hartshorn, Heegaard splittings of Haken manifolds have bounded distance, Pacific J. Math. 204(2002) 61-75.
\[4\] J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40(2001) 631-657.
\[5\] T. Kobayashi, A construction of arbitrarily high degeneration of tunnel numbers of knots under connected sum, J. Knot Theory Ramifications 3(1994) 179-186.
\[6\] T. Kobayashi and R. Qiu, The amalgamation of high distance Heegaard splittings is always efficient, Math. Ann. 341(2008) 707-715.
\[7\] T. Kobayashi, R. Qiu, Y. Rieck and S. Wang, Separating incompressible surfaces and stabilizations of Heegaard splittings, Math. Proc. Cambridge Philos. Soc. 137(2004) 633-643.
\[8\] T. Kobayashi and Y. Rieck, Heegaard genus of the connected sum of $m$-small knots, Commu. Anal. Geom. 14(2006), 1037-1077.
\[9\] M. Lackenby, The Heegaard genus of amalgamated 3-manifolds, Geom. Dedicata 109(2004) 139-145.
\[10\] Tao Li, On the Heegaard splittings of amalgamated 3-manifolds, Geom. Topol. Monographs 12(2007) 157-190.
\[11\] K. Morimoto, On the super additivity of tunnel number of knots, Math. Ann. 317(2000), 489-508.
\[12\] R. Qiu, K. Du, J. Ma and M. Zhang, Distance and the Heegaard genera of annular 3-manifolds, Preprint.
\[13\] M. Scharlemann, Proximity in the curve complex: boundary reduction and bicompressible surfaces, Pacific J. Math. 228(2006), 325-348.
\[14\] M. Scharlemann and A. Thompson, Thin position for 3-manifolds. Geometric Topology (Haifa, 1992) 231-238, Contemp. Math., 164, Amer. Math. Soc., Providence, RI, 1994.
\[15\] M. Scharlemann and A. Thompson, Heegaard splittings of ${\rm
Surfaces}\times I$ are standard, Math. Ann., 295(1993), 549-564.
\[16\] M. Scharlemann and M. Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10(2006) 593-617.
\[17\] J. Schultens, Additivity of tunnel number for small knots, Comment. Math. Helv. 75(2000) 353-367.
\[18\] J. Schultens and R. Weidman, Destabilizing amalgamated Heegaard splittings, Geom. Topol. Monographs 12(2007) 319-334.
\[19\] J. Souto, The Heegaard genus and distance in curve complex, Preprint.
Department of Applied Mathematics, Dalian University of Technology, Dalian, China
qiurf@dlut.edu.cn
School of Mathematics, Peking University, Beijing, China
wangsc@math.pku.edu.cn
Department of Applied Mathematics, Dalian University of Technology, Dalian, China
zhangmx@dlut.edu.cn
[^1]: Supported by a grant of NSFC (No. 10625102)
[^2]: Supported by a grant of NSFC (No. 10631060)
|
\#1 \#1[\#1]{} \#1[/\#1]{}
[**On the Color Dipole Picture**]{} [^1]
[**Dieter Schildknecht$^{1,2,a)}$**]{}\
[*$^1$Universität Bielefeld, Fakultät für Physik Universit[ä]{}tsstra[ß]{}e 25, D-33615 Bielefeld*]{}\
[*$^2$Max-Planck-Institut für Physik Föhringer Ring 6, D-80805 München*]{}\
$^{a)}$Corresponding author: schild@physik.uni-bielefeld.de\
URL: http://www.physik.uni-bielefeld.de
[ We give a brief representation of the theoretical results from the color dipole picture, covering the total photoabsorption cross section, high-energy $J/\psi$ photoproduction with respect to recent experimental data from the LHCb Collaboration at CERN, and ultra-high energy neutrino scattering, relevant for the ICE-CUBE experiment.]{}\
**DEEP INELASTIC SCATTERING**
In terms of the imaginary part of the (virtual) forward Compton-scattering amplitude, deep inelastic electron-proton scattering at low values of $x \cong Q^2/W^2 \lsim 0.1$ proceeds via $q \bar q$ forward scattering, compare Figure 1.
The total photoabsorption cross section is determined by (color-dipole picture, CDP) \_[\^\*\_[T,L]{}p]{} (W\^2 , Q\^2) = dz d\^2 r\_ | \_[T,L]{} (r\_, z(1-z), Q\^2) |\^2 \_[(q |q)\^[J=1]{}\_[T,L]{}p]{} (r\_, W\^2). \[1\] Compare e.g. refs. [@bib1] for a presentation of the CDP and a list of literature. In standard notation, $\vert \psi_{T,L} (r_\bot , z(1-z), Q^2) \vert^2$ denotes the probability for the photon of virtuality $Q^2$ to couple to a $(q \bar
q)^{J=1}_{T,L}$ state specified by the transverse size $\vec r_\bot$ and the longitudinal momentum partition $0 \le z \le 1$, and $\sigma_{(q \bar q)^{J=1}_{T,L}p} (\vec
r_\bot \sqrt{z(1-z)}, W^2)$ denotes the color-dipole-proton cross section at the total $\gamma^*p$ center-of-mass energy $W$. The gauge-invariant two-gluon coupling of the $q \bar q$ dipole in Figure 1 requires a representation of the $(q \bar q)$-color-dipole-proton cross section of the form \_[(q |q)\^[J=1]{}\_[T,L]{}p]{} (r\_, W\^2) = d\^2 l\^ \_|\_[(q |q)\^[J=1]{}\_[T,L]{}p]{} (l\^[ 2]{}\_, W\^2)(1 - e\^[-l\^[ ]{}\_r\_]{}) . \[2\] Note the factorization in (\[1\]) into the $Q^2$-dependent “photon wave function”, and the $W^2$-dependence $(q \bar q)p$ cross section. The photon wave function is known from quantum electrodynamics. It implies that at sufficiently large values of $Q^2$ only small dipoles of transverse size $\vec
r^{~2}_\bot \sim 1/Q^2$ contribute to the interaction.
Concerning the $\vec r^{~2}_\bot$ dependence of the dipole cross section in (\[2\]), with an energy-dependent upper limit, $\vec l^{~2}_\bot \le
\vec l^{~2}_{\bot Max} (W^2)$ in (\[2\]), for any fixed dipole size $\vec r^{~2}_\bot$, we either have i) $\vec l^{~\prime 2}_{\bot Max} (W^2) \vec r^{~2}_\bot \ll 1$, implying $\sigma_{(q \bar q)p} \sim \vec r^{~2}$, (“color transparency”), or ii) $\vec l^{~\prime 2}_{\bot Max} (W^2) \vec r^{~\prime 2}_\bot \gg 1$, implying $\sigma_{(q \bar q)p} \sim \sigma^{(\infty)} (W^2)$, (“saturation”), compare Figure 2.
Evaluation of the photoabsorption cross section (\[1\]), upon inserting the dipole cross section in the limits i) and ii), translates color transparency (c.tr.) and saturation (sat.) into specified limits of photoabsorption [@bib1; @bib2]. \_[\^\*p]{} (W\^2,Q\^2) = \_[\^\* p]{} ((W\^2,Q\^2)) \~\^[()]{} (W\^2) {
[l@[,]{}l]{} & [for]{} (W\^2,Q\^2) 1, [ c. tr.]{}\
& [for]{} (W\^2,Q\^2) 1, [ sat.]{}
. \[5\] With $\sigma^{(\infty)} (W^2) \approx const$, the photoabsorption cross section only depends on the single low-$x$ scaling variable $
\eta (W^2,Q^2) = (Q^2 + m^2_0)/(\Lambda^2_{sat} (W^2)).
$ The “saturation scale”, $\Lambda^2_{sat} (W^2)$ is determined by the first moment of $\bar \sigma_{(q \bar q)^{J=1}_{T,L}p} (\vec l^{~\prime 2}_\bot,
W^2)$ in (\[2\]), and $m^2_0 \lsim m^2_\rho$ for light quarks is fixed by quark-hadron duality. Actually, $\sigma^{(\infty)} \sim \ln W^2$, i.e. logarithmic violation of $\eta$-scaling.
We summarize: the gauge-invariant two-gluon coupling of the color dipole implies color transparency and saturation, which translate into $\sigma_{\gamma^*p}\sim 1/\eta (W^2,Q^2)$ and $\sigma_{\gamma^*p} \sim \ln
(1/\eta (W^2,Q^2))$, respectively. No specific free-parameter-dependent ansatz is necessary to arrive at this conclusion. The experimental results agree with the above prediction, compare Figure 3. From (\[5\]), for $W^2 \to \infty$ with $Q^2 > 0$ fixed, $\sigma_{\gamma^*p}
(\eta (W^2,Q^2))$ approaches [@bib2] the $Q^2=0$ photoproduction limit. Compare Ref. [@bib1] for results on the longitudinal-to-transverse ratio. In Figure 4, we show the proton structure function. In the color-transparency region, as a consequence from $\sigma_{\gamma^*p} \sim 1/\eta (W^2,Q^2)$, we have $F_2(x, Q^2) = F_2 (W^2)$.
**PHOTOPRODUCTION AND ELECTROPRODUCTION OF THE $J/\psi$ VECTOR MESON**
As depicted in Figure 5, the diffractive production of $q \bar q$ pairs, in distinction from the total photoabsorption cross section in (\[1\]), depends on the square of the $(q \bar q)$-proton cross section. Employing quark-hadron duality, diffractive vector-meson production is obtained by integration of $q \bar q$ production over the mass interval that it determined by the level spacing of the vector meson states under consideration.
\[Figure5\]
In view of recent $J/\psi$-photoproduction data [@LHCb] from the LHCb collaboration, we concentrate [@Unique] on $J/\psi$ production. Approximating $(c \bar c)$-production by the cross section at the production threshold simplifies the integration over the level spacing $\Delta M^2_{J/\psi}$. This integration yields a factor $\Delta F^2 (m^2_c, \Delta M^2{J/\psi})$, compare details in ref. [@Ku-Schi]. Upon suppressing an overall constant factor, the $Q^2$ dependence and the $W^2$ dependence are given by [@Unique] |\_[t 0]{} { . \[10\] where the first line on the right-hand side refers to $\eta_{c \bar c} (W^2,Q^2) \equiv
(Q^2 + M^2_{J/\psi})/\Lambda^2_{sat} (W^2) \gg 1$, while the second line refers to $\eta_{c \bar c} (W^2, Q^2) \equiv
(Q^2 + M^2_{J/\psi})/\Lambda^2_{sat} (W^2) \ll 1$.
At any fixed value of $Q^2$, for the $W$-dependence in (\[10\]), we have the significant limits of $\eta_{c \bar c} (W^2,Q^2) \gg 1$ and $\eta_{c \bar c} (W^2,Q^2) \ll 1$, where $\eta_{c \bar c} (W^2,Q^2) $ generalizes the low-$x$ scaling variable $\eta (W^2,Q^2)$. The limits in (\[10\]), respectively, correspond to color transparency and saturation for $J/\psi$ production. Note that (\[10\]) yields a parameter-free prediction for $J/\psi$ production, once $\sigma^{(\infty)}
(W^2)$ and $\Lambda^2_{sat} (W^2)$ are known from the measurements of the total photoabsorption cross section (\[1\]).
For the comparison of the $Q^2$ dependence and the $W^2$ dependence in (\[10\]) with the experimental data in the HERA energy range of $W \lsim 300$ GeV, we refer to ref. [@Ku-Schi]. The figures in ref. [@Ku-Schi] show good agreement of the prediction (\[10\]) with the experimental data.
Turning to photoproduction at $W \gsim 100$ GeV, from (\[10\]), we obtain \_[p J/p]{} (W\^2) & = & \_[p J/p]{} (W\^2\_1 = (100 [GeV]{})\^2),\
& & F\_A (\^2\_[sat]{} (W\^2)) F\_B (W\^2) \_[p J/p]{} (W\^2\_1 = (100 [GeV]{})\^2), \[12\] where $\sigma_{\gamma p \to J/\psi} (W^2_1 = (100~{\rm GeV})^2) = 80 nb$ is to be inserted on the right-hand side. From (\[12\]), one finds the numerical results [@Unique] given in the last column of Table 1.
[$W [{\rm GeV}]$]{} [$\Lambda^2_{sat} (W^2) [{\rm GeV}^2]$ ]{} [$\frac{M^2_{J/\psi}}{\Lambda^2_{sat} (W^2)}$]{} [$F_A (\Lambda^2_{sat} (W^2))$]{} [$F_B (W^2)$]{} [$\sigma_{\gamma p \to J/\psi p} (W) [nb]$]{}
--------------------- -------------------------------------------- -------------------------------------------------- ----------------------------------- ----------------- -----------------------------------------------
100 4.32 2.22 1 1 80
300 7.92 1.21 2.12 1.02 173
1000 15.4 0.624 3.93 1.11 349
2000 22.6 0.425 5.11 1.16 474
: The $W$-dependence of $J/\psi$ photoproduction.
\[Figure6\]
Comparison with the experimental data in Figure 6, taken from ref. [@LHCb], shows agreement with our predictions in the last column of Table 1. According to Table 1, the experimental data from the LHCb collaboration, with $\eta_{c \bar c} (W^2,Q^2=0) =
M^2_{J/\psi}/\Lambda^2_{sat} (W^2)$, lie in the range of $2.2 \gsim
\eta_{c \bar c} (W^2,Q^2) \gsim 0.43$. This region of $\eta_{c \bar c}$, according to (\[10\]), covers the transition from color transparency to saturation; the deviation from the power-law fit in Figure 6 is to be interpreted as a transition from color transparency to saturation. We stress that the frequently assumed proportionality of $J/\psi$ photoproduction to the square of the gluon distribution, corresponding to $\Lambda^4_{sat} (W^2) \sim (\alpha_s (Q^2) x g (x, Q^2))^2$, violates the necessary decent (logarithmic) high-energy saturation behavior that is naturally contained in our prediction from the CDP.
**THE NEUTRINO-NUCLEON CROSS SECTION AT ULTRA-HIGH ENERGIES**
Predictions of the neutrino-nucleon cross section at ultrahigh energies of the order of $E = 10^{6}~ {\rm GeV}$ and beyond are relevant and important for the interpretation of the search for cosmic neutrinos in e.g. the ICE-CUBE experiment. Due to the presence of the $W$-boson mass in conjunction with ultrahigh energy, the process is determined by $x \simeq Q^2/W^2
\ll 0.1$. For the flavor-independent interaction of $q \bar q$ pairs, the neutrino cross section is related to $\sigma_{\gamma^*p} (\eta (W^2,Q^2))$ by [@PRD88] \_[N]{}(E) = \^[s-M\_p\^2]{}\_[Q\^2\_[Min]{}]{} d Q\^2 \^[s-Q\^2]{}\_[M\_p\^2]{} (1 + (1-y)\^2) \_[\^\*p]{} ((W\^2, Q\^2)). \[13\] where $n_f/\sum_q Q^2_1 = 5/18$ for $n_f=4$ quark flavors. A careful evaluation of the cross section shows that the cross section, even at energies of the order of $E \sim 10^{10}~{\rm GeV}$ is dominated by the color-transparency region, where $\sigma_{\gamma^*p} \sim 1/\eta (W^2,Q^2)$.
The simple explicit expression for the photoabsorption cross section, upon substitution into (\[13\]) yields the results [@PRD88] in Figure 7. The extrapolation to ultrahigh energies in the CDP agrees with results from the perturbative-QCD improved parton model. It is interesting to note that the “Froissart-inspired” representation [@Block] of the available experimental data agrees with the results from the CDP below $E_\nu \simeq 10^{10}~{\rm GeV}$, but yields a suppression of the cross section for $E_\nu \gsim 10^{10}~{\rm GeV}$, compare Figure 8.
[99]{}
M. Kuroda and D. Schildknecht, Phys. Rev. [**D85**]{}, p. 094001 (2012); D. Schildknecht, Mod. Phys. Lett. [**A 29**]{}, p. 1430028 (2014); M. Kuroda and D. Schildknecht, Int. J. Mod. Phys. [**A31**]{} p. 1650157 (2016). D. Schildknecht, [*Deep inelastic scattering at low x: Generalized vector dominance and the color dipole picture*]{}, Nucl. Phys. Proc. Suppl. [**99A**]{}, pp. 121-125; D. Schildknecht, B. Surrow and M. Tentyukov, Phys. Lett. [**B 499**]{}, pp. 116-124 (2001); G. Cvetic, D. Schildknecht, B. Surrow and M. Tentyukov, EPJ [**C 20**]{}, pp 77-91 (2001); D. Schildknecht, [*Scaling in $\gamma^*p$ total cross-sections and the generalized vector dominance/color dipole picture*]{}, Sci. Cult. Ser.-Phys. [**21**]{}, pp. 798-803, edited by G. Bruni et al. (World Scientific, Singapore, 2002); D. Schildknecht, B. Surrow and M. Tentyukov, Mod. Phys. Lett. [**A16**]{}, pp. 1829-1840 (2001). LHCb-CONF-2016-007, August 22, 2016. Dieter Schildknecht, arXiV: 1611.01382 M. Kuroda and D. Schildknecht, Phys. Lett. [**B638**]{}, pp. 473-479 (2006); M. Kuroda and D. Schildknecht, EPJ [**C 37**]{}, pp. 205-222 (2004); M. Kuroda and D. Schildknecht, EPJ [**C 44**]{}, p. 613 (2005) Erratum. M. Kuroda and D. Schildknecht, Phys. Rev. [**D88**]{}, p. 053007 (2013). M.M. Block, P. Ha and D. McKay, Phys. Rev. [**D82**]{}, p. 077302 (2010).
[^1]: Presented at Diffraction 2016, Acireale (Catania, Sicily) September 2-8, 2016, AIP Conference Proceedings, ed. by A. Papa , to be published
|
---
abstract: 'Rings and radial gaps are ubiquitous in protoplanetary disks, yet their possible connection to planet formation is currently subject to intense debates. In principle, giant planet formation leads to wide gaps which separate the gas and dust mass reservoir in the outer disk, while lower mass planets lead to shallow gaps which are manifested mainly on the dust component. We used the Atacama Large Millimeter/submillimeter Array (ALMA) to observe the star HD169142, host to a prominent disk with deep wide gaps that sever the disk into inner and outer regions. The new ALMA high resolution images allow for the outer ring to be resolved as three narrow rings. The HD169142 disk thus hosts both the wide gaps trait of transition disks and a narrow ring system similar to those observed in the TW Hya and HL Tau systems. The mass reservoir beyond a deep gap can thus host ring systems. The observed rings are narrow in radial extent (width/radius of 1.5/57.3, 1.8/64.2 and 3.4/76.0, in [au]{}) and have asymmetric mutual separations: the first and middle ring are separated by 7 [au]{} while the middle and outermost ring are distanced by $\sim$12 [au]{}. Using hydrodynamical modeling we found that a simple explanation, involving a single migrating low mass planet (10 M$_\oplus$), entirely accounts for such an apparently complex phenomenon. Inward migration of the planet naturally explains the ring’s asymmetric mutual separation. The isolation of HD169142’s outer rings thus allows a proof of concept to interpret the detailed architecture of the outer region of protoplanetary disks with low mass planet formation of mini-Neptune’s size, i.e. as in the protosolar nebula.'
author:
- Sebastián Pérez
- Simon Casassus
- Clément Baruteau
- Ruobing Dong
- Antonio Hales
- Lucas Cieza
title: 'DUST UNVEILS THE FORMATION OF A MINI-NEPTUNE PLANET IN A PROTOPLANETARY RING'
---
Introduction {#sec:intro}
============
Planet formation theories describe two main pathways for the formation of protoplanet embryos: gravitational instability followed by fragmentation, which forms gas giant planets, or (sub-)stellar mass companions [e.g., @Boss1997; @Kratter2010]; and bottom-up growth by core accretion, which forms terrestrial and icy planets, or the cores of giant planets [e.g., @Pollack1996]. However, the timescale of core accretion increases dramatically with stellocentric distance due to the longer dynamical timescale and low surface densities at large radii [e.g., @Goldreich2004], which poses a problem for the formation of planets before the protoplanetary disk dissipates. Mechanisms such as pebble accretion [@Ormel2010] are thought to alleviate the tension required to form the cores of Jupiter, Saturn, and the icy giants in the Solar System (all within 30[au]{}) and the lifetime of the protosolar disk [@Lambrechts2012].
The formation and evolution of pebbles and planetesimals, and hence planetary cores, relies upon radial pressure bumps that curb the catastrophic drift of solids towards the star due to aerodynamic drag [e.g., @Pinilla2012]. These radial dust traps have been associated with the ring systems observed in disks in the thermal emission from cold dust grains at radio wavelengths. Forming planets are thought to open gaps in the dust distribution, and the masses of such putative protoplanets are commonly inferred from the gap’s depth and width [@Rosotti2016; @DongFung2017]. It is frequent to associate one planet per gap [@Dipierro2015; @Mentiplay2018; @Clarke2018], multiple planets to one (wide) gap [@Dong2015], and one planet to multiple (narrow) gaps [e.g., a 0.1$M_{\rm
Jup}$ planet can explain the multiple gaps in AS 209, @Zhang2018].
In the so-called ‘gapped disks’, sometimes called transitional disks, dust is evacuated from central cavities or deep radial gaps, resulting in one or two bright rings. Accreting giant planets of masses comparable to or larger than that of Jupiter are able to open deep gaps in the gas and accumulate dust into an outer ring [@Crida2006; @Pinilla2012; @Dipierro2016]. By contrast, in disks without a deep gap, dust accumulation in mild pressure maxima produces sequences of several [*shallow*]{} rings [e.g., the TW Hya disk, @Andrews2016]. Clear signs of low-mass planet formation, as hinted by these fine shallow rings [@Dong2017; @Dong2018], have been absent in the outer regions of ‘gapped disks’.
The 1.7 $M_\odot$ star HD169142 [estimated age 6$^{+6}_{-3}$ Myr, @Grady2007] hosts a prominent disk with deep wide gaps imaged in the near-infrared [@Quanz2013; @Momose2015; @Pohl2017; @Monnier2017; @Ligi2018; @Bertrang2018], mid-infrared [@Honda2012], 1.3 millimeter [@Fedele2017], and centimetre [@Osorio2014; @Macias2018] wavelengths. Circum-planetary features related to giant protoplanets have tentatively been reported in HD169142 from radio/IR imaging [@Reggiani2014; @Biller2014; @Osorio2014; @Gratton2019], but the nature of these signals remains to be confirmed [@Ligi2018].
In this paper, we present new 1.3 mm observations of the dust structures around HD169142 at $\sim$2 [au]{} resolution (Sec. \[sec:obs\]). In these new observations, the outer disk shows an intricate system of fine rings and narrow gaps (described in Sec. \[sec:res\]). Using hydrodynamical modeling (Sec. \[sec:model\]), a simple connection between the fine ring structure and a single, [*migrating*]{}, mini-Neptune protoplanet can be made, which we present and discuss in Sec. \[sec:model2\]. Implications are discussed in Sec. \[sec:summary\].
Observations and data reduction {#sec:obs}
===============================
We obtained 1.3 millimeter observations by combining ALMA 12-m array extended (C40-9) and more compact (C40-6) configurations, resulting in baselines ranging from 19 meters to up to 13.9 kilometers and a total of 40-46 antennas. The combined observations are sensitive to spatial scales of up 20, with a spatial resolution of $\sim$20 mas. The long baseline observations were acquired between Sept. and Nov. 2017 (Cycle 4) in four different blocks of $\sim$40 min each. Precipitable water vapor ranged between 0.7 and 1.8 mm. Observations of a phase calibrator (J1826-2924) were alternated with the science to calibrate the time-dependent variation of the complex gains. The cycling time for phase calibration was set to 8 minutes and 54 seconds for the compact and extended configurations, respectively. The ALMA correlator was configured in Frequency Division Mode (FDM). Two spectral windows with 1.875 GHz bandwidth were set up for detecting the dust continuum, centred at 232.0 GHz and 218.0 GHz, respectively.
All data were calibrated by the ALMA staff using the ALMA Pipeline version 40896 in the CASA package [@McMullin2007], including offline Water Vapor Radiometer (WVR) calibration, system temperature correction, as well as bandpass, phase, and amplitude calibrations. The short baseline and long baseline datasets were calibrated independently. Self-calibration of the data was performed to improve coherence (a single iteration of phase-only), after which they were combined using the CASA task [concat]{}. A positional offset of (-20,-40) mas between the short and long baseline resulting images was measured and corrected prior to combining the datasets. This offset is consistent with the (-2.8,-38.0) mas/yr proper motion listed for this star (SIMBAD) and with the expected astrometric accuracy of ALMA (typically a 10th of the synthesized beam).
ALMA continuum imaging
----------------------
Image reconstruction was initially performed using the CLEAN algorithm (CASA version 5.2, task [tclean]{}). As the source is relatively bright, we super-resolve the continuum data using our Maximum Entropy Method (MEM) package [uvmem]{}, which is part of the family of algorithms based on maximum-entropy regularization. Here we used the publicly-available GPU adaptation <span style="font-variant:small-caps;">gpuvmem</span>[^1] [@Carcamo2018], and regularized by minimizing the Laplacian of the model image with an objective function. The MEM reconstruction provides the highest resolution without compromising on sensitivity. We adopted the MEM image for our analysis. The final image has a peak of 67 mJy beam$^{-1}$ and an rms ($1\sigma$) of 14.7$\mu$Jy beam$^{-1}$ for a synthesized beam of 27$\times$20 mas. The total flux density over the whole image is 178$\pm$18 mJy (10% flux calibration uncertainty). The flux density of the central source at 1.3 millimeter is 194$\pm$20$\mu$Jy. The bright ring at $\sim$022 has an integrated flux of 59$\pm$6 mJy, and the flux density of the outer region is 126$\pm$13 mJy. A side-by-side comparison of the MEM versus CLEAN image reconstructions is shown in Fig.\[supp:clean\] (Appendix \[app:clean\]).
{height="44.00000%"} {width="49.00000%"}
A triple ring system in the disk’s outer region {#sec:res}
-----------------------------------------------
The ALMA image (Fig.\[fig:obs\]a) reveals a prominent bright ring (B1, see labelling in Fig.\[fig:obs\]b) followed by substructure in the outer region in the form of a system of three fine rings. In previous observations the three rings (B2, B3, and B4) were seen as a single structure (we refer to this structure as the ‘outer region’), with evidence of ripples at the last scattering surface of micron-sized grains [@Pohl2017; @Gratton2019]. In these new observations, we see that the outer region is a packed system of three rings, with mutual separations of $\sim$7 [au]{} between B2 and B3, and $\sim$12 [au]{} between B3 and B4, for a distance to HD169142 of 113.9$\pm$0.8 pc [@Gaia2018].
Rings B2 and B3 are detected at a 5 to 20$\sigma$ level (1$\sigma$=15$\mu$Jybeam$^{-1}$), while emission from B4 reaches 4-5$\sigma$ over the image. For a quantitative characterization of the rings, we deproject the image by the disk inclination angle $i$=$12.5^\circ$$\pm$0.5$^\circ$ and transform to polar coordinates centred on the compact millimeter emission detected at the location of the star (see Appendix \[app:depro\]). The polar deprojections zoomed on each ring are shown in Fig.\[fig:depro\]. The disk azimuthally-averaged surface brightness profile is shown in Fig.\[fig:obs\]b.
The locations and widths of the fine rings are determined from Gaussian fits to the average radial profile. Three Gaussian components were fit simultaneously plus a common low-order polynomial to account for low level background emission. Thus, the observed locations of B2, B3 and B4 are 0503$\pm$0005 (57.3 [ au]{}), 0563$\pm$0005 (64.2 [au]{}), and 0667$\pm$0009 (76.0 [au]{}), respectively. Uncertainties only represent the error in fitting the Gaussian centroids. Since the rings are not circular (see below), the radii of each ring depend on azimuth, varying within $\pm$001 around the mean locations quoted here.
The deconvolved width of each ring is determined by subtracting the beam width $\sigma_{\rm beam} = {\rm FWHM}/\sqrt{8\log{2}}$ from the best fit Gaussian width in quadrature. Rings B2 and B3 have deconvolved widths of only 1.5 and 1.8 [au]{}, respectively, while B4 is more radially extended with a deconvolved width of 3.4 [ au]{}. Rings B2 and B3 in HD169142 are likely some of the narrowest structures in protoplanetary disks reported so far [see DSHARP ring characterization in @Dullemond2018 where the narrowest ring is $\sim$3.4 [ au]{} in deconvolved width].
Fig.\[fig:depro\] shows evidence for azimuthal dips along ring B2 spanning azimuthal angles $\sim$50$^\circ$ to 100$^\circ$ and $\sim$180$^\circ$ to 270$^\circ$, measured counter clockwise from the disk PA. Interestingly, all three outer rings have a finite eccentricity (0.09$\pm$0.03) and share a common focus. This is demonstrated in Fig.\[fig:depro\] which shows ellipses fit to the rings’ radii as a function of azimuthal angle (see Appendix \[app:depro\] for details).
A structured ring and two deep wide gaps in the disk’s inner region {#sec:B1}
-------------------------------------------------------------------
The brightest feature at 1.3 millimeter is the previously known inner ring B1, whose remarkable radial and azimuthal structure is further emphasized here at finer angular resolutions (Fig.\[fig:depro\]). Previous hydrodynamical modelling have suggested that this structure is consistent with dynamical interactions with forming giant planets located inside the gaps D1 and D2 [@Bertrang2018]. The narrowness of ring B1 is explained by efficient radial trapping of dust in a pressure maximum, which puts an abrupt halt to the inward radial drift of dust particles $\geq$1mm [@Pinilla2012]. Interestingly, as seen in Figs.\[fig:obs\]a and \[fig:depro\], the structure in B1 may well be interpreted as a multiple ring. B1 is not split in the same way as the outer ring, rather it shows a very irregular and faint inner ring.
In addition, compact continuum emission is detected at the star’s location. We interpret this emission as coming from thermal emission from an inner disk of $\leq$1[au]{} in radius, compatible with its near-infrared excess [@Chen2018] as free-free emission is not expected to be significant in HD169142 [@Osorio2014].
The massive protoplanets required to explain the perturbed morphology of B1 are unlikely to split the outer dust disk into the observed triple ring structure [@Bae2018]. Instead, their dynamical interaction with the disk should lead to the clearing of deep and wide gaps such as D1 and D2, which are both $\sim$20 [au]{} in width and heavily depleted of dust [*and*]{} gas [@Pohl2017; @Fedele2017; @Ligi2018; @Bertrang2018]. Yet, with the fine angular resolution provided by these new ALMA observations, we find that the outer region of HD169142 is reminiscent of the ring systems in TW Hya [@Andrews2016] and HL Tau [@ALMA2015], both of which lack the presence of a deep wide gap in the dust continuum.
{width="40.00000%"} {width="35.00000%"}
Hydrodynamical simulations and radiative transfer calculations {#sec:model}
==============================================================
Theoretical background {#sec:tb}
----------------------
Several models have been proposed to explain disks with rings. Enhanced dust growth at condensation fronts can produce fine rings [@Banzatti2015; @Zhang2015; @Okuzumi2016], however, the location of rings and gaps are uncorrelated with the expected locations of snowlines [@Long2018; @Huang2018; @vandermarel2019], and these rings are unlikely to be eccentric. Sharp variations in the gas viscosity associated with dead-zones [@Flock2015; @Miranda2017] and radially variable magnetic disk winds [@Suriano2018] also yield ring-like structures, but they are driven by MHD instabilities that occur mostly within the central $\sim$20[au]{}.
Theory of disk-planet interactions predicts that a low-mass planet can produce multiple dust rings by the propagation and dissipation of spiral density waves [@Goodman2001; @Dong2017; @Dong2018]. The dissipation of waves progressively moves dust away from the planet’s orbit, with the consequence that two narrow dust gaps form, one on each side of the planet’s orbit. The expelled dust thus forms two rings exterior to the gaps. Also, dust can remain at the planet’s (co-)orbital radius, forming a third ring between the two dust gaps [@Dong2017]. Previous hydrodynamic simulations have explored this scenario and found that, in a low-viscosity disk ($\alpha$ viscosity of 10$^{-4}$ or lower), a single low-mass planet indeed produces multiple rings in the dust distribution separated by annular gaps [@Dong2017; @Dong2018; @Bae2018], observable at ALMA resolutions. The resulting ring system depends mainly on disk viscosity, disk temperature (aspect ratio), planet mass, and the elapsed time, which will dictate the number of additional spiral arms and hence the potential number of gaps [@Bae2018].
Given the sharpness and small widths of the rings in HD169142 (the deconvolved width of ring B2 is only $\sim$1.5 [au]{}), the $\alpha$ turbulent viscosity should be fairly small allowing a single low-mass planet to produce multiple rings and gaps. Assuming standard radial profiles for the disk structure, a relation that connects the distance between the two dust gaps, relative to the orbit of the planet, and the planet’s mass and disk scale-height can be found. Informed by hydrodynamic calculations [assuming $\alpha \la
10^{-4}$, @Dong2018], the relation reads $$\frac{r_{\rm OG} - r_{\rm IG}}{r_{\rm p}} \approx 2.9 \left (
\frac{\gamma + 1 }{12/5} \frac{M_{\rm p}}{M_{\rm th}}\right)^{-2/5}
\left( \frac{h}{r} \right),$$ where $r_{\rm IG}$ and $r_{\rm OG}$ are the locations of the inner and outer gaps around the planet, respectively, and $r_{\rm p}$ is the planet’s orbit. $M_{\rm th}=M_\star(h/r)^3$ is the disk thermal mass, $\gamma$ is the polytropic index (equal to 1 for isothermal gas), and $h/r$ is the disk aspect ratio. Fig.\[fig:obs\]b shows that the narrow gaps D3 and D4 are separated by $\sim$9 [au]{}. In the outer region, the observed flux density of the rings translates into a brightness temperature of 8-12 K, consistent with a disk aspect ratio of 3%. Thus, for HD169142’s stellar mass ($M_\star\!\approx\!1.7\,M_\odot$), disk aspect ratio ($h/r\!\approx\!0.03$), and approximating $r_{\rm p}$ by B3’s location at $\sim$64 [au]{}, a planet of $\la$10$M_\oplus$ could produce gaps separated by $r_{\rm D4} - r_{\rm
D3}\approx\,9$[au]{}.
Physical model and numerical setup
----------------------------------
We perform hydrodynamic simulations of an outer disk composed of gas and a distribution of dust particles, to test whether the three bright rings (B2, B3, B4) unveiled by our ALMA observations can be explained by a single forming planet. The main observables are the distance between the gaps, the mutual separation between the rings, and the observed fluxes. In this study, we do not attempt to model the previously known inner features (D1 and B1) but rather focus on the new substructure of the outer region.
{width=".7\textwidth"}
### Hydrodynamical simulations
Our 2D hydrodynamical simulation of the gas and dust in the HD169142 disk has been carried out with the code Dusty FARGO-ADSG. It is an extended version of the publicly available code FARGO-ADSG, which solves the gas hydrodynamics equations on a polar mesh [@Masset2000; @Baruteau2008a; @Baruteau2008b], and which models dust as Lagrangian test particles[^2] [@Baruteau2016; @Fuente2017]. The simulation has been designed to test whether the three bright rings (B2, B3, B4) unveiled by our ALMA observations can be explained by a single forming planet. In our simulation the star’s mass is assumed to be 2 $M_\odot$. We perform a set of simulations sampling the main parameter space {planet mass, disk scale-height, disk alpha viscosity}, although no exhaustive exploration is intended as finding a perfect fit is not in the scope of this work.
The gas momentum and continuity equations are solved on a polar mesh with 900 cells logarithmically spaced between 35 and 105 [au]{} in radius, and 1,200 cells evenly spaced between 0 and 2$\pi$ in azimuth. Non-reflecting boundary conditions are used to avoid reflections of the planet wakes at the radial edges of the computational grid.
The disk has an initial surface density profile decreasing with radius $r$ as $r^{-1}$ with an exponential cutoff beyond 100 [au]{}. The initial surface density is 2.9 g cm$^{-2}$ at 100 [au]{}, very close to the radiative transfer model which fits the CO istotopologue and continuum observations at 1.3 mm [@Fedele2017]. This large surface density means that Type I migration cannot be discarded, especially for the low planet masses we are exploring. A locally isothermal equation of state is used with the disk temperature decreasing with radius as $r^{-1/2}$, and equal to 8 K at 70 [au]{} (motivated by the observed brightness temperature), slightly lower than previous models. This low temperature, which is required to match the small mutual separations between the bright rings, translates into a disk aspect ratio profile in $r^{1/4}$ and equal to 0.033 at 69 [ au]{}. Gas self-gravity is included since the disk’s Toomre $Q$-parameter varies from $\sim$6 to $\sim$9 throughout the computational grid. The $\alpha$ turbulent viscosity is set to $10^{-5}$. This low viscosity is motivated by the sharpness of the observed rings. Accretion onto the planet is not included.
The code solves the equations of motion for 200,000 dust particles with radii between 10 $\mu$m and 3 mm. Dust particles feel the gravity of the star, the planet, the disk gas (since gas self-gravity is included) and gas drag. Dust turbulent diffusion is also included as stochastic kicks on the particles position vector [@Charnoz2011]. However, dust self-gravity, dust drag, growth and fragmentation are not taken into account. Dust particles (assumed to be compact spheres of 2 g cm$^{-3}$ internal density) are inserted at the beginning of the simulation with a number density profile decreasing as $r^{-2}$ between 59 and 90 [au]{}. This corresponds to an initial dust surface density decreasing as $r^{-4}$. Although quite steep, the surface density evolves into a profile which increases with $r$ between the rings after 150 orbits. See the perturbed gas density as well as the distribution of dust particles shown in Fig.\[fig:hydro\]. Dust feedback onto the gas is discarded as the dust’s surface density along the B2, B3, and B4 rings remains comfortably smaller than the gas surface density in our model (see Fig.\[fig:hydro\]), for the gas-to-dust mass ratio assumed in the radiative transfer calculation (see Section \[sec:rt\]).
A low-mass planet of a mini-Neptune mass (motivated by Sec. \[sec:tb\]) is inserted at 69 [au]{} at the beginning of the simulation. The planet-to-star mass ratio is set to 1.7$\times$10$^{-5}$, which for a 2$M_\odot$ star translates into 11 $M_{\oplus}$[^3]. The planet migrates due to disk-planet interactions and reaches $\sim$64 [au]{} after 150 orbits, which is very close to the location of the B3 ring. Higher mass planets (here we considered planet-to-star mass ratio of 6$\times$10$^{-5}$, i.e., a $\sim$33$M_\oplus$ planet) produce significantly wider gaps even at higher viscosities (a simulation with $\alpha=10^{-4}$ was performed, not shown here).
The time that best reproduces our observations of the outer rings is approximately at 150 orbits of the mini Neptune, i.e., $\sim$0.05Myr. We note that the rings are present since orbit $\sim$100 onwards (see Fig. \[fig:hydro\]). The model is stable for at least 1000 planet orbits ($\sim$0.4Myr). After this, a global $m$=$1$ mode grows in the disk, leading both the planet and the disk to develop some eccentricities. After this, the co-rotating ring gradually diffuses with time as the gas surface density decreases.
### Radiative transfer calculation {#sec:rt}
We compute the continuum emission from our dust simulations using the public radiative transfer (RT) code [radmc3d]{} (version 0.41). Twenty logarithmically-spaced bins are used to allocate the 200,000 dust particles from the simulation. The dust sizes ($s$) follow a power-law distribution $n(s)\propto s^{-3.5}$, with minimum and maximum sizes of $1\,\mu{\rm m}$ and $1{\rm mm}$, respectively. Assuming a gas-to-dust ratio of 33 yields a total dust mass in the outer region of $\sim$100$M_\oplus$.
Each 2D distribution of dust particles is turned into a surface density and expanded in the vertical direction assuming hydrostatic equilibrium. The dust’s scale height $h_{i,{\rm d}}$ is size-dependent and follows $$h_{i, {\rm d}} = h \times \sqrt{\frac{D_{\rm z}}{D_{\rm z} + {\rm St}_i}},
\label{eq:Hd}$$ where $h$ is the initial gas pressure scale-height, ${\rm
St}_i$ is the averaged Stokes number for the $i^{\rm th}$ dust size bin, and $D_{\rm z}$ is a turbulent diffusion coefficient in the vertical direction which depends on the level of turbulent activity across the vertical extent of the disk [@Yang2018]. We assume $D_z$ to be proportional to the $\alpha$ turbulent viscosity in our 2D hydrodynamic simulation [see @Baruteau2019 for more details]. A $D_z$ value equal to 10$\alpha$ yields dust temperatures which match the observed brightness temperature of rings B2 and B3. In the RT calculation, we use 24 cells logarithmically-spaced to sample 3 scale-heights in colatitude (with finer sampling towards the midplane).
We expand the hydrodynamic simulation’s grid to include the inner regions and then artificially add the inner disk and the bright ring B1. Their parameterisation follows previous RT modelling which fits the 1.3 millimeter data [@Fedele2017]. The presence of these structures that block stellar radiation are needed to recover the low temperatures of the outer regions. A 7320 K star with a radius of 1.5 $R_\odot$ [@Gaia2018] is placed at the centre of the grid and $10^8$ photon packages are propagated to compute the temperature of each dust bin size via the Monte Carlo [radmc3d]{} [mctherm]{} task. The temperatures range between $\sim$13 and $\sim$20 K in the outer rings, consistent with previous multi-wavelength modelling [@Fedele2017]. These values are 2$\times$ higher than the locally isothermal prescription used in the hydrodynamical simulation, which is representative of the gas scale-height, rather than a physical temperature of dust grains. Ray-tracing is performed to solve the radiative transfer equation for continuum emission assuming the dust is a mix of 30% amorphous carbons and 70% silicates. The resulting synthetic image is convolved with the MEM resolution beam and it is shown in Fig.3. The high fidelity of the ALMA image allows direct comparison with the data in the image plane. The hydrodynamical model reproduces the locations of B2, B3, and B4 as well as the level of flux along rings B2 and B3, and to a lesser extend ring B4’s. This can be best appreciated in Fig.\[supp:depro\] (Appendix \[app:giant\]).
Results {#sec:model2}
-------
{height="46.00000%"} {height="46.00000%"}
A hydrodynamical model including a 10 $M_\oplus$ planet embedded in a disk with an aspect ratio $h=0.033$ reproduces the observations (Fig.\[fig:model\]). The initial condition is such that the planet is introduced at 69[au]{}. After about fifty thousand years, the planet migrates inwards to a $\sim$64 [au]{} orbit, shepherding ring B3 and “pushing” rings B2 and B4, thus explaining the asymmetry in the rings mutual separations. This can be appreciated in Fig.\[fig:hydro\]. A simulation with a planet on a fixed orbit (not included) produces rings that are too equidistant for the disk and planet parameters assumed here. If the 10$M_\oplus$ core formed closer to the star and then migrated outwards, it would not be consistent with the rings locations: the observed asymmetry reveals the direction of migration. This migration could have distinct observational signatures in multiwavelength observations, for example, by probing the spatial segregation of larger grains with longer wavelength observations. Such predictions have recently been reported by @Meru2019 for a planet migrating in a high viscosity disk. Despite the little knowledge we have about the initial conditions of the dust density distribution, the model reproduces the observed fluxes along B2 and B3, albeit with a slightly brighter ring B4. The initial amount of large particles near B4’s location is likely overestimated in our simulations.
As the planet continues to interact dynamically with the dust reservoir, it also starts clearing an [*azimuthal*]{} opening in its vicinity. This produces a dip in azimuth along the middle ring at the planet’s location. Whether this dip is observable at 1.3 millimeters depends on the planet’s mass, the maximum size in the dust distribution, and the elapsed time. The more massive the planet, the faster and deeper is the opening of the azimuthal dip. On the other hand, a large maximum particle size produces a more pronounced dip. Dust growth and fragmentation, which are not taken into account in our simulations, probably also have an effect on the observability of this azimuthal opening. The lack of a clear azimuthal dip along ring B3 indicates that the bulk of particles we observe have sizes $<$1mm, the planet is rather small ($\la$10$M_\oplus$), or that it may have formed less than a few hundred orbits ago ($\sim$50 kyr), or most likely a combination of these options. Note that the signal-to-noise ratio at the rings location might not be sufficient to probe the azimuthal structure along B3. Longer wavelength observations might be able to trace the location of the mini-Neptune as the larger grains pile-up behind the planet (note mm particles in Fig.\[fig:hydro\]).
Discussion
----------
### Could rings B2, B3 and B4 be shaped by two low-mass planets?
A scenario where two low mass planets (one for each narrow gap) results in the triple ringed structure is unlikely for several reasons. The small separation between the gaps (only 9 [au]{}) requires scale-heights at the location of the planets which would be too small for a realistic disk temperature, these translate into $<$5 K at 64 [au]{} [following eq. 16 in @DongFung2017]. In other words, if you put two planets into the two gaps, they will likely open one common gap instead of two. Moreover, the similarity between D3 and D4 suggests that if they are produced by two planets, these ought to have formed at similar times and with similar masses. The dynamical interactions between the closely orbiting planets might also make them unstable.
### How does an inner giant planet impact the structure of rings B2, B3 and B4?
In recent years, it has been shown that planet perturbations can extend far beyond the Hill radius if the disk viscosity is low [@Bae2018]. Since the model presented here assumes a very low viscosity, any inner giant planet (in D2, for example) could have an impact on the outer triple ring system. In a debris disk or a protoplanetary disk with low gas surface density, giant planets could potentially produce fine structures by mean motion resonances with the dust. However, HD169142 is gas rich and the dust in the outer region is still coupled to the gas distribution. To further test the possibility that the fine structure is somehow related to inner giant planets, we included a giant planet in our simulation, of roughly a Jupiter mass located in the middle of D2 at 38 [au]{} (see Appendix \[app:giant\]). The giant indeed opens a deep gap in the gas, but fails to generate any additional narrow rings in the outer disk. However, the spiral wakes excited by the giant and the vortices at the gap edge (produced by the Rossby wave instability) induce eccentricity on the outer triple rings associated with the mini-Neptune (Fig. \[supp:giant\]). Recent scattered light imaging indeed shows faint spiral features at the disk surface which can be associated to putative giant planets in the wide gaps [@Gratton2019]. The dust particles around the mini-Neptune can acquire non-circular trajectories either (i) by direct gravitational interaction with the mini-Neptune, whose eccentricity varies on account of the inner giant’s wakes depositing energy and angular momentum in the planet’s horseshoe region (Fig. \[supp:orbit\]), and/or (ii) by being deflected by the (shock) wakes upon crossing them.
The addition of a giant planet adds significant complexities as the final configuration of rings would strongly depend on the initial conditions for the dust distribution and the precise timing of the formation of the mini Neptune and the giant planet. As protoplanetary disk observations grow in sensitivity and resolution, more and new ingredients need to be considered in the modeling efforts. This paper is meant as a proof of concept to show that a simple one-planet simulation can account for seemingly complex sub-structure in the outer ring of a transition disk. Finding a perfect fit to the data is not in the scope of this work. Indeed, some important aspect have been left aside such as: dust growth [e.g., @Bae2018] and fragmentation, dust back-reaction onto the gas [e.g., @Gonzalez2017; @Dipierro2018], dust torque due to scattered pebbles [@Benitez2018], 3-D and MHD effects [e.g., @Flock2015; @Miranda2017], and possibly many more. The inclusion of planet migration was needed here to explain the asymmetry in the rings’ mutual separations, a step forward towards understanding how low-mass planet migration can be studied from observations of dust radial structures. However, a proper account of dust dynamics, planet accretion and thermodynamics, in a self-consistent way, is necessary to build the migration history of these low-mass protoplanets [@Benitez2018].
### What can these observations tell us about the planet formation process?
The low mass of the putative protoplanet in HD169142, in addition to the lack of clear evidence for structures associated with gravitational instabilities in (abundant) multi-wavelength observations of this source, suggests that a bottom-up process such as core accretion could be responsible for its formation. This implies that core accretion can potentially operate at $\sim$65 [au]{} within the age of the system [$\sim$6 Myr, @Grady2007], possibly assisted by pebble accretion as the outer region banks more than 100$M_\oplus$ worth of pebble-sized solids (subject to uncertainties of the dust opacities). The modeling presented here also shows that the mini-Neptune should have formed well after the giant planets carved gaps D1 and D2. The presence of a giant planet in D2 would have produced a pressure maxima in the outer region which could have enhanced core accretion via a dust trap [@Pinilla2012].
Concluding remarks {#sec:summary}
==================
The new ALMA observations presented here show that a transition disk with wide deep gaps can also host narrow-ring structures in its outer region, similar to those observed in the HL Tau and the TW Hya systems.
The HD 169142 observation allows to link the architecture of protoplanetary disks with low mass planets. The interpretation via hydrodynamics is a proof of concept that links the structure of closely packed double gaps and tripple rings with a single and migrating low-mass planet. Planetary migration naturally explains the distinct mutual separations between the narrow rings. The connection was made possible in HD169142 thanks to the isolation of its outer region. In the absence of a clear gap that separates an outer ring, the superposition of multiple rings due to several planets hampers simple and clear explanations such as that found for HD169142.
In HD169142, we have thus found evidence that suggests that low-mass planet formation can occur in the outer regions of disks bearing evidence for giant protoplanets. The planet formation mechanism, likely core accretion or any bottom-up process, can thus produce planet embryos at $\sim$65[au]{} and outside the orbit of inner giants, at least in certain disks.
We thank Ed Fomalont, Anya Yermakova and Philipp Weber for useful discussions, as well as our anonymous referee for their constructive comments. Financial support was provided by the government of Chile grants Millennium Scientific Initiative RC130007, CONICYT-Gemini 32130007, and CONICYT-FONDECYT grant numbers 1171624, 1171246 and 1191934. S.P acknowledges support from the Joint Committee of ESO and the Government of Chile. The data analysis and some of the simulations were carried out in the Brelka cluster, hosted at DAS/U. de Chile (Fondequip EQM140101). Numerical simulations were carried out on the CalMip machine of the Centre Interuniversitaire de Calcul de Toulouse, which is gratefully acknowledged. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2016.1.00344.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), 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 has made use of data from the European Space Agency (ESA) mission [*Gaia*]{}, processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement.
Comparison between CLEAN and MEM algorithm {#app:clean}
==========================================
A side-by-side comparison of HD169142 image synthesis using CLEAN and MEM algorithms is shown in Fig. \[supp:clean\]. The triple ring structure in the outer region is robust in all image reconstructions. The ‘uniform’ weighting CLEAN image has a comparable resolution to MEM but at the cost of decreasing sensitivity.
![HD169142 image synthesis. CASA [tclean]{} with Briggs weighting, with robust parameter 0.0, yielding a resolution of 39$\times$23 mas (a), uniform weight with robust -2.0 and resolution 32$\times$18 mas (b), and the [gpuvmem]{} deconvolution (c). The effective resolution of the MEM deconvolved image is 27$\times$20 mas and is measured directly from a 2D Gaussian fit to the stellar component in the centre of the field. []{data-label="supp:clean"}](figA1a.pdf "fig:"){height=".365\textwidth"} ![HD169142 image synthesis. CASA [tclean]{} with Briggs weighting, with robust parameter 0.0, yielding a resolution of 39$\times$23 mas (a), uniform weight with robust -2.0 and resolution 32$\times$18 mas (b), and the [gpuvmem]{} deconvolution (c). The effective resolution of the MEM deconvolved image is 27$\times$20 mas and is measured directly from a 2D Gaussian fit to the stellar component in the centre of the field. []{data-label="supp:clean"}](figA1b.pdf "fig:"){height=".365\textwidth"} ![HD169142 image synthesis. CASA [tclean]{} with Briggs weighting, with robust parameter 0.0, yielding a resolution of 39$\times$23 mas (a), uniform weight with robust -2.0 and resolution 32$\times$18 mas (b), and the [gpuvmem]{} deconvolution (c). The effective resolution of the MEM deconvolved image is 27$\times$20 mas and is measured directly from a 2D Gaussian fit to the stellar component in the centre of the field. []{data-label="supp:clean"}](figA1c.pdf "fig:"){height=".365\textwidth"}
Polar deprojection and ellipse fitting {#app:depro}
======================================
To study the eccentricity of the rings, we extract the radial profile of each ring as a function of azimuth, $R(\theta)$, by fitting gaussian functions in radius (using the [Python]{} [iminuit]{} optimization library). Only signal above $5\sigma$ is included in the process. Ring B4 is not considered as it only has a small number of points with high signal-to-noise. B1 is also left aside as its perturbed morphology cannot be reproduced by a simple elliptical curve. The optimization yields the values for $R(\theta)$ and their error bars $\Delta R(\theta)$, for each ring. The ellipse equation is then fitted to $R(\theta)$ for a given ring using a direct least square procedure [@Fitzgibbon1996]. The fitting procedure yields the ellipse centre, its angle of rotation and its eccentricity ($e=\sqrt{1-(b/a)^2}$ where $a$ and $b$ are the semi-major and semi-minor axes, respectively). The best fit ellipse for each ring are plotted on the polar projection in Fig.\[fig:depro\]. The uncertainties in the eccentricity values are calculated using a Monte-Carlo approach where data points are modified by random numbers of the order of $3\Delta\theta$ and then fitted following the aforementioned procedure. This is repeated a thousand times and the final errors are drawn from the standard deviation. Correlated noise is accounted for by multiplying the centroids’ error by a factor $\sqrt{N_{\rm pix}}$, where $N_{\rm pix}=5$ is the number of pixels along the synthesized beam’s major axis.
The uncertainty of the eccentricity is dominated by our imprecise knowledge of the inclination angle of the disk. In order to determine the significance of the fitted values, we explore a range of inclination angles ranging between 9$^\circ$ and 16$^\circ$, repeating the fitting procedure described above for each inclination value. The results are shown in Fig.\[fig:depro\]b. Rings B2 and B3 show eccentricities which indeed vary with inclination angle, with a minimum eccentricity of $e$$\approx$$0.085$. Interestingly, the rings have different eccentricities for a given inclination, except at $i$$\approx$$12.5^\circ$, where both rings have the same eccentricity. This inclination angle is close to the standard value of $i$$\sim$$13^\circ$ used in previous multi-wavelength modelling [e.g. @Fedele2017; @Bertrang2018]. The intrinsic position angles of the ellipses also coincides at $i$$\approx$$12.5^\circ$. This fitting procedure thus suggests a common inclination for both B2 and B3 of $i$$\approx$$12.5^\circ\pm0.5\deg$ (1$\sigma$). At this inclination, the eccentricity of B2 and B3 share the common value of 0.09$\pm$0.02 (2$\sigma$ uncertainty, same as the shaded area in Fig.\[fig:depro\]). A disk inclination angle different from $12.5^\circ$ yields different eccentricity values for B2 and B3, which would suggest the occurrence of warping in the outer regions.
Addition of a giant planet associated with gap D2 {#app:giant}
=================================================
![Effect of an inner giant planet on the triple ring structure over 100 kyr. The figure shows the perturbed gas density and the distribution of dust particles after 100 ([*left*]{}), 150 ([*middle*]{}), and 300 ([*right*]{}) orbits of the mini-Neptune. The onset of the ring structure around the mini-Neptune happens near 100 orbits, while at 300 orbits (100 kyr) the ring B3 develops an azimuthal asymmetry, similar as in the case without the giant planet. In the presence of the giant, the outer rings associated with the mini-Neptune interact with the giant’s wakes, acquiring some eccentricities. The gas distribution in the outer region also becomes eccentric. []{data-label="supp:giant"}](figA2.pdf){width="\textwidth"}
The impact that an inner giant could have on the outer narrow rings is explored by including a 0.5 $M_{\rm Jup}$ planet fixed at 38 [au]{}, in the middle of the observed gap D2, in addition to the mini-Neptune. The general setup inherits the same parameters as the single mini-Neptune simulation, with some changes. The initial radius of the orbit of the mini-Neptune planet is decreased to 68 [au]{}, this is to account for a slightly lower migration rate in the presence of the inner giant. The grid resolution is lowered to 600$\times$900 cells in radius and azimuth, respectively. The grid’s inner edge is extended down to 20 [au]{} as to include gap D2 in the radial domain. The smaller inner edge translates into a significant computational cost. The number of particles is decreased by half (i.e only 100,000 particles are used) so as to make the computation less expensive. Dust particles are distributed over the same region as in the previous simulation. Dust around the giant planet is not included as this would require increasing the number of particles, making the simulation too expensive. As mentioned in the main text, fitting the already known features B1 and D1 is beyond the scope of this work. Modeling all features in HD169142 system requires an extensive search of a large parameter space, but the main difficulty lies on the little knowledge available of the initial conditions and the relative age of each planet. In our simple two-planet setup, both companions are introduced simultaneously at the beginning of the simulation.
\
![Orbital radius (a) and eccentricity (b) of the migrating Mini-Neptune in the presence of an inner giant planet. The amplitude of the oscillations in eccentricity and orbital radius reflects the impact of the giant planet’s wakes on the mini Neptune’s orbit. The planet’s initial eccentricity arises because of the disk’s gravity being felt by the planet in addition to that of the star. The planet’s eccentricity is calculated assuming a two-body problem with only the star and the planet, not the disc. The value of the planet’s eccentricity thus reflects the disc-to-star mass ratio. []{data-label="supp:orbit"}](figA3a.jpg "fig:"){height="0.25\textheight"} ![Orbital radius (a) and eccentricity (b) of the migrating Mini-Neptune in the presence of an inner giant planet. The amplitude of the oscillations in eccentricity and orbital radius reflects the impact of the giant planet’s wakes on the mini Neptune’s orbit. The planet’s initial eccentricity arises because of the disk’s gravity being felt by the planet in addition to that of the star. The planet’s eccentricity is calculated assuming a two-body problem with only the star and the planet, not the disc. The value of the planet’s eccentricity thus reflects the disc-to-star mass ratio. []{data-label="supp:orbit"}](figA3b.jpg "fig:"){height="0.25\textheight"}
Fig. \[supp:giant\] shows that the presence of a giant planet at 38 [au]{} carves a wide gap in the gas at the location of the D2 dust gap. At the same time, the mini-Neptune shapes the outer region into the triple ring system, consistent with the single planet simulation. The presence of the giant planet makes the morphology of the outer rings appear eccentric. The asymmetry in the outer rings mutual separation is also reproduced. The migration of the mini-Neptune is slightly affected by the gas giant’s wakes (see Fig. \[supp:orbit\]). Vortices develop at the outer edge of the giant’s gap but these start decaying shortly after a couple of hundred orbits of the mini-Neptune (see right panel in Fig. \[supp:giant\]). The lack of observational evidence for these vortices may suggest that the mini-Neptune formed long after the giant planet carved the gap D2, at a time where the vortices had already decayed.
[ ]{}\
![Comparison between polar deprojection of the ALMA observation (a) and synthetic predictions from hydrodynamic models after 150 orbits of the mini-Neptune. Panel b shows the result of the hydrodynamic simulations for a single mini-Neptune (same as Fig. \[fig:model\]), while panel c shows the mini-Neptune’s outer rings under the influence of an inner giant planet at 38 [au]{}. All panels have linear color stretch between 0.0 and 0.25 mJy/beam. []{data-label="supp:depro"}](figA4a.pdf "fig:"){width="30.00000%"} ![Comparison between polar deprojection of the ALMA observation (a) and synthetic predictions from hydrodynamic models after 150 orbits of the mini-Neptune. Panel b shows the result of the hydrodynamic simulations for a single mini-Neptune (same as Fig. \[fig:model\]), while panel c shows the mini-Neptune’s outer rings under the influence of an inner giant planet at 38 [au]{}. All panels have linear color stretch between 0.0 and 0.25 mJy/beam. []{data-label="supp:depro"}](figA4b.pdf "fig:"){width="30.00000%"} ![Comparison between polar deprojection of the ALMA observation (a) and synthetic predictions from hydrodynamic models after 150 orbits of the mini-Neptune. Panel b shows the result of the hydrodynamic simulations for a single mini-Neptune (same as Fig. \[fig:model\]), while panel c shows the mini-Neptune’s outer rings under the influence of an inner giant planet at 38 [au]{}. All panels have linear color stretch between 0.0 and 0.25 mJy/beam. []{data-label="supp:depro"}](figA4c.pdf "fig:"){width="30.00000%"}
ALMA Partnership, Brogan, C. L., P[é]{}rez, L. M., et al. 2015, , 808, L3. Andrews, S. M., Wilner, D. J., Zhu, Z., et al. 2016, , 820, L40. Andrews, S. M., Huang, J., P[é]{}rez, L. M., et al. 2018, , 869, L41. Bae, J., Zhu, Z., & Hartmann, L. 2017, , 850, 201. Bae, J., Pinilla, P., & Birnstiel, T. 2018, , 864, L26. Baruteau, C., Barraza, M., P[é]{}rez, S., et al. 2019, , Boss, A. P. 1997, Science, 276, 1836. Banzatti, A., Pinilla, P., Ricci, L., et al. 2015, , 815, L15. Baruteau, C., & Zhu, Z. 2016, , 458, 3927. Baruteau, C., & Masset, F. 2008, , 672, 1054. Baruteau, C., & Masset, F. 2008, , 678, 483. Ben[í]{}tez-Llambay, P., & Pessah, M. E. 2018, , 855, L28. Bertrang, G. H.-M., Avenhaus, H., Casassus, S., et al. 2018, , 474, 5105. Biller, B. A., Males, J., Rodigas, T., et al. 2014, , 792, L22. C[á]{}rcamo, M., Rom[á]{}n, P. E., Casassus, S., et al. 2018, Astronomy and Computing, 22, 16. Charnoz, S., Fouchet, L., Aleon, J., et al. 2011, , 737, 33. Chen, L., K[ó]{}sp[á]{}l, [Á]{}., [Á]{}brah[á]{}m, P., et al. 2018, , 609, A45. Clarke, C. J., Tazzari, M., Juhasz, A., et al. 2018, , 866, L6. Crida, A., Morbidelli, A., & Masset, F. 2006, , 181, 587. Dipierro, G., Price, D., Laibe, G., et al. 2015, , 453, L73. Dipierro, G., Laibe, G., Price, D. J., et al. 2016, , 459, L1. Dipierro, G., Laibe, G., Alexander, R., et al. 2018, , 479, 4187. Dong, R., Zhu, Z., & Whitney, B. 2015, , 809, 93. Dong, R., Li, S., Chiang, E., et al. 2017, , 843, 127. Dong, R., & Fung, J. 2017, , 835, 146. Dong, R., Liu, S.-. yuan ., Eisner, J., et al. 2018, , 860, 124. Dong, R., Li, S., Chiang, E., et al. 2018, , 866, 110. Dullemond, C. P., Juhasz, A., Pohl, A., et al. 2012, RADMC-3D: A multi-purpose radiative transfer tool, ascl:1202.015. Dullemond, C. P., Birnstiel, T., Huang, J., et al. 2018, , 869, L46. Fedele, D., Carney, M., Hogerheijde, M. R., et al. 2017, , 600, A72. Flock, M., Ruge, J. P., Dzyurkevich, N., et al. 2015, , 574, A68. A. W. Fitzgibbon, M. Pilu, R. B. Fisher, [*Proceedings of 13th International Conference on Pattern Recognition*]{} (1996), vol. 1, pp. 253–257 vol.1. Fuente, A., Baruteau, C., Neri, R., et al. 2017, , 846, L3. Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, , 616, A1. Goldreich, P., & Tremaine, S. 1980, , 241, 425. Goldreich, P., Lithwick, Y., & Sari, R. 2004, Annual Review of Astronomy and Astrophysics, 42, 549. Gonzalez, J.-F., Pinte, C., Maddison, S. T., et al. 2012, , 547, A58. Gonzalez, J.-F., Laibe, G., & Maddison, S. T. 2017, , 467, 1984. Goodman, J., & Rafikov, R. R. 2001, , 552, 793. Grady, C. A., Schneider, G., Hamaguchi, K., et al. 2007, , 665, 1391. Gratton, R., Ligi, R., Sissa, E., et al. 2019, arXiv e-prints, arXiv:1901.06555. Honda, M., Maaskant, K., Okamoto, Y. K., et al. 2012, , 752, 143. Huang, J., Andrews, S. M., Dullemond, C. P., et al. 2018, , 869, L42 Kley, W., & Dirksen, G. 2006, , 447, 369. Kratter, K. M., Murray-Clay, R. A., & Youdin, A. N. 2010, , 710, 1375. Lambrechts, M., & Johansen, A. 2012, , 544, A32. Ligi, R., Vigan, A., Gratton, R., et al. 2018, , 473, 1774. Long, F., Pinilla, P., Herczeg, G. J., et al. 2018, , 869, 17. Mac[í]{}as, E., Espaillat, C. C., Ribas, [Á]{}., et al. 2018, , 865, 37. Masset, F. 2000, Astronomy and Astrophysics Supplement Series, 141, 165. Meru, F., Rosotti, G. P., Booth, R. A., et al. 2019, , 482, 3678. McMullin, J. P., Waters, B., Schiebel, D., et al. 2007, Astronomical Data Analysis Software and Systems XVI, 127. Mentiplay, D., Price, D. J., & Pinte, C. 2018, , L210. Miranda, R., Li, H., Li, S., et al. 2017, , 835, 118. Momose, M., Morita, A., Fukagawa, M., et al. 2015, Publications of the Astronomical Society of Japan, 67, 83. Monnier, J. D., Harries, T. J., Aarnio, A., et al. 2017, , 838, 20. Okuzumi, S., Momose, M., Sirono, S.-. iti ., et al. 2016, , 821, 82. Ormel, C. W., & Klahr, H. H. 2010, , 520, A43. Osorio, M., Anglada, G., Carrasco-Gonz[á]{}lez, C., et al. 2014, , 791, L36. Pinilla, P., Benisty, M., & Birnstiel, T. 2012, , 545, A81. Pohl, A., Benisty, M., Pinilla, P., et al. 2017, , 850, 52. Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, , 124, 62. Quanz, S. P., Avenhaus, H., Buenzli, E., et al. 2013, , 766, L2. Reggiani, M., Quanz, S. P., Meyer, M. R., et al. 2014, , 792, L23. Rosotti, G. P., Juhasz, A., Booth, R. A., et al. 2016, , 459, 2790. Suriano, S. S., Li, Z.-Y., Krasnopolsky, R., et al. 2018, , 477, 1239. van der Marel, N., Dong, R., di Francesco, J., et al. 2019, arXiv e-prints , arXiv:1901.03680. Yang, C.-C., Mac Low, M.-M., & Johansen, A. 2018, , 868, 27 Zhang, K., Blake, G. A., & Bergin, E. A. 2015, , 806, L7. Zhang, S., Zhu, Z., Huang, J., et al. 2018, , 869, L47.
[^1]: The [ gpuvmem]{} code is publicly available and can be found here: [ https://github.com/miguelcarcamov/gpuvmem]{}.
[^2]: The FARGO-ADSG code is publicly available at the address [ http://fargo.in2p3.fr/-FARGO-ADSG-]{}. The version including a Lagrangian treatment of the dust particles, Dusty FARGO-ADSG, can be made available upon request to co-author Clement Baruteau.
[^3]: The stellar mass of HD169142 is closer to $M_\star=1.7~M_\odot$, which yields a planet mass of 9.6$M_{\oplus}$ in our simulation. Hence we round the mini-Neptune’s mass off to $\sim$10$M_\oplus$ throughout the paper.
|
Zinc oxide (ZnO) is one of the most promising semiconductors for the next generation of electronic and optelectronic devices. It has already been applied to transducers, phosphors and varistors, due to its unique piezoelectric, optical, and electrical properties. In these applications, polycrystalline material has mainly been used. Moreover, recent progress in single crystal growth [@dcl:98] has opened up new possibilities, like bright blue and uv light emitters. Optical uv lasing has already been observed even at room temperature [@dmb:97].
For applications to optoelectrical devices, it is crucial to control the bulk electronic conductivity of crystalline ZnO. However, it is notoriously difficult to obtain intrinsic ZnO, ending up with materials showing strong $n$ type conductivity. In spite of more than 20 years of investigations, the origin of this unintentional carrier doping is still controversial. It has long been speculated that the dominant donor is a native defect, either oxygen vacancy, or zinc interstitial [@ghe:90; @dcl:99]. Unfortunately, recent theoretical investigations have revealed that none of those native defects behave as shallow donors [@afk:00].
Recently, it was theoretically pointed out that hydrogen (H), which is quite difficult to remove from the crystal growth environment, is an excellent candidate for such a shallow donor[@cgv:00]. As shown in Fig. 1, ZnO crystallizes in the wurtzite structure corresponding to an elongated zinc blend structure with hexagonal symmetry around the \[0001\] axis. The lattice parameters are known by experiments as $a=0.325$ nm, $c/a=1.602$ and $u=0.382$ in normalized coordinates. From a first-principle calculation, the lowest energy configurations for hydrogen are predicted to be at the BC$_{\perp}$ site, with a nearly equivalent formation energy for the BC$_{\parallel}$, AB$_{O, \perp}$, and AB$_{O, \parallel}$ sites [@cgv:00]. Experimental evidence for this scenario has been claimed in several reports [@emo:54; @dgt:56; @jil:57; @sjb:97; @kwk:94], where an increase in the conductivity was observed upon introducing H into ZnO.
In this Letter we report on a determination of the electronic structure and the location of a muonium (Mu, an analogue of isolated hydrogen whose proton is substituted by a positive muon) as a shallow donor in ZnO. By using single-crystalline ZnO, two species of muonium have been clearly distinguished. The muonium center is readily observed in a wide variety of semiconductors after positive muon implantation, and has been serving as a unique source of information on the electronic structure of [*isolated*]{} hydrogen centers [@bdp:88]. While the dynamical aspect (e.g., diffusion property) may be considerably different between Mu and H due to the light mass of Mu ($\simeq\frac{1}{9} m_p$), the local electronic structure of Mu is virtually equivalent to that of H after a small correction due to the difference in the reduced mass ($\sim4$%). It is now well established in elemental and III-V compound semiconductors that there are two stable (and metastable) sites, one at the center of the matrix bond (i.e., BC-site, Mu$_{BC}^{0}$) with a large outward relaxation of the nearest-neighbor (nn) host atoms, and the other around the center of a tetrahedron cage (i.e., $T_d$ site, Mu$_{T}^{0}$). While Mu$_{T}^{0}$ has a large isotropic hyperfine parameter (almost the same order of the vacuum value, $A_\mu=4463$ MHz), the hyperfine parameter of Mu$_{BC}^{0}$ has a value about one order of magnitude smaller with a large unpaired spin density distributed on the nn host atoms. Recently, a novel muonium state having an extremely small hyperfine parameter ($10^{-4}\times A_\mu$) has been reported in a II-VI compound semiconductor, CdS [@jmg:99], suggesting that such a shallow Mu center (and H center as well) might be present in ZnO to serve as a donor.
The experiment was performed at the Meson Science Laboratory (located in KEK) which provides a pulsed (50 ns pulse width and 20 Hz repetition) beam of 100% spin-polarized muons with a beam energy of 4 MeV. The muon beam with longitudinal polarization was implanted into a single-crystalline wafer (40 mm diameter, 0.5mm thickness, \[0001\] orientation) of ZnO obtained from Eagle-Picher Industries, Inc. The conventional time differential muon spin rotation ([$\mu$SR]{}) measurements were performed under a magnetic field applied in two different orientations: one in the transverse direction (TF, $\vec{B}$ in Fig. 1) and the other in a tilted direction ($\vec{B}'$) with respect to the initial muon spin polarization $\vec{P}_\mu$. To obtain the tilted field, both transverse and longitudinal (LF) magnetic fields were applied simultaneously. In the case of hyperfine parameter measurements, the specimen was placed on a cold finger with the \[0001\] axis parallel with $\vec{P}_\mu$ (O-face up). As shown in Fig. 1, the \[11$\bar{2}$0\] axis was set either perpendicular to $\vec{B}$ (Fig. 1a) or parallel with $\vec{B}$ (Fig. 1b) to examine the angular dependence of the hyperfine constants. For the temperature dependence measurements of the muonium fraction, the \[0001\] axis was tilted by 45$^\circ$ to $\vec{P}_\mu$ (i.e.,$\vec{B}\angle[0001]=45^\circ$) while $\vec{B}\perp[11\bar{2}0]$.
It has been inferred from TF (=2.00 mT, 4.00 mT and 30.0 mT) measurements that only a single diamagnetic muon state is present above 40 K. The relaxation rate is almost independent of temperature with a rate of $\simeq0.022(6)$ $\mu$s$^{-1}$ for Gaussian damping, which is consistent with the dipole-dipole interaction of muons with $^{67}$Zn nuclei (natural abundance 4.1%). On the other hand, the muon spin rotation signal changes drastically below 40 K. A typical [$\mu$SR]{} spectrum is shown in Fig. 2 with fit errors, where the data were obtained under 30.0 mT with $\vec{B}\angle[0001]=45^\circ$. Fig. 3 shows the temperature dependence of the FFT spectrum, in which two pairs of satellite lines are seen with their position situated symmetrically around the central line corresponding to the precession of diamagnetic muons (with the gyromagnetic ratio $\gamma_\mu=2\pi\times135.53$ MHz/T) at 5 K. The splitting of these satellites remained unchanged when the applied field was changed to 2.00 mT or 4.00mT. Moreover, a nearly equivalent frequency spectrum was observed when the specimen was rotated by 90$^\circ$ around the \[0001\] axis (i.e., between (a) and (b) in Fig. 1). These results strongly suggest that two muonium centers with extremely small anisotropic hyperfine parameters exist in ZnO. The hyperfine parameters are about $10^{-4}$ times smaller than the vacuum value and they are symmetric to the \[0001\] axis.
Provided that the hyperfine interaction has an axial symmetry, we expect two muonium precession signals for the high-field limits with frequencies $$\begin{aligned}
\nu_-(\theta)&\simeq&\nu_0-\frac{1}{2}\Delta\nu(\theta),\\
\nu_+(\theta)&\simeq&\nu_0+\frac{1}{2}\Delta\nu(\theta),\\
\Delta\nu(\theta) &=& A(\theta) = \mid A_{\parallel}\cos^2(\theta) +
A_{\perp}\sin^2(\theta) \mid,\end{aligned}$$ where $2\pi\nu_0=\gamma_\mu B$ with $B=|\vec{B}|$ being the applied field ($B\gg 2\pi A/\gamma_e$, where $\gamma_e=2\pi\times28.024$ GHz/T is the gyromagnetic ratio of electron), $\theta$ is the angle between $\vec{B}$ and the symmetry axis \[0001\], and $ A_{\parallel}$ and $ A_{\perp}$ are the hyperfine parameters parallel and normal to \[0001\], respectively. It was revealed upon preliminary analysis that the fitting of [$\mu$SR]{} time spectra assuming the three frequency components ($\nu_0$, $\nu_-$, and $\nu_+$) did not reproduce the time spectrum, yielding a large fraction ($\sim20$ %) of fit errors and poor reduced $\chi^2$ ($\simeq2.40$). On the other hand, as suggested in Fig. 3a, fitting analysis with two sets of satellites including five components ($\nu_0$, $\nu_{i-}$, and $\nu_{i+}$, $i=1,2$) turned out to yield a satisfactory result with drastically improved $\chi^2$ ($\simeq1.52$). This indicates that there are two species of Mu centers with respective fractional yields in this compound. From the spectrum with $\vec{B}\perp[11\bar{2}0]$ (Fig. 1a), the hyperfine parameters are deduced to be $$\begin{aligned}
A _{1} (90^\circ) &=& \mid A_{1\perp}\mid = 358(4)\:{\rm kHz}, \\
A _{2} (90^\circ) &=& \mid A_{2\perp}\mid = 150(4)\:{\rm kHz}.\end{aligned}$$ Combining this result with the data under a tilted field $\vec{B}'$ (where $\theta
=54.0^{\circ}, \Delta\nu_{1} = 495(2)$kHz, $\Delta\nu_{2} = 298(4)$kHz), the rest of the hyperfine parameters are deduced as $$\begin{aligned}
\mid A_{1\:\parallel}\mid &=& 756(13)\:{\rm kHz}, \\
\mid A_{2\:\parallel}\mid &=& 579(19)\:{\rm kHz}.\end{aligned}$$ As shown in TABLE I, the angular dependence of the frequencies ($\Delta\nu_{i}$ ) calculated by the above parameters is in excellent agreement with the experimental observation.
The possibility that these Mu centers have a hyperfine tensor with the symmetry axis parallel to BC$_{\perp}$ is eliminated by the fact that the observed precession frequency is independent of the rotation of the crystal around the \[0001\] axis by 90$^\circ$. Another attempt to explain this by resorting to a sufficiently small anisotropy with the symmetry axis parallel to BC$_{\perp}$ fails to account for the difference between $A_{\parallel}$ and $A_{\perp}$ consistently with the data. Thus, we conclude that there are two species of Mu centers, both of which have axially symmetric hyperfine structure along with the \[0001\] axis. Hereafter, we denote these two centers as Mu$_{I}$ and Mu$_{II}$ with the corresponding hyperfine parameters, $A_{1}(\theta)$ and $A_{2}(\theta)$, respectively. The static dielectric constants in ZnO are reported to be 7.8(3) for perpendicular and 8.75(40) for parallel to the \[0001\] axis [@dvd;01]. The degree of obtained anisotropy for the muonium hyperfine tensor ($\sim$50%) is much larger than that of the dielectric constant ($\sim$10%), indicating that the anisotropy is determined by the local electronic structure with the BC$_{\parallel}$ and AB$_{O,\parallel}$ sites (see Fig. 1) being the most probable candidates for the sites of those Mu centers. Considering the magnitude of anisotropy in the hyperfine tensors, it would be reasonable to presume that Mu$_{I}$ is located at the AB$_{O, \parallel}$ site and Mu$_{II}$ at the BC$_{\parallel}$ site. Let us compare our results to a simple model of shallow level centers in a dielectric medium. In this model, the hyperfine parameter is inversely proportional to the cube of the Bohr radius ($a_d$) of the bound electrons. The isotropic part of the hyperfine parameter, $A_{\rm iso}$ ($=\frac{1}{3}A_{\parallel}+\frac{2}{3}A_{\perp}$), is 491 kHz for Mu$_{I}$ and 293 kHz for Mu$_{II}$. Compared with $A_\mu=4463$ MHz, one obtains $a_d=21a_{0}=1.1$ nm for Mu$_{I}$ and $a_d=25a_{0}=1.3$ nm for Mu$_{II}$ (where $a_0$ is the Bohr radius of the free Mu). On the other hand, the Bohr radius for a hydrogen-like defect is calculated from the average dielectric constant, $\epsilon=8.12$, and the electron effective mass, $m^{\star}=0.318m_{e}$, of ZnO [@dlr:75], i.e. $a_d =(\epsilon/m_{e}/m^{\star})a_0=25.5a_0$. This value is qualitatively in good accord with those of Mu$_{I}$ and Mu$_{II}$.
The temperature dependence of the amplitudes of Mu$_{I}$, Mu$_{II}$, and diamagnetic muon are plotted in Fig. 4. The total yield of all states are almost independent of temperature, suggesting that Mu$_{I}$ and Mu$_{II}$ are ionized to a diamagnetic muon above the transition temperature($\sim40$ K). It is unlikely that these Mu energy levels are just above the valence band. Otherwise, the temperature dependence of the muonium charge state would not be expected due to the $n$ type conductivity of the present specimen where the Fermi level is much higher than the mid-gap level. These results indicate that the Mu centers act as shallow level donors. Thus, since Mu centers simulate the electronic structure of H in ZnO, our result provides convincing evidence that the hydrogen centers in ZnO are shallow donors, leading to $n$ type conductivity in ZnO.
The activation energies of Mu$_{I}$ and Mu$_{II}$ were obtained to be 3 meV and 25 meV, respectively from the data in Fig. 4. According to the analysis in , the relation $E_d=2E_a$ is satisfied between the defect level energy ($E_d$) and the activation energy ($E_a$), which leads to the respective defect level energies of Mu$_{I}$ and Mu$_{II}$ to be 6 meV and 50 meV. The latter is fairly consistent with the calculated value of the hydrogen-like impurity model, $13.6(m^*/m_e/\epsilon^2)=66$ meV, and the observed value of 61meV attributed to H in an earlier report[@dcl:98]. Considering the large ambiguity in determining the defect level energy for another donor at 31 meV which has a much lower concentration in , Mu$_{I}$ may correspond to this shallower donor.
The reason for the absence of Mu centers at other interstitial sites is yet to be understood. Another issue is that a large fraction of diamagnetic muons (about 50%) exists even at the lowest temperature. One of the possibilities is that muon-oxygen bounding is formed, which has been commonly observed in various oxides. The other is that the diamagnetic centers may correspond to those at the BC$_\perp$ or AB$_\perp$ sites, where their defect energy levels are in the conduction band and/or their hyperfine parameters are too small to observe in our experiment. Further experiments, including [$\mu$SR]{} measurements at different geometry, would be helpful to address these issues. Meanwhile, more accurate theoretical investigations are strongly required to unambiguously identify the observed Mu centers.
In summary, we have demonstrated that two species of muonium centers are formed in ZnO below 40 K with extremely small hyperfine parameters. These centers have an axially symmetric hyperfine interaction around the \[0001\] axis. The temperature dependence of their fractional yields indicates that they act as shallow donors, strongly suggesting that hydrogen is the primary origin of unintentional $n$ type conductivity in ZnO.
We would like to thank the staff of KEK-MSL for their technical support. In particular, special thanks are due to K. Nagamine for his continuous encouragement and support for this work. We also appreciate helpful discussions with S. Tsuneyuki on theoretical aspects of this work and communications with G. Cantwell, D.C. Look and D. Eason on the bulk property of the ZnO specimen.
Note added: After submission of this manuscript, the presence of a muonium state in the powder sample of ZnO was reported by a separate group[@cox:01].
Also at School of Mathematical and Physical Science, The Graduate University for Advanced Studies
D. C. Look, D. C.Reynolds, J. R.Sizelove, C. W.Litton, G. Cantwell and W. C. Harsch, Solid State commun. [**105**]{}, 399 (1998).
D. M. Bagnall, Y. F. Chen, Z. Zhu, T. Yao, S. Koyama, M. Y. Shen and T. Goto, Appl. Phys. Lett. [**70**]{}, 2230 (1997).
For a review, G. Heiland, E. Mollwo and F. Stöckmann, in Solid State Physics (Academic, New York. 1959), vol. [**8**]{}. p.191.
D. C. Look, J. W. Hemsky and J. R. Sizelove, Phys. Rev. Lett. [**82**]{}, 2552 (1999).
A. F. Kohan, G. Ceder, D. Morgan and C. G. Van de Walle, Phys. Rev. B [**61**]{}, 15019(2000).
C. G. Van de Walle, Phys. Rev. Lett. [**85**]{}, 1012 (2000).
E. Mollwo, Z.Phys. [**138**]{}, 478 (1954).
D. G. Thomas and J. J. Lander, J. Chem. Phys. [**25**]{}, 1136 (1956).
J. I. Lander, Chem. Solids [**3**]{}, 87 (1957).
S. J. Baik, J. H. Jang, C. H. Lee, W. Y. Cho and K. S. Lim, Appl. Phys. Lett. [**70**]{}, 3516 (1997).
S. Kohiki, M. Nishitani, T. Wada and T. Hirao, Appl. Phys. Lett. [**64**]{}, 2876(1994).
For reviews, B.D.Patterson, Rev. Mod. Phys. [**60**]{},69(1988); R.F.Kiefl and T.L.Estle, in Hydrogen in Semiconductors (Academic,Sandiego,1991).
J. M. Gil, H. V. Alberto, R.C. Viläo, J. Piroto Duarte, P. J. Mendes, L. P.Foreira, N. Ayres de Campos, A. Weidinger, J. Krause, Ch. Niedermayer and S.F.J. Cox, Phys. Rev. Lett. [**83**]{}, 5294 (1999).
D. Eason (Eagle-Pichier Technologies) ; private communications.
D. L. Rode, Semiconductors and Semimetals, [**10**]{}, 1(1975).
S.F.J.Cox, E.A.Davis, S.P.Cottrell, P.J.C.King, J.S.Lord, J.M.Gil, H.V.alberto, R.C.Viläo, J. Piroto Duarte, N. Ayres de Campos, A. Weidinger, R.L. Lichti and S.J.C. Irvine, Phys. Rev. Lett. [**86**]{}, 2601 (2001).
configuration $\theta$ $\Delta\nu_{1cal}$ $\Delta\nu_{1exp}$ $ \Delta\nu_{2cal}$ $\Delta\nu_{2exp}$
---------------------------------------- ---------------- -------------------- -------------------- --------------------- --------------------
$B\perp [11\bar{2}0]$ (Fig.1a) $90.0^{\circ}$ 358(4) 358(4) 150(4) 150(4) kHz
$B\perp [11\bar{2}0]$ (Fig.1a) $54.0^{\circ}$ 495(7) 495(2) 298(8) 298(4) kHz
$B\parallel [11\bar{2}0]$ (Fig.1b) $90.0^{\circ}$ 358(4) 356(2) 150(4) 153(4) kHz
$B\angle [11\bar{2}0]=42.4^\circ$ (1b) $47.6^{\circ}$ 539(8) 541(4) 345(9) 350(4) kHz
: Hyperfine splitting of muonium centers in ZnO, where the measured values are compared with those calculated from a particular set of hyperfine parameters.
|
---
abstract: 'One important feature of the mammalian immune system is the highly specific binding of antigens to antibodies. Antibodies generated in response to one infection may also provide some level of cross immunity to other infections. One model to describe this cross immunity is the notion of antigenic space, which assigns each antibody and each virus a point in $\mathbb{R}^n$. Past studies have suggested the dimensionality of antigenic space, $n$, may be small. In this study we show that data from hemagglutination assays suggest a high dimensional random walk (or self avoiding random random walk). The discrepancy between our result and prior studies is due to the fact that random walks can appear low dimensional according to a variety of analyses. including principal component analysis (PCA) and multidimensional scaling (MDS).'
author:
- James Moore
- Hasan Ahmed
bibliography:
- 'PCA.bib'
title: 'High dimensional random walks can appear low dimensional: application to influenza H3N2 evolution'
---
Introduction
============
Antigenic Space
---------------
During a viral infection, antibodies bind to viral antigens by recognizing specific epitopes on their surface. The same antibody may provide protection against other strains of the same virus if the antigens are not too dissimilar. In 1979, Alan Perelson and George Oster defined the idea of antigenic space [@perelson1979theoretical]. They supposed that each antibody and antigen might be described by a vector in $\mathbb{R}^n$. This idea was used as a basis to study the dynamics of the immune response, with $n$ assumed to be a small number [@segel1989shape; @de1992pattern]. Subsequent work to estimate $n$ based on the frequency of cross reactivity resulted in the conclusion that $n$ was around five to eight [@smith1997deriving].
The notion of antigenic space has proven particularly popular for understanding the evolution of influenza H3N2 [@fonville2014antibody]. This strain has been circulating in the human population since 1968 and gradually mutating. These mutations can in principle be represented as the movement of the virus through antigenic space. As the antigen moves it can evade the antibodies elicited by older strains and thus reinfect individuals. The distance between a viral strain and an antibody can be measured via the hemagglutination inhibition (HAI) assay, in which a viral strain and a serum of antibodies are both added to a culture of red blood cells. If the antibodies are ineffective against the viral strain then the virions stick to the red blood cells causing them to cluster together (hemagglutinate). However if the antibodies are effective, they will neutralize the virions and inhibit their hemagglutination of the red blood cells. In the former case, the strain and the serum are distant antigenically, whereas in the latter case they are close. By performing serial dilutions of the antibody serum, one can quantify just how close a serum and antibody are.
Points in antigenic space can be inferred from a distance matrix via multidimensional scaling (MDS). Low dimensional reconstructions of antigenic space can reproduce the HAI data with high fidelity, and adding new dimensions beyond $n=5$ does not improve the quality of the fit [@lapedes2001geometry; @smith2004mapping]. Thus it may be tempting to conclude that influenza is evolving in an antigenic space of no more than five dimensions.
Outline of results
------------------
In this work we will argue that influenza H3N2 is evolving in a very high dimensional space, and that it may appear to be low dimensional due to the nature of random walks. Our argument consists of three parts.
1. High dimensional random walks contain most of their variance in a small number of dimensions. Specifically, one would expect at least 60% of the variance to occur in a single dimension. We show this via principal component analysis. However, we note that the true dimensionality of random walks can be revealed by out of sample PCA.
2. This apparent low dimensionality also occurs with multidimensional scaling — the method used to analyze HAI titer data. We simulate HAI data generated using a high dimensional random walk and find we can accurately reproduce the data with points taken from a low dimensional space; increasing the dimensionality of our representation does not improve our fit. We also show that non-metric MDS can be used to reconstruct an infinite dimensional random walk in a single dimension.
3. Finally, we show that H3N2 data has characteristics of a high dimensional random walk, indicating that it was unlikely to result from a random walk of dimensionality less than $n=10$. This is even the case when we consider that the random walk of H3N2 is likely self avoiding.
Why a random walk?
------------------
Throughout this paper we argue for a high dimensional random walk as a model for influenza evolution. A random walk may seem *a priori* to be a poor model for viral evolution, as immunological memory should prevent a virus from revisiting areas of antigenic space. Therefore we should expect the path of viral evolution to be self avoiding. In high dimensions an unbiased random walk and self avoiding random walk will behave very similarly, because a high dimensional random walk is already extremely unlikely to cross itself. In an $n$ dimensional random walk the distance between points $i$ and $j$ is a random variable $D_{ij}$. Its distribution is $$\begin{aligned}
D_{ij}^2\sim\alpha |i-j|\chi^2_n\end{aligned}$$ where $\alpha$ is a constant of proportionality. This means that for large $n$ the distances increase in a very predictable manner as the $\chi_n^2$ distribution narrows. The probability of the random walk approaching a previous point is essentially zero, so we need not include any further tendency for self avoidance. However, in the latter part of the paper we will address the question as to whether low dimensional self avoiding random walk could also be consistent with the data.
True dimensionality vs effective dimensionality
-----------------------------------------------
Let $x_i \in \mathbb{R}^n$ represent distinct viral strains and/or antisera. The antigenic dissimilarity of the two different strains $x_i$ and $x_j$ is the euclidean distance $D_{ij}=\|x_i-x_j\|_2$. $D_{ij}$ represents the true distance between these two strains and $n$ is the true dimensionality of antigen space.
If we can represent each point $x_i$ with a corresponding point $y_i \in\mathbb{R}^k$ such that the distances from $y_i$ to $y_j$ is approximately $D_{ij}$, then we say that the effective dimensionality of $\{x_i\}$ is $k$.
There are several possible ways to obtain the points $y_i$ and to evaluate how closely the reconstructed distances match the true or measured distances. In this paper we shall make use of three: classical MDS, metric MDS and nonmetric MDS. Classical MDS is also commonly known as principal component analysis (PCA). The points $y_i$ are found by orthogonally projecting the points $x_i$ onto the dimensions that contain the most variance. Metric MDS simply minimizes the residual between the true distances and the reconstructed distances. Non-metric MDS is similar to metric MDS but the relationship between the true distances and the reconstructed distances is only assumed to be monotonically increasing as opposed to directly proportional.
Results
=======
Truly high dimensional random walks have low effective dimensionality according to PCA
--------------------------------------------------------------------------------------
We will now show that random walks with high true dimension can have a very low effective dimension. Our analysis will focus on infinite dimensional random walks, but we will show via simulations that finite random walks have similar behavior. Let $x_i\in\mathbb{R}^n$ be the $i$th step in our random walk in $n$ dimensions. Let $X$ be an $m$ by $n$ matrix whose $i$th row is $x_i$. The rows of $x_i$ are determined by a seres of random steps $b_i \in \mathbb{R}^n$, each entry of which is an independent sample from $\mathcal{N}(0,1/\sqrt{n})$ $$\begin{aligned}
x_1&=b_1\\
x_{i}-x_{i-1}&=b_i \quad i>1\end{aligned}$$ We can state this relationship compactly as $LX=B$, where the rows of $B$ are $b_i$ and $L$ is an $m$ by $m$ matrix with one on the diagonal and negative one on the subdiagonal.
To perform PCA, we first center each column of $X$ by premultiplying be the projection matrix $P=I-{\mathbf{1}}^T{\mathbf{1}}$. Note that centering each column is just a translation of the data, so all pairwise distances are preserved. We then compute the eigenvalues of $A=PXX^{T}P$, $\lambda_1>\lambda_2>\cdots \lambda_m$. These eigenvalues indicate the variance in each principal component of $X$. For this section we shall define effective dimension as $$\begin{aligned}
\quad \text{ED}_a=\min \left\{d:\sum_{i=1}^k \lambda_k\geq a~ \text{tr}(A)\right\}
\label{qeqn}\end{aligned}$$ where $0<a<1$ is the fraction of variance that we require within the first $k$ dimensions.
The random matrix $A=PL^{-1}BB^{T}L^{-T}P$ is full rank and has a wishart distribution if the number of steps is no greater than the number of dimensions, i.e. $m \leq n$. If number of steps is greater than the number of dimension ($n>m$), $A$ is singular and has a pseudo-wishart distribution.
### Infinite dimensional random walks
The matrix $W=BB^T$ is an uncorrelated Wishart matrix. The joint probability distribution of the eigenvalues of Wishart matrices is known, but the formula is cumbersome [@johnstone2001distribution; @kang2003largest; @zanella2009marginal]. Therefore we will focus our analysis on the limiting behavior as $n\rightarrow \infty$, i.e. infinite dimensional random walks. As $n$ increases, $W$ approaches the $m$ by $m$ identity matrix. (To see this, either note that the $i,j$th entry of $W$ is $<b_i,b_j>$). Using this simplification, the matrix $A$ approaches $$\begin{aligned}
A=PL^{-1}L^{-T}P\end{aligned}$$ Noting that $L^{-1}$ is an upper triangular matrix with all ones, we can calculate $tr(A)=(m-1)(m+1)/6$.
We can first observe that this matrix has one eigenvalue $v={\mathbf{1}}$ with corresponding eigenvalue $\lambda_1=0$. To find the remaining eigenvalues we use the pseudo-inverse of $A$, ${A^{\dagger}}$ which has the simple form $$\begin{aligned}
{A^{\dagger}}=
\begin{pmatrix}
1 & -1 & 0 &\cdots & 0\\
-1 & 2 & -1 & \ddots &\vdots\\
0 & \ddots & \ddots & \ddots & 0\\
\vdots & \ddots & -1 &2 & -1\\
0 & \cdots &0& -1 & 1
\end{pmatrix}\end{aligned}$$ The eigenvalues of this matrix are $\mu_s=2-2\cos(\pi s/m)$. Therefore the eigenvalues of $A$ are $$\begin{aligned}
\lambda_k=1/(2-2\cos(\pi s/m))\end{aligned}$$
The fraction of the variance in the first $k$ dimensions is $$\begin{aligned}
\frac{\sum_{s=1}^k \lambda_s}{\text{tr}(A)}&=\frac{\sum_{s=1}^k (1-\cos(\pi s/m))^{-1}}{(m-1)(m+1)} &\text{Finite $m$}\\
&=\frac{6}{\pi^2}\sum_{s=1}^k 1/s^2 &\text{Infinite $m$}\end{aligned}$$ For infinite dimensional random walks, the first principal component contains at least $6/\pi^2$ or roughly 60% of the total variance. The first two components contain 80% of the variance, but the subsequent convergence is slow as 12 components are needed to account for 95% of the variance (Fig. \[fig1\]A,B). These numbers are the limiting behavior as the length of the random walk grows. For shorter walks, more of the variance is in the first few components (Fig. \[fig1\]C, E) and thus the effective dimensionality may be even lower.
{width="\textwidth"} \[fig1\]
### Finite dimensional random walks
In a finite number of dimensions $n$, the matrix $A$ is a random variable with a correlated Wishart or Pseudo-Wishart distribution. We therefore use simulation to investigate the effective dimensionality. As the true dimensionality increases, the effective dimensionality asymptotically approaches the value for the infinite case, in which 12 dimensions contain 95% of the variance. However, the effective dimensionality for finite random walks is stochastic and in a few cases we find that it may exceed 12, although the mean is lower (Fig \[fig1\]A).
Like the effective dimensionality, the percentage of variance in the first principal component is a random variable, $\Psi_1$. Regardless of the length of the walk or the true dimensionality, the mean of $\Psi_1$ is very similar to the infinite dimensional case, and the variance of $\Psi_1$ decreases with the true dimensionality but is independent on the length of the walk (Fig \[fig1\]C,D). This gives the somewhat counter intuitive result that lower dimensional random walks may have less variance in their first principal component than higher dimensional walks. When we consider the percentage of variance in the first 5 components $\Psi_5$, this pattern disappears (Fig. \[fig1\]E,F). The values of $\Psi_5$ tend to decrease monotonically as both the true dimensionality and the number of steps increase, asymptotically approaching the prediction of roughly 89% from the infinite dimensional case.
### True dimensionality of random walks can be revealed by subsampling
We have demonstrated that when we perform PCA on a random walk, we find most of the variance in only one dimensoin. Therefore, it is be difficult to tell whether the data originate from a high or low dimensional random walk using PCA. In this section we demonstrate a way around this problem using out of sample PCA.
We consider two scenarios: a uniform random walk in which movement in every direction is equally likely, and a nonuniform random walk in which movement is ten times greater — and thus the variance is 100 times greater — in 14 dimensions compared to the other 86. PCA finds the majority of variance is in the first dimension for both of these scenarios (Fig \[fig5\]). However if we take the projection matrix derived from performing PCA on the first half of the random walk and apply it to the second half, we see that the bias towards the first dimension largely disappears. The approximation improves further if we divide the second half into subsequences, apply the projection to each subsequence and then average the results (see methods for further details). This latter method shows clearly that in the uniform random walk, each principal component has roughly the same variance. It also captures the fact that 14 dimensions contain significantly more variance in the nonuniform random walk. Therefore out of sample PCA can in principle allow one to recover the true dimensionality of a random walk.
{width="\textwidth"} \[fig5\]
Truly high dimensional random walks look low dimensional according to MDS
-------------------------------------------------------------------------
### H3N2 evolution
Derek Smith, Alan Lapedes and colleagues have used dimensional reduction techniques to reconstruct two-dimensional antigenic maps of the evolution of Influenza H3N2 [@smith2004mapping]. In these maps, some points represent viruses whereas others represent antisera. Distances can only be computed between a virus and an antisera as opposed to between different viruses directly. Thus classic MDS, which requires measured distances between all pairs of points, cannot be used for this problem. Instead, they use metric MDS with a slight modification to handle sub-threshold values. Using this method, they have reconstructed a two-dimensional map of antigenic space. To validate this map, they challenged it to predict the antigenic distance between strains and antisera that were not used as inputs to build the map. They report a good correlation (roughly 80%) between the predicted and actual values, and furthermore that the accuracy of their predictions is similar for a two dimensional map and for higher dimensions. This is consistent with prior work showing that no more than five dimensions is required to satisfactorily reconstruct HAI data [@lapedes2001geometry].
Although these authors make no claims about the true dimensionality of the underlying space, it may be tempting to conclude that antigenic space has a relatively low dimensionality. Our work, however, suggests that the antigenic space of H3N2 may appear to have low dimensionality even though the true dimensionality is very high. We therefore created artificial data to to mimic the evolution of H3N2 via a high dimensional random walk. We then followed the procedure for evaluating the quality of reconstructions outlined in [@smith2004mapping].
1. We partitioned the real and artificial distance matrices into a training set (90% of the data) and a test set (10%)
2. Using only the distances in the training set we reconstructed points in antigenic space corresponding to each virus and serum. We varied the dimensionality of antigenic space ($k$) between 2 and 20. Note our artificial data was created using points in 100 dimensional space.
3. We then uses the reconstructed points to predict the antigenic distances in the test set. We evaluated the quality of these fits both by the root-mean-square error and the correlation between the actual and predicted distance. These were the same criteria used in [@smith2004mapping].
We find that increasing $k$ beyond 2, i.e. adding more dimensionality to our reconstructed antigen space, did not improve the quality of the distance estimates (Fig \[fig2\]A,B). This suggests that even data originating from a high dimensional space can be approximated in two dimensions.
{width="\textwidth"} \[fig2\]
### An extreme case: High dimensional random walks can look one dimensional in non-metric MDS
The goal of metric MDS is to find points $x \in \mathbb{R}^k$ whose pairwise euclidean distances approximately match a set of given distances. In particular, the points $x$ must minimize the stress $$\begin{aligned}
\text{Metric Stress}=\sqrt{\frac{(\|x_i-x_j\|-d_{ij})^2}{\sum d^2_{ij}}}\end{aligned}$$ where $d_{ij}$ is the distance between point $i$ and $j$.
In practice, the $d_{ij}$ used as an input to MDS may not represent physical distances but rather a generic measure of dissimilarity that does not behave like a metric. For example, consider three points with dissimilarities $d_{12}=1$, $d_{23}=2$ and $d_{13}=100$. The dissimilarity cannot be a metric as it violates the triangle inequality. However, there may be an underlying metric space that generated the points, and the underlying dissimilarity represents a transformation, $f$, of that metric. The goal of non-metric MDS is to find both $f$ and values for $x$ that minimize $$\begin{aligned}
\text{Non-metric Stress}=\sqrt{\frac{(f(\|x_i-x_j\|)-d_{ij})^2}{\sum d^2_{ij}}}
\label{nonmet}\end{aligned}$$
The distance-squared between steps in an $n$ dimensional random walk follows a $\chi^2_n$ distribution. $$\begin{aligned}
D_{ij}^2\sim\alpha |i-j|\chi^2_n\end{aligned}$$ where $\alpha$ is a constant of proportionality. In this case we let the dissimilarity $d_{ij}$ be the euclidean distance $D_{ij}$. As $n \rightarrow \infty$, the dissimilarity becomes exactly proportional to $\sqrt{|i-j|}$. Thus we can construct a one dimensional map of our random walk via $$\begin{aligned}
\begin{split}
\text{Dissimilarity between point $i$ and point $j$} \,\,\quad d_{ij}^2 &=\alpha |i-j|\\
\text{Location of points} \qquad x_i &=\alpha i\\
\text{Monotonic Transform} \quad f(\cdot) &=\sqrt{\cdot}
\end{split}
\label{interpointd}\end{aligned}$$ Plugging into we find that the stress is identically equal to zero. Therefore an infinite dimensional random walk can be exactly represented in one dimension using non-metric MDS.
Distinguishing between low dimensional and high dimensional random walks using distance measures
------------------------------------------------------------------------------------------------
So far we have demonstrated that high dimensional random walks can appear low dimensional using methods such as MDS or PCA. We therefore ask whether there is any good way to distinguish a low dimensional from a high dimensional random walk. One characteristic of high dimensional random walks is the curve that forms when the first two principal components are plotted against each other [@bookstein2012random]. This pattern emerges as the number of dimensions approaches infinity. Fig. \[fig3\]A shows the first two principal components by year, whose path is reminiscent of the quadratic shape predicted in [@bookstein2012random]. This pattern is suggestive of a high (FIg \[fig3\]B), rather than low dimensional (Fig \[fig3\]C) random walk.
{width="\textwidth"} \[fig3\]
We next sought an objective measure to distinguish a low dimensional from high dimensional walk. Once again we take advantage of the fact that the square of the distance between the $i$th and $j$th step in an $n$ dimensional random walk is proportional to a $\chi^2$ distribution with $n$ degrees of freedom. $$\begin{aligned}
D_{ij}^2\sim\alpha |i-j|\chi^2_n\end{aligned}$$ Let $\bar{D}^2_s$ denote the average distance between two points separated by $s$ steps, i.e. $$\begin{aligned}
\bar{D}^2_s=\frac{1}{m-s}\sum_{i=1}^{i-m+s}D^2_{i, i+s}\end{aligned}$$ where $m$ is the number of steps. As $n\rightarrow \infty$, the coefficient of variation of $\chi^2_n \rightarrow 0$, so we should expect $\bar{D}^2_s \approx \alpha s$ for some $\alpha$. In other words, for high dimensional random walks plotting $\bar{D}^2_s$ vs $s$ should give a linear relationship. We therefore grouped all antigenic map points in the H3N2 data by year, representing both viral strains and anti-sera, and computed the mean of each group. Computing the distance-squared between the annual means and plotting versus the number of years elapsed does indeed produce a linear pattern. In this regard, the H3N2 data once again qualitatively resembles a high dimensional random walk rather than a low dimensional one.(Fig \[fig3\]D-F).
To quantify this resemblance, let $\tau$ be the coefficient of variation of $x_s=\frac{1}{s}\bar{D}^2_s$, i.e $$\begin{aligned}
\tau=\sqrt{\frac{\sum_{s=1}^{m-1} x^2_s}{\left(\sum_{s=1}^{m-1} x_s\right)^2}-1}\end{aligned}$$ For high dimensional random walks, we expect $\tau$ to be close to zero. We simulated $\tau$ for large numbers of random walks and compared to the value computed for the H3N2 data (Fig \[fig4\]). We find that the H3N2 data is unlikely to have been produced by a random walk of less than ten dimensions ($n<10$). Note also that the value of $\tau$ for the data may be inflated due to any number of sources of variability not accounted for in the model, such as measurement error or a heavy tailed distribution in annual step size. Therefore $d=10$ is the lower bound for the dimensionality antigenic space and dimensionality of 30 or more is quite likely. These findings extend to self avoiding random walks, as decribed in the methods section, which behave similarly in this regard.
![**True dimensionality of antigenic is at least $n=10$** We measured deviation from a linear relationship between number of steps and distance squared between points by dividing the latter by the former and taking the coefficient of variation. The horizontal line shows the value found for the H3N2 antigenic map, which is compared to the values found for 10000 simulated random walks of each of the indicated dimensions. We find that the consistency of the linear trend as depicted in \[fig3\]D is inconsistent with a random walk of dimension less than 10.](pcafig4.eps "fig:"){width="\textwidth"} \[fig4\]
Discussion
==========
High dimensional random walks can appear to be low dimensional. We have shown that most of their variance will appear in a single principle component, and that their pointwise distances can be well approximated by embedding in a two dimensional space. Therefore when using principal component analysis (PCA) or multidimensional scaling (MDS) we may erroneously find that only a few dimensions are important. We demonstrate that by splitting the data set into two, we can use out of sample PCA to potentially recover the true dimensionality of the random walk. If only MDS is possible, perhaps due to missing data, we demonstrate that high dimensional and low dimensional random walks can be distinguished by testing the relationship between distance and step number.
We apply these ideas to the evolution of influenza H3N2. Using data from the HAI assay different strains from 1968 to 2003 can be readily represented in a two dimensional antigenic map. The distances on this map represent the level of cross immunity that an antibody response to one strain may provide against another. Given this it may be tempting to conclude that influenza is constrained to move in a two dimensional space or at least is biased to move in two dimensions. However, we caution that antigenic space may in fact be very high dimensional or even infinite dimensional and we would still expect to be able to reconstruct it in two dimensions. In fact, we find that a low dimensional random walk or self avoiding random walk is very unlikely to have produced the H3N2 HAI titer data.
One interesting aspect of influenza H3N2 evolution is that despite the annual variation in strain, the strains have not been diverging. Different strains circulating in a given year tend to be close together antigenically. There have been many models proposed to explain this phenomenon [@gog2002dynamics; @ferguson2003ecological; @koelle2006epochal; @bedford2012canalization; @wikramaratna2013antigenic]. Here we have only shown that the data is consistent with a high dimensional random walk or self avoiding random walk. We assume that the direction of the next step is equally likely to be in any direction, provided that this does not bring the walk back to an already visited region. Therefore our model is consistent with antigenic drift hypotheses as described in [@gog2002dynamics; @ferguson2003ecological; @koelle2006epochal; @bedford2012canalization] but not the antigenic thrift hypothesis described in [@wikramaratna2013antigenic].
Methods
=======
Out of sample PCA
-----------------
1. Start with a length $2m$ random walk in $n$ dimensions.
2. Split the random walk into two halves.
3. Apply PCA to the first half. This will yield an orthogonal projection matrix $P\in\mathbb{R}^{n\times n}$ which projects any point onto the principal component axis.
4. Split the second half of the walk into subsequences with a short length, $\mu$. These subsequences consist of steps $m+1$ to $m+\mu$, $m+\mu+1$ to $m+2\mu$, etc. Let $S_i \in\mathbb{R}^{n\times \mu}$ denote the $i$th such subsequence.
5. Apply $P$ to each subsequence $S_i$, i.e. compute $B_i=PS_i$.
6. Let $\mathbf{\sigma^2}_i$ be the variance of each row of $B_i$.
7. Let $\mathbf{\sigma^2}=\frac{1}{\nu}\sum_{i=1}^{\nu}\mathbf{\sigma^2}_i$ be the sum of all $i$, where $\nu$ is the number of subsequences (i.e. $\nu\mu=m$).
Generating simulated data
-------------------------
We want to generate $N_v$ viral strains and $N_s$ antisera. We will represent each strain and antisera as a point in $\mathbb{R}^n$, where $d$ is our chosen dimensionality of antigenic space. The generation of the artificial data makes repeated use of random vectors in $\mathbb{R}^n$. We draw these random vectors from $N(0,I_d)$, i.e. their entries are i.i.d unit normal random variables. We first generate the viral strains via a random walk. Let $V_i \in \mathbb{R}^n$ represent the $i$th viral strain in the walk, then $$\begin{aligned}
V_1&=b_1\\
V_{i}-V_{i-1}&=b_i \quad i>1\end{aligned}$$ where $b_i~N(0,I_n)$
Next we use these strains to randomly generate antisera. Let $S_j \in \mathbb{R}^n$ be the $j$th antiserum, then $$\begin{aligned}
S_j&=V_{p_j}+b\end{aligned}$$ where $p_j$ is a random integer between 1 and $N_v$ and $c_j~N(0,I_n)$.
Multi Dimensional Procedure
---------------------------
In this paper we use the form of multidimensional scaling described in [@smith2004mapping]. We perform this scaling in two distinct cases: starting from a matrix of measured HI and starting from artificial data generated as described in the previous section.
Not every viral strain is measured against each serum. Let $N_{\text{Meas}}$ be the number of measurements and let $0<l\leq N_{\text{Meas}}$ index those measurements. We can then define $$\begin{aligned}
i_l & \quad \text{Strain corresponding to the $l$th measurement}\\
j_l & \quad \text{Serum corresponding to the $l$th measurement}\\
d_l & \quad \text{Distance between strain $i_l$ and serum $j_l$}\end{aligned}$$
In the case of real HI titer data, we convert the titer to a distance using the method described in [@smith2004mapping]. When using artificial data, we simply compute $d_l=\|V_{i_l}-S_{j_l}\|$.
Next we seek a set of points $\hat{V}_i\in \mathbb{R}^k$ and $\hat{S}_j\in \mathbb{R}^k$ that minimizes the objective function $$\begin{aligned}
F\left(\{\hat{V}_i\},\{\hat{S}_j\}|\{d_{l}\}, \{i_l\}, \{j_l\}\right)=\sum_{l=0}^{N_{\text{Meas}}} \left(d_l-\|\hat{V}_{i_l}-\hat{S}_{j_l}\|\right)^2\end{aligned}$$ We use values of $k$ between 2 and 20.
We minimize $F$ numerically using the optim function in R. We use the BFGS method and provide the analytically derived gradient of $F$ for efficient computation. This is an iterative algorithm that requires an initial guess. We generate these guesses using classic MDS, which is equivalent to principal component analysis. The classic MDS algorithm requires a complete distance matrix. In the case of the artificial data, we can calculate this distance matrix directly. In the case of the real data, we construct a plausible distance matrix by finding the shortest path between any two points whose distance was not directly measured. Note that although this plausible distance matrix is likely incorrect, it is only used to find a good starting guess for our algorithm. The solution found using this method compared favorably to the solution found in [@smith2004mapping]
Self avoiding random walk
-------------------------
The self avoiding random walk behaves similarly to an unbiased random walk accept that the new steps are excluded from getting to close to previous ones. Each new step in the walk is determined according to the following algorithm.
1. Given a current step $x_i$, propose a new step $\hat{x}=x_i+b\in \mathbb{R}^n$, where the entries of $b$ are i.i.d with distribution $\mathcal{N}(0,1/\sqrt{n}$.
2. Calculate the probability of acceptance via $$\begin{aligned}
P&=\prod_{j=0}^i g(\hat{x}|x_i)\\
g(\hat{x}|x_i)&=1-e^{-\|\hat{x}-x_i\|^2}\end{aligned}$$
3. With probability $P$ accept the new step and let $x_{i+1}=\hat{x}$. With probability $1-P$ return to step one and propose a new step.
|
---
abstract: 'In the present paper we show a link between bistochastic quantum channels and classical maps. The primary goal of this work is to analyse the multiplicative structure of the Birkhoff polytope of order 3 (the simplest non-trivial case). A suitable complex parametrization of the Birkhoff polytope is proposed, which reveals several its symmetries and characteristics, in particular: (i) the structure of Markov semigroups inside the Birkhoff polytope, (ii) the relation between the set of Markov time evolutions, the set of positive definite matrices and the set of divisible matrices. A condition for Markov time evolution of semigroups in the set of symmetric bistochastic matrices is then derived, which leads to an universal conserved quantity for all Markov evolutions. Finally, the complex parametrization is extended to the Birkhoff polytope of order 4.'
author:
- Mateusz Snamina
- 'Emil J. Zak'
bibliography:
- 'Refs.bib'
title: Dynamical semigroups in the Birkhoff polytope of order 3 as a tool for analysis of quantum channels
---
Introduction
============
Recent experimental and theoretical developments in the field of quantum information theory attracted a considerable attention to the dynamics of quantum open systems. The density operator, as a central object in the quantum theory of open systems undergoes time evolution, which is in general non-unitary, due to interaction with the environment [@GeomOfQState; @TheTheoryOfOpenQuantumSystems; @Nielsen]. Instead, completely positive (CP) trace preserving (TP) linear maps need to be introduced, which transform the initial state of the system (initial density operator) into an arbitrary time-advanced state; these maps are identified with quantum channels, which remain of great importance both for the physics of quantum open systems and the information theory. A comprehensive review on CPTP maps was given by Kraus [@Kraus].
Time evolution of a quantum system may be analysed at different levels of approximation. By considering time independent Hamiltonians, as well as neglecting memory effects (mediated by the environment), one obtains the *homogeneous Markov evolution* [@EvolOfAttainableStructuresOfHMS; @HMS_AsAnElasticMedium; @Rivas2012]. The corresponding dynamical semigroup of quantum channels is then described by the Kossakowski – Lindblad master equation [@kossakowski1976; @Lindblad1976; @Havel]. In a general case, memory effects are present, imposing the necessity for a more complicated approach based on the Nakijama-Zwanzig equation [@Rivas2012; @chruscinski2012; @chruscinski2014]. Despite the existence of general formulas, the problem of finding, among all possible channels, these realizing Markov evolution still brings much effort [@Rivas2012; @Wolf2007; @Aniello2013]. This issue has been analysed by *Wolf, et al.* in Ref. [@DividingQuCh], where a classification of quantum channels is introduced (in particular with respect to their ability to represent a simple Markov dynamics).
The intrinsic structure of the set of quantum channels is non-trivial even in the simplest case of the evolution of a qubit. Despite numerous classical approaches [@HMS_AsAnElasticMedium; @A_structure_of_doubly_stochastic_Markov_chains], to the best of our knowledge, the problem of classification of quantum channels (in terms introduced in Ref. [@DividingQuCh]) defined in a three-dimensional Hilbert space remains open. Therefore, the present paper aims in a preliminary analysis and classification of quantum channels in the 3D Hilbert space.
A considerable complexity of the problem creates a demand for selection of a subset of quantum channels which exhibit physical significance. An important example is a class of channels for which the steady-state corresponds to the maximal entropy state. Such channels are called *unital* [@Wolf2009; @Evans1984]. In the 2D case the unital channels are often named *Pauli channels*, which contain operations such as: bit flip, phase flip or depolarizing channel [@Nielsen]. For a comprehensive review see Refs. [@GeomOfQState; @Nielsen; @Ruskai2002].
The method employed in this work utilizes a quantum-classical analogy introduced in chapter \[sec:Analogia\]. Classical counterparts of quantum open systems are called stochastic systems [@hj2]. Consequently, in terms of this analogy one can link general quantum channels with stochastic matrices and unital quantum channels with bistochastic matrices. In the standard approach bistochastic transformations represent the classical limit for quantum evolution [@duality2004; @Landau; @Grabert1979; @Meara2013; @Linial]. However, in the present work we target in setting a non-asymptotic equivalence.
Apart from quantum considerations, the analysis of classical stochastic systems represent a worthwhile branch. Classical bistochastic matrices have been found very useful in practical applications, such as financial risk models [@Jarrow1997] or medical sciences [@Charitos2008]. Some preliminary results on stochastic roots of stochastic matrices of order three were reported in Ref. [@He2003]. Nevertheless, ideas presented there need to be extended for our purposes.
Chapter \[sec:Analogia\] explains on the qubit example, a peculiar link between the dynamics of quantum systems and stochastic systems. Chapters \[sec:Parameterization\] and \[sec:DynamicalSemigroups\] refer to stochastic systems in the three-element space. In chapter \[sec:Parameterization\] we propose a parametrization for the set of $3\times3$ bistochastic matrices. The following chapter \[sec:DynamicalSemigroups\] contains a detailed analysis of the set of bistochastic matrices. In particular we target Markovian dynamics in stochastic systems. Finally, in chapter \[sec:Parameterization4\] we generalize the approach applied in chapter \[sec:Parameterization\] to $4\times4$ matrices.
Quantum-classical analogy {#sec:Analogia}
=========================
The main goal of this paper is analysis of the time evolution of quantum open systems with three degrees of freedom. It is however convenient to introduce basic formalism and methodology on a simpler example of a *qubit*, a quantum system with two degrees of freedom. This is the simplest quantum system capable of propagating information. Ideas presented for the qubit are generalized in section \[sec:Parameterization\] onto systems with three and four degrees of freedom.
Description of a qubit
----------------------
The *Qubit* can be described by the density operator in the two-dimensional Hilbert space [@Nielsen; @duality2004]. A choice of an orthonormal basis in the Hilbert space $\mathcal{H}$ provides a matrix representation for the density operator, called the density matrix. For the *qubit* the density matrix is given by $$\label{eq:postac_rho}
\rho = \begin{pmatrix} p_1 & c \\ \bar c & p_2\end{pmatrix}
\qquad
\begin{array}{c}
p_1,p_2\in\mathbb{R}_+ , ~ c \in \mathbb{C}, \\
p_1 + p_2 = 1,\; p_1p_2 - |c|^2 \geq 0.
\end{array}$$
A change in state of the quantum system is determined by a map, which can be formally written as $\rho'=\mathcal{E}[\rho]$, and is called the *quantum channel* [@TheTheoryOfOpenQuantumSystems; @Nielsen; @duality2004; @Landau]. Here we consider a class of *unital quantum channels* only, which in the case of *qubits* is denoted as the class of *Pauli channels* [@Ruskai2002]. Following Kraus [*et al.*]{} [@Kraus; @Rivas2012], the matrix representation for the *Pauli channel* can be given by $$\label{def_operacji_Pauliego}
\rho
\mapsto \mathcal{E}(\vec a)[\rho] = a_0 \rho + \sum_{\gamma\in\{x,y,z\}} a_\gamma \sigma_\gamma \rho \sigma_\gamma
\qquad a_0 \equiv 1 - \sum_{\gamma\in\{x,y,z\}} a_\gamma ,$$ where $\vec a = (a_x, a_y, a_z) \in\mathbb{R}_+^3$ are coefficients parametrizing the map (channel), and $\vec \sigma = (\sigma_x, \sigma_y, \sigma_z)$ is the vector of *Pauli matrices*.
The set of all *Pauli channels*, denoted as $\mathcal{P}$, constitutes a tetrahedron which can be symbolically expressed as $\operatorname{span}( \mathcal{E}_{\text{id}} , \mathcal{E}_{x}, \mathcal{E}_{y}, \mathcal{E}_{z})$ [@GeomOfQState; @Ruskai2002], where: $\mathcal{E}_{\text{id}} [\rho] = \rho$ and $\mathcal{E}_\gamma[\rho] = \sigma_\gamma \rho \sigma_\gamma$ ($\gamma\in\{x,y,z\}$). Figure \[fig:foliacja:bezfolii\] displays an example of such tetrahedron. Motivation for a graphical representation of Pauli channels is supported by a classical-quantum analogy introduced in section \[sec:DynamicalSemigroups\].
![Geometrical representation of the set of all Pauli channels. Vertices of the tetrahedron denoted as $\mathcal{E}_{\text{id}} , \mathcal{E}_{x}, \mathcal{E}_{y}, \mathcal{E}_{z}$ correspond to Pauli channels defined by eq. \[def\_operacji\_Pauliego\].[]{data-label="fig:foliacja:bezfolii"}](CP_Wid75mm.pdf)
Classical part of description of a qubit {#classical-quantum}
----------------------------------------
The state of the qubit can be represented by the density matrix or equivalently by a vector of *classical probabilities* ($p_1$ and $p_2$) completed with *quantum coherences* ($c$). In the classical picture, quantum coherences are neglected, and the $\mathbf{p}$ vector fully characterizes the state of the qubit. This correspondence is marked with a wiggled arrow in eq. \[eq:QuantumClassicalAnalogy\_state\]. $$\label{eq:QuantumClassicalAnalogy_state}
\rho
\rightsquigarrow
\mathbf{p}
=
\begin{pmatrix} p_1 \\ p_2\end{pmatrix},$$
Thus, for every Pauli channel, there exists an associated classical evolution $ \mathcal{E}[\rho] \rightsquigarrow \mathbf{B}\mathbf{p}$, which is determined by a stochastic matrix $\mathbf{B}$. Such an assignment is legitimized by the fact, that *classical probabilities* characterizing the final state (after the evolution) only depend on *classical probabilities*, which refer to the initial state, i.e. classical probabilities of the final state are independent of the *quantum coherences*.
A general form of $\mathbf{B}$ is given by $$\label{eq:identificationEB}
\mathcal{E}[.]
\rightsquigarrow
\mathbf{B}
=
\begin{pmatrix}
a_0 + a_z & a_x + a_y \\
a_x + a_y & a_0 + a_z \\
\end{pmatrix}
.$$ where $a_x, a_y, a_z $ are coefficients defining the Pauli channel in eq. \[def\_operacji\_Pauliego\]. For this reason, classical bistochastic matrices are directly linked to the evolution of the quantum system. A chosen classical evolution can be realized by many quantum channels. This gives rise to an equivalence relation in the set of Pauli channels. Two Pauli channels are in a relation if and only if they generate identical classical evolution. This equivalence relation imposes foliation of the set of Pauli channels. Respective layers, denoted in Figure \[fig:foliacja:zfolia\] with $\mathcal{P}_\lambda$, contain all Pauli channels associated with a single bistochastic matrix. Eq. \[eq:foliation\] presents the situation where $\lambda$ parametrizes a $2\times 2$ bistochastic matrix associated with the classical evolution:
$$\label{eq:foliation}
\mathcal{P}_\lambda
\leftrightsquigarrow
\mathbf{B}(\lambda)
=
\frac{1}{2} \begin{pmatrix} 1+\lambda & 1-\lambda \\ 1-\lambda & 1+\lambda \end{pmatrix}
.$$
![Foliation of the set of Pauli channels. Respective layers, denoted as $\mathcal{P}_\lambda$, contain all quantum evolutions associated with single classical evolution.[]{data-label="fig:foliacja:zfolia"}](CP_foil_Wid75mm.pdf)
Dynamics
--------
In general, the classical picture of time evolution of the system is far more intuitive and more accessible, than the full quantum description. At the same time, quantum dynamics of a number of systems can be analysed and interpreted by means of their classical properties. As shown in section \[classical-quantum\], the qubit is an example of such system. Thus, in the present section we will focus on the classical part of the dynamics of the qubit. From this perspective, the qubit is a two dimensional classical stochastic system, which has only one dynamical semigroup associated with it, $$\label{eq:BiSt22Range_izo}
\left\{
\frac{1}{2} \begin{pmatrix} 1 + e^{-v_z t} & 1 - e^{-v_z t}\\ 1 - e^{-v_z t} & 1 + e^{-v_z t}\end{pmatrix}
\middle|
t\in \mathbb{R}_+
\right\},$$ where $v_z > 0$ is a parameter, which identifies the parametrization of the unique dynamical semigroup. This parametrization can be viewed as an isomorphism between the semigroup and $(\mathbb{R}_+,+)$. A necessary condition for a group of quantum evolutions to be well defined, is the existence of a group of classical time evolutions associated with the quantum group. In the present case, this means that the quantum group (parametrized by $\vec a(t)$) has to satisfy the following condition: $$\label{eq:comesDown_matrix_condition}
\begin{pmatrix}
a_0(t) + a_z(t) & a_x(t) + a_y(t) \\
a_x(t) + a_y(t) & a_0(t) + a_z(t)
\end{pmatrix}
=
\frac{1}{2}
\begin{pmatrix}
1 + e^{-v_z t} & 1 - e^{-v_z t} \\
1 - e^{-v_z t} & 1 + e^{-v_z t}
\end{pmatrix} ;$$ which, can be rewritten in a compact scalar form $$\label{eq:comesDown_scalar_condition}
a_0(t) + a_z(t) = \frac{1}{2} \big( 1 + e^{-v_z t} \big) .$$ From the geometric point of view, eq. \[eq:comesDown\_scalar\_condition\] provides information about the dynamics in the resolution of a single layer. Dynamics inside the layer is completely undetermined.
The set of Pauli channels is a specific case, where it is straightforward to provide a general formula for all Markov groups of time evolutions. $$\Big\{ \mathcal{E}\big( \vec a(\vec v; t))[.] \Big| t\in \mathbb{R}_+ \Big\}$$ where $$\label{eq:PauliAfine_izo}
\begin{pmatrix} a_x(\vec v; t) \\ a_y(\vec v; t) \\ a_z(\vec v; t) \end{pmatrix}
=
\frac{1}{4}
\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}
+
\frac{1}{4}
\begin{pmatrix} +1 & -1 & -1 \\ -1 & +1 & -1 \\ -1 & -1 & +1 \end{pmatrix}
\begin{pmatrix} e^{-v_x t} \\ e^{-v_y t} \\ e^{-v_z t} \end{pmatrix} .$$ Direction of the $\vec v$ vector identifies the semigroup, and the length of $\vec v$ determines the parametrization of the semigroup - an isomorphism between the semigroup and $(\mathbb{R}_+,+)$. In other words, the set of all dynamical semigroups is two dimensional, and can be parametrized by the unit vector $\vec v / |\vec v|$. Comparison of eq. \[eq:PauliAfine\_izo\] and eq. \[eq:BiSt22Range\_izo\], with the help of the formula in eq. \[eq:identificationEB\], suggests that $v_z$ introduced in eq. \[eq:BiSt22Range\_izo\] and $v_z$ introduced in eq. \[eq:PauliAfine\_izo\] refer to the same quantity. This means that the $v_z$ component of the vector $\vec v$ is directly linked to the classical evolution. As a consequence, the dynamics of quantum channels can be investigated by means of the classical-quantum equivalence presented above. In the first step, the classical-quantum equivalence is established, as it was done in eq. \[eq:identificationEB\]. At this stage, a choice of the quantization axis is required (here, the $z$-direction was chosen). In the next step, a detailed analysis of classical Markov semigroups is performed (see eq. \[eq:BiSt22Range\_izo\]). Equations mentioned in steps one and two allow to derive a condition for a quantum evolution to be consistent with an associated classical evolution (see eq. \[eq:comesDown\_scalar\_condition\]).
The consistency condition is insufficient for complete reproduction of the quantum dynamics in $\mathcal{P}$. Nonetheless, it provides a constraint on Markov semigroups existing in $\mathcal{P}$. The condition is associated with the chosen basis in the Hilbert space. Fortunately, one is free to chose different bases. For example, in the case of the Pauli channels, the choice of three different quantization axis $x$, $y$ and $z$ results in three independent conditions, which if solved, deliver a general formula for any semigroup in $\mathcal{P}$.
Mathematical objects and notation
=================================
Motivation
----------
The quantum-classical equivalence introduced in the previous section, was shown to provide a simple way for describing the dynamics of the quantum system, with the use of classical dynamical groups. The form of the classical dynamical group for the qubit can be straightforwardly postulated, as done in eq. \[eq:BiSt22Range\_izo\], without any derivation, due to simplicity of the two-dimensional system. The knowledge of dynamical groups in this generic case allowed to conclude, that unital quantum evolutions correspond to subgroups of the group of bistochastic matrices. In systems with more than two degrees of freedom, the explicit form of the classical dynamical groups requires a more rigorous approach, which is developed in this section.
Introduction of mathematical objects
------------------------------------
For the sake of self-consistency, key mathematical objects are briefly introduced below.
The set of all bistochastic matrices of size $N$ is denoted with $\mathcal{B}_N$. Bistochastic matrices from $\mathcal{B}_N$ describe transformations of *classical stochastic systems* with $N$ degrees of freedom, for which the maximum entropy state is the steady state. This fact puts constraints on bistochastic matrices, written as
$$\begin{aligned}
\label{bistochastic}
\mathcal{B}_N
&=
\bigg\{ \mathbf{B} \in \mathbb{R}^{N\times N} \bigg| \sum_i B_{ij}=1, ~ \sum_j B_{ij}=1, ~B_{ij}\geq0 \bigg\} .
\end{aligned}$$
The geometry of the $\mathcal{B}_N$ set is analysed in the Birkhoff–von Neumann theory [@Birkhoff], where $\mathcal{B}_N$ is represented as a polyhedron, which constitutes convex hull of the set of $N!$ permutation matrices. This polyhedron is also called the $N$-dimensional Birkhoff polytope [@Landau; @CourInConv]. $\mathcal{B}_N$ forms a group, and in this work, we will focus on dynamical subgroups of $\mathcal{B}_N$, i.e. group structures in the interior of the Birkhoff polytope.
Definition in eq. \[bistochastic\] contains two conditions: summation to unity of elements in every column and every row, and non-negativity of individual matrix elements. Here, it is useful to introduce an auxiliary definition, which contains only the former condition:
$$\begin{aligned}
\label{bistochastic2}
\mathcal{W}_N
&=
\bigg\{ \mathbf{W} \in \mathbb{R}^{N\times N} \bigg| \sum_i W_{ij}=1, ~ \sum_j W_{ij}=1 \bigg\} ,
\end{aligned}$$
Let us denote with $\mathcal{B}_{N\text{sym}}$ and $\mathcal{W}_{N\text{sym}}$, respective sets of symmetric matrices in $\mathcal{B}_N$ and $\mathcal{W}_N$. Below we define subsets of $\mathcal{B}_N$, which will be used further on.
(set $\mathcal{B}_{3\text{sym}}^\text{Markov}$, set $\mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$ and set $\mathcal{B}_{3\text{sym}}^{\infty \text{root}}$) We denote:
- the set of matrices describing classical Markov evolution: $$\mathcal{B}_{3\text{sym}}^\text{Markov}
=\\
\Big\{
e^{\mathbf{L}} \in \mathcal{B}_{3\text{sym}}
\Big|
\mathbf{L} \in \mathbb{R}^{3\times3} :
\big\{ e^{t\mathbf{L}} \big| t>0 \big\} \subset \mathcal{B}_{3\text{sym}}
\Big\} ;$$
- the set of matrices describing Markov evolution extended with feasible asymptotic evolutions (infinite time evolutions): $$\mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}
=
\mathcal{B}_{3\text{sym}}^\text{Markov}
\bigcup
\Big\{
\lim_{t\rightarrow\infty} e^{t\mathbf{L}} \in \mathcal{B}_{3\text{sym}}
\Big|
\mathbf{L} \in \mathbb{R}^{3\times3} :
\big\{ e^{t\mathbf{L}} \big| t>0 \big\} \subset \mathcal{B}_{3\text{sym}}
\Big\} ;$$
- the set of divisible bistochastic matrices: $$\mathcal{B}_{3\text{sym}}^{\infty \text{root}}
=
\left\{
\mathbf{B} \in \mathcal{B}_{3\text{sym}}
\middle|
\begin{array}{c}
\forall n\in\mathbb{N} ~ \exists \mathbf{B}_n \in \mathcal{B}_{3\text{sym}} : \\
\mathbf{B} = (\mathbf{B}_n)^n
\end{array}
\right\} .$$
Parametrization of the Birkhoff polytope of order 3 {#sec:Parameterization}
===================================================
Definition of the parametrization
---------------------------------
A $3\times3$ matrix contains nine independent elements. However, the condition in eq. \[bistochastic2\], narrows the number of parameters for a $3\times3$ matrix to four independent elements. For this reason, a more convenient representation of the bistochastic matrix can be obtained by a suitable parametrization. The choice of such parametrization should lead to an isomorphism with another group, which has a simpler structure. Here we postulate, the geometry of the set of bistochastic matrices is related to the geometry of complex numbers. Accordingly, a complex parametrization of the set of bistochastic matrices can be proposed:
\[obs:parametryzacja\] Every matrix contained in $\mathcal{W}_3$ can be expressed as $$\label{FormulaMaster}
\mathbf{W}(u,w)
= \mathbf{B}_\star
+ \frac{2}{3} \operatorname{Re} \left[ u \mathbf{M}_1 \right]
+ \frac{2}{3} \operatorname{Re} \left[ w \mathbf{M}_2 \right]$$ for any $u \in \mathbb{C}, w \in \mathbb{C}$, with the respective matrices defined as below $$\mathbf{B}_\star = \frac{1}{3} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} , \qquad
\mathbf{M}_1 = \begin{pmatrix} 1 & \Omega & \Omega^2 \\ \Omega^2 & 1 & \Omega \\ \Omega & \Omega^2 & 1 \end{pmatrix} , \qquad
\mathbf{M}_2 = \begin{pmatrix} \Omega^2 & 1 & \Omega \\ 1 & \Omega & \Omega^2 \\ \Omega & \Omega^2 & 1 \end{pmatrix}$$ where $\Omega = \exp\big(i\tfrac{2}{3}\pi\big)$.
Equation \[FormulaMaster\] establishes 1:1 correspondence between the set $\mathcal{W}_3$ and pairs $(u,w) \in \mathbb{C}^2$. Table \[tab:special\_points\] lists appropriate $(u,w)$ values for six permutation matrices – vertices of the Birkhoff polytope of order 3.
$$\begin{array}{|cc|cc|}
\hline
\text{matrix} & \text{coordinate} &
\text{matrix} & \text{coordinate} \\
\hline
\hline
\begin{array}{c} \mathbf{P}_e =\\= \mathbf{W}(1,0) \end{array}
&
\left\{
\begin{array}{l} u=1 \\ w=0 \end{array}
\right.
&
\begin{array}{c} \mathbf{P}_{(12)} =\\= \mathbf{W}(0,1) \end{array}
&
\left\{
\begin{array}{l} u=0 \\ w=1 \end{array}
\right.
\\ \hline
\begin{array}{c} \mathbf{P}_{(123)} =\\= \mathbf{W}(\Omega^2,0) \end{array}
&
\left\{
\begin{array}{l} u=\Omega^2 \\ w=0 \end{array}
\right.
&
\begin{array}{c} \mathbf{P}_{(13)} =\\= \mathbf{W}(0,\Omega^2) \end{array}
&
\left\{
\begin{array}{l} u=0 \\ w=\Omega^2 \end{array}
\right.
\\ \hline
\begin{array}{c} \mathbf{P}_{(132)} =\\= \mathbf{W}(\Omega,0) \end{array}
&
\left\{
\begin{array}{l} u=\Omega \\ w=0 \end{array}
\right.
&
\begin{array}{c} \mathbf{P}_{(23)} =\\= \mathbf{W}(0,\Omega) \end{array}
&
\left\{
\begin{array}{l} u=0 \\ w=\Omega \end{array}
\right.
\\ \hline
\end{array}$$
Properties of the parametrization
---------------------------------
The main advantage of the chosen parametrization is a significant simplification of analysis of the Birkhoff polytope. Indeed, nine real matrix elements are transformed into four independent real parameters arranged in a pair of complex numbers. These pairs can be further arrayed into $2\times2$ matrices, which form a representation of the $(\mathcal{W}_3,\cdot)$ group.
An isomorphism defined as $$\mathbf{W}(u,w)
\sim
\begin{pmatrix}
u & \bar w \\
w & \bar u \\
\end{pmatrix} .$$ \[prop:2Dw3\] constitutes a two-dimensional representation of the $(\mathcal{W}_3,\cdot)$ group.
Below we list characteristics of selected important subgroups of $\mathcal{W}_{3\text{sym}}$.
\[prop:Charakt\_Mac\_sym\] $\mathbf{W}(u,w) \in \mathcal{W}_{3\text{sym}} \Leftrightarrow u \in \mathbb{R}$.
Let us define half-planes in $\mathcal{W}_{3\text{sym}}$, which are closed under multiplication of matrices. The analysis of the dynamical semigroups in $\mathcal{B}_{3\text{sym}}$ is largely simplified in terms of these half-planes.
\[prop:half-planes\] There exist two-dimensional subsets of $\mathcal{W}_{3\text{sym}}$ closed under multiplication of matrices. These subsets define half-planes $\mathcal{W}_\phi$, $\phi \in [0,2\pi)$: $$\mathcal{W}_\phi = \big\{ \mathbf{W} (u,w) \big| u\in\mathbb{R}, \operatorname{arg} (w) = \phi \big\}.$$ Of course $\bigcup \mathcal{W}_\phi = \mathcal{W}_{3\text{sym}}$.
As a consequence, the half plane $\mathcal{W}_\phi$ can be described in the two-dimensional space of parameters $\big(u, |w| \big) \in \mathbb{R} \times \mathbb{R}_+$. Within the introduced half-planes $\mathcal{W}_\phi$ two regions are of special importance: a region of positive definite matrices and a region of bistochastic matrices. Both regions are characterized by the following relations:
Characterization of the set of positive definite matrices $$\mathcal{W}_\phi \cap \big\{ \mathbf{M} \in \mathbb{R}^{3\times3} \big| \mathbf{M} >0 \big\} =
\big\{ \mathbf{W} (u, |w| e^{i\phi}) \big| u > |w| \big\} .$$
Characterization of the set of bistochastic matrices $$\label{eq:MarkovInW}
\mathcal{W}_\phi \cap \mathcal{B}_{N\text{sym}}
=
\operatorname{span}
\Big\{
\mathbf{W} \big(-1/2,0\big) ,
\mathbf{W} \big(1,0\big),
\mathbf{W} \big(0,f(\phi)e^{i\phi} \big)
\Big\}$$
where $f(\phi)$ is an auxiliary function defined as $$\label{eq:fOdPhi}
f(\phi) =
\left\{
\begin{array}{ll}
+\frac{1}{2}\sec(\phi-\pi/3) & \text{for: } \phi \in [0, 2\pi/3], \\
-\frac{1}{2}\sec(\phi) & \text{for: } \phi \in [2\pi/3, 4\pi/3], \\
+\frac{1}{2}\sec(\phi+\pi/3) & \text{for: } \phi \in [4\pi/3, 2\pi].
\end{array}
\right.$$
Graphical representation
------------------------
![Representation of the Birkhoff polytope in terms of the parametrization from observation \[obs:parametryzacja\]. The blue triangle $(P_e,P_{(123)},P_{(132)})$ is spanned by points with constant $w$ coordinate ($w=0$), while red triangles represent points with constant $u$ coordinate ($u=0$, $u=1$, $u=\Omega$, $u=\Omega^2$ respectively). Vertices of the triangles correspond to permutation matrices. Edges of the polytope are marked with three blue, three red and six green lines. (Note that the origin of the coordinate system is located in the center of the blue triangle).[]{data-label="fig:tesseract_one"}](tesseract_one.png)
The complex parametrization given in observation \[obs:parametryzacja\] enables a simple graphical representation of the Birkhoff polytope. Indeed, the parametrization conserves the structure of the affine combination of points in the Birkhoff polytope, thus any polytope contained in $\mathcal{W}_3$ corresponds to a polytope in $\mathbb{C}^2$. Vertices of the former and the latter polytope overlap. For this reason, the Birkhoff polytope is represented by a polytope in $\mathbb{C}^2$, which is spanned by points collected in Table \[tab:special\_points\]. This situation is displayed in Figure \[fig:tesseract\_one\].
As shown in property $\ref{prop:Charakt_Mac_sym}$, symmetric bistochastic matrices are represented by points $(u,w) \in \mathbb{R}\times\mathbb{C}$. Then, $\mathcal{B}_{3\text{sym}}$ can be visualized as the intersection of the polytope shown in Figure $\ref{fig:tesseract_one}$ with $\mathbb{R}\times\mathbb{C}$. In other words, the representation of $\mathcal{B}_{3\text{sym}}$ is directly obtained from the polytope in Figure $\ref{fig:tesseract_one}$, by neglecting the $\text{Im}(u)$ dimension. In this way, $\mathcal{B}_{3\text{sym}}$ is represented by the trigonal bipiramid, whereas vertices of $\mathcal{B}_{3\text{sym}}$ correspond to the permutation matrices $\mathbf{P}_{e}, \mathbf{P}_{(13)},\mathbf{P}_{(12)}, \mathbf{P}_{(23)}$ and $\tfrac{1}{2}(\mathbf{P}_{(123)}+\mathbf{P}_{(132)})$ (see Figure \[fig:BirkhoffSym\]).
![Graphical representation of $\mathcal{B}_{3\text{sym}}$. The color code used in this figure is consistent with the color code used in fig \[fig:tesseract\_one\]. Three green segments, three red segments and three brown segments are the edges of the trigonal bipiramid representing $\mathcal{B}_{3\text{sym}}$. All of them but the brown segments are also edges of the polygon representing $\mathcal{B}_{3}$. []{data-label="fig:BirkhoffSym"}](BirkhoffSym_Wid75mm.pdf)
Example half-planes ($\mathcal{W}_{0}$, $\mathcal{W}_{\pi/6}$, $\mathcal{W}_{\pi/3}$), defined by property \[prop:half-planes\], are displayed in Figure \[fig:BsymPhiPlaneS\]. These three half-planes will be used in the next section in discussion of the dynamical semigroups.
![Representation of example half-planes $\mathcal{W}_\phi$ (corresponding to $\phi$ equal to $0$, $\pi/6$ and $\pi/3$) together with the trigonal bipiramid representing $\mathcal{B}_{3\text{sym}}$.[]{data-label="fig:BsymPhiPlaneS"}](PhiPlaneS_Wid75mm.pdf)
Dynamical semigroups in the set of bistochastic matrices of order 3 {#sec:DynamicalSemigroups}
===================================================================
Symmetric matrices case
-----------------------
Every dynamical semigroup in $\mathcal{W}_{3\text{sym}}$ belongs to a half-plane $\mathcal{W}_\phi$. It is therefore convenient to formulate results in terms of dynamical semigroups in an arbitrary half-plane $\mathcal{W}_\phi$, as done in eq. \[eq:MarkovSemigroupsInWphi\].
$\mathcal{W}_\phi$ includes the following semigroups $$\label{eq:MarkovSemigroupsInWphi}
\Big\{
\mathbf{W}\big( u(\theta;t) , \big|w(\theta;t)\big| e^{i\phi} \big)
\Big|
t \in [0,\infty)
\Big\}
\subset \mathcal{W}_\phi$$ parametrized as follows $$\left\{
\begin{array}{rcl}
u(\theta;t) &=& \operatorname{exp}(-t\cos\theta) \cosh(t\sin\theta) \\
\big|w(\theta;t)\big| &=& \operatorname{exp}(-t\cos\theta) \sinh(t\sin\theta) \\
\end{array}
\right.$$ where $\theta\in[0,\pi]$ is a parameter characterizing the semigroup.
A semigroup characterized by the $\theta$ parameter is tangent (for its identity representative) to a segment connecting identity (unit matrix) with $\mathbf{W}( 1-\cos\theta , \sin\theta e^{i\phi} )$. Examples of such segments are depicted in Figure \[fig:Wphi\].
Note that both $u(\theta;t)$ and $\big|w(\theta;t)\big|$ are independent of the complex argument $\phi$. Thus, the considered semigroups conserve full rotational symmetry along the axis defined by affine combinations of the $\mathbf{B}_\star$ and $\mathbf{P}_e$ matrices. This symmetry is reduced to the three fold axis for the $\mathcal{B}_3$ set.
![Analysis of half-planes $\mathcal{W}_\phi$ for (a) $\phi=0$, (b) $\phi=\pi/6$ and (c) $\phi=\pi/3$. Grey shadowed region represents the subset of bistochastic matrices, and the orange region (with dashed edge) encloses the subset of all positive definite matrices. Purple lines in the orange region give examples of one-parameter semigroups in $\mathcal{W}_{3\text{sym}}$.[]{data-label="fig:Wphi"}](PhiPlaneS_50mm.png)
Characterization of $\mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$ and $\mathcal{B}_{3\text{sym}}^{\infty \text{root}}$
-------------------------------------------------------------------------------------------------------------------------
The previous section delivered tools necessary to formulate a criterion for a matrix belonging to the semigroup of bistochastic symmetric Markov $3\times3$ matrices.
(Criterion for the $\mathbf{W}(a,b e^{i\phi})$, $a\in\mathbb{R}$, $b\in\mathbb{R}_+$, $\phi\in[0,2\pi)$ matrix to be contained in $\mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$).
1. if $a<b$, then $\mathbf{W}(a,b e^{i\phi})$ does not belong to any of the semigroups described by the formula . In this case $\mathbf{W} \not\in \mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$.
2. if $a>b$, then the $\mathbf{W}(a,b e^{i\phi})$ matrix belongs to the semigroup described by the formula . The $\theta$ parameter describes a semigroup containing matrices which satisfy one of the conditions above and is given by the following formula: $$\tan \theta
=
\frac { \ln(a-b) - \ln(a+b) }
{ \ln(a-b) + \ln(a+b) }
.$$ Limitation on $\tan \theta$: $$0 \leq \tan \theta \leq f(\phi)$$ implies $\mathbf{W} \in \mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$. In the opposite case: $\mathbf{W} \not\in \mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$.
3. in the $a=b$ case, the $\mathbf{W}(a,b e^{i\phi})$ matrix belong to the semigroup described by the formula only if $(a,b)=(0,0)$ or $(a,b)=(\tfrac{1}{2},\tfrac{1}{2})$. Hence, $\mathbf{W} \in \mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$ in four exclusive cases: $$\begin{aligned}
&(a,b) = (0,0), &
&(a,b,\phi) = \big( \tfrac{1}{2}, \tfrac{1}{2}, 0 \big) , \\
&(a,b,\phi) = \big( \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{2}{3}\pi \big),&
&(a,b,\phi) = \big( \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{4}{3}\pi \big).
\end{aligned}$$
The above conditions provide a procedure for finding matrices, which describe classical Markov evolution of 3-state systems. Note that the set of infinitely divisible matrices $\mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$ is included in $\mathcal{B}_{3\text{sym}}^{\infty \text{root}}$, but inclusion in the opposite direction is not generally satisfied. The difference between $\mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$ and $\mathcal{B}_{3\text{sym}}^{\infty \text{root}}$ is quite subtle. Interiors of the two sets are identical. At the boundaries, $\mathcal{B}_{3\text{sym}}^{\infty \text{root}}$ is a closed set, unlike $\mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$. More precisely, the difference between these sets occurs for three segments:
$$\label{difference}
\mathcal{B}_{3\text{sym}}^{\infty \text{root}}
\setminus
\mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}
=
\bigcup_{\phi \in \{0,\frac{2\pi}{3},\frac{4\pi}{3}\}}
\left\{
\mathbf{W}(x,xe^{i\phi})
\middle|
x \in \bigg(0,\frac{1}{2} \bigg)
\right\}
.$$
The relation in eq. \[difference\] is rationalized by the observation that $\mathcal{B}_{3\text{sym}}$ contains one-parameter semigroups, in which the neutral element is a non-unit matrix, i.e. $$\bigg\{
\mathbf{W}\bigg( \frac{1}{2} e^{-t} , \frac{1}{2} e^{-t} e^{i\phi} \bigg)
\bigg|
t \in [0,\infty)
\bigg\}
\subset \mathcal{B}_{3\text{sym}} \\$$ for $\phi = \big\{ 0,\frac{2\pi}{3},\frac{4\pi}{3} \big\}$, with the neutral element given by $\mathbf{W}\big(\frac{1}{2},\frac{1}{2}e^{i\phi} \big)$. Each of these subgroups is associated to a single segment in $\mathcal{B}_{3\text{sym}}^{\infty \text{root}} \setminus \mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}$.
At the end of this section, let us introduce a relation between $\mathcal{B}_{3\text{sym}}^\text{Markov}$ and the set of positive $3\times3$ matrices. Figure \[fig:BsymPhiPlaneS\] shows that: $\mathcal{B}_{3\text{sym}}^\text{Markov}
\subsetneq
\big\{ \mathbf{B} \big| \mathbf{B} > 0 \big\}$ and $\mathcal{B}_{3\text{sym}}^{\text{MarkovLimit}}
\subsetneq
\big\{ \mathbf{B} \big| \mathbf{B} \geq 0 \big\}$.
However, it is possible to find symmetric bistochastic matrices, which are simultaneously (a) contained in the Markov semigroup (included in the Birkhoff polytope) and (b) not positive definite. Below is an example of such matrix: $$\exp
\left[
\frac{2}{\sqrt{3}}
\left(
\begin{smallmatrix}
-1 & 0 & +1 \\
+1 & -1 & 0 \\
0 & +1 & -1
\end{smallmatrix}
\right)
t
\right]
=
\mathbf{W} \Big(e^{-\sqrt{3}t}e^{it}, 0\Big)
\in
\mathcal{B}_{3}$$ together with its representative for $t:=\pi$ $$\frac{1}{3}
\left(
\begin{smallmatrix}
1 - 2e^{-\sqrt{3} \pi } & 1 + e^{-\sqrt{3} \pi } & 1 + e^{-\sqrt{3} \pi } \\
1 + e^{-\sqrt{3} \pi } & 1 - 2e^{-\sqrt{3} \pi } & 1 + e^{-\sqrt{3} \pi } \\
1 + e^{-\sqrt{3} \pi } & 1 + e^{-\sqrt{3} \pi } & 1 - 2e^{-\sqrt{3} \pi } \\
\end{smallmatrix}
\right)
=
\mathbf{W} \big(-e^{-\sqrt{3}\pi}, 0\big)
\in
\mathcal{B}_{3\text{sym}} .$$
General case
------------
The $\mathcal{B}_{3}$ set is characterized by four independent parameters, which means that the 4-dimensional geometry of this set can only be partially visualized (see Figure \[fig:tesseract\_one\]). A graphical representation for one-parameter Markov semigroups occurs even more challenging due an additional parameter. Nonetheless, in the introduced framework of the complex parametrization this issue becomes largely simplified. A support for this statement is given by the following lemma:
\[lemma\_markov\] The $\mathcal{W}_{3}$ set includes the following groups: $$\Big\{ \mathbf{W}\big(u (a,b;t), w (a,b;t) \big) \Big| t\in[0,\infty) \Big\}$$ where $b\in\mathbb{C},a\in\mathbb{R}$ denotes parameters characterizing the semigroup, and formulas for $u (a,b;t)$, $w (a,b;t)$ are given below:
- for $a = 0$, $b = 0$ $$\left\{
\begin{array}{rcl}
u (a,b;t)
&=&
e^{-t}
\\
w (a,b;t) &=& 0
\end{array}
\right.
;$$
- if $\big|b\big|^2 - a^2 > 0$ $$\left\{
\begin{array}{rcl}
u (a,b;t)
&=&
\left(
\cosh \big[\Delta t \big]
+ i \frac{a}{\Delta} \sinh \big[\Delta t \big]
\right)
e^{-t}
\\
w (a,b;t)
&=&
\frac{ b }{\Delta} \sinh \big[\Delta t \big]
e^{-t}
\end{array}
\right.
\qquad
\text{where: }\Delta = \sqrt{ \big|b\big|^2 - a^2} ;$$
- if $a^2 - \big|b\big|^2 > 0$ $$\left\{
\begin{array}{rcl}
u (a,b;t)
&=&
\left(
\cos \big[\Gamma t \big]
+ i \frac{a}{\Gamma} \sin \big[\Gamma t \big]
\right)
e^{-t}
\\
w (a,b;t)
&=&
\frac{ b }{\Gamma} \sin \big[\Gamma t \big]
e^{-t}
\end{array}
\right.
\qquad
\text{where: }\Gamma = \sqrt{ a ^2 - \big|b\big|^2 } .$$
Lemma \[lemma\_markov\] indicates that every curve representing a semigroup characterized by parameters $b\in\mathbb{C},a\in\mathbb{R}$, is tangent (for identity element) to a segment connecting identity with the point $(u,w) = \big(i a, b \big)$. This allows to conclude:
$\arg \big[ w(a,b;t) \big]$ remains constant for all matrices contained in a single semigroup. $$\arg \big[ w(a,b;t) \big] = \arg \big[ b \big] \in \text{cons}(t).$$ Furthermore, the values $u (a,b;t)$ and $\big| w (a,b;t) \big|$ depend only on $a$ and $|b|$, hence are independent of $\arg [ b ]$. As a consequence, the non-dynamic variable $\arg \big[ w(a,b;t) \big]$ separates from the rest of the problem.
We conclude that the analysis of semigroups in $\mathcal{W}_3$ should be performed in reduced subsets of $\mathcal{W}_3$ with constant $\arg[w]$. Such subsets represent tetrahedrons with height dependent on the value of $\arg[w]$. An interactive applet, which generates semigroups as a function of input parameters $a$ and $|b|$ and draws their representations in 3D subsets of $\mathcal{W}_3$ with constant $\arg[w]$ may be found in the supplementary materials.
Parameterization of the Birkhoff polytope of order $N=4$ {#sec:Parameterization4}
========================================================
The parametrization procedure given in observation \[obs:parametryzacja\] can be extended to $4\times4$ bistochastic matrices: $$\mathbf{W}(u,w_2,w_3,w_4,x)
=
\mathbf{B}_\star + x \mathbf{X} +
+ 2\operatorname{Re} \Big[ u \mathbf{D}_1 + w_2 \mathbf{D}_2 + w_3 \mathbf{D}_3 + w_4 \mathbf{D}_4 \Big]
\in \mathcal{W}_4$$
($u,w_2,w_3,w_4 \in \mathbb{C}$, $x\in\mathbb{R}$) where: $$\begin{aligned}
\mathbf{B}_\star
&=
\frac{1}{4}
\left(
\begin{smallmatrix}
+1 & +1 & +1 & +1 \\
+1 & +1 & +1 & +1 \\
+1 & +1 & +1 & +1 \\
+1 & +1 & +1 & +1 \\
\end{smallmatrix}
\right),
&
\mathbf{X}
&=
\frac{1}{4}
\left(
\begin{smallmatrix}
+1 & -1 & +1 & -1 \\
-1 & +1 & -1 & +1 \\
+1 & -1 & +1 & -1 \\
-1 & +1 & -1 & +1 \\
\end{smallmatrix}
\right),
&&\\
\mathbf{D}_1
&=
\frac{1}{4}
\left(
\begin{smallmatrix}
+1 & +i & -1 & -i \\
-i & +1 & +i & -1 \\
-1 & -i & +1 & +i \\
+i & -1 & -i & +1 \\
\end{smallmatrix}
\right),
&
\mathbf{D}_2
&=
\frac{1}{4}
\left(
\begin{smallmatrix}
-i & +1 & +i & -1 \\
+i & -1 & -i & +1 \\
-i & +1 & +i & -1 \\
+i & -1 & -i & +1 \\
\end{smallmatrix}
\right),
&&\\
\mathbf{D}_3
&=
\frac{1}{4}
\left(
\begin{smallmatrix}
-1 & -i & +1 & +i \\
-i & +1 & +i & -1 \\
+1 & +i & -1 & -i \\
+i & -1 & -i & +1 \\
\end{smallmatrix}
\right),
&
\mathbf{D}_4
&=
\frac{1}{4}
\left(
\begin{smallmatrix}
+i & -i & +i & -i \\
+1 & -1 & +1 & -1 \\
-i & +i & -i & +i \\
-1 & +1 & -1 & +1 \\
\end{smallmatrix}
\right) .\end{aligned}$$
This particular parametrization enables to introduce a lower-dimensional representation of $\mathcal{W}_4$ (cf. observation \[obs:parametryzacja\]) $$\mathbf{W}(u,w_2,w_3,w_4,x)
\sim
\begin{pmatrix}
u & \bar w_3 & w_4 \\
w_3 & \bar u & \bar w_4 \\
w_2 & \bar w_2 & x \\
\end{pmatrix}
.$$
Future studies will focus on multiplicative structure of the set of $4\times 4$ bistochastic matrices.
Concluding remarks
==================
The present work demonstrates how the analysis of the dynamics of classical stochastic systems supports understanding of its quantum analogue. As shown in section 2, the strategy for investigation of the set of bistochastic matrices can be understood as a preliminary step towards solution of the fully quantum problem in the three dimensional *Hilbert space*. Next we focused on the description of Markovian dynamics in classical systems, which were defined by appropriately chosen semigroups in the set of bistochastic matrices. The complex parametrization introduced in section 4 significantly simplified the analysis of the multiplicative structure of semigroups in $\mathcal{W}_3$. In consequence, multiplication of matrices in $\mathcal{W}_3$ was reduced to few operations on complex numbers (2 complex conjugates, four number multiplications and two additions). This representation led to formulation of several theorems, indicating however that the topic is nothing but saturated. A key point of the present study was extraction of an invariant coordinate associated with the Markov evolution in $\mathcal{W}_3$. The $4\times4$ case opens field for generalizations. Finally, a significant result from the quantum-information point of view, was the proof for no equality between the set of infinitely divisible time evolutions and the set of Markov evolutions.
We would like express our gratitude to K. Życzkowski, for his encouragement to challenge this problem, and valuable remarks.
|
---
abstract: 'We argue that a Bose-Einstein condensate can be transformed into a Floquet condensate, that is, into a periodically time-dependent many-particle state possessing the coherence properties of a mesoscopically occupied single-particle Floquet state. Our reasoning is based on the observation that the denseness of the many-body system’s quasienergy spectrum does not necessarily obstruct effectively adiabatic transport. Employing the idealized model of a driven bosonic Josephson junction, we demonstrate that only a small amount of Floquet entropy is generated when a driving force with judiciously chosen frequency and maximum amplitude is turned on smoothly.'
author:
- Christoph Heinisch
- Martin Holthaus
date: 'December 24, 2015'
title: Adiabatic preparation of Floquet condensates
---
Introduction {#sec:1}
============
The study of ultracold atoms in optical lattices under the influence of time-periodic external forcing has gained tremendous momentum recently. Activities in this field include the realization of tunable gauge potentials [@StruckEtAl12] and topological insulators [@HaukeEtAl12] with ultracold atoms in periodically shaken optical lattices [@Eckardt15], the simulation of effective ferromagnetic domains [@ParkerEtAl13] and of the roton-maxon dispersion known from superfluid helium [@HaEtAl15], the realization of the topological Haldane model with ultracold fermions [@JotzuEtAl14], the observation of Bose-Einstein condensation in strong synthetic magnetic fields [@KennedyEtAl15], and the detection of multiphoton-like transitions with quantum gases in driven optical lattices [@WeinbergEtAl15]. Further theoretical proposals have addressed the creation of Majorana fermions [@JiangEtAl13; @LiuEtAl13] and of topologically protected edge states [@ReichlMueller14] in driven cold-atom quantum systems. The diversity of this list, which is still far from complete, suggests that the addition of time-periodic forcing to the toolbox of ultracold-atoms physics may well constitute a decisive step towards efficient quantum simulation.
The common theme underlying this development is the use of the Floquet picture for periodically time-dependent quantum systems [@Holthaus15]. The formal content of this approach is easy to formulate: Consider a quantum system defined on some Hilbert space ${\mathcal H}$, be it a single-particle or a many-body system, the dynamics of which are governed by an explicitly time-dependent Hamiltonian $H(t)$ which is periodic in time with period $T$, $$H(t) = H(t+T) \; .
\label{eq:HOT}$$ Then Floquet’s theorem [@Floquet83] suggests the existence of a set of particular solutions to the time-dependent Schrödinger equation possessing the form $$| \psi_n(t) \rangle = | u_n(t) \rangle \exp(-{{\mathrm i}}\varepsilon_n t/\hbar)
\; ,
\label{eq:FLS}$$ where the Floquet functions $| u_n(t) \rangle$ inherit the $T$-periodicity of the Hamiltonian, so that $$| u_n(t) \rangle = | u_n(t+T) \rangle \; ;$$ the phase factors accompanying their time-evolution are determined by the quasienergies $\varepsilon_n$ [@Zeldovich66; @Ritus66]. Actually the existence of square-integrable Floquet states (\[eq:FLS\]) is subject to severe mathematical complications in the general case of systems possessing an infinite-dimensional Hilbert space [@Howland92]. Fortunately, in many cases of practical interest the dynamics remain confined to an effectively finite-dimensional ${\mathcal H}$, so that these complications may be neglected. Then the Floquet states form a complete system in ${\mathcal H}$ at each instant $t$, and [*every*]{} solution $|\psi(t) \rangle$ to the time-dependent Schrödinger equation can be expanded with respect to this basis, $$|\psi(t) \rangle = \sum_n a_n | u_n(t) \rangle
\exp(-{{\mathrm i}}\varepsilon_n t/\hbar) \; ,
\label{eq:EXP}$$ with coefficients $a_n$ remaining constant in time, since the periodic time-dependence of the Hamiltonian has already been incorporated into the basis states themselves. In other words, under conditions of perfect isolation, guaranteeing unitary evolution, the Floquet states (\[eq:FLS\]) are equipped with constant occupation probabilities $|a_n|^2$.
However, in an actual experiment the time-periodic influence will not last forever. Rather, it has to be turned on at some point, and will be turned off later. That is, instead of a perfectly time-periodic Hamiltonian (\[eq:HOT\]) one is more likely to encounter a Hamiltonian of the form $$H(t) = H_0 + H_1(t) \; ,$$ where $H_0$ describes the isolated, ultracold quantum gas as long as it is left to itself, while $H_1(t)$ models an external force that is initially absent, then switched on in some way or other, is time-periodic only for a finite number of periods $T$, and is finally switched off. In such cases the expansion (\[eq:EXP\]) holds in the middle interval, after the periodic forcing has been turned on and before it is turned off again, which may coincide with the interval during which the measurements are performed. But then the precise manner in which the forcing has been turned on is of crucial importance for the entire experiment: Unless there is a further relaxation mechanism, the preserved occupation amplitudes $a_n$ of the individual Floquet states during the action of the periodic force are determined solely by its turn-on.
This observation sets the stage for the current paper. We will demonstrate that it is possible to prepare a [*Floquet condensate*]{}, that is, a bosonic many-body Floquet state possessing the coherence properties of a macroscopically occupied single-particle Floquet state [@GertjerenkenHolthaus14a; @GertjerenkenHolthaus14b], if the external force is switched on in an effectively adiabatic manner, provided the forcing strength does not exceed a critical value which depends on the number of particles. Although we employ the idealized model of a periodically driven bosonic Josephson junction [@HolthausStenholm01] for numerical demonstration purposes, the main qualitative features derived from that model may be valid in more general settings.
We proceed as follows: In Sec. \[sec:2\] we recapitulate the adiabatic principle for Floquet states [@BreuerHolthaus89a; @BreuerHolthaus89b; @DreseHolthaus99], and show that the adiabatic transport of Floquet states is accompanied by a Berry phase. In Sec. \[sec:3\] we then discuss numerical model calculations which clarify how this principle works in practice in a bosonic many-body system, and elucidate why it has a limited regime of applicability under conditions of rather strong forcing. In the final Sec. \[sec:4\] we then formulate our conclusions, aiming at model-independent predictions.
The adiabatic principle for Floquet states {#sec:2}
==========================================
Let us assume that the Hamiltonian of an externally forced ultracold-atoms system depends on a set of slowly changing parameters $$\bm P (t) = \big( P_1(t), P_2(t), \ldots \big) \; ,$$ such that it is strictly periodic in time when these parameters are kept [*fixed*]{} at instantaneous values $\bm P$, $$H^{\bm P}(t) = H^{\bm P}(t+T) \; .
\label{eq:HPT}$$ For instance, $P_1(t)$ may denote the slowly changing envelope of a sinusoidal force with angular frequency $\omega = 2\pi/T$, as in the example considered later in Sec. \[sec:3\]; the term “slow” then means “slow compared to the cycle time $T$”. In principle, also the driving frequency $\omega$ may be varied in an adiabatic manner [@DreseHolthaus99]. The task now is to solve the time-dependent Schrödinger equation with moving parameters, $${{\mathrm i}}\hbar\frac{{{\mathrm d}}}{{{\mathrm d}}t} | \psi(t) \rangle =
H^{\bm P (t)}(t) | \psi(t) \rangle \; .
\label{eq:SGL}$$ Because this parameter motion is supposed to occur slowly, it is useful to invoke the [*instantaneous Floquet states*]{} $$| \psi_n^{\bm P}(t) \rangle = | u_n^{\bm P}(t) \rangle
\exp(-{{\mathrm i}}\varepsilon_n^{\bm P} t/\hbar)
\label{eq:IFS}$$ associated with the Hamiltonian operators (\[eq:HPT\]) for each fixed set $\bm P$ encountered in the course of time. These states (\[eq:IFS\]) obviously obey the equation $$\begin{aligned}
& &
{{\mathrm i}}\hbar \frac{{{\mathrm d}}}{{{\mathrm d}}t} | \psi_n^{\bm P}(t) \rangle
\nonumber \\ & = &
\left( {{\mathrm i}}\hbar \frac{{{\mathrm d}}}{{{\mathrm d}}t} | u_n^{\bm P}(t) \rangle
+ \varepsilon_n^{\bm P} | u_n^{\bm P}(t) \rangle \right)
\exp(-{{\mathrm i}}\varepsilon_n^{\bm P} t/\hbar)
\nonumber \\ & = &
H^{\bm P}(t) | u_n^{\bm P}(t) \rangle
\exp(-{{\mathrm i}}\varepsilon_n^{\bm P} t/\hbar) \; ,
\phantom{\frac{{{\mathrm d}}}{{{\mathrm d}}t}}\end{aligned}$$ giving $$\left( H^{\bm P}(t) - {{\mathrm i}}\hbar \frac{{{\mathrm d}}}{{{\mathrm d}}t} \right)
| u_n^{\bm P}(t) \rangle =
\varepsilon_n^{\bm P} | u_n^{\bm P}(t) \rangle \; .
\label{eq:EVP}$$ This is an [*eigenvalue equation*]{} for the time-periodic Floquet functions $| u_n^{\bm P}(t) \rangle$, providing the quasienergies $\varepsilon_n^{\bm P}$ as their eigenvalues, quite similar to a stationary Schrödinger equation which yields the energy eigenvalues and eigenfunctions of a time-independent Hamiltonian. However, this eigenvalue problem (\[eq:EVP\]) is no longer posed in the system’s physical Hilbert space ${\mathcal H}$, because one has to incorporate the periodic boundary conditions $$| u_n^{\bm P}(t) \rangle = | u_n^{\bm P}(t+T) \rangle \; .
\label{eq:PBC}$$ To this end, one introduces the [*extended Hilbert space*]{} $L_2[0,T] \otimes {\mathcal H}$, consisting of $T$-periodic square-integrable functions, in which the time $t$ is treated on the same footing as the spatial coordinates. Accordingly, the scalar product in this extended space is given by [@Sambe73] $${{\langle\!\langle}}u | v {{\rangle\!\rangle}}= \frac{1}{T}\int_0^T \! {{\mathrm d}}t \,
\langle u(t) | v(t) \rangle \; ,$$ naturally involving integration over the “time coordinate”. Following Sambe [@Sambe73], one writes $| u_n^{\bm P}(t) {{\rangle\!\rangle}}$ with a “double ket” symbol if a Floquet function is no longer regarded as an element of ${\mathcal H}$, but rather of the extended space $L_2[0,T] \otimes {\mathcal H}$. Next, one introduces the [*quasienergy operators*]{} $$K^{\bm P} = H^{\bm P}(t) + p_t \; ,$$ where $$p_t = \frac{\hbar}{{{\mathrm i}}} \frac{{{\mathrm d}}}{{{\mathrm d}}t}$$ denotes the momentum operator conjugate to the $t$-coordinate; observe that the periodic boundary conditions (\[eq:PBC\]) make sure that this operator is hermitian on $L_2[0,T] \otimes {\mathcal H}$ . With these conventions, the eigenvalue equation (\[eq:EVP\]) takes its proper form $$K^{\bm P} | u_n^{\bm P}(t) {{\rangle\!\rangle}}=
\varepsilon_n^{\bm P} | u_n^{\bm P}(t) {{\rangle\!\rangle}}\; .
\label{eq:PEV}$$ The use of adiabatic techniques for obtaining approximate solutions to the Schrödinger equation (\[eq:SGL\]) in terms of instantaneous Floquet states nows rests on the following observation [@BreuerHolthaus89a; @BreuerHolthaus89b; @DreseHolthaus99]: The instantaneous Floquet states are obtained by “freezing” the slowly moving parameters, while retaining the fast, periodic $t$-dependence of the operators (\[eq:HPT\]). Hence, we require a further, time-like variable $\tau$ in order to monitor the protocol $\bm P (\tau)$ according to which these parameters are changed, and consider the Schrödinger-like evolution equation $${{\mathrm i}}\hbar\frac{{{\mathrm d}}}{{{\mathrm d}}\tau} | \Psi(\tau,t) {{\rangle\!\rangle}}= K^{\bm P(\tau)} | \Psi(\tau,t) {{\rangle\!\rangle}}\; ;
\label{eq:EVO}$$ note that its “Kamiltonian” $ K^{\bm P(\tau)}$ remains periodic in time $t$ for [*any*]{} protocol $\bm P (\tau)$. From the solutions to this evolution equation (\[eq:EVO\]) one then finds the desired solutions to the actual Schrödinger equation (\[eq:SGL\]) by restricting the “extended” functions $| \Psi(\tau,t) {{\rangle\!\rangle}}$ to the diagonal, [*i.e.*]{}, by equating $\tau$ and $t$: Requiring $$| \psi(t) \rangle = | \Psi(\tau,t) {{\rangle\!\rangle}}\Big|_{\tau = t} \; ,$$ one immediately has [@BreuerHolthaus89a; @BreuerHolthaus89b] $$\begin{aligned}
{{\mathrm i}}\hbar\frac{{{\mathrm d}}}{{{\mathrm d}}t} | \psi (t) \rangle & = &
{{\mathrm i}}\hbar\frac{{{\mathrm d}}}{{{\mathrm d}}\tau} | \Psi (\tau,t) {{\rangle\!\rangle}}\Big|_{\tau = t} +
{{\mathrm i}}\hbar\frac{{{\mathrm d}}}{{{\mathrm d}}t} | \Psi (\tau,t) {{\rangle\!\rangle}}\Big|_{\tau = t}
\nonumber \\ & = &
\left( H^{\bm P (\tau)}(t) - {{\mathrm i}}\hbar\frac{{{\mathrm d}}}{{{\mathrm d}}t} \right)
| \Psi (\tau,t) {{\rangle\!\rangle}}\Big|_{\tau = t}
\nonumber \\ & &
+ {{\mathrm i}}\hbar\frac{{{\mathrm d}}}{{{\mathrm d}}t} | \Psi (\tau,t) {{\rangle\!\rangle}}\Big|_{\tau = t}
\nonumber \\ & = &
H^{\bm P (t)}(t) | \psi(t) \rangle \; ,
\phantom{\frac{{{\mathrm d}}}{{{\mathrm d}}t}}
\label{eq:IDE}\end{aligned}$$ having exploited Eq. (\[eq:EVO\]) in the second step.
After these somewhat painstaking preparations we are now in a position to make use of the standard quantum adiabatic theorem [@BornFock28; @Kato50]: Let us stipulate that the system is initially, at $\tau = 0$, in a Floquet state corresponding to the parameter set $\bm P (0)$, as expressed by $$| \Psi(\tau = 0,t) {{\rangle\!\rangle}}= | u_n^{\bm P (\tau=0)}(t) {{\rangle\!\rangle}}\; ,
\label{eq:INC}$$ and let us assume that the technical propositions required by the adiabatic theorem are met. Then the adiabatic solution to the evolution equation (\[eq:EVO\]) takes the form $$| \Psi(\tau,t) {{\rangle\!\rangle}}= \exp\left( -\frac{{{\mathrm i}}}{\hbar} \int_0^\tau \!
{{\mathrm d}}\tau' \, \varepsilon_n^{\bm P(\tau')} \right)
{{\mathrm e}}^{{{\mathrm i}}\gamma_n(\tau)} | u_n^{\bm P (\tau)}(t) {{\rangle\!\rangle}}\; ,
\label{eq:ADI}$$ where the eigenfunctions $| u_n^{\bm P}(t) {{\rangle\!\rangle}}$ are determined by solving the instantaneous eigenvalue equations (\[eq:PEV\]); note that these equations (\[eq:PEV\]) do not fix the phases of the eigenfunctions. Thus, when writing down this expression (\[eq:ADI\]) a certain (arbitrary, but differentiable) choice of these phases has implicitly been made for each value of $\bm P$. On the other hand, the overall phase of $| \Psi(\tau,t) {{\rangle\!\rangle}}$ is uniquely fixed by the requirement that this function be a solution to the initial-value problem posed by Eqs. (\[eq:EVO\]) and (\[eq:INC\]). Therefore, following Berry [@Berry84], we have introduced a phase $\gamma_n(\tau)$ to ensure the equality of the total phase on both sides of Eq. (\[eq:ADI\]). This phase $\gamma_n(\tau)$ then has to obey the equation $$\dot \gamma_n(\tau) = - {\rm Im} \; {{\langle\!\langle}}u_n^{\bm P(\tau)} |
\nabla_{\bm P} u_n^{\bm P(\tau)} {{\rangle\!\rangle}}\cdot \dot{\bm P}(\tau) \; ,
\label{eq:EFP}$$ as is confirmed by inserting the proposed solution (\[eq:ADI\]) into Eq. (\[eq:EVO\]); note that the normalization ${{\langle\!\langle}}u_n^{\bm P} | u_n^{\bm P} {{\rangle\!\rangle}}= 1$ implies that ${{\langle\!\langle}}u_n^{\bm P} | \nabla_{\bm P} u_n^{\bm P} {{\rangle\!\rangle}}$ is imaginary. Finally, implementing the general philosophy implied by the identity (\[eq:IDE\]), the desired adiabatic solution to the original Schrödinger equation (\[eq:SGL\]) reads $$| \psi(t) \rangle = \exp\left( -\frac{{{\mathrm i}}}{\hbar} \int_0^t \!
{{\mathrm d}}t' \, \varepsilon_n^{\bm P(t')} \right) {{\mathrm e}}^{{{\mathrm i}}\gamma_n(t)}
| u_n^{\bm P (t)}(t) \rangle \; ,
\label{eq:SOL}$$ stating that, indeed, a system starting out in a Floquet state tends to remain in the “connected” Floquet state if its parameters are varied sufficiently slowly.
These rather formal considerations require two clarifications. First, there obviously is a geometrical Berry phase if the parameters $\bm P$ are led along a closed contour ${\mathcal C}$: In perfect analogy to Berry’s original work [@Berry84], Eq. (\[eq:EFP\]) here yields [@BreuerHolthaus89b] $$\gamma_n({\mathcal C}) = - {\rm Im} \; \oint_{\mathcal C}
{{\langle\!\langle}}u_n^{\bm P} | \nabla_{\bm P} u_n^{\bm P} {{\rangle\!\rangle}}\cdot {{\mathrm d}}{\bm P} \; ,$$ depending only on the loop ${\mathcal C}$ itself, but not on the way it is traversed. As has been pointed out by Simon [@Simon83], the standard adiabatic theorem provides a way of transporting a system’s eigenstate along a curve in parameter space, [*i.e.*]{}, a connection; Berry’s phase therefore is an expression of the (an)holonomy associated with this connection. In the same sense, we now obtain a connection in $L_2[0,T] \otimes {\mathcal H}$ by [*parallel transport of Floquet states*]{}, formally given by the requirement that the phases of the Floquet functions occurring in Eq. (\[eq:ADI\]) be chosen such that $${{\langle\!\langle}}u_n^{\bm P} | \nabla_{\bm P} u_n^{\bm P} {{\rangle\!\rangle}}= 0 \; ,$$ so that one may set $\gamma_n(t) \equiv 0$; note that the assignment of Floquet functions to the parameters ${\bm P}$ may not be single-valued then.
The second, possibly more serious clarification concerns the propositions required for the validity of the formal “solution” (\[eq:SOL\]). Namely, the standard adiabatic theorem [@BornFock28; @Kato50] requires that the eigenvalue of the state to be transported be separated by a certain gap from all others; this proposition [*cannot*]{} be fulfilled in most cases of Floquet transport. This is due to the Brillouin zone structure of the quasienergy spectrum: Suppose that we have found one solution $|u_n^{\bm P}(t) {{\rangle\!\rangle}}$ to the eigenvalue equation (\[eq:PEV\]), and define $\omega = 2\pi/T$. Then one has $$K^{\bm P} | u_n^{\bm P}(t) {{\mathrm e}}^{{{\mathrm i}}m\omega t} {{\rangle\!\rangle}}=
(\varepsilon_n^{\bm P} + m\hbar\omega) | u_n^{\bm P}(t)
{{\mathrm e}}^{{{\mathrm i}}m\omega t}{{\rangle\!\rangle}}\; ,$$ where $| u_n^{\bm P}(t) {{\mathrm e}}^{{{\mathrm i}}m\omega t}{{\rangle\!\rangle}}$ again is a $T$-periodic Floquet function if $m$ is any integer number, be it positive, zero, or negative. On the other hand, all these different solutions to the eigenvalue problem (\[eq:PEV\]) amount to the same Floquet state (\[eq:FLS\]) in the system’s physical Hilbert space ${\mathcal H}$, since $$\begin{aligned}
& &
| u_n^{\bm P}(t) {{\mathrm e}}^{{{\mathrm i}}m\omega t} \rangle
\exp(-{{\mathrm i}}[\varepsilon_n^{\bm P} + m\hbar\omega] t/\hbar)
\phantom{\sum}
\nonumber \\ & = &
| u_n^{\bm P}(t) \rangle \exp(-{{\mathrm i}}\varepsilon_n^{\bm P} t/\hbar) \; .
\phantom{\sum}\end{aligned}$$ Therefore, for each ${\bm P}$ the quasienergy spectrum consists of identical Brillouin zones of width $\hbar\omega$, with each Floquet state $| u_n^{\bm P}(t) \rangle \exp(-{{\mathrm i}}\varepsilon_n^{\bm P} t/\hbar)$ leaving precisely one representative of its quasienergies $\{ \varepsilon_n + m\hbar\omega \, | \, m =0,\pm 1,\pm 2, \ldots\}$ in each zone. Hence, even if we assume that a periodically driven ultracold-atoms system actually possesses square-integrable many-body Floquet states, which may be enforced by a suitable confinement, its quasienergy spectrum will generally cover the energy axis densely, leaving no gap that could be exploited for strictly adiabatic transport.
Because of this lack of a quasienergy gap, the question whether or not effectively adiabatic transport of a periodically driven Bose-Einstein condensate could actually be exploited in a laboratory experiment is far from trivial. In the following section we will present model calcuations which indicate that there may be a window of opportunity when the parameters are varied at speeds which, on the one hand, are so low that the usual non-adiabatic transitions are suppressed, while they still remain sufficiently high on the other hand, such that the unusual processes associated with the denseness of the quasienergy spectrum do not yet figure.
Numerical experiments {#sec:3}
=====================
We consider the two-site model of a bosonic Josephson junction defined by the Hamiltonian [@LMG65; @ScottEilbeck86; @MilburnEtAl97; @Leggett01] $$\begin{aligned}
H_0 & = & -\frac{\hbar\Omega}{2}
\left( {a^{\phantom{\dagger}}}_1{a^{\dagger}}_2 + {a^{\dagger}}_1{a^{\phantom{\dagger}}}_2 \right)
\nonumber \\ & &
+ \hbar\kappa \left( {a^{\dagger}}_1{a^{\dagger}}_1{a^{\phantom{\dagger}}}_1{a^{\phantom{\dagger}}}_1
+ {a^{\dagger}}_2{a^{\dagger}}_2{a^{\phantom{\dagger}}}_2{a^{\phantom{\dagger}}}_2 \right) \; ,
\label{eq:UJJ}\end{aligned}$$ where the operators ${a^{\phantom{\dagger}}}_j$ and ${a^{\dagger}}_j$ effectuate the annihilation and creation of a Bose particle at the $j$th site, obeying the commutation relations ($j,k = 1,2$) $$\left[ {a^{\phantom{\dagger}}}_j, {a^{\phantom{\dagger}}}_k \right] = 0 \; , \quad
\left[ {a^{\dagger}}_j, {a^{\dagger}}_k \right] = 0 \; , \quad
\left[ {a^{\phantom{\dagger}}}_j, {a^{\dagger}}_k \right] = \delta_{jk} \; .$$ Both sites are coupled by a tunneling contact with single-particle tunneling frequency $\Omega$, while the particles are interacting repulsively, with each pair of Bosons sitting on a common site contributing the amount $2\hbar\kappa$ to the total energy. In a typical experimental realization [@GatiOberthaler07] the scaled interaction strength $N\kappa/\Omega$ is on the order of unity for $N = 10^3$ particles.
We assume that this system is subjected to external driving of the form [@HolthausStenholm01; @GertjerenkenHolthaus15] $$H_1(t) = \hbar \mu(t) \sin(\omega t)
\left( {a^{\dagger}}_1{a^{\phantom{\dagger}}}_1 - {a^{\dagger}}_2{a^{\phantom{\dagger}}}_2 \right) \; ,
\label{eq:HFT}$$ so that the driving amplitude $\hbar\mu(t)$ here plays the role of the parameter $P_1(t)$ considered in the previous section; all other parameters will be held constant. When occupied with $N$ particles, this system lives in a merely $(N+1)$-dimensional Hilbert space ${\mathcal H}$, and thus is ideally suited for numerical experiments.
In Fig. \[F\_1\] we display parts of the quasienergy spectrum obtained when the system is driven with constant scaled amplitude $\mu/\Omega$, while $N = 100$, $N\kappa/\Omega = 0.95$, and $\omega/\Omega = 1.0$; these parameters will be kept fixed in the following. The upper panel shows the quasienergies emerging from the three lowest energy eigenstates $n = 0,1,2$ of the undriven junction (\[eq:UJJ\]), reduced to the fundamental quasienergy Brillouin zone $-1/2 \le \varepsilon/(\hbar\omega) < +1/2$, for small scaled driving amplitudes $0 \le \mu/\Omega \le 0.5$. These quasienergy lines still appear to be smooth, providing favorable conditions for adiabatic transport. The middle panel then shows all 101 quasienergies of the system in the interval $0.78 \le \mu/\Omega \le 0.86$; the representative associated with the ground state $n = 0$ of the junction (\[eq:UJJ\]) has been highlighted. Actually, the corresponding Floquet state “feels” ([*i.e.*]{}, interacts with) the “background” provided by all other states: For constant driving amplitude, the quasienergy operator $$K^\mu = H_0 + \hbar\mu\sin(\omega t)
\left( {a^{\dagger}}_1{a^{\phantom{\dagger}}}_1 - {a^{\dagger}}_2{a^{\phantom{\dagger}}}_2 \right)
+ \frac{\hbar}{{{\mathrm i}}} \frac{{{\mathrm d}}}{{{\mathrm d}}t}$$ remains unchanged when swapping the two sites by interchanging the indices $1$ and $2$, and simultaneously shifting the time by half a period, $t \to t + \pi/\omega$. The Floquet functions are even or odd under this generalized parity, and eigenvalues belonging to the same parity should not cross each other, according to the von Neumann-Wigner theorem [@NeumannWigner29]. Therefore, about half of the apparent crossings observed in the middle panel of Fig. \[F\_1\] actually are non-resolved anticrossings. This deduction is confirmed in the lower panel, which magnifies two of these avoided crossings. “Relevant” avoided crossings with a sizeable gap only occur in the strong-driving regime; avoided crossings affecting the Floquet state emerging from the ground state with a gap larger than about $\delta\varepsilon/(\hbar\omega) = 10^{-3}$ have been indicated in the middle panel. Here we encounter, albeit in a relatively small system only, a pertinent consequence of the Brillouin zone structure of the quasienergy spectrum: The denseness of the quasienergies gives rise to a plethora of avoided crossings, corresponding to multiphoton-like resonances which endanger adiabatic transport. However, as long as the driving amplitude remains sufficiently small these anticrossings remain possibly even undetectably narrow, so that the Floquet state emerging from the undriven system’s many-body ground state is not strongly affected, and its quasienergy function $\varepsilon_0(\mu/\Omega)$ may be regarded as smooth and isolated in a coarse-grained sense [@Holthaus15].
A useful means to characterize the individual many-body Floquet states is to compute their respective one-particle reduced density matrix $$\varrho_n = \left( \begin{array}{cc}
\langle {a^{\dagger}}_1 {a^{\phantom{\dagger}}}_1 \rangle & \langle {a^{\dagger}}_1 {a^{\phantom{\dagger}}}_2 \rangle \\
\langle {a^{\dagger}}_2 {a^{\phantom{\dagger}}}_1 \rangle & \langle {a^{\dagger}}_2 {a^{\phantom{\dagger}}}_2 \rangle
\end{array} \right) \; ,$$ and to determine the “degrees of simplicity” [@Leggett01] $$\eta_n = 2 N^{-2} \, {\rm tr} \, \varrho_n^2 - 1 \; .
\label{eq:ETA}$$ Obviously $\eta_n = 1$ when $|u_n(t)\rangle$ corresponds to an $N$-fold occupied, periodically time-dependent single-particle state, [*i.e.*]{}, to a Floquet condensate, whereas $\eta_n = 0$ when the state is maximally fractionalized. Hence, the numerical value $0 \le \eta_n \le 1$ provides a measure for the coherence of $|u_n(t)\rangle$.
In Fig. \[F\_2\] we depict $\eta_n$ for $n = 0, 1, \ldots, 12$, again for $N = 100$, $N\kappa/\Omega = 0.95$, and $\omega/\Omega = 1.0$. Evidently the Floquet state emerging from the undriven ground state corresponds to a periodically time-dependent Bose-Einstein condensate up to roughly $\mu/\Omega \approx 0.8$. For higher driving amplitudes the coarse-graining approach mentioned above does no longer work, the web of resonances starts to make itself felt, and the coherence is lost.
Remarkably, here the ordering of the system’s Floquet states with respect to their degree of coherence follows the quantum number $n$ of the eigenstates of the undriven junction (\[eq:UJJ\]). This is due to the fact that our driving frequency $\omega = \Omega$, times $\hbar$, is somewhat lower than the energy level spacing of the system (\[eq:UJJ\]) in the vicinity of its ground state; for other choices of $\omega$ the condensate-carrying Floquet ground state may not be connected to the unperturbed ground state $n = 0$ [@GertjerenkenHolthaus14a; @GertjerenkenHolthaus14b].
Next, we explore the expected possibility of an adiabatic preparation of a Floquet condensate: We choose a Gaussian switch-on function $$\mu(t) = \left\{
\begin{array}{ll}
\mu_{\rm max} \exp\left( -\frac{t^2}{2\sigma^2} \right)
& , \; t \le 0 \\
\mu_{\rm max}
& , \; t > 0
\end{array} \right.
\label{eq:SOF}$$ with steepness parameter $\sigma$. The larger the dimensionless ratio $\sigma/T$, the longer is the time during which the system’s wave function can adjust to the changing driving amplitude.
We then populate the ground state $n = 0$ of the undriven junction (\[eq:UJJ\]) at large negative times $t_0$, when $H_1(t)$ is still negligible, and solve the time-dependent Schrödinger equation with this initial condition to obtain $| \psi(t \rangle$ for $t_0 \le t \le 0$, employing switch-on functions (\[eq:SOF\]) with various values of the steepness $\sigma$; all calculations reported in the following start at $t_0 = -5\sigma$. At suitable moments $t > t_0$ the solution $| \psi(t) \rangle$ is expanded with respect to the corresponding instantaneous Floquet states, thus obtaining their occupation probabilities $|a_n(t)|^2$. Figure \[F\_3\] depicts the resulting deviation from perfect adiabaticity at $t = 0$ when the final amplitude $\mu_{\rm max}/\Omega = 0.8$ has been reached: If $1 - |a_0(0)|^2$ were equal to zero, the initial ground state would have been transferred without loss into the connected Floquet state, in the sense of Eq. (\[eq:SOL\]). Instead, one observes large deviations from adiabaticity when $\sigma$ is not longer than a few cycles; for such sudden turn-ons the wave function simply has no time to adjust itself to the forcing. As expected, these deviations are diminished rapidly when the turn-on proceeds more slowly, being smallest when the steepness parameter $\sigma$ is about $20$ cycles. However, when is $\sigma$ increased further, so that the steepness is [*reduced*]{} even more, the deviations from adiabaticity [*increase*]{} again: Now the system’s wave function does not “jump” more or less entirely over the small avoided crossings exemplified in the lower panel of Fig. \[F\_1\], but undergoes sizeable Landau-Zener transitions to the anticrossing Floquet states [@BreuerHolthaus89b; @DreseHolthaus99]. That is, the system becomes able to resolve the multitude of avoided crossings if it is given sufficient time. Therefore, in a system with a truly macroscopic number of particles, and hence with an uncomputably dense web of anticrossing quasienergies, an “adiabatic limit” in the mathematical sence, [*i.e.*]{}, for $\sigma/T \to \infty$, cannot be attained. Our key point is that this absence of a proper adiabatic limit does [*not*]{} obstruct an effectively adiabatic controllability for reasonably chosen, finite parameter speed, and moderate maximum driving amplitude.
To substantiate this claim, we introduce the [*Floquet entropy*]{} $$S_F(t) = - \sum_n |a_n(t)|^2 \ln |a_n(t)|^2
\label{eq:FEN}$$ which is zero if only one single Floquet state is populated, and takes on its maximum value $\ln(N+1) \approx \ln N$ when all $N+1$ Floquet states are populated equally. In Fig. \[F\_4\] we show the normalized entropy $S_F(0)/\ln N$ resulting from turn-ons with $\mu_{\rm max}/\Omega = 0.6$, $0.8$, and $0.9$, respectively: As witnessed by the previous Fig. \[F\_2\], for maximum driving amplitude $\mu_{\rm max}/\Omega = 0.6$ the many-body wave function evolving from the undriven system’s ground state still remains in the regime where the quasienergy anticrossings cannot be resolved, so that its coherence is well preserved and the final entropy is still decreasing with increasing $\sigma$ even for “creeping” turn-ons with $\sigma/T \approx 50$; of course it has to increase eventually when $\sigma/T$ is made much larger. The curve for $\mu_{\rm max}/\Omega = 0.8$ corresponds to the one plotted in Fig. \[F\_3\]; this case falls into the parameter regime where the system starts to “feel” the anticrossings. But for $\mu_{\rm max}/\Omega = 0.9$ these anticrossings become so wide that the final Floquet entropy $S_F(0)$ already is comparable to $\ln N$, indicating that so many Landau-Zener transitions have taken place during the turn-on that a quite substantial fraction of the Floquet states has been populated to an appreciable degree, and the system’s coherence is destroyed more or less completely.
These three regimes of response — the [*effectively adiabatic regime*]{} in which the final Floquet entropy $S_F(0)$ is small, and decreases with increasing $\sigma$; the [*transition regime*]{}; and the [*chaotic regime*]{} in which $S_F(0)$ is large, and almost independent of $\sigma$ — are clearly discernible in Fig 5: Here we show $S_F(0)/\ln N$ as function of $\mu_{\rm max}/\Omega$ for several steepnesses $\sigma$. The upper panel, computed again for merely $N = 100$ particles, allows one to identify the transition regime $0.75 < \mu_{\rm max}/\Omega < 0.85$; the lower panel, obtained for $N = 1000$, reveals a much sharper transition at $\mu_{\rm max}/\Omega \approx 0.85$. The fact that the main features remain unchanged when going from $N = 100$ to $N = 1000$, while keeping $N\kappa/\Omega = 0.95$ at a constant value, is not trivial, because there are two opposing tendencies [@GertjerenkenHolthaus14b]: On the one hand, the quasienergy density in the Brillouin zone increases with $N$, leading to more avoided crossings; on the other hand, the reduction of $\kappa$ with $1/N$ implies that the individual anticrossings become more narrow. In combination, these trends still allow for an extended effectively adiabatic regime even for $N = 1000$ and larger: In this regime the $N$-particle ground state of the undriven Josephson junction (\[eq:UJJ\]) can be adiabatically shifted, by means of a well designed, smooth turn-on of a time-periodic driving force, into the connected many-body Floquet state; according to Fig. \[F\_2\], this state has the coherence properties of an $N$-fold occupied, $T$-periodic single-particle state. In short, with judiciously chosen parameters the adiabatic preparation of a Floquet condensate is possible.
Discussion {#sec:4}
==========
The exact numerical calculations presented in Sec. \[sec:3\] refer to a highly idealized model system. Yet, they have revealed some features which we believe to be generic, and which are likely to persist in actual laboratory set-ups not necessarily involving a condensate in a driven double well. We surmise the existence of windows of opportunity, [*i.e.*]{}, of parameter regimes enabling one to adiabatically transform a static Bose-Einstein condensate into a dynamic Floquet condensate without appreciable generation of Floquet entropy, although the denseness of the system’s quasienergy spectrum seems to forbid a naive application of the standard adiabatic theorem. This prediction, which is substantiated by Fig. \[F\_3\], could be verified experimentally by subjecting a trapped Bose-Einstein condensate a to an oscillating drive with a smooth turn-on, and by swiching off the trapping potential suddenly after the maximum driving amplitude has been reached: Time-of-flight absorption images taken in the adiabatic regime then should reveal a high degree of coherence, despite the previous action of the possibly strong force.
Another observation of interest, deduced from Fig. \[F\_5\], concerns the possibility that the adiabatic regime may have a rather sharp border; this is related to the recently discussed “sudden death” of a macroscopic wave function under strong forcing [@GertjerenkenHolthaus15]. Here we encounter a dynamically induced (instead of thermal) destruction of a condensate’s coherence which can be traced to a change of the nature of the system’s quasienergy spectrum: While the quasienergy of the state to be transported, when viewed as a function of the slowly changing parameter, is effectively smooth (that is, broken only by unresolvably narrow anticrossings) in the adiabatic regime, it becomes disrupted by a multitude of large avoided crossings in the coherence-killing chaotic regime. Thus, characteristic properties of spectra which have been investigated in great detail in the context of single-particle quantum chaos [@Haake10; @Stoeckmann07] may find further, unforeseen applications in the realm of forced many-particle systems. Again, the existence of such a “chaos border” should have a clear-cut experimental signature: In a sequence of measurements of the type sketched above, performed with subsequently enhanced maximum driving strengths, one should observe a sharp coherence drop within a relatively small interval of these maximum driving amplitudes.
We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) through grant No. HO 1771/6-2. The computations were performed on the HPC cluster HERO, located at the University of Oldenburg and funded by the DFG through its Major Research Instrumentation Programme (INST 184/108-1 FUGG), and by the Ministry of Science and Culture (MWK) of the Lower Saxony State.
[99]{}
J. Struck, C. Ölschläger, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewenstein, K. Sengstock, and P. Windpassinger, Phys. Rev. Lett. [**108**]{}, 225304 (2012).
P. Hauke, O. Tieleman, A. Celi, C. Ölschläger, J. Simonet, J. Struck, M. Weinberg, P. Windpassinger, K. Sengstock, M. Lewenstein, and A. Eckardt, Phys. Rev. Lett. [**109**]{}, 145301 (2012).
For a review, see A. Eckardt, [*Periodically driven lattices: From dynamic localization to artificial magnetism*]{} (in preparation).
C. V. Parker, L.-C. Ha, and C. Chin, Nature Physics [**9**]{}, 769 (2013).
L.-C. Ha, L. W. Clark, C. V. Parker, B. M. Anderson, and C. Chin, Phys. Rev. Lett. [**114**]{}, 055301 (2015).
G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature [**515**]{}, 237 (2014).
C. J. Kennedy, W. C. Burton, W. C. Chung, and W. Ketterle, Nature Physics [**11**]{}, 859 (2015).
M. Weinberg, C. Ölschläger, C. Sträter, S. Prelle, A. Eckardt, K. Sengstock, and J. Simonet, Phys. Rev. A [**92**]{}, 043621 (2015).
L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Phys. Rev. Lett. [**106**]{}, 220402 (2011).
D. E. Liu, A. Levchenko, and H. U. Baranger, Phys. Rev. Lett. [**111**]{}, 047002 (2013).
M. D. Reichl and E. J. Mueller, Phys. Rev. A [**89**]{}, 063628 (2014).
For a tutorial review, see: M. Holthaus, J. Phys. B: At. Mol. Opt. Phys. [**49**]{}, 013001 (2016).
G. Floquet, Annales de l’ École Normale Supérieure [**12**]{}, 47 (1883).
Ya. B. Zel’dovich, J. Exptl. Theoret. Phys. (U.S.S.R.) [**51**]{}, 1492 (1966) \[Sov. Phys. JETP [**24**]{}, 1006 (1967)\].
V. I. Ritus, J. Exptl. Theoret. Phys. (U.S.S.R.) [**51**]{}, 1544 (1966) \[Sov. Phys. JETP [**24**]{}, 1041 (1967)\].
J. S. Howland, in: [*Schrödinger operators: The quantum mechanical many-body problem.*]{} Lecture Notes in Physics [**403**]{}, p. 100 (Springer, Berlin, 1992).
B. Gertjerenken and M. Holthaus, New J. Phys. [**16**]{}, 093009 (2014).
B. Gertjerenken and M. Holthaus, Phys. Rev. A [**90**]{}, 053614 (2014).
M. Holthaus and S. Stenholm, Eur. Phys. J. B [**20**]{}, 451 (2001).
H. P. Breuer and M. Holthaus, Z. Phys. D [**11**]{}, 1 (1989).
H P. Breuer and M. Holthaus, Phys. Lett. A [**140**]{}, 507 (1989).
K. Drese and M. Holthaus, Eur. Phys. J. D [**5**]{}, 119 (1999).
H. Sambe, Phys. Rev. A [**7**]{}, 2203 (1973).
M. Born and V. Fock, Z. Phys. [**51**]{}, 165 (1928).
T. Kato, J. Phys. Soc. Japan [**5**]{}, 435 (1950).
M. V. Berry, Proc. R. Soc. Lond. A [**392**]{}, 45 (1984).
B. Simon, Phys. Rev. Lett. [**51**]{}, 2167 (1983).
H. J. Lipkin, N. Meshkov, and A. J. Glick, Nucl. Phys. [**62**]{}, 188 (1965).
A. C. Scott and J. C. Eilbeck, Phys. Lett. A [**119**]{}, 60 (1986).
G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A [**55**]{}, 4318 (1997).
A. J. Leggett, Rev. Mod. Phys. [**73**]{}, 307 (2001).
R. Gati and M. K. Oberthaler, J. Phys. B: At. Mol. Opt. Phys. [**40**]{}, R61 (2007).
B. Gertjerenken and M. Holthaus, Annals of Physics [**362**]{}, 482 (2015).
J. von Neumann and E. P. Wigner, Phys. Z. [**30**]{}, 467 (1929).
F. Haake, [*Quantum Signatures of Chaos*]{}, 3rd edition (Springer-Verlag Berlin Heidelberg, 2010).
H.-J. Stöckmann, [*Quantum Chaos – An Introduction*]{} (Cambridge University Press, 2007).
|
---
abstract: 'The most distant galaxies known are at $z\sim10-11$, observed $400-500$ Myr after the Big Bang. The few $z\sim10-11$ candidates discovered to date have been exceptionally small– barely resolved, if at all, by the *Hubble Space Telescope*. Here we present the discovery of SPT0615-JD, a fortuitous $z\sim10$ ($z_{\rm phot}$=$9.9\pm0.6$) galaxy candidate stretched into an arc over $\sim2.5\arcsec$ by the effects of strong gravitational lensing. Discovered in the Reionization Lensing Cluster Survey (RELICS) *Hubble* Treasury program and companion S-RELICS *Spitzer* program, this candidate has a lensed $H$-band magnitude of $25.7\pm0.1$ AB mag. With a magnification of $\mu\sim4-7$ estimated from our lens models, the de-lensed intrinsic magnitude is $27.6\pm0.3$ AB mag, and the half-light radius is $r_e<0.8$ kpc, both consistent with other $z>9$ candidates. The inferred stellar mass ($\log [M_\star /\rm{M}_\Sun]=9.7^{+0.7}_{-0.5}$) and star formation rate ($\log [\rm{SFR}/{\rm{M}}_\Sun$ ${\rm{yr}}^{-1}]=1.3^{+0.2}_{-0.3}$) indicate that this candidate is a typical star-forming galaxy on the $z>6$ SFR–$M_\star$ relation. We note that three independent lens models predict two counterimages, at least one of which should be of a similar magnitude to the arc, but these counterimages are not yet detected. Counterimages would not be expected if the arc were at lower redshift. However, the only spectral energy distributions capable of fitting the *Hubble* and *Spitzer* photometry well at lower redshifts require unphysical combinations of $z\sim2$ galaxy properties. The unprecedented lensed size of this $z\sim10$ candidate offers the potential for the *James Webb Space Telescope* to study the geometric and kinematic properties of a galaxy observed 500 Myr after the Big Bang.'
author:
- 'Brett Salmon$^{1,\dagger}$, Dan Coe$^{1}$, Larry Bradley$^{1}$, Marusa Brada[č]{}$^{2}$, Kuang-Han Huang$^{2}$, Victoria Strait$^{2}$, Pascal Oesch$^{3}$, Rachel Paterno-Mahler$^{4}$, Adi Zitrin$^{5}$, Ana Acebron$^{5}$, Nathália Cibirka$^{5}$, Shotaro Kikuchihara$^{6,7}$, Masamune Oguri$^{7,8,9}$, Gabriel B. Brammer$^{1}$, Keren Sharon$^{4}$, Michele Trenti$^{10}$, Roberto J. Avila$^{1}$, Sara Ogaz$^{1}$, Felipe Andrade-Santos$^{11}$, Daniela Carrasco$^{10}$, Catherine Cerny$^{4}$, William Dawson$^{12}$, Brenda L. Frye$^{13}$, Austin Hoag$^{2}$, Christine Jones$^{11}$, Ramesh Mainali$^{13}$, Masami Ouchi$^{6,8}$, Steven A. Rodney$^{14}$, Daniel Stark$^{13}$, Keiichi Umetsu$^{15}$'
bibliography:
- 'ms.bib'
title: 'A Candidate $z\sim10$ Galaxy Strongly Lensed into a Spatially Resolved Arc'
---
Introduction
============
With its high resolution and sensitivity, observations using the *Hubble Space Telscope* ([*HST*]{}) have sharpened our understanding of the high-$z$ universe. Deep and wide extragalactic imaging surveys with ACS and WFC3 have uncovered thousands of galaxies at $z > 6$ in blank fields [see @Finkelstein16; @Stark16 for reviews], including the most distant galaxy found to-date at $z=11.1$ [GN-z11; @Oesch16]. In addition, we have prioritized [*HST*]{} to observe the most massive galaxy clusters, taking advantage of the natural telescopes they create via strong gravitational lensing (CLASH, PI Postman; Frontier Fields, PI Lotz; RELICS, PI Coe). This investment in lensing fields has proven fruitful. We have discovered highly magnified (MACS1149-JD, @Zheng12, @Hoag17; MACS1115-JD and MACS1720-JD, @Bouwens14; MACS0416-JD, @Infante15) and multiply-imaged galaxies (MACS0647-JD, @Coe13; A2744-JD, @Zitrin14) at redshifts up to $z \sim10.8$, which have allowed us to study faint UV metal lines [@Stark14; @Rigby15; @Mainali17], nebular emission lines [@Smit17a; @Stark15b; @Hoag17; @Laporte17], and the star formation rate density deep into the epoch of reionization [@Oesch14; @Oesch17].
However, little is known in detail about the $z>9$ universe, and the handful of candidates found so far exhibit peculiar properties. At $z\sim11$, MACS0647-JD has a radius smaller than 100 pc, the size of Giant Molecular Clouds in the local universe. GN-z11 is three times brighter than the characteristic UV luminosity ($L_*$) of galaxies at that distance, surprisingly bright given the CANDELS search area. Both z$\sim$10 candidates MACS1149-JD and M0416-JD appear to have an evolved stellar population of $\approx 340$ Myr (due to red \[3.6 \]$-$\[4.5 \] [*Spitzer*]{} colors), when the age of the universe was only $\approx500$ Myr [@Hoag17]. *JWST* NIRCam will better sample the rest-frame UV-to-optical colors which will break some parameter degeneracies and challenge these initial inferences. However, with typical $z\sim10$ effective radii of $<0.2$ and a NIRCAM PSF FWHM[^1] of $\sim0.05$ at 1.5 , it will still be difficult resolve these galaxies spatially. Ideally, we can use the help of strong lensing to study the kinematics and intrinsic stellar populations at $z \sim10$ in detail.
In this Letter we present a galaxy gravitationally lensed into an arc with a photometric redshift of $z_{\rm{phot}}=9.9\pm0.6$. Discovered in the Reionization Lensing Cluster Survey (RELICS) *Hubble* ([*HST*]{}) and *Spitzer Space Telescope* imaging, the arc features of this candidate extend across $\sim$2.5, allowing unprecedented physical resolution deep in the epoch of reionization. This new candidate has an [*HST*]{} F160W $H$-band magnitude of $H$=25.7$\pm0.1$ AB, bright enough for follow-up spectroscopic or grism observations. In this work, we present the supporting evidence that this candidate is indeed at $z\sim10$, and discuss the remaining uncertainties. Throughout, we assume concordance cosmology with $H_0$=70 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\Lambda,0}$=0.7 and $\Omega_{\rm M,0}$=0.3.
Data and Photometry {#sec:Data}
===================
The galaxy cluster SPT-CL J0615-5746 (hereafter SPT0615-57; also known as PLCK G266.6-27.3) was discovered independently by the South Pole Telescope survey [@Williamson11] and the [@Planck11]. It is exceptionally massive ($M_{500} = 6.8\times10^{12}\ M_\Sun$) for its high redshift ($z=0.972$). The SPT and Planck teams obtained [*HST*]{} imaging (GO 12477 and 12757) of the cluster with the ACS/WFC F606W filter $V$ (1-orbit depth) and F814W filter $I$ (combined 2-orbit depth). RELICS (GO 14096) obtained ACS/WFC imaging (1 orbit) in F435W $B$ and WFC3/IR imaging (2 orbits) in F105W $Y$, F125W $J$, F140W $JH$, and F160W $H$.
RELICS obtained similar [*HST*]{} imaging with WFC3/IR and ACS as needed on a total of 41 clusters. The details of the image reduction, SExtractor [version 2.8.6; @Bertin96] object selection, and [*HST*]{} photometry are described by [@Salmon17] and Coe et al. (in prep). SPT0615-57 was the second highest high-$z$-producing cluster field out of the 41 RELICS fields, revealing 25 new candidate galaxies over the redshift range ${5.5<z<8.5}$ [@Salmon17].
[l @l l l]{}\
\[-0.2cm\] Field & SPT0615-57 & PLCKG138-10 & RXC0018+16\
RELICS ID & 336 & 748 & 1107\
$\alpha_{J2000}$ & 06:15:55.03 & 02:27:00.86 & 00:18:33.84\
$\delta_{J2000}$ & $-$57:46:19.56 & 49:00:22.68 & 16:25:18.84\
$B_{435}$ & $>$28.7 & $>$26.8 & $>$28.8\
$V_{606}$ & $>$28.4 & $>$28.4 & $>$28.8\
$I_{814}$ & $>$29.5 & $>$27.0 & $>$29.4\
$Y_{105}$ & $>$27.3 & $>$27.3 & $>$28.4\
$J_{125}$ & $>$26.5 & $>$26.5 & $>$26.9\
$JH_{140}$ & $>$26.3 & 26.0$\pm$0.2 & $>$26.6\
$H_{160}$ & 25.7$\pm$0.1 & 25.2$\pm$0.1 & 26.1$\pm$0.1\
& 25.2$\pm$0.3 & 23.4$\pm$0.1 & 23.1$\pm$0.1\
& 24.4$\pm$0.3 & 22.9$\pm$0.1 & 22.8$\pm$0.1\
$z_{\rm phot,{\textit{HST}}\ only}$$^{\rm a}$ & 9.6$^{+0.7}_{-7.4}$ & 10.0$^{+0.6}_{-7.5}$ & 9.9$^{+0.7}_{-1.0}$\
$z_{\rm phot,{\textit{HST}}+{\textit{Spitzer}}}$ & 9.9$^{+0.6}_{-0.6}$ & 2.7$^{+0.1}_{-0.1}$ & 3.6$^{+0.2}_{-0.2}$\
\[0.2 cm\]
[**Notes:**]{} $^{a}$Photometric redshifts found using BPZ. The two $z\sim3$ interlopers from PLCKG138-10 and RXC0018+16 were initially identified as $z\sim10$ candidates prior to including the [*Spitzer*]{} data, whereas the candidate in SPT0615-57 remained at $z\sim10$.
\[fig:cluster\]
Table \[tab:candidates\] shows the three $z>9$ candidates found in RELICS after fitting photometric redshifts to only [*HST*]{} data. We then vetted these candidates by checking [*Spitzer*]{} data from the S-RELICS programs (PI Brada[č]{}; PI Soifer). The IRAC channel 1 and 2 bands (3.6 and 4.5 respectively, with $\approx$5-hour depth per band), correspond to rest-frame optical flux at $z\sim9-10$ and are invaluable for distinguishing between intrinsically bluer $z\sim10$ star-forming galaxies and intrinsically redder $z\sim$3 interloper galaxies. The [*Spitzer*]{} fluxes were extracted using [@Merlin16] which uses the higher-resolution [*HST*]{} imaging as a prior to extract photometry from the lower resolution [*Spitzer*]{} images. First, we produce PSF convolution kernels based on all available [*HST*]{} images. We manually sharpen the PSF to minimize residuals between the convolved image and the [*Spitzer*]{} images. Then, we run on the entire cluster field to extract the [*Spitzer*]{} photometry.
After obtaining the [*Spitzer*]{} photometry and re-running the photometric redshifts, we rule out two candidates as low-$z$ interlopers, leaving one $z\sim10$ candidate. Fig. \[fig:cluster\] shows image cutouts of this candidate, hereafter named SPT0615-JD (“JD" for [*HST*]{} F125W $J$-band dropout), in each of the available [*HST*]{} and [*Spitzer*]{} bands, as well as a WFC3/IR color composite. SPT0615-JD has an AB magnitude of $25.7\pm0.1$ in F160W detected with S/N$\sim$11 (the F160W exposures were in two epochs 44 days apart and each detected the source with S/N$\sim$5). The extended arc shape is consistent with the direction of lensing shear expected from the cluster (see §\[sec:lens\]). The bands blueward of F140W are undetected with S/N$\lesssim$2, and F140W is just undetected (S/N=2.9). Importantly, we emphasize that observed-frame size of SPT0516-JD is rather large ($\sim$2.5 long), and can easily be spatially resolved by [*JWST*]{} (see §\[sec:compare\]).
We note that these image cutouts reveal an important caveat to the [*Spitzer*]{} fluxes of SPT0615-JD. reports a maximum covariance between SPT0615-JD and all other sources fit simultaneously (max CV ratio) of $\sim1.4$ for both the 3.6 and 4.5 images. This implies a covariance between the [*Spitzer*]{} photometry of SPT0615-JD and a nearby source. The 8 x8 [*HST*]{} image in Fig. \[fig:cluster\] shows an IR-bright nearby $z\sim3$ galaxy. We conclude that the [*Spitzer*]{} fluxes for SPT0615-JD are biased by this source and are likely over-estimated. Even so, the fluxes in each [*Spitzer*]{} band are already several magnitudes fainter than typical low-$z$ interlopers. This is critical because while all $z \sim10$ solutions could have lower *Spitzer* fluxes, the $z\sim2$ solution *requires* them to be high, especially at 4.5 . As we will discuss in §\[sec:SED\], the inflated [*Spitzer*]{} fluxes also increase the inferred $z\sim$10 UV dust attenuation, which should be considered an upper limit. Upcoming deeper [*Spitzer*]{} imaging (PI Brada[č]{}) of this cluster will improve constraints on the flux and derived properties of this candidate.
Lens Models {#sec:lens}
===========
We identify three sets of multiply-imaged galaxies: two with spectroscopic redshifts ($z=1.358$ and $z=4.013$) and one whose redshift is free to vary in the modeling. (Paterno-Mahler et al., in preparation). Based on these, we produce three lens models using [@Jullo07], [@Oguri10], and the [@Zitrin15b] Light Traces Mass (LTM) method. Based on these models, we estimate the magnification of SPT0615-JD to be $\mu\sim4-7$.
All three models predict two counterimages at the positions shown in Fig. \[fig:cluster\]. Our results using (Kikuchihara et al., in preparation) and (Paterno-Mahler et al., in preparation) predict the upper-right counterimage is $\approx$1 magnitude fainter than the original arc, and therefore below the detection limit. All three models predict the lower-left counterimage to be of similar magnification and magnitude of the original arc, and predicts the counterimages to have the same magnitude as the original arc. Given these models, we would have expected to see an image near the lower-left position, but none are yet detected. We note that the WFC3 limiting depths are $\sim$26 AB mag, and the counterimages may be fainter. Conversely, all lens models predict no counterimages if SPT0615-JD is at $z\sim2$. Deeper imaging of this field is required to properly search for the $z \sim10$ counterimages and yield geometric support as in [@Coe13], [@Zitrin14], and [@Chan17].
\[fig:SEDfit\]
\[fig:MagVsRedshift\]
SED Fitting {#sec:SED}
===========
Thanks to the [*Spitzer*]{} data that probes the rest-frame optical and near-ultraviolet (UV, ${\sim2900-4500}$ Å), we can infer upper-limits on physical parameters like stellar mass and dust attenuation to test if the high and low-redshift solutions are sensible. We use a Bayesian SED-fitting procedure originally described by [@Papovich01] and updated by [@Salmon15]. In short, we sample the posterior using a grid of SEDs that represent a range of stellar population ages (${10\ {\rm{Myr}}<t_{\rm{age}}<t_{\rm universe}}$, logarithmically spaced), attenuation (${0<A_{\rm{UV}}<7.4}$), metallicity (${0.02Z_\Sun<Z<Z_\Sun}$), and rising star-formation histories ($\Psi (t) =\Psi_0 \exp(t/\tau_{\rm SFH})$, where the $e$-folding timescale $\tau_{\rm SFH}$ can be 0.3, 0.5, 0.7, 1, 3, 5, 7, 10, 30, 50, 70, or 100 Gyr). We use [@Bruzual03] stellar population synthesis models with a [@Chabrier03] IMF[^2] and include the effects of nebular emission lines following [@Salmon15]. We assume the dust-attenuation law derived by [@Salmon16] that varies in shape from a steep law at low attenuation (similar in shape to the extinction law of the Small Magellanic Cloud) to a grey law at high attenuation (similar in shape to the starburst curve of [@Calzetti00]).
The results of our SED fitting are summarized in Fig. \[fig:SEDfit\]. For all SED fitting, we correct for lensing magnification assuming $\mu=7$, and do not further correct the [*Spitzer*]{} fluxes despite likely contamination (see §\[sec:Data\] and Fig. \[fig:cluster\]). The fits assuming the $z\sim10$ redshift show a moderately high stellar mass of $M_\star=10^{9.7^{+0.7}_{-0.5}}\ \rm{M}_\Sun$ and star-formation rate of SFR=21$^{+34}_{-11}$ $\rm{M}_\Sun$/yr. However, the stellar mass, star-formation rate, age, and UV dust attenuation will be lower if the rest-frame optical fluxes are over-estimated, as implied by the [*Spitzer*]{} contaminant shown in Fig. \[fig:cluster\]. Therefore, we consider these preliminary estimates to be upper limits. Nevertheless, the stellar mass and SFR of SPT0615-JD are indicative of a typical star-forming galaxy at $z\sim10$ [@Oesch14] and would lie on the SFR- relation at $z\sim6$ [@Salmon15].
The SED fit assuming $z\sim2$ is quite different. The median, marginalized results, which account for the full probability density, imply a low stellar mass ${M_\star=10^{8.7^{+0.2}_{-0.3}}\ \rm{M}_\Sun}$, low star-formation rate SFR= ${0.14^{+0.13}_{-0.06}\
\rm{M}_\Sun }$/yr, and evolved stellar population age $t$=1714$^{+773}_{-872}$ Myr, with high uncertainty. Similarly, the best-fit SED has the same stellar mass, but a higher SFR (SFR$\sim{3\ \rm{M}_\Sun}$/yr), a slightly younger stellar population age ($t$=1585 Myr), a very dusty SED ($A_{\rm UV}=6.1$ mag), and high nebular emission (\[OIII\]+H$_\beta$ equivalent width EW=780 Å, or 1671 Åto match the $H$-band magnitude).
This $z\sim2$ SED solution is unphysical for several reasons. Its dust attenuation is dramatically high for its low stellar mass [@Pannella09], its size is too large and SED too dusty compared to other extreme \[OIII\] emitting galaxies at $z\sim2$ [@Malkan17], and it has too high EW compared to \[OIII\] emitters at $z\sim2$ of similar mass [@Maseda13]. Such a rare high EW and high dust interloper was spectroscopically ruled out for a similar $z\sim11$ candidate, MACS0646-JD [@Pirzkal15]. Importantly, the $z\sim2$ SED necessitates such strong \[OIII\] emission to match the observations with appreciable likelihood; a dusty SED template alone cannot match both the bright H-band flux and the relatively faint optical flux. Finally, unlike the $z\sim10$ solution, the $z\sim2$ solution *requires* the already overestimated [*Spitzer*]{} fluxes to be high, and it becomes increasingly harder to justify a $z\sim2$ SED with lower 3.6 and 4.5 fluxes.
We caution the reader that the best-fit SEDs in Fig. \[fig:SEDfit\] (and best-fit SED solutions in general) are not necessarily representative of the full probability density of the posterior [@Leja17]. A better indicator of the goodness-of-fit than the best-fit $\chi^2$ is the unconditional marginal likelihood of the data, or the Bayesian evidence [see e.g., @Salmon16 for definitions], which describes probability of seeing the data given all parameters. The ratio of two Bayesian evidences[^3] under different model assumptions, like whether we assume the SED lies at $z\sim2$ or $z\sim$10, is called the Bayes Factor and describes the relative evidence between two model assumptions. We find a Bayes-Factor evidence of 10.7 in favor of the $z\sim10$ solution, which is considered “very strong" evidence [@Kass95]. Our interpretation is that there are more SEDs that fit the data well assuming the $z\sim10$ redshift than the select few (and justifiably unphysical) SEDs that fit the data well assuming the $z\sim2$ redshift.
Comparison with Other High-Redshift Candidates {#sec:compare}
==============================================
Fig. \[fig:MagVsRedshift\] shows the $H$-band magnitude versus redshift for all high-$z$ ($z>5.5$) candidate galaxies discovered in RELICS [@Salmon17] and many other deep and wide surveys. The lensed, observed-frame size of SPT0615-JD stands out as spatially much larger than other $z\sim10$ candidates (other candidates at these redshifts have similar point-like sizes to those found by @Coe13 and @Oesch16, see below). The intrinsic (de-lensed) magnitude of SPT0615-JD is similar to that of the $z\sim11$ candidate MACS0647-JD [@Coe13].
\[fig:Sizes\]
An independent way to test high and low-$z$ solutions for SPT0615-JD is to calculate its physical size and compare to other known interlopers. Moreover, the sizes of galaxies can give us great physical insight into the initial conditions of early disk evolution [@Ferguson04]. Broadly, the $z>5$ size evolution at fixed luminosity scales as (1+$z$)$^{-m}$ where $m=1-2$ [@Shibuya15]. [@Holwerda15] demonstrated that a combination of UV-to-optical color, sampled by the F160W and 3.6 bands, and physical size can be used to identify obvious low-$z$ contaminants. They summarized that the sizes of $z>9$ galaxy candidates have typical half-light radii of $r_e<0.8$ kpc.
To calculate the size of SPT0615-JD, we used our lens models to reconstruct its image in the source plane. The LTM lens model finds a relatively mild tangential magnification, or shear, of a factor of $\sim3$, leaving the full width of the de-lensed source to be about 3–3.5 kpc. If we assume the light distribution is uniform, we can take the half light radius to be about $\sim$1/4 of the full size and find $r_e\approx0.7-0.8$ kpc. The statistical error on this size (from the lens model) is only a couple of percent, so we are dominated by systematic errors ($\sim$10%). Curiously, the reconstructed source’s axis ratio is still about 2:1 in the same direction as the lensing shear, which could mean that the shear is underestimated and the size is in fact smaller.
Fig. \[fig:Sizes\] shows that the inferred size of SPT0615-JD is typical compared to other high-$z$ candidates. This provides crucial evidence in support of the $z\sim10$ solution that is independent of the galaxy SED. While the uncertainty in the $z\sim10$ UV dust attenuation should be considered as an upper limit, the candidate is still within the range of $M_{\rm UV}$ and SFR surface density of known $z>9$ candidates.
Conclusions
===========
We present SPT0615-JD, a promising $z\sim10$ galaxy candidate that appears to be stretched into the shape of an arc by the effects of strong gravitational lensing. Out of all combined lensing fields from RELICS, CLASH, and the Frontier Fields, there is no other galaxy candidate spatially stretched by lensing as distant as SPT0615-JD. While our three independent lens models predict at least one detectable counterimage, we do not see one in the current data. No counterimages are expected if the candidate is at lower redshift. However, the only $z\sim2$ SED that fits the data well is unphysical based on the required combination of its size, mass, dust attenuation, and \[OIII\]+H$_\beta$ EW. In addition, we find very strong Bayesian evidence that the SED-inferred physical properties of this candidate are of a $z\sim10$ typical star-forming galaxy. Finally, the source-plane size of SPT0615-JD is similar to other $z=9-10$ galaxies, while the observed-frame image offers unprecedented spatial resolution. This galaxy candidate offers the unique opportunity for resolving stellar populations deep in the epoch of reionization, especially with the higher resolution of [*JWST*]{}.
Acknowledgements {#acknowledgements .unnumbered}
================
This paper uses observations from NASA/ESA [*HST*]{}. STScI is operated by AURA under NASA contract NAS 5- 26555. ACS under NASA contract NAS 5-32864, and *Spitzer* by JPL. These observations are associated with program GO-14096 and archival data with GO-9270, GO-12166, GO-12477, GO-12253. Some data were obtained from MAST. This work was performed under the auspices of the U.S. Department of Energy by LLNL under contract DE-AC52-07NA27344. F.A.-S. acknowledges support from Chandra grant G03-14131X.
[^1]: see <https://jwst-docs.stsci.edu>
[^2]: Switching from a Chabrier to a [@Salpeter55] IMF would result in higher derived stellar mass and star-formation rate (SFR) by 0.25 dex.
[^3]: The Bayes-factor evidence is typically described as $\zeta=2\ln B_{12}$, where $B_{12}$ is the ratio of the Bayesian evidence under model assumption 1 to that of model assumption 2. Larger positive numbers favor the assumptions in model 1, and negative favor the assumptions in model 2 [@Kass95].
|
---
abstract: 'The initial states which minimize the predictability loss for a damped harmonic oscillator are identified as quasi-free states with a symmetry dictated by the environment’s diffusion coefficients. For an isotropic diffusion in phase space, coherent states (or mixtures of coherent states) are selected as the most stable ones.'
---
=-10mm =-5mm -5mm
[**Classical States via Decoherence**]{}\
Gh.–S. Paraoanu[^1]\
[*Department of Physics, University of Illinois at Urbana-Champaign,\
1110 W. Green St., Urbana, IL 61801, USA*]{}\
H. Scutaru[^2]\
[*Department of Theoretical Physics, Institute of Atomic Physics\
Bucharest-Magurele, POB MG-6, Romania*]{}\
PACS numbers: 03.65.Bz, 05.30.-d, 05.40.+j
The emergence of classical reality from the underlying quantum description of the world is one of the most fascinating unsolved problems of present-day physics. Decoherence was proposed as a mechanism for the selection of classical (preferred) states: due to the interaction with an environment (external degrees of freedom) classical states are singled out as the most stable ones.
Zurek, Habib and Paz [@z] invented a criterion, called “predictability sieve”, for distinguishing the preferred set of states from the rest of the Hilbert space: classical states are characterized by the least increase in entropy. They addressed this problem in the context of the Caldeira-Leggett model (a particle moving in a potential and linearly coupled with a bath of harmonic oscillators, [@z; @cl; @uz; @zp]). They found out that coherent states are selected, via predictability sieve, as the most “classical” ones.
In contradistinction to this approach, we will study the problem of classicality in the framework of Lindblad’s theory, in which the structure of the master equation is derived from very general assumptions concerning the mathematical description of the evolution of an open quantum system [@l]. This strategy leads to general results (since Lindblad equations work for a large class of physically interesting systems [@ss]), and also avoids any problems related to the non-preservation of the positivity of the reduced density matrix, which was perceived by some authors [@am] as an inconvenient of the Caldeira-Leggett type of evolution. We will identify the states with classical behavior as the states which minimize the rate of entropy increase. We will apply this criterion in an identical fashion for both pure and mixed initial states, using a formalism that doesn’t require one to discriminate between them.
Lindblad’s result shows that a general form for the generator of a completely positive dynamical semigroup in the Schrödinger picture is [@l]
$$L(\rho)=-\frac{i}{\hbar}[H,\rho]+\frac{1}{2\hbar}
\sum_{j}([V_{j}
\rho,V_{j}^{+}]+[V_{j},\rho V_{j}^{+}]),\label{l}$$
where for the operators $V_{j}$, we will take [@ss] $
V_{j}=a_{j}p+b_{j}q,\label{ec}
$ where $j=1,2$ while $a_{j}$ and $b_{j}$ are complex numbers. For the Hamiltonian $H$ of the system we will consider [@ss] a second-order polynomial operator in coordinate and position written as a sum of a harmonic-oscillator part and a mixed term $$H=\frac{m\omega^{2}}{2}q^{2} + \frac{p^{2}}{2m} + {\mu\over 2}(pq+qp).$$ We denote the diffusion coefficients by $$D_{qq}={\hbar\over2}\sum_{j=1}^{2}|a_{j}|^{2},~~D_{pp}={\hbar\over2}\sum
_{j=1}^{2}|b_{j}|^{2},~~D_{pq}=D_{qp}=-{\hbar\over2}Re\sum_{j=1}^{2}
a_{j}^{*}b_{j},$$ and the friction constant is $$\lambda=-Im\sum_{j=1}^{2}a_{j}^{*}b_{j}.$$ It is easy to show [@ss] that the following inequality must be satisfied $$D_{pp}D_{qq}-D_{pq}^{2}\geq{\lambda^{2}\hbar^{2}\over 4}.\label{s}$$ With the above notation, the master equation which governs the evolution of the system takes the form $$\begin{aligned}
\frac{d\rho(t)}{dt}&=&-{i\over\hbar}[H,\rho(t)]
-{i\lambda\over\hbar}([q,\rho(t) p]-[p,\rho(t)
q]) \nonumber\\ & &
-{D_{qq}
\over\hbar^2}[p,[p,\rho(t)]]
-{D_{pp}\over\hbar^2}[q,[q,\rho(t)]]
+{2D_{pq}\over
\hbar^2}[p,[q,\rho(t)]]
.\label{ll} \end{aligned}$$
For the correlations between two operators $C$ and $C'$ we will use the definition $$\sigma_{C,C'}(t)=Tr\left(\rho(t){{CC'+C'C}\over 2}
\right)-\sigma_{C}(t)\sigma_{C'}(t)
,$$ where $\sigma_{C}(t)=Tr(\rho(t) C)$ and $\sigma_{C'}(t)=Tr(\rho(t) C')$. $\Sigma(t)$ will denote the dispersion matrix $$\Sigma(t)
=\left(\matrix{m \omega \sigma_{qq}(t)&\sigma_{pq}(t)\cr\sigma_{pq}(t)&
{\sigma_{pp}(t)\over m \omega}\cr}\right).$$ With the notations above, Heisenberg’s inequality reads $det\Sigma(t)\geq
{\hbar^2\over4}$. For $\Sigma(t)$ the time-evolution is known [@ss] $${d\Sigma(t) \over dt}=Y\Sigma(t)+\Sigma(t) Y^{T}+2\cal{D}\label{ev},$$ where $$Y=\left(\matrix{-(\lambda-\mu) & \omega\cr-\omega &
-(\lambda+\mu)\cr}\right),$$ $Y^{T}$ is the transposed matrix of $Y$ and $\cal D $ is a $2\times 2$ matrix with elements ${\cal D}_{qq}=m\omega
D_{qq}$, ${\cal D}_{pq}={\cal D}_{qp}=D_{qp}$ and ${\cal D}_{pp}={D_{pp}\over m\omega}$.
For the case of the damped harmonic oscillator, it is known that a certain class of states (quasi-free states [@s]) has the property of invariance under the action of the quantum semigroup. This can be seen by writing the Fokker-Planck equation which corresponds to (\[ll\]) in the form [@i] $${\partial f_{W}(x_{1}, x_{2}, t)\over\partial t}=
\sum_{i,j=1}^{2}Y_{ij}{\partial\over
\partial x_{i}}
(x_{j}f_{W}(x_{1}, x_{2}, t))
+{1\over 2\hbar}\sum_{i,j=1}^{2}{\cal D}_{ij}{\partial^{2}\over\partial x_{i}
\partial x_{j}} f_{W}(x_{1}, x_{2}, t),\label{f}$$ where $f_{W}$ is the Wigner function of a quasi-free state $$f_{W}(x_{1}, x_{2}, t)=
[(2\pi)^2det\Sigma(t)]^{-{1\over 2}}\exp\left\{-{1\over2}X^{T}(t)\Sigma^{-1}X(t)
\right\},\label{w}$$ and $X(t)=\left(\matrix{x_{1}-\sigma_{q}(t)\cr x_{2}-\sigma_{p}(t)}\right)$. It is now easy to verify that the Gaussian Wigner function (\[w\]), with the time-dependence of the mean values $\sigma_{p}(t)$, $\sigma_{q}(t)$ and of dispersions $\sigma_{qq}(t)$, $\sigma_{pp}(t)$ , $\sigma_{pq}(t)$ given by equations (3.26 - 3.27) from [@ss], is a solution of the equation (\[f\]). Thus, quasi-free states are preserved during the evolution of the system.
Starting with such a state, we are interested to calculate the rate of linear entropy variation. The initial states with the lowest rate of linear entropy increase will be identified as the most stable ([*i.e.*]{} the most classical-like) states.
The linear entropy is a convenient measure of the purity of a quantum state and is defined by $${\it s}(\rho) = 1 - Tr(\rho ^2).$$ For a quasi-free state we have $${\it s}(\rho) = 1 - {1 \over A }.$$ where $A(t)$ is the “area” in phase space, $A(t)={2\over\hbar}\sqrt{det\Sigma(t)}$. The condition for a quasi-free state to be a pure one is $A(t)=1$, that is $det\Sigma(t)={\hbar^{2}\over4}$ (the equality case in the Heisenberg relation). The time-derivative of $s(t)$ can be calculated by making use of the following relation $${d(\ln det\Sigma(t)) \over dt}~=~Tr \left({d \Sigma(t) \over
dt} \Sigma ^{-1}(t) \right),\label{ff}$$ where $$\Sigma ^{-1}(t)~=~{1 \over det\Sigma(t)} \left(
\matrix{{\sigma_{pp}(t)\over m\omega} & -\sigma_{pq}(t)
\cr -\sigma_{pq}(t) & m\omega\sigma_{qq}(t) \cr} \right).$$ Then we obtain from (\[ev\]) and (\[ff\]) that $${dA(t) \over dt}~=~{{2 \over \hbar} \sqrt{det\Sigma(t)}}
( TrY~+~Tr({\cal D}\Sigma ^{-1}(t))). \label{eon}$$
But the rate of the linear entropy increase is given by $${d{\it s(t)} \over dt} = {1\over A(t)^2}
{dA(t)\over dt},\label{linentr}$$ so the behavior of the rate of linear entropy increase is given entirely by the time derivative (\[eon\]) of the area $A(t)$ for all states (including pure states). In the following we shall find the squeezing parameter of the initial state for which the rate of increase of $A(t)$ is minimized.
Let us notice, for the beginning, that the positive $2\times 2$ matrix $\Sigma(t)$ can be diagonalized, at any particular instant $t$, in the form (see also [@s1; @b]) $$\Sigma(t)={\hbar A(t)\over 2}O^{T}(t)\left(\matrix{\aleph^{2}(t)&0\cr 0&
\aleph^{-2}(t)}\right)O(t).\label{dec}$$ where $\aleph (t)$ is a real positive number (the squeezing parameter) $A(t)$ is the area occupied by the system in phase space and $O(t)$ is an orthogonal symplectic matrix for which we will employ the usual form $$O(t)=\left(\matrix{\cos\theta(t)&-\sin\theta(t)\cr\sin\theta(t)&
\cos\theta(t)\cr}\right).$$
A similar formula holds for the ${\cal D}$-matrix $${\cal D}=\frac{\hbar\Delta}{2}O_{D}^{T}\left(\matrix{d^{2}&0\cr 0&
d^{-2}}\right)O_{D},$$ with $$O_{D}=\left(\matrix{\cos\varphi &-\sin\varphi \cr\sin\varphi &
\cos\varphi \cr}\right).$$ Here, $\Delta$ is a parameter controlling the intensity of diffusion, $d$ characterizes the degree of anisotropy and $\varphi$ is the rotation angle.
Now, (\[eon\]) and (\[linentr\]) imply $$\frac{ds}{dt}\begin{array}{|c} \\ t=0 \end{array} = \frac{1}{A(0)}\left[ -2\lambda +Tr\left(\Sigma^{-1}(0){\cal D}\right)\right].\label{yeye}$$
This result is even more general when the system is in a pure state at t=0 (so that $A(0)=1$), in the sense that it does not depend on the kind of initial states we are starting with (quasi-free states or not). Indeed, in [@ss] it was shown that $${d Tr(\rho ^2) \over dt} = 2 Tr(\rho L(\rho)) =
{2 \over \hbar} \sum_{j}(Tr(\rho V_{j} \rho V_{j}^{*})-
Tr(\rho ^2 V_{j}^{*}V_{j})) \geq 0 .$$ For a pure state $\rho ^2 = \rho$ and $\rho O \rho = Tr(\rho O) \rho$ for any selfadjoint operator $O$. Then $${dTr(\rho ^2) \over dt} = {2 \over \hbar}
\sum_{j} (\vert Tr(\rho V_{j})\vert ^2 -
Tr(\rho V{j}^{*}V_{j})) \geq 0 .$$ We have $Tr(\rho V_{j}) = a_{j} \sigma_{p} + b_{j}\sigma_{q}$. Hence we get $$\begin{aligned}
\nonumber
{d\over dt}(1-Tr(\rho^{2}))\begin{array}{|l}\\t=0\end{array}
&=&-2\lambda+{4\over\hbar^{2}}(D_{qq}\sigma_{pp}(0)
+D_{qq}\sigma_{pp}(0)-2D_{pq}\sigma_{pq}(0))\\ \nonumber
&=&-2\lambda +Tr\left(\Sigma^{-1}(0)\cal{D}\right),\end{aligned}$$ which has the same form as the rate of linear entropy increase calculated before (see (\[yeye\])) for pure ($A(0)=1$) initial quasi-free states.
We are interested in finding the states which produce the least increase of the phase-space area at the initial moment $t=0$. By minimizing the expression $$\begin{aligned}
\nonumber
-2\lambda +Tr\left(\Sigma^{-1}(0)\cal{D}\right)&=& -2\lambda + \frac{\Delta}{A(0)}\{\cos^{2}(\theta (0) -\varphi )[\aleph^{2}(0)d^{-2}+\aleph^{-2}(0)d^{2}] \\ \nonumber
& &+\sin^{2}(\theta (0) -\varphi )[\aleph ^{2}(0)d^{2}+\aleph^{-2}(0)d^{-2}]\},\nonumber\end{aligned}$$
with respect to $\aleph (0)$ and $\theta(0)$, one gets $$min\left[\frac{ds}{dt}\right]\begin{array}{|c} \\ t=0 \end{array}
=2\frac{\Delta - A(0)\lambda }{A(0)^{2}} .$$ corresponding to $$\begin{array}{c}\aleph^{*}(0)=d ,\\ \theta^{*}(0)=\varphi .\end{array}$$
So, in the general case of an anisotropic diffusion, the minimum variation of the area in phase space is obtained when the squeezing parameter of the state equals d, the degree of anisotropy of the diffusion, and the characteristic rotation angle of the correlation and diffusion matrices are equal. For an isotropic diffusion, ${\cal D}_{pp}={\cal D}_{qq}$, ${\cal D}_{pq}=0$, [*i.e.*]{} d=1 (and the rotation angle vanishes trivially from all the relations, since now the system has rotation symmetry in phase space); we obtain $\aleph^{*}(0)=1$. This case corresponds to many models of dissipation, especially from quantum optics (see [@ik]); the same result was obtained by Zurek, Habib and Paz [@z] in the context of the Caldeira-Leggett environment model. In other words, a phase-space isotropic environment favors a symmetric state, while an anisotropic environment selects states with the same degree of anisotropy.
When the initial state is pure, $$min\left[\frac{ds}{dt}\right]\begin{array}{|c} \\ t=0 \end{array}
=2\left(\Delta - \lambda\right)\geq 0,$$ where the last inequality comes from (\[s\]) and expresses the fact that the linear entropy of pure states in an environment always increases, because the pure initial states become more and more mixed.
Our result shows that the values of $\aleph^{*}(0)$ and $\theta^{*}(0)$ are independent of the overall magnitude $\Delta$ of the diffusion. They are also independent of $A(0)$. Thus, the same degree of squeezing is singled out, irrespective of the purity of the initial state, thus confirming previous insights [@z; @zp] regarding the structure of the mixed preferred states: they can be seen as the thermalization of the selected pure states.
We conclude by emphasizing the main results of this paper. In general, the pure or mixed state which produces the minimum rate of increase in the area occupied by the system in phase–space is a quasi-free state which has the same symmetry as that induced on the evolution in phase-space by the diffusion coefficients. For isotropic phase-space diffusion, the selected pure states are the coherent states.
[99]{}
W. H. Zurek, S. Habib and J. P. Paz, [*Phys. Rev. Lett.*]{} [**70**]{}, 1187 (1993).
A. O. Caldeira and A. J. Leggett, [*Physica (Amsterdam)*]{} [**121A**]{}, 587 (1983).
W.G.Unruh and W.H. Zurek, [*Phys. Rev.*]{} D [****]{}40, 1071 (1989).
W. H. Zurek, [*Progr. Theor. Phys.*]{} [**89**]{}, 281 (1993).
G. Lindblad, [*Commun. Math. Phys.*]{} [**48**]{}, 119 (1976).
A. Sandulescu and H. Scutaru, [*Ann. Phys.*]{} (N.Y.) [**173**]{}, 277 (1987).
V. Ambegaokar, [*Phys. Today*]{} [**46**]{} (4), 82 (1993).
H. Scutaru, [*Phys. Lett.*]{} A [**141**]{}, 223 (1989).
A. Isar, [*Helv. Phys. Acta.*]{}, [**67**]{}, 436 (1994).
G. S. Agarwal, [*Phys. Rev. A*]{} [**3**]{}, 828 (1971).
R. S. Ingarden and A. Kossakowski, [*Ann. Phys.*]{} (N.Y.) [**89**]{}, 451 (1975).
H. Scutaru, [*Phys. Lett. A*]{} [**200**]{}, 91 (1995).
R. Balian, C. De Dominicis, and C. Itzykson, [*Nuclear Physics*]{} [**67**]{}, 609 (1965).
[^1]: On leave from the [*Department of Theoretical Physics, Institute of Atomic Physics, Bucharest-Magurele, PO Box MG-6, Romania*]{}. Electronic address: paraoanu@physics.uiuc.edu .
[^2]: Electronic address: scutaru@theor1.ifa.ro .
|
---
abstract: 'We present the first results from the deep and wide 5 GHz radio observations of the Great Observatories Origins Deep Survey (GOODS)-North ($\sigma = 3.5 \, \mu$Jy beam$^{-1}$, synthesized beam size $\theta = 1.47\arcsec \times1.42\arcsec$, and 52 sources over 109 arcmin$^{2}$) and GOODS-South ($\sigma = 3.0\, \mu$Jy beam$^{-1}$, $\theta = 0.98\arcsec \times0.45\arcsec$, and 88 sources over 190 arcmin$^{2}$) fields using the Karl G. Jansky Very Large Array. We derive radio spectral indices $\alpha$ between 1.4 and 5 GHz using the beam-matched images and show that the overall spectral index distribution is broad even when the measured noise and flux bias are considered. We also find a clustering of faint radio sources around $\alpha=$0.8, but only within $S_{5GHz} < 150\, \mu$Jy. We demonstrate that the correct radio spectral index is important for deriving accurate rest frame radio power and analyzing the radio-FIR correlation, and adopting a single value of $\alpha=$0.8 leads to a significant scatter and a strong bias in the analysis of the radio-FIR correlation, resulting from the broad and asymmetric spectral index distribution. When characterized by specific star formation rates, the starburst population (58%) dominates the 5 GHz radio source population, and the quiescent galaxy population (30%) follows a distinct trend in spectral index distribution and the radio-FIR correlation. Lastly, we offer suggestions on sensitivity and angular resolution for future ultra-deep surveys designed to trace the cosmic history of star formation and AGN activity using radio continuum as a probe.'
author:
- 'Hansung B. Gim'
- 'Min S. Yun'
- 'Frazer N. Owen'
- Emmanuel Momjian
- 'Neal A. Miller'
- Mauro Giavalisco
- Grant Wilson
- 'James D. Lowenthal'
- Itziar Aretxaga
- 'David H. Hughes'
- 'Glenn E. Morrison'
- Ryohei Kawabe
title: 'Nature of Faint Radio Sources in GOODS-North and GOODS-South Fields – I. Spectral Index and Radio-FIR Correlation'
---
Introduction
============
Stellar mass build-up and central massive black-hole growth are two key observational constraints for understanding galaxy evolution in modern astronomy. A significant fraction of these activities are heavily obscured by dust over the cosmic history [@lefloch05; @caputi07; @magnelli11a; @whitaker17], and we need another tracer that can penetrate deep into column densities exceeding $N_{HI} > 10^{24}$ cm$^{-2}$ ($A_{V} \gg 100$). The completion of the NSF’s Karl G. Jansky Very Large Array[^1] (VLA) with a more than 100 times larger spectral bandwidth and a new powerful digital correlator translates to more than an order of magnitude improvement in sensitivity to probe star formation and black hole activities at cosmological distances [@perley11].
The low-frequency ($\nu \lesssim 10$ GHz) radio sky is dominated by synchrotron emission [@condon92], which mainly comes from star-forming galaxies (SFGs) and active galactic nuclei (AGN). In SFGs, synchrotron emission is generated through cooling of cosmic rays accelerated by shocks associated with Type II supernovae. In AGN, synchrotron radiation is produced by relativistic charged particles in radio cores and jets. Different origins of the observed synchrotron radiation are encoded in radio spectral index $\alpha$, which is defined as $S \propto \nu^{-\alpha}$, where $S$ is the flux density and $\nu$ is the frequency. Star-forming regions are optically thin to synchrotron radiation, which yields a steep, characteristic radio spectral index of $\alpha \approx$ 0.8 [@condon92]. Synchrotron emission in AGN is produced in two different ways. Radio core AGN are optically thick enough to absorb synchrotron emission and re-emit, which makes the slope of the synchrotron radiation flatter (“synchrotron self-absorption”), $\alpha \ll 0.8$ [@debruyn76]. In jets, relativistic electrons lose their energy over time while traveling down the length of the jets, and the resulting radio spectral index is steeper (“synchrotron aging”), $\alpha > 0.8$ [e.g., @burch79].
Radio spectral indices have been used to study the nature of radio sources. In particular, the emergence of flat spectrum sources in the sub-mJy regime has been reported by several authors [e.g., @donnelly87; @prandoni06; @randall12], although others have reported no flattening in the mean spectral index [@fomalont91; @ibar09]. Deeper radio observations with $\mu$Jy sensitivity have shown that the fraction of steep spectrum sources increases with decreasing flux density, suggesting the emergence of SFGs at the sub-mJy level [@ibar09; @huynh15; @murphy17], in agreement with the interpretation of the normalized number counts [@owen08; @condon12] and the analysis of the polarization [@rudnick14]. A radio study of sub-millimeter galaxies (SMG) has showed that their radio spectral index distribution is a skewed Gaussian with a peak near $\alpha\sim0.7$ and a tail towards flatter spectrum [@ibar10]. These studies indicate a promising potential for the radio spectral index as a tracer of underlying physical activity in distant galaxies.
We show here that obtaining [*correct*]{} measurements of radio spectral indices is critically important in calculating the rest-frame radio power and for understanding the cosmic evolution of the faint radio population. Any uncertainty in radio spectral index translates directly to the uncertainty in derived radio power, and this in turn affects the accuracy of the radio-far infrared (FIR) correlation analysis [@gim15; @delhaize17]. Radio AGNs with jets are often resolved by interferometric observations, and even normal SFGs show spatially resolved structures at arcsecond scales [e.g., @chapman04; @barger17].
In this paper, we present the analysis of radio spectral indices between 1.4 and 5 GHz derived with matched beams, for a large sample of faint radio sources identified from the deep and wide 5 GHz radio observations on the GOODS-North (GN) and -South (GS) fields. We examine the correlations among radio spectral index, radio-FIR correlation, and star formation properties. We also discuss the limitations of radio observations tracing normal SFGs, the importance of correct derivation of radio spectral index, and the constraints provided by radio spectral index to classifying radio SFGs. Throughout this paper we adopt the cosmological parameters, $H_{0}=$ 67.8 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{m}=$ 0.308, and $\Omega_{\Lambda}=$ 0.692 [@pdg18].
Observations \[OBS\]
====================
Radio Observations \[RADIO\]
----------------------------
### GOODS-North \[RADIO\_GN\]
Our observations of the GN field were conducted in February and March of 2011, for a total of 22 hours at 5 GHz in the B-configuration of the VLA under the program code [**10C-225**]{}. As summarized in Table \[observations\], we observed two fields with the VLA’s Wideband Interferometric Digital Architecture (WIDAR) correlator which was configured to deliver two 128 MHz sub-bands in full polarization. The sub-bands were further split into 64$\times$2 MHz channels each, and centered at 4896 and 5024 MHz, respectively. The correlator integration time was 3 seconds.
[ccccc]{} Field & R.A. (J2000) & Dec. (J2000) & Date & Duration\
& & & 2011 Feb 28 & 5.5 hrs\
& & & 2011 Mar 10 & 5.5 hrs\
& & & 2011 Mar 15 & 5.5 hrs\
& & & 2011 Mar 20 & 5.5 hrs\
& 03$^{h}$ 32$^{m}$ 30$^{s}$.00 & -27$^{\circ}$4345.0& 2012 Dec 16 & 2.5 hrs\
& 03$^{h}$ 32$^{m}$ 13$^{s}$.33 & -27$^{\circ}$4552.5& 2012 Dec 23 & 2.5 hrs\
& 03$^{h}$ 32$^{m}$ 13$^{s}$.33 & -27$^{\circ}$5007.5& 2012 Dec 31 & 2.5 hrs\
& 03$^{h}$ 32$^{m}$ 30$^{s}$.00 & -27$^{\circ}$5215.0& 2013 Jan 01 & 2.5 hrs\
& 03$^{h}$ 32$^{m}$ 46$^{s}$.67 & -27$^{\circ}$5007.5& 2013 Jan 05 (1) & 2.5 hrs\
& 03$^{h}$ 32$^{m}$ 46$^{s}$.67 & -27$^{\circ}$4552.5& 2013 Jan 05 (2) & 2.5 hrs\
The calibration and reduction of the VLA data were carried out using the standard data reduction package Astronomical Image Processing System (AIPS). The flux calibrator 3C286 was used for the calibrations of delay, flux density scale, and polarization while the gain calibrator J1400+6210 was used for the bandpass and gain calibration. The radio quasar J1400+6210 is bright enough (1.72 Jy at 5 GHz) to be used for the bandpass calibration.
Imaging of the visibility data was performed using the Common Astronomy Software Applications [CASA, @mcmullin07]. The wide field imaging of each field was carried out using nine facets, each with 4096$\times$4096 pixels with a cell size of 0.35, down to 3$\sigma$. The Clark point spread function (PSF) model is adopted, and the Briggs function is used to weight the data with a robust value of $R=1$. The Briggs weighting function is intermediate between natural (lowest noise, poorest resolution) and uniform (highest noise, best resolution) weighting functions, and the robust factor of $R=1$ gives an optimal compromise between sensitivity and resolution. The final mosaic and sensitivity images incorporating the primary beam correction are produced using the AIPS tasks LTESS and STESS, respectively. The final mosaic image has a size of 5120$\times$5120 pixels, centered at \[12$^{h}$ 36$^{m}$ 49$^{s}$.4, 62$^{\circ}$ 12 50.5\] (J2000), with a synthesized beam of 1.47$\times$1.42. The effective central frequency of the image is 4.959 GHz (hereafter 5 GHz) with a total bandwidth of 240 MHz. The final noise is $\sigma=3.5$ $\mu$Jy beam$^{-1}$ in the image center. The survey coverage map for the GN field is shown in panel (A) of Figure \[coverage\].
### GOODS-South \[RADIO\_GS\]
The GS field was observed at 5 GHz for a total of 15 hours in the A-configuration of the VLA under the project code of [**12B-274**]{}. The coordinates of the six pointing centers and observation dates are listed in Table \[observations\]. The WIDAR correlator was configured to deliver sixteen 128 MHz sub-bands, each with 64$\times$2 MHz channels and full polarization products. The frequency span was from 4488 to 6512 MHz. Correlator integration time was 1 second to minimize the time smearing effect. The observations were executed in six different sessions, each with 2.5 hrs long.
Data reduction and imaging were performed using CASA. The flux density scale calibrator 3C48 was used for the calibrations of delay, flux scale, and polarization, while the gain calibrator J0240$-$2309 (2.33Jy at 5 GHz) was used for calibrations of bandpass, phase, and delay. Severe radio-frequency interference (RFI) dominated the last four SPWs (12 to 15), and they are excluded in the analysis. Self-calibration was carried out successfully to improve the overall dynamic range of the image using bright sources ($>$ 1 mJy) in each field.
Initial imaging was done in CASA for each field and each SPW exploiting the wide-field imaging with 36 facets that are each 10240$\times$10240 pixels in size and using a cell size of 0.1, down to 3$\sigma$. The Clark PSF and the Briggs weighting function with a robust value of $R=0.8$ are adopted for imaging. The synthesized beam depends on the frequency, and all images are convolved to match the largest beam at the lowest frequency SPW before the final mosaic image is constructed. Using the weights of $w_{i}=(beam\; area)_{new}/(beam\; area)_{old}$, all images were convolved to have beam sizes of 0.98$\times$0.45. The mosaic image of each SPW is produced first using the AIPS tasks LTESS and STESS with primary beam correction. The final band-merged image is produced by averaging the SPW mosaic images using the $1/\sigma^{2}$ weight, where $\sigma$ is an RMS noise of each mosaic. The final band-merged mosaic image is 16384$\times$16384 pixels in size with the central frequency of 5.245 GHz (hereafter 5 GHz) and a total bandwidth of 1.486 GHz. The RMS noise in the center of the mosaic is $\sigma = 3.0\, \mu$Jy beam$^{-1}$, and the coverage map centered on \[03$^{h}$ 32$^{m}$ 30$^{s}$.0, $-$27$^{\circ}$ 48 00\] is shown in panel (B) of Figure \[coverage\].
### Source Catalogs \[CATALOG\]
The 5 GHz sources are extracted from primary-beam corrected images using the AIPS task SAD. Since radio-frequency interference is time-dependent and the primary beam response is not uniform, the final noise distribution is not uniform or symmetric across the mosaic. Therefore, we limit the source search for generating the catalogs to the central regions with up to twice the RMS noise in primary-beam corrected maps, i.e., $7\, \mu$Jy beam$^{-1}$ for the GN field and $6\, \mu$Jy beam$^{-1}$ for the GS fields as shown with inner red contours in Figure \[coverage\]. We also minimized the impact of the effective frequency shift to lower frequency toward the edge of the final image. Since the coverage of the image is different at each SPW due to the frequency-dependent primary beam correction, the effective frequency moves to the lower frequency toward the edge of the frequency-stacked image. We have created a matching sensitivity map to track the frequency-dependent effects in the final mosaic. We also limited our catalog to the more central region reasonably far away from the edges. Sources detected with a peak signal-to-noise ratio (SNR) $>$ 5 are selected for the final catalogs, and the measured flux densities are corrected for bandwidth smearing by setting the AIPS adverb BWSMEAR as the fraction of channel width with respect to the central frequency in the SAD. However, the time averaging effect is not taken into account since its impact on the flux density is small ($<$0.1%) enough to be neglected within our catalog regions [@bridle99]. The final catalogs include 52 & 88 sources in the GN & GS fields covering 109 & 190 arcmin$^{2}$ areas, respectively. These catalogs are shown in Appendix \[CAT\].
### Comparisons with previous results \[PREVOBS\]
There are recent radio continuum observations of both GN & GS fields with comparable or higher sensitivity and at a higher angular resolution, and they offer an interesting and complementary view on the nature of the faint radio source population. @guidetti17 have studied the GN field at 5.5 GHz with an RMS noise of $3\, \mu$Jy beam$^{-1}$ and a synthesized beam size of 0.5, and they reported a total of 94 sources ($\ge5\sigma$) over their 154 arcmin$^{2}$ survey area. This is about 80% larger number of sources over a 50% larger area with a similar flux density sensitivity compared to our survey. At least part of this difference must be due to their 3 times smaller beam (9 times worse surface brightness sensitivity), which can fragment some of the resolved star-forming galaxies and jet sources into multiple components. @guidetti17 also suggested this surface brightness sensitivity effect as the root cause for their unexpectedly large (80%) AGN fraction.
Earlier surveys of the GS field by @kellermann08 at 4.9 GHz using the VLA and by @huynh15 5.5 GHz using the Australia Telescope Compact Array were both about a factor of 2 shallower in sensitivity ($\sigma\approx8\, \mu$Jy) and 2-3 times lower in angular resolution ($\theta\approx 4\arcsec$) compared to our survey. @huynh15 reported finding 212 source components over their 0.34 deg$^2$ survey area down to a flux density of $\sim50$ $\mu$Jy ($\ge5\sigma$). @kellermann08 did not report the source count in their 4.9 GHz VLA survey, but @huynh15 reported their data to be consistent because of their similar resolution and sensitivity. The 5 GHz source density derived from these surveys with $\sim3$ times shallower depth is 2.6 times lower than our survey.
More recently, @rujopakarn16 have observed the Hubble Ultra Deep Field (HUDF) within the GOODS-South at 6 GHz with an RMS noise of $0.32\, \mu$Jy beam$^{-1}$ at an angular resolution of 0.61$\times$0.31. A direct comparison of the source density is difficult in this case because these authors report two source counts that are not fully reflective of the true source density: (1) a total of 68 “bright" ($\ge8\sigma$) sources within the 61 arcmin$^2$ survey region extending beyond the primary beam; and (2) a total of 11 sources detected at $\ge5\sigma$ among the 13 sources detected by ALMA inside the 40.7 arcmin$^{2}$ ALMA survey area. The former number offers a more useful comparison, and corresponds to about 2.5 times higher source density at 6-8 times better sensitivity compared with our survey. The latter number is strictly a lower limit since it includes only ALMA-detected sources at $z=1-3$. The resulting source density is only 60% of the source density we derive, despite their 10 times better flux density sensitivity.
In summary, the source density we derive is consistent with those of the past surveys. A striking trend seen is that the derived source density increases relatively slowly with improved sensitivity. There are potentially important systematic differences in how the catalogs are generated, and these source counts are not corrected for completeness in a consistent way. Nevertheless, the rise in source density with improving depth of the survey is far flatter than the Euclidean case. Along with the improving sensitivity, subsequent observations have also employed higher angular resolution, and this might play an important role in the derived source statistics, as discussed further below in § \[RESOLUTION\]. This also serves as one of our motivations for using beam-matched data for our spectral index analysis (see § \[ALPHA\]).
Multi-wavelength data \[MULTI\]
-------------------------------
### VLA 1.4 GHz \[DATA14\]
1.4 GHz data are needed to calculate the radio spectral index with our 5 GHz data. For the GN field, we use the deep 1.525 GHz (hereafter 1.5 GHz) imaging data obtained by @owen18 with RMS noise of 2.2$\, \mu$Jy beam$^{-1}$ and an angular resolution of 1.6$\times$1.6(FWHM). @owen18 have used different beam sizes (2, 3, 6, and 12) to measure the flux densities of extended sources because those sources were resolved out with the original beam size, which resulted in the prevention of the loss of flux densities. All of our 5 GHz sources have a matching counterpart in the 1.5 GHz source catalog.
For the GS field, we use the 1.4 GHz VLA data by @miller13, which has RMS noise of $\sim$6 $\mu$Jy beam$^{-1}$ at the image center with a beam size of 2.8$\times$1.6. Since the beam area of these 1.4 GHz data is about ten times larger than our 5 GHz data and the depth of the 1.4 GHz data is significantly shallower than in the GN field, matching the counterparts to the 5 GHz sources is more complicated. We convolve the 5 GHz images for each field and SPW to yield a beam size of 2.8$\times$1.6 using the AIPS task CONVL, and the final mosaic is produced by summing over all pointings and SPW using the AIPS tasks LTESS and STESS.[^2] The RMS noise of the convolved 5 GHz image is slightly higher, 6.4 $\mu$Jy beam$^{-1}$. We generated the 3$\sigma$ catalog from the convolved image using the AIPS task SAD. For the 38 sources that were not found in this 3$\sigma$ catalog due to increased noise and low completeness at low SNR, we manually performed aperture photometry on the convolved image centered on the source coordinates from the original, full resolution image. A total of 83 sources are identified in the final convolved 5 GHz mosaic image with a beam size of 2.8$\times$1.6, as eight of the sources in the original catalog are now blended into three sources. Matching the 1.4 GHz catalog with this beam-matched 5 GHz data yields 64 counterparts among the 83 sources. A total of 19 sources lack a 1.4 GHz counterpart because the 1.4 GHz data are too shallow (5$\sigma$ $\geq$ 30 $\mu$Jy beam$^{-1}$ at the image center) to detect 5 GHz sources with a flat or inverted spectrum which is a characteristic of some of the radio AGNs (see the panels (D) and (E) of Figure \[alpha\_flux\]). Throughout this paper, we analyze only the GS sources that have a unique 1.4 GHz counterpart to avoid the uncertainty introduced by the upper limits.
### Chandra X-ray Observatory \[XRAY\]
We use X-ray data taken from the [*Chandra X-ray Observatory*]{} survey with full band (0.5-7 keV), soft band (0.5-2 keV), and hard band (2-7 keV) catalogs. We make use of 2 Ms observations for the GN field [@xue16] and 7 Ms observations for the GS field [@luo17]. The limiting fluxes for the GN field are $3.5 \times 10^{-17}$, $1.2 \times 10^{-17}$, $5.9 \times 10^{-17}$ erg cm$^{-2}$ s$^{-1}$ at full band, soft band, and hard band respectively. For the GS field, the limiting fluxes are $1.9 \times 10^{-17}$ at full band, $6.4 \times 10^{-18}$ at soft band, and $2.7 \times 10^{-17}$ erg cm$^{-2}$ s$^{-1}$ at hard band. To calculate the X-ray luminosity, we assume a photon index of $\Gamma=1.8$ for X-ray detected radio sources [@tozzi06] but $\Gamma=1.4$ for X-ray undetected radio sources [@luo17]. The full band X-ray luminosity at \[0.5-7 keV\] is converted to the luminosity at \[0.5-8 keV\] using the relation of $L_{[0.5-8 keV]} = 1.066 \times L_{[0.5-7keV]}$ for the assumed $\Gamma=1.8$ [@xue16].
### Spitzer Space Telescope \[SPITZER\]
We exploit publicly released $Spitzer$ $Space$ $Telescope$ ($Spitzer$) IRAC catalogs of the GN [@wang10] and GS [@damen11] fields. The GN field IRAC catalog has a sensitivity (1$\sigma$) of 0.15$\mu$Jy at 3.6$\mu$m, while the GS field IRAC catalog by the $Spitzer$ IRAC/MUSYC Public Legacy Survey in the Extended Chandra Deep Field-South (ECDFS) has a sensitivity (1$\sigma$) of 0.22$\mu$Jy at 3.6$\mu$m. We make use of the high angular resolutions of our radio observations to find counterparts within the beam sizes, i.e., 1.47 for the GN and 0.98 for the GS fields.
### Herschel Space Observatory \[HERSCHEL\]
The comparison FIR data are constructed using the public archival data for the Photodetector Array Camera and Spectrometer (PACS) and the Spectral and Photometric Imaging REceiver (SPIRE) of the $Herschel$ $Space$ $Observatory$[^3]. The PACS photometry data at 70, 100, and 160 $\mu$m are taken from the combination of PACS Evolutionary Probe [@lutz11 PEP] and GOODS-Herschel [@elbaz11] programs described by @magnelli13. The SPIRE 250, 350, and 500 $\mu$m photometry data are taken from the Herschel Multi-tiered Extragalactic Survey (HerMES) DR 3 and 4 [@roseboom10; @magnelli11b; @roseboom12]. We adopt the catalogs extracted using the $Spitzer$ MIPS 24 $\mu$m position priors for the PACS bands by the GOODS-Herschel collaboration[^4]. As for the SPIRE bands, we used the catalogs extracted at the SPIRE 250 $\mu$m source positions (HerMES DR4)[^5].
To identify FIR counterparts to the radio sources, we apply the likelihood ratio technique [@sutherland92]. The search radius adopted is three times the combined positional uncertainties of the radio and Herschel sources. Sources with the reliability of $Rel_{i}>$0.8[^6] are accepted as formal counterparts. We consider an FIR source to be the counterpart to a radio source if it is detected in at least one band in both PACS and SPIRE, with a SNR$>4$ in at least one band.
We have compiled the observed 24, 100, 160, 250, 350, and 500 $\mu$m band fluxes of 40 GN and 44 GS sources. The best-fit FIR SED models are identified using a widely used SED fitting code $Le$ $Phare$[^7] [@arnouts99; @ilbert06] with various SED templates for SFGs [@chary01; @dale01; @lagache03] and QSOs [@polletta07]. This analysis yielded a good SED model for 39 GN and 42 GS sources. For the radio sources undetected at FIR or with a poor-fit SED, we calculate IR luminosity with 4$\sigma$ flux limits adopting the average $z=1$ SFG SED template [@kirkpatrick12].
### Spectroscopic redshifts \[SPECZ\]
Spectroscopic redshifts are compiled from the published surveys: GN [@cowie04a; @donley07; @barger08; @wirth15] and GS [@szokoly04; @zheng04; @mignoli05; @ravikumar07; @vanzella08; @popesso09; @straughn09; @balestra10; @silverman10; @cooper12; @kurk13; @lefevre14; @skelton14; @morris15], respectively. From these compilations, we have 45 (out of 52) sources with spectroscopic redshifts for the GN and 55 (out of 64) for the GS field. In particular, all 55 GS sources with a spectroscopic redshift are in the subsample of 64 sources with both 1.4 GHz and 5 GHz photometry used for the spectral index analysis.
Even though reliable photometric redshifts from well-sampled photometry data exist in both fields, we limit our analysis to only those with a spectroscopic redshift because errors in redshift translate directly to a large scatter and systematic biases in the derived quantities such as the rest frame radio power, radio-FIR correlation, and star formation rate (SFR). A detailed evaluation of the accuracy of the existing photometric redshifts and a quantitative analysis on the magnitude of error introduced by using photometric redshifts using this spectroscopic subsample are presented in Appendix \[ZPHOT\]. Adding those sources with only photometric redshifts to our statistical analysis can in principle expand our sample by up to 16%, but we have elected to remove this major source of scatter in our statistical analyses presented here for now.
### 3D-HST \[3DHST\]
We adopt physical parameters such as stellar mass, SFR, and effective radius for our 5 GHz sources that also appear in the 3D-HST [^8] [@brammer12; @skelton14; @momcheva16] database. Stellar mass is estimated by the FAST code [@kriek09] with the @chabrier03 initial mass function, and the @bc03 stellar population synthesis library [@skelton14]. The SFR is computed through the conversion of UV+IR luminosity, where UV luminosity is derived from the rest-frame luminosity at 2800Å, and IR \[8-1000$\mu$m\] luminosity is derived from $Spitzer$ MIPS 24$\mu$m flux density by assuming the log average of @dh02 templates [see @whitaker14]. Effective radius (R$_{eff}$) is the semi-major axis of the ellipse containing one half of the total flux of the best Sérsic model given by GALFIT [@vanderwel12].
The spectroscopic redshifts given in the 3D-HST database are not as complete as our compilation, and we have to match our spectroscopic redshifts with the best redshifts in the 3D-HST database, which ranks them by spectroscopic, grism, or photometric redshift. A comparison of the best 3D-HST redshifts with our spectroscopic redshifts is shown in Figure \[specz\_3dhst\]. We choose the 3D-HST counterparts with best redshifts satisfying $|z_{spec}-z_{best,3D-HST}|/(1+z_{spec}) < 0.05$, which is shown with dashed lines in Figure \[specz\_3dhst\]. Spectroscopic redshifts of the best redshifts in 3D-HST are mostly the same as ours while there are some small to significant offsets in grism and photometric redshifts. Through matching the redshifts, we have 3D-HST counterparts for 39 GN and 45 GS radio sources.
![Comparison of our spectroscopic redshifts and the best redshifts in 3D-HST for the GN (square) and the GS (triangle) fields. The color represents the type of redshift measurements, e.g. spectroscopy (blue), grism (green), and photometry (red). The solid line is the one-to-one line and dashed lines show the selection limits of $\pm 0.05$ in $|z_{spec}-z_{best,3D-HST}|/(1+z_{spec}) < 0.05$. \[specz\_3dhst\]](f2.pdf)
Selection Function and Rest Frame 5 GHz Radio Power \[P5GHz\]
=============================================================
In Figure \[lumz\], we show the selection function of our radio sources with rest-frame 5 GHz radio power as a function of redshift. The rest-frame radio power is calculated using the measured spectral index as $$P_{5GHz}=4 \pi d_{L}^{2} S_{5GHz} (1+z)^{\alpha-1} [{\rm W\, Hz}^{-1}],$$ where $d_L$ is the luminosity distance, $S_{5GHz}$ is the measured 5 GHz flux density of the original map, and $\alpha$ is the measured radio spectral index between 1.4 GHz and 5 GHz using the convolved map (see § \[ALPHA\]). The strong positive k-correction associated with radio sources translates to a significant selection bias in favor of flat spectrum sources ($\alpha=0$, dashed line) with lower intrinsic radio power, but such flat spectrum sources are rare in our sample, as shown in this plot (also see Fig. \[alpha\_flux\]). The selection functions of the two fields are similar with comparable mean and median values of 5 GHz radio power and redshifts, and a joint analysis of the combined sample is reasonable as long as the slight difference in the catalog depth is properly taken into account.
The majority of the detected sources have rest frame 5 GHz radio power between $10^{22}$ and $10^{24}$ W Hz$^{-1}$, which is the range of radio power associated with intense starburst systems (LIRGs, ULIRGs) and Seyfert nuclei in the local universe. However, gas content and SFR of star forming main sequence (MS) galaxies are known to increase rapidly with increasing redshift by an order of magnitude to $z\ge1$ [e.g., @speagle14; @scoville17], and a large fraction of these galaxies at higher redshifts are likely powered by star formation, as discussed below. Only four sources (two in each field) have a radio power high enough to be classified as “radio-loud" with $P_{5GHz}\ge 10^{25}$ W Hz$^{-1}$ [@miller90].
Radio Spectral Index \[ALPHA\]
==============================
Radio spectral index $\alpha$ is a measure of the shape of a radio spectrum characterized as a power-law, $S \sim \nu^{-\alpha}$. We compute the spectral index between 1.4 and 5 GHz using the flux densities derived from the 5 GHz images beam-matched to the 1.4 GHz images as described in Section \[DATA14\]. In principle, the radio spectral index can be estimated using only the 5 GHz data with its wide bandwidth of 1.5 GHz through the multi-frequency synthesis. The algorithm that can produce in-band spectral index calculation for mosaic observations was not available in CASA when the data were being analyzed. The significant changes in the size of both the primary beam and the synthesized beam across the bandwidth make this in-band spectral index calculation challenging, especially away from the pointing center. These difficulties result in the errors of the in-band spectral index which are not competitive with those using the full 1.4-5 GHz spectral baseline. It is empirically shown that the majority of radio sources in a wide range of redshifts show radio spectra that are fit well with a simple power-law [e.g., @klamer06]. In the frequency range between 1.4 and 5 GHz, the contribution by free-free emission is generally negligibly small [@condon92].
The distributions of radio spectral index as a function of flux density are shown in Figure \[alpha\_flux\]. Panels (A), (B), and (C) are for the GN field while the panels (D), (E), and (F) are for the GS field. Since the sensitivity for radio spectral index (dotted line) depends on the flux density limit of the second band (dashed lines), it is not uniform as a function of flux density, and this is a common but important feature for all flux-limited surveys. Specifically, this non-uniform completeness limits our study to a narrower range of radio spectral indices at fainter flux densities. For our 5 GHz selected sample analyzed here, the depth of the existing 1.4 GHz data restricts the observable range of radio spectral index. We can see this effect clearly in panel (D), where the range of the radio spectral indices is limited to $\alpha > 0$ even at $S_{5GHz}=30\, \mu$Jy (10$\sigma$), and this can potentially lead to missing sources with inverted spectra at flux densities of $S_{5GHz} < 30\, \mu$Jy. In practice, however, few inverted spectrum sources with $S_{5GHz} < 35\, \mu$Jy ($10\sigma$) are found in the GN field (panel A), and the actual impact of this potential bias may be limited.
The uncertainties in the derived radio spectral indices are mainly attributed to the larger uncertainties of flux densities at 5 GHz for the GN field and flux densities at 1.4 GHz for the GS field. The radio spectral index distribution in the GS field is broader and smoother than that in the GN field, and this can be attributed to the shallow depth of the 1.4 GHz data and the noisier 5 GHz photometry as a result of the convolution with a larger Gaussian kernel. Another source of the uncertainty is the wide bandwidth of the VLA. The effective frequency of each flux density measurement depends on the bandwidth and the spectral shape of the source, and this could lead to a significant offset of the effective frequency from the instrumental frequency. For the steepest spectrum source with $\alpha=$1.64 in the GS field, we estimate that this effect can lead to a maximum frequency offset of 0.1 GHz and a maximum deviation of 0.02 in the derived radio spectral index. Thus, we conclude that this effect has only a minor impact on our radio spectral index calculation. When these systematic effects are taken into account, the distributions of radio spectral indices in these two fields are consistent with each other.
Panel (A) in Figure \[alpha\_flux\] shows a clustering of radio sources at $\alpha \sim$0.75 and $S_{5GHz} \le 150\, \mu$Jy, leading to a prominent peak in the histogram in panel (C). The peak of the radio spectral index histogram for the GS field (panel F) occurs at the same $\alpha$ value, but the clustering is not as pronounced, possibly diluted and broadened by the larger uncertainties in the measured radio spectral indices (see panels B & E). This peak in the $\alpha$ of steep spectrum radio sources at $S_{5GHz} \le 150\, \mu$Jy has not been reported by earlier studies [e.g., @donnelly87; @fomalont91], but their small sample size (30 in @donnelly87 and 41 in @fomalont91) likely contributed to their poor statistics. A more recent study of a larger sample by @huynh15, who measured radio spectral indices of 5.5 GHz selected sources above $S_{5.5GHz} \gtrsim 50\mu$Jy in the Extended Chandra Deep Field-South (ECDFS) using the 1.4 GHz catalog of @miller13, did report a spectral index distribution with a clear peak near $\alpha\sim0.7$, as long expected of the star forming galaxy population (see the discussion below). We note that @huynh15 computed their radio spectral index without matching the beam sizes (about a factor of 2.2 in diameter), and this might be a source of an important systematic error – see further discussions in § \[DISC\_SI\].
A natural explanation for the peak near $\alpha \sim 0.7$ is the contribution by the SFG population. Synchrotron emission is optically thin when it is produced by the shocks associated with supernovae in SFGs [@condon92; @seymour08]. The flattening or upturn in the number counts of radio sources seen around $S_{20cm} \leq 100-200\, \mu$Jy [@owen08; @condon12] is explained by the emergence of this population of SFGs at faint flux density levels, exceeding those of the radio-loud AGN population that is dominant at flux densities $\geq 1\,$mJy. The increase of fractional polarization and the change of slope in the polarized number count at polarized flux densities $\leq$1 mJy also imply the increasing contribution of SFGs [@rudnick14]. The broad radio spectral index distributions for the GS and GN fields shown in Figure \[alpha\_flux\] suggests the existence of both steep spectrum ($\alpha=0.5-1.0$) and flatter or inverted spectrum ($\alpha < 0.5$) sources at $S_{5GHz} < 150\, \mu$Jy, supporting the conclusions of the more recent analyses indicating that the faint $\mu$Jy radio population consist of both SFGs and radio-quiet AGN [@padovani09; @bonzini13; @rudnick14]. A detailed study of a small sample of 14 local SFGs by @klein18 has shown that there is also some scatter in the observed radio spectral index in the GHz range due to a varying degree of free-free emission and opacity effects. What our study further indicates is that a larger sample with higher quality radio spectral index measurements are needed to characterize the relative contribution by these two populations.
Star Formation Properties of Radio Sources \[RADIO\_SFMS\]
==========================================================
In the previous section, we have shown and discussed the distributions of radio spectral indices derived between 1.4 and 5 GHz from the beam matched images. In this section, we investigate how the radio spectral index correlates with star formation properties by utilizing the SFRs and stellar masses derived by the 3D-HST project [@skelton14; @momcheva16].
$\Delta_{SFR}$ as a measure of SF activity \[DELTASFR\]
-------------------------------------------------------
The distributions of SFR and stellar mass of radio sources in GN (squares) and GS (triangles) are shown in four redshift bins in Figure \[sfr\_mass\_z\]. The dashed lines indicate the SFR-stellar mass relation of the star forming MS at a mean redshift in each panel, and the shaded regions represent dispersions of SFR-stellar mass relation at the MS with $log_{10} SFR - log_{10} SFR(MS) = \pm 0.2$ [@speagle14]. As @speagle14 and others noted, the MS evolves strongly with redshift, and it is not clear whether the SFRs measured at different redshifts can be compared directly in a meaningful way. A more insightful measure might be the level of SF activity normalized by that of the MS at the same redshift. Therefore, we define “$\Delta_{SFR}$", the logarithm of the ratio of SFR with respect to that of the MS, as
$$\Delta_{SFR} \equiv log_{10} SFR - log_{10} SFR(MS),$$
where $SFR(MS)$ is the $SFR$ for the star forming MS galaxy at a given stellar mass and redshift calculated using Equation (28) by @speagle14.
![Offset of SFR from the MS ($\Delta_{SFR}$) and stellar mass with a color code according to radio spectral index in GN (squares) and GS (triangles) fields. Small gray points are the 3D-HST galaxies without radio counterparts in GN and GS fields for a comparison. The dashed lines of $\Delta_{SFR}= \pm 0.2$ indicate the selection of SFGs. SBs are sources above a line of $\Delta_{SFR} > 0.2$ while quiescent galaxies are those below a line of $\Delta_{SFR} < -0.2$. The distribution of radio spectral index show that SFG+SB have mainly steep spectra while quiescent galaxies have flatter spectra even though both have wide distributions. \[sfms\_selection\]](f6.pdf)
Following @speagle14, we define “SFGs" as galaxies with $-0.2 \le \Delta_{SFR} \le 0.2$, “starbursts (SBs)" as those with $\Delta_{SFR} > 0.2$, and “quiescent galaxies" as those with $\Delta_{SFR} < -0.2$. In total, we have 49 SBs (58%), 10 SFGs (12%), and 25 quiescent galaxies (30%). The dominance of the SB population among the $\mu$Jy radio population identified by one of the deepest surveys thus far is somewhat surprising, but this reflects the selection bias driven by the survey depth as discussed further below (also see § \[RADIO\_OBS\]).
In Figure \[sfms\_selection\], we show the distribution of $\Delta_{SFR}$ as a function of stellar mass, color-coded by radio spectral index, $\alpha$. Quiescent galaxies detected in radio continuum are on average more massive than the SFG+SB while the SFG+SB show a wider range of stellar masses as shown in Figure \[sfms\_selection\]. The median stellar masses are 3.8$\times$10$^{10} M_{\bigodot}$ for SFG+SB and 9.3$\times$10$^{10} M_{\bigodot}$ for quiescent galaxies, respectively. The two-sided Kolmogorov-Smirnov test for two samples in R [@rcite] indicates that stellar mass distributions in both populations are substantially different with a p-value of $< 4.3 \times 10^{-5}$. This significant difference in mass distributions is consistent with the mass quenching scenario for quiescent galaxies [e.g., @kauffmann03].
The majority of our radio sources (58%) show intense star formation activity with $\Delta_{SFR} > 0.2$ while only 12% of radio sources fall within the range of MS SFGs with $-0.2 < \Delta_{SFR} < 0.2$. For comparison, we show the 3D-HST galaxies without radio counterparts (light gray) in Figure \[sfms\_selection\]. In the same stellar mass range as the radio sources (log $M_{*} \ge 9.08$), the 3D-HST galaxies undetected in radio are classified into SBs (25%), SFGs (44%), and quiescent galaxies (31%). The fraction of quiescent galaxies among source undetected in radio is the same as radio detected sources. Therefore, the main difference is in the fraction of SBs. In all cases, the radio detected galaxies trace the high stellar mass envelope for all types of galaxies, independent of $\Delta_{SFR}$, and this is a natural consequence of a flux-limited survey as demonstrated by our selection function shown in Figure \[lumz\]. Since our radio observations trace the synchrotron emission from star formation and AGN activities, these statistics imply that our radio survey is not deep enough to detect the star formation activity in the star forming MS galaxies in the full range of redshift probed, even with $\mu$Jy sensitivity. We discuss this finding in more detail in § \[RADIO\_OBS\].
Star Formation Activity and Radio Spectral Index \[SFRSI\]
----------------------------------------------------------
An apparent correlation between radio spectral index and star formation property ($\Delta_{SFR}$) is hinted in the color-coded data for radio spectral index in Figure \[sfms\_selection\]. Steep spectrum sources with $\alpha > 0.5$ (green and blue) appear predominantly in the $\Delta_{SFR} > -0.2$ region while sources with a flat or inverted spectrum ($\alpha < 0.5$, yellow and orange) appear mostly in the region below $\Delta_{SFR} = -0.2$. This might be an indication that steep spectrum sources are abundant among SFG+SB galaxies with $\Delta_{SFR} > -0.2$ while few steep spectrum sources are in the quiescent galaxy region with $\Delta_{SFR} < -0.2$.
This apparent trend is examined more directly in Figure \[alpha\_delsfr\] by plotting the radio spectral index as a function of $\Delta_{SFR}$. What is apparent now is that the SFG+SB galaxies are more tightly clustered around $\alpha \sim 0.8$, while the quiescent galaxies ($\Delta_{SFR} < -0.2$) are distributed more uniformly, spanning a nearly twice as large range in spectral index $\alpha$ – the SFG+SB galaxies have a tighter distribution with a higher mean (0.72$\pm$0.05) than the quiescent galaxies (0.22$\pm$0.11). The histograms in the panel (B) of Figure \[alpha\_delsfr\] show these trends clearly with different peak positions – the SFG+SB galaxies (blue) have a peak at $\alpha \approx 0.8$, but the quiescent galaxies (red) have a peak at $\alpha \approx 0.13$. The two-sided Kolmogorov-Smirnov test for the two samples in R indicates that the null hypothesis of their radio spectral index distributions drawn from the same parent population is rejected with a p-value of 0.0015. This result is consistent with the expectation that star formation yields steep radio spectra with $\alpha \sim 0.8$ through optically thin synchrotron emission produced by supernova shocks [@condon92] while AGN are associated with flat or inverted radio spectra with $\alpha \ll 0.8$ through synchrotron self-absorption [e.g., @debruyn76].
It is tempting to speculate that there is a weak trend of decreasing $\alpha$ with decreasing $\Delta_{SFR}$ if the handful of sources with $\alpha \ge1$ in the upper left corner of Figure \[alpha\_delsfr\] are ignored. These ultra-steep spectrum sources are generally jet-dominated AGNs, and one could separate them out morphologically, but that kind of handpicking is not generally possible for a study without the necessary spatial information.[^9] The large spread in $\alpha$ at a given value of $\Delta_{SFR}$ also makes such a generalization difficult to trust. What seems to be more certain is that this spread is real and essentially independent of star forming activity $\Delta_{SFR}$, and this has an important consequence for understanding and modeling the nature of faint radio population and their evolution, as we discuss further below.
![Radio spectral index distribution as a function of $\Delta_{SFR}$. The panel (A) shows that the radio spectral index distribution of SFG+SB ($\Delta_{SFR} > -0.2$) is more tightly clustered around $\alpha \sim 0.8$, in comparison with the quiescent galaxies ($\Delta_{SFR} < -0.2$), which are distributed more uniformly and widely in spectral index $\alpha$. These trends are easily seen in the histograms of SFG+SB (blue) and quiescent galaxies (red) in panel (B). The Kolmogorov-Smirnov test indicates that the radio spectral index distributions of the two populations are different from each other with a p-value of 0.0015. \[alpha\_delsfr\]](f7.pdf)
Radio-FIR Correlation of Radio Sources \[QFIR\_RADIO\]
======================================================
The radio-FIR correlation is one of the robust indicators of star formation and black hole activities [@helou85; @condon92; @yun01; @bell03]. In particular, the radio-FIR correlation of SFGs is a tight correlation with a less than 0.3 dex scatter over five orders of magnitudes in luminosity [@yun01], and this obviously indicates that a strong coupling exists between dust-reprocessed emission of ultraviolet radiation from massive young stars and synchrotron radiation by cosmic rays accelerated in type II supernovae [@condon92]. In this section, we examine the radio-FIR correlation of the $\mu$Jy radio sources identified in the GN & GS fields as a function of their star formation properties and their measured radio spectral index.
The rest-frame radio-FIR correlation parameter, $q_{FIR}$ is defined as $$q_{FIR} = log_{10}\left( {{L_{FIR} [W] } \over {3.75 \times 10^{12} Hz}} \right) - log_{10} P_{1.4GHz} [W Hz^{-1} ] ,$$ where $L_{FIR}$ is a rest-frame FIR luminosity from 40 to 120 $\mu$m [@helou85; @yun01]. The radio-FIR correlations of radio sources as a function of redshift are shown in Figure \[rfc\_p5\] for GN (squares) and GS (triangles), color-coded by $\Delta_{SFR}$. The overwhelming majority of the SFG+SB population (86%) follow the local radio-FIR correlation for SFGs [@yun01], and galaxies near the star-forming MS ($-0.5\le \Delta_{SFR}\le +0.5$) nearly exclusively fall within the grey band shown in the left panel of Figure \[rfc\_p5\]. On the other hand, only $\sim$30% of the quiescent galaxies have $q_{FIR}$ of local SFGs, and their radio continuum emission likely has an origin other than star formation. Most of the quiescent galaxies (76%=19/25) are not detected in the far-IR, and they are marked with a down arrow in Figure \[rfc\_p5\].
A statistical analysis of the radio-FIR correlation for each subpopulation distinguished by its star formation properties shows a clear difference between the SFG+SB galaxies and the quiescent galaxies. We have applied the Kaplan-Meier estimator for $q_{FIR}$ of the two subpopulations with the subroutine [**cenfit**]{} of the statistical package NADA[^10] in R [@rcite]. This analysis shows that the SFG+SB galaxies have a median $q_{FIR}$ value of 2.26$\pm$0.09, in good agreement with the local canonical value $<q_{FIR}>\approx 2.3$ [@yun01], while the quiescent galaxies have a median value of 1.10$\pm$0.10. The difference in these median values is quite substantial with a significance of $\sim 8.8 \sigma_{c}$ (the combined uncertainty for both populations is $\sigma_{c}=0.13$). To quantify the difference of radio-FIR correlation distributions between SFG+SB and quiescent galaxies further, we perform the Log-rank test with left-censored data using the [**cendiff**]{} function in the NADA in R [@rcite]. This test indicates that the SFG+SB galaxies and the quiescent galaxies have entirely different distributions of $q_{FIR}$ with a p-value of $< 2 \times 10^{-6}$. These statistical tests confirm the results of previous studies that the radio-FIR correlation is a powerful tracer of star formation activity [@yun01; @bell03].
An obvious trend seen in the left panel of Figure \[rfc\_p5\] is the decreasing $q_{FIR}$ with increasing 5 GHz radio power. A straightforward interpretation is that radio AGN contribution is increasing both fractionally and in absolute value for the most radio luminous objects at $P_{5GHz}\ge 10^{24}$ W Hz$^{-1}$. A somewhat surprising fact is that the majority of these “radio-excess" objects with $P_{5GHz}\ge 10^{24}$ W Hz$^{-1}$ are also intensely starbursting galaxies with $\Delta_{SFR}\gtrsim 1$. Similar objects found in the local Universe are mostly Seyfert AGNs associated with a nuclear starburst, but they are exceedingly rare, accounting for only 1% of the $IRAS$ 2 Jy Sample studied by @yun01. One might conclude a sharp increase (up to $\sim$5%) of such AGN+SB hybrid objects at $z>1$, but our sample size is too small to be highly quantitative. Furthermore, survey depth and sample definition might have a strong influence in such an inference as even our $\mu$Jy sensitivity is not sufficient to probe the MS star forming galaxies (see below § \[RADIO\_OBS\]). Indeed, both the AGN fraction and the radio-excess fraction reported by the deeper survey of the COSMOS field by @smolcic17b are much higher, $\sim$20%, at the $S_{1.4GHz}=50\, \mu$Jy and rising up to $\sim$50% at $S_{1.4GHz}=100\, \mu$Jy (see their Figure 12). A similar result was also reported by a study with a different AGN identification using the VLBA observations on the same field, where the AGN fraction is $>$40$-$55% at 100 $< S_{1.4GHz} < $ 500 $\mu$Jy [@herrera-ruiz18].
The dependence of radio-FIR correlation on radio spectral index is examined on the right panel of Figure \[rfc\_p5\], and the quiescent galaxies with $\Delta_{SFR}\le$ -0.2 show systematically lower $q_{FIR}$ (on average by 0.6-0.8) compared with the SFG+SB population, nearly independent of radio spectral index $\alpha$. An in-depth analysis of the similarities and differences among these different subpopulations is discussed in our next paper (Paper II), but this is another indication that quiescent galaxies are indeed a distinct population in their radio and IR properties as well. It is interesting that the extreme steep spectrum quiescent galaxies identified in Figure \[alpha\_delsfr\] and discussed in § \[SFRSI\] are [*not*]{} extreme outliers and instead nearly follow the normal radio-FIR correlation. A real outlier in the distribution is again the radio-excess SBs with $\Delta_{SFR}\gtrsim 1$ discussed above, and their radio spectral index is typically around $\alpha\sim +0.9$, indistinguishable from the bulk of the normal SFGs and SBs. Intense starbursts associated with massive galaxies in the local universe, such as luminous infrared galaxies (LIRGs) and ultraluminous infrared galaxies (ULIRGs), are associated with high free-free opacity, leading to the flattening of radio spectrum [e.g., @klein18] and even obscuring a radio AGN altogether at longer wavelengths (e.g., Mrk 231). Therefore, the distribution and geometry of starburst activity in these $z>1$ luminous radio-excess SBs are somehow different from local examples. And they certainly cannot be identified from their luminosity and radio spectral index alone. Future higher resolution observations that can resolve the star-forming structures and kinematics are required to yield deeper insight on these sources.
Discussion \[DISCUSSION\]
=========================
Importance of Survey Sensitivity \[RADIO\_OBS\]
-----------------------------------------------
What makes deep radio continuum imaging attractive as a tool for studying galaxy evolution is the high angular resolution of an interferometer like the VLA to deliver spatial information at much better than 1, free from the fundamental limits of source confusion that restrict the usefulness of current infrared facilities such as [*Herschel*]{}. Advances in sensitivity through increased bandwidth and collecting area also enable us to probe star forming galaxies and AGN population at cosmological distances directly. One of the main goals of this VLA study of the GOODS cosmology fields is to analyze the nature of the faintest radio continuum sources detectable with the current technology and establish technical specifications for future surveys for galaxy evolution using facilities such as MeerKAT, ASKAP, and eventually the Square Killimeter Array.
The plot of rest-frame 5 GHz radio power versus spectroscopic redshift shown in Figure \[lumz\] and the analysis of their star formation properties discussed in § \[RADIO\_SFMS\] clearly demonstrate that our deep 5 GHz continuum data indeed probes star forming galaxies out to $z\sim 3$. On the other hand, our detailed examination of their specific star formation rate shown in Figure \[sfms\_selection\] finds that the fraction of SBs (58%) in our radio sources are more than twice the fraction among the parent general galaxy population in the 3D-HST survey. Since there are no reasons for radio-selected SFGs to be fundamentally different from optical or UV selected SFGs, this statistical difference is likely the result of the combined effects of our survey depth and the strong evolution of cosmic star formation rate density [see review by @madau14].
![Detectability of MS SFGs and a sensitivity of radio observations. We show the observable galaxies with a certain SFR and stellar mass as a function of redshift. We show SFGs with SFR of MS (solid lines) and $5 \times$SFR of MS (dashed lines) as a function of redshift with respect to the stellar masses of $10^{10} M_{\odot}$ (blue), $10^{11} M_{\odot}$ (green), and $10^{12} M_{\odot}$ (red). The survey limits ($5\sigma$) of our radio observations are indicated by the horizontal lines, i.e. 15$\mu$Jy for GS. As examples, we marked the maximum redshifts of detecting M82-like (red diamond) and Arp220-like (red star) galaxies at the survey limits. \[detection\]](f9.pdf)
To explore this further, we show the calculated 5 GHz radio flux density of SFGs with SFR of MS (solid lines) and $5 \times$SFR (dashed lines) for stellar masses of $10^{10} M_{\odot}$ (blue), $10^{11} M_{\odot}$ (green), and $10^{12} M_{\odot}$ (red) in Figure \[detection\]. We assume that SFR scales with 1.4 GHz radio power following the radio-total IR correlation with $q_{TIR}=2.64$ [@murphy11] and a single average radio spectral index of +0.8 (but see the discussion on potential bias below). In general, angular resolution and source size are important considerations for survey sensitivity. Here, we make a simplifying assumption that most sources detected in a deep survey like this are at high redshifts are unresolved or marginally resolved [@owen08; @murphy17; @owen18].[^11] At our 15$\, \mu$Jy ($5\sigma$) survey limit for the GS field (black horizontal line), the maximum observable redshifts for star forming MS galaxies (dashed lines) are $z$=0.13 for $10^{10} M_{\odot}$ (solid blue), $z$=0.32 for $10^{11} M_{\odot}$ (solid green), and $z$=2.55 for $10^{12} M_{\odot}$ (solid red). SFGs with $5\times$SFR of the MS can be detected out to $z$=0.41 for $10^{10} M_{\odot}$ (dashed blue), $z$=2.19 for $10^{11} M_{\odot}$ (dashed green), and $z >$3 for $10^{12} M_{\odot}$ (dashed red). In terms of well-known local SFGs, we can detect M82-like galaxy out to $z$=0.34 and Arp220-like galaxy out to $z$=1.63, respectively. Therefore, even with the $\mu$Jy sensitivity we achieved in these two GOODS fields, we can probe a main sequence SFG with a stellar mass of $10^{11} M_{\odot}$ only out to $z\sim$0.3, and our survey is strongly biased to ULIRG-like starbursts and AGN-host galaxies at $z>1$.
This same plot also demonstrates that directly probing the evolution of the star forming MS galaxies will require a [*much*]{} deeper survey. To probe a MS SFG with $SFR=10 M_{\odot}$ yr$^{-1}$ at the Cosmic Noon ($z=2.5$) at $5\sigma$, a 5 GHz radio survey needs to reach a survey sensitivity of 28 nJy with the Next Generation VLA or Square Kilometer Array. This required sensitivity is $\sim$11.5 times deeper than the existing deepest 5 GHz continuum survey of the Hubble Ultra Deep Field by @rujopakarn16 and [*more than 100 times deeper*]{} than our own surveys presented here.
Importance of Angular Resolution \[RESOLUTION\]
-----------------------------------------------
In the previous section, we discussed the importance of sensitivity in probing star forming galaxies at cosmological distances and the requirement for future surveys to improve the sensitivity by more than an order of magnitude to probe the evolution of the main sequence SFGs. However, another surprising outcomes of our deep VLA 5 GHz surveys is that simply obtaining a deeper data itself does not guarantee probing much deeper into the luminosity function. As discussed in § \[PREVOBS\], the comparison of the past and recent deep surveys seems to suggest that the rise in source density is [*apparently*]{} much flatter than the Euclidean case. Obviously this is not an entirely fair and rigorous comparison, and the situation is quite a bit more complex.
![Measured flux density comparison among the radio sources in the GS field with those reported by previous studies with different angular resolution. Those by @kellermann08 and @huynh15 with $\sim3$ times larger beams are on average $\sim$30 percent larger. The higher resolution survey by @rujopakarn16 with 0.61$\times$0.31beam has only one detected source in common [the 6 GHz source flux densities are actually reported by @dunlop17] that agrees well with ours. The dotted line is the unity ratio line to guide the eye. \[GSCompare\]](f10.pdf)
A potentially important experimental parameter here is angular resolution. Both statistical [e.g., @windhorst90; @morrison10] and direct imaging [e.g., @chapman04] studies have shown that faint radio sources have an intrinsic size of $1\arcsec-2\arcsec$. Resolving sources with an angular resolution higher than the intrinsic size can negatively impact deep surveys of star forming galaxies in two ways: (1) by fragmenting individual radio sources into multiple components, especially in the low SNR regime; and (2) loss of surface brightness sensitivity and the resulting loss of extended emission. The former is a well-known phenomenon for nearly all deep radio surveys, and most previous studies have produced catalogs of “source components" as well as integrated source catalogs. In analyzing the $0.5\arcsec$ imaging data of the GN field, @guidetti17 identified the loss of surface brightness sensitivity and their bias toward compact sources as the primary cause for their extra-ordinarily high AGN fraction. Only a modest (a factor of $\sim3$) increase in the source density reported by @rujopakarn16 in their ultra-deep imaging of the GS field with nearly 10 times better sensitivity than our survey is likely driven by the loss of flux density and surface brightness sensitivity resulting from their using very high angular resolution (0.61$\times$0.31).
We explore the impact of angular resolution on flux recovery further by comparing the measured flux density of the faint radio sources in the GS field reported by different surveys with varying angular resolution in Figure \[GSCompare\]. The flux densities reported by @kellermann08 at 4.85 GHz and by @huynh15 at 5.5 GHz were both measured using a $\theta\approx4\arcsec$ beam, and these flux densities are systematically higher when compared with our measurements obtained with a 1.5 beam. The average flux ratio between the @kellermann08 flux density to our flux density is 1.34, with a median ratio of 1.14. Similarly, the average and median ratios of the @huynh15 flux density to our flux density is 1.26 and 1.18, respectively. A small correction due to intrinsic spectral index is neglected, as both low angular resolution measurements are significantly larger (about 30%) than our measurements with an effective center frequency of 5.25 GHz. These measured differences are much larger than the expected absolute calibration uncertainties ($\lesssim$10%) associated with the standard flux density bootstrapping calibration. The comparison with the higher resolution (0.61$\times$0.31) imaging by @rujopakarn16 does not provide much new insight as there is only one source in common.
In summary, observing angular resolution smaller than the expected intrisic radio source size of $1\arcsec-2\arcsec$ can lead to a significant systematic bias in deep radio surveys. Carefully accounting for this resolution effect and surface brightness sensitivity is an important consideration for all future ultra-deep surveys with nJy sensitivity.
Importance of Accurate Radio Spectral Index \[DISC\_SI\]
--------------------------------------------------------
Obtaining accurate radio spectral indices is an important step in studying the radio-FIR correlation and its evolution over the cosmic time because computing the rest-frame radio-FIR correlation requires a correction with a “$log_{10} \left[(1+z)^{1-\alpha} \right]$" dependence on radio spectral index, associated with the $k$-correction for the observed radio power. This has the largest impact at the highest redshifts, where the evidence for any evolution in the radio-FIR correlation is expected to be the most pronounced.
Many previous studies of faint radio source population have applied only a partial correction for this spectral index effect, largely because of practical constraints, but the magnitude of the resulting error may have been under-appreciated. Ideally, one should obtain observations at two different frequencies with matched beams and depths to derive correct radio spectral index. However, conducting observations in [*two*]{} frequency bands can be prohibitively expensive in telescope time, especially for deep surveys that require tens to hundreds of hours of integration time in each band. Instead, a common practice is to take advantage of existing survey at another frequency, as we have done using the existing 1.4 GHz surveys by @miller13 and by @owen18. If the complementary archival data are not readily available in raw format as is often the case, however, radio spectral index has to be computed without the beam correction [e.g., @ivison10a; @bourne11; @magnelli15; @delhaize17]. Alternatively, a number of other studies have resorted to adopting a single average radio spectral index of 0.7-0.8 instead [e.g., @appleton04; @ibar08; @murphy09a; @sargent10; @ivison10b; @mao11]. Because even SFGs at $z\ge1$ are resolved at $\sim1\arcsec$ scales, ignoring this resolution effect can lead to significant systematic errors in computing the total radio power and the radio spectral index. Similarly, the radio spectral index distribution is intrinsically broad as discussed in § \[ALPHA\], and adopting a single value of $\alpha$ can introduce significant errors in the derived source properties. Here, we analyze both of these issues quantitatively using our GN and GS deep survey data with and without the appropriate corrections.
### Importance of Beam-matching for the Radio Spectral Index Calculation \[NON\_ALPHA\]
A measured radio spectral index is a direct indicator of the primary radiation mechanism for the observed radio power. In this section, we compare the radio spectral index estimated without matching beam sizes ($\alpha_{non}$) and with those with matched beams ($\alpha_{beam}$), to quantify the importance of the beam effect. The ratio of beam areas is mostly between 1.2 and 1.9 for the GN sources while the GS sources have an average beam area ratio of 10.2, requiring a much larger correction.
The impact of ignoring the beam size difference is clearly shown in the plot of the deviation of radio spectral index $\alpha_{non}$ from $\alpha_{beam}$ ($\Delta \alpha \equiv \alpha_{non} - \alpha_{beam}$) as a function of total 5 GHz flux density in Figure \[alpha\_diff\_flux\]. In the GN field (left panel) where the synthesized beams of 5 GHz and 1.4 GHz data are closely matched, the change is small for most objects as expected. A few sources still show a large deviation with a large positive $\Delta \alpha$ value, indicating that extended or blended sources can lead to large errors in derived spectral indices even when the beam size difference is relatively small. Otherwise the observed scatter is consistent with the expected increase in the noise of the 5 GHz data by the larger photometry aperture. The scatter in the derived spectral index is much larger in the GS field (right panel), and this reflects the impact of a much larger beam difference. As in the GN field, the source distribution is biased to the large positive $\Delta \alpha$ values with a mean of $0.054$, especially among $S_{5GHz}\ge$ 1 mJy sources that are usually associated with extended radio jet sources.
This analysis clearly demonstrates that a small but non-negligible fraction of radio sources are resolved at 1 scale by our 5 GHz beam, and beam-matching is critically important in deriving a correct radio spectral index. This analysis also indicates that our deep 5 GHz data might suffer from loss of flux density due to spatial filtering, even after the beams are matched by smoothing. These combined effects lead to a systematic bias to a steeper (more positive) spectral index and smearing of the overall spectral index distribution, as seen in Figure \[alpha\_flux\] and discussed in section § \[ALPHA\]. Indeed, all interferometric observations are subject to loss of flux density, and matching the resolution to source size is the best that can be done without obtaining additional data.
### Impact of Spectral Index on Radio-FIR Correlation
The rest-frame radio-FIR correlation depends on the radio spectral index through the k-correction for the rest-frame radio power, and there are two common ways which incorrect radio spectral index has impacted the radio-FIR correlation analysis in the literature: (a) not matching beams; and (b) adopting a single value of $\alpha$. Here, we demonstrate how both of these errors in radio spectral index can lead to systematic deviations in the derived radio-FIR correlation parameters $q_{FIR}$ using our data. The deviation of radio-FIR correlation is defined as $\Delta q_{FIR} \equiv q_{FIR} (\alpha_{non}) - q_{FIR} (\alpha_{beam})$ \[for the unmatched beam case\], and they are shown as a function of redshift, color-coded by $\Delta_{SFR}$, in Figure \[qdiff\].
As discussed in the previous section, the net effect of not correcting for the beam size difference is over-estimating radio spectral indices (for this study, because of the higher angular resolution of the 5 GHz data), which in turn leads to a larger k-correction and an over-estimation of the rest frame radio power. As shown on the left panel of Figure \[qdiff\], the overall scatter in $\Delta q_{FIR}$ resulting from not matching the beams is not large, less than 0.1 in dex. However, all sources with a significant deviation in $q_{FIR}$ are [*nearly uniformly and systematically towards a lower value with a mean scatter of -0.019, and this bias is larger in magnitude at a higher redshift*]{} because of a larger k-correction.
The common practice of adopting a single “average" value (e.g., $\alpha=0.8$) leads to an even greater scatter and a stronger bias in $q_{FIR}$ than the unmatched beam case, as shown on the right panel of Figure \[qdiff\]. The magnitude of the scatter in $\Delta q_{FIR}$ is now nearly 0.2 in dex, approaching the [*total*]{} intrinsic scatter in the observed radio-FIR correlation for the local SFGs [@yun01]. In addition, $\Delta q_{FIR}$ is heavily biased towards the negative values with a mean of -0.061 (and growing with redshift), as is the case for the unmatched beam. Both of these trends are the direct results of the large and asymmetric spread in the measured radio spectral index distribution shown in Figure \[alpha\_flux\].
The fact that both of these common errors in radio spectral index can lead to a significant scatter and a strong bias in the derived $q_{FIR}$ is a serious concern for the study of the faint radio source population in general and the study of radio-FIR correlation specifically. The magnitude of the error grows systematically with redshift and is more biased to a lower value of $q_{FIR}$, and this has an important consequence for the evaluation of possible evolution of the radio-FIR correlation. We will discuss this effect in the context of radio-FIR correlation evolution in Paper II.
Conclusions \[CONCLUSION\]
==========================
We reported the first results from our deep and wide VLA 5 GHz surveys of the GN and GS fields with the resolution and sensitivity of $\theta=1.47\arcsec\times1.42\arcsec$ & $\sigma=3.5\, \mu$Jy beam$^{-1}$ and $\theta=0.98\arcsec \times0.45\arcsec$ & $\sigma=3.0\, \mu$Jy beam$^{-1}$, respectively. The central deep cosmology fields with HST and other multi-wavelength data are covered with a nearly uniform sensitivity and resolution, and a total of 52 & 88 sources are identified at $\ge5\sigma$ significance in the 109 & 190 arcmin$^{2}$ survey areas, respectively. We have carefully derived their radio spectral indices by utilizing the existing 1.4 GHz images and catalogs by @owen18 and by @miller13 and examined the radio spectral index distribution and radio-FIR correlation using only a subset of 84 sources with a reliable spectroscopic redshift to minimize introducing additional scatter. Some of the main results from our analyses of these data include:
1. The radio spectral index is measured from beam-matched images of 1.4 & 5 GHz, and its distributions show the clustering of faint radio sources with $S_{5GHz} \lesssim 150 \mu$Jy at around the steep radio spectral index of $\alpha \sim$0.8, which has not seen in previous studies. The associated peak in the GN field is more distinct than in the GS field where the distribution is more smeared out by higher noise. The overall spectral index distribution derived is quite broad, ranging $-0.5 \le \alpha \le 1.4$, as many earlier studies have reported.
2. The star formation activity is characterized by the distance from the “star formation main sequence" [@speagle14], taking into account the strong evolution of SFR with redshift. The majority of faint radio sources are identified as SBs (58%) while only 12% is identified as star forming MS galaxies with $|\Delta_{SFR}| \le 0.2$. The remaining 30% are quiescent galaxies with $\Delta_{SFR} \le -0.2$. This high frequency of SBs is traced to the relatively poor sensitivity of even this deep continuum survey to normal MS SFGs at $z\ge 0.5$, and [*future surveys with up to 100 times better sensitivity ($\sigma_{5GHz} \lesssim 30$ nJy) are needed in order to trace the evolution of the star forming MS at the Cosmic Noon ($z=2.5$).*]{} Our comparison of flux density measurements and source density at different angular resolution support the $\sim$1 extent of intrinsic radio source size reported by previous studies [e.g., @windhorst90; @chapman04; @morrison10], and future ultra-deep surveys should carefully consider the resolution effects, e.g., such as surface brightness sensitivity as well.
3. The SFG+SB population shows a significantly tighter distribution of spectral index than the quiescent galaxies, as shown in Figure \[alpha\_delsfr\], suggesting a systematically different origin for their radio emission. The overwhelming majority of the SFG+SB population (86%) follow the local radio-FIR correlation for SFGs [@yun01] with a median $q_{FIR}$ value of $2.26\pm0.09$. Only $\sim$30% of quiescent galaxies follow the same trend, with a median $q_{FIR}$ value of $1.10\pm0.10$ – most of the quiescent galaxies (76%) are not detected in any of the $Herschel$ far-IR bands. The fraction of radio-excess objects with $q_{FIR} \le 1.6$ increases with increasing 5 GHz radio power, especially for objects at $z\ge1$ with $P_{5GHz}\ge 10^{24}$ W Hz$^{-1}$, and the majority of these objects are intense starburst galaxies with $\Delta_{SFR}\gtrsim1$. This may indicate a sharp rise in the AGN+SB hybrid population at these redshifts, as suggested by previous studies.
4. Determining and applying correct radio spectral indices is important for deriving accurate radio power and analyzing the radio-FIR correlation. Using our own survey data, we demonstrate that the common practice of not matching the beams carefully can lead to a significant and strongly bias estimation of $\alpha$ and over-estimation of radio power for high redshift sources. More importantly, as shown in Figure \[qdiff\], the widely used practice of adopting a single “characteristic" value of spectral index ($\alpha \approx 0.7-0.8$) leads to a much greater scatter matching or exceeding the intrinsic scatter seen in the local population and also a strong systematic bias to negative $q_{FIR}$ values, resulting from the broad width and the asymmetry in the intrinsic radio spectral index distribution.
Lastly, analyzing our data using the photometric redshifts from the 3D-HST project leads to an additional scatter of 0.112 dex in the derived radio-FIR correlation – see Appendix \[ZPHOT\]. The resulting scatter is nearly symmetric, unlike the errors in spectral index discussed above, and analyzing a much larger sample with high quality photometric redshifts might be acceptable for future studies requiring much better statistics.
We are grateful to Ryan Cybulski, Stéphane Arnouts, Olivier Ilbert for the use of Le Phare, Katherine E. Whitaker for help in using the 3D-HST, and Daniel Q Wang for a discussion of X-ray AGN and HMXBs. We also thank Urvashi Rau for a discussion about the radio imaging and Ken Kellermann for a valuable discussion. Hansung B. Gim acknowledges special thanks to NRAO employees for their hospitality when he was visiting NRAO at Socorro, NM, and for valuable helps offered through the NRAO helpdesk. We appreciate the anonymous referees to help us improving this paper.\
A. Spectroscopic Redshifts versus Photometric Redshifts \[ZPHOT\]
=================================================================
As discussed in § \[SPECZ\], we limit our analysis only to the subsample of GN and GS radio sources with a spectroscopic redshift because we aim to remove any additional and possibly systematic noise introduced by adopting photometric redshifts, at the expense of reducing the total sample size by up to 16%. As shown in Figure \[qdiff\_photoz\], photometric redshifts reported by the 3D-HST project, derived using the well-sampled and deep UV-to-NIR photometry available in these fields, are quite good in general, with a few catastrophic outliers. When these redshift errors are propagated into the derivation of $q_{FIR}$ as shown on the right panel, the magnitude of additional scatter introduced by using photometric redshifts is 0.112 in dex. This is about 50% of the intrinsic scatter measured among the local sample of IR-selected SFGs by @yun01 and thus is substantial in magnitude. Fortunately, the redshift error and the resulting changes in $\Delta q_{FIR}$ seem random and not systematic, and using photometric redshifts might be acceptable in future studies if the analysis requires a much larger sample size for improved statistics.
B. Catalog of 5 GHz flux densities and spectral indices of our radio sources \[CAT\]
====================================================================================
The final radio source catalog is presented in Table \[tab:catalog\]. It includes all 52 GN and 88 GS sources cataloged from images with original beam sizes. The 5 GHz flux densities listed in Table \[tab:catalog\] are measured with the original beam sizes, but the spectral index is derived with the beam-matched catalogs as shown in § \[DATA14\]. Eight GS radio sources with original beam sizes are merged into three sources in the image with the beam size matched to that of 1.4 GHz image (refer to § \[DATA14\]). Positions of three merged sources (GS-15, GS-44 and GS-73) are found in the beam matched catalog, but their 5 GHz flux densities are measured from the image with the original beam size. We also list the eight GS sources below the merged sources as GS-15a, -15b, -15c, GS-44a, -44b, -44c, GS-73a, and -73b. The merged sources are not Gaussian-like shapes in the image with the original beam size, so their flux densities are poorly measured by AIPS tasks SAD or JMFIT which utilize the 2D Gaussian fitting function. For this reason, the flux densities of three merged sources are measured with the AIPS task TVSTAT which is appropriate for measuring the flux density of the irregular shaped source. The flux density measured with TVSTAT are larger in general than summation over flux densities of individual sources, because the TVSTAT traces flux densities of regions among individual sources.
Data columns of Table \[tab:catalog\] are summarized as follows: (1) Source ID (ID), (2) Right Ascension (RA J2000), a unit of \[hour, minute, second\], (3) uncertainty of RA, a unit of second, (4) Declination (DEC J2000), a unit of \[$^{\circ}$ \], (5) uncertainty of DEC, a unit of , (6) peak flux density (S$_{peak}$) and its uncertainty, a unit of $\mu$Jy beam$^{-1}$, (7) integrated flux density (S$_{int}$) and its uncertainty, a unit of $\mu$Jy, and (8) radio spectral index ($\alpha$) and its uncertainty.
[rccccccc]{}
\
& [**RA J2000**]{} & [**eRA**]{} & [**DEC J2000**]{} & [**eDEC**]{} & [**S$_{peak}$**]{} & [**S$_{int}$**]{}$^{10}$ & [**$\alpha$**]{}$^{11}$\
& \[h m s\] & \[s\] & \[$^{\circ}$ \] & \[\] & \[$\mu$Jy beam$^{-1}$ \] & \[$\mu$Jy\] &\
[[** – continued from previous page**]{}]{}\
& [**RA J2000**]{} & [**eRA**]{} & [**DEC J2000**]{} & [**eDEC**]{} & [**S$_{peak}$**]{} & [**S$_{int}$**]{}$^{10}$ & [**$\alpha$**]{}$^{11}$\
& \[h m s\] & \[s\] & \[$^{\circ}$ \] & \[\] & \[$\mu$Jy beam$^{-1}$ \] & \[$\mu$Jy\] &\
\
GS-01 & 3 31 59.619 & 0.034 & -27 47 32.87 & 0.07 & 27.8 $\pm$4.9 & 27.8 $\pm$ 4.9 & 0.265 $\pm$ 0.138\
GS-02 & 3 31 59.843 & 0.011 & -27 45 40.88 & 0.02 & 96.2 $\pm$ 5.2 & 96.2 $\pm$ 5.2 & 0.727 $\pm$ 0.051\
GS-03 & 3 32 1.547 & 0.006 & -27 46 47.84 & 0.01 & 550.4 $\pm$ 4.0 & 9338.2 $\pm$ 78.7 & 0.903 $\pm$ 0.001\
GS-04 & 3 32 3.667 & 0.015 & -27 46 3.98 & 0.03 & 63.8 $\pm$ 4.1 & 66.3 $\pm$ 7.3 & 0.189 $\pm$ 0.061\
GS-05 & 3 32 6.446 & 0.054 & -27 47 28.96 & 0.08 & 18.2 $\pm$ 3.5 & 25.1 $\pm$ 7.4 & 0.901 $\pm$ 0.083\
GS-06 & 3 32 8.538 & 0.042 & -27 46 48.55 & 0.06 & 26.7 $\pm$ 3.2 & 55.8 $\pm$ 9.3 & 1.088 $\pm$ 0.044\
GS-07 & 3 32 8.673 & 0.000 & -27 47 34.68 & 0.00 & 4030.0 $\pm$ 3.0 & 4030.0 $\pm$ 3.0 & -0.521 $\pm$ 0.002\
GS-08 & 3 32 9.716 & 0.003 & -27 42 48.43 & 0.01 & 329.5 $\pm$ 4.8 & 329.5 $\pm$ 4.8 & -0.168 $\pm$ 0.019\
GS-09 & 3 32 10.734 & 0.060 & -27 48 7.49 & 0.08 & 19.0 $\pm$ 3.0 & 41.9 $\pm$ 9.0 & 0.408 $\pm$ 0.086\
GS-10 & 3 32 10.797 & 0.008 & -27 46 28.11 & 0.01 & 92.5 $\pm$ 3.2 & 99.2 $\pm$ 5.8 & 0.518 $\pm$ 0.028\
GS-11 & 3 32 10.923 & 0.001 & -27 44 15.26 & 0.00 & 1589.5 $\pm$ 4.0 & 1740.9 $\pm$ 7.0 & 0.449 $\pm$ 0.003\
GS-12 & 3 32 11.501 & 0.017 & -27 48 15.90 & 0.04 & 39.8 $\pm$ 3.1 & 51.4 $\pm$ 6.3 & 0.108 $\pm$ 0.081\
GS-13 & 3 32 11.532 & 0.014 & -27 47 13.31 & 0.02 & 57.5 $\pm$ 3.1 & 72.9 $\pm$ 6.3 & 0.889 $\pm$ 0.033\
GS-14 & 3 32 11.615 & 0.048 & -27 50 27.54 & 0.09 & 16.2 $\pm$ 3.2 & 16.2 $\pm$ 3.2 & $<$ 0.347\
GS-15 & 3 32 13.104 & 0.020 & -27 43 50.95 & 0.21 & & 368.1 $\pm$ 28.5 & 1.312 $\pm$ 0.022\
15a & 3 32 13.047 & 0.095 & -27 43 50.60 & 0.09 & 25.6 $\pm$ 3.3 & 159.2 $\pm$ 23.2 &\
15b & 3 32 13.115 & 0.056 & -27 43 51.63 & 0.05 & 32.5 $\pm$ 3.3 & 90.6 $\pm$ 12.2 &\
15c & 3 32 13.139 & 0.029 & -27 43 50.62 & 0.04 & 42.7 $\pm$ 3.4 & 105.5 $\pm$ 11.1 &\
GS-16 & 3 32 13.247 & 0.033 & -27 42 41.31 & 0.06 & 30.0 $\pm$ 4.3 & 30.0 $\pm$ 4.3 & 0.751 $\pm$ 0.097\
GS-17 & 3 32 13.490 & 0.008 & -27 49 53.11 & 0.02 & 87.3 $\pm$ 3.0 & 103.4 $\pm$ 5.9 & -0.604 $\pm$ 0.052\
GS-18 & 3 32 13.898 & 0.013 & -27 50 0.88 & 0.02 & 56.4 $\pm$ 3.1 & 56.4 $\pm$ 3.1 & $<$ -0.483\
GS-19 & 3 32 14.164 & 0.051 & -27 49 10.53 & 0.08 & 17.9 $\pm$ 2.9 & 33.8 $\pm$ 7.8 & 0.959 $\pm$ 0.070\
GS-20 & 3 32 14.213 & 0.053 & -27 46 34.89 & 0.08 & 16.6 $\pm$ 3.0 & 24.4 $\pm$ 6.7 & $<$ 0.338\
GS-21 & 3 32 14.992 & 0.033 & -27 42 25.49 & 0.07 & 24.8 $\pm$ 4.2 & 24.8 $\pm$ 4.2 & $<$ 0.238\
GS-22 & 3 32 15.267 & 0.053 & -27 50 19.76 & 0.12 & 15.0 $\pm$ 2.9 & 32.0 $\pm$ 8.5 & $<$ -0.143\
GS-23 & 3 32 15.338 & 0.043 & -27 50 37.72 & 0.09 & 16.4 $\pm$ 3.0 & 20.9 $\pm$ 6.1 & 0.349 $\pm$ 0.114\
GS-24 & 3 32 17.157 & 0.019 & -27 43 3.70 & 0.04 & 40.2 $\pm$ 3.6 & 40.2 $\pm$ 3.6 & 0.461 $\pm$ 0.108\
GS-25 & 3 32 17.183 & 0.032 & -27 52 21.10 & 0.05 & 32.0 $\pm$ 3.3 & 54.1 $\pm$ 8.1 & 0.452 $\pm$ 0.059\
GS-26 & 3 32 18.023 & 0.002 & -27 47 18.77 & 0.00 & 375.9 $\pm$ 3.0 & 384.9 $\pm$ 5.2 & 0.220 $\pm$ 0.009\
GS-27 & 3 32 18.563 & 0.044 & -27 51 34.82 & 0.07 & 18.2 $\pm$ 3.1 & 22.6 $\pm$ 6.1 & $<$ 0.048\
GS-28 & 3 32 19.052 & 0.048 & -27 52 14.99 & 0.09 & 18.2 $\pm$ 3.1 & 32.4 $\pm$ 8.0 & 0.737 $\pm$ 0.115\
GS-29 & 3 32 19.310 & 0.019 & -27 52 19.52 & 0.04 & 37.7 $\pm$ 3.2 & 44.4 $\pm$ 6.2 & -0.033 $\pm$ 0.103\
GS-30 & 3 32 19.316 & 0.003 & -27 54 6.58 & 0.00 & 352.4 $\pm$ 4.3 & 2432.7 $\pm$ 60.0 & 0.923 $\pm$ 0.007\
GS-31 & 3 32 19.514 & 0.012 & -27 52 17.87 & 0.02 & 63.0 $\pm$ 3.2 & 69.3 $\pm$ 6.0 & 0.693 $\pm$ 0.039\
GS-32 & 3 32 19.817 & 0.012 & -27 41 23.10 & 0.02 & 83.1 $\pm$ 4.6 & 83.1 $\pm$ 4.6 & 0.594 $\pm$ 0.047\
GS-33 & 3 32 21.285 & 0.016 & -27 44 35.90 & 0.03 & 43.6 $\pm$ 2.9 & 43.6 $\pm$ 2.9 & 1.102 $\pm$ 0.042\
GS-34 & 3 32 22.159 & 0.058 & -27 49 36.76 & 0.09 & 14.5 $\pm$ 2.9 & 23.3 $\pm$ 6.9 & 0.673 $\pm$ 0.114\
GS-35 & 3 32 22.281 & 0.032 & -27 48 4.83 & 0.10 & 15.5 $\pm$ 3.0 & 15.5 $\pm$ 3.0 & 0.713 $\pm$ 0.162\
GS-36 & 3 32 22.514 & 0.017 & -27 48 4.99 & 0.03 & 38.0 $\pm$ 3.0 & 38.0 $\pm$ 3.0 & 0.343 $\pm$ 0.095\
GS-37 & 3 32 22.597 & 0.028 & -27 44 26.11 & 0.04 & 30.3 $\pm$ 2.9 & 41.5 $\pm$ 6.1 & 0.809 $\pm$ 0.056\
GS-38 & 3 32 22.723 & 0.037 & -27 41 26.79 & 0.07 & 28.5 $\pm$ 4.1 & 44.8 $\pm$ 9.7 & 0.095 $\pm$ 0.112\
GS-39 & 3 32 24.262 & 0.039 & -27 41 26.81 & 0.06 & 31.9 $\pm$ 4.0 & 47.9 $\pm$ 9.1 & $<$ -0.859\
GS-40 & 3 32 24.670 & 0.045 & -27 53 34.37 & 0.09 & 19.5 $\pm$ 3.5 & 24.4 $\pm$ 7.1 & 0.895 $\pm$ 0.108\
GS-41 & 3 32 25.174 & 0.051 & -27 54 50.31 & 0.09 & 24.1 $\pm$ 4.6 & 30.1 $\pm$ 9.1 & 0.795 $\pm$ 0.086\
GS-42 & 3 32 25.180 & 0.035 & -27 42 19.15 & 0.06 & 23.1 $\pm$ 3.4 & 27.3 $\pm$ 6.6 & 1.347 $\pm$ 0.193\
GS-43 & 3 32 26.769 & 0.037 & -27 41 45.98 & 0.08 & 23.9 $\pm$ 3.6 & 36.9 $\pm$ 8.4 & $<$ 0.084\
GS-44 & 3 32 26.974 & 0.001 & -27 41 7.16 & 0.01 & & 5390.7 $\pm$ 33.0 & 0.958 $\pm$ 0.002\
44a & 3 32 26.953 & 0.001 & -27 41 7.88 & 0.00 & 1069.0 $\pm$ 4.0 & 3613.0 $\pm$ 17.0 &\
44b & 3 32 27.011 & 0.001 & -27 41 5.44 & 0.00 & 1079.0 $\pm$ 4.0 & 1290.0 $\pm$ 8.0 &\
44c & 3 32 27.060 & 0.044 & -27 41 3.69 & 0.03 & 77.6 $\pm$ 3.9 & 463.6 $\pm$ 27.2 &\
GS-45 & 3 32 27.018 & 0.072 & -27 42 18.66 & 0.14 & 16.4 $\pm$ 3.2 & 30.1 $\pm$ 8.6 & $<$ -0.020\
GS-46 & 3 32 27.728 & 0.031 & -27 50 41.24 & 0.05 & 18.9 $\pm$ 2.9 & 18.9 $\pm$ 2.9 & 1.311 $\pm$ 0.177\
GS-47 & 3 32 28.002 & 0.024 & -27 46 39.65 & 0.04 & 30.0 $\pm$ 2.9 & 39.5 $\pm$ 6.1 & 0.592 $\pm$ 0.060\
GS-48 & 3 32 28.425 & 0.037 & -27 43 44.85 & 0.08 & 15.1 $\pm$ 2.9 & 15.1 $\pm$ 2.9 & $<$ 0.740\
GS-49 & 3 32 28.513 & 0.030 & -27 46 58.48 & 0.06 & 22.9 $\pm$ 3.0 & 22.9 $\pm$ 3.0 & 0.864 $\pm$ 0.098\
GS-50 & 3 32 28.742 & 0.008 & -27 46 20.60 & 0.01 & 94.7 $\pm$ 2.9 & 127.8 $\pm$ 6.1 & 0.534 $\pm$ 0.022\
GS-51 & 3 32 28.826 & 0.005 & -27 43 55.94 & 0.01 & 127.5 $\pm$ 2.8 & 244.2 $\pm$ 8.8 & 1.554 $\pm$ 0.027\
GS-52 & 3 32 28.886 & 0.026 & -27 41 29.76 & 0.04 & 38.6 $\pm$ 3.9 & 38.6 $\pm$ 3.9 & $<$ -0.464\
GS-53 & 3 32 29.876 & 0.036 & -27 44 25.26 & 0.14 & 28.2 $\pm$ 2.5 & 226.1 $\pm$ 22.7 & 1.099 $\pm$ 0.025\
GS-54 & 3 32 29.986 & 0.101 & -27 44 5.39 & 0.14 & 15.6 $\pm$ 2.6 & 71.7 $\pm$ 14.2 & 1.140 $\pm$ 0.056\
GS-55 & 3 32 31.489 & 0.055 & -27 46 23.51 & 0.09 & 15.4 $\pm$ 2.8 & 27.4 $\pm$ 7.3 & 1.067 $\pm$ 0.082\
GS-56 & 3 32 31.546 & 0.008 & -27 50 29.00 & 0.01 & 89.8 $\pm$ 2.9 & 110.9 $\pm$ 5.8 & -0.578 $\pm$ 0.075\
GS-57 & 3 32 33.007 & 0.033 & -27 46 6.64 & 0.07 & 16.1 $\pm$ 2.9 & 16.1 $\pm$ 2.9 & $<$ 0.597\
GS-58 & 3 32 33.446 & 0.057 & -27 52 28.55 & 0.07 & 19.0 $\pm$ 2.9 & 38.4 $\pm$ 8.3 & 0.981 $\pm$ 0.062\
GS-59 & 3 32 36.185 & 0.053 & -27 49 32.17 & 0.08 & 15.1 $\pm$ 2.9 & 20.3 $\pm$ 6.2 & 1.105 $\pm$ 0.107\
GS-60 & 3 32 37.734 & 0.030 & -27 50 0.71 & 0.05 & 28.3 $\pm$ 2.9 & 38.0 $\pm$ 6.1 & 0.908 $\pm$ 0.084\
GS-61 & 3 32 37.768 & 0.027 & -27 52 12.63 & 0.05 & 29.5 $\pm$ 3.1 & 36.6 $\pm$ 6.2 & 0.631 $\pm$ 0.061\
GS-62 & 3 32 37.890 & 0.069 & -27 53 17.86 & 0.15 & 17.1 $\pm$ 3.4 & 30.5 $\pm$ 8.8 & $<$ 0.277\
GS-63 & 3 32 38.791 & 0.033 & -27 44 49.28 & 0.05 & 22.4 $\pm$ 2.9 & 26.4 $\pm$ 5.5 & 0.633 $\pm$ 0.103\
GS-64 & 3 32 38.838 & 0.076 & -27 49 56.60 & 0.07 & 15.0 $\pm$ 2.8 & 28.0 $\pm$ 7.5 & 0.136 $\pm$ 0.093\
GS-65 & 3 32 39.193 & 0.053 & -27 53 57.94 & 0.10 & 22.4 $\pm$ 3.8 & 48.5 $\pm$ 11.5 & 0.384 $\pm$ 0.081\
GS-66 & 3 32 39.488 & 0.024 & -27 53 1.87 & 0.04 & 40.7 $\pm$ 3.4 & 62.1 $\pm$ 7.8 & 0.607 $\pm$ 0.049\
GS-67 & 3 32 43.320 & 0.034 & -27 46 47.01 & 0.06 & 19.4 $\pm$ 2.9 & 19.4 $\pm$ 2.9 & $<$ 0.256\
GS-68 & 3 32 43.542 & 0.045 & -27 54 55.05 & 0.07 & 29.1 $\pm$ 5.8 & 29.1 $\pm$ 5.8 & $<$ 0.271\
GS-69 & 3 32 44.051 & 0.062 & -27 51 43.90 & 0.19 & 20.9 $\pm$ 2.9 & 105.2 $\pm$ 17.4 & 1.072 $\pm$ 0.039\
GS-70 & 3 32 44.275 & 0.009 & -27 51 41.31 & 0.02 & 85.1 $\pm$ 3.2 & 106.9 $\pm$ 6.4 & 0.741 $\pm$ 0.024\
GS-71 & 3 32 45.401 & 0.036 & -27 43 49.36 & 0.08 & 17.2 $\pm$ 3.4 & 17.2 $\pm$ 3.4 & $<$ 0.502\
GS-72 & 3 32 45.967 & 0.038 & -27 53 16.25 & 0.08 & 25.0 $\pm$ 4.2 & 25.0 $\pm$ 4.2 & 1.641 $\pm$ 0.146\
GS-73 & 3 32 46.802 & 0.008 & -27 42 14.40 & 0.14 & & 93.5 $\pm$ 13.0 & -0.265 $\pm$ 0.078\
73a & 3 32 46.770 & 0.039 & -27 42 12.50 & 0.05 & 34.2 $\pm$ 4.6 & 42.8 $\pm$ 9.2 &\
73b & 3 32 46.884 & 0.045 & -27 42 15.56 & 0.07 & 29.4 $\pm$ 4.6 & 38.8 $\pm$ 9.4 &\
GS-74 & 3 32 47.494 & 0.040 & -27 42 43.97 & 0.10 & 21.9 $\pm$ 4.3 & 21.9 $\pm$ 4.3 & $<$ 0.737\
GS-75 & 3 32 47.902 & 0.047 & -27 42 33.12 & 0.10 & 24.1 $\pm$ 4.3 & 45.2 $\pm$ 11.5 & 1.155 $\pm$ 0.074\
GS-76 & 3 32 48.185 & 0.031 & -27 52 57.02 & 0.06 & 31.7 $\pm$ 4.1 & 37.7 $\pm$ 8.0 & 0.066 $\pm$ 0.120\
GS-77 & 3 32 48.566 & 0.040 & -27 49 34.63 & 0.05 & 24.8 $\pm$ 3.0 & 39.4 $\pm$ 7.2 & 0.636 $\pm$ 0.086\
GS-78 & 3 32 49.440 & 0.002 & -27 42 35.54 & 0.00 & 599.6 $\pm$ 4.7 & 716.9 $\pm$ 9.1 & 1.159 $\pm$ 0.008\
GS-79 & 3 32 51.838 & 0.020 & -27 44 37.09 & 0.03 & 53.7 $\pm$ 3.7 & 72.2 $\pm$ 7.7 & 0.218 $\pm$ 0.059\
GS-80 & 3 32 52.077 & 0.008 & -27 44 25.57 & 0.01 & 151.8 $\pm$ 3.8 & 214.6 $\pm$ 8.2 & -0.279 $\pm$ 0.030\
GS-81 & 3 32 52.326 & 0.055 & -27 45 42.24 & 0.07 & 19.0 $\pm$ 3.4 & 26.1 $\pm$ 7.3 & 0.445 $\pm$ 0.133\
GS-82 & 3 32 53.863 & 0.045 & -27 51 36.91 & 0.10 & 21.4 $\pm$ 4.1 & 29.3 $\pm$ 8.6 & $<$ -0.035\
GS-83 & 3 32 59.386 & 0.050 & -27 47 58.50 & 0.08 & 22.7 $\pm$ 4.4 & 28.8 $\pm$ 8.8 & $<$ 0.040\
GN-01 & 12 36 0.117 & 0.144 & 62 10 46.92 & 0.16 & 29.0 $\pm$ 5.4 & 46.1 $\pm$ 13.0 & 0.796 $\pm$ 0.101\
GN-02 & 12 36 1.803 & 0.111 & 62 11 26.34 & 0.12 & 32.7 $\pm$ 5.4 & 32.7 $\pm$ 5.4 & 1.034 $\pm$ 0.064\
GN-03 & 12 36 3.238 & 0.070 & 62 11 10.67 & 0.07 & 43.9 $\pm$ 5.2 & 43.9 $\pm$ 5.2 & 1.042 $\pm$ 0.049\
GN-04 & 12 36 6.607 & 0.054 & 62 9 50.91 & 0.06 & 63.0 $\pm$ 4.7 & 90.8 $\pm$ 10.6 & 0.665 $\pm$ 0.044\
GN-05 & 12 36 8.122 & 0.018 & 62 10 35.70 & 0.02 & 158.2 $\pm$ 4.5 & 169.6 $\pm$ 8.2 & 0.205 $\pm$ 0.018\
GN-06 & 12 36 8.790 & 0.295 & 62 11 43.57 & 0.15 & 21.6 $\pm$ 4.2 & 60.7 $\pm$ 15.6 & -0.149 $\pm$ 0.098\
GN-07 & 12 36 12.513 & 0.158 & 62 11 40.22 & 0.16 & 21.4 $\pm$ 4.0 & 39.3 $\pm$ 10.7 & 0.626 $\pm$ 0.099\
GN-08 & 12 36 17.096 & 0.068 & 62 10 11.35 & 0.06 & 38.0 $\pm$ 3.9 & 38.0 $\pm$ 3.9 & 0.222 $\pm$ 0.052\
GN-09 & 12 36 19.453 & 0.078 & 62 12 52.47 & 0.09 & 31.9 $\pm$ 4.1 & 31.9 $\pm$ 4.1 & 0.930 $\pm$ 0.054\
GN-10 & 12 36 20.284 & 0.022 & 62 8 44.12 & 0.02 & 122.9 $\pm$ 4.3 & 133.7 $\pm$ 7.9 & -0.054 $\pm$ 0.023\
GN-11 & 12 36 21.217 & 0.122 & 62 11 8.68 & 0.17 & 18.2 $\pm$ 3.5 & 25.8 $\pm$ 7.8 & 0.865 $\pm$ 0.112\
GN-12 & 12 36 22.536 & 0.012 & 62 6 53.70 & 0.01 & 325.8 $\pm$ 6.4 & 325.8 $\pm$ 6.4 & -0.158 $\pm$ 0.008\
GN-13 & 12 36 31.266 & 0.038 & 62 9 57.66 & 0.04 & 56.5 $\pm$ 3.5 & 56.5 $\pm$ 3.5 & 0.806 $\pm$ 0.028\
GN-14 & 12 36 32.480 & 0.063 & 62 11 5.19 & 0.07 & 30.2 $\pm$ 3.4 & 30.2 $\pm$ 3.4 & 0.100 $\pm$ 0.073\
GN-15 & 12 36 34.456 & 0.043 & 62 12 13.01 & 0.05 & 55.8 $\pm$ 3.3 & 85.0 $\pm$ 7.6 & 0.761 $\pm$ 0.036\
GN-16 & 12 36 34.505 & 0.040 & 62 12 41.00 & 0.04 & 59.8 $\pm$ 3.4 & 78.1 $\pm$ 7.1 & 0.726 $\pm$ 0.036\
GN-17 & 12 36 35.608 & 0.115 & 62 14 23.97 & 0.14 & 23.0 $\pm$ 3.9 & 33.0 $\pm$ 8.7 & 0.718 $\pm$ 0.104\
GN-18 & 12 36 37.042 & 0.074 & 62 8 52.16 & 0.09 & 31.3 $\pm$ 4.0 & 31.3 $\pm$ 4.0 & 0.946 $\pm$ 0.055\
GN-19 & 12 36 40.742 & 0.100 & 62 10 11.33 & 0.18 & 21.9 $\pm$ 3.4 & 44.1 $\pm$ 9.5 & 0.065 $\pm$ 0.116\
GN-20 & 12 36 41.563 & 0.077 & 62 9 48.16 & 0.08 & 29.7 $\pm$ 3.7 & 29.7 $\pm$ 3.7 & 0.967 $\pm$ 0.052\
GN-21 & 12 36 42.093 & 0.016 & 62 13 31.29 & 0.02 & 137.8 $\pm$ 3.5 & 147.3 $\pm$ 6.3 & 0.980 $\pm$ 0.020\
GN-22 & 12 36 42.187 & 0.057 & 62 15 45.22 & 0.07 & 46.3 $\pm$ 4.3 & 54.5 $\pm$ 8.4 & 1.018 $\pm$ 0.058\
GN-23 & 12 36 44.390 & 0.003 & 62 11 33.05 & 0.00 & 641.0 $\pm$ 3.4 & 963.0 $\pm$ 6.3 & 0.471 $\pm$ 0.018\
GN-24 & 12 36 46.074 & 0.100 & 62 14 48.58 & 0.09 & 28.3 $\pm$ 3.6 & 42.6 $\pm$ 8.3 & 0.726 $\pm$ 0.072\
GN-25 & 12 36 46.331 & 0.012 & 62 14 4.58 & 0.01 & 177.7 $\pm$ 3.5 & 177.7 $\pm$ 3.5 & 0.380 $\pm$ 0.014\
GN-26 & 12 36 46.334 & 0.082 & 62 16 29.25 & 0.08 & 47.2 $\pm$ 4.3 & 95.9 $\pm$ 12.4 & 1.196 $\pm$ 0.046\
GN-27 & 12 36 46.660 & 0.104 & 62 8 33.15 & 0.09 & 33.2 $\pm$ 4.6 & 41.7 $\pm$ 9.2 & 0.710 $\pm$ 0.083\
GN-28 & 12 36 49.663 & 0.027 & 62 7 37.97 & 0.03 & 130.6 $\pm$ 5.9 & 130.6 $\pm$ 5.9 & 0.723 $\pm$ 0.021\
GN-29 & 12 36 50.181 & 0.190 & 62 8 44.80 & 0.22 & 22.0 $\pm$ 4.4 & 59.6 $\pm$ 15.6 & 0.289 $\pm$ 0.092\
GN-30 & 12 36 51.091 & 0.082 & 62 10 30.91 & 0.08 & 32.3 $\pm$ 3.7 & 45.0 $\pm$ 8.0 & 0.568 $\pm$ 0.067\
GN-31 & 12 36 51.721 & 0.078 & 62 12 21.36 & 0.08 & 22.6 $\pm$ 3.4 & 22.6 $\pm$ 3.4 & 0.910 $\pm$ 0.066\
GN-32 & 12 36 52.814 & 0.088 & 62 18 7.95 & 0.10 & 44.9 $\pm$ 5.6 & 66.9 $\pm$ 12.7 & 0.670 $\pm$ 0.070\
GN-33 & 12 36 52.888 & 0.012 & 62 14 43.97 & 0.01 & 188.1 $\pm$ 3.5 & 205.8 $\pm$ 6.4 & 0.028 $\pm$ 0.018\
GN-34 & 12 36 53.372 & 0.089 & 62 11 39.33 & 0.16 & 19.7 $\pm$ 3.5 & 23.3 $\pm$ 6.8 & 0.806 $\pm$ 0.109\
GN-35 & 12 36 55.800 & 0.111 & 62 9 17.32 & 0.11 & 30.4 $\pm$ 4.6 & 45.0 $\pm$ 10.4 & 0.375 $\pm$ 0.087\
GN-36 & 12 36 59.317 & 0.003 & 62 18 32.46 & 0.00 & 1106.0 $\pm$ 6.0 & 1122.0 $\pm$ 10.0 & 1.202 $\pm$ 0.012\
GN-37 & 12 36 59.926 & 0.110 & 62 14 49.80 & 0.15 & 18.4 $\pm$ 3.4 & 18.4 $\pm$ 3.4 & 0.316 $\pm$ 0.117\
GN-38 & 12 37 0.260 & 0.030 & 62 9 9.76 & 0.03 & 114.2 $\pm$ 5.3 & 119.7 $\pm$ 9.5 & 0.766 $\pm$ 0.032\
GN-39 & 12 37 1.558 & 0.090 & 62 11 46.40 & 0.12 & 28.3 $\pm$ 3.6 & 47.4 $\pm$ 9.0 & 0.593 $\pm$ 0.071\
GN-40 & 12 37 2.106 & 0.115 & 62 17 34.32 & 0.16 & 26.7 $\pm$ 4.5 & 46.4 $\pm$ 11.4 & -0.286 $\pm$ 0.091\
GN-41 & 12 37 8.211 & 0.128 & 62 16 59.05 & 0.13 & 21.6 $\pm$ 4.1 & 21.6 $\pm$ 4.1 & 0.514 $\pm$ 0.129\
GN-42 & 12 37 8.287 & 0.144 & 62 10 56.17 & 0.18 & 23.4 $\pm$ 4.4 & 43.0 $\pm$ 11.7 & 0.348 $\pm$ 0.098\
GN-43 & 12 37 11.327 & 0.106 & 62 13 30.91 & 0.07 & 30.5 $\pm$ 3.5 & 46.6 $\pm$ 8.1 & 0.769 $\pm$ 0.067\
GN-44 & 12 37 13.854 & 0.011 & 62 18 26.27 & 0.01 & 321.0 $\pm$ 5.8 & 321.0 $\pm$ 5.8 & 0.564 $\pm$ 0.013\
GN-45 & 12 37 16.375 & 0.015 & 62 15 12.32 & 0.01 & 153.0 $\pm$ 3.7 & 153.0 $\pm$ 3.7 & 0.126 $\pm$ 0.016\
GN-46 & 12 37 16.672 & 0.027 & 62 17 33.39 & 0.03 & 108.3 $\pm$ 4.8 & 118.4 $\pm$ 8.8 & 0.869 $\pm$ 0.030\
GN-47 & 12 37 21.271 & 0.008 & 62 11 29.91 & 0.01 & 416.1 $\pm$ 5.3 & 429.3 $\pm$ 9.4 & -0.129 $\pm$ 0.015\
GN-48 & 12 37 25.962 & 0.024 & 62 11 28.59 & 0.01 & 314.8 $\pm$ 5.6 & 1174.7 $\pm$ 26.8 & 1.270 $\pm$ 0.014\
GN-49 & 12 37 30.818 & 0.066 & 62 12 58.75 & 0.07 & 43.1 $\pm$ 5.2 & 43.1 $\pm$ 5.2 & 0.924 $\pm$ 0.050\
GN-50 & 12 37 34.503 & 0.173 & 62 17 23.45 & 0.14 & 32.3 $\pm$ 6.2 & 55.8 $\pm$ 15.6 & 0.442 $\pm$ 0.102\
GN-51 & 12 37 36.922 & 0.092 & 62 14 29.51 & 0.13 & 28.4 $\pm$ 5.4 & 28.4 $\pm$ 5.4 & 0.652 $\pm$ 0.076\
GN-52 & 12 37 42.331 & 0.091 & 62 15 18.19 & 0.11 & 46.6 $\pm$ 6.4 & 62.1 $\pm$ 13.4 & 0.397 $\pm$ 0.084\
The integrated flux density is the same as the peak flux density for a point source.
$^{13}$The spectral index $\alpha$ is estimated between 1.4 and 5 GHz using 1.4 GHz images (@owen18 for the GN and @miller13 for the GS fields) and 5 GHz images with same beam sizes as those of 1.4 GHz images.
natexlab\#1[\#1]{}\[1\][[\#1](#1)]{} \[1\][doi: [](http://doi.org/#1)]{} \[1\][[](http://ascl.net/#1)]{} \[1\][[](https://arxiv.org/abs/#1)]{}
, P. N., [Fadda]{}, D. T., [Marleau]{}, F. R., [et al.]{} 2004, , 154, 147,
, S., [Cristiani]{}, S., [Moscardini]{}, L., [et al.]{} 1999, , 310, 540,
, I., [Mainieri]{}, V., [Popesso]{}, P., [et al.]{} 2010, , 512, A12,
, A. J., [Cowie]{}, L. L., [Owen]{}, F. N., [Hsu]{}, L.-Y., & [Wang]{}, W.-H. 2017, , 835, 95,
, A. J., [Cowie]{}, L. L., & [Wang]{}, W.-H. 2008, , 689, 687,
, E. F. 2003, , 586, 794,
, M., [Padovani]{}, P., [Mainieri]{}, V., [et al.]{} 2013, , 436, 3759,
, N., [Dunne]{}, L., [Ivison]{}, R. J., [et al.]{} 2011, , 410, 1155,
, G. B., [van Dokkum]{}, P. G., [Franx]{}, M., [et al.]{} 2012, , 200, 13,
, A. H., & [Schwab]{}, F. R. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 180, Synthesis Imaging in Radio Astronomy II, ed. G. B. [Taylor]{}, C. L. [Carilli]{}, & R. A. [Perley]{}, 371
, G., & [Charlot]{}, S. 2003, , 344, 1000,
, S. F. 1979, , 186, 519,
, K. I., [Lagache]{}, G., [Yan]{}, L., [et al.]{} 2007, , 660, 97,
, G. 2003, , 115, 763,
, S. C., [Smail]{}, I., [Windhorst]{}, R., [Muxlow]{}, T., & [Ivison]{}, R. J. 2004, , 611, 732,
, R., & [Elbaz]{}, D. 2001, , 556, 562,
, J. J. 1992, , 30, 575,
, J. J., [Cotton]{}, W. D., [Fomalont]{}, E. B., [et al.]{} 2012, , 758, 14,
, M. C., [Yan]{}, R., [Dickinson]{}, M., [et al.]{} 2012, , 425, 2116,
, L. L., [Barger]{}, A. J., [Hu]{}, E. M., [Capak]{}, P., & [Songaila]{}, A. 2004, , 127, 3137,
, D. A., & [Helou]{}, G. 2002, , 576, 159,
, D. A., [Helou]{}, G., [Contursi]{}, A., [Silbermann]{}, N. A., & [Kolhatkar]{}, S. 2001, , 549, 215,
, M., [Labb[é]{}]{}, I., [van Dokkum]{}, P. G., [et al.]{} 2011, , 727, 1,
, A. G. 1976, , 52, 439
, J., [Smol[č]{}i[ć]{}]{}, V., [Delvecchio]{}, I., [et al.]{} 2017, , 602, A4,
, J. L., [Rieke]{}, G. H., [P[é]{}rez-Gonz[á]{}lez]{}, P. G., [Rigby]{}, J. R., & [Alonso-Herrero]{}, A. 2007, , 660, 167,
, R. H., [Partridge]{}, R. B., & [Windhorst]{}, R. A. 1987, , 321, 94,
, J. S., [McLure]{}, R. J., [Biggs]{}, A. D., [et al.]{} 2017, , 466, 861,
, D., [Dickinson]{}, M., [Hwang]{}, H. S., [et al.]{} 2011, , 533, A119,
, E. B., [Windhorst]{}, R. A., [Kristian]{}, J. A., & [Kellerman]{}, K. I. 1991, , 102, 1258,
, H. B., [Hales]{}, C. A., [Momjian]{}, E., & [Yun]{}, M. S. 2015, in American Astronomical Society Meeting Abstracts, Vol. 225, American Astronomical Society Meeting Abstracts, 143.40
, D., [Bondi]{}, M., [Prandoni]{}, I., [et al.]{} 2017, , 471, 210,
, G., [Soifer]{}, B. T., & [Rowan-Robinson]{}, M. 1985, , 298, L7,
, N., [Middelberg]{}, E., [Deller]{}, A., [et al.]{} 2018, , 616, A128,
, M. T., [Bell]{}, M. E., [Hopkins]{}, A. M., [Norris]{}, R. P., & [Seymour]{}, N. 2015, , 454, 952,
, E., [Ivison]{}, R. J., [Best]{}, P. N., [et al.]{} 2010, , 401, L53,
, E., [Ivison]{}, R. J., [Biggs]{}, A. D., [et al.]{} 2009, , 397, 281,
, E., [Cirasuolo]{}, M., [Ivison]{}, R., [et al.]{} 2008, , 386, 953,
, O., [Arnouts]{}, S., [McCracken]{}, H. J., [et al.]{} 2006, , 457, 841,
, R. J., [Alexander]{}, D. M., [Biggs]{}, A. D., [et al.]{} 2010, , 402, 245,
, R. J., [Magnelli]{}, B., [Ibar]{}, E., [et al.]{} 2010, , 518, L31,
, G., [Heckman]{}, T. M., [White]{}, S. D. M., [et al.]{} 2003, , 341, 54,
, K. I., [Fomalont]{}, E. B., [Mainieri]{}, V., [et al.]{} 2008, , 179, 71,
, A., [Pope]{}, A., [Alexander]{}, D. M., [et al.]{} 2012, , 759, 139,
, I. J., [Ekers]{}, R. D., [Bryant]{}, J. J., [et al.]{} 2006, , 371, 852,
, U., [Lisenfeld]{}, U., & [Verley]{}, S. 2018, , 611, A55,
, M., [van Dokkum]{}, P. G., [Labb[é]{}]{}, I., [et al.]{} 2009, , 700, 221,
, J., [Cimatti]{}, A., [Daddi]{}, E., [et al.]{} 2013, , 549, A63,
, G., [Dole]{}, H., & [Puget]{}, J.-L. 2003, , 338, 555,
, O., [Cassata]{}, P., [Cucciati]{}, O., [et al.]{} 2013, , 559, A14,
, E., [et al.]{} 2005, , 632, 169,
, B., [Brandt]{}, W. N., [Xue]{}, Y. Q., [et al.]{} 2017, , 228, 2,
, D., [Poglitsch]{}, A., [Altieri]{}, B., [et al.]{} 2011, , 532, A90,
, P., & [Dickinson]{}, M. 2014, , 52, 415,
, B., [Elbaz]{}, D., [Chary]{}, R. R., [et al.]{} 2011, VizieR Online Data Catalog, 352
, B., [et al.]{} 2011, , 528, A35,
, B., [Popesso]{}, P., [Berta]{}, S., [et al.]{} 2013, , 553, A132,
, B., [Ivison]{}, R. J., [Lutz]{}, D., [et al.]{} 2015, , 573, A45,
, M. Y., [Huynh]{}, M. T., [Norris]{}, R. P., [et al.]{} 2011, , 731, 79,
, J. P., [Waters]{}, B., [Schiebel]{}, D., [Young]{}, W., & [Golap]{}, K. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 376, Astronomical Data Analysis Software and Systems XVI, ed. R. A. [Shaw]{}, F. [Hill]{}, & D. J. [Bell]{}, 127
, M., [Cimatti]{}, A., [Zamorani]{}, G., [et al.]{} 2005, , 437, 883,
, L., [Peacock]{}, J. A., & [Mead]{}, A. R. G. 1990, , 244, 207
, N. A., [Bonzini]{}, M., [Fomalont]{}, E. B., [et al.]{} 2013, , 205, 13,
, I. G., [Brammer]{}, G. B., [van Dokkum]{}, P. G., [et al.]{} 2016, , 225, 27,
, A. M., [Kocevski]{}, D. D., [Trump]{}, J. R., [et al.]{} 2015, , 149, 178,
, G. E., [Owen]{}, F. N., [Dickinson]{}, M., [Ivison]{}, R. J., & [Ibar]{}, E. 2010, , 188, 178,
, E. J., [Chary]{}, R.-R., [Alexander]{}, D. M., [et al.]{} 2009, , 698, 1380,
, E. J., [Momjian]{}, E., [Condon]{}, J. J., [et al.]{} 2017, , 839, 35,
, E. J., [Condon]{}, J. J., [Schinnerer]{}, E., [et al.]{} 2011, , 737, 67,
, F. N. 2018, , 235, 34,
, F. N., & [Morrison]{}, G. E. 2008, , 136, 1889,
, P., [Mainieri]{}, V., [Tozzi]{}, P., [et al.]{} 2009, , 694, 235,
, R. A., [Chandler]{}, C. J., [Butler]{}, B. J., & [Wrobel]{}, J. M. 2011, , 739, L1,
, M., [Tajer]{}, M., [Maraschi]{}, L., [et al.]{} 2007, , 663, 81,
, P., [Dickinson]{}, M., [Nonino]{}, M., [et al.]{} 2009, , 494, 443,
, I., [Parma]{}, P., [Wieringa]{}, M. H., [et al.]{} 2006, , 457, 517,
. 2013, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria
, K. E., [Hopkins]{}, A. M., [Norris]{}, R. P., [et al.]{} 2012, , 421, 1644,
, C. D., [Puech]{}, M., [Flores]{}, H., [et al.]{} 2007, , 465, 1099,
, I. G., [Oliver]{}, S. J., [Kunz]{}, M., [et al.]{} 2010, , 409, 48,
, I. G., [Ivison]{}, R. J., [Greve]{}, T. R., [et al.]{} 2012, , 419, 2758,
, L., & [Owen]{}, F. N. 2014, , 785, 45,
, W., [Dunlop]{}, J. S., [Rieke]{}, G. H., [et al.]{} 2016, , 833, 12,
, M. T., [Schinnerer]{}, E., [Murphy]{}, E., [et al.]{} 2010, , 186, 341,
, N., [Lee]{}, N., [Vanden Bout]{}, P., [et al.]{} 2017, , 837, 150,
, N., [Dwelly]{}, T., [Moss]{}, D., [et al.]{} 2008, , 386, 1695,
, J. D., [Mainieri]{}, V., [Salvato]{}, M., [et al.]{} 2010, , 191, 124,
, R. E., [Whitaker]{}, K. E., [Momcheva]{}, I. G., [et al.]{} 2014, , 214, 24,
, V., [Delvecchio]{}, I., [Zamorani]{}, G., [et al.]{} 2017, , 602, A2,
, J. S., [Steinhardt]{}, C. L., [Capak]{}, P. L., & [Silverman]{}, J. D. 2014, , 214, 15,
, A. N., [Pirzkal]{}, N., [Meurer]{}, G. R., [et al.]{} 2009, , 138, 1022,
, W., & [Saunders]{}, W. 1992, , 259, 413
, G. P., [Bergeron]{}, J., [Hasinger]{}, G., [et al.]{} 2004, , 155, 271,
Tanabashi, M., Hagiwara, K., Hikasa, K., [et al.]{} 2018, Phys. Rev. D, 98, 030001,
, P., [Gilli]{}, R., [Mainieri]{}, V., [et al.]{} 2006, , 451, 457,
, A., [Bell]{}, E. F., [H[ä]{}ussler]{}, B., [et al.]{} 2012, , 203, 24,
, E., [Cristiani]{}, S., [Dickinson]{}, M., [et al.]{} 2008, , 478, 83,
, W.-H., [Cowie]{}, L. L., [Barger]{}, A. J., [Keenan]{}, R. C., & [Ting]{}, H.-C. 2010, , 187, 251,
, K. E., [Pope]{}, A., [Cybulski]{}, R., [et al.]{} 2017, , submitted
, K. E., [Franx]{}, M., [Leja]{}, J., [et al.]{} 2014, , 795, 104,
, R., [Mathis]{}, D., & [Neuschaefer]{}, L. 1990, in Astronomical Society of the Pacific Conference Series, Vol. 10, Evolution of the Universe of Galaxies, ed. R. G. [Kron]{}, 389–403
, G. D., [Trump]{}, J. R., [Barro]{}, G., [et al.]{} 2015, , 150, 153,
, Y. Q., [Luo]{}, B., [Brandt]{}, W. N., [et al.]{} 2016, , 224, 15,
, M. S., [Reddy]{}, N. A., & [Condon]{}, J. J. 2001, , 554, 803,
, W., [Mikles]{}, V. J., [Mainieri]{}, V., [et al.]{} 2004, , 155, 73,
[^1]: The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
[^2]: Since the final radio image is a combination of cleaned components with flux density scaled by clean beam and residuals with flux densities weighted by dirty beam, the convolution of the radio image with the clean beam includes the convolution of the residuals scaled by the dirty beam in addition to the convolution of the clean components scaled by the clean beam. The former contributes on the uncertainty of the convolved images, but it is not easy to estimate its contributions because it involves many parameters such as clean thresholds, PSF shape, and the convolution kernel size. This is a subtle but notable systematic effect that we have decided to ignore for the moment.
[^3]: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
[^4]: Data are available at http://www.mpe.mpg.de/ir/Research/PEP/DR1
[^5]: Data are available at http://hedam.lam.fr/HerMES/index/dr4
[^6]: Reliability is defined as $Rel_{i} = LR_{i}/(\Sigma LR_{i} + (1-q_{0}))$ for the likelihood ratio $LR_{i}$ and the fraction of true counterparts above the detection limit, $q_{0}$
[^7]: $Le$ $Phare$ is available at http://www.cfht.hawaii.edu/ $\sim$arnouts/lephare.html
[^8]: This work is based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
[^9]: The identification of AGN among the faint radio source population and their impact on observed properties are presented exclusively in Paper II.
[^10]: Nondetects and Data Analysis for Environmental Data.
[^11]: Median of radio source sizes reported at 1.4 GHz by @owen08 and @owen18 are 1.2- 1.5while the median source size at 10 GHz reported by @murphy17 is 0.17 $\pm$ 0.03. The apparent difference in these median radio source sizes is likely attributable to the structures present in these radio sources and the differences in the surface brightness sensitivity achieved.
|
---
abstract: 'We study the motion of electrons in a periodic background potential (usually resulting from a crystalline solid). For small velocities one would use either the non-magnetic or the magnetic Bloch hamiltonian, while in the relativistic regime one would use the Dirac equation with a periodic potential. The dynamics, with the background potential included, is perturbed either through slowly varying external electromagnetic potentials or through a slow deformation of the crystal. In either case we discuss how the Hilbert space of states decouples into almost invariant subspaces and explain the effective dynamics within such a subspace.'
author:
- Gianluca Panati
- Herbert Spohn
- Stefan Teufel
title: Motion of electrons in adiabatically perturbed periodic structures
---
${\left(}
\def$[)]{}
Introduction {#PST_sec.1}
============
In a crystalline solid the conduction electrons move in the potential created by the ions and the core electrons. Somewhat mysteriously and linked to the Pauli exclusion principle, the Coulomb repulsion between conduction electrons may be ignored, within a good approximation. Thereby one arrives at a fundamental model of solid state physics, namely an ideal Fermi gas of electrons subject to a periodic crystal potential. Let $\Gamma$ be the periodicity lattice. It is a Bravais lattice and generated through the basis $\{\gamma_1,\gamma_2,\gamma_3\}$, $\gamma_j\in\mathbb{R}^3$, as $$\label{PST_1.1}
\Gamma=\{\gamma=\sum^3_{j=1}\alpha_j\gamma_j\quad \textrm{with
}\alpha\in\mathbb{Z}^3\}\,.$$ The crystal potential $V_\Gamma$ is then $\Gamma$-periodic, i.e., $V_\Gamma:\mathbb{R}^3\to\mathbb{R}$ and $V_\Gamma(x+\gamma)=V_\Gamma(x)$ for all $\gamma\in\Gamma$, and the electrons are governed by the one-particle hamiltonian $$\label{PST_1.2}
H_{\mathrm{SB}}=-\frac{1}{2}\Delta_x+V_\Gamma\,.$$ $H_{\mathrm{SB}}$ is the (Schrödinger)–Bloch hamiltonian. A wave function $\psi_t\in L^2(\mathbb{R}^3)$ evolves in time according to the Schrödinger equation $$\label{PST_1.3}
i\frac{\partial}{\partial t}\psi_t=H_{\mathrm{SB}}\psi_t\,.$$ We have chosen units such that the mass of an electron $m_\mathrm{e}=1$ and $\hbar=1$. The electron charge, $e$, is absorbed in $V_\Gamma$. Since $V_\Gamma$ is periodic, electrons move ballistically with an effective dispersion relation given by the Bloch energy bands $E_n$, see below for a precise definition. $E_n$ is periodic with respect to the lattice $\Gamma^\ast$ dual to $\Gamma$, $E_n(k+\gamma^\ast)=E_n(k)$ for all $\gamma^\ast\in\Gamma^\ast$, $k\in\mathbb{R}^3$. This feature makes the dynamical properties of a Bloch electron very different from a massive particle with dispersion $E_{\textrm{free}}(k)=\frac{1}{2}k^2$ valid in case $V_\Gamma=0$.
The thermodynamics of the electron gas is studied taking $H_\mathrm{SB}$ as a starting point. Dynamically, however, one wants to probe the response of the electrons to external forces which very crudely come in two varieties.\
(i) *External electromagnetic potentials*. Electrostatic potentials manufactured in a lab have a slow variation on the scale of the lattice $\Gamma$. Therefore we set $V_\mathrm{ext}(x)=e\phi(\varepsilon x)$, $e$ the charge of the electron, with $\varepsilon$ a dimensionless parameter and $\phi$ independent of $\varepsilon$. $\varepsilon\ll 1$ means that the potential $V_\mathrm{ext}$ has a slow variation when measured with respect to the lattice spacing of $\Gamma$. Note that the electrostatic force is $\mathcal{O}(\varepsilon)$ and thus weak. External magnetic fields on the other hand can be so strong that the radius of gyration is comparable to the lattice spacing. It then makes sense to split the vector potential as $A_0+A_\mathrm{ext}$, where $A_0(x)=-\frac{1}{2}B_0 \wedge x$ with $B_0\in\mathbb{R}^3$ a constant magnetic field. Included in $H_\mathrm{SB}$, this yields the magnetic Bloch hamiltonian $$\label{PST_1.4}
H_\mathrm{MB}=\frac{1}{2}(-i\nabla_x- A_0)^2+V_\Gamma\,.$$ $A_\mathrm{ext}$ is a probing vector potential in addition to $A_0$. $A_\mathrm{ext}$ is slowly varying on the scale of the lattice, $A_\mathrm{ext}(x)=A(\varepsilon x)$ with $A$ independent of $\varepsilon$, and the corresponding magnetic field is small of order $\varepsilon$. Including all electromagnetic potentials, for simplicity with the electric charge absorbed into $A$ and $\phi$, the hamiltonian becomes $$\label{PST_1.5}
H=\frac{1}{2}\big(-i\nabla_x-A_0(x)-A(\varepsilon x)\big)^2
+V_\Gamma(x)+\phi(\varepsilon x)\,.
\medskip$$ (ii) *Mechanical forces*. The crystal lattice can be deformed through external pressure and shear. Thereby an electric polarization is induced, an effect which is known as piezoelectricity. If charges are allowed to flow, in this way mechanical pressure can be transformed into electric currents. The mechanical forces are time-dependent but slow on the typical time-scale of the electrons. Therefore in (\[PST\_1.2\]) $V_\Gamma(x)$ is replaced by $V_{\Gamma(\varepsilon
t)}(x,\varepsilon t)$. $\Gamma(\varepsilon t)$ is the instantaneous periodicity lattice and is defined as in (\[PST\_1.1\]). $V_{\Gamma(t)}$ is space-periodic, i.e. $V_{\Gamma(t)}(x+\gamma,t)=V(x,t)$ for all $\gamma\in\Gamma(t)$. The special case of a time-independent lattice, $\Gamma(t)=\Gamma$, but a still slowly in time varying crystal potential is also of interest. For example, one may imagine a unit cell with two nuclei. If the two nuclei are displaced relative to each other, then $\Gamma$ remains fixed while the crystal potential in the unit cell changes with time. The resulting piezoelectric hamiltonian reads $$\label{PST_1.6}
H_\mathrm{PE}(t)=-\frac{1}{2}\Delta_x+V_{\Gamma(\varepsilon
t)}(x,\varepsilon t)\,.$$
Our general goal is to understand, in each case, the structure of the solution of the time-dependent Schrödinger equation for small $\varepsilon$. Obviously, $H$ in (\[PST\_1.5\]) is a space-adiabatic problem, while (\[PST\_1.6\]) corresponds to a time-adiabatic problem. However in the latter case it turns out to be profitable to transform to a time-independent lattice, say $\Gamma(0)$. Then also terms varying slowly in space are generated. Thus, in the general case the full power of the space-adiabatic perturbation theory [@1; @2] will be needed. A word of caution must be issued here for the magnetic Bloch hamiltonian. To use the methods from [@1] in this context, the unperturbed Hamiltonian must be periodic, which is the case only if the magnetic flux per unit cell is rational. One can then define an enlarged magnetic unit cell such that $H_\mathrm{MB}$ is invariant with respect to magnetic translations. If the magnetic flux is not rational, the crutch is to include in $A_0$ a nearby rational flux part of the magnetic field, with a small denominator, and to treat the remainder as $A_\mathrm{ext}$.
To achieve our goal, depending on the context we use one of the periodic hamiltonians as backbone. The periodic hamiltonian is denoted by $H_\mathrm{per}$ with $H_\mathrm{per}$ either $H_\mathrm{SB}$, or $H_\mathrm{MB}$, or $H_\mathrm{PE}$ at fixed $t$, or $H_\mathrm{LS}$ from (\[PST\_1.8\]), or $H_\mathrm{DB}$ from (\[PST\_1.9\]). As explained below, the Hilbert space $\mathcal{H}=L^2(\mathbb{R}^3)$ then splits as $\mathcal{H}=\bigoplus^\infty_{n=0}\mathcal{H}_n$, where $n$ is the band index. Each subspace $\mathcal{H}_n$ is invariant under $\exp[-itH_\mathrm{per}]$ and $H_\mathrm{per}$ restricted to $\mathcal{H}_n$ is unitarily equivalent to multiplication by $E_n(k)$. $E_n(k)$ is the effective hamiltonian associated to the $n$-th band. The complexity of the full problem has been reduced substantially, since only a single band dynamics has still to be studied. Modifying $H_\mathrm{per}$ such that it becomes slowly varying in space-time is, vaguely speaking, a small perturbation. Thus one would expect that the invariant subspace $\mathcal{H}_n$ is to be substituted by a slightly tilted subspace. On this subspace $E_n(k)$ will turn into a more complicated effective hamiltonian. The difficulty is that, while the dynamics generated by $H_\mathrm{per}$ can be computed by solving a purely spectral problem, none of the perturbed hamiltonians can be understood in this way. In particular, one has to spell out carefully over which time scale the slightly tilted subspace associated to $\mathcal{H}_n$ remains approximately invariant and in what sense the dynamics generated by the effective hamiltonian approximates the true time evolution.
To lowest order the effective hamiltonian can be guessed from elementary considerations and belongs to a standard tool of solid state physics [@3]. The guess provides however little hint on the validity of the approximation. There one needs a mathematical theorem which states precise conditions on the initial wave function and provides an error bound, from which the time scale for validity can be read off.
Under the header “geometric phase” physicists and quantum chemists have realized over the past twenty years, say, that the first order correction to the effective hamiltonian carries a lot of interesting physics, see [@4] for a recent comprehensive overview. For the magnetic Bloch hamiltonian the first order correction yields a Hall current proportional to the Chern number of the magnetic Bloch vector bundle. Similarly, the modern theory of piezoelectricity, expresses the piezocurrent as an integral of the Berry connection over the Brillouin zone, see King-Smith, Vanderbilt [@5] and Resta [@6]. First order effective Hamiltonians are no longer guessed so easily and it is convenient to have the systematic scheme [@1] available.
In nature electrons are spin $\frac{1}{2}$ particles. The wave function is thus $\mathbb{C}^2$-valued and the hamiltonian in (\[PST\_1.5\]) is modified to $$\label{PST_1.7}
H=\frac{1}{2}\big(-i\nabla_x-A_0(x)-A(\varepsilon x)\big)^2
+V_\Gamma(x)+\phi(\varepsilon
x)-\frac{1}{2}\sigma\cdot\big(B_0+\varepsilon B(\varepsilon x)\big)$$ with $B=\nabla\wedge A$ for the slowly varying part of the magnetic field. Here $\sigma=(\sigma_1,\sigma_2,\sigma_3)$ is the 3-vector of Pauli spin matrices. Besides the term proportional to the uniform magnetic field $B_0$, $H$ acquires a subleading term of order $\varepsilon$. More accurately one may want to include the spin-orbit coupling. The periodic piece of the hamiltonian reads then $$\label{PST_1.8}
H_\mathrm{LS}=-\frac{1}{2}\Delta_x+V_\Gamma(x)+\frac{1}{4}\sigma
\cdot\big(\nabla V_\Gamma(x)\wedge(-i\nabla_x)\big)$$ and the slowly varying potential is added as in (\[PST\_1.7\]) with the additional subleading term $\varepsilon\frac{1}{4}\sigma\cdot (\nabla \phi(\varepsilon
x)\wedge(-i\nabla_x))$.
Depending on the crystalline solid, the conduction electrons can move so fast that relativistic corrections become important. On the one-particle level an obvious choice is then the Dirac equation with a periodic potential $V_\Gamma$. Wave functions are $\mathbb{C}^4$-valued and the hamiltonian reads $$\label{PST_1.9}
H_\mathrm{DB}=\beta m_\mathrm{e}c^2+c\alpha\cdot p+V_\Gamma\,,\quad
p=-i\nabla_x\,.$$ We introduced here the mass, $m_\mathrm{e}$, of the electron and the speed of light, $c$. The $4\times 4$ matrices $\beta,\alpha_1,\alpha_2,\alpha_3$ are standard and defined in [@7; @8], for example. Note that the Lorentz frame is fixed by the solid, i.e. by $V_\Gamma$.
In fact, the non-relativistic limit for $H_\mathrm{DB}$ yields the spin-orbit hamiltonian $H_\mathrm{LS}$ [@7; @8]. If $\|V_\Gamma\|$ is bounded, for sufficiently large $c$, the Dirac hamiltonian $H_\mathrm{DB}$ has a spectral gap, which widens as $c\to\infty$. Projecting onto the electron subspace, to leading order in $1/c$ one obtains the Pauli-Bloch hamiltonian $-(1/2
m_\mathrm{e})\Delta_x+V_\Gamma$ with the spin-orbit coupling in (\[PST\_1.8\]) as a correction of strength $1/(m_\mathrm{e}c)^2$. In addition the crystal potential is corrected by $-\Delta_xV_\Gamma(x)/8(m_\mathrm{e}c)^2$.
In our contribution we will provide some background on how to establish, including error bounds, the validity of the approximate dynamics as generated by an effective hamiltonian, including order $\varepsilon$ corrections, for most of the models mentioned in the introduction. For this purpose it is necessary to briefly recall the spectral theory for the periodic hamiltonian, which is done in the following section. In the subsequent sections we deal with particular cases in more detail. We start with the non-magnetic Bloch hamiltonian, see (\[PST\_1.5\]) with $B_0=0$. For the magnetic Bloch hamiltonian we explain how $B_0\to 0$ and $B_0\to\infty$ may be viewed as particular adiabatic limits. Piezoelectricity is discussed in the last section.\
**Remark**. Our contribution is one part of the research project jointly with S. Bauer and M. Kunze within the Schwerpunkt. Their part will be covered in [@9]. Both contributions appear now as almost disjoint, which only reflects that we wanted to present a coherent story. The unifying aspect is an adiabatic limit for wave-type evolution equations. In this contribution we stay on the level of effective hamiltonians while in [@9] one pushes the scheme to the first dissipative correction.
The periodic hamiltonians {#PST_sec.2}
=========================
We consider a general dimension, $d$, and assume that the periodicity lattice $\Gamma$ is represented as $$\label{PST_2.1a}
\Gamma =\Big\{ x\in\mathbb{R}^d: x=
\textstyle{\sum_{j=1}^d}\alpha_j\,\gamma_j \,\,\,\mbox{for
some}\,\,\alpha \in \mathbb{Z}^d \Big\}\,,$$ where $\{\gamma_1,\ldots,\gamma_d \}$ are vectors spanning $\mathbb{R}^d$. We denote by $\Gamma^*$ the dual lattice of $\Gamma$ with respect to the standard inner product in $\mathbb{R}^d$, i.e. the lattice generated by the dual basis $\{\gamma_1^*,\ldots,\gamma_d^*\}$ determined through the conditions $\gamma_i^* \cdot \gamma_j = 2\pi
\delta_{ij}$, $i,j\in\{1,\ldots,d\}$. The centered fundamental domain $M$ of $\Gamma$ is defined by $$\label{PST_2.2}
M = \Big\{ x\in\mathbb{R}^d: x=
\textstyle{\sum_{j=1}^d}\beta_j\,\gamma_j
\,\,\,\mbox{for}\,\,\beta_j\in
[-\textstyle{\frac{1}{2},\frac{1}{2}}]
\Big\}\,,$$ and analogously the centered fundamental domain $M^\ast$ of $\Gamma^\ast$. The set $M^\ast$ is the [*first Brillouin zone*]{} in the physics parlance.
**Assumption 1.** *The crystal potential $V_\Gamma:\mathbb{R}^d\to\mathbb{R}$ satisfies $V_\Gamma(x+\gamma)=V_\Gamma(x)$ for all $\gamma\in\Gamma$, $x\in\mathbb{R}^d$. $V_\Gamma$ is infinitesimally bounded with respect to $-\Delta $.*
It follows from Assumption 1 that the periodic hamiltonians discussed below are self-adjoint on the domain of $-\Delta$.
The Bloch hamiltonian {#PST_sec.2a}
---------------------
We consider $$\label{PST_2a.1a}
H=-\frac{1}{2}\Delta+V_\Gamma\,.$$ The periodicity of $H$ is exploited through the Bloch-Floquet-Zak transform, or just the Zak transform for sake of brevity [@10]. The advantage of such a variant is that the fiber at $k$ of the transformed Hamiltonian operator has a domain which does not depend on $k$.
The Zak transform is defined as $$\label{PST_2a.1}
(\mathcal{U}_\mathrm{Z}\psi)(k,x):=\sum_{\gamma\in\Gamma}
\mathrm{e}^{-\mathrm{i}k \cdot (x+\gamma)} \psi(x+\gamma)\,, \qquad
(k,x)\in\mathbb{R}^{2d},$$ initially for a fast-decreasing function $\psi\in\mathcal{S}(\mathbb{R}^d)$. One directly reads off from (\[PST\_2a.1\]) the following periodicity properties $$\label{PST_2a.2}
\big(\mathcal{U}_\mathrm{Z}\psi\big) (k, y+\gamma) = \big(
\mathcal{U}_\mathrm{Z}\psi\big) (k,y)\quad \mbox{ for all} \quad
\gamma\in\Gamma\,,$$ $$\label{PST_2a.3}
\big(\mathcal{U}_\mathrm{Z}\psi\big) (k+\lambda, y) =
\mathrm{e}^{-\mathrm{i}y\cdot\lambda}\,\big(\mathcal{U}_\mathrm{Z}\psi\big)
(k,y) \quad\mbox{ for all} \quad \lambda\in\Gamma^*\,.$$ From (\[PST\_2a.2\]) it follows that, for any fixed $k\in{\mathbb{R}^d}$, $\big(\mathcal{U}_\mathrm{Z}\psi \big)(k,\cdot)$ is a $\Gamma$-periodic function and can thus be regarded as an element of $\mathcal{H}_\mathrm{f} = L^2(M)$. $M=\mathbb{R}^d/\Gamma$ and it has the topology of the $d$-dimensional torus $\mathbb{T}^d$. On the other side, Equation (\[PST\_2a.3\]) involves a unitary representation of the group of lattice translations on $\Gamma^*$ (isomorphic to $\Gamma^*$ and denoted as $\Lambda$), given by $$\label{PST_2a.4}
\tau:\Lambda \to \mathcal{U}(\mathcal{H}_\mathrm{f})\,,\quad\lambda
\mapsto \tau(\lambda)\,, \quad \big(\tau(\lambda)\varphi \big)(y) =
\mathrm{e}^{\mathrm{i}y\cdot\lambda}\varphi(y).$$ It is then convenient to introduce the Hilbert space $$\begin{aligned}
\label{PST_2a.5}
&&\hspace{-40pt}\mathcal{H}_\tau =\big\{ \psi\in L^2_{\rm
loc}(\mathbb{R}^d, \mathcal{H}_\mathrm{f}):\,\, \psi(k - \lambda) =
\tau(\lambda)\,\psi(k) \qquad \textrm{for all } \lambda \in \Lambda
\big\}\nonumber\\
&&\hspace{-18pt}= L^2_\tau(\mathbb{R}^d\,,\;\mathcal{H}_\mathrm{f})\,,\end{aligned}$$ equipped with the inner product $$\label{PST_2a.6}
\langle \psi,\,\varphi\rangle_{\mathcal{H}_\tau} = \int_{M^{*}} dk\,
\langle \psi(k),\,\varphi(k)\rangle_{\mathcal{H}_\mathrm{f}}\,.$$ Obviously, there is a natural isomorphism between $\mathcal{H}_\tau$ and $L^2(M^{*}, \mathcal{H}_\mathrm{f})$ given by restriction from $\mathbb{R}^d$ to $M^{*}$, and with inverse given by $\tau$-equivariant continuation, as suggested by (\[PST\_2a.3\]). Equipped with these definitions, one checks that the map in (\[PST\_2a.1\]) extends to a unitary operator $$\label{PST_2a.7}
\mathcal{U}_\mathrm{Z}: L^2(\mathbb{R}^d)\to \mathcal{H}_\tau \cong
L^2(M^{*}, L^2 (M)),$$ with inverse given by $$\label{PST_2a.8}
(\mathcal{U}_\mathrm{Z}^{-1}\varphi)(x) = \int_{M^*} dk \,
\mathrm{e}^{\mathrm{i} k \cdot x} \varphi(k, [x]),$$ where $[\, \cdot \, ]$ refers to the a.e. unique decomposition $x =
\gamma_x + [x]$, with $\gamma_x \in \Gamma$ and $[x] \in M$.
As already mentioned, the advantage of this construction is that the transformed hamiltonian is a fibered operator over $M^*$. Indeed, for the Zak transform of the hamiltonian operator (\[PST\_2a.1a\]) one finds $$\label{PST_2a.9}
\mathcal{U}_\mathrm{Z}H \mathcal{U}_\mathrm{Z}^{-1} =
\int_{M^{*}}^\oplus dk\,H(k)$$ with fiber operator $$\label{PST_2a.10}
H(k) = \frac{1}{2}\big( -{\mathrm{i}} \nabla_y + k\big)^2 +
V_\Gamma(y)\,, \quad k\in M^{*} \,.$$ By Assumption 1, for fixed $k\in M^{*}$, the operator $H(k)$ acts on $L^2(M)$ with the Sobolev space $H^2(M)$ as domain independently of $k\in M^{*}$. Each fiber operator $H(k)$ has pure point spectrum accumulating at infinity. For definiteness the eigenvalues are enumerated according to their magnitude $E_0(k) \leq E_1(k) \leq
E_2(k) \leq \ldots$ and repeated according to their multiplicity. $E_n:M^\ast\to
\mathbb{R}$ is the $n$-th energy band function. It is continuous on $M^\ast$ when viewed as a $d$-torus. Generically the eigenvalues $E_n(k)$ are non-degenerate. Of course, there may be particular points in $k$-space where particular energy bands touch each other and the corresponding eigenvalue becomes degenerate. The normalized eigenfunction corresponding to $E_n(k)$ is the Bloch function and denoted by $\varphi_n(k)\in H^2(M)$. It is determined only up to a $k$-dependent phase factor. A further arbitrariness comes from points where energy bands touch. We denote by $P_n(k)$ the projection along $\varphi_n(k)$ and set $$\label{PST_2a.11}
P_n=\int^\oplus_{M^\ast}dk P(k)\,,\qquad
\mathcal{H}_n=P_nL^2(\mathbb{R}^d)\,.$$
Through the Zak transform we have achieved the product structure $$\label{PST_2a.12}
\mathcal{H}=\mathcal{H}_\mathrm{s}\otimes\mathcal{H}_\mathrm{f}\,,\qquad
\mathcal{H}_\mathrm{s}=L^2(M^\ast)\,,\;\mathcal{H}_\mathrm{f}=L^2(M)\,.$$ $\psi\in\mathcal{H}_\mathrm{n}$ is of the form $\phi(k)\varphi_n(k,y)$. The band index $n$ fixes the local pattern of the wave function $\psi$ while $\phi(k)$ provides the slow variation. Therefore $L^2(M^\ast)$ is the Hilbert space of the slow degrees of freedom. On the other hand for fixed $k$, one has oscillations in time determined by the eigenvalues $E_n(k)$. On long time scales, these become fast oscillations. Therefore $\mathcal{H}_\mathrm{f}=L^2(M)$ is the Hilbert space of the fast degrees of freedom.
Since $[P_n,H]=0$, the subspaces $\mathcal{H}_n$ are invariant under $\mathrm{e}^{-\mathrm{i}Ht}$. $P_n\mathrm{e}^{-\mathrm{i}Ht}P_n$ is unitarily equivalent to multiplication by $\mathrm{e}^{-\mathrm{i}E_n(k)t}$ on $L^2(M^\ast)$. Note that, in general, $\mathcal{H}_n$ is not a spectral subspace for $H$. The band functions generically have overlapping ranges. Therefore, if slowly varying terms are added to the hamiltonian, the dynamics can no longer be captured so easily by a spectral analysis of the perturbed hamiltonian.
The magnetic Bloch hamiltonian {#PST_sec.2b}
------------------------------
We consider $d=3$. The hamiltonian reads $$\label{PST_2b.1}
H=\frac{1}{2}\big(-i\nabla_x-A(x)\big)^2+V_\Gamma(x)\,,\quad
x\in\mathbb{R}^3\,,$$ with $A(x)=-\frac{1}{2}B\wedge x$, $B\in\mathbb{R}^3$. Physically the most relevant case is $d=2$. It is included here by setting $x=(x_1,x_2,0)$ and $B=(0,0,B_0)$. Following Zak [@11], see also [@12], one introduces the magnetic translations $$\label{PST_Magnetic translations}
(T_\alpha\psi)(x)=\big(e^{-i\alpha\cdot(-i\nabla_x+A(x))}\psi\big)(x)=
e^{i\alpha A(x)}\psi(x-\alpha)$$ with $\alpha\in\mathbb{R}^3$. They satisfy the Weyl relations $$\label{PST_2b.3}
T_\alpha T_\beta=e^{-\frac{i}{2}B\cdot(\alpha\wedge\beta)}
T_{\alpha+\beta}=e^{-i B\cdot(\alpha\wedge\beta)}T_\beta T_\alpha\,.$$ To have a commuting subfamily we need
**Assumption 2.** *The magnetic field $B$ is such that $B\cdot(\gamma\wedge\gamma')\in 2\pi\mathbb{Q}$ for all $\gamma,\gamma'\in\Gamma$.*
In the two-dimensional case our assumption requires that the magnetic flux per unit cell, $B_0\cdot(\gamma_1\wedge\gamma_2)$, is a rational multiple of $2\pi$.
Under the Assumption 2 there exists a sublattice $\Gamma_0\subset\Gamma$ such that $B\cdot(\gamma\wedge\gamma')\in 2\pi\mathbb{Z}$ for every $\gamma,\gamma'\in\Gamma_0$. $\Gamma_0$ is not unique. The set $\{T_\alpha\}_{\alpha\in\Gamma_0}$ is a family of commuting operators, which commute with $H$. Since $T_\alpha T_\beta=\pm T_{\alpha+\beta}$, the magnetic translations still form only a projective group. It becomes a group by an even smaller sublattice $\Gamma_1\subset \Gamma_0$ such that $B\cdot(\gamma\wedge\gamma')\in 4\pi\mathbb{Z}$ for all $\gamma,\gamma'\in\Gamma_1$. Another common choice is a further modification of the phase through $$\label{PST_2b.3bis}
\mathcal{T}_\alpha=e^{-\frac{i}{2}\varphi(\alpha)}T_\alpha$$ with $\varphi(\alpha)=B_1\alpha_2\alpha_3+B_3\alpha_1\alpha_2-B_2\alpha_1\alpha_3$. Then $\mathcal{T}_\alpha\mathcal{T}_\beta=\mathcal{T}_{\alpha+\beta}$ for all $\alpha,\beta\in\Gamma_0$.
We can now proceed as in the non-magnetic case. The Zak transform becomes $$\label{PST_2b.4}
(\mathcal{U}_\mathrm{Z}\psi)(k,x)=\sum_{\gamma\in\Gamma_0}e^{-ik\cdot(x+\gamma)}
\mathcal{T}_\gamma\psi(x)\,,\quad (k,x)\in\mathbb{R}^6\,.$$ The properties of $\mathcal{U}_\mathrm{Z}\psi$ are as in (\[PST\_2.2\]), (\[PST\_2a.1a\]) provided $\Gamma$ is replaced by $\Gamma_0$, and $\mathcal{H}_\tau$ is replaced by $\mathcal{H}^B_\tau=\{u\in
L^2_\mathrm{loc}(\mathbb{R}^d,\mathcal{H}_\mathrm{f}): $ (\[PST\_2b.7\]) below holds true$\}$. In particular, $H$ of (\[PST\_2b.1\]) admits the fiber decomposition $$\label{PST_2b.5}
\mathcal{U}_\mathrm{Z}H\mathcal{U}_\mathrm{Z}^{-1}=\int^\oplus_{M^\ast}dk
H(k)$$ with $M^\ast$ the first Brillouin zone of $\Gamma^\ast_0$ and with the fiber operator $$\label{PST_2b.6}
H(k)=\frac{1}{2}(-i\nabla_y+\frac{1}{2}B\wedge y+k)^2+V_\Gamma(y)\,.$$ The domain of $H(k)$ is independent of $k$ but, in contrast to $H(k)$ from (\[PST\_2a.10\]), a function $u$ in the domain has to satisfy the more complicated boundary condition $$\label{PST_2b.7}
e^{-\frac{i}{2} y\cdot(\alpha\wedge B)}u(y-\alpha)=u(y)\,.$$
Dirac hamiltonian, spin-orbit coupling {#PST_sec.2c}
--------------------------------------
The Dirac hamiltonian with periodic potential reads $$\label{PST_2c.1}
H=\beta-i \alpha\cdot\nabla_x+V_\Gamma(x)\,,$$ where we have set $m_\mathrm{e}=1$, $c=1$. As for the Bloch hamiltonian, $H$ admits the fiber decompositon $$\label{PST_2c.2}
H=\int^\oplus_{M^\ast} dk H(k)$$ with fiber hamiltonian $$\label{PST_2c.3}
H(k)=\beta+\alpha\cdot(-i\nabla_y+k)+V_\Gamma(y)\,.$$ $H(k)$ acts on $L^2(M,\mathbb{C}^4)$ with periodic boundary conditions (\[PST\_2a.2\]). The free Dirac operator has a spectral gap of size 2, in our units, between the electron and positron subspace. If we assume $\|V_\Gamma\|<1$, then this gap persists and the eigenvalues can be labelled as $E_0(k)\leq
E_1(k)\leq\ldots$ in the electron subspace and as $E_{-1}(k)\geq
E_{-2}(k)\geq\ldots$ in the positron subspace. One has $E_{-1}(k)<E_0(k)$ for all $k\in M^\ast$. (In fact the labelling can be achieved without a restriction on $\|V_\Gamma\|$, see [@13]).
For $V_\Gamma=0$, the eigenvalue $E(k)$ is two-fold degenerate reflecting the spin $\frac{1}{2}$ of the electron, resp. positron. This degeneracy persists if the periodic potential is inversion symmetric, see [@13] for details.
\[PST\_2c.prop1\] Let $H$ be given by (\[PST\_2c.1\]) with $\|V_\Gamma\|<1$. Let there exist $a\in\mathbb{R}^3$ such that $V_\Gamma(x+a)=V_\Gamma(-x+a)$. Then each $E_n(k)$ is at least two-fold degenerate.
Without loss of generality we may assume $a=0$. We use the standard basis for the $\alpha$-matrices, see [@7]. In this basis time-reversal symmetry is implemented by the anti-unitary operator $$\label{PST_2c.4}
T\psi(y)= -i\alpha_3\alpha_1\psi^\ast(y)\,,$$ where the complex conjugation is understood component-wise. Using that $\alpha_\ell\alpha_3\alpha_1=
-\alpha_3\alpha_1\overline{\alpha}_\ell$, $\ell=1,2,3$, where $\overline{\alpha}_\ell$ refers to matrix element-wise complex conjugation, one checks that $$\label{PST_2c.5}
-i\nabla_y\alpha_\ell T=-i T\nabla_y\alpha_\ell\,,\quad k\alpha_\ell
T=-Tk\alpha_\ell$$ and therefore $$\label{PST_2c.6}
T^{-1}H(k)T=H(-k)\,.$$
Secondly we use space inversion as $$\label{PST_2c.7}
R\psi(y)=\beta\psi(-y)\,.$$ Then $$\label{PST_2c.8}
R^{-1}H(k)R=H(-k)\,.$$ Combining both symmetries implies $$\label{PST_2c.9}
T^{-1}R^{-1}H(k)RT=H(k)\,.$$
If $H(k)\psi=E\psi$, then also $RT\psi$ is an eigenfunction with the same eigenvalue. Thus our claim follows from $\langle\psi,RT\psi\rangle=0$. To verify this identity we note that $-i\alpha_3\alpha_1=\textrm{diag }(\sigma_2,\sigma_2)$ and $\langle
\chi,R\sigma_2\chi^\ast\rangle=0$ for an arbitrary two-spinor $\chi$.
\[PST\_2c.2a\] The eigenvalue $E_n(0)$ of $H(0)$ is at least two-fold degenerate.
Since $T^\ast H(0)T=H(0)$ by (\[PST\_2c.6\]) and $\langle\psi,(-i\alpha_3\alpha_1)\psi^\ast\rangle=0$, the claim follows.
If $V_\Gamma$ is not inversion symmetric, generically an energy band is two-fold degenerate at $k=0$ and then splits into two non-degenerate bands. Note that a non-degenerate eigenvalue $E_n(k)$ has an associated eigenvector with a definite spin orientation.
The Pauli equation with spin-orbit coupling has the hamiltonian $$\label{PST_2c.12}
H=-\frac{1}{2}\Delta_x+V_\Gamma(x)+\frac{1}{4}\sigma\cdot\big(\nabla
V_\Gamma(x)\wedge(-i\nabla_x)\big)\,.$$ After Zak transform the corresponding fiber hamiltonian becomes $$\label{PST_2c.13}
H(k)=\frac{1}{2}(-i\nabla_y+k)^2+V_\Gamma(y)+
\frac{1}{4}\sigma\cdot\big(\nabla
V_\Gamma(y)\wedge(-i\nabla_y+k)\big)$$ with periodic boundary conditions. $H$ of (\[PST\_2c.12\]) is bounded from below. But otherwise the band structure is similar to the periodic Dirac operator. Proposition \[PST\_2c.prop1\] and Corollary \[PST\_2c.2\] hold as stated. In the proof one only has to use the appropriate time-reversal operator, which is $T\psi=\sigma_2\psi^\ast$ in the $\sigma_3$-eigenbasis.
Gap condition and smoothness {#PST_sec.2d}
----------------------------
Let us consider one of the periodic hamiltonians, $H_\mathrm{per}$, with fiber decomposition $H(k)$. $H_\mathrm{per}$ is adiabatically perturbed to $H^\varepsilon$. Very crudely the corresponding unitary groups should be close. To make such a notion quantitative a gap condition must be imposed. We denote by $\sigma(H)$ the spectrum of the self-adjoint operator $H$.\
**Gap condition:** *We distinguish a family of $m$ physically relevant energy bands $\{E_j(k)\,,\;n\leq j\leq
n+m-1\}=\sigma_0(k)$. This family satisfies the gap condition if* $$\label{PST_2d.1}
\textrm{dist}\{\sigma_0(k)\,,\;\sigma(H(k))\setminus\sigma_0(k)\}\geq
g>0\quad \textrm{for all }k\in M^\ast\,.$$ We repeat that the gap condition is not a spectral condition for $H_\mathrm{per}$. Let us set $P^0=\bigoplus^{n+m-1}_{j=n} P_j$.
Under the gap condition the projector $P(k)$ depends smoothly, in many cases even (real) analytically, on $k$. $\mathrm{Ran}\,P^0(k)$ is spanned by the basis $\{\varphi_j(k)\}_{j=n,\ldots,n+m-1}$. If the $m$ relevant energy bands have no crossings amongst each other, then $\varphi_j$ is necessarily an eigenvector of $H(k)$ satisfying $H(k)\varphi_j(k)$$=E_j(k)\varphi_j(k)$. But if there are band crossings, it can be convenient not to insist on $\varphi_j(k)$ being an eigenvector of $H(k)$. Thus, while $P^0(k)$ is unique, the spanning basis is not. In applications it is of importance to know whether there is at least some choice of $\varphi_j(k)$, $j=n,\ldots,n+m-1$, such that they have a smooth $k$-dependence. Locally, this can be achieved. However, since $M^\ast$ has the topology of a torus, a global extension might be impossible. In fact this will generically happen for the magnetic Bloch hamiltonian, see [@14; @Novikov; @16] for examples. Somewhat surprisingly, a reasonably general answer has been provided only recently [@17]. For the case of the Bloch hamiltonian, analyticity has been proved before in cases $d=1$, $m$ arbitrary, and $d$ arbitrary, $m=1$, see Nenciu [@18; @19] and Helffer, Sjöstrand [@20]. They rely on analytical techniques. In [@17] topological methods are developed.
\[PST\_2d.prop3\] In case of the non-magnetic Bloch hamiltonian let either $d\leq 3$, $m\in\mathbb{N}$ or $d\geq 4$, $m=1$. Then there exists a collection of smooth maps $\mathbb{R}^d\ni k\mapsto\varphi_j(k)\in L^2(M)$, $j=n,\ldots,n+m-1$, with the following properties\
(i) the family $\{\varphi_j(k)\}_{j=n,\ldots,n+m-1}$ is orthonormal and spans the range of $P^0(k)$.\
(ii) each map is equivariant in the sense that $$\label{PST_2d.2}
\varphi_j(k)=\tau(\lambda)\varphi(k+\lambda)\quad \textrm{for all
}k\in\mathbb{R}^d\,,\;\lambda\in \Lambda\,,$$ where $\tau(\lambda)$ is multiplication by $e^{i\lambda\cdot y}$. The same property holds for the non-magnetic periodic Dirac operator and Pauli operator with spin-orbit coupling.
**Remark**. The proof uses the first Chern class of the vector bundle whose fiber at $k$ is the span of the family $\{\varphi_j(k)\}_{j=n,\ldots,n+m-1}$ i.e. $\mathrm{Ran}\,P^0(k)$. To establish continuity, and thus smoothness, this first Chern class has to vanish, a property, which does not hold for a magnetic Bloch hamiltonian except for some particular energy bands.
If $\varepsilon$ is small, excitations across the energy gap are difficult to achieve. More precisely to $P^0$ one can associate a projection operator $\Pi^\varepsilon$ such that for arbitrary $\ell,\ell'\in \mathbb{N}$, $\tau\in \mathbb{R}_+$, it holds $$\label{PST_2d.1a}
\|(1-\Pi^\varepsilon)e^{-iH^\varepsilon t}\Pi^\varepsilon\psi\|\leq
c_{\ell,\ell'}(\tau) \varepsilon^\ell\|\psi\|$$ for $0\leq t\leq\varepsilon^{-\ell'}\tau$ with suitable constants $c_{\ell,\ell'}(\tau)$ independent of $\varepsilon$. In other words that the subspaces $\Pi^\varepsilon\mathcal{H}$ and $(1-\Pi^\varepsilon)\mathcal{H}$ almost decouple, i.e. decouple at any prescribed level of precision and over any polynomial length of the time span under consideration. For the specific case of the Bloch hamiltonian more quantitative details on the decoupling are provided in Section \[PST\_sec.3\].
If the gap condition is not satisfied, the dynamical properties are much more model dependent. Firstly the gap condition can be violated in various ways. In our context, since $H(k)$ has discrete spectrum, the violation comes through band crossings. The behavior close to a band crossing has to be studied separately [@23; @24]. In other models the energy band sits at the bottom of the continuous spectrum of $H(k)$ without gap [@25]. Then an assertion like Equation (\[PST\_2d.1a\]) holds only under a suitable restriction to small $\ell,\ell'$, usually $\ell,\ell'=1$ or perhaps $\ell=2$, $\ell'=1$.
The inequality (\[PST\_2d.1a\]) makes no assertion about the dynamics inside the almost invariant subspace $\Pi^\varepsilon\mathcal{H}$. While there is a general theory available [@1], we prefer to discuss the examples separately in the subsequent sections.
Nonmagnetic Bloch hamiltonians: Peierls substitution and geometric phase corrections {#PST_sec.3}
====================================================================================
We discuss in more detail the effective dynamics for the Schrödinger equation with a periodic potential. For concreteness we fix the spatial dimension to be 3. Under Zak transform the nonmagnetic Bloch hamiltonian becomes $$\label{PST_3.1}
\mathcal{U}_Z\Big(\frac{1}{2}\big(-i\nabla_x -A(\varepsilon
x)\big)^2 +V_\Gamma(x)+\phi(\varepsilon
x)\Big)\mathcal{U}^{-1}_Z=H^\varepsilon_Z$$ with $$\label{PST_3.2}
H^\varepsilon_Z=\frac{1}{2}\big(-i\nabla_y +k
-A(i\varepsilon\nabla^\tau_k)\big)^2
+V_\Gamma(y)+\phi(i\varepsilon\nabla^\tau_k)\,.$$ Here $\nabla^\tau_k$ is differentation with respect to $k$ and satisfying the $y$-dependent boundary conditions (\[PST\_2a.3\]). $H^\varepsilon_Z$ is a self-adjoint operator on $L^2_\tau(\mathbb{R}^3,H^2(M))$, compare with (\[PST\_2a.5\]).
In (\[PST\_3.2\]) we observe that the external potentials couple the fibers. To emphasize this feature we think of (\[PST\_3.2\]) as being obtained through Weyl quantization from the operator valued function $$\label{PST_3.3}
H_0(k,r)=\frac{1}{2}\big(-i\nabla_y +k -A(r)\big)^2
+V_\Gamma(y)+\phi(r)$$ as defined on $(r,k)\in\mathbb{R}^6$ and acting on $\mathcal{H}_\mathrm{f}$ with fixed domain $H^2(M)$, see [@21] for details. In this form one is reminded of the Weyl quantization of the classical hamiltonian function $h_{\mathrm{cl}}(q,p)=\frac{1}{2}p^2+V(q)$ which yields the semiclassical hamiltonian $$\label{PST_3.4}
H_\mathrm{sc}=\frac{1}{2}(-i\varepsilon\nabla_x)^2+V(x)$$ acting in $L^2(\mathbb{R}^3)$. The analysis of (\[PST\_3.4\]) yields that on the time-scale $\varepsilon^{-1}t$ the wave packet dynamics governed by $H_\mathrm{sc}$ well approximates the flow generated by $h_\mathrm{cl}$. In contrast, the adiabatic analysis deals with operator valued symbols, as in (\[PST\_3.3\]), and has as a goal to establish that the dynamics decouples into almost invariant subspaces and to determine the approximate dynamics within each such subspace.
To be specific, let us then fix throughout one band index $n$ and let us assume that the band energy $E_n$ is nondegenerate and satisfies the gap condition. Therefore we know that $E_n:M^\ast\to\mathbb{R}$ is smooth and we can choose the family of Bloch functions $\varphi_n(k)$, with $H(k)\varphi_n(k)=E_n(k)\varphi_n(k)$, such that $\varphi_n$ depends smoothly on $k$. For each $\ell\in\mathbb{N}=\{0,1,\ldots\}$ there exists then an orthogonal projection $\Pi^\varepsilon_\ell$ on $\mathcal{H}_\tau$ such that $$\label{PST_3.5}
\|[H^\varepsilon_Z\,,\,\Pi^\varepsilon_\ell]\|\leq c_\ell\varepsilon^{\ell+1}$$ for some constants $c_\ell$. Integrating in time one concludes that the subspaces $\Pi^\varepsilon_\ell\mathcal{H}_\tau$ are almost invariant in the sense that $$\label{PST_3.6}
\|(1-\Pi^\varepsilon_\ell)e^{-i\varepsilon^{-\ell'}tH^\varepsilon_Z}\Pi^\varepsilon_\ell\psi\|\leq
\|\psi\|(1+|t|)c_{\ell}\,\varepsilon^{\ell+1}\varepsilon^{-\ell'}$$ for any $\ell, \ell'\in\mathbb{N}$. Note that the adiabatic time scale, order $\varepsilon^{-\ell'}$, can have any power law increase, at the expense of choosing the order of the projection $\Pi^\varepsilon_\ell$ sufficiently large. Only for times of order $e^{1/\varepsilon}$ one observes transitions away from the almost invariant subspace. The zeroth order projection is attached to the $n$-th band under consideration, while the higher orders are successively smaller corrections to $\Pi^\varepsilon_0$. To construct $\Pi^\varepsilon_0$ one considers the projection onto the $n$-th band, $|\varphi_n(k)\rangle\langle \varphi_n(k)|$, as an operator valued function with values in $B(\mathcal{H}_\mathrm{f})$. From it we obtain the minimally substituted projection $|\varphi_n(k-A(r))\rangle\langle
\varphi_n(k-A(r))|$. Its Weyl quantization is $\varepsilon$-close to the orthogonal projection $\Pi^\varepsilon_0$.
The second task is to determine the approximate time-evolution on $\Pi^\varepsilon_\ell\mathcal{H}_\tau$. Since the subspace changes with $\varepsilon$, it is more convenient to unitarily map $\Pi^\varepsilon_\ell\mathcal{H}_\tau$ to an $\varepsilon$-independent reference Hilbert space, which in our case is simply $L^2(M^\ast)$. The dynamics on $L^2(M^\ast)$ is governed by an effective hamiltonian. It is written down most easily in terms of a hamiltonian function $h^\varepsilon_\ell:M^\ast\times\mathbb{R}^3\to\mathbb{R}$. $h^\varepsilon_\ell$ is a smooth function. We also may regard it as defined on $\mathbb{R}^3\times\mathbb{R}^3$ and $\Gamma^\ast$-periodic in the first argument. $h^\varepsilon_\ell$ admits the power series $$\label{PST_3.7}
h^\varepsilon_\ell=\sum^\ell_{j=0}\varepsilon^j h_j$$ with $\varepsilon$-independent functions $h_j$. The effective quantum hamiltonian is obtained from $h^\varepsilon_\ell$ through Weyl quantization. The index $\ell$ regulates the time scale over which the approximation is valid and the size of the allowed error.
In [@21] we provide an iterative algorithm to compute $h_j$. In practice only $h_0$ and $h_1$ can be obtained, at best $h_2$ under simplifying assumptions. While this may look very restrictive, it turns out that already $h_1$ yields novel physical effects as compared to $h_0$. Even higher order corrections seem to be less significant.
To lowest order one obtains $$\label{PST_3.8}
h_0(k,r)=E_n(k-A(r))+\phi(r)\,,$$ which Weyl-quantizes to $$\label{PST_3.9}
\mathcal{W}^\varepsilon
[h_0]=E_n(k-A(i\varepsilon\nabla_k))+\phi(i\varepsilon\nabla_k)$$ acting on $L^2(M^\ast)$, where $i\nabla_k$ is the operator of differentiation with periodic boundary conditions. (The twisted boundary conditions appearing in (\[PST\_3.2\]) are absorbed into the unitary map of $\Pi^\varepsilon_0\mathcal{H}$ to $L^2(M^\ast)$.) In solid state physics the Weyl quantization (\[PST\_3.9\]) is referred to as [*Peierls substitution*]{}. (\[PST\_3.8\]), (\[PST\_3.9\]) have a familiar form. The periodic potential merely changes the kinetic energy $\frac{1}{2}k^2$ of a free particle to $E_n(k)$. The main distinctive feature is the periodicity of the kinetic energy. For example, in presence of a linear potential $\phi$, $\phi(x)=-E\cdot x$, an electron, initially at rest, will start to accelerate along $E$ but then turns back because of periodicity in $k$.
To first order the effective hamiltonian reads $$\label{PST_3.10}
h_1(k,r)=\big(\nabla \phi(r)-\nabla E_n(\widetilde{k})\wedge
B(r)\big)\cdot\mathcal{A}_n(\widetilde{k})-B(r)\cdot
\mathcal{M}_n(\widetilde{k})\,,$$ with the kinetic wave number $\widetilde{k}=k-A(r)$. The coefficients $\mathcal{A}_n$ and $\mathcal{M}_n$ are the geometric phases. They carry information on the Bloch functions $\varphi_n(k)$. $\mathcal{A}_n$ is the Berry connection given through $$\label{PST_3.11}
\mathcal{A}_n(k)=i\langle \varphi_n(k)\,,\; \nabla_k
\varphi_n(k)\rangle_{\mathcal{H}_f}$$ and $\mathcal{M}_n $ is the Rammal-Wilkinson phase given trough $$\label{PST_3.12}
\mathcal{M}_n(k)=\frac{1}{2}i\langle\nabla_k
\varphi_n(k),\wedge(H(k)-E_n(k))\nabla_k \varphi_n(k)\rangle_{\mathcal{H}_f}\,.$$
The Bloch functions $\varphi_n$ are only determined up to a smooth phase $\alpha(k)$, i.e. instead of $\varphi_n(k)$ one might as well use $e^{-i\alpha(k)}\varphi_n(k)$ with smooth $\alpha:M^\ast\to \mathbb{R}$. Clearly $\mathcal{M}_n$ is independent of the gauge field $\alpha$. On the other hand, $\mathcal{A}$ is gauge-dependent while its curl $$\label{PST_3.13}
\Omega_n=\nabla\wedge\mathcal{A}_n$$ is gauge independent. From time-reversal one concludes that $$\label{PST_3.14}
\Omega_n(-k)=-\Omega_n(k)\,.$$ In particular, in dimension $d=2$ for the first Chern number of the Bloch vector bundle one obtains $$\label{PST_3.15}
\int_{M^\ast}dk\Omega_n(k)=0\,.$$
For the magnetic Bloch hamiltonian, (\[PST\_3.14\]) is violated in general, see Section \[PST\_sec.4\]. The integral in (\[PST\_3.15\]) can take only integer values (in the appropriate units) and the first Chern number may be different from zero. Physically this leads to the quantization of the Hall current [@21; @22].
We still owe the reader precise a statement on the error in the approximation. At the moment we work in the representation space at precision level $\ell=1$. Let $H_\mathrm{eff}$ be the Weyl quantization of $h_0+\varepsilon h_1$, see (\[PST\_3.8\]) and (\[PST\_3.10\]). There is then a unitary $U^\varepsilon:\Pi^\varepsilon_1\mathcal{H}_\tau\to L^2(M^\ast)$ such that for all $\psi\in \mathcal{H_\tau}$ $$\label{PST_3.16}
\|\big(e^{-iH^\varepsilon_Z t}-U^{\varepsilon\ast}
e^{-iH_{\mathrm{eff}}t}U^\varepsilon\big)
\Pi^\varepsilon_1\psi\|\leq c\|\psi\|(1+|\tau|)\varepsilon^2$$ with $|t|\leq\varepsilon^{-1}\tau$ and some constant $c$ independent of $\|\psi\|$, $\tau$, and $\varepsilon$.
Magnetic Bloch hamiltonians: the Hofstadter butterfly {#PST_sec.4}
=====================================================
We turn to a magnetic Bloch hamiltonian in the form (\[PST\_1.4\]), in dimension $d=2$ and with a transverse constant magnetic field $B_0$. We want to explain how the limits $B_0 \to \infty$ and $B_0 \to 0$ can be understood with adiabatic methods. As a remark, it is worthwhile to recall that, when the physical constants are restored, the dimensionless parameter $B_0$ is given by $$\label{PST_B magnitude}
B_0 = \frac{\mathcal{B}_0 S}{2 \pi \hbar c / e}\,,$$ where $S$ is the area of the fundamental cell of $\Gamma$ and $\mathcal{B}_0$ the strength of the magnetic field, both expressed in their dimensional units. Thus $B_0$ corresponds physically to the magnetic flux per unit cell divided by $hc/e$, as the fundamental quantum of magnetic flux. This section is based essentially on [@FaurePanati], which elaborates on previous related results [@Bel1986; @20].
Adiabatic limits are always related to separation of time-scales. In the present case, one indeed expects that as $B_0 \to \infty$ the cyclotron motion induced by $B_0$ is faster than the motion induced by $V_{\Gamma}$, while in the limit $B_0 \to
0$ the microscopic variations of the wave function induced by $V_{\Gamma}$ are expected to be faster than the cyclotron motion.
Let us focus first on the Landau regime $B_0 \to \infty$. In order to make quantitative the previous claim, one introduces the operators $$\label{PST_L operators}
\left\{\begin{array}{ll}
L_1 = \frac{1}{\sqrt{B_0}}\(p_1 + \frac{1}{2}B_0 \, x_2 \)\,,\\
& \qquad \qquad [L_1,L_2]=i \1, \\
L_2 = \frac{1}{\sqrt{B_0}}\(p_2 - \frac{1}{2}B_0 \, x_1 \)\,, \\
\end{array}\right.$$ and the complementary pair of operators $$\label{PST_G operators}
\left\{\begin{array}{ll}
G_1 = \frac{1}{B_0}\(p_1 - \frac{1}{2}B_0 \, x_2 \)\,,\\
& \qquad \qquad [G_1,G_2]= \frac{i}{B_0} \1, \\
G_2 = \frac{1}{B_0}\(p_2 + \frac{1}{2}B_0 \, x_1 \)\,, \\
\end{array}\right.$$ where the relative sign is chosen such that $[L_i, G_j]=0$, for $i,j=1,2$.
If $V_{\Gamma} =0$, then $H_{\rm MB}$ describes a harmonic oscillator, with eigenfunctions localized on a scale $|B_0|^{-1/2}$; this corresponds to the cyclotron motion in classical mechanics. Since $[G_i, H_{\rm MB}] =0$, the operators $G_1$ and $G_2$ describe conserved quantities, which semiclassically correspond to the coordinates of the center of the cyclotron motion.
If $V_{\Gamma} \neq 0$, but the energy scale $ \| V_{\Gamma}\|$ is smaller than the cyclotron energy $\approx B_0$, then the operators $G_i$ have a non-trivial but slow dynamics. By introducing the adiabatic parameter $\s = 1/B_0$ the hamiltonian reads $$H_{\rm MB} = \frac{1}{2 \s} \( L_1^2 + L_2^2 \) + V_{\Gamma}\( G_2 - \sqrt{\s}L_2,
\, - G_1 + \sqrt{\s} L_1\).$$
In view of the commutator $[G_1,G_2] = i \s \1$, one can regard $\s
\, H_{\rm MB}$ as the $\s$-Weyl quantization (in the sense of the mapping $(q,p)\mapsto(G_1,G_2)$) of the operator-valued symbol $$h_{\rm MB}(q,p) = \frac{1}{2} \( L_1^2 + L_2^2 \) + \s \, V_{\Gamma}\( p -
\sqrt{\s}L_2, - q + \sqrt{\s}L_1\).$$ For each fixed $(q,p) \in \R^2$, $h_{\rm MB}(q,p)$ is an operator acting in the Hilbert space $\Hf \cong L^2(\R)$ corresponding to the fast degrees of freedom. If $\| V_{\Gamma}\|_{\B(\Hi)} < \infty$, then $h_{\rm MB}(q,p)$ has purely discrete spectrum, with eigenvalues $$\lambda_{n,\, \s}(q,p) = (n + \frac{1}{2}) + \s V_{\Gamma}(p,-q) +
\Or(\s^{3/2}), \qquad n \in \N,$$ as $\s \downarrow 0$. The index $n \in \N$ labels the *Landau levels*. For $\s$ small enough, each eigenvalue band is separated from the rest of the spectrum by an uniform gap. Thus we can apply space-adiabatic perturbation theory to show that the band corresponds to an almost-invariant subspace $\Pi_{n, \s}
L^2(\R^2)$. Let us focus on a specific $n \in \N$. One can prove that the dynamics inside $\mathrm{Ran}\, \Pi_{n, \s} L^2(\R^2)$ is described by an effective hamiltonian, which at the first order of approximation in $\s$ reads $$\label{PST_Landau effective}
h^{\s}_1 = (n + \frac{1}{2}) + \s V_{\Gamma}(G_1,-G_2).$$
The first term in (\[PST\_Landau effective\]) is a multiple of the identity, and as such does not contribute to the dynamics as far as the expectation values of observables are concerned. Leading-order dynamics is thus described by the second term, which does not depend on the Landau level $n \in \N$. Since $V_{\Gamma}$ is a biperiodic function and $(G_1, G_2)$ a canonical pair, the second term is a Harper-like operator. The spectrum of such operators exhibit a complex fractal behavior (*Hofstadter butterfly*) sensitively depending on the diophantine properties of $\alpha = \frac{B_0}{2 \pi}$ (notice that $V_{\Gamma}(G_1, G_2)$ depends on $\alpha$ through the commutator $[G_1,G_2] = i B_0^{-1} \1$). The Cantor structure of the spectrum was proven first in [@BeS] for the case $V_{\Gamma}(x_1,x_2) = \lambda \cos x_1 + \cos x_2$ (Harper model), for a dense set of the parameter values. Later Helffer and Sjöstrand accomplished a detailed semiclassical analysis of the Harper operator [@HS_Harper]. As a final step the Cantor spectrum has been proven by Puig ($\lambda \neq 0$, $\alpha$ Diophantine) [@Puig], and by Avila and Jitomirskaya [@AvilaJitomirskaya] for all the conjectured values of the parameters: $\lambda \neq 0$, $\alpha$ irrational (the *Ten Martini conjecture*, as baptized by B. Simon).
Secondly we turn to the opposite limit $B_0 \to 0$, where the slow part of the dynamics is still described by the magnetic momentum operators $\widetilde{L}_j =
\sqrt{B_0} L_j$ $(j=1,2)$, with commutator of order $\Or(B_0)$. However the decomposition given by (\[PST\_L operators\]) and (\[PST\_G operators\]) is no longer convenient.
Since $A_0$ is a linear function, $A_0(\epsi x) = \frac{1}{2} \epsi B_0 \wedge x $, the slow variation limit $\epsi \to 0$ agrees with the weak field limit $B_0 \to 0$. We then pose $\epsi=B_0$ and we regard $H_{\rm MB}$ in (\[PST\_1.4\]) as an adiabatic perturbation of the periodic hamiltonian (\[PST\_2a.1a\]). Thus we are reduced to the situation described in Section 3: to each isolated Bloch band of the unperturbed hamiltonian there corresponds a subspace $\Pi_{n, \epsi}L^2(\R^2)$ which is approximately invariant under the dynamics as $\epsi \downarrow 0$. The dynamics inside this subspace is described by Peierls substitution (\[PST\_3.9\]), which now reads $$\label{PST_Peierls magnetic}
\mathcal{W}^{\epsi}[h_0] = E_n(k - \frac{1}{2} e_3 \wedge (i \epsi \nabla_k)),$$ as an operator acting in $L^2(\T^2, dk)$. Here $B_0=(0,0,\epsi)$ and $e_i$ is the unit vector in the $i$-th direction.
Formula (\[PST\_Peierls magnetic\]) shows that the leading order effective hamiltonian depends only on the operators $(K_1,K_2)=K$, $$K = k - \frac{1}{2} e_3 \wedge (i \epsi \nabla_k),$$ which roughly speaking are the Fourier transform of the pair $(\widetilde{L}_1,
\widetilde{L}_2)$, and not on the complementary pair of operators. The same property holds true for the effective hamiltonian $h^{\epsi}_\ell$, at any order of approximation $\ell \in \N$, see [@FaurePanati], with important consequences on the splitting of magnetic subbands at small but finite $B_0$.
An operator in the form (\[PST\_Peierls magnetic\]), shortly written $E_n(K_1,K_2)$, is *isospectral* to an Harper-like operator, namely $E_n(G_1,G_2)$ acting in $L^2(\R)$. Indeed the first numerical evidence of the butterfly-like Cantor structure of the spectrum of Harper-like operators appeared when Hofstadter investigated the spectrum of $\cos K_1 + \cos K_2$ as a function of $\epsi$ [@Hofstadter1976]. On the other side, an operator of the form $E_n(K_1,K_2)$ is not *unitarily equivalent* to the Harper operator $E_n(G_1,G_2)$. The important geometric and physical consequences of this fact are developed in [@FaurePanati].
Having explained the two extreme cases, $B_0 \to 0$ and $B_0 \to \infty$, the reader may wonder about the intermediate values of the magnetic field, $B_0 \approx 1$. As explained already in Section \[PST\_sec.2c\] it is convenient to introduce the magnetic translations $$\mathcal{T}_{\alpha}= e^{-\frac{i}{2}\varphi(\alpha)}\, \exp(i B_0 \, \alpha \cdot
G), \qquad \alpha \in \Gamma_0,$$ see (\[PST\_Magnetic translations\]) and (\[PST\_2b.3bis\]). If $B_0$ satisfy Assumption 2, then $\{ \mathcal{T}_{\alpha} \}$ is a commutative group, thus leading to the magnetic Zak transform (\[PST\_2b.4\]). $H_{\rm MB}$ is then a fibered operator over the magnetic Bloch momentum $\kappa \in \T^2$. At each $\kappa$ the spectrum of $H_{\rm MB}(\kappa)$ is pure point and the corresponding eigenvalues $\mathcal{E}^{B_0}_{n}$ are the *magnetic Bloch bands*.
In view of this structure, one might argue that the adiabatic perturbation of the hamiltonian which includes, on top of the constant magnetic field $B_0$, a slowly varying magnetic potential $A(\epsi x)$ as in (\[PST\_1.5\]) can be treated with the methods of Section 3. There is however one crucial element missing. Indeed one can still associate to each magnetic Bloch band $\mathcal{E}^{B_0}_{n}$, isolated from the rest of the spectrum, an almost-invariant subspace ${\rm Ran}\,
\Pi^{B_0}_{n}$. On the other side the construction of the effective hamiltonian relies on smoothness which may be impeded for topological reasons. Indeed the analogue of Proposition \[PST\_2d.prop3\] is generically false for magnetic Bloch hamiltonians, as well-understood [@14; @Novikov; @16]. In geometric terminology this fact is rephrased by saying that the magnetic Bloch bundle is generically non-trivial (in technical sense). This important fact has sometimes been overlooked. For example, Assumption B in [@DGR04] is equivalent to the triviality of the magnetic Bloch bundle. Under this assumption the magnetic case is already covered by the results in [@21]. Thus the problem of adiabatic perturbation of a generic magnetic Bloch hamiltonian appears to be an open, in our view challenging, problem for the future.
Piezoelectricity {#PST_sec.5}
================
In the year 1880 the brothers Jacques and Pierre Curie discovered that some crystalline solids (like quartz, tourmaline, topaz, …) exhibit a macroscopic polarization if the sample is strained.
It turns out that also this effect can be understood in the framework of adiabatically perturbed periodic hamiltonians, cf.[@27; @28]. The perturbation is now slowly in time, $$\label{PST_1.6'}
H_\mathrm{PE}(t)=-\frac{1}{2}\Delta_x+V_{\Gamma(\varepsilon
t)}(x,\varepsilon t)\,.$$ If the potential $V_\Gamma(x,\epsi t)$ has no center of inversion, i.e. there is no point with respect to which the potential has space-reflection symmetry, then the slow variation of the periodic potential is expected to generate a non-zero current and can be shown to do so for particular examples [@avron1997]. By translation invariance this current if averaged over a unit cell is everywhere the same and we denote the average current by $J^\epsi(t)$. For the following discussion we assume that $V_\Gamma$ varies only for times in the finite interval $[0,T]$. Integrating the current per volume over the relevant time interval yields the average polarization, $$\Delta {\bf P}^\epsi = \int_0^T\D t\, J^\epsi(t)\,.$$ In this section we discuss results that relate the current $J^\epsi(t)$ directly to the quantum mechanics of non-interacting particles governed by the hamiltonian , without the detour via the semiclassical model. For this we need to solve the Schrödinger equation with initial state $\rho(0)=P (0)$ being the spectral projection of $H_{\rm PE}(0)$ below the Fermi energy $E(0)$. Since the piezo effect occurs only for insulators, we can assume that $E(0)$ lies in a gap of the spectrum of $H_{\rm PE}(0)$ and, in order to simplify the discussion, we also assume that this gap does not close in the course of time. Hence there is a continuous function $E:[0,T]\to\R$ such that $E(t)$ lies in a gap of $H_{\rm PE}(t)$ for all $t$. The state at time $t$ is given by $$\rho^\epsi(t) = U^\epsi(t,0)\, P (0)\, U^\epsi(t,0)^*\,,$$ where the unitary propagator $U^\epsi(t,0)$ is the solution of the time-dependent Schrödinger equation $$\I\epsi\frac{\D}{\D t}\,U^\epsi(t,0) = H_{\rm PE}(t) \,U^\epsi(t,0)
\qquad\mbox{with}\quad U^\epsi(0,0)= {\bf 1}\,.$$ With the current operator given by $$\label{PST_curr}
j^\epsi := \frac{ \I }{\epsi} \, [ H(t), x] =
-\frac{\I}{\epsi}\nabla_x\,,$$ and the trace per volume defined as $$\label{PST_TraceperVolume}
\mathcal T (A) := \lim_{\Lambda_n\to\mathbb{R}^3}\frac{1}{|\Lambda_n|}
\re\, \Tr ({\bf 1}_{\Lambda_n} A)\,,$$ with ${\bf 1}_{\Lambda_n}$ being the characteristic function of a $3$-dimensional box $\Lambda_n$ with finite volume $|\Lambda_n|$, the average current in the state $\rho^\epsi(t)$ is $$J^\epsi(t) = \mathcal{T}(\rho^\epsi(t)\,j^\epsi)\,.$$ Finally the average polarization is $$\label{PST_defp}
\Delta \p^\epsi = \int_{0}^{T} \!\!\!\D t \,\,\mathcal T (\rho^\epsi(t)\,J^\epsi)\, ,$$ which is the main quantity of physical interest. The given framework allows us to describe the macroscopic polarization of a solid by a pure *bulk property*, i.e. independently of the shape of the sample.
In the simplest but most important case (see Paragraph (ii) in Section 1 for a discussion of the model), the periodic potential $V_\Gamma(x,\epsi t)$ is periodic with respect to a time-[*independent*]{} lattice $\Gamma$. For this case King-Smith and Vanderbilt [@5] derived a formula for $\Delta\p$ based on linear response theory, which turned out to make accurate predictions for the polarization of many materials. Their by now widely applied formula reads $$\label{PST_KSV0}
\Delta \p = \frac{1}{(2\pi)^3} \sum_{n=0}^{N_{\rm c}} \int_{M^*} \D k \,\, \big(
\A_n(k,T) - \A_n(k,0) \big),$$ where the sum runs over all the occupied Bloch bands and $\A_n(k,t)$ is the Berry connection coefficient for the $n$-th Bloch band at time $t\in \R$, $$\A_n(k,t)= \I \langle \varphi_n(k,t) , \nabla_k \varphi_n (k,t) \rangle_{L^2(M)}\,.$$ Although $\mathcal{A}_n$ depends on the choice of the Bloch function $\varphi_n$, the average polarization defines a gauge invariant quantity, i.e. it is independent of the choice of Bloch functions.
In [@27] we show that $\Delta \p^\epsi$ defined in approaches $\Delta \p$ as given by the King-Smith and Vanderbilt formula with errors smaller than any power of $\epsi$, whenever the latter is well defined. More precisely we show that under suitable technical conditions on $V_\Gamma(t)$ the average polarization is well defined and that for any $N\in\N$ $$\label{PST_KSV1}
\Delta \p^\epsi = -\frac{1}{(2\pi)^d} \int_{0}^{T} \!\!\!\D t \int_{M^*} \D k \,\,
\Theta (k,t) + \mathcal{O}(\epsi^{N})\, ,$$ where $$\label{PST_theta}
\Theta(k,t):=-\I \,\tr \left( P(k,t)\,[\partial_t P(k,t),\,\nabla_k P(k,t)\,]\,\right)\,,$$ and $P(k,t)$ is the Bloch-Floquet fiber decomposition of the spectral projector $P(t)={\bf 1}_{(-\infty,E(t)]}(H_{\rm PE}(t))$. Whenever all Bloch bands within Ran$P(k,t)$ are isolated, the explicit term in agrees with . Note however that is more general, since it can be applied also to situations where band crossings occur within the set of occupied bands.
From the point of view of adiabatic approximation, this result is actually quite simple, since one just needs the standard time-adiabatic theory. At time $t=0$ the state $\rho(0)$ is just the projection $P(0)$ onto the subspace of the isolated group of occupied bands. Since these bands remain isolated during time evolution, this subspace is adiabatically preserved according to the original adiabatic theorem of Kato [@26], i.e.$$\rho^\epsi(t) = P(t) + \mathcal{O}(\epsi)\,,$$ and one can compute the higher order corrections to $\rho^\epsi(t)$ using the higher order time-adiabatic approximation due to Nenciu [@Nenciu1993]. In order to get explicit results, one has to do the adiabatic approximation for each fixed $k\in
M^*$ separately. This is possible since $H_{\rm PE}(t,k)$ is still fibered in $k$, due to translation invariance with respect to a time-independent lattice. However, since we need to differentiate with respect to $k$ in order to compute the current, as suggested by formula , the expansion needs to be done uniformly on spaces of suitable equivariant functions. This makes the proof more technical than expected at first sight.
Alternatively one can derive also for $H_{\rm PE}(t)$ the semiclassical equations of motion including first order corrections: $$\label{PST_sceqPE} \left\{
\begin{array}{lcl}
\dot q &=& \nabla_k E_n(k,t) - \epsi\, \Theta_n(k,t), \\[2mm]
\dot k &=& 0\,.
\end{array}
\right.$$ And again averaging the velocity over the first Brillouin zone yields the correct quantum mechanical average current that is the contribution from the $n$-th band.
Note the striking similarity between the semiclassical corrections in and the electromagnetic field. If we define the geometric vector potential $$\A_n(k,t)= \I \langle \varphi_n(k,t) , \nabla_k \varphi_n (k,t) \rangle_{L^2(M)},$$ and the geometric scalar potential $$\phi_n(k,t) = - \I \langle \varphi_n(k,t) , \partial_t \varphi_n (k,t)
\rangle_{L^2(M)},$$ in terms of the Bloch function $\varphi_n(k,t)$ of some isolated band, then in complete analogy to the electromagnetic fields we have $$\label{PST_Piezocurvature2}
\Theta_n(k,t)= -\partial_t \A_n(k,t) - \nabla_k \phi_n(k,t),$$ and $$\Omega_n(k,t) = \nabla_k \wedge \A_n(k,t)\,.$$
Time-dependent deformations of a crystal generically also lead to a time-dependent periodicity lattice $\Gamma(t)$, see (\[PST\_1.6’\]). This more general situation is considered in [@28; @31]. Now the lattice momentum $k$ is no longer a conserved quantity and the full space-adiabatic perturbation theory is required in order to compute the corresponding piezoelectric current. As a result an additional term appears in the semiclassical equations of motion, reflecting the deformation of the lattice of periodicity.
[**Acknowlegdments.**]{} We thank Ulrich Mauthner, Max Lein, and Christof Sparber for most informative discussions. This work has been supported by the DFG Priority Program 1095 “Analysis, Modeling and Simulation of Multiscale Problems” under Sp 181/16-3.
|
---
abstract: 'As machine learning becomes more widely used, the need to study its implications in security and privacy becomes more urgent. Research on the security aspects of machine learning, such as adversarial attacks, has received a lot of focus and publicity, but privacy related attacks have received less attention from the research community. Although there is a growing body of work in the area, there is yet no extensive analysis of privacy related attacks. To contribute into this research line we analyzed more than 40 papers related to privacy attacks against machine learning that have been published during the past seven years. Based on this analysis, an attack taxonomy is proposed together with a threat model that allows the categorization of the different attacks based on the adversarial knowledge and the assets under attack. In addition, a detailed analysis of the different attacks is presented, including the models under attack and the datasets used, as well as the common elements and main differences between the approaches under the defined threat model. Finally, we explore the potential reasons for privacy leaks and present an overview of the most common proposed defenses.'
author:
- Maria Rigaki
- Sebastian Garcia
bibliography:
- 'references.bib'
title: A Survey of Privacy Attacks in Machine Learning
---
<ccs2012> <concept> <concept\_id>10002978</concept\_id> <concept\_desc>Security and privacy</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257</concept\_id> <concept\_desc>Computing methodologies Machine learning</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Introduction
============
Fueled by large amounts of available data and hardware advances, machine learning has experienced tremendous growth, both in terms of academic research and of real world applications. At the same time, the impact of machine learning in security, privacy, and fairness is receiving increasing attention. In terms of privacy, our personal data are being harvested by almost every online service and are used to train models that power machine learning based applications. When these applications are presented as black-box models, it is expected that they should not reveal information about the data used for their training. If a model was trained using sensitive data such as location, health records, or identity information, then an attack that allows an adversary to extract this information is highly undesirable. At the same time, if private data have been used without their owners’ consent, the same type of attack could be used as a way to determine unauthorized use and thus work in favor of the user’s privacy.
The security of machine learning and the impacts of adversarial attacks in the performance of the models have been widely studied in the community, with several surveys highlighting the major advances in the area [@papernot2018sok; @wang2019security; @biggio2018wild; @maiorca2019towards]. Some of these surveys also provide a partial coverage on the topic of privacy attacks, but there is no overall survey that considers privacy attacks against machine learning models as its main focus. This paper is, as far as we know, the *first comprehensive survey of privacy-related attacks against machine learning*. This survey focuses on leaks of information from the training data and also leaks of information about the models themselves. In this sense, an attack that extracts information about the model structure is, strictly speaking, an attack against model confidentiality. The decision to include model extraction attacks was made because (i) these attacks are an important part of the threat model presented in Section \[sec:threat\_model\] and (ii) because in the existing literature, attacks against model confidentiality are usually grouped together with privacy attacks [@papernot2018sok; @biggio2018wild]. In addition, Veale et al. [@veale2018algorithms] made the argument that privacy attacks such as membership inference (Section \[subsec:mi\_attacks\]) increase the risk of machine learning models being classified as personal data under the General Data Protection Regulation (GDPR) because they can render a person identifiable. Although models are currently not covered by the GDPR, if they are potentially considered as personal data, then attacks against them may fall on the same scope as attacks against personal data. This may be further complicated by the fact that model extraction attacks can be used as a stepping stone for other privacy based attacks.
This survey present and summarize research about privacy-based attacks on machine learning that has been published in top tier conferences and journals during 2014-2020 in the areas of security, privacy, and machine learning. An initial set of papers was selected in Google Scholar using keyword searches related to “privacy”, “machine learning” and the names of attacks themselves. After the initial set of papers was selected, backward searches based on their references as well as forward searches based on papers that cited them, were used to generated the final list.
The main contributions of this paper are:
- The first comprehensive study of attacks against privacy and confidentiality of machine learning systems.
- A threat model and a taxonomy of attacks against machine learning privacy (Sections \[sec:threat\_model\] and \[sec:taxonomy\]).
- An in-depth comparison of similarities and differences of the design of the attacks (Section \[sec:attack\_design\]).
- A discussion on the probable causes of the privacy leaks in machine learning systems (Section \[sec:why\_leak\]).
- An overview of the different defensive measures tested to protect against the attacks (Section \[sec:defenses\]).
Machine Learning
================
Machine learning (ML) is a field that studies the problem of learning from data without being explicitly programmed. This section provides a very high level overview of machine learning in order to facilitate the discussion in the subsequent chapters and to introduce the relevant notation. Several textbooks such as [@goodfellow2016deep; @bishop2006pattern; @murphy2012machine; @shalev2014understanding] provide a more thorough coverage of the topic.
Types of Machine Learning
-------------------------
At a very high level ML is usually split into three major areas: *supervised*, *unsupervised* and *reinforcement* learning. Deep Learning is a subset of ML that focuses on deep neural network (DNN) models. It has grown in popularity during the past decade and has applications in all ML areas.
### Supervised Learning
In a supervised learning setting, a model $f$ with parameters $\mathbf{\theta}$ is a mapping function between inputs $\mathbf{x}$ and outputs $\mathbf{y}=f(\mathbf{x};\mathbf{\theta})$, where $\mathbf{x}$ is a vector of attributes or features with dimensionality $n$ and the output or response can assume different dimensions depending on the learning task. A training set $\mathcal{D}$ used for training the model is a set of data points $\mathcal{D} = \{(\mathbf{x}_{i}, \mathbf{y}_{i})\}_{i=1}^{m}$, where $m$ is the number of the input-output pairs. The most common supervised learning tasks are *classification* and *regression*. The vast majority of the attack papers presented in this work are focused in supervised learning.
### Unsupervised Learning
In unsupervised learning there are no labels $\mathbf{y}$. The training set $D$ consists only of the inputs $\mathbf{x}_{i}$. Unsupervised algorithms aim to find structure or patterns in the data without having access to labels. Usual tasks in unsupervised learning are *clustering*, *feature learning* and *dimensionality reduction*. Generative tasks that aim to learn how to generate samples from the underlying data distribution, such as Generative Adversarial Networks (GANs) [@goodfellow2014gans] and Variational Autoencoders (VAEs) [@diederik2014auto] are also considered a part of unsupervised learning.
**Generative Adversarial Networks** consist of two neural networks, a generator and a discriminator. The generator $\mathcal{G}$ maps a latent variable $z$ typically sampled from a Gaussian or Uniform distribution, to the output $x$. The discriminator $\mathcal{D}$ is trying to learn the difference between the generated output and the real data. The generic architecture of GANs is depicted in Figure \[fig:gan\]. The training between the two components is adversarial in nature and in its initial formulation it was expressed as a zero-sum game [@goodfellow2014gans]. Since its inception, several hundreds of papers have been published proposing formulations that improve not only the quality of generated data, but also address problems in GAN training such as mode collapse.
**Variational Autoencoders** consist also of two components, an encoder $\mathcal{E}nc$ and a decoder $\mathcal{D}ec$. The encoder maps the input $x$ to a latent variable $\mathbf{z}s$ while the decoder takes $z$ as input and tries to reconstruct $\mathbf{x}$. VAEs are constructed and trained in such a way so that the latent variable $\mathbf{z}$ is sampled from a known distribution, typically a Gaussian [@diederik2014auto]. The VAE architecture is depicted in Figure \[fig:vae\].
Attacks against unsupervised learning are until now focused mostly on GANs and VAEs.
### Reinforcement Learning
Reinforcement learning concerns itself with agents that make observations of the environment and use these to take actions with the goal of maximizing a reward signal. In the most general formulation the set of actions is not predefined and the rewards are not necessarily immediate but can occur after a sequence of actions [@sutton2018reinforcement]. At the moment, no privacy related attacks against reinforcement learning have been reported but it has been used to mount model extraction attacks [@orekondy2019knockoff].
Training and Inference
----------------------
Training of supervised ML models usually follows the Empirical Risk Minimization (ERM) approach, where the objective is to find the parameters $\mathbf{\theta}^*$ that minimize the *risk* or *objective function*, which is calculated as an average over the training dataset: $$\mathcal{J}(\mathcal{D};\mathbf{\theta}) = \frac{1}{m}\sum_{i=1}^{m}l(f(x_{i}; \mathbf{\theta}), y_{i})$$
where $l(\cdot)$ is a loss function such as cross entropy loss and $m$ is the number of data points in the dataset $\mathcal{D}$.
The idea behind ERM is that the training dataset is a subset drawn from the unknown true data distribution for the learning task. Since we have no knowledge of the true data distribution we cannot minimize the true objective function but instead we minimize the estimated objective over the data samples that we have. In some cases a regularization term is added to the objective function in order to reduce overfitting and stabilize the training process.
The training process usually involves an iterative optimization algorithm such as gradient descent which aims to minimize the objective function by following the path induced by its gradients. When the dataset is large, as is often the case with deep neural networks, taking one gradient step becomes too costly. In that case, a variant of gradient descent which involves steps taken over smaller batches of data is preferred. This optimization method is called Stochastic Gradient Descent (SGD): $$\label{eq:theta_update}
\mathbf{\theta}_{t+1} = \mathbf{\theta}_{t} - \eta \textbf{g}$$
$$\label{eq:sgd_grad}
\textbf{g} = \frac{1}{m'} \nabla_{\theta} \sum_{i=1}^{m'} l(f(\mathbf{x}_{i}; \mathbf{\theta}), \mathbf{y}_{i})
$$
where $\eta$ is the learning rate and the gradient $\mathbf{g}$ of the loss function with respect to parameters $\mathbf{\theta}$ is calculated over the batch of data that has size $m'$.
Once models are trained, they can be used to make inferences or predictions over previously unseen data. At this stage, the assumption is that the model parameters are fixed.
Threat Model {#sec:threat_model}
============
In order to understand and defend against attacks to machine learning from a privacy perspective, it is useful to have a general model of the environment, the different actors, and the assets to protect.
From a threat model perspective, the assets that are sensitive and are potentially under attack, are the training dataset $\mathcal{D}$ and the model itself; its parameters $\mathbf{\theta}$, its hyper-parameters, and architecture. The actors identified in this threat model are
1. The **data owners** whose data may be sensitive.
2. The **model owners** which may or may not own the data and may or may not want to share information about their models.
3. The **model consumers** that use the services that the model owner exposes, usually via some sort of programming or user interface.
4. The **adversary** may also have access to these interfaces as a normal consumer does. If the model owner allows, they may have access to the model itself.
Figure \[fig:threat\_model\] depicts the assets and the identified actors under the threat model, as well as the information flow and possible actions. This threat model is a logical model and it does not preclude the possibility that some of these assets may be collocated or spread in multiple locations.
![Threat Model of privacy and confidentiality attacks against machine learning systems. The human figure represents actors and the symbols represent the assets. Dashed lines represent data and information flow, while full lines represent possible actions. In red are the adversarial actions available under the threat model.[]{data-label="fig:threat_model"}](images/actor_assets.png){width="10cm"}
Since the interest of this survey is in the privacy attacks based on unintentional information leakage with regards to the data or the machine learning model, there is no coverage of *security-based* attacks, such as model poisoning or evasion attacks, or attacks against the infrastructure that hosts the data, the models or the provided services.
The different attack surfaces against machine learning models can be modelled in terms of **adversarial knowledge**. The range of knowledge varies from limited e.g., having access to a machine learning API, to having knowledge of the full model parameters and training settings. In between these two extremes there is a range of possibilities such as partial knowledge of the model architecture, its hyper-parameters or training setup. The knowledge of the adversary can also be considered from a dataset point of view. In the majority of the works reviewed, the authors assume the adversary has no knowledge of the training data samples, but some knowledge of the underlying data distribution.
From a taxonomy point of view, the attacks where the adversary has no knowledge of the model parameters, architecture or training data are called **black-box** attacks. An example of a black-box system is Machine Learning as a Service (MLaaS) where the users usually provide some input and receive either a prediction vector or a class label from a pre-trained model hosted in the cloud. Most black-box papers assume the existence of a prediction vector. In a similar fashion, **white-box** are the types of attacks where the adversary has either complete access to the target model parameters or their loss gradients during training. This is the case for example, in most distributed modes of training. In between the two extremes, there are also attacks that make stronger assumptions than the black-box ones, but do not assume full access to the model parameters. We refer to these attacks as **partial white-box** attacks. It is important to add here, that the majority of works assumes full knowledge of the expected input, although some form of preprocessing might be required.
The time of the attack is another parameter to consider from a taxonomy point of view. The majority of the works in the area are dealing with attacks during **inference**, however most white-box attacks assume access to the model parameters and gradients during **training**. Attacks during the training phase of the model open up the possibilities for different types of adversarial behavior. A **passive** or *honest-but-curious* attacker does not interfere with the training process and they are only trying to infer knowledge during or after the training. If the adversary interferes with the training in any way, they are considered an **active** attacker.
Taxonomy of Threats {#sec:taxonomy}
===================
In privacy related attacks an adversary’s goal is related to gaining knowledge that was not intended to be shared, such as knowledge about the training data $\mathcal{D}$ or information about the model, or even extracting information about properties of the data such as unintentionally encoded biases. In our taxonomy, the attacks studied are categorized into four types: **membership inference**, **reconstruction**, **property inference**, and **model extraction**.
Membership Inference Attacks {#subsec:mi_attacks}
----------------------------
Membership inference tries to determine whether an input sample $\mathbf{x}$ was used as part of the training set $\mathcal{D}$. This is the most popular category of attacks and was first introduced by Shokri et al. [@shokri2017membership]. The attack assumes only knowledge of the model’s prediction vector (black-box) and was carried against supervised machine learning models. White-box attacks are also a threat especially in a collaborative setting, where an adversary can mount both passive and active attacks. Access to model parameters and gradients allows for more effective white-box membership inference attacks in terms of attack accuracy [@nasr2019comprehensive].
Apart from supervised models, generative models such as GANs and VAEs are also susceptible to membership inference attacks [@hayes2019logan; @hilprecht2019monte; @chen2019gan]. The goal of the attack in this case is to retrieve information about the training data using varying degrees of knowledge of the data generating elements.
Finally, while we are mostly focused on these attacks from a negative perspective, they can also be used from a positive viewpoint. One such example is the ability to audit black-box models in order to see if data have been used without the data owner’s authorization [@song2019auditing; @hishamoto2020embership].
Reconstruction Attacks
----------------------
Reconstruction attacks try to recreate one or more training samples and / or their respective training labels. The reconstruction can be partial or full. Previous work have also used the terms **attribute inference** or **model inversion** to describe attacks that, given output labels and partial knowledge of some features, try to recover sensitive features or the full data sample. For the purpose of this survey, all these attacks are considered as part of the larger set of reconstruction attacks. The term **attribute inference** has been used in other parts of the privacy related literature to describe attacks that infer sensitive “attributes” of a targeted user by leveraging publicly accessible data [@gong2016you; @jia2018attriguard]. These attacks are not part of this review as they are mounted against the individual’s data directly and not against ML models.
A major distinction between the works of this category is between those that create an actual reconstruction of the data [@zhu2019dlg; @he2019collaborative; @wang2019beyondclass; @yang2019neural; @zhang2020secret] and the ones that create class representatives or probable values of sensitive features that do not necessarily belong to the training dataset [@fredrikson2014pharma; @hitaj2017deep; @yang2019neural; @hidano2017model]. In classification models, the latter case is limited to scenarios were classes are made up of one type of object, e.g., faces of the same person. While this limits the applicability of the attack, it can still be an interesting scenario in some cases.
Property Inference Attacks
--------------------------
The ability to extract dataset properties which were not explicitly encoded as features or were not correlated to the learning task, is called **property inference**. An example of property inference is the extraction of information about the ratio of women and men in a patient dataset when this information was not an encoded attribute or a label of the dataset. Or having a neural network that performs gender classification and can be used to infer if people wear glasses or not. In some settings this type of leak can have privacy implications. These types of properties can also be used to get more insight about the training data, which can lead to adversaries using this information to create similar models [@ateniese2015hacking] or even have security implications when the learned property can be used to detect vulnerabilities of a system [@ganju2018property].
Property inference aims to extract information that was learned from the model unintentionally and that is not related to the training task. Even well generalized models may learn properties that are relevant to the whole input data distribution and sometimes this is unavoidable or even necessary for the learning process. What is more interesting from an adversarial perspective are properties that may be inferred from the specific subset of data that was used for training, or eventually about a specific individual.
Property inference attacks so far target either class wide properties [@ateniese2015hacking; @ganju2018property] or the emergence of properties within a batch of data [@melis2019exploiting]. The latter attack was performed against collaborative training of a model.
Model Extraction Attacks {#subsec:model_extraction}
------------------------
**Model extraction** is a class of black-box attacks where the adversary tries to extract information and potentially fully reconstruct a model or create a substitute model $\hat{f}$ that behaves very similarly to the model under attack $f$. When it comes to substitute models the focus is on creating models that either match the accuracy of $f$ in some test set that is drawn from the input data distribution related to the learning task [@milli2019model; @orekondy2019knockoff; @tramer2016stealing; @krishna2020Thieves] or to create a model $\hat{f}$ that matches $f$ at a set of input points that are not necessarily related to the learning task [@tramer2016stealing; @juuti2019prada; @Correia-Silva-IJCNN2018; @jagielski2020high]. Jagielski et al. [@jagielski2020high] referred to the former attack as **task accuracy** extraction and the latter as **fidelity** extraction. In task accuracy extraction the adversary is interested in creating a substitute that learns the same task as the target model equally well or better. In the latter case the adversary aims to create a substitute that replicates the decision boundary of $f$ as faithfully as possible. This type of attack can be later used as a stepping stone before mounting other types of attacks such as adversarial attacks [@papernot2017practical; @juuti2019prada] or membership inference attacks [@nasr2019comprehensive]. In both cases, it is assumed that the adversary wants to be as efficient as possible, i.e., to use as few queries as possible. Knowledge of the target model architecture is assumed in some works but it is not strictly necessary if the adversary selects a substitute model that has the same or higher complexity than the model under attack [@orekondy2019knockoff; @juuti2019prada; @krishna2020Thieves].
Apart from creating substitute models there are also approaches that focus on recovering information from the target model such as hyper-parameters in the objective function [@wang2018stealing] or information about various neural network architectural properties such as activation types, optimisation algorithm, number of layers, etc [@joon2018towards].
Design of the Attacks {#sec:attack_design}
=====================
To study the design of these attacks, more than 40 papers were analyzed in relation to privacy attacks against machine learning. This section describes in some detail the techniques used in most of these attacks by tracing the most common design elements as well as essential differences between the various techniques. The papers are discussed in three sections: attacks on centralized supervised learning, attacks on distributed modes of learning, and attacks on generative models.
Attacks Against Centralized Supervised Learning
-----------------------------------------------
### Shadow training
A common design pattern for a lot of supervised learning attacks is the use of **shadow models** and **meta-models** or **attack-models** [@ateniese2015hacking; @shokri2017membership; @ganju2018property; @joon2018towards; @rahman2018membership; @jayaraman2019evaluating; @Salem0HBF019; @truex2019demystifying; @sablayrolles2019plmr; @hishamoto2020embership]. The general shadow training architecture is depicted in Figure \[fig:shadow\]. The main intuition behind this design is that models behave differently when they see data that do not belong to the training dataset. This difference is captured in the model outputs as well as in their internal representations. In most designs there is a target model and a target dataset. The adversary is trying to infer either membership or properties of the training data. They train a number of shadow models using shadow datasets $\mathcal{D}_{shadow} = \{\textbf{x}_{shadow,i}, \textbf{y}_{shadow,i}\}_{i=1}^{n}$ that usually are assumed to come from the same distribution as the target dataset. After the shadow models’ training, the adversary constructs an attack dataset $\mathcal{D}_{attack}=\{f_i(\textbf{x}_{shadow, i}), \textbf{y}_{shadow, i}\}_{i=1}^{n}$, where $f_i$ is the respective shadow model. The attack dataset is used to train the meta-model which essentially performs inference based on the outputs of the shadow models. Once the meta-model is trained it is used for testing using the outputs of the target model.
{width="12cm"}
### Membership inference attacks {#membership-inference-attacks}
In *membership inference* black-box attacks the output of the shadow models is usually a prediction vector [@shokri2017membership; @Salem0HBF019; @rahman2018membership; @truex2019demystifying; @jayaraman2019evaluating]. The labels used for the attack dataset come from the test and training splits of the shadow data, where data points that belong to the test set are labelled as non-members of the training set. The meta-model is trained to recognize patterns in the prediction vector output of the target model. These patterns allow the meta-model to infer whether a data point belongs to the training dataset or not. The number of shadow models affects the attack accuracy but it also incurs cost to the attackers. Salem et al. [@Salem0HBF019] showed that membership inference attacks are possible with as little as one shadow model.
Shadow training can be further reduced to a threshold-based attack, where instead of training a meta-model, one can calculate a suitable threshold function that indicates whether a sample is a member of the training set. The threshold can be learned from multiple shadow models [@sablayrolles2019plmr] or even without using any shadow models [@yeom2018privacy]. Sablayrolles et al. [@sablayrolles2019plmr] showed that a Bayes optimal membership inference attack depends only on the loss and their attack outperformed previous attacks such as [@shokri2017membership; @yeom2018privacy]. In terms of attack accuracy, they reported up to 90.8% attack accuracy against large neural network models such as VGG16 which were performing classification on the Imagenet dataset.
In addition to relaxations on the number of shadow models, attacks have been shown to be transferable i.e., an attack to one target model transfers to another target if the training dataset was the same [@truex2019demystifying].
Shadow model training requires a shadow dataset. One of the main assumptions of membership inference attacks against supervised learning models is that the adversary has no or limited knowledge of the training samples used. However the adversary knows something about the underlying data distribution of the training data. If the adversary does not have access to a suitable dataset, they can try to generate one [@shokri2017membership; @truex2019demystifying]. Access to statistics about the probability distribution of several features allows an attacker to create the shadow dataset using sampling techniques. If a statistics based generation is not possible, a query based approach using the target models’ prediction vectors is possible. If the adversary manages to find input data that generate predictions with a high confidence, then no prior knowledge of the data distribution is required for a successful attack [@shokri2017membership]. Salem et al. [@Salem0HBF019] went so far as to show that it is not even necessary to train the shadow models using data from the same distribution as the target, making the attack more realistic since it does not assume any knowledge of the training data.
The previous discussion is mostly relevant to supervised classification or regression tasks. The efficacy of membership inference attacks against sequence-to-sequence models training for machine translation was studied by [@hishamoto2020embership]. The authors used shadow models that try to mimic the target model’s behavior and then used a meta-model to infer membership. They found that sequence generation models are much harder to attack compared to other types of models such as image classification. However, membership of *out-of-domain* and data was easier to infer.
### Reconstruction attacks
The initial reconstruction attacks were based on the assumption that the adversary has access to the model $f$, the priors of the sensitive and non-sensitive features and the output of the model for a specific input $x$. The attack was based on estimating values of sensitive features given values of non-sensitive features and the output label [@fredrikson2014pharma]. This method used a maximum a posteriori (MAP) estimate of the attribute that maximizes the probability of observing the known parameters. Hidano et al. [@hidano2017model] used a similar attack but they made no assumption about knowledge of the non-sensitive attributes. In order for their attack to work, they assumed that the adversary can perform a *model poisoning* attack during training.
Both of the previous attacks worked against linear regression models, but as the number of features and their range increases, attack feasibility decreases. In order to overcome the limitations of the MAP attack, Fredrikson et al. [@fredrikson2015model] proposed another inversion attack which recovers features using target labels and optional auxiliary information. The attack was formulated as an optimization problem where the objective function is based on the observed model output and uses gradient descent in the input space in order to recover the input data point. The method was tested on image reconstruction. The result was a class representative image which in some cases was quite blurry even after denoising. A formalization of the model inversion attacks in [@fredrikson2014pharma; @fredrikson2015model] was later proposed by Wu et al. [@wu2016methodology].
Since the optimization problem in [@fredrikson2015model] is quite hard to solve, Zhang et al. [@zhang2020secret] proposed to use a GAN in order to learn some auxiliary information of the training data and produce better results. The auxiliary information in this case is the presence of blurring or masks in the input images. The attack first uses the GAN in order to learn to generate realistic looking images from masked or blurry images using public data. The second step is a GAN inversion that calculates the latent vector $\hat{z}$ which generates the most likely image:
$$\hat{z}=\arg\min_z L_{prior}(z) + \lambda L_{id}(z)$$
where the prior loss $L_{prior}$ is ensuring the generation of realistic images and $L_{id}$ ensures that the images have a high likelihood in the target network. The attack is quite successful, especially against masked images.
The only black-box reconstruction attack until now was proposed by Yang et al. [@yang2019neural]. This attack employs an additional classifier that performs an inversion from the output of the target model $f(x)$ to a candidate output $\hat{x}$. The setup is similar to that of an autoencoder, only in this case the target network that plays the role of the encoder is a black-box and it is not trainable. The attack was tested in different types of target model outputs: the full prediction vector, a truncated vector and the target label only. When the full prediction vector is available the attack performs a good reconstruction, but with less available information, the produced data point looks more like a class representative.
### Property inference attacks
In *property inference* the shadow datasets are labelled based on the properties that the adversary wants to infer, so the adversary needs access to data that have the property and data that do not have it. The meta-model is then trained to infer differences in the output vectors of the data that have the property versus the ones that they don’t have it. In white-box attacks, the meta-model input can be other feature representations such as support vectors [@ateniese2015hacking] or transformations of neural network layer outputs [@ganju2018property].
### Model extraction attacks {#model-extraction-attacks}
When the adversary has access to the inputs and prediction outputs of a model, it is possible to view these pairs of inputs and outputs as a system of equations where the unknowns are the model parameters [@tramer2016stealing] or hyper-parameters of the objective function [@wang2018stealing]. In the case of a linear binary classifier, the system of equations is linear and only $d + 1$ queries are necessary to retrieve the model parameters, where $d$ is the dimension of the parameter vector $\theta$. In more complex cases, such as multi-class linear regression or multi-layer perceptrons the systems of equations are no longer linear. Optimization techniques such as Broyden–Fletcher–Goldfarb–Shanno (BFGS) [@nocedal2006numerical] or stochastic gradient descent are then used in order to approximate the model parameters [@tramer2016stealing].
Lack of prediction vectors or a high number of model parameters renders equation solving attacks inefficient. A strategy is required in order to select the inputs that will provide the most useful information for model extraction. From this perspective, model extraction is quite similar to *active learning* (AL) [@chandrasekaran2020exploring]. Active learning makes use of an external oracle that provides labels to input queries. The oracle can be a human expert or a system. The labels are then used to train or update the model. In the case of model extraction, the target model plays the role of the oracle.
Following the AL approach, several papers propose an adaptive training strategy. They start with some initial data points or *seeds* which they use to query the target model and retrieve labels or prediction vectors which they use to train the substitute model $\hat{f}$. For a number of subsequent rounds they extend their dataset with new synthetic data points based on some adaptive strategy that allows them to find points close to the decision boundary of the target model [@tramer2016stealing; @juuti2019prada; @papernot2017practical; @chandrasekaran2020exploring]. Chandrasekaran et al. [@chandrasekaran2020exploring] provided a more query efficient method of extracting non-linear models such as kernel SVMs, with slightly lower accuracy than the method proposed by Tramer et al. [@tramer2016stealing], while the opposite was true for Decision Tree models.
Several other strategies for selecting the most suitable data for querying the target model use: (i) data that are not synthetic but belong to different domains such as images from different datasets [@Correia-Silva-IJCNN2018; @orekondy2019knockoff], (ii) semi-supervised learning techniques such as rotation loss [@zhai2019s4l] or MixMatch [@berthelot2019mixmatch] to augment the dataset [@jagielski2020high] or (iii) randomly generated input data [@tramer2016stealing; @juuti2019prada; @krishna2020Thieves]. In terms of efficiency, unsupervised methods such as MixMatch require much fewer queries than fully supervised extraction methods in order to perform similarly or better in terms of task accuracy and fidelity, against models trained for classification using CIFAR-10 and SVHN datasets [@jagielski2020high]. For larger models, trained for Imagenet classification, even querying a 10% of the Imagenet data, gives a comparable performance to the target model [@jagielski2020high]. Against a deployed MLaaS service that provides facial characteristics, Orekondy et al. [@orekondy2019knockoff] managed to create a substitute model that performs at 80% of the target in task accuracy, spending as little as \$30.
Some, mostly theoretical, work has demonstrated the ability to perform direct model extraction beyond linear models [@milli2019model; @jagielski2020high]. Full model extraction was shown to be theoretically possible against two-layer fully connected neural networks with rectified linear unit (ReLU) activations by Milli et al. [@milli2019model]. However, their assumption was that the attacker has access to the loss gradients with respect to the inputs. Jagielski et al. [@jagielski2020high] managed to do a full extraction of a similar network without the need of gradients. Both approaches take into account that ReLUs transfomrs the neural network into a piece-wise linear function of the inputs. By probing the model with different inputs it is possible to identify where the linearity breaks and use this knowledge to calculate the network parameters. In a hybrid approach that uses both a learning strategy and direct extraction, Jagielski et al. [@jagielski2020high], showed that they can extract a model trained on MNIST with almost 100% fidelity by using $2^{19.2}$ to $2^{22.2}$ queries against models that contain up to 400,000 parameters. However, this attack assumed access to the loss gradients similarly to [@milli2019model].
Finally, apart from learning substitute models directly, there is also the possibility of extracting model information such as architecture, optimization methods and hyper-parameters using shadow models [@joon2018towards]. The majority of attacks were performed against neural networks trained on MNIST. Using the shadow models’ prediction vectors as input, the meta-models managed to learn to distinguish whether a model has certain architectural properties. An additional attack by the same authors, proposed to generate adversarial samples which were created by models that have the property in question. The generated samples were created in a way that makes a classifier output a certain prediction if they have the attribute in question. The target model’s prediction on this adversarial sample is then used to establish if the target model has a specific property. The combination of the two attacks, proved to be the most effective approach. Some properties such as activation functions, presence of dropout and max-pooling where the most successfully predicted.
Attacks Against Distributed Learning
------------------------------------
In centralized modes of learning, attacks are focused on one machine learning model and its properties. The adversaries are assumed to be able to query the model or to have access to its parameters. Distributed modes of learning such as federated or collaborative learning, introduce different spatial models of adversaries. In a federated learning setting, the adversary can be collocated with the global model but it can also be a local attacker that **actively** or **passively** tries to attack (Figure \[fig:federated\_threat\]). The presence of multiple actors allows also the possibility of *colluding* adversaries that join forces.
{width="9cm"}
Federated learning (FL) is a form of decentralized training where the goal is to learn one **global** model from data stored in multiple remote devices / locations [@li2019federated]. The main idea is that the data do not leave the remote devices. They are processed **locally** and only the intermediate updates are sent to the central server that hosts the global model. The most popular learning algorithm for FL is Federated Averaging [@pmlr-v54-mcmahan17a], where each remote device, calculates one step of the gradient descent locally and then shares the updated model weights with the parameter server. The parameter server averages the weights of all the remote participants and updates the global model which is subsequently shared again with the remote devices:
$$\label{eq:fed_avg}
\theta_{t+1} = \frac{1}{K} \sum_{k=1}^{K} \theta_{t}^{(k)}$$
where K is the number of remote participants and the parameters $\theta_{t}^k$ of participant $k$ have been calculated locally based on Equations \[eq:theta\_update\] and \[eq:sgd\_grad\].
Another approach that comes from the area of distributed computing is Downpour (or synchronized) SGD [@dean2012large], which proposes to share the loss gradients of the distributed devices with the parameter server that aggregates them and then performs one step of gradient descent:
$$\label{eq:SSGD}
\theta_{t+1} = \theta_{t} - \eta \sum_{k=1}^{K} \frac{m^{(k)}}{M}\mathbf{g}_t^{(k)}$$
where $\mathbf{g}_t^{(k)}$ is the gradient computed by participant $k$ based on Equation \[eq:sgd\_grad\] using their local data, $m^{(k)}$ is the number of data points in the remote participant and $M$ is the total number of data points in the training data. After the calculation of Equation \[eq:SSGD\], the parameter server shares the updated model parameters $\theta_{t+1}$ with the remote participants.
Both Federated Averaging and Synchronous SGD can be problematic from a privacy perspective because, in essence, each remote device has access to the model parameters. The same applies to the parameter server that obtains the model parameters or their loss gradients from multiple devices.
### Membership inference attacks {#membership-inference-attacks-1}
Nasr et al. [@nasr2018machine] showed that a membership inference attack is more effective than the black-box one, under the assumption that the adversary has some auxiliary knowledge about the training data, i.e., has access to some data from the training dataset, either explicitly or because they are part of a larger set of data the adversary possesses. The adversary can use the model parameters and the loss gradients as inputs to another model which is trained to distinguish between members and non-members. The white-box attack accuracy against various neural network architectures was up to 75.1%, however, all models had a high generalization error.
In the active attack scenario, the attacker which is also a local participant, alters the gradient updates to perform a gradient ascent instead of descent for the data whose membership is under question. If some other participant uses the data for training, then their local SGD will significantly reduce the gradient of the loss and the change will be reflected in the updated model, allowing the adversary to extract membership information. Attacks from a local active participant reached attack accuracy of 76.3% and in general attack active attack accuracy was higher than the passive accuracy in all tested scenarios. However, as the number of participants increases, it has adverse effects in the attack accuracy which drops significantly after five or more participants. A global active attacker which is in a more favourable position, can isolate the model parameter updates they receive from each participant. Such an active attacker reached attack accuracy of 92.1%.
### Property inference attacks
Passive property inference requires access to some data that possess the property and some that do not. The attack applies to both federated average and synchronized SGD settings, where each remote participant receives the parameter updates from the parameter server after each training round [@melis2019exploiting]. The initial dataset is of the form $\mathcal{D'}=\{(\mathbf{x}, \mathbf{y}, \mathbf{y'})\}$, where $\mathbf{x}$ and $\mathbf{y}$ are the data used for training the distributed model and $\mathbf{y}'$ are the property labels. Every time the local model is updated, the adversary calculates the loss gradients for two batches of data. One batch that has the property in question and one that does not. This allows the construction of a new dataset that consists of gradients and property labels $(\nabla L, \mathbf{y}')$. Once enough labeled data have been gathered, a second model $f'$ is trained to distinguish between loss gradients of data that have the property versus those that do not. This model is then used to infer whether subsequent model updates were made using data that have the property. The model updates are assumed to be done in batches of data. The attack reaches an attack area under the curve (AUC) score of 98% and becomes increasingly more successful as the number of epochs increases. Attack accuracy also increases as the fraction of data with the property in question, also increases. However, as the number of participants in the distributed model increases, the attack performance decreases significantly.
### Reconstruction attacks
Some data reconstruction attacks in a federated learning setting make use of generative models and specifically GANs [@hitaj2017deep; @wang2019beyondclass]. When the adversary is one of the participants they can force the victims to release more information about the class they are interested in reconstructing [@hitaj2017deep]. This attack works as follows: The potential victim has data for a class “A” that the adversary wants to reconstruct. The adversary trains an additional GAN model. After each training round the adversary uses the target model parameters for the GAN discriminator, whose purpose is to decide whether the input data come from class “A” or are generated by the generator. The aim of the GAN is to create a generator that is able to generate faithful class “A” samples. In the next training step of the target model, the adversary generates some data using the GAN and labels them as class “B”. This forces the target model to learn to discriminate between classes “A” and “B” which in turn improves the GAN training and its ability to generate class “A” representatives.
If the adversary has access to the central parameter server, they have direct access to model updates of each remote participant. This makes it possible to mount more successful reconstruction attacks [@wang2019beyondclass]. In this case the GAN discriminator is again using the shared model parameters and learns to distinguish between real and generated data, as well as the identity of the participant. Once the generator is trained, the reconstructed samples are created using an optimization method that minimizes the distance between the real model updates and the updates due to the generated data. Both GAN based methods assume access to some auxiliary data that belong to the victims. However, the former method generates only class representatives.
In a synchronized SGD setting, an adversary with access to the parameter server has access to the loss gradients of each participant during training. Using the loss gradients is enough to produce a high quality reconstruction of the training data samples, especially when the batch size is small [@zhu2019dlg]. The attack is utilizing a second “dummy” model. Starting with random dummy inputs $x'$ and and labels $y'$, the adversary tries to match the dummy model’s loss gradients $\nabla_{\theta} \mathcal{J'}$ to the participant’s loss gradients $\nabla_{\theta} \mathcal{J}$. This gradient matching is formulated as an optimization task that seeks to find the optimal $x'$ and $y'$ that minimize the gradients’ distance:
$$\label{eq:zhu}
x^*, y^* = \arg\min_{x',y'} \| \nabla_{\theta} \mathcal{J'}(\mathcal{D'};\theta) - \nabla_{\theta} \mathcal{J}(\mathcal{D};\theta) \|^2$$
The minimization problem in Equation \[eq:zhu\] is solved using limited memory BFGS (L-BFGS) [@liu1989limited]. The size of the training batch is an important factor in the speed of convergence in this attack.
Data reconstruction attacks are also possible during the inference phase in a collaborative inference scenario [@he2019collaborative]. This is a setup relevant to situations where remote or edge devices are connected to a central cloud server but they have limited resources. This scenario is typical with internet of things (IoT) devices. In collaborative inference the trained model is split into two or more parts. The edge devices keep the initial layers of the deep learning model and the centralized server keeps the final layers [@hauswald2014hybrid; @kang2017neuro]. The reason for the split is mainly to lower communication costs by sending intermediate model outputs instead of the input data.
When the local nodes process new data, they perform inference on these initial layers and then send their outputs to the centralized server. In this attack, the adversary is placed in the centralized server and their goal is to try to reconstruct the data used for inference. He et al. [@he2019collaborative] cover a range of scenarios: (i) white-box, where the adversary has access to the initial layers and uses them to reconstruct the images, (ii) black-box where the adversary has no knowledge of the initial layers but can query them and thus re-create the missing layers and (iii) query-free where the adversary cannot query the remote participant and tries to create a substitute model that allows data reconstruction. The latter attack produces the worst results, as expected, since the adversary is the weakest. The split of the layers between edge device and centralized server is also affecting the quality of reconstruction. Fewer layers in the edge neural network allow for better reconstruction in the centralized server.
Attacks Against Generative Models
---------------------------------
Previous sections dealt with attacks on discriminative learning and specifically on supervised learning. However, attacks on privacy may be relevant for generative models, too. An adversary may be interested in information about the training dataset used in a generative model or interested in the model itself. While there is prior research in many different types of generative models, there are two types that are currently the most popular; Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs). Most research on attacking generative models has been focused on these two types of models so far.
Since generative models have more than one component (generator/discriminator, encoder/decoder), adversarial knowledge needs to take them into account. For these type of models, the taxonomy proposed by Chen et al. [@chen2019gan] is partially followed. We consider black-box access to the generator as the ability to access generated samples and partial black-box access, the ability to provide inputs $z$ and generate samples. Having access to the generator model and its parameters is considered a white-box attack. The ability to query the discriminator is also a white-box attack. This scenario, addressed by Hayes et al. [@hayes2019logan] is considered unrealistic, since these component are not likely to be published. Similarly, the VAE white-box attack requires access to the full VAE model which is also not a realistic scenario [@hilprecht2019monte]. However, it is important to understand what is possible and what is not and to establish an upper bound of potential leakage.
So far, the focus of privacy related attacks against GANs and VAEs has been membership inference [@hayes2019logan; @hilprecht2019monte; @chen2019gan]. The full white-box attacks with access to the GAN discriminator are based on the assumption that if the GAN “overfitted”, then data points used for its training will receive higher confidence values as output in the discriminator [@hayes2019logan]. In addition to the previous attack, Hayes et al. proposed a set of attacks in the partial black-box setting. These attacks are applicable to both GANs and VAEs or any generative model. If the adversary has no auxiliary data, they can attempt to train an auxiliary GAN whose discriminator distinguishes between data generated by the target generator and data generated by the auxiliary GAN. Once the auxiliary GAN is trained, its discriminator can be used for the white-box attack. The authors considered also scenarios where the adversary may have auxiliary information such as knowledge of training and test data. Using the auxiliary data they can train another GAN whose discriminator would be able to distinguish between members of the original training set and non-members.
An attack applicable to both GANs and VAEs in a partial black-box setting was proposed by Hilprecht et al. [@hilprecht2019monte]. Assuming that an “overfitted” generator will have memorized the training data, a way to establish membership for a new data sample, is to measure how likely it is to be close to the training points generated by the generator or encoder using a metric such as Euclidean distance. The estimation of the probability of a data point being a member of the training set is performed using Monte Carlo integration while the Euclidean distance was measured based on the top 40 principal components derived by principal component analysis.
Another distance based attack over the nearest neighbors of a data point, was proposed by Chen et al. [@chen2019gan] for the full black-box model. In this case a data point $\mathbf{x}$ is a member of the training set if within its k-nearest neighbors there is at least one point that has a distance lower than a threshold $\epsilon$. The authors proposed more complex attacks as the level of knowledge of the adversary increases, based on the idea that the reconstruction error between the real data point $x$ and a sample generated by the generator given some input $z$ should be smaller if the data point is coming from the training set. In both the partial black-box attack and the white-box attack with access to the generator, the authors propose to calculate the value of $z$ which provides the smallest distance between a generated data point and the original one:
$$z^* = \arg\min_z \mathcal{L}(x, G(z))$$
where $\mathcal{L}$ is a distance metric. In the image generation domain $\mathcal{L}$ calculated based on pixel-to-pixel similarity, a regularization term and the Learned Perceptual Image Patch Similarity (LPIPS) metric [@zhang2015beyond]. Once $z^*$ is calculated, if the distance between $x$ and $G(z^*)$ is below a certain threshold, the data point in question is assumed to belong to the training set $\mathcal{D}$. The white-box setting provides the best results because access to the model allows differentiation and the use of optimization methods such as L-BFGS.
A property of a successful generative model is to produce high quality samples that cover the data distribution as much as possible. One of the problems, especially with the training of GANs is that it is not that easy to measure the generated data quality and distribution support in conjunction with the training metrics. One of most popular metrics of the quality of a GAN generator is the Frechet Inception Distance (FID) which scores both the quality and the diversity of the generated data. Unfortunately only [@chen2019gan] offers FID measurements for the models under test.
These are early stages of attacks against generative models. While the attacks are promising, they suffer as the size of the training set increases. The only attack that seems to withstand the increase of the dataset is the one that assumes access to the discriminator, which is the strongest possible adversary.
Design Summary
--------------
To summarize the attacks proposed against machine learning privacy, Table \[table:attack\_summary\] presents the 40 papers analyzed in terms of adversarial knowledge, model under attack, attack type and timing of the attack.
[@ l c c c >c >c >c >c >c>c c c c c >c >c @ ]{} Reference & Year & & & &\
& & & & & & & & & & & & & & &\
Fredrikson et al. [@fredrikson2014pharma] & 2014 & & $\bullet$ & $\bullet$ & & & & & & & $\bullet$ & & & & $\bullet$\
Fredrikson et al. [@fredrikson2015model] & 2015 & $\bullet$ & $\bullet$ & & $\bullet$ & & & $\bullet$& & & $\bullet$ & & & & $\bullet$\
Ateniese et al. [@ateniese2015hacking] & 2015 & & $\bullet$ & & & $\bullet$ & $\bullet$& & & & & $\bullet$ & & & $\bullet$\
Tramer et al. [@tramer2016stealing]& 2016 & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & $\bullet$ & &$\bullet$ & & & & & $\bullet$ & & $\bullet$\
Wu et al. [@wu2016methodology] & 2016 & $\bullet$ & $\bullet$ & & $\bullet$ & & & $\bullet$ & & & $\bullet$ & & & & $\bullet$\
Hidano et al. [@hidano2017model] & 2017 & & $\bullet$ & $\bullet$& & & & & & & $\bullet$ & & & & $\bullet$\
Hitaj et al. [@hitaj2017deep]& 2017 & & $\bullet$ & & & & & $\bullet$ & & & $\bullet$ & & & $\bullet$ &\
Papernot et al. [@papernot2017practical] & 2017 & $\bullet$ & & & & & &$\bullet$ & & & & & $\bullet$ & & $\bullet$\
Shokri et al. [@shokri2017membership] & 2017 & $\bullet$ & & & & & &$\bullet$ & & $\bullet$ & & & & & $\bullet$\
Correia-Silva et al. [@Correia-Silva-IJCNN2018] & 2018 & $\bullet$ & & & & & &$\bullet$ & & & & & $\bullet$ & & $\bullet$\
Ganju et al. [@ganju2018property] & 2018 & & $\bullet$ & & & & & $\bullet$& & & & $\bullet$ & & & $\bullet$\
Oh et al. [@joon2018towards] & 2018 & $\bullet$ & & & & & & $\bullet$ & & & & & $\bullet$ & & $\bullet$\
Long et al. [@long2018understanding] & 2018 & $\bullet$ & & & & & & $\bullet$ & & $\bullet$ & & & & & $\bullet$\
Rahman et al. [@rahman2018membership] & 2018 & & $\bullet$ & & & & & $\bullet$ & & $\bullet$ & & & & & $\bullet$\
Wang & Gong [@wang2018stealing] & 2018 & & $\bullet$ & $\bullet$ & & $\bullet$ & & & & & & & $\bullet$ & & $\bullet$\
Yeom et al. [@yeom2018privacy] & 2018 & $\bullet$ & $\circ$ & $\bullet$ & $\bullet$ & & &$\bullet$ & & & $\bullet$ & & & & $\bullet$\
Carlini et al. [@carlini2019secret] & 2019 & $\bullet$ & & & & & & $\bullet$ & & & $\bullet$ & & & & $\bullet$\
Chen et al. [@chen2019gan] & 2019 & $\bullet$ & $\bullet$ & & & & & & $\bullet$ & $\bullet$ & & & & & $\bullet$\
Hayes et al. [@hayes2019logan] & 2019 & $\bullet$ & $\bullet$ & & & & & & $\bullet$ & $\bullet$ & & & & & $\bullet$\
He et al. [@he2019collaborative] & 2019 & $\bullet$ & $\bullet$ & & & & & $\bullet$ & & & $\bullet$ & & & & $\bullet$\
Hilprecht et al. [@hilprecht2019monte] & 2019 & $\bullet$ & & & & & & & $\bullet$ & $\bullet$ & & & & & $\bullet$\
Jayaraman & Evans [@jayaraman2019evaluating] & 2019 & $\bullet$ & $\bullet$ & & & & & $\bullet$ & & $\bullet$ & $\bullet$ & & & & $\bullet$\
Juuti et al. [@juuti2019prada] & 2019 & $\bullet$& & & & & & $\bullet$ & & & & & $\bullet$& & $\bullet$\
Milli et al. [@milli2019model] & 2019 & $\bullet$& & $\bullet$ & & & & $\bullet$ & & & & & $\bullet$ & & $\bullet$\
Nasr et al. [@nasr2019comprehensive] & 2019 & & $\bullet$ & & & & & $\bullet$& & $\bullet$ & & & & $\bullet$ &\
Melis et al. [@melis2019exploiting] & 2019 & & $\bullet$ & & & & & $\bullet$ & & $\bullet$ & & $\bullet$ & & $\bullet$ &\
Orekondy et al. [@orekondy2019knockoff] & 2019 & $\bullet$ & & & & & &$\bullet$ & & & & & $\bullet$ & & $\bullet$\
Sablayrolles et al. [@sablayrolles2019plmr] & 2019 & & $\circ$ & & & & & $\bullet$ & & $\bullet$ & & & & &$\bullet$\
Salem et al. [@Salem0HBF019] & 2019 & $\bullet$ & & & & & & $\bullet$ & & $\bullet$ & & & & & $\bullet$\
Song L. et al. [@song2019privacy] & 2019 & $\bullet$ & & & & & & $\bullet$& & $\bullet$ & & & & & $\bullet$\
Truex, et al. [@truex2019demystifying] & 2019 & $\bullet$ & & & & & & $\bullet$ & & $\bullet$ & & & & & $\bullet$\
Wang et al. [@wang2019beyondclass] & 2019 & & $\bullet$ & & & & & $\bullet$ & & & $\bullet$ & & & $\bullet$ &\
Yang et al. [@yang2019neural] & 2019 & $\bullet$ & & & & & & $\bullet$ & & & $\bullet$ & & & & $\bullet$\
Zhu et al. [@zhu2019dlg] & 2019 & & $\bullet$ & & & & & $\bullet$ & & & $\bullet$ & & & $\bullet$ &\
Chandrasekaran et al. [@chandrasekaran2020exploring] & 2020 & $\bullet$ & & $\bullet$ & $\bullet$ &$\bullet$ & &$\bullet$ & & & & & $\bullet$ & & $\bullet$\
Hishamoto et al. [@hishamoto2020embership] & 2020 & $\bullet$ & & & & & &$\bullet$ & & $\bullet$ & & & & & $\bullet$\
Jagielski et al. [@jagielski2020high] & 2020 & $\bullet$ & & & & & &$\bullet$ & & & & & $\bullet$ & & $\bullet$\
Krishna et al. [@krishna2020Thieves] & 2020 & $\bullet$ & & & & & &$\bullet$ & & & & & $\bullet$ & & $\bullet$\
Pan et al. [@pan2020privacy] & 2020 & & $\bullet$ & & & & &$\bullet$ & & & $\bullet$ & & & & $\bullet$\
Zhang et al. [@zhang2020secret] & 2020 & & $\bullet$ & & & & &$\bullet$ & & & $\bullet$ & & & & $\bullet$\
\[table:attack\_summary\]
In terms of model types, 92.5% of the papers dealt with attacks against neural networks, with linear models being the second most popular model to attack at 17.5% (some papers covered attacks against multiple model types). The concept of neural networks groups together both shallow and deep models, as well as multiple architectures, such as convolutional neural networks, recurrent neural networks, GANs, and VAEs.
The most popular attack types are membership inference and reconstruction attacks (35% of the papers, respectively) with model extraction the next most popular (27.5%). The majority of proposed attacks are mounted during the inference phase (87.5%). Attacks during training are mainly against distributed forms of learning. Black-box and white-box attacks were studied in 65% and 55% of the papers, respectively (some papers covered both settings). In the white-box category we also include partial white-box attacks.
While there is a diverse set of works presented, it is possible to discern some high-level patterns in the proposed attacking techniques. Figure \[fig:heatmap2\] shows the number of papers in relation to the attacking technique and attack type. Most notably, nine papers used shadow training mainly for membership and property inference attacks. Active learning was quite popular in model extraction attacks with four papers, while four papers used GANs and another three used gradient matching techniques. It should be noted here, that the “Learning” technique includes a number of different approaches, spanning from using model parameters and gradients as inputs to classifiers [@nasr2018machine; @melis2019exploiting] to using input-output queries for substitute model creation [@orekondy2019knockoff; @Correia-Silva-IJCNN2018; @jagielski2020high] and learning classifiers from language models for reconstruction attacks [@pan2020privacy]. In “Threshold” based attacks we categorized the attacks proposed in [@yeom2018privacy] and [@sablayrolles2019plmr] and subsequent papers that used them for membership and property inference.
![Number of papers that used an attacking technique for each attack type. Darker gray means higher number of papers.[]{data-label="fig:heatmap2"}](images/heatmap_technique.png){width="12cm"}
Some attacks may be applicable against multiple learning tasks and datasets, however, this is not the case universally. Dataset size and complexity might also be a factor for the success of certain attacks, especially since most of them are empirical. Table \[table:datasets\_summary1\] is a summary of the datasets used in all attack papers along with the data types of their features, the learning task they were used for and the dataset size. The datasets were used during the training of the target models and in some cases as auxiliary information during the attacks. The table contains 49 unique datasets used across 40 papers, an indication of the variation of different approaches.
[m[3cm]{} m[2cm]{} m[3.2cm]{} m[2.5cm]{} r]{} **Name** & **Data Type** & **Learning Task** & **Reference(s)** & **Size (samples)**\
538 Steak Survey[@538steak] & mixed features & multi-class classification & [@fredrikson2015model; @tramer2016stealing; @chandrasekaran2020exploring; @hidano2017model] & 332\
AT&T Faces [@attfaces] & images & multi-class classification & [@fredrikson2015model; @hitaj2017deep; @wang2019beyondclass] & 400\
Bank Marketing [@dua2019] & mixed features & multi-class classification & [@wang2018stealing] & 45,210\
Bitcoin prices & time series & regression & [@tramer2016stealing] & 1,076\
Breast Cancer [@dua2019] & numerical feat. & binary classification & [@tramer2016stealing; @chandrasekaran2020exploring; @long2018understanding] & 699\
Caltech 256 [@griffin2007caltech] & images & multi-class classification & [@orekondy2019knockoff] & 30,607\
Caltech birds [@wah2011caltech] & images & multi-class classification & [@orekondy2019knockoff] & 6,033\
CelebA [@liu2017oblivious] & images & binary classification & [@ganju2018property; @chen2019gan; @yang2019neural; @zhang2020secret] & 20-202,599\
CIFAR-10 [@krizhevsky2009learning] & images & image generation, multi-class classification & [@hayes2019logan; @rahman2018membership; @Salem0HBF019; @shokri2017membership; @song2019privacy; @yeom2018privacy; @sablayrolles2019plmr; @he2019collaborative; @truex2019demystifying; @hilprecht2019monte; @milli2019model; @jagielski2020high; @yang2019neural] & 60,000\
CIFAR-100 [@krizhevsky2009learning] & images & multi-class classification & [@nasr2019comprehensive; @Salem0HBF019; @shokri2017membership; @yeom2018privacy; @zhu2019dlg; @jayaraman2019evaluating] & 60,000\
CLiPS stylometry [@verhoeven2014clips] & text & binary classification & [@melis2019exploiting] & 1,412 reviews\
Chest X-ray [@wang2017chest] &images & multi-class classification & [@zhang2020secret] & 10,000\
Diabetes [@dua2019] & time series & binary class., regression & [@tramer2016stealing; @wang2018stealing; @chandrasekaran2020exploring] & 768\
Diabetic ret. [@diabeticretinopathydata] & images & image generation & [@hayes2019logan; @orekondy2019knockoff] & 88,702\
Enron emails & text & char-level language model & [@carlini2019secret] & -\
Eyedata [@scheetz2006regulation] & numerical feat.& regression & [@yeom2018privacy] & 120\
FaceScrub [@ng2014data] & images & binary classification & [@melis2019exploiting; @yang2019neural] & 18,809-48,579\
Fashion-MNIST [@xiao2017fashion] & images & multi-class classification & [@song2019privacy; @hilprecht2019monte; @jagielski2020high] & 60,000\
Foursquare [@yang2015nationtelescope] & mixed features & binary classification & [@melis2019exploiting; @shokri2017membership; @Salem0HBF019] & 528,878\
Geog. Orig. Music [@dua2019] & numerical feat. & regression & [@wang2018stealing] & 1,059\
German Credit [@dua2019] & mixed features & binary classification & [@tramer2016stealing] & 1,000\
GSS marital survey [@gssmarital] & mixed features & multi-class classification & [@fredrikson2015model; @tramer2016stealing; @chandrasekaran2020exploring] & 16127\
GTSRB [@stallkamp2011german] & images & multi-class classification & [@juuti2019prada; @papernot2017practical] & 51839\
HW Perf. Counters & numerical feat. & binary classification & [@ganju2018property] & 36,000\
Imagenet [@imagenet_cvpr09] & images & multi-class classification & [@joon2018towards; @sablayrolles2019plmr; @jagielski2020high] & 14,000,000\
Instagram [@backes2017walk2friends] & location data & vector generation & [@chen2019gan] & -\
Iris [@fisher1936use] & numerical feat. & multi-class classification & [@tramer2016stealing; @chandrasekaran2020exploring] & 150\
IWPC [@international2009estimation] & mixed features & regression & [@fredrikson2014pharma; @yeom2018privacy] & 3497\
IWSLT Eng-Vietnamese & text & neural machine translation & [@carlini2019secret] & -\
LFW [@huang2008labeled]& images & image generation & [@hayes2019logan; @melis2019exploiting; @zhu2019dlg] & 13233\
Madelon [@dua2019] & mixed features & multi-class classification & [@wang2018stealing] & 4,400\
MIMIC-III [@johnson2016mimic] & binary features & record generation & [@chen2019gan] & 41,307\
Movielens 1M [@movielens] & numerical feat. & regression & [@hidano2017model] & 1,000,000\
MNIST [@yann1998mnist] & images & multi-class classification & [@ganju2018property; @hitaj2017deep; @joon2018towards; @rahman2018membership; @Salem0HBF019; @shokri2017membership; @tramer2016stealing; @yeom2018privacy; @zhu2019dlg; @he2019collaborative; @truex2019demystifying; @hilprecht2019monte; @wang2019beyondclass; @juuti2019prada; @papernot2017practical; @milli2019model; @chandrasekaran2020exploring; @jagielski2020high; @yang2019neural; @zhang2020secret; @long2018understanding] & 70,000\
Mushrooms [@dua2019] & categorical feat. & binary classification & [@tramer2016stealing; @chandrasekaran2020exploring] & 8,124\
Netflix [@netflixdataset] & binary features & binary classification & [@yeom2018privacy] & 2,416\
Netflows & network data & binary classification & [@ateniese2015hacking] & -\
PTB [@marcus1993building] & text & char-level language model & [@carlini2019secret] & 5 MB\
PiPA [@zhang2015beyond] & images & binary classification & [@melis2019exploiting] & 18,000\
Purchase-100 [@purchase100] & binary features & multi-class classification & [@nasr2019comprehensive; @shokri2017membership; @truex2019demystifying; @jayaraman2019evaluating] & 197,324\
SVHN [@netzer2011reading] & images & multi-class classification & [@zhu2019dlg; @jagielski2020high] & 60,000\
TED talks [@tedtalksiwslt] & text & machine translation & [@carlini2019secret] & 100,000 pairs\
Texas-100 [@texashealthdata] & mixed features & multi-class classification & [@nasr2019comprehensive; @shokri2017membership] & 67,330\
UJIndoor [@dua2019] & mixed features & regression & [@wang2018stealing] & 19,937\
UCI / Adult [@dua2019] & various & binary classification & [@ganju2018property; @Salem0HBF019; @shokri2017membership; @tramer2016stealing; @truex2019demystifying; @chandrasekaran2020exploring; @long2018understanding] & 48,842\
Voxforge [@voxforgedata] & audio & speech recognition & [@ateniese2015hacking] & 11,137 rec.\
Wikitext-103 [@merity2016pointer] & text & word-level language model & [@carlini2019secret; @krishna2020Thieves] & 500 MB\
Yale-Face [@georghiades2001few] & images & multi-class classification & [@song2019privacy] & 2,414\
Yelp reviews [@yelpopendataset] & text & binary classification & [@melis2019exploiting] & 16-40,000\
\[table:datasets\_summary1\]
This high variation is both a blessing and a curse. On one hand it is highly desirable to use multiple types of datasets to test the different hypotheses and the majority of the reviewed research follows that approach. However, these many options make it harder to compare methods. As it is evident from Table \[table:datasets\_summary1\], some of the datasets are quite popular. MNIST, CIFAR-10, CIFAR-100 and UCI Adult have been used by more than six papers while 24 datasets have been used by only one paper.
The number of model parameters varies based on the model, task and dataset used in the experiments. As it can be seen in Table \[table:datasets\_summary1\], most datasets are not extremely large, hence the models under attack are not extremely large. Given that most papers deal with neural networks this might indicate that most attacks focused on smaller datasets and models which might not be representative of realistic scenarios. However, privacy attacks do not necessarily have to target large models with extreme amounts of data and neural networks however popular, are not necessarily the most used models in the “real world”.
Another dimension that could be interesting to analyze is the types of learning tasks that have been the target of attacks so far. Figure \[fig:heatmap\] presents information about the number of papers in relation to the learning task and the attack type. By learning task we refer to the task in which the target model initially trained. As the figure clearly shows, the majority of the attacks are against models that were trained for classification tasks, both binary and multi-class. This is the case across all four attack types.
![Number of papers used against each learning task and attack type. Classification includes both binary and multi-class classification. Darker gray means higher number of papers. This figure shows in which attack types were still not studied for some learning tasks.[]{data-label="fig:heatmap"}](images/heatmap3.png){width="12cm"}
Why Do Machine Learning Models Leak? {#sec:why_leak}
====================================
The connection between overfitting and black-box membership inference was initially investigated by Shokri etal. [@shokri2017membership]. The authors showed experimentally that overfitting can lead to privacy leakage but also noted that it is not the only condition. This finding was later corroborated by Yeom et al. [@yeom2018privacy] where they showed that overfitting is a sufficient condition for performing membership inference attacks but not a necessary one. Additionally, Long et al. [@long2018understanding] showed that even in well generalized models, it is possible to perform membership inference for a subset of the training data which they named *vulnerable records*.
Other factors, such as the model architecture, the model type, and the dataset structure, affect attack accuracy. Models with smaller generalization error on the same dataset were shown to leak more in some MLaaS cases [@shokri2017membership]. This means that model type and complexity are an important factor in membership inference. Similarly, in the white-box setting, Nasr et al. [@nasr2019comprehensive] showed that two models with the same generalization error showed different degrees of leakage. More specifically the most complex model in terms of numbers of parameters exhibited higher attack accuracy, showing that model complexity is also an important factor.
Truex et al. [@truex2019demystifying] ran different types of experiments to establish the significance of model and data complexity. They found that certain model types such as Naive Bayes are less susceptible to membership inference attacks than decision trees or neural networks. They also found that the complexity of the data increases the potential of membership leaks. The higher number of classes led to highest attack accuracy [@truex2019demystifying].
Securing machine learning models against adversarial attacks can also have an adverse effect on the model’s privacy as shown by Song et al. [@song2019privacy]. Current state of the art proposals for robust model training, such as projective gradient descent (PGD) adversarial training [@madry2018towards], increases the model’s susceptibility to membership inference attacks. This is not unexpected since robust training methods (both empirical and provable defenses) tend to increase the generalization error. As previously discussed, the generalization error is related to the success of the attacks. Furthermore, the authors of [@song2019privacy] argue that robust training may lead to increased model sensitivity to the training data, which can also affect membership inference.
The generalization error is easily measurable in supervised learning under the assumption that the test data can capture the nuances of the real data distribution. In generative models and specifically in GANs this is not the case, hence the notion of overfitting is not directly applicable. All three papers that deal with membership inference attacks against GANs, mention overfitting as an important factor behind successful attacks [@hayes2019logan; @hilprecht2019monte; @chen2019gan]. In this case, overfitting means that the generator has memorized and replays part of the training data. This is further corroborated in the ablation study in [@chen2019gan], where their attacks are shown to be less successful as the training data size increases.
While black-box membership inference attacks are connected to overfitting this is not necessarily the case with other types of attacks. Model extraction is possible even when the models under attack have 98% or higher accuracy rate in the test set [@joon2018towards]. Property inference is also possible against well-generalized models [@melis2019exploiting; @ganju2018property].
When it comes to reconstruction attacks, Yeom et al. [@yeom2018privacy] showed that a higher generalization error can lead to higher probability in inferring data attributes, but also that the influence of the target feature to the model is an important factor. However, they assume that the adversary has knowledge of the prior distribution of the target features and labels. Using weaker assumptions about the adversary’s knowledge, Zhang et al. [@zhang2020secret] showed theoretically and experimentally that a model that has high predictive power is more susceptible to reconstruction attacks. Finally, similarly to vulnerable records in membership inference, memorization and retrieval of data which are *out-of-distribution* was shown to be the case even for models that do not overfit [@carlini2019secret].
Defending Machine Learning Privacy {#sec:defenses}
==================================
Leaking personal information such as medical records or credit card numbers is usually an undesirable situation. The purpose of studying attacks against machine learning models is to be able to explore the limitations and assumptions of machine learning and to anticipate the adversaries’ actions. Most of the analyzed papers propose and test mitigations to counter their attacks. One of the most popular proposed countermeasure is differential privacy (DP). This section presents a non exhaustive overview of differential privacy as it is used in privacy preserving machine learning (PPML), as well as other defensive measures proposed in the reviewed literature.
Differential Privacy
--------------------
Differential privacy started as a privacy definition for data analysis and it is based on the idea of “learning nothing about an individual while learning useful information about a population” [@Dwork2013]. Its definition is based on the notion that if two databases differ only by one record and are used by the same algorithm (or mechanism), the output of that algorithm should be similar. More formally,
A randomized mechanism $\mathcal{M}$ with domain $\mathcal{R}$ and output $\mathcal{S}$ is ($\epsilon,\delta$)-differentially private if for any adjacent inputs $D, D' \in \mathcal{R}$ and for any subsets of outputs $\mathcal{S}$ it holds that:
$$Pr[\mathcal{M}(D) \in \mathcal{S} ] \leq e^{\epsilon} Pr[\mathcal{M}(D') \in \mathcal{S} ] +\delta$$
where $\epsilon$ is the privacy budget and $\delta$ is the failure probability.
The original definition of DP did not include $\delta$ which was introduced as a relaxation that allows some outputs not to be bounded by $e^{\epsilon}$.
The usual application of DP is to add Laplacian or Gaussian noise to the output of a query or function over the database. The amount of noise is relevant to the *sensitivity* which gives an upper bound on how much we must perturb the output of the mechanism in order to preserve privacy [@Dwork2013]:
$l_{1}$ (or $l_{2}$)-Sensitivity of a function $f$ is defined as $$\Delta f = \underset{D, D', \|D-D'\|=1}{\max} \|f(D) - f(D')\|$$
where $\|.\|$ is the $l_1$ or the $l_2$-norm and the max is calculated over all possible inputs $D, D'$.
From a machine learning perspective, $D$ and $D'$ are two datasets that differ by one training sample and the randomized mechanism $\mathcal{M}$ is the machine learning training algorithm. In deep learning the noise is added at the gradient calculation step. Because it is necessary to bound the gradient norm, gradient clipping is also applied [@Abadi2016].
Differential privacy offers a trade-off between privacy protection and utility or model accuracy. Evaluations of differentially private machine learning models against membership inference attacks concluded that the models could offer privacy protection only when they considerably sacrifice their utility [@rahman2018membership; @jayaraman2019evaluating]. Jayaraman et al. [@jayaraman2019evaluating] evaluated several relaxations of DP in both logistic regression and neural network models against membership inference attacks. They showed that these relaxations have an impact to the utility-privacy trade off. While they reduce the required added noise they also increase the privacy leakage.
Distributed learning scenarios require additional considerations when it comes to differential privacy. In a centralized model the focus is on sample level DP, i.e., in protecting the privacy at the individual data point level. In a federated learning setting where we have multiple participants, we not only care about the individual training data points they use but we care about ensuring privacy at the participant level. A proposal which applies DP at the participant level was introduced by McMahan et al. [@brendan2018learning] however it requires a large number of participants. When it was tested with a number as low as 30 the method was deemed unsuccessful [@melis2019exploiting].
Other Defensive Approaches
--------------------------
While differential privacy is one of the most popular countermeasures proposed in attack oriented papers, there are several other defences that have been explored.
1. **Regularization**. Most often in the form of dropout, regularization, is proposed by multiple papers with varying levels of success [@carlini2019secret; @hayes2019logan; @melis2019exploiting; @Salem0HBF019; @shokri2017membership; @song2019privacy]. Given that black-box membership inference attacks are connected to overfitting, it is a sensible approach against this type of attack.
2. **Prediction vector tampering**. As many models assume access to the prediction vector during inference, one of the countermeasures proposed was the restriction of the output to the top-k classes or predictions of a model [@shokri2017membership]. However, this restriction, even to the strictest form (outputting only the class label) did not seem to fully mitigate membership inference attacks, since information leaks can still happen due to model misclassifications. The level of prediction vector truncation affects also reconstruction attacks, but it does not stop them completely [@yang2019neural]. Another option is to lower the precision of the prediction vector which leads to less information leakage [@shokri2017membership]. Adding noise calculated using adversarial learning, also thwarts membership inference attacks [@jia2019memguard].
3. **Model compression**. Setting all the loss gradients which are below a certain threshold to zero, was proposed as a defence against reconstruction attacks in deep learning. This technique proved quite effective with as little as 20% of the gradients set to zero and with negligible effects in model performance [@zhu2019dlg].
4. **Ensemble** methods, such as model stacking were tested in [@Salem0HBF019] and produced positive results against membership inference.
5. **Noisy data** addition. Randomly flipping labels on 5% of the training data had moderate success on preventing property inference attacks [@ganju2018property].
6. **Weight Quantization** or using **half-precision floating points** for neural network weights did not seem to deter the attacks in [@carlini2019secret] and [@zhu2019dlg], respectively.
7. **Selective sharing** of gradients. In a distributed learning system that uses synchronized SGD, Shokri and Shmatikov proposed that the participants can partially share their gradients with the parameter server [@shokri2015privacy]. While this did not impact the model performance, it was later shown to be an inadequate measure [@phong2017privacy].
8. **Protecting against DNN Model Stealing Attacks (PRADA)**. Detecting model stealing attacks based on the model queries that are used by the adversary was proposed by Juuti et al. [@juuti2019prada]. The detection is based on the assumption that model queries that try to explore decision boundaries will have a different distribution than the normal ones. While the detection was successful, the authors noted that it is possible to be evaded if the adversary adapts their strategy.
9. **Membership inference**. The idea of using membership inference in order to defend against model extraction was studied by Krishna et al. [@krishna2020Thieves]. It is based on the premise that using membership inference the model owner can distinguish between legitimate user queries and nonsensical ones whose only purpose is to extract the model. The authors note that this type of defence has limitations such as potentially flagging legitimate but out-of-distribution queries made by legitimate users, but more importantly that they can be evaded by adversaries that make adaptive queries.
Discussion
==========
Attacks against machine learning privacy have been increasingly brought to light. However, we are still at an exploratory stage. Many of the attacks are applicable only under specific sets of assumptions or do not scale to larger training data sets, number of classes, number of participants, etc. The attacks will keep improving and in order to successfully defend against them, the community needs to answer fundamental questions about why they are possible in the first place. While progress has been made in the theoretical aspects of some of the attacks, there is still a long way to go when it comes to achieve better theoretical understanding of privacy leaks in machine learning.
As much as we need answers about why leaks happen at a theoretical level, we also need to know how well privacy attacks work against real deployed systems. Adversarial attacks against realistic systems brought to light the issue of additional constraints that need to be in place for the attacks to work. When creating glasses that can fool a face recognition system, Sharif et al. [@sharif2016accessorize], had to pose constraints that had to do with physical realizations, e.g., that the color of the glasses should be printable. In privacy related attacks, the most realistic attacks come from the model extraction area, where attacks against MLaaS systems have been demonstrated in multiple papers. For the majority of other attacks, it is certainly an open question of how well they would perform against deployed models and what kind of additional requirements need to be in place for them to succeed.
At the same time, the main research focus up to now has been supervised learning. Even within supervised learning, there are areas and learning tasks that have been largely unexplored, such as recurrent models. In unsupervised learning, the focus is mainly in generative models and only just recently papers started exploring areas such as representation learning. Some attacks against image classifiers, do not transfer that well against natural language processing tasks [@hishamoto2020embership] while others do, but may require different sets of assumptions and design considerations [@pan2020privacy].
Beyond expanding the focus to different learning tasks there is the question of datasets. The impact of data complexity in the attack success has been demonstrated by several papers. Yet, currently, we lack a common approach as to which datasets are best suited to demonstrate privacy attacks, or constitute the minimum requirement for a successful attack. Several questions are worth considering: do we need standardized datasets and if yes how do we go about and create them? Are all data worth protecting and if some are more interesting than others, shouldn’t we be testing attacks beyond popular image datasets?
Finally, as we strive to understand the privacy implications of machine learning, we also realize that several research areas are connected and affect each other. We know, for instance, that adversarial training adversely affects membership inference [@shokri2019privacy] and that model censoring can still leak private attributes [@Song2020Overlearning]. Property inference attacks can deduce properties of the training dataset that were not specifically encoded or were not necessarily correlated to the learning task. This can be understood as a form of bias detection which means that relevant literature in the area of model fairness should be reviewed as potentially complementary. Looking at adjacent areas of machine learning research might help us improve our understanding of privacy attacks, too.
Conclusion
==========
As machine learning becomes ubiquitous, the scientific community becomes increasingly interested in its impact and side-effects in terms of security, privacy, fairness, and explainability. This survey conducted a comprehensive study of the state-of-the-art privacy-related attacks to create a new threat model and taxonomy of the different types of attacks and their characteristics. An in-depth examination of the different types of attacks allowed us to perform a further analysis which revealed common design patterns and differences between them. Our analysis revealed a somewhat narrow focus of the research conducted so far. At the same time, a thorough theoretical understanding of the reasons behind privacy leaks is still under-developed and this affects both the proposed defensive measures and our understanding of the limitations of privacy attacks. While the community is still in exploratory mode in regards to privacy leaks of machine learning systems, we hope that this survey will provide the necessary background to both the interested readers as well as the researchers that wish to continue working in this topic.
This work was partially supported by Avast Software and the OP RDE funded project Research Center for Informatics No.: CZ.02.1.01/0.0./0.0./16\_019/0000765.
|
---
abstract: 'World Health Organization (WHO) characterized the novel coronavirus (COVID-19) as a global pandemic on March 11th, 2020. Before this and in late January, more specifically on January 27th, while the majority of the infection cases were still reported in China and a few cruise ships, we began crawling social media user postings using the Twitter search API. Our goal was to leverage machine learning and linguistic tools to better understand the impact of the outbreak in China. Unlike our initial expectation to monitor a local outbreak, COVID-19 rapidly spread across the globe. In this short article[^1] we report the preliminary results of our study on automatically detecting the positive reports of COVID-19 from social media user postings using state-of-the-art machine learning models.'
author:
- |
Negin Karisani\
Purdue University\
[nkarisan@purdue.edu]{}\
Payam Karisani\
Emory University\
[payam.karisani@emory.edu]{}\
bibliography:
- 'coling2020.bib'
title: 'Mining Coronavirus (COVID-19) Posts in Social Media'
---
Introduction and Motivation {#sec:intro}
===========================
According to a tally by Johns Hopkins University 566,269 persons are tested positive and 25,423 persons have died around the globe as of today, March 27th. Approximately a third of the world’s population is impacted by COVID-19. The United States became the epicenter of the virus pandemic on March 26th–yesterday–and New York City with 23,112 confirmed cases is the epicenter of the US outbreak. The US House passed a \$2 trillion stimulus bill to combat the negative impact of COVID-19 on the country’s economy. Despite the devastating global impact of COVID-19, WHO has announced that the current pandemic would be the first pandemic in human history that could be controlled.
The impact of COVID-19 on societies is unprecedented. Numerous countries in Asia and the EU, including Iran, Italy, and Spain are under a lock-down. In the US, states such as California and New York are experiencing the same situation. People are ordered to stay home, and are encouraged to practice social distancing. Psychologists advise the residents of the affected areas to practice certain routines to maintain their mental well-being. With people staying at home more often, the role of the internet, as a means of communication, has become even more critical. For instance, NextDoor, a hyperlocal social network, recently announced that the daily rate of its active users has increased by 80%.
It has long been known that social networks are effective media for public health monitoring. Despite the well-understood limitations and biases present in the conclusions drawn from social media data [@biases], they are proven to be invaluable resources [@survey-1]. In this article, we report the preliminary results of our study on automatically mining the user postings related to COVID-19 on Twitter. Our goal is to find the extent in which machine learning models can distill the user generated data. As pointed out by previous studies [@wespad], this can facilitate the related institutions’ responses to the outbreak. In the next section, we focus on automatically detecting the positive reports of COVID-19 infections in the data that we have been collecting since January 27th, 2020.
Dataset
=======
We started collecting Twitter data on January 27th, 2020. As of March 26th, we have collected 5,621,048 tweets. We used the Twitter search API to crawl the data, and our search keywords were initially “coronavirus” and “corona virus”. On mid-March we added the keywords “COVID-19” and “COVID 19” to the search criteria. We only collected the English user postings, and omitted retweets and replies.
In order to evaluate the machine learning models we also manually inspected the data, and used stratified sampling to construct a dataset. Thus, for the tweets posted in February we randomly selected between 100 and 300 examples a day and collected the total of 6,090 tweets. This set of tweets constitute our training set. Additionally, we also randomly selected a set of 200 examples a day between March 3rd and March 12th, and collected the total of 2,000 tweets, which constitute our test set. We hired an annotator and annotated the data based on the following criteria: 1) If a tweet mentions individuals infected with COVID-19, and also explicitly or implicitly contains a time reference then it was labeled positive. 2) If a tweet does not mention any individual or lacks a time reference it was labeled negative. To validate the quality of the labels we hired a second annotator and randomly relabeled 10% of the tweets. The inter-agreement between the two annotators was 0.70 based on Cohen Kappa coefficient, which indicates a substantial correlation [@kappa]. Table \[tbl-stat\] summarizes the training and test sets, and their corresponding percentage of the positive and negative examples. In the next section, we discuss the methods that we implemented and report their results.
[|p[0.5in]{} |p[0.5in]{} |p[0.5in]{} |\[1pt\] p[0.5in]{} |p[0.5in]{} |p[0.5in]{} |]{} &\
& [Negative]{} & [Positive]{} & [Count]{} & [Negative]{} & [Positive]{}\
6090 & 90.8% & 9.2% & 2000 & 94.1% & 5.9%\
Experiments
===========
We begin this section by describing the methods that we implemented, then we briefly discuss the training procedure, and finally report the results.
Methods Compared
----------------
We included seven methods in our experiments. One classic generative model (Naive Bayes), one classic discriminative model (Logistic Regression), one widely used neural network model (fasttext), and four models based on the state-of-the-art model Bidirectional Encoder Representations from Transformers (BERT). Below we briefly describe each one.
***NB*.** We included the Naive Bayes classifier. We incorporated the MALLET implementation [@mallet] of this classifier, and used the tweet unigrams and bigrams as features.
***LR*.** We included the Logistic Regression as the discriminative counterpart of Naive Bayes. We incorporated the MALLET implementation, and again used the tweet unigrams and bigrams as features.
***Fasttext*.** We included the neural model introduced in [@fasttext]. This model is a shallow wide network, capable of updating the input word embeddings during the training. We used the pretrained word2vec vectors [@word2vec] as input features. The learning rate was empirically set to 0.5, and the window size was set to 2.
***BERT-BASE*.** We included the state-of-the-art model introduced in [@bert]. This model is based on a multi-layer transformer encoder [@transformer]. We used the pre-trained base variant, followed by one layer fully connected network. We applied the default model settings recommended in [@bert]. We used the pytorch implementation of BERT introduced in [@bert-impl].
***BERT-Twitter*.** Since our classification problem is defined on social media posts, we can expect that a model specifically exposed to the social media language model (through the Masked Language model task) would perform better than regularly pre-trained ones. Thus, we used a corpus of 35 million tweets collected between 2018 and 2019 through the Twitter streaming API to further pre-train *BERT-BASE*. We set the maximum window size to 160 and batch-size to 32–the rest of the settings were set to the default values, as suggested in [@bert]–and pre-trained this model for 4.5 million steps–about 5 epochs.
***BERT-Corona*.** We hypothesized that if a model is already familiar with the contexts in which the word coronavirus is used in, it would perform better during the classification phase. Thus, we used the tweets that we collected between January 27th and March 3rd to pre-train *BERT-BASE*. We pre-trained this model for 400K steps–approximately 5 epochs. The pre-training settings were identical to *BERT-Twitter*.
***BERT-Corona-BiLSTM*.** Even though BERT already utilizes a sophisticated attention mechanism, we still experimented with sequence encoding models. Thus, we used a Bidirectional Long Short Term Memory Network [@lstm] on top of *BERT-Corona*, followed by one layer fully connected network. We empirically observed that if we set the size of the hidden dimensions of the BiLSTM to a half of the size of the hidden dimensions of BERT (i.e., 768) we would get the best performance.
Experimental Setup
------------------
We trained *Fasttext* for 100 iterations. The models based on BERT, i.e., *BERT-BASE*, *BERT-Twitter*, and *BERT-Corona*, were trained for 2 iterations–these models are already based on a pre-trained model. We trained *BERT-Corona-BiLSTM* for 3 iterations, since it has more parameters than the other BERT based baselines. For training the models, we used the default model optimizers proposed by the references. In none of the experiments we did any text pre-processing. Since there is a randomness in model initialization and drop-out regularization, we carried out all of the neural network experiments for five times. The results reported in the next section are the average over these experiments.
The task that we defined is a binary classification problem–detecting the positive reports of COVID-19 from Twitter data. Since the class distribution is highly skewed, following the previous studies [@metrics], we report the F1, Precision, and Recall of the models in the positive class.
Results
-------
Table \[tbl-result\] summarizes the performance results. We see that the baselines based on the pre-trained models show the best results. The experiments validate our hypothesis about the effectiveness of pre-training BERT on domain specific data. We see that *BERT-Corona* has achieved the best F1 value. By comparing *BERT-Twitter* and *BERT-Corona-BiLSTM* we can also hypothesize that model initialization–through pre-training–can be potentially more effective than increasing model complexity. Even though validating this hypothesis require more comprehensive experiments.
**Model** [**F1**]{} [**Precision**]{} [**Recall**]{}
---------------------- ------------ ------------------- ----------------
*NB* 0.109 **0.700** 0.059
*LR* 0.470 0.662 0.364
*Fasttext* 0.530 0.650 0.447
*BERT-BASE* 0.645 0.589 0.715
*BERT-Twitter* 0.666 0.613 **0.730**
*BERT-Corona* **0.676** 0.632 0.727
*BERT-Corona-BiLSTM* 0.659 0.666 0.654
: Average F1, precision, and recall in detecting positive reports of COVID-19 from Twitter data.[]{data-label="tbl-result"}
We believe a robust social media surveillance system can be immensely helpful. Although the results are encouraging, there are still a lot of challenges to be addressed to build such a system for COVID-19. Automatically detecting positive reports, or even following up on the mental well-being of patients through social media posts can greatly enhance the concerned institutions’ endeavor to monitor the public health and respond in timely manner.
Conclusions
===========
In this short article we reported the preliminary results of our study on the capability of machine learning models to distill social media posts related to COVID-19, namely we focused on automatically detecting the positive reports of this illness. We constructed a manually annotated dataset, and showed that state-of-the-art classifiers have encouraging results. Our pre-trained model and unlabeled data can be accessed through our Github [^2]. We will also release our labeled data, along a more comprehensive analysis soon.
[^1]: This paper is a short version of a longer study.
[^2]: Available at <https://github.com/nkarisan/Covid19_Research>
|
---
author:
- 'J. Álvarez-Márquez'
- 'L. Colina'
- 'R. Marques-Chaves'
- 'D. Ceverino'
- 'A. Alonso-Herrero'
- 'K. Caputi'
- 'M. García-Marín'
- 'A. Labiano'
- 'O. Le Fèvre'
- 'H. U. Norgaard-Nielsen'
- 'G. Östlin'
- 'P. G. Pérez-González'
- 'J. P. Pye'
- 'T. V. Tikkanen'
- 'P. P. van der Werf'
- 'F. Walter'
- 'G. S. Wright'
bibliography:
- 'Bibliography.bib'
title: 'Investigating the physical properties of galaxies in the Epoch of Reionization with MIRI/JWST spectroscopy'
---
[The [*James Webb Space Telescope*]{} (JWST) will provide deep imaging and spectroscopy for sources at redshifts above 6, covering the entire Epoch of Reionization (EoR, 6$< z <$10), and enabling the detailed exploration of the nature of the different sources during the first 1 Gyr of the history of the Universe. The Medium Resolution Spectrograph (MRS) of the mid-IR Instrument (MIRI) will be the only instrument on board JWST able to observe the brightest optical emission lines H$\alpha$ and \[OIII\]0.5007$\mu$m at redshifts above 7 and 9, respectively, providing key insights into the physical properties of sources during the early phases of the EoR. This paper presents a study of the H$\alpha$ fluxes predicted by state-of-the-art FIRSTLIGHT cosmological simulations for galaxies at redshifts of 6.5 to 10.5, and its detectability with MIRI. Deep (40 ksec) spectroscopic integrations with MRS will be able to detect (S/N $>$ 5) EoR sources at redshifts above 7 with intrinsic star formation rates (SFR) of more than 2 M$_{\odot}$ yr$^{-1}$, and stellar masses above 4-9 $\times$ 10$^7$ M$_{\odot}$. These limits cover the upper end of the SFR and stellar mass distribution at those redshifts, representing $\sim$ 6% and $\sim$1% of the predicted FIRSTLIGHT population at the 6.5-7.5 and 7.5-8.5 redshift ranges, respectively. In addition, the paper presents realistic MRS simulated observations of the expected rest-frame optical and near-infrared spectra for some spectroscopically confirmed EoR sources recently detected by ALMA as \[OIII\]88$\mu$m emitters. The MRS simulated spectra cover a wide range of low metallicities from about 0.2 to 0.02 Z$_{\odot}$, and different \[OIII\]88$\mu$m/\[OIII\]0.5007$\mu$m line ratios. The simulated 10ks MRS spectra show S/N in the range of 5 to 90 for H$\beta$, \[OIII\]0.4959,0.5007 $\mu$m, H$\alpha$ and HeI1.083$\mu$m emission lines of the currently highest spectroscopically confirmed EoR (lensed) source MACS1149-JD1 at a redshift of 9.11, independent of metallicity. In addition, deep 40 ksec simulated spectra of the luminous merger candidate B14-65666 at 7.15 shows the MRS capabilities of detecting, or putting strong upper limits on, the weak \[NII\]0.6584$\mu$m, \[SII\]0.6717,0.6731$\mu$m, and \[SIII\]0.9069,0.9532$\mu$m emission lines. These observations will provide the opportunity of deriving accurate metallicities in bright EoR sources using the full range of rest-frame optical emission lines up to 1$\mu$m. In summary, MRS will enable the detailed study of key physical properties such as internal extinction, instantaneous star formation, hardness of the ionizing continuum, and metallicity in bright (intrinsic or lensed) EoR sources.]{}
Introduction {#Int}
============
Deep imaging surveys with the [*Hubble Space Telescope*]{} (HST) have detected galaxies at very high redshifts ($z$ > 5) in large numbers; there are hundreds of them at photometric redshifts of about 7, and about 200 candidates at redshifts of 8-10, well within the Epoch of Reionization (EoR) of the universe [@Bouwens2015; @Oesch2015; @Roberts-Borsani2016; @Stefanon2017; @Oesch2018]. The combination of [*HST*]{} and [*Spitzer*]{} deep imaging has further identified these galaxies as potential strong optical line emitters based on the flux excess in the IRAC 3.6 and 4.5 $\mu$m bands [e.g. @Schaerer2009; @Labbe2013; @Stark2013; @Smit2015; @Bouwens2016b; @Rasappu2016; @Roberts-Borsani2016]. The H$\beta$+\[OIII\] and the H$\alpha$ lines have large equivalent widths with values of up to (rest-frame) 1000-2000$\AA$ (e.g. @Faisst2016 [@Marmol-Queralto2016; @Rasappu2016; @Smit2016; @Caputi2017; @Lam2019arXiv]), and a non-linear dependency with redshift (1+$z$)$^\alpha$ with $\alpha$ $\sim$ 1.8 and $\sim$ 1.3 for sources at redshifts $z < 2.5$ and $2.5 < z < 6$, respectively (@Faisst2016 [@Marmol-Queralto2016]).
The spectroscopic confirmation of EoR sources remains very limited. It is based mostly on the Ly$\alpha$ detection (@Stark2017 for a recent compilation; @Zitrin2015 [@Oesch2015b; @Jung2019arXiv]), which becomes very inefficient at $z>7$ as only the brightest sources exhibit Ly$\alpha$ emission [@Pentericci2011]. Additionally, detection of far-infrared \[CII\]158$\mu$m and \[OIII\]88$\mu$m line emitters at redshifts of up to 9.11 have recently been reported with ALMA [@Inoue2016; @Carniani2017; @Tamura2018; @Hashimoto2019; @Hashimoto2018b; @Smit2018].
The subarcsec imaging and spectroscopic capabilities of the [*James Webb Space Telescope*]{} (JWST), combined with its broad spectral range coverage (0.6 to 28 $\mu$m) and its increased sensitivity of one to two orders of magnitude better than that of previous space observatories such as HST and [*Spitzer*]{} will provide exquisite data to investigate in detail the physical nature and properties of EoR sources at redshifts above 6. Among the JWST instruments, the Mid-infrared Instrument (MIRI) Medium Resolution Spectrograph (MRS) covering the 5 to 28 $\mu$m spectral range, will be the only instrument capable of detecting the strongest optical lines, H$\alpha$ and \[OIII\]0.5007$\mu$m at redshifts well above 6.6 and 9, respectively (@Wright2015 [@Wells2015]; and references therein). In addition, while the JWST near-infared spectrograph (NIRSpec) will be covering the UV and blue rest-frame range [@Chevallard2019], the MRS extends the observed spectral range above the rest-frame \[OIII\] lines, and well into the 1 $\mu$m region, where internal extinction is less relevant, and other less explored lines such as \[SIII\]0.907,0.953 $\mu$m and HeI1.083$\mu$m are present.
The rest-frame optical and near-IR spectral range covered by the MRS is key to developing a full understanding of the physical properties and mechanisms involved in the earliest stages in the formation of galaxies during EoR. Of the main optical diagnostic lines, the H$\alpha$ is the least affected by internal extinction, and therefore the cleanest tracer of the instantaneous star formation rate (SFR, @Kennicutt1998 for review) in EoR sources, even if a measurement of internal extinction is not available. Moreover, the combination of H$\alpha$ with Ly$\alpha$ and UV continuum measurements will provide more accurate values for the Ly$\alpha$ and the ionizing escaping fractions. In addition, the ratios \[NII\]0.6584$\mu$m/H$\alpha$ and \[SII\]0.6717,0.6731$\mu$m/H$\alpha$ trace the metallicity (Z) in pure star-forming galaxies (@Maiolino2019 for a review), and combined with the ratio \[OIII\]/H$\beta$ trace the nature of the ionizing source [@Kewley2013a]. Several other less explored combinations of line ratios involving the \[NII\], \[SII\], and \[SIII\] lines become available as additional metallicity tracers (@Maiolino2019 and references therein). Finally, the HeI 1.083 $\mu$m, which is the strongest HeI line in the entire optical and near-IR range [@Porter2005], can provide in combination with the H$\alpha$ line, a measurement of the hardness of the ionizing continuum, and therefore information on the nature of the ionizing source, as the hydrogen and HeI lines are sensitive to the total amount of photons with energies above 13.6 eV (912$\AA$) and 24.6 eV (504$\AA$), respectively.
Predictions of the nebular spectra of EoR sources from state-of-the-art cosmological simulations [@Barrow2017; @Ceverino2019; @Katz2019] are now available for a direct comparison with future JWST observations. These simulations follow the physical processes associated with the early formation and evolution of galaxies during the first 1 Gyr of the universe. These simulations predict galaxies in the early universe as strong line emitters, confirmed by the detection of Ly$\alpha$ and, more recently, \[OIII\]88$\mu$m line emitters at redshifts $\sim$ 7-9. Therefore, the prospects of investigating the nature, evolution, and physical properties of early galaxies with the MIRI spectrograph should be explored in detail.
This paper presents a study of the detectability of FIRSTLIGHT simulated galaxies at redshifts of 6.5 to 10.5, and realistic MIRI/JWST spectra of the newly discovered high-z \[OIII\]88$\mu$m emitters detected with ALMA. The paper is structured as follows. The most relevant features of the FIRSTLIGHT simulations and the apparent fluxes of the two strongest optical emission lines (\[OIII\]0.5007$\mu$m and H$\alpha$) for FIRSLIGHT galaxies at redshifts 6.5 to 10.5 as a function of their SFR, stellar mass (M$_{*}$), and specific star formation (sSFR) are presented in Sect. \[Met:FirstLight\], together with a discussion of the detectability of the population of FIRSTLIGHT H$\alpha$ emitters with MRS. Specific examples of MRS simulated spectra for two of the recently detected \[OIII\]88$\mu$m emitters, MACS1149-JD1 [@Zheng2012; @Hashimoto2018b] and B14-65666 [@Bowler2014; @Bowler2017; @Hashimoto2019] at a respective redshift of 9.11 and 7.15, are presented in Sect. \[Met:FullSim\], together with the possibilities that MRS opens for the detail studies of their physical properties, such as internal extinction, instantaneous star formation, hardness of ionizing continuum, and metallicity. A summary of the results and future work is presented in Sect. \[Conc\]. Throughout this paper we use a standard cosmology with matter and dark energy density $\Omega_{\rm m} = 0.3$ and $\Omega_{\Lambda} = 0.7$, the Hubble constant $H_{0} = 70$ km s$^{-1}$ Mpc$^{-1}$, and the AB magnitude system.
FIRSTLIGHT: EoR line emitters from cosmological simulations {#Met:FirstLight}
===========================================================
FIRSLIGHT overview
------------------
We use the zoom-in cosmological simulations of galaxies of the FIRSTLIGHT project [@Ceverino2017; @Ceverino2018; @Ceverino2019]. Briefly, this consists of a complete mass-selected sample of 289 halos, selected at $z = 5$ in two cosmological boxes of 10 and 20 Mpc h$^{-1}$ with halo masses between 10$^{9}$ - 10$^{11}$ M$_{\odot}$ (see details in @Ceverino2017). The maximum spatial resolution is 10 pc. The dark matter particle mass resolution is m$_{\rm DM}$ = 10$^{4}$ M$_{\odot}$ and the minimum star particle mass is 100 M$_{\odot}$.
These high-resolution simulations are performed with the ART code [@Kravtsov1997; @Kravtsov2003; @Ceverino2014; @CeverinoKlypin2009; @Ceverino2019]. They follow the evolution of a gravitating system and the Eulerian gas hydrodynamics, and incorporate other astrophysical processes, such as gas cooling radiation, photoionization heating by the cosmological UV background, a stochastic star formation model, and a model that includes thermal, kinetic and radiative feedback (see details in @Ceverino2017).
The FIRSTLIGHT database[^1] includes several properties for all snapshots of the main galaxy progenitor of the 289 zoom-in simulations, such as the virial, stellar and gas masses, and its SFR, in steps of 10 Myr. The database starts when the galaxy reaches the halo mass of M$_{vir} = 10^{9}$ M$_{\odot}$ and ends in the last available snapshot at $z \geq 5$. In general, these galaxies show non-uniform star formation histories, spending most of their time (70%) in bursts of star formation [@Ceverino2018], consistent with cosmological gas accretion events. In this work, we use all snapshots within the redshift range $6.5 \leq z \leq 10.5$. This sample is composed of 10,064 snapshots, and covers a wide range of stellar masses ($\sim 10^{5-9}$ M$_{\odot}$), SFRs ($\sim 0 - 30$ M$_{\odot}$ yr$^{-1}$), and metallicities ($Z = 3 \times 10^{-5} - 8 \times 10^{-3}$).
In addition to the physical properties mentioned above, spectral energy distributions (SEDs) are also publicly available for all these snapshots [@Ceverino2019]. Stellar SEDs are generated using the Binary Population and Spectral Synthesis model [BPASS: @Eldridge2017] and assume a [@Kroupa2001] initial mass function (IMF). The contribution of nebular emission is also available and assumes the stellar metallicity, and a gas covering factor of one with an electron density of 100 cm$^{-3}$ [see @Ceverino2019 for details].
MRS detectability of FIRSTLIGHT EoR line emitters {#Res:FirstLight}
-------------------------------------------------
Luminosities of the two strongest optical emission lines, \[OIII\]0.5007$\mu$m and H$\alpha$, are extracted for each nebular SED component and converted to observable fluxes (in units of erg s$^{-1}$ cm$^{-2}$) using the equation
$$F_{\rm obs} (\rm [OIII], H\alpha) = \frac{L (\rm [OIII], H\alpha)}{4 \pi D_{L}^2},$$
where $D_{L}$ is the luminosity distance at a given redshift for the adopted cosmology. Figure \[fig:FS\_Halpha\_\[OIII\]\] shows the relation between the ratio \[OIII\]/H$\alpha$ the H$\alpha$ emission line fluxes of the simulated galaxies. The most luminous FIRSTLIGHT galaxies present similar H$\alpha$ and \[OIII\]0.5007$\mu$m fluxes (\[OIII\]0.5007$\mu$m/H$\alpha$ $\ge$ 1), whereas for fainter galaxies H$\alpha$ tends to be brighter than \[OIII\]0.5007$\mu$m (\[OIII\]0.5007$\mu$m/H$\alpha$ < 1), similar to the values found in metal-deficient, low-$z$ galaxies [e.g. @Izotov2011; @Hirschauer2016; @Izotov2018]. We note, however, that these simulations do not include the effect of dust attenuation, although it is expected to be negligible in low-mass, low-metallicity, high-$z$ galaxies [e.g. @Hashimoto2018b].
$\begin{array}{rl}
\includegraphics[width=0.48\textwidth]{Figure_Halpha_OIII_v2.pdf}
\end{array}$
Figure \[fig:FS\_galaxies\] shows the derived H$\alpha$ fluxes of all snapshots of the main galaxy progenitor of the FIRSTLIGHT simulations with redshifts between 6.5 and 10.5 as a function of their SFRs, stellar masses, and specific SFRs. Overall, these galaxies show a linear relation between H$\alpha$ fluxes and SFRs, as expected, since FIRSTLIGHT SFRs are computed using stellar particles younger than $\sim 10$ Myr that produce copious amounts of ionizing photons that ionize the surrounding gas. The relation between H$\alpha$ flux and stellar mass, and therefore sSFR, is nevertheless much broader due to the stochastic star formation histories in the simulations [for details, see @Ceverino2018]. This sample is dominated by numerous low-mass galaxies with extremely faint H$\alpha$ emission for JWST spectroscopy, characterized by median values of $\widetilde{F} (\rm H\alpha) = 3.8$, $2.4$, and $1.5 \times 10^{-20}$ erg s$^{-1}$ cm$^{-2}$ in the redshift intervals of 6.5-7.5, 7.5-8.5, and $z > 8.5$, respectively. However, a fraction of the galaxies show much higher fluxes of around $F (\rm H\alpha) \sim 10^{-18}$ - $10^{-17}$ erg s$^{-1}$ cm$^{-2}$ that are accessible to observation with MIRI/JWST spectroscopy.
In order to study the detectability of H$\alpha$ in such galaxy population, we use the expected MRS limiting sensitivity curves of [@Glasse2015]. We note that these sensitivity curves refer to point-like sources with spectrally unresolved lines. By using medium deep (10 ks) and deep (40 ks) on-source MRS spectroscopic observations we find limiting H$\alpha$ fluxes of $ \simeq 5.8 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$ ($10\sigma$ in 10 ks) and $\simeq 1.4 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$ ($5\sigma$ in 40 ks), respectively.[^2] As shown in Figure \[fig:FS\_galaxies\], this means that for the entire $6.5 < z < 10.5$ FIRSTLIGHT sample, only a small fraction of about 6.2, 1.1, and $0.4 \%$ of FIRSTLIGHT galaxies in the redshift range of 6.5-7.5, 7.5-8.5, and $z > 8.5$, respectively, would be detected (S/N $\geq$ 5) in deep 40 ks observations. This indicates that only the most luminous FIRSTLIGHT simulated galaxies, those with star formation rates higher than $1.6$, $1.9$, and $3.9$ M$_{\odot}$ yr$^{-1}$, and stellar masses higher than $ 4$, $9$, and $14 \times 10^{7}$ M$_{\odot}$ in the redshift intervals of 6.5-7.5, 7.5-8.5, and $z > 8.5$, respectively, will be accessible for detailed studies with MRS spectroscopy in a moderate amount of observing time (40 ks).
$\begin{array}{rl}
\includegraphics[width=1.0\textwidth]{Figure_Halpha_sfr_ms_ssfr_v3.pdf}
\end{array}$
It should be noted, however, that the FIRSTLIGHT simulations are limited to halo masses of a few times $10^{11} M_{\odot}$ within a cosmological volume of $\sim2 \times 10^{4}$ Mpc$^{3}$. As shown in the middle panel of Figure \[fig:FS\_galaxies\], this limits simulated galaxies to have stellar masses above $2 \times 10^9$ M$_{\odot}$. However, massive EoR galaxies have been recently detected by ALMA as \[OIII\]$88\mu$m emitters [@Inoue2016; @Hashimoto2019; @Tamura2018]. These galaxies are relatively massive, $M_{*} = (2-5) \times 10^{9}$ M$_{\odot}$, and show very strong \[O III\] $88\mu$m line fluxes, $\simeq (0.6 - 17.5) \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$. The UV-luminous, high EW\[H$\beta$+OIII\], Ly$\alpha$-emitter sources [@Roberts-Borsani2016; @Stark2017], could also belong to the same class of EoRs. Assuming a wide range of \[OIII\]0.5007$\mu$m/\[OIII\]88$\mu$m (hereafter R\[OIII\]) and \[OIII\]$0.5007\mu$m/H$\alpha$ line ratios (R\[OIII\]$= 6.5 - 10$ and \[OIII\]$0.5007\mu$m/H$\alpha = 0.59 - 1.93$, see Table \[tab\_MRS\_sim\]), these galaxies will show H$\alpha$ fluxes of about $(0.2 - 30) \times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ (see Figure \[fig:FS\_galaxies\]), well above the detection limits even for medium-deep (10 ks) MRS observations. On the other hand, for more typical, less luminous galaxies, the power of strong gravitational lensing may add the required boost in the apparent fluxes necessary to reach the MRS sensitivity. Therefore, high S/N optical ($\sim$ 0.5 to 1 $\mu$m) emission line spectra will become available with MRS for the first time at such early cosmic times, providing the opportunity of characterizing several of the physical properties of these sources. An exploration of these possibilities is presented in the following section with two specific examples.
MIRI/JWST spectroscopy: EoR \[OIII\]88$\mu$m line emitters {#Met:FullSim}
==========================================================
In the previous section we conclude that all ALMA detected \[OIII\]88$\mu$m sources, and also known UV-luminous Lyman-alpha emitters (LAEs) [@Stark2017], in the EoR will be easily studied using the H$\alpha$ emission line with a medium-deep (10ks) and deep (40ks) MRS observations. In the following we present realistic MRS simulated observations of the rest-frame optical and near-IR spectrum ($\sim$0.5 - 1.2 $\mu$m) for two recently ALMA detected \[OIII\]88$\mu$m emitters, MACS1149-JD1 [@Zheng2012; @Hashimoto2018b] and B14-65666 [@Bowler2014; @Bowler2017; @Hashimoto2019]. MACS1149-JD1 is a lensed galaxy with a magnification factor of $\sim$10 at a redshift of 9.11, being the highest-$z$ spectroscopically confirmed galaxy based on an emission line. Its derived intrinsic SFR of 4.2M$_{\odot}$ yr$^{-1}$, sSFR of 4 Gyr$^{-1}$, stellar mass of 1.1$\times$ 10$^9$ M$_{\odot}$, and observed \[OIII\] 88$\mu$m flux of $3 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$ places it within the range of H$\alpha$ fluxes clearly detectable with medium-deep MRS spectroscopy. On the other hand, B14-65666 is a Lyman-break galaxy system of two sources likely interacting or merging at redshift 7.15, identified as UV bright with an absolute magnitude of M$_{UV} \sim -22.3$, which places it in the range of luminous LAEs [@Roberts-Borsani2016; @Stark2017]. The global system has a derived SFR of 200 M$_{\odot}$ yr$^{-1}$, sSFR of 259 Gyr$^{-1}$, stellar mass of 7.7 $\times$ 10$^8$ M$_{\odot}$, and \[OIII\] 88$\mu$m flux of $21.8 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$. Table \[tab\_gen\_prop\] summarizes the intrinsic properties of the two sources.
$^{(1)}$ [@Hashimoto2018b], $^{(2)}$ [@Hashimoto2019]\
$^{(*)}$ Intrinsic physical properties (after magnification correction of $\mu \sim$ 10)
Generating MRS simulated spectra
--------------------------------
The process to build a final calibrated 1D MRS simulated spectrum has four different phases. First, a variety of spectral templates that cover the expected range of metallicities and excitation conditions of the ionized gas for galaxies in the EoR are built (Sect. \[Met:Templates\]). Second, we take advantage of the MIRI instrument simulator (MIRISim)[^3] to generate simulated MRS observations where the spectral template, astronomical scene, and instrumental and observational configurations are set up (Sect. \[MeT:MIRISim\]). Third, the official JWST calibration pipeline is used to calibrate the simulated MRS observations and derive the 3D spectral cubes (Sect. \[Met:pipeline\]). Finally, we extract the final 1D calibrated spectrum for each simulated MRS observation and calculate the emission line fluxes (Sect. \[Met:emi\_flux\]).
### Low-metallicity spectral templates {#Met:Templates}
[Note: Line ratios are normalized to the flux of \[OIII\]0.5007$\mu$m emission line.\
References: [@Izotov2011]$^{(a)}$ and [@Izotov2011]$^{(b)}$\
$^{(c)}$ Average of O/H values from II Zw 40, Mrk 71, Mrk 930, and Mrk 996. Their metallicity range is 7.85<12+$\log$(O/H)<8.10.\
$^{(d)}$The template TM\_0.02\_solar does not include emission lines redder than \[SII\] 0.6731$\mu$m because only optical spectral lines are available. The HeI1.087$\mu$m flux have been calculated from HeI 0.5876$\mu$m, using the line ratio (HeI 1.087$\mu$m/HeI 0.5876$\mu$m) derived using the templates TM\_0.2\_solar and TM\_0.02\_solar.]{}
{width="\hsize"}
The spectral templates consist of only rest-frame optical/near-IR emission lines (i.e. no stellar continuum included), where the line ratios are based on observed spectra of low-$z$, low-metallicity, dwarf galaxies (Figure \[fig:Templates\] and Table \[tab\_MRS\_templates\]). To cover the range of metallicities expected in EoR sources, three different templates are constructed according to metallicity: one low-metallicity ($\sim$0.2 Z$_{\odot}$) and two metal-poor (0.04 and 0.02 Z$_{\odot}$). For the low-metallicity template (METAL\_0.2\_SOLAR) the emission lines are taken as the average ratios derived for a sample of well-measured, low-metallicity (0.2 Z$_{\odot}$), dwarf galaxies [@Izotov2011]. For the metal-poor templates the spectra of the metal-deficient, blue compact dwarf SBS0335-052E (@Izotov2011, METAL\_0.04\_SOLAR), and the lowest metallicity J0811+4130 dwarf star-forming galaxy (@Izotov2018, METAL\_0.02\_SOLAR) are used. The selected values cover the wide range of metallicities derived for FIRSTLIGHT EoR galaxies [@Ceverino2019]. The spectral templates do not include any contribution from a low-luminosity active galactic nucleus (AGN). In the optical range the presence of an AGN increases the luminosity of the metallic lines relative to hydrogen, and therefore would help to detect the presence of an AGN [@Kewley2013].
The templates are further normalized in flux using the R\[OIII\], and the observed \[OIII\]88$\mu$m flux. The line ratios between the optical and far-infrared \[OIII\] lines have a well-known and strong dependency on the electron density and temperature of the ionized gas [@Dinerstein1985; @Keenan1990]. According to these authors, the R\[OIII\] ratio has a value of $\sim$3 to $\sim$15 for an electron density of 100 cm$^{-3}$, and electron temperatures in the (1$-$2) $\times$ 10$^4$ K range. For a given temperature in this range, the R(\[OIII\]) ratio would further increase with density by factors of up to 2 for electron densities of up to 1000 cm$^{-3}$.
Studies of a large representative sample of giant HII regions, nearby HII galaxies, and green peas covering the 8.5 to 7.2 (12+$\log$\[O/H\])[^4] metallicity range show a well-defined relation between the metallicity and the \[OIII\] electron temperature given by the expression [@Amorin2015] 12+log(O/H) = 9.22 - 0.89$\times$T$_e$(\[OIII\], in units of 10$^4$ K). According to this expression, the electron temperature of the \[OIII\] ionized gas ranges from 1.4 to 2.3 $\times$ 10$^4$ K for metallicities $\sim$8.4 to $\sim$7.2 (@Amorin2015). These temperatures agree well with those measured in a sample of high-ionization, metal-deficient, blue dwarf galaxies [@Thuan2005]. Blue dwarfs have \[OIII\] temperatures in the range of 1.3-1.4, 1.4-1.8, and 1.8-2.0 $\times$ 10$^4$ K for metallicities above 8.0, between 7.5 and 7.9, and below 7.5, respectively. On the other hand, the electron density derived from the \[SII\] doublet line ratio has values in the 110 to 1310 cm$^{-3}$ range, with an average of 410 cm$^{-3}$. These densities are similar to those measured in some of the most metal-deficient, low-$z$ galaxies known, such as SBS0335-052E [@Izotov2009], J0811+4730 [@Izotov2018], or A198691 [@Hirschauer2016]. In summary, the physical conditions of the \[OIII\] emitting gas in low-metallicity and metal-poor galaxies favour electron densities above 100 cm$^{-3}$, and temperatures well above 10$^4$ K, and closer to 2$\times$ 10$^4$K. Therefore, following the dependence of R(\[OIII\]) with temperature and density, the spectral templates are normalized in flux with two different R(\[OIII\]) ratios, R(\[OIII\]) = 6.5 for the low-metallicity template (i.e. 0.2 Z$_{\odot}$) and R(\[OIII\])= 10 for the metal-poor template (i.e. 0.04-0.02 Z$_{\odot}$).
The width of the emission lines in the templates is simulated by a Gaussian with a full width at half maximum (FWHM) of the \[OIII\]88$\mu$m flux measured in each galaxy (i.e. 154 km s$^{-1}$ for MACS1149-JD1, and 300 and 267 km s$^{-1}$ for the components of the B14-65666 system). Finally, the templates are normalized to the \[OIII\]0.5007$\mu$m flux derived from the \[OIII\]88$\mu$m flux, and redshifted to the corresponding observed wavelengths. Galaxies at redshifts above 6 show a steep UV continuum slope ($\beta < -2$, @Bouwens2016), in other words an optical extinction A$_V < 0.3$ mag, and therefore no internal extinction correction is applied to the line fluxes in the templates.
Finally, the UV-brightest sources at $z > 7$ [@Roberts-Borsani2016; @Stark2017] have continuum fluxes of 0.2-0.4 $\mu$Jy at 4.5$\mu$m. The 10$\sigma$ sensitivity for a 10ks observation with the MRS Channel 1 is $\sim$35-55 $\mu$Jy [@Glasse2015], depending of the wavelength. The continuum emission is well bellow the detection limit of the MRS in the exposure time used here. Then the templates only contain the main optical and near-IR emission lines in the H$\beta$ to Pa$\beta$ spectral range without continuum emission.
### MIRI instrument simulator: MRS raw observations {#MeT:MIRISim}
We use MIRISim (Klaassen et al. in prep.), public release 2.0.0,[^5] to perform simulated MRS observations of the EoR sources, MACS1149-JD1 and B14-65666. MIRISim is the MIRI instrument simulator able to reproduce realistic observations with the MRS and with other MIRI observational modes. It takes advantage of the full information collected during the cryogenic test and calibration campaigns of MIRI to simulate realistic point spread function (PSF), detector read noise, Poisson noise, dark current, detector non-linearity, flat-fielding, cosmic rays, fringing, and other observational and instrumental effects. MIRISim allows modelling of astronomical targets, combining SEDs and emission line information with different morphologies, and with user-provided astronomical images. It produces the raw uncalibrated data that are input into the MIRI JWST calibration pipeline to obtain the calibrated data cube.
The lensed galaxy detected at $z=9.11$ with ALMA, MACS1149-JD1, presents \[OIII\] 88$\mu$m observed flux of $3 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$ with a line width of 154 km s$^{-1}$ (FWHM, @Hashimoto2018b). Its strongest optical and near-IR lines (H$\beta$, \[OIII\] 0.4959,0.5007$\mu$m, H$\alpha$, and HeI 1.087$\mu$m) fall in MRS Channels 1 and 2.[^6] In order to investigate their detectability as a function of metallicity and electron temperature and density, we simulate three medium-deep (10 ks) MRS observations with different spectral templates and R\[OIII\] ratios (see Table \[tab\_MRS\_sim\] for details). We consider MACS1149-JD1 as an unresolved source for the MRS, and located in the centre of the Channel 1 field of view. The solar activity, which is related with the frequency of cosmic rays events, and the instrument and sky backgrounds are set to low. A four-point dither pattern is used to generate the MRS observations. Each of the dither pointings consists of 35 groups, three integrations, and one exposure in SLOW read-out mode, which gives 2.5 ks of integration per pointing, for a total of 10ks on-source integration time per MRS spectral setting[^7]. We note that a MRS spectral setting (SHORT, MEDIUM, or LONG) covers one-third of the available wavelength range in each channel; therefore, the three different spectral settings are needed for full spectral coverage. For MACS1149-JD1 simulations, we use two spectral settings, SHORT and LONG.
The interacting or merging system at redshift of $z=7.15$, B14-65666, is composed of two UV-bright sources with a projected separation of 2-4 kpc. The system presents a total integrated \[OIII\] 88$\mu$m flux of $21.8 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$. We simulated a deep (40 ks) Channel 1 and 2 MRS observation to investigate the possibility of detecting H$\alpha$ and other weak optical and near-IR emission lines (\[NII\]0.6583$\mu$m, \[SII\] 0.6716,0.6731$\mu$m, \[SIII\] 0.9069,0.9532$\mu$m, Paschen series). B14-65666 is simulated combining two unresolved sources with a separation of 1“. The full extension in \[OIII\]88$\mu$m is 0.84”, and the separation between clumps in rest-frame UV is around 0.5". Since the optimal deblending of two sources in the MRS observations is beyond the scope of this paper, the separation between clumps has been increased to reduce the confusion. B14-65666\_0.2\_solar is simulated at a redshift of 7.153, with \[OIII\] 88$\mu$m flux of $13.5 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$, R\[OIII\] of 6.5, and a line width of $\sim$325 km s$^{-1}$ (FWHM). B14-65666\_0.04\_solar is simulated at a redshift of 7.1482, with \[OIII\] 88$\mu$m flux of $8.3 \times 10^{-18}$ erg s$^{-1}$ cm$^{-2}$, R\[OIII\] of 10, and a line width of $\sim$267 km s$^{-1}$ (FWHM). We note that an offset in velocity between the two components has been included, as presented in [@Hashimoto2019]. A different spectral template and R\[OIII\] ratio is used for each component to analyse the detectability of the lines with different metallicity and physical conditions (see Table \[tab\_MRS\_sim\]). The solar activity and the instrument and sky background are set to low. An eight-point dither pattern is used to generate the MRS observations. Each of the dither pointings consists of 35 groups, three integrations, and two exposures in SLOW read-out mode, which gives 5 ks of integration per pointing, and a total of 40 ks on-source integration time per MRS spectral setting (SHORT, MEDIUM, and LONG).
Epoch of Reionization sources are expected to have sizes of less than 1 kpc [@Shibuya2019], and therefore are point-like sources for the MRS PSF[^8]; galaxies at lower redshifts would be larger in size, with a median radius of 2.2 kpc [@Ribeiro2016]. This would imply a dilution of the observed flux over a larger number of spaxels, and therefore would require these galaxies to be treated as extended sources with a specific light profile and clumpiness in the simulations.
Alternatively, the SED-fitting SFRs could be used to derive the H$\alpha$ emission [@Kennicutt1998]. MACS1149-JD1 and B14-65666 system have a intrinsic SFRs of 4.2 and 200 M$_{\odot}$ yr$^{-1}$ that is equivalent to observed H$\alpha$ fluxes of 8 and 67 $\times~10^{-18}$ erg s$^{-1}$ cm$^{-2}$, respectively. The predicted H$\alpha$ fluxes are in close agreement with the low-metallicity templates derived using the methodology presented in Sect. \[Met:Templates\].
$^{(1)}$ R\[OIII\] = F(\[OIII\]0.5007$\mu$m)/F(\[OIII\]88$\mu$m)\
$^{(2)}$ Flux given in units of $10^{-18}$ erg s$^{-1}$ cm$^{-2}$
### Calibration of MRS observations {#Met:pipeline}
The MACS1149-JD1 and B14-65666 MIRISim simulated MRS observations are calibrated with the JWST calibration pipeline (release 0.9.6).[^9] The pipeline is divided into three different processing stages. The first stage performs a detector-level correction, where the MRS observations are corrected for saturation, linearity, and dark current. It also applies the jump detection and ramp-fitting modules to transform the raw MRS ramps observations to slope detector products. We use a rejection threshold of 1.75$\sigma$ to identify the jumps between adjacent frames and correct the cosmic ray events. The selected rejection threshold is optimized to produce the best S/N on the final calibrated spectrum. The modification of the rejection threshold from 4$\sigma$ to 1.75$\sigma$ produces variations of S/N in factors of $\sim$1.5 and $\sim$1.25 for Channel 1 and 2, respectively. These variations are likely to be relevant during on-orbit operations as the solar activity, and therefore changes in the density and energy of cosmic rays, could have different residual effects in the calibrated data. The second stage corrects the slope products from flat-fields and fringes, assigns the coordinate system, and produces a photometric calibration at individual exposure levels. We note that the pipeline and MIRISim use the same reference file to simulate and calibrate the effect of the fringes. It could underestimate the fringe residuals in the final MRS simulated spectra, which are expected to be lower than 2%. The third stage combines the different dither exposures to create a 3D spectral cube. The final cubes have a spatial and spectral resolution of 0.196“ $\times$ 0.196” $\times$ 0.001 $\mu$m for Channel 1, and 0.196“ $\times$ 0.196” $\times$ 0.002 $\mu$m for Channel 2. Figure \[fig:2dimageMRS\] shows an example of the MRS calibrated 3D spectral cubes, and illustrates the integrated H$\alpha$ map of the simulated B14-65666 system (see detailed explanations and caveats in Sect. \[MeT:MIRISim\]).
![Simulated B14-65666 system. It illustrates the integrated H$\alpha$ map for a deep (40ks) MRS observations, where two components are simulated as point-like source separated by 1", and with metallicities of 0.2 and 0.04 Z$_{\odot}$.[]{data-label="fig:2dimageMRS"}](Figure_2d_v4.pdf){width="\hsize"}
### Extraction and analysis of 1D MRS spectra {#Met:emi_flux}
The 1D spectra are obtained by performing circular aperture photometry with a radius equal to the PSF FWHM (r $\sim$ 0.31“-0.42” depending on the Channel). The subtracted background is obtained in an annulus from 0.78“ to 1.37” centred in the source. An aperture correction is applied to obtain the final 1D calibrated spectra. The aperture correction is calculated by combining simulated bright point sources on MIRISim and the PSF model obtained during the test and calibration campaigns of MIRI. The aperture correction is calculated in each wavelength of the spectral cube, Channel 1 presents values from 1.59 to 1.69 and Channel 2 from 1.64 to 1.89.
The emission line fluxes are derived by fitting a single Gaussian model to the line profile. The fit is performed within a spectral range equal to 0.1$\mu$m and 0.14$\mu$m for Channel 1 and 2, respectively. If the defined spectral range contains more than one emission line, we use a multiple Gaussian model to simultaneously fit the different line profiles. To estimate the flux error of each emission line, we implement a Monte Carlo method. We measure the noise of the spectra as the root mean square (rms) of the residuals after subtracting the derived Gaussian profile. The noise is used to generate N ($N=3000$) new spectra, where a random Gaussian noise with a sigma equal to the rms is added to the original spectrum and the lines are again fitted. The error of the measurements is obtained as the standard deviation of the N derived fluxes. Tables \[tab\_fluxes\_MACS\] and \[tab\_fluxes\_B14\] contain the derived integrated fluxes and uncertainties for the optical and near-IR emission lines analysed in the simulated MACS1149-JD1 and B14-65666 MRS observations.
----------------------------- -------------- ------------- -------------- -------------- -------------
Simulated\_Spectrum$^{(a)}$ H$\beta$ \[OIII\] \[OIII\] H$\alpha$ HeI
0.4861 0.4959 0.5007 0.6563 1.087
MACS1149\_0.2\_solar 3.1$\pm$0.7 5.4$\pm$0.7 18.8$\pm$0.7 9.2$\pm$0.5 2.0$\pm$0.4
MACS1149\_0.04\_solar 9.1$\pm$0.7 9.5$\pm$0.7 28.6$\pm$0.7 25.0$\pm$0.5 2.2$\pm$0.4
MACS1149\_0.02\_solar 16.7$\pm$0.7 9.7$\pm$0.7 30.0$\pm$0.7 44.8$\pm$0.5 3.9$\pm$0.4
----------------------------- -------------- ------------- -------------- -------------- -------------
Note: The fluxes and noise for all emission lines and metallicities correspond to an exposure time of 10ks.\
$^{(a)}$ flux given in units of $10^{-18}$ erg s$^{-1}$ cm$^{-2}$
{width="\hsize"}
The absolute fluxes of the emission lines detected with high significance (S/N > 10) are in agreement with the input values with average deviations lower than 10%[^10]. The same emission lines, those with S/N > 10, are also used to investigate the S/N differences between the JWST exposure time calculator (ETC)[^11] and the MRS simulated observations based on the combination of MIRISim and JWST pipeline. The ETC provides mean S/N values of 25% and 6% lower than those derived from respectively the medium-deep (10ks) and deep (40ks) MRS simulated observations for Channel 1 and 2. As we note in Sect. \[Met:pipeline\], the tuning of the configuration parameters of the JWST pipeline produces variations in the S/N of the final 1D spectrum. This level of difference is expected as the ETC, MIRISim, and JWST pipelines approximate our current best knowledge and understanding of the performance of MIRI, and the remaining uncertainties associated with noise properties, cosmic ray effects, and pipeline processing are still under study, and will be revised with in orbit commissioning data.
Exploring the physical properties of EoR \[OIII\]88$\mu$m line emitters
-----------------------------------------------------------------------
The MRS simulated 1D spectra of the \[OIII\]-emitters MACS1149-JD1 and B14-65666 were analysed to investigate the detectability of their main optical and near-IR emission lines, and the prospects of inferring key physical properties such as the instantaneous star formation rates, ionization, Ly$\alpha$ escape fractions, shape and hardness of the ionizing continuum, metallicity, among other physical properties.
### EoR lensed sources: MACS1149-JD1 {#Res:MACS}
As already mentioned in Sect. \[Met:FullSim\] and Table \[tab\_gen\_prop\], MACS1149-JD1 is a lensed galaxy recently detected in \[OIII\]88$\mu$m at a redshift of 9.11. The intrinsic properties, SFR of 4.2 M$_{\odot}$ yr$^{-1}$, and sSFR of 4 Gyr$^{-1}$ place it in the upper SFR range of FIRSTLIGHT galaxies at a redshift of 9, but at the lower end of the sSFR as the total estimated stellar mass is 1.1$\times$ 10$^9$ M$_{\odot}$ [@Hashimoto2018b]. Figure \[fig:MACSspectra\] shows the simulated (10 ks) MRS spectra for a MACS1149-JD1-like source using three different metallicities (0.2, 0.04, and 0.02 Z$_{\odot}$) and R\[OIII\] values, covering the expected range of metallicities and excitation conditions in the ionized gas at a redshift of 9.11. The 1D extracted spectra containing the brightest optical emission lines (H$\beta$, \[OIII\]0.4959,0.5007$\mu$m, H$\alpha$, and HeI1.087$\mu$m) show the detection of all lines at a significance level higher than 4$\sigma$ at different metallicities. In particular, we obtain S/N $\sim$ 5-24, 8-42, 18-90, and 5-10 for the integrated fluxes of H$\beta$, \[OIII\]0.4959,0.5007$\mu$m, H$\alpha$, and HeI1.083$\mu$m emission lines, respectively.
Note: The fluxes and noise for all emission lines and metallicities correspond to an exposure time of 40ks.\
$^{(a)}$ flux given in units of $10^{-18}$ erg s$^{-1}$ cm$^{-2}$\
$^{(b)}$ 3$\sigma$ upper-limits.
{width="\hsize"}
Thanks to the additional magnification factor due to lensing, the MRS spectra illustrate important results for strong \[OIII\] 88$\mu$m line emitters at the highest redshifts (i.e. z $>9$). First, for a given \[OIII\] 88 $\mu$m luminosity, the \[OIII\]0.5007$\mu$m and H$\alpha$ lines will be most luminous for the lowest metallicity, and therefore would be detected with the highest significance at 0.02Z$_{\odot}$. This effect is mainly due to the expected increase in the electron temperature of the ionized gas, and therefore the R\[OIII\] decreases with metallicity from subsolar to metal poor (see Sect. \[Met:Templates\]). Second, additional detection of the H$\beta$ emission line provides the opportunity to set direct quantitative constraints in key physical aspects of these galaxies like the internal extinction (H$\alpha$/H$\beta$ ratio) and the total instantaneous star formation rate (H$\alpha$). Third, the detection of both H$\alpha$ and HeI1.083$\mu$m, the strongest HeI in the entire UV to near-IR spectral range, will provide unique information on the hardness of the ionizing source, even for the lowest metallicity sources. The ratio of ionizing photons can be derived as $$\dfrac{Nph~[>13.6eV]}{Nph~[>24.6eV]} = (0.89-2.10) \times \dfrac{L~[H\alpha]}{L~[HeI1.083\mu m]}$$ after extinction correction, and assuming emissivities for hydrogen [@Osterbrock1989book], and HeI [@Benjamin1999; @Porter2005] for electron densities of 100 cm$^{-3}$ and temperatures of 1-2 $\times$ 10$^4$ K, similar to those measured in low-metallicity, low-z galaxies [@Izotov2014]. However, as the HeI 1.083$\mu$m emissivity has a strong dependence with the electron density (factors 6 to 8 for densities in the $10^2 - 10^4$ cm$^{-3}$ range), a measure of the electron density (e.g. \[SII\]0.6717+0.6731$\mu$m optical lines or \[OIII\]52,88$\mu$m far-IR lines) is required to get an accurate value for the hardness of the ionizing source. For an instantaneous starburst and a given metallicity, the ratio of ionizing photons is a strong function of age with a drop in photons with energies above 24.6 eV relative to 13.6 eV for ages older than 6 Myr. Binaries [@Eldridge2017] could also be playing a relevant role in changing the hardness of the ionizing spectrum in these galaxies, in particular at low metallicities. The presence of a low-luminosity AGN could also produce an ionizing spectrum harder than the predicted from stars only.
In addition, if Ly$\alpha$ measurements are available, the MIRI-MRS H$\alpha$ observed flux and the H$\alpha$/H$\beta$ derived internal extinction measurement, will provide a measurement of the Ly$\alpha$ escaping fraction, $$F_{esc}[Ly\alpha]= \dfrac{F_{obs}[Ly\alpha]}{F_{int}[Ly\alpha]}=\dfrac{F_{obs}[Ly\alpha]}{R(Ly\alpha,H\alpha) \times F_{int}[H\alpha]},$$ where F$_{obs}$\[Ly$\alpha$\] is the observed Ly$\alpha$ flux (at $z>6$),[^12] F$_{int}$\[H$\alpha$\] is the intrinsic H$\alpha$ emission after correction for internal extinction, and R(Ly$\alpha$,H$\alpha$) is the theoretical recombination case B value assumed to be 8.7 for the typical electron densities (a few $\times$ 10$^{2}$ cm$^{-3}$ and temperatures $< 2 \times 10^4$ K). Likewise, as the H$\alpha$ line is the least affected by extinction of all the optical hydrogen recombination lines, it provides a more accurate estimate of the escape fraction of ionizing photons (e.g. @Matthee2017) when combined with the observed rest-frame $<$912$\AA$ photometry from existing HST or future NIRCam/JWST imaging.
Finally, the high S/N of \[OIII\]0.5007$\mu$m and H$\alpha$ emission lines open the possibility of detecting the presence of ionized gas outflows. Although beyond the scope of the present paper, preliminary simulations show that massive ionized outflows ($>$ 10$^7$ M$_{\odot}$, blueshifted by $\sim$300 kms$^{-1}$, and with terminal velocities of 650-700 km s$^{-1}$) could by traced by the H$\alpha$ line in metal-poor sources similar to MACS1149-JD1 (Colina et al. in prep.)
### UV-bright and massive EoR sources: B14-65666 {#Res:B14}
B14-65666, as already mentioned in Sect. \[Met:FullSim\] and Table \[tab\_gen\_prop\], is a strong \[OIII\]88$\mu$m line emitter at a redshift of 7.15 detected with ALMA also as a \[CII\]158$\mu$m source [@Hashimoto2019]. This UV-bright source (M$_{UV} \sim -22.3$), is identified with a system of two galaxies, likely interacting or merging. Its derived global properties with a total SFR of 200 M$_{\odot}$ yr$^{-1}$, a stellar mass of 7.7 $\times$ 10$^8$ M$_{\odot}$, a low visual extinction (A$_V$=0.3 mag), and sSFR of 259 Gyr$^{-1}$ place it among the most massive star-forming galaxies known at a redshift above 7, excluding quasi-stellar objects (QSOs). As such, it provides with an extraordinary opportunity for the detection of faint metallic lines (\[NII\]0.6584$\mu$m, \[SII\]0.6717,0.6731$\mu$m, and \[SIII\]0.9069,0.9532$\mu$m), and therefore establishes strong constraints on the metallicity of the ionized gas in addition to the physical properties already mentioned in Sect. \[Res:MACS\]. Figure \[fig:TDspectra\] shows the deep (40 ks) MRS simulated spectra of B14-65666 assuming, for the purpose of this simulation, that one of the components of the system has a metallicity of 0.2 Z$_{\odot}$ (upper panel), while the metallicity for the second component is 0.04 Z$_{\odot}$ (bottom panel). For the 0.2 Z$_{\odot}$ spectrum, the weak \[NII\]0.6584$\mu$m and \[SII\]0.6717,0.6731$\mu$m integrated emission lines are detected at about the 3$\sigma$ level, while the \[SIII\]0.9069,0.9532$\mu$m integrated lines are detected at 6$\sigma$ and 17$\sigma$, respectively. On the other hand, the 0.04 Z$_{\odot}$ spectrum shows no detection (at the 3$\sigma$ level) of \[NII\]0.6584$\mu$m or \[SII\]0.6717,0.6731$\mu$m, while the \[SIII\]0.9069,0.9532$\mu$m lines are detected at 3$\sigma$ and 7$\sigma$, respectively. Thus, the metallicity of luminous \[OIII\]88$\mu$m emitters detected by ALMA or other JWST instruments (e.g. NIRSpec) could be explored in full, using the standard R23, and all the different optical tracers as well, including the N2, S2, N2S2H$\alpha$, N2S2 ratios, as well as the combined O3N2, O3S2, and S23 ratios (see @Maiolino2019 for a review).
Conclusions {#Conc}
===========
This paper has presented a study of the H$\alpha$ fluxes predicted by state-of-the-art FIRSTLIGHT cosmological simulations for galaxies at redshifts of 6.5 to 10.5, covering the Epoch of Reionization, and of its detectability with the Medium Resolution Spectrograph (MRS) of the mid-IR Instrument (MIRI) on JWST. The paper has investigated the MRS detectability of the FIRSTLIGHT sources as a function of redshift, star formation rate, stellar mass, and specific star formation. In addition, it has presented realistic MRS simulated observation of the rest-frame optical and near-IR spectra of EoR sources recently detected by ALMA as \[OIII\]88$\mu$m emitters. These include the lensed source MACS1149-JD1 and the interacting-merger candidate B14-65666 at a redshift of 9.11 and 7.15, respectively. These simulations cover different metallicities and emission line ratios, and are based on medium-deep (10ks) and deep (40ks) MRS observations using the current versions of the MIRI instrument simulator (MIRISim), and of the official JWST calibration pipeline. The main conclusions are as follows:
1. All currently ALMA detected \[OIII\]88$\mu$m emitters at redshifts above 7 can be detected in the H$\alpha$ line with MRS spectroscopy in a few hours (10 ks) with a high significance (i.e. with S/N $>$ 5$\sigma$).
2. Deep integrations (40 ksec) with MRS will detect (at least at the 5$\sigma$ level) H$\alpha$ emission line in EoR sources at redshifts above 7 with a SFR above $\sim$ 2 M$_{\odot}$ yr$^{-1}$, stellar masses above $\sim$ 4-9 $\times$ 10$^{7}$ M$_{\odot}$, and specific star formation above 4 Gyr$^{-1}$. These limits cover the upper end of the SFR and stellar mass distribution at those redshifts, representing $\sim$ 6% and $\sim$1% of the predicted FIRSTLIGHT population in the 6.5-7.5 and 7.5-8.5 redshift ranges, respectively.
3. The FIRSTLIGHT population is dominated by numerous low-mass galaxies with faint H$\alpha$ emission for JWST spectroscopy, characterized by median values of $\widetilde{F} (\rm H\alpha) = 3.8$, $2.4$, and $1.5 \times 10^{-20}$ erg s$^{-1}$ cm$^{-2}$ in the redshift intervals of 6.5-7.5, 7.5-8.5, and $z > 8.5$, respectively. However, a fraction of galaxies show much higher fluxes around $F (\rm H\alpha) \sim 10^{-18}$ - $10^{-17}$ erg s$^{-1}$ cm$^{-2}$ and are accessible to observation with MIRI/JWST spectroscopy.
4. The MRS will provide a good S/N H$\beta$ (5-24$\sigma$) - H$\alpha$ (18-90$\sigma$) emission line spectra of sources similar to the MACS1149-JD1 at a redshift of 9.11 in exposures of a few hours ($\sim$ 10ks) for metallicity 0.2-0.02 Z$_{\odot}$. This example clearly illustrates the possibility of performing detailed studies of intrinsically bright or lensed sources, even at the beginning of the Epoch of Reionization.
5. The MRS will be able to establish and put strong limits on the metallicity of bright EoR sources, as demonstrated by the simulated B14-65666 system at 7.15 with metallicities 0.2 and 0.04 Z$_{\odot}$. This will be achieved by adding the optical metallicity tracers (N2, S2, N2S2H$\alpha,$ and N2S2) to the standard R23.
6. A measure of the hardness of the ionizing spectrum, Nph(>912$\AA$)/Nph(>504$\AA$), can be derived directly from the L(H$\alpha$)/L(HeI1.083$\mu$m) line ratio if the electron density is known. This measure of the hardness will constrain the nature of the ionization source, i.e. the age and IMF upper mass limit of the stellar population, or the presence of a low luminosity AGN.
As shown in this paper, the prospects of detecting the H$\alpha$ emission line with very high S/N (>50) at least in bright (intrinsic or lensed) sources at redshifts of 7 to 9, opens the opportunity of investigating the presence and properties of outflows of ionized gas in galaxies during the Epoch of Reionization.
The authors gratefully thank the Referee for the constructive comments and recommendations that helped to improve the quality of the paper, and the EC MIRI test team and MIRISim developers for providing a great and useful tool, the MIRI instrument simulator (MIRISim). The authors also acknowledge the STScI and the developer team of the official JWST calibration pipeline. This work was supported by the Spanish Ministry for Science, Innovation and Universities project number ESP2017-83197. D.C. acknowledges the Gauss Center for Supercomputing for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (Project ID: pr92za). D.C. is supported by the state of Baden-Württemberg through bwHPC. D.C. is a DAWN fellow. A.L. acknowledges funding from the Comunidad de Madrid, Spain, under Atracción de Talento Investigador Grant 2017-T1/TIC-5213. J.P.P. and T.V.T. acknowledge financial support from UK Space Agency grants. A.A.-H. acknowledges support from the Spanish Ministry of Science, Innovation and Universities through grants AYA2015-64346-C2-1-P and PGC2018-094671-B-I00, which were party funded by the FEDER program and from CSIC grant PIE201650E36. K.I.C. acknowledges funding from the European Research Council through the award of the Consolidator Grant ID 681627-BUILDUP.
[^1]: Data retrieval is available from <http://www.ita.uni-heidelberg.de/~ceverino/FirstLight/index.html>
[^2]: The sensitivity is roughly constant within the wavelength range covered by the MRS Channel 1, from 4.9 to 7.6 $\mu$m, i.e. H$\alpha$ redshifted to $6.5 < z < 10.5$ [see @Glasse2015].
[^3]: It is part of MIRICLE python environment (<http://www.miricle.org>)
[^4]: Throughout the paper a value of 8.69 is assumed for the solar abundance, as given by [@Asplund2009]
[^5]: Public and stable MIRISim releases are available at <http://miri.ster.kuleuven.be/bin/view/Public/MIRISim_Public>.
[^6]: MRS has wavelength ranges in Channel 1 (4.89 < $\lambda_{obs}$\[$\mu$m\] < 7.66) and Channel 2 (7.49 < $\lambda_{obs}$\[$\mu$m\] < 11.71), and its resolving power ranges are 2750 < $\lambda/\Delta\lambda$ < 3610.
[^7]: Information about wavelength coverage, spectral setting, spatial resolution, dithering pattern, detector read-out mode, and exposure time for the MRS can be found at <https://jwst-docs.stsci.edu/display/JTI/MIRI+Medium-Resolution+Spectroscopy>
[^8]: FWHM $\sim$ 0.31“-0.42” depending on the Channel; see @Wells2015 for an extensive explanation of the PSF dependence with the wavelength
[^9]: For more information about the JWST pipeline, see <https://jwst-docs.stsci.edu/jwst-data-reduction-pipeline>
[^10]: The absolute photometric calibration uncertainties reported by the MIRISim team are can be found at <http://miri.ster.kuleuven.be/bin/view/Public/MIRISimPublicReleases>
[^11]: <https://jwst.etc.stsci.edu/>
[^12]: F$_{obs}$\[Ly$\alpha$\] also depends on the intergalactic medium (IGM) transmission, so that F$_{obs}$\[Ly$\alpha$\]=F$_{em}$\[Ly$\alpha$\]$\times$T$^{\rm IGM}_{\rm Ly \alpha}$, where F$_{em}$\[Ly$\alpha$\] is the emitted Ly$\alpha$ flux and T$^{\rm IGM}_{\rm Ly \alpha}$ is the IGM transmission to Ly$\alpha$ photons (e.g. @Inoue2014)
|
---
abstract: 'In the presence of a strong uniform magnetic field, we study the influence of space noncommutativity on the electromagnetic waves propagating through a quasi-static homogeneous plasma. In this treatment, we have adopted a physical model which considers plasma as quasi-neutral single fluid. By using noncommutative Maxwell theory, the ideal magnetohydrodynamics (MHD) equations are established, in which new equilibrium conditions are extracted. As an empirical study, some attractive features of MHD waves behavior are investigated. Furthermore, it is shown that the presence of space noncommutativity enhances slightly the phase velocity of the incompressive shear Alfvén waves. In a compressible plasma, the noncommutativity plays the role of an additional compression on the medium, in which its relevant effect on the fast mode occurs for highly oblique branchs, while the low effect appears when the propagations are nearly parallel or anti-parallel. In addition, it turned out that the influence of space deformation on the slow modes is $\sim 10^{3}$ times smaller than that on the fast modes. The space noncommutativity effect on the slow waves is negligible in low plasma $\beta $ value, and could appear when $\beta $ is higher than $0.1,$ thus the extreme modification occurs for oblique slow waves propagating with angles between $30^{\circ }$ and $60^{\circ }$. Finally, we comment on the possible effect of such waves on CMB spectrum in photon-baryon plasma.'
author:
- 'S. Bourouaine[^1]'
- 'A. Benslama'
title: MHD waves within noncommutative Maxwell theory
---
Introduction:
=============
It is well known that magnetohydrodynamics (MHD) waves $\left[
1-3\right] $ play an important role in the field of space plasma. In astrophysics, it remained for a long time the preferred theory in the description of the dynamics of various astrophysical plasma systems such as the formation of the solar corona which is associated with the problem of the plasma heating and the solar wind acceleration $\left[ 4-6\right] $. Indeed, many examples could be given in the application of the MHD theory namely in space plasma, like as the study of the linear properties of the fast magnetosonic propagating in inhomogeneous plasma which is done by several authors $\left[ 7-9\right] $ in order to model these waves in coronal loop. T. K. Suzuki et al. $\left[ 10\right] $ proposed a collisionless plasma in which the damped fast MHD waves are responsible of the heating and acceleration of winds from rotating stars due to the observational evidence for locally strong magnetic fields in stellar atmospheres.
In cosmology, after the observation of the cosmic microwaves background (CMB) radiations in 1965, it is believed that MHD played a major role in shaping the radiation spectrum during the so-called plasma epoch. In fact, the evidence that the plasma was magnetized has been confirmed after the measurements of background magnetic fields which are of the order of $\mu $Gauss. Moroever, most of the theories predict that these magnetic fields are the amplified remnants of a seed cosmological magnetic field generated in the early Universe \[11\]. The presence of such field in the primordial plasma influences the acoustic waves pattern of the CMB anisotropy power spectrum $[12-14]$, e.g. Adams et al. $[13]$ argued that the primordial density fluctuations that are generated in inflationary Universe enter the horizon before the last photon scattering, and initiate magneto-acoustic oscillations in the photon-baryon plasma due to the presence of primordial magnetic fields. These oscillations distort the primordial spectrum of fluctuations and affect CMB anisotropy. Therefore, dealing with the features of MHD wave dynamics is a significant part of cosmological plasma, and because of the magnetic forces, the theory of MHD is more complicated and fascinating than hydrodynamics itself due to the influence of the magnetic field on the traditional sound waves which consequently transform to slow and fast magneto-acoustic waves. Furthermore, a new kind of wave appears, called Alfvén wave, which arises from magnetic tension and propagates along the field lines without disturbing the thermal pressure or density of the plasma.
On the other hand, there has been a large interest in the study of physical phenomena on a non commutative (NC) spacetime. The idea of spacetime noncommutativity is not new and was first discussed by Snyder in 1947 $[15].$ At this time the theory of renormalization was not yet well established and the goal was to introduce a natural cut-off to deal with infinities in quantum field theory. However this theory was plagued with several problems such as the violation of unitarity and causality which make people abandoning it. The appearance of such theory, baptized noncommutative geometry, as a limit of string theory has generated a revival of interest for this theory $[16,17]$.
In the framework of noncommutative geometry the position vector $x^{\mu }$ is promoted to an operator $\hat{x}^{\mu }$ satisfying the relation $$\left[ \hat{x}^{\mu },\hat{x}^{\nu }\right] =i\theta ^{\mu \nu },$$
where $\theta ^{\mu \nu }$ is a real, antisymmetric constant matrix which has the dimension of area with elements of order ($\Lambda
_{NC})^{-2}$ in system unit $(\hbar =c=1)$. $\Lambda _{NC}$ is the energy scale where the effects on the noncommutativity of spacetime will be relevant.
The role of $\theta ^{\mu \nu }$ can be compared to that of the Planck constant $\hbar $ which quantifies in quantum mechanics the level of noncommutativity between space and momentum. In Moyal algebra, the product of two arbitrary fields is defined by $\ast $ product (the star or Moyal product) $[18,19]:$$$(f\ast g)(x)=\left[ \exp (\frac{i}{2}\theta ^{\mu \nu }\partial
_{x_{\mu }}\partial _{y_{\nu }})f(x)g(y)\right] _{x=y}.$$
Some cosmological effects of noncommutativity have been studied. When spacetime is noncommutative on short distance scales, this may have an imprint on early Universe physics, and leads to an interesting consequence at a microscopic level. Indeed, it could be one of possible scenarios that may cause the generation of the density perturbations and primordial magnetic fields in the inflationary Universe $[20]$. Such studies allow to predict some bounds on noncommutativity scale $\Lambda _{NC}$ which may have a temperature dependence \[21\].
Our work is devoted to study the MHD waves by taking into account the space modification, and focusing on the description of a homogeneous plasma in a non-smooth space. The aim of this work is to make a theoretical background in the field of NCMHD waves and seek some future works to study this topic. Opening this new window inevitably leads to deal with the interactions of space deformation and plasma waves, which could be considered as a new experimental area for testing space noncommutativity contribution.
This paper is organized as follows. In the second section we start with a brief review of noncommutative classical electrodynamics, from which we derive the NCMaxwell equations. Then in the third section, and by assuming a small $\theta $ matrix, we establish the modified MHD equations to first $\theta -$parameter for a single conductor medium. As an application, in the fourth section we deal with a particularly interesting case of a high conductor $\left( \sigma \rightarrow \infty \right) $ fluid which is considered as plasma medium. New equilibrium conditions are deduced as well as the main attractive features related to the NCMHD waves are studied for a homogeneous plasma around an equilibrium state. Finally, the obtained results are discussed and compared with those known in usual space.
Noncommutative Maxwell equations:
=================================
It is understood that NC gauge field theories cause the violation of Lorentz invariance when $\theta $ is considered as a constant matrix, except if this matrix is promoted to a tensor related to the contracted Snyder’s Lie algebra $[22] .$ The problem of unitarity appears also with time-space noncommutativities ($\theta ^{0i}\neq
0$) $\left[ 9,10\right] $. In particular, NC Maxwell theory loses the causality due to the appearance of derivative couplings in the Lagrangian with the Lorentz invariance exhibited by plane wave solutions $[23].$
The free Maxwell action on noncommutative space is given by $$S=-\frac{1}{4}\int dx\hat{F}_{\mu \nu }\ast \hat{F}^{\mu \nu }$$where $\hat{F}_{\mu \nu }$ is the noncommutative strength field $$\hat{F}_{\mu \nu }=\partial _{\mu }\hat{A}_{\nu }-\partial _{\nu }\hat{A}%
_{\mu }-ie\left[ \hat{A}_{\mu },\hat{A}_{\nu }\right] _{\ast },$$where $e$ is the electric charge, and $\left[ \hat{A}_{\mu },\hat{A}_{\nu }%
\right] _{\ast }$ is the Moyal bracket defined as $$\left[ \hat{A}_{\mu },\hat{A}_{\nu }\right] _{\ast }=\hat{A}_{\mu }\ast \hat{%
A}_{\nu }-\hat{A}_{\nu }\ast \hat{A}_{\mu }.$$
According to the Seiberg-Witten map to the first $\theta $ order of the NC gauge and strength fields $[17] $ , we get$$\begin{aligned}
\hat{A}_{\mu } &=&A_{\mu }-\frac{e}{2}\theta ^{\alpha \beta
}A_{\alpha
}\left( \partial _{\beta }A_{\mu }+F_{\beta \mu }\right) \notag \\
\hat{F}_{\mu \nu } &=&F_{\mu \nu }+e\theta ^{\alpha \beta }F_{\mu
\alpha }F_{\nu \beta }-e\theta ^{\alpha \beta }A_{\alpha }\partial
_{\beta }F_{\mu \nu },\end{aligned}$$with $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu
}$ is the usual strength electromagnetic field and $A_{\mu }$ is the vector-potential.
Hence, from action $\left( 3\right) $ the Lagrangian in four-dimensional spacetime is$$\begin{aligned}
\mathfrak{L} &\mathit{=}&-\frac{1}{4}F_{\mu \nu
}^{2}+\frac{e}{8}\theta
^{\alpha \beta }F_{\alpha \beta }F_{\mu \nu }^{2} \notag \\
&&-\frac{e}{2}\theta ^{\alpha \beta }F_{\mu \alpha }F_{\nu \beta
}F^{\mu \nu }+\mathit{O}\left( \theta
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
\right) +A_{\mu }J^{\mu },\end{aligned}$$where we have added the external vector-current $J^{\mu }$ $=4\pi
(\rho _{q},\mathbf{j})$ and taken into consideration the integration of the term $A_{\mu }\ast J^{\mu }$ over the whole spacetime which leads to $%
A_{\mu }J^{\mu }$ due to the integral property of star product $[18]$.
By using the expressions of the electric field $E^{i}=F^{0i}$ and the magnetic induction field $B_{k}=\frac{1}{2}\epsilon
_{ijk}F^{ij}$ $(\epsilon ^{123}=1),$ and considering $e\theta
^{ij}=\epsilon ^{ijk}\theta _{k}$ with $\theta ^{0i}=0$ (space-space noncommutativity), we can extract a non-linear equation, one of the most important NC Maxwell equations in Gaussian units as follows $[24] $ $$\begin{aligned}
\text{\ }\mathbf{\nabla .}\mathcal{E} &=&4\pi \rho _{q} \\
\text{ }\frac{\partial }{\partial t}\mathcal{E}\mathbf{-\mathbf{\nabla }%
\wedge }\mathcal{H} &=&-4\pi \mathbf{j},\end{aligned}$$ where $\mathbf{j}$ is the current and $\rho _{q}$ is the charge density. Eq. $\left( 8\right) $ represents the modified Ampere’s law. The displacement $\mathcal{E}$ and magnetic $\mathcal{H}$ fields are also given by$$\begin{aligned}
\mathcal{E} &=&\mathbf{E+d}, \notag \\
\mathcal{H} &\mathbf{=}&\mathbf{B+h,} \notag\end{aligned}$$with$$\begin{aligned}
\mathbf{d} &=&\left( \mathbf{\theta }.\mathbf{B}\right)
\mathbf{E-}\left(
\mathbf{\theta }.\mathbf{E}\right) \mathbf{B-}\left( \mathbf{E}.\mathbf{B}%
\right) \mathbf{\theta } \notag \\
\mathbf{h} &\mathbf{=}&\left( \mathbf{\theta }.\mathbf{B}\right) \mathbf{B+}%
\left( \mathbf{\theta }.\mathbf{E}\right)
\mathbf{E-}\frac{1}{2}\left(
\mathbf{E%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}-\mathbf{B%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}\right) \mathbf{\theta .}\end{aligned}$$
The dual tensor $\tilde{F}_{\mu \nu }=\frac{1}{2}\epsilon _{\mu \nu
\alpha \beta }F^{\alpha \beta }$ ($\epsilon _{\mu \nu \alpha \beta
}$ is an antisymmetric tensor Levi-Civita) always satisfies the equation $\partial
_{\mu }\tilde{F}^{\mu \nu }=0$ which implies that$$\begin{aligned}
\frac{\partial }{\partial t}\mathbf{B+\mathbf{\nabla }\wedge E} &=&0 \\
\text{\ }\mathbf{\nabla .B} &=&0,\end{aligned}$$where the symbol $\mathbf{\wedge }$ denotes the vector product.
The choice of the matrix $\theta ^{\mu \nu }$means that we are dealing with a preferred frame in which the background electromagnetic field related to $\theta $-matrix is only reduced to a constant background magnetic field. As it was shown in several works $\ [25-27]$, $\theta $ space-space noncommutativity preserves the unitarity and is compatible with most works done on NC theories.
It turned out from the paper of Kruglov $[24]$ that the electromagnetic waves solutions of the linear equations of the classical electrodynamics are the solutions of the nonlinear wave propagation equations of the electromagnetic fields derived from NC Maxwell theory at $\mathbf{j}=0=\rho . $ Also, more features of the classical waves propagating have been discussed by Z. Guralnik et al. $[28]$. The authors deduced that the phase speed of these waves is different from $c$ (with small modification) in case of a transverse propagation with respect to the background magnetic field induction, while the parallel propagation propagation is unchanged. Furthermore, Y. Abe et al. $[29] $ studied a more general case of the electric-magnetic duality symmetry within noncommutative Maxwell theory, in which the polarizations of the propagating waves have been discussed. T. Mariz et. al $[30] $ gave a detailed study on the dispersion relation for plane waves in the presence of a constant background electromagnetic field, the authors did not find any restriction on the plane waves solution in the Seiberg-Witten approach of noncommutative gauge theory which is not the case in strictly Moyal approach, where they deduced that no plane waves are allowed when time is noncommutative.
In our study, we focus on the classical behavior of the electromagnetic waves propagating through a plasma with high conductivity in the presence of both, a magnetic field, and NC space. This treatment is based on the classical MHD theory which is worked out in the framework of NC Maxwell theory.
Noncommutative MHD equations:
=============================
Let us start with the continuity equation that describes the flow motion $$\frac{\partial \rho }{\partial t}+\mathbf{\nabla }.\left( \rho \mathbf{V}%
\right) =0$$with $\rho $ is the mass density of the medium, $\mathbf{V}$ its velocity. If this fluid has the ability to carry a density current $\mathbf{J}$ (conductor), then the rising Ampere force is $\mathbf{J\times B,}$ once the magnetic field $\mathbf{B}$ is present, and the plasma is a quasi-neutral in large scale greater than Debye length, consequently the electric volume force $\rho
_{q}\mathbf{E}$ vanishes, and the moment fluid equation becomes$$\rho \left( \frac{\partial \mathbf{V}}{\partial t}\mathbf{+}\left( \mathbf{V}%
.\nabla \right) \mathbf{V}\right) =-\nabla p+\mathbf{j\wedge B},$$with $p$ is the pressure which acts on the boundaries of the infinitesimal fluid volume.
In case of a medium with high conductivity $\left( \sigma
\rightarrow \infty \right) $ (ideal plasma), the known Ohm’s law for a moving conductor which describes the current $\mathbf{J}$ in terms of magnetic and electric fields is reduced to the following simple relationship between these fields$$(\mathbf{E+V\wedge B})=0\mathbf{.}$$
Since the quasi-neutrality is assumed, the equation $\nabla .\mathcal{E}%
\approx 0$ does not constitute a dynamical evolution equation. Also the equation $\nabla .\mathbf{B}=0$ is only a constraint, not an evolution equation since it does not include time derivative. In case of non-relativistic MHD approximation where the conducting fluid moves very slowly, this term is absent$\left( \frac{\partial
\mathbf{E}}{\partial t}\sim 0\right) $ which is the limit of slow motion and large scale spatial derivatives, the displacement current is always negligible, and from Eq. $\left( 8\right) ,$ we get $$\mathbf{j=}\frac{1}{4\pi }\nabla \mathbf{\wedge }\mathcal{H}\mathbf{\mathbf{-%
}}\frac{1}{4\pi }\frac{\partial \mathbf{d}}{\partial t}.$$In order to express the relations of $\mathbf{d}$ and $\mathbf{h}$ given in Eq. $\left( 9\right) $ in terms of $\mathbf{\theta }$, $\mathbf{B}$ and $\mathbf{V}$, we use Eq. $(14)$ to extract the following relationships up to the first order $\theta $$$\begin{aligned}
\mathbf{d} &=&\left( \mathbf{-}\left( \mathbf{\theta
}.\mathbf{B}\right)
\left( \mathbf{V\wedge B}\right) \mathbf{+}\left( \left( \mathbf{V\wedge B}%
\right) .\mathbf{\theta }\right) \mathbf{B}\right) \notag \\
\mathbf{h} &\mathbf{=}&\mathbf{\left( \mathbf{\theta }.\mathbf{B}\right) B+}%
\left( \left( \left( \mathbf{V\wedge B}\right) .\mathbf{\theta
}\right)
\left( \mathbf{V\wedge B}\right) \right. \notag \\
&&\left. \mathbf{-}\frac{1}{2}\left( \left( \mathbf{V\wedge B}\right) ^{2}-%
\mathbf{B%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}\right) \mathbf{\theta }\right) .\end{aligned}$$
To establish the NCMHD equations, it is useful to express the current $%
\mathbf{j}$ as functions of $\mathbf{\mathbf{B}}$ and $\mathbf{V.}$ By using Eqs. $\left( 14\right) $ and $\left( 10\right) \mathbf{,}$ we then obtain$$\frac{\partial \mathbf{B}}{\partial t}\mathbf{=}\nabla \mathbf{\mathbf{%
\wedge }}\left( \mathbf{V\wedge B}\right).$$
When $\mathbf{\theta }$ goes to zero, Eq. $\left(
13\right) $ tends to the usual known momentum equation.
Based on the NCMHD equations, we study the electromagnetic waves behavior using a linear mode analysis around the equilibrium. First, we should determine the equilibrium condition of the considered ideal plasma in the presence of noncommutativity. Plasma is said to be in the equilibrium if $\mathbf{V}=0$ and none of the variables depend on time, so from Eq. $\left(
13\right) ,$ we get$$\nabla p=\mathbf{j}\wedge \mathbf{B.}$$Using the relationships $\left( 15\right) $ and $\left( 16\right) $, we deduce that$$\begin{aligned}
\nabla p &=&-\nabla \left( \frac{\left( \frac{1}{3}\mathbf{\left( \mathbf{%
\theta }.\mathbf{B}\right) +}1\right) B^{2}}{4\pi }\right) \notag \\
&&+\frac{1}{4\pi }B(1+\mathbf{\left( \mathbf{\theta }.\mathbf{B}\right) })%
\mathbf{\nabla }B+\frac{B^{2}}{4\pi }\mathbf{\nabla \left( \mathbf{\theta }.%
\mathbf{B}\right) } \notag \\
&&\frac{1}{4\pi }\left( \mathbf{B.\nabla }\right) \left(
\mathbf{B+\left(
\mathbf{\theta }.\mathbf{B}\right) B+}\frac{\mathbf{B%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}}{2}\mathbf{\theta }\right) ,\end{aligned}$$where the first term represents the modified magnetic pressure and the rest terms are the modified magnetic tension. In the equilibrium conditions, the evolution equation of the magnetic field in the noncommutative space is different from that given in usual space. Certainly, the homogeneous case of constant $\mathbf{B}$ and $p$ are one of the possible solutions of Eqs. $\left( 19\right) .$ In the next treatment of the electromagnetic waves propagation through a plasma around the equilibrium, we will assume that the component of the background field along the direction of the magnetic field, which means that $\mathbf{%
\theta }$ is parallel to $\mathbf{B}$.
Noncommutative MHD waves
========================
It is well known that the waves play an important role in the propagative phenomena related to the plasma physics. The following treatment is mainly devoted to the description of the wave properties of plasma in noncommutative space within MHD approximation by a linear mode analysis. Notice that, the equations of ideal NCMHD $\left( 12\right) ,$ $%
\left( 13\right) $ and $\left( 17\right) $ are highly nonlinear and self-consistent.
Let us study an homogeneous plasma (in equilibrium state) under a constant magnetic field $\mathbf{B}_{0}$, in which all the parameters do not depend on coordinates, i.e. $\rho _{0}$ and $p_{0}$ are constant with $%
\mathbf{V}_{0}=0$. Hence, all quantities $Q\left( \rho
,p,\mathbf{V}\text{ and }\mathbf{B}\right) $ can thus be written as a sum of an equilibrium term and a small first-order perturbation$$Q\left( r,t\right) =Q_{0}\left( r\right) +\epsilon Q_{1}\left(
r,t\right) .$$Higher order terms describing the perturbations are neglected. We linearize the NCMHD equations by inserting the Fourier transformation of all quantities $$Q_{1}\left( r,t\right) =\int dk\tilde{Q}_{1}\exp i\left(
\mathbf{k.r-}\omega t\right) \text{ }$$into eqs. $\left( 12\right) ,$ $\left( 13\right) $ and $\left(
17\right) $. Then, we extract the terms of first order $\epsilon $ with neglecting all remaining terms of order $\epsilon
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
$ and higher as follows $$\begin{aligned}
\tilde{\rho}_{1} &=&\frac{\rho _{0}}{\omega }\mathbf{k.\tilde{V}}_{1}, \\
\omega \rho _{0}\mathbf{\tilde{V}}_{1} &=&\frac{v_{s}^{2}\rho _{0}}{\omega }%
\mathbf{k}\left( \mathbf{k.\tilde{V}}_{1}\right) -\frac{e^{\theta B_{0}}}{%
4\pi }\mathbf{C}\wedge \mathbf{B}_{0} \notag \\
&&-\frac{\left( \mathbf{B}_{0}.\mathbf{\tilde{B}}_{1}\right) }{4\pi
}\left(
\mathbf{k\wedge \theta }\right) \wedge \mathbf{B}_{0}, \\
\omega \mathbf{C} &=&-\left( \mathbf{k\wedge \tilde{V}}_{1}\right)
\left(
\mathbf{k.B}_{0}\right) +\left( \mathbf{k\wedge B}_{0}\right) \left( \mathbf{%
k.\tilde{V}}_{1}\right)\end{aligned}$$with $1+\theta B_{0}\approx e^{\theta B_{0}}$ and $\mathbf{C=k\wedge \tilde{B%
}}_{1}$.
Notice that we have considered the static magnetic field $\mathbf{B}_{0}$ is parallel to $\mathbf{\theta }$ which is proportional to the background magnetic field$.$ A polytropic adiabatic law which gives the relationship between the pressure $p$ and the density $\rho $ has been taken into account $$p=p_{0}\left( \frac{\rho }{\rho _{0}}\right) ^{\gamma }.$$
As a consequence, $p_{1}=v_{s}^{2}\rho _{1}$, from which, we derive the expression of the square of sound speed $v_{s}^{2}=\frac{\gamma
p_{0}}{\rho _{0}},$ with $\gamma $ is the adiabaticity index.
The equations $\left( 22\right) ,\left( 23\right) $ and $\left(
24\right) $ are a homogeneous set of 6 independent equations for 6 variables since on the other hand, we have the constraint $\mathbf{k.\tilde{B}}_{1}=0$ which indicates a transverse propagation of electromagnetic waves$.$ From Eq. $\left( 22\right)
$, we deduce that the density variations are only related to the velocity components along the wave vector, and similarly to the commutative case, the study of the waves in an incompressible plasma $\left( \rho _{1}=p_{1}=0\right) $ shows that the velocity motion is transversal with respect to the wave vector $\mathbf{k}$ and consequently, the fluctuating magnetic field $\mathbf{\tilde{B}}_{1}$ is perpendicular to $\mathbf{\mathbf{%
B}_{0}.}$
In order to analyze the various modes arising from the perturbed homogeneous plasma, it is worth focusing on the dispersion relations which connect the wave phase velocity $\mathbf{v}_{ph}=\frac{\omega }{k%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}\mathbf{k}$ with the measured quantities related to the plasma features themselves. By substituting the vector product of eq. $\left( 23\right) $ with $\mathbf{k}$ into Eq. $\left( 24\right) ,$ we get$$\begin{aligned}
&&\left( 1-\frac{e^{\theta B_{0}}\left( \mathbf{k.B}_{0}\right)
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}{4\pi \rho _{0}\omega ^{2}}\right) \mathbf{C-}\frac{1}{\omega
}\left( \mathbf{k\wedge B}_{0}\right) \left(
\mathbf{k.\tilde{V}}_{1}\right) \notag
\\
&=&\frac{e\left( \mathbf{k.B}_{0}\right)
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}{4\pi \rho _{0}\omega ^{2}}\left( \mathbf{B}_{0}.\mathbf{\tilde{B}}%
_{1}\right) \left( \mathbf{k\wedge \theta }\right) \text{ }\end{aligned}$$
At this stage, it is worth analyzing the Alfvén and magnetosonic waves propagating in incompressible and compressible plasmas, and compare them with those obtained in commutative space. Let us separately treat the two different states of plasma when the propagation is not purely sonic $\left( v_{ph}\neq v_{s}\right) $. Firstly, we consider the incompressible case of plasma, i.e. $\mathbf{B}_{0}.\mathbf{B}_{1}=0$ which leads to$$\mathbf{C\wedge }\left( \mathbf{k\wedge \mathbf{B}_{0}}\right) \neq
0.$$From the vector product of Eq. $\left( 26\right) $ with $\left( \mathbf{%
k\wedge \mathbf{B}_{0}}\right) $, we obtain the first dispersion relation for an incompressible plasma
$$\left( 1-\frac{e^{\theta B_{0}}\left( \mathbf{k.B}_{0}\right)
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}{4\pi \rho _{0}\omega ^{2}}\right) =0,$$
where we can get the norm of wave phase velocity in the presence of noncommutativity$$v_{ph}^{2}=\left( v_{A}^{nc}\right) ^{2}\cos ^{2}\alpha \text{ }$$with $\left( v_{A}^{nc}\right) ^{2}=e^{\theta B_{0}}v_{A}^{2}$ is the modified Alfvén velocity and $v_{A}^{2}=B_{0}^{2}/4\pi \rho
_{0}$ is the usual Alfvén velocity. Indeed, in case of a parallel propagation, the modified incompressible Alfvén waves can be deduced, $$v_{ph}^{2}=\left( v_{A}^{nc}\right) ^{2}.$$
Note that the Alfvén mode undergoes a modification coming from space deformation. This modification plays a role of another velocity which is added to the usual Alfvén velocity, consequently, enhances the value of this latter. In fact, the influence of the noncommutativity is considerable when we deal with an incompressible plasma emerged in a strong magnetic field $\mathbf{B}_{0}.$
{width="8cm" height="70mm"}
{width="8cm" height="70mm"}
Secondly, if the plasma is compressible, this means that $\mathbf{B}_{0}.%
\mathbf{B}_{1}\neq 0,$ therefore, by performing the scalar product of Eq. $%
\left( 26\right) $ with $\left( \mathbf{k\wedge
\mathbf{B}_{0}}\right) $, we obtain the second dispersion relation for a compressible plasma $$\begin{aligned}
&&\left( 1-\frac{e^{\theta B_{0}}\left( \mathbf{k.B}_{0}\right)
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}{4\pi \rho _{0}\omega ^{2}}\right) \left( 1-\frac{k%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
v_{s}%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}{\omega
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}\right) = \notag \\
&&\frac{1}{4\pi \rho _{0}\omega
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}\left( \mathbf{k\wedge \mathbf{B}_{0}}\right) ^{2}\left[ e^{\theta B_{0}}+%
\frac{2}{k%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}\left( \mathbf{k\wedge \theta }\right) .\left( \mathbf{k\wedge B}%
_{0}\right) \right] \notag \\
&&+\frac{\left( \mathbf{k.B}_{0}\right)
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}{4\pi \rho _{0}\omega ^{2}k%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}\left[ \left( \mathbf{k}\wedge \mathbf{\theta }\right) .\left( \mathbf{k}%
\wedge \mathbf{\mathbf{B}_{0}}\right) \right] \left( 1-\frac{k%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
v_{s}%
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}{\omega
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
}\right) ,\end{aligned}$$
from which, we extract the following fast and the slow modes$$\begin{aligned}
\left( v^{nc}\right) _{ph\left( \pm \right) }^{2}
&=&\frac{1}{2}\left[ \pm \left( \left( \left( v_{A}^{nc}\right)
^{2}\exp \left( \theta B_{0}\sin
^{2}\alpha \right) +v_{s}^{2}\right) ^{2}\right. \right. \notag \\
&&\left. -4\left( v_{A}^{nc}\right) ^{2}v_{s}^{2}\exp \left( \theta
B_{0}\sin ^{2}\alpha \right) \cos
%TCIMACRO{\U{b2}}%
%BeginExpansion
{{}^2}%
%EndExpansion
\alpha \right) ^{1/2} \notag \\
&&\left. +\left( \left( v_{A}^{nc}\right) ^{2}\exp \left( \theta
B_{0}\sin ^{2}\alpha \right) +v_{s}^{2}\right) \right].\end{aligned}$$
The signs $(+)$ and $(-)$ in eq. $\left( 31\right) $ indicate the fast and slow magnetosonic (or magneto-acoustic) waves which arise from the coupling between magnetic compression (Alfvénic) and medium compression (sonic). We note that the influence of noncommutativity on the magneto-acoustic waves involves again the modified Alfvén velocity $\left( 29\right) $.
Let us define the quantity $\Delta _{\pm }$ $$\Delta _{\pm }=\frac{\left( v^{nc}\right) _{ph\left( \pm \right)
}^{2}-v_{ph\left( \pm \right) }^{2}}{\theta B_{0}v_{A}^{2}},$$that corresponds to the degeneracy rate of the fast and slow modes $\left( v^{nc}\right) _{ph\left( \pm \right) }^{2}$ in the presence of space noncommutativity from the usual modes $v_{ph\left( \pm
\right) }^{2}$ normalized to $\theta B_{0}$ quantity and the square of the usual Alfvén velocity $v_{A}^{2}$. Notice that $v_{A}^{2}$ and $v_{ph\left( \pm \right) }^{2}$ are respectively $\left( v_{A}^{nc}\right) ^{2}$ and $\left( v^{nc}\right) _{ph\left(
\pm \right) }^{2}$ in the absence of noncommutativity ($\theta =0).$
By considering a small value of $\theta B_{0}=10^{-5}$ which is a consequence of a strong value of the mean magnetic field, we plot $\Delta _{+}$ and $\Delta _{-}$ respectively in fig. 3 and fig. 4 in function of the propagation angle $\alpha $ for a different values of plasma $\beta =\frac{v_{s}^{2}}{v_{A}^{2}}$.
It is well known that the strength of the value of the magnetic field plays the major role in making the effect of the space deformation more relevant on the waves in plasma medium when the noncommutativity scale is relatively high.
{width="8cm" height="70mm"}
{width="8cm" height="70mm"}
The plasma $\beta $ variation in fixed mean magnetic field $B_{0}$ becomes proportional to the pressure $p_{0}$ by the adiabatic index $\gamma ,$ then $%
\beta $ increases when the plasma is initially strongly compressed. However in our study, we consider that the magnetic pressure is dominant in such way that $\beta <1$.
According to the fig. 3, it turns out that in compressible plasma, the phase velocity related to the fast waves is slightly enhanced due to space noncommutativity. This modification which is represented by the rate $\Delta _{+}$, has a very slow dependence on plasma $\beta $ variation. This extreme modification on the fast which is around $\sim 2$ which occurs when the propagation is perpendicular ($\alpha =90%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$) with respect to the mean magnetic field, and becomes smaller with $\Delta _{+}\sim 1$ when the fast waves is nearly Alfvénic in parallel or anti-parallel. Also, the noncommutativity effect on the fast mode is proportional to the phase speed of the wave which varies as a function of the propagation angle. In contrast, from fig. 4, the quantity $\Delta _{-}$ that corresponds to modification rate of the slow mode has a high dependence on plasma $\beta $ variation. Although this rate is much smaller comparing with $\Delta _{+}$, the noncommutative effect could slightly appear when plasma $%
\beta $ is higher than $0.04$, which means that the noncommutativity affects the slow mode as long as the plasma is strongly compressed (high $p_{0}$). The extreme modification on slow mode occurs mainly in oblique propagations between $30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ and $60%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$, and as it is expected from fig.2, the noncommutativity effect vanishes for $\alpha =90%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ at which originally there is no perpendicular propagation of the slow mode. In addition, no modification appears on parallel and anti-parallel slow waves which correspond to the maximum phase velocity of this mode. This means that no proportionality between noncommutativity effect and the phase velocity of the slow wave.
It is important to discuss the interesting physical case when $\beta
\rightarrow 0$ , which means a total domination of the magnetic pressure on compressible plasma due to the strength of the mean constant magnetic field $%
B_{0}$ or the plasma is not relatively enough compressed . In fact, according to fig. 1, in the absence of noncommutativity ($\theta
=0)$, the fast mode converges to the usual Alfvén mode ($v_{+}\approx v_{A}$) for any propagation angle, but when $\theta
\neq 0$, it is clear from fig.3 that the effect of space noncommutativity does not change and is nearly the same one for $\beta \neq 0.$ Therefore, this effect plays a role of medium compression which rises the fast mode from the Alfvénic one. While in fig.2, as it is expected, the slow mode vanishes in case of $\beta =0$ which is equivalent to the incompressible plasma case.
Conclusion and discussion
=========================
In this letter, we have studied the propagative phenomenon of the electromagnetic waves in a homogeneous plasma under the effect of a magnetic field and space-space uncertainty. By using a physical model which considers plasma as a high conductor single fluid medium, the ideal NCMHD equations are established with the help of NC Maxwell theory. Because of a quasi-static motion of the fluid, we neglected the relativistic effect which is very small and it is below the order of $(\theta B_{0})^{2}$. In this treatment, the behavior of the electromagnetic waves propagating in this medium are studied in both cases, incompressible and compressible plasmas. It turned out that in an incompressible plasma, the Alfvén mode undergoes a modification which appears as a small additional velocity which enhances the value of the usual Alfvén velocity in commutative space.
On the other hand, it is deduced that the influence of space noncommutativity on the fast waves in a compressible plasma is proportional to the phase velocity. Moreover, the high oblique fast waves undergo a strong modification which is about $2$ times higher than the perturbed parameter $\theta B_{0}$, and a weak modification occurs in case of nearly parallel and anti-parallel propagation. For the slow mode, the influence of space noncommutativity is very small especially when $\beta $ is low than $0.04$. This effect could slightly appear when $\beta $ is beyond the value of $0.1$, at which, the extreme modification on the slow mode occurs in oblique propagations between $30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ and $60%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$, and vanishes in the parallel one. In addition, for $\beta
\rightarrow 0$ there is no influence of space noncommutativity on slow mode while this influence on the fast mode is nearly identical to that for $\beta \neq 0$ which is higher in perpendicular propagation.
The impact of space noncommutativity on cosmological MHD waves properties may lead to new consequences on the study of the influence of such waves on CMB temperature spectrum. Although the effect of the space noncommutativity may decrease after the inflationary universe due to the decreasing in energy scale, the presence of a primordial magnetic field excites any possible effect of space-space uncertainty on the primordial photon-baryon plasma before the last scattering. Any role of space noncommutativity at that time depends on its scale $\Lambda_{NC}$. Several scenarios based on space noncommutativity, aimed to explain the mechanism behind the generation of the primordial magnetic field, hence, possible constraints on $\Lambda_{NC}$ parameter have been involved. In fact, the possibility that $\Lambda_{NC}$ has a temperature dependence has been discussed in \[20, 21\], where the authors mentioned that the world is commutative at low temperature but becomes more noncommutative once the temperature is higher than a certain threshold temperature $T_{0}$. In ref. \[21\], an intensive discussion on the temperature dependence of $\theta$-parameter based on the constraints on the primordial magnetic field which is $B(T=10Mev)= 10^{-8} Gev^{2}$ at the beginning of nucleosynthesis (see ref. \[31\]), the authors argued that the presence of noncommutativity may be not so efficient beyond nucleosynthesis scale. However, its effect cannot be omitted due to high temperature especially when the radiations dominate the plasma. Hence, this drives our attention, that this effect can be treated as a perturbative correction in the study of the distortion of the primordial spectrum of fluctuations by MHD waves before the last scattering. As we have seen in this letter, the space noncommutativity influences the compressible oscillations depending on their propagation angle, on the other hand, the fact that the velocity of the fast waves depends on the propagation angle between the wave-number and the magnetic field, the CMB anisotropy would be affected $[13]$. This could reduce the impact of noncommutativity on some branches of the MHD waves, therefore, this leaves imprints on the primordial spectrum of the fluctuations anisotropy. Furthermore, such an influence may also correct the magnitude of the primordial magnetic field, this is because the noncommutativity can play a role of an additional magnetic field applied on the medium. In addition, dealing with the influence of the NCMHD waves on CMB radiations may provide a better estimation of the scale of $\theta$ parameter during plasma epoch. More details will be given in our future works.
[31]{}
E. R. Priest, Solar Magnetohydrodynamics (D. Reidel Publishing Company, 1982).
M. Goossens, An introduction to Plasma Astrophysics and Magnetohydrodynamics (Kluwer Academic Publishers, 2003).
V. M, Nakariakov, L. Ofman, E. E. Deluca, B. Roberts, and J. M. Davila, Science **285** (1999) 862.
J. A. McLaughlin and A. W. Hood, Astron. Astrophys. **420** (2004) 1129.
U. Narain, P. Ulmschneider, Space. Sci. Rev. **75** (1996) 453.
G. Haerendel, Nature. **360** (1992) 241.
P. S. Cally, Solar. Phys. **108** (1986)183.
E M. Edwin and B. Roberts, Astron. Astrophys. **192** (1988) 343.
V.M. Nakariakov and B. Roberts, Solar. Phys. **159** (1995) 399.
T. K. Suzuki, H. Yan, A. Lazarian and J. P. Cassinelli. Astrophys.J.** 640** (2006) 1005.
L. M. Widrow, Rev. Mod. Phys.**74** (2002) 775; M. Giovannini, Int. J. Mod. Phys. D **13** , (2004) 391.
T. Kahniashvili and B. Ratra, Phys. Rev. D **75**, (2007) 023002; M. Giovannini, Classical Quantum Gravity **23** , (2006) R1.
J. Adams, U. H. Danielsson, D, Grasso, and H. Rubinstein, Phys. Lett. B**388**, (1996) 253; D. Grasso and H. Rubinstein, Phys. Rep. **348**, (2001) 163.
D. G. Yamasaki, K. Ichiki, T. Kajino, and G. J. Mathews, Astrphys. J. **646**, (2006) 719; S. Koh and C. H. Lee, Phys. Rev. D **62**, (2000) 083509.
H. S. Synder, Phys. Rev. **71** (1947) 38.
A. Connes, M. R. Douglas and A. Schwarz, JHEP **9802** (1998) 003.
N. Seiberg and E. Witten, JHEP **9909** (1999) 032; arXiv: hep-th/9908142.
R. J. Szabo, Phys. Rep. **378** (2003) 207; arXiv: hep-th/0109162; M. R. Douglas and N. Nekrasov, Rev. Mod. Phys. **73** (2001) 977; arXiv: hep-th/0106048.
S. Bourouaine and A. Benslama, Mod. Phys. Lett. A**20** (2005) 1997; arXiv: hep-th/0507060; S. Bourouaine and A. Benslama, J. Phys. A: Math. Gen. **38** (2005) 7389.
C. Chu, B. Greene, and G. Shiu, Mod. Phys. Lett. A**16** (2001) 38; arXiv: hep-th/0011241; K. Bamba and J. Yokoyama, Phys. Rev. D **70** (2004) 083508; S. Tsujikawa and R. Maartens, Phys. Lett. B**574** (2003) 141.
A. Mazumdar and M. Sheikh-Jabbari, Phys. Rev. Lett. **87** (2001) 011301.
C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D**66** (2002) 075001; arXiv: hep-th/0206035.
H. Bozkaya, P. Fischer, H. Grosse, M. Pitschmann, V. Putz, M. Schweda and R. Wulkenhaar, Eur. Phys. J. C. **26** (2002) 139; arXiv: hep-th/0205153.
S. I Kruglov, Annales de la foundation Louis de Broglie, **27** (2002) 343; arXiv: hep-th/0110059.
J. Gomis and T. Mehen, Nucl. Phys. B**591** (2000) 265; arXiv: hep-th/0005129.
O. Aharony, J. Gomis and T. Mehen, JHEP **0009** (2000) 23; arXiv: hep-th/0006236.
C. Rim and J. H. Yee, Phys. Lett. B**574** (2003) 111; arXiv: hep-th/0205193.
Z. Guralnik, R. Jackiw, S.Y. Pi and A.P. Polychronakos, Phys. Lett. B**517** (2001) 450; arXiv: hep-th/0106044.
Y. Abe , R. Banerjee and I. Tsutsui, Phys. Lett. B**517** (2001) 450; arXiv: hep-th/0306272.
T. Mariz, J. R. Nascimento and V. O. Rivelles, Phys. Lett. B**517** (2001) 450; arXiv: hep-th/0609132.
D. Grasso and H. R. Rubinstein, Phys. Lett. B**379** (1996) 73.
[^1]: On leave of absence from Department of Physics, Faculty of Science, Mentouri University, Constantine, Algeria. Email: bourouaine@mps.mpg.de
|
---
abstract: 'We present a comparative study of different probabilistic forecasting techniques on the task of predicting the electrical load of secondary substations and cabinets located in a low voltage distribution grid, as well as their aggregated power profile. The methods are evaluated using standard KPIs for deterministic and probabilistic forecasts. We also compare the ability of different hierarchical techniques in improving the bottom level forecasters’ performances. Both the raw and cleaned datasets, including meteorological data, are made publicly available to provide a standard benchmark for evaluating forecasting algorithms for demand-side management applications.'
author:
-
-
- '\'
title: Hierarchical Demand Forecasting Benchmark for the Distribution Grid
---
Forecasting, benchmark, hierarchical forecasting, electric demand.
Introduction
============
The increasing monitoring capacity in low voltage (LV) and medium voltage (MV) distribution systems allows operators to gather power measurements from different levels of aggregation within the power grid. For instance, smart meters provide measurements from single households or buildings, dedicated power meters or phasor measurement units from secondary substations, and remote terminal units from primary substations at the interface between distribution and (sub)transmission systems. Real measurements of the power-flows “naturally” embed the notion of hierarchy. E.g., in a radial distribution system, the power flow at the grid connection point is, at the net of grid losses, the sum of the downstream elements. In the case of forecasts, however, the forecasted top-level series computed by using the information at that level of aggregation does not necessarily correspond to the sum of the bottom-level forecasts, thus invalidating the principle of hierarchy. The process of re-establishing coherency between upper and aggregated bottom-level predictions is called reconciliation.
In current power systems operational practices, forecasts of the demand for a given aggregation level are generally computed by using measurements from that same level. Computing a top-level forecast by aggregating series at the bottom level is generally not pursued because bottom-level measurements are affected by higher levels of volatility, that impact negatively on forecasting performance. Moreover, the separation of concerns between different grid operators and data ownership conflicts do not encourage the exchange of data and the use of reconciliation strategies.
However, future operational paradigms in active distribution networks will require tighter coupling between operations at different aggregation levels. The operator of an active distribution network will control distributed energy resources (i.e., demand response, storage, distributed renewable generation) to respect operational and physical constraints of the local power network (i.e., assure adequate voltage levels and respect line ampacity constraints) as well as providing ancillary services to the upper-level grid (i.e., dispatch, reserve, frequency control) through aggregation. In this context, the operator can take advantage of both disaggregated measurements and measurements at the grid connection point to compute coherent forecasts which satisfy the principle of aggregation to feed into optimal scheduling algorithms for the flexible resources.
In this paper, we first perform a comparison between different forecasters of the electrical demand. Then, based on the best performing method, we assess the effect of reconciliation on the forecasting performance. The analyses are carried out using data from an urban LV distribution network in Switzerland. The adopted data also includes numerical weather predictions (NWP) from a commercial provider. Compared to existing analyses and data sets in the literature that consider measurements at a 1-hour time resolution, e.g. [@Hong2014a] and [@Hong2019], we use a resolution of 10 minutes, that is more in-line with the targets for real-time market operations. To enable benchmarking with other algorithms, we make the data used for this research publicly available in a repository [@public_dataset].
The structure of the paper is as follows. Section II introduces the problem statement, Section III describes the adopted forecasting models, Section IV describes the tested reconciliation strategies, Section V presents and discusses results, and Section VI draws the conclusions.
Problem formulation and case study
==================================
Problem Statement
-----------------
To illustrate the problem, we consider the grid in Fig. 1. It is a radial system that interfaces five nodes to the grid connection point (GCP), which is connected to the upper-level grid through a transformer. Each node corresponds to a specific active power injection (e.g., demand or generation), which is measurable. In this paper, we assume that the grid losses are negligible, so the active power at the grid connection is the algebraic sum of the nodal injections. Based on the historical measurements, the operator can determine forecasts for all the nodes, including those at the GCP. While real power measurements will embed the hierarchical structure imposed by the grid topology (i.e., the power at the GCP will match the sum of individual nodes), forecasts will not as they are estimated individually. The problem that we tackle in this paper is how to forecasts the nodal injections individually, and secondly, how to reconcile them.
{width="0.5\columnwidth"}
Input data
----------
The input data consists of power measurements and meteorological forecasts relative to a set of power meters located in Rolle (Switzerland). The available measurements are thoroughly described in Appendix 1. In total, we consider 24 nodes, with an average power of 81 kW, and 7 synthetic aggregated series that are created by partial aggregations of the original data, as explained in section \[sec:hier\]. The complete dataset can be downloaded at [@public_dataset].
Forecast formulation {#sec:formulation}
--------------------
We compare the accuracy of different parametric and non-parametric forecasting techniques using k-fold cross-validation (CV). As the power generation gets more decentralized and uncertain due to the presence of renewable energy, system operators have moved from day-ahead optimization (as for standard unit commitment problem) to shorter clearing times, solving the optimization problem in a receding horizon fashion [@Zheng2015a].
For this reason, we tested the methods using the same sliding window concept. We applied a preliminary causal embedding of the explanatory variables and the target time series, which we explain in the following. Starting from the original time series $s\in \mathcal{S}$, a predictors (or regressors) matrix $X$ and a target matrix $Y$ are obtained. Given a dataset with $T$ observations, a prediction horizon of $h$ steps ahead, and an history embedding of $e$ steps, we obtain the Hankel matrix of targets $Y\in \rm I\!R^{ (T-h-e) \times h}$, and the Hankel matrix of the past regressors, $X_p\in \rm I\!R^{ (T-h-e) \times n_x e}$, where $n_x$ is the number of regressors. Verbosely, $X_p$ and $Y$ can be written as: $$X_p = \left[\begin{smallmatrix}
x_{1,t-e}&x_{1,t-e+1}&...&x_{1,t}&x_{2,t-e}&...&x_{n_x,t}\\
&&&...&&&\\
x_{1,t-e+1}&x_{1,t-e+2}&...&x_{1,t+1}&x_{2,t-e+1}&...&x_{n_x,t+1}\\
x_{1,T-2h}&x_{1,T-2h+1}&...&x_{1,T-h}&x_{2,T-2h}&...&x_{n_x,T-h}\\
\end{smallmatrix}\right]$$ $$Y = \left[\begin{smallmatrix}
y_{t+1}&y_{t+2}&...&y_{1,t+h}\\
&&&...&&&\\
y_{T-h+1}&y_{T-h+2}&...&y_{T}\\
\end{smallmatrix}\right]$$ where $x_{1,t}$ stands for the first regressor at time $t$. In hour case, we fixed $h=144$, corresponding to a prediction horizon of 24 hours ahead. The past regressors matrix $X_p$ is then augmented with categorical time features, e.g. day of week, and NWP variables, to obtain the final regressors matrix $X$. Rows of the $X$ and $Y$ matrices are then used to create the cross-validation training and testing dataset folds, $\{\left(D_{tr,f},D_{te,f}\right), f = 1,2,\dots,k \}$, where $k$ is the number of folds. Since we are dealing with time series forecasting, in order to avoid having very similar entries in some of the $D_{tr,f}$ and $D_{te,f}$ rows, we built them such that they are always separated by the embeddig length. The procedure we have used to build the different folds is the following. Each fold is divided into 10 days sequences, whose first 7 belong to the training set $D_{tr,f}$. Since for most of the regressors we have adopted a 24 hours embedding, for each sequence in $D_{tr,f}$ we discarded the 8-th and 10-th day, while the 9-th day is assigned to the testing dataset $D_{te,f}$.
An example of the division of a data sequence in training and testing days is shown in Fig. \[fig:cv\]. We then adopted a 10 fold CV ($k=10$), for each of which we shifted the start of the sequences by one day. In this way, by stacking the prediction of all the folds, it is possible to obtain forecasts for the whole period of the original dataset.
![Cross validation segment. Green squares: training days. Red square: test day. During testing, due to the adopted 24 hours embedding, the algorithms only see data contained in the 8-th day, avoiding overlapping of training and testing datasets.[]{data-label="fig:cv"}](CV2.pdf){width="2.5in"}
Forecasting models {#sec:methods}
==================
ARMAX
-----
ARMAX are state-full models, i.e. they require past values of both target variables and prediction error to perform a prediction for the next timestep, and have been largely applied to time series forecasting. The model is a regression where the covariates are a set of exogenous inputs $x \in \mathds{R}^{n_x}$, some past values of the target variable $y$, and the model error is assumed to be a white noise process $\epsilon$. The model can be written as: $$\phi(q) y = \beta x + \theta(q) \epsilon$$ where $q$ is the backshift operator, $\phi(q) = 1 - \sum_{i=1}^{n_p} \phi_i q $, $\theta(q) = 1 - \sum_{i=1}^{n_q} \theta_i q $ and $n_p$ and $n_q$ are the auto regressive and moving average orders of the model. Due to the stateful nature of the models, it is not possible to use all the observations in the CV data folds $D_{tr,f}$, since those are discontinuous. To overcome this issue, for each segment of 10 days (see Fig. \[fig:cv\]) of a given training data fold, we fit a different ARMAX model. The fitted models are then used to form an ensemble, i.e. the final prediction is given by: $$\hat{y}_t = \frac{1}{n_s} \sum_{i=1}^{n_s} \hat{y}_{t,i}$$ where $\hat{y}_{t,i}$ is the prediction of the $i$-th model and $n_s$ is the number of segments in the current CV data fold $D_{tr,f}$. The ensemble process invalidates the assumptions upon which the confidence interval of the predictions are usually derived. For this reason, we have obtained the prediction quantiles a-priori, for given combination of time of the day and step ahead. Denoting as $q_{\alpha_i}$ as the empirical $\alpha$-quantile: $$\label{eq:emp_quant}
\hat{q}_{\alpha_i,h,d} = q_{\alpha_i}(e_{h,d})$$ where $e_{h,d}$ is the set of training errors obtained on $D_{tr,f}$ related to the $h$-th step ahead and to the $d$-th step of the day. The optimal values of the autoregressive and moving average orders, $n_p$ and $n_q$, are obtained using the `autoarima` R package using random samples from the time series. The resulting values, respectively 6 and 5, were kept fixed during the CV. Note that the actual models’ parameters have still been properly fitted in CV; in our case, the ARMAX’s orders represent hyper-parameters of the overall model, and this procedure can be seen as fixing them to an ‘educated guess’.
Detrended Holt Winters
----------------------
The Holt-Winters (HW) model [@Holt2004] is a special class of the exponential smoothing [@Gardner1985], which consists of three smoothing equations, such that the final prediction is a combination of the level $a$, trend $b$ and seasonality $s$. We tested different flavors of the HW families and based on performance, we adopted a double seasonality additive HW: $$\label{eq:HW}
\begin{aligned}
\hat{y}_{t+h} &= (a_t +hb_t) +s_{1,t-p(1)+1+(h-1)\backslash p_1}+s_{2,t-p_2+1+(h-1)\backslash p_2}\\
a_t &= \alpha(y_t-s_{1,t-p_1}-s_{2,t-p_2}) + (1-\alpha)(a_{t-1} + b_{t-1})\\
b_t &= \beta(a_t - a_{t-1}) + (1-\beta)b_{t-1}\\
s_{1,t} &= \gamma_1(y_t-a_t-s(2,t-p_2)) + (1-\gamma_1)s_{1,t-p_1}\\
s_{2,t} &= \gamma_2(y_t-a_t-s(1,t-p_1)) + (1-\gamma_1)s_{2,t-p_2}\\
\end{aligned}$$ where $\alpha$, $\beta$, $\gamma_1$ and $\gamma_2$ are parameters to be learned from data, while $p_1=96$ and $p_2=672$ are the periods of the seasonalities, and $\backslash$ is the modulo operator. The values for $p_1$ and $p_2$ correspond to a daily and weekly period. The model , and HW in general, do not include exogenous inputs. Since quantities like external temperature and irradiance are important explanatory variables in load forecasting, we included them with an a-priori linear detrend, such that the new target is $y = y - X\beta_d $, where $X$ is a three column matrix containing $GHI$, $T$ and the unit vector (for the intercept), and $\beta_d$ is the vector of linear coefficients. Usually, a single set of $\alpha$, $\beta$, $\gamma_1$ and $\gamma_2$ values is fitted, and the prediction of each step ahead is obtained applying equations \[eq:HW\] recursively, as usually done for state-space systems. To increase the accuracy of the method, we instead fitted 144 sets of $\alpha$, $\beta$ and $\gamma$ parameters, based on the step ahead. As done for the ARMAX models, we used random samples from the bottom time series to fit these parameters. Due to the linear detrend we applied to the target, the fitted $\beta$ values were close to $0$ for all the steps ahead, and thus we decided to fix this parameter to $0$. Also, in this case, the prediction quantiles are obtained a priori, using equation \[eq:emp\_quant\].
K-nearest neighbors
-------------------
The K-nearest neighbours [@Hastie2009] regressor is based on a simple but effective technique. The method selects the most similar K points in the training set, based on the features at the given prediction time. A weighted average of the target value of the selected points is then used to obtain the final prediction. $$\hat{y}_t = \sum_{i=1}^k \omega_i y_i$$ where $\omega_i$ and $y_i$ are the weight and the target variable of the $i$-th neighbour, respectively. In our case, we have used the Euclidean distance as a similarity measure to select the neighbours, and the inverse distance as weights, as implemented in the `KNeighborsRegressor` class of `scikit-learn` Python package. Forecast quantiles are obtained estimating them from the distribution of the k nearest neighbours predictions. We adopted a multiple-input single-output (MISO) strategy, in which different models are trained for different steps ahead, for a total of 144 models for each fold.
Gradient Boosting
-----------------
Tree boosting is a widely used machine learning technique, both for classification and regression tasks. The method relies on repeatedly fitting regression trees on the residual of the predicted variable. In order to reduce overfitting, the well-known implementation of `XGBoost` [@Chen2016] includes a penalization on the number of parameters in the fitting process. In this comparison, we relied on the LightGBM implementation described in [@Ke2017], which is characterized by a highly parallelizable algorithm for the construction of the trees, tailored to big datasets. Also in this case, we adopted a MISO strategy.
Hierarchical forecasting {#sec:hier}
------------------------
Hierarchical forecasting aims to increase the accuracy of the prediction of signals organized in a hierarchical structure with increasing levels of aggregations, with respect to the case in which the aggregated signals are forecasted directly. Secondly, it aims at providing aggregate-consistent forecats, which can be obtained by encoding the hierarchical structure in a learning algorithm. This is done exploiting the forecasts of the bottom series $y_b \in \mathds{R}^{T \times n_b}$: usually an optimization technique is used to find a latent variable $\tilde{y}_b \in \mathds{R}^{T \times n_b}$, which can be used to approximate the whole set of original forecasts $y = \left[y_u^T ,y_b^T \right]^T \in \mathds{R}^{T \times n}$, where $n = n_b + n_u$ and $n_b$ and $n_u$ are the number of the bottom and upper time series. Formally, the following must hold for $\tilde{y}_b$: $$\tilde{y}^T = S \tilde{y}_b^T$$ where $S \in \mathds{R}^{n\times n_b}$ is a summation matrix and $\tilde{y}$ is the set of corrected forecasts. In this paper we have fictitiously aggregated the bottom time series in order to provide two levels of aggregation, such that $S$ is: $$S = \left[
\begin{matrix}
&\mathds{1}_{n_b}\\
&I_{2} \otimes \mathds{1}_{n_b/2}\\
&I_{4} \otimes \mathds{1}_{n_b/4}\\
&I_{n_b}
\end{matrix}
\right]$$ where $I_{k}$ is the identity matrix of dimension $k\times k$, $\mathds{1}_{k}$ is the unit raw vector of dimension $k$ and $\otimes$ is the Kronecker product. Following this approach, in [@Hyndman2011] the authors used ordinary least squares (OLS) regression to reconcile the forecasts in the hierarchy. Elaborating on this approach, [@Wickramasuriya2017a; @Wickramasuriya2018] proposed a trace minimization method (called minT) in which the covariance matrix of the forecasters error is estimated to perform a weighted least squares regression. In [@Taieb2017b], an elastic net penalization was proposed in order to induce sparseness in the forecasters adjustments, and the benefit was shown on the reconciliation of the forecasts for the power consumption of residential consumers. A probabilistic hierarchical reconciliation through empirical copulas is proposed in [@Taieb2017a]. Another probabilistic reconciliation approach has been recently proposed in [@Corani]: under the hypothesis of a joint Gaussian distribution for the base forecasts, this method exploits Bayes rule to obtain a closed-form solution to the probabilistic reconciliation.
Numerical results
=================
Evaluation KPIs
---------------
The results have been compared by means of standard key performance indicators (KPIs) for regression tasks. For the point forecast evaluation we have used the root mean squared error (RMSE) and the mean absolute percentage error (MAPE). The two aforementioned metrics have been evaluated using two levels of aggregation: we retrieved the expected value over the cross validation, as a function of the step ahead and hour of the day; secondly we have further aggregated the KPIs with respect to the hour of the day. Formally, we have evaluated:
$$\begin{aligned}
\mathrm{RMSE}_{d,h} &= \frac{1}{n_f}\sum_{f=1}^{n_f} \left( \frac{1}{\vert \mathcal{J}_{h,d,f} \vert} \sum_{j\in\mathcal{J}_{h,d,f}} e_{j}^2 \right)^{1/2}\label{eq:RMSE_map}\\
\mathrm{MAPE}_{d,h} &= \frac{100}{n_f}\sum_{f=1}^{n_f} \left( \frac{1}{\vert \mathcal{J}_{h,d,f} \vert} \sum_{j\in\mathcal{J}_{h,d,f}} \frac{abs(e_{j})}{y_{j}} \right) \label{eq:MAPE_map}\end{aligned}$$
where $\mathcal{J}_{h,d,f}$ is the set of observations relative to the $h$-th step ahead, $d$-th step of the day and $f$-th CV fold, $e$ is the forecast error, $n_f$ is the number of folds.
The resulting $\mathds{R}^{n_d, h}$ matrices are then normalized with the values of the same KPIs obtained using the persistence model. The probabilistic forecasts have been evaluated by means of continuous ranked probability score (CRPS) [@Gneiting2011], normalized with the values of the persistence model, and quantile score (QS), also known as pinball loss [@Koenker2006]: $$\begin{aligned}
\epsilon_{\alpha} &= \hat{q}_{\alpha} - y \\
QS_{\alpha} &= \epsilon_{\alpha} \left(\mathds{I}_{\epsilon_{\alpha}\geq 0} -\alpha \right)\\
CRPS &= \int_0^1 QS_{\alpha} \mathrm{d} \alpha \end{aligned}$$ where $\hat{q}_{\alpha}$ is the predicted $\alpha$-quantile, while $y$ is the observed ground truth.
Single time series forecasting
------------------------------
We performed day ahead forecasts for all the time series in the hierarchy previously described, applying the methods introduced in section \[sec:methods\] and following the CV approach introduced in section \[sec:formulation\]. An example of day-ahead forecasts for the whole aggregate is shown in Fig. \[fig:example\_ts\], along with eleven evenly spaced quantiles in the $\left[0.05,0.95\right]$ interval. From the picture, it can be noticed that the prediction of the ARMAX model is not centred in its quantile prediction during the central part of the day. This means that, during these hours, the model consistently underestimated the load in the training sets, and the empirical estimation of quantiles using equation reports this effect, which couldn’t be visible using the standard Gaussian process assumption.
![Example of day-ahead forecasts for the whole power aggregate, and for the different forecasters methods.[]{data-label="fig:example_ts"}](example_ts.pdf){width="3.5in"}
In Fig. \[fig:MAPE\_maps\] and \[fig:RMSE\_maps\] the normalized RMSE and MAPE matrices of equation and are reported for the tested forecasters. The regions for which the values exceed the unity, that is, where the persistence method achieves better performances, are enclosed in a violet line. For all the methods, we can see that the combination of step-ahead and step of the day close to the antidiagonal present the highest values of normalized KPIs. This means that in a time window of a few hours centred around midnight, the persistence method is already very accurate. The KNN and LightGBM models are strictly better than the persistence model for all the steps ahead and for all the times of prediction.
![Average MAPE from CV, normalized with the persistence forecaster MAPE, plotted as a function of day hour (vertical axis) and hour ahead time of the prediction (horizontal axis). The regions inside violet contours are the ones in which the persistence model has a better MAPE w.r.t. the considered forecaster (nMAPE$\geq1$) .[]{data-label="fig:MAPE_maps"}](mape_map.pdf){width="3.5in"}
![The same kind of plot of figure $\ref{fig:MAPE_maps}$ is shown, with RMSE values.[]{data-label="fig:RMSE_maps"}](rmse_map.pdf){width="3.5in"}
Fig. \[fig:nMAPEnRMSE\] shows the raw average of the normalized RMSE and MAPE matrices of equations and , i.e. the sample expectations of these KPIs with respect to the prediction horizon. The dashed lines represent the average over all the bottom series, while the continuous lines refer to the whole aggregate. It can be seen how, while all the methods are strictly better than the persistence model in the first few step-ahead, the ARMAX model rapidly worsen its performance, especially when considering the bottom series. On the contrary, the HW model shows better performances on the bottom series, while being strictly better than the persistence model in terms of RMSE for all the steps ahead. The KNN and LightGBM models are consistently better for all the prediction horizons for both the bottom and top series. However, we can see how the KNN method is not able to obtain low scores for the first prediction steps. This is because the KNN model does not include dynamics and is not able to discriminate the importance of the covariates based on the prediction step, while this is the case for LightGBM, being a tree-based model.
![nMAPE and nRMSE of different forecasters. Continuous lines: values refer to the top series (whole aggregate). Dashed lines: values refer to the average on the bottom time series.[]{data-label="fig:nMAPEnRMSE"}](nMAPE_nRMSE_ts.pdf){width="3.5in"}
![Top: average normalized $crps$ as a function of step ahead for different forecasters. Bottom: quantile scores as a function of the quantile for different forecasters.[]{data-label="fig:quantile_scores"}](quantile_scores.pdf){width="3.5in"}
In Fig. \[fig:quantile\_scores\] both the normalized CRPS as a function of step ahead and the quantile score as a function of the predicted quantile, are shown for the whole aggregate. The upper part of the figure shows that the HW method presents less reliable predicted quantiles after 10 hours ahead, while the other methods present a lower CRPS with respect to the persistence model, for all the prediction horizons. The lower part of the figure shows the QS mediated on all the prediction horizon. In this case, all the methods achieve better results compared to the persistence model. Once again, the ranking of the forecasters is unchanged, with the non-parametric models achieving better results. Table \[table2\] summarizes the average scores for the different forecasters, reporting the time-averaged MAPE and RMSE. The number on the left refers to the aggregated power profile, while the one on the right refers to the average score over the bottom time series. We can see that the best scores are always obtained using the LightGBM model.
[cc|c|c|c|c|l]{} & &\
& & armax & hw & knn & lgb\
& & 8.2 / 21.8 & 7.2 / 14.9 & 4.9 / 13.8 & **3.0** / **9.8**\
& & 67.1 / 6.1 & 60.4 / 4.7 & 46.2 / 4.4 & **26.2** / **3.1**\
Hierarchical reconciliation
---------------------------
For this analysis, we use the LGBM forecaster, that was the best performing model on the single time series under all KPIs, as discussed above. We retrieve the base forecasts for the whole dataset for all the 24 bottom series and the additional 7 aggregations using the CV method explained in section \[sec:formulation\]. As introduced in Section \[sec:hier\], we test the minT [@Wickramasuriya2017a] and Bayesian [@Corani] methods in combination with two different techniques for the estimation of the error covariance matrix, on which both the methods rely to obtain the reconciled time series. We tested both the Ledoit-Wolf shrinkage approach[@Ledoit2004] and the graphical Lasso method [@Friedman2008] using the implementation in the `scikit-learn` Python package. Figures \[fig:HR\_bottom\_boxplots\] and \[fig:reconciliation\_rRMSE\] show the relative reduction of RMSE compared to the base forecasts for the bottom and aggregated time series, respectively. The results are presented by means of temporal aggregations of 4 hours each, with respect to the step ahead. As it can be seen from Fig. \[fig:HR\_bottom\_boxplots\], on the one hand, the combination of minT with the shrunk covariance estimation score the worst performance and it even leads to increasing the RMSE on the bottom time series. On the other hand, minT with graphical Lasso covariance estimation provides the best results. The Bayesian reconciliation showed less sensitivity to the adopted covariance estimation method. However, the contribution of the reconciliation on the reduction of RMSE is marginal since it is lower than 1% in all cases.
As Fig. \[fig:reconciliation\_rRMSE\] shows, the reconciliation displayed a higher reduction on the RMSE on the aggregated series forecasts. Also in this case, minT with shrunk covariance scores the worst, whereas minT with graphical Lasso is the best performing model.
The average relative change of RMSE over all the time series, as well as for the whole aggregate, as a function of the step-ahead are shown in Fig. \[fig:reconciliation\_rRMSE\_ts\]. Once again, it is clear that the reconciliation affects the aggregated time series positively, while it has a lower impact on the bottom one.
![Boxplots of the RMSE reduction for the bottom time series, using different reconciliation techniques, as a function of step ahead. The values are normalized with the RMSE of the base forecasters, and aggregated using 4 hours bins. Positive values indicate an improvement.[]{data-label="fig:HR_bottom_boxplots"}](reconciliation_rRMSE_bottom.pdf){width="3.5in"}
![The same kind of plot of figure \[fig:HR\_bottom\_boxplots\], referred to the aggregated time series.[]{data-label="fig:reconciliation_rRMSE"}](reconciliation_rRMSE_aggregated.pdf){width="3.5in"}
![Relative RMSE reduction as a function of step ahead. Continuous lines: average value over the whole hierarchy. Dashed lines: top series (whole aggregate).[]{data-label="fig:reconciliation_rRMSE_ts"}](reconciliation_rRMSE_ts.pdf){width="3.5in"}
Conclusions
===========
We have discussed the performance of different forecasters and reconciliation methods in forecasting the active power demand at different levels of aggregation in an LV distribution network. We considered 24 time series (with an average power consumption of 81 kW) from a real distribution system, which were aggregated synthetically to form 7 series with two levels of aggregation. The forecaster adopted to the test the reconciliation performance was LightGBM, that, as presented in the paper, was selected as the best performing forecaster among Holt-Winters, ARMAX, KNN, and LightGBM. The reconciliation strategies that were tested are minT and Bayesian, each coupled with two methods for the estimation of the covariance matrix, i.e., graphical Lasso and the shrunk method, for a total of 4 models. Results showed that the best performing model using both upper and lower level measurements is minT with the graphical Lasso covariance estimation method. Upper-level forecasts showed the largest margin of improvements, with a reduction of the RMSE of up to 10%. Reconciliation marginally improved the forecasting performance for the lower time series, that improved by less than 1% on average.
Dataset
=======
The dataset consists of measurements coming from 62 IEC 61000-4-30 Class A power quality meters manufactured by DEPsys (Puidoux, Switzerland) installed in secondary substations and LV cabinets of the distribution grid of the city of Rolle (Basel, Switzerland). The dataset has been enriched with numerical weather predictions from commercial provider Meteoblue (Switzerland), updated every 12 hours. The series available in the data set along with their sampling time are reported in Table \[table1\]. The power measurements include mean active and reactive power, voltage magnitude and maximum total harmonic distortion (THD) for each phase, voltage frequency $\omega$ and the average power over the three phases, $P_{mean}$. The latter one has been used as target variable in this paper. The meteorological forecasts include the temperature, global horizontal and normal irradiance (GHI and GNI, respectively), the relative humidity (RH) pressure and wind speed and direction ($W_s$ and $W_{dir}$, respectively).
[cc|c|c|l]{} & &\
& & variables & sampling time\
& & $\begin{matrix}
P,Q, \vert V\vert, THD \ \mathrm{(each \ phase)} \\
\omega, P_{mean}
\end{matrix}$ & 10 min\
& & $\begin{matrix}
T,GHI,GNI &\\
RH,p,W_{s},W_{dir}&
\end{matrix}$ & 1 h, 12 h updates\
![Time periods where the original series are present. Series are ranked by the number of available data points in descending order. Color: logarithm of the ratio of missing values from the timestamp of the first available measurement, normalized with the highest value among all the series. The red cross indicates the selected time series.[]{data-label="fig:observations"}](observations.png){width="3.5in"}
Since the meters have been progressively installed in the grid (see Fig. \[fig:observations\]), in order to obtain a complete dataset, the meters with less than one year data have been discarded, as well as the meters presenting more than six consecutive missing values (corresponding to 1 hour). The remaining missing values were completed with PCHIP interpolation, which uses non-overshooting splines [@Fritsch2005]. Two of the selected meters presented a single sudden change of sign in the power measurements, which has been manually corrected. The final data set spans one entire year, with measurements from 13 January 2018 to 19 January 2019.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[ l@\#1 =l@\#1 \#2]{}]{}
T. Hong, P. Pinson, and S. Fan, “[Global energy forecasting competition 2012]{},” *International Journal of Forecasting*, vol. 30, no. 2, pp. 357–363, 2014.
T. Hong, J. Xie, and J. Black, “[Global energy forecasting competition 2017: Hierarchical probabilistic load forecasting]{},” *International Journal of Forecasting*, 2019.
“[https://zenodo.org/record/3463137[\#]{}.XY3GqvexWV4]{}.”
Q. P. Zheng, J. Wang, S. Member, and A. L. Liu, “[Stochastic Optimization for Unit Commitment — A Review]{},” *IEEE Transactions on Power Systems*, vol. 30, no. 4, pp. 1913–1924, 2015.
C. C. Holt, “[Forecasting seasonals and trends by exponentially weighted moving averages]{},” *International Journal of Forecasting*, 2004.
E. S. Gardner, “[Exponential smoothing: The state of the art]{},” *Journal of Forecasting*, 1985.
T. Hastie, R. Tibshirani, and J. Friedman, “[The Elements of Statistical Learning]{},” *Elements*, vol. 1, pp. 337–387, 2009.
T. Chen and C. Guestrin, “[XGBoost : Reliable Large-scale Tree Boosting System]{},” *arXiv*, 2016.
G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye, and T.-Y. Liu, “[LightGBM: A Highly Efficient Gradient Boosting Decision Tree]{},” *Nips ’17*, no. Nips, p. 9, 2017. \[Online\]. Available: <https://github.com/Microsoft/LightGBM.>
R. J. Hyndman, R. A. Ahmed, G. Athanasopoulos, and H. L. Shang, “[Optimal combination forecasts for hierarchical time series]{},” *Computational Statistics and Data Analysis*, 2011.
S. L. Wickramasuriya and G. Athanasopoulos, “[Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization]{},” *Journal of the American Statistical Association*, 2017.
S. L. Wickramasuriya, G. Athanasopoulos, and R. J. Hyndman, “[Forecasting hierarchical and grouped time series through trace minimization]{},” *Journal of the American Statistical Association*, no. November, 2018.
S. B. Taieb, R. Rajagopal, S. [Ben Taieb]{}, J. Yu, M. [Neves Barreto]{}, and R. Rajagopal, “[Regularization in Hierarchical Time Series Forecasting With Application to Electricity Smart Meter Data]{},” *Aaai*, no. 2011, pp. 4474–4480, 2017.
S. B. Taieb, J. W. Taylor, and R. J. Hyndman, “[Coherent Probabilistic Forecasts for Hierarchical Time Series]{},” *Proceedings of the 34th International Conference on Machine Learning*, vol. 70, no. April, pp. 3348–3357, 2017.
G. Corani, D. Azzimonti, and M. Zaffalon, “[Reconciling Hierarchical Forecasts via Bayes’ Rule]{},” *arXiv*, 2019.
T. Gneiting and R. Ranjan, “[Comparing density forecasts using thresholdand quantile-weighted scoring rules]{},” *Journal of Business and Economic Statistics*, 2011.
R. Koenker and G. Bassett, “[Regression Quantiles]{},” *Econometrica*, 2006.
O. Ledoit and M. Wolf, “[A well-conditioned estimator for large-dimensional covariance matrices]{},” *Journal of Multivariate Analysis*, 2004.
J. Friedman, T. Hastie, and R. Tibshirani, “[Sparse inverse covariance estimation with the graphical lasso]{},” *Biostatistics*, 2008.
F. N. Fritsch and R. E. Carlson, “[Monotone Piecewise Cubic Interpolation]{},” *SIAM Journal on Numerical Analysis*, 2005.
|
---
abstract: 'In a [*realistic*]{} application of the SPA + RPA theory for calculation of the nuclear level densities we find that quantal fluctuation corrections (RPA) are important even up to temperature $T = 2.0$ $MeV$. This leads to a good agreement between calculated numbers and the available experimental data for $^{104}Pd$ and $^{114}Sn$, particularly the excitation energy ($E^*$) dependence. Furthermore, we also argue that $a=S^2/4E^*$ is the only correct definition of the level density parameter in the present context which is also consistent with the Bethe like level density formula.'
---
=5.8 true in =8.5 true in
[**Level density and level density parameter in medium heavy**]{}\
[**nuclei including thermal and quantal fluctuation effects**]{}\
B. K. Agrawal and A. Ansari\
Institute of Physics, Bhubaneswar 751 005, India
In the last a few years there have been considerable efforts to develop microscopic methods for the calculation of accurate values of level densities as a function of excitation energies[@Chang; @anp66]-[@puddu; @anp206]. One of these methods which is of present interest is based on the auxiliary field path integral representation of the partition function for a given nuclear Hamiltonian. The path integral representation can be obtained in two ways: (a) the so called as shell model Monte Carlo (SMMC) method and (b) the SPA$+$RPA approach[@puddu; @anp206] which includes the thermal fluctuations through static path approximation (SPA) and the quantal fluctuations about static paths are included using random phase approximaton (RPA). It is now well established that SMMC approach can be applied even at very low temperatures. On the otherhand, SPA$+$RPA approach is computationally faster than SMMC approach and can be used for moderately low to high temperatures, typically, for $T \, \geq\, 0.2$ $MeV$. However, in most of these studies so far the main emphasis has been to demonstrate, through simple model studies, the relative defferences between the SPA and SPA$+$RPA results for the temperature dependence of the energy and the level density.
Recently[@drc; @prc], the level densities as well as the level density parameters have been extracted for the medium mass nuclei $^{104}Pd$ through the measurement of proton yields in reaction $^{93}Nb(^{12}C,p)^{104}Pd$ and that of $^{114}Sn$ through $^{103}Rh(^{12}C,p)^{114}Sn$ reaction. These data are available for the excitation energy ($E^*$) ranging from $5 - 25$ $MeV$ or equivalently $T\simeq 0.5 - 1.5$ $MeV$ which is well suited to test the feasibility of the SPA$+$RPA approach. In this letter we calculate the level density and the level density parameter as a function of $E^*$ for $^{104}Pd$ and $^{114}Sn$ using SPA$+$RPA approach with a quadrupole-quadrupole interaction model Hamiltonian $$H=H_0-\frac{1}{2}\chi\sum_{\mu=-2}^{2}(Q_\mu)^2.
\label{H}$$ In the above, $H_0$ represents the spherical part, $Q_0=
Q'_0$, $Q_{+\mu}=\frac{1}{\sqrt {2}} (Q'_\mu+Q'^\dagger_\mu)$ and $Q_{-\mu}=\frac{i}{\sqrt{2}}(Q'_\mu-Q'^\dagger_\mu)$ with $\mu =$ 1 and 2 and $Q's$ stand for the usual quadrupole moment operators. Value of the quadrupole interaction strength $\chi=120 A^{-5/3} f_c$ $MeV$ ($A$ denotes the mass number) is taken from Ref. [@aberg; @plb] where $f_c = 1 - 2 $ is a core polarization factor.
The grand canonical partition function in SPA$+$RPA takes the following form [@Lauritzen; @plb246] $$\begin{aligned}
{\cal Z}_{RPA}=4\pi^2\left (\frac{\alpha}{2\pi T}\right )^{5/2}
\int \beta^4 d\beta \int \mid sin 3\gamma\mid d\gamma
e^{-\frac{\alpha\beta^2}{2T}}\nonumber\\
\times Tr\left [ e^{-H'/T} \right ] {\cal C}_{\scriptstyle {RPA}}.
\label{zrpa}\end{aligned}$$ where, $\alpha=(\hbar\omega_0)^2/\chi$ with $\hbar\omega_0=41 A^{-1/3}$ $MeV$. The quantities $H'$ and ${\cal C}_{
\scriptstyle {RPA}}$ are the single-particle Hamiltonian and the RPA correction factor, respectively, given by $$H'=H_0-\hbar\omega_0 \beta \left (Q_0\,cos\gamma + Q_{+2}\, sin\gamma
\right )
\label{H'}$$ and $${\cal C}_{\scriptstyle {RPA}}=\left (\prod_{m\not = 0}^{N_m}
Det \mid C^m\mid \right ]^{-1}
\label{crpa}$$ with $$\label{cm}
C^m_{\mu\nu}=\delta_{\mu\nu} + \chi\sum_{ij}
\frac{\langle i\mid
Q_\mu\mid j\rangle \langle j\mid Q_\nu \mid i\rangle}
{\Delta_{ij}^2 + (2\pi m T)^2}
f_{ij}\Delta_{ij}$$ where, $f_{ij}=f_i-f_j$ and $\Delta_{ij}=\epsilon_i-\epsilon_j$ with $f_i$ being the Fermi distribution function and $\epsilon_i$ is the eigenvalue of $H'$. In the above, $\mid i \rangle$ represents an eigenstate of $H'$ . The grand canonical trace in eq. (\[zrpa\]) can simply be performed using $$\begin{aligned}
Tr e^{-\beta' H'}=\left (
\prod_i[1+e^{-\beta'\epsilon_i+\alpha_p}]\right )
\left (\prod_j[1+e^{-\beta'\epsilon_j+\alpha_n}]\right )\end{aligned}$$ where, $\beta'=1/T$ and $\alpha_p(\alpha_n)$ is the Lagrange multiplier required to adjust the proton(neutron) numbers.
The SPA representation of the partition function can be obtained by putting ${\cal C}_{\scriptstyle {RPA}} = 1$. It is, therefore, clear from eqs. (\[crpa\],\[cm\]) that for higher temperatures ${\cal
C}_{\scriptstyle {RPA}}
\longrightarrow 1$ or in other words $Z_{\scriptstyle {RPA}}
\longrightarrow Z_{\scriptstyle{SPA}}$.
Once the partition function is known, the level densiy or more precisely the state density $W(E)$ can be calculated using saddle point approximation which gives $$W(E)=\frac{e^{S}}{(2\pi)^{3/2} {\cal D}^{1/2}}
\label{rhoe}$$ where, $$S=lnZ+\beta' E - \alpha_p N_p - \alpha_n N_n
\label{s}$$ is the entropy with ${\cal Z}$ being the grand canonical partition function in the SPA or SPA+RPA approach. The quantities $\alpha_{p,n}$ and $\beta'$ (or $T^{-1}$) are so chosen that the saddle point conditions $$E = - \frac{\partial}{\partial \beta'}ln{\cal Z}
\label{et}$$ and $$N_{p,n}=\frac{\partial}{\partial\alpha_{p,n}}ln{\cal Z}$$ are satisfied. The quantity ${\cal D}$ in eq. (7) is a determinant of $3\times 3$ matrix defined by the elements $$d_{ij}=\frac{\partial^2}{\partial x_i\partial x_j}
ln {\cal Z}$$ where, $x\equiv (\beta',-\alpha_p,-\alpha_n)$ and the second derivatives are evaluated at saddle points. For the sake of comparison with the experiment we present below the results for $\rho(E)$, instead of $W(E)$, which can be obtained as[@Gilbert; @cjp], $$\rho(E)=\frac{W(E)}{\sqrt{2\pi\sigma^2}}$$ where, $\sigma^2=I^{rig}/\hbar^2$ is the spin cut-off factor with $I^{rig}$ being the rigid - body value of the moment of inertia.
The model space used to perform the numerical calculations is given in Table 1. For the range of temperature $T\le 2.0$ $MeV$ this basis space should be adequate. The choice of the values of the spherical single particle energies is a rather difficult task. Due to the quantal nature of the nucleus there is no smooth $A$-dependence for a large range of $A$. From our experience in the $pf$-shell and rare earth region we have finally chosen these numbers with the help of Fig. 1 (suitable for $A\approx 100$ ) in Ref. [@Skalski; @npa617] and Fig. 1 (suitable for $^{108}Sn$) in Ref. [@Wads; @npa559] which are actually obtained as solutions of Woods-Saxon potential. The neutron core with N=40 ensures sufficient number of active valance neutrons. The value of the core polarization factor, $f_c$, is taken to be 1.5 . With this the value of the quadrupole deformation parameter, $\beta$, in the ground state of $^{104}Pd$ comes out to be about 0.1 which is quite reasonable [@dan; @npa557]. $^{114}Sn$ is spherical in the ground state. The pairing correlations are also expected to be small for these nuclei.
For the sake of compactness most of our results and discussions will be presented for $^{104}Pd$ only. For $^{114}Sn$ results on level densities will be presented towards the end , before conclusions. In Fig. 1 we display the SPA as well as SPA$+$RPA results for the variation of energy as a function of temperature for $^{104}Pd$ . To obtain $E(T)$ within the SPA$+$RPA we first of all check the convergence of the RPA correction factor $\cal {C}_{\scriptscriptstyle {RPA}}$ by choosing various values of $N_m$ in eq. (4). We find that $N_m = 40 $ is sufficient but we have used $N_m=80$ in the present calculation so that at very low temperatures it should be sufficiently accurate. We see that, as in Refs. , the RPA or quantal fluctuation corrections lower the value of $E(T)$ at lower temperatures. As temperature increases, quantal fluctuation decreases or $\cal {C}_{\scriptscriptstyle {RPA}}\rightarrow 1$ yielding the value of $E(T)$ close to the one obtained within SPA. However, it shows that the RPA corrections are important up to about $T=2$ MeV. We then next show in Fig. 2 the variation of the level density as a function of $E^*$. The values of $E(0)$ needed to calculate $E^*$ are obtained in the case of SPA as well as SPA$+$RPA by extrapolating the corresponding curves of Fig. 1 to $T=0$. These values come out to be equal to $-310.603$ and $-315.357$ $MeV$ in the case of SPA and SPA$+$RPA, respectively. We see that SPA$+$RPA results for the level density are in good agreement with the ones extracted recently [@drc; @prc] through the measurements of proton yields in the reaction $^{93}Nb(^{12}C, p) ^{104}Pd$. On the other hand, SPA level densities are higher compared to the measured values. For instance, the ratio $\rho_{\scriptscriptstyle{SPA}}/\rho_{exp}\approx 10^2$ at $E^*=10$ $MeV$ which reduces to about 10 at $E^*= 24$ $MeV$. Whereas, $\rho_{\scriptscriptstyle{SPA+RPA}}/\rho_{exp}$ varies from 1.0 $-$ 1.2 over the entire range of $E^*$ for which experimental data are available. It should be mentioned that we have not put the error bars on the experimental data. The experimental data have errors of the order of 20 - 30$\%$ at the lower end and of a few percent at the other end.
Next, we now discuss about the parameterization of the SPA$+$RPA level densities in order to extract a value of the level density parameter, $a$. We have shown in our earlier work[@Agrawal; @plb339] that SPA level density can be reproduced by Bethe’s formula provided an appropriate value of the parameter ’$a$’ is used. Normally there are two relations which are used to compute the value of $a$, namely, $$E^*=a_e T^2
\label{ae}$$ and $$S=2 a_s T
\label{as}$$ where, the suffices $e$ and $s$ are used to distinguish the value of $'a'$ determined using $E^*$ or $S$, respectively. Treating $a_e$ and $a_s$ equal, usually a third relation $a=S^2/4E^*$ is also derived. However, in reality there is no rigorous reason to do so. Usually all the three relations yield different values of $a$ when computed numerically. Hence, it becomes a natural question to ask as to which ’$a$’ is appropriate for the use in Bethe’s formula for the level densities[@Gilbert; @cjp]:
$$\rho(E)=\frac{\sqrt{\pi}}{12}\frac{e^{2\sqrt{aE^*}}}
{a^{1/4}(E^*)^{5/4}} \frac{1}{\sqrt{2\pi\sigma^2}}.
\label{rb}$$
It is important to realise from eqs. (7 - 9) that the constant part of the prefactor[@Lau; @prc39] in eq. (2) which normalizes the ’measure’ does not contribute to the calculation of $E^*$ but certainly contributes to the value of $S$. In any mean field approach the effect of this prefactor would be missing. Now using eqs. (\[ae\]) and (\[as\]) we can write $$S=2\sqrt{\frac{a_s^2 E^*}{a_e}}=2\sqrt{aE^*}
\label{aes}$$ where, $$a=a_s^2/a_e = S^2/4E^*
\label{aef}$$ We find that the values of $a$ required in eq. (\[rb\]) to get $\rho_{\scriptscriptstyle{SPA+RPA}}$ is quite close to the one given by eq. (\[aef\]). For instance, at $T = $ 0.5, 1.0, 1.5 and 2.0 $MeV$ we get $a (a_{fit})$ = 10.46(11.04), 12.92(13.12), 12.42(12.49), 11.89(11.94), respectively. In Fig. 3 we have displayed the variation of the inverse level density parameter ($K=A/a$) as a function of the excitation energy per particle, $\epsilon = E^* / A$ [@Shlomo; @prc44] which correspond roughly to $T = $ 0 - 2.0 MeV. The values of $K_e$, $K_s$ and $K_{es}$ are obtained using $'a'$ calculated from eqs. (\[ae\]), (\[as\]) and (\[aef\]), respectively. We see that in comparison to $K_e$ and $K_s$ the values of $K_{es}$ are increasing very slowly with $\epsilon$ or temperature for $T \ge 1.0$ $MeV$. Also, we would like to point out that the values of $K_{es}$ are quite close to the experimental values [@drc; @prc] available for $E^* = 16$ $-$ $22$ $MeV$. Behaviour of $K$ in the very low temperature region($T \leq 0.5 MeV$) is reflecting the well known effects of shell structures.
Finally in Fig. 4 we have displayed the variation of the level density as a function of $E^*$ for $^{114}Sn$. The theoretical curves are drawn after dividing the actual numbers by a factor 20.827 such that the upper most point (with minimum uncertainty) of the experimental data matches with the one calculated in the SPA$+$RPA approach. The variation is rather very well reproduced by the solid curve keeping in mind the fact that the experimental values have also large inherent uncertainties [@drc; @prc]. The SPA curve is roughly parallel to the SPA$+$RPA one (the ratio $\rho_{\scriptstyle{SPA}}/\rho_{\scriptstyle{SPA+RPA}} \approx
4.5$ at $E^* = 5.0$ $MeV$ and 2.6 at $E^*=30$ $MeV$) emplying that here the variation is approximately reproduced.
In conclusion, we have studied the excitation energy dependence of level densities for $^{104}Pd$ and $^{114}Sn$ including thermal as well as quantal fluctuations of the nuclear quadrupole shape parameters. We find that inclusion of quantal fluctuations is essential to reproduce the experimental data even upto the excitation energy of about 25 $MeV$ (or $T\approx 1.5$ $MeV$). We have also shown that the value of the level density parameter ’$a$’ to be used in the Bethe’s formula (eq. (\[rb\])) should be computed from eq. (\[aef\]) only. Also, this value of $a$ can not be used to relate temperature to get $S$ or $E^*$, particularly in the SPA or SPA$+$RPA approach. Actually the calculation of $a$ from $a_s^2/a_e$ is essentially equivalent to fitting the value of the level densities to the Bethe’s formula.
[99]{} F. S. Chang, J. B. French and T. H. Thio, Ann. Phys. (N.Y.) [**66**]{}, 137 (1971). J. B. French and V. K. B. Kota, Phys. Rev. Lett. [**51**]{}, 2183 (1983). P. Arve, G. Bertsch, B. Lauritzen, and G. Puddu,Ann. of Phys. [**183**]{}, 309 (1988). G. H. Lang, C. W. Johnson, S. E. Koonin and W. E. Ormand, Phys. Rev. [**C 48**]{}, 1518 (1993) and references therein. S. E. Koonin, D.J. Dean and K. Langanke, Phys. Rep. [**278**]{}, 1 (1997). G. Puddu, P. F. Bortignon and R. A. Broglia, Ann. of Phys. [**206**]{}, 409 (1991). D. R. Chakrabarty, V. M. Datar, Suresh Kumar, T. Mirgule H. Oza and U. K. Pal, Phys. Rev. [**C 51**]{}, 2942 (1995). S. Aberg, Phys. Lett. [**B157**]{}, 9 (1985). B. Lauritzen, G. Puddu, P.F. Bortignon and R.A. Broglia, Phys. Lett. [**B246**]{}, 329 (1990). A. Gilbert and A. G. W. Camaron, Can. J. Phys. [**43**]{}, 1446 (1965). J. Skalski, S. Mizutori and W. Nazarewicz, Nucl. Phys. [**A617**]{}, 282 (1997). R. Wadsworth [*et al*]{}, Nucl. Phys. [**A559**]{}, 461 (1993). Dan Jerrestam [*et al*]{}, Nucl. Phys. [**A557**]{}, 411c (1993). G. Puddu, Phys. Rev. [**C 47**]{}, 1066 (1993).
B. K. Agrawal and A. Ansari, Phys. Lett. [**B339**]{}, 7 (1994).
B. Lauritzen and G. Bertsch, Phys. Rev. [**C 39**]{}, 2412 (1989). S. Shlomo and J. B. Natowitz, Phys. Rev. [**C 44**]{}, 2878 (1991).
------------- ----------- ------------- ----------
$1p_{3/2}$ $-1.376$ $0g_{9/2}$ -0.975
$0f_{5/2}$ $-1.374$ $1d_{5/2}$ $-0.484$
$1p_{1/2}$ $-1.171 $ $0g_{7/2}$ $-0.30$
$0g_{9/2}$ $-0.975$ $2s_{1/2}$ $-0.216$
$1d_{5/2}$ $-0.484$ $1d_{3/2}$ $-0.122$
$0g_{7/2}$ $-0.30$ $0h_{11/2}$ $-0.122$
$2s_{1/2}$ $-0.216$ $0h_{9/2}$ $ 0.358$
$1d_{3/2}$ $-0.122$ $1f_{7/2}$ $0.405$
$0h_{11/2}$ $-0.122$ $---$ $--$
------------- ----------- ------------- ----------
: Spherical single-particle energies (in units of $\hbar\omega_0$) with $Z=28$ and $N=40$ as a core for proton and neutron, respectively.
[**Figure Captions**]{}\
|
---
abstract: 'We explore some integrals associated with the Riesz function and establish relations to other functions from number theory that have appeared in the literature. We also comment on properties of these functions.'
author:
- Alexander E Patkowski
title: On some Integrals associated with the Riesz function
---
Introduction
============
A now well-known criterion for the Riemann hypothesis was offered by Riesz \[10\] (see also \[12, pg.382\] and \[14\]), who stated that, a necessary and sufficient condition for the Riemann hypothesis is $$x\sum_{n\ge1}\frac{\mu(n)}{n^2}e^{-x/n^2}=O(x^{\frac{1}{4}+\epsilon}).$$ Throughout this paper $\mu(n)$ will denote the M$\ddot{o}$bius function \[12\], and $\dot{R}(x)$ will denote the Riesz series on the left side of (1.1). In \[2, 3\] Bartz gave an explicit formula for Merten’s function \[12, pg.372, Theorem 14.27\] $\sum_{n\le x}\mu(n)$ without the assumption of the Riemann hypothesis, and subsequently described the analytic character of two functions defined when there are no multiple zeros (non-trivial) of the Riemann zeta function. One of these functions is defined by $$m(z)=\lim_{n\rightarrow\infty}\sum_{0<\Im\rho<T_n}\frac{e^{\rho z}}{\zeta'(\rho)}.$$ Bartz further offers that \[2, Theorem 2\], this function can be continued analytically to a meromorphic function on the whole complex plane, and satisfies the functional equation $$m(z)+\bar{m}(\bar{z})=-2\sum_{n\ge1}\frac{\mu(n)}{n}\cos(\frac{2\pi}{n}e^{-z})=A(z).$$ (Here the bar designates the complex conjugate.) In the next section we will offer a criteria for the Riemann hypothesis for a Laplace transform involving $$A(-\frac{1}{2}\log(z))=-2\sum_{n\ge1}\frac{\mu(n)}{n}\cos(\frac{2\pi}{n}\sqrt{z}).$$
main integrals
==============
An explicit formula (which is known \[11\]) for the Riesz function that will become central in this section is given by $x>0$ $$\dot{R}(x)=\sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}-\sum_{n\ge1}\frac{n!x^{-n}}{\zeta'(-2n)}.$$ To see this formula, one may apply the Residue theorem (using a positively oriented circle of radius $\frac{1}{2}+M$ centered at the origin) to $$g(z)=\frac{x^{z/2}\Gamma(1-\frac{z}{2})}{\zeta(z)},$$ after noting simple poles at the complex zeros of the Riemann zeta function, $\rho=\frac{1}{2}+i\gamma,$ $\gamma\in\mathbb{R},$ the trivial zeros at $z=-2j,$ $j\in\mathbb{N},$ and the zeros of the gamma function at $z=2i+2,$ $i\in\mathbb{N}_0.$ Further, using the formula \[1\] $$\frac{1}{\zeta'(-2n)}=\frac{\pi^{2n}2^{2n+1}}{(-1)^n\zeta(2n+1)(2n)!},$$ for positive integers $n,$ we may write the series on the far right hand side of (2.1) as $$\sum_{n\ge1}\frac{n!x^{-n}}{\zeta'(-2n)}=\frac{1}{2}\sum_{n\ge1}\frac{(-1)^nn!}{\zeta(2n+1)(2n)!}\left(\frac{2\pi}{\sqrt{x}}\right)^{2n}.$$ The series (2.2) might be realized as a Fourier cosine integral in the following way. First, note that $$\int_{0}^{\infty}xe^{-ax^2}\cos(xb)dx=\frac{1}{2}\int_{0}^{\infty}e^{-ax}\cos(\sqrt{x}b)dx,$$ and that $$\int_{0}^{\infty}e^{-ax}\cos(\sqrt{x}b)dx=\frac{1}{a}\sum_{n\ge0}\frac{n!}{(2n)!}\left(-\frac{b^2}{a}\right)^n.$$ Consequently, by uniform convergence, we may write the series in (2.2) as $$\int_{0}^{\infty}t\left(x\sum_{n\ge1}n\mu(n)e^{-x(nt)^2}\right)\cos(t2\pi)dt.$$ We may also use (2.3) to write this as $$\frac{x}{(2\pi)^2}\int_{0}^{\infty}e^{-xt/(2\pi)^2}\left(\sum_{n\ge1}\frac{\mu(n)}{n}\cos(\frac{\sqrt{t}}{n})\right)dt,$$ which involves the function of Bartz \[2, eq.(2.8)\].
A necessary and sufficient condition for the Riemann hypothesis, is (for all $\epsilon>0$) $$\sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}-\frac{x}{(2\pi)^2}\int_{0}^{\infty}e^{-xt/(2\pi)^2}\left(\sum_{n\ge1}\frac{\mu(n)}{n}\cos(\frac{\sqrt{t}}{n})\right)dt=O(x^{\frac{1}{4}+\epsilon}).$$
We now turn our attention to a useful study on Fourier integrals employed by Csordas \[4\]. The following definition, along with applicable theorems, was used there (see also that papers’ references) to determine the nature of the zeros and other properties of the function represented by the Fourier cosine integral $$f(x):=\int_{0}^{\infty}k(t)\cos(xt)dt.$$ These type of results had its beginnings with that of Pólya \[9\].\
\
[**Definition 2.2**]{} *A function $k:\mathbb{R}\rightarrow\mathbb{R}$ is said to be an *’admissible kernel,’ if it satisfies (i) $k(t)\in C^{\infty}(\mathbb{R}),$ (ii) $k(t)>0$ for $t\in\mathbb{R},$ (iii) $k(t)=k(-t)$ for $t\in\mathbb{R},$ (iv) $\frac{d}{dt}k(t)<0$ for $t>0,$ and (v) for some $\epsilon>0,$ $$k^{(n)}(t)=O(e^{-|t|^{2+\epsilon}}),$$ as $t\rightarrow\infty.$\
Some interesting properties are known about $f(x)$ when $k(t)$ satisfies Definition 2.2 \[4\]. Namely, by the Riemann-Lebesgue Lemma, $f(x)\rightarrow0$ as $|x|\rightarrow\infty.$ Additionally, $f(x)$ is then an entire function of order $\frac{2+\epsilon}{1+\epsilon}<2.$\
**
We may observe that we may make the the change of variables in (2.4) with $t$ replaced by $t/\sqrt{x}$ to get that $$\int_{0}^{\infty}t\left(\sum_{n\ge1}n\mu(n)e^{-(nt)^2}\right)\cos(t2\pi/\sqrt{x})dt,$$ still represents the series in (2.2). However, our $k(t)$ function here is odd and subsequently it is not the case that the function $f(x)$ represented by our integral can have only real zeros. Our $k(t)$ implies that our $f(x)$ has infinitely many non-real zeros, and finitely many real zeros. To see this, we need only observe that if we let $s(t)=\sum_{n\ge1}n\mu(n)e^{-(nt)^2},$ then $s(t)>0$ when $t>0,$ $s(t)=s(-t),$ and $s'(t)<0$ when $t>0.$ Comparing these properties with the work of Csordas \[4\] gives our claim. In fact, it was already shown in Rieszs’ study \[10\] that $\dot{R}(x)$ has infinitely many imaginary zeros using a different approach. See also \[13\] for more properties on the zeros of the Riesz function.
Remarks on a recent generalization
==================================
Recently, Dixit et. al. \[5, 6\] studied a more general series than a function considered by Hardy and Littlewood, with a similar condition to (1.1). Their function is given by $$P_{z}(y)=\sum_{n\ge1}\frac{\mu(n)}{n}e^{-y/n^2}\cosh(\frac{\sqrt{y}z}{n}).$$ They offer the condition that the Riemann hypothesis implies $P_{z}(y)=O_{z,\epsilon}(y^{-\frac{1}{4}+\epsilon})$ as $y\rightarrow\infty$ for all $\epsilon>0.$ We offer some more comments on $P_{z}(y)$ herein. It is well-known \[7\] that (for $y>0$ and $\Re(\beta)>0$) $$\int_{0}^{\infty}e^{-t^2/{4\beta}}\cosh(\alpha t)\cos(yt)dt=\sqrt{\pi/\beta}e^{\alpha^2\beta}e^{-\beta y^2}\cos(2\alpha\beta y).$$ Or equivalently, $$e^{-t^2/{4\beta}}\cosh(\alpha t)=\sqrt{\pi/\beta}e^{\alpha^2\beta}\int_{0}^{\infty}e^{-\beta y^2}\cos(2\alpha\beta y)\cos(yt)dy.$$ Put $\beta=n^2,$ and $\alpha=1/n,$ where $n\in\mathbb{N}.$ Then sum over $\mu(n)/n$ to get, by uniform convergence, $$\sum_{n\ge1}\frac{\mu(n)}{n}e^{-t^2/{4n^2}}\cosh(\frac{t}{n})=\sqrt{\pi}e\int_{0}^{\infty}\sum_{n\ge1}\frac{\mu(n)}{n^2}e^{-n^2 y^2}\cos(2n y)\cos(yt)dy.$$ We need to check that the function $$\bar{k}(t)=\sum_{n\ge1}\frac{\mu(n)}{n^2}e^{-n^2 t^2}\cos(2n t),$$ is an admissible kernel according to Definition 2.2. The condition (ii) that $\bar{k}(t)>0$ for $t\in\mathbb{R},$ can not be met for all $t\in\mathbb{R}.$ For example, in considering $\bar{k}(t)$ partial sums, we see that $t=\pi/2$ gives $$\sum_{1\le n \le 3}\frac{\mu(n)e^{-n^2\pi^2/4}\cos(n\pi)}{n^2}=-\frac{1}{36}e^{-\frac{9\pi^2}{4}}\left(-4+9e^{5\pi^2/4}+36e^{2\pi^2}\right),$$ which is $<0.$ Subsequently (ii) only holds for a subset of $\mathbb{R}.$ If instead we choose $\alpha=i/n,$ initially in our computations from (3.3), we find that applying our same analysis leads to an admissible kernel for the function $P_{iz'}(y),$ $z'\in\mathbb{R},$ which implies it would have only real zeros if $z'\in\mathbb{R}.$
Riesz \[10\] noted that $|\dot{R}(x)|<|x|e^{|x|},$ since $x\in\mathbb{R},$ and hence $\dot{R}(x)$ has order one.
Recall the Hermite polynomials are generated by $$e^{2xt-t^2}=\sum_{n\ge0}\frac{H_n(x)t^n}{n!},$$ and the Hermite numbers are $H_n:=H_n(0).$
The function $\dot{R}(x^2)/x^2$ may take the form $$\sum_{n\ge0}\frac{H_nx^n}{n!\zeta(n+2)},$$ where $H_n$ is the $n$th Hermite number, and therefore is an entire function of order $\lambda,$ given by $$\lambda=\lim_{n\rightarrow\infty}\sup\frac{n\log{n}}{\log{\frac{|n!\zeta(n+2)|}{|H_n|}}}.$$
More on the Riesz Criterion and other possible directions
=========================================================
We make some further comments on Theorem 2.1 and offer some possible further directions. First if we first note that the series on the far right hand side of (2.1) is $O(\frac{1}{x}),$ we may write $$\dot{R}(x)=\sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}+O(\frac{1}{x}).$$ Let $$\sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}=O(w(x)).$$ Then $$\dot{R}(x)=O(\max\{w(x),\frac{1}{x}\}).$$ This, together with Riesz criterion (1.1), tells us that proving $w(x)=x^{\frac{1}{4}+\epsilon},$ $\epsilon>0,$ would imply the Riemann Hypothesis. This is equivalent to the observation that (4.1) says $\dot{R}(x)\sim \sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}.$
A possible direction for further research would be to consider the following integral, which we produce from some observations. We start with Riesz’s integral \[10\] $$\frac{\Gamma(1-\frac{s}{2})}{\zeta(s)}=\int_{0}^{\infty}x^{-(\frac{s}{2}+1)}\dot{R}(x)dx.$$ Now if we assume a suitable test function $T(x)$ has a Mellin transform $\bar{T}(s),$ which exists in the region $\frac{1}{2}+\eta\le\Re(s)\le 2-\eta,$ ($\eta>0$), then $$\int_{0}^{\infty}x^{s-1}\left(\int_{0}^{\infty}T(x\sqrt{t})\frac{\dot{R}(t)}{t}dt\right)dx=\frac{\bar{T}(s)\Gamma(1-\frac{s}{2})}{\zeta(s)}.$$ So we may write $$\int_{0}^{\infty}T(x\sqrt{t})\frac{\dot{R}(t)}{t}dt=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\bar{T}(s)\Gamma(1-\frac{s}{2})}{\zeta(s)}x^{-s}ds.$$ If we choose our test function to be the dirac delta function $T(t)=\delta(t-a)$ and choosing $c=\frac{1}{2}+\epsilon,$ we can obtain the Riesz condition after noting that Littlewood \[8, pg.161\] showed that a criterion for the Riemann Hypothesis is that $\sum_{n\ge1}\mu(n)n^{-\frac{1}{2}-\epsilon}$ converges for every $\epsilon>0.$ It would be interesting to see some further examples by choosing other test functions $T(x).$ For example, choosing a function $T_{\epsilon_1}(t)$ whose limit is $\lim_{\epsilon_1\rightarrow0}T_{\epsilon_1}(t)=T(t)=\delta(t-a),$ and estimating the integral in (4.6) involving $T_{\epsilon_1}(t)$ may lead to improvements or new criteria.
[**Acknowledgement.**]{} We thank Professor Dixit and Professor Wolf for helpful comments.
[9]{} G. Andrews, R. Askey, and R. Roy. *Special Functions,* volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, New York, 1999. K. M. Bartz, *On some complex explicit formulae connected with the M$\ddot{o}$bius function, I,* Acta Arithmetica 57, p.283–293 (1991). K. M. Bartz, *On some complex explicit formulae connected with the M$\ddot{o}$bius function, II,* Acta Arithmetica 57, p.295–305 (1991). G. Csordas, *Fourier Transforms of Positive Definite Kernels and the Riemann $\xi$-function,* Computational Methods and Function Theory, Volume 15, Issue 3, pp 373–391 (2015). A. Dixit, *Analogues of the general theta transformation formula,* Proc. Roy. Soc. Edinburgh, Sect. A, 143 (2013), 371–399 A. Dixit, A. Roy and A. Zaharescu, *Riesz-type criteria and theta transformation analogues,* J. Number Theory 160, p. 385–408 (2016). I. S. Gradshteyn and I. M. Ryzhik, eds., Table of Integrals, Series, and Products, 7th ed., Academic Press, San Diego, 2007. G. H. Hardy and J. E. Littlewood, *Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes,* Acta Math., 41 (1916), 119–196. G. Pólya, *Uber trigonometrische Integrale mit nur reellen Nullstellen,* J. Reine Angew. Math. 158 (1927), 6–18. M. Riesz, *Sur l’hypoth’ese de Riemann,* Acta Math., 40 (1916), 185–190. G.W. Smith, *On a function of Marcel Riesz,* http://arxiv.org/abs/1209.5652, September 2012. E. C. Titchmarsh, *The Theory of the Riemann Zeta Function,* Clarendon Press, Oxford, 1986. H.Wilf, *On the zeros of Riesz’ function in the analytic theory of numbers,* Illinois J. Math., 8 (1964), pp. 639–641 M. Wolf, *Evidence in favor of the Baez-Duarte criterion for the Riemann Hypothesis,* Computational Methods in Science and Technology, v.14 (2008) pp.47–54
1390 Bumps River Rd.\
Centerville, MA 02632\
USA\
E-mail: alexpatk@hotmail.com
|
---
abstract: |
For *single-commodity* networks, the increase of the price of anarchy is bounded by a factor of $(1+{\varepsilon})^p$ from above, when the travel demand is increased by a factor of $1+{\varepsilon}$ and the latency functions are polynomials of degree at most $p$. We show that the same upper bound holds for *multi-commodity* networks and provide a lower bound as well.\
**Keywords** Wardrop equilibria; Selfish routing; Price of Anarchy; Sensitivity analysis
author:
- Mahdi Takalloo
- 'Changhyun Kwon[^1]'
bibliography:
- 'sensitivity.bib'
date: 'February 5, 2020'
title: 'Sensitivity of Wardrop Equilibria: Revisited'
---
Introduction and Notation
=========================
We study Wardrop’s traffic equilibria [@wardrop1952some] and how the price of anarchy changes with demand increases. Wardrop’s traffic equilibria is an example of nonatomic congestion games. Nonatomic games [@schmeidler1973equilibrium] involve a continuum of players and congestion games [@rosenthal1973class] are a class of noncooperative Nash games where the utility of each player is a function of the number of total players who choose the same or overlapping strategies. The price of anarchy measures the inefficiency of equilibria [@koutsoupias1999worst; @papadimitriou2001algorithms] by comparing the worst-case social cost of equilibria to the social cost of the system optimal solution. The price of anarchy for nonatomic congestion games have been well studied in the literature [@roughgarden2002bad; @roughgarden2005selfish; @correa2008geometric].
For *single-commodity* networks, @englert2010sensitivity have provided the upper bound on the change of the price of anarchy when the demand increases. A commodity is the travel demand for an origin-destination (O-D) pair and we assume that there is only one commodity for each O-D pair. In this paper, we show that the same upper bound is also valid for *multi-commodity* networks.
We also provide a lower bound on the change of the price of anarchy when the demand increases for multi-commodity networks. We utilize a classical sensitivity analysis approach, which has not been well recognized in the price of anarchy literature. By making connections between classical and modern approaches, we derive both upper and lower bounds on the changes of the price of anarchy, which were not straightforward using the methods available in the literature.
Preliminaries
=============
For a given directed graph $G=(V,E)$, we consider non-decreasing latency functions $\ell_e : \Rb_{\geq 0} \mapsto \Rb_{\geq 0}$ for each edge $e\in E$. For each commodity $i\in[k]=\{1,2,...,k\}$, the flow demand is $d_i$. We let $\Pc_i$ denote the available paths for commodity $i$ and $\Pc = \cup_{i\in[k]} \Pc_i$. Note that $\Pc_i \cap \Pc_j = \emptyset$ for any commodities $i \neq j$. Let $(G,(d_i),\ell)$ denote an instance of Wardrop equilibrium problems.
A feasible path flow vector $f$ is feasible when $\sum_{P\in\Pc_i} f_P = d_i$ for all $i\in[k]$ and $f_p \geq 0$ for all $p\in\Pc$. A path flow vector $f$ can also be written for each edge $e$, such that $f_e = \sum_{i\in[k]} \sum_{P\in\Pc_i:e\in P} f_P$. The path latency is defined as $\ell_P(f) = \sum_{e\in P} \ell_e(f_e)$. The total cost is defined as $C(f) = \sum_{P\in\Pc} \ell_P(f)f_P = \sum_{e\in E} \ell_e(f_e)f_e$. An optimal flow $f$ minimizes $C(f)$.
For each commodity $i \in [k]$, we define $$\mu_i(f) = \min_{P\in\Pc_i} \ell_P(f_P).$$ We consider the latency function for each edge $e\in E$ of the following form: $$\ell_e(f_e) = \sum_{m=0}^p {b}_{em} f_e ^m$$ for constants ${b}_{em}\geq 0$ for all $e\in E$ and $m=0,1,...,p$.
A feasible flow vector $f$ is at Wardrop equilibrium if $$f_P > 0 \implies \ell_P(f) = \mu_i(f)$$ for all $P\in\Pc_i$ and $i\in[k]$.
It is well-known [@smith1979existence; @dafermos1980traffic] that at any equilibrium flow $f$ for instance $(G, (d_i), \ell)$, we have $$\label{VIP}
\sum_{e\in E} \ell_e(f_e) ( {\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu}_e - f_e ) \geq 0$$ for all feasible ${\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu}$ for instance $(G, (d_i), \ell)$.
Considering the demand increase, @roughgarden2005selfish derived the following bound for general non-decreasing latency functions:
\[thm:roughgarden\] Let $\ell_e(\cdot)$ be a non-decreasing function for all $e\in E$. Let $C'_{\mathrm{opt}}$ be the cost of an optimal flow for instance $(G, ((1+{\varepsilon})d_i), \ell)$ and let $f$ be equilibrium flow for instance $(G,(d_i),\ell)$. $$\label{vi}
C(f) \leq \frac{1}{{\varepsilon}} C'_{\mathrm{opt}}.$$
If we assume polynomial functions for the latency, we can improve the bound in Theorem \[thm:roughgarden\] using the following lemma:
\[lem:chris\] Let $\ell_e(\cdot)$ be a polynomial with nonnegative coefficients of degree $p$ for all $e\in E$. The inequality $$\ell_e(f_e) f'_e \leq \frac{p}{(p+1)^{1+1/p}} \ell_e(f_e) f_e + \ell_e(f'_e) f'_e$$ holds for any $f_e \geq 0 $ and $f'_e \geq 0$ for all $e\in E$.
The improved bound follows:
\[thm:chris\_bound\] Let $C'_{\mathrm{opt}}$ be the cost of an optimal flow for instance $(G, ((1+{\varepsilon})d_i), \ell)$ and let $f$ be equilibrium flow for instance $(G,(d_i),\ell)$. For both instances, suppose polynomial latency functions of degree at most $p$ with nonnegative coefficients. We have $$C(f) \leq \Bigg[ (1+{\varepsilon}) - \frac{p}{(p+1)^{1+1/p}} \Bigg]^{-1} C'_{\mathrm{opt}}.$$ for all ${\varepsilon}\geq 0 $.
For any feasible ${\widehat}{f}$ for instance $(G, ((1+{\varepsilon})d_i), \ell)$, we let ${\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu} = \frac{{\widehat}{f}}{1+{\varepsilon}}$ so that ${\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu}$ is feasible for instance $(G, (d_i), \ell)$. From , we have $$\sum_{e\in E} \ell_e(f_e) f_e \leq \frac{1}{(1+{\varepsilon})} \sum_{e\in E} \ell_e(f_e) {\widehat}{f}_e.$$ The right-hand-side can be bounded using Lemma \[lem:chris\] as follows: $$\sum_{e\in E} \ell_e(f_e) {\widehat}{f}_e \leq \frac{p}{(p+1)^{1+1/p}} \sum_{e\in E} \ell_e(f_e) f_e + \sum_{e\in E} \ell_e({\widehat}{f}_e) {\widehat}{f}_e$$ Therefore, we have $$\Bigg( 1 - \frac{\frac{p}{(p+1)^{1+1/p}}}{(1+{\varepsilon})} \Bigg) \sum_{e\in E} \ell_e(f_e) f_e \leq \frac{1}{(1+{\varepsilon})} \sum_{e\in E} \ell_e({\widehat}{f}_e) {\widehat}{f}_e$$ and consequently $$C(f) \leq \frac{ 1 } { (1+{\varepsilon}) - \frac{p}{(p+1)^{1+1/p}} } C({\widehat}{f}).$$
Since $\frac{p}{(p+1)^{1+1/p}} < 1$ for all $p \geq 0$, Theorem \[thm:chris\_bound\] certainly improves Theorem \[thm:roughgarden\] by considering a specific form of latency functions. Note that we obtain Theorem \[thm:chris\_bound\] by comparing an equilibrium flow with any feasible flow. Using the following result, however, we will compare the performances of equilibrium flows and obtain a tighter bound.
\[thm:dafermos\] Let $\ell_e(\cdot)$ be a non-decreasing function for all $e\in E$. Let $f$ and $f'$ be equilibrium flows for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$, respectively. Then the following inequality holds: $$\sum_{i\in [k]} ( \mu_i(f') - \mu_i(f) ) ( d'_i - d_i ) \geq 0$$ where $d'_i = (1+{\varepsilon}) d_i$.
Theorem \[thm:dafermos\] shows that the minimum path latency function $\mu_i(\cdot)$ exhibits the characteristics of monotone functions with respect to the travel demand changes. In Theorem \[thm:equ\_bound\], we show that the ratio between the performances of equilibrium flows is bounded by $\frac{1}{1+{\varepsilon}}$. See Figure \[fig:compare\] for comparison.
Changes of the Price of Anarchy
===============================
We consider the demand changes from $d_i$ to $(1+{\varepsilon})d_i$ for all commodity $i\in[k]$ for some ${\varepsilon}\geq 0$. We first compare the performances of system optimum flows in the two instances.
\[thm:opt\_bound\] Let $C_{\mathrm{opt}}$ and $C'_{\mathrm{opt}}$ be the cost of an optimal flow for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$ with polynomial latency functions of degree at most $p$ with nonnegative coefficients, respectively. Then we can show $$\begin{aligned}
(1+{\varepsilon}) C_{\mathrm{opt}} &\leq C'_{\mathrm{opt}} \leq (1+{\varepsilon})^{p+1} C_{\mathrm{opt}}.\end{aligned}$$
We let $f'^{*}$ be the optimal flow for instance $(G, ((1+{\varepsilon})d_i), \ell)$ and ${\widehat}{f} = \frac{f'^*}{1+{\varepsilon}}$. Then, $$\begin{aligned}
C'_{\mathrm{opt}}
& = \sum_{e\in E} \ell_e( (1+{\varepsilon}) {\widehat}{f}_e) (1+{\varepsilon}) {\widehat}{f}_e \\
& = (1+{\varepsilon}) \sum_{e\in E}\ell_e( (1+{\varepsilon}){\widehat}{f}_e) {\widehat}{f}_e \\
& \geq (1+{\varepsilon}) \sum_{e\in E}\ell_e({\widehat}{f}_e) {\widehat}{f}_e \\
& \geq (1+{\varepsilon}) C_{\mathrm{opt}}.\end{aligned}$$ Also, we let $f^{*}$ be the optimal flow for instance $(G, (d_i), \ell)$ and then $(1+{\varepsilon}) f^*$ is feasible to instance $(G, ((1+{\varepsilon})d_i), \ell)$. We have $$\begin{aligned}
C'_{\mathrm{opt}}
& \leq \sum_{e\in E} \ell_e( (1+{\varepsilon}) f^*_e) (1+{\varepsilon}) f^*_e \\
& = \sum_{e\in E} \bigg( \sum_{m=0}^p {b}_{em} ( (1+{\varepsilon}) f^*_e )^m \bigg) (1+{\varepsilon}) f^*_e\\
& \leq \sum_{e\in E} \bigg(\sum_{m=0}^p {b}_{em} (1+{\varepsilon})^p (f^*_e)^m \bigg) (1+{\varepsilon}) f^*_e\\
& = (1+{\varepsilon})^{p+1} \sum_{e\in E} \ell_e(f^*_e) f^*_e \\
& = (1+{\varepsilon})^{p+1} C_{\mathrm{opt}}.\end{aligned}$$
Next, we compare the performances of equilibrium flows in the two instances. Although Theorem 3 of @englert2010sensitivity considers single-commodity networks and focuses on path latency, the same technique is valid for showing the following theorem for multi-commodity networks. While the bound on the path latency does not hold in multi-commodity networks as noted by @englert2010sensitivity, it still provides a bound on the total cost. Using Theorem \[thm:dafermos\], we can also provide a lower bound.
\[thm:equ\_bound\] Let $f$ and $f'$ be equilibrium flows for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$ with polynomial latency functions of degree at most $p$ with nonnegative coefficients, respectively. Then we can show $$\begin{aligned}
(1+{\varepsilon}) C(f) &\leq C(f') \leq (1+{\varepsilon})^{p+1} C(f).\end{aligned}$$
By Theorem \[thm:dafermos\], we have $$\begin{aligned}
0
& \leq \sum_{i\in [k]} ( \mu_i(f') - \mu_i(f) ) ( (1+{\varepsilon}) d_i - d_i ) \\
& = {\varepsilon}\sum_{i\in [k]} ( \mu_i(f') - \mu_i(f) ) d_i \\
& = \frac{{\varepsilon}}{1+{\varepsilon}} \sum_{i\in [k]} \mu_i(f') (1+{\varepsilon}) d_i - {\varepsilon}\sum_{i\in [k]} \mu_i(f) d_i \\
& = \frac{{\varepsilon}}{1+{\varepsilon}} \sum_{i\in [k]} \mu_i(f') d'_i - {\varepsilon}\sum_{i\in [k]} \mu_i(f) d_i \\
& = \frac{{\varepsilon}}{1+{\varepsilon}} C(f') - {\varepsilon}C(f),\end{aligned}$$ which leads to $(1+{\varepsilon}) C(f) \leq C(f')$.
The upper bound, $C(f') \leq (1+{\varepsilon})^{p+1} C(f)$, is already proved by the proof of Theorem 3 in @englert2010sensitivity.
In Theorems \[thm:opt\_bound\] and \[thm:equ\_bound\], both lower and upper bounds are tight. The lower bound happens when the latency functions are constant in all edges. The upper bound happens when the latency function is a monomial of degree $p$ in a single-edge network with a single commodity.
For any $p\geq 0$, we have $$\frac{1}{(1+{\varepsilon})} \leq \frac{ 1 } { (1+{\varepsilon}) - \frac{p}{(p+1)^{1+1/p}} } < \frac{1}{{\varepsilon}}.$$ Therefore the bound in Theorem \[thm:equ\_bound\] is tighter than the bounds in Theorems \[thm:roughgarden\] and \[thm:chris\_bound\], as seen in Figure \[fig:compare\].
When the demand increases, from Theorems \[thm:opt\_bound\] and \[thm:equ\_bound\], we can observe that the cost of both the optimal flow and the equilibrium flow increases at least by factor of $1+{\varepsilon}$. We obtain both lower and upper bounds on the change of price of anarchy as follows:
\[thm2\] Let $\rho$ and $\rho'$ denote the Price of Anarchy (PoA) for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$ with polynomial latency functions of degree at most $p$ with nonnegative coefficients, respectively. Then $\frac{1}{(1+{\varepsilon})^{p}} \leq \frac{\rho'}{\rho} \leq (1+{\varepsilon})^p $.
We can show that $$\frac{\rho'}{\rho}
= \frac{ C(f')/C'_{\textrm{opt}} }{ C(f)/C_{\textrm{opt}} }
= \frac{C(f')}{C(f)} \cdot \frac{C_{\textrm{opt}}}{C'_{\textrm{opt}}}
\leq (1+{\varepsilon})^{p+1} \cdot \frac{1}{1+{\varepsilon}}
= (1+{\varepsilon})^p$$ where the inequality holds by Theorem \[thm:opt\_bound\]. Similarly, $$\frac{\rho}{\rho'}
= \frac{ C(f)/C_{\textrm{opt}} }{ C(f')/C'_{\textrm{opt}} }
= \frac{C(f)}{C(f')} \cdot \frac{C'_{\textrm{opt}}}{C_{\textrm{opt}}}
\leq \frac{1}{1+{\varepsilon}} \cdot (1+{\varepsilon})^{p+1}
= (1+{\varepsilon})^p$$
The upper bound is identical to the result of @englert2010sensitivity, but holds for multi-commodity networks. @o2016mechanisms study how the price of anarchy may decay as the demand increases. When the price of anarchy decreases, the lower bound in Theorem \[thm2\] provides useful information.
Examples and Insights
=====================
\(s) at (0,0) [$s$]{}; (t2) at (5,0) [$t_2$]{}; (t1) at (5,3) [$t_1$]{}; (s) edge \[above\] node [$x$]{} (t1); (t1) edge \[right\] node [$k^2-1$]{} (t2); (s) edge \[below\] node [$k\cdot x$]{} (t2);
Consider an example in Figure \[fig:example\], originally considered in @englert2010sensitivity. The edge latency function $\ell_e(x)$ is written on each edge as a function of edge flow $x$. There are two commodities with demand $d_1=1$ and $d_2=k$ for constant $k\geq 1$. Suppose we increase the travel demand by factor $(1+{\varepsilon})$ for both commodities. At the equilibrium, the path latency for $d_1$ increases from $1$ to $1+k{\varepsilon}$ and the path latency for $d_2$ increases from $k^2$ to $k^2+k{\varepsilon}$; by a multiplicative factor of $(1+k{\varepsilon})$ and $(1+\frac{{\varepsilon}}{k})$, respectively. For $d_1$, the multiplicative increase factor exceeds the bound $(1+{\varepsilon})^p$, while it is below the bound for $d_2$. Although the increase in the travel demand for commodities is uniform, the increase in the resulting path latency is not.
The total cost, however, is still bounded as indicated in Theorem \[thm:equ\_bound\]. Before the increase, the total cost at the equilibrium is $C(f) = 1 + k^3$, while after the increase, it is $C(f') = (1+{\varepsilon})( 1+2k{\varepsilon}+ k^3)$. Since $k\geq 1$, it is easy to show that the ratio $C(f')/C(f)$ is bounded below and above as follows: $$(1+{\varepsilon}) \leq \frac{(1+{\varepsilon})( 1+2k{\varepsilon}+ k^3)}{1+k^3} \leq (1+{\varepsilon})^2.$$ Note that when $k=1$, the ratio is equal to the upper bound. The lower bound becomes tight only when ${\varepsilon}=0$. It is easy to see, however, that the lower bound in Theorem \[thm:equ\_bound\] becomes tight when all edge latency functions are constant functions.
We can obtain an insight from this example for the upper bound. At the equilibrium $f$, the total cost is the sum of the path latency, weighted by the travel demand: $$C(f) = \sum_{i\in[k]} \mu_i(f) d_i.$$ Since the increase in $C(f)$ is bounded, an over-increase in the path latency for a certain commodity is alleviated by under-increases in the path latency for other commodities. Those commodities with under-increases likely have more total travel demands than those with over-increases. In the example in Figure \[fig:example\], $d_1=1$ and $d_2=k$ with $k\geq 1$; therefore the travel demand for the second commodity with under-increase in the path latency exceeds the travel demand for the first commodity with over-increase.
\(s) at (0,0) [$s$]{}; (t) at (5,0) [$t$]{}; (s) edge \[bend left\] node [$x^p$]{} (t); (s) edge \[bend right\] node \[below\] [$1$]{} (t);
We consider the nonlinear Pigou example in Figure \[fig:pigou\_example\], which consists of a single commodity from $s$ to $t$ through two edges. In this example, the PoA decreases as ${\varepsilon}$ increases. When the travel demand is $1$, the PoA is at its upper bound, $\rho = \frac{(p+1)^{(1+1/p)}}{(p+1)^{(1+1/p)} - p}$. If the travel demand increases to $1+{\varepsilon}$, then the total cost at equilibrium is $1+{\varepsilon}$ and the optimal total cost is $1+{\varepsilon}- \frac{p}{1+p} \big(\frac{1}{1+p} \big)^{1/p}$. Therefore, the PoA becomes $$\rho'
=
\frac{1+{\varepsilon}}{1+{\varepsilon}- \frac{p}{1+p} \big(\frac{1}{1+p} \big)^{1/p}}
=
\bigg[ 1 - \frac{1}{1+{\varepsilon}} \frac{p}{1+p} \Big(\frac{1}{1+p} \Big)^{1/p} \bigg]^{-1},$$ which monotonically decreases as ${\varepsilon}$ increases. In Figure \[fig:pigou\], $\rho'/\rho$ is compared with the lower bound in Theorem \[thm2\].
We test the Sioux Falls network for a single commodity following @o2016mechanisms. The Sioux Falls network, shown in Figure \[fig:siouxfalls\], has been popularly used in transportation research and the dataset is available online [@TNTP]. The edge latency functions are polynomials of order $p=4$. We choose a single commodity from node 20 to node 3 and set the initial travel demand as 1,000. Then we increase the demand by 10% each time by multiplying $(1+{\varepsilon})$ with ${\varepsilon}=0.1$. We repeat 50 times. We compute both Wardrop equilibrium and system optimal flows. The resulting price of anarchy (PoA) is presented in Figure \[fig:PoA\_single\]. As the travel demand increases, PoA tends to increase and then decrease, but not monotonically. There are several up-and-down points. In each travel demand increase, the ratio between two PoA values, namely $\rho'/\rho$ as in Theorem \[thm2\] is computed and presented in Figure \[fig:ratio\_single\]. While the ratio is bounded between $(1+{\varepsilon})^{-4}=0.683$ and $(1+{\varepsilon})^4=1.464$ as given in Theorem \[thm2\], the ratios remain near 1.0 mostly.
[0.49]{}
[0.49]{}
In the original dataset of the Sioux Falls network, there are 528 commodities with non-zero travel demand. In this time, we consider all 528 commodities. We set the initial travel demand as 5% of the original travel demand given in the dataset and then start increasing demands by factor of $(1+{\varepsilon})$ with ${\varepsilon}=0.1$. We repeat this process 40 times. The results are presented in Figure \[fig:full\]. Similar observations can be made for both single-commodity and 528-commodity cases.
[0.49]{}
[0.49]{}
To observe the effect of ${\varepsilon}$ on the PoA ratio, we present the PoA ratio $\frac{\rho'}{\rho}$ for various ${\varepsilon}$ values for single-commodity and 528-commodity Sioux Falls network in Figure \[fig:eps3\]. For the single OD pair Sioux Falls network, we set the initial demand to 3000 units. For the full Sioux Falls network, we set the initial demand to the original travel demand. As presented in Figure \[fig:eps1\], for the single commodity network, the PoA ratio is not monotone and it has a breakpoint. The PoA ratio initially increases with ${\varepsilon}$ with an increasing rate until the breakpoint, after which it decreases with a decreasing rate up to a local minimum point and then increases again. For the full Sioux Falls network, shown in Figure \[fig:eps2\], the PoA ratio decreases with a non-increasing rate.
[0.49]{}
[0.49]{}
Discussion on the Non-tightness of the Bounds
=============================================
In the examples, we observe that the bounds in Theorem \[thm2\] are not tight. Indeed, it cannot be tight for large ${\varepsilon}$. Clearly, the PoA $\rho$ is bounded as follows [@roughgarden2005selfish]: $$1 \leq \rho \leq \frac{(p+1)^{(1+1/p)}}{(p+1)^{(1+1/p)} - p}$$ and so is $\rho'$. Therefore, $$\frac{(p+1)^{(1+1/p)} - p}{(p+1)^{(1+1/p)}} \leq \frac{\rho'}{\rho} \leq \frac{(p+1)^{(1+1/p)}}{(p+1)^{(1+1/p)} - p}$$ also holds regardless ${\varepsilon}$. This indicates that the bounds in Theorem \[thm2\] will never be tight for large ${\varepsilon}$ such that $$\label{large_e}
{\varepsilon}\geq \bigg[ \frac{(p+1)^{(1+1/p)}}{(p+1)^{(1+1/p)} - p} \bigg]^{\frac{1}{p}} - 1.$$ This effective ${\varepsilon}$ is plotted in Figure \[fig:large\_e\].
Figure \[fig:eps6\] represents the PoA ratio, theoretical bound $(1+{\varepsilon})^p$ and the clear bound $\frac{(p+1)^{(1+1/p)}}{(p+1)^{(1+1/p)} - p}$ based on @roughgarden2005selfish for different ${\varepsilon}$ values for single commodity and full Sioux Falls network ($p=4$). As Figure \[fig:eps4\] and \[fig:eps5\] represent the PoA ratio is always less than the theoretical bounds. The two values, $(1+{\varepsilon})^p$ and $\frac{(p+1)^{(1+1/p)}}{(p+1)^{(1+1/p)} - p}$, become the same at about ${\varepsilon}\approx 0.2110$. When ${\varepsilon}< 0.2110$, $(1+{\varepsilon})^p$ is smaller compare to $\frac{(p+1)^{(1+1/p)}}{(p+1)^{(1+1/p)} - p}$; for larger ${\varepsilon}$ values, it is greater than $\frac{(p+1)^{(1+1/p)}}{(p+1)^{(1+1/p)} - p}$. Hence, the obtained bound $(1+{\varepsilon})^p$ can be tight only for small ${\varepsilon}$ values.
[0.49]{}
[0.49]{}
In general, we make a series of observations to see how hard it is to achieve the bounds in Theorem \[thm2\], while we do not make definitive conclusion. First, we obtain the following two lemmas regarding the upper bound in Theorem \[thm2\]. Note that the upper bound is obtained only if $C'_{\mathrm{opt}} = (1+{\varepsilon}) C_{\mathrm{opt}}$ and $C(f') = (1+{\varepsilon})^{p+1} C(f)$.
\[lem:upper\_opt\] Let $C_{\mathrm{opt}}$ and $C'_{\mathrm{opt}}$ be the cost of an optimal flow for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$ with polynomial latency functions of degree at most $p$ with nonnegative coefficients, respectively. Let $f^*$ and $f^{'*}$ be the optimal flows for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$, respectively. Suppose $C'_{\mathrm{opt}} = (1+{\varepsilon}) C_{\mathrm{opt}}$. Then, there exist edges with constant latency functions; in particular those edges $e \in E$ with $f_e^{'*} > 0$.
We let $f'^{*}$ be the optimal flow for instance $(G, ((1+{\varepsilon})d_i), \ell)$ and ${\widehat}{f} = \frac{f'^*}{1+{\varepsilon}}$. Then, from (i), we have $$\begin{aligned}
(1+{\varepsilon}) C_{\mathrm{opt}} = C'_{\mathrm{opt}}
& = \sum_{e\in E} \ell_e( (1+{\varepsilon}) {\widehat}{f}_e) (1+{\varepsilon}) {\widehat}{f}_e \\
& = (1+{\varepsilon}) \sum_{e\in E}\ell_e( (1+{\varepsilon}){\widehat}{f}_e) {\widehat}{f}_e \\
& \geq (1+{\varepsilon}) \sum_{e\in E}\ell_e({\widehat}{f}_e) {\widehat}{f}_e \\
& \geq (1+{\varepsilon}) C_{\mathrm{opt}}.\end{aligned}$$ Therefore, we must have $$\sum_{e\in E}\ell_e( (1+{\varepsilon}){\widehat}{f}_e) {\widehat}{f}_e = \sum_{e\in E}\ell_e({\widehat}{f}_e) {\widehat}{f}_e.$$ Consequently, $$\sum_{e\in E}[\ell_e( (1+{\varepsilon}){\widehat}{f}_e) - \ell_e({\widehat}{f}_e)]{\widehat}{f}_e = 0$$ Since $\ell_e(\cdot)$ is monotone, we have $$\begin{aligned}
{\widehat}{f}_e > 0
& \implies \ell_e((1+{\varepsilon}){\widehat}{f}_e) = \ell_e({\widehat}{f}_e) \end{aligned}$$ for any $e\in E$. Therefore, $\ell_e(\cdot)$ is constant on $[{\widehat}{f}_e, (1+{\varepsilon}){\widehat}{f}_e]$ for any $e\in E$ such that ${\widehat}{f}_e > 0$. Since we consider polynomial latency functions, we conclude that there exist constant latency functions.
\[lem:upper\_equ\] Let $f$ and $f'$ be equilibrium flows for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$ with polynomial latency functions of degree at most $p$ with nonnegative coefficients, respectively. Suppose $C(f') = (1+{\varepsilon})^{p+1} C(f)$. Then, on all edge $e\in E$ with $f_e>0$ or $f'_e>0$, the latency function $\ell_e(\cdot)$ is of order $p$.
Adopting the approach used in Theorem 3 of @englert2010sensitivity, we consider monic monomial latency functions only: $\ell_e(f_e) = f_e^{p_e}$ for each $e\in E$. It is well known that equilibrium flows $f$ and $f'$ minimize the potential function $$\Phi(x) = \sum_{e\in E} \int_0^{x_e} \ell_e(u) {\mathop{}\!\mathrm{d}u}$$ on their respective feasible set. Therefore, we have $\Phi(f) \leq \Phi(f'/(1+{\varepsilon}))$ and $\Phi(f') \leq \Phi((1+{\varepsilon})f)$, which can be written as follows: $$\begin{aligned}
(1+{\varepsilon})^{p+1} \sum_{e\in E} \frac{1}{p_e+1} f_e^{p_e+1}
&\leq
\sum_{e\in E} \frac{(1+{\varepsilon})^{p-p_e}}{p_e+1} f_e^{'p_e+1}, \label{A} \\
\sum_{e\in E}\frac{1}{p_e+1} f_e^{'p_e+1}
&\leq
\sum_{e\in E} \frac{(1+{\varepsilon})^{p_e+1}}{p_e+1} f_e^{p_e+1}. \label{B}\end{aligned}$$ The condition $C(f') = (1+{\varepsilon})^{p+1} C(f)$ can be written as follows: $$\label{C}
(1+{\varepsilon})^{p+1} \sum_{e\in E} f_e^{p_e+1} = \sum_{e\in E} f_e^{'p_e+1}.$$ We can express $p\cdot\eqref{A} + ((p+1)(1+{\varepsilon})^p-1)\cdot\eqref{B} + ((1+{\varepsilon})^p-1)\cdot\eqref{C}$ as follows: $$\label{ABC}
\sum_{e\in E} c_e f_e^{p_e+1} \leq
\sum_{e\in E} c'_e f_e^{'p_e+1}$$ where $$\begin{aligned}
c_e & = p \cdot \frac{(1+{\varepsilon})^{p+1}}{p_e+1}
- ((p+1)(1+{\varepsilon})^p-1)\cdot \frac{(1+{\varepsilon})^{p_e+1}}{p_e+1}
+ ((1+{\varepsilon})^p-1)\cdot (1+{\varepsilon})^{p+1} \\
c'_e & = p \cdot \frac{(1+{\varepsilon})^{p-p_e}}{p_e+1}
- ((p+1)(1+{\varepsilon})^p - 1) \cdot \frac{1}{p_e+1}
+ ((1+{\varepsilon})^p - 1)\end{aligned}$$ @englert2010sensitivity show that $c_e\geq 0$ and $c'_e\leq 0$ for all $e\in E$. Therefore, for to hold, we must have the following condition for all $e\in E$: if either $f_e$ or $f'_e$ is nonzero, then $c_e = c'_e = 0$. It is easy to see that $c_e=c'_e=0$ implies $p_e=p$.
Lemma \[lem:upper\_opt\] claims that the path latency function $\ell_P(\cdot)$ along any actively used path used by the optimal flow $f^{'*}$ must be a constant function. Lemma \[lem:upper\_equ\], on the other hand, indicates that $\ell_P(\cdot)$ along any path used by equilibrium flow either $f$ or $f'$ must be polynomials of order $p$. Therefore, if the upper bound in Theorem \[thm2\] is obtained, then we must have $f_e = f'_e = 0$ on any edge used by $f^{'*}$.
We make similar observations regarding the lower bound $(1+{\varepsilon})^{-p}$.
\[lem:lower\_opt\] Let $C_{\mathrm{opt}}$ and $C'_{\mathrm{opt}}$ be the cost of an optimal flow for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$ with polynomial latency functions of degree at most $p$ with nonnegative coefficients, respectively. Let $f^*$ and $f^{'*}$ be the optimal flows for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$, respectively. Suppose $C'_{\mathrm{opt}} = (1+{\varepsilon})^{p+1} C_{\mathrm{opt}}$. Then, on all edge $e\in E$ with $f_e^{*} > 0$, the latency function $\ell_e(\cdot)$ is of order $p$.
We let $f^{*}$ be the optimal flow for instance $(G, (d_i), \ell)$ and then $(1+{\varepsilon}) f^*$ is feasible to instance $(G, ((1+{\varepsilon})d_i), \ell)$. From (i), we have $$\begin{aligned}
(1+{\varepsilon})^{p+1} C_{\mathrm{opt}} = C'_{\mathrm{opt}}
& \leq \sum_{e\in E} \ell_e( (1+{\varepsilon}) f^*_e) (1+{\varepsilon}) f^*_e \\
& = \sum_{e\in E} \bigg( \sum_{m=0}^p {b}_{em} ( (1+{\varepsilon}) f^*_e )^m \bigg) (1+{\varepsilon}) f^*_e\\
& \leq \sum_{e\in E} \bigg(\sum_{m=0}^p {b}_{em} (1+{\varepsilon})^p (f^*_e)^m \bigg) (1+{\varepsilon}) f^*_e\\
& = (1+{\varepsilon})^{p+1} \sum_{e\in E} \ell_e(f^*_e) f^*_e \\
& = (1+{\varepsilon})^{p+1} C_{\mathrm{opt}},\end{aligned}$$ where all inequalities must hold as equalities. From the second inequality, we must have $$\sum_{e\in E} \bigg(\sum_{m=0}^p {b}_{em} \Big[ (1+{\varepsilon})^p (f^*_e)^m - ( (1+{\varepsilon}) f^*_e )^m \Big] \bigg) (1+{\varepsilon}) f^*_e =0.$$ For any $e\in E$, if $f^*_e >0$, then we observe that $b_{em}=0$ for $m\neq p$, which implies that $\ell_e(\cdot)$ is a monomial of order $p$. From the first inequality, we also obtain $f^{'*}_e = (1+{\varepsilon}) f^*_e$ for all $e\in E$.
\[lem:lower\_equ\] Let $f$ and $f'$ be equilibrium flows for instances $(G,(d_i),\ell)$ and $(G, ((1+{\varepsilon})d_i), \ell)$ with polynomial latency functions of degree at most $p$ with nonnegative coefficients, respectively. Suppose $C(f') = (1+{\varepsilon}) C(f)$. Then, on all edge $e\in E$ with $f'_e\neq f_e$, the latency functions $\ell_e(\cdot)$ is constant.
The bound $C(f') = (1+{\varepsilon}) C(f)$ is based on Theorem \[thm:dafermos\], which is derived from the monotonicity of latency functions; that is, $$\sum_{e\in E} [\ell_e(f^1_e) - \ell_e(f^2_e)] (f^1_e - f^2_e) \geq 0$$ for any $f^1, f^2 \geq 0$. If $C(f') = (1+{\varepsilon}) C(f)$ holds, then, by backtracking the proof of Theorem \[thm:dafermos\], given in @dafermos1984sensitivity, we can easily show that $$\sum_{e\in E} [\ell_e(f') - \ell_e(f)] (f'_e - f_e) = 0.$$ Since $\ell_e(\cdot)$ is monotone, we must have $[\ell_e(f') - \ell_e(f)] (f'_e - f_e) = 0$ for each $e \in E$. If $f'_e \neq f_e$, then we must have $\ell_e(f'_e) = \ell_e(f_e)$, which implies that $\ell_e(\cdot)$ is a constant function. There must exist edges with $f'_e \neq f_e$; otherwise, either $f'$ or $f$ is infeasible.
Lemma \[lem:lower\_opt\] states that the latency functions are of order $p$ on all paths used by $f^*$. Lemma \[lem:lower\_equ\] indicates that the latency functions are constant on edges with $f'_e \neq f_e$. Therefore, if the lower bound in Theorem \[thm2\] is obtained, then we must have $f'_e = f_e$ on any edge used by $f^{*}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was partially supported by the National Science Foundation under grant CMMI-1558359.
[^1]: Corresponding author: `chkwon@usf.edu`
|
---
abstract: 'The purity of the reduced state for spins which is pure in the rest frame will most likely appear to degrade because spin and momentum become mixed when viewed by a moving observer. We show that such boost-induced decrease in spin purity observed in a moving reference frame is intrinsically related to the spatial localization properties of the wave package observed in the rest frame. Furthermore, we prove that, for any localized pure state with separable spin and momentum in the rest frame, its reduced density matrix for spins inevitably appears to be mixed whenever viewed from a moving reference frame.'
author:
- Hui Li
- Jiangfeng Du
title: Spatial Localization and Relativistic Transformation of Quantum Spins
---
Introduction
============
One of the most nontrivial and striking observations of relativistic thermodynamics [@b1] is that probability distributions can depend on frames. Consequently, entropy and information may change if viewed from different reference frames [@r0]. Recently, relativistic quantum information theory has attracted particular interests [@r1; @r2; @r3; @r4; @r5; @r6; @r7]. Current investigations show that a single quantum spin is not covariant under Lorentz transformations [@r1], and maximal entanglement between spins in the rest frame will most likely degrade due to mixing with the momentum if viewed from a moving frame [@r2; @r3], depending on the initial momentum wave function. For many quantum information protocols [@b5], coherence and entanglement are extremely important and expensive. The observation that coherence and entanglement of the reduced state for spins may degrade when viewed in moving frames implies that there are particular problems for relativistic quantum information processing, particularly for relativistic quantum communication.
It is known that, for a single quantum spin, if and only if we consider the momentum eigenstates (plane waves), can the reduced density matrix for the spin be covariant under Lorentz transformations. But momentum eigenstates are not localized so they may be difficult in feasible applications [@r1]. The similar difficulties exist for multipartite states [@ft1].
In this paper, we investigate the Lorentz boost-induced decrease in the purity of the reduced density matrix for spins, when a state which has pure reduced state for spins in the rest frame is viewed from a moving reference frame. Taken to the leading order, the decrease in spin purity observed in the moving frame is linear with respect to the momentum mean square deviation observed in the rest frame, which according to the position-momentum uncertainty relationship can be reasonably regarded as a measure of how much the spacial wave package is localized. We also present numerical studies as instance of our general analysis. Furthermore, we prove that, for any localized pure state with separable spin and momentum in the rest frame, its reduced density matrix for spins cannot be covariant under any Lorentz boosts, i.e. it inevitably appears to be mixed when viewed from a moving reference frame. Considering that in practical applications states should be localized, our results may have important consequences for relativistic quantum information processing.
General Relationship between the Boost-induced Spin Depurification and the Spatial Localization
===============================================================================================
A Simple Example: Single Spin-$1/2$ Massive Particle
----------------------------------------------------
We start by briefly recalling Peres et al.’s paper [@r1]. Consider a spin half massive particle (of mass $m$) that is prepared with spin in the $z$ direction. The spin state can be represented by the Bloch vector $\mathbf{n}=(n_{x},n_{y},n_{z})$ with $n_{x}=n_{y}=0$ and $n_{z}=1$. The momentum wave function is a Gaussian $g(\mathbf{p})\propto\exp(-\mathbf{p}%
^{2}/2w^{2})$. When viewed by an observer moving in the $x$ direction, the Lorentz-transformed Bloch vector is $\mathbf{n}^{\prime}=(n_{x}^{\prime}%
,n_{y}^{\prime},n_{z}^{\prime})$ with $n_{x}^{\prime}=n_{y}^{\prime}=0$ yet $n_{z}^{\prime}<1$. It is shown in Ref. [@r1] that $1-n_{z}^{\prime
}\propto w^{2}$ to the leading order of $w/m\ll1$. By denoting the Lorentz-transformed density matrix for spin as $\rho^{\prime}$, the Lorentz boost-induced decrease in its purity is $1-\operatorname{tr}(\rho^{\prime
2})\propto w^{2}$. Meanwhile, for this particular case the momentum mean square deviation is $\langle\Delta p_{\mu}^{2}\rangle\propto w^{2}$ ($\mu=x,y,z$), hence $1-\operatorname{tr}(\rho^{\prime2})\propto\langle\Delta
p_{\mu}^{2}\rangle$. According to the position-momentum uncertainty relationship $\langle\Delta x_{\mu}^{2}\rangle\langle\Delta p_{\mu}^{2}%
\rangle\geqslant\hbar^{2}/4$, the smaller $\langle\Delta x_{\mu}^{2}\rangle$ is, the larger $\langle\Delta p_{\mu}^{2}\rangle$ is and so is $1-\operatorname{tr}(\rho^{\prime2})$. This suggests that the more the wave package is localized in space, the more the boost-induced decrease in spin purity is when viewed in a moving frame.
Generalization to Multiple Massive Particles with Arbitrary Spins
-----------------------------------------------------------------
The above observation can indeed generalize to states of multiple massive particles with arbitrary spin quantum number. Consider a pure quantum state with separable spin and momentum in the rest frame. The system consists of $N$ massive particles, labelled by $k=1,\cdots,N$. The spin quantum number of particle $k$ is $s_{k}$, and mass $m_{k}>0$. The reduced density matrix for spins viewed from the rest frame is denoted by $\rho$, which is pure with $\operatorname{tr}(\rho^{2})=1$. The normalized momentum wave function in the rest frame is denote by $g(\mathcal{P})$, where $\mathcal{P}:=(p_{1x}%
,p_{1y},p_{1z},\cdots)$ for compactness of notation. To a moving observer Lorentz transformed by $\Lambda^{-1}$, the state appears to be transformed by $\Lambda^{\otimes N}$. Because the Lorentz-transformed state viewed in the moving frame differs from rest-frame one by unitary transformations, the purity will not change provided we do not trace out a part of the state. However, in looking at spins, tracing out over the momentum degrees of freedom is implied. To the Lorentz-transformed observer, the spin and momentum may appear to be entangled, thus the purity of spins may appear to degrade viewed by this observer.
The reduced density matrix for spins viewed by the moving observer is [@r1; @r2; @r3]$$\rho^{\prime}=\int\left\vert g(\mathcal{P})\right\vert ^{2}U_{\Lambda
}(\mathcal{P})\rho U_{\Lambda}^{\dag}(\mathcal{P})\widetilde{\mbox{d}}%
\mathcal{P},\label{eq 1}%$$ where, for compactness, we define $U_{\Lambda}(\mathcal{P}):=U^{s_{1}}%
(\Lambda,\mathbf{p}_{1})\otimes\cdots\otimes U^{s_{N}}(\Lambda,\mathbf{p}%
_{N})$ with $U^{s}(\Lambda,\mathbf{p})$ being the spin-$s$ representation of the Wigner rotation $R(\Lambda,\mathbf{p})$ [@b3]. $\widetilde
{\mbox{d}}\mathcal{P}:=\widetilde{\mbox{d}}\mathbf{p}_{1}\cdots\widetilde
{\mbox{d}}\mathbf{p}_{N}$ and $\widetilde{\mbox{d}}\mathbf{p}_{k}\ $is the Lorentz invariant integration measure (defined in Ref. [@b3]). We represent the Lorentz-transformed spin purity by $\operatorname{tr}%
(\rho^{\prime2})$, which can be calculated by$$\operatorname{tr}(\rho^{\prime2})=\iint\left\vert g(\mathcal{P})\right\vert
^{2}\left\vert g(\mathcal{P}^{\prime})\right\vert ^{2}\Gamma_{\rho}^{\Lambda
}(\mathcal{P},\mathcal{P}^{\prime})\widetilde{\mbox{d}}\mathcal{P}%
\hspace{\stretch{4}}\widetilde{\mbox{d}}\mathcal{P}^{\prime}.\label{eq 2}%$$ where$$\Gamma_{\rho}^{\Lambda}(\mathcal{P},\mathcal{P}^{\prime}):=\operatorname{tr}%
[U_{\Lambda}(\mathcal{P})\rho U_{\Lambda}^{\dag}(\mathcal{P})U_{\Lambda
}(\mathcal{P}^{\prime})\rho U_{\Lambda}^{\dag}(\mathcal{P}^{\prime
})].\label{eq 2-2}%$$ Denoting $\langle\mathbf{\cdot}\rangle$ as the mean value (observed in the rest frame), we define $\Delta p_{k\mu}^{(\prime)}:=p_{k\mu}^{(\prime
)}-\langle p_{k\mu}^{(\prime)}\rangle$ ($\mu=x,y,z$) and $\Delta
\mathcal{P}^{(\prime)}=\mathcal{P}^{(\prime)}-\langle\mathcal{P}^{(\prime
)}\rangle$. Then $\Gamma_{\rho}^{\Lambda}(\mathcal{P},\mathcal{P}^{\prime})$ can be expanded into power series with respect to $\Delta\mathcal{P}$ and $\Delta\mathcal{P}^{\prime}$, noting that $\Gamma_{\rho}^{\Lambda}%
(\mathcal{P},\mathcal{P}^{\prime})=\Gamma_{\rho}^{\Lambda}(\mathcal{P}%
^{\prime},\mathcal{P})$:$$\Gamma_{\rho}^{\Lambda}(\mathcal{P},\mathcal{P}^{\prime})=1-\frac{1}{2}%
(\Delta\mathcal{P},\Delta\mathcal{P}^{\prime})\left(
\begin{array}
[c]{cc}%
\mathcal{U} & \mathcal{V}^{T}\\
\mathcal{V} & \mathcal{U}%
\end{array}
\right) \left(
\begin{array}
[c]{c}%
\Delta\mathcal{P}^{T}\\
\Delta\mathcal{P}^{\prime T}%
\end{array}
\right) +\cdots,\label{eq 3}%$$ where $3N$-dimensional matrices $\mathcal{U}$ and $\mathcal{V}$ are real functions of $\Lambda$ and $\rho$. When $\Delta p_{k\mu}=\Delta p_{k\mu
}^{\prime}=0$, $\Gamma_{\rho}^{\Lambda}(\mathcal{P},\mathcal{P}^{\prime})$ reaches its maximum, i.e. $\Gamma_{\rho}^{\Lambda}(\langle\mathcal{P\rangle
},\langle\mathcal{P\rangle})=1$. So in the r.h.s. of Eq. (\[eq 3\]), the zero-order term is $1$, the first-order terms vanish, and $\mathcal{U}$ is positive-semidefinite. Terms of higher than second orders are not explicitly presented here. Using $\langle\Delta p_{k\mu}\rangle=\langle\Delta p_{k\mu
}^{\prime}\rangle=0$ and $\langle\Delta\mathcal{P}\cdot\mathcal{U}\cdot
\Delta\mathcal{P}^{T}\rangle=\langle\Delta\mathcal{P}^{\prime}\cdot
\mathcal{U}\cdot\Delta\mathcal{P}^{\prime T}\rangle$, we see that the boost-induced decrease in spin purity, to the leading order, is$$1-\operatorname{tr}(\rho^{\prime2})\simeq\langle\Delta\mathcal{P}%
\cdot\mathcal{U}\cdot\Delta\mathcal{P}^{T}\rangle.$$ The matrix $\mathcal{U}$ can be diagonalized as $\mathcal{U}=\mathcal{M}%
\cdot\mathcal{D}\cdot\mathcal{M}^{T}$, where $\mathcal{M}$ is real and orthogonal, $\mathcal{D}=\operatorname*{diag}(\mathcal{D}_{1},\cdots
,\mathcal{D}_{3N})$ with $\mathcal{D}_{\lambda}\geqslant0$ ($\lambda
=1,\cdots,3N$) due to that $\mathcal{U}$ is positive-semidefinite. By denoting $\mathcal{Q}=\mathcal{P}\cdot\mathcal{M}$ and $\Delta\mathcal{Q}%
=\mathcal{Q}-\langle\mathcal{Q}\rangle=\Delta\mathcal{P\cdot M}$, we have$$1-\operatorname{tr}(\rho^{\prime2})\simeq\sum_{\lambda=1}^{3N}\mathcal{D}%
_{\lambda}\langle\Delta\mathcal{Q}_{\lambda}^{2}\rangle.\label{eq 4}%$$ Inspired by $[\widehat{x}_{k\mu},\widehat{p}_{k^{\prime}\mu^{\prime}}%
]=i\hbar\delta_{kk^{\prime}}\delta_{\mu\mu^{\prime}}$, we define $\mathcal{X}=(x_{1x},x_{1y},x_{1z},\cdots)\cdot\mathcal{M}$ so that $[\mathcal{\widehat{X}}_{\lambda},\mathcal{\widehat{Q}}_{\lambda^{\prime}%
}]=i\hbar\delta_{\lambda\lambda^{\prime}}$, leading to the uncertainty relationship $\langle\Delta\mathcal{X}_{\lambda}^{2}\rangle\langle
\Delta\mathcal{Q}_{\lambda}^{2}\rangle\geqslant\hbar^{2}/4$. Hence from Eq. (\[eq 4\]) we obtain$$\operatorname{tr}(\rho^{\prime2})\alt1-\frac{\hbar^{2}}{4}\sum_{\lambda
=1}^{3N}\frac{\mathcal{D}_{\lambda}}{\langle\Delta\mathcal{X}_{\lambda}%
^{2}\rangle}.\label{eq 6}%$$
Here we shall note that both $\langle\Delta\mathcal{Q}_{\lambda}^{2}\rangle$ and $\langle\Delta\mathcal{X}_{\lambda}^{2}\rangle$ are observed in the rest frame, while $\operatorname{tr}(\rho^{\prime2})$ is the purity observed in the moving frame. In addition, $\mathcal{D}$ and $\mathcal{M}$ are functions of $\Lambda$ and $\rho$. For any $\rho$, when $\Lambda$ degenerate to a pure three-dimensional rotation, we can see in Eq. (\[eq 2-2\]) that $\Gamma_{\rho}^{\Lambda}(\mathcal{P},\mathcal{P}^{\prime})=1$ and then in Eq. (\[eq 4\]) that $\mathcal{D}_{\lambda}=0$ for all $\lambda$, hence obviously $\operatorname{tr}(\rho^{\prime2})=1$. Equation (\[eq 6\]) also indicates that the purity of the Lorentz-transformed reduced spin state is (to the leading order) bounded by its spatial localization properties.
Equations (\[eq 4\]) and (\[eq 6\]) dictate the intrinsic relation between the boost-induced decrease in spin purity (observed in the moving frame) and the spatial localization of the wave package (observed in the rest frame). Indeed, $\mathcal{X}_{\lambda}$ is a linear combination of $(x_{1x}%
,x_{1y},x_{1z},\cdots)$. The more the wave package is localized (the less $\langle\Delta\mathcal{X}_{\lambda}^{2}\rangle$ is), the more mixed the reduced state for spins becomes when viewed in a moving reference frame. According to the uncertainty relationship $\langle\Delta\mathcal{X}_{\lambda
}^{2}\rangle\langle\Delta\mathcal{Q}_{\lambda}^{2}\rangle\geqslant\hbar^{2}/4
$, $\sum_{\lambda=1}^{3N}\mathcal{D}_{\lambda}\langle\Delta\mathcal{Q}%
_{\lambda}^{2}\rangle$ can be reasonably regarded as a measure of how much the wave package is localized in space.
We shall note that $\mathcal{D}$ and $\mathcal{M}$ do not explicitly depend on the momentum wave function (the implicit dependence is through the mean value $\langle\mathcal{P}\rangle$ because $\Delta\mathcal{P}=\mathcal{P}%
-\langle\mathcal{P}\rangle$). When the state is not strongly localized, whatever the momentum wave function as long as the localization is the same (i.e. $\sum_{\lambda=1}^{3N}\mathcal{D}_{\lambda}\langle\Delta\mathcal{Q}%
_{\lambda}^{2}\rangle$ is the same), the same reduced state for spins suffers the same amount of boost-induced decrease in spin purity when viewed from the moving frame. This provides a general and feasible method to possibly estimate how mixed the reduced spin state would appear by the spatial localization properties, which might be useful in practical relativistic quantum information processing. In addition, for multipartite states whose spins are not entangled, position (momentum) coordinates of different particles will not be mixed in $\mathcal{X}_{\lambda}$ ($\mathcal{Q}_{\lambda}$). The r.h.s. of Eq. (\[eq 4\]), as well as that of Eq. (\[eq 6\]), turns out to be a sum over each individual particles. While interestingly, if the spins are entangled, position (momentum) coordinates of different particles will in general be mixed in $\mathcal{X}_{\lambda}$ ($\mathcal{Q}_{\lambda}$). This implies that, for the present case, in measuring how much the spatial wave package is localized, the correlation (entanglement) between the spins needs also to be taken into account.
A Further Illustration: Two Spin-$1/2$ Massive Particles
--------------------------------------------------------
![Lorentz boost-induced decrease of spin purity (observed in the moving frame) versus the momentum mean square deviation (observed in the rest frame), for two states, $|\psi ^{-}\rangle$ (at left) and $|\psi^{+}\rangle$ (at right), with momentum wave function in Eq. (\[eq 5\]). For each set of $(\xi,x)$, $\sigma/m$ takes value from $0.025$ to $0.5$ with a step $0.025$. Here we adopt the natural units: $c=\hbar=1$, and $m=1$ so that the quantities are all dimensionless.[]{data-label="figure"}](Figure.eps){width="\columnwidth"}
As an illustrative example, we numerically study the case of two spin half particles (of mass $m$) with momentum wave function being the entangled Gaussian as presented in Ref. [@r3]:$$\begin{aligned}
& g\left( \mathbf{p}_{1},\mathbf{p}_{2}\right) \nonumber\\
& =\sqrt{\frac{1}{\mathsf{N}}\exp\left[ -\frac{\mathbf{p}_{1}^{2}%
+\mathbf{p}_{2}^{2}}{4\sigma^{2}}\right] \exp\left[ -\frac{\mathbf{p}%
_{1}^{2}+\mathbf{p}_{2}^{2}-2x\mathbf{p}_{1}\cdot\mathbf{p}_{2}}{4\sigma
^{2}\left( 1-x^{2}\right) }\right] },\label{eq 5}%\end{aligned}$$ where $\mathsf{N}$ is the normalization, $\sigma\geqslant0$ is the width and $x\in\left[ 0,1\right) $. The Lorentz transformation is chosen to be a pure boost $L(\boldsymbol{\xi})$ in the $z$ direction, where $\boldsymbol{\xi}$ is the rapidity and denote $\xi=|\boldsymbol{\xi}|$, as defined in Ref. [@r3]. For this particular momentum wave function, $\langle\Delta p_{k\mu}^{2}\rangle$ is the same for any $k$ and $\mu$. Moreover, when $\langle\Delta p_{k\mu}^{2}\rangle$ and $\langle\Delta p_{k^{\prime}\mu^{\prime}}^{2}\rangle$ are relatively small, $\langle\Delta p_{k\mu}\Delta p_{k^{\prime}\mu^{\prime}}\rangle$ is (if not zero) proportional to $[\langle\Delta p_{k\mu}^{2}\rangle\langle\Delta
p_{k^{\prime}\mu^{\prime}}^{2}\rangle]^{1/2}$, with the proportion depending upon $x$. Thus for this particular case, Eq. (\[eq 4\]) reduces to $1-\operatorname{tr}(\rho^{\prime2})\propto\langle\Delta\mathbf{p}_{1}%
^{2}\rangle$. Figure \[figure\] depicts the relation between $1-\operatorname{tr}(\rho^{\prime2})$ and $\langle\Delta\mathbf{p}_{1}%
^{2}\rangle$, for the spin parts being $|\psi^{-}\rangle$ and $|\psi ^{+}\rangle$. When the momentum mean square deviation is relatively small, the relation can be well described by linearity, confirming the validity of Eq. (\[eq 4\]). As the momentum mean square deviation increases, corresponding to stronger localization, the boost-induced decrease in spin purity increases monotonously. The deviation from linearity is due to terms of higher than second orders, such as $\langle\Delta
p_{k\mu}^{3}\rangle$ and $\langle\Delta p_{k\mu}^{4}\rangle$ etc. However, such terms can also be regarded in some sense as measures of the spatial localization of the wave package.
A Theorem
---------
In the remaining part of this paper, we prove the following theorem.
**Theorem:** *When a pure state with separable spin and momentum in the rest frame is viewed from a moving reference frame, its reduced density matrix for spins necessarily appears to be mixed if its spatial wave package is localized.*
Here we shall first specify the meaning of being localized. We regard a state localized in the sense that its position wave function could be *commonly normalized*, i.e. it is square-integrable. This requirement excludes nonlocalized states, such as momentum eigenstates and the singular ones presented in Ref. [@r3]. Since momentum wave function is the Fourier transformation of position wave function, being localized means that, in our case, $g(\mathcal{P})$ is square-integrable, i.e. $G(\mathcal{P}%
):=|g(\mathcal{P})|^{2}\widetilde{\mbox{d}}\mathcal{P}/\mbox{d}\mathcal{P}%
\geqslant0$ is *integrable* on $\mathbb{R}^{3N}$ in the sense of *Lebesgue integration*, and it is normalized as $\int G(\mathcal{P}%
)\mbox{d}\mathcal{P}=\int_{\operatorname*{supp}(G(\mathcal{P}))}%
G(\mathcal{P})\mbox{d}\mathcal{P}=1$, where $\operatorname*{supp}%
(G(\mathcal{P})):=\{\mathcal{P}\mid G(\mathcal{P})>0\}$ is the support of $G(\mathcal{P})$.
We denote $\mathcal{K}=(\mathcal{P},\mathcal{P}^{\prime})\in\mathbb{R}^{6N} $, $\mathcal{T}(\mathcal{K})=\Gamma_{\rho}^{\Lambda}(\mathcal{P},\mathcal{P}%
^{\prime})\in\lbrack0,1]$, and $\mathcal{G}(\mathcal{K})=G(\mathcal{P}%
)G(\mathcal{P}^{\prime})\geqslant0$ for compactness. Let $\Omega
_{g}=\operatorname*{supp}(\mathcal{G}(\mathcal{K}))=\{\mathcal{K}%
\mid\mathcal{G}(\mathcal{K})>0\}=\operatorname*{supp}(G(\mathcal{P}%
))\times\operatorname*{supp}(G(\mathcal{P}^{\prime}))$, $\Omega_{t}%
=\{\mathcal{K}\mid\mathcal{T}(\mathcal{K})=1\}$, and $m(\cdot)$ be the Lebesgue measure in $\mathbb{R}^{6N}$. The integral in Eq. (\[eq 2\]) now turns to be the Lebesgue integral over $\Omega_{g}$: $\operatorname{tr}%
(\rho^{\prime2})=\int_{\Omega_{g}}\mathcal{G}(\mathcal{K})\mathcal{T}%
(\mathcal{K})\mbox{d}\mathcal{K}.$
Since the state is localized, $\mathcal{G}(\mathcal{K})$ must be integrable. In addition, $\int_{\Omega_{g}}\mathcal{G}(\mathcal{K})\mbox{d}\mathcal{K}%
=[\int_{\operatorname*{supp}(G(\mathcal{P}))}G(\mathcal{P})\mbox{d}\mathcal{P}%
]^{2}=1$ implies that $\mathcal{G}(\mathcal{K})$ is bounded almost everywhere. Hence one must have $m(\Omega_{g})>0$ [@b4]. Otherwise supposing $m(\Omega_{g})=0$, one necessarily encounters the contradiction that $\int_{\Omega_{g}}\mathcal{G}(\mathcal{K})\mbox{d}\mathcal{K}=0$ and $\int_{\operatorname*{supp}(G(\mathcal{P}))}G(\mathcal{P})\mbox{d}\mathcal{P}%
=0$.
Denote $x=(\mathbf{p}_{2},\cdots,\mathbf{p}_{N},\mathbf{p}_{1}^{\prime}%
,\cdots,\mathbf{p}_{N}^{\prime})$ and $\Omega_{t}(x)=\{\mathbf{p}_{1}%
\mid(\mathbf{p}_{1},x)\in\Omega_{t}\}$. It can be easily verified that $\Omega_{t}(x)\ $has Lebesgue measure zero in $\mathbb{R}^{3}$ for any given $x$ when $\Lambda$ does not degenerate to a pure three-dimensional rotation. Then, using the Tonelli’s theorem [@b4], one obtains $m(\Omega_{t})=\int
m^{\prime}(\Omega_{t}(x))\mbox{d}x=0$ \[$m^{\prime}(\cdot)$ is the Lebesgue measure in $\mathbb{R}^{3}$\]. Physically, this observation is intuitive. If there is $\mathcal{K}_{0}=(\mathcal{P}_{0},\mathcal{P}_{0}^{\prime})\in
\Omega_{t}$, then $U_{\Lambda}^{\dag}(\mathcal{P}_{0})U_{\Lambda}%
(\mathcal{P}_{0}^{\prime})$ must have an eigenstate to be exactly $\rho$ (the rest-frame reduced density matrix for spins). All such $(\mathcal{P}%
_{0},\mathcal{P}_{0}^{\prime})$ occupy only a low dimensional subset in $\mathbb{R}^{6N}$ for any pure $\rho$ and nondegenerate $\Lambda$, thus $m(\Omega_{t})=0$.
**Proof of the Theorem:** Because $m(\Omega_{g})>0$ while $m(\Omega
_{t})=0$, we have $\int_{\Omega_{g}}(\cdot)\mbox{d}\mathcal{K}\equiv
\int_{\Omega_{g}/\Omega_{t}}(\cdot)\mbox{d}\mathcal{K}$. Therefore $$\begin{aligned}
\operatorname{tr}(\rho^{\prime2}) & =\int_{\Omega_{g}}\mathcal{G}%
(\mathcal{K})\mathcal{T}(\mathcal{K})\mbox{d}\mathcal{K}=\int_{\Omega
_{g}/\Omega_{t}}\mathcal{G}(\mathcal{K})\mathcal{T}(\mathcal{K}%
)\mbox{d}\mathcal{K}\\
& <\int_{\Omega_{g}/\Omega_{t}}\mathcal{G}(\mathcal{K})\mbox{d}\mathcal{K}%
=\int_{\Omega_{g}}\mathcal{G}(\mathcal{K})\mbox{d}\mathcal{K}=1,\end{aligned}$$ where the inequality is due to that $\mathcal{T}(\mathcal{K})$ is strictly less than $1$ on $\Omega_{g}/\Omega_{t}$. Now that $\operatorname{tr}%
(\rho^{\prime2})<1$ immediately gives that $\rho^{\prime}$ is mixed.$\blacksquare$
The fact that for localized states $m(\Omega_{g})>0$ is essential. Its meanings can be further clarified by reviewing Eq. (\[eq 4\]). If $\operatorname{tr}(\rho^{\prime2})=1$, we must have $\mathcal{D}_{\lambda
}\langle\Delta\mathcal{Q}_{\lambda}^{2}\rangle=0$ for all $\lambda$. Supposing there is $\lambda_{0}$ so that $\mathcal{D}_{\lambda_{0}}\neq0$, consequently we have $\langle\Delta\mathcal{Q}_{\lambda_{0}}^{2}\rangle=0$. Due to that $\mathcal{M}$ is orthogonal, there is $\sigma_{0}$ so that $\mathcal{M}%
_{\sigma_{0}\lambda_{0}}\neq0$. Supposing $\sigma_{0}$ corresponds the position coordinate labelled by $\mu_{0}$ of particle $k_{0}$, we obtain that $[\widehat{x}_{k_{0}\mu_{0}},\mathcal{\widehat{Q}}_{\lambda_{0}}%
]=i\hbar\mathcal{M}_{\sigma_{0}\lambda_{0}}$, which leads to that $\langle\Delta x_{k_{0}\mu_{0}}^{2}\rangle\langle\Delta\mathcal{Q}%
_{\lambda_{0}}^{2}\rangle\geqslant\hbar^{2}\mathcal{M}_{\sigma_{0}\lambda_{0}%
}^{2}/4$. However because $\langle\Delta\mathcal{Q}_{\lambda_{0}}^{2}%
\rangle=0$ here, we see that $\langle\Delta x_{k_{0}\mu_{0}}^{2}%
\rangle\rightarrow\infty$. On the one hand, $\langle\Delta\mathcal{Q}%
_{\lambda_{0}}^{2}\rangle=0$ implies that $g(\mathcal{P})$ is an eigenstate of $\mathcal{\widehat{Q}}_{\lambda_{0}}$ and consequently $m(\Omega_{g})=0$. On the other hand, $\langle\Delta x_{k_{0}\mu_{0}}^{2}\rangle\rightarrow\infty$ implies that the (reduced) wave package of particle $k_{0}$ is nonlocalized. Inversely, $m(\Omega_{g})>0$ guarantees that $g(\mathcal{P})$ is not an eigenstate of any $\mathcal{\widehat{Q}}_{\lambda}$, so the state can be localized because all $\langle\Delta x_{k\mu}^{2}\rangle$ can be finite.
Discussion and Conclusion
=========================
We would like to note that the present paper adopts the same notion of spins as in Ref. [@r1; @r2; @r3]. Beside, there are other possible notions (e.g. see Ref. [@r4]). It would be interesting to see that when spin is defined with respect to projection of Pauli-Lubanski’s vector in a principal null direction of the Lorentz transformation, the reduced density matrix for spins viewed by the moving observer does not depolarize [@r4]. However, since establishing a perfect shared reference frame requires infinite communication even in non-relativistic situations [@srf], it might be extremely difficult to acquire precise information about a Lorentz transformation. We argue that in practical applications spin may be defined independently of the particular Lorentz transformation that defines the relative motion between the observers, and the transformation law of such spins would then in general depend upon momentum. Indeed, our results would hold for all such notions of spin, including those adopted in Refs. [@r1; @r2; @r3], but Ref. [@r4].
In conclusion, states one can prepare in real experiments are necessarily localized. Nonlocalized states, e.g. momentum eigenstates and the singular ones presented in Ref. [@r3], are not practical in reality, although they are useful in theories. We show that, in relativistic applications reduced spin state which is pure in the rest frame unavoidably appears to be mixed whenever viewed from moving reference frames. How much such boost-induced decrease in purity is depends on how much the spatial wave package is localized. The more the spatial wave package is localized, the more the purity of the reduced spin state decreases when viewed from moving frames. This observation may be important for relativistic quantum information processing, particularly for relativistic quantum communication.
Although our investigations are based on massive particles, the generalize to massless cases, such as to photons [@r6], should be analogous. This may be of interest since most of current experiments in quantum communication are based on photons [@b5]. Another interesting problem might be to determine how our results generalize to accelerated frames [@r7].
We thank S.X. Yu and Z.B. Chen for useful discussion. This work was supported by the Nature Science Foundation of China (Grant No. 10075041), the National Fundamental Research Program (Grant No. 2001CB309300), and the ASTAR Grant No. 012-104-0040 & R-144-000-071-305.
[99]{} R.C. Tolman, *Relativity, Thermodynamics, and Cosmology* (Oxford University Press, Oxford, 1934); R.M. Wald, *Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics* (University of Chicago Press, Chicago, 1994).
P.T. Landberg and G.E.A. Matsas, Phys. Lett. A **223**, 401 (1996); P.J.B. Peebles and D.T. Wilkinson, Phys. Rev. **174**, 2168 (1968).
A. Peres *et al.*, **88**, 230402 (2002).
R.M. Gingrich and C. Adami, **89**, 270402 (2002).
H. Li and J. Du, **68**, 022108 (2003).
M. Czachor and M. Wilczewski, **68**, 010302(R) (2003).
M. Czachor, **55**, 72 (1996); J. Rembieliński and K.A. Smoliński, **66**, 052114 (2002); P.M. Alsing and G.J. Milburn, Quantum Inf. and Comput. **2**, 487 (2002); D. Ahn *et al.*, **67**, 012103 (2003).
R.M. Gingrich et al., **68**, 042102 (2003); A. Peres and D.R. Terno, J. Mod. Opt. **50**, 1165 (2003).
P.M. Alsing and G.J. Milburn, **91**, 180404 (2003).
*The Physics of Quantum Information*, edited by D. Bouwmeester, A. Ekert, and A. Zeilinger (Springer, New York, 2000).
In Ref. [@r3], it is shown that a class of states of two spin half particles, other than momentum eigenstates, do have covariant reduced density matrix for spins. However, it can be proved that for such states the reduced state for a single particle is the convex combination of momentum eigenstates. Taking the momentum wave function $g(\mathbf{p}%
_{1},\mathbf{p}_{2})=[f(\mathbf{p}_{1})\delta^{3}(\mathbf{p}_{1}%
-\mathbf{p}_{2})]^{1/2}$ (see Ref. [@r3]) as an example, and noting that $g(\mathbf{p}_{1},\mathbf{p}_{2})=g(\mathbf{p}_{2},\mathbf{p}_{1})$ and $\delta(x)^{1/2}\equiv\delta(x)/[\delta(0)^{1/2}]$, the reduced state of either particle can be written as a density matrix with elements being $\varrho(\mathbf{p},\mathbf{p}^{\prime})=\int g^{\ast}(\mathbf{p}%
,\mathbf{q})g(\mathbf{p}^{\prime},\mathbf{q})\mbox{d}\mathbf{q}=|f(\mathbf{p}%
)|\delta^{3}(\mathbf{p}-\mathbf{p}^{\prime})/[\delta(0)^{3}]=|f(\mathbf{p}%
)|\delta_{\mathbf{p},\mathbf{p}^{\prime}}$. The similar holds for other singular states presented in Ref. [@r3].
S. Weinberg, *The Quantum Theory of Fields* (Cambridge University Press, Cambridge, England, 1996).
S.K. Berberian, *Measure and Integration* (The Macmillan Company, New York, 1962).
A. Peres and P. F. Scudo, **86**, 4160 (2001), **87**, 167901 (2001); E. Bagan et al., **87**, 257903 (2001).
|
---
abstract: 'The causal interpretation of quantum mechanics is applied to a homogeneous and isotropic quantum universe, whose matter content is composed by non interacting dust and radiation. For wave functions which are eigenstates of the total dust mass operator, we find some bouncing quantum universes which reachs the classical limit for scale factors much larger than its minimum size. However these wave functions do not have unitary evolution. For wave functions which are not eigenstates of the dust total mass operator but do have unitary evolution, we show that, for flat spatial sections, matter can be created as a quantum effect in such a way that the universe can undergo a transition from an exotic matter dominated era to a matter dominated one.'
author:
- 'N. Pinto-Neto'
- 'E. Sergio Santini'
- 'F. T. Falciano'
title: 'Quantization of Friedmann cosmological models with two fluids: dust plus radiation'
---
Introduction
============
The Bohm-de Broglie (BdB) interpretation [@bohm1][@bohm2][@hol] has been sucessfully applied to quantum minisuperspace models [@vink; @bola1; @kow; @hor; @bola2; @fab; @fab2], and to full superspace [@must] [@cons] [@tese]. In the first case, it was discussed the classical limit, the singularity problem, the cosmological constant problem, and the time issue. It was shown in scalar field and radiation models for the matter content of the early universe that quantum effects driven by the quantum potential can avoid the formation of a singularity through a repulsive quantum force that counteract the gravitational attraction. The quantum universe usually reach the classical limit for large scale factors. However, it is possible to have small classical universes and large quantum ones: it depends on the state vector and on initial conditions [@fab]. It was also shown that the quantum evolution of homogeneous hypersurfaces form the same four-geometry independently on the choice of the lapse function [@bola1].
In the present work we study the minisuperspace model given by a quantum Friedmann-Lemaître-Robertson-Walker (FLRW) universe filled with dust and radiation decoupled from each other. We write down the hamiltonian that comes from the velocity potential Schutz formalism [@schutz1]. After implementing a canonical transformation, the momentum associated to the radiation fluid $p_{T}$ and to the dust fluid $p_{\varphi}$ appear linearly in the superhamiltonian constraint. Both can be associated to time parameters, but physical reasons and mathematical simplicity led us to choose the coordinate $T$ associated with $p_{T}$ as the time parameter. This is equivalent to choose the (reversed) conformal time. We quantize this system obtaining a Schrödinger-like equation. We analyze its time dependent solutions applying the BdB interpretation in order to study the scale factor quantum dynamics.
We first consider an initial quantum state given by a gaussian superposition of the scale factor which is an eigenstate of the total dust mass operator (matter is not being created nor destroyed in such states), and we compute the solution at a general subsequent time by means of the propagator approach. We calculate the bohmian trajectories for the scale factor. For flat and negative curvature spatial sections, we find that the quantum solutions for the scale factor reach the classical behaviour for long times, but do not present any initial singularities due to quantum effects. In the same way, in the case of positive curvature spatial sections, the classical initial and final singularities are removed due to quantum effects, and the scale factor oscillates between a minimum and a maximum size. For large scale factor, the classical behaviour is recovered. However, such eigenfunctions of the total dust mass operator do not have unitary evolution. This led us to consider an initial state given by gaussian superpositions of the total dust matter content. In this situation, dust and radiation can be created and destroyed. We calculate general solutions for flat, negative and positive curvature spatial sections. In particular, for flat spatial sections, we construct a wave packet whose quantum trajectories represent universes which begin classically in an epoch where the dust matter has negative energy density (exotic dust matter), evolving unitarily to a configuration where quantum effects avoid the subsequent classical big crunch singularity, performing a graceful exit to an expanding classical model filled with conventional matter and radiation. There is thus a transition from an exotic matter era to a conventional matter one due to quantum effects.
This paper is organized as follows. In section \[bdbs\] we synthesize the basic features of the Bohm-de Broglie interpretation of quantum mechanics, which will be necessary to interpret our quantum model studied in other sections. In section \[drs\], we briefly summarize the velocity potential Schutz formalism, and we apply it to construct the hamiltonian of the FLRW universe filled with two perfect fluids, which are dust and radiation. We then review and analyze the classical features of the two perfect fluids FLRW model in order to have the results to be contrasted with the quantum models of the following sections. In section \[1f\], we present some new results concerning the existence of singularities in the quantization of the one fluid case. We show that, when the fluid is radiation, all quantum solutions do not present singularities. In section \[quantum\], we quantize the model with two fluids, and we compute the solutions of the Schrödinger like equation for two different initial conditions: the first being an eigenstate of the total dust matter operator, and the second a gaussian superposition of total dust matter eigenstates. We interpret the solutions according to the BdB view and we develop the main results of the paper. Section \[conclu\] is for discussion and conclusions.
The Bohm-de Broglie interpretation of quantum mechanics {#bdbs}
=======================================================
In this section, we briefly review the basic principles of the Bohm-de Broglie (BdB) interpretation of quantum mechanics. According to this causal interpretation, an individual physical system comprises a wave $\Psi(x,t)$, which is a solution of the Schrödinger equation, and a point particle that follows a trajectory ${x}(t)$, independent of observations, which is solution of the Bohm guidance equation
$$\label{bohmg}
p=m\dot{x}=\nabla S(x,t)|_{x=x(t)} ,$$
where $S(x,t)$ is the phase of $\Psi$. In order to solve Eq.(\[bohmg\]), we have to specify the initial condition $x(0)=x_0$. The uncertainty in the initial conditions define an ensemble of possible motions, [@bohm1][@bohm2][@hol].
It is sufficient for our purposes to analyze the Schrödinger equation for a non relativistic particle in a potential $V(x)$, which, in coordinate representation, is
$$\label{s}
i\frac{\partial\Psi(x,t)}{\partial t}=
\biggl[-\frac{1}{2m}\nabla^2 +V(x)\biggr]\Psi(x,t) .$$
Substituting in (\[s\]) the wave function in polar form, $\Psi=A \exp (iS)$, and separating into real and imaginary parts, we obtain the following two equations for the fields $A$ and $S$
$$\label{equacaoHJ}
\frac{\partial {S}}{\partial t}+\frac{\left(\nabla S\right)^2}{2m} +
V-\frac{1}{2m}\frac{\nabla^2 A}{A}=0 ,$$
$$\frac{\partial A^2}{\partial t}+\nabla.\left(A^2\frac{\nabla S}{m}\right)=0 .$$
Equation (\[equacaoHJ\]) can be interpreted as a Hamilton-Jacobi type equation for a particle submitted to a potential, which is given by the classical potential $V(x)$ plus a [*quantum potential*]{} defined as $$\label{qpote}
Q\equiv -\frac{1}{2m}\frac{\nabla^2 A}{A} .$$ It is possible to verify that the particle trajectory $x(t)$ satisfies the equation of motion
$$m\frac{d^2 x}{dt^2}=-\nabla V - \nabla Q .$$
The classical limit is obtained when $Q=0$. The BdB interpretation does not need a classical domain outside the quantized system to generate the physical facts out of potentialities. In a real measurement, we do not see superpositions of the pointer apparatus because the measurement interaction causes the wave function to split up into a set of non overlapping packets. The particle will enter in one of them, the rest being empty, and it will be influenced by the unique quantum potential coming from the sole non zero wave function of this region. The particle cannot pass to another branch of the superposition because they are separated by regions where $\Psi=0$, nodal regions.
In section \[quantum\], the FLRW minisuperspace model containing dust and radiation as two perfect decoupled fluids will be quantized. A preferred time variable can be chosen as one of the velocity potencials associated to the fluids (radiation), yielding a Schrödinger like equation. Then the BdB interpretation of our quantum model runs like in Ref. [@bola2], in close analogy to the non relativistic particle model described above. In the present case, however, the scale factor of the universe will not be the only degree of freedom: the velocity potential associated with the dust field and its canonical momentum, interpreted as the dust total mass, are also present. They satisfy a Hamilton-Jacobi equation modified by an extra term, the quantum potential, so that their time evolution will be different from the classical one. The main features of this classical model we describe in the following section.
Classical dust plus radiation model in the velocity potential Schutz formalism {#drs}
==============================================================================
We start by considering a perfect fluid in a FLRW universe model. The line element is given by
$$\label{metric}
ds^{2}=-N^{2}dt^{2}+a^{2}\left(t\right)\gamma_{ij}dx^{i}dx^{j}$$
where $N$ is the lapse function, $a$ is the scale factor, and $\gamma_{ij}$ is the metric of the three-dimensional homogeneous isotropic static spatial section of constant curvature $\kappa=1, 0,$ or $-1$.
Following the Schutz’s canonical formalism to describe the relativistic dynamics of a perfect fluid in interaction with the gravitational field [@schutz1], we introduce the five velocity potentials, $ \alpha, \beta, \theta, \varphi$ and $s$. The potentials $\alpha$ and $\beta$, which describe vortex motion, vanish in the FLRW model because of its symmetry. The potential $s$ is the specific entropy and $\theta$ can be related with the temperature of the fluid. By now $\varphi$ works only as a mathematical tool.
The four-velocity of the fluid is obtained from the velocity potentials as
$$\mathit{U}_{\nu}=\frac{1}{\mu}\left(\varphi,_{\nu} +\theta\,s,_{\nu}\right),$$
where $\mu$ stands for the specific enthalpy. The four velocity is normalized as $$g_{\alpha\,\beta}\mathit{U}^{\alpha}\mathit{U}^{\beta}=-1 .$$ Using this equation, it is possible to write the specific enthalpy $\mu$ as a function of the velocity potentials.
The action for a relativistic perfect fluid and the gravitational field in the natural units $c=\hbar=1$ is given by
$$\label{A}
I = -\frac{1}{6l_p^2}\int_{M}d^{4}x\sqrt{-g}\, ^{4}{\cal R}+
\int_{M}d^{4}x\sqrt{-g}\, p+
\frac{1}{3l_p^2}\int_{\partial M} d^{3}x\sqrt{h}h_{ij}K^{ij},$$
where $l_p\equiv(8\pi G/3)^{-1/2}$, $G$ is the Newton’s constant (hence $l_p$ is the Planck length in the natural units), $^{4}{\cal R}$ is the scalar curvature of the spacetime, $p$ is the pressure of the fluid, $h_{ij}$ is the three metric on the boundary $\partial M$ of the 4-dimensional manifold $M$, and $K^{ij}$ its extrinsic curvature. The velocity potentials are supposed to be functions of $t$ only, in accordance with the homogeneity of spacetime. The perfect fluid follows the equation of state $p=\lambda \rho$.
Substituting the metric (\[metric\]) into the action (\[A\]), using the formalism of Schutz [@schutz1] to write the pressure of the fluid as $$p= p_{0r}\left[ \frac{\dot{\varphi}+\theta\dot{s}}
{N(\lambda+1)}
\right]^{\frac{\lambda+1}{\lambda}}
\exp{\left(-\frac{s}{s_{0r}\lambda} \right)},$$ with $p_{0r}$ and $s_{0r}$ constants, computing the canonical momenta $p_{\varphi},p_{s}, p_{\theta}$ for the fluid and $p_a$ for the gravitational field, using the two constraints equations $p_{\theta}=0, \,\,\, \theta p_{\varphi}=p_s $, and performing the canonical transformation $$T=-\frac{p_s}{6^{1-3\lambda}}\exp \left( -\frac{s}{s_{0r}}\right)
p_\varphi^{-(\lambda+1)}\rho_{0r}^{\lambda}s_{0r},
\label{can1}$$ and $$\varphi_N=\varphi + (\lambda + 1)s_{0r} \frac{p_s}{p_\varphi},
\label{can3}$$ leading to the momenta $$p_{_T}=6^{1-3\lambda}
\frac{p_\varphi^{(\lambda+1)}}{\rho_{0r}^{\lambda}}
\exp\left(\frac{s}{s_{0r}}\right),
\label{can2}$$ and $$p_{\varphi_N}=p_{\varphi},
\label{can4}$$ we obtain for the final Hamiltonian (see Ref. [@Lapshinskii] for details), $$\label{superh}
H\equiv N{\cal H}=N\biggl(-\frac{p_{a}^2}{24a}-6\kappa a+
\frac{p_T}{a^{3\lambda}}\biggr),$$ where $N$ plays the role of a Lagrange multiplier whose variation yields the constraint equation $$\label{constr}
{\cal H}\approx 0,$$ where $\approx$ means ‘weakly zero’ (this phase space function is constrained to be zero, but its Poisson bracket to other quantities is not). We have redefined $\tilde{a}=\sqrt{V/(16\pi l_p^2)}\; a$ in order for $\tilde{a}$ be dimensioless, and $\tilde{N}=\sqrt{6}N$, where $V$ is the total comoving volume of the spatial sections. The tilda have been omitted. Considering now two decoupled fluids, one being radiation ($\lambda_r=1/3$), and the other dust matter ($\lambda_d=0$), the Hamiltonian reads: $$\label{hrm}
H \equiv N{\cal H}=N\biggl(-\frac{p_{a}^2}{24a}-6\kappa a+
\frac{p_{T}}{a}+p_{\varphi}\biggr)$$
The classical Hamilton equations are:
$$\label{aponto}
\dot{a}=\left\{a,H\right\}=-\frac{N}{12a}p_{a}
\Rightarrow p_{a}=-\frac{12a \dot{a}}{N} ,$$
$$\label{13}
\dot{p_{a}}=\left\{p_{a},H\right\}=
N\biggl(-\frac{p_{a}^2}{24 a^2}+6\kappa+
\frac{p_{T}}{a^2}\biggr),
\label{pr14}$$
$$\label{conf}
\dot{T}=\frac{N}{a} ,$$
$$\label{cosm}
\dot{\varphi}=\left\{\varphi,H\right\}=N ,$$
$$\label{15}
\dot{p}_{T}=\dot{p}_{\varphi}=0 \Rightarrow \mbox{$p_{T}$, $p_{\varphi}$ are constants}.$$
The superhamiltonian is constrained to vanish due to variation of the Hamiltonian with respect to the lapse function $N$, ${\cal H}\approx0$,
$$\label{ham0}
-\frac{p_{a}^2}{24a}-6\kappa a + \frac{p_{T}}{a} + p_{\varphi} = 0 .$$
The constraint (\[ham0\]) combined with Eqs. (\[aponto\]) and (\[15\]) yield the Friedmann’s equation
$$\label{friedmann}
\left(\frac{\dot{a}}{a}\right)^{2}=N^{2}\left[-\frac{\kappa}{a^{2}}+\frac{1}{6}
\left(\frac{p_{T}}{a^{4}}+
\frac{p_{\varphi}}{a^{3}}\right)\right]$$
Note that the conjugate momenta $p_T$ and $p_{\varphi}$, classical constants of motion, can be identified to the total content of dust and radiation in the universe:
$$p_{\varphi}=16\pi Ga^{3} \rho_{m} ,$$
$$p_{T}=16\pi Ga^{4} \rho_{r}.$$
Note also that Eq.(\[cosm\]) implies that $d\varphi=Ndt$, hence $\varphi$ is cosmic time, while Eq.(\[conf\]) yields $adT=Ndt$ so $T$ is conformal time. Consequently, choosing $N=1$ means taking coordinate time $t$ as cosmic time $\varphi$, while choosing $N=a$ imposes coordinate time to be conformal time $T$. Explicit analytic solutions of Eqs.(\[aponto\],\[pr14\],\[15\],\[friedmann\]) can be obtained only in the gauge $N=a$. In this gauge, besides the constraint (\[friedmann\]) with $N=a$, we obtain the simple second order equation, $$\label{2order}
a''+\kappa a=\frac{p_{\varphi}}{12},$$ where a prime means differentiation with respect to conformal time, which we denote $\eta$ from now on. The solutions read:
$$\label{cdr}
a = \left\{ \begin{array}{ll}
\left(\frac{2a_{eq}}{\eta_{eq}^{2}}\right)\left[1-\cos( \eta )+
\eta_{eq}\sin( \eta)\right] & \;\; ; \kappa=1 ,\\
\\
a_{eq}\left[2\frac{\eta }{\eta_{eq}}+\left(\frac{\eta }{\eta_{eq}}\right)^{2}\right]
& \;\; ; \kappa=0 ,\\
\\
\left(\frac{2a_{eq}}{\eta_{eq}^{2}}\right)\left[\cosh( \eta )+
\eta_{eq}\sinh( \eta)-1\right] & \;\; ; \kappa=-1 .
\end{array} \right.$$
The quantity $a_{eq}$ is defined to be the value of the scale factor at the equilibrium time where $\rho_{m}=\rho_{r}$, and $\eta_{eq}^{2}=3/\left(2\pi\, G \, \rho_{r}a^4\right)=24\,
a_{eq}/\mid p_{\varphi}\mid $.
As we will see in section (\[quantum\]), the presence of quantum effects can create exotic dust matter content. Hence, for comparison, we analyze a classical universe filled with exotic dust, which means $\rho_m<0$, i.e $p_{\varphi}<0$. For simplicity, let us focus on the flat spatial case.
In the presence of exotic dust, the behaviour of the scale factor is drastically different. From the Friedmann Eq.(\[friedmann\]), since $p_{\varphi}<0$, the radiation density must always be equal or greater then the dust density, otherwise the Friedmann equation $$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{1}{6}\left(\frac{p_{\eta}}{a^{2}}-
\frac{\mid p_{\varphi}\mid }{a}\right)$$ has no solution. For small values of the scale factor, the radiation term dominates. As the scale factor grows, the exotic dust term begins to be comparable to the radiation term up to the critical point where both are equal and $\dot{a}=0$. >From this point, the scale factor decreases until the universe recollapses. Note that Eq.(\[cdr\]) for $\kappa=0$ and $p_{\varphi}<0$ implies that $a''<0$ at all times. Hence, contrary to the normal dust matter case where after the big bang the universe expands forever \[see Eq.(\[cdr\]) for $\kappa=0$\], in the exotic case the universe recollapses in a big crunch. The qualitative evolution of the scale factor is plotted in figure 1:
The deceleration parameter in conformal time is given by $$\label{q}
q=-\frac{{a''}a}{{a'}^{2}}+1 .$$ It diverges when the scale factor reaches its maximum value (${a'}=0$ and $a''<0$).
FLRW Quantum Model with Radiation {#1f}
=================================
In this section, we present a general result concerning the presence of singularities in the quantization of a FLRW model with radiation. The hamiltonian constraint in this case is $$\label{hconstr}
{\cal H}=-\frac{p_{a}^2}{24a}-6\kappa a+\frac{p_\eta}{a} \approx 0,$$ and $\eta$ is conformal time, as discussed above.
Using the Dirac quantization procedure, the hamiltonian constraint phase space function $\cal{H}$ becomes an operator which must annihilate the quantum wave function: $\hat{\cal{H}}\Psi=0$. One then obtains in natural units the Wheeler-De Witt equation for the minisuperspace FLRW metric with radiation: $$\label{sch27}
i \frac{\partial }{\partial \eta}\Psi\left(a,\eta\right)=
\left(-\frac{1}{24}
\frac{\partial^{2}}{\partial a^{2}}+6\kappa a^{2}\right)\Psi \left(a,\eta\right).$$ Note that a particular factor ordering has been chosen and, because $p_\eta$ appears linearly in Eq.(\[hconstr\]), $\eta\rightarrow -\eta$ is chosen to be the time label in which the wave function evolves (the sign reversing was done in order to express this quantum equation in a familiar Schröedinger form [@Lapshinskii]).
The scale factor is defined only in the half line $[0,\infty)$, which means that the superhamiltonian (\[hconstr\]) is not in general hermitian. Hence, if one requires unitary evolution, the Hilbert space is restricted to functions in $L^{2}(0,\infty)$ satisfying the condition $$\label{cond1}
\frac{\partial\Psi}{\partial a}(0,\eta)=\alpha\Psi(0,\eta),$$ where $\alpha$ is a real parameter [@JMP371449].
We will now show that condition (\[cond1\]), together with the assumption that $\Psi(a,\eta)$ is analytic in $\eta$ at $a=0$, implies that general quantum solutions of Eq.(\[sch27\]), when interpreted using the BdB interpretation, yield quantum cosmological models without any singularity.
We can rearrange Eq.(\[sch27\]) in order to isolate the second spatial derivative: $$\label{d2psi}
\frac{\partial^{2}}{\partial a^{2}}\Psi \left(a,\eta\right)=
24\left[-i\frac{\partial }{\partial \eta}\Psi\left(a,\eta \right)+
6 \, \kappa \, a^{2}\Psi \left(a,\eta\right)\right].$$ Using the BdB interpretation, the scale factor equation of motion is given by the gradient of the phase $S\left(a,\eta\right)$ of the wave function $$a'=\frac{1}{12}S_a\left(a,\eta\right)=-\frac{i}{24}\frac{(\Psi \Psi_a^{ \ast }-\Psi_a
\Psi^{\ast })}{\Psi \Psi^{\ast }}\equiv f\left(a,\eta\right),$$ where the index $a$ means derivative with respect to $a$. Taking the boundary condition (\[cond1\]) at $a=0$, the velocity function $f\left(0,\eta\right)$ vanishes. Hence, if there is a time $\eta_0$ where $a(\eta_0)=0$, then $a'(\eta_0)=0$. For $a''$ one has: $${a''}=\frac{\partial f}{\partial a}a'+
\frac{\partial f}{\partial \eta}=\frac{\partial f}{\partial a}f+
\frac{\partial f}{\partial \eta}.$$ This is also zero at $a=0$ unless $\partial f/\partial a$ diverges there. However, $$\begin{aligned}
\frac{\partial f}{\partial a} & = & -\frac{i}{24}\frac{(\Psi \Psi_{aa}^{ \ast }-
\Psi_{aa} \Psi^{\ast })}{\Psi \Psi^{\ast }}+\frac{i}{24}
\frac{\biggl[\left(\Psi \Psi_a^{ \ast }\right)^{2}-
\left(\Psi_a \Psi^{\ast }\right)^{2}\biggr]}
{\left(\Psi \Psi^{\ast }\right)^{2}}\\
& = &\frac{(\Psi
\frac{\partial \Psi^{ \ast }}{\partial t}+\frac{\partial \Psi}{\partial t})
\Psi^{\ast }}{\Psi \Psi^{\ast }}
-\frac{i}{2}\frac{\biggl[\left(\Psi \Psi_a^{ \ast }\right)^{2}-
\left(\Psi_a \Psi^{\ast }\right)^{2}\biggl]}{\left(\Psi \Psi^{\ast }\right)^{2}}\\\end{aligned}$$ is obviously finite if condition (\[cond1\]) and analyticity of $\Psi$ in $\eta$ is satisfied at $a=0$. The case when $\left(\Psi \Psi^{\ast }\right)^{2}=0$ does not need to be analyzed because bohmian trajectories cannot pass through nodal regions of the wave function.
The same reasoning can be used for all higher derivatives $d^n a/d\eta^n$ at $a=0$ to show that they are all zero: one just have to use equation (\[d2psi\]) to substitute $\partial ^2\Psi/\partial a^2$ for $\partial \Psi/\partial \eta$ and condition (\[cond1\]) to substitute $\partial ^2\Psi/\partial a\partial \eta$ for $\alpha\partial \Psi/\partial \eta$ at $a=0$ whenever they appear, and then use analyticity of $\Psi$ in $\eta$ at $a=0$.
With these results, if there is a time $\eta_0$ where $a(\eta_0)=0$, expanding $a(\eta)$ in Taylor series around $\eta_0$ shows that $a(\eta)\equiv 0$. This means that the only singular bohmian trajectory is the trivial one of not having a universe at all! All non trivial quantum solutions have to be non singular.
Quantum behaviour of a FLRW Model With Dust and Radiation {#quantum}
=========================================================
As we have seen in section \[drs\], the superhamiltonian constraint for a FLRW model with non interacting dust and radiation is given by Eq.(\[ham0\]):
$$\label{ham27}
{\cal{H}}\equiv -\frac{p_{a}^2}{24a}-6\kappa a + \frac{p_{\eta}}{a} + p_{\varphi}\approx 0,$$
We see that both $p_\eta$ and $p_{\varphi}$ appear linearly in $\cal{H}$, and their canonical coordinates $\eta$ and $\varphi$ are, respectively, conformal and cosmic time. As in the preceeding section, from the Dirac quantization procedure one obtains the quantum equation $\hat{\cal{H}}\Psi=0$, which reads
$$\label{hamo}
\left(\frac{1}{24a}\frac{\partial^{2}}{\partial a^2}-6\kappa a
-\frac{i}{a}\frac{\partial }{\partial \eta}-
i\frac{\partial }{\partial \varphi}\right)\Psi(a,\varphi , \eta)=0,$$
where we have used the usual coordinate representation $\hat{p}=-i\partial/\partial q$.
Either $\eta$ or $\varphi$ can be chosen as time parameters on which $\Psi$ evolves. However, the classical solutions can be expressed explicitly only in conformal time $\eta$ \[see Eq.(\[cdr\])\]. Furthermore, cosmic time $\varphi$ depends on the constants characterizing each particular solution through $\varphi=\int d\eta a(\eta)$, and it is not the same parameter for all classical solutions (see Ref.[@Tipler] for deatils). Hence, we will take $\eta$ (in fact $-\eta$, for the reasons mentioned in the previous section) as the time parameter of the quantum theory[^1]. With this choice, and for a particular factor ordering, Eq.(\[hamo\]) can be written as: $$\label{hamo2}
i\frac{\partial }{\partial \eta}\Psi(a,\varphi , \eta)=
\left(-\frac{1}{24}\frac{\partial^{2}}{\partial a^2}+6\kappa a^2
+i a\frac{\partial }{\partial \varphi}\right)\Psi(a,\varphi, \eta).$$
Eigenstates of total matter content {#the}
-----------------------------------
In this subsection we only consider initial states $|\Psi(\eta_0)\rangle$ which are eigenstates of the total dust matter operator $\hat{p}_{\varphi}$. It follows that these states at time $\eta$, $|\Psi(\eta)\rangle$ will also be eigenstates of $\hat{p}_{\varphi}$ with the same eigenvalue because $[\hat{H},\hat{p}_{\varphi}]=0$. In other words, we consider that dust matter is not created nor destroyed. In such a way, we have $\hat{p}_{\varphi}|\Psi(\eta)\rangle=p_{\varphi}|\Psi(\eta)\rangle$ and the wave function in the $a$, $\varphi$ representation, $\langle a,\varphi|\Psi(\eta)\rangle=\Psi(a,\varphi,\eta)$, is given by
$$\label{mom}
\Psi(a,\varphi,\eta)=\Psi(a,\eta)e^{ip_{\varphi} \varphi}.$$
>From the Schrödinger’s equation (\[hamo2\]), we have for $\Psi(a,\eta)$ $$\label{hamoeig}
i\frac{\partial }{\partial \eta}\Psi(a,\eta)=\left(-\frac{1}{24}
\frac{\partial^{2}}{\partial a^2}+ 6\kappa a^2-p_{\varphi}a\right)\Psi(a,\eta),$$ which is the Schrödinger equation for a particle of mass $m=12$ in a one dimensional forced oscilator with frequency $w=\sqrt{\kappa}$ and constant force $p_{\varphi}$, which we write as
$$\label{hamof}
i\frac{\partial}{\partial \eta}\Psi(a,\eta)=\left(-\frac{1}{2m}
\frac{\partial^{2}}{\partial a^2}+\frac{mw^2}{2}a^2-p_{\varphi}a\right)\Psi(a,\eta)$$
The scale factor is defined only in the half line $[0,\infty)$, which means that the hamiltonian (\[ham27\]) is not in general hermitian. Hence, if one requires unitary evolution, the Hilbert subspace is resctricted to functions on $L^{2}(0,\infty;-\infty,\infty)$ satisfying the condition: $$\label{condition}
\int_{-\infty}^{\infty}{d\varphi \left[\frac{\partial \Psi^{\ast }_{2}
\left(a,\varphi , \eta\right)}{\partial a}\,
\Psi_{1}\left(a,\varphi , \eta\right)\right]_{a=0}}=\int_{-\infty}^{\infty}{d\varphi
\left[\frac{\partial
\Psi_{1}\left(a,\varphi , \eta\right)}{\partial a}\, \Psi^{\ast
}_{2}\left(a,\varphi , \eta\right)\right]_{a=0}}$$ for any $\Psi_{1}(a,\varphi , \eta), \Psi_2(a,\varphi , \eta) \in L^{2}(0,\infty;-\infty,\infty)$. In the special case considered in this section, this condition is reduced to
$$\label{cond}
\frac{\partial\Psi}{\partial a}(0,\eta)=\alpha\Psi(0,\eta),$$
where $\alpha$ is a real parameter [@JMP371449]. We will analyze the two extreme cases: $\alpha=0$ and $\alpha=\infty$, which are the simpler and usually studied in the literature on quantum cosmology [@bola2; @Lapshinskii; @alvarenga; @dewitt; @gotay; @JMP371449].
For the case $\alpha=0$ we have that $$\label{alfa0}
\frac{\partial\Psi}{\partial a}(0,\eta)=0,$$ which is satisfied for even functions of $a$. For the case $\alpha=\infty$, we have $$\Psi(0,\eta)=0 ,$$ which is satisfied for odd functions of $a$. Both of them address the boundary conditions of the wave packet near the singularity at $a=0$.
In order to develop the BdB interpretation, we substitute into the Schrödinger’s equation (\[hamof\]), the wave function in the polar form $\Psi=Ae^{iS}$, obtaining for the real part
$$\label{equacaoH-J1}
\frac{\partial S}{\partial t}+\frac{1}{2m}\left(\frac{\partial S}{\partial
a}\right)^{2}-a\, p_{\varphi}+
\frac{m\,w^{2}}{2}\,a^{2}+Q=0,$$
where
$$Q\equiv -\frac{1}{2m\,{A}}\frac{\partial^{2}{A}}{\partial a^{2}}$$
is the quantum potential. The Bohm guidance equation reads $$\label{bgr}
ma'=\frac{\partial S}{\partial a}.$$
A solution $\Psi(a,\eta)$ of Eq.(\[hamof\]) can be obtained from an initial wave function $\Psi_{0}(a)$ using the propagator of a forced harmonic oscillator. Let us do it for the two boundary conditions just presented.
### **The case of boundary condition $\alpha=0$** {#case1}
We denote the propagator $K^{\alpha=0}(2,1)\equiv K^{\alpha=0}(\eta_2,a_2;\eta_1,a_1)$, where $1$ stands for the initial time and initial scale factor $\eta_1, a_1$ respectively, and $2$ stands for their final values.
The propagator when the Hilbert space is restricted to $a>0$ can be obtained from the usual one (i.e with coordinate $-\infty<a<\infty$) which is associated to a particle in a forced oscilator $K(2,1)\equiv K(\eta_2,a_2;\eta_1,a_1)$ by means of $$\label{Kpar}
K^{\alpha=0}(2,1)=K(\eta_2,a_2;\eta_1,a_1)+K(\eta_2,a_2;\eta_1,-a_1)$$ This symmetry condition is necessary to consistently eliminate the contribution of the negative values of the scale factor [@IJMPA53029]. The usual propagator associated to a particle in a forced oscilator is [@feynman]:
$$K(2,1)=\sqrt{\frac{mw}{2 \pi i \sin(w\eta)}} \exp(i \, S_{cl})$$
where $\eta\equiv \eta_2 - \eta_1$ . The classical action $S_{cl}$ is given by
$$\begin{aligned}
S_{cl}&=&\frac{mw}{2\sin(w\eta)} \biggl\{\cos(w\eta)(a_{2}^2+a_{1}^2)-2a_{2}a_{1}+(a_{2}+
a_{1})\frac{2p_{\varphi}}{m w^2}[1-\cos(w\eta)]- \nonumber \\
&&[1-\cos(w\eta)]\frac{2 p_{\varphi}^2}{m^2 w^4}+\frac{p_{\varphi}^2}{m^2 w^4}\sin(w\eta)
w\eta \biggr\} .\end{aligned}$$
We assume that for $\eta_1=0,$ the initial wave function is given by $$\label{initwave}
\Psi_0(a)=\biggl(\frac{8\sigma}{\pi}\biggr)^{1/4}\exp(-\sigma a^2),$$ where $\sigma>0$. The wave function in a future time $\eta_2$ is
$$\Psi(a_2, \eta_2)=\int_{0}^{\infty} K^{\alpha=0}(2,1)\Psi_0(a_{1})da_{1}=
\int_{-\infty}^{\infty} K(2,1)\Psi_0(a_{1})da_{1},$$
where the even caracter of $\Psi(a,0)$ has been taken into account to extend the integral. Integrating and renaming $\eta\equiv\eta_2,\,a\equiv a_2$ we have
$$\begin{aligned}
\label{psi1t}
\Psi^{\alpha=0}(a,\eta)=\biggl(\frac{8 \sigma}{\pi}\biggr)^{1/4}
\sqrt{\frac{mw}{i\cos(w\eta)
[2 \sigma \tan(w\eta)-imw]}} \exp\biggr\{\frac{imw}{2 \tan(w\eta)}
\biggl[a^2 && + \nonumber \\
i\frac{mw}{\cos^2(w\eta)[2 \sigma \tan(w\eta)-imw]}
\biggl(-a+\frac{p_{\varphi}}{m w^2}[1-\cos(w\eta)]\biggr)^2+\frac{2ap_{\varphi}}{m}
\frac{[1-\cos(w\eta)]}{w^2 \cos(w\eta)}+\nonumber \\
\frac{2 p_{\varphi}^2}{m^2} \biggl(\frac{[\cos(w\eta)-1]}{w^4 \cos(w\eta)} +
\eta\frac{\tan(w\eta)}{w^3}\biggr)\biggr]\biggr\} .\end{aligned}$$
\[0\]
We consider the case $\kappa=0$, which is obtained by taking the limit of the wave function given by Eq. (\[psi1t\]) for $w\rightarrow 0$. We compute its phase $S$ from $\Psi\equiv {\it A} e^{iS}$ and calculate the derivative $\partial S/\partial a$. In this way we have, for the Bohm guidance equation Eq.(\[bgr\]), $$a'-\frac{4\sigma^2 \eta}{4\sigma^2 \eta^2+m^2}a=
\frac{1}{m}\frac{(2\sigma^2 \eta^2+m^2)}{(4\sigma^2 \eta^2+m^2)}p_{\varphi} \eta .$$ Comparing with the radiation case studied in [@bola2], we see that here it appears a term proportional to $p_{\varphi}$ in the RHS of the Bohm equation. The general solution is:
$$\label{gk0}
a(\eta)=C_0 \sqrt{4\sigma^2 \eta^2+m^2}+\frac{p_{\varphi}}{2m}\eta^2 ,$$
where $C_0$ is a positive integration constant. We can see that, contrary to the classical solution (\[cdr\]), there is no singularity at $\eta=0$. The quantum effects avoid it. Furthermore, for long times $\eta\gg m/2\sigma$, Eq. (\[gk0\]) reproduces the classical behaviour (\[cdr\]) for the scale factor.
For the case in which the evolution starts from a shifted gaussian wave function
$$\Psi_0(a)=\biggl(\frac{8\sigma}{\pi}\biggr)^{1/4}\exp\left[-\sigma (a-a_0)^2\right],$$
the Bohm guidance relation contains an additional term yielding the general solution $$a(\eta)=C_0 \sqrt{4\sigma^2 \eta^2+m^2}+\frac{p_{\varphi}}{2m}\eta^2+\frac{a_0}{2},$$ which has exactly the same behaviour, apart from the shift on the minimal value of the scale factor by the $a_0/2$ term.
Setting $w=\sqrt{\kappa}=1$ in the wave function given by Eq. (\[psi1t\]) and computing its phase $S$, we obtain for the bohmian trajectories $$\label{gk1}
a(\eta)=C_0 \sqrt{4\sigma^2\sin ^2(\eta)+m^2\cos ^2(\eta)}+
\frac{p_{\varphi}}{2m}[1-\cos(\eta)],$$ where $C_0$ is a positive integration constant. This is a non singular cyclic universe, see figure 2, which presents classical behaviour for $\eta$ such that $\mid \tan (\eta)\mid\gg m/2$ \[see Eq. (\[cdr\])\]. Quantum effects avoid the classical big bang and big crunch.
Setting now $w=\sqrt{\kappa}=i$ in the wave function (\[psi1t\]) yields the bohmian trajectories:
$$\label{gk-1}
a(\eta)=C_0 \sqrt{4\sigma^2\sinh ^2(\eta)+m^2\cosh ^2(\eta)}+
\frac{p_{\varphi}}{2m}[\cosh(\eta)-1].$$
Again, $C_0$ is a positive integration constant. This is a non singular ever expanding universe which presents classical behaviour for $\eta$ such that $\mid \tanh (\eta)\mid\gg m/2$ \[see Eq. (\[cdr\])\]. Quantum effects avoid the classical big bang.
As in the $\kappa =0$ case, a shift in the center of the initial gaussian will not modify these solutions qualitatively.
For the boundary conditions $\alpha =\infty$, or $\Psi(0,t)=0$, the propagator $K^{\alpha=\infty}(2,1)$ can be obtained from the usual (i.e, with coordinate $-\infty<a<\infty$) propagator associated to a particle in a forced oscilator $K(2,1)$ by means of
$$\label{Kimpar}
K^{\alpha=\infty}(2,1)=K(\eta_2,a_2;\eta_1,a_1)-K(\eta_2,a_2;\eta_1,-a_1) .$$
In order to satisfiy the condition $\Psi(0,\eta)=0$, we take as the initial wave function a wave packet given by
$$\Psi_0(a)=\biggl(\frac{128 \sigma^3}{\pi}\biggr)^{1/4}a \exp(-\sigma a^2) ,$$
where $\sigma>0$. Following a similar procedure as in the case $\alpha=0$, we calculate the wave function by propagating the initial wave function as
$$\Psi(a_{2}, \eta_{2})=\int_{0}^{\infty} K^{\alpha=\infty}(2,1)\Psi_0(a_{1})da_{1}=
\int_{-\infty}^{\infty} K(2,1)\Psi_0(a_1)da_{1} ,$$
where the odd caracter of $\Psi$ has been used in order to extend the integral. Integrating this expression and renaming $a\equiv a_{2}$ and $\eta \equiv \eta_{2}$ with $\eta_1=0$, we have
$$\Psi^{\alpha=\infty}(a,\eta)=\biggl(\frac{-C}{2 D}\biggr)\Psi^{\alpha=0}(a,\eta)$$
where
$$C\equiv \frac{imw}{\sin(w\eta)}\biggl[-a+\frac{p_{\varphi}}{mw^2}(1-\cos(w\eta)\biggr]$$
and
$$D\equiv \frac{imw}{2\tan(w\eta)}-\sigma$$
The phase of $\Psi^{\alpha=\infty}(a,\eta)$ can be expressed as the sum :
$${\rm phase}[\Psi^{\alpha=\infty}(a,\eta)]=
{\rm phase}\biggl(\frac{-C}{2 D}\biggr) + {\rm phase}
[\Psi^{\alpha=0}(a,\eta)],$$
and it is easy to see that the phase of $(-C/2 D)$ is independent of $a$. Then, $[\partial {\rm phase}(\Psi^{\alpha=\infty}(a,\eta))]/\partial a=
[\partial{\rm phase} [\Psi^{\alpha=0}(a,\eta)]]/\partial a$, and the Bohm guidance relations are the same as in the previous cases. Therefore, the solutions are the same.
The quantum cosmological models obtained in this subsection have the nice properties of being non singular and presenting classical behaviour for large $a$. However, they suffer from a fundamental problem: the wave function (\[psi1t\]) from which they are obtained does not have an unitary evolution. The reason is that propagators constructed from Eq.’s (\[Kpar\]) and (\[Kimpar\]) do not in general preserve the hermiticity condition (\[cond\]) imposed on the wave functions: it depends on the classical potential. In Ref.[@IJMPA53029], there are obtained the potentials which allow propagators in the half line ($a>0$) to preserve unitary evolution. The potentials of the previous section are some of them but the potentials of the present one are not. Hence, even though the initial wave function Eq.(\[initwave\]) satisfies the hermiticity condition, the wave function (\[psi1t\]) does not. Let us then explore the more general case of initial superpositions of the total dust mass operator eigenstates.
Analysis of wave packets given by superpositions of total dust mass eigenstates
-------------------------------------------------------------------------------
In this subsection we consider the case of a general solution of Eq.(\[hamo2\]) which is not necessarily one of the eigenstates of $\hat{p}_{\varphi}$, the total dust mass operator.
Following the BdB interpretation of quantum mechanics, we substitute in Eq.(\[hamo2\]) the wave function in polar form: $\Psi = A\left(a,\varphi,\eta \right)
\exp\left\{{i}S\left(a,\varphi,\eta \right)\right\}$. The dynamical equation splits in two real coupled equation for the two real functions $S$ and $ A$ (recall that $w=\sqrt{\kappa}$ and $m=12$). $$\label{equacaoH-J}
\frac{\partial S}{\partial \eta}+\frac{1}{2m}\left(\frac{\partial S}{\partial
a}\right)^{2}-a\frac{\partial S}{\partial \varphi}+
\frac{m\, w^{2}}{2} a^{2}+ Q = 0 ,$$
$$\label{equacaocontinuidade}
\frac{\partial A^{2}}{\partial \eta}+\frac{\partial}{\partial \varphi}\left(a\,
A^{2}\right)+\frac{\partial}{\partial
a}\left( A^{2}\frac{1}{m}\frac{\partial S}{\partial a}\right)=0 ,$$
where $$Q\equiv -\frac{1}{2m\, A}\frac{\partial^{2}{A}}{\partial a^{2}} .$$
Equation (\[equacaoH-J\]) is the modified Hamilton-Jacobi equation where $Q\left(a,\varphi,\eta\right)$ is the quantum potential which is responsible for all the peculiar non classical behaviours. When the quantum potential is zero, the equation is exactly the classical Hamilton-Jacobi equation. The momenta are given by the Bohm’s guidance equations $$\begin{aligned}
p_{a}&\equiv & \frac{\partial S\left(a,\varphi,\eta\right)}{\partial a},\\
p_{\varphi}&\equiv& \frac{\partial S\left(a,\varphi,\eta\right)}{\partial \varphi} .
\label{bphi}\end{aligned}$$ Note also that $$\label{rad}
p_\eta=\frac{\partial S\left(a,\varphi,\eta\right)}{\partial \eta}$$ is the total ‘energy’ of the system, which is interpreted, from its classical meaning, as the total amount of radiation in the universe model.
In the causal interpretation, equation (\[equacaocontinuidade\]) is a continuity equation where ${A}^{2}$ is a probability density. The generalised velocities can easily be identified as $$\begin{aligned}
{a'} &\equiv & \frac{1}{m}\frac{\partial S\left(a,\varphi,\eta\right)}{\partial a} ,
\label{ba}\\
{\varphi'}&\equiv & a .\label{velocphi}\end{aligned}$$ Consider now the classical limit ($Q=0$). Then the solution of the principal Hamilton function ($S$) is just $S=W\left(a\right)-E\eta+p_{\varphi}\varphi$, where $E$ and $p_{\varphi}$ are constants. Since $p_{\varphi}$ is proportional to the total amount of dust matter in the universe, and $E$ to the total amount of radiation, there is no creation or annihilation of dust matter nor radiation. However, in the presence of a quantum potential, this solution is no longer valid, opening the possibility of non conservation of matter and radiation due to quantum effects.
### Formal Solutions
We now turn to the problem of solving the Schrödinger’s equation (\[hamo2\]).
For the case of flat spatial section ($\kappa=0$), equation (\[hamo2\]) simplify to
$$i\frac{\partial \Psi \left(a,\varphi ,\eta\right)}{\partial \eta}=
-\frac{1}{2m}\frac{\partial^{2}
\Psi \left(a,\varphi ,\eta\right)}{\partial
a^{2}} +i a\frac{\partial \Psi \left(a,\varphi ,\eta\right)}{\partial \varphi}
\label{eqkzero}$$
To solve this equation we make the ansatz $$\Psi \left(a,\varphi ,\eta\right)=\chi\left(a\right) \exp \left(-\frac{i}{2m}\beta \,
\eta\right) \exp\left(\frac{i}{2m}\upsilon \, \varphi\right) ,$$ where $\chi\left(a\right)$ must satisfy the differential equation[^2]
$$\frac{\partial^{2} \chi\left(a\right)}{\partial a^{2}}+\upsilon a
\chi\left(a\right)+\beta\chi\left(a\right)=0.$$
This is essentially an Airy equation with solution given by
$$\chi\left(a\right)=\sqrt{a+\frac{\beta}{\upsilon}}\left\{A\,Z_{\frac{1}{3}}
\left[\frac{2\sqrt{\upsilon}}{3}\left(a+\frac{\beta}{\upsilon}
\right)^{\frac{3}{2}}\right]+ B\,Z_{-\frac{1}{3}} \left[\frac{2\sqrt{\upsilon}}{3}
\left(a+\frac{\beta}{\upsilon}\right)^{\frac{3}{2}}\right]\right\}$$
The $Z_{\frac{1}{3}}$ function is the first kind Bessel function of degree $\frac{1}{3}$, and the $A$ and $B$ can be any functions of $\upsilon$ and $\beta$.
The general solution is a superposition given by $$\begin{aligned}
\Psi \left(a,\varphi,\eta\right)= \int{d\beta\, d\upsilon
\exp\left\{-\frac{i}{2m}\beta\,\eta\right\}\exp\left\{\frac{i}{2m}\upsilon\,
\varphi\right\}\sqrt{a+\frac{\beta}{\upsilon}}} \times \\
\times \left\{ A\left(\beta,\upsilon\right) \, Z_{\frac{1}{3}} \left[
\frac{2\sqrt{\upsilon}}{3}
\left(a+\frac{\beta}{\upsilon}\right)^{\frac{3}{2}}\right] +B\left(\beta,\upsilon\right)
\,Z_{-\frac{1}{3}}\left[\frac{2\sqrt{\upsilon}}{3}
\left(a+\frac{\beta}{\upsilon}\right)^{\frac{3}{2}}\right]\right\}\\\end{aligned}$$
In the positive curvature case ($\kappa=1$), Eq.(\[hamo2\]) reads
$$\label{wheeler-dewittk1}
i\frac{\partial \Psi \left(a,\varphi ,\eta\right)}{\partial \eta}=
-\frac{1}{2m}\frac{\partial^{2}
\Psi \left(a,\varphi ,\eta\right)}{\partial
a^{2}} +\frac{m}{2}a^{2} \Psi \left(a,\varphi,\eta\right)+i a\frac{\partial \Psi
\left(a,\varphi ,\eta\right)}{\partial\varphi} .$$
There is a canonical transformation which simplifies the problem. Let us define new variables given by $$\begin{aligned}
\xi \equiv \sqrt{m}\, a - \frac{p_{\varphi} }{\sqrt{m}} &; & \sigma \equiv
-\sqrt{m}\, \varphi + \frac{p_{a}
}{\sqrt{m w}} ,\\
p_{\xi} \equiv \frac{p_{a}}{\sqrt{m}} &; & p_{\sigma} \equiv
-\frac{p_{\varphi}}{\sqrt{m}} .\end{aligned}$$ Using these new variables, the hamiltonian decouples in two parts, one describing a harmonic oscillator and the other a free particle:
$$\label{schk1}
\hat{H}=\underbrace{\frac{1}{2}\left(\hat{p}_{\xi}^{2}+
\hat{\xi}^{2}\right)}_{\mbox{\it harmonic
oscillator}}-\underbrace{\frac{1}{2}\hat{p}_{\sigma}^{2}}_{\mbox{ \it free particle}} .$$
Decomposing the wave function as
$$\Psi\left(\xi,\sigma,\eta \right)=
\chi\left(\xi\right)\exp\left\{-i\left(\epsilon\,\eta+\sqrt{2\,k}\,\sigma\right)\right\},$$
we immediately recognize that $\chi\left(\xi\right)=
\exp\left\{-\frac{\xi^{2}}{2}\right\}h_{n}\left(\xi\right)$, where $h_{n}$ are the Hermite polinomial of degree $n$. Just as for the harmonic oscillator, the index $\epsilon$ is constrained to take the values $$\label{landau}
\epsilon_{n}=k+\left(n+\frac{1}{2}\right) ,$$ where $k$ can take any real positive value while $n$ is a positive integer. Eq. (\[landau\]) determines a set of [*Landau levels*]{} for the cosmological model [@landaulevels]. The most general solution is a superposition given by
$$\begin{aligned}
\label{solucaogeral}
\Psi \left( \xi ,\sigma ,\eta \right) &=& \sum_{n=0}^{\infty}{
\int{dk\,\chi_{n}\left(\xi\right)\left[D_{n}\left(k\right)\exp\left\{i\sigma\,
\sqrt{2\,k}\right\}+ \right. }} \nonumber \\
&& \left. G_{n}\left(k\right)\exp\left\{-i\sigma\,\sqrt{2\,k}\right\}\right]\times
\exp\left\{-i\, \epsilon_{n}\, \eta\right\}.\end{aligned}$$
The quantities $D_{n}\left(k\right)$ and $G_{n}\left(k\right)$ are arbritary coefficients that can depend on the parameter $k$. Recall that we have performed a canonical transformation that mix coordinates and momenta, and these are not the proper variables to apply the causal interpretation. Instead, it is imperatif to apply the inverse transformation to the coordinate basis before using the guidance relations. This is a necessary requirement to maintain the consistency of the causal interpretation of quantum mechanics [@CQG141993]-[@PR89319B].\
For the negative curvature spatial section ($\kappa=-1$), the general solutions are hypergeometric functions whose asymptotic behaviours are rather complicated to study in order to obtain reasonable boundary conditions. Hence, we will not treat this case here. We proceed to the analysis of an interesting particular solution.
### Transition from exotic dust to dust in the flat case
The quantum states of the matter and radiation FLRW universe studied in section \[0\] are eigenstates of the total dust matter operator ${\hat{p}}_{\varphi}$. The total wave function is given by $\Psi(a,\varphi,\eta)= \Psi(a,\eta) \exp(i\varphi p_{\varphi})$ where $\Psi(a,\eta)$ is given by Eq.(\[psi1t\]). Taking the limit $w \rightarrow 0$ in that equation, we obtain the wave function $\Psi(a,\eta)$ for the case of flat spatial section, $\kappa=0$ which, after renaming the eigenvalues of total mass by $\upsilon\equiv p_{\varphi}$, is given by
$$\begin{aligned}
\Psi_{\upsilon}\left(a,\eta\right)= \left(\frac{8\sigma m^{2}}{\pi\,\mu}
\right)^{\frac{1}{4}}\exp\left\{
-\frac{m^{2}\sigma}{\mu}\left(a-\frac{\upsilon\,\eta^{2}}{2m}\right)^{2}-
i\frac{\upsilon^{2}\eta^{3}}{6m}-i\frac{\theta}{2}+\right.& \nonumber \\
\left.+i\frac{m}{2 \eta}\left[
\left(a+\frac{\upsilon\,\eta^{2}}{2m}\right)^{2}-\frac{m^{2}}{\mu}
\left(a-\frac{\upsilon\,\eta^{2}}{2m}\right)^{2}\right] \right\} & ,\end{aligned}$$
where $$\begin{aligned}
&&\mu= 4\sigma^{2}\eta^{2}+m^{2} ,\\
&&\theta= \arctan \left(\frac{2\sigma \eta}{m}\right) .\end{aligned}$$
Now we consider a more general situation than in section \[0\]. We suppose an initial state at $\eta=0$ which is given by a gaussian superposition of eigenstates of total matter
$$\label{superposicaoguassiana0}
\Psi \left(a,\varphi,0\right)=\int_{-\infty}^{\infty}{d \upsilon\,
\exp^{-\gamma\left(\upsilon-\upsilon_{0}\right)^{2}}\Psi_{\upsilon}\left(a,0\right)\,
\exp\{-i\,\varphi\,\upsilon\}} .$$
Then, the state at time $\eta$ is given by $$\label{superposicaoguassiana}
\Psi \left(a,\varphi,\eta \right)=\int^{\infty}_{-\infty}{d \upsilon \,
\exp^{-\gamma\left(\upsilon-\upsilon_{0}\right)^{2}}\Psi_{\upsilon}\left(a,\eta \right)\,
\exp\{-i\,\varphi\,\upsilon\}} .$$ In this way, we have a square-integrable wave function. We find
$$\begin{aligned}
\Psi \left(a,\varphi,\eta\right)= \left(\frac{8\sigma \pi
m^{2}}{\mu\,\nu}\right)^{\frac{1}{4}}\exp\left\{\left(\frac{\Re\left(F\right)}{4\nu}-\frac{\sigma
m^{2}}{\mu}\right)a^{2}+\frac{\Re\left(G\right)}{4\nu}\,a\,\varphi
+\frac{\Re\left(J\right)}{4\nu}\varphi^{2}+\frac{\Re\left(L\right)}{4\nu}\,a+\right. & \\
\left. +\frac{\Re\left(M\right)}{4\nu}\varphi +\frac{\Re\left(P\right)}{4\nu}\; + i\left[ \,
\left(\frac{\Im\left(F\right)}{4\nu}+\frac{m}{2\mu \eta}\left(\mu-m^{2}\right)
\right)a^{2}+\frac{\Im\left(G\right)}{4\nu}a\, \varphi +\right. \right. & \\
\left.\left. +\frac{\Im\left(J\right)}{4\nu}\varphi^{2}+\frac{\Im\left(L\right)}{4\nu}a
+\frac{\Im\left(M\right)}{4\nu}\varphi+\frac{\Im\left(P\right)}{4\nu} \right]-
i\frac{\theta+\tau}{2} ,
\right\} &\end{aligned}$$
where we defined
$$\begin{aligned}
&&\nu= \left(\gamma+\frac{\sigma \eta^{4}}{4\mu}\right)^{2}+\frac{\eta^{6}}{\left(24m \mu
\right)^{2}}\left(\mu +3m^{2}\right)^{2}\\
&&\tau= \arctan \left[\frac{\eta^{3}(\mu+3m^{2})}{24m(\gamma
\mu+\sigma \eta^4)}\right]\\
&&F= \left[\frac{m\sigma \eta^{2}}{\mu}+i\frac{\eta}{2\mu }\left(\mu+m^{2}\right)
\right]^{2}\left[\gamma+\frac{\sigma \eta^{4}}{4\mu}-i\frac{\eta^{3}}{24m\mu
}\left(\mu+3m^{2}\right)\right]\\
&&G= -2\, i\left[\frac{m\sigma \eta^{2}}{\mu}+i\frac{\eta}{2\mu }
\left(\mu+m^{2}\right)
\right]\left[\gamma+\frac{\sigma \eta^{4}}{4\mu}-i\frac{\eta^{3}}{24m\mu
}\left(\mu+3m^{2}\right)\right]\\
&&J= -\left[\gamma+\frac{\sigma \eta^{4}}{4\mu}-i\frac{\eta^{3}}{24m\mu
}\left(\mu+3m^{2}\right)\right]\\
&&L= -2\,i\,\gamma\upsilon_{0}\,G \\
&&M= 4\,i\,\gamma\upsilon_{0} \,J \\
&&P= -4\gamma^{2}\upsilon_{0}^{2} \,J\end{aligned}$$
and $\Re$ and $\Im$ stands for the real and imaginary part, respectively.
If one calculates the squared norm of the wave function, one obtains $$\int_{0}^{\infty}{da}\int_{-\infty}^{\infty}{d\varphi}\left\|\Psi\right\|^{2}=
\sqrt{\frac{8\pi^{3}}{\gamma}}\left[1+\frac{1}{\sqrt{\pi}} {\rm{erf}}
\left(\frac{\upsilon_{0}
\eta^{2}}{2m}\right)\right],$$ where ${\rm{erf}}(x)$ is the error function. The only dependence on time can be eliminated by choosing the gaussian to be centered at $\upsilon_{0}=0$. With this choice we garantee unitary evolution of the total wave function. >From equations (\[bphi\])-(\[velocphi\]), the trajectories can be computed by solving the given system of equations
$$\begin{aligned}
& &{a'}= \frac{2}{m} \left[\frac{\Im\left(F\right)}{4\nu}+\frac{m}{2\mu
\eta}\left(\mu-m^{2}\right)\right]\,a\,+ \frac{\Im\left(G\right)}{4m\nu}\,
\varphi \label{aevol}\\
& &{\varphi'}= a \\
& &p_{\varphi} = \left[ 2\frac{\Im\left(J\right)}{4\nu}\, \varphi
+\frac{\Im\left(G\right)}{4\nu}\,a\right]\end{aligned}$$
Note that $p_{\varphi}$ is no longer constant. We integrated numerically these equations with the renormalisation condition $a\left(0\right)=1$. The quantum potential $Q\equiv
-\frac{1}{2m\, A}\frac{\partial^{2} A}{\partial a^{2}}$ is non zero only close to the origin as shown on figure 3. Hence, we expect that quantum effects be relevant only in this region. Far form the origin, the scale factor must behave classically.
The behaviour of $p_{\varphi}$ is plotted on figure 4.
>From this plot we can see that far from the origin $p_{\varphi}$ is constant. This is in accordance with classical behaviour as long as the quantum potential is zero in this region. The surprising feature is that in the far past the universe was filled with a classical exotic dust ($p_{\varphi} < 0$).
>From Eq.(\[rad\]), one can also compute the amount of radiation. Figure 5 shows the result. Again, far from the origin, radiation is conserved while in the origin, due to quantum effects, it is not conserved.
For the evolution of the scale factor, numerical integration of equation (\[aevol\]) yields the plot of figure 6.
In the far positive region, the scale factor behaves classically, as expected, and matter is conserved. On the other hand, in the far negative region the scale factor also behaves classically but with a universe filled with exotic dust, and here again matter is conserved (compare this region with figure 1). Both regions have a consistent classical behaviour. Hence, the universe begins classically from a big bang filled with exotic dust and conventional radiation. It evolves until it reaches a configuration when quantum effects avoid the classical big crunch while transforming exotic dust into normal dust. From this point on, the universe expands classically filled with conventional dust and radiation.
Conclusions {#conclu}
===========
In the present work we studied some features of the minisuperspace quantization of FLRW universes with one and two fluids. For the one fluid case (radiation), we have generalized results in the literature by showing that all bohmian trajectories coming from reasonable general solutions of the wave equation obtained through the assumptions of unitarity and analyticity at the origin, do not present any singularity. Hence, this quantum minisuperspace theory is free of singularities.
For the two fluids case (non interacting radiation and dust), we first obtained bohmian quantum universes free of singularities reaching the classical limit for large scale factors. However, these trajectories arise from eigenfunctions of the total dust mass operator whose time evolution is not unitary. When considering the general case, we managed to obtain a wave solution presenting unitary evolution with some surprising effects. Now dust and radiation can be created but the new feature is the possibility of creation of exotic fluids. We have shown that dust matter can be created as a quantum effect in such a way that the universe can undergo a transition from an exotic dust matter era to a conventional dust matter one. In this transition, one can see from figure 5 that radiation also becomes exotic due to quantum effects, helping the formation of the bounce.
The fluid approach is not fundamental, but we expect that it can be quite accurate in describing quantum aspects of the Universe, in the same way the Landau description of superfluids in terms of fluid quantization was capable of showing many quantum features of this system [@landau2]. After all, creation and annihilation of particles as well as quantum states with negative energy are usual in quantum field theory. The formalism developed in the present paper seems to be a simple and calculable way to grasp these features of quantum field theory. Their physical applications may be important: exotic fluids are relevant not only in causing cosmological bounces and avoiding cosmological singularities [@peter], but also for the formation of wormholes [@thorn; @matt] and for superluminal travels [@27]. These are some developments of the present paper we want to explore in future works.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We would like to thank [*Conselho Nacional de Desenvolvimento Científico e Tecnológico*]{} (CNPq) of Brazil and [*Centro Latinoamericano de Física*]{} (CLAF) for financial support. We would also like to thank ‘Pequeno Seminario’ of CBPF’s Cosmology Group for useful discussions.
[99]{}
D. Bohm, Phys. Rev. [**85**]{} (1952) 166. D. Bohm, Phys. Rev. [**85**]{} (1952) 180. P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, (Cambridge University Press, Cambridge, 1993). J. C. Vink, Nucl. Phys. [**B369**]{} (1992) 707. J. A. de Barros and N. Pinto-Neto, Int. J. of Mod. Phys. [**D7**]{} (1998) 201. J. Kowalski-Glikman and J. C. Vink, Class. Quantum Grav. [**7**]{} (1990) 901. E. J. Squires, Phys. Lett. [**A162**]{}, (1992) 35. J. A. de Barros, N. Pinto-Neto and M. A. Sagioro-Leal, Phys. Lett. [**A241**]{} (1998) 229. R. Colistete Jr., J. C. Fabris and N. Pinto-Neto, Phys. Rev. [**D57**]{} (1998) 4707. R. Colistete Jr., J. C. Fabris and N. Pinto-Neto, Phys. Rev. [**D62**]{} (2000) 83507. N. Pinto-Neto and E. Sergio Santini, Phys. Rev. [**D59**]{} (1999) 123517. N. Pinto-Neto and E. Sergio Santini, Gen. Relativ. Gravit. 34 (2002) 505. E. Sergio Santini, PhD Thesis, CBPF, Rio de Janeiro, May 2000, gr-qc/0005092. B.F. Schutz, Phys. Rev. [**D2**]{} (1970) 2762 ; [**4**]{} (1971) 3559. C. W. Misner, in: Magic Without Magic: John Archibald Wheeler, ed. J.R. Klauder, (Freeman, San Fransisco, CA, 1972). V. G. Lapshinskii and V. A. Rubakov, Theor. Math. Phys. [**33**]{} (1977) 1076. F. J. Tipler, Phys. Rep. [**137**]{} (1986) 231. F. G. Alvarenga, J. C. Fabris, N. A. Lemos and G. A. Monerat, gr-qc/0106051. Bryce S. DeWitt, Phys. Rev.[**D**]{} [**160**]{} 5 (1967) 1113. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, New York , 1965. M. J. Gotay and J. Demaret, Phys. Rev. [**D28**]{} (1983) 2402. L. D. Landau, Z. Phys. [**64**]{} (1930) 629, Reprinted and translated in [*Collected papers of Landau*]{}, Paper 4, Edited by D. Ter Haar, (Pergamon Press Ltd and Gordon and Breach, Science Publishers, Oxford, 1965). M.Abramowitz, I.A. Stegun- “Handbook of Mathematical Functions”, (National Bureau of Standards, Washington D.C., 1964). N. Pinto-Neto, in: Cosmology and Gravitation II, Proceedings of the VIII Brazilian School of Cosmology and Gravitation, Edited by Mário Novello, (Editions Frontieres 1995). N.A. Lemos, J. Math. Phys. [**37**]{} (1996) 1449. E. Farhi, Int. J. Mod. Phy. A [**5**]{} (1990) 3029. J. Acacio de Barros, N. Pinto-Neto; Class. Quantum Grav. [**14**]{} (1997) 1993. D. Bohm; Phys. Rev. [**89**]{} (1952) 319. I. M. Khalatnikov; An Introduction to the Theory of Super-fluidity (W. A. Benjamin, New York, 1965). P. Peter and N. Pinto-Neto, Phys. Rev. [**D65**]{} (2001) 023513. M. S. Morris and K. S. Thorne, Am. J. Phys. [**56**]{} (1988) 395. D. Hochberg, M. Visser, Phys. Rev. Lett. [**81**]{} (1998) 746. K. D. Olum, Phys. Rev. Lett [**81**]{} (1998) 3567.
[^1]: The choice of $\varphi$ will probably yield a different theory, with a different Hilbert space. The kinetic term is more complicate and the measure is not the trivial one. We will not study this possibility here.
[^2]: As in the following we will make superpositions of eigenfunctions of the total dust matter operator, we will use from now on the letter $\upsilon$ in order to not confuse it with the beable $p_{\varphi}=\partial S/\partial\varphi$. We did not make this distinction before because they coincide for eigenfunctions of the total dust matter operator.
|
---
abstract: 'We find that electron states at the bottom of the conduction bands of covalent semiconductors are distributed mainly in the interstitial channels and that this [*floating*]{} nature leads to the band-gap variation and the anisotropic effective masses in various polytypes of SiC. We find that the channel length, rather than the hexagonality prevailed in the past, is the decisive factor for the band-gap variation in the polytypes. We also find that the [*floating nature*]{} causes two-dimensional electron and hole systems at the interface of different SiC polytypes and even one-dimensional channels near the inclined SiC surface.'
author:
- 'Yu-ichiro Matsushita'
- Atsushi Oshiyama
title: Interstitial Channels that Control Band Gaps and Effective Masses in Tetrahedrally Bonded Semiconductors
---
Most semiconductors, elemental or compound, have the four-fold coordinated tetrahedral structure caused by the hybridization of atomic orbitals. It is written in textbooks [@phillips] that the resultant hybridized $sp^3$ bonding orbitals constitute valence bands, whereas the anti-bonding counter parts do conduction bands. This is not necessarily true, however: We have recently found that the wave-functions of the conduction-band minima (CBM) of the semiconductors are distributed not near atomic sites but in the interstitial channels [@Matsushita_floating], as shown in Fig. \[float\]. The wave-functions [*float*]{} in the internal space, i.e., the channels, inherent to the $sp^3$-bonded materials. Another structural characteristic in the semiconductor is the stacking of atomic bilayers along the bond axis direction such as AB (wurtzite) or ABC (diamond or zincblende). The different stacking sequence leads to the different polytype [@Dissertation10] generally labeled by the periodicity of the sequence $n$ and its symmetry, hexagonal ($H$) or cubic ($C$), as in 2$H$(AB), 3$C$(ABC), 4$H$(ABCB) and so on. These differences in the stacking sequence have been assumed to be minor in the electronic properties. However, the sequence determines the lengths and the directions of the interstitial channels, hereby affecting the shapes of the wave-functions of CBMs. The internal space overlooked in the past may be closely related to the electronic properties of the semiconductors, that we discuss in this Letter.
![(Color online). Energy bands and the contour plot of the Kohn-Sham orbital of the conduction-band minimum at $M$ in $3C$-SiC shown on $(0\bar{1}1)$ plane. The orbital is distributed along the \[110\] channel. The calculation has been done with the hybrid exchange-correlation functional, HSE [@hse; @matsushita]. White and burgundy balls depict C and Si atoms, respectively. Simple localized-orbital basis sets are incapable of describing the floating nature of the conduction-band states [@Matsushita_floating; @matsushita_th] []{data-label="float"}](Fig1.eps){width="0.9\linewidth"}
Silicon carbide (SiC) is a promising material in power electronics due to its superior properties which are suitable to the operations under harsh environment [@Matsunami]. From science viewpoints, SiC is a manifestation of the polytypes explained above: Dozens of polytypes of SiC are observed and the band gaps vary by 40 %, from 2.3 eV in 3$C$ to 3.3 eV in 2$H$ despite that the structures are locally identical to each other in the polytypes [@Harris]. This mysterious band-gap variation has been discussed in terms of an empirical quantity, hexagonality [@Choyke], for a half century: A bilayer sandwiched by the two same stacking indexes, as in 2H structure, is called a hexagonal layer and the ratio of the hexagonal layers in whole stacking sequence is called hexagonality; the band-gap variation in the polytypes is argued to be linear with respect to the hexagonality. Yet, the linearity is not satisfactory (see below) and moreover the underlying physics is totally lacking.
In this Letter, we find, on the basis of the density-functional calculations [@cal], that the extent of the internal space, i.e., the length of the interstitial channel, in covalent semiconductors is decisive in the nano-scale shapes of the wave-functions of the CBM and hereby explains the mysterious variation of the band gap in SiC polytypes. We also find that the observed anisotropy of the effective masses in SiC, and the pressure dependence of the band gaps generally observed in most semiconductors, are naturally explained in terms of the channel length. Further, we find that the stacking control dramatically modifies the electronic properties, leading to generation of low-dimensional electron and hole systems in three-dimensional SiC.
The sequence of the atomic bilayers determines the length and direction of the interstitial channels: e.g., in the 3$C$ polytype the channel along $\langle 110 \rangle$ extends infinitely, whereas in the 6$H$ polytype the cannel along $\langle 2\bar{2}01 \rangle$ has a finite length of 7$a_0$ / 2 $\sqrt{2}$ ($a_0$: lattice constant). To examine the relation between the extent of the internal space and the band gap, we consider 24 representative stacking sequences in 3$C$ and $nH$ (2 $\le n \le$ 12) polytypes. Details of the 24 polytypes are listed in Supplement Material [@SM]. In the 2$H$, 3$C$, 4$H$ and 5$H$ polytypes, the sequence of the bilayer stacking is unique. In the $6H$, $8H$, $10H$, and $12H$ polytypes, there are 2, 6, 18, and 58 possibilities in the stacking sequence, respectively [@Iglesias]. The possible values of the hexagonality in the $10H$ polytype are 20, 40, 60, and 80%, whereas those in the $12H$ polytype are 16.7, 33.3, 50, 66.7, and 83.3%. Our 24 representatives include all the possible hexagonality in the the $6H$, $8H$, $10H$, and $12H$ polytypes [@Kobayashi].
![(Color online). Band gaps for 24 representative SiC polytypes calculated in GGA as a function of the hexagonality (left panel) and a function of the channel length (right panel). In each panel, the fitting function (see text) is also shown. The corresponding variance is 355 meV for the left (hexagonality) and 85 meV for the right (channel length). []{data-label="channel_length"}](Pic1_1.eps){width="1.0\linewidth"}
Calculated band gaps for these representative polytypes of SiC are plotted as a function of either the hexagonality or the channel length in Fig.\[channel\_length\]. Here the channel length is defined as the number of bilayers along the longest interstitial channels. The left panel in Fig. \[channel\_length\] shows a positive correlation between the band gap and the hexagonality. However, the linearity between the two is poor: The best fitted function we have obtained is $\epsilon_g=1.425+0.0124 h$ in eV (h: hexagonality) with the large variance of 355 meV. On the other hand, the band gap as a function of the channel length shows monotonic decrease, indicating that the channel length is a proper quantity to describe the band gap. We have indeed found that the calculated band gaps $\varepsilon_{g}$ are nicely fitted to a single function of the channel length $l$ as, $$\varepsilon_g = 1.425 + \frac{17.63}{( l + 1.268)^2} \ ,
\label{fitting}$$ in eV with the variance of 85 meV.
As stated above, the wavefunction of the CBM [*floats*]{} in the interstitial channel the length $l$ of which is determined by the way of stacking of atomic bilayers. Hence, the CBM state is regarded as being confined in the one-dimensional quantum tube with the length $l$. The energy level $\varepsilon_l$ thus confined is given by [@matsushita_th], $$\varepsilon_l = \varepsilon_{3C} + \frac{\pi^2 \hbar^2}{2m^* (l + \Delta)^2} \ ,
\label{quantum_well}$$ where $\varepsilon_{3C}$ is the energy level of 3$C$-SiC which has the infinite channel length and $m^*$ is the effective mass along the channel direction. The $\Delta$ in the second term represent the spill of the wave function from the quantum tube with the length $l$. Since the valence-band top has a character of the bonding $sp^3$ orbitals common to all the polytypes [@Matsushita_floating; @matsushita_th], the variation in (\[quantum\_well\]) corresponds to that in the band gap in the polytypes expressed in (\[fitting\]). Our GGA calculation indeed provides $\varepsilon_{3C}$ = 1.419 eV, showing agreement with the first term in (\[fitting\]). By further comparing the second terms in (\[fitting\]) and (\[quantum\_well\]), we obtain the effective mass of $m^* = 0.326 m_0$ ($m_0$: bare electron mass) which shows agreement with the experimental value of $m^* = 0.363 m_0$ [@3C-SiC]. The factor 1.268 in unit of the bilayer length in the denominator of the fitting function (\[fitting\]) indicates that the confinement is imperfect and the wavefunction spills from the interstitial tube by about a single bilayer.
![(Color online). Contour plots on ($11\bar{2}0$) plane of the Kohn-Sham (KS) orbitals of the conduction band minimum at $M$ for $10H$-SiC with the ABCABCABAB stacking (left panel) and with the ABCACBCACB stacking (right panel). The value for each contour color is relative to the corresponding maximum absolute value. White and burgundy balls depict C and Si atoms, respectively. The broken lines represent the interface where the longest channel in the cubic region is blocked by the hexagonal region. []{data-label="distribution"}](Fig3-improved.eps){width="1.0\linewidth"}
We have now clarified that the channel length rather than the hexagonality is the principal quantity to determine the band-gap variation. This situation becomes visible by examining the wavefunction of the CBM. Figure \[distribution\] shows the Kohn-Sham orbital of the CBM for the two 10$H$-SiC polytypes where the stacking sequences are ABCABCABAB and ABCACBCACB, respectively. The hexagonality of the two polytypes is identical, i.e., 40 %. However, the calculated band gap is 1.59 eV for the former and 2.07 eV for the latter. This difference in the gap beautifully corresponds to that in their channel lengths, 8 and 4, respectively [@SM]. The wavefunction of CBM [*floats*]{} along the $\langle 10\ \bar{10}\ 0 3 \rangle$ for both polytypes as is shown in Fig. \[distribution\]. Yet the length of the channel and consequently the extension of the wavefunction is substantially longer for the former polytype, leading to the narrower band gap.
![(Color online). (a) Local density of states, $D(\varepsilon, z)$, near the band gap for the $12H$-SiC (ABCACACACACB) with its side view of the atomic structure. The shaded region in the side view depicts the cubic stacking region. The lower panel is the extension of $D(\varepsilon, z)$ near the valence-band top. The ordinate $\varepsilon$ is the electronic energy, where the Fermi energy is set to be 0, and the abscissa $z$ is the coordinate along the stacking direction. The value of $D ( \varepsilon, z)$ is represented by the color code (Yellow: high, black: low). The black region corresponds to the energy gap. (b) Band gap variation as a function of the thickness of hexagonal-stacking region of SiC polytypes with the thickness of the cubic stacking region fixed with 5 bilayers. []{data-label="SP"}](Fig4-improved.eps){width="1.0\linewidth"}
The calculated band gaps in the right panel of Fig. \[channel\_length\] show small but sizable variance from the fitting function described above. This variance is a consequence of spontaneous polarization (SP) in the region of the hexagonally stacked bilayers (hexagonal stacking). Let us consider the polytypes with the 4-bilayer channel length in Fig. \[channel\_length\]: A $12H$ polytype whose stacking sequence is ABCACACACACB shows narrower band gap by 0.1 eV than those of other polytypes with $l$ = 4. Figure \[SP\] (a) shows calculated local density of states (LDOS) near the energy gap for this $12H$-SiC. The spiky contrast below the energy gap (black region) manifests atomic positions along the stacking direction. It is clearly shown that the conduction bands in the region with the cubic stacking (ABC) are located at lower positions in energy than in the hexagonal stacking region. More importantly, SP takes place in the hexagonally stacked region due to the lack of inversion symmetry and renders the band lineup slanted in real space along the stacking direction. Further the counter polarization in the cubic region makes it slanted in the reverse direction as in Fig. \[SP\] (a). We have found that the slanted band lineup causes downward (upward) shift of the conduction (valence) band edge and the band gap becomes narrower.
We have indeed calculated the band-gap variation by increasing the thickness of the hexagonal bilayers with the thickness of the cubic region fixed at 5 bilayer \[Fig. \[SP\] (b)\]. The calculated band gaps decreases monotonically. This is a consequence from the enhanced band-slanting induced by the polarization. The estimated band-gap decrease by adding a single bilayer in the hexagonally stacked region is 10 meV. By using this quantity, the estimated band gap of the $12H$-SiC (ABCACACACACB) [*without*]{} SP is 2.07 eV, which is just on the fitting function in Fig.\[channel\_length\]. This finding opens a possibility of the band-gap tuning by changing the thickness of the hexagonally stacked region.
We have revealed that the floating nature of the CBM causes the band-offset at the interface between the cubic-stacking and the hexagonal-stacking regions. The SP in the hexagonal region combined with the counter polarization in the cubic region generates two-dimensional electron and hole gases at the interface. It may be evident from LDOS slanted in real space, as is shown in Fig.\[SP\] (a). It is further quantified by calculating the effective masses of the CBM and the valence-band maximum along the stacking direction in the polytypes: It is found that the effective mass along the direction increases to a hundred of $m_0$, whereas that in the lateral plane keeps its value of 0.67 $m_0$ for electron and 2.20 $m_0$ for hole (not shown). This finding of the carrier confinement is the generalization of the hetero-crystalline superlattice of SiC first proposed by Bechstedt and Käckel [@bechstedt] and later pursued theoretically [@ke; @miao; @iwata]. We here emphasize that the underlying physics, unrevealed in the past, is the floating nature of the CBM states controlled by the nanoshapes of the interstitial channels. The anisotropic effective mass observed in 6$H$-SiC polytype [@6H-SiC] is one of the fingerprints of such floating nature.
![(color online) Calculated CBM energy at several $k$ points as a function of reduced lattice constant $a/a_0$ ($a_0$: lattice constant without pressure) in 3C-SiC (a) and in GaAs (c). The corresponding pressures (GPa) at $a/a_0$ = 0, 0.95, 0.9, 0.85, 0.80, and 0.75 are 0 (0), 31.0 (5.9), 73.7 (12.5), 128.7 (21.3), 202.6 (33.6), and 306.5 (51.5) in SiC (GaAs). (b) The kinetic-energy contribution $\epsilon_{\rm kin}=\left<\phi_i\left|-\nabla^2/2\right|\phi_i\right>$ to the orbital energy of each Kohn-Sham (KS) state at $M$ point in 3C-SiC. The abscissa represents the $i$th KS state from the valence-band bottom and the 25th state is the CBM. (d) Contour plots of the KS orbital of the CBM in GaAs at $a=0.75a_0$. The gray and green balls represent Ga and As atoms, respectively. []{data-label="pressure_dependence"}](pressure_dependence_improved5.eps){width="1.0\linewidth"}
Another noteworthy feature of the floating state is its pressure dependence. Fig. \[pressure\_dependence\] (a) shows the pressure dependence of the CBM energy at several high-symmetry $k$ points. The $M$ point energy shifts downward with increasing the pressure, whereas the $\Gamma$ point energy shifts upward. The latter is easily understood by the enhancement of the bonding-antibonding splitting. The former is a consequence of the floating nature. As shown in Fig. \[pressure\_dependence\] (b), the reduction of the kinetic-energy contribution to the orbital energy is one of the characteristics of the floating state. This is due to the extended distribution in the internal space of the floating state. When the lattice constant is reduced under the pressure, the kinetic energy generally increases but in the floating state such increase is minor. This causes the lack of the upward shift or even the downward shift at the $M$ point. The floating nature is not restricted to SiC. Figure \[pressure\_dependence\] (c) shows the CBM energy variation in pressurized GaAs. The CBM at $M$ point, which is folded from the $X$ point in the cubic BZ, shifts downward with increasing pressure. We have actually found that the KS orbital at $M$ point has floating character in pressurized circumstances \[see Fig. \[pressure\_dependence\] (d)\]. The direct- and indirect-gap transition in the pressurized GaAs well established in experiments [@GaAs] is a manifestation of the floating nature of the CBM states.
![(color online) One dimensional electron channel on the inclined SiC surface (xylophone channel). (a) Schematic side view of $5H$-SiC near the inclined surface (the inclined angle $\theta$ ) on $( 11\bar{2}0)$ plane. The five arrows represent channels with each number discriminating the channel length. (b) and (d): Contour plots on the $( 11\bar{2}0)$ plane of the CBM KS orbitals near the surface with the inclined angle of $\theta=30.39^\circ$ and $\theta=16.32^\circ$. (c) Top view of the CBM KS orbitals as an iso-value surface at its value of 20% of the maximum value. Blue and green iso-value surfaces represent the positive and negative signs of the KS orbital. White and burgundy balls depict C and Si atoms, respectively. []{data-label="xylophonemodel"}](Fig6-improved.eps){width="1.0\linewidth"}
Nanofabrication of the semiconductor surfaces introduces further modification of wavefunctions of the floating states by controlling the internal space. Suppose, e.g., $5H$-SiC which has the interstitial channels along the $\langle 5\bar{5}03 \rangle$ direction with the finite length of 5 bilayers. When the surface is inclined relative to the $\langle 0001 \rangle$ direction with the angle of $\theta$, the lengths of the channels near the surface vary depending on the lateral positions like a xylophone \[Fig. \[xylophonemodel\] (a)\]. In this case, the lowest CBM state is distributed along the longest channel \[Fig.\[xylophonemodel\] (b), and (c) for $\theta = 30.39^\circ$\]. It is clearly seen that one-dimensional (1D) electron channel appears near the surface. In fact, the effective mass along the 1D channel is 0.34 $m_0$ whereas the mass along the perpendicular direction is 71 $m_0$. By changing the inclined angle $\theta$, the width of the 1D channel and its separation from the adjacent channels are controlled, as is demonstrated in Fig. \[xylophonemodel\] (d) for the case of $\theta = 16.32^\circ$.
We have clarified that channel structure plays important roles in understanding of the band gap variation in SiC polytypes. This variation is not limited to SiC since CBMs of most tetrahedrally-bonded materials have floating nature [@Matsushita_floating]. The band-gap engineering through the control of the nanoshapes of the interstitial channels comes true when syntheses of the polytypes in other materials are realized.
In summary, we have found that the internal nanospace plays a decisive role in determining the band gaps, the effective masses and then the electronic properties of the covalent semiconductors. This is a consequence of the floating nature of the conduction-band minima where the wave-functions are distributed along the interstitial channels in the semiconductors. We have shown that the band-gap variation in various polytypes in SiC is quantitatively explained in terms of the channel length. We have found that the length and the direction of the interstitial channel are the principal quantities to determine the anisotropic effective masses, leading to the two dimensional electron and hole gases in heterocrystalline SiC and also to the one-dimensional electron channel by the surface fabrication.
This work was supported by the Grants-in-Aid for scientific research conducted by MEXT, Japan, under Contract No. 22104005. Computations were performed mainly at the Supercomputer Center in ISSP, The University of Tokyo.
[99]{}
J. C. Phillips and G. Lucovsky, [*Bonds and Bands in Semiconductors*]{} (2nd ed. Momentum Press, 2009).
Y. -i. Matsushita, S. Furuya, and A. Oshiyama, Phys. Rev. Lett. [**108**]{}, 246404 (2012).
J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. [**118**]{}, 8207 (2003); [**124**]{}, 219906(E) (2006).
Y. -i. Matsushita, K. Nakamura and A. Oshiyama, Phys. Rev. B [**84**]{}, 075205 (2011) and references therein.
Y.-i. Matsushita, Thesis, The University of Tokyo, 2013, http://hdl.handle.net/2261/55684. A. R. Verma and Krishna, Polymorphism and Polytypism in Crystals (Wiley, New York, 1966
H. Matsunami, and T. Kimoto, Mater. Sci. and Eng., R [**20**]{}, 125 (1997).
For a review, G. L. Harris (ed.) ,$Properties$ $of$ $Silicon$ $Carbide$, (INSPEC, London, 1995).
W. J. Choyke, D. R. Hamilton, and L. Patrick, Phys. Rev. [**133**]{}, A1163 (1964).
See Supplemental Material at http://link.aps.org/\
supplemental/10.1103/PhysRevLett.112.136403 for details of our calculations based on the density functional theory [@Hohenberg] with the generalized gradient approximation [@Perdew] and with the hybrid approximation [@hse; @matsushita].
P. Hohenberg and W. Kohn, Phys. Rev. [**136**]{}, B864 (1964); W. Kohn and L. J. Sham, Phys. Rev. [**140**]{}, A1133 (1965).
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996).
See the Supplemental Material at http://link.aps.org/\
supplemental/10.1103/PhysRevLett.112.136403 for details of the 24 representative SiC polytypes.
J. E. Iglesias, Acta Crystallogr. A [**62**]{}, 178 (2006).
K. Kobayashi, and S. Komatsu, J. Phys. Soc. Jpn., [**81**]{}, 024714 (2012).
R. Kaplan, R. J. Wagner, H. J. Kim, and R. F. Davis, Solid State Commun. [**55**]{}, 67 (1985).
F. Bechstedt and P. Käckell, Phys. Rev. Lett. [**75**]{}, 2180 (1995).
S. H. Ke, J. Zi, K. M. Zhang, and X. D. Xie, Phys. Rev. B [**54**]{}, 8789 (1996).
M. S. Miao and W. R. L. Lambrecht, Phys. Rev. B [**68**]{}, 155320 (2003).
H. Iwata, U. Lindefelt, S. Öberg, and P. R. Briddon, Phys. Rev. B [**65**]{}, 033203 (2001).
N. T. Son, O. Kordina, A. O. Konstantinov, W. M. Chen, E. Sorman, B. Monemar, and E. Janzen, Appl. Phys. Lett. [**65**]{}, 3209 (1994).
N. Lifshitz, A. Jayaraman, R. A. Logan, and H. C. Card, Phys. Rev. B, [**21**]{}, 670 (1980).
|
---
abstract: 'Aspect Based Sentiment Analysis (ABSA) is the task of identifying sentiment polarity of a text given another text segment or aspect. In ABSA, a text can have multiple sentiments depending upon each aspect. Aspect Term Sentiment Analysis (ATSA) is a subtask of ABSA, in which aspect terms are contained within the given sentence. Most of the existing approaches proposed for ATSA, incorporate aspect information through a different subnetwork thereby overlooking the advantage of aspect terms’ presence within the sentence. In this paper, we propose a model that leverages the positional information of the aspect. The proposed model introduces a decay mechanism based on position. This decay function mandates the contribution of input words for ABSA. The contribution of a word declines as farther it is positioned from the aspect terms in the sentence. The performance is measured on two standard datasets from SemEval 2014 Task 4. In comparison with recent architectures, the effectiveness of the proposed model is demonstrated.'
author:
- Avinash Madasu
- Vijjini Anvesh Rao
bibliography:
- 'nldb.bib'
title: |
A Position Aware Decay Weighted Network\
for Aspect based Sentiment Analysis
---
Introduction {#sec:intro}
============
Text Classification deals with the branch of Natural Language Processing (NLP) that involves classifying a text snippet into two or more predefined categories. Sentiment Analysis (SA) addresses the problem of text classification in the setting where these predefined categories are sentiments like positive or negative [@pang2002thumbs]. Aspect Based Sentiment Analysis (ABSA) is proposed to perform sentiment analysis at an aspect level [@hu2004mining].
There are four sub-tasks in ABSA namely Aspect Term Extraction (ATE), Aspect Term Sentiment Analysis (ATSA), Aspect Category Detection (ACD), Aspect Category Sentiment Analysis (ACSA). In the first sub-task (ATE), the goal is to identify all the aspect terms for a given sentence. Aspect Term Sentiment Analysis (ATSA) is a classification problem where given an aspect and a sentence, the sentiment has to classified into one of the predefined polarities. In the ATSA task, the aspect is present within the sentence but can be a single word or a phrase. In this paper, we address the problem of ATSA. Given a set of aspect categories and a set of sentences, the problem of ACD is to classify the aspect into one of those categories. ACSA can be considered similar to ATSA, but the aspect term may not be present in the sentence. It is much harder to find sentiments at an aspect level compared to the overall sentence level because the same sentence might have different sentiment polarities for different aspects. For example consider the sentence, *“The taste of food is good but the service is poor”*. If the aspect term is *food*, the sentiment will be *positive*, whereas if the aspect term is *service*, sentiment will be *negative*. Therefore, the crucial challenge of ATSA is modelling the relationship between aspect terms and its context in the sentence. Traditional methods involve feature engineering trained with machine learning classifiers like Support Vector Machines (SVM) [@kiritchenko2014nrc]. However, these methods do not take into account the sequential information and require a considerable struggle to define the best set of features. With the advent of deep learning, neural networks are being used for the task of ABSA.
For ATSA, LSTM coupled with attention mechanism [@bahdanau2014neural] have been widely used to focus on words relevant to certain aspect. Target-Dependent Long Short-Term Memory (TD-LSTM) uses two LSTM networks to model left and right context words surrounding the aspect term [@tang2015effective]. The outputs from last hidden states of LSTM are concatenated to find the sentiment polarity. Attention Based LSTM (ATAE-LSTM) uses attention on the top of LSTM to concentrate on different parts of a sentence when different aspects are taken as input [@wang2016attention]. Aspect Fusion LSTM (AF-LSTM) [@tay2018learning] uses associative relationship between words and aspect to perform ATSA. Gated Convolution Neural Network (GCAE) [@xue2018aspect] employs a gated mechanism to learn aspect information and to incorporate it into sentence representations.
However, these models do not utilize the advantage of the presence of aspect term in the sentence. They either employ an attention mechanism with complex architecture to learn relevant information or train two different architectures for learning sentence and aspect representations. In this paper, we propose a model that utilizes the positional information of the aspect in the sentence. We propose a parameter-less decay function based learning that leverages the importance of words closer to the aspect. Hence, evading the need for a separate architecture for integrating aspect information into the sentence. The proposed model is relatively simple and achieves improved performance compared to models that do not use position information. We experiment with the proposed model on two datasets, restaurant and laptop from SemEval 2014.
Related Work
============
Aspect Term Sentiment Analysis
------------------------------
Early works of ATSA, employ lexicon based feature selection techniques like Parts of Speech Tagging (POS), unigram features and bigram features [@kiritchenko2014nrc]. However, these methods do not consider aspect terms and perform sentiment analysis on the given sentence. Phrase Recursive Neural Network for Aspect based Sentiment Analysis (PhraseRNN) [@nguyen-shirai-2015-phrasernn] was proposed based on Recursive Neural Tensor Network [@socher-etal-2013-recursive] primarily used for semantic compositionality. PhraseRNN uses dependency and constituency parse trees to obtain aspect representation. An end-to-end neural network model was introduced for jointly identifying aspect and polarity [@schmitt-etal-2018-joint]. This model is trained to jointly optimize the loss of aspect and the polarity. In the final layer, the model outputs one of the sentiment polarities along with the aspect. [@AAAI1816570] introduced Aspect Fusion LSTM (AF-LSTM) for performing ATSA.
Model
=====
In this section, we propose the model Position Based Decay Weighted Network (PDN). The model architecture is shown in Figure \[fig:architecture\]. The input to the model is a sentence $S$ and an Aspect $A$ contained within it. Let $n$ represent the maximum sentence length considered.
Word Representation
-------------------
Let V be the vocabulary size considered and $X$ $\in$ $\mathbb{R}^{V \times d_{w}}$ represent the embedding matrix[^1], where for each word $X_{i}$ is a $d_{w}$ dimensional word vector. Words contained in the embedding matrix are initialized to their corresponding vectors whereas words not contained are initialized to 0’s. $I$ $\in$ $\mathbb{R}^{n \times d_{w}}$ denotes the pretrained embedding representation of a sentence where $n$ is the maximum sentence length.
Position Encoding
-----------------
In the ATSA task, aspect $A$ is contained in the sentence $S$. A can be a word or a phrase. Let $k_{s}$ denote the starting index and $k_{e}$ denote the ending index of the aspect term(s) in the sentence. Let $i$ be the index of a word in the sentence. The position encoding of words with respect to aspect are represented using the formula $$p(i) =\left\{
\begin{array}{@{}ll@{}}
k_{s}-i+1, & \ k_{s}>i \\
1, & \ i \in k_{s},k_{s+1},..,k_{e-1},k_{e} \\
i-k_{e}+1, & \ i>k_{e}
\end{array}\right.$$ The position encodings for the sentence “granted the space is smaller than most it is the best service" where “space" is the aspect is shown in Figure \[fig:architecture\]. This number reflects the relative distance of a word from the closest aspect word. The position embeddings from the position encodings are randomly initialized and updated during training. Hence, $P$ $\in$ $\mathbb{R}^{n \times d_{p}}$ is the position embedding representations of the sentence. $d_{p}$ denotes the number of dimensions in the position embedding.
Architecture
------------
As shown in Figure \[fig:architecture\], PDN comprises of two sub-networks: Position Aware Attention Network(PDN) and Decay Weighting Network (DWN).
### Position Aware Attention Network (PAN) {#position-aware-attention-network-pan .unnumbered}
An LSTM layer is trained on $I$ to produce hidden state representation $h_{t}$ $\in$ $\mathbb{R}^{d_{h}}$ for each time step $t$ $\in$ $\{1,n\}$ where $d_{h}$ is the number of units in the LSTM. The LSTM outputs contain sentence level information and Position embedding contain aspect level information. An attention subnetwork is applied on all $h$ and $P$ to get a scalar score $\alpha$ indicating sentiment weightage of the particular time step to the overall sentiment. However, prior to concatenation, the position embeddings and the LSTM outputs may have been output from disparate activations leading to different distribution. Training on such values may bias the network towards one of the representations. Therefore, we apply a fully connected layer separately but with same activation function Scaled Exponential Linear Unit (SELU)[@klambauer2017self] upon them. Two fully connected layers follow this representation. Following are the equations that produce $\alpha$ from LSTM outputs $h$ and position embeddings $P$. $$P_{t}^{\prime} = selu(W_{p} \cdot P_{t} + b_{p})$$ $$h_{t}^{\prime} = selu(W_{h} \cdot h_{t} + b_{h})$$ $$H_{t} = relu(W_{a} \cdot [h_{t}^{\prime}P_{t}^{\prime}] + b_{a})$$ $$e_{t} = \tanh( \mathbf{v_{}}^\intercal \cdot H_{t})$$ $$\alpha_{t} = \frac{\exp(e_{t})}{\sum_{i=1}^{n}\exp(e_{i})}$$
### Decay Weighting Network (DWN) {#decay-weighting-network-dwn .unnumbered}
In current and following section, we introduce decay functions. The decay function for scalar position encoding $p(i)$ is represented as the scalar $d(p(i))$. These functions are continuously decreasing in the range $[0,\infty)$. The outputs from the LSTM at every time step are scaled by the decay function’s output. $$Z_{t} = h_{t} \cdot d(p(t)) \: \forall \: t \in \{1,n\}$$ A weighted sum $O$ is calculated on the outputs of Decay Weighted network using the attention weights from PAN. $$O = \alpha \cdot Z$$ A fully connected layer is applied on $O$ which provides an intermediate representation $Q$. A softmax layer is fully connected to this layer to provide final probabilities.
{width="4cm"}
It is paramount to note that the DWN does not contain any parameters and only uses a decay function and multiplication operations. The decay function provides us with a facility to automatically weight representations closer to aspect as higher and far away as lower, as long as the function hyperparameter is tuned fittingly. Lesser parameters makes the network efficient and easy to train.
{width="12cm"}
------------------------------------- -- --
**Model & **Restaurant & **Laptop\
Majority &65.00 &53.45\
NBOW &67.49 &58.62\
LSTM & 67.94& 61.75\
TD-LSTM & 69.73 &62.38\
AT-LSTM & 74.37 & 65.83\
ATAE-LSTM & 70.71 & 60.34\
DE-CNN & 75.18 & 64.67\
AF-LSTM & 75.44 & 68.81\
GCAE & 76.07 & 67.27\
Tangent-PDN & 78.12 & 68.82\
Inverse-PDN & **78.9** & **70.69**\
Expo-PDN & 78.48 & 69.43\
******
------------------------------------- -- --
: \[tab:acc\] Accuracy Scores of all models. Performances of baselines are cited from [@tay2018learning]
### Decay Functions {#decay-functions .unnumbered}
We performed experiments with the following decay functions.\
[**Inverse Decay**]{}:\
Inverse decay is represented as: $$d(x) = \frac{\lambda}{x}$$ [**Exponential Decay**]{}:\
Exponential decay is represented as: $$d(x) = e^{-\lambda * x}$$ [**Tangent Decay**]{}:\
Tangent decay is represented as: $$d(x) = 1-tanh(\lambda*x)$$ $\lambda$ is the hyper-parameter in all the cases.[^2]
Experiments
===========
Datasets
--------
We performed experiments on two datasets, Restaurant and Laptop from SemEval 2014 Task 4 [@pontiki-etal-2014-semeval]. Each data point is a triplet of sentence, aspect and sentiment label. The statistics of the datasets are shown in the Table \[statistics\]. As most existing works reported results on three sentiment labels $\textit{positive,negative,neutral}$ we performed experiments by removing *conflict* label as well.
Compared Methods
----------------
We compare proposed model to the following baselines:
### Neural Bag-of-Words (NBOW)
NBOW is the sum of word embeddings in the sentence [@tay2018learning].
### LSTM
Long Short Term Memory (LSTM) is an important baseline in NLP. For this baseline, aspect information is not used and sentiment analysis is performed on the sentence alone. [@tay2018learning].
### TD-LSTM
In TD-LSTM, two separate LSTM layers for modelling the preceding and following contexts of the aspect is done for aspect sentiment analysis [@tang2015effective].
### AT-LSTM
In Attention based LSTM (AT-LSTM), aspect embedding is used as the context for attention layer, applied on the sentence [@wang2016attention].
### ATAE-LSTM
In this model, aspect embedding is concatenated with input sentence embedding. LSTM is applied on the top of concatenated input [@wang2016attention].
------------ ------- ------ ------- ------ ------- ------
Train Test Train Test Train Test
Restaurant 2164 728 805 196 633 196
Laptop 987 341 866 128 460 169
------------ ------- ------ ------- ------ ------- ------
### DE-CNN
Double Embeddings Convolution Neural Network (DE-CNN) achieved state of the art results on aspect extraction. We compare proposed model with DE-CNN to see how well it performs against DE-CNN. We used aspect embedding instead of domain embedding in the input layer and replaced the final CRF layer with MaxPooling Layer. Results are reported using author’s code[^3] [@xu2018double].
### AF-LSTM
AF-LSTM incorporates aspect information for learning attention on the sentence using associative relationships between words and aspect [@tay2018learning].
### GCAE
GCAE adopts gated convolution layer for learning aspect representation which is integrated into sentence representation through another gated convolution layer. This model reported results for four sentiment labels. We ran the experiment using author’s code[^4] and reported results for three sentiment labels [@xue2018aspect].
Implementation
--------------
Every word in the input sentence is converted to a 300 dimensional vector using pretrained word embeddings. The dimension of positional embedding is set to 25 which is initialized randomly and updated during training. The hidden units of LSTM are set to 100. The number of hidden units in the layer fully connected to LSTM is 50 and the layer fully connected to positional embedding layer is 50. The number of hidden units in the penultimate fully connected layer is set to 64. We apply a dropout [@srivastava2014dropout] with a probability 0.5 on this layer. A batch size 20 is considered and the model is trained for 30 epochs. Adam [@kingma2014adam] is used as the optimizer with an initial learning rate 0.001.
Results and Discussion
======================
The Results are presented in Table \[tab:acc\]. The Baselines Majority, NBOW and LSTM do not use aspect information for the task at all. Proposed models significantly outperform them.
The Role of Aspect Position
---------------------------
The proposed model outperforms other recent and popular architectures as well, these architectures use a separate architecture which takes the aspect input distinctly from the sentence input. In doing so they loose the positional information of the aspect within the sentence. We hypothesize that this information is valuable for ATSA and our results reflect the same. Additionally since proposed architecture does not take any additional aspect inputs apart from position, we get a fairer comparison on the benefits of providing aspect positional information over the aspect words themselves.
The Role of Decay Functions
---------------------------
Furthermore, while avoiding learning separate architectures for weightages, decay functions act as good approximates. These functions rely on constants alone and lack any parameters thereby expressing their efficiency. The reason these functions work is because they consider an assumption intrinsic to the nature of most natural languages. It is that description words or aspect modifier words come close to the aspect or the entity they describe. For example in Figure \[fig:architecture\], we see the sentence from the Restaurant dataset, “granted the space is smaller than most, it is the best service you can...”.The proposed model is able to handle this example which has distinct sentiments for the aspects “space” and “service” due to their proximity from “smaller” and “best” respectively.
Conclusion
==========
In this paper, we propose a novel model for Aspect Based Sentiment Analysis relying on relative positions on words with respect to aspect terms. This relative position information is realized in the proposed model through parameter-less decay functions. These decay functions weight words according to their distance from aspect terms by only relying on constants proving their effectiveness. Furthermore, our results and comparisons with other recent architectures, which do not use positional information of aspect terms demonstrate the strength of the decay idea in proposed model.
[^1]: https://nlp.stanford.edu/data/glove.840B.300d.zip
[^2]: In our experiments we took $\lambda$ = 0.45 for Tangent-PDN, 1.1333 for Inverse-PDN and 0.3 for Expo-PDN
[^3]: https://github.com/howardhsu/DE-CNN
[^4]: https://github.com/wxue004cs/GCAE
|
---
abstract: |
AgapeZ1 is the brightest and the shortest duration microlensing candidate event found in the Agape experiment. It occurred only $42''$ from the center of M31 at $RA=0^h 42^m 41.47^s$ and $Dec=41^\circ 16' 39.1''$ (J2000). Our photometry shows that the half intensity duration of the event is 4.8days and at maximum brightness we measure a stellar magnitude of $R = 18.0$ ($M_R \sim -6$) with $B-R=0.80$ mag color. A search on HST archives produced a single resolved star within the projected event position error box. Its magnitude is $R=22$, and its color is compatible with that of the event at the $2 \sigma$ level.
If the identification with the HST star is real, it implies for the event an amplification of about 4 magnitudes or 40 in brightness. This would lead to an Einstein crossing time radius of about 55 days. AgapeZ1 could be a bulge/bulge microlensing event involving a binary star.
The photometric properties of the object exclude classical M31 variable stars such as miras, novae, dwarf-novae, and bumpers. However, we cannot rule out the possibility that AgapeZ1 is in fact an odd variable star.
author:
- 'R. Ansari'
- 'M. Auri[è]{}re'
- 'P. Baillon'
- 'A. Bouquet'
- 'G. Coupinot'
- 'Ch. Coutures'
- 'C. Ghesqui[è]{}re'
- 'Y. Giraud-H[é]{}raud'
- 'P. Gondolo'
- 'J. Hecquet'
- 'J. Kaplan'
- 'A. Kim'
- 'Y. Le Du'
- 'A.L. Melchior'
- |
\
M. Moniez
- 'J.P. Picat'
- 'G. Soucail'
bibliography:
- 'mnemoaa.bib'
- 'lensing.bib'
- 'newagp.bib'
date: submitted
nocite: '[@dellavalle96]'
title: ' AgapeZ1: a large amplification microlensing event or an odd variable star towards the inner bulge of M31 [^1]'
---
Introduction
============
The Agape experiment [@agape1] is devoted to the search of dark matter towards M31. It looks for gravitational microlensing effects on unresolved stars by the so-called pixel method [@BBGK1; @BBGK2]. In the active field of MACHO microlensing searches [@Paczynski96] only two groups explore the promising M31 direction [@crottsm31; @BBGK1], and though several events with light curves compatible with microlensing have been presented [@crotts96; @YLD98], all have lacked the strong supporting evidence needed for them to be classified as true microlensing events. The present work describes the properties of AgapeZ1 (hereafter called Z1), our brightest and shortest duration candidate which occurred only $42''$ from the center of M31 in our central so-called “Z” field. The observational interest of our central field is manifold. It was observed at least once each of the 79 observing nights in R, and 30 nights in B. The central region of M31 has been observed by HST allowing for high-resolution archival searches for the quiescent sources of detected events. The central region of M31 is also of great astrophysical interest since it contains a huge number of stars. It is thus in this direction that the greatest number of microlensing events is expected to occur. However, because of the high star background level, only those with the highest amplification parameters will be resolved. A large number of variable stars, including exotic objects, may also be expected.
TBL observations and photometry \[obsphot\]
===========================================
The Agape observations were made at the 2m Bernard Lyot telescope (TBL) of the Pic du Midi Observatory with the F/8 spectro-reducer ISARD. A thin Tektronik 1024x1024 CCD was used with a useful field of $4' \times 4.5'$ with $0.3 ''$ pixels. The exposure times were generally 1min in both the B and R passbands for the (Z) central field and 30(20)min in the B(R) passband for the 6 other fields investigated by the experiment [@agape1]. We have 93 (30) R (B) exposures for field “Z” and 70 (33) R (B) exposures on the edge of a second field. The observing campaign ran from 1994 to 1996.
The Agape detection procedure is described in Ansari et al. [-@agape1]. It is based on the photometry in super-pixel (grouping of 7x7 pixels) with sides roughly two times the standard seeing. These super-pixels are photometrically normalised to a reference frame and corrected for seeing variations. The light curves for each super-pixel are analysed, yielding over 2000 variable objects. Of these, the 61 with only a single bump are then fitted with degenerate Paczyński curves [@wozniak97]. Selecting those with $\chi^2/dof < 1.5$ leaves only 19 light-curves. After a cut on the color and the event duration (simulations described in section 4.1 show that 70% of expected microlensing events should have half intensity duration shorter than 40 days) we are left with only two candidates.
Z1 is brighter, the shorter time-scale event and is located in the central bulge (Fig. \[figszb\]). In our non central field containing Z1, it lies close to the edge where shadowing occurs which causes some systematic photometric uncertainty. However, because of their larger exposure times, the precision of the second field observations is greater than for the Z field ones. Fig. \[figszblc\] shows the super-pixel light curve found by Agape for the object Z1. The days correspond to $J-2449624.5$ where J is the julian date.
To study the selected candidates, we developed a sophisticated photometry which will be described in a forthcoming paper [@PB98]. Our procedure belongs to the so-called image subtraction technics, already used by Tomaney and Crotts [-@tomaney96] and Alard and Lupton [-@alard98]. It is based on a global fit of one PSF (10x10 parameters) for each image and a unique reference background field (200x200 parameters). As it takes seeing effects more efficiently into account than does the super-pixel photometry described in Ansari et al. [-@agape1] we found its results more accurate, with however still $15\%$ systematic uncertainty .
Fig. \[figszblcz\] shows an enlargement of the improved photometry light curve of Z1 at the time of the event, averaging the star fluxes measured on the 2 fields when available. Apart from a (significant) bump at day $\approx 428$, a good fit of a Paczyński curve ($t_{1/2}$=4.8days) is obtained (data for 1994 and 1996 seasons are also used for the fit). At maximum, on 16 december 1995, the R magnitude is found to be $R$=18.0 and the color $B-R=0.80$. We have four color measurements during the event. The color at maximum has accurate precision (0.05 mag. stat.) since the corresponding B image is a 30 min exposure. Because of the faintness of the star or short exposures, the three other measurements cannot be used to tightly constrain achromaticity.
HST observations of the Z1 field
================================
The Z1 event positional error box lies on a series of HST WFPC2 archive taken on 9 September 1994 as part of a single observing program[^2] (The PI was R. Bohlin from STScI). There were no change in pointing between each exposure. We studied one 2300s image taken with the F656N filter and two co-added images with an effective exposure of 1200s taken with the F547M filter.
We have computed the spatial transformation between the Agape and HST fields with a least square reduction based on 10 common stars. The standard deviation for the projection accuracy of the standard stars is [$0.06''$]{} and is mainly due to the uncertainties on the position of the stars in the Agape fields. We have projected the position of Z1 onto the HST images with a 3$\sigma$ uncertainty of $0.18''$ and found a faint star ($R \sim 22$ mag) [$0.14''$]{} away from the projected position. It is the only resolved star on the HST image nearer than $0.4''$ from the Z1 projection. We call this star HST1. Fig. \[fighst\] shows a negative print of the HST field for the Z1 projected region.
We use the DAOPHOT package [@stetson87] to perform PSF fitting on the star. Since the field is crowded and there are no bright stars near our candidate, we used the theoretical PSF of TINYTIM. The standard magnitudes from the PSF photometry are $m_{F656N}=21.6\pm
0.2$ and $m_{F547M}=21.9\pm 0.1$ which are consistent within error from the results obtained from aperture photometry.
Interpretation \[interpretation\]
=================================
AgapeZ1 as a microlensing event
--------------------------------
On Fig. \[figsevtmc\] we show two plots obtained from Monte-Carlo simulations which include the known characteristics of the two galaxies (M31 and Milky Way), and, for each, an isothermal halo filled with 0.5$M_\odot$ machos. With respect to simulations described in [@agape1], 0.6$M_\odot$ M31 bulge lenses are added. Simulations give the distributions of the V magnitude of lensed stars and of the effective duration ($t_{1/2}$ defined as the FWHM of the amplification peak in the lightcurve) of the event expected for a microlensing effect detected with the same criteria as in our selection process.
If Z1 is interpreted as being due to a microlensing amplification of HST1, the event characteristics ($V \sim 22$ and $t_{1/2}=4.8$days) are fully compatible with those expected considering these simulations. In this case the magnification is of 4 magnitudes or 40 in brightness and the Einstein radius crossing time is about 55 days . This is typical for a microlensing event between bulge-bulge stars with a mass of 0.6$M_\odot$ which is expected whatever is the nature of the halo or for a microlensing event between a halo macho of 0.5$M_\odot$ and a M31 star.
Now, in this hypothesis, HST1 and Z1 must have the same color and the same spectral type. After correcting for the galactic extinction of $E_{B-V}=0.08$ [@vandenbergh91] with the extinction model of Cardelli, Clayton & Mathis [-@cardelli89], we find that Z1 has the $B-R$ color of an F5 star [@allen73]. The color and magnitude of HST1 are consistent also with an F5II star at the $2 \sigma$ level. However F5II stars are rare and only some tens are found in huge spectroscopic catalogs [@houk78]. They correspond to a very short stage in stellar evolution of massive stars. For example, using a “Geneve" model [@schaller92] we find that it would correspond to a subgiant of 4 $M_\odot$ ( between the main sequence and the helium flash).
The color and magnitude of HST1 can also be attributed to a highly reddened supergiant. However, such a large extinction is unlikely in M31 considering Han [-@han96] measurement of a uniform $A_{V}=0.24$ disk extinction. We thus prefer the identification of HST1 as an F5II star.
The fit with a Paczyński model ($t_{1/2}$=4.8days) is good for all points except for a statistically significant bump two days before the rapid rise to maximum. The shape of this lightcurve could be explained by the presence of a binary source [@griest92] or a binary lens [@distefano97]. The binary source hypothesis could explain the odd color of Z1/HST1 and the possible difference of color between the two objects.
Finally, Z1 could be a microlensing amplification of a fainter star, blended or not with HST1. In this case the source would be at least 1 magnitude fainter than HST1, and the amplification greater than 100.
AgapeZ1 as a variable star:
----------------------------
M31 variable stars could mimic microlensing events. For example, Crotts and Tomaney [-@crotts96] point out the possible pollution of their sample of candidates by very long period Miras, some of which they have already discarded. In the case of Z1, its blue color definitively excludes this hypothesis.
Della Valle and Livio (1996) explored the possibility that dwarf novae could contaminate microlensing survey samples. For observed dwarf novae, colour ranges between $B-V = -0.1$ and $B-V = +0.6$ and main outburst amplitude ranges between 2 and 5 magnitudes [@warner95]. Thus the colors of HST1 and Z1 as well as the amplitude of the event are consistent with a dwarf nova outburst. However, the quiescent absolute magnitude of dwarf novae is around 7 with a rather large range [@warner95]. If HST1 and Z1 correspond to a dwarf nova, the object would be in the foreground, well outside M31 and in the Galactic halo within 10kpc from the sun . The existence of such an object exactly projected towards the inner bulge of M31 is rather unlikely. There exists a broad relation between outburst amplitude and outburst interval for dwarf novae [@warner95]: for a 4 magnitude amplitude, one can expect outbursts to occur with intervals smaller than 100 days. In this case, the repetition of the Z1 event could be observable with follow-up observations.
Bumpers are variable stars which were detected by the MACHO experiment [@alcock96]. These objects have small amplitudes, unlike Z1 event.
Although the overall appearance of Z1 is similar to that of a nova, its faint magnitude would imply a long rate of decline while we observe a rapid one (0.25 magnitude per day observed, for 0.02-0.04 magnitude per day expected from the relation established by Capaccioli et al. [-@capaccioli89] for M31 novae). Reconciliation with the Capaccioli et al. trend would require Z1 to have a reddening of around 2 magnitudes in the visible which would imply a $E_{B-R}$ reddening of about 1 magnitude. The Agape experiment observed ten M31 novae, two being in the “Z field", and nine having $B-R$ colors. All the novae for which the respective relevant data are available follow the Capaccioli et al. [-@capaccioli89] trend and/or they have a $B-R$ color near maximum in the range 0.4-0.6 (apart from one which is strongly reddened near maximum). The color of $B-R=0.80$ at maximum for Z1 is thus not what would be expected for an M31 nova reddened by 2 magnitudes in the visible.
Conclusion \[conclusion\]
=========================
Our work shows that the AgapeZ1 event could be due to the gravitational amplification of a F5II color binary object corresponding to HST1 with an Einstein radius crossing time of 55 days. On the other hand the photometric properties of Z1 are incompatible with those of a classical M31 variable star. The foreground dwarf novae hypothesis appears unlikely. However, the inner bulge of M31 may be the site for rather odd objects. We have thus compared the position of Z1 with those of already known exotic objects including the 1885 supernovae [@devaucouleurs85], the X-ray sources [@primini93; @trinchieri91], and novae observed up to the inner bulge by [@ciardullo87]. Z1 is located about 6” from the position of the transient X-ray source E47 but the chance of association is weak. However, Z1 could be an unknown kind of cataclysmic variable. New Z field Agape observations are on the way to monitor for a recurrence of Z1 and to search for similar objects in the bulge of M31.
[^1]: Based on data collected with the 2m Bernard Lyot Telescope (TBL) operated by INSU-CNRS and Pic-du-Midi Observatory (USR 5026). The experiment was funded by IN2P3 and INSU of CNRS
[^2]: Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under the NASA contract NAS 5-26555.
|
---
abstract: 'We study linear programming and general LP-type problems in several big data (streaming and distributed) models. We mainly focus on low dimensional problems in which the number of constraints is much larger than the number of variables. Low dimensional LP-type problems appear frequently in various machine learning tasks such as robust regression, support vector machines, and core vector machines. As supporting large-scale machine learning queries in database systems has become an important direction for database research, obtaining efficient algorithms for low dimensional LP-type problems on massive datasets is of great value. In this paper we give both upper and lower bounds for LP-type problems in distributed and streaming models. Our bounds are almost tight when the dimensionality of the problem is a fixed constant.'
author:
- 'Sepehr Assadi[^1]'
- 'Nikolai Karpov[^2]'
- Qin Zhang
bibliography:
- 'paper.bib'
- 'general.bib'
title: Distributed and Streaming Linear Programming in Low Dimensions
---
Acknowledgments {#acknowledgments .unnumbered}
===============
Qin Zhang would like to thank Yufei Tao for introducing the problem (as well as the two-curve intersection problem as a means toward proving a lower bound for linear programming).
[^1]: Department of Computer Science, Princeton University. Supported in part by the Simons foundation Algorithms and Geometry collaboration. Majority of the work done while the author was a graduate student at University of Pennsylvania. Email: `sassadi@princeton.edu`.
[^2]: Department of Computer Science, Indiana University Bloomington. Supported in part by NSF CCF-1525024, IIS-1633215 and CCF-1844234. Email: `{nkarpov,qzhangcs}@indiana.edu`.
|
---
abstract: 'Restricted Boltzmann machines are undirected neural networks which have been shown to be effective in many applications, including serving as initializations for training deep multi-layer neural networks. One of the main reasons for their success is the existence of efficient and practical stochastic algorithms, such as contrastive divergence, for unsupervised training. We propose an alternative deterministic iterative procedure based on an improved mean field method from statistical physics known as the Thouless-Anderson-Palmer approach. We demonstrate that our algorithm provides performance equal to, and sometimes superior to, persistent contrastive divergence, while also providing a clear and easy to evaluate objective function. We believe that this strategy can be easily generalized to other models as well as to more accurate higher-order approximations, paving the way for systematic improvements in training Boltzmann machines with hidden units.'
author:
- 'Marylou Gabrié$^{1,2}$, Eric W. Tramel$^1$ and Florent Krzakala$^{1,3}$'
title: 'Training Restricted Boltzmann Machines via the Thouless-Anderson-Palmer Free Energy'
---
Introduction {#sec:intro}
============
Training restricted Boltzmann machines {#sec:basic}
======================================
Extended mean field theory of RBMs {#sec:theory}
==================================
\[GY\]
Numerical experiments {#sec:experiments}
=====================
Conclusion {#sec:conclusion}
==========
Acknowledgment {#acknowledgment .unnumbered}
==============
The research leading to these results has received funding from the European Research Council under the European Union’s $7^{th}$ Framework Programme (FP/2007-2013/ERC Grant Agreement 307087-SPARCS). We thank Lenka Zdeborová for numerous discussion.
|
---
author:
- Lucia Di Vizio
bibliography:
- 'difftransc.bib'
- 'qG.bib'
title: Approche galoisienne de la transcendance différentielle
---
Introduction {#introduction .unnumbered}
============
Le théorème de Hölder dit que la fonction Gamma d’Euler n’est pas solution d’une équation différentielle algébrique à coefficients dans le corps $\C(x)$ des fonctions rationnelles à coefficients complexes. Une littérature relativement vaste est dédiée à cet énoncé, à la fois dans le but d’en donner une nouvelle preuve et de le généraliser.
Le théorème de Hölder a été le premier résultat de *transcendance différentielle* ou d’*hypertranscendance*. En théorie de la transcendance, on s’intéresse à la transcendance des fonctions sur $\Q(x)$, souvent pour en déduire la transcendance de leurs valeurs spéciales. En hypertranscendance on s’intéresse à la transcendance simultanée des fonctions et de toutes leurs dérivées, donc à la propriété d’un ensemble de fonctions de ne pas être solution d’une équation différentielle algébrique. Depuis le théorème de Hölder, les mathématiciens ont étudié l’hypertranscendance pour les raisons le plus diverses. On en évoquera quelques unes tout au long de ce papier, et en particulier dans le dernier paragraphe.
Récemment une théorie de Galois paramétrée des équations fonctionnelles a vu le jour ([@cassisinger] et [@HardouinSinger]), avec le bout de fournir une approche systématique à la transcendance différentielle, en contraste avec la littérature plus ancienne où on traite les différentes fonctions spéciales “au cas par cas”. On va donner un bref survol de cette théorie et en montrer quelques applications en révisitant des résultats classiques, tel le théorème de Hölder, par exemple.
*Remerciements.* C’est un plaisir de remercier W. Bergweiler, D. Bertrand, X. Buff, C. Hardouin, B.Q. Li, Pierre Nguyen, M.F. Singer, Z. Ye pour leurs suggestions et commentaires. Je suis particulièrement reconnaissante à D. Bertrand, Z. Djadli, D. Harari, C. Hardouin, M.F. Singer pour leur relecture attentive du manuscrit et leur remarques et corrections et a‘ J.-P. Allouche pour son invitation.
Fonction Gamma et théorème de Hölder {#sec:Holder}
====================================
En 1729, Euler, dans une lettre à Goldbach [@eulerletter], définit la fonction Gamma grâce aux limites suivantes, convergentes pour tout $x\in\C\smallsetminus\Z_{<0}$: $$\begin{array}{rcl}
\Ga(x)
&=&\ds\frac{1}{x}\prod_{n=1}^\infty
\l[\l(1+\frac{1}{n}\r)^x\l(1+\frac{x}{n}\r)^{-1}\r]\\~\\
&=&\ds\lim_{n\to\infty}\frac{(n-1)!}{x(x+1)\cdots(x+n-1)}n^x.
\end{array}$$ Weierstrass en a donné une autre caractérisation qui montre que $1/\Ga$ est une fonction analytique entière: $$\frac{1}{\Ga(x)}=xe^{\ga x}\prod_{n=1}^\infty
\l(1+\frac{x}{n}\r)e^{-\frac x n},$$ où $\gamma$ est la constante d’Euler-Mascheroni: $$\ga=\lim_{m\to\infty}
\l(1+\frac{1}{2}+\dots+\frac{1}{m}-\log m\r)=0,5772157....$$ Une propriété fondamentale de la fonction $\Gamma$ est celle de vérifier l’équation aux différences \[eq:ga\] y(x+1)=xy(x) qui, compte tenu du fait que $\Ga(1)=1$, implique immédiatement que $\Ga(n)=(n-1)!$, pour tout entier positif $n$. D’après le théorème de Bohr-Mollerup, la fonction $\Gamma$ est l’unique solution de l’équation $y(x+1)=xy(x)$, logarithmiquement convexe et telle que y(1)=1. L’équation est linéaire, donc l’ensemble de toutes ses solutions méromorphes forme un espace vectoriel engendré par $\Ga$ sur le corps des fonctions $1$-périodiques. [^1]
Le célèbre théorème de Hölder [@holderGamma] affirme que la fonction $\Gamma$ est *différentiellement transcendante* (ou *hypertranscendante*) sur $\C(x)$, c’est-à-dire:
La fonction $\Gamma$ n’est pas solution d’une équation différentielle algébrique à coefficients dans $\C(x)$.
Il existe nombreuses preuves de ce résultat (voir par exemple [@owstroskiGamma1], [@owstroskiGamma2], [@hausdorffGamma], [@BankKaufamannGamma2], [@MooreGamma], [@Nielsen]) et aussi des nombreuses généralisations dans des directions différentes (voir par exemple [@BankKaufamanGamma], [@BankGammaHyptrFunctions], [@BankKaufamnnGammaNevalinna], [@GrossOsgood], [@Nishizawa1], [@Nishizawa2], [@Miller], [@Pastro]). Parmi ces démonstrations on citera celle de Bank et Kaufmann [@BankKaufamannGamma2], qui se déduit du théorème suivant:
\[theo:BankKaukmannGamma\] Soit $\cF$ un sous-corps du corps $\cM er(\C)$ des fonctions méromorphes sur $\C$, contenant $\C(x)$ et fermé par rapport à l’opérateur de translation $\tau:f(x)\mapsto f(x+1)$ et à la dérivation par rapport à $x$. Si la fonction $\Gamma$ d’Euler est solution d’une équation différentielle algébrique à coefficients dans $\cF$, alors, il existe $g,f_0,f_1,\dots,f_n\in \cF$, avec $f_0,f_1,\dots,f_n$ périodiques de période $1$, non tous nuls, telles que $$\sum_{i=0}^n f_i(x)\frac{d^i}{dx^i}\l(\frac{1}{x}\r)=g(x)-g(x+1).$$
\[rema:implicationfacile\] La réciproque de cet énoncé est quasiment vraie. Soit $\cF\langle\Gamma(x)\rangle_{\frac{d}{dx}}$ la plus petite extension de $\cF(\Gamma(x))$ contenue dans $\cM er(\C)$, fermée par rapport à la dérivation $\frac{d}{dx}$. Si on suppose que le corps différentiel $\cF\langle\Gamma(x)\rangle_{\frac{d}{dx}}$ ne contient pas plus de fonctions périodiques que $\cF$, alors la réciproque du théorème ci-dessus est immédiate. En effet, soit $\psi(x):=\frac{\Gamma^\p(x)}{\Gamma(x)}$ la dérivée logarithmique de $\Gamma(x)$, qu’on appelle usuellement *fonction digamma*. On a: $$\begin{array}{rcl}
\lefteqn{\tau\l(\sum_{i=0}^n f_i(x)\frac{d^i\psi}{dx^i}(x)+g(x)\r)}\\
&=&\ds\sum_{i=0}^n f_i(x)\frac{d^i\psi}{dx^i}(x)
+\sum_{i=0}^n f_i(x)\frac{d^i}{dx^i}\l(\frac{1}{x}\r)+g(x+1)\\
&=&\ds\sum_{i=0}^n f_i(x)\frac{d^i\psi}{dx^i}(x)+g(x).
\end{array}$$ Ceci implique que $\sum_{i=0}^n f_i(x)\frac{d^i\psi}{dx^i}(x)+g(x)$ est une fonction périodique de $\cF$ et nous fournit gratuitement une relation différentielle algébrique sur $\cF$ pour $\psi$, et, donc, pour $\Gamma$.
La démonstration du Théorème \[theo:BankKaukmannGamma\] donnée dans [@BankKaufamannGamma2] est assez élémentaire. On en donnera une preuve galoisienne plus loin. Le théorème de Hölder s’en déduit aisément en raisonnant sur les pôles de $g(x)-g(x+1)$, compte tenu du fait que les seules fonctions périodiques contenues dans $\C(x)$ sont les constates.
Notons que, comme dans le cas des extensions algébriques, il est équivalent de démontrer que $\Gamma$ ne satisfait à aucune équation différentielle algébrique à coefficients dans $\C(x)$ ou dans $\C$. En effet, sans trop formaliser les définitions (qui sont très intuitives et pour lesquels on peut se reporter à [@RittDifferentialAlgebra], [@Kolchin:differentialalgebraandalgebraicgroups] ou à [@MarkusZetaGamma §2], pour un résumé rapide), le corps $\C(x)$ est différentiellement algébrique sur $\C$, car $\frac{d}{dx}(x)\in\C$. On peut aussi se limiter à démontrer la transcendance différentielle de $\Gamma$ sur la *clôture différentielle* de $\C(x)$. Cette dernière est une extension différentielle de $(\C(x),\frac{d}{dx})$ contenant une solution de tout système d’équations différentielles algébriques à coefficients dans $(\C(x),\frac{d}{dx})$, qui a une solution dans une extension différentielle quelconque de $(\C(x),\frac{d}{dx})$. C’est bien l’analogue différentiel de la clôture algébrique.
Considérons la fonction $\zeta$ de Riemann, la fonction obtenue par prolongement analytique de $$\zeta(x)=\sum_{n=1}^\infty \frac{1}{n^x},
\hbox{~pour tout $x\in\C$, $\Re(x)>0$.}$$ Elle satisfait à l’équation fonctionnelle $$\zeta(x)=2(2\pi)^{x-1}\Gamma(1-x)\sin\l(\frac{\pi x}{2}\r)\zeta(1-x).$$ Comme le terme $(2\pi)^{x-1}\sin\l(\frac{\pi x}{2}\r)$ est différentiellement algébrique, le théorème de Hölder implique immédiatement la transcendance différentielle de $\zeta$, car si $\zeta$ était différentiellement algébrique la fonction $\Gamma(x)=\zeta(1-x)\frac{1}{2}(2\pi)^{1-x}\l(\sin\frac{\pi(1-x)}{2}\r)^{-1}\zeta(x)^{-1}$ devrait l’être aussi:
La fonction $\zeta$ de Riemann est différentiellement transcendante sur $\C(x)$.
En 1920 Ostrowski [@OstrowskiZeta] prouve aussi la transcendance différentielle sur $(\C(x),\frac{d}{dz},\frac{d}{dx})$ de la fonction obtenue par prolongement analytique de la série $$\zeta(z,x)=\sum_{n=1}^\infty \frac{z^n}{n^x},$$ en répondant à une question posée par Hilbert. Les résultats sur la fonction zeta de Riemann ont été généralisés aussi dans plusieurs directions, souvent à l’aide de la théorie de Nevanlinna [@Laine]. Par contre, la question de l’indépendance différentielle de $\Gamma$ et $\zeta$, c’est-à-dire de la propriété de $\Gamma$ et de $\zeta$ de ne pas être solutions d’une équation différentielle algébrique en deux fonctions inconnues à coefficients dans $\C(x)$, est ouverte. Pour les résultats sur la fonction $\zeta$ de Riemann, on renverra plutôt aux travaux de B.Q. Li et Z. Ye, qui fournissent un survol de la littérature sur le sujet (voir [@LiYeActa], [@LiYeNevalinna], [@LiYeSurvey]). On peut démontrer alors le corollaire suivant (qui généralise et simplifie le Théorème 3 dans [@MarkusZetaGamma]):
\[coro:markuszetagamma\] Soient $\Psi$ et $\Omega$ deux fonctions méromorphes sur $\C$ qui vérifient respectivement les équations fonctionnelles $$\Psi(x+1)=\Psi(x)
\hbox{~et~}
\Omega(x+1)=x\Omega(x).$$ Si $\Psi(x)$ est différentiellemet transcendante sur $\C$ (ou, de façon équivalente sur $\C(x)$), $\Psi(x)$ et $\Omega(x)$ sont différentiellement indépendantes sur $\C$ (ou sur $\C(x)$).
Il existe une fonction $1$-périodique $\Pi(x)$ telle que $\Omega(x)=\Pi(x)\Gamma(x)$. Soit $\cC=\C\langle \Psi(x),\Pi(x)\rangle_{\frac{d}{dx}}\subset\cM er(\C)$ le corps différentiel engendré par $\C$, $\Pi(x)$ et $\Psi(x)$ dans $\cM er(\C)$. Le corps $\cF=\cC(x)$ vérifie les hypothèses du théorème précédent. De plus, le sous-corps des éléments $1$-périodiques de $\cF$ coïncide avec $\cC$.
Si on démontre que $\Omega$ est différentiellement transcendante sur $\cC(x)$ on pourra conclure que $\Psi$ et $\Omega$ sont différentiellement indépendantes sur $\C$. Si la fonction méromorphe $\Omega$ vérifiait une équation différentielle algébrique à coefficients dans $\cF$, il en serait de même pour $\Gamma$ et, donc, il existerait $g,f_0,f_1,\dots,f_n\in \cF$, avec $f_0,f_1,\dots,f_n$ périodiques de période $1$, telles que $$\sum_{i=0}^n f_i(x)\frac{d^i}{dx^i}\l(\frac{1}{x}\r)=
\frac{f_0(x)}{x}+\sum_{i=1}^n \frac{(-1)^{i}\,(i-1)!f_i(x)}{x^{i+1}}=g(x)-g(x+1).$$ On remarque que $x$ est nécessairement transcendante sur $\cC$, car, si le polynôme $P(T)\in\cC[T]$ s’annulait en $x$, il devrait s’annuler sur l’ensemble infini $x+\Z$. On en déduit que la formule ci-dessus fournit une décomposition en éléments simples de $g(x)-g(x+1)$ dans le corps des fonctions rationnelles $\cC(x)$. Ceci est impossible car, si $g(x)-g(x+1)$ a un pôle en $x=0$, il doit aussi avoir au moins un autre pôle en quelque $x\in\Z\smallsetminus\{0\}$.
Les fonctions méromorphes $x\mapsto \zeta(\sin(2\pi x))$ et $\Gamma$ (resp. $x\mapsto \Gamma(\sin(2\pi x))$ et $\Gamma$) sont différentiellement indépendantes sur $\C(x)$.
On démontre seulement les cas de $\zeta(\sin(2\pi x))$ et $\Gamma$. Pour pouvoir appliquer le Corollaire \[coro:markuszetagamma\], il suffit de démontrer que $\zeta(\sin(2\pi x))$ est différentiellement transcendante. On sait que $\zeta$ est différentiellement transcendante sur $\C(x)$, c’est-à-dire que la famille de fonctions $\l\{\frac{d^i\zeta}{dx^i}(x)\r\}_{i\geq 0}$ est algébriquement indépendante sur $\C(x)$. Il s’ensuit que la famille $\l\{\frac{d^i\zeta}{dx^i}(\sin(2\pi x))\r\}_{i\geq 0}$ est algébriquement indépendante sur $\C(\sin(2\pi x))$ et donc sur son extension algébrique $\C(\sin(2\pi x),\cos(2\pi x))$. Donc $\zeta(\sin(2\pi x))$ est différentiellement transcendante sur $\C(\sin(2\pi x),\cos(2\pi x))$ et donc sur $\C$, car $\C(\sin(2\pi x),\cos(2\pi x))$ est une extension différentiellement algébrique de $\C$.
Théorie de Galois paramétrée
============================
La théorie de Galois paramétrée des équations différentielles et aux différences est étudiée dans [@cassisinger] et [@HardouinSinger]. Le cadre plus général est celui décrit dans ce dernier papier. Les auteurs considèrent un corps $F$ équipé de deux familles finies de dérivations, $\Delta$ et $\Pi$, et d’une famille finie d’automorphismes $\Sg$ et ils supposent que les éléments de $\Delta\cup\Pi\cup\Sg$ commutent deux à deux, en tant qu’opérateurs agissant sur $F$. Ils se donnent un système *intégrable* d’équations matricielles[^2] \[eq:syshs\] ł{
[l]{} Y=A\_Y\
Y=B\_Y
. avec $A_\sg$ et $B_\partial$ matrice carrées à coefficients dans $F$, et $A_\sg$ inversible pour tout $\sg\in\Sg$. Moralement, il faut considérer $\Pi$ comme l’ensemble des dérivations associées à des paramètres du système. À partir de cela, ils construisent un groupe qui donne des informations sur les relations différentielles vérifiées par les solutions de par rapport aux paramètres. Dans le but de simplifier les notations de l’exposition qui suit, sans que cela simplifie vraiment les preuves, on se placera dans un cadre moins général.
Théorie de Picard-Vessiot paramétrée {#subsec:DPV}
------------------------------------
Considérons un corps différentiel aux différences, un triplet $(F,\sg,\partial)$, où $F$ est un corps, $\sg$ un automorphisme de $F$ et $\partial$ une dérivation de $F$, telle que $\partial\sg=\sg\partial$. On suppose que $\sg$ n’est pas un automorphisme cyclique, bien que cette hypothèse ne soit nécessaire qu’à quelques endroits. On dira que $F$ est un $(\sg,\partial)$-corps (et on utilisera sans les définir les concepts, très intuitifs, de $(\sg,\partial)$-anneau, $(\sg,\partial)$-algèbre, ...; voir [@Levin:difference] et [@Cohn:difference] pour une exposition systématique de la théorie).
La donnée initiale est celle d’un système aux différences \[eq:sys\] (Y)=AY, où $A\in GL_\nu(F)$ est une matrice inversible à coefficients dans $F$.
Typiquement on peut considérer le corps $\C(x)$ des fonctions rationnelles à coefficients complexes avec les opérateurs suivants:
- $\tau: f(x)\mapsto f(x+1)$ et $\partial=\frac{d}{dx}$;
- $\sgq: f(x)\mapsto f(qx)$, pour un $q\in\C$, $q\neq 0$ fixé, et $\partial=x\frac{d}{dx}$.
\[defn:PV\] On appelle *$(\sg,\partial)$-extension de Picard-Vessiot pour* un $(\sg,\partial)$-anneau $\cR$, extension de $F$, muni d’une extension de $\sg$ et $\partial$, préservant la commutativité, $[\sg,\partial]=0$, tel que:
1. $\cR$ est un $(\sg,\partial)$-anneau simple, il n’a pas d’idéaux propres invariants par $\sg$ et $\partial$;
2. $\cR$ est engendré, en tant que $\partial$-anneau, par une matrice inversible $Z \in GL_\nu(\cR)$ et $\frac{1}{det(Z)}$, avec $Z$ solution de .
Il est possible de construire formellement un tel objet. Considérons l’anneau de $\partial$-polynômes $$F\{X,\det X^{-1}\}_{\partial}:=F\l[X_{i,j}^{(k)};\,i,j=1,\dots,n;\, k\geq 1\r]\l[\frac{1}{\det(X^{(1)}_{i,j})}\r],$$ où $X_{i,j}^{(k)}$ sont des variables algébriquement indépendantes, telles que $\partial(X_{i,j}^{(k)})=X_{i,j}^{(k+1)}$. Soient $X=(X_{i,j}^{(1)})$ et $X^{(k)}=\partial^k X$. On définit sur $F\{X,\det X^{-1}\}_{\partial}$ une structure de $(\sg,\partial)$-algèbre, en posant $\sg(X)=A X$ et \[eq:sigmastructure\]
[rcl]{} ( X\^[(k)]{}) &=&(\^kX)=\^k((X))=\^k(AX)\
&=&\_[h=0]{}\^k[kh]{}\^[h]{}(A) X\^[(k-h)]{},
Le quotient $\cR$ de $F\{X,\det X^{-1}\}_{\partial}$ par un idéal invariant par $\sg$ et $\partial$ et maximal par cette propriété (donc par un *$(\sg,\partial)$-idéal maximal*) est bien sûr une $(\sg,\partial)$-extension de Picard-Vessiot pour .
Soit $K=F^\sg$ le sous-corps de $F$ des éléments invariants par $\sg$. La commutativité de $\sg$ et $\partial$ implique que $K$ est un corps différentiel par rapport à $\partial$.
Si $(K,\partial)$ est différentiellement clos alors:
1. Le sous-anneau des constantes $\cR^\sg$ d’une $(\sg,\partial)$-extension de Picard-Vessiot $\cR$ pour coïncide avec $K$, c’est-à-dire que $\cR$ ne contient pas de nouvelles constantes par rapport à $F$.
2. Deux $(\sg,\partial)$-extensions de Picard-Vessiot pour sont isomorphes en tant que $(\sg,\partial)$-anneaux.
\[rema:descente\] Si $K$ est seulement algébriquement clos et la $(\sg,\partial)$-extension de Picard-Vessiot $\cR$ est en plus un $\sg$-anneau simple, alors le point 1 de la proposition ci-dessus est encore vrai. Par contre il faut en général procéder à une extension des constantes pour avoir un isomorphisme entre deux $(\sg,\partial)$-extensions de Picard-Vessiot. M. Wibmer a affiné la construction donnée ci-dessus pour obtenir une $(\sg,\partial)$-extension de Picard-Vessiot qui est aussi un $\sg$-anneau simple, [@Wibmchev] et [@wibmer2011existence] (son argument est aussi repris dans [@diviziohardouinPacific]). Pour cela il construit de façon fine un $(\sg,\partial)$-idéal maximal de $F\{X,\det X^{-1}\}_{\partial}$, qui est aussi un $\sg$-idéal maximal, en partant d’un $\sg$-idéal maximal de $F[X,\det X^{-1}]$ qu’il prolonge en le dérivant. Ces questions de descente sont traitées en toute généralité, par des méthodes tannakiennes, dans [@GilletGorchinskyOvchinnikov].
Groupe de Picard Vessiot paramétré {#subsec:difPVgr}
----------------------------------
Supposons, pour simplifier, que le corps des $\sg$-constantes $(K,\partial)$ est différentiellement clos.
Soit $\cR$ une $(\sg,\partial)$-extension de Picard-Vessiot pour . Comme dans la théorie de Galois des équations aux différences non paramétrées (voir [@vdPutSingerDifference]), $\cR$ n’est pas, en général, un anneau intègre, mais il est la somme directe de copies d’un anneau intègre, de façon qu’on peut considérer son corps total des fractions $L$, qui est isomorphe à une somme directe de copies d’un même corps ([@HardouinSinger]).
\[defn:pvgr\] Le groupe $Gal^{\partial} (A)$ (qu’on note aussi $Aut^{\sg,\partial}(L / F)$) des automorphismes de $L$, qui fixent $F$ et commutent avec $\sg$ et $\partial$, est le *groupe de Galois paramétré* de . On l’appellera aussi *$\partial$-groupe de Galois* de .
Le groupe $Gal^{\partial}(A)$ agit sur une matrice fondamentale $Z\in GL_\nu(L)$ de solutions de . Pour tout $\varphi\in Gal^{\partial}(A)$, la matrice $\varphi(Z)$ est encore une solution de , donc il existe $U\in GL_\nu(K)$ telle que $\varphi(Z)=ZU$, avec $\sg(ZU)=\sg(Z)U=AZU$. Cette action fournit une représentation fidèle de $Gal^{\partial}(A)$ dans $GL_\nu(K)$, dont l’image est formée des $K$-points d’un $\partial$-groupe algébrique linéaire de $GL_\nu(K)$, dans le sens de Kolchin. C’est-à-dire que c’est un sous-groupe de $GL_\nu(K)$ définit par un $\partial$-idéal de $K\l\{X,\det X^{-1}\r\}_{\partial}$, donc un lieu de zéros d’un ensemble fini d’équations différentielles à coefficients dans $K$. Comme $(K,\partial)$ est un corps différentiellement clos, nous pouvons nous contenter ici d’une description naïve de ce groupe, via son ensemble de points $K$-rationnels. On aura tendance à ne pas faire très attention à distinguer les groupes de Galois et leur représentations en tant que sous-groupes de $GL_\nu$.
On reconnaîtra dans la proposition ci-dessous le c[œ]{}ur de la correspondance de Galois, qu’on n’énoncera pas en entier. On n’aura pas de difficulté à en imaginer les énoncés en s’inspirant de la théorie de Galois classique.
1. L’anneau $L^{Gal^{\partial} (A)}$ des éléments de $L$ fixés par $Gal^{\partial} (A)$ coïncide avec $F$.
2. Soit $H$ un $\partial$-sous-groupe algébrique de $Gal^{\partial}(A)$. Si $L^H = F$, alors $H = Gal^{\partial} (A)$.
Le groupe de Galois (non paramétré) $Gal(A)$ de sur K est construit de la façon suivante: on considère le quotient de l’algèbre de polynômes $F[X,\det X^{-1}]$, munie de l’action de $\sg$ définie par $\sg(X)=A(X)$, par un $\sg$-idéal maximal, et son corps total des franctions $L$; alors $Gal(A)$ est le groupe d’automorphismes de $L/F$ qui commutent avec $\sg$ (voir [@vdPutSingerDifference]). Nous avons:
Le groupe algébrique $Gal(A)$ est la clôture de Zariski de $Gal^{\partial} (A)$ (dans $GL_\nu(K)$).
Si $F$ a un corps des constantes $K$ algébriquement clos et si on considère une $(\sg,\partial)$-extension de Picard-Vessiot de $F$, en suivant la construction de [@wibmer2011existence], on peut construire un schéma en $\partial$-groupes défini sur $K$, dont les points rationnels sur la clôture différentielle de $K$ peuvent être identifiés avec $Gal^\partial(A)$. Évidemment, pour définir $Gal^\partial(A)$ il faut considérer la clôture différentielle $\wtilde K$ de $K$ et travailler sur le corps des fractions de $F\otimes_K \wtilde K$, avec $\sg$ agissant sur $\wtilde K$ comme l’identité ($F$ et $\wtilde K$ étant linéairement disjoints). Pour plus de détails voir [@diviziohardouinPacific §1.2].
Dépendance différentielle {#sec:difdependency}
-------------------------
Une $(\sg,\partial)$-extension de Picard-Vessiot $\cR$ de $F$ pour est un $Gal^{\partial} (A)$-torseur, dans le sens de Kolchin. Cela implique, en particulier, que toutes les relations différentielles par rapport à la dérivation $\partial$, satisfaites par une matrice fondamental de solutions de , sont entièrement déterminées par le groupe $Gal^\partial(A)$:
\[Proposition 6.29 dans [@HardouinSinger]\]\[prop:degtrs\] Le degré de $\partial$-transcendance de $\cR$ sur $F$ est égal à la $\partial$-dimension de $Gal^\partial(A)$.
Les notions de $\partial$-transcendance et $\partial$-dimension sont celles intuitives, notamment le degré de $\partial$-transcendance de $\cR/F$ est égal au nombre maximal d’élément différentiellement indépendants de $\cR$ sur $F$ et la $\partial$-dimension de $Gal(A)$ est égal au degré de $\partial$-transcendance de son algèbre de Hopf différentielle sur le corps des constantes $K$. En gros, ce résultat dit que plus le groupe est petit, plus il y a des relations différentielles entre les solutions de dans $\cR$.
Équations aux différences linéaires d’ordre $1$
-----------------------------------------------
Il n’est pas difficile de se convaincre que les sous-groupes différentiels de ${\mathbb G}_a^n$ sont définis par des équations différentielles linéaires (voir [@cassdiffgr]). On déduit du Théorème \[prop:degtrs\] le critère:
\[prop:gatrans\] Soient $a_1, ..., a_n$ des éléments non nuls de $F$ et $S$ une $(\sg,\partial)$-extension de $F$ telle que $S^\sg=F^\sg =K$. Si $z_1,...,z_n \in S$ sont solutions des équations aux différences $\sg (z_i) -z_i =a_i$, pour $i=1,...,n$, alors $z_1,...,z_n \in S$ satisfont à une $\partial$-relation différentielle non banale sur $F$ si et seulement s’il existe un polynôme différentiel linéaire homogène non nul $L(Y_1,...,Y_n)$ à coefficients dans $K$ et un élément $f \in F$ tels que $L(a_1, ..., a_n) =\sg (f)-f$.
On remarquera la similitude entre cet énoncé et le Théorème \[theo:BankKaukmannGamma\]. En effet, si on considère la dérivée logarithmique de l’équation de la fonction Gamma $$z(x+1)=z(x)+\frac{1}{x},$$ on en déduit facilement un énoncé analogue sur le corps $F$, ayant un corps des constantes $K$ différentiellement clos par rapport à $\partial$. Cette dernière hypothèse n’est pas vérifiée dans le cas des fonctions méromorphes. Il est néanmoins possible de prouver un critère de ce type pour les solutions méromorphes. On reviendra de nouveau sur ce point.
Intégrabilité
-------------
La proposition suivante établit le lien entre la structure du $\partial$-groupe de Galois et l’intégrabilité du système aux différences par rapport à l’opérateur différentiel. Ce genre de problématique se retrouve très naturellement lorsque, par exemple, on cherche une paire de Lax pour une équation qui mélange opérateur différentiels et aux différences. Ce type d’équations est appelé *équations à retard* (*delay equations* dans la littérature en anglais), ou bien, dans le cas spécifique des équations aux $q$-différences, équations du pantographe. Elles se retrouvent naturellement lorsque l’équation décrit un système dépendant de la variable libre, disons le temps $t$, à la fois de façon continue et discrète. La définition de $\partial$-groupe constant est expliquée immédiatement après l’énoncé.
\[prop:integrabilite\] Les assertions suivantes sont équivalentes:
1. Le $\partial$-groupe de Galois $Gal^\partial (A)$ est conjugué sur $K$ avec un $\partial$-groupe constant.
2. Il existe $B\in M_n(F)$ tel que le système $$\left\{\begin{array}{l} \sg(Y)=AY \\
\partial Y=BY
\end{array} \right.$$ est intégrable, c’est-à-dire que les matrices $B$ et $A$ satisfont à l’équation fonctionnelle suivante, induite par la commutativité entre $\sg$ et $\partial$: $$\sg(B)A= \partial(A) +A B.$$
Soient $K$ un $\partial$-corps et $C$ son sous-corps des $\partial$-constantes. On dit qu’un $\partial$-groupe linéaire $G \subset GL_\nu$ défini sur $K$ est un *$\partial$-groupe constant* (ou, plus brièvement, qu’il est $\partial$-constant) si son idéal de définition dans $K\{X,\frac{1}{\det\left(X\right)}\}_\partial$ contient les polynômes différentiels $\partial(X_{i,j})$, pour tout $i,j=1,\dots,\nu$. Puisque $K$ est différentiellement clos, cela est équivalent au fait que les points $K$-rationnels de $G$ coïncident avec les points $C$-rationnels d’un groupe linéaire défini sur $C$. On en déduit le corollaire suivant:
Considérons un système à coefficients dans $F$ et son $\partial$-groupe de Galois $Gal^\partial(A)$. S’il existe une représentation fidèle $\varrho:Gal^\partial(A)\hookrightarrow GL_\mu(K)$ et une matrice dans l’image de $\varrho$ dont le polynôme minimal n’est pas à coefficients dans $C$, alors n’est pas intégrable au sens de la proposition précédente.
Il existe un critère d’intégrabilité analogue pour des équations aux différences dépendant de plusieurs paramètres. Dans le cas des équations différentielles d’ordre 2, dépendant de plusieurs paramètres, il est possible de vérifier l’intégrabilité paramètre par paramètre pour conclure à l’intégrabilité globale [@dreyfus2011kovacic]. Ce résultat a été prouvé aussi pour les équations différentielles d’ordre quelconque dans [@gorchinskiy2012isomonodromic]. La preuve repose sur des théorèmes de structure des groupes algébriques différentiels et donc un résultat analogue devrait être vrai aussi pour les équations aux différences.
Selon [@cassdiffgr], si $H$ un $\partial$-groupe sur un corps différentiellement clos $K$, dont la clôture de Zariski est un groupe algébrique linéaire simple $G$ sur $K$, alors soit $H=G$ soit $H$ est conjugué sur $K$ à un $\partial$-groupe constant. Compte tenu de la Proposition \[prop:degtrs\], on obtient:
Si $Gal(A)$ est un groupe algébrique simple, soit nous sommes dans la situation de la Proposition \[prop:integrabilite\] soit il n’existe aucune relation différentielle non banale entre les éléments d’une matrice fondamentale de solutions de $\sg(Y)=AY$ à coefficients dans $L$.
Si des relations algébriques entre les éléments d’une matrice fondamentale de solutions existent, on peut toujours en déduire des relations différentielles par dérivation. On peut considérer que celles-ci sont des relations différentielles banales.
Transcendance différentielle des solutions méromorphes
======================================================
On a vu que la théorie de Galois fournit des critères de transcendance différentielle pour des solutions abstraites d’une équation aux différences. Dans le cas de la fonction Gamma d’Euler, par exemple, en s’inspirant de la construction plus haut, nous pourrions considérer l’anneau $$\cR_\Gamma=\mathcal P(x)\l[\Gamma(x), \Gamma^\p(x),\Gamma^{(2)}(x),\dots,\frac{1}{\Gamma(x)}\r],$$ où $\cP$ est le corps des fonctions méromorphes sur $\C$ et $1$-périodiques. L’anneau $\cR$ est bien un $(\tau,\partial)$-anneau, par rapport à l’opérateur $\tau:f(x)\mapsto f(x+1)$ et à la dérivation $\partial=\frac{d}{dx}$, et, puisque $\cR_\Gamma\subset\cM er(\C)$, ses constantes coïncident avec $\cP$. Par contre il est assez difficile, en général, d’établir si $\cR_\Gamma$ est un $(\tau,\partial)$-anneau simple, donc toute la discussion précédente tombe (ou risque de tomber) à l’eau. Pour s’en sortir, il suffit de considérer que la clôture différentielle $\wtilde\cP$ de $\cP$ par rapport à $\partial$. Il n’est pas difficile de voir que le corps $\cM er(\C)$ des fonctions méromorphes sur $\C$ et $\wtilde\cP$ sont linéairement disjoints sur $\cP$ (voir le Lemme \[lemm:lindisj\] ci-dessus). On peut alors considérer l’anneau $\cR_\Gamma\otimes_\cP\wtilde\cP$ et le comparer à une $(\tau,\partial)$-extension de Picard-Vessiot, au sens de la Définition \[defn:PV\]. On va formaliser ces considérations.
On appellera $(\cF,\sg,\partial)$ l’un des deux $(\sg,\partial)$-corps[^3] suivants:
1. Le corps $(\cF,\sg,\partial)$ est une extension de $(\cP(x),\tau,\frac{d}{dx})$ contenue dans $(\cM,\sg,\partial):=(\cM er(\C),\tau,\frac{d}{dx})$.
2. Pour $q\in\C$, $|q|\neq 1$, on considère le corps des fonctions elliptiques $\cE_q$, autrement dit le sous-corps du corps $\cM er (\C^*)$ des fonctions méromorphes sur $\C^*$ des fonctions invariantes par $\sg:=\sgq:f(qx)\mapsto f(x)$. Dans ce cas on considère une extension $(\cF,\sg,\partial)$ de $(\cE_q(x),\sgq,x\frac{d}{dx})$ contenue dans $\cM=\cM er(\C^*)$.
Ces deux situations ont beaucoup en commun, mais diffèrent par la nature différentielle du corps des constantes. En effet, le corps $\cK$ des éléments $\sg$-invariants de $\cF$ coïncide avec celui de $\cM$, donc $\cK=\cP$ pour $\sg=\tau$ et $\cK=\cE_q$ si $\sg=\sgq$. Il est bien connu que le corps des fonctions elliptiques $\cE_q$ est différentiellement algébrique. Pour le voir il est suffisant de passer de la notation multiplicative à la notation additive et de se souvenir du fait que la fonction $\wp(x)$ de Weierstrass satisfait à une équation différentielle d’ordre $2$. D’un autre côté, on a vu que $\cP$ contient au moins $x\mapsto\zeta(\sin(2\pi x))$, qui est différentiellement transcendant. Néanmoins on a:
\[lemm:lindisj\] La clôture différentielle $\wtilde\cK$ de $\cK$ et le corps $\cM$ (resp. $\cF$) sont linéairement disjoints sur $\cK$.
Soit $\{\a_i\}_{i\in I}$ une famille finie d’éléments de $\wtilde\cK$ linéairement indépendants sur $\cK$, mais qui deviennent liés sur $\cM$ (resp. $\cF$) en tant qu’éléments de $\cF\otimes_\cK\wtilde\cK$. On suppose qu’elle est minimale, c’est-à-dire que pour tout $\iota\in I$ la famille $\{\a_i\}_{i\in I,i\neq\iota}$ reste linéairement indépendante sur $\cM$ (resp. $\cF$). Soit $\sum_i\la_i\a_i=0$ une combinaison linéaire non banale des $\a_i$ sur $\cM$ (resp. $\cF$). On peut supposer qu’il existe $\iota\in I$ tel que $\la_\iota=1$. On obtient une contradiction en considérant $\sum_i(\la_i-\sg(\la_i))\a_i=0$.
Soit $\sg Y=AY$ un système aux différences tel que $A(x)\in GL_\nu(\cF)$, ayant une matrice fondamentale de solutions $U\in GL_\nu(\cM)$. On appelle $\cR_{\cM}$ l’anneau $\cF\{U, \det U^{-1}\}_\partial\subset\cM$ et $\cR^\p_{\cM}$ un quotient de l’anneau des polynômes différentiels $\cF\{X, \det X^{-1}\}_\partial$ par un $(\sg,\partial)$-idéal maximal. On note aussi $\cR$ la $(\sg,\partial)$-extension de Picard-Vessiot sur $\wtilde\cF=Frac(\cF\otimes_\cK\wtilde\cK)$ associée à $\sg Y=AY$.
$\cR_\cM\otimes_\cK\wtilde\cK
\cong\cR_\cM^\p\otimes_\cK\wtilde\cK
\cong\cR$.
Voir le Corollaire 3.3 et la Proposition 3.4 dans [@diviziohardouinComp] (et [@ChatHardouinSinger] pour le cas non paramétré), dans le cas $(\cF,\sg,\partial)=(\cE_q(x),\sgq,x\frac{d}{dx})$. La preuve se généralise sans difficulté.
Moralement, la proposition précédente dit que les groupes $Aut^{\sg,\partial}(\cR_\cM/\cF)$, $Aut^{\sg,\partial}(\cR_\cM^\p/\cF)$ et $Aut^{\sg,\partial}(\cR/\wtilde\cF)$ coïncident. Cette affirmation n’a pas vraiment de sens car les deux premiers groupes peuvent ne pas avoir beaucoup d’éléments, à cause du fait que $\cK$ n’est pas différentiellement clos. Il est par contre possible de donner un sens rigoureux à cette affirmation en utilisant les schémas en groupes et les catégories tannakiennes différentielles, introduite dans [@ovchinnikovdifftannakian] (voir aussi [@kamensky2009model], [@kamensky2011tannakian] and [@GilletGorchinskyOvchinnikov]). En effet, chacun de ces anneaux détermine un foncteur fibre pour la catégorie tannakienne différentielle engendrée par le module aux différences associé à $\sg Y=AY$. Les schémas en groupes des automorphismes tensoriels de ces foncteurs deviennent tous isomorphes deux à deux sur $\wtilde\cK$. On en déduit:
Il existe un $\partial$-groupe algébrique $G_\cK$ défini sur $\cK$ tel que $Aut^{\sg,\partial}(\cR_\cM/\cF)$ est le groupe de $\cK$-points de $G_\cK$ et que $G_\cK\otimes\wtilde\cK\cong Gal^\partial(A)$.
Ceci nous permet de donner une preuve d’un analogue de la Proposition \[prop:gatrans\] sur un corps de fonctions méromorphes, qui est cachée entre la Proposition 3.1 et le Corollaire 3.2 de [@HardouinSinger] (voir aussi [@hardouincompositio]). Une fois de plus, on utilise de façon cruciale la classification des sous-groupes différentiels de $\mathbb G_a^n$ dans [@cassdiffgr].
\[prop:gatransmero\] Soient $a_1, ..., a_n$ des éléments non nuls de $\cF$. Si $z_1,...,z_n \in\cM$ satisfont aux équations aux différences $\sg (z_i) -z_i =a_i$, pour $i=1,...,n$, alors $z_1,...,z_n$ satisfont à une $\partial$-relation différentielle sur $\cF$ si et seulement s’il existe un polynôme différentiel linéaire homogène non nul $L(Y_1,...,Y_n)$ à coefficients dans $\cK$ et un élément $f \in\cF$ tels que $L(a_1, ..., a_n) =\sg (f)-f$.
Pour $\sg=\tau$, on retrouve une preuve du Théorème \[theo:BankKaukmannGamma\], avec l’hypothèse supplémentaire que $\cP(x)\subset\cF$. On reviendra sur le problème de descente de $\cP$ à $\C(x)$.
On en déduit aussi immédiatement que toute solution méromorphe de l’équation $\Omega(x+1)=x\Omega(x)$ est différentiellement transcendante sur $\cP$, ce qui prouve le Corollaire \[coro:markuszetagamma\].
Une implication a déjà été prouvée dans la Remarque \[rema:implicationfacile\]. Considérons l’anneau $\cR_\cM$ associé au système aux différences $\sg Y=AY$, où $A$ est une matrice diagonale par blocs: $$A=diag\l(\begin{pmatrix}1&a_1\\0&1\end{pmatrix},\dots,
\begin{pmatrix}1&a_n\\0&1\end{pmatrix}\r).$$ Une matrice fondamentale de solutions de $\sg Y=AY$ est donnée par: $$U=diag\l(\begin{pmatrix}1&z_1\\0&1\end{pmatrix},\dots,
\begin{pmatrix}1&z_n\\0&1\end{pmatrix}\r)\in GL_{2n}(\cM).$$ Il s’ensuit que $Gal^\partial(A)$ est un $\partial$-sous-groupe de $\mathbb G_a^n$ défini sur $\cK$. Par hypothèse, c’est un sous-groupe propre (Proposition \[prop:degtrs\]). Il existe donc un polynôme différentiel linéaire homogène non nul $L(Y_1,...,Y_n)$ à coefficients dans $\cK$, contenu dans l’idéal de définition de $Gal^\partial(A)$. On pose $f=L(z_1,\dots,z_n)\in\cR_\cM$. Un argument galoisien montre que $f$ est invariant par l’action de $Gal^\partial(A)$ et donc que $f\in\cF$. On en déduit que $$0=\sg(L(z_1,\dots,z_n)-f)-(L(z_1,\dots,z_n)-f)
=L(a_1,\dots,a_n)-(\sg(f)-f).$$ Pour plus de détails voir la Proposition 3.1 dans [@HardouinSinger].
Soient $a_1, ..., a_n$ des éléments non nuls de $\C(x)$ et $z_1,...,z_n \in\cM$ des solutions méromorphes des équations aux différences $\sg (z_i) -z_i =a_i$, pour $i=1,...,n$. Les assertions suivantes sont équivalentes:
1. Les fonctions $z_1,...,z_n$ satisfont à une $\partial$-relation différentielle sur $\cK(x)$.
2. Il existe un polynôme différentiel linéaire homogène non nul $L(Y_1,...,Y_n)$ à coefficients dans $\cK$ et un élément $f \in \cK(x)$ tels que $L(a_1, ..., a_n) =\sg (f)-f$.
3. Les fonctions $z_1,...,z_n$ satisfont à une $\partial$-relation différentielle sur $\C(x)$.
4. Il existe un polynôme différentiel linéaire homogène non nul $L(Y_1,...,Y_n)$ à coefficients dans $\C$ et un élément $f \in \C(x)$ tels que $L(a_1, ..., a_n) =\sg (f)-f$.
La proposition précédente donne l’équivalence entre 1. et 2. L’implication $4.\Rightarrow 3.$ se prouve comme la Remarque \[rema:implicationfacile\] et l’implication $3.\Rightarrow 1.$ est tautologique. Il ne nous reste qu’a démontrer que $2.\Rightarrow 4.$ Pour cela on va utiliser un argument de descente classique. On considère un polynôme différentiel linéaire homogène $\wtilde L$ et une fonction rationnelle $\wtilde f$ en $x$, obtenus des $L$ et $f$ en remplaçant leurs coefficients (dans $\cK$) par des coefficients génériques. L’identité $L(a_1, ..., a_n) =\sg (f)-f$ se traduit en une série d’équations algébriques en les coefficients de $\wtilde L$ et $\wtilde f$, à coefficients dans $\C$. Ces équations ont une solution dans $\cK$, car $L$ et $f$ existent par hypothèse. On conclut qu’elles doivent avoir une solution dans $\C$, puisque $\C$ est algébriquement clos. Ceci termine la preuve.
Le cas des équations aux $q$-différences {#sec:qdiff}
========================================
Les résultats du paragraphe précédent s’appliquent aussi bien aux équations aux différences finies qu’aux équations aux $q$-différences. Considérons un nombre complexe $q$ tel que $|q|>1$ et la fonction Theta de Jacobi $$\theta_q(x)=\sum_{n\in\Z}q^{-n(n-1)/2} x^n.$$ Elle vérifie l’équation aux $q$-différences $y(qx)=qxy(x)$. La dérivée logarithmique $\ell_q(x)$ de $\theta_q(x)$ par rapport à la dérivation $\partial=x\frac{d}{dx}$ vérifie l’équation $$\ell_q(qx)=\ell_q(x)+1.$$ Il s’ensuit que $\partial(\ell_q)\in\cE_q$ et que, sans surprise, la fonction Theta de Jacobi est différentiellement algébrique. Si on avait voulu appliquer la Proposition \[prop:gatransmero\] à l’équation de $\ell_q(x)$ il aurait suffit de poser $f=\partial(\ell_q)$ et $L=\partial$.
L’algébricité différentielle de $\Theta_q$ est équivalente au fait que le $\partial$-groupe de Galois $Gal^\partial(qx)$ est un sous-groupe différentiel propre de $\mathbb G_m$. Dans le cas différentiel, $\mathbb G_m$ se plonge dans $\mathbb G_a$ grâce à la dérivée logarithmique $z\mapsto \partial(z)/z$. On peut prouver que les sous-groupes différentiels propres non finis de $\mathbb G_m$, définis sur $\cE_q$, ont un idéal de définition engendré par un nombre fini d’équations différentielles $\cL(\partial(z)/z)=0$, où $\cL$ est un opérateur différentiel linéaire dans $\cE_q[\partial]$. Il n’est pas difficile de voir que $Gal^\partial(qx)\subset\l\{\partial\l(\frac{\partial z}{z}\r)=0\r\}\subset\mathbb G_m$.
Une problématique propre aux équations aux $q$-différences est celle liée à la dépendance différentielle en $q$ des solutions, lorsque $q$ est un paramètre (voir [@diviziohardouinPacific]). Par exemple, si on pose $\partial_q=q\frac{d}{dq}$ et $\partial_x=x\frac{d}{dx}$, la fonction Theta de Jacobi vérifie l’équation aux dérivées partielles $$2\partial_q\theta_q=-\partial_x^2\theta_q+\partial_x\theta_q,$$ laquelle est, à un changement de variable près, l’équation de la chaleur. Il est possible de déduire des arguments ci-dessus qu’il n’y a guère que la fonction $\theta_q$ qui vérifie une équation aux $q$-différences d’ordre $1$ à coefficients dans $\C(x)$ et qui satisfait à des relations différentielles par rapport à $\partial_q,\partial_x$.
Commençons par formaliser le cadre. On considère le corps $\C(q)$ avec la norme $q^{-1}$-adique, c’est-à-dire qu’on fixe un réel $d>1$ et pour tout $f(q),g(q)\in\C[q]$, avec $g(q)\neq 0$ on pose: $$\l|\frac{f(q)}{g(q)}\r|=d^{\deg_q f-\deg_q g}.$$ Ceci définie une norme ultramétrique sur $\C(q)$ qui s’étend à la plus petite extension normée $\cC$ de $\C(q)$, complète et algébriquement close. On peut alors considérer les fonctions méromorphes $\cM$ sur $\cC^*=\cC\smallsetminus\{0\}$, qui sont les quotients de séries entières à coefficients dans $\cC$, ayant un rayon de convergence infini. Les opérateurs $\sgq,\partial_q,\partial_x$ s’étendent naturellement à $\cM$ et on peut considérer le corps $\cE_q$ des fonctions elliptiques, $\sgq$-invariantes, de $\cM$.
Nous allons considérer le corps des fonctions méromorphes $\cF=\cE_q(x,\ell_q(x))\subset\cM$. Puisque $\ell_q(qx)=\ell_q(x)+1$, le corps $\cF$ est stable par $\sgq$. Évidemment le triplet $(\cF,\sgq,\partial_x)$ se comporte exactement comme les corps considérés dans la section précédente, bien que la nature des fonction méromorphes dans ce contexte soit un peu différente. Bien sûr, le corps des $\sgq$-invariants de $\cF$ coïncide avec $\cE_q$.
Si on pose $\de=\ell_q(x)\partial_x+\partial_q$, on peut vérifier que $\de$ commute avec $\sgq$ (voir Lemme 2.1 dans [@diviziohardouinPacific]), que $\de(\ell_q)\in\cE_q$ et que, donc, elle laisse $\cF$ stable dans $\cM$. Il s’ensuit qu’aussi le triplet $(\cF,\sgq,\de)$ est de la même nature que les corps différentiels/aux différences considérés précédemment. Son sous-corps des $\sgq$-invariants est toujours $\cE_q$.
On déduit de l’équation $\ell_q(qx)=\ell_q(x)+1$ que $\de\ell_q(x)\in\cE_q$, ce qui prouve que $\theta_q(x)$ vérifie une équation différentielle non banale en $\de$. Comme on l’a déjà remarqué, ceci est équivalent au fait que le $\de$-groupe de Galois $Gal^\de(qx)$ est un sous-groupe différentiel propre de $\mathbb G_m$. Le calcul du $\de$-groupe de Galois $Gal^\de(qx)$ est étroitement lié à l’équation de la chaleur (voir (2-3) dans [@diviziohardouinPacific]).
La Proposition \[prop:gatransmero\] est valable pour $(\cF,\sgq,\partial_x)$ et pour $(\cF,\sgq,\de)$, avec exactement la même preuve (voir le Corollaire 2.5 dans [@diviziohardouinPacific]). Puisque $\cF$ est une extension purement transcendante de $\cE_q$, on peut en déduire, par un argument élémentaire de décomposition en éléments simples, la proposition suivante:
Soient $a(x)\in\C(q,x)$ et $u\in\cM$ une solution de $y(qx)=a(x)y(x)$. Les affirmations suivantes sont équivalentes:
1. Il existe $r\in\Z$, $g(x)\in\C(q,x)$ et $\mu\in\C(q)$ tels que $a(x)=\mu x^rg(qx)/g(x)$.
2. La fonction $u$ est solution d’une équation différentielle algébrique non triviale sur $(\cF,\partial_x)$ (et donc sur $\cC(x)$).
3. La fonction $u$ est solution d’une équation différentielle algébrique non triviale sur $(\cF,\de)$ (et donc sur $\cC(x)$).
L’équivalence entre la première et la deuxième assertion est le Théorème 1.1 dans [@HardouinSinger], alors que l’équivalence entre la première et la troisième affirmation est prouvée dans la Proposition 2.7 de [@diviziohardouinPacific].
Une solution méromorphe de $y(qx)=a(x)y(x)$, avec $a(x)=\mu x^rg(qx)/g(x)$, est donnée par: $$\theta_q(\mu x/q^r)\theta_q(x)^{r-1}g(x)\in\cM.$$ Il n’y a, donc, guère que la fonction Theta de Jacobi, qui soit solution d’une équation aux $q$-différences d’ordre $1$ et qui ait des propriétés d’algébricité différentielle non banales par rapport a $\partial_x,\partial_q$.
Signalons le fait qu’on peut aussi étudier l’intégrabilité des systèmes aux $q$-différences d’ordre $>1$ par rapport à $\partial_x$ et $\partial_q$ (voir le Corollaire 2.9 dans [@diviziohardouinPacific]).
Quelques mots sur ce que ce survol ne contient pas
==================================================
Ce survol est un introduction à des thématiques galoisiennes liées aux équations aux différences et à la transcendance différentielle. On a rapidement dû renoncer à la velléité de donner une liste de références relativement complète sur le sujet de la transcendance différentielle, car la littérature est tentaculaire. L’article de survol de Rubel [@Rubelsurvey], ainsi que [@Rubelpbs1] et [@Rubelpbs2], fournissent une jolie vue panoramique des travaux plus classiques. On renvoie le lecteur à ces articles et à leur bibliographie. On signale aussi:
- Dans \[a\], \[b\], \[c\] on trouvera un approche effectif à la transcendance différentielle, dans un style diophantien.
- Dans [@MarkusZetaGamma] on trouve une allusion aux liens entre transcendance différentielle et dynamique holomorphe. Sur ce point la littérature semble se limiter aux articles [@Bergweilerpbrubel; @BeckerBergweiler]
- En combinatoire, il arrive qu’on se demande si des séries qui proviennent d’un problème énumératif, et qui en général sont solutions d’une équation aux différences, sont aussi solutions d’une équation différentielle, linéaire ou pas. Ceci a pour but d’obtenir des informations sur les récurrences qui engendrent les séries en question. On pourra citer à titre d’exemple [@BousquetMishna], [@BousquetPetk] et [@BostanKauers] et [@singerletter].
Pour conclure on se limitera à faire une liste, quasiment en vrac, de quelques résultats en relation avec le sujet principal de ce texte.
Dans §\[sec:Holder\], on a déjà beaucoup parlé d’équations aux différences finie, associées à la translation $x\mapsto x+1$. En ce qui concerne les équations aux $q$-différences, associées à l’homothétie $x\mapsto qx$, nous avons d’un côté les résultats de rationalité des séries formelles solutions des systèmes d’équations aux $q$-différences/différentiels [@RamisToulouse] et des systèmes d’équations aux $q$-différences/$q^\p$-différences [@BezivinBoutabaa]. La rationalité des solutions est aussi étudiée dans [@DVInv] et [@diviziohardouinqGroth], par des méthodes arithmétiques inspirées de la conjecture de Grothendieck sur les $p$-courbures. De l’autre, on a le résultat de Ishizaki [@Ishizaki] sur l’hypertranscendance des solutions méromorphes des équations de la forme $y(qx)=a(x)y(x)+b(x)$. Une premier approche galoisienne à ce sujet se trouve dans [@hardouincompositio], suivi par le travail [@HardouinSinger], sur lequel on s’est longuement étendu.
La transcendance différentielle des fonctions de Mahler $f(x)=\sum_{n\geq 0}x^{k^n}$ est étudiée dans [@mahler] et [@Loxtonpoorten]. La fonction $f$ est solution de l’équation fonctionnelle $f(x^k)=f(x)-x$. La question de la transcendance différentielle des solutions de ce type d’équation fonctionnelle est étudiée, toujours par des méthodes galoisiennes, dans la thèse de P. Nguyen, dont les résultats sont annoncés dans la note [@pierre]. M. Singer a aussi prouvé des résultat dans cette direction [@singerletter].
Pour ce qui concerne les travaux en théorie de Galois paramétrée, il faut signaler que le point de départ a été la théorie paramétrée des équations différentielles, développée dans [@cassisinger]. Le problème inverse a été étudié par M. Singer [@singer2011linear]. Pour cette théorie on dispose d’une description du groupe de Galois dans le cas analytique [@dreyfus2012density], dans l’esprit du théorème de densité de Ramis, et d’un algorithme de Kovacic pour les équations différentielles d’ordre $2$ [@dreyfus2011kovacic]. Signalons aussi l’étude de l’intégrabilité dans [@gorchinskiy2012isomonodromic].
La théorie de Galois paramétrée est liée aux catégories tannakiennes différentielles, introduites par A. Ovchinnikov [@ovchinnikovdifftannakian; @ovchinnikovDiffAlgPara] et par M. Kamesky [@kamensky2009model]. Les questions liées à la descente peuvent être traitées via la théorie de Picard-Vessiot [@wibmer2011existence] ou bien l’approche tannakienne [@GilletGorchinskyOvchinnikov]. Par ailleurs, l’analogue de la conjecture de Grothendieck sur les $p$-courbures permet de donner une caractérisation arithmétique du groupe de Galois intrinsèque [@diviziohardouinqGroth] et de son analogue paramétré [@diviziohardouinqMalg] et de le comparer avec les différentes théorie de Galois dans la littérature [@diviziohardouinComp], en complétant le travail de comparaison commencé dans [@ChatHardouinSinger].
Il est naturel de se demander si la théorie de Galois peut aider à analyser la transcendance d’une fonction et de ses itérées par rapport à un automorphisme. Ceci fait l’objet de travaux en cours par l’auteur de ce texte, C. Hardouin et M. Wibmer, d’un côté, et par A. Ovchinnikov, D. Trushin et M. Wibmer, de l’autre. La géométrie des variétés aux différences étant plus compliquée que la géométrie des variétés différentielles (au sens de Kolchin), il y a beaucoup de difficultés. Dans cette direction, on citera aussi le travail de M. Kamesky [@kamensky2011tannakian].
De façon un peu surprenante, la théorie de Galois non linéaire [@MalgGGF; @UmemuranonlinearGalois], a été développée bien avant la théorie de Galois paramétrée. Elle a été généralisée au cas des équations aux différences non linéaires dans [@GranierFourier]. La théorie de Galois non linéaire généralise plutôt la théorie de Galois paramétrée que la théorie de Galois “classique” des équations aux différences (voir Corollaire 4.10 dans [@diviziohardouinqMalg; @diviziohardouinCRAS]). Les deux papiers [@casaleroques; @casaleroquesCrelle] mélangent la théorie linéaire des équations fonctionnelles et la théorie non linéaire de Malgrange pour traiter des problèmes d’intégrabilité.
Enfin, ils existe plusieurs approches différentes à la théorie de Galois des équations aux différences: voir par exemple [@SauloyENS] et [@andreens]. Dans le cas particulier des équations aux $q$-différences, les travaux de J.-P. Ramis, J. Sauloy et C. Zhang étudient des questions galoisiennes d’un point de vue beaucoup plus analytique [@RamisSauloySauvageI; @RamisSauloySauvageII; @ramis2009local].
[^1]: Pour une présentation des formules classiques sur la fonction Gamma et pour des références précises à la littérature plus ancienne, voir [@whittakerwatson §XII].
[^2]: Le fait que le système est intégrable signifie que les matrices $A_\sg$ et $B_\partial$ satisfont à des équations fonctionnelles liées à la commutativité des opérateurs; Proposition \[prop:integrabilite\].
[^3]: En réalité, nous n’avons pas besoin de fixer un choix pour $\partial$: les propositions qui suivent sont vraies pour toute dérivation commutant avec les deux choix de $\sg$ ci-dessous. Ca sera le cas dans §\[sec:qdiff\].
|
---
author:
- '**Ricardo Lima Alves and Mariana Reis**'
title: About existence and regularity of positive solutions for a Quasilinear Schrödinger equation with singular nonlinearity
---
[^1]
[**Abstract**]{}
[*2010 Mathematics Subject Classification*]{}. [35J62, 35J20, 55J75]{}.\
[*Key words*]{}. [Strong singularity, Variational methods, Regularity.]{}
**Introduction**
================
In this article we are concerned with the existence of solution for the following quasilinear Schrödinger equation $$(P)\left\{
\begin{array}{l}
-\Delta u -\Delta (u^{2})u=h(x) u^{-\gamma} + f(x,u)~\mbox{in} ~ \Omega,\\
u> 0~\mbox{in}~ \Omega,~~ u(x)=0~\mbox{on}~\partial \Omega,
\end{array}
\right.$$ where $1<\gamma, \Omega \subset \mathbb{R}^{N} (N\geq 3)$ is a bounded smooth domain, $0<h\in L^{1}(\Omega)$, $f:\Omega \times \mathbb{R}\longrightarrow \mathbb{R}$ is a measurable function and satisfies one of the following conditions
- $f(x,s)=b(x)s^{p}$ with $p\in (0,1), b\in L^{\infty}(\Omega)$ and $b^{+}\not\equiv 0$ in $\Omega$,
- $f(x,s)=-b(x)s^{22^{\ast}-1}$ with $0\leq b\in L^{\infty}(\Omega)$.
Recently, some papers have worked with equation of the form $$\label{2}
-\Delta u -\Delta (u^{2})u=f(x,u)~\mbox{in} ~ \Omega,$$ where $\Omega \subset \mathbb{R}^{N}$ is a bounded smooth domain and $f$ is non-singular. See for example [@AGS; @LLL; @LP; @CRK] and its references, where the authors used variational methods to prove the existence of solution. In these works the nonlinearity is non-singular and therefore the functional energy associated to the problem has a good regularity to use the usual techniques for functional of class $C^{1}$.
When $f$ is singular, problems of type (\[2\]) was studied by Do Ó-Moameni [@DM], Liu-Liu-Zhao [@JDP] and Wang [@W]. In [@DM] the authors studied the problem $$\label{3}
-\Delta u -\frac{1}{2}\Delta (u^{2})u=\lambda |u|^{2}u-u-u^{-\gamma}~\mbox{in} ~ \Omega, u>0~\mbox{in} ~ \Omega~,$$ where $\Omega$ is a ball in $\mathbb{R}^{N}$ centered at the origin, $0<\gamma<1$ and $N\geq 2$. They show, using the Nehari manifold method, that problem (\[3\]) has a radially symmetric solution $u\in H_{0}^{1}(\Omega)$ for all $\lambda \in I$, where $I$ is a open and bounded interval.
In 2016 Liu-Liu-Zhao in [@JDP] considered the problem $$\label{4}
-\Delta_{s} u -\frac{s}{2^{s-1}}\Delta (u^{2})u=h(x)u^{-\gamma}+\lambda u^{p}~\mbox{in} ~ \Omega, u>0~\mbox{in} ~ \Omega~,$$ where $\Delta_{s}$ is the $s$-Laplacian operator, $s>1$,$\gamma>0$, $\Omega\subset \mathbb{R}^{N}(N\geq 3)$ is a bounded smooth domain, $2s<p+1<\infty$ and $h(x) \geq 0$ is a nontrivial measurable function satisfying the following condition: there exists a function $\phi_{0}\geq 0$ in $C_{0}^{1}(\overline{\Omega})$ and $q>N$ such that $h(x)\phi_{0}^{-\gamma}\in L^{q}(\Omega)$. Combining the sub and supersolution method, truncation argument and variational methods, they proved the existence of $\lambda_{\ast}>0$ such that the problem (\[4\]) has at least two solutions for $\lambda \in (0,\lambda_{\ast})$.
Recently Wang in [@W] proved the existence and uniqueness of solution to the following quasilinear Schrödinger equation $$-\Delta u -\Delta (u^{2})u=h(x) u^{-\gamma}-u^{p-1}~\mbox{in} ~ \Omega, u>0~\mbox{in} ~ \Omega~,$$ where $\Omega\subset \mathbb{R}^{N}(N\geq 3)$ is a bounded smooth domain, $\gamma \in (0,1), p\in [2,22^{\ast}]$ and $0<h\in L^{\frac{22^{\ast}}{22^{\ast}-1+\gamma}}(\Omega)$. The author used global minimization arguments to prove the existence of solution. Here after the use of variable change developed in Colin-Jeanjean [@CJ] the functional associated to the dual problem is well defined in $H^{1}_{0}(\Omega)$ and continuous.
Before stating our results we would like to cite here the work of Sun [@Y]. In this work the following problem was considered $$\label{1}
-\Delta u=h(x)u^{-\gamma}+b(x)u^{p}~\mbox{in}~\Omega, ~u=0~\mbox{on}~\partial \Omega,$$ where $\Omega\subset \mathbb{R}^{N}$ is a bounded smooth domain, $b\in L^{\infty}(\Omega)$ is a non-negative function, $0<p<1$, $\gamma > 1$ and $0<h\in L^{1}(\Omega)$. The author has proved, using variational methods, that the existence of positive solution in $H^{1}_{0}(\Omega)$ of the problem (\[1\]) is related to a compatibility hypothesis between on the couple $(h(x),\gamma)$, more precisely it has been proved that the problem $(\ref{1})$ has a solution in $H^{1}_{0}(\Omega)$ if and only if there is $v_{0}\in H^{1}_{0}(\Omega)$ such that $$\label{C}
\int_{\Omega} h(x)|v_{0}|^{1-\gamma}<\infty.$$
The main difficulty there was because of the strong silgularity that causes a serious loss of regularity of the functional energy associated with which it is not continuous. In order to deal with these difficulty she worked with appropriate constrainsts sets to restore the integrability of singular term. Moreover, it was essential in its approach that nonlinearity was homogeneous.
Motivated by [@Y] a natural question arises: the hypothesis of compatibility between on the couple $(h(x),\gamma)$ given by (\[C\]) remains necessary and sufficient for the existence of solution for our class of problems $(P)$? In this paper we will give a positive answer to this question. Also requesting more regularity in the function $h$ we prove that the solution have $C^{1,\alpha}$ regularity and as a consequence of this the solution is unique.
Our main results are
\[T1\] Assume that $(f)_{1}$ is satisfied. Then:
- the problem $(P)$ admits an solution $u\in H_{0}^{1}(\Omega)$ if and only if there exists $v_{0}\in H_{0}^{1}(\Omega)$ such that (\[C\]) is satisfied.
- if $b\geq 0$ and there exist constants $c>0$ and $\beta \in (0,1)$ such that $$\label{D}
h(x)\leq c d^{\gamma-\beta}(x,\partial \Omega), \forall x\in \Omega,$$ then the solution $u$ given in $a)$ belongs to $C^{1,\alpha}(\overline{\Omega})$ for some $\alpha \in (0,1)$. In particular the problem $(P)$ has a unique solution in $H_{0}^{1}(\Omega)$.
\[T2\] If $(f)_{2}$ is satisfied, the problem $(P)$ admits an unique solution $ u\in H_{0}^{1}(\Omega)$ if and only if there exists $v_{0}\in H_{0}^{1}(\Omega)$ such that (\[C\]) is satisfied.
To prove our main result let us use the method of changing variables developed by Colin-Jeanjean [@CJ]. After this the functional associated with the dual problem is not homogeneous and this causes several difficulties. For example, the techniques used by previous work do not apply directly. To cover this difficulty we will make a careful analysis of the fiber maps associated to the functional of the dual problem and will approach the problem in a new way to prove the existence of a solution to the problem $(P)$.
Now let us mention some contributions from our work. In this work we consider the most general potentials, for instance the potential $ b $ can change signal. The regularity of solution (and hence uniqueness) for problem with strong singularity has not been treated yet. Theorem \[T2\] completes the study made by Wang in [@W] in the sense that we now consider the case $\gamma>1$, while [@W] consider the case $0<\gamma<1$. Moreover in our work we consider the more general potentials also.
This paper is organized as follows: In the next section we reformulate the problem $(P)$ into a new one which finds its natural setting in the Sobolev space $H_{0}^{1}(\Omega)$ and we will present some preliminary lemmas. In section $3$, we give the proof of Theorem \[T1\] and in section $4$ the proof of Theorem \[T2\]. The last section consists of an appendix to which we will study the problem $(P)$ with $b(x)\equiv \lambda b(x), \lambda \geq 0$ and prove that the solutions found in the Theorem \[T1\] vary continuously with respect to $\lambda$ and the norm from $H_{0}^{1}(\Omega)$.
$Notation$. In the rest of the paper we make use of the following notation:
- $c,C$ denote positive constants, which may vary from line to line,
- $H_{0}^{1}(\Omega)$ denote the Sobolev Space equipped with the norm $||u|| \!\!=\!\! \left(\displaystyle\int_{\Omega}|\nabla u|^{2}dx\right)^{2}$,
- $L^{s}(\Omega)$ denotes the Lebesgue Space with the norms $||u||_{s}= \left(\displaystyle\int_{\Omega}|\nabla u|^{s}dx\right)^{1/s}$, for $1\leq p<\infty$ and $||u||_{\infty}=\inf \left\{C>0: |u(x)|\leq C~\mbox{a.s. in}~ \Omega\right\}$,
- for each set $B\subset \mathbb{R}^{N}$ the characteristic function of $B$ is denoted by $\chi_{B}$.
Variational framework and Preliminary Lemmas
============================================
In this section we provide preliminary results wich will be used to prove the existence of a solution of the problem $(P)$. By solutions we mean here weak solutions in $H_{0}^{1}(\Omega)$, that is $u \in H_{0}^{1}(\Omega)$ satisfying $u(x) > 0$, in $\Omega$ and $$\displaystyle \int_{\Omega} [(1+u^{2})\nabla u \nabla \varphi+2u\vert \nabla u\vert^{2}\varphi - h(x)u^{-\gamma} \varphi-f(x,u)\varphi]dx=0,$$ for every $\varphi \in H^{1}_{0}(\Omega)$, which is formally the variational formulation of the following functional $J:D(J)\subset H_{0}^{1}(\Omega)\rightarrow \mathbb{R}$ $$J(u)=\frac{1}{2}\displaystyle\int_{\Omega} (1+2u^{2})|\nabla u|^{2}+\frac{1}{\gamma-1}\int_{\Omega} h(x)|u|^{1-\gamma}-\int_{\Omega} F(x,u),$$ where $D(J)=\left\{u\in H_{0}^{1}(\Omega): \displaystyle\int_{\Omega} h(x)|u|^{1-\gamma}<\infty \right\}$ if $D(J)\neq \emptyset$ and $F(x,s)=\displaystyle\int_{0}^{s}\!\!\!f(x,t)dt$.
However, this functional is not well-defined, because $\displaystyle\int_{\Omega} u^{2}|\nabla u|^{2}dx$ is not finite for all $u\in H^{1}_{0}(\Omega) $, hence it is difficult to apply variational methods directly. Firstly we use the method developed in [@CJ] introducing the unknown variable $v:=g^{-1}(u),$ where $g$ is defined by $$g^{\prime}(t)=\frac{1}{\left(1+2|g(t)|^{2}\right)^{\frac{1}{2}}},~\forall t\in [0,\infty), ~
g(t)=-g(-t),~\forall t\in (-\infty,0].$$
It is easy to see that if $v$ is solution of $$(P_{A})\left\{
\begin{array}{l}
-\Delta v=\left[ h(x) (g(v))^{-\gamma} + f(x,g(v))\right]g^{\prime}(v) ~\mbox{in} ~ \Omega,\\
v> 0~\mbox{in}~ \Omega,~~ v(x)=0 ~\mbox{on}~ \partial \Omega,
\end{array}
\right.$$ if and only if $u = g(v)$ is solution of $(P)$. We will call the problem $(P_{A})$ of dual problem to $(P)$.
The weak form of the equation $(P_A)$ is $$\int_{\Omega} \nabla v \nabla \phi dx=\int_{\Omega} h(x) (g(v))^{-\gamma}g^{\prime}(v)\phi dx+\int_{\Omega} f(x,g(v))g^{\prime}(v)\phi dx,$$ for every $\phi \in H_{0}^{1}(\Omega)$ and therefore $v$ is a critical point of functional $$\Phi(v)=\frac{1}{2}\int_{\Omega} |\nabla v|^{2}+\frac{1}{\gamma-1}\int_{\Omega} h(x)|g(v)|^{1-\gamma}-\int_{\Omega} F(x,g(v)),$$ which is defined in $D(\Phi)=\left\{v\in H_{0}^{1}(\Omega): \displaystyle\int_{\Omega} h(x)|g(v)|^{1-\gamma}<\infty \right\}$ if $D(\Phi)\neq \emptyset$ and $F(x,s)=\displaystyle\int_{0}^{s}f(x,t)dt$. Now, we list some properties of $g$, whose proofs can be found in Liu [@L].
\[L1\] The function $g$ satisfies the following properties:
- $g$ is uniquely defined, $C^{\infty}$ and invertible;
- $g(0)=0$;
- $0<g^{\prime}(t)\leq 1$ for all $t\in \mathbb{R}$;
- $\frac{1}{2}g(t)\leq tg^{\prime}(t)\leq g(t)$ for all $t>0$;
- $|g(t)|\leq |t|$ for all $t\in \mathbb{R}$;
- $|g(t)|\leq K_{0}|t|^{\frac{1}{2}}$ for all $t\in \mathbb{R}$;
- $(g(t))^{2}-g(t)g^{\prime}(t)t\geq 0$ for all $t\in \mathbb{R}$;
- There exists a positive constant $C$ such that $|g(t)|\geq C|t|$ for $|t|\leq 1$ and $|g(t)|\geq C|t|^{\frac{1}{2}}$ for all $|t|>1$;
- $g^{\prime \prime}(t)<0$ when $t>0$ and $g^{\prime \prime}(t)>0$ when $t<0$;
- the functions $(g(t))^{1-\gamma}$ and $(g(t))^{-\gamma}$ are decreasing for all $t>0$;
- the function $(g(t))^{p}t^{-1}$ is decreasing for all $t>0$;
- $|g(t)g^{\prime}(t)|<1/ \sqrt[]{2}$ for all $t\in \mathbb{R}$.
We only prove $(10),(11)$. Since $g(t),g^{\prime}(t)>0 $ for each $t>0$ and $\gamma>1$ follows that $$\left[(g(t))^{1-\gamma}\right]^{\prime}=(1-\gamma)(g(t))^{-\gamma}g^{\prime}(t) < 0, ~\forall t>0.$$ Hence the function $g^{1-\gamma}:(0,\infty)\longrightarrow \mathbb{R}$ is decreasing. Similarly we have that the function $g^{-\gamma}:(0,\infty)\longrightarrow \mathbb{R}$ is decreasing.
Let us prove $(11)$. To do this, note that $$\begin{aligned}
\left[(g(t))^{p}t^{-1}\right]^{\prime}=&p(g(t))^{p-1}g^{\prime}(t)t^{-1}-(g(t))^{p}t^{-2}\\
=&p(g(t))^{p-1}(g^{\prime}(t)t)t^{-2}-(g(t))^{p}t^{-2}\\
<& t^{-2}\left[(g(t))^{p-1}g(t)-(g(t))^{p}\right]<0,
\end{aligned}$$ where we use the item $(4)$ of this Lemma and $p<1$. Therefore the function $(g(t))^{p}t^{-1}$ is decreasing for all $t>0$.
The next lemma gives us a relation of duality between the compatibility hypothesis for the problems $(P)$ and $(P_{A})$.
\[L2\] Let $v>0$ in $\Omega$. The following conditions are equivalent:
- $\displaystyle\int_{\Omega} h(x)|v|^{1-\gamma}<\infty$;
- $\displaystyle\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v<\infty$;
- $\displaystyle\int_{\Omega} h(x)(g(v))^{1-\gamma}<\infty$.
Firstly let us prove that $(a)\Rightarrow (b)$. We denote by $A_{1}=\left\{x\in \Omega:|v(x)|\leq 1\right\}$ and $A_{2}=\left\{x\in \Omega:|v(x)|> 1\right\}$. By the Lemma \[L1\] $(4),(8)$ we have $$|h(x)(g(v))^{-\gamma}g^{\prime}(v)v|\leq C^{1-\gamma} h(x)|v|^{1-\gamma},~\forall x\in A_{1},$$ and $$\begin{array}{rl}
|h(x)(g(v))^{-\gamma}g^{\prime}(v)v| \leq & h(x)|(g(v))^{1-\gamma}|\\
\leq & C^{1-\gamma}h(x)|v|^{\frac{1-\gamma}{2}}\\
\leq & C^{1-\gamma}h(x),~\forall x\in A_{2},
\end{array}$$ and this implies that $$\label{100}
h(g(v))^{-\gamma}g^{\prime}(v)v\in L^{1}(A_{1})\cap L^{1}(A_{2}),$$ because $h|v|^{1-\gamma},h\in L^{1}(\Omega)$.
Now, using we conclude that $h(g(v))^{-\gamma}g^{\prime}(v)v\in L^{1}(\Omega)$ because $$h(x)(g(v))^{-\gamma}g^{\prime}(v)v=h(x)(g(v))^{-\gamma}g^{\prime}(v)v\chi_{A_{1}}+h(x)(g(v))^{-\gamma}g^{\prime}(v)v\chi_{A_{2}}.$$
To prove that $(b)\Rightarrow (c)$ note that by Lemma \[L1\] $(4)$, $$\frac{1}{2}\int_{\Omega} h(x)(g(v))^{1-\gamma}\leq \int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v <\infty.$$
Finally to prove that $(c)\Rightarrow (a)$ we use the Lemma \[L1\] $(5)$ to obtain that $$\int_{\Omega} h(x)|v|^{1-\gamma}\leq \int_{\Omega} h(x)(g(v))^{1-\gamma}<\infty,$$ and the proof is completed.
Note that if $v_{0}$ satisfies the condintion (\[C\]) then, since $\rvert v_{0}\rvert \in H_{0}^{1}(\Omega)$ we have that $\rvert v_{0}\rvert$ satisfies the condition (\[C\]). Hence we may assume that $v_{0}\geq 0$. Also as we are interested in positive solution let us work on the following subset of $H_{0}^{1}(\Omega)$ $$V_{+}=\left\{v\in H_{0}^{1}(\Omega)\setminus\{0\}: v\geq 0\right\}.$$
Assume that $v\in V_{+}$ and $$\label{CD}
\int_{\Omega} h(x)|v|^{1-\gamma}<\infty (~\mbox{and therefore}~ \int_{\Omega} h(x)(g(v))^{1-\gamma}< \infty),$$ and consider the fiber map $\phi_{v}:(0,\infty)\rightarrow \mathbb{R}$ $$\phi_{v}(t):=\Phi(tv)=\frac{t^{2}}{2}\int_{\Omega} |\nabla v|^{2}+\frac{1}{\gamma-1}\int_{\Omega} h(x)(g(tv))^{1-\gamma}-\int_{\Omega} F(x,g(tv)).$$
The understanding of the fibering maps will be extremely important in the next section. Let us start by proving that for each $v$ satisfying (\[CD\]) the fiber map associated to $v$ has a good regularity.
\[C1\] We have that $\phi_{v}\in C^{1}((0,\infty),\mathbb{R})$ for each $v$ satisfying (\[CD\]).
We have that prove just that $\Gamma:(0,\infty )\longrightarrow \mathbb{R}$ defined by $$\Gamma (t)= \displaystyle\int_{\Omega} h(x)(g(tv))^{1-\gamma },$$ is of classe $C^{1}$. To do this we fix $t>0$ and note that for each $s>0$ by Mean Value Theorem there exits a mensurable function $\theta=\theta(s,x) \in (0,1)$ such that, $$\Gamma(t+s)-\Gamma(t)=(1-\gamma)\displaystyle \int_{\Omega} h(x)(g((t+\theta s)v))^{-\gamma}g^{\prime}((t+\theta s)v) sv$$ and $t+\theta(s,x)s\longrightarrow t$ as $s\longrightarrow 0$.
As the function $(g(t))^{-\gamma}g^{\prime}(t),t>0$ is decreasing (by Lemma \[L1\]$(9),(10)$) follows that $(g((t+\theta s)v))^{-\gamma}g^{\prime}((t+\theta s)v)\leq (g(tv))^{-\gamma}g^{\prime}(tv)$. Furthermore, as consequence of the Lemma \[L2\], $h(g(tv))^{-\gamma}g^{\prime}(tv)v\in L^{1}(\Omega)$. Hence we are able to apply the Lebesgue’s dominated convergence theorem to infer that $$\Gamma^{\prime}(t)=\displaystyle \lim_{s\to 0}\frac{\Gamma(t+s)-\Gamma(t)}{s}=(1-\gamma)\displaystyle \int_{\Omega} h(x)(g(tv))^{-\gamma}g^{\prime}(tv)v,$$ that is, the derivative $\Gamma^{\prime}(t)$ there exists for all $t>0$ and is given by the last expression above. Now, using the Lemma \[L2\] and the Lebesgue’s dominated convergence theorem we have that $\Gamma^{\prime}:(0,\infty)\longrightarrow \mathbb{R}$ is a continuous function.
The next lemma guarantees that for each $v$ satisfying (\[CD\]) the fiber map $\phi_{v}$ assumes its minimum value and therefore $\phi_{v}$ has a critical point.
\[L3\] For each $v$ satisfying (\[CD\]) there holds $$\displaystyle \lim_{t\to 0}\phi_{v}(t)=\infty~\mbox{and}~\displaystyle \lim_{t\to \infty}\phi_{v}(t)=\infty,$$ and therefore there exists $t(v)>0$ such that $$\phi_{v}(t(v))=\displaystyle \inf_{t>0}\phi_{v}(t).$$
Firstly we will consider the sublinear case, that is $(f)_{1}$ with $p\in (0,1)$. By the Lemma \[L1\] $(5)$ we have that $$\int_{\Omega} h(x) (g(tv))^{1-\gamma}dx\geq t^{1-\gamma}\int_{\Omega} h(x) |v|^{1-\gamma},$$ and $$t^{p+1}\int_{\Omega} |b(x)||v|^{p+1}\geq \rvert \int_{\Omega} b(x)(g(tv))^{p+1}\rvert \geq 0,$$ which implies that $$\displaystyle \lim_{t\to 0}\int_{\Omega} h(x) (g(tv))^{1-\gamma}dx=\infty ~\mbox{and}~\displaystyle \lim_{t\to 0}\int_{\Omega} b(x)(g(tv))^{p+1}=0.$$
Since $\gamma>1$ we have that $\displaystyle \lim_{t\to 0}\phi_{v}(t)=\infty$.
By other side $$\displaystyle \lim_{t\to \infty}\phi_{v}(t)\geq \lim_{t\to \infty}t^{2}\left[ ||v||^{2}-t^{p-2}\frac{\rVert b\rVert_{\infty}}{p+1}\int_{\Omega} |v|^{p+1}dx\right]=\infty.$$
The continuity of the function $\phi_{v}$ and the limits $\displaystyle \lim_{t\to 0}\phi_{v}(t)=\infty$ and $\displaystyle \lim_{t\to \infty}\phi_{v}(t)=\infty$ implies that there exists $t(v)>0$ such that $\phi_{v}(t(v))=\displaystyle \inf_{t>0}\phi_{v}(t).$
If $(f)_{2}$ is satisfied the proof is similar.
The following picture give the possible graph of the fiber map.
(-1, 3) – (6, 3); (0, 0) – (0, 7); (0.5, 6.8) .. controls (0, 1) and (4, -2) ..(5,6.8); (0,7) node\[left\][$\phi_{v}$]{}; (6,3) node\[below\][$t$]{}; (0, 3) node\[below left\][$0$]{}; (2.5,2.8) node\[above\][$t(v)$]{}; (2.5, 3) – (2.5,1.3); (3,0) node\[below\];
(-1, 3) – (6, 3); (0, 0) – (0, 7); (0.5, 6.8) .. controls (1, 4) and (3.7, 2) ..(5.5,6.8); (0,7) node\[left\][$\phi_{v}$]{}; (6,3) node\[below\][$t$]{}; (0, 3) node\[below left\][$0$]{}; (3.5,1) node\[above\][$t(v)$]{}; (2.9, 3) – (2.9,4); (3,0) node\[below\];
When $(f)_{1}$ is satisfied the graph of $\phi_{v}$ can be as in Figures $1$ and $2$. On the other hand if $(f)_{2}$ is satisfied then the graph of $\phi_{v}$ can only be as in Figure $2$.
Motivated by [@Y] we define the following constraint sets for the problem $(P_{A})$ $$\mathcal{N}_{1}=\left\{v\in V_{+}: ||v||^{2}-\int_{\Omega} f(x,g(v))g^{\prime}(v)v\geq \int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v \right\},$$ and $$\mathcal{N}_{2}=\left\{v\in V_{+}: ||v||^{2}-\int_{\Omega} f(x,g(v))g^{\prime}(v)v= \int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v \right\}.$$
We note that if $v$ is a solution to the problem $(P_{A})$ then $v\in \mathcal{N}_{2}$.
It should be noted that for $\gamma>1$, $\mathcal{N}_{2}$ is not closed as usual (certainly not weakly closed).
Next lemma ensures that any function $v\in V_{+}$ satisfying the following condition of dual compatibility $$\label{E1}
\int_{\Omega} h(x) (g(v))^{1-\gamma}<\infty,~v\in V_{+},$$ can be projected over $\mathcal{N}_{2}$.
\[L5\] Assume that there exists $v\in V_{+}$ such that $(\ref{E1})$ is satisfied. Then there exists $t(v)>0$ such that $t(v)v\in \mathcal{N}_{2}$.
As $v$ satisfied it follows from the Lemma \[L2\] that $v$ satisfied also and therefore by Lemma \[C1\] we have that $\phi_{v}\in C^{1}((0,\infty),\mathbb{R})$. Follows from the Lemma \[L3\] that there exists $t(v)>0$ such that $$\phi_{v}(t(v))=\displaystyle \inf_{t>0}\phi_{v}(t),$$ that is $t(v)$ is a critical point of $\phi_{v}$ and hence $\phi_{v}^{\prime}(t(v))=0$, which implies that $$\begin{aligned}
||t(v)v||^{2}-\int_{\Omega} h(x) (g(t(v)v))^{-\gamma}g^{\prime}(t(v)v)(t(v)v)-\int_{\Omega} f(x,t(v)v)g^{\prime}(t(v)v)(t(v)v)\\=t(v)\phi_{v}^{\prime}(t(v))=0.
\end{aligned}$$
So $t(v)v\in \mathcal{N}_{2}\subset \mathcal{N}_{1}$.
The following lemmas will be used to prove the regularity of the solution.
\[BN\] Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial \Omega$. Let $u\in L^{1}_{loc}(\Omega)$ and assume that, for some $k\geq 0$, $u$ satisfies, in the sense of distributions, $$\left\{
\begin{array}{c}
-\Delta u+ku\geq 0~\mbox{in}~\Omega\\
u\geq 0~~\mbox{in}~~\Omega.\\
\end{array}
\right.$$ Then either $u\equiv 0$, or there exists $\epsilon>0$ such that $u(x)\geq \epsilon d(x,\partial \Omega),~x\in \Omega.$
See Brezis-Nirenberg \[[@BN], Theorem 3\].
\[Ht\] Let $a\in L^{1}(\Omega)$ and suppose that there exist constants $\delta \in (0,1)$ and $C>0$ such that $|a(x)|\leq C\phi_{1}^{-\delta}(x),$ for a.e. $x\in \Omega$. Then, the problem $$\left\{
\begin{array}{c}
-\Delta u= a ~\mbox{in}~\Omega\\
u=0 ~~\mbox{on}~~\partial\Omega,\\
\end{array}
\right.$$ has a unique solution $u\in H_{0}^{1}(\Omega)$. Furthermore, there exist constants $\alpha \in (0,1)$ and $M>0$ depending only on $C,\alpha, \Omega $ such that $u\in C^{1,\alpha}(\overline{\Omega})$ and $|u|_{1,\alpha}<M$.
See Hai \[[@H], Lemma 2.1, Remark 2.2\].
For a later use we recall that there exist constants $l_{1},l_{2}>0$ such that $$l_{1}d(x,\partial \Omega)\leq \phi_{1}(x)\leq l_{2}d(x,\partial \Omega),~x\in \Omega,$$ where $\phi_{1}$ is the first eigenfunction of $(-\Delta,H_{0}^{1}(\Omega))$.
\[M\] Let $\psi_{j}:\Omega \times (0,\infty)\longrightarrow [0,\infty), j=1,2$ are measurable functions such that $$\psi_{1}(x,s)\leq \psi_{2}(x,s)~\mbox{for all}~(x,s)\in \Omega \times (0,\infty),$$ and for each $x\in \Omega$, the function $s\longmapsto \psi_{1}(x,s)s^{-1}$ is decreasing on $(0,\infty).$ Furthermore let $u,v\in H^{1}(\Omega)$, with $u\in L^{\infty}(\Omega), u>0, v>0$ on $\Omega$ are such that $$-\Delta u\leq \psi_{1}(x,u)~\mbox{and}~ -\Delta v\geq \psi_{2}(x,v)~\mbox{on}~\Omega.$$ If $u\leq v$ on $\partial \Omega$ and $\psi_{1}(\cdot,u)$ (or $\psi_{2}(\cdot,u)$) belongs to $L^{1}(\Omega)$, then $u\leq v$ on $\Omega$.
See Mohammed \[[@M], Theorem 4.1\].
Proof of Theorem \[T1\].
========================
In this section we will show the Theorem \[T1\]. First we wil given some preliminary lemmas.
\[LL1\] The set $\mathcal{N}_{1}\neq \emptyset$ and the functional $\Phi$ is coercive in $\mathcal{N}_{1}$.
Since is satisfied it follows from Lemmas \[L2\], \[L5\] that $\mathcal{N}_{1}\neq \emptyset$. Now, let us prove that $\Phi$ is coercive. Indeed for every $v\in \mathcal{N}_{1}$, $$\begin{array}{rl}
\Phi(v)= & \displaystyle\frac{1}{2}\int_{\Omega} |\nabla v|^{2}+\frac{1}{\gamma-1}\int_{\Omega} h(x)(g(v))^{1-\gamma}-\frac{1}{p+1}\int_{\Omega} b(x)(g(v))^{p+1}\\
\geq & \displaystyle\frac{1}{2}\int_{\Omega} |\nabla v|^{2}-\frac{\rVert b\rVert_{\infty}}{p+1}\int_{\Omega} (g(v))^{p+1},
\end{array}$$ and by Lemma \[L1\] $(5)$ and Sobolev embedding we have $$\Phi(v)\geq \frac{1}{2}\int_{\Omega} |\nabla v|^{2}-\frac{\rVert b\rVert_{\infty}}{p+1}\int_{\Omega} |v|^{p+1}\geq \frac{||v||^{2}}{2}-C\frac{||v||^{p+1}}{p+1},$$ for some constant $C>0$. Since $p\in (0,1)$ follows that $\Phi$ is coercive.
By the Lemma \[LL1\] we have that $$J_{1}=\displaystyle \inf_{v\in \mathcal{N}_{1}}\Phi(v)~\mbox{and}~J_{2}=\displaystyle \inf_{v\in \mathcal{N}_{2}}\Phi(v),$$ are well defined with $J_{1},J_{2}\in \mathbb{R}$ and $J_{2}\geq J_{1}$.
\[LL2\] There exists $v\in \mathcal{N}_{2}$ such that $J_{1}=\Phi(v)=J_{2}.$
By Lemma \[LL1\] there exists a sequence $\left\{v_{n}\right\}\subset \mathcal{N}_{1}$ such that $\Phi(v_{n})\longrightarrow J_{1},$ and may assume that $\left\{v_{n}\right\}$ is bounded and exist $v \in H_{0}^{1}(\Omega)$ such that $$\left\{
\begin{array}{l}
v_{n}\rightharpoonup v~\mbox{in}~H_{0}^{1}(\Omega),\\
v_{n}\longrightarrow v~\mbox{in}~L^{s}(\Omega)~\mbox{for all}~s\in (0,2^{\ast}),\\
v_{n}\longrightarrow v~a.s.~\Omega.\\
\end{array}
\right.$$
Since that $v_{n}>0$ follows that $v\geq 0$ in $\Omega$. Moreover, there exists a constant $C>0$ such that $||v_{n}||\leq C$ for every $n\in \mathbb{N}$. By definition of $\mathcal{N}_{1}$ and Lemma \[L1\] $(3),(4),(5)$ we have that
$$\begin{array}{rl}
\dfrac{1}{2} & \displaystyle\int_{\Omega} h(x)(g(v_{n}))^{1-\gamma} \leq \displaystyle\int_{\Omega} h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})v_{n} \\
& \leq ||v_{n}||^{2}-\displaystyle\int_{\Omega} b(x)(g(v_{n}))^{p}g^{\prime}(v_{n})v_{n}\\
& \leq ||v_{n}||^{2}+\displaystyle\int_{\Omega} |b(x)||v_{n}|^{p+1}\\
& \leq ||v_{n}||^{2}+c||v_{n}||^{p+1} \leq C^{2}+cC^{p+1}:=C,
\end{array}$$ where we used Sobolev embedding. Therefore using the Fatou’s lemma in the last inequality we have that $\displaystyle\int_{\Omega} \theta(x)\leq C<\infty,$ where $$\theta(x)=\left\{
\begin{array}{ccc}
h(x)(g(v(x)))^{1-\gamma}, & \mbox{if} & v(x)\neq 0 \\
\infty, & \mbox{if} & v(x)=0.\\
\end{array}
\right.$$ Since that $g(0)=0$ (by Lemma \[L1\] $(2)$) and $\displaystyle\int_{\Omega} \theta(x)<\infty$ follows that $v>0$ in $\Omega$. Once again by Fatou’s lemma we obtain $$0<\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v\leq C< \infty.$$
Consequently by the Lemma \[L3\] and Lemma \[L5\] there exists $t(v)>0$ such that $\phi_{v}(t(v))=\displaystyle \inf_{t>0} \phi_{v}(t)$ and $t(v)v\in \mathcal{N}_{2}$. Taking advantage of this information it follows that $$\begin{array}{l}
J_{1} = \displaystyle\lim_{n\to \infty} \Phi(v_{n})= \displaystyle \liminf_{n\to \infty} \Phi(v_{n})\\
= \displaystyle \liminf_{n\to \infty} \left[ \displaystyle\frac{1}{2}\int_{\Omega} |\nabla v_{n}|^{2}+\frac{1}{\gamma-1}\int_{\Omega} h(x)g(v_{n})^{1-\gamma}-\frac{1}{p+1}\int_{\Omega} b(x)(g(v_{n}))^{p+1}\right]\\
\geq \displaystyle \liminf_{n\to \infty} \left[\displaystyle\frac{1}{2}\int_{\Omega} |\nabla v_{n}|^{2}\right]+\displaystyle \liminf_{n\to \infty} \left[\frac{1}{\gamma-1}\int_{\Omega} h(x)(g(v_{n}))^{1-\gamma}\right]-\frac{1}{p+1}\int_{\Omega}b(x) (g(v))^{p+1}\\
\geq \displaystyle\frac{1}{2}\int_{\Omega} |\nabla v|^{2}+\frac{1}{\gamma-1}\int_{\Omega} h(x)(g(v))^{1-\gamma}-\frac{1}{p+1}\int_{\Omega} b(x)(g(v))^{p+1}=\phi_{v}(1)\\
\geq \phi_{v}(t(v))=\Phi(t(v)v)\geq J_{2}\geq J_{1},
\end{array}$$ which implies that $$J_{1}=\phi_{v}(1)=\Phi(v)=J_{2},$$ and $\phi_{v}(1)=\phi_{v}(t(v))= \displaystyle\inf_{t>0}\phi_{v}(t)$. Hence $\phi^{\prime}_{v}(1)=0$ and consequently $v\in \mathcal{N}_{2}\subset \mathcal{N}_{1}$.
Now let us prove the Theorem \[T1\].
Firstly we will to prove $a)$. Suppose that $(P)$ has a solution $v_ {0}$. Taking $v_{0}$ as test function is easy to see that is satisfied.
Now assume that is satisfied. Let $v$ as in the Lemma \[LL2\]. Let us prove that $v$ is a solution of problem $(P_{A})$. Consider $\varphi \in H_{0}^{1}(\Omega)$ such that $\varphi\geq 0$ in $\Omega$ and let $\epsilon>0$. By the Lemma \[L1\] $(10)$ $$\int_{\Omega}h(x) (g(v+\epsilon \varphi))^{1-\gamma}\leq \int_{\Omega} h(x)(g(v))^{1-\gamma}<\infty,$$ and consequently by Lemmas \[L3\], \[L5\] there exist $t(\epsilon)>0$ such that $\phi_{v+\epsilon \varphi}(t(\epsilon)) =\displaystyle\inf_{t>0}\phi_{v+\epsilon \varphi}(t)$ and $t(\epsilon)(v+\epsilon \varphi)\in \mathcal{N}_{2}$. Therefore $$\Phi(v+\epsilon\varphi)=\phi_{v+\epsilon \varphi}(1)\geq \phi_{v+\epsilon \varphi}(t(\epsilon))=\Phi(t(\epsilon)(v+\epsilon\varphi))\geq J_{2}=\Phi(v),$$ which implies that $$\begin{aligned}
\nonumber &\int_{\Omega}\frac{h(x)(g(v+\epsilon \varphi))^{1-\gamma}-h(x)(g(v))^{1-\gamma}}{1-\gamma}\\
\nonumber &\leq \frac{||v+\epsilon \varphi ||^{2}-||v||^{2}}{2}-\int_{\Omega}\frac{b(x)(g(v+\epsilon \varphi))^{p+1}-b(x)(g(v))^{p+1}}{p+1}.
\end{aligned}$$
Thus, dividing the last inequality by $\epsilon>0$ and passing to the $\liminf$ as $\epsilon\longrightarrow 0$, by Fatou’s Lemma we have $$\begin{aligned}
\label{110}
\nonumber\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi=& \int_{\Omega} \liminf \frac{h(x)(g(v+\epsilon \varphi))^{1-\gamma}-h(x)(g(v))^{1-\gamma}}{1-\gamma} \\
\leq & \int_{\Omega} \nabla v \nabla \varphi -\int_{\Omega} b(x)(g(v))^{p}g^{\prime}(v)\varphi.~~~~~~~~~~~~~~
\end{aligned}$$
To end the proof of item $a)$ we will use an argument inspired by Graham-Eagle [@GE]. Since that $v\in \mathcal{N}_{2}$ we have $$||v||^{2} - \displaystyle\int_{\Omega} b(x) (g(v))^{p}g^{\prime}(v)v-\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v=0.$$
For $\varphi\in H_{0}^{1}(\Omega)$ and $\epsilon>0$ define $\Psi=(v+\epsilon \varphi)^{+}$. Put $$\Omega_{1}^{\epsilon}=\left\{x\in \Omega:b(x)< 0 ~\mbox{and}~v(x)+\epsilon \varphi(x)<0 \right\}.$$
Taking $\Psi$ as a test function in we have $$\begin{array}{l}
0\leq \displaystyle\int_{\Omega} \nabla v \nabla \Psi -\int_{\Omega} b(x)(g(v))^{p}g^{\prime}(v)\Psi-\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)\Psi \\
= \displaystyle\int_{\left[v+\epsilon \varphi\geq 0\right]}\nabla v \nabla (v+\epsilon \varphi) - b(x)(g(v))^{p}g^{\prime}(v)(v+\epsilon \varphi)- h(x)(g(v))^{-\gamma}g^{\prime}(v)(v+\epsilon \varphi) \\
= \left(\displaystyle\int_{\Omega}-\int_{\left[v+\epsilon \varphi< 0\right]}\right)\nabla v \nabla (v+\epsilon \varphi) - b(x)(g(v))^{p}g^{\prime}(v)(v+\epsilon \varphi)- h(x)(g(v))^{-\gamma}g^{\prime}(v)(v+\epsilon \varphi)\\
= ||v||^{2} -\displaystyle\int_{\Omega} b(x) (g(v))^{p}g^{\prime}(v)v-\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v\\
+\epsilon \left[\displaystyle\int_{\Omega }\nabla v \nabla \varphi - b(x)(g(v))^{p}g^{\prime}(v) \varphi- h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi\right]\\
-\displaystyle\int_{\left[v+\epsilon \varphi< 0\right]}\nabla v \nabla (v+\epsilon \varphi) - b(x)(g(v))^{p}g^{\prime}(v)(v+\epsilon \varphi)- h(x)(g(v))^{-\gamma}g^{\prime}(v)(v+\epsilon \varphi)\\
\leq \epsilon \left[\displaystyle\int_{\Omega }\nabla v \nabla \varphi - b(x)(g(v))^{p}g^{\prime}(v) \varphi- h(x)(g(u))^{-\gamma}g^{\prime}(v)\varphi\right]\\
-\epsilon \displaystyle\int_{\left[v+\epsilon \varphi< 0\right]}\nabla v \nabla\varphi +\epsilon\displaystyle\int_{\Omega_{1}^{\epsilon}}b(x)(g(v))^{p}g^{\prime}(v) \varphi.
\end{array}$$ Since the measures of the domains of integration $\left[v+\epsilon \varphi< 0\right]$ and $\Omega_{1}^{\epsilon}$ tends to zero as $\epsilon \rightarrow 0$, we then divide the last expression above by $\epsilon>0$ to obtain $$0\leq \int_{\Omega }\nabla v \nabla \varphi - b(x)(g(v))^{p}g^{\prime}(v) \varphi- h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi,$$ as $\epsilon \rightarrow 0$. Replacing $\varphi$ by $-\varphi$ we conclude: $$\int_{\Omega }\nabla v \nabla \varphi - b(x)(g(v))^{p}g^{\prime}(v) \varphi- h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi=0,~\forall \varphi \in H_{0}^{1}(\Omega),$$ and therefore $v$ is a solution of $(P_{A})$.
Defining $u=g(v)$ we have that $u$ is a solution of problem $(P)$.
Now, let us prove $b)$. Suppose that $v$ is a solution of the problem $(P_{A})$. We will show that $v\in C^{1,\alpha}(\overline{\Omega})$. Hence as $g\in C^{\infty}$ we obtain that $u=g(v)\in C^{1,\alpha}(\overline{\Omega})$. Note that the function $v$ satisfies in the sense of distributions, $$\arraycolsep=1pt \medmuskip = 4mu \left\{ {\begin{array}{*{20}{rcl}}
-\Delta v & \geq & 0~\mbox{in}~\Omega\\
v & \geq & 0~~\mbox{in}~~\Omega.\\
\end{array}}\right.$$
Since that $v\in H_{0}^{1}(\Omega)$ and $v\not\equiv 0$ by Lemma \[BN\] there exists $\epsilon>0$ such that $$v(x)\geq \epsilon d(x,\partial \Omega),~x\in \Omega.$$
Consider $\epsilon> 0$ such that $$\label{12}
\epsilon d(x,\partial \Omega) < 1,$$ for all $x\in \Omega$. Then by (\[D\]) and Lemma \[L1\] $(3),(8), (10)$ there exist constants $c,C>0$ and $\beta \in (0,1)$ such that $$\begin{aligned}
\label{111}
\nonumber|h(x)(g(v))^{-\gamma}g^{\prime}(v)| & \leq h(x)(g(\epsilon d(x,\partial \Omega)))^{-\gamma} \leq h(x)C(\epsilon d(x,\partial \Omega))^{-\gamma}\\
\nonumber& \leq C c d^{\gamma-\beta}(x,\partial \Omega)d^{-\gamma}(x,\partial \Omega) \\
&= C d^{-\beta}(x,\partial \Omega)\leq C \phi_{1}^{-\beta}(x)\end{aligned}$$ for $x\in \Omega$, and since $\beta \in (0,1)$ follows that $h(g(v))^{-\gamma}g^{\prime}(v)\in L^{1}(\Omega)$. By Lemma \[Ht\] there exists a solution $\Psi_{1}\in C^{1,\alpha_{1}}(\overline{\Omega})$, for some $\alpha_{1}\in (0,1)$ of the problem $$\left\{
\arraycolsep=1pt
\medmuskip = 4mu
\begin{array}{rl}
-\Delta w & = h(x)(g(v))^{-\gamma}g^{\prime}(v)~\mbox{in}~\Omega,\\
w & > 0 ~\mbox{in}~ \Omega \quad w=0, ~\mbox{on}~ \partial \Omega.\\
\end{array}
\right.$$
Now, let us prove that the problem $$\label{15}
\left\{
\arraycolsep=1pt
\medmuskip = 4mu
\begin{array}{rl}
-\Delta w & =b(x)(g(v))^{p}g^{\prime}(v)~\mbox{in}~\Omega,\\
w & > 0 ~\mbox{in}~ \Omega \quad w=0, ~\mbox{on}~ \partial \Omega,\\
\end{array}
\right.$$ has a unique solution $\Psi_{2}\in C^{1,\alpha_{2}}(\overline{\Omega})$ for some $\alpha_{2} \in (0,1)$.
To do this, let $\delta:=1-p\in (0,1)$ and note that from and Lemma \[L1\] $(8),(12)$ we have $$|b(x)g^{p}(v(x))g^{\prime}(v(x))|\leq||b||_{\infty}g^{-\delta}(v(x))(g(v(x))g^{\prime}(v(x)))\leq C \phi_{1}^{-\delta}(x),$$ that is $$\label{14}
|b(x)g^{p}(v(x))g^{\prime}(v(x))|\leq C \phi_{1}^{-\delta}(x),$$ for every $x\in \Omega$ and some constant $C>0$. Therefore, by Lemma \[Ht\] we have that the problem has a unique solution $\Psi_{2}\in C^{1,\alpha_{2}}(\overline{\Omega})$ for some $\alpha_{2} \in (0,1)$.
The existence of $\Psi_{1}$ and $\Psi_{2}$ and the fact that $v$ is a solution of $(P_{A})$ implies that $$\int_{\Omega} \nabla v \nabla \varphi=\int_{\Omega} \left[h(x)(g(v))^{-\gamma}g^{\prime}(v)+b(x)(g(v))^{p}g^{\prime}(v)\right]\varphi=\int_{\Omega} \nabla(\Psi_{1}+\Psi_{2})\nabla \varphi,$$ for every $\varphi \in H_{0}^{1}(\Omega)$ and therefore $v=\Psi_{1}+\Psi_{2}$, which implies that $v\in C^{1,\alpha}(\overline{\Omega})$, where $\alpha:=\min \left\{\alpha_{1},\alpha_{2}\right\}\in (0,1)$.
Now, suppose that $v_{1},v_{2}$ are solutions of the problem $(P_{A})$. Let $\psi_{1}(x,s)=\psi_{2}(x,s):=h(x)(g(s))^{-\gamma}g^{\prime}(s)+b(x)(g(s))^{p}g^{\prime}(s)$. By Lemma \[L1\] $(9),(10),(11)$ follows that for each $x\in \Omega$ the function $s \longmapsto \psi_{j}(x,s)s^{-1},j=1,2,$ is decreasing on $(0,\infty)$. Moreover by $$0\leq \psi_{j}(v_{i})\leq C\phi_{1}^{-\beta}(x)+b(x)(g(v_{i}(x)))^{p}g^{\prime}(v_{i}(x)),~x\in \Omega,$$ and therefore $\psi_{j}(x,v_{i})\in L^{1}(\Omega),j=1,2,i=1,2$. We apply the Lemma \[M\] with $u=v_{1}$ and $v=v_{2}$ to conclude that $v_{1}\leq v_{2}$ in $\Omega$. Once more applying the Lemma \[M\] with $u=v_{2}$ and $v=v_{1}$ we have that $v_{2}\leq v_{1}$ in $\Omega$. Therefore $v_{1}=v_{2}$ and the uniqueness follows.
Let us end this section by making some remarks.
- If is satisfied, then problem $(P)$ has solution. In fact we have $h|\phi_{1}|^{1-\gamma}\leq c|\phi_{1}|^{1-\beta}\in L^{1}(\Omega)$ and by the item $a)$ of the Theorem \[T1\] the problem $(P)$ has solution.
- Lazer-Mckenna [@LM] proved that the solution of the following semilinear problem $$\left\{
\arraycolsep=1pt
\medmuskip = 4mu
\begin{array}{rl}
-\Delta u & = h(x)u^{-\gamma}~\mbox{in}~\Omega,\\
u>0&\mbox{in}~\Omega,~u=0~\mbox{on}~\partial \Omega,
\end{array}
\right.$$ does not belong to $C^{1}(\overline{\Omega})$, if $0<h\in C^{\alpha}(\overline{\Omega}),\alpha \in (0,1)$ and $1<\gamma$. Note that in this case $\displaystyle \inf_{\Omega}h>0$. When $b\equiv 0$ the proof of item $b)$ of Theorem \[T1\] applies to the semilinear case also, showing that the solution of these problem belong to $C^{1}(\overline{\Omega})$, if the assumption (\[D\]) is satisfied. Note that in this case $\displaystyle\inf_{\Omega} h=0$, unlike of the case in [@LM].
Proof of Theorem.2.
===================
In this section we assume that $f(x,s)=-b(x)s^{22^{\ast}-1}$ with $0\leq b\in L^{\infty}(\Omega), b\not\equiv 0$. Since that the embedding $H_{0}^{1}(\Omega)\hookrightarrow L^{2^{\ast}}(\Omega)$ is not compact, the proof of the Theorem \[T2\] can not be applied directly. To cover this difficulty we use the Brezis-Lieb Theorem (see [@BL]). Now the functional associated with the problem $(P_{A})$ is $$\Phi(v)=\frac{1}{2}\int_{\Omega} |\nabla v|^{2}+\frac{1}{\gamma-1}\int_{\Omega} h(x)(g(v))^{1-\gamma}+\frac{1}{22^{\ast}}\int_{\Omega} b(x)(g(v))^{22^{\ast}}.$$
If is satisfied, using the Lemmas \[L2\], \[L5\] we have $\mathcal{N}_{1}\neq \emptyset$. So it follows that.
\[LLL1\] The functional $\Phi$ is coercive in $\mathcal{N}_{1}$
Indeed, for every $v\in \mathcal{N}_{1}$, we have $\Phi(v)\geq \dfrac{||v||^{2}}{2},$ and hence $\Phi$ is coercive.
Let us prove the Theorem \[T2\].
**(Theorem \[T2\])** Suppose that $(P)$ has a solution $v_ {0}$. Taking $v_{0}$ as test function is easy to see that is satisfied.
Now assume that is satisfied. Define $J_1 = \displaystyle\inf_{v \in \mathcal{N}_1} \Phi(v)$ and let $\left\{v_{n}\right\}\subset \mathcal{N}_{1}$ such that $\Phi(v_{n}) \longrightarrow J_{1}.$
By Lemma \[LLL1\], $J_{1}\in \mathbb{R}$ and we may assume that $\left\{v_{n}\right\}$ is bounded in $H_{0}^{1}(\Omega)$ and in $L^{2^{\ast}}(\Omega)$ and there exists $v\in H_{0}^{1}(\Omega)$ such that $$\left\{
\begin{array}{l}
v_{n}\rightharpoonup v~\mbox{in}~H_{0}^{1}(\Omega),\\
v_{n}\longrightarrow v~\mbox{in}~L^{s}(\Omega)~\mbox{for all}~s\in (0,2^{\ast}),\\
v_{n}\longrightarrow v~a.s.~\Omega.\\
\end{array}
\right.$$
Similarly to the Theorem \[T1\] we may show that $\displaystyle\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v<\infty$ and consequently there exists $t(u)>0$ such that $t(v)v\in \mathcal{N}_{2}$. By Lemma \[L1\] $(6)$ there exists a constant $C>0$ such that $$\int_{\Omega} b(x)(g(v_{n}))^{22^{\ast}}=\int_{\Omega} \left[ b^{\frac{1}{2^{\ast}}}\right]^{2^{\ast}}\left[(g(v_{n}))^{2}\right]^{2^{\ast}}\leq ||b||_{\infty}K_{0}^{22^{\ast}}\int_{\Omega} |v_{n}|^{2^{\ast}}\leq C.$$
Moreover $b(x)(g(v_{n}))^{22^{\ast}}\longrightarrow b(x)(g(v))^{22^{\ast}}$ a.s. in $\Omega$. Hence by Brezis-Lieb Theorem (see [@BL]) $$\label{5}
\begin{array}{rl}
\displaystyle\int_{\Omega} b(x)(g(v_{n}))^{22^{\ast}} & = \displaystyle\int_{\Omega} b(x)(g(v))^{22^{\ast}}+\int_{\Omega} b(x)|(g(v_{n}))^{22^{\ast}}-(g(v))^{22^{\ast}}|+o(1)\\
& \geq \displaystyle\int_{\Omega} b(x)(g(v))^{22^{\ast}}+o(1).
\end{array}$$ Using the inequality (\[5\]) and the Fatou’s lemma we get $$\begin{array}{rl}
J_{1} & = \lim \Phi(v_{n})\\
& =\liminf \left[\frac{1}{2}\displaystyle\int_{\Omega} |\nabla v_{n}|^{2}+\frac{1}{\gamma-1}\int_{\Omega} h(x)(g(v_{n}))^{1-\gamma}+\frac{1}{22^{\ast}}\int_{\Omega} b(x)(g(v_{n}))^{22^{\ast}}\right]\\
& \geq \frac{1}{2}\displaystyle\int_{\Omega} |\nabla v|^{2}+\frac{1}{\gamma-1}\int_{\Omega} h(x)(g(v))^{1-\gamma} + \frac{1}{22^{\ast}}\int_{\Omega} b(x)(g(v))^{22^{\ast}}\\
& =\phi_{v}(1)\geq \phi_{v}(t(v))=\Phi(t(v)v) \geq J_{2}\geq J_{1},
\end{array}$$ that is $$J_{1}=\phi_{v}(1)=\Phi(v)=J_{2},$$ and $1$ is a critical point of $\phi_{v}$. Therefore $v\in \mathcal{N}_{2}$ and $$||v||^{2} +\int_{\Omega} b(x) (g(v))^{22^{\ast}-1}g^{\prime}(v)v-\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v=0.$$
Let $\varphi \in H_{0}^{1}(\Omega), \varphi\geq 0$ and $\epsilon >0$. Then similarly to the Theorem \[T1\] we have $h(\cdotp)(g(v+\epsilon \varphi))^{1-\gamma}\in L^{1}(\Omega)$ and therefore, by Lemmas \[L3\],\[L5\] there exists $t(\epsilon)>0$ such that $\phi_{v+\epsilon \varphi}(t(\epsilon))=\displaystyle \inf_{t>0}\phi_{v+\epsilon \varphi}(t)$ and $t(\epsilon)(v+\epsilon \varphi)\in \mathcal{N}_{2}$.
Since that $$\Phi(v+\epsilon \varphi)=\phi_{v+\epsilon \varphi}(1)\geq \phi_{v+\epsilon \varphi}(t(\epsilon))\geq \phi_{v}(1)=\Phi(v),$$ again, similar to the Theorem \[T1\] we may show that $$\begin{aligned}
\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi
\leq \int_{\Omega} \nabla v \nabla \varphi +\int_{\Omega} b(x)(g(v))^{22^{\ast}-1}g^{\prime}(v)\varphi,~~~~~~~~~~~~~~
\end{aligned}$$ for every $\varphi\geq 0$ and $$\arraycolsep=1pt
\medmuskip = 4mu
\begin{array}{rl} 0 \leq & ||v||^{2} +\displaystyle\int_{\Omega} b(x) (g(v))^{22^{\ast}-1}g^{\prime}(v)v-\int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v\\
& +\epsilon \left[\displaystyle\int_{\Omega }\nabla v \nabla \varphi + b(x)(g(v))^{22^{\ast}-1}g^{\prime}(v) \varphi- h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi\right]\\
& -\displaystyle\int_{\left[v+\epsilon \varphi< 0\right]}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \nabla v \nabla (v+\epsilon \varphi) + b(x)(g(v))^{22^{\ast}-1}g^{\prime}(v)(v+\epsilon \varphi)- h(x)(g(v))^{-\gamma}g^{\prime}(v)(v+\epsilon \varphi)\\
\leq & \epsilon \left[\displaystyle\int_{\Omega }\nabla v \nabla \varphi + b(x)(g(v))^{22^{\ast}-1}g^{\prime}(v) \varphi- h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi\right]\\
& -\epsilon \displaystyle\int_{\left[v+\epsilon \varphi< 0\right]} \!\!\!\!\!\!\!\!\!\! \nabla v \nabla \varphi+b(x)(g(v))^{22^{\ast}-1}g^{\prime}(v)\varphi,
\end{array}$$ for every $\varphi \in H_{0}^{1}(\Omega)$.
Since the measure of the domain of integration $\left[v+\epsilon \varphi< 0\right]$ tends to zero as $\epsilon \rightarrow 0$, we then divide the last expression above by $\epsilon>0$ to obtain $$0\leq \int_{\Omega }\nabla v \nabla \varphi + b(x)(g(v))g^{\prime}(v) \varphi- h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi,$$ as $\epsilon \rightarrow 0$. Replacing $\varphi$ by $-\varphi$ we conclude: $$\int_{\Omega }\nabla v \nabla \varphi + b(x)(g(v))^{22^{\ast}-1}g^{\prime}(v) \varphi- h(x)(g(v))^{-\gamma}g^{\prime}(v)\varphi=0,~\forall \varphi \in H_{0}^{1}(\Omega),$$ and therefore $v$ is a solution of $(P_{A})$.
Defining $u=g(v)$ we have that $u$ is a solution of problem $(P)$.
To prove that the solution is unique, let us denote by $$j(x,t)=-b(x)(g(t))^{22^{\ast}-1}g^{\prime}(t)+ h(x)(g(t))^{-\gamma}g^{\prime}(t),$$ for $x\in \Omega, t>0$. Note that $j(.,t)$ is decreasing by Lemma \[L1\] $(9),(10)$. Suppose that $v_{1}$ and $v_{2}$ are solutions from $(P_{A})$. Then, $$\rVert v_{1}-v_{2}\rVert^{2}=\displaystyle \int_{\Omega} (j(x,v_{1})-j(x,v_{2}))(v_{1}-v_{2}) <0,$$ which implies that $v_{1}=v_{2}$. Therefore the solution is unique.
Appendix
========
In this section we will study the stabilty of the solutions of the following problem with parameter $$(P_{\lambda})\left\{
\begin{array}{l}
-\Delta u -\Delta (u^{2})u=h(x) u^{-\gamma} + \lambda b(x)u^{p}~\mbox{in} ~ \Omega,\\
u> 0~\mbox{in}~ \Omega,~ u(x)=0~\mbox{on}~,\partial \Omega,
\end{array}
\right.$$ where $\lambda \geq 0, 0<p<1$ and $0\lneq b \in L^{\infty}(\Omega)$. The main result of this section is
\[A\] Let $\lambda\geq 0$. Suppose that (\[C\]) is satisfied e let $u_{\lambda}\in H_{0}^{1}(\Omega)$ the solution from $(P_{\lambda})$, which there exists by Theorem \[T1\]. There holds the following:
- $u_{\lambda}\geq u_{0}$ for all $\lambda >0$,
- $u_{\lambda}\longrightarrow u_{0}$ in $H_{0}^{1}(\Omega)$ when $\lambda \longrightarrow 0$.
To prove Theorem \[A\] firstly we will consider the dual problem associated to $(P_{\lambda})$, that is for $\lambda \geq 0, 0<p<1$ and $0\lneq b \in L^{\infty}(\Omega)$ consider the following family of problems dual to $(P_{\lambda})$ $$(D_{\lambda})\left\{
\begin{array}{l}
-\Delta v =h(x) (g(v))^{-\gamma}g^{\prime}(v) + \lambda b(x)(g(v))^{p}g^{\prime}(v)~\mbox{in} ~ \Omega,\\
v > 0~\mbox{in}~ \Omega,~~ v(x)=0~\mbox{on}~,\partial \Omega,
\end{array}
\right.$$ and let $\Phi_{\lambda}$ the energy functional associated to $(D_{\lambda})$. For each $\lambda\geq 0$ let us denote by $$\mathcal{N}_{\lambda}=\left\{v\in V_{+}:||v||^{2}-\int_{\Omega} \lambda b(g(v))^{p}g^{\prime}(v)v\geq \int_{\Omega} h(x)(g(v))^{-\gamma}g^{\prime}(v)v \right\},$$ the constrained set associated to $(D_{\lambda})$.
The proof of Theorem \[A\] is a consequence of the following lemma.
\[AA\] Suppose that (\[C\]) is satisfied and let $v_{\lambda}$ the solution from $(D_{\lambda})$ obtained in the Theorem \[T1\]. Then
- $v_{\lambda}\geq v_{0} $, where $v_{0}$ is a unique solution from $(D_{0})$,
- $v_{\lambda}\longrightarrow v_{0}$ in $H_{0}^{1}(\Omega)$ when $\lambda \longrightarrow 0$,
- $\displaystyle\lim_{\lambda \to 0}\Phi_{\lambda}(v_{\lambda})=\Phi_{0}(v_{0})>0$,
- if the conditions of Theorem \[T1\] $b)$ are satisfied, then the function $[0,\infty )\ni \lambda \longmapsto \Phi_{\lambda}(v_{\lambda})$ is continuous and decreasing.
First we will to prove $a)$. Taking $(v_{\lambda}-v_{0})^{-}=\max\left\{-(v_{\lambda}-v_{0}),0\right\}$ as test function we have $$\arraycolsep=1pt
\medmuskip = 4mu
\begin{array}{rl}
-\rVert (v_{\lambda}-& v_{0})^{-}\rVert^{2} \\
& = \displaystyle \int_{\Omega} ((g(v_{\lambda}))^{-\gamma}g^{\prime}(v_{\lambda})-(g(v_{0}))^{-\gamma}g^{\prime}(v_{0})+\lambda b(x)(g(v_{\lambda}))^{p}g^{\prime}(v_{\lambda}))(v_{\lambda}-v_{0})^{-}\\
&\geq \displaystyle \int_{\Omega} ((g(v_{\lambda}))^{-\gamma}g^{\prime}(v_{\lambda})-(g(v_{0}))^{-\gamma}g^{\prime}(v_{0}))(v_{\lambda}-v_{0})^{-}\\
&=\displaystyle \int_{\left\{v_{\lambda}<v_{0}\right\}} \!\!\!\!\!\! ((g(v_{\lambda}))^{-\gamma}g^{\prime}(v_{\lambda})-(g(v_{0}))^{-\gamma}g^{\prime}(v_{0}))(v_{\lambda}-v_{0})^{-}\geq 0,
\end{array}$$ where the last inequality is holds because the function $(g(t))^{-\gamma}g^{\prime}(t),t> 0$ is decreasing (see Lemma \[L1\] $(9),(10)$). As a consequence of the last inequality above we have $\rVert (v_{\lambda}-v_0)^{-}\rVert=0$, which implies that $v_{\lambda}\geq v_0$ in $\Omega$.
To prove $b)$ let $\left\{\lambda_{n}\right\}\subset (0,\infty)$ such that $\lambda_{n}\to 0$ and denote $v_{\lambda_{n}}=v_{n}$. Let us show that $\left\{v_{n}\right\}$ is bounded. Indeed, since that $\left\{v_{n}\right\}\subset \mathcal{N}_{\lambda_{n}}$ we have $$\label{I}
\rVert v_{n}\rVert^{2}=\displaystyle \int_{\Omega} h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})v_{n}+\lambda_{n}\displaystyle \int_{\Omega} b (g(v_{n}))^{p}g^{\prime}(v_{n})v_{n},$$ and using the Lemma \[L1\] $(4), (5), (10)$, the item $a)$ of this Lemma and Sobolev embedding we get the following inequality $$\begin{array}{rl}
\rVert v_{n}\rVert^{2} & \leq \displaystyle \int_{\Omega} h(x)(g(v_{n}))^{1-\gamma}+\lambda_{n}\displaystyle \int_{\Omega} b(x)(g(v_{n}))^{p+1}\\
& \leq \displaystyle \int_{\Omega} h(x) (g(v_{0}))^{1-\gamma}+\lambda_{n}\displaystyle \int_{\Omega} b(x) |v_{n}|^{p+1}\\
& \leq \displaystyle \int_{\Omega} h(x) (g(v_0))^{1-\gamma}+\lambda_{n}C\displaystyle \rVert v_{n}\rVert^{p+1},~~~~~~
\end{array}$$ and since that $0<p<1$ the last inequality implies that $\left\{v_{n}\right\}$ is bounded.
Now we may assume that there exists $0 \leq \psi \in H_{0}^{1}(\Omega)$ such that $$\label{convergencia}
\left\{
\begin{array}{l}
v_{n}\rightharpoonup \psi ~\mbox{in}~ H_{0}^{1}(\Omega),\\
v_{n} \to \psi ~\mbox{in}~ L^{s}(\Omega)~ \mbox{for all}~ s\in (0,2^{\ast}),\\
v_{n}\to \psi ~ \mbox{a.s. in}~ \Omega.
\end{array}
\right.$$
Using the Fatou’s lemma in (\[I\]) we can proceed as in the proof of the Theorem \[T1\] to show that $\psi>0$ in $\Omega$. This considerations implies that $$h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})(v_{n}-\psi)\to 0~\mbox{a.s.}~ \Omega,$$ and from Lemma \[L1\] $(4),(9), (10)$ and $v_{n}\geq v_0$ in $\Omega$ we have $$\begin{aligned}
\vert h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})(v_{n}-\psi)\vert\leq h(x)(g(v_{n}))^{1-\gamma}+h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})\psi\\
\leq h(x)(g(v_0))^{1-\gamma}+h(x)(g(v_0))^{-\gamma}g^{\prime}(v_0)\psi~~~
\end{aligned}$$ and $h(x)(g(v_0))^{1-\gamma}+h(x)(g(v_0))^{-\gamma}g^{\prime}(v_0)\psi\in L^{1}(\Omega)$, because $v_{0} $ is a solution from $D_{0}$. Hence by the Lebesgue Dominated Convergence Theorem we get $$\label{II}
\displaystyle \int_{\Omega} h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})(v_{n}-\psi) \longrightarrow 0.$$
As a consequence of (\[II\]) and that $v_{n}$ is a solution from $D_{\lambda_{n}}$ we have $$\arraycolsep=1pt
\medmuskip = 4mu
\begin{array}{l}
\displaystyle\lim_{n\to \infty} (v_{n}, v_{n}-\psi)=\displaystyle\lim_{n\to \infty} \int_{\Omega}\nabla v_{n}\nabla (v_{n}-\psi)=\\
= \displaystyle\lim_{n\to \infty}\left[ \int_{\Omega} h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})(v_{n}-\psi)+\displaystyle \lambda_{n}\int_{\Omega} b(x)(g(v_{n}))^{p}g^{\prime}(v_{n})(v_{n}-\psi)\right]\\
= 0,
\end{array}$$ and since that $v_{n}\rightharpoonup \psi$, follows that $$\lim_{n\to \infty}\rVert v_{n}-\psi \rVert^{2}=\lim_{n\to \infty}(v_{n},v_{n}-\psi)+\lim_{n\to \infty}(\psi,v_{n}-\psi)=0,$$ which implies that $v_{n}\longrightarrow \psi $ in $H_{0}^{1}(\Omega) $ as $n\rightarrow \infty$.
To finish the proof is suficient show that $\psi=v_{0}$. Indeed, note that we have the equation $$\label{130}
\displaystyle \int_{\Omega} \nabla v_{n}\nabla \varphi=\displaystyle \int_{\Omega} h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})\varphi+\displaystyle \lambda_{n}\int_{\Omega} b(x)(g(v_{n}))^{p}g^{\prime}(v_{n})\varphi,$$ being satisfied for all $\varphi \in H_{0}^{1}(\Omega)$. So, by (\[convergencia\]) and the Lemma \[L1\] $(9),(10)$ we have that $$h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})\varphi \longrightarrow h(x)(g(\psi))^{-\gamma}g^{\prime}(\psi)\varphi ~\mbox{a.s.}~\Omega,$$ and $$|h(x)(g(v_{n}))^{-\gamma}g^{\prime}(v_{n})\varphi|\leq h(x)(g(v_{0}))^{-\gamma}g^{\prime}(v_{0})|\varphi|\in L^{1}(\Omega),$$ were we use that $v_{0}\leq v_{n}$ in $\Omega$ and $v_{0}$ is a solution from $D_{0}$.
Therefore, using we may apply the dominated convergence theorem to conclude that $$\displaystyle \int_{\Omega} \nabla \psi\nabla \varphi= \int_{\Omega} h(x)(g(\psi))^{-\gamma}g^{\prime}(\psi)\varphi,$$ for every $\varphi \in H_{0}^{1}(\Omega)$, that is $\psi$ is a solution from $D_{0}$. By uniqueness of solutions from $D_{0}$ we have $\psi=v_{0}$.
Let us prove $c)$. To do this, note that $v_{\lambda}\geq v_{0}$ for all $\lambda>0$ and by the item $b)$ $v_{\lambda}\longrightarrow v_{0}$ in $H^{1}_{0}(\Omega)$. Thus we can proceed as in the proof of item $b)$ and apply the Lebesgue dominated convergence theorem for get that $$\displaystyle \lim_{\lambda \to 0}\Phi_{\lambda}(v_{\lambda})=\Phi_{0}(v_{0}).$$
Finally we will prove $d)$. We will give the summary proof, since it is very similar to proof of item $b)$. Let $\lambda \in [0,\infty)$ and consider $\left\{\lambda_{n}\right\} \subset [0,\infty)$ such that $\lambda_{n}\longrightarrow \lambda $. Using (\[I\]) follows that $\left\{v_{n}\right\}$ is bounded in $H^{1}_{0}(\Omega)$ and there exists $\psi \in H^{1}_{0}(\Omega)$ such that $\psi>0$, $v_{n}\rightharpoonup \psi$ in $H^{1}_{0}(\Omega)$ and $v_{n}\longrightarrow \psi$ in $L^{s}(\Omega)$ for all $s\in (0,2^{\ast})$. Since $v_{n}\geq v_{0}$ we may use the Lebesgue dominated convergence Theorem to conclude that $$\displaystyle \lim_{n\to \infty}(v_{\lambda_{n}},v_{\lambda_{n}}-\psi)= 0,$$ which implies that $v_{\lambda_{n}} \longrightarrow \psi$ in $H^{1}_{0}(\Omega)$. Again by Lebesgue dominated convergence theorem we get that $\psi $ is solution from $D_{\lambda}$ and by the Theorem \[T1\] $b)$ we have $\psi=v_{\lambda}$ and consequently
$$\displaystyle \lim_{n \to \infty}\Phi_{\lambda_{n}}(v_{\lambda_{n}})=\Phi_{\lambda}(v_{\lambda}).$$
Therefore the function $[0,\infty )\ni \lambda \longmapsto \Phi_{\lambda}(v_{\lambda})$ is continuous.
To prove that the function $[0,\infty )\ni \lambda\longmapsto \Phi_{\lambda}(v_{\lambda})$ is decreasing consider $0\leq \lambda<\mu $. Then we have $$\Phi_{\lambda}(v_{\lambda})>\Phi_{\mu}(v_{\lambda})\geq \Phi_{\mu}(t_{\mu}(v_{\lambda})v_{\lambda})\geq \Phi_{\mu}(v_{\mu})$$ and the proof is complete.
By the Lemma \[AA\] we have the following picture
(-1, 2) – (8, 2); (0, -2) – (0, 7); (0, 5) .. controls (1, 2) and (2, 0) ..(4,-1); (8,2) node\[below\][$\lambda$]{}; (0, 2) node\[below left\][$0$]{}; (3,4) node\[below\][$\Phi_{\lambda}(v_{\lambda})$]{}; (-1.9,6) node\[below\][$\Phi_{0}(v_{0})$]{}; (3,-1.8) node\[below\];
Now we will prove the Theorem \[A\].
**Theorem \[A\]**. Firstly we prove the item **a)**. Let $v_{\lambda}$ and $v_{0}$ as in the Lemma \[AA\]. Then $u_{\lambda}=g(v_{\lambda})$, $u_{0}=g(v_{0})$ and $v_{\lambda}\geq v_{0}$ by Lemma \[AA\] $a)$. So $$u_{\lambda}=g(v_{\lambda})\geq g(v_{0})=u_{0},$$ because the function $g(t)$ is increasing for $t\geq 0$ (see Lemma \[L1\] $(9)$).
To prove **b)** note that $\nabla u_{\lambda} =g^{\prime}(v_{\lambda})\nabla v_{\lambda}$ for each $\lambda\geq0$ and by inequality $(x+y)^{2}\leq 2(x^{2}+y^{2})$ for $x,y\geq 0$ we get $$\begin{array}{rl}
\displaystyle \int_{\Omega} \rvert \nabla u_{\lambda}-\nabla u_{0}\rvert^{2} & =\displaystyle \int_{\Omega} \rvert g^{\prime}(v_{\lambda})\nabla v_{\lambda}-g^{\prime}(v_{0})\nabla v_{0}\rvert^{2} \\
& \leq \displaystyle \int_{\Omega} (g^{\prime}(v_{\lambda})\rvert \nabla v_{\lambda}-\nabla v_{0}\rvert +\rvert g^{\prime}(v_{\lambda})-g^{\prime}(v_{0})\rvert \rvert \nabla v_{0}\rvert )^{2}\\
& \leq 2\displaystyle \int_{\Omega} (g^{\prime}(v_{\lambda}))^{2}\rvert \nabla v_{\lambda}-\nabla v_{0}\rvert^{2} +2\displaystyle \int_{\Omega} \rvert g^{\prime}(v_{\lambda})-g^{\prime}(v_{0})\rvert^{2} \rvert \nabla v_{0}\rvert^{2}\\
& \leq 2\displaystyle \int_{\Omega} \rvert \nabla v_{\lambda}-\nabla v_{0}\rvert^{2} +2\displaystyle \int_{\Omega} \rvert g^{\prime}(v_{\lambda})-g^{\prime}(v_{0})\rvert^{2} \rvert \nabla v_{0}\rvert^{2},~~~~~~~~~~~
\end{array}$$ where we use that $g^{\prime}(t)\leq 1$ for all $t\geq 0$ (see Lemma \[L1\]$(3)$). By Lemma \[AA\] $b)$ we have $v_{\lambda}\longrightarrow v_{0}$ in $H_{0}^{1}(\Omega)$ as $\lambda \rightarrow 0$, therefore since $g^{\prime}(t)\leq 1$ for all $t\geq 0$ from the Lebesgue dominated convergence Theorem follows $$\int_{\Omega} \rvert g^{\prime}(v_{\lambda})-g^{\prime}(v_{0})\rvert^{2} \rvert \nabla v_{0}\rvert^{2}\longrightarrow 0,$$ as $\lambda \longrightarrow 0$. This convergence together with the last inequality above implies that $u_{\lambda}\longrightarrow u_{0}$ in $H_{0}^{1}(\Omega)$ as $\lambda \rightarrow 0$.
[D]{} C.O. Alves, G.M. Figueiredo, U.B. Severo, [*A result of multiplicity of solutions for a class of quasilinear equations*]{}, Proc. Edinburgh Math. Soc. 55 (2012) 291–309. H. Brezis, E. Lieb, [*A relation between pointwise convergence of functions and convergence of functionals*]{}, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486-490. H. Brezis, L. Nirenberg, [*$H^{1}$ versus $C^{1}$ local minimizers*]{}, C.R. Acad. Sci. Paris 317 (1993) 465-472. M. Colin, L. Jeanjean, [*Solutions for a quasilinear Schrödinger equation: a dual approach*]{}, Nonlinear Anal. 56 (2004) 213–226. M.G. Crandall, P.H. Rabinowitz, L. Tatar, [*On a Dirichlet problem with a singular nonlinearity*]{}, Comm. Partial Differential Equations 2 (1977) 193–222. J.M.B. do Ó, A. Moameni, [*Solutions for singular quasilinear Schrödinger equations with one parameter*]{}, Commun. Pure Appl. Anal. 9 (2010), no. 4, 1011-1023. W. Fulks, J.S. Maybee, [*A singular non-linear equation*]{}, Osaka Math. J. 12 (1960) 1–19. J. Graham-Eagle, [*A variational approach to upper and lower solutions*]{}, IMA J. Appl. Math. 44 (1990) 181-184. D.D. Hai, [*On an asymptotically linear singular boundary value problems*]{}, Topo. Meth. in Nonlin. Anal. 39 (2012), 83-92. A. C. Lazer, P. J. Mckenna, [*On a singular nonlinear elliptic boundary value problem*]{}, Proc. Am. Math. Soc. 111 (1991) 720-730. D. Liu, [*Soliton solutions for a quasilinear Schrödinger equation*]{}, Electron. J. Differ. Equ. 267 (2013) 1–13. J. Liu, D. Liu, P. Zhao, [*Soliton solutions for a singular schcrodinger equation with any groth exponents*]{}, Acta Appl. Math. 148 (2017), 179-199. J. Liu, D. Liu, [*Multiple soliton solutions for a quasilinear Schrödinger equation*]{}, Indian Journal of Pure and Applied Mathematics 48 (2017) 75-90. D. Liu, P. Zhao, [*Soliton solutions for a quasilinear Schrödinger Equation via Morse theory*]{}, Proc. Indian Acad. Sci. Math. Sci., 125 (2015), 307-321. A. Mohammed, [*Positive solutions of the $p$-Laplace equation with singular nonlinearity*]{}, J. Math. Anal. Appl. 352 (2009) 234–245. C. A. Santos, R. L. Alves, K. Silva [*Multiplicity of negative-energy solutions for singular-superlinear Schrödinger equations with indefinite-sign potential*]{}, https://arxiv.org/abs/1811.03365v1. Y.J. Sun [*Compatibility phenomena in singular problems*]{}, Proc. R. Soc. Edinb. A 143 (2013) 1321-1430. L.L. Wang, [*Existence and uniqueness of solutions to singular quasilinear Schrödinger equations*]{}, Electron. J. Differ. Equ. 38 (2018) 1–9.
[*Ricardo Lima Alves*]{}\
[*Departamento de Matemática*]{}\
[*Universidade de Brasília* ]{}\
[*70910-900 Brasília*]{}\
[*DF - Brasil*]{}\
[*e-mail: ricardoalveslima8@gmail.com*]{}\
[*Mariana Reis*]{}\
[*Departamento de Matemática*]{}\
[*Universidade Federal da Integração Latino-Americana* ]{}\
[*85867-970 Foz do Iguaçu*]{}\
[*PR - Brasil*]{}\
[*e-mail: mariana.reis@unila.edu.br*]{}
[^1]: Ricardo Lima Alves acknowledges the support of CNPq/Brazil
|
---
author:
- Koushik Chatterjee
- 'Alexander Yu. Sokolov'
title: 'Second-Order Multi-Reference Algebraic Diagrammatic Construction Theory for Photoelectron Spectra of Strongly Correlated Systems'
---
Introduction {#sec:intro}
============
Recently, there has been a significant progress in increasing tractability of strong electron correlation problem. New methods enable computations of systems with a large number of strongly correlated electrons in the ground or excited electronic states.[@Olsen:1988p2185; @Malmqvist:1990p5477; @White:1999p4127; @Legeza2008; @Booth:2009p054106; @Kurashige:2009p234114; @Marti:2011p6750; @Chan:2011p465; @Wouters:2014p272; @Zhang:2016p4326; @Schriber:2017p5354; @Holmes:2016p3674; @Sharma:2017p1595; @Holmes:2017p164111] These approaches usually start by computing a multi-configurational wavefunction that describes strong correlation in a subset of frontier (active) molecular orbitals with near-degeneracies.[@Werner:1980p2342; @Werner:1981p5794; @Knowles:1985p259] The remaining (dynamic) correlation effects outside of the active orbitals are usually captured by multi-reference perturbation theory (MRPT),[@Wolinski:1987p225; @Hirao:1992p374; @Werner:1996p645; @Finley:1998p299; @Andersson:1990p5483; @Andersson:1992p1218; @Angeli:2001p10252; @Angeli:2001p297; @Angeli:2004p4043; @Li:2015p2097] configuration interaction,[@Buenker:1974p33; @Siegbahn:1980p1647; @Werner:1988p5803; @Saitow:2013p044118; @Saitow:2015p5120] or coupled cluster (CC) methods.[@Mukherjee:1977p955; @Lindgren:1978p33; @Jeziorski:1981p1668; @Mahapatra:1999p6171; @Evangelista:2007p024102; @Datta:2011p214116; @Evangelista:2011p114102; @Kohn:2012p176; @Datta:2012p204107; @Nooijen:2014p081102; @Huntington:2015p194111; @Kirtman:1981p798; @Hoffmann:1988p993; @Yanai:2006p194106; @Yanai:2007p104107; @Chen:2012p014108; @Li:2016p164114; @Evangelista:2018p030901] In particular, low-order MRPT methods have been very successful at computing accurate energies of large strongly correlated systems, due to their relatively low computation cost and ability to treat large active spaces with up to $\sim$ 30 orbitals.[@Kurashige:2011p094104; @Kurashige:2014p174111; @Guo:2016p1583; @Sharma:2017p488; @Yanai:2017p4829; @Freitag:2017p451; @Sokolov:2017p244102; @Schriber:2018p6295]
Despite significant advances, application of conventional MRPT methods to a wider range of problems, such as simulating excited-state or spectroscopic properties, is hindered by a number of limitations. For example, computation of transition intensities in MRPT is not straightforward due to complexity of the underlying response equations.[@MacLeod:2015p051103] Another limitation is that MRPT methods do not describe electronic transitions involving orbitals outside active space that are important for simulating broadband spectra or core-level excitations in X-ray spectroscopies. Furthermore, for computations involving many electronic states of the same symmetry, MRPT methods rely on using state-averaged reference wavefunctions, which introduce dependence of their results on the number of states and weights used in state-averaging. This motivates the development of new efficient multi-reference theories that are not bound by these limitations.
We have recently proposed a multi-reference formulation of algebraic diagrammatic construction theory (MR-ADC) for simulating spectroscopic properties of strongly correlated systems.[@Sokolov:2018p204113] MR-ADC is a generalization of the conventional (single-reference) ADC theory proposed by Schirmer in 1982.[@Schirmer:1982p2395] Rather than computing energies and wavefunctions of individual electronic states, in MR-ADC excitation energies and transition intensities are directly obtained from poles and residues of a retarded propagator approximated using multi-reference perturbation theory. In contrast to conventional MRPT, MR-ADC describes electronic transitions involving all orbitals (i.e., core, active, and external), enables simulations of various spectroscopic processes (e.g., ionization or two-photon excitation), and provides direct access to spectral properties. In this regard, MR-ADC is related to multi-reference propagator theories,[@Banerjee:1978p389; @Yeager:1979p77; @Dalgaard:1980p816; @Yeager:1984p85; @Graham:1991p2884; @Yeager:1992p133; @Nichols:1998p293; @Khrustov:2002p507; @HelmichParis:2019p174121] but has an advantage of a Hermitian eigenvalue problem and including dynamic correlation effects beyond single excitations. For electronic excitations, MR-ADC can also be considered as a low-cost alternative to multi-reference equation-of-motion (MR-EOM) theories, such as MR-EOM-CC,[@Datta:2012p204107; @Nooijen:2014p081102; @Huntington:2015p194111] and internally-contracted linear-response theories, such as ic-MRCC.[@Samanta:2014p134108]
In this work, we present a second-order formulation of MR-ADC (MR-ADC(2)) for photoelectron spectra of multi-reference systems. We begin by describing the derivation of MR-ADC(2) (\[sec:theory\]) and discuss details of its implementation (\[sec:implementation\]), demonstrating that it has a lower computational scaling with the number of active orbitals compared to conventional MRPT methods. Next, we describe computational details (\[sec:computational\_details\]) and test the performance of MR-ADC(2) for computing photoelectron energies and transition intensities of small molecules, carbon dimer, as well as equally-spaced hydrogen chains and (\[sec:results\]). Finally, we present our conclusions (\[sec:conclusions\]) and outline future developments.
Theory {#sec:theory}
======
Multi-Reference Algebraic Diagrammatic Construction Theory (MR-ADC) {#sec:theory:mr_adc_overview}
-------------------------------------------------------------------
We begin with a brief overview of MR-ADC. In Ref. , we have described the derivation of MR-ADC using the formalism of effective Liouvillean theory.[@Mukherjee:1989p257] Here, we only summarize the main results. Our starting point is a general expression for the retarded propagator[@Fetter2003; @Dickhoff2008] that describes response of a many-electron system to an external perturbation with frequency $\omega$: $$\begin{aligned}
\label{eq:g_munu}
G_{\mu\nu}(\omega)
& = G_{\mu\nu}^+(\omega) \pm G_{\mu\nu}^-(\omega) \notag\\
& = \bra{\Psi}q_\mu(\omega - H + E)^{-1}q^\dag_\nu\ket{\Psi} \notag \\
&\pm \bra{\Psi}q^\dag_\nu(\omega + H - E)^{-1}q_\mu\ket{\Psi}\end{aligned}$$ Here, $G_{\mu\nu}^+(\omega)$ and $G_{\mu\nu}^-(\omega)$ are the forward and backward components of the propagator, $\ket{\Psi}$ and $E$ are the eigenfunction and eigenvalue of the electronic Hamiltonian $H$, and the frequency $\omega \equiv \omega' + i\eta$ is written in terms of its real component ($\omega'$) and an infinitesimal imaginary number ($i\eta$). Depending on the form of operators $q^\dag_\nu$, the propagator $G_{\mu\nu}(\omega)$ can describe various spectroscopic processes. Choosing $q^\dag_\nu = {a^\dagger_{p}}{a_{q}} - \braket{\Psi|{a^\dagger_{p}}{a_{q}}|\Psi}$, where ${a^\dagger_{p}}$ and ${a_{p}}$ are the usual creation and annihilation operators, corresponds to polarization propagator that provides information about electronic excitations in optical (e.g., UV/Vis) spectroscopy. Alternatively, a propagator with $q^\dag_\nu = {a^\dagger_{p}}$ describes electron attachment and ionization processes. The number of creation and annihilation operators in $q^\dag_\nu$ (odd or even) determines the sign ($+$ or $-$) of the second term in \[eq:g\_munu\].
Evaluation of the exact propagator is very expensive computationally. For this reason, many approximate methods[@Goscinski:1980p385; @Weiner:1980p1109; @Prasad:1985p1287; @Datta:1993p3632; @Lowdin:1970p231; @Nielsen:1980p6238; @Sangfelt:1984p3976; @Bak:2000p4173; @Nooijen:1992p55; @Nooijen:1993p15; @Nooijen:1995p1681; @Moszynski:2005p1109; @Korona:2010p14977; @Kowalski:2014p094102; @Schirmer:1982p2395; @Schirmer:1991p4647; @Mertins:1996p2140; @Schirmer:2004p11449; @Schirmer:1983p1237; @Schirmer:1998p4734; @Trofimov:2005p144115; @Dempwolff:2019p064108; @Liu:2018p244110; @Hedin:1965p796; @Faleev:2004p126406; @vanSchilfgaarde:2006p226402; @Cederbaum:1975p290; @VonNiessen:1984p57; @Ortiz:2012p123; @Georges:1996p13; @Kotliar:2006p865; @Phillips:2014p241101; @Lan:2015p241102; @Banerjee:1978p389; @Yeager:1979p77; @Dalgaard:1980p816; @Yeager:1984p85; @Graham:1991p2884; @Yeager:1992p133; @Nichols:1998p293; @Khrustov:2002p507] have been developed to compute $G_{\mu\nu}(\omega)$ for realistic systems. A common assumption in most of these approaches is that the eigenfunction $\ket{\Psi}$ can be well approximated by a single Slater determinant. Although this assumption significantly simplifies the underlying equations, such [*single-reference*]{} methods do not provide reliable results when strong correlation is important and the wavefunction $\ket{\Psi}$ becomes multi-configurational.
![Orbital energy diagram showing the index convention used in this work.[]{data-label="fig:mo_diagram"}](mo_diagram){width="45.00000%"}
To efficiently and accurately compute $G_{\mu\nu}(\omega)$ for strongly correlated systems, in MR-ADC we consider an expansion of \[eq:g\_munu\] using [*multi-reference*]{} perturbation theory, where the zeroth-order (reference) wavefunction $\ket{\Psi_0}$ is obtained by solving the complete active space configuration interaction (CASCI) or self-consistent field (CASSCF) variational problem in a set of active molecular orbitals (\[fig:mo\_diagram\]). The eigenfunction $\ket{\Psi}$ is related to $\ket{\Psi_0}$ via a unitary transformation[@Kirtman:1981p798; @Hoffmann:1988p993; @Yanai:2006p194106; @Yanai:2007p104107; @Chen:2012p014108; @Li:2015p2097; @Li:2016p164114] $$\begin{aligned}
\label{eq:mr_adc_wfn}
\ket{\Psi} &= e^{A} \ket{\Psi_0} = e^{T - T^\dag} \ket{\Psi_0} , \quad T = \sum_{k=1}^N T_k \\
\label{eq:mr_adc_t_amplitudes}
T_k &= \frac{1}{(k!)^2} {\sum_{i'j'a'b'\ldots}} t_{i'j'\ldots}^{a'b'\ldots} {a^\dagger_{a'}}{a^\dagger_{b'}}\ldots{a_{j'}}{a_{i'}}, \ t_{xy\ldots}^{wz\ldots} = 0\end{aligned}$$ where $T$ generates all internally-contracted excitations between core, active, and external orbitals (see \[fig:mo\_diagram\] for orbital index notation). Defining the zeroth-order Hamiltonian to be the Dyall Hamiltonian[@Dyall:1995p4909; @Angeli:2001p10252; @Angeli:2001p297; @Angeli:2004p4043] $$\begin{aligned}
\label{eq:h_dyall_general}
H^{(0)} &\equiv C + \sum_{i} {\ensuremath{\varepsilon_{i}}} {a^\dagger_{i}}{a_{i}} + \sum_{a} {\ensuremath{\varepsilon_{a}}} {a^\dagger_{a}}{a_{a}} + H_{act} \\
\label{eq:h_act}
H_{act} &= \sum_{xy}({h_{{x}}^{{y}}} + \sum_{i} {{v}_{{xi}}^{{yi}}}) {a^\dagger_{x}} {a_{y}}
+ \frac{1}{4} \sum_{xywz} {{v}_{{xy}}^{{zw}}} {a^\dagger_{x}} {a^\dagger_{y}} {a_{w}} {a_{z}} \\
C &= \sum_i {h_{{i}}^{{i}}} + \frac{1}{2}\sum_{ij}{{v}_{{ij}}^{{ij}}} - \sum_i {\ensuremath{\varepsilon_{i}}} \\
\label{eq:f_gen}
{f_{{p}}^{{q}}} &= {h_{{p}}^{{q}}} + \sum_{rs} {{v}_{{pr}}^{{qs}}} {\gamma_{{s}}^{{r}}} \ , \quad {\gamma_{{q}}^{{p}}} = \braket{\Psi_0|{a^\dagger_{p}}{a_{q}}|\Psi_0}\end{aligned}$$ expressed in the basis of diagonal core and external generalized Fock operators (${f_{{i}}^{{j}}} = {\ensuremath{\varepsilon_{i}}}{\delta_{{i}}^{{j}}}$, ${f_{{a}}^{{b}}} = {\ensuremath{\varepsilon_{a}}}{\delta_{{a}}^{{b}}}$), we expand the propagator in \[eq:g\_munu\] in perturbative series with respect to the perturbation $V = H - H^{(0)}$: $$\begin{aligned}
\label{eq:g_pt_series}
\mathbf{G}(\omega) & = \mathbf{G}^{(0)}(\omega) + \mathbf{G}^{(1)}(\omega) + \ldots + \mathbf{G}^{(n)}(\omega) + \ldots\end{aligned}$$ Truncating \[eq:g\_pt\_series\] at the $n$th order in perturbation theory corresponds to the propagator of the MR-ADC(n) approximation.
An important property of MR-ADC (along with that of its single-reference variant)[@Mukherjee:1989p257] is that the forward and backward components of the propagator in \[eq:g\_munu\] are decoupled and, thus, perturbative expansion can be performed for $G_{\mu\nu}^+(\omega)$ and $G_{\mu\nu}^-(\omega)$ separately. The MR-ADC(n) $G_{\mu\nu}^+(\omega)$ and $G_{\mu\nu}^-(\omega)$ contributions are expressed in the matrix form $$\begin{aligned}
\label{eq:Gn_matrix}
\mathbf{G}_{\pm}(\omega) & = \mathbf{T}_{\pm} \left(\omega \mathbf{S}_{\pm} - \mathbf{M}_{\pm}\right)^{-1} \mathbf{T}_{\pm}^{\dag}\end{aligned}$$ where $\mathbf{M}_{\pm}$, $\mathbf{T}_{\pm}$, and $\mathbf{S}_{\pm}$ are the effective Liouvillean, transition moment, and overlap matrices, respectively, each evaluated up to $n$th order in perturbation theory. The $\mathbf{M}_{\pm}$ matrix contains information about transition energies, which are obtained by solving the Hermitian generalized eigenvalue problem $$\begin{aligned}
\label{eq:adc_eig_problem}
\mathbf{M}_{\pm} \mathbf{Y}_{\pm} = \mathbf{S}_{\pm} \mathbf{Y}_{\pm} \boldsymbol{\Omega}_{\pm}\end{aligned}$$ where $\boldsymbol{\Omega}_{\pm}$ is a diagonal matrix of eigenvalues. The eigenvectors $\mathbf{Y}_{\pm}$ are used to compute spectroscopic amplitudes $$\begin{aligned}
\label{eq:spec_amplitudes}
\mathbf{X}_{\pm} = \mathbf{T}_{\pm} \mathbf{S}_{\pm}^{-1/2} \mathbf{Y}_{\pm}\end{aligned}$$ which are related to transition intensities. Combining the eigenvalues $\boldsymbol{\Omega}_{\pm}$ and spectroscopic amplitudes $\mathbf{X}_{\pm}$, we obtain expressions for the MR-ADC(n) propagator and spectral function $$\begin{aligned}
\label{eq:g_mr_adc}
\mathbf{G}_{\pm}(\omega) &= \mathbf{X}_{\pm} \left(\omega - \boldsymbol{\Omega}_{\pm}\right)^{-1} \mathbf{X}_{\pm}^\dag \\
\label{eq:spec_function}
T(\omega) &= -\frac{1}{\pi} \mathrm{Im} \left[ \mathrm{Tr} \, \mathbf{G}_{\pm}(\omega) \right]\end{aligned}$$
Second-Order MR-ADC for Ionization Energies and Spectra {#sec:theory:mr_adc_ip}
-------------------------------------------------------
### Overview {#sec:theory:mr_adc_ip:overview}
In this work, we consider the MR-ADC(2) approximation for photoelectron spectra, which incorporates all contributions to $\mathbf{G}(\omega)$ up to the second order in perturbation theory. A propagator of choice for the description of electron ionization processes is the backward component of the one-particle Green’s function $\mathbf{G}_{-}(\omega)$, which can be defined by specifying $q^\dag_\nu = {a^\dagger_{p}}$ in the second term of \[eq:g\_munu\]. To simplify our notation, we will drop the subscript $-$ everywhere in the equations. Thus, matrices $\mathbf{M}$, $\mathbf{T}$, and $\mathbf{S}$ will refer to the components of $\mathbf{G}_{-}(\omega)$ in \[eq:Gn\_matrix\]. Following the effective Liouvillean approach,[@Mukherjee:1989p257; @Sokolov:2018p204113] we express the $n$th-order MR-ADC matrices as: $$\begin{aligned}
\label{eq:M_matrix}
M_{\mu\nu}^{(n)} & = \sum_{klm}^{k+l+m= n} \braket{\Psi_0|[h_{\mu}^{(k)\dag}, [\tilde{H}^{(l)},h_{\nu}^{(m)}]]_{+}|\Psi_0} \\
\label{eq:T_matrix}
T_{\mu\nu}^{(n)} & = \sum_{kl}^{k+l=n} \braket{\Psi_0|[\tilde{q}_{\mu}^{(k)}, h_{\nu}^{(l)}]_{+}|\Psi_0} \\
\label{eq:S_matrix}
S_{\mu\nu}^{(n)} & = \sum_{kl}^{k+l=n} \braket{\Psi_0|[h_{\mu}^{(k)\dag}, h_{\nu}^{(l)}]_{+}|\Psi_0}\end{aligned}$$ where $[\ldots]$ and $[\ldots]_+$ denote commutator and anticommutator, respectively. In \[eq:M\_matrix,eq:T\_matrix,eq:S\_matrix\], $\tilde{H}^{(k)}$ and $\tilde{q}_{\mu}^{(k)}$ are the $k$th-order contributions to the effective Hamiltonian $\tilde{H} = e^{-A} H e^{A}$ and observable $\tilde{q}_{\mu} = e^{-A} q_{\mu} e^{A}$ operators. These contributions can be obtained by expanding $\tilde{H}$ and $\tilde{q}_{\mu}$ using the Baker–Campbell–Hausdorff (BCH) formula and collecting terms at the $k$th order. The low-order components of these operators have the form $$\begin{aligned}
\label{eq:H_bch_0}
\tilde{H}^{(0)} &= H^{(0)} \\
\label{eq:H_bch_1}
\tilde{H}^{(1)} &= V + [H^{(0)}, A^{(1)}] \\
\label{eq:H_bch_2}
\tilde{H}^{(2)} &= [H^{(0)}, A^{(2)}] + \frac{1}{2}[V + \tilde{H}^{(1)}, A^{(1)}] \\
\label{eq:q_bch_0}
\tilde{q}^{(0)}_\mu &= q_\mu = {a_{p}} \\
\label{eq:q_bch_1}
\tilde{q}^{(1)}_\mu &= [{a_{p}}, A^{(1)}] \\
\label{eq:q_bch_2}
\tilde{q}^{(2)}_\mu &= [{a_{p}}, A^{(2)}] + \frac{1}{2} [[{a_{p}}, A^{(1)}], A^{(1)}]\end{aligned}$$ where $A^{(k)} = T^{(k)} - T^{(k)\dag}$ as shown in \[eq:mr\_adc\_wfn\]. The operators $h_{\mu}^{(k)\dag}$ compose the $k$th-order ionization operator manifold that is used to construct a set of internally-contracted (ionized) basis states $\ket{\Psi_\mu^{(k)}} = h_{\mu}^{(k)\dag} \ket{\Psi_0}$ necessary for representing the eigenstates in \[eq:adc\_eig\_problem\].
Introducing shorthand notations[@Mukherjee:1989p257] for the matrix elements of arbitrary operator sets $\mathbf{A} = \{ A_\mu \}$ and $\mathbf{B} = \{ B_\mu \}$ $$\begin{aligned}
\{\textbf{A}|\textbf{B}\} &= \braket{\Psi_0|[A_\mu,B_\nu^\dag]_+|\Psi_0} \\
\{\textbf{A}|\tilde{\mathcal{H}}|\textbf{B}\} &= \braket{\Psi_0|[A_\mu,[\tilde{H},B_\nu^\dag]]_+|\Psi_0}\end{aligned}$$ we express contributions to the MR-ADC(2) matrices in the following form $$\begin{aligned}
\label{eq:mr_adc_2_M_matrix}
\mathbf{M} & \approx \{\mathbf{h}^{(0)\dag}|\tilde{\mathcal{H}}^{(0)}|\mathbf{h}^{(0)\dag}\} + \{\mathbf{h}^{(1)\dag}|\tilde{\mathcal{H}}^{(0)}|\mathbf{h}^{(0)\dag}\} \notag \\
&+ \{\mathbf{h}^{(0)\dag}|\tilde{\mathcal{H}}^{(1)}|\mathbf{h}^{(0)\dag}\} + \{\mathbf{h}^{(0)\dag}|\tilde{\mathcal{H}}^{(0)}|\mathbf{h}^{(1)\dag}\} \notag \\
&+ \{\mathbf{h}^{(1)\dag}|\tilde{\mathcal{H}}^{(1)}|\mathbf{h}^{(0)\dag}\} + \{\mathbf{h}^{(1)\dag}|\tilde{\mathcal{H}}^{(0)}|\mathbf{h}^{(1)\dag}\} \notag \\
&+ \{\mathbf{h}^{(0)\dag}|\tilde{\mathcal{H}}^{(1)}|\mathbf{h}^{(1)\dag}\} + \{\mathbf{h}^{(0)\dag}|\tilde{\mathcal{H}}^{(2)}|\mathbf{h}^{(0)\dag}\} \\
\label{eq:mr_adc_2_T_matrix}
\mathbf{T} & \approx \{\mathbf{\tilde{q}}^{(0)}|\mathbf{h}^{(0)\dag}\} + \{\mathbf{\tilde{q}}^{(1)}|\mathbf{h}^{(0)\dag}\} + \{\mathbf{\tilde{q}}^{(0)}|\mathbf{h}^{(1)\dag}\} \notag \\
&+ \{\mathbf{\tilde{q}}^{(1)}|\mathbf{h}^{(1)\dag}\} + \{\mathbf{\tilde{q}}^{(2)}|\mathbf{h}^{(0)\dag}\} \\
\label{eq:mr_adc_2_S_matrix}
\mathbf{S} & \approx \{\mathbf{h}^{(0)\dag}|\mathbf{h}^{(0)\dag}\} + \{\mathbf{h}^{(1)\dag}|\mathbf{h}^{(0)\dag}\} \notag \\
&+ \{\mathbf{h}^{(0)\dag}|\mathbf{h}^{(1)\dag}\} + \{\mathbf{h}^{(1)\dag}|\mathbf{h}^{(1)\dag}\}\end{aligned}$$ Computing matrix elements in \[eq:mr\_adc\_2\_M\_matrix,eq:mr\_adc\_2\_T\_matrix,eq:mr\_adc\_2\_S\_matrix\] requires solving for amplitudes of the excitation operators ($T^{(1)}$ and $T^{(2)}$) and determining the ionization operator manifolds ($h_{\mu}^{(k)\dag}$, $k = 0, 1$).
### Amplitudes of the Excitation Operators {#sec:theory:mr_adc_ip:amplitudes}
To solve for amplitudes of the $T^{(k)}$ $(k = 1,2)$ operators, we express these operators in a general form $$\begin{aligned}
\label{eq:excit_op_tensor}
T^{(k)} = \mathbf{t^{(k)}} \, \boldsymbol{\tau} = \sum_\mu t_\mu^{(k)} \tau_\mu\end{aligned}$$ where $t_\mu^{(k)}$ are the $k$th-order coefficients and $\tau_\mu$ are the corresponding excitation operators (\[eq:mr\_adc\_t\_amplitudes\]). The first-order operator $T^{(1)}$ includes up to two-body terms ($T^{(1)} = T^{(1)}_1 + T^{(1)}_2$) parametrized using three classes of single excitation and eight classes of double excitation amplitudes $$\begin{aligned}
\label{eq:t1_amp_tensor}
\mathbf{t^{(1)}} = &\left\{t_{i}^{a(1)};\ t_{i}^{x(1)};\ t_{x}^{a(1)};\ t_{ij}^{ab(1)};\ t_{ij}^{ax(1)};\ t_{ix}^{ab(1)}; \right. \notag \\
&\left. t_{ij}^{xy(1)};\ t_{xy}^{ab(1)};\ t_{ix}^{ay(1)};\ t_{ix}^{yz(1)};\ t_{xy}^{az(1)}\right\}\end{aligned}$$ Defining $a_{q}^{p} \equiv {a^\dagger_{p}}{a_{q}}$ and $a_{rs}^{pq} \equiv {a^\dagger_{p}}{a^\dagger_{q}}{a_{s}}{a_{r}}$, the corresponding excitation operators are $$\begin{aligned}
\label{eq:tau_tensor}
\boldsymbol{\tau} = &\left\{a_{i}^{a};\ a_{i}^{x};\ a_{x}^{a};\ a_{ij}^{ab};\ a_{ij}^{ax};\ a_{ix}^{ab}; \right. \notag \\
&\left. a_{ij}^{xy};\ a_{xy}^{ab};\ a_{ix}^{ay};\ a_{ix}^{yz};\ a_{xy}^{az}\right\}\end{aligned}$$ To compute $\mathbf{t^{(1)}}$, we consider a system of projected linear equations $$\begin{aligned}
\label{eq:proj_amplitude_equations_1}
\braket{\Psi_0|\tau^\dag_\mu\tilde{H}^{(1)}|\Psi_0} = 0\end{aligned}$$ Using the definition of $\tilde{H}^{(1)}$ from \[eq:H\_bch\_1\], this system of equations can be expressed in the matrix form[@Sokolov:2018p204113] $$\begin{aligned}
\label{eq:proj_amplitude_equations_1_matrix}
\mathbf{H^{(0)}} \mathbf{t^{(1)}} = - \mathbf{V^{(1)}}\end{aligned}$$ where the zeroth-order Hamiltonian and perturbation matrix elements are defined as $$\begin{aligned}
\label{eq:H_zero_matrix}
H_{\mu\nu}^{(0)} &= \braket{\Psi_0|\tau^\dag_\mu(H^{(0)} - E_0)\tau_\nu|\Psi_0} \\
\label{eq:V_first_order_matrix}
V_{\mu}^{(1)} &= \braket{\Psi_0|\tau^\dag_\mu V|\Psi_0}\end{aligned}$$ and $E_0$ is the zeroth-order (reference) energy. \[eq:proj\_amplitude\_equations\_1\_matrix\] is identical to equation that defines the first-order wavefunction in the standard Rayleigh–Schrödinger perturbation theory. Since $H^{(0)}$ is the Dyall Hamiltonian, the first-order MR-ADC reference wavefunction $\ket{\Psi^{(1)}} = T^{(1)} \ket{\Psi_0}$ is equivalent to the first-order wavefunction in internally-contracted second-order $N$-electron valence perturbation theory (NEVPT2).[@Angeli:2001p10252; @Angeli:2001p297; @Angeli:2004p4043] Importantly, this suggests that solutions of \[eq:proj\_amplitude\_equations\_1\] do not suffer from intruder-state problems, provided that $\ket{\Psi_0}$ is the ground-state reference wavefunction. The $\mathbf{t^{(1)}}$ amplitudes can be used to compute the second-order correlation correction to the reference energy $$\begin{aligned}
\label{eq:E_reference_second_order}
E^{(2)} &= \braket{\Psi_0| V |\Psi^{(1)}} = \braket{\Psi_0| V T^{(1)} |\Psi_0}\end{aligned}$$ which is equivalent to the NEVPT2 correlation energy. We note that \[eq:proj\_amplitude\_equations\_1\_matrix,eq:E\_reference\_second\_order\] have been recently derived in the context of perturbation expansion of internally-contracted multi-reference coupled cluster theory.[@Aoto:2019p2291]
Evaluating the MR-ADC(2) matrices in \[eq:mr\_adc\_2\_M\_matrix,eq:mr\_adc\_2\_T\_matrix\] also requires semi-internal amplitudes of the second-order excitation operator $T^{(2)}$ $$\begin{aligned}
\label{eq:t2_amp_tensor}
\mathbf{t^{(2)}} = \left\{t_{i}^{a(2)};\ t_{i}^{x(2)};\ t_{x}^{a(2)};\ t_{ix}^{ay(2)};\ t_{ix}^{yz(2)};\ t_{xy}^{az(2)}\right\}\end{aligned}$$ These parameters are obtained by solving the second-order linear equations $$\begin{aligned}
\label{eq:proj_amplitude_equations_2_matrix}
\mathbf{H^{(0)}} \mathbf{t^{(2)}} = - \mathbf{V^{(2)}}\end{aligned}$$ where the matrix elements of $\mathbf{V^{(2)}}$ are defined as $$\begin{aligned}
\label{eq:V_second_order_matrix}
V_{\mu}^{(2)} &= \frac{1}{2}\braket{\Psi_0|\tau^\dag_\mu [V + \tilde{H}^{(1)}, A^{(1)}]|\Psi_0}\end{aligned}$$ \[eq:proj\_amplitude\_equations\_2\_matrix\] is analogous to the first-order \[eq:proj\_amplitude\_equations\_1\_matrix\] with r.h.s. modified by the second-order matrix $\mathbf{V^{(2)}}$ and, thus, can be solved in a similar way. In practice, only a small number of terms in \[eq:mr\_adc\_2\_M\_matrix,eq:mr\_adc\_2\_T\_matrix\] depend on the $\mathbf{t^{(2)}}$ amplitudes and their contributions have a very small effect on the ionization energies and spectral intensities. We will discuss solution of the first- and second-order amplitude equations in more detail in \[sec:implementation:amplitude\_equations\].
\[sec:theory:mr\_adc\_ip:ionization\_operators\]
{width="80.00000%"}
### Ionization Operator Manifolds
To determine the ionization operators $h_{\mu}^{(k)\dag}$ ($k = 0, 1$), we use the fact that these operators must satisfy two requirements:[@Mukherjee:1989p257; @Sokolov:2018p204113] (i) at the $k$th order, the particle-hole rank of $h_{\mu}^{(k)\dag}$ must not exceed that of $\tilde{q}_{\mu}^{(k)\dag}$ or $\tilde{q}_{\mu}^{(k)}$ for the forward or backward components of the propagator, respectively; (ii) $h_{\mu}^{(k)\dag}$ must fulfill the vacuum annihilation condition (VAC)[@Goscinski:1980p385; @Weiner:1980p1109; @Prasad:1985p1287; @Datta:1993p3632] with respect to the reference state, i.e. $h_{\mu}^{(k)}\ket{\Psi_0} = 0$, which ensures decoupling of the forward and backward components of the propagator in \[eq:g\_munu\].[@Mukherjee:1989p257; @Sokolov:2018p204113] To obtain $h_{\mu}^{(0)\dag}$, we recall that $\tilde{q}_{\mu}^{(0)} = {a_{p}}$, where the annihilation operator can be of three different types: ${a_{i}}$, ${a_{x}}$, or ${a_{a}}$ (core, active, or external). Out of these three classes, only the core operator ${a_{i}}$ satisfies VAC with respect to $\ket{\Psi_0}$ (${a^\dagger_{i}}\ket{\Psi_0} = 0$) and, thus, can be added to $h_{\mu}^{(0)\dag}$. Since $\ket{\Psi_0}$ does not contain electrons in the active space, the external operator ${a_{a}}$ is redundant (${a_{a}}\ket{\Psi_0} = 0$) and cannot be included in $h_{\mu}^{(0)\dag}$. Although the active-space operator ${a_{x}}$ does not fulfill VAC (${a^\dagger_{x}}\ket{\Psi_0} \ne 0$), it can be expanded[@Sokolov:2018p204113] in the form ${a_{x}} = \sum_{I} Z^\dag_I c_{I,x}$, where $Z^\dag_I$ is a complete set of active-space eigenoperators,[@Freed:1977p401; @Lowdin:1985p285; @Kutzelnigg:1998p5578] defined as: $$\begin{aligned}
\label{eq:z_ketbra}
Z^\dag_I &= \ket{\Psi_I^{N-1}}\bra{\Psi_0}\end{aligned}$$ Here, $\ket{\Psi_I^{N-1}}$ are the CASCI states of the ionized system with $N-1$ electrons computed using the active space and one-electron basis of the reference state $\ket{\Psi_0}$. We note that in the context of propagator theory the configurational operators $Z^\dag_I$ were first used by Freed and Yeager[@Freed:1977p401] and have two important properties: they are linearly-independent and include all types of active-only ionization operators (${a_{x}}$, ${a^\dagger_{x}}{a_{y}}{a_{z}}$, $\ldots$). Incidentally, these operators also satisfy VAC with respect to $\ket{\Psi_0}$ and can be added to $h_{\mu}^{(0)\dag}$. Although we have assumed that the set of operators $Z^\dag_I$ is complete, only a subset of these operators corresponding to CASCI states in the spectral region of interest need to be included in practice. We summarize that the MR-ADC(2) zeroth-order manifold $h_{\mu}^{(0)\dag}$ consists of two sets of operators: $$\begin{aligned}
\label{eq:zeroth_order_manifold}
\mathbf{h}^{(0)\dag} = \left\{{a_{i}}; Z^\dag_I \right\}\end{aligned}$$
Following a similar strategy, we determine that the first-order operators $h_{\mu}^{(1)\dag}$ have a general form $a_{qr}^{p} \equiv {a^\dagger_{p}}{a_{r}}{a_{q}}$ and can be further divided into five classes $$\begin{aligned}
\label{eq:first_order_manifold}
\mathbf{h}^{(1)\dag} = \left\{a_{ij}^{x}; a_{ij}^{a}; a_{ix}^{y}; a_{ix}^{a}; a_{xy}^{a}\right\}\end{aligned}$$ describing ionization in the core or active spaces accompanied by core-active, active-external, or core-external single excitations, as shown in \[fig:excitations\]. The all-active operators $a_{zy}^{x}$ do not appear in $\mathbf{h}^{(1)\dag}$, since they are already included in the $\mathbf{h}^{(0)\dag}$ manifold by the $Z^\dag_I$ operators.
\[fig:M\_S\_matrices\] illustrates perturbative structure of the MR-ADC(2) effective Liouvillean ($\mathbf{M}$) and overlap ($\mathbf{S}$) matrices. The $\{\mathbf{h}^{(0)\dag}|\tilde{\mathcal{H}}^{(k)}|\mathbf{h}^{(0)\dag}\}$ block of the $\mathbf{M}$ matrix includes all contributions up to $k$ $=$ 2, while the coupling block $\{\mathbf{h}^{(1)\dag}|\tilde{\mathcal{H}}^{(k)}|\mathbf{h}^{(0)\dag}\}$ is evaluated to first order, as given by \[eq:mr\_adc\_2\_M\_matrix\]. In the manifold of first-order ionized states, the $\{\mathbf{h}^{(1)\dag}|\tilde{\mathcal{H}}^{(0)}|\mathbf{h}^{(1)\dag}\}$ sector is block-diagonal with non-zero elements for the $h_{\mu}^{(1)\dag}$ excitations from the same class (\[eq:first\_order\_manifold\]). Overall, the general perturbative structure of the MR-ADC(2) matrices closely resembles that of non-Dyson SR-ADC(2)[@Schirmer:1998p4734; @Trofimov:2005p144115; @Dempwolff:2019p064108] and the two methods become equivalent in the limit of single-determinant reference wavefunction $\ket{\Psi_{0}}$.
Implementation {#sec:implementation}
==============
General Algorithm {#sec:implementation:general_algorithm}
-----------------
In this section, we describe a general algorithm of our MR-ADC(2) implementation for complete active space (CAS) reference wavefunctions. Although in this work we always employ the ground-state CASSCF wavefunction of a neutral system as a reference, in MR-ADC other choices of reference orbitals are possible (e.g., Hartree-Fock, state-averaged, or unrestricted natural orbitals).[@Bofill:1998p3637] The main steps of the MR-ADC(2) algorithm are summarized below:
1. Choose active space, compute the reference orbitals and CAS wavefunction $\ket{\Psi_0}$ for the neutral system with $N$ electrons.
2. Using reference orbitals, compute the CASCI energies $E_I^{N-1}$ and wavefunctions $\ket{\Psi_I^{N-1}}$ for $N_{\mathrm{CI}}$ lowest-energy states of the ionized system with $(N-1)$ electrons.
3. Compute active-space reduced density matrices (RDMs) for the reference state $\ket{\Psi_0}$, transition RDMs between $\ket{\Psi_0}$ and ionized states $\ket{\Psi_I^{N-1}}$, and transition RDMs between two ionized states $\ket{\Psi_I^{N-1}}$.
4. Solve linear amplitude \[eq:proj\_amplitude\_equations\_1\_matrix,eq:proj\_amplitude\_equations\_2\_matrix\] to compute $\mathbf{t^{(1)}}$ and $\mathbf{t^{(2)}}$.
5. Solve the generalized eigenvalue problem to obtain ionization energies $\boldsymbol{\Omega}$.
6. Compute spectroscopic amplitudes and (if necessary) spectral function .
As discussed in \[sec:theory:mr\_adc\_ip\], the number of active-space ionized states ($N_{\mathrm{CI}}$) should be sufficiently large to include all important CASCI states in the spectral region of interest. Implementation of the algorithm outlined above requires derivation of equations for contributions to the $\textbf{M}$, $\textbf{T}$, and $\textbf{S}$ matrices (\[eq:mr\_adc\_2\_M\_matrix,eq:mr\_adc\_2\_T\_matrix,eq:mr\_adc\_2\_S\_matrix\]). Although most of these contributions have compact expressions, matrix elements of the second-order effective Hamiltonian (e.g., $\{\mathbf{h}^{(0)\dag}|\tilde{\mathcal{H}}^{(2)}|\mathbf{h}^{(0)\dag}\}$) are very complicated containing $\sim$ 250-300 terms for each matrix block. Such algebraic complexity is a common feature of many internally-contracted multi-reference theories.[@Neuscamman:2009p124102; @Datta:2012p204107; @Saitow:2013p044118; @MacLeod:2015p051103; @Sharma:2017p488]
To speed up tedious derivation and implementation of MR-ADC(2), we have developed a Python program that automatically generates equations and code for arbitrary-order MR-ADC(n) approximation. Our code generator is a modified version of the <span style="font-variant:small-caps;">SecondQuantizationAlgebra</span> (SQA) program developed by Neuscamman and co-workers.[@Neuscamman:2009p124102] We use SQA to define and normal-order all active-space creation and annihilation operators in \[eq:mr\_adc\_2\_M\_matrix,eq:mr\_adc\_2\_T\_matrix,eq:mr\_adc\_2\_S\_matrix\] with respect to the physical vacuum. Next, we additionally normal-order core creation and annihilation operators relative to the Fermi vacuum and evaluate expectation values with respect to the active-space states $\ket{\Psi_0}$ and $\ket{\Psi_I^{N-1}}$. The resulting equations, written as contractions of the one- and two-electron integrals, $\mathbf{t^{(1)}}$ and $\mathbf{t^{(2)}}$ amplitudes, and RDMs, are used to generate code and can be implemented using any available tensor contraction engine. We present working equations for all matrix elements in \[eq:mr\_adc\_2\_M\_matrix,eq:mr\_adc\_2\_T\_matrix,eq:mr\_adc\_2\_S\_matrix\] in the Supporting Information.
In \[sec:implementation:amplitude\_equations,sec:implementation:rdms,sec:implementation:generalized\_eigenvalue\_problem\], we provide more details about the solution of amplitude equations, efficient computation of terms that depend on high-order RDMs, and solution of the generalized eigenvalue problem.
Amplitude Equations {#sec:implementation:amplitude_equations}
-------------------
General form of the first- and second-order amplitude equations has been discussed in \[sec:theory:mr\_adc\_ip:amplitudes\]. Since the Dyall Hamiltonian (\[eq:h\_dyall\_general\]) does not contain terms that couple excitations outside of the active space, its matrix representation $\mathbf{H^{(0)}}$ (\[eq:H\_zero\_matrix\]) is block-diagonal and the amplitude equations and can be solved for each block separately. Using the standard notation for classifying excitations adopted in N-electron valence perturbation theory,[@Angeli:2001p10252; @Angeli:2001p297; @Angeli:2004p4043] operators $\boldsymbol{\tau}$ in \[eq:tau\_tensor\] are split into eight groups $\boldsymbol{\tau}^{\mathbf{[i]}}$ ($i$ $\in$ $\{0; +1; -1; +2; -2; +1'; -1'; 0'\}$), where $i$ is the number of electrons added to ($i > 0$) or removed from ($i < 0$) active space upon excitation. The operator classes with $i$ $\in$ $\{+1'; -1'; 0'\}$ are used to represent three coupled sets of single and semi-internal double excitations: $\boldsymbol{\tau}^{\mathbf{[+1']}} = \{ a_{i}^{x}; \ a_{ix}^{yz}\}$, $\boldsymbol{\tau}^{\mathbf{[-1']}} = \{ a_{x}^{a};\ a_{xy}^{az} \}$, and $\boldsymbol{\tau}^{\mathbf{[0']}} = \{ a_{i}^{a};\ a_{ix}^{ay}\}$.
Separating the $\mathbf{H^{(0)}}$, $\mathbf{t^{(1)}}$, and $\mathbf{V^{(1)}}$ matrices in \[eq:proj\_amplitude\_equations\_1\_matrix\] into blocks according to excitation classes $\boldsymbol{\tau}^{\mathbf{[i]}}$ (denoted as $\mathbf{K^{[i]}}$, $\mathbf{t^{[i](1)}}$, and $\mathbf{V^{[i](1)}}$, respectively), we express the first-order amplitude equations in the following form $$\begin{aligned}
\label{eq:proj_amplitude_equations_1_subblock}
\mathbf{K^{[i]}} \mathbf{t^{[i](1)}} = - \mathbf{V^{[i](1)}}\end{aligned}$$ To solve \[eq:proj\_amplitude\_equations\_1\_subblock\] for each excitation class, we consider the generalized eigenvalue problem for the matrix $\mathbf{K^{[i]}}$ $$\begin{aligned}
\label{eq:K_eig_problem}
\mathbf{K^{[i]}} \mathbf{Z^{[i]}} = \mathbf{S^{[i]}} \mathbf{Z^{[i]}} \boldsymbol{\epsilon}^{\mathbf{[i]}}\end{aligned}$$ which allows to obtain expression for the first-order amplitudes[@Sokolov:2018p204113] $$\begin{aligned}
\label{eq:proj_amplitude_equations_1_subblock_solution}
\mathbf{t^{[i](1)}} = - (\mathbf{S^{[i]}})^{-1/2}\, \mathbf{\tilde{Z}^{[i]}}\, (\boldsymbol{\epsilon}^{\mathbf{[i]}})^{-1}\, \mathbf{\tilde{Z}^{[i]\dag}} \,(\mathbf{S^{[i]}})^{-1/2} \, \mathbf{V^{[i](1)}}\end{aligned}$$ where $K_{\mu\nu}^{[i]} = \braket{\Psi_0|\tau^{[i]\dag}_\mu(H^{(0)} - E_0)\tau^{[i]}_\nu|\Psi_0}$, $S^{[i]}_{\mu\nu} = \braket{\Psi_0|\tau^{[i]\dag}_\mu \tau^{[i]}_\nu|\Psi_0}$, and $\mathbf{\tilde{Z}^{[i]}} = (\mathbf{S^{[i]}})^{1/2}\, \mathbf{Z^{[i]}}$. Computing the $\mathbf{t^{[i](1)}}$ amplitudes in \[eq:proj\_amplitude\_equations\_1\_subblock\_solution\] requires diagonalizing $\mathbf{K^{[i]}}$ and $\mathbf{S^{[i]}}$ and removing linear dependencies corresponding to eigenvectors of $\mathbf{S^{[i]}}$ with small eigenvalues. Since the matrix elements $K_{\mu\nu}^{[i]}$ and $S^{[i]}_{\mu\nu}$ are zero when the operators $\tau^{[i]\dag}_\mu$ and $\tau^{[i]}_\nu$ do not share the same core and external indices, diagonalization of $\mathbf{K^{[i]}}$ and $\mathbf{S^{[i]}}$ can be performed very efficiently. For the semi-internal amplitudes $\mathbf{t^{[i](1)}}$ ($i$ $\in$ $\{+1'; -1'; 0'\}$), removing redundancies in the overlap matrix may introduce small size-consistency errors of the MR-ADC energies due to the appearance of disconnected terms in the amplitude equations that become non-zero when linear dependencies are eliminated.[@Sokolov:2018p204113; @Hanauer:2011p204111] To restore full size-consistency of the MR-ADC energies, we use the approach developed by Hanauer and Köhn[@Hanauer:2012p131103] that removes the disconnected terms by transforming the excitation operators $\boldsymbol{\tau}^{\mathbf{[i]}}$ ($i$ $\in$ $\{+1'; -1'; 0'\}$) to a generalized normal-ordered form. We will demonstrate size-consistency of the MR-ADC(2) ionization energies in \[sec:results:size\_consistency\].
We use \[eq:proj\_amplitude\_equations\_1\_subblock\_solution\] to compute $\mathbf{t^{[i](1)}}$ for all double ($i$ $\in$ $\{0; +1; -1; +2; -2\}$) and one class of semi-internal ($i$ $=$ $0'$) excitations. For the $\mathbf{t^{[+1'](1)}}$ and $\mathbf{t^{[-1'](1)}}$ amplitudes, diagonalization of $\mathbf{K^{[+1']}}$ and $\mathbf{K^{[-1']}}$ requires the four-particle reduced density matrix (4-RDM) of the reference state $\ket{\Psi_0}$, which is expensive to compute and store in memory for large active spaces (see \[sec:implementation:rdms\] for details). To avoid computation of 4-RDM, we evaluate $\mathbf{t^{[+1'](1)}}$ and $\mathbf{t^{[-1'](1)}}$ using imaginary-time algorithm developed in Ref. , which employs a Laplace transform[@Sokolov:2016p064102; @Sokolov:2017p244102] to evaluate the operator resolvent $(H^{(0)} - E_0)^{-1}$ without explicitly constructing and inverting the $\mathbf{K^{[+1']}}$ and $\mathbf{K^{[-1']}}$ matrices.
The second-order amplitude equations need to be solved only for the semi-internal amplitudes $\mathbf{t^{[+1'](2)}}$, $\mathbf{t^{[-1'](2)}}$, and $\mathbf{t^{[0'](2)}}$ (\[eq:t2\_amp\_tensor\]). Among these, only $\mathbf{t^{[+1'](2)}}$ enter equations for the $\textbf{M}$ matrix, while all three sets of semi-internal amplitudes are necessary to compute the $\textbf{T}$ matrix elements. The second-order amplitudes can be obtained in a similar way as their first-order counterparts $\mathbf{t^{[i](1)}}$, i.e. by expressing $\mathbf{t^{[i](2)}}$ in the form of \[eq:proj\_amplitude\_equations\_1\_subblock\_solution\] (with $\mathbf{V^{[i](1)}}$ replaced by $\mathbf{V^{[i](2)}}$ defined in \[eq:V\_second\_order\_matrix\]) or using the imaginary-time algorithm. Although solving the second-order equations is straightforward, matrix elements of the perturbation operator $\mathbf{V^{[i](2)}}$ contain $\sim$ 600 terms and are rather tedious to evaluate. On the other hand, since the primary role of $\mathbf{t^{[i](2)}}$ ($i$ $\in$ $\{+1'; -1'; 0'\}$) is to describe relaxation of the orbitals, their contributions are expected to have a small effect on the results of the MR-ADC(2) method that already incorporates orbital relaxation via the first-order amplitudes $\mathbf{t^{[i](1)}}$ and ionization operators $\mathbf{h}^{(1)\dag}$. To test this, we considered an approximation where we neglect contributions of $\mathbf{t^{[+1'](2)}}$ and $\mathbf{t^{[-1'](2)}}$ and approximate $\mathbf{t^{[0'](2)}}$ by setting $t_{ix}^{ay(2)} \approx 0$ and neglecting all terms that depend on active-space RDMs in $\mathbf{V^{[0'](2)}}$ to obtain $t_{i}^{a(2)}$ (see the Supporting Information). The resulting amplitude equations ensure that MR-ADC(2) is equivalent to SR-ADC(2) in the single-reference limit. As demonstrated in the Supporting Information, approximating the $\mathbf{t^{(2)}}$ terms has a very small effect on the MR-ADC(2) results with errors of $\le$ 0.005 eV and $\le$ $3 \times 10^{-4}$ in ionization energies and spectroscopic factors, respectively. For this reason, we adopted this approximation in our implementation of MR-ADC(2).
Avoiding High-Order Reduced Density Matrices {#sec:implementation:rdms}
--------------------------------------------
As other internally-contracted multi-reference perturbation theories, MR-ADC(2) contains terms that depend on high-order reduced density matrices (e.g., 4-RDM) in its equations. In this section, we will demonstrate that these terms can be efficiently evaluated without computing and storing 4-RDMs in memory. There are two sources of high-order RDMs in the MR-ADC(2) equations: (i) $\mathbf{t^{(1)}}$ and $\mathbf{t^{(2)}}$ amplitude equations and (ii) second-order contributions to the effective Liouvillean matrix $\mathbf{M}$. As discussed in \[sec:implementation:amplitude\_equations\], using the imaginary-time algorithm[@Sokolov:2018p204113] allows to completely avoid computation of 4-RDM in the amplitude equations.
For the $\mathbf{M}$ matrix, 4-RDMs appear in expectation values of the second-order effective Hamiltonian $\tilde{\mathcal{H}}^{(2)}$ with respect to the reference ($\braket{\Psi_0|\tilde{\mathcal{H}}^{(2)}|\Psi_0}$) and ionized ($\braket{\Psi_I^{N-1}|\tilde{\mathcal{H}}^{(2)}|\Psi_J^{N-1}}$) wavefunctions. In particular, the latter matrix elements depend on transition 4-RDMs between all CASCI ionized states (e.g., $\braket{\Psi_I^{N-1}|{a^\dagger_{w}}{a^\dagger_{x}}{a^\dagger_{y}}{a^\dagger_{z}}{a_{z'}}{a_{y'}}{a_{x'}}{a_{w'}}|\Psi_J^{N-1}}$), which have a high $\mathcal{O}(N_{\mathrm{det}} N_{\mathrm{CI}}^2 N^8_{\mathrm{act}})$ computational scaling, where $N_{\mathrm{det}}$ is the dimension of CAS Hilbert space, $N_{\mathrm{CI}}$ is the number of CASCI ionized states, and $N_{\mathrm{act}}$ is the number of active orbitals. To demonstrate how to avoid computation of 4-RDMs, we consider one of the contributions to the $\braket{\Psi_I^{N-1}|\tilde{\mathcal{H}}^{(2)}|\Psi_J^{N-1}}$ matrix elements $$\begin{aligned}
\label{eq:avoiding_4rdm}
\frac{1}{8} \sum_{\substack{awxyzu\\vy'w'z'}} v^{z w}_{x y} t^{x u}_{a v} t^{a y'}_{z' w'}\langle\Psi_I\lvert{a^\dagger_{z}}{a^\dagger_{w}}{a^\dagger_{u}}{a^\dagger_{y'}}{a_{y}}{a_{v}}{a_{w'}}{a_{z'}}\rvert\Psi_J\rangle\end{aligned}$$ where we use shorthand notation for the first-order amplitudes $t_{xy}^{az(1)}\equiv t_{xy}^{az}$ and CASCI states $
\ket{\Psi_I^{N-1}} \equiv \ket{\Psi_I}$. Changing the order of creation and annihilation operators, we express \[eq:avoiding\_4rdm\] in the following form $$\begin{aligned}
\label{eq:avoiding_4rdm_2}
-\frac{1}{8} \sum_{\substack{awxyzu\\vy'w'z'}} v^{z w}_{x y} t^{x u}_{a v} t^{a y'}_{z' w'}\langle\Psi_I\lvert{a^\dagger_{z}}{a^\dagger_{w}}{a_{y}}{a^\dagger_{u}}{a_{v}}{a^\dagger_{y'}}{a_{w'}}{a_{z'}}\rvert\Psi_J\rangle + \ldots\end{aligned}$$ where the remaining terms involve contractions of transition 2- and 3-RDMs. Computing intermediate states $$\begin{aligned}
\label{eq:avoiding_4rdm_3}
\rvert t^{J}_{a}\rangle &= \frac{1}{2} \sum_{y'w'z'} t^{a y'}_{z' w'} {a^\dagger_{y'}}{a_{w'}}{a_{z'}}\rvert\Psi_J\rangle \\
\label{eq:avoiding_4rdm_4}
\rvert v^{I}_{x}\rangle &= \frac{1}{2} \sum_{ywz} v^{x y}_{z w} {a^\dagger_{y}}{a_{w}}{a_{z}}\rvert\Psi_I\rangle\end{aligned}$$ we evaluate the first term in \[eq:avoiding\_4rdm\_2\] using a compact expression $$\begin{aligned}
\label{eq:avoiding_4rdm_5}
-\frac{1}{2} \sum_{\substack{axuv}} t^{x u}_{a v} \langle v^{I}_{x} \lvert{a^\dagger_{u}}{a_{v}}\rvert t^{J}_{a}\rangle\end{aligned}$$ Using \[eq:avoiding\_4rdm\_3,eq:avoiding\_4rdm\_4,eq:avoiding\_4rdm\_5\] allows us to significantly lower the cost of computing transition 4-RDM terms from $\mathcal{O}(N_{\mathrm{det}} N_{\mathrm{CI}}^2 N^8_{\mathrm{act}})$ to $\mathcal{O}(N_{\mathrm{det}} N_{\mathrm{CI}}^2 N^3_{\mathrm{act}} N_{\mathrm{ext}})$, where $N_{\mathrm{ext}}$ is the number of external orbitals. We use the same technique to efficiently evaluate all 4-RDM terms that appear in the $\braket{\Psi_I^{N-1}|\tilde{\mathcal{H}}^{(2)}|\Psi_J^{N-1}}$ and $\braket{\Psi_0|\tilde{\mathcal{H}}^{(2)}|\Psi_0}$ matrix elements. We note that similar techniques have been used to avoid computation of 4-RDM in implementations of complete active space second-order perturbation theory (CASPT2) and NEVPT2 in combination with matrix product state wavefunctions.[@Sokolov:2016p064102; @Wouters:2016p054120; @Sokolov:2017p244102]
The $\mathbf{M}$ matrix elements also depend on transition RDMs of the form $\braket{\Psi_0|{a^\dagger_{w}}{a^\dagger_{x}}{a^\dagger_{y}}{a^\dagger_{z}}{a_{z'}}{a_{y'}}{a_{x'}}|\Psi_I^{N-1}}$, which we denote as 3.5-RDMs. These RDMs contribute to the second-order matrix elements $\braket{\Psi_0|{a^\dagger_{i}} \tilde{\mathcal{H}}^{(2)}|\Psi_I^{N-1}}$, as well as some elements of the first-order off-diagonal blocks $\{\mathbf{h}^{(1)\dag}|\tilde{\mathcal{H}}^{(1)}|\mathbf{h}^{(0)\dag}\}$ and $\{\mathbf{h}^{(0)\dag}|\tilde{\mathcal{H}}^{(1)}|\mathbf{h}^{(1)\dag}\}$ in \[eq:mr\_adc\_2\_M\_matrix\]. For example, a 3.5-RDM contribution to $\braket{\Psi_0|{a^\dagger_{i}} \tilde{\mathcal{H}}^{(2)}|\Psi_I^{N-1}}$ has a form $$\begin{aligned}
\frac{1}{8} \sum_{\substack{awxyz\\uvu'w'}} v^{x y}_{z w} t^{i z}_{a u} t^{v u'}_{a w'} \langle\Psi_{0}\lvert{a^\dagger_{w}}{a^\dagger_{u}}{a^\dagger_{v}}{a^\dagger_{u'}}{a_{y}}{a_{x}}{a_{w'}}\rvert\Psi_{I}\rangle\end{aligned}$$ To evaluate this term, we reorder creation and annihilation operators, contract $v^{x y}_{z w}$ and $t^{v u'}_{a w'}$ with ${a^\dagger_{x}}{a^\dagger_{y}}{a_{w}}\rvert\Psi_{0}\rangle$ and ${a^\dagger_{v}}{a^\dagger_{u'}}{a_{w'}}\rvert\Psi_{I}\rangle$ to form intermediate states ($\rvert v^{z} \rangle$ and $\rvert t_{I}^{a} \rangle$), and contract $t^{i z}_{a u}$ with their overlap matrix element ($\langle v^{z} \lvert{a^\dagger_{u}}\rvert t^{a}_{I} \rangle$). As in the case of 4-RDM, using intermediate states allows to completely avoid computation and storage of 3.5-RDMs for all terms of the $\mathbf{M}$ matrix, lowering computational scaling from $\mathcal{O}(N_{\mathrm{det}} N_{\mathrm{CI}} N^7_{\mathrm{act}})$ to $\mathcal{O}(N_{\mathrm{det}} N_{\mathrm{CI}} N^2_{\mathrm{act}} N_{\mathrm{ext}})$.
Combining efficient algorithms for the solution of amplitude equations and evaluation of high-order RDM terms, our MR-ADC(2) implementation has $\mathcal{O}(N_{\mathrm{det}} N^2_{\mathrm{CI}} N^6_{\mathrm{act}})$ computational scaling, which is significantly lower than the $\mathcal{O}(N_{\mathrm{det}} N^8_{\mathrm{act}})$ scaling of the conventional multi-reference perturbation theories (e.g., CASPT2 or NEVPT2) with the number of active orbitals. Although the scaling of our current MR-ADC(2) algorithm originates from computing transition 3-RDMs ($\braket{\Psi_I^{N-1}|{a^\dagger_{w}}{a^\dagger_{x}}{a^\dagger_{y}}{a_{y'}}{a_{x'}}{a_{w'}}|\Psi_J^{N-1}}$) for all ionized states, we note that using intermediate states the computational cost can be further lowered to $\mathcal{O}(N_{\mathrm{det}} N_{\mathrm{CI}} N^6_{\mathrm{act}})$. We did not take advantage of it in our present implementation.
Solution of the Generalized Eigenvalue Problem {#sec:implementation:generalized_eigenvalue_problem}
----------------------------------------------
Finally, we briefly discuss solution of the MR-ADC(2) generalized eigenvalue problem in \[eq:adc\_eig\_problem\]. Since the $\mathbf{M}$ and $\mathbf{S}$ matrices are computed in the non-orthogonal basis of internally-contracted ionized states, we transform the eigenvalue equation to the symmetrically-orthogonalized form $$\begin{aligned}
\label{eq:adc_eig_problem_orthogonal}
\mathbf{\tilde{M}} \mathbf{\tilde{Y}} = \mathbf{\tilde{Y}} \boldsymbol{\Omega}\end{aligned}$$ where $\mathbf{\tilde{M}} = \mathbf{S}^{-1/2} \mathbf{M} \mathbf{S}^{-1/2}$ and $\mathbf{\tilde{Y}} = \mathbf{S}^{1/2} \mathbf{Y}$. Here, the overlap matrix $\mathbf{S}$ contains four non-diagonal blocks corresponding to ionized states $\ket{\Psi_\mu} = \{a_{ij}^{x} \ket{\Psi_0}$; $a_{ix}^{a} \ket{\Psi_0}$; $a_{xy}^{a} \ket{\Psi_0}$; ${a_{i}} \ket{\Psi_0}$; $a_{ix}^{y} \ket{\Psi_0}\}$ (\[fig:S\_matrix\]). Conveniently, the $\mathbf{S}^{-1/2}$ matrix can be constructed together with the $(\mathbf{S^{[i]}})^{-1/2}$ matrices used for solution of the amplitude equations (\[sec:implementation:amplitude\_equations\]). As an example, we consider non-zero elements of $\mathbf{S}$ for $a_{ij}^{x}\ket{\Psi_0}$ that have the form $S_{ijx,ijy} = \braket{\Psi_0|a_{y}^{ij}a_{ij}^{x}|\Psi_0} = \braket{\Psi_0|{a_{y}}{a^\dagger_{x}}|\Psi_0}$. These elements are equal to the $\mathbf{S^{[+1]}}$ matrix elements $S_{ijay,ijax}^{[+1]} = \braket{\Psi_0|a_{ay}^{ij}a_{ij}^{ax}|\Psi_0} = \braket{\Psi_0|{a_{y}}{a^\dagger_{x}}|\Psi_0}$. Thus, by diagonalizing the density matrix $\braket{\Psi_0|{a_{y}}{a^\dagger_{x}}|\Psi_0}$ and removing linearly-dependent eigenvectors corresponding to small eigenvalues ($<$ $\eta_{d}$, where $\eta_{d}$ is a user-defined truncation parameter), we simultaneously obtain elements of $(\mathbf{S^{[+1]}})^{-1/2}$ and $\mathbf{S}^{-1/2}$ for the $a_{ij}^{x}\ket{\Psi_0}$ ionized wavefunctions. Similarly, we construct $(\mathbf{S^{[-1]}})^{-1/2}$ and $(\mathbf{S^{[-2]}})^{-1/2}$ together with $\mathbf{S}^{-1/2}$ for $a_{ix}^{a}\ket{\Psi_0}$ and $a_{xy}^{a}\ket{\Psi_0}$, respectively.
For the $a_{ij}^{x}\ket{\Psi_0}$, $a_{ix}^{a}\ket{\Psi_0}$, and $a_{xy}^{a}\ket{\Psi_0}$ states, numerical instabilities due to linear dependencies are completely eliminated when using small truncation parameters ($\eta_{d}$ $\sim$ $10^{-10}$). Except for very small active spaces ($N_{\mathrm{act}} < 6$), orthogonalization of these ionized states does not require discarding any eigenvectors of the overlap matrix. The zeroth-order ${a_{i}} \ket{\Psi_0}$ and first-order $a_{ix}^{y} \ket{\Psi_0}$ ionized states exhibit much stronger linear dependencies in their overlap matrix. To remove these linear dependencies, we project out ${a_{i}} \ket{\Psi_0}$ from $a_{ix}^{y} \ket{\Psi_0}$ using the projection approach developed by Hanauer and Köhn[@Hanauer:2011p204111] and subsequently orthogonalize $a_{ix}^{y} \ket{\Psi_0}$ between each other. Importantly, this ensures that the zeroth-order states ${a_{i}} \ket{\Psi_0}$, which are already orthogonal, are not affected by removing redundancies in the first-order $a_{ix}^{y} \ket{\Psi_0}$ ionization manifold. To discard linearly-dependent eigenvectors of the $a_{ix}^{y} \ket{\Psi_0}$ overlap matrix, we use a larger truncation parameter ($\eta_{s}$ $\sim$ $10^{-6}$) than the one used for other ionized states ($\eta_{d}$).
We solve the eigenvalue problem using a multi-root implementation of the Davidson algorithm,[@Davidson:1975p87; @Liu:1978p49] which avoids storing the full $\textbf{M}$ and $\textbf{S}$ matrices, significantly reducing the memory requirements. Since the second-order block $\{\mathbf{h}^{(0)\dag}|\tilde{\mathcal{H}}^{(2)}|\mathbf{h}^{(0)\dag}\}$ of $\textbf{M}$ is small (with $(N_{\mathrm{CI}} + N_{\mathrm{act}})^2$ elements) and its computation is the most time-consuming step of the MR-ADC(2) implementation, we precompute this block, store it memory, and use it for the efficient evaluation of matrix-vector products in the Davidson procedure.
Computational Details {#sec:computational_details}
=====================
We implemented MR-ADC(2) for photoelectron spectra in our pilot code <span style="font-variant:small-caps;">Prism</span>, which was interfaced with <span style="font-variant:small-caps;">Pyscf</span>[@Sun:2018pe1340] to obtain integrals and CASCI/CASSCF reference wavefunctions. Our implementation follows the general algorithm outlined in \[sec:implementation:general\_algorithm\]. All MR-ADC(2) computations used the CASSCF reference wavefunctions with molecular orbitals optimized for the ground electronic state of each (neutral) system. To remove linear dependencies in the solution of amplitude equations and generalized eigenvalue problem, we truncated eigenvectors of the overlap matrices using two parameters: $\eta_{s}$ = $10^{-6}$ and $\eta_{d}$ = $10^{-10}$ (see \[sec:implementation:generalized\_eigenvalue\_problem\] for details). The $\eta_{s}$ parameter was used to orthogonalize the $a_{ix}^{y} \ket{\Psi_0}$ ionized states and to compute the semi-internal $\mathbf{t^{[i](1)}}$ ($i$ $\in$ $\{+1'; -1'; 0'\}$) amplitudes (\[sec:implementation:amplitude\_equations\]), while $\eta_{d}$ was employed for other amplitudes and ionized states. To efficiently compute $\mathbf{t^{[+1'](1)}}$ and $\mathbf{t^{[-1'](1)}}$, our implementation used imaginary-time algorithm,[@Sokolov:2018p204113; @Sokolov:2016p064102; @Sokolov:2017p244102] where propagation in imaginary time was performed using the embedded Runge-Kutta method that automatically determines time step based on the accuracy parameter $\Delta_{it}$.[@Press:2007] In all computations, we used $\Delta_{it}$ = $10^{-7}$ [$E_\mathrm{h}$]{}, which allows to obtain very accurate amplitudes and reference NEVPT2 correlation energy. All MR-ADC(2) results were converged with respect to the number of CASCI ionized states ($N_{\mathrm{CI}}$). For most of the systems employed in this study, using $N_{\mathrm{CI}}$ = 20 was enough to obtain well-converged results.
We benchmarked the accuracy of MR-ADC(2) for a set of small molecules (, , , , , , , and ), carbon dimer (), and hydrogen chains ( and ). For small molecules, equilibrium and stretched geometries were considered. The equilibrium structures were taken from Ref. . For diatomic molecules, the stretched geometries were obtained by increasing the bond length by a factor of two. For the , , and stretched geometries, we doubled the , , and bond distances, respectively. The bond length in was set to 1.2425 , which is very close to its equilibrium geometry. Unless noted otherwise, all computations employed the aug-cc-pVDZ basis set.[@Kendall:1992p6796] For and , the cc-pVDZ basis set was used for the hydrogen atoms, as employed in Ref. . We denote active spaces used in CASCI/CASSCF as ($n$e, $m$o), where $n$ is the number of active electrons and $m$ is the number of active orbitals. Active spaces of small molecules included 10 orbitals with $n$ = 8, 14, 10, 10, 8, 10, 12, and 10 active electrons for , , , , , , , and , respectively. For , the (8e, 12o) active space was used. For the hydrogen chains, we employed the (10e, 10o) active space.
The MR-ADC(2) results were compared to results of single-reference non-Dyson ADC methods (SR-ADC(2) and SR-ADC(3)),[@Schirmer:1998p4734; @Trofimov:2005p144115; @Dempwolff:2019p064108] equation-of-motion coupled cluster theory for ionization energies with single and double excitations (EOM-CCSD),[@Sinha:1989p544; @Mukhopadhyay:1991p441; @Nooijen:1992p55] quasi-degenerate strongly-contracted second-order N-electron valence perturbation theory (QD-NEVPT2),[@Angeli:2004p4043] as well as full configuration interaction (FCI). All methods employed the same geometries and basis sets as those used for MR-ADC(2). SR-ADC(2) and SR-ADC(3) were implemented by our group as a module in the development version of <span style="font-variant:small-caps;">Pyscf</span>. The FCI results were computed using the semistochastic heat-bath configuration interaction algorithm (SHCI) implemented in the <span style="font-variant:small-caps;">Dice</span> program.[@Holmes:2016p3674; @Sharma:2017p1595; @Holmes:2017p164111] The SHCI electronic energies were extrapolated using a linear fit according to procedure described in Ref. . We estimate that errors of the computed SHCI energy differences relative to FCI do not exceed 0.03 eV. For and , the $1s$ atomic orbitals of carbon and oxygen were not correlated in the SHCI computations. For all other methods, all electrons were correlated in all computations. The EOM-CCSD and QD-NEVPT2 results were obtained using <span style="font-variant:small-caps;">Q-Chem</span>[@qchem:44] and <span style="font-variant:small-caps;">Orca</span>,[@Neese:2017pe1327] respectively. For the ground state of each neutral system, QD-NEVPT2 used the same active spaces and CASSCF reference wavefunctions as those employed in MR-ADC(2). The QD-NEVPT2 computations of ionized states used the state-averaged CASSCF reference wavefunctions, where state-averaging included four electronic states for each abelian subgroup irreducible representation of the full symmetry point group.
Intensities of photoelectron transitions were characterized by computing spectroscopic factors $$\begin{aligned}
\label{eq:spec_factors}
P_{\mu} = \sum_{p} |X_{p,\mu}|^2\end{aligned}$$ where $X_{p,\mu}$ are elements of the spectroscopic amplitude matrix $\mathbf{X}_{\pm}$ defined in \[eq:spec\_amplitudes\]. Spectroscopic factors in \[eq:spec\_factors\] correspond to intensities of photoelectron transitions under the approximation that only single-electron detachment contributes to the spectrum. More rigorous simulation of photoelectron intensities require computation of Dyson orbitals with explicit treatment of the wavefunction of injected free electron and will be one of the subjects of our future work.[@Gozem:2015p4532]
Results {#sec:results}
=======
Size-Consistency of Energies and Properties {#sec:results:size_consistency}
-------------------------------------------
[L[3.5cm]{}L[1cm]{}C[2.75cm]{}C[2.75cm]{}]{} System & State & $\Delta \Omega$ & $\Delta P$\
($r_e$) & $1b_{1}$ & $-$2.4 $\times$ $10^{-5}$ & 6.0 $\times$ $10^{-8}$\
& $3a_{1}$ & 9.4 $\times$ $10^{-6}$ & $-$2.7 $\times$ $10^{-7}$\
& $1b_{2}$ & 2.8 $\times$ $10^{-6}$ & 2.5 $\times$ $10^{-7}$\
($2 r_e$) & $1b_{1}$ & $-$1.3 $\times$ $10^{-5}$ & 3.4 $\times$ $10^{-7}$\
& $3a_{1}$ & 1.5 $\times$ $10^{-5}$ & 2.3 $\times$ $10^{-6}$\
& $1b_{2}$ & 1.3 $\times$ $10^{-5}$ & 1.8 $\times$ $10^{-6}$\
($r_e$) & $1\pi$ & 1.2 $\times$ $10^{-4}$ & $-$1.0 $\times$ $10^{-6}$\
& $3\sigma$ & 5.4 $\times$ $10^{-5}$ &2.4 $\times$ $10^{-6}$\
($2 r_e$) & $1\pi$ & 1.0 $\times$ $10^{-4}$ & $-$4.5 $\times$ $10^{-6}$\
& $3\sigma$ & 5.9 $\times$ $10^{-5}$ & $-$9.1 $\times$ $10^{-7}$\
We begin by testing size-consistency of the MR-ADC(2) ionization energies and spectroscopic factors. As for single-reference ADC, the MR-ADC equations are fully connected, which guarantees size-consistency of the MR-ADC energies and transition properties. In practice, however, removing redundancies in the overlap matrix during the solution of the MR-ADC amplitude equations may result in small size-consistency errors.[@Sokolov:2018p204113] As we discussed in \[sec:implementation:amplitude\_equations\], in this work we employ a technique developed by Hanauer and Köhn that restores size-consistency of the MR-ADC results. \[tab:size\_consistency\] shows deviations from size-consistency of the MR-ADC(2) ionization energies ($\Delta \Omega$) and spectroscopic factors ($\Delta P$) for the and systems, each composed of two noninteracting monomers with near-equilibrium ($r_e$) and stretched geometries ($2\times r_e$). The computed size-consistency errors are very small: $\Delta \Omega$ $\sim$ $10^{-5}$ eV and $\Delta P$ $\sim$ $10^{-6}$ on average, with the largest errors of eV and . These remaining errors originate from a finite time step used in the imaginary-time algorithm for solving the semi-internal amplitude equations and become increasingly smaller with a tighter $\Delta_{it}$ parameter (see \[sec:computational\_details\] for details). Overall, our numerical results demonstrate size-consistency of the MR-ADC(2) results in the present implementation.
Small Molecules {#sec:results:small_molecules}
---------------
----------------------------- --------------- ---------- ------ ---------- ------ ---------- ----------- ---------- ------ ---------- ---------- ----------
System State EOM-CCSD QD-NEVPT2 FCI
$\Omega$ $P$ $\Omega$ $P$ $\Omega$ $P$ $\Omega$ $P$ $\Omega$ $\Omega$ $\Omega$
HF $1\pi$ 14.41 0.89 16.79 0.93 16.41 0.93 16.35 0.93 15.85 16.00 16.07
$3\sigma$ 18.69 0.90 20.65 0.94 20.30 0.94 20.38 0.94 19.88 20.04 20.06
F$_2$ $1\pi_{g}$ 13.90 0.87 16.03 0.89 15.87 0.90 16.55 0.88 15.40 15.38 15.64
$1\pi_{u}$ 17.06 0.84 19.25 0.75 19.11 0.81 19.86 0.80 18.77 18.58 18.83
$3\sigma_{g}$ 20.25 0.89 21.26 0.89 21.01 0.88 22.08 0.87 21.16 20.88 21.15
CO $5\sigma$ 13.78 0.91 13.57 0.90 13.80 0.89 14.07 0.92 13.99 13.53 13.74
$1\pi$ 16.24 0.89 17.16 0.90 16.88 0.90 17.38 0.90 16.93 16.75 16.90
$4\sigma$ 18.28 0.85 20.46 0.76 20.10 0.79 20.15 0.85 19.67 19.48 19.56
N$_2$ $3\sigma_{g}$ 14.79 0.88 15.42 0.91 15.60 0.91 15.76 0.91 15.43 15.21 15.30
$1\pi_{u}$ 16.98 0.91 16.60 0.92 16.77 0.92 17.33 0.92 17.11 16.75 16.83
$2\sigma_{u}$ 17.96 0.85 18.79 0.82 18.93 0.82 19.00 0.83 18.71 18.44 18.50
H$_2$O $1b_{1}$ 11.23 0.89 12.99 0.92 12.78 0.92 12.74 0.93 12.38 12.55 12.53
$3a_{1}$ 13.53 0.89 15.28 0.92 15.08 0.93 15.07 0.93 14.66 14.85 14.81
$1b_{2}$ 17.95 0.90 19.34 0.93 19.16 0.93 19.28 0.94 18.89 19.05 18.98
CS $7\sigma$ 10.99 0.86 10.99 0.85 11.33 0.85 11.59 0.85 11.36 10.95 11.13
$2\pi$ 12.84 0.91 12.67 0.90 12.66 0.90 13.43 0.91 12.94 12.74 12.83
$6\sigma$ 16.88 0.85 15.53 0.18 15.51 0.19 16.83 0.40 17.02 15.83 15.88
H$_2$CO $2b_{2}$ 9.46 0.87 11.11 0.91 10.87 0.91 11.23 0.92 10.62 10.28 10.72
$1b_{1}$ 13.73 0.88 14.54 0.88 14.30 0.88 15.14 0.90 14.47 14.07 14.48
$5a_{1}$ 14.62 0.86 16.61 0.90 16.20 0.90 16.70 0.90 15.95 15.64 16.01
$1b_{2}$ 16.67 0.88 17.04 0.69 17.32 0.65 17.76 0.88 17.21 16.50 16.86
C$_2$H$_4$ $1b_{1u}$ 10.14 0.91 10.47 0.91 10.46 0.91 11.01 0.90 10.58 10.41 10.58
$1b_{1g}$ 12.79 0.91 13.22 0.91 13.19 0.91 13.75 0.92 13.22 13.05 13.21
$3a_{g}$ 13.78 0.89 14.34 0.91 14.36 0.91 14.74 0.89 14.31 14.12 14.25
$1b_{2u}$ 16.13 0.87 16.50 0.74 16.49 0.79 17.10 0.84 16.61 16.35 16.45
[$\Delta_{\mathrm{MAE}}$]{} 0.83 0.30 0.21 0.56 0.17 0.17
[$\Delta_{\mathrm{STD}}$]{} 0.68 0.32 0.22 0.23 0.28 0.14
----------------------------- --------------- ---------- ------ ---------- ------ ---------- ----------- ---------- ------ ---------- ---------- ----------
Non-Dyson SR-ADC(3) incorporating high-order self-energy corrections from Ref. .
In this section, we benchmark the MR-ADC(2) accuracy for predicting ionization energies of small molecules. \[tab:ionization\_energies\_re\] compares vertical ionization energies ($\Omega$) and spectroscopic factors ($P$) of MR-ADC(2) with those obtained by single-reference non-Dyson ADC methods (SR-ADC), equation-of-motion coupled cluster theory with single and double excitations (EOM-CCSD), quasi-degenerate NEVPT2 (QD-NEVPT2), and full configuration interaction (FCI) for a set of eight molecules near their equilibrium geometries (see \[sec:computational\_details\] for computational details). In addition to strict second- and third-order SR-ADC (SR-ADC(2) and SR-ADC(3)), \[tab:ionization\_energies\_re\] also presents results of SR-ADC(3) incorporating high-order self-energy corrections, reported in Ref. , which we denote as SR-ADC(3+). Out of six approximate methods, the best agreement with FCI is shown by SR-ADC(3+), EOM-CCSD, and QD-NEVPT2. These three methods produce similar mean absolute errors in vertical ionization energies ([$\Delta_{\mathrm{MAE}}$]{}$\sim$ 0.2 eV) with standard deviations from the mean signed error ([$\Delta_{\mathrm{STD}}$]{}) ranging from $\sim$ 0.15 to 0.3 eV, as illustrated in \[fig:mae\_std\_re\]. The MR-ADC(2) method shows a similar [$\Delta_{\mathrm{STD}}$]{}error (0.23 eV), but a larger [$\Delta_{\mathrm{MAE}}$]{}error (0.56 eV), which is lower than [$\Delta_{\mathrm{MAE}}$]{}of SR-ADC(2) (0.83 eV), but higher than that of SR-ADC(3) (0.30 eV), indicating that including high-order effects in MR-ADC(2) improves its accuracy relative to SR-ADC(2). For all systems, the MR-ADC(2) ionization energies systematically overestimate energies computed using FCI, showing a good agreement with FCI for energy spacings between electronic states of the ionized systems ([$\Delta_{\mathrm{MAE}}$]{}of 0.11 eV and [$\Delta_{\mathrm{STD}}$]{}of 0.10 eV). The QD-NEVPT2 method shows the best agreement with FCI for energy spacings ([$\Delta_{\mathrm{MAE}}$]{}and [$\Delta_{\mathrm{STD}}$]{}of 0.03 eV), while EOM-CCSD shows larger errors compared to MR-ADC(2) ([$\Delta_{\mathrm{MAE}}$]{}= 0.16 eV, [$\Delta_{\mathrm{STD}}$]{}= 0.27 eV). The MR-ADC(2) spectroscopic factors agree well with those computed using SR-ADC(3) and SR-ADC(3+), with two exceptions observed for the $6\sigma$ state of and the $1b_{2}$ state of . In these cases, the computed spectroscopic factors vary significantly depending on the order of the ADC approximation, suggesting that properties of these photoelectron transitions are significantly affected by electron correlation effects.
----------------------------- --------------- ---------- ------ ---------- ---------- ----------- ------ ---------- ---------- -------
System State EOM-CCSD QD-NEVPT2 FCI
$\Omega$ $P$ $\Omega$ $P$ $\Omega$ $P$ $\Omega$ $\Omega$
HF $1\pi$ 9.84 0.77 16.15 0.84 13.86 0.60 13.67 13.61 13.65
$3\sigma$ 13.30 0.84 14.68 0.76 14.98 0.73 14.76 14.83 14.84
F$_{2}$ $1\pi_{g}$ 10.63 0.64 17.55 0.88 18.12 0.74 16.86 16.81 17.13
$1\pi_{u}$ 10.66 0.64 17.69 0.89 18.16 0.82 16.95 16.87 17.19
N$_{2}$ $3\sigma_{g}$ 15.70 0.63 $-$2.60 1.69 14.00 0.69 14.36 13.06 13.38
$1\pi_{u}$ 17.50 0.55 $-$5.24 2.16 14.17 0.51 14.77 13.21 13.49
H$_{2}$O $1b_{1}$ 6.53 0.71 12.24 0.66 11.31 0.64 10.65 10.99 11.07
$3a_{1}$ 10.49 0.75 12.78 0.67 13.22 0.67 12.69 12.99 13.02
$1b_{2}$ 11.18 0.75 13.01 0.72 13.78 0.71 13.26 13.53 13.56
H$_{2}$CO $2b_{2}$ 10.65 0.85 8.31 0.21 11.51 0.39 9.85 10.24 10.37
$1b_{1}$ 10.69 0.86 8.35 0.22 11.21 0.48 9.66 10.38 10.55
$5a_{1}$ 10.60 0.91 10.97 0.88 13.16 0.57 10.97 12.32 13.16
C$_{2}$H$_{4}$ $1b_{1u}$ 9.37 0.76 6.87 0.83 9.69 0.53 9.41 9.26 9.25
$3a_{g}$ 11.38 0.79 8.74 0.91 11.36 0.73 11.17 10.94 10.93
[$\Delta_{\mathrm{MAE}}$]{} 2.70 3.66 0.50 0.56 0.18
[$\Delta_{\mathrm{STD}}$]{} 3.10 6.28 0.36 0.81 0.23
----------------------------- --------------- ---------- ------ ---------- ---------- ----------- ------ ---------- ---------- -------
To assess performance of MR-ADC(2) when strong correlation is important, we computed its ionization energies and spectroscopic factors for molecules with stretched geometries, where at least one of the bonds is elongated by a factor of two (see \[sec:computational\_details\] for details). The MR-ADC(2) results are shown in \[tab:ionization\_energies\_2re\], along with those computed using SR-ADC(2), SR-ADC(3), EOM-CCSD, QD-NEVPT2, and FCI. Due to the difficulty of obtaining the FCI energies, we show results only for a few lowest-energy transitions of six molecules. Importance of strong electron correlation for these non-equilibrium geometries is demonstrated by the poor performance of SR-ADC(2) and SR-ADC(3), which show very large deviations from the FCI reference values with [$\Delta_{\mathrm{MAE}}$]{}$>$ 2.5 eV and [$\Delta_{\mathrm{STD}}$]{}$>$ 3 eV. Although SR-ADC(3) shows moderate $\sim$ 0.5 eV errors for single-bond stretching in HF and , these errors drastically increase when multiple bonds are elongated, leading to unphysical values of ionization energies that significantly underestimate the FCI results. EOM-CCSD significantly improves prediction of ionization energies over SR-ADC(2) and SR-ADC(3), but still exhibits large errors ([$\Delta_{\mathrm{MAE}}$]{}= 0.56 and [$\Delta_{\mathrm{STD}}$]{}= 0.81 eV, \[fig:mae\_std\_2re\]). MR-ADC(2) shows performance similar to that for equilibrium geometries (\[tab:ionization\_energies\_re\]), with [$\Delta_{\mathrm{MAE}}$]{}(0.50 eV) and [$\Delta_{\mathrm{STD}}$]{}(0.36 eV) smaller than the corresponding errors for the single-reference methods. The best agreement with FCI is again shown by QD-NEVPT2 with [$\Delta_{\mathrm{MAE}}$]{}= 0.18 eV and [$\Delta_{\mathrm{STD}}$]{}= 0.23 eV. As for the equilibrium geometries, the MR-ADC(2) ionization energies for stretched geometries are systematically overestimated relative to FCI, reproducing energy spacings between ionized states within 0.1 eV for all systems except , where $\sim$ 0.5 eV errors are observed. QD-NEVPT2 shows a similar performance to MR-ADC(2) for energy spacings with a large error of $\sim$ 0.7 eV for the difference of the $1b_{1}$ and $5a_{1}$ ionization energies of .
---------------------------------------- ----------------- ---------- -------- ----------- -------- ---------- ----------
Configuration State QD-NEVPT2 FCI
$\Omega$ $P$ $\Omega$ $P$ $\Omega$ $\Omega$
$(2\sigma_u)^2(1\pi_u)^3(3\sigma_g)^0$ $1^2\Pi_u$ 11.69 0.9215 12.50 0.8986 12.28 12.34
$(2\sigma_u)^2(1\pi_u)^2(3\sigma_g)^1$ $1^2\Delta_g$ 11.17 0.0002 14.31 0.0002 13.92 13.94
$(2\sigma_u)^2(1\pi_u)^2(3\sigma_g)^1$ $1^2\Sigma_g^-$ 14.55 0.0000 14.12 14.15
$(2\sigma_u)^2(1\pi_u)^2(3\sigma_g)^1$ $1^2\Sigma_g^+$ 11.43 0.0004 14.60 0.0047 14.26 14.29
$(2\sigma_u)^1(1\pi_u)^4(3\sigma_g)^0$ $1^2\Sigma_u^+$ 13.95 0.8738 15.33 0.7190 15.07 15.09
$(2\sigma_u)^2(1\pi_u)^1(3\sigma_g)^2$ $2^2\Pi_u$ 14.77 0.0183 15.35 15.43
---------------------------------------- ----------------- ---------- -------- ----------- -------- ---------- ----------
State is absent in SR-ADC(3).
An important advantage of MR-ADC(2) over conventional multi-reference perturbation theories (such as QD-NEVPT2) is that it provides efficient access to spectroscopic properties. We demonstrate this by computing the photoelectron spectrum of at equilibrium and stretched geometries in the range between 8.5 and 20 eV, shown in \[fig:c2h4\_spectrum\]. The spectrum at equilibrium geometry exhibits five very intense well-separated peaks corresponding to vertical ionizations in five highest occupied molecular orbitals. All of the computed peaks are systematically shifted by $\sim$ 0.5 eV relative to FCI. The computed spacings between the main peaks are in a good agreement with FCI (\[tab:ionization\_energies\_re\]), as well as experimental photoelectron spectrum.[@Branton:1970p802] At the stretched geometry, the MR-ADC(2) spectrum shows four main peaks with significantly decreased intensities, along with several satellite peaks originating from shake-up transitions that involve ionization and simultaneous excitation in the valence orbitals.
Carbon Dimer {#sec:results:carbon_dimer}
------------
![Simulated photoelectron spectrum of carbon dimer for $r$ = 1.2425 computed using the MR-ADC(2) method by broadening peaks centered at vertical ionization energies with a half width of 0.03 eV (aug-cc-pVDZ basis set). Vertical dashed lines indicate FCI ionization energies for the main peaks corresponding to the $1^2\Pi_u$ and $1^2\Sigma_u^+$ states of (\[tab:c2\]).[]{data-label="fig:c2_spectrum"}](C2.pdf){width="45.00000%"}
Next, we investigate performance of MR-ADC(2) for simulating photoelectron spectrum of , which is a challenging test for [*ab initio*]{} methods, since electronic states of both and require very accurate description of static and dynamic correlation.[@Roos:1987p399; @Bauschlicher:1987p1919; @Watts:199p6073; @Abrams:2004p9211; @Wouters:2014p1501; @Holmes:2017p164111; @Rosmus:1986p289; @Kraemer:1987p345; @Watts:1998p6073; @Petrongolo:1998p4594; @Ballance:2001p1201; @Shi:2013p2020] \[tab:c2\] compares results of SR-ADC(3), MR-ADC(2), and QD-NEVPT2 with those from FCI. The MR-ADC(2) photoelectron spectrum, shown in \[fig:c2\_spectrum\], exhibits two very intense peaks for ionizations in the $1\pi_u$ and $2\sigma_u$ orbitals, corresponding to the $1^2\Pi_u$ and $1^2\Sigma_u^+$ electronic states of , respectively. For both peaks, MR-ADC(2) is in a good agreement with FCI, showing errors in vertical ionization energies (0.16 and 0.24 eV) within [$\Delta_{\mathrm{MAE}}$]{}and [$\Delta_{\mathrm{STD}}$]{}of small molecules computed in \[sec:results:small\_molecules\]. The MR-ADC(2) results show significant improvement over SR-ADC(3), which underestimates the $1\pi_u$ and $2\sigma_u$ ionization energies from FCI by 0.65 and 1.14 eV, respectively, indicating that description of multi-reference effects is important for these ionization processes. The best agreement with FCI is demonstrated by QD-NEVPT2, with errors smaller than 0.1 eV.
In addition to the intense peaks, the photoelectron spectrum also exhibits several much weaker (satellite) peaks, which involve ionization in the $1\pi_u$ orbital accompanied by single and double $(1\pi_u)^3$ $\rightarrow$ $(3\sigma_g)^0$ excitations (\[tab:c2\]). Out of four satellite transitions, only two are predicted by SR-ADC(3), with large errors ($>$ 2 eV). For the singly-excited shake-up states of ($1^2\Delta_g$, $1^2\Sigma_g^-$, and $1^2\Sigma_g^+$), the largest MR-ADC(2) error is 0.37 eV. However, for the doubly-excited $2^2\Pi_u$ state, MR-ADC(2) produces a larger 0.66 eV error. The QD-NEVPT2 ionization energies for all four electronic states are within 0.1 eV from the reference FCI values. The large error of MR-ADC(2) for $2^2\Pi_u$ may be attributed to the importance of differential dynamic correlation effects between this state and the ground state of , since in MR-ADC(2) the first-order amplitudes of the effective Hamiltonian are preferentially determined for the latter state (\[sec:theory:mr\_adc\_ip:amplitudes\]), while in QD-NEVPT2 the first-order wavefunction is constructed for each electronic state separately. The description of these differential correlation effects is expected to improve for higher-order MR-ADC approximations and will be a subject of our future research.
Hydrogen Chains {#sec:results:hydrogen_chains}
---------------
Finally, we use MR-ADC to study equally-spaced hydrogen chains and . Hydrogen chains are one-dimensional models for understanding strong electron correlation in molecules and materials, as well as the hydrogen phase diagram at high pressures.[@Sinitskiy:2010p014104; @Stella:2011p245117; @Lin:2011p096402; @Tsuchimochi:2009p121102; @Rusakov:2016p054106; @Motta:2017p031059; @Rusakov:2019p229] An important property of a hydrogen chain is its band gap, which can be calculated as the difference between ionization potential and electron affinity. For equally-spaced chains in the thermodynamic limit, this band gap is believed to be zero at short distances ($r$), corresponding to a metallic phase, and non-zero for long distances, corresponding to an insulator. Recently, Ronca [*et al.*]{} computed local density of states (LDOS) of the H$_n$ chains ($n$ = 10, 30, and 50) at the central hydrogen atom using density matrix renormalization group (DMRG) method with the minimal STO-6G basis set.[@Ronca:2017p5560] They demonstrated that for near-equilibrium and stretched geometries ($r$ = 1.8 and 3.6 [$a_0$]{}) LDOS converges to thermodynamic limit already for , while for compressed chains ($r$ = 1.4 [$a_0$]{}) finite size effects are still significant. Although in this study all valence electrons of hydrogen atoms were correlated, importance of dynamic correlation effects beyond those in the minimal one-electron basis was not investigated.
Here, we use MR-ADC to study effect of dynamic correlation and basis set on the density of occupied states in and . \[fig:h10\_spectrum\] shows LDOS of for $r$ = 1.4, 1.8, and 3.6 [$a_0$]{}computed at the central hydrogen atom using the MR-ADC(0) and MR-ADC(2) methods. We use the full valence (10e, 10o) active space for the CASSCF reference wavefunction and combine MR-ADC with the STO-6G[@Hehre:1969p2657] and cc-pVTZ basis sets, plotting LDOS for two broadening parameters: 0.05 [$E_\mathrm{h}$]{}(\[fig:h10\_spectrum\]) and 0.003 [$E_\mathrm{h}$]{}(Supporting Information). For the minimal STO-6G basis, results of MR-ADC(0) and MR-ADC(2) are equivalent to FCI. The computed LDOS are in a very good agreement with LDOS obtained by Ronca [*et al.*]{} employing the dynamical DMRG algorithm for all three geometries.[@Ronca:2017p5560] Next, we consider LDOS computed using MR-ADC(0) with the larger cc-pVTZ basis set. Increasing the basis set shifts LDOS to higher ionization energies, relative to LDOS from FCI/STO-6G. For short bond distances ($r$ = 1.4 and 1.8 [$a_0$]{}), the largest shifts are observed for the lowest-energy peaks corresponding to the ionization potential of the system ($\sim$ 0.03 and 0.05 [$E_\mathrm{h}$]{}, respectively). For the stretched chain ($r$ = 3.6 [$a_0$]{}), increasing the basis set compresses LDOS and shifts the position of its maximum by $\sim$ 0.04 [$E_\mathrm{h}$]{}. Incorporating dynamic correlation effects from MR-ADC(0) to MR-ADC(2) shifts the computed LDOS further to lower energies. For most of the peaks at short bond distances, the computed shifts are $\le$ 0.02 [$E_\mathrm{h}$]{}. For $r$ = 3.6 [$a_0$]{}, including dynamic correlation does not significantly change position of the first band in the spectrum. Overall, our results suggest that increasing the one-electron basis set and incorporating dynamic correlation effects are similarly important and should be both taken into account in accurate computations of LDOS for hydrogen chains.
An attractive feature of MR-ADC is that it is not limited to describing ionization processes only in active orbitals. We demonstrate this by computing total density of occupied states (DOS) for the chain using MR-ADC(2) with the active space. Since for this system we do not include all valence orbitals in the active space, we do not consider the stretched $r$ = 3.6 [$a_0$]{}geometry. \[fig:h30\_spectrum\_full\] shows MR-ADC(2) DOS computed using the STO-6G and cc-pVDZ basis sets. For both geometries, DOS computed using MR-ADC(2) with the STO-6G basis closely resembles LDOS of the same system from the DMRG study of Ronca [*et al*]{}.[@Ronca:2017p5560] \[fig:h30\_spectrum\_contr\] plots contributions to MR-ADC(2)/STO-6G DOS from core and active orbitals separately. For the compressed chain ($r$ = 1.4 [$a_0$]{}), contributions from active orbitals dominate the low-energy part of the spectrum, whereas, for equilibrium geometry ($r$ = 1.8 [$a_0$]{}), core and active orbitals have similar contributions to DOS already for low ionization energies. Increasing the basis set from STO-6G to cc-pVDZ shifts peaks in DOS to higher energies. As for the chain, the largest shifts are observed for the peak at the first ionization potential.
Conclusions {#sec:conclusions}
===========
We presented derivation and implementation of second-order multi-reference algebraic diagrammatic construction theory (MR-ADC(2)) for simulating ionization energies and transition properties of strongly correlated systems. In MR-ADC(2), ionization energies and spectral properties are determined from poles and residues of the one-electron Green’s function that is evaluated to second order in multi-reference perturbation theory with respect to a complete active space (CAS) reference wavefunction. In contrast to conventional second-order multi-reference perturbation theories (such as multi-state CASPT2 or NEVPT2), MR-ADC(2) describes ionization in all orbitals (e.g., core and active), does not require using state-averaged wavefunctions to compute higher-energy ionized states, and provides direct access to spectroscopic properties. Although equations of MR-ADC(2) depend on four-particle reduced density matrices, we demonstrated that computation of these large matrices can be completely avoided by constructing efficient intermediates, without introducing any approximations. The resulting MR-ADC(2) implementation has a lower $\mathcal{O}(N_{\mathrm{det}} N^6_{\mathrm{act}})$ computational scaling with respect to the number of active orbitals ($N_{\mathrm{act}}$), compared to the $\mathcal{O}(N_{\mathrm{det}} N^8_{\mathrm{act}})$ scaling of conventional multi-reference perturbation theories.
We benchmarked accuracy of MR-ADC(2) for predicting ionization energies of eight small molecules, carbon dimer (), and hydrogen chains ( and ), against results from full configuration interaction (FCI). For small molecules, MR-ADC(2) shows consistent performance for equilibrium and stretched geometries, with mean absolute errors of $\sim$ 0.5 eV in ionization energies and 0.1 eV in energy separations between ionized states. For , MR-ADC(2) predicts energies of the main and singly-excited satellite peaks within 0.4 eV from the FCI reference values, but has a large $\sim$ 0.7 eV error for the doubly-excited satellite transition. The QD-NEVPT2 method shows smaller ($\sim$ 0.1 eV) errors than MR-ADC(2) for all ionized states of , providing an improved description of differential dynamic correlation effects, which are important for this system. We expect that these effects will be better described using the higher-order MR-ADC approximations, which will be one of the directions of our future work. Finally, we used MR-ADC(2) to investigate density of occupied states (DOS) in and . For , our results provide numerical evidence that including dynamic correlation effects beyond those incorporated in a full valence CAS and increasing single-particle basis set have a similar effect on the computed local DOS. Since dynamic correlation is a local phenomenon, we expect that its effect will be similar for longer hydrogen chains as well. For , we showed that DOS computed using MR-ADC(2) combined with a small (10e, 10o) active space is in a very good agreement with previously reported results from density matrix renormalization group, incorporating 30 electrons and orbitals in the active space.
Overall, our results suggest that MR-ADC is a promising theoretical approach for computing ionization energies and spectral densities of multi-reference systems and encourage its further development. Future work will be directed towards more efficient implementation of MR-ADC(2) for systems with a large number of electrons and active orbitals, as well as the development of more accurate MR-ADC approximations that will incorporate description of higher-order dynamic correlation effects. We also plan extending our MR-ADC methods to simulations of core-level ionizations in X-ray photoelectron spectroscopy, which has become a widely used tool for experimental investigations of molecules and materials.
This work was supported by start-up funds provided by the Ohio State University. Computations were performed at the Ohio Supercomputer Center under projects PAS1317 and PAS1583.[@OhioSupercomputerCenter1987] The authors would like to thank Sandeep Sharma for useful suggestions about carrying out SHCI computations.
Comparison of MR-ADC(2) results with exact and approximate second-order amplitudes. Equations of the MR-ADC(2) method for the $\mathbf{M}$, $\mathbf{T}$, and $\mathbf{S}$ matrices. Density of states for with a small broadening parameter.
@ifundefined
[150]{}
Olsen, J.; Roos, B. O.; J[ø]{}rgensen, P.; Jensen, H. J. A. [Determinant based configuration interaction algorithms for complete and restricted configuration interaction spaces]{}. *J. Chem. Phys.* **1988**, *89*, 2185 Malmqvist, P. [Å]{}.; Rendell, A.; Roos, B. O. [The restricted active space self-consistent-field method, implemented with a split graph unitary group approach]{}. *J. Phys. Chem.* **1990**, *94*, 5477–5482 White, S. R.; Martin, R. L. [Ab initio quantum chemistry using the density matrix renormalization group]{}. *J. Chem. Phys.* **1999**, *110*, 4127 Legeza, [Ö]{}.; Noack, R.; S[ó]{}lyom, J.; Tincani, L. *Computational Many-Particle Physics*; Springer, 2008; pp 653–664 Booth, G. H.; Thom, A. J. W.; Alavi, A. [Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space]{}. *J. Chem. Phys.* **2009**, *131*, 054106 Kurashige, Y.; Yanai, T. [High-performance ab initio density matrix renormalization group method: Applicability to large-scale multireference problems for metal compounds]{}. *J. Chem. Phys.* **2009**, *130*, 234114 Marti, K. H.; Reiher, M. [New electron correlation theories for transition metal chemistry]{}. *Phys. Chem. Chem. Phys.* **2011**, *13*, 6750–6759 Chan, G. K.-L.; Sharma, S. [The Density Matrix Renormalization Group in Quantum Chemistry]{}. *Annu. Rev. Phys. Chem.* **2011**, *62*, 465–481 Gunst, K.; Wouters, S.; De Baerdemacker, S.; Van Neck, D. [Block product density matrix embedding theory for strongly correlated spin systems]{}. *Eur. Phys. J. D* **2014**, *68*, 272 Zhang, T.; Evangelista, F. A. [A Deterministic Projector Configuration Interaction Approach for the Ground State of Quantum Many-Body Systems]{}. *J. Chem. Theory Comput.* **2016**, *12*, 4326–4337 Schriber, J. B.; Evangelista, F. A. [Adaptive Configuration Interaction for Computing Challenging Electronic Excited States with Tunable Accuracy]{}. *J. Chem. Theory Comput.* **2017**, *13*, 5354–5366 Holmes, A. A.; Tubman, N. M.; Umrigar, C. J. [Heat-Bath Configuration Interaction: An Efficient Selected Configuration Interaction Algorithm Inspired by Heat-Bath Sampling]{}. *J. Chem. Theory Comput.* **2016**, *12*, 3674–3680 Sharma, S.; Holmes, A. A.; Jeanmairet, G.; Alavi, A.; Umrigar, C. J. [Semistochastic Heat-Bath Configuration Interaction Method: Selected Configuration Interaction with Semistochastic Perturbation Theory]{}. *J. Chem. Theory Comput.* **2017**, *13*, 1595–1604 Holmes, A. A.; Umrigar, C. J.; Sharma, S. [Excited states using semistochastic heat-bath configuration interaction]{}. *J. Chem. Phys.* **2017**, *147*, 164111 Werner, H.-J.; Meyer, W. [A quadratically convergent multiconfigurationself-consistent field method with simultaneous optimization of orbitals and CI coefficients]{}. *J. Chem. Phys.* **1980**, *73*, 2342–2356 Werner, H.-J. [A quadratically convergent MCSCF method for the simultaneous optimization of several states]{}. *J. Chem. Phys.* **1981**, *74*, 5794–5801 Knowles, P. J.; Werner, H.-J. [An efficient second-order MC SCF method for long configuration expansions]{}. *Chem. Phys. Lett.* **1985**, *115*, 259–267 Wolinski, K.; Sellers, H. L.; Pulay, P. [Consistent generalization of the M[ø]{}ller-Plesset partitioning to open-shell and multiconfigurational SCF reference states in many-body perturbation theory]{}. *Chem. Phys. Lett.* **1987**, *140*, 225–231 Hirao, K. [Multireference M[ø]{}llerPlesset method]{}. *Chem. Phys. Lett.* **1992**, *190*, 374–380 Werner, H.-J. [Third-order multireference perturbation theory. The CASPT3 method]{}. *Mol. Phys.* **1996**, *89*, 645–661 Finley, J. P.; Malmqvist, P. [Å]{}.; Roos, B. O.; Serrano-Andr[é]{}s, L. [The multi-state CASPT2 method]{}. *Chem. Phys. Lett.* **1998**, *288*, 299–306 Andersson, K.; Malmqvist, P. [Å]{}.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. [Second-order perturbation theory with a CASSCF reference function]{}. *J. Phys. Chem.* **1990**, *94*, 5483–5488 Andersson, K.; Malmqvist, P. [Å]{}.; Roos, B. O. [Second-order perturbation theory with a complete active space self-consistent field reference function]{}. *J. Chem. Phys.* **1992**, *96*, 1218–1226 Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P. P. [Introduction of n-electron valence states for multireference perturbation theory]{}. *J. Chem. Phys.* **2001**, *114*, 10252–10264 Angeli, C.; Cimiraglia, R.; Malrieu, J.-P. P. [N-electron valence state perturbation theory: a fast implementation of the strongly contracted variant]{}. *Chem. Phys. Lett.* **2001**, *350*, 297–305 Angeli, C.; Borini, S.; Cestari, M.; Cimiraglia, R. [A quasidegenerate formulation of the second order n-electron valence state perturbation theory approach]{}. *J. Chem. Phys.* **2004**, *121*, 4043–4049 Li, C.; Evangelista, F. A. [Multireference Driven Similarity Renormalization Group: A Second-Order Perturbative Analysis]{}. *J. Chem. Theory Comput.* **2015**, *11*, 2097–2108 Buenker, R. J.; Peyerimhoff, S. D. [Individualized configuration selection in CI calculations with subsequent energy extrapolation]{}. *Theor. Chem. Acc.* **1974**, *35*, 33–58 Siegbahn, P. E. M. [Generalizations of the direct CI method based on the graphical unitary group approach. II. Single and double replacements from any set of reference configurations]{}. *J. Chem. Phys.* **1980**, *72*, 1647 Werner, H.-J.; Knowles, P. J. [An efficient internally contracted multiconfigurationreference configuration interaction method]{}. *J. Chem. Phys.* **1988**, *89*, 5803–5814 Saitow, M.; Kurashige, Y.; Yanai, T. [Multireference configuration interaction theory using cumulant reconstruction with internal contraction of density matrix renormalization group wave function]{}. *J. Chem. Phys.* **2013**, *139*, 044118 Saitow, M.; Kurashige, Y.; Yanai, T. [Fully Internally Contracted Multireference Configuration Interaction Theory Using Density Matrix Renormalization Group: A Reduced-Scaling Implementation Derived by Computer-Aided Tensor Factorization]{}. *J. Chem. Theory Comput.* **2015**, *11*, 5120–5131 Mukherjee, D.; Moitra, R. K.; Mukhopadhyay, A. [Applications of a non-perturbative many-body formalism to general open-shell atomic and molecular problems: calculation of the ground and the lowest $\pi$-$\pi$\* singlet and triplet energies and the first ionization potential of trans-butadiene]{}. *Mol. Phys.* **1977**, *33*, 955–969 Lindgren, I. [A coupled-cluster approach to the many-body perturbation theory for open-shell systems]{}. *Int. J. Quantum Chem.* **1978**, *14*, 33–58 Jeziorski, B.; Monkhorst, H. J. [Coupled-cluster method for multideterminantal reference states]{}. *Phys. Rev. A* **1981**, *24*, 1668–1681 Mahapatra, U. S.; Datta, B.; Mukherjee, D. [A size-consistent state-specific multireference coupled cluster theory: Formal developments and molecular applications]{}. *J. Chem. Phys.* **1999**, *110*, 6171 Evangelista, F. A.; Allen, W. D.; Schaefer, H. F. [Coupling term derivation and general implementation of state-specific multireference coupled cluster theories]{}. *J. Chem. Phys.* **2007**, *127*, 024102 Datta, D.; Kong, L.; Nooijen, M. [A state-specific partially internally contracted multireference coupled cluster approach]{}. *J. Chem. Phys.* **2011**, *134*, 214116 Evangelista, F. A.; Gauss, J. [An orbital-invariant internally contracted multireference coupled cluster approach]{}. *J. Chem. Phys.* **2011**, *134*, 114102 K[ö]{}hn, A.; Hanauer, M.; M[ü]{}ck, L. A.; Jagau, T.-C.; Gauss, J. [State-specific multireference coupled-cluster theory]{}. *WIREs Comput. Mol. Sci.* **2013**, *3*, 176–197 Datta, D.; Nooijen, M. [Multireference equation-of-motion coupled cluster theory]{}. *J. Chem. Phys.* **2012**, *137*, 204107 Nooijen, M.; Demel, O.; Datta, D.; Kong, L.; Shamasundar, K. R.; Lotrich, V.; Huntington, L. M.; Neese, F. [Communication: Multireference equation of motion coupled cluster: A transform and diagonalize approach to electronic structure]{}. *J. Chem. Phys.* **2014**, *140*, 081102 Huntington, L. M. J.; Nooijen, M. [Application of multireference equation of motion coupled-cluster theory to transition metal complexes and an orbital selection scheme for the efficient calculation of excitation energies]{}. *J. Chem. Phys.* **2015**, *142*, 194111 Kirtman, B. [Simultaneous calculation of several interacting electronic states by generalized Van Vleck perturbation theory]{}. *J. Chem. Phys.* **1998**, *75*, 798–808 Hoffmann, M. R. R.; Simons, J. [A unitary multiconfigurational coupled-cluster method: Theory and applications]{}. *J. Chem. Phys.* **1988**, *88*, 993 Yanai, T.; Chan, G. K.-L. [Canonical transformation theory for multireference problems.]{} *J. Chem. Phys.* **2006**, *124*, 194106 Yanai, T.; Chan, G. K.-L. [Canonical transformation theory from extended normal ordering]{}. *J. Chem. Phys.* **2007**, *127*, 104107 Chen, Z.; Hoffmann, M. R. R. [Orbitally invariant internally contracted multireference unitary coupled cluster theory and its perturbative approximation: Theory and test calculations of second order approximation]{}. *J. Chem. Phys.* **2012**, *137*, 014108 Li, C.; Evangelista, F. A. [Towards numerically robust multireference theories: The driven similarity renormalization group truncated to one- and two-body operators]{}. *J. Chem. Phys.* **2016**, *144*, 164114 Evangelista, F. A. [Perspective: Multireference coupled cluster theories of dynamical electron correlation]{}. *J. Chem. Phys.* **2018**, *149*, 030901 Kurashige, Y.; Yanai, T. [Second-order perturbation theory with a density matrix renormalization group self-consistent field reference function: Theory and application to the study of chromium dimer]{}. *J. Chem. Phys.* **2011**, *135*, 094104 Kurashige, Y.; Chalupsk[ý]{}, J.; Lan, T. N.; Yanai, T. [Complete active space second-order perturbation theory with cumulant approximation for extended active-space wavefunction from density matrix renormalization group]{}. *J. Chem. Phys.* **2014**, *141*, 174111 Guo, S.; Watson, M. A.; Hu, W.; Sun, Q.; Chan, G. K.-L. [N-Electron Valence State Perturbation Theory Based on a Density Matrix Renormalization Group Reference Function, with Applications to the Chromium Dimer and a Trimer Model of Poly(p-Phenylenevinylene)]{}. *J. Chem. Theory Comput.* **2016**, *12*, 1583–1591 Sharma, S.; Knizia, G.; Guo, S.; Alavi, A. [Combining Internally Contracted States and Matrix Product States To Perform Multireference Perturbation Theory]{}. *J. Chem. Theory Comput.* **2017**, *13*, 488–498 Yanai, T.; Saitow, M.; Xiong, X.-G.; Chalupsk[ý]{}, J.; Kurashige, Y.; Guo, S.; Sharma, S. [Multistate Complete-Active-Space Second-Order Perturbation Theory Based on Density Matrix Renormalization Group Reference States]{}. *J. Chem. Theory Comput.* **2017**, *13*, 4829–4840 Freitag, L.; Knecht, S.; Angeli, C.; Reiher, M. [Multireference Perturbation Theory with Cholesky Decomposition for the Density Matrix Renormalization Group]{}. *J. Chem. Theory Comput.* **2017**, *13*, 451 Sokolov, A. Y.; Guo, S.; Ronca, E.; Chan, G. K.-L. [Time-dependent N-electron valence perturbation theory with matrix product state reference wavefunctions for large active spaces and basis sets: Applications to the chromium dimer and all-trans polyenes]{}. *J. Chem. Phys.* **2017**, *146*, 244102 Schriber, J. B.; Hannon, K. P.; Li, C.; Evangelista, F. A. [A Combined Selected Configuration Interaction and Many-Body Treatment of Static and Dynamical Correlation in Oligoacenes]{}. *J. Chem. Theory Comput.* **2018**, *14*, 6295–6305 MacLeod, M. K.; Shiozaki, T. [Communication: Automatic code generation enables nuclear gradient computations for fully internally contracted multireference theory]{}. *J. Chem. Phys.* **2015**, *142*, 051103 Sokolov, A. Y. [Multi-reference algebraic diagrammatic construction theory for excited states: General formulation and first-order implementation]{}. *J. Chem. Phys.* **2018**, *149*, 204113 Schirmer, J. [Beyond the random-phase approximation: A new approximation scheme for the polarization propagator]{}. *Phys. Rev. A* **1982**, *26*, 2395–2416 Banerjee, A.; Shepard, R.; Simons, J. [One-particle Green’s function with multiconfiguration reference states]{}. *Int. J. Quantum Chem.* **1978**, *14*, 389–404 Yeager, D. L.; J[ø]{}rgensen, P. [A multiconfigurational time-dependent Hartree-Fock approach]{}. *Chem. Phys. Lett.* **1979**, *65*, 77–80 Dalgaard, E. [Time-dependent multiconfigurational HartreeFock theory]{}. *J. Chem. Phys.* **1980**, *72*, 816–823 Yeager, D. L.; Olsen, J.; J[ø]{}rgensen, P. [Generalizations of the multiconfigurational time-dependent Hartree-Fock approach]{}. *Faraday Symp. Chem. Soc.* **1984**, *19*, 85–95 Graham, R. L.; Yeager, D. L. [The multiconfigurational particleparticle propagator method for directly determining vertical double ionization potentials and double electron affinities]{}. *J. Chem. Phys.* **1991**, *94*, 2884–2893 Yeager, D. L. *Applied Many-Body Methods in Spectroscopy and Electronic Structure*; Springer, Boston, MA: Boston, MA, 1992; pp 133–161 Nichols, J. A.; Yeager, D. L.; J[ø]{}rgensen, P. [Multiconfigurational electron propagator (MCEP) ionization potentials for general open shell systems]{}. *J. Chem. Phys.* **1998**, *80*, 293–314 Khrustov, V. F.; Kostychev, D. E. [Multiconfigurational Green’s function approach with quasidegenerate perturbation theory]{}. *Int. J. Quantum Chem.* **2002**, *88*, 507–518 Helmich-Paris, B. [CASSCF linear response calculations for large open-shell molecules]{}. *J. Chem. Phys.* **2019**, *150*, 174121 Samanta, P. K.; Mukherjee, D.; Hanauer, M.; K[ö]{}hn, A. [Excited states with internally contracted multireference coupled-cluster linear response theory]{}. *J. Chem. Phys.* **2014**, *140*, 134108 Mukherjee, D.; Kutzelnigg, W. *Many-Body Methods in Quantum Chemistry*; Springer, Berlin, Heidelberg: Berlin, Heidelberg, 1989; pp 257–274 Fetter, A. L.; Walecka, J. D. *Quantum theory of many-particle systems*; Dover Publications, 2003 Dickhoff, W. H.; Van Neck, D. *Many-body theory exposed!: propagator description of quantum mechanics in many-body systems*; World Scientific Publishing Co., 2005 Goscinski, O.; Weiner, B. [The Role of Algebraic Formulations of Approximate Green’s Functions for Systems With a Finite Number of Electrons]{}. *Phys. Scr.* **1980**, *21*, 385–393 Weiner, B.; Goscinski, O. [Self-consistent approximation to the polarization propagator]{}. *Int. J. Quantum Chem.* **1980**, *18*, 1109–1131 Prasad, M. D.; Pal, S.; Mukherjee, D. [Some aspects of self-consistent propagator theories]{}. *Phys. Rev. A* **1985**, *31*, 1287–1298 Datta, B.; Mukhopadhyay, D.; Mukherjee, D. [Consistent propagator theory based on the extended coupled-cluster parametrization of the ground state]{}. *Phys. Rev. A* **1993**, *47*, 3632–3648 L[ö]{}wdin, P. O. [Some properties of inner projections]{}. *Int. J. Quantum Chem.* **1970**, *5*, 231–237 Nielsen, E. S.; J[ø]{}rgensen, P.; Oddershede, J. [Transition moments and dynamic polarizabilities in a second order polarization propagator approach]{}. *J. Chem. Phys.* **1980**, *73*, 6238–6246 Sangfelt, E.; Kurtz, H. A.; Elander, N.; Goscinski, O. [Excited states via the AGP polarization propagator. I. Application to Li$_{2}$]{}. *J. Chem. Phys.* **1984**, *81*, 3976–3986 Bak, K. L.; Koch, H.; Oddershede, J.; Christiansen, O.; Sauer, S. P. A. [Atomic integral driven second order polarization propagator calculations of the excitation spectra of naphthalene and anthracene]{}. *J. Chem. Phys.* **2000**, *112*, 4173–4185 Nooijen, M.; Snijders, J. G. [Coupled cluster approach to the single-particle Green’s function]{}. *Int. J. Quantum Chem.* **1992**, *44*, 55–83 Nooijen, M.; Snijders, J. G. [Coupled cluster Green’s function method: Working equations and applications]{}. *Int. J. Quantum Chem.* **1993**, *48*, 15–48 Nooijen, M.; Snijders, J. G. [Second order many-body perturbation approximations to the coupled cluster Greens function]{}. *J. Chem. Phys.* **1995**, *102*, 1681–1688 Moszynski, R.; [Ż]{}uchowski, P. S.; Jeziorski, B. [Time-Independent Coupled-Cluster Theory of the Polarization Propagator]{}. *Collect. Czech. Chem. Commun.* **2005**, *70*, 1109–1132 Korona, T. [XCC2a new coupled cluster model for the second-order polarization propagator]{}. *Phys. Chem. Chem. Phys.* **2010**, *12*, 14977–14984 Kowalski, K.; Bhaskaran-Nair, K.; Shelton, W. A. [Coupled-cluster representation of Green function employing modified spectral resolutions of similarity transformed Hamiltonians]{}. *J. Chem. Phys.* **2014**, *141*, 094102 Schirmer, J. [Closed-form intermediate representations of many-body propagators and resolvent matrices]{}. *Phys. Rev. A* **1991**, *43*, 4647 Mertins, F.; Schirmer, J. [Algebraic propagator approaches and intermediate-state representations. I. The biorthogonal and unitary coupled-cluster methods]{}. *Phys. Rev. A* **1996**, *53*, 2140–2152 Schirmer, J.; Trofimov, A. B. [Intermediate state representation approach to physical properties of electronically excited molecules]{}. *J. Chem. Phys.* **2004**, *120*, 11449–11464 Schirmer, J.; Cederbaum, L. S.; Walter, O. [New approach to the one-particle Green’s function for finite Fermi systems]{}. *Phys. Rev. A* **1983**, *28*, 1237–1259 Schirmer, J.; Trofimov, A. B.; Stelter, G. [A non-Dyson third-order approximation scheme for the electron propagator]{}. *J. Chem. Phys.* **1998**, *109*, 4734 Trofimov, A. B.; Schirmer, J. [Molecular ionization energies and ground- and ionic-state properties using a non-Dyson electron propagator approach]{}. *J. Chem. Phys.* **2005**, *123*, 144115 Dempwolff, A. L.; Schneider, M.; Hodecker, M.; Dreuw, A. [Efficient implementation of the non-Dyson third-order algebraic diagrammatic construction approximation for the electron propagator for closed- and open-shell molecules]{}. *J. Chem. Phys.* **2019**, *150*, 064108 Liu, J.; Asthana, A.; Cheng, L.; Mukherjee, D. [Unitary coupled-cluster based self-consistent polarization propagator theory: A third-order formulation and pilot applications]{}. *J. Chem. Phys.* **2018**, *148*, 244110 Hedin, L. [New method for calculating the one-particle Green’s function with application to the electron-gas problem]{}. *Phys. Rev. A* **1965**, *139*, 796–823 Faleev, S. V.; van Schilfgaarde, M.; Kotani, T. [All-Electron Self-Consistent GW Approximation: Application to Si, MnO, and NiO]{}. *Phys. Rev. Lett.* **2004**, *93*, 126406 van Schilfgaarde, M.; Kotani, T.; Faleev, S. [Quasiparticle Self-Consistent GW Theory]{}. *Phys. Rev. Lett.* **2006**, *96*, 226402 Cederbaum, L. S. [One-body Greens function for atoms and molecules]{}. *J. Phys. B: At. Mol. Phys.* **1975**, *8*, 290–303 Von Niessen, W.; Schirmer, J.; Cederbaum, L. S. [Computational methods for the one-particle Green’s function]{}. *Computer Physics Reports* **1984**, *1*, 57–125 Ortiz, J. V. [Electron propagator theory: an approach to prediction and interpretation in quantum chemistry]{}. *WIREs Comput. Mol. Sci.* **2012**, *3*, 123–142 Georges, A.; Kotliar, G.; Krauth, W.; Rozenberg, M. J. [Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions]{}. *Rev. Mod. Phys.* **1996**, *68*, 13–125 Kotliar, G.; Savrasov, S. Y.; Haule, K.; Oudovenko, V. S.; Parcollet, O.; Marianetti, C. A. [Electronic structure calculations with dynamical mean-field theory]{}. *Rev. Mod. Phys.* **2006**, *78*, 865–951 Phillips, J. J.; Zgid, D. [Communication: The description of strong correlation within self-consistent Green’s function second-order perturbation theory]{}. *J. Chem. Phys.* **2014**, *140*, 241101 Lan, T. N.; Kananenka, A. A.; Zgid, D. [Communication: Towards ab initio self-energy embedding theory in quantum chemistry]{}. *J. Chem. Phys.* **2015**, *143*, 241102 Dyall, K. G. [The choice of a zeroth-order Hamiltonian for second-order perturbation theory with a complete active space self-consistent-field reference function]{}. *J. Chem. Phys.* **1995**, *102*, 4909–4918 Aoto, Y. A.; Bargholz, A.; Kats, D.; Werner, H.-J.; K[ö]{}hn, A. [Perturbation Expansion of Internally Contracted Coupled-Cluster Theory up to Third Order]{}. *J. Chem. Theory Comput.* **2019**, *15*, 2291–2305 Freed, K. F.; Yeager, D. L. [A wavefunction approach to equations of motionGreen’s function methods]{}. *Chem. Phys.* **1977**, *22*, 401–414 L[ö]{}wdin, P. O. [Some Aspects on the Hamiltonian and Liouvillian Formalism, the Special Propagator Methods, and the Equation of Motion Approach]{}. *Adv. Quant. Chem.* **1985**, *17*, 285–334 Kutzelnigg, W.; Mukherjee, D. [Time-independent theory of one-particle Greens functions]{}. *J. Chem. Phys.* **1989**, *90*, 5578–5594 Bofill, J. M.; Pulay, P. [The unrestricted natural orbitalcomplete active space (UNOCAS) method: An inexpensive alternative to the complete active spaceself-consistent-field (CASSCF) method]{}. *J. Chem. Phys.* **1998**, *90*, 3637–3646 Neuscamman, E.; Yanai, T.; Chan, G. K.-L. [Quadratic canonical transformation theory and higher order density matrices]{}. *J. Chem. Phys.* **2009**, *130*, 124102 Hanauer, M.; K[ö]{}hn, A. [Pilot applications of internally contracted multireference coupled cluster theory, and how to choose the cluster operator properly]{}. *J. Chem. Phys.* **2011**, *134*, 204111 Hanauer, M.; K[ö]{}hn, A. [Communication: Restoring full size extensivity in internally contracted multireference coupled cluster theory]{}. *J. Chem. Phys.* **2012**, *137*, 131103 Sokolov, A. Y.; Chan, G. K.-L. [A time-dependent formulation of multi-reference perturbation theory]{}. *J. Chem. Phys.* **2016**, *144*, 064102 Wouters, S.; Van Speybroeck, V.; Van Neck, D. [DMRG-CASPT2 study of the longitudinal static second hyperpolarizability of all-trans polyenes]{}. *J. Chem. Phys.* **2016**, *145*, 054120 Davidson, E. R. [The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices]{}. *J. Comput. Phys.* **1975**, *17*, 87 Liu, B. *[The Simultaneous Expansion Method for the Iterative Solution of Several of the Lowest-Lying Eigenvalues and Corresponding Eigenvectors of Large Real-Symmetric Matrices]{}*; 1978; pp 49–53 Sun, Q.; Berkelbach, T. C.; Blunt, N. S.; Booth, G. H.; Guo, S.; Li, Z.; Liu, J.; McClain, J. D.; Sayfutyarova, E. R.; Sharma, S.; Wouters, S.; Chan, G. K.-L. [PySCF: the Python-based simulations of chemistry framework]{}. *WIREs Comput. Mol. Sci.* **2018**, *8*, e1340 Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. *Numerical Recipes: The Art of Scientific Computing*; Cambridge University Press: New York, 1986 Kendall, R. A.; Dunning Jr, T. H.; Harrison, R. J. [Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions]{}. **1992**, *96*, 6796–6806 Sinha, D.; Mukhopadhyay, S. K.; Chaudhuri, R.; Mukherjee, D. [The eigenvalue-independent partitioning technique in Fock space: An alternative route to open-shell coupled-cluster theory for incomplete model spaces]{}. *Chem. Phys. Lett.* **1989**, *154*, 544–549 Mukhopadhyay, D.; Mukhopadhyay, S.; Chaudhuri, R.; Mukherjee, D. [Aspects of separability in the coupled cluster based direct methods for energy differences]{}. *Theoret. Chim. Acta* **1991**, *80*, 441–467 Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X.; Ghosh, D.; Goldey, M.; Horn, P. R.; Jacobson, L. D.; Kaliman, I.; Khaliullin, R. Z.; Kuś, T.; Landau, A.; Liu, J.; Proynov, E. I.; Rhee, Y. M.; Richard, R. M.; Rohrdanz, M. A.; Steele, R. P.; Sundstrom, E. J.; III, H. L. W.; Zimmerman, P. M.; Zuev, D.; Albrecht, B.; Alguire, E.; Austin, B.; Beran, G. J. O.; Bernard, Y. A.; Berquist, E.; Brandhorst, K.; Bravaya, K. B.; Brown, S. T.; Casanova, D.; Chang, C.-M.; Chen, Y.; Chien, S. H.; Closser, K. D.; Crittenden, D. L.; Diedenhofen, M.; Jr., R. A. D.; Do, H.; Dutoi, A. D.; Edgar, R. G.; Fatehi, S.; Fusti-Molnar, L.; Ghysels, A.; Golubeva-Zadorozhnaya, A.; Gomes, J.; Hanson-Heine, M. W.; Harbach, P. H.; Hauser, A. W.; Hohenstein, E. G.; Holden, Z. C.; Jagau, T.-C.; Ji, H.; Kaduk, B.; Khistyaev, K.; Kim, J.; Kim, J.; King, R. A.; Klunzinger, P.; Kosenkov, D.; Kowalczyk, T.; Krauter, C. M.; Lao, K. U.; Laurent, A. D.; Lawler, K. V.; Levchenko, S. V.; Lin, C. Y.; Liu, F.; Livshits, E.; Lochan, R. C.; Luenser, A.; Manohar, P.; Manzer, S. F.; Mao, S.-P.; Mardirossian, N.; Marenich, A. V.; Maurer, S. A.; Mayhall, N. J.; Neuscamman, E.; Oana, C. M.; Olivares-Amaya, R.; O’Neill, D. P.; Parkhill, J. A.; Perrine, T. M.; Peverati, R.; Prociuk, A.; Rehn, D. R.; Rosta, E.; Russ, N. J.; Sharada, S. M.; Sharma, S.; Small, D. W.; Sodt, A.; Stein, T.; Stück, D.; Su, Y.-C.; Thom, A. J.; Tsuchimochi, T.; Vanovschi, V.; Vogt, L.; Vydrov, O.; Wang, T.; Watson, M. A.; Wenzel, J.; White, A.; Williams, C. F.; Yang, J.; Yeganeh, S.; Yost, S. R.; You, Z.-Q.; Zhang, I. Y.; Zhang, X.; Zhao, Y.; Brooks, B. R.; Chan, G. K.; Chipman, D. M.; Cramer, C. J.; III, W. A. G.; Gordon, M. S.; Hehre, W. J.; Klamt, A.; III, H. F. S.; Schmidt, M. W.; Sherrill, C. D.; Truhlar, D. G.; Warshel, A.; Xu, X.; Aspuru-Guzik, A.; Baer, R.; Bell, A. T.; Besley, N. A.; Chai, J.-D.; Dreuw, A.; Dunietz, B. D.; Furlani, T. R.; Gwaltney, S. R.; Hsu, C.-P.; Jung, Y.; Kong, J.; Lambrecht, D. S.; Liang, W.; Ochsenfeld, C.; Rassolov, V. A.; Slipchenko, L. V.; Subotnik, J. E.; Voorhis, T. V.; Herbert, J. M.; Krylov, A. I.; Gill, P. M.; Head-Gordon, M. Advances in molecular quantum chemistry contained in the Q-Chem 4 program package. *Mol. Phys.* **2015**, *113*, 184–215 Neese, F. [Software update: the ORCA program system, version 4.0]{}. *WIREs Comput. Mol. Sci.* **2017**, *8*, e1327 Gozem, S.; Gunina, A. O.; Ichino, T.; Osborn, D. L.; Stanton, J. F.; Krylov, A. I. [Photoelectron Wave Function in Photoionization: Plane Wave or Coulomb Wave?]{} **2015**, *6*, 4532–4540 Branton, G. R.; Frost, D. C.; Makita, T.; McDowell, C. A.; Stenhouse, I. A. [Photoelectron Spectra of Ethylene and Ethylene-d4]{}. *J. Chem. Phys.* **1970**, *52*, 802–806 Roos, B. O. [The Complete Active Space Self-Consistent Field Method and its Applications in Electronic Structure Calculations]{}. *Adv. Chem. Phys.* **1987**, *69*, 399–445 Bauschlicher Jr, C. W.; Langhoff, S. R. [Ab initio calculations on C$_{2}$, Si$_{2}$, and SiC]{}. *J. Chem. Phys.* **1987**, *87*, 2919 Watts, J. D.; Bartlett, R. J. [Coupled-cluster calculations on the C$_{2}$ molecule and the C$^{+}_{2}$ and C$^{-}_{2}$ molecular ions]{}. *J. Chem. Phys.* **1992**, *96*, 6073 Abrams, M. L.; Sherrill, C. D. [Full configuration interaction potential energy curves for the X$^{1}$$\Sigma$$_{g}^{+}$, B$^{1}$$\Delta$$_{g}$, and B$^{'1}$$\Sigma$$_{g}^{+}$ states of C$_{2}$: A challenge for approximate methods]{}. *J. Chem. Phys.* **2004**, *121*, 9211 Wouters, S.; Poelmans, W.; Ayers, P. W.; Van Neck, D. [CheMPS2: A free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry]{}. *Comput. Phys. Commun.* **2014**, *185*, 1501–1514 Rosmus, P.; Werner, H.-J.; Reinsch, E. A.; Larsson, M. [Multireference CI calculations of radiative transition probabilities between low lying quartet states of the C$_{2}^{+}$ ion]{}. *J. Electron. Spectrosc. Relat. Phenom.* **1986**, *41*, 289–296 Kraemer, W. P.; Roos, B. O. [A CAS SCF CI study of the $^{1}$$\Sigma$$^{+}_{g}$ and $^{3}$$\Pi$$_{u}$ states of the C$_{2}$ molecule and the $^{4}$$\Sigma$$^{-}_{g}$ and $^{2}$$\Pi$$_{u}$ states of the C$^{+}_{2}$ ion]{}. *Chem. Phys.* **1987**, *118*, 345–355 Watts, J. D.; Bartlett, R. J. [Coupled-cluster calculations on the C$_{2}$ molecule and the C$^{+}_{2}$ and C$^{-}_{2}$ molecular ions]{}. *J. Chem. Phys.* **1998**, *96*, 6073–6084 Petrongolo, C.; Bruna, P. J.; Peyerimhoff, S. D.; Buenker, R. J. [Theoretical prediction of the potential curves for the lowest-lying states of the C$_{2}^{+}$ molecular ion]{}. *J. Chem. Phys.* **1998**, *74*, 4594–4602 Ballance, C. P.; McLaughlin, B. M. [Valence configuration interaction study of the C$_{2}^{+}$ cation]{}. *J. Phys. B: At. Mol. Opt. Phys.* **2001**, *34*, 1201–1212 Shi, D.; Niu, X.; Sun, J.; Zhu, Z. [Potential Energy Curves, Spectroscopic Parameters, and SpinOrbit Coupling: A Theoretical Study on 24 $\Lambda$-S and 54 $\Omega$ States of C$^{+}_{2}$ Cation]{}. *J. Phys. Chem. A* **2013**, *117*, 2020–2034 Sinitskiy, A. V.; Greenman, L.; Mazziotti, D. A. [Strong correlation in hydrogen chains and lattices using the variational two-electron reduced density matrix method]{}. *J. Chem. Phys.* **2010**, *133*, 014104 Stella, L.; Attaccalite, C.; Sorella, S.; Rubio, A. [Strong electronic correlation in the hydrogen chain: A variational Monte Carlo study]{}. *Phys. Rev. B* **2011**, *84*, 245117 Lin, N.; Marianetti, C. A.; Millis, A. J.; Reichman, D. R. [Dynamical Mean-Field Theory for Quantum Chemistry]{}. *Phys. Rev. Lett.* **2011**, *106*, 096402 Tsuchimochi, T.; Scuseria, G. E. [Strong correlations via constrained-pairing mean-field theory]{}. *J. Chem. Phys.* **2009**, *131*, 121102 Rusakov, A. A.; Zgid, D. [Self-consistent second-order Greens function perturbation theory for periodic systems]{}. *J. Chem. Phys.* **2016**, *144*, 054106 Motta, M.; Ceperley, D. M.; Chan, G. K.-L.; Gomez, J. A.; Gull, E.; Guo, S.; Jim[é]{}nez-Hoyos, C. A.; Lan, T. N.; Li, J.; Ma, F.; Millis, A. J.; Prokofev, N. V.; Ray, U.; Scuseria, G. E.; Sorella, S.; Stoudenmire, E. M.; Sun, Q.; Tupitsyn, I. S.; White, S. R.; Zgid, D.; Zhang, S. [Towards the Solution of the Many-Electron Problem in Real Materials: Equation of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods]{}. *Phys. Rev. X* **2017**, *7*, 031059 Rusakov, A. A.; Iskakov, S.; Tran, L. N.; Zgid, D. [Self-Energy Embedding Theory (SEET) for Periodic Systems]{}. *J. Chem. Theory Comput.* **2019**, *15*, 229–240 Ronca, E.; Li, Z.; Jim[é]{}nez-Hoyos, C. A.; Chan, G. K.-L. [Time-Step Targeting Time-Dependent and Dynamical Density Matrix Renormalization Group Algorithms with ab Initio Hamiltonians]{}. *J. Chem. Theory Comput.* **2017**, *13*, 5560–5571 Hehre, W. J.; Stewart, R. F.; Pople, J. A. [Self-Consistent Molecular-Orbital Methods. I. Use of Gaussian Expansions of Slater-Type Atomic Orbitals]{}. *J. Chem. Phys.* **1969**, *51*, 2657–2664 Ohio Supercomputer Center. 1987; <http://osc.edu/ark:/19495/f5s1ph73>
|
---
abstract: 'We study the statistical limits of testing and estimation for a rank one deformation of a Gaussian random tensor. We compute the sharp thresholds for hypothesis testing and estimation by maximum likelihood and show that they are the same. Furthermore, we find that [the]{} maximum likelihood [estimator]{} achieves the maximal correlation with the planted vector [among measurable estimators]{} above the estimation threshold. In this setting, the maximum likelihood estimator exhibits a discontinuous BBP-type transition: below the critical threshold the estimator is orthogonal to the planted vector, but above the critical threshold, it achieves positive correlation which is uniformly bounded away from zero.'
address:
- 'Department of Mathematics, Harvard University, Department of Applied Math, University of Waterloo, Department of Statistics and Actuarial Sciences, University of Waterloo'
- 'Department of Mathematics, Harvard University'
- 'INRIA and Département d’Informatique de l’École Normale Supérieure, Paris'
author:
- Aukosh Jagannath
- Patrick Lopatto
- Léo Miolane
bibliography:
- 'TensorPCA.bib'
- 'references.bib'
title: Statistical thresholds for Tensor PCA
---
Introduction
============
Suppose that we are given an observation, $Y$, which is a $k$-tensor of rank $1$ in dimension $N$ subject to additive Gaussian noise. That is, $$\label{eq:spiked-tensor-def}
Y=\lambda\sqrt{N}X^{{\otimes}k}+W,$$ where $X\in\mathbb{S}^{N-1}$, the unit sphere in ${\mathbb{R}}^{N}$, $W$ is an i.i.d. Gaussian $k$-tensor with $W_{i_{1}\ldots i_{k}}\sim{\mathcal{N}}(0,1)$, and $\lambda\geq0$ is called the signal-to-noise ratio.[[^1]]{} Throughout this paper, we assume that $X$ is drawn from an uninformative prior, namely the uniform distribution on $\mathbb S^{N-1}$. We study the fundamental limits of two natural statistical tasks. The first task is that of hypothesis testing: for what range of $\lambda$ is it statistically possible to distinguish the law of $W$, the null hypothesis, from the law of $Y$, the alternative? The second task is one of estimation: for what range of $\lambda$ does the maximum likelihood estimator of X, $\widehat{x}^{\rm ML}_{\lambda}(Y)$, achieve asymptotically positive inner product with $X$?
When $k=2$, this amounts to hypothesis testing and estimation for the well-known spiked matrix model. Here, maximum likelihood estimation corresponds to computing the top eigenvector of $Y$. This problem was proposed as a natural statistical model of [principal]{} component analysis [@johnstone2001distribution]. It is a fundamental result of random matrix theory that there is a critical threshold below which the spectral theory of $Y$ and $W$ are asymptotically equivalent, but above which the maximum likelihood estimator achieves asymptotically positive inner product with $X$[—called the [correlation]{} of the estimator with $X$—]{}where the correlation increases continuously from $0$ to $1$ as $\lambda$ tends to infinity after $N$ [@edwards1976eigenvalue; @baik2005phase; @peche2006largest; @baik2006eigenvalues; @feral2007largest; @capitaine2009largest; @benaych2011eigenvalues]. This transition is called the BBP transition after the authors of [@baik2005phase] and has received a tremendous amount of attention in the random matrix theory community. Far richer information is also known, such as universality, large deviations, and fluctuation theorems. For a small sample of work in this direction, see [@maida2007; @BGM12; @bloemendal2013limits; @bloemendal2016principal]. More recently, it has been shown that the BBP transition is also the transition for hypothesis testing [@montanari2015limitation]. See also [@deshpande2016asymptotic; @barbier2016mutual; @perry2016optimality; @banks2016information; @lelarge2016fundamental; @lesieur2017constrained; @alaoui2017finite] for analyses of the testing and estimation problem with different prior distributions.
Our goal in this paper is to understand the case $k\geq3$, which is called the spiked tensor problem. This was introduced [@richard2014statistical] as a natural generalization of the above to testing and estimation problems where the data has more than two indices or requires higher moments, which occurs throughout data science [@li2010tensor; @duchenne2011tensor; @anandkumar2014tensor]. In this setting, it is known that there is an order 1 lower bound on the threshold for hypothesis testing which is asymptotically tight in $k$ [@richard2014statistical; @perry2016statistical] and an order 1 upper bound on the threshold for estimation via the maximum likelihood [@montanari2015limitation; @perry2016statistical]. On the other hand, if one imposes a more informative, product prior distribution, i.e., $X\sim\mu_{0}^{{\otimes}N}$ for some $\mu_{0}\in\Pr({\mathbb{R}})$, the threshold for minimal mean-square error estimati[on]{} (MMSE) has been computed exactly [@korada2009exact; @lesieur2017statistical; @barbier2017stochastic] as has the threshold for hypothesis testing under the additional assumption that $\mu_{0}$ is compactly supported [@perry2016statistical; @chen2017phase; @chen2018phase]. We note that by a standard approximation argument (see below), the results of [@lesieur2017statistical; @barbier2017stochastic] also imply a sharp threshold $\lambda_{c}$ for which the MMSE achieves non-trivial correlation for the uniform prior considered here.
The authors of [@BMMN17] and [@BBCR18] began a deep geometric approach to studying this problem by studying the geometry of the sub-level sets of the log-likelihood function. [In [@BMMN17], the authors compute the (normalized) logarithm of the expected number of local minima below a certain energy level via the Kac-Rice approach and show that there is a transition at a point $\lambda_s$ such that for $\lambda<\lambda_s$ this is negative for any strictly positive correlation, and for $\lambda>\lambda_s$ it has a zero with correlation bounded away from zero. [The work]{} [@BBCR18] study the (normalized) logarithm of the (random) number of local minima via a novel (but non rigorous) replica theoretic approach.]{} In particular, they predict that this problem exhibits a much more dramatic transition than the BBP transition. They argued that there are in fact two transitions for the log-likelihood, called $\lambda_{s}$ and $\lambda_{c}$. First, for $\lambda<\lambda_{s}$, all local maxima of the log-likelihood only achieve asymptotically vanishing correlation. For $\lambda_s<\lambda<\lambda_{c}$, there is a local maximum of the log-likelihood with non-trivial correlation but the maximum likelihood estimator still has vanishing correlation. Finally, for $\lambda_{c}<\lambda$ the maximum likelihood estimator has strictly positive correlation. In particular, if we let $m(\lambda)$, denote the limiting value of the correlation of the maximum likelihood estimator and $X$, they predict that $m(\lambda)$ has a jump discontinuity at $\lambda_{c}$. Finally, they predict that $\lambda_{c}$ should correspond to the hypothesis testing threshold. We verify several of these predictions.
We obtain here the sharp threshold for hypothesis testing and estimation by maximum likelihood and show that they are equal to $\lambda_c$. Furthermore, we compute the asymptotic correlation between $\widehat{x}^{\rm ML}_\lambda$ and $X$, $(\widehat{x}^{\rm ML}_\lambda,X)$, where $(\cdot,\cdot)$ denotes the Euclidean inner product. We find that the maximum likelihood estimator achieves the maximal correlation among measurable estimators, and that it is discontinuous at $\lambda_c$. This is in contrast to the matrix setting ($k=2)$, where this transition is continuous. As a consequence of these results, the threshold $\lambda_c$ is also the threshold for multiple hypothesis testing: the maximum likelihood is able to distinguish between all of the hypotheses $\lambda>\lambda_c$. [Finally, as a consequence of our arguments, we compute the maximum of the log-likelihood for fixed correlation $m$, call it $E_\lambda(m)$, and find that, for $\lambda_s<\lambda<\lambda_c$, $E_\lambda(m)$ has a local maximum at some $m_s>0$.]{}
These testing and estimation problems have received a tremendous amount of attention recently as they are expected to be an extreme example of statistical problems that admit a *statistical-to-algorithmic gap*: the thresholds for estimation and detection are both order $1$ in $N$; on the other hand, the thresholds for efficient testing and estimation are expected to diverge polynomially in $N$, $\lambda_{\mathrm{alg}} = O(N^\alpha)$. Indeed, this problem is known to be NP-hard for all $\lambda$ [@HiLi13]. Sharp algorithmic thresholds have been shown for semi-definite and spectral relaxations of the maximum likelihood problem [@hopkins2015tensor; @hopkins2016fast; @kim2017community] as well as optimization of the likelihood itself via Langevin dynamics [@BGJ18]. Upper bounds have also been obtained for message passing and power iteration [@richard2014statistical], as well as gradient descent [@BGJ18]. Our work complements these results by providing sharp statistical thresholds for maximum likelihood estimation and hypothesis testing.
Let us now discuss our main results and methods. We begin this paper by computing the sharp threshold for hypothesis testing. There have been two approaches to this in the literature to date. One is by a modified second moment method [@montanari2015limitation; @perry2016statistical], which yields sharp results in the limit that $k$ tends to infinity after $N$. The other approach, which we take here, is to control the fluctuations of the log-likelihood and yields sharp results for finite $k$. The key idea behind this approach is to prove a correspondence between the statistical threshold for hypothesis testing and a phase transition, called the “replica symmetry breaking” transition, in a corresponding statistical physics problem. For more on this connection see below.
Previous approaches to making this connection precise apply to the bounded i.i.d. prior setting [@korada2009exact; @perry2016statistical; @chen2017phase; @chen2018phase]. There one may apply a techincal, inductive argument of Talagrand [@talagrand2010meanfield1] related to the “cavity method” [@mezard1987spin] to control these fluctuations. This approach uses the boundedness and product assumption on $\mu_0$ in an essential way, neither of which hold in our setting (though we note here the work [@panchenko2009cavity] which applies for $\lambda$ sufficiently small). Our main technical contribution in this direction is a simpler, large deviations based approach which allows us to obtain the sharp threshold without using the cavity method. This argument applies with little modification to the product prior setting as well, though we do not investigate this here[. For more on this see .]{}
We then turn to computing the threshold for maximum likelihood estimation. We begin by directly computing the almost sure limit of the normalized maximum likelihood, which is an immediate consequence of the results of [@jagannath2017low; @chen2017parisi]. Combining this with of the results of [@lesieur2017statistical] (and a standard approximation argument), we then obtain a sharp estimate for the correlation between the MLE and $X$ for $\lambda>\lambda_c$ and find that it matches that of the Bayes-optimal estimator, confirming a prediction from [@gillin2000p]. The fact that the MLE has non-trivial correlation down to the information-theoretic threshold $\lambda_c$ is surprising in this setting as it is not expected to be true for all prior distributions. See, e.g., [@gillin2001multispin].
Main results
------------
Let us begin by stating our first result regarding hypothesis testing. Consider an observation $Y$ of a random tensor. Let $P_\lambda^N$ denote the law of . The null hypothesis is then $P_0^N$ and the alternative $P_\lambda^N$. Define for $t \in [0,1)$, $$\label{eq:f-lambda-def}
f_{\lambda}(t) = \lambda^2 t^k + \log(1-t) + t$$ and let $$\label{eq:lambda_c-def}
\lambda_c = \sup \Big\{ \lambda \geq 0 \, \Big| \, \sup_{t \in [0,1)} f_{\lambda}(t) \leq 0 \Big\}.$$ Our goal is to show that $P_\lambda^N$ and $P_0^N$ are mutually contiguous when $\lambda<\lambda_c$ and that for $\lambda>\lambda_c$ there is a sequence of tests $T_N$ which asymptotically distinguish these distributions. More precisely, we obtain the following stronger result regarding the total variation distance between these hypotheses which we state in the case $k$ even for simplicity.
\[thm:main-thm\] For $k\geq6$ even, $$\lim_{N\to\infty}d_{TV}(P_{0}^{N},P_{\lambda}^{N})
=\begin{cases} 0 & \text{if} \quad \lambda <\lambda_c\\
1 & \text{if} \quad \lambda >\lambda_c.
\end{cases}$$
The preceding result shows us that the transition for hypothesis testing occurs at $\lambda_c$. Let us now turn to the corresponding results regarding maximum likelihood estimation.
It is straightforward to show that maximizing the log-likelihood is equivalent to
maximizing $( Y,x^{{\otimes}k})$ over the sphere, $x \in \mathbb S^{N-1}$. The maximum likelihood estimator (MLE) ${\widehat{x}}^{\rm ML}_{\lambda}$ is then defined as
$${\widehat{x}}^{\rm ML}_{\lambda} = \operatorname{arg\,max}_{x\in\mathbb S^{N-1}} ( Y,x^{{\otimes}k}).$$
Our second result is that the preceding transition is also the transition for which maximum likelihood estimation yields an estimator which achieves positive correlation with $X$.
Let $q_*(\lambda)$ be defined by $$\label{eq:q-star-def}
q_*(\lambda) = \begin{cases}
0 & \text{if} \quad \lambda <\lambda_c\\
\operatorname{arg\,max}_{t\in[0,1)} f_\lambda(t) & \text{if} \quad \lambda>\lambda_c.
\end{cases}$$ As shown in Lemma \[lem:gauss\_scalar\], the function $f_{\lambda}$ admits a unique positive maximizer on $[0,1)$ when $\lambda > \lambda_c$, so that this is well-defined. Let $z_k$ denote the unique zero on $(0,+\infty)$ of $$\label{eq:def_varphi}
\varphi_k(z) = \frac{1+z}{z^2} \log(1+z) - \frac{1}{z} - \frac{1}{k}.$$ Finally, let $$\label{eq:def_GS}
{\rm GS}_k = \frac{\sqrt k}{\sqrt{1+z_k}} \left(1+\frac{z_k}{k}\right).$$
We then have the following.
\[thm:max\_likelihood\] Let $\lambda \geq 0$ and $k\geq 3$. The following limit holds almost surely $$\label{eq:lim_max_likelihood}
\lim_{N \to \infty}
\frac{1}{\sqrt N} \max_{x \in \mathbb S^{N-1}} \big( x^{\otimes k}, Y \big)
\ = \
\begin{cases}
{\rm GS}_k & \text{if} \ \ \lambda \leq \lambda_c \\
\displaystyle\sqrt{k}\frac{1 + \lambda^2 q_*(\lambda)^{k-1}}{\sqrt{1 + \lambda^2 k q_*(\lambda)^{k-1}}} & \text{if} \ \ \lambda > \lambda_c.
\end{cases}$$ Furthermore, we have that [for $\lambda\neq \lambda_c$]{} $$\label{eq:correlation}
\lim_{N\to\infty} \Big| \big( {\widehat{x}}_{\lambda}^{\rm ML}, X \big) \Big|
= \sqrt{q_*(\lambda)}.$$
As a consequence of , the maximum likelihood estimator [achieves maximal correlation]{}. Unlike the case $k=2$, the transition in $q_*(\lambda)$ is not continuous. See Figure \[fig:1\].
![ Asymptotic correlation $\lim\limits_{N \to \infty} \frac{1}{N} | \langle {\widehat{x}}^{\rm ML}_{\lambda}, X \rangle | = \sqrt{q_*(\lambda)}$ as a function of the signal-to-noise ratio $\lambda$, for different values of $k$. []{data-label="fig:1"}](./q_star-crop.pdf){width=".5\textwidth"}
Let us pause for a moment to comment on the importance of the assumption that the prior is a uniform spherical prior, as opposed to, e.g., a bounded i.i.d. prior as in earlier works.
\[rem:LDP\] The argument for the main hypothesis testing result does not use in an essential way the spherical nature of the prior. The key idea is to show that a certain rate function has a locally quadratic lower bound near its zero, implying a CLT-type upper bound on fluctuations of the corresponding random variable. This estimate extends to rather broad families of problems which satisfy the “positive replicon eigenvalue” condition. See [@arous2018spectral] for more on this condition and how it implies such bounds. However, the sphericity assumption is used in an essential way in the estimation section. The point here is that the variational problem that arises in the spherical setting has a very explicit form. As a result, one can obtain exact expressions for the transition as well as the optimizers. Using these exact expressions, we can verify that the maximum likelihood estimator is information-theoretically optimal, which plays a vital role in our argument. One can obtain a corresponding variational representation for other priors, but the variational problem is in general substantially less transparent.
Regarding the second threshold {#regarding-the-second-threshold .unnumbered}
------------------------------
While the regime $\lambda_s<\lambda<\lambda_c$ and the expected transition at $\lambda_s$ [are]{} not relevant for testing and estimation, [it has]{} a natural interpretation from the perspective of the landscape of the maximum likelihood. In [@BMMN17; @BBCR18], this is explained in terms of the complexity. There is also an explanation in terms of the optimization of the maximum likelihood. We end this section with a brief discussion of this phase. Let $\lambda_s$ be given by $$\label{eq:lambda_s-def}
\lambda_s = \sqrt{\frac{(k-1)^{k-1}}{k (k-2)^{k-2}}}.$$ Consider the constrained maximum likelihood, $$\label{eq:def_E}
E_{\lambda}(m) = \lim_{N \to \infty} \frac{1}{N} \max_{x \in \mathbb{S}^{N-1}, \, (X,x) = m} \Big\{ \lambda N (X,x)^k + \sqrt N (W,x^{{\otimes}k}) \Big\}.$$ This limit exists and is given by an explicit variational problem (see below). For $\lambda>\lambda_s$, let ${\sqrt{q_s(\lambda)}}$ be the (unique) positive, strict local maximum of $f_\lambda$. By , this is well-defined and satisfies $q_s(\lambda)=q_*(\lambda)$ for $\lambda>\lambda_c$. In [@BBCR18], it is argued by the replica method that $E_\lambda(m)$ has a local maximum at $\sqrt {q_s(\lambda)}$ for all $\lambda>\lambda_s$. Establishing this rigorously is a key step in our proof of . In particular, we prove the following.
For $\lambda>\lambda_s$, the function $E_\lambda$ has a strict local maximum at ${\sqrt{q_s(\lambda)}}$. [ It is a global maximum of $E_{\lambda}$ if and only if $\lambda \geq \lambda_c$. For $\lambda \leq \lambda_c$ the global minimum of $E_{\lambda}$ is achieved at $m=0$. ]{}
[The proof of this result immediate[ly]{} follows by combining below.]{}
It is easy to verify (by direct differentiation) that the map $\lambda \mapsto E_\lambda({\sqrt{q_s(\lambda)}})$ is strictly increasing on $(\lambda_s, +\infty)$. We have also that $E_{\lambda_c}(\sqrt{q_s(\lambda_c)}) = {\rm GS}_k$ by and , so we get that for $\lambda_s < \lambda < \lambda_c$ the strict local maximum at $\sqrt{q_s(\lambda)}$ has $E_{\lambda}(\sqrt{q_s(\lambda)})$ strictly less than the maximum likelihood. In fact, can be solved numerically, as it can be shown that one may reduce this variational problem, in the setting we consider here, to a two-parameter family of problems in three real variables. This is discussed in below. In particular, see Figure \[fig:2\] for an illustration of these two transitions in the case $k=4$.
![ Asymptotic constrained maximum likelihood $E_\lambda(m)$ for $p=4$ with $\lambda= 1, 1.299,1.35,1.405,1.5$. Here $\lambda_s\approx 1.299$ and $\lambda_c\approx 1.405$. For $\lambda<\lambda_s$, the function is (numerically) seen to be monotone. A secondary maximum occurs at the transition $\lambda=\lambda_s$. This local maximum is bounded away from $m=0$. Finally, at the information theoretic threshold $\lambda_c$, the maximum likelihood is now maximized at this second point.[]{data-label="fig:2"}](./Em-lambda-varied.pdf){width=".5\textwidth"}
Let us now compare this with the complexity based approach in [@BMMN17; @BBCR18]. In [@BMMN17], the authors computed the expected number of local maxima of a fixed likelihood and correlation (called the annealed 0-complexity) on the exponential in $N$ scale. They showed that for $\lambda<\lambda_s$ this expectation scales like $O(e^{-cN})$ for correlations $q>0$, and $O(\exp(cN))$ for a suitable range of likelihoods when $q=0$, whereas for $\lambda_s<\lambda$, the show that at $q_s$, the logarithm of this quantity is $o(N)$. Furthermore, they find that once $\lambda>\lambda_c$, this exponent is maximized at a value of likelihood that is larger than the value for correlation $q=0$. While this argument does not show that this behavior is typical, it was argued in [@BBCR18], via a novel (but non-rigorous) replica method, that the same result holds for the log-number of local maxima (called the quenched $0$-complexity) with high probability. In contrast, in this paper we bypass the analysis of critical points and instead obtain a similar picture by directly computing the almost sure limit of the constrained maximum likelihood, .
Acknolwedgements {#acknolwedgements .unnumbered}
----------------
A.J. would like to thank G. Ben Arous for encouraging the preparation of this paper. A.J. and L.M. would like to thank the organizers of the BIRS workshop “Spin Glasses and Related Topics" where part of this research was conducted. This work was conducted while A.J. was supported by NSF OISE-1604232 and [P.L. was partially supported by the NSF Graduate Research Fellowship Program under grant DGE-1144152.]{}
Proof of and connection to spin glasses {#sec:connection-to-sg}
=======================================
In this section, we prove . In particular, we connect the phase transition for the hypothesis testing problem to a phase transition in a class of models from statistical physics, which is proved in the remaining sections.
Let us begin by explaining this connection. First note that the null hypothesis is a centered Gaussian distribution on the space of $k$-tensors in ${\mathbb{R}}^N$, whereas the alternative corresponds to one with a random mean $\lambda \sqrt N X^{{\otimes}k}$. Thus by Gaussian change of density, the likelihood ratio, $dP_\lambda/dP_0$, satisfies $$L(Y) = \frac{dP_\lambda}{dP_0}(Y) = \int\exp\left( \lambda \sqrt N( Y,x^{{\otimes}k}) - N\frac{\lambda^2}{2}\right) dx,$$ where $dx$ denotes the uniform measure on $\mathbb S^{N-1}$. Observe that the total variation distance satisfies $$\label{eq:total-identity}
d_{TV}(P_\lambda,P_0) ={\mathbb{E}}_{P_0}\left((1-L(Y)){\mathbbm{1}_{\{L(Y)\leq 1\}}}\right) = \int_0^1 P_0(L(Y) \leq s)ds.$$ We will show that this probability tends to zero almost everywhere when $\lambda < \lambda_c$.
Let us now make the following change of notation, motivated by statistical physics. [ For $x\in \mathbb S^{N-1}$ and ${\lambda}\geq0$, define $$\label{eq:ham-def}
H(x)=\sqrt{N}(W,x^{{\otimes}k}), \quad Z({\lambda})=\int\exp(-{\lambda} H(x))\, dx.$$]{} We view $H$ as a function on $\mathbb S^{N-1}$, which is called the *Hamiltonian* of the *spherical ${k}$-spin glass model* in the statistical physics literature [@crisanti1992sphericalp]. The log-likelihood ratio then has an interpretation in terms of what is called a “free energy" in the statistical physics literature. More precisely, define the *free energy at temperature ${\lambda}$* for the spherical ${k}$-spin model by $$F_N({\lambda}) =\frac{1}{N}\log Z_N({\lambda})$$ and observe that under the null hypothesis, $$\frac{1}{N}\log L(Y) = F_N(\lambda) - \frac{\lambda^2}{2}.$$ The key conceptual step in our proof is to connect the phase transition for hypothesis testing to what is called the “replica symmetry breaking" transition in statistical physics. While it is not within the scope of this paper to provide a complete description of this transition, we note that one expects this transition to be reflected in the limiting properties of $F_N$: if $\lambda$ is small $F_N$ should fluctuate around $\lambda^2/2$, but for large $\lambda$ it should be much smaller than $\lambda^2/2$. A sharp transition is expected to occur at $\lambda_c$. For an in-depth discussion of replica symmetry breaking transitions see [@mezard1987spin]. In the remainder of this section we reduce the proof of our main result to the proof that the phase transition for the fluctuations of $F_N$ does in fact occur at $\lambda_c$. We then prove this phase transition exists in the next two sections.
Let us turn to this reduction. By and the equivalence noted above, $$\label{eq:tv-to-fe}
d_{TV}(P_{0},P_{\lambda})=\int_{0}^{1}P_0\left(F_N(\lambda)-\frac{\lambda^{2}}{2}<\frac{\log(x)}{N}\right)dx.$$ We have the following theorem of Talagrand, which we state in a weak form for the sake of exposition. [Here and in the following, unless otherwise specified $\mathbb P$ and $\mathbb E$ will always denote integration with respect to the law of the Gaussian random tensor $W$.]{}
\[thm:parisi2\] For every ${\lambda}>0$, ${\mathbb{E}}F_N({\lambda})$ is a convergent sequence. Furthermore, $$\lim_{N\to\infty} {\mathbb{E}}F_{N} \leq \frac{{\lambda}^{2}}{2},$$ with equality if and only if ${\lambda}\leq\lambda_c$.
With this in hand, it suffices to show the following.
\[thm:FE-decay\] For $k\geq 4$ even, ${\epsilon}>0$ and ${\lambda}<\lambda_{c}$, there is a [constant]{} $C>0$ such that for every $N\geq 1$ and $x>0$, $${\mathbb{P}}\left( \left| F_N({\lambda})-\frac{{\lambda}^{2}}{2} \right| > \frac{x}{N}\right)\leq C\frac{1}{x^2N^{{\frac{k-4}{4}} -{\epsilon}}}.$$
The proof of this theorem will constitute the next two sections. Let us begin by making the following elementary observations, which will reduce the theorem to certain fluctuation theorems. To this end, observe that by Chebyshev’s inequality, $$\label{eq:chebyshev}
{\mathbb{P}}\left(\left|F_N({\lambda})-\frac{{\lambda}^{2}}{2}\right|>\eta/N\right)\leq\frac{N^{2}}{\eta^{2}}\left({\operatorname{Var}}(F_{N})+\left({\mathbb{E}}F_{N}-\frac{{\lambda}^{2}}{2}\right)^{2}\right).$$ The key point in the following will be to quantify the rate of convergence in Talagrand’s theorem when $\lambda<\lambda_c$. This rate of convergence will also allow us to control the variance of $F_{N}$. More precisely, in the subsequent sections we will prove the following two theorems.
\[thm:conv-means\] [Fix $k\geq 4 $ even]{} and ${\lambda}<\lambda_{c}$. For any $\epsilon>0$ there is a $C({\lambda},\epsilon)>0$ such that for $N\geq 1$ $$\left| {\mathbb{E}}F_{N}-\frac{{\lambda}^{2}}{2}\right|\leq CN^{-{k/4}+\epsilon}.$$
\[thm:variance-bound\] [Fix $k\geq 4$ even.]{} For ${\lambda}<\lambda_c$ and ${\epsilon}>0$, for $N\geq 1$ $${\operatorname{Var}}(F_N)\leq\frac{C({\lambda})}{N^{{k/4}+1-{\epsilon}}}.$$
The desired result then follows upon combining and with .
We can now prove the main theorem.
Suppose first that $\lambda <\lambda_c$. Then by combined with , the total variation distance vanishes for $k\geq 6$ even.
Suppose now that $\lambda> \lambda_c$. Note $${\mathbb{P}}( F_N \leq {\mathbb{E}}F_N -\epsilon) \leq \exp (-N\epsilon^2/2)$$ by Gaussian concentration (see, e.g., [@boucheron2013concentration Theorem 5.6]). By Jensen’s inequality ${\mathbb{E}}F_N(\lambda) \leq \lambda^2/2$, and by Talagrand’s theorem, we know that $${\mathbb{E}}F_N\to F\neq {\lambda}^2/2,$$ so for some ${\epsilon}>0$, [ $$\lim_{N\to\infty}\mathbb{P}(F_N(\lambda)-\frac{\lambda^2}{2}<-\epsilon)=1.$$ ]{} The desired result then follows by using this to lower bound the right side of .
Rate of convergence of the mean and Decay of variance.
======================================================
In this section, we prove . In the following we will make frequent use of the measure $$\pi_{{\lambda}}(dx)\propto\exp(-{\lambda} H(x))\, dx$$ where $H$ is as in and $0<{\lambda}$. We call this the *Gibbs measure*, which we normalize to be a probability measure. Observe that this normalization constant is given exactly by $\log Z({\lambda})$. Here and in the remaining sections, we will let $\left\langle \cdot\right\rangle $ denote expectation with respect to the (random) measure $\pi_{{\lambda}}$. We will suppress the dependence on ${\lambda}$ whenever it is unambiguous as it will always be fixed. Throughout this section, $\lambda$ will always be fixed and less than $\lambda_{c}$. It will also be useful to define the quantity $$F_{2,N}(u,\eta;{{\lambda}})=\frac{1}{N}\log\int\int_{{\lvert{\lvert(x,y)\rvert}-u\rvert}<\eta}\exp(-{\lambda} H(x)-{\lambda} H(y))\, dx\, dy,$$ where $(x,y)$ denotes the Euclidean inner product. Evidently this is related to the large deviations rate function for the event $(x,y) \approx u$. To simplify notation, for $X^1,X^2\sim \pi_{{\lambda}}$, we let $$R_{12} = (X^1,X^2).$$
The starting point for our analysis is the estimate of the rate of convergence of ${\mathbb{E}}F_N$ to ${\lambda}^2/2$.
In the following, let $$\psi({\lambda})=\frac{{\lambda}^{2}}{2}-{\mathbb{E}}F_{N}.$$ By Jensen’s inequality, $\psi({\lambda})\geq0$.
Let us now turn to an upper bound. Recall $H$ from . Observe that $H$ is centered and has covariance $${\mathbb{E}}H(x^1)H(x^2) = N(x^1,x^2)^k.$$ It then follows that $$\frac{d}{d{\lambda}} \psi({\lambda}) = {\lambda} -\frac{1}{N} {\mathbb{E}}{\left\langleH\right\rangle} = {{\lambda}}{\mathbb{E}}{\left\langleR_{12}^k\right\rangle},$$ where the first equality is by definition of the Gibbs measure and the second follows by Gaussian integration by parts for Gibbs expectations, . We now claim that $$\label{eq:gronwall-step}
\frac{d}{d{\lambda}}\psi({\lambda})={{\lambda}}{\mathbb{E}}\left\langle R_{12}^{k}\right\rangle \leq C\psi({\lambda})+\frac{C}{N^{{k/4-k\delta/2}}}.$$ [for some constant $C>0$]{} and $\delta>0$ sufficiently small. With this claim in hand, we may apply Gronwall’s inequality and the lower bound from above to obtain $$\begin{aligned}
0 & \leq\psi({\lambda})\leq\left(\psi(0)+\frac{1}{N^{k/2- k\delta}}\right)\exp\left(C{\lambda}\right)=\frac{C({\lambda})}{N^{{k/2-k\delta/2}}},\end{aligned}$$ as desired. Let us now turn to the proof of this claim.
Observe that the maps $W\mapsto F_{2,N}(u,\frac{1}{N};{\lambda})$ and $W\mapsto F_{N}({\lambda})$ are uniformly ${{\lambda}}/\sqrt{N}$-Lipschitz, so that Gaussian concentration of measure [(see for instance [@boucheron2013concentration], Theorem 5.6)]{} implies that there are constants $C,c {>0}$ [depending only on $\lambda$ and $k$]{} such that for any $\delta \in (0,1/2)$, with probability at least $1-C\exp(-cN^{2\delta}),$ $$\begin{aligned}
F_{2,N}(u,\frac{1}{N};{\lambda})-{\mathbb{E}}F_{2,N}(u,\frac{1}{N};{\lambda}) \leq \frac{1}{N^{1/2-\delta}},\qquad
F_{N}({\lambda})-{\mathbb{E}}F_{N}({\lambda}) \leq\frac{1}{N^{1/2-\delta}}.\end{aligned}$$ Thus, on this event, call it $A(u,\delta)$, $$\label{eq:ldp-vs-conv}
\begin{aligned}
\frac{1}{N}\log\pi^{{\otimes}2}\left({\lvertR_{12}\rvert}\in(u-\frac{1}{N},u+\frac{1}{N})\right) & = F_{2,N}(u,\frac{1}{N},{\lambda})-2F_{N}({\lambda}).\\
& \leq2\left(\frac{{\lambda}^{2}}{2}-{\mathbb{E}}F_{N}({\lambda})\right)+\left({\mathbb{E}}F_{2,N}(u,\frac{1}{N},{{\lambda}})-{\lambda}^{2}\right)+\frac{C}{N^{1/2-\delta}}.
\end{aligned}$$ As we shall show in Corollary \[cor:parisi\], for every $\delta>0$ there is some $c>0$ such that for $N\geq 1$ and for all $N^{-1/2+\delta}\leq u\leq1$[;]{} $${\mathbb{E}}F_{2,N}(u,\frac{1}{N},{{\lambda}})-{\lambda}^{2}\leq-cu^{2}.$$ Let $$v=\frac{1}{c}\left(2\psi({\lambda})+\frac{C}{N^{1/2-\delta}}+\frac{1}{\sqrt N}\right).$$ Then for $u^2\geq v$, on $A(u,\delta)$, $$\pi^{{\otimes}2}(|R_{12}|\in(u-\frac{1}{N},u+\frac{1}{N}))\leq\exp(-\sqrt{N}).\label{eq:decay-sqrt-n-ovlp}$$ Consequently, if we take $\{u_i\}_{i=1}^L$ to be the centers of a partition of the interval $[v,1]$ into intervals of size [$2/N$ ]{}, then if we let $A(\delta)=\cap_i A(u_i,\delta)$, $$\begin{aligned}
{\mathbb{E}}\left\langle R_{12}^k\right\rangle & \leq{\mathbb{E}}\left(\left\langle R_{12}^k{\mathbbm{1}_{R_{12}^{2}>v}}\right\rangle {\mathbbm{1}_{A(\delta)}}\right)+{\mathbb{E}}\left(\left\langle R_{12}^k{\mathbbm{1}_{R_{12}^2\leq v}}\right\rangle {\mathbbm{1}_{A(\delta)}}\right)+Ne^{-cN^{{2}\delta}}.\\
& \leq N\exp(-\sqrt{N})+v^{{k/2}}+Ne^{-cN^{{2}\delta}},\end{aligned}$$ where we use that $L\leq N$. From this it follows, by the inequality $(x+y)^k \leq 2^{k-1}(x^k+y^k)$, that $${\mathbb{E}}\left\langle R_{12}^k\right\rangle \leq C\left(\psi({\lambda})^k+\frac{1}{N^{{k/4-k\delta/2}}}\right)\label{eq:fe-conv-ovlp-gronwall}$$ for some $C>0$, $\delta$ small enough and $N\geq 1$. The claim then follows since [$\psi(\lambda) \leq \lambda_c^2$ for all $\lambda \leq \lambda_c$]{}. For this last claim, observe that ${\mathbb{E}}F_N(\lambda)$ is convex in $\lambda$ with ${\mathbb{E}}F_N(0)= 0$ and right derivative $\frac{d}{d\lambda}{\mathbb{E}}F_N(0^+)=0$, so that ${\mathbb{E}}F_N\geq0$. As a result, $\psi(\lambda)\leq \lambda^2\leq \lambda_c^2$.
Notice that by the above argument, we also have the following.
\[cor:overlap\] For [any $k\geq 4$ even and]{} ${\lambda}<\lambda_{c}$ and $\eta>0$, there is a $C({\lambda},\eta)>0$ such that for $N$ sufficiently large, $${\mathbb{E}}\left\langle {\lvertR_{12}\rvert}^k\right\rangle \leq\frac{C({\lambda}{,\eta})}{N^{{k/4 -k\eta/2}}}.$$
By , we have, [for $N$ large enough]{}, $$\psi({\lambda}) \leq \frac{C({\lambda}{,\eta})}{N^{{k/4-k\eta/2}}}.$$ Combining this with yields the desired [inequality]{}.
We are now in a position to prove the variance decay.
By the Gaussian Poincaré inequality [(see for instance [@boucheron2013concentration Theorem 3.20])]{}, $$\begin{aligned}
{\operatorname{Var}}(F_{N}({\lambda})) & \leq\frac{1}{N^{2}}{\mathbb{E}}\sum_{1 \leq i_{1},\ldots,i_{k} \leq N}\left(\partial_{W_{i_{1}\ldots i_{k}}}\log Z({\lambda})\right)^{2}
=\frac{{\lambda}^{2}}{N}{\mathbb{E}}\sum_{1 \leq i_{1},\ldots,i_{k} \leq N}\left\langle x_{i_{1}}\cdots x_{i_{k}}\right\rangle ^{2}
=\frac{{\lambda}^{2}}{N}{\mathbb{E}}\left\langle R_{12}^k\right\rangle \end{aligned}$$ The result then follows by combining this with Corollary \[cor:overlap\].
The Parisi functional and large deviations
==========================================
The main technical tool we need is a bound on the following expected value, which is related to large deviations of ${\lvertR_{12}\rvert}$ from its mean: $${\mathbb{E}}\frac{1}{N}\log\pi^{{\otimes}2}({\lvertR_{12}\rvert}\in(u-\eta,u+\eta))={\mathbb{E}}F_{2,N}(u,\eta)-2{\mathbb{E}}F_N.$$ We relate the quantities ${\mathbb{E}}F_{2,N}(u,\eta)$ and ${\mathbb{E}}F_{N}$ to explicit *Parisi-type* formulas. In the following, let $\xi(t)=\lambda^{2}t^k$ and $\theta(t) = t \xi'(t) - \xi(t)$. For $u, \Lambda \in [ 0 ,1]$ and $m\in [1,2]$, define $$\begin{aligned}
\mathcal P (u, m, \Lambda) &= \xi(1) + (1-m)\theta(u) - \Lambda u + \frac{1}{m} \log \frac{1 + \xi'(u) - \Lambda}{1 + (1-m) \xi'(u) - \Lambda} \\& - \frac{1}{2}\left( \log( 1 + \xi'(u) + \Lambda) + \log ( 1 + \xi'(u) -\Lambda) \right).\end{aligned}$$ Then we have the following from [@TalSphPF06]. See also [@PanchTal07; @ko2018free] for alternative presentations.
For [$k\geq 2$ even]{}, there exists a constant $C( k)>0$ such that for every $N\geq1$, $\eta>0$, $m \in [1,2]$, $\Lambda \in [0,1]$, and $0<u<1$, we have $$\begin{aligned}
{\mathbb{E}}F_{2,N}(u,\eta;\lambda) & \leq \mathcal P (u,m, \Lambda )+\mathcal{R}\label{eq:bound-1}\end{aligned}$$ where $ | \mathcal{R} | \le C\eta+\frac{C\log N}{N}$.
We first observe that by symmetry of $H(x)$, it suffices to prove the same estimate for $$\tilde F_{2,N}(u,\eta) = \frac{1}{N}\log\pi^{{\otimes}2}( R_{12} \in (u-\eta,u+\eta)).$$ We apply [@PanchTal07 Eq. 2.22] with the choice of parameters $$\begin{aligned}
Q^{0} =Q^{1}=Q^2=0,\quad
Q^{3} =\left(\begin{array}{cc}
u & u\\
u & u
\end{array}\right),\quad
Q^{4} =\left(\begin{array}{cc}
1 & u\\
u & 1
\end{array}\right),\end{aligned}$$ $$\mathbf{m}=(0,1/2,m/2,1), \quad A_3 = \left(\begin{array}{cc}
1 + \xi'(u) & -\Lambda\\
-\Lambda & 1 + \xi'(u)
\end{array}\right),$$ to obtain $${\mathbb{E}}{\tilde{F}}_{2,N}(u,\eta) \leq {\mathcal{P}}(u,m,\Lambda) + {\mathcal{R}},$$ where the error term $\mathcal{R}$ in [@PanchTal07 Eq. 2.22] satisfies $$\mathcal R = {C \eta} - 2 \left ( \frac{1 }{N}\log \mathbb P \left(\sum_{i=1}^N X_i^2 \geq N\right) + \frac{b - 1 - \log b}{2} \right),$$ where $b=(1 + \xi'(u))$, the $X_i$ are i.i.d. gaussian random variables with variance $1/b$, as given in , and $C>0$ is universal. Using the elementary bound of Lemma \[lem:chi\_lb\], it follows that Modifying $C$ appropriately yields the desired.
\[lem:parisi-1\] For [$k\geq 4$ even,]{} $\lambda<\lambda_c$ and $\epsilon>0$, there are constants $C,c>0$ such that for every $N\geq1$, and $c>u\geq N^{-1/2+\epsilon}$, we have $${\mathbb{E}}F_{2,N}\left(u,\frac{1}{N}{;\lambda}\right)\leq\lambda^{2}-Cu^{2}.$$
Observe that $\mathcal{P}(u,1, \Lambda)$ is $C^{2}$ in $(u,\Lambda)$ and $(0,0)$ is a critical point with Hessian $$\operatorname{Hess}({\mathcal{P}})(0,0) =
\left(\begin{array}{cc}
0 & -1 \\
-1 & {1}
\end{array}\right).$$
This has an eigenvector of the form $(1,x)$ for some $x>0$ with strictly negative eigenvalue $-\mu<0$. It follows that for $(u,\Lambda)=(u,ux)$ we have $$\mathcal P (u,1,ux)\leq {\mathcal{P}}(0,1, 0)-K u^{2}=\lambda^{2}-K u^{2}$$ for $u\leq c$ for some $K,c>0$ independent of $N$. Combining this with , we obtain $${\mathbb{E}}F_{2,N}(u,\eta{;\lambda}) \leq \lambda^2 - K u^2 +C\left(\eta+\frac{\log N}{N}\right).$$ If we choose $\eta = 1/N$ and decrease $K$, the result follows since $u^2 \geq N^{-1+{\epsilon}}> \log N/N$ for ${\epsilon}>0$ and $N\geq 1$.
\[lem:parisi-2\] [For $\lambda<\lambda_{c}$ and $\epsilon>0$, there are $K,C>0$]{} such that for every $u>\epsilon$, and [$N\geq 1$]{} $${\mathbb{E}}F_{2,N}( u ,\frac{1}{N})\leq\lambda^2-K + C\cdot\frac{\log N}{N}.$$
[Notice that ]{} $$\begin{aligned}
\frac{d}{dm}\bigg|_{m=1} {\mathcal{P}}(s ,m,0) &= -(s \xi'(s ) - \xi (s ))+\xi'(s )-\log(1+\xi'(s ))
{= \phi_{\lambda}(s),}\end{aligned}$$ where $\phi_{\lambda}$ is defined by . By Lemma \[lem:gauss\_scalar\] we have $\phi_{\lambda}(s) < 0$ for all $\lambda < \lambda_c$ and $s \in (0,1]$.
Note that ${\mathcal{P}}(u,1,0)=\lambda^2$. Thus for every $0<u \leq 1$, ${\mathcal{P}}(u,m ,{0})< \lambda^2$ for some $m> 1$. [Observe that $\Phi(u) = \inf_m {\mathcal{P}}(u,m,0)$ is upper-semicontinuous. Thus for any ${\epsilon}>0$, there exists $K({\epsilon}){>0}$ such that for all $u\in [{\epsilon},1]$, $$\Phi(u) <\lambda^2 -K({\epsilon}).$$ In particular, for such $u$,]{} it follows that $${\mathbb{E}}F_{2,N}(u,{\frac{1}{N}})\leq \lambda^2 - K(\epsilon) + C\left( \frac{\log N}{N}\right)$$ for $u > \epsilon$, which implies the desired result.
Combining these two results, we obtain the following.
\[cor:parisi\] For $\lambda<\lambda_c$ and ${\epsilon}>0$ sufficiently small, there is a $c>0$ such that for $N\geq 1$, $${\mathbb{E}}F_{2,N}(u,\frac{1}{N};\lambda)\leq \lambda^2 - c u^2,$$ [for all $N^{-1/2+{\epsilon}}<u\leq 1$.]{}
[Fix $\lambda$ and ${\epsilon}>0$.]{} By , there is some $c_1,c_2>0$ such that for [all $N\geq 1$ and ]{} $N^{-1/2+\epsilon}< u<c_1$, $${\mathbb{E}}F_{2,N}(u,\eta)\leq \lambda^2 - c_2 u^2$$ Now for $c_1<u<1$, let $K(c_1)$ be as in . Then $K(c_1)u^2<K(c_1)$, so that, if we take $c = \min\{c_2, K(c_1)\}$ the result follows.
Estimation
==========
In this section, we prove Theorem \[thm:max\_likelihood\]. We begin by providing a lower bound for the maximum likelihood for every $\lambda\geq 0$ using results on the ground state of the mixed $p$-spin model recently proved in [@jagannath2017low; @chen2017parisi]. We then use the information-theoretic bound on the maximal correlation achievable by any estimator from [@lesieur2017statistical] to obtain the matching upper bound. We end by proving the desired result for the correlation $({\widehat{x}}^{\rm ML}_{\lambda},X)$. In the remainder of this paper, for ease of notation, we let $$\label{eq:def_Hl}
H_{\lambda}(x) = H(x) + \lambda N (x,X)^k,$$ where $H(x)$ is as in .
Variational formula for the ground state of the mixed $p$-spin model
--------------------------------------------------------------------
We begin by recalling the following variational formula for the ground state of the mixed $p$-spin model. Consider the Gaussian process indexed by $x \in \mathbb{S}^{N-1}$: $$Y_N(x) = {\sqrt{N}} \sum_{p \geq 1} a_p \sum_{1 \leq i_1, \dots, i_p \leq N} g_{i_1, \dots, i_p} x_{i_1} \dots x_{i_p},$$ where $g_{i_1,\ldots,i_p}$ are i.i.d. standard Gaussian random variables and $\sum_{p \geq 1} 2^p a_p^2 < \infty$. The covariance of $Y_N$ is given by $${\mathbb{E}}\big[Y_N(x) Y_N(y)\big] = {N}\xi((x,y)),$$ where $\xi(t) = \sum_{p \geq 1} a_p^{{2}} t^p.$ Let ${\mathcal{C}}$ denote the subset of $C([0,1])$ of functions that are positive, non-increasing and concave. For any $h \geq 0$, we let $P_{h}:{\mathcal{C}}\to {\mathbb{R}}$ be $$P_h(\phi) = \int \xi''(x)\phi(x) + \frac{1}{\phi(x)}dx + (h^2+\xi'(0)) \phi(0).$$ Set $$\mathcal{G}(\xi,h)=\frac{1}{2} \min_{\phi\in {\mathcal{C}}} P_h(\phi).$$ Let us recall the following variational formula. For $x \in \mathbb S^{N-1}$, we write $x = (x_1, \dots , x_N)$.
\[thm:GS\] For all $h \geq 0$, $${\lim_{N\to\infty}\frac{1}{N}\max_{x \in \mathbb{S}^{N-1}} \Big\{ Y_N(x) + h \sqrt N \sum_{i=1}^N x_i \Big\}}
= \mathcal{G}(\xi,h),$$ almost surely and in $L^1$.
While the results of [@chen2017parisi; @jagannath2017low] are stated with $\xi'(0)=0$, they still hold when $\xi'(0)>0$ by replacing $\xi\mapsto\xi(t)-\xi'(0) t$ and $h^2\mapsto h^2+\xi'(0)$ . To see this, simply note that the Crisanti-Sommers formula still holds in this setting by the main result of [@chen2013aizenman]. The reformulation from [@jagannath2017low Eq. (1.0.1)] is then changed by this replacement by simply repeating the integration by parts argument from [@jagannath2017low Lemma 6.1.1]. From here the arguments are unchanged under the above replacement.
The lower bound
---------------
By Borell’s inequality, the constrained maximum likelihood concentrates around its mean with sub-Gaussian tails. In particular, combining this with Borell-Cantelli we see that $$\label{eq:def_E2}
E_{\lambda}(m) = \lim_{N \to \infty} \frac{1}{N} {\mathbb{E}}\Big[ \max_{x \in \mathbb{S}^{N-1}, \, (x,X) = m} \Big\{ \lambda N (x,X)^k + H(x) \Big\} \Big].$$ Clearly, ${\varliminf}\frac{1}{N} {\mathbb{E}}\big[\max_{\mathbb{S}^{N-1}} H_{\lambda}\big] \geq E_{\lambda}(m)$ for all $m \in [-1,1]$. [Recall the definition of $\lambda_s$ from and $q_s(\lambda)$, see, e.g., Lemma \[lem:gauss\_scalar\].]{} If we apply this for $\lambda > \lambda_s$ and $m = \sqrt{q_s(\lambda)} > \sqrt{1-\frac{1}{k-1}}$ (by Lemma \[lem:gauss\_scalar\]), Lemma \[lem:up\_qs\] below will immediately yield the following lower bound.
\[lem:lower\_bound\] For all $\lambda > \lambda_s$, $$\label{eq:lower_bound}
{\varliminf}_{N \to \infty} {\mathbb{E}}\Big[\frac{1}{N}\max_{x \in \mathbb{S}^{N-1}} H_{\lambda}(x) \Big] \geq \sqrt{k}\frac{1 + \lambda^2 q_s(\lambda)^{k-1}}{\sqrt{1 + \lambda^2 k q_s(\lambda)^{k-1}}}.$$
We now turn to the proof of . We begin by observing the following explicit representation for $E_\lambda$.
For all $m \in [-1,1]$ the limit in exists and $$\label{eq:E-rs}
E_{\lambda}(m)
={\lambda m^k }+ {\mathcal{G}}(\xi_m,0),$$ where $\xi_m(t) = (m^2 + (1-m^2)t )^k - m^{2k}$.
We begin by observing that by rotational invariance, $$\frac{1}{N} {\mathbb{E}}\Big[ \max_{x \in \mathbb{S}^{N-1}, \, (x,X) = m} \Big\{ \lambda N (x,X)^k + H(x) \Big\}\Big]
= \lambda m^k + \frac{1}{N} {\mathbb{E}}\Big[ \max_{x \in \mathbb{S}^{N-1}, \, x_1 = m} H(x) \Big].$$
Let $x \in \mathbb{S}^{N-1}$ such that $x_1 = m$. Then $$\begin{aligned}
H(x) {\stackrel{(d)}{=}}\sqrt N m^k g_{1, \dots, 1} + \sqrt N \sum_{j=0}^{k-1} \binom{k}{j}^{1/2} m^j \sum_{2 \leq i_1, \dots, i_{k-j} \leq N} g_{i_1, \dots, i_{k-j}} x_{i_1} \dots x_{i_{k-j}},\end{aligned}$$ where $\big( (g_{i_1, \dots, i_p})_{1 \leq i_1, \dots, i_p \leq N} \big)_{p \leq k}$ are i.i.d.\
standard Gaussians.
So that ${\mathbb{E}}\big[ \max_{x \in \mathbb{S}^{N-1}, \, x_1 = m} H(x) \big] = {\mathbb{E}}\big[ \max_{x \in \mathbb{S}^{N-2}} H_m(x)\big]$, where $H_m$ is given by: $$H_m(x)
=\sqrt N \sum_{j=0}^{k-1} \binom{k}{j}^{1/2} m^j (1-m^2)^{(k-j)/2}\sum_{1 \leq i_1, \dots, i_{k-j} \leq N-1} g_{i_1, \dots, i_{k-j}} x_{i_1} \dots x_{i_{k-j}}.$$ The function $H_m$ is a Gaussian process with covariance $${\mathbb{E}}\big[ H_m(x)H_m(y)\big] = N \xi_m( (x,y)),$$ where $\xi_m$ is given by $$\label{eq:xi-m}
\xi_m(t)
= \sum_{j=0}^{k-1}\binom{k}{j} m^{2j} (1-m^2)^{k-j} t^{k-j}
=
(m^2 + (1-m^2)t )^k - m^{2k}.$$ We conclude using Theorem \[thm:GS\] to obtain the result.
We now observe that for [$m$]{} large enough, this formula has a particularly simple form.
For all $|m| \geq \sqrt{1 - \frac{1}{k-1}}$ we have: $$\label{eq:E}
E_{\lambda}(m) = \lambda m^k + \sqrt{k(1-m^2)}.$$
In the setting of Theorem \[thm:GS\] it was also shown in [@jagannath2017low; @chen2017parisi] that if $\xi'(1) + h^2 \geq \xi''(1)$ then $\mathcal{G}(\xi,h) = \sqrt{\xi'(1) + h^2}$. Since $$\begin{aligned}
\xi_m'(t) &= k (1-m^2)(m^2 + (1-m^2)t )^{k-1}
\\
\xi_m''(t) &= k(k-1) (1-m^2)^2(m^2 + (1-m^2)t )^{k-2},\end{aligned}$$ the condition $\xi_m'(1) \geq \xi_m''(1)$ corresponds to $(k-1) (1-m^2) \leq 1$, i.e. $|m| \geq \sqrt{1 - \frac{1}{k-1}}$. When this holds, we get that $$E_{\lambda}(m) = \lambda m^k + {\mathcal{G}}(\xi_m,0)=\sqrt{\xi_m'(1)} = \lambda m^k + \sqrt{k (1-m^2)}$$ by .
We end with the desired explicit formula for $E_\lambda(\sqrt q_s(\lambda))$.
\[lem:up\_qs\] For all $\lambda > \lambda_s$, $\sqrt{q_s(\lambda)}$ is a local maximizer of $E_{\lambda}$ and if we write $x(\lambda) = \lambda^2 k q_s^{k-1}(\lambda)$, $$E_{\lambda}\big(\sqrt{q_s(\lambda)}\big) = \frac{\sqrt{k}}{\sqrt{1 + x(\lambda)}} \Big(1 + \frac{x(\lambda)}{k}\Big).$$
Differentiating the expression for $m \geq \sqrt{1 - \frac{1}{k-1}}$ yields $$\begin{aligned}
E_{\lambda}'(m)
&= \lambda k m^{k-1} - \frac{\sqrt{k}m}{\sqrt{1-m^2}}
=
k \frac{1}{1-m^2}
\Big(\lambda k m^{k-1} + \frac{\sqrt{k}m}{\sqrt{1-m^2}} \Big)^{-1}
\Big(\lambda^2 k m^{2k-2} - \lambda^2 k m^{2k} - m^2 \Big)
\end{aligned}$$ so that the functions $\phi_{\lambda}$, $f_{\lambda}$ and $m^2 \mapsto E_{\lambda}(m)$ have precisely the same monoticity on $[1 - \frac{1}{k-1},1)$ (recall the expression of the derivatives $f_{\lambda}'$ and $\phi_{\lambda}'$ given by ). Lemma \[lem:gauss\_scalar\] gives that $q_s(\lambda)$ is a local maximum of $f_{\lambda}$ and $\phi_{\lambda}$ for $\lambda > \lambda_s$, $\sqrt{q_s(\lambda)}$ is therefore a local maximum of $E_{\lambda}$.
Let us now compute $E_{\lambda}(\sqrt{q_s(\lambda)})$. By Lemma \[lem:gauss\_scalar\], $q_s(\lambda) = \frac{x(\lambda)}{1+x(\lambda)}$. Consequently, $$\begin{aligned}
E_{\lambda}(q_s(\lambda)^{1/2})
&= \lambda q_s(\lambda)^{k/2} + \sqrt{k(1 - q_s(\lambda))}
= \frac{\sqrt{k}}{\sqrt{1 + x(\lambda)}} \Big(1 + \frac{x(\lambda)}{k}\Big).\qedhere
\end{aligned}$$
The upper bound
---------------
We prove here the upper bound.
\[lem:upper\_bound\] For all $\lambda \geq 0$, $$\label{eq:upper_bound}
{\varlimsup}_{N \to \infty} {\mathbb{E}}\Big[\frac{1}{N}\max_{x \in \mathbb{S}^{N-1}} H_{\lambda}(x) \Big] \leq {\rm GS}_k + \int_0^\lambda q_*(t)^{k/2}\, dt.$$
We defer the proof of this momentarily to observe the following information-theoretic bounds which will be useful in its proof.
\[prop:IT\] Assume that $X$ is uniformly distributed over ${\mathbb S}^{N-1}$, independently from $W$. Then for all $\lambda \in (0, +\infty) \setminus \{\lambda_c \}$ $$\lim_{N \to \infty}
{\mathbb{E}}\Big[
\Big\| X^{\otimes k} - {\mathbb{E}}\big[ X^{\otimes k} \big| Y \big] \Big\|^2
\Big]
=
1 - q_*(\lambda)^k.$$
This result follows from [@lesieur2017statistical; @barbier2017stochastic] by approximating the uniform measure on $\mathbb S^N$ by an i.i.d. Gaussian measure. For the completeness, we provide a proof in . As a consequence of this, we have the following.
\[cor:upper\_IT\] Assume that $X$ is uniformly distributed over ${\mathbb S}^{N-1}$, independently from $W$. Then for all measurable functions ${\widehat{x}}: ({\mathbb{R}}^N)^{\otimes k} \to \mathbb{S}^{N-1}$ and for all $\lambda \neq \lambda_c$ we have $${\varlimsup}_{N \to \infty} {\mathbb{E}}\Big[ \big( {\widehat{x}}(Y), X \big)^k \Big] \leq q_*(\lambda)^{k/2}.$$
Compute $$\begin{aligned}
{\mathbb{E}}\Big[\Big\| X^{\otimes k} - \big(\sqrt{q_*(\lambda)} {\widehat{x}}(Y)\big)^{\otimes k} \Big\|^2 \Big]
&=
{\mathbb{E}}\big[\big\| X^{\otimes k}\big\|^2 \big]
+
q_*(\lambda)^{k} {\mathbb{E}}\big[\big\| {\widehat{x}}(Y)^{\otimes k}\big\|^2 \big]
- 2 q_*(\lambda)^{k/2} {\mathbb{E}}\Big[ \big( {\widehat{x}}(Y), X \big)^k \Big]
\\
&= 1 + q_*(\lambda)^{k}
- 2 q_*(\lambda)^{k/2}{\mathbb{E}}\Big[ \big( {\widehat{x}}(Y), X \big)^k \Big].
\end{aligned}$$ [Recall that the posterior mean, ${\mathbb{E}}( X^{{\otimes}k}\vert Y)$, uniquely achieves the minimal mean squared error over all square-integrable tensor-valued estimators, ${\widehat{T}}(Y)$, for $X^{{\otimes}k}$.]{} The proposition follows then from Proposition \[prop:IT\] which gives $${\varliminf}_{N \to \infty}
{\mathbb{E}}\Big[\Big\| X^{\otimes k} - \big(\sqrt{q_*(\lambda)} {\widehat{x}}(Y)\big)^{\otimes k} \Big\|^2 \Big]
\geq {\varliminf}_{N \to \infty}
{\mathbb{E}}\Big[
\Big\| X^{\otimes k} - {\mathbb{E}}\big[ X^{\otimes k} \big| Y \big] \Big\|^2
\Big]
= 1 - q_*(\lambda)^k.\qedhere$$
With this in hand we may now prove .
By Proposition \[prop:unique\_ML\] and an application of an envelope-type theorem (see, e.g., Proposition \[prop:envelope\_compact\]), [the map]{} $\lambda \mapsto \frac{1}{N}{\mathbb{E}}\big[\max_{\mathbb{S}^{N-1}} H_{\lambda}(x) \big]$ [is differentiable for $\lambda\geq 0$]{}, with derivative $$\label{eq:M_N-diff}
\frac{\partial}{\partial \lambda} {\mathbb{E}}\Big[\frac{1}{N} \max_{x \in \mathbb{S}^{N-1}} H_{\lambda}(x)\Big]
={\mathbb{E}}\Big[
{({\widehat{x}}_{\lambda}^{\rm ML}, X )^k
\Big].}$$ By [@jagannath2017low; @chen2017parisi] we know that $\frac{1}{N} {\mathbb{E}}[\max_{\mathbb{S}^{N-1}} H_0\big] \rightarrow {\rm GS}_k$. The reverse Fatou lemma gives then $$\begin{aligned}
{\varlimsup}_{N \to \infty} {\mathbb{E}}\Big[\frac{1}{N} \max_{x \in \mathbb{S}^{N-1}} H_{\lambda}(x)\Big]
\leq \int_0^{\lambda} {\varlimsup}_{N \to \infty}{\mathbb{E}}\Big[
\big( {\widehat{x}}_{\gamma}^{\rm ML}, X \big)^k
\Big]\, d\gamma + {\rm GS}_k \leq \int_0^{\lambda} q_*(\gamma)^{k/2}\, d\gamma + {\rm GS}_k,\end{aligned}$$ where the second inequality follows from .
Proof of first part of Theorem \[thm:max\_likelihood\] {#sec:proof_th_max_likelihood}
-------------------------------------------------------
By an elementary but tedious calculation (see ) the right sides of and are equal for $\lambda\geq \lambda_c$ (recall that $q_*(\lambda)=q_s(\lambda)$ for such $\lambda$). Thus for all $\lambda > \lambda_c$, $$\label{eq:cv_up}
{\mathbb{E}}\Big[\frac{1}{N}\max_{x \in \mathbb{S}^{N-1}} H_{\lambda}(x) \Big] \xrightarrow[N \to \infty]{} \sqrt{k}\frac{1 + \lambda^2 q_*(\lambda)^{k-1}}{\sqrt{1 + \lambda^2 k q_*(\lambda)^{k-1}}}.$$
We will now prove that for $\lambda \leq \lambda_c$, $\widebar{M}_N(\lambda) \xrightarrow[N \to \infty]{} {\rm GS}_k$, where $\widebar{M}_N(\lambda)$ is defined by $$\widebar{M}_N (\lambda) = {\mathbb{E}}\left[\frac{1}{N} \max_{x \in \mathbb{S}^{N-1}} H_{\lambda}(x)\right].$$ Notice that $\widebar{M}_N(\lambda)$ is convex as an expectation of a maximum of linear functions. By , it follows that $\widebar{M}_N'(0^+)\geq 0$. (When $k$ is odd, we use rotational invariance to see that it is in fact zero.) Consequently, $\widebar{M}_N$ is non-decreasing on $[0,+\infty)$.
By [@jagannath2017low; @chen2017parisi] (see ), $\lim_{N \to \infty} \widebar{M}_N(0) = {\rm GS}_k$. By and Lemma \[lem:key\_identities\], $$\lim_{\lambda \to \lambda_c^+} \lim_{N \to \infty} \widebar{M}_N(\lambda) = {\rm GS}_k.$$ Consequently, we obtain that for all $\lambda \in [0,\lambda_c]$, $\lim_{N \to \infty} \widebar{M}_N(\lambda) = {\rm GS}_k$.
The almost sure convergence of follows then from the convergence of the expectation $\widebar{M}_N(\lambda)$, combined with Borell’s inequality for suprema of Gaussian processes (see for instance [@boucheron2013concentration Theorem 5.8]) and the Borel-Cantelli Lemma.
\[rem:rigorous\] By , $E_\lambda$ is given by a variational problem over the space ${\mathcal{C}}$. We first observe that one can easily solve this variational problem numerically due to the following simple reductions. First note that if we let $\xi_m$ be as in , then $(1/\sqrt{\xi_m''})''$ is strictly positive, where the prime denotes differentiation in $t$. Thus by [@jagannath2017low Theorem 1.2.4], the minimizer $\phi$ must be of the form $\phi(s)=\int_s^1 d\nu_s$ where $\nu_s = \theta_1 \delta_a + \theta_2 \delta_b$, where $a,b\in [0,1]$ and $\theta_i\geq 0$. Thus the variational problem is a variational problem over 4 parameters which can be solved numerically. These observations then rigorously justify the starting point of the discussion in [@BBCR18 Section 4], namely the “RS" and “1RSB" calculation in [@BBCR18 Sect. 4.B] in the regime they analyize, called the “$T=0$" regime there. We refer the reader there for a more in-depth discussion, see [@BBCR18 Sect. 4.C].
Proof of second part of Theorem \[thm:max\_likelihood\] {#sec:proof_cor_max_likelihood}
-------------------------------------------------------
We now turn to the second part of , namely . Let $M_N(\lambda)$ denote $$M_N(\lambda) = \frac{1}{N} \max_{x \in \mathbb{S}^{N-1}} H_{\lambda}(x).$$ Fix $\lambda > 0$. By and Lemma \[lem:key\_identities\] $$\lim_{N\to\infty}M_N(\lambda) = \ell(\lambda) =
\begin{cases}
{\rm GS}_k & \text{if} \quad \lambda \leq \lambda_c \\
{\rm GS}_k + \int_{0}^{\lambda} q_*(\gamma)^{k/2} \, d\gamma & \text{if} \quad \lambda > \lambda_c.
\end{cases}$$ Let $\lambda \in (0, +\infty) \setminus \{ \lambda_c \}$. By Proposition \[prop:unique\_ML\] and the Milgrom-Segal envelope theorem (see ), $M_N$ is differentiable in $\lambda$ with derivative $$M_N'(\lambda) = \big({\widehat{x}}^{\rm ML}_{\lambda}, X \big)^k,$$ almost surely. As $M_N$ is convex in $\lambda$ (it is a maximum of linear functions), we see that for any $0<h<\lambda$, $$\frac{M_N(\lambda - h) - M_N(\lambda)}{h} \leq M_N'(\lambda) \leq \frac{M_N(\lambda + h) - M_N(\lambda)}{h}.$$ By taking the $N \to \infty$ limit, we get that almost surely $$\label{eq:limsup_as}
\frac{\ell(\lambda - h) - \ell(\lambda)}{h} \leq
{\varliminf}_{N \to \infty} ( {\widehat{x}}^{\rm ML}_{\lambda}, X )^k
\leq
{\varlimsup}_{N \to \infty} ( {\widehat{x}}^{\rm ML}_{\lambda}, X )^k
\leq \frac{\ell(\lambda + h) - \ell(\lambda)}{h}.$$ Since $\ell$ is differentiable for $\lambda \neq \lambda_c$, we may take $h\to0$ to obtain $
\lim_{N \to \infty}
( {\widehat{x}}^{\rm ML}_{\lambda}, X )^k = q_*(\lambda)^{k/2}
$ almost surely, which proves .
Appendix {#sec:appendix}
========
Uniqueness of minimizers and envelope theorems {#app:elementary}
----------------------------------------------
This section gathers some basic lemmas that will be useful for the analysis.
\[prop:unique\_ML\] Recall the definition of $H_{\lambda}$. We have the following
- If $k$ is odd, then $H_{\lambda}$ has almost surely one unique maximizer over $\mathbb{S}^{N-1}$.
- If $k$ is even, then $H_{\lambda}$ has almost surely two maximizers over $\mathbb{S}^{N-1}$, $x^*$ and $-x^*$.
We note the following basic fact from the theory of Gaussian processes, see, e.g. [@kim1990cube].
\[lem:unique-min-gaussian\] Let $(Z(t))_{t \in T}$ be a Gaussian process indexed by a compact metric space $T$ such that $t \mapsto Z(t)$ is continuous almost surely. If the intrinsic quasi-metric, $d(s,t)^2 = {\operatorname{Var}}\big(Z(s) - Z(t)\big)$, is a metric, i.e., $d(s,t) \neq 0$ for $s\neq t$, then $Z$ admits a unique maximizer on $T$ almost surely.
Observe $H_{\lambda}$ is continuous on the compact $\mathbb{S}^{N-1}$. For $x_1,x_2 \in \mathbb{S}^{N-1}$, we have $${\operatorname{Var}}\big(H_{\lambda}(x^1) - H_{\lambda}(x^2)\big)
= 2 N \big(1 - ( x^1, x^2 )^k \big).$$ If $k$ is odd, then the proposition follows directly from the Lemma. If $k$ is even, we apply the Lemma on the quotient space $\mathbb{S}^{N-1} / \sim$ where $\sim$ denotes the equivalence relation defined by $x^1 \sim x^2 \Leftrightarrow \big(x^1 = x^2 \ \text{or} \ x^1 = - x^2\big)$.
We recall the following envelope theorem of Milgrom and Segal [@milgrom2002envelope]. Let $X$ be a set of parameters and consider a function $f: X \times [0,1] \to {\mathbb{R}}$. Define, for $t \in [0,1]$ $$\begin{aligned}
V(t) &= \sup_{x \in X} f(x,t) \,,\\
X^*(t) &= \big\{ x \in X \, \big| \, f(x,t) = V(t) \big\} \,.
\end{aligned}$$
\[prop:envelope\_compact\] Suppose that $X$ is nonempty and compact. Suppose that for all $t\in [0,1]$, $f(\cdot,t)$ is continuous. Suppose also that $f$ admits a partial derivative $f_t$ with respect to $t$ that is continuous in $(x,t)$ over $X \times [0,1]$. Then
- $\displaystyle V'(t^+) = \max_{x^* \in X^*(t)} f_t(x^*,t)$ for all $t\in [0,1)$ and $\displaystyle V'(t^-) = \min_{x^* \in X^*(t)} f_t(x^*,t)$ for all $t\in (0,1]$.
- $V$ is differentiable at $t \in (0,1)$ is and only if $\displaystyle \Big\{ f_t(x^*,t) \, \Big| \, x^* \in X^*(t) \Big\}$ is a singleton. In that case $V'(t) = f_t(x^*,t)$ for all $x^* \in X^*(t)$.
Study of the asymptotic equations {#app:asymptotic}
---------------------------------
Define, for all $q \in [0,1]$ $$\label{eq:def_phi}
\phi_{\lambda}(q) = \lambda^2 k q^{k-1} - \log(1 + \lambda^2 k q^{k-1}) - \lambda^2 (k-1) q^k.$$
\[lem:gauss\_scalar\] We have for all $\lambda > 0$, $$\max_{q \in [0,1)} f_{\lambda}(q) = \max_{q \in [0,1]} \phi_{\lambda}(q)$$ Furthermore, if we let $\lambda_s = \sqrt{\frac{(k-1)^{k-1}}{k (k-2)^{k-2}}}$:
- For $\lambda < \lambda_s$, then the functions $f_{\lambda}$ and $\phi_{\lambda}$ are decreasing on $[0,1)$.
- For $\lambda > \lambda_s$, the functions $f_{\lambda}$ and $\phi_{\lambda}$ have a strict local minimum at $q_u(\lambda)$ and a strict local maximum at $q_s(\lambda)$ where $0<q_u< \frac{k-2}{k-1}<q_s<1$, [and both functions are]{} strictly monotone on the intervals $(0,q_u)$, $(q_u,q_s)$ and $(q_s,1)$. Moreover, $q_s(\lambda)$ [is strictly increasing in $\lambda$]{} and satisfies: $$\label{eq:q_SE}
q_s(\lambda) = \frac{\lambda^2 k q_s(\lambda)^{k-1}}{1 + \lambda^2 k q_s(\lambda)^{k-1}} \,.$$
Finally, for $\lambda > \lambda_c$, $q_*(\lambda) = q_s(\lambda)$ is the unique maximizer of $f_{\lambda}$ and $\phi_{\lambda}$ over $[0,1)$.
We have for $q \in [0,1)$ $$\begin{aligned}
\label{eq:der_phi0}
\phi_{\lambda}'(q) =
\frac{ k(k-1) \lambda^2 q^{k-1}}{1 + \lambda^2 k q^{k-1}}
h(q)
\qquad \text{and} \qquad
f_{\lambda}'(q) = \frac{h(q)}{1-q}
\end{aligned}$$ where $h(q) = \lambda^2 k q^{k-1} - \lambda^2 k q^{k} - q$. It suffices therefore to study the variations of $f_{\lambda}$. Notice also that $$\phi_{\lambda}(q) = f_{\lambda}(q) + h(q) - \log(1 + h(q)).$$ [Since $f_\lambda'(q) = 0$ implies $h(q)=0$,]{} this implies that $$\max_{q \in [0,1]} \phi_{\lambda}(q) = \max_{q \in [0,1)} f_{\lambda}(q)$$ and that these maxima are achieved at the same points. Let us now study the sign of the polynomial $h(q)$: $$\label{eq:si}
h(q) = qk \lambda^2 \big(q^{k-2} - q^{k-1} - \frac{1}{k \lambda^2}\big).$$ One verifies easily that the polynomial $q^{k-2} - q^{k-1}$ achives its maximum at $\frac{k-2}{k-1}$ and that the value of this maximum is $\frac{(k-2)^{k-2}}{(k-1)^{k-1}}$. We get that for $\lambda < \lambda_s$, $h'(q) < 0$ for all $q>0$. For $\lambda > \lambda_s$ we get that $h$ admits exactly 3 zeros on ${\mathbb{R}}$: $0< q_u( \lambda ) < q_s(\lambda)<1$. Since the maximum of $h$ is achieved at $\frac{k-2}{k-1}$ we get that $q_u(\lambda) < \frac{k-2}{k-1} < q_s(\lambda)$. [ $q_u<q_s$ are the positive roots of $q^{k-2} - q^{k-1} = \frac{1}{k \lambda^2}$: $q_u$ is therefore strictly decreasing and $q_s$ is strictly increasing in $\lambda$. ]{} This proves the two points of the lemma; simply follows from the fact that $h(q_s(\lambda)) = 0$. The last statement of Lemma \[lem:gauss\_scalar\] is then an immediate consequence of the definition of $\lambda_c$.
Recall that $z_k$ is defined as the unique zero of $\varphi_k(z) = \frac{1+z}{z^2} \log(1+z) - \frac{1}{z} - \frac{1}{k}$ on $(0,+\infty)$.
\[lem:x\_k\] The mapping $\lambda \mapsto q_s(\lambda)$ is ${\mathcal{C}}^{\infty}$ on $(\lambda_s, + \infty)$. Moreover $\lambda^2 k q_s(\lambda_c)^{k-1} = z_k$.
The first part follows from a straightforward application of the implicit function theorem.
We get in particular that the mapping $\lambda \mapsto q_s(\lambda)$ is continuous for $\lambda > \lambda_s $. So by definition of $\lambda_c$ and Lemma \[lem:gauss\_scalar\], $\phi_{\lambda_c}(q_s(\lambda_c)) = 0$. Let us write $x = \lambda^2 k q_s(\lambda_c)^{k-1}$. $$\begin{aligned}
0 = \phi_{\lambda_c}(q_s(\lambda_c)) =
x - \log(1+ x) - \frac{k-1}{k} x q_s(\lambda_c)
=
x - \log(1+ x) - \frac{k-1}{k} \frac{x^2}{1+x} \,,
\end{aligned}$$ because $q_s(\lambda_c) = \frac{x}{1+x}$ (see ). This gives that $\varphi_k(x) = 0$ and thus $x=z_k$.
\[lem:key\_identities\] Let $\lambda > \lambda_c$ and write $x(\lambda) = \lambda^2 k q_s(\lambda)^{k-1}$. Then we have $$\frac{\sqrt{k}}{\sqrt{1+x(\lambda)}} \Big(1 + \frac{x(\lambda)}{k}\Big)
= {\rm GS}_k + \int_{\lambda_c}^{\lambda} q_s(\gamma)^{k/2} d\gamma
\,.$$
Let us write $g(\lambda) = \frac{\sqrt{k}}{\sqrt{1+x(\lambda)}} \big(1 + \frac{x(\lambda)}{k}\big)$. By Lemma \[lem:up\_qs\], $\sqrt{q_s(\lambda)}$ is a local maximizer of $E_{\lambda}$ and thus a critical point of $E_{\lambda}$. This gives $$g'(\lambda) = \partial_{\lambda} \Big[E_{\lambda}(\sqrt{q_s(\lambda)})\Big]
= \partial_{\lambda} E_{\lambda}(\sqrt{q_s(\lambda)}) = q_s(\lambda)^{k/2}.$$ The lemma follows then from the fact that $x(\lambda_c) = z_k$ by Lemma \[lem:x\_k\] and the definition of ${\rm GS}_k$.
Proof of {#app:proof-prop-IT}
---------
For $P_0$ a probability distribution over $({\mathbb{R}}^N)^{\otimes k}$ with finite second moment, we define the free energy $$F_{P_0}(\gamma) =
\frac{1}{N} {\mathbb{E}}\log \int P_0(dx) \exp\Big(
\sqrt{\gamma N}( x , W )
+ {\gamma}N ( x, X_0 )
- \frac{1}{2} {\gamma} N \| x\|^{2}
\Big)$$ where $X_0 \sim P_0$ and ${W}_{i_1,\dots,i_k} \sim {\mathcal{N}}(0,1)$ are independent. Proposition A.1 from [@miolane2018phase] states that for two probability distributions $P_1$, $P_2$ on $({\mathbb{R}}^N)^{\otimes k}$ with finite second moment, we have $$\big| F_{P_1}(\gamma) - F_{P_2}(\gamma) \big|
\leq \frac{\gamma}{2} \Big(\sqrt{{\mathbb{E}}_{P_1} \|X_1\|^2} + \sqrt{{\mathbb{E}}_{P_2} \| X_2 \|^2}\Big) W_2(P_1,P_2),$$ where $W_2(P_1,P_2)$ denotes the Wasserstein distance of order $2$ between $P_1$ and $P_2$. Let $\mu_N$ be the distribution of $X^{\otimes k}$ when $X \sim {\rm Unif}(\mathbb{S}^{N-1})$ and let $\nu_N$ be the distribution of $X^{\otimes k}$ when $X \sim {{\mathcal{N}}(0,\frac{1}{N}Id_N)}$. Let us compute a bound on $W_2(\mu_N, \nu_N)$. Let [$X $ be drawn uniformly over $\mathbb S^{N-1}$]{}, and $G \sim {{\mathcal{N}}(0,Id)}$, independently from $X$. Then [$(X^{{\otimes}k}, (\|G\| X/\sqrt{N})^{{\otimes}k} )$]{} is a coupling of $\mu_N$ and $\nu_N$, so that, by definition $W_2$, $$\begin{aligned}
W_2(\mu_N, \nu_N)^2 \leq
{\mathbb{E}}\big\| X^{\otimes k} - \big( X \|G\| /\sqrt{N} \big)^{\otimes k} \big\|^2
= {\mathbb{E}}\Big[\Big(\Big( \frac{1}{N} \sum_{i=1}^N G_i^2\Big)^{k/2} - 1 \Big)^2 \Big]\end{aligned}$$ [where we use that ${\mathbb{E}}{\lvert\lvertX\rvert\rvert}^k=1$. By the law of large numbers, it then follows that $$\lim_{N\to\infty} {\lvertF_{\mu_N}(\gamma)-F_{\nu_N}(\gamma)\rvert}\to0.$$]{} Recall the definition of $\phi_{\lambda}(q)$ and define $L(\gamma) = \frac{1}{2}\max_{q \in [0,1]} \phi_{{\sqrt{\gamma}}}(q) = \frac{1}{2} \max_{q \in [0,1)} f_{{\sqrt{\gamma}}}(q)$, where the equality comes from Lemma \[lem:gauss\_scalar\]. Now, [@lesieur2017statistical Theorem 1] gives that for all $\lambda \geq 0$, $F_{\nu_N}(\gamma) \rightarrow L(\gamma)$ as $N\rightarrow \infty$, which implies $F_{\mu_N}(\gamma) \rightarrow L(\gamma)$.
The “I-MMSE Theorem” from [@guo2005mutual] (see [@miolane2018phase Proposition 1.4] for a statement [of this result]{} closer to the notations used here) gives that $\gamma \mapsto F_{\mu_N}(\gamma)$ is convex and differentiable over $[0, + \infty)$ and $$F_{\mu_N}'(\lambda^2) = \frac{1}{2} \left(
1 -
{\mathbb{E}}\Big[
\big\| X^{\otimes k} - {\mathbb{E}}\big[ X^{\otimes k} \big| Y \big] \big\|^2
\Big]
\right).$$ [By Griffith’s lemma for convex functions, $F_{\mu_N}'$ converges to [$L'$ ]{} for each $\lambda > 0$ at which $L$ is differentiable.]{} For [$\gamma < \lambda_c^2$]{}, $L(\gamma) = 0$, so $L$ is differentiable on [$(0,\lambda_c^2)$]{} with derivative equal to $0$. For [$\gamma > \lambda_c^2$]{}, we know by Lemma \[lem:gauss\_scalar\] that $f_{{\sqrt{\gamma}}}$ admits a unique maximizer $q_*({\sqrt{\gamma}})$ on $[0,1]$. Proposition \[prop:envelope\_compact\] gives that $L$ is differentiable at $\gamma$ with derivative $$\begin{aligned}
L'(\gamma)
&= \frac{1}{2} (\partial_{\gamma} f_{\sqrt{\gamma}}) (q_*(\sqrt \gamma))
= \frac{1}{2} q_*(\sqrt \gamma)^k.\end{aligned}$$ We conclude that $$\lim_{N \to \infty} \frac{1}{2} \left(
1 -
{\mathbb{E}}\Big[
\big\| X^{\otimes k} - {\mathbb{E}}\big[ X^{\otimes k} \big| Y \big] \big\|^2
\Big]
\right)
=
\lim_{N \to \infty}
F_{\mu_N}'(\lambda^2) =
\begin{cases}
0 & \text{if} \ \ \lambda < \lambda_c \\
\frac{1}{2} q_*(\lambda)^k & \text{if} \ \ \lambda > \lambda_c.
\end{cases}\qed$$
Elementary lemmas
-----------------
We collect here the following elementary lemmas which are used in the above.
\[lem:chi\_lb\] Let $\{X_i\}$ be standard normal random variables, and let $S_N = \frac{1}{bN} \sum_{i=1}^N {X^2_i}.$ [There is a $C>0$ such that for $N\geq 1$ and $b> 1$ (possibly varying in $N$)]{}, $$\label{eq:chi-square-lower-bound}
\frac{1}{N } \log {\mathbb{P}}( S_N \ge 1 ) \ge - \frac{1}{2}(b-1-\log b) - \frac{2 \log b}{N} - C \cdot\frac{\log N}{N} .$$
If we let $K=N/2$, then $${\mathbb{P}}( S_N \ge 1 ) = \frac{1 }{\Gamma(K)} \int_{bK}^\infty y^{K-1} e^{-y}\, dy.$$ In the integrand, we may bound $ y^{K-1} \ge (bK)^{K-1}$, yielding $${\mathbb{P}}( S_N \ge 1 ) \ge \frac{(bK)^{K-1}}{\Gamma(K)} \int_{bK}^\infty e^{-y}\, dy = \frac{(bK)^{K-1}}{\Gamma(K)} e^{-bK} .$$ By Stirling’s approximation, it then follows that $$\begin{aligned}
\frac{1}{K} \log {\mathbb{P}}( S_N \ge 1 )
&{\geq } - (b-1-\log b) - \frac{\log b}{K} {- } C\cdot\frac{\log K}{K}, \end{aligned}$$ for some $C>0$ from which the result follows.
The following result is an elementary consequence of Gaussian integration by parts. For a proof in the discrete setting, see, e.g., [@PanchSKBook Lemma 1.1]. The proof in our setting then follows by an elementary approximation argument.
Let $a(x)$ and $b(x)$ be centered Gaussian processes on $\mathbb S^{N-1}$ for any $N\geq 1$, with smooth covariances, continuous mutual covariance $$C(x^1,x^2) = {\mathbb{E}}a(x^1)b(x^2),$$ which is assumed to be smooth and such that ${\mathbb{E}}\max a(x)$ is finite. Then if we let ${\pi(dx) = \exp(b(x)){dx}/Z}$, where $Z$ is chosen so that this is a probability measure, $$\label{eq:GGIBP}
{\mathbb{E}}\int a(x)\, d\pi = {\mathbb{E}}\int\int C(x^1,x^1)- C(x^1,x^2)\, d\pi^{{\otimes}2}$$
By the assumption on the covariances, the processes $a(x)$ and $b(x)$ are a.s. smooth [@AdlerTaylor]. By the law of large numbers, there is a collection of points $(y^\ell)\in\mathbb S^{N-1}$, such that the empirical measure $${\mathcal{E}}_n = \frac{1}{n}\sum_{\ell=1}^n \delta_{y^\ell}$$ converges weak-\* to the uniform measure. Evidently, if we let $$\pi_n = \frac{\sum_{\ell=1}^n \exp(b(y^\ell))\delta_{y^\ell}}{\sum_{\ell=1}^n \exp(b(y^\ell))},$$ then $\pi_n\to\pi$ weak-\* a.s. By Gaussian integration by parts, [@PanchSKBook Lemma 1.1], $${\mathbb{E}}\int a(x) d\pi_n = {\mathbb{E}}\int \int C(x^1,x^1)- C(x^1,x^2)d\pi_n^{{\otimes}2}.$$ Since, $\max a(x)$, has bounded mean. The result then follows by applying the dominated convergence theorem to each side of this equality.
[^1]: We note here that none of our results are changed if one symmetrizes $W$, i.e., if we work with the symmetric Gaussian $k$-tensor.
|
Spatial optical solitons i.e. light beams propagating without transverse spreading arise when diffraction is balanced by a nonlinear process such as self-focussing in a nonlinear dispersive or reactive medium. Light propagating inside an optical resonator filled with a nonlinear medium can thus form stable filaments, or localized structures (spatial solitons). These are free to move in the resonator cross section (or move by themselves [@tag:1]) which implies their bistability, and ability to carry information. The mobility of spatial solitons, however, makes them different from arrangements of fixed binary elements, so that new types of information processing have been considered, making use of spatial resonator solitons.
Early realisations of such resonator solitons in slow materials were given in [@tag:2; @tag:3]. We investigated in the past spatial resonator solitons of phase-type [@tag:4] and intensity type [@tag:5], including experiments demonstrating large simultaneous collections of solitons [@tag:6] and their manipulation [@tag:5] as is required for practical applications. These experiments were conducted using slow nonlinear materials for the sake of easy observeability of the complex 2D space-time dynamics. For practical purposes, however, speed is of prime importance and compatibility with semiconductor technology is desirable. Spatial solitons and their switching in semiconductor microresonators have therefore been predicted theoretically recently [@tag:7; @tag:9]. With the aim of realizing spatial solitons in semiconductor resonators, experiments were conducted recently, addressing passive resonators [@tag:10] and resonators with population inversion [@tag:11]. We showed the spontaneous formation of bright and dark spatial solitons in [@tag:12]. We confirm here the bistable nature of the bright spatial semiconductor resonator solitons by the results of local switching experiments and demonstrate the incoherent writing and erasing of the bright solitons.
The experimental arrangement (FIG. 1) was essentially as described in [@tag:10] and [@tag:12]. Light of a Ti:Al$_{2}$O$_{3}$-laser around 855 nm wavelength illuminates the semiconductor resonator sample. This consists of two Bragg mirrors of about 99,5 $\%$ reflectivity and 18 pairs of GaAs/GaAlAs-quantum-wells between them. The band edge and the wavelength of the exciton line is at 849 nm. Observations are done in reflection because the substrate material (GaAs) is opaque at the working wavelength.
The laser light is modulated by a mechanical chopper to limit illumination to durations of a few $\mu$s, in order to avoid thermal nonlinear effects. The repetition rate of the illuminations is 1 kHz, permitting stroboscopic recordings of the dynamics or signal averaging. Part of the laser light is split away from the main beam, with orthogonal polarisation, for local injection into the illuminated sample area. The injection is applied in pulses of several 10 ns duration using an electro-optical modulator (EOM). The light reflected from the sample is imaged onto a CCD camera. For time-resolved observations the reflected light is passed through another EOM (50 ns aperture time), which is opened with a variable delay with respect to the start of the illumination. The 2D intensity can thus be recorded for arbitrary moments during the illumination. Further, the intensity in a particular point can be monitored by a small area photodiode PD. In the observations reported, the intensity of the main beam was chosen so that a bright soliton (dark in reflection) would appear only at the center of the main beam which has Gaussian intensity profile with a width of 30 $\mu$m. The small area photodiode PD measures the intensity at this point.
FIG. 2 shows the switch-on of a bright soliton. The illuminating intensity rises initially due to the mechanical chopper opening. The maximum intensity is below the switching intensity for the bistable resonator. At t $\approx$ 3.9 $\mu$s the injection beam (orthogonal polarisation with respect to the illumination, width 12 $\mu$m) is opened for 70 ns to switch the resonator. During the switch initiated by the injecting pulse a switching front travels radially outward [@tag:10] and forms a switched area surrounded by the switching front (dark in left inset in FIG. 2). The switching area then collapses into a spot about 10 $\mu$m diameter (see right inset in FIG. 2), the expected size of a soliton for this resonator (details see [@tag:12]). This collapse takes place from 4 to 5 $\mu$s. After 5 $\mu$s a stationary soliton exists, recorded in the right inset of FIG. 2. When the incident illumination (dotted trace) is finally decreased (chopper closes) the soliton switches off.
Although we do not understand presently the mechanism by which the relatively slow collapse of the switching front occurs, FIG. 2 demonstrates that the bright soliton can be switched on by an external control beam, implying the bistability of the soliton. The switch-on of the soliton would presumably proceed more directly if the injection beam size, intensity, phase and polarization were matched to the final soliton dimensions.
FIG. 3 shows conversely the switching off of a bright soliton (dark in reflection). The illumination intensity in FIG. 3 is chosen above the switching intensity of the resonator so that a soliton forms spontaneously as described in [@tag:12] during the transient phase from about 2.4 to 3.5 $\mu$s. At about 3.5 $\mu$s a stationary soliton is existing (see central inset in FIG. 3). At about 3.9 $\mu$s the injection beam (same properties and alignment as for Fig. 2) is opened. This switches the soliton off and returns the whole resonator to the unswitched state (see the right inset in FIG. 3). We mention that the state after the switch-back is stable here, although at its intensity it was unstable in the beginning (t $\approx$ 2.4 $\mu$s). Measurements showed that after the switch-off the threshold of instability was a few percent higher than initially. Raising the background intensity slightly led to renewed spontaneous appearance of the soliton with the pronounced feature of critical slowing. One might tentatively ascribe the small increase of the instability threshold to heating of the material during the time of formation and existence of the soliton, during which the intensity and dissipation in the resonator is high.
FIG. 4 shows that the switching off of the soliton requires a minimum intensity in the external beam. Here the external beam is opened at t $\approx$ 3.9 $\mu$s with an intensity 10 $\%$ smaller than in FIG. 3. The soliton in this case is transiently perturbed, but remains stably switched-on.
In summary, we have shown that bright spatial solitons of a semiconductor resonator can be switched on and off by an external incoherent address beam. Thus we demonstrate that such solitons are controllable as required for applications. The bistable nature of the solitons is unambiguously demonstrated. The switch-on mechanism observed presently is too slow for fast processing applications. We attribute this to an insufficient match of the injection beam field with the soliton and suppose that a matched injection beam should directly switch on the bright solitons, without the long transient soliton formation phase. The details of the incoherent switching mechanism observed here will be clarified in the near future.\
Acknowledgement
This work was supported by ESPRIT LTR project PIANOS. The quantum-well semiconductor sample was provided by R.Kuszelewicz, CNET, Bagneux, France.
[99]{} K.Staliunas, V.B.Taranenko, G.Slekis, R.Viselga, C.O.Weiss, Phys. Rev. A[**57**]{}, 599 (1998). M.Kreuzer, H.Gottschling, T.Tschudi, R.Neubecker, Mol. Cryst. Liq. Cryst. [**207**]{}, 219 (1991). B.Fischer, O.Werner, M.Horowitz, Appl. Phys. Lett. [**58**]{}, 2729 (1991). V.B.Taranenko, K.Staliunas, C.O.Weiss, Phys. Rev. Lett. [**81**]{}, 2236 (1998). V.B.Taranenko, K.Staliunas, C.O.Weiss, Phys. Rev. A[**56**]{}, 1582 (1997). G.Slekis, K.Staliunas, C.O.Weiss, Opt. commun. [**149**]{}, 113 (1998). M.Brambilla, L.A.Lugiato, F.Prati, L.Spinelli, W.J.Firth, Phys. Rev. Lett. [**79**]{}, 2042 (1997). D.Michaelis, U.Peschel, F.Lederer, Phys. Rev. A[**56**]{}, R3366 (1997). D.Michaelis, U.Peschel, F.Lederer, Opt. Lett. [**23**]{}, 337 (1998). V.B.Taranenko, I.Ganne, R.Kuszelewicz, C.O.Weiss, “Patterns and localized structures in bistable semiconductor resonators”, in print, Phys. Rev. A (2000); also Los Alamos Preprint Server nlin.PS/0001055. T.Ackemann, S.Barland, M.Cara, M.Giudici, S.Balle, in Nonlinear guided waves and their applications, 1999, OSA Technical Digest, p.53. V.B.Taranenko, I.Ganne, R.Kuszelewicz, C.O.Weiss, “Spatial solitons in a semiconductor microresonator”, submitted to Phys. Rev. Lett. (1999); also Los Alamos Preprint Server nlin. PS/0001056.
|
---
abstract: |
There has been extensive recent progress in X-ray observations of clusters of galaxies with the analysis of the entire [*ASCA*]{} database and recent new results from [*Beppo-SAX*]{}, [*Chandra*]{}, and [*XMM-Newton*]{}. The temperature profiles of most clusters are isothermal from 0.05–0.6 $R_{\rm viral}$, contrary to theoretical expectations and early results from [*ASCA*]{}. Similarly, the abundance profiles of Fe are roughly constant outside the central regions. The luminosity-temperature relation for a very large sample of clusters show that $L_{\rm X}\propto T^3$ over the whole observable luminosity range at low redshift, but the variance increases at low luminosity, explaining the previously claimed steepening at low luminosity. Recent accurate cluster photometry in red and infrared passbands have resulted in much better correlations of optical and X-ray properties, but there is still larger scatter than one might expect between total light and X-ray temperature and luminosity. The velocity dispersion and the X-ray temperature are strongly correlated, but the slope of the relation is somewhat steeper than expected. The surface brightness profiles of clusters are very well fit by the isothermal $\beta$ model out to large radii and show scaling relations, outside the central regions, consistent with a $\Lambda$-dominated Universe.
At high masses the gas mass fraction of clusters is quite uniform and is consistent with the low WMAP value of $\Omega_{\rm m}$. The recent analysis of cluster mass-to-light ratio and the mass-to-light ratio of stars indicates that the ratio of gas to stellar mass is $\sim$10:1 in massive clusters. There is an apparent decrease in gas mass fraction and increase in stellar mass fraction at lower mass scales, but the very flat surface brightness of the X-ray emission makes extension of this result to large scale lengths uncertain. The normalization of the scaling of mass with temperature, derived from measurements of density and temperature profiles and assuming hydrostatic equilibrium, is lower than predicted from simulations that do not include gas cooling or heating and has a slightly steeper slope. Detailed [*Chandra*]{} and [*XMM-Newton*]{} imaging spectroscopy of several clusters show that the form of the potential is consistent with the parameterization of Navarro, Frenk, & White (1997) over a factor of 100 in length scale and that there is no evidence for a dark matter core. [*Chandra*]{} X-ray images have revealed rather complex internal structures in the central regions of some clusters, which are probably due to the effects of mergers; however, their nature is still not completely clear.
There are now more than 100 clusters with well-determined Fe abundance, several with accurate values at redshifts $z \approx 0.8$, with little or no evidence for evolution in the Fe abundance with redshift. There is real variance in the Fe abundance from cluster to cluster, with a trend for clusters with higher gas densities to have higher Fe abundances. The Si, S, and Ni abundances do not follow patterns consistent with simple sums of standard Type Ia and Type II supernova, indicating that the origin of the elements in clusters is different from that in the Milky Way. The Si/Fe abundance rises with cluster mass, but the S/Fe ratio does not. The high Ni/Fe ratio indicates the importance of Type Ia supernovae. [*XMM-Newton*]{} grating spectra of the central regions of clusters have derived precise O, Ne, Mg, and Fe abundances. [*XMM-Newton*]{} CCD data are allowing O abundances to be measured for a large number of clusters.
author:
- |
RICHARD F. MUSHOTZKY\
NASA/Goddard Space Flight Center
---
å[[A&A]{}]{}
Clusters of Galaxies: An X-ray Perspective
==========================================
Introduction
------------
Clusters of galaxies are the largest and most massive collapsed objects in the Universe, and as such they are sensitive probes of the history of structure formation. While first discovered in the optical band in the 1930’s (for a review, see Bahcall 1977a), in same ways the name is a misnomer since most of the baryons and metals are in the hot X-ray emitting intracluster medium and thus they are basically “X-ray objects.” Studies of their evolution can place strong constraints on all theories of large-scale structure and determine precise values for many of the cosmological parameters. As opposed to galaxies, clusters probably retain all the enriched material created in them and being essentially closed boxes they provide an unbiased record of nucleosynthesis in the Universe. Thus, measurement of the elemental abundances and their evolution provide fundamental data for the origin of the elements. The distribution of the elements in clusters reveals how the metals were removed from stellar systems into the intergalactic medium (IGM). Clusters should be fair samples of the Universe, and studies of their mass and their baryon fraction reveal the gross properties of the Universe as a whole. Since most of the baryons are in the gaseous phase and clusters are dark matter dominated, the detailed physics of cooling and star formation are much less important than in galaxies. This makes clusters much more amenable to detailed simulations than galaxies or other systems in which star formation has been an overriding process. Detailed measurements of their density and temperature profiles allow an accurate determination of the dark matter profile and total mass. While gravity is clearly dominant in massive systems, much of the entropy of the gas in low-mass systems maybe produced by nongravitational processes. Clusters are luminous, extended X-ray sources and are visible out to high redshifts with present-day technology. The virial temperature of most groups and clusters corresponds to $kT \approx (2-100) \times
10^6$ K (velocity dispersions of 180–1200 km s$^{-1}$), and while lower mass systems certainly exist, we usually call them galaxies. Most of the baryons in groups and clusters of galaxies lie in the hot X-ray emitting gas, which is in virial equilibrium with the dark matter potential well \[the ratio of gas to stellar mass is $\sim$(2–10):1; Ettori & Fabian 1999\]. This gas is enriched in heavy elements (Mushotzky et al. 1978) and is thus the reservoir of stellar evolution in these systems. The presence of heavy elements is revealed by line emission from H and He-like transitions, as well as L-shell transitions of the abundant elements. Most clusters and groups are too hot to have significant line emission from C or N, but all abundant elements with $Z > 8$ (O) have strong lines from H and He-like states in the X-ray band, and their abundances can be well determined.
Clusters of galaxies were discovered as X-ray sources in the late 1960’s (see Mushotzky 2002 for a historical review), and large samples were first obtained with the [*Uhuru*]{} satellite in the early 1970’s (Jones & Forman 1978). Large samples of X-ray spectra and images were first obtained in the late 1970’s with the [*HEAO*]{} satellites (see Forman & Jones 1982 for an early review). The early 1990’s brought large samples of high-quality images with the [*ROSAT*]{} satellite and good quality spectra with [*ASCA*]{} and [*Beppo-SAX*]{}. In the last three years there has been an enormous increase in the capabilities of X-ray instrumentation due to the launch and operation of [*Chandra*]{} and [*XMM-Newton*]{}. Both [*Chandra*]{} and [*XMM-Newton*]{} can find and identify clusters out to $z > 1.2$, and their morphologies can be clearly discerned to $z > 0.8$ (Fig. 1.1). The cluster temperatures can be measured to $z \approx 1.2$, and [*XMM-Newton*]{} can determine their overall chemical abundances to $z
\approx 1$ with sufficiently long exposures (very recently the temperature and abundance of a cluster at $z$ = 1.15 was measured accurately in a 1 Ms [*XMM-Newton*]{} exposure; Hasinger et al. 2004). Temperature and abundance profiles to $z \approx 0.8$ can be well measured and large samples of X-ray selected clusters can be derived. [*Chandra*]{} can observe correlated radio/X-ray structure out to $z > 0.1$ and has discovered internal structure in clusters. The [*XMM-Newton*]{} grating spectra can determine accurate abundances for the central regions of clusters, in a model independent fashion, for O, Ne, Mg, Fe, and Si.
{width="1.00\columnwidth"}
It is virtually impossible to give a balanced review of the present observational state of X-ray cluster research, with more than 100 papers published each year. I will not say much about those issues for which we have had detailed talks at this meeting: cooling flows, high-redshift clusters and evolution, X-ray data and the Sunyaev-Zel’dovich effect, radio source interaction, X-ray selected active galaxies in clusters, X-ray emission from groups, and detailed comparison of masses derived from lensing and X-ray observations. Other areas, such as the presence of nonthermal emission and the existence of very soft components, were not discussed. Even limiting the talk this much results in an abundance of material. However, for the purposes of continuity, I have included some material that overlaps with the reviews on chemical abundance given by Renzini (2004) and on groups by Mulchaey (2004). This review does not consider work published after February 2003.
Temperature Structure of Clusters
---------------------------------
As discussed in detail by Evrard (2004), we now have a detailed understanding of the formation of the dark matter structure for clusters of galaxies. If gravity has completely controlled the formation of structure, one predicts that the gas should be in hydrostatic equilibrium with the vast majority of the pressure being due to gas pressure. If this is true, its density and temperature structure provide a detailed measurement of the dark matter distribution in the cluster. Recent theoretical work has also taken into account other process such as cooling and turbulence, which can be important. The fundamental form of the Navarro, Frenk, & White (1997; hereafter NFW) dark matter potential and the assumption that the fraction of the total mass that is in gas is constant with radius result in a prediction, both from analytic (Komatsu & Seljak 2001) and numerical modeling (Loken et al. 2002), that the cluster gas should have a declining temperature profile at a sufficiently large distance from the center (in units of $R/R_{\rm virial}$). The size of the temperature drop in the outer regions is predicted to be roughly a factor of 2 by $R/R_{\rm virial} \approx 0.5$, which is consistent with the [*ASCA*]{} results of Markevitch (1998). However, there is considerable controversy about the analysis and interpretation of temperature profiles before [*XMM-Newton*]{} and [*Chandra*]{}. Results from both [*ASCA*]{} (Kikiuchi et al. 1999; White & Buote 2000) and [*Beppo-SAX*]{} (Irwin & Bregman 2000; De Grandi & Molendi 2002), indicate either isothermal gas or a temperature gradient in the outer regions of some “cooling flow” clusters. [*XMM-Newton*]{} is perfect for resolving this controversy, having a much better point spread function than [*ASCA*]{} and much more collecting area than [*Beppo-SAX*]{} and [*Chandra*]{}, and having a larger field of view than [*Chandra*]{}. However, there is a selection effect due to the smaller [*XMM-Newton*]{} field of view than [*ASCA*]{}, and in order to go out to the virial radius in one pointing one must observe clusters at $z> 0.1$.
There are several published temperature profiles from [*XMM-Newton*]{} (Tamura et al. 2001; Majerowicz, Neumann, & Reiprich 2002; Pratt & Arnaud 2002) and I have analyzed several other moderate redshift clusters and others were presented at this conference (Jones et al. 2004). With the exception of one object (A1101S; Kaastra et al. 2001) all the published [*XMM-Newton*]{} profiles are consistent with isothermal profiles out to $R/R_{\rm virial} \approx 0.5$ (Fig. 1.2), which is in strong disagreement with the numerical and analytic modeling. This sample of $\sim$12 objects is highly biased to smooth, centrally condensed clusters (with the exception of Coma, which has been known to be isothermal from the early work of Hughes et al. 1988). The data for A2163 are consistent with a temperature drop at even larger radii (Pratt, Arnaud, & Aghanim 2002), but the relatively high [*XMM-Newton*]{} background makes the results somewhat uncertain. The origin of the difference between some of the [*Beppo-SAX*]{}, [*ASCA*]{}, and [*XMM-Newton*]{} results is not clear. It is possible that there is a difference between the low-$z$ systems studied by [*Beppo-SAX*]{} and [*ASCA*]{} and the higher-redshift systems studied by [*XMM-Newton*]{} and/or a selection effect in the objects so far analyzed with [*XMM-Newton*]{}. While the [*Chandra*]{} data do not go out to very large length scales (Allen, Schmidt, & Fabian 2002), analysis of 2 $z \approx 0.7$ clusters with [*Chandra*]{} (Ettori & Lombardi 2003) also show isothermal profiles.
We must now take seriously the disagreement between theory and observation in the temperature profiles in comparing cluster properties with simulations. Another serious issue is the inability of theoretical models to match the observed temperature drops in the centers of the “cooling flow” clusters. The question is then, what is the origin of the discrepancy? Several possibilities are that the form of the theoretical potential is incorrect, that the gas distribution is not calculated correctly, or that physics other than gravity needs to be included.
{width="0.80\columnwidth"}
As shown below (§1.8) the form of the potential from X-ray imaging spectroscopy agrees quite well with the NFW potential, which is consistent with the analytic work. [*ROSAT*]{} and [*XMM-Newton*]{} analysis of X-ray surface brightness distributions (§1.5) shows that the $\beta$ model is a good description of the X-ray surface brightness at large radii. This leaves us with the possibility that additional physics is needed. Recent analysis of [*Chandra*]{} data (cf. Markevitch et al. 2003) strongly constrains the effects of conduction, which will tend to make isothermal spectra, while the inclusion of cooling and heating in the theoretical models (Loken et al. 2002) does not seem to affect the temperature profile significantly. Thus, the origin of this severe discrepancy is not currently known.
{width="0.85\columnwidth"}
Luminosity-Temperature Relation for Clusters
--------------------------------------------
As pointed out by Kaiser (1986), simple scaling relations predict that the cluster luminosity should scale as the temperature squared ($T^2$). To see this, note that the X-ray luminosity should scale as the density squared times the volume times the gas emissivity, $L_{\rm
X} \propto \rho^2 V \Lambda$. The mass of gas scales like $\sim \rho
V$, and it is assumed that the total mass $M_{\rm T}$ scales as $M_{\rm gas}$. Since the emissivity for bremsstrahlung, the prime cooling mechanism in gas hotter than 2 keV, scales as $T^{0.5}$ (Sutherland & Dopita 1993), one has $L_{\rm X} \propto M \rho
T^{0.5}$. Finally, since, theoretically, the total mass scales as $T^{1.5}$, one has $L_{\rm X} \propto \rho T^2$. The other free parameter, the average density, is related to the mass and collapse epoch of the cluster.
It has been known for 20 years (Mushotzky 1984) that the actual relationship between temperature and luminosity is steeper than the simplest theoretical prediction. Recently, Horner et al. (2004) have examined the $L_{\rm X}-T$ relation using the largest sample of clusters to date (270 clusters taken from the [*ASCA*]{} database). In this sample one finds that, over a factor of $10^4$ in luminosity, the luminosity scales as $T^3$. As one goes to lower luminosities there is a wider range of luminosity at a fixed temperature (Fig. 1.3), but there is no need to change the scaling law. This increase in variance probably explains the steeper fit at low luminosity found by Helsdon & Ponman (2000). This continuity is rather strange, since at $T<2$ keV the cooling function changes sign and scales more like $T^{-1}$, and thus the theoretical relation between $L_{\rm X}$ and $T$ changes slope.
There have been many papers written about the origin of the discrepancy, but the main conclusion is that it is due to the breaking of scaling laws via the inclusion of physics other than gravity. The same physics that helps to explain the deviation of entropy in groups, such as heating and cooling, can also explain the slope and normalization of the $L_{\rm X}-T$ relation (see Mulchaey 2004 and Borgani et al. 2002). Another indication of this scale breaking is the relative low level of evolution in the $L_{\rm
X}-T$ relation out to $z\approx 1$ (Borgani et al. 2002) which is not what is predicted in simple theories of cluster evolution, since objects at $z\approx 1$ are predicted to be denser and have a higher temperature for a fixed mass. Simple scaling predicts that $T \propto
M^{1.5} (1+z)$, and thus one predicts $L \propto T^2 (1+z)^{0.5}$ at a fixed mass, which is not seen (but see Vikhlinin et al. 2002 for a different opinion).
It was pointed out by Fabian et al. (1994) that high central density, short cooling time clusters (alias “cooling flow” clusters) have a considerably higher luminosity for their temperature than non-cooling flow systems. This result is confirmed in the larger Horner et al. (2004) sample. Markevitch (1998) removed the high-central surface brightness central regions from these clusters and found that the scatter in the $L_{\rm X}-T$ relationship was much reduced and the fit was flatter than $T^3$. If the scatter in the $L_{\rm X}-T$ relationship was due to cool gas in the center of the cooling flow clusters, one should expect that the [*ROSAT*]{} luminosities, which are very sensitive to low-temperature gas, would be systematically larger than the luminosities calculated from isothermal fits to the [*ASCA*]{} data. However, Horner et al. (2004) find that the bolometric luminosities obtained by [*ASCA*]{} are in very good agreement with the [*ROSAT*]{} results. This indicates that the central luminosity “excess” is not due to cool gas, as was originally shown in the [*ASCA*]{} data for the Centaurus cluster (Ikebe et al. 1999) and recently shown in detail by [*XMM-Newton*]{} spectroscopy of many cooling flow clusters (Peterson et al. 2003). Horner et al. (2004) find that the most reasonable explanation for the higher luminosity of the cooling flow clusters is due to their higher central density in the core. This result is consistent with the detailed analysis of cluster surface brightness profiles by Neumann & Arnaud (2001) (see §1.5). It thus seems that the scatter in the $L_{\rm X}-T$ relation at high temperatures is due to differing cluster central gas densities, while the scatter at low temperatures is due to different “amounts” of additional, nongravitational physics.
{width="1.00\columnwidth"}
Optical Light, Velocity Dispersion, and X-ray Properties
--------------------------------------------------------
It has been known since the early [*Uhuru*]{} results (Jones & Forman 1978) that there is a great degree of scatter in the correlation between cataloged optical properties, such as Abell richness, and X-ray properties, such as luminosity and temperature. The best correlations between optical and x-ray properties seen in the early data were between central galaxy density and X-ray luminosity (Bahcall 1977b), and between X-ray temperature and optical velocity dispersion (Edge & Stewart 1991). The wide scatter is nicely illustrated in Figure 5 of Borgani & Guzzo (2001)which shows that the Abell counts are only weakly related to total mass, while the x-ray luminosity is strong correlated.
Bird, Mushotzky, & Metzler (1995) showed that much of the scatter in the temperature - velocity dispersion correlation was due to undersampled optical data and velocity substructure in the clusters. More recent optical and X-ray work (Girardi et al. 1998; Horner et al. 2004) shows that when the velocities of a sufficient number of galaxies in a cluster are measured (one needs more than 30 galaxies) (Fig. 1.4) there is a tight relation between velocity dispersion and temperature of the form $\sigma
\propto T^{0.59\pm0.03}$, consistent with the work of Bird et al. (1995) and close to the theoretical slope of 0.5. This has been confirmed in an infrared-selected sample by Kochanek et al. (2003). The normalization of this relation at high temperatures agrees with theoretical work (Evrard 2004), and thus one has to conclude that low-velocity dispersion clusters are too hot for their dispersion, or that low-temperature clusters have too low a dispersion for their temperature. The fact that clusters have very small radial velocity dispersion gradients (Biviano & Girardi 2003) or temperature gradients (§1.2) makes comparison of the average temperature and velocity dispersion meaningful. This variation with temperature of the velocity dispersion to temperature ratio will also change the effective X-ray vs. optically determined mass by a factor of 50% over the full mass range of clusters.
Recent 2MASS work by Kochanek et al. (2003) shows that, if the “optical” data are handled carefully (e.g., accurate photometry, well-defined selection criteria, observing in a red passband, etc.), there is a strong relation between the total light in a cluster and the X-ray temperature and luminosity (also see Yee & Ellingson 2003). However, while the correlations are much better than in previous work, the scatter in the relation is large, almost a factor of 10 in light at a fixed X-ray temperature or luminosity, or, alternatively, a factor of $\sim$2 in temperature at a fixed optical luminosity. Thus, one expects that optical and X-ray catalogs of clusters might differ considerably depending on where the cuts are made. There is no evidence for either optically or X-ray quiet clusters, but there is evidence for relatively optically or X-ray bright objects. The nature and origin of this variance is not understood at present, but, given the quality of modern data, this variance seems to be real, rather than due to measurement uncertainties. Assuming that the X-ray properties accurately trace mass, the $K$-band light is a mass indicator accurate to 50% (Lin, Mohr, & Stanford 2003). The converse test, estimating the mass from the optical data and comparing it to the X-ray data, shows large scatter (Yee & Ellingson 2003), where the temperature data are taken from the literature. If it is indeed the case that there is a large variation in the ratio of optical light to X-ray temperature, this indicates that there is a considerable variance in cluster mass-to-light ratio at a fixed mass. This would be a major challenge to structure formation theories.
Surface Brightness Profiles
---------------------------
It has been known since the pioneering work of Jones & Forman (1984) that the surface brightness profiles of most clusters can be well fit, at large radii, by the “isothermal” $\beta$ model, $S(r)=S_0(1+(r/a)^2)^{(-3\beta + 0.5)}$, with a central excess above the $\beta$ model in cooling flow clusters. As seen in [*ROSAT*]{} data for high-redshift systems (Vikhlinin, Forman, & Jones 1999), the $\beta$ model fits amazingly well out to the largest radii measurable for massive clusters. The fitted values of $\beta$ are smaller for low-mass systems (Helsdon & Ponman 2000; Mulchaey et al. 2003), but there are two selection effects that make the interpretation of this result difficult. First of all, because of their low surface brightness, the group profile hits the background at relatively small distances from the center, and thus one does not detect low-mass systems out to large fractions of the virial radius. This can introduce a bias to the fitted values of $\beta$. Secondly, the effects of the central galaxy on the surface brightness is often not well determined from [*ROSAT*]{} PSPC data (Helsdon & Ponman 2003)and thus, frequently, the structural parameters are not well constrained. This latter effect is not present in [*XMM*]{} or [*Chandra*]{} data.
Neumann & Arnaud (2001) have pointed out that the surface brightness profiles of high-temperature clusters remain self-similar as a function of mass and redshift, as expected from cold dark matter models (see also Vikhlinin et al. 1999). Since the conversion from angle to distance depends on the cosmology, they have been able to show that the change of profile with redshift is most consistent with a $\Lambda$ dominated cosmology. The homology of the profiles is only applicable outside of the central 100 kpc, as inside this radius there are often large deviations from the scaling laws. However, in order to achieve the scaling they require that the relationship of gas mass to temperature be $M_{\rm gas} \propto T^2$, steeper than the theoretical scaling between total mass and temperature (i.e., $M_{\rm
total}\propto T^{1.5}$). Since the surface brightness profiles scale according to the predicted evolution from the cold dark matter models, the lack of evolution in the luminosity-temperature law must be a cosmic conspiracy between the cosmological model and the change of density with redshift. The prediction is that the emission measure of the gas scales as $EM \propto \beta f^2_{\rm
gas}\Delta^{1.5}(1+z)^{9/2}(kT)^{0.5} h^3$, where $\Delta$ is the overdensity of the cluster and $f_{\rm gas}$ is the fraction of mass that is in gas (Arnaud, Aghanim, & Neumann 2002).
There are “single” clusters that are not well fit by the $\beta$ model. The most obvious example is MS 1054$-$0321 at $z$ = 0.82 (Jeltema et al. 2001), which is much more concentrated than a $\beta$ model. This is not a function of redshift, since many clusters at $z>0.6$ are well fit by the $\beta$ model.
Mass of Baryons and Metals and How They Are Partitioned
-------------------------------------------------------
The two main baryonic components of clusters are the X-ray emitting gas and the stars, since the total contribution from cold gas and dust is very small. The major uncertainty in the relative baryonic contribution is due to the uncertainty in the transformation from light to mass for the stars. Recent work from large optical surveys (Bell et al. 2003) shows that the mass-to-light ratio of stars changes as a function of galaxy but is $\sim$3.5 in the Sloan $g$ band for a bulge-dominated population. Using this value and the mean mass-to-light ratio of clusters $\sim$240 (Girardi et al. 2002), the stars have $\sim$0.015 of the total mass. The gas masses have been well determined from [*ROSAT*]{} data (Ettori & Fabian 1999; Allen et al. 2002) and scatter around $f_{\rm gas} \approx 0.16 h_{70}^{-0.5}$. Thus, the gas-to-stellar mass ratio is $\sim$10:1, and the total baryon fraction is almost exactly consistent with the recent [*WMAP*]{} results for the Universe as a whole. Since it is thought that clusters are representative of the Universe as a whole, this suggests that the vast majority of baryons in the Universe do not lie in stars. Turning this around, one can use the baryonic fraction in clusters as a bound on $\Omega_{\rm m}$ (White et al. 1993). The most recent analysis using this technique finds $\Omega_{\rm m} < 0.38
h_{70}^{-0.5}$ (Allen et al. 2002), in excellent agreement with the [*WMAP*]{} data. It is interesting to note that the high baryonic fraction in clusters has been known for over 10 years and was one of the first strong indications of a low $\Omega_{\rm m}$ Universe. Since it is thought that the baryonic fraction in clusters should not evolve with redshift, derivation of the baryonic abundance in high-$z$ clusters, which depends on the luminosity distance, provides a strong constraint on cosmological parameters (Ettori & Tozzi and Rosati 2003).
The mean metallicity of the gas in clusters is $\sim$1/3 solar (see §1.10), while that of the stars may be somewhat larger. If we assume 1/2 solar abundance for the stars, than $\sim$85% of the metals are in the gas phase. Since all the metals are made in stars, which lie primarily in galaxies, this implies that most of metals have either been ejected or removed from the galaxies. Since the stellar mass is dominated by galaxies near $L^{\star}$, which have a mean escape velocity, today, of $>$300 km s$^{-1}$, this implies very strong galactic winds at high redshift. This scenario is consistent with the results of Adelberger et al. (2003) on the high-redshift, rapidly star-forming $U$ and $B$-band drop-out galaxies, which all have large-velocity winds.
Analysis of the gas mass fraction in groups and clusters (Sanderson et al. 2003) indicates that the fraction apparently drops at lower masses by a factor of 2–3, with the reduction setting in at a mass scale corresponding to 1–3 keV at 0.3 $R_{200}$. In addition, the stellar mass-to-light ratio decreases by 60% over the same mass range (Marinoni & Hudson 2002), and thus in groups the gas-to-stellar mass ratio is only (1–2):1 at 0.3 $R_{200}$, considerably smaller than in clusters. However, there is a serious problem for groups in evaluating both the gas and stellar masses at large radii (see Fig. 10 in Mulchaey et al. 2003), and this result should be taken with some caution. In particular, the X-ray surface brightness distribution of groups is often very flat, and extrapolating from 0.3 $R_{200}$ to $R_{200}$ is rather risky. However, if these trends are real, this would indicate that groups are truly baryon poor, that the baryons have been pushed out of the group, or that the gas has been puffed up. If the gas has been puffed up, this is consistent with the somewhat high temperatures of groups compared to their optical galaxy velocity dispersions, indicative of extra heat deposited in the gas, which both heats it and “puffs” it up (see discussion in the review by Mulchaey 2004).
Mass Scaling Laws
-----------------
Detailed theoretical work has verified that clusters should satisfy the virial theorem, and thus their mass should scale as $M \propto T R$, with $R \propto
T^{1/2}$, and thus $(1+z)M^{2/3} \propto T$ (e.g., Eke, Navarro, & Frenk 1998), with the normalization being set by theory and the value of the cosmological parameters (Evrard 2003). The first test of this relation (Horner, Mushotzky, & Scharf 1999) found a scaling that was somewhat steeper, with $M \propto T^{1.7}$, and a normalization that was 40% lower than predicted. Finoguenov, Reiprich, & Böhringer (2001) and Reiprich & Böhringer (2002) have confirmed these results with more uniform samples, and higher quality, spatially resolved spectra. Recent [*Chandra*]{} results (Allen, Schmidt, & Fabian 2001) are also consistent with the Horner et al. (1999) finding. [*XMM-Newton*]{} data for A1413 (Pratt & Arnaud 2002) show that the normalization scaling is not only violated by the sample, but by individual objects. The normalization in the Reiprich & Böhringer (2002) sample agrees with theoretical expectations at the high-mass end. This indicates that lower-temperature clusters are less massive than expected on the basis of their temperature, consistent with the trend seen in the velocity dispersion-temperature relation. Recently, it has been pointed out (Shimizu et al. 2003) that the combination of the scaling of mass by $M \propto T^{1.7}$ and the gas mass fraction scaling as $T^{1/3}$ (a reasonable fit to the Sanderson et al. 2003 data) can reproduce the observed $L_{\rm X}
\propto T^3$ relationship. Theoretical calculations that include the effects of cooling (Thomas et al. 2002) seem to be consistent with the lower normalization, but so far the slope difference has not been explained.
Form of the Potential
---------------------
As discussed extensively in this conference, the form of the potential in clusters should be determined by the distribution of dark matter. Recent numerical work seems to validate the NFW potential, and much has been made of the fact that low-mass and low-surface brightness galaxies do not seem to follow this form in their central regions. Recent [*Chandra*]{} and [*XMM-Newton*]{} observations (Allen et al. 2002; Arabadjis, Bautz, & Garmire 2002; Pratt & Arnaud 2002) have been able to determine extremely accurate mass profiles via spatially resolved X-ray spectroscopy and the assumption of hydrostatic equilibrium. Perhaps the best documented of these examples are the [*Chandra*]{} data for Abell 2029 (Lewis, Buote, & Stocke 2003), in which the profile is determined over a factor of 100 in length scale, from 0.001–0.1 characteristic lengths of the NFW profile, with essentially no deviation from the NFW prediction. This striking result is also seem in other [*Chandra*]{} results in the cores of clusters. The data show that the central regions of clusters tend to have rather steep density profiles in the innermost radii, indicating that whatever causes the deviation of the form of the potential in dwarf galaxies does not occur in clusters. This results strongly constrains interacting dark matter models (Bautz & Arabadjis 2004). A survey of [*Chandra*]{} central mass profiles is made somewhat difficult because of the possibly complex nature of the IGM in the central regions of many clusters, and the exact slope and normalization of the mass depends on the details of the thermal model used. However, if the data are of sufficiently high signal-to-noise ratio, the form of the mass profile can be determined precisely. I anticipate quite a few exciting new results in this area; preliminary results, presented in several conferences, indicate a predominance of steep mass profiles with slopes close to the NFW level, but with some scatter.
{width="1.00\columnwidth"}
Merges, Structures, etc.
------------------------
The early [*Einstein*]{} Observatory images of clusters (Henry et al. 1979) showed that a substantial fraction of the X-ray images were not simple, round systems, but often complex in form and sometimes even double. This observation is consistent with the idea that clusters form in a hierarchical fashion via mergers, and that the complex systems are in the process of merging. The fact that mergers are actually occurring, rather than the complex structures in the images being simply projection effects, was indicated by complexity in the temperature structure of many of these systems shown by the [*ROSAT*]{} (Briel & Henry 1994) and [*ASCA*]{} (Markevitch 1996) data. The details of the nature of this process have had to wait until the precise [*Chandra*]{} spectral images showed the full range of complexity. While “textbook” examples of merger shocks have been seen (e.g., 1E 0657$-$56; Markevitch et al. 2002), many of the objects show only subtle temperature variations (e.g., Sun et al. 2002). These variations have only shown up in the most recent, very high-resolution numerical simulations, indicating the non-intuitive nature of these data.
The recent spectacular [*XMM-Newton*]{} temperature image of the Perseus cluster (Churazov et al. 2003) illustrates the wealth of detail that is now possible to obtain. It is interesting that these spectral images do not show the numerous “cold spots” that are predicted in cluster simulations that include cooling (Motl et al. 2004). The ability to obtain spectral images has also revealed “hidden mergers.” Both the Coma and Ophiuchus clusters, the hottest nearby systems, show smooth, regular X-ray images; however, X-ray temperature maps show strong spatial variations (Arnaud et al. 2001; Watanabe et al. 2001). So far the data on abundance variations in the mergers is sparse, but the abundances seem uniform, within errors, in Coma and may vary by less than a factor of $\sim$2 in Ophiuchus. It seems as if many of the large-scale length, non-cooling flow clusters are recent mergers.
One of the surprises of the [*Chandra*]{} data was the discovery of surface brightness discontinuities in the surface brightness — the so-called cold fronts (Fig. 1.5; Vikhlinin & Markevitch 2002). These cold fronts are apparently contact discontinuities, across which the pressure is smoothly varying but the density and temperature change discontinuously by factors of $\sim$2. They can occur in “pure hydro” numerical simulations (Bialek, Evrard, & Mohr 2002). Their relative frequency is a indication of the merger rate. However, the details (e.g., temperature drop, size of region, etc.) and their relation to merger dynamics are not certain (Fujita et al. 2002). The stability of the cold fronts, their sizes, and shapes are indications of the strength of the magnetic field, velocity vector of the merger, and the amount of turbulence (Mazzota, Fusco-Femiano, & Vikhlinin 2002) in the cluster gas. It is clear that there is much to learn from further studies of these unexpected structures, but they already confirm that the gas is usually not strongly shocked, nor highly turbulent.
Abundances
----------
As indicated above, most of the metals in the cluster lie in the hot, X-ray emitting gas. Thus, in order to understand the formation and evolution of the elements one must determine accurate abundances, the abundance distribution in the gas, and its evolution with cosmic time. Before giving the results it is important to remind the reader that the measurement of abundances in the cluster gas via X-ray imaging spectroscopy is a robust process. Most of the baryons and metals are in the hot gas, and the spectral signature of the heavy elements are relatively strong H and He-like lines (Fig. 1.6). This is a well-understood emission mechanism, with little or no radiative transfer difficulties. Because of the high temperature and short spallation times, dust is destroyed rapidly and thus is not a problem. The deep potential well captures an integrated record of all the metals produced, and thus the derived abundances are true averages of the metal production process. All the abundant elements from oxygen to nickel can have their abundances determined. Direct measurement of the electron temperature from the form of the continuum and from ratios of H to He-like lines ensures small systematic errors in the abundances. The strongest lines in the spectrum of hot clusters are due to Fe and Si, followed by O, S and Ni. The emission from Ne and Mg is blended with Fe L-shell lines from Fe XVII–XXIV at the resolution of X-ray CCDs, and the lines from Ca and Ar are weak. With present-day technology, one can measure Fe to $z\approx 1$ and Si to $z \approx
0.4$, and can thus obtain a true measure of the metal formation mechanism and its evolution. For much of the rationale and background for cluster abundance measurements, see Renzini’s (2004) review in this volume.
{width="1.05\columnwidth"}
{width="0.80\columnwidth"}
Recently (Baumgartner et al. 2004; Horner et al. 2004), a uniform analysis of the [*ASCA*]{} cluster database of 270 clusters has been performed, which updates previous work (e.g., Mushotzky et al. 1996; Fukazawa et al. 2000) on cluster abundances. Horner et al. (2004) and Baumgartner et al. (2004) measure the average cluster Fe, Si, S, and Ni abundances with no spatial information. They find (Fig. 1.7) that the Fe abundance is not the same for all clusters, but shows a small spread of a factor of $\sim$2. In agreement with Fabian et al. (1994), the cooling flow clusters show, on average, a higher Fe abundance. There is no evidence for any evolution in the Fe abundance out to $z
\approx 0.5$ on the basis of [*ASCA*]{} data. Recent [*XMM-Newton*]{} and [*Chandra*]{} results (Jones et al. 2004; Mushotzky, private communication) show no evolution in the Fe abundance to $z \approx
0.8$. This lack of evolution indicates that the metals are created at $z > 1.3$ for a $\Lambda$ cosmology ( I have added in the lifetime of the A stars that would be visible for the massive amount of star formation necessary to produce the observed metals). Since the vast majority of the metals are in the gas, the rate of specific star formation (e.g., the rate per unit visible stars) would have to be enormous to produce the elements if it were to occur at $z<2$. The Fe abundance is weakly correlated with the temperature, reaching a maximum at $kT \approx 2-3$ keV, but is more or less constant for $kT>4.5$ keV clusters. Given the accuracy of recent plasma codes, the “peak” in abundance, which occurs at a temperature range where both Fe L and K lines contribute to the abundance determination, is almost certainly a real effect. The physical origin of the variance in Fe abundance and the trends are unknown. However, since there are trends in the apparent ratio of starlight to gas (§1.6), this may be the cause. Further progress in this area requires an enhancement of the original work of Arnaud et al. (1992), which found a correlation between the light in elliptical galaxies and the total mass of Fe in the cluster.
The distribution of the elements in a cluster determines the total amount of material and gives clues as to how the material was deposited in the IGM. The previous generation of X-ray satellites ([*ASCA*]{}, [*ROSAT*]{}, and [*Beppo-SAX*]{}) derived abundance profiles of Fe in $>$20 clusters (Finoguenov, David, & Ponman 2000; Irwin & Bregman 2000; White & Buote 2000; De Grandi et al. 2003). However, these results did not always agree (different analysis of the same data and comparison of data from the same object from different satellites produced different results). There was a tendency for cooling flow clusters to have high central Fe abundances and larger total abundances, suggesting a different origin of IGM enrichment in the central regions, the effects of mixing by mergers on the Fe abundance profile, or a physical difference in the origin of the metals in cooling flow clusters.
[*XMM-Newton*]{} and [*Chandra*]{} data have much smaller systematic errors and much better signal-to-noise ratios than the data from the earlier observatories. Early results are available for $\sim$15 systems — most are isochemical at large radii, with several having gradients in the central 100 kpc. The [*Chandra*]{} and [*XMM-Newton*]{} data are well resolved and show that the abundance gradients are quite concentrated toward the center (cf. David et al. 2001; Tamura et al. 2001). For a few objects the profiles reach to near the virial radius (e.g., Zw 3146, Cl 0016 (Mushotzky priv. comm.) , and A 1835; Majerowicz et al. 2002), two of which (Zw 3156 and A 1835) are massive cooling flows do not show abundance gradients outside 100 kpc. Numerical evaluation of the observed Fe abundance gradients (De Grandi et al. 2003) shows that most of the variation in the average Fe abundance between the cooling flow and non-cooling flow clusters is not due to differences in the Fe gradients. The “excess” amount of Fe in the central regions seen in the cooling flow systems is correlated with the presence of a cD galaxy, and the mass of “excess” Fe is roughly consistent with its being produced in the stars in the central cD galaxy. This is rather unexpected, since isolated elliptical galaxies have only $\sim$1/5 of the Fe that should have been produced by the stars (Awaki et al. 1994).
The fact that gradients do not dominate the average abundance allows a direct interpretation of the [*ASCA*]{} average abundances. The [*ASCA*]{} database of $\sim$270 X-ray spectra allow determination of average Fe, Si , S, and Ni in clusters of galaxies (Baumgartner et al. 2004). However, the signal-to-noise ratio for most of the individual clusters is not adequate to derive robust S or Ni abundances, and 20–40 objects in each temperature bin must be added together to derive average values and their variation with temperature. Since cluster mass is directly related to the temperature and line strength is also directly connected to temperature, this is the natural space for averaging. As originally pointed out by Fukazawa et al. (2000), as $T$ increases, Si/Fe increases. However, the new data show that S remains roughly constant versus temperature. Baumgartner et al. (2004) also find that the Ni/Fe ratio is approximately 3 times solar. While these are very surprising results, they are similar to previous analysis of smaller [*ASCA*]{} and [*XMM-Newton*]{} data sets. The S/Fe, Si/Fe, and Ni/Fe ratios depend on the relative abundance of the types of supernovae (SNe). Type Ia SNe produce mostly Fe and Ni, while Type II SNe produce a wide range of elements but large ratios of the $\alpha$ elements (O, Ne, and Si) to Fe. Si and S are produced via very similar mechanisms, and at first sight it is hard to understand how they could have different abundance patterns. In addition, in the Milky Way, S almost always directly tracks Si. The fact that both Si/Fe and S/Fe drop as Fe increases shows that there is indeed a difference in the mechanisms producing the metals as a function of mass scale. It seems rather unexpected that the ratio of Type II to Type Ia SNe in the stars that live in cluster galaxies should change with the mass scale of the cluster. However, the high Ni/Fe ratio indicates that Type Ia SNe are important in the production of Fe, at least in the central regions of clusters (Dupke & White 2000), and this high ratio does not allow a simple variation in SN type with cluster mass to readily explain the abundances patterns seen in the ASCA data.
[*XMM-Newton*]{} data allow the measurement of O abundances for a reasonable sample of objects for the first time. The best sample published to date is based on the high-resolution RGS data (Peterson et al. 2003). They find that the O/Fe ratio varies by a factor of $\sim$2 from cluster to cluster, with no apparent correlation with temperature. Analysis of [*XMM-Newton*]{} CCD data taken over a larger scale (the RGS data sample only the central $1^{\prime}-2^{\prime}$ of the cluster) confirm this variance. As noted in Gibson, Loewenstein, & Mushotzky (1997), the elemental abundance ratios averaged over the cluster do not agree with any simple ratio of Type Ia to Type II SNe. However, it is clear that over 90% of the O, Ne, and Mg must originate in Type II SNe.
The new [*XMM-Newton*]{} O abundances further strengthen this conclusion. However, some of the difficulties may be caused by differential abundance gradients of different elements. There are strong indications from [*ASCA*]{} data (Fukazawa et al. 2000; Finoguenov et al. 2001) that the Fe/Si ratio rises in the cluster centers, consistent with the cD galaxy being a source of Fe-rich material, probably due to Type Is SNe. However, the new [*XMM-Newton*]{} data show that O does not follow this pattern. It is clear that more work is necessary with larger samples and abundance profiles before we can obtain a clear picture of the metal enrichment process in clusters.
Conclusion
----------
The progress in this field in the last 10 years has been amazing. The X-ray properties of objects at redshift $z \approx 0.8$ are routinely measured, and clusters are now X-ray detected at $z>1.15$. The use of clusters for cosmology, an area covered in the volume by Freedman (2004), is exploding. The physics of clusters and groups holds the key to understanding the origin and evolution of structure and the origin of the elements. It was the cluster data that first showed that most of the baryons and metals in the Universe are in the hot phase, and that the baryonic Universe, as seen by our eyes, is only a shadow of the real Universe. In the next few years we will continue to obtain vast amounts of new data from [*Chandra*]{} and [*XMM-Newton*]{}, and much of the present observations will be analyzed, interpreted, and new patterns found. There are over 400 [*Chandra*]{} and [*XMM-Newton*]{} observations of clusters and groups in the database so far, with many more to be observed over the lifetimes of these telescopes. I anticipate many major new discoveries based on these instruments. Furthermore, the launch of [*Astro-E2*]{} in 2005 will allow detailed measurements of cluster turbulence, accurate abundances of many elements outside the cluster cores, and direct measures of the thermodynamic properties of the gas.
The field has benefited enormously from the synergistic interaction of theory and observation. Most theorists and observers are now aware of the major issues and the current observational capabilities. Looking beyond the next few years, I anticipate that a major new X-ray survey, perhaps 30 times better than [*ROSAT*]{}, will fly, producing an extremely large and uniform cluster catalog complete out to $z
\approx 0.7$. In the more distant future, the [*Constellation-X*]{} mission will provide precision temperatures and abundances out to the highest redshifts that clusters exist.
. I would like to thank my long-time collaborators and students at Goddard for their major contribution to this work: Keith Arnaud, Wayne Baumgartner, Don Horner, Mike Loewenstein, and John Mulchaey. I would like to thank the [*Chandra*]{} and [*XMM-Newton*]{} projects for their major efforts in developing, launching, and operating these amazing instruments. I would also like to thank the [*ASCA*]{} team for their pioneering efforts in the first X-ray imaging spectroscopy mission. I thank M. Arnaud and D. Neumann for communicating results ahead of publication. I also thank the organizers, especially John Mulchaey, for an exciting and stimulating meeting.
Adelberger, Kurt L., Steidel, C. C., Shapley, A. E., & Pettini, M. 2003, , 584, 45
Allen, S. W., Schmidt R. W., & Fabian A. C. 2001, MNRAS, 328, L37
——. 2002, MNRAS, 334, L11
Arabadjis, J. S., Bautz, M. W., & Garmire, G. P. 2002, , 572, 66
Arnaud, M., et al. 2001, å, 365, 67
Arnaud, M., Aghanim, N., & Neumann, D. M. 2002, å, 389, 1
Arnaud, M., Rothenflug, R., Boulade, O., Vigroux, L., & Vangioni-Flam, E. 1992, å, 254, 49
Awaki, H., et al. 1994, PASJ, 46, L65
Bahcall, N. A. 1977a, , 15, 505
——. 1977b, , 218, 9
Baumgartner, W., et al. 2004, in preparation
Bautz, M. W., & Arabadjis, J. S. 2004, in Carnegie Observatories Astrophysics Series, Vol. 3: Clusters of Galaxies: Probes of Cosmological Structure and Galaxy Evolution, ed. J. S. Mulchaey, A. Dressler, & A. Oemler (Pasadena: Carnegie Observatories, http://www.ociw.edu/ociw/symposia/series/symposium3/proceedings.html)
Bell, E. F., McIntosh, D. H., Katz, N., & Weinberg, M. D. 2003, , 585, L117
Bialek, J. J., Evrard, A. E., & Mohr, J. J. 2002, , 578, L9
Bird, C. M., Mushotzky, R. F., & Metzler, C. A. 1995, , 453, 40
Biviano, A., & Girardi, M. 2003, , 585, 205
Borgani S., Governato F., Wadsley, J., Menci, N., Tozzi, P., Quinn, T., Stadel, J., & Lake, G. 2002, MNRAS, 336, 409
Borgani, S., & Guzzo, L. 2001, Nature, 409, 39
Briel, U. G., & Henry, J. P. 1994, Nature, 372, 439
Churazov, E., Forman, W., Jones, C., & Böhringer, H. 2003, , 590, 225
David, L. P., Nulsen, P. E. J., McNamara, B. R., Forman, W., Jones, C., Ponman, T., Robertson, B., & Wise, M. 2001, , 557, 546
De Grandi, S., et al. 2003, Ringberg Conference Proceedings (XXX)
De Grandi, S., & Molendi, S. 2002, , 567, 163
Dupke, R. A., & White, R. E., III 2000, , 528, 139
Edge, A. C., & Stewart, G. C. 1991, MNRAS, 252, 428
Eke, V. R., Navarro, J. F., & Frenk, C. S. 1998, , 503, 569
Ettori, S., & Fabian, A. C 1999, MNRAS, 305, 834
Ettori, S., & Lombardi, M. 2003, å, 398, L5
Ettori, S., Tozzi, P., & Rosati, P. 2003, å, 398, 879
Evrard, A. E. 2004, in Carnegie Observatories Astrophysics Series, Vol. 3: Clusters of Galaxies: Probes of Cosmological Structure and Galaxy Evolution, ed. J. S. Mulchaey, A. Dressler, & A. Oemler (Cambridge: Cambridge Univ. Press), in press
Fabian, A. C., Crawford, C. S., Edge, A. C., & Mushotzky, R. F. 1994, MNRAS, 267, 779
Finoguenov, A., David, L. P., & Ponman, T. J. 2000, , 544, 188
Finoguenov, A., Reiprich, T. H., & Böhringer, H., 2001, å, 330, 749
Forman, W., & Jones, C. 1982, , 20, 547
Freedman, W. L., ed. 2004, Carnegie Observatories Astrophysics Series, Vol. 2: Measuring and Modeling the Universe (Cambridge: Cambridge Univ. Press), in press
Fujita, Y., Sarazin, C. L., Nagashima, M., & Yano, T. 2002, , 577, 11
Fukazawa, Y., Makishima, K., Tamura, T., Nakazawa, K., Ezawa, H., Ikebe, Y., Kikuchi, K., & Ohashi, T. 2000, MNRAS, 313, 21
Gibson, B. K., Loewenstein, M., & Mushotzky, R. F. 1997, MNRAS, 290, 623
Girardi, M., Giuricin, G., Mardirossian, F., Mezzetti, M., & Boschin, W. 1998, , 505, 74
Girardi, M., Manzato, P., Mezzetti, M., Giuricin, G., & Limboz, F. 2002, , 569, 720
Hasinger, G., et al. 2004, in preparation
Helsdon, S. F., & Ponman, T. J. 2000, , 315, 356
——. 2003, , 340, 485
Henry, J. P., Branduardi, G., Briel, U., Fabricant, D., Feigelson, E., Murray, S., Soltan, A., & Tananbaum, H. 1979, , 234, L15
Horner, D. J. 2001, Ph.D. Thesis, Univ. Maryland
Horner, D. J., et al. 2004, in preparation
Horner, D. J., Mushotzky, R. F., & Scharf, C. A. 1999, , 520, 78
Hughes, J. P., Yamashita, K., Okumura, Y., Tsunemi, H., & Matsuoka, M. 1988, , 327, 615
Ikebe, Y., Makishima, K., Fukazawa, Y., Tamura, T., Xu, H., Ohashi, T., & Matsushita, K. 1999, , 525, 58
Irwin, J. A., & Bregman, J. N. 2000, , 538, 543
Jeltema, T. E., Canizares, C. R., Bautz, M. W., Malm, M. R., Donahue, M., & Garmire, G. P. 2001, , 562, 124
Jones, C., & Forman, W. 1978, , 224, 1
——. 1984, , 276, 38
Jones, L. R., et al. 2004, in Carnegie Observatories Astrophysics Series, Vol. 3: Clusters of Galaxies: Probes of Cosmological Structure and Galaxy Evolution, ed. J. S. Mulchaey, A. Dressler, & A. Oemler (Pasadena: Carnegie Observatories, http://www.ociw.edu/ociw/symposia/series/symposium3/proceedings.html)
Kaastra, J. S., Ferrigno, C., Tamura, T., Paerels, F. B. S., Peterson, J. R., & Mittaz, J. P. D. 2001, å, 365, L99
Kaiser, N. 1986, MNRAS, 222, 323
Kikuchi, K., Furusho, T., Ezawa, H., Yamasaki, N. Y., Ohashi, T., Fukazawa, Y., & Ikebe, Y. 1999, PASJ, 51, 301
Kochanek, C. S., White, M., Huchra, J., Macri, L., Jarrett, T. H., Schneider, S. E., & Mader, J. 2003, , 585, 161
Komatsu, E., & Seljak, U. 2001, MNRAS, 327, 1353
Lewis, A. D., Buote, D. A., & Stocke, J. T. 2003, , 586, 135
Lin, Y.-T., Mohr, J. J., & Stanford, S. A. 2003, , in press (astro-ph/0304033)
Loken, C., Norman, M. L., Nelson, E., Burns, J., Bryan, G. L., & Motl, P. 2002, , 579, 571
Majerowicz, S., Neumann, D. M., & Reiprich, T. 2002, å, 394, 77
Marinoni, C., & Hudson, M. J. 2002, , 569, 101
Markevitch, M. 1996, , 465, L1
——. 1998, , 504, 27
Markevitch, M., et al. 2003, , 586, L19
Markevitch, M., Gonzalez, A. H., David, L., Vikhlinin, A., Murray, S., Forman, W., Jones, C., & Tucker, W. 2002, , 567, L27
Mazzotta, P., Fusco-Femiano, R., & Vikhlinin, A. 2002, , 569, L31
Motl, P. M., Burns, J. O., Loken, C., Norman, M. L., & Bryan, G. 2004, , in press (astro-ph/0302427)
Mulchaey, J. S. 2004, in Carnegie Observatories Astrophysics Series, Vol. 3: Clusters of Galaxies: Probes of Cosmological Structure and Galaxy Evolution, ed. J. S. Mulchaey, A. Dressler, & A. Oemler (Cambridge: Cambridge Univ. Press), in press
Mulchaey, J. S., Davis, D. S., Mushotzky, R. F., & Burstein, D. 2003, , 145, 39
Mushotzky, R. F. 1984, Physica Scripta, 7, 157
——. 2002, in A Century of Space Science, ed. J. A. M. Bleecker, J. Geis, & M. Huber (Dordrecht: Kluwer), 473
Mushotzky, R. F., Loewenstein, M., Arnaud, K. A., Tamura, T., Fukazawa, Y., Matsushita, K., Kikuchi, K., & Hatsukade, I. 1996, , 466, 686
Mushotzky, R. F., Serlemitsos, P. J., Boldt, E. A., Holt, S. S., & Smith, B. W. 1978, , 225, 21
Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, , 490, 493 (NFW)
Neumann, D. M., & Arnaud, M. 2001, å, 373, L33
Peterson, J. R., Kahn, S. M., Paerels, F. B. S., Kaastra, J. S., Tamura, T., Bleeker, J. A. M., Ferrigno, C., & Jernigan, J. G. 2003, , 590, 207
Pratt, G. W., & Arnaud, M. 2002, å, 394, 375
Pratt, G. W., Arnaud, M., & Aghanim, N. 2002, in Tracing Cosmic Evolution with Galaxy Clusters, ed. S. Borgani, M. Mezzetti, & R. Valdarnini (San Francisco: ASP), 433
Reiprich, T. H., & Böhringer H. 2002, , 567, 716
Renzini, A. 2004, in Carnegie Observatories Astrophysics Series, Vol. 3: Clusters of Galaxies: Probes of Cosmological Structure and Galaxy Evolution, ed. J. S. Mulchaey, A. Dressler, & A. Oemler (Cambridge: Cambridge Univ. Press), in press
Sanderson, A. J. R., Ponman, T. J., Finoguenov, A., Lloyd-Davies, E. J., & Markevitch, M. 2003, , 340, 989
Shimizu, M., Kitayama, T., Sasaki, S., & Suto, Y. 2003, , 590, 197
Stanford, S. A., Holden, B., Rosati, P., Tozzi, P., Borgani, S., Eisenhardt, P. R., & Spinrad, H. 2001, , 552, 504
Sun, M., Murray, S. S., Markevitch, M., & Vikhlinin, A. 2002, , 565, 867
Sutherland, R. S., & Dopita, M. A. 1993, , 88, 253
Tamura, T., Bleeker, J. A. M., Kaastra, J. S., Ferrigno, C., & Molendi, S. 2001, å, 379, 107
Thomas, P. A., Muanwong, O., Kay, S. T., & Liddle, A. R. 2002, MNRAS, 330, L48
Vikhlinin, A., Forman, W., & Jones, C. 1999, , 525, 47
Vikhlinin, A. A., & Markevitch, M. L. 2002, Astron. Lett., 28, 495
Vikhlinin, A., VanSpeybroeck, L., Markevitch, M., Forman, W. R., & Grego, L. 2002, , 578, L107
Watanabe, M., Yamashita, K., Furuzawa, A., Kunieda, H., & Tawara, Y. 2001, PASJ, 53, 605
White, D. A., & Buote, D. A. 2000, , 312, 649
White, S. D. M., Navarro, J. F., Evrard, A. E., & Frenk, C. S. 1993, Nature, 366, 429
Yee, H. K. C., & Ellingson, E. 2003, , 585, 215
|
---
abstract: 'We consider the problem of communicating over a channel for which no mathematical model is specified. We present achievable rates as a function of the channel input and output known a-posteriori for discrete and continuous channels, as well as a rate-adaptive scheme employing feedback which achieves these rates asymptotically without prior knowledge of the channel behavior.'
author:
- |
Yuval Lomnitz, Meir Feder\
Tel Aviv University, Dept. of EE-Systems\
Email: {yuvall,meir}@eng.tau.ac.il
title: Communication over Individual Channels
---
=1
Introduction
============
The problem of communicating over a channel with an individual, predetermined noise sequence which is not known to the sender and receiver was addressed by Shayevitz and Feder [@Ofer_BSC] [@Ofer_EMP] and Eswaran et al [@Eswaran][@Eswaran_conf]. The simple example discussed in [@Ofer_BSC] is of a binary channel $y_n=x_n \oplus e_n$ where the error sequence $e_n$ can be any unknown sequence. Using perfect feedback and common randomness, communication is shown to be possible in a rate approaching the capacity of the binary symmetric channel (BSC) where the error probability equals the empirical error probability of the sequence (the relative number of ’1’-s in $e_n$). Subsequently both authors extended this model to general discrete channels and modulu-additive channels ([@Eswaran], [@Ofer_EMP] resp.) with an individual state sequence, and showed that the empirical mutual information can be attained.
Now we take this model one step further. We consider a channel where no specific probabilistic or mathematical relation between the input and the output is assumed. In order to define positive communication rates without assumptions on the channel, we characterize the achievable rate using the specific input and output sequences, and we term this channel an *individual channel*. This way of treating with unknown channels is different from other concepts of dealing with the problem, such as compound channels and arbitrarily varying channels, in the fact that the later require a specification of the channel model up to some unknown parameters, whereas the current approach makes no a-priori assumptions about the channel behavior. We usually assume the existence of a feedback link in which the channel output or other information from the decoder can be sent back to the encoder. Without this feedback it would not be possible to match the rate of transmission to the quality of the channel so outage would be inevitable.
Although one may not be fully convenient with the mathematical formulation of the problem, there is no question about the reality of this model: this is the only channel model that we know for sure exists in nature. This point of view is similar to the approach used in universal source coding of individual sequences where the goal is to asymptotically attain for each sequence the same coding rate achieved by the best encoder from a model class, tuned to the sequence.
Just to inspire thought, let’s ask the following question: suppose the sequence $\{x_i\}_{i=1}^n$ with power $P = \frac{1}{n}\sum_{i=1}^{n}{x_i^2}$ encodes a message and is transmitted over a continuous real-valued input channel. The output sequence is $\{y_i\}_{i=1}^n$. One can think of $v_i = y_i - x_i$ as a noise sequence and measure its power $N=\frac{1}{n}\sum_{i=1}^{n}{v_i^2}$. Is the rate $R=\half \log \left( 1 + \frac{P}{N} \right)$ which is the Gaussian channel capacity, achievable in this case, under appropriate definitions ?
The way it was posed, the answer to this question would be “no”, since this model predicts a rate of $\half$ bit/use for the channel whose output is $\forall i: y_i=0$ which cannot convey any information. However with the slight restatement done in the next section the answer would be “yes”.
We consider two classes of individual channels: discrete input and output channels and continuous real valued input and output channels, and two communication models: with feedback and without feedback. In both cases we assume common randomness exists. The case of feedback is of higher interest, since the encoder can adapt the transmission rate and avoid outage. The case of no-feedback is used as an intermediate step, but the results are interesting since they can be used for analysis of semi probabilistic models. The main result is that with a small amount of feedback, a communication at a rate close to the empirical mutual information (or its Gaussian equivalent for continuous channels) can be achieved, without any prior knowledge, or assumptions, about the channel structure.
The paper is organized as follows: in section \[sec:overview\] we give a high level overview of the results. In section \[sec:definitions\] we define the model and notation. Section \[sec:nonadaptive\] deals with communication without feedback where the results pertaining to discrete and continuous case are formalized and proven, and the choice of the rate function and the Gaussian prior for the continuous case is justified. Section \[sec:rate\_adaptive\] deals with the case where feedback is present. After reviewing similar results we state the main result and the adaptive rate scheme that achieves it, and delay the proof to section \[sec:analysis\]. Here, the error probability and the achieved rate are analyzed and bounded. Section \[sec:examples\] gives several examples, and section \[sec:comments\] is dedicated to comments and highlights areas for further study.
Overview of main results {#sec:overview}
========================
We start with a high level overview of the definitions and results. The definitions below are conceptual rather than accurate, and detailed definitions follow in the next sections.
A rate function is a function $\Remp : \mathcal{X}^n \times \mathcal{Y}^n \to \mathbb{R}$ of the input and output sequences. In communication without feedback we say a given rate function is achievable if for large block size $n \rightarrow \infty$, it is possible to communicate at rate $R$ and an arbitrarily small error probability is obtained whenever $\Remp$ exceeds the rate of transmission, i.e. whenever $\Remp(\vr{x}, \vr{y}) > R$. In communication with feedback we say a given rate function is achieved by a communication scheme if for large block size $n$, data at rate close to or exceeding $\Remp(\vr{x}, \vr{y})$ is decoded successfully with arbitrarily large probability for every output sequence and almost every input sequence. Roughly speaking, this means that in any instance of the system operation, where a specific $\vr x$ was the input and a specific $\vr y$ was the output, the communication rate had been at least $\Remp(\vr x, \vr y)$. Note that the only statistical assumptions are related to the common randomness, and we consider the rate and error probability *conditioned* on a specific input and output, where the error probability is averaged over common randomness. We say that a rate function $\Remp$ is *an* optimal (but not *the* optimal) function if any $\Remp' \geq \Remp $ which is strictly larger than $\Remp$ at at least one point, is not achievable.
The definition of achievability is not complete without stating the input distribution, since it affects the empirical rate. For example, by setting $\vr x = 0$ one can attain every rate function where $\Remp(0,\vr y)=0$ in a void way, since other $\vr x$ sequences will never appear. Different from classical results in information theory, we do not use the input distribution only as a means to show the existence of good codes: taking advantage of the common randomness we require the encoder to emit input symbols that are random and distributed according to a defined prior (currently we assume i.i.d. distribution).
The choice of the rate functions is arbitrary in a way: for any pair of encoder and decoder, we can tailor a function $\Remp(\vr{x}, \vr{y})$ as a function equaling the transmitted rate whenever the error probability given the two sequences (averaged over messages and the common randomness) is sufficiently small, and 0 otherwise. However it is clear that there are certain rates which cannot be exceeded uniformly. Our interest will focus on simple functions of the input and output, and specifically in this paper we focus on functions of the instantaneous (zero order) empirical statistics. Extension to higher order models seems technical.
For the discrete channel we show that a rate $$\label{R_emp_discrete}
\Remp = \hat{I} (\vr{x}; \vr{y})$$ is achievable with any input distribution $P_X$ where $\hat{I}(\cdot;\cdot)$ denotes the empirical mutual information [@Goppa] (see definition in section \[sec:definitions\], and Theorems \[theorem:discrete\_nonadaptive\], \[theorem:discrete\_adaptive\]). For the continuous (real valued) channel we show that a rate
$$\label{R_emp_continuous}
\Remp = \half \log \left( \frac{1}{1-\hat\rho(\vr x, \vr y)^2} \right)$$
is achievable with Gaussian input distribution $\Normal(0,P)$, where $\hat \rho$ is the empirical correlation factor between the input and output sequences (see Theorems \[theorem:continuous\_nonadaptive\], \[theorem:continuous\_adaptive\]). These results pertain both to the case of feedback and of no-feedback according to the definitions above.
Throughout the current paper we define correlation factor in a slightly non standard way as $\rho = \frac{E(XY)}{\sqrt{E(X^2)E(Y^2)}}$ (that is, without subtracting the mean). This is done only to simplify definitions and derivations, and similar claims can be made using the correlation factor defined in the standard way. Although the result regarding the continuous case is less tight, we show that this is the best rate function that can be defined by second order moments, and is tight for the Gaussian additive channel (for this channel $\rho^2 = \frac{P}{P+N}$ therefore $\Remp = \half \log \left( 1 + \frac{P}{N}\right)$)
We may now rephrase our example question from the introduction so that it will have an affirmative answer: given the input and output sequences, describe the output by the virtual additive channel with a gain $y_i = \alpha x_i + v_i$, so the effective noise sequence is $v_i = y_i - \alpha x_i$. Chose $\alpha$ so that $\vr v \perp \vr x$, i.e. $\frac{1}{n} \sum_i{v_i x_i}=0$. An equivalent condition is that $\alpha$ minimizes $\lVert \vr v \rVert^2$. The resulting $\alpha$ is the LMMSE coefficient in estimation of $\vr y$ from $\vr x$ (assuming zero mean), i.e. $\alpha = \frac{\vr x ^T \vr y}{\lVert \vr x \rVert^2}$. Define the effective noise power as $N=\frac{1}{n}\sum_{i=1}^{n}{v_i^2}$, and the effective $\textit{SNR} \equiv \frac{\alpha^2 P}{N}$. It is easy to check that $\textit{SNR}=\frac{\hat\rho^2}{1-\hat\rho^2}$ where $\hat\rho=\frac{\vr x ^T \vr y}{\lVert \vr x \rVert \cdot \lVert \vr y \rVert}$ is the empirical correlation factor between $\vr{x}$ and $\vr{y}$. Then according to Eq.(\[R\_emp\_continuous\]) the rate $R=\half \log \left( 1 + \textit{SNR} \right)$ is achievable, in the sense defined above. Reexamining the counter example we gave above, in this model if we set $\vr y = 0$ we obtain $\hat\rho=0$ and therefore $\Remp=0$, or equivalently the effective channel has $\vr v = 0$ and $\alpha=0$, therefore $\textit{SNR}=0$ (instead of $\vr v = -\vr x$, $\alpha=1$ and $\textit{SNR}=1$).
As will be seen, we achieve these rates by random coding and universal decoders. For the case of feedback we use iterated instances of rateless coding (i.e. we encode a fixed number of bits and the decision time depends on the channel). The scheme is able to operate asymptotically with “zero rate” feedback (meaning any positive capacity of the feedback channel suffices). A similar although more complicated scheme was used in [@Eswaran] (see a comparison in the appendix).
Before the detailed presentation we would like to examine the differences between the model used here and two proximate models: the arbitrarily varying channel (AVC) and the channel with individual noise sequence.
In the AVC (see for example [@Lapidoth_AVC][@Csiszar_AVC]), the channel is defined by a probabilistic model which includes an unknown state sequence. Constraints on the sequence (such as power, number of errors) may be defined, and the target is to communicate equally well over all possible occurrences of the state sequence. In AVC, the capacity depends on the existence of common randomness and on whether the average or maximum error probability (over the messages) is required to approach $0$, yet when sufficient common randomness is used, the capacities for maximum and average error probability are equal. The notes in [@Lapidoth_AVC] regarding common randomness and randomized encoders (see p.2151) are also relevant to our case.
A treatment of AVC-s which is similar in spirit to our results exists in watermarking problems. For example a rather general case of AVC is discussed in [@Agarwal_RD]. They consider communication over a black box (representing the attacker) which is only limited to a given level $D$ of distortion according to a predefined metric, but has otherwise a block-wise undefined behavior. They show that it is possible to achieve a rate equal to the rate-distortion function of the input $R_X(D)$, if the black box guarantees a given level of average distortion in high probability. This result is similar to our Theorem \[theorem:discrete\_nonadaptive\]. The remarkable distinction from other results for AVC is that the rate is determined using a constraint on the channel inputs and outputs, rather than the channel state sequence. We note that for the Gaussian additive channel the above result is suboptimal since the rate is $R_X(N)=\half \log(P/N)$ and our results improve this result by using the correlation factor yields rather than the mean squared error. See further discussion of these results in the proof Lemma \[lemma:pairwise\_discrete\] and the discussion following Theorem \[theorem:discrete\_adaptive\].
Channels with individual noise (or state) sequence are treated by Shayevitz and Feder [@Ofer_BSC][@Ofer_EMP] and Eswaran et al [@Eswaran]. The probabilistic setting is the same as in the AVC, and the difference is that instead of achieving a uniform (hence worst-case) rate, the target is to achieve a variable rate which depends on the particular sequence of noise, using a feedback link. In this setup, prior constraints on the state sequence can be relaxed. As opposed to AVC where the capacity is well defined, the target rate for each state sequence is determined in a somewhat arbitrary way (since many different constraints on the sequence can be defined). As an example, in the binary channel of [@Ofer_BSC], a rate of 0 would be obtained for the sequence $\vr{e}='01010101...'$ since the empirical error probability is $\half$, although obviously a scheme which favors this specific sequence and achieves a rate of 1 can be designed. On the other hand, with the AVC approach communication over this channel would not be possible without prior constraints on the noise sequence. Channels with individual noise sequence can be thought of as compound-AVCs (i.e. an AVC with unknown parameter, in this case, the constraint). As in AVC, existence of common randomness as well as the definition of error probability affect the achievable rates.
In the individual channel model we use here, since no equation with state sequence connecting the input and output is given, the achievable rates cannot be defined without relating to the channel input. Therefore the definitions of achieved rates depend in a somewhat circular way on the channel input which is determined by the scheme itself. Currently we circumvent this difficulty by constraining the input distribution, as mentioned above.
In many aspects the model used in this paper is more stringent than the AVC and the individual noise sequence models, since it makes less assumptions on the channel, and the error probability is required to be met for (almost) every input and output sequence (rather than on average). In other aspects it is lenient since we may attribute ’bad’ channel behavior to the rate rather than suffer an error, therefore the error exponents are better than in probabilistic models. This is further explained in section \[sec:discrete\_nonadaptive\].
The model we propose suggests a new approach for the design of communication systems. The classical point of view first assumes a channel model and then devises a communication system optimized for it. Here we take the inverse direction: we devise a communication system without assumptions on the channel which guarantees rates depending on channel behavior. This change of viewpoint does not make probabilistic or semi probabilistic channel models redundant but merely suggests an alternative. By using a channel model we can formalize questions relating to optimality such as capacity (single user, networks) and error exponent as well as guarantee a communication rate a-priori. Another aspect is that we pay a price for universality. Even if one considers an individual channel scheme that guarantees asymptotically optimum rates over a large class of channels, it can never consider all possible channels (block-wise), and for a finite block size it will have a larger overhead (a reduction in the amount of information communicated with same error probability) compared to a scheme optimized for the specific channel.
Following our results, the individual channel approach becomes a very natural starting point for determining achievable rates for various probabilistic and arbitrary models (AVC-s, individual noise sequences, probabilistic models, compound channels) under the realm of randomized encoders, since the achievable rates for these models follow easily from the achievable rates for specific sequences, and the law of large numbers. We will give some examples later on.
Definitions and notation {#sec:definitions}
========================
Notation {#sec:notation}
--------
In general we use uppercase letters to denote random variables, respective lowercase letters to denote their sample values and boldface letters to denote vectors, which are by default of length n. However we deviate from this practice when the change of case leads to confusion, and vectors are always denoted by lowercase letters even when they are random variables.
$\lVert \vr x \rVert \equiv \sqrt{\vr x^T \vr x}$ denotes $L_2$ norm. We denote by $P \circ Q$ the product of conditional probability functions e.g. $(P \circ Q)(x,y) = P(x) \cdot Q(y|x)$. A hat ($\hat{\square}$) denotes an estimated value.
We denote the empirical distribution as $\hat{P}$ (e.g. $\hat{P}_{(\vr x, \vr y)}(x,y) \equiv \frac{1}{n} \sum_{i=1}^n {\delta_{(\vr x_i - x), (\vr y_i - y)}}$). The source vectors $\vr x, \vr y$ and/or the variables $x,y$ are sometimes omitted when they are clear from the context. We denote by $\hat{H}(\cdot)$, $\hat{I}(\cdot;\cdot)$, $\hat{\rho}(\cdot;\cdot)$ the empirical entropy, the empirical mutual information and the empirical correlation factor, which are the respective values calculated for the empirical distribution. All expressions such as $\hat H(\vr x)$, $\hat H(\vr x | \vr y)$, $\hat I (\vr x; \vr y)$, $\hat I (\vr x; \vr y | \vr z)$, $\hat I (\vr x; \vr y | \vr z = z_0)$ are interpreted as their respective probabilistic counterparts $H(X)$, $H(X|Y)$, $I (X;Y)$, $I (X;Y|Z)$, $I (X;Y|Z=z_0)$ where $(X,Y,Z)$ are random variables distributed according to the empirical distribution of the vectors $\hat{P}_{(\vr x, \vr y, \vr z)}$, or equivalently are defined as a random selection of an element of the vectors i.e. $(X,Y,Z)=(x_i, y_i, z_i), i \sim \mathrm{U}\{1,\ldots,n\}$. It is clear from this equivalence that relations on entropy and mutual information (e.g. positivity, chain rules) are directly translated to relations on their empirical counterparts.
We apply superscript and subscript indices to vectors to define subsequences in the standard way, i.e. $\vr x_i^j \equiv (x_i, x_{i+1}, ... , x_j)$, $\vr x^i \equiv \vr x_1^i$
We denote $I(P,W)$ the mutual information $I(X;Y)$ when $(X,Y) \sim P(x)\cdot W(y|x)$. $\mathrm{U}(A)$ denotes a uniform distribution over the set $A$. $Ber(p)$ denotes the Bernoulli distribution, and $h_b(p) \equiv H(Ber(p)) = -p \log p - (1-p) \log (1-p) $ denotes the binary entropy function. The indicator function $\Ind(E)$ where $E$ is a set or a probabilistic event is defined as $1$ over the set (or when the event occurs) and $0$ otherwise.
The functions $\log(\cdot)$ and $\exp(\cdot)$ as well as information theoretic quantities $H(\cdot), I(\cdot;\cdot), D(\cdot || \cdot)$ refer to the same, unspecified base. We use the term “information unit” as the unit of these quantities (equals $\frac{1}{\log(2)}$ bits).
The notation $f_n = O(g_n)$ and $f_n < O(g_n)$ (or equivalently $O(f_n) = O(g_n)$ and $O(f_n) < O(g_n)$) means $\frac{f_n}{g_n} {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty}$}} \const > 0$ and $\frac{f_n}{g_n} {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty}$}} 0$ respectively.
Throughout this paper we use the term “continuous” to refer to the continuous *real valued* channel $\mathbb{R} \to \mathbb{R}$, although this definition does not cover all continuous input - continuous output channels. By the term “discrete” in this paper we always refer to finite alphabets (as opposed to countable ones).
Definitions {#sec:definitions}
-----------
A channel is defined by a pair of input and output alphabets $\mathcal{X,Y}$, and denoted $\mathcal{X} \to \mathcal{Y}$
A randomized block encoder and decoder pair for the channel $\mathcal{X \to Y}$ with block length $n$ and rate $R$ without feedback is defined by a random variable $S$ distributed over the set $\mathcal{S}$, a mapping $\phi : \{1,2,\ldots\exp(nR)\} \times \mathcal{S} \to \mathcal{X}^n$ and a mapping $\bar\phi : \mathcal{Y}^n \times \mathcal{S} \to \{1,2,
\ldots\exp(nR)\}$. The error probability for message $w \in \{1,2,\ldots\exp(nR)\}$ is defined as $$P_e^{(w)}(\vr x, \vr y) = \Pr \left(\bar\phi(\vr y, S) \neq w \big\vert \phi(w,S) = \vr x \right)$$
where for $\vr x$ such that the condition cannot hold, we define $P_e^{(w)}(\vr x, \vr y)=0$.
Note that the encoder rate must pertain to a discrete number of messages $\exp(nR) \in \mathbb{Z}_+$, but the empirical rates defined in the following theorems may be any positive real numbers.
(140,30) (23,16)[Encoder]{}(20,10)[(1,0)[20]{}]{}(40,10)[(0,1)[15]{}]{}(40,25)[(-1,0)[20]{}]{}(20,25)[(0,-1)[15]{}]{} (63,16)[Channel]{}(60,15)[(1,0)[20]{}]{}(80,15)[(0,1)[5]{}]{}(80,20)[(-1,0)[20]{}]{}(60,20)[(0,-1)[5]{}]{} (103,16)[Decoder]{}(100,10)[(1,0)[20]{}]{}(120,10)[(0,1)[15]{}]{}(120,25)[(-1,0)[20]{}]{}(100,25)[(0,-1)[15]{}]{} (0,17.5)[(1,0)[20]{}]{}(0,18.5)[$\vr w$ (message)]{} (40,17.5)[(1,0)[20]{}]{}(45,18.5)[$x_i \in \mathcal{X}$]{} (80,17.5)[(1,0)[20]{}]{}(85,18.5)[$y_i \in \mathcal{Y}$]{} (120,17.5)[(1,0)[20]{}]{}(125,18.5)[$\hat{\vr w}$ (message)]{} (30,5)[(0,1)[5]{}]{}(30,0)[$S$ (common randomness)]{} (110,5)[(0,1)[5]{}]{}(110,0)[$S$]{}
A randomized block encoder and decoder pair for the channel $\mathcal{X \to Y}$ with block length $n$, adaptive rate and feedback is defined as follows:
- The message $\vr w$ is expressed by the infinite sequence $\vr w_1^{\infty} \in \{0,1\}^{\infty}$
- The common randomness is defined as a random variable $S$ distributed over the set $\mathcal{S}$
- The feedback alphabet is denoted $\mathcal{F}$
- The encoder is defined by a series of mappings $ x_k = \phi_k(\vr w, s, \vr f^{k-1})$ where $\phi_k : \{0,1\}^{\infty} \times \mathcal{S} \times \mathcal{F}^{k-1} \to \mathcal{X}$.
- The decoder is defined by the feedback function $\varphi_k: \mathcal{Y}^{k-1} \times \mathcal{S} \to \mathcal{F}$, the decoding function $\bar\phi : \mathcal{Y}^n \times \mathcal{S} \to \{0,1\}^{\infty}$ and the rate function $r : \mathcal{Y}^n \times \mathcal{S} \to \mathbb{R}^+$ (where the rate is measured in bits), applied as follows:
$$\begin{aligned}
f_k &=& \varphi_k(\vr y^k, S) \\
\hat {\vr w} &=& \bar\phi (\vr y, S) \\
R &=& r (\vr y, S)\end{aligned}$$
The error probability for message $\vr w$ is defined as
$$P_e^{(\vr w)}(\vr x, \vr y) = \Pr \left({\hat {\vr w}}_1^{\lceil nR \rceil} \neq {{\vr w}}_1^{\lceil nR \rceil} \big\vert \vr x, \vr y \right)$$
In other words, a recovery of the first $\lceil nR \rceil$ bits by the decoder is considered a successful reception. For $\vr x$ such that the condition cannot hold, we define $P_e^{(\vr w)}(\vr x, \vr y)=0$. The conditioning on $\vr y$ is mainly for clarification, since it can be treated as a fixed vector. This system is illustrated in figure \[fig:system\_adaptive\].
(140, 30) (23,16)[Encoder]{}(20,10)[(1,0)[20]{}]{}(40,10)[(0,1)[15]{}]{}(40,25)[(-1,0)[20]{}]{}(20,25)[(0,-1)[15]{}]{} (63,20)[Channel]{}(60,19)[(1,0)[20]{}]{}(80,19)[(0,1)[5]{}]{}(80,24)[(-1,0)[20]{}]{}(60,24)[(0,-1)[5]{}]{} (103,16)[Decoder]{}(100,10)[(1,0)[20]{}]{}(120,10)[(0,1)[15]{}]{}(120,25)[(-1,0)[20]{}]{}(100,25)[(0,-1)[15]{}]{} (0,17.5)[(1,0)[20]{}]{}(6,18.5)[$\vr w$]{}(0,14)[(message)]{} (40,21.5)[(1,0)[20]{}]{}(42,22.5)[$x_i \in \mathcal{X}$]{} (80,21.5)[(1,0)[20]{}]{}(82,22.5)[$y_i \in \mathcal{Y}$]{} (100,15)[(-1,0)[60]{}]{}(55,10)[$f_i \in \mathcal{F}$ (feedback)]{} (120,21.5)[(1,0)[20]{}]{}(125,22.5)[$R$ (rate)]{} (120,15)[(1,0)[20]{}]{}(125,16)[$\hat{\vr w}$ (message)]{} (30,5)[(0,1)[5]{}]{}(30,0)[$S$ (common randomness)]{} (110,5)[(0,1)[5]{}]{}(110,0)[$S$]{}
Note that if we are not interested in limiting the feedback rate, and perfect feedback can be assumed, the definition of feedback alphabet and feedback function is redundant (in this case $\mathcal{F} = \mathcal{Y}$ and $f_k=y_k$). The model in which the decoder determines the transmission rate is lenient in the sense that it gives the flexibility to exchange rate for error probability: the decoder may estimate the error probability and decrease it by reducing the decoding rate. In the scheme we discuss here the rate is determined during reception, but it’s worth noting in this context the posterior matching scheme [@Ofer_Posterior_analysis] for the known memoryless channel. In this scheme the message is represented as a real number $\theta \in [0,1)$ and the rate for a given error probability $P_e$ can be determined *after* the decoding by calculating $\Pr(\theta \vert \vr y)$ and finding the smallest interval with probability at least $1-P_e$.
Communication without feedback {#sec:nonadaptive}
==============================
In this section we show that the empirical mutual information (in the discrete case) and its Gaussian counterpart (in the continuous case) are achievable in the sense defined in the overview. For the continuous case we justify the choice of the Gaussian distribution as the one yielding the maximum rate function that can be defined by second order moments.
The discrete channel without feedback {#sec:discrete_nonadaptive}
-------------------------------------
The following theorem formalizes the achievability of rate $\hat{I} (\vr{x}; \vr{y})$ without feedback:
\[theorem:discrete\_nonadaptive\] Given discrete input and output alphabets $\mathcal{X,Y}$, for every $P_e>0$, $\delta>0$, prior $Q(x)$ over $\mathcal{X}$ and rate $R>0$ there exists $n$ large enough and a random encoder-decoder pair of rate $R$ over block size $n$, such that the distribution of the input sequence is $\vr{x} \sim Q^n$ and the probability of error for any message given an input sequence $\vr{x} \in \mathcal{X}^n$ and output sequence $\vr{y} \in \mathcal{Y}^n$ is not greater than $P_e$ if $\hat I (\vr{x},\vr{y})> R + \delta$.
Theorem \[theorem:discrete\_nonadaptive\] follows almost immediately from the following lemma, which is proven in the appendix using simple a calculation based on the method of types [@MethodOfTypes]:
\[lemma:pairwise\_discrete\] For any sequence $\vr y \in \mathcal{Y}^n$ the probability of a sequence $\vr x \in \mathcal{X}^n$ drawn independently according to $Q^n$ to have $\hat{I}(\vr x; \vr y) \geq t$ is upper bounded by: $$Q^n \left( \hat{I}(\vr x; \vr y) \geq t \right) \leq \exp \left( -n \left(t - \delta_n \right) \right)$$ where $\delta_n = |\mathcal{X}||\mathcal{Y}|\frac{\log(n+1)}{n} \to 0$.
Following notations in [@MethodOfTypes], $Q^n(A)$ denotes the probability of the event $A$ or equivalently the set of sequences $A$ under the i.i.d. distribution $Q^n$. Remarkably this bound does not depend on $Q$.
To prove Theorem \[theorem:discrete\_nonadaptive\], the codebook $\{\vr x_m\}_{m=1}^{\exp(nR)}$ is randomly generated by i.i.d. selection of its $L = \exp(nR) \cdot n$ letters, so that the common randomness $S \in \mathcal{X}^{L}$ may be defined as the codebook itself and is distributed $Q^{L}$. The encoder sends the $w$-th codeword, and the decoder uses maximum mutual information decoding (MMI) i.e. chooses: $$\hat{w} = \bar\phi (\vr y,\{\vr x_m\}) = {\raisebox{-1.2ex}{$\stackrel{\textstyle \mathrm{argmax}}{\scriptscriptstyle m}$}} \left[ \hat I (\vr x_m ; \vr y) \right]$$ where ties are broken arbitrarily. By Lemma \[lemma:pairwise\_discrete\], the probability of error is bounded by: $$\begin{gathered}
P_e^{(w)}(\vr x_w, \vr y) \leq \Pr \left\{ \bigcup_{m \neq w} \left( \hat I (\vr x_m ; \vr y) \geq \hat I (\vr x_w ; \vr y) \right) \right\} \leq \\
\leq \exp(nR) \exp \left( -n \left(\hat I (\vr x_w ; \vr y) - \delta_n \right) \right)
=\\= \exp \left( -n \left(\hat I (\vr x_w ; \vr y) - R - \delta_n \right) \right)\end{gathered}$$ For any $\delta$ there is $n$ large enough such that $\frac{-\log(P_e)}{n} + \delta_n < \delta$. For this $n$, whenever $\hat I (\vr x ; \vr y) > R + \delta$ we have $$P_e^{(w)}(x,y) \leq \exp \left( -n \left(\delta - \delta_n \right) \right) < P_e$$ which proves the theorem.
Note that the MMI decoder used here is a popular universal decoder (see [@Goppa][@MethodOfTypes][@Tchamkerten]), and was shown to achieve the same error exponent as the maximum likelihood decoder for fixed composition codes. The error exponent obtained here is better than the classical error exponent (slope of -1), and the reason is that the behavior of the channel is known, and therefore no errors occur as result of non-typical channel behavior. Comparing for example with the derivation of the random coding error exponent for the probabilistic DMC based on the method of types (see [@MethodOfTypes]), in the later the error probability is summed across all potential “behaviors” (conditional types) of the channel accounting for their respective probabilities (resulting in one behavior, usually different from the typical behavior, dominating the bound), while here the behavior of the channel (the conditional distribution) is fixed, and therefore the error exponent is better. This is not necessarily the best error exponent that can be achieved (see [@Tchamkerten][@Burnashev] which discuss error exponent with random decision time and feedback for probabilistic and compound models).
Note that the empirical mutual information is always well defined, even when some of the input and output symbols do not appear in the sequence, since at least one input symbol and one output symbol always appear. For the particular case of empirical mutual information measured over a single symbol, the empirical distributions become unit vectors (representing constants) and their mutual information is 0.
In this discussion we have not dealt with the issue of choosing the prior $Q(x)$. Since the channel behavior is unknown it makes sense to choose the maximum entropy, i.e. the uniform, prior which was shown to obtain a bounded loss from capacity [@Shulman_Prior].
The continuous channel without feedback {#sec:continuous_nonadaptive}
---------------------------------------
When turning to define empirical rates for the real valued alphabet case, the first obstacle we tackle is the definition of the empirical mutual information. A potential approach is to use discrete approximations. We only briefly describe this approach since it is somewhat arbitrary and less elegant than in the discrete case. The main focus is on empirical rates defined by the correlation factor. Although the later approach is pessimistic and falls short of the mutual information for most channels, it is much simpler and elegant than discrete approximations. We believe this approach can be further extended to obtain results closer to the (probabilistic) mutual information.
### Discrete approximations {#sec:continuous_by_quantization}
Define the continuous input and output alphabets $\mathcal{X},\mathcal{Y}$. Suppose $Q$ is an arbitrary (continuous) prior. Define input and output quantizers to discrete alphabets $A_n : \mathcal{X} \to \mathcal{\tilde{X}}_n$ and $B_n : \mathcal{Y} \to \mathcal{\tilde{Y}}_n$ where $\mathcal{\tilde{X}}_n$, $\mathcal{\tilde{Y}}_n$ are discrete alphabets of growing size, chosen to grow slowly enough so that $\delta_n = |\mathcal{\tilde{X}}_n||\mathcal{\tilde{Y}}_n|\frac{\log(n+1)}{n} {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty}$}} 0$. Define the empirical mutual information between continuous vectors as the empirical mutual information between their quantized versions (quantized letter by letter): $$\hat I_{A,B}(\vr x, \vr y) \equiv \hat I(A_n(\vr x), B_n(\vr y))$$ Then based on Lemma \[lemma:pairwise\_discrete\], by using a random codebook drawn according to $Q$ and applying a maximum mutual information decoder using the above definition, we could asymptotically achieve the rate function $\Remp = \hat I_{A,B}(\vr x, \vr y)$ based on the definitions of Theorem \[theorem:discrete\_nonadaptive\]. The main issue with this approach is that determining $A_n, B_n$ is arbitrary, and especially $B_n$ is difficult to define when the output range is unknown. Therefore in the following we focus on the suboptimal approach using the correlation factor.
### Choosing the input distribution and rate function {#sec:choosing_continunous_rate_function}
First we justify our choice of the Gaussian input distribution and the aforementioned rate function. We take the point of view of a compound (probabilistic, unknown) channel. If a rate function cannot be attained for compound channel model, it cannot be attained also in the more stringent individual model. It is well known that for a memoryless additive noise channel with constraints on the transmit power and noise variance, the Gaussian noise is the worst noise when the prior is Gaussian, and the Gaussian prior is the best prior when the noise is Gaussian. Thus by choosing a Gaussian prior we choose the best prior for the worst noise, and can we guarantee the mutual information will equal, at least, the Gaussian channel capacity. See the “mutual information game” (problem 9.21) in [@Cover]. For the additive noise channel [@ErezGaussian] shows the loss from capacity when using Gaussian distribution is limited to $\half$ a bit. However the above is true only for additive noise channels. For the more general where no additivity is assumed case we show below (Lemma \[lemma:R\_rho\_best\]) that the rate function $R=-\half\log(1-\rho^2)$ is the best rate function that can be defined by second order moments, and attained universally. Of course, this proof merely supplies the motivation to use a Gaussian distribution and does not rid us from the need to prove this rate is achievable for specific, individual sequences.
\[lemma:gaussian\_mi\_bound\] Let X,Y be two continuous random variables with correlation factor $\rho \equiv \frac{E(XY)}{\sqrt{E(X^2)E(Y^2)}}$, where $X$ is Gaussian $X \sim \Normal(0,P)$. Then $I(X;Y)\geq -\half \log(1 -\rho^2)$
\[corollary1\_gaussian\_mi\_bound\] Equality holds iff X,Y are jointly Gaussian
\[corollary2\_gaussian\_mi\_bound\] The lemma does not hold for general X (not Gaussian)
The proof is given in the appendix. Note that $-\half \log(1 -\rho^2)$ is the mutual information of two Gaussian r.v-s ([@Cover], example 8.5.1). Also note the relation to Theorem 1 in [@Hassibi] dealing with an additive channel with uncorrelated, but not necessarily independent noise. The following lemma justifies our selection of $R(\rho) = -\half \log(1 - \rho^2)$:
\[lemma:R\_rho\_best\] Let $Q(x)$ be an input prior, $W(y|x)$ be an unknown channel, $\Lambda(Q,W)$ be the correlation matrix $\Lambda \equiv E \left( \substack{X \\ \\ Y} \right) \left( \substack{X \\ \\ Y} \right)^T$ between $X,Y$ induced by the joint probability $Q \circ W$ and $\rho(Q,W)$ be the correlation factor induced by $Q,W$ ($\rho = \frac{\Lambda_{12}}{ \sqrt{\Lambda_{11}\Lambda_{22}}}$). We say a function $R(\Lambda)$ is an attainable second order rate function if there exists a $Q(x)$ such that for every channel $W(y|x)$ inducing correlation $\Lambda$ the mutual information is at least $R(\Lambda)$ (in other words can carry the rate $R(\Lambda)$). Then $R(\Lambda) = -\half \log(1 - \rho^2)$ is the largest attainable second order rate function.
Alternatively this can be stated as: $$R(\Lambda) \equiv \max_{Q} \min_{W: \Lambda(Q,W) = \Lambda} I(Q,W) = -\half \log(1 - \rho^2)$$
*Proof of lemma \[lemma:R\_rho\_best\]*: $R(\Lambda) = -\half \log(1 - \rho^2)$ is attainable by selecting an input prior $Q = \Normal(0,\sigma_x^2)$ and by lemma \[lemma:gaussian\_mi\_bound\] the mutual information is at least $R(\Lambda)$ for all channels. $R(\Lambda)$ is the maximum attainable function since by writing the condition of the lemma for the additive white gaussian noise (AWGN) channel $W^*$ (a specific choice of $W$) and any $Q$, we have $R(\Lambda) \leq I(Q,W^*) \leq I(\Normal(0,E_P(X^2),W^*)) = -\half \log(1-\rho^2)$, where the inequalities follow from the conditions of the lemma on $R$ and from the fact the Gaussian prior achieves the AWGN capacity.
### Communication scheme for the empirical channel (without feedback) {#sec:nonadaptive_continuous_achievability}
The following theorem is the analogue of Theorem \[theorem:discrete\_nonadaptive\] where the expression $-\half \log(1 - \rho^2)$ (interpreted as the Gaussian effective mutual information) plays the role of mutual information.
\[theorem:continuous\_nonadaptive\] Given the channel $\mathbb{R} \to \mathbb{R}$ for every $P_e>0$, $\delta>0$, power $P>0$ and rate $R>0$ there exists $n$ large enough and a random encoder-decoder pair of rate $R$ over block size $n$, such that the distribution of the input sequence is $\vr{x} \sim \Normal^n(0,P)$ and the probability of error for any message given an input sequence $\vr{x}$ and output sequence $\vr{y}$ with empirical correlation $\hat\rho$ is not greater than $P_e$ if $\Remp = \half \log \left( \frac{1}{1 - \hat\rho^2} \right) > R + \delta$
As before, the theorem will follow easily from the following lemma, proven in the appendix.
\[lemma:pairwise\_continuous\] Let $\vr x, \vr y \in \mathbb{R}^n$ be two sequences, and $\hat\rho \equiv \frac{\vr x^T \vr y}{\lVert \vr x \rVert \lVert \vr y \rVert }$ be the empirical correlation factor. For any $\vr y$, the probability of $\vr x$ drawn according to $\Normal^n(0,\sigma_x^2)$ to have $|\hat\rho| \geq t$ is bounded by: $$\Pr ( |\hat\rho| \geq t ) \leq 2 \exp\left(-(n-1) R_2(t)\right)$$ where $$R_2(t) \equiv \half \log \left( \frac{1}{1 - t^2} \right)$$
To prove Theorem \[theorem:continuous\_nonadaptive\], the codebook $\{\vr x_m\}_{m=1}^{\exp(nR)}$ is randomly generated by Gaussian i.i.d. selection of its $L = \exp(nR) \cdot n$ letters, and the common randomness $S \in \mathcal{X}^{L}$ is defined as the codebook itself and is distributed $\Normal^{L}(0,P)$. The encoder sends the $w$-th codeword, and the decoder uses maximum empirical correlation decoder i.e. chooses: $$\hat{w} = \bar\phi (\vr y,\{\vr x_m\}) = {\raisebox{-1.2ex}{$\stackrel{\textstyle \mathrm{argmax}}{\scriptscriptstyle m}$}} \rvert \hat \rho (\vr x_m ; \vr y) \lvert =
{\raisebox{-1.2ex}{$\stackrel{\textstyle \mathrm{argmax}}{\scriptscriptstyle m}$}} \left[ \frac{\lvert \vr x_m^T \vr y \rvert }{\lVert \vr x_m \rVert} \right]$$ where ties are broken arbitrarily. By Lemma \[lemma:pairwise\_continuous\], the probability of error is bounded by: $$\begin{gathered}
P_e^{(w)}(\vr x_w, \vr y) \leq \Pr \left\{ \bigcup_{m \neq w} \left( |\hat \rho (\vr x_m ; \vr y)| \geq |\hat \rho (\vr x_w ; \vr y)| \right) \right\} \leq \\
\leq \exp(nR) \cdot 2 \exp\left(-(n-1) R_2(\hat \rho (\vr x_w ; \vr y)))\right)
=\\= 2 \exp(R) \cdot \exp \left( -(n-1) \left(R_2(\hat \rho) - R\right) \right)\end{gathered}$$
Choosing $n$ large enough so that $\frac{1}{n-1} \left( R+\log \left(\frac{2}{P_e}\right) \right) \leq \delta$ (where $P_e$ is from Theorem \[theorem:continuous\_nonadaptive\]) we have that when $R_2(\hat \rho) > R + \delta$: $$P_e^{(w)}(x,y) \leq 2 \exp(R) \cdot \exp \left( -(n-1) \delta \right) \leq P_e$$ which proves the theorem.
A note is due regarding the definition of $\hat\rho$ in singular cases where $\vr x$ or $\vr y$ are 0. The limit of $\hat\rho$ as $\vr y \to 0$ is undefined (the directional derivative may take any value in \[0,1\]), however for consistency we define $\hat\rho=0$ when $\vr y=0$. Since $\vr x$ is generated from a Gaussian distribution we do not worry about the event $\vr x=0$ since the probability of this event is 0.
It’s worth spending a few words on the connections between the receivers used for the discrete and the continuous cases. Since the mutual information between two Gaussian r.v-s is $-\half \log(1 - \rho^2)$, one can think of this value as a measure of mutual information under Gaussian assumptions. Thus, using this metric as an effective mutual information, since the mutual information is an increasing function of $|\rho|$ the MMI decoder becomes a maximum empirical correlation decoder. On the other hand, the receiver we used can be identified as the GLRT (generalized maximum likelihood ratio test) for the AWGN channel $Y=\alpha X + \Normal(0,\sigma^2)$ with $\alpha$ an unknown parameter, resulting from maximizing the likelihood of the codeword and the channel simultaneously: $$\begin{gathered}
\hat w = {\raisebox{-1.2ex}{$\stackrel{\textstyle \mathrm{argmax}}{\scriptscriptstyle \vr x_m}$}} \max_{\alpha} \log \Pr(\vr y|\vr x;\alpha)
=\\= {\raisebox{-1.2ex}{$\stackrel{\textstyle \mathrm{argmin}}{\scriptscriptstyle m}$}} \min_{\alpha} \lVert \vr y - \alpha \vr x_m \rVert^2 = {\raisebox{-1.2ex}{$\stackrel{\textstyle \mathrm{argmax}}{\scriptscriptstyle m}$}} \frac{(\vr x_m^T \vr y)^2}{\lVert \vr x_m \rVert^2}
=\\= {\raisebox{-1.2ex}{$\stackrel{\textstyle \mathrm{argmax}}{\scriptscriptstyle m}$}} \left[ \hat\rho^2(\vr x_m, \vr y) \right]\end{gathered}$$ The choice of the GLRT is motivated by considering the individual channel as an effective additive channel with unknown gain (as presented in section \[sec:overview\]), combined with the fact Gaussian noise is the worse. For discrete memoryless channels it is easy to show that the GLRT (where the group of channels consists of all DMC-s) is synonymous with the MMI decoder (see [@Lapidoth_AVC]). Thus, we can identify the two decoders as GLRT decoders, or equivalently as variants of MMI decoders. In the sequel we sometimes use the term “empirical mutual information” in a broad sense that includes also the metric $-\half \log(1 - \hat\rho^2)$.
Regarding the receiver required to obtain the rates of Theorem \[theorem:continuous\_nonadaptive\], it is interesting to consider the simpler maximum projection receiver ${\raisebox{-1.2ex}{$\stackrel{\textstyle \mathrm{argmax}}{\scriptscriptstyle \vr x_m}$}} |\vr x_m^T \vr y|$. This receiver seems to differ from the maximum correlation receiver only in the term $\lVert \vr x_m \rVert$ which is nearly constant for large $n$ due to the law of large numbers. However surprisingly, the maximum rate achievable with the projection receiver is only $\half \hat\rho^2$ as can be shown by a simple calculation equivalent to Lemma \[lemma:pairwise\_continuous\] (simpler, since $z = \vr x^T \vr y$ is Gaussian). The reason is that when $\vr x$ is chosen independently of $\vr y$, a large value of the projection (non typical event) is usually created by a sequence with power significantly exceeding the average (another non typical event). When one non-typical event occurs there is no reason to believe the sequence is typical in other senses thus the approximation $\lVert \vr x_m \rVert \approx \sqrt {n P}$ is invalid. The correlation receiver normalizes by the power of $\vr x$ and compensates this effect. An alternative receiver which yields the rates of Theorem \[theorem:continuous\_nonadaptive\] and is similar to the AEP receiver looks for the codeword with the maximum absolute projection subject to power limited to $\frac{1}{n} \lVert \vr x_m \rVert^2 < P+\epsilon$. This can be shown by Sanov theorem [@MethodOfTypes] or by using the Chernoff bound. The maximum correlation receiver was chosen because of its elegance and the simplicity of the proof of Lemma \[lemma:pairwise\_continuous\].
Combining this lemma with the law of large numbers provides a simple proof for the achievability of the AWGN capacity ($\half\log(1+\textit{SNR})$), which uses much simpler mechanics than the popular proof based on AEP or error exponents. This receiver has the technical advantage, compared to the AEP receiver, that it does not declare an error for codewords which have power deviating from the nominal power. This technical advantage is important in the context of rateless decoding since the power condition needs to be re-validated each symbol, thus increasing its contribution to the overall error probability.
Lapidoth [@Lapidoth_Nearest] showed that the nearest neighbor receiver achieves a rate equal to the Gaussian capacity $\half \log (1 + P/N)$ over the additive channel $Y=X+V$ with arbitrary noise distribution (with fixed noise power). This result parallels the result that the random code capacity of the AVC $Y=X+V$ with a power constraint on $V$ equals the Gaussian capacity [@Hughes_AVC] (this stems directly from the characterization of the random code capacity of the AVC as $\max_{P_X(x)}\min_{P_S(s)} I(X;Y)$, cf.[@MethodOfTypes] Eq.(V.4)). Our result is stronger since it does not assume the channel is additive (nor any fixed behavior), but considering the former results it is not surprising, if one assumes (1) that any channel can be modeled as $Y=\alpha X+V$ with $V \perp X$, (2) that the dependence of $V$ on $X$ does not increase the error probability due to orthogonality (see [@Hassibi]) and (3) that the loss from the single unknown parameter $\alpha$ is asymptotically small.
Another related result is Agarwal et al’s [@Agarwal_RD] result that it is possible to communicate with a rate approaching the rate-distortion function $R_X(D)$ over an arbitrarily varying channel with unknown block-wise behavior satisfying a distortion constraint $\hat E d(\vr x, \vr y) \leq D$ in high probability. This relation is further discussed in the proof of Lemma \[lemma:pairwise\_discrete\]. Their result is similar to ours in the fact they define the rate in terms of the input and output alone. The result is similar to obtaining the rate function $\Remp \approx R_X(\hat E d(\vr x, \vr y))$ in the sense of Theorems \[theorem:discrete\_nonadaptive\],\[theorem:continuous\_nonadaptive\]. However their result is not tight even for the Gaussian channels: for the gaussian channel $Y=X+V$ with noise $V$ limited to power $N$ and the Gaussian prior $X \sim \Normal(0,P)$ this rate function equals $R_X(N) = \half \log \left( \frac{P}{N} \right)$ which is smaller than this channel capacity, whereas with Theorem \[theorem:continuous\_nonadaptive\] we would obtain $\half \log \left( 1 + \frac{P}{N} \right)$. Agarwal’s result is tight in the sense that this is the maximum rate that can be guaranteed given this distortion. There exists a channel with the same distortion $N$ whose capacity is only $\half \log \left( \frac{P}{N} \right)$: the channel $Y = \alpha X + \beta V$ with $\alpha = \beta^2 = 1-\frac{N}{P}$. The reason for the sub-optimality of the result is that the squared distance between the input and output, in contrast with the correlation factor, does not yield a tight representation of all memoryless linear Gaussian channels (in the sense of Lemma \[lemma:R\_rho\_best\]).
Communication with feedback {#sec:rate_adaptive}
===========================
Overview and background {#sec:rate_adaptive_overview}
-----------------------
In this section we present the rate-adaptive counterparts of Theorems \[theorem:discrete\_nonadaptive\], \[theorem:continuous\_nonadaptive\], and the scheme achieving them. The proof is delayed to the next section. The scheme we use in order to adaptively attain these rates is by iterating a rateless coding scheme. In other words, in each iteration we send a fixed number of bits $K$, by transmitting symbols from an $n$ length codebook, until the receiver has enough information to decode. Then, the receiver sends an indication that the block is over and a new block starts. Before developing the details we give some background regarding the evolution of rateless codes, and the differences between the proposed techniques. The earliest work is of Burnashev [@Burnashev] who showed that for known channels, using feedback and a random decision time (i.e. decision time which depends on the channel output) yields an improved error exponent, which is attained by a 3 step protocol (best described in [@Tchamkerten]) and shown to be optimal. Shulman [@Shulman] proposed to use random decision time as a means to deal with sending common information over broadcast channels (static broadcasting), and for unknown compound channels (which are treated as broadcast). In this scheme later described as “rateless coding” (or Incremental Redundancy Hybrid ARQ) a codebook of $\exp(K)$ infinite sequences is generated, and the sequence representing the message is sent to the receiver symbol by symbol, until the receiver decides to decode (and turn off, in case of a broadcast channel). Tchamkerten and Telatar [@Tchamkerten] connect the two results by showing that for some, but not all compound channels Burnashev error exponent can be attained universally using rateless coding and the 3 step protocol. Eswaran, Sarwate, Sahai and Gastpar [@Eswaran] used iterated rateless coding to achieve the mutual information related to the empirical noise statistics on channels with individual noise sequences. The scheme we use here is most similar to the one used in [@Eswaran] but less complicated. We do not use training symbols to learn the channel in order to decide on the decoding time but rely on the mutual information itself as the criterion (based on Lemmas \[lemma:pairwise\_discrete\],\[lemma:pairwise\_continuous\]) and the partitioning into blocks and the decision rules are simpler. The result in [@Eswaran] is an extension of a result in [@Ofer_BSC] regarding the binary channel to general discrete channels with individual noise sequence. The original result in [@Ofer_BSC] was obtained not by rateless codes but by a successive estimation scheme [@Ofer_Posterior] which is a generalization of the Horstein [@Horstein] and Schalkwijk-Kailath [@Schalkwijk] schemes. The same authors extend their results to discrete channels [@Ofer_EMP] using successive schemes (where the target rate is the capacity of the respective modulu-additive channel). The two concepts in achieving the empirical rates differ in various factors such as complexity and the amount of feedback and randomization required. The successive schemes require less common randomness but assume perfect feedback, while the schemes based on rateless coding require less (asymptotically 0 rate) feedback but potentially more randomness.
As noted the technique we use here is similar to that of [@Eswaran] in its high level structure, while the structure of the rateless decoder is similar to [@Shulman]’s (chapter 3). The application of this scheme to individual inputs and outputs and the extension to real-valued models requires proof and especially issues such as abnormal behavior of specific (e.g. last) symbols have to be treated carefully. The result of [@Eswaran] cannot be applied directly to individual channels since the channel model cannot be extracted based on the input and output sequences alone, and in the later both the model and the sequence are assumed to be fixed (over common randomness).
Statement of the main result {#sec:rate_adaptive_theorems}
----------------------------
In this section we prove the following theorems, relating to the definitions given in section \[sec:definitions\]:
\[theorem:discrete\_adaptive\] Given discrete input and output alphabets $\mathcal{X,Y}$, for every $P_e>0$, $P_A>0$, $\delta>0$ and prior $Q(x)$ over $\mathcal{X}$ there is $n$ large enough and random encoder and decoder with feedback and variable rate over block size $n$ with a subset $J \subset\mathcal{X}^n$, such that:
- The distribution of the input sequence is $\vr{x} \sim Q^n$ independently of the feedback and message
- The probability of error is smaller than $P_e$ for any $\vr x, \vr y$
- For any input sequence $\vr x \not\in J$ and output sequence $\vr{y} \in \mathcal{Y}^n$ the rate is $R \geq \hat I (\vr{x},\vr{y}) - \delta$
- The probability of $J$ is bounded by $\Pr(\vr x \in J) \leq P_A$
\[theorem:continuous\_adaptive\] Given the channel $\mathbb{R} \to \mathbb{R}$ for every $P_e>0$, $P_A>0$, $\delta>0$, $\bar R>0$, and power $P>0$ there is $n$ large enough and random encoder and decoder with feedback and variable rate over block size $n$ with a subset $J \subset\mathbb{R}^n$, such that
- The distribution of the input sequence is $\vr{x} \sim \Normal(0,P)^n$ independently of the feedback and message
- The probability of error is smaller than $P_e$ for any $\vr x, \vr y$
- For any input sequence $\vr x \not\in J$ and output sequence $\vr{y} \in \mathbb{R}^n$ the rate is $R \geq \min \left[ \half \log \left( \frac{1}{1-\hat\rho(\vr x, \vr y)^2} \right) - \delta, \bar R \right] $
- The probability of $J$ is bounded by $\Pr(\vr x \in J) \leq P_A$
![Illustration of $\Remp$ lower bound of theorem \[theorem:continuous\_adaptive\] ($\RLBTWO$) and the lower bound $\RLBONE$ shown in the proof in section \[sec:rate\_analysis\_continuous\], as a function of $\rho$ (top) and the effective SNR $= \frac{\rho^2}{1-\rho^2}$ (bottom). Parameters appear in table \[table:rate\_in\_continuous\_theorem\_params\] in the appendix[]{data-label="fig:rate_in_continuous_theorem"}](rate_in_continuous_theorem "fig:"){width="8cm"} ![Illustration of $\Remp$ lower bound of theorem \[theorem:continuous\_adaptive\] ($\RLBTWO$) and the lower bound $\RLBONE$ shown in the proof in section \[sec:rate\_analysis\_continuous\], as a function of $\rho$ (top) and the effective SNR $= \frac{\rho^2}{1-\rho^2}$ (bottom). Parameters appear in table \[table:rate\_in\_continuous\_theorem\_params\] in the appendix[]{data-label="fig:rate_in_continuous_theorem"}](rate_in_continuous_theorem_esnr "fig:"){width="8cm"}
Note that in the last theorem we do not have uniform convergence of the rate function in $\vr x, \vr y$. Unfortunately our scheme is limited by having a maximum rate for each $n$, and although the maximum rate tends to infinity as $n \to \infty$, we cannot guarantee uniform convergence for each $n$ in the continuous case, where the target rate may be unbounded. The rates in the theorems are the minimal rates, and in certain conditions (e.g. a channel varying in time) higher rates may be achieved by the scheme proposed below. Regarding the set $J$ as we shall see in the sequel there are some sequences for which poor rate is obtained, and since we committed to an input distribution we cannot avoid them (one example is the sequence of $\half n$ zeros followed by $\half n$ ones, in which at most one block will be sent). However there is an important distinction between claiming for example that “for each $\vr y$ the probability of $R < \Remp$ is at most $P_A$” and the claim made in the theorems that “$R < \Remp$ only when $\vr x$ belongs to a subset $J$ with probability at most $P_A$”. The first claim is weaker since smartly chosen $\vr y$ may increase the probability (see figure \[fig:Bad\_x\_sequences\]). This is avoided in the second claim. A consequence of this definition is that the probability of $R < \Remp$ is bounded by $P_A$ for any conditional probability $\Pr(\vr y | \vr x)$ on the sequences. This issue is further discussed in section \[sec:rate\_adaptive\_perliminaries\].
Note that the probability $P_A$ could be absorbed into $P_e$ by a simple trick, but this seems to make the Theorem less insightful. After reception the receiver knows the input sequence in probability of at least $1-P_e$ and may calculate the empirical mutual information $\hat I(\vr x, \vr y)$. If the rate achieved by the scheme we will describe later falls short of $\hat I(\vr x, \vr y)$ it may declare a rate of $R=\hat I(\vr x, \vr y)$ (which will most likely result in a decoding error). This way the receiver will never declare a rate which is lower than $\hat I(\vr x, \vr y)$ unless there is an error, and we could avoid the restriction $\vr x \not\in J$ required for achieving $\Remp$, but on the other hand, the error probability becomes conditioned on the set $J$. The question whether the set $J$ itself is truly necessary (i.e. is it possible to attain the above Theorems with $J=\emptyset$) is still open.
Figure (\[fig:rate\_in\_continuous\_theorem\]) illustrates the lower bound for $\Remp$ presented by Theorem \[theorem:continuous\_adaptive\] ($\RLBTWO$) as well as a (higher) lower bound $\RLBONE$ for the rate achieved by the proposed scheme (see section \[sec:rate\_analysis\_continuous\], Eq.(\[eq:cont\_rate\_analysis9\])). The parameters generating these curves appear in table \[table:rate\_in\_continuous\_theorem\_params\] in the appendix.
We prove the two theorems together. First we define the scheme, and in the next section we analyze its error performance and rate and show it achieves the promise of the theorems. Throughout this section and the following one we use $n$ to denote the length of a complete transmission, and $m$ to denote the length of a single block.
A proposed rate adaptive scheme {#sec:rate_adaptive_scheme}
-------------------------------
The following communication scheme sends $B$ indices from $\{1,\ldots,M\}$ over $n$ channel uses (or equivalently sends the number $\theta \in [0,1)$ in resolution $M^{-B}$), where $M$ is fixed, and $B$ varies according to empirical channel behavior. The building block is a rateless transmission of one of $M$ codewords ($K \equiv \log(M)$ information units), which is iterated until the $n$-th symbol is reached.
The transmit distribution $Q$ is an arbitrary distribution for the discrete case and $Q = \Normal(0,P)$ for the continuous case. We define the decoding metric as the empirical rate: $$\label{eq:def_mu}
\Remp(\vr x, \vr y) \equiv \left\{ \begin{array}{ll}
\hat I (\vr x, \vr y) & \textrm{discrete}\\
\half \log \left( \frac{1}{1 - \hat\rho^2(\vr x, \vr y)} \right) & \textrm{continuous}\end{array} \right.$$ The codebook $C_{M \times n}$ consists of $M$ codewords of length $n$, where all $M \times n$ symbols are drawn i.i.d. $\sim Q$ and known to the sender and receiver. For brevity of notation we denote $\Remp^m(\vr x, \vr y)$ instead of $\Remp(\vr x_1^m, \vr y_1^m)$. $k$ denotes the absolute time index $1 \leq k \leq n$. Block $b$ starts from index $k_b$, where $k_1=1$. $m = k - k_b + 1$ denotes the time index inside the current block.
In each rateless block $b=1,2,\ldots$, a new index $i = i_b \in \{1,\ldots,M\}$ is sent to the receiver using the following procedure:
1. The encoder sends index $i$ by sending the symbols of codeword $i$: $$x_k = C_{i,k}$$ Note that different blocks use different symbols from the codebook.
2. The encoder keeps sending symbols and incrementing $k$ until the decoder announces the end of the block through the feedback link.
3. The decoder announces the end of the block after symbol $m$ in the block if for any codeword $x_i$ : $$\label{eq:termination_condition}
\Remp^m(\vr x_i, \vr y) \equiv \Remp \left( (\vr x_i)_{k_b}^k, \vr y_{k_b}^k \right) \geq \mu^*_m$$ where $\mu^*_m$ is a fixed threshold per symbol defined in Eq.(\[eq:termination\_threshold\]) below.
4. When the end of block is announced one of the $i$ fulfilling Eq.(\[eq:termination\_condition\]) is determined as the index of the decoded codeword $\hat i_b$ (breaking ties arbitrarily).
5. Otherwise the transmission continues, until the $n$-th symbol is reached. If symbol $n$ is reached without fulfilling Eq.(\[eq:termination\_condition\]), then the last block is terminated without decoding.
After a block ends, $b$ is incremented and if $k < n$ a new block starts at symbol $k_b = k+1$. After symbol $n$ is reached the transmission stops and the number of blocks sent is $B=b-1$.
The threshold $\mu^*_m$ is defined as: $$\begin{gathered}
\label{eq:termination_threshold}
\mu^*_m = \frac{K}{m-s} + \frac{1}{m-s}\log \left( \frac{n}{P_e} \right) + \delta_m
=\\=
\left\{ \begin{array}{ll}
\frac{K + \log \left( \frac{n}{ P_e} \right) + |\mathcal{X}||\mathcal{Y}| \log(m+1)}{m} & \textrm{discrete}\\
\frac{K + \log \left( \frac{2n}{P_e} \right)}{m-1} & \textrm{continuous}\end{array} \right.\end{gathered}$$ where $s=0$ for the discrete case and $1$ for the continuous case and $\delta_m$ is defined in Lemma \[lemma:pairwise\_discrete\] for the discrete case and equals $\frac{\log(2)}{m-1}$ for the continuous case. The threshold $\mu^*_m$ is tailored to achieve the designated error probability and is composed of 3 parts. The first part requires that the empirical rate $\Remp$ would approximately equal the transmission rate of the block $\frac{K}{m}$, which guarantees there is approximately enough mutual information to send $K$ information units. The second part is an offset responsible for guaranteeing error probability bounded by $P_e$ over all the blocks in the transmission. The third part $\delta_m$ compensates the overhead terms in Lemmas \[lemma:pairwise\_discrete\],\[lemma:pairwise\_continuous\].
The scheme achieves the claims of Theorems \[theorem:discrete\_adaptive\],\[theorem:continuous\_adaptive\] with a proper choice of the parameters (discussed in section \[sec:rate\_analysis\]). Note that the scheme uses feedback rate of $1$ bit/use however it is easy to show any positive feedback rate is sufficient (see section \[sec:rate\_analysis\]), therefore we can claim the theorems hold with “zero rate” feedback.
We devote the next section to the analysis of the error probability and rate of the scheme, showing it attains Theorems \[theorem:discrete\_adaptive\],\[theorem:continuous\_adaptive\]. Unfortunately although the scheme is simple, the current analysis we have is somewhat cumbersome.
Proof of the main result {#sec:analysis}
========================
In this section we analyze the adaptive rate scheme presented and show it achieves Theorems \[theorem:discrete\_adaptive\],\[theorem:continuous\_adaptive\]. Before analyzing the scheme we develop some general results pertaining to the convexity of the mutual information and correlation factors over sub-vectors. The proof of the error probability is simple (based on the construction of $\mu^*_m$) and common to the two cases. The proof of the achieved rate is more complex and performed separately for each case.
Preliminaries {#sec:rate_adaptive_perliminaries}
-------------
### Likely convexity of the mutual information {#sec:likely_convexity_of_MI}
A property which would be useful for the analysis is $\cup$-convexity of the empirical mutual information with respect to joint empirical distributions $\hat P_{(\vr x, \vr y)}(x,y)$ measured over different sub-vectors, so for example we would like to have for $0 \leq m \leq n$: $$\label{eq:erronous_convexity}
\hat I(\vr x_1^n; \vr y_1^n) \leq \frac{m}{n} \cdot \hat I(\vr x_1^m; \vr y_1^m) + \left(1-\frac{m}{n}\right) \cdot \hat I(\vr x_{m+1}^n; \vr y_{m+1}^n)$$ which would guarantee that if we achieve a rate equal to the empirical mutual information over the two sections $0 \leq k \leq m$ and $m < k \leq n$, then we would achieve the empirical mutual information over the entire vector $0 \leq k \leq n$. However this property does not hold in general since the mutual information is not convex with respect to the joint distribution. The mutual information $I(P,W)$ is known to be convex $\cup$ with respect to $W$ and concave $\cap$ with respect to $P$, so if, for example, the conditional distributions over the sections $[1,m]$ and $[m+1, n]$ are equal and only the distribution of $\vr x$ differs, the condition would in general not hold. On the other hand should the empirical distributions of $\vr x_1^{m}$ and $\vr x_{m+1}^n$ be equal, then the empirical mutual information expressions appearing in Eq.(\[eq:erronous\_convexity\]) would differ only in the conditional distributions of $\vr y$ w.r.t $\vr x$ and the assertion would hold. Since we generate $\vr x$ by i.i.d. drawing of its elements the empirical distributions converge to the prior $Q$, and we would expect that if the size of both regions $m$ and $m-n$ is large enough the convexity would hold up to a fraction $\epsilon$ in high probability. We show below that such convexity holds under even milder conditions. The cases in which this approximate convexity is used later on can serve as examples of the difference between the individual model used here and probabilistic models (including the individual noise sequence). We use the lemma to:
1. Bound the loss due to insufficient utilization of the last symbol in each rateless encoding block.
2. Bound the loss due to not completing the last rateless encoding block.
3. Show that the average rate (empirical mutual information) over multiple blocks equals at least the mutual information measured over the blocks together
Had the rate been averaged over multiple sequences $\vr x$ rather than obtained for a specific sequence, the regular convexity of the mutual information with respect to the channel distribution would have been sufficient. The property is formalized in the following lemma:
\[lemma:likely\_convexity\_of\_MI\] Let $\{A_i\}_{i=1}^p$ define a disjoint partitioning of the index set $\{1, \ldots, n\}$, i.e. $\bigcup_i A_i = \{1, \ldots, n\}$ and $A_i \cap A_j = \emptyset \text{ for } i \neq j$. $\vr x$ , $\vr y$ are n-length sequences, and $\vr x_A$, $\vr y_A$ define the sub-sequences of $\vr x$, $\vr y$ (resp.) over the index set $A$. Let the elements of $\vr x$ be chosen i.i.d. with distribution $Q$. Then for any $\Delta>0$ there is a subset $J_{\Delta} \subset \mathcal{X}^n$ such that: $$\forall \vr x \not\in J_{\Delta}, \vr y \in \mathcal{Y}^n : \hspace{3ex}
\sum_{i=1}^p \frac{|A_i|}{n} \hat I (\vr x_{A_i}; \vr y_{A_i}) \geq \hat I (\vr x; \vr y)-\Delta$$ And $$Q^n \left\{J_{\Delta} \right\} \leq \exp \left( -n \left(\Delta - \tilde{\delta}_n \right) \right)$$ With $\tilde{\delta}_n = p |\mathcal{X}| \cdot \frac{\log(n+1)}{n} \to 0$.
The lemma does not claim that convexity holds with high probability, but rather that any positive deviation from convexity may happen only on a subset of $\vr x$ with vanishing probability. It is surprising that the bound does not depend on $\vr y$, $Q$ and the size of the subsets, and only weakly depends on the number of subsets.
Before proving the lemma we emphasize a delicate point: the lemma does not only claim that for each $\vr y$ the probability of deviation from convexity is small, but makes a stronger claim that apart from a subset of the $\vr x$ sequences with vanishing probability, convexity always holds independently of $\vr y$. This distinction is important since this lemma defines a set of “bad” input sequences that fail our scheme. In these sequences there exists a partitioning that yields an excessive deviation from the distribution $Q$ between rateless blocks. As an example of such a sequence consider the binary channel and the input sequence $0^{n/2}1^{n/2}$ ($n/2$ zeros followed by $n/2$ ones). This sequence is bad since it guarantees that on one hand at most one block will be received (since at most one block includes both $0$-s and $1$-s at the input), but on the other hand the zero order empirical input distribution is good ($Ber(\half)$), so potentially we have the combination of high empirical mutual information with low communication rate. The sequences that deviate from convexity are a function of the output $\vr y$. Had we only bounded the probability of deviation from convexity to occur for each $\vr y$ individually, then a potential adversary could have increased this probability by determining $\vr y$ (given $\vr x$) such that $\vr x$ will be a bad sequence with respect to this $\vr y$. To avoid this, we claim that there is a fixed group of $\vr x$ such that if the sequence is not in the group, approximate convexity holds regardless of $\vr y$. This is illustrated in fig.(\[fig:Bad\_x\_sequences\]) where the dark spots mark the pairs $(\vr x, \vr y)$ for which convexity does not hold.
![Illustration of bad sequences and lemma \[lemma:likely\_convexity\_of\_MI\][]{data-label="fig:Bad_x_sequences"}](Bad_x_sequences "fig:"){width="8cm"}\
*Proof of lemma \[lemma:likely\_convexity\_of\_MI\]*: Define the vector $\vr u$ denoting the subset number of each element $\vr u_k = i \hspace{1em} \forall k \in A_i$. Then $\hat I (\vr x_{A_i}; \vr y_{A_i}) = \hat I (\vr x, \vr y | \vr u = i)$, and $\hat P_{\vr u}(i) = \frac{|A_i|}{n}$, therefore we can write the weighted sum of empirical mutual information over the partitions, as a conditional empirical mutual information: $$\begin{gathered}
\sum_{i=1}^p \left( \frac{|A_i|}{n} \hat I (\vr x_{A_i}; \vr y_{A_i}) \right) =
\sum_{i=1}^p \hat P_{\vr u}(i) \hat I (\vr x; \vr y | \vr u = i)
=\\= \hat I(\vr x; \vr y | \vr u)\end{gathered}$$ Using the chain rule for mutual information (see [@Cover] section 2.5): $$\begin{gathered}
\hat I (\vr x; \vr y) - \hat I(\vr x; \vr y | \vr u)
= \hat I (\vr x; \vr y) - \left(\hat I(\vr x; \vr y \vr u) - \hat I(\vr x; \vr u)\right)
=\\=
\hat I(\vr x; \vr u) - \hat I(\vr x; \vr u | \vr y) \leq \hat I(\vr x; \vr u)\end{gathered}$$ Define the set $J_{\Delta} = \{\vr x : \hat I(\vr x; \vr u) > \Delta \}$, then $$\forall \vr x \not\in J_{\Delta}, \vr y: \hat I (\vr x; \vr y) - \hat I(\vr x; \vr y | \vr u) \leq \hat I(\vr x; \vr u) \leq \Delta$$ And since $\vr x$ is chosen iid and $\vr u$ is a fixed vector, we have from Lemma \[lemma:pairwise\_discrete\]: $$\Pr \left( \vr x \in J_{\Delta} \right) \leq \exp \left( -n \left(\Delta - \tilde{\delta}_n \right) \right)$$ with $\tilde{\delta}_n = |\mathcal{X}||\{1,\ldots,p\}|\frac{\log(n+1)}{n}$.
Note that if the distribution of $\vr x$ is the same over all partitions then $\hat H(\vr x| \vr u) = \hat H(\vr x)$ therefore $\hat I(\vr x; \vr u) = 0$ and the empirical mutual information will be truly convex.
### Likely convexity of the correlation factor {#sec:likely_convexity_of_rho}
For the continuous case we use the following property which somewhat parallels Lemma \[lemma:likely\_convexity\_of\_MI\]. The reasons for not following the same path as the discrete case will be explained in the sequel (subsection \[sec:rate\_analysis\]). Unfortunately the proof is very technical and less elegant and will therefore be expelled to the appendix (appendix-\[appendix:likely\_convexity\_of\_rho\]). Note that again the bound does not depend on the size of the subsets.
\[lemma:likely\_convexity\_of\_rho\] Define $\{A_i\}_{i=1}^p$ as in Lemma \[lemma:likely\_convexity\_of\_MI\]. Let $\vr x$ , $\vr y$ be $n$-length sequences and define the correlation factors of the sub-sequences, and the overall correlation factor as $$\hat\rho_i = \frac{\lvert \vr x_{A_i}^T \vr y_{A_i} \rvert}{\lVert \vr x_{A_i} \rVert \cdot \lVert \vr y_{A_i} \rVert} \hspace{2em} \text{and} \hspace{2em} \hat\rho = \frac{\lvert \vr x^T \vr y \rvert}{\lVert \vr x \rVert \cdot \lVert \vr y \rVert}$$ respectively. Let $\vr x$ be drawn i.i.d from a Gaussian distribution $\vr x \sim \Normal(0,P)$. Then for any $0<\Delta\leq \frac{1}{7}$ there is a subset $J_{\Delta} \subset \mathbb{R}^n$ such that: $$\forall \vr x \not\in J_{\Delta}, \vr y \in \mathbb{R}^n : \hspace{3ex}
\sum_{i=1}^p \frac{|A_i|}{n} \hat\rho_i^2 \geq \hat\rho^2 - \Delta$$ And $$\Pr \left\{\vr x \in J_{\Delta} \right\} \leq 2^p e^{ - n \Delta^2/8}$$ I.e. there is a subset with high probability on which the mean of the correlation factors does not fall considerably below the overall correlation factor.
### Likely convexity with dependencies {#sec:likely_convexity_with_dependency}
The properties of likely convexity defined in the previous sections pertain to a case where the partition of the $n$ block is fixed and $\vr x$ is drawn i.i.d. However in the transmission scheme we described, the partition varies in a way that depends on the value of $\vr x$ (through the decoding decisions and the empirical mutual information), which may, in general, change the probability of the convexity property with a given $\Delta$ to occur. Although it stands to reason that the variability of the block sizes in the decoding process reduces the probability to deviate from convexity since it tends to equalize the amount of mutual information in each rateless block, for the analysis we assume an arbitrary dependence, and assume that the size of the set $J$ increases by factor of the number of possible partitions, as explained below.
Denote a partition by $\pi = \{A_i\}_{i=1}^p$ (as defined in Lemmas \[lemma:likely\_convexity\_of\_MI\],\[lemma:likely\_convexity\_of\_rho\]) and the group of all possible partitions (for a given encoder-decoder) by $\Pi$. For each partition $\pi$ from Lemmas \[lemma:likely\_convexity\_of\_MI\],\[lemma:likely\_convexity\_of\_rho\] there is a subset $J(\pi)$ with probability bounded by $p_J$ outside which approximate convexity (as defined in the lemmas) holds. Then approximate convexity is guaranteed to hold for $\vr x \not\in J \equiv \displaystyle \bigcup_{\pi \in \Pi} J(\pi)$, where the probability of the set $J$ is bounded by the union bound: $$\Pr(\vr x \in J) = \Pr \left( \bigcup_{\pi \in \Pi} (\vr x \in J(\pi)) \right) \leq |\Pi| \cdot p_J$$
Now we bound the number of partitions. In the two cases we will deal with in section \[sec:rate\_analysis\] the number of subsets can be bounded by $p_{\max}$, and all subsets but one contain continuous indices. Therefore the partition is completely defined by the start and end indices of $p_{\max}-1$ subsets (allowed to overlap if there are less than $p_{\max}$ subsets), thus $|\Pi| \leq n^{2p_{\max}-2} < n^{2p_{\max}}$ and we have $$\label{eq:likely_convexity_with_dependency}
\Pr (J) \leq n^{2 p_{\max}} \cdot p_J = \exp( 2 p_{\max} \log(n) ) \cdot p_J$$ where $p_J$ is defined in the previous lemmas. So for our purposes we may say that these lemmas hold even if the partition depends on $\vr x$ with an appropriate change in the probability of $J$.
Item Discrete case Continuous case
----------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------
Input distribution Any $Q$ $Q = \Normal(0,P)$
Decoding metric $\Remp(\vr x, \vr y) \equiv \hat I (\vr x, \vr y)$ $\Remp(\vr x, \vr y) \equiv \half \log \left( \frac{1}{1 - \hat\rho^2(\vr x, \vr y)} \right)$
Decoder maximize $\Remp(\vr x, \vr y)$ $\Leftrightarrow$ maximize $\hat I (\vr x, \vr y)$ maximize $\Remp(\vr x, \vr y)$ $\Leftrightarrow$ maximize $|\hat \rho(\vr x, \vr y)|$
Pairwise error probability $\Pr(\Remp \geq t)$ $\leq \exp(-n (t - \delta_n))$ (Lemma \[lemma:pairwise\_discrete\]) $\leq 2 \exp(-(n-1) t)$ (Lemma \[lemma:pairwise\_continuous\])
Likely convexity condition ($\forall \vr x \not\in J_{\Delta}, \vr y \in \mathcal{Y}^n$ with $\lambda_i \equiv \frac{1}{n}|A_i|$) $\sum_{i=1}^p \lambda_i \hat I (\vr x_i; \vr y_i) \geq \hat I (\vr x; \vr y)-\Delta$ (Lemma \[lemma:likely\_convexity\_of\_MI\]) $\sum_{i=1}^p \lambda_i \hat\rho_i^2 \geq \hat\rho^2 - \Delta $ (Lemma \[lemma:likely\_convexity\_of\_rho\])
Likely convexity probability ($\Pr(\vr x \not\in J_{\Delta})$, fixed partitioning) $\geq 1 - \exp \left( -n \left(\Delta - \tilde{\delta}_n \right) \right)$ $\geq 1 - 2^p e^{ - n \Delta^2/8}$
Error probability analysis {#sec:error_analysis}
--------------------------
In this subsection we show the probability to decode incorrectly any of the $B$ indices is smaller than $P_e$.
With $\Remp$ defined in Eq.(\[eq:def\_mu\]), we have from Lemma \[lemma:pairwise\_continuous\] that under the conditions of the lemma $\Pr(\Remp \geq t) = \Pr(|\hat\rho| \geq R_2^{-1}(t)) \leq 2 \exp(-(n-1) t)$. Then combining Lemmas \[lemma:pairwise\_discrete\] and \[lemma:pairwise\_continuous\], we may say that for any $\vr y_1^m$ the probability of $\vr x_1^m$ generated i.i.d. from the relevant prior to have $\Remp \geq t$ is bounded by: $$\label{eq:unified_pairwise}
Q^m(\Remp(\vr x_1^m, \vr y_1^m) \geq t) \leq \exp \left( -(m-s) (t - \delta_m) \right)$$ where $$\label{eq:unified_pairwise_delta}
\delta_m = \left\{ \begin{array}{ll}
|\mathcal{X}||\mathcal{Y}|\frac{\log(m+1)}{m} & \textrm{discrete}\\
\frac{\log 2}{m-1} & \textrm{continuous}\end{array} \right.$$ And $$s = \left\{ \begin{array}{ll}
0 & \textrm{discrete}\\
1 & \textrm{continuous}\end{array} \right.$$
An error might occur if at any symbol $1 \leq k \leq n$ an incorrect codeword meets the termination condition Eq.(\[eq:termination\_threshold\]). The probability that codeword $j \neq i$ meets Eq.(\[eq:termination\_threshold\]) at a specific symbol $k$ which is the $m$-th symbol of a rateless block is bounded by: $$\begin{gathered}
\Pr(\Remp^m(\vr x_j, \vr y) \geq \mu^*_m) \leq \exp \left( -(m-s) (\mu^*_m - \delta_m) \right)
=\\= \exp \left( -\left[ K + \log \left( \frac{n}{P_e} \right) \right] \right) = \frac{P_e}{n \exp(K)} = \frac{P_e}{M n}\end{gathered}$$
The probability of any erroneous codeword to meet the threshold at any symbol is bounded by the union bound: $$\begin{gathered}
\Pr(\textrm{error}) \leq \Pr \left\{ \bigcup_{k=1}^n \bigcup_{j \neq i} \left(\mu_m(\vr x_j, \vr y)\geq \mu^*_m \right) \right\}
\leq \\ \leq
n (M-1) \frac{P_e}{M n} < P_e\end{gathered}$$
The first inequality is since the correct codeword might be decoded even if an erroneous codeword met the threshold. Although the index $m$ in the expression above depends on $k$ and the specific sequences $\vr x, \vr y$ in an unspecified way, the assertion is true since the probability of the event in the union has an upper bound independent of $m$.
Rate analysis {#sec:rate_analysis}
-------------
Roughly speaking, since $\mu^*_m \approx \frac{K}{m}$, if no error occurs, the correct codeword crossed the threshold when $\Remp^m(\vr x_i, \vr y) \approx \frac{K}{m}$ therefore the rate achieved over a rateless block is $R_b = \frac{K}{m} \approx \Remp^m(\vr x_i, \vr y)$, and due to the approximate convexity by achieving the above rate on each block separately we meet or exceed the rate $\Remp(\vr x, \vr y)$ over the entire transmission. However in a detailed analysis we have the following sources of rate loss:
1. The offsets inserted in $\mu^*_m$ to meet the desired error probability
2. The offset from convexity (Lemma \[lemma:likely\_convexity\_of\_MI\]) introduced by the slight differences in empirical distribution of $\vr x$ between the blocks
3. Unused symbols:
1. The last symbol of each block is not fully utilized, as explained below
2. The last (unfinished) block is not utilized
Regarding the last symbol of each block, note that after receiving the previous symbol the empirical mutual information is below the threshold, and at the last symbol it meets or exceeds the threshold. However the proposed scheme does not gain additional rate from the difference between the mutual information and the threshold, and thus it loses with respect to its target (the mutual information over the block) when this difference is large. Here a “good” channel works adversely to our worse. Since we operate under an individual channel regime, the increase of the mutual information at the last symbol is not bounded to the average information contents of a single symbol. This is especially evident in the continuous case where the empirical mutual information is unbounded. A high value of $y$ together with high value of $x$ at the last symbol causes an unbounded increase in $\Remp$: if we choose $\vr x_m,\vr y_m \to \infty$ then $\rho \to 1$ regardless of the history $\vr x_1^{m-1}, \vr y_1^{m-1}$. Therefore over a single block we might have an arbitrarily low rate ($|\hat\rho|$ is small over the $m-1$ first symbols) and arbitrarily large $\Remp$. In the discrete case this phenomenon exists but is less accented (consider for example the sequences $\vr x = \vr y = 0^{n-1}1 = (0,\ldots,0,1)$) Similarly regarding the last block, the fact that the length of the block may be bounded does not mean the increase in the empirical mutual information can be bounded as well. We use the approximate convexity (Lemma \[lemma:likely\_convexity\_of\_MI\]) to show the last two losses are bounded for most $\vr x$ sequences.
Note that by the same argument that shows the loss from not utilizing the last symbol vanishes asymptotically, it is easy to show that feeding back the block success information only once every $1/\epsilon$ symbols thereby decreasing the feedback rate to $\epsilon$ does not decrease the asymptotical rate, since this is equivalent to having $1/\epsilon$ unused symbols instead of one. Hence the scheme can be modified to operate with “zero rate” feedback. Similarly the scheme can operate with a noisy feedback channel by introducing in the feedback link a delay suitable to convey the decoder decisions with sufficiently low error rate over the noisy channel.
In addition to having rate losses the scheme also has a minimal rate and a maximal rate for each block length. The minimal rate is $\frac{K}{n}$ resulting from sending a single block. If channel conditions are worse ($\Remp < \frac{K}{n}$), no information will be sent. A maximal rate exists since at best $K$ information units could be sent every $2$ symbols (since for the continuous case $\mu^*_1 = \infty$ and for the discrete case $\Remp^1(x,y) = 0$ thus the decoding never terminates at the first symbol of the block), hence the maximum rate is $\frac{K}{2}$. As $n \to \infty$ we increase $K$ so that the minimum rate (and the rate offsets) tend to 0 and the maximum rate tends to $\infty$. The maximum rate is the reason that the scheme cannot approach the target rate $\Remp(\vr x_i, \vr y)$ uniformly in $\vr x,\vr y$ in the continuous case, since for some pairs of sequences the target rate (which is unbounded) may be much higher than the maximum rate. The rate $\bar R$ that we achieve in the proof of Theorem \[theorem:continuous\_adaptive\] is much smaller than the absolute maximum $\frac{K}{2}$. Note that successive schemes (such as Schalkwijk’s [@Schalkwijk]) do not suffer from the problem of maximum rate. For the discrete case the target rate is bounded by $\max (|\mathcal{X}|,|\mathcal{Y}|)$ therefore for sufficiently large $n$ the maximal rate $\frac{K}{2}$ exceeds $\max (|\mathcal{X}|,|\mathcal{Y}|)$ and we are able to show uniform convergence.
Although our target is the empirical mutual information over the $n$-block, an artifact of the partitioning to smaller blocks is that higher rates can be attained when the empirical conditional channel distribution varies over time, since by the convexity of mutual information with respect to the channel law the convex sum of mutual information over blocks exceeds the overall mutual information if these are not constant.
We now turn to prove the achieved rate. The total amount of information sent (with or without error) is $B \cdot K$ therefore the actual rate is $$R_{\mathrm{act}} = \frac{B K}{n}$$ We now endeavor to show this rate is close to or better than the empirical mutual information in probability of at least $P_A$ over the sequences $\vr x$, regardless of $\vr y$ and of whether a decoding error occurred.
The following definition of index sets in $\{1,\ldots,n\}$ is used for both the discrete and the continuous cases: $U_b = \{k\}_{k=k_b}^{k_{b+1}-2}$ denotes the channel uses of block $b$ *except* the last one, $L_0$ collects the last channel uses of all the blocks $L_0 = \{k_b-1: b>1\}$, and $U_{B+1}$ denotes the indices of the un-decoded (last) block $U_{B+1} = \{k\}_{k=k_{B+1}}^{n}$ (including its last symbol), and is an empty set if the last block is decoded. The sets $\{U_b\}_{b=1}^{B+1}, L_0$ are disjoint and their union is $\{1,\ldots,n\}$. We denote the length of each block not including the last symbol by $m_b \equiv |U_b|$. From this point on we split the discussion and we start with the discrete case which is simpler.
### Rate analysis for the discrete case {#sec:rate_analysis_discrete}
We write $\mu^*_m$ as $\mu^*_m = \frac{K+\Delta_m}{m} \leq \frac{K+\Delta_{\mu}}{m}$ with $$\begin{gathered}
\Delta_m = \log \left( \frac{n}{ P_e} \right) + m\delta_m
=\log \left( \frac{n}{ P_e} \right) + |\mathcal{X}||\mathcal{Y}| \log(m+1)
\leq \\ \leq
\log \left( \frac{n}{P_e} \right) + |\mathcal{X}||\mathcal{Y}| \log(n+1) \equiv \Delta_{\mu}\end{gathered}$$
From Lemma \[lemma:likely\_convexity\_of\_MI\] and Eq.(\[eq:likely\_convexity\_with\_dependency\]) we have that the following equation: $$\label{eq:convexity_in_rate_analysis}
\hat I (\vr x; \vr y) - \Delta \leq \sum_{b=1}^{B+1} \left( \frac{m_b}{n} \hat I (\vr x_{B_b}; \vr y_{B_b}) \right) +
\frac{|L_0|}{n} \hat I (\vr x_{L_0}; \vr y_{L_0})$$ is satisfied when $\vr x$ is outside a set $J_{\Delta}$ with probability of at most $\exp \left( -n \left(\Delta - \tilde{\delta}_n\right) \right)$ where $\tilde{\delta}_n = (B+2) |\mathcal{X}| \cdot \frac{\log(n+1)}{n} + 2 B_{\max} \frac{\log(n)}{n}$. We shall find $B_{\max}$ later on. To make sure the probability of $J$ is less than $P_A$ we require $\exp \left( -n \left(\Delta - \tilde{\delta}_n \right) \right) \leq P_A $ therefore $$\begin{gathered}
\Delta \geq \tilde{\delta}_n - \frac{1}{n} \log \left( P_A \right)
=\\= (B+2) |\mathcal{X}| \cdot \frac{\log(n+1)}{n} + 2 B_{\max} \frac{\log(n)}{n} - \frac{1}{n} \log \left( P_A \right)\end{gathered}$$ and we choose $$\label{eq:rate_analysis_delta}
\Delta = (3 B_{\max}+2) |\mathcal{X}| \cdot \frac{\log(n+1)}{n} - \frac{1}{n} \log \left( P_A \right)$$
We now bound each element of Eq.(\[eq:convexity\_in\_rate\_analysis\]). Consider block $b$ with $m_b+1$ symbols. At the last symbol before decoding (symbol $m_b \equiv |U_b|$) none of the codewords, including the correct one crosses the threshold $\mu_m^*$, therefore: $$\label{eq:rate_analysis2}
\mu^*_{m_b} = \frac{K+ \Delta_{m_b}}{m_b} > \hat I (\vr x_{U_b}; \vr y_{U_b})$$ Specifically for the unfinished block we have at symbol $n$: $$\label{eq:rate_analysis3}
\mu^*_{m_{B+1}} = \frac{K+ \Delta_{m_{B+1}}}{m_{B+1}} > \hat I (\vr x_{U_{B+1}}; \vr y_{U_{B+1}})$$ The way to understand these bounds is as guarantee on the shortness of the blocks given sufficient mutual information. On the other hand, at the end of each block *including* the last symbol (symbols $(k_b, k_b+m_b)$), since one of the sequences was decoded we have: $$\begin{gathered}
\label{eq:rate_analysis2a}
\mu^*_{m_b+1} = \frac{K+ \Delta_{m_b+1}}{m_b+1}
\leq \\ \leq \max_{i} \hat I \left( \left( \vr x_i \right)_{k_b}^{k_b+m_b}; \vr y_{k_b}^{k_b+m_b} \right)
\leq
\log \min (|\mathcal{X}|,|\mathcal{Y}|) \equiv h_0\end{gathered}$$ Which we can use to bound the number of blocks, since $m_b+1 \geq \frac{K}{h_0}$ therefore $$\label{eq:rate_analysis2b}
B \leq \sum_{b=1}^B \left( \frac{h_0}{K}(m_b+1) \right) \leq \frac{h_0 \cdot n}{K} \equiv B_{\max}$$ As for the unused last-symbols we bound: $$\label{eq:rate_analysis4}
\hat I (\vr x_{L_0}; \vr y_{L_0}) \leq h_0$$ Combining Eq.(\[eq:rate\_analysis2b\]) and Eq.(\[eq:rate\_analysis\_delta\]) we have: $$\label{eq:rate_analysis_delta2}
\Delta \leq \left(\frac{3 h_0}{K} + \frac{2}{n} \right) |\mathcal{X}| \cdot \log(n+1) - \frac{1}{n} \log \left( P_A \right)$$ Combining Eq.(\[eq:rate\_analysis2\]),(\[eq:rate\_analysis3\]),(\[eq:rate\_analysis4\]) with Eq.(\[eq:convexity\_in\_rate\_analysis\]) and substituting $\Delta_m \leq \Delta_{\mu}$ yields: $$\begin{gathered}
\label{eq:rate_analysis5}
\hat I (\vr x; \vr y) <
\Delta + \sum_{b=1}^{B+1} \frac{m_b}{n} \left( \frac{K+ \Delta_{m_b}}{m_b} \right) +
\frac{B}{n} h_0
\leq \\ \leq
\Delta + \sum_{b=1}^{B+1} \frac{1}{n} \left( K + \Delta_{\mu} \right) +
\frac{B}{n} h_0
=\\=
\Delta + \frac{B+1}{n} \left( K + \Delta_{\mu} \right) +
\frac{B}{n} h_0\end{gathered}$$
From Eq.(\[eq:rate\_analysis5\]) $B$ and consequently $R_{\mathrm{act}}$ can be lower bounded: $$\begin{gathered}
\label{eq:rate_analysis6}
R_{\mathrm{act}} = \frac{B}{n} \cdot K
> \frac{\hat I (\vr x; \vr y) - \Delta - \frac{1}{n} \left( K + \Delta_{\mu} \right)}{K + \Delta_{\mu} + h_0} \cdot K
=\\= \frac{\hat I (\vr x; \vr y) - \Delta - \frac{K}{n} \left( 1 + \frac{\Delta_{\mu}}{K} \right)}{1 + \frac{\Delta_{\mu} + h_0}{K}}\end{gathered}$$
Now if we increase $K$ with $n$ such that $O(\log(n))<O(K)<O(n)$ (for example by choosing $K=n^\alpha$, $0<\alpha<1$), then $\frac{K}{n} \to 0$ as $n \to \infty$, since $\Delta_{\mu} = O(\log(n))$ we have $\frac{\Delta_{\mu}}{K} \to 0$ and from Eq.(\[eq:rate\_analysis\_delta2\]) we have $\Delta \to 0$ thus for any $\epsilon$ we have $n$ large enough so that: $$\begin{gathered}
R_{\mathrm{act}} > \frac{\hat I (\vr x; \vr y) - \epsilon}{1 + \epsilon}
> \left( \hat I (\vr x; \vr y) - \epsilon \right) (1 - \epsilon)
>\\>
\hat I (\vr x; \vr y) - (1+h_0) \epsilon
\equiv \Remp\end{gathered}$$ Outside the set $J$, where the last inequality is due to the fact $\hat I$ is bounded. Hence we proved our claim that the rate exceeds a rate function which converges uniformly to the empirical mutual information and the proof of Theorem \[theorem:discrete\_adaptive\] is complete.
### Rate analysis for the continuous case {#sec:rate_analysis_continuous}
The continuous case is more difficult from several reasons. One is that the error probability exponent has a missing degree of freedom ($\approx \exp((n-1)t)$). This results in a rate loss (through $s$ in the definition of $\mu^*_m$), which is larger for small blocks, and can be bounded only when assuming the number of blocks does not grow linearly with $n$. Since the effective mutual information $\Remp(\vr x, \vr y)$ is unbounded we cannot simply bound the loss of mutual information over the unused symbols. Specifically for a single symbol, $\hat\rho=1$ and $\Remp=\infty$. Therefore we use the convexity of the correlation factor and the fact it is bounded by 1. As a result, the loss introduced in order to attain convexity (over the rateless blocks) is in the correlation factor rather than the empirical mutual information. A loss in the correlation factor induces unbounded loss in the rate function for $\rho \approx 1$, leading to a maximum rate. In order to cope with these difficulties we use a threshold $T$ on the number of symbols in a block ($T$ is chosen to grow slower than $n$), and treat large and small blocks differently: the large blocks are analyzed through their correlation factor and for the small blocks the correlation factor is upper bounded by 1 and only the number of blocks is accounted for.
We denote $\hat\rho_b \equiv \hat\rho(\vr x_{U_b}, \vr y_{U_b})$ and $\hat\rho \equiv \hat\rho(\vr x, \vr y)$ the correlation factor measured on a rateless block and on the entire transmission block, respectively. We denote by $B_S = \{b : m_b \leq T\}$ and $B_L = \{b: m_b > T\}$ the indices of the small and the large blocks respectively (the last unfinished block included). The total number of symbols in the large blocks is denoted $m_L \equiv \sum_{b \in B_L} m_b$. The number of large blocks is bounded by $|B_L| < \frac{n}{T}$.
The decoding threshold is written as $$\mu^*_m = \frac{K}{m-1} + \frac{1}{m-1}\log \left( \frac{n}{P_e} \right) + \frac{\log(2)}{m-1} = \frac{K + \Delta_{\mu}}{m-1}$$ where we denoted $\Delta_{\mu} \equiv \log \left( \frac{2n}{P_e} \right)$. We consider the partitioning of the index set $\{1,\ldots,n\}$ into at most $p=\frac{n}{T}$ sets: the first $\frac{n}{T}-1$ (or less) sets are the large blocks except their last symbol $\bigcup_{b \in B_L} U_b$ (each with at least $T+1$ symbols by definition), and the last set denoted $L_1$ includes the rest of the symbols (last symbols of these blocks and all symbols of small blocks), and has $|L_1| = n - m_L$. Since this partitioning has a bounded number of sets, by applying Lemma \[lemma:likely\_convexity\_of\_rho\] and Eq.(\[eq:likely\_convexity\_with\_dependency\]) with $p = \frac{n}{T}$ we have that Eq.\[eq:convexity\_in\_cont\_rate\_analysis\] below is satisfied when $\vr x$ is outside a set $J$ with probability at most: $$\begin{gathered}
\label{eq:convexity_prob_in_cont_rate_analysis}
\Pr(J) \leq n^{2p} \cdot 2^p e^{ - n \Delta^2/8}
= \left( \sqrt{2} n \right)^{2\frac{n}{T}} e^{ - n \Delta^2/8}
=\\=
\exp \left[ - n \left( \log(e) \Delta^2/8 - \frac{2}{T} \log \left( \sqrt{2} n \right) \right) \right]\end{gathered}$$ For any $0 < \Delta \leq \frac{1}{7}$. This bound tends to 0 if $T > O(\log(n))$ (since $\log(e) \Delta^2/8 - \frac{2}{T} \log \left( \sqrt{2} n \right) \to \log(e) \Delta^2/8 > 0$) therefore for any such $\Delta$ there is $n$ large enough such that this probability falls below the required $P_A$. The convexity condition is: $$\begin{gathered}
\label{eq:convexity_in_cont_rate_analysis}
\hat \rho^2 - \Delta \leq \sum_{b \in B_L} \frac{m_b}{n} \hat \rho_b^2 +
\frac{|L_1|}{n} \hat \rho(\vr x_{L_1}; \vr y_{L_1})^2
\leq \\ \leq
\sum_{b \in B_L} \frac{m_b}{n} \hat \rho_b^2 + \frac{n - m_L}{n}
\end{gathered}$$ where $\Delta$ can be made arbitrarily close to 0. We define a factor $\eta_1<1$ and apply the function $\smallminushalf \log(1-\eta_1 t)$ to both sides of the above equation. Since the function is monotonically increasing and convex $\cup$ over $t \in [0,1)$ (stemming from concavity $\cap$ of $\log(t)$), we have: $$\begin{gathered}
\label{eq:cont_rate_analysis3}
r_0 \equiv \smallminushalf \log(1 - \eta_1 \cdot (\hat \rho^2 - \Delta))
\leq \\ \stackrel{\text{(\ref{eq:convexity_in_cont_rate_analysis})}}{\leq} \smallminushalf \log \left[ 1 - \eta_1 \left( \sum_{b \in B_L} \frac{m_b}{n} \hat \rho_b^2 + \frac{n - m_L}{n} \cdot 1\right) \right]
\leq \\ \leq
\sum_{b \in B_L} \frac{m_b}{n} \smallminushalf \log \left( 1 - \eta_1 \hat \rho_b^2 \right)
+\\+ \frac{n - m_L}{n} \smallminushalf \log \left( 1 - \eta_1 \cdot 1 \right)
\end{gathered}$$
We start by bounding the terms related to the large blocks. At the last symbol before decoding in each block (or symbol $n$ for the unfinished block) none of the codewords, including the correct one crosses the threshold $\mu_m^*$, therefore we have for $b=1,\ldots,B+1$: $$\label{eq:cont_rate_analysis2}
\mu^*_{m_b} = \frac{K + \Delta_{\mu}}{m_b-1} > \Remp(\vr x_{U_b}, \vr y_{U_b}) = -\half \log (1 - \hat\rho_b^2)$$ and since $m_b \geq T+1$: $$\begin{gathered}
\label{eq:cont_rate_analysis4}
\frac{m_b}{n} \smallminushalf \log \left( 1 - \eta_1 \hat \rho_b^2 \right) \leq
\frac{m_b}{n} \smallminushalf \log \left( 1 - \hat \rho_b^2 \right)
< \\ \stackrel{\text{(\ref{eq:cont_rate_analysis2})}}{<}
\frac{m_b}{n} \cdot \frac{K + \Delta_{\mu}}{m_b-1} = \left(1 + \frac{1}{m_b-1} \right) \frac{K + \Delta_{\mu}}{n}
\leq \\ \leq
\left(1 + \frac{1}{T}\right) \frac{K + \Delta_{\mu}}{n}
\end{gathered}$$ For the small blocks we use $n \leq \sum_{b \in B_L} (m_b + 1) + \sum_{b \in B_S} (m_b + 1) \leq m_L + |B_L| + (T+1) |B_S|$ (where the inequality is since the unterminated block has length $m_b$) to bound $ n - m_L \leq |B_L| + (T+1) |B_S|$.
Combining Eq.(\[eq:cont\_rate\_analysis3\]) with these bounds we have: $$\begin{gathered}
\label{eq:cont_rate_analysis5}
r_0 \leq
|B_L| \left(1 + \frac{1}{T}\right) \frac{K + \Delta_{\mu}}{n}
+\\+
\frac{|B_L| + (T+1) |B_S|}{n} \left[ - \half \log \left( 1 - \eta_1 \right) \right]\end{gathered}$$ The last equation is a lower bound on a linear combination of $|B_L|$ and $|B_S|$. Since the total information sent depends on $|B_L| + |B_S|$ we equalize the coefficients multiplying $|B_L|$ and $|B_S|$ by determining $\eta_1$ so that: $$\label{eq:cont_rate_analysis6}
- \half \log \left( 1 - \eta_1 \right) = \left(1 + \frac{1}{T}\right) \frac{K + \Delta_{\mu}}{T}$$ This is always possible since the RHS is positive and the LHS maps $\eta_1 \in (0,1)$ to $(0, \infty)$. Then $$\begin{gathered}
\label{eq:cont_rate_analysis7}
r_0 \leq \left(|B_L| + \frac{|B_L| + (T+1) |B_S|}{T} \right) \left(1 + \frac{1}{T}\right) \frac{K + \Delta_{\mu}}{n}
=\\=
\left(|B_L| + |B_S| \right) \frac{K + \Delta_{\mu}}{n} = \left(B+1 \right) \frac{K + \Delta_{\mu}}{n}\end{gathered}$$ Extracting a lower bound on $B$ from Eq.(\[eq:cont\_rate\_analysis7\]) yields a bound on the empirical rate: $$\begin{gathered}
\label{eq:cont_rate_analysis8}
R_{\mathrm{act}} = \frac{K}{n} \cdot B
\geq \\ \geq \frac{K}{n} \cdot \left( \frac{r_0 \cdot n}{K + \Delta_{\mu}} -1 \right) =
\frac{r_0}{ 1 + K^{-1}\Delta_{\mu}} -\frac{K}{n}
=\\=
\frac{\smallminushalf \log(1 - \eta_1 (\hat \rho^2 - \Delta))}{ (1 + K^{-1}\Delta_{\mu})} -\frac{K}{n}
\equiv \RLBONE\end{gathered}$$ Equation (\[eq:cont\_rate\_analysis8\]) may be optimized with respect to $T$ to obtain a tighter bound, but this is not necessary to prove the theorem. Recall that $\Delta_{\mu} = O(\log(n))$. By choosing $O(\log(n)) < K < O(n)$ the factor $(1 + K^{-1}\Delta_{\mu})$ in Eq.(\[eq:cont\_rate\_analysis8\]) can be made arbitrarily close to 1 and $\frac{K}{n}$ can be made arbitrarily close to 0. As we saw above choosing $O(\log(n)) < T < O(n)$ enables us to have $P_A \to 0$ with $\Delta$ arbitrarily close to 0, and finally if $K > O(T)$ then the RHS of Eq.(\[eq:cont\_rate\_analysis6\]) tends to $\infty$ and therefore we can choose $\eta_1$ arbitrarily close to 1. Summarizing the above, by selecting $O(\log(n)) < O(T) < O(K) < O(n)$ we can write the rate as $$\label{eq:cont_rate_analysis9}
R_{\mathrm{act}} \geq \RLBONE = \smallminushalf \log(1 - \eta_1 \cdot (\hat \rho^2 - \Delta)) \cdot \eta_2 - \epsilon_1$$ With $\eta_1, \eta_2 {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty}$}} 1^{-}$ and $\epsilon_1,\Delta {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty}$}} 0^+$. $\RLBONE$ tends to the target rate $ R_2(\hat\rho) \equiv \half \log \left( \frac{1}{1 - \hat\rho^2} \right)$ for each point $\hat\rho \in [0,1)$ (but not uniformly), and it remains to show that for any $\bar{R}, \epsilon$ there is $n$ large enough such that $\RLBONE \geq \RLBTWO \equiv \min(R_2(\hat\rho)-\epsilon, \bar{R})$.
The functions $R_2(\rho)$ and $\RLBONE(\rho)$ are monotonically increasing (for fixed $\eta_1, \eta_2$ and $\epsilon_1$) and it is easy to verify by differentiation that the difference $R_2(\rho) - \RLBONE(\rho)$ is also monotonically increasing. Given $\bar{R}, \epsilon$, choose $\rho_0$ such that $R_2(\rho_0) = \bar{R} + \epsilon$. Since $\RLBONE(\rho_0) {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty}$}} R_2(\rho_0)$, for $n$ large enough we have $R_2(\rho_0) - \RLBONE(\rho_0) \leq \epsilon$, and therefore $\RLBONE(\rho_0) \geq R_2(\rho_0) - \epsilon = \bar{R}$. For this $n$, for any $\rho \leq \rho_0$ from the monotonicity of the difference we have that $R_2(\rho) - \RLBONE(\rho) \leq \epsilon$, and for any $\rho \geq \rho_0$ we have from the monotonicity of $\RLBONE(\rho)$ that $\RLBONE(\rho) \geq \bar{R}$, therefore $\RLBONE \geq \RLBTWO$, which completes the proof of Theorem \[theorem:continuous\_adaptive\].
Examples {#sec:examples}
========
In this section we give some examples to illustrate the model developed in this paper. In this section we use a slightly less formal notation.
Constant outputs and other illustrative cases {#sec:paradox}
---------------------------------------------
The statement that a rate which is determined by the input and output sequences can be attained without assuming any dependence between them may seem paradoxical at first. Some insight can be gained by looking at the specific case where the output sequence is fixed and does not depend on the input. In this case, obviously, no information can be transferred. Since the encoder uses random sequences, the result of fixing the output is that the probability to have an empirical mutual information larger than $\epsilon>0$ tends to $0$, therefore most of the time the rate will be $0$. Infrequently, however, the input sequence accidentally has empirical mutual information larger than $\epsilon>0$ with the output sequence. In this case the decoder will set a positive rate, but very likely fail to decode. These cases occur in vanishing probability and constitute part of the error probability. So in this case we will transmit rate $R=0$ with probability of at least $1-P_e$ and $R>0$ with probability at most $P_e$. Conversely, if the channel appears to be good according to the input and output sequences (suppose for example $y_k=x_k$), the decoder does not know if it is facing a good channel or just a coincidence, however it takes a small risk by assuming it is indeed a good channel and attempting to decode, since the chances of high mutual information appearing accidentally are small (and uniformly bounded for all output sequences).
Another point that appears paradoxical at first sight is that the decoder is able to determine a rate $R \geq \Remp$ without knowing $\vr x$ for any $\vr x \not\in J$. First observe that although it is an output of the decoder, the rate $R$ is not controlled by the encoder and therefore cannot convey information. Since the decoder knows the codebook, and given the codebook the sequence $\vr x$ is limited to a number of possibilities (determined by the possible messages and block locations), it is easy to find an $R(\vr y) \geq \Remp(\vr x, \vr y)$ by maximizing $\Remp$ over all possible sequences $\vr x$. Vaguely speaking, the decoding process is indeed a maximization of $\Remp$ over multiple $\vr x$ sequences and by Lemmas \[lemma:pairwise\_discrete\], \[lemma:pairwise\_continuous\] such a decoding process guarantees small probability of error.
Applying the continuous alphabet scheme to other input alphabets {#sec:example_adaptation_func}
----------------------------------------------------------------
The scheme used for the continuous case can be adapted to peak limited or even discrete input, by using an adaptation function, i.e. the channel input will be $x_k' = f(x_k)$. In this case the modified codebook $C' = f(C)$ will be generated by passing the Gaussian codebook through the adaptation function, but for analysis purposes the adaptation function $f(\cdot)$ may be considered part of the channel and the correlation factor is calculated with respect to $\vr x$ which is used to generate the codebook. In order to write the rate guaranteed by this approach as a function of $\vr x'$ rather than $\vr x$, the law of large numbers has to be utilized (in general) with respect to the distribution $\Pr(x_k \vert x_k')$.
Non linear channels {#sec:example_non_linear}
-------------------
In analyzing probabilistic channels, the correlation model determines the rate $\half \log \left( \frac{1}{1-\rho^2} \right)$ is always achievable using Gaussian code (no randomization is needed if the channel is probabilistic as can be shown by the standard argument about the existence of a good code). This is actually a result of Lemma \[lemma:gaussian\_mi\_bound\].
This expression is useful for analyzing channels in which the noise is not additive or non linearities exist. As an example, transmitter noise is usually modeled as an additive noise. However large part of this noise is due to distortion (e.g. in the power amplifier), and therefore depends on the transmitted signal and is inversely correlated to it. Consider the non linear channel $Y = f(X) + V$ with $V \sim \Normal(0,N)$. In this case if we define the effective SNR as $\textit{SNR}=\frac{\rho^2}{1-\rho^2}$ then rate $R=\half \log \left( 1 + \textit{SNR} \right)$ is achievable. The correlation factor is: $$\rho^2 = \frac{E(XY)^2}{E(X^2) E(Y^2)} = \frac{E(X f(X))^2}{E(X^2) (E(f(X)^2) + N)}$$ Therefore the effective *SNR* is: $$\begin{gathered}
\label{eq:eff_noise}
\textit{SNR}=\frac{\rho^2}{1-\rho^2}
=\\= \frac{E(X f(X))^2}{E(X^2) (E(f(X)^2) + N) - E(X f(X))^2} =
\frac{\Peff}{ N + \Neff}\end{gathered}$$ where we defined the effective gain $\gamma$, the effective power $\Peff$ and the effective noise $\Neff$ as: $$\begin{aligned}
\gamma &\equiv& \frac{E(X f(X))}{E(X^2)} \\
\Peff &\equiv& \frac{(E[(X f(X)])^2}{E(X^2)} = E\left[(\gamma X)^2\right] \\
\Neff &\equiv& E(f(X)^2) - \frac{(E[X f(X)])^2}{E(X^2)} \nonumber \\
&=& E \left[(f(X) - \gamma X)^2 \right]\end{aligned}$$ This yields a simple characterization of the degradation caused by the non linearity, which is independent of the noise power and is tight if the non linearity is small. This model enables to characterize the transmitter distortions by the two parameters $\Peff, \Neff$, a characterization which is more convenient and practical to calculate than the channel capacity, and on the other hand guarantees that transmitter noise evaluated this way never degrades the channel capacity in more than determined by Eq.(\[eq:eff\_noise\]).
Another interesting application of this bound is in treating receiver estimation errors, since it is simpler to calculate the loss in the correlation factor induced due to the imperfect knowledge of the channel parameters than the loss in capacity. For example, the bound in [@Hassibi] for the loss due to channel estimation from training, when specialized to single input single output (SISO) channels, may be computed using the correlation factor bound.
Employing continuous channel scheme over a BSC {#sec:example_continuous_over_BSC}
----------------------------------------------
When operated over a channel different than the Gaussian additive noise channel, the rates achieved with the scheme we described in the continuous case are suboptimal compared to the channel capacity. The loss depends on the channel in question. As an example, suppose the communication system is used over a BSC with error probability $\epsilon$, i.e. the continuous input value $X$ is translated to a binary value by $\sign(X)$, and the output is $Y = \sign(X) \cdot (-1)^{Ber(\epsilon)}$. The capacity of this channel is $C = 1_{\mathrm{bit}}-h_b({\epsilon})$ and we are interested to calculate the rate which would be achieved by our scheme (which does not know the channel) for this channel behavior. For this channel with Gaussian $\Normal(0,P)$ input we have (through a simple calculation): $$E(XY) = (1-2\epsilon) \sqrt{\frac{2P}{\pi}}$$
Hence $$\rho^2 = \frac{E(XY)^2}{P \cdot E(1^2)} = \frac{2}{\pi} (1-2\epsilon)^2$$ And $$R = \half \log \left( \frac{1}{1-\frac{2}{\pi} (1-2\epsilon)^2} \right)$$ The comparison between $C$ and $R$ is presented in fig.(\[fig:rate\_in\_bsc\]). It can be shown that $R \geq \frac{2}{\pi} C$, thus the maximum loss is 36%.
![Comparison of C,R for the BSC[]{data-label="fig:rate_in_bsc"}](rate_in_bsc){width="50.00000%"}
Channels that fail the zero order and the correlation model {#sec:example_failure_models}
-----------------------------------------------------------
Although we did not assume anything about the channel, and specifically we did not assume the channel is memoryless, the fact we used the zero-order empirical distribution means the results are less tight for channels with memory. Specifically if delay is introduced then the scheme would fail completely. For example, for the channel $y_k = x_k + \half x_{k-1} + v_k$ we would obtain positive rates and the intersymbol interference (ISI) $\half x_{k-1}$ would be treated (suboptimally) as noise, but for the error free channel $y_k = x_{k-1}$ the achieved rate would be 0 (with high probability). Similarly we can find a memoryless channel with infinite capacity but for which the correlation model we used for the continuous alphabet scheme fails: if $y_k = x_k^2$ then $\rho=0$. Another example of practical importance is the fading channel (with memory) $y_n=h_n x_n+v_n$, where $h_n$ is slowly fading with mean $0$. All these examples result from the simplicity of the models used, and can be solved by schemes employing higher order empirical distributions (over blocks, or by using Markov models), and by employing tighter approximations of the empirical statistics (e.g. by higher order statistics) in the continuous case.
Using individual channel model to analyze adversarial individual sequence {#sec:example_adversary}
-------------------------------------------------------------------------
As we noted in the overview, the results obtained for the individual channel model constitute a convenient starting point for analyzing channel models which have a full or partial probabilistic behavior. It is clear that results regarding achievable rates in fully probabilistic, compound, arbitrarily varying and individual noise sequence models can be obtained by applying the weak law of large numbers to the theorems discussed here (limited, in general, to the randomized encoders regime).
E.g. for a compound channel model $W_{\theta}(y|x)$ with an unknown parameter $\theta$ since $\hat P(\vr x; \vr y) {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty}$}} P_{\theta}(x,y) = W_{\theta}(y|x) Q(x) $ in probability for every $\theta$ and since $I(\cdot ; \cdot)$ is continuous $\hat I(\vr x; \vr y) {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty}$}} I_{\theta}(X;Y)$. Hence from Theorem \[theorem:discrete\_nonadaptive\] rate $\min_{\theta} I_{\theta}(X;Y)$ can be obtained without feedback, and from Theorem \[theorem:discrete\_adaptive\] rate $I_{\theta}(X;Y)$ can be obtained with feedback. These results are not new (see [@Blackwell_compound][@Lapidoth_compound] for the first and the second is obtained as a special case of the results in [@Eswaran] and [@Ofer_EMP] since the individual noise sequence model can be degenerated into a compound model) and are given only to show the ease of using the individual model once established.
To show the strength of the model we analyze a problem considered also in [@Ofer_EMP] of an individual sequence which is determined by an adversary and allowed to depend in a fixed or randomized way on the past channel inputs and outputs. For simplicity we start with the binary channel $y_k = x_k \oplus e_k$ where $e_k$ is allowed to depend on $\vr x_1^{k-1}$ and $\vr y_1^{k-1}$ (possibly in a random fashion), and the target is to show the empirical capacity is still achievable in this scenario. Note that here $E_k$ is a random variable but not assumed to be i.i.d. We denote the relative number of errors by $\hat \epsilon \equiv \frac{1}{n} \sum_{k=1}^n e_k$. We would like to show the communication scheme achieves a rate close to $1_{\mathrm{bit}} - h_b(\hat\epsilon)$ in high probability, regardless of the adversary’s policy. Note that both the achieved rate and the target $1_{\mathrm{bit}} - h_b(\hat\epsilon)$ are random variables and the claim is that they are close in high probability (i.e. that the difference converges in probability to $0$ when $n \to \infty$) Applying the scheme achieving Theorem \[theorem:discrete\_adaptive\] with $Q = Ber(\half)$ we can asymptotically approach (or exceed) the rate: $$\begin{gathered}
\hat I (\vr x; \vr y) = \hat H (\vr y) - \hat H (\vr y | \vr x) = \hat H (\vr y) - \hat H (\vr e | \vr x)
\geq \\ \geq
\hat H (\vr y) - \hat H (\vr e) = \hat H (\vr y) - h_b(\hat \epsilon)\end{gathered}$$ Note that unlike in the probabilistic BSC where we have $I(X;Y)=H(Y)-H(E)$, here the empirical distribution of $\vr e$ is not necessarily independent of $\vr x$, therefore the entropies are only related by the inequality $\hat H (\vr e | \vr x) \leq \hat H (\vr e)$ (conditioning reduces entropy). In order to show a rate of $1_{\mathrm{bit}} - h_b(\hat\epsilon)$ is achieved, we only need to show $\hat H (\vr y) {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty, prob.}$}} 1_{\mathrm{bit}}$. Since $X_k$ is independent of $X_1^{k-1}, Y_1^{k-1}$ and therefore also of $E_k$ we have: $$\begin{gathered}
\Pr(Y_k=0 | Y_1^{k-1}) = \sum_{e_k} \Pr(Y_k=0 | Y_1^{k-1}, e_k) \Pr(e_k)
=\\= \sum_{e_k} \Pr(X_k = e_k | Y_1^{k-1}, e_k) \Pr(e_k)
=\\= \sum_{e_k} \Pr(X_k = e_k) \Pr(e_k)
= \sum_{e_k} \half \Pr(e_k) = \half\end{gathered}$$ Therefore $Y_1^n$ is distributed i.i.d. $Ber(\half)$ and from the law of large numbers and the continuity of $H(\cdot)$ we have the desired result. This result is a special case of the results in [@Ofer_EMP].
We can extend the example above to general discrete channels and perform a consolidation of the adversarial sequence model considered in [@Ofer_EMP] (for modulu additive channels) with the general discrete channel with fixed sequence considered in [@Eswaran]. We address the channel $W_s(y|x)$ with state sequence $s_k$ potentially determined by an adversary knowing all past inputs and outputs. We would like to show that the rate $I(Q, \sum_s W_s(y|x) \hat P_{\vr s}(s) )$ (the mutual information of the state-averaged channel) can be asymptotically attained in the sense defined above.
This result is a superset of the results of [@Eswaran] and [@Ofer_EMP]. It overlaps with [@Eswaran] in the case $\vr s$ is a fixed sequence and with [@Ofer_EMP] for the case of modulu-additive channel (or when the target rate is based on the modulu additive model).
Since Theorem \[theorem:discrete\_adaptive\] shows the rate $\hat I (\vr x; \vr y) \equiv I(\hat P(\vr x), \hat P(\vr y | \vr x))$ can be approached or exceeded asymptotically, it remains to show that the empirical distribution $\hat P(\vr x, \vr y)$ is asymptotically close to the state-averaged distribution $P_{avg}(x,y) \equiv \sum_s W_s(y|x) \hat P_{\vr s}(s) Q(x) = \frac{1}{n} \sum_k W_{S_k}(y|x) Q(x) $, and the result will follow from continuity of the mutual information. Note that the later value is a random variable (function) depending on the behavior of the adversary. Here we do not use the law of large numbers because of the interdependencies between the signals $\vr x, \vr y$ and $\vr s$.
Our purpose is to prove that the difference $\Delta(t,r)$ defined below converges in probability to 0 for every $t,r$: $$\begin{gathered}
\Delta(t,r) \equiv \hat P_{(\vr x, \vr y)}(t,r) - P_{avg}(t,r)
=\\= \frac{1}{n} \sum_k \Ind(X_k = t, Y_k = r) - \frac{1}{n} \sum_k W_{S_k}(r|t) Q(t)
\equiv \\ \equiv
\frac{1}{n} \sum_k \varphi_k(t,r)\end{gathered}$$ where $\varphi_k(t,r) \equiv \Ind(X_k = t, Y_k = r) - W_{S_k}(r|t) Q(t)$. For brevity of notation we omit the argument $(t,r)$ from $\varphi_k(t,r)$ since from this point on it takes a fixed value. Then $$\begin{gathered}
\label{eq:example_adversary1}
E (\Ind(X_k = t, Y_k = r) \vert X^{k-1},Y^{k-1},S^{k})
=\\= \Pr(X_k = t, Y_k = r \vert X^{k-1},Y^{k-1},S^{k})
=\\= \Pr(X_k = t \vert X^{k-1},Y^{k-1},S^{k})
\cdot \\ \cdot \Pr(Y_k = r \vert X_k = t, X^{k-1},Y^{k-1},S^{k})
=\\ \stackrel{(a)}{=}
\Pr(X_k = t) \cdot \Pr(Y_k = r \vert X_k = t, S_k)
\stackrel{(b)}{=} Q(t) W_{S_k}(r|t)\end{gathered}$$ where (a) is due to the independent drawing of $X_k$ (when not conditioned on the codebook), the fact $S^k$ is independent of $X_k$, and the memoryless channel (defining the Markov chain $(X^{k-1}, Y^{k-1}, S^{k-1}) \leftrightarrow (X_k,S_k) \leftrightarrow Y_k$), and (b) is due to the i.i.d drawing of $X_k$ from $Q$ and the definition of $W$. From Eq.(\[eq:example\_adversary1\]) we have that: $$E(\varphi_k \vert X^{k-1},Y^{k-1},S^{k}) = 0$$
By the smoothing theorem we also have that $\varphi_k$ has zero mean $E(\varphi_k)=0$. We now show that $\varphi_k$ are uncorrelated. Consider two different indices $j < k$ (without loss of generality) then $$\begin{gathered}
E(\varphi_k\cdot \varphi_j) = E \left[ E(\varphi_k \cdot \varphi_j \vert X^{k-1},Y^{k-1},S^{k}) \right]
=\\=
E \left[ \varphi_j \cdot E(\varphi_k \vert X^{k-1},Y^{k-1},S^{k}) \right] = 0\end{gathered}$$ where we used the smoothing theorem and the fact $\varphi_j$ is completely determined by $X_j,Y_j,S_j$ which are given. In addition since by definition $-1 \leq \varphi_k \leq 1$, $E(\varphi_k^2) \leq 1$. Therefore $$E (\Delta^2) = \frac{1}{n^2} \sum_{j,k=1}^n E( \varphi_k \cdot \varphi_j) \leq \frac{1}{n^2} \sum_{j,k=1}^n \delta_{jk} = \frac{1}{n}$$ and by Chebychev inequality for any $\epsilon>0$: $$\Pr(|\Delta(t,r)| > \epsilon) \leq \frac{E (\Delta^2)}{\epsilon^2} \leq \frac{1}{n \epsilon^2} {\raisebox{-1.0ex}{$\stackrel{\textstyle \longrightarrow}{\scriptscriptstyle n \to \infty}$}} 0$$ which proves the claim.
This result is new, to our knowledge, however the main point here is the relative simplicity in which it is attained when relying on the empirical channel model (note that most of the proof did not require any information-theoretic argument).
Comments and further study {#sec:comments}
==========================
Limitations of the model
------------------------
The scheme presented here is suboptimal when operated over channels with memory or, in the continuous case over non AWGN channels, and in section \[sec:example\_failure\_models\] we discussed several cases where the communication fails completely. Obviously the solution is to extend the time order of the model. A simple extension is by using the super-alphabets $\mathcal{X}^p$ and $\mathcal{Y}^p$ and treating a block of channel uses as one symbol. A more delicate extension is by considering a Markov model (the $p$-th order empirical conditional probability $\hat P (x_k,y_k | \vr x_{k-p}^{k-1}, \vr y_{k-p}^{k-1})$).
For the continuous channel we focused on a specific class of continuous channels where the alphabet is the real numbers (we have not considered vectors as in MIMO channels), and we did not achieve the full mutual information. A possible extension is to find measures of empirical mutual information for the continuous channels which are also attainable and approach the probabilistic mutual information for probabilistic channels. The current paper exhibits a considerable similarity between the continuous case and the discrete case which is not fully explored here, and a unifying theory which will include the two as particular cases is wanting.
We conjecture that the following definition of empirical mutual information may achieve these goals: given a family of joint distributions (not necessarily i.i.d) $\{ P_\theta(\vr x, \vr y), \theta \in \Theta\}$ define the entropy with respect to the family $\Theta$ as the entropy of the closest member of the family (in maximum likelihood sense): $\hat H_{\Theta}(\vr x) = \min_{\theta \in \Theta} - \frac{1}{n} \log P_{\theta}(\vr x)$ and likewise $\hat H_{\Theta}(\vr x|\vr y) = \min_{\theta \in \Theta} - \frac{1}{n} \log P_{\theta}(\vr x | \vr y)$, and define the relative mutual information as $\hat I_{\Theta}(\vr x;\vr y) = \hat H_{\Theta}(\vr x) - \hat H_{\Theta}(\vr x|\vr y)$. This definition corresponds to our target rates for the discrete case (with $\Theta$ as the family of all DMC-s) and continuous case (with $\Theta$ the family of all joint Gaussian zero-mean distributions $\Normal(0,\Lambda_{XY})$).
Overhead and error exponent
---------------------------
Another aspect is the overhead associated with extending the empirical distribution (“channel”) family which is considered (both in considering time dependence and in increasing the accuracy with which the distribution is estimated or described). This overhead is related to the redundancy or regret associated with universal distributions (see [@Barron_MDL]). Although we haven’t performed a detailed analysis of the overheads and considered only the asymptotically achievable rates, it is obvious from comparing Lemmas \[lemma:pairwise\_discrete\] and \[lemma:pairwise\_continuous\] that the tighter rates we obtained for the discrete channel come at the cost of additional overhead ($O(\log(n))$ compared to $O(1)$ in the continuous case) which is associated with the richness of the channel family (describing a conditional probability as opposed to a single correlation factor). Thus for example for a discrete channel with a large alphabet and a small block size $n$ we would sometimes be better off using the “continuous channel model” version of our scheme (gaining only from the correlation) rather than the scheme of the discrete case (gaining the empirical mutual information). The issue of overheads requires additional analysis in order to determine the bounds on the overheads and the tradeoff between richness of the channel family and the rate, for a finite $n$. As we noted in section \[sec:rate\_analysis\_continuous\] the bounds we currently have for the rate-adaptive continuous case are especially loose and call for improvement.
Since rate can be traded off for error probability, a related question is the error exponent. Here, a good definition is still lacking for variable rate schemes, and the error exponents are not known for individual channels. The scheme we described does not endeavor to attain a good error exponent. Specifically, since the block of $n$ channel uses is broken into multiple smaller blocks, it is probably not an efficient scheme in terms of error rate. We note, however, that for rate adaptive schemes with feedback a good error exponent does not necessarily relate to the capability of sending a message with small probability of error, but rather to the capability to detect the errors. A similar situation occurs in the setting of random decision time considered by Burnashev [@Burnashev]. In the later, an uncertainty of the decoder with respect to the message is mitigated by sending an acknowledge / unacknowledge (ACK/NACK) message and possibly repeating the transmission with small penalty in the average rate (see a good description in [@Tchamkerten] sec IV.B). A similar approach can be used in our setting (fixed decoding time, variable rate), by sending an ACK/NACK over a fixed portion of the block and setting $R=0$ when the decoder is not certain of the received message. However we did not perform a detailed analysis. Note also that the analysis of the probability $P_A$ to transmit at a rate lower than the target rate function is entangled with the error analysis, since by such schemes it is possible to trade off rate for error, and reduce the error probability at the expense of increasing the probability to fall short of the target rate.
Determining the behavior of the transmitted signal (prior)
----------------------------------------------------------
In this work we assumed a fixed prior (input probability distribution) and haven’t dealt with the question of determining the prior, or more generally, how the encoder should adapt its behavior based on the feedback. Had the channel been a compound one, it stands to reason that a scheme using feedback may estimate the channel and adjust the input prior, and may asymptotically attain the channel capacity. However in the scope of individual channels (as well as individual sequence channels and AVC-s) it is not clear whether the approach of adjusting to the input distribution to the measured conditional distribution is of merit, if the empirical channel capacity can be attained for every sequence, and even the definition of achievability is unclear if the input distribution is allowed to vary.
Another related aspect is what we require from a communications system when considered under the individual channel framework. This question is relevant to all the requirements defined in the theorems (for example is the existence of the failure set $J$ necessary ?), however the most outstanding requirement is related to the prior.
Currently we constrained the input sequence to be a random i.i.d. sequence chosen from a fixed prior, which seems to be an overly narrow definition. The rationale behind this choice is that without any constraint on the input, the theorems we presented can be attained in a void way by transmitting only bad (e.g. fixed) sequences that guarantee zero empirical rate. Furthermore, without this constraint, attainability results for probabilistic models, and in general any attainable rates which are not conditioned on the input sequence could not be derived from our individual sequence theorems. A weaker requirement from the encoder is to be able to emit any possible sequence, however this requirement is not sufficient, since from the existence of such encoders we could not infer the existence of encoders achieving any positive rate over a specific channel. Consider for example the encoder satisfying the requirement by transmitting bad sequences in probability $1-\epsilon$ and good sequences in probability $\epsilon \to 0$. Theorems \[theorem:discrete\_nonadaptive\],\[theorem:continuous\_nonadaptive\],\[theorem:discrete\_adaptive\] and \[theorem:continuous\_adaptive\] are existence theorems, i.e. they guarantee the existence of at least one system satisfying the conditions. Had we removed the requirement for fixed input prior we saw these theorems would be attained by encoders that are unsatisfactory in other aspects. Once the theorem is satisfied by one encoder it cannot guarantee the existence of other (satisfactory) encoders, thus making it un-useful. Therefore the requirement for fixed prior is necessary in the current framework. Although in the scope of the theorems presented here, this requirement only strengthens the theorems (since it reveals additional properties of the encoder attaining the other conditions of the theorem), we are still bothered by the question what should be the minimal requirements from a communication system, and these hopefully will not include a constraint on the input distribution.
This issue relates to a fundamental difficulty which aries in communication over individual channels: unlike universal source coding in which the sequence is given a-priori, here the sequences are given a-posteriori, and the actions of the encoder affect the outcome in an unspecified way. Currently we broke the tie by placing a constraint on the encoder, but we seek a more general definition of the problem.
Amount of randomization
-----------------------
We have assumed so far there is no restriction on the amount of common randomness available and have not attempted to minimize the amount of randomization required (while maintaining the same rates). It is shown in [@Ofer_EMP] that less than $O(n)$ of randomization information is required in some cases and $O(n)$ is enough for others (see section V.5 therein), whereas we have used at least $O(M\cdot n) > O(n^2)$ random drawings to produce the codebook.
Practical aspects
-----------------
The scheme described in this work is a theoretical one, but the concept appears to be extendable to practical coding systems. Below we focus on the continuous case and merely give the motivation (without proof). One may replace the correlation receiver (GLRT) by a receiver utilizing training symbols to learn the channel effective gain, and then apply maximum likelihood (or approximate, e.g. iterative) decoding. The randomization of the codebook may be replaced by using a fixed code with random interleaving, since with random interleaving only the empirical distribution of the (effective) noise sequence affects the error probability, and we may conjecture that the property that Gaussian noise distribution is the worst is approximately true for practical codes (such as turbo codes and LDPC). When using a random interleaver the training symbols as well as the part of the coded symbols can be interleaved together, and the decoding attempts (which occur every symbol in the theoretical scheme) occur only at the end of each interleaving block. The rateless code is replaced by an incremental redundancy scheme, i.e. by sending each time part of the symbols of the codeword, and repeating the codeword if all symbols were transmitted without successful decoding. The decision when to decode can be simply replaced by decoding and using a CRC check. Finally the common randomness (required only for the generation of the interleaver permutation) can be replaced by pseudo-randomness. Such a scheme may not be able to attain the promise of Theorem \[theorem:continuous\_adaptive\] for every individual sequence but may be able to adapt to every natural and man-made channel.
Random decision time
--------------------
In our discussion we have described two communication scenarios: fixed rate without feedback and variable rate with feedback, and in both we assumed a fixed block size $n$. Another scenario is that of random decision time or rateless coding (as in [@Burnashev] [@Shulman]) in which the block size is not fixed but determined by the decoder. We did not include this scenario since the achievability result is less elegant in a way: the decoder indirectly affects the target rate (mutual information) through the block size. On the other hand this case may be of practical interest. Clearly the mutual information can be asymptotically attained for this communication scenario as well and its analysis is merely a simpler version of the rate analysis performed in section \[sec:rate\_analysis\], since convexity is not required.
Bounds
------
In this paper we focused on achievable rates and did not show a converse. An almost obvious statement is that any continuous rate function which depends only on the zero-order empirical statistics / correlation (respectively) cannot exceed asymptotically the rate functions of Theorems \[theorem:discrete\_adaptive\], \[theorem:continuous\_adaptive\] respectively with vanishing error probability. To show the statement for the discrete case determine $\vr y$ using a memoryless channel $W(y|x)$. Then by the law of large numbers the empirical distribution converges to the channel distribution and from the continuity of the rate function the empirical rate converges to the rate function taken at the channel distribution. Since by Theorem \[theorem:discrete\_adaptive\] the actual rate asymptotically meets or exceeds the rate function, and by the converse of the channel capacity theorem the actual rate cannot exceed (asymptotically) the mutual information, we have that the rate function cannot exceed the mutual information ($\Remp \leq R_{act} \leq I(P,W)$), up to asymptotically vanishing factors. For the continuous case the analogue claim is shown by taking a Gaussian additive channel and replacing “distribution” by “correlation” and “empirical mutual information” by $-\half \log (1 - \hat\rho^2)$. The same applies also to rate functions obeying the conditions of Theorems \[theorem:discrete\_nonadaptive\], \[theorem:continuous\_nonadaptive\]. More general bounds are yet to be studied.
Item Eswaran et al [@Eswaran] Current Paper Comments
-------------------------- ----------------------------------------------------------------------------- -------------------------------------------------------------- ---------------------------------------------------------------------------------------------------
Channel model Individual sequence Individual channel
Mechanism for adaptivity Repeated instanced of rateless coding Repeated instanced of rateless coding
Transmit format Total time divided to rounds (=rateless blocks) which are divided to chunks Total time divided to rateless blocks Chunks in [@Eswaran] used as feedback instances and expurgated code has constant type over chunks
Feedback Ternary (Bad Noise/Decoded/Keep Going), once per chunk Binary (Decoded/Not Decoded) per symbol Easy to generalize to once every $1/\epsilon$ symbols (see \[sec:rate\_analysis\])
Alphabet Discrete Discrete or Real valued
Training Known symbols in random locations in each chunk None
Randomness Full ($O(\exp(nR))$) Full ($O(\exp(nR))$) Might be reduced by selection from a smaller collection of codebooks (in both cases)
Codebook construction Constant composition + expurgation + training insertion Random i.i.d.
Stopping condition Threshold over mutual information of channel estimated from training Threshold over empirical mutual information of best codeword
Decoding Maximum (empirical) mutual information Maximum (empirical) mutual information
Stopping location End of Chunk Any symbol
Comparison of the rate adaptive scheme with the similar scheme in [@Eswaran]
----------------------------------------------------------------------------
As noted the rate adaptive scheme we use is similar to the scheme of [@Eswaran] in its high level structure. Table \[table:comparison\_with\_Eswaran\] compares some attributes of the schemes.
Another important factor is the overhead (i.e. the loss in number of bits communicated with a given error exponent, compared to the target rate), which we were unable to compare. We conjecture that the current scheme may have a lower overhead due to its simplicity which results in a smaller number of parameters and constraints on their order of magnitude (compared to the scheme of [@Eswaran] where relations between factors such as number of pilots and the minimum size of a chunk may require a large value of $n$).
Conclusion
==========
We examined achievable transmission rates for channels with unspecified models, and focused on rates determined by a channel’s a-posteriori empirical behavior, and specifically on rate functions which are determined by the zero-order empirical distribution. This communication approach does not require a-priori specification of the channel model. The main result is that for discrete channels the empirical mutual information between the input and output sequences is attainable for any output sequence using feedback and common randomness, and for continuous real valued channels an effective “Gaussian capacity” $-\half(1-\hat\rho^2)$ can be attained. This generalizes results obtained for individual noise sequences and is a useful model for analyzing compound, arbitrarily varying, and individual noise sequence channels.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank the reviewers of the ISIT 2009 conference paper on the subject for their helpful comments and references.
Proof of Lemma 1 {#appendix:pairwise_discrete}
----------------
The proof is a rather standard calculation using the method of types. We use the notations of [@MethodOfTypes]. We divide the sequences according to their joint type $\mathcal{T}_{XY}$. The type $\mathcal{T}_{XY}$ is defined by the probability distribution $T_{XY} \in \mathcal{P}_n(\mathcal{XY})$. For notational purposes we define the dummy random variables $(\tilde{X},\tilde{Y}) \sim T_{XY}$ and $T_X$, $T_Y$, $T_{Y|X}$ as the marginal and conditional distributions resulting from $T_{XY}$. Following [@MethodOfTypes], the conditional type is defined as $\mathcal{T}_{X|Y}(\vr y) \equiv \left\{\vr y : (\vr x, \vr y) \in \mathcal{T}_{XY}\right\}$. The empirical mutual information of sequences in the type $\mathcal{T}_{XY}$ is simply $I(\tilde{X};\tilde{Y}) = I(T_Y , T_{Y|X})$. Define $T_t \equiv \{ T_{XY} \in \mathcal{P}_n(\mathcal{XY}): I(T_Y,T_{Y|X}) \geq t \}$. Since all sequences in the conditional type have the same (marginal) type, we can write: 1 $$\begin{gathered}
Q^n \left( \hat{I}(\vr x; \vr y) \geq t \right) = \sum_{T_t} Q^n \left( \mathcal{T}_{X|Y}(\vr y) \right) = \\ \stackrel{(a)}{=}
\sum_{T_t} \lvert \mathcal{T}_{X|Y}(\vr y) \rvert \exp \left\{-n \left[H(T_X) + D(T_X || Q) \right]\right\}
\leq \\ \stackrel{(b)}{\leq}
\sum_{T_t} \exp \left\{ n H(\tilde{X} | \tilde{Y}) \right\} \exp \left\{-n \left[H(\tilde{X}) + D(T_X || Q) \right]\right\}
=\\=
\sum_{T_t} \exp \left\{-n \left[I(\tilde{X};\tilde{Y}) + D(T_X || Q) \right]\right\}
\leq \\ \leq
|\mathcal{P}_n(\mathcal{XY})| \cdot \exp \left\{-n \left( \min_{T_t} \left[I(T_Y,T_{X|Y}) + D(T_X || Q) \right] \right) \right\}
\\ \stackrel{(c)}{\leq}
(n+1)^{|\mathcal{X}||\mathcal{Y}|} \cdot \exp \left(-n t \right)
=\\= \exp \left\{ -n \left(t - |\mathcal{X}||\mathcal{Y}|\frac{\log(n+1)}{n} \right) \right\}\end{gathered}$$ where (a) is due to [@MethodOfTypes] Eq.(II.1), (b) results from eq.(\[eq:size\_of\_condtype\]) below which is an extension of (II.4) there to conditional types (and is a stronger version of Lemma II.3), based on the fact that in the conditional type $\mathcal{T}_{X|Y}(\vr y)$ the values of $\vr x$ over the $n_a = n T_Y(a)$ indices for which $y_i = a$ have empirical distribution $T_{X|Y}$ and therefore the number of such sequences is limited to $\exp \left( n_a H(\tilde{X} | \tilde{Y}=a) \right) $, hence: $$\begin{gathered}
\label{eq:size_of_condtype}
\lvert \mathcal{T}_{X|Y}(\vr y) \rvert \leq \prod_a{\exp \left( n T_Y(a) H(\tilde{X} | \tilde{Y}=a) \right) }
=\\= \exp \left( n H(\tilde{X} | \tilde{Y}) \right)\end{gathered}$$ (c) is based on bounding the number of types (see [@Cover], Theorem 11.1.1), and the fact that in the minimization region $I(T_Y,T_{X|Y}) \geq t$ and $D(T_X || Q) \geq 0$ therefore the result of the minimum is at least $t$.
Discussion of Lemma 1
---------------------
### An alternative proof for the exponential rate
For the proof of Theorem \[theorem:discrete\_nonadaptive\] we do not need the strict inequalities and equality in the error exponent would be sufficient, however these will be useful later for the rateless coding. An explanation for the fact that the result does not depend on $Q$ can be obtained by showing that the above probability can be bounded for each type of $\vr x$ separately. I.e. if $\vr x$ is drawn uniformly over the type $\mathcal{T}_X$ the probability of the above condition is: $$\begin{gathered}
\label{eq:pairwise_discrete_approx}
\frac{\displaystyle \sum_{T_{XY} \in T_t} \lvert \mathcal{T}_{X|Y}(\vr y) \rvert}{\lvert \mathcal{T}_X \rvert} \doteq
\frac{\displaystyle \sum_{T_{XY} \in T_t} \exp(n H(\tilde{X} | \tilde{Y}) )}{\exp(n H(\tilde{X}))}
=\\=
\sum_{T_{XY} \in T_t} \exp(-n I(\tilde{X} ; \tilde{Y})) \doteq \exp(-n t)\end{gathered}$$ where $T_t \equiv \big\{ T_{XY} \in \mathcal{P}_n(\mathcal{XY}): (T_{XY})_X = T_X, (T_{XY})_Y = T_Y, I(T_Y,T_{Y|X}) \geq t \big\} $ and since drawing $\vr x \sim Q^n$ is equivalent to first drawing the type of $\vr x$ and then drawing $\vr x$ uniformly over the type, the bound holds when $\vr x \sim Q^n$.
### Extension to alpha receivers
Following we discuss an extension of the bound and relate it to Agarwal’s [@Agarwal_RD] coding theorem using the rate distortion function. Consider a communication system similar to that of Theorem \[theorem:discrete\_nonadaptive\], where the codebook is a constant composition code, consisting of randomly selected sequences of type $Q$, and the receiver is an $\alpha$ receiver (see [@CsiszarNarayan_mismatch95]), i.e. selects the received codeword by maximizing a function $\hat\alpha(\vr x, \vr y)$ depending only on the joint empirical distribution of the sequences $\vr x, \vr y$. The function $\alpha(\tilde{X}, \tilde{Y}) = \alpha(T_{XY})$ is defined as the respective function of the distribution of $\tilde{X}, \tilde{Y}$. Then, the pairwise error probability may be bounded similarly to eq. (\[eq:pairwise\_discrete\_approx\]) by replacing the condition the condition $I(T_Y,T_{Y|X}) \geq t$ in the definition of $T_t$ by $\alpha(T_{XY}) \geq t$, and obtaining: $$\begin{gathered}
\label{eq:generalized_pairwise_discrete_approx}
\Pr(\hat \alpha(\vr x, \vr y) \geq t) \leq P_\alpha
\doteq \\ \doteq \exp \bigg[ -n \bigg(\min_{ \begin{array}{c} \scriptstyle P_{\tilde{X}\tilde{Y}}: \tilde{X} \sim Q \\ \scriptstyle \tilde{Y} \sim \hat P(\vr y) \\ \scriptstyle \alpha(\tilde{X}, \tilde{Y})\geq t \end{array}} I(\tilde{X} ; \tilde{Y}) \bigg) \bigg]
\leq \\ \leq \exp \bigg[ -n \bigg(\min_{\begin{array}{c} \scriptstyle P_{\tilde{X}\tilde{Y}}: \tilde{X} \sim Q \\ \scriptstyle \alpha(\tilde{X}, \tilde{Y})\geq t \end{array}} I(\tilde{X} ; \tilde{Y}) \bigg) \bigg]\end{gathered}$$ Following the proof of Theorem \[theorem:discrete\_nonadaptive\], the RHS of eq.(\[eq:generalized\_pairwise\_discrete\_approx\]) determines the following achievable rate: $$\label{eq:rate_from_pairwise_generalized_approx}
\Remp(\vr x, \vr y) \approx \min_{\begin{array}{c} \scriptstyle \tilde{X} \sim Q,\\ \scriptstyle \alpha(\tilde{X}, \tilde{Y})\geq \hat \alpha(\vr x, \vr y) \end{array}} I(\tilde{X} ; \tilde{Y}) \hspace{2ex} \stackrel{\approx}{\leq} \hat I(\vr x, \vr y)$$ Where the approximate inequality stems from substituting the empirical distribution of $\vr x, \vr y$ as a particular distribution of $\tilde{X}, \tilde{Y}$ meeting the minimization constraints. The above expression is similar to the one obtained in mismatch decoding with random codes. Eq.(\[eq:generalized\_pairwise\_discrete\_approx\]) allows a larger (but still limited) scope of empirical rate functions, but also shows that within this scope the best function is still the empirical mutual information. On the other hand, an advantage of this expression is that under some continuity conditions it can be extended from discrete to continuous vectors (as performed in [@Agarwal_RD]).
When substituting $\alpha$ with the distortion function $\alpha(\tilde{X}, \tilde{Y}) = - E d(\tilde{X}, \tilde{Y})$, we would obtain: $$\begin{gathered}
\label{eq:rate_distortion_from_pairwise_generalized_approx}
\Remp(\vr x, \vr y) \approx \min_{\begin{array}{c} \scriptstyle \tilde{X} \sim Q,\\ \scriptstyle E d(\tilde{X}, \tilde{Y})\geq \hat E d(\vr x, \vr y) \end{array}} I(\tilde{X} ; \tilde{Y}) \hspace{2ex}
=\\= R_X(\hat E d(\vr x, \vr y)) = R_X(\hat D)\end{gathered}$$ where $R_X(D)$ is the rate distortion function of an i.i.d. source $X \sim Q$ with the distortion metric $d$. The later relation can be used to show the result that communication at the rate $R_X(D)$ is possible where $D$ is the empirical or the maximum guaranteed distortion of the channel as shown in [@Agarwal_RD]. On the other hand, when using the correlation function $\alpha(\tilde{X}, \tilde{Y}) = \frac{E (\tilde{X}\tilde{Y})}{E (\tilde{X}^2) E(\tilde{Y}^2)} = \rho$, we would obtain from eq.(\[eq:rate\_from\_pairwise\_generalized\_approx\]) and Lemma \[lemma:gaussian\_mi\_bound\]: $\Remp(\vr x, \vr y) \approx - \frac{1}{2} \log(1 - \hat\rho^2)$. Note that although the later expression is the same as the one obtained in Theorem \[theorem:continuous\_nonadaptive\], the above derivation only proves it for discrete vectors.
Proof of Lemma 2 {#appendix:gaussian_mi_bound}
----------------
For random variables $X$ and $Y$ where $X$ is continuous (not necessarily Gaussian) we have the following bound on the conditional differential entropy ($\tilde{Y}$ denotes a dummy variable with the same distribution as $Y$ and used for notational purposes): $$\begin{gathered}
\label{eq:gaussian_mi_bound_eq1}
h(X|Y) = E_{\tilde{Y}} \left[ h \left(X \big\vert Y=\tilde{Y} \right) \right]
\leq \\ \stackrel{(a)}{\leq}
E \left[ \half \log \left( 2\pi e VAR(X|Y) \right) \right]
\leq \\ \stackrel{(b)}{\leq}
\half \log \left(2\pi e E \left[VAR(X|Y) \right] \right)
=\\= \half \log \left(2\pi e E \left[VAR(X - \alpha \cdot Y|Y)\right] \right)
\leq \\ \stackrel{(c)}{\leq}
\half \log \left(2\pi e E(X - \alpha \cdot Y)^2 \right) =_{\alpha:=\frac{E(XY)}{E(Y^2)}} \\
= \half \log \left(2\pi e \left(E(X^2) - \frac{E(XY)^2}{E(Y^2)} \right) \right)
=\\= \half \log \left(2\pi e E(X^2) (1 - \rho^2)\right)
=\\= \half \log \left(2\pi e E(X^2)\right) + \half \log \left(1 - \rho^2\right)\end{gathered}$$ where the (a) is based on Gaussian bound for entropy and (b) on concavity of the $\log$ function (see also [@Cover] Eq.(17.24)) (c) is based on $VAR(X) = E(X^2)-(E X)^2 \leq E(X^2)$ and is similar to the assertion that $E[VAR(X|Y)]$ which is the MMSE estimation error is not worse than the LMMSE estimation error (except our disregard for the mean).
Therefore for a Gaussian $X$: $$\begin{gathered}
I(X;Y) = h(X) - h(X|Y)
=\\= \half \log(2\pi e E(X^2)) - h(X|Y) \stackrel{(\ref{eq:gaussian_mi_bound_eq1})}{\geq} -\half \log(1 - \rho^2)\end{gathered}$$
*Proof of corollary \[corollary1\_gaussian\_mi\_bound\]*: Equality (a) holds only if $X|Y$ is Gaussian for every value of $Y$, (b) holds if $X$ has fixed variance conditioned on every $Y$, and (c) if $E(X- \alpha \cdot Y|Y)=0 \Longrightarrow E(X|Y)=\alpha \cdot Y$, therefore it results in $X|Y \sim \Normal(\alpha Y, \const)$ which implies $X,Y$ are jointly Gaussian (easy to check by calculating the pdf).
Note that if $X,Y$ are jointly Gaussian then $Y$ can be represented as a result of an additive white Gaussian noise channel (AWGN) with gain operating on $X$: $$Y \sim E(Y|X) + \Normal(0,VAR(Y|X)) = \tilde{\alpha} \cdot X + \Normal(0,\sigma^2) + \const$$
To show corollary \[corollary2\_gaussian\_mi\_bound\] consider $X=Y=Ber(\half)$, in which case $I(X;Y)=1$ and $\rho=1$, therefore the assertion doesn’t hold.
Proof of Lemma 4 {#appendix:pairwise_continuous}
----------------
Write the empirical correlation as $$\hat\rho \equiv \frac{\vr x^T \vr y}{\lVert \vr x \rVert \lVert \vr y \rVert } =
\left( \frac{\vr x}{\lVert \vr x \rVert} \right) ^T \left( \frac{\vr y}{\lVert \vr y \rVert}\right)$$ From the expression above we can infer that $\hat\rho$ does not depend on the amplitude of $\vr x$ and $\vr y$ but only on their direction. Since $\vr x$ is isotropically distributed, the result does not depend on the direction of $\vr
y$ (unless $\vr y = 0$ in which case it is trivially correct), therefore it is independent of $\vr y$ and we can conveniently choose $\vr y = (1,0,0,\ldots,0)$. To put the claim above more formally, for any unitary $n \times n$ matrix $\mt U$ we can write: $$\begin{gathered}
\hat\rho = \frac{\vr x^T \vr y}{\sqrt{(\vr x^T \vr x)(\vr y^T \vr y)}} =
\frac{\vr x^T \mt U^T \mt U \vr y}{\sqrt{(\vr x^T \mt U^T \mt U \vr x)(\vr y^T \mt U^T \mt U \vr y)}}
=\\= \left( \frac{\mt U \vr
x}{\lVert \mt U \vr x \rVert}\right)^T \left(\frac{\mt U \vr y}{\lVert \mt U
\vr y \rVert}\right)\end{gathered}$$ Since $\vr x$ is Gaussian, $\mt U \vr x$ has the same distribution of $\vr x$, thus the probability remains unchanged if we remove $\mt U$ from the left side and remain with $\hat\rho' = \left( \frac{\vr
x}{\lVert \vr x \rVert}\right)^T \left(\frac{\mt U \vr y}{\lVert \mt U
\vr y \rVert}\right) $. For $\vr y \neq 0$, we may choose the unitary matrix $\mt U$ whose first row is $\frac{\vr y}{\lVert \vr y \rVert}$ and the other rows complete it to an orthonormal basis of the linear space $\mathbb{R}^n$. Then $\mt
U \vr y = (\lVert \vr y \rVert,0,0,..0)$ and therefore $\left(\frac{\mt U \vr y}{\lVert \mt U \vr y
\rVert}\right) = (1,0,0,..0)$. Thus the distribution of $\hat\rho' = (1,0,0,\ldots,0) \cdot \left( \frac{\vr
x}{\lVert \vr x \rVert}\right) = \frac{x_1}{\lVert \vr x \rVert}$ equals the distribution of $\hat\rho$. Assuming without loss of generality that $\vr x \sim \Normal^n(0,1)$ we have: $$\begin{gathered}
\Pr (|\hat\rho| \geq t) = \Pr \left( \frac{x_1}{\lVert \vr x \rVert} \geq t \right)
=\\=
\Pr \left( x_1^2 \geq t^2 (\lVert \vr x_2^n \rVert^2 + x_1^2) \right)
= \\ =
\Pr \left( x_1^2 \geq \frac{t^2}{1-t^2} \lVert \vr x_2^n \rVert^2 \right)
=\\=
E \left[ \Pr \left( x_1^2 \geq \frac{t^2}{1-t^2} \lVert \vr x_2^n \rVert^2 \right) \bigg\vert \vr x_2^n \right]
=\\=
E \left[ 2 Q \left( \sqrt {\frac{t^2}{1-t^2} \lVert \vr x_2^n \rVert^2 } \right) \right] \leq
E \left[ 2 e ^{-\half \frac{t^2}{1-t^2} \lVert \vr x_2^n \rVert^2} \right]
=\\=
\int_{\mathbb{R}^{n-1}} \left( 2 e ^{-\half \frac{t^2}{1-t^2} \lVert \vr x_2^n \rVert^2} \right) \left( \frac{1}{(2 \pi)^{(n-1)/2}} e^{- \half \lVert \vr x_2^n \rVert^2} \right) d \vr x_2^n
=\\=
2 \int_{\mathbb{R}^{n-1}} \frac{1}{(2 \pi)^{(n-1)/2}} e ^{-\half \frac{1}{1-t^2} \lVert \vr x_2^n \rVert^2}
\cdot d \vr x_2^n
=\\=
2 (1-t^2)^{\frac{n-1}{2}} \int_{\mathbb{R}^{n-1}} f_{\Normal^{n-1}(0,1-t^2)}(\vr x_2^n) \cdot d \vr x_2^n
=\\=
2 (1-t^2)^{\frac{n-1}{2}} = 2 \exp \left( - (n-1) R_2(t) \right)\end{gathered}$$ where we used the rough upper bound of the Gaussian error function $Q(x) \equiv \Pr(\Normal(0,1)\geq x) \leq e^{-x^2/2}$, and $f_{\Normal^{n}(\mu,\sigma^2)}$ denotes the pdf of a Gaussian i.i.d. vector.
![A geometric interpretation of Lemma \[lemma:pairwise\_continuous\][]{data-label="fig:Geometric_interpretation_of_gaussian_pairwise"}](Geometric_interpretation_of_gaussian_pairwise "fig:"){width="8cm"}\
*Discussion:* A geometrical interpretation of Lemma \[lemma:pairwise\_continuous\] relates this probability to the solid angle of the cone $\{\vr x: |\hat\rho| > t\}$. Since $\vr x$ is isotropically distributed, the probability to have $|\hat\rho| > t$ equals the relative surface determined by vectors having $|\hat\rho| > t$ on the unit $n$-ball (termed the solid angle). Since $\hat\rho$ is the cosine of the angle between $\vr x$ and $\vr y$ the points where $|\hat\rho| > t$ generate a cone with inner angle $2\alpha$ where $\cos(\alpha)=t$ and their intersection with the unit $n$-ball is a spherical cap (dome), shown in figure \[fig:Geometric\_interpretation\_of\_gaussian\_pairwise\]. We can obtain a similar bound as above using geometrical considerations. Write the volume of an $n$ dimensional ball as $V_n r^n$ where $V_n$ is a fixed factor $V_n = \frac{\pi^{n/2}}{\Gamma(1+n/2)}$ [@Ball:Mathworld], and accordingly the surface of an $n$ dimensional ball is (the derivative) $n V_n r^{n-1}$, then the relative surface of the spherical cap can be computed by integrating the surfaces of the $n-1$ dimensional balls with radius $\sin(\theta)$ that have a fixed angle $\theta$ with respect to $\vr y$, and can be bounded as follows: $$\begin{gathered}
\Pr (|\hat\rho| \geq t) = \frac{\textrm{Surface of cap}}{\textrm{Surface of ball}}
=\\=
\frac{1}{n V_n} \cdot \int_{\theta=0}^{\alpha} (n-1) V_{n-1} \sin^{n-2}(\theta) d\theta
\leq \\ \leq
\frac{V_{n-1}}{V_n} \cdot \sin^{n-3}(\alpha) \int_{\theta=0}^{\alpha} \sin(\theta) d\theta
=\\=
\frac{V_{n-1}}{V_n} \cdot \sin^{n-3}(\alpha) (1 - \cos(\alpha))
\leq \\ \stackrel{\alpha\leq\frac{\pi}{2}}{\leq} \frac{V_{n-1}}{V_n} \cdot \sin^{n-3}(\alpha) (1 - \cos^2(\alpha))
= \\ =
O(\sqrt{n}) \cdot \sin^{n-1}(\alpha) = O(\sqrt{n}) \cdot \sqrt{1-\cos^2(\alpha)}^{n-1}
=\\=
O(\sqrt{n}) \cdot (1-t^2)^{(n-1)/2}\end{gathered}$$ where the asymptotic ratio $\frac{V_{n-1}}{\sqrt{n} V_n} \to 1$ is based on [@Gamma:Mathworld] Eq.(99). An interesting observation is that the assumption of Gaussian distribution is not necessary and this bound is true for all isotopical distributions.
Item Referrence Parameter set 1 of figure \[fig:rate\_in\_continuous\_theorem\] Parameter set 2
---------------------- -------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------
Transmission scheme section \[sec:rate\_adaptive\_scheme\] $n=1e+008, K=1e+006, P_A=0.001, P_e=0.001$ $n=1e+020, K=1e+017, P_A=0.001, P_e=0.001$
$\RLBONE$ parameters section \[sec:rate\_analysis\_continuous\], Eq.(\[eq:cont\_rate\_analysis9\]) $T=2.5e+005, \Delta_{\mu} = 37.5412, \Delta = 0.0345958, \eta_1 = 0.996007, \eta_2 = 0.999962, \epsilon_1 =0.01$ $T=7.5e+015, \Delta_{\mu} = 77.4043, \Delta = 3.14616e-007, \eta_1 = 1, \eta_2 = 1, \epsilon_1 =0.001$
$\RLBTWO$ parameters section \[sec:rate\_analysis\_continuous\], Theorem \[theorem:continuous\_adaptive\] $\rho_0=0.9, \epsilon=0.139438, \bar R = 1.05173$ $\rho_0=0.99998, \epsilon=0.0068209, \bar R = 7.29818$
Proof of Lemma 6 {#appendix:likely_convexity_of_rho}
----------------
We denote $\vr x_i, \vr y_i$ as the sub-vectors over $A_i$ (i.e. $\vr x_i \equiv \vr x_{A_i}, \vr y_i \equiv \vr y_{A_i}$), their length by $n_i \equiv |A_i|$ and their relative length by $\lambda_i = n_i/n$. We are interested to find a subset $J$ of $\vr x$ with bounded probability such that outside the set $\sum_i \lambda_i \hat \rho_i^2 \geq \hat \rho^2 - \Delta$ for any $\vr y$. Consider the following inequality: $$\begin{gathered}
\label{eq:rho_convexity_c1}
\lVert \vr x \rVert^2 \cdot \lVert \vr y \rVert^2 \cdot \hat\rho^2 = \left( \vr x^T \vr y \right)^2 =
\left( \sum_i \vr x_i^T \vr y_i \right)^2
=\\=
\left( \sum_i \hat\rho_i \lVert \vr x_i \lVert \cdot \lVert y_i \lVert \right)^2
\stackrel{(a)}{\leq}
\left( \sum_i \hat\rho_i^2 \lVert \vr x_i \lVert^2 \right)\cdot \left( \sum_i \lVert \vr y_i \lVert^2 \right)
=\\=
\left( \sum_i \lambda_i \hat\rho_i^2 + \sum_i \hat\rho_i^2 \left( \frac{\lVert \vr x_i \lVert^2}{\lVert \vr x \lVert^2} - \lambda_i \right) \right) \cdot \lVert \vr x \lVert^2 \cdot \lVert \vr y \rVert^2
\leq \\ \stackrel{(b)}{\leq}
\left( \sum_i \lambda_i \hat\rho_i^2 +
\sum_i \max \left( \frac{\lVert \vr x_i \lVert^2}{\lVert \vr x \lVert^2} - \lambda_i , 0 \right) \right) \cdot \lVert \vr x \rVert^2 \cdot \lVert \vr y \rVert^2\end{gathered}$$ where (a) is from Cauchy-Swartz inequality (b) is since $\hat\rho_i z_i \leq z_i$ for $z_i \geq 0$ and $\hat\rho_i z_i \leq 0$ for $z_i \leq 0$ therefore always $\hat\rho_i z_i \leq \max(z_i,0)$ (attained for $\hat\rho_i = \Ind(z_i > 0)$). Both inequalities are tight in the sense that for each $\vr x$ there is a sequence $\vr y$ (equivalent to choosing $\{\lVert \vr y_i \rVert^2\}$ and $\{\hat\rho_i\}$) that meets them in equality. Dividing by $\lVert \vr x \rVert^2 \cdot \lVert \vr y \rVert^2$ we have that $$\label{eq:rho_convexity_c2}
\hat\rho^2 - \sum_i \lambda_i \hat\rho_i^2 \leq
\sum_i \max \left( \frac{\lVert \vr x_i \lVert^2}{\lVert \vr x \lVert^2} - \lambda_i , 0 \right)$$ where the RHS depends only on $\vr x$ and should be bounded by $\Delta$. Thus the minimal set $J_{\Delta}$ is: $$\label{eq:rho_convexity_c2}
J_{\Delta} \equiv \left\{ \vr x: \sum_i \max \left( \frac{\lVert \vr x_i \lVert^2}{\lVert \vr x \lVert^2} -
\lambda_i , 0 \right) > \Delta \right\}$$ The set is minimal in the sense that none of its elements can be removed while meeting the conditions of the lemma. We would like to bound the probability of $J_{\Delta}$. The result of $\sum_i \max (z_i, 0)$ is a partial sum of $z_i$, and since negative $z_i$ are not summed, it is easy to see this is the maximal partial sum, i.e. we can write this sum alternatively as $$\sum_i \max (z_i, 0) = \max_{I \in \mathcal{P}} \sum_{i \in I} z_i$$ where $\mathcal{P} \equiv 2^{\{1,\ldots,p\}} \setminus \emptyset$ denotes all non empty sub-sets of $\{1,\ldots,p\}$, and its size is $2^p - 1$. Therefore from the union bound we have: $$\begin{gathered}
\label{eq:rho_convexity_c3}
\Pr \{ J_{\Delta} \} = \Pr \left\{ \max_{I \in \mathcal{P}} \sum_{i \in I} \left( \frac{\lVert \vr x_i \lVert^2}{\lVert \vr x \lVert^2} -
\lambda_i \right) > \Delta \right\}
\leq \\ \leq
\sum_{I \in \mathcal{P}} \Pr \left\{ \sum_{i \in I} \left( \frac{\lVert \vr x_i \lVert^2}{\lVert \vr x \lVert^2} -
\lambda_i \right) > \Delta \right\}\end{gathered}$$
To bound the above probability we first develop bound on the probability $\Pr \left( \sum_i a_i \lVert \vr x_i \rVert^2 \leq 0 \right)$ for some coefficients $a_i$:
\[lemma:rho\_convexisy\_sum\_bound\] Let $\vr x \sim \Normal(0,P)^n$. For coefficients $\{a_i\}_{i=1}^p$ with $\sum_i \lambda_i a_i = \bar a > 0$ and $|a_i| \leq A$ where $|\bar a| \leq \frac{1}{8} A$, we have $$\label{eq:rho_convexisy_sum_bound}
\Pr \left( \sum_i a_i \lVert \vr x_i \rVert^2 \leq 0 \right) \leq e^{ - n E}$$ where $$\label{eq:rho_convexisy_sum_bound_E}
E = \frac{\bar a^2}{6 A^2}$$
Now we apply the bound to the events in Eq.(\[eq:rho\_convexity\_c3\]): $$\begin{aligned}
& \displaystyle \sum_{i \in I} \left( \frac{\lVert \vr x_i \lVert^2}{\lVert \vr x \lVert^2} - \lambda_i \right) > \Delta & \\
& \Updownarrow & \\
& \displaystyle \sum_{i \in I} \lVert \vr x_i \lVert^2 - \sum_{i \in I} \lambda_i \sum_{i=1}^p \lVert \vr x_i \lVert^2 > \Delta \sum_{i=1}^p \lVert \vr x_i \lVert^2 \\
& \Updownarrow & \\
& \displaystyle \sum_{i=1}^p \underbrace{ \left( \Delta + \sum_{i \in I} \lambda_i - \Ind(i \in I) \right)}_{\equiv a_i} \lVert \vr x_i \lVert^2 < 0 &\end{aligned}$$ We have: $$\begin{gathered}
\bar a = \sum_{i=1}^p \lambda_i a_i = \Delta \cdot \sum_{i=1}^p \lambda_i + \sum_{i \in I} \lambda_i \cdot \sum_{i=1}^p \lambda_i
-\\- \sum_{i=1}^p \Ind(i \in I)\lambda_i = \Delta\end{gathered}$$ And $|a_i| \leq 1+\Delta \equiv A$, therefore for $\Delta \leq 1/7$ we have $\bar a \leq \frac{1}{8} A$ and by Lemma \[lemma:rho\_convexisy\_sum\_bound\]: $$\Pr \left\{ \sum_{i \in I} \left( \frac{\lVert \vr x_i \lVert^2}{\lVert \vr x \lVert^2} - \lambda_i \right) > \Delta \right\}
\leq e^{ - n E} \leq e^{ - n E_0}$$ where $$E = \frac{\bar a^2}{6 A^2} = \frac{\Delta^2}{6 (1+\Delta)^2} \geq \frac{\Delta^2}{6 (1+1/7)^2} \geq \frac{\Delta^2}{8} \equiv E_0$$ and from Eq.(\[eq:rho\_convexity\_c3\]) we have: $$\begin{gathered}
\label{eq:rho_convexity_c4}
\Pr \{ J_{\Delta} \} \leq
\sum_{I \in \mathcal{P}} \Pr \left\{ \sum_{i \in I} \left( \frac{\lVert \vr x_i \lVert^2}{\lVert \vr x \lVert^2} -
\lambda_i \right) > \Delta \right\}
\leq \\ \leq
|\mathcal{P}| \cdot e^{ - n E_0} \leq 2^p e^{ - n E_0}\end{gathered}$$ which proves the lemma. Note that different bounds can be obtained by applying the bound on $m$ smaller sets in $\{1,\ldots,p\}$ and requiring that the sum over each set will be bounded by $\Delta/m$ (as an example we could bound each $\max(z_i,0)$ separately by $\Delta/p$), however this bound is most suitable for our purpose since when $p << n$ the element $2^p$ becomes negligible.
*Proof of Lemma \[lemma:rho\_convexisy\_sum\_bound\]*: We assume without loss of generality that $\vr x \sim \Normal(0,1)^n$. For Gaussian r.v. $X \sim \Normal(0,1)$ and $a < \half$ we have: $$\begin{gathered}
E(e^{a x^2}) = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} e^{(a-\half)x^2} dx
=\\=
\frac{1}{\sqrt{1-2a}} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi (1-2a)^{-1}}} e^{-\frac{x^2}{2(1-2a)^{-1}}} dx
=\\=
\frac{1}{\sqrt{1-2a}}\end{gathered}$$
For coefficients $\{a_i\}_{i=1}^p$ with $\sum_i \lambda_i a_i = \bar a > 0$ and $|a_i| \leq A$, $w>0$ a positive constant of our choice, and $\vr x \sim \Normal(0,1)^n$ we have: $$\begin{gathered}
\ln \Pr \left( \sum_i a_i \lVert \vr x_i \rVert^2 \leq 0 \right) \leq
\ln E e^{-\half w \cdot \sum_i a_i \lVert \vr x_i \rVert^2} =\\=
\ln E e^{-\half w \cdot \sum_i a_i \sum_{j \in A_i} x_j^2}
=
\ln \prod_i \prod_{j \in A_i} E e^{-\half w \cdot a_i \cdot x_j^2}
=\\=
\sum_i \sum_{j \in A_i} \ln \left( (1 + w \cdot a_i)^{-\half} \right) =
-\half n \sum_i \lambda_i \ln (1+ w \cdot a_i)
= \\ \stackrel{(a)}{=}
-\half n \sum_i \lambda_i \left( (w \cdot a_i) - \half \frac{1}{(1+w\cdot t_i)^2} (w \cdot a_i)^2 \right)
\leq \\ \stackrel{(b)}{\leq}
-\half n \sum_i \lambda_i \left( (w \cdot a_i) - \half \frac{1}{(1-w \cdot A)^2} (w \cdot A)^2 \right)
= \\ =
-\half n \left( \bar a w - \frac{A^2 w^2}{2 (1-w \cdot A)^2} \right)\end{gathered}$$ where (a) is based on the second order Tailor series of $\ln(1+wt)$ around $t=0$ with some $t_i \in [0,a_i] \cup [a_i,0]$ and (b) is since $|t_i| \leq |a_i| \leq A$. For simplicity we choose a sub-optimal $w^* = \frac{\bar a}{A^2}$ (which is obtained by assuming small $a,w$ and optimizing the bound with respect to $w$ ignoring the denominator) and obtain: $$\begin{gathered}
\bar a w^* - \frac{A^2 {w^*}^2}{2 (1-w^* \cdot A)^2}
= \frac{\bar a^2}{A^2} - \frac{\bar a^2 / A^2 }{2(1-\bar a / A)^2}
=\\= \frac{\bar a^2}{A^2} \left( 1 - \frac{A^2}{2(A-\bar a)^2} \right)\end{gathered}$$ To simplify the bound, we make a further assumption that $|\bar a| \leq \frac{1}{8} A$ therefore: $$\begin{gathered}
\frac{\bar a^2}{A^2} \left( 1 - \frac{A^2}{2(A-\bar a)^2} \right)
\geq \frac{\bar a^2}{A^2} \left( 1 - \frac{A^2}{2 \cdot (7/8)^2 \cdot A^2} \right)
=\\=
\frac{\bar a^2}{A^2} \cdot \frac{17}{49}
\geq \frac{\bar a^2}{3A^2}\end{gathered}$$ Therefore we can write the following bound: for $|\bar a| \leq \frac{1}{8} A$ we have $$\Pr \left( \sum_i a_i \lVert \vr x_i \rVert^2 \leq 0 \right) \leq e^{ - n E}$$ where $E = \frac{\bar a^2}{6 A^2}$. Note that the bound is true for any $\vr x \sim \Normal(0,P)^n$.
Parameters of adaptive rate scheme used for figure \[fig:rate\_in\_continuous\_theorem\]
----------------------------------------------------------------------------------------
Table \[table:rate\_in\_continuous\_theorem\_params\] lists two sets of parameters for the continuous alphabet adaptive rate scheme. The first set was used for the curves in figure \[fig:rate\_in\_continuous\_theorem\], and the second set shows the convergence of $\epsilon, \bar R$, for higher values of $n,K$. Note that the values of $n,K$ are extremely high, and this is due to the looseness of the bounds used in the continuous case: specifically the exponent of Lemma \[lemma:likely\_convexity\_of\_rho\] which yields a relatively slow convergence of the ill-convexity probability in equation \[eq:convexity\_prob\_in\_cont\_rate\_analysis\].
O. Shayevitz and M. Feder, “Communicating using Feedback over a Binary Channel with Arbitrary Noise Sequence”, International Symposium on Information Theory (ISIT), Adelaide, Australia, September 2005. Ofer Shayevitz and Meir Feder, “Achieving the Empirical Capacity Using Feedback Part I: Memoryless Additive Models”, Dept. of Electrical Engineering Systems, Tel Aviv University, Tel Aviv 69978, Israel Krishnan Eswaran, Anand D. Sarwate, Anant Sahai, and Michael Gastpar, “Limited feedback achieves the empirical capacity,” Department of Electrical Engineering and Computer Sciences, University of California, arXiv:0711.0237v1 \[cs.IT\] 2 Nov 2007. K. Eswaran and A.D. Sarwate and A. Sahai and M. Gastpar, “Using zero-rate feedback on binary additive channels with individual noise sequences,” Proceedings of the 2007 Infernational Symposium on Information Theory (ISIT 2007), Nice, France, June 2007 V. D. Goppa, “Nonprobabilistic mutual information without memory,” Probl. Contr. Inform. Theory, vol. 4, pp. 97-102, 1975 Lapidoth, A.; Narayan, P., “Reliable communication under channel uncertainty,” Information Theory, IEEE Transactions on , vol.44, no.6, pp.2148-2177, Oct 1998 I. Csiszár and P. Narayan, “The Capacity of the Arbitrarily Varying Channel Revisited : Positivity, Constraints”, IEEE Transactions On Information Theory, Vol. 34, No. 2, March 1988 Mukul Agarwal, Anant Sahai, Sanjoy Mitter, “Coding into a source: a direct inverse Rate-Distortion theorem,” arXiv:cs/0610142v1 \[cs.IT\]. Orginally presented at Allerton 06 Shayevitz, O.; Feder, M., “The posterior matching feedback scheme: Capacity achieving and error analysis,” Information Theory, 2008. ISIT 2008. IEEE International Symposium on , vol., no., pp.900-904, 6-11 July 2008 Csiszár, I., “The method of types \[information theory\],” Information Theory, IEEE Transactions on , vol.44, no.6, pp.2505-2523, Oct 1998 Aslan Tchamkerten and I. Emre Telatar, “Variable Length Coding Over an Unknown Channel”, IEEE Transactions On Information Theory, Vol. 52, No. 5, May 2006 M. V. Burnashev, “Data transmission over a discrete channel with feedback: Random transmission time,” Probl. Inf. Transm., vol. 12, no. 4, pp. 250-265, 1976 Shulman, N.; Feder, M., “The uniform distribution as a universal prior,” Information Theory, IEEE Transactions on , vol.50, no.6, pp. 1356-1362, June 2004 T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley-Interscience, Second edition, 2006. Zamir, R.; Erez, U., “A Gaussian input is not too bad,” Information Theory, IEEE Transactions on , vol.50, no.6, pp. 1362-1367, June 2004 Hassibi, B.; Hochwald, B.M., “How much training is needed in multiple-antenna wireless links?,” Information Theory, IEEE Transactions on , vol.49, no.4, pp. 951-963, April 2003 Lapidoth, A., “Nearest neighbor decoding for additive non-Gaussian noise channels ,” Information Theory, IEEE Transactions on , vol.42, no.5, pp.1520-1529, Sep 1996 Hughes, B.; Narayan, P., “Gaussian arbitrarily varying channels,” Information Theory, IEEE Transactions on , vol.33, no.2, pp. 267-284, Mar 1987 Nadav Shulman, “Communication over an Unknown Channel via Common Broadcasting,” Ph.D. dissertation, Tel Aviv University, 2003 Shayevitz, Ofer; Feder, Meir, “Communication with Feedback via Posterior Matching,” Information Theory, 2007. ISIT 2007. IEEE International Symposium on , vol., no., pp.391-395, 24-29 June 2007 M. Horstein, “Sequential transmission using noiseless feedback,” IEEE Trans. Info. Theory, pp. 136–143, July 1963. J. P. M. Schalkwijk, “A coding scheme for additive noise channels with feedback part II: Band-limited siganls,” IEEE Trans. Info. Theory, vol. IT-12, pp. 183 – 189, 1966. D. Blackwell, L. Breiman, and A. J. Thomasian, “The capacities of certain channel classes under random coding,” Ann. Math. Statist., vol. 31, pp. 558-567, 1960 Lapidoth, A.; Telatar, I.E., “The compound channel capacity of a class of finite-state channels ,” Information Theory, IEEE Transactions on , vol.44, no.3, pp.973-983, May 1998 Barron, A.; Rissanen, J.; Bin Yu, “The minimum description length principle in coding and modeling,” Information Theory, IEEE Transactions on , vol.44, no.6, pp.2743-2760, Oct 1998 Csiszar, I.; Narayan, P., “Channel capacity for a given decoding metric,” Information Theory, IEEE Transactions on , vol.41, no.1, pp.35-43, Jan 1995 Weisstein, Eric W. “Ball.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Ball.html Weisstein, Eric W. “Gamma Function.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaFunction.html
|
---
abstract: 'Lagrangian systems represent a wide range of robotic systems, including manipulators, wheeled and legged robots, and quadrotors. Inverse dynamics control and feedforward linearization techniques are typically used to convert the complex nonlinear dynamics of Lagrangian systems to a set of decoupled double integrators, and then a standard, outer-loop controller can be used to calculate the commanded acceleration for the linearized system. However, these methods typically depend on having a very accurate system model, which is often not available in practice. While this challenge has been addressed in the literature using different learning approaches, most of these approaches do not provide safety guarantees in terms of stability of the learning-based control system. In this paper, we provide a novel, learning-based control approach based on Gaussian processes (GPs) that ensures both stability of the closed-loop system and high-accuracy tracking. We use GPs to approximate the error between the commanded acceleration and the actual acceleration of the system, and then use the predicted mean and variance of the GP to calculate an upper bound on the uncertainty of the linearized model. This uncertainty bound is then used in a robust, outer-loop controller to ensure stability of the overall system. Moreover, we show that the tracking error converges to a ball with a radius that can be made arbitrarily small. Furthermore, we verify the effectiveness of our approach via simulations on a 2 degree-of-freedom (DOF) planar manipulator and experimentally on a 6 DOF industrial manipulator.'
author:
- 'Mohamed K. Helwa, Adam Heins, and Angela P. Schoellig'
title: '**Provably Robust Learning-Based Approach for High-Accuracy Tracking Control of Lagrangian Systems [^1]**'
---
Introduction {#sec:introd}
============
High-accuracy tracking is an essential requirement in advanced manufacturing, self-driving cars, medical robots, and autonomous flying vehicles, among others. To achieve high-accuracy tracking for these complex, typically high-dimensional, nonlinear robotic systems, a standard approach is to use inverse dynamics control [@spong] or feedforward linearization techniques [@fflin] to convert the complex nonlinear dynamics into a set of decoupled double integrators. Then, a standard, linear, outer-loop controller, e.g., a proportional-derivative (PD) controller, can be used to make the decoupled linear system track the desired trajectory [@spong]. However, these linearization techniques depend on having accurate system models, which are difficult to obtain in practice.
To address this problem, robust control techniques have been used for many decades to design the outer-loop controllers to account for the uncertainties in the model [@robust_con]. However, the selection of the uncertainty bounds in the robust controller design is challenging. On the one hand, selecting high bounds typically results in a conservative behavior, and hence, a large tracking error. On the other hand, relatively small uncertainty bounds may not represent the true upper bounds of the uncertainties, and consequently, stability of the overall system is not ensured. Alternatively, several approaches have been proposed for learning the inverse system dynamics from collected data where the system models are not available or not sufficiently accurate; see [@thesis; @peters1; @invlearn1; @invlearn2]. Combining a-priori model knowledge with learning data has also been studied in [@thesis; @invlearn_model1]. However, these learning approaches typically neglect the learning regression errors in the analysis, and they do not provide a proof of stability of the overall, learning-based control system, which is crucial for safety-critical applications such as medical robots. The limitations of the robust control and the learning-based techniques show the urgent need for novel, robust, learning-based control approaches that ensure both stability of the overall control system and high-accuracy tracking. This sets the stage for the research carried out in this paper.
In this paper, we provide a novel, robust, learning-based control technique that achieves both closed-loop stability and high-accuracy tracking. In particular, we use Gaussian processes (GPs) to approximate the error between the commanded acceleration to the linearized system and the actual acceleration of the robotic system, and then use the predicted mean and variance of the GP to calculate an upper bound on the uncertainty of the linearization. This uncertainty bound is then used in a robust, outer-loop controller to ensure stability of the overall system (see Figure \[fig:structure\]). Moreover, we show that using our proposed strategy, the tracking error converges to a ball with a radius that can be made arbitrarily small through appropriate control design, and hence, our proposed approach also achieves high-accuracy tracking. Furthermore, we verify the effectiveness of the proposed approach via simulations on a 2 DOF planar manipulator using MATLAB Simulink and experimentally on a UR10 6 DOF industrial manipulator. This paper is organized as follows. Section \[sec:related\] provides a summary of some recent related work. Section \[sec:problem\] describes the considered problem, and Section \[sec:alg\] provides the proposed approach. Section \[sec:theory\] derives theoretical guarantees for the proposed approach. Section \[sec:sim\] and \[sec:exp\] include the simulation and experimental results, and Section \[sec:con\] concludes the paper. *Notation and Basic Definitions:* For a set $S$, $\bar{S}$ denotes its closure and $S^{\circ}$ its interior. The notation $B_{\delta}(y)$ denotes a ball of radius $\delta$ centered at a point $y$. A matrix $P$ is *positive definite* if it is symmetric and all its eigenvalues are positive. For a vector $x$, $\|x\|$ denotes its Euclidean norm. A function $f(x)$ is *smooth* if its partial derivatives of all orders exist and are continuous. The solutions of $\dot{x}=f(t,x)$ are *uniformly ultimately bounded with ultimate bound $b$* if there exist positive constants $b,~c$, and for every $0<a<c$, there exists $T(a,b)\geq 0$ such that $\|x(t_0)\|\leq a$ implies $\|x(t)\|\leq b$, for all $t\geq (T+t_0)$, where $t_0$ is the initial time instant. A *kernel* is a symmetric function $k:{{\mathcal A}}\times {{\mathcal A}}\rightarrow {{\mathbb R}}$. A *reproducing kernel Hilbert space (RKHS)* corresponding to a kernel $k(.,.)$ includes functions of the form $f(a)=\sum_{j=1}^{m}\alpha_jk(a,a_j)$ with $m\in {{\mathbb N}}$, $\alpha_j\in {{\mathbb R}}$ and representing points $a_j\in {{\mathcal A}}$.
Related Work {#sec:related}
============
The study of safe learning dates back to the beginning of this century [@general_safe_learning]. In [@LyapRL1] and [@LyapRL2], Lyapunov-based reinforcement learning is used to allow a learning agent to safely switch between pre-computed baseline controllers. Then, in [@risk_rl], risk-sensitive reinforcement learning is proposed, in which the expected return is heuristically weighted with the probability of reaching an error state. In several other papers, including [@safe_exp1], [@safe_exp2] and [@GP_felix], safe exploration methods are utilized to allow the learning modules to achieve a desired balance between ensuring safe operation and exploring new states for improved performance. In [@general_safe_learning], a general framework is proposed for ensuring safety of learning-based control strategies for uncertain robotic systems. In this framework, robust reachability guarantees from control theory are combined with Bayesian analysis based on empirical observations. The result is a safety-preserving, supervisory controller of the learning module that allows the system to freely execute its learning policy almost everywhere, but imposes control actions to ensure safety at critical states. Despite its effectiveness for ensuring safety, the supervisory controller in this approach has no role in reducing tracking errors. Focusing our attention on safe, learning-based inverse dynamics control, we refer to [@adaptive1; @Schaal; @Kulic]. In [@adaptive1], a model reference adaptive control (MRAC) architecture based on Gaussian processes (GPs) is proposed, and stability of the overall control system is proved. While the approach in [@adaptive1] is based on adaptive control theory, our approach is based on robust control theory. In particular, in [@adaptive1], the mean of the GP is used to exactly cancel the uncertainty vector, while in our approach, we use both the mean and variance of the GP to learn an upper bound on the uncertainty vector to be used in a robust, outer-loop controller. Hence, unlike [@adaptive1], in our approach, the uncertainty of the learning module is not only incorporated in the stability analysis but also in the outer-loop controller design. Intuitively, the less certain our GPs are, the more robust the outer-loop controller should be for ensuring safety. When more data is collected and the GPs are more certain, the outer-loop controller can be less conservative for improved performance. While the results of [@adaptive1] are tested in simulations on a two-dimensional system, we test our results experimentally on a 6 DOF manipulator.
In [@Schaal; @Kulic], GPs are utilized to learn the errors in the output torques of the inverse dynamics model online. In [@Schaal], the GP learning is combined with a state-of-the-art gradient descent method for learning feedback terms online. The main idea behind this approach is that the gradient descent method would correct for fast perturbations, while the GP is responsible for correcting slow perturbations. This allows for exponential smoothing of the GP hyperparameters, which increases the robustness of the GP at the cost of having slower reactiveness. Nevertheless, [@Schaal] does not provide a proof of the robust stability of the closed-loop system. In [@Kulic], the variance of the GP prediction is utilized to adapt the parameters of an outer-loop PD controller online, and the uniform ultimate boundedness of the tracking error is proved under some assumptions on the structure of the PD controller (e.g., the gain matrix was assumed to be diagonal, which imposes a decentralized gain control scheme). The results of [@Kulic] are verified via simulations on a 2 DOF manipulator.
Our approach differs from [@Kulic] in several aspects. First, we do not use an adaptive PD controller in the outer loop, but add a robustness term to the output of the outer-loop controller. Second, while [@Kulic] uses the GP to learn the error in the estimated torque from the nominal inverse dynamics, in our approach, we learn the error between the commanded and actual accelerations. This can be beneficial in two aspects: *(i)* This makes our approach applicable to industrial manipulators that have onboard controllers for calculating the torque and allow the user to only send commanded acceleration/velocity; *(ii)* this makes our approach also applicable beyond inverse dynamics control of manipulators; indeed, our proposed approach can be applied to any Lagrangian system for which feedforward/feedback linearization can be used to convert the nonlinear dynamics of the system to a set of decoupled double integrators, such as a quadrotor under a feedforward linearization, see Section 5.3 of [@HS17]. Third, while [@Kulic] shows uniform ultimate boundedness of the tracking error, it does not provide discussions on the size of the ultimate ball. In this work, we show that using our proposed approach, the size of the ball can be made arbitrarily small through the control design. Fourth, in our approach, we do not impose any assumption on the structure of the outer-loop PD controller and decentralized, outer-loop control is not needed for our proof. Finally, we verify our approach experimentally on a 6 DOF manipulator.
Problem Statement {#sec:problem}
=================
In this paper, we consider Lagrangian systems, which represent a wide class of mechanical systems [@Murray]. In what follows, we focus our attention on a class of Lagrangian systems represented by: $$\label{eq:sys_man}
M(q(t))\ddot{q}(t)+C(q(t),\dot{q}(t))\dot{q}(t)+g(q(t))=u(t),$$ where $q=(q_1,\cdots,q_N)$ is the vector of generalized coordinates (displacements or angles), $\dot{q}=(\dot{q}_1,\cdots,\dot{q}_N)$ is the vector of generalized velocities, $u=(u_1,\cdots,u_N)$ is the vector of generalized forces (forces or torques), $N$ is the system’s degree of freedom, $M$, $C$, and $g$ are matrices of proper dimensions and smooth functions, and $M(q)$ is a positive definite matrix. Fully-actuated robotic manipulators are an example of Lagrangian systems that can be expressed by . Despite focusing our discussion on systems represented by , we emphasize that our results in this paper can be easily generalized to a wider class of nonlinear Lagrangian systems for which feedback/feedforward linearization can be utilized to convert the dynamics of the system into a set of decoupled double integrators plus an uncertainty vector. For the nonlinear system with uncertain matrices $M$, $C$, and $g$, we aim to make the system positions and velocities $(q(t),\dot{q}(t))$ track a desired smooth trajectory $(q_d(t),\dot{q}_d(t))$. For simplicity of notation, in our discussion, we drop the dependency on time t from $q,~q_d$, their derivatives, and $u$. Our goal is to design a novel, learning-based control strategy that is easy to interpret and implement, and that satisfies the following desired objectives:
- *Robustness:* The overall, closed-loop control system satisfies robust stability in the sense that the tracking error has an upper bound under the system uncertainties.
- *High-Accuracy Tracking:* For feasible desired trajectories, the tracking error converges to a ball around the origin that can be made arbitrarily small through the control design. For the ideal case, where the pre-assumed system parameters are correct, the tracking error should converge exponentially to the origin.
- *Adaptability:* The proposed strategy should incorporate online learning to continuously adapt to online changes of the system parameters and disturbances.
- *Generalizability of the Approach:* The proposed approach should be general enough to be also applicable to industrial robots that have onboard controllers for calculating the forces/torques and allow the user to send only commanded acceleration/velocity.
Methodology {#sec:alg}
===========
We present our proposed methodology, and then in the next sections, we show that it satisfies objectives (O1)-(O4). A standard approach for solving the tracking control problem for is inverse dynamics control. Since $M(q)$ is positive definite by assumption, it is invertible. Hence, it is evident that if the matrices $M$, $C$, and $g$ are all known, then the following inverse dynamics control law $$\label{eq:inv_ideal}
u=C(q,\dot{q})\dot{q}+g(q)+M(q)a_q$$ converts the complex nonlinear dynamic system into $$\label{eq:lin_sys_ideal}
\ddot{q}=a_q,$$ where $a_q$ is the commanded acceleration, a new input to the linearized system to be calculated by an outer-loop control law, e.g., a PD controller (see Figure \[fig:structure\]). However, the standard inverse dynamics control heavily depends on accurate knowledge of the system parameters. In practice, the matrices $M$, $C$, and $g$ are not perfectly known, and consequently, one has to use estimated values of these matrices $\hat{M}$, $\hat{C}$, and $\hat{g}$, respectively, where $\hat{M}$, $\hat{C}$, and $\hat{g}$ are composed of smooth functions. Hence, in practice, the control law should be replaced with $$\label{eq:inv_prac}
u=\hat{C}(q,\dot{q})\dot{q}+\hat{g}(q)+\hat{M}(q)a_q.$$ Now by plugging into the system model , we get $$\label{eq:lin_sys_prac}
\ddot{q}=a_q+\eta(q,\dot{q},a_q),$$ where $\eta(q,\dot{q},a_q)=M^{-1}(q)(\tilde{M}(q)a_q+\tilde{C}(q,\dot{q})\dot{q}+\tilde{g}(q))$, with $\tilde{M}=\hat{M}-M$, $\tilde{C}=\hat{C}-C$, and $\tilde{g}=\hat{g}-g$. It can be shown that even if the left hand side (LHS) of has a smooth, unstructured, added uncertainty $E(q,\dot{q})$, e.g., unmodeled friction, is still valid with modified $\eta$. Because of $\eta$, the dynamics resulting from the inverse dynamics control are still nonlinear and coupled. To control the uncertain system , on the one hand, robust control methods are typically very conservative, while on the other hand, learning methods do not provide stability guarantees.
Hence, in this paper, we combine ideas from robust control theory with ideas from machine learning, particularly Gaussian processes (GPs) for regression, to provide a robust, learning-based control strategy that satisfies objectives (O1)-(O4). The main idea behind our proposed approach is to use GPs to learn the uncertainty vector $\eta(q,\dot{q},a_q)$ in online. Following [@Kulic], we use a set of $N$ independent GPs, one for learning each element of $\eta$, to reduce the complexity of the regression. It is evident that conditioned on knowing $q,~\dot{q}$, and $a_q$, one can learn each element of $\eta$ independently from the rest of the elements of $\eta$. A main advantage of GP regression is that it does not only provide an estimated value of the mean $\mu$, but it also provides an expected variance $\sigma^2$, which represents the accuracy of the regression model based on the distance to the training data. The punchline here is that one can use both the mean and variance of the GP to calculate an upper bound $\rho$ on $\|\eta\|$ that is guaranteed to be correct with high probability, as we will show later in this section. One can then use this upper bound to design a robust, outer-loop controller that ensures robust stability of the overall system. Hence, our proposed strategy consists of three parts: [***(i)* *Inner-Loop Controller:***]{} We use the inverse dynamics control law , where $\hat{M}$, $\hat{C}$, and $\hat{g}$ are estimated values of the system matrices from an a-priori model.
[***(ii)* *GPs for Learning the Uncertainty:***]{} We use a set of $N$ GPs to learn the uncertainty vector $\eta$ in . We start by reviewing GP regression [@GP_conf; @GP_felix]. A GP is a nonparametric regression model that is used to approximate a nonlinear function $J(a):{{\mathcal A}}\rightarrow {{\mathbb R}}$, where $a\in {{\mathcal A}}$ is the input vector. The ability of the GP to approximate the function is based on the assumptions that function values $J(a)$ associated with different values of $a$ are random variables, and that any finite number of these variables have a joint Gaussian distribution. The GP predicts the value of the function, $J(a^*)$, at an arbitrary input $a^*\in {{\mathcal A}}$ from a set of $n$ observations $D_{n}:=\{a_j,\hat{J}(a_j)\}_{j=1}^{n}$, where $\hat{J}(a_j)$, $j\in\{1,\cdots,n\}$, are assumed to be noisy measurements of the function’s true values. That is, $\hat{J}(a_j)=J(a_j)+\omega'$, where $\omega'$ is a zero mean Gaussian noise with variance $\sigma_\omega^2$. Assuming, without loss of generality (w.l.o.g.), a zero prior mean of the GP and conditioned on the previous observations, the mean and variance of the GP prediction are given by: $$\label{eq:GP_mean}
\mu_n(a^*)={\bf k}_n(a^*)({\bf K}_n+{\bf I}_n\sigma_\omega^2)^{-1}{\bf \hat{J}}_n,$$ $$\label{eq:GP_variance}
\sigma_n^2(a^*)=k(a^*,a^*)-{\bf k}_n(a^*)({\bf K}_n+{\bf I}_n\sigma_\omega^2)^{-1}{\bf k}_n^T(a^*),$$ respectively, where ${\bf \hat{J}}_n=[\hat{J}(a_1),\cdots,\hat{J}(a_n)]^T$ is the vector of observed, noisy function values. The matrix ${\bf K}_n\in {{\mathbb R}}^{n\times n}$ is the covariance matrix with entries $[{\bf K}_n]_{(i,j)}=k(a_i,a_j)$, $i,~j\in\{1,\cdots,n\}$, where $k(a_i,a_j)$ is the covariance function defining the covariance between two function values $J(a_i),~J(a_j)$ (also called the kernel). The vector ${\bf k}_n(a^*)=[k(a^*,a_1),\cdots,k(a^*,a_n)]$ contains the covariances between the new input and the observed data points, and ${\bf I}_n\in {{\mathbb R}}^{n\times n}$ is the identity matrix. The tuning of the GP is typically done through the selection of the kernel function and the tuning of its hyperparameters. For information about different standard kernel functions, please refer to [@GP_conf]. We next discuss our implementation of the GPs. The GPs run in discrete time with sampling interval $T_s$. At a sampling instant $k$, the inputs to each GP regression model are the same $(q(k),\dot{q}(k),{a_q}(k))$, and the output is an estimated value of an element of the $\eta$ vector at $k$. For the training data for each GP, $n$ observations of $(q,\dot{q},a_q)$ are used as the labeled input together with $n$ observations of an element of the vector $\ddot{q}-a_q+{\bf \omega}_v$ as the labeled output, where ${\bf \omega}_v\in {{\mathbb R}}^N$ is Gaussian noise with zero mean and variance $diag(\sigma_{\omega_1}^2,\cdots,\sigma_{\omega_N}^2)$; see . For selecting the $n$ observations, we use the oldest point (OP) scheme for simplicity; this scheme depends on removing the oldest observation to accommodate for a new one [@adaptive1]. We use the squared exponential kernel $$\label{eq:GP_kernel1}
k(a_i,a_j)=\sigma_{\eta}^2e^{-\frac{1}{2}(a_i-a_j)^T\bar{M}^{-2}(a_i-a_j)},$$ which is parameterized by the hyperparameters: $\sigma_\eta^2$, the prior variance, and the positive length scales $l_1,\cdots,l_{3N}$ which are the diagonal elements of the diagonal matrix $\bar{M}$. Hence, the expected mean and variance of each GP can be obtained through equations -. Guidelines for tuning the GP hyperparameters $\sigma_\omega^2,\sigma_\eta^2,l_1,\cdots,l_{3N}$ can be found in [@GP_felix].
As stated before, a main advantage of GP regression is that the GP provides a variance, which represents the accuracy of the regression model based on the distance between the new input and the training data. One can then use the predicted mean and variance of the GP to provide a confidence interval around the mean that is guaranteed to be correct with high probability. There are several comprehensive studies in the machine learning literature on calculating these confidence intervals. For completeness, we review one of these results, particularly Theorem 6 of [@GP_conf]. Let $a_{aug}=(q,\dot{q},a_q)$, and $\eta_i(a_{aug})$ denote the $i$-th element of the unknown vector $\eta$.
\[assum0\] The function $\eta_i(a_{aug})$, $i\in\{1,\cdots,N\}$, has a bounded RKHS norm $\|\eta_i\|_{k}$ with respect to the covariance function $k(a,a')$ of the GP, and the noise $\omega_i$ added to the output observations, $i\in\{1,\cdots,N\}$, is uniformly bounded by $\bar{\sigma}$.
The RKHS norm is a measure of the function smoothness, and its boundedness implies that the function is well-behaved in the sense that it is regular with respect to the kernel [@GP_conf]. Intuitively, Assumption \[assum0\] does not hold if the uncertainty $\eta$ is discontinuous, e.g., discontinuous friction.
\[lemma1\] Suppose that Assumption \[assum0\] holds. Let $\delta_{p}\in (0,1)$. Then, $\Pr\{\forall a_{aug} \in {{\mathcal A}},\|\mu(a_{aug})-\eta_i(a_{aug})\|\leq \beta^{1/2}\sigma(a_{aug})\}\geq 1-\delta_p$, where $\Pr$ stands for the probability, ${{\mathcal A}}\subset {{\mathbb R}}^{3N}$ is compact, $\mu(a_{aug}),~\sigma^2(a_{aug})$ are the GP mean and variance evaluated at $a_{aug}$ conditioned on $n$ past observations, and $\beta=2\|\eta_i\|_{k}^2+300\gamma \ln^3((n+1)/{\delta}_p)$. The variable $\gamma\in {{\mathbb R}}$ is the maximum information gain and is given by $\gamma=\max_{\{a_{aug,1},\cdots,a_{aug,n+1}\}\in {{\mathcal A}}}0.5~\log(\det(I+\bar{\sigma}^{-2}{\bf K}_{n+1}))$, where $\det$ is the matrix determinant, $I\in {{\mathbb R}}^{(n+1)\times (n+1)}$ is the identity matrix, ${\bf K}_{n+1}\in {{\mathbb R}}^{(n+1)\times (n+1)}$ is the covariance matrix given by $[{\bf K}_{n+1}]_{(i,j)}=k(a_{aug,i},a_{aug,j})$, $i,~j\in\{1,\cdots,n+1\}$.
Finding the information gain maximizer can be approximated by an efficient greedy algorithm [@GP_conf]. Indeed, $\gamma$ has a sublinear dependence on $n$ for many commonly used kernels, and can be numerically approximated by a constant [@Kulic]. The punchline here is that we know from Lemma \[lemma1\] that one can define for each GP a confidence interval around the mean that is guaranteed to be correct for all points $a_{aug}\in {{\mathcal A}}$, a compact set, with probability higher than $(1-\delta_p)$, where $\delta_p$ is typically picked very small. Let $\mu_{k,i}$ and $\sigma_{k,i}^2$ represent the expected mean and variance of the $i$-th GP at the sampling instant $k$, respectively, and let $\beta_i$ denote the $\beta$ parameter in Lemma \[lemma1\] of the $i$-th GP, where $i\in\{1,\cdots,N\}$. We select the upper bound on the absolute value of $\eta_i$ at $k$ to be $$\label{eq_rho_i}
\rho_{k,i}(\mu_{k,i},\sigma_{k,i})=\max(|\mu_{k,i}-\beta_i^{1/2}\sigma_{k,i}|,|\mu_{k,i}+\beta_i^{1/2}\sigma_{k,i}|).$$ Then, a good estimate of the upper bound on $\|\eta\|$ at $k$ is $$\label{eq_rho}
\rho_k=\sqrt{\rho_{k,1}(\mu_{k,1},\sigma_{k,1})^2+\cdots+\rho_{k,N}(\mu_{k,N},\sigma_{k,N})^2}.$$
[***(iii)* *Robust, Outer-Loop Controller:***]{} We use the estimated upper bound $\rho_k$ to design a robust, outer-loop controller. In particular, for a smooth, bounded desired trajectory $q_d(t)$, we use the outer-loop control law $$\label{eq:outer_controller}
a_q(t)=\ddot{q}_d(t)+K_P(q_d(t)-q(t))+K_D(\dot{q}_d(t)-\dot{q}(t))+r(t),$$ where $K_P\in {{\mathbb R}}^{N\times N}$ and $K_D\in {{\mathbb R}}^{N\times N}$ are the proportional and derivative matrices of the PD control law, respectively, and $r\in {{\mathbb R}}^N$ is an added vector to the PD control law that will be designed to achieve robustness. Let $e(t):=(q(t)-q_d(t),\dot{q}(t)-\dot{q}_d(t))$ denote the tracking error vector. From and , it can be shown that the tracking error dynamics are $$\label{eq:tracking_error}
\dot{e}(t)=Ae(t)+B(r(t)+\eta(q(t),\dot{q}(t),a_q(t))),$$ where $$\label{eq:AandB}
A=\left[
\begin{array}{cc}
0 & I \\
-K_P & -K_D
\end{array}
\right]\in {{\mathbb R}}^{2N\times2N},~~
B=\left[
\begin{array}{rr}
0 \\ I \end{array}
\right]\in {{\mathbb R}}^{2N\times N},$$ and $I\in {{\mathbb R}}^{N\times N}$ is the identity matrix. From and , it is clear that the controller matrices $K_P$ and $K_D$ should be designed to make $A$ a Hurwitz matrix.
We now discuss how to design the robustness vector $r(t)$. To that end, let $P\in {{\mathbb R}}^{2N\times 2N}$ be the unique positive definite matrix satisfying $A^TP+PA=-Q$, where $Q\in {{\mathbb R}}^{2N\times 2N}$ is a positive definite matrix. We define $r(t)$ as follows $$\label{eq:robust_term}
r(t)=
\begin{cases}
-\rho(t)\frac{B^TPe(t)}{\|B^TPe(t)\|} \text{~~~~$\|B^TPe(t)\|>\epsilon$,}
\\
-\rho(t)\frac{B^TPe(t)}{\epsilon} \text{~~~~~~$\|B^TPe(t)\|\leq \epsilon$,}
\end{cases}$$ where $\rho(t)\in {{\mathbb R}}$ is the last received upper bound on $\|\eta\|$ from the GPs, i.e., we use $$\label{rho_zoh}
\rho(t)=\rho_k, \forall t\in [kT_s,(k+1)T_s),$$ and $\epsilon$ is a small positive number. It should be noted that $\epsilon$ is a design parameter that can be selected to ensure high-accuracy tracking, as we will discuss in the next section.
Theoretical Guarantees {#sec:theory}
======================
After discussing the proposed strategy, we now justify that it satisfies both robust stability and high-accuracy tracking. To that end, we require the following reasonable assumption:
\[assum2\] The GPs run at a sufficiently fast sampling rate such that the calculated upper bound on $\|\eta\|$ is accurate between two consecutive sampling instants.
We impose another assumption to ensure that the added robustness vector $r(t)$ will not cause the uncertainty vector norm $\|\eta(q(t),\dot{q}(t),a_q(t))\|$ to blow up. It is easy to show that the uncertainty function $\eta(q,\dot{q},a_q)$ is smooth, and so $\|\eta\|$ attains a maximum value on any compact set in its input space $(q,\dot{q},a_q)$. However, since from and , $a_q$ is a function of $\rho(t)$, an upper bound on $\|\eta\|$, one still needs to ensure the boundedness of $\|\eta\|$ for bounded $q,~\dot{q}$ or bounded tracking error $e$. Hence, we present the following assumption.
\[assum3\] For a given, smooth, bounded desired trajectory $(q_d(t),\dot{q}_d(t))$, there exists $\bar{\rho}>0$ such that $\|\eta\|\leq \bar{\rho}$ for each $e\in D$, where $D$ is a compact set containing $\{e\in {{\mathbb R}}^{2N}:e^TPe\leq e(0)^TPe(0)\}$, and $e(0)$ is the initial tracking error.
We now justify that Assumption \[assum3\] is reasonable. In particular, we show that the assumption is satisfied for small uncertainties in the inertia matrix $M(q)$ [@spong]. In this discussion, we suppose that $\rho(t)$ in satisfies $\rho(t)\leq \|\eta(t)\|+c$, where $c$ is a positive scalar, and study whether imposing $r(t)$ into can make $\|\eta(t)\|$ blow up. Recall that $\eta(q,\dot{q},a_q)=M^{-1}(q)(\tilde{M}(q)a_q+\tilde{C}(q,\dot{q})\dot{q}+\tilde{g}(q))$. From , we have $\eta(q,\dot{q},a_q)=M^{-1}(q)(\tilde{M}(q)r+\tilde{M}(q)(\ddot{q}_d+K_P(q_d-q)+K_D(\dot{q}_d-\dot{q}))+\tilde{C}(q,\dot{q})\dot{q}+\tilde{g}(q))$. It is evident that $\|\eta\|\leq \|M^{-1}(q)\tilde{M}(q)\|\|r\|+\|T_b(q,\dot{q},q_d,\dot{q}_d,\ddot{q}_d)\|$, where $T_b=M^{-1}(q)(\tilde{M}(q)(\ddot{q}_d+K_P(q_d-q)+K_D(\dot{q}_d-\dot{q}))+\tilde{C}(q,\dot{q})\dot{q}+\tilde{g}(q))$. From , it is easy to verify $\|r(t)\|\leq \|\rho(t)\|=\rho(t)$, and so $\|r(t)\|\leq \|\eta(t)\|+c$. Hence, $\|\eta\|\leq \|M^{-1}(q)\tilde{M}(q)\|\|\eta\|+ \|M^{-1}(q)\tilde{M}(q)\|c+\|T_b\|$. Now if the uncertainty in the matrix $M(q)$, $\tilde{M}(q)$, is sufficiently small such that $\|M^{-1}(q)\tilde{M}(q)\|<1$ is satisfied, then $\|\eta\| \leq \frac{1}{1-\|M^{-1}(q)\tilde{M}(q)\|}(\|M^{-1}(q)\tilde{M}(q)\|c+\|T_b(q,\dot{q},q_d,\dot{q}_d,\ddot{q}_d\|)$. Since $q_d,~\dot{q}_d,~\ddot{q}_d$ are all bounded by assumption, if $e(t)\in D$, a compact set, then $q(t),~\dot{q}(t)$ are also bounded. It is easy to show that there exists a fixed upper bound $\bar{\rho}$ on $\|\eta\|$ that is valid for each $e\in D$, and Assumption \[assum3\] is satisfied.
We have shown that if the uncertainty in the matrix $M(q)$, $\tilde{M}(q)$, is sufficiently small such that $\|M^{-1}(q)\tilde{M}(q)\|<1$ is satisfied, then Assumption \[assum3\] holds. This argument is true even if we have large uncertainties in the other system matrices, $C(q,\dot{q})$ and $g(q)$. As indicated in Chapter 8 of [@spong], if the bounds on $\|M\|$ are known ($\underline{m} \leq \|M\| \leq \overline{m}$), then one can always select $\hat{M}$ such that $\|M^{-1}(q)\tilde{M}(q)\|<1$ is satisfied. In particular, by selecting $\hat{M}=\frac{\overline{m}+\underline{m}}{2}I$, where $I$ is the identity matrix, it can be shown that $\|M^{-1}(q)\tilde{M}(q)\|\leq \frac{\overline{m}-\underline{m}}{\overline{m}+\underline{m}}<1$. Consequently, it is not difficult to satisfy the condition $\|M^{-1}(q)\tilde{M}(q)\|<1$ in practice, and Assumption \[assum3\] is not restrictive.
From Assumption \[assum3\], we know that $\|\eta(t)\|\leq \bar{\rho}$ if $e(t) \in D$, and consequently, it is reasonable to saturate any estimate of $\rho(t)$ beyond $\bar{\rho}$. Hence, we suppose that the estimation of $\rho$ is slightly modified to be $$\label{eq:rho_new}
\rho(t)=\min(\rho_{GP}(t),\bar{\rho}),$$ where $\rho_{GP}(t)$ is the upper bound on the uncertainty norm, $\|\eta\|$, calculated from the GPs in , , and . It is straightforward to show that with the choice of $\rho(t)$ in and for bounded smooth trajectories, the condition $e(t)\in D$ for all $t\geq 0$ implies that $a_q(t)$ in is always bounded, and so $a_{aug}=(q,\dot{q},a_q)$ always lies in a compact set. To be able to provide theoretical guarantees, we also assume w.l.o.g. that the small positive number $\epsilon$ in is selected sufficiently small such that $$\label{eq:epsilon}
\sqrt{\frac{\epsilon \bar{\rho}}{2\lambda_{min}(Q)}}\ll \delta_1,$$ where $\delta_1>0$ is such that $B_{\delta_1}(0)\subset \{e\in {{\mathbb R}}^{2N}:e^TPe<e(0)^TPe(0)\}$, and $\lambda_{min}(Q)>0$ is the smallest eigenvalue of the positive definite matrix $Q$.
Based on Assumptions \[assum0\], \[assum2\] and \[assum3\], we provide the following main result.
\[thm:main\] Consider the Lagrangian system and a smooth, bounded desired trajectory $(q_d(t),\dot{q}_d(t))$. Suppose that Assumptions \[assum0\], \[assum2\], and \[assum3\] hold. Then, the proposed, robust, learning-based control strategy in , , and , with the uncertainty upper bound $\rho$ calculated by and the design parameter $\epsilon$ satisfying , ensures with high probability of at least $(1-\delta_p)^N$ that the tracking error $e(t)$ is uniformly ultimately bounded with an ultimate bound that can be made arbitrarily small through the selection of the design parameter $\epsilon$.
From Assumption \[assum3\], we know that $\|\eta(t)\|\leq \bar{\rho}$ when $e(t)\in D$, where $D$ is a compact set containing $\{e\in {{\mathbb R}}^{2N}:e^TPe\leq e(0)^TPe(0)\}$. In the first part of the proof, we assume that the upper bound $\rho_{GP}(t)$ calculated by , and is a correct upper bound on $\|\eta(t)\|$ when $e(t)\in D$. Thus, in the first part of the proof, we know that $\rho(t)$ calculated by is a correct upper bound on $\|\eta(t)\|$ when $e(t)\in D$, and we use Lyapunov stability analysis to prove that $e(t)$ is uniformly ultimately bounded. Then, in the second part of the proof, we use Lemma \[lemma1\] to evaluate the probability of satisfying the assumption that $\rho_{GP}(t)$ is a correct upper bound on $\|\eta(t)\|$ when $e(t)\in D$, and hence, the probability that the provided guarantees hold. The first part of the proof closely follows the proof of the effectiveness of the robust controller in Theorem 3 of Chapter 8 of [@spong], and we include the main steps of the proof here for convenience. Consider a candidate Lyapunov function $V(e)=e^TPe$. From , it can be shown that $\dot{V}=-e^TQe+2w^T(\eta+r)$, where $w=B^TPe$. Then, from , we need to study two cases.
For the case where $\|w\|>\epsilon$, we have $$\begin{aligned}
w^T(\eta+r)&=w^T(\eta-\rho\frac{w}{\|w\|})=w^T\eta-\rho\|w\| \\ &\leq \|\eta\|\|w\|-\rho\|w\| \end{aligned}$$ from the Cauchy-Schwartz inequality. Since $\{e\in {{\mathbb R}}^{2N}:e^TPe\leq e(0)^TPe(0)\}\subset D$ by definition and from Assumption \[assum3\], we know that $\|\eta\|\leq \bar{\rho}$. Also, by our assumption in this part of the proof, $\|\eta\|\leq \rho_{GP}$. Then, from , $\|\eta\|\leq \rho$, and $w^T(\eta+r)\leq 0$. Thus, for this case, $\dot{V}\leq -e^TQe$, which ensures exponential decrease of the Lyapunov function.
Next, consider the case where $\|w\|\leq \epsilon$. If $w=0$, then $\dot{V}=-e^TQe<0$. Then, for $\|w\|\leq \epsilon$ and $w\neq 0$, it is easy to show $$\dot{V}=-e^TQe+2w^T(\eta+r)\leq-e^TQe+2w^T(\rho\frac{w}{\|w\|}+r).$$ From , we have $$\dot{V}\leq-e^TQe+2w^T(\rho\frac{w}{\|w\|}-\rho\frac{w}{\epsilon}).$$ It can be shown that the term $2w^T(\rho\frac{w}{\|w\|}-\rho\frac{w}{\epsilon})$ has a maximum value of $(\epsilon \rho)/2$ when $\|w\|=\epsilon/2$. Thus, $\dot{V}\leq -e^TQe+(\epsilon \rho)/2$. From , $\rho\leq \bar{\rho}$, and consequently $\dot{V}\leq -e^TQe+(\epsilon \bar{\rho})/2$. If the condition $e^TQe>(\epsilon \bar{\rho})/2$ is satisfied, then $\dot{V}<0$. Since $Q$ is positive definite by definition, then $e^TQe\geq \lambda_{min}(Q)\|e\|^2$, where $\lambda_{min}(Q)>0$ is the smallest eigenvalue of $Q$. Hence, if $\lambda_{min}(Q)\|e\|^2>(\epsilon \bar{\rho})/2$, then $\dot{V}<0$. Thus, the Lyapunov function is strictly decreasing if $\|e\|>\sqrt{\frac{\epsilon \bar{\rho}}{2\lambda_{min}(Q)}}$. Let $B_{\delta}$ be the ball around the origin of radius $\delta:=\sqrt{\frac{\epsilon \bar{\rho}}{2\lambda_{min}(Q)}}$, $S_\delta$ be a sufficiently small sublevel set of the Lyapunov function V satisfying $\bar{B}_\delta\subset S_\delta^{\circ}$, and $B_c$ be the smallest ball around the origin satisfying $S_\delta\subset \bar{B}_c$. Since the Lyapunov function $V$ is strictly decreasing outside $\bar{B}_\delta$, the tracking error $e(t)$ eventually reaches and remains in $S_{\delta}\subset \bar{B}_c$, and so the tracking error $e(t)$ is uniformly ultimately bounded, and its ultimate bound is the radius of $B_c$. Note that from , $B_{\delta}\subset \{e\in {{\mathbb R}}^{2N}:e^TPe<e(0)^TPe(0)\}\subset D$, and $\rho$ is a correct upper bound on $\|\eta\|$. One can see that $\delta$ and hence the radius of $B_c$ depend on the choice of the design parameter $\epsilon$. Indeed, $\epsilon$ can be selected sufficiently small to make $B_{\delta}$ and $B_c$ arbitrarily small. In the second part of the proof, we calculate the probability of our assumption in the first part that $\rho_{GP}(t)$ is a correct upper bound on $\|\eta(t)\|$ when $e(t)\in D$. Recall that $e(t) \in D$ implies that $a_{aug}(t)$ is in a compact set, as discussed immediately after . From Assumption \[assum2\], our problem reduces to calculating the probability that $\rho_{GP}$ is a correct upper bound on $\|\eta\|$ for all the sampling instants. Using the confidence region proposed in Lemma \[lemma1\] for calculating the upper bound on the absolute value of each element of $\eta$, and under Assumption \[assum0\], the probability that this upper bound is correct for all samples is higher than $(1-\delta_p)$ from Lemma \[lemma1\]. Since the $N$ GPs are independent and the added noise to the output observations ${\bf \omega}_v$ is uncorrelated, then the probability that the upper bounds on the absolute values of all the elments of $\eta$, and hence the upper bound on $\|\eta(t)\|$, are correct is higher than $(1-\delta_p)^N$.
Although in practice it is difficult to estimate the upper bound $\bar{\rho}$ on $\|\eta\|$ used in , one can be conservative in this choice. Unlike robust control techniques that keep this conservative bound unchanged, would relax the upper bound $\bar{\rho}$ when the GPs learn a lower upper bound from collected data. Having a less-conservative upper bound $\rho$ results in a lower tracking error. It can be shown that if $\rho(t)\leq \rho'<\bar{\rho}$ for all $t$, then the tracking error will converge to an ultimate ball $B_{c'}$ smaller than $B_c$.
In theory, $\epsilon$ can be selected sufficiently small to ensure arbitrarily accurate tracking as shown in the proof of Theorem \[thm:main\]. Achieving that for cases with large uncertainties may be limited by the actuation limits of the robots. Incorporating the actuation limits in the theoretical analysis is an interesting point for future research.
Simulation Results {#sec:sim}
==================
The proposed approach is first verified via simulations on a 2 DOF planar manipulator using MATLAB Simulink We use the robot dynamics for the system, where $M$, $C$, and $g$ are as defined in Chapter 7 of [@spong]. For the system parameters, a value of $1~\mathrm{kg}$ is used for each link mass and $1~
\mathrm{kg\cdot{m^2}}$ for each link inertia. The length of the first link is $2~ \mathrm{m}$ and that of the second link is $1~\mathrm{m}$. The joints are assumed to have no mass and are not affected by friction. Then, it is assumed that these parameters are not perfectly known. Thus, in the inverse dynamics controller , we use parameters with different levels of uncertainties. The desired trajectories are sinusoidal trajectories with different amplitudes and frequencies. All the simulation runs are initialized at zero initial conditions.
We use $2$ GPs to learn the uncertainty vector $\eta$ in . Each GP uses the squared exponential kernel parameterized with $\sigma_{\eta,{i}}=1$, $\sigma_{\omega,{i}}=0.001$, and $l_{j,i}=0.5$, for all $j\in\{1,\cdots,6\}$ and $i\in \{1,2\}$. The GPs run at $10~\mathrm{Hz}$ and use the past $n=20$ observations for prediction. To generate confidence intervals, we use $[\mu_k-3\sigma_k,\mu_k+3\sigma_k]$, which is simple to implement and found to be effective in practice [@GP_felix]. For the robust controller, we use $\epsilon=0.001$. We set the upper bound $\bar{\rho}$ in to be a very high positive number to evaluate the effectiveness of the upper bound estimated by the GPs. A sequence of *12 trajectories* was run for *3 different cases of model uncertainty*. Each of the three cases makes the $\hat{M}$ matrix differ from the $M$ matrix by using values for the estimated link masses that differ from the true link mass values. In particular, in the three uncertainty cases, the estimated mass differs from the actual mass by $10\%$, $20\%$, and $30\%$ for each link, respectively.
The tracking performance was compared between four controllers: a nominal controller with no robust control, a robust controller with a fixed upper bound on the uncertainty norm $\rho=1000$, a learning-based inverse dynamics controller in which GPs are used to learn the error of the nominal inverse model at the torque level $\Delta u$ and a non-robust outer-loop controller is used, and our proposed robust learning controller. The root-mean-square (RMS) error of the joint angles was averaged over the 12 trajectories, and is presented for each controller and uncertainty case in Table \[table:2dof\_rmse\].
Uncertainty Nominal Learning $\Delta u$
------------- --------- -------- --------------------- --------
$10\%$ 0.1554 0.0476 0.0190 0.0082
$20\%$ 0.2793 0.0498 0.0319 0.0103
$30\%$ 0.3768 0.0519 0.0539 0.0141
: Average RMS Tracking Error (in rad) Over 12 Trajectories for Different Controllers on a 2 DOF Manipulator[]{data-label="table:2dof_rmse"}
It is clear that while the robust controller with a high, fixed value for the upper bound on the uncertainty improves the tracking performance compared to the nominal controller, it is conservative, and thus, still causes considerable tracking errors. The tracking errors are significantly reduced by our proposed robust learning controller, which is able to learn a less conservative upper bound on the uncertainty. On average, our proposed controller reduces the tracking errors by $95.8\%$ compared to the nominal controller, by $78.2\%$ compared to the fixed, robust controller, and by $66\%$ compared to the non-robust learning controller that learns $\Delta u$.
Experimental Results {#sec:exp}
====================
The proposed approach is further tested on a UR10 6 DOF industrial manipulator (see Figure \[fig:ur10\]) using the Robot Operating System (ROS).
Experimental Setup
------------------
The interface to the UR10 does not permit direct torque control. Instead, only position and velocity control of the joints are available. Thus, for our proposed approach, we need to implement only the GP regression models and the robust, outer-loop controller. The commanded acceleration $a_q$ calculated by the outer-loop controller in is integrated to obtain a velocity command that can be sent to the UR10. To test our approach for various uncertainties, we introduce artificial model uncertainty by adding a function $\eta(q, \dot{q},
a_q)$ to our calculated acceleration command $a_q$. The PD gains of the outer-loop controller are tuned to achieve good tracking performance on a baseline desired trajectory in a nominal scenario with no added uncertainty. A desired trajectory of $q_d=0.25(1-\cos(2.0t))$ for each joint is used for this purpose, with gains selected to produce a total joint angle RMS error less than $0.01~\mathrm{rad}$. This resulted in $K_p=7.0I$ and $K_d=1.0I$, where $I$ is the identity matrix.
We use 6 GPs to learn the uncertainty vector $\eta$, each of which uses the squared exponential kernel. The prior variance and length scale hyperparameters are optimized by maximizing the marginal likelihood function, while each noise variance is set to $0.001$. Hyperparameter optimization is performed offline using approximately 1000 data points collected while tracking sinusoidal trajectories under uncertainty $\eta=0.5\dot{q}$. Implementation and tuning of the GPs are done with the Python library GPy. Each GP runs at $125~\mathrm{Hz}$, and uses the past $n=50$ observations for prediction. For the confidence intervals, we use $[\mu_k-3\sigma_k,\mu_k+3\sigma_k]$ for simplicity [@GP_felix]. For the robust controller, we use $\epsilon=0.1$.
Results
-------
The performance of the proposed robust learning controller is initially compared to that of the nominal, outer-loop PD controller using a single trajectory and various cases of model uncertainty. Ten different cases of uncertainty of the form $\eta(q, \dot{q}, a_q)$ are tested over the desired trajectory $q_d=0.25(1-\cos(2.0t))$ for each joint, with the results displayed in Figure \[fig:many\_eta\]. The average RMS error of the nominal controller is $0.111~\mathrm{rad}$ and that of the proposed controller is $0.068~\mathrm{rad}$, yielding an average improvement of $41.5\%$.
Further experiments were performed to verify the generalizability of the proposed approach for different desired trajectories that cover different regions of the state space. A single case of uncertainty, $\eta=0.3\dot{q}+0.01q\odot \dot{q}$ where $\odot$ is the entrywise product, is selected and the performance of the proposed and nominal controllers under this uncertainty is compared on five additional trajectories. The results are presented in Table \[table:many\_traj\], with an average overall improvement of $39.9\%$ compared to the nominal controller. The six trajectories are shown in a demo video at <http://tiny.cc/man-traj>
Trajectory Nominal [**Robust Learning**]{} Improvement
------------ --------- ------------------------- -------------
1 0.070 0.037 47.1%
2 0.058 0.037 36.2%
3 0.092 0.058 37.0%
4 0.085 0.043 49.4%
5 0.050 0.029 42.0%
6 0.029 0.021 27.6%
Average 0.064 0.038 39.9%
: RMS Tracking Error (in rad) for Various Trajectories with a Fixed Uncertainty Function $\eta=0.3\dot{q}+0.01q\odot \dot{q}$, where $\odot$ is the entrywise product[]{data-label="table:many_traj"}
To verify the reliability of the proposed method, experiments for six combinations of uncertainty and trajectory are repeated five times each with both the nominal and proposed robust learning controllers. The results are summarized in Figure \[fig:boxplot\]. The figure shows that the performance under our proposed controller is highly repeatable and that it outperforms the nominal controller in all 30 cases.
Conclusions {#sec:con}
===========
We have provided a novel, learning-based control strategy based on Gaussian processes (GPs) that ensures stability of the closed-loop system and high-accuracy tracking of smooth trajectories for an important class of Lagrangian systems. The main idea is to use GPs to estimate an upper bound on the uncertainty of the linearized model, and then use the uncertainty bound in a robust, outer-loop controller. Unlike most of the existing, learning-based inverse dynamics control techniques, we have provided a proof of the closed-loop stability of the system that takes into consideration the regression errors of the learning module. Moreover, we have proved that the tracking error converges to a ball with a radius that can be made arbitrarily small. Furthermore, we have verified the effectiveness of our approach via simulations on a planar manipulator and experimentally on a 6 DOF industrial manipulator.
[99]{}
M. W. Spong, S. Hutchinson, M. Vidyasagar. *Robot Modeling and Control*. John Wiley & Sons, Inc., 2006.
V. Hagenmeyer, E. Delaleau. Exact feedforward linearization based on differential flatness. *International Journal of Control*, 76(6), pp. 537-556, 2003.
C. Abdallah, D. M. Dawson, P. Dorato, M. Jamshidi. Survey of robust control for rigid robots. *IEEE Control Systems Magazine*, 11(2), pp. 24-30, 1991.
J. Sun de la Cruz. *Learning Inverse Dynamics for Robot Manipulator Control*. M.A.Sc. Thesis, University of Waterloo, 2011.
R. Calandra, S. Ivaldi, M. P. Deisenroth, E. Rueckert, J. Peters. Learning inverse dynamics models with contacts. *IEEE International Conf. on Robotics and Automation*, Seattle, 2015, pp. 3186-3191.
A. S. Polydoros, L. Nalpantidis, V. Kruger. Real-time deep learning of robotic manipulator inverse dynamics. *IEEE/RSJ International Conf. on Intelligent Robots and Systems*, Hamburg, 2015, pp. 3442-3448.
S. Zhou, M. K. Helwa, A. P. Schoellig. Design of deep neural networks as add-on blocks for improving impromptu trajectory tracking. *IEEE Conf. on Decision and Control*, Melbourne, 2017, pp. 5201-5207.
D. Nguyen-Tuong, J. Peters. Using model knowledge for learning inverse dynamics. *IEEE International Conf. on Robotics and Automation*, Anchorage, 2010, pp. 2677-2682.
J. F. Fisac, A. K. Akametalu, M. N. Zeilinger, S. Kaynama, J. Gillula, C. J. Tomlin. A general safety framework for learning-based control in uncertain robotic systems. *arXiv preprint arXiv:1705.01292*, 2017.
T. J. Perkins, A. G. Barto. Lyapunov design for safe reinforcement learning. *The Journal of Machine Learning Research*, 3, pp. 803-832, 2003.
J. W. Roberts, I. R. Manchester, R. Tedrake. Feedback controller parameterizations for reinforcement learning. *IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning*, Paris, 2011, pp. 310-317.
P. Geibel, F. Wysotzki. Risk-sensitive reinforcement learning applied to control under constraints. *Journal of Artificial Intelligence Research*, 24, pp. 81-108, 2005.
M. Turchetta, F. Berkenkamp, A. Krause. Safe exploration in finite markov decision processes with Gaussian processes. *Advances in Neural Information Processing Systems*, Barcelona, 2016, pp. 4312-4320.
F. Berkenkamp, R. Moriconi, A. P. Schoellig, A. Krause. Safe learning of regions of attraction for uncertain, nonlinear systems with Gaussian processes. *IEEE Conf. on Decision and Control*, Las Vegas, 2016, pp. 4661-4666.
F. Berkenkamp, A. P. Schoellig, A. Krause. Safe controller optimization for quadrotors with Gaussian processes. *IEEE International Conf. on Robotics and Automation*, Stockholm, 2016, pp. 491-496.
G. Chowdhary, H. A. Kingravi, J. P. How, P. A. Vela. Bayesian nonparametric adaptive control using gaussian processes. *IEEE trans. on neural networks and learning systems*, 26(3), pp. 537-550, 2015.
F. Meier, D. Kappler, N. Ratliff, S. Schaal. Towards robust online inverse dynamics learning. *IEEE/RSJ International Conf. on Intelligent Robots and Systems*, Daejeon, 2016, pp. 4034-4039.
T. Beckers, J. Umlauft, D. Kulić, S. Hirche. Stable Gaussian process based tracking control of Lagrangian systems. *IEEE Conf. on Decision and Control*, Melbourne, 2017, pp. 5580-5585.
M. K. Helwa, A. P. Schoellig. On the construction of safe controllable regions for affine systems with applications to robotics. *arXiv preprint arXiv:1610.01243*, 2016.
R. M. Murray. Nonlinear control of mechanical systems: a Lagrangian perspective. *Annual Reviews in Control*, 21, pp. 31-42, 1997.
N. Srinivas, A. Krause, S. M. Kakade, M. W. Seeger. Information-theoretic regret bounds for Gaussian process optimization in the bandit setting. *IEEE Trans. on Information Theory*, 58(5), pp. 3250-3265, 2012.
[^1]: The authors are with the Dynamic Systems Lab (www.dynsyslab.org), Institute for Aerospace Studies, University of Toronto, Canada. E-mail: mohamed.helwa@robotics.utias.utoronto.ca, adam.heins@robotics.utias.utoronto.ca, schoellig@utias.utoronto.ca. This research was supported by NSERC grant RGPIN-2014-04634 and OCE/SOSCIP TalentEdge Project \#27901.
|
---
title: Precise Determination of the Charm Quark Mass
---
The determination of the charm quark mass is now possible to 1% from QCD, with lattice QCD pushing the error down below 1%. I will describe the ingredients of this approach and how it can achieve this accuracy. Results for quark mass ratios, $m_c/m_s$ and $m_b/m_c$, can also be determined to 1% from lattice QCD, allowing accuracy for the heavy quark masses to be leveraged into the light quark sector. I will discuss the prospects for, and importance of, improving results in future calculations.
Introduction {#sec:intro}
============
Quark masses are important parameters of the Standard Model but cannot be obtained directly from experiment because quarks are never seen as free particles. Instead they must be inferred from experimental results for hadrons. The accuracy of the determination of quark masses is a topical issue because of the need to test the couplings to quarks of the newly discovered Higgs boson [@lhcwg1; @lhcwg2; @snowmasswg]. The Standard Model rate for decay of a Higgs to $c\overline{c}$ or $b\overline{b}$ is sensitive to the charm/bottom quark mass.
The quark mass parameter in the QCD Lagrangian is a well-defined quark mass but it is scheme- and scale-dependent (i.e. it ‘runs’). Lattice QCD has a clear advantage here when determining quark masses, because the calculations start from the QCD Lagrangian and the parameters of that Lagrangian are readily tuned. To do this, quark mass parameters are chosen, at a given value of the lattice spacing, to reproduce the experimental result for the mass of a hadron containing that quark. This gives the quark mass in the lattice scheme very accurately. However, most calculations (such as those for Higgs decay) need quark masses in a continuum renormalisation scheme such as $\overline{MS}$. A key source of error is then the conversion from the lattice quark mass to the $\overline{MS}$ scheme.
Continuum methods for determining the quark mass rely on evaluating a quantity from experiment that can also be calculated accurately in QCD perturbation theory in terms of, say, the $\overline{MS}$ quark mass. As discussed below, accurate values for $c$ and $b$ masses can be obtained in this way using experimental results derived from $\sigma(e^+e^- \rightarrow \mathrm{hadrons})$ [@karlsruhe1; @karlsruhe2]. Very similar methods can be used with lattice QCD results [@hpqcdkarlsruhe; @hpqcd10] effectively to convert the lattice quark mass to the $\overline{MS}$ scheme, and it is these methods that give the most accurate results from lattice QCD also.
I will describe both methods and their results in Section \[sec:currcurr\] but first give a brief introduction to lattice QCD.
Lattice QCD Calculations {#sec:lattice}
========================
Lattice QCD calculations proceed by a standard recipe [@DeGrandDeTar] which starts with setting up a 4-d space-time volume, discretised into a set of points with lattice spacing, $a$. Configurations of gluon fields (one SU(3) matrix for every link joining two points on the lattice) are generated by Monte Carlo methods according to the probability distribution required in the QCD Feynman Path Integral. This probability distribution is $\exp(-S_{QCD})$ where $S_{QCD}$ is the sum over the configuration of the Lagrangian of QCD. The probability distribution is for the gluon fields but, in modern lattice QCD calculations, it includes the effect of sea quarks that are generated in the ‘soup’ of particles that make up the QCD vacuum. The parameters of QCD enter in specifying the QCD Lagrangian. These are the bare coupling constant and the quark masses. It is important to realise that the lattice spacing is [*not*]{} specified at this point - it must be determined from calculations performed on these configurations.
![Left: a $c\overline{c}$ meson correlation function in QCD and also the charm quark vacuum polarisation. Charm quark propagators connect the two currents, $J$. Right: The correlation function for a $c\overline{c}$ pseudoscalar meson multiplied by $e^{Mt}$ (where $M$ is the fitted ground-state mass) and plotted against time $t/a$ in lattice units. The ground-state clearly dominates the correlation function at large $t$. The statistical errors from the lattice calculation are shown, but are so small as to be barely visible. []{data-label="fig:masses"}](curr-curr.pdf "fig:"){width="6.0cm"} ![Left: a $c\overline{c}$ meson correlation function in QCD and also the charm quark vacuum polarisation. Charm quark propagators connect the two currents, $J$. Right: The correlation function for a $c\overline{c}$ pseudoscalar meson multiplied by $e^{Mt}$ (where $M$ is the fitted ground-state mass) and plotted against time $t/a$ in lattice units. The ground-state clearly dominates the correlation function at large $t$. The statistical errors from the lattice calculation are shown, but are so small as to be barely visible. []{data-label="fig:masses"}](mplot.pdf "fig:"){width="8.0cm"}
Once sets of gluon field configurations have been generated, we can calculate quark propagators on them by solving the Dirac equation. In this equation the gluon field appears in the covariant derivative term and the quark mass is a parameter. Combining a quark and antiquark propagator together (making sure the colours match at both ends and the spins are combined appropriately) makes a meson correlation function. This is the amplitude to create a meson at one point and destroy it at some other point. Averaging the meson correlation functions obtained over all the gluon field configurations generated gives us a Monte Carlo estimate of the result for this amplitude from the QCD Feynman Path Integral. The meson correlation function is illustrated in Figure \[fig:masses\] (left). It shows the meson being created and destroyed by an operator $J$, which is implemented when the quark propagators are tied together. At intermediate points the charm quark and antiquark interact with each other via the gluon fields and sea quarks in the background configuration.
The meson mass is determined by fitting the average meson correlation function as a function of time on the lattice (we sum the end-points over $x$, $y$, $z$, at fixed $t$ to project onto zero spatial momentum for the meson). Because we are working with Euclidean time, the expected behaviour at large times is as an exponential (rather than the more normal phase factor): $$\langle 0 | J^{\dag}(t_0+t) J(t_0) | 0 \rangle \equiv G(t) \stackrel{t \rightarrow \infty}{=} Ae^{-Mt} = Ae^{-Ma\times(t/a)}.
\label{eq:corr}$$ The exponent is the mass of the lowest mass meson with the quantum numbers of the operator, $J$. The last piece of the equation above shows that, in fitting the correlation function in terms of time on the lattice $t/a$ (i.e. the number of lattice spacings between two points in time) we will be able to determine the mass of the meson also in lattice units, i.e. the dimensionless combination $Ma$. We need to obtain a value for $a$ in order to convert this to physical, GeV, units. This is done by using another hadron mass (preferably one that is rather insensitive to quark masses) and setting the lattice result equal to the experimental value [@pdg]. Quantities used for this include the radial excitation energy in the $\Upsilon$ system [@Dowdallups] and the $\pi$ decay constant [@DowdallpiK].
![Left: the gold-plated heavy meson spectrum from lattice QCD (points) compared to experiment [@pdg] (red lines). Light cyan crosses denote those masses used to fix the parameters of QCD; green squares indicate postdictions and dark blue circles indicate predictions ahead of experiment. Recent lattice results are from [@Dowdallups; @Dowdallhl; @Daldrop; @Donaldpsi] Right: Moments of vector current-current correlators from lattice QCD plotted against the square of the lattice spacing [@Donaldpsi]. The dashed line shows the continuum extrapolation. The black points at $a=0$ correspond to values extracted from experiment for the charm contribution to $R_{e^+e^-}$ [@karlsruhe2]. []{data-label="fig:gold"}](goldmesonpdg13.pdf "fig:"){width="7.0cm"} ![Left: the gold-plated heavy meson spectrum from lattice QCD (points) compared to experiment [@pdg] (red lines). Light cyan crosses denote those masses used to fix the parameters of QCD; green squares indicate postdictions and dark blue circles indicate predictions ahead of experiment. Recent lattice results are from [@Dowdallups; @Dowdallhl; @Daldrop; @Donaldpsi] Right: Moments of vector current-current correlators from lattice QCD plotted against the square of the lattice spacing [@Donaldpsi]. The dashed line shows the continuum extrapolation. The black points at $a=0$ correspond to values extracted from experiment for the charm contribution to $R_{e^+e^-}$ [@karlsruhe2]. []{data-label="fig:gold"}](vecmoments.pdf "fig:"){width="7.0cm"}
Figure \[fig:masses\] (right) shows the correlator for the $c\overline{c}$ pseudoscalar meson, for which the lowest mass meson is the $\eta_c$. The quantity plotted is $G(t)e^{Mt}$, where $Ma$ is the value obtained for the ground-state mass from a fit to the correlator. This value is $Ma=1.32724(3)$ which corresponds to 2.982(3) GeV at this value of the lattice spacing ($a=0.08784(9) \mathrm{fm} \equiv 1/2.2466(23) \,\mathrm{GeV}^{-1}$). This shows how accurately the $\eta_c$ mass can be obtained. The calculation required fixing the charm quark mass, also in lattice units. The value here, using the Highly Improved Staggered formalism [@HISQ] for the $c$ quarks, was $m_ca=0.432$. The 0.1% accuracy obtainable on the $\eta_c$ mass, means that the lattice $c$ quark mass (on which it is linearly dependent) can be tuned to a similar level of accuracy [@fds].
Figure \[fig:masses\] also demonstrates the behaviour of the correlator. At large values of $t$ it is dominated by the ground-state $\eta_c$, so that $G(t)e^{Mt}$ is a constant. At shorter times this is not true. Then higher mass states (for example, radial excitations) contribute and their masses can be determined with a careful calculation (as discussed elsewhere in these Proceedings [@prelcharm2013]). This region merges seamlessly with the region where the correlator is controlled by perturbative QCD. It is the short time region that we use to match the lattice $m_c$ to that in a continuum scheme, as described in the next Section.
The $\eta_c$ is only one of a range of meson masses that can be accurately determined from lattice QCD. Figure \[fig:gold\] shows a summary plot of the spectrum of ‘gold-plated’ mesons containing $c$ and $b$ quarks from lattice QCD and its comparison with experiment. A gold-plated meson is one that has no strong Zweig-allowed decay mode and so has a very narrow width. The accuracy of many of these masses from lattice QCD is now at the few MeV level where we need to worry about and estimate electromagnetic effects missing from our pure QCD calculations [@fds]. The agreement with experiment is excellent, providing a stringent test of QCD. Indeed, some of the masses were predicted ahead of experiment.
Handling $c$ and $b$ quarks presents some difficulties in lattice QCD because they are relatively heavy. When the Dirac equation is discretised onto a lattice of points the covariant derivative is replaced by a finite difference and this is only correct up to systematic errors of $\mathcal{O}(a^2)$. The question is, what sets the scale for these errors? For hadrons made of light quarks, this will typically be the scale of QCD, i.e. a few hundred MeV. For heavy quarks it can be the quark mass itself. For $c$ quarks, $m_ca$ is around 0.4 for typical lattice spacing values of around 0.1 fm. An error of $\mathcal{O}([m_ca]^2)$ could then be of size 20%. Working with ‘improved’ discretisations raises the power of $ma$ in the error and improves the situation. For the Highly Improved Staggered Quark (HISQ) action [@HISQ] that is used here, the leading errors are $\alpha_s^2 (m_ca)^2$ and $(m_ca)^4$, which give errors of a few % at $a=$ 0.1fm. It is important to obtain results at multiple values of the lattice spacing and extrapolate to $a=0$ to remove the discretisation errors. This extrapolation is relatively benign if a highly improved action is used and therefore the error in the final result from this extrapolation is small.
The current-current correlator method {#sec:currcurr}
=====================================
The production of a $c\overline{c}$ pair occurs directly in the real world in $e^+e^-$ collisions. Figure 1 (left) can also illustrate this case by representing the ‘heavy quark vacuum polarisation’. Then $J$ is the $c\overline{c}$ vector current which couples to the photon produced in $e^+e^-$. If we cut the diagram down the centre we expose a lot of quark-antiquark pairs and gluons produced from the original $c\overline{c}$ pair which, by unitarity, will end up as hadrons in the final state. Information about the charm quark vacuum polarisation can then be extracted from $\sigma(e^+e^- \rightarrow \mathrm{hadrons})$ if we can isolate the piece of the cross-section that corresponds to $c$ quark pair production. Because $R_{e^+e^-}=\sigma(e^+e^- \rightarrow \mathrm{hadrons})/\sigma_{\mathrm{point}}$ has step-like behaviour as a function of centre-of-mass energy $\sqrt{s}$ with well-separated heavy quark regions, this can be done using a mixture of theory and experiment. The contribution from $u$, $d$ and $s$ quarks can be calculated and subtracted, as illustrated in Figure \[fig:R\] from [@karlsruhe2]. The basic tree-level QED calculation from textbooks [@PeskinSchroeder] gives $R_{e^+e^-}=3\sum_i Q^2_{q_i}$ for $i$ flavours of quarks with $Q_{q_i}$ the electric charge of that quark flavour in units of $e$, ignoring quark mass effects. QCD corrections can be incorporated that are impressively known up to and including $\alpha_s^3$ terms [@Harlander]. The ‘natural’ scale for $\alpha_s$ is $\sqrt{s}$, so this gives an accurate picture for $R$ in the region of a few GeV (below the charm threshold), and the agreement with experiment is good. Higher order electromagnetic contributions can be determined and they are very small; effects from the $Z$ are negligible.
![$R_{e^+e^-}$ as a function of centre-of-mass energy, $\sqrt{s}$, around the charm threshold region. The solid line, with uncertainties given by the dashed lines, gives the prediction in perturbative QCD below and above the charm threshold. Figure from [@karlsruhe2].[]{data-label="fig:R"}](R.pdf){width="12.0cm"}
The $c$ quark contribution to $R_{e^+e^-}$, $R_c(s)$, then has pieces corresponding to the charm resonances (modelled as narrow peaks using the experimental information about each state), the charm threshold region (obtained from experiment after subtraction for $u$, $d$, and $s$ quarks) and the higher $s$ region above the charm threshold obtained from perturbation theory, again compared to experiment [@karlsruhe2] or directly from experiment [@Hoang]. Around the charm threshold region $R_c(s)$ is very sensitive to the charm quark mass and this can be used to determine $m_c$.
The determination of $m_c$ uses analyticity properties to obtain the (dispersion) relationship between $s$-inverse moments of $R_c(s)$ and $q^2$-derivative moments of the charm quark vacuum polarisation function evaluated at $q^2=0$ [@PeskinSchroeder]: $$\mathcal{M}_{k,expt} \equiv \int \frac{ds}{s^{k+1}}R_c(s) = \mathcal{M}_{k,th} \equiv \left.\frac{12\pi^2}{k!}\left(\frac{d}{dq^2}\right)^k \Pi_c(q^2)\right|_{q^2=0}.
\label{eq:smom}$$ $\mathcal{M}_{k,expt}$ is evaluated from $R_c(s)$ and the numbers are shown as the black points on the right-hand plot of Figure \[fig:masses\] (where $n = 2k+2$). Errors are 1% or better. The contribution from the resonances dominates.
$\mathcal{M}_{k,th}$, i.e. the $q^2$ derivatives of $\Pi_c$, needs evaluation of the behaviour of $\Pi_c$ at small $q^2$, i.e. in a very different kinematic region to that for $R_c(s)$. For heavy quarks, $q^2=0$ is well below the threshold to produce real quarks (so the $c$ quarks in Figure \[fig:masses\] (left) would be virtual). The expansion of $\Pi_c$ about $q^2=0$ can then be evaluated in QCD perturbation theory and the derivatives obtained, giving for the vector current case $$\mathcal{M}_{k,th} = Q_c^2 \frac{9}{4} C_{k,V} \left( \frac{1}{4m_c^2}\right)^k; \,\, C_{k,V} = C_{k,V}^{{0}} + \alpha_s C_{k,V}^{(1)} + \ldots .
\label{eq:moments}$$ This exposes clearly the sensitivity of $\mathcal{M}_k$ to the $c$ quark mass. $m_c$ in the equation above can be, for example, the $c$ quark mass in the $\overline{MS}$ scheme evaulated at the scale $\mu$. The perturbative series, $C_k$, is a power series expansion in $\alpha_s$. Its coefficients will reflect the scheme and scale chosen for $m_c$ so that the final result (to all orders) for $\mathcal{M}_k$ is scheme and scale invariant, and has the value obtained from experiment via $R_c$ in equation \[eq:smom\]. The ‘natural’ scale for $\alpha_s$ here is $2m_c$, which is large enough for reasonably good control of the perturbative expansion.
In fact, the QCD perturbation theory for $C_k$ has reached an extremely impressive level of calculation. Values for $C_k^{(3)}$ are known for the first few values of $k$, which corresponds to NNNLO [@karlsruhepert; @Boughezal; @Maier]. Small values of $k$, 1 to 4, are preferred because larger values of $k$, although more sensitive to $m_c$, start to receive significant contributions from operators such as the gluon condensate which increase the uncertainty. Matching $\mathcal{M}_{k,th}$, with an input value of $\alpha_s$, to $\mathcal{M}_{k,expt}$ described above, Chetyrkin et al [@karlsruhe2] obtain, in the $\overline{MS}$ scheme with number of flavours, $n_f=4$, $m_c(m_c)$ = 1.279(13) GeV. This is obtained from using the lowest moment, $k=1$, in equation \[eq:moments\]. The error is dominated by the experimental error in $R_c(s)$ and by the uncertainty taken in the value of $\alpha_s$ (3 times the current PDG uncertainty [@pdg]). The uncertainty coming from unknown higher order terms in the perturbation theory is estimated in the standard way by varying the scale, $\mu$, at which $\alpha_s$ is evaluated (varying the coefficients $C_{k,V}$ appropriately). The central value used here is 3 GeV, which is the same as the scale used for the central value of $m_c$ (subsequently iteratively run to the scale of $m_c$). The variation taken is $\pm$ 1 GeV [@karlsruhe2]. Perturbative error estimates are always somewhat subjective and these have been criticised in [@Hoang] as being too small, in particular pointing out that larger $\mu$ dependence can be seen when the $\mu$ in $\alpha_s$ and that in $m_c$ are decoupled.
In the lattice QCD analysis described below we use the same perturbation theory but take a somewhat different approach to the perturbative errors, estimating directly the effect on $m_c$ of missing higher order terms using a Bayesian analysis. We are also able to fit multiple moments simultaneously and extract at the same time a value for $\alpha_s$. These features improve the accuracy with which $m_c$ can be determined. They require the use of pseudoscalar current-current correlators which are not accessible from experiment but, as discussed in Section \[sec:lattice\], can be calculated very accurately in lattice QCD.
![Time moments of pseudoscalar $c\overline{c}$ correlators calculated in lattice QCD as a function of the square of the lattice spacing [@hpqcd10]. The result extrapolated to $a=0$ can be used with continuum QCD perturbation theory to determine the $c$ quark mass in the $\overline{MS}$ scheme. []{data-label="fig:rvasq"}](Rvsa2.pdf){width="8.0cm"}
In lattice QCD we can substitute for $M_{k,expt}$ values of $\mathcal{M}_{k,latt}$ obtained by taking time-moments of the $c\overline{c}$ meson correlation functions described in Section \[sec:lattice\]. The correlation functions (at zero spatial momentum) are the Fourier transform from energy to time-space of the charm quark vacuum polarisation function. Thus $q^2$-derivative moments become (squared) time-moments and we use [@hpqcdkarlsruhe] $$G_n = \sum_t (t/a)^n G(t); \,\, n=2k+2 .$$ $G(t)$ is a meson correlation function averaged over gluon field configurations, as in equation \[eq:corr\]. To compare to the continuum QCD perturbation theory of equation \[eq:moments\] we need to extrapolate $G_n$ to $a=0$ to obtain a continuum value. Thus we have to define $G(t)$ to be well-defined in that limit. For the HISQ formalism used here we have a PCAC relation (as in continuum QCD) that enables us to define an absolutely normalised pseudoscalar current operator: $J=(am_c) \overline{c}\gamma_5c$ and we use this to create and destroy pseudoscalar $c\overline{c}$ states in our correlation function.
To perform the analysis [@hpqcdkarlsruhe; @hpqcd10] we calculate meson correlation functions at multiple values of the lattice spacing, fixing $am_c$ at each value of $a$ to be the value which gives the correct $\eta_c$ mass from the long-time behaviour of the correlator. We then calculate the time-moments as above for $n=$ 4, 6, 8 and 10 (corresponding to $k=$ 1, 2, 3 and 4 in equation \[eq:moments\]). Taking time-moments emphasises the small, but non-zero, $t$ region of the correlation function since it is falling approximately exponentially (see Figure \[fig:masses\] right). Having a result for each moment at each value of $a$ then allows us to extrapolate to the continuum limit. The result for doing this for $c$ quarks is shown in Figure \[fig:rvasq\]. To reduce discretisation errors we have actually plotted and extrapolated the ratio of $G_n$ to its value in the absence of gluon fields, $G_n^{(0)}$ (readily calculated by simply omitting the coupling to gluons in the Dirac equation). In fact we take: $$R_{n,latt} = G_4/G_4^{(0)},\,n=4;\,\, R_{n,latt} = \frac{aM_{\eta_c}}{2am_c}\left(\frac{G_n}{G_n^{(0)}}\right)^{1/(n-4)}, \,n=6,8,10 \ldots .
\label{eq:rdef}$$ Here $aM_{\eta_c}$ and $am_c$ are lattice values. From Figure \[fig:rvasq\] we see that the extrapolation is relatively benign, especially as $n$ increases from 4. We include results from 4 values of the lattice spacing from 0.12 fm down to 0.045 fm. Values in the continuum limit have errors of order 0.1%.
At $a=0$ we can compare to the same ratio determined perturbatively: $$R_{4,cont} = \frac{C_{1,PS}}{C_{1,PS}^{(0)}};\,\,R_{n,cont} = \frac{M_{\eta_c}}{2m_c(\mu)}\frac{C_{k,PS}}{C_{k,PS}^{(0)}};\,\, n=2k+2
\label{eq:rcont}$$ where $C_{k,PS}$ is the full perturbative series for the pseudoscalar moment and $C_{k,PS}^{(0)}$ is the leading ($\alpha_s^0$) term. So $C_{k,PS}/C_{k,PS}^{(0)} = 1+c_1\alpha_s + \ldots$. For $n=4$ ($k=1$) in the pseudoscalar case we have no factor of masses in front of the series. This means that the $n=4$ moment is insensitive to the charm quark mass (it appears only in the scale for $\alpha_s$) and can be used to determine $\alpha_s$. The higher moments ($n=$ 6, 8, 10) can be used to determine $m_c(\mu)$ in terms of the (experimental) $\eta_c$ mass from equation \[eq:rcont\].
We match the lattice results at $a=0$ to the continuum perturbation theory, simultaneously fitting $n=$ 4, 6, 8 and 10 (including correlations between them and allowing for gluon condensate contributions) to extract $\alpha_s(\mu)$ and $M_{\eta_c}/m_c(\mu)$. The result we obtain for $m_c$ in the $\overline{MS}$ scheme with $n_f=4$ is $m_c(m_c)=1.273(6)$ GeV. Note that the calculation is done with 3 flavours of sea quarks and QCD perturbation theory is used to convert to 4 flavours. A complete error budget is given in [@hpqcd10]. The error is dominated by the unknown higher order terms in the perturbative expansion of the moments and is estimated by including such terms with coefficients that are constrained by Bayesian priors. Information about the known $\mu$ dependence of the coefficients from the renormalisation group can be included this way. Our perturbative error is then about half the combined perturbative-$\alpha_s$ error in [@karlsruhe1]. The statistical error from $\mathcal{M}_{k,latt}$ is much smaller in this case than that from $\mathcal{M}_{k,expt}$. The place in which experiment enters into the lattice calculation is in the tuning of the lattice $c$ quark mass using the experimental $\eta_c$ mass. For this we estimate the effect of missing electromagnetism from the lattice calculation; it is a tiny effect [@fds]. There is no further error from missing electromagnetism because we are comparing a lattice QCD calculation to continuum QCD perturbation theory.
A further test of this approach is to calculate the vector-vector correlator and compare the moments to those extracted from experiment via $R_c(s)$, $\mathcal{M}_{k,expt}$, described above [@Donaldpsi]. To extrapolate the lattice vector charmonium correlator to $a=0$ we first have to renormalise the vector current. This we do using the continuum QCD perturbation theory for the $n=4$ ($k=1$) moment. Figure \[fig:masses\] (right) shows the comparison of lattice QCD vector moments against $a^2$ with the moments determined from experiment via $R_c(s)$ as the black points at $a=0$. The extrapolated lattice QCD results agree well with experiment, with the lattice QCD results having approximately double the error (including a small contribution allowing for higher order QED effects which are not included in the lattice calculation but are present in experiment). This then represents an impressive 1% test of QCD and adds confidence to the determination of $m_c$ from the pseudoscalar moments. Using the lattice QCD vector moments to determine $m_c$ would give a result in agreement with that from the pseudoscalar but with a larger error. The long-time behaviour of the vector correlators simultaneously gives accurate results for the $J/\psi$ mass and leptonic width [@Donaldpsi].
Because the HISQ action has small discretisation errors we can push to higher masses than $m_c$ and this was done in [@hpqcd10]. By extrapolating up in mass we can also determine $m_b$ in the $\overline{MS}$ scheme from the same method: ${m}^{(5)}_b({m}_b)=4.164(23) \mathrm{GeV}$. Here the error is dominated by the extrapolation to the $b$ quark mass/$a=0$. The physical curve for the ratio of heavyonium mass to heavy quark mass is obtained on solving equation \[eq:rcont\], and this is shown in Figure \[fig:massrat\] (left).
![Left: the ratio of pseudoscalar heavyonium mass to heavy quark mass in the $\overline{MS}$ scheme at scale $\mu$ as a function of heavyonium meson mass and for 3 different values of $\mu$ [@hpqcd10]. Notice how flat the curve is for ${m}_h({m}_h)$. Right: the ratio of $c$ to $s$ quark masses determined from lattice QCD plotted against the square of the lattice spacing. Extrapolation to $a=0$ gives the physical result 11.85(16) [@mcms].[]{data-label="fig:massrat"}](moverM.pdf "fig:"){width="7.5cm"} ![Left: the ratio of pseudoscalar heavyonium mass to heavy quark mass in the $\overline{MS}$ scheme at scale $\mu$ as a function of heavyonium meson mass and for 3 different values of $\mu$ [@hpqcd10]. Notice how flat the curve is for ${m}_h({m}_h)$. Right: the ratio of $c$ to $s$ quark masses determined from lattice QCD plotted against the square of the lattice spacing. Extrapolation to $a=0$ gives the physical result 11.85(16) [@mcms].[]{data-label="fig:massrat"}](mcms.pdf "fig:"){width="7.0cm"}
Mass ratios {#sec:massratios}
===========
Lattice QCD enables us to determine the ratios of quark masses fully nonperturbatively. Provided that we have used the same lattice discretisation for both quarks, the $Z$ factors that connect the lattice quark mass to the $\overline{MS}$ quark mass at a given scale will cancel in the ratio. We then have, extrapolating to the continuum $$\left(\frac{m_{q1,latt}}{m_{q2,latt}}\right)_{a=0} = \frac{m_{q1,\overline{MS}}(\mu)}{m_{q2,\overline{MS}}(\mu)}.$$ Using the HISQ action for both $c$ and $s$ quarks, fixing $am_c$ from $M_{\eta_c}$ and $am_s$ from $M_K$ (via an unphysical $s\overline{s}$ pseudoscalar particle called the $\eta_s$), gives the results shown in Figure \[fig:massrat\] (right) as a function of lattice spacing. The extrapolated result, $m_c/m_s = 11.85(16)$ could not be obtained with this accuracy by any other method. It enables us to convert the accurate value for $m_c$ discussed in the previous Section into a 1% determination of $m_s$. Running $m_s$ up from $m_c$ to the conventional 2 GeV gives $m_s(2\mathrm{GeV})$ = 92.2(1.3) MeV. In [@hpqcd10] we determine nonperturbatively the ratio of $m_b/m_c$, obtaining 4.51(4), and this acts as a check on the determination using moments and perturbation theory. Using mass ratios in this way we can leverage the accuracy in the heavy quark masses across the full set from $u$ to $b$ [@hpqcd10]. Amusingly we can use this to test the Georgi-Jarlskog expectation from GUTs that $m_b/m_s = 3m_{\tau}/m_{\mu}$ [@GeorgiJarlskog]. For $m_b/m_s$ we have 53.4(1.1) (allowing for some statistical correlation between $m_b/m_c$ and $m_c/m_s$). This is only in marginal agreement with $3m_{\tau}/m_{\mu}$ = 50.450(5) [@pdg]. As lattice QCD calculations become more accurate there will be more tension in simple relationships of this kind, including those between quark masses and CKM elements [@McNeile].
Conclusions {#sec:conclusions}
===========
Both continuum and lattice QCD determinations of the $\overline{MS}$ $c$ quark mass have reached a level of accuracy around 1%. The most accurate result is ${m}_c^{(4)}({m}_c)=1.273(6)$ GeV using lattice QCD [@hpqcd10]. This calculation used 3 flavours of sea quarks; future work is underway by both the ETM and HPQCD Collaborations to determine $m_c$ including $c$ quarks directly in the sea, thus removing any worries about the 3 to 4 flavour matching. To improve the accuracy on $m_c$ will be hard without having yet another order in QCD perturbation theory. It could be done by using a nonperturbative determination of $m_b/m_c$ and a more accurate result for $m_b$ from lattice QCD (using for finer lattices) because $m_b$ has a smaller perturbative error. It is important for phenomenologists to use these accurate values for quark masses in, for example, determination of Higgs cross-sections if they are to estimate reliably the uncertainty in the Standard Model cross-section. Currently the error on the value of $m_c$ being used [@lhcwg1; @lhcwg2; @snowmasswg] is inflated by a factor of 3.
I am grateful to Steve King, Peter Lepage, Paul Mackenzie, Craig McNeile and Marcus Petschlies for useful discussions. Thanks also to the organisers for a very interesting meeting.
[99]{}
S. Heinemeyer [*et al.*]{} \[LHC Higgs Cross Section WG\], arXiv:1307.1347 \[hep-ph\]. S. Dittmaier [*et al.*]{} \[LHC Higgs Cross Section WG\], arXiv:1201.3084 \[hep-ph\]. S. Dawson [*et al.*]{} \[Snowmass 2013 Higgs WG\], arXiv:1310.8361 \[hep-ex\]. K. G. Chetyrkin [*et al*]{}, Phys. Rev. D [**80**]{}, 074010 (2009) \[arXiv:0907.2110 \[hep-ph\]\]. J. H. Kuhn, M. Steinhauser and C. Sturm, Nucl. Phys. B [**778**]{}, 192 (2007) \[arXiv:hep-ph/0702103\]. I. Allison [*et al.*]{} \[HPQCD Collaboration + Karlsruhe\], Phys. Rev. D [**78**]{}, 054513 (2008) \[arXiv:0805.2999 \[hep-lat\]\]. C. McNeile [*et al*]{} \[HPQCD Collaboration\], Phys. Rev. D [**82**]{}, 034512 (2010) \[arXiv:1004.4285 \[hep-lat\]\]. T. DeGrand and C. E. Detar, “Lattice methods for quantum chromodynamics,” New Jersey, USA: World Scientific (2006) 345 p
J. Beringer [*et al.*]{} \[Particle Data Group Collaboration\], Phys. Rev. D [**86**]{}, 010001 (2012). R. J. Dowdall [*et al.*]{} \[HPQCD Collaboration\], Phys. Rev. D [**85**]{}, 054509 (2012) \[arXiv:1110.6887 \[hep-lat\]\]. R. J. Dowdall [*et al*]{} \[HPQCD Collaboration\], Phys. Rev. D [**88**]{}, 074504 (2013) \[arXiv:1303.1670 \[hep-lat\]\]. E. Follana [*et al.*]{} \[HPQCD and UKQCD Collaborations\], Phys. Rev. D [**75**]{}, 054502 (2007) \[arXiv:hep-lat/0610092\]. C. T. H. Davies [*et al*]{} \[HPQCD Collaboration\], Phys. Rev. D [**82**]{}, 114504 (2010) \[arXiv:1008.4018 \[hep-lat\]\]. S. Prelovsek, arXiv:1310.4354 \[hep-lat\]. R. J. Dowdall [*et al*]{} \[HPQCD Collaboration\], Phys. Rev. D [**86**]{}, 094510 (2012) \[arXiv:1207.5149 \[hep-lat\]\]. J. O. Daldrop [*et al.*]{} \[HPQCD Collaboration\], Phys. Rev. Lett. [**108**]{}, 102003 (2012) \[arXiv:1112.2590 \[hep-lat\]\]. G. C. Donald [*et al*]{} \[HPQCD Collaboration\], Phys. Rev. D [**86**]{}, 094501 (2012) \[arXiv:1208.2855 \[hep-lat\]\]. M. E. Peskin and D. V. Schroeder, “An Introduction to quantum field theory,” Reading, USA: Addison-Wesley (1995) 842 p
R. V. Harlander and M. Steinhauser, Comput. Phys. Commun. [**153**]{}, 244 (2003) \[arXiv:hep-ph/0212294\]. B. Dehnadi [*et al*]{}, JHEP [**1309**]{}, 103 (2013) \[arXiv:1102.2264 \[hep-ph\]\]. K. G. Chetyrkin, J. H. Kuhn and C. Sturm, Eur. Phys. J. C [**48**]{}, 107 (2006) \[arXiv:hep-ph/0604234\]. R. Boughezal, M. Czakon and T. Schutzmeier, Phys. Rev. D [**74**]{}, 074006 (2006) \[arXiv:hep-ph/0605023\]. A. Maier [*et al*]{}, Nucl. Phys. B [**824**]{}, 1 (2010) \[arXiv:0907.2117 \[hep-ph\]\]. C. T. H. Davies [*et al.*]{} \[HPQCD Collaboration\], Phys. Rev. Lett. [**104**]{}, 132003 (2010) \[arXiv:0910.3102 \[hep-ph\]\]. H. Georgi and C. Jarlskog, Phys. Lett. B [**86**]{}, 297 (1979). C. McNeile, arXiv:1004.4985 \[hep-lat\].
|
---
abstract: 'Recently, Bieñ \[A. Bieñ, The problem of singularity for planar grids, [*Discrete Math.*]{} 311 (2011), 921–931\] obtained a recursive formula for the determinant of a grid. Also, recently, Pragel \[D. Pragel, Determinants of box products of paths, [*Discrete Math.*]{} 312 (2012), 1844–1847\], independently, obtained an explicit formula for this determinant. In this paper, we give a short proof for this problem. Furthermore, applying the same technique, we get explicit formulas for the determinant of a torus, a cylinder, and a Möbius ladder.'
author:
- Khodakhast Bibak$^1$
- Roberto Tauraso$^2$
title: |
Determinants of grids, tori, cylinders\
and Möbius ladders
---
\[theorem\][Corollary]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Conjecture]{} \[theorem\][Definition]{}
\[theorem\][Example]{}
\[theorem\][Remark]{}
Introduction and results
========================
We denote by $A(G)$ the adjacency matrix of a graph $G$. A path (cycle) on $n$ vertices is denoted by $P_n$ (resp., $C_n$). Given two graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$, their [*Cartesian product*]{} $G_1\; \Box \;G_2$ is the graph with vertex set $V_1\times V_2$ and edge set $$\Big\{\big((u,v),(u',v)\big):(u,u')\in E_1, v\in V_2\Big\}\bigcup \Big\{\big((u,v),(u,v')\big): u \in V_1, (v,v')\in E_2\Big\}.$$ Cartesian product produces many important classes of graphs. For example, a [*grid*]{} (also called [*mesh*]{}) is the Cartesian product of two paths, a [*torus*]{} (also called [*toroidal grid*]{} or [*toroidal mesh*]{}) is the Cartesian product of two cycles, and a [*cylinder*]{} is the Cartesian product of a path and a cycle. One can generalize these definitions to more than two paths or cycles. These classes of graphs are widely used computer architectures (e.g., grids are widely used in multiprocessor VLSI systems) [@HL].
The [*nullity*]{} of a graph $G$ of order $n$, denoted by $\eta(G)$, is the multiplicity of 0 in the spectrum of $G$. Clearly, $\eta(G) = n - r(A(G))$, where $r(A(G))$ is the rank of $A(G)$. The nullity of a graph is closely related to the minimum rank problem of a family of matrices associated with a graph (see, e.g., [@FAHO] and the references therein). Nullity of a (molecular) graph (specifically, determining whether it is positive or zero) has also important applications in quantum chemistry and Hückel molecular orbital (HMO) theory (see, e.g., [@GUBO] and the references therein). A famous problem, posed by Collatz and Sinogowitz in 1957 [@COSI], asks to characterize all graphs with positive nullity. Clearly, $\det
A(G)=0$ if and only if $\eta(G)>0.$ So, examining the determinant of a graph is a way to attack this problem. But there seems to be little published on calculating the determinant of various classes of graphs.
Recently, Bieñ [@BIE] obtained a recursive formula for the determinant of a grid. Also, recently, Pragel [@PRA], independently, obtained an explicit formula for this determinant (see below). Here, using trigonometric identities, we give a short proof for this problem. Furthermore, applying the same technique, we get explicit formulas for the determinant of a torus, and a cylinder.
\[thm:detcar\] Let $m>1$ and $n>1$ be integers. Then $$\begin{aligned}
\label{T1}
\det A(P_{m-1}\; \Box \; P_{n-1})&=
\begin{cases}
(-1)^{\frac{(m-1)(n-1)}{2}} & \text{if}\; \gcd(m,n) = 1;\\
0 & \text{otherwise}.
\end{cases}\\\label{T2}
\det A(C_{m}\; \Box \; C_{n})&=
\begin{cases}
4^{\gcd(m,n)}\qquad & \text{if $m$ and $n$ are odd};\\
0\qquad & \text{otherwise}.
\end{cases}\\\label{T3}
\det A(P_{m-1}\; \Box \; C_{n})&=
\begin{cases}
m & \text{if $n$ is odd and $\gcd(m,n)=1$};\\
(-1)^{m-1}m^2 & \text{if $n$ is even and $\gcd(m,n/2)=1$};\\
0 & \text{otherwise}.
\end{cases}\end{aligned}$$
Note that for $m=2$, and give the following well-known determinants. $$\begin{aligned}
\det A(P_{n-1})=
\begin{cases}
(-1)^{\frac{n-1}{2}} & \text{if $n$ is odd};\\
0 & \text{otherwise}.
\end{cases}
\quad\mbox{and}\quad
\det A(C_{n})=
\begin{cases}
2 & \text{if $n$ is odd};\\
-4& \text{if $n\equiv 2 \pmod{4}$};\\
0 & \text{otherwise}.
\end{cases}\end{aligned}$$ Our proof techniques or its modifications may be useful in other situations with similar flavor (see, e.g., [@BISH]). For example, let us consider the [*Möbius ladder*]{} $M_{2n}$, the graph on $2n$ vertices whose edge set is the union of the edge set of $C_{2n}$ and $\{(v_i,v_{n+i}):i=1,\dots, n\}$. We prove that
\[thm:mob\] Let $n>1$ be an integer. Then $$\begin{aligned}
\label{T4}
\det A(M_{2n})&=
\begin{cases}
-3 & \text{if $n\equiv \pm 2 \pmod{6}$;}\\
-9 & \text{if $n\equiv \pm 1 \pmod{6}$;}\\
0 & \text{otherwise}.
\end{cases}\end{aligned}$$
Techniques and Proofs
=====================
The starting point of our calculations is the following well-known theorem which gives the eigenvalues of the Cartesian product of two graphs (see, e.g., [@ROS p. 587]).
\[thm:eigcar\] Let $G_1$ be a graph of order $m$, and $G_2$ be a graph of order $n$. If the eigenvalues of $A(G_1)$ and $A(G_2)$ are, respectively, $\lambda_{1}(G_1),\ldots,\lambda_{m}(G_1)$ and $\lambda_{1}(G_2),\ldots,\lambda_{n}(G_2)$, then the eigenvalues of $A(G_1\; \Box \;G_2)$ are precisely the numbers $
\lambda_{i}(G_1)+\lambda_{j}(G_2),$ for $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$.
We also need the following trigonometric identities, which might be of independent interest.
\[lem:triiden\] Let $n$ be a positive integer and let $a\in\mathbb{Z}$ such that $\gcd(a,n)=1$. Then for any real number $x$, $$\label{eq1}
\sin(nx)=2^{n-1}(-1)^{\frac{(a-1)(n-1)}{2}}\prod_{j=0}^{n-1}\sin\left(x+{a j\pi \over n}\right).$$ Moreover, $$\label{eq2}
\prod_{j=1}^{n-1}\sin\left({a j\pi \over n}\right)=(-1)^{\frac{(a-1)(n-1)}{2}}\cdot{n\over 2^{n-1}}$$ and $$\label{eq3}
\prod_{j=1}^{n-1}\cos\left({a j\pi \over n}\right)=
\begin{cases}
(-1)^{\frac{a(n-1)}{2}}\cdot{1\over 2^{n-1}} & \mbox{if $n$ is odd};\\
0 & \mbox{otherwise}.
\end{cases}$$
Let $\omega=e^{\pi aI/n}$, where $I=\sqrt{-1}$. Then, since $\{\omega^{-2j}:j=0,\dots,n-1\}$ are all the $n$-th roots of unity, it follows that $$\prod_{j=0}^{n-1}(z-\omega^{-2j})=z^n-1.$$ Hence, $$\begin{aligned}
\prod_{j=0}^{n-1}\sin\left(x+{a j\pi \over n}\right)&=
\prod_{j=0}^{n-1}\frac{e^{Ix}\omega^{j}-e^{-Ix}\omega^{-j}}{2I}\\
&=\frac{e^{-nIx}\omega^{n(n-1)/2}}{(2I)^n}\prod_{j=0}^{n-1}(e^{2Ix}-\omega^{-2j})\\
&=\frac{I^{(a-1)(n-1)}}{2^{n-1}}\cdot\frac{e^{nIx}-e^{-nIx}}{2I}\\
&=\frac{(-1)^{\frac{(a-1)(n-1)}{2}}}{2^{n-1}}\cdot\sin(nx).\end{aligned}$$ Moreover, $$\prod_{j=1}^{n-1}\sin\left({a j\pi \over n}\right)
={(-1)^{\frac{(a-1)(n-1)}{2}}\over 2^{n-1}}\lim_{x\to
0}\frac{\sin(nx)}{\sin x} =(-1)^{\frac{(a-1)(n-1)}{2}}\cdot{n\over
2^{n-1}}.$$ Finally, $$\prod_{j=1}^{n-1}\cos\left({a j\pi \over n}\right)
=(-1)^{n-1}\prod_{j=1}^{n-1}\sin\left(-{\pi\over 2}+{a j\pi \over
n}\right)
=(-1)^{n-1}\sin(n\pi/2)\cdot{(-1)^{\frac{(a-1)(n-1)}{2}}\over
2^{n-1}},$$ which easily yields the required formula.
Now, we are ready to prove Theorem \[thm:detcar\].
It is known (see, e.g., [@ROS p. 588]) that the eigenvalues of $A(P_{m-1})$ and $A(C_{n})$ are, respectively, $$\left\{2\cos\left(\frac{i\pi}{m}\right)\;:\; 1\leq i\leq m-1\right\}\quad\mbox{and}\quad
\left\{2\cos\left(\frac{2j\pi}{n}\right)\;:\; 1\leq j\leq n\right\}.$$
The proof is done by a direct combination of Theorem \[thm:eigcar\] and Lemma \[lem:triiden\].
We start with . Using the identity $\cos(a+b)+\cos(a-b)=2\cos(a)\cos(b)$, $$\begin{aligned}
\det A(P_{m-1}\; \Box \; P_{n-1})&=\prod_{i=1}^{m-1}\prod_{j=1}^{n-1}\Big(2\cos\left(\frac{i\pi}{m}\right)+2\cos\left(\frac{j\pi}{n}\right)\Big)\\
&=2^{(m-1)(n-1)}\prod_{i=1}^{m-1}\prod_{j=1}^{n-1}2\cos\Big(\frac{i\pi}{2m}+\frac{j\pi}{2n}\Big)\cos\Big(\frac{i\pi}{2m}-\frac{j\pi}{2n}\Big)\\
&=2^{(m-1)(n-1)}\prod_{i=1}^{m-1}\prod_{j=1}^{n-1}2\cos\Big(\frac{i\pi}{2m}+\frac{j\pi}{2n}\Big)\cos\Big(\frac{i\pi}{2m}-\frac{(n-j)\pi}{2n}\Big)\\
&=2^{(m-1)(n-1)}\prod_{i=1}^{m-1}\prod_{j=1}^{n-1}2\cos\Big(\frac{i\pi}{2m}+\frac{j\pi}{2n}\Big)\sin\Big(\frac{i\pi}{2m}+\frac{j\pi}{2n}\Big)\\
&=2^{(m-1)(n-1)}\prod_{i=1}^{m-1}\prod_{j=1}^{n-1}\sin\Big(\frac{i\pi}{m}+\frac{j\pi}{n}\Big)\\
&=2^{(m-1)(n-1)}\prod_{i=1}^{m-1}
\frac{\sin\left(\frac{ni\pi}{m}\right)}{2^{n-1}\sin\left(\frac{i\pi}{m}\right)}
=\frac{\prod_{i=1}^{m-1}\sin\left(\frac{ni\pi}{m}\right)}{\prod_{i=1}^{m-1}\sin\left(\frac{i\pi}{m}\right)},\end{aligned}$$ where in the last but one step we have used the identity . Clearly, if $\gcd(m,n)\not=1$ then $\prod_{i=1}^{m-1}\sin\left(\frac{ni\pi}{m}\right)=0$, otherwise we use .
Now, we show . In the case that $m$ or $n$ is even the proof is straightforward because one of the eigenvalues of $A(C_{m}\; \Box \; C_{n})$ is zero. Assume that $m$ and $n$ are odd and let $d=\gcd(m,n)$, with $m'=m/d$, $n'=n/d$. $$\begin{aligned}
\det A(C_{m}\; \Box \; C_{n})&=\prod_{i=1}^{m}\prod_{j=1}^{n}\left(2\cos\left(\frac{2i\pi}{m}\right)+2\cos\left(\frac{2j\pi}{n}\right)\right)\\
&=4^{mn}\prod_{i=0}^{m-1}\prod_{j=0}^{n-1}\cos\Big(\frac{i\pi}{m}+\frac{j\pi}{n}\Big)\cos\Big(\frac{i\pi}{m}-\frac{j\pi}{n}\Big)\\
&=4^{mn}\Bigg(\prod_{i=0}^{m-1}\prod_{j=0}^{n-1}
\cos\Big(\frac{i\pi}{m}+\frac{j\pi}{n}\Big)\cos\Big(\frac{i\pi}{m}-\frac{(n-j)\pi}{n}\Big)\Bigg)\\
&=4^{mn}\Bigg(\prod_{i=0}^{m-1}\prod_{j=0}^{n-1}
\cos\Big(\frac{i\pi}{m}+\frac{j\pi}{n}\Big)\Bigg)^2\\
&=4^{mn}\Bigg(\prod_{i=0}^{m-1}\prod_{j=0}^{n-1}
\sin\Big(-\frac{\pi}{2}+\frac{i\pi}{m}+\frac{j\pi}{n}\Big)\Bigg)^2\\
&=4^{m}\Bigg(\prod_{i=0}^{m-1}
\sin\left(n\left(-\frac{\pi}{2}+\frac{i\pi}{m}\right)\right)\Bigg)^2\\
&=4^{m}\Bigg(\prod_{i=0}^{m-1} \cos\left(\frac{ni\pi}{m}\right)\Bigg)^2
=4^{m}\Bigg(\prod_{i=1}^{m' d} \cos\left(\frac{n'i\pi}{m'}\right)\Bigg)^2=4^{m}\Big(\frac{1}{4^{m'-1}}\Big)^d=4^d,\end{aligned}$$ where in the last but one step we have used .
Finally, we prove . $$\begin{aligned}
\det A(P_{m-1}\; \Box \; C_{n})&=\prod_{i=1}^{m-1}\prod_{j=1}^{n}\left(2\cos\left(\frac{i\pi}{m}\right)+2\cos\left(\frac{2j\pi}{n}\right)\right)\\
&=4^{(m-1)n}\prod_{i=1}^{m-1}\prod_{j=0}^{n-1}
\cos\Big(\frac{i\pi}{2m}+\frac{j\pi}{n}\Big)\cos\Big(\frac{i\pi}{2m}-\frac{j\pi}{n}\Big)\\
&=4^{(m-1)n}\prod_{i=1}^{m-1}\prod_{j=0}^{n-1}
\cos\Big(\frac{(m-i)\pi}{2m}+\frac{j\pi}{n}\Big)\cos\Big(\frac{(m-i)\pi}{2m}-\frac{j\pi}{n}\Big)\\
&=(-4)^{(m-1)n}\prod_{i=1}^{m-1}\prod_{j=0}^{n-1}
\sin\Big(-\frac{i\pi}{2m}+\frac{j\pi}{n}\Big)\sin\Big(\frac{i\pi}{2m}+\frac{j\pi}{n}\Big)\\
&=(-4)^{(m-1)n}\prod_{i=1}^{m-1}\frac{1}{4^{n-1}}
\sin\Big(-\frac{ni\pi}{2m}\Big)\sin\Big(\frac{ni\pi}{2m}\Big)\\
&=(-1)^{(m-1)(n-1)}4^{m-1}\Bigg(\prod_{i=1}^{m-1}
\sin\Big(\frac{ni\pi}{2m}\Big)\Bigg)^2.\end{aligned}$$ If $n$ is even and $\gcd(m,n')=1$ where $n'=n/2$ then, by , $$(-1)^{(m-1)(n-1)}4^{m-1}\Bigg(\prod_{i=1}^{m-1}
\sin\Big(\frac{ni\pi}{2m}\Big)\Bigg)^2=
(-1)^{(m-1)}4^{m-1}\Bigg(\prod_{i=1}^{m-1}\sin\Big(\frac{n'i\pi}{m}\Big)\Bigg)^2=
(-1)^{(m-1)}m^2.$$ If $n$ is odd and $\gcd(m,n)=1$ then, $\gcd(2m,n)=1$ and by , $$\begin{aligned}
(-1)^{(m-1)(n-1)}4^{m-1}\Bigg(\prod_{i=1}^{m-1}
\sin\Big(\frac{ni\pi}{2m}\Big)\Bigg)^2
&=4^{m-1}\Bigg(\prod_{i=1}^{m-1}
\sin\Big(\frac{ni\pi}{2m}\Big)\Bigg)\Bigg(\prod_{i=m+1}^{2m-1}
\sin\Big(\frac{n(2m-i)\pi}{2m}\Big)\Bigg)\\
&=4^{m-1}\sin\Big(\frac{n\pi}{2}\Big)\prod_{i=1}^{2m-1}\sin\Big(\frac{ni\pi}{2m}\Big)=m\end{aligned}$$ It is easy to verify that the remaining cases yield zero.
Now, we prove Theorem \[thm:mob\].
The eigenvalues of $A(M_{2n})$ are (see, e.g., [@BIG p. 21]) $$\left\{(-1)^j + 2\cos\left(\frac{j\pi}{n}\right)\;:\; 1\leq j\leq 2n\right\}.$$ Hence, $$\begin{aligned}
\det A(M_{2n})&=\prod_{j=1}^{2n}\left((-1)^j + 2\cos\left(\frac{j\pi}{n}\right)\right)\\
&=\prod_{j=0}^{2n-1}\left(2\cos\left(\frac{(3j+1)\pi}{3}\right) + 2\cos\left(\frac{j\pi}{n}\right)\right)\\
&=4^{2n}\prod_{j=0}^{2n-1}\left(\cos\left(\frac{(3j+1)\pi}{6}+\frac{j\pi}{2n}\right)\cos\left(\frac{(3j+1)\pi}{6}-\frac{j\pi}{2n}\right) \right)\\
&=4^{2n}\prod_{j=0}^{2n-1}\sin\left(\frac{\pi}{3}-\frac{(n+1)j\pi}{2n}\right)\prod_{j=0}^{2n-1}\sin\left(\frac{\pi}{3}-\frac{(n-1)j\pi}{2n}\right).\end{aligned}$$ If $n$ is even then $\gcd(n+1,2n)=1$, and then by , $$\det A(M_{2n})=-4\sin^2\left(\frac{2n\pi}{3}\right)=\begin{cases}
-3 & \text{if $n\equiv \pm 2 \pmod{6}$;}\\
0 & \text{if $n\equiv 0 \pmod{6}$.}
\end{cases}$$ If $n$ is odd then $\gcd(n'+1,n)=1$, where $n'=(n-1)/2$, and then by , $$\begin{aligned}
\det A(M_{2n})&=
-4^{2n}\left(\prod_{j=0}^{n-1}\sin\left(\frac{\pi}{3}-\frac{(n'+1)j\pi}{n}\right)\right)^4\\
&=-16\sin^4\left(\frac{n\pi}{3}\right)=\begin{cases}
-9 & \text{if $n\equiv \pm 1 \pmod{6}$;}\\
0 & \text{if $n\equiv 3 \pmod{6}$.}
\end{cases}\end{aligned}$$
[99]{}
K. Bibak and M. H. Shirdareh Haghighi, [*The number of spanning trees in some classes of graphs*]{}, Rocky Mountain J. Math. [**42**]{}(4) (2012), 1183–1195.
A. Bieñ, [*The problem of singularity for planar grids*]{}, Discrete Math. [**311**]{} (2011), 921–931.
N. Biggs, [*Algebraic graph theory*]{}, 2ed. ed., Cambridge University Press, (1994).
L. Collatz and U. Sinogowitz, Spektren endlicher Grafen, [*Abh. Math. Sem. Univ. Hamburg*]{} 21 (1957), 63–77.
S. M. Fallat and L. Hogben, The minimum rank of symmetric matrices described by a graph: A survey, [*Linear Algebra Appl.*]{} 426 (2007), 558–582.
I. Gutman and B. Borovićanin, Nullity of graphs: An updated survey, in [*Selected topics on applications of graph spectra*]{}, Math. Inst., Belgrade, (2011), pp. 137–154.
L.-H. Hsu and C.-K. Lin, [*Graph theory and interconnection networks*]{}, CRC Press, (2008).
D. Pragel, [*Determinants of box products of paths*]{}, Discrete Math. [**312**]{} (2012), 1844–1847.
K. H. Rosen, [*Handbook of discrete and combinatorial mathematics*]{}, CRC Press, (1999).
|
---
date: 'July 13, 2005'
---
amssym.def amssym.tex
\[subsection\]
\[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Claim]{} \[thm\][Conjecture]{} \[thm\][Definition]{} \[thm\][Proposition]{} \[thm\][Condition]{}
\[thm\][Remark]{} \[thm\][Example]{} \[thm\][Definition]{}
This is the first in a series of articles devoted to deformation quantization of gerbes. We introduce basic definitions, interpret deformations of a given stack as Maurer-Cartan elements of a differential graded Lie algebra (DGLA), and classify deformations of a given gerbe in terms of Maurer-Cartan elements of the DGLA of Hochschild cochains twisted by the cohomology class of the gerbe. We also classify all deformations of a given gerbe on a symplectic manifold, as well as provide a deformation-theoretic interpretation of the first Rozansky-Witten class.
Introduction {#s:introduction}
============
The notion of deformation quantization, as well as the term, was first introduced in [@BFFLS]. Both became standard since then. A deformation quantization of a manifold $M$ is a multiplication law on the ring of functions on $M$ which depends on a formal parameter $\hbar$. This multiplication law is supposed to satisfiy certain properties, in particular its value at $\hbar=0$ must be equal to the usual multiplication. A deformation quantization defines a Poisson structure on $M$; therefore it is natural to talk about deformation quantization of Poisson manifolds. In the case when $M$ is a symplectic manifold, deformation quantizations of $C^{\infty}(M)$ were classified up to isomorphism in [@DWL], [@Fe], [@D]. In the case of a complex manifold $M$ with a holomorphic symplectic form, deformation quantizations of the sheaf of algebras ${{\cal O}}_M$ are rather difficult to study. They were classified, under additional cohomological assumptions, in [@NT] (Theorem \[thm:classification of sheaves\] of the present paper; cf. also [@BK] for the algebraic case). If one moves away from symplectic to general Poisson manifolds, the problem becomes much more complicated. All deformation quantizations of ${{\cal O}}_M$ were classified by Kontsevich in [@K1]. For the algebraic case, cf. [@Y].
In this paper we start a program of studying deformation quantization of stacks and gerbes. Stacks are a natural generalization of sheaves of algebras. They appear in geometry, microlocal analysis and mathematical physics, cf. [@Gi], [@Br], [@DP], [@Ka], [@MMS], [@MR1], [@MR2], [@PS], and other works.
The main results of this paper are as follows.
1\) We prove that deformations of every stack (in the generality adopted by us here) are classified by Maurer-Cartan elements of a differential graded Lie algebra, or DGLA (Theorems \[thm:deformations of stacks via dgla\], \[thm:deformations of stacks via dgla II\]). This generalizes the results of Gerstenhaber [@Ge] for associative algebras and of Hinich [@H] for sheaves of associative algebras.
2\) We show that the DGLA controlling deformations of a gerbe on a manifold is equivalent to the Hochschild cochain complex of this manifold, twisted by the cohomology class of the gerbe (Theorem \[thm:classification of deformations of a gerbe\]).
3\) We classify deformation quantizations of all gerbes on a symplectic manifold (Theorems \[thm:classification of deformations of the trivial gerbe, symplectic case\] and \[thm:symplectic classification\]). This generalizes the classification results for deformation quantizations of $C^{\infty}$ symplectic manifolds [@DWL], [@D], [@Fe], [@Fe1].
4\) We show that the first Rozansky-Witten class of a holomorphic symplectic manifold is an obstruction for a canonical stack deformation quantization to be a sheaf of algebras (Theorem \[thm:RW\]).
We start by defining stacks, gerbes and their deformations in the generality suited for our purposes (section \[s:stacks and cocycles\]). We then recall (in subsections \[ss:definitions of deformations\], \[ss:Descent data for Deligne two-groupoids\] ) the language of differential graded Lie algebras (DGLAs) in deformation theory, along the lines of [@GM], [@Ge], [@S], [@SS], [@Dr], [@HS]. Then we pass to a generality that suits us better, namely to the case of cosimplicial DGLAs (subsection \[ss:cosimplicial DGLAs\]). We define descent data for the Deligne two-groupoid (cf. [@G], [@G1] and references thereof) of a cosimplicial DGLA and prove that the set of isomorphism classes of such data does not change if one passes to a quasi-isomorphic cosimplicial DGLA (Proposition \[prop:quis of cosimplicial dglas\]). Next, we recall the construction of totalization of a cosimplicial DGLA (subsection \[ss:Totalization of cosimplicial DGLAs\]). We prove that isomorphism classes of descent data of a cosimplicial DGLA are in one-to-one correspondence with isomorphism classes of Maurer-Cartan elements of its totalization.
After that, given a gerbe on a Poisson manifold, we define its deformation quantization. We first classify deformations of the trivial gerbe, i.e. deformations of the structure sheaf as a stack, on a symplectic manifold $M$, $C^{\infty}$ or complex (Theorem \[thm:classification of deformations of the trivial gerbe, symplectic case\]; this result is very close to the main theorem of [@P]). More precisely, we first reduce the classification problem to classifying certain [*$Q$-algebras*]{}, using the term of A. Schwarz (or [*curved DGAs*]{}, as they are called in [@Bl]). (Similar objects were studied in several contexts, in particular in [@C]). The link between these objects and gerbes was rather well understood for some time; for example, it is through such objects that gerbes appear in [@Kapu]). We also give a new proof of the classification theorem for deformations of the sheaf of algebras of functions (Theorem \[thm:classification of sheaves\]). Then we show how the first Rozansky-Witten class [@RW], [@Kap], [@K2]) can be interpreted as an obstruction for a certain canonical deformation of the trivial gerbe to be a sheaf, not just a stack. This canonical stack is very closely related to stacks of microdifferential operators defined in [@Ka] and [@PS].
Next, we show how to interpret deformations of any gerbe in the language of DGLAs (Theorems \[thm:classification of deformations of a gerbe\] and \[thm:classification of deformations of a holomorphic gerbe\]). The proof is based on a DGLA interpretation of the deformation theory of any stack (within our generality); such an interpretation is provided by Theorem \[thm:deformations of stacks via dgla\]. We show there that deformations of a stack are classified by the DGLA of De Rham-Sullivan forms with coefficients in [*local Hochschild cochains of the twisted matrix algebra*]{} associated to this stack.
Note that De Rham-Sullivan forms were used in [@Y] to classify deformation quantizations of algebraic varieties.
(The DGLA above is actually a DGLA of Hochschild cochains of a special kind of an associative DGA; the cyclic homology of this DGA is the natural recipient of the Chern character of a twisted module over a stack. We will study this in the sequel).
Afterwards we prove a classification theorem for deformation quantizations of any gerbe on a symplectic manifold (Theorems \[thm:symplectic classification\] and \[thm:symplectic holomorphic classification\]). This can be viewed as an adaptation of Fedosov’s methods [@Fe], [@Fe1] to the case of gerbes. Note that some ideas about deformation quantization of gerbes appeared already in Fedosov’s work; cf. also [@K], as well as [@Ka] and [@PS].
This paper was motivated by the index theory, in particular by index theorems for Fourier integral operators or by index theorems such as in [@MMS]. Among the applications other than index theory, we would like to mention dualities between gerbes and noncommutative spaces, as in [@Kapu], [@Bl], [@BBP], [@MR1], [@MR2]. The deformation-theoretical role of the first Rozansky-Witten class is also quite intriguing and worthy of further study.
The research of A. G. and B. T. was partially supported by NSF grants.
Stacks and cocycles {#s:stacks and cocycles}
===================
Let $M$ be a smooth manifold ($C^{\infty}$ or complex). In this paper, by a stack on $M$ we will mean the following data:
1\) an open cover $M=\cup U_i$;
2\) a sheaf of rings ${{\cal{A}}}_i$ on every $U_i$;
3\) an isomorphism of sheaves of rings $G_{ij}: {{\cal{A}}}_j|(U_i \cap U_j) {\overset{\sim}{\to}}{{\cal{A}}}_i |(U_i \cap U_j)$ for every $i,\,j$;
4\) an invertible element $c_{ijk} \in {{\cal{A}}}_i (U_i \cap U_j \cap U_k)$ for every $i,\,j,\,k$ satisfying $$\label {eq:2-cocycle 1}
G_{ij}G_{jk}={\operatorname{ Ad}}(c_{ijk})G_{ik}$$ such that, for every $i,\,j,\,k,\,l$, $$\label {eq:2-cocycle 2}
c_{ijk}c_{ikl}=G_{ij}(c_{jkl})c_{ijl}$$
If two such data $(U'_i,\; {{\cal{A}}}' _i, \; G'_{ij},\; c'_{ijk})$ and $(U''_i,\; {{\cal{A}}}'' _i, \; G''_{ij},\; c''_{ijk})$ are given on $M$, an isomorphism between them is an open cover $M=\cup U_i$ refining both $\{U'_i\}$ and $\{U''_i\}$ together with isomorphisms $H_i: {{\cal{A}}}' _i {\overset{\sim}{\to}}{{\cal{A}}}''_i$ on $U_i$ and invertible elements $b_{ij}$ of ${{\cal{A}}}' _i (U_i \cap U_j)$ such that $$\label{eq:equivalence of stacks 1}
G''_{ij}=H_i {\operatorname{Ad}}(b_{ij})G'_{ij}H_j ^{-1}$$ and $$\label{eq:equivalence of stacks 2}
H_i^{-1}(c''_{ijk})=b_{ij}G'_{ij}(b_{jk})c'_{ijk}b_{ik}^{-1}$$
[*A gerbe*]{} is a stack for which ${{\cal{A}}}_i = {\cal O}_{U_i}$ and $G_{ij}={\operatorname {id}}$. In this case $c_{ijk}$ form a two-cocycle in $Z^2(M,{\cal O}_M^*)$.
Categorical interpretation {#ss:categorical}
--------------------------
Here we remind the well-known categorical interpretation of the notions introduced above. Though not used in the rest of the paper, this interpretation provides a very strong motivation for what follows.
A stack defined as above gives rise to the following categorical data:
1\) A sheaf of categories ${{\cal C}}_i$ on $U_i$ for every $i$;
2\) an invertible functor $G_{ij}: {{\cal C}}_j |(U_i \cap U_j) {\overset{\sim}{\to}}{{\cal C}}_i |(U_i \cap U_j)$ for every $i,\,j$;
3\) an invertible natural transformation $$c_{ijk}: G_{ij}G_{jk}|(U_i \cap U_j \cap U_k) {\overset{\sim}{\to}}G_{ik}|(U_i \cap U_j \cap U_k)$$ such that, for any $i,\,j,\,k,\,l$, the two natural transformations from $G_{ij}G_{jk}G_{kl}$ to $G_{il}$ that one can obtain from the $c_{ijk}$’s are the same on $U_i \cap U_j \cap U_k \cap U_l$.
If two such categorical data $(U'_i,\; {{\cal C}}' _i, \; G'_{ij},\; c'_{ijk})$ and $(U''_i,\; {{\cal C}}'' _i, \; G''_{ij},\; c''_{ijk})$ are given on $M$, an isomorphism between them is an open cover $M=\cup U_i$ refining both $\{U'_i\}$ and $\{U''_i\}$, together with invertible functors $H_i: {{\cal C}}' _i {\overset{\sim}{\to}}{{\cal C}}''_i$ on $U_i$ and invertible natural transformations $b_{ij}:H_i G'_{ij}|(U_i \cap U_j) {\overset{\sim}{\to}}G''_{ij}H_j|(U_i \cap U_j)$ such that, on any $U_i \cap U_j \cap U_k$, the two natural transformations $H_i G'_{ij}G'_{jk} {\overset{\sim}{\to}}G''_{ij}G''_{jk}H_k$ that can be obtained using $H_i$’s, $ b_{ij}$’s, and $c_{ijk}$’s are the same. More precisely: $$\label{eq:equivalence of categorical data for stacks 1}
((c''_{ijk})^{-1}H_k)(b_{ik})(H_i c'_{ijk})=(G''_{ij}b_{jk})(b_{ij} G'_{jk})$$
The above categorical data are defined from $({{\cal{A}}}_i, G_{ij}, c_{ijk})$ as follows:
1\) ${{\cal C}}_i$ is the sheaf of categories of ${{\cal{A}}}_i$-modules;
2\) given an ${{\cal{A}}}_i$-module ${\cal M}$, the ${{\cal{A}}}_j$-module $G_{ij}({\cal M})$ is the sheaf ${\cal M}$ on which $a \in {{\cal{A}}}_i$ acts via $G_{ij}^{-1}(a)$;
3\) the natural transformation $c_{ijk}$ between $G_{ij}G_{jk}({\cal M})$ and $G_{jk}({\cal M})$ is given by multiplication by $G_{ik}^{-1}(c_{ijk}^{-1})$.
From the categorical data defined above, one defines a sheaf of categories on $M$ as follows. For an open $V$ in $M$, an object of ${{\cal C}}(V)$ is a collection of objects $X_i$ of ${{\cal C}}_i (U_i \cap V)$, together with isomorphisms $g_{ij}: G_{ij}(X_j) {\overset{\sim}{\to}}X_i$ on every $U_i \cap U_j \cap V$, such that $$g_{ij}G_{ij}(g_{jk})=g_{ik}c_{ijk}$$ on every $U_i \cap U_j \cap U_k \cap V$. A morphism between objects $(X'_i, g'_{ij})$ and $(X''_i, g''_{ij})$ is a collection of morphisms $f_i: X'_i \to X''_i$ (defined for some common refinement of the covers), such that $f_i g'_{ij}=g''_{ij}G_{ij}(f_j)$.
\[rmk:terminology of stacks\] What we call stacks are what is referred to in [@DP] as descent data for a special kind of stacks of twisted modules (cf. Remark 1.9 in [@DP]). Both gerbes and their deformations are stacks of this special kind. We hope that our terminology, which blurs the distinction between stacks and their descent data, will not cause any confusion.
Deformations of stacks
----------------------
\[dfn:deformation1\] Let $k$ be a field of characteristic zero. Let ${\mathfrak{a}}$ be a local Artinian k-algebra with the maximal ideal ${\mathfrak{m}}$. A deformation of a stack ${{\cal A}}^{(0)}$ over ${\mathfrak{a}}$ is a stack ${{\cal A}}$ where all ${{\cal A}}_i$ are sheaves of ${\mathfrak{a}}$-algebras, free as ${\mathfrak{a}}$-modules, $G_{ij}$ are isomorphisms of algebras over ${\mathfrak{a}}$, and the induced stack ${{\cal A}}/{\mathfrak{m}}{{\cal A}}$ is equal to ${{\cal A}}^{(0)}$. An isomorphism of two deformations is an isomorphism of stacks which is identity modulo ${\mathfrak{m}}$ and such that $H_i$ are isomorphisms of algebras over ${\mathfrak{a}}.$
Consider the filtration of ${\mathfrak{a}}$ by powers of ${\mathfrak{m}}.$ Choose a splitting of the filtered $k$-vector space $${\mathfrak{a}}=\oplus_{m=0}^N {\mathfrak{m}}_m$$ where ${\mathfrak{m}}_m={\mathfrak{m}}^m/{\mathfrak{m}}^{m+1}.$
Given a deformation, we can identify ${{\cal A}}_i={{\cal A}}^{(0)}_i\otimes {\mathfrak{a}};$ the multiplication on ${{\cal A}}_i$ is determined by $$f*_i g = fg + \sum_{m=1}^N P_i^{(m)}(f,g)$$ with $P_i^{(m)}: {{{\cal A}}^{(0)}_i}^{\otimes 2}\to {{\cal A}}^{(0)}_i\otimes {\mathfrak{m}}_m.$ Similarly, $G_{ij}$ is determined by $$G_{ij}(f) = f + \sum_{m=1}^{N} T_{ij}^{(m)}(f)$$ with $T_{ij}^{(m)}:{{\cal A}}^{(0)}\to {{\cal A}}^{(0)}_i\otimes {\mathfrak{m}}_m,$ and $$c_{ijk} = \sum _{m=0}^{N} c^{(m)}_{ijk}$$ with $c^{(m)}_{ijk}\in {\mathfrak{m}}_m.$ For an isomorphism of two stacks, $H_i$ is determined by $$H_{i}(f) = f + \sum_{m=1}^{N} R_{i}^{(m)}(f)$$ with $H_{i}^{(m)}:{{\cal A}}^{(0)}\to {{\cal A}}^{(0)}_i\otimes {\mathfrak{m}}_m;$ $$b_{ij} = \sum _{m=0}^{N} b^{(m)}_{ij}$$ with $b^{(m)}_{ij}\in {\mathfrak{m}}_m.$
\[dfn:deformation\] Consider a gerbe ${{\cal A}}^{(0)}$ given by a two-cocycle $c^{(0)}_{ijk}$. A deformation of ${{\cal A}}^{(0)}$ is by definition its deformation as a stack, such that $P_i^{(m)}(f,g)$ are (holomorphic) bidifferential expressions and $T_{ij}^{(m)}$are (holomorphic) differential expressions.
An isomorphism between two deformations is an isomorphism $(H_i, b_{ij})$ where $R_i^{(m)}$ are (holomorphic) differential expressions.
Differential graded Lie algebras and deformations {#s:Differential graded Lie algebras and deformations}
=================================================
{#ss:definitions of deformations}
Here we give some definitions that lie at the foundation of the deformation theory program along the lines of [@Ge], [@GM], [@S], [@SS], [@Dr], [@HS], as well as of the notions such as Deligne two-groupoid (cf. [@G], [@G1] and references thereof). Let $${{\cal L}}= \bigoplus_{m\geq -1} {{\cal L}}^m$$ be a differential graded Lie algebra (DGLA). Let ${\mathfrak{a}}$ be a local Artinian k-algebra with the maximal ideal ${\mathfrak{m}}$. We call [*a Maurer-Cartan element*]{} an element ${\lambda}$ of ${{\cal L}}^1 \otimes {\mathfrak{m}}$ satisfying $$\label{eq:MC}
d\lambda + \frac{1}{2}[\lambda,\lambda]=0$$ [*A gauge equivalence*]{} between two Maurer-Cartan elements $\lambda $ and $\mu$ is an element $G={\operatorname {exp}}\,X$ where $X\in {{\cal L}}^0 \otimes {\mathfrak{m}}$ such that $$\label{eq:MC equivalence}
d+\mu={\operatorname {exp}}\,{\operatorname {ad}}X \,(d+\lambda)$$ The latter equality takes place in the cross product of the one-dimensional graded Lie algebra $kd$ concentrated in dimension one and ${{\cal L}}^0 \otimes {\mathfrak{m}}$. Given two gauge transformations $G={\operatorname{exp}}\,X,\,H={\operatorname{exp}}\, Y$ between $\lambda$ and $\mu$, [*a two-morphism*]{} from $H$ to $G$ is an element $c= {\operatorname{exp}}\,t$ of ${{\cal L}}^{-1} \otimes{\mathfrak{m}} $ such that $$\label{eq:MC 2-morphism}
{\operatorname {exp}}(X)={\operatorname {exp}}(dt+[\mu,t]) {\operatorname {exp}}Y$$ in the unipotent group ${\operatorname {exp}} ({{\cal L}}^0\otimes {\mathfrak{m}})$. The composition of gauge transformations $G$ and $H$ is the product $GH$ in the unipotent group ${\operatorname {exp}} ({{\cal L}}^0)\otimes {\mathfrak{m}}$. The composition of two-morphisms $c_1$ and $c_2$ is the product $c_1c_2$ in the prounipotent group ${\operatorname {exp}}({{\cal L}}^{-1}\otimes {\mathfrak{m}})$. Here ${{\cal L}}^{-1}\otimes {\mathfrak{m}}$ is viewed as a Lie algebra with the bracket $$\label{eq:mu-bracket}
[a,\,b]_{\mu}=[a,\,\delta b+[\mu, \,b]]$$ We denote the above pronilpotent Lie algebra by $({{\cal L}}^{-1}\otimes {\mathfrak{m}})_{\mu}.$ The above definitions, together with the composition, provide the definition of [*the Deligne two-groupoid*]{} of ${{\cal L}}\otimes {\mathfrak{m}}$ (cf. [@G1])..
\[rmk:ezra\] Recently Getzler gave a definition of a Deligne $n$-groupoid of a DGLA concentrated in degrees above $-n$, cf. [@G].
Descent data for Deligne two-groupoids {#ss:Descent data for Deligne two-groupoids}
--------------------------------------
Let ${{\cal L}}$ be a sheaf of DGLAs on $M$. A descent datum of the Deligne two-groupoid of ${{\cal L}}\otimes {\mathfrak{m}}$ are the following:
1\) A Maurer-Cartan element $\lambda _i \in {{\cal L}}^{1} \otimes {\mathfrak{m}}$ on $U_i$ for every $i$;
2\) a gauge transformation $G_{ij}: \lambda_j |(U_i \cap U_j) {\overset{\sim}{\to}}\lambda_i |(U_i \cap U_j)$ for every $i,\,j$;
3\) a two-morphism $$c_{ijk}: G_{ij}G_{jk}|(U_i \cap U_j \cap U_k) {\overset{\sim}{\to}}G_{ik}|(U_i \cap U_j \cap U_k)$$ such that, for any $i,\,j,\,k,\,l$, the two two-morphisms from $G_{ij}G_{jk}G_{kl}$ to $G_{il}$ that one can obtain from the $c_{ijk}$’s are the same on $U_i \cap U_j \cap U_k \cap U_l$.
If two such data $(U'_i,\; \lambda' _i, \; G'_{ij},\; c'_{ijk})$ and $(U''_i,\; \lambda'' _i, \; G''_{ij},\; c''_{ijk})$ are given on $M$, an isomorphism between them is an open cover $M=\cup U_i$ refining both $\{U'_i\}$ and $\{U''_i\}$, together with gauge transformations $H_i: \lambda' _i {\overset{\sim}{\to}}\lambda''_i$ on $U_i$ and two-morphisms $b_{ij}:H_i G'_{ij}|(U_i \cap U_j) {\overset{\sim}{\to}}G''_{ij}H_j|(U_i \cap U_j)$ such that, on any $U_i \cap U_j \cap U_k$, the two two-morphisms $H_i G'_{ij}G'_{jk} {\overset{\sim}{\to}}G''_{ij}G''_{jk}H_k$ that can be obtained using $H_i$’s, $ b_{ij}$’s, and $c_{ijk}$’s are the same.
Finally, given two isomorphisms $(H'_i, b'_{ij})$ and $(H''_i, b''_{ij})$ between the two data $(U_i,\; \lambda' _i, \; G'_{ij},\; c'_{ijk})$ and $(U_i,\; \lambda'' _i, \; G''_{ij},\; c''_{ijk})$, define a [*two-isomorphism*]{} between them to be a collection of two-morphisms $a_i: H'_i \to H''_i$ such that $$b''_{ij}\circ (a_i \circ G'_{ij})=(G''_{ij}\circ a_i)\circ b'_{ij}$$ as two-morphisms from $H'_i\circ G'_{ij}\to G''_{ij}\circ H''_{j}$.
Cosimplicial DGLAs and descent data {#ss:cosimplicial DGLAs}
-----------------------------------
The notion of a descent datum above, as well as an analogous notion for simplicial sheaves of DGLAs that we use below, is a partial case of a more general situation that we are about to discuss. Recall that [*a cosimplicial object*]{} of a category ${{\cal C}}$ is a functor $X: \Delta \to {{\cal C}}$ where $\Delta$ is the category whose objects are sets $[n]=\{0,\,\ldots , \, n\}$ with the standard linear ordering ($n\geq 0$), and morphisms are nondecreasing maps. We denote $X([n])$ by $X^n.$ For $0\leq i\leq n,$ let $d_i:[n]\to [n+1]$ be the only injective map such that $i$ is not in the image, and $s_i: [n+1]\to [n]$ the only surjection for which every element of $[n-1]$ except $i$ has exactly one preimage. For a cosimplicial Abelian group ${{\cal A}}$, one defines the standard differential $$\partial=\sum_{i=0}^n(-1)^id_i:{{\cal A}}^n\to {{\cal A}}^{n+1}.$$
For a cosimplicial set $X,$ let $x\in X^k.$ Let $n\geq k$ and $0\leq i_0<\ldots <i_k\leq n.$ By $x_{i_0\ldots i_k}$ we denote the object of $X^n$ which is the image of $x$ under the map in $\Delta$ which embeds $[k]$ into $[n]$ as the subset $\{i_0,\,\ldots,\, i_k\}.$
Let ${{\cal L}}$ be a cosimplicial DGLA. We will denote by ${{\cal L}}^{n,p}$ the component of degree $p$ of the DGLA ${{\cal L}}^n$, $n\geq 0.$
Let ${\mathfrak{a}}$ be a local Artinian algebra over $k$ with the maxiamal ideal ${\mathfrak{m}}$. Consider a cosimplicial DGLA ${{\cal L}}$ such that ${{\cal L}}^{n,p}=0$ for $p<-1.$ [*A descent datum for the Deligne two-groupoid of ${{\cal L}}\otimes {\mathfrak{m}}$*]{} is the following:
1\) A Maurer-Cartan element $\lambda \in {{\cal L}}^{0,1} \otimes {\mathfrak{m}};$
2\) a gauge transformation $G: \lambda_1 {\overset{\sim}{\to}}\lambda_0$ in ${\operatorname{exp}}({{\cal L}}^{1,0});$
3\) a two-morphism $$c: G_{01}G_{12} {\overset{\sim}{\to}}G_{02}$$ in ${\operatorname{exp}}({{\cal L}}^{2,-1}_{\lambda_0})$ such that, for any $i,\,j,\,k,\,l$, the two two-morphisms from $G_{01}G_{12}G_{23}$ to $G_{03}$ that one can obtain from the $c_{ijk}$’s are the same.
An isomorphism between two data $(\lambda' , \; G',\; c')$ and $( \lambda'' , \; G'',\; c'')$ is a pair of a gauge transformation $H: \lambda' {\overset{\sim}{\to}}\lambda''$ and a two-morphism $b_{01}:H_0 G'_{01} {\overset{\sim}{\to}}G''_{01}H_1$ such that the two two-morphisms $H_0 G'_{01}G'_{12} {\overset{\sim}{\to}}G''_{01}G''_{12}H_2$ that can be obtained using $H_i$’s, $ b_{ij}$’s, and $c$ are the same.
For two isomorphisms $(H', b')$ and $(H'', b'')$ between the two data $ \lambda' , \; G',\; c')$ and $(U,\; \lambda'' , \; G'',\; c'')$, define a [*two-isomorphism*]{} between them to be a collection of two-morphisms $a: H' \to H''$ such that $$b''_{01}\circ (a_0 \circ G'_{01})=(G''_{01}\circ a_0)\circ b'_{01}$$ as two-morphisms from $H'_0\circ G'_{01}\to G''_{01}\circ H''_{1}$.
\[prop:quis of cosimplicial dglas\] a). A morphism $f:{{\cal L}}_1 \to {{\cal L}}_2$ of cosimplicial DGLAs induces a map from the set of isomorphism classes of descent data of the Deligne two-groupoid of ${{\cal L}}_1\otimes {\mathfrak{m}}$ to the set of isomorphism classes of descent data of the Deligne two-groupoid of ${{\cal L}}_2\otimes {\mathfrak{m}}$.
b). Assume that $f$ induces a quasi-isomorphism of total complexes of the double complexes ${{\cal L}}_1^{n,p} \to {{\cal L}}_2^{n,p} $. Then the map defined in a) is a bijection.
c). Under the assumptions of b), let ${{\cal A}}$ be a descent datum of the Deligne two-groupoid of ${{\cal L}}_1\otimes {\mathfrak{m}}$, and let $f({{\cal A}})$ be its image under the map from a). The morphism $f$ induces a bijection $$\frac{{\operatorname{Iso}}({{\cal A}}, {{\cal A}}')}{2-{\operatorname{Iso}}}{\overset{\sim}{\to}}\frac{{\operatorname{Iso}}(f({{\cal A}}), f({{\cal A}}'))}{2-{\operatorname{Iso}}}.$$
d). For two isomorphisms $\phi, \psi:{{\cal A}}\to {{\cal A}}'$, denote their images under the above bijection by $f(\phi), f(\psi).$ Then $f$ induces a bijection $$2-{\operatorname{Iso}}(\phi, \psi){\overset{\sim}{\to}}2-{\operatorname{Iso}}(f(\phi), f(\psi))$$
In other words, $f$ induces [*an equivalence of two-groupoids of descent data*]{}, compare to [@G], [@G1].
[**Proof.**]{} What follows is essentially a standard deformation theoretical proof. We start by establishing a rigorous expression of the following intuitive statement. First, a descent datum $(\lambda, G, c)$ is a non-Abelian version of a two-cocycle of the double complex ${{\cal L}}^{\bullet, \bullet}, \partial+d;$ second, if one takes an arbitrary datum $(\lambda, G, c)$ and measures its deviation from being a descent datum, the result will be a non-Abelian version of a three-cocycle. This is, in essence, what enables us to study deformations of descenta data by homological methods.
### {#sss:a}
Consider a triple $(\lambda, G, t)$ with $\lambda \in {{\cal L}}^{0,1}\otimes {\mathfrak m},$ $G \in {\operatorname{exp}}({{\cal L}}^{1,0}\otimes {\mathfrak m}),$ and $t \in {{\cal L}}^{2,-1}\otimes {\mathfrak m}.$ Define the operation $a\cdot _{\lambda} b$ on ${{\cal L}}^{-1}\otimes {\mathfrak m}$ to be the Campbell-Dynkin-Hausdorff series corresponding to the bracket $[a,\,b]_{\lambda}=[a,\,db+[\lambda,b]].$ If $\lambda$ is a Maurer-Cartan element, this is a group multiplication. If not, one can still define the operation which is no longer associative; zero is the neutral element, and every element is invertible. Denote the set ${{\cal L}}^{-1}\otimes {\mathfrak m}$ with the operation $a\cdot _{\lambda} b$ by ${\operatorname{exp}}(({{\cal L}}^{-1}\otimes {\mathfrak m})_{\lambda})$. For $t \in {{\cal L}}^{-1}\otimes {\mathfrak m},$ we will denote by ${\operatorname{exp}}(t)$ the element $t$ viewed as an element of ${\operatorname{exp}}(({{\cal L}}^{-1}\otimes {\mathfrak m})_{\lambda})$.
[**A notation convention.**]{} For $G\in {\operatorname{exp}}({{\cal L}}^{1,0}\otimes {\mathfrak m})$ and $X\in {{\cal L}}\otimes {\mathfrak m}),$ we will denote ${\operatorname{Ad}}_G(X)$ simply by $G(X).$ For $\lambda \in {{\cal L}}^{0,1}\otimes {\mathfrak m},$, $G(d+\lambda)$ will stand for $d+\lambda'$ where $\lambda'$ is the image of $\lambda$ under the gauge transformation by $\lambda.$
Given $(\lambda, G, t)$ as above; let $c={\operatorname{exp}}(t)$ and $\gamma={\operatorname{exp}}(dt+[\lambda_0,t])$ in ${\operatorname{exp}}({{\cal L}}^{1,0}\otimes {\mathfrak m}).$ Define
$$\label{eq:R,Z,T,Phi 1}
R=d\lambda+\frac{1}{2}[\lambda,\lambda]\in {{\cal L}}^{0,2}\otimes {\mathfrak m};$$
$$\label{eq:R,Z,T,Phi 2}
Z=G(d+\lambda_1)-(d+\lambda_0) \in {{\cal L}}^{1,1}\otimes {\mathfrak m};$$
$$\label{eq:R,Z,T,Phi 3}
G_{02}=T\gamma G_{01} G_{12}\in {\operatorname{exp}}({{\cal L}}^{2,0}\otimes {\mathfrak m})$$
(this is a definition of $T$); $$\label{eq:R,Z,T,Phi 4}
\Phi=((G_{01}(c_{123})^{-1}c_{013}^{-1})c_{023})c_{012}\in {\operatorname{exp}}(({{\cal L}}^{3,-1}\otimes {\mathfrak m})_{\lambda_0})$$ (the order of parentheses is in fact irrelevant for our purposes).
Define ${\mathcal I}$ to be the cosimplicial ideal of ${{\cal L}}\otimes{\mathfrak m}$ generated by $[R_{i}, {{\cal L}}\otimes{\mathfrak m}]$, $[Z_{ij}, {{\cal L}}\otimes{\mathfrak m}]$, $({\operatorname{Ad}}(T_{ijk})-{\operatorname{Id}}) ({{\cal L}}\otimes{\mathfrak m})$. Note that the operation $a\cdot _{\lambda} b$ becomes a group law modulo ${\operatorname{exp}}({\mathcal{I}})$.
\[lemma:nonabelian d2=0\] 1) (The Bianchi identity): $dR+[\lambda, R]=0;$
2\) (Gauge invariance of the curvature): $$R_0+dZ+[\lambda_0, Z]+\frac{1}{2}[Z,Z]-G(R_1)=0;$$
3\) $T\gamma(d+\lambda_0)-(d+\lambda_0)+Z_{01}+G_{01}(Z_{12})-Z_{02}=0;$
4\) $T_{013}(\gamma_{013}G_{01})(T_{123})\gamma_{013}G_{01}(\gamma_{123})=T_{023}\gamma_{023}(T_{012})\gamma_{023}\gamma_{012}$
modulo ${\operatorname{exp}}({\mathcal{I}})$;
5\) (The pentagon equation): $$G_{01}(\Phi _{1234}){{\operatorname{Ad}}_{G_{01}(c_{123})^{-1}}}(\Phi _{0134})\Phi _{0123}={{\operatorname{Ad}}_{G_{01}G_{12}(c_{234})^{-1}}}(\Phi _{0124})\Phi _{0234}$$ modulo ${\operatorname{exp}}({\mathcal{I}})$.
[**Proof.**]{} The first equality is straightforward. The second follows from $(G(d+\lambda_1))^2=G((d+\lambda_1)^2).$ The third is obtained by applying both sides of to $d+\lambda_2.$ The fourth can be seen by transforming $G_{01}(G_{12}G_{23})=(G_{01}G_{12})G_{23}$ in two different orders, using . The fifth equation compares two two-morphisms from $((G_{01}G_{12})G_{23})G_{34}$ to $G_{01}(G_{12}(G_{23}G_{34}))$ corresponding to the two different routes along the perimeter of the Stasheff pentagon. (This is just a motivation for writing the formula which is then checked directly. We could not think of a reason for this formula to be true [*a priori*]{}).
\[cor:non-abelian d2=0, II\] Let $(\lambda, \,G, t)$ be as in the beginning of \[sss:a\]. Assume that they define a descent datum modulo ${\mathfrak{m}}^{n+1}.$ Then $(R^{(n+1)},$ $ Z^{(n+1)},$ $ T^{(n+1)},$ $ -\Phi^{(n+1)})$ is a $d+\partial$-cocycle of degree three.
### {#sss:aa}
We need analogues of the above statements for isomorphisms and two-morphisms. Let $(\lambda, G, c)$ and $(\lambda', G', c')$ be two descent data. Consider a pair $(H, s)$ where $H\in {\operatorname{exp}}({{\cal L}}^{0,0}\otimes {\mathfrak{m}})$ and $s\in {{\cal L}}^{1,-1}\otimes {\mathfrak{m}}.$ Put $b={\operatorname{exp}}(s)$ in ${\operatorname{exp}}(({{\cal L}}^{1,-1}\otimes {\mathfrak{m}})_{\lambda'_0})$. Define also $\beta={\operatorname{exp}}((ds+[{\lambda'}_0,s])$ in ${\operatorname{exp}}(({{\cal L}}^{1,0}\otimes {\mathfrak{m}}))$.
As above, we measure the deviation of the pair $(H,b)$ from being an isomorphism of descent data. Put $$\label{eq:C,S,Psi 1}
C=H(d+\lambda)-(d+{\lambda'}) \in {{\cal L}}^{0,1}\otimes {\mathfrak m};$$ $$\label{eq:C,S,Psi 2}
H_0G=S\beta G'H_1\in {\operatorname{exp}}({{\cal L}}^{1,0}\otimes {\mathfrak m})$$ (this is a definition of $S$); $$\label{eq:C,S,Psi 3}
\Psi=b_{02}^{-1}{c'}_{012}{G'}_{01}(b_{12})b_{01}H_0(c_{012})$$ in ${\operatorname{exp}}(({{\cal L}}^{2,-1}\otimes {\mathfrak m})_{\lambda'_0}).$ The pair $(H,b)$ is an isomorphism between the two descent data if and only if $C=0, $ $S=1,$ $\Psi=1.$
Denote by $\mathcal J$ the cosimplicial ideal of ${{\cal L}}\otimes{\mathfrak m}$ generated by $[C_i,{{\cal L}}\otimes{\mathfrak m}]$ and $({\operatorname{Ad}}(S_{ij})-{\operatorname{Id}}) ({{\cal L}}\otimes{\mathfrak m}).$
\[lemma:nonabelian d2=0 I\] 1) $dC+[\lambda', C]+\frac{1}{2}[C,C]=0;$
2\) $S\beta (d+\lambda'_0)-(d+\lambda'_0)+C_0-G'_{01}(C_1)=0;$
3\) $S_{01}\beta_{01}G'_{01}(S_{12}\beta_{12})=H_0(\gamma_{012})S_{02}\beta_{02}{\gamma'}_{012}^{-1}$ modulo ${\operatorname{exp}}({\mathcal{J}})$;
4\) $\Psi_{023}{\operatorname{Ad}}_{H_0(c_{023})}(\Psi_{012})={\operatorname{Ad}}_{{b_{03}}^{-1}{c_{013}}G'_{01}(b_{13})}(G'_{01}(\Psi_{123}))\Psi_{012}$
modulo ${\operatorname{exp}}({\mathcal{J}})$.
[**Proof.**]{} The first equality follows from $(H(d+\lambda))^2=0;$ the second from comparing the action of both sides of on $d+\lambda'_1;$ the third is obtained by comparing two different expressions for $H_0G_{01}G_{12}$ that can be obtained from . The fourth equality compares two different two-morphisms from $H_0G_{03}$ to itself. If one passes to two-morphisms from $H_0G_{01}G_{12}G_{03}$ to itself, it becomes the pentagon equation which compares two different routes from $((H_0G_{01})G_{12})G_{03}$ to $H_0(G_{01}(G_{12}G_{03}))$. One side of the pentagon, namely the edge between $H_0((G_{01}G_{12})G_{03})$ and $H_0(G_{01}(G_{12}G_{03}))$, degenerates into a point.
\[cor:non-abelian d2=0, III\] Let $(H, s)$ be as in the beginning of \[sss:aa\]. Assume that they define an isomorphism of descent data $(\lambda, G, c)$ and $(\lambda', G', c')$ modulo ${\mathfrak{m}}^{n+1}.$ Then $(C^{(n+1)},$ $ S^{(n+1)},$ $-\Psi^{(n+1)})$ is a $d+\partial$-cocycle of degree two.
### {#sss:aaa}
Finally, we need an analogous statement for two-morphisms. Let $(H,b)$ and $({\widetilde{H}}, {\widetilde{b}})$ be isomorphisms between the descent data $(\lambda, G, c)$ and $(\lambda', G', c')$. Let $r\in {{\cal L}}^{0,{-1}}\otimes {\mathfrak m}$ and $a={\operatorname{exp}}(r)$ in ${\operatorname{exp}}(({{\cal L}}^{0,-1}\otimes{\mathfrak m})_{\lambda'}).$ Define $P$ and $\Omega$ by $$\label{eq:P,Omega 1}
{\widetilde{H}}=P\alpha H$$ where $\alpha={\operatorname{exp}}((d+\lambda')a).$ Let ${\mathcal{K}}$ be the cosimplicial ideal generated by all $({\operatorname{Ad}}_{P_i}-{\operatorname{Id}})({{\cal L}}\otimes {\mathfrak m}).$ $$\label{eq:P,Omega 2}
{\widetilde{b}}_{01}a_0=\Omega G'_{01}(a_1)b_{01}.$$ $a: (H,b)\to ({\widetilde{H}}, {\widetilde{b}})$ is a two-morphism if and only if $P=1$ and $\Omega=1.$
\[lemma:nonabelian d2=0 II\] 1) $(P\alpha)(d+\lambda')=0;$
2\) ${\operatorname{Ad}}_{\widetilde{\beta}}(P_0^{-1})G'(P_1)=({\widetilde{\beta}}\alpha_0)(G'(\alpha_1)\beta)^{-1}$ where $\beta={\operatorname{exp}}((d+\lambda')b)$ and ${\widetilde{\beta}}={\operatorname{exp}}((d+\lambda'){\widetilde{b}});$
3\) ${\operatorname{Ad}}_{G'_{01}({\widetilde{b}}_{12})}(\Omega_{01})G'_{01}(\Omega_{12}){\operatorname{Ad}}_{c'_{012}}(\Omega_{02}^{-1})=1$ modulo ${\operatorname{exp}}({\mathcal{K}})$.
[**Proof.**]{} The first equality follows from the fact that $H$ and ${\widetilde{H}}$ both preserve $d+\lambda'.$ The second is obtained by compairing the equalities $H_0G=\beta G'H_1,$ ${\widetilde{H}}_0G={\widetilde{\beta}}G'{\widetilde{H}}_1,$ and . The third equality is obtained by comparing two different expressions for $G'_{01}({\widetilde{b}}_{12}){\widetilde{b}}_{01}a_0$ using .
\[cor:non-abelian d2=0, IV\] Let $r$ be as in the beginning of \[sss:aaa\]. Assume that it defines a two-isomorphism $(H,b)\to ({\widetilde{H}}, {\widetilde{b}})$ modulo ${\mathfrak{m}}^{n+1}.$ Then $(P^{(n+1)},$ $-\Omega^{(n+1)})$ is a $d+\partial$-cocycle of degree one.
### End of the proof of Proposition \[prop:quis of cosimplicial dglas\]
The statement a) is obvious. Let us prove the surjectivity of b). Let $(\mu,$ $G,$ $c$ be a descent datum for ${{\cal L}}_2.$ Let $G={\operatorname{exp}}(y)$ in ${\operatorname{exp}}({{\cal L}}^{1,-0}_2\otimes{\mathfrak m})$ and $c={\operatorname{exp}}(t)$ in ${\operatorname{exp}}(({{\cal L}}^{2,-1}_2\otimes{\mathfrak m})_\mu).$ We write $$y=\sum y^{(k)};\;t=\sum t^{(k)}$$ [*etc.*]{}, where $y^{(k)},\,t^{(k)}\in {{\cal L}}_2\otimes{\mathfrak m}_k.$ Note that the triple $(\mu^{(1)},$ $y^{(1)},$ $t^{(1)}$) is a two-cocycle. By our assumption, $$(\mu^{(1)},y^{(1)},t^{(1)})=f(\lambda^{(1)},x^{(1)},s^{(1)})+(d+\partial)(u^{(1)},r^{(1)})$$ for some cocycle $(\lambda^{(1)},x^{(1)},s^{(1)})$ and some cochain $(u^{(1)},r^{(1)}).$ Apply the gauge transformation $H={\operatorname{exp}}(u^{(1)}),$ $b={\operatorname{exp}}(r^{(1)})$ to $(\mu,$ $G,$ $c).$ We may assume that $(\mu^{(1)},y^{(1)},t^{(1)})=f(\lambda^{(1)},x^{(1)},s^{(1)})$ where $(\lambda^{(1)},x^{(1)},s^{(1)}$ is a cocycle.
By induction, we can replace $(\mu,$ $G,$ $c)$ by an isomorphic descent datum and assume that, modulo ${\mathfrak m}^{n+1},$ it is equal to $f(\lambda, F, a)$ where $(\lambda, G, a)$ is a descent datum modulo ${\mathfrak m}^{n+1}.$ By Corollary \[cor:non-abelian d2=0, II\], the cochain $(R^{(n+2)},$ $ Z^{(n+2)},$ $ T^{(n+2)},$ $ -\Phi^{(n+2)})$ is a cocycle. It is a coboundary, because its image under $f$ is (since $f(\lambda,F,a)$ is a descent datum), and $f$ is a quasi-isomorphism. Therefore, one can modify $(\lambda,F,a)$ in the component ${\mathfrak m}_{n+1},$ so that it will become a descent datum modulo ${\mathfrak m}^{n+2}.$ Furthermore, $$(d+\partial)(\mu^{(n+1)},y^{(n+1)},t^{(n+1)})-f(\lambda^{(n+1)}, x^{(n+1)},s^{(n+1)})=0,$$ therefore $$(\mu^{(n+1)},y^{(n+1)},t^{(n+1)})-f(\lambda^{(n+1)}, x^{(n+1)},s^{(n+1)})=$$ $$(d+\partial)(u^{(n+1)}, r^{(n+1)})+f({\lambda'}^{(n+1)},{ x'}^{(n+1)},{s'}^{(n+1)})$$ where $({\lambda'}^{(n+1)},$ $ {x'}^{(n+1)},$ ${s'}^{(n+1)})$ is a cocycle. Replace $(\lambda^{(n+1)},$ $ x^{(n+1)},$ $s^{(n+1)}))$ by $(\lambda^{(n+1)}+{\lambda'}^{(n+1)},$ $ x^{(n+1)}+{x'}^{(n+1)},$ $s^{(n+1)}+{s'}^{(n+1)}),$ then apply the gauge transformation $H={\operatorname{exp}}(u^{(n+1)}),$ $b={\operatorname{exp}}(r^{(n+1)})$ to $(\mu,$ $G,$ $c.$ We get a new $(\lambda, G, a)$ which is a descent datum modulo ${\mathfrak m}^{n+2},$ and $f(\lambda, G, a)=(\mu, G, c)$ modulo ${\mathfrak m}^{n+2}.$
Now let us prove the injectivity in b). Let $(\lambda,$ $F,$ $a)$ and $(\lambda',$ $F',$ $a')$ be two descent data whose images under $f$ are isomorphic. Denote the isomorphism by $(H,$ $b).$ Let $F={\operatorname{exp}}(y),$ $F'={\operatorname{exp}}(y')$, $a={\operatorname{exp}}(s),$ $a'={\operatorname{exp}}(s'),$ $H={\operatorname{exp}}(x),$ $b={\operatorname{exp}}(r).$ We have $$f(\lambda ^{(1)},y^{(1)}, s^{(1)})-f({\lambda'} ^{(1)},{y'}^{(1)},{ s'}^{(1)})=(d+\partial)(u^{(1)}, r^{(1)});$$ therefore, since $f$ is a quasi-isomorphism, the cocycle $$(\lambda ^{(1)},y^{(1)}, s^{(1)})-({\lambda'} ^{(1)},{y'}^{(1)},{ s'}^{(1)})$$is a coboundary. After replacing the datum $(\lambda',$ $F',$ $a')$ by a datum which is isomorphic to it and identical to it modulo ${\mathfrak m}^2,$ we may assume that $f(\lambda,$ $F,$ $a)=f(\lambda',$ $F',$ $a')$ modulo ${\mathfrak m}^2.$ By induction, we may assume that $(\lambda,$ $F,$ $a)$ and $(\lambda',$ $F',$ $a')$ coincide modulo ${\mathfrak m}^{n+1}$ and that their images are isomorphic, the isomorphism being equal to identity modulo ${\mathfrak m}^{n}$. Apply Corollary \[cor:non-abelian d2=0, III\] to study the failure of $(H=1,$ $b=1)$ to be an isomorphism between $(\lambda,$ $F,$ $a)$ and $(\lambda',$ $F',$ $a')$. The corresponding cocycle is a coboundary because its image under $f$ is. Therefore we can act upon $(H=1,$ $b=1)$ by a two-morphism and obtain a new $(H,b)$ which is an isomorphism between $(\lambda,$ $F,$ $a)$ and $(\lambda',$ $F',$ $a')$ modulo ${\mathfrak m}^{n+1}$. Now one can assume that $(\lambda,$ $F,$ $a)$ and $(\lambda',$ $F',$ $a')$ coincide modulo ${\mathfrak m}^{n+1}$ and that their images are isomorphic, the isomorphism being equal to identity modulo ${\mathfrak m}^{n+1}.$ We have $$f(\lambda ^{(n+1)},y^{(n+1)}, s^{(n+1)})-f({\lambda'} ^{(n+1)},{y'}^{(n+1)},{ s'}^{(n+1)})=(d+\partial)(u^{(n+1)}, r^{(n+1)});$$ since $f$ is a quasi-isomorphism, the cocycle $(\lambda ^{(n+1)},y^{(n+1)}, s^{(n+1)})-({\lambda'} ^{(n+1)},{y'}^{(n+1)},{ s'}^{(n+1)})$ is a coboundary. After replacing the datum $(\lambda',$ $F',$ $a')$ by a datum which is isomorphic to it and identical to it modulo ${\mathfrak m}^{n+2},$ we may assume that $f(\lambda,$ $F,$ $a)=f(\lambda',$ $F',$ $a')$ modulo ${\mathfrak m}^{n+2}.$
This proves the statement b). The proofs of c) and d) are very similar, and we leave them to the reader.
Totalization of cosimplicial DGLAs {#ss:Totalization of cosimplicial DGLAs}
----------------------------------
Here we recall how one can construct a DGLA from a cosimplicial DGLA by the procedure of totalization. We then prove that isomorphism classes of descent data for a cosimplicial DGLA are in one-to one correspondence with isomorphism classes of Maurer-Cartan elements of its totalization. This is a two-groupoid version of a theorem of Hinich [@H1].
Define for $p\geq 0$ $${\mathbb Q}[\Delta ^p]={\mathbb Q}[t_{0}, \ldots, t_{p}]/(t_{0}+ \ldots + t_{p}-1)$$ and $$\Omega ^{\bullet} [\Delta ^p]={\mathbb Q}[t_{0}, \ldots, t_{p}]\{dt_{0}, \ldots, dt_{p}\}/(t_{0}+ \ldots + t_{p}-1,dt_{0}+ \ldots + dt_{p} )$$ The collection $\{\Omega ^{\bullet} [\Delta ^p]\}, p\geq 0,$ is a simplicial DGA.
Let ${{\cal M}}$ be the category whose objects are morphisms $f:[p]\to [q]$ in $\Delta$ and a morphism from $f:[p]\to [q]$ to $f':[p']\to [q']$ is a pair $a: [p']\to [p]$, $b:[q]\to [q']$ such that $f'=bfa.$ Given $(a,b):f\to f'$ and $(a',b'): f'\to f'',$ define their composition to be $(a'a,bb').$
Given a cosimplicial DGLA ${{\cal L}},$ we can construct a functor from ${{\cal M}}$ to the category of vector spaces by assigning to the object $f:[p]\to [q]$the space $\Omega ^{\bullet} [\Delta ^p]\otimes {{\cal L}}^q.$ Set $${{\operatorname{Tot}}}({{\cal L}})={{\operatorname{lim}}\,{\operatorname{dir}}}_{{{\cal M}}} \Omega ^{\bullet} [\Delta ^p]\otimes {{\cal L}}^q$$ This is a DGLA (with the differential being induced by $d_{\operatorname{DR}}.$
\[prop:Hinich\] a). There is a bijection between the set of isomorphism classes of descent data of the Deligne two-groupoid of ${{\cal L}}$ and the set of Maurer-Cartan elements of ${{\operatorname{Tot}}}({{\cal L}}).$
b). For a descent datum ${{\cal A}}$ of ${{\cal L}}$, denote by $\lambda({{\cal A}})$ a Maurer-Cartan element from the isomorphism class given by a). Then there is a bijection $$\frac{{\operatorname{Iso}}({{\cal A}}, {{\cal A}}')}{2-{\operatorname{Iso}}}{\overset{\sim}{\to}}\frac{{\operatorname{Iso}}(\lambda({{\cal A}}), \lambda({{\cal A}}'))}{2-{\operatorname{Iso}}}.$$ c). For two isomorphisms $\phi, \psi:{{\cal A}}\to {{\cal A}}'$, denote their images under the above bijection by $G(\phi), G(\psi).$ Then $f$ induces a bijection $$2-{\operatorname{Iso}}(\phi, \psi){\overset{\sim}{\to}}2-{\operatorname{Iso}}(G(\phi), G(\psi))$$
[**Proof.**]{} Recall that for every small category ${{\cal M}}$ and for every functor $C:{{\cal M}}\to {\operatorname{Vect}}_k$ one can define a cosimplicial space $$({\bf R}{{\operatorname{lim}}\,{\operatorname{inv}}}_{{{\cal M}}}C)^n=\prod _{f_0\stackrel{\alpha_1}{\rightarrow} f_1\stackrel{\alpha_2}{\rightarrow}\ldots \stackrel{\alpha_n}{\rightarrow}f_n} C(f_n)$$ with the standard maps $d_i$ and $s_i.$ The product is taken over all composable chains of morphisms in ${{\cal M}}.$ If $C$ is a functor from ${{\cal M}}$ to the category of $DGLAs$ then ${\bf R}{{\operatorname{lim}}\,{\operatorname{inv}}}_{{{\cal M}}}C$ is a cosimplicial DGLA.
Consider the cosimplicial DGLA ${\bf R}{{\operatorname{lim}}\,{\operatorname{inv}}}_{{{\cal M}}}\Omega ^{\bullet} [\Delta ^p]\otimes {{\cal L}}^q,$ together with the constant cosimplicial DGLA ${{\operatorname{Tot}}}({{\cal L}})$ and the cosimplicial DGLA ${\bf R}{{\operatorname{lim}}\,{\operatorname{inv}}}_{\Delta}({{\cal L}})$. The second and the third DGLAs embed into the first, and these embeddings are quasi-isomorphisms with respect to the differentials $d+\partial.$ By Proposition \[prop:quis of cosimplicial dglas\], our statement is true if we replace the cosimplicial DGLA ${{\cal L}}$ by ${\bf R}{{\operatorname{lim}}\,{\operatorname{inv}}}_{\Delta}({{\cal L}})$. But these two cosimplicial DGLA are quasi-isomorphic, whence the statement.
The Hochschild complex
----------------------
\[ex:Hochschild complex\] For any associative algebra $A$, let ${{\cal L}}^H (A)$ be the Hochschild cochain complex equipped with the Gerstenhaber bracket [@Ge]. The standard Hochschild differential is denoted by $\delta$. For a sheaf of algebras ${{\cal A}},$ let ${{\cal L}}^H({{\cal A}})$ denote the sheafification of the presheaf of DGLA $U\mapsto {{\cal L}}^H(A(U)).$ For the sheaf of algebras $C^{\infty}_M$ on a smooth manifold, resp. ${\cal O}_M$ on a complex analytic manifold, let ${{{\cal L}}} ^H_M$ be the sheaf of Hochschild cochains $D(f_1, \ldots, f_n)$ which are given by multi-differential, resp. holomorphic multi-differential, expressions in $f_1, \ldots, f_n$.
One gets directly from the definitions the following
\[lemma:defs of triv gerbe through DGLA\] The set of isomorphism classes of deformations over ${\mathfrak{a}}$ of a sheaf of $k$-algebras ${{\cal A}}$ as a stack is in one-to-one correspondence with the set of isomorphism classes of descent data of the Deligne two-groupoid of ${{\cal L}}^H({{\cal A}})\otimes{\mathfrak{m}}$. Similarly, the set of isomorphism classes of deformations of the trivial gerbe on $M$ is in one-to-one correspondence with the set of isomorphism classes of descent data of the Deligne two-groupoid of ${{\cal L}}^H _M\otimes{\mathfrak{m}}$.
Hochschild cochains at the jet level {#ss:Hochschild cochains at the jet level}
------------------------------------
For a manifold $M$, let $J$, or $J_M$, be the bundle of jets of smooth, resp. holomorphic, functions on $M$. By $\nabla _{{\operatorname {can}}}$ we denote the canonical flat connection on the bundle $J$. Let $C^{\bullet}(J,J)$ be the bundle of Hochschild cochain complexes of $J$. More precisely, the fibre of this bundle is the complex of jets of multi-differential multi-linear expressions $D(f_1, \ldots, f_n)$. We denote by $\delta$ the standard Hochschild differential.
\[prop:defs of triv gerbe through DGLA of jets\] The set of isomorphism classes of deformations of the trivial gerbe on $M$ is in one-to-one correspondence with the set of isomorphism classes of Maurer-Cartan elements of the DGLA ${{\cal L}}^{H,J} (M)\otimes{\mathfrak m}$ where $${{\cal L}}^{H,J} (M)= A ^{\bullet} (M,C^{\bullet +1} (J,J))$$ with the differential ${\nabla _{{\operatorname {can}}}}+ \delta$. Here by $A^{\bullet}$ we mean $C^{\infty}$ forms with coefficients in a bundle.
[**Proof.**]{} We have an embedding of sheaves of DGLA: $${{\cal L}}^H_M \to A ^{\bullet} _M (C^{\bullet +1} (J,J))$$ which is a quasi-isomorphism, and the sheaf on the right hand side has zero cohomology in positive degrees. The proposition follows from Proposition \[prop:quis of cosimplicial dglas\].
Deformation quantization of the trivial gerbe on a symplectic manifold {#s:Deformations of the trivial gerbe on a symplectic manifold}
======================================================================
Deformation quantization of gerbes
----------------------------------
\[dfn:deformation quantization\] A deformation quantization of a gerbe ${{\cal A}}^{(0)}$ on a manifold $M$ is a collection of deformations ${{\cal A}}^{(N)}$ over ${\mathfrak{a}}={\mathbb{C}}[\hbar]/(\hbar^{N+1}),$ $N\geq 0$ (cf. Definition \[dfn:deformation1\]), such that ${{\cal A}}^{(N)}/\hbar^{N}={{\cal A}}^{(N-1)}.$ An isomorphism of two deformation quantizations is a collection of isomorphisms of deformations $\varphi _N:{{\cal A}}^{(N)}\to {{{\cal A}}'}^{(N)}$ such that $\varphi _N {=}\varphi _{N-1} \;{\operatorname{mod}}\hbar ^N.$
Given a deformation quantization of a gerbe, one can define a stack of ${\mathbb{C}}[\hbar]$-algebras ${{\cal A}}={{\operatorname{lim}}\,{\operatorname{inv}}}{{\cal A}}^{(N)}.$ Usually we will not distinguish between the deformation quantization and this stack.
{#section-1}
Let $(M, \omega)$ be a symplectic manifold ($C^{\infty}$ or complex analytic with a holomorphic symplectic form). In this section, we extend Fedosov’s methods from [@Fe] to deformations of the trivial gerbe. We say that a deformation quantization of the trivial gerbe on $M$ corresponds to $\omega$ if, on every $U_k$, $f*g-g*f = {\sqrt{-1}}\hbar \{f,g\} + o(\hbar)$ where $\{\,,\,\}$ is the Poisson bracket corresponding to $\omega$.
Let us observe that the group $H^2 (M, \hbar {{\Bbb C}}[[\hbar]])$ acts on the set of equivalence classes of deformations of any stack: a class $\gamma$ acts by multiplying $c_{ijk}$ by ${\operatorname {exp}}\,\gamma _{ijk}$ where $\gamma _{ijk}$ is a cocycle representing $\gamma$.
\[thm:classification of deformations of the trivial gerbe, symplectic case\] Denote by $\operatorname{Def}(M,\omega)$ the set of isomorphism classes of deformation quantizations of the trivial gerbe on $M$ compatible with the symplectic structure $\omega$. The action of $H^2 (M, \hbar {{\Bbb C}}[[\hbar]])$ on $\operatorname{Def}(M,\omega)$ is free. The space of orbits of this action is in one-to-one correspondence with an affine space modelled on the vector space $ H^2 (M, {{\Bbb C}})$ (in the $C^{\infty}$ case) or $H^1(M,{{{\cal O}}_M}/{\Bbb C})$ (in the complex case).
[**Proof.**]{} As in [@Fe1], we will reduce the proof to a classification problem for certain connections in an infinite-dimensional bundle of algebras.
Let us observe that the Proposition \[prop:defs of triv gerbe through DGLA of jets\] is true if we replace deformations over Artinian rings by deformation quantizations. Indeed, the proof of Proposition \[prop:quis of cosimplicial dglas\] works verbatim for the DGLAs that are needed for Proposition \[prop:defs of triv gerbe through DGLA of jets\], since one can start with a good cover, and all cohomological obstructions are zero already in the Čech complex of this cover; one has no need of refining the cover, and therefore one can carry out the induction procedure infinitely many times. Next, note that in Proposition \[prop:defs of triv gerbe through DGLA of jets\] we can replace the bundle of algebras $J$ by the bundle of algebras $${\operatorname{gr}} J = \prod S^m(T^*_M).$$ Indeed, a standard argument shows that they are isomorphic as $C^{\infty}$ bundles of algebras.
Under this isomorphism, the canonical connection ${\nabla _{{\operatorname {can}}}}$ becomes a connection $\nabla _0$ on ${\operatorname{gr}} J$. We are reduced to classifying up to isomorphism those Maurer-Cartan elements of $(A^{\bullet}(M, C^{\bullet +1}({\operatorname{gr}} J, {\operatorname{gr}} J)), \nabla _0 +\delta)$ whose component in $A^0(M, C^2)$ is equal to $\frac{1}{2}{\sqrt{-1}}\hbar \{f,g\}$ modulo $\hbar$. In other words,these components must be, pointwise, deformation quantizations of $\prod S^m(T^*_M)$ corresponding to the symplectic structure. But all such deformations are isomorphic to the standard Weyl deformation from the definition below:
\[dfn:Weyl algebra\] The Weyl algebra of $T^*_M$ is the bundle of algebras $$W = {\operatorname{gr}} J[[\hbar]] = \prod S^m(T^*_M)[[\hbar]]$$ with the standard Weyl product $*$.
Moreover, a smooth field of such deformations on $M$ admits a smooth gauge transformation making it the standard Weyl deformation. Therefore, we have to classify up to isomorphism those Maurer-Cartan elements of $A^{\bullet}(M, C^{\bullet +1}({\operatorname{gr}} J, {\operatorname{gr}} J))$ whose component in the subspace $A^0(M, C^2)$ is equal to $f*g - fg$. Here $*$ is the product in the standard Weyl deformation.
\[prop:Fedosov for stacks\] Deformations of the trivial gerbe on $M$ compatible with a symplectic structure $\omega$ are classified up to isomorphism by pairs $(A,c)$ where $$\label{eq:Fedosov 1}
A\in \hbar A^1 (M, {\operatorname{hom}}({\operatorname{gr}} J, {\operatorname{gr}} J))[[\hbar]];$$ $$\label{eq:Fedosov 2}
c\in \hbar A^2 (M, {\operatorname{gr}} J)[[\hbar]],$$ such that, if $$\nabla = \nabla _0 + A,$$ then $$\label{eq:Fedosov 3}
\nabla (f*g) = \nabla(f) *g + f*\nabla(g);$$ $$\label{eq:Fedosov 4}
\nabla ^2 = {\operatorname{ad}}(c);\;\nabla (c) = 0$$ Two pairs $(A,c)$ and $(A',c')$ are equivalent if one is obtained from the other by a composition of transformations of the following two types. a) $$\label{eq:Fedosov 5}
(A,\,c) \mapsto ({\operatorname{exp}}({\operatorname{ad}}(X))(A),\, {\operatorname{exp}}({\operatorname{ad}}(X))(c))$$ where $X \in \hbar{\operatorname{Der}}(W);$
b\) $$\label{eq:Fedosov 6}
(A,\,c) \mapsto (A+B,\,c+{\nabla}B +{\frac{1}{2}}[B,\,B])$$ where $B \in \hbar W$.
It is straightforward that the set of Maurer-Cartan elements discussed above, up to isomorphism, is in one-to-one correspondence with the set of pairs $(A,\,c)$ up to equivalence. Indeed, given $(A,c)$, the Maurer-Cartan element is constructed as follows: the component in $A^0(M, C^2)$ is the difference between the Weyl product and the commutative product; the component in $A^1(M, C^1)$ is $\nabla-\nabla _0$, and the component in $A^2(M, C^0)$ is $c$. It remains to show that the pairs $(A,\,c)$ are classified as in Theorem \[thm:classification of deformations of the trivial gerbe, symplectic case\].
Let us start with notation. Let $${\widetilde{{{\frak{g}}}}}^0={\operatorname{ gr}}{J}$$ be the bundle of Lie algebras of formal power series with the standard Poisson bracket. Let ${{\frak{g}}}^0 = {\operatorname{ gr}}{J}/{\Bbb C}$ be the quotient bundle of Lie algebras. In other words, the fibre of ${{\frak{g}}}^0$ is the Lie algebra of formal Hamiltonian vector fields on the tangent space. Also, put $${\widetilde{{{\frak{g}}}}}={\frac{1}{\hbar}}W$$ with the bracket $a*b-b*a$ where $*$ is the Weyl product, and $${{\frak{g}}}= {\widetilde{{{\frak{g}}}}}/{\frac{1}{\hbar}}{\Bbb C}[[\hbar]]$$ This is the Lie algebra of continuous derivations of the Weyl algebra. It maps surjectively to ${{\frak{g}}}^0$ via ${\frac{1}{\hbar}}(f_0 + \hbar f_1 + \cdots)\mapsto f_0$. Put $|a|=m$ for $a\in S^m(T^*_M)$ and $|\hbar|=2$. This defines the degree of any monomial in $S^m(T^*_M)[\hbar].$ By ${\widetilde{{{\frak{g}}}}}^{0}_m$ we denote the subspace $S^{m+2}(T^*_M)$, and by ${{\widetilde{{{\frak{g}}}}}}_{m}$ the set of ${\frac{1}{\hbar}}f$ where $f$ is a polynomial from $S^{m+2}(T^*_M)[\hbar].$ Then $$[{\widetilde{{{\frak{g}}}}}^0_{m},\,{\widetilde{{{\frak{g}}}}}^{0}_{r}]\subset {\widetilde{{{\frak{g}}}}}^{0}_{m+r};\;\;[{\widetilde{{{\frak{g}}}}}_{m},\,{\widetilde{{{\frak{g}}}}}_{r}]\subset {\widetilde{{{\frak{g}}}}}_{m+r};$$ $${\widetilde{{{\frak{g}}}}}^{0}=\prod_{m\geq -2} {\widetilde{{{\frak{g}}}}}^0_{m};\;{\widetilde{{{\frak{g}}}}}=\prod_{m\geq -2} {\widetilde{{{\frak{g}}}}}_{m}$$ One defines ${{\frak{g}}}^0_{m}$ and ${{\frak{g}}}_m$ accordingly. We have $${{{{\frak{g}}}}}^{0}=\prod_{m\geq -1} {{{{\frak{g}}}}}^{0}_{m};\;{{{{\frak{g}}}}}=\prod_{m\geq -1} {{{{\frak{g}}}}}_{m}$$ In particular, the bundle ${\widetilde{g}}^0_{-1}={{\frak{g}}}^{0}_{-1}={\widetilde{g}}_{-1}={{\frak{g}}}_{-1}$ is the cotangent bundle $T^*_M$. The symplectic form identifies this bundle with $T_M$.
\[df:A\_-1\] By $A_{-1} $ we denote the canonical form ${\operatorname{id}}\in A^1(M,T_M)$ which we view as a form with values in ${\widetilde{g}}^{0}_{-1},$ etc. under the identifications above.
The form $A_{-1}$ is smooth in the $C^{\infty}$ case and holomorphic in the complex case.
The connection $\nabla _0$ can be expressed as $$\label{eq:nabla_0}
\nabla _0=A_{-1}+\nabla _{0,0}+\sum_{k=1}^{\infty} A_k=\nabla _{0,0}$$ where $\nabla _{0,0}$ is an ${\frak{sp}}_n$-valued connection in the tangent bundle $T_M$ and $A_k \in A^1(M, {{\frak{g}}}^{0}_{k})$. Define $$A^{(-1)}=\sum_{k=1}^{\infty} A_k$$ (Here $n=\frac{1}{2}{\operatorname{dim}}(M)$). The form $A_{-1}$ is in fact the canonical form from the above definition. In the case of a complex manifold, locally $\nabla _{0,0}=\partial +{\overline {\partial}}+A_{0,0}$ where $A_{0,0}$ is a $(1,0)$-form with values in ${\frak{sp}}_n$. The form $A^{(-1)}$ can be viewed as a ${\widetilde{{{\frak{g}}}}}^0$-valued one-form: $$\label{eq:nabla_0, I}
A^{(-1)}\in A^1(M, {\widetilde{{{\frak{g}}}}}^0)$$ Let us look for $\nabla$ of the form $$\label{eq:nabla in general}
\nabla={\nabla_0}+ \sum_{m=0}^{\infty}({\sqrt{-1}}\hbar)^m A^{(m)}$$ where $A^{(m)} \in A^1(M, {{\frak{g}}}^0)$. The condition $\nabla ^2 = o(\hbar)$ is equivalent to $$\label{eq:flatness mod h}
{\nabla_0}A^{(0)}+{\frac{1}{2}}[A^{(-1)},A^{(-1)}]_{2}=0$$ Here we use the notation $$a*b-b*a=\sum_{m=1}^{\infty} ({\sqrt{-1}}\hbar)^m [a,b]_{m}$$ (in particular, $[\,,\,]_0$ is the Poisson bracket); we then extend the brackets $[a,b]_{m}$ to forms with values in the Weyl algebra. Since $[{\nabla _{{\operatorname {can}}}},[{\nabla _{{\operatorname {can}}}},{\nabla _{{\operatorname {can}}}}]]=0$ and $[{\nabla_0},{\nabla_0}]=0,$ we conclude that $${\nabla_0}[A^{(-1)}, A^{(-1)}]_2 =0$$ in $A^2 (M, {\widetilde{{{\frak{g}}}}}^0)$. Moreover, observe that the left hand side lies in fact in $A^2 (M, \prod _{m\geq 0}{\widetilde{{{\frak{g}}}}}^0_{m}).$
\[lemma:acyclicity of A-1\] If $c\in A^p(M, {\widetilde{{{\frak{g}}}}}^0_{m})$, $m\geq -1,$ satisfies $[A_{-1},c]=0,$ then $c=[A_{-1},c']$ for $c'\in A^{p-1}(M, {\widetilde{{{\frak{g}}}}}^0_{m+1}).$
[**Proof.**]{} Indeed, the complex $A^{\bullet}(M, {\widetilde {{\frak{g}}}}^0)$ with the differential $[A_{-1},\;]$ is isomorphic to the complex of smooth sections of, resp, $A^{0,{\bullet}}$ forms with coefficients in, the bundle of complexes $S[[T^*_M]]\otimes \wedge (T^*_M)$ with the standard De Rham differential.
We now know that pairs $(\nabla, c)$ exist. The theorem is implied by the following lemma (we use the notation of -).
\[lemma:classification of connections\] 1) For any two connections $\nabla$ and $\nabla ',$ $A^{(0)}-{A'}^{(0)}$ is a cocycle in $A^1(M, J/{{{\Bbb C}}});$ a pair $(\nabla, c)$ is equivalent to a pair $(\nabla ', c')$ for some $c'$ by some transformation $(X, B)$ if and only if $A^{(0)}-{A'}^{(0)}$ is a coboundary;
2\) for any two pairs $(\nabla, c)$ and $(\nabla, c')$ with the same $\nabla,$ $c-c'$ is a closed form in $A^2 (M, \hbar {{\Bbb C}}[[\hbar]]);$ two such pairs are equivalent if and only if $c-c'$ is exact.
[**Proof.**]{} 1) The first statement of 1) follows from . To prove the second, note that $$\nabla ' = {{\operatorname{exp}}\,{\operatorname{ad}}}(X) (\nabla) + {\operatorname ad}(B),$$ $$B \in A^1 (M, \hbar {\widetilde {{{\frak{g}}}}})$$ with $$X=\sum_{m=0}^{\infty}({\sqrt {-1}}\hbar)^m X^{(m)}$$ and $X^{(m)}\in A^0(M, {{\frak{g}}}^0),$ is possible if and only if $${\nabla_0}X^{(0)} + A^{({0})}-{A'}^{({0})}=0.$$ 2) The first statement of 2) follows from . To prove the second, consider a lifting of $\nabla$ to a ${\widetilde{g}}$-valued connection ${{\widetilde{\nabla}}}.$ We have $$c={{\widetilde{\nabla}}}^2 + \theta$$ where $\theta \in A^2 (M, \hbar {{\Bbb C}}[[\hbar]])$. One has $$\nabla = {{\operatorname{exp}}\,{\operatorname{ad}}}(X)(\nabla)+B$$ if and only if the following two equalities hold: $${{\widetilde{\nabla}}}= {{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}})+B+\alpha$$ for some $\alpha \in A^1 (M, {{\Bbb C}}[[\hbar]]);$ $$c' = {{\operatorname{exp}}\,{\operatorname{ad}}}(X)(c)+{{\operatorname{exp}}\,{\operatorname{ad}}}(X)(B)+{\frac{1}{2}}[B,B].$$ But in this case $$c'={{\operatorname{exp}}\,{\operatorname{ad}}}(X) ({{\widetilde{\nabla}}}^2 + \theta)+[{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}}), {{\widetilde{\nabla}}}-{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}})-\alpha]+$$ $${\frac{1}{2}}[{{\widetilde{\nabla}}}-{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}}),{{\widetilde{\nabla}}}-{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}})]=$$ $${\frac{1}{2}}[{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}}),{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}})]+$$ $$\theta + [{{\operatorname{exp}}\,{\operatorname{ad}}}(X) {{\widetilde{\nabla}}}, {{\widetilde{\nabla}}}]- {\frac{1}{2}}[{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}}),{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}})] -d\alpha +{\frac{1}{2}}[{{\widetilde{\nabla}}},{{\widetilde{\nabla}}}]-$$ $$[{{\widetilde{\nabla}}}, {{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}})]+
{\frac{1}{2}}[{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}}),{{\operatorname{exp}}\,{\operatorname{ad}}}(X)({{\widetilde{\nabla}}})]={{\widetilde{\nabla}}}^2 +\theta -d\alpha$$ $$=c-d\alpha$$ This proves the theorem.
The characteristic class of a deformation and the Rozansky-Witten class
=======================================================================
The characteristic class
------------------------
Given a deformation of the trivial gerbe on a symplectic manifold $(M,\omega)$, one defines its characteristic class $$\theta= \frac{1}{{\sqrt{-1}}\hbar}\omega+\sum_{k=0}^{\infty}({{\sqrt{-1}}\hbar})^k\theta_k\in \frac{1}{{\sqrt{-1}}\hbar}\omega+H^2(M)[[\hbar]]$$ as follows. Represent the deformation by a pair $(\nabla, c)$ as in Proposition \[prop:Fedosov for stacks\]. Choose a lifting ${{\widetilde{\nabla}}}$ of $\nabla$ to a ${\mathfrak{g}}$-valued connection; define $$\theta ={{\widetilde{\nabla}}}^2-c.$$ It is easy to see that: i) $\theta\in A^2(M, {\frac{1}{\hbar}}{\mathbb C}[[\hbar]]);$
ii\) $d\theta=0,$ and the cohomology class of $\theta$ is invariant under the equivalence and independent of the lifting.
The above construction generalizes Fedosov’s Weyl curvature. It is easy to see that the class of$ \theta_0$ coincides with the image of the class from Theorem \[thm:classification of deformations of the trivial gerbe, symplectic case\] under the morphism $\partial: H^1(M, {{\cal O}}_M/{\mathbb C})\to H^2(M, {\mathbb C}).$ In particular, if this map is not injective, there may be non-isomorphic deformations with the same class $\theta$.
Deformation quantization of the sheaf of functions
--------------------------------------------------
Here we recall a theorem from [@NT] (cf. [@BK] for the algebraic case).
Let $(M, \omega)$ be either a symplectic $C^{\infty}$ manifold or a complex manifold with a holomorphic symplectic structure. By ${{\cal O}}_M$ we denote the sheaf of smooth, resp. holomorphic, functions.
In what follows we will study deformation quantization of ${{\cal O}}_M$ [*as a sheaf*]{}. In the language adopted in this article, these are deformation quantizations of the trivial gerbe such that $c_{ijk}=1$. An isomorphism is by definition an isomorphism of deformation quantizations such that $b_{ij}=1$.
\[thm:classification of sheaves\] Assume that the maps $H^i (M, {{\Bbb C}}) \to H^i (M, {{\cal O}}_M)$ are onto for $i=1,\,2$. Set $$H^2_F (M,{{\Bbb C}})={\operatorname {ker}}(H^2(M, {{\Bbb C}}) \to H^2(M, {{\cal O}}_M)).$$ Choose a splitting $$H^2(M, {{\Bbb C}}) = H^2(M, {{\cal O}}_M) \oplus H^2_F (M,{{\Bbb C}}).$$ The set of isomorphism classes of deformation quantizations of ${{\cal O}}_M$ as a sheaf which are compatible with $\omega$ is in one-to-one correspondence with a subset of the affine space $${\frac{1}{{\sqrt{-1}}\hbar}}\omega + H^2 (M, {{\Bbb C}})[[\hbar]]$$ whose projection to $${\frac{1}{{\sqrt{-1}}\hbar}}\omega + H^2 _F (M, {{\Bbb C}})[[\hbar]]$$ is a bijection.
The first Rozansky-Witten class {#s:the RW class}
-------------------------------
We have seen in the previous section that, under the assumptions of Theorem \[thm:classification of sheaves\], deformations of the sheaf of algebras ${{\cal O}}_M$ are classified by cohomology classes $\theta$ as in where $\theta _{-1}=\frac{1}{{\sqrt {-1}}\hbar }\omega;$ the (non-natural) projection of the set of all possible classes $\theta$ to $\frac{1}{{\sqrt {-1}}\hbar }\omega + H^2_F(M,{{\Bbb C}}[[\hbar]])$ is a bijection. More precisely, the (natural) projection of $\theta _{n+1}$ to $H^2 (M, {{\cal O}}_M)$ is a nonlinear function in $\theta _i ,\;0\leq i \leq n.$ We are going to describe this function for the case $n=0.$
Let $M$ be a complex manifold with a holomorphic symplectic structure $\omega$. We start by describing two ways of constructing cohomology classes in $H^2 (M, {{\cal O}}_M) .$ The first one was invented by Rozansky and Witten, cf. [@RW], [@Kap], [@K2]. Let $\nabla _{0,0}$ be a torsion-free connection in the tangent bundle which is locally of the form $d+A_0$ for $A_0 \in A^{1,0}(M, {\mathfrak{sp}}).$ Let $R={\overline{\partial}}A_0$ be the $(1,1)$ component of the curvature of $\nabla _{0,0}$. We can view $R$ as a $(1,1)$ form with coefficients in $S^2 (T^*_M)$. Let $z^i$ be holomorphic coordinates on $M.$ By ${\widehat{z}}^i$ we denote the corresponding basis of $T^*_M.$ We write $$\label{eq:R}
R=\sum R_{abi{\overline{j}}}{\widehat{z}}^a{\widehat{z}}^bdz^id{\overline{z}}^j$$ Put $$\label{eq:RW}
{\operatorname{RW}}_{\Gamma _0}(M,\omega)=\sum R_{abi{\overline{j}}} R_{cdk{\overline{l}}}\omega^{ac}\omega^{bd}\omega^{ik}d{\overline{z}}^jd{\overline{z}}^l$$ Here $\Gamma _0$ refers to the graph with two vertices and three edges connecting them. In fact a similar form ${\operatorname{RW}}_{\Gamma }(M,\omega)$ can be defined for any finite graph $\Gamma$ for which every vertex is adjacent to three edges; the cohomology class of this form is independent of the connection [@RW].
The other way of obtaining $(0,2)$ classes is as follows. For $\alpha = \sum \alpha _{i{\overline{j}}}dz^i d{\overline{z}}^j$ and $\beta = \sum \beta _{i{\overline{j}}}dz^i d{\overline{z}}^j,$ put $$\label{eq:def of pairing}
\omega(\alpha, \beta)=\sum \alpha _{i{\overline{j}}} \beta _{k{\overline{l}}}\omega_{ik}d{\overline{z}}^jd{\overline{z}}^l$$ It is straightforward that the above operation defines a symmetric pairing $$\omega: H^{1,1}(M)\otimes H^{1,1}(M) \to H^{0,2}(M).$$ Combined with the projection $H^2_F(M)\to H^{1,1}(M),$ this gives a symmetric pairing $$\omega: H^2_F(M)\otimes H^2_F(M) \to H^2(M,{{\cal O}}_M).$$
\[thm:RW\] Under the assumptions of Theorem \[thm:classification of sheaves\], let a deformation of the sheaf of algebras ${{\cal O}}_M$ correspond to a cohomology class $$\theta=\sum ({\sqrt{-1}}\hbar)^m \theta _m, \; \theta _m \in H^2 (M).$$ Then the projection of the class of $\theta _1$ to $H^2 (M,{{\cal O}}_M)$ is equal to $${\operatorname{RW}}_{\Gamma _0}(M,\omega)+\omega(\theta _0,\theta _0)$$
[**Proof.**]{} First, observe that Lemma \[lemma:defs of triv gerbe through DGLA\] and Proposition \[prop:defs of triv gerbe through DGLA of jets\] have their analogs for deformations of the structure sheaf as a sheaf of algebras. The only difference is that the Hochschild complex $C^{\bullet +1}$ is replaced everywhere by $C^{\bullet +1}, {\bullet \geq 0}.$ Similarly to -, one has
\[lemma:Fedosov for sheaves\] Deformations of the sheaf of algebras ${\cal{O}}_M$ which are compatible with a symplectic structure $\omega$ are classified by forms $A\in \hbar A^1 (M, {\operatorname{hom}}({\operatorname{gr}} J, {\operatorname{gr}} J))[[\hbar]]$ such that, if $$\nabla = \nabla _0 + A,$$ then $$\label{eq:Fedosov 33}
\nabla (f*g) = \nabla(f) *g + f*\nabla(g);$$ and $\nabla ^2 = 0$. Two such forms are equivalent if, for $X\in A^0 (M, \hbar {\operatorname{Der}}(W),$ $$\nabla ' ={{\operatorname{exp}}\,{\operatorname{ad}}}(X)\nabla$$
The proof is identical to the proof of Lemma \[prop:Fedosov for stacks\].
Let us now classify pairs $(\nabla, c)$.
We start by constructing a flat connection $\nabla$. We use a standard proof from the homological perturbation theory. One has to solve recursively $$\label{eq:curvature}
R_n +{\nabla_0}A^{(n+1)}=0$$ where $$R_n=\frac{1}{2}\sum_{i,j\geq 0\;;i+j+m=n+1} [A^{(i)},A^{(j)}]_m$$ At every stage ${\nabla_0}R_n = 0$; the class of $R_n$ is in the image of the map $$H^2(M, {{\cal O}}_M)\to H^2(M, {{\cal O}}_M /{{\Bbb C}})$$ which is zero under our assumptions.
We have shown that flat connections $\nabla$ exist. For any such connection we can consider its lifting to a ${\widetilde {{{\frak{g}}}}}$-valued connection ${{\widetilde{\nabla}}}.$ Put $$\label{eq:theta}
{{\widetilde{\nabla}}}^2 = \theta = \sum _{m=-1}^{\infty}({\sqrt{-1}}\hbar)^m \theta _m\in A^2 (M, {\frac {1}{\hbar}}{{\Bbb C}}[[\hbar]])$$ Let us try to determine all possible values of $\theta.$
\[lemma:nablas via tetas\] Under the assumptions of Theorem \[thm:classification of sheaves\], the map $\nabla\mapsto \theta$ establishes a one-to-one correspondence between the set of equivalence classes of connections $\nabla$ and a subset of the affine space $${\frac{1}{{\sqrt{-1}}\hbar}}\omega + H^2 (M, {{\Bbb C}})[[\hbar]]$$ whose projection to $${\frac{1}{{\sqrt{-1}}\hbar}}\omega + H^2 _F (M, {{\Bbb C}})[[\hbar]]$$ is a bijection.
First of all, $\theta _{-1}=\frac{1}{\sqrt{-1}\hbar}\omega.$ There exists ${{\widetilde{\nabla}}}$ with $\theta _0 =0$ (see and the argument after it). To obtain other possible $\theta _0$ we have to add to ${{\widetilde{\nabla}}}$ a form ${A'}^{(0)}-{A}^{(0)}$ whose image in $A^1(M, J/{{\Bbb C}})$ is ${{\widetilde{\nabla}}}$-closed. Therefore, the cohomology class of a possible $\theta _0$ must be in the image of the map $$H^1 (M, {{\cal O}}_M/{{\Bbb C}})\to H^2 (M, {{\Bbb C}}),$$ which is precisely $H^2_F(M, {{\Bbb C}})$ under our assumptions.
Proceeding by induction, we see that, having constructed $\theta _i, $ $i\leq n,$ and ${{\widetilde{\nabla}}}_{(n)}$ such that $$\label{eq:teta}
{{\widetilde{\nabla}}}_{(n)}^2 = \sum _{m=-1}^{n}({\sqrt{-1}}\hbar)^m \theta _m +o(\hbar ^n),$$ we can find $\theta _{n+1}$ and ${{\widetilde{\nabla}}}_{(n+1)}={{\widetilde{\nabla}}}^{(n)}+o(\hbar ^n)$ such that $${{\widetilde{\nabla}}}_{(n+1)}^2 = \sum _{m=-1}^{n+1}({\sqrt{-1}}\hbar)^m \theta _m +o(\hbar ^{n+1}).$$ The cohomology class of such $\theta_{n+1}$ can be changed by adding any element of $H^2_F(M)$.
Proceeding by induction, we see that we can construct unique ${{\widetilde{\nabla}}}$ with any given projection of $\theta$ to $H^2_F(M)[[\hbar]].$ Now observe that, if $\nabla' = {{\operatorname{exp}}\,{\operatorname{ad}}}(X)\nabla,$ then ${{\widetilde{\nabla}}}' = {{\operatorname{exp}}\,{\operatorname{ad}}}(X){{\widetilde{\nabla}}}+ \alpha$ for $\alpha \in A^1 (M, {{\Bbb C}}[[\hbar]])$ and therefore $\theta ' = {{\operatorname{exp}}\,{\operatorname{ad}}}(X) (\theta) + d\alpha.$ Therefore two connections with non-cohomologous curvatures are not equivalent. An inductive argument, similar to the ones above, shows that two connections with cohomologous curvatures are equivalent. Indeed, by adding an $\alpha$ we can arrange for $\theta '$ and $\theta$ to be equal. Then we find $X=\sum ({\sqrt {-1}}\hbar)^m X_m$ by induction. At each stage we will have an obstruction in the image of the map $$H^1(M,{{\cal O}}_M)\to H^1(M,{{\cal O}}_M/{{\Bbb C}}).$$ But this image is zero under our assumptions.
### End of the proof of Theorem \[thm:RW\]
Let us start by observing that one can define the projection $$\label{eq:Proj}
{\operatorname {Proj}}:( A^{\bullet, \bullet}(M, {\operatorname {gr}}\,J), {\nabla_0}) \to (A^{0, \bullet}(M), {\overline{\partial}})$$ as follows: if ${\cal I}$ is the DG ideal of the left hand side generated by $dz ^i$ and by the augmentation ideal of ${\operatorname {gr}}\,J$ then the right hand side is identified with the quotient of the left hand side by ${\cal I.}$ It is straightforward that ${\operatorname {Proj}}$ is a quasi-isomorphism.
Using the notation introduced in and after Definition \[df:A\_-1\], we can write $$\label{eq:flatness mod h again}
{\nabla_0}A^{(0)}+{\frac{1}{2}}[A^{(-1)},A^{(-1)}]_{2}=\theta _0$$ and $$\label{eq:flatness mod h^2}
{\nabla_0}A^{(1)}+{\frac{1}{2}}[A^{(-1)},A^{(-1)}]_{3}+[A^{(-1)},A^{(0)}]_{2}+[A^{(-0)},A^{(0)}]_{1}=\theta _1.$$ Observe that:
a\) ${\operatorname {Proj}}[A^{(-1)},A^{(-1)}]_{2}={\operatorname {Proj}}[A^{(-1)},A^{(-0)}]_{2}=0;$
b\) ${\operatorname {Proj}}[A^{(-1)},A^{(-1)}]_{3}$ depends only on the $(0,1)$ component of the form $A^{(-1)}_1;$
c\) ${\operatorname {Proj}}[A^{(0)},A^{(0)}]_{1}$ depends only on the $(0,1)$ component of the form $A^{(0)}_{-1}.$
The connection ${\nabla_0}$ can be chosen in such a way that the form from b) is equal to $$\label{eq:A-1+101}
\sum R_{ijk{\overline{l}}}{\widehat{z}}^i{\widehat{z}}^j{\widehat{z}}^kd{\overline{z}}^l;$$ therefore for this connection $${\frac{1}{2}}{\operatorname {Proj}}[A^{(-1)},A^{(-1)}]_{3}={\operatorname RW}_{\Gamma _0}(M,\omega).$$ Since $[A^{(-1)},A^{(-1)}]_{2}\in A^2 (M,{\widetilde{{{\frak{g}}}}}_{\geq 0}),$ we can choose $A^{(0)}\in A^1 (M,{\widetilde{{{\frak{g}}}}}_{\geq 1});$ we conclude, because of b) and c), that there exists ${{\widetilde{\nabla}}}$ with $\theta _0=0$ such that the projection of $\theta _1$ to $H^2 (M, {{\cal O}}_M)$ is equal to ${\operatorname RW}_{\Gamma _0}(M,\omega).$
Now we can produce a connection with a given $\theta _0$ by adding to the above connection a form $A'-A;$ for this new connection, the form from c) may be chosen as $$\sum \alpha _{i{\overline j}}{\widehat{z}}^i d{\overline{z}}^j$$ where $$\alpha = \sum \alpha _{i{\overline j}}{d{z}}^i d{\overline{z}}^j$$ is the $(1,1)$ component of a form representing the class $\theta.$ This implies $${\operatorname Proj}[A^{(0)},A^{(0)}]_{1} = \omega (\theta _0, \theta _0).$$
\[rmk:canonical deformation\] In [@NT1], 4.8, we defined the canonical deformation of the trivial gerbe on a symplectic manifold. It is easy to see that the characteristic class $\theta$ of this deformation is equal to $\frac{1}{{\sqrt{-1}}\hbar}\omega.$ We see from Theorem \[thm:RW\] that the first Rozansky-Witten class is an obstruction for the canonical stack deformation to be a sheaf of algebras.
Deformation complex of a stack as a DGLA
========================================
In this section we will construct a DGLA whose Maurer-Cartan elements classify deformations of any stack (Theorem \[thm:deformations of stacks via dgla\]). In order to that, we will start by noticing that a stack datum can be defined in terms of the simplicial nerve of a cover; if we replace the nerve by its first barycentric subdivision, we arrive at a notion of a descent datum for ${{\cal L}}$ where ${{\cal L}}$ is a simplicial sheaf of DGLAs (Definitions \[dfn:simplicial sheaf of dglas\], \[dfn:simplicial sheaf of dglas 1\]). We reduce the problem to classifying such descent data in Proposition \[lemma:iso classes as L-stacks\]. Then we replace our simplicial sheaf of DGLAs by a quasi-isomorphic acyclic simplicial sheaf of DGLAs. For the latter, classifying descent data is the same as classifying Maurer-Cartan elements of the DGLA of global sections, whence Theorem \[thm:deformations of stacks via dgla\]. It states that deformations of a stack are classified by Maurer-Cartan elements of [*De Rham-Sullivan forms with values in local Hochschild cochains of the twisted matrix algebra*]{}.
Twisted matrix algebras {#ss:Twisted matrix algebras}
-----------------------
For any simplex $\sigma$ of the nerve of an open cover $M=\cup U_i$ corresponding to $U_{i_0}\cap \ldots \cap U_{i_p}$, put $I_{\sigma}=\{i_o, \ldots, i_p\}$ and $U_{\sigma}=\cap_{i\in I}U_i.$ Define the algebra ${\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}})$ whose elements are finite matrices $$\sum_{i,j\in I_{\sigma}} a_{ij}E_{ij}$$ such that $a_{ij} \in {{\cal A}}_i(U_{\sigma}). $ The product is defined by $$a_{ij}E_{ij}\cdot a_{lk}E_{lk} = \delta_{jl} a_{ij}G_{ij}(a_{jk})c_{ijk}E_{ik}$$
We call a Hochschild $k$-cochain $D$ of ${\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}})$ [*local*]{} if:
a\) For $k=0,$ $ D=\sum_{i\in I_{\sigma}} a_{i}E_{ii};$
b\) for $k>0,$ $D(E_{i_1 j_1}, \ldots ,E_{i_k j_k}) = 0$ whenever $j_p \neq i_{p+1}$ for some $p$ between $1$ and $k-1;$
c\) for $k>0,$ $D(E_{i_1 j_1}, \ldots ,E_{i_k j_k}) $ is a product of an element of $E_{i_1 j_k}$ and an element of ${{\cal A}}$.
Local cochains form a DGL subalgebra of all Hochschild cochains $C^{\bullet +1}({\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}}),\,{\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}}))$. Denote it by ${{\cal L}}^{H, \operatorname{local}}({\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}})). $
\[rmk:local cochains and categories\] It is easy to define a sheaf of categories on $U_{\sigma}$ whose complex of Hoschild cochains is exactly the complex of local Hochschild cochains above.
De Rham-Sullivan forms. {#De Rham-Sullivan forms}
-----------------------
For any $p$-simplex $\sigma$ of the nerve of an open cover $M=\cup U_i$ corresponding to $U_{i_0}\cap \ldots \cap U_{i_p}$, let $${\mathbb Q}[\Delta _{\sigma}]={\mathbb Q}[t_{i_0}, \ldots, t_{i_p}]/(t_{i_0}+ \ldots + t_{i_p}-1)$$ and $$\Omega ^{\bullet} [\Delta _{\sigma}]={\mathbb Q}[t_{i_0}, \ldots, t_{i_p}]\{dt_{i_0}, \ldots, dt_{i_p}\}/(t_{i_0}+ \ldots + t_{i_p}-1,dt_{i_0}+ \ldots + dt_{i_p} )$$
As usual, given a sheaf ${{\cal L}}$ on $M$, define [*De Rham-Sullivan forms*]{} with values in ${{\cal L}}$ as collections $\omega _{\sigma}\in \Omega ^{\bullet} [\Delta _{\sigma}]\otimes {{\cal L}}(U_{\sigma})$ where $\sigma$ runs through all simplices, subject to $\omega _{\tau}|\Delta _{\sigma}=\omega _{\sigma}$ on $U_{\tau}$ whenever $\sigma \subset \tau$. De Rham-Sullivan forms form a complex with the differential $(\omega_{\sigma})\mapsto (d_{\operatorname{DR}}\omega_{\sigma}).$ We denote the space of all $k$-forms by ${\Omega_{\operatorname{DRS}}}^k({\mathfrak U}, {{\cal L}})$, or simply by ${\Omega_{\operatorname{DRS}}}^k({\mathfrak U})$ in the case when ${{\cal L}}={\mathbb C}.$ The complex $({\Omega_{\operatorname{DRS}}}^{\bullet}({\mathfrak U}, {{\cal L}}), d_{\operatorname{DR}})$ computes the Čech cohomology of $M$ with coefficients in ${{\cal L}}$. Finally, put $${\Omega_{\operatorname{DRS}}}^k(M, {{\cal L}})={{\operatorname{lim}}\,{\operatorname{dir}}}_{\mathfrak U}{\Omega_{\operatorname{DRS}}}^k({\mathfrak U}, {{\cal L}})$$ where the limit is taken over the category of all open covers.
We need to say a few words about the functoriality of Hochschild cochains. Usually, given a morphism of algebras $A\to B$, there is no natural morphism between $C^{\bullet}(A,A)$ and $C^{\bullet}(B,B)$ (both map to $C^{\bullet}(A,B).$ Nevertheless, in our special case, there are maps ${\operatorname{Matr}}_{\operatorname{tw}}^{\sigma}\to {\operatorname{Matr}}_{\operatorname{tw}}^{\tau}$ on $U_{\tau}$ if $\sigma\subset\tau.$ These maps do induce morphisms of sheaves of [*local*]{} cochains on the open subset $U_{\tau}$ in the opposite direction; we call these morphisms [*the restriction maps*]{}. And, as before, we consider Hochschild cochain complexes already as sheaves of complexes. For example, in all the cases we are interested in, Hochschild cochains are given by multidifferential maps.
\[dfn:DRS local\] Let ${\Omega_{\operatorname{DRS}}^{\bullet}}({\mathfrak U}, {{\cal L}}^{H,{\operatorname{local}}}({{\operatorname{Matr}}_{\operatorname{tw}}({{\cal A}})}))$ be the space of all collections $$D_{\sigma}\in {{\cal L}}^{H,{\operatorname{local}}}({\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}}))\otimes \Omega ^k(\Delta _{\sigma})$$ such that for $\sigma\subset\tau$ the restriction of the cochain $D_{\tau}|{\Delta _\sigma}$ to ${\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}})$ is equal to $D_{\sigma}$ on $U_{\tau}.$ These spaces form a DGLA with the bracket $[(D_{\sigma}),\, (E_{\sigma})]=([D_{\sigma},\,E_{\sigma}])$ and the differential $(D_{\sigma})\mapsto ((d_{\operatorname{DR}}+\delta)D_{\sigma}).$ We put $${\Omega_{\operatorname{DRS}}^{\bullet}}(M, {{\cal L}}^{H,{\operatorname{local}}}({{\operatorname{Matr}}_{\operatorname{tw}}({{\cal A}})}))={{\operatorname{lim}}\,{\operatorname{dir}}}_{\mathfrak U}{\Omega_{\operatorname{DRS}}^{\bullet}}({\mathfrak U}, {{\cal L}}^{H,{\operatorname{local}}}({{\operatorname{Matr}}_{\operatorname{tw}}({{\cal A}})}))$$
\[thm:deformations of stacks via dgla\] Isomorphism classes of deformations of any stack ${{\cal A}}$ are in one-to-one correspondence with isomorphism classes of Maurer-Cartan elements of the DGLA ${\Omega_{\operatorname{DRS}}^{\bullet}}(M, {{\cal L}}^{H,{\operatorname{local}}}({{\operatorname{Matr}}_{\operatorname{tw}}({{\cal A}})}))$.
The DGLAs above are examples of a structure that we call [*a simplicial sheaf of DGLAs.*]{}
\[dfn:simplicial sheaf of dglas\] A simplicial sheaf ${{\cal L}}$ is a collection of sheaves ${{\cal L}}_{\sigma}$ on $U_{\sigma}$, together with morphisms of sheaves $r_{\sigma\tau}:{{\cal L}}_{\tau}\to{{\cal L}}_{\sigma}$ on $U_{\tau}$ for all $\sigma \subset \tau$, such that $r_{\sigma\tau}r_{\tau\theta}=r_{\sigma\theta}$ for any $\sigma \subset \tau\subset \theta.$ A simplicial sheaf of DGLAs ${{\cal L}}$ is a simplicial sheaf such that all ${{\cal L}}_{\sigma}$ are DGLAs and all $r_{\sigma\tau}$ are morphisms of DGLAs.
\[dfn:simplicial sheaf of dglas 1\] For a simplicial sheaf of DGLAs ${{\cal L}}$, [*a descent datum*]{} is a collection of Maurer-Cartan elements $\lambda _{\sigma} \in \hbar{{\cal L}}^1(U_{\sigma}[[\hbar]]),$ together with gauge transformations $G_{\sigma\tau}:r_{\sigma\tau}\lambda_{\tau}\to \lambda_{\sigma}$ on $U_{\tau}$ and two-morphisms $c_{\sigma\tau\theta}:G_{\sigma\tau}r_{\sigma\tau}(G_{\tau\theta})\to G_{\sigma\theta}$ on $U_{\theta}$ for any $\sigma \subset \tau\subset \theta,$ subject to $$c_{\sigma\tau\omega}G_{\sigma\tau}(r_{\sigma\tau}(c_{\tau\theta\omega}))=c_{\sigma\theta\omega}c_{\sigma\tau\theta}$$ for any $\sigma \subset \tau\subset \theta\subset \omega.$
We leave to the reader the definition of isomorphisms (and two-isomorphisms) of descent data. Given a simplicial sheaf ${{\cal L}}$, and denoting the cover by ${\mathfrak U}$, one defines the cochain complex $$C^{p}({\mathfrak U}, {{\cal L}})=\prod _{\sigma _{0}\subset \ldots \subset\sigma_{p}} {{\cal L}}_{\sigma _0}(U_{\sigma_p})$$ Put $$(d_0s)_{\sigma_{0}\ldots\sigma_{p+1}}=s _{\sigma_{1}\ldots\sigma_{p+1}};$$ $$(d_is)_{\sigma_{0}\ldots\sigma_{p+1}}=s _{\sigma_{0}\ldots{\widehat{\sigma _{i}}}\ldots\sigma_{p+1}},$$ $1\leq i \leq p;$ $$(d_{p+1}s)_{\sigma_{0}\ldots\sigma_{p+1}}=r_{\sigma_{p},\sigma_{p+1}} s_{\sigma_{0}\ldots\sigma_{p}}$$ We leave to the reader the definition of the maps $s_i$. We see that $C^{\bullet}({\mathfrak U}, {{\cal L}})$ is a cosimplicial space. It is a cosimplicial DGLA if ${{\cal L}}$ is a simplicial sheaf of DGLAs.
Finally, note that, if a cover ${\mathfrak V}$ is a refinement of the cover ${\mathfrak U},$ then there is a morphism of cosimplicial spaces (DGLAs) $$C^{\bullet}({\mathfrak U}, {{\cal L}})\to C^{\bullet}({\mathfrak V}, {{\cal L}}).$$ Let $$C^{\bullet}( {{\cal L}})={{\operatorname{lim}}\,{\operatorname{dir}}}_{\mathfrak U}C^{\bullet}({\mathfrak U}, {{\cal L}}).$$ We say that ${{\cal L}}$ is [*acyclic*]{} if for every $q$ the cohomology of this complex is zero for $p>0$.
\[dfn:Cech complex\] The cochain complex $(C^{\bullet}({\mathfrak U}, {{\cal L}}), \partial +d)$ where $\partial=\sum_{i=0}^n (-1)^id_i$ is called the Čech complex of ${{\cal L}}$ with respect to the cover ${\mathfrak U}.$
The collection of sheaves ${{\cal L}}^{H, \operatorname{local}}({\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}})) $ forms a simplicial sheaf of DGLAs if one sets $r_{\sigma\tau}(\omega)$ to be the restriction of the $\omega $ to the algebra ${\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}})$. We denote this simplicial sheaf of DGLAs by ${{\cal L}}^{H, \operatorname{local}}({\operatorname {Matr}}_{\operatorname{tw}}({{\cal A}})) $.
\[lemma:iso classes as L-stacks\] Isomorphism classes of deformations over ${\mathfrak a}$ of any stack ${{\cal A}}$ are in one-to-one correspondence with isomorphism classes of descent data of the Deligne two-groupoid of ${{\cal L}}^{H, \operatorname{local}}({\operatorname {Matr}}_{\operatorname{tw}}({{\cal A}}))\otimes {\mathfrak a}.$
[**Proof**]{}. Given a deformation, it defines a Maurer-Cartan element of ${{\cal L}}^{H, \operatorname{local}}({\operatorname {Matr}}_{\operatorname{tw}}^{\sigma}({{\cal A}})) $ for every $\sigma$, namely the Hochschild cochain corresponding to the deformed product on ${\operatorname {Matr}}_{\operatorname{tw}}({{\cal A}})$. It is immediate that this cochain is local. The restriction $r_{\sigma\tau}$ sends these cochains to each other, so a deformation of ${{\cal A}}$ does define a descent datum for the Deligne two-groupoid of ${{\cal L}}^{H, \operatorname{local}}$. Conversely, to have such a descent datum is the same as to have a deformed stack datum ${\tilde {{\cal A}}} _{\sigma}$ on every $U_{\sigma}$ (with respect to the cover by $U_i \cap U_{\sigma}= U_{\sigma},$ $i\in I_{\sigma}$), together with an isomorphism ${\tilde {{\cal A}}} _{\tau}\to {\tilde {{\cal A}}} _{\sigma}$ on $U_{\tau}$ for $\sigma \subset \tau$ and a two-isomorphism on $U_{\theta}$ for every $\sigma \subset \tau \subset \theta .$ But the cover consists of several copies of the same open set, which coincides with the entire space. All stack data with respect to such a cover are isomorphic to sheaves of rings; all stack isomorphisms are two-isomorphic to usual isomorphisms of sheaves. Trivializing the stacks ${\tilde {{\cal A}}} _{\sigma}$ on $U_{\sigma}$ according to this, we see that isomorphism classes of such data are in one-to-one correspondence with isomorphism classes of the following:
1\) a deformation ${\mathbb A}_{\sigma}$ of the sheaf of algebras ${{\cal A}}_{i_0}$ on $U_{\sigma}$ where $I_\sigma = \{i_0, \ldots,i_p\}$;
2\) an isomorphism of deformations $G_{\sigma \tau}:{\mathbb A}_{\tau}\to {\mathbb A}_{\sigma}|U_{\tau}$ for every $\sigma \subset \tau$;
3\) an invertible element of $c_{\sigma \tau \rho} \in {\mathbb A}_{\sigma} (U_{\theta})$ for every $\sigma \subset \tau \subset \theta, $
satisfying the equations that we leave to the reader. Finally, one can establish a one-to-one correspondence between isomorphism classes of the above data and isomorphism classes of deformations of ${{\cal A}}$. This is done using an explicit formula utilizing the fact that sequences $\sigma_0 \subset \ldots \subset \sigma_p$ are numbered by simplices of the barycentric subdivision of $\sigma _p$ (cf., for example, [@Seg]). More precisely, given a datum ${\mathbb A}_{\sigma}, G_{\sigma \tau}, c_{\sigma \tau \rho},$ we would like to construct a stack datum ${\mathbb A}_{i}, G_{ij}, c_{ijk}.$ We start by putting ${\mathbb A}_{i}={\mathbb A}_{(i)}$ and $G_{ij}=G_{(i),(ij)}G_{(j),(ij)}^{-1}.$ Now we want to guess a formula for $c_{ijk}.$ For that, observe that $$G_{(i),(ij)}={\operatorname{Ad}}(c_{(i), (ij), (ijk)})G_{(i), (ijk)}G_{(ij), (ijk)}^{-1}$$ and $$G_{(j),(ij)}={\operatorname{Ad}}(c_{(j), (ij), (ijk)})G_{(j), (ijk)}G_{(ij), (ijk)}^{-1},$$ therefore $$G_{ij}={\operatorname{Ad}}(c_{(i), (ij), (ijk)})G_{(i), (ijk)}G_{(j), (ijk)}^{-1}{\operatorname{Ad}}(c_{(j), (ij), (ijk)}^{-1})$$ We see that $$G_{ij}G_{jk}={\operatorname{Ad}}(c_{ijk})G_{ik}$$ where $$c_{ijk}=c_{(i), (ij), (ijk)}(G_{(i), (ijk)}G_{(j), (ijk)}^{-1})(c_{(j), (ij), (ijk)}^{-1}c_{(j), (jk), (ijk)})\times$$ $$\times (G_{(i), (ijk)}G_{(k), (ijk)}^{-1})(c_{(k), (jk), (ijk)}^{-1}c_{(k), (ik), (ijk)})c_{(i), (ik), (ijk)}^{-1}$$ (as one would expect, this is an alternated product of terms corresponding to the six faces of the first barycentric subdivision of the simplex $(ijk)$, in the natural order). One checks directly that the cocyclicity condition on the $c_{ijk}$’s holds. Furthermore, given an isomorphism $H_{\sigma}, b_{\sigma \tau}$ of the data ${\mathbb A}_{\sigma}, G_{\sigma \tau}, c_{\sigma \tau \rho}$ and ${\mathbb A}'_{\sigma}, G'_{\sigma \tau}, c'_{\sigma \tau \rho},$ one defines $$H_i=H_{(i)}, \; b_{ij}=b_{(i),(ij)} b_{(j),(ij)}^{-1}$$ and checks that this is indeed an isomorphism of the corresponding data ${\mathbb A}_{i}, G_{ij}, c_{ijk}$ and ${\mathbb A}'_{i}, G'_{ij}, c'_{ijk}.$ This ends the proof of Lemma \[lemma:iso classes as L-stacks\].
### End of the proof of Theorem \[thm:deformations of stacks via dgla\]
Define the simplicial sheaf of DGLAs as follows. Put $${{\cal L}}_{\sigma}= {{\cal L}}^{H,{\operatorname{local}}}({\operatorname {Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}}))\otimes \Omega ^k(\Delta _{\sigma}),$$ with the differential $d_{\operatorname{DR}}+\delta$ and transition homomorphisms $$r_{\sigma\tau}(D_{\tau})=D_{\tau}|\Delta _{\sigma}\; {\operatorname{restricted}}\;{\operatorname{to}} \;{\operatorname{Matr}}^{\sigma}_{\operatorname{tw}}({{\cal A}}).$$ We denote this simplicial sheaf of DGLAs by $${\underline {\Omega}} ^{\bullet}_{\operatorname{DRS}} (M, {{\cal L}}^{H,{\operatorname{local}}}({{\operatorname{Matr}}_{\operatorname{tw}}({{\cal A}})})).$$ It is acyclic as a simplicial sheaf. Therefore, by Proposition \[prop:quis of cosimplicial dglas\], isomorphism classes of descent data of its Deligne two-groupoid are in one-to-one correspondence with isomorphism classes of Maurer-Cartan elements of the DGLA ${\Omega_{\operatorname{DRS}}^{\bullet}}(M, {{\cal L}}^{H,{\operatorname{local}}}({{\operatorname{Matr}}_{\operatorname{tw}}({{\cal A}})}))$, because the latter is its zero degree Čech cohomology. Now, the embedding $${{\cal L}}^{H,{\operatorname{local}}}({{\operatorname{Matr}}_{\operatorname{tw}}({{\cal A}})}))\to {\underline {\Omega}} ^{\bullet}_{\operatorname{DRS}} (M, {{\cal L}}^{H,{\operatorname{local}}}({{\operatorname{Matr}}_{\operatorname{tw}}({{\cal A}})}))$$ is a quasi-isomorphism of simplicial sheaves of DGLAs (the left hand side is the zero degree De Rham cohomology, and the higher De Rham cohomology vanishes locally). Again by Proposition \[prop:quis of cosimplicial dglas\], isomorphism classes of descent data are in one-to-one correspondence for the two simplicial sheaves of DGLAs above.
### Another version of Theorem \[thm:deformations of stacks via dgla\]
The language the previous subsection allows one to classify deformations of a given stack in terms of another DGLA which is a totalization of a cosimplicial DGLA. This is perhaps a little bit more consistent with the framework of [@H1].
\[thm:deformations of stacks via dgla II\] Isomorphism classes of deformations of a stack ${{\cal A}}$ are in one-to-one correspondence with isomorphism classes of Maurer-Cartan elements of the DGLA ${{\operatorname{Tot}}}C^{\bullet}({{\cal L}}^{H, \operatorname{local}}({\operatorname {Matr}}_{\operatorname{tw}}({{\cal A}})))\otimes {\mathfrak a}.$
Deformations of a given gerbe {#s:Deformations of a given gerbe}
=============================
{#section-2}
The aim of this section is to classify deformations of a given gerbe, trivial or not. As above, let ${{\cal A}}$ be a gerbe on $M$; by ${{\cal O}}_M$ we will denote the sheaf of smooth functions (in the $C^{\infty}$ case) or the holomorphic functions (in the complex analytic case).
The two-cocycle $c_{ijk}$ defining the gerbe belongs to the cohomology class in $H^2(M, {{{\cal O}}}_M /2\pi i{\Bbb Z})$. Project this class onto $H^2(M, {{{\cal O}}}_M / {\Bbb C})$.
\[dfn:R\] We denote the above class in $H^2(M, {{{\cal O}}}_M / {\Bbb C})$ by $R({{\cal A}})$ or simply by $R$.
The class $R$ can be represented by a two-form $R$ in ${\Omega_{\operatorname{DRS}}}^2({{\cal O}}_M/{\mathbb C})$, cf. \[De Rham-Sullivan forms\].
\[thm:classification of deformations of a gerbe\] Given a gerbe ${{\cal A}}$ on a manifold $M$, the set of deformations over ${\mathfrak a}$ of ${{\cal A}}$ up to isomorphism is in one-to-one correspondence with the set of equivalence classes of Maurer-Cartan elements of the DGLA ${\Omega_{\operatorname{DRS}}}^{\bullet}(M, C^{\bullet +1}({{\cal O}}_M, {{\cal O}}_M))\otimes{\mathfrak m}$ with the differential $d_{\operatorname{DR}} + \delta + i_R.$
Here $C^{\bullet +1}({{\cal O}}_M, {{\cal O}}_M)$ is the sheaf of complexes of multi-differential Hochschild cochains of the jet algebra; $R \in {\Omega_{\operatorname{DRS}}}^2(M, {{\cal O}}_M/{\mathbb C})$ is a form representing the class from Definition \[dfn:R\]; $i_R$ is the Gerstenhaber bracket with the Hochschild zero-cochain $R$. Explicitly, if $R$ is an element of an algebra $A$, $$i_R D(a_1, \ldots, a_n)=\sum_{i=0}^n (-1)^i D(a_1, \ldots,a_i,\, R, \ldots, a_n).$$ In Theorem \[thm:classification of deformations of a gerbe\] this operation is combined with the wedge multiplication on forms.
If the manifold $M$ is complex, we can formulate the theorem in terms of Dolbeault complexes, without resorting to De Rham-Sullivan forms.
\[thm:classification of deformations of a holomorphic gerbe\] Given a holomorphic gerbe ${{\cal A}}$ on a complex manifold $M$, the set of deformations of ${{\cal A}}$ over ${\mathfrak a}$ up to isomorphism is in one-to-one correspondence with the set of equivalence classes of Maurer-Cartan elements of the DGLA $A^{0, \bullet}(M, C^{\bullet +1}({{{\cal O}}}_M, {{{\cal O}}}_M))\otimes{\mathfrak m}$ with the differential ${\overline{\partial}} + \delta + i_R.$
Here $R \in A^{0,2} (M, {{{\cal O}}}_M/{\Bbb C})$ is a form representing the class from Definition \[dfn:R\]; $i_R$ is the Gerstenhaber bracket with the Hochschild zero-cochain $R$.
We start with a coordinate change that replaces twisted matrices by usual matrices, at a price of making the differential and the transition isomorphisms more complicated (Lemma \[lemma:first coordinate change\]). The second coordinate change ( and up) allows to get rid of matrices altogether.
The rest of this section is devoted to the proof of the theorems above.
The plan of the proof is the following. Having reduced the problem of classifying deformations of a gerbe to the problem of classifying Maurer-Cartan elements of a DGLA (Theorem \[thm:deformations of stacks via dgla\]), we will now simplify this DGLA.
### First coordinate change: untwisting the matrices {#ss:first coordinate change}
Recall that we are working on a manifold M with an open cover $\{U_i\}_{i\in I}$ and a Čech two-cocycle $c_{ijk}$ with coefficients in ${{\cal O}}^* _M$.
In what follows, we will denote by $\Omega ^k(\Delta _{\sigma}, {{\cal O}}(U_{\sigma}))$, etc. the space of forms on the simplex $\Delta _{\sigma}$ with values in ${{\cal O}}(U_{\sigma})$, etc.
We start by observing that in the definition of De Rham-Sullivan forms one can replace [*algebraic*]{} ${{\cal L}}$-valued forms $\Omega^{\bullet}(\Delta_{\sigma})\otimes {{{\cal L}}}$ by [*smooth*]{} ${{\cal L}}$-valued forms $\Omega^{\bullet}(\Delta_{\sigma},{{{\cal L}}})$ where ${{\cal L}}$ is the DGLA of local Hochschild cochains. Indeed, one DGLA embeds into the other quasi-isomorphically, and one can apply Proposition \[prop:quis of cosimplicial dglas\].
Locally, $c$ can be trivialized. Indeed, as in the proof of Proposition \[lemma:iso classes as L-stacks\], $c$ is a cocycle on $U_{\sigma}$ with respect to the cover of $U_{\sigma}$ by several copies of itself. We write $$\label{eq:first trivialization, 1}
c_{ijk} = h_{ij}(\sigma) h_{ik}(\sigma)^{-1} h_{jk}(\sigma)$$ on $U_{\sigma}$ for a simplex $\sigma $, where $h_{ij}$ are elements of $\Omega ^0(\Delta _{\sigma}, {{\cal O}}(U_{\sigma}))$. As a consequence, $$\label{eq:first trivialization, 2}
d_{\operatorname{DR}}{\operatorname{log}} h_{ij}(\sigma)-d_{\operatorname{DR}} {\operatorname{log}} h_{ik}(\sigma)+d_{\operatorname{DR}} {\operatorname{log}} h_{jk}(\sigma)=0$$
\[rmk:dependence\] At this stage the cochains $h_{ij}(\sigma)$, $a_{i}(\sigma, \tau)$ can be chosen to be constant as functions on simplices. But later they will be required to satisfy Lemma \[lemma:choice of ai\], and for that they have to be dependent on the variables $t_i$.
Note that two local trivializations of the two-cocycle $c$ differ by a one-cocycle which is itself locally trivial (by the same argument as the one before ). Therefore $$\label{eq:first trivialization, 3}
h_{ij}(\sigma) = a_{i}(\sigma, \tau) h_{ij}(\tau)a_{j}(\sigma, \tau)^{-1}$$ on $ U_{\tau}$ where $a_i$ are some invertible elements of $\Omega ^0(\Delta _{\sigma}, {{\cal O}}(U_{\tau})).$ We have another local trivialization: $$\label{eq:first trivialization, 4}
d_{\operatorname{DR}} {\operatorname{log}} h_{ij}(\sigma) = \beta _i(\sigma)-\beta _j(\sigma)$$ on $U_{\sigma}, $ where $\beta _i(\sigma)$ are elements of $\Omega ^1(\Delta _{\sigma}, {{\cal O}}(U_{\sigma}))$. Now introduce the coordinate change $$\label{eq:first coordinate change}
a_{ij}E_{ij} \mapsto a_{ij}h_{ij}(\sigma)E_{ij}$$
\[dfn:matrices\] By ${\operatorname {Matr}}_{\sigma}({{\cal A}})$ we denote the sheaf on $U_{\sigma}$ whose elements are finite sums $\sum a_{ij}E_{ij}$ where $a_{ij}\in {{\cal A}}_i.$ The multiplication is the usual matrix multiplication.
One gets immediately
\[lemma:first coordinate change\] Put $$a(\sigma , \tau)={\operatorname {diag}}\,a_{i}(\sigma, \tau)$$ and $$\beta(\sigma)={\operatorname {diag}}\,\beta_{i}(\sigma)$$
Consider the spaces of all collections $$D_{\sigma}\in \Omega ^k(\Delta _{\sigma}, {{\cal L}}^{H,{\operatorname{local}}}({\operatorname {Matr}}^{\sigma}({{\cal O}})) )$$ such that for $\sigma\subset\tau$ the restriction of the cochain $D_{\tau}|{\sigma}$ to ${\operatorname {Matr}}^{\sigma}({{\cal A}})$ is equal to ${\operatorname{Ad}}(a(\sigma,\tau))(D_{\sigma})$ on $U_{\tau}.$ These spaces form a DGLA with the bracket $[(D_{\sigma}),\, (E_{\sigma})]=([D_{\sigma},\,E_{\sigma}])$ and the differential $(D_{\sigma})\mapsto ((d_{\operatorname{DR}}+\delta+{\operatorname{ad}}(\beta(\sigma)))D_{\sigma}).$ The coordinate change provides an isomorphism of this DGLA and the DGLA ${\Omega_{\operatorname{DRS}}^{\bullet}}(M, {{\cal L}}({{\operatorname{Matr}}_{\operatorname{tw}}({{\cal A}})}))$ from Definition \[dfn:DRS local\] (modified as in the beginning of \[ss:first coordinate change\]).
### Second coordinate change {#ss:second coordinate change}
We have succeeded in replacing the sheaf of DGLAs of Hochschild complexes of twisted matrices by the sheaf of DGLAs of Hochschild complexes of usual matrices, at a price of having more complicated differential and transition functions. Both involve conjugation (or commutator) with a diagonal matrix. Our next aim is to make these diagonal matrices have all the entries to be the same. This will allow us eventually to get rid of matrices altogether.
We already have one such diagonal matrix. Indeed, from one concludes that $$\label{eq:nabla beta is scalar}
d_{\operatorname{DR}} \beta _i(\sigma)=d_{\operatorname{DR}} \beta _j(\sigma)$$ and therefore $$d_{\operatorname{DR}} \beta (\sigma) \in \Omega ^2(\Delta _{\sigma},{{\cal O}}(U_{\sigma}) )$$ is well-defined. The other one is $$\label{eq:definition of gamma}
\gamma(\sigma, \tau)= d_{\operatorname{DR}} {\operatorname{log}}a_i(\sigma, \tau)-\beta _i (\sigma) + \beta _i(\tau)$$ To see that this expression does not depend on $i$, apply $d_{\operatorname{DR}} {\operatorname{log}}$ to and compare the result with . Thus, we have a well-defined element $$\gamma(\sigma, \tau) \in \Omega ^1(\Delta _{\sigma},{{\cal O}}(U_{\tau}) ) .$$ Also, from we observe that $$\label{eq:definition of s}
s(\sigma, \tau, \theta)=a_i(\sigma, \tau)a_i(\sigma, \theta)^{-1}a_i(\tau, \theta)$$ does not depend on $i$ and therefore defines an invertible element $$s(\sigma , \tau , \theta) \in \Omega ^0(\Delta _{\sigma},{{\cal O}}(U_{\theta}) ).$$ The above cochains form a cocycle in the following sense: $$\label{eq:cech-nabla cocycle, 1}
d_{\operatorname{DR}} (d_{\operatorname{DR}} \beta)=0;$$ $$\label{eq:cech-nabla cocycle, 2}
d_{\operatorname{DR}} \beta(\sigma) - d_{\operatorname{DR}} \beta(\tau) = -d_{\operatorname{DR}} \gamma (\sigma, \tau);$$ $$\label{eq:cech-nabla cocycle, 3}
\gamma (\sigma , \tau) - \gamma (\sigma , \theta) + \gamma (\tau , \theta) = d_{\operatorname{DR}} {\operatorname {log}} s(\sigma , \tau , \theta);$$ $$\label{eq:cech-nabla cocycle, 4}
s(\sigma, \tau, \theta) s(\rho, \tau, \theta)^{-1}s(\rho, \sigma, \theta)s(\rho,\sigma, \tau )^{-1}=1$$
\[lemma:s,nablabeta,gamma\]
The cohomology of the Čech bicomplex of the complex of simplicial sheaves $$\sigma\mapsto\Omega^0(\Delta_{\sigma}, {{\cal O}}(U_{\sigma})) ^*\stackrel{d_{\operatorname{DR}}{\operatorname{log}}}{\longrightarrow} \Omega^1(\Delta_{\sigma}, {{\cal O}}(U_{\sigma})) \stackrel{d_{\operatorname{DR}}}{\longrightarrow} \Omega^2(\Delta_{\sigma}, {{\cal O}}(U_{\sigma})) \stackrel{d_{\operatorname{DR}}}{\longrightarrow}\ldots$$ is isomorphic to the Čech cohomology $H^{\bullet}(M, {\mathfrak U}; {{\cal O}}^*_M)$ with respect to the cover ${\mathfrak U}.$ Under this isomorphism, the cohomology class of the cocycle $( d_{\operatorname{DR}} \beta, \gamma, s)$ of this complex becomes the cohomology class of the cocycle $c_{ijk}$.
The proof is straightforward, using the fact that sequences $\sigma_0 \subset \ldots \subset \sigma_p$ are numbered by simplices of the barycentric subdivision of $\sigma _p$ (cf. [@Seg]; compare with the proof of Proposition \[lemma:iso classes as L-stacks\]) where a nonlinear version of the same argument is used).
From now on, we assume that the cover ${\mathfrak U}=\{U_i\}$ is good. We need another lemma to prooceed.
\[lemma:choice of ai\] The cochains $a_i (\sigma, \tau)$ can be chosen as follows: $$a_i (\sigma, \tau)=a_0 (\sigma, \tau){\widetilde{a}}_i (\sigma, \tau)$$ where $a_0 (\sigma, \tau)$ does not depend on $i$ and ${\widetilde{a}}_i (\sigma, \tau)$ take values in the subgroup $\Omega ^0(\Delta _{\sigma}, {{{\Bbb C}}} \cdot 1)^*$.
[**Proof**]{}. Choose local branches of the logarithm. We have from $${\operatorname {log}}a_i (\alpha, \sigma)-{\operatorname {log}}a_i (\alpha, \tau)+{\operatorname {log}}a_i ( \sigma , \tau)-{\operatorname {log}}s (\alpha, \sigma, \tau)=2\pi {\sqrt{-1}} N_i(\alpha, \sigma, \tau)$$ where $N_i(\alpha, \sigma, \tau)$ are constant integers. The Čech complex of the simplicial sheaf $\sigma \mapsto \Omega ^0(\Delta _{\sigma}, {{\cal O}}_{U_{\sigma}}) $ is zero in positive degrees. Let $S$ be a contracting homotopy from this complex to its zero cohomology. Put $$b_i (\sigma)= {\operatorname{exp}}(S({\operatorname {log}}a_i (\alpha, \sigma)));$$ then $$b_i(\sigma)b_i(\tau)^{-1} = a_i(\sigma, \tau)^{-1} {\widetilde{a}}_i(\sigma, \tau) a(\sigma, \tau)$$ where $${\widetilde{a}}_i(\sigma, \tau)={\operatorname{exp}}(2\pi {\sqrt{-1}} S(N_i (\alpha, \sigma, \tau)))$$ and $$a(\sigma, \tau)={\operatorname{exp}}(S(s(\alpha, \sigma, \tau)))$$ Therefore we can, from the start, replace $h_{ij}(\sigma)$ by $b_i(\sigma)h_{ij}(\sigma)b_j(\sigma)^{-1}$ in , and $a_i (\sigma, \tau)$ by $ {\widetilde{a}}_i(\sigma, \tau) a(\sigma, \tau)$ in . This proves the lemma.
Now consider the operator $$i _{\beta(\sigma)}: {\Omega}^{\bullet} (\Delta _{\sigma}, C^{\bullet+1 }({\operatorname{Matr}}({{\cal O}}))) \to {\Omega}^{\bullet+1} (\Delta _{\sigma}, C^{\bullet}({\operatorname{Matr}}({{\cal O}})))$$ This operator acts by the Gerstenhaber bracket (at the level of $C^{\bullet})$, combined with the wedge product at the level of $\Omega^{\bullet}$, with the cochain $\beta (\sigma) \in {\Omega}^{1} (\Delta _{\sigma}, C^0 ({\operatorname{Matr}}({{\cal O}})))$. One has $$[\delta, i _{\beta(\sigma)}] = {\operatorname{ad}} _{\beta(\sigma)}:{\Omega}^{\bullet} (\Delta _{\sigma}, C^{\bullet }({\operatorname{Matr}}({{\cal O}}))) \to {\Omega}^{\bullet+1} (\Delta _{\sigma}, C^{\bullet}({\operatorname{Matr}}({{\cal O}})))$$ and $$[d_{\operatorname{DR}}, i _{\beta(\sigma)}] = i _{d_{\operatorname{DR}}\beta(\sigma)}$$ which is an operator $${\Omega}^{\bullet} (\Delta _{\sigma}, C^{\bullet+1 }({\operatorname{Matr}}({{\cal O}}))) \to {\Omega}^{\bullet+2} (\Delta _{\sigma}, C^{\bullet}({\operatorname{Matr}}({{\cal O}})))$$ Now define the second coordinate change as $$\label{eq:second coordinate change}
{\operatorname{exp}}(i _{\beta(\sigma)})$$ on ${\Omega}^{\bullet} (\Delta _{\sigma}, C^{\bullet}({\operatorname{Matr}}({{\cal O}})))$. This coordinate change turns the DGLA from Lemma \[lemma:first coordinate change\] into the following DGLA. Its elements are collections of elements $$\label{eq:new DGLA 1}
\omega _{\sigma}\in {\Omega}^{\bullet} (\Delta _{\sigma}, C^{\bullet}({\operatorname{Matr}}^{\sigma}({{\cal O}}({U_{\sigma}}))))$$ such that the restriction of $D_{\tau}|\Delta_{\sigma}$ to the subalgebra ${\operatorname{Matr}}^{\sigma}({{\cal O}}({U_{\sigma}}))$ is equal to $$\label{eq:new DGLA 3}
{\operatorname{exp}}(i _{\beta(\sigma)}-i_{\beta(\tau)}){\operatorname{Ad}}(a(\sigma, \tau))D_{\sigma};$$ the differential is $$\label{eq:new DGLA 2}
d_{\operatorname{DR}} + \delta +i _{d_{\operatorname{DR}} \beta(\sigma)}$$ We can replace by $$\label{eq:new DGLA 4}
{\operatorname{exp}}(i _{\gamma(\sigma, \tau)}-i_{{d_{\operatorname{DR}}}{\operatorname{log}}{a_0}(\sigma, \tau)} - i_{{d_{\operatorname{DR}}}{\operatorname{log}}{\widetilde{a}}(\sigma, \tau)})){\operatorname{Ad}}(a_0(\sigma, \tau))D_{\sigma}$$ where ${\widetilde{a}}(\sigma, \tau)={\operatorname{diag}}\,{\widetilde{a}}_i(\sigma, \tau)$ (cf. Lemma \[lemma:choice of ai\]).
Getting rid of matrices
-----------------------
Consider the morphism $$C^{\bullet}({{\cal O}}_{U_{\sigma}}) \to C^{\bullet}({\operatorname{Matr}}^{\sigma}({{\cal O}}_{U_{\sigma}})$$ defined as follows. Put ${\overline{{{\cal O}}}}={{\cal O}}/{\Bbb C}$. Then for $D\in C^p({{\cal O}},{{\cal O}}),$ $D:{\overline{{{\cal O}}}}^{\otimes p} \to {{{{\cal O}}}},$ define $${\widetilde{D}}(m_1a_1, \ldots, m_pa_p)= m_1\ldots m_p D(a_1, \ldots, a_p)$$ where $a_i \in {{\cal O}}$ and $m_i \in M({\Bbb{C}}).$ The following is true:
a\) the cochains ${\widetilde{D}}$ are invariant under isomorphisms ${\operatorname{Ad}}(m)$ for $m\in GL({\Bbb C})$;
b\) the cochains ${\widetilde{D}}$ become zero after substituting and argument from $M({\Bbb C})$.
It is well known that the map $D\mapsto {\widetilde{D}}$ is a quasi-isomorphism with respect to the Hochschild differential $\delta$. Therefore this map establishes a quasi-isomorphism of the DGLA from , , , with the following DGLA: its elements are collections $D_{\sigma}\in \Omega ^{\bullet} (\Delta _{\sigma}, C^{\bullet +1}({{\cal O}}({U_{\sigma}})))$ such that $$\label{eq:neznayu 2}
D_{\tau}|{\Delta _{\sigma}}={\operatorname{exp}}(i_{\gamma (\sigma, \tau)} - i_{d{\operatorname{log}}\,{a_0}(\sigma, \tau)})D_{\sigma}$$ on $U_{\tau}$, with the differential $$\label{eq:neznayu 1}
d_{\operatorname{DR}} + \delta + i_{ d_{\operatorname{DR}}\beta(\sigma)}.$$
Now consider any cocycle $r(\sigma)\in {\Omega }^2 (U_{\sigma}, {{\cal O}}/{\Bbb C})$, $t(\sigma, \tau)\in \Omega ^1 ( U_{\tau}, {{\cal O}}/{\Bbb C});$ $$r(\sigma)-r(\tau) + t(\sigma, \tau) = 0;$$ $$t(\sigma, \tau)-t(\sigma, \theta)+t(\tau, \theta)=0$$ Such a cocycle defines a of DGLA of collections $D_{\sigma}$ as above, where gets replaced by $$\label{eq:neznayu 21}
D_{\tau}|{\Delta _{\sigma}}={\operatorname{exp}}(i_{t(\sigma, \tau)})D_{\sigma}$$ and the differential is $d_{\operatorname{DR}} + \delta + i_{r(\sigma)}$ If two cocycles differ by the differential of $u(\sigma) \in \Omega^1(\Delta ^{\sigma}, {{\cal O}}(U_{\sigma})/{\Bbb C})$, then operators ${\operatorname {exp}}(i_{u(\sigma)})$ define an isomorphism of DGLAs. Finally, put $r(\sigma)= \beta(\sigma)$ and $t(\sigma, \tau)=\gamma (\sigma, \tau)-d{\operatorname{log}}{\,a_0}(\sigma, \tau)$. This is a cocycle of Č$^{\bullet} (M, {{{\cal A}}}_M({{\cal O}}/{\Bbb C}))$. It lies in the cohomology class of the cocycle $({\operatorname{log}}\,s, \; \gamma, \;d_{\operatorname{DR}} \beta)$ from Lemma \[lemma:s,nablabeta,gamma\]. Now replace this cocycle by a cohomologous cocycle which has $t=0$.
This proves that isomorphism classes of deformations of a gerbe ${{\cal A}}$ are in one-to-one correspondence with isomorphism classes of Maurer-Cartan elements of the DGLA of collections of cochains $$D_{\sigma}\in \Omega^{\bullet}(\Delta _{\sigma}, C^{\bullet +1}({{\cal O}}_{U_{\sigma}}, {{\cal O}}_{U_{\sigma}}))$$ such that $D_{\sigma}|U_{\tau}=D_{\tau};$ the differential is $d_{\operatorname{DR}}+\delta+i_R$ where $R\in \Omega ^2 _{\operatorname{DRS}}(M,{{\cal O}}/{\mathbb C})$ represents the class $R$ as defined in the beginning of this section. To pass to the DGLA of Dolbeault forms (Theorem \[thm:classification of deformations of a holomorphic gerbe\]), we apply Proposition \[prop:quis of cosimplicial dglas\].
### The jet formulation
Theorem \[thm:classification of deformations of a gerbe\] also admits a formulation in the language of jets. As above, let $J_M$ be the bundle of algebras whose fiber at a point is the algebra of jets of $C^{\infty}$, resp. holomorphic, functions on $M$ at this point; this bundle has the canonical flat connection ${\nabla _{{\operatorname {can}}}}$. Horizontal sections of $J_M$ correspond to smooth, resp. holomorphic, functions.
The two-cocycle $c_{ijk}$ defining the gerbe belongs to the cohomology class in $H^2(M, {{{\cal O}}}_M /2\pi i{\Bbb Z})$. Project this class onto $H^2(M, {{{\cal O}}}_M / {\Bbb C})$ and denote the result by $R$ (as in Definition \[dfn:R\]). The class $R$ can be represented by a two-form $R$ in $A^2 (M, J_M/{\Bbb C})$.
\[thm:classification of deformations of a gerbe, jet version\] Given a gerbe ${{\cal A}}$ on a manifold $M$, the set of deformations of ${{\cal A}}$ over ${\mathfrak a}$ up to isomorphism is in one-to-one correspondence with the set of equivalence classes of Maurer-Cartan elements of the DGLA $A^{\bullet}(M, C^{\bullet +1}(J_M, J_M))\otimes{\mathfrak m}$ with the differential ${\nabla _{{\operatorname {can}}}}+ \delta + i_R.$
Here $C^{\bullet +1}(J_M, J_M)$ is the complex of vector bundles of Hochschild cochains of the jet algebra; $R \in A^2 (M, J_M/{\Bbb C})$ is a form representing the class from Definition \[dfn:R\]; $i_R$ is the Gerstenhaber bracket with the Hochschild zero-cochain $R$. The proof follows from a simple application of Proposition \[prop:quis of cosimplicial dglas\].
Deformations of gerbes on symplectic manifolds {#s:Deformations of gerbes on symplectic manifolds}
==============================================
{#section-3}
For a gerbe on $M$ defined by a cocycle $c$, we denote by $c$ the class of this cocycle in $H^2(M, {{{\cal O}}} _M /2\pi i{\Bbb Z})$ and by $\partial c$ its boundary in $H^3(M, 2\pi i{\Bbb Z})$.
\[thm:symplectic classification\] Let ${{\cal A}}$ be a gerbe on a symplectic manifold $(M, \omega)$. The set of isomorphism classes of deformations of ${{\cal A}}$ compatible to $\omega$:
a\) is empty if the image of the class $\partial c$ under the map $H^3(M, 2\pi i{\Bbb Z})\to H^3(M, {\Bbb C})$ is non-zero;
b\) is in one-to-one correspondence with the space $Def(M,\omega)$ (Theorem \[thm:classification of deformations of the trivial gerbe, symplectic case\]) if the image of the class $\partial c$ under the map $H^3(M, 2\pi i{\Bbb Z})\to H^3(M, {\Bbb C})$ is zero.
Let $R$ be the projection of $c$ to $H^2(M, {{{\cal O}}} _M /{\Bbb C})$, as in Definition \[dfn:R\].
\[thm:symplectic holomorphic classification\] Let ${{\cal A}}$ be a gerbe on a complex symplectic manifold $(M, \omega)$. The set of isomorphism classes of deformations of ${{\cal A}}$ compatible to $\omega$ is:
a\) is empty if $R\neq 0$;
b\) is in one-to-one correspondence with the space $Def(M,\omega)$ if $R=0$.
[**Proof.**]{} The arguments from the proof of Theorem \[thm:classification of deformations of the trivial gerbe, symplectic case\] show that deformations of a gerbe are classified exactly as in -, with one exception: equation should be replaced by the requirement that the class of $c$ modulo $A^2 (M, {{\Bbb C}}+ \hbar{\operatorname{gr}} J)[[\hbar]]$ should coincide with $R$ where $R$ is a form defined before Theorem \[thm:classification of deformations of a gerbe\]. Therefore, if $R=0,$ the classification goes unchanged; if $R\neq 0$ in $H^2(M, {{{\cal O}}} _M /{\Bbb C})$, then $$\label{eq:flatness mod h with R}
{\nabla_0}A^{(0)}+{\frac{1}{2}}[A^{(-1)},A^{(-1)}]_{2}=R$$ shows that no connection $\nabla$ exists.
[ABCDEF]{} O. Ben-Bassat, J. Block, T. Pantev, [*Noncommutative tori and Fourier-Mukai duality*]{}, QA/0509161. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, [*Deformation theory and quantization*]{}, Ann. Phys. [**111**]{} (1977). p. 61–151. J. Block, [*Duality and equivalence of module categories in noncommutative geometry, I*]{}, QA/0509284. A. K. Bousfield, V. K. A. M. Gughenheim, [*On $PL$ De Rham theory and rational homotopy type*]{}, Memoirs of the A.M.S. vol. 8, number 179 (1976). A. K. Bousfield, D. M. Kan, [*Homotopy limits, completions and localizations*]{}, Lecture Notes in Math. [**304**]{}, Springer Verlag, 1972. R. Bezrukavnikov, A. Kaledin, [*Fedosov quantization in algebraic context*]{}, Moscow Mathematical Journal, [**4**]{}, 3, 592–616. J. L. Brylinski, [*Loop groups, Chern classes and geometric quantization*]{}, Progress in Mathematics, Birkhäuser, [**107**]{}. A. Connes, [*Noncommutative differential geometry*]{}, Inst. Hautes Études Sci. Publ. Math. No. 62 (1985), p. 257–360. V. Chekhov, V. Fock, [*Quantum Teichm*]{}[*ü*]{}[*ller spaces*]{}, (Russian) Teoret. Mat. Fiz. [**120**]{} (1999), no. 3, p. 511–528; translation in Theoret. and Math. Phys. [**120**]{} (1999), no. 3, p. 1245–1259. P. Deligne, [*Déformations, de l’algèbre de fonctions d’une varieté symplectique*]{}, Selecta Math. [**1**]{} (1995), 667–698. V. Drinfeld, [*Letter to Schechtman*]{}, September 1988. A. D’Agnolo, P. Polesello, [*Stacks of twisted modules and integral transforms*]{}, Geometric Aspects of Dwork’s Theory, Walter de Gruyter, Berlin (2004), 461–505. M. De Wilde, P. Lecomte, [*Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds*]{}, Lett. Math.Phys. [**7**]{} (1983), no. 6, p.486–493. B. Fedosov, [*Deformation Quantization and Index Theorem*]{}, Akademie Verlag, 1996. B. Fedosov, [*A simple geometric construction for deformation quantization*]{}, Journal of Differential Geometry [**40**]{} (1994), p.213–238. E. Getzler, [*Lie theory for nilpotent $L_{\infty}$ algebras*]{}, math.AT/0404003. E. Getzler, [*A Darboux theorem for Hamiltonian operators in the formal calculus of variations*]{}, Duke Mathematical Journal [**111**]{} (2002), no. 3, 535–560. M. Gerstenhaber, [*The cohomology structure of an associative ring*]{}, Ann. Math. [**[78]{}**]{} (1963), 267–288. J. Giraud, [*Cohomologie non abélienne*]{}, Gründlehren [**179**]{}, Springer Verlag (1971). W. Goldman, J. Milson, [*The deformation theory of representations of fundamental groups of compact Kähler manifolds*]{}, Publ. Math. IHES, [**67**]{} (1988), 43–96. V. Hinich, [*Deformations of sheaves of algebras*]{}, Adv. Math. [**195**]{} (2005), 1, 102-164. V. Hinich, [*Descent on Deligne groupoids*]{}, IMRN 1997, no. 5, 223–239. V. Hinich, V. Schechtman, [*Deformation theory and Lie algebra homology*]{}, I, II, Algebra Colloq. [**4**]{} (1997), 2, 213–240; 3, 291-316. M. Kapranov, [*Rozansky-Witten invariants via Atiyah classes*]{}, Compositio Math. [**115**]{} (1999), no. 1, 71–113. M. Kashiwara, [*Quantization of symplectic manifolds*]{}, Publ. Res. Inst. Math. Sci. [**32**]{} no.1 (1996), 1–7. M. Kontsevich, [*Deformation quantization of algebraic varieties*]{}, Euroconférence Moshe Flato, Part III (Dijon, 2000), Lett. Math. Phys. [**56**]{} no.3 (2001), 271–294. M. Kontsevich, [*Deformation quantization of Poisson manifolds*]{}, Lett. Math. Phys. [**66**]{} (2003), no. 3, 157–216. M. Kontsevich, [*Rozansky-Witten invariants via formal geometry*]{}, Compositio Math. [**115**]{} (1999), no. 1, 115–127. A. Kapustin, [*Topological strings on noncommutative manifolds*]{}, hep-th/0310057. V. Mathai, R. Melrose, I. Singer, [*The index of projective families of elliptic operators*]{}, Geom. Topol. [**9**]{} (2005), p. 341–373 (electronic). V. Mathai, J. Rosenberg, [*$T$-duality for torus bundles with $H$-fluxes via noncommutative topology*]{}, Comm. Math. Phys. [**253**]{} (2005), 3, 705–721. V. Mathai, J. Rosenberg, [*T-duality for torus bundles with $H$-fluxes via noncommutative topology, II, The higher-dimensional case and the $T$-duality group*]{}, Adv. Theor. Math. Phys. [**10**]{} (2006), 1, 123–158. R. Nest, B. Tsygan, [*Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems*]{}, Asian J. Math. [**5**]{} (2001), no. 4, 599–635. R. Nest, B. Tsygan, [*Remarks on modules over deformation quantization algebras*]{}, Moscow Matehmatical Journal [**4**]{} (2004), 911–940. N. O’Brian, [*Geometry of twisting cochains*]{}, Compositio Math. [**63**]{} (1987), 1, 41–62. N. O’Brian, D. Toledo, Y. L. Tong, [*Hierzebruch-Riemann-Roch for coherent sheaves*]{}, Amer.J.Math. [**103**]{} (1981), 2, 253–271. N. O’Brian, D. Toledo, Y. L. Tong, [*The trace map and characteristic classes for coherent sheaves*]{}, Amer. J. Math. [**103**]{} (1981), 2, 225–252. P. Polesello, [*Classification of deformation-quantization algebroids on complex symplectic manifolds*]{}, math.AG/0503400. P. Polesello, P. Schapira, [*Stack of quantization deformation modules on complex symplectic manifolds*]{}, IMRN (2004), no. 49, p.2637–2664. L. Rozansky, E. Witten, [*Hyper-Kähler geometry and invariants of three-manifolds*]{}, Selecta Math. (N.S.) [**3**]{} (1997), no. 3, 401–458. M. Schlessinger, [*Functors on Artinian rings*]{}, Transactions AMS [**130**]{} (1968), 208–222. M. Schlessinger, J. Stasheff, [*The Lie algebra structure on tangent cohomology and deformation theory*]{}, Journal of Pure and Applied Algebra [**38**]{} (1985), 2-3, 313–322. G. Segal, [*Classifying spaces and spectral sequences*]{}, Inst. Hautes Études Sci. Publ. Math. No. 34 (1968), p. 105–112. P. Ševera, [*Quantization of Poisson families and of twisted Poisson structures*]{}, Lett. Math. Phys. [**63**]{} (2003), no. 2, p. 105–113. P. Ševera, [*On deformation quantization of Dirac structures*]{}, math.QA/0511403. P. Ševera, A. Weinstein, [*Poisson geometry with a 3-form background*]{}, Noncommutative geometry and string theory (Yokohama, 2001), Progr. Theoret. Phys. Suppl. No. 144 (2001), 145–154. A. Yekutieli,[*Deformation Quantization in Algebraic Geometry*]{}, Advances in Mathematics [**198**]{} (2005), p. 383–432
|
---
abstract: 'Modeling the structure formation in the universe, we extend the spherical collapse model in the context of MOND starting with the linear Newtonian structure formation followed by the MONDian evolution. In MOND the formation of structures speed up without a need for dark matter. Starting with the top-hat over-dense distribution of the matter, the structures virialize with a power–law profile of the distribution of matter. We show that the virialization process takes place gradually from the center of the structure to the outer layers. In this scenario the smaller structures enter to the MONDian regime earlier and evolve faster, hence they are older than larger structures. We also show that the virialization of the structures occur in the MONDian regime, in which the smaller structures have stronger gravitational acceleration than the larger ones. This feature of the dynamical behavior of the structures is in agreement with this fact that the smaller structures as the globular clusters or galactic bulges have been formed earlier and need less dark matter in CDM scenario.'
author:
- 'M. Malekjani, S. Rahvar , H. Haghi'
title: 'Spherical Collapse in Modified Newtonian Dynamics (MOND)'
---
Introduction
============
The conjecture for the existence of dark matter dates back to Zwicky in 1933, who failed to explain the dynamics of Coma cluster by the virial theorem through the distribution of visible matter [@zwi]. To interpret the dynamics of the structure, the concept of missing mass or dark matter entered the Astrophysical studies since then. In addition to the cosmological scales, in the smaller galactic scales, the flat rotation curve of the spiral galaxies requires the existence of the dark matter halo [@bos81]. On the other hand studying the dynamics of the universe as a whole reveals that the universe is dominated by the dark matter and dark energy [@spe03]. This hypothetical matter neither emits light nor interacts with the ordinary matter and only shows its presence through its gravitational interaction. The advantage of the Cold Dark Matter ($CDM$) model is that, it can successfully explain the rotation curve of the spiral galaxies and lensing by galaxies and cluster of galaxies. Within the framework of general relativity (GR) it also provides a reasonable description for the hierarchy in the structure formation. Although currently $CDM$ and $\Lambda CDM$ models are remarkably successful in large scales [@spe03], they cannot explain Tully-Fisher and Freeman laws [@bosch00]. On the other hand, the high-resolution N-body simulations are still in contradiction with the observations on sub-galactic scales where they predict orders of magnitude more substructures than what is observed [@moo99; @kly99]. They also provide incompatible spatial distribution of the sub-halos [@kro04].
The other alternative to the dark matter is the modification to the gravity law by means of taking a generic form of action for the gravity rather than that of the Einstein-Hilbert action. This approach has been introduced to be an alternative model to the dark energy [@car04]. $f(R)$ gravity also is used to interpret the rotation curve of the spiral galaxies [@sob07; @saf08]. There is also recent efforts on modifying the gravity law by using a simple kinetic Lagrangian whether the pressure can bend space-time sufficiently to replace the roles of dark energy, cold dark matter and heavy neutrinos in explaining anomalous accelerations at all scales [@zhao07]. Halle et al (2008) also proposed a generalized lagrangian with non-uniform cosmological constant for the vacuum field within the framework of the Einstein gravity.
Finally the third approach, which we are concerned with in this work is the modification of the conventional Newtonian law is so-called MOdified Newtonian Dynamics (MOND). The dynamics of a structure in MOND under the gravitational field is given by [@mil83]: $$\mu(g/a_0)\bf{g} = \bf{g_N}, \label{mond}$$ where $g_N$ is the Newtonian gravitational acceleration, $a_0
=1.2\times 10^{-10}ms^{-2}$ is a fundamental acceleration parameter and $\mu(x)$ is a function for the transition from the Newtonian to the MONDian regime (e.g. $\mu(x) = x/\sqrt{1+x^2}$). The dynamics of the structure for $g<a_0$ deviates from the Newtonian law by $\mu(x)$ and for $a_0\ll g $ we recover the Newtonian dynamics (i.e. $\mu =1$). On the other hand for $g \ll a_{0}$, so-called deep MOND regime, $\mu(x) = x$ and the effective acceleration is given by $g=\sqrt{g_{N}a_{0}}$.
Due to confusion in the definition of the dynamical concepts in MOND, this model can be interpreted as a modification to the gravity law instead of dynamics. Bekenestein and Milgrom (1984) used a non-conventional Newtonian action for the gravity to extract the modified Poisson equation as follows: $$\nabla.\left[\mu(\nabla\phi/a_0)\bf{\nabla\phi}\right] = 4\pi G
\rho, \label{mond2}$$ where in the spherical symmetric systems this equation reduces to equation (\[mond\]). One of the problems with MOND is that it is not a covariant gravity model. Bekenestein (2004) proposed a covariant formulation of this model. This theory in addition to the metric has a scalar field as well as a 4-vector field called TeVeS, where in the limit of non-relativistic, small accelerations, the field equation reduces to that in MOND. The total potential in this theory is given by the sum of the Newtonian potential $\phi_N$ and the potential due to the scalar field, $\phi_s$: $$\Phi = \Phi_{N} + \phi_s ,$$ where the added scalar field plays the role of the dark matter. TeVeS theory has also a Newtonian limit for the non-relativistic dynamics with significant acceleration. In the non-relativistic limit we can simplify the gravity inside the spherical system as $$g(r) = -\nabla\Phi \simeq \left\{
\begin{array}{ll}
\sqrt{a_0 g_N}, & g_N \leqslant a_0; \\
g_N, & otherwise.
\end{array}
\right.
\label{eqn:eqmotionTeves2}$$ The advantage of MOND is that it could provide a successful fit to the rotation curve of spiral galaxies and dispersion velocity of elliptical galaxies [@mil03; @san96]. It has been tested against the Cosmic Background Radiation [@sko06], gravitational lensing [@che06; @zha06; @ang07], stellar systems and galactic dynamics [@hagh06; @lon07; @tir07], solar system [@bek06; @ser06] and Tully-Fisher and Freeman laws [@Mcg99; @Mcg00]. MOND also decreases the mass discrepancy in the cluster of galaxies [@poi05] but yet in clusters it remains necessary to invoke the undetected matter, possibly in the form of a massive neutrino [@san99; @san03; @agu02; @san02].
Studying the cosmology and formation of the large structures in the universe is another tool to examine MOND. Studying the MONDian scenario of the structure formation has been started by Felten (1984) and Sanders (1998). In the paper by Sanders (1998), it is shown that a patch of universe in the MONDian regime smaller than the horizon size can evolve with a different rate than the background, hence the structures naturally can be formed through this scale dependent dynamics. In this scenario small structures form before the larger ones and this provides a bottom-up hierarchical procedure for the formation of the structures in the universe. The problem with this model is that the center of collapse is not identified and every point in the space depending on the choice of the coordinate system can be considered as the center of the structure. For solving this problem one can consider MOND formula to be applied to a peculiar acceleration developing from density fluctuation rather than Hubble expansion [@san01; @nus01]. Nusser (2002) used the amplitude of the CMB anisotropies as the initial condition in the N-body simulation to simulate the large scale structures in the universe. However to have a compatible result, one has to either decrease $a_0$ by one order of magnitude or reduce the amplitude of the fluctuations at the initial condition. Recent studies in the formation of the galaxies by Sanders (2008) shows that massive elliptical galaxies may be formed at $z>10$, as a consequence of the monolithic dissipation-less collapse. Applying MOND with cooling mechanism put an upper limit to the stellar clustering in the form of the galaxy. Extending the structure formation in TeVeS theory has been done by linear perturbation of metric, vector and scalar fields. The predictions are compatible with the observations of the structures [@dod]. Skordis (2008) also used a generalized TeVeS theory to construct the primordial adiabatic perturbations of a general family of the scalar field kinetic functions.
In this work we extend the spherical collapse model in MOND for studying the general behavior of the structures during their formation. The initial baryonic density contrast for the structures is taken from the CMB anisotropy. We obtain the dynamics of the baryonic structures in the early epoches with the linear Newtonian structure formation until the entering of the structure to the MONDian regime and follow the evolution with the MOND. The dynamical evolution with MOND shows that the structures virialize with power-law distribution of matter. We show that while all of the structures re-collapse and virialize in the MONDian regime, the gravitational acceleration of the structures in this stage inversely depends on the size of the structure. This dynamical behavior of the structure formation is compatible with the less-existence of the dark matter in the globular clusters and central parts of the galaxies in CDM scenario.
The organization of the paper is as follows: In section \[str\] we give a brief review of the spherical collapse model in MOND and extend it by looking at the dynamics of each layer in the onion model. In section \[evol\] we discuss about the results of the calculation, showing that the density profile inside the structure is a time varying function during the evolution and the structure at the final stage, virialize from the center to the outer areas. In section \[conc\] we summarize and discuss the results.
Structure Formation in MOND {#str}
===========================
In this section we introduce the MONDian cosmology and apply MOND in the spherical collapse model for studying the formation of the structures. Here we take the onion model for the spherical structure, dividing the sphere to the co-centric shells and studying the evolution of each layer separately and the structure as a whole.
MONDian cosmology {#initial}
-----------------
\[mondcos\] In the standard cosmology the dynamics of the universe in the matter dominated regime can be derived from the Newtonian gravity. Sanders (1998) used this approach to obtain the dynamics of universe in the MONDian scenario. In MODian cosmology, a patch of universe can evolve with a different rate than the background as soon as the acceleration fulfills the condition of $g<a_0$. The result is dynamical decoupling of the smaller scales from the background which causes the production of the over dense regions in the universe. The reason for this feature of cosmological dynamics in MOND is that unlike the Newtonian mechanics, the acceleration in the comoving frame depends on the length scale.
Let us take a spherical region with radius $r$ from the background in which $a_0 \ll g(r)$. Using the Newtonian dynamics the acceleration is given by $$\ddot{r}=-\frac{GM}{r^{2}}, \label{Newt_equation}$$ where $M$ is the active gravitational mass and to have a compatible relation with the relativistic results, we define it to be composed of the relativistic and non-relativistic matter as $$M=\frac{4\pi r^{3}}{3}(\rho+3p), \label{mass}$$ where $\rho$ and $p$ are the density and pressure of the cosmic fluid. Substituting equation (\[mass\]) in equation (\[Newt\_equation\]), the acceleration is given by: $$\ddot{r}=-\frac{4\pi G}{3}(\rho + 3p) r. \label{nc}$$ In equation (\[nc\]), the gravitational acceleration increases linearly with $r$. This implies that there should be a critical radius $r_{c}$ where inside it the acceleration is smaller than the MOND threshold $a_{0}$ and the dynamics is given by the MOND where outside that radius it is Newtonian. This critical length scale obtain by equaling the left hand side of equation (\[nc\]) with $\ddot{r} = - a_0$ as $$r_{c}=\frac{3 a_0}{4\pi G(\rho+3p)}. \label{rc}$$ This length scale separates the Newtonian $(r>r_c)$ and the MONDian $(r<r_c)$ domains. As the density of universe changes with the expansion of the universe, the critical radius also changes with time. From the continuity equation, the matter and the radiation densities vary as $\rho=\rho_0 a^{-3}$ and $p = p_0 a^{-4}$. Using the definition of the critical density $\rho_c = 3 H_0^2/8 \pi G$, equation (\[rc\]) can be written in terms of the density parameters, $\Omega$, and the scale factor $$r_{c}=\frac{2a_{0}}{H_{0}^{2}|\Omega_{b}^{(0)}a^{-3}+2\Omega_{r}^{(0)}a^{-4}-2\Omega_{\Lambda}^{(0)}|},
\label{rc2}$$ where we adapt $H_{0}=75 km s^{-1} Mpc^{-1}$, $\Omega_{b}^{(0)}=0.02$, $\Omega_{r}^{(0)}=5\times10^{-5}$ and $\Omega_{\Lambda}^{(0)}=0$.
Now we do comparison of the size of a structure with $r_c$. From equation (\[rc2\]) the critical radius $r_c$ changes with the scale factor as $r_c\propto a^{4}$ in the radiation and $r_c\propto
a^{3}$ in the matter dominant epoches. On the other hand the size of the structure is proportional to the scale factor (i.e. $\lambda
\propto a$). So we expect that the Newtonian structures eventually will enter the MONDian regime as $r_c$ grows faster than $\lambda$. Using the adapted cosmological parameters, the critical radius at the present time is obtained $r_c\simeq 10 H_0^{-1}$ which means that the whole observable universe resides in a MONDian domain. Comparing these two length scales at the last scattering surface results in $r_c/H^{-1}\simeq 10^{-4}$. The mass corresponding to this critical radius at last scattering surface is about $M_{c}
\simeq 10^4 M_\odot$ and for $M<M_c$ the dynamics is given by MOND. While we expect to have density contrast growth for the scales $r<r_c$ at the decoupling, comparing the critical mass $M_c \simeq
10^4 M_\odot$ with the Jeans mass of $M_J \simeq 10^{5}M_\odot$ at this time indicates that the structures at these scales should be washed out by the pressure [@san98].
The other feature of MONDian cosmology is that unlike the standard cosmology where decoupling redshift is smaller than the equality redshift ($z_{de}<z_{eq}$), in MOND the equality epoch is much after than the decoupling time. This feature results from this fact that decoupling is related to the baryonic density of the universe and the temperature and both parameters depend only on the scale factor, independent of the dynamics of the universe. So we expect to have the same decoupling redshift in the MOND as the standard cosmology. However since the dark matter does not exist in the MONDian cosmology, the equality will be shifted to the lower redshifts. For our adopted cosmological parameters the equality redshift is obtained $z_{eq} = 400$.
Structure Formation: Spherical Collapse Model
---------------------------------------------
In this part we model the evolution of a structure with an over-dense spherical region in MOND. To calculate the evolution of this spherical patch, for simplicity we take this over-dense region with a top-hat distribution of matter. As the acceleration of this structure depends on the distance from the center, we expect to have different dynamics for each radius. Hence, we divide the structure into the co-centric spherical shells like an onion model in cosmology [@lem33; @tol34; @bon47] and calculate the dynamics of each shell separately. The initial density contrast for this over-dense region is taken from the fluctuations of the last scattering surface. In the Newtonian treatment of the structure formation $(r_c<\lambda)$, the density contrast grows linearly $(\delta \propto a)$ from the decoupling epoch up to the entrance of the structure to $r_c$. For the MONDian regime $(\lambda<r_c)$, we switch the dynamics to MOND an calculate the evolution of the structure. As an example let us take a sphere with a mass of $M=10^{11}M_{\odot}$ and find its acceleration with respect to the center. In Figure (\[N-MOND\]) we compare the acceleration of this spherical structure in MONDian and Newtonian dynamics as a function of the redshift. We note that the redshift is defined according to the dynamics of the scale factor at the background. At the early epoches, the difference between these two dynamics is small as $r_c<\lambda$, but after entering the structure to the MONDian regime $\lambda \lesssim r_c$, the evolution of the structure by the Newtonian dynamics and MOND start to diverge. This deviation of the dynamics from that of Newtonian plays the role of the dark matter in the standard scenario of the structure formation.
For calculating the dynamics of each shell in the onion model, we take the following notation: $r^i(t)$ is the radius of $i$th shell as a function of time and $t^i_{ent}$ and $r^i_{ent}$ are the entering time and radius of the $i$th shell to the MONDian domain, respectively. The velocity of $i$th shell in terms of the Hubble parameter at entrance time is given by $v^{i}_{ent}= H_{ent}
r^i_{ent}(1-\delta^i_{ent})$, where $H_{ent}$ is the Hubble parameter of the Newtonian background and $\delta^i_{enter}$ is the density contrast of the sphere enveloped by $r^i$. As we discussed in the previous section, all the shells will eventually enter the MONDian regime and we take this time as the initial condition for each shell in the MONDian evolution of the structures. Table (\[tab1\]) shows the initial density contrast, radius and the corresponding redshift of the entrance of each shell to the MONDian regime. In the MONDian regime, the acceleration approximately is $\sqrt{g_{N}a_0}$ and the evolution of each shell is as follows: $$\ddot{r^i}=-\frac{\sqrt{GM^i a_0}}{r^i}, \label{monddyn}$$ where $M^i$ is the mass of the structure enveloped by the $i{\it
th}$ shell. Using the initial conditions given by Table (1), we obtain the evolution of each shell as shown in Figure (\[R-G\]). To visualize the evolution of the shells, we divide the sphere into ten equidistant shells when all the structure is in the Newtonian regime and obtain their evolution as a function of background redshift. The dynamics of shells shows that the inner shells evolve faster, reaching to a maximum radius and then collapse earlier than the outer ones. Here the initial radius of outermost shell is about $14 kpc$ at the entrance time of $z_{enter} \sim 146$ to the MOND regime. This shell expands up to a maximum radius of $\sim49 kpc$ at $z_{max}\sim 28$. Eventually, the shell starts to collapse and virialize at $z\simeq 18$ with the radius of $\sim28.5 kpc$. The maximum radius of each shell, $r_{max}$, is obtained from integrating equation (\[monddyn\]), letting $\dot{r}(t)=0$ as follows: $$\label{alpha}
r^i_{max}=r^i_{ent}e^{\alpha}, ~~~
\alpha=\frac{{v^i_{ent}}^{2}}{2\sqrt{GM^i a_0}}.$$ The next phase of the evolution of shells after reaching to a maximum radius is re-collapsing. Similar to the standard scenario of the spherical collapse models we expect that the global radial velocity of the structure during the free fall collapse convert to the dispersion velocity and prevent the structure from a catastrophic collapse. This steady stage of the structure is given by the virial theorem. The corresponding radius that fulfill the virial condition is called the virial radius and is calculated from: $$\label{energy_cons}
\frac{1}{2}r^i\frac{d V(r^i)}{dr^i}+ V(r^i) = E, \label{vir}$$ where $V(r^i)$ is the gravitational potential at the $i$th shell. The potential in the Newtonian or MONDian regimes is given by $$V(r^i) = \left\{
\begin{array}{ll}
\sqrt{GM^ia_0}\ln(r^i), & MOND; \\
-\frac{GM^i}{r^i}+\frac{GM^i}{r^i_{ent}} + \sqrt{GM^ia_0}\ln(r^i_{ent}). & Newt.
\end{array}
\right.
\label{pot2}$$ For the non-dissipative evolution of the structure, the total energy is conserved and we substitute the right hand side of equation (\[vir\]) by the energy of the system at the enterance time to the MONDian regime, $E = \frac{1}{2}H_{ent}^2 {r^i_{ent}}^2+\sqrt{GM^i
a_0}\ln r^i_{ent}$. Using the potentials given by equation (\[pot2\]) at the left hand side of the equation (\[vir\]), the virial radius for the MOND and Newtonian regimes obtain as $$r^i_{vir} = \left\{
\begin{array}{ll}
r^i_{ent}e^{\alpha-1/2}, & MOND; \\
r^i_{ent}(2-\frac{H_{ent}^2{r^i_{ent}}^2}{\frac{G M^i}{r^i_{ent}}})^{-1}. & Newt.
\end{array}
\right.
\label{vir2}$$ If the virialization of the structure takes place in the MONDian regime, $r^i_{vir}>r^i_{ent}$ implies $\alpha>1/2$ which results in $2T^i_{ent}/W^i_{ent}>1$ where $T^i_{ent}$ is the kinetic energy for a unit mass and $W^i_{ent} = r^i {dV^i}/{dr^i}$. On the other hand if the structure virializes in the Newtonian regime, $
r^i_{vir}<r^i_{ent}$ condition from the equation (\[vir2\]) implies $2T^i_{ent}/W^i_{ent}<1$. We calculate the expression of $2T^i_{ent}/W^i_{ent}$ for each shell (see Table \[tab1\]) and show that all the shells of the structure virialize in the MONDian regime (i.e. $r_{vir}^i>r_{ent}^i$).
Table (\[tab2\]) shows the parameters of shells for the moment of maximum radius and the virialization stage and Fig.(\[R-G\]) visualizes quantitative behavior of the shells during their evolution. In this figure the spots on the evolution curves shows the two critical stages of the maximum radius and the virialization radius, reported in Table (\[tab2\]). The evolution lines of the shells show that the inner shells evolve faster and virialize at the higher redshifts, while the outer shells evolve slower.
Predictions of the Model {#evol}
========================
In this section we discuss the evolution of the density contrast and the profile of matter distribution inside the structure. We use the definition of the density contrast of the shells in our model: $$\delta^i(t)=\frac{\rho^i(t)-\bar\rho(t)}{\bar\rho(t)},$$ where $\rho^i$ is the density of $i$th shell and $\bar\rho$ is the density of the background. As the collapsing of the shells starts from the inner to the outer parts of the structure, we will not have a shell crossing during the evolution of the structure and from the conservation of the mass, we can perform the Jacobian transformation from the initial distribution of matter inside the sphere to the evolved distribution as follows: $$\rho^i=\rho^i_{ent}(\frac{r^i_{ent}}{r^i})^2\frac{\delta
r^i_{ent}}{\delta r^i}, \label{rho}$$ where $\delta r^i$ represents the thickness of the $i{\it th}$ shell. Substituting the dynamics of each shell from the previous section in equation (\[rho\]), we obtain the evolution of density contrast for the shells up to the virialization stage as shown in Figure (\[dens\]). The inner shells evolve faster and reach the non-linear regime $(\delta>1)$ at the higher redshifts while the outer ones evolve slower. Figure (\[f4\]) shows the dependence of the corresponding non-linear redshift to the mass of structure. Comparing a small scale structure with the mass of $\sim 10^8
M_\odot$ with a galaxy having the mass of $\sim 10^{11} M_\odot$ shows that the former structure enters the non-linear regime at $z_{nl}\simeq 106$ while the later one becomes non-linear at $z_{nl}\simeq 33$. More details on the characteristic redshifts of the structures in terms of their masses is reported in Table (\[tab2\]). From this table we extrapolate the dependence of the non-linear redshift, maximum radius redshift and virialization redshift to the mass of structure with the following functions (see Fig. \[f4\]): $$\begin{aligned}
\log(z_{nl}) &=& -0.167 \log(\frac{M}{M_{\odot}}) + 3.38, \nonumber\\
\log(z_{max}) &=& -0.192\log(\frac{M}{M_{\odot}}) + 3.56, \nonumber\\
\log(z_{vir}) &=& -0.203 \log(\frac{M}{M_{\odot}}) + 3.48.\end{aligned}$$ In what follows we describe the qualitative predictions of this simple model.
Density Profile
---------------
In this part we compare the evolution of the density profile of the structures in the spherical collapse model for the Newtonian and MONDian regimes. In the Newtonian regime $(g>a_0)$ we have seen that the dynamics of the shells in the structure depends only on time and transforming to a comoving frame, the dynamics is scale independent (see equation \[nc\]). This means that the dynamics is invariant under the scale transformation and the result is preserving the initial profile of the structure. In MOND $(g<a_0)$, since the dynamics depends on the scale unlike the Newtonian case the density profile will change with time. In Fig. (\[dens-prof\]) we plot the spatial variation of density from equation (\[rho\]) for a galaxy mass structure in four different stages of $z = 400$, $87$, $23$ and $18$. The initial stage at the redshift of $z = 400$ is taken a top-hat distribution for the density when the innermost shell enters to the MONDian regime. We continuously do Jacobian transformation from this stage to the later times (i.e. $z<400$) and obtain the density of each shell. While the structure evolve, the distance of shells change with time as shown with a point on the profile representing the position of each shell (Fig. \[dens-prof\]). As the inner shells enter to the MONDian regime earlier we expect the density profile deviates from the homogeneous distribution. Fig.(\[dens-prof\]) shows a small deviation of the density profile from the homogenous one at $z = 87$. At this moment all the shells are in the MONDian domain, however as the inner shells have been evolved faster, they are more dense than the outer shells. Each shell after virialization freezes and preserves its density. We fit the density profile of the structure when the outermost shell, the latest part of the structure, virializes and the result is a power-law function of $\rho\propto r^{\beta}$ with the index of $\beta \simeq -1.22$.\
Age of a structure
------------------
A question that may be answered in this simple model is the dependence of the age of the structure to the mass. Observations show that small isolated structures as the globular clusters or the center of galaxies are older than the spiral arms [@cha98]. In this model we have seen that the smaller structures enter the MONDian regime earlier, evolve faster and virialize at the higher redshifts compared to the larger structures. As an example, in Table (\[tab2\]) it is shown that a structure with the mass of $10^{8}
M_\odot$ virializes at $z \simeq 71$ while the whole of galaxy virializes at $ z \simeq 18$. The age of a structure from the virialization up to now is given by $t_{age} = \int^{t_0}_{t_{vir}}
dt$, where $t_0$ is the present age of the Universe. Changing the variable to the redshift results in: $$t_{age} =
H_0^{-1}\int_{0}^{z_{vir}}\frac{dz}{(1+z)\sqrt{\Omega_b^{(0)}(1+z)^3
+ \Omega_r^{(0)}(1+z)^4 + (1-\Omega_t^{(0)})(1+z)^2}},$$ where we have adopted the cosmological parameters in (\[mondcos\]). The age of structures depending on their masses is given in Table (\[tab2\]).
Contribution of dark matter in the smaller structures
-----------------------------------------------------
The other question in the formation of the structures is that why the smaller structures are mainly made of the baryonic matter and have a less dark matter compared to the larger ones. This feature of the structures can be explained if we can show that in addition to the fast evolution of the smaller structures, the acceleration of the structure in terms of $a_0$, ($g_N/a_0$) gets larger. To show this property of the structures in MOND, we calculate the gravitational acceleration at the virialization time of each shell and compare it with $a_0$. As the density of a structure at this stage changes with $\rho\propto r^{\beta}$, the gravitational acceleration will depend on $r$ as $g_N(r) \propto r^{\beta +1}$. For $\beta = - 1.22$ the smaller radii should have larger acceleration compare to the larger ones.
In Figure (\[na\]) we plot the gravitational acceleration for each shell at the time of virilization in terms of the virialization radius. For the outer shells the gravitational acceleration, $g_N/a_0$ is smaller than the inner shells. This means that the smaller structures after virialization need less dark matter compare to the larger ones in the standard CDM scenario. Finally we plot the mass of the structure after virialization in terms of its size in Figure (\[f7\]). This feature reveals the general property of structures in MOND and an accurate result to compare with the observation may be obtained by N-body simulation of the structure formation.
Conclusion {#conc}
==========
Summarizing this work, we extend the spherical collapse model to study the generic properties of the structure formation in MOND. We showed that in the MONDian scenario, structures can evolve without a need to the dark matter and have the following three main features: ([**a**]{}) MONDian spherical collapse unlike to that of CDM does not preserve the initial density profile of the structure. In MOND starting with an initial homogenous density profile the structure virializes with a power-law distribution of the matter, having singularity at the center. ([**b**]{}) We showed that the small scale structures enter the MONDian regime earlier than the larger ones, evolve faster and virialize at higher redshifts. This picture from the spherical collapse model in MOND provides a bottom-top scenario for the structure formation in which the smaller structures formed before the larger ones. This result is compatible with the observations where the old stars are located in the smaller structures such as the globular clusters or center of galaxies. ([**c**]{}) Finally we showed that while the smaller structures enter the MONDian regime before the larger ones, they have larger gravitational acceleration at the virialization time and hence need a less dark matter in CDM scenario.
We would like to thank anonymous referee for useful comments. Also we would like to thank M. Nouri-Zonoz for reading the manuscript and giving useful comments.
[50]{} Aguirre A., Schaye J., Quataert E., 2002, AJ, 561, 550.
Angus, G.W., [*et al.*]{}, 2007, ApJ, 654, L13.
Bekenestein, J. D., 2004, Phys. Rev. D 70, 083509.
Bekenstein, J., Magueijo, J., 2006, Phys. Rev. D 73, 103513.
Bekenestein, J. D., Milgrom, M., 1984, ApJ, 286, 7.
Bennett, C. L., [*et al.*]{}, 2003, ApJS, 148, 1B.
Bondi, H., 1997, MNRAS, 107, 343.
Bosch, V. D., Dalcanton, J. J., 2000, ApJ. 534, 146.
Bosma, A. 1981, AJ, 86, 1825.
Carroll, S. M., [*et al.*]{}, 2004, Phys. Rev. D 70, 043528.
Chanoyer, B., [*et al.*]{}, 1998, ApJ, 494, 98.
Chen, D. M., Zhao H. S., 2006, ApJ, 650, L9.
Dalcanton, J. J., Spergel, D. N. and Summers, F. J., 1997 ApJ. 482, 659.
Dodelson, S., & Liguori, M. 2006, Phys. Rev. Lett., 97, 231301.
Felten, J.E. 1984 ApJ , 286, 3.
Haghi, H., Rahvar, S., Hasani-Zonooz, A., 2006, ApJ, 652, 354.
Halle, A., Zhao, H. S., Li, B., 2008, ApJS, 177, 1.
Klypin, A. [*et al.*]{}. 1999, ApJ, 522, 82.
Kroupa, P., Theis, C., & Boily, C. 2004, Astronomische Nachrichten Supplement, 325, 55.
Lemaitre, G., 1933, Annales Soc. Sci. Brux. Ser. I Sci. Math. Astron. Phys. A 53, 51. For an English translation, see: Lemaitre, G., 1997, Gen. Rel. Grav. 29, 641.
McGaugh S. S. and de Blok W. J. G., 1998 ApJ, 499, 66.
McGaugh S. S., [*et al.*]{}, 2000, ApJ, 533, L99.
Milgrom, M., 1983, ApJ, 270, 365.
Milgrom, M., Sanders, R. H., 2003, ApJ, 599, L25.
Moore, B. [*et al.*]{} 1999, ApJ, 524, L19.
Nipoti, C., Londrillo, P., Ciotti, L., 2007, ApJ, 660, 256.
Nusser A., 2002, MNRAS, 331, 909.
Nusser A., Pointecouteau E., 2006, MNRAS, 366, 969.
Pointecouteau, E., Silk, J., 2005, MNRAS, 364, 654.
Sanders R. H., 1996, ApJ, 473, 117.
Sanders R. H., 1998, MNRAS, 296, 1009.
Sanders R. H., 1999, ApJ, 512, L23.
Sanders R. H., 2001 APJ, 560, 1.
Sanders R. H., 2003, MNRAS, 342, 90.
Sanders, R. H. 2008, MNRAS, 386, 1588.
Sanders R. H., McGaugh S., 2002, ARA&A, 40, 263.
Sereno M., Jetzer Ph., 2006, MNRAS, 371, 626.
Skordis C., [*et al.*]{}, 2006, Phys. Rev. Lett., 96, 011301.
Skordis C., [*et al.*]{}, 2008, Phys. Rev. D 77, 123502.
Sobouti, Y., 2007, A&A, 464, 921.
Saffari, R., Rahvar, S., 2008, Phys. Rev. D 77, 104028.
Spergel, D. N., [*et al.*]{}, 2003, ApJS 148, 175.
Stachniewicz S., Kutschera M., 2001, Acta Phys. Pol. B, 32, 3629.
Stachniewicz S., Kutschera M., 2005, MNRAS, 362, 89.
Tiret, O.,Combes, F., 2007, A&A, 464, 517.
Tolman, R. C., 1934, Proc. Natl. Acad. Sci. U.S.A. 20, 410.
Zhao, H.S., [*et al.*]{}, 2006, MNRAS, 368, 171.
Zhao, H. S., 2007, ApJ, 671, L1.
Zwicky, F., 1933, Helvetica Physica Acta 6, 110.
![Comparison of the accelerations in the Newtonian (dashed-line) and MONDian (solid-line) regimes for a region with a galaxy mass scale ($M=10^{11}M_\odot$) as a function of redshift. For the early universe the difference between the two dynamics is small but increases at the later times. The corresponding redshift for entering of the structure to the critical radius is $z_{enter}
= 146$.[]{data-label="N-MOND"}](f1.eps)
![The evolution of shells inside the spherical structure with a galaxy mass $( 10^{11}M_{\odot})$ as a function of redshift, $r_i
= r_i(z)$. We take ten equidistant shells at the initial condition for representing the evolution of each shell. The trend of the evolution is reaching to a maximum radius, recollapsing and virializing . The dashed line connects the maximum radii of the shells and the dotted line indicates the virialized line, connecting the virilize points of each shell.[]{data-label="R-G"}](f2.eps)
![A log-log plot of the density contrast evolution for each shell as a function of the background redshift. Curves from the left to the right corresponds the inner to the outer shells of the structure. The horizon line represents $\delta=1$, separate the linear and non-linear regimes. The inner layers reach to the non-linear regime at the higher redshifts while the outer ones become non-linear at the lower redshifts.[]{data-label="dens"}](f3.eps)
![Dependence of the non-linear redshift (solid line), maximum radius redshift (dashed-line) and virialized redshift (dotted-dashed line) as a function of the mass of structure in logarithmic scale. Smaller structures enter to the non-linear regime at higher redshifts and the larger ones enter at lower redshifts. This feature of dependence of mass to the characteristic redshifts reveals that the smaller structure in this model form before the larger ones.[]{data-label="f4"}](f4.eps)
![ The evolution of density profile in logarithmic scale normalized to the initial density $y= \rho/\rho_{initial}$, as a function of distance from the center of structure in four different redshifts. The initial density profile is taken top-hat distribution with ten equidistant shells for representing the evolution of the density. The inner most shell enters to the MOND domain at $z= 400$. Each point on the curves notifies the position of the shell. We plot density profile of the structure for $z=400$ and $z=87$ at the left panel and $z=23$ and $z=18$ at the right panel for comparison. At $z=17$ all the shells virialize and the density profile freezes with a power-low function of $\rho \propto r^{\beta}$ where $\beta\simeq
-1.22$.[]{data-label="dens-prof"}](f5.eps)
![The Newtonian gravitational acceleration of each shell normalized to $a_0$ as a function of shell size at the virialization time. Spots on the curve represent the acceleration and the position of each shell. The gravitational acceleration in the inner shells is larger than the outer shells.[]{data-label="na"}](f6.eps)
\[Newt\_acceleration\]
![The mass of structure in terms of size of structure after virialization. Spots on the curve represent the the corresponding value for each shell. Structures with the mass smaller than $\simeq10^{10}M_{\odot}$ are virializad with the radius smaller than $\sim10 kpc$.[]{data-label="f7"}](f7.eps)
[crrrrr]{} i &$M_i [10^{11}M_{\odot}]$ & $r_{i}(t_{enter})[kpc]$ & $z_{eneter}$ & $\delta_{enter}\times 10^{-5}$ & ${2T^i_{ent}}/{W^i_{ent}}$\
1&$0.001$&$0.45$ & $400$ & $2.71 $ & 2.24\
2&$0.008$&$1.25$ & $297$ &$3.70 $ & 2.28\
3&$0.027$&$2.40$ & $242$ & $4.52 $ & 2.32\
4&$0.064$&$3.55$ & $218$ & $5.03 $ & 2.36\
5&$0.124$&$5.03$ & $195$ & $5.61 $ & 2.41\
6&$0.215$&$6.35$ & $184$ & $ 5.97 $& 2.48\
7&$0.341$&$8.17$ & $170$ & $6.47 $ & 2.51\
8& $0.510$&$9.77$ & $161$ & $6.81 $& 2.53\
9& $0.725$&$11.44$ & $154$ & $7.12 $& 2.55\
10 & $1.000$&$13.57$& $146$ & $7.51 $& 2.57\
[crrrrrrrrrrr]{} $i$ &$M[10^{11}M_\odot]$&$r_{max}[kpc]$&$z_{max}$ &$\delta_{max}$&$r_{vir} [kpc]$ &$z_{vir}$& $\delta_{vir}$&$z_{nl}$ & age\[Gyr\]\
$1$ &$0.001$ &$1.32$ &$106.52$ &$1.01$ &$0.79$& $71.46$ &$28.51$& $106.52$& 12.48\
$2$ &$0.008$ &$3.72$ &$73.94$ &$1.25$ &$2.21$ &$49.84$ &$34.37$& $78.57$& 12.43\
$3$ &$0.027$ &$7.43$ &$58.88$ &$1.44$ &$4.45$ &$38.52$ &$36.82$& $65.66$& 12.38\
$4$ &$0.064$ &$11.24$ &$49.53$ &$1.52$ &$6.85$ &$32.57$ &$38.32$& $54.28$& 12.34\
$5$ &$0.124$ &$16.24$ &$43.44$ &$1.65$ &$9.79$ &$28.41$ &$40.30$& $48.75$& 12.29\
$6$ &$0.215$ &$20.77$ &$39.26$ &$1.72$ &$12.46$ &$25.57$ &$42.55$& $44.58$& 12.25\
$7$ &$0.341$ &$27.72$ &$35.23$ &$1.84$ &$16.01$ &$22.80$ &$44.49$& $40.66$& 12.19\
$8$ &$0.510$ &$32.78$ &$32.65$ &$1.89$ &$19.79$ &$21.11$ &$45.05$& $38.19$& 12.17\
$9$ &$0.725$ &$38.63$ &$30.75$ &$1.96$ &$24.01$ &$19.83$ &$46.21$& $36.32$& 12.15\
$10$&$1.000$ &$46.98$ &$28.02$ &$2.06$ &$28.51$ &$17.99$ &$48.72$& $33.66$& 12.09\
|
---
abstract: 'We describe the time-dependent restricted-active-space self-consistent-field (TD-RASSCF) method for a system of interacting bosons. We provide the theory of the method and discuss its numerical implementation. The method provides a general wavefunction based approach to solve the time-dependent and time-independent Schrödinger equation for a system of bosons. It is based on the time-dependent variational principle to optimize at each instant of time a set of time-dependent coefficients and time-dependent orbitals used to describe the total wavefunction. Including the concept of a restricted-active-space, the exponential growth of the configurational space, resulting from all possible distributions of $N$ bosons in $M$ orbitals, can be controlled trough a specific excitation scheme. We show, by illustrative time-independent and time-dependent examples, that the method provides an accurate description of the system with a substantially smaller configurational space than the one required in the multi-configurational time-dependent Hartree method for bosons (MCTDHB). The TD-RASSCF method can also tackle problems beyond the reach of the MCTDHB method when a large number of orbitals are required.'
author:
- Camille Lévêque
- Lars Bojer Madsen
bibliography:
- 'biblio.bib'
title: 'Time-dependent restricted-active-space self-consistent-field theory for bosonic many-body systems'
---
Introduction
============
Since the first realizations of Bose-Einstein condensates (BEC) [@Anderson95; @Bradley95; @Davis95], the experimental and theoretical investigation of trapped cold atoms has attracted much attention. It is nowadays experimentally possible to design and control systems with a specific number of atoms [@Preiss15; @Kaufman14] trapped in various potential shapes [@Bloch05; @Jotzu14] and dimensions [@Greiner02], with tunable inter-particle interactions [@Courteille98; @Inouye98], and to provide a controllable transition from a few- to a many-particle system. Such a detailed control of cold atom systems has opened the possibility to simulate various physical systems [@Bloch12] from solid-state physics [@Anderson98] to black-holes analogs [@Steinhauer16] through matter-light interaction [@Sala13] and electrons dynamics in molecules [@Sengstock15].
Various theoretical models [@Bloch08] have been used so far to describe static and dynamical properties of many-boson systems, among which, a handful are exactly solvable. One of the most prominent models was introduced by Lieb and Liniger [@Liniger63; @Lieb63], to describe a system of spinless bosons interacting through a two-body contact interaction: using the Bethe *Ansatz* and periodic boundary conditions the resulting Schrödinger equation can be solved exactly for any interaction strength and an arbitrary number of bosons. Unfortunately, this model is exactly solvable only without a trapping potential. In the limit of infinite interaction strength, the Tonks-Girardeau model use the Fermi-Bose mapping to map the wavefunction of bosons into a fermonic wavefunction of non-interacting fermions with frozen parallel spins [@Girardeau60]. This mapping provides the exact solution for the ground-state of the system for arbitrary trapping potentials and remains valid also for the excited states, as well as non-equilibrium solutions also for any external potential [@Yukalov05]. In the case of non-interacting bosons or more generally in the Gross-Pitaevskii (GP) limit, i.e., $N \rightarrow \infty $ and $N\lambda = constant$, with $N$ the number of bosons and $\lambda$ the interaction strength, the GP equation or its time-dependent (TD-GP) analog provides the exact description of the system. In this situation, the exact wavefunction of the system is described by a single product of single-particle functions and the interactions between the particles are correctly described by the mean-field approach. The above models assume that the bosons interact through a pair-wise contact potential. Considering other types of interaction potentials between the particles, other models can be solved exactly with an external potential. One model uses an inverse-harmonic interaction between the particles and can be solved exactly with a harmonic trapping potential [@Calogero69], while an other model considers a harmonic interaction potential [@Cohen85; @Yan03]. The latter model has the peculiarity that it can be solved exactly numerically also for time-dependent Hamiltonians with a time-dependent trapping potential or a time-dependent interaction potential [@Lode12].
These exact models, unfortunately, do not cover the large variety of interaction or trapping potentials that are encountered in experiments. Nonetheless, they are of primary interest as they provide a unique way to benchmark numerical methods and approximations. The GP equation can be simplified when the potential and interaction energies are much larger than the kinetic energy, giving rise to the Thomas-Fermi approximation when the kinetic energy is neglected [@Baym96]. On an other hand, to overcome the lack of correlation in the GP theory and to take into account a small depletion of the BEC, i.e., to account for atoms which are not in the condensate, a perturbative expansion of the particle number in the condensate leads to Bogoliubov theory [@Bogoliubov47; @Lee57; @Lee57_2]. In the specific case of periodic trapping potentials, such as optical lattices [@Greiner02], for weak contact interactions and deep lattices the Bose-Hubbard model (BHM) [@Fisher89] is obtained by expanding the Bose field operator in term of the Wannier functions of the lowest Bloch band and neglecting the tunneling between nonconsecutive sites and interactions between different sites. The BHM and its various extensions have been extensively and successfully used to describe the ground state of trapped atoms in optical lattices and their dynamics [@Dutta15]. A more general and efficient numerical approach to deal with optical lattices is the density-matrix renormalization group (DMRG) method [@White92; @White92_2; @White93] based on the matrix product states *Ansatz* [@Schollwock05]. The method has been used to provide accurate results for ground and exited states of the system, and more recently has been used to investigate time-dependent systems [@Vidal04; @Daley04; @White04]. The second wide-spread and promising numerical method to study trapped atoms is the quantum Monte Carlo (QMC) approach. It includes, among others, the variational Monte Carlo (VMC) [@McMillan65] and the diffusion Monte Carlo (DMC) [@Anderson75; @Reynolds82; @Blume2001; @Astrakharchik04] methods, which used a Bijl-Jastrow decomposition of the wavefunction [@Bijl40; @Jastrow55], but are, however, not applicable to time-dependent systems.
Along with the above theory developments it has been a long standing idea to explore quantum chemistry methodologies to describe a *time-independent* system of trapped cold atoms. This idea was, to the best of our knowledge, introduced by the work of Ersy [@Esry97], applying the mean-field Hartree-Fock (HF) theory and the configuration interaction (CI) method up to double excitations (CISD) to harmonically trapped bosons. The HF method for bosons can be viewed has a variant of the GP theory but has the advantage that it provides a set of optimized virtual orbitals, i.e., non-occupied orbitals, that can be subsequently used in a CI expansion of the wavefunction. The CI expansion corrects the lack of correlation between the particles, not included at the HF level. The CI method is in principle exact but requires a severe truncation of the CI expansion to be numerically tractable. Later, Streltsov et *al* [@Streltsov06] introduced the multiconfigurational Hartree theory for bosons (MCHB), which is an extension of the multiconfiguration *self-consistent* field (MCSCF) method introduced for fermions and widely used in electronic-structure calculations in atoms and molecules [@Lowdin55]. The MCHB method uses a CI expansion *Ansatz* for the many-body wavefunction in which both the coefficients of the expansion and the orbitals are variationally optimized, providing better accuracy with substantially less configurations and orbitals. The coupled-cluster (CC) method was originally introduced in nuclear physics [@Coester58; @Coester60] and subsequently extended to describe electronic wavefunctions in atoms and molecules [@Cizek66]. This framework was also extended to bosons up to double excitations (CCSD) by Cederbaum et *al*, and successfully applied to various particle numbers and interaction strengths [@Cederbaum06].
Over the past decade, numerous numerical methods have been developed [@Lysaght09; @Hochstuhl12; @Pabst12; @Kvaal12; @Sato13; @Bauch14] to tackle the problem of *time-dependent* multi-electron dynamics induced by laser pulses that are strong or short or both [@Popmintchev10; @Calegari14; @Kraus15]. In short, the various successful methods used so far to investigate static properties of atoms and molecules have been extended to solve the time-dependent Schrödinger equation including a time-dependent operator. Among these methods, the multiconfigurational time-dependent Hartree-Fock method [@Zanghellini03; @Kato04; @Nest05; @Haxton11] variationally optimizes a set of time-dependent orbitals and CI coefficients, following the idea of the multiconfigurational time-dependent Hartree (MCTDH) method [@Meyer90; @Beck00], originally introduced to describe molecular dynamics. The MCTDHF method has been extended to identical bosons, within the framework of the MCTDH for bosons MCTDHB [@Alon08], in which the indistinguishability is taken into account using permanents instead of Slater determinants. Further development includes the case of particle mixtures of different type of bosons and fermions [@Alon07; @Alon12]. The fundamental concept of using a set of *time-dependent* single-particle functions or orbitals to expand the total wavefunction offers the possibility to use substantially less orbitals than in the case of time-independent orbitals, because the former basis optimally adapts during the evolution of the system. Recently, the framework of the multi-layer (ML) MCTDH method [@Wang03; @Manthe08; @Vendrell11] was extended to systems of bosons and mixtures of them [@Kronke13; @Cao13]. The method uses a ML expansion to reduce the size of the wavefunction in comparison to the MCTDHB method for multi-species or multi-dimensional systems. It is particularly effective for systems which can be subdivided in strongly interacting subsystems while the individual subsystems interact only weakly with each other. In the case of a one-dimensional system consisting of only a single type of particles, the ML-MCTDHB and MCTDHB wavefunctions are identical [@Schmitz13].
The MCTDHB method shed new light on the dynamics of trapped cold atoms, especially when fragmentation occurs and more than one orbital is populated - a situation which can not be describe by the TD-GP theory. Fragmentation occurs in different systems such as during the dynamics at a Josephson junction [@Sakmann09], which is a universal phenomenon [@Sakmann14], and can not be described, even qualitatively, using the TD-GP or BH theories. In double-well trapping potentials, fragmentation of the BEC is also obtained for the ground-state [@Sakmann08] for large barrier height between the two wells and the GP theory fails to describe the variance of position and momentum operators [@Klaiman15]. Multiconfigurational methods are also required to accurately describes the formation and dynamics of fragmented states with repulsive or attractive interactions between the particles [@Streltsov08; @Streltsov09; @Streltsova14] and tunneling of a many boson system to open space [@Lode12_2] or tunneling of trapped vortices [@Beinke15]. Using time-dependent orbitals reduce the number of orbitals and thus the number of configurations required to describe accurately time-evolving systems in comparison with methods with time-independent orbitals. Nevertheless, simulations using such full-configurational wavefunctions remain a difficult task due to the exponential scaling of the configurational space, i.e., the dimensionality determined by the number of ways to arrange $N$ particles in $M$ orbitals, especially for bosons.
This challenge leads us to the quest for a method which maintains the appealing properties of the time-dependent orbitals based methods mentioned above, but is free from the exponential scaling problem. One such method uses the concept of a restricted active-space (RAS), well-known in quantum chemistry, where it has been applied with time-independent molecular orbitals [@Olsen88]. The RAS based method was successfully extended to time-dependent orbitals in the time-dependent restricted active-space self-consistent-field method (TD-RASSCF) to deal with electron dynamics in atoms [@Haru13; @Haru14_1]. Introducing a RAS scheme by fixing the promotion of the electrons between three sets of orbital spaces can considerably reduced the number of configurations. In addition, the theory has the specificity to include, as limiting cases, the TD Hartree-Fock (HF), the TD complete active space self-consistent field (TD-CASSCF) [@Sato13] and the MCTDHF frameworks, as a particular RAS schemes are applied to the MCTDHF wavefunction. Successful applications of the TD-RASSCF method include calculations of the ground-states (GS) of atoms, and time-dependent dynamics in the presence of strong laser fields to describe, for instance, high-order harmonic generation [@Haru13; @Haru14_1]. The aim of this work is to extend the TD-RASSCF method to systems of spinless interacting bosons. To follow the generic naming introduced for the MCTDH methods, we call this method TD-RASSCF-B where the additional B stands for bosons and we will refer to the original TD-RASCSF method for fermions [@Haru13; @Haru14_1; @Haru14_2] as TD-RASSCF-F to avoid any confusion concerning the particles considered. As a main finding, we derive the working equations of the TD-RASSCF-B method and we present the general set of working equations for the TD-RASSCF method where the type of particles plays a role in the symmetry of the creation and annihilation operators, only. The applications of the method to compute the GS energy of trapped bosons show that the TD-RASSCF-B theory provides accurate results, in comparison to MCTDHB, while the expansion of the wavefunction is considerably reduced. Moreover, the MCTDHB accuracy can be overtaken by using large numbers of time-dependent orbitals while the number of configurations remains small thanks to the RAS schemes. The investigation of the breathing dynamic of a BEC illustrates how the TD-GP theory fails to describe the time-evolution of the system, while various examples of the TD-RASSCF-B method *qualitatively* or *quantitatively* reproduce the exact dynamics obtained using the MCTDHB method, depending of the choice of the excitation scheme.
The paper is organized as follows. In Sec. \[wavefunc\] we introduce the TD-RASSCF-B *Ansatz* for the wavefunction and in Sec. \[EOM\_TDRAS\] we derive the equations of motion for the set of coefficients and orbitals. In Sec. \[time\_indpdt\] the method is applied and compared to the MCTDHB method to study the static properties of a system consisting of $N=100$ bosons trapped in a harmonic potential. The applicability of the method to time-dependent systems is illustrated by two examples of a breathing dynamics following a sudden quenching of the two-body interaction in Sec. \[time\_dpdt\]. Finally, in Sec. \[conclusion\] we conclude and provide perspectives to future work. In the Appendices \[wf\_representation\] to \[efficiency\_TDRAS\], we provide the key ingredients for the numerical implementation of the method and discuss the numerical effort in comparison to the MCTDHB method.
Theoretical framework {#theory}
=====================
Ansatz for the many-body wavefunction {#wavefunc}
-------------------------------------
For the energy regime of interest, the time evolution of a system composed of $N$ bosons is governed by the time-dependent Schrödinger equation: $$i \hbar \frac{\partial}{\partial t}|\Psi(t)\rangle= \hat{H}(t)|\Psi(t)\rangle$$ with $\hat{H}(t)$ the many-body Hamiltonian of the system and $|\Psi(t)\rangle$ the $N$-particle wavefunction. Hereafter we set $\hbar=1$, unless explicitly specified. We can approximate the wavefunction using linear combinations of suitably symmetrized sets of products of time-dependent single-particle functions $\{\phi_{i}(\bf{r},t)\}$. In the following, the single-particle functions are denoted *orbitals*. To take into account the indistinguishability of the bosons, the total wavefunction is expressed in terms of permanents. For a given number of bosons and orbitals the multi-configurational wavefunction is constructed by taking into account all the possible arrangement of the particles in the given orbitals, each arrangements being called a configuration $|\Phi_{I}(t)\rangle$, $$|\Psi(t)\rangle=\sum_{I\in {\cal V}_{\text{FCI}}}{ C_{I}(t)|\Phi_{I}(t)}\rangle.$$ This *Ansatz* converges to the exact wavefunction when the number of orbitals increases to infinity. The configurational space increases exponentially with respect to the number of orbitals and often makes a numerical treatment impossible, even for a small number of orbitals. In the case of a system of $N$ bosons and $M$ orbitals, the dimension of the full-configurational Fock space ${\cal V}_{\text{FCI}}$ can be evaluated as, $$\label{dim_wf_MCTDHB}
dim({\cal V}_{\text{FCI}})=\begin{pmatrix}
N+M-1 \\
N
\end{pmatrix}
= \frac{(N+M-1)!}{N!(M-1)!}.$$ In the case of the TD-RASSCF-B method, we introduce two orbital spaces, ${\cal P}_{1}$ and ${\cal P}_{2}$, such that $M_{1}+M_{2} = M$, with $M_{1}$ and $M_{2}$ the number of orbitals in ${\cal P}_{1}$ and ${\cal P}_{2}$, respectively (see Fig. \[General\_orbtial\_space\]). The ${\cal P}_{1}$ subspace must include enough orbitals such that it can accommodate all the particles. For bosons one orbital, i.e., $M_{1}=1$ is the lower bound, and there is no restriction concerning the upper bound. In this subspace all the configurations are used to construct the total wavefunction. Concerning the ${\cal P}_{2}$-space, particles are promoted from ${\cal P}_{1}$ to ${\cal P}_{2}$ according to a specific excitation scheme, which is based on the highest number of particles that can be promoted. This number is chosen at will and the *restriction* of the configurational space provides a way to constrain its size, defining the *Ansatz* for the TD-RASSCF method as, $$\label{RAS_wf}
|\Psi(t)\rangle=\sum_{I\in {\cal V}} C_{I}(t)|\Phi_{I}(t)\rangle,$$ where the configurations are drawn from the space $\cal{V}$ subject to restrictions. To evaluate the size of the $\cal{V}$, we introduce $N_{\text{max}}$ the highest number of bosons in ${\cal P}_{2}$ and consider a RAS scheme allowing all occupations of ${\cal P}_{2}$ from $0$ to $N_{\text{max}}$. The dimension of the $\cal{V}$ of Eq. (\[RAS\_wf\]), is then given by $$\label{config_general_RAS}
\begin{split}
dim({\cal V})& =\begin{pmatrix}
N+M_{1}-1 \\
N
\end{pmatrix} + \\
& \sum_{k=1}^{N_{\text{max}}}
\begin{pmatrix}
k+M_{2}-1 \\
k
\end{pmatrix}
\begin{pmatrix}
(N-k)+M_{1}-1 \\
N-k
\end{pmatrix} .
\end{split}$$ The first term is the total number of configurations obtained with the $N$ bosons in the $M_{1}$ orbitals and no particle in ${\cal P}_{2}$. The sum takes into account the configurations resulting from the excitation of $k$ bosons in ${\cal P}_{2}$, with $1\le k\le N_{\text{max}}$. The total number of configurations with $k$ bosons in ${\cal P}_{2}$ is obtained as a product of the possible arrangements of $k$ bosons in $M_{2}$ orbitals and $(N-k)$ bosons in $M_{1}$ orbitals, see also Appendix \[wf\_representation\].
The TD-RASSCF-B *Ansatz* holds some interesting specificities. First, if only ${\cal P}_{1}$ orbitals are used, i.e., $M_{1}=M$ and $M_{2}=0$, then the TD-RASSCF-B and MCTDHB *Ansätze* are equivalent with the same number of configurations, as seen be replacing $M_{1}$ by $M$ in Eq. (\[config\_general\_RAS\]). Note that this is also true for $M_{2}\ne 0$ and $N_{\text{max}}=N$. Moreover, if only a single time-dependent orbital is considered, i.e., $M_{1}=1$ and $M_{2}=0$, the RAS wavefunction includes a single configuration with all particles in one orbital, which is equivalent the to time-dependent GP wavefunction. Thus the theoretical framework of TD-RASSCF-B is very general and holds, as limiting cases, the GP and MCTDHB theories. The TD-RASSCF-B wavefunction is built from a set of time-dependent coefficients $\{C_{I}(t)\}$ and orbitals $\{|\phi_{i}(t)\rangle\}$. To describe its dynamics, we need a set of equations of motion (EOM), which provides the time-evolution of the coefficients and orbitals through their time-derivatives $\{\dot{C}_{I}(t)\}$ and $\{|\dot{\phi}_{i}(t)\rangle\}$. The $\cal{P}$-space is a subset of the total single-particle Hilbert space and we can define its orthogonal complement, $\cal{Q}$, collecting the virtual orbitals, as depicted in Fig. \[General\_orbtial\_space\]. While in the case of time-independent orbitals these two subspaces remain fixed, in the case of time-dependent orbitals the $\cal{P}$-space is variationally optimized at each time and both $\cal{P}$- and $\cal{Q}$-space are time-dependent. We can define $\hat{P}$ and $\hat{Q}$, the time-dependent projectors onto the subspaces $\cal{P}$ and $\cal{Q}$, respectively, with the property $\hat{P}+\hat{Q}=\hat{1}$, the identity operator. The role of the $\cal{Q}$-space emanates from the time-derivative of the $\cal{P}$-space orbitals, that can be written as, $$\label{general_time_deriv_Porb}
|\dot{\phi}_{i}(t)\rangle=(\hat{P}+\hat{Q})|\dot{\phi}_{i}(t)\rangle = \hat{P}|\dot{\phi}_{i}(t)\rangle+\hat{Q}|\dot{\phi}_{i}(t)\rangle,$$ with one contribution from the $\cal{P}$-space and one contribution from the $\cal{Q}$-space. In the following we establish the EOM of the TD-RASSCF-B theory, providing the time-derivative of the expansion coefficients in Sec. \[Sec\_EOM\_Coeff\] and the time-derivative of the orbitals (Sec. \[Sec\_EOM\_Orb\]) through the $\cal{Q}$- and $\cal{P}$-space contributions in Secs. \[Sec\_EOM\_OrbQ\] and \[Sec\_EOM\_OrbP\], respectively.
Derivation of the working equations {#EOM_TDRAS}
-----------------------------------
The EOM for the TD-RASSCF-F theory have been already established in Refs. [@Haru13; @Haru14_1]. In the following we provide the derivation of the EOM in the case of the TD-RASSCF-B method, and highlight the differences with respect to the TD-RASSCF-F theory. Starting from the Lagrangian formulation of the time-dependent Schrödinger equation [@Kramer81], we define the action functional using the TD-RASSCF-B *Ansatz*, Eq. (\[RAS\_wf\]), as, $$\label{action_func}
\begin{split}
S [\{C_{I}(t)\},&\{ |\phi_{i}(t)\rangle\}, \{\epsilon_{j}^{i}(t)\}] = \int_{t_{1}}^{t_{2}} \Bigg[ \langle\Psi(t)| \hat{K} |\Psi(t)\rangle \\
& \left.+ \sum_{ij}\epsilon_{j}^{i}(t)\bigg( \langle\phi_{i}(t)|\phi_{j}(t)\rangle - \delta_{ij} \bigg)\right] dt,
\end{split}$$ with $\hat{K}\equiv i\partial/\partial t -\hat{H}$ and $\delta_{ij}$ the Kronecker delta function. The Lagrange multipliers, $\epsilon_{j}^{i}(t)$, ensure that the orbitals remain orthonormal for all time $t$. In the following the indexes $i, j, k, \cdots$ are used to denote the orbitals of the $\cal{P}$-space, the indexes $a, b, c, \cdots$ denote the orbitals of the $\cal{Q}$-space and $p, q, r, \cdots$ are used for either $\cal{P}$- or $\cal{Q}$-space orbitals, see also Fig. \[General\_orbtial\_space\]. For our purpose, we consider only one- and two-body operators, such that the Hamiltonian can be expressed in the framework of second quantization as $$\label{Hamiltonian}
\hat{H}(t)=\sum_{pq}h_{q}^{p}(t)\hat{b}_{p}^{\dag}\hat{b}_{q}+\frac{1}{2}\sum_{pqrs}{v_{qs}^{pr}(t) \hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q}},$$ with $\hat{b}_{p}$ ($\hat{b}_{p}^{\dag}$) the annihilation (creation) operator of a particle in the orbital $|\phi_{p}(t)\rangle$ \[see also Appendix \[apply\_op\]\]. These operators satisfy the commutation relation, $[ \hat{b}_{p},\hat{b}_{q}^{\dag} ] = \hat{b}_{p}\hat{b}_{q}^{\dag}- \hat{b}_{q}^{\dag}\hat{b}_{p} = \delta_{qp}$, for bosons and the anti-commutation relation, $\{\hat{b}_{p},\hat{b}_{q}^{\dag} \}= \hat{b}_{p}\hat{b}_{q}^{\dag}+\hat{b}_{q}^{\dag}\hat{b}_{p} = \delta_{qp}$, for fermions, see for instance Ref. [@Cederbaum16]. The matrix elements of the one-body and two-body operators in the basis of the time-dependent orbitals, are expressed as $$\label{one-bod}
h_{q}^{p}(t)=\int \phi_{p}^{*}(\textbf{r},t) h(\textbf{r},t) \phi_{q}(\textbf{r},t) d\textbf{r},$$ and $$\label{two-bod}
\begin{split}
v_{qs}^{pr} (t) =\int \int &\phi_{p}^{*}({\bf r},t) \phi_{r}^{*}({\bf r'},t) \\
&\times W({\bf r},{\bf r'},t) \phi_{q}({\bf r},t)\phi_{s}({\bf r'},t) d{\bf r}d{\bf r'},
\end{split}$$ respectively. In the following, the explicit time dependence of the operators, coefficients and orbitals is dropped for brevity.
According to the time-dependent variational principle [@Kramer81; @Broeckhove88], the best approximation using the wavefunction *Ansatz* is obtained by seeking stationarity of the action $S$, i.e., $\delta S = 0$, for any variation of the parameters and with the boundary condition $|\delta\Psi(t_{1})\rangle = |\delta\Psi(t_{2})\rangle=0$. The variation of the action gives, $$\label{var_action_func}
\begin{split}
\delta &S = \int_{t_{1}}^{t_{2}} \Bigg[ \langle\delta\Psi| \hat{K} \Psi\rangle + \langle \hat{K}\Psi| \delta\Psi\rangle \\
&+ \sum_{ij}\left[\epsilon_{j}^{i}\left( \langle\delta\phi_{i}|\phi_{j}\rangle + \langle\phi_{i}|\delta\phi_{j}\rangle\right) + \delta\epsilon_{j}^{i} (\langle\phi_{i}|\phi_{j}\rangle-\delta_{ij}) \right] \Bigg] dt,
\end{split}$$ where the boundary condition is used to remove the additional term $i\partial_{t} \langle\Psi|\delta\Psi\rangle$ resulting from the action of $\hat{K}$ on $\langle\Psi|$ instead of $|\delta\Psi\rangle$, see Ref. [@Broeckhove88]. The variation of the wavefuntion is explicitly written as [@Haru13], $$\label{var_wavefunc}
|\delta\Psi\rangle = \sum_{I\in {\cal V}} \delta C_{I}|\Phi_{I}\rangle + \sum_{pq} \hat{b}_{p}^{\dag}\hat{b}_{q}|\Psi\rangle\langle\phi_{p}|\delta\phi_{q}\rangle.$$ We can now proceed with the stationarity condition of the action with respect to the parameters $\{C_{I}\}$, $\{|\phi_{i}\rangle\}$ and $\{\epsilon_{j}^{i}\}$ to obtain the EOM of the TD-RASSCF-B method.
### Equations of motion for the coefficients {#Sec_EOM_Coeff}
The variation w.r.t. the Lagrange multipliers leads to the conservation of the orthonormality of the orbitals. We then consider the variation of the action functional with respect to the expansion coefficients. The action $S$ depends on the expansion coefficient $C_{I}^{*}$ only through the *bra* $\langle \delta \Psi|$ of the first expectation value in Eq. (\[var\_action\_func\]). Thus, the stationarity condition, $\delta S/\delta C_{I}^{*}=0$, readily leads to $\langle\Phi_{I}|i\partial_{t}-\hat{H}|\Psi\rangle = 0$. Moreover, the derivative of the wavefunction with respect to time reads, $$\label{TD_WF_RAS}
\frac{\partial}{\partial t}|\Psi\rangle=\sum_{I\in {\cal V}} \dot{C}_{I}|\Phi_{I}\rangle + \left[ \sum_{pq}\eta_{q}^{p} \hat{b}_{p}^{\dag}\hat{b}_{q} \right]|\Psi\rangle,$$ with $\eta_{q}^{p}\equiv\langle \phi_{p}|\dot{\phi}_{q}\rangle$, which results from the time-derivative of the orbitals used to build the configurations in $|\Psi\rangle$. Hereafter, the operator in bracket in Eq. (\[TD\_WF\_RAS\]), $\sum_{pq}\eta_{q}^{p} \hat{b}_{p}^{\dag}\hat{b}_{q}$, will be called $\hat{D}$, for brevity. We can now rewrite the stationary condition $\delta S/\delta C_{I}^{*}=0$ using the explicit form of $\partial_{t}|\Psi\rangle$, Eq. (\[TD\_WF\_RAS\]), as $$i\dot{C}_{I}+\langle\Phi_{I}|(i\hat{D}-\hat{H})|\Psi\rangle=0, \forall I \in {\cal V} \label{EOM_C_general},$$ or equivalently, using the expressions of $\hat{H}$ and $\hat{D}$, $$\label{EOM_C_general_expand}
\begin{split}
i\dot{C}_{I}=\sum_{ij}&\left(h_{j}^{i}-i\eta_{j}^{i}\right)\langle\Phi_{I}|\hat{b}_{i}^{\dag}\hat{b}_{j}|\Psi\rangle\\
&+\frac{1}{2}\sum_{ijkl}v_{jl}^{ik}\langle\Phi_{I}|\hat{b}_{i}^{\dag}\hat{b}_{k}^{\dag}\hat{b}_{l}\hat{b}_{j}|\Psi\rangle.
\end{split}$$ The indexes in the summations are now restricted to the $\cal{P}$-space. It is clear that if either annihilation or creation operators act on an orbital of $\cal{Q}$, the inner product with all RAS configurations $\langle\Phi_{I}|$ vanishes. The EOM for the expansion coefficients, the amplitude equations (\[EOM\_C\_general\_expand\]), are identical to those obtained for fermions in the TD-RASSCF-F theory, see Refs. [@Haru13; @Haru14_1], and those of the MCTDHB [@Alon08] and MCTDHF [@Caillat05] theories. It is worthwhile to keep in mind that the action of the creation and annihilation operators differs for fermions and bosons. The $\eta_{j}^{i}$ matrix elements in the amplitude equations \[Eq. (\[EOM\_C\_general\_expand\])\] describe the rotation of the orbitals into one another and are also present in the EOM of the MCTDHB/F methods. In these latter cases, besides to be elements of an anti-Hermitian matrix, there are no constraints on the $\eta_{j}^{i}$ and their values are usually set to zero. The same is true in the TD-RASSCF-B/F methods for equivalent orbitals, i.e., for pairs of orbitals which belong to the same ${\cal P}_{i}$-space (i=1, 2). For orbitals which do not belong to the same ${\cal P}_{i}$-space, the $\eta_{j}^{i}$ matrix elements must be evaluated, as discussed in Refs. [@Haru13; @Haru14_1] and in Sec. \[even\_exci\] and \[all\_exci\].
### Equations of motion for the orbitals {#Sec_EOM_Orb}
Seeking stationarity of $S$ with respect to a variation of an orbital $\langle\phi_{i}|$, i.e., $\delta S/ \delta\langle\phi_{i}|=0$, gives $$\label{EOM_orb}
\begin{split}
\sum_{q} |\phi_{q}\rangle \langle\Psi_{i}^{q}| &\left[\sum_{I\in {\cal V}} i\dot{C}_{I}Ê|\Phi_{I}\rangle + ( i\hat{D}-\hat{H})|\Psi\rangle \right] \\
&+ \sum_{j}\epsilon_{j}^{i}|\phi_{j}\rangle= 0,
\end{split}$$ with $\langle\Psi_{i}^{q}| \equiv \langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{q}$. The index $q$ in the above equation runs over all the orbitals, i.e., the orbitals of the ${\cal P}$-space and the ${\cal Q}$-space, see Fig. \[General\_orbtial\_space\]. The EOM for the orbitals of the ${\cal P}$- and ${\cal Q}$-space are obtained by projecting Eq. (\[EOM\_orb\]) on either an orbital of the ${\cal P}$-space, $\langle\phi_{j}|$, or of the ${\cal Q}$-space, $\langle\phi_{a}|$, as done in the following.\
#### Equations of motion for the ${\cal Q}$-space orbitals
\[Sec\_EOM\_OrbQ\]\
Starting with the EOM for the ${\cal Q}$-space orbitals, we multiply Eq. (\[EOM\_orb\]) from the left with an orbital $\langle\phi_{a}|$ belonging to the ${\cal Q}$-space and obtain, $$\label{EOM_orb_Q}
\sum_{I\in {\cal V}} i\dot{C}_{I} \langle\Psi_{i}^{a}Ê|\Phi_{I}\rangle + \langle\Psi_{i}^{a}Ê| (i\hat{D}-\hat{H})|\Psi\rangle= 0,$$ where we used the orthogonality between the orbitals of the ${\cal P}$ and ${\cal Q}$ spaces to get rid of the Lagrange multipliers. Moreover the inner product $\langle\Psi_{i}^{a}Ê|\Phi_{I}\rangle = \langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{a}|\Phi_{I}\rangle$ vanishes because in all configurations $|\Phi_{I}\rangle$ the orbital $|\phi_{a}\rangle$ is unoccupied. Using the explicit expression of the Hamiltonian, Eq. (\[Hamiltonian\]), and for the operator $\hat{D}$, Eq. (\[TD\_WF\_RAS\]), we obtain $$\label{EOM_with_H_Q}
\begin{split}
\sum_{pq}\left( i\eta_{q}^{p}-h_{q}^{p} \right) &\langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{a}\hat{b}_{p}^{\dag}\hat{b}_{q}|\Psi\rangle \\
&=\frac{1}{2}\sum_{pqrs}v_{qs}^{pr}\langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{a}\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q}|\Psi\rangle.
\end{split}$$ Using the commutation relation for the creation/annihilation operators for bosons (fermions), we can reestablish the normal ordering of the chains of operators, $$\begin{aligned}
\hat{b}_{i}^{\dag}\hat{b}_{a}\hat{b}_{p}^{\dag}\hat{b}_{q}&=\hat{b}_{i}^{\dag}\hat{b}_{q}\delta_{pa} \pm \hat{b}_{i}^{\dag}\hat{b}_{p}^{\dag}\hat{b}_{q}\hat{b}_{a} \label{four_op} \\
\hat{b}_{i}^{\dag}\hat{b}_{a}\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q}&= \hat{b}_{i}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q}\delta_{pa} \pm \hat{b}_{i}^{\dag}\hat{b}_{p}^{\dag}\hat{b}_{s}\hat{b}_{q}\delta_{ra}+\hat{b}_{i}^{\dag}\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q}\hat{b}_{a}, \label{six_op} \end{aligned}$$ with the upper sign holding for bosons and the lower for fermions. The chain of four operators in Eq. (\[four\_op\]) and six operators in Eq. (\[six\_op\]) both annihilate a particle in orbital $|\phi_{a}\rangle$, from the ${\cal Q}$-space, which is not include in $|\Psi\rangle$ and thus vanish. The l.h.s. of Eq. (\[EOM\_with\_H\_Q\]) now reads, $$\sum_{q}\left( i\eta_{q}^{a}-h_{q}^{a} \right) \langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{q}|\Psi\rangle = \sum_{j}\left( i\eta_{j}^{a}-h_{j}^{a} \right) \langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{j}|\Psi\rangle,$$ where we restrict the summation over $j\in {\cal P}$, the summation over the ${\cal Q}$-space orbitals being zero. In the same way, inserting Eq. (\[six\_op\]) in the r.h.s. of Eq. (\[EOM\_with\_H\_Q\]) simplifies its expression to $$\sum_{j}\left( i\eta_{j}^{a}-h_{j}^{a} \right) \langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{j}|\Psi\rangle =\sum_{jkl}v^{ak}_{jl}\langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{k}^{\dag}\hat{b}_{l}\hat{b}_{j}|\Psi\rangle.$$ Here we used that $v^{ak}_{jl}=v^{ka}_{lj}$ \[Eq. (\[two-bod\])\]. Interestingly, this equation is exactly the same for bosons and fermions. The time-derivative of the orbitals, included in the term $\eta_{j}^{a}$, requires the explicit consideration of the ${\cal Q}$-space orbitals. This issue is circumvented by using the projector $\hat{Q}$ onto the subspace spanned by the ${\cal Q}$-space orbitals, $$\begin{aligned}
\hat{Q}&=&\sum_{a}|\phi_{a}\rangle\langle\phi_{a}| \nonumber \\
&=& \hat{1}-\sum_{i}|\phi_{i}\rangle\langle\phi_{i}| \nonumber \\
&=& \hat{1}-\hat{P},\end{aligned}$$ with $\hat{P}$ the projector onto the ${\cal P}$-space. Introducing the one-body density matrix, $\rho_{i}^{j}= \langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{j}|\Psi\rangle$ and the two-body density matrix $\rho_{ik}^{jl}=\langle\Psi|\hat{b}_{i}^{\dag}\hat{b}_{k}^{\dag}\hat{b}_{l}\hat{b}_{j}|\Psi\rangle$, we obtain, $$\label{EOM_Qspace}
i\sum_{j} \hat{Q}|\dot{\phi}_{j}\rangle \rho_{i}^{j} = \hat{Q}\left[\sum_{j} \hat{h}|\phi_{j}\rangle \rho_{i}^{j} + \sum_{jkl} \hat{W}_{l}^{k}|\phi_{j}\rangle \rho_{ik}^{jl}\right],$$ with, $$\hat{W}_{l}^{k}({\bf r})=\int{\phi_{k}^{*}({\bf r'})W({\bf r},{\bf r'})\phi_{l}({\bf r'})d{\bf r'}},$$ the mean-field operator, which describes the interaction between the particles. The role of the $\cal{Q}$-space appears in the time-derivative of the orbitals of the $\cal{P}$-space through the term $\hat{Q}|\dot{\phi}_{i}\rangle$, see Eq. (\[general\_time\_deriv\_Porb\]). We rearrange Eq. (\[EOM\_Qspace\]), see appendix \[Ap\_Num\_imple\], to uncouple the contribution of each $\hat{Q}|\dot{\phi}_{i}\rangle, \forall i\in\cal{P}$, and obtain $$\label{final_Qspace}
\hat{Q}|\dot{\phi}_{i}\rangle = -i(\hat{1}-\hat{P})\left[ \hat{h}|\phi_{i}\rangle + \sum_{jklm} (\underline{\underline{\bm{\rho}}}^{-1})_{i}^{m} \rho_{mk}^{jl}\hat{W}_{l}^{k}|\phi_{j}\rangle\right],$$ with $\underline{\underline{\bm{\rho}}}^{-1}$ the inverse of the one-body density matrix. The MCTHB theory leads also to Eq. (\[final\_Qspace\]), see Ref. [@Alon08], but the l.h.s. is subsequently simplified thanks to the choice of the matrix elements $\eta_{j}^{i}=0$ and using $\hat{Q}=\hat{1}-\hat{P}$, see Eq. (\[time\_deriv\_Porb\]) below. As discussed in \[Sec\_EOM\_Coeff\], such a fixed choice of $\eta_{j}^{i}$ is not possible in the TD-RASSCF theory. The derivation of $\cal{Q}$-space EOM differ slightly for bosons and fermions, see Eqs. (\[four\_op\]) and (\[six\_op\]), but the final result, Eq. (\[final\_Qspace\]), is the same for both types of particles.\
#### Equations of motion for the ${\cal P}$-space orbitals
\[Sec\_EOM\_OrbP\]\
Going back to the stationary condition for the variation of the action functional with respect to an orbital, Eq. (\[EOM\_orb\]), we multiply this latter on the left by an orbital of the ${\cal P}$-space, $\langle\phi_{j}|$, leading to, $$\label{EOM_orb_P_1}
\sum_{I\in {\cal V}} i\dot{C}_{I} \langle\Psi_{i}^{j}Ê|\Phi_{I}\rangle + \langle\Psi_{i}^{j}Ê| ( i\hat{D}-\hat{H})|\Psi\rangle + \epsilon_{j}^{i} = 0.$$ This equation still contains the Lagrange multiplier $ \epsilon_{j}^{i}$. A variation of $S$ with respect to the orbital $|\phi_{j}\rangle$ and its projection onto the orbital $\langle\phi_{i}|$, leads to an equation containing the same Lagrange multiplier, $$\label{EOM_orb_P_2}
\sum_{I\in {\cal V}} -i\dot{C}_{I}^{*} \langle\Phi_{I}|\Psi_{j}^{i}Ê\rangle + \langle\Psi | ( i\hat{D}-\hat{H})|\Psi_{j}^{i}Ê\rangle + \epsilon_{j}^{i} = 0,$$ and subtracting Eq. (\[EOM\_orb\_P\_1\]) and Eq. (\[EOM\_orb\_P\_2\]) gives the EOM for the $\cal{P}$-space orbitals, i.e., $$\label{P_space_general}
\langle\Psi | ( i\hat{D}-\hat{H})|\Psi_{j}^{i}Ê\rangle - \langle\Psi_{i}^{j}Ê| ( i\hat{D}-\hat{H})|\Psi\rangle = i\dot{\rho}_{i}^{j},$$ where we have introduced $\dot{\rho}_{i}^{j}\equiv\sum_{I\in {\cal V}}(\dot{C}^{*}_{I}\langle\Phi_{I}|\Psi_{j}^{i}\rangle+\langle\Psi_{i}^{j}|\Phi_{I}\rangle\dot{C}_{I})$. The $\cal{P}$-space EOM provide the contribution of the $\cal{P}$-space in the time-derivative of the orbitals, see Eq. (\[general\_time\_deriv\_Porb\]), $$\label{time_deriv_Porb}
\hat{P}|\dot{\phi}_{i}\rangle = \sum_{j}|\phi_{j}\rangle\eta_{i}^{j},$$ through the evaluation of the matrix elements $\eta_{i}^{j}$ included in the operator $\hat{D}$. Nonetheless, solving Eq. (\[P\_space\_general\]) is not a trivial task because of the presence of $\dot{\rho}_{i}^{j}$, which couples the amplitude and ${\cal P}$-space orbitals equations. In the case of the wavefunction based on the RAS *Ansatz*, a freedom in the choice of the elements $\eta_{i}^{j}$ is still possible for pairs of orbitals which belong to the same ${\cal P}_{i}$-space ($i=1,2$), and we use $\eta_{i}^{j}=0, \forall \{i,j\}\in{\cal P}_{1}\text{ or }{\cal P}_{2}$. The $\cal{P}$-space equation \[Eq. (\[P\_space\_general\])\] remains to be solved only for pairs of orbitals $\{i',j''\}$, which belong to different ${\cal P}_{i}$-spaces, $$\label{P_space_pair}
\langle\Psi | ( i\hat{D}-\hat{H})|\Psi_{j''}^{i'}Ê\rangle - \langle\Psi_{i'}^{j''}Ê| ( i\hat{D}-\hat{H})|\Psi\rangle = i\dot{\rho}_{i'}^{j''},$$ but remains coupled to the amplitude equations through $\dot{\rho}_{i'}^{j''}$. In the meantime it is noted that *if* Eq. (\[P\_space\_pair\]) is solved for $\eta_{i}^{j}$, the r.h.s. of Eq. (\[time\_deriv\_Porb\]) can be constructed. Moreover Eq. (\[final\_Qspace\]) can be solved, and hence $|\dot{\phi}_{i}(t)\rangle$ of Eq. (\[general\_time\_deriv\_Porb\]) can be evaluated. In the derivation of the TD-RASSCF-F method, a way to circumvent the difficulty of solving Eq. (\[P\_space\_general\]) was proposed [@Haru13; @Haru14_1]. This approach will be used in the following also for bosons.\
#### Even excitation RAS scheme
\[even\_exci\]\
First we suggest to consider the case in which only an even number of particles is promoted from ${\cal P}_{1}$ to ${\cal P}_{2}$, see Fig. \[Ras\_Schemes\]a. In this case, $\dot{\rho}_{i'}^{j''}$ explicitly reads, $$\dot{\rho}_{i'}^{j''}=\sum_{I\in {\cal V}}\left(\dot{C}^{*}_{I}\langle\Phi_{I}|\hat{b}_{i'}^{\dag}\hat{b}_{j''}|\Psi\rangle+\langle\Psi|\hat{b}_{i'}^{\dag}\hat{b}_{j''}|\Phi_{I}\rangle\dot{C}_{I}\right).$$ The action of $\hat{b}_{i'}^{\dag}\hat{b}_{j''}$ on the wavefunction $|\Psi\rangle$ annihilates one particle in ${\cal P}_{2}$ and creates one in ${\cal P}_{1}$. Since only an even number of particles is present in ${\cal P}_{2}$, $\hat{b}_{i'}^{\dag}\hat{b}_{j''}|\Psi\rangle$ would contain only configurations with an odd number of particles in ${\cal P}_{2}$, which makes the inner product with $\langle\Phi_{I}|, \forall I \in \text{RAS}$ vanish. In the same manner $\hat{b}_{i'}^{\dag}\hat{b}_{j''}$ acting on $|\Phi_{I}\rangle$ is either zero, if $|\phi_{j''}\rangle$ is unoccupied in the configuration $|\Phi_{I}\rangle$, or gives an odd number of particles in ${\cal P}_{2}$. In this specific excitation scheme, $\dot{\rho}_{i'}^{j''}=0$, for all pairs of orbitals $\{i',j''\}$, leaving the amplitudes and the ${\cal P}$-space orbitals equations uncoupled. Using the explicit expressions of the Hamiltonian \[Eq. (\[Hamiltonian\])\] and the operator $\hat{D}$, \[Eq. (\[TD\_WF\_RAS\])\], Eq. (\[P\_space\_pair\]) reads, $$\label{eta_eq_even}
\begin{split}
\sum_{pq}&\left(h_{q}^{p}-i\eta_{q}^{p}\right)\left[\langle\Psi|\hat{b}_{i'}^{\dag}\hat{b}_{j''}\hat{b}_{p}^{\dag}\hat{b}_{q}-\hat{b}_{p}^{\dag}\hat{b}_{q}\hat{b}_{i'}^{\dag}\hat{b}_{j''}|\Psi\rangle\right]+\\
&\frac{1}{2}\sum_{pqrs}v_{qs}^{pr}\left[\langle\Psi|\hat{b}_{i'}^{\dag}\hat{b}_{j''}\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q}-\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q}\hat{b}_{i'}^{\dag}\hat{b}_{j''}|\Psi\rangle\right]=0.
\end{split}$$ We can simplify this expression, starting with $$\begin{aligned}
\langle\Psi|&\hat{b}_{i'}^{\dag}\hat{b}_{j''}\hat{b}_{p}^{\dag}\hat{b}_{q}-\hat{b}_{p}^{\dag}\hat{b}_{q}\hat{b}_{i'}^{\dag}\hat{b}_{j''}|\Psi\rangle\nonumber \\
&= \langle\Psi|\hat{b}_{i'}^{\dag}\hat{b}_{q} \delta_{pj''} \pm \hat{b}_{i'}^{\dag}\hat{b}_{p}^{\dag}\hat{b}_{j''}\hat{b}_{q}-\hat{b}_{p}^{\dag}\hat{b}_{j''}\delta_{i'q}\mp \hat{b}_{i'}^{\dag}\hat{b}_{p}^{\dag}\hat{b}_{j''}\hat{b}_{q}|\Psi\rangle \nonumber \\
&=\langle\Psi|\hat{b}_{i'}^{\dag}\hat{b}_{q} \delta_{pj''} -\hat{b}_{p}^{\dag}\hat{b}_{j''}\delta_{i'q}|\Psi\rangle \nonumber \\
&=\rho_{i}^{q}\delta_{pj''} - \rho_{p}^{j''}\delta_{i'q}\nonumber \\
&\equiv A_{pi'}^{qj''}. \end{aligned}$$ Now we turn to the chains of six operators in the last term in Eq. (\[eta\_eq\_even\]). The first product of operators is expressed as $$\begin{split}
\hat{b}_{i'}^{\dag}\hat{b}_{j''}\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q} =\ &\hat{b}_{i'}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q}\delta_{pj''} \\
&\pm \hat{b}_{i'}^{\dag}\hat{b}_{p}^{\dag}\hat{b}_{s}\hat{b}_{q}\delta_{rj''}+\hat{b}_{i'}^{\dag}\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{j''}\hat{b}_{s}\hat{b}_{q} \label{simply_1}
\end{split}$$ and the second product of operators as $$\begin{split}
-\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{q}\hat{b}_{i'}^{\dag}\hat{b}_{j''}= &-\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{s}\hat{b}_{j''}\delta_{i'q} \\
&\mp\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{q}\hat{b}_{j''}\delta_{i's}-\hat{b}_{i'}^{\dag}\hat{b}_{p}^{\dag}\hat{b}_{r}^{\dag}\hat{b}_{j''}\hat{b}_{s}\hat{b}_{q}. \label{simply_2}
\end{split}$$ The sum of Eqs. (\[simply\_1\]) and (\[simply\_2\]) enters Eq. (\[eta\_eq\_even\]), and we see that only chains of four operators remain. Using the fact that $v^{ik}_{jl}=v^{ki}_{lj}$ \[Eq. (\[two-bod\])\] and that $\eta_{q}^{p}$ must be evaluated for orbitals which belong to different ${\cal P}_{i}$ space, $\{l',k''\}$, Eq. (\[eta\_eq\_even\]) can be rewritten, $$\label{eta_eq_even_final}
\sum_{k''l'}(h_{l'}^{k''}-i\eta_{l'}^{k''})A_{k''i'}^{l'j''}+\sum_{klm}(v_{kl}^{j''m}\rho_{i'm}^{kl}-v_{i'm}^{kl}\rho_{kl}^{j''m})=0.$$ This equation, used to determine $\eta_{l'}^{k''}$, is identical for fermions and bosons, only the evaluation of the one- and two-body reduced density matrices depends on the kind of particles. The coefficients and the ${\cal P}$-space orbitals equations are separable and can now be solved. The $\eta_{l'}^{k''}$ are obtained using Eq. (\[eta\_eq\_even\]), and their values are used to determine the time-derivative of the coefficients from Eq. (\[EOM\_C\_general\_expand\]) and the time-derivative of the ${\cal P}$-space orbitals, Eq. (\[general\_time\_deriv\_Porb\]), is obtained from Eq. (\[time\_deriv\_Porb\]) in addition to the $\cal{Q}$-space equations \[Eq. (\[final\_Qspace\])\].\
#### General RAS scheme
\[all\_exci\]\
Considering only even excitations provides an efficient and simple way to uncouple the equations of the TD-RASSCF-B method. Nonetheless, it is also possible to consider both even and odd excitations in the configurational space. In the following, we specifically consider a RAS scheme with all successive numbers of particles occupying ${\cal P}_{2}$ from $0$ to $N_{\text{max}}$, where $N_{\text{max}}$, defined in Sec. \[wavefunc\], is the highest number of particles allowed in ${\cal P}_{2}$, see Fig. \[Ras\_Schemes\]b. Note that $N_{\text{max}}$ must fulfill the condition $N_{\text{max}}\le N$. For instance, taking $N_{\text{max}}=4$, we consider the promotion of 0,1,2,3 and 4 particles from ${\cal P}_{1}$ to ${\cal P}_{2}$. In this way, the configurational space is span by the direct sum of $N_{\text{max}}+1$ subspaces, $$\label{Fock_decomp}
\mathcal{V} = \mathcal{V}_{0}\oplus\mathcal{V}_{1}\oplus \cdots \oplus \mathcal{V}_{N_{\text{max}}}.$$ Using the expression of the time derivative of the $\{C_{I}\}$ coefficients, Eq. (\[EOM\_C\_general\]), the time derivative of the one-body density matrix, present in Eq. (\[P\_space\_general\]), can be expressed as, $$\begin{split}
i\dot{\rho}_{i'}^{j''} = \sum_{I\in\mathcal{V}} &\left(\langle\Psi|(i\hat{D}-\hat{H})|\Phi_{I}\rangle\langle\Phi_{I}|\Psi_{j''}^{i'}\rangle\right.\\
&\left.-\langle\Psi_{i'}^{j''}|\Phi_{I}\rangle\langle\Phi_{I}|(i\hat{D}-\hat{H})|\Psi\rangle\right).
\end{split}$$ We introduce the projector onto the RAS space $\mathcal{V}$ as $\hat{\Pi} = \sum_{I\in\mathcal{V}} |\Phi_{I}\rangle\langle\Phi_{I}|$. Using the above expression of $\dot{\rho}_{i}^{j}$, we obtain a new formulation of the ${\cal P}$-space orbital equation, $$\label{P_space_EOM_proj}
\langle\Psi|(i\hat{D}-\hat{H})(\hat{1}-\hat{\Pi})|\Psi_{j''}^{i'}\rangle-\langle\Psi_{i'}^{j''}|(\hat{1}-\hat{\Pi})(i\hat{D}-\hat{H})|\Psi\rangle=0.$$ For $|\phi_{i'}\rangle\in P_{1}$ and $|\phi_{j''}\rangle\in P_{2}$, we note that $|\Psi_{j''}^{i'}\rangle$ belongs to $\mathcal{V}$, with one particle from ${\cal P}_{2}$ being annihilated and one particle in ${\cal P}_{1}$ created, leading to $(\hat{1}-\hat{\Pi})|\Psi_{j''}^{i'} \rangle= 0$. On the other hand, $\langle\Psi_{i'}^{j''}| = \langle\Psi|\hat{b}_{i'}^{\dag}\hat{b}_{j''}$, provides configurations with a creation of an additional particle in ${\cal P}_{2}$, which may lie in $\mathcal{V}_{N_{\text{max}}+1}$, not included in $\mathcal{V}$. In this case, $\langle\Psi_{i'}^{j''}|(\hat{1}-\hat{\Pi})\ne 0$ and Eq. (\[P\_space\_EOM\_proj\]) simplifies to, $$\label{Eq_P_space_proj_RAS}
\begin{split}
\langle\Psi_{i'}^{j''}|(\hat{1}&-\hat{\Pi})(i\hat{D}-\hat{H})|\Psi\rangle=0, \textnormal{ with } \\
& Ê\langle\Psi_{i'}^{j''}|(\hat{1}-\hat{\Pi})= \sum_{I\in \mathcal{V}_{N_{\text{max}}}} C_{I}^{*}\langle\Phi_{I}|\hat{b}_{i'}^{\dag}\hat{b}_{j''}.
\end{split}$$ Using the expression of the Hamiltonian, \[Eq. (\[Hamiltonian\])\], of the operator $\hat{D}$, \[Eq. (\[TD\_WF\_RAS\])\], and keeping in mind that $\eta_{l}^{k}$ has only to be determined for pairs of orbitals $\{l',k''\}$ which belong to different ${\cal P}_{i}$ subspaces, Eq. (\[Eq\_P\_space\_proj\_RAS\]) is equivalent to $$\sum_{k''l'}(i\eta_{l'}^{k''}-h_{l'}^{k''}) \zeta_{k''i'}^{l'j''} = \frac{1}{2}\sum_{klmn}v_{ln}^{km}\zeta_{kmi'}^{lnj''},Ê\label{Pspace_General_RAS}$$ where the fourth- and sixth-order tensors are defined by $$\begin{aligned}
\zeta_{k''i'}^{l'j''} &=& \langle\Psi_{i'}^{j''}|(\hat{1}-\hat{\Pi})\hat{b}_{k''}^{\dag}\hat{b}_{l'}|\Psi\rangle \label{four_order_tens} \\
\zeta_{kmi'}^{lnj''}&=&\langle\Psi_{i'}^{j''}|(\hat{1}-\hat{\Pi})\hat{b}_{k}^{\dag}\hat{b}_{m}^{\dag}\hat{b}_{n}\hat{b}_{l}|\Psi\rangle. \label{six_order_tens}
\end{aligned}$$ Here again, the $\cal{P}$-space EOM for the determination of the $\eta_{l'}^{k''}$, Eq. (\[Pspace\_General\_RAS\]), are identical for bosons and fermions [@Haru13; @Haru14_1]. These equations are solved to determine the $\eta_{l'}^{k''}$ for each pairs of orbitals belonging to different ${\cal P}_{i}$-space. The value of $\eta_{l'}^{k''}$ is subsequently used to solve the amplitudes equations \[Eq. (\[EOM\_C\_general\_expand\])\] and to evaluate the time-derivative of the $\cal{P}$-space orbitals from Eqs. (\[time\_deriv\_Porb\]) and (\[final\_Qspace\]), as for the case of the even excitation scheme.
The *general* excitations scheme and the *only even* excitations schemes were originally introduced in the case of fermions in Refs. [@Haru13; @Haru14_1]. We mention that recently Haxton et *al*. [@Haxton15] derived a general RAS scheme for fermions, in the sense that the configurational space can be build from of any arbitrary configurations. For both excitation schemes presented in this work, the time-derivative of the coefficients and orbitals are obtained by solving the amplitudes equations, Eq. (\[EOM\_C\_general\_expand\]), the $\cal{Q}$-space equations, Eq. (\[final\_Qspace\]) and the $\cal{P}$-space equations Eq. (\[Pspace\_General\_RAS\]) and Eq. (\[eta\_eq\_even\_final\]) for the general RAS scheme and the only even excitation scheme, respectively. In the case of the MCTDHB method, the amplitude \[Eq. (\[EOM\_C\_general\_expand\])\] and the $\cal{Q}$-space equations \[Eq. (\[final\_Qspace\])\] are also solved to obtain the time-derivative of the wavefunction, see Appendix \[Ap\_Num\_imple\]. The numerical efficiency to solve these equations scale differently with the number of configurations and the number of orbitals, as detailed in Appendix \[efficiency\_TDRAS\]. For a given number of orbitals, the TD-RASSCF-B method is more efficient to solve Eqs. (\[EOM\_C\_general\_expand\]) and (\[final\_Qspace\]), irrespectively of the excitation scheme used. Nonetheless, in the TD-RASSCF-B framework one additional system of equations needs to be solved, namely the $\cal{P}$-space equations \[Eq. (\[eta\_eq\_even\_final\]) or (\[Pspace\_General\_RAS\])\]. For only even excitations, the number of operations required to obtain the time-derivative of the wavefunction is always smaller in the case of the TD-RASSCF-B method than in the MCTDHB method. In the case of the general excitation scheme, the evaluation of the sixth-order tensor, Eq. (\[six\_order\_tens\]), requires a significantly large number of operations. Thus, the TD-RASSCF-B method may require more operations than MCTDHB for large values of $N_{\text{max}}$ and large numbers of orbitals. As shown in Appendix \[efficiency\_TDRAS\], this happens only for large values of $N_{\text{max}}$, for instance for $N_{\text{max}} > 38$ with $N=50$ or $N_{\text{max}} > 909$ for $N=1000$ bosons. Except for these high excitation schemes, the TD-RASSCF-B method is numerically more efficient than the MCTDHB method, but more importantly the exponential grows of the configurational space with respect to the number of orbitals can be controlled thanks to the RAS $\textit{Ansatz}$. In addition, we have shown that the TD-RASSCF equations of motion are the same for bosons and fermions, which means that the TD-RASSCF theory is a general framework including as limiting cases the TD-GP (TD-HF) and the MCTDHB (MCTDHF) theories for bosons (fermions). This result is reminiscent to the work of Alon et *al* [@Alon07_2] where a unified set of EOM for the MCTDH theory for both bosons and fermions was derived.
Application to a time-independent system: Ground state energy {#time_indpdt}
=============================================================
In this section, we consider a system of $N=100$ bosons trapped in a 1-dimensional (1D) harmonic potential. Experimentally, quasi-1D systems have been obtained by using a tight confinement in the transversal coordinates, freezing in that way the transversal dynamics of the system [@Gorlitz01; @Moritz03; @Laburthe04; @Kinoshita05; @Hofferberth07]. In the following, we consider an anisotropic harmonic trap such that the longitudinal frequency ($\omega_{x}$) is much smaller than the transversal frequency ($\omega_{\perp}$), i.e., $\omega_{\perp}\gg \omega_{x}$, such that the transverse part of the wavefunction can be assumed to be energetically frozen to the ground state and be integrated out. The resulting 1D Hamiltonian for the $N$ boson system reads, $$\label{H_relax}
\hat{H}=\frac{1}{2}\sum_{i=1}^{N} \left( -\frac{\partial^2}{\partial x_{i}^{2}}+x_{i}^{2}\right) + \lambda\sum_{i<j}\delta(x_{i}-x_{j}),$$ using the unit of length $l_{0}=\sqrt{\hbar(m\omega_{x})^{-1}}$ and the unit of energy $E_{0}=\hbar\omega_{x}$, with $m$ the mass of the particles. Assuming no confinement induced resonances [@Olshanii98], the interaction strength, $\lambda$, is related to the 3D s-wave scattering length of the particles, $a_{s}$, through $\lambda=2a_{s}l_{0}l^{-2}_{\perp}$, with $l_{\perp}$ the transversal harmonic oscillator length. Experimentally, the 1D interaction strength can be tuned either by controlling the longitudinal and transversal frequencies or using an external magnetic field [@Courteille98; @Inouye98].
To solve numerically the EOM of the MCTDH and TD-RASSCF-B theories, the time-dependent orbitals are expanded on a time-independent basis or *primitive* basis, which consists of a sine discrete variable representation (DVR), see Ref. [@Beck00]. We use $101$ basis functions in a box $[-8,8]$ and compare the results with larger basis sets to ensure the convergence of the energies presented in Tables I and II. We numerically integrate the EOM using different integration algorithms, namely the 4th order runge-kutta (RK), the adaptive time-step 5th order RK [@NumRecipe] and the Adams-Bashforth-Moulton (ABM) predictor-corrector as implemented in the Heidelberg MCTDH package [@MCTDH]. The different integration schemes were tested against each other and we report the results obtained using the ABM integrator to the 7th order, as it is the most efficient. We calculate the GS energy using imaginary time propagation [@Kosloff86] of the EOM and give its energy in units of $E_{0}$.
To assess the accuracy of the GP, MCTDHB and TD-RASSCF-B methods we compare the GS energies and by virtue of the variational principle (see for instance Ref. [@Szabo96]), the lower the energy the higher the accuracy. First, as a general remark, for any value of $\lambda \ne 0$ the energy obtained with the MCTDHB method systematically decreases with increasing number of orbitals and subsequently for increasing number of configurations, see for instance the first line of Table I where the numbers of configurations are indicated in parentheses. Concerning the TD-RASSCF-B method, for a given excitation scheme the energy also decreases when the number of orbitals is increased. In addition, for a given number of orbitals the energy decreases when we increase the highest number of allowed particles in ${\cal P}_{2}$, $N_{\text{max}}$. To simplify the following discussion, we introduce some quantities to help the comparison between the MCTDHB and TD-RASSCF-B methods. Firstly, we define the correlation energy as the difference between the energy obtained with a given method and the mean-field GP energy, $$\label{Ecorr}
{\cal E}_{\text{corr}}=E_{\text{GP}}-E_{\text{method}},$$ where $E_{\text{method}}$ designates the energy obtained with a given method. By definition, the GP correlation energy is equal to zero and is considered as uncorrelated. We use as a reference, ${\cal E}_{\text{ref}}$, the correlation energy obtained for the MCTDHB method with $5$ orbitals, i.e., $$\label{Eref}
{\cal E}_{\text{ref}}=E_{\text{GP}}-E^{5}_{\text{MCTDHB}},$$ where the superscript $5$ denotes for the number of orbitals. Using this reference, we can easily compare the results obtained for different numbers of orbitals by expressing the correlation energy in percent of ${\cal E}_{\text{ref}}$. Secondly, we define the *relative* correlation energy, ${\cal E}_{\text{rel}}^{X}$, as the difference between the energy obtained from a MCTDHB calculation with $X$ orbitals and the GP energy, i.e., $$\label{Erel}
{\cal E}^{X}_{rel}=E_{\text{GP}}-E^{X}_{\text{MCTDHB}}.$$ This quantity is particularly useful to compare the results of different RAS schemes within a given number of orbitals. Indeed, the TD-RASSCF-B *Ansatz*, with restrictions on the active space, is an approximation to the MCTDHB wavefunction. Thus, when the correlation energy of a RAS scheme with $X$ orbitals is equal to ${\cal E}_{\text{rel}}^{X}$ the calculation is converged.
We first focus on the results obtained with $\lambda=0.01$, the weakest interaction strength considered and we report the results in Table I. The reference for the correlation energy is ${\cal E}_{\text{ref}}=-2.9 \times 10^{-2}$ (the GP result is obtained from MCTDHB with a single orbital). Increasing the number of orbitals in the MCTDHB calculations from $2$ to $5$ allows us to account for more and more of ${\cal E}_{\text{ref}}$. Specifically we obtain $51\%$, $78\%$ and $91\%$ of ${\cal E}_{\text{ref}}$ for 2, 3, and 4 orbitals, respectively. The $\simeq 10\%$ variation of the correlation energy between $4$ and $5$ orbitals indicates that the results are not fully converged with respect to the number of orbitals, and more than 5 orbitals are required to converge the energy below $10^{-3}$, see Table I. Unfortunately, the MCTDHB wavefunction with $5$ orbitals includes already $\sim 4.6 \times 10^{6}$ configurations and using 6 (8) orbitals leads to $\sim 96 \times 10^{6}$ ($\sim 26 \times 10^{9}$) configurations, far beyond the scope of any practical numerical implementation.
The TD-RASSCF-B method provides more flexibility to describe the wavefunction in the sense that we can choose different RAS schemes, different numbers of orbitals and their partitions into ${\cal P}_{1}$ and ${\cal P}_{2}$ spaces. In Table I we report the results obtained for $M=2$ to $8$ orbitals with a single ${\cal P}_{1}$ orbital, i.e., $M_{1}=1$ and $M_{2}=M-M_{1}$ ${\cal P}_{2}$ orbitals, and a few specific cases of the *general* RAS scheme. We indicate the excitation schemes with the usual notation. For example -SD denotes that single and double excitations are allowed from ${\cal P}_{1}$ to ${\cal P}_{2}$. We follow this notation up to -SDTQ56789 and for larger excitations, we just indicate the value of $N_{\text{max}}$ (e.g., “-10” means that all excitations from ${\cal P}_{1}$ to ${\cal P}_{2}$ up to $10$ are included). For each number of orbitals, when we increase the excitation scheme the energy becomes closer to the MCTDHB result and converges to this latter for the -SDTQ5678 RAS scheme, as indicated by the underlined digits in Table I. Thus, ${\cal E}_{\text{ref}}$ is recovered for the -SDTQ5678 scheme with 5 orbitals, but the expansion of wavefunction includes *only* $495$ configurations, i.e., $9 \times 10^{3}$ times fewer configurations than the MCTDHB expansion for $5$ orbitals. It is worthwhile to note that this RAS scheme converged for all number of orbitals, and always for much fewer configurations than with the MCTDHB. The least accurate TD-RASSCF-B calculation, presented in Table I, consists of $2$ orbitals and the -SD RAS scheme. The correlation energy includes $98.7\%$ of ${\cal E}_{\text{rel}}^{2}$ and interestingly when we increase the number of orbitals from $3$ to $5$, a similar amount of correlation is obtained ($98.8\%$ for all values) in comparison to the respective ${\cal E}_{\text{rel}}^{X=3,4,5}$ correlation energies. Thus, using the -SD scheme with $5$ orbitals $98.8\%$ of ${\cal E}_{\text{ref}}$ is obtained but the TD-RASSCF-B wavefunction includes only $15$ configurations while the MCTDHB wavefunction includes more than $4.5Ê\times 10^{6}$ configurations, i.e., $\sim 3 \times 10^{5}$ times more configurations. Moreover, the energy difference between the -SD scheme and MCDTHB method is systematically below $5 \times 10^{-4}$, lower than the convergence obtained with respect to the number of orbitals. Concerning the correlation energy of the -SDTQ and -SDTQ56 schemes, we find that they include $99.99\%$ and $99.9999\%$ of ${\cal E}_{\text{rel}}^{X}$, with $X = 2$ to $5$. It is remarkable that the correlation energy remains almost constant while the difference between the number of configurations between the MCTDHB and TD-RASSCF-B increases exponentially with the number of orbitals. These results show that the correlation energy does not strongly depend on the number of configurations used in the wavefunction expansion, as the configurational space of the -SD RAS scheme increases only from $3$ to $15$ configurations for $M=2$ to $M=5$ but captures $98.8\%$ of ${\cal E}_{\text{rel}}^{X}$, with $X=2$ to $5$. Thus, the correlation depends more critically on the number of orbitals than the number of configurations used in the calculation. To illustrate this point, we compute the GS energy with $6$ to $8$ orbitals with the TD-RASSCF-B method, see Table I, and we obtain energies lower than the energy of the MCTDHB with $5$ orbitals for all excitation schemes used here. It means that the -SD scheme with $8$ orbitals and $36$ configurations is more accurate than the MCTDHB method with $5$ orbitals and $\sim 4.6 \times 10^{6}$ configurations. Moreover, comparing the energies obtained for the -SDTQ5678 and -10 excitation schemes, we can conclude that the GS energy has converged with respect to the number of excitations. Thus, the TD-RASSCF-B method, thanks to the restriction imposed on the configurational space, can provide more accurate results than the MCTDHB method, whoes practical applicability is limited by the exponential growth of the number of configurations.
We also consider RAS schemes with only even excitations (see Table II), for which the numerical effort is always reduced in comparison to the MCTDHB method, see Appendix \[efficiency\_TDRAS\]. The energy difference between the -D and the -SD schemes is below $1.3 \times 10^{-6}$ for all numbers of orbitals, indicating that more than $98.7 \%$ of the relative correlation energy is obtained with slightly fewer configurations. The same conclusion holds for the comparison of the -DQ and -SDTQ schemes with an energy difference below $1.5 \times 10^{-5}$, including more than $99.95 \%$ of the relative correlation energy. The number of configurations is slightly smaller in the case of the RAS schemes with only even excitations but the numerical efficiency is better as the $\cal{P}$-space EOM, Eq. (\[eta\_eq\_even\_final\]), does not require the update of a sixth-order tensor at each time-step as it is the case of the general RAS schemes, see Eq. (\[six\_order\_tens\]). For values of $N_{\text{max}}\ge6$ and $M\ge3$, the energy converges with respect to $N_{\text{max}}$, as the energy does not change by increasing $N_{\text{max}}$ further, but with energy slightly larger than the MCTDHB ones. The energy difference between the -DQ68 scheme and the MCTDHB calculation, with $5$ orbitals for both methods, is $\sim1.1 \times 10^{-5}$ and includes $99.96 \%$ of ${\cal E}_{\text{ref}}$. In the case of only even excitations, we also find that for $M>5$ the energy for all schemes is below the energy of the best MCTDHB calculation performed. The comparison of the converged -DQ68 and -SDTQ5678 RAS schemes, show that the energy difference remains below $1.5 \times 10^{-5}$ for $M>5$, which is two orders of magnitude small then the convergence obtain with respect to the number orbitals $\sim 10^{-3}$.
We perform the same analysis for an interaction strength $\lambda=0.1$ and we obtain, as a reference for the correlation energy, ${\cal E}_{\text{ref}}=-1.34$, see Table I. This value is much larger than the one obtained previously and can be explained by the stronger interaction between the particles. Indeed, for a stronger interaction strength, the energy of the system is lowered by allowing the particles to occupied higher orbitals, i.e., orbitals leading to higher kinetic and potential energies, such that the interaction energy is reduced. In the mean-field GP theory, this possibility is not possible as only one orbital is used to describe the wavefunction. The orbitals that diagonalized the reduced density matrix and their respective eigenvalues, or *population*, can be used to characterized the system. If the largest eigenvalue is of the same order as $N$, the system is condensed [@Penrose56]. As the GP wavefunction includes a single orbital, it can only describe condensed systems. If more than one eigenvalue is of the order of $N$, then the system is fragmented (see Ref. [@Nozieres82] and the discussion in Ref. [@Sakmann08]). We find that, indeed, the occupation of the lowest natural orbital in the MCTDHB calculation using $5$ orbitals decreases from a population of $99.987 \%$ for $\lambda=0.01$ to a population of $99.465 \%$ for $\lambda=0.1$. This slightly larger depletion of the condensate has a strong impact on the correlation energy, as the mean-field GP theory provides a less accurate description of the system. We point out that increasing the number of orbitals from $4$ to $5$ in the MCTDHB calculations gives an energy difference $\sim 10^{-1}$, see Table I, which means that the energy does not converge below this value. The relative correlation energy ${\cal E}_{\text{rel}}^{2}$, ${\cal E}_{\text{rel}}^{3}$ and ${\cal E}_{\text{rel}}^{4}$ include $40.1\%$, $68.8\%$ and $86.7\%$ of ${\cal E}_{\text{ref}}$, respectively. Starting the discussion with the *general* RAS schemes, see Table I, we note that to converge to the MCTDHB energies and thus include $100\%$ of the relative correlation energies, ${\cal E}_{\text{rel}}^{X}$ with $X=2$ to $5$, large values of $N_{\text{max}}$ are required. For the -SDTQ5 RAS scheme we find that the correlation energy includes $\sim 97\%$ of ${\cal E}_{\text{rel}}^{X}$, the -10 RAS scheme includes $\sim 99.9\%$ of ${\cal E}_{\text{rel}}^{X}$, the -15 RAS scheme includes $\sim 99.997\%$ of ${\cal E}_{\text{rel}}^{X}$ and the -20 RAS scheme is converged with more than $99.9999\%$ of ${\cal E}_{\text{rel}}^{X}$. These results are obtained irrespectively of the number of orbitals, i.e., $X=2$ to $5$. Even if large values of $N_{\text{max}}$ are used, the expansion of the wavefunction using the -20 RAS scheme includes $\sim 10^{4}$ configurations for $5$ orbitals while the MCTDHB wavefunction includes $\sim 4.6\times 10^{6} $ configurations. We also use RAS schemes with *only* even excitations and we report the results in Table II. Similarly to the case with $\lambda=0.01$, we find that, except for $2$ orbitals, the energy does not converge to the MCTDHB energy, irrespectively to the value of $N_{\text{max}}$ used. Thus the energies obtained with $3$, $4$ and $5$ orbitals include $99.7\%$, $99.0\%$ and $98.8\%$ of the respective ${\cal E}_{\text{rel}}^{X}$ with the -20 RAS scheme, which is converged. For similar numbers of configurations the *general* RAS scheme provides more accurate results but remains more demanding in term of computation, see Appendix \[efficiency\_TDRAS\]. It is important to keep in mind that the convergence with respect to the number of orbitals is $\sim 10^{-1}$ and a convergence one order of magnitude below is achieved with the -SDTQ5 excitation scheme (Table I) and the -DQ excitation scheme (Table II). As previously, the configurational space of the MCTDHB wavefunction becomes unworkable for more than $5$ orbitals, but the TD-RASSCF-B method can include more orbitals. At the level of the -SDT scheme and $8$ orbitals and for higher excitation schemes with $6$ to $8$ orbitals, the GS energy is always below the energy obtained with the MCTDHB method with $5$ orbitals, see Table I. In the same way, using only even excitations provides more accurate results for excitation schemes higher than -D. As a remark, we obtain an energy $\sim 0.3$ below the energy of the MCTDHB method with $5$ orbitals by using the -15 RAS scheme with 8 orbitals, see Table I.
These preceding examples show that the TD-RASSCF-B method provides an efficient approach for computing the GS energy of trapped cold atoms. This wavefunction based approach gives access to quantities of interest such as the one and two-body reduced densities and the fragmentation using the population analysis of the natural orbitals. The accuracy was compared with the MCTDHB results and we showed that the TD-RASSCF-B method converges for relatively low excitation schemes. Moreover, the possibility to constrain the growth of the configurational space gives the possibility to use more orbitals than in the MCTDHB calculations and better results were systematically obtained using the TD-RASSCF-B method. This result can be understand as follows, the MCTDHB wavefunction for a small number of orbitals generates a large number of configurations, as all orbitals can be equally populated. The main part of these configurations, however, do not contribute to lower the energy of the system as they describe states with many particles occupying the same spatial orbital, which induced a large interaction energy for a large value of $\lambda$. As a limiting case, we know that in the Tonks-Girardeau model [@Girardeau60], obtained for an infinite value of $\lambda$, each boson occupies a different orbital. Thus, using a larger number of orbitals in the TD-RASSCF-B method introduces configurations for which a small number of particles occupy a larger number of different orbitals, and thus describes more efficiently the system. This flexibility of the TD-RASSCF-B method of choosing more orbitals opens a new possibility to explore the static properties of trapped cold atoms in systems with hundreds of particles and large numbers of orbitals, which are for the moment beyond the possibility of the MCTDHB method.
Application to a time-dependent system: Dynamics of bosons with harmonic interaction {#time_dpdt}
====================================================================================
As an illustration of an application of the TD-RASSCF-B method to a truly time-dependent problem, we simulate the dynamics of an ensemble of $N=10$ bosons in a 1D harmonic trap interacting through a harmonic interaction potential. We consider an initial system of non-interacting bosons, for which the Hamiltonian $\hat{H}_{0}$ reads, $$\label{H0_dyn}
\hat{H}_{0}=\frac{1}{2}\sum_{i=1}^{N} \left( -\frac{\partial^2}{\partial x_{i}^{2}}+x_{i}^{2}\right),$$ where we use the units described in Sec. \[time\_indpdt\] and the time is expressed in units of $t_{0}=\omega_{x}^{-1}$. The analytical ground state wavefunction and energy are used to ensure the convergence of the imaginary time propagation and, as expected, are the same for all methods. The dynamics is initiated at $t=0$ by quenching instantaneously the strength of the two-body interaction, as performed in Ref. [@Lode12], leading to the evolution of the system under the action of the following Hamiltonian, $$\label{H_quench}
\hat{H}=\frac{1}{2}\sum_{i=1}^{N} \left( -\frac{\partial^2}{\partial x_{i}^{2}}+x_{i}^{2}\right) + \sum_{i<j}\lambda (x_{i}-x_{j})^2,$$ with $\lambda$ the strength of the two-body interaction. This sudden change in the interaction between the bosons leads to a breathing dynamics of the BEC with frequencies $\Omega_{n}=2n\sqrt{\omega_{x}^{2}+2N\lambda}$, with $\omega_{x}$ the frequency of the harmonic trap, see Ref. [@Lode12]. For positive values of $\lambda$ the two-body interaction is *attractive*, while for negative values the interaction is repulsive and leads to unbound dynamics for $\lambda<-\omega_{x}/2N$. Note that we use the same parameters than in Sec. \[time\_indpdt\] for the numerical resolution of the EOM.
Breathing dynamics with $\lambda = 0.1$
---------------------------------------
We first consider the dynamics following a quenching of the interaction strength from $\lambda=0$ to $\lambda=0.1$ \[Eq. (\[H\_quench\])\]. We find that the MCTDHB method with $M=4$ orbitals and $286$ configurations is numerically exact for the propagation time considered here, i.e., $0\le t\le 15$, see Fig. \[gene\_RAS\_0\_1\]. The time evolution of the system is characterized by the one-particle density, $\rho(x=0,t)$, at the center of the trap $x=0$ and exhibits a periodic evolution with a frequency $\omega_{\text{MCTDHB}}=3.46$. This value agrees perfectly with the analytical prediction $\Omega_{n=1}=3.46$ meaning that the first excited state is mainly responsible for the dynamics. Nonetheless, the discrepancy with a pure cosine function $\sim \cos(\Omega_{1}t)$ indicates the role of higher excited states with higher harmonic frequencies [@Lode12]. The mean-field GP fails to describe the system evolution, even at short time ($t<1$) as depicted in Fig. \[gene\_RAS\_0\_1\] (a) and we obtain a lower frequency $\omega_{\text{GP}} = 3.35$. First, we perform TD-RASSCF-B simulations using a single ${\cal P}_{1}$ orbital ($M_{1}=1$) and $M_{2}=3$ ${\cal P}_{2}$ orbitals for different RAS schemes reported in Fig. \[gene\_RAS\_0\_1\] (a). For a short time, i.e., $0\le t\le 5$, all RAS schemes describe accurately the dynamics of the system, in contrast to the GP result. On the scale of the figure, the convergence to the MCTDHB result is obtained by using the -SDTQ excitation scheme including $35$ configurations, a reduction by a factor of $\sim 8$. For a longer time, $10\le t\le 15$, the -SD RAS scheme substantially differs from the MCTDHB result with a shift in the frequency and a smaller amplitude of the oscillations. The -SDTQ scheme only slightly differs by a smaller amplitude for $t>11.5$ and convergence is achieved for the -SDTQ56 excitation scheme with $84$ configurations. We also investigate the role of the ${\cal P}_{1}$ orbitals on the accuracy of the computations by considering $M_{1}=2$ and $M_{2}=2$, such as the total number of orbitals, $M=4$, remains unchanged. The -SD excitation scheme converged for short time, see Fig. \[gene\_RAS\_0\_1\] (b), and provides a better description of long time dynamics than the -SDTQ scheme used previously \[Fig. \[gene\_RAS\_0\_1\] (a)\] but includes $58$ configurations. This number of configurations is similar to the $56$ configurations obtained by using the -SDTQ5 RAS scheme with a single ${\cal P}_{1}$ orbital, which differs from the MCTDHB results for $t>11.5$ (not shown) while the -SD scheme with $2$ ${\cal P}_{1}$ orbitals differs for $t>13.5$, see Fig. \[gene\_RAS\_0\_1\] (b). We obtain converged results for the -SDT RAS scheme with $90$ configurations. In the previous section, we showed that the RAS schemes with *only* even excitations provide accurate results for GS energy and reduce the numerical effort (see Appendix \[efficiency\_TDRAS\]). We apply the -D and -DQ schemes with $M_{1}=1$ and $M_{1}=2$ Fig. \[even\_RAS\_0\_1\] (a) and (b), respectively, and keep $M=4$. For both $M_{1}=1$ and $M_{1}=2$, the results do not converge to the MCTDHB results and do not significantly improve for larger excitation schemes. For $M_{1}=1$, Fig. \[even\_RAS\_0\_1\] (a), the results obtained with the -D and the -DQ schemes are in very good agreement with the MCTDHB results concerning both the frequency and the amplitude and start to deviate only for $t>11$. In both cases, the number of configurations used to expand the wavefunction is substantially smaller than the expansion of the MCTDHB wavefunction with $7$ and $22$ configurations, respectively. When we use $2$ orbitals in the ${\cal P}_{1}$-space both -D and -DQ provide the same results for the dynamics, see Fig. \[even\_RAS\_0\_1\] (b). For short time dynamics, the results are similar to the ones obtained previously with $M_{1}=1$, but for a longer time, the oscillations remain in phase with the MCTDHB result, only the amplitude deviates for $t>13$. The wavefunction of the -D scheme includes $38$ configurations. Thus, the TD-RASSCF-B method provides an access to describe accurately the dynamics of the interacting system, while the mean-field GP theory failed even for short time. We obtain a good agreement in comparison to the MCTDHB method with a substantial reduction of the configurational space, using few tens instead of the few hundreds of configurations with the MCTDHB method. Moreover, the different parameters of TD-RASSCF-B method which define the wavefunction can be used to converge the results to the MCTDHB calculations. The implications of this reduction on the CPU time are discussed at the end of this Section.
Breathing dynamics with $\lambda = 0.5$
---------------------------------------
We pursue the illustration of the TD-RASSCF-B method by considering a quenching from $\lambda=0$ to $\lambda=0.5$ \[Eq. (\[H\_quench\])\]. This interaction strength was used in Ref. [@Lode12] to benchmark the MCTDHB method. For the time interval considered here, $0\le t\le 15$, we find that the result obtained with the MCTDHB method using $8$ orbitals and $19448$ configurations is numerically exact, in agreement with Ref. [@Lode12]. As previously, the one-body density exhibits oscillations as a function of the time with a period $\omega_{\text{MCTDHB}} = 6.63$, in perfect agreement with the analytical frequency $\Omega_{n=1}=6.63$. In comparison to the previous results, the shape of the oscillations indicates that the role of higher excited states is stronger as a large deviation from a simple cosine function, $\sim \cos(\Omega_{1}t)$, is obtained. This is not surprising since the value of $\lambda$ is now $5$ times larger than the one of the previous example. Along with the MCTDHB result we report, for comparison, the result obtained using the mean-field GP theory, see Fig. \[RAS\_0\_5\] (a), which strongly deviates from the MCTDHB result for $t>0.3$ with a larger amplitude in the oscillations and gives a lower frequency for the oscillations, $\omega_{\text{GP}}=6.32$. We start the TD-RASSCF-B simulations using $M=8$ orbitals with one single ${\cal P}_{1}$ orbital and $M_{2}=7$ ${\cal P}_{2}$ orbitals, Fig. \[RAS\_0\_5\] (a). For times between $0$ and $2.5$, the excitation schemes larger or equal to -SDTQ provide an accurate description of the dynamics, while the -SDTQ scheme includes $330$ configurations in the wavefunction, i.e., a factor of $\sim 60$ less than the MCTDHB. For longer times, the -SDTQ5678 RAS scheme with $6435$ configurations is required to accurately describe the MCTDHB results, the lower excitation schemes give substantially different results. To improve the results of the TD-RASSCF-B method, we increase the number of orbital in ${\cal P}_{1}$ and keep constant the total number of orbitals, $M=8$. In Fig. \[RAS\_0\_5\] (b), (c) and (d) we report the results with $M_{1}=2$, $3$ and $4$ ${\cal P}_{1}$ orbitals, respectively. In the case of $M_{1}=2$, the -SDTQ56 RAS scheme, with $5412$ configurations, provides a very accurate description of the dynamics for $0\le t\le15$. For lower excitation schemes, the results are accurate for $0\le t\le 3$ and using the -SDTQ scheme, the frequency of the oscillation at a longer time are correctly obtained, see Fig. \[RAS\_0\_5\] (b). For $M_{1}=3$, the results in Fig. \[RAS\_0\_5\] (c) show that the -SDTQ5 RAS scheme including $6882$ configurations converged to the MCTDHB results, while the -SDT scheme with $2276$ configurations gives the correct frequency for the oscillation but with a smaller amplitude. Finally, Fig. \[RAS\_0\_5\] (d) report converged results for $M_{1}=4$ using the -SDT schemes, which includes $5216$ configurations. Nonetheless, the -SD scheme with $2816$ configurations is accurate for the considered time of propagation in comparison to the MCTDHB result.
Using different numbers of ${\cal P}_{1}$ orbitals and different RAS schemes points out that the number of configurations required to accurately describe the evolution of the system change substantially. For instance, a similar accuracy is achieved for the -SDTQ5678 scheme with $M_{1}=1$, the -SDTQ56 scheme with $M_{1}=2$, the -SDT scheme with $M_{1}=3$ and the -SD scheme with $M_{1}=4$. For each case, a smaller amplitude of the oscillations is observed in comparison to the MCTDHB for $t >12$. But the numbers of configurations used in the wavefunction expansions are $6435$, $5412$, $2276$ and $2816$, respectively. Thus, for a comparable accuracy, the number of configurations can be divided by a factor $\sim 2$ by choosing adequately the size of the ${\cal P}_{1}$-space and a factor $\sim 10$ in comparison to the MCTDHB configurational space. The reduction of the configurational space impacts strongly the required CPU times of the simulations. For instance, the -SDTQ5678 RAS scheme with $M_{1}=1$ took $\sim 18.7$ CPU hours, while the -SDTQ56 scheme with $M_{1}=2$, the -SDT scheme with $M_{1}=3$ and the -SD scheme with $M_{1}=4$ took $\sim 19.8$, $\sim 6.9$ and $\sim 11.1$ CPU hours, respectively, on a 2.4 GHz Intel E5-2680 CPU. Using these RAS schemes, the CPU time is drastically reduced in comparison to the $\sim 113.4$ CPU hours on a 2.5 GHz Intel E5-2680 CPU needed to perform the MCTDHB simulation, but the dynamics is accurately described. Moreover, the -SDTQ5 RAS scheme with $M_{1}=3$ converged to the exact solution with $\sim 4$ times less CPU time. Albeit the drastic reduction in the size of configuration space, the CPU time needed to perform the TD-RASSCF-B calculations is also substantially reduced.
We briefly summarize the findings concerning the dynamical evolution of trapped atoms after a quenching of the interaction strength of an attractive harmonic interaction. In the case of $\lambda=0.1$, the MCTDHB theory converged to the numerically exact result for $4$ orbitals. The TD-RASSCF-B method using different size of the ${\cal P}_{1}$-space and different RAS schemes can accurately describe the dynamics of the system characterized by $\rho(x=0,t)$ with $\sim 4$ times fewer configurations. Moreover, converged TD-RASSCF-B results were obtained with substantially less configurations, for instance with $M_{1}=2$ and the -SDT excitation scheme. In the case of $\lambda=0.5$, the exact solution was obtained with $8$ orbitals using the MCTDHB method leading to $19448$ configurations. A larger number of orbitals is needed for the stronger interaction between the particles. Accurate results were obtained with the TD-RASSCF-B method, reducing by a factor $\sim 10$ the size of the configurational space and converged results were obtained with $4$ times less configurations, for instance considering $M_{1}=4$ and the -SDT RAS scheme. Moreover, all calculations performed with the TD-RASSCF-B method were better than the mean-field GP theory, which failed to describe both scenarios.
Conclusion and outlook {#conclusion}
======================
In this work, we presented a general formalism for the time-dependent restricted active-space self-consistent field (TD-RASSCF) method, which includes the first derivation obtained for fermions (TD-RASSCF-F) [@Haru13; @Haru14_1; @Haru14_2] and extended it for systems of spinless bosons (TD-RASSCF-B). This TD-RASSCF-B method includes, as limiting cases, the (TD)-GP and the MCTDHB theories and provides a way to tackle the exponential growth of the configurational space in the MCTDHB method. The EOM were derived for two families of RAS schemes, which give the possibility to restrict the full-configurational description of the MCTDHB wavefunction. Through a set of numerical examples, we have shown that the method can provide an accurate description of the static properties of the ground-state of the system. In the case of hundreds of particles, the method can lead to results beyond the reach of the MCTDHB method, providing a better accuracy by including more orbitals while constraining the number of configurations. In this sense, the TD-RASSCF-B method paves the way for numerical investigation of intermediate system sizes with a few tens to hundreds of bosons with a better accuracy than what was possible with the MCTDHB method. We also provided a comparison between the MCTDHB and TD-RASSCF-B method in the case of breathing dynamics induced by a sudden quenching of the interaction strength with two different initial conditions. As for the MCTDHB method, the TD-RASSCF-B method does not have any restriction on the choice of the two-body interaction potential used, as was illustrated by the use of a non-contact harmonic interaction between the bosons. Using as a reference the numerically exact result obtained from the MCTDHB method, we showed that the TD-RASSCF-B method is always more accurate than the mean-field TD-GP theory to describe the dynamics of the system. Moreover, using different RAS schemes and partitions of the ${\cal P}$-space we obtained very accurate results for substantially less configurations and thus less CPU time than with the MCTDHB method. This reduction of the configurational space can be efficiently exploited to solve numerically the TDSE beyond the mean-field approach. Dynamical effects such as the four wave mixing (FWM) process [@Hilligsoe05] used to produce correlated atoms beams [@Bonneau13] or the dynamics of bright [@Khaykovich02; @Strecker02] and dark [@Burger99; @Denschlag97] solitons can be investigated *ab-initio* beyond the mean-field TD-GP theory. In addition, the dynamics induced by a time-dependent Hamiltonian can also be explored using the TD-RASSCF-B method, such as in the case of periodically driven optical lattices [@Aidelsburger13; @Goldman14].\
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are indebted to Dr. Haruhide Miyagi for useful discussions. This work was supported by the ERC-StG (Project No. 277767-TDMET), and the VKR center of excellence, QUSCOPE.
Appendices {#appendices .unnumbered}
==========
In these Appendices we provide a brief description of the implementation the TD-RASSCF-B method. The implementation is rather similar for bosons and fermions in the sense that the set of equations that we have to solve, i.e. Eqs. (\[EOM\_C\_general\]), (\[EOM\_Qspace\]) and (\[eta\_eq\_even\_final\]) or (\[Pspace\_General\_RAS\]), only depend on the type of particles trough the creation and annihilation operators and the set of configurations $\{|\Phi_{I}\rangle\}$.
Compact representation of the wavefunction {#wf_representation}
==========================================
Our implementation is based on the general mapping of bosonic operators in Fock space introduced in Ref. [@Streltsov10] and implemented, for instance, for bosons [@Streltsov11] and fermions [@Fasshauer16] in the framework of multi-configurational TD methods. Assuming $M$ orbitals and $N$ bosons, the configurations are expressed using the occupation number formalism, where $|n_{1},n_{2},\cdots,n_{M}\rangle$, represents a configuration with $n_{1}$ bosons in the orbital $|\phi_{1}\rangle$, $n_{2}$ bosons in the orbital $|\phi_{2}\rangle$, etc. Such a configuration is indexed by an unique integer $J$ defined as, $$\label{indexing_conf}
J =1+\sum_{k=1}^{M}
\begin{pmatrix}
N+M -1 -k - \sum_{l=1}^{k} n_{l} \\
M-k
\end{pmatrix}.$$ Thus, for each configuration we store its complex coefficient $C_{J}$ in an array according to the index $J$ provided by the above mapping. In Ref. [@Streltsov11], this mapping was employed to avoid the storage of the configuration vectors $|n_{1},n_{2},\cdots,n_{M}\rangle$, which can be prohibitively memory consuming in the case of the MCTDHB method. Thus, to access the coefficient of the configurations, a set of M-nested loops over the occupation number $n_{i}$ is used to span the full configurational space and to compute, using Eq. (\[indexing\_conf\]), the respective indexes. In the case of the TD-RASSCF-B method only a selected number of configurations are used to expand the wavefunction, and the scheme of Ref. [@Streltsov11] is not readily applicable in the sense that we want to avoid explicit use of the full configurational space. Instead we follow a different strategy for indexing the RAS configurations. We introduce $M_{1}$ and $M_{2}$ the number of orbitals in the ${\cal P}_{1}$ and ${\cal P}_{2}$ spaces, respectively. For $\mathcal{V}_{0}$, i.e., configuration with particles only in ${\cal P}_{1}$, see Eq. (\[Fock\_decomp\]), we enumerate all possible configurations, evaluate their index, using Eq. (\[indexing\_conf\]), and store them. Then for the excited configurations, i.e., configurations with one or more particles in ${\cal P}_{2}$, we introduce the excitation number $n_{exc}$ which is equivalent to the number of particles in ${\cal P}_{2}$. The number of remaining particles in ${\cal P}_{1}$ is $N-n_{exc}$. For each excitation we enumerate the configurations of $N-n_{exc}$ particles in the ${\cal P}_{1}$ orbitals and compute their index. The same is performed for the configurations with $n_{exc}$ in ${\cal P}_{2}$ orbitals. Then the permanents for the total number of bosons are obtained by combining them, $$\begin{split}
|\textbf{n}^{n_{exc}}\rangle = &\left(\sum_{i=1}^{N_{1}^{n_{exc}}}|n_{1},n_{2},\cdots,n_{M_{1}}\rangle \right) \\
&\otimes \left(\sum_{i=1}^{N_{2}^{n_{exc}}}|n_{M_{1}+1},\cdots,n_{M}\rangle\right),
\end{split}$$ where $|\textbf{n}^{n_{exc}}\rangle$ is an array of dimension $M \times (N_{1}^{n_{exc}}\times N_{2}^{n_{exc}})$, with $N_{1}^{n_{exc}}$ ($N_{2}^{n_{exc}}$) the total number of configurations obtained from arranging the $N_{b}-n_{exc}$ ($n_{exc}$) particles in the ${\cal P}_{1}$ (${\cal P}_{2}$) orbitals. We evaluate the indexes of the configurations resulting from the ${\cal P}_{1}$ subsystem, $J_{P_{1}}^{n_{exc}}$, and from the ${\cal P}_{2}$ subsystem, $J_{P_{2}}^{n_{exc}}$, applying Eq. (\[indexing\_conf\]) for the subsystems, separately. The index of the configuration with the total number of bosons is build as a three components array defined by $\{J_{P_{1}}^{n_{exc}},J_{P_{2}}^{n_{exc}},n_{exc}\}$, which stores the position of the configuration in the configurational vector. This scheme is thus applied for all excitations, $n_{exc}$, included in the RAS scheme. With this storage or construction of the wavefunction, for a given configuration we have access to its coefficient in the following way. (i) We evaluate the index $J_{P_{1}}^{n_{exc}}$ for the ${\cal P}_{1}$ subsystem, (ii) we evaluate the index $J_{P_{2}}^{n_{exc}}$ for the ${\cal P}_{1}$ subsystem, (iii) we know or evaluate the excitation, $n_{exc}$, of the configuration (iv) we access to the index of the coefficient which is stored and the three component array at position $\{J_{P_{1}}^{n_{exc}},J_{P_{2}}^{n_{exc}},n_{exc}\}$. This scheme has, as a draw back, the requirement to store the list of occupation numbers and the indexes to be efficient for numerical evaluation. But the idea behind the TD-RASSCF-B method is to reduced the size of the configurational space, which makes such a storage manageable for applications done so far.
Applying operators in second quantization {#apply_op}
=========================================
In second quantization, the action of the Hamiltonian of Eq. (\[Hamiltonian\]) on the wavefunction requires the application of annihilation and creation operators and multiplication by the matrix elements of the one- and two-body operators. To know the action of the Hamiltonian, we first need to know the action of the creation-annihilation operators [@Streltsov11]. Concerning the one-body term, we have, $$\begin{split}
b_{i}^{\dag}b_{j} &|n_{1},\cdots,n_{j},\cdots,n_{i},\cdots,n_{M}\rangle \\
&= \sqrt{n_{j}}\sqrt{n_{i}+1}|n_{1},\cdots,n_{j}-1,\cdots,n_{i}+1,\cdots,n_{M}\rangle.
\end{split}$$ For a full-configurational wavefunction, the resulting configuration belongs to the configurational space and its index can be determined as described in Appendix \[wf\_representation\]. Now, considering the action on the wavefunction, $$\label{reord_conf}
b_{i}^{\dag}b_{j} \left[ \sum_{I\in {\cal V}}C_{I}|\Phi_{I}\rangle \right]= \sum_{I\in {\cal V}}C_{I}\sqrt{n_{j}}\sqrt{n_{i}+1}|\Phi_{I'}\rangle,$$ with $|\Phi_{I'}\rangle$ the new configuration resulting from the action of $b_{i}^{\dag}b_{j}$ on the initial configuration $|\Phi_{I}\rangle$. The result can be interpreted as a reordering of the configuration in the wavefunction, as in Eq. (\[reord\_conf\]) or inversely to a reordering of the coefficients with a new factor ($\sqrt{n_{j}}\sqrt{n_{i}+1}$) if the configuration are reorganized in the initial order, i.e., $$b_{i}^{\dag}b_{j} \left[ \sum_{I\in {\cal V}}C_{I}|\Phi_{I}\rangle \right]= \sum_{I\in {\cal V}}C_{I'}|\Phi_{I}\rangle.$$ In the basis of the configurational states, $\{|\Phi_{I}\rangle\}$, the wavefunction is characterized by its coefficients only, and is stored as a vector. The new set of coefficients, $\{C'_{I}\}$, resulting from the action of $b_{i}^{\dag}b_{j}$, is obtained as, $$C'_{I} = \langle \Phi_{I}| b_{i}^{\dag}b_{j} \left[ \sum_{I\in {\cal V}}C_{I}|\Phi_{I}\rangle \right].$$ In practice, we apply $b_{i}^{\dag}b_{j}$ on the bra $ \langle \Phi_{I}|$, which provides a new configurational state $\langle \Phi_{J}|$. The configurational vectors are orthonormal, and only the configuration $|\Phi_{J}\rangle$ in the sum remains from the projection. Thus, we evaluate the index of $\langle \Phi_{J}|$ to directly access the coefficient $C_{J}$, i.e., $$C'_{I} =C_{J}\sqrt{n_{i}}\sqrt{n_{j+1}}.$$ The action on the total wavefunction is obtained by repeating these operations for each configuration, providing a new coefficient vector. In the case of the RAS wavefunction, the configuration obtained from the successive application of the annihilation and creation operators may not belong to the configurational space. Nonetheless, the scheme applied above can be applied with, in addition, a test to check if the resulting configuration remains in the RAS space. This naive approach can be easily improved thanks to the representation of the wavefunction used and its indexing (see Appendix \[wf\_representation\]). The orbitals $\{i,j\}$, on which the operators $b_{i}^{\dag}b_{j}$ act, can (i) belong to ${\cal P}_{1}$ only $\{i',j'\}$, (ii) belong to ${\cal P}_{2}$ only $\{i'',j''\}$, or belong to ${\cal P}_{1}$ and ${\cal P}_{2}$, (iii) $\{i',j''\}$ and (iv) $\{i'',j'\}$. For (i) and (ii) the excitation, or the number of particle in ${\cal P}_{2}$ orbital, do not change in the resulting configuration. For the situation (iii) one particle is removed from ${\cal P}_{2}$ and added in ${\cal P}_{1}$ and the opposite happens for (iv). The excitation of the final configuration is thus known without counting the number of particles in ${\cal P}_{2}$, which is required to determine the index of the configuration. Moreover, to always remain in the RAS configurational space, the case (iv) is never applied to the configuration with the maximum excitation allowed for the general RAS scheme (Sec. \[all\_exci\]), and both (iii) and (iv) are not used for the scheme with only even excitations (Sec. \[even\_exci\]). The action of the one-body operator of the Hamiltonian is now straightforward, the coefficient vectors obtained by applying the $b_{i}^{\dag}b_{j}$ operators are multiplied by the corresponding matrix element $h_{j}^{i}$ \[Eq. (\[one-bod\])\] and summed for each couple of $\{i,j\}$, with the restriction mentioned above for the RAS wavefunction. The two-body operator, see Eq. (\[two-bod\]), included in the Hamiltonian of Eq. (\[Hamiltonian\]) and the four- and six-order tensors specific to the RAS schemes \[Eqs. (\[four\_order\_tens\]) and (\[six\_order\_tens\])\] can be evaluated using the same strategy as the one detailed for the one-body operator. We mention that using the commutation relation for bosonic creation and annihilation operators can substantially reduced the numerical cost. For instance, if we consider the chain of operators $b^{\dag}_{i}b^{\dag}_{j}b_{k}b_{l}$, we have the equalities $b^{\dag}_{i}b^{\dag}_{j}b_{k}b_{l}=b^{\dag}_{i}b^{\dag}_{j}b_{l}b_{k}=b^{\dag}_{j}b^{\dag}_{i}b_{k}b_{l}=b^{\dag}_{i}b^{\dag}_{j}b_{l}b_{k}$.
Numerical implementation for the TD-RAS equations {#Ap_Num_imple}
=================================================
The EOM for the TD-RASSCF-B and F methods, Eqs. (\[EOM\_C\_general\]), (\[EOM\_Qspace\]) and (\[eta\_eq\_even\_final\]) or (\[Pspace\_General\_RAS\]), are solved to obtain the time derivative of the coefficients and orbitals. The main difference with the MCTDHB and F methods results from the evaluation of the matrix elements $\eta_{i'}^{j''}$. These elements are evaluated from Eqs. (\[eta\_eq\_even\_final\]) or (\[Pspace\_General\_RAS\]) depending of the RAS scheme used, but both are solved in the same way. We recall that the matrix $\underline{\underline{\bm{\eta}}}$, with elements $\eta_{i'}^{j''}$, is anti-hermitian and thus $\eta_{j''}^{i'} = -(\eta_{i'}^{j''})^{*}$, and $i'$ and $j''$ hold for orbitals of the ${\cal P}_{1}$ and ${\cal P}_{2}$ subspace, respectively. Writing down Eqs. (\[eta\_eq\_even\_final\]) or (\[Pspace\_General\_RAS\]) for any set of orbitals $\{i',j''\}$, provides a system of $M_{1}\times M_{2}$ linear equations with $M_{1}\times M_{2}$ unknowns. Introducing a composite index for the couple of $\{i',j''\}$, this system can be written in a matrix form, $$\label{sol_eta}
\underline{\underline{\bf{A}}} . \underline{\bf{X}}=\underline{\bf{B}},$$ where the matrix $\underline{\underline{\bf{A}}}$, of dimension ($M_{1}\times M_{2}$, $M_{1}\times M_{2}$), contains the values of $A_{k''i'}^{l'j''}$ or $\zeta_{k''i'}^{l'j''}$, for all set of $\{i',j''\}$ and $\{l',k''\}$. The vector $\underline{\bf{B}}$, of dimension ($M_{1}\times M_{2}$), contains the r.h.s. of Eq. (\[eta\_eq\_even\_final\]) or (\[Pspace\_General\_RAS\]) for each set of $\{i',j''\}$ and the vector $\underline{\bf{X}}$ with the same dimension as $\underline{\bf{B}}$ contains the unknown values of $\eta_{i'}^{j''}$ and the matrix elements of the one-body operator. The system of linear equations, Eq. (\[sol\_eta\]), can be solved using a standard numerical routine included, for instance, in the LAPACK library [@lapack]. The values for $\eta_{i'}^{j''}$ are trivially obtained from the elements of the vector $\underline{\bf{X}}$, $$\eta_{i'}^{j''} = \left\{
\begin{split}
i(X\{i',j''\}-h_{i'}^{j''}) \ \ \ \ \text{for Eq. (\ref{eta_eq_even_final}),}\\
-i(X\{i',j''\}+h_{i'}^{j''}) \ \ \ \ \text{for Eq. (\ref{Pspace_General_RAS}).}
\end{split}
\right.$$ After evaluating the matrix elements $\eta_{i'}^{j''}$, the time derivative of the coefficients $\{\dot{C}_{I}\}$ can be computed from Eq. (\[EOM\_C\_general\_expand\]) and the contribution of the ${\cal P}$-space orbitals to the time derivative of the orbitals is obtained from, $$\hat{P}|\dot{\phi_{i}}\rangle= \sum_{j}^{M}|\phi_{j}\rangle\eta_{i}^{j}.$$ It remains to evaluate the contribution from the orbitals of the ${\cal Q}$-space, i.e. $\hat{Q}|\dot{\phi_{i}}\rangle$ from Eq. (\[EOM\_Qspace\]). This latter can be expressed in a matrix form,
$$\label{mat_form_Q_space}
i\hat{Q}\underline{\underline{\bm{\rho}}}\underline{\dot{\bm{X}}}= \hat{Q}\left[ \underline{\underline{\bm{\rho}}}\underline{\bm{\tilde{h}}} + \underline{\bm{\tilde{W}}}\right],$$
with $\underline{\underline{\bm{\rho}}}$ the one-body reduced density matrix, $\underline{\dot{\bm{X}}}$ a vector collecting the time derivative of the orbitals, $\underline{\bm{\tilde{h}}}$ and $\underline{\bm{\tilde{W}}}$ are both vectors with elements $h(\bm{r},t)|\phi_{i}\rangle$ and $\sum_{jlk}\hat{W}_{l}^{k}|\phi_{j}\rangle\rho_{ik}^{jl}$, respectively. To obtain Eq. (\[mat\_form\_Q\_space\]), we used the fact that $\underline{\underline{\bm{\rho}}}$ commutes with the projector $\hat{Q}$, as easily seen from the equality $\hat{Q}=\hat{1}-\hat{P}$. Multiplying on the left by the inverse of the one-body density matrix, $\underline{\underline{\bm{\rho}}}^{-1}$, we have, for the orbital $|\phi_{i}\rangle$, $$\begin{aligned}
\hat{Q}|\dot{\phi}_{i}\rangle &= -i\hat{Q}\left[ \hat{h}|\phi_{i}\rangle + \sum_{jklm} (\underline{\underline{\bm{\rho}}}^{-1})_{i}^{m} \rho_{mk}^{jl}\hat{W}_{l}^{k}|\phi_{j}\rangle\right] \nonumber\\
&=-i(\hat{1}-\hat{P})\left[ \hat{h}|\phi_{i}\rangle + \sum_{jklm} (\underline{\underline{\bm{\rho}}}^{-1})_{i}^{m} \rho_{mk}^{jl}\hat{W}_{l}^{k}|\phi_{j}\rangle\right].\end{aligned}$$
The right hand side of the above equation is similar to the one that is solved in the MCTDH-based methods [@Meyer90; @Alon08; @Haxton11] and we follow the numerical implementation used for the MCTDH method [@Beck97] to avoid singularities in the inverse of the one-body reduced density matrix and in Eq. (\[sol\_eta\]) for the matrix $\underline{\underline{{\bm{A}}}}$, as well as for the projector onto the ${\cal P}$-space orbitals.
Numerical efficiency of the method {#efficiency_TDRAS}
==================================
Comparing the efficiency between different methods is a difficult task as it depends of the specific implementation and integration schemes used. Nonetheless, we can roughly estimate the number of operations required to evaluate the time derivative of the orbitals and coefficients and compare the MCTDHB and TD-RASSCF-B methods in this way. We denote by $N_{grid}$ the number of grid points that are used to describe the time-dependent orbital in the time-independent basis, usually a DVR [@Heather83], which is the same for both methods. Starting with the MCTDHB method, at each evaluation of the time derivative the matrix elements of the two-body operator $v_{kl}^{ij}$ \[Eq. (\[two-bod\])\] and the two-body reduced density matrix $\rho_{ik}^{jl}$ \[see text above Eq. (\[EOM\_Qspace\])\] are computed. These updates require $M^{4}N_{grid}^{2}$ and $M^{4}\mathcal{V}_{\text{FCI}}$ operations, respectively, where $\mathcal{V}_{\text{FCI}}$ is the size of the configurational space of the MCTDHB wavefunction evaluated from Eq. (\[dim\_wf\_MCTDHB\]). Then computing the time derivative of the coefficients and the orbitals require $M^{4}\mathcal{V}_{\text{FCI}}$ and $M^{4}N_{grid}^{2}$ operations, respectively. The total cost is thus, approximatively, $2M^{4}(N_{grid}^{2}+\mathcal{V}_{\text{FCI}})$. Considering now the case of the TD-RASSCF-B method. The evaluation of the matrix elements of two-body operator and the calculation of the ${\cal Q}$-space equations for the time derivative of the orbitals require the same number of operations as with the MCTDHB method, i.e., $M^{4}N_{grid}^{2}$ operations for each. The evaluation of the time derivative of the coefficients and the matrix elements of the two-body reduced density matrix scale as $M^{4}\mathcal{V}$, with $\mathcal{V}$ the size of the configurational RAS space. In addition, we also need to solve the ${\cal P}$-space equations, which requires $M^{4}$ operations for excitation schemes with only even excitations and $M^{4}\mathcal{V}_{N_{\text{max}}}$ for the general RAS scheme, with $\mathcal{V}_{N_{\text{max}}}$ the number of configuration including $N_{\text{max}}$ particles in the ${\cal P}_{2}$-space. The total number of configurations included in the RAS wavefunction for the general excitation scheme is evaluated using Eq. (\[config\_general\_RAS\]) and $\mathcal{V}_{N_{\text{max}}}$ is the last term of the summation. The dimension of the configurational space including only even excitations can be evaluated in a similar way, $$\label{config_even_RAS}
\begin{split}
dim({\cal V})& =\begin{pmatrix}
N+M_{1}-1 \\
N
\end{pmatrix} \\
&+ \sum_{k=1}^{N_{\text{max}}/2}
\begin{pmatrix}
2k+M_{2}-1 \\
2k
\end{pmatrix}
\begin{pmatrix}
(N-2k)+M_{1}-1 \\
N-2k
\end{pmatrix}.
\end{split}$$ Combining the results for the general RAS scheme, the number of operations required to evaluate the time derivative of the coefficients and orbitals scales as $2M^{4}(N_{grid}^2+\mathcal{V}+M^{2}\mathcal{V}_{\text{N}_{\text{max}}})$ and in the case of only even excitations it scales as $2M^{4}(N_{grid}^2+\mathcal{V}+1/2)$. To compare the numerical cost between the MCTDHB and TD-RASSCF-B methods, we can introduce $\Delta(Op)$, the difference between the MCTDHB and TD-RASSCF-B operations to remove the constant number of operation resulting from $N_{grid}$, $$\label{delta_Op}
\small{
\Delta(Op)= \left\{
\begin{split}
& 2M^{4}(\mathcal{V}_{\text{FCI}}-\mathcal{V}-1/2)\quad \quad \quad \quad \text{\small{even excitation,}}\\
& 2M^{4}(\mathcal{V}_{\text{FCI}}-\mathcal{V}-M^{2}\mathcal{V}_{\text{N}_{\text{max}}}/2)\ \ \text{\small{general scheme.}}
\end{split}
\right.
}$$ From the expression of $\Delta(Op)$, a positive value represents a computational gain with the TD-RASSCF-B in comparison to the MCTDHB method, while a negative value is obtained when the MCTDHB method is more efficient. In the case of a scheme with only even excitations, $\Delta(Op)$ is proportional to the size difference of the MCTDHB and TD-RASSCF-B configurational spaces and is always positive, which means that the TD-RASSCF-B method is always more efficient. In the case of the general excitation scheme the six-order tensor of the ${\cal P}$-space equations, Eq. (\[six\_order\_tens\]), can provide an overhead for the computation. To illustrate the computational efficiency we evaluate $\Delta(Op)$ for $10$, $50$ and $100$ bosons in $M=2$ to $8$ orbitals, see Fig. \[scaling\_op\]. For the TD-RASSCF-B, we consider the case of a single ${\cal P}_{1}$ orbital and $M-1$ orbitals in ${\cal P}_{2}$. The case with only even excitations reduces the computational cost almost exponentially for increasing number of orbitals for any number of particles, which results from the efficiency of solving the ${\cal P}$-space equation. In the case of the general RAS scheme, there is always a value of $N_{\text{max}}$ which leads to more operations in the TD-RASSCF-B than in the MCTDHB method, due to the evaluation of the six-order tensor in the ${\cal P}$-space equation. But as shown in Fig. \[scaling\_op\], this value is rather large, i.e. $N_{\text{max}}=6$ for 10, $N_{\text{max}}=40$ for 50 particles and $N_{\text{max}}=90$ for 100 particles for the schemes depicted in Figs. \[scaling\_op\] (a), (b) and (c), respectively.\
TABLES {#tables .unnumbered}
======
---------------- -- ------------- ------------- ------------- ------------- ------------- ------------- -------------- --------------- --
Method $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
$\lambda=0.01$
MCTDHB 68.76816487 68.75335446 68.74538390 68.74152088 68.73891122 - - -
(1) (101) (5151) (176851) (4598126) (96560646) (1705904746) (26075972546)
-SD - 68.753 68.745 68.741 68.73 68.73761231 68.736360917 68.73545355
(3) (6) (10) (15) (21) (28) (36)
-SDTQ - 68.75335 68.74538 68.74152 68.73891 68.73724366 68.73598073 68.73506372
(5) (15) (35) (70) (126) (210) (330)
-SDTQ56 - 68.7533544 68.7453839 68.741520 68.738911 68.73723959 68.73597655 68.73505943
(7) (28) (84) (210) (462) (924) (1716)
-SDTQ5678 - 68.75335446 68.74538390 68.74152088 68.73891122 68.73723955 68.73597651 68.73505938
(9) (45) (165) (495) (1287) (3003) (6435)
-10 - 68.75335446 68.74538390 68.74152088 68.73891122 68.73723955 68.73597651 68.73505938
(11) (66) (286) (1001) (3003) (8008) (19448)
$\lambda=0.1$
MCTDHB 193.5509587 193.0154216 192.6308389 192.3920265 192.2138048 - - -
(1) (101) (5151) (176851) (4598126) (96560646) (1705904746) (26075972546)
-SDT - 193.0 192. 192. 192. 192.3153377 192.2315684 192.1681159
(4) (10) (20) (35) (56) (84) (120)
-SDTQ5 - 193.0 192.6 192. 192.2 192.1396169 192.0434375 191.9701535
(6) (21) (56) (126) (252) (462) (792)
-SDTQ567 - 193.01 192.63 192. 192.2 192.1000115 192.0013608 191.9259461
(8) (36) (120) (330) (792) (1716) (3432)
-SDTQ56789 - 193.01 192.63 192.39 192.21 192.0903987 191.9911974 191.9152346
(10) (55) (220) (715) (2002) (5005) (11440)
-10 - 193.015 192.63 192.39 192.21 192.0890211 191.9894655 191.9133983
(11) (66) (286) (1001) (3003) (8008) (19448)
-15 - 193.01542 192.6308 192.3920 192.2138 192.0872814 191.9879161 191.9117429
(16) (136) (816) (3876) (15504) (54264) (170544)
-20 - 193.01542 192.63083 192.39202 192.21380 192.0872403 191.9878729 -
(21) (231) (1771) (10626) (53130) (230230) (888030)
-23 - 193.0154216 192.6308389 192.3920265 192.213804 192.0872393 191.9878720 -
(24) (300) (2600) (17550) (98280) (475020) (2035800)
-25 - 193.0154216 192.6308389 192.3920265 192.2138048 192.0872393 191.9878719 -
(26) (351) (3276) (23751) (142506) (736281) (3365856)
---------------- -- ------------- ------------- ------------- ------------- ------------- ------------- -------------- --------------- --
: Ground-state energy \[in units of $E_{0}$ see text after Eq. (\[H\_relax\])\] of $100$ bosons trapped in a 1D harmonic potential interacting through a contact potential with a strength $\lambda = 0.01$ and $0.1$. The TD-RASSCF-B calculations were performed with a single ${\cal P}_{1}$ orbital, $M_{1}=1$ and $M_{2}=M-1$ ${\cal P}_{2}$ orbitals, with $M$ the total number of orbitals. The results were obtained using the general RAS scheme, which includes both even and odd excitations. The excitation schemes are indicated with the usual notations -S, -SD, $\cdots$, up to $N_{\text{max}}=9$ and the RAS schemes are labeled by the value of $N_{\text{max}}$ for larger excitations, e.g. -10, -20. In addition MCTDHB calculations were carried out to compare the accuracy of the TD-RASSCF-B method and the efficiency. The number of configurations used in the wavefunction expansion are indicated in parentheses. The result obtained with a single orbital is equivalent to the GP method. To highlight the difference between the TD-RASSCF-B and MCTDHB results, the digits that differ are underlined.
---------------- -- ------------- ------------- ------------- ------------- ------------- ------------- -------------- --------------- --
Method $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
$\lambda=0.01$
MCTDHB 68.76816487 68.75335446 68.74538390 68.74152088 68.73891122 - - -
(1) (101) (5151) (176851) (4598126) (96560646) (1705904746) (26075972546)
-D - 68.753 68.745 68.741 68.73 68.73761353 68.73636218 68.73545481
(2) (4) (7) (11) (16) (22) (29)
-DQ - 68.75335 68.74538 68.7415 68.7389 68.73725598 68.73599408 68.73507803
(3) (9) (22) (46) (86) (148) (239)
-DQ6 - 68.7533544 68.74538 68.74152 68.7389 68.73725217 68.73599018 68.73507404
(4) (16) (50) (130) (296) (610) (1163)
-DQ68 - 68.75335446 68.74538 68.74152 68.7389 68.73725213 68.73599014 68.73507400
(5) (25) (95) (295) (791) (1897) (4166)
-10 - 68.75335446 68.74538 68.74152 68.7389 68.73725213 68.73599014 68.73507400
(6) (36) (161) (581) (1792) (4900) (12174)
$\lambda=0.1$
MCTDHB 193.5509587 193.0154216 192.6308389 192.3920265 192.2138048 - - -
(1) (101) (5151) (176851) (4598126) (96560646) (1705904746) (26075972546)
-D - 193. 192. 192. 192. 192.4050140 192.3243169 192.2636708
(2) (4) (7) (11) (16) (22) (29)
-DQ - 193.0 192.6 192. 192.2 192.1793448 192.0861393 192.0157234
(3) (9) (22) (46) (86) (148) (239)
-DQ6 - 193.0 192.6 192. 192.2 192.1259116 192.0303689 191.9579281
(4) (16) (50) (130) (296) (610) (1163)
-DQ68 - 193.01 192.63 192. 192.2 192.1129247 192.0169500 191.9440484
(5) (25) (95) (295) (791) (1897) (4166)
(6) (36) (161) (581) (1792) (4900) (12174)
-20 - 193.0154216 192.63 192. 192.2 192.1089342 192.0128774 191.9398292
(11) (121) (946) (5786) (29458) (129844) (508937)
-30 - 193.0154216 192.63 192. 192.2 192.1089338 192.0128771 -
(16) (256) (2856) (24616) (174624) (1061208) (5678340)
---------------- -- ------------- ------------- ------------- ------------- ------------- ------------- -------------- --------------- --
: Same as Table I but for ground-state energies obtained using RAS schemes with *only* even excitations. The excitation schemes are indicated with the notations -D, -DQ, $\cdots$, up to $N_{\text{max}}=8$ and the RAS schemes are labeled by the value of $N_{\text{max}}$ for larger excitations, e.g. -10, -20.
FIGURES {#figures .unnumbered}
=======
![ []{data-label="General_orbtial_space"}](General_orbtial_space.pdf)
![ []{data-label="Ras_Schemes"}](RAS_Schemes.pdf)
![ []{data-label="gene_RAS_0_1"}](results_K_0_to_0_1_general_RAS.pdf)
![ []{data-label="even_RAS_0_1"}](results_K_0_to_0_1_only_even_RAS.pdf)
![ []{data-label="RAS_0_5"}](results_K_0_to_0_5.pdf)
![[]{data-label="scaling_op"}](Results_scaling_op.pdf)
|
---
abstract: |
We introduce a new geometric spanner, $\delta$-*Greedy*, whose construction is based on a generalization of the known *Path-Greedy* and *Gap-Greedy* spanners. The $\delta$-Greedy spanner combines the most desirable properties of geometric spanners both in theory and in practice. More specifically, it has the same theoretical and practical properties as the Path-Greedy spanner: a natural definition, small degree, linear number of edges, low weight, and strong $(1+\varepsilon)$-spanner for every $\varepsilon>0$. The $\delta$-Greedy algorithm is an improvement over the Path-Greedy algorithm with respect to the number of shortest path queries and hence with respect to its construction time. We show how to construct such a spanner for a set of $n$ points in the plane in $O(n^2 \log n)$ time.
The $\delta$-Greedy spanner has an additional parameter, $\delta$, which indicates how close it is to the Path-Greedy spanner on the account of the number of shortest path queries. For $\delta = t$ the output spanner is identical to the Path-Greedy spanner, while the number of shortest path queries is, in practice, linear.
Finally, we show that for a set of $n$ points placed independently at random in a unit square the expected construction time of the $\delta$-Greedy algorithm is $O(n \log n)$. Our analysis indicates that the $\delta$-Greedy spanner gives the best results among the known spanners of expected $O(n \log n)$ time for random point sets. Moreover, the analysis implies that by setting $\delta = t$, the $\delta$-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected $O(n \log n)$ time.
author:
- 'Gali Bar-On'
- Paz Carmi
title: '$\delta$-Greedy $t$-spanner '
---
Introduction {#sec:Intro}
============
Given a set $P$ of points in the plane, a Euclidean $t$-spanner for $P$ is an undirected graph $G$, where there is a $t$-spanning path in $G$ between any two points in $P$. A path between points $p$ and $q$ is a $t$-spanning path if its length is at most $t$ times the Euclidean distance between $p$ and $q$ (i.e., $t|pq|$).
The most known algorithm for computing $t$-spanner is probably the *Path-Greedy* spanner. Given a set $P$ of $n$ points in the plane, the Path-Greedy spanner algorithm creates a $t$-spanner for $P$ as follows. It starts with a graph $G$ having a vertex set $P$, an empty edge set $E$ and $ {n \choose 2} $ pairs of distinct points sorted in a non-decreasing order of their distances. Then, it adds an edge between $p$ and $q$ to the set $E$ if the length of the shortest path between $p$ and $q$ in $G$ is more than $t|pq|$, see Algorithm \[alg:pathGreedy\] for more details. It has been shown in [@Chandra; @Chandra94; @Das1; @DasHN93; @GudmundssonLN02; @Soares1994] that for every set of points, the Path-Greedy spanner has $O(n)$ edges, a bounded degree and total weight $O(wt(MST(P)))$, where $wt(MST(P))$ is the weight of a minimum spanning tree of $P$. The main weakness of the Path-Greedy algorithm is its time complexity – the naive implementation of the Path-Greedy algorithm runs in near-cubic time. By performing $n \choose 2$ shortest path queries, where each query uses Dijkstra’s shortest path algorithm, the time complexity of the entire algorithm reaches $O(n^3 \log n)$, where $n$ is the number of points in $P$. Therefore, researchers in this field have been trying to improve the Path-Greedy algorithm time complexity. For example, the *Approximate-Greedy* algorithm generates a graph with the same theoretical properties as the Path-Greedy spanner in $O(n \log n)$ time [@DBLP97; @DBLP02]. However, in practice there is no correlation between the expected and the unsatisfactory resulting spanner as shown in [@DBLP07; @FarshiG09]. Moreover, the algorithm is complicated and difficult to implement.
Another attempt to build a $t$-spanner more efficiently is introduced in [@FarshiG05; @DBLP07]. This algorithm uses a matrix to store the length of the shortest path between every two points. For each pair of points, it first checks the matrix to see if there is a $t$-spanning path between these points. In case the entry in the matrix for this pair indicates that there is no $t$-spanning path, it performs a shortest path query and updates the matrix. The authors in [@DBLP07] have conjectured that the number of performed shortest path queries is linear. This has been shown to be wrong in [@BCFMS08], as the number of shortest path queries may be quadratic. In addition, Bose et al. [@BCFMS08] have shown how to compute the Path-Greedy spanner in $O(n^2\log n)$ time. The main idea of their algorithm is to compute a partial shortest path and then extend it when needed. However, the drawback of this algorithm is that it is complex and difficult to implement. In [@AlewijnseBBB15], Alewijnse et al. compute the Path-Greedy spanner using linear space in $O(n^2\log^2n)$ time by utilizing the Path-Greedy properties with respect to the Well Separated Pair Decomposition (WSPD). In [@Alewijnse2016], Alewijnse et al. compute a $t$-spanner in $O(n \log^2 n\log^2\log n)$ expected time by using bucketing for short edges and by using WSPD for long edges. Their algorithm is based on the assumption that the Path-Greedy spanner consists of mostly short edges.
A set $P$ of points in the plane and a constant $t > 1$ A $t$-spanner $G(V,E)$ for $P$ sort the $n \choose 2$ pairs of distinct points in non-decreasing order of their distances and store them in list $L$ $E \longleftarrow \emptyset$ $ \pi \longleftarrow$ length of the shortest path in $G$ between $p$ and $q$ $E:=E\cup|pq|$ $G=(P,E)$
Additional effort has been put in developing algorithms for computing $t$-spanner graphs, such as $\theta$-Graph algorithm [@Clarkson87; @Kei88], Sink spanner, Skip-List spanner [@AryaMS94], and WSPD-based spanners [@Callahan93; @CallahanK92]. However, none of these algorithms produces a $t$-spanner as good as the Path-Greedy spanner in all aspects: size, weight and maximum degree, see [@DBLP07; @FarshiG09].
Therefore, our goal is to develop a simple and efficient algorithm that achieves both the theoretical and practical properties of the Path-Greedy spanner. In this paper we introduce the $\delta$-Greedy algorithm that constructs such a spanner for a set of $n$ points in the plane in $O(n^2 \log n)$ time. Moreover, we show that for a set of $n$ points placed independently at random in a unit square the expected running time of the $\delta$-Greedy algorithm is $O(n \log n)$.
$\delta$-Greedy {#sec:delta-Greedy}
================
In this section we describe the $\delta$-Greedy algorithm (Section \[sec:algDes\]) for a given set $P$ of points in the plane, and two real numbers $t$ and $\delta$, such that $1 < \delta \leq t$. Then, in Section \[subSec:SR\] we prove that the resulting graph is indeed a $t$-spanner with bounded degree. Throughout this section we assume that $\delta < t$ (for example, $\delta = t^{\frac{4}{5}}$ or $\delta = \frac{1 + 4t}{5}$), except in Lemma \[lemma:equal\], where we consider the case that $\delta=t$.
Algorithm description {#sec:algDes}
---------------------
For each point $p \in P$ we maintain a collection of cones $C_p$ with the property that for each point $q \in P$ that lies in $C_p$ there is a $t$-spanning path between $p$ and $q$ in the current graph. The main idea of the $\delta$-Greedy algorithm is to ensure that two cones of a constant angle with apexes at $p$ and $q$ are added to $C_p$ and to $C_q$, respectively, each time the algorithm runs a shortest path query between points $p$ and $q$. The algorithm starts with a graph $G$ having a vertex set $P$, an empty edge set, and an initially empty collection of cones $C_p$ for each point $p \in P$. The algorithm considers all pairs of distinct points of $P$ in a non-decreasing order of their distances. If $p \in C_q$ or $q \in C_p$, then there is already a $t$-spanning path that connects $p$ and $q$ in $G$, and there is no need to check this pair. Otherwise, let $d$ be the length of the shortest path that connects $p$ and $q$ in $G$ divided by $|pq|$. Let $c_p(\theta,q)$ denote the cone with apex at $p$ of angle $\theta$, such that the ray $\stackrel{\rightarrow}{pq}$ is its bisector. The decision whether to add the edge $(p, q)$ to the edge set of $G$ is made according to the value of $d$. If $d > \delta$, then we add the edge $(p,q)$ to $G$, a cone $c_p (2 \theta,q)$ to $C_p$, and a cone $c_q (2 \theta,p)$ to $C_q$, where $\theta = \frac{\Pi}{4} - \arcsin(\frac{1}{\sqrt 2 \cdot t})$. If $d \leq \delta$, then we do not add this edge to $G$, however, we add a cone $c_p (2 \theta,q)$ to $C_p$ and a cone $c_q (2 \theta,p)$ to $C_q$, where $\theta = \frac{\Pi}{4} - \arcsin(\frac{d}{\sqrt 2 \cdot t})$.
![The three scenarios of the $\delta$-Greedy algorithm. (a) $v \in C_p$; (b) $u \notin C_p$ and $d \leq \delta$; (c) $w \notin C_p$ and $d > \delta$. []{data-label="fig:C_p"}](C_p.pdf){width="80.00000%"}
In Algorithm \[alg:deltaGreedy\], we give the pseudo-code description of the $\delta$-Greedy algorithm. In Figure \[fig:C\_p\], we illustrate a cone collection $C_p$ of a point $p$ and how it is modified during the three scenarios of the algorithm. The figure contains the point $p$, its collection $C_p$ colored in gray, and three points $v$, $u$, and $w$, such that $|pv| < |pu| < |pw|$. Point $v$ lies in $C_p$ representing the first case, where the algorithm does not change the spanner and proceeds to the next pair without performing a shortest path query. The algorithm runs a shortest path query between $p$ and $u$, since $u \notin C_p$ (for the purpose of illustration assume $p \notin C_u$). Figure \[fig:C\_p\](b) describes the second case of the algorithm, where the length of the shortest path between $p$ and $u$ is at most $\delta|pu|$. In this case the algorithm adds a cone to $C_p$ without updating the spanner. Figure \[fig:C\_p\](c) describes the third case of the algorithm, where the length of the shortest path between $p$ and $w$ is more than $\delta|pw|$. In this case the algorithm adds a cone to $C_p$ and the edge $(p,w)$ to the spanner.
A set $P$ of points in the plane and two real numbers $t$ and $\delta$ s.t. $1 < \delta \leq t$ A $t$-spanner for $P$ sort the $n \choose 2$ pairs of distinct points in non-decreasing order of their distances (breaking ties arbitrarily) and store them in list $L$ $E \longleftarrow \emptyset$ /\* E is the edge set \*/ $C_p \longleftarrow \emptyset \ \ \forall p \in P$ /\* $C_p$ is set of cones with apex at $p$ \*/ $G \longleftarrow (P,E)$ /\* G is the resulting $t$-spanner \*/ \[alg:edgeIteration\] \[Alg:shortestPath\] $ d \longleftarrow $ length of the shortest path in $G$ between $p$ and $q$ divided $|pq|$ \[StepAddingEdges\] $E \longleftarrow E \cup \{ (p,q) \}$ $ d \longleftarrow 1 $ $\theta \longleftarrow \frac{\Pi}{4} - \arcsin(\frac{d}{\sqrt 2 \cdot t})$ /\* $\frac{1}{\cos \theta - \sin \theta} = \frac{t}{d}$ \*/ \[alg:addConesP\] $c_p (2 \theta,q) \longleftarrow$ cone of angle $2 \theta$ with apex at $p$ and bisector $ \stackrel{\rightarrow}{pq}$ $c_q (2\theta ,p) \longleftarrow$ cone of angle $2 \theta$ with apex at $q$ and bisector $ \stackrel{\rightarrow}{qp}$
$C_p \longleftarrow C_p \cup c_p (2 \theta,q) $ $C_q \longleftarrow C_q \cup c_q (2 \theta,p)$ $G=(P,E)$
Algorithm analysis {#subSec:SR}
------------------
In this section we analyze several properties of the $\delta$-Greedy algorithm, including the spanning ratio and the degree of the resulting graph.
The following lemma is a generalization of Lemma 6.4.1. in [@GiriSmid07].
\[lemma:theta\] Let $t$ and $\delta$ be real numbers, such that $1 \leq \delta \leq t$. Let $p$, $q$, and $r$ be points in the plane, such that
1. $p \neq r$,
2. $ |pr| \leq |pq|$,
3. $\frac {1} {\cos \theta - \sin \theta} \leq \frac{t}{\delta}$, where $\theta$ is the angle $\angle rpq$ $($i.e., $\angle rpq = \theta \leq \frac{\Pi}{4} - \arcsin(\frac{\delta}{\sqrt 2 \cdot t}) )$.
Then $\delta|pr|+ t|rq| \leq t|pq|$.
Let $r'$ be the orthogonal projection of $r$ onto segment $\overline{pq}$. Then, $|rr'| = |pr| \sin \theta$, $|pr'| = |pr| \cos \theta$, and $|r'q| = |pq| - |pr'|$. Thus, $|r'q| = |pq| - |pr| \cos \theta$. By triangle inequality $$\begin{aligned}
|rq| & \leq |rr'| + |r'q| \\
& \leq |pr| \sin \theta + |pq| - |pr| \cos \theta \\
& = |pq| - |pr|( \cos \theta - \sin \theta).\end{aligned}$$ $$\begin{aligned}
\text{We have, \ } \delta|pr|+ t|rq| &\leq& \delta |pr| + t (|pq| - |pr|( \cos \theta - \sin \theta) ) \\
&=& t|pq| - t|pr| ( \cos \theta - \sin \theta) + \delta |pr| \\
&\leq& t|pq| - t|pr| ( \cos \theta - \sin \theta) + t ( \cos \theta - \sin \theta) |pr| \\
&\leq& t|pq|.\end{aligned}$$
\[lemma:shortest-path\] The number of shortest path queries performed by $\delta$-Greedy algorithm for each point is $O(\frac{1}{t/\delta -1})$.
Clearly, the number of shortest path queries performed for each point is at most $n-1$. Thus, we may assume that $t/\delta > 1 + 1/n$. Consider a point $p\in P$ and let $(p,q)$ and $(p,r)$ be two pairs of points that $\delta$-Greedy algorithm has run shortest path queries for. Assume w.l.o.g. that the pair $(p,r)$ has been considered before the pair $(p,q)$, i.e., $|rp| \leq |pq|$. Let $d$ be the length of the path computed by the shortest path query for $(p,r)$ divide by $|pr|$. If $d \leq \delta$, then the cone added to the collection $C_p$ has an angle of at least $\frac{\Pi}{4} - \arcsin(\frac{\delta}{\sqrt 2 \cdot t})$. Otherwise, the algorithm adds the edge $(p,r)$ to $G$ and a new cone to the collection of cones $C_p$, where the angle of this cone is $\frac{\Pi}{4} - \arcsin(\frac{1}{\sqrt 2 \cdot t})$. Thus, after the shortest path query performed for the pair $(p,r)$, the collection $C_p$ contains a cone $c_p (\theta, r)$, where $\theta$ is at least $\frac{\Pi}{2} - 2\arcsin(\frac{\delta}{\sqrt 2 \cdot t})$. The $\delta$-Greedy algorithm performs a shortest path query for $(p,q)$ only if $p \notin C_q$ and $q \notin C_p$. Thus, the angle $\angle rpq$ is at least $\frac{\Pi}{4} - \arcsin(\frac{\delta}{\sqrt 2 \cdot t})$, and we have at most $k= \frac{2 \pi}{\theta}$ shortest path queries for a point.
Let us consider the case where $t>1$ and $\frac{t}{\delta} \rightarrow 1$. The equation $\theta = \frac{\Pi}{4} - \arcsin(\frac{\delta}{\sqrt 2 \cdot t})$ implies that $\frac {1} {\cos \theta - \sin \theta} = \frac{t}{\delta}$. Then, we have $$\theta \rightarrow 0 , \ \frac{t}{\delta} \sim 1 + \theta, \ \text{and} \ \theta \sim \frac{t}{\delta} -1.$$ Thus, we have $k \sim \frac{2\pi}{ \frac{t}{\delta} -1} = O(\frac{1}{t / \delta -1})$.
For $\delta = t^{\frac{x-1}{x}} $, where $x>1$ is a fixed integer, the number of shortest path queries performed by $\delta$-Greedy algorithm for each point is $O(\frac{x}{t -1})$.
As in Lemma \[lemma:shortest-path\], let us consider the case where $t>1$ and $\frac{t}{\delta} \rightarrow 1$. Then, we have $$\theta \rightarrow 0, \ \ \frac{t}{\delta} \sim 1 + \theta, \ \
\frac{t}{t^{(\frac{x-1}{x})}} \sim 1 + \theta, \ \ t^{( \frac{1}{x})} \sim 1 + \theta,$$ $$t \sim (1+\theta)^x, \ \ t \sim 1 + x \cdot \theta, \ \text{and} \ \theta \sim \frac{t- 1}{x}.$$ Thus, we have $k \sim \frac{2\pi x}{ t -1} = O(\frac{x}{t -1})$.
The running time of $\delta$-Greedy algorithm is $O(\frac{n^2 \log n}{(t/\delta -1)^2})$.
First, the algorithm sorts the $n \choose 2$ pairs of distinct points in non-decreasing order of their distances, this takes $O(n^2 \log n)$ time. A shortest path query is done by Dijkstra’s shortest path algorithm on a graph with $O(\frac{n}{t/\delta -1})$ edges and takes $O(\frac{n}{t/\delta -1} + n \log n)$ time. By Lemma \[lemma:shortest-path\] each point performs $O(\frac{1}{t/\delta -1})$ shortest path queries. Therefore, we have that the running time of $\delta$-Greedy algorithm is $O( (\frac{n}{t/\delta -1})^2 \log n )$.
\[lemma:cone\] The number of cones that each point has in its collection along the algorithm is constant depending on $t$ and $\delta$ ($O(\frac{1}{t/\delta -1})$).
As shown in Lemma \[lemma:shortest-path\], the number of shortest path queries for each point is $O(\frac{1}{t/\delta -1})$. The subsequent step of a shortest path query is the addition of two cones, meaning that for each point $p$ the number of cones in the collection of cones $C_p$ is $O(\frac{1}{t/\delta -1})$.
The additional space for each point $p$ for the collection $C_p$ is constant.
\[lemma:Spanner\] The output graph $G=(P,E)$ of $\delta$-Greedy algorithm (Algorithm \[alg:deltaGreedy\]) is a $t$-spanner for $P$ (for $1< \delta < t$).
Let $G=(P,E)$ be the output graph of the $\delta$-Greedy algorithm. To prove that $G$ is a $t$-spanner for $P$ we show that for every pair $(p,q) \in P$, there exists a $t$-spanning path between them in $G$. We prove the above statement by induction on the rank of the distance $|pq|$, i.e., the place of $(p,q)$ in a non-decreasing distances order of all pairs of points in $P$.
**Base case:** Let $(p, q) $ be the first pair in the ordered list (i.e., the closest pair). The edge $(p,q)$ is added to $E$ during the first iteration of the loop in step \[StepAddingEdges\] of Algorithm \[alg:deltaGreedy\], and thus there is a $t$-spanning path between $p$ and $q$ in $G$.
**Induction hypothesis:** For every pair $(r,s) $ that appears before the pair $(p,q)$ in the ordered list, there is a $t$-spanning path between $r$ and $s$ in $G$.
**The inductive step:** Consider the pair $(p,q)$. We prove that there is a $t$-spanning path between $p$ and $q$ in $G$. If $p \notin C_q$ and $q \notin C_p$, we check whether there is a $\delta$-spanning path in $G$ between $p$ and $q$. If there is a path which length is at most $\delta |pq|$, then $ \delta|pq| \leq t|pq|$, meaning there is a $t$-spanning path between $p$ and $q$ in $G$. If there is no path of length of at most $\delta|pq|$, we add the edge $( p,q)$ to $G$, which forms a $t$-spanning path.
Consider that $p \in C_q$ or $q \in C_p$, and assume w.l.o.g. that $q \in C_p$. Let $(p,r)$ be the edge handled in Step \[alg:edgeIteration\] in Algorithm \[alg:deltaGreedy\] when the cone containing $q$ has been added to $C_p$ (Step \[alg:addConesP\] in Algorithm \[alg:deltaGreedy\]). Notice that $|pr| \leq |pq|$. Step \[Alg:shortestPath\] of Algorithm \[alg:deltaGreedy\] has computed the value $d$ for the pair $(p,r)$. In the algorithm there are two scenarios depending on the value of $d$.
The first scenario is when $d > \delta$, then the algorithm has added the edge $(p,r)$ to $G$ and a cone $c_p (\theta,r)$ to $C_p$, where $\theta = 2(\frac{\Pi}{4} - \arcsin(\frac{1}{\sqrt 2 \cdot t}))$. Thus, the angle between $(p,q)$ and $(p, r)$ is less than $\theta /2$. Hence, $|rq| < |pq|$ and by the induction hypothesis there is a $t$-spanning path between $r$ and $q$. Consider the shortest path between $p$ and $q$ that goes through the edge $(p,r)$. The length of this path is at most $|pr| + t|rq|$. By Lemma \[lemma:theta\], we have $|pr|+ t|rq| \leq \delta |pr|+ t|rq| \leq t|pq|$ for $\delta = 1$. Therefore, we have a $t$-spanning path between $p$ and $q$. The second scenario is when $d \leq \delta$, then the algorithm has added a cone $c_p (\theta,r)$ to $C_p$, where $\theta = 2(\frac{\Pi}{4} - \arcsin(\frac{d}{\sqrt 2 \cdot t}))$. Thus, the angle between $(p,q)$ and $(p, r)$ is less than $\theta / 2$. Hence, $|rq| < |pq|$ and by the induction hypothesis there is a $t$-spanning path between $r$ and $q$. Consider the shortest path between $p$ and $q$ that goes through $r$. The length of this path is at most $d|pr| + t|rq|$. By Lemma \[lemma:theta\], we have $d|pr|+ t|rq| \leq t|pq|$. Therefore, we have a t-spanning path between $p$ and $q$.
The $\delta$-Greedy algorithm computes a $t$-spanner for a set of points $P$ with the same properties as the Path-Greedy $t$-spanner, such as degree and weight, in $O( (\frac{n}{t/\delta -1})^2 \log n )$ time.
Clearly, the degree of the $\delta$-Greedy is at most the degree of the Path-Greedy $\delta$-spanner. The edges of the $\delta$-Greedy spanner satisfy the $\delta$-leap frog property, thus, the weight of the $\delta$-Greedy is as Path-Greedy $t$-spanner. Hence, we can pick $\delta$ close to $t$, such that we will have the required bounds.
\[lemma:equal\] If $t=\delta$, the result of the $\delta$-Greedy algorithm is identical to the result of the Path-Greedy algorithm.
Assume towards contradiction that for $t=\delta$ the resulting graph of the $\delta$-Greedy algorithm, denoted as $G=(P,E)$, differs from the result of the Path-Greedy algorithm, denoted as $G'=(P,E')$. Assuming the same order of the sorted edges, let $(p,q)$ be the first edge that is different in $G$ and $G'$. Notice that $\delta$-Greedy algorithm decides to add the edge $(p,q)$ to $G$ when there is no $t$-spanning path between $p$ and $q$ in $G$. Since until handling the edge $(p,q)$ the graphs $G$ and $G'$ are identical, the Path-Greedy algorithm also decides to add the edge $(p,q)$ to $G'$. Therefore, the only case we need to consider is $(p,q) \in E'$ and $(p,q) \notin E$. The $\delta$-Greedy algorithm does not add an edge $(p,q)$ to $G$ in two scenarios:
- there is a $t$-spanning path between $p$ and $q$ in the current graph $G$ – which contradicts that the Path-Greedy algorithm adds the edge $(p,q)$ to $G'$;
- $p \in C_q$ or $q \in C_p$ – the $\delta$-Greedy algorithm does not perform a shortest path query between $p$ and $q$. Assume w.l.o.g., $q \in C_p$, and let $(p,r)$ be the edge considered in Step \[alg:edgeIteration\] in Algorithm \[alg:deltaGreedy\] when the cone containing $q$ has been added to $C_p$. The angle of the added cone is $\theta = \frac{\Pi}{2} - 2\arcsin(\frac{d}{\sqrt 2 \cdot t}) $, where $d$ is the length of the shortest path between $p$ and $r$ divided $|pr|$. Thus, we have $ |pr| \leq |pq|$ and $\frac {1} {\cos \alpha - \sin \alpha} \leq \frac{t}{d}$, where $\alpha \leq \theta$ is the angle $\angle rpq $. Then, by Lemma \[lemma:theta\], $\delta|pr|+ t|rq| \leq t|pq|$, and since there is a path from $p$ to $r$ of length at most $\delta |pr|$, we have that there is $t$-spanning path between $p$ and $q$ in the current graph. This is in contradiction to the assumption that the Path-Greedy algorithm adds the edge $(p,q)$ to $E'$.
$\delta$-Greedy in Expected $O(n \log n)$ Time for Random Set {#subSec:calc-nlogn}
=============================================================
In this section we show how a small modification in the implementation improves the running time of the $\delta$-Greedy algorithm. This improvement yields an expected $O(n \log n)$ time for random point sets. The first modification is to run the shortest path query between points $p$ to $q$ up to $\delta |pq|$. That is, running Dijkstra’s shortest path algorithm with source $p$ and terminating as soon as the minimum key in the priority queue is larger than $\delta |pq|$.
Let $P$ be a set of $n$ points in the plane uniformly distributed in a unit square. To prove that $\delta$-Greedy algorithm computes a spanner for $P$ in expected $O(n \log n)$ time, we need to show that:
- each point runs a constant number of shortest path queries – follows from Lemma \[lemma:shortest-path\];
- the expected number of points visited in each query is constant – The fact that the points are randomly chosen uniformly in the unit square implies that the expected number of points at distance of at most $r$ from point $p$ is $\Theta(r^2 \cdot n)$. A shortest path query from a point $p$ to a point $q$ terminates as soon as the minimum key in the priority queue exceeds $\delta |pq|$, thus, it is expected to visit $O(n \cdot (\delta|pq|)^2)$ points.
By Lemma \[lemma:shortest-path\] the number of shortest path queries performed by the algorithm for a point $p$ is $O(\frac{1}{t/\delta -1})$. Each such query defines a cone with apex at $p$ of angle $\Omega(t/\delta -1)$, such that no other shortest path query from $p$ will be performed to a point in this cone. By picking $k=\frac{1}{t/\delta -1}$ and $r= \frac{k}{\sqrt n}$, we have that the expected number of points around each point in a distance of $r$ is $\Theta (k^2) = \Theta ( \frac{1}{(t/\delta -1)^2} )$.
Assume we partition the plane into $k$ equal angle cones with apex at point $p$. The probability that there exists a cone that does not contain a point from the set of points of distance $\frac{k}{\sqrt n}$ is at most $k \cdot (1- \frac{1}{k})^{k^2}$. Let $Q$ be the set of points that $p$ computed a shortest path query to, and let $q \in Q$ be the farthest point in $Q$ from $p$. Then, the expected Euclidean distance between $p$ and $q$ is less than $\frac{k}{\sqrt n}$. Thus, the expected number of points visited by the entire set of shortest path queries from a point is $O(\frac{\delta^2 k^2}{t/\delta -1}) = O(\frac{\delta^2}{(t - \delta)^3})$;
- the next pair to be processed can be obtained in expected $O(\log n)$ time without sorting all pairs of distinct points – Even-though this is quite straight forward, for completeness we give a short description how this can be done. Divide the unit square to $n \times n$ grid cells of side length $1/n$. A hash table of size $3n$ is initialized, and for each non-empty grid cell (at most $n$ such cells) we map the points in it to the hash table. In addition, we maintain a minimum heap $H_p$ for each point $p \in P$ (initially empty), and one main minimum heap $H$ that contains the top element of each $H_p$. Each heap $H_p$ contains a subset of the pairs that include $p$. For each point $p \in P$, all the cells of distance at most $\frac{k}{\sqrt n}$ from $p$ are scanned (using the hash table) to find all the points in these cells, where $k$ is a parameter that we fix later. All the points found in these cells are added to $H_p$ according to their Euclidean distance from $p$.
The heap $H$ holds the relevant pairs in an increasing order, therefore the pairs are extracted from the main heap $H$. After extracting the minimum pair in $H$ that belongs to a point $p$, we add to $H$ the next minimum in $H_p$. To insure the correctness of the heaps, when needed we increase the distance to the scanned cells. Observe that there may be a pair $(p,q)$ such that $|pq| < |rw|$, where the pair $(r,w)$ is the top pair in $H$. This can occur only when the pair $(p,q)$ has not been added to $H_p$ nor $H_q$, and this happens when $p \in C_q$ or $q \in C_p$. However, in this case we do not need to consider the pair $(p,q)$.
Notice that the only cells that are not contained in $C_p$ are scanned to add more pairs to $H_p$. Thus, points that are in $C_p$ are ignored.
Therefore, the total expected running time of the algorithm is $O( \frac{\delta^2}{(t - \delta)^3} n \log n )$. Since both $t$ and $t/\delta $ are constants bigger than one, the expected running time of the $\delta$-Greedy algorithm is $O( n \log n )$.
A very nice outcome of $\delta$-Greedy algorithm and its analysis can be seen when $\delta$ is equal to $t$. Assume that $\delta$-Greedy algorithm (for $\delta = t$) has computed a shortest path query for two points $p$ and $q$ and the length of the received path is $d|pq|$. If the probability that $ t/d > 1 +\varepsilon $ is low (e.g, less than 1/2), for some constant $\varepsilon >0$, then $\delta$-Greedy algorithm computes the Path-Greedy spanner with linear number of shortest path queries. Thus $\delta$-Greedy algorithm computes the Path-Greedy spanner for a point set uniformly distributed in a square in expected $O(n \log n)$ time.
Not surprisingly our experiments have shown that this probability is indeed low (less than 1/100), since most of the shortest path queries are performed on pairs of points placed close to each other (with respect to Euclidean distance), and thus with a high probability their shortest path contains a constant number of points. Moreover, it seems that for a “real-life" input this probably is low. Thus, there is a very simple algorithm to compute the Path-Greedy spanner in expected $O(n^2 \log n)$ time for real-life inputs, based on the $\delta$-Greedy algorithm
For real-life input we mean that our analysis suggests that in the current computers precision (Memory) one cannot create an instance of points set with more than 1000 points, where the Path-Greedy spanner based on the $\delta$-Greedy algorithm has more than $O(n^2 \log n)$ constructing time.
Experimental Results {#sec:res}
====================
In this section we discuss the experimental results by considering the properties of the graphs generated by different algorithms and the number of shortest path queries performed during these algorithms. We have implemented the Path-Greedy, $\delta$-Greedy, Gap-Greedy, $\theta$-Graph, Path-Greedy on $\theta$-Graph algorithms. The Path-Greedy on $\theta$-graph $t$-spanner, first computes a $\theta$-graph $t'$-spanner, where $t'< t$, and then runs the Path-Greedy $t/t'$-spanner on this $t'$-spanner. The shortest path queries criteria is used for an absolute running time comparison that is independent of the actual implementation. The known theoretical bounds for the algorithms can be found in Table \[table:bounds\].
**Algorithm** **Edges** $\frac{\textbf{Weight}}{wt(MST)}$ **Degree** **Time**
----------------- ---------------------------- ----------------------------------- ---------------------------- ------------------------------------------- -- --
Path-Greedy $O(\frac{n}{t-1})$ $O(1)$ $O(\frac{1}{t-1})$ $O(n^3 \log n)$
Gap-Greedy $O(\frac{n}{t-1})$ $O(\log n)$ $O(\frac{1}{t-1})$ $O(n \log^2n)$
$\theta$-Graph $O(\frac{n}{\theta})$ $O(n)$ $O(n)$ $O(\frac{n}{\theta}\log n)$
$\delta$-Greedy $O(\frac{n}{t/\delta -1})$ $O(1)$ $O(\frac{1}{t/\delta -1})$ $O(\frac{1}{t/\delta -1}\cdot n^2\log n)$
: Theoretical bounds of different $t$-spanner algorithms[]{data-label="table:bounds"}
The experiments were performed on a set of $8000$ points, with different values of the parameter $\delta$ (between 1 and $t$). We have chosen to present the parameter $\delta$ for the values $t, t^{0.9} $ and $\sqrt t $. This values do not have special properties, they where chosen arbitrary to present the behavior of the spanner.
To avoid the effect of specific instances, we have run the algorithms several times and taken the average of the results. However, in all the cases the difference between the values is negligible. Table \[table:res-1.1\]–\[table:res-2\] show the results of our experiments for different values of $t$ and $\delta$. The columns of the weight (divided by $wt(MST)$) and the degree are rounded to integers, and the columns of the edges are rounded to one digit after the decimal point (in $k$).
-------------------------- -------------- ------------------ ---------------------- ------------ --------------------
**Algorithm** **$\delta$** **Edges** (in K) **Weight** **Degree** **Shortest path**
$\overline{wt(MST)}$ **queries** (in K)
Path-Greedy - 35.6 10 17 $31996$
$\delta$-Greedy 1.1 35.6 10 17 254
$\delta$-Greedy $1.0896$ 37.8 12 18 242
$\delta$-Greedy $1.048$ 51.6 19 23 204
$\theta$-Graph - 376.6 454 149 -
Greedy on $\theta$-Graph $1.0896$ 37.8 12 18 3005
Greedy on $\theta$-Graph $1.048$ 52 19 23 693
Gap-Greedy - 51.6 19 23 326
-------------------------- -------------- ------------------ ---------------------- ------------ --------------------
: Comparison between several $t$-spanner algorithms for $t=1.1$[]{data-label="table:res-1.1"}
\[table:res-1.5\]
-------------------------- -------------- ------------------ ---------------------- ------------ --------------------
**Algorithm** **$\delta$** **Edges** (in K) **Weight** **Degree** **Shortest path**
$\overline{wt(MST)}$ **queries** (in K)
Path-Greedy - $15.1$ $3$ 7 $31996$
$\delta$-Greedy $1.5$ $15.1$ $3$ 7 $82$
$\delta$-Greedy $1.44$ $16$ $3$ 8 $77$
$\delta$-Greedy $1.224$ $22.5$ $5$ 11 $63$
$\theta$-Graph - $118.6$ $76$ 53 -
Greedy on $\theta$-Graph $1.44$ $16$ $3$ 8 $817$
Greedy on $\theta$-Graph $1.224$ $22.5$ $6$ 11 $198$
Gap-Greedy - $22.6$ $5$ 11 $95$
-------------------------- -------------- ------------------ ---------------------- ------------ --------------------
: Comparison between several $t$-spanner algorithms for $t=1.5$
-------------------------- -------------- ------------------ ---------------------- ------------ --------------------
**Algorithm** **$\delta$** **Edges** (in K) **Weight** **Degree** **Shortest path**
$\overline{wt(MST)}$ **queries** (in K)
Path-Greedy - 11.4 2 5 $31996$
$\delta$-Greedy $2$ 11.4 2 5 55
$\delta$-Greedy $1.866$ 11.9 2 5 52
$\delta$-Greedy $1.414$ 16.3 3 8 44
$\theta$-Graph - 85.3 48 42 -
Greedy on $\theta$-Graph $1.866$ 11.9 3 6 493
Greedy on $\theta$-Graph $1.414$ 16.5 3 8 129
Gap-Greedy - 16 3 8 63
-------------------------- -------------- ------------------ ---------------------- ------------ --------------------
: Comparison between several $t$-spanner algorithms for $t=2$[]{data-label="table:res-2"}
Implementation details {#subSec:algo-details}
----------------------
All the algorithms mentioned above were implemented in Java using JGraphT and JGraph libraries. The experiments were performed on an Intel ® Xeon® CPU E5-2680 v2 $@$ 2.80 GHz (2 processors) and 128 GB RAM on Windows Server 2012 Standard OS using ECJ for compilation. The sample point sets were generated by Java.util.Random pseudo random number generator.
Results analysis {#subSec:res-analysis}
----------------
The experiments indicate that the $\delta$-Greedy algorithm achieves good results in practice as expected. The outcome of the $\delta$-Greedy algorithm for all values of $\delta$, that have been checked, is roughly the same as the results of the Path-Greedy algorithm for all parameters. Compared to other algorithms, the $\delta$-Greedy graphs are superior to the graphs produced by the $n^2$-Gap algorithm, and are as good as Path-Greedy on $\theta$-Graph, with significantly a lower number of shortest path queries. The theoretical complexity of the Path-Greedy on $\theta$-Graph is $O(n^2 \log n)$, same as the $\delta$-Greedy algorithm. However in practice the $\delta$-Greedy algorithm computes considerably less shortest path queries. Hence, the $\delta$-Greedy algorithm has the same results in weight, size and degree as the Path-Greedy on $\theta$-Graph algorithm with better running time.
In addition, Farshi and Gudmundsson in [@FarshiG09] have implemented various spanner algorithms and shown that the Path-Greedy algorithm for $t=1.1$ and for $t=2$ on random graphs are almost identical to ours experimental results in weight, size and degree. Moreover, they have shown that Path-Greedy spanner is the highest quality geometric spanner in terms of edge count, degree and weight. They have presented the results for $t=1.1$ and for $t=2$ on random point set with 8000 points. Moreover, they have shown that the $\theta$-Graph spanner achieves in practice the best results after the Path-Greedy spanner for all parameters that have been tested (size, weight and degree) comparing to other spanners that they have implemented (such as the Approximate-Greedy, the WSPD-spanner, Skip-list and Sink-Spanner). Our experiments show that the $\delta$-Greedy spanner achieves better results than the $\theta$-Graph spanner. Thus, combining this with the results in [@FarshiG09], we conclude that the $\delta$-spanner achieves the highest quality geometric spanner with respect to $\theta$-Graph, Approximate-Greedy, the WSPD-spanner, Skip-list, Sink-Spanner, and Gap-Greedy spanners.
The experiments reinforce the analysis that picking $\delta$ very close to $t$ (for example $\delta= t^{0.9}$), the results are very close to the Path-Greedy spanner, and the number of the performed shortest paths queries is still small. Moreover, the experiments show that the number of shortest path queries is linear while selecting $\delta =t$ and obtaining the $\delta$-Greedy spanner identical to the Path-Greedy $t$-spanner.
The experiments presented in this paper were performed on set of points placed independently at random in a unit square. However, we conjecture that the $\delta$-Greedy algorithm computes a $t$-spanner in expected $O(n \log n) $ time for almost all realistic inputs, that is, the $\delta$-Greedy algorithm computes a $t$-spanner in expected $O(n \log n) $ time for point sets that are not deliberately hand-made to cause a higher number of shortest path queries.
Acknowledgments {#sec:ack}
===============
We would like to thank Rachel Saban for implementing the algorithms.
[10]{}
S. P. A. Alewijnse, Q. W. Bouts, A. P. ten Brink, and K. Buchin. Computing the greedy spanner in linear space. , 73(3):589–606, 2015.
S. P. A. Alewijnse, Q. W. Bouts, Alex P. ten Brink, and K. Buchin. Distribution-sensitive construction of the greedy spanner. , pages 1–23, 2016.
S. Arya, D. M. Mount, and M. H. M. Smid. Randomized and deterministic algorithms for geometric spanners of small diameter. In [*FOCS*]{}, pages 703–712, 1994.
Prosenjit Bose, Paz Carmi, Mohammad Farshi, Anil Maheshwari, and Michiel H. M. Smid. Computing the greedy spanner in near-quadratic time. In [*SWAT*]{}, pages 390–401, 2008.
P. B. Callahan and S. R. Kosaraju. A decomposition of multi-dimensional point-sets with applications to k-nearest-neighbors and n-body potential fields. In [*STOC*]{}, pages 546–556, 1992.
Paul B. Callahan. Optimal parallel all-nearest-neighbors using the well-separated pair decomposition. In [*FOCS*]{}, pages 332–340, 1993.
B. Chandra, G. Das, G. Narasimhan, and J. Soares. New sparseness results on graph spanners. , 5:125–144, 1995.
Barun Chandra. Constructing sparse spanners for most graphs in higher dimensions. , 51(6):289–294, 1994.
Kenneth L. Clarkson. Approximation algorithms for shortest path motion planning. In [*STOC*]{}, pages 56–65, 1987.
G. Das, P. J. Heffernan, and G. Narasimhan. Optimally sparse spanners in 3-dimensional [E]{}uclidean space. In [*SoCG*]{}, pages 53–62, 1993.
G. Das and G. Narasimhan. A fast algorithm for constructing sparse [E]{}uclidean spanners. , 7(4):297–315, 1997.
Gautam Das and Giri Narasimhan. A fast algorithm for constructing sparse [E]{}uclidean spanners. , 7(4):297–315, 1997.
M. Farshi and J. Gudmundsson. Experimental study of geometric t-spanners: [A]{} running time comparison. In [*WEA*]{}, pages 270–284, 2007.
Mohammad Farshi and Joachim Gudmundsson. Experimental study of geometric *t*-spanners. In [*ESA*]{}, pages 556–567, 2005.
Mohammad Farshi and Joachim Gudmundsson. Experimental study of geometric *t*-spanners. , 14, 2009.
Joachim Gudmundsson, Christos Levcopoulos, and Giri Narasimhan. Fast greedy algorithms for constructing sparse geometric spanners. , 31(5):1479–1500, 2002.
J. Mark Keil. Approximating the complete [E]{}uclidean graph. In [*SWAT*]{}, pages 208–213, 1988.
Christos Levcopoulos, Giri Narasimhan, and Michiel H. M. Smid. Improved algorithms for constructing fault-tolerant spanners. , 32(1):144–156, 2002.
Giri Narasimhan and Michiel Smid. . Cambridge University Press, New York, NY, USA, 2007.
Jos[é]{} Soares. Approximating [E]{}uclidean distances by small degree graphs. , 11(2):213–233, 1994.
|
---
abstract: 'We describe two-flavor QCD lattice data for the pressure at nonzero temperature and vanishing chemical potential within a quasiparticle model. Relying only on thermodynamic consistency, the model is extended to nonzero chemical potential. The results agree with lattice calculations in the region of small chemical potential.'
author:
- 'A. Peshier'
- 'B. Kämpfer'
- 'G. Soff'
title: From QCD lattice calculations to the equation of state of quark matter
---
Introduction
============
One of the fundamental issues which triggered, and has influenced since, heavy ion physics is the question of the phase structure and the thermodynamic properties of strongly interacting matter at energy densities above 1GeV/fm$^3$. Under such conditions, exceeding the energy density in nuclei but still far away from the asymptotic regime, the coupling strength $\alpha_s$ is large, which makes the theoretical description of the many-body problem challenging.
In the recent past the understanding of this field has become much more detailed. The phase diagram for QCD with $n_{\!f} = 2$ massless flavors, which is the case we will consider in the following, can be briefly described as follows (we refer to [@RajaW] for a recent review). At zero quark chemical potential, $\mu = 0$, the broken chiral symmetry of hadron matter is restored within the quark-gluon plasma, at a critical temperature $T_c \approx 170\,$MeV. It is thought that this second order transition persists also for nonzero $\mu$, thus defining a critical line, which changes to a first order transition line at the tricritical point. For small temperatures and $\mu {\,\raise-0.3ex\hbox{$\sim$}\kern-.7em\hbox{$^>$}\,}\mu_c$ one anticipates a color-superconducting phase of quark matter. The value of $\mu_c$ is expected to be 100...200MeV larger than the quark chemical potential $\mu_n = 307\,$MeV in nuclear matter. Quantitative results for large $\alpha_s$ can be obtained from first principles by lattice calculations which were, however, restricted to nonzero temperature and $\mu = 0$ until very recently. Therefore, the described picture for $\mu \not= 0$ is mainly based on general arguments combined with results from various models, including extrapolations of perturbative QCD.
As a phenomenological description of the thermodynamics of deconfined strongly interacting matter we proposed a quasiparticle model [@PKPS96; @PKS00]. Its parameters are fixed by the lattice data at $\mu = 0$. We then use the fact that within the model the thermodynamic potentials at zero chemical potential and $\mu \not= 0$ are related by thermodynamic consistency. In [@PKS00] we analyzed lattice data for $n_{\!f}=2$ flavors [@Karsc], and $n_{\!f}=4$ [@Engel], which were, however, still derogated by sizable lattice artifacts which have an effect on the absolute scaling of the data. We therefore introduced a constant effective number of degrees of freedom of the quasiparticles as an additional model parameter to obtain first qualitative estimates. Later we considered in [@PKS01] the lattice data [@Karsch00], where also the physical case of (2+1) flavors was simulated. As the absolute scaling of the lattice data enters as important information in particular near $T_c$, we pragmatically applied the continuum extrapolation of the data, which was proposed in [@Karsch00] for $T > 2\, T_c$, also for smaller temperatures. The results of this prescription can now be compared to new lattice data [@CPPACS]. Meanwhile, there are other lattice calculations which allow us to test directly the assumptions underlying the quasiparticle model as well as, for the first time, some of its predictions for nonzero chemical potential.
We will therefore consider here the presently available lattice data for $n_{\!f}=2$. Based on that, we will fit and discuss the quasiparticle parameters at $\mu=0$ in Section 3. In Section 4, we will briefly summarize how to extend the model to nonzero chemical potential, and compare our findings with the results [@AlltHKKLSS] from lattice simulations studying the region of small $\mu$. Section 5 concludes with the discussion of some physical implications.
Finite temperature lattice data
===============================
The simulations [@CPPACS] are performed on lattices with spatial extent $N_\sigma=16$ and temporal sizes $N_\tau = 4$ and $N_\tau =
6$, with an improved Wilson quark action and renormalized quark masses corresponding to fixed ratios $m_{ps}/m_v$ of the pseudoscalar to vector meson masses. We first consider the data for two light flavors, corresponding to $0.6 \le m_{ps}/m_v \le 0.75$. Although this is larger than the physical value, the results are almost insensitive to the ratio, which suggests that they are not too far from the chiral limit. As expected for the rather small lattice sizes, the results for $N_\tau=4$ and 6 differ. However, we observe that normalizing the pressure data by $p_0^{\rm cont}/p_0^{N_\tau}$, the ratio of the free limits in the continuum and on the lattice, improves considerably the consistency between the data sets. As a matter of fact, the normalized $N_\tau=4$ data are in agreement with the normalized $N_\tau=6$ data after rescaling by a constant of $1.14$. This simple scaling behavior for large coupling is rather remarkable. Based on this observation we suggest the continuum estimate for the pressure shown in Fig. \[fig:latt dat p\].
![Compilation of $n_{\!f}=2$ lattice data for the pressure in units of the free pressure $p_0$. Shown are the scaled (see text) data [@CPPACS] for light quarks corresponding to meson mass ratios of $0.65 \le m_{ps}/m_v \le 0.75$ (open circles: $N_\tau=4$; open squares: $N_\tau=6$), and the continuum estimate [@Karsch00] (grey band). The full line is the quasiparticle result. The full symbols represent the data [@CPPACS] for large quark masses, with $m_{ps}/m_v = 0.95$. For comparison, the hatched band shows the SU(3) lattice data (dotted line: [@Boyd]; dashed line: [@Okamo]) normalized to the corresponding free pressure. \[fig:latt dat p\]](comp_p.eps)
We assume here that the normalized $N_\tau=6$ data are already close to the continuum limit. This is supported by the fact that the thus interpreted data match the aforementioned continuum estimate from the staggered quark simulations [@Karsch00][^1]. Therefore, a consistent picture forms for the thermodynamics of QCD with $n_{\!f} = 2$ light flavors.
In Fig. \[fig:latt dat s\], the corresponding data for the entropy are shown. It is noted that since the slope of the continuum extrapolated pressure [@Karsch00] is slightly larger than that from the data [@CPPACS] (see Fig. \[fig:latt dat p\]), the upper part of the error band is already for $T \sim 3\, T_c$ very close to the free limit. This would be in contrast to the pure gauge case, where the uncertainty due to lattice artifacts has become small, so we will assume that the lower side of this estimate is more relevant.
![The lattice data for the entropy corresponding to the data for the pressure shown in Fig. \[fig:latt dat p\], and the quasiparticle fit. \[fig:latt dat s\]](comp_s.eps)
Quasiparticle model
===================
For completeness, we briefly recall here the main ideas of the quasiparticle model [@PKPS96; @PKS00] of the QCD plasma.
For weak coupling $g$, the thermodynamic behavior of the system is dominated by its excitations with momenta $\sim T$. While hard collective modes (the longitudinal plasmon and the quark hole excitation) are exponentially suppressed, the transverse gluons and the quark particle excitations propagate predominantly on simple mass shells, $\omega_i^2(k) \approx m_i^2+k^2$ [@leBel]. In the chiral limit the so-called asymptotic masses are given by $$\begin{aligned}
m_g^2
&=&
\frac16
\left[
\left( N_c+ \frac12\, n_{\!f} \right) T^2
+ \frac{N_c}{2\pi^2} \sum_q \mu_q^2
\right] g^2 \, ,
\nonumber \\
m_q^2
&=&
\frac{N_c^2-1}{8N_c}\, \left[ T^2+\frac{\mu_q^2}{\pi^2} \right] g^2 \, ,
\label{m2}\end{aligned}$$ where $\mu_q$ denotes the quark chemical potential, and $N_c=3$. Interpreting the relevant excitations as quasiparticles, the thermodynamic potential is p(T,) = \_i p\_i(T, \_i(); m\_i\^2) - B(m\_j\^2) , \[p\_eff\] where $p_i = \pm d_i\,T \int d^3k/(2\pi)^3\, \ln(1 \pm
\exp\{-(\omega_i-\mu_i)/T\})$ are the contributions of the gluons (with vanishing chemical potential) and the quarks (for the antiquarks, the chemical potential differs in the sign), and $d_g=2(N_c^2-1)$ and $d_q=2N_c$ count the degrees of freedom. As shown in [@GoreY], thermodynamic consistency requires the derivative, with respect to the $m_j^2$, of the right-hand side of Eq. (\[p\_eff\]) to vanish, i.e., the contribution $B$ is related to the $T$ and $\mu$ dependent masses by = . \[B’\] This implies that the entropy and the particle densities are simply given by the sum of the individual quasiparticle contributions, s\_i = . |\_[m\_i\^2]{} , n\_i = . |\_[m\_i\^2]{} , \[s,n\] while the energy density has the form $e = \sum_i e_i + B$.
Expanded in the coupling $g$, the above approach reproduces the leading-order perturbative results. The full expressions, however, represent a thermodynamically consistent resummation of terms of all orders in $g$. This suggests pondering the application of the model also in the strong coupling regime.[^2] Considering first the case $\mu=0$, it indeed turns out that the lattice data for the entropy shown in Fig. \[fig:latt dat s\] can be described by the model with the ansatz \_s(T,=0) = \[eq:alpha(T)\] for $g^2/(4\pi)$. This is the leading order perturbative result at a momentum scale determined by the temperature: $T_c/\lambda$ is related to the QCD scale $\Lambda$, while $T_s$ parameterizes the behavior in the infrared. For the parameters we obtain[^3] = 17.1 , T\_s = 0.89 T\_c . The resulting quasiparticle masses are large; near $T_c$ they reach several times the value of the temperature.[^4] The existence of such heavy excitations, which we have inferred from the thermodynamic bulk properties, has meanwhile been confirmed directly by lattice calculations of the propagators [@Petre]. Finally, since the derivative of the ‘bag’ function $B$ is related to the quasiparticle masses by Eq. (\[B’\]), the model is completely defined by fixing B\_0 = B(T\_c) = 1.1 T\_c\^4 , which enters the fit in Fig. \[fig:latt dat p\] as the third parameter.
Since all the information about the coupling is encoded in the parameters $T_s$ and $\lambda$, it is interesting to look at their flavor dependence. Comparing to the pure gauge plasma, it is recalled that in this case the pressure becomes very small close to the transition since there it has to match the pressure of the heavy glue balls in the confined phase. Similarly, the entropy is small at $T \sim T_c$, which implies a large coupling there. For $n_{\!f}=2$ the scaled entropy for $T \sim T_c$ is somewhat larger, thus close to the transition the coupling has to be smaller than for pure SU(3). However, for fixed parameters $\lambda$ and $T_s$, the coupling (\[eq:alpha(T)\]) would increase with increasing number of active flavors. Therefore, a difference of the parameters for $n_{\!f} = 2$ to those for the pure gauge plasma [@PKS00], \^[\_[SU(3)]{}]{} = 4.9 , T\_s\^[\_[SU(3)]{}]{} = 0.73 T\_c , is not unexpected. Interestingly, the parameter $T_s$ does not change by much compared to the case of $n_{\!f} = 2$.
Nonzero chemical potential
==========================
The quasiparticle model as applied in the previous section can be generalized to nonzero quark chemical potential $\mu_q = \mu$. The quasiparticle masses now depend also on $\mu$ – explicitly by the dimensionful coefficients of the coupling in Eq. (\[m2\]), and implicitly by the coupling itself. As shown in [@PKS00], Maxwell’s relation, $\partial s / \partial \mu =
\partial n / \partial T$, directly implies a partial differential equation for $\alpha_s(T,\mu)$. It is of first order and linear in the derivatives of the coupling (but nonlinear in $\alpha_s$), c\_T +c\_ = C , \[eq: flow eq\] where the coefficients $c_T$, $c_\mu$ and $C$ depend on $T$, $\mu$ and $\alpha_s$. It can easily be solved by reduction to a system of coupled ordinary differential equations, = c\_T , = c\_ , = C , which determines the so-called characteristic curves $T(s)$, $\mu(s)$, and the evolution of $\alpha_s$ along such a curve, given an initial value.
With regard to the underlying physics it is worth pointing out some properties of the flow equation (\[eq: flow eq\]). The coefficients are combinations of products of a derivative of the quasiparticle entropy or density with respect to the quasiparticle mass, and a derivative of the quasiparticle mass with respect to $T$, $\mu$ or $\alpha_s$. Writing down the explicit expressions, it is easy to see that the flow equation is elliptic. In particular, one finds c\_T(T, =0) = 0 , c\_(T=0, ) = 0 . The coefficient $c_\mu$, e.g., vanishes because not only the entropy goes to zero as $T \rightarrow 0$, but also its derivative with respect to the mass. Therefore, the characteristics are perpendicular to both the $T$ and the $\mu$ axes. This guarantees that specifying the coupling on some interval on the $T$ axis sets up a valid initial condition problem. From the temperature dependence of the effective coupling as obtained from the lattice data at $\mu =
0$, e.g. in the physically motivated parameterization (\[eq:alpha(T)\]), we can therefore determine numerically the coupling from Eq. (\[eq: flow eq\]), and hence the equation of state, in other parts of the $\mu\,T$ plane.
It is instructive to consider the asymptotic limit, $\alpha_s
\rightarrow 0$, of Eq. (\[eq: flow eq\]), where the coefficient $C$ vanishes. Then the coupling is constant along the characteristics, which become ellipses in the variables $T^2$ and $\mu^2$, leading to the mapping T ( )\^[1/4]{} . This holds approximately also for larger coupling, see Fig. \[fig: char\], so the lattice data at $\mu=0$ are mapped in elliptic strips into the $\mu\,T$ plane. On the other hand, an ansatz analog to Eq. (\[eq:alpha(T)\]) to parameterize $\alpha_s(T=0,\mu)$ is quantitatively less satisfactory than in the case $\mu=0$.
![Represented by the full lines are the characteristics of the flow equation (\[eq: flow eq\]). The characteristic through $T_c$ coincides for small $\mu$ with the critical line (dashed, with a hatched error band) obtained in the lattice calculation [@AlltHKKLSS]. In the region under the dash-dotted line the resulting quasiparticle pressure is negative – a transition to another phase has to happen somewhere outside. Therefore, the narrow grey region under the $p=0$ line, where the solution of the flow equation is not unique, is physically irrelevant. Indicated by the symbol (assuming, for the scaling, $T_c = 170\,$MeV) is the chemical potential $\mu_n$ in nuclear matter. \[fig: char\]](char.eps)
A closer look at the characteristics emanating from the interval $[T_c,1.06T_c]$ reveals that they intersect in a narrow half-crescent region, which indicates that there the solution of the flow equation is not unique. This, however, is only an ostensible ambiguity. It so happens that the extrapolation of the pressure becomes negative in a larger region, see Figs. \[fig: char\] and \[fig: p\_muT\]. This implies that a transition to another phase, at a certain positive pressure, happens already outside this region, so the encountered ambiguity of the flow equation is of no physical relevance.[^5]
At this point we emphasize again that this extrapolation of the quasiparticle model relies only on the requirement of thermodynamic consistency. Of course, it implicitly assumes also that the quasiparticle structure does not change, i.e., that deconfined quarks and gluons are the relevant degrees of freedom. For small enough $\mu$ and temperatures above (or near, as $\mu$ gets larger) $T_c$ this is a justified assumption. However, the quasiparticle structure will change in the hadronic phase, when both $T$ and $\mu$ are small, as well as for sufficiently cold and dense systems where the color-superconducting phase is expected. Although the present quasiparticle model cannot make any statements about these phases, it is interesting to observe that it ‘anticipates’ the existence of another phase only from the lattice input at $T>T_c$ and $\mu=0$. An interpretation of the apparent similarity of the line of vanishing pressure in Fig. \[fig: char\] with the expected transition line from the hadron to the superconducting quark matter phase, see Ref. [@RajaW], remains, of course, a speculation.
There is, however, a related question which we can address with the quasiparticle model without knowing details about the other phases, just based on the fact that for nonzero chemical potential the transition from the deconfined to the confined phase occurs at the critical line $T_c(\mu)$. The critical line is expected to be perpendicular to the $T$ axis, which has been confirmed in a recent lattice calculation [@AlltHKKLSS] where also its curvature at $\mu=0$ has been calculated[^6], $T_c\, d^2 T_c(\mu)/d\mu^2|_{\mu=0} \approx -0.14$. Within the quasiparticle model it is natural to relate, at least for small $\mu$, the critical line to the characteristic through $T_c(\mu=0)$, which, as shown above, is also perpendicular to the $T$ axis. For small $\mu$ where only the quadratic terms are relevant (practically even for $\mu$ as large as $2\,T_c$), we indeed find the $T_c$ characteristic in a striking agreement with the critical line from [@AlltHKKLSS], see Fig. \[fig: char\]. Another argument supporting the above interpretation of the $T_c$ characteristic comes from considering the case where the quark flavors have opposite chemical potentials, $\mu_u = -\mu_d = \tilde\mu$. With this isovector chemical potential the fermion determinant is positive definite, and standard Monte Carlo techniques can be applied to study this system on the lattice [@AlfKW]. The lattice result [@AlltHKKLSS] obtained for the curvature of the critical line in that case agrees with the value quoted above for the isoscalar potential $\mu$. Within the quasiparticle model, the equality of these two numbers is immediately evident.
In Ref. [@AlltHKKLSS] it was furthermore mentioned that the quadratic behavior, with the same curvature as at $\mu = 0$, of the critical line is not likely to extrapolate down to small transition temperatures since $T_c(\mu)$ would then vanish only at $\mu_c \sim
650\,$MeV. Phenomenologically, however, $\mu_c$ is expected to be not very much larger, say at most by 200MeV, than the quark chemical potential $\mu_n = 307\,$MeV in nuclear matter. In the quasiparticle model, from the chemical potential where the extrapolated pressure vanishes at $T=0$, we estimate $\mu_c \approx 3\,T_c \sim 500\,$MeV.
This value is in the expected ball park, which encourages us to consider the extrapolation of the model down to smaller temperatures. Although for $T \rightarrow 0$ quark matter will be in the superconducting phase, it is still possible to give an estimate of its equation of state in that region from the quasiparticle model. The quark pairing influences thermodynamic bulk properties at the order of $(\Delta\,\mu)^2$, with the gap energy $\Delta$ being at most 100MeV [@RajaW]. This has little effect on the energy density $e = \sum_i e_i + B$ as both the quasiparticle contributions and the function $B$ are parametrically of the order ${\cal
O}(\mu^4)$. For the pressure, on the other hand, the pairing effects become comparable to our expression $p = \sum_i p_i - B$ only when the latter becomes small. Since the pressure of the thermodynamically favored superconducting phase is larger than that of the plasma phase, the relation $e(p)$ as shown in Fig. \[fig: e(p)\] is therefore an upper estimate of the equation of state of cold quark matter.
![The estimate for the equation of state of quark matter at $T=0$. \[fig: e(p)\]](e_p.eps)
For $p \ge 5\, T_c^4$, we obtain $e(p) \approx 13\, T_c^4 + 3.2\, p$, where the slope is mainly determined by the fact that the pressure at $T=0$ essentially scales as $\mu^4$. For smaller pressure, the slope is only slightly larger, and the energy at $p=0$ is approximately[^7] $11\, T_c^4$. Assuming $T_c \approx 170\,$MeV, this translates into an energy density of 1GeV/fm$^3$. Bearing in mind that this is an upper estimate, and comparing to the bag model equation of state, $e^{\rm bag}(p) = 4\tilde B + 3p$, this result is still considerably larger than estimates with commonly assumed values of the bag constant $\tilde B$.
Coming back to the region of the phase space where the quasiparticle model is well grounded, we finally address the question of the behavior of the pressure and the energy density along the critical line near $\mu=0$. In the lattice simulations [@AlltHKKLSS] both quantities have been found to be constant within the numerical errors. This is compatible with our result for small $\mu$, p(T\_c(),)-p(T\_c(0),0) -0.02 \^2 T\_c\^2 . The corresponding change in the energy density is about three times larger. These results differ notably from the estimate from the bag model which, although the critical line has a similar shape for small $\mu$, would yield coefficients larger by a factor of four.
Conclusions
===========
Within our quasiparticle model [@PKPS96; @PKS00] we analyze recent $n_{\!f} = 2$ QCD lattice calculations [@CPPACS] of the equation of state at nonzero temperature and $\mu = 0$, and then extend the quasiparticle model to nonzero baryon density. The resulting elliptic flow equation for the coupling relates the thermodynamic potential along the characteristic curves in the $\mu\,T$ plane. We argue that the characteristic line through $T_c(\mu=0)$ is related to the critical line in the phase diagram. This is confirmed by comparing our results for the curvature of the critical line at $\mu = 0$, and the variation of the equation of state along it, with recent lattice simulations [@AlltHKKLSS] exploring the region of small $\mu$.
We give an estimate for the equation of state of cold quark matter. Energy density and pressure are almost linearly related, as in the bag model, however with parameters obtained from the lattice data at $\mu=0$. The relevant physical scale is given by the transition temperature $T_c$, and the parameter corresponding to the bag constant turns out to be large compared to conventional estimates, ${\,\raise-0.3ex\hbox{$\sim$}\kern-.7em\hbox{$^>$}\,}250\,$MeV$^4$.
We have restricted ourselves to the case $n_{\!f}=2$, for which the lattice data for the equation of state at $\mu = 0$ appear to be established best. However, we expect similar results for other numbers of flavors since the pronounced decrease of the ratio $p/p_0$ as $T$ approaches $T_c$, which indicates a large coupling strength, seems to be generic. This universality is then echoed at nonzero $\mu$ because for all $n_{\!f}$ the flow equation behaves for strong coupling similarly as in the perturbative limit, where $\alpha_s$ is constant along the elliptic-like characteristics. Indeed, the shape of the phase boundary calculated in [@FodorK] for the physically relevant case $n_{\!f}=2+1$, although now being a crossover near $T_c$, is very similar to the shape for $n_{\!f}=2$. With the same reasoning, we remark that our estimates are robust with respect to remaining uncertainties of the underlying lattice data. Indeed, the equation of state at $\mu \not= 0$ is not very sensitive to the precise values of the model parameters as long as they reasonably describe the gross features of the equation of state at $\mu = 0$. Therefore, the large energy density at small pressure seems to be a general feature of the equation of state.
As shown in [@PKS00; @PKS01], this would allow for pure quark stars with masses $\le 1 M_\odot$ and radii $\le 10$ km. Similar small and light quark stars have also been obtained within other approaches, cf. [@Blasch]. Such objects are of interest in the ongoing discussion of the data of the quark star candidate RXJ1856.5-3754 [@Pons]. It should be emphasized, however, that the outermost layers of such pure quark stars are metastable with respect to hadronic matter with a larger pressure at $\mu \sim \mu_c$. The details of the star structure depend sensitively on the hadronic equation of state [@BK]. However, as discussed in [@Fraga], a stable branch of hybrid stars with a dense quark core and a thin hadronic mantle could indeed be possible.
[**Acknowledgments:**]{} We would like to thank M. Alford, E. Fraga, R. Pisarski, and D. Rischke for discussions. This work is supported by BMBF.
[99]{} K. Rajagopal and F. Wilczek, in [*At the Frontier of Particle Physics*]{}, edited by M. Shifman, (World Scientific, Singapore, 2001), Vol. 3, p. 2061, hep-ph/0011333. A. Peshier, B. Kämpfer, O.P. Pavlenko, and G. Soff, Phys. Rev. D54, 2399 (1996). A. Peshier, B. Kämpfer, and G. Soff, Phys. Rev. C61, 045203 (2000). F. Karsch, Nucl. Phys. B (Proc. Suppl.) 83, 14 (2000). J. Engels et al., Phys. Lett. B396, 210 (1997). A. Peshier, B. Kämpfer, and G. Soff, hep-ph/0106090. F. Karsch, E. Laermann, and A. Peikert, Phys. Lett. B478, 447 (2000). A. AliKhan et al., Phys. Rev. D64, 074510 (2001). C.R. Allton et al., hep-lat/0204010. G. Boyd et al., Nucl. Phys. B469, 419 (1996). M. Okamoto et al., Phys. Rev. D60, 094510 (1999). M. LeBellac, [*Thermal Field Theory*]{} (Cambridge University Press, Cambridge, England, 1996). M.I. Gorenstein and S.N. Yang, Phys. Rev. D52, 5206 (1995). J.P. Blaizot, E. Iancu, and A. Rebhan, Phys. Rev. D63, 065003 (2001); A. Peshier, ibid. 63, 105004 (2001). A. Peshier, hep-ph/9809379. R.A. Schneider and W. Weise, Phys. Rev. C64, 055201 (2001). P. Petreczky et al., Nucl. Phys. B (Proc. Suppl.) 106, 513 (2002). M.G. Alford, A. Kapustin, and F. Wilczek, Phys. Rev. D59, 054502 (1999). Z. Fodor and S.D. Katz, J. High Energy Phys. 03, 014 (2002). D. Blaschke et al., Phys. Lett. B450, 207 (1999); E.S. Fraga, R.D. Pisarski, and J. Schaffner-Bielich, Phys. Rev. D63, 121702 (2001). J.A. Pons et al., Astrophys. J. 564, 981 (2002); J.J. Drake et al., ibid. 572, 996 (2002); F.M. Walter and J. Lattimer, astro-ph/0204199. B. Kämpfer, Phys. Lett. B101, 366 (1981); J. Phys. A14, L471 (1981). E.S. Fraga, R.D. Pisarski, and J. Schaffner-Bielich, Nucl. Phys. A702, 217 (2002).
[^1]: In these calculations $m_q = 0.1\,T$ was assumed, corresponding to $m_{ps}/m_v = 0.7$ at $T_c$. From the weak quark mass sensitivity observed in [@CPPACS], both results should indeed be comparable.
[^2]: A formal reason supporting this attempt is the stationarity of the thermodynamic potential with respect to variation of the self-energies around the physical value; see [@sca] and the references given there. Moreover, there are heuristic arguments that resummation improved leading order results might be more appropriate at large coupling than high order perturbative results [@Pesh98].
[^3]: For the fit we considered only the normalized data [@CPPACS]. The result then reproduces the extrapolated data [@Karsch00] on the lower side of the estimated error band; see the remark at the end of the last section.
[^4]: In an alternative approach, instead of attributing the deviations from the free limit at smaller temperatures to the mass of the quasiparticles, a variable number of degrees of freedom is proposed in [@SchneW].
[^5]: We remark that the region where the solution of the flow equation is not unique is determined only by $\alpha_s(\mu=0,T)$, i.e. by the parameters $\lambda$ and $T_s$ fitted from the entropy, whereas the $p=0$ line depends also on $p(\mu=0,T_c)$ and thus on the third parameter $B_0$. Therefore, the fact that the potential ambiguity is irrelevant is based in a nontrivial way on the underlying lattice data for the equation of state at $\mu=0$.
[^6]: In passing we note the amusing fact that the result agrees with the value from the bag model assuming free massless pions for the hadronic phase.
[^7]: This value renders more precisely the rough estimate [@PKS01], which was about 40% larger. Based on the pragmatic extension of the continuum extrapolation of the lattice data [@Karsch00] shown in Fig. \[fig:latt dat p\] near $T_c$, the fit led to a similar value for $T_s$, but to $\lambda \approx 11$. This demonstrates that details of the underlying lattice data are important for quantitative predictions at $\mu \not= 0$ but, on the other hand, that the estimates are rather robust.
|
---
abstract: 'Transport in weighted networks is dominated by the minimum spanning tree (MST), the tree connecting all nodes with the minimum total weight. We find that the MST can be partitioned into two distinct components, having significantly different transport properties, characterized by centrality — number of times a node (or link) is used by transport paths. One component, the [*superhighways*]{}, is the infinite incipient percolation cluster; for which we find that nodes (or links) with high centrality dominate. For the other component, [*roads*]{}, which includes the remaining nodes, low centrality nodes dominate. We find also that the distribution of the centrality for the infinite incipient percolation cluster satisfies a power law, with an exponent smaller than that for the entire MST. The significance of this finding by showing that one can improve significantly the global transport by improving a very small fraction of the network, the superhighways.'
author:
- Zhenhua Wu
- 'Lidia A. Braunstein'
- Shlomo Havlin
- 'H. Eugene Stanley'
title: |
Transport in weighted networks:\
Partition into superhighways and roads
---
Recently much attention has been focused on the topic of complex networks, which characterize many natural and man-made systems, such as the internet, airline transport system, power grid infrastructures, and the world wide web [@Barabasi_rmp_review; @vespignani_book; @Mendes_book]. Besides the static properties of complex networks, dynamical phenomena such as transport in networks are of vital importance from both theoretical and practical perspectives. Recently much effort has been focused on weighted networks [@Barrat_WAN; @Macdonald_ecoli], where each link or node is associated with a weight. Weighted networks yield a more realistic description of real networks. For example, the cable links between computers in the internet network have different weights, representing their capacities or bandwidths.
In weighted networks the minimum spanning tree (MST) is a tree including all of the nodes but only a subset of the links, which has the minimum total weight out of all possible trees that span the entire network. Also, the MST is the union of all “strong disorder” optimal paths between any two nodes [@FN_sd; @Barabasi_ibp; @Dobrin_mst; @Cieplak:op1; @Porto; @Lidia_op_prl; @Zhenhua]. The MST which plays a major role for transport is widely used in different fields, such as the design and operation of communication networks, the traveling salesman problem, the protein interaction problem, optimal traffic flow, and economic networks [@Khan_tech; @Skiena_book_mst; @Fredman_mst; @Kruskal_mst; @Macdonald_ecoli; @mst_eco_1; @mst_eco_2].
An important quantity that characterizes transport in networks is the betweenness centrality, $C$, which is the number of times a node (or link) used by the set of all shortest paths between all pairs of nodes [@Newman_centrality; @Goh_load_prl; @Kim_bc]. For simplicity we call the “betweenness centrality” here “centrality” and we use the notation “nodes” but similar results have been obtained for links. The centrality, $C$, quantifies the “importance” of a node for transport in the network. Moreover, identifying the nodes with high $C$ enables, as shown below, to improve their transport capacity and thus improve the global transport in the network. The probability density function (pdf) of $C$ was studied on the MST for both scale-free (SF) [@Barabasi_sf] and Erdős-Rényi (ER) [@erdos] networks and found to satisfy a power law, $${\cal P}_{\rm MST}(C) \sim C^{-\delta_{\rm MST}}$$ with $\delta_{\rm MST}$ close to $2$ [@Goh_centrality; @Kim_bc].
Here we show that a sub-network of the MST [@FN_IIC_isIn_MST], the infinite incipient percolation cluster (IIC) has a significantly higher average $C$ than the entire MST — i.e., the set of nodes inside the IIC are typically used by transport paths more often than other nodes in the MST. — In this sense the IIC can be viewed as a set of [*superhighways*]{} (SHW) in the MST. The nodes on the MST which are not in the IIC are called [*roads*]{}, due to their analogy with roads which are not superhighways (usually used by local residents). We demonstrate the impact of this finding by showing that improving the capacity of the superhighways (IIC) is surprisingly a better strategy to enhance global transport compared to improving the same number of links of the highest $C$ in the MST, although they have higher $C$ [@FN_highest_bc_IIC]. This counterintuitive result shows the advantage of identifying the IIC subsystem, which is very small compared to the full network [@FN_iic_mst_mass_ratio]. Our results are based on extensive numerical studies for centrality of the IIC, and comparison with the centrality of the entire MST. We study ER, SF and square lattice networks.
To generate a ER network of size $N$ with average degree $\langle k \rangle$, we pick at random a pair of nodes from all possible $N (N-1) /2$ pairs, link this pair, and continue this process until we have exactly $\langle k \rangle
N/2$ edges. We disallow multiple connections between two nodes and self-loops in a single node. To construct SF networks with a prescribed power law distribution ${\cal P}(k) \sim k^{-\lambda}$ with $k \ge k_{\rm
min}$ [@Barabasi_sf], we use the Molloy-Reed algorithm [@Molloy_Reed; @Molloy_Reed_book]. We assign to each node $i$ a random number $k_i$ of links drawn from this power law distribution. Then we choose a node $i$ and connect each of its $k_i$ links with randomly selected $k_i$ different nodes.
To construct a [*weighted*]{} network, we next assign a weight $w_i$ to each link from a uniform distribution between $0$ and $1$. The MST is obtained from the weighted network using Prim’s algorithm [@network_flow_book]. We start from any node in the largest connected component of the network and grow a tree-like cluster to the nearest neighbor with the minimum weight until the MST includes all the nodes of the largest connected component. Once the MST is built, we compute the value of $C$ of each node by counting the number of paths between all possible pairs passing through that node . We normalize $C$ by the total number of pairs in the MST, $N(N-1)/2$, which ensures that $C$ is between 0 and 1 [@FN_sd_mst].
To find the IIC of ER and SF networks, we start with the fully connected network and remove links in descending order of their weights. After each removal of a link, we calculate $\kappa \equiv \langle k^2 \rangle /\langle k
\rangle$, which decreases with link removals. When $\kappa < 2 $, we stop the process because at this point, the largest remaining component is the IIC [@Cohen_random_attack]. For the two dimensional (2D) square lattice we cut the links in descending order of their weights until we reach the percolation threshold, $p_c$ ($=0.5$). At that point the largest remaining component is the IIC [@Bunde_book].
To quantitatively study the centrality of the nodes in the IIC, we calculate the pdf, ${\cal P}_{\rm IIC}(C)$ of $C$. In Fig. \[graph\_Pcent\_node\] we show for nodes that for all three cases studied, ER, SF and square lattice networks, ${\cal P}_{\rm IIC}(C)$ satisfies a power law $${\cal P}_{\rm IIC}(C) \sim C^{-\delta_{\rm IIC}},
\label{BC_IIC_scaling}$$ where $$\delta_{\rm IIC} \approx \left\{ \begin{array}{ll}
1.2 & {\rm [ER, SF] } \\
1.25 & {\rm [square~lattice]}
\end{array}\right..$$ Moreover, from Fig. \[graph\_Pcent\_node\], we find that $\delta_{\rm IIC} <
\delta_{\rm MST}$, implying a larger probability to find a larger value of $C$ in the IIC compared to the entire MST. Our values for $\delta_{\rm MST}$ are consistent with those found in Ref. [@Goh_centrality]. We obtain similar results for the centrality of the links. Our results thus show that the IIC is like a network of [*superhighways*]{} inside the MST. When we analyze centrality for the entire MST, the effect of the high $C$ of the IIC is not seen since the IIC is only a small fraction of the MST. Our results are summarized in Table \[table\_para\].
To further demonstrate the significance of the IIC, we compute for each realization of the network the average $C$ over all nodes, $\langle C
\rangle$. In Fig. \[graph\_ave\_bc\_node\], we show the histograms of $\langle
C \rangle$ for both the IIC and for the other nodes on the MST. We see that the nodes on the IIC have a much larger $\langle C \rangle$ than the other nodes of the MST.
Figure \[graph\_schematic\_of\_hw\] shows a schematic plot of the SHW inside the MST and demonstrates its use by the path between pairs of nodes. The MST is the “skeleton” inside the network, which plays a key role in transport between the nodes. However, the IIC in the MST is like the “spine in the skeleton”, which plays the role of the superhighways inside a road transportation system. A car can drive from the entry node ${\rm A}$ on roads until it reaches a superhighway, and finds the exit which is closest to the exit node ${\rm B}$. Thus those nodes which are far from each other in the MST should use the IIC superhighways more than those nodes which are close to each other. In order to demonstrate this, we compute $f$, the average fraction of pairs of nodes using the [IIC]{}, as a function of $\ell_{\rm
MST}$, the distance between a pair of nodes on the MST (Fig. \[graph\_ave\_frac\]). We see that $f$ increases and approaches one as $\ell_{\rm MST}$ grows. We also show that $f$ scales as $\ell_{\rm MST} /
N^{\nu_{\rm opt}}$ for different system sizes, where $\nu_{\rm opt}$ is the percolation connectedness exponent [@Lidia_op_prl; @Zhenhua].
The next question is how much the IIC is used in transport on the MST? We define the IIC [*superhighway usage*]{}, $$u \equiv \frac{\ell_{\rm IIC}}{\ell_{\rm MST}},$$ where $\ell_{\rm IIC}$ is the number of the links in a given path of length $\ell_{\rm MST}$ belonging to the [IIC]{} superhighways. The average usage $\langle u \rangle$ quantifies how much the IIC is used by the transport between all pairs of nodes. In Fig. \[graph\_ratio\_and\_flow\](a), we show $\langle u \rangle$ as a function of the system size $N$. Our results suggest that $\langle u \rangle$ approaches a constant value and becomes independent of $N$ for large $N$. This is surprising since the average value of the ratio between the number of nodes on the IIC and on the MST, $\langle N_{\rm IIC} /
N_{\rm MST} \rangle$, approaches zero as $N \to
\infty$ [@FN_iic_mst_mass_ratio], showing that although the IIC contains only a tiny fraction of the nodes in the entire network, its usage for the transport in the entire network is constant. We find that $\langle u \rangle
\approx 0.3$ for ER networks, $\langle u \rangle \approx 0.2$ for SF networks with $\lambda = 4.5$, and $\langle u \rangle \approx 0.64$ for the square lattice. The reason why $\langle u \rangle$ is not close to $1.0$ is that in addition to the IIC, the optimal path passes through other percolation clusters, such as the second largest and the third largest percolation clusters. In Fig. \[graph\_ratio\_and\_flow\], we also show for ER networks, the average usage of the two largest and the three largest percolation clusters for a path on the MST and we see that the average usage increases significantly and is also independent of $N$. However, the number of clusters used by a path on MST is relatively small and proportional to $\ln
N$ [@Sameet_mst], suggesting that the path on the MST uses only a few percolation clusters and a few jumps between them ($\sim \ln N$) in order to get from an entry node to an exit node on the network. When $N \to \infty$ the average usage of all percolation clusters should approach $1$.
Can we use the above results to improve the transport in networks? It is clear that by improving the capacity or conductivity of the highest $C$ links one can improve the transport (see Fig. \[graph\_ratio\_and\_flow\](b) inset). We hypothesize that improving the IIC links (strategy I), which represent the superhighways is more effective than improving the same number of links with the highest $C$ in the MST (strategy II), although they have higher centrality [@FN_highest_bc_IIC]. To test the hypothesis, we study two transport problems: (i) current flow in random resistor networks, where each link of the network represents a resistor and (ii) the maximum flow problem well known in computer science [@Algorithm_book]. We assign to each link of the network a resistance/capacity, $e^{ax}$, where $x$ is an uniform random number between 0 and 1, with $a = 40$. The value of $a$ is chosen such as to have a broad distribution of disorder so that the MST carries most of the flow [@Zhenhua; @Sameet_mst]. We randomly choose $n$ pairs of nodes as sources and other $n$ nodes as sinks and compute flow between them. We compare the transport by improving the conductance/capacity of the links on the IIC (strategy I) with that by improving the same number of links but those with the highest $C$ in the MST (strategy II). Since the two sets are not the same and therefore higher centrality links will be improved in II [@FN_highest_bc_IIC], it is tempting to suggest that the better strategy to improve global flow would be strategy II. However, here we demonstrate using ER networks as an example that counterintuitively strategy I is better. We also find similar improvements of strategy I compared to strategy II for SF networks with $\lambda = 3.5$. In Fig. \[graph\_ratio\_and\_flow\](b), we compute the ratio between the flow using strategy I ($F_{s\rm I}$) and the flow using strategy II ($F_{s\rm
II}$) as a function of the factor of improving conductivity/capacity of the links. The figure clearly shows that strategy I is better than strategy II. Since the number of links in the IIC is relatively very small comparing to the number of links in the whole network [@FN_iic_mst_mass_ratio], it could be a very efficient strategy.
In summary, we find that the centrality of the IIC for transport in networks is significantly larger than the centrality of the other nodes in the MST. Thus the IIC is a key component for transport in the MST. We demonstrate that improving the capacity/conductance of the links in the IIC is useful strategy to improve transport.
We thank ONR, Israel Science Foundation, European NEST project DYSONET, FONCyT (PICT-O2004/370) and Israeli Center for Complexity Science for financial support.
[99]{}
R. Albert and A.-L. Barabási, Rev. Mod. Phys. **74**, 47 (2002).
R. Pastor-Satorras and A. Vespignani, [ *Evolution and Structure of the Internet : A Statistical Physics Approach*]{} (Cambridge University Press, Cambridge, 2004).
S. N. Dorogovtsev and J. F. F. Mendes, [*Evolution of Networks: From Biological Nets to the Internet and WWW*]{} (Oxford University Press, Oxford, 2003).
A. Barrat, et al., PNAS, **101**, 3747 (2004).
P. J. Macdonald, E. Almaas, and A.-L. Barabási, Europhys. Lett., **72**(2), 308 (2005).
By strong disorder it is meant that the weight of the path is determined by the largest weight along the path.
M. Cieplak, et al., Phys. Rev. Lett. **72**, 2320 (1994).
M. Porto, et al., Phys. Rev. E **60**, 2448 (1999).
L. A. Braunstein, et al., Phys. Rev. Lett. **91**, 168701 (2003).
Z. Wu, et al., Phys. Rev. E. **71**, 045101(R) (2005).
A.-L. Barabási, Phys. Rev. Lett. **76**, 3750 (1996).
R. Dobrin, et al., Phys. Rev. Lett. **86**, 5076 (2001).
M. Khan, G. Pandurangan, and B. Bhargava, Tech. Rep. CSD TR 03-013, Dept. of Computer Science, Purdue University (2003).
S. Skiena, [*Implementing Discrete Mathematics: Combinatorics and Graph Theory With Mathematica*]{} (Addison-Wesley, NY, 1990).
M. L. Fredman and R. E. Tarjan, J. ACM **34**, 596 (1987).
J. B. Kruskal, Proc. Amer. Math. Soc. **7**, 48 (1956).
G. Bonanno, G. Caldarelli, F. Lillo, and R. N. Mantegna, Phys. Rev. E **68**, 046130 (2003).
J.-P. Onnela, et al., Phys. Rev. E **68**, 056110 (2003).
M. E. J. Newman, Phys. Rev. E **64**, 016131 (2001); **64** 016132 (2001).
K.-I. Goh, et al., Phys. Rev. Lett. **87**, 278701 (2001).
D.-H. Kim, et al., Phys. Rev. E **70**, 046126 (2004).
A.-L. Barabási and R. Albert, Science **286**, 509 (1999).
P. Erdős and A. Rényi, Publ. Math. (Debrecen) **6**, 290 (1959).
K.-I. Goh, et al., Phys. Rev. E **72**, 017102 (2005).
The IIC contains loops in lattices in dimension $d$ below 6. However, for networks ($d = \infty$), the IIC contains no loops and in this case for large $N$, the IIC must be a subset of the MST. In our simulations, we tested this and found that more than $99\%$ links of IIC belong to MST. For lattices, we only choose the part of the IIC that belongs to the MST.
The overlap between the two groups is about $30\%$ for ER networks.
The ratio between the mass of the IIC, $N_{\rm IIC}$ and the system size $N \equiv N_{\rm MST}$ approaches zero for large $N$ due to the fractal nature of the IIC. Indeed, $N_{\rm IIC}
\sim N^{2/3}$ both for ER [@erdos] and for SF with $\lambda >
4$ [@Cohen_hdbook]. For SF with $\lambda = 3.5$, $N_{\rm IIC} \sim
N^{0.6}$ [@Cohen_hdbook] and for the $L \times L$ lattice $N_{\rm
IIC} \sim L^{91/48} \sim N^{91/96}$ [@Bunde_book].
M. Molloy and B. A. Reed, Comb. Probab. Comput. **7**, 295 (1998).
M. Molloy and B. A. Reed, Random Structures and Algorithms **6**, 161 (1995).
R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, [*Network Flows: Theory, Algorithms and Applications*]{} (Prentice-Hall, Inc. Englewood Cliffs, 1993).
This $C$ measurement is equivalent to counting the number of times a node (link) is used by the set of optimal paths linking all pairs of nodes, in the limit of strong disorder.
R. Cohen, et al., Phys. Rev. Lett. **85**, 4626 (2000).
A. Bunde and S. Havlin, eds., [*Fractals and Disordered Systems*]{} (Springer, New York, 1996).
S. Sreenivasan, et al., Phys. Rev. E **70**, 046133 (2004).
R. Cohen, et al., chap. 4 in [*Handbook of Graphs and Networks*]{}, S. Bornholdt and H. G. Schuster, eds., (Wiley-VCH, 2002).
T. H. Cormen, C. E. Leiserson and R. L. Rivest, [*Introduction to Algorithms*]{} (The MIT Press, Cambridge, Massachusetts, 1990).
ER SF ($\lambda = 4.5$) SF ($\lambda = 3.5$) square lattice
--------------------- -------- ---------------------- ---------------------- ----------------
$\delta_{\rm IIC}$ 1.2 1.2 1.2 1.25
$\delta_{\rm MST}$ 1.6 1.7 1.7 1.32
$\nu_{\rm opt}$ 1/3 1/3 0.2 0.61
$\langle u \rangle$ $0.29$ $0.20$ $0.13$ $0.64$
: Results for the IIC and the MST[]{data-label="table_para"}
![The pdf of the centrality of nodes for (a) ER graph with $\langle
k \rangle = 4$, (b) SF with $\lambda = 4.5$, (c) SF with $\lambda = 3.5$ and (d) $90 \times 90$ square lattice. For ER and SF, $N = 8192$ and for the square lattice $N = 8100$ . We analyze $10^4$ realizations. For each graph, the filled circles show ${\cal P}_{\rm IIC}(C)$; the unfilled circles show ${\cal P}_{\rm MST}(C)$.[]{data-label="graph_Pcent_node"}](Measure_1_Pcent.eps){width="\textwidth"}
![The normalized pdf for superhighway and roads of $\langle C
\rangle$, the $C$ averaged over all nodes in one realization. (a) ER network, (b) SF network with $\lambda = 4.5$, (c) SF network with $\lambda = 3.5$ and (d) square lattice network. To make each histogram, we analyze 1000 network configurations.[]{data-label="graph_ave_bc_node"}](Measure_2_ave_bc.eps){width="\textwidth"}
![Schematic graph of the network of connected superhighways (heavy lines) inside the MST (shaded). A, B and C are examples of possible entry and exit nodes, which connect to the network of superhighways by “roads” (thin lines). The middle size lines indicates other percolation clusters with much smaller size compared to the IIC.[]{data-label="graph_schematic_of_hw"}](highway_schematic.eps){width="60.00000%"}
![The average fraction, $\langle f \rangle$, of pairs using the SHW, as a function of $\ell_{\rm MST}$, the distance on the [MST]{}. (a) ER graph with $\langle k \rangle = 4$, (b) SF with $\lambda = 4.5$, (c) SF with $\lambda = 3.5$ and (d) square lattice. For ER and SF: ($\bigcirc$)$N = 1024$ and ($\Box$)$N = 2048$ with $10^4$ realizations. For square lattice: ($\bigcirc$)$N = 1024$ and ($\Box$)$N =
2500$ with $10^3$ realizations. The $x$ axis is rescaled by $N^{\nu_{\rm
opt}}$, where $\nu_{\rm opt} = 1/3$ for ER and for SF with $\lambda > 4$, and $\nu_{\rm opt} = (\lambda - 3)/(\lambda -1)$ for SF networks with $3
< \lambda < 4$ [@Lidia_op_prl]. For the $L \times L$ square lattice, $\ell_{\rm MST} \sim L^{d_{\rm opt}}$ and since $L^2 = N$, $\nu_{\rm opt}
= d_{\rm opt} / 2 \approx 0.61$ [@Cieplak:op1; @Porto].[]{data-label="graph_ave_frac"}](Measure_3_ave_frac.eps){width="\textwidth"}
![(a) The average usage $\langle u \rangle \equiv \langle \ell_{\rm
IIC} / \ell_{\rm MST} \rangle$ for different networks, as a function of the number of nodes $N$. $\bigcirc$ (ER with $\langle k \rangle = 4$), $\Box$ (SF with $\lambda = 4.5$), $\Diamond$ (SF with $\lambda = 3.5$), $\bigtriangleup$ ($L \times L$ square lattice). The symbols ($\rhd$) and ($\lhd$) represent the average usage for ER with $\langle k \rangle = 4$ when the two largest percolation clusters and the three largest percolation clusters are taken into account, respectively. (b) The ratio between the flow using strategy I, $F_{s\rm I}$, and that using strategy II, $F_{s\rm II}$, as a function of the factor of improving conductivity/capacity. The inset is the ratio between the flow using strategy I and the flow in the original network, $F_{\rm 0}$. The data are all for ER networks with $N = 2048$, $\langle k \rangle = 4$ and $n = 50$($\bigcirc$), $n =
250$($\Diamond$) and $n = 500$($\Box$). The unfilled symbols are for current flow and the filled symbols are for maximum flow.[]{data-label="graph_ratio_and_flow"}](HW_ratio_and_flow.eps){width="\textwidth"}
|
---
abstract: 'We investigate the formation of a Bose polaron when a single impurity in a Bose-Einstein condensate is quenched from a non-interacting to an attractively interacting state in the vicinity of a Feshbach resonance. We use a beyond-Fröhlich Hamiltonian to describe both sides of the resonance and [a coherent-state]{} variational ansatz to compute the time evolution of boson density profiles in position space. We find that on the repulsive side of the Feshbach resonance, [the Bose polaron performs long-lived oscillations, which is surprising given that the two-body problem has only one bound state coupled to a continuum. They arise due to interference between multiply occupied bound states and therefore can be only found with many-body approaches such as the coherent-state ansatz. This is a distinguishing feature of the Bose polaron compared to the Fermi polaron where the bound state can be occupied only once. We derive an implicit equation for the frequency of these oscillations and show that it can be approximated by the energy of the two-body bound state. Finally, we consider an impurity introduced at non-zero velocity and find that, on the repulsive side,]{} it is periodically slowed down or even arrested before speeding up again.'
author:
- Moritz Drescher
- Manfred Salmhofer
- Tilman Enss
bibliography:
- 'references.bib'
title: 'Real space dynamics of attractive and repulsive polarons in Bose-Einstein condensates'
---
\#1[|\#1]{}\#1[\#1|]{}\#1\#2[\#1|.\#2]{}\#1\#2\#3[\#1|\#2|\#3]{}\#1[{ \#1} ]{}\#1[(\#1)]{}\#1\#1[|\#1|]{}\#1[\#1]{}
Introduction
============
The polaron is a general concept of many-body physics that naturally arises in different fields like solid state physics and the theory of ultracold gases. While it has long been used to describe electrons in a crystal lattice, only recent experimental advances allowed one to realize polarons in ultracold gases. Here, the Feshbach resonance allows for a high level of control and in particular for realizing the strong-coupling regime, which could not be done before. This gives access to interesting phenomena such as self-localization and bubble formation. Moreover, the interaction can be changed abruptly, which allows for the investigation of the dynamics. Combined with the possibility of direct imaging, this allows to view polaron formation in position space, which is crucial for the physical intuition and interpretation of the time evolution.
The concept of polarons was originally invented by Landau [@Landau1933]. He showed that an electron in a crystal lattice interacts with the surrounding atoms in such a way that it can be described as a quasiparticle with a higher effective mass, moving through free space. Describing the lattice deformations induced by the electron as phonons, the polaron can be imagined as an electron carrying a cloud of phonons around it. A very similar picture arises in ultracold bosonic gases: According to Bogoliubov theory, the elementary excitations of a BEC are phonons as well, so when an impurity is moving through the gas, the situation is analogous to that of an electron in a crystal. But in an ultracold gas, it is possible to tune the interaction between the particles via a Feshbach resonance and in particular to investigate the regime of strong coupling between impurity and host bosons.
A number of different theoretical approaches has been used to investigate different aspects of the Bose polaron. In 1954, Fröhlich introduced a Hamiltonian which is commonly used to study polarons [@Frohlich1954]. It can be recovered from Bogoliubov theory with one further approximation [@Girardeau1961]. This was first done for ultracold gases in [@Tempere2009], where the ground state properties were studied using a variational ansatz due to Feynman [@Feynman1955]. This ansatz works well for all couplings in the original case of electrons in a lattice, but in the ultracold gas, the regularization of the contact interaction leads to errors when the coupling becomes strong. This was discussed in [@Vlietinck2015] with Diagrammatic Monte Carlo calculations. These give access to the ground-state properties and are computationally intensive but numerically exact and provide valuable benchmarks for other methods. A coherent-state variational ansatz originally due to Lee, Low and Pines (LLP) [@Lee1953] has been used to study dynamical properties [@Shashi2014; @Shchadilova2016]. It neglects entanglement in momentum space and is considered best for heavy impurities and weak couplings. More quantum fluctuations have been taken into account by a renormalization group technique for the ground state [@Grusdt2015; @Grusdt2016all] and the dynamics [@Grusdt2018] and by the correlated gaussian wave function ansatz [@Shchadilova2016a], as well as a Hartree-Fock-Bogoliubov description [@Kain2016]. There are some more works related to the Fröhlich Hamiltonian [@Casteels2010; @Casteels2012; @Nielsen2018]; for a review, see [@Grusdt2015a]. The Bose polaron exhibits characteristic signatures also at finite temperature [@Levinsen2017; @Guenther2018].
The interaction term in the Fröhlich Hamiltonian is, however, just an approximation in the case of ultracold gases and higher order terms become important in the regime of strong coupling. This was first observed in [@Rath2013], where a T-matrix approximation was used [(for a real-time version, see [@Volosniev2015]). Subsequently, a number of approaches have been applied to the fully interacting model within Bogoliubov theory [@Shchadilova2016; @Grusdt2017; @Li2014; @Christensen2015; @Schmidt2018].]{} In the one-dimensional case, some analytical results for heavy impurities have been obtained [@Kain2018] and phonon-phonon interactions beyond Bogoliubov theory have been considered [@Grusdt2017a].
Approaches not based on Bogoliubov theory are more limited in number: Quantum Monte Carlo calculations [@Ardila2015] provide exact ground states for a limited number of parameters. Coupled Gross-Pitaevskii equations [@Astrakharchik2004; @Bruderer2008; @Blinova2013] can describe the spatial deformation of the BEC and the phenomena of self-localization and the bubble polaron but work on a mean-field level. A variational approach which treats the molecular state as an independent quasi-particle has been used to investigate three-body bound states [@Levinsen2015; @Christensen2015]. [In one dimension, the Bose polaron problem can be solved exactly in certain limiting cases [@McGuire1965] and the general case has been addressed by related techniques [@Volosniev2017], but these methods do not carry over to three dimensions.]{}
Experimentally, Bose polarons in ultracold gases have been observed with a focus on absorption spectra and decoherence [@Catani2012; @Scelle2013; @Hu2016; @Jorgensen2016; @Camargo2018], for which some theoretical predictions have been made. Direct imaging experiments on the other hand are still in preparation and there have been few theoretical results concerning the real space dynamics of the bose polaron: the Monte Carlo calculations in [@Ardila2015] include density profiles but only statically for ground states while the Gross-Pitaevskii method in [@Blinova2013] considered a repulsive interaction. This is different from an attractive interaction with a positive scattering length in that it does not feature a bound state.
In this paper, we investigate the dynamics of polaron formation when an initially non-interacting impurity is quenched to an attractively interacting state. This situation has been studied before to compute radio-frequency absorption spectra [@Shashi2014; @Shchadilova2016] and, on the attractive side of the Feshbach resonance, polaron trajectories [@Grusdt2018], as well as pre-thermalization dynamics [@Lausch2018]. Here, we focus on two new aspects: We compute the density profile of the BEC around the impurity as a function of time and thus view the formation of the polaron in position space. This can be directly measured with current imaging technologies, and corresponding experiments are in preparation. On the other hand, we investigate the repulsive side of the Feshbach resonance where the scattering length is positive. Here, a two-body bound state exists and its interplay with the polaron leads to new effects, in particular characteristic oscillations and a depletion of the boson density in a halo around the impurity. These were inaccessible to many previous works based on the Fröhlich Hamiltonian, which depends only on the modulus of the scattering length and cannot describe the bound states. Our study, instead, uses the extended Hamiltonian including higher-order terms in the interaction[. The dynamics are computed by applying a coherent-state ansatz. We find oscillations on the repulsive side of the Feshbach resonance which arise as a result of multiply bound states. This demonstrates the necessity to use a truly many-body ansatz such as the coherent-state ansatz.]{}
The paper is organized as follows. In Sec. \[sec:Model\], we review the construction of the Hamiltonian and the variational ansatz starting from Bogoliubov theory and discuss the stationary solution. Section \[sec:Results\] contains the results for the time evolution after a quench: we start with the case of an impurity initially at rest and compute boson density profiles as well as the total number of bosons gathering around the impurity. [We then present an analytical study that demonstrates the reason for the long-lived oscillations that occur on the repulsive side of the Feshbach resonance and provide a way to compute their frequencies.]{} Finally, we investigate the influence of a non-zero initial velocity and compute polaron trajectories.
Model\[sec:Model\]
==================
Our starting point is the Hamiltonian of a single impurity in a bath of bosons $$\begin{aligned}
H= & \frac{\hat{p}_{I}^{2}}{2m_{I}}+\sum_{\boldsymbol{k}}\frac{k^{2}}{2m_{B}}a_{\boldsymbol{k}}^{\dagger}a_{\boldsymbol{k}}\\
& +\frac{1}{2V}\sum_{\boldsymbol{k},\boldsymbol{q},\boldsymbol{p}}V_{BB}\k{\boldsymbol{p}}\,a_{\boldsymbol{k+p}}^{\dagger}a_{\boldsymbol{q-p}}^{\dagger}a_{\boldsymbol{k}}a_{\boldsymbol{q}}\\
& +\int d^{3}\boldsymbol{x}\,V_{IB}\k{\boldsymbol{x}-\hat{\boldsymbol{x}}_{I}}n_{B}\k{\boldsymbol{x}}\end{aligned}$$ where $m_{I}$ and $m_{B}$ are the masses of impurity and bosons, $\boldsymbol{\hat{p}_{I}}$ and $\boldsymbol{\hat{x}_{I}}$ the impurity momentum and position operators and $a_{\boldsymbol{k}}^{\k{\dagger}}$ the bosonic creation and annihilation operators. $n_{B}(\boldsymbol{x})=a_{\boldsymbol{x}}^{\dagger}a_{\boldsymbol{x}}$ is the boson density and $V_{BB}$ and $V_{IB}$ are the boson-boson and impurity-boson interaction potentials. Our derivation follows Shchadilova et al. [@Shchadilova2016].
Since we are dealing with just one impurity, it is convenient to go to relative coordinates. This is achieved by the [exact]{} canonical transformation $\exp\k{iS}$ where $$S=\boldsymbol{\hat{x}_{I}}\cdot\sum_{\boldsymbol{k}}\boldsymbol{k}a_{\boldsymbol{k}}^{\dagger}a_{\boldsymbol{k}}\,.$$ It is known as the Lee-Low-Pines (LLP) transformation [@Lee1953], see also [@Girardeau1961]. Its effect on the operators is $$\begin{aligned}
e^{iS}\boldsymbol{\hat{p}_{I}}e^{-iS} & =\boldsymbol{\hat{p}}-\sum_{\boldsymbol{k}}\boldsymbol{k}a_{\boldsymbol{k}}^{\dagger}a_{\boldsymbol{k}}\,, & e^{iS}a_{\boldsymbol{k}}^{\dagger}e^{-iS} & =e^{i\boldsymbol{\hat{x}_{I}}\cdot\boldsymbol{k}}a_{\boldsymbol{k}}^{\dagger}\,,\\
e^{iS}a_{\boldsymbol{x}}^{\dagger}e^{-iS} & =a_{\boldsymbol{x}+\boldsymbol{\hat{x}_{I}}}^{\dagger}\,, & e^{iS}a_{\boldsymbol{k}}e^{-iS} & =e^{-i\boldsymbol{\hat{x}_{I}}\cdot\boldsymbol{k}}a_{\boldsymbol{k}}\,.\end{aligned}$$ Note that formally $\boldsymbol{\hat{p}}=\boldsymbol{\hat{p}_{I}}$, but we have dropped the index after the transformation since the physical meaning is not the impurity but the total momentum. It is, of course, conserved and can be replaced by the initial impurity momentum $\boldsymbol{p_{0}}$ such that the transformed Hamiltonian reads $$\begin{aligned}
H_{\text{LLP}} & =\frac{\k{\boldsymbol{p_{0}}-\sum_{\boldsymbol{k}}\boldsymbol{k}a_{\boldsymbol{k}}^{\dagger}a_{\boldsymbol{k}}}^{2}}{2m_{I}}+\sum_{\boldsymbol{k}}\frac{k^{2}}{2m_{B}}a_{\boldsymbol{k}}^{\dagger}a_{\boldsymbol{k}}\\
& +\frac{1}{2V}\sum_{\boldsymbol{k},\boldsymbol{q},\boldsymbol{p}}V_{BB}\k{\boldsymbol{p}}\,a_{\boldsymbol{k+p}}^{\dagger}a_{\boldsymbol{q-p}}^{\dagger}a_{\boldsymbol{q}}a_{\boldsymbol{k}}\\
& +\int d^{3}\boldsymbol{x}\,V_{IB}\k{\boldsymbol{x}}n_{B}\k{\boldsymbol{x}}\,.\end{aligned}$$ This transformation has simplified the interaction term and replaced the impurity momentum by the difference of total momentum and boson momentum. Here, fourth-order terms in the boson operators appear unless the impurity is taken to be infinitely heavy, i.e., stationary. A delocalized impurity thus induces effective interactions between the bosons.
Bogoliubov Theory {#bogoliubov-theory .unnumbered}
-----------------
We use Bogoliubov theory which pre-supposes Bose-Einstein-Condensation in the $\boldsymbol{k}=0$ mode and approximates the low-temperature behaviour by discarding terms in 3rd and 4th order of boson operators with $\boldsymbol{k}\ne0$. The resulting bosonic part of the Hamiltonian is diagonalized by the Bogoliubov transformation $b_{\boldsymbol{k}}^{\dagger}=\cosh\k{\varphi_{k}}a_{\boldsymbol{k}}^{\dagger}-\sinh\k{\varphi_{k}}a_{-\boldsymbol{k}}$ with $\exp\k{4\varphi_{k}}=\xi^{2}k^{2}/(2+\xi\text{\texttwosuperior}k\text{\texttwosuperior})$ and the healing length $$\xi=\frac{1}{\sqrt{8\pi a_{BB}n_{0}}}\,.$$ Up to a constant energy offset, $$\begin{aligned}
H_{\text{Bog}}= & \frac{\k{\boldsymbol{p_{0}}-\sum_{\boldsymbol{k}}\boldsymbol{k}b_{\boldsymbol{k}}^{\dagger}b_{\boldsymbol{k}}}^{2}}{2m_{I}}+\sideset{}{^{\prime}}\sum_{\boldsymbol{k}}\omega_{k}b_{\boldsymbol{k}}^{\dagger}b_{\boldsymbol{k}}\\
& +\int d^{3}\boldsymbol{x}\,V_{IB}\k{\boldsymbol{x}}n_{B}\k{\boldsymbol{x}}\end{aligned}$$ with phonon dispersion $$\begin{aligned}
\omega_{k} & =\frac{k}{2m_{B}\xi}\sqrt{2+\xi\text{\texttwosuperior}k\text{\texttwosuperior}}\,.\end{aligned}$$ $a_{BB}$ and $a_{IB}$ are the scattering lengths of the potentials $V_{BB}$ and $V_{IB}$. Finally, $n_{0}$ is the condensate density, which is a free parameter in Bogoliubov theory. $\sum'$ means that the sum runs over $\boldsymbol{k}\ne0$.
Contact interaction {#contact-interaction .unnumbered}
-------------------
In a dilute ultracold gas, the range of interactions is small compared to all other length scales. The effect of the interaction can therefore be described by a single number, the scattering length, while the precise shape of the potential does not matter and can be chosen arbitrarily. The most convenient choice is a zero-range pseudopotential. Taken literally, the Fourier transform of a delta potential would correspond to a potential in momentum space that is constant over an unbounded region, which does not make sense. Instead, one constructs it as a scaling limit, first cutting off all momentum sums at some large $\Lambda$, and then tuning the interaction strength in the limit $\Lambda\to\infty$ such that the scattering length remains fixed at the desired value. The correctly regularized interaction strength is then given by $$\int d^{3}\boldsymbol{x}\,V_{IB}\k{\boldsymbol{x}}n_{B}\k{\boldsymbol{x}}=\frac{g_{IB}}{V}\sum_{\boldsymbol{k},\boldsymbol{q}}a_{\boldsymbol{k}}^{\dagger}a_{\boldsymbol{q}}\label{eq:Contact Interaction}$$ $$g_{IB}^{-1}=m_{\text{red}}\k{\frac{1}{2\pi a_{IB}}-\frac{2}{V}\sum_{\boldsymbol{k}}^{\Lambda}\frac{1}{k\text{\texttwosuperior}}}$$ with $m_{\text{red}}$ being the reduced mass of impurity and bosons, $m_{\text{red}}^{-1}=m_{I}^{-1}+m_{B}^{-1}$. Note that such a cutoff effectively corresponds to an interaction with a non-zero range of order $1/\Lambda$. The cutoff $\Lambda$ will be used implicitly in all sums and integrals throughout the paper. Also note that instead of a “hard” cutoff, one can also multiply the integrands by a decaying function such as $\exp\k{-2k{{}^2}/\Lambda{{}^2}}$. This leads to smoother results when the cutoff is not large enough for perfectly converged behaviour.
In eq. (\[eq:Contact Interaction\]), we still have to express the boson operators $a_{\boldsymbol{k}}$ by phonon operators $b_{\boldsymbol{k}}$. The result is $$\begin{aligned}
\sum_{\boldsymbol{k},\boldsymbol{q}}a_{\boldsymbol{k}}^{\dagger}a_{\boldsymbol{q}}= & N_{0}+\sqrt{N_{0}}\sideset{}{^{\prime}}\sum_{\boldsymbol{k}}W_{k}\k{b_{\boldsymbol{k}}^{\dagger}+b_{\boldsymbol{k}}}\nonumber \\
& +\sideset{}{^{\prime}}\sum_{\boldsymbol{k},\boldsymbol{q}}\cosh\k{\varphi_{k}+\varphi_{q}}b_{\boldsymbol{k}}^{\dagger}b_{\boldsymbol{q}}\nonumber \\
& \phantom{\sideset{}{^{\prime}}\sum_{\boldsymbol{k},\boldsymbol{q}}}+\sinh\k{\varphi_{k}+\varphi_{q}}\frac{b_{\boldsymbol{k}}^{\dagger}b_{\boldsymbol{q}}^{\dagger}+b_{\boldsymbol{k}}b_{\boldsymbol{q}}}{2}\label{eq:density_expanded}\end{aligned}$$ where $W_{k}=\exp\k{\varphi_{k}}$. Here, $N_{0}=n_{0}V$ is the number of condensed bosons and we have approximated $\bok 0{\hat{N}}0\approx N_{0}$, i.e. neglected the ground state depletion, which gives a constant density shift [@PitaevskiiStringari] $\frac{1}{V}\bok 0{\hat{N}}0-n_{0}=\frac{\sqrt{2}}{12\pi^{2}}\xi^{-3}\approx0.01\xi^{-3}$.
Inserting (\[eq:density\_expanded\]) into $H_{\text{Bog}}$, we obtain the final Hamiltonian
$$\begin{aligned}
H= & \frac{\k{\boldsymbol{p_{0}}-\sum_{\boldsymbol{k}}'\boldsymbol{k}b_{\boldsymbol{k}}^{\dagger}b_{\boldsymbol{k}}}^{2}}{2m_{I}}+\sideset{}{^{\prime}}\sum_{\boldsymbol{k}}\omega_{k}b_{\boldsymbol{k}}^{\dagger}b_{\boldsymbol{k}}+g_{IB}n_{0}\nonumber \\
& +g_{IB}\sqrt{\frac{n_{0}}{V}}\sideset{}{^{\prime}}\sum_{\boldsymbol{k}}W_{k}\k{b_{\boldsymbol{k}}^{\dagger}+b_{\boldsymbol{k}}}\nonumber \\
& +\frac{g_{IB}}{V}\sideset{}{^{\prime}}\sum_{\boldsymbol{k},\boldsymbol{q}}\cosh\k{\varphi_{k}+\varphi_{q}}b_{\boldsymbol{k}}^{\dagger}b_{\boldsymbol{q}}\nonumber \\
& \hphantom{\frac{g_{IB}}{V}\sideset{}{^{\prime}}\sum_{\boldsymbol{k},\boldsymbol{q}}}+\sinh\k{\varphi_{k}+\varphi_{q}}\frac{b_{\boldsymbol{k}}^{\dagger}b_{\boldsymbol{q}}^{\dagger}+b_{\boldsymbol{k}}b_{\boldsymbol{q}}}{2}\,.\label{eq:Final Hamiltonian}\end{aligned}$$
The first two lines of (\[eq:Final Hamiltonian\]) correspond to a Fröhlich Hamiltonian, which is often used to study polarons. It has also been used for polarons in ultracold gases, even though for such systems the quadratic terms in the last two lines of eq. (\[eq:Final Hamiltonian\]) are present. The so obtained results are still valid as long as the coupling between impurity and host atoms is sufficiently weak. One needs to take care however, that when using the Fröhlich Hamiltonian, the regularized contact interaction $g_{IB}$ may not be used and needs to be replaced by the result from the Born approximation $g_{IB}^{Fr}=2\pi a_{IB}/m_{\text{red}}$. Near the Feshbach resonance, the quadratic terms become important as pointed out in [@Rath2013; @Shchadilova2016; @Li2014; @Christensen2015].
Coherent state ansatz {#coherent-state-ansatz .unnumbered}
---------------------
Also by Lee, Low and Pines [@Lee1953], a variational ansatz for the Fröhlich model was suggested, which approximates the ground state by a coherent state. In the limit of infinitely heavy impurities, this ansatz becomes exact. In [@Shashi2014], a time dependent version of this ansatz has been applied to the Bose polaron described by a Fröhlich Hamiltonian. Subsequently, the same time-dependent ansatz has been applied to the full Hamiltonian (\[eq:Final Hamiltonian\]) in [@Shchadilova2016].
Specifically, one considers wave functions of the form
$$\ket{\alpha(t)}=\exp\k{\frac{1}{\sqrt{V}}\sideset{}{'}\sum_{\boldsymbol{k}}\alpha_{k}(t)b_{\boldsymbol{k}}^{\dagger}-h.c.}\ket 0$$ and projects the Schrödinger equation onto the submanifold spanned by these functions. Equivalent to this projection is the stationarity of the functional $$\int dt\,\mathcal{L}(\alpha(t),\dot{\alpha}(t)):=\int dt\,\bok{\alpha}{i\partial_{t}-H}{\alpha}\,.$$ with respect to the $\alpha_{k}$. Setting up the Euler-Lagrange equations [^1] $\frac{\partial\mathcal{L}}{\partial\overline{\alpha}_{k}}-\frac{\partial}{\partial t}\frac{\partial\mathcal{L}}{\partial\dot{\overline{\alpha}}_{k}}=0$ results in the following differential equations, where we also took the limit $V\rightarrow\infty$, replacing $\frac{1}{V}\sum'_{\boldsymbol{k}}$ by $\int\frac{d^{3}\boldsymbol{k}}{\k{2\pi}^{3}}$: $$i\dot{\alpha}_{\boldsymbol{k}}=\k{\Omega_{k}-\frac{\boldsymbol{k}\cdot\boldsymbol{p_{I}}[\alpha]}{m_{I}}}\alpha_{\boldsymbol{k}}+W_{k}C_{1}[\alpha]+iW_{k}^{-1}C_{2}[\alpha]\label{eq:DiffEq}$$ where $$\begin{aligned}
\Omega_{k} & =\frac{k^{2}}{2m_{I}}+\omega_{k}\\
\boldsymbol{p_{I}}[\alpha] & =\boldsymbol{p_{0}}-\int\frac{d^{3}\boldsymbol{k}}{\k{2\pi}^{3}}\,\boldsymbol{k}\abs{\alpha_{k}}^{2}\\
C_{1}[\alpha] & =g_{IB}\sqrt{n_{0}}+g_{IB}\int\frac{d^{3}\boldsymbol{k}}{\k{2\pi}^{3}}\,W_{k}\Re\alpha_{\boldsymbol{k}}\\
C_{2}[\alpha] & =g_{IB}\int\frac{d^{3}\boldsymbol{k}}{\k{2\pi}^{3}}\,W_{k}^{-1}\Im\alpha_{\boldsymbol{k}}\,.\end{aligned}$$ The initial value $\alpha_{\boldsymbol{k}}(0)=0$, i.e., $\ket{\alpha(0)}=\ket 0$, corresponds to the situation of a quench from the phonon vacuum.
If the impurity is initially at rest, $\boldsymbol{p_{0}}=0$, then $\boldsymbol{p_{I}}[\alpha]=0$ for all times due to spherical symmetry. In this case, the equation becomes $\mathbb{R}$-linear and can be written in the form $$\begin{pmatrix}\Re\dot{\alpha}_{k}\vphantom{\alpha_{k}^{(s)}}\\
\Im\dot{\alpha}_{k}\vphantom{\alpha_{k}^{(s)}}
\end{pmatrix}=\begin{pmatrix}0 & \vphantom{\alpha_{k}^{(s)}}H^{(2)}\\
-H^{(1)} & \vphantom{\alpha_{k}^{(s)}}0
\end{pmatrix}\begin{pmatrix}\Re\big(\alpha_{k}-\alpha_{k}^{(s)}\big)\\
\Im\big(\alpha_{k}-\alpha_{k}^{(s)}\big)
\end{pmatrix}\label{eq:time evolution matrix}$$ with a constant offset $\alpha^{(s)}$ (the stationary solution, see below).
Our results are based on solving (\[eq:DiffEq\]) numerically with Verner’s 8th-order Runge-Kutta scheme [@Verner2010], using the julia language [@Bezanson2014a] and the DifferentialEquations.jl package [@Rackauckas2017]. In the $\boldsymbol{p_{0}}=0$ case, we also diagonalize the matrix in (\[eq:time evolution matrix\]) for comparison.
![\[fig:Regimes\]The three regimes of the coherent state ansatz, exhibiting an attractive, unstable or repulsive polaron, separated by the curves $a_{-}^{-1}$ and $a_{+}^{-1}$.](regimes){width="0.8\columnwidth"}
Stationary solution {#stationary-solution .unnumbered}
-------------------
Before turning to the dynamical solutions, it is instructive to look at the stationary solution obtained from $\dot{\alpha}_{\boldsymbol{k}}=0$ or equivalently $\frac{\partial\bok{\alpha}H{\alpha}}{\partial\alpha_{\boldsymbol{k}}}=0$. One finds $$\alpha_{\boldsymbol{k}}^{(s)}=-C_{1}\frac{W_{k}}{\Omega_{k}-\frac{\boldsymbol{k}\cdot\boldsymbol{p_{I}}}{m_{I}}}\label{eq:station=0000E4re L=0000F6sung}$$ where $C_{1}$ and $\boldsymbol{p_{I}}$ are determined by the implicit equations $$\begin{aligned}
C_{1} & =\sqrt{n_{0}}\k{g_{IB}^{-1}+\int\frac{d^{3}\boldsymbol{k}}{\k{2\pi}^{3}}\,\frac{W_{k}^{2}}{\Omega_{k}-\frac{\boldsymbol{k}\cdot\boldsymbol{p_{I}}}{m_{I}}}}^{-1}\\
\boldsymbol{p_{I}} & =\boldsymbol{p_{0}-}C_{1}^{2}\int\frac{d^{3}\boldsymbol{k}}{\k{2\pi}^{3}}\,\boldsymbol{k}\frac{W_{k}^{2}}{\k{\Omega_{k}-\frac{\boldsymbol{k}\cdot\boldsymbol{p_{I}}}{m_{I}}}^{2}}\,.\end{aligned}$$ Note that these quantities are UV convergent: For large $k$, one has $W_{k}^{2}=1+\mathcal{O}(k^{-2})$ and $\Omega_{k}=\frac{k^{2}}{2m_{\text{red}}}+\mathcal{O}(1)$. The [$C_1$-]{}integrand is thus $\frac{1}{\Omega_{k}}+\frac{\boldsymbol{k}\cdot\boldsymbol{p_{I}}}{\Omega_{k}^{2}m_{I}}+\mathcal{O}(k^{-4})$. The first term cancels with the divergence of $g_{IB}^{-1}$, the second vanishes by antisymmetry. The momentum integrand is $\frac{\boldsymbol{k}}{\Omega_{k}^{2}}+\mathcal{O}(k^{-4})$ and again, the first term is antisymmetric.
The integrals exist if and only if $p_{I}/m_{I}<c$, i.e., the stationary impurity velocity must always be below the speed of sound $c=1/\sqrt{2m_{B}\xi}$. For too large initial momenta, no stationary solution exists (the same is true in the Fröhlich model where $C_{1}=2\pi a_{IB}\sqrt{n_{0}}/m_{\text{red}}$ is a constant, see [@Shashi2014]).
{width="80.00000%"}
In the special case $\boldsymbol{p_{0}}=0$, one obtains $\boldsymbol{p_{I}}=0$, $$C_{1}=\frac{\sqrt{n_{0}}}{\frac{m_{\text{red}}}{2\pi}\k{a_{IB}^{-1}-a_{+}^{-1}}}$$ and the stationary energy $$E^{(s)}=\frac{n_{0}}{\frac{m_{\text{red}}}{2\pi}\k{a_{IB}^{-1}-a_{+}^{-1}}}\,.$$ Here, $a_{+}$ is one of two critical scattering lengths defined by $$a_{\pm}^{-1}=\frac{2\pi}{m_{\text{red}}}\int\frac{d^{3}\boldsymbol{k}}{\k{2\pi}^{3}}\,\k{\frac{2m_{\text{red}}}{k^{2}}-\frac{W_{k}^{\pm2}}{\Omega_{k}}}\,,$$ which satisfy $a_{-}<0<a_{+}$. They were found in [@Grusdt2017] and delimit three regions of different stability of the stationary solution. This will be reflected in the convergence behaviour of observables in our dynamical analysis:
1. $a_{IB}^{-1}<a_{-}^{-1}$: The stationary point is a minimum, observables converge. This is the region where the attractive Bose polaron is expected to form.
2. $a_{-}^{-1}<a_{IB}^{-1}<a_{+}^{-1}$: The stationary point behaves like a saddle point, the system is dynamically unstable. This is well understood as coming from phase fluctuations growing without bounds [@Grusdt2017]. [This behaviour is unphysical and means that the approach cannot cover very strong couplings. ]{}
3. $a_{+}^{-1}<a_{IB}^{-1}$: The stationary point behaves like a maximum, observables are oscillating. In this region, the stationary solution is usually interpreted as a repulsive polaron due to its positive energy, while at negative energies, a molecular state is expected. [We will explain the reason for these oscillations and provide an estimate of the frequency. ]{}
In Fig. \[fig:Regimes\] we show how the boundaries of the three regimes change with the reduced mass. For the case of light impurities, $m_{\text{red}}\rightarrow0$, the unstable region grows. Here, quantum fluctuations become especially important as the impurity is delocalized.
Applicability of the method {#applicability-of-the-method .unnumbered}
---------------------------
Two approximations were involved in the derivation.
The Bogoliubov approximation neglects third and fourth order terms in the Bose-Bose interaction of non-condensed modes. This is justified if most of the particles are condensed since then, the coupling of excited to condensed modes outweighs the coupling between different excited modes.
The coherent state ansatz, on the other hand, is a product state ansatz and as such, it neglects correlations between different phonon modes. This is as well justified if the number of excited particles is small. Note that the coherent state ansatz is closely related to Gross-Pitaevskii theory since it corresponds to a replacement of a quantum field with a classical field and a coherent state in momentum space is equivalent to one in real space.
In a weakly interacting Bose gas without an impurity, the condensate depletion is indeed very small. If the impurity is added, there is, however, the unstable region in which the theory predicts attraction of an unlimited number of bosons. In reality, fourth order terms in the Bose-Bose interaction would prevent this. Both in the attractive and repulsive regimes, however, the number of bosons attracted by the impurity will remain on the order of only one to ten, as we show below, such that the theory is valid here.
[ For an estimate of the time scale on which the results can be trusted, observe that beyond Bogoliubov theory, the decay time of phonons due to phonon-phonon interactions is proportional to the inverse square root of the gas parameter: $\tau \sim (n_0a_{BB}^3)^{-{1 \over 2}} {m_B \xi^2 \over \hbar} \approx
300 {m_B \xi^2 \over \hbar}$ for a typical gas parameter of $n_0a_{BB}^3 = 10^{-5}$, where the prefactor depends on the number of excited modes. We find below that all interesting effects occur for short times up to $10 {m_B \xi^2 \over \hbar}$. For these times the beyond-Bogoliubov corrections are negligible even for local boson excitation numbers of order 10. ]{}
Results\[sec:Results\]
======================
Time evolution of density profiles
----------------------------------
Fourier transforming the numerical solution of (\[eq:DiffEq\]) back to position space, we can compute the boson density at distance $r$ from the impurity. More precisely, since the impurity is itself a quantum particle, the quantity to consider is correlation function $$\begin{aligned}
n(\boldsymbol{x}) & =\dk{\hat{n}_{B}(\boldsymbol{\hat{x}_{I}}+\boldsymbol{x})}\\
& =\left<\hat{n}_{B}(\boldsymbol{x})\hat{n}_{I}(\boldsymbol{x})\right> & \text{(original frame)}\\
n(\boldsymbol{x}) & =\left<\hat{n}_{B}(\boldsymbol{x})\right>\,. & \text{(LLP frame)}\end{aligned}$$
Expressing the boson density by phonon operators and applying the variational ansatz, $n(\boldsymbol{x})$ takes the following form in terms of the coefficients $\alpha_{\boldsymbol{k}}$: $$\begin{aligned}
n(\boldsymbol{x})= & \k{\sqrt{n_{0}}+\Re\mathscr{F}^{-1}\left(\alpha W\right)(\boldsymbol{x})}^{2}\nonumber \\
& +\k{\Im\mathscr{F}^{-1}\left(\alpha W^{-1}\right)(\boldsymbol{x})}^{2}\label{eq:density}\end{aligned}$$ where $\mathscr{F}^{-1}f(\boldsymbol{x})=\int\frac{d{{}^3}\boldsymbol{k}}{(2\pi){{}^3}}e^{i\boldsymbol{k}\cdot\boldsymbol{x}}f(\boldsymbol{k})$ denotes the transformation to position space.
Figure \[fig:Density Profiles\] shows the results for the three different regimes:
1. Attractive regime: Bosons are gathering around the impurity and the profile quickly converges to form the attractive Bose polaron. The final shape matches precisely that of the stationary solution.
2. Unstable regime: The impurity keeps pulling in more and more bosons. As mentioned before, this unphysical behaviour reflects the failure of the Bogoliubov approximation when interactions are too strong. Including phonon interactions, i.e., higher-order terms in the bosonic operators $a_{\boldsymbol{k}}$, might prevent this.
3. Repulsive regime: Close to the impurity, the boson density is strongly increased but there is a halo of reduced density around it. There is no convergence to a ground state profile, but instead, the solution keeps oscillating between two states of the coupled system of impurity and surrounding condensate: At some times, the bath is completely depleted at a certain distance while at other times, there is still about half the original density left.
![\[fig:Density Heatmap\]Stationary density profiles for $a_{IB}^{-1}>a_{+}^{-1}$. The point of minimum density is at the distance of the impurity-boson scattering length $a_{IB}$. Parameters are as in Fig. \[fig:Density Profiles\] except that $\Lambda=600\xi^{-1}$ (to ensure $a_{IB}\ll\Lambda^{-1}$ even for weak coupling).](density_heatmap){width="1\columnwidth"}
Comparing these results with the quantum Monte Carlo calculations of the ground state profile in [@Ardila2015], the results are qualitatively similar, even though quantitatively slightly different (our parameters correspond to $a_{IB}/a_{BB}=-20.94$, $\infty$ and $20.94$).
{width="80.00000%"}
The complete depletion is a feature that is present even in the stationary solution and for all scattering lengths above $a_{+}$: Since $\alpha^{(s)}$ is real, the last term in (\[eq:density\]) vanishes and one can always find an $r$ so that the first term vanishes as well. The length scale on which the depletion takes place is given by the scattering length, as shown in Fig. \[fig:Density Heatmap\]. This is not surprising since this is the scale of the two-body bound state. The return to the condensate density then happens on the order of the healing length (not shown in the figure).
Boson Number
------------
From the momentum space coefficients, we can compute the total change in the number of bosons. This is not zero because the Bogoliubov theory does not preserve particle number. It can be seen as a measure of how many particles the impurity attracts in total. The formula is $$\begin{aligned}
\Delta N_{B}(t) & =\bok{\psi(t)}{\sum_{\boldsymbol{k}}n_{k}}{\psi(t)}-\bok 0{\sum_{\boldsymbol{k}}n_{k}}0\\
& =\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\Big(\cosh(2\varphi_{k})\abs{\alpha_{\boldsymbol{k}}}^{2}+\\
& \hphantom{=\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\Big(}\sinh(2\varphi_{k})\Re\alpha_{\boldsymbol{k}}\alpha_{-\boldsymbol{k}}\Big)\,.\end{aligned}$$ Results are shown in Fig. \[fig:Boson Number Combined\]a. The characteristics of the three regimes - convergence, unbounded growth and oscillations - are clearly visible. Note that in the repulsive case, the maxima of the boson number correspond to the more extreme density profiles, i.e., those with full depletion. Doing the same computations for many different scattering lengths, we arrive at Fig. \[fig:Boson Number Combined\]b. As the critical scattering lengths $a_{+}$ and $a_{-}$ are approached from the attractive or repulsive regime, the total boson number grows rapidly as well as the time to convergence. In the figure, this is indicated by the fact that in these areas, the curves are still washed out, therefore not yet converged.
Discussion of the repulsive regime
----------------------------------
The presence of oscillations that do not decay is surprising to the physical intuition, given that the two-particle problem features only one bound state and a continuum of scattering states and one may wonder if this an artifact of one of the approximations involved. In [@Li2014], decaying oscillations were predicted instead by applying a trial wave function of one impurity and at most one phonon excitation. In [@Shchadilova2016], this was contrasted with the same approach that we use in this paper and which predicts stable oscillations. These were interpreted as ocurring between a many-body polaron branch and few-body bound states. Here, we take a different point of view and claim that the repulsive polaron branch plays no role. Instead, these oscillations occur between different multiply bound states. This is a new feature of the Bose polaron in contrast to the Fermi polaron, where the bound state can be occupied at most once.
We demonstrate this by considering the simplified case of an infinitely heavy impurity in a non-interacting BEC. But we emphasize that the latter restriction is not necessary and undamped oscillations occur even in an exact solution of the full Bogoliubov-impurity Hamiltonian. This is shown in appendix \[sec:OscillationMechanism\]. Here, we restrict to the non-interacting case, since the expressions are much simpler and the basic mechanism stands out clearer, which is the same with and without Bose-Bose interactions.
For the situation considered here,
the following Hamilonian is exact (in the thermodynamic limit, to justify the substitution of $a_0$, $a_0^\dagger$ with $\sqrt{n_0}$): $$H = \sideset{}{^{\prime}}\sum_k {k^2 \over 2m_B} a_k^\dagger a_k + g_{IB} \sqrt{n_0 \over V}
\left( a_k^\dagger + a_k \right) + {g_{IB} \over V} \sideset{}{^{\prime}}\sum_{k,q} a_k^\dagger
a_q$$ The quadratic part can be easily diagonalized and yields the two-body spectrum. Even though it has only one bound state, above Hamiltonian can lead to stable oscillations in observables. This is surprising from the two-body point of view where a single eigenstate is coupled only to the continuum, [such that oscillations dephase]{}. The difference lies in the linear terms in the Hamiltonian, which lead to a shift of the creation and annihilation operators. Assume we have diagonalized the quadratic part in terms of new operators $c_E = A_{Ek} a_k$, i.e. switched to the basis of two-body eigenstates: $$\begin{aligned}
H &= \sum_E E c_E^\dagger c_E + E v_E c_E^\dagger + E \overline{v_E} c_E\end{aligned}$$ for some $A_{Ek}$ and $v_E$. The linear terms can be eliminated by the shift $d_E = c_E + v_E$: $$\begin{aligned}
H &= \sum_E E d_E^\dagger d_E + const.\end{aligned}$$ This shift leads to a transformation of both the initial state $\ket{0} = \ket{0}_c$ and observables, for instance the total particle number: $$\begin{aligned}
\ket{0}_c &= \exp \left( \sum_E v_E d_E^\dagger - \overline{v_E} d_E \right) \ket{0}_d \\
\hat N &= \sum_E c_E^\dagger c_E = \sum_E (d_E^\dagger -\overline{v_E}) (d_E - v_E) \, .\end{aligned}$$ The time evolution of this operator is easily computed in the Heisenberg picture: $$\begin{aligned}
\hat{N}(t) &= \sum_E \left( e^{iEt} d_E^\dagger - \overline{v_E} \right) \left( e^{-iEt} d_E -
v_E \vphantom{d_E^\dagger} \right) \, .\end{aligned}$$
Now, if the quadratic part of the Hamiltonian has one bound state and a continuum of scattering states—as is the case in the repulsive regime—the continuum part $E>0$ will dephase but oscillations with the frequency of the bound state energy remain at long times: $$\begin{aligned}
{}_c \bra{0} \hat{N}(t) \ket{0}_c &= \sum_E 2|v_E|^2 (1 - \cos(Et)) \\
& \xrightarrow[]{t\rightarrow \infty} 2|v_{E_B}|^2 (1 - \cos(E_B t)) + \sum_{E>0} 2|v_E|^2\end{aligned}$$ where $E_B = -1/2m_{\text{red}}a_{IB}^2$.
[As we demonstrate in appendix \[sec:OscillationMechanism\], including Bose-Bose interaction within Bogoliubov theory does not destroy this mechanism, since the linear terms are still present]{} while the quadratic terms can be diagonalized by means of a generalized Bogoliubov transformation. On the other hand, third and fourth order terms beyond Bogoliubov theory would likely lead to a damping of the oscillations. But importantly, this damping rate is determined entirely by properties of the BEC while the oscillation frequency is determined by the impurity-boson interaction. In experiments, these two time scales can be controlled independently and for weak Bose-Bose interactions, they will be well distinguishable.
In this sense, we predict a damping reminiscent of the few-body calculations [@Li2014], but for a different reason and on different time scales: In an ansatz with at most one phonon, the bound state couples only to the continuum and rapidly dephasing oscillations are obtained. In an ansatz allowing an arbitrary number of excitations, coherent bound states can be formed which decay only slowly because of Boson-Boson interactions.
This argument also shows that while our approach (and the Bogoliubov approximation in general) is expected to be valid for all times on the attractive side of the Feshbach resonance, it is valid on the repulsive side only as long as the damping has not set in, which is, however, a large time scale for a weakly interacting Bose gas.
Oscillation Frequencies
-----------------------
In the case of an impurity initially at rest, the frequencies of the oscillations in the repulsive regime can be predicted by making an ansatz for the long-time solution. The coefficients $C_{1}$ and $C_{2}$ from the differential equation (\[eq:DiffEq\]) show the same qualitative behaviour as the other observables: convergence, divergence or oscillations, according to the regime. We therefore make an asymptotic ansatz $$\begin{aligned}
C_{1} & =\sum_{\lambda}A_{\lambda}e^{\lambda t} \\
C_{2} & =\sum_{\lambda}B_{\lambda}e^{\lambda t}\end{aligned}$$ where the coefficients $\lambda$ can take finitely many complex values with $\Re\lambda\ge0$. This covers all of the three cases: convergence if only $\lambda=0$ is present, exponential growth if a $\lambda>0$ exists and oscillations for imaginary $\lambda$. The case $\Re\lambda<0$ would be interesting as well to describe the speed of convergence, but the restriction to $\Re\lambda\ge0$ will prove necessary for the calculation. Since $C_{1}$ and $C_{2}$ are real, we must have $A_{\overline{\lambda}}=\overline{A_{\lambda}}$ and $B_{\overline{\lambda}}=\overline{B_{\lambda}}$ (the bar denotes complex conjugation).
Our aim is to derive conditions on $\lambda$ to be able to predict the exponential growth rate or oscillation frequency of the physical observables. We thus insert the ansatz into the differential equation (\[eq:DiffEq\]) with $\boldsymbol{p_{I}}[\alpha]=0$ and find the solution
$$\alpha_{k}(t)=s_{k}e^{-i\Omega_{k}t}+\sum_{\lambda}b_{k\lambda}e^{\lambda t}$$
with the coefficients
$$b_{k\lambda}=-\frac{W_{k}A_{\lambda}+iW_{k}^{-1}B_{\lambda}}{\Omega_{k}-i\lambda}$$ and unknown $s_{k}$, which depend on the full history of the time evolution.
This ansatz solves the projected Schrödinger equation asymptotically only if the values of $\lambda$ are restricted to either $\lambda=0$ or the solutions of the implicit equation $$\begin{aligned}
& \left(\Delta_{+}-\lambda^{2}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{W_{k}^{2}}{\Omega_{k}(\Omega_{k}^{2}+\lambda^{2})}\right)
\nonumber \\
\times &
\left(\Delta_{-}-\lambda^{2}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{W_{k}^{-2}}{\Omega_{k}(\Omega_{k}^{2}+\lambda^{2})}\right)
\nonumber \\
= & -\lambda^{2}\left(\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{1}{\Omega_{k}^{2}+\lambda^{2}}\right)^{2}\label{eq:lambda}\end{aligned}$$ where we abbreviated $$\Delta_\pm := {\mu \over 2\pi} \left(a_{IB}^{-1} - a_\pm^{-1}\right) \,.$$ The detailed derivation is reported in appendix \[sec:FrequencyDerivation\]. It contains also a discussion of the case $\lambda^2<0$, where principal value integrals have to be used.
Solving (\[eq:lambda\]) numerically for different parameters, we find that it has
1. no solution in the attractive regime, so only $\lambda=0$ is possible here;
2. one solution for positive real $\lambda^{2}$ in the unstable regime;
3. one solution for negative real $\lambda^{2}$ in the repulsive regime.
[These values]{} give predictions of the exponential growth rate or frequency, respectively, which are in perfect agreement with the numerical simulations.
![\[fig:Lambda\]Predictions for exponential growth rate $\Re\lambda$ [ (dash-dotted line)]{} and oscillation frequency $\Im\lambda$ [ (solid line)]{} in the long-time limit for the unstable and repulsive regime. The oscillation frequency is well approximated by $1/2m_{\text{red}}a_{IB}^{2}$, the energy of the two-body bound state. As the small figures show, this remains true for different mass ratios (note the difference in vertical scale). Parameters are as in Fig. \[fig:Density Profiles\] with a hard momentum cutoff at $\Lambda=1000\xi^{-1}$.](frequencies){width="1\columnwidth"}
Figure \[fig:Lambda\] shows $\lambda$ over a range of different scattering lengths. [ In the last section, we have shown that for an infinitely heavy impurity in a non-interacting BEC, the oscillation frequency is given by the energy of the two-body bound state, $1/2m_{\text{red}}a_{IB}^{2}$. We find that this is still a very good approximation for the general case away from the resonance, i.e. in the repulsive region where our theory applies. In particular, the frequency depends only on quantities of the impurity-boson scattering problem. Only close to the resonance, deviations become visible. Here, the many-body environment leads to a shift of the bound state energy and consequently, the oscillation frequency starts to depend on properties of the BEC as well. ]{}
{width="80.00000%"}
{width="80.00000%"}
Moving Impurity
---------------
We now turn to the case where the impurity has an initial velocity $\boldsymbol{v_{0}}=\boldsymbol{p_{0}}/m_{I}$. This case has been investigated for the Fröhlich Hamiltonian in [@Shashi2014] using the coherent-state variational ansatz and in [@Grusdt2018] with a time-dependent renormalization group method. In the latter reference, also the full Hamiltonian was investigated on the attractive side of the Feshbach resonance and it was argued that in this case, the second order terms lead only to a shift of the inverse scattering length. On the repulsive side, this is not true since the Fröhlich Hamiltonian cannot describe molecule formation.
In Fig. \[fig:Momentum and Position\] we show the time evolution of the impurity velocity $\boldsymbol{v_{I}}(t)=\boldsymbol{v_{0}}-\frac{1}{m_{I}}\int\frac{d^{3}\boldsymbol{k}}{\k{2\pi}^{3}}\boldsymbol{k}\abs{\alpha_{\boldsymbol{k}}}^{2}$ and position $\boldsymbol{x_{I}}(t)=\int_{0}^{t}\,\boldsymbol{v_{I}}(t')dt'$ (according to Ehrenfest’s theorem). In the attractive regime, the behaviour is simple: If the total momentum is not too high, it converges to a non-zero final value, which agrees with the stationary solution. [ (Note that a slow-down of the impurity even when its velocity is already below the speed of sound does not contradict Landau’s theory, which makes predictions for the stationary state. Here, the quench into a far-from equilibrium state introduces enough interaction energy to excite phonons even when the impurity is slower than the speed of sound.)]{} This matches the picture of a polaron with an increased effective mass. If the total momentum is too large such that no stationary solution exists, the velocity converges to the speed of sound. Note, however, that close to the resonance, it has been predicted that quantum fluctuations beyond the coherent-state ansatz lead to an enhanced damping or even recoil effects, cf. [@Grusdt2018].
On the repulsive side, the behaviour is different: the velocity is oscillating with the same frequency as the density profile and boson number. Indeed, the velocity is smallest when the boson number is largest, which corresponds to a high effective mass of the impurity. The effect is most striking close to the critical scattering length $a_{+}^{-1}$: Here the impurity velocity quickly reaches zero but has periodic revivals.
On both sides of the resonance, the initial velocity does not matter much as long as it is below or close to the speed of sound: it leads only to a rescaling of the velocity at later times. This is also reflected in the density profiles (Fig. \[fig:density profiles moving\]), which are still symmetric around the impurity. Above the speed of sound, however, the number of attracted bosons is increased, leading to a faster decay of the velocity. The density profiles now become asymmetric with some depletion in front of the impurity.
Discussion
==========
We investigated the dynamics of polaron formation in a BEC after a quench, focusing on real space density profiles and the behaviour for positive scattering lengths. These could not be investigated in previous works that used the Fröhlich Hamiltonian.
We found that three regions of qualitatively different behaviour exist, where the strong-coupling region is unstable, as expected from the stationary analysis in [@Grusdt2017]. The fact that the instability persists even in the limit of heavy impurities is a hint that Bogoliubov theory is not adequate to investigate the strong-coupling regime: the deformation of the BEC is too important for the Bogoliubov approximation to hold. [Our results are thus most reliable away from this critical region.]{}
For positive scattering lengths, oscillations can be observed in the expectation values of many observables and we presented a way to compute their frequency. [For an infinitely heavy impurity in a non-interacting BEC, it is exactly given by the energy of the two-body bound state, $1/2m_{\text{red}}a_{IB}^{2}$, while for the general case, this is still a very good approximation. In contrast to the case of the Fermi polaron, these oscillations do not dephase due to coupling to the continuum because they occur in a coherent bound state. Nevertheless, Bose-Bose interactions beyond Bogoliubov theory likely lead to damping, but on an independent time scale, that is determined by properties of the BEC only. In the experimentally relevant case of a weakly interacting BEC, it will be slow and many oscillations are expected to be observable. A quantitative estimate is, unfortunately, not possible from within the theory and requires beyond-Bogoliubov methods. ]{}
Remarkably, these oscillations are present even in the impurity velocity, leading to striking “stop-and-go” polaron trajectories. The effect is most pronounced for strong coupling when oscillations are slow compared to the velocity relaxation: Here the position is advanced in steps. It will be interesting to see if this can be observed in experiments.
In position space, the positive scattering length leads to a halo of reduced condensate density around the impurity, whose size corresponds to a scattering length and is independent of the mass. This is a version of a bubble polaron, where the impurity has, however, still a core of increased density around it and the profile oscillates in intensity. At certain times, the depletion is even perfect, which was not visible in ground state calculations [[@Ardila2015]]{}.
Experimentally, the spatial structure of the Bose polaron could be either detected by direct imaging [on a scale of $a_{IB}$]{}. Alternatively, both the impurity RF spectra and Ramsey spectroscopy of the contrast [@Mistakidis2018] are sensitive to [oscillations in ]{}the local density. [ In the case of a ${}^6$Li impurity in a BEC of $^{133}$Cs atoms [@Ulmanis2016], typical parameters are on the order of $n_0 a_{BB}^3 = 1.5\cdot 10^{-5}$, $a_{IB} = 100 \si{nm}$ and $a_{BB}=8\si{nm}$. The time scale of the oscillations is then $2\pi{2m_\text{red}a_{IB}^2 \over \hbar} \simeq 10 \si{\mu s}$. On the other hand, the time scale of phonon decay due to beyond-Bogoliubov terms is a subleading effect of order of $(n_0a_{BB}^3)^{-{1 \over 2}}$ slower than the BEC time scale ${m_B \xi^2 \over \hbar}$ and therefore of order $10\si{ms}$ for typical parameters. We therefore expect that a large number of oscillations can be observed. ]{}
It will be interesting for future work to investigate how the system behaves for strong coupling where the coherent state ansatz becomes unstable. However, suitable techniques still have to be developed. Within Bogoliubov theory, the so-called correlated gaussian wave functions are a promising way since they should become exact in the limit of heavy impurities. On the other hand, it will be important to find out in which region Bogoliubov theory is not reliable any more and how the system can be described in this region.
We thank Richard Schmidt and Matthias Weidemüller for useful discussions. This work is part of the DFG Collaborative Research Centre SFB 1225 ISOQUANT.
Undamped oscillations in Bogoliubov Theory\[sec:OscillationMechanism\]
======================================================================
In the main text, we have shown that an infinitely heavy impurity in a non-interacting BEC is subject to undamped oscillations due to the presence of coherently bound states. Here, we argue that this remains true if Bose-Bose interactions are included within Bogoliubov approximation, i.e., up to second order in the boson operators with non-zero momentum. Consequently, any decay of oscillations that one may physically expect can only be due to beyond-Bogoliubov terms. Since these are proportional to the Bose-Bose coupling strength $g_{BB}$, the time scale of the decay will be given by properties of the BEC only. In this appendix we go beyond the rest of the paper in that we do not restrict the Hilbert space to coherent states but consider the exact dynamics of the Bogoliubov theory.
The Hamiltonian reads $$\begin{aligned}
H_{\text{Bog}} = & \sumprime_{\boldsymbol k, \boldsymbol q}
\left[ \left({k^2 \over 2m_B} + g_{BB}n_0\right) \delta_{\boldsymbol k, \boldsymbol q} +
{g_{IB} \over V} \right] a_{\boldsymbol k}^\dagger a_{\boldsymbol q} \\
+ & g_{BB} n_0 \sumprime_{\boldsymbol k} a_{\boldsymbol k}^\dagger a_{- \boldsymbol
k}^\dagger + a_{\boldsymbol k} a_{- \boldsymbol k} \\
+ & g_{IB} \sqrt{n_0 \over V} \sumprime_{\boldsymbol k} a_{\boldsymbol k}^\dagger + a_{\boldsymbol k}\end{aligned}$$ after substituting $a_0, a_0^\dagger \rightarrow \sqrt{n_0}$ and dropping third and fourth order interaction terms but before applying the Bogoliubov transformation.
The first two lines, i.e., the quadratic parts, can be diagonalized by a generalized Bogoliubov transformation, see [@Kain2018]. (In this reference, the transformation is applied on top of the usual Bogoliubov transformation. We found it simpler to use only one transformation in total.) This defines new Bogoliubov quasiparticle operators $c$: $$\begin{aligned}
c_E &= A_{Ek} a_k + B_{Ek} a_k^\dagger \\
a_k &= C_{kE} c_E + D_{kE} c_E^\dagger \\\end{aligned}$$ where $$\begin{aligned}
C_{kE} A_{Eq} + D_{kE} \overline{B_{Eq}} &= \delta_{kq} \\
C_{kE} B_{Eq} + D_{kE} \overline{A_{Eq}} &= 0 \, ,\end{aligned}$$ such that $$H_{\text{Bog}} = \sum_E E c_E^\dagger c_E + E v_E c_E^\dagger + E \overline{v_E} c_E \, .
\label{eq:c Hamiltonian}$$ The values of $E$ can be thought of as two-body energies that are shifted by the presence of the many-body environment. Our analysis is not rigorous in that we do not prove the existence of such a transformation—while one will be able to find matrices $A$ and $B$ such that (\[eq:c Hamiltonian\]) holds, the real question is if the so defined operators $c$ admit a vacuum state $\ket{0}_c$ with $c_E\ket{0}_c = 0$ for all $E$. We expect that this is the case at least outside the unstable region and that one eigenstate with negative energy exists on the repulsive side, c.f. [@Kain2018]. In fact, above transformation might also be non-unitary, but in this case, an analogous expression to (\[eq:c Hamiltonian\]) holds after transforming to a non-orthogonal basis, which poses no problem.
As for the non-interacting case, the Hamiltonian is of the form $$\begin{aligned}
H &= \sum_E d_E^\dagger d_E + const. \\\end{aligned}$$ for shifted quasi-particles $d_E = c_E + v_E$. This allows us to compute the time-evolution of the Boson operators $a$ by writing them in terms of $d$, applying the time evolution operator, and writing the result again in terms of $a$. We obtain $$\begin{aligned}
e^{iHt} a_k e^{-iHt} =
\sum_{E,q} & C_{kE} \left( A_{Eq} a_q + B_{Eq} a_q^\dagger \right) e^{-iEt} \\
+ & D_{kE} \left( \overline{A_{Eq}} a_q^\dagger + \overline{B_{Eq}} a_q \right) e^{iEt} \\
+ & C_{kE} v_E \left( e^{-iEt} - 1 \right) \\
+ & D_{kE} \overline{v_E} \left(e^{iEt} - 1\right) \, .\end{aligned}$$ Again, the continuum part dephases in the long-time limit, leaving only terms with $E = E_B'$ where $E_B'$ is the negative energy eigenvalue in (\[eq:c Hamiltonian\]). As for the non-interacting case, the time-evolution does not just lead to an oscillating overall phase, but to terms of the form $\left( e^{-iEt} - 1 \right)$ that remain visible as interferences in the expectation values of observables. Moreover, terms with $\exp(-i E t)$ can combine to give oscillations with a frequency of $2E_B'$. These might, of course, be rather small in amplitude for a weakly interacting Bose gas. Crucially, the inclusion of Bose-Bose interactions in Bogoliubov approximation has not led to a damping of the oscillations already present in the non-interacting case, but only to a frequency shift $E_B \rightarrow E_B'$.
Derivation of the Oscillation Frequencies\[sec:FrequencyDerivation\]
====================================================================
In this appendix, we derive equation (\[eq:lambda\]) for the oscillation frequencies in the repulsive regime and discuss its poles for $\lambda^2 < 0$. This is done by finding the asymptotic solutions of the projected Schrödinger equation (\[eq:DiffEq\]).
As already stated in the main text, we use the ansatz $$\begin{aligned}
C_{1} & =\sum_{\lambda}A_{\lambda}e^{\lambda t} \\
C_{2} & =\sum_{\lambda}B_{\lambda}e^{\lambda t}\end{aligned}$$ with finitely many complex $\lambda$, subject to the condition $\Re \lambda \ge 0$. The prefactors must fulfill $A_{\overline{\lambda}}=\overline{A_{\lambda}}$ and $B_{\overline{\lambda}}=\overline{B_{\lambda}}$ to ensure that $C_1$ and $C_2$ are real. Inserting into (\[eq:DiffEq\]) yields $$\alpha_{k}(t)=s_{k}e^{-i\Omega_{k}t}+\sum_{\lambda}b_{k\lambda}e^{\lambda t} \, .$$ The coefficients $\lambda$ are fixed by $$b_{k\lambda}=-\frac{W_{k}A_{\lambda}+iW_{k}^{-1}B_{\lambda}}{\Omega_{k}-i\lambda}$$ while the $s_{k}$ depend on the full history of the system and are therefore not determined by the asymptotic solution.
Re-inserting the expression for $\alpha$ into the definitions of $C_{1}$ and $C_{2}$ leads to $$\begin{aligned}
\sum_{\lambda}A_{\lambda}e^{\lambda t}= & g_{IB}\sqrt{n_{0}}\nonumber \\
+ & g_{IB}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}W_{k}^{\phantom{-1}}\Re\Big(s_{k}e^{-i\Omega_{k}t}+\sum_{\lambda}b_{k\lambda}e^{\lambda t}\Big)\nonumber \\
\sum_{\lambda}B_{\lambda}e^{\lambda t}= & g_{IB}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}W_{k}^{-1}\Im\Big(s_{k}e^{-i\Omega_{k}t}+\sum_{\lambda}b_{k\lambda}e^{\lambda t}\Big)\,.\label{eq:ABsum}\end{aligned}$$
These equations should be regarded only as determining the solution asymptotically because the integrals over $e^{-i\Omega_{k}t}$ will decay while no finite sum of exponentials $e^{\lambda t}$ with all $\Re\lambda\ge0$ can ever be decaying. But since we are interested in the long-time limit, we can ignore the oscillatory integrals. The need to drop these terms simply reflects the fact that a system never exactly reaches its asymptotic solution but only comes arbitrarily close.
We want to use the linear independence of $e^{\lambda t}$ with different $\lambda$, but first, the $\Re$ and $\Im$ need to be expanded as $2\Re\sum_{\lambda}b_{k\lambda}e^{\lambda t}=\sum_{\lambda}(b_{k\lambda}e^{\lambda t}+\overline{b_{k\lambda}}e^{\bar{\lambda}t})=\sum_{\lambda}(b_{k\lambda}+\overline{b_{k\overline{\lambda}}})e^{\lambda t}$ and similarly for $\Im$. We then find $$\begin{aligned}
\lambda=0: & & A_{0} & =g_{IB}\left(\sqrt{n_{0}}+\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}W_{k}\Re b_{k0}\right)\\
& & B_{0} & =g_{IB}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}W_{k}^{-1}\Im b_{k0}\\
\lambda\ne0: & & 2A_{\lambda} & =g_{IB}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}W_{k}\left(b_{k\lambda}+\overline{b_{k\overline{\lambda}}}\right)\\
& & 2iB_{\lambda} & =g_{IB}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}W_{k}^{-1}\left(b_{k\lambda}-\overline{b_{k\overline{\lambda}}}\right)\,.\end{aligned}$$ Recall that $A_{\lambda}$ and $B_{\lambda}$ need not be real even though the sums in (\[eq:ABsum\]) are. Also note that these equations would not be true for $\Re\lambda<0$ because the oscillating integrals could not be ignored.
Using the expressions for $b_{k\lambda}$ and the relations $$\begin{aligned}
g_{IB}^{-1}+\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{W_{k}^{\pm2}}{\Omega_{k}} & =\frac{\mu}{2\pi}\left(a_{IB}^{-1}-a_{\pm}^{-1}\right)\\
& =:\Delta_{\pm}\end{aligned}$$ we arrive at
$$\begin{aligned}
\lambda=0: & & A_{0} & =\frac{\sqrt{n_{0}}}{\Delta_{+}}\\
& & B_{0} & =0\\
\lambda\ne0: & & A_{\lambda}\left(\Delta_{+}-\lambda^{2}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{W_{k}^{2}}{\Omega_{k}(\Omega_{k}^{2}+\lambda^{2})}\right) & =\lambda B_{\lambda}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{1}{\Omega_{k}^{2}+\lambda^{2}}\\
& & B_{\lambda}\left(\Delta_{-}-\lambda^{2}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{W_{k}^{-2}}{\Omega_{k}(\Omega_{k}^{2}+\lambda^{2})}\right) & =-\lambda A_{\lambda}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{1}{\Omega_{k}^{2}+\lambda^{2}}\,.\end{aligned}$$
Written in this way, all integrals are UV convergent. Multiplying the last two equations finally leads to equation (\[eq:lambda\]) for $\lambda^{2}$, independent of $A_{\lambda}$ and $B_{\lambda}$: $$\left(\Delta_{+}-\lambda^{2}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{W_{k}^{2}}{\Omega_{k}(\Omega_{k}^{2}+\lambda^{2})}\right)
\left(\Delta_{-}-\lambda^{2}\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{W_{k}^{-2}}{\Omega_{k}(\Omega_{k}^{2}+\lambda^{2})}\right)
=-\lambda^{2}\left(\int\frac{d^{3}\boldsymbol{k}}{(2\pi)^{3}}\frac{1}{\Omega_{k}^{2}+\lambda^{2}}\right)^{2}
\, . \label{eq:lambdaAppendix}$$
The case of $\lambda^2 < 0$ {#the-case-of-lambda2-0 .unnumbered}
---------------------------
In the above expressions, many of the integrals do in fact not exist if $\lambda$ is purely imaginary since the integrands have a pole at $k=k_{c}$ in this case. What does exist, however, are the Cauchy principal value (PV) integrals $$\mathcal{P}\int_{0}^{\Lambda}\ldots=\lim_{\epsilon\rightarrow0}\k{\int_{0}^{k_{c}-\epsilon}\ldots+\int_{k_{c}+\epsilon}^{\Lambda}\ldots}\,.$$ Such integrals are not invariant under coordinate transformations because the way in which the pole is approached is crucial, so it is not immediately clear how to make sense of (\[eq:lambdaAppendix\]) for negative $\lambda^{2}$. This becomes clearer if, instead of making an ansatz for an asymptotic solution, one considers the time evolution operator in (\[eq:time evolution matrix\]). In fact, the product $H^{(2)}H^{(1)}$ determines the dynamics completely, so it is sufficient to compute its spectrum. One obtains, once again, equation (\[eq:lambdaAppendix\]), where now $\lambda^{2}$ are the eigenvalues of $H^{(2)}H^{(1)}$. But in the case $\lambda^{2}<0$, one finds that (\[eq:lambdaAppendix\]) must hold with the integrals replaced by PV integrals in any choice of coordinates, as long as the same is used in all three integrals. Therefore, a coordinate transformation will change the value of the individual integrals, but when it is applied to all of them, the equation must stay true.
[ As stated in the main text, one finds no solution in the attractive regime and one solution in the unstable and repulsive regime. Using PV integrals in a particular choice of coordinates, one may find a second solution in the two latter cases, but they are not valid because they change when different coordinates are used. The valid solution is therefore unique and leads to figure \[fig:Lambda\]. ]{}
[^1]: $\frac{\partial}{\partial\overline{z}}=\frac{1}{2}\k{\frac{\partial}{\partial\Re z}+i\frac{\partial}{\partial\Im z}}$ denotes a Wirtinger derivative, which can be used instead of $\frac{\partial}{\partial\Re z}$ and $\frac{\partial}{\partial\Im z}$.
|
---
abstract: 'Contact effects in devices incorporating strongly-correlated electronic materials are comparatively unexplored. We have investigated the electrically-driven phase transition in magnetite (100) thin films by four-terminal methods. In the lateral configuration, the channel length is less than 2 $\mu$m, and voltage-probe wires $\sim$100 nm in width are directly patterned within the channel. Multilead measurements quantitatively separate the contributions of each electrode interface and the magnetite channel. We demonstrate that on the onset of the transition contact resistances at both source and drain electrodes and the resistance of magnetite channel decrease abruptly. Temperature dependent electrical measurements below the Verwey temperature indicate thermally activated transport over the charge gap. The behavior of the magnetite system at a transition point is consistent with a theoretically predicted transition mechanism of charge gap closure by electric field.'
author:
- 'A. A. Fursina$^{1}$, R. G. S. Sofin$^{2}$, I. V. Shvets$^{2}$, D. Natelson$^{3, 4}$'
title: 'Interplay of bulk and interface effects in the electric-field driven transition in magnetite'
---
The complex iron oxide, magnetite, Fe$_3$O$_4$, is an example of strongly correlated 3d-electron systems [@1998_Tokura_MIT_review]. It has been known for decades that bulk magnetite undergoes a first-order metal-insulator transition (two-order-of-magnitude change in electrical resistivity) at the so-called Verwey temperature, $T_{\mathrm{V}}\sim$120 K, accompanied by a structural transformation [@1939_Verwey_first_Nature].
Efforts on magnetite characterization are numerous in the seventy years since the discovery of the Verwey transition, including thorough investigations of its electrical properties [@2002_Walz_review; @1954_mag_el_lowT; @2007_planar_Hall; @1982_el_trans_MO; @2007_spin_PES_review] supported by theoretical calculations of electronic structure [@1987_polaron_cond; @2004_CO_OO; @1984_highT_band_struc_calc]. Recent advances in nanofabrication and film growth allow electrical characterization at previously inaccessible scales, leading to the recent discovery of an electric field driven transition: Magnetite films or nanoparticles below $T_{\mathrm{V}}$ experience a transition from an insulating state to a state with much lower resistance upon application of a sufficiently high voltage [@Our_magnetite_2008; @2008_APL_HAR_Cr; @2009_PRB_hyster]. The switching voltage scales linearly with the channel length suggesting an electric-field driven transition.
The key point of these experiments was an examination of magnetite films or nanoparticles between two electrodes separated by only several hundreds of nanometers or less. In this configuration the electric field needed to drive the transition was accessible at relatively low voltages, thus preventing both excessive heating and damaging of the sample. We proved the observed switching not to be an artifact of heating [@Our_magnetite_2008; @2009_PRB_hyster], in contrast to previously observed transitions in magnetite driven by Joule heating of the samples above $T_{\mathrm{V}}$ under bias [@1969_Tinduced_MIT_Fe3O4_1; @1969_Tinduced_MIT_Fe3O4_2].
The downside of such small channel length experiments is an unavoidable, dominant contribution of the contacts, which prevents direct insight into the properties of magnetite before and after transition. By fitting our data for two-terminal devices with different channel lengths it was demonstrated that contact resistance of Au/magnetite interfaces comprises more than 70 % of the total resistance [@Our_magnetite_2008]. Upon testing several different contact metals (Au, Pt, Cu, Fe and Al), copper showed the lowest contact resistance with magnetite film [@2009_PRB_hyster]. Even with a Cu contacting layer, however, the contribution of the contacts to the total two-terminal device resistance cannot be neglected.
One of the most effective ways to differentiate between bulk and interface effect is to make multilead measurements. To date no such experiments have been performed to study the recently discovered electrically driven transition in magnetite. In this paper we perform four-terminal experiments in a lateral electrode configuration using magnetite thin films. The channel length is less than 2 $\mu$m and voltage-probe wires $\sim$100 nm in width are directly inserted into the channel. These multilead experiments quantitatively and unambigously separate the role of each interface and the magnetite channel. For the first time we study the changes in contact and channel resistance contributions at the onset of electric-field-driven transition in magnetite. Results indicate that at the transition point [*both*]{} contact resistances and the resistance of the magnetite channel decrease abruptly. By doing temperature-dependent electrical measurements below $T_{\mathrm{V}}$ we trace the thermally activated transport over the charge gap in magnetite and provide an insight into the transition mechanism in this system.
The Fe$_{3}$O$_{4}$ (100) thin films (thickness: 50-100nm) used in the present study were grown on (100) oriented MgO single crystal substrates using oxygen plasma assisted molecular beam epitaxy system (DCA MBE M600) with a base pressure 2$\times$10$^{-10}$ Torr. The substrates were cleaned in-situ at 873 K in 5$\times$10$^{-6}$ Torr oxygen for two hours.
Reflection high energy electron diffraction, RHEED, (STAIB Instruments) was used to monitor the growth mode and growth rate (0.3 $\mathrm{\AA}$/s). Room temperature Raman spectroscopy (performed in the backscattering configuration using Rainshaw 1000 Micro Raman system), High resolution X-Ray diffraction measurements using a multi-crystal high-resolution X-ray diffractometer (HRXRD, Bede-D1, Bede, UK), and low temperature four probe resistance measurements were performed to establish the crystal structure and stoichiometry of the Fe$_{3}$O$_{4}$ phase [@Shvets_backscat; @Shvets_high_res_Xray].
Devices for two- and four-terminal measurements were prepared by electron beam lithography (Jeol-6500 SEM). A channel length of 400 nm - 1.9 $\mu$m is defined by two 10 $\mu$m wide source and drain leads. One or two pairs of voltage-probe leads were directly patterned within the channel (Fig. \[fig1\]a). The contacts were fabricated by the electron-beam deposition. In a typical experiment 6 nm of Cu (the best adhesion to magnetite film and lowest contact resitance out of studied contact metals [@2009_PRB_hyster]) and 10-20 nm cover layer of Au were deposited. The leads were connected to micrometer-size pads (300 $\times$ 300 $\mu$m) to which Au wires are attached by In soldering (Fig. \[fig1\]b) and then connected to external contacts of the puck. The puck was placed into the chamber of a Quantum Design Physical Property Measurement System (PPMS model 6000) for variable temperature measurements (300 K - 80 K). The lower temperature bound of these measurements is limited by the increasing switching voltages and concerns about device damage as $T$ is further decreased.
![(color online) (a) SEM image of the device for four-probe measurements showing source and drain leads and two pairs of voltage probes within the channel. (b) Colored SEM image demonstrating electrical contacts to $\mu$m-size Au pads with further In soldering to attach Au wires. (c) Schematics of electrical circuit of four-probe measurements. Letters S and D denote source and drain contact, respectively. Contacts are made of 6 nm Cu adhesion layer (reddish) and 10-20 nm cover layer of Au (yellow). (d) Temperature dependence of the low-bias resistance of magnetite channel ($R_{\mathrm{DEV}}$) and corresponding contact resistances ($R_{\mathrm{C}}$) at source and drain electrodes. []{data-label="fig1"}](fig1.eps){width="8cm"}
Electrical characterization of the devices was performed by standard four-terminal methods using a semiconductor parameter analyzer (HP 4155A). The schematic of device electrical connections is presented in Fig. \[fig1\]c. The voltage, $V_{\mathrm{out}}$, is applied to the source lead with the drain grounded, and current flowing through the channel is recorded. The pair of voltage probes, directly inserted into the channel between source and drain leads, senses voltages $V_1$ and $V_2$. A voltage drop in the channel without the contact contribution is then calculated as $\Delta V=V_{1}-V_{2}$. Only one pair of voltage probes (either left or right, see Fig. \[fig1\]a) is active in a given measurement, with the second pair being intact. Having two pairs of voltage probes allows two independent sets of measurements (one for each pair of voltage probes) in a given channel, to verify data consistency in these four-terminal devices. As was demonstrated in our previous paper, sweeping voltage in a continuous staircase mode leads to the overheating of the sample and appearance of a hysteresis in forward and reverse bias sweeps [@2009_PRB_hyster]. To minimize Joule heating of the channel, the voltage was always swept in a [*pulsed*]{} regime with the shortest available pulse duration (500 $\mu$s) and with pulse period $>$ 5 ms. This pulse measurement procedure greatly reduced apparent hysteresis in the transition as a function of bias sweep [@2009_PRB_hyster].
At any $V_{\mathrm{out}}$, voltage first drops at source electrode/Fe$_3$O$_4$ interface ($V_{\mathrm{C}}$(source)). We assume that at low source-drain biases the contact interface contributions are dominated by an Ohmic contribution, $R_{\mathrm{C}}$(source)$\equiv V_{\mathrm{C}}$(source)/$I$. Then, in the assumption of a homogeneous film (medium) between the electrodes, voltage [*linearly*]{} drops across the channel, and two values are recorded at the two locations of the voltage probes. The remaining potential drop to zero volts (grounded drain electrode) occurs at Fe$_3$O$_4$/drain electrode interface ($V_{\mathrm{C}}$(drain)$\rightarrow$ $R_{\mathrm{C}}$(drain)). Conventionally, the total device may be represented as a voltage drop over three resistors in series, $R_{\mathrm{C}}$(source), $R_{\mathrm{DEV}}$ and $R_{\mathrm{C}}$(drain). By knowing the geometrical characteristics of our devices from SEM images (i.e., $\ell_1$, $\Delta \ell$, and $\ell_2$, see Fig. \[fig1\]c) we can calculate the values of all three voltage drops and, by dividing over measured current, corresponding resistances.
An example of the temperature dependence of $R_{\mathrm{DEV}}$, $R_{\mathrm{C}}$(source) and $R_{\mathrm{C}}$(drain), calculated this way at $V_{\mathrm{out}}$ = 100mV, is presented in Figure \[fig1\]d in the temperature range around $T_{\mathrm{V}}$. The Verwey temperature is inferred for each device as an inflection point in $R_{\mathrm{DEV}}$(T) dependence; and for the various devices studied here $T_{\mathrm{V}}$ range from 100 K to 110 K. For source and drain electrodes made of the same metal (Cu in this case), $R_{\mathrm{C}}$(source) $\approx$ $R_{\mathrm{C}}$(drain) , besides, $R_{\mathrm{DEV}}$ and $R_{\mathrm{C}}$ have nearly identical temperature dependence (Fig. \[fig1\]d ).
![(color) (a) Examples of two-terminal resistance dependence on the channel length at three different temperatures (85 K, 90 K and 95 K) with corresponding linear fits. Extrapolation to zero channel length gives the total contact resistance, R$_C$(total), at each temperature. (b) and (c) show SEM images of devices with different channel lengths. (d) and (e) plot calculated values of device resistances, R$_{DEV}$, and contact resistances, R$_C$(source) and R$_C$(drain), around zero source voltage for devices in (b) and (c), respectively.[]{data-label="fig2"}](fig2.eps){width="8cm"}
Let us consider the relative contributions of contacts and magnetite channel to the total voltage drop. Our assumption of linearity in the channel conduction presumes that $R_{\mathrm{DEV}}$ linearly scales with the channel length, $L$, while $R_{\mathrm{C}}$(source) and $R_{\mathrm{C}}$(drain) should remain independent of $L$. This is supported experimentally. To demonstrate this, we made a set of devices on the same piece of magnetite film with different channel lengths while all other geometrical parameters (film thickness and the width of source and drain leads) remained exactly the same for all devices. Two representive SEM images of such devices with $L$=1.1 $\mu$m and $L$=1.9 $\mu$m are shown in Fig. \[fig2\]b and c, respectively. The total (two-terminal) resistance at each temperature linearly depends on the channel length as demonstrated in Fig. \[fig2\]a at several temperatures (85 K, 90 K and 95 K).
The calculations of contact resistances based on $\ell_{1}$, $\Delta \ell$ and $\ell_2$ for each device show that at each temperature $R_{\mathrm{C}}$(source) and $R_{\mathrm{C}}$(drain) are equal to each other and are the same for devices with different lengths, $L$. It is worth mentioning that $R_{\mathrm{C}}$(source) and $R_{\mathrm{C}}$(drain) do not change upon switching the grounds, [*i.e.*]{}, exchanging the place of injecting and grounded electrodes. Calculated resistances in Fig. \[fig2\]d and e represent the data for the devices shown in Fig. \[fig2\]b and \[fig2\]c, respectively. While $R_{\mathrm{C}}$(source) and $R_{\mathrm{C}}$(drain) remain independent of channel length, $R_{\mathrm{DEV}}$ increases as $L$ increases which is obvious from comparison of Fig. \[fig2\]d and Fig. \[fig2\]e. All three resistances increase significantly with decreasing the temperature (compare data at 85 K, 90 K and 95 K in Fig. \[fig2\]d, e), as will be discussed below in detail. The contact resistance, $R_{\mathrm{C}}$(source) + $R_{\mathrm{C}}$(drain), contributes from 20% to 13% of the total two-terminal $R$ for devices with channel lengths ranging from 1 $\mu$m to 2 $\mu$m.
The increase in total two-terminal resistance with channel length (Fig. \[fig2\]a) is caused by the increased contribution of $R_{\mathrm{DEV}}$ in longer devices. Moreover, the extrapolation of a two-terminal $R$ vs $L$ linear fit to zero channel length gives a resistance value very close to the sum of calculated $R_{\mathrm{C}}$(source) and $R_{\mathrm{C}}$(drain) at each temperature. The latter proves the consistency of our calculations and independence of contact resistances on the channel length within $L$ range investigated in this work.
![(color online) Examples of $I$-$V$ and corresponding $I$-$\Delta V$ curves at 80 K and 85 K at the voltage ranges above switching voltage at each temperature. Arrows indicate the direction of the voltage sweep. The pulse sweep parameters are those to demonstrate the small hysteresis in forward and reverse voltage sweeps. Top inset shows $I$-$\Delta V$ curves at the same temperatures, but at the voltage range below switching voltage. Bottom inset zooms in to $I$-$\Delta V$ curve at 80 K around transition point to demonstrate the discontinuity in measured $\Delta V$ value.[]{data-label="fig3"}](fig3.eps){width="8cm"}
At temperatures below $T_{\mathrm{V}} \sim 105K$, the current-voltage characteristics, $I$-$V$, show Ohmic behavior at low source voltage range ($<$ 1V), while start to exhibit nonlinearities at higher voltages, symmetrical for positive and negative source voltages. Examples at two selected temperatures (80 K and 85 K) are shown in the top inset of Figure \[fig3\]. Upon further increasing source voltage $I$-$V$ curves show a sharp jump in current (Fig. \[fig3\]) as soon as the source voltage reaches a critical switching value, $V_{SW}$, at a certain temperature as described in detail in [@Our_magnetite_2008; @2009_PRB_hyster]. This is a transition from high resistance (Off) state to a state with much lower resistance (On) state. Note the two-order of magnitude difference in current after transition by comparing $I$-$V$ curves before (top inset) and after transition in Fig. \[fig3\].
Corresponding $I$ vs $\Delta V=V_{1}-V{_2}$ plots (Fig. \[fig3\]) have much lower switching $\Delta V_{SW}$ values and reveal at a transition point not only a discontinuity in current, but also in the measured $\Delta V$ value, which decreases in absolute magnitude (Fig. \[fig3\] bottom inset). Since $\Delta V=V_{1}-V_{2}$, in general, reflects properties of magnetite channel without contact effects, the discontinuity (jump) in $\Delta V$ at a transition point can be explained as a sudden decrease in device resistance, $R_{\mathrm{DEV}}$.
Now let us turn to the quantitative description of contact effects at the onset of the field-induced transition. Calculations of $R_{\mathrm{C}}$(source), $R_{\mathrm{DEV}}$ and $R_{\mathrm{C}}$(drain) show that at a transition point the voltage drops at the contacts, $V_{\mathrm{C}}$(source) and $V_{\mathrm{C}}$(drain), increase in absolute value, while $\Delta$V decreases. Compare the blue open squares (at the transition point) and red closed squares (at the next point after transition) in Fig. \[fig4scheme\]a, which depict voltage distribution over the channel length. From this sketch the decrease in $\Delta$V value is also clearly visible.
![(color online) (a) Schematic diagram of voltage distribution along the channel at a transition point (blue open squares) and right after transition (red closed squares). V$_{\mathrm{C}}$ denotes the voltage drop at the interfaces. (b) A fragment of $I$-$V$ curve at 85 K in the vicinity of the transition. Blue open square marks the transition point and red closed square shows a point right after transition; blue and red squares in (a) corresond to the voltage distribution over the channel at these points. (c) and (d) are the voltage dependences of $R_{\mathrm{C}}$(source), $R_{\mathrm{C}}$(drain) and $R_{\mathrm{DEV}}$, respectively, demonstrating the abrupt decreases (jumps) in all three resistances at the transition point.[]{data-label="fig4scheme"}](fig4.eps){width="8.5cm"}
Although $V_{\mathrm{C}}$(source) and $V_{\mathrm{C}}$(drain) increase, due to the overall increased current, $R_{\mathrm{C}}$(source) and $R_{\mathrm{C}}$(drain) actually [*decrease*]{} upon passing through the transition point. Fig. \[fig4scheme\]b,c, and d explicitly demonstrates these decreases (jumps) in $R_{\mathrm{DEV}}$ and $R_{\mathrm{C}}$ at a transition point, denoted further as $R^{jump}_{\mathrm{DEV}}$ and $R^{jump}_{\mathrm{C}}$. For source and drain electrodes made of Cu, $R_{\mathrm{C}}$(source) and $R_{\mathrm{C}}$(drain) jumps at the transition point are equal to each other and remain unchanged in the experiments on exchanging the grounds.
Note that at the transition point [*both*]{} device and source and drain contact resistances decrease abruptly. This behavior is distinct from the one for other systems exhibiting voltage-driven transitions (such as manganites and doped SrTiO$_{3}$). For these systems the leading role of oxygen vacancies drift under applied bias was demonstrated[@2006_Nature_SrTiO3_Waser; @2007_oxygen_diff_Ignatiev], and source and drain contact resistances show variations of opposite sign [@2007_PRL_mechanism]; [*i.e.*]{}, while source contact resistance increases, the drain contact resistance decreases. Since this is inconsistent with our observations, magnetite clearly must exhibit a different switching mechanism.
![(color online) Temperature dependences of (a) the jumps in $R_{\mathrm{C}}$(source), $R_{\mathrm{C}}$(drain) and in $R_{\mathrm{DEV}}$ at a transition point; each point represents an average over 8 independent measurements (left/right voltage pairs, different grounds, positive/negative switching voltages), standard deviation is within the symbol size (b) jumps in $\Delta$V (squares) at a transition point and its exponential fit (solid line) (c) $V_{SW}$ and $\Delta V_{SW}$ with corresponding exponential fits and (d) conductance ($1/R_{\mathrm{DEV}}$) of magnetite channel in Arrhenius coordinates and its linear fit.[]{data-label="fig5"}](fig5.eps){width="8cm"}
As temperature decreases, the jumps in contact resistance, $R^{jump}_{\mathrm{C}}$, and device resistance, $R^{jump}_{\mathrm{DEV}}$, remain negative, but increase in absolute magnitude (Fig. \[fig5\]a). Upon approaching the temperature when switching is not observed (T $\sim$ $T_{\mathrm{V}}$ [@Our_magnetite_2008; @2009_PRB_hyster]), $R^{jump}_{C}$ and $R^{jump}_{DEV}$ approach zero. The jump in $\Delta V$ is also temperature dependent and its magnitude exponentially decays with the temperature, approaching zero at $T_{\mathrm{V}}$. $V_{SW}$ and $\Delta V_{SW}$ depend on temperature exponentially as well, as demonstrated in Fig. \[fig5\]c.
To explain the temperature dependence of the parameters related to the observed transition, [*i.e.*]{}, jumps in current, $\Delta V$, $R_{\mathrm{C}}$ and $R_{\mathrm{DEV}}$ (Fig. \[fig5\]), we should review the properties of magnetite below Verwey temperature, since the transition is only observed below $T_{\mathrm{V}}$. We will consider the Verwey transition physics in magnetite from the electronic structure point of view. While magnetite has strong electron correlations, it is believed that a band-type description of its electronic structure is a reasonable approximation [@2007_spin_PES_review], with transport being dominated by low-lying electronic states near an effective Fermi energy, $E_{\mathrm{F}}$.
The electronic structure of magnetite has been probed extensively by photoelectron and scanning tunneling spectroscopies and band-structure was calculated using different methods [@1995_PES_gap_closed; @1984_highT_band_struc_calc; @2006_PRB_STM_film_surface]. Above $T_{\mathrm{V}}$ there is a finite (non-zero) density of states (DOS) around $E_{\mathrm{F}}$, which is dominated by Fe 3$d$ states of the $B$-site sublattice of cubic structure. Below $T_{\mathrm{V}}$ the DOS near $E_{\mathrm{F}}$ exhibits a clear gap, causing two orders of magnitude increase in resistivity at $T_{\mathrm{V}}$ [@1995_PES_gap_closed; @2005_PES_EuroPhys].
Recently, it has been theoretically predicted that a gap to charged excitations (charge gap) in correlated insulators can be closed by applying external electric field, resulting in field-induced metal-insulator transition [@2003_PRL_Mott_break; @2008_PRB_theory]. Several systems exhibit this behavior, for example, charge-ordered state of complex manganese oxides [@1997_Xray_MIT_Tokura; @1997_Tokura_first], as well as 1D cuprates [@2001_PRB_1D_Mott_break] and 2D nickelates [@1999_PRL_2D_Mott_break].
Our data on magnetite seem to be another experimental observation consistent with this sort of gap closure by electric field. The absence of hysteresis in forward and reverse bias sweeps, meaning that metallic state persists only if applied voltage (electric field) exceeds a critical value, $|V|>|V_{SW}|$ , is expected from this mechanism and is indeed observed in Fe$_{3}$O$_{4}$ (see Ref. [@2009_PRB_hyster] for details). The jump in $|R_{\mathrm{DEV}}|$ at the transition point is a natural consequence of the gap closure. The increase in the absolute value of $R^{\mathrm{jump}}_{\mathrm{DEV}}$ as $T$ decreases (Fig. \[fig5\]a) is also easily explained, as at lower temperatures there is a transition from more insulating state to the same metallic state with zero-size gap. Accompanying drops of contact resistances, $R^{jump}_{C}$, are direct consequences of gap closure and, thus, the change in the position of magnetite $E_{\mathrm{F}}$ relative to the $E_{\mathrm{F}}$ of the contact metal.
The exponential dependences of the above parameters, particularly that shown in Fig. \[fig5\]d, imply thermally activated transport below $T_{\mathrm{V}}$. Indeed, plotting the inverse of $R_{\mathrm{DEV}}$ (in the Ohmic regime near zero bias) in Arrhenius coordinates gives a straight line in a given $T$ range (Fig.\[fig5\]d). The activation energy, $E_{\mathrm{a}}$, inferred from these data on the magnetite channel (device) lies in 85-89 meV range for several devices. These values match well with the size of the gap below $T_{\mathrm{V}}$, inferred from photoemission [@1995_PES_gap_closed; @2005_PES_EuroPhys] and optical [@1998_optical_gap_Tokura; @2005_THz_cond] spectroscopies data. This suggests that transport below $T_{\mathrm{V}}$ involves charge carriers thermally activated over the gap.
In conclusion, by doing four-terminal experiments at magnetite thin films below $T_{\mathrm{V}}$ we quantitatively separate the contributions of each electrode and the magnetite channel before and after the electric field driven transition. For devices of increasing channel lengths we demonstrate the increase in total resistance to be caused by increased contribution of the magnetite channel, while contact resistances are unchanged for all channel lengths within 1 to 2 $\mu$m range. At all temperatures the transition is observed ($T< T_{\mathrm{V}}$), contact resistances of [*both*]{} source and drain electrodes and the resistance of magnetite channel decrease abruptly at the transition point. This behavior is consistent with the mechanism of charge gap closure by electric field predicted in theory [@2003_PRL_Mott_break; @2008_PRB_theory]. To further explore the field-driven switching mechanism in magnetite, the effect of contact metals with different work functions is currently under study. In the framework of the charge gap closure mechanism, the magnitude of contact resistance jumps at a transition point, $R^{jump}_{C}$, are expected to be dependent on the work function of the contact metal according to the relative alignment of metal Fermi level and effective Fermi level of magnetite.
This work was supported by the US Department of Energy grant DE-FG02-06ER46337. DN also acknowledges the David and Lucille Packard Foundation and the Research Corporation. RGSS and IVS acknowledge the Science Foundation of Ireland grant 06/IN.1/I91.
[10]{}
M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. [**70**]{}, 1039 (1998).
E. J. W. Verwey, Nature [**144**]{}, 327 (1939).
F. Walz, J. Phys.: Condens. Matter. [**14**]{}, R285 (2002).
B. A. Calhoun, Phys. Rev. [**94**]{}, 1577.
Y. Bason, L. Klein, H. Q. Wang, J. Hoffman, X. Hong, V. E. Henrich, and C. H. Ahn, J. Appl. Phys. [**101**]{}, 09J507 (2007).
J. M. Honig, J. Solid State Chem. [**45**]{}, 1 (1982).
M. Fonin, Yu. S. Dedkov, R. Pentcheva, U. Rüdiger, and G. Güntherodt, J. Phys.: Condens. Matter [**19**]{}, 315217 (2007).
L. Degiorgi, P. Wachter, and D. Ihle, Phys. Rev. B [**35**]{}, 9259.
I. Leonov, A. N. Yaresko, V. N. Antonov, M. A. Korotin, and V. I. Anisimov, Phys. Rev. Lett. [**93**]{}, 146404 (2004).
A. Yanase and K. Siratori, J. Phys. Soc. Jpn. [**53**]{}, 312 (1984).
S. Lee, A. Fursina, J. T. Mayo, C. T. Yavuz, V. L. Colvin, R. G. S. Sofin, I. V. Shvets, and D. Natelson, Nature Mater. [**7**]{}, 130 (2008).
A. Fursina, S. Lee, R. G. S. Sofin, I. V. Shvets, and D. Natelson, Appl. Phys. Lett. [**92**]{}, 113102 (2008).
A. A. Fursina, R. G. S. Sofin, I. V. Shvets, and D. Natelson, Phys. Rev. B [**79**]{}, 245131 (2009).
P. J. Freud and A. Z. Hed, Phys. Rev. Lett. [**23**]{}, 1440 (1969).
T. Burch, P. P. Craig, C. Hedrick, T. A. Kitchens, J. I. Budnick, J. A. Cannon, M. Lipsicas, and D. Mattis, Phys. Rev. Lett. [**23**]{}, 1444 (1969).
A. Koblischka-Veneva, M. R. Koblischka, Y. Zhou, S. Murphy, F. Mücklich, U. Hartmann, I. V. Shvets, J. Magn. Magn. Mater. [**316**]{}, e663 (2007).
S. K. Arora, R. G. S. Sofin, I. V. Shvets, and M. Luysberg, J. Appl. Phys. [**100**]{}, 073908 (2006).
K. Szot, W. Speier, G. Bihlmayer, and R. Waser, Nat. Mater. [**5**]{}, 312 (2006).
Y. B. Nian, J. Strozier, N. J. Wu, X. Chen, and A. Ignatiev, Phys. Rev. Lett. [**98**]{}, 146403 (2007).
M. Quintero, P. Levy, A. G. Leyva, and M. J. Rozenberg, Phys. Rev. Lett. [**98**]{}, 116601 (2007).
A. Chainani, T. Yokoya, T. Morimoto, T. Takahashi, and S. Todo, Phys. Rev. B [**51**]{}, 17976 (1995).
K. Jordan, A. Cazacu, G. Manai, S. F. Ceballos, S. Murphy, and I. V. Shvets, Phys. Rev. B [**74**]{}, 085416 (2006).
D. Schrupp, M. Sing, M. Tsunekawa, H. Fujiwara, S. Kasai, A. Sekiyama, S. Suga, T. Muro, V. A. M. Brabers, and R. Claessen, Europhys. Lett. [**70**]{}, 789 (2005).
T. Oka, R. Arita, and H. Aoki, Phys. Rev. Lett. [**91**]{}, 066406 (2003).
N. Sugimoto, S. Onoda, and N. Nagaosa, Phys. Rev. B [**78**]{}, 155104 (2008).
V. Kiryukhin, D. Casa, J. P. Hill, B. Keimer, A. Vigliante, Y. Tomioka and Y. Tokura, Nature [**386**]{}, 813 (1997).
A. Asamitsu, Y. Tomioka, H. Kuwahara, and Y. Tokura, Nature [**388**]{}, 50 (1997).
Y. Taguchi, T. Matsumoto, and Y. Tokura, Phys. Rev. B [**62**]{}, 7015 (2000).
S. Yamanouchi, Y. Taguchi, and Y. Tokura, Phys. Rev. Lett. [**83**]{}, 5555 (1999).
S. K. Park, T. Ishikawa, and Y. Tokura, Phys. Rev. B [**58**]{}, 3717 (1998).
A. Pimenov, S. Tachos, T. Rudolf, A. Loidl, D. Schrupp, M. Sing, R. Claessen, and V. A. M Brabers, Phys. Rev. B [**72**]{}, 035131 (2005).
|
---
abstract: 'In this paper, we propose a deep learning-based method, deep Euler method (DEM) to solve ordinary differential equations. DEM significantly improves the accuracy of the Euler method by approximating the local truncation error with deep neural networks which could obtain a high precision solution with a large step size. The deep neural network in DEM is mesh-free during training and shows good generalization in unmeasured regions. DEM could be easily combined with other schemes of numerical methods, such as Runge-Kutta method to obtain better solutions. Furthermore, the error bound and stability of DEM is discussed.'
author:
- |
Xing Shen\
School of Mathematical Sciences\
Zhejiang University\
Hangzhou, Zhejiang, China\
`shenxingsx@zju.edu.cn`\
Xiaoliang Cheng\
School of Mathematical Sciences\
Zhejiang University\
Hangzhou, Zhejiang, China\
`xiaoliangcheng@zju.edu.cn`\
Kewei Liang[^1]\
School of Mathematical Sciences\
Zhejiang University\
Hangzhou, Zhejiang, China\
`matlkw@zju.edu.cn`\
bibliography:
- 'sample.bib'
title: 'Deep Euler method: solving ODEs by approximating the local truncation error of the Euler method'
---
Introduction
============
Many problems in science and engineering can be modeled into a set of ordinary differential equations (ODEs) $$G(x,y,y^{\prime},y^{\prime\prime},\cdots) = 0, \quad x\in [a,b]\subset \mathbb{R}.$$ In most cases, it can not be easy to obtain the analytic solution and so one must typically rely on a numerical scheme to accurately approximate the solution. The important issues confronting the numerical study appear in the initial value problems since higher-order ODEs can be converted into the system of the first-order ODEs. Basic methods for initial value problems are the extremely popular Euler method or the Runge-Kutta method. However, numerical methods have often to balance the discretization step size and computation time. Furthermore, the class of stiff ordinary differential equations may still present a more serious challenge to numerical computation.
In recent years, there has been a growing interest in solving the differential equations and the inverse problems by deep learning. The works include numerical solutions of ODEs and PDEs ([@rudy2019deep], [@raissi2018deep], [@sun2019neupde], [@farimani2017deep]), recovery of the involving systems ([@both2019deepmod], [@khoo2017solving], [@khoo2019switchnet]), overcoming the curse of dimension of high-dimensional PDEs ([@han2017overcoming], [@hutzenthaler2018overcoming]), uncertainty quantification ([@tripathy2018deep], [@winovich2019convpde]) etc. Besides, several works have focused on the combination of traditional numerical methods and deep neural networks. ([@sirignano2018dgm]) proposed a merger of Galerkin methods and deep neural networks (DNNs) to solve high-dimensional partial differential equations (PDEs). They trained DNNs to satisfy the differential operator, initial condition, and boundary conditions. ([@raissi2019physics]) introduced physics-informed neural networks (PINNs), which is a deep learning framework for the synergistic combination of mathematical models and data. Following the physical laws of the control dynamics system, PINNs can deduce the solution of PDE and obtain the surrogate model. ([@weinan2018deep]) presented a deep Ritz method for the numerical solution of variational problems based on the Ritz method. ([@he2018relu]) theoretically analyzed the relationship between DNN and finite element method(FEM). They explored the ReLU DNN representation of a continuous piecewise linear basis function in the finite element method. ([@long2019pde]) proposed PDE-Net to predict the dynamics of complex systems. The underlying PDEs can be discovered from the observation data by establishing the connections between convolution kernels in CNNs and differential operators. Based on the integral form of the underlying dynamical system, ([@qin2019data]) considered ResNet block as a one-step method and recurrent ResNet and recursive ResNet as multi-step methods. ([@wu2020data]) approximated the evolution operator by a residual network to solve and recover unknown time-dependent PDEs. ([@raissi2018multistep]) blended the multi-step schemes with deep neural networks to identify and forecast nonlinear dynamical systems from data. ([@regazzoni2019machine]) proposed neural networks based Model Order Reduction technique to solve dynamical systems arising from differential equations. ([@wang2019learning]) used reinforcement learning to empower Weighted Essentially Non-Oscillatory Schemes(WENO) for solving 1D scalar conservation laws.
It is well known that the forward Euler method is very easy to implement but it can’t give accurate solutions. The main reason is that the Euler method has only one order approximation accuracy, which requires a very small step size for any meaningful result. This makes the Euler method rarely used in practical applications and motivates us to propose a new Euler method combined with DNNs. We call the new method as deep Euler method (DEM). DEM only has the most general structure of a fully connected neural network, without any special designs in its structure, such as residual connections. As with some other deep neural network models, DEM also learns its representation using supervised pre-training. After the neural network gets trained satisfactorily, we post-process it to predict the solution of the ODE. The key difference is that in DEM, we explicitly capture information of the local truncation error of the Euler method instead of directly approaching the solution of the ODE. DEM has achieved state-of-the-art performance in solving ODEs, which is much better than the conventional numerical method, especially than the classical Euler method. This success can be attributed to the ability of the deep neural network in learning very strong hierarchical nonlinear representation. In particular, breakthroughs in supervised learning training are essential for deep neural networks to effectively and robustly predict.
The paper is organized as follows. We introduce the main idea of DEM in section 2 and give theoretical results of DEM in section 3. Based on DEM, we also derive other schemes of the single-step method for solving ODEs in section 4. In Section 5, numerical examples are given to demonstrate the capability and effectiveness of DEM. Finally, we conclude the paper in section 6.
Deep Euler Method
=================
Formulation
-----------
Considering the following initial value ordinary differential equation: $$\label{eq:1order_original_ODE}
\left\{
\begin{array}{l}
\frac{d y}{d x}=f(x, y), \quad x \in I = [a, b], \\ y(a)= c,
\end{array}
\right.$$ where the solution $y(x) : I \rightarrow \Omega \subset \mathbb{R}^n$ and $f$ satisfies the Lipschitz condition in $y$, i.e., $$\| f(x,y_1) -f(x,y_2)\| < L\|y_1-y_2\|.$$ We introduce the discretization mesh (or sampling points) in $x$, $$a = x_0 < x_1 < \cdots < x_{M} = b.$$ Let $h_{m} = x_{m+1} - x_{m}$ be the mesh size and $y_{m}$ be the numerical approximation of $y(x_{m})$. The forward Euler method for (\[eq:1order\_original\_ODE\]) is $$\left\{
\begin{array}{l}
y_{m} = y_{m-1} + h_{m-1} f(x_{m-1},y_{m-1}), \quad m=1,\cdots,M \\
y_0= c.
\end{array}
\right.$$ Note that in most cases, we always adopt the uniform mesh for the forward Euler method, $h = h_m $ and $x_m = a + mh$, $m=0,1,\cdots,M-1$. The local truncation error and the global error are defined as $$\label{eq:local_truncation_error_define}
R_m = y(x_{m+1}) - y(x_m) - hf(x_m,y(x_m)) = \int_{x_m}^{x_{m+1}}f(x,y(s)) ds - h f(x_m,y(x_m)).$$ and $$e = |y(x_m) - y_m |,$$ respectively. It is well known that $R_m = \mathcal O (h^2)$ and $e = \mathcal O (h)$ ([@Stig2008book]). To obtain higher accuracy than the Euler method, a direct scheme is to separate a part of $R_m$ to improve the Euler step $y_{m+1}$, so as to reduce the local truncation error and the global error of Euler method. To this end, we introduce a feedforward neural network in DEM that infers the update of an Euler step. As universal approximators, multilayer fully connected feedforward neural networks can approximate any continuous function arbitrarily ([@leshno1993multilayer]). From (\[eq:local\_truncation\_error\_define\]), we could consider $R_m$ as a continuous function of variables $x_m,x_{m+1},y_m$. Thus, we utilize the fully connected neural network trained with enough measured data to approximate $\frac{1}{h_m^2}R_m$.
Let $\mathcal{N}(x_i,x_j,y_i;\theta) : \mathbb{R}^{n+2} \rightarrow \mathbb{R}^{n}$ be the nonlinear operator defined by a multilayer fully connected neural network. The parameter $\theta$ includes all the weights and the biases in the neural network. DEM for (\[eq:1order\_original\_ODE\]) can be written as $$\label{eq:ODE_DEM_formula}
\left\{
\begin{array}{l}
y_{m+1} = y_m + h_m f(x_m,y_m) + h_m^2 \mathcal{N}(x_m,x_{m+1},y_m;\theta), \quad m=0,\cdots, M-1,\ \\
y_0 = c.
\end{array}
\right.$$ Formula (\[eq:ODE\_DEM\_formula\]) consists of Euler approximation and neural network approximation. The first part makes full use of the information of $f$ to express the linearity of ODE. The latter corrects the results of Euler approximation to obtain higher accuracy and express nonlinear features. Abstractly, ${\mathcal {N}}$ can be thought of as a parametric function that learns how to represent the local truncation error so that their most salient characteristics can be reconstructed from its inputs and outputs. The output of ${\mathcal{N}}(x_{m},x_{m+1},y_{m};\theta)$ contains all features extracted in training to update the formula of the Euler method. Compared with using a neural network to approximate the solution of ODE directly, DEM separates the nonlinear part from the numerical scheme and then makes full use of neural networks to approximate the local truncation error of Euler method. Moreover, ${\mathcal {N}}$ has the same function as the nonlinear denoising process. This provides a very powerful and flexible method for solving ODEs because we can impose high order error correction and reduce the constrain of the step size $h$ in the Euler method. Hence, DEM can either improve the accuracy of the Euler method or speed up the computations of ODEs.
There are underlying principles for designing neural network architecture. In fact, we have another design of neural network $\mathcal{N}(x_i,x_j;\theta) : \mathbb{R}^{2} \rightarrow \mathbb{R}^{n}$. The output of the neural network is still an approximation of the local truncation error, while the input only has $x_{m}$ and $x_{m+1}$. In this case, if $n >> 2$, the neural network becomes very difficult to train. Because the dimension of the output is much larger than that of the input, it is almost impossible for the neural network to predict the sophisticated target with such few features.
Details of DEM
--------------
DEM is only a standard multilayer fully connected neural network, without any special designs in its structure, such as residual connections. With the input ${\bf{x}} = (x_i,x_j,y_i) \in \mathbb{R}^{n+2}$, the neural network in DEM can be written as $$\mathcal{N} ({\bf{x}} ;\theta) = L_{K} \circ \sigma \circ L_{K-1} \cdots \sigma \circ L_{1} ({\bf{x}}).$$ The nonlinear activation function $\sigma(t) = \max\{0,t\}$ is rectified linear units (ReLU) function. The $k$-th hidden layer has the following form $$L_{k}(z) = \boldsymbol{W}_{k} z + \boldsymbol{b}_{k}, \quad 1\leq k\leq K,$$ where the weight matrix $\boldsymbol{W}_{k} \in \mathbb{R}^{p_k\times p_{k-1}}$, the bias $\boldsymbol{b}_{k}\in\mathbb{R}^{p_k}$, $p_{k}$ is the number of neurons in the $k$-th layer.
We assume that the measurement data is contaminated by noise, so that the training dataset $\mathbf D = \{(x_j,z_j)\}_{j=1}^{N}$ has the form $z_j = y(x_j) + \delta_j$ and satisfies $$\frac{1}{N} \sum_{j=0}^{N} \delta_j^2 \leq \delta^2,$$ where the scalar $\delta$ is called the noise level.
For any pair of measurements $\{(x_i,z_i), (x_j,z_j)\}$ ($x_i<x_j$), we introduce the local truncation error function $$\label{eq:R_define}
R(x_i,x_j,z_i,z_j) = \frac{1}{(\Delta x)^2} \left[ z_j - z_i - \Delta x f(x_i,z_i) \right],$$ where $\Delta x = x_j - x_i$. With the following supervised loss $$\label{eq:loss_define}
J(\theta) = \frac{2}{N(N-1)} \sum_{1\leq i,j \leq N} \| {\mathcal{N}}(x_i,x_j,z_i;\theta) - R(x_i,x_j,z_i,z_j) \|_{L^1},$$ DEM learns to approximate the local truncation error of the Euler method. The coefficient comes from $ {\rm C}_N^2 = \frac{N(N-1)}{2}$ the number of pairs in dataset $D$.
Note that for any input pair $\{x_i, x_j\}$, each of the training captures the features in the local truncation error, which have close relations with $f$. Once features extractors corresponding to all pairs are trained and the strong hierarchical non-linear representations are generated, any new $y(x)$ ($x\neq x_{i}\in D$) is then represented by (\[eq:ODE\_DEM\_formula\]). On the other hand, DEM is mesh-free because all training data can be generated randomly and not necessary to locate at mesh points. Moreover, recalling the local truncation error of the Euler method is proportional to the square of the step size, we have $\frac{1}{h^2}R_m = \mathcal{O}(1)$. Hence, the neural network of DEM approaches a non-linear continuous function of $\mathcal{O}(1)$, which is much easier than directly approximating the solution of the ODE. This makes DEM easier to train and requires fewer data in training.
Theoretical Analysis
====================
Error Bound
-----------
\[lemma\] Assume that the trained neural network ${\cal N}$ satisfies $$\left|\mathcal{N}(x_m,x_{m+1},z_m;\theta)-R(x_m,x_{m+1},z_m,z_{m+1})\right|<\mathcal{O}(\eta).$$ If $\delta<\eta$ and $h > \sqrt{\frac{\delta}{\eta}}$, then $$\left|{\mathcal{N}}(x_{m},x_{m+1},y(x_m);\theta) - \dfrac{1}{h^{2}}R_{m}\right| < \mathcal{O}(\eta).$$
From Lemma 4 in [@xu2012robustness], we have the conclusion that the neural network in DEM is Lipschitz continuous. That is, for any $\mathbf{x}_1,\mathbf{x}_2$, $$|\mathcal{N}(\mathbf{x}_1;\theta)- \mathcal{N}(\mathbf{x}_2;\theta)| \leq L_{\mathcal{N}} \|\mathbf{x}_1-\mathbf{x}_2\| ,$$ where $L_{\mathcal{N}}=\alpha^K \beta^K $, $\alpha = \max_{1 \leq k \leq K} \|W_k \|_{\infty}$, $\beta = \max _{a, b \in \mathbb{R}, a \neq b} \frac{|\sigma(a)-\sigma(b)|}{|a-b|}$, $K$ is the number of layers of the neural network. From $$\begin{aligned}
\left| \mathcal{N}(x_m,x_{m+1},y(x_m);\theta)-\mathcal{N}(x_m,x_{m+1},z_m;\theta) \right|
& \leq & L_{\mathcal{N}}|y(x_m)-z_m| \\
& \leq & C L_{\mathcal{N}} \delta < \mathcal{O}(\eta), \quad (C\ \text{is\ a\ constant})
\end{aligned}$$ and $$\begin{aligned}
&&\left|R(x_m,x_{m+1},z_m,z_{m+1}) - \frac{1}{h^2}R_m\right| \\
&=& \frac{1}{h^2}\left|[z_{m+1}-y(x_{m+1})]-[z_{m}-y(x_{m})]- \dfrac{}{} h[f(x_m,z_m)-f(x_m,y(x_m))] \right|\\
& \leq & \frac{2C\delta}{h^2}+ \frac{L\delta}{h}\\
& \leq & \mathcal{O}(\eta),
\end{aligned}$$ we have $$\begin{aligned}
\left|{\mathcal{N}}(x_{m},x_{m+1},y(x_m);\theta) - \dfrac{1}{h^{2}}R_{m}\right| &<&\left| \mathcal{N}(x_m,x_{m+1},y(x_m);\theta)-\mathcal{N}(x_m,x_{m+1},z_m;\theta) \right|\\
&+& \left|R(x_m,x_{m+1},z_m,z_{m+1}) - \frac{1}{h^2}R_m\right|\\
&+& \left|\mathcal{N}(x_m,x_{m+1},z_m;\theta)-R(x_m,x_{m+1},z_m,z_{m+1})\right|\\
&\leq & \mathcal{O}(\eta)
\end{aligned}$$
For each pair of measurements $\{(x_i,z_i),(x_j,z_j)\}$, we use (\[eq:R\_define\]) to construct the training samples of the neural network. In the proof of Lemma \[lemma\], we have known that $\left|R(x_m,x_{m+1},z_m,z_{m+1}) - \frac{1}{h^2}R_m\right| $ is smaller than a quantity which contains a factor $\frac{1}{h}$. This indicates the big $h$ is a good choice. Therefore, we will only select the measurement pair with the large $h=x_j-x_i$ to construct training samples. 0.5cm
\[thm:local\_error\_DEM\] Under the assumptions of Lemma \[lemma\], the local truncation error of DEM is $\mathcal O(\eta h^2)$ and the global truncation error is $\mathcal O(\eta h)$.
From (\[eq:ODE\_DEM\_formula\]), the local truncation error (LTE) of DEM is $$\begin{aligned}
LTE &=& \left| y(x_{m+1}) - y(x_m) - hf(x_m,y(x_m)) - h^{2}{\mathcal{N}}(x_{m},x_{m+1},y(x_{m});\theta) \right| \\
&\leq & h^2 \left|{\mathcal{N}}(x_{m},x_{m+1},y(x_{m});\theta) - \dfrac{1}{h^{2}}R_{m}\right| \\
&<& \mathcal{O}(\eta h^2).\end{aligned}$$ Hence, we can also conclude that the global truncation error is $\mathcal O(\eta h)$.
Compared with the Euler method, the errors of DEM are reduced by $\eta$ times. The solution with high accuracy can be obtained. Besides, the size constrains of $h$ in the Euler method can be relaxed to speed up the computation of the solutions. For example, if the global error should be $O(10^{-6})$, the step size in the Euler method is at most $10^{-6}$. Starting from the initial $y(0)$, it takes $10^{6}$ Euler steps to get the solution $y(1)$. If $\eta = 10^{-4}$, the step size in DEM can be $10^{-2}$ and the number of steps can be reduced to $10^2$, which is much smaller than the Euler method.
Numerical stability
-------------------
Under the assumptions of Theorem \[thm:local\_error\_DEM\], DEM is stable.
For the initial values $y_0$ and $z_0$ ($y_0\neq z_0$), DEM generates two approaches $\{y_m\}$ and $\{z_m\}$ with $$\begin{array}{l}
{y_{m} = y_{m-1} + h_{m-1} f(x_{m-1},y_{m-1}) + h_{m-1}^2\mathcal{N}(x_{m-1},x_{m},y_{m-1};\theta)},\\
{z_{m} = z_{m-1} + h_{m-1} f(x_{m-1},z_{m-1}) + h_{m-1}^2\mathcal{N}(x_{m-1},x_{m},z_{m-1};\theta)}.\\
\end{array}$$ Since $\sum_{m=0}^{M-1} h_m = b-a$ and $| \mathcal{N}(x_{m-1},x_{m},y_{m-1};\theta) - \mathcal{N}(x_{m-1},x_{m},z_{m-1};\theta)| \leq L_{\mathcal{N}} |y_{m-1}-z_{m-1}|$, we have $$\begin{aligned}
|y_{m} - z_{m}| &\leq & |y_{m-1} - z_{m-1}| + h_{m-1}|f(x_{m-1},y_{m-1})-f(x_{m-1},z_{m-1})| \\
&& \qquad + h_{m-1}^2 L_{\mathcal{N}} |y_{m-1}-z_{m-1}|\\
&\leq & (1+h_{m-1}L+h_{m-1}^2L_{\mathcal{N}}) |y_{m-1} - z_{m-1}|\\
&\leq & \prod_{n=0}^{m-1} (1+h_{n}L+h_{n}^2L_{\mathcal{N}})|y_{0} - z_{0}|\\
&\leq & C|y(0)-z(0)|\end{aligned}$$ where $C$ is constant.
Considering the stiff equation $\frac{dy}{dx} = \lambda y$, where $\lambda<0$ and the initial value $y(a) = c$. The stability domain of DEM is $\{ h\in \mathbb{C} | \space |1+ h\lambda + h^2L_{\mathcal{N}}|\leq 1\}$, while the Euler method is $\{ h\in \mathbb{C} | \space |1+ h\lambda|\leq 1\}$. Although it can not be proved theoretically that the former must be larger than the latter, DEM can use a large step in numerical experiments. Under such a step size, the forward Euler method is certainly unstable. On the other hand side, a large stability domain can be obtained by adjusting $L_{\mathcal{N}}$. Recall that $L_{\mathcal{N}} = \alpha^k\beta^K$, where $\beta$ is determined by the activation function. If ReLU activation function is adopted then $\beta=1$. We can change the norm of weight matrix $\alpha$ by using the techniques of weight clipping ([@salimans2016weight]) and weight normalization ([@arjovsky2017wasserstein]), to adjust $L_{\mathcal{N}}$. For example, when $\lambda=-5 $ and $L_{\mathcal{N}} = 6$, The stability domain of DEM is $0<h\leq \frac{5}{6}$, while the Euler method is $0<h\leq \frac{2}{5}$. Hence we can use a larger step size to solve the equation than the Euler method.
Single Step methods
===================
Based on the idea of approximating the local truncation error with a deep neural network, DEM could be generalized to other linear single-step methods. For instance, Heun’s method $$y_{m+1}=y_{m}+\frac{h}{2}\left[f\left(x_{m}, y_{m}\right)+f\left(x_{m+1}, y_{m}+h f\left(x_{m}, y_{m}\right)\right)\right]$$ is a second-order Runge-Kutta method. We can also add a deep neural network in it and get $$\label{eq:Heun_method_NN_ODE}
y_{m+1}=y_{m}+\frac{h}{2}\left[f\left(x_{m}, y_{m}\right)+f\left(x_{m+1}, y_{m}+h f\left(x_{m}, y_{m}\right)\right)\right]
+ h_m^3{\cal N}(x_m,x_{m+1},y_m;\theta).$$ More generally, a $p$ order single-step method of (1) can be written as: $$\label{eq:p_order_single_step_ODE}
\left\{
\begin{array}{l}{
y_{m+1} = y_m + h \psi(x_m,y_m,h) }, \\
{y_0 = c}.
\end{array}
\right.$$ The local truncation error is $$R_m = y_{m+1} - y_m - h\psi(x_m,y_m,h) = \mathcal{O}(h^{p+1}).$$ The method (\[eq:p\_order\_single\_step\_ODE\]) can also be modified as $$\label{eq:p_order_NN_ODE}
\left\{
\begin{array}{l}
y_{m+1} = y_m + h \psi(x_m,y_m,h) + h^{p+1}{\cal N}(x_m,x_{m+1},y_m;\theta), \\
{y_0 = c}.
\end{array}
\right.$$ For (\[eq:Heun\_method\_NN\_ODE\]) and (\[eq:p\_order\_NN\_ODE\]), the local truncation errors are $\mathcal{O}(\eta h^3)$ and $\mathcal{O}(\eta h^{p+1})$, respectively. The corresponding global truncation errors are $\mathcal{O}(\eta h^2)$ and $\mathcal{O}(\eta h^p)$.
Numerical Example
=================
Example 1
---------
Considering the following initial value problem: $$\label{eq:the_first_example}
\left\{
\begin{array}{l}
{\frac{dy}{dx} = \frac{3}{2}\frac{y}{x+1}+\sqrt{x+1}}, \space x\in[0,10]\\
{y(0) = 0}
\end{array}
\right.$$ where the exact solution is $y =(x+1)^{3/2} \log(x+1)$.
At first, we highlight the performance of DEM with different step sizes with noise-free measured data. We generate 200 random noise-free measured data $\left\{ (t_i, y(t_i) )\right\}_{i=1}^{200}$, by sampling from a uniform distribution ${\cal F}_{t} = U(0,5)$, then we train the deep neural network ${\cal N}$ by minimizing the loss function of (\[eq:loss\_define\]). All norms used in this paper are $\mathcal{L}_1$ norm. The neural network, with $8$ layers, $80$ neurons and the ReLU activation function in each layer, is trained for 50 epochs, optimized with Adam. All the learning rate in this paper is set to be $5\times 10^{-3}$. The same neural network architecture is used in Deep Heun’s method (\[eq:Heun\_method\_NN\_ODE\]) for comparison. Figure \[fig:1\] shows the evolution of the trained ${\cal N}(x_{m}, x_{m+1},y_{m};\theta)$ in DEM and the local truncation error function $R(x_{m}, x_{m+1},y_{m},y_{m+1})$ in the Euler method. The four different step sizes, that is $h = 0.01,$ $0.1$, $1.0$ and $2.0$ are displayed. Since it is trained in $(0,5)$, ${\cal N}(x_{m},x_{m+1},y_{m};\theta)$ almost coincides with $R(x_{m},x_{m+1},y_{m},y_{m+1})$ in $(0,5)$.
Figure \[fig:2\] shows the exact solution and four approximations of (\[eq:the\_first\_example\]). The four different step sizes are also displayed. It is observed that only in the small step $h$ can the Euler method and the Huen obtain a more accurate approximation of the solution. We also note the fact that ${\cal N}$ is only trained in $(0,5)$. However, in $(5,10)$, DEM and DHM can get the accurate approximation of the solution even for the bigger step size $h=2.0$. This indicates the efficient prediction of the deep neural network.
In Table \[table:the\_first\_example\], we discuss the results of the comparison among four methods for the different step sizes. The Euler method, the Heun’s method, DEM and DHM are adopted for solving (\[eq:the\_first\_example\]). The first four columns of Table \[table:the\_first\_example\] show the prediction errors between the exact solution and the approximation in $L_1$ norm, i.e. $e= \max_{m}|y(x_m) - y_m|$. Since the global truncation error of Deep Euler Method and Deep Heun’s method are $\mathcal{O}(\eta h)$ and $\mathcal{O}(\eta h^2)$. That is the reason that when $h\geq 1$, DEM and DHM get more accuracy than the Euler method and the Heun method. Our goal is to get the estimates of $\eta$. In view of Lemma \[lemma\], $\left| {\cal N}(x_{m},x_{m+1},z_{m};\theta) - \dfrac{1}{h^{2}}R_{m} \right| \approx \left| {\cal N}(x_{m},x_{m+1},z_{m};\theta) - R(x_{m},x_{m+1},z_{m},z_{m+1}) \right| = \mathcal{O}(\eta)$. In the column of $\varepsilon_{mean}$, we present the mean of the difference between ${\cal N}$ and $R$, which is $\varepsilon_{mean}=\dfrac{1}{M}\sum_{m} \left| {\cal N} - R \right| = \mathcal{O}(\eta)$. In the last column, we present the results of DEM (the fourth column) divided by the result of the Euler method (the second column), i.e., ${e_{DEM}}/{e_{Euler}} = \mathcal{O}(\eta)$. From the last two columns, we can conclude that $\eta = {\cal O} (0.001)$, which also indicates the efficiency of DEM.
Table \[table:Example1\_layers\_neurals\_points\] shows $\varepsilon_{mean}$ for different network architectures (the number of hidden layers and neurons) and the different number of random measured points. We evolution the neural networks with $h=0.1$. To avoid uncertainty during the training process, we simulate each case ten times and take the average value of them. It can be observed that the prediction accuracy increase with the number of measured points. If the networks with too small layers and neurons per layer (such as $2$ layers and $20$ neurons per layer), it is not suitable in the case of a small number of measured data since it has a high bias in the training region $[0,5]$ and a high variance in the testing region $(5,10]$. If the networks with the more layers and neurons per layer (such as $16$ layers and $160$ neurons per layer), it may be overfitted in the case of the small number of measured data since it has low bias and high variance. More measured data can reduce the variance. We choose the networks with $8$ layers and $80$ neurons per layer for all experiments. Because no matter in a small amount or a large number of measured data, it has low variance and low bias than others.
[0.48]{} ![The evolution of the trained neural network ${\cal N}(x_{m}, x_{m+1},y_{m};\theta)$ in DEM and of the local truncation error function $R(x_{m}, x_{m+1},y_{m},y_{m+1})$ in the Euler method. The four different step sizes are $0.01$, $0.1$, $1.0$ and $2.0$. The red line is the result of ${\cal N}$. The blue line is the result of $R$.[]{data-label="fig:1"}](./exp1_ep_delta_0_01.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The evolution of the trained neural network ${\cal N}(x_{m}, x_{m+1},y_{m};\theta)$ in DEM and of the local truncation error function $R(x_{m}, x_{m+1},y_{m},y_{m+1})$ in the Euler method. The four different step sizes are $0.01$, $0.1$, $1.0$ and $2.0$. The red line is the result of ${\cal N}$. The blue line is the result of $R$.[]{data-label="fig:1"}](./exp1_ep_delta_0_1.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The evolution of the trained neural network ${\cal N}(x_{m}, x_{m+1},y_{m};\theta)$ in DEM and of the local truncation error function $R(x_{m}, x_{m+1},y_{m},y_{m+1})$ in the Euler method. The four different step sizes are $0.01$, $0.1$, $1.0$ and $2.0$. The red line is the result of ${\cal N}$. The blue line is the result of $R$.[]{data-label="fig:1"}](./exp1_ep_delta_1_0.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The evolution of the trained neural network ${\cal N}(x_{m}, x_{m+1},y_{m};\theta)$ in DEM and of the local truncation error function $R(x_{m}, x_{m+1},y_{m},y_{m+1})$ in the Euler method. The four different step sizes are $0.01$, $0.1$, $1.0$ and $2.0$. The red line is the result of ${\cal N}$. The blue line is the result of $R$.[]{data-label="fig:1"}](./exp1_ep_delta_2_0.jpg "fig:"){width="\linewidth"}
step size Euler method Heun’s method DEM DHM $\varepsilon_{mean}$ ${e_{DEM}}/{e_{Euler}}$
----------- -------------- --------------- ------------ -------------- ---------------------- -------------------------
0.01 [0.42]{} [0.0017]{} [0.0014]{} [0.000053]{} [0.0086]{} [0.0033]{}
[0.1]{} [4.05]{} [0.15]{} [0.013]{} [0.0051]{} [0.0089]{} [0.0032]{}
[1]{} [28.42]{} [8.10]{} [0.073]{} [0.32]{} [0.012]{} [0.0026]{}
[2]{} [43.16]{} [18.78]{} [0.083]{} [1.03]{} [0.016]{} [0.0019]{}
: The results of the comparison among four methods for different step sizes. Prediction errors between the exact solution and the approximation in $L_1$ norm are listed, i.e., $e= \max_{m}|y(x_m) - y_m|$. The column of $\varepsilon_{mean}$ is $\sum_{m} \left| {\cal N} - R \right|/M$, where $M$ is the number of the steps. The last column is the result of DEM (the fourth column) divided by the result of the Euler method (the second column). []{data-label="table:the_first_example"}
[0.48]{} ![The exact solution and four approximations of (\[eq:the\_first\_example\]). The four different step sizes are $0.01$, $0.1$, $1.0$ and $2.0$. The green plus is the result of the Euler method. The yellow plus is the result of the Heun method. The blue circle is the result of DEM. The black circle is the result of DHM. []{data-label="fig:2"}](./exp1_y_delta_0_01.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The exact solution and four approximations of (\[eq:the\_first\_example\]). The four different step sizes are $0.01$, $0.1$, $1.0$ and $2.0$. The green plus is the result of the Euler method. The yellow plus is the result of the Heun method. The blue circle is the result of DEM. The black circle is the result of DHM. []{data-label="fig:2"}](./exp1_y_delta_0_1.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The exact solution and four approximations of (\[eq:the\_first\_example\]). The four different step sizes are $0.01$, $0.1$, $1.0$ and $2.0$. The green plus is the result of the Euler method. The yellow plus is the result of the Heun method. The blue circle is the result of DEM. The black circle is the result of DHM. []{data-label="fig:2"}](./exp1_y_delta_1_0.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The exact solution and four approximations of (\[eq:the\_first\_example\]). The four different step sizes are $0.01$, $0.1$, $1.0$ and $2.0$. The green plus is the result of the Euler method. The yellow plus is the result of the Heun method. The blue circle is the result of DEM. The black circle is the result of DHM. []{data-label="fig:2"}](./exp1_y_delta_2_0.jpg "fig:"){width="\linewidth"}
-------- ------------ ---------- ------------ ----------- ------------ ----------- ------------ -----------
points
10 [0.21]{} [0.69]{} [0.067]{} [0.36]{} [0.033]{} [0.11]{} [0.071]{} [0.17]{}
25 [0.20]{} [0.81]{} [0.03]{} [0.16]{} [0.014]{} [0.061]{} [0.075]{} [0.16]{}
50 [0.081]{} [0.41]{} [0.024]{} [0.36]{} [0.022]{} [0.073]{} [0.049]{} [0.12]{}
100 [0.017]{} [0.21]{} [0.0093]{} [0.14]{} [0.011]{} [0.045]{} [0.014]{} [0.052]{}
200 [0.0096]{} [0.28]{} [0.0056]{} [0.080]{} [0.0093]{} [0.030]{} [0.0084]{} [0.035]{}
500 [0.0066]{} [0.22]{} [0.0024]{} [0.072]{} [0.0028]{} [0.048]{} [0.0039]{} [0.048]{}
-------- ------------ ---------- ------------ ----------- ------------ ----------- ------------ -----------
: The results of $\varepsilon_{mean}$, i.e., the average error of between $\mathcal{N}$ and $R$, for the different number of hidden layers and neurons per layer, as well as the different number of measured points. The number on the left is the error in the training region $[0, 5]$, and the error in the test region $(5, 10]$ is on the right. All the step size is $h=0.1$. []{data-label="table:Example1_layers_neurals_points"}
[0.48]{} ![ For comparison, the DEMs and the local truncation error function $R$ are listed together. Each ${\cal N}$ of DEM is trained for one case of noise levels ($\delta = 0\%, 1\%,5\%,10\%$). Four subfigures show the evolutions of the four ${\cal N}(x_{m}, x_{m+1},y_{m};\theta)$ and $R$ for $h=0.01$, $0.1$, $1.0$ and $2.0$, respectively. []{data-label="fig:noise"}](./noise_ep_delta_0_01.jpg "fig:"){width="\linewidth"}
[0.48]{} ![ For comparison, the DEMs and the local truncation error function $R$ are listed together. Each ${\cal N}$ of DEM is trained for one case of noise levels ($\delta = 0\%, 1\%,5\%,10\%$). Four subfigures show the evolutions of the four ${\cal N}(x_{m}, x_{m+1},y_{m};\theta)$ and $R$ for $h=0.01$, $0.1$, $1.0$ and $2.0$, respectively. []{data-label="fig:noise"}](./noise_ep_delta_0_1.jpg "fig:"){width="\linewidth"}
[0.48]{} ![ For comparison, the DEMs and the local truncation error function $R$ are listed together. Each ${\cal N}$ of DEM is trained for one case of noise levels ($\delta = 0\%, 1\%,5\%,10\%$). Four subfigures show the evolutions of the four ${\cal N}(x_{m}, x_{m+1},y_{m};\theta)$ and $R$ for $h=0.01$, $0.1$, $1.0$ and $2.0$, respectively. []{data-label="fig:noise"}](./noise_ep_delta_1_0.jpg "fig:"){width="\linewidth"}
[0.48]{} ![ For comparison, the DEMs and the local truncation error function $R$ are listed together. Each ${\cal N}$ of DEM is trained for one case of noise levels ($\delta = 0\%, 1\%,5\%,10\%$). Four subfigures show the evolutions of the four ${\cal N}(x_{m}, x_{m+1},y_{m};\theta)$ and $R$ for $h=0.01$, $0.1$, $1.0$ and $2.0$, respectively. []{data-label="fig:noise"}](./noise_ep_delta_2_0.jpg "fig:"){width="\linewidth"}
Figure \[fig:noise\] shows the DEMs and the local truncation error function $R$ together for comparison. Each ${\cal N}$ of DEM is trained for one case of noise levels ($\delta = 0\%, 1\%,5\%,10\%$). The four different step sizes, that is $h= 0.01$, $0.1$, $1.0$ are displayed. It can be observed from the subfigures that the change of step size has little effect on the result. In fact, from Table \[table:noise\], we also noted that. When the step size $h$ changes from $0.01$ to $2.0$, $\varepsilon_{mean}$ under various noise levels ($\delta = 0, 1 \%, 5 \%, 10 \%$) are almost the same. However, obviously, $e_{DEM}$ is the smallest in the case of the smallest $h$ and the lest noise level (noise-free).
------ ------- ------ ------ ------ ------- ------- ------ ------
0% 1% 5% 10% 0% 1% 5% 10%
0.01 0.005 0.02 0.04 0.07 0.001 0.001 0.01 0.02
0.1 0.005 0.02 0.03 0.07 0.01 0.01 0.01 0.02
0.5 0.005 0.02 0.03 0.07 0.06 0.09 0.35 0.58
1.0 0.006 0.03 0.03 0.07 0.12 0.27 0.50 0.71
2.0 0.01 0.03 0.02 0.07 0.24 0.55 0.45 0.50
------ ------- ------ ------ ------ ------- ------- ------ ------
: The performance of DEM for the different noise levels and the corresponding numerical solutions. []{data-label="table:noise"}
Example 2
---------
We now consider a system of first-order nonlinear differential equations, the Lotka-Volterra equation ([@lotka1925elements]), $$\label{eq:Example2_Lotka_Volterra}
\left\{
\begin{aligned}
\frac{dy_1}{{dx}} &=\alpha y_1- \beta {y_1y_2}, \\
\frac{{dy_2}}{{dx}}& = -\gamma y_2+ \delta {y_1y_2}.
\end{aligned}
\right.$$ This equation describes the dynamics of the populations of two species, one as a predator and the other as prey. In (\[eq:Example2\_Lotka\_Volterra\]), $y_1$ and $y_2$ are the number of prey and predator, respectively. Let $\boldsymbol{y} = [y_1,y_2]^T$, $x $ represent time and $\alpha,\beta,\gamma,\delta$ be the parameters describing the relationship of two species. In this work, we take $\alpha = \beta = \gamma = \delta = 1 $, and the initial conditions $\boldsymbol{y} = [y_1(0),y_2(0)]^T=[2,1]^T.$ For comparison, the exact solution is gotten by the numerical method of $RK45$ in scipy.integrate [@2020SciPy-NMeth] with $10^{-6}$ relative tolerances. We sample 1000 random points from a uniform distribution $\mathcal{F}_t = U(0,15)$ to construct the noise-free training dataset $\{(t_i,y(t_i)\}_1^{1000}$. The deep neural network with $8$ hidden layers and $80$ neurons per layer is trained with super parameters the same as example 1.
[0.48]{} ![The evolution of the trained neural networks $\mathcal{N}(x_m,x_{m+1},\boldsymbol{y_m})$ in DEM and the local truncation error function $R(x_m,x_{m+1},\boldsymbol{y_m},\boldsymbol{y_{m+1}})$ for different step sizes $h=0.01,0.1,0.5,1.0$ []{data-label="fig:4"}](./exp_lk_ep_delta_0_01.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The evolution of the trained neural networks $\mathcal{N}(x_m,x_{m+1},\boldsymbol{y_m})$ in DEM and the local truncation error function $R(x_m,x_{m+1},\boldsymbol{y_m},\boldsymbol{y_{m+1}})$ for different step sizes $h=0.01,0.1,0.5,1.0$ []{data-label="fig:4"}](./exp_lk_ep_delta_0_1.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The evolution of the trained neural networks $\mathcal{N}(x_m,x_{m+1},\boldsymbol{y_m})$ in DEM and the local truncation error function $R(x_m,x_{m+1},\boldsymbol{y_m},\boldsymbol{y_{m+1}})$ for different step sizes $h=0.01,0.1,0.5,1.0$ []{data-label="fig:4"}](./exp_lk_ep_delta_0_5.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The evolution of the trained neural networks $\mathcal{N}(x_m,x_{m+1},\boldsymbol{y_m})$ in DEM and the local truncation error function $R(x_m,x_{m+1},\boldsymbol{y_m},\boldsymbol{y_{m+1}})$ for different step sizes $h=0.01,0.1,0.5,1.0$ []{data-label="fig:4"}](./exp_lk_ep_delta_1_0.jpg "fig:"){width="\linewidth"}
[0.4]{} ![ The evolutions of the Euler method, the Huen method and DEM are displayed for noise-free measured data. The four different step sizes ($h=0.01, 0.1, 0.5$ and $1.0$) are considered. []{data-label="fig:5"}](./exp_lk_y_delta_0_01_0_15.jpg "fig:"){width="\linewidth"}
[0.4]{} ![ The evolutions of the Euler method, the Huen method and DEM are displayed for noise-free measured data. The four different step sizes ($h=0.01, 0.1, 0.5$ and $1.0$) are considered. []{data-label="fig:5"}](./exp_lk_y_delta_0_01_15_25.jpg "fig:"){width="\linewidth"}
[0.4]{} ![ The evolutions of the Euler method, the Huen method and DEM are displayed for noise-free measured data. The four different step sizes ($h=0.01, 0.1, 0.5$ and $1.0$) are considered. []{data-label="fig:5"}](./exp_lk_y_delta_0_1_0_15.jpg "fig:"){width="\linewidth"}
[0.4]{} ![ The evolutions of the Euler method, the Huen method and DEM are displayed for noise-free measured data. The four different step sizes ($h=0.01, 0.1, 0.5$ and $1.0$) are considered. []{data-label="fig:5"}](./exp_lk_y_delta_0_1_15_25.jpg "fig:"){width="\linewidth"}
[0.4]{} ![ The evolutions of the Euler method, the Huen method and DEM are displayed for noise-free measured data. The four different step sizes ($h=0.01, 0.1, 0.5$ and $1.0$) are considered. []{data-label="fig:5"}](./exp_lk_y_delta_0_5_0_15.jpg "fig:"){width="\linewidth"}
[0.4]{} ![ The evolutions of the Euler method, the Huen method and DEM are displayed for noise-free measured data. The four different step sizes ($h=0.01, 0.1, 0.5$ and $1.0$) are considered. []{data-label="fig:5"}](./exp_lk_y_delta_0_5_15_25.jpg "fig:"){width="\linewidth"}
[0.4]{} ![ The evolutions of the Euler method, the Huen method and DEM are displayed for noise-free measured data. The four different step sizes ($h=0.01, 0.1, 0.5$ and $1.0$) are considered. []{data-label="fig:5"}](./exp_lk_y_delta_1_0_0_15.jpg "fig:"){width="\linewidth"}
[0.4]{} ![ The evolutions of the Euler method, the Huen method and DEM are displayed for noise-free measured data. The four different step sizes ($h=0.01, 0.1, 0.5$ and $1.0$) are considered. []{data-label="fig:5"}](./exp_lk_y_delta_1_0_15_25.jpg "fig:"){width="\linewidth"}
Figure \[fig:4\] shows the neural network ${\cal N}$ of DEM and the local truncation error function $R$ in both the training region $[0,15]$ and the testing region $(15,25]$. The four different cases of $h=0.01, 0.1, 0.5, 1.0$ are displayed. It shows that DEM can accurately approximate the solution in the whole region $(15,25]$ even for $h=1.0$. In Figure \[fig:5\], we show the evolutions of the Euler method, the Huen method and DEM for comparison. For the cases of $h=0.5$ and $h=1.0$, the results of the Euler method over $[0,15]$ are displayed in the left sub-figure. For the case of $h = 0.5$, the solutions of the Euler method diverge from the exact value very much near $x=15$. When $h = 1.0$, the solution $y_2$ of the Euler method is close to $4$, but the solution of $y_1$ is near $0$, both of which are far away from the exact value. The Huen method has similar defects. When $h=1.0$, the solutions of the Huen method are close to $30000$, which is far away from the exact value. It can be observed that no matter the small $h$ or the big $h$, the results of DEM and the curve of the exact solution almost coincide.
Example 3
---------
Considering the Kepler problem : $$\label{eq:Example3_Kepler_eq}
\left\{
\begin{aligned}
\frac{d y_{1}}{d x} &=y_{3}, \\
\frac{d y_{2}}{d x} &=y_{4}, \\
\frac{d y_{3}}{d x} &=-\frac{y_{1}}{\left(y_{1}^{2}+y_{2}^{2}\right)^{3 / 2}}, \\
\frac{d y_{4}}{d x} &=-\frac{y_{2}}{\left(y_{1}^{2}+y_{2}^{2}\right)^{3 / 2}}.
\end{aligned}
\right.$$
It describes the motion of the sun and a single planet which is a special case of the two-body problem. We denote the time by$x$. Let $(y_1(x), y_2(x)) $ be the positions of the planet in rectangular coordinates centered at the sun and $y_3(x), y_4(x)$ be the velocity components in the $y_1$ and $y_2$ directions. The initial value are $y(0)=[1,0,0,1]^T$, and the exact solution is $y(x)=[\cos (x), \sin (x),-\sin (x), \cos (x)]^{T}$.
Similar to example 2, we use the uniform distribution to generate $1000$ noise-free data points and select the same values of the super parameters as in Example 1. In Figure \[fig:6\], four different cases of $h$ are displayed. It can be observed that $\mathcal{N}(x_m,x_{m+1},\boldsymbol{y}_m)$ of DEM well approximates $R(x_m,x_{m+1},\boldsymbol{y}_m,\boldsymbol{y}_{m+1})$ for all of $h$. We can also note that the approximations of DEM (circles in the figure) almost coincides with the curve of the exact solution in Figure \[fig:7\], even for $h=1.0$.
Conclusion
==========
In this work, we proposed a Deep Euler Method with the idea of approximating the truncation error in the Euler method via deep learning. When deep neural network $\mathcal{N}$ is trained to approximate the local truncation error function with accuracy $\mathcal{O}(\eta)$, the global truncation error of DEM with the step size $h$ would be $\mathcal{O}(\eta h)$ while the Euler method is only $\mathcal{O}(h)$. Since $\eta$ can be small enough, it could achieve high accuracy solutions even with a big step size ($h\geq 1$). DEM significantly improves the accuracy of the Euler method and reduces the constrain of the step size in the Euler method. On the other hand, since the training objective function of the deep neural network in DEM is always $\mathcal{O}(1)$, the deep neural network can be easily trained and fast to converge, even if only the simplest architecture of the fully connected network and only a few training data are used. Moreover, DEM shows good robustness with the noise of the measured data.
[0.48]{} ![The evolutions of the neural network $\mathcal{N}(x_m,x_{m+1},\boldsymbol{y}_m)$ and the local truncation error function $R(x_m,x_{m+1},\boldsymbol{y}_m,\boldsymbol{y}_{m+1})$ of the equation (\[eq:Example3\_Kepler\_eq\]) for $h = 0.1,0.2,0.5,1.0$, respectively.[]{data-label="fig:6"}](./exp_3_ep_delta_0_1.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The evolutions of the neural network $\mathcal{N}(x_m,x_{m+1},\boldsymbol{y}_m)$ and the local truncation error function $R(x_m,x_{m+1},\boldsymbol{y}_m,\boldsymbol{y}_{m+1})$ of the equation (\[eq:Example3\_Kepler\_eq\]) for $h = 0.1,0.2,0.5,1.0$, respectively.[]{data-label="fig:6"}](./exp_3_ep_delta_0_2.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The evolutions of the neural network $\mathcal{N}(x_m,x_{m+1},\boldsymbol{y}_m)$ and the local truncation error function $R(x_m,x_{m+1},\boldsymbol{y}_m,\boldsymbol{y}_{m+1})$ of the equation (\[eq:Example3\_Kepler\_eq\]) for $h = 0.1,0.2,0.5,1.0$, respectively.[]{data-label="fig:6"}](./exp_3_ep_delta_0_5.jpg "fig:"){width="\linewidth"}
[0.48]{} ![The evolutions of the neural network $\mathcal{N}(x_m,x_{m+1},\boldsymbol{y}_m)$ and the local truncation error function $R(x_m,x_{m+1},\boldsymbol{y}_m,\boldsymbol{y}_{m+1})$ of the equation (\[eq:Example3\_Kepler\_eq\]) for $h = 0.1,0.2,0.5,1.0$, respectively.[]{data-label="fig:6"}](./exp_3_ep_delta_1_0.jpg "fig:"){width="\linewidth"}
[0.4]{} ![The exact solution and the approximation of DEM of the equation (\[eq:Example3\_Kepler\_eq\]) on region $(15,20]$ for $h=0.1, 0.2, 0.5, 1.0$, respectively.[]{data-label="fig:7"}](./exp_3_y12_delta_0_1_15_20.jpg "fig:"){width="\linewidth"}
[0.4]{} ![The exact solution and the approximation of DEM of the equation (\[eq:Example3\_Kepler\_eq\]) on region $(15,20]$ for $h=0.1, 0.2, 0.5, 1.0$, respectively.[]{data-label="fig:7"}](./exp_3_y34_delta_0_1_15_20.jpg "fig:"){width="\linewidth"}
[0.4]{} ![The exact solution and the approximation of DEM of the equation (\[eq:Example3\_Kepler\_eq\]) on region $(15,20]$ for $h=0.1, 0.2, 0.5, 1.0$, respectively.[]{data-label="fig:7"}](./exp_3_y12_delta_0_2_15_20.jpg "fig:"){width="\linewidth"}
[0.4]{} ![The exact solution and the approximation of DEM of the equation (\[eq:Example3\_Kepler\_eq\]) on region $(15,20]$ for $h=0.1, 0.2, 0.5, 1.0$, respectively.[]{data-label="fig:7"}](./exp_3_y34_delta_0_2_15_20.jpg "fig:"){width="\linewidth"}
[0.4]{} ![The exact solution and the approximation of DEM of the equation (\[eq:Example3\_Kepler\_eq\]) on region $(15,20]$ for $h=0.1, 0.2, 0.5, 1.0$, respectively.[]{data-label="fig:7"}](./exp_3_y12_delta_0_5_15_20.jpg "fig:"){width="\linewidth"}
[0.4]{} ![The exact solution and the approximation of DEM of the equation (\[eq:Example3\_Kepler\_eq\]) on region $(15,20]$ for $h=0.1, 0.2, 0.5, 1.0$, respectively.[]{data-label="fig:7"}](./exp_3_y34_delta_0_5_15_20.jpg "fig:"){width="\linewidth"}
[0.4]{} ![The exact solution and the approximation of DEM of the equation (\[eq:Example3\_Kepler\_eq\]) on region $(15,20]$ for $h=0.1, 0.2, 0.5, 1.0$, respectively.[]{data-label="fig:7"}](./exp_3_y12_delta_1_0_15_20.jpg "fig:"){width="\linewidth"}
[0.4]{} ![The exact solution and the approximation of DEM of the equation (\[eq:Example3\_Kepler\_eq\]) on region $(15,20]$ for $h=0.1, 0.2, 0.5, 1.0$, respectively.[]{data-label="fig:7"}](./exp_3_y34_delta_1_0_15_20.jpg "fig:"){width="\linewidth"}
[^1]:
|
---
address: |
$^{(a)}$ P.L. Kapitza Institute for Physical Problems, Kosygin Str. 2, 117334 Moscow, Russian Federation\
$^{(b)}$ Institut de Physique, Université de Neuchâtel, Rue A.L. Breguet 1, 2000 Neuchâtel, Switzerland
author:
- 'Maxim Yu. Kagan$^{(a)}$, Raymond Frésard$^{(b)}$, Massimiliano Capezzali$^{(b)}$ and Hans Beck$^{(b)}$'
title: |
One-Electron Spectral Functions of\
the Attractive Hubbard Model for Intermediate Coupling
---
(a) Introduction {#a-introduction .unnumbered}
================
The Hubbard model involving electrons on a lattice, subject to an attractive interaction when they are on the same site, is one of the simplest models for describing superconductivity. Despite of its simplicity, it has turned out to be very challenging for the theoreticians to give a simple description of its properties which is valid in the various regimes of coupling strength. In the weak coupling regime, the link with BCS theory of superconductivity has been done by Nozières and Schmitt-Rink [@nozi1]. At sufficiently low $T$, an instability of the Fermi sea towards superconductivity occurs. In three dimensions, the transition is essentially mean-field in character. In the opposite strong coupling limit ($|U|\rightarrow\infty$), the electrons form bound pairs which are immobile since they can only move via virtual ionization with an infinite energy barrier. However, for large but finite $U$, those bound pairs essentially behave like heavy hard core bosons (with an effective mass $m^{*}\sim m{U\over t}$) which are undergoing Bose-Einstein condensation at sufficiently low $T$. On the lattice, $T_{C}$ vanishes in the limit $|U|\rightarrow\infty$, while in the continuum limit, it remains finite [@Zwerger]. This difference is due to the absence of a pair hopping term when working on the lattice.\
In the intermediate coupling regime, the physics will be dominated by the interplay between the quasi-particles and the bound pairs, which may lead to non-trivial behavior.\
Even though it is still lacking a microscopical derivation, the model is interesting in its own right, since it allows for studying various routes leading to superconductivity. Since the interaction is local, it will be $s$-wave superconductivity, but the generalization to non-local interaction can be considered [@meintrup]. In the weak coupling regime, perturbation theory is expected to work, and this has been worked out by a series of authors [@galitski], some of them focusing on 2-d systems [@randeria]. Of special interest is the low-density regime where chances of obtaining meaningful results are better, since the ratio of the scattering length to the average inter-particle distance can be used as a small parameter. Unfortunately those calculations are quickly becoming very involved since the simplest conserving approximation is the self-consistent T-matrix approach [@fresard; @rodriguez; @haussmann]. Alternatively, Variational-Monte-Carlo (VMC) calculations, based on the Gutzwiller wave-function [@dent] and Quantum-Monte-Carlo (QMC) simulations have been performed [@singer]. These methods are providing results which then generate a need for a qualitative analytical understanding. To that aim simpler calculations based on Hubbard-Stratonovitch decoupling of the interaction [@morten], slave-boson mean-field calculations (see for instance Ref. [@bulka] and references therein), or on the moment calculation of the electronic spectral function have been performed [@micnas]. Unfortunately the latter does not account for the damping of the quasi-particles.\
The aim of this paper is to treat analytically the intermediate coupling regime, which is the most delicate. This allows us to give an analytical account of the results obtained with QMC simulations. We first review the self-consistent T-matrix approximation. As pointed out by several authors [@schafroth], the corresponding numerical calculations typically yield a superconducting instability at a finite $T$, even in two dimensions. This contradicts the Mermin-Wagner theorem. We then propose an alternative scheme which complies with this theorem. We then proceed to the calculation of the electronic structure.
(b) Theoretical framework {#b-theoretical-framework .unnumbered}
=========================
We study the Hubbard model on the square lattice : $$\label{ham}
H=\sum_{i,j}^{}{\sum_{\left\langle \sigma \right\rangle}^{}{t_{ij}
c^{\dagger}_{i,\sigma}c_{j,\sigma}}}+U\sum_{i}{}{n_{i,\uparrow}n_{i,\downarrow}}.$$ We consider an attractive interaction ($U<0$) in the intermediate coupling regime ($|U|{\ \lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }W$), $W$ being the band width. In 2 dimensions, any attractive potential has a bound state. In the case $|U|=W$, the binding energy $E_{b}$ has been found to be $E_{b}\approx
0.2W$ [@rodriguez], namely $E_{b}\ll W$. We are thus in a situation where bound pairs exist and have a strong influence on the physics via the splitting of the non-interacting band into 2 sub-bands. In this regime, the pairs are extended. They become purely local only in the $|U|=\infty$ limit since for any finite $U$ they can move via virtual ionization [@nozi1]. We also note that the BCS theory successfully describes the weak coupling regime. However there does not exist any analytical theory in the intermediate coupling regime, and most results are obtained out of numerical simulations [@randeria; @singer]. In the low-density regime, the self-consistent T-matrix approximation is expected to be exact and has been solved by a variety of authors [@fresard; @rodriguez; @haussmann]. Unfortunately, numerical difficulties prevented those authors from obtaining results for arbitrary $U$. We also note that the numerical solutions may lead to unphysical results such as a finite critical temperature for Bose condensation of the pairs in two dimensions, which is contradicting the Mermin-Wagner theorem. We believe (see below) that this is due to the use of an inappropriate expression for the particle density. That however does not discredit the scheme, and we are basing our approach on it. It amounts to solving $$\label{tfrai}
T(\vec{q},i\nu_{n})={-U\over 1+
U\chi(\vec{q},i\nu_{n})} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;$$ $$\label{chi}
\chi(\vec{q},i\nu_{n})=\beta^{-1}\sum_{\vec{p},i\omega_{n}}^{}{
G(\vec{p},i\omega_{n})G(\vec{q}-\vec{p},i\nu_{n}-i\omega_{n})} \;$$ $$\label{s1tot}
\Sigma(\vec{q},i\omega_{n})=-\beta^{-1}\sum_{\vec{p},i\nu_{n}}^{}{
T(\vec{p},i\nu_{n})G(\vec{p}-\vec{q},i\nu_{n}-i\omega_{n})}$$ $$\label{g1tot}
G(\vec{q},i\omega_{n})={1\over i\omega_{n}-t_{\vec{q}}+\mu-
\Sigma(\vec{q},i\omega_{n})}\, .\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;$$ Here, $\omega_{n}$ are Fermionic, and $\nu_{n}$ Bosonic Matsubara frequencies. This set of equations is valid [*above*]{} $T_{C}$, as no anomalous Green’s function enters. Otherwise one can resort to the scheme obtained by Pedersen [*et al.*]{}, by functional derivative techniques [@morten]. This approximation is conserving and it diagrammatically corresponds to summing up the dressed particle-particle ladder which includes the leading order in an expansion in $k_{F}a$ [@galitski; @haussmann]. Another important quantity is the two-particle Green’s function which is defined by : $$\begin{aligned}
\label{g2b}
G^{(2)}(\vec{q},i\nu_{n})=\int_{0}^{\beta}{e^{i\nu_{n}\tau}
\left\langle T_{\tau}\left[ Q(\vec{q},\tau)Q^{\dagger}(-\vec{q},0) \right] \right\rangle d\tau},\end{aligned}$$ where $T_{\tau}$ is the usual time-ordering operator and the operator $$\begin{aligned}
\label{creaop}
Q^{+}(\vec{q})={1\over N}\sum_{\vec{k}}^{}{c_{-\vec{k},\uparrow}^{\dagger}
c_{\vec{k}-\vec{q},\downarrow}^{\dagger}}\end{aligned}$$ creates a pair having (center-of-mass) wave vector $\vec{q}$. $G^{(2)}(\vec{q},i\nu_{n})$ is related to the T-matrix by : $$\begin{aligned}
\label{g2}
G^{(2)}(\vec{q},i\nu_{n})={U+T(\vec{q},i\nu_{n})\over U^{2}}.\end{aligned}$$ We calculate $G^{(2)}(\vec{q},i\nu_{n})$ by inserting the free-electron Green’s function into expression (\[chi\]) for $\chi(\vec{q},i\nu_{n})$. For simplicity, we approximate the density of states (DOS), $\rho(\epsilon)$, of the tight-binding band resulting from the Hamiltonian (\[ham\]) by the square DOS (i.e. $\rho(\epsilon)={1\over W}$ for $|\epsilon|\leq{W\over 2}$ and $\rho(\epsilon)=0$ otherwise).\
For small momenta, $G^{(2)}(\vec{q},i\nu_{n})$ is given by : $$\begin{aligned}
\label{gexp}
G^{(2)}(\vec{q},i\nu_{n})={1\over 2W\left(1-{q^{2}\over 16} \right)}\times
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \nonumber \\
{\ln{\left( {i\nu_{n}+\mu_{B}-|E_{b}|-2W+{q^{2}t\over 2}\over
i\nu_{n}+\mu_{B}-|E_{b}|-{q^{2}t\over 2}} \right)}\ln{\Phi}
\over
\ln{\Phi}-\left(1+{q^{2}\over 16} \right)
\ln{\left( {i\nu_{n}+\mu_{B}-|E_{b}|-2W+{q^{2}t\over 2}\over
i\nu_{n}+\mu_{B}-|E_{b}|-{q^{2}t\over 2}}\right)}},\end{aligned}$$ where $\mu_{B}=2\mu+W+|E_{b}|$, $\Phi={2W+|E_{b}|\over |E_{b}|}$ and $|E_{b}|$ is the binding energy of a pair. The binding energy is obtained as a solution of : $$\label{bind}
-{1\over U}=\chi(\vec{q}=\vec{0}, \omega=E_{b})|_{\mu=-{W\over 2}},$$ which yields $$\label{bind2}
|E_{b}|=2W\left( {1\over e^{{-2W\over U}}-1} \right).$$ The form (\[gexp\]) has the correct behavior for $\nu_{n}$ going to infinity in the low density regime, i.e. $G^{(2)}(\vec{q},i\nu_{n})\rightarrow{1\over i\nu_{n}}$.\
The spectrum of $G^{(2)}(\vec{q},i\nu_{n})$ presents two features : (i) a sharp quasi-particle peak, which can be found by expanding (\[gexp\]) with respect to $i\nu_{n}+\mu_{B}-{q^{2}t\over 2}$; (ii) a continuous spectrum which extends over energies above the one of the quasi-particle. Correspondingly, the lowest order form of the T-matrix, valid for small wave vector and frequency is given by : $$\label{t0}
T_{0}(\vec{q},i\nu_{n})={-|E_{b}|^{2}\Phi\over i\nu_{n}-{\vec{q}\,^{2}\over
4m_{0}^{*}}+\mu_{B}}.$$ The mass renormalization factor of a pair is given by : $$\begin{aligned}
\label{renorm}
Z\equiv{m\over m_{0}^{*}}={W+|E_{b}|\over W}-{|E_{b}|^{2}\over 2W^{2}}\Phi\ln{\Phi}.\end{aligned}$$ In the intermediate coupling regime, the mass is only weakly renormalized while in the strong coupling regime, there is a strong renormalization of the order ${W\over|U|}$. Due to the relationship between the two-particle T-matrix and the two-particle Green’s function, the quantity $\mu_{B}$ that we defined above does represent the chemical potential of a pair, which has bosonic character.\
For $\vec{q}$ ’s close to the nesting vector $\vec{Q}=(\pi,\pi)$, we obtain: $$\label{teta}
T_{0}(\vec{q},i\nu_{n})={-U^{2}\over i\nu_{n}+{(\vec{q}-\vec{Q})^{2}\over 4m^{*}}+2\mu+|U|}.$$ In the vicinity of the zone corner, the renormalization of the pair-mass is different from the one close to the zone center. Even in the intermediate coupling regime, it is strongly renormalized to be ${m^{*}\over m}\approx{|U|\over t}$. At $\vec{q}=\vec{Q}$, the form (9) of the T-matrix is actually exact, related to the fact that the creation operator $$\eta^{\dagger}=\sum_{\vec{p}}^{}{c_{\vec{p}+\vec{Q}, \uparrow}^{\dagger}
c_{-\vec{p}, \downarrow}^{\dagger}}$$ of an “$\eta$-pair” with center of mass momentum $\vec{Q}$, satisfies the simple commutation relation [@yang; @nowak; @demler] : $$\left[ H,\eta^{\dagger} \right]=(U-2\mu)\eta^{\dagger}.$$ Using the above found expressions (\[t0\]) and (\[teta\]), we can calculate the self-energy. To lowest order, we insert the free-electron Green’s function in Eqn. (\[s1tot\]). The first contribution to the self-energy arises from the poles of the T-matrix. Due to the statistical factors we obtain (to that order of approximation) that the contribution of the $\eta$-resonance is exponentially small, as well as those following from the poles of the Green’s function. After performing analytical continuation, we are left with : $$\Sigma_{1}(\vec{k},\omega)={U^{2}n_{d}\over\omega+t_{\vec{k}}
-\mu+\mu_{B}+i0^{+}}.$$ The quantity $n_{d}$ will be defined below, in Eqns. (\[doubleocc\]) and (\[doubleocc2\]). $\Sigma_{1}(\vec{k},\omega)$ yields then the Green’s function as : $$\begin{aligned}
\label{g2p}
G(\vec{k},\omega)=\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\nonumber \\
{1\over 2}\left(1+
{2\left(t_{\vec{k}}-\mu \right)-\mu_{B}\over
x_{\vec{k}}}\right)
{1\over\omega+{1\over 2}\mu_{B}-{1\over 2}x_{\vec{k}}+i0^{+}} \;\,\nonumber \\
+{1\over 2}\left(1-{2\left(t_{\vec{k}}-\mu \right)+\mu_{B}\over
x_{\vec{k}}}\right)
{1\over\omega+{1\over 2}\mu_{B}+{1\over 2}x_{\vec{k}}+i0^{+}}, \end{aligned}$$ where $x_{\vec{k}}=\sqrt{\left( 2\left( t_{\vec{k}}-\mu \right)+\mu_{B} \right)^2
+4U^{2}n_{d}}$. We immediately note the two limiting behaviors, with respect to momentum $\vec{k}$ : $$x_{\vec{k}}\approx \Delta
+2t\gamma k^{2},$$ with $\gamma={|E_{b}|\over \Delta}$ and $$\begin{aligned}
\Delta=\sqrt{|E_{b}|^2+4U^{2}n_{d}}\end{aligned}$$ for small momenta; respectively, $$x_{\vec{k}}\approx 2\left( t_{\vec{k}}-\mu \right)+\mu_{B}+
{2U^{2}n_{d}\over 2\left( t_{\vec{k}}-\mu \right)+\mu_{B}},$$ for large momenta.\
At this stage of the calculation, the Green’s function has a two-pole structure. The lower excitation branch corresponds to quasi-bound fermions (hereafter denoted as “bosonic” band), while the upper branch describes the unpaired fermions (fermionic band). At small momenta, we obtain: $$\begin{aligned}
\label{g111}
G(\vec{k},\omega)={\Delta+|E_{b}|\over 2\Delta}
{1\over \omega+{1\over 2}\left( \mu_{B}-\Delta \right)-
\gamma tk^{2}+i0^{+}} \;\;\;\;\;\nonumber \\
+{\Delta-|E_{b}|\over 2\Delta}
{1\over \omega+{1\over 2}\left( \mu_{B}+\Delta \right)+\gamma tk^{2}+i0^{+}}\;\;\;\;\;\;\end{aligned}$$ with the spectral weight mainly located in the unpaired fermion band (first contribution in Eqn. (\[g111\])). At large momenta, the Green’s function results into : $$\begin{aligned}
G(\vec{k},\omega)={1-{2U^{2}n_{d}
\over \left(2\left(t_{\vec{k}}-\mu \right)+\mu_{B} \right)^{2}}
\over \omega-\left( t_{\vec{k}}-\mu \right)+i0^{+}} \;\;\;\;\;\;\;\nonumber \\
+{{2U^{2}n_{d}
\over \left(2\left(t_{\vec{k}}-\mu \right)+\mu_{B} \right)^{2}}
\over \omega+t_{\vec{k}}-\mu+\mu_{B}+i0^{+}}, \;\end{aligned}$$ where the weight of the paired fermion band is even smaller than for small momenta. The form of the Green’s function Eqn. (\[g2p\]) differs from the one of Ref.[@vilk2] because the chemical potential $\mu$ is located below the fermionic band in our problem.\
We note that there are two equivalent expressions for the particle density operator $\hat{n}$: $$\label{equiv}
\hat{n}_{i}=\sum_{\sigma}^{}{\left(\hat{n}_{i,\sigma}(1-\hat{n}_{i,-\sigma})+
\hat{n}_{i,\sigma}\hat{n}_{i,-\sigma}\right)}$$ On one hand, we can use the left hand side to express the particle density $n$ as $n_{1}$, where the subscript 1 indicates that the density is calculated out of the one-particle Green’s function : $$\label{densi}
n_{1}=\beta^{-1}\sum_{i\omega_{n},\vec{k}}^{}{\sum_{\sigma}^{}{
G_{\sigma}(\vec{k},i\omega_{n})e^{i\omega_{n}0^{+}}}} \quad .$$ Alternatively, we may use the r.h.s. of Eqn. (\[equiv\]) by separating explicitly the contributions from the unpaired fermions (first term) and the doubly occupied sites (second term). The total density $n_{d}$ of the latter is given by $$\begin{aligned}
\label{doubleocc}
n_{d}={1\over \beta}\sum_{\vec{q}}^{}{\sum_{i\nu_{n}}^{}{
G^{(2)}(\vec{q},i\nu_{n})e^{i\nu_{n}0^{+}}
}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\nonumber \\
=\sum_{\vec{q}}^{}{\int_{-\infty}^{\infty}{{d\omega\over 2\pi}Im\left\{
G^{(2)}(\vec{q},\omega+i0^{+}) \right\}N_{B}(\omega)
}},\end{aligned}$$ $N_{B}(\omega)$ being the Bose-Einstein distribution function. Owing to the latter, at low temperatures, only the low-energy part of the two-particle spectrum, $Im\left\{
G^{(2)}(\vec{q},\omega+i0^{+}) \right\}$, will contribute to $n_{d}$. According to the discussion following Eqn. (\[bind2\]), this low-energy part has two contributions : the sharp quasi-particle excitation, given by (\[t0\]), which dominates for small momenta, and the low-frequency tail of the continuous spectrum of $G^{(2)}(\vec{q},i\nu_{n})$. Since it does not have a sharp structure, it would only contribute a featureless background to the one-electron spectral function; thus, we have neglected it for the calculation of the self-energy $\Sigma_{1}(\vec{k},\omega)$. Neglecting the continuum also in calculating $n_{d}$, we find, according to Eqns. (\[g2\]), (\[t0\]) and (\[doubleocc\]) : $$\begin{aligned}
\label{doubleocc2}
n_{d}=R\sum_{\vec{q}}^{}{N_{B}\left( {q^{2}\over 4m}Z-\mu_{B} \right)
}.\end{aligned}$$ Thus, in our approximation, the number of doubly occupied sites $n_{d}$ is given by the “number of bosons”, $n_{B}$ $$\label{bosons}
n_{B}\equiv\sum_{\vec{q}}^{}{N_{B}\left( {q^{2}\over 4m}Z-\mu_{B} \right)
},$$ weighted by a factor $$\begin{aligned}
\label{residue}
R=\left( {|E_{b}|\over 2W} \right)^{2}\Phi\left( \ln{\Phi} \right)^{2},\end{aligned}$$ which is the residue of the two-particle Green’s function at the bottom of the two-particle band. These bosons, having energy $E_{B}={q^{2}\over 4m}Z$ and chemical potential $\mu_{B}$, represent pairs of electrons being (virtually) bound by the on-site attraction. These results correspond to the observation of other authors (see, for example Ref. [@haussmann]) that, for sufficiently strong attraction, the two-particle Green’s function, (respectively the T-matrix) can be interpreted as a “bosonic Green’s function”. However, there is a weight factor between the two. For large $|U|$, this weight factor $R$ goes to one : in the strong coupling limit, all the double occupancy is due to coherently propagating quasi-bound pairs. In the intermediate coupling regime, which we want to consider ($|U| {\ \lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }W$), the weight factor $R$ is roughly 0.5 : only about one half of the two-particle spectrum in Eqn. (\[gexp\]) is resulting from coherent excitations. In the weak coupling regime, $R$ vanishes, thus rendering our approximation invalid in this limit.\
Since, in our approach, there is gap in the one-electron spectrum, separating the “bosonic” band from the fermionic one, there is no difficulty to obtain the number of unpaired electrons $n_{F}$ as : $$\label{densf}
n_{F}=\sum_{\vec{k},\sigma}^{}{\Xi_{\vec{k}}f_{F}
(\varepsilon_{\vec{k},\sigma})},$$ where $f_{F}(\varepsilon_{\vec{k},\sigma})$ is the usual Fermi distribution function. The dispersion $\varepsilon_{\vec{k},\sigma}$ of the unpaired fermions and the spectral weight $\Xi_{\vec{k}}$ entering Eqn. (\[densf\]) are : $$\begin{aligned}
\label{gksi}
\varepsilon_{\vec{k},\sigma}={1\over 2}\left( x_{\vec{k}}-\mu_{B} \right),
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \nonumber \\
\Xi_{\vec{k}}={1\over 2}\left(1+{2\left( t_{\vec{k}}-\mu \right)-\mu_{B}\over x_{\vec{k}}} \right).
%\nonumber\end{aligned}$$ The total particle density results into : $$\label{densto}
n=n_{F}+2n_{d}.$$ For non-interacting particles, we may equally well use both r.h.s. and l.h.s of Eqn. (\[equiv\]) to calculate the density, since Wick’s theorem applies. However, the identity (\[equiv\]) may be violated in an approximate treatment like perturbation theory. This is the reason why the self-consistent T-matrix calculation breaks down at low $T$ in two dimensions. By calculating the density by means of Eqn. (\[densi\]), there is nothing preventing $T(\vec{q}=\vec{0},\omega=0)$ from diverging at finite $T$, signaling a phase transition, while using Eqn. (\[densto\]) would definitively keep $T(\vec{0},0)$ finite for any finite temperature. By making use of Eqn. (\[densto\]), we make sure that Bose condensation can only take place at $T=0$, in agreement with the Mermin-Wagner theorem. Indeed, according to the expressions (\[doubleocc2\]) and (\[bosons\]) for $n_{d}$ and $n_{B}$, respectively, the bosonic chemical potential $\mu_{B}$, for a given $n_{d}$ (or $n_{B}$) and for $d=2$, is different from zero at any finite temperature, which inhibits Bose condensation, except for $T=0$. In principle, for $d=2$ one should see a Kosterlitz-Thouless (KT) transition. However, since our approximation does not treat the bosonic phase fluctuations in an adequate way, we cannot expect to see the KT-scenario. On the other hand, $T_{C}$ may well be finite for a 3-d system. Actually our procedure is similar in spirit to the two-particle self-consistent approach to the repulsive Hubbard model by Vilk [*et al.*]{} [@vilk; @vilk2].\
Now for a two-dimensional system, we can explicitly evaluate the number of pairs. We obtain : $$\label{nb}
n_{d}={2R\over\beta WZ}\ln{\left( {e^{-\beta(W{Z\over 2}-\mu_{B})}-1\over
e^{\beta\mu_{B}}-1 }\right)}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$ $$\label{nf}
n_{F}={1\over\beta W\gamma}
\left(|E_{b}|+\Delta\over\Delta \right)\ln{\left( {e^{{\beta\over 2}\left( \mu_{B}-\Delta \right)}
+1\over e^{{\beta\over 2}\left( \mu_{B}-\Delta-\gamma W \right)}+1}\right)},$$ where $\Delta$ and $\gamma$ have been given above.\
We are now ready to (numerically) solve Eqns. (\[densto\]), (\[nb\]) and (\[nf\]) for the chemical potential, as a function of temperature. Hereafter, the density $n$ is fixed to be $n=0.1$. The resulting $\mu_{B}(T)$ vanishes exponentially at $T=0$. From Eqns. (\[g2\]) and (\[t0\]), we obtain that the range of the two-particle Green’s function, $\xi$, is given by $${\xi\over a}=\sqrt{{t\over 2|\mu_{B}|}},$$ where $a$ is the lattice spacing. It is displayed on fig. 1, for three different values of the coupling strength $U$. We note that it shows a strong dependence on $U$. At very low temperatures, it is given by : $$\begin{aligned}
{\xi\over a}=\sqrt{{\beta t\over2}}e^{{\beta WZn\over 8R}}.\end{aligned}$$ Oppositely, in the intermediate temperature range, $\xi$ becomes independent of $U$. We also find that $\xi$ decreases with increasing $|U|$. In the strong coupling limit the ratio ${Z\over R}$ tends to zero and thus the exponential divergence of $\xi$ in ${1 \over T}$ is suppressed (there remains only the power-law dependence $\xi\sim{1\over\sqrt{T}}$).\
We may now define a coherence temperature $T_{coh}$ as the temperature at which the range of the two-particle Green’s function exceeds 10 lattice spacings. We obtain $T_{coh}\approx 0.16t$ for $U=-4t$ and $T_{coh}\approx 0.1t$ for $U=-6t$, which may be compared to $T_{C}$ as obtained from the numerical simulations [@singer]. We see that they compare favorably and that, moreover, $T_{coh}$ decreases with increasing $U$. Finally, we note that $\xi$ becomes of the order of the lattice spacing at temperatures well below $|E_{b}|/2$. We now turn to the temperature dependence of the number of pairs. It is displayed on fig. 2, as a function of $T$, for three different values of $U$. At low temperatures, it is independent of $T$. Since the binding energy of the pair (and the gap) decreases as $U$ gets smaller, $n_{d}$ begins to decrease at a lower $T$ for $U=-6t$ than for $U=-10t$, for example. In all cases $n_{d}$ decreases by a factor 2 at $T\approx\Delta/2$.\
In order to reach a better self-consistency, we now calculate how the quasi-bound states affect the two-particle propagator. Introducing $G_{1}(\vec{q},\omega)$ as the lower branch of $G(\vec{q},\omega)$ given by Eqn. (\[g2p\]), we can calculate the first correction to the two-particle propagator as $\chi_{1}$, with : $$\begin{aligned}
\chi_{1}(\vec{q},i\nu_{n})=\sum_{i\omega_{n},\vec{p}}^{}{
G_{1}(\vec{q}-\vec{p},i\nu_{n}-i\omega_{n})
G_{0}(\vec{p},i\omega_{n})}\nonumber \\
\;\;\; +\sum_{i\omega_{n},\vec{p}}^{}{
G_{0}(\vec{q}-\vec{p},i\nu_{n}-i\omega_{n})G_{1}(\vec{p},i\omega_{n})}.\end{aligned}$$ Carrying out the summation over Matsubara frequencies, we obtain that the statistical factors are exponentially small and thus there is no correction to $\chi$ to that order. Thus, the T-matrix is still given by Eqn. (\[t0\]). We can now proceed to the calculation of the second-order correction to the self-energy. It is given by : $$\Sigma_{2}(\vec{q},i\omega_{n})=\sum_{i\nu_{n},\vec{p}}^{}{
T_{0}(\vec{p},i\nu_{n})G_{1}(\vec{p}+\vec{q},i\nu_{n}+i\omega_{n})},$$ yielding, after performing analytical continuation, $$\begin{aligned}
\Sigma_{2}(\vec{q},\omega)=\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\nonumber\end{aligned}$$ $$\begin{aligned}
\left( {|E_{b}|^{2}\Phi t\over W\gamma} \right)\left({x_{\vec{q}}-2\left( t_{\vec{q}}-
\mu \right)-\mu_{B}\over 2x_{\vec{q}}}\right)\nonumber \;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{aligned}$$ $$\begin{aligned}
\times\left({1\over \omega-{1\over 2}\left( x_{\vec{q}}-\mu_{B} \right)+i0^{+}}\right)
\nonumber\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{aligned}$$ $$\begin{aligned}
+{U^{2}\omega_{c}\over W}
\left({x_{\vec{Q}-\vec{q}}-2\left(t_{\vec{Q}-\vec{q}}-\mu \right)-\mu_{B}\over 2x_{\vec{Q}-\vec{q}}}\right) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \nonumber\end{aligned}$$ $$\begin{aligned}
\label{sig2}
\times\left({1\over \omega+2\mu+|U|-{1\over 2}\left(x_{\vec{Q}-\vec{q}}+\mu_{B}
\right)+i0^{+}}\right),\;\;\;\;\;\;\;\;\;\;\;\;\end{aligned}$$ where $\omega_{c}$ is a frequency cut-off needed by the assumption that the $\eta$-resonance is sharp for ${\left( \vec{Q}-\vec{q} \right)^{2}\over 4m^{*}}
\leq\omega_{c}
\approx {U^{2}\over W}$, with $m^{*}=m{|U|\over t}$. This corresponds to $|\vec{Q}-\vec{q}|{\ \lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{U \over W}$, as it is borne out by numerical calculations of the two-particle Green’s function $G^{(2)}$ covering the whole Brillouin zone [@morten; @demler]. The precise value of the cut-off has little influence in the following (numerical) results. We note that the first contribution in Eqn. (\[sig2\]) is following from the long-wavelength behavior of the T-matrix, while the second is due to the $\eta$-resonance. Even though both $\Sigma_{1}$ and the first contribution to $\Sigma_{2}$ originate from the pole of $T_{0}(\vec{q},i\nu_{n})$, given in Eqn. (\[t0\]), they are found to have opposite dispersions. The total self-energy results as : $$\begin{aligned}
\label{self1}
\Sigma(\vec{q},\omega)=\Sigma_{1}(\vec{q},\omega)+\Sigma_{2}(\vec{q},\omega).\end{aligned}$$ At this point of the calculation, we can recalculate the particle density $n_{1}$ by evaluating explicitly Eqn. (\[densi\]), using the Green’s function resulting from Eqn. (\[self1\]). Since there is no a priori reason that this would yield a result comparable to what is following from Eqn. (\[densto\]), this is a consistency check of the framework we are using. The result is displayed on fig. 3. It is obvious that this is in very good agreement with the expected value $n=0.1$ for all values of $U$. This means that our calculation is consistent. It also implies that, on one hand a small change in the chemical potential corresponds to a small change in the particle density, but on the other hand, it may induce a big change in the pair density.
(c) Excitation spectrum {#c-excitation-spectrum .unnumbered}
=======================
We can now summarize our findings by plotting the various pieces of the spectral function. This is done on fig. 4a, for $U=-6t$ ($E_{b}=-t$) and on fig. 4b for $U=-10t$ ($E_{b}=-4.2t$) for momenta along the diagonal of the Brillouin zone. In both cases, the spectrum consists of four branches. The spectral weights of the various branches are displayed on fig. 5a for $U=-6t$ and fig. 5b for $U=-10t$.\
These results can be commented as follows :\
(i) There is a branch at negative energies following from the quasi-bound states. It is centered around $-{W\over 2}-{\Delta+|E_{b}|\over 4}$ with a width $W-{\Delta-|E_{b}|+2\mu_{B}\over 2}<W$. The dependence in $U$ is weak, and the dispersions for $|U|=6t$ and $|U|=10t$ are essentially the same, up to a small shift; moreover, we note that they are opposite to the free fermion dispersion. The weight decreases with increasing momentum and gets mostly negligible for $q_{x}\sim 2$. It decreases with increasing $|U|$, at small $\vec{q}$ ’s, but the total weight remains constant.\
(ii) The fermionic band that was coming out of the first approximation (Eqn. (\[g2p\])) is now resulting as a superposition of two branches which would merge into a single one, were damping taken into account. This superposition is done out of the two branches which are most free-electron-like. This is somewhat arbitrary, especially in the domain where the hybridization with the other branch, which lies at positive energies too (see below), is strong. Consequently, we do not show the spectral weights in this range. The width of the fermionic band is slightly larger than the original bandwidth, and shows little dependence on $U$. It lies at slightly larger energies for $|U|=10t$ than for $|U|=6t$, but again the dependence in $U$ is weak. Its dispersion is parallel to the free fermion one.\
(iii) There is a third branch resulting from the $\eta$- resonance; its dependence in $U$ is weak and both curves (for $|U|=6t$ and $|U|=10t$) are parallel. It comes down in energy with increasing $|U|$. The contribution of the $\eta$-resonance band is very small, barely accounting for 5 percent of the spectral weight at its maximum, which is located at $\vec{q}=\vec{Q}$.\
It is interesting to compare our results with Quantum Monte Carlo (QMC) simulations [@singer; @singer2]. The calculations of Ref.[@singer] have been performed for $n=0.4$ and for $|U|=-4t, -8t, -12t$. This is likely to be outside the realm of densities for which the T-matrix approximation is really valid. Nevertheless, the following features coincide with our findings : there are two distinct excitation branches which get more and more clearly separated by a gap when $|U|$ increases. The width of the two bands is smaller than in our calculation, in particular the lower (bosonic) band has very little dispersion. However, the weights (the fermionic band has large weight for large wave vectors, whereas the weight of the negative energy bosonic band is concentrated near the zone center) is similar to our figures 4 and 5.\
Ref.[@singer2] presents new results for the one-electron spectral function for $n=0.1$, the same value that we have used, and $|U|=8t$. Here again, there is a fermionic branch at positive energies $E$ (positive with respect to the chemical potential), separated by a “pseudo-gap” from the excitations at $E<0$. Weight and width of the fermionic band is similar to our result. The excitations at $E<0$ have most weight near the zone center, as we find it. However, their structure seems to be more complex : with increasing wave number they split into two “sub-branches”. The lower part has downward dispersion and seems to correspond to our bosonic branch. The other sub-branch produces weight near the chemical ($E\simeq 0$) potential for large wave numbers. This might correspond to our $\eta$-resonance, provided that its energy near the zone corner, lying at a positive energy in our calculation, would in reality be lower. In this respect, the spectral functions for $n=0.4$ and $|U|=-6t$, also shown in Ref.[@singer2], are particularly interesting : for $E>0$, besides the strong fermionic branch, there is a second branch of excitations the dispersions of which is very similar to the $\eta$-peak in our figures 4 and 5.\
Thus, the main features of our spectral functions seem to be present in the Monte Carlo results, although the latter show a more complex structure. A detailed and more quantitative comparison with QMC should also take into account the fact the “Maximum Entropy method” used there in order to extract spectral functions from data obtained as functions of the (imaginary) Matsubara frequencies does not easily allow for an unambiguous identification of excitations with small weight.\
Finally, we note that we do not obtain any spectral weight at zero frequency, signaling the presence of a correlation induced gap. We also checked that including particle-hole excitations in the calculation does not affect this conclusion. The appearance of a true gap in our calculation is, at least partly, due to the fact that our spectral lines have no width (except that the “doubling” of the fermionic branch gives a hint to a broadening of the latter). Spectral functions with finite line width would be obtained either by doing the $\vec{q}$-sums in the expression (\[s1tot\]) for the self-energy more precisely, and/or by evaluating $G$ and $\Sigma$ by solving Eqns. (\[tfrai\]) to (\[g1tot\]) self-consistently.\
In summary, we have determined the excitation spectrum of the attractive Hubbard model at intermediate coupling out of a simple analytical calculation. We first pointed out an intrinsic problem of perturbation theory relative to the implementation of the Mermin-Wagner theorem. We made use of an alternative expression for the density to obtain a qualitatively correct theory which does not break down at low $T$ in two dimensions, in contrast to previous self-consistent calculations. We obtain an analytical expression for the Green’s function which reproduces the qualitative features of the QMC simulations in the low density regime.\
We acknowledge valuable discussions with T. Schneider, J.M. Singer, M.H. Pedersen and J.J. Rodríguez-Núñez. We thank J.M. Singer for providing us with his numerical data prior to publication. This work was financially supported by the Swiss National Science Foundation, under grants 20-43111.95 and 20-47149.96. M. Yu. K. acknowledges the University of Neuchâtel, where part of this work has been performed, for hospitality and the Swiss National Science Foundation for partial funding.
P. Nozières and S. Schmitt-Rink, J. Low Temp. Phys. [**59**]{}, 195 (1985). M. Drechsler and W. Zwerger, Ann. Phys. [**1**]{}, 15 (1992). Th. Meintrup, T. Schneider and H. Beck, Europhys. Lett. [**31**]{}, 231 (1995). V.M. Galitskii, Sov. Phys. JETP [**7**]{}, 104 (1958); D.M. Eagles, Phys. Rev. [**186**]{}, 456 (1969); P. Bloom, Phys. Rev. B [**12**]{}, 125 (1975) J.R. Engelbrecht and M. Randeria, Phys. Rev. Lett [**65**]{}, 1032 (1990); H. Fukuyama, Y. Hasegawa and O. Narikiyo, J. Low Temp. Phys. [**60**]{}, 2013 (1991). R. Frésard, B. Glaser and P. Wölfle, J. Phys.: Cond. Mat. [**4**]{}, 8565 (1992). R. Micnas, M.H. Pedersen, S. Schafroth, T. Schneider, J.J. Rodríguez-Núñez and H. Beck, Phys. Rev. B [**52**]{}, 16223 (1995). R. Haussmann, Z. Phys. B [**91**]{}, 291 (1993). P.J.H. Denteneer, G. An and J.M.J. van Leeuwen, Phys. Rev. B [**47**]{}, 6256 (1993). J.M. Singer, M.H. Pedersen, T. Schneider, H. Beck and H.G. Matuttis, Phys. Rev. B [**54**]{}, 1286 (1996); J.M. Singer, M.H. Pedersen and T. Schneider, submitted to Physica C. M.H. Pedersen, Ph.D. Thesis, University of Zurich (1996); M.H. Pedersen, J.J. Rodríguez-Núñez, H. Beck, T. Schneider and S. Schafroth, accepted for publication in Z. Phys. B and cond-mat/9702173. B. Bulka, cond-mat/9703040. T. Schneider, M.H. Pedersen and J.J. Rodríguez-Núñez, Z. Phys. B [**100**]{}, 263 (1996). See for example, S. Schafroth, J. J. Rodríguez-Núñez and H. Beck, J. Phys.: Cond. Mat. [**9**]{}, L111 (1997).; J.J. Rodríguez-Núñez, private communication. C.N. Yang, Phys. Rev. Lett. [**63**]{}, 2144 (1989) E. Nowak, Z. Phys. B [**45**]{}, 173 (1981) E. Demler, S. C. Zhang, N. Bulut and D.J. Scalapino, Int. J. of Mod. Phys. B [**10**]{}, 2137 (1996). Y.M. Vilk, L. Chen and A.-M.S. Tremblay, Phys. Rev. B [**49**]{}, 13267 (1994) and Physica C [**235-240**]{}, 2235 (1994). Y.M. Vilk and A.-M.S. Tremblay, Europhys. Lett [**33**]{}, 159 (1996) and cond-mat/9702188. J.M. Singer and T. Schneider, private communication.
|
---
address:
- |
Grup de F[í]{}sica Te[ò]{}rica, Departament de F[í]{}sica and IFAE,\
Edifici Cn, Universitat Aut[ò]{}noma de Barcelona\
E-08193 Bellaterra (Barcelona) Spain
- |
Department of Mathematics and Statistics,\
University of Plymouth, Drake Circus,\
Plymouth, Devon PL4 8AA, UK
author:
- 'E. BAGAN, M. LAVELLE, B. FIOL, N. ROY'
- 'D. McMULLAN'
title: |
CONSTITUENT QUARKS FROM QCD:\
PERTURBATION THEORY AND THE INFRA-RED [^1]
---
epsf.sty
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
\#1\#2[[(+ (\#1\_2-\#2\_2)\^2+ (\#1\_3-\#1\_3)\^2)]{}]{}
The physical content of our theories of the fundamental interactions is profoundly affected by the gauge symmetries that lie at their heart. Such theories are in fact examples of systems with constraints and it is well known that a consequence of this for QED is that only two of the four initial $A^\mu$ potentials are actually physical. The implications of the gauge symmetries of QED for charged fields (such as electrons) are less well understood and for non-abelian theories, such as QCD, the extraction of the physical degrees of freedom has not been performed. This talk reports recent progress in understanding these fundamental issues. In particular a gauge invariant description of charged fields in electrodynamics and a physical interpretation is provided. This leads to predictions which are then tested in perturbation theory. The gauge structure of scalar QED is so similar to that of standard QED that exactly the same predictions may be made for it. They are also verified here.
Any description of a physical charge must be gauge invariant and Gauß’s law implies an intimate link between charges and a chromo-(electro-)magnetic cloud. Such a description in terms of a so-called dressed field, $\psi_f$, was proposed by Dirac in the 50’s: $
%\begin{equation}
\psi_{f}(x)=\exp\left\{ ie\int d^4zf_\mu(z,x)A^\mu(z)\right\}
\psi(x)
%\,.
%\label{eq:fcond}
%\end{equation}
$. This can be seen to be gauge invariant if $f_\mu(z,x)$ satisfies $\partial_\mu^z f^\mu(z,x)=
\delta^{(4)}(z-x)$.
Not all gauge invariant descriptions are, however, physically relevant. In Fig.\[fig\] two such possible clouds are shown.
Clearly, (a) is not stable in QED and for a static charge it will decay into the Coulomb cloud in (b). Our claim [@LaMcMu96a; @slow; @fast] is that the latter, and a more general version corresponding to a charge moving with velocity $\vec v$, are suitable for constructing physical asymptotic states. This in either QED or scalar QED since the magnetic field associated with the magnetic moment of an electron falls off rapidly away from the charge and is thus infra-red safe. To be explicit, for an electron moving with velocity $\vec v$ we dress the fermion as follows $$\psi_v=\exp\left\{
ie{g^{\mu\nu}-(\eta+ v )^\mu(\eta- v)^\nu\over\partial^2-
(\eta\cdot\partial)^2+(
v\cdot\partial)^2}\partial_\nu A_\mu
\right\}
\psi ,
\label{eq:boos}$$ where $v=(0,\vec v)$, and $\eta=(1,\vec 0)$. Rather than giving the arguments underlying this statement, we now report a perturbative calculation [@slow; @fast] which verifies the following. Recall that the usual gauge-dependent fermion propagator in QED or QCD is plagued with infra-red divergences in an on-shell scheme. These reflect the fact that the fermion is not a good physical state —the chromo-(electro-)magnetic field it generates is missing. If our dressing has a physical significance, we should be able to perform an [*infra-red finite*]{} on-shell renormalization for the propagator of the dressed charge defined by (\[eq:boos\]).
For this non covariant description, we find [@fast] a multiplicative [*matrix*]{} renormalization $$\psi^{({\rm bare})}_v=\sqrt{Z_2} \exp\left\{
-i { Z'\over Z_2}\sigma^{\mu\nu}\eta_\mu v_\nu \right\}\psi_v
\quad\mbox{\rm and}\quad m^{({\rm bare})}=m-
\delta m
\label{eq:Zs}$$ necessary. This is reminiscent of a naive Lorentz boost upon a fermion. The mass shift renormalization from demanding that the pole is at $m$ yields the standard gauge-invariant result found in any textbook. The residue renormalization condition is where the infra-red divergences are usually found and we find here for our non covariant case three equations for only two unknowns, $Z'$ and $Z_2$. It is highly gratifying that at the expected physical momentum, $p=m\gamma(1,\vec v)$, we can consistently solve these three equations and further that [*the renormalization constants are gauge-invariant and infra-red finite*]{}.
The matrix renormalization (\[eq:Zs\]) is forced upon us by the fermion structure of QED. We have checked that in scalar QED where such a scheme is not possible, a straightforward multiplicative renormalization also yields infra-red finite results — this despite exactly the same non-covariant dressing also being used in the scalar theory.
The next step is the non-abelian theory. Quarks and gluons are believed to be confined inside colourless hadrons and yet the success of the constituent quark model and the jet structures observed in experiments show that it must be possible to attach some physical meaning to quarks and gluons. The Lagrangian fields are, however, gauge dependent —like their QED counterparts— and we need to find the physical degrees of freedom of such a non-abelian gauge theory. We note here that the action of the gauge dependent colour charge operator is only gauge invariant on locally gauge invariant objects and so the colour statistics of the constituent quark model require a gauge invariant description [@LaMcMu96a]. Perturbative calculations are possible here, but, it can be shown that a non-perturbative obstruction, the Gribov ambiguity, prevents quarks and gluons from being true observables. This is because the QCD equivalent of the dressing function $f_\mu(z,x)$ can be used to fix the gauge. This obstruction sets the fundamental hadronic scale and the limits of the quark model.
References {#references .unnumbered}
==========
[99]{} A simulation of the evolution of an initial, gauge-invariant state made up of two fixed opposite electric charges linked by a string and a more detailed description of the underlying physics may be found at [http://www.ifae.es/\~roy/]{} M. Lavelle and D. McMullan, [ *Constituent Quarks from QCD*]{}, to appear in Physics Reports C. E. Bagan, M. Lavelle and D. McMullan, . E. Bagan, M. Lavelle and D. McMullan, [*A Class of Physically Motivated Gauges with an Infra-Red Finite Electron Propagator*]{}, submitted for publication. (hep-th/9602083).
[^1]: Talk presented by E. Bagan
|
---
abstract: 'Infrared (IR) videos are presented which show a warm water surface undergoing convective processes. These fluid dynamics videos show the water surface with: 1) no surfactant monolayer material present, 2) a liquid-phase monolayer of oleyl alcohol, and 3) a solid-phase monolayer of cetyl alcohol.'
author:
- |
S. M. Bower & J. R. Saylor\
Department of Mechanical Engineering\
Clemson University, Clemson, SC 29634, USA
title: Infrared video of a warm water surface in the presence and absence of surfactant monolayers
---
Introduction
============
This [video](http://ecommons.library.cornell.edu/bitstream/1813/14124/2/DFDV2.mpg) ([low quality version](http://ecommons.library.cornell.edu/bitstream/1813/14124/3/DFDVideoLowQualV1.mpeg)) provides IR visualizations of a warm water surface (T $\approx$ 45$^{\circ}$C) under several air/water-interfacial conditions: namely with a clean surface (no surfactant material present), a liquid-phase monolayer present, and a solid-phase monolayer present. The clean water surface exhibits fine-scale structures and an average surface temperature that is slightly less than the bulk water temperature.
When the liquid-phase monolayer of oleyl alcohol is spread across the surface, it imposes a constant-elasticity boundary condition. The average surface temperature is noticeably less than that of the clean surface and the fine-scale structures vanish when the liquid-phase monolayer is present.
A solid-phase monolayer of cetyl alcohol is then compared to the oleyl alcohol surfactant by depositing talc powder onto both monolayer-contaminated water surfaces. Because the solid-phase surfactant imposes a nearly rigid boundary condition at the interface, the monolayer both supports the talc powder at the surface and is not deformed by the subsurface motion of the water bulk.
Lastly, air is blown across both the liquid-phase and solid-phase surfactants. While wind shear cools and deforms the surface with the liquid-phase monolayer, the solid-phase monolayer tends to resist surface motion and does not cool noticeably. When considering that these monolayers are only one molecule thick, these visualizations are impressive and reveal the significance that these monolayers have on transport processes.
|
---
abstract: |
This study concerns the use of e-learning in the educational system shedding the light on its advantages and disadvantages, and analyzing its applicability either partially or totally. From mathematical perspectives, theories are developed to test the courses tendency to online transformation. This leads to a new trend of learning, the offline-online-offline learning (fnf-learning), it merges e-learning mode with the traditional orientation of education. The derivation of the new trend is based on the learning approaches and the study levels, this makes the new trend flexible and applicable for all mathematical courses.\
author:
- Hasan Almanasreh
title: 'e-Learning Mathematics'
---
Introduction.
=============
Technology rapidly continues transforming our aspects of life, improving the health care, the military developments, the financial affairs, and even the educational system, etc., ending without end. Many advantages, profits, and benefits for both users and developers are a consequence of the technology revolution, also, to that, many interesting and important (and some time frightening) problems were solved or being arisen as a side-resulted of deploying new strategies and equipments. Thus, careful investigations should intensively be devoted to test the efficiency of these technologies and to determine to which extents and scopes they can be applied.
Like others, education is affected by technology, and learning is bit by bit departing from the traditional mode, letting the online electronic-learning (e-learning) occupy a large place of the educational process. The information and communication technology plays a crucial (and even, in some situations, an irreplaceable) role nowadays, where most attentions are being devoted to transform the conventional learning system to (what is expected and hoped to be a peer orientation) e-learning by a help of the technology sagacity.
In a topic field of education, using e-learning as an alternative to the conventional learning is not simple per se. Indeed, it needs deep study to determine the efficiency of the new method and its consequences. Also, for a course in a given field, further study is needed to investigate the readiness of that course for the online transformation either entirely (which exempts the offline strategy) or partially, at the same time to determine whether all courses admit the same portion of e-learning applicability.
In this work, we study e-learning from mathematical point of view, showing the efficiency and deficiency of the method, criticizing and commenting on some views on e-learning in general and particularly e-learning mathematics. Also, we will discuss the capability of mathematics courses to e-learning applicability. Finally, we will analyze which parts of a course admit e-learning and which parts can only be treated by the conventional technique. This leads to an alternative proposed learning strategy, the so-called offline-online-offline learning (fnf-learning). For this approach, a mathematical formula is developed to control the time period for the online mode in teaching a mathematical course. This formula is not in its final shape and needs to be more studied and improved, but it is still a good starting step towards controlling the parts of the online and the traditional modes in teaching mathematics courses. The derived formula depends on the study level and the learning approach level of each course, which makes it applicable for any mathematics course.\
The paper is arranged as follows; In Section 2, we will discuss the advantages and disadvantages of e-learning dwelling upon the excessive use of e-learning from mathematical perspectives. Theories of e-learning mathematics are developed in Section 3. In Section 4, the approach fnf-learning is discussed, and a mathematical formula is modelled for this approach which naturally provides theorems about e-learning mathematics for any course.
Advantages and Disadvantages of e-learning
==========================================
e-Learning is a form of distance learning, where the time and geographical flexibility are preserved. Students can any time any where pursue their study, which has great beneficial effects especially for those who have part-time jobs or have families. Beside that, the students can take as many courses as they can manage. This flexibility works for teachers as well, the teacher can instruct many courses at the time without geographical restrictions. Moreover, e-learning gives the students more self-independence and self-confidence. Over that, it gives opportunities for shy students to ask questions where they hesitate to deliver questions directly to the lecturer in the face-to-face (f2f) interaction, and others else.
Nevertheless, despite of the above positive features, where many of pro-e-learning are mentioning them repeatedly, there are also disadvantages of this trend. Many students learn better in the traditional class which is a result of live and direct interaction with the tutors and other students. In e-learning, the need arises to specific level of skills in order to follow the course, which varies from one student to another; hence a weak knowledgeable student of these skills will get himself lost in some parts of the course. Also, e-learning needs continuous accessibility to the online updated instructions, i.e., every student should have a computer with internet connection to follow the course.
These are not only the disadvantages of the system, below we will dwell upon some others, where it is worthwhile to mention each of which separately as critics for some views. Also we will elaborate on some of them from a generic point of view and some others from mathematical perspectives.
Lack of controlling student-student interaction
-----------------------------------------------
Once the online course has been started, the instructor needs to encourage certain range of social talk, to give his course a slot of attraction to interaction; this is suggested by L. Jonsson and R. Säljö [@2]. They caution also from the dominancy of the social interaction in the communications and they stipulate the instructor to control that. As teachers, we fully agree that we should allow, from time to time, some sociality in our course to reactivate and attract the students. We do believe this can be done and easily be controlled in the traditional courses, where the lecturer can manage that. But, in e-learning, the teacher is not always there (on the course webpage) to get the students under his eyes. Even more, if we controversially assume that the teacher all the time is available online, then how he can control the students social interactions and in which way. As a result, there is no clear strategy to be used to efficiently control the initiated social talk. We believe here (in the online mode) that, there is no need to create social aspects where it is known there is no efficient way to get them controlled, taking into account that there are many social websites for this purpose. Also the students of the online mode are not sitting, as those who are continuously sitting for two or more hours where they need to be refreshed to keep them awake and active, in the classroom.
What is learning mathematics
-----------------------------
What is needed is, to adapt the technology for the purpose of learning, not the opposite. To this end, learning is not about gathering information, it is about interpreting it. We would exactly repeat, and completely agree, what Andrew Hart [@5 Ch.9, p. 151] said in this issue\
*”Becoming rich in information but poor in knowledge. The spread of information is dangerously entropie. It may lead to uncertainty and insecurity rather than confidence and self-assurance. What we need from educational technology is forms of knowledge which may lead to understanding, rather than information overload and confusion.”*\
As mathematicians, we teach the students how and why to choose this method or that, to interpret the facts, to relate them to real-life problems, and to teach them the way, and even the most powerful and comprehensible way, of proving theorems. In other words, we teach them the way of thinking and the hidden logic of mathematics, which the students need most to acquire rather than just collecting information and remembering facts which, with time passage, are susceptible to oblivion. We do not down the latter, but in mathematics, not like most other fields, it is of the least importance. To this end, we do not see how students achieve, gain, and develop the properties above if the course is given completely online. Of course, they can learn the ways of solving exercises, they can learn the proofs of theorems, and they can remember facts and ideas (which might be sufficient learning outcomes of a course in some other fields, but not mathematics). But they will miss the most crucial part, the mathematical logical thinking.
In the traditional class, the teacher tries not only to solve exercises or to prove theorems, but also, through out that, he provides the students ways of thinking, makes them challenge the problems and be enthusiastic, gives them the opportunity to comment on some obstacles (that naturally arise or deliberately created as a result of the direct interaction and discussion) and on some new and further assumptions. We do not want to overstate the words here, but one more thing can be addressed; for a practiced teacher, a direct feedback can be obtained from the students reactions (simply by reading their faces), so that he can judge if they get interested or bored, whether they do or do not understand the topic, and if they are keeping up with or getting lost.
Poor deep understanding
-----------------------
For an online mathematics course, the claim of being providing deep understanding is stripped of truth. Some authors [@1; @6] claim that e-learning may induce deep understanding and strong retention. They support their views by a study run at the University of Helsinki in fall 2004: The basic course in calculus was offered in the traditional way and as a fully online course. The students of the online trend did all of their study through the web, but the examination was only as usual, traditionally. The results of the studentss achievement are given in Table 1 below\
---------------------------------------------------- -------- ---------
Online Offline
\[0.1ex\] Retention Rate 66% 62%
Average of students passing the course (out of 24) 12.74 11.74
\[1ex\]
---------------------------------------------------- -------- ---------
: the result of the basic course in calculus taken by two groups, online and offline at the University of Helsinki in fall 2004.
\[table:nonlin\]
From our point of view, the result is not surprising. On the contrary, the online students might be expected to perform even better. The courses in calculus focus on collecting information, solving exercises, verifying facts, and at most proving some simple theorems. So, as much the students are acquainted with the materials (reading and practicing) as much they perform better, which is the main feature of the online courses where more study materials can be distributed. Besides that, the students have more time compared to those who are enrolled in the traditional courses to devote to the course materials. To be more precise, let us classify the calculus course corresponding to learning approaches. In the figure below, we divide the approaches of learning in an inverted hierarchy diagram. The first two from above are what we call the surface levels, the third is the first intermediate level, the two coming after is the second intermediate level, and the last two are the deep levels. The calculus course is at the surface levels, and if it extends deeper it mostly touches the first intermediate level. The students there are not required to interpret or analyze, instead they are assumed to remember and practice, and as much they do so, as much they perform better. So, as a result, the major performance of such course is proportional to how much time it is being devoted. The role of the teacher here, in the courses with surface levels of learning, is to solve as many examples as he can, rather than going deeper in the material, simply because there are no goals of such courses that touch the second intermediate level.
{width="11cm" height="8cm"}
For advanced (high levels) courses, the picture is different, the intermediate and deep levels of learning are emphasized, and the tutor is required to come up with the students to higher levels of thinking which can not be achieved by distributing the course materials. As we mentioned before, the lecturer role here is not only to solve or to prove some facts, but also to provide ways of thinking and strategies of treatment as he can, e.g., assume some artificial obstacles and let the students provide their own views before he directs them to the right approach, which may be not unique. Consequently, the students are challenged and become enthusiastic when they share and directly examine their views with the teacher in an interesting environment using the simplest and the most preferable materials, the shocks and the board. Parallel to that, the teacher can change, modify, and simplify his way of explanation depending on the direct feedback he obtains by noting the students reactions.
The theories of e-learning mathematics courses
==============================================
There is no resort of avoiding the technology facilities in the educational system; they ease handling many processes and provide simpler and faster trends. Meanwhile, careful awareness should be stressed from passive usage of them. So, the use of technology in the learning system should be in a fluent way, keeping it as simple as possible while providing advanced functionalities. Given that, taking into account the individual differences of the students and freezing all other factors of learning, still a question remains: Do all courses undergo online transformation, totally, partially?\
Concerning mathematics courses, the capability of the online applicability varies depending on the levels of learning approaches and the study levels. Below are some theories formulated according to our perspectives.\
**Theory 1:** There are courses that can be completely transformed to online mode.\
All courses that are not exceeding the first intermediate level of learning approaches can be completely given online. The academic achievement is proportional to the time devoted to reading and practicing the course materials. Such as courses are Calculus (I,II,III), the introductory courses to probability, and the first course in ordinary differential equations.\
**Theory 2:** Problem-based courses and project-oriented courses can be fully given online.\
These courses are based on the fact that learning mathematics is achieved by interpreting not by absorbing information.\
In the above two theories, the teacher should convert his role from a lecturer to a coach.\
**Theory 3:** Most courses have no tendency to totally online transformation.\
This is actually the feature of mathematics courses, which arises from the fact that the learning approaches of most mathematics courses are exceeding the second intermediate level.\
**Theory 4:** There exists no course that does not admit at least online partial transformation.\
This fact is obvious per se.\
The most powerful way to let the online strategy play a role in the learning process is to apply it partially. Major parts of most mathematics courses undergo the advanced levels of learning approaches, where the role of the traditional lectures is crucial and can not be replaced. To this end, the lecturer can vary (or more appropriately can merge) between the two modes in the same course, i.e., using blended-learning (b-learning), see e.g. [@1; @8].
What is Blended Learning
========================
The educational system is part of our life that can not be annexed from the revolution of technology and of one of the most benefited from the technological achievements. Thus, it is not possible to ignore the crucial role of the rapid acceleration of the technological advancement in the learning system. Consequently, it is wise to mix between the two modes of the learning system, the online and offline, which is the well-known blended learning. According to Singh [@7]\
*”Blended learning mixes various event-based activities, including face-to-face classrooms, live e-learning, and self-paced learning. This often is a mix of traditional instructor-led training, synchronous online conferencing or training, and asynchronous self-paced study.”*\
By using blended learning (online partial transformation), the disadvantages of both learning modes can be get rid of, at the same time, most of their advantages are preserved ; whereas *”a single mode of instructional delivery may not provide sufficient choices, engagement, social contact, relevance, and context needed to facilitate successful learning and performance”* according to Singh [@7] . More about blended learning can be found in ,e.g., [@3; @4; @8]\
For online partially transformed courses, it is quite important to decide when to use the traditional classes (lecturing) and when to convert to the online mode. Also, to meet the beneficial goals of the online transformation, it is not recommended to over alternate between the two strategies. For that, we recommend to start the course with the conventional mode, then switch to e-learning, finally switch back to the starting mode. We will abbreviate it as fnf-learning (offline-online-offline learning). Before proceeding, to study this system in appropriate way we will introduce some notations and definitions. By the gap, we mean the middle stage of fnf-learning, i.e., the part of the course that is given online. Let the size of the gap, denoted by $S(gap)$, be the size of the course part that is given online. Let also $L$ denote the movement from the lowest level towards the deepest level of the learning approaches, and $T_L$ be the transformation of the interval $(\text{lowest level}\,,\, \text{deepest level})$ into the interval $(0\,,\,\infty)$, so that the movement from the lowest level to the deepest is equivalent to the movement from $0$ to $\infty$. Further, let $E$ be the length measure, clearly that $E(T_L)\to\infty$ as $L\to \text{deepest level}$. In the same way we introduce the variable $s$ to denote the movement toward higher levels in the study (i.e., say $s$ moves from the first year of the Bachelor program and moving higher). Let $T_s$ be another transformation (with functionality similar to $T_L$) connects the interval of $s$ with $[0\,,\,\infty)$. One can also find that $E(T_s)\to\infty$ as $s\to\text{highest study level}$.
As more the level in the learning approaches getting higher as much we need to use the offline mode of learning. This means that $S(gap)$ is inversely proportional to $E(T_L)$, which is denoted by $S(gap)\;\alpha\;\frac{1}{E(T_L)}$. Also, as the study levels getting higher, the learner will have acquired enough techniques and ways of thinking to independently pursue his study. This means that we can completely switch to the online mode regardless the learning levels, thus, $S(gap)$ is proportional to $E(T_s)$). Taking into account the above formulations, together with Theory 4, one can derive a formula for $S(gap)$. Using the notations $S_g:=S(gap)$, $x=E(T_s)$, and $y=E(T_L)$, then such formula might be in the form $$S_g(x,y)=c\displaystyle\frac{e^{rx}}{y^m}+k$$ The constant $c$ is a positive real number called the difficulty coefficient of mathematics, $k$ is a positive real number represents the minimum size of the online part that the course can admit, $m$ is a positive integer, and $r\in(0\,,\,1)$. By the formula above we do not intend to give $S_g$ the exact form, but we want to indicate the possibility of giving it a mathematical expression that can be applied for all courses. Note that each mathematics course admits at least a partial online transformation that is the value $k$ and this is harmonizing with Theory 4, on the other hand, each mathematics course has a particular $k$. Also, the difficulty coefficient, $c$, of each mathematics course depends on the topics and contents of that course, which means that each mathematics course has a different value of $c$. The constants $r$ and $m$ are independently chosen for each mathematics course.\
Based on the above formula for $S_g$, we can formulate the following theorems
Regardless the learning level $(y)$, if the study approaches the highest level $(x\to\infty)$, then the course can completely be given online.
Regardless the study level, if learning approaches the lowest level $(y\to\infty)$, then the course can completely be given online.
If learning tends to the lowest level, and if the study becomes closed to the lowest level, then the course admits the $k$ value of online transformation.
Therefore, fnf-learning is flexible and can be applied for almost all courses. It gives the opportunity to efficiently match the rapidly accelerated digital technology, while keeping a live and direct interaction in the learning system. Moreover, particularly in mathematics, fnf-learning maintains the traditional mode which is of the most importance to feed the students with the logical mathematical thinking, at the same time it gives the students opportunity to develop their own thinking, consequently providing them with self-dependence and confidence.\
**Conclusion.**
For any course in the educational system, efforts and deep study are needed to investigate the capability of the online transformation. According to the study, for mathematics courses that can be completely given online, we do admit that it is not wise to void the course from the traditional orientation. Nevertheless, in applying online partial transformation, other aspects should be considered such as individual differences (the students preferential to online or traditional modes, and the student ability of using the required technology), the availability and accessibility of the technology, the time and cost, and the social aspects. The latter is of significant importance which is being neglected. In the regions of the world, where the social activities are dying out, we should be aware of the intense and numerous digital transformation of our system of life. Thus, by complete transformation to the online mode, we freeze, and even kill, the opportunity of students meeting, which is a purpose of the educational process that also aims to increase the social interactions (f2f and not online interactions) among the people. Thus we exaggerate the problem instead of solving it.
[References]{} S. Descamps, H. Bass, G.B. Evia, R. Seiler, and M. Seppälä, *e-Learning mathematics*, Conference of Spanish mathematics deans, 2006. L-E. Jonsson and R. Säljö, *The Online Seminar as Enacted Practice*, Conference paper, ECER, 2008, Göteborg. B. Khan, *Web-based training*, Educational Technology Pubns, 2001. B. Khan, *Web-based instruction*, Educational Technology Publications Inc, 1997. P. Ramsden, *Learning to teach in higher education*, Routledge Falmer, London and New York, 2003. M. Seppälä, O. Caprotti, and S. Descamps, *Using web technologies to teach mathematics*, Departaments de Matemàtica Aplicada. Journal papers, 2005. H. Singh, *Building effective blended learning programs*, Educational Technology, 43(2003), pp. 51-54. B. Yushau, *The effects of blended e-learning on Mathematics and computer attitudes in pre-calculus algebra*, The Montana mathematics enthusiast, 3(2006), pp. 176-183.
|
---
author:
- |
\
[*Theoretical Solid State Research Group of the Hungarian Academy of Sciences*]{}\
\
[*Department of Chemical Information Technology, BUTE, Hungary*]{}\
\
[*Department of Theoretical Physics, BUTE, Hungary*]{}\
\
[*Department of Physics, University Duisburg-Essen, Germany*]{}\
bibliography:
- 'powderng.bib'
title: |
Computer simulation of three dimensional shearing of granular materials:\
Formation of shear bands
---
INTRODUCTION
============
Spontaneous symmetry breaking in granular materials occurs in many different forms. Here we focus on strain localization and subsequent development of shear bands. Shear bands appear nearly always if dry granular material is subjected to shear. Its first study dates back to the nineteenth century and since then it was investigated in many different geometries and specially designed laboratory tests (e.g. plane strain, biaxial, and triaxial tests). Here we present numerical studies of axisymmetric triaxial tests, the most common laboratory tests in Geomechanics.
In a simplified picture, a triaxial test typically consists of a cylindrical specimen enclosed between two end platens and surrounded by a rubber membrane. An external pressure is applied on the membrane, either by placing the system into a pressurized fluid, or creating a relative vacuum inside the system. The end platens are pressed against each other in a controlled way, either with constant velocity (strain control) or with constant force (stress control). The force resulting on the platens, or the displacement rate of the platens is recorded, as well as the volume change of the specimen.
The triaxial test is an elementary test, performed to obtain mechanical properties of soils. Antifriction devices (lubricated end platens) were designed in order to suppress strong heterogeneous responses, such as barreling and localization of deformation along failure planes. In the past 20 years the study of localization patterns gained more attention and strain localization became an important research field, as experimental tools as Computed Tomography (CT) became available to study the internal structure of strained specimens . Such studies revealed complex localization patterns and shear band morphologies depending on the test conditions.
SIMULATION METHOD
=================
We used standard three-dimensional Distinct Element Method (also known as Molecular Dynamics) to perform simulations of strain controlled triaxial shear tests. As an advantage, contrary to the Finite Element Method (FEM) commonly used in simulations of soils, the Distinct Element Method (DEM) does not require any macroscopic *constitutive model*, instead it is based on a microscopic *contact model* (for a review see and references therein). We used the Hertz contact model with appropriate damping , combined with a frictional spring-dashpot model .
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Simulation setup. A cylindrical specimen placed between rigid horizontal platens and surrounded by an elastic membrane is subjected to axial load and confining pressure. []{data-label="fig:simmod"}](cylinder2 "fig:") ![ Simulation setup. A cylindrical specimen placed between rigid horizontal platens and surrounded by an elastic membrane is subjected to axial load and confining pressure. []{data-label="fig:simmod"}](bound2 "fig:") ![ Simulation setup. A cylindrical specimen placed between rigid horizontal platens and surrounded by an elastic membrane is subjected to axial load and confining pressure. []{data-label="fig:simmod"}](wires2 "fig:")
a b c
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The simulation setup can be seen on [Fig. \[fig:simmod\]]{}. An initially axisymmetric system of spherical grains (particles) was placed between two horizontal platens. The bottom platen was fixed. On the upper platen, having mass $M=10^{-3}\ kg$, an axial load was applied (as described later). The upper platen could not rotate along the vertical axis. The rotational inertia of the upper platen in tilting was $I=10^{-7}\ kg\:m^2$. The particles were surrounded by an “elastic membrane” composed of overlapping spheres having equal diameter $d_m=10^{-3}\ m$ and mass density ${\rho}_m=0.1 \cdot 10^3\ kg/m^3$, and initially forming a triangular lattice (see part (b) of [Fig. \[fig:simmod\]]{}). The rotational degree of freedom of the “membrane nodes” was frozen (i.e. they could not rotate).
The “membrane nodes” were interconnected with linear springs having zero base length and initial elongation $l_0=0.5 \cdot 10^{-3}\ m$ (equal to the initial distance of neighboring “membrane nodes”). The force $F_s$, acting between two “membrane nodes” connected with a spring, was calculated as $$F_s = {\kappa}_s l_s - {\gamma}_s v_s,$$ where ${\kappa}_s=0.5\ N/m$ and ${\gamma}_s=10^{-3}\ N\:s/m$ are stiffness and damping coefficients, $l_s$ is the spring’s elongation and $v_s$ is the relative velocity of the nodes. At any time the spring’s elongation is equal to the relative distance of the nodes. The stiffness of the springs was chosen such that the particles could not escape by passing through the membrane. Additionally a confining pressure ${\sigma}_c=0.5 \cdot 10^{3}\ N/m^2$ was applied on the membrane, by calculating the forces acting on the triangular facets formed by neighboring “membrane nodes” (see part (c) of [Fig. \[fig:simmod\]]{}). The effective external pressure in this setup is larger than ${\sigma}_c$, because the “membrane springs” have their own contribution as well, however, with the used parameters, this contribution is small compared to ${\sigma}_c$.
For simplicity we made no difference between particle-particle, particle-platen, and particle-membrane contacts. The normal $F_n$ and tangential ${\bf F}_t$ components of the contact force were calculated as $$\begin{aligned}
F_n &=&
{\kappa}_n {\delta}_n^{3/2}
- {\gamma}_n {\delta}_n^{1/2} {v_n}, \\
{\bf F}_t &=&
{\kappa}_t {\bmath{\delta}}_t
- {\gamma}_t {\bf v}_t,\end{aligned}$$ where ${\kappa}_{n}=10^{6}\ N/m^{3/2}$, ${\kappa}_{t}=10^{4}\ N/m$, ${\gamma}_{n}=1\ N\:s/m^{3/2}$, and ${\gamma}_{t}=1\ N\:s/m$ are the normal and tangential stiffness and damping coefficients, ${\delta}_n$ and ${\bmath{\delta}}_t$ are normal and tangential displacements, and ${v}_n$ and ${\bf v}_t$ are the normal and tangential relative velocities. The normal displacement ${\delta}_n$, the normal velocity ${v}_n$, and the normal force $F_n$ are one-dimensional quantities measured along to the normal vector of the contact plane, while the tangential displacement ${\bmath{\delta}}_t$, the tangential velocity ${\bf v}_t$, and the tangential force ${\bf F}_t$ are two-dimensional vectors embedded in the contact plane.
Knowing the relative position, the shape, and the size of the bodies in contact, the normal displacement can be calculated directly. Calculating the tangential displacement is much more complicated: The tangential velocity must be integrated during the lifetime of the contact. This integration must be performed in the contact plane. In our simulation program the tangential displacement ${\bmath{\delta}}_t$ was implemented as a 3D vector in the observational space. During integration, this vector was rotated as the local configuration changed, keeping it always in the contact plane.
The Coulomb friction law limits the frictional force to $\mu F_n$, where $\mu=0.5$ is the coefficient of friction. To allow for sliding contacts, we also limited the length of the tangential displacement to $\mu F_n / {\kappa}_t$, shortening the displacement vector accordingly. This frictional spring-dashpot model implements both sliding and static friction.
Our simulations are run at zero gravity. After calculating the interaction forces, and adding the external load and confining pressure, the motion of bodies (grains, “membrane nodes”, and upper platen) is calculated by solving numerically the Newton equations using a given $\Delta t=10^{-6}\ s$ integration time step. The translational motion is calculated with Verlet’s leap-frog method. The rotational state of bodies (given in quaternion representation) is integrated with Euler’s method.
In quasi-static processes as the one simulated by us, the vibration introduced by grain (spring) elasticity is basically undesired noise. We checked the noise level in our simulations and set the parameters to keep it low. The inverse of the eigenfrequency of all contacts, in both normal and tangential direction, is more than one order of magnitude larger than the integration time step.
In the first part of the simulation (preparation phase) a hard cylinder touching the internal side of the membrane was introduced, and thus the membrane was neglected. The particle system was built by randomly placing spheres into this cylinder. The maximum allowed initial grain overlap was $1\%$. To assure reasonable execution time, we started with a sufficiently tall system (usually $3$ time taller than the final system size). The particles (grains) were given equal mass density ${\rho_p}=7.5 \cdot 10^{3}\ kg/m^3$. The diameter of the spherical grains was taken from a Gaussian distribution with mean value $\left\langle d \right\rangle=0.9 \cdot 10^{-3}\ m$ and standard deviation $\Delta d=0.025 \cdot 10^{-3}\ m$, and cut at $4 \Delta d$ around the mean value.
[>l>l>l]{} & Upper platen velocity & Upper platen tilting\
(a) & Base & Enabled\
(b) & Base & Disabled\
(c) & Two times faster & Enabled\
(d) & Two times faster & Disabled\
\[tab:simrun\]
In the preparation phase the rotational degree of freedom of the upper platen was frozen and the coefficient of friction was set to zero. Each particle and the upper platen were given a velocity proportional to a contraction velocity $v_c=80 \cdot 10^{-2}\ m/s$ and their distance from the bottom. The system contracted until the upper platen’s velocity in grain-platen collisions decreased to zero. At this point an axial load $F_0=200 \cdot 10^{-3}\ N$ was switched on, which further contracted the system. After the system relaxed, the cylinder was removed, letting the membrane and the confining pressure carry the load. The axial load $F_0$ and the confining pressure ${\sigma}_c$ were chosen such that the system came to equilibrium without barreling. We prepared one sample having diameter $D=22 \cdot 10^{-3}\ m$ and height $H=46 \cdot 10^{-3}\ m$, containing $N_p=27000$ particles and $N_m=14904$ membrane nodes. The sample’s geometry factor $H/D \approx 2$, is similar to the typical geometry factors used in experiments.
For preparing sphere packings many different methods exist (e.g. ). Using the deposition method described above, the volume fraction at the end of the preparation phase was $f_0=0.643$, which is slightly larger than the random close packing value of identical spheres, as expected for an ensemble of spheres with size dispersion.
After the preparation phase the sample was compressed by moving the upper platen downward in vertical direction with constant velocity $u_c$ (strain controlled experiment). In the executed simulation runs we used two different velocities: $u_c=u_0=10^{-2}\ m/s$ (base value), and $u_c=2 u_0$ (two times faster). During compression, tilting of the upper platen was either enabled or disabled. We executed four different runs (see [Tab. \[tab:simrun\]]{}) starting from the same initial condition.
During the runs we measured the stress $\sigma$ on the upper platen, and calculated the stress ratio $\sigma/\sigma_0$, where $\sigma_0$ denotes the initial stress. The identification of the shear bands is a non-trivial task. One possibility is to generalize the method used in two dimensions , i.e., to calculate the shear intensity around the particles from the local deformation tensor. After doing this, we have realized that monitoring the rotational state of the particles is sufficient for the purpose of shear band identificiation, and it leads to the same results. Our observations are presented in the next section.
SIMULATION RESULTS
==================
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Stress-strain relation. The stress ratio ($\sigma/\sigma_0$, where $\sigma_0$ denotes the initial stress) measured on the upper platen is shown as function of the axial strain, for the executed simulation runs (see [Tab. \[tab:simrun\]]{}). (See inset for low stress ratios.) For the lower two curves (a, c) tilting of the upper plate was enabled, and for the upper two (b, d) it was disabled. []{data-label="fig:strainfunc"}](stressratiobw2a "fig:")
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
As the axial strain increases, the response of the granular sample (the stress ratio) increases until it reaches a peak value (see [Fig. \[fig:strainfunc\]]{}). This is a basic observation of triaxial shear tests of dense granular specimens . Strain localization in granular materials is followed by a decrease in load bearing capacity. Dense granular materials dilate during shear. Due to this, the load bearing particle chains collapse. Shear bands occur after peak failure and result in further decrease in strength.
According to [Fig. \[fig:strainfunc\]]{}, up to $15\%$ axial strain there is no significant difference in the stress-strain relation measured in the different simulation runs, indifferent of the strain rate and tilting of the upper platen. However, we observe a change after this point. At large axial strain values, when tilting of the upper platen was disabled, the stress ratio increases again.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Vertical cross sections. The snapshots were taken at the middle of the sample at $10\%$ axial strain from the (c) (left) and the (d) (right) simulations. The darkness (red content of color) is proportional to the rotational energy. []{data-label="fig:vercrosec"}](m3p46a "fig:") ![ Vertical cross sections. The snapshots were taken at the middle of the sample at $10\%$ axial strain from the (c) (left) and the (d) (right) simulations. The darkness (red content of color) is proportional to the rotational energy. []{data-label="fig:vercrosec"}](m8p46a "fig:")
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Taking cross sections of the sheared samples, and coloring the grains according to their rotational state we could visualize the shear bands (see [Fig. \[fig:vercrosec\]]{}). On these renderings the larger the rotational energy, the more red (or darker in black-and-white version) a particle gets. The fact that the rotational state of the grains identifies the shear bands, is due to the fact that the shear bands are characterized not only by dilation but also by rotation of the grains, which is known from both experiments and simulations .
Another important result is that internal instabilities can develop into a symmetry-breaking localized deformation along a failure plane when tilting of the upper platen is enabled, while nontilting platens act as a stabilizing factor resulting in two axisymmetric conical surfaces and complex localization patterns around them . This result is confirmed by similar experiments executed in micro-gravity , and proves that in the absence of reinforced axisymmetry, spontaneous symmetry breaking can take place as a result of internal instabilities .
CONCLUSIONS
===========
We executed triaxial shear test simulations using DEM. Different shear band morphologies known from experiments could be reproduced. To our knowledge it is the first time that these localization patterns were reproduced in DEM simulations. We showed that the shear bands can be identified with the rotational state of the grains, and symmetry breaking strain localization can develop if the symmetry is not enforced with nontilting platens. The agreement of our results with the experimental results is very good, even if the system size (number of particles) in our simulations is much smaller than in experiments.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This research was carried out within the framework of the “Center for Applied Mathematics and Computational Physics” of the BUTE, and it was supported by BMBF, grant HUN 02/011, and Hungarian Grant OTKA T035028, F047259.
|
---
abstract: 'We present a family of self-consistent, spherical, lowered isothermal models, consisting of one or more mass components, with parameterized prescriptions for the energy truncation and for the amount of radially biased pressure anisotropy. The models are particularly suited to describe the phase-space density of stars in tidally limited, mass-segregated star clusters in all stages of their life-cycle. The models extend a family of isotropic, single-mass models by Gomez-Leyton and Velazquez, of which the well-known Woolley, King and Wilson (in the non-rotating and isotropic limit) models are members. We derive analytic expressions for the density and velocity dispersion components in terms of potential and radius, and introduce a fast model solver in (), that can be used for data fitting or for generating discrete samples.'
date: 'Accepted 2015 August 7. Received 2015 August 6; in original form 2015 June 26'
title: A family of lowered isothermal models
---
methods: analytical – methods: numerical – stars: kinematics and dynamics – globular clusters: general – open clusters and associations: general – galaxies: star clusters: general
Introduction {#Sect:Intro}
============
The evolution of globular clusters (GCs) is the result of an interplay between stellar astrophysics (stellar and binary evolution, stellar mergers, etc.), dynamical two-body relaxation and the interaction with the tidal field of their host galaxy [@2003gmbp.book.....H]. Despite this plethora of physical processes at work on their respective time-scales, the [*instantaneous*]{} surface brightness profiles and kinematics of GCs are well described by relatively simple distribution function (DF) based models that need very few assumptions .
The relative simplicity of GC properties is owing to the absence of gas and non-baryonic dark matter and the collisional nature of their evolution, which drives them to tractable properties, such as spherical symmetry, isotropy and (quasi-)equipartition between different mass species [e.g. @1987degc.book.....S]. Because the relaxation time-scale of GCs is much longer than their dynamical time, their instantaneous properties can be described by models that satisfy the collisionless Boltzmann equation [see e.g. Chapter 8 in @2014dyga.book.....B].
Two-body interactions in GCs evolve the velocity distribution of stars towards a Maxwell-Boltzmann distribution, at least in the core, where the relaxation time-scale is short. Models with isothermal cores are therefore a good choice for fitting properties of GCs. An obvious starting point for a discussion on model choice is, therefore, the isothermal model. This model has an infinite spatial extent and infinite mass [@1939isss.book.....C] and to make the model applicable to real star clusters, the assumption of the idealized Maxwell-Boltzmann distribution of velocities needs to be relaxed. This can be done by changing the model such that stars have a finite escape velocity. @1954MNRAS.114..191W developed such a model by simply ‘lowering’ the (specific) energy $E$ by a constant. The DF, which describes the density in six-dimensional phase-space as a function of $E$, is then simply $f(E) = A\exp[-(E-\phi(\rt))/s^2]$, for $E\le\phi(\rt)$, and $f(E)=0$ for $E>\phi(\rt)$. Here $s$ is a velocity scale, which in the isothermal model equals the one-dimensional velocity dispersion and $E$ is reduced by the specific potential at the truncation radius $\rt$, $\phi(\rt)$. This truncation in energy mimics the role of tides due to the host galaxy, which makes it easier for stars to escape by reducing the escape velocity. The resulting models are nearly isothermal in the core, and have a finite mass and extent.
The DF of these models is discontinuous at $E=\phi(\rt)$. @1963MNRAS.125..127M and @1966AJ.....71...64K avoided this by subtracting a constant from the DF introduced by Woolley, which makes the models continuous at $E=\phi(\rt)$. Compared to the Woolley models, the density of stars near the escape energy is reduced in these models (hereafter referred to as King models), and they display a more gentle truncation of their density profile. The resulting, more extended, low-density envelopes make these models resemble real GCs more [for an in depth discussion on the effect of the truncation on the density profiles see @1977AJ.....82..271H]. The spherical, non-rotating limit of the models introduced by @1975AJ.....80..175W, hereafter called Wilson models, are models that are continuous both in the DF and its derivative. This is achieved by subtracting an additional term linear in $E$ from the DF. These models are yet more spatially extended than King models. For some GCs in Local Group galaxies, the Wilson models provide a better description of the observed surface brightness profiles compared to the King models (@2005ApJS..161..304M; @2012MNRAS.419...14C also show that models that are more extended than King models better describe the surface brightness profiles of some GCs).
An additional outcome of the two-body relaxation process is that it drives the velocity distribution of the stars towards isotropy. Isotropic models, defined by a DF that only depends on $E$, are therefore a natural choice for clusters that are in late stages of their evolution, near dissolution. At early phases, however, the velocity distribution in the outer parts is expected to be radially anisotropic. This is, first, because the (incomplete) violent relaxation process that takes place during their formation results in a halo of radial orbits [@1967MNRAS.136..101L]. Secondly, two-body ejections from the dense core populate the halo with radial orbits on a two-body relaxation time-scale [@1972ApJ...173..529S]. @1963MNRAS.125..127M proposed a separable DF, dependent on $E$ and on the (specific) angular momentum $J$ to introduce radial anisotropy (hereafter referred to as Michie-King models). The DF of the Michie-King models is the product of the isotropic DF with an exponential term with a $J^2$ dependent argument. This is similar to Eddington’s method of including radial anisotropy in the isothermal model [@1915MNRAS..75..366E]. As a result, the inner parts of the models remain approximately isothermal and isotropic, which is appropriate to GCs because there the relaxation time is short, and anisotropy becomes important at larger distances from the centre. Near the truncation radius the models become isotropic again as a result of the energy truncation. The latter property has a somewhat coincidental resemblance to GCs, because near the Jacobi radius the orbits of stars gain angular momentum due to the interaction with the (tri-axial) tidal potential [@1992ApJ...386..519O], therewith suppressing the amount of radial anisotropy near the truncation energy. A review of the effect of anisotropy on model properties can be found in .
In real GCs, which contain multiple mass components, the relaxation process drives the systems towards equipartition, resulting in the heavier components being more centrally concentrated, a state which is often referred to as mass segregated. King models with different mass species were first introduced by @1976ApJ...206..128D and have since been applied to take into account the effects of mass segregation in mass-modelling efforts of Galactic GCs (e.g. M3: @1979AJ.....84..752G, Omega Cen: and larger samples of GCs: @1993ASPC...50..357P [@2012ApJ...755..156S]). Mass segregation is important for almost all of the Galactic GCs, given their short relaxation time-scales, relative to their ages [@H61; @2011MNRAS.413.2509G]. Approximating multimass systems by single-mass models can lead to severe biases in the inferred properties of GCs [@2015MNRAS.448L..94S; @sollima15] and it is, therefore, desirable to have the ability to include multiple mass components in a dynamical model of a GC.
It is our aim to develop a family of models that capture the general behaviour of collisional systems discussed above, and whose properties can be varied by parameters that can be constrained by observational data. showed that the expressions for the DF of the isotropic Woolley, King and Wilson models can be generalized by a DF in which the exponential function of $E$ is reduced by the leading orders of its series expansion. This approach was further generalized by @2014JSMTE..04..006G [hereafter GV14], who showed that solutions [*in between*]{} these models can be obtained (these models are briefly reviewed in Section \[ssec:iso\]). In this paper we extend the models of GV14 to allow for the presence of (radially biased) pressure anisotropy and multiple mass components. We present an efficient Poisson solver in to facilitate the use of these models in fitting observational data, and in drawing samples from the models, which can be used as initial conditions for numerical simulations.
The paper is organized as follows: in Section \[sec:model\], we define the models and in Section \[sec:properties\], we illustrate their main properties. In Section \[sec:limepycode\], we present the code [^1] and our conclusions and a discussion are presented in Section \[sec:conclusion\]. Supporting material can be found in the appendices.
Model definition and scaling {#sec:model}
============================
Single-mass models
------------------
### Distribution function (DF) {#ssec:df}
The DF of the single-mass family of models is $$f(E,J^2) =
\displaystyle A\exp\left(-\frac{J^2}{2\ra^2 s^2}\right)\Eg\left(g, -\frac{E-\phi(\rt)}{s^2}\right)
\label{eq:dfani}$$ for $E\le\phi(\rt)$, and 0 for $E>\phi(\rt)$. The DF depends on two integrals of motion: the specific energy $E = v^2/2 + \phi(r)$, with $v$ the velocity and $\phi(r)$ the specific potential at distance $r$ from the centre, and the norm of the specific angular momentum vector $J = |{\textbf{\emph{r}}}\times {\textbf{\emph{v}}}|=rv\sin\vartheta$, where ${\textbf{\emph{r}}}$ and ${\textbf{\emph{v}}}$ are the position vector and velocity vector, respectively, and $\vartheta$ is the angle between them. The energy $E$ is lowered by the potential at the truncation radius $\phi(\rt)$.
In equation (\[eq:dfani\]) we introduced the function $$\begin{aligned}
\Eg(a, x) =
\begin{cases}
\exp(x) & a=0 \\
\displaystyle\exp(x) P(a, x) & a>0,
\end{cases}
\label{eq:eg}\end{aligned}$$ where $P(a, x) \equiv \gamma(a, x)/\Gamma(a)$ is the regularized lower incomplete gamma function (see Appendix \[app:gamma\] for the definition of this function and its properties). Combining the exponential and the incomplete gamma function into a single function $\Eg(a, x)$ has advantages in deriving the model properties (see GV14 and Appendix \[AppD:Eg\] for details on the behaviour of this function).
A model is specified by three parameters: the central potential, which is a required boundary condition for solving Poisson’s equation and defines how concentrated the model is; the anisotropy radius $\ra$, which determines the amount of anisotropy present in the system (for increasing $\ra$ the models are more isotropic); the truncation parameter $g$, which controls the sharpness of the truncation of the model (this parameter is called $\gamma$ in GV14). The physical units of a model are defined by two scales: the velocity scale $s$, and the normalization constant $A$, which sets the phase-space density and therewith the total mass $M$. For more information regarding scales and parameters of the models we refer the reader to Section \[ssec:scaling\].
The isotropic models ($\ra\rightarrow\infty$) and their properties are discussed in detail in GV14. For these models, and integer values of $g$, three well-known families of models are recovered: when $g=0$ we retrieve the @1954MNRAS.114..191W models, for $g=1$ we recover the King models [@1963MNRAS.125..127M; @1966AJ.....71...64K], and for $g=2$ we find the (isotropic, non-rotating) Wilson models [@1975AJ.....80..175W][^2]. In practice, the models defined by equation (\[eq:dfani\]) are radially anisotropic for $\ra\lesssim\rt$, because of the $J^2$ dependence in the first exponential. When $g=1$, the DF is the Michie-King model [@1963MNRAS.125..127M], which is often used to fit GC data .
The potential $\phi(r)$ is found by solving Poisson’s equation. For the self-consistent problem we consider here, the potential is completely determined by the density $\rho$ associated with the DF. This problem is non-linear, because the DF depends on the potential. Since the models defined by equation (\[eq:dfani\]) are spherically symmetric, Poisson’s equation is $$\begin{aligned}
\frac{1}{r^2}\frac{\dr}{\dr r}\left(r^2\frac{\dr\phi}{\dr r}\right)& = 4\pi G\rho,\end{aligned}$$ where the density is obtained by means of an integration of the DF over all velocities $$\rho = \int \dr^3 v \, f(E, J^2).
\label{eq:rho}$$ In Sections \[ssec:iso\] and \[ssec:ani\], we derive analytic expressions for $\rho$ as a function of $\phi$ and $r$. Note that only in the anisotropic case the dependence on the radial coordinate $r$ is both implicit (through $\phi$, as in the isotropic case), and explicit, i.e. $\rho(\phi, r)$. Having analytic expressions for $\rho(\phi, r)$, avoids the need of solving a double integral at each radial step, making it significantly faster to obtain the solution to Poisson’s equation. In the next section we introduce a convenient set of units to solve the model.
### Scaling and units {#ssec:scaling}
To solve Poisson’s equation, we use a dimensionless (positive) energy $\ehat = \phihat - \khat$, with dimensionless potential $\phihat =
(\phi(\rt) - \phi)/s^2$, and $\khat \equiv v^2/(2 s^2)$. As in @1966AJ.....71...64K, we consider the dimensionless density by normalizing $\rho$ to its central value, i.e. $\rhohat
= \rho/\rho_0$. In this way, Poisson’s equation in dimensionless form reads $$\frac{1}{\rhat^2}\frac{\dr}{\dr\rhat}\left(\rhat^2\frac{\dr\phihat}{\dr\rhat}\right) = -9\rhohat.$$ The dimensionless radius is now defined by the other scales: $\rhat =
r/\rs$, with $\rs^2 = 9 s^2/(4\pi G\rho_0)$. This radial scale was introduced in @1966AJ.....71...64K and is often referred to as the King radius. The factor of 9 was introduced to give $\rs$ the meaning of a core radius, because for models with moderately high central concentration, the projected density at $\rs$ is about one half of its central value.
The Poisson equation is solved by assuming the boundary conditions at $\rhat = 0$: $\phihat = \phihat_0$ and $\dr\phihat/\dr r = 0$. As mentioned in Section \[ssec:df\], the central potential $\phihat_0$ is one of the parameters that define the model[^3].
### Isotropic models {#ssec:iso}
We first briefly review the isotropic version of these models, as introduced by GV14. Many quantities can be derived from the DF. The density $\rho$ is found by integrating the DF over all velocities (equation \[eq:rho\]) and the pressure is found by taking the second velocity moment of the DF [^4] $$\begin{aligned}
\rho &= (2\pi s^2)^{3/2}A\rhoint, \label{eq:rhodef} \\
\rho\mvsq &= (2\pi s^2)^{3/2}s^2A \vsqint.\end{aligned}$$ Here $\sigma^2=3\sigmaoned^2$ is the mean-square velocity, $\sigmaoned$ is the one-dimensional velocity dispersion and we introduce a dimensionless density integral ($\rhoint$) and a dimensionless pressure integral ($\vsqint$) $$\begin{aligned}
\rhoint &= \frac{2}{\sqrt{\pi}}\int_0^{\phihat} \dr \khat \, \khat^{1/2}\Eg(g, \phihat - \khat) = \Eg(\gthree, \phihat), \label{eq:rhointiso}\\
\vsqint &= \frac{4}{\sqrt{\pi}}\int_0^{\phihat} \dr \khat \, \khat^{3/2}\Eg(g, \phihat - \khat) = 3\Eg(\gfive, \phihat). \label{eq:vsqintiso}\end{aligned}$$ The results of these integrations follow straightforwardly from the convolution formula of the $\Eg(a,x)$ function (equation \[EG\_convolution\]). An alternative derivation by means of fractional calculus is presented in Appendix \[app:fractional\]. The dimensionless density that appears in Poisson’s equation is therefore $\rhohat=\rhoint/\rhointnull$, where $\rhointnull$ is the result of equation (\[eq:rhointiso\]) evaluated at $\phihat=\phihat_0$. The dimensionless mean-square velocity is found from $\sigmahat^2 = \sigma^2/s^2 = \vsqint/\rhoint$.
### Anisotropic models {#ssec:ani}
Here we present the relevant quantities for the anisotropic case. The details of the derivations can be found in Appendix \[app:series\], and the derivations by means of fractional calculus can be found in Appendix \[app:fractional\]. To solve the anisotropic models, we introduce $t=\cos\theta$, such that we can write the integral over the angles as $4\pi\int_0^1 \dr t$. We further introduce $\rrahat= \rhat/\rahat$ such that the density integral becomes
$$\begin{aligned}
\rhoint&=\frac{2}{\sqrt{\pi}}\!\int_0^{\phihat}\!\dr\khat\int_{0}^{1}\! \dr t \, \exp\left[\khat \phat^2 (t^2-1)\right] \khat^{1/2}\Eg(g, \phihat - \khat) \nonumber \\
&=\frac{2}{\sqrt{\pi}} \int_0^{\phihat}\dr\khat\, \frac{F(\phat\khat^{1/2})}{\phat}\Eg(g, \phihat - \khat).
\label{eq:rhointani}\end{aligned}$$
Here $F(x)$ is Dawson’s integral and we refer to Appendix \[App:Dawson\] for some properties of this function. To first order, $F(x) \propto x$, and we thus find that for large $\rahat$, i.e. small $\phat$, equation (\[eq:rhointani\]) converges to the integral of the isotropic model (equation \[eq:rhointiso\]). The solution of the integration gives $\rhoint$ as a function of $\phihat$ and $\rrahat$ $$\rhoint \!=\! \frac{\Eg( \gthree, \phihat)}{1+\phat^2}+\frac{\phat^2}{1+\phat^2} \frac{\phihat^{g+{\tfrac{3}{2}}}{_1}F_1(1, \gfive, -\phihat\phat^2)}{\Gamma(g+{\tfrac{5}{2}})}.
\label{eq:rhointanires}$$ Here $\hyp(a,b, x)$ is the confluent hypergeometric function whose properties are given in Appendix \[1F1\]. For small $\phat$, the second term on the right-hand-side goes to zero and the solution converges to the isotropic result of equation (\[eq:rhointiso\]). This expression for the density integral allows for fast computations of the right-hand-side of Poisson’s equation and facilitates efficient solving of the anisotropic models.
For the anisotropic models, we need to calculate both the radial and the tangential[^5] components of the pressure tensor, as well as the total pressure. The radial and tangential component of the velocity vector are defined as $\vr =
v\cos\theta$ and $\vt = v\sin\theta$ and for the corresponding integrals we find $$\begin{aligned}
\displaystyle\vrsqint&\hspace{-0.cm}=\hspace{-0.cm}\frac{4}{\sqrt{\pi}}\!\int_0^{\phihat}\!\!\!\dr\khat\!\int_{0}^{1}\!\!\!\dr t\,\exp\!\left[\rrahat^2\khat(t^2\!\!-\!1)\right]t^2\khat^{3/2}\Eg(g, \phihat-\khat),\\
\displaystyle\vtsqint&\hspace{-0.cm}=\hspace{-0.cm}\frac{4}{\sqrt{\pi}}\!\int_0^{\phihat}\!\!\!\dr\khat\!\int_{0}^{1}\!\!\!\dr t\,\exp\!\left[\rrahat^2\khat(t^2\!-\!1)\right](1\!-t^2)\khat^{3/2}\Eg(g, \phihat-\khat),\\
\displaystyle\vsqint&\hspace{-0.cm}=\hspace{-0.cm}\frac{4}{\sqrt{\pi}}\!\int_0^{\phihat}\!\!\!\dr\khat\!\int_{0}^{1}\!\!\!\dr t\,\exp\!\left[\rrahat^2\khat(t^2\!-\!1)\right]\khat^{3/2}\Eg(g, \phihat-\khat).\end{aligned}$$ By carrying out these integrals as described in Appendices \[sapp:sigma\_deriv\] and \[sapp:sigma\_deriv\_frac\], we obtain $$\begin{aligned}
\hspace{-0.25cm}\vrsqint &\!=\!\frac{\Eg(\gfive, \phihat)}{1+\phat^2}\!+\!\frac{\phat^2}{1+\phat^2} \frac{\phihat^{g+{\tfrac{5}{2}}}\hyp(1, g\!+\!{\tfrac{7}{2}}, \!-\phihat\phat^2)}{\Gamma(g+{\tfrac{7}{2}})}, \label{eq:vrsqintres} \\
\hspace{-0.25cm}\vtsqint &= \frac{\Eg(\gfive, \phihat)}{1+\phat^2}\frac{2}{(1+\phat^2)} + \frac{2\phat^2}{1+\phat^2}\frac{\phihat^{g+{\tfrac{5}{2}}}}{\Gamma(g+{\tfrac{7}{2}})} \nonumber\\
&\times\left[ \frac{\hyp(1, g+{\tfrac{7}{2}}, -\phihat\phat^2)}{1+\phat^2} + \hyp(2, g+{\tfrac{7}{2}}, -\phihat\phat^2) \right], \label{eq:vtsqintres} \\
\hspace{-0.25cm}\vsqint&\!=\!\frac{\Eg(\gfive, \phihat)}{1+\phat^2}\frac{(3+\phat^2)}{(1+\phat^2)} +\frac{\phat^2}{1+\phat^2} \frac{\phihat^{g+{\tfrac{5}{2}}}}{\Gamma(g+{\tfrac{7}{2}})} \nonumber\\
\times&\left[\!\frac{3+\phat^2}{1+\phat^2}{_1}F_1(1, g\!+\!{\tfrac{7}{2}},\!-\phihat\phat^2)\!+\!2{_1}F_1(2, g\!+\!{\tfrac{7}{2}},\!-\phihat\phat^2)\right]\!. \label{eq:vsqintres} \end{aligned}$$ Note that the expression for $\vrsqint$ resembles the expression for $\rhoint$ of equation (\[eq:rhointanires\]), in the sense that the functional form is the same, but all arguments and the power index that include $g$ are increased by 1. We already saw a similar resemblance between $\rhoint$ and $\vsqint$ in the isotropic case (equations \[eq:rhointiso\] and \[eq:vsqintiso\], respectively).
With these expressions for the density and pressure integrals, we defined most of the properties of these models that are of direct relevance for comparison to data. In Section \[ssec:project\], we discuss how the projected quantities can be derived.
### Limits {#sssec:limits}
In this section we consider some limits of the models. In the core, where $\phat$ is small ($\rhat\ll\rahat$), the model is isotropic. This is because the second terms in equations (\[eq:rhointanires\]), (\[eq:vrsqintres\]), (\[eq:vtsqintres\]) and (\[eq:vsqintres\]) vanish due to the multiplication by $\phat^2$.
Near the truncation radius the models behave like polytropes and are, therefore, also isotropic, because $$\begin{aligned}
\lim_{\phihat\rightarrow0} \hyp(1, a\!+\!1,-\phat^2\phihat) &= 1,\\
\lim_{\phihat\rightarrow0} \Eg(a, \phihat) &= \frac{\phihat^a}{\Gamma(a+1)},\end{aligned}$$ and the $\rrahat$ dependence disappears. In this regime, we find $$\begin{aligned}
\lim_{\phihat\rightarrow0} \rhohat & = \frac{\phihat^{g+3/2}}{\Gamma(g+5/2)}, \label{eq:poly}\\
\lim_{\phihat\rightarrow0} \rhohat \sigmahat^2 & = 3\frac{\phihat^{g+5/2}}{\Gamma(g+7/2)},\\
\lim_{\phihat\rightarrow0} \rhohat \sigmarhat^2 & = \frac{1}{3}\lim_{\phihat\rightarrow0} \rhohat \sigmahat^2 = \frac{1}{2} \lim_{\phihat\rightarrow0} \rhohat \sigmathat^2.\end{aligned}$$ This suppression of the velocity anisotropy near the truncation radius results naturally from the mathematical definition of the truncation, and is appropriate for tidally truncated systems [@1992ApJ...386..519O]. In $N$-body models a tangentially biased anisotropy is observed near $\rt$ [@sollima15], which cannot be reproduced by the models presented here. However, it is likely that most of the stars with tangentially biased velocities are above the escape energy, so-called potential escapers and these are not considered by these models, nor any other model we are aware off.
Models with $\phihat_0\rightarrow0$ are close to pure polytropes over their entire radial range. In this regime, and for $g=7/2$ (i.e. a polytropic index $n=5$, equation \[eq:poly\]), we recover the @1911MNRAS..71..460P model, which is infinite in extent ($\rho \propto r^{-5}$ at large radii), but finite in mass. Polytropes with $n\ge 5$ (i.e. $g\ge7/2$) are infinite in extent and will not be considered here. For $g<7/2$ models can have a finite $\rt$ depending on both $\phihat_0$ and $\ra$ (see GV14 and Section \[sec:properties\]).
In the cores of models with $\phihat_0\gg0$ the DF approaches the isothermal sphere, because $$\begin{aligned}
\lim_{\phihat\rightarrow\infty} \Eg(a,\phihat) & =\exp(\phihat).\end{aligned}$$ Models with $g\rightarrow\infty$ also approach the isothermal sphere. To conclude, these models approach the isothermal sphere in the limit of $\phihat_0\rightarrow\infty$, independent of $g$, but also in the limit of $g\rightarrow\infty$, independent of $\phihat_0$.
Multimass models {#ssec:multimassmodel}
----------------
It is possible to consider models with multiple mass components, by considering the DF as the sum of DFs of the form of equation (\[eq:dfani\]), each of which describes a different mass component with a mass-dependent velocity scale parameter. The first to do this were @1976ApJ...206..128D, who calculated multimass King models. For a multimass model with $\Ncomp$ mass components, $2\Ncomp+2$ parameters are required in addition to the ones introduced in Section \[ssec:df\] for single-mass models. These additional parameters are the values for the component masses $m_j$, the amount of mass in each component $M_j$, $\delta$ and $\eta$. The latter two parameters set the mass dependence of the velocity scale $s_j$ and the anisotropy radius of each component $\rajhat$, for which we adopt power-law relations $$\begin{aligned}
s_j &= s\mu_j^{-\delta}, \label{eq:delta} \\
\rajhat &= \rahat\mu_j^{\eta}. \label{eq:eta}\end{aligned}$$ Here $\mu_j= m_j/\bar{m}$ is the dimensionless mass of component $j$ and $\bar{m}$ is the central density weighted mean-mass $$\bar{m} = \frac{\sum_j m_j\rho_{0j}}{\sum_j \rho_{0j}}.$$ Note that in the multimass models, the values of $s$ and $\rahat$ are the velocity scale and anisotropy radius corresponding to $\bar{m}$. The definitions of $\delta$ and $\eta$ are such that the anisotropy profiles are approximately mass independent when $\delta=\eta$ (see equation \[eq:dfani\]). The typical values considered for these parameters are $\delta = 1/2$ and $\eta = 0$.
We notice that in the limit of infinite $\phihat_0$ the velocity scale $s_j$ approaches the one-dimensional velocity dispersion of mass component $j$, $\sigma_{{\rm 1d},j}$, hence the traditional assumption for $\delta=1/2$ implies equipartition ($m_j s_j^2 =\bar{m}\sigma_{{\rm 1d},j}^2=$ constant). However, it is important to keep in mind that for multimass models with typical and realistic values of $\phihat_0$, the velocity dispersion of each component in the centre is smaller than $s_j$ and, therefore, there is no equipartition (see Section \[ssec:delta\] and @1981AJ.....86..318M [@2006MNRAS.366..227M]).
To solve a multimass model self-consistently, we compute the density for each mass component as in equation (\[eq:rho\]) and add all components on the right-hand-side of Poisson’s equation. The detailed procedure is described in @1979AJ.....84..752G, and here we only briefly summarize the required steps. The dimensionless Poisson equation to solve is $$\hat{\nabla}^2\phihat = -9\sum_j\alpha_j\rhohat_j,$$ where $\alpha_j$ is the ratio of the central density of the $j$-th mass component to the total central density, such that $$\sum_j \alpha_j = \sum_j \frac{\rho_{0j}}{\rho_0} = 1$$ and $$\rhohat_j = \frac{\rho_j}{\rho_{0j}} = \frac{\rhoint(\mu^{2\delta}\phihat,\hat{r})}{{\rhoint(\mu^{2\delta}\phihat_0, 0)}}.
\label{eq:rhointratio}$$ By considering multiple mass components, we introduce an eigenvalue problem in the solution of Poisson’s equation, because the values of $\rho_{0j}$ that yield the desired ${M}_j$ values are not known a priori. Therefore, as a first step to solve the model, we assume that $\alpha_j = {M}_j/\sum_j M_j$, and we obtain the solution by iteration (see Section \[sec:limepycode\] for details).
Normalization and potential energy
----------------------------------
In solving the models we have chosen to define the dimensionless quantities in terms of the density scale $\rho_0$ and the velocity scale $s$ (Section \[ssec:scaling\]). In some cases it is useful to have an expression for the normalization constant $A$ in the DF (equation \[eq:dfani\]), for example, when fitting models to discrete data. From equation (\[eq:rhodef\]) we find that $A$ relates to the other scales as
$$\begin{aligned}
A & = \frac{\rho_0}{(2\pi s^2)^{3/2}\rhointnull}.\end{aligned}$$
For the multimass models there is a normalisation for each component, $A_j$. The relation with the mass scale $\ms=M/\mhat$ is $\ms=\rs^3\rho_0=\rs^3(2\pi s^2)^{3/2} A\rhointnull$, where we introduced $\mhat = \int\rhohat\dr^3\rhat$.
The total dimensionless (positive) gravitational energy $\Uhat$ of the model is calculated from integrating the potential [@1966AJ.....71...64K] $$\begin{aligned}
\Uhat &= \frac{1}{2} \int_{0}^{\mhat} \dr \hat{m} \, \phihat + \frac{\hat{G}\mhat^2}{2\rthat}.
\label{eq:Uhat}\end{aligned}$$ The second term has to be added because $\phihat$ is a lowered potential. Note that this integration of $\phihat$ over mass is readily obtained from solving Poisson’s equation.
Model properties {#sec:properties}
================
Single-mass models
------------------
### Density and velocity dispersion profiles
In Fig. \[fig:rhovel\], we show the density profiles, the velocity dispersion profiles, and the anisotropy profiles for isotropic and anisotropic models with different values of the truncation parameter $g$. The anisotropy profile is computed from $\sigmat^2$ and $\sigmar^2$ as $$\beta = 1-\frac{\sigmat^2}{2 \sigmar^2}.
\label{eq:beta}$$ In the case of isotropy $\beta = 0$, $0<\beta\le1$ indicates radially biased anisotropy (with $\beta=1$ implying fully radial orbits) and for tangentially biased anisotropy $\beta<0$. Because $\beta$ is a measure of anisotropy locally, we also quantify the total amount of anisotropy with $$\kappa = \frac{2K_{\rm r}}{K_{\rm t}},
\label{eq:kappa}$$ introduced by @1981SvA....25..533P. Here $K_{\rm r}$ and $K_{\rm t}$ are the radial and tangential components of the kinetic energy, respectively. For isotropic models $\kappa =1$, and for radially biased anisotropic models $\kappa>1$. @1981SvA....25..533P found that for $\kappa>1.7\pm0.25$ radial orbit instability occurs. We use this criterion to check the stability of the anisotropic models we calculate.
![Dimensionless density profile (top), velocity dispersion profile (middle) and anisotropy profile (bottom) for models with different truncation parameters $g$ (different colours). Isotropic models are shown with solid lines, anisotropic models with $\rahat/\rhhat=1.5$ with dashed lines.[]{data-label="fig:rhovel"}](rho){width="\columnwidth"}
In Fig. \[fig:rhovel\] we show anisotropic models characterized by $\rahat/\rhhat=1.5$. Because the (dimensionless) half-mass radius $\rhhat$ is not known before solving the model, we find the value of $\rahat$ that gives the correct ratio $\rahat/\rhhat$ iteratively. We see that all models are approximately isothermal in the centre. When increasing $g$, the models become more extended. Including radial anisotropy also results in a larger truncation radius.
Note that, with this choice of $\rahat/\rhhat$, the maximum value assumed by the anisotropy function for $g = 0$ (Woolley model) is about $0.4$, while for $g = 2$ (Wilson model) it is possible to achieve $\beta\simeq1$ in the outer parts of the model. This dependence of the maximum value of $\beta$ on $g$ does not imply that there are differences in the total amount of anisotropy: for all the anisotropic models shown in Fig. \[fig:rhovel\], indeed, we find $\kappa\simeq1.2$. The ability to calculate models with more radial orbits (larger $\beta$) without increasing the radial component of the total kinetic energy is important to keep in mind when considering other physical effects that can enhance or suppress the amount of radial orbits, such as the presence of a dark matter halo [@2013MNRAS.428.3648I] and the galactic tides [@1992ApJ...386..519O]. In a forthcoming study, we quantify the presence of radial orbits in direct $N$-body models of tidally limited clusters [@2016MNRAS.462..696Z].
### DF, density of states and differential energy distribution
In the top panels of Fig. \[fig:dmde\_comp\], we show the DF as a function of $\ehat$, for isotropic models, with different values of $g$ and $\phihat_0$. In the middle panels we show the density of states $\gdos(\ehat)$, which is the phase-space volume per unit of energy (see equation \[densityofstates\] for a definition). The bottom panels display the differential energy distribution $\dr \mhat/\dr \ehat$, which is the amount of mass per unit energy. For the isotropic models it is simply the product of $f(\ehat)$ and $\gdos(\ehat)$ (equation \[eq:dmde\]). Details on how this was derived for the models presented here, and on the procedure for anisotropic models, are given in Appendix \[app:dmde\]. A general discussion on the differential energy distribution can be found in chapter 4 of @BT1987.
{width="\textwidth"}
In the first and third columns (linear $x$-scale), we recognize the exponential behaviour of $f(\ehat)$ for the $g=0$ model, and the exponential behaviour at high $\ehat$ for $g>0$ models. From the second and fourth column, we see that at low $\ehat$, the DF scales as $f(\ehat) \propto \ehat^g$, which corresponds to the regime where the models behave as polytropes. From Fig. \[fig:dmde\_comp\] it is also evident that when $\ehat \simeq \phihat_0$, the model behaviour is independent of $g$.
From the differential energy distribution, we see that only for $g=0$ there is a non-zero mass at $\ehat=0$. For models with $g>0$, the truncation is such that $f(\ehat=0)=\dr \hat{M}/\dr \ehat\left|_{\ehat=0}\right.=0$. These models give rise to more realistic looking density profiles, but in real GCs the number of particles with the escape energy is not zero [@2001MNRAS.325.1323B], because of the gradual scattering of particles over the critical energy for escape by two-body relaxation, and because of the finite time for stars to escape from the Jacobi surface imposed by the galactic tidal field [@2000MNRAS.318..753F].
### Finite and infinite models
As discussed in Section \[sssec:limits\], there are no models with finite extent if $g\ge3.5$. GV14 showed that the maximum value $g_{\rm
max}$ to get models with a finite extent depends on $\phihat_0$, and $g_{\rm max}=3.5$ holds in the limit of $\phihat_0\rightarrow0$. GV14 show that all their isotropic models are finite for $g \lesssim 2.1$.
We note that there is a class of isotropic models that are finite in extent, but are not relevant to star clusters, and that are not discussed in GV14. This is illustrated in Fig. \[fig:rho\_multi\], where we show density profiles for models with different $\phihat_0$ and $g=2.75$. The model with $\phihat_0=3$ converges to a finite $\rthat$ and has a density profile comparable in shape to the ones shown in Fig. \[fig:rhovel\]. The model with $\phihat_0=9$ is infinite in extent, and only plotted up to $\log\,\rhat=10$. The models with $\phihat_0=5$ and $\phihat_0=7$ are finite, but show a sharp upturn in the density profile at large radii, which causes them to have a lot of mass in the envelope, but little energy, which makes these models inapplicable to real stellar systems. Their extreme density contrast between the core and the extended halo makes these models perhaps applicable to red giant stars [see the density profiles for red giants in @2012ApJ...744...52P]. To quantify the boundary between models with, and without the core-halo structure, we compute the ratio of the dimensionless virial radius $\rvhat=-G\mhat^2/(2\Uhat)$ over $\rhhat$ for a grid of models with $0\leq\phihat_0\leq20$ and $0\leq g\leq3.5$, and we show the result as contours in Fig. \[fig:rvrh\_phi0\]. We find that for a given $g(\phihat_0)$, when increasing $\phihat_0(g)$, the change in $\rvhat/\rhhat$ is large and abrupt once the models develop the core-halo structure. We identify the value of $\rvhat/\rhhat\simeq0.64$ as the one separating the two classes of models. In the remaining discussion, we only consider models with $\rvhat>0.64\rhhat$.
When considering anisotropic models, we find that for each $\phihat$ and $g$, there is a minimum value of $\rahat$ that can be used to obtain a model that has a finite extent. We note that models with infinite extent can have a finite total mass, but because we envision an application of these models to tidally limited systems we do not consider them here. In Fig. \[fig:ramin\], we show the minimum $\rahat$ for which models are finite in extent. The lines show, as a function of $\phihat_0$, and for different $g$, the values of $\rahat$ that are needed to get $\rthat = 10^7$. Note that this minimum for $\rahat$ goes up approximately exponentially with $\phihat_0$, and also increases with $g$.
![Density profiles for isotropic models with truncation parameter $g=2.75$. Models with $\phihat_0 = 3$, $5$, and $7$ (blue, green, and red line, respectively) have a finite truncation radius, but only the model with $\phihat_0=3$ is relevant when describing GCs; the model with $\phihat_0=9$ (light blue line) is infinite in extent.[]{data-label="fig:rho_multi"}](rho_multi){width="\columnwidth"}
![Ratio of dimensionless virial radius to half-mass radius, $\rvhat/\rhhat$, for models with different $\phihat_0$ and $g$. We consider models with $\rvhat/\rhhat\ge0.64$ as relevant to describe star clusters. Models that have an infinite $\rthat$ are plotted as $\rvhat/\rhhat=0$ (i.e. they correspond to the white region in the plot).[]{data-label="fig:rvrh_phi0"}](rvrh_phi0){width="\columnwidth"}
![Minimum $\rahat$ for finite sized models, for different $\phihat_0$ and $g$.[]{data-label="fig:ramin"}](ramin){width="8cm"}
### Entropy
@1966AJ.....71...64K suggested that in the process of core collapse, clusters evolve along a sequence of models with increasing central concentration. He also noted that his models are probably not able to describe the late stages of core collapse, because for large central concentration the variation in energy due to a change in the central concentration occurs in the envelope, and not in the core. Further support for this idea comes from @1968MNRAS.138..495L, who showed that a maximum in entropy occurs at $\phihat_0 \simeq 9$ for both Woolley and King models at constant mass and energy. The entropy of a self-gravitating system is obtained from the DF as
$$S = -\int \dr^3 r \, \dr^3v \, f \ln f \ .
\label{eq:entropy}$$
Because two-body encounters continuously increase the total entropy of the system, we do not expect King models to be able to describe a system in the late stages of core collapse (i.e. $\phihat_0\gtrsim9$). This was confirmed by Fokker-Planck models of isolated star clusters going into core collapse [@1980ApJ...242..765C], for which the entropy increase follows that of King models with increasing central concentration, up to a value of $\phihat_0 \simeq 9$, but then it continues to rise during the gravothermal catastrophe. Cohn concluded that in this regime, the isotropic King models are not able to describe the entropy evolution in his simulations.
In Fig. \[fig:entropy\], we show the entropy $S$, computed as in equation (\[eq:entropy\]), for the isotropic King models (black solid line), which shows a maximum at $\phihat_0 \simeq 9$. We also show the entropy curves for different values of $g$, and for selected anisotropic models. All models are scaled to the same $M$ and total energy $\Etot$, in the conventional $N$-body units: $G=M=-4\Etot=1$ . For $0\lesssim \phihat_0 \lesssim 1$, the anisotropic models are similar to their corresponding isotropic models, and therefore they have similar entropy. From this plot it is apparent that evolution at constant mass and energy, and with increasing entropy is possible beyond $\phihat\gtrsim9$ if $g$ is increased, and/or $\ra$ is decreased (i.e. including more anisotropy). A local maximum in entropy is seen near $\phihat_0\simeq17$. Similar oscillating behaviour of the entropy was found for isothermal models in a non-conducting sphere and we refer to @1968MNRAS.138..495L and @1989ApJS...71..651P for detailed discussions. A study of equilibria in lowered isothermal models of the Woolley and King-type can be found in @1978ApJ...223..299K; for a discussion on the evolutionary sequence of quasi-equilibrium states in $N$-body systems we refer to @2005MNRAS.364..990T. It would be of interest to compare the models discussed here to the phase-space density of particles in an $N$-body system undergoing core collapse.
![Entropy curves for isotropic and anisotropic models with different truncation prescriptions (i.e. different values of $g$). All models are scaled to the same mass and energy. The anisotropic models are shown as dashed lines and for these models we used $\ra=\rv$. For $g\ge1.5$ the anisotropic models are not finite for all $\phihat_0$, and the corresponding curves are therefore not plotted. This figure shows that the entropy can be increased by increasing $g$, and/or by decreasing $\ra$.[]{data-label="fig:entropy"}](entropy){width="\columnwidth"}
In Fig. \[fig:gphi0\], we illustrate the dependence of the entropy on $g$ and $\phihat_0$ for isotropic models. For a model with $g=1$ and a low concentration, the entropy can be increased by moving to the right in this diagram, and near $\phihat_0\simeq9$ the entropy can be increased by increasing $g$.
![Entropy contours for isotropic models, all scaled to $G=M=-4\Etot=1$, with different $\phihat_0$ and $g$. Contours of constant $\rthat$ are shown as black lines. Moving to the right at constant $g$ leads to an increase of entropy up to $\phihat_0\simeq9$ [@1968MNRAS.138..495L; @1980ApJ...242..765C]. The entropy can grow further by increasing $g$ at constant $\phihat_0\simeq9$. The maximum entropy is found near $\phihat_0 \simeq9$ and $g\simeq2.2$.[]{data-label="fig:gphi0"}](gphi0){width="\columnwidth"}
In Fig. \[fig:rarh\_phi0\], we show the dependence of entropy on anisotropy, expressed here in terms of $\ra/\rh$, for models with $g=0$. We see that for constant $\ra/\rh\gtrsim1$, the entropy can increase by increasing $\phihat_0$, up to about $\phihat_0\simeq9$ [this was also found by @1998MNRAS.301...25M in a study of anisotropic Woolley, King and Wilson models]. The entropy can be increased further by decreasing the anisotropy radius. A maximum is found near $\phihat_0\simeq9$ and $\ra\simeq\rh$.
![Entropy for models, all scaled to $G=M=-4\Etot=1$, with $g=0$ and different concentrations and different amounts of anisotropy, quantified here in terms of $\ra/\rh$. Contours of constant $\rthat$ are shown as black lines. A maximum in entropy is found at $\phihat_0\simeq9$ and $\ra\simeq\rh$.[]{data-label="fig:rarh_phi0"}](rarh_phi0){width="\columnwidth"}
multimass models {#ssec:multimass}
----------------
Multimass models with $\Ncomp$ mass bins require, in addition to the parameters of the single-mass models, $2\Ncomp+2$ parameters (Section \[ssec:multimassmodel\]). There is, therefore, a large variety of models that can be considered, and many properties that we can chose from to illustrate the behaviour of these models. We decide to focus on two properties that highlight important features of these multimass models in relation to mass segregation. In a follow-up study (Peuten et al., in preparation) we present a detailed comparison between the multimass models and $N$-body simulations of clusters with different mass functions.
### The role of $\delta$ {#ssec:delta}
In Fig. \[fig:vrms0\] we show the dimensionless central velocity dispersion of each mass component, $\hat{\sigma}_{{\rm 1d},j0}$, as a function of its mass $m_j$ for isotropic, 20-component models with different $\phihat_0$ and $g$. The mass bins are logarithmically spaced between $0.1\,\msun$ and $1\,\msun$ (note that the units are not important, because the model behaviour depends only on the dimensionless values $m_j/\bar{m}$), and $M_j\propto m_j^{0.7}$, which corresponds to a power-law mass function $\dr N/\dr m_j \propto m_j^{-1.3}$ (i.e. a GC-like mass function). The mass segregation parameter was set to $\delta=1/2$ (for the definition of $\delta$, see equation \[eq:delta\]).
Despite the fact that $m_j s_j^2$ is constant for all mass bins, there is no equipartition between the different mass species, i.e. $\sigmaonedj$ does not scale as $m_j^{-1/2}$ for the different mass components. This is because only in the limit of infinite central concentration $\phihat_0\rightarrow\infty$, $s_j = \sigmaonedj$, but for realistic values of $\phihat_0$, the ratio $\sigmaonedj/s_j<1$. Because the central potential for the lower mass components is smaller than the global $\phihat_0$ that defines the model, the truncation in energies reduces $\sigmaonedj$ more for low-mass components [@1981AJ.....86..318M; @2006MNRAS.366..227M]. This is illustrated by the $\phihat_0=16$ model in Fig. \[fig:vrms0\], for which a constant $m_j\sigmaonedj^2$ only holds for the most massive bins.
@2013MNRAS.435.3272T recently observed very similar trends between $\sigmaonedj$ and $m_j$ in $N$-body models of GCs (see their fig. 1) as those shown in Fig. \[fig:vrms0\]. They concluded that modelling techniques that assume equipartition, such as multimass Michie-King models, are ‘approximate at best’. We stress that multimass models that are widely used in literature, i.e. those with $\delta=1/2$ [@1976ApJ...206..128D; @1979AJ.....84..752G], are [*not*]{} in a state of equipartition, as is illustrated in Fig. \[fig:vrms0\] and has been stated previously [@1981AJ.....86..318M; @2006MNRAS.366..227M]. In fact, from a comparison of the model behaviour in Fig. \[fig:vrms0\] and the $N$-body models of Trenti and van der Marel we conclude that the most commonly chosen flavour of multimass models (i.e. King models with $\delta = 1/2$) do a good job in reproducing the degree of mass segregation in evolved stellar system [see also @sollima15].
![Dimensionless central velocity dispersion for each mass component of multimass models with a power-law mass function, and different $\phihat_0$ and $g$. The value of the mass segregation parameter is $\delta=1/2$. Equipartition in energy is only reached for large values of $m_j$ for the model with $\phihat_0=16$. The dash-dotted lines show the velocity dispersion each of the models would have in the case of equipartition.[]{data-label="fig:vrms0"}](vrms0){width="\columnwidth"}
### The role of $\eta$
In Fig. \[fig:beta\], we illustrate the effect of the parameter $\eta$ that sets the anisotropy radius of the different mass components (for the definition of $\eta$ see equation \[eq:eta\]). We show the anisotropy profiles for three-component models with $m_j = [0.2, 0.4,
0.8]$, and the same mass function as before (i.e. $M_j \propto
m_j^{0.7}$), and for different values of $\eta$. All models have $\phihat_0=9$, $g=1.5$, $\delta=1/2$ and $\rahat = 20$.
In the multimass models used in the literature $\eta$ is implicitly assumed to be $0$. From Fig. \[fig:beta\] we can see that this implies that the $\beta$ profile of the high-mass stars rises to larger values. It is tempting to interpret this as that massive stars are on more radial orbits. However, the more massive stars are also more centrally concentrated, where the velocity distribution is more isotropic. To quantify the importance of this effect, we show in each panel the values of the parameter $\kappa_j$ for each mass component (equation \[eq:kappa\]). From this we can see that in fact for the $\eta=0$ models the intermediate mass component is the most anisotropic. The relation between $\beta_j$ and $\kappa_j$ depends on the mass function, $\phihat_0$, and $g$ and this is therefore not a general property of $\eta=0$ models.
We note that for $\eta=\delta=1/2$ the $\beta_j$ profiles are nearly mass independent. Again, this does not mean that the kinetic energy in radial orbits relative to that in tangential orbits is constant, as can be seen from the values of $\kappa_j$. When considering a value of $\eta > 1/2$ we observe that the component for which $\beta_j$ assumes the largest values is the least massive one.
![Anisotropy profiles for three-component models ($m_j = [0.2,
0.4, 0.8]$) with different values for the anisotropy parameter $\eta$, that sets the anisotropy radius of the individual components as a function of their mass. All models have $\phihat_0=9$, $g=1.5$, $\delta=1/2$ and $\rahat=20$. The values of $\kappa_j$ are shown for each component in the individual panels.[]{data-label="fig:beta"}](beta){width="\columnwidth"}
The code {#sec:limepycode}
=========
General implementation {#ssec:generalimplementation}
----------------------
We introduce a -based code that solves the models and allows the user to compute some useful quantities from the DF. The code is called [ Lowered Isothermal Model Explorer in PYthon]{} (), and is available from: <https://github.com/mgieles/limepy>.
One of the main features of the code is its flexibility: the user can easily solve isotropic or anisotropic models, and include one or more mass components. The type of model to calculate is determined by the input parameters:
1. the dimensionless central potential $\phihat_0$;
2. the truncation parameter $g$;
3. the anisotropy radius $\rahat$ (for anisotropic models);
4. two arrays $m_j$, $M_j$ and $\delta$ and $\eta$ (for multimass models).
By default, the model is solved in the dimensionless units described in Section \[ssec:scaling\]. There we pointed out that the scales of the models are set by $A$ and $s$, which correspond to a mass density (in six-dimensional phase space) and a velocity scale. These two scales, combined with the gravitational constant $G$ then define the radial scale. To allow a user to scale a model to physical units, we decided to use the mass and radial scale as input, and from this the velocity scale is computed internally. The reason for this is that we foresee that an important application of the code is to recover the GC mass and radius from a comparison of the models to data. It is possible to scale the model to physical units by specifying $M$ in $\msun$ and a radial scale (either $\rv$ or $\rh$) in $\pc$. The resulting unit of velocity is then $\kms$, with $G=0.004302\,\pc\,(\kms)^2/\msun$. Alternative units, such as the units $G=\rv=M=1$ , can be considered by redefining the scales. After solving the model, the values of all typical radii are available: the King radius $\rs$, the half-mass radius $\rh$, the truncation radius $\rt$, the anisotropy radius $\ra$, and the virial radius $\rv$.
The code solves Poisson’s equation with the ‘dopri5’ integrator [@hairer1993solving], which is a Runge-Kutta integrator with adaptive step-size to calculate fourth and fifth order accurate solutions. It is supplied by the sub-package . The relative and absolute accuracy parameters are chosen as a compromise between speed and accuracy and can be adjusted by the user. The basic version of the code allows us to obtain, as a result of this integration, only the potential as a function of radius. The full model solution contains, in addition to the potential, the density, the radial and tangential components of the velocity dispersion, the global velocity dispersion profile, and the anisotropy profile (equation \[eq:beta\]). It is possible to use the potential calculated in this way to compute the value of the DF as a function of input $E$ (isotropic models), or $E$ and $J$ (anisotropic models), or positions and velocities.
After solving a model, the code carries out a simple test to see whether it is in virial equilibrium: $2\hat{K} - \Uhat = 0$, where $\hat{K}$ is the dimensionless total kinetic energy (recall that $\Uhat$ is defined to be positive, equation \[eq:Uhat\]). For models that are infinite in extent, the solver stops at a large radius, the virial equilibrium assertion fails and the lack of convergence is flagged.
For multimass models the central densities of the components need to be found by iteration (Section \[ssec:multimass\]). @1979AJ.....84..752G proposed a recipe in which the ratios of central densities over the total density, $\alpha_j$, are set equal to $M_j/\sum_j M_j$ in the first iteration. Because for $\delta>0$ the more massive components are more centrally concentrated, the amount of mass in these components is underestimated in the first iteration, while the mass in low-mass stars is overestimated. After each iteration, $\alpha_j$ is multiplied by the ratio of $M_j/M_j^\prime$, where $M_j^\prime$ is the array of masses obtained in the previous step, and then normalized again. This is repeated until convergence. However, we found that for models with low $\phihat_0$ and a wide mass spectrum the mass function does not always converge with this method. We found that multiplying $\alpha_j$ by $\sqrt{M_j/M_j^\prime}$, instead, is more reliable and results in a similar number of iterations for models that do converge with the method proposed by Gunn and Griffin.
Solutions are not numerically stable when considering large values of the arguments of the hypergeometric functions. To stabilize the calculations, we adopt the asymptotic behaviour of the hypergeometric function $\hyp(1, b, -x)$ and $\hyp(2, b, -x)$ for $x\ge700$ (see equations \[eq:1f1asym1\] and \[eq:1f1asym2\]). For multimass models with a wide mass spectrum (e.g. when stellar mass black holes are considered in addition to the stellar mass function), the central potential of the massive component can be too large for the computation of $\rhoint(\mu^{2\delta}\phihat, \rhat)$ (see equation \[eq:rhointratio\]) in the first iteration. We therefore use the approximation $\rhohat_j
= \exp[\mu^{2\delta}(\phihat-\phihat_0)]$ if $\mu^{2\delta}\phihat
>700$.
Model properties in projection {#ssec:project}
------------------------------
In order to compare the models to observations of GCs, it is necessary to compute the model properties in projection. For a spherically symmetric system, it is straightforward to compute the projected properties as a function of the projected radial coordinate $R$ [for a more detailed discussion, see for example @BT1987].
The projected surface mass density is found from the intrinsic mass density as $$\Sigma(R) = 2 \int_0^{\rt} \dr z \, \rho (r),
\label{Proj_Sigma}$$ where $r^2 = R^2+z^2$, and $z$ is along the direction of the line-of-sight. The velocity dispersion along the line of sight is given by the following integral $$\sigma_{\rm LOS}^2(R) = \frac{2}{\Sigma(R)} \int_0^{\rt} \dr z \, \rho (r) \sigma_z^2(r),
\label{Proj_Disp1}$$ where $\sigma_z^2$ is the contribution of the velocity dispersion tensor to the $z$-direction. For isotropic models, $\sigma_z^2
= \sigma^2/3$. For anisotropic models, it is possible to calculate $$\sigma_z^2(r) = \sigma_{\rm r}^2 \cos ^2 \xi + \sigma_{\theta}^2 \sin ^2 \xi,
\label{eq:sigma2zr}$$ where $\sin \xi = R/r$. We recall that, for the anisotropic models considered here, $\sigma_{\theta}^2 = \sigma_{\varphi}^2
= \sigmat^2/2$. The component $\sigma_{\varphi}^2$ does not contribute to $\sigma_z^2$, because it is always perpendicular to the line of sight. We can use the anisotropy profile $\beta$ (see equation \[eq:beta\]) to rewrite equation (\[eq:sigma2zr\]) as $$\sigma_z^2(r) = \sigma_{\rm r}^2 \left[ 1 - \beta(r) \frac{R^2}{r^2} \right].$$
The quantity $\sigma_{\rm LOS}(R)$ is useful when comparing the models to the velocity dispersion profiles that are calculated from radial (i.e. line-of-sight) velocities. Now that proper motions data are becoming available for an increasing number of GCs [@2014ApJ...797..115B], it is also interesting to compare the velocity dispersion components that can be measured on the plane of the sky with those calculated from the models. This comparison is particularly important because it is a direct way to detect the presence of anisotropy in the systems. We calculate, therefore, the radial and tangential projected components of the velocity dispersion as $$\begin{aligned}
\sigma_{\rm R}^2(R) &= \frac{2}{\Sigma(R)}\int_0^{\rt} \dr z \, \rho (r) \sigma_{\rm S}^2(r), \\
\sigma_{\rm T}^2(R) &= \frac{2}{\Sigma(R)}\int_0^{\rt} \dr z \, \rho (r) \sigma_{\rm \varphi}^2(r),\end{aligned}$$ where $\sigma_{\rm S}^2$ is given by $$\sigma_{\rm S}^2(r) = \sigma_{\rm r}^2 \left[ 1 - \beta(r) \left(\frac{1 - R^2}{r^2}\right) \right].$$
In the case of multimass models, the projected quantities introduced above are calculated separately for each mass component, by replacing every quantity in the equations above with the respective $j$th profile.
Generating discrete samples from the DF
---------------------------------------
A separate sampling routine [<span style="font-variant:small-caps;">limepy.sample</span>]{} is provided that generates discrete samples from the models. The routine takes a object containing a model as input and the number of points $N$ that need to be sampled. In the case of a multimass models the input $N$ is ignored and computed from the total mass $M$ and the pair $m_j, M_j$. Radial positions are sampled by generating numbers between 0 and 1 and interpolating the corresponding $r$ values from the (normalized) cumulative mass profile(s).
To obtain velocities, we first sample values of $x$, where $x=\khat^{3/2}=(\hat{v}^2/2)^{3/2}$. The probability density function (PDF) for $x$ can be written as $$P(x) = \frac{F(\phat x^{1/3})}{\phat
x^{1/3}}\Eg(g, \phihat(\rhat)-x^{2/3}),$$ where $\phat=\rhat/\rahat$. The function $P(x)$ has a maximum at $x=0$, and declines monotonically to 0 at $x=\phihat(\rhat)^{3/2}$. These properties make it easier to efficiently sample values for $x$ from $P(x)$, than sampling values of $v$ from $v^2 f(r,v)$. To make the rejection sampling more efficient, we adapt a supremum function $F(x)$, which consists of 10 segments between 11 values $x_i$ which are linearly spaced between $0$ and $\phihat(\rhat)^{3/2}$, and for each segment $x_i< x<x_{i+1}$, $F_i(x) = P(x_i)$. We then sample values from the function $F(x)$, reject the points that are above $P(x)$ and resample the rejected points until all points are accepted. Typically, a handful of iterations are needed.
For anisotropic models we also need to sample angles $\theta$. We do this by sampling values for $t=\cos\theta$. From the DF it follows that the PDF for $t$ is $$P(t) = \exp\left[\phat^2\khat(t^2-1)\right].
\label{eq:Pt}$$ By integrating equation (\[eq:Pt\]) we find that the cumulative DF is the imaginary error function. This function cannot be inverted analytically, hence the values for $t$ need to be found by numerically inverting this function, which can be done accurately with built-in routines.
When values for $r, \vt$ and $\vr$ are obtained, these are converted to Cartesian coordinates by generating three additional random angles.
Conclusions and discussion {#sec:conclusion}
==========================
In this study we present a family of lowered isothermal models, with the ability to consider multiple mass components and a variable amount of radially biased pressure anisotropy. The models extend the single-mass family of isotropic models recently developed by GV14. The new additions we propose here make the models ideally suited to be compared to data of resolved GCs.
The models are characterized by an isothermal and isotropic core, and a polytropic halo. The shape of the halo is set by the truncation parameter $g$, that controls the sharpness of the energy truncation, i.e. the prescription of lowering the isothermal model. For integer values of $g$, several well-known isotropic models are recovered: for $g=0$ we recover the @1954MNRAS.114..191W models, for $g=1$ the @1963MNRAS.125..127M, or @1966AJ.....71...64K models and $g=2$ corresponds to the non-rotating, isotropic @1975AJ.....80..175W models. The DF proposed by GV14, with the introduction of the continuous parameter $g$ to determine the truncation, allows us to consider models [*in between*]{} these models. The advantage of this prescription for the truncation is that it is now possible to control the sharpness of the truncation by means of a parameter.
We present [ Lowered Isothermal Model Explorer in PYthon]{} ($\limepy$), a -based code that solves the models, and computes observable quantities such as the density and velocity dispersion profile in projection. In addition, the code can be used to draw random positions and velocities from the DF, which can be used to generate initial conditions for numerical simulations.
It is interesting to discuss possible extensions of, and improvements to the models. One obvious pitfall is that the tidal field is not included in a self-consistent way. To quantify the effect of the omission of the tidal field, we can consider the specific energy $E$ at $\rt$. In our models $E(\rt)=\phi(\rt)=-GM/\rt$, whilst inclusion of the tidal terms would give (for a cluster on a circular orbit, in a reference frame corotating with the galactic orbit) a specific Jacobi energy of $E_{\rm J}(\rt) = -(3/2)GM/\rt$. Therefore, the properties of stars near the critical energy for escape are described only approximately by these models, because in this energy range the galactic tidal potential is of comparable importance as the cluster potential. Another simplification of the models is that the galactic tidal potential is triaxial, whilst our models are spherical. Both of these points could be improved upon by including a galactic tidal potential in the solution of Poisson’s equation, following the methods described by @1995MNRAS.272..317H or @2008ApJ...689.1005B, @2009ApJ...703.1911V.
The models do not include a prescription for rotation, which can be an important factor to take into account when describing real GCs . Self-consistent models with realistic rotation curves exist and have been successful in describing the rotational properties of several Galactic GCs [@2013ApJ...772...67B]. It is feasible to include rotation in the models presented in this paper, for example, in the way it is done in the @1975AJ.....80..175W model, by multiplying the DF in equation (\[eq:dfani\]) by a $J_{\rm z}$ dependent exponential term. Including the rotation, and a description of the galactic tidal field, would make the models more realistic and, therefore, a worthwhile exercise for future studies.
Lastly, we note that our models could be useful in modelling nuclear star clusters. Despite the fact that these systems are not tidally truncated in the same way as clusters on an orbit around the galaxy centre, their profiles are well described by lowered isothermal models [e.g. @2014MNRAS.441.3570G]. For a general application to nuclear star clusters, it is desirable to include the effect of the presence of a black hole in the centre, which generates a point-mass potential. provided a method to self-consistently solve King models with an external point-mass potential: this recipe could be used to include the effect of a massive black hole in the models presented here, to make them more versatile in describing nuclear star clusters.
The aim of this project was to introduce models that can be used to describe the phase-space density of stars in tidally limited, mass-segregated star clusters, in any stage of their life-cycle. At early stage, GCs are dense with respect to their tidal density [e.g. @2013MNRAS.432L...1A] and at the present day about half of the GCs is still much denser than their tidal density [@2010MNRAS.401.1832B; @2011MNRAS.413.2509G]. These GCs ought to have a population of stars with radial orbits in their envelopes, either as a left-over of the violent relaxation process during their formation [@1967MNRAS.136..101L], and/or because of two-body ejections from the core [@1972ApJ...173..529S]. In this phase we expect models with high values of $g$, and small $\ra$ to describe GCs well. These models can thus describe GCs with large Jacobi radii, relative to $\rh$. This applies to a large fraction of the Milky Way GC population, and these objects are beyond the reach of King models [@2010MNRAS.401.1832B]. In later stages of evolution, GCs will be more tidally limited, and isotropic, hence we expect $g$ to reduce and $\ra$ to increase during the evolution (up to a value that practically corresponds to having isotropic models). Capturing these variations in GCs properties with continuous parameters has the advantage that these parameters can be inferred from data. This avoids the need of a comparison of goodness-of-fit parameters of different models.
When only surface brightness data are available, it is challenging to distinguish between models with different truncation flavours and pressure anisotropy, because their role has an impact mostly on the low-density outer parts, far from the centre of the cluster, where foreground stars and background stars are dominating. The addition of kinematical data of stars in the outer region of GCs greatly aids in discriminating between models, but this is challenging at present. Precise proper motions ($\lesssim1\,\kms$) can be obtained with the [*Hubble Space Telescope*]{} [[*HST*]{}; e.g. @2006ApJS..166..249M; @2015arXiv150200005W], but the field of view of [*HST*]{} limits observations to the central parts of Milky Way GCs. Radial velocity measurements of stars in the outer parts of GCs are expensive because of the contamination of non-member stars [@2012ApJ...751....6D]. The upcoming data of the ESA-[*Gaia*]{} mission will improve this situation: the availability of all-sky proper motions and photometry measurements will facilitate membership selection, and for several nearby GC the proper motions will be of sufficient quality that they can be used for dynamical modelling and to unveil the properties of the hidden low-energy stars [@2012MNRAS.420.2562A; @2013MmSAI..84...83P; @sollima15]. The models presented in this paper allow for higher level of inference of physical properties of GCs from these upcoming data.
In two forthcoming studies we will compare the family of models to a series of direct $N$-body simulations of the long term evolution of single-mass star clusters [@2016MNRAS.462..696Z] and multimass clusters (Peuten et al., in preparation) evolving in a tidal field.
Acknowledgements {#acknowledgements .unnumbered}
================
MG acknowledges financial support from the European Research Council (ERC-StG-335936, CLUSTERS) and the Royal Society (University Research Fellowship) and AZ acknowledges financial support from the Royal Society (Newton International Fellowship). This project was initiated during the [*Gaia*]{} Challenge (<http://astrowiki.ph.surrey.ac.uk/dokuwiki>) meeting in 2013 (University of Surrey) and further developed in the follow-up meeting in 2014 (MPIA in Heidelberg). The authors are grateful for interesting discussions with the [*Gaia*]{} Challenge participants, in particular Antonio Sollima, Anna Lisa Varri, Vincent Hénault-Brunet, and Adriano Agnello. Miklos Peuten and Eduardo Balbinot are thanked for doing some of the testing of and Maxime Delorme for suggestions that helped to improve the code. We thank Giuseppe Bertin for comments on an earlier version of the manuscript and the anonymous referee for constructive feedback. Our model is written in the programming language and the following open source modules are used for the code and for the analyses done for this paper: [^6], [^7], [^8].
P. E. R., [Gieles]{} M., 2013, , 432, L1
J., [Evans]{} N. W., [Deason]{} A. J., 2012, , 420, 2562
H., 2001, , 325, 1323
H., [Parmentier]{} G., [Gieles]{} M., [Vesperini]{} E., 2010, , 401, 1832
M., [Bragaglia]{} A., [Carretta]{} E., [Gratton]{} R. G., [Lucatello]{} S., [Catanzaro]{} G., [Leone]{} F., 2012, , 538, A18
A., [Anderson]{} J., [van der Marel]{} R. P., [Watkins]{} L. L., [King]{} I. R., [Bianchini]{} P., [Chanam[é]{}]{} J., [Chandar]{} R., [Cool]{} A. M., [Ferraro]{} F. R., [Ford]{} H., [Massari]{} D., 2014, , 797, 115
G., 2014, [Dynamics of Galaxies]{}. Cambridge University Press, Cambridge
G., [Varri]{} A. L., 2008, , 689, 1005
P., [Varri]{} A. L., [Bertin]{} G., [Zocchi]{} A., 2013, , 772, 67
J., 1982, , 20, 399
J., [Tremaine]{} S., 1987, [Galactic Dynamics]{}. Princeton University Press, Princeton, NJ
L., [Idczak]{} D., 2014, preprint (arXiv:1402.0319)
J. A., [Gieles]{} M., [Sollima]{} A., [Koposov]{} S., [Mart[í]{}nez-Delgado]{} D., [Pe[ñ]{}arrubia]{} J., 2012, , 419, 14
S., 1939, [An Introduction to the Study of Stellar Structure]{}. [Chicago Univ. Press, Chicago, IL]{}
H., 1980, , 242, 765
G. S., 2012, , 751, 6
G. S., [Freeman]{} K. C., 1976, , 206, 128
E., 1977, , 61, 391
A. S., 1915, , 75, 366
T., [Heggie]{} D. C., 2000, , 318, 753
I. Y., [B[ö]{}ker]{} T., 2014, , 441, 3570
M., [Heggie]{} D. C., [Zhao]{} H., 2011, , 413, 2509
Y. J., [Velazquez]{} L., 2014, J. Stat. Mech.: Theory Exp., 4, 6 (GV14)
J. E., [Griffin]{} R. F., 1979, , 84, 752
Hairer E., N[ø]{}rsett S., Wanner G., 1993, Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag, Berlin
D., [Hut]{} P., 2003, [The Gravitational Million-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics]{}. Cambridge Univ. Press, Cambridge
D. C., [Ramamani]{} N., 1995, , 272, 317
M., 1961, Ann. Astrophys., 24, 369; English translation: preprint (arXiv:1103.3499)
M. H., 1971, , 14, 151
C., 1977, , 82, 271
R., [Nipoti]{} C., [Sollima]{} A., [Bellazzini]{} M., [Chapman]{} S. C., [Dalessandro]{} E., 2013, , 428, 3648
J., [Horwitz]{} G., [Dekel]{} A., 1978, , 223, 299
I. R., 1966, , 71, 64
D., 1967, , 136, 101
D., [Wood]{} R., 1968, , 138, 495
D. E., [Anderson]{} J., [Meylan]{} G., [Gebhardt]{} K., [Pryor]{} C., [Minniti]{} D., [Phinney]{} S., 2006, , 166, 249
D. E., [van der Marel]{} R. P., 2005, , 161, 304
M., [Pucacco]{} G., [Vesperini]{} E., 1998, , 301, 25
D., 1981, , 86, 318
G., 1987, , 184, 144
G., [Heggie]{} D. C., 1997, A&AR, 8, 1
R. W., 1963, , 125, 127
P., 2006, , 366, 227
P., 2010, , 514, A52
K. S., [Lin]{} D. N. C., 1992, , 386, 519
T., 1989, , 71, 651
E., [Bellazzini]{} M., [Marinoni]{} S., 2013, , 84, 83
J.-C., [De Marco]{} O., [Fryer]{} C. L., [Herwig]{} F., [Diehl]{} S., [Oishi]{} J. S., [Mac Low]{} M.-M., [Bryan]{} G. L., [Rockefeller]{} G., 2012, , 744, 52
H. C., 1911, , 71, 460
V. L., [Shukhman]{} I. G., 1981, , 25, 533
C., [Meylan]{} G., 1993, in [Djorgovski]{} S. G., [Meylan]{} G., eds, ASP Conf. Ser. Vol. 50, [Structure and Dynamics of Globular Clusters]{}. Astron. Soc. Pac. San Francico, p. 357
Samko S. G., Kilbas A. A., Marichev O. I., 1993, Fractional integrals and derivatives: theory and applications. Gordon & Breach, Philadelphia, PA
R. L., [Gieles]{} M., 2015, , 448, L94
A., [Baumgardt]{} H., [Zocchi]{} A., [Balbinot]{} E., [Gieles]{} M., [Henault-Brunet]{} V., [Varri]{} A. L., 2015, MNRAS, 451, 2185
A., [Bellazzini]{} M., [Lee]{} J.-W., 2012, , 755, 156
L., 1987, [Dynamical Evolution of Globular Clusters]{}. Princeton, NJ, Princeton University Press, 1987
Jr. L., [Shapiro]{} S. L., 1972, , 173, 529
A., [Sakagami]{} M.-a., 2005, , 364, 990
M., [van der Marel]{} R., 2013, , 435, 3272
A. L., [Bertin]{} G., 2009, , 703, 1911
A. L., [Bertin]{} G., 2012, , 540, A94
L. L., [van der Marel]{} R. P., [Bellini]{} A., [Anderson]{} J., 2015, , 803, 29
C. P., 1975, , 80, 175
R. V. D. R., 1954, , 114, 191
A., [Bertin]{} G., [Varri]{} A. L., 2012, , 539, A65
Zocchi A., Gieles M., H[é]{}nault-Brunet V., Varri A. L. 2016, , 462, 696
Derivations {#app:series}
===========
The DF introduced in equation (\[eq:dfani\]) can be expressed as a function of the dimensionless quantities $\phihat$, $\khat$ and $\phat$ defined in Sections \[ssec:scaling\] and \[ssec:ani\] as $$f = A \exp\left(-\khat \phat^2 \sin^2\theta\right)\Eg\left(g, \phihat - \khat\right).$$ We want to calculate, for these models, the density and velocity dispersion components. We recall that these quantities can be obtained from the DF in the following way:[^9] $$\begin{aligned}
\rho &= \int \dr^3 v \, f, \\
\sigma^2_{i} &= \frac{1}{\rho} \int \dr^3 v \, f v^2_i, \label{DF_vel_disp_i}\end{aligned}$$ where the subscript $i$ denotes the $i$-th component of the velocity vector. To carry out these integrals of the DF in the three-dimensional velocity volume we can use the dimensionless variable $\khat$ and the variable $t = \cos \theta$ $$\dr^3 v = \dr v \dr\theta \dr\varphi \, v^2 \sin\theta = - \dr \khat \dr t \dr\varphi \, \sqrt{\khat} (2 s^2)^{3/2}.$$ In calculating the relevant quantities mentioned above, we encounter the following integrals with respect to the variable $t$ $$\begin{aligned}
&\int_0^{1} \dr t \, \exp\left[-\khat \phat^2 (1-t^2)\right]
= \frac{F\left(\sqrt{\khat}\phat\right)}{\sqrt{\khat}\phat} , \label{Itheta1} \\
&\int_0^{1} \dr t \, t^2 \exp\left[-\khat \phat^2 (1-t^2)\right]
= \frac{1}{2 \khat\phat^2} - \frac{F\left(\sqrt{\khat}\phat\right)}{2 (\sqrt{\khat}\phat)^3}, \label{Itheta2} \end{aligned}$$ where $F(x)$ is the Dawson integral, whose properties are presented in Section \[App:Dawson\]. We use the above results to proceed and derive the density and velocity components.
Density profile
---------------
The density is calculated as $$\begin{aligned}
\rho &= \int \dr^3v \, f \nonumber \\
&= \frac{\tilde{A}}{\sqrt{\pi}} \int_0^{\phihat} \dr\khat \, \int_0^{1} \dr t \, \khat^{1/2}\exp\left[\khat \phat^2 (t^2-1)\right] \Eg\left(g, \phihat-\khat\right) \nonumber \\
&= \tilde{A} \frac{2}{\sqrt{\pi}}\int_0^{\phihat} \dr\khat \, \frac{F\left(\sqrt{\khat} \phat\right)}{\phat }\Eg\left(g, \phihat-\khat\right) \nonumber\\
&= \tilde{A} \rhoint,\end{aligned}$$ where we replaced $\Gamma(3/2)$ by $\sqrt{\pi}/2$, and we introduced $\tilde{A} = A \left(2 \pi s^2 \right)^{3/2}$ and we solved the integral over $t$ as shown in equation (\[Itheta1\]). The integral $\rhoint$ can be solved by first doing an integration by parts (by using the results in equations \[Egintegral\] and \[DawsDeriv\]) and by then using the convolution formula of equation (\[EG\_convolution\]) and the recurrence relation of equation (\[RecurrencyEg\]) in the following way: $$\begin{aligned}
\rhoint &= \frac{2}{\sqrt{\pi}}\int_0^{\phihat} \dr\khat \, \Eg\left(g, \phihat-\khat\right) \frac{F\left(\sqrt{\khat} \phat\right)}{\phat} \label{Irho} \\
&= \frac{2}{\sqrt{\pi}}\int_0^{\phihat} \dr\khat \, \Eg\left(g+1, \phihat-\khat\right)\times\left[ \frac{1}{2\sqrt{\khat}} - \phat F\left(\sqrt{\khat} \phat\right) \right]\nonumber\\
&= \Eg\left(g+{\tfrac{3}{2}}, \phihat\right)
- \phat^2 \frac{2}{\sqrt{\pi}}\int_0^{\phihat} \dr\khat \, \Eg\left(g, \phihat-\khat \right) \frac{F\left(\sqrt{\khat} \phat\right)}{\phat} \nonumber \\
& \hspace{1cm} +\phat^2 \frac{2}{\sqrt{\pi}}\int_0^{\phihat} \dr\khat \, \frac{\left(\phihat-\khat\right)^g}{\Gamma(g+1)} \frac{F\left(\sqrt{\khat} \phat\right)}{\phat} \nonumber \\
&= \Eg\left(g+{\tfrac{3}{2}}, \phihat \right) - \phat^2 \rhoint + \phat^2 \fintone.\end{aligned}$$ The integral $\fintone$ can be calculated by substituting the Dawson function for its series representation (see equation \[DawsonSeries\]), by changing variable to $y
= \khat/\phihat$, by using the Beta function of equation (\[Beta\]), and by recognizing the expression in equation (\[1F1DefSer\]) $$\begin{aligned}
\fintone &= \frac{2}{\sqrt{\pi}}\int_0^{\phihat} \dr\khat \, \frac{\left(\phihat-\khat\right)^g}{\Gamma(g+1)} \frac{F\left(\sqrt{\khat} \phat\right)}{\phat } \nonumber \\
&= \frac{\phihat^{g+\frac{3}{2}}\hyp\left(1,g+{\tfrac{5}{2}},-\phat^2\phihat\right)}{\Gamma\left(g+\frac{5}{2}\right)},
\label{IntegralDawson1}\end{aligned}$$ where $ \hyp\left(a,b,x\right)$ is the confluent hypergeometric function (see Section \[1F1\]). Therefore, we can finally write the density integral as $$\rhoint \!= \!\frac{\Eg(g+{\tfrac{3}{2}}, \phihat )}{1+\rrahat^2} + \frac{\phat^2}{1+\phat^2} \frac{\phihat^{g+\frac{3}{2}}\hyp(1,g+\frac{5}{2},-\phat^2\phihat)}{\Gamma\left(g+\frac{5}{2}\right)}.
\label{DENSITY_Anis}$$
Velocity dispersion profiles {#sapp:sigma_deriv}
----------------------------
The velocity dispersion profile can be computed in a similar way as the density, by using again the result in equation (\[Itheta1\]) $$\begin{aligned}
\sigma^2\rho &= \int \dr^3v \, v^2 f \nonumber \\
&= \frac{2\tilde{A} s^2}{\sqrt{\pi}} \int_0^{\phihat} \dr\khat\!\int_0^{1} \dr t \, \exp\left[{-\khat \phat^2 (1-t^2)}\right] \khat^{3/2} \Eg(g, \phihat-\khat ) \nonumber \\
&= \frac{4 \tilde{A} s^2}{\sqrt{\pi} } \int_0^{\phihat} \dr\khat \,\frac{F(\sqrt{\khat} \phat)}{\phat} \khat \Eg(g, \phihat-\khat ) \nonumber \\
&= \tilde{A} s^2 \vsqint.\end{aligned}$$ The integral $\vsqint$ can be solved with an integration by parts, then by using equation (\[EG\_convolution\]), and finally, after having used the recurrence relation of equation (\[RecurrencyEg\]), by recognizing the presence of the integral $\rhoint$ found when calculating the density $$\begin{aligned}
\vsqint &= \frac{4}{\sqrt{\pi}}\int_0^{\phihat} \dr\khat \, \khat \Eg\left(g, \phihat-\khat \right) \frac{F\left(\sqrt{\khat} \phat\right)}{\phat }\label{DefVSQint}\\
&= \frac{4}{\sqrt{\pi}}\int_0^{\phihat} \dr\khat \, \Eg\left(g+1, \phihat-\khat \right) \nonumber \\
& \hspace{1cm}\times \left[\frac{\sqrt{\khat}}{2} + \frac{F\left(\sqrt{\khat} \phat\right)}{\phat } -\khat\phat^2 \frac{F\left(\sqrt{\khat} \phat\right)}{\phat} \right] \nonumber \\
&= \Eg\left(g+{\tfrac{5}{2}}, \phihat \right) + 2 \rhoint_{g+1} + \phat^2 \left(\finttwo - \vsqint \right),\end{aligned}$$ where $\rhoint_{g+1} = \rhoint(g+1, \rrahat, \phihat)$. We then solve the integral $\finttwo$ in a similar way as we did for $\fintone$, to get $$\begin{aligned}
\finttwo &= \frac{4}{\sqrt{\pi}}\int_0^{\phihat} \dr\khat \, \khat \frac{F\left(\sqrt{\khat} \phat\right)}{\phat } \frac{\left(\phihat-\khat\right)^g}{\Gamma(g+1)} \nonumber \\
&= \frac{\phihat^{g+\frac{5}{2}}}{\Gamma\left(g+{\tfrac{7}{2}}\right)} \left[\hyp\left(1,g+{\tfrac{7}{2}},-\phat^2\phihat\right) \right. \nonumber \\
& \left. + 2 \hyp\left(2,g+{\tfrac{7}{2}},-\phat^2\phihat\right) \right].
\label{IntegralDawson2}\end{aligned}$$ Therefore, we finally have $$\begin{aligned}
\sigma^2\rho &= \frac{\tilde{A} s^2}{(\phat^2+1)} \left\lbrace \Eg\left(g+{\tfrac{5}{2}}, \phihat \right)\left(\frac{3+\phat^2}{1+\phat^2}\right) \right. \nonumber \\
& \left. + \frac{\phat^2 \phihat^{g+{\tfrac{5}{2}}}}{\Gamma\left(g+\frac{7}{2}\right)} \left[ 2 \, \hyp\left(2,g+{\tfrac{7}{2}},-\phat^2\phihat\right) \right.\right. \nonumber \\
& \left.\left. + \hyp\left(1,g+{\tfrac{7}{2}},-\phat^2\phihat\right) \left(\frac{3+\phat^2}{1+\phat^2} \right) \right] \right\rbrace.
\label{DISP_Anis}\end{aligned}$$ The radial component of the velocity dispersion is given by $$\begin{aligned}
\sigmar^2\rho &= \int \dr^3 v \, (v \cos\theta)^2 f \nonumber \\
&= \frac{2\tilde{A} s^2}{\sqrt{\pi}} \int_0^{\phihat} \dr\khat \, \khat^{3/2} \Eg\left(g, \phihat-\khat \right) \left[ \frac{1}{\khat\phat^2} - \frac{F\left(\sqrt{\khat}\phat\right)}{(\sqrt{\khat}\phat)^3} \right] \nonumber \\
&= \frac{\tilde{A} s^2}{\phat^2} \left[ \Eg\left(g+{\tfrac{3}{2}}, \phihat \right) - \rhoint \right],
\label{DISP_r_Anis_int}\end{aligned}$$ where we solved the integral by using equations (\[Itheta2\]) and (\[EG\_convolution\]). We point out that we can express this quantity by means of the density integrals of the isotropic and the anisotropic case (see also equations \[eq:rhointiso\] and \[eq:rhointani\] in Section \[sec:model\]). We finally obtain $$\sigmar^2\rho = \tilde{A} s^2 \left[ \frac{\Eg\left(g+{\tfrac{3}{2}}, \hat{\phi} \right)}{(1+\hat{p}^2)} - \frac{\hat{\phi}^{g+{\tfrac{3}{2}}} \hyp\left(1,g+{\tfrac{5}{2}},-\hat{p}^2\hat{\phi}\right)}{(1+\hat{p}^2) \Gamma\left(g+\frac{5}{2}\right)} \right],
\label{DISP_r_Anis}$$ which, by using equations (\[RecurrencyEg\]) and (\[RecurrencyHyp2\]), can be rewritten as $$\thickmuskip=2mu
\medmuskip=2mu
\sigmar^2\rho = \tilde{A} s^2 \left[ \frac{\Eg\left(g+{\tfrac{5}{2}}, \hat{\phi} \right)}{(1+\hat{p}^2)} + \frac{\phat^2 \hat{\phi}^{g+{\tfrac{5}{2}}} \hyp\left(1,g+\frac{7}{2},-\hat{p}^2\hat{\phi}\right)}{(1+\hat{p}^2) \Gamma\left(g+\frac{7}{2}\right)} \right].
\label{DISP_r_Anis_REW}$$ To calculate the tangential component of the velocity dispersion we solve the integral over $t$ by expressing it as the difference between equation (\[Itheta1\]) and equation (\[Itheta2\]), and we carry out an integration by parts $$\begin{aligned}
\sigmat^2\rho &= \int \dr^3 v \, (v \sin\theta)^2 f \nonumber \\
&= \frac{2 \tilde{A} s^2}{\sqrt{\pi}} \int_0^{\phihat} \dr\khat \, \khat^{3/2} \Eg\left(g, \phihat-\khat \right) \nonumber\\
&\hspace{0.25cm}\times\left[\frac{2 F\left(\sqrt{\khat}\phat\right)}{\sqrt{\khat}\phat} - \frac{1}{\khat\phat^2} + \frac{F\left(\sqrt{\khat}\phat\right)}{(\sqrt{\khat}\phat)^3} \right] \nonumber \\
&= \frac{\tilde{A} s^2}{\phat^2} \left[\phat^2 \vsqint - \Eg\left(g+{\tfrac{3}{2}}, \phihat \right) + \rhoint \right] .
\label{DISP_T_Anis_int}\end{aligned}$$ After recognizing the integrals we solved above, we can finally write $$\begin{aligned}
\sigmat^2\rho &= \frac{\tilde{A} s^2}{(1+\phat^2)} \left\lbrace \Eg\left(\gfive, \phihat \right)\frac{2}{(1+\phat^2)} \right. \nonumber \\
& \left. + \, \frac{2\phat^2 \phihat^{g+{\tfrac{5}{2}}}}{\Gamma(g+{\tfrac{7}{2}})} \left[ \frac{\hyp\left(1, g+{\tfrac{7}{2}}, -\phihat\phat^2\right)}{(1+\phat^2)} \right.\right. \nonumber \\
& \left.\left. + \, \hyp\left(2, g+{\tfrac{7}{2}}, -\phihat\phat^2\right) \right] \right\rbrace \ .
\label{DISP_T_Anis_REW}\end{aligned}$$
Derivations using fractional calculus {#app:fractional}
=====================================
Fractional calculus is a branch of mathematics that considers real numbers for the orders of derivatives and integration. Because the integrals that need to be solved contain terms like $\khat^{1/2}$ and $\khat^{3/2}$, we can use semi-derivatives and semi-integrals and integration by parts to solve them. By following @2014arXiv1402.0319B, we define the left- and right-sided Riemann-Liouville fractional integrals of order $\alpha >
0$ of a function $q \in L^1$ as $$\begin{aligned}
I^{\alpha}_{a+} q(t) &= \frac{1}{\Gamma(\alpha)} \int_{a}^{t} \dr x \, q(x) (t-x)^{\alpha-1}, \label{leftFracInt} \\
I^{\alpha}_{b-} q(t) &= \frac{1}{\Gamma(\alpha)} \int_{t}^{b} \dr x \, q(x) (x-t)^{\alpha-1}. \label{rightFracInt}\end{aligned}$$ In the remainder of this section, we will use the result illustrated by @2014arXiv1402.0319B in their Proposition 2 [for a proof see @samko1993] $$\begin{aligned}
\int_{a}^{b} \dr t \, \left(I^{\alpha}_{a+} q_1\right)(t) q_2(t) = \int_{a}^{b} \dr t \, q_1(t) \left(I^{\alpha}_{b-} q_2\right)(t).
\label{Prop2}\end{aligned}$$
Density
-------
When considering the isotropic limit of the DF, the integral to be solved to calculate the density is: $$\rhoint = \frac{2}{\sqrt{\pi}} \int_0^{\phihat}\dr\khat\, \khat^{1/2} \Eg\left(g, \phihat-\khat \right).
\label{densISOFrac1}$$ We can use fractional calculus to solve it, by changing variable of integration (using $x = \phihat-\khat$) and by considering that $$\begin{aligned}
q_1(x) &= \Eg\left(g, x \right), \\
q_2(x) &= 1, \\
I^{1/2}_{0+} q_1(x) &= \Eg\left(g + {\tfrac{1}{2}}, x \right), \\
I^{1/2}_{\phihat-} q_2(x) &= \frac{2(\phihat-x)^{1/2}}{\sqrt{\pi}},\end{aligned}$$ thus obtaining $$\begin{aligned}
\rhoint &= \int_0^{\phihat}\dr x\, \Eg(g +{\tfrac{1}{2}}, x) = \Eg(g +{\tfrac{3}{2}}, \phihat),\end{aligned}$$ which was solved by using equation (\[Egintegral\]).
The density of the anisotropic models is calculated by solving the integral $\rhoint$ introduced in equation (\[Irho\]). To do this, we can use the result shown in equation (\[Prop2\]) by noticing (see equations \[EG\_convolution\] and \[DawsonDef\]) that $$\begin{aligned}
q_1(\khat) &= \exp\left(-\khat\phat^2\right), \label{q11}\\
q_2(\khat) &= \Eg\left(g, \phihat-\khat\right), \\
I^{1/2}_{0+} q_1(\khat) &= \frac{2F(\sqrt{\khat}\phat)}{\sqrt{\pi} \phat}, \label{q1i1} \\
I^{1/2}_{\phihat-} q_2(\khat) &= \Eg\left(g + {\tfrac{1}{2}}, \phihat-\khat \right),\end{aligned}$$ and by rewriting the integral of equation (\[Irho\]) as $$\rhoint = \int_0^{\phihat} \dr \khat \, \exp\left(-\khat\phat^2\right) \Eg\left(g + {\tfrac{1}{2}}, \phihat-\khat \right).
\label{densFrac1}$$ This integral can be solved by parts, by using the expression of the derivative of the lower incomplete gamma function (equation \[GammaDeriv\]) and by using equation (\[IntDefHyp\]) to obtain $$\rhoint = \frac{\Eg\left(g+{\tfrac{1}{2}}, \phihat \right)}{1+\phat^2} - \frac{\phihat^{g+{\tfrac{1}{2}}}\hyp(1,g+{\tfrac{3}{2}}, -\phihat\phat^2)}{(1+\phat^2)\Gamma(g+{\tfrac{3}{2}})}.$$ The last step can be rewritten as equation (\[DENSITY\_Anis\]) by using the recurrence relations shown in equations (\[RecurrencyEg\]) and (\[RecurrencyHyp2\]).
Velocity dispersion {#sapp:sigma_deriv_frac}
-------------------
In the isotropic limit of the DF, the velocity dispersion is calculated by means of an integral with the same structure as the one found in equation (\[densISOFrac1\]). The velocity dispersion of the anisotropic models is calculated by solving the integral $\vsqint$ introduced in equation (\[DefVSQint\]). We can use the result shown in equation (\[Prop2\]) also in this case, by considering the function $q_1$ introduced in equation (\[q11\]), its fractional integral (equation \[q1i1\]), and $$\begin{aligned}
\thickmuskip=2mu
\medmuskip=2mu
q_2(\khat) &= \khat \, \Eg\left(g, \phihat-\khat \right), \\
I^{1/2}_{\phihat-} q_2(\khat) &= \frac{1}{2} \Eg\left(g+{\tfrac{3}{2}}, \phihat-\khat \right) + \khat \Eg\left(g+{\tfrac{1}{2}}, \phihat-\khat \right),\end{aligned}$$ and by rewriting the integral $\vsqint$ as $$\begin{aligned}
\vsqint &= \int_0^{\phihat} \dr \khat \exp\left(-\khat\phat^2\right) \Eg\left(g +{\tfrac{3}{2}}, \phihat-\khat \right) \nonumber\\
& +2 \int_0^{\phihat} \dr \khat \exp\left(-\khat\phat^2\right) \khat \Eg\left(g +{\tfrac{1}{2}}, \phihat-\khat \right).\end{aligned}$$ The first integral is in the same form as the one we found for the density, and the second one can be reduced to something similar with an integration by parts. By solving the integrals, we obtain: $$\begin{aligned}
\vsqint &= \frac{1}{1+\phat^2 } \left[ \Eg\left(g+{\tfrac{3}{2}}, \phihat\right)\left(\frac{3+\phat^2}{1+\phat^2}\right) - \frac{2 \phihat^{g+{\tfrac{3}{2}}}}{\Gamma\left(g+{\tfrac{3}{2}}\right)} \right. \nonumber\\
& \left. +\frac{2 \phihat^{g+{\tfrac{3}{2}}}\hyp(1,g+\frac{5}{2}, -\phihat\phat^2)}{\Gamma\left(g+\frac{5}{2}\right)} \left( g + \phat^2\phihat + \frac{\phat^2}{1+\phat^2} \right) \right],\end{aligned}$$ which can be rewritten as equation (\[DISP\_Anis\]) by using the recurrence relations shown in equations (\[RecurrencyEg\]), (\[RecurrencyHyp1\]), and (\[RecurrencyHyp2\]).
By inspecting equations (\[DISP\_r\_Anis\_int\]) and (\[DISP\_T\_Anis\_int\]), it is immediate to notice that the radial and tangential components of the velocity dispersion are calculated with integrals that can be written as a combination of those solved in this section by means of fractional calculus.
Differential energy distribution {#app:dmde}
================================
The differential energy distribution gives the amount of mass per units of energy [@BT1987]. Here we briefly recall how to calculate it for isotropic and anisotropic systems.
For a DF that only depends on $E$, the differential energy distribution can be calculated as: $$\frac{\dr M}{\dr E} \equiv f(E)\gdos(E),
\label{eq:dmde}$$ where $f(E)$ is the DF, and $\gdos(E)$ is the density of states, which is the volume of phase space per unit energy and is defined as $${\gdos}(E) = \int \dr^3r \, \dr^3v \, \delta(E-H),
\label{densityofstates}$$ where $\delta(x)$ is the Dirac delta function. For a spherically symmetric systems, this integral can be expressed as $$\thickmuskip=2mu
\medmuskip=2mu
\gdos(E) = 16 \pi^2 \, \int_0^{\rmm(E)} \dr r \, r^2 \, \int \dr v \, v^2 \, \delta\left(\frac{1}{2} v^2 + \phi - E\right),
\label{densityofstates_1}$$ where $\rmm(E)$ is the radius at which $\phi = E$. By changing variable (using $y = v^2/2$), we finally obtain $$\gdos(E) = 16 \pi^2 \, \int_0^{\rmm(E)} \dr r \, r^2 \, \sqrt{2(E - \phi)},
\label{densityofstates_2}$$ and the differential energy distribution is therefore $$\frac{\dr M}{\dr E} = 16 \pi^2 \, f(E) \, \int_0^{\rmm(E)} \dr r\, r^2 \, \sqrt{2(E-\phi)}.
\label{N_E_iso}$$
When considering anisotropic systems, for which the DF depends also on the angular momentum $J$, the differential energy distribution is obtained as $$\frac{\dr M}{\dr E} = \int \dr^3r \, \dr^3v \, \delta(E-H) f(H,J).
\label{differentialenergydistribution}$$ This integral can be expressed as $$\frac{\dr M}{\dr E}= 8 \pi^2 \int \dr r \, r^2 \int \dr \vr \, \dr \vt \, \vt \, \delta(E-H) f(H,J),$$ and it can then be rearranged by changing variable and introducing $J = r \, \vt$ in the following way $$\frac{\dr M}{\dr E} = 8 \pi^2 \int \dr r \int \dr \vr \, \dr J \, J \, \delta(E-H) f(H,J).$$ The integral over $\vr$ is solved by using the fact that $$\vr^2 = 2(E-\Phi) - \frac{J^2}{r^2},$$ to obtain $$\begin{aligned}
\frac{\dr M}{\dr E} = 16 \pi^2 \, \int \dr r \int \dr J \, \frac{J f(E,J)}{\sqrt{2(E-\Phi) - J^2/r^2}}.\end{aligned}$$ By using the expression for the DF of the models presented in this paper (equation \[eq:dfani\]) we can perform the integration over $J$ in this last equation and write the differential energy distribution as a function of the part of the DF that depends on energy only, $f(E)$: $$\begin{aligned}
\frac{\dr M}{\dr E}= 16 \pi^2 \, f(E) \, \int_0^{\rmm(E)}\dr r\, \sqrt{2} \, \ra s \, r \, F\left(\frac{r \sqrt{E-\phi}}{\ra s}\right),\end{aligned}$$ where $F(x)$ is the Dawson integral (see Appendix \[App:Dawson\]). In the limit of $\ra\rightarrow\infty$ this reduces to the result for the isotropic case shown in equation (\[N\_E\_iso\]), which follows from substituting the leading term of equation (D14) in equation (C11).
Useful properties of mathematical functions {#app:functions}
===========================================
Useful properties of the gamma functions {#app:gamma}
----------------------------------------
The gamma function of a positive integer $n$ is defined as $$\Gamma(n) = (n-1)!,$$ while for non-integer arguments $a$, it can be written as an integral $$\Gamma(a) = \int_0^{\infty} \dr t \, t^{a-1}\exp(-t).$$ The lower incomplete gamma function is given by $$\gamma(a, x) = \int_0^x \dr t \, t^{a-1}\exp(-t),$$ and its derivative is $$\frac{\dr \gamma(a,x)}{\dr x} = x^{a-1}\exp(-x).
\label{GammaDeriv}$$
Useful properties of the $\Eg(a,x)$ function {#AppD:Eg}
--------------------------------------------
The exponential function $\Eg(a,x)$ is defined as $$\Eg(a,x) = \frac{1}{\Gamma(a)} \int_0^x \dr t \, t^{a-1} \exp(x-t),$$ and an alternative expression is given by means of the lower incomplete gamma function $$\Eg(a,x) = \frac{\exp(x) \gamma(a,x)}{\Gamma(a)}.
\label{E_gamma_exp}$$ The series representation of this function is $$\Eg(a,x) = \sum_{i=0}^{\infty} \frac{x^{i+a}}{\Gamma(i+a+1)}.
\label{Eg_series_repr}$$ The following recurrence relation holds $$\Eg(a,x) = \Eg(a+1, x) + \frac{x^a}{\Gamma(a+1)}.
\label{RecurrencyEg}$$ The derivative and the integral of $\Eg(a,x)$ are given by $$\begin{aligned}
\frac{\dr \Eg(a, x)}{\dr x} &= \Eg(a-1,x), \\
\int \dr x \, \Eg(a, x) &= \Eg(a+1,x) + \mathrm{constant}. \label{Egintegral}\end{aligned}$$ A proof of these equations can be easily obtained by writing $\Eg(a,x)$ as in equation (\[E\_gamma\_exp\]), and by considering equation (\[GammaDeriv\]) and the recurrence relation (equation \[RecurrencyEg\]). The convolution formula $$\frac{1}{\Gamma(b)} \int_0^x \dr y \, \Eg(a,x-y) y^{b-1} = \Eg(a+b,x)
\label{EG_convolution}$$ can be obtained by using the series representation of $\Eg(a,x)$ (equation \[Eg\_series\_repr\]) and by changing variable, to express the integral with a form that allows us to recognize the Beta function: $$B(m,n) = \int_0^1 \dr y \, (1-y)^{m-1}y^{n-1} = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}.
\label{Beta}$$ The identity of equation (\[EG\_convolution\]) accounts for the fractional integration of $\Eg(a,x)$ (see equation \[leftFracInt\]).
Useful properties of the Dawson integral {#App:Dawson}
----------------------------------------
The Dawson integral (sometimes called the Dawson function) is defined as $$F(x) = \exp(-x^2) \int_0^x \dr y \exp(y^2).
\label{DawsonDef}$$ It is also possible to express $F(x)$ as a sum as $$F(x) = \sum_{i=0}^{\infty} \frac{(-1)^i x^{2i+1} \Gamma\left({\tfrac{3}{2}}\right)}{\Gamma\left(i+{\tfrac{3}{2}}\right)} \ .
\label{DawsonSeries}$$ The Dawson integral is an odd function, and its derivative is $$\frac{\dr F(x)}{\dr x} = 1 - 2 x F(x).
\label{DawsDeriv}$$
Useful properties of the confluent hypergeometric function {#1F1}
----------------------------------------------------------
The confluent hypergeometric function is defined as $$\begin{aligned}
\hyp(a,b,x) &= \displaystyle\sum_{i=0}^{\infty} \frac{\Gamma(a+i)}{\Gamma(a)}\frac{\Gamma(b)}{\Gamma(b+i)}\frac{x^i}{\Gamma(i+1)}.
\label{1F1DefSer}\end{aligned}$$ It can also be defined by means of an integral, as $$\hyp(a,b,x) = \frac{\Gamma(b)}{\Gamma(a) \Gamma(b-a)} \int_{0}^{1} \dr y \, \exp(xy) y^{a-1} (1-y)^{b-a-1},
\label{IntDefHyp}$$ which holds for ${\rm Re}(b)>{\rm Re}(a)>0$. The following recurrence relations hold $$\begin{aligned}
\hyp(2,b,x) &= (2-b+x) \, \hyp(1,b,x) + b - 1, \label{RecurrencyHyp1} \\
x \, \hyp(1,b+1,x) &= b \, \hyp(1,b,x) - b. \label{RecurrencyHyp2}\end{aligned}$$ We also note that this function is related to the exponential function, and for $b=a$ we have $\hyp(a,a,x)=\exp(x)$. Another useful property of this function is that $$\hyp(a, b, 0) = 1.
\label{Hyp1}$$
The density (equation \[eq:rhointanires\]) and the velocity moments (equations \[eq:vrsqintres\] - \[eq:vsqintres\]) of anisotropic models are expressed by means of the function $\hyp(a,b,x)$ with $a=1$. When considering $a = 1$ in equation (\[IntDefHyp\]), we obtain $$\begin{aligned}
\hyp(1,b,x) &= \frac{(b-1)}{x^{b-1}} \, \exp(x) \, \gamma(b-1,x) .
\label{1F1_alt}\end{aligned}$$ We point out that when $x<0$ two of the quantities appearing in equation (\[1F1\_alt\]) are imaginary. This is the reason why in general this expression cannot be used to speed up the code by expressing the equations mentioned above in a more compact way. When considering integer values of $b$, however, we can simplify the hypergeometric function, and express it by means of exponentials and polynomials. The smallest value of $b$ we use is $g +{\tfrac{5}{2}}$, therefore $b$ assumes the smallest integer value when $g = {\tfrac{1}{2}}$ for $b=3$ we calculate $$\begin{aligned}
\hyp(1,3,x) &= \frac{2}{x^{2}} \left[ \exp(x) -1 -x\right].\end{aligned}$$ Combined with the recurrence relation mentioned earlier, the results of the anisotropic models and half-integer values of $g$ can be expressed by means of these elementary functions.
The asymptotic series expansion for the confluent hypergeometric function when $|x| \rightarrow \infty$ is given by $$\begin{aligned}
\hyp(a,b,x) & \propto \frac{\Gamma(b)}{\Gamma(b-a)} (-x)^{-a} \left[ 1 + \mathcal{O}\left( \frac{1}{z} \right) \right] \nonumber \\
& +\frac{\Gamma(b)}{\Gamma(a)} \exp(x) x^{a-b} \left[ 1 + \mathcal{O}\left( \frac{1}{z} \right) \right].\end{aligned}$$ This is useful to compute the density and velocity moments, because for very large $|x|$ the evaluation of this function becomes inaccurate in (see section \[ssec:generalimplementation\]). In particular, the functions that are needed to compute these quantities, $\hyp(1,b,-x)$ and $\hyp(2,b,-x)$, have the following behaviour for large $|x|$: $$\begin{aligned}
\lim_{x\rightarrow \infty}\hyp(1,b,-x) &= \frac{b-1}{x},\label{eq:1f1asym1}\\
\lim_{x\rightarrow \infty}\hyp(2,b,-x) &= \frac{(b-1)(b-2)}{x^2}.\label{eq:1f1asym2}\end{aligned}$$
[^1]: is available from <https://github.com/mgieles/limepy.>
[^2]: The Woolley, King and Wilson DFs follow straightforwardly from equations (\[eq:dfani\]) and (\[eq:eg\]), because $\Eg(0,x) = \exp(x)$, $P(1,x) =
1-\exp(-x)$, such that $\Eg(1,x) = \exp(x)-1$ and $P(2,x) =
1-\exp(-x)-x\exp(-x)$, such that $\Eg(2,x) = \exp(x) - 1 - x$.
[^3]: This parameter is called $W_0$ in @1966AJ.....71...64K.
[^4]: By considering the first velocity moment of the DF we find the mean velocity: for these models, this quantity vanishes everywhere. We also note that expressions for higher order moments of the velocity distribution can be derived, but these are beyond the scope of this paper.
[^5]: The tangential velocity comprises the two components $\vt^2 = \vtheta^2+\vphi^2$, where $\vtheta =
v\sin\theta\sin\varphi$ and $\vphi = v\sin\theta\cos\varphi$. The corresponding components of the velocity dispersion tensor are equal to each other, and each of them accounts for half of the tangential component: $\sigmat^2 = 2 \sigmatheta^2 = 2\sigmaphi^2$.
[^6]: <http://www.numpy.org>
[^7]: <http://www.scipy.org>
[^8]: <http://matplotlib.sourceforge.net>
[^9]: As noticed already in Section \[ssec:iso\], equation (\[DF\_vel\_disp\_i\]) holds because the mean velocity for the systems described by these models is zero everywhere.
|
---
abstract: 'The first author introduced the circuit-cocircuit reversal system of an oriented matroid, and showed that when the underlying matroid is regular, the cardinalities of such system and its variations are equal to special evaluations of the Tutte polynomial (e.g., the total number of circuit-cocircuit reversal classes equals $t(M;1,1)$, the number of bases of the matroid). By relating these classes to activity classes studied by the first author and Las Vergnas, we give an alternative proof of the above results and a proof of the converse statements that these equalities fail whenever the underlying matroid is not regular. Hence we extend the above results to an equivalence of matroidal properties, thereby giving a new characterization of regular matroids.'
address:
- 'Emeric Gioan: CNRS, LIRMM, Université de Montpellier, France'
- |
Chi Ho Yuen: School of Mathematics, Georgia Institute of Technology\
Atlanta, Georgia 30332-0160, USA
author:
- Emeric Gioan
- Chi Ho Yuen
title: 'On the Number of Circuit-cocircuit Reversal Classes of an Oriented Matroid'
---
Introduction
============
The [*cycle-cocycle reversal system*]{} of a graph, introduced in [@gioan2007enumerating], consists of equivalence classes of orientations of the graph with respect to the [*cycle-cocycle reversal relation*]{}, that is, two orientations are equivalent if they differ by successive reversals of directed (co)cycles. As proven in the same paper, the number of cycle-cocycle reversal classes of $G$ equals the evaluation $t(G;1,1)$ of its Tutte polynomial, which is also the number of spanning trees of $G$. This result can be thought as a “linear algebra free” formulation of Kirchhoff’s Matrix-Tree Theorem. In particular, the cycle-cocycle reversal system is related to various combinatorial objects associated to the graph Laplacian, notably the [*sandpile group*]{} (also known as the [*critical group*]{} or [*Jacobian group*]{} in the literature). The notion of [*chip-firing*]{} (related to the [*abelian sandpile model*]{}) can also be partially interpreted by [*cycle-cocycle reversals*]{}. We refer the reader to [@gioan2007enumerating Section 5] and [@backman2014riemann] for details.
The above setup, and further results involving Tutte polynomial evaluations for related reversal classes, were generalized to [*regular matroids*]{} in [@gioan2008circuit] (in terms of a [*circuit-cocircuit reversal system*]{}). This provides an approach to generalize the theory of sandpile groups and chip-firing to regular matroids (different from the approach in [@MerinoDissertation]). Such topic has been investigated since then in [@bby2017geometric] in a unifying way.
The circuit-cocircuit reversal system is actually defined for general oriented matroids, and it was shown in [@gioan2008circuit] that the aforementioned Tutte polynomial evaluations are not available for $U_{2,k}$, $k\geq 4$. Since $U_{2,4}$ is the excluded minor for the class of regular matroids within oriented matroids, it was expected that these enumerative results are not available when the oriented matroid is not regular; we will prove this rigorously in this note. In particular, we extend the results in [@gioan2008circuit] to an equivalence of (oriented) matroid properties. We use a noteworthy general relation between circuit-cocircuit reversal classes and activity classes of (re)orientations, which are known to be enumerated using the same Tutte polynomial evaluations, and have been introduced in the context of [*active bijections*]{} [@gioan2002thesis; @gioan2005activity; @gioanlasvergans2018AB; @gioanlasvergans2018AB2]. We also use this to give a short proof of results in [@gioan2008circuit].
Preliminaries
=============
We assume that the reader is familiar with the basic theory of oriented matroids [@bjorner1999oriented]. Given an oriented matroid $M$ on $E$, we identify the set of its reorientations with $2^E$ via the bijection associating $A\subseteq E$ with $_{-A}M$.
Let $M$ be an oriented matroid on $E$. Following [@gioan2008circuit], let us write $M \sim -_CM$ if $C$ is a positive circuit or cocircuit of $M$ (we say that $-_CM$ is obtained from $M$ by a [*circuit or cocircuit reversal*]{}, respectively). Applying the same rule to reorientations $-_{A}M$ for $A\subseteq E$ (i.e., writing $-_{A}M\sim-_{C\triangle A}M$ when $C$ is a positive circuit or cocircuit of $-_{A}M$) and taking the transitive closure of the relation, we obtain an equivalence relation, whose equivalence classes are called [*circuit-cocircuit reversal classes*]{} of reorientations of $M$. Allowing only the use of positive circuits, resp. positive cocircuits, yields by the same way the *circuit reversal classes*, resp. the *cocircuit reversal classes*. As observed in [@gioan2008circuit], circuit reversals act on the totally cyclic part of $M$ (the union of positive circuits of $M$) and cocircuit reversals act on the acyclic part of $M$ (the union of positive cocircuits of $M$).
It was shown and geometrically illustrated in [@gioan2008circuit Proposition 2 and Figure 1] that, for any integer $k$, the number of acyclic cocircuit reversal classes of a uniform oriented matroid $U_{2,k}$ equals $1$, or $2$, if $k$ is even, or odd, respectively. On the other hand, it is well-known that an oriented matroid $M$ is regular if and only if $U_{2,4}$ is not a minor of $M$. Indeed, regular matroids are precisely the orientable binary matroids [@bjorner1999oriented Theorem 7.9.3], so the claim follows from [@oxley2006matroid Theorem 6.5.4].
Now, let $M$ be an oriented matroid on a linearly ordered set $E$. Consider a reorientation $-_AM$ of $M$ such that $A$ does not contain the minimum element of a positive circuit or cocircuit of $-_AM$, we call such a reorientation *circuit-cocircuit minimal* (with respect to $M$); the terminology here is from [@backman2014riemann], and it is called *active fixed and dual-active fixed* in [@gioanlasvergans2018AB; @gioanlasvergans2018AB2]. Similarly, we can define a *circuit minimal* (or *active-fixed*), resp. a *cocircuit minimal* (or *dual-active-fixed*), reorientation $-_AM$ of $M$ when $A$ does not contain the minimum element of a positive circuit, resp. cocircuit. Let us denote by $t(M;x,y)$ the Tutte polynomial of $M$. From the works on [*active bijections*]{} [@gioan2002thesis; @gioan2005activity; @gioanlasvergans2018AB; @gioanlasvergans2018AB2], we have:
\[CCMO\_main\] Let $M$ be an oriented matroid on a linearly ordered set. Then
1. $t(M;1,1)=\#$ circuit-cocircuit minimal reorientations of $M$,
2. $t(M;1,2)=\#$ cocircuit minimal reorientations of $M$,
3. $t(M;2,1)=\#$ circuit minimal reorientations of $M$,
4. $t(M;1,0)=\#$ (circuit-)cocircuit minimal acyclic reorientations of $M$,
5. $t(M;0,1)=\#$ circuit(-cocircuit) minimal totally cyclic reorientations of $M$.
Let us explain this briefly; details can be found in [@gioan2002thesis; @gioan2005activity; @gioanlasvergans2018AB; @gioanlasvergans2018AB2]. The [*active partition*]{} of $M$ is a partition of its ground set induced by taking differences of unions of positive circuits/cocircuits whose the minimum element is greater than a given element. Therefore, the minimum elements of the parts are precisely the minimum elements of some positive circuits/cocircuits (called [*active/dual-active elements*]{}). Activity classes of reorientations are the sets of reorientations obtained from a given reorientation by arbitrarily reorienting parts of its active partition. It turns out that all reorientations obtained by this way share the same active partition. Hence activity classes partition the set of reorientations. By choosing a suitable reorientation for each part, each activity class contains a unique circuit-cocircuit minimal reorientation, which can be thought of as a representative of the class. Finally, using a classical formula of the Tutte polynomial in terms of orientation activities [@lasvergnas1984activity], one enumerates activity classes and gets the above evaluations.
Results
=======
We first give a noteworthy property relating reversal classes and activity classes.
\[prop:CCMO\_rep\] Let $M$ be an oriented matroid on a linearly ordered ground set. Every circuit-cocircuit reversal class of $M$ contains at least one circuit-cocircuit minimal reorientation.
The following proof is essentially given in [@backman2018partial] for graphs, but for the sake of interest and completeness, we include it here. A corollary of the proof is that a minimal reorientation can be obtained greedily. Moreover, we note that there is an interpretation using combinatorial commutative algebra [@backman2018partial Section 4].
Start with an arbitrary reorientation of $M$, and greedily reorient any positive (co)circuit whose minimal element is in the set of reoriented elements with respect to $M$. Once the procedure stops, we will have a circuit-cocircuit minimal reorientation equivalent to the starting reorientation, so it suffices to show the procedure always terminates. If this is not the case, then, since the number of reorientations is finite, without loss of generality, we must return to the starting reorientation. Let $e$ be the minimal element that was reoriented (which must occur at least twice) in the process. When $e$ was reoriented for the first time, we must have reoriented it to remove it from the set of reoriented elements with respect to $M$, so the second reorientation is not valid, a contradiction.
By combining Theorem \[CCMO\_main\] and Proposition \[prop:CCMO\_rep\], we can use the set of circuit-cocircuit minimal reorientations (and variations thereof) as an intermediate object, and get the following corollary concerning the enumeration of reversal classes in terms of the Tutte polynomial.
\[coro:CCMO\_red\] Let $M$ be an oriented matroid on a linearly ordered ground set. The number of circuit-cocircuit reversal classes is at most $t(M;1,1)$, with equality if and only if no two circuit-cocircuit minimal reorientations are contained in the same class. The number of acyclic cocircuit reversal classes is at most $t(M ; 1, 0)$, with equality if and only if no two acyclic cocircuit minimal reorientations are contained in the same class. Analogous statements hold for each of the other settings in Theorem \[CCMO\_main\].
The first statement follows from comparing Equation (1) of Theorem \[CCMO\_main\] and Proposition \[prop:CCMO\_rep\]. The variations follow from comparing the other equations of Theorem \[CCMO\_main\] and the corresponding counterparts of Proposition \[prop:CCMO\_rep\], since circuit and cocircuit reversals preserve the acyclic and totally cyclic parts of reorientations.
Now we prove the main theorem of this note, which includes the original results of [@gioan2008circuit] (the enumerations when $M$ is regular, with a new proof) and their converses.
\[thm\] Let $M$ be an oriented matroid. Consider the following six statements:
1. $M$ is regular,
2. $t(M;1,1)=\#$ circuit-cocircuit reversal classes of $M$,
3. $t(M;1,2)=\#$ cocircuit reversal classes of $M$,
4. $t(M;2,1)=\#$ circuit reversal classes of $M$,
5. $t(M;1,0)=\#$ acyclic (circuit-)cocircuit reversal classes of $M$,
6. $t(M;0,1)=\#$ totally cyclic circuit(-cocircuit) reversal classes of $M$.
Then we have the following implications: (1) implies all other statements; (2), (3), (4) each implies (1); (5) implies (1) if $M$ has no loops; (6) implies (1) if $M$ has no coloops. In particular, if $M$ has no loops nor coloops, then all statements are equivalent. Moreover, if any of the equalities fail, then the left hand side is larger.
Let us separate implications.
$\bullet$ $(1)\Rightarrow (2)$. We give an alternative proof to that of [@gioan2008circuit]. By Corollary \[coro:CCMO\_red\], it suffices to show that every circuit-cocircuit reversal class contains a unique circuit-cocircuit minimal reorientation. We claim that any two reorientations within a reversal class differ by a disjoint union of positive circuits and cocircuits, which will imply that at most one of them can be minimal, thus proving the implication.
By induction and restricting to the totally circuit part (the acyclic part follows from duality), it suffices to show that if $C$ is a positive circuit of $M$ and $D$ is a positive circuit of $-_{C}M$, then $C\triangle D$ is a disjoint union of positive circuits of $M$. By [@bjorner1999oriented Corollary 7.9.4], we may assume that some totally unimodular matrix $Q$ realizes $M$. By total unimodularity, if $C$ is a positive circuit of $M$, then the characteristic vector $\rchi_C\in\{0,1\}^E$ of $C$ is in the kernel $\ker(Q)$ of $Q$; similarly, since $D$ is a positive circuit of $-_CM$, $\rchi_{D\setminus C}-\rchi_{C\cap D}\in\ker(Q)$. Since their sum $\rchi_{C\triangle D}$ is in $\ker(Q)$, $C\triangle D$ is a positive vector and contains some positive circuit $C_1$ of $M$, thus $\rchi_{(C\triangle D)\setminus C_1}=\rchi_{C\triangle D}-\rchi_{C_1}\in\ker(Q)$. Proceeding by induction, we can write $C\triangle D$ as a disjoint union of positive circuits.
$\bullet$ $(1)\Rightarrow (3),(4),(5),(6)$. Alternatively to [@gioan2008circuit], the proofs are similar to the above one by restricting to circuit reversals only, then restricting to totally cyclic reorientations only, and then taking duals.
$\bullet$ $(5)\Rightarrow (1)$ assuming $M$ has no loops. Suppose $M$ is not regular. Then it has a minor $M/A\setminus B$ that is isomorphic to $U_{2,4}$. By Corollary \[coro:CCMO\_red\], it suffices to show that there are two cocircuit minimal acyclic orientations in the same (circuit-)cocircuit reversal acyclic class. Up to enlarging $A$, we can assume that $M/A$ is loopless and that the rank of $M/A$ equals $2$. Since $M$ is loopless, up to reorientation, we can assume that $M$ and $M/A$ are acyclic. Let $C$ and $D$ be the two positive cocircuits of $M/A$ (which are also cocircuits of $M$). Thus, $-_CM$ and $-_DM$ are in the same acyclic reversal class. Denote $S=C\triangle D$. Since the rank of $M/A$ is $2$, any cocircuit of $M/A$ is the union of all parallel classes but one. Since $S$ is the union of two parallel classes ($C\setminus D$ and $D\setminus C$) of $M/A$, no cocircuit of $M$ is contained in $S$ (otherwise, the complement of $S$ is a parallel class in $M/A$, and reducing parallel classes of $M/A$ yields $U_{2,3}$, a contradiction). Choose a linear ordering of $E$ such that elements of $S$ are greater than elements of $E\setminus S$, then $S$ does not contain the minimum element of a cocircuit (otherwise it would contain a cocircuit), thus $-_CM$ and $-_DM$ are both cocircuit minimal.
$\bullet$ $(2)\Rightarrow (1)$. Suppose $M$ is not regular. Then $M'$, the oriented matroid obtained from removing all loops of $M$, is also not regular. By the implication $(5)\Rightarrow (1)$ for $M'$, there exist distinct acyclic reorientations $-_{A}M'$ and $-_{B}M'$ that are (circuit-)cocircuit reversal equivalent and both (circuit-)cocircuit minimal. Now $-_{A}M$ and $-_{B}M$ are circuit-cocircuit reversal equivalent reorientations of $M$ that are both circuit-cocircuit minimal. The implication follows from Corollary \[coro:CCMO\_red\].
$\bullet$ $(3)\Rightarrow (1)$. The proof is the same as for $(2)\Rightarrow (1)$ except that, at the end, $-_{A}M$ and $-_{B}M$ are cocircuit reversal equivalent reorientations of $M$ that are both cocircuit minimal. The implication again follows from Corollary \[coro:CCMO\_red\].
$\bullet$ $(4)\Rightarrow (1)$, and $(6)\Rightarrow (1)$ assuming $M$ has no coloops. The two implications are the dual statements of $(3)\Rightarrow (1)$ and $(5)\Rightarrow (1)$, respectively.
Finally, let us mention that the relations between the implication $(5)\Rightarrow (1)$ and the other ones could also be handled from the decomposition into acyclic and totally cyclic parts, along with the convolution formula for the Tutte polynomial, similarly as in [@gioan2008circuit].
We end with an open question. By direct computation, the number of circuit-cocircuit reversal classes and the number of bases differ by a rather large margin for small non-regular oriented matroids. So, does there exist an absolute constant $K>1$ such that the number of bases of a non-regular oriented matroid is at least $K$ times the number of circuit-cocircuit reversal classes?
Acknowledgements {#acknowledgements .unnumbered}
================
The second author, Chi Ho Yuen, would like to thank Matthew Baker for suggesting the problem, and Spencer Backman for introducing him the work of the first author, Emeric Gioan. The role of the first author of this paper has been to simplify and extend a preprint written at the initiative of the second author.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.