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+Why do Nearest Neighbor Language Models Work?
+Frank F. Xu
+Uri Alon
+Graham Neubig
+Language Technologies Institute
+Carnegie Mellon University
+{fangzhex,ualon,gneubig}@cs.cmu.edu
+Abstract
+Language models (LMs) compute the probability of a text by sequentially computing
+a representation of an already-seen context and using this representation to predict the
+next word. Currently, most LMs calculate these representations through a neural network
+consuming the immediate previous context. However recently, retrieval-augmented LMs
+have shown to improve over standard neural LMs, by accessing information retrieved from a
+large datastore, in addition to their standard, parametric, next-word prediction. In this paper,
+we set out to understand why retrieval-augmented language models, and specifically why
+k-nearest neighbor language models (kNN-LMs) perform better than standard parametric
+LMs, even when the k-nearest neighbor component retrieves examples from the same
+training set that the LM was originally trained on. To this end, we perform a careful
+analysis of the various dimensions over which kNN-LM diverges from standard LMs, and
+investigate these dimensions one by one. Empirically, we identify three main reasons
+why kNN-LM performs better than standard LMs: using a different input representation
+for predicting the next tokens, approximate kNN search, and the importance of softmax
+temperature for the kNN distribution. Further, we incorporate these insights into the
+model architecture or the training procedure of the standard parametric LM, improving
+its results without the need for an explicit retrieval component. The code is available at
+https://github.com/frankxu2004/knnlm-why.
+1
+Introduction
+Language modeling is the task of predicting the probability of a text (often conditioned on context), with
+broad-spanning applications across natural language processing (Bengio et al., 2003; Merity et al., 2018;
+Baevski and Auli, 2018; Brown et al., 2020). This modeling is usually done by sequentially encoding a context
+ct using a trained neural network function f, and computing the probability of the next word wt according to
+f (ct) and a vector representation of wt.
+Recently, retrieval-augmented LMs have shown a series of impressive results (Grave et al., 2017; Guu et al.,
+2018; He et al., 2020; Khandelwal et al., 2020b; Borgeaud et al., 2022; Alon et al., 2022). Retrieval-augmented
+LMs compute next token distributions based not only on the immediately preceding context ct and the model
+parameters, but also on an external datastore, from which examples are retrieved and incorporated into the
+base LM’s prediction.
+One retrieval-augmented model that is notable for both its simplicity and efficacy is the k-nearest neighbor
+language model (kNN-LM; Khandelwal et al., 2020b). It extends a trained base LM by linearly interpolating
+the output word distribution with a kNN model. The nearest neighbors are retrieved according to the distances
+between the current context embedding of the base LM and all the context embeddings in the datastore. The
+datastore is created by encoding all contexts from any text collection, including the original LM training data.
+One of the most surprising results from Khandelwal et al. (2020b) is that kNN-LM reduces the perplexity of
+the base LM even when the kNN component is retrieving examples from the same training set that the LM
+was originally trained on, indicating that the kNN-LM improves the ability to model the training data and is
+Preprint. Under review.
+arXiv:2301.02828v1  [cs.CL]  7 Jan 2023
+
+Multi Headed 
+Attention
+Feed Forward 
+Network
+Layer Norm
+ℎ𝑠𝑚
+𝑊𝑠𝑚
+𝐷
+𝑉
+ℎ𝑑𝑠
+𝑊𝑑𝑠
+𝐷
+𝑁𝑑𝑠
++
+𝑃𝐿𝑀 parametric component
+𝑃𝑘𝑁𝑁 non-parametric component
+In 𝑘NN-LM:
+𝑁𝑑𝑠: up to 5000𝑉
+𝐷
+𝐷
+mask-to-k()
+In 𝑘NN-LM: 
+top-𝑘() 
+FFN
+ATT
+softmax()
+softmax()
+Figure 1: An illustration of the generalized formulation of kNN-LM in Equation 5.
+not simply benefiting from access to more data. Intrigued by this, we ask questions like, could kNN-LM be
+improving because of capacity issues in the parametric base LM? In this paper, we set out to understand why
+kNN-LMs work even in this setting.
+In the following sections, we first elucidate connections between the added kNN component and the standard
+LM component. Specifically, we note that word distributions from the two components are both calculated
+using a softmax function, based on the similarity of the current context embedding with a set of embeddings
+that corresponds to different next words. With this intuition, we formalize and generalize the non-parametric
+distribution calculation with the softmax layer and word embedding layer used in parametric LMs. We then
+show that this generalized form exposes a variety of design choices, e.g., the number of context embeddings
+in the datastore, the input representation used in softmax layer, different similarity functions, as well as the
+approximation and sparsification implementations in the kNN search. This provides a general framework for
+analyzing kNN-LM and similar models and allows us to perform ablation studies that test the importance of
+various design decisions.
+We proceed to propose multiple hypotheses for why kNN-LM works, which are testable by adjusting the
+various parameters exposed by our generalized formulation. Based on these hypotheses, we perform ablation
+experiments and analyze the nuances between different implementations of the generalized version of PkNN.
+As the answer to our question, “why kNN-LMs work”, we eventually show that the most probable reasons are
+threefold:
+1. Ensembling the output of softmax using two representations from different layers of the transformer
+is important; in our experiments, this accounts for 55% of the performance gain of kNN-LM, or 6.5%
+relative perplexity improvement compared to the base LM.
+2. kNN-LM uses approximate nearest neighbor search to handle the large number of candidates, and
+the lack of this preciseness in this algorithm actually helps kNN-LM to generalize better than using
+exact nearest neighbor search and distance calculation, possibly due to a regularization effect. The
+relative perplexity improvement from this factor is about 2.6%.
+3. Depending on the design decisions that are chosen for modeling, adding a temperature term to
+the kNN non-parametric component can become crucial to the success of modeling (although
+coincidentally, in the original settings of Khandelwal et al. (2020b), a temperature of 1.0 is close to
+optimal, which hid the importance of this term). In some settings, the relative perplexity gap between
+the default and optimal temperature can be as high as 3.7%.
+Finally, one significant drawback to the current kNN-LM is the inefficiency of kNN search performed at each
+step (He et al., 2021; Borgeaud et al., 2022; Alon et al., 2022; Wang et al., 2022). Because of the similarity
+between kNN-LM and the parametric LM’s last layers and the many design choices, we also demonstrate that
+we are able to make kNN-LM more efficient by substituting the kNN search with another matrix operation
+that can fit in accelerator memory while maintaining more than half the perplexity improvement, or more than
+6.5% relative improvement compared to the base LM.
+2
+
+2
+Formalizing and Generalizing kNN-LM
+kNN-LM (Khandelwal et al., 2020b) is a linear interpolation between a base LM and a kNN model. Given a
+set of contexts ci and their corresponding next token wi as a pair (ci, wi) ∈ D, kNN-LMs create a datastore
+(K, V) = {(ki, vi)}, as a set of key-value pairs:
+(K, V) = {(f (ci) , wi) | (ci, wi) ∈ D}
+(1)
+During inference, the parametric component of the LM generates the output distribution pLM(wt|ct; θ) over
+the next tokens and produces the corresponding context representation f(ct), given the test input context ct.
+Then the non-parametric component of the LM queries the datastore with the f(ct) representation to retrieve
+its k-nearest neighbors N according to a distance function d(·, ·). Next, the kNN-LM computes a probability
+distribution over these neighbors using the softmax of their negative distances, and aggregates the probability
+mass for each vocabulary item across all of its occurrences in the retrieved targets:
+pkNN(wt|ct) ∝
+�
+(ki,vi)∈N
+1wt=vi exp(−d(ki, f(ct)))
+(2)
+Finally, this distribution is interpolated with the parametric LM distribution pLM to produce the final kNN-LM
+distribution:
+p(wt|ct; θ) = (1 − λ)pLM(wt|ct; θ) + λpkNN(wt|ct)
+(3)
+where λ is a scalar that controls the weights of the interpolation between two components, with higher λ
+putting more weight on the non-parametric component.
+Looking closely at Equation 2, we can notice a similarity between the calculation of PkNN and the standard
+PLM. The kNN distribution is based on the distances between the current context and the nearest neighbors
+from the datastore, normalized by a softmax function. Recall that in (standard) parametric language models,
+the distribution over the vocabulary is also based on a measure of distance, the inner product between the
+current context embedding and the word embeddings of every token in the vocabulary. Because each context
+embedding in the datastore (K, V) corresponds to a target token, we can also view this datastore as a large
+word embedding matrix with multiple word embeddings for each of the vocabulary words. Theoretically,
+given unlimited computation, we could calculate the distribution based on the distances to every embedding in
+the datastore, and aggregate by vocabulary items, making it more closely resemble PLM. In this case, k = |D|,
+the size of the entire datastore, and Equation 2 becomes the following, based on the distances to every context
+in the datastore D instead of a subset of nearest neighbors N.
+pD(wt|ct) ∝
+�
+(ki,vi)∈D
+1wt=vi exp(−d(ki, f(ct)))
+(4)
+In practice, we use kNN search as a way of approximation, by limiting the calculation to only k nearest
+neighbors to avoid the computational cost of calculating the distribution over the entire datastore.
+If we re-write and generalize Equation 2, both the kNN-LM of Khandelwal et al. (2020b) and a large number
+of related models can be expressed through the following equation:
+Pinterp = (1 − λ) softmax(Wsm · hsm)
+�
+��
+�
+PLM parametric component
++λ Msoftmax(mask-to-k(Wds ⊗ hds)/τ)
+�
+��
+�
+PkNN non-parametric component
+.
+(5)
+Figure 1 provides an illustration of Equation 5. The first term of the equation is the standard parametric
+language model, whereas the second represents a generalized version of utilizing an external datastore. The
+first component, the output layer of a common parametric language model, is relatively straightforward. Wsm
+of size V × D is the embedding matrix of the output token, and hsm is the context vector used to calculate the
+distribution of the output token, usually the output of the final feedforward layer in the transformer.
+In the second component, Wds represents the datastore, of size Nds × D. Nds is the number of entries in
+the datastore, and D is the size of each context vector. hds represents the context vector used to query the
+datastore. As shown in Figure 1, these vectors can come from different layers of the transformer architecture.
+⊗ represents the operation type used to calculate the similarity between context vectors and the query vector,
+which also has several alternatives that we discuss below.
+mask-to-k(·) represents a function to sparsify similarity scores across the datastore, setting all but k similarity
+scores to −∞, which results in probabilities of zero for all masked similarity scores after the softmax.
+3
+
+Practically, this is necessary for kNN-LMs because the size of the datastore Nds makes it infeasible to
+calculate all outputs at the same time. With masked logits, we apply a more generalized version of softmax
+with temperature τ. Intuitively adding the temperature can adjust the peakiness or confidence of the softmax
+probability distribution output. After the softmax, the matrix M of dimension V × Nds sums the probability of
+the Nds datastore entries corresponding to each of the V vocabulary entries. Each column in this matrix consists
+of a one-hot vector with a value of 1 and the index corresponding to the vocabulary item wi corresponding to
+the datastore entry for ci.
+Within this formulation, it becomes obvious that there are many design choices for kNN-LM-like models. One
+important thing to note is that the right side of Equation 5 is actually very similar to the left side representing
+the standard parametric language model, with a few additional components: M, mask-to-k, and ⊗. More
+specifically, some of the design decisions that go into the kNN-LM, and parallels with standard parametric
+models are:
+1. Size of Wds: In the standard parametric model, the size of Wsm is V embedding vectors, each with
+D dimensions. In the kNN-LM it is very large: Nds, the size of the datastore, usually the number of
+tokens in the entire training corpus.
+2. Input representation: In the parametric model, hsm is the output from the feedforward layer in the
+last transformer block, which we abbreviate “ffn”. In contrast, Khandelwal et al. (2020b) rather use
+as hds the output from the multi-headed attention layer of the last transformer block (before running
+the representations through the feed-forward network, and after the LayerNorm (Ba et al., 2016)),
+which we abbreviate as “att”.
+3. Similarity & Temperature: In the parametric model, the functional form of ⊗ is the inner product
+(abbreviated IP), whereas Khandelwal et al. (2020b) use negative squared L2 distance (abbreviated
+L2) as a similarity function between Wds and hds. As the similarity scores are turned into probability
+distributions with the softmax function, the choice of softmax temperature (τ) can control the scaling
+of the similarity scores and thus the peakiness of the non-parametric component distribution.
+4. Approximation & Sparsification: In the parametric model, k = V , and no values are masked,
+but in the kNN-LM, k ≪ V , and most of the datastore entries are pruned out. The definition of
+the mask-to-k(·) function, i.e. how to select the important datastore embeddings to include in the
+similarity calculation (in kNN-LM’s case the k nearest neighbors), is a crucial open design choice.
+In the following sections, we set out to better understand how each of these design decisions contributes to the
+improvement in accuracy due to the use of kNN-LMs.
+3
+Baseline kNN-LM Results
+First, we evaluate the kNN-LM baseline on the Wikitext-103 dataset (Merity et al., 2016), and examine the
+importance of two design choices: the input representation hds and the similarity function ⊗.
+In models examined in this paper, the parametric model is a transformer language model with mostly the
+same architecture as in Khandelwal et al. (2020b). However, We do make modifications to the original base
+LM (Baevski and Auli, 2018) to accommodate our experimentation need. We using BPE tokenization (Sennrich
+et al., 2015) to train a smaller vocabulary (33K) than the original (260K) on the training corpus of Wikitext-103,
+as subword tokenization is ubiquitous in many state-of-the-art language models (Radford et al., 2019; Devlin
+et al., 2018; Liu et al., 2019; Brown et al., 2020). Using subword tokenization also eliminates the need for
+adaptive softmax (Joulin et al., 2017). It makes the output layer more generalized, sharing more resemblance
+to the kNN component as described in Section 2, and facilitates the ablation studies in this paper.1 This base
+LM has 268M parameters. To get a perspective on how large the datastore is, it is built on the training data
+that contains nearly 150M BPE tokens, each paired with a context vector of size 1024. This datastore has a
+total memory consumption of about 300GB. At every retrieval step, we take the top 1024 nearest neighbors,
+i.e., k = 1024, following Khandelwal et al. (2020b). The interpolated perplexity is computed with optimal
+interpolation parameter λ tuned according to the perplexity on the development set. λ is fixed during the
+inference for all predictions, the same as the standard kNN-LM.
+1By training our own version of the base LM from scratch with BPE tokenization and a standard output softmax layer,
+our LM’s perplexity is worse than that used in the original kNN-LM paper. However, we observe similar relative gains
+from the additional kNN component. We argue that the base LM’s performance is orthogonal to the study of the factors
+behind kNN-LM’s improvements.
+4
+
+hds
+⊗
++#params
+PPL
+Interp. PPL
+Oracle
+Base LM
+-
+-
+0
+21.750
+-
+-
+kNN-LM-L2
+att
+L2
+Nds × D
+∞
+19.174
+14.230
+kNN-LM-IP
+att
+IP
+Nds × D
+∞
+19.095
+14.077
+kNN-LM-L2
+ffn
+L2
+Nds × D
+∞
+20.734
+15.594
+kNN-LM-IP
+ffn
+IP
+Nds × D
+∞
+21.101
+16.254
+Table 1: Performance of the parametric language model and several kNN-LM variants.
+Results comparing multiple kNN-LM variants are shown in Table 1. The first row represents the base
+parametric language model’s perplexity. The second is a formulation analogous to that of Khandelwal et al.
+(2020b), and in the remaining rows, we vary the input representation hds and distance function ⊗ from
+Equation 5. All of them use a large datastore with size Nds, approximately 5000 times the size of the
+vocabulary V , as also reflected in “+#params”, the number of additional parameters other than the base LM.
+We report several important quantities with respect to each model.
+• “PPL” shows the perplexity of only the kNN component of the model pkNN(). This is ∞ for all kNN-
+LM models in all cases, as when the kNN search does not retrieve any datastore entries corresponding
+to the true target word wt the probability of the target word will be zero.
+• “Oracle” shows the lower bound of the interpolation performance by choosing the best λ for each
+token in the evaluation dataset, which will either be λ = 0 or λ = 1 depending on whether
+PLM(wt|ct) > Pknn(wt|ct) or not, respectively.
+From the table, we can see that:
+1. Using the output of the multi-headed attention layer (“att”) as hds (instead of the standard “ffn” layer)
+is crucial for better performance of kNN-LM.
+2. In general, using negative squared L2 distance or inner product as a similarity function does not result
+in a large and consistent difference, although in our setting, IP provides slightly better performance
+when using the “att” inputs, and slightly worse when using “ffn” inputs.
+3. Interestingly, when using “ffn” and “IP”, the same input and distance metric used in the parametric
+model, the results are the worst, indicating that the kNN-LM is particularly benefiting when the
+kNN-LM achieves a different view of the data from the parametric model.
+We found in preliminary experiments that kNN-LM is generalizable to other base language models as well,
+ranging from small models with 82M parameters to larger models with 774M parameters. The gain from
+kNN-LM is more significant when used with a smaller, less capable base language model, as expected. The
+details are shown in Appendix A. In this paper, we are mainly focused on the factors contributing to the
+relative improvements from kNN-LM, instead of the absolute performance, so we use the 268M model for the
+remainder of the paper.
+In the next sections, we perform further experiments with ablations on the general formulation Equation 5 to
+elucidate the key elements contributing to the performance improvements in kNN-LM.
+4
+Effect of Different Wds Formulations
+4.1
+Replacing the Datastore with Trainable Embeddings
+From the observation in Section 3, we see that the choice of hds has a large impact on the performance of
+kNN-LM. This intrigues us to explore if one key to the improvements afforded by kNN-LM lies in the use
+of different input representations together, namely the attention output (hds = att) and feedforward output
+(hds = ffn). However, from only the experiments above, it is not possible to disentangle the effect of the
+choice of hds and that of other design choices and factors in Equation 5.
+To test the effect of hds in a more controlled setting, we remove the non-parametric datastore entirely, and
+initialize Wds in Equation 5 with a randomly initialized word embedding matrix with the same size (Nds = V )
+5
+
+as the LM’s output embedding Wsm, and train Wds with all other parameters fixed.2 The loss function for
+training is the cross-entropy loss of softmax(Wds · hds) with respect to the ground-truth tokens, identically
+to how the base LM is trained. We compare how using hds = att or hds = ffn affects the interpolated
+performance. The results are shown in Table 2, and we also show results from kNN-LMs using these two
+varieties of input representation for reference.
+From these experiments we can find several interesting conclusions:
+Effectiveness of re-training Wds: In the case of “Learned Wds w/ FFN”, we are essentially re-learning the
+weights feeding into the softmax function separately from the underlying LM encoder. Despite this fact, we
+can see the model achieves a PPL of 20.920, which is 0.83 points better than the base model. This suggests
+that there is some benefit in learning the parameters of Wds after freezing the body of the transformer encoder.
+Effectiveness of ensembling two predictors: In both cases for Wds, the interpolated perplexity is significantly
+better than that of using a single predictor. This is particularly the case when using the “att” representation for
+hds, suggesting that the utility of ensembling predictions from two views of the data is not only useful when
+using kNN-LM, but also in standard parametric models as well.
+Parametric ensembles as an alternative to kNN-LM?: Overall, by using a separate word embedding matrix
+with size V × D as an alternative to kNN, we can recover about 55% of the performance gain achieved by
+kNN-LM, with only a limited number of parameters and without the necessity for slow kNN retrieval every
+time a token is predicted. This suggests that the majority of the gain afforded by kNN-LM could be achieved
+by other more efficient means as well.
+hds
+Nds
+⊗
++#params
+PPL
+Interp.
+Oracle
+Base LM
+-
+-
+-
+0
+21.750
+-
+-
+kNN-LM w/ ATT
+att
+Big
+IP
+Nds × D
+∞
+19.095
+14.077
+Learned Wds w/ ATT
+att
+1x
+IP
+V × D
+22.584
+20.353
+16.954
+kNN-LM w/ FFN
+ffn
+Big
+IP
+Nds × D
+∞
+21.101
+16.254
+Learned Wds w/ FFN
+ffn
+1x
+IP
+V × D
+20.920
+20.694
+18.772
+Table 2: Performance comparison how the choice of hds, input representation, affects kNN baselines and
+models with learnable embeddings as datastore alternative. hds is the attention layer output.
+4.2
+Increasing the Softmax Capacity
+One premise behind kNN-LM is that the large datastore is the key reason for the model working well: the
+larger the softmax capacity, the better the performance. Naturally, as a first step, we need to check whether
+such a big datastore is warranted and whether the high rank of Wds leads to better performance. We test
+the effect of the datastore size for kNN retrieval on kNN-LM interpolated perplexity. If a bigger datastore
+(a high rank Wds) is better in kNN-LM than a smaller datastore, then the hypothesis of softmax capacity is
+more probable. We randomly subsample the full datastore in varying percentages and the results are shown
+in Figure 2. The full datastore contains more than 150M entries and storing them takes 293GB when using
+half-precision floating points (fp16). We can see that whether or not approximate kNN is used, the final
+perplexity decreases almost linearly with more percentage of the original datastore. Even with just 5% of
+the datastore size (15G), kNN-LM still provides a benefit over just using the base LM. However, even when
+the subsampling percentage reaches 90%, having more entries in the datastore still provides benefits without
+having significant diminishing returns, suggesting that a large datastore is beneficial.
+One possible reason why a larger datastore is helpful is that words can be difficult to predict. There are several
+reasons: (1) They are rare, or (2) they are frequent, but they have multiple meanings and appear in different
+contexts. The softmax bottleneck (Yang et al., 2017) suggests that the final dot product of language model
+Wsm · hsm limits the expressivity of the output probability distributions given the context; that is, a single
+output vector of a fixed (1024) size cannot express all the possible mappings between 100M training examples
+and 33K vocabulary outputs. We hypothesize that kNN-LM improves performance by alleviating the problem,
+since Wds ⊗ hds has a higher rank and is more expressive than just Wsm · hsm. In other words, kNN is a
+sparse approximation of the full softmax over all the embeddings in the datastore Wds. To test this hypothesis,
+2Because we previously found little difference between IP and L2 as similarity functions, we use IP in the experiments.
+For simplicity, we set temperature τ = 1.
+6
+
+we disentangle the effect of the high rank in Wds from the actual saved context embeddings in Wds, by training
+an embedding matrix of the same desired size to test from scratch.
+Ratio to Full Datastore Size
+Interpolated Perplexity
+19.000
+20.000
+21.000
+22.000
+0.00
+0.25
+0.50
+0.75
+1.00
+Figure 2: The effect of the size of the datastore used for kNN retrieval on final interpolated perplexity.
+We explore several potential solutions for increasing the capacity of softmax, and examine if they can achieve
+a similar effect of kNN-LM. The first and easiest solution is to increase the embedding matrix size by adding
+more embedding vectors for each word type in the vocabulary. To test this, we replace Wsm with a much
+smaller matrix of size nV × D, where we allocate n embedding vectors for each word type. When calculating
+the probability from this component, we compute the softmax over nV items and sum the probabilities for
+each vocabulary entry to calculate the final probability. mask-to-k(·) is no longer needed, as this formulation
+is small enough to fit the entire matrix in the GPU. We then finetune the new Wds on the training data until
+convergence.
+Figure 3 compares the base LM and the original kNN-LM versus using either attention layer output (“att”)
+or feedforward layer output (“ffn”) as hds. We plot the number of embeddings for each word type (nV total
+embeddings in Wds) versus the interpolated perplexity, with full details found in Appendix B. In both cases,
+comparing with the top horizontal line which represents the perplexity of the base LM, replacing the datastore
+with a much smaller weight matrix (from Nds to nVds) by assigning only a few more embeddings for each
+word helps, although only about half as effective as kNN-LM. To give a perspective, the original datastore
+size is about 5000V . Surprisingly, we find that increasing n does not always bring better performance, even
+though a larger datastore is better than using a small datastore in kNN-LM. We can see that when hds = ffn,
+over-parameterization provides very limited improvements, while for hds = att it does not bring consistent
+improvements at all. Comparing the trend of increasing the embeddings in Wds, with the bottom horizontal line
+in the plot, which represents the perplexity of the standard kNN-LM using the full datastore (Wds with approx.
+5000V embeddings), we can see no clear trend that more trainable embeddings result in better perplexity, and
+that the gap between using trained embeddings and using full datastore is still significant. This suggests that
+simply over-parameterizing Wds is not an effective method of achieving accuracy gains similar to kNN-LM.
+We hypothesize that this is because by just adding more embeddings, while still using the same training
+procedure as the original LM, the multiple embeddings for each word type after learning could still be very
+close to each other, and thus do not increase the softmax capacity much. This suggests that some regularization
+terms may be needed during training to make the multiple embeddings not converge to the same vector,
+rendering over-parameterization useless.
+Besides simply increasing the number of embedding vectors equally for each word type, we also propose
+other alternatives to increase softmax capacity. First, we hypothesize that different word types have different
+difficulties for the language model to predict. For those words that appear very frequently, they may appear
+in many different contexts. As a result, instead of adding an equal number of additional embeddings to
+each word type, we propose to adaptively increase the number of embeddings for word types based on word
+frequency, or total training loss for the word. Second, we try to break the softmax bottleneck with a Mixture
+of Softmax. Yang et al. (2017) proposes a solution to the problem using a Mixture of Softmax (MoS) to
+produce more linearly independent probability distributions of words given different contexts. Last, opposite
+to training the word embeddings of increased size, we also consider ways to compress the datastore down to a
+similar-sized embedding matrix for softmax computation by clustering the whole datastore and allowing for
+further finetuning of the embedding matrix consisting of cluster centroids. However, none of these alternative
+methods provided additional benefits over the simple multi-embedding approach. More details on these
+attempts can be found in Appendix C.
+7
+
+Number of Trained Embeddings (nV)
+Interpolated Perplexity
+19
+20
+21
+22
+2
+4
+6
+8
+att
+   
+ffn
+ 
+Figure 3: The number of embeddings per word type (nV total embeddings in Wds) versus interpolated
+perplexity. The horizontal line at the top represents the perplexity of the base LM. The horizontal line at the
+bottom represents the interpolated perplexity using a full datastore with kNN-LM.
+5
+Approximate kNN Search & Softmax Temperature
+5.1
+Comparing Approximate kNN Search
+To calculate PkNN of the non-parametric component in Equation 5, it is usually prohibitive to use exhaustive
+kNN search, and thus Khandelwal et al. (2020a) use approximate kNN search using the FAISS library (Johnson
+et al., 2019). The use of FAISS (similarly to other approximate search libraries) results in two varieties of
+approximation.
+• Approximate Neighbors: Because the search for nearest neighbors is not exact, the set of nearest
+neighbors might not be equivalent to the actual nearest neighbors. Recall the function mask-to-k(·) in
+Equation 5, it is the function where we select the kNN entries from the datastore Wds. We denote
+“real mask” as the accurate nearest neighbors for mask-to-k(·) selection, and “FAISS mask” as the
+approximate nearest neighbors returned by the FAISS library.3
+• Approximate Scores: In addition, FAISS makes some approximations in calculating the distances
+between the query and the retrieved neighbors for efficiency purposes. We denote “real score” as the
+scores calculated from ground truth distances between the embeddings, and “FAISS score” as the
+distances returned by FAISS approximate search.
+The comparison of the different approximation settings is shown in Table 3. Quite surprisingly, we actually
+find that the interpolated perplexity with approximate search is better than that with exact search, both with
+respect to the mask and the score calculation. Intrigued by this counter-intuitive result, we explore the effect of
+kNN search approximation.
+hds
+⊗
++#params
+PPL
+λ
+Interp. PPL
+Oracle
+Base LM
+-
+-
+0
+21.750
+-
+-
+-
+kNN-LM w/ FAISS mask, FAISS score
+att
+L2
+Nds × D
+∞
+0.271
+19.174
+14.230
+kNN-LM w/ FAISS mask, real score
+att
+L2
+Nds × D
+∞
+0.176
+19.672
+14.393
+kNN-LM w/ real mask, real score
+att
+L2
+Nds × D
+∞
+0.172
+19.735
+14.480
+Table 3: Performance of the parametric language model and comparison of kNN-LMs using the approximate
+versus ground truth kNN.
+First, we plot the subsampled size of the datastore with the interpolated perplexity Figure 4, a similar plot
+to Figure 2, but showcasing the comparison between approximate and real masks, between approximate and
+real scores in both the full datastore as well as a small subsampled datastore setting. We find that using an
+approximate FAISS mask to find nearest neighbors is better than using the ground truth nearest neighbors and
+that using the approximate score returned by FAISS is better than recomputing the ground truth distances
+3To calculate the real mask over a large datastore, we shard the datastore into several smaller datastores, calculate the
+nearest neighbors for each of the smaller datastores, and combine them back together to get the final result.
+8
+
+between embeddings for the kNN distribution at different levels of datastore size, both at 5% or 100%.
+Interestingly, the gap between using an approximate score or real score given the same approximate nearest
+neighbors (“FAISS mask, FAISS score” vs. “FAISS mask, real score”) is larger than that between using
+approximate or real nearest neighbors given the same ground truth method of calculating the distance (“real
+mask, real score” vs. “FAISS mask, real score”), for reasons we will elucidate in the next section.
+Ratio to Full Datastore Size
+Interpolated Perplexity
+19.000
+20.000
+21.000
+22.000
+0.00
+0.25
+0.50
+0.75
+1.00
+FAISS mask, FAISS score
+FAISS mask, real score
+real mask, real score
+Figure 4: The differences between using approximate and accurate kNN search on varying size of the datastore.
+5.2
+Adding Softmax Temperature to kNN Distribution
+Because the number of retrieved nearest neighbors, k is usually much smaller than the vocabulary size V ,
+intuitively, the kNN distribution PkNN used for interpolation tends to be more peaky than the standard LM
+output distribution. When k = 1024 and V = 33000, as in our experiments, PkNN will only have a few
+vocabulary items with a non-zero probability. Furthermore, many of the retrieved neighbors share the same
+target token and thus make the kNN distribution even peakier. One way to control the entropy, or peakiness of
+the distribution is to add temperature to the logits that go into the softmax function (Holtzman et al., 2019).
+We calculate the probability of non-parametric component PkNN with the following equation where t is the
+softmax temperature:
+PkNN = Msoftmax(mask-to-k(Wds ⊗ hds)/t)
+(6)
+In general, the higher the temperature, the less “peaky” the distribution would become. We experiment with
+both the 5% as well as the full datastore using different temperatures ranging from 0 to 3 at 0.1 intervals. The
+results are shown in Figure 5a and Figure 5b respectively.
+(a) On 5% subsampled datastore.
+(b) On full datastore.
+Figure 5: The interpolated perplexity varies with different softmax temperature values.
+We can see that the default temperature t = 1 does not always result in the best-interpolated perplexity and
+tuning softmax temperature is desirable for all sizes of datastore. The lesson learned here is that tuning the
+9
+
+real mask, real score
+21.70
+FAISS mask, FAISS score
+FAlSS mask, real score
+21.65
+21.60
+21.55
+21.50
+21.45
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0real mask, real score
+20.6
+FAISS mask, FAISS score
+FAiss mask, real score
+20.4
+20.2
+20.0 
+19.8
+19.6
+19.4
+19.2
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0softmax temperature for the kNN distribution is crucial for getting optimal results from each setting. Only
+coincidentally, a temperature of 1.0 was close to optimal in the original settings of Khandelwal et al. (2020b),
+which hid the importance of this hyperparameter.
+In both the 5% subsampled datastore and the full datastore scenarios, temperature t = 1 is close to optimal
+when using “FAISS mask, FAISS score”. When using either “real mask” or “real score”, this is not true
+anymore. Even at the optimal temperature for each setting, “real mask, real score” somewhat underperforms
+“FAISS mask, real score”. It is consistent with the counter-intuitive phenomenon discussed in Section 5.1.
+There are also differences between the two scenarios of different datastore sizes. With the full datastore, using
+“real score” outperforms “FAISS score” given the same “FAISS mask”. However, the opposite is true when
+using the 5% datastore. This suggests that as the datastore size grows, using accurate distance values are better
+than the approximate ones. The relatively small gap between using “real score” and “FAISS score” in both
+datastore settings shows that the main contributor to the improvements is using approximate nearest neighbors
+(“FAISS mask”) rather than using approximate distance values (“FAISS score”).
+We hypothesize that this is related to regularization for preventing overfitting, and approximate search provides
+fuzziness that functions as a regularizer. We can think of the non-parametric part in kNN-LM, the kNN
+component as a model, where the datastore size is its model capacity, and the datastore is its training data.
+Considering that the kNN component uses the exact same training data as the base parametric LM, having
+ground truth, accurate kNN search may cause the kNN component to overfit the training data. Comparing the
+small datastore with only 5% with the original datastore, we see that a small datastore means a small training
+set for the kNN “model” and it thus it benefits more from this regularization, both both through using the
+FAISS mask and FAISS score (at optimal temperature settings). From these experiments, we can see that,
+surprisingly, one of the important ingredients in kNN-LM seems to be approximate kNN search, which likely
+prevents overfitting to the datastore created from the same training set. We further analyze this unexpected
+result in Appendix D, where we find that longer words and words that appear in many different contexts have
+slightly better results with approximate nearest neighbors.
+Notably, similar effects, where an approximation component lead to better generalization, have been reported in
+other NLP tasks as well, and are sometimes referred to as “beneficial search bias”, when modeling errors cause
+the highest-scoring solution to not be the correct one: Meister et al. (2020b) suggest that “quite surprisingly,
+beam search often returns better results than exact inference due to beneficial search bias for NLP tasks.”
+Stahlberg and Byrne (2019) also conclude that “vanilla NMT in its current form requires just the right amount
+of beam search errors, which, from a modeling perspective, is a highly unsatisfactory conclusion indeed, as
+the model often prefers an empty translation”.
+6
+Probably Wrong Hypotheses for Why kNN-LMs Work
+The results in the previous sections are the result of extensive analysis and experimentation, in which we also
+tested a number of hypotheses that did not turn out to have a significant effect. Additional details of these
+hypotheses are detailed in Appendix E, and we hope that they may provide ideas for future improvements of
+retrieval-based LMs.
+Ensemble of Distance Metrics
+We hypothesized that the ensemble of two distance metrics: the standard
+inner product distance (which the LM uses) and the L2 distance (which the kNN component uses), is the key
+to the improvement. However, we found that similar gains can be achieved using the inner-product metric for
+the retrieved kNN. More details can be found in Appendix E.1.
+Ensembling of Two Models
+We hypothesized that the kNN component merely provides another model
+for ensembling. The improvement from kNN-LM is purely due to the ensembling effect of different models.
+However, we found that kNN-LM’s improvement is orthogonal to ensembling with a different base LM. More
+details can be found in Appendix E.5.
+Sparsification
+The mask-to-k(·) used by kNN retrieval induces sparsity in the distribution over the vocab-
+ulary, due to a small k (typically 1024) compared to the size of the vocabulary V (33K in our experiments
+and 260K in the original settings of Khandelwal et al. (2020b)). We hypothesized that kNN-LM increases
+the probability of the top-k entries while taking “probability mass” from the long tail of unlikely word types.
+However, we could not gain any benefits solely from sparsifying the output probability of a standard LM and
+interpolating it with the original LM. More details can be found in Appendix E.2.
+10
+
+Stolen Probabilities
+The stolen probabilities effect (Demeter et al., 2020) refers to the situation where the
+output embeddings of an LM are learned such that some words are geometrically placed inside the convex
+hull that is formed by other word embeddings and can thus never be “selected” as the argmax word. We
+hypothesized that kNN-LM solves the stolen probabilities problem by allowing to assign the highest probability
+to any word, given a test context that is close enough to that word’s datastore key. However, we found that
+none of the vectors in our embedding matrix and in the original embedding matrix of Khandelwal et al. (2020b)
+is located in the convex hull of the others, which is consistent with the findings of Grivas et al. (2022). More
+details can be found in Appendix E.4.
+Memorization
+We hypothesized that the kNN component simply provides memorization of the training set.
+However, we could not improve a standard LM by interpolating its probability with another standard LM that
+was further trained to overfit the training set. More details can be found in Appendix E.6.1.
+Soft Labels
+We hypothesized that kNN-LM’s improvement lies in reducing the “over-correction” error
+when training with 1-hot labels, as hypothesized by Yang et al. (2022), and that retrieving neighbors is not
+important. If only “soft labels” are the key, we could hypothetically improve the performance of another
+fresh LM with the same model architecture but trained with the soft labels from the base LM, instead of from
+kNN-LM. This separates the effect of “soft labeling” from the additional guidance provided by kNN. However,
+this does not help with the interpolated perplexity at all. More details can be found in Appendix E.6.2.
+Optimizing Interpolated Loss
+We hypothesized that the standard LM cross-entropy training loss does
+not emphasize the examples where base LM performs badly which could benefit from kNN, and directly
+optimizing the interpolated loss of standard LM and a separate trainable softmax layer could be a better
+alternative. However, we could not gain any benefits by training an additional softmax layer together with a
+base LM using the interpolated loss. More details can be found in Appendix E.6.3.
+7
+Conclusion
+In this paper, we investigate why kNN-LM improves perplexity, even when retrieving examples from the same
+training data that the base LM was trained on. By proposing and testing various hypotheses and performing
+extensive ablation studies, we find that the key to kNN-LM’s success is threefold:
+1. Ensembling different input representations – the feedforward layer output and the attention layer
+output – can recover 55% of the performance, even without retrieval.
+2. One of the most unexpected discoveries in the paper is that using approximate nearest neighbor
+search allows kNN-LMs to generalize better than exact nearest neighbor search, possibly due to a
+regularization effect.
+3. Tuning the softmax temperature for the kNN distribution is crucial to adjust the standard LM output
+distribution with the distribution created by the retrieved neighbors’ distances.
+We performed extensive experiments which ruled out other hypotheses as to why kNN-LMs work, such as
+over-parameterization, datastore clustering, sparsification, overfitting, ensembling of distance metrics, and
+alternative training methods.
+We believe that this work unlocks a variety of exciting research directions for efficient kNN alternatives.
+For example, exploring methods that replace the kNN component with trainable parameters and achieve
+comparable results without the latency burden of kNN-LM.
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+https://aclanthology.org/2022.naacl-main.406.
+13
+
+A
+kNN-LM Generalization to Other LMs
+#params
+Base LM PPL
+kNN-LM PPL
+Absolute PPL Gain
+Ours
+268M
+21.75
+19.17
+2.58
+Distilled-GPT2
+82M
+18.25
+14.84
+3.41
+GPT2-small
+117M
+14.84
+12.55
+2.29
+GPT2-medium
+345M
+11.55
+10.37
+1.18
+GPT2-large
+774M
+10.56
+9.76
+0.80
+Table 4: Performance of kNN-LM applied to other pretrained language models of different sizes.
+To test the generalizability of kNN-LM, we follow the same experimental setup as used in Section 3. We
+select several pretrained models from the GPT2 family (Radford et al., 2019) of various parameter counts,
+plus a distilled version of GPT2, DistillGPT2. (Sanh et al., 2019) We take the pretrained model checkpoint,
+build the datastore and evaluate on the Wikitext-103 dataset splits. The results are shown in Table 4. We can
+see that kNN-LMs has good generalizability on other models. It improves the perplexity of all the base LMs
+tested. However, the larger the model is, and usually the better the base LM’s perplexity is, the less gain can
+be achieved from adding kNN. Note that our model is trained from scratch on Wikitext-103 dataset and thus
+even with a relatively large model size, the perplexity and perplexity gain from adding kNN is still less than
+models with pretraining. Without loss of generalizability, we will use our own trained-from-scratch model as
+the base LM in the following sections for ablation study.
+B
+Detailed Results for Increasing the Softmax Capacity
+hds
+Nds
+⊗
++#params
+PPL
+Interp.
+Oracle
+-
+-
+-
+0
+21.750
+-
+-
+att
+Big
+IP
+Nds × D
+∞
+19.095
+14.077
+att
+1x
+IP
+V × D
+22.584
+20.353
+16.954
+att
+2x
+IP
+2V × D
+21.903
+20.529
+17.432
+att
+3x
+IP
+3V × D
+22.434
+20.395
+17.132
+att
+4x
+IP
+4V × D
+21.936
+20.521
+17.423
+att
+5x
+IP
+5V × D
+22.025
+20.643
+17.560
+att
+6x
+IP
+6V × D
+21.972
+20.519
+17.422
+att
+9x
+IP
+9V × D
+22.084
+20.696
+17.631
+ffn
+Big
+IP
+Nds × D
+∞
+21.101
+16.254
+ffn
+1x
+IP
+V × D
+20.920
+20.694
+18.772
+ffn
+2x
+IP
+2V × D
+20.889
+20.646
+18.701
+ffn
+3x
+IP
+3V × D
+20.829
+20.603
+18.717
+ffn
+4x
+IP
+4V × D
+20.769
+20.629
+18.876
+ffn
+5x
+IP
+5V × D
+20.720
+20.594
+18.878
+ffn
+6x
+IP
+6V × D
+20.726
+20.599
+18.902
+ffn
+9x
+IP
+9V × D
+20.687
+20.567
+18.887
+Table 5: Performance comparison of kNN baselines and models with learnable embeddings as datastore
+alternative. hds is either attention layer output (att) or feedforward layer output (ffn).
+C
+Alternative Methods for Increasing Softmax Capacity
+C.1
+Adaptive Increasing Embedding Size
+We hypothesize that different word types have different difficulties for the language model to predict. For
+those words that appear very frequently, they may appear in many different contexts. As a result, instead
+of adding equal number of additional embeddings to each word type, we propose to adaptively increase the
+number of embeddings for word types based on word frequency, or total training loss for the word. Based on
+the intuition of Zipf’s law (Clauset et al., 2009), we assign 1 + logb fv for each word type v ∈ V , based on
+14
+
+either the frequency or the total training loss of the word, fv. The b is a hyperparameter that could be tuned.
+To ensure fair comparison, we tune b so that for each experiment the total number of embeddings matches:
+�
+v∈V 1 + logb fv = nV . The results are shown in Table 6. We can see that although nice in paper, given the
+same number of total embeddings, adaptively increasing the number of embeddings assigned for each word
+type does not make a significant difference in the final perplexity, when compared with the models that use
+equal number of embeddings for each word type.
+hds
+Nds
+⊗
++#params
+PPL
+λ
+Interp. PPL
+Oracle
+Base LM
+-
+-
+-
+0
+21.750
+-
+-
+-
+KNN
+att
+Big
+L2
+Nds × D
+∞
+0.271
+19.174
+14.230
+KNN
+att
+Big
+IP
+Nds × D
+∞
+0.266
+19.095
+14.077
+Equal Per Word
+att
+3x
+IP
+3V × D
+22.434
+0.417
+20.395
+17.132
+Loss Weighted
+att
+3x
+IP
+3V × D
+21.948
+0.437
+20.440
+17.303
+Freq. Weighted
+att
+3x
+IP
+3V × D
+22.507
+0.412
+20.387
+17.105
+KNN
+ffn
+Big
+L2
+Nds × D
+∞
+0.065
+20.734
+15.594
+KNN
+ffn
+Big
+IP
+Nds × D
+∞
+0.050
+21.101
+16.254
+Equal Per Word
+ffn
+3x
+IP
+3V × D
+20.829
+0.622
+20.603
+18.717
+Loss Weighted
+ffn
+3x
+IP
+3V × D
+20.764
+0.713
+20.659
+18.978
+Freq. Weighted
+ffn
+3x
+IP
+3V × D
+20.757
+0.658
+20.572
+18.782
+Table 6: Performance comparison of kNN baselines and several configurations that adaptively increase the
+embedding size with training loss or word frequency.
+C.2
+Mixture of Softmaxes
+Yang et al. (2017) proposes a solution to the problem using a Mixture of Softmax (MoS) to produce more
+linearly independent probability distributions of words given different contexts. Suppose that there are a
+total of R mixture components. MoS first uses R linear layers with weight wr to transform the current query
+context vector hds into wrhds. With a shared word embedding matrix Wsm, we can calculate each softmax
+component’s probability distribution with softmax(Wsm · wrhds). The mixture distribution is then given by:
+PMoS =
+R
+�
+r
+πr,hdssoftmax(Wsm · wrhds)
+(7)
+The prior weights are calculated using another linear layer with weight wπ, as πr,hds = softmax(wπhds).
+The softmax ensures that �R
+r πr,hds
+= 1.
+Comparing the MoS with the first term in Equation 5,
+Msoftmax(mask-to-k(Wds ⊗ hds)), we can see that there are some connections between the two. MoS
+eliminates the mask-to-k(·) operation, and replaces the single softmax across a very large vector (size of
+datastore), into multiple smaller softmaxes, each across only a vector of the size of vocabulary. As a result,
+the huge Wds is replaced by several linear layers to project the word embedding matrix. Now the first term
+becomes:
+M(⊕R
+r softmax(Wsm · wrhds))
+(8)
+Mir = πr,hds, ∀i ≤ V
+(9)
+where ⊕ represents the vector concatenation operation, and the aggregation matrix M now contains the mixture
+weights for each softmax being concatenated. We perform experiments with a varying number of mixtures (R),
+different definitions hds, and whether to fine-tune the output word embeddings Wsm. We allow fine-tuning the
+word embedding when we use attention layer output as context vector, since the word embedding matrix is
+trained with feedforward layer output originally. The results for this formulation are shown in Table 7. MoS
+models on its own increase the performance of the language model marginally. When compared with Table 5,
+we find that these models are worse than those that simply increases the number of embeddings. This is
+expected because MoS has fewer added parameters compared to those, as it only requires several additional
+linear projection layers for the embeddings.
+C.3
+Clustering Datastore
+Opposite to training the word embeddings of an increased size, we also consider ways to compress the datastore
+down to a similar-sized embedding matrix for softmax computation. The intuition is that the datastore contains
+15
+
+hds
+R
+⊗
++#params
+PPL
+λ
+Interp. PPL
+Oracle
+Base LM
+-
+-
+-
+0
+21.750
+-
+-
+-
+KNN
+att
+-
+L2
+Nds × D
+∞
+0.271
+19.174
+14.230
+KNN
+att
+-
+IP
+Nds × D
+∞
+0.266
+19.095
+14.077
+KNN
+ffn
+-
+L2
+Nds × D
+∞
+0.065
+20.734
+15.594
+KNN
+ffn
+-
+IP
+Nds × D
+∞
+0.050
+21.101
+16.254
+Ft. MoS+embed
+att
+2
+IP
+V D + 2D2 + 2D
+21.986
+0.437
+20.720
+17.573
+Ft. MoS+embed
+att
+3
+IP
+V D + 3D2 + 3D
+22.106
+0.422
+20.779
+17.609
+Ft. MoS Only
+att
+2
+IP
+2D2 + 2D
+22.552
+0.371
+21.011
+17.796
+Ft. MoS Only
+att
+3
+IP
+3D2 + 3D
+22.573
+0.371
+21.024
+17.812
+Ft. MoS Only
+ffn
+2
+IP
+2D2 + 2D
+21.351
+0.843
+21.338
+20.258
+Ft. MoS Only
+ffn
+3
+IP
+3D2 + 3D
+21.495
+0.733
+21.460
+20.322
+Ft. MoS Only
+ffn
+4
+IP
+4D2 + 4D
+21.321
+0.994
+21.321
+20.396
+Ft. MoS Only
+ffn
+5
+IP
+5D2 + 5D
+21.371
+0.909
+21.367
+20.406
+Table 7: Performance comparison of kNN baselines and several MoS configurations. R is the number of
+mixtures.
+redundant context vectors, and thus compression could make the datastore smaller without sacrificing too
+much performance gain. He et al. (2021) shows that we can safely compress the datastore by clustering to 50%
+of the original size without losing performance. We test this idea further by clustering the entire datastore
+into a size that could fit in GPU memory (e.g. 2V , 3V ) and thus could be easily fine-tuned further and use the
+resulting centroids to replace Wds. Within each cluster, there will be a distribution of different words with
+contexts, and we use the frequency of words within each cluster to calculate the aggregation matrix M in
+Equation 5. This would have the added benefit of “multi-sense” embedding, which allows similar meanings to
+be clustered to form a new “meta word” while the same word with different meanings would form different
+“meta words”. A notable example is bank, shore, and financial institution. However, this does not work, mostly
+because of the high compression loss after clustering and the imbalanced distribution of word types among
+each cluster.
+D
+Which Words Benefit from Approximation?
+To further understand the unexpected results when using the different kNN approximate retrieval settings
+in Section 5.1 and Section 5.2, we analyze on a token level, based on how many times each ground truth
+token’s probability in the evaluation set are helped by each kNN setting. It means that for each ground truth
+token in the evaluation, we count the times when the kNN distribution is higher than the base LM distribution
+PLM, i.e., PkNN > PLM.
+Since we found previously that approximate kNN provides an additional performance boost compared to
+ground truth kNN, we thus compare “real mask, real score” versus “FAISS mask, real score” in this analysis.
+To prevent outliers, we filter out words with less than 10 occurrences in the evaluation set. For each setting, we
+calculate the percentage of occurrences in the evaluation set where each token in the vocabulary where the
+kNN module achieves a better probability than base LM. We then plot the absolute difference between the
+percentages of the two settings, with respect to various possible attributes of the token that achieves better
+probability using each setting.
+Figure 6 shows that the longer the token is, which usually suggests proper nouns and harder and less common
+words in English, are better with approximate neighbors than ground truth ones, and vice versa. We hypothesize
+that this is due to longer words are more prone to overfitting in kNN-LM and thus using approximate kNN
+provides an effect similar to smoothing and regularization.
+We also compare words that could appear in more diverse contexts with words that co-occur with few distinct
+contexts. To measure how diverse the contexts of each word in the vocabulary is, we calculate both the forward
+and backward bigram entropy for each word in the evaluation set that has more than 10 occurrences. The
+bigram entropy is a simple yet good indicator of context diversity for a given word, as used in Kneser–Ney
+smoothing (Ney et al., 1994). We calculate both the forward and backward bigram entropy for each word w as
+16
+
+Figure 6: The effect of the token character length on how much accurate nearest neighbors are better than
+approximate FAISS neighbors. Negative values mean worse. The trend line of the scatter points is shown.
+follows, where wafter and wbefore represent the word after and before the given word w.
+Hforward(w) = −
+�
+wafter
+p(wafter|w) log p(wafter|w)
+(10)
+Hbackward(w) = −
+�
+wbefore
+p(wbefore|w) log p(wbefore|w)
+(11)
+Forward and backward entropy represents how diverse the context after and before the given word is. Intuitively,
+bigram entropy is supposed to indicate words that can appear in lots of different contexts. The higher the
+entropy of a word, the more diverse its context is, and vice versa. For example, words like “Francisco” would
+have a low entropy because it mostly comes after “San”.
+Figure 7: The effect of the forward and backward entropy of words on how accurate nearest neighbors are
+better than approximate FAISS neighbors. Negative values mean worse. The trend line of the scatter points are
+shown.
+The comparison is shown in Figure 7. We can see that the higher the entropy in both forward and backward
+cases, the better using approximate nearest neighbor search becomes. This suggests that words that appear
+in many different contexts are better off with an approximate kNN, and “easy-to-predict” examples such
+as “Jersey” and “Fransisco” is better with accurate kNN, possibly because these examples are less prone to
+overfitting errors and thus requires less regularization from approximation.
+17
+
+E
+Failed Hypotheses
+E.1
+Distance Metric
+We hypothesize that the key to kNN-LM’s performance gain is the ensemble of two distance metrics: the
+standard dot product distance (which the LM uses) with the L2 distance (which the kNN component uses as
+⊗). We tried to replace the kNN component with a component that just takes the tokens retrieved by the kNN
+search and returns their L2 distance to the LM output word embeddings: Wsm ⊗ hds instead of Wds ⊗ hds,
+where ⊗ represents the negative L2 distance. We tried this with both variants of hds, attention layer output,
+and feedforward layer output. None of these helped.
+E.2
+Sparsification
+In Equation 5, mask-to-k(·) used by kNN retrieval induces sparsity in the distribution over the vocabulary,
+due to a small k compared to the number of vocabulary V . We hypothesize that the in kNN-LM, the kNN
+distribution is sparse, practically increasing the probability of the top-k entries. The kNN distribution has
+up to 1024 entries that are non-zero, concentrating more probability mass over the most likely tokens. This
+effect is similar to the redistribution of probability mass for text generation in Holtzman et al. (2019). We
+test this hypothesis only by taking top 32, 64, 128, 512, or 1024 tokens in the parametric LM probability and
+zeroing out the probabilities of the rest of the tokens. To compensate, we experiment with different softmax
+temperatures and then interpolate with the parametric LM probability. This isolates the effect of the datastore
+and retrieval at all, and this does not help at all, suggesting that sparsification of the output probability alone is
+not enough.
+Another attempt is to hypothesize that the key in kNN-LM is that it selects “which tokens to include” in the
+kNN distribution, and not their distances. The intuition behind is that maybe the selection of the top tokens
+according to the kNN search is better than that from the dot-product distance between the language model’s
+output vector and all the vocabulary embeddings. We perform experiments similar to the previous attempt,
+sparsifying the output probability with the tokens retrieved by the kNN search (but ignoring the distances
+provided by the kNN search) rather than the top k tokens of the LM, with and without removing duplicates. In
+the best case, they manage to reduce the perplexity by 0.5 (whereas kNN-LM reduces by nearly 2).
+E.3
+Location within Context Window
+Supposedly, words in the beginning of the “context window” of the transformer at test time have less contextual
+information than words toward the end of context window.
+We hypothesized that maybe the base LM performs worse in one of these (beginning vs. end of the context
+window), and maybe kNN-LM provides a higher improvement in one of these. We measured the per-token
+test perplexity with respect to the location of each token in the context window. However, we did not find any
+significant correlation between the performance of the base LM and the location, and no significant correlation
+between the difference between kNN-LM and the base LM and the location.
+We also hypothesized that maybe the beginning of every Wikipedia article is more “predictable”, and the text
+becomes more difficult to predict as the article goes into details. However, we also did not find any correlation
+with the location of the word within the document it appears in.
+E.4
+Stolen Probabilities
+The stolen probabilities effect (Demeter et al., 2020) refers to the situation where the output embeddings of
+an LM are learned such that some words are geometrically placed inside the convex hull that is formed by
+other word embeddings. Since language models generate a score for every output word by computing the
+dot product of a hidden state with all word embeddings, Demeter et al. (2020) prove that in such a case, it is
+impossible for words inside the convex hull to be predicted as the LM’s most probable word (the “argmax”).
+We hypothesized that kNN-LM solves the stolen probabilities problem by allowing to assign the highest
+probability to any word, given a test hidden state that is close enough to that word’s datastore key. Nevertheless,
+as shown by Grivas et al. (2022), although this problem might happen in small RNN-based language models,
+in modern transformers it rarely happens in practice. Using the code of Grivas et al. (2022), we checked the
+embeddings matrix of our model and of the checkpoint provided by Khandelwal et al. (2020b). Indeed, we
+found that in both models – no word is un-argmaxable.
+18
+
+E.5
+Are kNN-LM Just Ensembling?
+Our hypothesis is that kNN component only provides another model for ensembling. The interpolation
+process is basically an ensemble model. Technically it is unsurprising that kNN-LM will have the benefit
+from ensembling, but we perform experiments to see how it compares to other ensembling. We trained
+another language model with the same architecture as the base LM we used throughout the experiments,
+with some variants having more than one embedding vector for each word (similar to Section 4.2). We
+interpolate the models with the original base LM, and the results are shown in Table 8. We can see that even
+just ensembling the base LM with another identical model, but trained with a different random seed, provides
+a huge performance boost, both on interpreted perplexity and on oracle perplexity.
+Prev. Layers
+hds
+Nds
+⊗
++#params
+PPL
+Interp.
+Oracle
+same
+-
+-
+-
+0
+21.750
+-
+-
+same
+att
+Big
+L2
+Nds × D
+∞
+19.174
+14.230
+same
+att
+Big
+IP
+Nds × D
+∞
+19.095
+14.077
+same
+ffn
+Big
+L2
+Nds × D
+∞
+20.734
+15.594
+same
+ffn
+Big
+IP
+Nds × D
+∞
+21.101
+16.254
+diff
+ffn
+1x
+IP
+F + V × D
+21.569
+18.941
+14.980
+diff
+ffn
+2x
+IP
+F + 2V × D
+21.914
+18.948
+14.885
+diff
+ffn
+3x
+IP
+F + 3V × D
+22.206
+18.981
+14.853
+Table 8: Performance comparison of kNN baselines and models with different size output embeddings
+re-trained from scratch.
+However, just because ensembling two LMs of the same architecture provides better performance than
+interpolating the base LM with kNN does not necessarily suggest that kNN’s performance improvement can
+be fully replaced by model ensembling. In other words, we are interested in whether the kNN performance
+improvements are orthogonal to that of model ensembling. To test this, we compare the performance of the
+ensemble of K multiple LMs versus the ensemble of K − 1 multiple LMs plus the kNN component. The
+comparison is fair because we have the same number of models in the ensemble, and the only difference is
+whether the kNN component is included. The results are shown in Figure 8. For the “LM” series, each point
+is K LMs ensemble, and for the “kNN” series, each point is K − 1 LMs plus kNN. We can see that even at
+4-ensemble, the ensemble that contain kNN as a component still have a considerable edge over the 4-ensemble
+that contain just LMs.
+Ensemble Components
+16
+18
+20
+22
+1
+2
+3
+4
+LM
+KNN
+LM and KNN
+Figure 8: Ensembling effect comparison, between multiple base LMs and multiple base LMs plus kNN
+component.
+E.6
+Are kNN-LM Just Alternative Training Methods?
+E.6.1
+Overfitting
+Since kNN-LM improves perplexity even with the same training dataset as datastore, we are curious if
+kNN-LM works by only “memorizing” the training data. The hypothesis is that the datastore and the kNN
+19
+
+Prev. Layers
+hds
+Nds
+⊗
++#params
+PPL
+Interp.
+Oracle
+Base LM
+same
+-
+-
+-
+0
+21.750
+-
+-
+KNN
+same
+att
+Big
+L2
+Nds × D
+∞
+19.174
+14.230
+KNN
+same
+att
+Big
+IP
+Nds × D
+∞
+19.095
+14.077
+KNN
+same
+ffn
+Big
+L2
+Nds × D
+∞
+20.734
+15.594
+KNN
+same
+ffn
+Big
+IP
+Nds × D
+∞
+21.101
+16.254
+Overfit@92
+diff
+ffn
+V
+IP
+F + V × D
+1702.806
+21.732
+17.764
+Overfit@129
+diff
+ffn
+V
+IP
+F + V × D
+8966.508
+21.733
+17.814
+Table 9: Performance comparison of several baselines with two overfitted models, at 92 and 129 additional
+epochs.
+search are trying to memorize the training data. In other words, the parametric LM is under-fitting some
+tokens. The intuition behind this is that the kNN component retrieves examples directly from the training set.
+What if we could retrieve the same examples using an overfitted LM? We took the trained LM, removed the
+dropout, and continued training until almost perfect fit (very small training loss). We then interpolated the
+overfitted transformer with the original LM. The results are shown in Table 9. F represents the number of
+parameters in the base LM, minus the output embedding matrix. We can see that overfitting can provide very
+little help after interpolation. Looking at the oracle performance, we think that the overfitted model memorizes
+some rare contexts and tokens in the training set where it could be useful during evaluation. However, the
+overfitting hurts the performance on other tokens too much so that even interpolation is not able to balance the
+performance.
+E.6.2
+Soft-Label Training
+Yang et al. (2022) claims that using “soft labels” during training is the key to kNN’s success, that interpolates
+the ground truth labels with kNN-LM model outputs, effectively “distilling” kNN-LM. It is based on the
+hypothesis that the room for kNN-LM’s improvement over base LM lies in the “over-correction” when training
+with a 1-hot labels. This is related to the effect from label smoothing methods (Szegedy et al., 2016; Pereyra
+et al., 2017; Meister et al., 2020a). However, we believe that this explanation is not satisfactory. If the key is
+training with soft-labels, why do these soft labels must be provided specifically by a kNN search? If soft labels
+were the key, then soft-label training where the labels come from the base LM itself should have worked as
+well. To separate the effect of soft labeling from the kNN’s additional guidance, we train another LM with the
+same model architecture as the base LM, with the soft labels from the base LM. This teacher-student training
+is to distill the knowledge from the base LM (Hinton et al., 2015). We find that by just training with “soft
+labels“ from the base LM to alleviate the alleged “over-correction” problem is not the key, as this does not help
+with the interpolated perplexity at all. This suggests that even with the same training data, kNN still provides
+valuable additional guidance.
+E.6.3
+Training to Optimize Interpolated Loss
+In Section 4.2, we discover that using over-parameterization with standard LM training loss does not further
+close the gap towards kNN-LM. This suggests that some regularization term may be needed during training to
+make the multiple embeddings not converge to the same vector, rendering over-parameterization useless.
+From Table 2, we see that a better interpolated perplexity may not require a very low perplexity when measured
+only with the extra input representation. However, we still use a standard LM loss to only train the additional
+embedding matrix, that directly minimizes the perplexity using only the extra input representation. This
+discrepancy between training and the evaluation with interpolation suggests that training with an alternative
+loss function that interpolates the base LM’s output with the output using the extra input representation may
+be beneficial.
+To test the hypothesis that standard LM training loss do not emphasize the examples where base LM performs
+badly, we train the extra model’s parameter Wds, with interpolated loss L:
+L = CrossEntropy(λsoftmax(Wds · hds) + (1 − λ)softmax(Wsm · hsm), y)
+(12)
+y represents the ground truth label for each context. We only learn the parameter Wds while freezing all
+other parameters, similar to all other experiments. We choose λ = 0.25 as it is the best hyper-parameter for
+kNN-LM experiments and our goal for this training is to mimic the loss of kNN-LM after interpolation. This
+training loss effectively assigns a higher value to the training examples where the base LM’s loss is high,
+20
+
+suggesting the need for the extra Wds to help with these hard cases. However, for either “att” for “ffn” for hds,
+either V or 3V for the number of embeddings in Wds, we are unable to achieve a better perplexity than just
+the base LM. This suggests that, while nice on paper, the interpolated loss optimization process is not trivial.
+21
+

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+Additive 3D photonic integration that is CMOS compatible
+Adria Grabulosa,1 Johnny Moughames,1 Xavier Porte,1, 2 Muamer Kadic,1 and Daniel Brunner1, a)
+1)FEMTO-ST Institute/Optics Department, CNRS & University Franche-Comt´e,
+15B avenue des Montboucons, Besan¸con Cedex, 25030, France
+2)Now with: Institute of Photonics, Department of Physics, University of Strathclyde, Glasgow G1 1RD,
+UK
+(Dated: 4 January 2023)
+Today, continued miniaturization in electronic integrated circuits (ICs) appears to have reached its funda-
+mental limit at ∼2 nm feature-sizes, from originally ∼1 cm. At the same time, energy consumption due
+by communication becomes the dominant limitation in high performance electronic ICs for computing, and
+modern computing concepts such a neural networks further amplify the challenge. Communication based
+on co-integrated photonic circuits is a promising strategy to address the second. As feature size has leveled
+out, adding a third dimension to the predominantly two dimensional integrated circuits appears the most
+promising future strategy for further IC architecture improvement. Crucial for efficient electronic-photonic
+co-integration is CMOS compatibility of the associated photonic integration fabrication process. Here, we
+review our latest results obtained in the FEMTO-ST RENATECH facilities on using additive photo-induced
+polymerization of a standard photo-resin for truly 3D photonic integration according to these principles.
+Based on one- and two-photon polymerization and combined with direct-laser writing, we 3D-printed air-
+and polymer-cladded photonic waveguides. An important application of such circuits are the interconnects of
+optical neural networks, where 3D integration enables scalability in terms of network size versus its geometric
+dimensions. In particular via flash-TPP, a fabrication process combining blanket one- and high-resolution
+two-photon polymerization, we demonstrated polymer-cladded step-index waveguides with up to 6 mm length,
+low insertion (∼0.26 dB) and propagation (∼1.3 dB/mm) losses, realized broadband and low loss (∼0.06 dB
+splitting losses) adiabatic 1 to M couplers as well as tightly confining air-cladded waveguides for denser in-
+tegration. By stably printing such integrated photonic circuits on standard semiconductor samples, we show
+the concept’s CMOS compatibility. With this, we lay out a promising, future avenue for scalable integration
+of hybrid photonic and electronic components.
+I.
+INTRODUCTION
+The backbone behind most of today’s cutting-edge
+technology is dense integration of two dimensional (2D)
+electronic circuits. However, by now these do experience
+several challenges. Further pushing the performance of
+2D computing chips becomes increasingly difficult, while
+new applications, in particular neural networks (NNs),
+challenge the hegemony of such 2D circuits - and this on a
+fundamental level1,2. New integration concepts and fab-
+rication technologies are needed if we are to continue the
+astonishing technological progress of the past decades.
+Crucially, these integration concepts need to take the es-
+sential features behind the success of 2D electronic inte-
+grated circuits (ICs) into account.
+Elevating a new integration technology even close to
+the level of 2D electronic ICs is a daunting and certainly
+a long-term challenge. Since the first demonstration of
+a planar, i.e. 2D, monolithic IC at Fairchild, this clas-
+sical integration has continuously been advanced for 60
+years plus in an almost world-wide effort. The concept’s
+success is a testimony to what can be achieved when
+previously individual components are integrated inside
+a single, monolithic circuit. It typically led to substan-
+tial miniaturization and increased reliability as well as
+a)Electronic mail: daniel.brunnerfemto-st.fr
+robustness, all while fabrication costs plummeted. Com-
+bined, these factors enabled decades of exponential scal-
+ing for electronic ICs: around every two years the num-
+bers of transistors per chips doubled (Moore’s law) while
+the power consumption per component halves (Dennard
+scaling).
+Monolithic ICs comprising different compo-
+nents and functionalities are therefore also indispensable
+for 3D photonic integration.
+While still far from the levels of today’s electronic
+IC, photonic integration also has considerably advanced.
+In order to maximize compatibility and synergy with
+electronics, photonic integration based on silicon sub-
+strates emerged in the 1980s with the demonstration of
+the silicon waveguide3,4, the photonic equivalent to a
+metallic or polysilicon wire in integrated electronics ICs.
+Electronic ICs are almost exclusively based on comple-
+mentary metal–oxide–semiconductor (CMOS) technol-
+ogy that uses mostly silicon as semiconductor host lever-
+aging boron, gallium, indium, phosphorus, arsenic and
+bismuth as dopants, and CMOS compatibility is consid-
+ered fundamentally important for photonic ICs.
+By a vast majority, both, electronic and photonic inte-
+gration leverages fabrication concepts developed for pla-
+nar, 2D substrates. The layout of a circuit’s single layer is
+etched into a thin surface of either mostly metal or semi-
+conductor materials, which is the process of 2D lithogra-
+phy. Typically, coating said surface with a photo-resist
+protects certain surface-areas from etching, which is de-
+termined by photo-resist illumination that is structured
+arXiv:2301.00983v1  [physics.optics]  3 Jan 2023
+
+2
+by a photo-mask. The appeal of such 2D lithography is
+that each of the involved process steps, photo-resist ap-
+plication, exposure by photo-mask, etching and several
+washing sequences, can be carried out in a single pro-
+cedure for a large area or even an entire wafer, which
+strongly reduces fabrication costs.
+A new challenge to classical electronics computers
+based on 2D substrates arose with the breakthrough of
+NN computing around a decade ago. Conceptually, NNs
+link a large number of neurons through the network’s
+connections, c.f.
+Fig. 1 (a).
+In an physical hardware
+implementation that mirrors this topology, these con-
+nections correspond to electronic or photonic signaling
+’wires’. Currently, these connections are emulated, which
+creates substantial energy and speed overheads. Future
+NN circuits that abolish this overhead require ICs with
+a far higher degree of connectivity, i.e. much more wires
+to communicate signals across the chip. This causes sev-
+eral problems.
+Energetically speaking, electronic com-
+munication is the factor limiting performance even for
+classical computing concepts; communicating a floating
+point number costs around 80-times more energy than
+creating a new floating point number5. NN computation
+dramatically escalates this problem, as the number of a
+NN’s connections by far out-scale the number of neurons.
+Photonic and 3D integration provide promising solutions,
+see Fig. 1 (b). Optical communication is (i) energetically
+superior for ever shorter distances and (ii) mitigates heat
+dissipation challenges that arise for volumetric circuits,
+while (iii) 3D integration shortens the length of commu-
+nication links. Most importantly, in many NN topologies
+the number of connections, i.e. wires, increases quadratic
+or faster with the number of neurons. Consequently, in-
+tegrating a NN’s interconnect in 2D results in a quadratic
+scaling (or worse) of chip-area with the size of a neural
+network. Recently, the number of neurons in a NN has
+turned into the parameter of fundamental relevance, and
+alternative strategies for integrating NNs are of funda-
+mental importance for the field.
+In this review for the RENATECH special issue, we
+describe our recent work addressing such photonic ICs
+based on standard techniques and fabrication infrastruc-
+ture available in our local RENATECH cleanroom. In
+those efforts, we have demonstrated additive, 3D pho-
+tonic integration, which importantly is using concepts
+and materials that make the entire fabrication and result-
+ing photonic IC CMOS compatible. Based on additive
+two-photon polymerization (TPP) in a direct-laser writ-
+ing (DLW) system, combined with rapid and large area
+one-photon polymerization (OPP), we integrated large
+3D photonic waveguide circuits. We demonstrate indi-
+vidual waveguides as well as optical splitters and net-
+works of splitter6 based on (i) air-cladded waveguides
+comprising polymer cores7, and (ii) step-index waveg-
+uides where we induce the refractive index difference
+between core and cladding required for guiding by dy-
+namically controlling the optical power used for printing
+our 3D structures8. Finally, we substantially accelerate
+FIG. 1. 3D photonic integration and optical waveguide basics.
+(a) Schematics of a typical neural network where a large num-
+ber of neurons are highly interconnected through a network.
+(b) Integrating a large number connections in 2D leads to an
+exponential growth of the number of channels over the chip’s
+area; whereas leveraging integration in 3D results in a efficient
+and linear scalability of optical interconnects. (c) In photonic
+waveguides, the light is confined within the core of diameter d
+due to total internal reflection. For this, the refractive index
+of the core ncore must be larger than the cladding’s ncladding,
+and hence ∆n = ncore − ncladding > 0. All the waveguide’s
+optical properties relies on the parameters ∆n and d.
+the fabrication process by developing the flash-TPP con-
+cept, which combines TPP-DLW with ultraviolet (UV)
+blanked illumination to efficiently polymerize an IC’s
+non-light guiding volume in a single step9. We achieve
+very symmetric splitting ratios in optical couplers, and
+(for a first proof of concept) low propagation losses of
+∼ 1.3 dB/mm and insertion losses of ∼ 0.26 dB. Fi-
+nally, we printed optical waveguides on semiconductor
+substrates hosting micro-lasers, demonstrating that our
+concept is CMOS compatible.
+II.
+BASICS OF ADDITIVE FABRICATION
+In the past 15 years, DLW and TPP have become
+a versatile fabrication tool of polymer structures with
+sub-micron dimensions10–12.
+In contrast to 2D pla-
+nar methods such as electron-beam lithography or
+mask based lithography,
+DLW allows for fabricat-
+ing three-dimensional structures13.
+DLW has played
+a crucial role for many proof-of-concept designs in
+optics7, acoustics14,15, elasticity13,16–18, robotics19 and
+even electric transport20.
+Major challenges such as
+inclusion of conductive resins21, quantum-dots doped
+resins22, liquid-crystals doped resins23 are still in the
+development phase.
+Recently, great progress towards
+parallel direct-laser writing has been made,
+which
+enables a substantially accelerated fabrication process24.
+Finally, different polymerization concepts are constantly
+being developed, some of which use novel approaches
+to high-resolution 3D printing based on polymer resins25.
+
+3
+III.
+PHOTONIC INTEGRATION VIA
+PHOTO-INDUCED POLYMERIZATION
+Standard photonic waveguides covered in this review
+rely the guiding element called the core having a higher
+refractive index ncore than the refractive index of the
+confining part called the cladding ncladding, i.e. ∆n =
+ncore − ncladding > 0.
+As schematically illustrated in
+Fig. 1 (c), in such a configuration optical rays imping-
+ing on the core-cladding interface with an angle smaller
+than the critical angle θc = arcsin(1−(∆n/ncore)) exhibit
+total internal reflection. As a consequence, they are con-
+fined to the waveguide’s core and propagate along this
+structure, allowing to direct optical propagation along
+pre-designed paths via an integrated and solid core.
+Refractive index contrast ∆n combined with the core
+diameter d are a waveguide’s determining characteris-
+tics, which determine a waveguide’s numerical aperture
+NA =
+�
+n2core − n2
+cladding. The same holds for the num-
+ber of spatial modes allowed to propagated through the
+waveguide M ≈ V 2/2 = (4πd/λ)NA for large M, where λ
+is the optical wavelength.. Here, V is the normalized fre-
+quency a central indirect property of optical waveguides;
+for V ≤ 2.405 a waveguide is single-mode, otherwise it
+allows for higher modes to propagate. Finally, ∆n also
+determines the minimal bending radius for which light
+can be directed without exceedingly high losses. This in
+turn is the limiting factor for integration density inside a
+photonic IC.
+In work covered in this review, we used the commer-
+cial 3D direct-laser writing Nanoscribe GmbH (Photon-
+ics Professional GT) system, which is equipped with a
+femtosecond (fs) laser operating at 780 nm, and galvo-
+mirrors for rapid beam movement in the lateral direc-
+tions. The fs-laser is usually tightly focused into the resin
+through an objective lens of high numerical aperture. Af-
+ter finishing the TPP-DLW step, the unpolymerized resin
+was removed in a two-step development process, immers-
+ing the structure first in propylene-glycol-methyl-ether-
+acetate (PGMEA) acting as a developer for 20 minutes,
+followed by rinsing in isopropyl alcohol (2-propanol) for
+3-5 minutes. For OPP, we deposited samples in the com-
+mercial UV-chamber Rolence Enterprise Inc., LQ-Box
+model, 405 nm wavelength, 150 mW/cm2 average light
+intensity.
+A.
+Two-photon polymerization
+Two-photon polymerization is a maskless direct-laser
+writing technique26. A highly focused pulsed laser beam
+in the femtosecond regime is used to induce the absorp-
+tion of two-photons in the exposed volume inside the
+photo-resist (which is a monomer in its liquid phase),
+c.f Fig. 2 (a). This two-photon activated polymerization
+creates long-chained polymer molecules that in turn form
+a solid volume due to molecule interlinkage. Forming al-
+most arbitrary 3D structures can then be achieved by
+translating the laser through the overall volume of the
+photo-resist along all three spatial dimensions.
+Grav-
+ity can impose limitations on attainable shapes, yet this
+aspect usually does not have a too strong impact: the
+polymer and the original monomer resin have very simi-
+lar mass densities, and thus the Archimedes forces keep a
+polymerized voxel locked in its position due to the resin’s
+viscosity.
+FIG. 2. Principle of direct-laser writing (DLW). (a) The fs-
+writing laser is scanned through the photo-resist through the
+monomer resin using high-speed galvo-mirrors for the dis-
+placement in the (x, y)-plane, while a piezo controls the z-
+position. (b) The resin is two-photon polymerized only inside
+a small voxel volume, and voxels are placed on a grid deter-
+mined by hatching distance h in the (x, y)-plane, and slicing
+distance s in the z-direction. The laser power (LP) as well as
+s, h determine the overlap of neighboring voxels and through
+this the minimum feature size and the smoothness of printed
+surfaces. (c) In our work we use the ’dip-in’ technique, where
+a drop of resin is located between the microscope objective
+and the substrate. The printing direction is downwards, and
+the maximum size of 3D-printed structures is around 6 mm
+in height.
+Originally,
+the writing laser spot was translated
+through the resin using piezo stages.
+This approach
+is highly accurate as the stages readily have nanomet-
+ric precision. However, it does not allow for large dis-
+placement, is very slow and hence cannot be used for
+large printing areas/volumes.
+A major breakthrough
+resulted from using galvo-mirrors for moving the writ-
+ing laser’s focal spot through the resin (see Fig. 2 (a)).
+As a consequence, printing speed increased by orders of
+magnitude27, and fabricating large-scale 2.5 metasurfaces
+or 3D volumes became possible.
+Crucial for the quality of 3D structures and for inte-
+gration in general is the feature size of a single, polymer-
+ized voxel relative to the the scanning speed of the print-
+ing laser. The photoinitiation of the chemical reaction
+which essentially is instantaneous relative to the the writ-
+ing speed, and hence the writing-volume directly follows
+laser’s scanning. However, polymerization is a chemical
+reaction with an associated time scale, like any diffusion
+phenomenon. Typically, this timescale is orders of mag-
+nitude slower than the galvo-controlled laser scanning28.
+This aspect is crucial, since as a consequence polymer-
+ization is taking place for several neighboring voxels at
+overlapping times. It makes the polymerization process
+
+(a)
+(q)
+h
+LP1 < LP2
+LP1
+Sample holder and substrate
+3D Scanning piezo
+LP2
+Positioning stage
+S
+High-resolution objective
+Zm
+(c)
+DiLL (Dip-in)
+substrate
+resin
+Galvo high-speed 2D scanning
+Femtosecond laser
+Xg4
+more homogeneous, and the obtained structures do not
+suffer from (unintended) variations of material properties
+resulting from stitching countless small voxels together
+to form a large structure. As schematically illustrated
+in Fig. 2 (b), the writing laser power (LP), the hatch-
+ing h and slicing s distances as well as the scan speed
+modify the overlap between neighboring polymer voxels.
+Through this, the smoothness of surfaces and the ho-
+mogeinity of the polymer-medium can be controlled to a
+good degree. For much faster polymerization, the peri-
+odic voxels would results in a photonic crystal like struc-
+ture, thus introduce scattering and all related phenomena
+inside the produced polymer. Thanks to diffusion, this
+aspect is almost not observable, yet it potentially is a
+source of optical losses in long waveguides.
+A powerful technique, called ’dip-in’ mode, c.f. Fig. 2
+(c), where the liquid resin is held between the substrate
+and the microscope objective, was introduced in 2013.
+This avoids having to print through the substrate (con-
+trary to immersion-oil techniques), which reduces aber-
+rations and removes the thickness of the substrate as a
+limitation of the maximal height of printed structures.
+Importantly for CMOS compatibility, it enables printing
+on materials that are not transparent at fs-laser’s wave-
+length. Piezo actuators and/or the writing field (deter-
+mined by the microscope objective of the printer) are
+usually quite limited in area, usually below mm-scales.
+For printing larger structures stitching various writing
+fields together is required, and in that it is not dissimilar
+to the stepper-process used in 2D semiconductor lithog-
+raphy. One can select a lower NA microscope objective to
+increase the writing field, however, this can only be em-
+ployed on the cost of a reduced printing low-resolution29.
+Generally, 3D printing via direct-laser writing creates
+structures of high quality, and their optical and ellastical
+properties have been characterized with high accuracy
+using Brillouin light scattering30. In this paper, the au-
+thors demonstrate an excellent quality check of the poly-
+mer in the GHz regime for elastic waves. For example,
+the 3D-printed samples can have an elastic quality fac-
+tor only ten times smaller than fused silica at hypersonic
+frequencies.
+Importantly for printing photonic waveguides, the de-
+gree of polymerization and through the Clausius rela-
+tionship also the refractive index n, is mainly determined
+by the type of photo-resist and the dose parameters D
+of the fs-laser, i.e. scanning speed and LP. Within the
+window between the TPP-threshold and the breakdown
+point above which the polymerized voxel contains defects,
+the so-called dynamic power range of the photo-resist26,
+the size of the TPP-voxel can be further modified by
+adapting D and fabrication parameters distances h and
+s.
+Figure 3 (a-b) depicts the experimental optimization
+of the dynamic power range of the liquid negative-
+tone IP-S photo-resist, with n ≈ 1.51 when fully TPP-
+polymerized31,32 and using a 25X magnification NA = 0.8
+microscope objective for writing. We printed, on a fused
+h = 0.3 μm 5 μm
+5 μm
+LP = 11 mW
+LP = 7mW
+5 μm
+LP = 15 mW5 μm
+LP = 17 mW5 μm
+LP = 19 mW5 μm
+h = 0.4 μm 5 μm
+h = 0.5 μm 5 μm
+h = 0.6 μm 5 μm
+5 μm
+h = 0.7 μm
+(a)
+(b)
+FIG. 3. Dynamic power range characterization of waveguide
+cores printed via TPP using the IP-S photo-resist.
+Image
+taken with permission from9. (a) SEM micrograph of pilars,
+printed to reassemble the cores of waveguides, with 20 µm
+height and d = 5 µm, with laser power LP ∈ {7, . . . , 19} mW,
+using hatching h = 0.4 µm and slicing distance s = 1 µm.
+(b) Impact of hatching distance h ∈ {0.3 : 0.1 : 0.7} µm, with
+fixed LP = 15 mW and s = 1 µm.
+silica substrate, a set of five free-standing pillars to em-
+ulate waveguide cores with 20 µm height and diame-
+ter d
+=
+5 µm using a range of TPP laser power LP
+∈ {7, . . . , 19} mW and hatching distances h ∈ {0.3 : 0.1 :
+0.7} µm. As globally fixed parameters in all our fabrica-
+tions we use a scanning speed of 10 mm/s and a slicing
+distance of s = 1 µm. The scanning electron microscopy
+(SEM) micrograph in Fig. 3 (a) shows the effect of grad-
+ually modifying the LP with a hatching distance constant
+h = 0.4 µm. Structures printed with LP = 7 mW and
+LP = 11 mW have ondulated surfaces, whereas when in-
+creasing the laser power to LP = 15 mW results in larger
+TPP voxels and therefore smoother surfaces. Exceeding
+LP = 15 mW leads to overpolymerization of the IP-S
+photo-resist (see two last micrographs of Fig 3 (a)). We
+therefore select LP = 15 mW and proceed to optimize
+the second fabrication parameter by scanning the hatch-
+ing distance from h ∈ {0.3 : 0.1 : 0.7} µm, and Fig. 3 (b)
+shows the results. We found that for h = 0.3 µm results
+are not always reproducible since smaller hatching dis-
+tance increases local exposure dose D and hence moves
+the process above the available power range.
+B.
+One-photon polymerization
+One-photon polymerization is widely used to process
+thin material layers in the current 2D photo-lithography
+technology used for electronic semiconductor ICs. The
+process is based on the exposure of a photosensitive resin,
+usually at the UV range, through a photo-mask including
+specific design patterns.
+Repeating this process layer-
+by-layer is possible to process and stack different thin
+material layers and fabricate 3D structures33. For highly
+structured patterns like SD memory cards, this has led to
+
+HV
+curr
+usecase
+det
+mag
+贝
+WD
+tilt 
+20μm
+5.00kv
+25pA
+Standard
+ETD
+2500x
+10.0mm
+40.0
+FEMTO-STHV
+curr
+use case
+det
+mag
+贝
+WD
+tilt 
+20μm
+5.00kv
+25pA
+Standard
+ETD
+2500x
+10.0mm
+40.0
+FEMTO-STHV
+curr
+use case
+det
+mag
+贝
+WD
+tilt 
+20μm
+5.00kv
+25pA
+Standard
+ETD
+2500x
+10.0mm
+40.0
+FEMTO-STHV
+curr
+use case
+det
+mag
+WD
+tilt 
+20μm
+5.00kv
+25pA
+Standard
+ETD
+2500x
+10.0mm
+40.0
+FEMTO-STHV
+curr
+use case
+det
+mag只WD
+tilt 
+50um
+5.00 kV
+0.25 nA
+ Standard
+ETD
+800x
+10.0mm40.0
+FEMTO-STHV
+curr
+det
+mag 只
+WD
+tilt
+50μm
+use case
+5.00kv
+0.25nA
+Standard
+ETD
+800x
+10.0mm
+40.0°
+FEMTO-STHV
+curr
+use case
+det
+mag贝
+WD
+tilt
+20 μm
+5.00kV
+25pA
+Standard
+ETD
+2500x
+10.0mm
+40.0
+FEMTO-STHV
+curr
+usecase
+det
+mag
+贝
+WD
+tilt 
+20μm
+5.00kv
+25pA
+Standard
+ETD
+2500x
+10.0mm
+40.0
+FEMTO-STHV
+curr
+use case
+det
+mag
+贝
+WD
+tilt 
+20μm
+5.00kv
+25pA
+Standard
+ETD
+2500x
+10.0mm
+40.0
+FEMTO-STHV
+curr
+use case
+det
+mag
+贝
+WD
+tilt 
+20μm
+5.00kv
+25pA
+Standard
+ETD
+2500x
+10.0mm
+40.0
+FEMTO-ST5
+ICs with up 100 or more circuit layers2. However, such
+stacking of layers created via a generically 2D fabrication
+concept has several severe drawbacks. For one, it requires
+to precisely align the photo-mask multiple times in each
+photo-lithographic step, which is challenging and time-
+consuming. Secondly, one of the strongest features of 2D
+lithography is its economic appeal. Between each layer,
+each of the process step have to be repeated in a loop-
+like manner. A process where the entire IC’s volume is
+created during few of such process steps will potentially
+have the upper hand economically speaking. Still, such
+stacked 2D lithography has also been used of complex 3D
+photonic integration, c.f. Fig. 4.
+(a)                                                       (b)
+                                                           (c)
+FIG. 4. Multilayer 3D waveguide fabrication using OPP. Im-
+age taken with permission from34. (a) Schematic diagram of
+the fabrication sequence for the stacking waveguide using spin
+coating and simple direct UV photolithography curing (s1);
+UV irradiation of the waveguides using a mask (s2); develop-
+ment (s3); UV irradiation of the cladding (s4). (b) Layout
+of the 3D interconnect polymer structure with an array of
+4x8 waveguides. (c) Cross-section microscope optical image
+of 4x8 stack waveguides.
+Just as with TPP, the refractive index of the poly-
+merized resin is a function of the optical exposure does
+D31,35–37. However, in OPP the refractive index of the
+resin is modified for substantially larger volumes, and in
+particular volumes outside the intended plane of exposure
+do strongly accumulate unintended irradiation doses. It
+is therefore a formidable challenge to precisely control a
+3D refractive index distribution, i.e. a volume hologram,
+with high spatial resolution.
+OPP is therefore better
+suited for simultaneous polymerization of, either, large
+areas like in classical 2D lithography, or for large uni-
+form volumes.
+C.
+Flash-TPP: combining one- and two-photon
+polymerization for photonic integration
+One can combine one- and two-photon polymeriza-
+tion as an hybrid configuration to accelerate the fabri-
+cation of 3D photonic chips.
+Several approaches com-
+50 μm
+Waveguide core
+TPP
+OPP
+(a)
+Mechanical supports
+(a)
+(b)
+(c)
+FIG. 5. Flash-TPP printing concept for 3D integrated pho-
+tonics.
+Image taken with permission from9.
+(a) Classical
+’dip-in’ process for the DLW-TPP fabrication of 3D photonic
+waveguides. (b) UV chamber that polymerizes the unexposed
+regions of the 3D structure via OPP. (c) SEM micrograph
+of a 3D-printed cuboid cross-section embedding 16 photonic
+waveguides. The waveguide cores (mechanical supports) are
+printed with large (small) hatching distances, which defines
+the resolution of each component of the 3D photonic circuit.
+Red colour represents regions polymerized via TPP, while
+blue colour regions via OPP.
+bining UV lithography with DLW-TPP have been pre-
+viously demonstrated in38 and39 for the fabrication of
+high resolution 3D optical microcomponents. However,
+those methodologies require the polymerization of multi-
+ple photo-resists in two separated fabrication steps and
+become time-consuming if used for 3D fabrication due to
+the layer-by-layer approach.
+We demonstrated a novel lithographic strategy that
+combines OPP and TPP, flash-TPP9, where we combine
+high resolution and quality TPP with unstructured and
+uniform OPP in order to accelerate the fabrication pro-
+cess by one order of magnitude when compared to us-
+ing TPP-only. Importantly, the concept only requires a
+single resin and adding the OPP step does not add ad-
+ditional development and washing steps. In flash-TPP,
+TPP and OPP are used for the fabrication of the dif-
+ferent sections of a photonic circuit, Fig. 5 illustrates
+the working principle, here for the liquid negative-tone
+IP-S photo-resist.
+Waveguide cores accommodate the
+large majority of an optical signal’s electromagnetic field,
+hence cores are printed via TPP with a precisely opti-
+mized laser power and fine resolution in the (x, y)-plane,
+i.e. small hatching distance. This ensures smooth core-
+cladding interfaces and hence low propagation losses.
+Mechanical supports, i.e. surfaces that define the outer
+limits of the volumetric circuit, are printed with larger
+hatching distance and high LP.
+Figure 5 (a) depicts the typically ’dip-in’ DLW-TPP
+printing procedure. After development, the photonic cir-
+cuit is transferred to a UV chamber, c.f. Fig. 5 (b), and
+the OPP dosage D of the 3D circuit’s volume is con-
+
+a
+s1)
+buffer layer
+silicon
+substrate
+s2)
+mask
+spin-
+coated
+waveguide
+layer
+s3)
+s4)
+506
+trolled via the duration of the UV exposure, through
+which we tailor the refractive index of the waveguides’
+cladding ncladding and hence ∆n. The SEM micrograph
+from Fig. 5 (c) shows the cross-section of an exemplary
+3D photonic chip fabricated via flash-TPP consisting of
+a cuboid integrating 16 waveguides. The cores and me-
+chanical supports, printed via TPP, are highlighted in
+red region, while the cladding volume, polymerized via
+OPP, is highlighted in blue.
+Via flash-TPP, we fabricated photonic waveguides
+with a refractive index contrast between core and
+cladding in the order of ∆n ≈ 5·10−39.
+Figure 6 (a)
+shows the evolution of the the average numerical aperture
+<NA> and refractive index of the cladding < ncladding >
+polymerized via OPP versus D. We used UV exposure
+doses D of 0, 750, 3000 and 9000 mJ/cm2, respectively.
+Assuming a constant ncore ≈ 1.51, we can precisely con-
+trol, both, <NA> and < ncladding >. Waveguides are
+single-mode for d ≤ 4.9 µm, which are feasible to fab-
+ricate via standard DLW-TPP processes. We obtained
+1.3 dB/mm (0.26 dB) propagation (injection) losses for
+the fundamental LP01 mode of waveguides printed via
+flash-TPP. Crucially, our 3D circuits did not degrade
+over time, and we evaluated the NA of waveguides under
+continuous operating condition across several months9.
+Overall, this demonstrates the reliability of the flash-
+TPP lithography methodology for an ultra-fast, single-
+step and high performance fabrication of 3D photonic
+components.
+Printing via flash-TPP consist in polymerizing only
+the sections vital for communication and mechanical in-
+tegrity. Importantly, the majority of a circuit’s area or
+volume is not involved in either, and they can hence be
+rapidly fabricated via UV blanket exposure. The print-
+ing times in flash-TPP is therefore drastically reduced,
+and in particular cases also scales different with the cir-
+cuit’s size9. This agrees with our experience; flash-TPP
+reduces the printing time to only 10% compared to only-
+TPP. As an example, printing a large structure that
+integrates waveguides with heights ranging from 0.1 to
+6 mm9, shown in Fig. 6 (b), requires ∼24 hours only
+using TPP but only ∼3 hours using flash-TPP.
+IV.
+AIR-CLADDED WAVEGUIDES
+Polymer waveguides with an air cladding have a rel-
+atively large ∆n ≈ 0.5 with ncore = 1.51. On the one
+hand, this leads to very strong confinement and a large
+NA = 1.13, which enables very small bending radii of
+25 µm (14 µm) at λ = 1550 nm (λ = 650 nm), and
+in turn dense photonic integration40–42. The large ∆n
+makes fabricating single-mode waveguide circuits chal-
+lenging. To be single-mode, air-cladded waveguides have
+to have a core diameter d ≤ 1 µm (d ≤ 0.43 µm) at
+λ = 1550 nm (λ = 650 nm). Printing waveguides with
+d ≤ 1 µm is possible7, and strongly confined photonic
+IC at λ = 1550 nm are within reach. For photonic 3D
+(a)                                                                (b)
+FIG. 6. Optical performance of waveguides printed via flash-
+TPP. Image taken with permission from9. (a) Average numer-
+ical aperture <NA> and cladding’s refractive index < n2 >
+over OPP dose D of photonic waveguides printed via flash-
+TPP. The <NA> (< n2 >) decreases (increases) over D,
+meaning that we can control the degree of polymerization of
+the cladding via the dosage of UV light.
+(b) Macroscopic
+structure scaled to a match that integrates waveguides with
+heights ranging from 0.1 to 6 mm.
+ICs close to the visible wavelength of light this remains
+a challenge.
+Recently, 3D optical splitter/combiners based on air-
+cladded waveguides with a 1 to 4, 1 to 9 and 1 to 16
+configuration were printed using TPP43,44. Figure 7 (a)
+shows an SEM image of the 1 to 4 fractal splitter/coupler,
+with its optical characterization at λ = 632 nm shown in
+Fig. 7 (b).
+There, the distance between output ports
+was scanned within the range D0 ∈ [10, 12, ..., 20] µm
+while keeping their height constant at 52 µm. Losses do
+not substantially increase for smaller distance between
+the output ports, which validates the estimated mini-
+mal bending radii given before. Furthermore, this per-
+formance was evaluated for two different LP settings. No
+clear difference can be seen between the two data-sets,
+and hence the printing power for air-cladded 3D polymer
+waveguides is not a critical parameter, as long one stays
+within the dynamic power range.
+For large-scale network interconnect, Moughames et al.
+demonstrated 3D parallel interconnects with high con-
+nectivity, shown in Figure 7 (c), by cascading two layers
+of 1 to 9 splitters and spatially multiplexing an arrays of
+such 1 to 81 splitters to allows for an array of 15x15 input
+waveguides. The entire circuits only occupies a volume
+of 460x460x300 µm3, in which an interconnect for 225
+inputs and 529 outputs is realized7. Figure 7 (d) shows
+a higher magnification of this interconnect. Individual
+wavegudies have a low surface roughness, and the incor-
+porated chirality of the fractal splitters/couplers avoids
+intersections of individual waveguides.
+V.
+STEP AND GRADED INDEX WAVEGUIDES
+Based on the previous discussed concepts and fabrica-
+tion technologies, we addressed step- (STIN) and graded-
+index (GRIN) waveguides. In STIN waveguides, the re-
+fractive index of the waveguide’s core is constant, while
+for GRIN waveguides it is a function of the radial distance
+to the core’s center. Usually, GRIN waveguides follow
+
+7
+FIG. 7. Air-cladded waveguides and couplers fabricated via
+DLW-TPP. Image taken with permission from7,43.
+(a) 2x2
+optical splitter/coupler with 1 input and 4 outputs with dis-
+tance D0 = 16 µm between waveguides, and 1.2 µm waveguide
+diameter43. (b) Optical losses of 2x2 splitters/couplers as a
+function of the distance D0 between waveguides, for hatching
+distances h = 0.1 µm (in blue) and h = 0.2 µm (in red). Data
+on top correspond to splitters/couplers written with laser
+power LP = 10.4 mW, and data at the bottom correspond to
+splitters/couplers written with laser power LP = 11.2 mW.
+(c) SEM micrographs of 3D-printed waveguides realizing par-
+allel interconnects with high connectivity7. (d) Zoom-in of
+(c).
+a parabolic refractive index distribution. For the STIN
+waveguides, all bound rays propagate at angles within
+the total internal reflection condition θc at any position
+in the core cross-section, while for GRIN waveguides, the
+range of angles varies with position45.
+We proposed a single-step additive fabrication tech-
+nique, (3+1)D printing8, by which we spatially modify
+the refractive index of a single resin over the TPP expo-
+sure dose during fabrication. Using the (3+1)D-printing
+concept, we constructed volume holograms and photonic
+waveguides with, both, STIN and GRIN profiles in a
+single-step, single-material fabrication with a commer-
+cially available process. This demonstrates the versatility
+of the 3D photonic integration approach based on DLW;
+optical manipulation based on integrated and monolithic
+3D structures can either rely on discrete components, i.e.
+waveguides, or leverage continuous manipulations of free
+optical propagation, i.e. holograms8. Both schemes can
+be exploited on the same photonic IC and be realized
+using the same fabrication concept and during the same
+fabrication step. We used the negative tone IP-Dip resin
+(n ≈ 1.547)36 and a 63X magnification NA = 1.4 micro-
+scope objective, c.f. Fig. 5 (a).
+The SEM micrograph of Fig. 8 (a) shows an exem-
+plary cuboid embedding 20 STIN waveguides fabricated
+via (3+1)D-printing. Contrary to flash-TPP, in (3+1)D-
+printing all the 3D photonic chip volume is fabricated
+via TPP-only. The refractive index contrast ∆n between
+core-cladding waveguides is achieved from the control
+over the TPP dosage D for individual writing voxels. For
+(a)
+100 µm
+(b)
+(c)
+FIG. 8. Step- (STIN) and graded-index (GRIN) waveguides
+fabricated via (3+1)D-printing.
+Image taken with permis-
+sion from8. (a) SEM micrograph of an exemplary 3D-printed
+cuboid integrating 20 STIN waveguides of 300 µm heigh.
+Waveguide cores (cladding) are printed via TPP with high
+(low) laser power, which ensures a refractive index contrast
+∆n ≈ 2.4·10−3. Panels (b) and (c) depict the output intensi-
+ties (triangles) and fundamental LP01 mode fits (dashed lines)
+of a 3 µm radius STIN and GRIN waveguide, respectively.
+a higher (lower) refractive index as needed for the waveg-
+uide cores (claddings), one requires an accordingly higher
+(lower) LP, i.e. D. STIN waveguides result from a con-
+stant LP all across their core, while for GRIN waveguides
+the writing power changes from high to low following a
+parabolic profile.
+To evaluate the optical performance, we fitted the ex-
+perimental output intensities for diameters d below the
+cut-off condition of the second propagation mode. The
+output intensity of the LP01 mode of a STIN waveguides
+is described by J2
+0(u r
+R) for | r | < R and K2
+0(v r
+R) for |
+r | > R, while for GRIN waveguides is given by an in-
+finite parabolic refractive index profile as exp − 1
+2V r2
+R2 45.
+Figure 8 (b-c) depicts the fit of fundamental LP01 mode
+to the normalized output of STIN and GRIN waveguides
+with radius R
+=
+3 µm, respectively. Considering the
+refractive index of the core constant (ncore ≈ 1.547),
+we obtained an averaged numerical aperture <NA> =
+0.08 ± 0.01 (i.e. ncore = ncladding + 2.4 · 10−3) for STIN
+and of <NA> = 0.18 ± 0.02 for GRIN waveguides. As
+expected, the core-confinement of GRIN waveguides is
+significantly higher than for STIN waveguides due to the
+inner core refractive index modification, which offers a
+crucial advantage for photonic integration schemes7.
+As seen, STIN waveguides with a polymer cladding
+have a refractive index contrast in the order of ∆n ≈
+2.4·10−3, with low NA ≈ 0.12. Contrary than for air-
+cladded waveguides, this leads to large bending radii of
+15 mm (7 mm) at λ = 1550 nm (λ = 650 nm), and in
+turn dense photonic integration is much more challeng-
+ing for STIN waveguides. However, the low ∆n allows to
+have single-mode propagation for waveguide diameters
+d ≤ 9.8 µm (d ≤ 4.2 µm) at λ = 1550 nm (λ = 650 nm),
+which is standard with the current DLW-TPP fabrica-
+tion technology. Future efforts include combining poly-
+mer and air-cladded waveguides, taking the strengths of
+each configuration in a single platform, i.e. air cladding
+waveguides providing highly-densed photonic integration
+
+P = 10.4 mW
+-5
+-7
+6-
+-11
+p = 11.2 mW
+-5
+-7
+-9
+30 μm
+.11
+200 μm
+50gmexp(cs291.2μmcs291.2μmcs291.2μmcs291.2μmcs291.2μmcs291.2μm8
+with their small bending radii, while STIN waveguides
+serving as tools for single-mode propagation with large
+waveguides diameters over wide distances.
+VI.
+FLASH-TPP PRINTED WAVEGUIDES
+Recently, we demonstrated the fabrication of large
+scale 3D integrated photonic components via flash-TPP.
+Several features of flash-TPP make it an enabling tech-
+nology for integration of larger circuits.
+Of primary
+importance is the substantial accelerated fabrication;
+without, fabrication of larger integrated circuits would
+quickly approach timescales beyond 24h9. Based on this
+approach, we demonstrated long (6 mm) single-mode
+waveguides, and we achieved exceptionally low injection
+(≈ 0.26 dB) and propagation (≈ 1.3 dB/mm) losses9.
+Next as the demonstration of optical splitters and com-
+biners based on this concept. These are the backbone of
+any photonic IC, and 3D integration enables interesting
+alternatives for creating 1 to M optical couplers without
+using sensitive optical interference units46. In 3D, 1 to M
+optical couplers can simply be realized by arranging nu-
+merous output waveguides around the input waveguide,
+something impossible to realize in a purely 2D integra-
+tion setting. We demonstrated broadband 1 to M split-
+ters leveraging adiabatic coupling6,47.
+Adiabatic cou-
+pling achieves low-loss single-mode optical transfer from
+1 to M waveguides through evanescent waves, where the
+optical mode adiabatically leaks from a tapered core of
+an input waveguide towards the cladding into inversely-
+tapered cores of the output waveguides48,49. All the pre-
+vious studies consider the 2D case of only one to one
+adiabatic coupling between optical components50.
+In our work, we showed efficient single-mode adiabatic
+transfer with 1 input and up to 4 outputs via a single
+component. Figure 9 (a) illustrates the design for the
+exemplary case of a 1 to 2 adiabatic couplers. The waveg-
+uide’s circular core cross-section continuously changes as
+a function of propagation direction z. The originally cir-
+cular core is reduced in size exclusively along the direc-
+tions where an output waveguide is located; the core is
+essentially cut along plane surfaces.
+These cut-planes
+move towards the input core’s center during the taper-
+length lt at equal rate d/lt along the (x, y)-plane in order
+to match their relative effective modal indices45. Output
+waveguides follow exactly the same concept, yet in an in-
+verted direction. We separated in and output waveguides
+via gap g and studied the evanescence coupling efficiency
+between coupled waveguides6. The same tapering strat-
+egy was applied to 1 to 3 and 1 to 4 as depicted in the
+output intensity profiles from Fig. 9 (b).
+We obtained record optical coupling losses of 0.06 dB
+for the optimal case of 1 to 2 adiabatic couplers, with
+a difference between the two outputs intensities of only
+∼ 3.4 %. We furthermore demonstrated broadband func-
+tionality from 520 nm to 980 nm during which losses re-
+main below 2 dB6. Importantly, these adiabatic couplers
+can be cascaded in order to exponentially increase the
+number of M outputs, c.f. Fig. 7 (c). We arranged a
+double-layer of 1 to 4 adiabatic couplers and the result-
+ing 1 to 16 single-mode output intensities can be seen in
+the last diagram of Fig. 9 (b). Importantly, the global
+losses of the entire device is only 1 dB , and the entire
+circuit was realized within (0.08 × 0.08 × 1.5) mm36.
+x
+y
+z
+Norm. Intensity
+0             1
+(a)                             
+                                 
+                                     (b)
+FIG. 9. Adiabatic 1 to M broadband-scalable couplers fabri-
+cated via flash-TPP. Image taken with permission from6. (a)
+Design of the 1 to 2 adiabatic couplers printed via flash-TPP.
+The same tapering strategy can be applied to higher-order
+couplers, i.e. 1 to 3 and 1 to 4 couplers. (b) Output intensity
+profiles of the 1 to 2, 3 and 4 adiabatic couplers. The last
+output intensity corresponds to a cascaded 1 to 16 adiabatic
+coupler.
+VII.
+TOWARDS A SCALABLE AND CMOS
+COMPATIBLE INTEGRATION OF PHOTONIC
+NETWORKS
+High-density photonic integration requires the inter-
+connection of several photonic platforms.
+Most of the
+current photonic devices are based on silicon-on-insulator
+(SOI) and CMOS technology. Combining the strength of
+multiple photonic and electronic systems in one hybrid
+and multi-chip platform can result in the diversification
+of specific computing tasks while increasing the overall
+performance.
+A versatile fabrication technology with low-losses is of
+vital importance for the scalability of free-form as well as
+integrated optical interconnects in three-dimensions. The
+polymer-based 3D printing technology based on DLW-
+TPP is excellently suited to address these challanges, and
+several proof-of-concept studies have been realized50–52.
+Figure 10 (a) shows photonic wire-bonding, realising a
+3D photonic waveguide forming a point to point com-
+munication for a chip-to-chip connection between SOI
+chips hosting individual waveguides. The photonic wire-
+bond was fabricated via DLW-TPP using the negative-
+tone MicroChem SU-8 2075 photo-resist (n ≈ 1.51 at
+1550 nm)53, and it connected two SOI waveguides sep-
+arated a distance of 100 µm on different CMOS chips.
+This demonstrated for the fist time the basic viability of
+TPP-based 3D printing as a tool for CMOS compatible,
+wafer-scale as well as chip-to-chip connections.
+A major challenge of the polymer-based 3D fabrication
+
+g
+C15.8
+Intensity
+11.9
+20
+Size (μm)
+0
+7.9
+4.0
+0.0
+4.0
+7.9
+11.9
+Size (μm)9
+and the CMOS technology is the interaction of the CMOS
+substrate with the photo-resist during the TPP printing
+process. In a standard fabrication setting, the interac-
+tion between the fs-pulsed laser and the glass substrate
+is negligible since the substrate material, i.e. fused silica,
+is transparent at the wavelength of the fs-laser (780 nm),
+and low specular reflection. However, the CMOS tech-
+nology is based on 2D stacking of multiple thin layers
+of semiconductor materials such as GaAs, InP or Sili-
+con. These often have a bandgap energy below that of
+the writing laser, and in that case printing through the
+semiconductor substrate is impossible; only the ’dip-in’
+concept is therefore a viable general approach for fabri-
+cating 3D photonic integrated circuits directly on top of
+a CMOS substrate based on DLW-TPP. Another chal-
+lenge is the higher specular reflection, as these semicon-
+ductor materials have a higher refractive index. The re-
+sulting optical reflection of the fs-laser laser at the semi-
+conductor substrate leads to a overpolymerization of the
+photo-resist if not compensated for. The LP therefore
+needs to be continuously adjusted at the vicinity of the
+CMOS/photonic circuit interface in order to achieve the
+intended degree of polymerization of the photo-resist. A
+further requirement is the precise alignment of the 3D
+photonic chip with the semiconductor device patterned
+on the CMOS substrate.
+(a) 
+(b)
+25 μm 
+IP-S
+GaAs
+IP-S
+FIG. 10. Polymer-based 3D printing and CMOS technology
+compatibility. (a) Chip-to-chip photonic wire bonding con-
+cept.
+A 3D polymer waveguide fabricated via DLW-TPP
+connects two SOI waveguides sitting on distant CMOS chips.
+SEM image taken with permission from 53. (b) SEM micro-
+graph of and exemplary 3D-cuboid integrating a cascaded 1
+to 16 adiabatic couplers printed via flash-TPP on top of a
+quantum dot micropillar laser array.
+Figure 10 (b) depicts an exemplary 3D-printed cuboid
+integrating a cascaded 1 to 16 adiabatic coupler (cf.
+Fig. 9 (b)) printed via flash-TPP on top of a semiconduc-
+tor substrate integrating quantum dot micropillar laser
+arrays. Each of the micropillar lasers consists of a cylin-
+drical microcavity (a vertical arrangement of highly re-
+flective distributed Bragg reflectors (DBR) alternating
+Al(Ga)As and GaAs mirror pairs) sandwiching a cen-
+tral gain section based on InGaAs self-assembled quan-
+tum dots (QDs). Further details about the fabrication
+and optical properties of the quantum dot micropillars
+laser arrays from Fig. 10 (b) can be found in54–56. We
+used IP-S photo-resist for the fabrication, with a lower
+laser power LP = 6.5 mW (compared to the previously
+LP = 15 mW) in order to avoid microexplosions of the
+photo-resist at the semiconductor-polymer interface dur-
+ing TPP printing. After development, the 3D photonic
+chip is then polymerized via OPP with a exposure dose
+D
+= 3000 mJ/cm2. The SEM micrograph shows the
+perfectly aligned 3D photonic structure with the angle
+of the periodic GaAs micropillar array. We checked the
+adherence of the polymer over time, and after a continu-
+ously observation over more than 4 months no deteriora-
+tion has been found. This confirms the reliability of in-
+tegrating our 3D printing technology with CMOS-based
+micro-laser arrays.
+VIII.
+CONCLUSION
+Here, we present a review over our recent work address-
+ing additive manufacturing towards future 3D photonic
+integration of optical components that is CMOS com-
+patible. Based on one- and two-photon polymerization
+processes combined with direct-laser writing systems, we
+demonstrated the fabrication of high performance indi-
+vidual photonic waveguides as well as scalabale optical
+splitters. All such 3D structures have been fabricated in
+our local FEMTO-ST RENATECH infrastructure.
+We demonstrated that using the commercial DLW-
+TPP Nanoscribe GmbH (Photonics Professional GT)
+system and the ’dip-in’ DLW strategy, we are able to
+the construct, both, air- and polymer-claddded photonic
+waveguides. For air-cladded waveguides, we used a TPP-
+only, a single-step and single resin (IP-Dip resist). A 3D
+IC comprising a network of fractal optical splitter with
+225 input and 529 output waveguides only occupies a
+volume of 460x460x300 µm3. Such air-cladded waveg-
+uide ICs are prime candidates for highly-dense photonic
+packaging thanks to their low bending-radii on 10s of µm
+scale. For polymer-cladded waveguides, we presented two
+different strategies in which we 3D-printed the waveguide
+cores via TPP while achieving a precise control over the
+refractive index contrast ∆n via, (i), the adjustment of
+the fs-laser dose D on an single-voxel level, i.e. (3+1)D-
+printing, and (ii), the duration of UV blanket exposure
+that determines the OPP dosage D to fix the index of the
+cladding material for the entire photonic IC in a single
+shot, i.e. flash-TPP. Noteworthy, both fabrication con-
+cepts require a single procedure writing step and a single
+resin (IP-S resist). Importantly, with flash-TPP fabri-
+cation times are reduced by up to ≈ 90 % compared to
+(3+1)D-printing thanks to the additional OPP process.
+Via flash-TPP, we achieved polymer-cladded waveguides
+with refractive index contrast ∆n ≈ 5·10−3, with low
+1.3 dB/mm (0.26 dB) propagation (injection) losses while
+printing waveguides up to 6 mm heigh. This allows to
+have single-mode propagation over large distances. We
+demonstrated the fabrication, via flash-TPP, of scalable-
+boadband couplers leveraging adiabatic transfer from 1
+input up to 4 outputs. Using a tapered/inversely-tapered
+waveguide sequence, we achieved record 0.06 dB optical
+coupling losses with very symmetric splitting ratios. We
+
+HV
+curr
+use case
+det
+mag
+只
+WD
+tilt
+50 μm
+5.00 kV
+0.20 nA
+Standard
+LVD
+1 000 x
+10.0 mm
+45.0°
+FEMTO-ST(a)
+(b)
+Photonic wire
+Photonicwire
+bond
+bond
+SOlwaveguide
+SOI
+25μm
+waveguides
+Chip1
+20 μm
+10 μm
+Chip2
+(c)
+Input fiber
+Qutput fiber
+Photonic wire bonds
+Chip1
+Chip.2
+Grating couplers10
+arranged a double-layer of 1 to 4 adiabatic couplers, re-
+sulting in a device with 16 single-mode outputs with only
+1 dB global losses.
+Importantly, we demonstrated the compatibility of
+our fabrication methodology based on DLW-TPP with
+CMOS substrates.
+As a proof-of-concept, we success-
+fully 3D-printed our cascaded 1 to 16 adiabatic couplers
+on top of a CMOS substrate integrating GaAs quantum
+dot micropillar laser arrays.
+Preliminary characteriza-
+tion of these structures shows encouraging performance
+in terms of losses and stability.
+Overall, in this review we have covered our novel 3D-
+printing technology, which represents a breakthrough
+with the potential to become a high-impact tool for the
+hybrid, highly-dense and hence compact packaging of,
+both, electronic and photonic devices.
+The concepts
+opens several potential avenues for future exploration.
+The combination of air- and polymer-cladded waveguides
+could enable dense integration with simultaneous precise
+control over optical signal properties such as mode num-
+ber, polarization and phase.
+As the concept leverages
+photo-polymerization, in principle the large-scale and
+exceptionally performing production facilities of CMOS
+electronic integration could be amended with 3D pho-
+tonic integration capability. Due to the excellent compat-
+ibility of standard photo-resins, the approach is largely
+agnostic to the underlying substrate. In this it is more
+flexible than integrated silicon photonics, and fabricat-
+ing additively on a already processed CMOS substrate
+removes many of the challenges compared to fabricating
+photonic ICs based on different process - such as DLW di-
+rectly into bulk dielectrics followed by bonding to CMOS.
+IX.
+ACKNOWLEDGMENT
+The authors would like to thank Stephan Reitzen-
+stein for his contribution through fabricating the semi-
+conductor laser sample used for producing the circuit
+shown in Fig.
+10 (b) and Erik Jung for the valuable
+help on the design of 3D waveguides.
+This work was
+partly supported by the french RENATECH network and
+its FEMTO-ST technological facility.
+The authors ac-
+knowledge the support of the Region Bourgogne Franche-
+Comt´e.
+This work was supported by the EUR EIPHI
+program (Contract No. ANR-17-EURE- 0002), by the
+Volkswagen Foundation (NeuroQNet II), by the French
+Investissements d’Avenir program, project ISITE-BFC
+(contract ANR-15-IDEX-03), by the European Union’s
+Horizon 2020 research and innovation programme un-
+der the Marie Sk�lodowska-Curie grant agreements No.
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+

3dE0T4oBgHgl3EQfvAER/content/tmp_files/2301.02611v1.pdf.txt

@@ -0,0 +1,897 @@
+Identifying Different Student Clusters in Functional
+Programming Assignments: From Quick Learners to Struggling
+Students
+Chuqin Geng
+McGill University
+Montreal, QC, Canada
+chuqin.geng@mail.mcgill.ca
+Wenwen Xu
+McGill University
+Montreal, QC, Canada
+wenwen.xu2@mail.mcgill.ca
+Yingjie Xu
+McGill University
+Montreal, QC, Canada
+yj.xu@mail.mcgill.ca
+Brigitte Pientka
+McGill University
+Montreal, QC, Canada
+bpientka@cs.mcgill.ca
+Xujie Si
+McGill University
+Montreal, QC, Canada
+xsi@cs.mcgill.ca
+ABSTRACT
+Instructors and students alike are often focused on the grade in
+programming assignments as a key measure of how well a student
+is mastering the material and whether a student is struggling. This
+can be, however, misleading. Especially when students have access
+to auto-graders, their grades may be heavily skewed.
+In this paper, we analyze student assignment submission data
+collected from a functional programming course taught at McGill
+university incorporating a wide range of features. In addition to the
+grade, we consider activity time data, time spent, and the number
+of static errors. This allows us to identify four clusters of students:
+"Quick-learning", "Hardworking", "Satisficing", and "Struggling"
+through cluster algorithms. We then analyze how work habits,
+working duration, the range of errors, and the ability to fix errors
+impact different clusters of students. This structured analysis pro-
+vides valuable insights for instructors to actively help different
+types of students and emphasize different aspects of their overall
+course design. It also provides insights for students themselves to
+understand which aspects they still struggle with and allows them
+to seek clarification and adjust their work habits.
+CCS CONCEPTS
+• Social and professional topics → Student assessment.
+KEYWORDS
+online programming platform; computer science education; cluster
+analysis
+1
+INTRODUCTION
+Online programming environments, such as RoboProf [8] for C++,
+DrScheme [13, 14] for Scheme or, more recently, Mumuki [4] , offer
+immense potential to enhance the students’ educational experience
+in large-scale programming-oriented courses. They not only lower
+the entry barrier for beginners but often feature auto-grading facili-
+ties that allow students to run and get feedback on their code while
+they are developing their programs, giving them the opportunity
+to fix bugs and address errors in their understanding right away.
+While having access to immediate feedback on their code has been
+recognized to significantly improve student learning outcomes and
+engagement (see, e.g., [15, 26, 30]), instructors and students alike
+are often too focused on the grade as a key measure of competency.
+Especially when students have access to auto-graders, the students’
+grades may be heavily skewed and misleading.
+This paper develops a data-driven approach to better understand
+students’ behavior when solving programming assignments in a
+functional programming course. In addition to the grade, we pro-
+pose to consider additional factors such as the number of static
+errors and total time spent on solving programming assignments to
+identify student clusters. Using this methodology, we analyze the
+assignment submission data collected in a functional programming
+course taught at McGill university which uses the Learn-OCaml
+online programming platform [5, 6, 17]. This allows us to identify
+four student clusters: "Quick-learning", "Hardworking", "Satisficing",
+and "Struggling". While the first two clusters can be characterized
+as maximizers, i.e. students strive to achieve the highest possible
+grades and continue to improve their work, they still differ in the
+amount of time and effort spent on completing a given homework.
+In contrast, satisficing1 students accept a possibly non-optimal out-
+come as ”good enough” allowing them to adequately achieve their
+goals by saving time and effort. We further analyze these clusters
+with respect to work habits and the number and kinds of errors
+that are prevalent. This leads to four key insights:
+• Leveraging the notion of chronotype - a circadian typology in
+humans and animals, we confirm that a work pattern where
+students tend to work in the morning is related to academic
+success. In particular, quick learners tend to work more in
+the morning, while other clusters of students rely more on
+afternoons and evenings.
+• In general, starting on the homework early is related to
+higher grades. However, we also noticed that satisficing stu-
+dents start relatively late but finish the earliest. This further
+emphasizes that satisficing students aim for satisfactory re-
+sults rather than the optimal one. At the same time, satisfic-
+ing students have one of the lowest numbers of programming
+errors suggesting that they struggle significantly less with
+static errors than for example hardworking students.
+• Our analysis of static errors shows that syntax and type
+errors are prevalent among all students. Further, students
+1The term ���satisficing” was introduced by H. Simon [27] to describe a decision-making
+process in which an individual makes a choice that is satisfactory rather than optimal.
+arXiv:2301.02611v1  [cs.CY]  6 Jan 2023
+
+continue to struggle with these errors throughout the se-
+mester. In addition, our analysis points to other common
+mistakes such as non-exhaustive case analysis and the use
+of unbound variables.
+• Taking into account students’ ability to fix static errors, i.e.
+how many tries a student needs to fix a particular error, we
+notice that the failure/success ratio is particularly high for
+hardworking students. This highlights both their desire and
+drive to get the best possible grade, but also that their path
+is full of small stumbling blocks.
+We believe our proposed set of features and data-driven analysis
+can provide instructors with a clearer and more detailed picture of
+students’ behaviours and performance. This in turn may be used
+to adjust how some concepts, such as how to avoid static errors,
+are taught. It may also be used to design different strategies for
+different students to enhance the students’ learning experience.
+Furthermore, this data may be interesting to students themselves
+to better understand how well they do in a class and identify areas
+where they can actively make changes and seek help early.
+2
+RELATED WORK
+Analyzing student data in programming courses is a central topic
+in learning analytics, and it is gaining increasing attention with
+the recent advances in storing and processing data. One of the core
+aims of analyzing student data is to understand student behaviours,
+and in turn, improve student learning experience [21].
+Over the past decade, there have been several studies that focus
+on identifying groups of students using cluster analysis. For exam-
+ple, Emerson et al. [12] use cluster algorithms to identify student
+misconceptions in a block-based programming environment for
+non-CS major students based on program structures. Wiggins et
+al. [29] finds five major clusters of hint requests in a block-based
+programming system equipped with an intelligent tutor. Hossein
+et al. [20] leverages clustering analysis to further investigate the
+correlation between students’ programming speed and program-
+ming behaviours by collecting programming snapshots whenever
+an action occurs. They then divide students into two clusters by
+comparing a student’s programming speed to the median speed.
+Lahtinen et al. [23] uses different levels of Bloom’s Taxonomy as fea-
+tures to identify six distinct student groups that instructor should
+be aware of when teaching introductory programming courses.
+In contrast to these existing works, our work considers multi-
+categorical features involving the grade, total time spent on the
+assignment, and the number of static errors encountered to identify
+clusters of students.
+Based on the identified clusters, we follow existing work in under-
+standing the work/rest patterns of students. In particular, Claes et al.
+[7] study programmers’ working patterns using clustering analysis
+on time stamps of committed activities of 86 large open-source
+software projects. Zavgorodniaia et al. [31] study the chronotypes
+of students through cluster algorithms using keystroke data. In our
+study, we use activity data (such as whether a student compiled or
+graded their homework) to study the work/rest patterns of students.
+It is the first study in the context of typed functional programming.
+We further analyze static errors in typed functional programming
+assignments and their impact on different student clusters. This
+is the first such study in this setting. Previous studies focus on
+compilation events in object-oriented programs written in Java.
+For example, Ahmadzadeh et al. [1] investigates compiler error
+frequencies of student programs and debugging activity patterns
+in Java. They suggest debugging skills should be emphasized in the
+teaching of programming. Altadmri et al. [2] collect a large dataset
+comprising compilation events of 250,000 students, which provides
+a robust analysis of error patterns and time for fixing different
+errors. Denny et al. [9] also study various syntax error frequencies
+and how long students spend fixing common syntax errors. They
+also found that certain types of errors remain challenging even for
+higher-ability students. Edwards et al. [11] analyze 10 million static
+analysis errors found in over 500 thousand program submissions
+made by students over a five-semester period. The experimental
+results suggest error frequencies made by CS major and non-major
+students are consistent.
+Our analysis is one of the first that investigates in more depth
+the frequency of various static errors in the typed functional pro-
+gramming assignments. Here, static errors go beyond syntax and
+simple type errors and include for example detection of missing
+branches in a program.
+3
+STUDY DESIGN
+This research aims to gain a deeper understanding of how students
+develop typed functional programs (TFP). We assume that the grade
+alone is not a good indicator of how well a student masters basic
+concepts and achieves competency. Instead, we propose that taking
+into account the time spent as well as the number of errors a student
+encounters can provide a more nuanced picture. Hence, the main
+question that we tackle in this paper is how can we best identify
+different clusters of students taking into account grades, time spent,
+and the number of errors. We then analyze our clusters with respect
+to five hypotheses:
+H1: Even students with a high grade in programming assign-
+ments may significantly struggle with a range of static errors.
+H2: Despite a lower grade, students who spend less time and
+have a low number of static errors do in fact well overall.
+H3: Work/rest patterns of students as well as the time a student
+spends on homework play a role in students achieving a high
+grade. It highlights how driven a student is.
+H4: Static errors in TFP range from syntax and type errors
+to detecting unbound variables and missing branches in
+programs. This wide range of static errors provides a fine-
+grained picture of concepts students find challenging.
+H5: Error fix ratio, i.e. how many tries a student needs to fix
+a static error, indicates how well students understand basic
+ideas in TFP and this is correlated to their understanding
+and performance.
+3.1
+Course Context
+Our study concerns students in a second-year undergraduate com-
+puter science course at McGill university. The course introduces
+concepts about functional programming and programming paradigms.
+It is offered every semester with more than 300 registered under-
+graduate students. In this study, all data is collected in the Fall 2021
+
+programming topics
+#tasks
+HW1
+basic expressions, recursion
+7
+HW2
+data types and pattern matching
+6
+HW3
+higher-order functions
+11
+HW4
+references, state, memorization
+5
+HW5
+exception, continuations
+5
+HW6
+lazy programming, toy language
+5
+Table 1: Overview of six programming assignments.
+Figure 1: Data collection pipeline. Grade and Compile and Eval
+events are handled by different servers, all submission data are
+stored in a MongoDB database. The components highlighted in light
+green are original components in the Learn-OCaml platform, while
+the components highlighted in light blue are newly introduced by
+us.
+semester when students could attend online Zoom or in-person
+sessions.
+The course had six bi-weekly programming assignments each
+worth 5% of the final grade. Each homework consists of several pro-
+gramming tasks to implement functions and test cases. Homework
+information is summarized in Table 1. All homework assignments
+were hosted on Learn-OCaml [6], an online programming platform
+for OCaml which allows students to edit, compile, test, and debug
+code all in one place. We used a modified version of Learn-OCaml
+by Hameer and Pientka [18] with additional features such as style
+checking and evaluation of test cases written by students.
+3.2
+Data Collection
+Our data collection pipeline is built on top of the Learn-OCaml
+platform and it can automatically log students’ actions. Specifically,
+we send local programming events like compile and evaluation (for
+testing and debugging) with asynchronous logging requests to our
+backend server. Figure 1 illustrates the process of collecting the
+data from the online environment Learn-OCaml.
+Around 52.81% (i.e., 169 out of 320) students gave us consent
+to access their data. We collect more than 270,000 programming
+events, and each event stores a snapshot of the code as well as
+feedback information (e.g., time-stamp, static errors, grades, etc.).
+3.3
+Feature
+For each homework, we collect a sequence of programming activity
+events. The activity events include grade, compile, and evaluation
+events. This allows us to create an activity density vector for each
+student. It is a four-element vector that represents the percentage
+of the student’s activity events that occurs in different ranges of
+hours [0-6, 6-12, 12-18, 18-0], which is the same choice of ranges
+suggested in [31].
+In addition, we design the following features based on the activity
+event sequence:
+• Start time. The day when a student starts actively working
+on an assignment based on the activity events collected.
+• End time. The day when a student finishes an assignment,
+which is the last Grade event.
+• Working session. Defined as the time window where ac-
+tivity events occur. If there is no activity event within 30min,
+then the working session is assumed to have ended.
+• Total time spent. Sum over the length of all working ses-
+sions.
+• Number of errors. The number of static errors that a stu-
+dent made while completing an assignment.
+• Grade. The final grade a student receives for an assignment.
+3.4
+Feature Engineering
+There are two challenges to applying clustering algorithms and sta-
+tistical tests to our study. The first one is skewed data . For instance,
+the grade is highly skewed as students can always improve their
+grades through interacting with the auto-grader. The second one
+is the difference between feature scales, which renders the clus-
+tering results incoherent. We use two approaches to address these
+challenges. First, we use non-parametric tests including Spearman
+correlations and Kruskal-Wallis H-Tests. Second, we apply the rank
+transformation on features to facilitate clustering algorithms.
+4
+IDENTIFYING STUDENT CLUSTERS
+To identify student clusters, we run the K-means[19] clustering al-
+gorithm on the aggregation (mean) of three most important features
+(i.e., grade, number of errors and time spent) over six homework. We
+use the elbow method to determine the optimal k (the number of
+clusters) to be 4. After determining the optimal k, we re-run the
+K-means algorithm and report the results in Table 2. We give the
+time in hours and note that all clusters have a similar size in terms
+of number of students (#𝑆𝑡𝑑).
+To determine whether the resulting four clusters are different, we
+run a Kruskal-Wallis H-Test, which is a nonparametric equivalent
+of an ANOVA, on the three features (time spent, #errors, and grade)
+of each cluster. The results are statistically significant with the
+statistics of 113.26, 100.87, and 123.02 respectively, and all p-values
+< 0.0001. This suggests the four clusters are statistically different.
+Students in cluster A have the highest average grade (95.24)
+while spending less than the expected 6h on solving the homework.
+This suggests that they achieve their goal with relative ease. In fact,
+Clusters
+#Std
+Time (Hours)
+# Error
+Grade
+A - Quick learning
+46
+5.30 (± 0.94)
+66.11 (± 26.95)
+95.24 (± 3.25)
+B - Hardworking
+46
+8.24 (± 1.52)
+148.67 (± 63.26)
+94.25 (± 3.90)
+C - Satisficing
+31
+4.47 (± 1.01)
+52.26 (± 21.89)
+74.43 (± 11.31)
+D - Struggling
+46
+6.49 (± 0.94)
+118.14 (± 35.32)
+72.81 (± 11.03)
+Table 2: Student clusters
+
+Feedback from
+Autograder
+programming
+Git repo
+autograder
+history
+Webserver
+Grade Event
+Grade data entry
+LearnOCaml
+[id, timestamp, code, grade]
+Client
+Compile and
+Eval Event
+Compilation
+Results
+MongoDB
+MongoDB
+Webserver
+Compile and Eval
+data entry
+[id, timstamp, code]students in this cluster outperform students in other clusters by a
+large margin. We characterize this cluster as quick learning.
+Students in cluster B have the second-highest average grade
+(94.25). However, they also have the highest average number of
+errors (148.67) and with 8.24h spend significantly more time on
+homework than any other group. In particular, they spend signifi-
+cantly more time than expected. This suggests that they face many
+difficulties which they manage to overcome by spending a signif-
+icant amount of time. These students are driven to improve their
+work and to achieve the highest possible grade. Hence, we charac-
+terize them as hardworking. This data supports our hypothesis
+H1.
+Cluster C has the lowest average number of errors (52.26) and
+spent the least amount of time (4.47h) on the homework. With an
+average grade of 74.43, they still achieve a “good enough” result.
+These students achieve their goals by saving time and effort. At the
+same time, these students reach a satisfying level of competency as
+evidenced by their low number of average errors. We describe these
+students as satisficing students. This supports our hypothesis H2.
+Students in Cluster D are in fact closely related to students in
+cluster B, which shows a similarly high average number of errors
+(118.14) and a significant amount of time (6.49h). However, com-
+pared to students in cluster B, they fail to overcome the difficulties
+along their path. These students are struggling.
+5
+UNDERSTANDING STUDENT CLUSTERS
+5.1
+How do work habits vary for different
+student clusters?
+To investigate our hypothesis H3, we consider when students are ac-
+tive based on our activity data. Prior research suggests that chrono-
+type, a person’s preference in carrying out activity at certain periods
+in a day, is governed by the circadian cycle which is controlled by
+clock genes [10, 25]. In this section, we are interested in investigat-
+ing the chronotypes, or in other words, the work habits of students.
+In particular, it has been observed that “morningness” is positively
+correlated with academic achievement [24, 31].
+To identify potential chronotypes, we run the K-means cluster-
+ing algorithm on the feature space spanned by activity density
+vectors. The elbow method yields 𝑘 = 3, suggesting three possible
+chronotypes, which is different from four chronotypes reported in
+[31]. We report centroids of each chronotype cluster in Table 3.
+Chrono clusters
+0 - 6
+6 - 12
+12 - 18
+18 - 0
+Chronotype
+Cluster 1
+8%
+14%
+26%
+52%
+Evening (Eve)
+Cluster 2
+4%
+26%
+20%
+50%
+Morning (Mor)
+Cluster 3
+2%
+19%
+37%
+42%
+Afternoon (Aft)
+Table 3: Centroids of each chronotype.
+As we can see, most activities occur from 18:00 - 00:00 for all
+three clusters. This is not surprising as most students may have
+classes during the day. Based on this observation, we aim to define
+chronotypes by considering secondary activity peaks as well. We
+notice that Cluster 2 has its secondary activity peak (26%) in 6:00 -
+12:00 whereas Cluster 3 has the secondary activity peak (37%) in
+12:00 - 18:00. Thus, we define Cluster 2 and 3 as the morning (Mor)
+and afternoon (Aft) type. Cluster 1 has only one activity peak in
+18:00 - 00:00, thus we define it as evening (Eve) type.
+Figure 2: Chronotype distribution in each student cluster.
+As Figure 2 suggests, quick-learning students usually tend to
+work in the morning and afternoon whereas satisficing students
+worked on their homework in the evening. This suggests quick-
+learning students were driven, motivated, and had possibly better
+time management skills. In general, satisficing students were the
+only group to have a strong incline to work in the evening. This
+could point to other commitments that students have or a high
+course load. The afternoon type occurs most frequently in strug-
+gling and hardworking clusters. This may be because they were
+seeking help during office hours that were offered during the day
+or they simply required more time in general. Overall, our results
+confirm previous findings that certain chronotypes are related to
+academic achievement[24, 31].
+Figure 3: Clustering result of different types of students The
+start of a time interval stands for the average start time whereas
+the end represents the average end time.
+5.2
+How long do different clusters of students
+work on their homework?
+To further investigate hypothesis H3, we investigate when students
+in a given cluster start and finish their homework. We report the
+average start time and end time for each cluster in Figure 3. In addi-
+tion, the Kruskal-Wallis H-Test suggests start date was statistically
+significantly different (stat = 22.59, p-value < 0.0001) whereas the
+end date was not (stat = 3.12, p-value = 0.37). Despite that, we can
+still observe some interesting patterns.
+
+25
+Mor
+Aft
+Eve
+20 
+T of Students
+15
+Number:
+10
+5
+0
+Quick learming
+Hardworking
+Satisficing
+StrugglingTime intervals of completing homework for each student cluster
+Quick learning
+6.38
+10.84
+Hardworking
+6.06
+11.08
+Satisficing
+7.51
+10.78
+Struggling
+7.22
+11.41
+6
+F7
+5
+8
+9
+10
+11
+12
+DaysafterhomeworkreleaseError Groups
+Error Categories
+HW1
+HW2
+HW2
+HW4
+HW5
+HW6
+A. General Static Errors
+1. Type Error
+38.12%
+30.94%
+40.93%
+32.65%
+36.90%
+34.83%
+2. Syntax Error
+42.33%
+21.54%
+21.79%
+32.68%
+17.80%
+25.66%
+3. Unbound value
+10.42%
+7.19%
+9.06%
+13.42%
+7.02%
+7.27%
+B. Imperative Thinking Errors
+4. Missing else branch
+1.92%
+0.75%
+0.43%
+0.08%
+1.03%
+1.07%
+5. Unused variable
+0.74%
+0.65%
+0.63%
+6.37%
+21.34%
+7.23%
+C. Pattern Matching Errors
+6. Pattern matching type error
+0.84%
+5.24%
+2.13%
+0.62%
+1.37%
+1.40%
+7. Non-exhaustive pattern matching
+1.02%
+16.78%
+15.74 %
+2.47%
+4.62%
+11.92%
+D. Function Applications Errors
+8. Wrong number of arguments
+1.67%
+2.19%
+3.38%
+1.17%
+2.09%
+1.89%
+9. Misuse of non-function values
+2.50%
+2.10%
+2.07%
+1.72%
+1.50%
+1.77%
+10. Others
+0.88%
+12.6%
+5.89%
+8.83%
+6.33%
+6.96%
+Total number of errors
+7,850
+27,519
+14,331
+19,859
+22,467
+26,681
+Table 4: Error Groups and error categories together with their distribution of HWs
+We note that both satisficing and struggling students start rela-
+tively late on their homework, at 7.51 and 7.22 average days respec-
+tively. However, satisficing students finish the earliest (10.78). This
+underscores the fact that they accept a “good enough” result rather
+than striving for better outcomes. Further, satisficing students had
+the shortest working duration. This substantiates our claim that
+these students achieve their goals by saving time and effort.
+Struggling students experienced many difficulties as evidenced
+by a high number of static errors that they encounter. These stu-
+dents finish indeed last (finish time (11.41)). This indicates that
+these students are struggling, although they do try their best until
+the very end. However, they lack the skills or support to overcome
+their difficulties.
+Hardworking students have the longest time interval. While they
+start the earliest (6.06), they finish the second latest (11.08). This
+shows the commitment and dedication they bring to their work.
+Quick-learning students tend to start quite earlier (6.38), al-
+though not as early as hardworking students. This suggests that
+these students have confidence in their abilities to finish the home-
+work smoothly.
+We ran Spearman correlations to examine the correlation be-
+tween start time and homework grade, the statistically significant
+result (correlation = -0.42, p-value < 0.0001) suggests procrastination
+affects negatively on student learning outcomes, which has been
+widely reported [3, 16, 22].
+5.3
+How do static errors affect students in
+different clusters?
+Compilers for typed functional programming languages such as
+OCaml provide a wealth of errors and feedback to programmers. It
+not only reports syntax and type errors but also reports, for example,
+unused variables, and missing branches in case-statements and if-
+expressions. This provides a basis for a better understanding of
+what basic concepts students struggle with the most.
+5.3.1
+Overview of static errors. To investigate our hypothesis H4,
+we analyze the types of errors of each failed compile event and
+group errors into four main categories: general static errors (eg.
+group A), errors due to imperative thinking (Group B), and errors
+related to pattern matching and function (eg. groups C and D). We
+also include how often particular errors occurred in assignment
+submissions (see Table 4).
+The first homework shows a significant spike (42.33%) in syntax
+errors encountered. This is unsurprising, as it is the first time that
+students attempt to write programs in a new language. However,
+it may be surprising that 20% to 30% of the errors encountered
+are related to syntax and type errors (Group A) throughout the
+semester. In fact, these errors constitute around 60% of errors for
+every homework assignment in Table 4. This may point to the fact
+that type errors in TFP catch conceptual errors in the programmer’s
+thinking early rather than later during testing. This may also sug-
+gest instructors dedicating more time to demystifying type error
+analysis.
+For some key concepts from typed functional programming such
+as pattern matching, our error analysis indicates that students do
+improve and gain a better understanding of it. When pattern match-
+ing is first introduced in HW2, pattern matching errors and non-
+exhaustive pattern matching errors (Group C) consist 22% of total
+static errors. After practicing HW2 and HW3, the proportion of
+Error Group C drops greatly, which suggests that students gain a
+deeper understanding with more programming practice.
+One of the prerequisites of this course is taking an introduc-
+tory CS course, which is taught in Java or Python at our univer-
+sity. This implies that all of the participants had experience in
+programming before and had to deal with conceptual transfer from
+imperative/object-oriented programming (Python or Java) to func-
+tional programming (OCaml). Students usually report transition-
+ing smoothly between procedural language and object-oriented
+language for concepts such as if-conditionals and functions and
+scope[28]. From our observations, students struggle more when tran-
+sitioning to functional programming. In particular, they struggle
+with the concept of bound or unbound variables, missing branches
+in if-expressions, and function application errors. Although these
+errors occur less frequently than syntax and type errors, we believe
+it highlights that students struggle with thinking recursively and
+considering all cases in such a recursive program (Error No.4,7).
+Therefore, if-else expression without an else branch also often leads
+to type errors in a language like OCaml.
+Moreover, imperative programming supports variables declared
+in the local or global state, while in functional languages, such
+as OCaml, we distinguish between stateful variables that can be
+updated and bound variables. While the concept of free variables
+and bound variables and the difference between stateful variables
+
+are discussed frequently in this course, students continue to en-
+counter errors related to variables. In particular, the unbound value
+error occurs throughout the semester. This seems to be a sign that
+the concept of stateful variable declarations as used in imperative
+programming is persisting in how students think about a given prob-
+lem. The most essential concept of functional programming is that
+functions are first-class citizens. Therefore, higher-order functions,
+which take a function as an argument, or return a function, are
+used frequently, especially in HW3 and subsequent assignments.
+If functions are not used correctly, it would most frequently be
+flagged as a type error. However, OCaml also provides other error
+reporting. In particular, it may report on the incorrect number of
+arguments (Error NO.8) and use a function value instead of apply-
+ing arguments on a non-function value (Error NO.9). These errors
+form a non-negligible class indicating where students stumble.
+5.3.2
+How efficiently do students in each cluster fix errors? Lastly,
+we investigate hypothesis H5 and aim to understand how students
+in different clusters vary in their ability to fix errors quickly. Table 5
+shows the average number of successful compile events and fail-
+ure ones experienced by different student clusters throughout the
+semester. The Failure/Success ratio x can be roughly interpreted
+as debugging efficiency or error fix rate that it on average costs a
+student x failure compile events to get a successful one.
+Quick-learning
+Hardworking
+Satisficing
+Struggling
+Success
+37.9
+60.4
+28.1
+40.7
+Failure
+85.7
+162.3
+66.9
+118
+F/S
+2.26
+2.67
+2.38
+2.90
+Table 5: Average success, failure and failure/success ratio
+(F/S) of compile events in each student cluster
+Struggling students have the most difficulty in fixing static errors,
+requiring 2.9 failure compilations to fix the error on average. By
+contrast, quick-learning students have the best ability to debug with
+only a 2.26 failure compilation to get a successful one. Furthermore,
+the gap between their debugging efficiency is more significant, if we
+look at their average failure and success. While the average success
+for struggling students (40.7) and quick learners (37.9 ) are close,
+their average failures have a substantial gap: a struggling student
+has around 30 more failure compilations than quick learners.
+Figure 4: Distribution of static errors in each student cluster.
+The row of Failure in Table 5 can be further represented by the
+average number of each group of static errors for four student clus-
+ters in Figure 4. Type and syntax errors (Group A) dominate for all
+clusters but there are noteworthy differences. Quick learners have
+fewer errors in all groups, not only general static errors but also
+errors specific to functional programming. Satisficing students have
+the fewest errors in Group B, C, and D which may indicate that
+they in fact achieve competency. Lastly, hardworking and strug-
+gling students have significantly more errors in all error groups. In
+particular, they struggle more with basic concepts such as bound or
+unused variables, missing branches, and the proper use of functions.
+6
+CONCLUSION
+In this study, we aim to understand how students develop func-
+tional programming assignments based on data collected through
+the Learn-OCaml programming platform. Our analysis considers
+grade, total time spent, and the total number of static errors to
+identify four student clusters: "Quick-learning", "Hardworking", "Sat-
+isficing", and "Struggling". Using statistical tests we validate our
+clustering results along with other analysis results. This provides
+a nuanced picture of students’ behaviours and also exposes differ-
+ent paths towards achieving academic success in the course. Our
+analysis of chronotypes confirms that students who work in the
+morning reach the highest grade most quickly and smoothly. The
+total amount of time students spend on the homework also high-
+lights the difference and similarities between the different student
+clusters. Although this part of the analysis was done in the context
+of a functional programming course, we expect our methodology
+to be applicable to other programming courses and help identify
+clusters of students who would benefit from additional support.
+Our detailed analysis of static errors in typed functional pro-
+gramming also highlights areas where instructors can adjust their
+course content and possibly revisit topics. We believe our analysis
+also provides insights for students themselves, in particular the
+hardworking students, to understand which aspects they still strug-
+gle with and to seek clarifications. This would possibly allow them
+to become more efficient debuggers, spend less time on homework
+assignments, and improve their conceptual understanding.
+REFERENCES
+[1] Marzieh Ahmadzadeh, Dave Elliman, and Colin Higgins. 2005. An analysis of
+patterns of debugging among novice computer science students. Proceedings
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+

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+Springer Nature 2021 LATEX template
+Accelerating Machine Learning Inference with GPUs in
+ProtoDUNE Data Processing
+Tejin Cai1, Kenneth Herner2*, Tingjun Yang2, Michael Wang2, Maria Acosta
+Flechas2, Philip Harris3, Burt Holzman2, Kevin Pedro2 and Nhan Tran2
+1Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto,
+M3J 1P3, ON, Canada.
+2Fermi National Accelerator Laboratory, Kirk Road and Pine Streets, Batavia, 60510, IL,
+USA.
+3Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
+Cambridge, 02139, MA, USA.
+*Corresponding author(s). E-mail(s): kherner@fnal.gov;
+Abstract
+We study the performance of a cloud-based GPU-accelerated inference server to speed up event
+reconstruction in neutrino data batch jobs. Using detector data from the ProtoDUNE experiment
+and employing the standard DUNE grid job submission tools, we attempt to reprocess the data by
+running several thousand concurrent grid jobs, a rate we expect to be typical of current and future
+neutrino physics experiments. We process most of the dataset with the GPU version of our processing
+algorithm and the remainder with the CPU version for timing comparisons. We find that a 100-GPU
+cloud-based server is able to easily meet the processing demand, and that using the GPU version of the
+event processing algorithm is two times faster than processing these data with the CPU version when
+comparing to the newest CPUs in our sample. The amount of data transferred to the inference server
+during the GPU runs can overwhelm even the highest-bandwidth network switches, however, unless
+care is taken to observe network facility limits or otherwise distribute the jobs to multiple sites. We
+discuss the lessons learned from this processing campaign and several avenues for future improvements.
+Keywords: machine learning, heterogeneous (CPU+GPU) computing, GPU (graphics processing unit),
+particle physics, cloud computing (SaaS), neutrino physics, distributed computing
+1 Introduction
+Machine learning (ML)-based algorithms have
+been widely used in the field of neutrino physics,
+for applications ranging from data acquisition to
+data reconstruction and analysis [1–4]. A detec-
+tor technology ideally suited for computer vision
+applications in neutrino physics is that of liquid
+argon time projection chambers (LArTPCs), which
+are employed by the Deep Underground Neutrino
+Experiment (DUNE) [5] and Short-Baseline Neu-
+trino [6] experiments. ML applications are now
+deeply integrated into the event reconstruction and
+data analyses for the LArTPC experiments [7–9].
+1
+arXiv:2301.04633v1  [hep-ex]  11 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+GPUaaS in ProtoDUNE data
+Event record sizes for the current generation of
+LArTPC experiments are typically ≤1 GB and are
+expected to increase in the next few years. With
+increased event size, the event reconstruction, espe-
+cially the inference of ML algorithms, will become
+a challenge. Additionally, neutrino detectors are
+sensitive to neutrinos from a core-collapse super-
+nova in or near the Milky Way. One of DUNE’s
+physics goals is to rapidly reconstruct detector
+trigger records from such a supernova to provide
+rapid localization information to optical telescopes,
+placing a premium on short event reconstruction
+times. We have demonstrated GPU-accelerated ML
+inference as a service, which significantly reduced
+the reconstruction time for simulated neutrino
+events in the ProtoDUNE experiment [10]. Later,
+we tested the same GPU-as-a-Service (GPUaaS)
+approach to process the entire ProtoDUNE Run
+I dataset to demonstrate the scalability of this
+method. This paper reports the results of those
+tests.
+2 Infrastructure setup and
+methods
+2.1 ProtoDUNE background
+The
+ProtoDUNE
+single
+phase
+detector
+(ProtoDUNE-SP) [11, 12] is a liquid argon time
+projection chamber (LArTPC) that serves as a
+prototype for the first far detector module of
+DUNE [5]. The ProtoDUNE-SP is installed at
+the CERN Neutrino Platform [13]. It has an
+active volume of 7.2 × 6.1 × 7.0 m3. The TPC
+wires are read out by 15,360 electric channels at
+a rate of 2 MHz. A typical event record consists
+of 6000 time samples, corresponding to a 3 ms
+time window. Between October 10 and November
+11, 2018, ProtoDUNE-SP was exposed to a beam
+that delivers charged pions, kaons, protons, muons
+and electrons with momenta in the range 0.3
+GeV/c to 7 GeV/c. After the beam runs ended,
+ProtoDUNE-SP continued to collect cosmic ray
+and calibration data until July 20, 2020, after
+which
+the
+detector
+decommissioning
+started.
+The total number of trigger records (also called
+“events”) during the beam period, which consist of
+both beam interactions and non-beam interactions
+such as cosmic rays, is approximately 7.2 million.
+A ProtoDUNE-SP TPC waveform recorded by
+a single electric channel consists of both signals
+and noise. There are typically three sources of sig-
+nals. During the beam runs, the beam particles
+can interact with the liquid argon inside the TPC
+and produce both ionization electrons and scintilla-
+tion light. Since ProtoDUNE-SP is located on the
+Earth’s surface, it is subject to a large flux of cos-
+mic ray muons, which induce signals over the entire
+detector. There are also radioactive backgrounds
+such as 39Ar that generate low energy signals on
+the scale of a few hundred keV to a few MeV.
+Figure 1 shows the event display of a 6 GeV/c pion
+interaction in the ProtoDUNE-SP detector.
+The first step in the reconstruction of events
+in the TPC is the signal processing. The goal of
+this stage is to produce distributions of charge
+arrival times and positions given the input TPC
+waveforms. The effects of induced currents due
+to drifting and collecting charge, as well as the
+response of the front-end electronics, are removed
+through de-convolution. The charge arrival distri-
+butions are used in subsequent reconstruction steps,
+starting with hit finding. The hit finding algorithm
+fits peaks in the wire waveforms, where a hit repre-
+sents a charge deposition on a single wire at a given
+time. Each hit corresponds to a fitted peak. The
+hits are input to pattern recognition algorithms
+such as Pandora [14–16]. This stage finds the high-
+level objects associated with particles, like tracks,
+showers, and vertices, and assembles them into a
+hierarchy of parent-daughter nodes that ultimately
+point back to the candidate neutrino interaction.
+
+Springer Nature 2021 LATEX template
+GPUaaS in ProtoDUNE data
+3
+0
+100
+200
+300
+400
+Wire Number
+3500
+3750
+4000
+4250
+4500
+4750
+5000
+Tick
+50 cm
+DUNE:ProtoDUNE-SP Run 5772 Event 15132
+2
+0
+2
+4
+6
+8
+10
+Charge/tick/channel (ke)
+Fig. 1: A 6 GeV/c beam π+ interaction in the ProtoDUNE-SP detector [11]. The x axis shows the
+wire number. The y axis shows the time tick in the unit of 0.5 µs. The color scale represents the charge
+deposition.
+More details on the reconstruction workflow are
+described in Ref. [11].
+In ProtoDUNE-SP, a novel algorithm is devel-
+oped based on a convolutional neural network
+(CNN) to perform the classification of each recon-
+structed hit as track-like or arising from electromag-
+netic cascades [9]. These hit-level classifications
+can be used alongside pattern recognition based
+reconstruction algorithms such as Pandora to refine
+the track or shower classification of reconstructed
+particles. The CNN model was trained using Ten-
+sorFlow [17]. Hereafter, we call this algorithm
+EmTrkMichelId.
+In order to improve the efficiency and speed
+of the inference of ML algorithms in a large-
+scale data processing, GPU acceleration specifically
+for the ProtoDUNE reconstruction chain has
+been integrated without disrupting the native
+computing workflow using the services for opti-
+mized network inference on coprocessors (SONIC)
+approach [10, 18]. With the integrated framework,
+the most time-consuming task, track and particle
+shower hit identification, runs faster by a factor of
+17. This results in a factor of 2.7 reduction in the
+total processing time when compared with CPU-
+only production. This initial test using a small
+number of simulated ProtoDUNE events showed
+a viable, cost-effective way to solve the comput-
+ing challenge facing the neutrino experiments. In
+this work, we report the results of reprocessing
+the entire 7 million ProtoDUNE events taken dur-
+ing the test beam runs with the SONIC-enabled
+framework.
+2.2 Inference server setup
+The Nvidia Triton™ Inference Server is an open-
+source inference serving software that helps stan-
+dardize model deployment and execution; its goal
+is to deliver fast and scalable AI in production [19].
+
+Springer Nature 2021 LATEX template
+4
+GPUaaS in ProtoDUNE data
+NVIDIA provides multiple ways to deploy the
+inference server on different cloud providers and
+infrastructure types, including both bare metal
+and containerized workloads.
+This study uses a cloud-based deployment of
+Nvidia Triton™ Inference Server within a Google
+Cloud Kubernetes Engine [20] cluster on virtual
+infrastructure provided by Google Cloud Platform.
+The use of this technology enables us to deploy
+a flexible GPUaaS model where a public end-
+point takes remote inference requests from various
+geographically distributed sources as depicted in
+Figure 2. The Triton™ server running on the Google
+cloud supports different backends. We use the Ten-
+sorFlow (version 1.15.5) backend for the inference
+of the EmTrkMichelId algorithm.
+In a similar way as Ref. [10], this study uses sev-
+eral Triton™ servers split into separate Kubernetes
+deployments with common services for network-
+ing and external load balancing in the form of
+ingress objects [21]. One significant improvement
+for the current study is the deployment of metrics
+and monitoring which provided us with observ-
+ability within the system in different states. In IT
+and cloud computing, observability is the ability
+to measure a system’s current state based on the
+data it generates, such as logs, metrics, and traces.
+It relies on telemetry derived from instrumenta-
+tion that comes from the endpoints and services in
+computing environments. Triton™ provides a built-
+in metrics endpoint [22] that publishes plain-text
+data in Prometheus format. Prometheus collects
+and stores data to be displayed by Grafana as seen
+in Figure 3.
+2.3 Methods
+The DUNE collaboration undertook a production
+campaign in 2021 to process ProtoDUNE-SP data
+using the LArSoft toolkit [23] version v09 30 00.
+Each production run during the beam period com-
+prises several data files, each containing between
+100 and 150 data records. In contrast to the previ-
+ous work, in which DUNE simulation events were
+processed by submitting jobs locally to a dedicated
+queue, we submit jobs to process each file via the
+current standard DUNE workflow management
+and job submission systems [24, 25], thus requir-
+ing no special treatment. Jobs may run either at
+Fermilab or one of several remote sites that we
+reach with opportunistic access enabled by the
+OSG Consortium [26].
+We begin from the existing reconstructed
+outputs and apply the updated EmTrkMichelId
+algorithm to produce new outputs. Of the 7.2 mil-
+lion ProtoDUNE events during the 2018 beam
+period, we process 6.4 million through the SONIC
+infrastructure, and 800k with the CPU-only ver-
+sion of the same algorithm for comparison. The
+OSG sites included in the SONIC runs were cho-
+sen to be geographically proximate to the location
+of the Google Cloud GPU servers (which were in
+Iowa, USA at the time) in order to minimize the
+latency in data transmissions.
+The difference in the time spent in the infer-
+ence step is the primary metric with which we
+assess the advantage of GPUaaS over traditional
+CPU processing. Each job produces a log file that
+statistically summarizes the time spent on each
+stage of the event reconstruction for the job as
+a whole. The log has no record of per-stage pro-
+cessing time at the individual event level, but we
+can closely approximate it by taking the difference
+between the start times of consecutive events. We
+estimate the per-event EmTrkMichelId duration
+by subtracting the median non-EmTrkMichelId
+duration from the total event duration, as the
+non-EmTrkMichelId stages display very little time
+variation across events. The CNN-based hit classi-
+fication occurs in the EmTrkMichelId stage and is
+
+Springer Nature 2021 LATEX template
+GPUaaS in ProtoDUNE data
+5
+Internet  
+(gRPC)
+t
+Google Kubernetes Engine - protoDUNE TritonRT
+Local Compute 
+FermiGrid farm
+Offsite Compute 
+University of Notre Dame
+Offsite Compute 
+Wayne State University  
+Offsite Compute 
+University of Wisconsin-Madison 
+Google Kubernetes Engine - protoDUNE monitoring
+Grafana
+Prometheus Server 
+TCP Network Load
+Balancer
+TritonRT Server
+Pod
+TritonRT Server
+Pod
+TritonRT Server
+Pod
+External Service 
+(https)  
+TCP Network Load
+Balancer
+Service 
+:8000 (http) 
+:8001 (gRPC) 
+:8002/metrics
+Internet
+(HTTPS) 
+User
+Real-time monitoring
+dashboard
+Offsite Compute  
+MWT2 - (U.Chicago, IU, U.of FL)
+Fig. 2: ProtoDUNE GPUaaS component diagram depicting local and remote batch inference runs
+submitted from Fermilab and OSG Grid sites.
+Fig. 3: A real-time monitoring view of a 100-GPU cluster run for ProtoDUNE (2021).
+the most time-consuming step in the event recon-
+struction, typically accounting for more than 90%
+of the processing time.
+3 Results
+3.1 CPU-only runs
+We process a set of 13 runs using CPU-based
+Tensorflow both at Fermilab and several off-site
+locations. The off-site locations are the University
+of Notre Dame, the University of Victoria, and
+the high performance computing center at Wayne
+
+ General / Nvidia GPU ★
++
+ 2021-09-30 09:34:31 to 2021-09-30 11:19:58
+Host
+All
+Average Utilization
+(3. 0%
+(320%
+(15.0%
+(18.0%
+(2.0%
+(22.0%
+(2.0%
+(9. 0% 
+( 25.0% 
+15.0%
+( 25.0%
+67.0%
+(1.0%
+(2.0
+2.0%
+(34.0
+(1.0%
+25.0
+32.0
+20.0
+5.000%
+26.760%
+38.000%
+10.000%
+22.6049
+32.0009
+60.00%
+2.756%
+3.000%
+8.6599
+18.000%
+.000%
+16.000%
+0%
+12.9179
+29.000%
+20.009
+75-ee634b3e8507
+0%
+1.676%
+0%
+13.012%
+25.000%
+0%09:35
+0%
+1.654%
+09:45
+09:50
+09:55
+GPU load - Power
+120.00%
+100.00%
+80.00%
+60.009
+fastp.d15@ 10.56.83:002Fermila.Springer Nature 2021 LATEX template
+6
+GPUaaS in ProtoDUNE data
+0
+100
+200
+300
+400
+500
+EmTrkMichelId Time (s)
+0
+2000
+4000
+6000
+8000
+# of Events / 2 sec
+CPU Series
+AMD 6376
+AMD EPYC 7502
+Intel E5-2650 v2
+Intel E5-2650 v3
+Intel E5-2670 v3
+Intel E5-2680 v4
+Intel Gold 6140
+non-FNAL
+Fig. 4: Timing distributions for CPU-only runs,
+broken down by CPU type.
+State University. The TensorFlow version used in
+the CPU-only runs is 2.3.1. Table 1 summarizes
+the number of events processed at each site and
+the median processing times. We did not request
+any specific CPU type when submitting these jobs
+since typical DUNE practice is to use any and all
+available CPU types.
+Table 1: List of CPU-only run sites and median
+processing time
+OSG Site
+N samples
+Median processing time (s)
+FermiGrid
+746603
+79
+Notre Dame
+36082
+68
+Victoria
+10944
+52
+Wayne State
+4242
+45
+There is a clear dependence on processor type
+in the EmTrkMichelId processing time distribution.
+In general, more recent CPUs process events faster.
+Figure 4 shows the CPU-based EmTrkMichelId
+timing for each of the CPU types currently avail-
+able on the Fermilab general purpose batch farm.
+We do not have access to CPU type information
+outside of Fermilab and thus group them together.
+3.2 GPU runs
+Our main processing effort uses the GPUaaS infras-
+tructure as described. Figure 5 shows the average
+EmTrkMichelId processing time when using the
+GPUaaS infrastructure for our entire running
+period. The first peak at approximately 20 s repre-
+sents a factor of two improvement with respect to
+the fastest CPU-only runs, and a factor of roughly
+11 over the slowest CPU runs. It is important to
+note that the EmTrkMichelId times we report here
+are wall times measured within the job, and thus
+include contributions from network latency to and
+from the server. There is another peak in the dis-
+tribution with a median of over 100 s, to which we
+now turn.
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Avg. EmTrkMchelID time (s)
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+Njobs
+All runs 9/30 - 10/20
+Fig. 5: Average EmTrkMichelId times for GPU
+runs during the period September 30, 2021 to Octo-
+ber 20, 2021. The double peak structure arises from
+periods during which the outbound network con-
+nection from the Fermilab grid processing center
+was saturated.
+
+Springer Nature 2021 LATEX template
+GPUaaS in ProtoDUNE data
+7
+3.2.1 Outbound network saturation
+During the first period of GPU running we
+averaged between 200 and 2000 concurrent jobs.
+Figure 6 shows the overlay of network traffic and
+event processing start rate during the period of
+September 30, 2021 to October 6, 2021. As the
+event start rate increases because of the rise in the
+number of concurrent jobs, we see that the 100
+Gb/s outbound network connection used by the
+Fermilab data center where the jobs run becomes
+saturated. While our jobs were not solely responsi-
+ble for the saturation (the connection serves the
+entire cluster), the saturation did result in a sig-
+nificant increase in the average EmTrkMichelId
+processing time as shown in Figure 7. The highest
+job concurrency levels were on October 5, when
+unusually low demand for computing resources
+from other Fermilab experiments resulted in a large
+number of opportunistic job slots being available
+at Fermilab. We were, without any direct interven-
+tion, thus able to scale up to approximately 6,000
+concurrent jobs. The monitoring does show switch
+saturation as early as October 1, however. After
+learning of the network saturation we implemented
+a concurrency limit on jobs of approximately 600;
+thereafter the jobs ran without incident and the
+EmTrkMichelId times returned to pre-saturation
+levels (see Figure 8).
+4 Discussion
+In order to understand the impact of ProtoDUNE
+jobs on the Fermilab network traffic, we plot the
+distribution of event processing start rate versus
+network traffic in Figure 9. Even though the net-
+work traffic has contributions from all grid jobs at
+Fermilab, there is a clear correlation between the
+number of ProtoDUNE concurrent jobs and the
+increase of network traffic. We fit a straight line
+to the data points below the network traffic of 80
+09/30 10/1
+10/2
+10/3
+10/4
+10/5
+10/6
+10/7
+Date
+0
+5
+10
+15
+20
+25
+Event Starting Rate/s
+0
+20
+40
+60
+80
+100
+Traffic (Gb/s)
+Event Rate
+Outbound Traffic
+Google Traffic
+Fig. 6: Overlay of network traffic and event pro-
+cessing start rate at FermiGrid as a function of
+time, which is a proxy for the number of concurrent
+jobs. The origin day is September 30, 2021. The
+solid line is the event start rate, the blue dot-dash
+line is the outbound network traffic rate through
+the 100 Gb/s switch at Fermilab used by the batch
+processing cluster, and the black dashed line is the
+ingress rate to the Google cloud server. We are
+unable to disambiguate traffic sources through the
+switch, so the blue dot-dash line represents the
+total traffic as opposed to only traffic generated
+by our processing campaign. We see that the net-
+work switch was effectively saturated in multiple
+instances, though Google ingress was not.
+Gb/s. The slope of the best fit line is 4.2 ± 0.2 Gb,
+which is the average outbound data transmission
+per event. The intercept is 44 ± 2 Gb/s, which is
+the average traffic from non-ProtoDUNE grid jobs.
+Based on the discussion of transmission time in
+Ref. [10], for 55,000 inferences per event, with each
+input a 48 × 48 image at 32 bits, the total amount
+of data transmitted is about 4.1 Gigabits per event.
+This is consistent with the slope of the best fit
+straight line. The spread in data with respect to
+the straight line could be caused by the variation
+in the number of non-ProtoDUNE grid jobs during
+this period.
+Figure 8 indicates that the average process-
+ing time is roughly 25 s/event for the GPU
+jobs. Assuming the entire 100 Gb/s bandwidth
+
+Springer Nature 2021 LATEX template
+8
+GPUaaS in ProtoDUNE data
+20
+30
+40
+50
+60
+70
+80
+90
+100
+FermiGrid Outbound Traffic (Gb/s)
+0
+50
+100
+150
+200
+250
+300
+EmTrkMichelId Time (s)
+EmTrkMichelId Time
+0
+2000
+4000
+6000
+8000
+10000
+12000
+No. of Events
+20
+30
+40
+50
+60
+70
+80
+90
+100
+FermiGrid Outbound Traffic (Gb/s)
+0
+50
+100
+150
+200
+250
+300
+EmTrkMichelId Time (s)
+EmTrkMichelId Time, normalized to max entry per column
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Events/Column Max
+Fig. 7: The average EmTrkMichelId duration
+before Oct. 7 as a function of the total network
+traffic through the 100 Gb/s network switch at Fer-
+milab used by the batch processing cluster. The
+top plot shows the real event rate. The bottom
+plot is the same as the left one, with each column
+scaled separately so the maximum amplitude is 1
+for each column.
+is available to the ProtoDUNE jobs, the max-
+imum number of concurrent ProtoDUNE jobs
+we can run without saturating the network is
+(100 Gb/s)/(4.1 Gb/event) · (25 s/event) ≃ 600.
+This is consistent with the concurrency limit of
+600 jobs that we implemented after October 7.
+Based on the above discussions, we conclude
+that, while overall computational time clearly
+decreases using GPUaaS, one does have to take
+particular care to understand what the expected
+data movement requirements will be for jobs using
+this architecture, and to set job concurrency limits
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Avg. EmTrkMchelID time (s)
+0
+1000
+2000
+3000
+4000
+5000
+6000
+Njobs
+All runs after Oct 8
+Fig. 8: The average time spent in the EmTrk-
+MichelId task for all GPU jobs after October 8,
+when the network saturation had subsided.
+appropriate to the capabilities of each local comput-
+ing site and input data source. HTCondor [27, 28]
+in particular has the ability to define an arbitrary
+kind of resource that each job requires; one could
+define a “bandwidth” resource for these jobs, for
+example. HTCondor additionally allows configur-
+ing the job submissions to prevent more jobs to
+start at a given site once the sum of consumed
+resources by running jobs at that site reaches a
+certain threshold. Therefore, if one knows the total
+network capacity of each site hosting jobs, one can
+configure per-site job limits and prevent network
+saturation in an automated way.
+4.1 Future improvements
+A number of improvements to overall scalability
+and ease of use are possible. In addition to auto-
+matic job concurrency limits to prevent network
+saturation as previously described, we are explor-
+ing the possibility of compressing the data sent to
+the GPU server to reduce the overall bandwidth
+requirements. While a reduced payload would obvi-
+ously increase job concurrency limits, that must
+be balanced against the additional run time that
+
+Springer Nature 2021 LATEX template
+GPUaaS in ProtoDUNE data
+9
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+Average Started Events (s
+1)
+40
+50
+60
+70
+80
+90
+100
+Network Traffic (Gb/s)
+y = mx + b
+slope: 4.2 ± 0.2 (Gb)
+intercept: 44 ± 2 (Gb/s)
+Outbound Traffic vs #Events Started Per Second
+Oct. 5
+Oct. 6
+Fit w/ Uncertainty
+0
+50
+100
+150
+200
+250
+EmTrkMichelId Time(s)
+Fig. 9: The outbound network traffic vs. the average event start rate per second in 2-minute sliding
+windows, on October 5 and October 6. Data from each day is denoted with a different marker type. The
+color coding corresponds to the median EmTrkMichelId time for events in each sliding window. The linear
+fit to the traffic below 80 Gb/s indicates that each event sends 4.2 ± 0.2 Gb of outbound traffic, on top of
+44 ± 2 Gb/s of baseline traffic from non-ProtoDUNE sources.
+would be introduced in compressing and decom-
+pressing the data on the worker node and server,
+respectively. Another desirable area of improve-
+ment is in overall ease of use and human effort
+requirements. In the current setup we make use
+of the standard DUNE Production job submission
+infrastructure, which allows for a high degree of
+automated job submission, but due to the current
+nature of the cloud server it requires an authorized
+individual to manually instantiate the GPU infer-
+ence server before we submit jobs. Establishing a
+method of automatically instantiating the server
+at job submission time and automatically ramp-
+ing it down when the associated jobs are complete
+would avoid a clear possible failure point should
+no authorized individuals be available when the
+infrastructure is needed.
+A second option to study is to use several geo-
+graphically distributed inference servers instead of
+a single server, while also spreading the job work-
+load over a much broader range of sites. Expanding
+the site pool has the advantage of making it much
+less likely that any single site would get enough
+work assigned to saturate its external connectivity,
+and using several inference servers spread around
+the world would help to mitigate the potential
+problem of network latency becoming comparable
+to the inference time. The cost changes in this sce-
+nario (for example, the relative cost of three cloud
+servers versus a single server three times the size)
+
+Springer Nature 2021 LATEX template
+10
+GPUaaS in ProtoDUNE data
+must be assessed and taken into account. Another
+consideration is how the overall event processing
+times would change if the worker nodes were much
+more geographically diffuse than they were for this
+study. Since we stream the input data over the
+network, longer network paths between the worker
+nodes and input data sources may lead to the non-
+EmTrkMichelId portions of the event processing
+taking longer, which in turn affects the total event
+processing time. DUNE is able to distribute data
+to various storage elements around the world via
+the Rucio framework [29], and pre-placing the data
+of interest at storage elements close to the sites to
+be used for processing may mitigate such concerns,
+though it is not required.
+Another potential avenue is to use the GPU
+server infrastructure, but to use sites with GPUs
+available on the worker nodes, and run an inde-
+pendent server on each worker node. Several
+high-performance computing sites have built or are
+building clusters with readily available GPUs, and
+in some cases with multiple GPUs on each worker
+node, that would naturally lend themselves to such
+a setup. If the jobs run on worker nodes with local
+GPUs, external network connectivity limitations
+become unimportant for carrying out the infer-
+ence calculations. In fact, Triton™ allows the use
+of shared memory for direct data transfer between
+CPU and GPU when the GPU is local. While it
+may not be necessary to retain the server infras-
+tructure in these cases, the advantage of doing so
+is that the experiment software does not have to
+be modified to directly access the GPU, making
+it maximally portable and easier to maintain. We
+plan to conduct a similar study using this type of
+setup in the future.
+5 Summary
+We have reprocessed approximately seven million
+data events from the ProtoDUNE detector installed
+at CERN. We use an Nvidia Triton™ inference
+server hosted on the Google Cloud Platform to
+run the most computationally expensive step of
+the workflow on a GPU, speeding up the required
+processing time by more than a factor of two, even
+comparing to the fastest CPU runs. Running at
+a scale similar to that expected during regular
+ProtoDUNE-II and DUNE operations, we see the
+expected performance improvement until the net-
+work switch through which the majority of our jobs
+communicate becomes saturated. Despite that, the
+cloud infrastructure easily kept up with demand
+and demonstrates the viability of the GPUaaS
+model at a level sufficient for current and future
+high-energy physics experiments, as long as the
+job concurrency levels at each site respect the
+site’s network resource limits. With several promis-
+ing avenues of improvement to explore, we expect
+that this computing model will become even more
+capable and easier to use in the future.
+Author Contributions
+All authors contributed to the study conception
+and design. Material preparation, data collection
+and analysis were performed by Tejin Cai, Ken-
+neth Herner, and Tingjun Yang. The first draft of
+the manuscript was prepared by Tejin Cai, Maria
+Acosta Flechas, Kenneth Herner, Kevin Pedro,
+Nhan Tran, and Tingjun Yang. All authors read
+and approved the final manuscript.
+Acknowledgments
+We acknowledge the Fast Machine Learning collec-
+tive as an open community of multi-domain experts
+and collaborators. This community was important
+for the development of this project. We acknowl-
+edge the DUNE collaboration for providing the
+ProtoDUNE-SP code base and data samples. The
+
+Springer Nature 2021 LATEX template
+GPUaaS in ProtoDUNE data
+11
+analysis is enabled in part by the Digital Research
+Alliance of Canada.
+Declarations
+Competing Interests
+The authors have no competing interests to declare
+that are relevant to the content of this article.
+Data Availability
+The datasets generated during and/or analysed
+during the current study are available from the
+corresponding author on reasonable request.
+Funding
+MF, KH, BH, KP, NT, MW, and TY are sup-
+ported by Fermi Research Alliance, LLC under
+Contract No. DE-AC02-07CH11359 with the U.S.
+Department of Energy, Office of Science, Office of
+High Energy Physics. NT is partially supported
+by the U.S. Department of Energy Early Career
+Award. KP is partially supported by the High
+Velocity Artificial Intelligence grant as part of the
+U.S. Department of Energy High Energy Physics
+Computational HEP program. PH is supported
+by NSF grants #1934700, #193146. Cloud credits
+for this study were provided by Internet2 man-
+aged Exploring Cloud to accelerate Science (NSF
+grant PHY-190444). TC is supported by NSERC
+Canada.
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+Frascati Physics Series Vol. 73 (2022)
+LFC22: Strong interactions from QCD to new strong dynamics at LHC and Future Colliders
+August 29 - September 2, 2022
+Non commutativity between massless and soft limit in processes with heavy quarks
+Andrea Ghira
+Dipartimento di Fisica, Universit`a degli Studi di Genova and INFN, Via Dodecaneso 33, 16146, Italy
+Abstract
+Processes involving heavy quarks can be computed in perturbation theory in two different ways: we
+can adopt a scheme in which the mass of the quark is considered only as a regulator of the collinear
+divergences because of the fact that the hard scale of the process is far bigger or we can consider the
+quark as a massive particle. Each picture has its own advantages and drawbacks: we investigate the
+differences between the two approaches with particular attention to the soft logarithmic structure. We
+examine the origin of this difference, focusing on different processes involving the Higgs boson . Finally
+we perform the threshold resummation of the Higgs boson decay rate into a b¯b pair at NLL accuracy in
+the massive scheme.
+1
+Introduction
+Quarks appear in the Quantum Chromo-Dynamics (QCD) lagrangian in different species, named flavours.
+From the point of view of strong interactions, different flavours are distinguished purely on the basis of the
+value of their masses. It is therefore natural to classify quark flavours according to their masses, compared
+to ΛQCD ≃ 300MeV. The masses of up, down and strange quarks, relevant for ordinary matter, are much
+smaller than ΛQCD, and can be taken to be zero for most applications in high-energy physics, on the
+other hand charm (c) and especially bottom (b) are heavy according to this definition. Heavy-flavour
+production cross-sections can be calculated in perturbative QCD because the mass of the b and c quarks
+sets the value of the coupling in the perturbative region and regulates collinear singularities. In order to
+compute processes involving heavy flavour two main approaches are employed. In the so-called massive
+scheme, the final-state heavy quarks are considered massive particles and we can compute order by order
+in perturbation theory the scattering amplitude. Within this approach the kinematics is treated correctly
+arXiv:2301.03985v1  [hep-ph]  10 Jan 2023
+
+h(q)
+b(p1)
+¯b(p2)
+g(k)
+1
+h(q)
+b(p1)
+¯b(p2)
+g(k)
+1
+Figure 1: Real-emission contributions to the decay of the Higgs boson into a b¯b pair at O (αs).
+but calculations become cumbersome at higher and higher perturbative orders. Another drawback is that
+large mass logarithms which arise due to the fact that the mass of the heavy quark is far smaller than
+hard scale of the process spoil the convergence of the perturbative series. Therefore another framework
+is employed which is the so called massless scheme. In the massless scheme, we treat the mass of the
+particle only as a regulator of the collinear divergences. Consequently we do not have control on the
+kinematics outside the collinear region, i.e. we consider only radiation emitted at small angle. This
+approach exploits the factorization theorem: the differential cross section can be written as a convolution
+product of a process dependent function times a fragmentation function, which is process independent
+and fulfills a first order linear equation that allows us to resum the mass logarithms (DGLAP). The
+initial condition of the DGLAP evolution equation is set at a scale µ2
+0 ≃ m2
+c,b ≫ Λ2
+QCD and therefore it
+is in the perturbative domain and it can be determined by matching the factorisation theorem with the
+massive scheme. It was determined to NLO in QCD for the b quark fragmentation function in
+1, 2)
+and to NNLO in 3, 4). The initial condition is affected by soft logarithms, that should be resummed
+to all-orders too 5, 6). The main problem we want to focus on is that the structure of soft logarithms
+in the initial condition of the fragmentation function cannot be always recovered by the massless limit
+of a massive-framework calculation: this strongly depends both on the considered process and on the
+specific observable that is computed. We will show this particular behaviour using a simple process as
+an example which is the decay of a Higgs boson in a b¯b pair. Secondly, we want to derive a resummed
+expression of the differential decay rate at NLL accuracy that fully take into account the heavy quark
+mass and outline also in this case the non commutativity of the massless and soft limit.
+2
+Interplay between soft and massless limit in H → b¯b
+In order to explain the aforementioned non commutativity of the limits we focus on the decay of the
+Higgs boson at NLO keeping the mass of the quarks:
+h(q) → b(p1) + ¯b(p2) + g(k)
+p2
+1 = p2
+2 = m2, k2 = 0.
+(1)
+We compute the differential decay rate dΓ
+dx, where x = 2p1·q
+q2
+is the energy of the quark in the CoM reference
+frame, and we are interested in the small mass limit necessary for the massless scheme ( m2
+|q2| ≡ ξ → 0)
+
+and in the soft limit (x → 1). Performing the soft limit and the massless in two different orders we find:
+lim
+ξ→0 lim
+x→1
+1
+Γ0
+dΓ
+dx = −2αsCF
+π
+�1 + log ξ
+1 − x
++ O(ξ0) + O
+�
+(1 − x)0��
+,
+(2)
+lim
+x→1 lim
+ξ→0
+1
+Γ0
+dΓ
+dx = −αsCF
+π
+� log ξ
+1 − x + log(1 − x)
+1 − x
++ 7
+4
+1
+1 − x + O(ξ0) + O
+�
+(1 − x)0��
+,
+where Γ0 is the Born level decay rate:
+Γ0 =
+�
+2q2GF m2β3NC
+8π
+,
+β =
+�
+1 − 4ξ,
+(3)
+with GF is the Fermi constant. In order to analyze the logarithmic structure of the previous equation,
+we introduce the Mellin transformation:
+M{f(x)}(N) =
+� 1
+0
+xN−1f(x) dx
+(4)
+We notice that in the first case of equation (2) we have a mass logarithm multiplied by a soft one
+(
+1
+1−x ↔ log N in Mellin space) whereas in the second one we have an additional term which corresponds
+to a log2 N after the Mellin transformation. We note also that the overall coefficient is halved in the
+second limit, as if the log(1 − x) contribution in the second line of (2) is playing the role of a mass
+logarithm.
+We would like to provide a physical interpretation to this fact: a measurment of x fixes
+the invariant mass (p2 + k)2 = m2
+g¯b thus screening one of the collinear (mass) logs and preventing the
+anti-quark propagator to go on-shell. In order to analyse the actual origin of the double logarithms, we
+have to look at the quark propagator: if we integrate it over the angle between the gluon and the quark
+in the ⃗p2 + ⃗k = 0 frame we find
+� 1
+−1
+1
+1 − β1 cos θ dcos θ = log
+x2
+ξ(1 − x) + O
+�
+(1 − x)0�
+,
+β1 = x
+�
+1 − 4ξ/x2
+x − 2ξ
+,
+(5)
+where β1 is the quark velocity in that reference frame. In this limit, collinear logarithms appear in two
+distinct ways: as explicit logarithm of the quark mass m or as logarithms of 1 − x. This consideration
+brings us to formulate a more general statement about double soft logs in processes with heavy quark. We
+expect this behaviour to arise if look at a differential distribution which is directly related to the virtuality
+of one of the propagators, here m2
+g¯b. Let us consider the differential distribution in ¯x = (p1+p2)2
+q2
+→ 1 as
+k → 0. Performing an explicit calculation:
+lim
+ξ→0 lim
+¯x→1
+1
+Γ0
+dΓ
+d¯x = lim
+¯x→1 lim
+ξ→0
+1
+Γ0
+dΓ
+d¯x = −2αsCF
+π
+1 + log ξ
+1 − ¯x
++ O(ξ0) + O
+�
+(1 − x)0�
+,
+(6)
+In this case we have only a single logarithmic enhancement and the two limits commute.
+2.1
+Higgs Production and Higgs DIS
+We test our statement by studying other processes related by crossing symmetry to the Higgs boson
+decay, i.e Higgs boson production and Higgs DIS. In the Higgs production b(p1) + ¯b(p2) → h(q) + g(k),
+we are differential in τ = (p1+p2)2
+q2
+, which is not related to the virtuality of the propagators. In this case
+we find that the limits commute, as expected:
+lim
+τ→1 lim
+ξ→0
+1
+σ0
+dσ
+dτ = lim
+ξ→0 lim
+τ→1
+1
+σ0
+dσ
+dτ = −2αsCF
+π
+1 + log ξ
+1 − τ
++ O(ξ0) + O
+�
+(1 − τ)0�
+,
+(7)
+σ0 =
+√
+2GF m2βπNC
+18s
+.
+
+Finally we study the differential distribution
+dσ
+dxB with xB =
+−q2
+2p1·q for the real emission corrections to the
+process b(p1) + h(q) → b(p2) + g(k). Due to the fact that xB is related to the virtuality of one of the
+propagator we expect that the limit do not commute. Indeed we find:
+lim
+xB→1 lim
+ξ→0
+1
+¯σ0
+dσ
+dxB
+= −αsCF
+π
+� log ξ
+1 − xB
++ log(1 − xB)
+1 − xB
++ 7
+4
+1
+1 − xB
++ O(ξ0) + O
+�
+(1 − xB)0��
+,
+(8)
+lim
+ξ→0 lim
+xB→1
+1
+¯σ0
+dσ
+dxB
+= −2αsCF
+π
+1 + log ξ
+1 − xB
++ +O(ξ0) + O
+�
+(1 − xB)0�
+,
+¯σ0 = π
+√
+2GF m2NCη
+−3q2
+,
+η =
+�
+1 + 4ξ.
+3
+Soft Resummation in the Massive Scheme
+In this section we want to give an explicit expression for the all-order soft resummation of the Higgs decay
+rate in a b¯b pair at NLL accuracy in the massive scheme. Since we look at the differential distribution
+over x, we are in class of process with the so called single-particle inclusive kinematics (see 7)). The
+main result of
+7) is that the resummed expression can be factorized as a product of a soft function
+times a hard function times a jet function for every massles particle n the final state. In our case the
+resummation formula simplifies considerably there are not massless particles. The resummed result of
+7) at NLL, adapted to the process we are considering, reads1
+�Γ(N, ξ) =
+�
+1 + αs
+π C(1)(ξ) + O
+�
+α2
+s
+��
+e
+−2
+� 1
+1/ ¯
+N
+dz
+z
+�
+αs(z2q2)
+π
+γ(0)
+soft(β)+
+�
+αs(z2q2)
+π
+�2
+γ(1)
+soft(β)+O(α3
+s)
+�
++ O
+� 1
+N
+�
+,
+(9)
+with ¯N = NeγE and γsoft the massive soft anomalous dimension. To this logarithmic accuracy we need
+the two loops expression of the running coupling, the coefficients γ(0)
+soft, γ(1)
+soft and C(1). The first order soft
+anomalous dimension can be obtained from the calculation of one gluon emission in the eikonal limit:
+γ(0)
+soft(β) = CF
+�1 + β2
+2β
+log
+�1 + β
+1 − β
+�
+− 1
+�
+,
+(10)
+while the second order was presented in 8)2:
+γ(1)
+soft =
+�K
+2 + CA
+2
+�
+−1
+3 log2 1 − β
+1 + β + log 1 − β
+1 + β − ζ2
+�
++(1 + β2)
+4β
+CA
+�
+Li2
+�(1 − β)2
+(1 + β)2
+�
++ 1
+3 log2 1 − β
+1 + β + ζ2
+��
+γ(0)
+soft(β)
++ CFCA
+�1
+2 + 1
+2 log 1 − β
+1 + β + 1
+3 log2 1 − β
+1 + β − (1 + β2)2
+8β2
+�
+−Li3
+�(1 − β)2
+(1 + β)2
+�
++ ζ3
+�
+− (1 + β2)
+2β
+�
+log 1 − β
+1 + β log (1 + β)2
+4β
+− 1
+6 log2 1 − β
+1 + β − Li2
+�(1 − β)2
+(1 + β)2
+���
+,
+(11)
+1We are not so sure about the argument of the running coupling, since in 7) αs(z2q2) is used, on the
+other hand it seems that in 8) αs(z2m2) is used.
+2It is worth to mention that there is a mismatch in the literature between 8) and 9)
+
+with K = CA
+� 67
+18 − ζ2
+�
+− 5nf
+9 . The coefficient C(1) is instead process-dependent, as it receives contri-
+butions from both the end-point of the real emission and from the virtual corrections (computed in the
+on-shell scheme). Writing the real emission differential decay rate as:
+dΓ(R)
+dx
+= αsCF
+π
+Γ(d)
+0
+fε
+�
+x, ξ, q2
+µ2
+�
+(1 − x)1+2ϵ ,
+Γ(d)
+0
+= Γ0
+π
+5−d
+2
+2d−3Γ
+� d−1
+2
+�
+� 4µ2
+q2β2
+� 4−d
+2
+,
+(12)
+the coefficient C(1) can be determined using the fact that virtual corrections are proportional to δ(1 − x)
+and the identity between distributions:
+fε
+�
+x, ξ, q2
+µ2
+�
+(1 − x)1+2ε = δ(1 − x)
+�
+−f0(1, ξ)
+2ε
++ f0(1, ξ) log(1 − 2
+�
+ξ) − 1
+2
+d
+dεfε
+�
+1, ξ, q2
+µ2
+� ���
+ε=0
+�
++ f0(x, ξ)
+(1 − x)+
++ O(ε) .
+(13)
+Summing up virtual and real contributions we obtain:
+C(1)(ξ) = CF
+2
+�
+− 2γ(0)
+soft(β)
+CF
+�
+−2 log
+�
+1 −
+�
+1 − β2
+�
++ log m2
+q2 + log
+�1 − β2
+4
+�
++ 1
+�
+− 2
++ 2L(β)
+�1 − β2
+β
+�
++ 1 + β2
+β
+�
+1
+2L(β) log
+�1 − β2
+4
+�
++ 2L(β)(1 − log β) + 2Li2
+�1 − β
+1 + β
+�
++ L(β)2 + L(β) log 1 − β
+2
++ 2
+3π2 − 1
+2
+�
+Li2
+�
+4β
+(1 + β)2
+�
+− Li2
+�
+−4β
+(1 − β)2
+�� ��
+,
+(14)
+with L(β) = log
+�
+1+β
+1−β
+�
+. We note that the non commutativity of the soft and massless limits has conse-
+quences for the resummed expression in the massive scheme: In the small ξ limit we find:
+αsC(1)(ξ) = αsCF
+�1
+2 log2 ξ + log ξ + O(ξ0)
+�
+.
+We have a double log of the mass in disagreement with DGLAP evolution equation. The problem is that
+equation (13) does not hold if we perform the massless limit because in this limit f0(1, ξ) is not defined.
+In a certain way we can say that double mass logarithms in the soft limit of the massive calculation and
+double soft logarithms of the massless scheme are connected. A well defined expression in the massless
+limit can be obtained rewriting the differential decay rate as:
+1
+Γ0
+dΓ
+dx = δ(1 − x) + αs
+π
+�
+CF
+�f0(x, ξ)
+1 − x
+�
++
++ A(ξ) δ(1 − x)
+�
+,
+(15)
+The delta coefficient has an expected behaviour for ξ → 0
+A(ξ) = CF
+3
+2 log ξ + O(ξ0).
+(16)
+4
+Conclusions
+We have considered observables with different kinematics in processes involving heavy quarks, and in all
+processes we have computed NLO corrections taking into account the mass dependence of the square
+amplitude.
+We have underlined that soft and massless do not always commute, in particular in the
+
+massless limit the structure of the distributions can radically change because of the presence of double
+logs of N. We have traced back the origin of this particular behaviour to the interplay between the
+observable we are computing and the fermionic propagators in the scattering amplitudes. Finally, we
+have focused on the massive scheme resummation of the process H → b¯b in the soft limit and we
+have found that within this approach double logarithms of the mass may appear, and the origin of this
+surprising behaviour can be lead back again to the non commutativity between the large N and small
+mass limit.
+An interesting phenomenological study, in the context of heavy-quark calculations, would be com-
+bine the massive scheme with the massless one where also soft logarithms are resummed. The merging
+of the two becomes far from trivial because of the lack of commutativity of the limits. One would like
+to design an all-order matching scheme that takes into account both the different logarithmic behaviour
+that arises in the two cases.
+5
+Acknowledgements
+We thank Simone Marzani and Giovanni Ridolfi for the aid in the drafting of this proceeding, which is
+entirely based on 10).
+References
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+[arXiv:2207.13567 [hep-ph]].
+

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+page_content=' 2022  Non commutativity between massless and soft limit in processes with heavy quarks  Andrea Ghira  Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Universit`a degli Studi di Genova and INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Via Dodecaneso 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' 16146,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Italy  Abstract  Processes involving heavy quarks can be computed in perturbation theory in two different ways: we  can adopt a scheme in which the mass of the quark is considered only as a regulator of the collinear  divergences because of the fact that the hard scale of the process is far bigger or we can consider the  quark as a massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Each picture has its own advantages and drawbacks: we investigate the  differences between the two approaches with particular attention to the soft logarithmic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' We  examine the origin of this difference, focusing on different processes involving the Higgs boson .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Finally  we perform the threshold resummation of the Higgs boson decay rate into a b¯b pair at NLL accuracy in  the massive scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  1  Introduction  Quarks appear in the Quantum Chromo-Dynamics (QCD) lagrangian in different species, named flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  From the point of view of strong interactions, different flavours are distinguished purely on the basis of the  value of their masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' It is therefore natural to classify quark flavours according to their masses, compared  to ΛQCD ≃ 300MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The masses of up, down and strange quarks, relevant for ordinary matter, are much  smaller than ΛQCD, and can be taken to be zero for most applications in high-energy physics, on the  other hand charm (c) and especially bottom (b) are heavy according to this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Heavy-flavour  production cross-sections can be calculated in perturbative QCD because the mass of the b and c quarks  sets the value of the coupling in the perturbative region and regulates collinear singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' In order to  compute processes involving heavy flavour two main approaches are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' In the so-called massive  scheme, the final-state heavy quarks are considered massive particles and we can compute order by order  in perturbation theory the scattering amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Within this approach the kinematics is treated correctly  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='03985v1  [hep-ph]  10 Jan 2023  h(q)  b(p1)  ¯b(p2)  g(k)  1  h(q)  b(p1)  ¯b(p2)  g(k)  1  Figure 1: Real-emission contributions to the decay of the Higgs boson into a b¯b pair at O (αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  but calculations become cumbersome at higher and higher perturbative orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Another drawback is that  large mass logarithms which arise due to the fact that the mass of the heavy quark is far smaller than  hard scale of the process spoil the convergence of the perturbative series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Therefore another framework  is employed which is the so called massless scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' In the massless scheme, we treat the mass of the  particle only as a regulator of the collinear divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Consequently we do not have control on the  kinematics outside the collinear region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' we consider only radiation emitted at small angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' This  approach exploits the factorization theorem: the differential cross section can be written as a convolution  product of a process dependent function times a fragmentation function, which is process independent  and fulfills a first order linear equation that allows us to resum the mass logarithms (DGLAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The  initial condition of the DGLAP evolution equation is set at a scale µ2  0 ≃ m2  c,b ≫ Λ2  QCD and therefore it  is in the perturbative domain and it can be determined by matching the factorisation theorem with the  massive scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' It was determined to NLO in QCD for the b quark fragmentation function in  1, 2)  and to NNLO in 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The initial condition is affected by soft logarithms, that should be resummed  to all-orders too 5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The main problem we want to focus on is that the structure of soft logarithms  in the initial condition of the fragmentation function cannot be always recovered by the massless limit  of a massive-framework calculation: this strongly depends both on the considered process and on the  specific observable that is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' We will show this particular behaviour using a simple process as  an example which is the decay of a Higgs boson in a b¯b pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Secondly, we want to derive a resummed  expression of the differential decay rate at NLL accuracy that fully take into account the heavy quark  mass and outline also in this case the non commutativity of the massless and soft limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  2  Interplay between soft and massless limit in H → b¯b  In order to explain the aforementioned non commutativity of the limits we focus on the decay of the  Higgs boson at NLO keeping the mass of the quarks:  h(q) → b(p1) + ¯b(p2) + g(k)  p2  1 = p2  2 = m2, k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  (1)  We compute the differential decay rate dΓ  dx, where x = 2p1·q  q2  is the energy of the quark in the CoM reference  frame, and we are interested in the small mass limit necessary for the massless scheme ( m2  |q2| ≡ ξ → 0)  and in the soft limit (x → 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Performing the soft limit and the massless in two different orders we find:  lim  ξ→0 lim  x→1  1  Γ0  dΓ  dx = −2αsCF  π  �1 + log ξ  1 − x  + O(ξ0) + O  �  (1 − x)0��  ,  (2)  lim  x→1 lim  ξ→0  1  Γ0  dΓ  dx = −αsCF  π  � log ξ  1 − x + log(1 − x)  1 − x  + 7  4  1  1 − x + O(ξ0) + O  �  (1 − x)0��  ,  where Γ0 is the Born level decay rate:  Γ0 =  �  2q2GF m2β3NC  8π  ,  β =  �  1 − 4ξ,  (3)  with GF is the Fermi constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' In order to analyze the logarithmic structure of the previous equation,  we introduce the Mellin transformation:  M{f(x)}(N) =  � 1  0  xN−1f(x) dx  (4)  We notice that in the first case of equation (2) we have a mass logarithm multiplied by a soft one  (  1  1−x ↔ log N in Mellin space) whereas in the second one we have an additional term which corresponds  to a log2 N after the Mellin transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' We note also that the overall coefficient is halved in the  second limit, as if the log(1 − x) contribution in the second line of (2) is playing the role of a mass  logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  We would like to provide a physical interpretation to this fact: a measurment of x fixes  the invariant mass (p2 + k)2 = m2  g¯b thus screening one of the collinear (mass) logs and preventing the  anti-quark propagator to go on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' In order to analyse the actual origin of the double logarithms, we  have to look at the quark propagator: if we integrate it over the angle between the gluon and the quark  in the ⃗p2 + ⃗k = 0 frame we find  � 1  −1  1  1 − β1 cos θ dcos θ = log  x2  ξ(1 − x) + O  �  (1 − x)0�  ,  β1 = x  �  1 − 4ξ/x2  x − 2ξ  ,  (5)  where β1 is the quark velocity in that reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' In this limit, collinear logarithms appear in two  distinct ways: as explicit logarithm of the quark mass m or as logarithms of 1 − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' This consideration  brings us to formulate a more general statement about double soft logs in processes with heavy quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' We  expect this behaviour to arise if look at a differential distribution which is directly related to the virtuality  of one of the propagators, here m2  g¯b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Let us consider the differential distribution in ¯x = (p1+p2)2  q2  → 1 as  k → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Performing an explicit calculation:  lim  ξ→0 lim  ¯x→1  1  Γ0  dΓ  d¯x = lim  ¯x→1 lim  ξ→0  1  Γ0  dΓ  d¯x = −2αsCF  π  1 + log ξ  1 − ¯x  + O(ξ0) + O  �  (1 − x)0�  ,  (6)  In this case we have only a single logarithmic enhancement and the two limits commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='1  Higgs Production and Higgs DIS  We test our statement by studying other processes related by crossing symmetry to the Higgs boson  decay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='e Higgs boson production and Higgs DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' In the Higgs production b(p1) + ¯b(p2) → h(q) + g(k),  we are differential in τ = (p1+p2)2  q2  , which is not related to the virtuality of the propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' In this case  we find that the limits commute, as expected:  lim  τ→1 lim  ξ→0  1  σ0  dσ  dτ = lim  ξ→0 lim  τ→1  1  σ0  dσ  dτ = −2αsCF  π  1 + log ξ  1 − τ  + O(ξ0) + O  �  (1 − τ)0�  ,  (7)  σ0 =  √  2GF m2βπNC  18s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  Finally we study the differential distribution  dσ  dxB with xB =  −q2  2p1·q for the real emission corrections to the  process b(p1) + h(q) → b(p2) + g(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Due to the fact that xB is related to the virtuality of one of the  propagator we expect that the limit do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Indeed we find:  lim  xB→1 lim  ξ→0  1  ¯σ0  dσ  dxB  = −αsCF  π  � log ξ  1 − xB  + log(1 − xB)  1 − xB  + 7  4  1  1 − xB  + O(ξ0) + O  �  (1 − xB)0��  ,  (8)  lim  ξ→0 lim  xB→1  1  ¯σ0  dσ  dxB  = −2αsCF  π  1 + log ξ  1 − xB  + +O(ξ0) + O  �  (1 − xB)0�  ,  ¯σ0 = π  √  2GF m2NCη  −3q2  ,  η =  �  1 + 4ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  3  Soft Resummation in the Massive Scheme  In this section we want to give an explicit expression for the all-order soft resummation of the Higgs decay  rate in a b¯b pair at NLL accuracy in the massive scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Since we look at the differential distribution  over x, we are in class of process with the so called single-particle inclusive kinematics (see 7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The  main result of  7) is that the resummed expression can be factorized as a product of a soft function  times a hard function times a jet function for every massles particle n the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' In our case the  resummation formula simplifies considerably there are not massless particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The resummed result of  7) at NLL, adapted to the process we are considering, reads1  �Γ(N, ξ) =  �  1 + αs  π C(1)(ξ) + O  �  α2  s  ��  e  −2  � 1  1/ ¯  N  dz  z  �  αs(z2q2)  π  γ(0)  soft(β)+  �  αs(z2q2)  π  �2  γ(1)  soft(β)+O(α3  s)  �  + O  � 1  N  �  ,  (9)  with ¯N = NeγE and γsoft the massive soft anomalous dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' To this logarithmic accuracy we need  the two loops expression of the running coupling, the coefficients γ(0)  soft, γ(1)  soft and C(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The first order soft  anomalous dimension can be obtained from the calculation of one gluon emission in the eikonal limit:  γ(0)  soft(β) = CF  �1 + β2  2β  log  �1 + β  1 − β  �  − 1  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  (10)  while the second order was presented in 8)2:  γ(1)  soft =  �K  2 + CA  2  �  −1  3 log2 1 − β  1 + β + log 1 − β  1 + β − ζ2  �  +(1 + β2)  4β  CA  �  Li2  �(1 − β)2  (1 + β)2  �  + 1  3 log2 1 − β  1 + β + ζ2  ��  γ(0)  soft(β)  + CFCA  �1  2 + 1  2 log 1 − β  1 + β + 1  3 log2 1 − β  1 + β − (1 + β2)2  8β2  �  −Li3  �(1 − β)2  (1 + β)2  �  + ζ3  �  − (1 + β2)  2β  �  log 1 − β  1 + β log (1 + β)2  4β  − 1  6 log2 1 − β  1 + β − Li2  �(1 − β)2  (1 + β)2  ���  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  (11)  1We are not so sure about the argument of the running coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' since in 7) αs(z2q2) is used,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' on the  other hand it seems that in 8) αs(z2m2) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  2It is worth to mention that there is a mismatch in the literature between 8) and 9)  with K = CA  � 67  18 − ζ2  �  − 5nf  9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The coefficient C(1) is instead process-dependent, as it receives contri-  butions from both the end-point of the real emission and from the virtual corrections (computed in the  on-shell scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Writing the real emission differential decay rate as:  dΓ(R)  dx  = αsCF  π  Γ(d)  0  fε  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' q2  µ2  �  (1 − x)1+2ϵ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  Γ(d)  0  = Γ0  π  5−d  2  2d−3Γ  � d−1  2  �  � 4µ2  q2β2  � 4−d  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  (12)  the coefficient C(1) can be determined using the fact that virtual corrections are proportional to δ(1 − x)  and the identity between distributions:  fε  �  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' q2  µ2  �  (1 − x)1+2ε = δ(1 − x)  �  −f0(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' ξ)  2ε  + f0(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' ξ) log(1 − 2  �  ξ) − 1  2  d  dεfε  �  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' q2  µ2  � ���  ε=0  �  + f0(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' ξ)  (1 − x)+  + O(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  (13)  Summing up virtual and real contributions we obtain:  C(1)(ξ) = CF  2  �  − 2γ(0)  soft(β)  CF  �  −2 log  �  1 −  �  1 − β2  �  + log m2  q2 + log  �1 − β2  4  �  + 1  �  − 2  + 2L(β)  �1 − β2  β  �  + 1 + β2  β  �  1  2L(β) log  �1 − β2  4  �  + 2L(β)(1 − log β) + 2Li2  �1 − β  1 + β  �  + L(β)2 + L(β) log 1 − β  2  + 2  3π2 − 1  2  �  Li2  �  4β  (1 + β)2  �  − Li2  �  −4β  (1 − β)2  �� ��  ,  (14)  with L(β) = log  �  1+β  1−β  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' We note that the non commutativity of the soft and massless limits has conse-  quences for the resummed expression in the massive scheme: In the small ξ limit we find:  αsC(1)(ξ) = αsCF  �1  2 log2 ξ + log ξ + O(ξ0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  We have a double log of the mass in disagreement with DGLAP evolution equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The problem is that  equation (13) does not hold if we perform the massless limit because in this limit f0(1, ξ) is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  In a certain way we can say that double mass logarithms in the soft limit of the massive calculation and  double soft logarithms of the massless scheme are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' A well defined expression in the massless  limit can be obtained rewriting the differential decay rate as:  1  Γ0  dΓ  dx = δ(1 − x) + αs  π  �  CF  �f0(x, ξ)  1 − x  �  +  + A(ξ) δ(1 − x)  �  ,  (15)  The delta coefficient has an expected behaviour for ξ → 0  A(ξ) = CF  3  2 log ξ + O(ξ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  (16)  4  Conclusions  We have considered observables with different kinematics in processes involving heavy quarks, and in all  processes we have computed NLO corrections taking into account the mass dependence of the square  amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  We have underlined that soft and massless do not always commute, in particular in the  massless limit the structure of the distributions can radically change because of the presence of double  logs of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' We have traced back the origin of this particular behaviour to the interplay between the  observable we are computing and the fermionic propagators in the scattering amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' Finally, we  have focused on the massive scheme resummation of the process H → b¯b in the soft limit and we  have found that within this approach double logarithms of the mass may appear, and the origin of this  surprising behaviour can be lead back again to the non commutativity between the large N and small  mass limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  An interesting phenomenological study, in the context of heavy-quark calculations, would be com-  bine the massive scheme with the massless one where also soft logarithms are resummed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' The merging  of the two becomes far from trivial because of the lack of commutativity of the limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content=' One would like  to design an all-order matching scheme that takes into account both the different logarithmic behaviour  that arises in the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
+page_content='  5  Acknowledgements  We thank Simone Marzani and Giovanni Ridolfi for the aid in the drafting of this proceeding, which is  entirely based on 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'}
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+January 13, 2023
+Tidal deformations of a binary system
+induced by an external Kerr black hole
+Filippo Camilloni†, Gianluca Grignani†, Troels Harmark‡,
+Roberto Oliveri∗, Marta Orselli† ‡, Daniele Pica† ‡
+† Dipartimento di Fisica e Geologia, Universit`a di Perugia, I.N.F.N. Sezione di Perugia,
+Via Pascoli, I-06123 Perugia, Italy
+‡ Niels Bohr Institute, Copenhagen University,
+Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
+∗ LUTH, Laboratoire Univers et Th´eories, Observatoire de Paris,
+CNRS, Universit´e PSL, Universit´e Paris Cit´e,
+5 place Jules Janssen, 92190 Meudon, France
+Abstract
+The dynamics of a binary system moving in the background of a black hole is affected by
+tidal forces. In this work, for the Kerr black hole, we derive the electric and magnetic
+tidal moments at quadrupole order, where the latter are computed for the first time in
+full generality.
+We make use of these moments in the scenario of a hierarchical triple
+system made of a Kerr black hole and an extreme-mass ratio binary system consisting of
+a Schwarzschild black hole and a test particle. We study how the secular dynamics of
+the test particle in the binary system is distorted by the presence of tidal forces from a
+much larger Kerr black hole. Our treatment includes strong gravitational effects beyond
+the post-Newtonian approximation both for the binary system and for the tidal forces since
+the binary system is allowed to be close to the event horizon of the Kerr black hole. We
+compute the shifts in the physical quantities for the secular dynamics of the test particle
+and show that they are gauge-invariant.
+In particular, we apply our formalism to the
+innermost stable circular orbit for the test particle and to the case of the photon sphere.
+Our results are relevant for the astrophysical situation in which the binary system is in the
+vicinity of a supermassive black hole.
+arXiv:2301.04879v1  [gr-qc]  12 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Tidal moments induced by a Kerr black hole
+3
+2.1
+Carter’s tetrad
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.2
+Marck’s tetrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.3
+Tidal tensors
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.4
+Electric and magnetic quadrupole moments . . . . . . . . . . . . . . . . . . . . .
+7
+3
+Hierarchical triple system
+8
+3.1
+Tidally deformed Schwarzschild spacetime
+. . . . . . . . . . . . . . . . . . . . .
+9
+3.2
+Tidal moments in spherical coordinates . . . . . . . . . . . . . . . . . . . . . . .
+10
+4
+Secular dynamics of binary system
+13
+4.1
+Secular Hamiltonian of test particle in binary system . . . . . . . . . . . . . . .
+13
+4.2
+Special case of circular equatorial geodesic in Kerr background . . . . . . . . . .
+15
+5
+Secular shifts for ISCO and photon sphere
+16
+5.1
+Gauge invariance of secular observables . . . . . . . . . . . . . . . . . . . . . . .
+18
+5.2
+Tidal effects around the ISCO orbit . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+5.3
+Tidal effects around the photon sphere . . . . . . . . . . . . . . . . . . . . . . .
+20
+6
+Conclusions and outlook
+21
+1
+Introduction
+The detection of gravitational waves from coalescing binary systems by the LIGO-Virgo-Kagra
+collaboration [1–3] has unsealed a new powerful and fascinating way of exploring our universe in
+a regime of strong gravitational field. This has made it increasingly relevant to investigate new
+types of strong gravitational phenomena analytically, to prepare for future experimental results.
+Indeed, with the next generation detectors such as the ground-based Einstein Telescope [4]
+and Cosmic Explorer [5], as well as the space-based LISA [6] and TianQin [7], the sensitivity
+and frequency band will be greatly expanded. This will make it possible to use black hole binary
+systems also as probes of their surrounding environment (see Ref. [8] for a comprehensive review).
+Examples of the effect of the environment include the presence of various types of energy and
+matter, such as an accretion disc [9–11] or dark matter [12–18]. Another example, relevant for
+this paper, is the presence of a third body, such as a nearby supermassive black hole [19–26]
+bound to the binary system.
+Moreover, the expansion in sensitivity and frequency band will make it possible to detect
+signals from new types of sources, such as for example extreme-mass-ratio (EMR) inspiraling
+systems. Among these systems, the ones that will typically be detectable in the LISA band
+[27, 28], are made of a stellar mass compact object of mass m and a black hole with a much
+larger mass M ≫ m, with mass ratios m/M ranging from 10−4 to 10−6.
+In this paper we are interested in the dynamical effects of having a binary black hole system
+immersed in a curved background spacetime. To access a scenario that at the same time is
+realistic, has strong gravitational effects included, and can be treated analytically, we consider
+the case of an EMR binary system, i.e. a black hole and a test particle, in the background of a
+third, larger black hole, affecting the binary system through tidal forces.
+We take the curved background spacetime to be the general case of a Kerr black hole of
+mass M∗. Instead the EMR binary system will consist of a Schwarzschild black hole of mass M
+1
+
+with a test particle of mass m, enabling us to use the tidally deformed Schwarzschild metric of
+Refs. [29,30] to describe the EMR binary system. For the test particle we consider it to move
+on a geodesic, neglecting higher order effects in m/M such as the self-force. As the size of the
+binary system will be set by the scale M, we need M ≪ R where R is the curvature length
+scale set by the background Kerr black hole. This ensures that the effects of the background
+can be described through tidal forces, with the condition M ≪ R known as the small-tide
+approximation [30].
+We will consider the quadrupole approximation to the tidal forces, being the leading order
+in M/R. This means we can consider the EMR binary as moving on a geodesic of the Kerr
+black hole geometry. A particularly interesting regime is when M∗ ≫ M thus corresponding
+to a hierarchical three body system. In this case, the binary system can be close to the event
+horizon of the Kerr black hole, even while the small-tide approximation is respected.
+Our setup is inspired by that of Ref. [25], while at the same time being a significant ex-
+tension. Their setup was restricted to a Schwarzschild black hole as the third body, and the
+EMR binary system was assumed to be at a large distance. Instead, we are able to consider the
+strong gravitational effects on the binary system when it moves in close vicinity to a Kerr black
+hole. This also means that we need to consider more carefully the relative orientation of the
+EMR binary system relative to the Kerr black hole. This is accomplished by introducing two
+independent rotation angles. Moreover, it is important to note that in our setup we are able to
+capture strong gravitational effects arising from curved spacetime, in contrast with most of the
+extensive literature on three body systems [31–35], as those works employ the approximation
+that all three bodies are small relative to their mutual distances.
+A significant part of our paper concerns the careful computation of the general quadrupole
+tidal forces due to the Kerr black hole, as these constitute the forces that can affect the binary
+system in our setup. These forces are described by the tidal tensors Cij and Cijk. The rank-2
+tidal tensors Cij were previously computed for a generic value of the Kerr angle ˆθ in a seminal
+paper by Marck [36], where he constructed the orthonormal tetrad that is parallel-transported
+along an arbitrary time-like geodesic in the Kerr spacetime. From the rank-2 tidal tensors Cij
+one can then compute the “electric” quadrupole moments Eij, which can be considered as “mass
+moments” produced by gravitational forces external to a certain region.
+A primary result of this paper, is the derivation of the general form of the rank-3 tidal
+tensors Cijk for all values of the angle ˆθ in the Kerr spacetime. This generalizes the results of
+Ref. [37] (later confirmed in Ref. [38]), where the tidal tensors Cijk were obtained only for the
+specific value ˆθ = π/2, namely in the equatorial plane of the Kerr spacetime. From the rank-3
+tidal tensors Cijk we moreover derive the “magnetic” quadrupole moments Bij, which can be
+considered as external “current moments” and generate velocity-dependent tidal forces on test
+bodies. This is another original result of this paper.
+We apply these tidal electric and magnetic quadrupole moments to the case described above,
+with an EMR binary system following a geodesic in the Kerr background. The effects induced by
+the tidal fields can be studied by computing the Hamiltonian of a test particle (the object of mass
+m) in the tidally deformed Schwarzschild spacetime. Specifically, starting from a circular orbit
+in the unperturbed Schwarzschild spacetime, we find that the geodesics in the tidally deformed
+spacetime acquire a small eccentricity proportional to the deformation parameter. The quasi-
+circular dynamics in the perturbed spacetime is governed by a secular Hamiltonian, which keeps
+into account the effects of the tidal deformation on circular orbits. It can be written as a sum of
+the unperturbed Hamiltonian in the Schwarzschild spacetime and an interaction term of order
+∼ M/M∗, which allows for example to compute perturbatively the effects of tides on the location
+and properties of the Innermost Stable Circular Orbit (ISCO) and of the photon sphere.
+Using the tidal moments we computed, we derive the effects of tides on the frequency, radius,
+energy and angular momentum of the ISCO of the binary system, by computing the shifts
+2
+
+induced by the small tides on these physical quantities. 1
+The case of tides generated by a
+Schwarzschild black hole was studied in Ref. [25, 40]. Here we derive the shifts in the case of
+tides induced by the Kerr geometry and we derive the expression of the parameter η entering
+these shifts. We find that η depends on the spin of the Kerr black hole, the Carter constant
+K, the Kerr angle ˆθ and the Boyer-Lindquist radius ˆr at which the black hole of mass M is
+located in the Kerr spacetime geometry. More generally, our result does not rely on the specific
+nature of the third body responsible for the tides. Indeed, the tidal parameter η in the secular
+Hamiltonian is shown to be proportional to the secular average of the scalar part of the electric
+tidal moment. This result holds in the quadrupole and in the secular approximation. We provide
+an expression for η in terms of arbitrary tides and specialize it to the case of a Kerr black hole.
+The paper is organized as follows. In Sec 2, we compute the tidal moments induced by a Kerr
+black hole. Following Ref. [36], we first recover the already known expression for the electric
+tidal moments and then we derive the most general expressions for the magnetic components of
+the tidal moments, generalising the computation done in Ref. [37]. In Sec. 3, we introduce the
+hierarchical triple system that we analyse in this paper. We write down the metric for a tidally
+deformed Schwarzschild black hole up to the quadrupole order. We moreover write down the
+explicit expression for the quadrupole electric and magnetic moments and we introduce the Euler
+angles which allow us to study any possible orientation of the binary system. In Sec. 4, we focus
+on the secular dynamics of the binary system in order to understand how the parameters which
+specify the orbits of the test particle around the Schwarzschild black hole, such as energy and
+angular momentum, are shifted by the tidal fields. In Sec. 5, we apply the results of the previous
+sections to the case in which the test particle is moving along the ISCO of the Schwarzschild
+black hole.
+In addition, we extend our computation also to the case of a massless particle
+studying how the photon sphere is deformed by the tidal fields. We furthermore discuss the
+gauge invariance of our results. Finally, Sec 6 contains our concluding remarks.
+Notation:
+Throughout this paper Greek indices run from 0 to 3, Latin lower-case indices
+(i, j, k, ...) run from 1 to 3, Latin upper-case indices (A, B, C, ...) label spherical coordinates.
+Indices in round brackets ((a), (b), (c), ...) label tensor components in the Carter’s tetrad. Sym-
+metric and tracefree (STF) tensors are denoted by angular brackets over their indices, e.g.,
+T⟨ij⟩ = T(ij) − 1
+3δijTklδkl. Hatted coordinates (ˆt, ˆr, ˆθ, ˆφ) are employed for the Kerr spacetime.
+Schwarzschild coordinates, used for the binary system, are instead denoted as (t, r, θ, φ). We use
+geometrized units with G = c = 1 and the Minkowski metric signature is η = diag(−1, 1, 1, 1).
+2
+Tidal moments induced by a Kerr black hole
+In this section we derive the general quadrupole tidal moments for geodesic motion around a
+Kerr black hole which we will use in Sections 3-5. In Sec. 2.1 we define the Carter’s tetrad,
+in terms of which the curvature tensor simplifies. In Sec. 2.2 we present an alternative inertial
+frame [36], parallel-transported along a generic geodesic in the Kerr spacetime, here called the
+Marck’s tetrad.
+This is the most suitable reference frame in which it is possible to extract
+analytic information concerning the tidal effects induced by the Kerr geometry on a system
+moving along its geodesics. The tidal effects are encoded in the rank-2 and rank-3 tidal tensors
+and in the set of electric and magnetic tidal moments, explicitly given in Sec. 2.3 and 2.4 at the
+quadrupole order. The expressions of the rank-3 tidal tensor and of the magnetic quadrupole
+moments outside the Kerr equatorial plane are derived here for the first time.
+1See Ref. [39] for similar treatments in the context of the self-force approximation.
+3
+
+2.1
+Carter’s tetrad
+The Kerr metric for a rotating black hole of mass M∗ and spin J∗, in Boyer-Lindquist (BL)
+coordinates ˆxµ = (ˆt, ˆr, ˆθ, ˆφ) takes the form
+dˆs2 = −
+�
+1 − 2M∗ˆr
+Σ
+�
+dˆt2 − 4M∗ˆr
+Σ
+a sin2 ˆθ dˆt dˆφ + A
+Σ sin2 ˆθ dˆφ2 + Σ
+∆dˆr2 + Σdˆθ2 ,
+(2.1)
+where a = J∗/M∗ is the specific angular momentum and
+Σ = ˆr2 + a2 cos2 ˆθ,
+∆ = ˆr2 − 2M∗ˆr + a2,
+A = (ˆr2 + a2)2 − a2∆ sin2 ˆθ .
+(2.2)
+We are interested in considering time-like geodesics around a Kerr black hole, specified by
+three constants of motion: the energy per unit mass ˆE, the angular momentum per unit mass
+ˆL and the Carter constant K. More specifically, the first integrals of the equations of motion
+read [41]
+˙ˆt = A ˆE − 2M∗ˆraˆL
+∆Σ
+,
+˙ˆr2 =
+� ˆE(ˆr2 + a2) − aˆL
+Σ
+�2
+− ∆
+Σ2(ˆr2 + K) ,
+˙ˆθ2 = 1
+Σ2
+�
+K − a2 cos ˆθ −
+�
+a ˆE sin ˆθ −
+ˆL
+sin ˆθ
+�2�
+,
+˙ˆφ = 1
+∆
+�
+2M∗ˆra ˆE
+Σ
++
+��
+1 − 2M∗ˆr
+Σ
+�
+ˆL
+sin2 ˆθ
+�
+,
+(2.3)
+where the dot denotes differentiation with respect to the proper time τ.
+A convenient tetrad for the Kerr geometry (2.1), such that dˆs2 = η(a)(b)ω(a)ω(b), was intro-
+duced in Ref. [42] and reads
+ω(0) =
+�
+∆
+Σ
+�
+dˆt − a sin2 ˆθdˆφ
+�
+,
+ω(1) =
+�
+Σ
+∆dˆr ,
+ω(2) =
+√
+Σdˆθ ,
+ω(3) = sin ˆθ
+√
+Σ
+�
+adˆt − (ˆr2 + a2)dˆφ
+�
+.
+(2.4)
+We dub this tetrad, the Carter’s tetrad. The curvature 2-form
+Ω(a)(b) = 1
+2C(a)(b)(c)(d)ω(c) ∧ ω(d) ,
+(2.5)
+with C(a)(b)(c)(d) being the components of the Weyl tensor, (Cµνρσ = Rµνρσ for the Kerr geometry
+(2.1)), projected along the Carter’s tetrad with the inverses of Eq. (2.4), ωµ
+(a), C(a)(b)(c)(d) =
+Cµνρσ ωµ
+(a)ων
+(b)ωρ
+(c)ωσ
+(d), reads [36,43]
+Ω(0)(1) = 2I1 ω(0) ∧ ω(1) + 2I2 ω(2) ∧ ω(3) ,
+Ω(0)(2) = −I1 ω(0) ∧ ω(2) + I2 ω(1) ∧ ω(3) ,
+Ω(0)(3) = −I1 ω(0) ∧ ω(3) − I2 ω(1) ∧ ω(2) ,
+Ω(1)(2) = −I1 ω(1) ∧ ω(2) + I2 ω(0) ∧ ω(3) ,
+Ω(1)(3) = −I1 ω(1) ∧ ω(3) − I2 ω(0) ∧ ω(2) ,
+Ω(2)(3) = 2I1 ω(2) ∧ ω(3) − 2I2 ω(0) ∧ ω(1) ,
+(2.6)
+where
+I1 = M∗ˆr
+Σ3
+�
+ˆr2 − 3a2 cos2 ˆθ
+�
+,
+I2 = aM∗ cos ˆθ
+Σ3
+�
+3ˆr2 − a2 cos2 ˆθ
+�
+.
+(2.7)
+4
+
+2.2
+Marck’s tetrad
+The orthonormal tetrad λ(a) =
+�
+λ(a)
+0 , λ(a)
+1 , λ(a)
+2 , λ(a)
+3
+�
+that is parallel-transported along an arbi-
+trary time-like geodesic was constructed in Ref. [36]. The tetrad component λ(a)
+0
+is a time-like
+unit vector tangent to the geodesics and λ(a)
+i
+are space-like unit vectors. They satisfy the fol-
+lowing conditions
+η(a)(b) λ(a)
+α λ(b)
+β = ηαβ ,
+λµ
+0∇µλν
+α = 0 ,
+(2.8)
+where λµ
+α = ωµ
+(a)λ(a)
+α
+and α, β = {0, 1, 2, 3} are the labels of the components of the tetrad. The
+first relation in Eq. (2.8) is the orthonormal condition, the second one is the parallel-transported
+requirement that implies the tetrad frame is inertial. Their explicit expressions in terms of the
+metric functions and the constants of motion are [36] 2
+λ(a)
+0
+=
+�
+1
+√
+∆Σ
+�
+ˆE(ˆr2 + a2) − aˆL
+�
+,
+�
+Σ
+∆
+˙ˆr,
+√
+Σ ˙ˆθ,
+1
+√
+Σ
+�
+a ˆE sin ˆθ −
+ˆL
+sin ˆθ
+��
+,
+λ(a)
+1
+= ˜λ(a)
+1 cos Ψ − ˜λ(a)
+2 sin Ψ ,
+λ(a)
+2
+= ˜λ(a)
+1 sin Ψ + ˜λ(a)
+2 cos Ψ ,
+λ(a)
+3
+=
+1
+√
+K
+�
+a cos ˆθλ(1)
+0 , a cos ˆθλ(0)
+0 , −ˆrλ(3)
+0 , ˆrλ(2)
+0 )
+�
+,
+(2.9)
+where
+˜λ(a)
+1
+=
+1
+√
+K
+�
+T
+S
+�
+ˆrλ(1)
+0 , ˆrλ(0)
+0 , S
+T a cos ˆθλ(3)
+0 , −S
+T a cos ˆθλ(2)
+0
+�
+,
+˜λ(a)
+2
+=
+�
+T
+S
+�
+λ(0)
+0 , λ(1)
+0 , S
+T λ(2)
+0 , S
+T λ(3)
+0
+�
+,
+(2.10)
+and
+S = ˆr2 + K ,
+T = K − a2 cos2 ˆθ .
+(2.11)
+Notice the identity Σ = S − T.
+In the second and third tetrad component of Eq. (2.9), we rotated the vectors ˜λ(a)
+1
+and ˜λ(a)
+2
+of an angle Ψ. This is necessary in order to ensure that the tetrad λ(a) =
+�
+λ(a)
+0 , λ(a)
+1 , λ(a)
+2 , λ(a)
+3
+�
+is parallel-transported along the geodesic motion [36]. In particular Ψ is an angle depending on
+the proper time along the Kerr geodesic. The equation satisfied by Ψ was derived in Ref. [36]
+and reads
+˙Ψ =
+√
+K
+Σ
+� ˆE(ˆr2 + a2) − aˆL
+S
++ a
+ˆL − a ˆE sin2 ˆθ
+T
+�
+.
+(2.12)
+A solution for this first order differential equation was provided in Ref. [36] and, more explicitly
+in terms of the Mino time, in Ref. [44].
+2.3
+Tidal tensors
+Tidal effects on test particles moving in the neighborhood of a geodesic in Kerr spacetime are best
+computed by evaluating the Weyl tensor on the parallel-transported tetrad λ(a) (see Eq. (2.9))
+with λ(a)
+0
+being the four-velocity. The explicit expressions for the tidal tensors are obtained once
+the Weyl tensor Cµνρσ is evaluated on the Kerr geodesic. In order to compute the electric and
+2We rename λ(a)
+2
+and ˜λ(a)
+3
+in Ref. [36] with our λ(a)
+3
+and ˜λ(a)
+2 , respectively. It is also important to stress that
+all the components of the space-like vectors λ(a)
+i
+can be written in terms of λ(a)
+0 .
+5
+
+magnetic quadrupole moments, we first need the following components of the rank-2 and rank-3
+tidal tensors in the basis of the tetrad λ(a) [30,36]
+Cij ≡ C(a)(b)(c)(d)λ(a)
+0 λ(b)
+i λ(c)
+0 λ(d)
+j
+,
+Cijk ≡ C(a)(b)(c)(d)λ(a)
+0 λ(b)
+i λ(c)
+j λ(d)
+k
+,
+(2.13)
+where we recall that C(a)(b)(c)(d) = Cµνρσ ωµ(a)ων(b)ωρ(c)ωσ(d). Note that, as a consequence of
+the symmetries of the Weyl tensor, Cij is an STF tensor, whereas Cijk is trace-free and anti-
+symmetric in (j, k) by definition.
+Morevoer, it obeys the condition Cijk + Cjki + Ckij = 0,
+implying that Cijk − Cjik = −Ckij and Cijk − Ckji = −Cjki.
+We compute now the explicit expression for the components of the Weyl tensor that are
+relevant for the calculations of the electric and magnetic quadrupole moments. Our expressions
+are valid for arbitrary time-like geodesics in the Kerr black hole spacetime. The Cij read
+C11 =
+�
+1 − 3ST
+KΣ2(ˆr2 − a2 cos2 ˆθ) cos2 Ψ
+�
+I1 + 6ST
+KΣ2aˆr cos ˆθ cos2 ΨI2 ,
+C12 = 3ST
+KΣ2
+�
+−
+�
+ˆr2 − a2 cos2 ˆθ
+�
+I1 + 2aˆr cos ˆθI2
+�
+sin Ψ cos Ψ ,
+C13 = −3
+√
+ST
+KΣ2
+�
+aˆr cos ˆθ(S + T)I1 +
+�
+ˆr2T − a2S cos2 ˆθ
+�
+I2
+�
+cos Ψ ,
+C22 =
+�
+1 − 3ST
+KΣ2(ˆr2 − a2 cos2 θ) sin2 Ψ
+�
+I1 + 6ST
+KΣ2aˆr cos ˆθ sin2 ΨI2 ,
+C23 = −3
+√
+ST
+KΣ2
+�
+aˆr cos ˆθ(S + T)I1 +
+�
+ˆr2T − a2S cos2 ˆθ
+�
+I2
+�
+sin Ψ ,
+C33 =
+�
+1 +
+3
+KΣ2(ˆr2T 2 − a2S2 cos2 ˆθ)
+�
+I1 − 6ST
+KΣ2aˆr cos ˆθI2 .
+(2.14)
+Note that Cij was already computed in Ref. [36] (with the label 2 renamed with 3 in this paper).
+As a new result, we provide also the general expression for the non-vanishing components of
+the rank-3 tidal tensor Cijk that enter the calculation of the magnetic moments which will be
+done in the next subsection. The non-vanishing components are given by
+C112 = 3
+√
+ST
+KΣ2
+��
+ˆr2T − a2S cos2 ˆθ
+�
+I1 − aˆr cos ˆθ(S + T)I2
+�
+cos Ψ ,
+C113 = 3ST
+KΣ2
+�
+2aˆr cos ˆθI1 +
+�
+ˆr2 − a2 cos2 ˆθ
+�
+I2
+�
+sin Ψ cos Ψ ,
+C123 = − 6ST
+KΣ2aˆr cos ˆθ cos2 ΨI1
++
+1
+KΣ2
+��
+ˆr2T + a2S cos2 ˆθ
+�
+(S − T) − 3ST
+�
+ˆr2 − a2 cos2 ˆθ
+�
+cos2 Ψ
+�
+I2 ,
+C212 = 3
+√
+ST
+KΣ2
+��
+ˆr2T − a2S cos2 ˆθ
+�
+I1 − aˆr cos ˆθ(S + T)I2
+�
+sin Ψ ,
+C213 = 6ST
+KΣ2aˆr cos ˆθ sin2 ΨI1
++
+1
+KΣ2
+�
+ˆr2T(2S + T) − a2 cos2 ˆθS(S + 2T) − 3ST
+�
+ˆr2 − a2 cos2 ˆθ
+�
+cos2 Ψ
+�
+I2,
+C312 = 6ST
+KΣ2aˆr cos ˆθI1 +
+1
+KΣ2
+�
+ˆr2T(S + 2T) − a2 cos2 ˆθS(2S + T)
+�
+I2 .
+(2.15)
+In addition, we observe that C223 = −C113, C312 = C213 − C123, C313 = −C212, C323 = C112.
+6
+
+If we specialize to geodesics in the equatorial plane ˆθ = π/2 of the Kerr black hole, the explicit
+expressions for the tidal tensors simplify considerably. We get, in agreement with Refs. [36,37,45],
+C11 =
+�
+1 − 3
+�
+1 + K
+ˆr2
+�
+cos2 Ψ
+� M∗
+ˆr3 ,
+C22 =
+�
+1 − 3
+�
+1 + K
+ˆr2
+�
+sin2 Ψ
+� M∗
+ˆr3 ,
+C12 = −3
+�
+1 + K
+ˆr2
+� M∗
+ˆr3 cos Ψ sin Ψ ,
+C33 =
+�
+1 + 3K
+ˆr2
+� M∗
+ˆr3 ,
+(2.16)
+and, for the rank-3 tidal tensor (in agreement with Ref. [37] and Ref. [38]),
+C121 = −C112 = C332 = −C323 = −3M∗
+√
+K
+ˆr4
+�
+1 + K
+ˆr2 cos Ψ ,
+C221 = −C212 = C313 = −C331 = −3M∗
+√
+K
+ˆr4
+�
+1 + K
+ˆr2 sin Ψ ,
+(2.17)
+where, for geodesics in the equatorial plane of the Kerr spacetime, the following expressions
+hold [46]
+ˆE =
+ˆr3/2 − 2M∗ˆr1/2 + σaM 1/2
+∗
+ˆr3/4
+�
+ˆr3/2 − 3M∗ˆr1/2 + 2σaM 1/2
+∗
+,
+ˆL =
+σM 1/2
+∗
+�
+ˆr2 + a2 − 2σa M 1/2
+∗
+ˆr1/2�
+ˆr3/4
+�
+ˆr3/2 − 3M∗ˆr1/2 + 2σaM 1/2
+∗
+,
+K =
+�
+a ˆE − ˆL
+�2
+,
+˙Ψ =
+√
+K
+ˆr2 + K
+�
+ˆE −
+a
+a ˆE − ˆL
+�
+= σ
+�
+M∗
+ˆr3 .
+(2.18)
+Above we introduced the parameter σ = ±1 that allows one to distinguish between prograde
+(+) and retrograde (−) orbits. A thorough analysis of the dynamics in the equatorial plane will
+be given in Sec. 4.2.
+2.4
+Electric and magnetic quadrupole moments
+The electric and magnetic quadrupole moments in Cartesian coordinates are defined as [30]
+Eij ≡ Cij ,
+Bij ≡ −1
+2ϵkl⟨iC
+kl
+j⟩
+,
+(2.19)
+with ϵijk the three-dimensional Levi-Civita symbol with ϵ123 = +1. We raise and lower Cartesian
+indices (i, j, k, ...) with the Kronecker delta δij. Being STF tensors, both the electric Eij and
+the magnetic Bij tensors have each five independent components thus, together, they account
+for the ten independent components of the Weyl tensor. In particular, the magnetic quadrupole
+moments in terms of the components of the rank-3 tidal tensor, read
+B11 = −C123 ,
+B12 = C113 ,
+B13 = −C112 ,
+B22 = C213 ,
+B23 = −C212 ,
+B33 = C123 − C213 ,
+(2.20)
+7
+
+where we used that C223 = −C113, C312 = C213 − C123, C313 = −C212 and C323 = C112.
+It is far more useful to decompose the rank-2 and rank-3 tensors by means of their irreducible
+representations of SO(3). Following Ref. [30], we introduce the radial unit vector Ωi ≡ xi/r,
+with r =
+�
+δijxixj being the Euclidean radius representing the distance from the geodesic. The
+projector to the space orthogonal to Ωi is given by γij = δij − ΩiΩj. The electric quadrupole
+moment Eij decomposes as follows
+Eij = Eq
+�
+ΩiΩj − 1
+2γij
+�
++ 2Eq
+(iΩj) + 1
+2Eq
+⟨ij⟩ ,
+(2.21)
+where the scalar Eq, the transverse vector Eq
+i (i.e. ΩiEq
+i = 0) and the transverse STF tensor Eq
+⟨ij⟩
+are given by
+Eq ≡ ΩiEijΩj = −γijEij ,
+Eq
+i ≡ γ j
+i EjkΩk ,
+Eq
+⟨ij⟩ ≡ 2γ k
+i γ l
+j Ekl − Eklγklγij = 2γ k
+i γ l
+j Ekl + Eqγij .
+(2.22)
+Similarly, for the magnetic quadrupole moment Bij, one has 3
+Bij = ϵlk
+(i
+�
+Bq
+l
+�
+Ωj)Ωk − γj)k
+�
++ 1
+4
+�
+Bq
+⟨j)l⟩Ωk − Bq
+⟨j)k⟩Ωl
+��
+,
+(2.25)
+with symmetrization w.r.t. the indices (i, j) and STF w.r.t. the indices ⟨jl⟩ and ⟨jk⟩. The
+transverse vector Bq
+i and the transverse STF tensor Bq
+⟨ij⟩ are
+Bq
+i ≡ ϵijkΩjBk
+lΩl ,
+Bq
+⟨ij⟩ ≡ 2ϵkl(iγm
+j)ΩkBl
+m .
+(2.26)
+3
+Hierarchical triple system
+In this section we apply the formalism introduced in Sec. 2 to an EMR binary system moving
+in the background of a Kerr black hole. The EMR binary system consists of a Schwarzschild
+black hole of mass M along with a test-particle of mass m ≪ M. We assume that the black
+hole with mass M∗ moves slowly relatively to the EMR binary system (M, m) and that one can
+describe the effect on the binary system to a good approximation by taking into account only
+the quadrupole tidal moments induced by M∗. This is valid provided
+M 2 ≪
+ˆr3
+M + M∗
+,
+(3.1)
+where ˆr is the Boyer-Lindquist radius at which M is located in the Kerr spacetime geometry
+induced by M∗ [30]. This arises from having two widely separated scales: one scale is the length
+scale of the Schwarzschild black hole M, the other is the curvature length scale R induced by
+3We used the decomposition of the rank-3 tidal tensor
+Cijk = Bq
+k (ΩiΩj − γij) − Bq
+j (ΩiΩk − γik) + 1
+2
+�
+Bq
+⟨ik⟩Ωj − Bq
+⟨ij⟩Ωk
+�
+,
+(2.23)
+with the inverse relations given by
+Bq
+i = CjkiΩjΩk ,
+Bq
+⟨ij⟩ = 2ΩkClk(iγl
+j) .
+(2.24)
+8
+
+the Kerr black hole M∗ at the location of M. We then require M ≪ R. This is called small-tide
+approximation [30] and it makes it possible to describe the motion of the binary system (M, m)
+in the external Kerr geometry, ensuring that the tidal deformation is weak. We can therefore
+describe the influence of M∗ on the binary system (M, m) using, to a first approximation, the
+quadrupole tidal moments induced by the Kerr black hole itself. Since R ∼
+�
+ˆr3/(M + M∗)
+this, combined with the condition M ≪ R, gives the condition (3.1).
+One natural way to achieve the condition (3.1) is that M is much smaller than M∗, here
+called the hierarchical regime
+M ≪ M∗ .
+(3.2)
+This implies (3.1) since ˆr ≳ M∗. In this case we have a hierarchical triple system of black holes
+m ≪ M ≪ M∗ (note that one could imagine both M and M∗ being a supermassive black hole,
+but with a mass hierarchy). The hierarchical triple system is the case that we shall primarily
+consider in this paper, since the dynamics of the triple system in general will depend on the full
+expressions of the quadrupole tidal moments of the Kerr black hole M∗.
+Another way to achieve the condition (3.1) is the case where M and M∗ are widely separated,
+here called the weak field regime
+M∗ ≪ ˆr ,
+(3.3)
+assuming as well that M ≲ M∗. This means one can consider two black holes M and M∗ of
+similar magnitude. In this case the expression of the tidal moments induced by the Kerr black
+hole simplifies considerably [25] due to the fact that frame-dragging effects induced by the Kerr
+black hole can be neglected (see discussion around and below Eq. (4.8) for further detail).
+It is also important to consider the time scales involved in our approximation. For simplicity,
+we consider the binary system having an orbit of m around the Schwarzschild black hole of mass
+M such that r = O(M). Then the time scale of the binary system is simply τbinary = O(M).
+Assuming ˆr = O(M∗) the time-scale associated with the orbit around the Kerr black hole of mass
+M∗ is τkerr = O(M∗). Indeed, one can see explicitly from Eq. (2.12) that we have ˙Ψ = O(1/M∗),
+which sets the rate of change of the angle Ψ. Thus, in the hierarchical regime (3.2), we have
+τkerr ≫ τbinary, which means that we can assume that the quadrupole moments and Ψ do not
+vary with time. Moreover, in the weak field regime (3.3), the time scale for the orbit around
+the Kerr black hole is even larger τkerr ≫ M∗ as the velocity will be non-relativistic. Thus, even
+if M is of same order as M∗, we find that τkerr ≫ τbinary, and we can again neglect the time
+dependence of Ψ and of the quadrupole moments.4
+3.1
+Tidally deformed Schwarzschild spacetime
+We can describe the black hole with mass M in the binary system using the tidally deformed
+Schwarzschild metric [30]. Concretely, we add to the background metric ¯gµν a tidal perturbation
+hµν
+ds2 = ¯gµνdxµdxν + hµνdxµdxν ,
+(3.4)
+where the tidal perturbation hµν is computed up to the first order in the small-tide approxima-
+tion. The background geometry (in spherical coordinates) is
+¯gµνdxµdxν = −fdt2 + dr2
+f
++ r2ΩABdθAdθB ,
+(3.5)
+with f = 1 − 2M/r and M being the black hole mass, θA = (θ, φ) and ΩABdθAdθB = dθ2 +
+sin2 θdφ2 being the metric of the unit sphere. By only retaining the quadrupole order terms in
+4A more general analysis can also take into account the regime M ≪ r ≪ R for which τbinary = O(
+�
+r3/M).
+9
+
+the tidal deformation hµν, one gets
+hµνdxµdxν = −r2Eq (fdt + dr)2 − 4
+3r3 (Eq
+A − Bq
+A) (fdt + dr) dθA
+− 1
+3r4
+��
+1 − 2M 2
+r2
+�
+Eq
+AB −
+�
+1 − 6M 2
+r2
+�
+Bq
+AB
+�
+dθAdθB.
+(3.6)
+The quadrupole moments are decomposed into the scalar Eq, vector Eq
+A, Bq
+A and tensor Eq
+AB,
+Bq
+AB components, following the decomposition in Eqs. (2.21)-(2.25), and are written in spherical
+coordinates. 5
+For an accurate description of our triple system, it is useful to identify the relative orientation
+between the orbital plane of the Kerr black hole – responsible for the tidal deformation – and
+the orbital plane where the dynamics of the EMR binary system (M, m) takes place; see Fig. 1
+illustrating four possible configurations in the special case when M∗ is a Schwarzschild black hole
+and the binary system is moving on a circular geodesic. To describe an arbitrary configuration,
+one first introduces the unit directional vector
+Ωi = (cos φ sin θ, sin φ sin θ, cos θ) ,
+(3.7)
+centered in the Schwarzschild black hole of mass M, and attached to the reference frame of
+the EMR system (M, m).
+One then sets, without loss of generality, the polar angle in the
+Schwarzschild reference system θ = π/2: this is because the orbital motion takes place on an
+orbital plane and we set it to be the equatorial plane. Any arbitrary orientation is therefore
+given by performing a rotation on the unit vector in Eq. (3.7), namely,
+⃗Ω′ = RχRβRα · ⃗Ω ,
+(3.8)
+with the Euler rotational matrices
+Rα =
+�
+�
+cos α
+sin α
+0
+− sin α
+cos α
+0
+0
+0
+1
+�
+� ,
+Rβ =
+�
+�
+1
+0
+0
+0
+cos β
+sin β
+0
+− sin β
+cos β
+�
+� ,
+Rχ =
+�
+�
+cos χ
+sin χ
+0
+− sin χ
+cos χ
+0
+0
+0
+1
+�
+� .
+(3.9)
+Note that Eq. (3.8) is only one among the 12 possible permutations of Euler matrices. Further-
+more, since we aim at describing an equatorial orbit in the binary system, it turns out that one
+of the Euler angle – α in our convention – can always be reabsorbed by a redefinition of the
+Schwarzschild azimuthal angle φ → φ + α. As a consequence, any orientation of a Schwarzschild
+orbit with respect to the Kerr perturber is specified only by the two angles β and χ.
+3.2
+Tidal moments in spherical coordinates
+The tidal moments also depend on the relative configuration between the binary system (M, m)
+and the Kerr pertuber. Here, we compute the explicit expression of the tidal quadrupole moments
+5For the sake of completeness, we write the change of coordinates from Cartesian to spherical coordinates:
+Eq
+i dxi = ∂xi
+∂xA Eq
+i dxA = Eq
+θ (rdθ) + Eq
+φ(rdφ) ,
+Eq
+⟨ij⟩dxi ⊗ dxj = ∂xi
+∂xA
+∂xj
+∂xB Eq
+⟨ij⟩dxA ⊗ dxB = Eq
+θθ(rdθ)2 + 2Eq
+θφr2dθdφ + Eq
+φφ(rdφ)2 .
+Similar considerations apply to the magnetic multipole moments Bq
+i and Bq
+⟨ij⟩.
+10
+
+I. Orthogonal Configuration
+β = 0, χ = 0
+II. Radial Configuration
+β = π
+2, χ = − π
+2
+III. Tangential Configuration
+β = − π
+2, χ = 0
+IV. Arbitrary Configuration
+β = − π
+4, χ = 5π
+6
+Figure 1: For illustrative purposes, we show four possible configurations for a hierarchical
+three-body system M∗ ≫ M ≫ m in the special case for which the perturber
+M∗ is a Schwarzschild black hole and the EMR binary system (M, m) is parallel-
+transported around a circular geodesic around M∗, whose orbital plane is depicted
+in gray and terminates at the ISCO. These configurations are altered significantly
+in more general cases with a Kerr perturber or non-circular geodesics. The names
+of the configurations refer to the orientation of the orbital angular momentum
+L of the binary system with respect to the gray orbital plane. The grey curve
+represents the orbit around M∗. The blue orbit marks a conventional “initial”
+orthogonal configuration for the binary system reference frame, with the Cartesian
+axis oriented according to the parallel transported tetrad (panel I). The red orbits
+in panels II, III and IV are obtained by Euler rotations with angles written in the
+bottom-left of each panel.
+associated to an arbitrary configuration. We recall that we set θ = π/2 because we start with an
+equatorial orbit around the Schwarzschild black hole. In Fig. 1 we have illustrated this and other
+configurations obtained by Euler rotations in the special case for which M∗ is a Schwarzschild
+black hole and the binary system moves on a circular geodesic. In spherical coordinates, the
+decomposition of the electric quadrupole moment in its scalar, transverse vector and STF tensor
+components is given by Eq. (2.22), where the unit directional vector Ωi is now replaced by the
+more general Ω′i defined in Eq. (3.8).
+11
+
+M*
+M
+M,
+M*The electric quadrupole moments read as
+Eq = −1
+8
+�
+C33 + T +
+2 + T +
+4
+�
++ 1
+8
+�
+4T +
+3 sin 2φ −
+�
+3(C33 + T +
+2 ) − T +
+4
+�
+cos 2φ
+�
+,
+Eq
+θ = 1
+4
+�
+2T −
+3 cos φ − T −
+4 sin φ
+�
+,
+Eq
+φ = 1
+8
+�
+4T +
+3 cos 2φ +
+�
+3(C33 + T +
+2 ) �� T +
+4
+�
+sin 2φ
+�
+,
+Eq
+θθ = −Eq
+φφ = Eq + 1
+2
+�
+C33 + T +
+2 + T +
+4
+�
+,
+Eq
+θφ = −1
+2
+�
+2T −
+3 sin φ + T −
+4 cos φ
+�
+,
+(3.10)
+where we defined the following rotations around χ of the components of Cij
+T +
+1 = C23 cos χ + C13 sin χ ,
+T −
+1 = C23 sin χ − C13 cos χ ,
+T +
+2 = 2C12 sin 2χ + (2C22 + C33) cos 2χ ,
+T −
+2 = 2C12 cos 2χ − (2C22 + C33) sin 2χ
+(3.11)
+and the rotations around β of T ±
+1,2
+T +
+3 = 2T −
+1 sin β + T −
+2 cos β ,
+T −
+3 = 2T −
+1 cos β − T −
+2 sin β ,
+T +
+4 = 4T +
+1 sin 2β + (3C33 − T +
+2 ) cos 2β ,
+T −
+4 = 4T +
+1 cos 2β − (3C33 − T +
+2 ) sin 2β .
+(3.12)
+Similarly for the magnetic quadrupole moments, whose decomposition is given in Eq. (2.26),
+we find that
+Bq
+θ = 1
+8
+�
+4S+
+3 cos 2φ +
+�
+3(C312 − S+
+2 ) − S+
+4
+�
+sin 2φ
+�
+,
+Bq
+φ = −1
+4
+�
+2S−
+3 cos φ − S−
+4 sin φ
+�
+,
+Bq
+θθ = −Bq
+φφ = −1
+2
+�
+2S−
+3 sin φ + S−
+4 cos φ
+�
+,
+Bq
+θφ = −3
+8
+�
+C312 − S+
+2 + S+
+4
+�
+− 1
+8
+�
+4S+
+3 sin 2φ −
+�
+3(C312 − S+
+2 ) − S+
+4
+�
+cos 2φ
+�
+,
+(3.13)
+where we defined the rotations around χ of the components of Cijk
+S+
+1 = C212 cos χ + C112 sin χ ,
+S−
+1 = C212 sin χ − C112 cos χ ,
+S+
+2 = 2C113 sin 2χ + (C123 + C213) cos 2χ ,
+S−
+2 = 2C113 cos 2χ − (C123 + C213) sin 2χ
+(3.14)
+and the rotations around β of S±
+1,2
+S+
+3 = 2S−
+1 sin β − S−
+2 cos β ,
+S−
+3 = 2S−
+1 cos β + S−
+2 sin β ,
+S+
+4 = 4S+
+1 sin 2β + (3C312 + S+
+2 ) cos 2β ,
+S−
+4 = 4S+
+1 cos 2β − (3C312 + S+
+2 ) sin 2β .
+(3.15)
+12
+
+The structure of the tidal quadrupole moments (3.10) and (3.13) is the following: the tidal
+deformations sourced by a generic third body over the EMR binary system (M, m) are fully
+encoded in the tidal tensors Cij and Cijk, while the angles β and χ, parametrizing the relative
+orientation between the third body and the binary system, affect the tidal effects over the binary
+system. We remark that the above expressions of the tidal quadrupole moments are general, and
+can also be employed to model environmental effects in numerical works. In the specific case of
+a Kerr black hole as a third body responsible for the tidal deformations, the explicit expressions
+of the tidal tensors Cij and Cijk are given, respectively, in Eqs. (2.14) and (2.15).
+We anticipate here another property of the tidal quadrupole moments. As we shall see in
+the next section, it is often useful to define the secular average over the azimuthal angle φ. The
+explicit dependence of the tidal quadrupole moments (3.10) and (3.13) implies that only Eq (and
+Eq
+θθ = −Eq
+φφ) as well as Bq
+θφ are relevant for physical observables upon secular averaging.
+4
+Secular dynamics of binary system
+In this section we focus on the secular dynamics of the binary system (M, m), i.e. the dynamics
+of the binary system after a large number of orbits of the test particle of mass m, and analyze
+how it is affected by the tidal fields induced by the Kerr perturber of mass M∗, in the hierarchical
+regime m ≪ M ≪ M∗. More specifically our goal is to understand how the orbital parameters
+of the test particle around the Schwarzschild black hole, such as the energy or the angular
+momentum, are shifted by the presence of an external tidal field.
+4.1
+Secular Hamiltonian of test particle in binary system
+Following the setup of the previous section, we focus on the orbital motion of the object of mass
+m, approximated as a test particle, taking place on the equatorial plane of the Schwarzschild
+black hole. This amounts to set θ = π/2. We approximate the four-velocity as
+uµ ≃ ¯uµ + uµ
+(1) ,
+(4.1)
+where ¯uµ is the 4-velocity of the unperturbed bound orbit, that can be taken as circular or elliptic,
+and uµ
+(1) is the leading correction due to the tidal perturbation hµν. In this work, we focus on
+perturbations of circular orbits ¯uµ = ( ¯E/f, 0, 0, ¯L/r2) on the Schwarzschild background metric
+¯gµν. Here ¯E = −¯uµ¯gµν(∂t)ν and ¯L = ¯uµ¯gµν(∂φ)ν are the conserved energy and angular momentum
+of the test particle in the Schwarzschild geometry. Tidal deformations to the four-velocity affect
+the gauge-independent photon red-shift measurements [47] (∼ ut
+(1)), are responsible for radial
+deviations (∼ ur
+(1)), tilt the orbital plane (∼ uθ
+(1)), and shift the orbital frequency (∼ uφ
+(1)).
+The Hamiltonian of a test particle moving around a tidally deformed Schwarzschild black
+hole (see Eq. (3.4)) is given by
+H = 1
+2uµuνgµν ≃ 1
+2 ¯uµ �
+¯uν + 2uµ
+(1)
+�
+¯gµν + 1
+2 ¯uµ¯uνhµν .
+(4.2)
+In the specific case of a circular orbit ¯uµ in the Schwarzschild background metric ¯gµν, radial and
+polar deviations affects the dynamics only at higher order [25,48]. Moreover, from Eq. (4.2), the
+tidal perturbations that enter the Hamiltonian are htt ∝ Eq, htφ ∝ Eq
+φ, Bq
+φ, and hφφ ∝ Eq
+φφ, Bq
+φφ.
+A further simplification, that is very common in celestial mechanics, is the secular averaging
+over a timescale much bigger than the orbital timescale. The effective dynamics of a test particle
+which follows a tidally-deformed geodesic γ′ at the first order in hµν can be well captured by
+replacing the physical trajectory γ′ with an averaged circular trajectory γ in the perturbed
+13
+
+spacetime.
+The averaged geodesic γ can be interpreted as a secular orbit in the perturbed
+background. We introduce the secular average of a quantity A as [25]
+⟨A⟩ = 1
+2π
+� 2π
+0
+A
+��
+γ dφ ,
+(4.3)
+where γ is the averaged circular orbit on gµν. In particular, if γ′ is quasi-circular, the averaged
+secular geodesic γ deviates from the physical orbit γ′ only starting from second order in hµν in
+the Hamiltonian (4.2).
+After averaging, from Eqs. (3.10) and (3.13), we get 6
+⟨htt⟩ = −r2f 2⟨Eq⟩,
+(4.4)
+⟨htφ⟩ = 0,
+(4.5)
+⟨hφφ⟩ = −r4
+�
+1 − 2M 2
+r2
+�
+⟨Eq⟩.
+(4.6)
+and therefore the secular average of the Hamiltonian (4.2) up to quadrupole order can be recast
+as 7
+⟨H⟩ ≃ −1
+2
+�⟨E⟩2
+f
+− ⟨L⟩2
+r2
+�
+− η
+�
+⟨E⟩2 +
+�
+1 − 2M 2
+r2
+� ⟨L⟩2
+r2
+� r2
+M 2 ,
+(4.7)
+where η is a parameter that encodes all the effects of the tidal deformations at the quadrupole
+order. E = −uµgµν(∂t)ν and L = −uµgµν(∂φ)ν are, respectively, the energy and angular mo-
+mentum with respect to the perturbed spacetime and the symbol ⟨·⟩ stands for secular average.
+We stress that ⟨E⟩ and ⟨L⟩ encode the kinematics (including the secular effects on the orbits),
+while the parameter η effectively depends on the secular tidal deformations (∝ Cij) and on the
+orientation (β, χ) of the binary system. More explicitly, we find that the tidal parameter η is
+proportional to the secular average of the electric scalar tidal field
+η = −M 2
+2 ⟨Eq⟩ = M 2
+16
+�
+C33 (1 + 3 cos 2β) + 4 (C13 sin χ + C23 cos χ) sin 2β
++ [2C12 sin 2χ + (2C22 + C33) cos 2χ] (1 − cos 2β)
+�
+.
+(4.8)
+Notice that this expression for η can also be used for other tidal tensors Cij than the one induced
+by the Kerr black hole in this paper. In fact, it is a general result for any EMR binary system
+consisting of a Schwarzschild black hole of mass M and a test particle of mass m, under the
+assumptions that: 1) it is immersed in a tidal environment, 2) only the quadrupole order is
+retained and 3) the secular approximation is valid.
+If we specialize Eq. (4.8) to the tidal tensors of a Kerr perturber that we presented in Sec. 2 in
+Eq. (2.14), it can be shown that the Marck’s angle Ψ appearing in the Cij’s, which is a constant
+in this approximation, can be reabsorbed by a simple shift of the angle χ , χ → χ − Ψ so that
+η is explicitly given by
+η = I1M 2
+16KΣ2
+�
+3ST(ˆr2 − a2 cos2 ˆθ)(1 − 4 sin2 β sin2 χ) + 6 cos 2β
+�
+ˆr2T 2 − a2S2 cos2 ˆθ
+�
+−3a cos ˆθ
+�
+aS2 cos ˆθ + 4ˆr sin 2β
+√
+ST(S + T) sin χ
+�
++ KΣ2 + 3ˆr2T 2�
+(4.9)
++ 3I2M 2√
+ST
+4KΣ2
+��
+a2S cos2 ˆθ − ˆr2T
+�
+sin 2β sin χ − 2aˆr
+√
+ST cos ˆθ
+�
+cos2 β − sin2 β sin2 χ
+��
+,
+6Our result differs from the one in Ref. [25] where ⟨htφ⟩ ̸= 0.
+7Notice that we used that ⟨uµuνgµν⟩ ≃ ⟨uµ⟩⟨uν⟩⟨gµν⟩ including corrections of order hµν.
+14
+
+where K is the Carter constant, and I1, I2, S and T are defined in Eqs. (2.7) and (2.11).
+In the weak field regime, where M⋆ ≪ ˆr, the leading order part of η is given by
+η = M 2
+4K
+M⋆
+ˆr3
+�
+3T(cos2 β − sin2 β sin2 χ) − K
+�
+2 − 3 sin2 β
+�
+− 3a
+√
+T cos ˆθ sin χ sin 2β
+�
+.
+(4.10)
+In the equatorial plane of the Kerr pertuber ˆθ = π/2, the parameter η takes the simpler form
+η = M 2
+4
+M⋆
+ˆr3
+�
+1 − 3 sin2 β sin2 χ
+�
+,
+(4.11)
+that depends only on the two Euler angles χ and β and not on the spin parameter a, so one
+cannot distinguish the effect of the tidal forces from the case of a Schwarzschild perturber (a = 0).
+This is reasonable in the sense that if one goes at large distances on the equatorial plane, one
+cannot feel the effect of the spin of the Kerr black hole. For χ = π/2, in particular, Eq. (4.11)
+coincides with the result of Ref. [25], provided one identifies β as the angle between the tidal
+symmetry axis, parallel to z, and the orbital plane: η = M2M⋆
+4ˆr3
+�
+1 − 3 sin2 β
+�
+.
+4.2
+Special case of circular equatorial geodesic in Kerr background
+We emphasize that neither the construction of the tidal quadrupole moments in Sec. 2, nor the
+discussion about the secular dynamics of the Schwarzschild binary system in the current section
+rely on any assumption concerning the geodesic motion followed by the Schwarzschild black hole
+of mass M around the Kerr black hole of mass M∗ ≫ M.
+However, in order to simplify the discussion, we now focus on solutions of the geodesic
+equations (2.3) describing circular ( ˙ˆr = 0) and equatorial geodesics (ˆθ = π/2 and ˙ˆθ = 0) in
+the Kerr spacetime. Under these assumptions, the parameters that characterise the geodesic –
+namely the energy, the angular momentum and the Carter’s constant – are written explicitly
+in Eq. (2.18). In this case the effective parameter η given in Eq. (4.9) reduces to the simple
+expression
+η = M∗M 2
+16ˆr3
+�
+1 + 3K
+ˆr2 − 3
+�K
+ˆr2 +
+�
+1 + K
+ˆr2
+�
+sin2 χ
+�
+sin2 β
+�
+.
+(4.12)
+Note that this is a general result, valid beyond the weak-field regime (M⋆ ≪ ˆr).
+For a circular equatorial geodesic it is moreover easy to express the Carter constant K in
+terms of the Kerr parameters (a, M∗) and the orbital radius ˆr, by means of the following relation
+K
+ˆr2 = −1
+2
+�
+1 − ˆr2 − ˆrM∗ − 2σa√ˆrM∗ + 2a2
+ˆr2 − 3ˆrM∗ + 2σa√ˆrM∗
+�
+.
+(4.13)
+We recall that σ = ±1 distinguishes whether a circular orbit is co-rotating or counter-rotating
+with respect to the Kerr black hole angular momentum.
+An intriguing observation is that, from the expression (4.12), one can see that there exist
+certain configurations for the EMR binary system (M, m) on the Kerr equatorial plane, such
+that η = 0, namely such that the dynamical contribution of the tidal effects vanishes in the
+secular approximation. For a given angle χ, this holds when the angle β = β∗(χ) with
+sin2 β∗(χ) =
+1 + 3K/ˆr2
+3
+�
+K/ˆr2 + (1 + K/ˆr2) sin2 χ
+� .
+(4.14)
+In the weak-field limit this relation reduces to sin2 β∗(χ) = (3 sin2 χ)−1, thus generalising the
+result obtained in Ref. [25], which is valid only for χ = π/2. Instead, the above result goes
+beyond the weak-field regime, and can be used also for circular geodesics close to the event
+horizon of Kerr.
+15
+
+Among all the time-like equatorial circular orbits, the Innermost Stable Circular Orbit
+(ISCO) stands out for its relevance in black hole astrophysics. We recall that two ISCOs ex-
+ist in the equatorial plane of a Kerr black hole, one which is co-rotating (σ = +1) and the
+other counter-rotating (σ = −1). As an illustrative example, previously not considered in the
+literature concerning hierarchical three-body systems, one can analyse the case where the circu-
+lar equatorial orbit, in which the binary system is located, is given by the Kerr ISCOs. More
+specifically, in the following we set
+ˆr ≡ ˆrσ
+ISCO = M∗
+�
+3 + Z2 − σ
+�
+(3 − Z1)(3 + Z1 + 2Z2)
+�
+,
+(4.15)
+where
+Z1 = 1 +
+�
+1 − a2
+M 2
+∗
+�1/3 ��
+1 + a
+M∗
+�1/3
++
+�
+1 − a
+M∗
+�1/3�
+,
+Z2 =
+�
+Z2
+1 + 3 a2
+M 2
+∗
+.
+(4.16)
+It is possible to show that the following relation implicitly defines the ISCOs in terms of the
+conserved Killing energy [46]
+ˆE2
+ISCO = 1 − 2
+3
+M∗
+ˆrσ
+ISCO
+,
+(4.17)
+so that, by combining the expression above with K = (a ˆE − ˆL)2 as in Eq. (2.18), one obtains
+that the Carter constant at the ISCOs takes the value K = 1/3 (ˆrσ
+ISCO)2. The expression for η
+in this limit considerably simplifies and it is given by
+η =
+M 2M∗
+2 (ˆrσ
+ISCO)3
+�
+1 − 1
+2(1 + 4 sin2 χ) sin2 β
+�
+.
+(4.18)
+Notice that, even if ˆrσ
+ISCO ∼ O(M∗), the small tide approximation Eq. (3.1) is still valid since
+M ≪ M∗. This means that one can still legitimately consider the quadrupole approximation
+for a hierarchical three-body system in which the binary system (M, m) is orbiting on the ISCO
+of the Kerr black hole of mass M⋆. It is interesting to notice that in the expression (4.18) the
+dependence on the spin parameter of the Kerr perturber is only contained in the prefactor,
+whereas the part inside square brackets specifies the configuration of the binary system. A plot
+of the prefactor showing the dependence on the spin of the Kerr black hole is shown in Fig. 2
+for different values of the ratio M/M∗.
+It is also interesting to observe that the expression for η at the ISCO remains well-defined
+even when the Kerr black holes is rotating close to extremality, namely for a → M∗. In this case
+one has ˆr+
+ISCO → M∗, so that the prefactor only depends on the ratio M 2/M 2
+∗. It is also evident
+by means of the plot in Fig. 2 that the extreme case represents the maximum value of η at the
+ISCO for a given configuration of the binary system.
+For the EMR binary system moving on the ISCO in the Kerr black hole spacetime, we can
+get the angle β = β∗(χ), as function of the angle χ, for which η = 0, at which the tidal effects
+vanish from the secular dynamics of the binary system. Using that K/(ˆrσ
+ISCO)2 = 1/3, one gets
+sin2 β∗(χ) =
+2
+1 + 4 sin2 χ .
+(4.19)
+In Fig. 3 we show the admissible values of β∗(χ) when the binary system is at the ISCO.
+5
+Secular shifts for ISCO and photon sphere
+In this section we investigate how the tidal deformations affect the secular motion of the charac-
+teristic orbits of a test-particle around a Schwarzschild black hole using the Hamiltonian given
+16
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-12
+-10
+-8
+-6
+-4
+a/M∗
+log10
+�
+M2M∗
+2(ˆrσ
+isco)
+3
+�
+Figure 2: The picture represents how η, when evaluated at the ISCO ˆr ≡ ˆrσ
+isco, depends on
+the black hole spin a. The logarithm of the prefactor in Eq. (4.18) is considered
+in order to have a clear distinction for the curves. Colours are used to represent
+different magnitudes for the ratio µ = M/M∗. In particular µ = 10−2 in blue,
+µ = 10−3 in purple, µ = 10−4 in red and µ = 10−5 in orange. Solid lines are
+representative for the co-rotating ISCO σ = 1, whereas dashed lines for counter-
+rotating ISCO σ = −1.
+0
+π
+6
+π
+2
+5 π
+6
+π
+7 π
+6
+3 π
+2
+11 π
+6
+2 π
+0
+π
+4
+π
+2
+χ
+β∗(χ)
+Figure 3: The red line identifies the configurations β∗(χ) for which the secular effect of tidal
+deformations vanishes under the assumption ˆr ≡ ˆrσ
+ISCO. The gray areas represent
+exclusion zones, namely values of the angle χ in which the relation (4.19) cannot
+be satisfied. More specifically, these corresponds to values of χ that would lead
+| sin2 β∗| > 1.
+17
+
+in Eq. (4.7). In particular, we consider two specific orbits in the case of general configurations of
+the three-body system, namely the ISCO and the photon sphere in the perturbed Schwarzschild
+spacetime. Before computing tidal effects on the orbital motion, we address the issue of gauge
+invariance of such effects.
+5.1
+Gauge invariance of secular observables
+We start by recalling that the energy E can be expressed in terms of the Killing vector ∂t,
+namely
+E = −uµgµνT ν ,
+(5.1)
+where in our coordinates T = ∂t and gµν and uν are the metric and four-velocity including tidal
+perturbations. Given that T is a Killing vector field, dE/dτ = 0 in any coordinate system when
+evaluated on a geodesic. Therefore, E is conserved and gauge-invariant.
+The angular momentum can be covariantly written as
+L = uµgµνJν ,
+(5.2)
+where in our coordinates J = ∂φ. However, as J is not a Killing vector field for the full metric
+gµν, L is not conserved along geodesics. The strategy here is to get a conserved quantity and
+show that it is also gauge-invariant. We assume that the angular momentum L can be expanded
+as
+L ≃ ¯L + ηL1 ,
+(5.3)
+where ¯L is the conserved angular momentum in the Schwarzschild background, while L1 is
+the correction induced by the tidal fields at the quadrupole order, which in general it is not
+conserved.
+The key observation is that the averaged metric field ⟨gµν⟩ does not depend on
+φ = φ(τ), implying that ⟨L⟩ is now a conserved quantity along the secular geodesic. Therefore,
+for a quasi-circular orbit we can write
+⟨L⟩ ≃
+� 2π
+0
+�¯L + ηL1
+�
+|γdφ = 2π ¯L + η
+� 2π
+0
+L1|γdφ .
+(5.4)
+We now consider a coordinate transformation which, up to the quadrupole order, is of the
+form
+φ → ˜φ ≃ φ + ηχ(φ) ,
+(5.5)
+such that χ is a periodic function of φ with a period of 2π, namely χ(φ) = χ(φ + 2π). Under
+this gauge transformation, the first term in Eq. (5.4) reads as
+� 2π
+0
+¯L|γd˜φ →
+� 2π
+0
+¯L|γdφ + η
+� 2π
+0
+¯L|γdχ = 2π ¯L ,
+(5.6)
+where we used the periodicity of χ and the fact that ¯L does not depend on φ. The second term
+in Eq. (5.4), under the gauge transformation in (5.5), transforms as
+� 2π
+0
+L1|γd˜φ →
+� 2π
+0
+L1|γdφ + η
+� 2π
+0
+L1|γdχ .
+(5.7)
+The second integral in the expression above does not vanish in general, since L1 depends on φ.
+However, we can neglect it because the second integral will be multiplied by η2 and therefore it
+is of higher order. Putting the pieces together we have
+⟨L⟩ ≃
+� 2π
+0
+�¯L + ηL1
+�
+|γd˜φ → 2π ¯L + η
+� 2π
+0
+L1|γdφ ,
+(5.8)
+18
+
+thus ⟨L⟩ is gauge-invariant under coordinate transformations of order O(η) which are 2π-periodic
+in φ.
+Along the same line of reasoning, one can prove the gauge invariance of ⟨uφ⟩ and ⟨ut⟩. Since
+the orbital frequency for a quasi-circular orbit is defined by
+Ω = uφ
+ut ,
+(5.9)
+we conclude that ⟨Ω⟩ is also gauge-invariant under coordinate transformations of order O(η)
+which are 2π-periodic in φ.
+As a side remark, we could extend the reasoning for the gauge invariance of secular quantities
+to certain classes of gauge transformations. For example, we can consider the case where the
+coordinate transformation involves a radial function
+˜φ ≃ φ + ηA (r) χ (φ) ,
+(5.10)
+where χ is still a function of φ with period 2π. In the averaging procedure, we would also have
+an integral over r that vanishes because the secular geodesic is circular. Another example is a
+gauge transformation depending on the polar coordinate θ, namely
+˜φ ≃ φ + ηA (θ) χ (φ) .
+(5.11)
+Once again, being any shift in θ of order O(η) and being the function A multiplied by η, we can
+neglect any contribution of A (θ) to the averaging procedure that goes beyond the first order in
+η.
+5.2
+Tidal effects around the ISCO orbit
+The innermost stable circular orbit (ISCO) for massive test-particles is completely characterised
+by three parameters: its radius, energy and angular momentum. It is defined as an extreme
+point of the Hamiltonian (4.7), namely
+⟨H⟩|r=rISCO = −1
+2 ,
+d⟨H⟩
+dr
+����
+r=rISCO
+= 0 ,
+∂2⟨H⟩
+∂r2
+����
+r=rISCO
+= 0 .
+(5.12)
+Using these conditions and keeping only terms proportional to η, it is possible to compute the
+secular effects caused by the tidal perturbations to the energy, angular momentum and radius
+of the Schwarzschild ISCO.
+We assume that observables are expanded around their unperturbed values. Physically, this
+is equivalent to assume that tidal (secular) effects are all proportional to the tidal parameter η. 8
+This assumption also defines the numerical values of the tidal corrections. Tidal corrections to
+the radius,9 the averaged energy and angular momentum read as 10
+rISCO ≃ r0 + η r1 ,
+EISCO ≃ E0 + η E1 ,
+LISCO ≃ L0 + η L1 .
+(5.13)
+By solving Eqs. (5.12) at leading order one can determine the value of (r0, E0, L0), respectively
+the value for the radius, the energy and the angular momentum of the ISCO for an unperturbed
+Schwarzschild black hole. They are
+r0 = 6 M ,
+E0 = 2
+√
+2
+3
+,
+L0 = 2
+√
+3 M .
+(5.14)
+8We recall that we consider only up to first order contributions in the small-tide approximation.
+9which is not a gauge-invariant quantity; see discussion at the end of this section.
+10From now on, we will drop the symbol of the secular average ⟨·⟩ for the sake of presentation.
+19
+
+At the first order in η, the first corrections to the ISCO quantities are given by
+r1 = 3072 M ,
+E1 = −152
+√
+2
+3
+,
+L1 = −348
+√
+3 M .
+(5.15)
+Note that we fixed our conventions for η in order to precisely reproduce the same numerical
+values of (r1, E1, L1) previously obtained in Ref. [25]. However, while the results of Ref. [25] are
+only valid in the weak-field approximation where ˆr ≫ M⋆ and on the equatorial plane ˆθ = π/2,
+our results are more general and hold for any value of ˆr and ˆθ, as we discussed earlier in Sec. 4.
+It is also possible to compute the shift in the ISCO orbital frequency. In general, for quasi-
+circular orbits, the orbital frequency can be determined by means of the ratio [25,47,49]
+Ω2 =
+�uφ
+ut
+�2
+=
+1
+2r2
+�2M
+r
+− (r − 3M) uµuν∂r⟨hµν⟩
+�
+,
+(5.16)
+where uµ are the components of the four-velocity (4.1). To first order in η, we obtain
+ΩISCO ≃ Ω0 + η Ω1 ,
+(5.17)
+where 11
+M Ω0 =
+1
+6
+√
+6,
+M Ω1 = −
+�
+2
+3
+491
+6
+.
+(5.18)
+This gives the shift induced by the tidal fields in the orbital frequency of the ISCO.
+Following Ref. [47], the angular frequency Ω can be used to compute a gauge-independent
+measure of the radial separation between the Schwarzschild black hole and the test particle. One
+defines
+RΩ =
+�M
+Ω2
+�1/3
+,
+(5.19)
+so that according to Eqs. (5.17) and (5.18)
+RΩ ≃ 22/3M
+Ω2/3
+0
+�
+1 − 2
+3ηΩ1
+Ω0
+�
+= 6M + 3928η M .
+(5.20)
+We notice that this gives a different radial shift than in Eq. (5.15). However, this is not surprising
+as the radial shift of Eq. (5.15), unlike the above, is not gauge-invariant.
+5.3
+Tidal effects around the photon sphere
+The photon sphere around a Schwarzschild black hole is composed by the last stable circular
+orbits for massless test-particles. Differently from the case of the ISCO, this orbit is only specified
+by two parameters: the photon sphere radius and the impact parameter b = L/E. A previous
+analysis of the photon sphere properties in a tidal environment can be found in Ref. [40], under
+more limited assumptions than the ones considered in this paper.
+From the secular Hamiltonian (4.7), one enforces the conditions
+⟨H⟩|r=rPS = 0 ,
+d⟨H⟩
+dr
+����
+r=rPS
+= 0 .
+(5.21)
+11Notice that this result agrees with Ref. [40] (but not with Ref. [25]), after a rescaling of -1/2 of the η parameter.
+For the ease of comparison, our radial configuration (see Fig. 1) is called polar companion configuration in
+Ref. [40]: this can be obtained in the weak-field limit ˆr ≫ M∗ and for β = π/2 and χ = −π/2.
+20
+
+By expanding the kinematic quantities in the tidal parameter η to retain only the leading
+contribution of the tidal secular effects in the small-tide approximation, one obtains
+rPS ≃ r0 + η r1 ,
+bPS ≃ b0 + η b1 ,
+(5.22)
+where the unperturbed values for the Schwarzschild black hole are obtained by solving (5.21) at
+the leading order
+r0 = 3 M ,
+b0 = 3
+√
+3 M .
+(5.23)
+Similarly, the tidal corrections are given by
+r1 = −30 M ,
+b1 = 30
+√
+3 M .
+(5.24)
+This results generalize the one obtained in Ref. [40] for the special configuration of polar com-
+panions (equivalent to our radial configuration), after a rescaling of η.
+Again, the orbital frequency at the photon sphere at first order in the tidal corrections can
+be computed in general from
+Ω = uφ
+ut = 1
+b ,
+(5.25)
+which at first order in η yields to
+ΩPS ≃ Ω0 + η Ω1 .
+(5.26)
+By means of Eqs. (5.23) and (5.24), one directly obtains the shift in the frequency of the photon
+sphere, given by
+M Ω0 =
+1
+3
+√
+3 ,
+M Ω1 = − 10
+3
+√
+3 .
+(5.27)
+6
+Conclusions and outlook
+We conclude by summarising our new results and discussing further developments.
+In Sec. 2, we retraced the computation performed in Ref. [36] for the construction of the
+Marck’s tetrad, defining a local inertial frame which is parallel-transported around a time-like
+geodesic in Kerr spacetime. Tidal effects induced by a Kerr black hole are obtained by projecting
+the Weyl tensor on certain components of the Marck’s tetrad. While the components of the rank-
+2 tensor Cij were computed in Marck’s paper [36], the components of the rank-3 tensor Cijk were
+previously known only on the equatorial plane of a Kerr black hole [37,38]. This paper therefore
+fills the gap in the literature: the explicit expressions for Cijk are given in Eq. (2.15). Our result
+is valid for generic angles ˆθ and for arbitrary time-like geodesics in the Kerr spacetime.
+In Sec. 3, we found a natural application of the tidal tensors computed in the previous sec-
+tion in the modeling of a hierarchical three-body system in General Relativity. We considered
+a 3-body system describing a supermassive rotating black hole of mass M∗ and an EMR bi-
+nary system, made of a non-rotating black hole of mass M ≪ M∗ and a smaller companion
+of mass m ≪ M, which gravitates around the supermassive black hole. In order to go be-
+yond the post-Newtonian approximation, in which the three bodies are sufficiently distant from
+each other to be treated as point-like masses, and capture strong general relativistic effects,
+one can model the region around the non-rotating black hole in terms of a tidally-deformed
+Schwarzschild spacetime. To this aim, it is convenient to decompose the tidal tensor in terms of
+irreducible representations of the rotation group, so as to construct “electric” E and “magnetic”
+B quadrupole tidal moments, that encode the leading-order deformations to the Schwzarschild
+metric immersed in a generic tidal environment [30]. By approximating the motion of the small-
+est body as that of a test-mass, it is possible to take into account all the possible configurations
+21
+
+of the binary system by introducing two Euler’s angles. Another new result obtained in this
+work is the explicit expressions for the electric and magnetic quadrupole tidal moments given
+in Eqs. (3.10)-(3.13), that take into account arbitrary orientations of the binary system with
+respect to the source of the tidal deformations. We remark that these expressions are valid for
+arbitrary sources of tidal effects. This can be of interest for numerical simulations and analytical
+study of binary systems immersed in a tidal environment. For the case of a supermassive Kerr
+black hole, the tidal moments (3.10) and (3.13) together with our result in Sec. 2 allow us to
+analytically compute tidal effects induced by a Kerr black hole in full generality.
+The hierarchy of masses makes it natural to study the dynamics of the binary system in the
+secular approximation. As first pointed out in Ref. [25], the tidal effects perturb the secular
+Hamiltonian for the binary system. Remarkably, at the quadrupole approximation, the tidal
+perturbation can be recast into an effective perturbative parameter η. The main result of Sec. 4
+is a general expression for η given in Eq. (4.8). It holds at the quadrupole order in the small-
+tide regime and in the secular approximation, and it models the deformed secular dynamics
+of a binary system. Our η generalises results obtained in Ref. [25] and Ref. [40] to arbitrary
+orientations of the binary system and tidal effects induced by a rotating black hole, including
+the strong gravity regime.
+Tidal deformations induce changes in certain gauge-invariant quantities characterising the
+EMR binary systems, such as the orbital frequency. Such tidal deformations induced by the
+environment are completely encoded in the effective perturbative parameter η.
+We devoted
+Sec. 5 to the study of such shifts in the case of marginally stable orbits for massive (ISCO shifts)
+and massless (photon sphere shifts) test-particles. We also addressed the issue of the gauge
+invariance of the shifts in the secular approximation. While we focus on the case of a Kerr
+black hole as the perturber, one can also use our expressions with general tidal moments. For a
+Kerr perturber, the expression for η (see Eq. (4.9)) shows the rich phenomenology of the triple
+system: it combines the parameters of the background Kerr metric (M∗ and a), the location of
+the geodesic where the binary system is located (ˆr, ˆθ, K), and the Euler angles that capture the
+geometric orientation of the binary system with respect to the Kerr perturber (β and χ). Our
+parameter η includes strong general relativistic effects of an EMR binary system which is affected
+by the presence of a large Kerr black hole, and considerably generalises the setup considered
+in Ref. [25] and Ref. [40] beyond the weak-field regime and for arbitrary configurations. As an
+example of a regime which was previously overlooked in the literature, in Sec. 4.2, we focused
+on the case in which the EMR system is placed on the ISCO of the Kerr background. We
+also derived configurations of the EMR system for which the tidal effects vanish in the secular
+approximation, generalising the findings of Ref. [25].
+There is a number of directions in which this work can be further extended, and for which
+the results obtained here can be of interest. In this paper, we analyze triple systems whose
+dynamics is stationary in time and restricted to circular orbits. This implies that we do not
+have gravitational waves in our setup. We also work in the leading quadrupole approximation
+for the tidal effects. The setup in this paper, though simplified, is useful to get analytic results
+and it should be considered as a first step towards a more realistic scenario that can be relevant
+for astrophysical interest.
+An extension of this work would include higher-order effects beyond the quadrupole approx-
+imation [50] and the stationary regime. It would be interesting to further develop waveforms
+from triple hierarchical systems [51,52] and approaches to effective description thereof [53,54].
+Another natural development would be extending this study where the primary companion of
+the EMR is a Kerr black hole. The metric for a rotating black hole deformed by tidal effects has
+been derived in full generality in Ref. [55] by solving the Teukolsky equation and using metric
+reconstruction techniques. Due to the very complicated structure of that metric, a simplified
+version obtained in the small-spin regime has been obtained in Ref. [56], explicitly written
+22
+
+in terms of tidal quadrupole moments.
+This is sufficient to capture all the main important
+features of spacetimes with non-vanishing angular momentum, and can lead to an even richer
+phenomenology – including couplings between the spins of the two black holes – possibly already
+at the level of the secular dynamics.
+A third interesting direction concerns the analysis of eccentric binary systems subject to
+tidal deformations. For this specific case it is probably more convenient to use the action-angle
+variables formalism [57–60]. This would allow us not only to extend our computation to the
+case of elliptic orbits for the test particle in the binary system, but also to study the precession
+of the orbits around the Schwarzschild black hole and the presence of possible resonances in the
+binary system [61,62].
+Acknowledgments
+We thank P. S. Cole, B. Liu and J. Samsing for interesting discussions. We thank V. Car-
+doso for useful comments on the manuscript. G.G. and M.O. acknowledge support from Fondo
+Ricerca di Base 2020 (MOSAICO) and 2021 (MEGA) of the University of Perugia. The work
+of T.H. is supported in part by the project “Towards a deeper understanding of black holes
+with non-relativistic holography” of the Independent Research Fund Denmark (grant number
+DFF-6108-00340). The work of R.O. is supported by the R´egion ˆIle-de-France within the DIM
+ACAV+ project SYMONGRAV (Sym´etries asymptotiques et ondes gravitationnelles). G.G. and
+R.O. thank the Niels Bohr Institute for hospitality at different stages of this project. T.H. thanks
+University of Perugia for hospitality.
+References
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+Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016), no. 6
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+Correlative mapping of local hysteresis properties in VO2
+Melissa Alzate Banguero,1 Sayan Basak,2, 3 Nicolas Raymond,1 Forrest Simmons,2, 3 Pavel Salev,4, 5
+Ivan K. Schuller,5 Lionel Aigouy,1, ∗ Erica W. Carlson,2, 3, 1, † and Alexandre Zimmers1, ‡
+1Laboratoire de Physique et d’´Etude des Mat´eriaux, ESPCI Paris,
+PSL Universit´e, CNRS, Sorbonne Universit´e, 75005 Paris, France
+2Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA
+3Purdue Quantum Science and Engineering Institute, West Lafayette, IN 47907, USA
+4Department of Physics and Astronomy, University of Denver, Denver, Colorado 80208, USA
+5Department of Physics and Center for Advanced Nanoscience,
+University of California San Diego, La Jolla, California 92093, USA
+(Dated: Thursday 12th January, 2023)
+We have developed a new optical microscopy technique able to track micron-sized surface clusters
+as temperature is varied. Potential candidates for study include phase separated metal-insulator
+materials, ferroelectrics, and porous structures. Several key techniques (including autofocus, step
+motor/cross correlation alignments, single-pixel thresholding, pair connectivity correlation length
+and image convolution) were implemented in order to obtain a time series of thresholded images.
+Here, we apply this new method to probe the archetypal phase separated insulator-metal transition in
+VO2. A precise time and temperature series of the insulator-metal transition was achieved, allowing
+us to construct for the first time in this material spatial maps of the transition temperature Tc.
+These maps reveal the formation of micron-sized patterns that are reproducible through multiple
+temperature sweeps within ∼0.6°C, although a few isolated patches showed Tc deviations up to
+±2°C. We also derive maps of the local hysteresis widths ∆Tc and local transition widths δTc.
+The hysteresis width maps show an average width of ∆Tc =4.3°C, consistent with macroscopic
+transport measurements, with, however, small regions as low as ∆Tc∼[0°C-1°C], and as high as
+8°C. The transition width δTc maps shows an average of 2.8°C and vary greatly (from 0°C to
+8°C), confirming the strong inhomogeneities of Tc in the subpixel structure. A positive correlation
+between Tc value and hysteresis width ∆Tc is observed by comparing the spatial distributions of each
+map. Finally, individual pixels with unique transition characteristics are identified and put forward.
+This unprecedented knowledge of the local properties of each spot along with the behavior of the
+entire network paves the way to novel electronics applications enabled by, e.g., addressing specific
+regions with desired memory and/or switching characteristics, as well as detailed explorations of
+open questions in the theory of hysteresis.
+I.
+INTRODUCTION
+Electronic phase separation commonly emerges in a
+wide variety of quantum materials such as high-Tc su-
+perconductors [1], colossal magnetoresistance mangan-
+ites [2], insulator-metal transition (IMT) materials [3],
+multilayer rhombohedral graphene [4],etc. An archety-
+pal example of a phase-separated material is vanadium
+dioxide, VO2, which undergoes a 1st order IMT at Tc
+∼68°C [5] (i.e., just above room temperature) accompa-
+nied by an abrupt several-order-of-magnitude resistivity
+decrease and monoclinic-to-tetragonal structural change.
+The exact nature of the transition, whether it is a Peierls
+transition driven by electron-phonon interactions or a
+Mott-Hubbard transition driven by electron-electron in-
+teractions, is still under debate [6]. In the vicinity of the
+transition, VO2 exhibits a spatial coexistence of metal
+and insulator domains that form intricate patterns [7].
+Analyzing the shape, characteristic size and scaling prop-
+erties of those patterns can yield valuable information
+∗ lionel.aigouy@espci.fr
+† ewcarlson@purdue.edu
+‡ azimmers@espci.fr
+about the fundamental interactions that drive the tran-
+sition [8]. Therefore, understanding and controlling the
+phase-separate state in quantum materials has become a
+major research field in recent years [9].
+Currently, phase separation imaging in quantum mate-
+rials reported in the literature mostly comes from scan-
+ning probe techniques such as STM [1, 2] and s-SNIM
+[7, 8].
+While these methods have a very high spatial
+resolution, fine temporal resolution remains hard to im-
+plement since scanning probes are very time-consuming.
+Moreover, STM lacks resolution at room temperature
+and loses registry as the temperature is changed [10]. To
+solve this we have developed a new microscopy method
+to map out clear and stabilized images of the IMT. This
+optical method allows the precise filming of the transi-
+tion with hundreds or even thousands of images taken
+in quick succession (∼10 seconds per final image). This
+allows us to not only follow fine details in the time evo-
+lution of the metal-insulating patches but also to filter
+out thermal noise if needed. We first describe the sam-
+ple preparation and optical response. We then describe
+the experimental steps necessary to achieve this map-
+ping.
+While most steps are straightforward, four new
+crucial steps were keys to this study: “Height z focusing”,
+“Single pixel time traces”, “Pair connectivity correlation
+arXiv:2301.04220v1  [cond-mat.str-el]  10 Jan 2023
+
+2
+length” and “Time domain convolution”. These techni-
+cal developments allowed us to acquire accurate spatial
+maps of transition temperature distribution, from which
+the phase separation patterns can be easily obtained at
+any given temperature. The Tc maps reveal multiple in-
+teresting features including the presence of spots with an
+extremely large or nearly absent hysteresis of the IMT, a
+positive correlation between the Tc value and the hystere-
+sis width, and high cycle-to-cycle reproducibility of the
+transition. The detailed knowledge of local properties is
+the necessary ingredient to develop and test basic phase
+separation and hysteresis theories, as well as to gain mi-
+croscopic understanding of the device performance for
+practical applications of quantum materials.
+II.
+METHODS
+A.
+VO2 thin film epitaxy, resistivity, and
+reflectivity
+Vanadium dioxide thin films were prepared by reactive
+RF magnetron sputtering of a V2O3 target (>99.7%, ACI
+Alloys, Inc.) on an r-cut sapphire substrate. Sample A is
+130nm thick and sample B is 300nm thick. A mixture of
+ultrahigh purity (UHP) argon and UHP oxygen was used
+for sputtering. The total pressure during deposition was
+4mTorr, and the oxygen partial pressure was optimized
+to 0.1mTorr (2.5% of the total pressure). The substrate
+temperature during deposition was 600oC while the RF
+magnetron power was kept at 100W. Grain size in these
+films is typically found to be 40-130nm in 100-150nm
+films [11]. Grain size is expected to typically be slightly
+larger in the 300nm film. The sample is found to have
+a relative 27% optical change in the visible range when
+passing the IMT (see SI Sec.S1 for details). Gold elec-
+trodes were deposited on top of the film, separated by
+10µm (sample A) and 30µm (sample B). Both samples
+showed a clear IMT (see Fig. S1) above 68oC as evidenced
+by a drop in resistivity of 4 orders of magnitude [12].
+B.
+Image/temperature recording
+The optical experimental setup consists of a VO2 thin
+film sample placed on a Peltier heater or a Linkam
+Thms350V temperature controller inside a Nikon opti-
+cal microscope in epi configuration (both the illumina-
+tion and reflection of light travel through the same objec-
+tive). Illumination in the visible range was used (halogen
+lamp, no filters) [13]. Two surface sample images (sample
+A 10µm×50µm and sample B 30µm×35µm) were mea-
+sured around the focal point of 1mm in the visible range
+using a ×150 magnification dry Olympus objective lens
+with an optical aperture of NA = 0.9. The theoretical
+lateral resolution is estimated to be δr= 1.22λ/(2 NA) =
+370nm in the visible range using the Rayleigh criterion
+[14].
+Temperature was measured using a Pt100 glued
+next to the sample. Temperature sweeps (35oC≪Tc to
+82oC≫Tc and back) spanning the entire IMT were per-
+formed multiple times at a rate of 1°C/min, temperature
+swept linearly, with temperature and images recorded ev-
+ery ∼0.17°C.
+C.
+Height z focusing and x-y drift correction
+Inevitable temperature dilation of the experimental
+system during temperature sweeps brings the sample out
+of focus during temperature sweeps. In order to com-
+pensate for this z drift, we employ a “fuzzy focusing”
+technique as follows. During the experiment, the sam-
+ple was continually moved up and down 10µm every 10
+seconds by a piezoelectric crystal placed under it, in or-
+der to bring the sample in and out of focus.
+A stack
+of 120 images was recorded this way for each tempera-
+ture. Over the years, various metrics have been evaluated
+for selecting the sharpest image in such a stack [16–18].
+Some studies focus explicitly on images that don’t have
+sharp contrast [19], like the raw images acquired here (see
+Fig. 2(m)). Most metrics reported perform well in select-
+ing the focused image. We have first chosen one using the
+compression rate of the recorded images [20]. This one is
+based on the intuitive idea that, when very out of focus,
+the sample surface will look homogeneously gray due to
+blurring. In this case, the raw recorded Bitmap (BMP)
+image can be highly compressed in lossless Tiff format
+using a standard Lempel-Ziv-Welch (LZW) compression
+protocol [21, 22], since nearly every pixel is the same.
+On the contrary, when the sample is in focus, the image
+contains much more information (since most pixels are
+different from their neighbors), and the raw BMP im-
+age cannot be compressed as much. Using this method,
+one can determine the most sharply focused image in the
+stack by selecting the one with the largest Tiff file size
+[23, 24]. Among the 62,000 images of sample A acquired
+during the 14 hour experiment (consisting of 3 major
+temperature loops and 10 subloops [25]), we retain the
+894 images that are in focus within 80nm.
+A recent update of the microscope has allowed us to
+select the best focused image of sample B during the
+experiment. In the live selection process we have used
+a computationally faster method based on image gra-
+dient using the Tenengrad function [19]. Both metrics
+cited above were vetted using micron-sized gold disks
+lithographed on a glass substrate where the sharpest im-
+age can be defined as the image with the sharpest step
+function (gold to substrate).
+Using the focusing stack
+technique, we have also compared the image height on
+the sample four corners. This allowed us to correct the
+tilt of the sample (due to sample positioning using ther-
+mal paste). The updated setup also uses a piezoelectric
+PI Pifoc PD72Z1x to move the objective up and down
+rather than moving the sample placed inside the Linkam
+stage. The current setup can thus output an image ev-
+ery 10s in focus on the full field of view as a function of
+
+3
+FIG. 1.
+Schematics of the microscope and image analysis created specifically to measure spatial maps of clusters in VO2
+during the IMT while recording resistivity R(T) simultaneously. The sample was positioned on a Peltier heater or Linkam
+Thms350V temperature controller to apply temperature ramps (bottom left). The sample height was varied by steps of 80nm
+via a piezoelectric actuator placed under it. The best-focused images were chosen post-experiment using an image compression
+method and Tenengrad function (described in Sec. II C). The height focus of the sample was thus controlled within 80nm
+throughout the experiment. Fine xy plane drift correction within a single pixel was performed post-experiment (described in
+Sec.II C). Camera sensitivity was normalized throughout the recording (described in Sec.S3 of the SI). Using this fully stabilized
+image series, black and white thresholds were applied for each pixel individually, accurately determining if it is in the metallic or
+insulating state (described in Sec. II D). We use this information to construct spatial maps of the local transition temperature
+Tc, hysteresis width ∆Tc and transition width δTc.
+temperature.
+As the temperature is cycled repeatedly, in addition
+to drifts along z-axis (perpendicular to the film), there
+are also drifts in the xy plane (the plane of the film).
+These thermal drifts were compensated: (i) live within
+1µm using step xy motors below the sample and (ii) post
+experiment using cross correlation to track and realign
+part of the gold leads which contain imperfections (spots)
+and rough edges with VO2 (see Fig. 5 (a)). Although
+the lateral image resolution is limited by diffraction and
+is estimated to be 370nm, the drift compensation tracks
+each pixel (≈ 37nm wide) on the sample throughout the
+whole experiment.
+The remaining spatial variations we observe in re-
+flected intensity from the VO2 region are primarily due
+to changes in local reflectivity due to the IMT. However,
+there can be other contributions to this spatial varia-
+tion, including effects such as surface height variations
+from sample warping, variations in film thickness, minor
+surface defects, and even shadows cast from the 150nm
+thick gold leads. There can even be differences in pixel
+sensitivity in the camera itself.
+Because each of these
+contributions is independent of temperature (i.e.
+con-
+stant in time), their effects can be distinguished from
+that of the temperature driven IMT, as described in the
+next section.
+D.
+Single pixel scaled and binary thresholded
+images
+In order to isolate the changes in local reflectivity
+which are due to the IMT, we introduce two novel image
+processing techniques. We use single pixel time traces to
+generate single pixel scaled images (panel (n) of Fig. 2),
+as well as binary thresholded images (panel (o) of Fig. 2,
+discussed in the following subsections). Both types of im-
+ages begin by considering a full warming or cooling sweep
+(i.e. from fully insulating to fully metallic, or vice versa)
+to follow the intensity and analyze each pixel individu-
+ally. As an example, Fig. 2 (a-l) shows the raw optical
+intensity time/frame traces of 12 different pixels during
+a cooling sweep. See S6 for the time traces of 1600 pix-
+els from the center of the sample. In order to construct
+
+CcD camera
+Height z focusing
+x-y drift correction
+Light
+Microscope
+Image sensitivity drift correction
+source
+R(T)
+Single pixel intensity time trace
+z piezoelectric
+heater
+Single pixel thresholded image
+stage
+VO2
+Temperature ramp
+9
+80
+Temperature (
+60
+40
+△Tc map
+STcmap
+1
+2
+3
+Tc map
+4
+Time (Hrs)4
+FIG. 2. Single pixel intensity normalization and thresholding process. (a-l) Representative single-pixel turn-on functions in
+sample A during cooling. Blue traces are the raw intensity in 8-bit grayscale where 0 is black and 255 is white. The orange
+traces are smoothed versions of the blue traces, in which we have applied an 11-point Gaussian convolution (σ=2.5). Purple
+curves are the difference between the raw (blue) curve and the smoothed version (orange curve). The green curve is a numerical
+derivative of the blue curve (discussed and used in SI Sec. S4), taken via a finite difference with a 10-point stencil [15]. (m)
+Raw optical image (frame 847) partway through cooling for VO2 sample A. (n) The same image after the intensity is scaled,
+pixel-by-pixel, such that light pixels are in the insulating phase and dark pixels are in the metallic phase. (o) The same image,
+with metal and insulator domains, clearly delineated as black and white. Images are 7.3µm wide.
+a single pixel scaled image, we normalize each individual
+pixel’s 8-bit grayscale intensity time trace with respect to
+itself, such that its maximum intensity is scaled to 1, and
+its minimum intensity is scaled to 0. The resulting single
+pixel scaled image is shown in Fig. 2(n). This type of im-
+age is a relatively quick way to study the temperature de-
+pendent IMT, as it eliminates temperature-independent
+spatial variations that are not due to the IMT.
+In order to construct a binary thresholded image which
+clearly delineates metal and insulator domains, we must
+define a criterion for when each pixel changes from metal
+to insulator or vice versa. The orange curve in each of
+the panels (a-l) in Fig. 2 is a Gaussian-smoothed version
+of the raw time trace, using an 11-point Gaussian convo-
+lution (σ=2.5). We use this smoothed time trace of the
+intensity in order to determine the midway point inten-
+sity for each individual pixel (shown by the red horizon-
+tal dotted lines). We use the pair connectivity correla-
+tion length to justify setting the threshold at midway, as
+described in the following subsections (Secs. II D 1 and
+II D 2).
+This allows us to construct binary black and
+white images of the metal and insulator domains at each
+measured temperature, as shown in Fig. 2(o). Different
+pixels go through the midway point at different frame
+numbers, and therefore at different temperatures. We use
+this information to construct spatial maps of the local
+transition temperature Tc recorded at each pixel reveal-
+ing the highly spatially-textured nature of the IMT in
+VO2 [7, 8]. These Tc maps, as well as hysteresis width
+∆Tc maps and transition width δTc maps, are presented
+in the experimental results Sec. III.
+1.
+Pair Connectivity Correlation Length
+As can be seen in the single pixel time traces shown in
+Fig. 2 (see SI Figures. S6 for many more examples), each
+pixel experiences a definite switch from metal to insula-
+tor or vice versa, consistent with the Ising-type model we
+have previously developed to describe the IMT in VO2
+thin films [8, 26]. While the Ising model was originally
+developed to describe magnetic domains of orientation
+“up” or “down”, here we map “up” and “down” to metal
+and insulator domains. While the metal-insulator tran-
+sition is first order, this transition ends in a critical point
+as a function of quenched disorder. The influence of that
+critical point is felt throughout a critical region, which
+includes part of the first order line in the vicinity of the
+critical end point.[8] We use the correlation length of the
+pair connectivity correlation function to determine the
+threshold between metal and insulator domains.
+Dur-
+ing the IMT, VO2 metal-insulator domains form intri-
+
+Horizontal pixel location
+[40]
+[80]
+[120]
+150
+Raw
+Convolved (11pt)
+[400]
+Derivative (1lpt)
+a)
+b)
+Convolved-Raw
+c)
+75
+(Min+Max)/2
+Max Slope
+0
+150
+Vertical pixel location
+[300]
+Pixel intensity
+d)
+e)
+f)
+75
+0
+150
+[200]
+g)
+h)
+75
+0
+150
+[100]
+j)
+k)
+D)
+75
+0.
+840
+880
+840
+880
+840
+880
+Frame NumberRaw
+Single Pixel Scaled
+Single Pixel Threshold
+Image
+[Min,Max]-->[0,1]
+(Min+Max)/2
+m)
+h
+a
+b
+400
+300
+9
+h
+200
+100
+40
+80
+120
+40
+80
+120
+40
+80
+1205
+FIG. 3. Pair connectivity correlation length ξpair vs. temper-
+ature during the warming branch of an extremal hysteresis
+loop, as a function of different threshold values for determin-
+ing metal and insulator domains in sample A. The correlation
+length diverges when the system is closest to criticality.
+cate patterns, often becoming fractal due to proximity
+to a critical point [8]. At criticality, correlation lengths
+diverge. Away from criticality, the divergence is muted,
+although the correlation length still displays a maximum
+at the point of closest approach to criticality. For exam-
+ple, changing the interaction strength between metal and
+insulator domains to be farther away from criticality, or
+changing the strength of various types of disorder farther
+from criticality causes the correlation length to go down.
+Similarly, changing the intensity threshold by which we
+identify metal and insulator domains also changes this
+correlation length. In disordered systems, setting an un-
+physical threshold will not move the system toward crit-
+icality, but only away.
+Therefore, one way to set the
+proper threshold between metal and insulator domains is
+to maximize the correlation length.
+The pair connectivity correlation function is familiar
+from percolation models, where the corresponding pair
+connectivity correlation length diverges at the critical
+point [27]. Coniglio and coworkers showed that the pair
+connectivity correlation length also diverges at the criti-
+cal temperature in the two-dimensional Ising model [28].
+We have recently shown that the pair connectivity corre-
+lation length also diverges at other Ising critical points,
+including that of the two-dimensional random field Ising
+model [29], as well as on slices of three dimensional mod-
+els at criticality, including the clean Ising model [30] and
+the random field Ising model [29]. Near a critical point,
+the correlation function is power law at distances less
+than the correlation length, in this case ξpair. This pair
+correlation length can be calculated directly from an im-
+age via [31]:
+ξ2
+pair =
+�
+i,j r2
+i,jpf
+i,j
+�
+i,j pf
+i,j
+(1)
+FIG. 4.
+(a) Single pixel time trace of intensity.
+The blue
+curve is the raw time trace of the measured optical intensity
+of pixel (127,734) in sample B. The orange curve is a Gaussian
+convolution (σ=2.5) of the same time trace over 3 frames. The
+double crossing at the midway is eliminated in the smoothed
+data set. (b) Binary black and white image (frame 260) of the
+sample generated by thresholding at midway the single pixel
+time traces as presented in (a).
+(c) Smoothed out binary
+black and white image (frame 260) of the sample generated
+by thresholding at midway the 3 frame convoluted single pixel
+time traces as presented in (a).
+where pf
+i,j is the likelihood that i and j are in the same
+finite cluster. Another way to view this is as:
+ξpair =
+�
+⟨R2
+G⟩f
+(2)
+where RG is the radius of gyration of each connected
+cluster, and the average is taken over the finite clusters.
+This quantity diverges at the percolation threshold as:
+ξpair ∝
+1
+|p − pc|νpair .
+(3)
+It diverges at clean Ising transitions as:
+ξpair ∝
+1
+|T − Tc|νpair ,
+(4)
+and it diverges at random field Ising transitions as:
+ξpair ∝
+1
+|R − Rc|νpair .
+(5)
+
+2.5
+Threshold
+-10%
+2.0
+-7.5%
+Correlation Length [μm]
+-5%
+-2.5%
+1.5
+Midway
++2.5%
++5%
+1.0
++7.5%
++10%
+0.5
+0.0
+45
+50
+55
+60
+65
+70
+75
+Temperature [oC]a
+Raw
+90
+Conv (3pt)
+(Min+Max)/2
+85
+80
+Pixel Intensity
+75
+70
+65
+60
+55
+0
+100
+200
+300
+400
+500
+600
+Frame Number
+3 point convoluted
+Raw binary image
+binary image6
+2.
+Setting Thresholds of Metal and Insulator Signal in
+Optical Data
+In order to know at what intensity to set the threshold
+between metal and insulator in each pixel, we calculate
+the pair connectivity correlation length in a series of im-
+ages, as a function of different intensity thresholds. For
+this we use the single pixel scaled images as described in
+the previous subsection. In Fig. 3, we plot the evolution
+of the pair connectivity correlation length (Eqn. 1) during
+the warming branch of a hysteresis loop. The blue circles
+in Fig. 3 have each pixel’s threshold set at the midway
+point of that particular pixel’s intensity. The black circles
+have each pixel’s threshold set higher by an amount that
+is +10% of the difference between the saturated metal
+and saturated insulator values of intensity. The pink cir-
+cles have each pixel’s threshold set higher by only +7.5%,
+and similarly for other colors as denoted in the figure leg-
+end. Similar to the way the theoretical threshold was set
+in Ref. [8], we set the threshold according to the longest
+correlation lengths. Since in Fig. 3 the longest correla-
+tion length happens for a threshold equal to the average
+between metal and insulator intensity (the blue circles
+in Fig. 3) we use this midway threshold throughout the
+paper.
+E.
+Time domain convolution
+One of the strong points of obtaining a series of 100-
+1000 images via this autofocus optical microscope is the
+possibility of filtering out high frequency noise. A simi-
+lar technique is used in resistivity experiments that probe
+samples thousands of times per second. Fig. 4 (a) com-
+pares a raw single pixel time trace to a smoothed ver-
+sion in which a 3-point Gaussian convolution (σ=2.5)
+has been applied in the time domain. In this example,
+the raw single pixel time trace crosses the midway point
+twice, whereas the 3-point convolved curve passes the
+midway point only once. Notice that this procedure of
+filtering high frequency noise in the time domain greatly
+suppresses the white noise evident in the spatial domain
+near the metal-insulator boundaries derived from the raw
+time traces (see Fig. 4 (b) and (c) for comparison). This
+smoothing is useful for studying spatial correlations from
+frame to frame. However, if filtering is not necessary, raw
+data is used throughout the analysis. This is the case for
+Tc maps in the section below and ramp reversal memory
+maps presented elsewhere [25]. High frequency noise was
+filtered in the temperature data taken using the Pt100
+by fitting a linear slope through the large temperature
+sweeps. This matched the internal temperature sensor
+slope of the Linkam Thms350V temperature controller.
+III.
+RESULTS
+Having described the various key steps in the previ-
+ous sections (including autofocusing, step motor/cross
+correlation aligning, single pixel scaling and threshold-
+ing, pair connectivity correlation length analysis, and
+time domain convolution) we now present the detailed
+spatially-resolved study of the IMT in VO2 films using
+our new optical mapping method.
+Maps
+Transition Temperature Tc maps: Fig. 5 (c) re-
+ports the local critical temperature Tc map in VO2 sam-
+ple B. These maps show a large spatial variation in Tc,
+with rich pattern formation over tens of microns, similar
+to s-SNIM sub-micron measurements [7], but acquired
+with a much faster procedure that allows for much finer
+time and temperature resolution. This large scale spatial
+variation, along with detailed spatial knowledge of the lo-
+cation of these variations, can potentially be exploited to
+optimize memory elements by addressing specific regions
+of the sample.
+Reproducibility of Tc maps:
+Previous reports on
+avalanches in this material showed jumps in resistivity
+randomly appearing during the transition in macroscopic
+transport measurements [33].
+This suggested that the
+metal-insulator patterns could be appearing randomly
+during each temperature sweep.
+At first glance, this
+appears to be at odds with the optical data reported
+in this study, where we find that the metal and insu-
+lator patterns are highly repeatable globally (occurring
+at the same location and with the same shape) during
+successive temperature sweeps (see Fig 6). The repeata-
+bility suggests that the patterns are strongly influenced
+by an underlying random field present in the thin film
+or its substrate [8, 26, 34]. The observed stochasticity
+of resistance jumps in transport measurements [33] could
+arise from small variations in the exact time at which
+avalanches are triggered. In addition, small changes in
+optical maps can potentially create large changes in re-
+sistance, when tiny “shorts” connect pre-existing larger
+metallic clusters.
+Transition Width δTc maps: The transition width
+δTc of each pixel can be accessed by fitting single pixel
+scaled intensity time traces to a hyperbolic tangent:
+− 1
+2(tanh( T−Tc
+δTc )-1). Because Tc is known from our time
+trace analysis, there is only one fitting parameter. The
+map of δTc distribution is shown in Fig. 5 (e). The aver-
+age transition width of the pixels as measured in optics
+is 2.8 ± 1.1°C with extremes from 0°C to 8°C. Moreover,
+a small number of pixels show more than one step dur-
+ing a transition (see for example first pixel (305,300) in
+Fig. S6). These cases could arise from an overlap be-
+tween multiple metal or insulator domains affecting a
+single pixel. This could be due to information from sur-
+rounding pixels affecting the signal at one pixel, since the
+
+7
+FIG. 5. (a) Optical image of VO2 sample B during the insulator (light gray) to metal (dark gray) transition (warming cycle),
+two gold leads are seen at the top and bottom.
+These electrodes also display some structure (spots) due to gold surface
+imperfections. Contrary to VO2 IMT structures seen in this image, gold imperfections do not change with time (see online
+movie [32]). Usually these imperfections are purposely washed away using strong image brightness. Here, on the contrary,
+brightness was set low to see and use these imperfections to autoalign within a pixel the images and thus compensate xy
+thermal drifts.
+Sapphire substrate is the dark surface.
+One can easily see the metal dark patches appearing.
+Scale bar
+is 10µm. (b) Single pixel intensity curve defining critical temperature Tc, hysteresis width ∆Tc and transition width δTc.
+Tc were determined at midways as explained in the main text. Hysteresis width was determine by taking the temperature
+differences between heating and cooling cycles Tc
+up-Tc
+down. Transition width was determined by fitting (smooth curve) the
+single time trace to a hyperbolic tangent: − 1
+2(tanh( T −Tc
+δTc )-1). (c) Local critical temperature Tc map, (d) ∆Tc maps, (e) δTc
+map (presented here for the temperature ramping up branch). Image are 27.6µm high. Histograms (with mean and standard
+deviation of maps a), b) and c) are shown in Fig. 7
+pixel size is ∼10 times smaller than the resolution. Or,
+it could arise from structures that are smaller than the
+pixel size. Indeed, s-SNIM has clearly observed inhomo-
+geneities on smaller length scales than the optical maps
+presented here [7, 8]. Interestingly, the standard devia-
+tion of local Tc’s across the sample, σTc(1.2°C), is smaller
+than the average transition width of pixels δTc(2.8°C). It
+remains an open question whether the self-similar metal-
+insulator domain patterns discussed in Ref. [8] could be
+the source of this difference.
+Hysteresis Width ∆Tc maps: By subtracting Tcup-
+Tcdown (see the caption of Fig.5 (b) for the definition) one
+can construct a hysteresis width ∆Tc map. The hystere-
+sis width ∆Tc map is shown in Fig. 5 (d) for sample B.
+The average width is found to be 4.3 ± 1.1°C as seen in
+macroscopic transport measurements. However, certain
+small regions have small ∆Tc, in the range [0°C - 1°C]
+(small blue clusters in Fig. 5 (d)). Probing these region
+with other local probes could shed light on whether this is
+an intrinsic property of these regions. These hysteresis-
+free patches could be very useful in multiple switching
+applications such as optical electronic devices. Indeed it
+has been shown that the presence of a large hysteresis
+in VO2 greatly complicates using it as an optical sensor
+[35].
+Correlations between maps
+With all of the maps above, one can check for cor-
+relations between these quantities. Fig. 8 plots Tc vs.
+∆Tc, ∆Tc vs. δTc and Tc vs. δTc for each pixel. A few
+horizontal and diagonal lines appear in these plots. The
+horizontal lines come from multiple pixels (spatially close
+by) switching at the same temperature (upon warming).
+The diagonal lines come from multiple pixels (spatially
+close by) switching at the same temperature (upon cool-
+ing). Although this is typically what one would expect
+
+1.2 
+b)
+a
+ Single pixel scaled intensity
+1.0
+Single pixel scaled
+0.8
+intensity time trace
+0.6
+△T
+T,down
+dn
+ 0.4 
+0.2 
+0.0
+2 STc
+-0.2
+50
+60
+70
+80
+Temperature °C
+T_map
+△T_map
+ST_map
+c)
+T [°C]
+d)
+T [°C]
+e)
+T [°C]
+C
+C
+72
+8
+12
+7
+71
+10
+70
+5
+8
+69
+4
+68
+6
+3
+67
+2
+4
+66
+2
+65
+0
+0
+648
+FIG. 6. a) Three Tc maps while cycling through the IMT (warming) at 1°C/min. b) Difference maps between cycles. Global
+patterns are generally reproducible (σTc/Tc = 0.6°C/68°C= 1%). However some small regions present deviations up to ±2°C.
+Full histograms (with mean and standard deviation) of maps in b) are shown in Fig. 7. Difference map between Tc3 and Tc1
+(the most separated, time wise, temperature sweeps in this study) and the corresponding histogram are presented in SI Fig. S4.
+Images are 33.6µm x 27.6µm.
+from avalanches, further analysis is needed to extract the
+full dynamics occurring. In the three correlation maps,
+no trend is seen in the last two, but Tc vs. ∆Tc shows a
+slight positive correlation. This means that pixels with
+low Tc tend to have low ∆Tc (i.e. close to zero) and vice
+versa. The positive correlation in Fig. 8(a) is not to be
+confused with the few diagonal lines present in this panel
+explained just above.
+Hand picking specific hysteric properties
+The wide range of behaviors contained in the three
+maps presented in the section above (Fig. 5 c, d and e),
+gives us the unprecedented opportunity to find individual
+pixels with desired properties. Fig. 9 shows the Tc map of
+the sample with six different types of pixels selected. The
+pixel labeled “std” for standard has a rounded transition
+with values of Tc, ∆Tc and δTc which are close to the
+average values found in the distribution of these three
+quantities (see Fig. 7 a, b and c).
+Pixels A and B show the most common type of local
+characteristics found in the maps: when Tc is high, ∆Tc
+is high; when Tc is low, ∆Tc is low. This positive correla-
+tion is evident at a global level in Fig. 8 (a). However, on
+a local level, individual pixels can have a large deviation
+from the global average behavior. Indeed pixel E shows
+a possibility of finding ∆Tc very low (0.3°C) with a Tc
+(66.3°C) low but closer to the mean value of the map.
+Pixels C and D illustrate the case where the width δTc
+of the transition is very sharp (0.5°C) or very wide (5°C).
+Pixel C shows a representative sharp pixel, where within
+the temperature steps of 0.17°C, the transition occurs in
+a sharp, avalanche mode. Further analysis to see where
+and how these avalanches occur will be pursued in future
+work.
+Finally pixel E shows a case where ∆Tc is within the
+lower values [0°C-1°C]. As mentioned previously, small
+hysteresis could be useful in opto-electronic devices or
+neuromorphic devices. In the first case, small hysteresis
+avoids optical detectors getting stuck in subloops [35];
+in the second case, small hysteresis allows lowering the
+voltage threshold needed for spiking [36].
+General remarks on the pixel selection procedure: (i)
+as mentioned previously in the δTc section above, some
+pixels in the map clearly present two steps during the
+IMT. These two-step pixels can potentially be detected
+in an automated way from their anomalously high error
+on the fit to the hyperbolic tangent function; (ii) the fea-
+tures put forward in these 6 pixels above are not unique
+to the 37nm square pixel location. These features usu-
+ally also hold for many pixels around the xy coordinates
+reported.
+
+Cycle # 2 - T.2map
+72
+a
+71
+69
+68
+67
+65
+64
+c2
+2
+b
+T [°C]
+3
+2
+1
+0
+-1
+-2
+39
+FIG. 7. Histograms of maps presented in in Fig. 5 and 6. (a) Tc maps (upon warming); (b) ∆Tc map; (c) δTc map and (d)
+and (e) two difference maps Tc2-Tc1 and Tc3-Tc2
+FIG. 8. Correlations between Tc (upon warming), ∆Tc and δTc. Each of the 666,000 pixels (900x740) is represented. Only
+Tc vs. ∆Tc (panel (a) shows a slight diagonal trend meaning that pixels with low Tc tend to have low ∆Tc (i.e. close to zero)
+and vice versa.
+IV.
+CONCLUSIONS
+We have reported the first Tc maps derived from sin-
+gle pixel optical imaging on VO2. Multiple new exper-
+imental steps were needed to align, focus and calibrate
+the raw grayscale images recorded. These experimental
+achievements allowed us to accurately track the spatial
+distribution of metal and insulator clusters. Binary black
+and white images, time traces, Tc maps, ∆Tc maps, and
+δTc maps were plotted and discussed. The sample shows
+micron-sized patterns that are found to be mostly repro-
+ducible through multiple temperature sweeps. The ∆Tc
+hysteresis width map exhibits, on average, the same av-
+erage hysteresis width of 4.3°C as macroscopic resistiv-
+ity hysteresis, but exhibits strong variation on a local
+scale, down to ∼[0°C-1°C] in certain small regions and
+as large as ∼ 8°C in other regions. These findings open
+an exciting opportunity to access local properties of VO2
+by, e.g., contacting specific parts of the sample electri-
+cally in order to select unique parameter combinations
+
+20
+80
+80
+b)
+a)
+c)
+75
+75 -
+15
+P0%
+p
+70
+70
+10
+ooo
+65
+65
+5
+60 
+60
+0 :
+55
+55
+0
+5
+10
+15
+20
+0
+10
+12
+14
+2
+4
+6
+8
+10
+12
+14
+2
+4
+6
+8
+0
+△T, [°C]
+[°C]
+ST
+ST
+Cx104
+x104
+a)
+c)
+μ= 2.8 [°C]
+μ= 68.2 [°C]
+μ= 4.3 [°℃]
+6
+5.
+= 1.1 [C]
+ = 1.2 [°℃]
+ = 1.1 [°℃]
+4
+ pixels
+5
+4
+4
+3
+3
+Number
+3
+2
+2
+2
+1
+1.
+1
+0
+0
+64　66　68　70　7274
+2
+４6
+234５
+62
+0
+8
+10
+0
+678
+△T, [°C]
+T,[°C]
+ST,[°C]
+x104
+x104
+d)
+e)
+10-
+μ= 0.0 [°℃]
+μ= 0.0 [C]
+ = 0.6 [°C]
+= 0.6 [℃C]
+8
+8.
+Number of pixels
+6
+6
+4
+4.
+2
+2
+0
+0+
+1234
+-4-3 -2 -1 0
+-4-3 -2 -1
+１234
+T.,-T., [C]
+.[°C]
+c210
+FIG. 9. Tc map with six pixels chosen to illustrate specific characteristics in the hysteresis loops. The table shows the numerical
+values of Tc, ∆Tc and δTc for each pixel. The numbers in bold highlight the unique characteristic of each pixel.
+for specific applications in electrical and optoelectronic
+devices.
+The observation of a positive correlation be-
+tween Tc value and hysteresis width could enable a new
+approach for tailoring the material’s response to exter-
+nal drives, in addition to providing a new perspective in
+studying open questions in the theory of hysteresis.
+ACKNOWLEDGEMENTS
+We thank M. J. Carlson for technical assistance with
+image stabilization, and acknowledge helpful conversa-
+tions with K. A. Dahmen. S.B., F.S., and E.W.C. ac-
+knowledge support from NSF Grant No. DMR-2006192
+and the Research Corporation for Science Advancement
+Cottrell SEED Award. S.B. acknowledges support from
+a Bilsland Dissertation Fellowship.
+E.W.C. acknowl-
+edges support from a Fulbright Fellowship, and thanks
+the Laboratoire de Physique et d’´Etude des Mat´eriaux
+(LPEM) at ´Ecole Sup´erieure de Physique et de Chimie
+Industrielles de la Ville de Paris (ESPCI) for hospital-
+ity. This research was supported in part through com-
+putational resources provided by Research Computing
+at Purdue, West Lafayette, Indiana [37]. The work at
+
+D 
+0.8
+0.6 
+std
+Single pixel scaled intensity
+0.4 
+0.8
+0.2 
+0.0
+0.0 
+0.4
+50
+60
+70
+80
+40
+50
+09
+70
+Temperature [°C]
+Temperature [°C]
+0.2
+0.0
+40
+50
+70
+700
+60
+T_[C]
+Temperature [°C]
+72
+600
+71
+70
+500
+B
+69
+400
+68
+300
+67
+0.4
+66
+0.0 
+200
+50
+60
+0
+65
+Temperature [°C]
+100
+ 64
+A
+01
+0.6
+200
+300
+500
+600
+100
+400
+700
+800 
+900
+0.4
+E
+0.0
+50
+40
+60
+Temperature [°C]
+ 0.2 
+0.0 
+40
+70
+50
+60
+Temperature [°C]
+Label
+(x,y) position
+Specific
+T. [°℃C]
+△T。 [°℃]
+STc[°C]
+characteristic
+std
+(85 , 285)
+68.0
+4.1
+2.6
+Tc,△Tc, STc
+(standard)
+close to mean value
+A
+(34, 135)
+Low T / Low △Tc
+64.8
+3.5
+1.6
+B
+(0, 213)
+High T. / High △T.
+71.7
+7.2
+1.5
+c
+(506 ,440)
+Low oT.
+65.7
+3.8
+0.4
+D
+(670, 547)
+High T.
+64.2
+2.9
+5.1
+E
+(880 , 425)
+Very low △T。
+66.3
+0.7
+1.911
+UCSD (PS, IKS) was supported by the Air Force Office
+of Scientific Research under award number FA9550-20-
+1-0242. The work at ESPCI (M.A.B., L.A., and A.Z.)
+was supported by Cofund AI4theSciences hosted by PSL
+University, through the European Union’s Horizon 2020
+Research and Innovation Programme under the Marie
+Sk�lodowska-Curie Grant No. 945304.
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+T. Hacker and B. Yang,
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+strategy-for-research-communities.
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+(EPL) 70, 225 (2005).
+
+13
+SUPPORTING INFORMATION: CORRELATIVE
+MAPPING OF LOCAL HYSTERESIS
+PROPERTIES IN VO2
+S1.
+VO2 Reflectivity
+The fact that the metallic reflectivity of VO2 is lower
+than that of the insulating phase in the visible range is
+counterintuitive.
+This is due to a subtle combination
+of a Drude response as well as intraband and interband
+transitions and thin film interferences in this material.
+The largest reported spectra in VO2 was measured by
+ellipsometry [38].
+Using the reported real part of the
+optical conductivity σ1, we have calculated the reflectiv-
+ity of the insulator and metallic states (see Fig. S2 and
+S3).
+This clearly shows that, as one would expect in
+the infrared, the sample becomes highly reflective when
+metallic.
+Above the plasma frequency (∼12000cm−1),
+interband transitions and spectral weight conservation
+make the reflectivity curves cross, leading to the metallic
+state having a lower reflectivity than the insulating state
+in this range. The relative optical contrast in the visible
+range (27%), is still more than sufficient in our setup
+to identify both states clearly (as seen in a raw image
+Fig. S1 (a)).
+S2.
+Key steps making this study possible
+The key step that have allowed us completing this
+study comes from the unique qualities of the VO2 ma-
+terial :
+- The IMT is above room temperature, which allows
+close optical microscopy (strong objective ×150 with a
+high numerical aperture 0.9 brought to 1mm focus above
+the sample surface). This setup would be much harder
+to achieve if cryogenic cooling (i.e.
+a cryostat with a
+window between the sample and objective) was needed.
+- Phase separation was observed by s-SNIM at sub-
+micron scales in this material [7, 8]. The fact that this
+phase separation is still found up to 30µm makes these
+optical microscopy surface maps possible.
+- In the visible range, a relative 27% drop in the thin
+film reflectivity is found in the metallic state Measuring
+in the visible range gave us results with a 400nm reso-
+lution. In the infrared, the contrast between metal and
+insulator is much larger, as expected, but only allows
+optical resolution up to the IR wavelength, i.e. 1-10µm.
+FIG. S1. (a) 35µm wide etched VO2 sample B image with
+30µm separated gap gold leads. The white square represents
+the 33.6µm x 27.6µm region where Tc maps (Fig.s 5). Scale
+bar is 10µm.(a) R(T) measurement of the IMT
+
+(a)
+Gold
+Substrate
+vO2
+1000
+(b)
+100
+Resistance (kΩ2)
+10
+0.1
+0.01
+40
+50
+60
+70
+80
+Temperature (°C)14
+FIG. S2.
+Simulated optical reflectivity of the insulating and metallic states in bulk VO2.
+Optical functions were derived
+by fitting standard Drude-Lorentz functions to ellipsometry measurements reporting the raw σ1 response in a large spectral
+range at low and high temperatures [38]. This procedure [39] allows other optical functions to be deduced, such as reflectivity,
+transmission, absorption, or dielectric constant. Reflectivities in this figure are not reported below 1000cm−1 as the fitting
+procedure was not precise enough in this low frequency/high σ1 region. On the other hand, reflectivities in the visible region
+(∼14000cm−1 to ∼25000cm−1) are in the middle of the spectral range and can be found with confidence.
+FIG. S3. Simulated optical reflectivity of the insulating and metallic states of a 130nm VO2 thin film on an r-cut sapphire
+substrate. Optical functions were found as described in Fig. S2. In contrast with the bulk reflectivity, a pronounced oscillation
+can be seen in the blue insulating spectrum. This is due to interference in the 130nm thin film (for example, constructive thin
+film interference creates a peak at ∼6700cm−1). Reflectivities are not reported below 1000cm−1 as the fitting procedure was
+not precise enough in this low frequency/high σ1 region. On the other hand, reflectivities in the visible region (∼14000cm−1
+to ∼25000cm−1) are in the middle of the spectral range and can be found with confidence.
+
+photon energy (eV)
+0
+1
+2
+3
+4
+1.0
+Metal
+T=360K ab0ve T
+0.8
+Insulator T=295K below T
+G
+Reflectivity
+0.6
+0.4
+Microscope spectral range
+0.2
+0.0
+0
+10000
+20000
+30000
+Wavenumber (cm-1)photon energy (eV)
+0
+1
+2
+3
+4
+1.0
+Metal (thin film)
+T=360K ab0ve T
+0.8
+Insulator (thin film)
+T=295K below
+Reflectivity
+0.6
+0.4
+Microscope spectral range
+0.2
+0.0
+0
+10000
+20000
+30000
+Wavenumber (cm-1)15
+S3.
+Image sensitivity drift correction
+Whereas the relative average intensity of VO2 increases
+almost 30% in changing from metal to insulator, the
+change in sapphire reflectance in this temperature range
+is negligible. We have used this fact to correct for any
+changes in incident light or CCD detector sensitivity
+throughout the experiment by dividing the average in-
+tensity in the VO2 region by the intensity in the sapphire
+region of the sample.
+Details: We assume that the input intensity is a func-
+tion of time I0(t) but spatially uniform. The reflected in-
+tensity from any region is IR(t, x, y) = I0(t) × R(t, x, y).
+Since the Sapphire’s reflectance does not vary signifi-
+cantly over the range of temperature the sample went
+through, it is assumed to be a constant. Let the spatially
+averaged sapphire reflectivity be RS. Then, the spatial
+average reflected intensity from the sapphire region is:
+IS
+R(t) = I0(t) × RS Any region of VO2 has a reflected
+intensity: IV
+R (t, x, y) = I0(t) × RV (t, x, y).
+Therefore,
+the ratio of reflected intensities from Sapphire and VO2
+is independent of input intensity:
+IV
+R (t, x, y)/IS
+R(t) =
+RV (t, x, y)/RS. We will use IS
+R(t) as a reference to cor-
+rect IV
+R (t) for any variation due to fluctuation of ambi-
+ent light. The quantity independent of input intensity:
+RV (t, x, y) = RSIV
+R (t, x, y)/IS
+R(t), Hence, setting the ref-
+erence input intensity I0(0), the corrected reflected in-
+tensity from VO2 would be:
+˜IV
+R (t, x, y) = I0(0)RV (t, x, y) = IV
+R (t, x, y)
+IS
+R(t)/IS
+R(0)
+S4.
+Single pixel thresholded images: inflection
+point
+In the main text, we have set the threshold between
+metal and insulator domains at the midway point of
+the intensity, based on the pair connectivity correlation
+length criterion described in Sec. II D. We also tested
+another method of setting the threshold based on the
+inflection point of the single pixel time traces.
+The
+green curves in panels (a-l) of Fig. 2 show a smoothed
+derivative of the raw time traces, achieved by using a
+finite difference with a 11-point Gaussian convolution
+(σ=2.5) [15]. The vertical dotted green line shows the
+extremum of this derivative, which locates the inflection
+point of the orange curves.
+Since the pixel switching
+curves (orange and blue traces) exhibit a relatively rapid
+change from metal to insulator, this inflection point at
+which the pixel brightness is changing most rapidly is
+the most natural place to assign a change from insulator
+to metal and vice versa. Because we have used a stencil
+with even number of 10, the inflection point happens
+between frames, and allows us to clearly identify frames
+which precede the inflection point (which are metallic)
+from frames which come after the inflection point (which
+are insulating). Notice that the frame number at which
+the solid orange curves cross the dotted orange lines
+coincides with the inflection point for each pixel. This
+means that both methods are equivalent for determining
+the frame number at which a pixel switches from metal
+to insulator or vice versa.
+
+16
+FIG. S4. Map and histogram of the difference between Tc3 and Tc1. Although these Tc maps are the most separated, time
+wise, in this study, they remain similar (mean and σ) to Tc2-Tc1 and Tc3-Tc2 presented in Fig.6.
+FIG. S5. Online movie[32] screenshot of the ∼1500 in focus consecutive spatial maps of a 33.6µm x 27.6µm VO2 surface.
+Central panels: raw, scaled and thresholded surface image (sample B) using “Single pixel scaled image” and “Single pixel
+intensity time trace and threshold” methods. Left panels: corresponding histogram changes during temperature ramps. Top
+right panel: average sample intensity (raw, scaled, thresholded) vs. frame number. Middle right panel: average sample intensity
+(raw, scaled, thresholded) vs. sample temperature. Bottom right panel: Temperature protocol - 3 major temperature loop
+spanning the entire IMT (36oC - 82oC),
+
+[。l 1
+c3
+X104
+3
+10
+μ= 0.0 [°C]
+ = 0.6 [℃C]
+Z
+8
+Number of pixels
+6
+0
+4
+-1
+2
+-2
+0
+¥-3-2
+-1
+0
+1
+2
+3
+4
+4
+-3
+T.,-T.,[°C]
+C3×105
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+1.0
+2.5 
+1.0 
+80
+0.8
+2.0
+0.8 
+75
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+0.4
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+0.0 
+0.0 
+0.0
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+50
+60
+70
+80
+90
+0
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+400
+600
+800
+1000
+12001400
+Framenumber
+×104
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+3.5
+1.0
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+3.0
+0.8
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+2.5.
+75
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+0.6
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+1.5
+0.4
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+1.0 -
+0.2 
+田
+raw
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+thresholded
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+0.0
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+70
+80
+Temperature[°C]
+×105
+thresholded
+thresholded
+1.0
+80
+6
+0.8
+5 
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+Framenumber17
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+11pt Conv
+(min+max)/2
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+FIG. S6. ML3 time trace of sample A in a patch of 40×40 pixels in the middle of the sample. Each pixel coordinates are
+indicated above the time trace. Description of the four curves in each mini panel is the same as the main text Figure 2.
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+Published in Transactions on Machine Learning Research (08/2022)
+Exploring Efficient Few-shot Adaptation for Vision Trans-
+formers
+Chengming Xu
+cmxu18@fudan.edu.cn
+School of Data Science, Fudan University
+Siqian Yang
+seasonsyang@tencent.com
+Yabiao Wang
+caseywang@tencent.com
+Youtu Lab, Tencent
+Zhanxiong Wang
+maxzxwang@tencent.com
+Tencent
+Yanwei Fu∗
+yanweifu@fudan.edu.cn
+Xiangyang Xue
+xiangyangxue@fudan.edu.cn
+School of Data Science, Fudan University
+Reviewed on OpenReview: https: // openreview. net/ forum? id= n3qLz4eL1l
+Abstract
+The task of Few-shot Learning (FSL) aims to do the inference on novel categories containing
+only few labeled examples, with the help of knowledge learned from base categories containing
+abundant labeled training samples. While there are numerous works into FSL task, Vision
+Transformers (ViTs) have rarely been taken as the backbone to FSL with few trials (Hu
+et al., 2022; Evci et al., 2022; Abnar et al.) focusing on naïve finetuning of whole backbone
+or classification layer.
+Essentially, despite ViTs have been shown to enjoy comparable
+or even better performance on other vision tasks, it is still very nontrivial to efficiently
+finetune the ViTs in real-world FSL scenarios. To this end, we propose a novel efficient
+Transformer Tuning (eTT) method that facilitates finetuning ViTs in the FSL tasks. The
+key novelties come from the newly presented Attentive Prefix Tuning (APT) and Domain
+Residual Adapter (DRA) for the task and backbone tuning, individually. Specifically, in APT,
+the prefix is projected to new key and value pairs that are attached to each self-attention
+layer to provide the model with task-specific information. Moreover, we design the DRA in
+the form of learnable offset vectors to handle the potential domain gaps between base and
+novel data. To ensure the APT would not deviate from the initial task-specific information
+much, we further propose a novel prototypical regularization, which maximizes the similarity
+between the projected distribution of prefix and initial prototypes, regularizing the update
+procedure. Our method receives outstanding performance on the challenging Meta-Dataset.
+We conduct extensive experiments to show the efficacy of our model. Our code is available
+at https://github.com/loadder/eTT_TMLR2022.
+1
+Introduction
+Modern computer vision models such as ResNet (He et al., 2016) and Faster R-CNN (Ren et al., 2015)
+are trained on large-scale training sets, and not well generalize to handle the long tail categories with few
+∗This paper is supported by the project NSFC(62076067).
+1
+arXiv:2301.02419v1  [cs.CV]  6 Jan 2023
+
+Published in Transactions on Machine Learning Research (08/2022)
+labeled samples. Few-shot Learning (FSL) has thus been studied to make inference on insufficiently-labeled
+novel categories typically with the transferable knowledge learned from base categories which are provided
+with abundant labeled training samples. Essentially, the FSL can be taken as representation learning, as its
+backbones should ideally extract features representative and generalizable to various novel tasks. Currently
+Convolutional Neural Networks (CNNs), especially ResNet, are the predominant backbone and widely utilized
+in most existing FSL works (Ravi & Larochelle, 2017; Finn et al., 2017; Nichol et al., 2018; Li et al., 2017;
+Sun et al., 2019).
+Recently, by taking the merits of Multi-headed Self-Attention (MSA) mechanism and Feed Forward Net-
+work (FFN), the transformers have been widely used in the recognition (Alexey et al.; Liu et al., 2021b),
+detection (Beal et al., 2020) and image editing (Cao et al., 2021). The general pipeline of Pretrain-(Meta-
+train)-Finetune has been explored in few ViTs on FSL (Hu et al., 2022; Evci et al., 2022; Abnar et al.),
+recently. Particularly, the ViT models are first pretrained/meta-trained on a large-scale dataset. Then
+a test-time finetune procedure is set up for each target task on novel data. The finetuning strategy can
+be generally categorized into linear classifier probing and backbone tuning: the former one optimizes the
+reasonable decision boundaries by the fixed embeddings, while the latter one considers the adaptation of
+both embedding space and classifier.
+In this paper we focus on the backbone tuning method. (Hu et al., 2022) shows that the naïve Pretrain-Meta-
+train-Finetune (P>M>F) baseline can generally have satisfactory performance in FSL. Unfortunately, it
+involves heavy computations and potential overfitting in FSL setting. Particularly, (1) It typically demands
+extraordinary computing power to formulate episodes from a large number of support classes to update
+the whole network parameters. Thus it is less efficient in many real-case applications. For example, the
+edge devices such as mobiles donot have enough computational power to adapt all model parameters by
+personalized/specialized data collected on these devices. (2) It is very subtle and difficult to directly fine-tune
+trained deep models on one or two labeled instances per class, as such few-shot models will suffer from severe
+overfitting (Snell et al., 2017; Fei-Fei et al., 2006; Brian et al.). By contrast, humans have the ability of
+conducting few-shot recognition from even single example of unseen novel category with very high confidence.
+Such problems may be the culprit of the phenomenon that their proposed finetune strategy only works on
+part of datasets and has less effect to the others. This suggests their limited usage of ViT backbone for any
+potential FSL applications. An alternative choice is to finetune specific layers in a ViT model with much
+smaller tunable parameters (ViT-s block in Fig. 1(a)). Such a strategy nevertheless can only finetune either
+low-level or high-level features, leading to inferior performance in many cases. Therefore it is desirable to
+have an efficient and light-weighted ViT tuning method that shall not only avoid overfitting to small training
+samples, but also achieve high performance of FSL.
+In this paper, we present a novel efficient Transformer Tuning (eTT) for few-shot learning task, which adopts
+a pretrain-finetune pipeline. To pretrain our transformer, we advocate utilizing the recent self-supervised
+method – DINO (Caron et al., 2021). Our key novelties are in the finetuning stage. As illustrated in Fig. 1(b),
+we propose Attentive Prefix Tuning (APT) and Domain Residual Adapter (DRA) as the key components to
+our eTT, to efficiently learn the newly-introduced tunable parameters over novel support sets. Specifically, we
+formulate the attentive prototypes by aggregating patch embeddings with the corresponding attention weights
+of the class token for each image, so as to provide the model with abundant task-specific information and
+guide each self-attention layer to aggregate more class-related features. To encourage the prefix to keep the
+prior knowledge from initial prototypes, we further propose a novel prototypical regularization which restricts
+the relationship between the prefix and prototypes by optimizing the similarity of their projected distributions.
+Moreover, we propose to additionally adopt a light-weighted domain residual adapter in the form of learnable
+offset to deal with the potential failure of APT on large domain gaps. Extensive experiments are conducted to
+evaluate our eTT: we use the ViT-tiny and ViT-small backbones on the large-scale Meta-Dataset (Triantafillou
+et al., 2019) consisting of ten sub-datasets from different domains; and the results show that our model can
+achieve outstanding performance with comparable or even much fewer model parameters. Thus our eTT is a
+promising method on efficiently finetuning ViTs on the FSL tasks.
+Our paper has the following contributions.
+1. In order to solve the problem of inefficiency and make better use of ViT in FSL, we propose a novel
+2
+
+Published in Transactions on Machine Learning Research (08/2022)
+Domain residual adapter
+ViT block
+…
+Attentive
+prototypes
+Visual
+prefix
+Domain residual adapter
+ViT block
+Few-shot
+episodes
+(a) Tunable parameters in Backbone Finetuning
+(b) Attentive Prefix Tuning in Task Tuning
+Support images
+Initialize
+Key/value 
+pairs
+project
+plug
+Figure 1: (a) Comparing with other backbones, we propose the Domain Residual Adapter (DRA) to tune much
+less parameters in our efficient Transformer Tuning (eTT); and effective for large-scale FSL. (b) The few-shot
+support samples are first processed into attentive prototypes which are used to initialize the task-specific
+visual prefix. Then the prefix together with the domain adapter are attached to each layer of the ViT to
+finetune our ViTs.
+finetuning method named efficient Transformer Tuning (eTT).
+2. Inspired by recent advance in language model, a novel attentive prefix tuning is presented utilizing the
+attentive prototypes to embed the task-specific knowledge into pretrained ViT model. Particularly, we propose
+a new initialization strategy tailored for FSL by leveraging prototypical information from the self-attention
+layers. Moreover, a novel domain residual adapter is repurposed to handle the various domain gaps between
+training and testing data.
+3. We introduce a prototypical regularization term which can constrain the update procedure of prefix during
+finetuning to maintain the initial task-specific knowledge.
+4.
+By utilizing the proposed eTT, our ViT models receive remarkable performance on Meta-Dataset,
+overpassing the existing ResNet-based methods without using additional training data. More importantly,
+both of the model scale and efficiency of our method are comparable with the other competitors, indicating
+the promising application of ViTs in FSL.
+2
+Related Works
+Few-shot recognition. FSL learns transferable knowledge from base classes and adapt it to a disjoint set
+(novel classes) with limited training data. Among those FSL tasks, few-shot image recognition is the one with
+most focus and researches. Existing works can be grouped into two main categories. One is optimization-based
+methods (Ravi & Larochelle, 2017; Finn et al., 2017; Nichol et al., 2018; Li et al., 2017; Sun et al., 2019),
+which learn parameters that can be better finetuned on few-shot support sets. The other is metric-based
+methods such as ProtoNet (Snell et al., 2017), RelationNet (Sung et al., 2018), CAN (Hou et al., 2019),
+DMF (Xu et al., 2021), COSOC (Luo et al., 2021) and CTX (Doersch et al., 2020), which solve FSL by
+applying an existing or learned metric on the extracted features of images. Particularly, CTX (Doersch et al.,
+2020) builds up a cross attention module which interacts between query and support images to adaptively
+aggregate better prototypes than simply averaging all support features. While these methods perform well on
+classical few-shot learning settings, most of them adopt convnet as backbone, especially ResNet (He et al.,
+2016). We, on the opposite, try to make full use of another widely-applied structure, i.e. ViT, in FSL, which
+requires extra design for training and finetuning strategy.
+Transformer in vision tasks. Transformers widely utilize the self-attention mechanism which originally
+are employed to process the feature sequence in Vaswani et al. (2017). Then large scale transformers become
+increasingly popular in NLP tasks to build complex language models, and also extend to vision tasks (Alexey
+3
+
+Published in Transactions on Machine Learning Research (08/2022)
+et al.; Yuan et al., 2021; Liu et al., 2021b) by formulating the token sequence with image patches processed
+with position embedding. It has been shown the efficacy in various applications, such as (Liu et al., 2021a) for
+image caption, (Sun et al., 2020) for multiple object tracking and (Esser et al., 2021; Cao et al., 2021) for image
+inpainting and editing. Critically, ViTs is typically trained by very large-scale dataset, and few effort has
+been dedicated in training or finetuning on few-shot supervision. We follow the pretrain-meta-train-finetune
+pipeline (Hu et al., 2022), while their method finetune the whole ViTs on few-shot examples, and thus has
+less efficiency and can easily overfit. In contrast, our proposed eTT has the key components of DRA and
+APT, demanding much less tunable parameters with much better performance.
+Finetuning algorithm for ViT. The idea of finetuning ViTs on small-scale datasets has been partly
+investigated in Natural Language Processing (NLP) communities. Houlsby et al. (2019) proposed to attach
+two learnable bottleneck adapters to each transformer layer. Other works (Xiang & Percy; Brian et al.)
+make use of the prompt which trains a small task-specific prompt for each task so that the prompt can guide
+the model with knowledge corresponding to the task. Such a prompting idea from NLP is inherited and
+repurposed to finetune a learnable prefix for each novel episode in this paper. However, these works (Xiang
+& Percy; Brian et al.; Houlsby et al., 2019) initialize the prefix or prompt with word embeddings which is
+not available in our problem. Instead, we propose an attentive prototype with regularization initializing the
+visual prefix with object-centric embeddings. Additionally, we notice that a very good concurrent technical
+report (Jia et al., 2022) also studies finetuning visual prompt for pretrained ViTs in downstream tasks. We
+highlight the two key differences from our eTT. The first is about the initialization. While initialization
+strategy does not matter in their method and the corresponding tasks, we show in our experiments that
+randomly initializing prefix does lead to sub-optimal performance in FSL, which leads to the necessity of a
+well-designed initialization. The second is that we further propose a regularization term to restrict the prefix,
+which has never been studied in existing works.
+Task-specific Adapter. The idea of task-specific adapter has been explored in several works like (Li et al.,
+2022; Rebuffi et al., 2017) to adapt CNNs to learn the whole information from support set. Besides, (Requeima
+et al., 2019; Bateni et al., 2020) adopt Feature-wise Linear Modulation (FiLM) layers (Perez et al., 2018) to
+adapt task-specific information into networks. In contrast, we repurpose the adapter as the domain residual
+to update transformer blocks in a more light-weighted way with less learnable parameters. Beyond different
+structures, our proposed DRA intrinsically serves as the domain adapter rather than meta-learner for the
+FSL in Rusu et al. (2018); Sun et al. (2019); Requeima et al. (2019). While these previous works require
+meta-training to optimize their adaptation modules, our method simply utilizes the novel support data to
+learn the DRA, thus reducing the training cost. Furthermore, our DRA is mostly tuned to bridge the visual
+domain gap between base and novel categories, thus improving the learning of APT on each episode task.
+3
+Methodology
+3.1
+Problem Setup
+We formulate few-shot learning in the meta-learning paradigm. In general, we have two sets of data, namely
+meta-train set Ds = {(Ii, yi) , yi ∈ Cs} and meta-test set Dt = {(Ii, yi) , yi ∈ Ct} which contain the base and
+novel data respectively and are possibly collected from different domains. Cs and Ct (Cs ∩ Ct = ∅) denote
+base and novel category sets. FSL aims to train a model on Ds which is generalizable enough on Dt. In the
+testing phase, the model can learn from few labelled data from each category of Ct.
+While most previous FSL works (Snell et al., 2017; Sung et al., 2018) utilize the setting of N-way K-shot in
+mini-ImageNet, i.e., K training samples from N class, we follow CTX (Doersch et al., 2020) to adopt the
+setting on the large-scale Meta-Dataset (Triantafillou et al., 2019). In each episode T , N is first uniformly
+sampled from [5, Nmax] where Nmax equals to min(50, |Ct|) or min(50, |Cs|) on training or testing stage,
+accordingly. N is supposed to be accessible knowledge during both training and testing. In the most naïve
+case, one can get N by directly counting the number of support classes. From each of the sampled category,
+M query samples per category are randomly selected, and thus constructing the query set Q = {(Iq
+i , yq
+i )}NQ
+i=1.
+After that random amount of samples are taken from the rest of samples belonging to these categories to
+form the support set S = {(Isupp
+i
+, ysupp
+i
+)}NS
+i=1. Note that compared to the classical N-way K-shot setting,
+4
+
+Published in Transactions on Machine Learning Research (08/2022)
+Patch embedding 
+layer 
+…
+transformer layer 
+…
+transformer layer 
+Patch 
+embeddings
+Linear
+ProtoNet
+aggregate
+ෝ𝒚
+𝜽𝒑
+෡𝑨
+MSA
++
+LN
+FFN
+LN
++
+projector
+attention
+Q
+K
+V
+𝜽𝒌 𝜽𝒗
+MSA
+𝜹𝒇
+𝜹𝒂
+𝜽𝒑
+Image 
+embedding
+Q
+K
+𝜽𝒌
+V
+𝜽𝒗
+𝒈
+Figure 2: Schematic illustration of our proposed model. For each image, we first fetch its patch embedding
+sequence and the attention score with regard to the last layer’s class token, from which the image embedding
+can be computed. Then the visual prefix is initialized as the attentive prototypes of image embeddings. The
+prefix, together with the proposed domain residual adapter are attached to the model. The final features are
+processed with an extra linear transformation layer and predicted with ProtoNet. Dashed arrows denote
+forward propagation before test-time finetuning.
+such a setting generates class-imbalanced support sets, and different episodes contain different numbers of
+support samples. This is much more challenging to the model and learning algorithms, as they shall handle
+both extremely large and small support sets.
+3.2
+Overview of Our Method
+To handle the optimization of various episodes on large-scale dataset, we present our novel finetuning model –
+efficient Transformer Tuning (eTT) as shown in Fig. 2. Our eTT follows the pipeline in Hu et al. (2022), and
+has key stages of the pretraining and finetuning. We employ DINO as pretraining, and conduct the task
+tuning by attentive prefix tuning (Sec. 3.4), and backbone tuning with domain residual adapter (Sec. 3.5).
+Pre-training. As previous work (Hu et al., 2022) shows the importance of self-supervised pre-training to
+learning vision transformer models, we adopt the same principle and introduce the self-supervised learning
+model to pre-train our ViT backbone on base data. Specifically, we utilize the recent State-of-the-art
+self-supervised ViT models – DINO (Caron et al., 2021) to pretrain our model. DINO builds up supervision
+based on a self-distillation framework by using the multi-crop strategy (Caron et al., 2020). As we will show
+in our experiments, such a pre-trained model shall have good cluster property even among cross domain
+images, potentially benefiting our following FSL stages. Note that different from (Hu et al., 2022) which
+takes an off-the-shelf model pretrained with DINO on full ImageNet, we strictly follow the FSL protocols to
+retrain the DINO models on the meta-train split in the target dataset to avoid the abuse of extra data.
+One would ask whether it is necessary to make use of the annotations for base data, since supervised pretrain
+has been proven to be effective in many previous FSL works (Ye et al., 2020; Hou et al., 2019). As we will
+show in the experiments, an additional finetuning with image labels on base data cannot bring consistent
+improvement and even makes it worse on most datasets, which may be caused by the overfitting on the
+image labels leads to less generalization ability across different domains. Moreover, compared with vanilla
+supervised training, the attention maps for models trained by DINO contain more semantic information,
+which we will utilize in the following context.
+3.3
+Preliminary: Vanilla Test-time Finetuning
+Before fully developing our fine-tuning contributions, we review the simple and effective finetuning method
+named LT+NCC (Li et al., 2021). The novel modules proposed by us in the following context are all adopted
+together with this simple baseline method. Given a ViT backbone fθ that is parameterized by θ and an
+episode T , the support features {xsupp
+i
+}NS
+i=1, where xsupp
+i
+= fθ(Isupp
+i
+), are extracted from the support set
+5
+
+Published in Transactions on Machine Learning Research (08/2022)
+{Isupp
+i
+}NS
+i=1. Then, a learnable linear transformation φ is added to the model to realize the adaptation, which
+results in the final support features used for classification {ˆxsupp
+i
+}NS
+i=1, where ˆxsupp
+i
+= φ(xsupp
+i
+). The prediction
+of these support images can thus be calculated based on the similarity between the transformed features and
+the aggregated prototypes as,
+¯xc =
+1
+�Ns
+i=1 1ysupp
+i
+=c
+Ns
+�
+i=1
+ˆxsupp
+i
+1ysupp
+i
+=c
+ˆysupp
+i
+(c) =
+exp(d(ˆxsupp
+i
+, ¯xc))
+�N
+c=1 exp(d(ˆxsupp
+i
+, ¯xc))
+(1)
+where d denotes cosine similarity, i.e., d(a, b) =
+aT b
+∥a∥∥b∥. We fix all of the parameters in the original backbone,
+and adopt the cross entropy loss to optimize the transformation φ. Precisely speaking, for each support image
+Isupp together with its annotation ysupp, the objective function is as following:
+ℓCE = −ysupp · log ˆysupp
+(2)
+After finetuning, φ is applied to query features and the same procedure as above is performed between the
+processed query features {ˆxq
+i } and prototypes {¯xc}N
+c=1 for the inference of each episode.
+3.4
+Task Tuning by Attentive Prefix Tuning
+We finetune the pre-trained ViT with support set via an attentive prefix tuning strategy. Specifically, a prefix
+matrix θP ∈ RNP ×d is first initialized, where NP denotes the number of prefix. Then a bottleneck g is added
+upon θP to produce ˆθP ∈ RNP ×(2Ld), where L denotes the number of backbone layers. The g plays the same
+role as the projector in each self-attention layer, except that all layers share the same module. The produced
+ˆθP can be reshaped and seen as L value and key pairs {θl
+v, θl
+k}L
+l=1, θl
+v, θl
+k ∈ RNP ×d. The MSA block in the
+L-th layer can then be reformed by attaching these new pairs to the original key and value sequences:
+Al = Attn(Q,
+�
+K; θl
+k
+�
+)
+output = Al �
+V ; θl
+v
+�
+(3)
+where [·; ·] denotes concatenation, Attn denotes the calculation of MSA matrices. In this way, the prefix can
+affect the attention matrix Al and result in different output features from the original ones.
+Remark. Compared with the naive strategy that finetunes specific layers in ViT (ViT-s block in Fig. 1(a))
+which can only adjust part of blocks, the prefix can evenly adapt each layer’s image embedding with almost
+the same parameter size as one transformer layer, as shown in Tab. 1(a). By fixing the model parameters and
+optimizing the prefix θP and the transformation module g, the support knowledge can be smoothly embedded
+into the prefix, which further helps the task adaptation.
+Attentive Prototype. The initialization of the prefix is very important to our APT, as it greatly boosts
+the performance. Critically, quite different from the prefix or prompt tuning in NLP and visual-context tasks
+that have task-specific instructions explicitly as word embedding sequences, each episode in our FSL only
+has the few support images and their labels. Thus, rather than steering the model with ’what should be
+done’ as in Xiang & Percy, our APT shall provide the model with ’what we have globally’ by leveraging the
+class-specific information. Thus, the attentive prototype is presented to aggregate the image embeddings
+with object-centric attention, as the initialization of the prefix. Particularly, each support image Isupp is first
+transformed to a patch embebdding sequence {˜xsupp
+m
+}P
+m=1 with the starting patch embedding layer,
+˜xsupp
+m
+= fθpe(Isupp
+m
+) + Epos
+m
+(4)
+where m = 1, · · · , P 2 is the patch index; fθpe denotes the patch embedding layer which is typically a
+convolutional layer whose kernel size equals to patch size; and Epos indicates the position embedding.
+Meanwhile, we can get unnormalized attention score A ∈ Rh×P between the class token and image patches
+from the last MSA layer, where h denotes number of heads in each MSA module. Such an attention vector
+can focus on the foreground in the image, especially for models trained with DINO (Caron et al., 2021), with
+each head indicating a particular part or an object. We can thus get the initial image-level representation
+ˆA = σ(A)
+˜xsupp = 1
+h
+h
+�
+n=1
+P 2
+�
+m=1
+ˆAnm˜xsupp
+m
+(5)
+6
+
+Published in Transactions on Machine Learning Research (08/2022)
+where σ is softmax function. Compared with simply averaging all patch embeddings, the attentive embeddings
+can highlight the objects of interest and suppress the background information. Then the prototypes ¯x can
+be calculated by averaging the attentive image embeddings belonging to each support category. We set the
+number of prefix as N, which is available during testing for each episode, and initialize the prefix with ¯x.
+Remark. In this way, commonly-used prototypes can provide the model with comprehensive information
+about the episode. Also such a first-order statistics is comparable with the normal patch features among
+the layers. This can benefit the training with more stability. When N is large, more prefix are required
+to fully learn the information included by each episode. On the other hand, when N is small so that the
+episode is relatively easy, fewer prefix can handle the support knowledge without trouble while decreasing the
+computing debt.
+3.5
+Backbone Tuning by Domain Residual Adapter
+Finetuning few-shot tasks by APT will make a good balance between performance and efficiency. To further
+improve the model generalization ability on different domains, we further propose the backbone tuning by
+leveraging the Domain Residual Adapters (DRA), as illustrated in Fig. 2. Specifically, for the l-th transformer
+layer, we attach two learnable offset vectors δl
+a, δl
+f ∈ Rd to the MSA and FFN. After features are processed
+with MSA and FFN, the corresponding offsets are added to them so that the extreme domain gap can be
+neutralized. These offsets are expected to represent the gap between source and target domains, and transfer
+the original manifold to a more appropriate one.
+3.6
+Loss Functions
+Prototypical Regularization. In addition to the cross entropy loss in Eq. 2, we propose a novel prototypical
+regularization to ensure the class-specific information, which is embedded in the prefix via initialization,
+can be maintained during update. The knowledge in attentive prototypes is distilled to the prefix during
+finetuning. Concretely, in each iteration, the prototypes ¯x and prefix θP are first projected to a latent space
+via a projector module ψ, which produces ¯x′ and θ′
+P respectively. Then the distillation loss is computed using
+these two embeddings as,
+ℓdist = 1
+N
+N
+�
+n=1
+H(¯x′n, θ′n
+P )
+(6)
+where H(a, b) = −a log b. The above objective function can ensure the prototype of each category and the
+corresponding prefix contain consistent information, which is indicated by the similarity between distributions
+after projection. To make training more stable and avoid collapse, for each episode we maintain an exponential
+moving average (EMA) of ¯x′ as the center variable ccenter. Before calculating ℓdist, we standardize ¯x′ as
+σ( ¯x′−xcenter
+τ
+), where σ denotes softmax function and τ is the temperature typically set as 0.04.
+Once having both of the above losses calculated, we can optimize the model parameters including the DRA,
+the prefix together with the transformation g and the projector ψ, with the following objective function:
+L = ℓCE + λℓdist
+(7)
+where the scalar weight λ controls the strength of the regularization.
+Remarks. For a ViT with L layers, nh heads and d feature dimension, the size of trainable parameters is
+(N + d′ + dproj + d)d + 2(d′ + 1)Ld, where d′ is the hidden dimension for transformation module g and dproj
+denotes output dimension for the projector ψ, which is much smaller than that of the whole backbone model.
+Specifically, the learnable modules during finetuning have only about 9% parameters with regard to the whole
+transformer model when using ViT-small and ViT-tiny.
+7
+
+Published in Transactions on Machine Learning Research (08/2022)
+4
+Experiments
+4.1
+Experimental Setup
+Datasets. We use Meta-Dataset (Triantafillou et al., 2019) – the most comprehensive and challenging
+large-scale FSL benchmark. It has 10 sub-datasets such as ImageNet (Deng et al., 2009) and Omniglot (Lake
+et al., 2015), with various domain gaps. Our experiments are conducted under the single training source
+setting, i.e. only ImageNet is used for training, and the meta-test split of all ten datasets for evaluation. Some
+of the test datasets such as CUB share similar or highly-related categories with ImageNet, while the others
+have greater domain gaps. Note that Hu et al. (2022) claims pretraining on all images in the training set of
+ImageNet is reasonable for introducing extra data and boosting the performance. However, such a strategy
+utilizes much more training samples (1.28M images, 1000 classes in ImageNet v.s. 0.9M images, 712 classes
+in meta-train split of ImageNet). Empirically so many additional images can greatly benefit generalization
+ability of self-supervised learning methods. Therefore to make a more fair comparison, we strictly follow
+the experiment protocol used in CTX (Doersch et al., 2020) and shall not use any extra data even in the
+unsupervised pretraining stage. We resize all images to 224 × 224 for ViT-small and 84 × 84 for ViT-tiny.
+Implementation details. We set the patch size as 8 for ViT-tiny (as it has small input image size), and
+keep the other hyper-parameters as default. We adopt standard ViT-small with 12 layers, 6 attention heads,
+feature dimension as 384 and patch size as 16. We strictly follow the hyper-parameter setting and data
+augmentation in DINO (Caron et al., 2021) for pretraining. In test-time finetuning, we empirically set the
+hidden dimension d′ of the transformation module as d/2, and output dimension dproj of the projector as 64
+for all datasets. We utilize AdamW optimizer finetuning, with learning rate set as 1e − 3 for TrafficSign and
+5e − 4 for other datasets. λ is set as 0.1. For simplicity, the selection of hyper-parameters is conducted on
+the meta-validation set of ImageNet, which is the only within-domain setting in Meta-Dataset.
+Evaluation benchmark. We report the accuracy of randomly sampled 600 episodes for each dataset and
+the average accuracy when comparing with the existing methods. The comprehensive comparison of both
+accuracy and 95% confidence interval is in Appendix.
+Backbone
+Image size
+Params(M)
+FLOPs(G)
+Res18
+84×84
+11.69
+1.82
+ViT-tiny
+84×84
+5.38
+0.72
+Res34
+224×224
+21.80
+3.68
+ViT-small
+224×224
+21.97
+4.61
+Table 1:
+Comparison of parameter size and FLOPs between different backbones.
+4.2
+Comparison with State-of-the-art Methods
+Before the comprehensive comparison, it is necessary to show the comparison between different backbone
+is fair enough since our backbone model is not the same as the existing method. Therefore we present the
+comparison of size of model parameters and FLOPs in Tab. 1, in which the FLOPs of all models are computed
+by fvcore1. The results show that (1) compared with Res18, ViT-tiny is a much smaller and efficient model,
+and (2) ViT-small is approximately comparable to Res34. In this way, the comparison of our proposed
+method with state-of-the-art methods is reasonable and fair.
+We compare our model with ProtoNet(Snell et al., 2017), CTX (Doersch et al., 2020), TSA (Li et al., 2022),
+etc. These methods take the backbones of ResNet18 or ResNet34. Also, the pretrain-meta-train-finetune
+baseline (P>M>F) (Hu et al., 2022) is not considered in computing average rank since extra data is used. As
+in Tab. 2, when using ViT-small as backbone whose parameter size is comparable to that of ResNet34, our
+model receives 1.6 average rank on all dataset. Specifically, on Texture and Fungi, our model outperforms the
+strongest competitors CTX and TSA by about 8% and 10%, while on other datasets the performance of our
+model is still comparable with or slight better than that of the existing methods. We notice that our model
+1https://github.com/facebookresearch/fvcore
+8
+
+Published in Transactions on Machine Learning Research (08/2022)
+Model
+Backbone ILSVRC Omni Acraft
+CUB
+DTD
+QDraw Fungi Flower
+Sign
+COCO
+Avg
+Rank
+Finetune
+Res18
+45.78
+60.85
+68.69
+57.31
+69.05
+42.60
+38.20
+85.51
+66.79
+34.86
+56.96
+10.2
+Proto
+50.50
+59.98
+53.10
+68.79
+66.56
+48.96
+39.71
+85.27
+47.12
+41.00
+56.10
+10.5
+Relation
+34.69
+45.35
+40.73
+49.51
+52.97
+43.30
+30.55
+68.76
+33.67
+29.15
+42.87
+14.6
+P-MAML
+49.53
+63.37
+55.95
+68.66
+66.49
+51.52
+39.96
+87.15
+48.83
+43.74
+57.52
+9.2
+BOHB
+51.92
+67.57
+54.12
+70.69
+68.34
+50.33
+41.38
+87.34
+51.80
+48.03
+59.15
+8.2
+TSA
+59.50
+78.20
+72.20
+74.90
+77.30
+67.60
+44.70
+90.90
+82.50
+59.00
+70.68
+4.3
+Ours
+ViT-t
+56.40
+72.52
+72.84
+73.79
+77.57
+67.97
+51.23
+93.30
+84.09
+55.68
+70.54
+4.1
+Proto
+Res34
+53.70
+68.50
+58.00
+74.10
+68.80
+53.30
+40.70
+87.00
+58.10
+41.70
+60.39
+7.4
+CTX
+62.76
+82.21
+79.49
+80.63
+75.57
+72.68
+51.58
+95.34
+82.65
+59.90
+74.28
+2.8
+TSA
+63.73
+82.58 80.13
+83.39
+79.61
+71.03
+51.38
+94.05
+81.71
+61.67
+74.93
+2.5
+P>M>F∗
+74.69
+80.68
+76.78
+85.04
+86.63
+71.25
+54.78
+94.57
+88.33
+62.57
+77.53
+—
+Ours
+ViT-s
+67.37
+78.11
+79.94
+85.93 87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+1.6
+Table 2:
+Test accuracies and average rank on Meta-Dataset. Note that different backbones are adopted by
+these methods. * denotes using extra data for training. The bolded items are the best ones with highest
+accuracies.
+is inferior to the best ones in Omniglot, while this is reasonable. Since Omniglot images represent simple
+characters with monotone color patterns, each image patches contain less information than images in other
+datasets. Vanilla ViTs have less efficiency in dealing with these image patches due to limited interaction
+among patch embeddings. This problem can be solved with much sophisticated variants of ViT like Swin (Liu
+et al., 2021b), and will be taken as future works. Moreover, our proposed method is better than P>M>F,
+which not only utilizes extra data for training but also finetunes all model parameters during testing, on more
+than half of the datasets, which strongly indicates the effectiveness of the proposed finetuning strategy in this
+paper. As for using ViT-tiny which has much less parameter than Res18, our model is still comparable to the
+state-of-the-art methods and outperforms many popular baselines. Particularly, compared with ProtoNet
+which is one of the most famous and efficient methods in FSL, our eTT shows significant boost on Aircraft
+by 19.74% and TrafficSign by 36.97%. The reason of the inferior results on several datasets against TSA can
+be two folds. Firstly, the ViT-tiny intrinsically has smaller capacity than Res18. On the other hand, while it
+is common to train ViT with large scale images and patches so that the images are splitted into abundant
+patches and each patch-level token can receive enough information. In contrast, we adopt 84 × 84 images
+with 8 × 8 patch size for ViT-tiny so that the comparison with Res18 is fair, which lead to less patches with
+smaller size and may have negative influence on the performance. In general, the results indicate that our
+proposed eTT can make ViT models a desirable choice for large scale FSL problems.
+4.3
+Model Analysis
+To further validate the effectiveness of our method, we conduct a series of ablation studies on Meta-Dataset
+using ViT-small below.
+4.3.1
+Design of Each Module
+Can finetuning on meta-train set boost the performance? One would ask whether it is necessary to
+make use of base annotations, as supervised pretraining is also effective in many FSL works (Ye et al., 2020;
+Hou et al., 2019). To verify it, we finetune DINO-pre-trained ViT-small on meta-train split of ImageNet, in
+which the options of all hyper-parameters and data augmentations follow DeiT (Touvron et al., 2021) using
+either way of class token features or averaged patch features as image representations. After such a supervised
+finetuning, we test the models both with the basic test-time finetuning method as in Sec. 3.3, which we
+denote as LT+NCC, and with our proposed eTT. The results are shown in Fig. 3, from which we find out
+that (1) Supervised finetuning does improve test accuracies on ImageNet, CUB and MSCOCO. Particularly,
+the token finetune model receives 89.83% accuracy on CUB when testing with our eTT, which is remarkably
+better than any other models. This is reasonable as similar images between ImageNet and these datasets. By
+9
+
+Published in Transactions on Machine Learning Research (08/2022)
+Model
+ILSVRC
+Omni
+Acraft
+CUB
+DTD
+QDraw
+Fungi
+Flower
+Sign
+COCO
+Avg
+Proto
+63.37
+65.86
+45.11
+72.01
+83.50
+60.88
+51.02
+92.39
+49.23
+54.99
+63.84
+LT+NCC
+65.96
+67.62
+64.03
+77.10
+83.46
+63.88
+57.79
+93.13
+66.91
+56.04
+69.59
+Last
+66.32
+71.04
+78.04
+86.25
+86.67
+64.22
+55.69
+94.44
+65.55
+55.94
+72.42
+First
+61.54
+50.46
+69.23
+79.17
+83.10
+68.69
+49.93
+93.50
+54.28
+58.45
+66.84
+LN
+66.22
+70.45
+69.41
+81.29
+86.37
+66.28
+58.38
+96.25
+71.09
+59.57
+72.53
+APT
+66.75
+75.16
+75.41
+84.25
+86.47
+69.55
+60.03
+96.38
+78.20
+61.10
+75.33
+Adapter
+66.53
+72.31
+73.75
+83.73
+86.86
+66.74
+58.49
+96.15
+82.65
+62.40
+74.93
+eTT
+67.37
+78.11
+79.94
+85.93
+87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+Random
+66.12
+76.33
+78.35
+84.77
+86.78
+70.13
+59.25
+96.00
+82.28
+59.59
+75.96
+Avg
+66.11
+75.06
+77.07
+85.16
+87.35
+70.72
+61.79
+96.54
+84.28
+62.18
+76.73
+Sampling
+67.81
+76.72
+77.96
+85.79
+87.25
+70.19
+60.73
+96.27
+83.72
+62.17
+76.86
+Full
+67.37
+78.11
+79.94
+85.93
+87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+Linear
+66.35
+74.26
+79.42
+83.65
+86.02
+71.11
+55.73
+95.89
+82.73
+59.90
+75.51
+Bottleneck
+67.29
+76.06
+79.72
+85.60
+87.21
+70.59
+61.59
+96.15
+85.00
+62.02
+77.12
+FiLM
+66.91
+75.32
+78.26
+85.78
+86.83
+70.29
+61.65
+96.50
+84.48
+61.75
+76.78
+Offset
+67.37
+78.11
+79.94
+85.93
+87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+w/o PR
+66.72
+74.20
+78.42
+85.06
+87.01
+70.34
+61.64
+96.51
+84.23
+61.08
+76.52
+w PR
+67.37
+78.11
+79.94
+85.93
+87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+w/o Stand
+67.09
+76.42
+78.87
+83.10
+86.50
+70.09
+61.02
+96.33
+82.88
+61.33
+76.36
+w Stand
+67.37
+78.11
+79.94
+85.93
+87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+Table 3:
+Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items
+are the best ones with highest accuracies.
+training on the image annotations of ImageNet, the model learns class-specific knowledge which cannot be
+obtained during self-supervised learning. Since the categories are highly correlated and overlapped among
+these datasets, the learned knowledge can also benefit the recognition on these novel datasets even though the
+specific novel classes do not appear in the meta-train set. (2) Despite the improvement on the three datasets,
+models with supervised finetuning degrade on the other datasets, especially on Traffic Sign and VGG Flower.
+This is due to fitting class labels weakens the effect of these features and makes it harder to generalize to
+novel domains. When taking into account the performance of all datasets, pretraining with DINO is generally
+the much more desirable choice for better generalization over different domains. (3) The improvement of
+our propose method against the basic LT+NCC is not consistent among three different kinds of pretraining
+strategy. For example, while our method can boost the performance of DINO pre-trained model by 9.47% on
+Aircraft and 4.83% on CUB, it can only bring much less advantage on models with supervised finetuning.
+Effectiveness of APT and DRA. We test the DINO pre-trained model with different kinds of testing
+strategies including (1) Proto: Directly generating predictions based on ProtoNet. The prototypes are
+computed using averaged class token features from each category.
+(2) LT+NCC: The basic test-time
+finetuning method in Sec. 3.3. (3) Last: Finetuning the last transformer layer during testing, together with
+LT+NCC. which has similar parameter size to our method. (4) First: Finetuning the first transformer layer
+during testing, together with LT+NCC. which has similar parameter size to our method. (5) LN: We try
+to finetune the affinity parameter in each layer normalization as an alternative finetune strategy, which is
+utilized in many cross-domain FSL works (Tseng et al.; Tsutsui et al., 2022). (6) APT: The model is finetuned
+using APT together with LT+NCC, using cross entropy loss and the proposed prototypical regularization.
+(7) Adapter: The model is finetuned using DRA together with LT+NCC, using cross entropy loss. (8) eTT:
+The model is finetuned using our proposed APT, DRA and LT+NCC. The results in Tab. 3 show that while
+LT+NCC can fundamentally improve the model which indicates the importance of test-time finetuning,
+adding our proposed modules to the finetuning procedure can consistently bring higher performance. Also,
+finetuning specific transformer layer can only bring limited improvement on few datasets: finetuning the last
+10
+
+Published in Transactions on Machine Learning Research (08/2022)
+Figure 3: Test accuracy of different training strategy if testing with (a) LT+NCC or (b) our eTT.
+Figure 4: Visualization of feature embeddings from a randomly sampled episode of TrafficSign.
+layer leads to good performance on Aircraft, CUB and Texture, while updating the first layer leads to good
+performance on Quickdraw and MSCOCO. However, this simple finetuning strategy cannot bring consistent
+improvement on all datasets. This indicates that different data requires different levels of adaptation, and the
+improvement is much smaller than that of our method. Moreover, we give the tSNE visualization of feature
+embeddings of a randomly sampled episode from TrafficSign in Fig. 4, which demonstrates that utilizing our
+proposed method can better regulate the feature embeddings into proper clusters.
+Is prototypical initialization necessary? One of the most important parts of our APT is the attentive
+prototypical initialization in which we use attentively aggregated patch embeddings to initialize the prefix
+matrix. To verify this strategy, we compare several different choices of initialization, including (1) Random:
+random initialization from normal distribution. (2) Avg: simply averaging all patch embeddings from each
+category. (3) Sampling: randomly sampling one image for each category, and then initializing the prefix
+matrix with the averaged patch embeddings of each image. (4) Full: computing prototypes with our proposed
+attentive prototype. Results in Tab. 3 show that random initialization performs the worst, which can be
+resulted from insufficient task-specific information provided by the prefix in this way. Meanwhile, among all
+other strategies, using the attention map to aggregate patch embeddings as in Eq. 5 is better than simply
+averaging, leading to about 1% improvement on average.
+Do we need a more complex adapter structure? One would argue that our DRA structure is too simple
+to learn the complex knowledge from support images. In Tab. 3 we compare three different instantiations of
+adapters including (1) Linear: As in Li et al. (2022), we use a linear layer for each adapter, whose output
+are than added to the original features in the MSA and FFN. (2) Bottleneck: We expand the linear layer
+11
+
+token
+token
+90
+pool
+90
+pool
+DINO
+DINO
+Test Accuracy
+80
+Test Accuracy
+80
+70
+70
+60
+60
+50
+50
+40
+40
+craft
+oraw
+Fung
+raw
+(a)TestAccuracy when using LT+NCC
+(b)TestAccuracywhenusing eTTPublished in Transactions on Machine Learning Research (08/2022)
+to a bottleneck structure where two linear layers are used for each adapters. (3) FiLM: We compare DRA
+with a FiLM-like variant, in which we add a scaling vector for each adapter as in FiLM layer Perez et al.
+(2018). Note that such a method is similar to MTL (Sun et al., 2019) in FSL. The difference lies in that we
+still use the original way to directly tune the parameters on the novel support sets, instead of using another
+meta-trained module to generate the parameters. (4) Offset: Only an offset vector is adopted for each adapter.
+The results reveals that the linear adapter performs the worst, which means such a structure is improper for
+ViT in finetuning. Moreover, we also find that using the bottleneck adapter will result in a dilemma. If we
+use small initial value for the adapter, the weights of each layer can only achieve gradient with extremely
+small values. As the result, these weights, except the bias term of the last layer, can hardly be updated based
+on the support knowledge, which means such an architecture almost equals to our design where only an offset
+vector is utilized. On the other hand, if large initial values are adopted to avoid gradient diminishing, then
+the output features from the adapters can make the predictions severely deviate from those without adapters,
+thus leading to worse performance. As for the FiLM-like DRA, it is worse than offset DRA by about 0.8% on
+average, while it doubles the parameter size based on offset DRA, leading to no significant additional efficacy.
+Effectiveness of prototypical regularization. We also validate this regularization. In Tab. 3 we present
+the test accuracy when finetuning with and without this loss function. We can find that by applying this
+objective function, the model can have higher results on most datasets. Besides, as described in Sec 3.6, we
+use a standardization technique when computing the prototypical regularization. To verify its efficacy, we
+compare the model with and without such a standardization. The results are shown in Tab. 3. When not
+using standardization, the results are generally worse given comparable confidence intervals (Tab. 11). The
+results verify that this strategy can help the model with more stable finetuning procedure.
+dproj
+ILSVRC
+Omni
+Acraft
+CUB
+DTD
+QDraw
+Fungi
+Flower
+Sign
+COCO
+Avg
+64
+67.18
+75.30
+78.88
+86.20
+87.09
+69.82
+61.61
+96.31
+82.24
+62.14
+76.68
+96
+66.23
+75.69
+78.26
+85.67
+87.28
+70.25
+61.97
+96.59
+84.10
+62.17
+76.82
+128
+67.31
+76.83
+78.81
+85.77
+87.36
+70.16
+60.81
+96.53
+84.29
+62.12
+77.00
+256
+66.83
+78.04
+78.38
+84.60
+86.68
+70.43
+61.03
+96.23
+85.33
+62.10
+76.97
+192
+67.37
+78.11
+79.94
+85.93
+87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+Table 4:
+Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items
+are the best ones with highest accuracies.
+4.3.2
+Comparison among Different Hyper-parameter Settings
+In additional to the ablation study about the proposed module, We further verify different choices of hyper-
+parameters in our model. Especially, dproj for the transformation module in APT and λ for the prototypical
+regularization are tested in Tab. 4 and Tab. 5 in the Appendix. In general, the improvement is not consistent.
+For dproj, we can find that using 192-d hidden dimension can get globally better results, which indicates
+that such a choice can make a good balance between the model capacity and scale so that the finetuning
+can be conducted both efficiently and effectively. As for λ, 0.1 seems to be a desirable choice. Intuitively,
+smaller λ leads to less control of the prefix from the proposed prototypical regularization. Therefore, the
+prefix may lose the desired information during the optimization on the support set. On the other hand, when
+λ is too large, the regularization overwhelms the label supervision, and thus the model can hardly adapt to
+the support knowledge, leading to worse performance especially on Omniglot and Aircraft.
+5
+Conclusion
+We propose a novel finetuning method named efficient Transformer Tuning (eTT) for few-shot learning with
+ViT as our backbone. By fixing the parameters in the backbone and utilizing attentive prefix tuning and
+domain residual adapter, our method can guide the ViT model with comprehensive task-specific information,
+which leads to better representations and performance. This is demonstrated by the fact that we establish
+new state-of-the-arts on the large-scale benchmark Meta-Dataset.
+12
+
+Published in Transactions on Machine Learning Research (08/2022)
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+A
+Appendix
+A.1
+Limitations and Future Work
+Despite the marginal effectiveness and efficiency of our proposed eTT, we mainly notice two points that should
+be explored in the future: (1) The plain ViT backbone utilized in this paper, may not be the best choice to
+the simple dataset, e.g., Omniglot, while a well-designed ViT backbone may potentially better improve the
+efficacy of our method on such dataset. (2) A flexible finetuning algorithm such as (Lee et al.) may have
+better generalization ability when facing episodes with various shots and ways, than the commonly-used
+methods that adopt fixed test-time finetuning iterations. However, it is non-trivial to directly merge such
+methods with our proposed eTT due to different network structures and tuning strategies. It can be taken as
+the future work to properly utilize these flexible finetuning algorithm to further improve the performance of
+ViT in FSL.
+15
+
+Published in Transactions on Machine Learning Research (08/2022)
+A.2
+Additional Experiment Results
+A.2.1
+Full Comparison with state-of-the-art methods
+We show the accuracies together with confidence interval in Tab. 6. Beyond the accuracies which is analyzed
+in the main context, the confidence interval of our eTT on both ViT-tiny and ViT-small is comparable with
+the other competitors, which reflects that our method is stable and robust enough among different testing
+episodes.
+A.2.2
+Influence of Training set
+As we have stated in the main context, our eTT is trained on the meta-train split of ImageNet to make fair
+comparison with other methods. To show to what extent training on the full ImageNet training set instead of
+the meta-train set can impact the performance, we test our eTT using off-the-shelf DINO ViT-s model. The
+results are shown in Tab. 7. We can find that (1) For those datasets on which DINO meta-train performs
+better than P>M>F, using full ImageNet to train DINO can bring further improvement. (2) With the help of
+more data, our eTT overpasses P>M>F on ILSVRC and MSCOCO. (3) While more data does improve the
+results on Omniglot and TrafficSign, the final results are still worse than those of P>M>F, which we think
+may be correlated with the limitations of our method as analyzed above. Given all these results, as a lighter
+model in that no meta-training phase is utilized and only few parameters are engaged in the test-time tuning,
+our method can still enjoy comparable performance with P>M>F when training on same amount of data.
+A.2.3
+Influence of DINO
+The DINO pretrain procedure is an important part of our method. To verify the effectiveness of DINO
+pretrain so that the comparison with other methods is fair enough, we present in Tab. 8 the results of TSA
+using DINO-pretrained ResNet-34 and eTT using supervised pretrained ViT-s. We can find that (1) The
+effect of DINO is not consistent on two backbones. While DINO benifits our eTT with about 5% accuracy on
+average, it severely weakens the performance of TSA with a large margin. It means that for FSL, different
+backbones require different pretrain strategy respectively. (2) While supervised pretrained ViT-s performs
+worse on most datasets, it is better on CUB and COCO, which indicates learning label information from
+ImageNet can help the model understanding novel knowledge from these two datasets.
+A.2.4
+Verification of potential overfitting in finetuning
+As we have stated in Sec. 1, finetuning the whole backbone model with few support data will meet potential
+overfitting problem. To reveal if such a problem exists in Meta-Dataset, we conduct an experiment as follow:
+during a normal testing phase, we select all episodes whose minimum shot (minimum number of support
+images for each class) is no larger than 2 (extremely small number of labelled instances), and compare the
+average accuracies of eTT and simple finetuning based on these episodes. Tab. 9 and Tab. 10 show the
+accuracies on support sets and query sets respectively. We can find that most of the datasets both methods
+receive nearly 100% accuracy, which means these two methods can well learn the support data. Given this
+fact, finetuning is much worse than eTT in terms of query accuracies. The overfitting can be reflected given
+high training accuracies and worse testing performance, and to some extent our proposed eTT can fix this
+problem.
+A.2.5
+More Visualization
+We visualize the self-attention map from models with and without DRA on ILSVRC and TraffignSign in
+Fig. 5. Specifically, we randomly sample an episode from each dataset and use our eTT to tune the model
+based on the support samples. Then we calculate the self-attention map of the last layer’s class token and
+highlight the areas with top 20% attention scores. We can find that for the in-domain ILSVRC episode, the
+model can attend to similar regions no matter whether DRA is used. In contrast, the model without DRA
+can easily attend to background regions with less valuable information, which reveals a potential reason that
+these two models has similar accuracies on ILSVRC but large performance gap on TrafficSign.
+16
+
+Published in Transactions on Machine Learning Research (08/2022)
+Futhermore, we visualize more episodes from Aircraft, TrafficSign and MSCOCO in Fig. 6, Fig. 7 and Fig. 8,
+which shows that our porposed eTT can remarkably improve the embedding space after test-time finetuning.
+A.3
+Broader Impact
+Our paper presents a more efficient and practical FSL pipeline utilizing ViT. We hope this work can shed
+light on the broader usage of ViT in FSL tasks. On the other hand, the proposed method can provide
+researchers with alternative choice for FSL applications in real-case scenarios with large-scale meta-train set
+and challenging various test episodes.
+dproj
+ILSVRC
+Omni
+Acraft
+CUB
+DTD
+QDraw
+Fungi
+Flower
+Sign
+COCO
+Avg
+0.01
+67.01
+76.56
+78.34
+85.53
+86.96
+70.03
+61.20
+96.17
+85.00
+62.67
+76.95
+0.05
+66.49
+77.40
+78.92
+85.80
+87.54
+70.23
+60.78
+96.28
+84.95
+62.38
+77.08
+0.5
+66.88
+77.73
+78.65
+86.00
+87.15
+70.48
+61.64
+96.23
+84.39
+63.39
+77.25
+0.9
+67.03
+76.55
+77.89
+85.78
+87.04
+70.08
+62.45
+96.20
+84.44
+62.83
+77.03
+0.1
+67.37
+78.11
+79.94
+85.93
+87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+Table 5:
+Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items
+are the best ones with highest accuracies.
+Model
+Backbone
+ILSVRC
+Omni
+Acraft
+CUB
+DTD
+QDraw
+Fungi
+Flower
+Sign
+COCO
+Rank
+Finetune
+Res18
+45.781.10
+60.851.58
+68.691.26
+57.311.26
+69.050.90
+42.601.17
+38.201.02
+85.510.68
+66.791.31
+34.860.97
+10.2
+Proto
+50.501.08
+59.981.35
+53.101.00
+68.791.01
+66.560.83
+48.961.08
+39.711.11
+85.270.77
+47.121.10
+41.001.10
+10.5
+Relation
+34.691.01
+45.351.36
+40.730.83
+49.511.05
+52.970.69
+43.301.08
+30.551.04
+68.760.83
+33.671.05
+29.151.01
+14.6
+P-MAML
+49.531.05
+63.371.33
+55.950.99
+68.660.96
+66.490.83
+51.521.00
+39.961.14
+87.150.69
+48.831.09
+43.741.12
+9.2
+BOHB
+51.921.05
+67.571.21
+54.120.90
+70.690.90
+68.340.76
+50.331.04
+41.381.12
+87.340.59
+51.801.04
+48.030.99
+8.2
+TSA
+59.501.10
+78.201.20
+72.201.00
+74.900.90
+77.300.70
+67.600.90
+44.701.00
+90.900.60
+82.500.80
+59.001.00
+4.3
+Ours
+ViT-t
+56.401.13
+72.521.36
+72.841.04
+73.791.09
+77.570.84
+67.970.88
+51.231.15
+93.300.57
+84.091.07
+55.681.05
+4.1
+Proto
+Res34
+53.701.07
+68.501.27
+58.000.96
+74.100.92
+68.800.77
+53.301.06
+40.701.15
+87.000.73
+58.101.05
+41.701.08
+7.4
+CTX
+62.760.99
+82.211.00
+79.490.89
+80.630.88
+75.570.64
+72.680.82
+51.581.11
+95.340.37
+82.650.76
+59.901.02
+2.8
+TSA
+63.730.99
+82.581.11
+80.131.01
+83.390.80
+79.610.68
+71.030.84
+51.381.17
+94.050.45
+81.710.95
+61.670.95
+2.5
+Ours
+ViT-s
+67.370.97
+78.111.22
+79.941.06
+85.930.91
+87.620.57
+71.340.87
+61.801.06
+96.570.46
+85.090.90
+62.330.99
+1.6
+Table 6:
+Test accuracies, confidence interval and average rank on Meta-Dataset. Note that different
+backbones are adopted by these methods. The bolded items are the best ones with highest accuracies.
+Model
+Train Set
+ILSVRC
+Omni
+Acraft
+CUB
+DTD
+QDraw
+Fungi
+Flower
+Sign
+COCO
+Avg
+eTT
+meta-train
+67.37
+78.11
+79.94
+85.93
+87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+eTT
+full
+74.76
+78.73
+80.10
+86.99
+87.72
+71.20
+61.95
+96.66
+85.83
+64.25
+78.82
+P>M>F
+full
+74.69
+80.68
+76.78
+85.04
+86.63
+71.25
+54.78
+94.57
+88.33
+62.57
+77.53
+Table 7:
+Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items
+are the best ones with highest accuracies. The highlighted rows denote the final model in our main paper.
+17
+
+Published in Transactions on Machine Learning Research (08/2022)
+Model
+Pretrain
+ILSVRC
+Omni
+Acraft
+CUB
+DTD
+QDraw
+Fungi
+Flower
+Sign
+COCO
+Avg
+TSA
+Sup.
+59.50
+78.20
+72.20
+74.90
+77.30
+67.60
+44.70
+90.90
+82.50
+59.00
+70.68
+TSA
+DINO
+48.18
+64.94
+56.74
+45.49
+69.06
+59.51
+31.13
+81.01
+48.70
+26.18
+53.09
+eTT
+Sup.
+65.17
+67.47
+73.30
+87.71
+84.50
+67.46
+55.51
+92.55
+64.08
+63.68
+72.14
+eTT
+DINO
+67.37
+78.11
+79.94
+85.93
+87.62
+71.34
+61.80
+96.57
+85.09
+62.33
+77.61
+Table 8:
+Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items
+are the best ones with highest accuracies. The highlighted rows denote the final model in our main paper.
+Model
+ILSVRC
+Omni
+Acraft
+CUB
+DTD
+QDraw
+Fungi
+Flower
+Sign
+COCO
+Avg
+eTT
+100.00
+99.99
+100.00
+100.00
+100.00
+100.00
+99.79
+100.00
+100.00
+99.15
+99.89
+FT
+100.00
+99.87
+100.00
+100.00
+100.00
+100.00
+96.95
+100.00
+100.00
+95.20
+99.20
+Table 9:
+Support set accuracies of eTT and Finetune on testing episodes whose minimum shots is no larger
+than 2.
+Model
+ILSVRC
+Omni
+Acraft
+CUB
+DTD
+QDraw
+Fungi
+Flower
+Sign
+COCO
+Avg
+FT
+29.19
+54.54
+35.10
+41.54
+53.66
+43.37
+38.53
+76.76
+72.90
+41.21
+48.68
+eTT
+40.22
+64.79
+41.33
+55.11
+66.20
+49.14
+56.33
+85.03
+75.29
+56.19
+58.96
+Table 10:
+Query set accuracies of eTT and Finetune on testing episodes whose minimum shots is no larger
+than 2. The bolded items are the best ones with highest accuracies.
+Model
+ILSVRC
+Omni
+Acraft
+CUB
+DTD
+QDraw
+Fungi
+Flower
+Sign
+COCO
+w/o stand
+1.06
+1.25
+1.05
+0.89
+0.64
+0.92
+1.05
+0.38
+0.96
+0.96
+w stand
+0.97
+1.22
+1.06
+0.91
+0.57
+0.87
+1.06
+0.46
+0.90
+0.99
+Table 11:
+The corresponding confidence intervals of models in ablation study on standardization.
+18
+
+Published in Transactions on Machine Learning Research (08/2022)
+w DRA
+w DRA
+w/o DRA
+w/o DRA
+Figure 5: Visualization of self-attention from model with and without DRA on ILSVRC (left) and TrafficSign
+(right). The white regions are those with top 20% attention scores
+19
+
++0STOPPublished in Transactions on Machine Learning Research (08/2022)
+Figure 6: More visualization of feature embeddings from a randomly sampled episode of TrafficSign.
+20
+
+Published in Transactions on Machine Learning Research (08/2022)
+Figure 7: More visualization of feature embeddings from a randomly sampled episode of Aircraft.
+21
+
+Published in Transactions on Machine Learning Research (08/2022)
+Figure 8: More visualization of feature embeddings from a randomly sampled episode of MSCOCO.
+22
+

E9AzT4oBgHgl3EQfUPzF/content/tmp_files/2301.01264v1.pdf.txt

@@ -0,0 +1,2590 @@
+Tunable intracellular transport on converging microtubule morphologies
+Niranjan Sarpangala,1 Brooke Randell,2 Ajay Gopinathan,1 and Oleg Kogan2
+1University of California, Merced, CA, 95343
+2California Polytechnic State University, San Luis Obispo, CA, 93407
+A common type of cytoskeletal morphology involves multiple converging microbutubules with their
+minus ends collected and stabilized by a microtubule organizing center (MTOC) in the interior of the
+cell. This arrangement enables the ballistic transport of cargo bound to microtubules, both dynein
+mediated transport towards the MTOC and kinesin mediated transport away from it, interspersed
+with diffusion for unbound cargo-motor complexes. Spatial and temporal positioning of the MTOC
+allows for bidirectional transport towards and away from specific organelles and locations within the
+cell and also the sequestering and subsequent dispersal of dynein transported cargo. The general
+principles governing dynamics, efficiency and tunability of such transport in the MTOC vicinity is
+not fully understood. To address this, we develop a one-dimensional model that includes advective
+transport towards an attractor (such as the MTOC), and diffusive transport that allows particles
+to reach absorbing boundaries (such as cellular membranes). We calculated the mean first passage
+time (MFPT) for cargo to reach the boundaries as a measure of the effectiveness of sequestering
+(large MFPT) and diffusive dispersal (low MFPT). We show that the MFPT experiences a dramatic
+growth in magnitude, transitioning from a low to high MFPT regime (dispersal to sequestering) over
+a window of cargo attachment/detachment rates that is close to in vivo values. Furthermore, we
+find that increasing either the attachment or detachment rate, while fixing the other, can result in
+optimal dispersal when the attractor is placed asymmetrically. Finally, we also describe a regime of
+rare events where the MFPT scales exponentially with advective velocity towards the attractor and
+the escape location becomes exponentially sensitive to the attractor positioning. Taken together,
+our results suggest that structures such as the MTOC allow for the sensitive control of the spatial
+and temporal features of transport and corresponding function under physiological conditions.
+Introduction
+The transport of material within eukaryotic cells is a
+critically important physiological process that cannot be
+achieved by passive diffusion alone. In these cells, cargo,
+including vesicles and organelles, are dragged along by
+a variety of molecular motors which utilize energy from
+ATP hydrolysis to power their directed stepping motion
+along cytoskeletal protein filaments with a well-defined
+polarity [1]. Motors from different families such as ki-
+nesins and myosins step along different filaments (micro-
+tubules and actin respectively) and others such as dynein
+move along the same microtubule filaments as kinesins
+but in the opposite direction. Transport at the cellular
+scale is therefore a complex process that involves phases
+of multiple motors effecting directed transport along cy-
+toskeletal filament networks interspersed with passive dif-
+fusion of the cargo [2, 3]. This process is essential for
+the transport of a variety of cargo between specific lo-
+cations and organelles within the cell. Examples include
+the transport of cargo in cilia [4], between the plasma
+membrane and Golgi apparatus [5], [6], between Endo-
+plasmic Reticulum and Golgi [7], [3], transport of viruses
+towards replication sites [8], [9], and the transport of
+many other vesicles and organelles for various functional
+purposes (see review [3]), [10].
+Much like the design of road networks affect traf-
+fic flow, the morphologies of the cytoskeletal networks
+in cells have been shown to have a significant effect
+on intracellular transport [11–14].
+This is particularly
+important as, even a single type of cytoskeletal fila-
+ment such as microtubules exhibit a wide diversity of
+morphologies within different cell types to enable dif-
+ferent functions[15].
+In some situations, such as in
+melanophores microtubules have a strongly orderly (in
+this case radial) - organization [16]. In others, the orien-
+tation or polarity of microtubule (MT) morphology can
+be broadly distributed. In pancreatic β cells, for exam-
+ple, MTs are arranged with both an orientational and
+polarity disorder [17], although there is an average po-
+larity. On the other hand, MTs in neuronal dendrites
+are essentially aligned with the long direction of the den-
+drite, but their polarity is not uniform [18] resulting in
+junctions of plus or minus ends along the dendrite.
+A common structural feature that governs these micro-
+tubule morphologies is the microtubule organizing center
+(MTOC) that is responsible for growing MTs and local-
+izing and stabilizing their minus ends leading to multi-
+ple MTs converging with their minus ends at the MTOC
+[15]. Dynein-driven transport along MTs will move cargo
+to the vicinity of MTOC, while kinesin mediated trans-
+port moves cargo away from it. These ballistic phases are
+interspersed with isotropic diffusion for unbound cargo-
+motor complexes. The spatial and temporal positioning
+of the MTOC therefore allows for bidirectional transport
+towards and away from specific organelles that can act
+as MTOCs as well as locations within the cell in the
+vicinity of the MTOC. Examples in which MTOC facil-
+itates direct transport to the destination of interest in-
+clude transport of cargo such as secretory vesicles away
+from the Golgi apparatus toward the cell membrane and
+endocytic vesicles towards the Golgi which is known to
+perform as an MTOC in many mammalian cells [5], [6].
+arXiv:2301.01264v1  [q-bio.CB]  3 Jan 2023
+
+2
++
++
++
++
++
++
++
++
++
++
++
++
++
+(a) 
+(b)
+FIG. 1: (a) A model of a cell in which microtubules have
+a strong central organization, with minus ends at the cen-
+trosome. A dark circle represents an organelle. Dynein mo-
+tors are shown moving on microtubules. (b) One dimensional
+morphology found in dentrites. Here the ends of the same po-
+larity from different microtubules can face each other. This
+schematic is based on [19].
+The dynein mediated transport of some viruses toward
+the nuclear envelope is also enabled by the presence of a
+MTOC in the vicinity of the nucleus [8, 9].
+In some cases, cargo need to traverse regions with
+convergent MT morphologies. Such cases occur in den-
+dritic processes of neuronal cells that have been shown
+to have regions of alternating polarity of MTs [18]. Di-
+rected transport of dynein (kinesin) carrying cargo at a
+junction of minus(plus) ends will have to overcome what
+is essentially a trap to maintain observed unidirectional
+transport towards or away from the main cell body [18].
+Finally, the location of MTOCs can also be tuned over
+time to accommodate different cellular functions such
+as sequestering and dispersal of cargo. For example, in
+melanophores [16, 20], a perinuclear MTOC produces a
+radial MT structure with minus ends in towards the nu-
+cleus and plus ends out toward the membrane.
+Cells
+achieve color change by aggregating and sequestering pig-
+ment containing melanosomes near the nucleus via bal-
+listic dynein mediated transport. Upon hormonal stimu-
+lation they can switch to a superdiffusive dispersal phase
+powered by a combination of kinesin and actin. Another
+example occurs in lymphocytes that enable cytotoxicity
+by secreting the contents of lysosomes (lytic granules) at
+the immunological synapse to kill the target cell. Here,
+dynein dependent sequestering of the lytic granules at
+the MTOC occurs rapidly followed by the gradual move-
+ment of the MTOC towards the synapse with subsequent
+secretion [21, 22].
+In all these cases, it is important to understand
+the dynamics of the transport and its sensitivity to
+biological parameters in order to understand functional
+efficiency and robustness. In particular, given the wide
+variety of functional contexts in which the converging
+MT geometry facilitates transport,
+it is critical to
+understand the general principles governing dynamics,
+efficiency and tunability of such transport in the MTOC
+vicinity.
+To address this gap,
+we develop a simple one-
+dimensional model that includes advective transport to-
+wards an attractor (such as the MTOC), and diffusive
+transport that allows particles to reach absorbing bound-
+aries (such as cellular membranes). This can be viewed
+as a 2-layer model consisting of an advective layer en-
+dowed with an attractor, a diffusive layer, and absorb-
+ing boundaries along the perimeter of the domain. We
+take the mean first passage time (MFPT) for cargo to
+reach the boundaries as a measure of the effectiveness
+of sequestering or directed transport (large MFPT) and
+diffusive dispersal (low MFPT). The number of indepen-
+dent control parameters in this problem can be reduced
+to four. These are the rates of attachment to and de-
+tachment from microtubules, advective velocity, and the
+placement of the attractor within the domain.
+Using this model we were able to make a series of tan-
+talizing predictions - on which we report here. A cen-
+tral calculation here is the residence time, or what is
+commonly called in the literature the mean first passage
+time (MFPT). Thus, given an initial location of the cargo
+within the domain (determined by organelle placement),
+this quantity tells the average time to reach either of the
+absorbing boundaries (i.e. escape the domain), or a spe-
+cific boundary (in one dimension, left or right). Another
+relevant quantity is the probability of escape through one
+or the other domain.
+Symmetric, or nearly symmetric attractor positions
+can give rise to a dramatic increase in the value of MFPT
+within a certain window of dimensionless coupling rates
+between the layers. Concurrently with this dramatic rise
+of MFPT, the probability to escape purely diffusively
+goes to zero in the same range of (dimensionless) cou-
+pling values. This means that for larger coupling values,
+any cargo particle will have to experience at least one
+episode of motion on microtubules. Crucially, we found
+that biophysical parameters in cells correspond precisely
+
+3
+to this range of dimensionless coupling rates. This sug-
+gests that parameter values in cells are optimized for the
+greatest sensitivity to small changes. With such parame-
+ters, a cell can achieve the largest change in functionality
+with smallest changes in parameter values.
+Second, we predict the existence of optimal coupling
+rates that minimize the MFPT. This minimal MFPT
+happens when the attractor is positioned asymmetrically
+(off center) in the domain. A similar phenomenon has
+been predicted in the study of diffusion with stochastic
+reset [23], [24]. Indeed, attachment to the microtubule,
+followed by a rapid transport to the attractor, followed
+by detachment from the microtubule back to diffusion in
+the cytoplasm is effectively a reset.
+When the coupling rates are much larger than all other
+rates in the problem, the model reduces to effectively
+one-layer. Here we demonstrate that even a slight asym-
+metry in the position of the attractor can lead to a very
+strong amplification of the preferred exit end. This pro-
+vides another example of sensitivity to small parameter
+changes - in this case asymmetric of the attractor place-
+ment. This effect happens at sufficiently large advective
+velocity, and corresponds to rare event physics. In the
+regime of rare events, a small fraction of particles escape
+quickly, while the majority advect to the attractor, and
+form a quasi-stationary distribution around it. They stay
+in the vicinity of the attractor for a time that scales expo-
+nentially with advective velocity (or inverse of diffusion
+coefficient).
+Methods
+Model
+We consider the minimal model in a one-dimensional
+domain of length L. It contains an advective layer (AL)
+that represents motion along microtubules, and a dif-
+fusive layer (DL) that represents diffusion in the cyto-
+plasm. We assume that attachment to and detachment
+from microtubules are Poisson processes, endowed with
+rates α and β respectively. This means, for example, that
+a motor spends on average a time 1/β since attaching to
+a microtubule. While advecting, particles move with a
+uniform velocity towards the attractor - which is an at-
+tracting fixed point located at some coordinate x = X0
+between x = 0 and x = L. Letting ρ(x) and θ(x) be
+probability densities of particles in the AL and DL re-
+spectively, the model reads
+∂ρ
+∂t = − ∂
+∂x (v(x)ρ) + αθ − βρ
+(1)
+∂θ
+∂t = −αθ + βρ + D ∂2θ
+∂x2
+(2)
+on 0 ≤ x ≤ L. The velocity field is given by
+v(x) =
+�
++v0 ... x < X0
+−v0 ... x > X0
+(3)
+The parameters are rates α and β, the diffusion coeffi-
+cient D, the advective velocity on microtubules v0, and
+the location of the attractor X0.
+There are absorbing
+BCs at x = 0 and x = L, i.e.
+ρ(0) = θ(0) = 0 and
+ρ(L) = θ(L) = 0. All together, there are six physical
+parameters.
+We will switch to dimensionless variables by rescaling
+the lengths by L and times by L2/D. Thus, x′ = x/L
+and t′ = tD/L2. The resulting equations will be
+∂ρ
+∂t′ = − ∂
+∂x′ (v′(x)ρ) + aθ − bρ
+(4)
+∂θ
+∂t′ = −aθ + bρ + ∂2θ
+∂x′2
+(5)
+on 0 < x′ < 1, with ρ(0) = θ(0) = 0 and ρ(1) = θ(1) = 0,
+the velocity field
+v′(x) =
+�
++v ... x < X
+−v ... x > X
+where X = X0/L and v = v0L
+D , and coupling rates a =
+αL2
+D
+and b = βL2
+D . From now on, we will drop primes.
+The model is depicted schematically in Fig. 2.
+𝐷 = 1
+𝑥 = 1
+FIG. 2: One-dimensional model with dimensionless parame-
+ters.
+Range of parameters
+Here we review the values of parameters from litera-
+ture. Both adsorption rate α and desorption rate β are
+expected to be of the order of 1 per second. For example,
+[11] cites α = 5 s−1 and β = 1 s−1. Microtubule lengths
+typically fall in the range of 1−10 µm [11]. However, the
+length of advective path may be much larger. For exam-
+ple, in neurons, a cargo that needs to be delivered from
+the soma to synapses on the ends of axons will travel a
+length of the order of a meter [3]. The velocity of molec-
+ular motors on MTs is on the order of 1 µm/s [3], [11],
+although this quantity also has a degree of variability
+[25]. Diffusion coefficient of vesicular organelles in the
+cytoplasm fall in the range 10−3 − 10−1 µm2/s [3].
+Given these physical parameters, our dimensionless pa-
+rameters a and b will take on values in the range [10, 105],
+and parameter v will take on values in the range [10, 104].
+
+v(x)
+Advective layer
+rate a
+rate b
+Diffusive layer
+D
+Junction point
+x=0
+x=L
+at x = X4
+There are four timescales in the problem: 1/a, 1/b,
+the advective timescale 1/v, and the diffusive timescale
+(which is of order 1 in dimensionless units).
+Different
+special cases or behavioral regimes emerge when one of
+these timescales differs significantly from others.
+The limit that is particularly amenable to analysis is
+one in which 1/a and 1/b are both much smaller than
+the advective time (which is of order 1/v in dimension-
+less units) and diffusive time (which is of order 1 in di-
+mensionless units). We will formally call it the a, b → ∞
+limit. In this regime, the model reduces to an advection-
+diffusion process in one single layer, which amenable to
+many analytical results.
+Analytical approach in the one-layer limit
+A very important special case is a = b. As a = b → ∞,
+the model reduces to an effeective one-layer model:
+∂P
+∂t = − ∂
+∂x
+�
+v(x)P(x) − ∂P
+∂x
+�
+where P(x, t) is the probability density (i.e. P describes
+both θ and ρ, which become identical). A general solution
+will be written as an eigenfunction expansion
+P(x, t) =
+�
+n
+cnpn(x)eσnt,
+(6)
+where pn(x) and σn is nth eigenfunction and eigenvalue,
+which satisfy Opn = σnpn, with the operator O given by
+O = − ∂
+∂x
+�
+v(x) − ∂
+∂x
+�
+,
+(7)
+with
+v(x) =
+�
++v ... x < X
+−v ... x > X
+(8)
+and a constant v. Thus, the one-layer model contains
+two parameters: dimensionless advective velocity v and
+dimensionless position of the attractor X, which can take
+on values between 0 and 1.
+The computation of eigenvalues σn and eigenfunctions
+pn(x) of the operator O, as well as the computation of
+the eigenfunctions qn(x) of the operator O† is given in
+Appendix B.
+Starting from the initial condition P(x, t = 0) = δ(x −
+x0), the probability density will be given by
+P(x, t; x0) =
+�
+n
+q∗
+n(x0)pn(x)
+� 1
+0 q∗n(x′)pn(x′) dx′ eσnt
+(9)
+Everything that we need to compute MFPT can be ex-
+tracted from this probability density.
+To calculate the MFPT τ(x0), we notice that the mag-
+nitude of the flux through the boundary is given by
+f(t) =
+�� ∂P
+∂x
+��
+bdry in dimensionless units. Then f(t)dt gives
+the fraction of initial particles that cross the boundary
+in [t, t + dt] = probability of crossing that boundary in
+[t, t + dt], since the initial condition is normalized to 1.
+So, p =
+� ∞
+0
+f(t) dt gives the probability of ever leav-
+ing through that boundary,
+f(t)dt
+p
+gives the probability
+that particles that leave through that boundary do so
+in [t, t + dt], and finally τ =
+� ∞
+0
+t f(t)
+p dt is the average
+time to leave through that boundary. In this problem,
+there are two boundaries, with τl and τr denoting MFPT
+to exit through the left and the right boundary respec-
+tively.
+We expect τl → 0 as x0 → 0 and τr → 0 as
+x0 → 1. Finally, MFPT in general - without condition-
+ing on a specific boundary - is the weighted average of
+the two: τ = τlpl+τrpr, which matches predictions using
+other methods [26].
+Analytical approach in the general case
+Analogously to the one-layer approach, we again seek
+a general solution to Eqs. (4)-(5) via an eigenfunction
+expansion of the form
+�
+ρ(x, t)
+θ(x, t)
+�
+=
+�
+n
+cn
+�
+Rn(x)
+Θn(x)
+�
+e−σnt
+(10)
+(we found it convenient to factor out the negative sign
+from σ here), where
+�
+Rn
+Θn
+�
+and σn is the nth (vector)
+eigenfunction and eigenvalue, which satisfy O
+�
+Rn
+Θn
+�
+=
+σn
+�
+Rn
+Θn
+�
+, with the operator O given by
+O =
+�
+∂
+∂xv(x) + b
+−a
+−b
+a −
+∂2
+∂x2
+�
+(11)
+with v(x) given by Eq. (8). The full model contains four
+parameters: dimensionless advective velocity v, dimen-
+sionless rates a and b, and dimensionless position of the
+attractor X, which can take on values between 0 and
+1. The computation of eigenvalues and eigenfunctions is
+given in Appendix A.
+Remarkably, there are only a finite number of eigen-
+functions and eigenvalues. In other words, the eigenset is
+not complete. As a = b → ∞, this number goes to infin-
+ity, while the lower-lying eigenvalues and eigenfunctions
+approach those of the one-layer model. The completeness
+is not guaranteed, since the operator O is not Hermitian.
+Thus, an expansion such as in Eq. (10) is of limited use,
+and cannot be used to fit a solution for an arbitrary initial
+condition - including a point-like δ function initial con-
+dition. This also implies that we cannot compute escape
+currents and MFPT from such initial conditions.
+However, we can always compute the ground state
+eigenvalue, σ1.
+Then the time 1/σ1, while not a true
+MFPT, is an estimate of a characteristic time for escape.
+
+5
+We found that this time alone agrees with MFPT com-
+puted in simulations quite well, so we will make MFPT
+arguments based on this estimate.
+Monte Carlo simulation method
+We considered a simple one dimensional computational
+model to simulate the transport process in a domain of
+length L with attractor formed by oppositely oriented mi-
+crotubules. Our computational model involves two lay-
+ers, an advective layer (AL) where the particle undergoes
+active transport and a diffusive layer where it does one
+dimensional random walk. We consider one particle at
+a time. To begin, we initialize the particle at position
+x = x0 within the domain x ∈ [0, L = 1] either in the
+diffusive or in advective layer as required. We consider
+that the particle can switch from diffusive layer to ad-
+vective layer with a rate a and from advective layer to
+diffusive layer with a rate b. When a particle switches to
+diffusive layer, a time td is drawn from the exponential
+distribution e−at and the particle is allowed to diffuse for
+n = td/∆t number of steps. ∆t is the time step in the
+simulation. In each step the position is updated as
+x(t + ∆t) = x(t) + r∆x,
+(12)
+where r is drawn from the set {−1, 0, 1} with the proba-
+bility p = 1/3. ∆x is the step size which is chosen such
+that the diffusion constant of the particle D = p∆x2
+∆t
+is
+1. Right after finishing a diffusive portion of a simula-
+tion run, the particle switches from diffusive to advective
+layer. In the advective layer, the particle stays for a time
+ta drawn from e−bt, i.e. n = ta/∆t number of steps. In
+the advective layer, the position of the particle is updated
+as
+x(t + ∆t) = x(t) + v(x)∆t,
+(13)
+where v(x) is the advective velocity given by Eq. (8).
+These alternative portions of a simulation run in diffu-
+sive and advective layers are continued until the particle
+reaches one of the boundaries (x = 0 or x = 1) or until
+maximum simulation time, Tmax is reached. We then re-
+peat with N particles to get enough statistics to calculate
+the overall MFPT, probabilities and MFPTs to exit out
+of specific boundaries, and other quantities.
+Trajectories
+To get the trajectories, we record the data of the x po-
+sition and the layer in which particle is located at regular
+time intervals during each simulation run. An example
+of trajectories is shown in Fig. 3.
+Diffusion
+1D Random Walk
+Molecular Motor 
+Based Transport
+FIG. 3: A sample trajectory generated by the Monte Carlo
+simulation.
+Diffusive motion is indicated with orange line,
+and advective motion with a magenta line. Grey colored lines
+indicate more sample trajectories. Here X = 0.5.
+Computation of Net MFPT
+To compute the net MFPT for a given parameter set,
+we perform simulation runs until the particle exits out of
+one of the boundaries (x = 0 or x = 1). We record the
+time of exit for each run and then compute the mean and
+standard error of the mean for all N runs.
+Computation of Conditional MFPT and escape probability
+To compute the MFPT for exit specifically through
+the left (or the right) boundary, we record the time as
+well as the boundary through which the particle exits.
+Then we filter out only those simulation runs where a
+particle exited out of the left (or right) boundary. Then
+we compute mean and standard error of the mean for
+those runs. We compute the escape probability through
+left (or right) boundary as the fraction of runs that exited
+out of the left (or right) boundary.
+Statistics of visits to the AL
+We measure the fraction of simulation runs in which
+a particle that started on the DL ended up making at
+least one visit to the AL. In each simulation run, we
+also compute the number of visits to the advection layer
+before exiting. To do this, we update a counter every
+time the particle switches its layer to get the number of
+times it switches layers prior to exiting the domain. We
+then compute the average over N runs.
+
+6
+Results and Discussion
+Variation of coupling rates can change escape times
+by orders of magnitude
+We begin our presentation of results with the symmet-
+ric case, X = 1/2. For simplicity we will set the particles’
+initial placement at x0 = 1/2 - this is the initial condi-
+tion (IC) in analytical calculations - and let a = b for
+now. Figure 4 (a) displays the mean first passage time
+(MFPT) as a function of a = b at different advective
+speeds v. To help understand the physics of the process,
+we also plot the fraction of times that particles visit the
+advective layer in panel (b) (for particles initially placed
+on the DL), as well as the number of times they do so in
+panel (c) (also when starting on DL).
+Two crossovers are evident from the plot of MFPT vs
+a (= b). The first crossover takes place around a = 10−2.
+As suggested by the plot of the fraction of visits to the
+AL, at this coupling rate the probability of visiting the
+AL becomes non-zero; below this crossover, the advective
+layer is not visited and the MFPT is a purely diffusive
+time ≈ 0.12. For a above this crossover value, the prob-
+ability of visiting the AL grows with increasing a. While
+the fraction of particles visiting the AL grows ∝ a, the
+time to remain in the AL (the longest time scale in this
+range of a) decreases ∝ 1/a, resulting in the plateau of
+MFPT vs. a.
+Because the probability (or fraction) to
+visits to the AL is less than 1 (for particles startin in the
+DL), a particle has a chance to escape purely diffusively
+for as in this plateau region.
+The MFPT is in dimensionless time units; to convert to
+time in seconds, multiply by L2/D expressed in physical
+units. For example, for L = 1 µm and D = 10−2 µm2/s,
+the MFPT of 10 dimensionless time units corresponds
+to 103 seconds. The MFPT for diffusive transport on a
+domain with two absorbing boundaries and a midpoint
+initial condition is 0.125 (in dimensionless time units),
+which is half of the first plateau value, and much lower
+than plateaus after the second crossover for v > 1.
+We continue our discussion of Fig. 4. The probabil-
+ity of visiting the AL (for particles starting in the DL)
+eventually reaches 1 at larger a; particles are now certain
+to visit the AL at least once. In other words, the prob-
+ability of a purely diffusive escape reaches zero and we
+encounter the second crossover. For v = 20, for example,
+this second crossover happens around a = b = 10, but its
+location - defined by the point of inflection - varies some-
+what with v. This crossover is broad - it can be several
+decades wide - and marked by a drastic growth of the
+MFPT, especially at larger v. In this second crossover
+regime, each particle experiences intermittent advection,
+punctuated by periods of diffusion. In other words, on a
+typical run from an initial location to one of the bound-
+aries, a particle’s trajectory will include multiple episodes
+of advection and diffusion following each other. Eventu-
+ally, we reach the second plateau, when the switching
+between the layers is so rapid that we now reach an ef-
+FIG. 4: Symmetric case: X = 0.5, the initial location of
+particles is also at x0 = 0.5. (a) MFPT vs. a = b. Dots: IC on
+the DL; crosses: IC on the AL. The solid curves are analytical
+estimations of MFPT given by 1/σ1, where σ1 is the ground
+state eigenvalue. The MFPT is in dimensionless time units;
+to convert to time in seconds, multiply by L2/D expressed in
+physical units. The dashed horizontal line has a value 0.25.
+The last two points (a = 105 and 106) required a smaller
+dt = 10−6
+3
+; dt = 10−4
+3
+was sufficient for the rest. Therefore,
+we used N = 103 for the last two points to optimize simulation
+time, and N = 104 for the rest. (b) Fraction of simulation
+runs that visit the AL at least once after starting in the DL.
+The dashed line is a fit, of the form 0.079a. Here N = 104
+and dt =
+10−4
+3
+.
+(c) Average number of visits for particles
+starting in the DL. Here N = 103, dt =
+10−4
+3
+(circles), and
+10−6
+3
+(diamonds). The x-axis is the same in all three plots; the
+plots are aligned. The shading guides the eye to the second
+crossover region.
+
+7
+fectively one-layer regime. This regime will be studied
+in the next section, where we examie a one layer model
+with advection and diffusion taking place simultaneously.
+MFPTs predicted by that model match the high a = b
+plateaus. Interestingly, there is a strong velocity depen-
+dence in the one-layer regime, but not in the range of
+a = b in the plateau below the second crossover.
+For a = b < 1/(simulation time), particles with IC in
+the advective layer (plus symbols in Fig. 4) will never
+enter the DL and therefore will not escape. MFPT will
+simply be limited by the simulation time - this is mani-
+fested in the saturation at MFPT = 500, since this was
+the simulation time.
+Appendix C displays examples of particle trajectories
+for a broad range of a = b that cover all of the behavioral
+regimes shown in Fig. 4. These figures demonstrate the
+change in the character of trajectories - from the types
+that contain advective periods long enough to arrive to
+the attractor at low a = b, to intermittent behavior in the
+second crossover region, to very rapid switching between
+layers for a = b beyond the second crossover - when the
+model is effectively in the one-layer regime.
+The region of the most sensitive behavioral tuning matches
+the biological parameters
+We now turn our attention to the biological significance
+of these results.
+Note that the second crossover takes
+place between a = 10 and a = 104. Remarkably, this
+is precisely the range of these parameters found in cells
+- see “Range of parameters” above. This might imply
+that these parameters evolved to have such values for
+an easy tunability. Indeed, the second crossover region
+is precisely where a change in the rates gives rise to the
+largest change in the outcome - especially at larger values
+of v.
+There is an optimal coupling rate between advective
+and diffusive behavior
+Placing the attractor asymmetrically can give rise to
+a decrease in MFPT with increasing coupling rates - see
+Fig. 5. This effect is only seen at larger v. The decrease
+happens over a range of 1/a that is comparable to the
+advective time, ∼ 1/v.
+For example, for v = 20, the
+time scale to travel advectively to the attractor is ∼ 0.05,
+while the decrease is seen for a between 1 and 100, which
+corresponds to the time scale between 1 and 0.01.
+We think that this decrease in the MFPT happens be-
+cause an increase in the interlayer coupling causes more
+material to congregate at the attractor, which is close to
+one of the ends - thus leading to an overall decrease in
+the MFPT.
+Fig. 6 shows an example of this phenomenon due to
+only the parameter a varied at fixed b. We mentioned in
+the discussion of the analytical approach in the general
+5
+1
+FIG. 5: Asymmetric case: X = 0.85. In this particular case,
+x0 = 0.7, but such dips are also present at other x0.
+case that a complete eigenset in the two-layer model does
+not exist, so the exact solution cannot be obtained as a
+sum of the modes. However, the MFPT can be estimated
+as τ = 1/σ1, where σ1 is the ground state eigenvalue.
+𝑎
+𝑣 = 13
+𝑏 = 169
+𝑿 = 𝟏/𝟐𝟔
+𝑣 = 13
+𝑏 = 169
+𝑿 = 𝟏/𝟐
+Net MFPT
+𝑎
+Net MFPT
+(a)
+(b)
+Out[ ]=
+0.01
+100
+0.1
+10
+1000
+105
+Out[ ]=
+0.01
+100
+0.1
+1
+10
+100
+1000
+FIG. 6: τ(a) at fixed b = 169. (a) X = 0.5, (b) X = 1/26.
+v = 13 for both. Lines: theory, dots: simulation. Red dots -
+IC on the diffusive layer, blue dots - IC on the advective layer.
+The numbers for the two types of initial conditions are not
+identical, but the difference is almost invisible. The analytical
+prediction is 1/σ1 - the inverse of the ground state eigenvalue,
+which is not a true MFPT. The IC in the simulation was at
+x0 = 0.5. The simulation time was 1000, which is the reason
+for flattening of the simulation data at large a in panel (a).
+
+8
+𝑏
+Net MFPT
+𝑏
+Net MFPT
+(a)
+(b)
+Out[ ]=
+0.1
+1
+10
+100
+1000
+104
+105
+0.1
+1
+10
+100
+1000
+Out[ ]=
+0.01
+100
+0.1
+1
+10
+100
+1000
+FIG. 7: τ(b) at fixed a = 169. (a) X = 0.5, (b) X = 1/26.
+v = 13 for both. Lines: theory, dots: simulation. Red dots
+- IC on the diffusive layer, blue dots - IC on the advective
+layer. The analytical prediction is 1/σ1 - the inverse of the
+ground state eigenvalue, which is not a true MFPT. The IC
+in the simulation was at x0 = 0.5. We again see saturation of
+simulation results at low b at the simulation time (here, 1000
+time units).
+The solid lines in Fig. 6 are values of 1/σ1.
+This es-
+timation should become more accurate as escape events
+become rare (MFPT ≫ than all other time scales); this is
+because higher eigenmodes contribute little to the prob-
+ability current in the rare event limit. Moreover, while
+this calculation does not give IC dependence, MFPT loses
+this dependence as escape events become rare. Some dis-
+cussion of rare events can be found in the next section,
+and a much more in-depth discussion will appear in [27].
+The dips in Fig. 6 happen, again, because increasing a
+causes particles to return back to the attractor, thus min-
+imizing the chance for them to wander too far to the right
+while diffusively exploring the long part of the domain.
+On the other hand, increasing a even further tends to
+keep the particles in the AL and therefore prevents them
+from escaping (particles cannot move in the direction of
+the ends when they are in the AL due to the advective
+flow being directed towards the attractor).
+These dips are somewhat counter-intuitive - an overall
+escape time is lowered by increasing the tendency to go
+towards the attractor inside the domain - as long as the
+attractor is placed asymmetrically.
+A similar phenomenon has been reported in connection
+to the problem of mean first passage time with a reset
+[23], [24], [28]. Here, in addition to diffusion, a particle
+experiences a reset back to some location, and resets form
+a Poisson process, endowed with a reset rate r. The au-
+thors of these sources found there exists an optimal rate,
+r∗ which minimizes the MFPT out of the semi-infinite
+domain.
+We note, however that these sources appear
+to return the particle back to the reset location once it
+has hit the absorbing end of the semi-infinite domain,
+thereby conserving the probability. This is different from
+our problem, in which the total probability inside the
+domain decreases with time, because once particles have
+reached one of the two absorbing ends, they are not re-
+turned back into the domain.
+This difference aside, the problem that we are ana-
+lyzing can be viewed as a version of a reset problem,
+although the time to reset is not instantaneous. More-
+over, the reset location is not necessarily the location
+of the attractor x = X, since a particle has a chance
+to return to the diffusive layer before reaching the at-
+tractor. The limit of infinite v would correspond to the
+instantaneous reset to the attractor, and the limit b → 0
+would cause the resetting to take particles back to x = X,
+i.e. approximating the standard reset problem (although,
+again, without returning of particles that have reached
+either of the domain ends).
+The dip phenomenon is also observed when b is varied
+at fixed a, see Fig. 7. At low b, MFPT is dominated by
+the waiting time 1/b to return from the attractor to the
+DL. A large b asymptote (for b ≫ a) is the regime of
+purely diffusive motion - the particles are forced into the
+DL. Evidently, having some acccess to the AL leads to
+a lowering of MFPT because it allows more material to
+congregate close to one end.
+It is interesting to ask what effect increasing the ad-
+vective velocity would have. The intuition - supported
+by the physics of the one-layer model - is that higher v
+should lead to either an increase of the MFPT or the
+disappearance of the dip, because with sufficiently large
+velocity, the density will be more and more localized near
+the attractor; so, even though the attractor is closer to
+one end than the other, it is no longer close to this end
+𝑎
+1/𝜎!
+Out[ ]=
+0.01
+0.10
+1
+10
+100
+1000
+104
+0.02
+0.05
+0.10
+0.20
+v=20, 40, 60, 80 top to bottom
+X=0.5
+FIG. 8: Top to bottom: v = 20, 40, 60, 80. Here X = 0.5,
+and b = 169.
+
+9
+Out[ ]=
+0.01
+0.10
+1
+10
+100
+1000
+104
+0.10
+1
+10
+100
+X=1/26, 2/26, 3/26, from bottom to top
+V=20
+1/𝜎!
+𝑎
+FIG. 9: Top to bottom: X = 3/26, 2/26, 1/26. Here v = 20,
+and b = 169.
+in comparison to the width of the density distribution.
+However, analytical calculations in fact predict the de-
+crease in the value of 1/σ1 at a fixed a with increasing v,
+see Fig. 8.
+An in-depth study of density distributions, which will
+be published elsewhere [27], sheds light on the reason for
+this counter-intuitive prediction. While the density pro-
+file in both layers does become more localized with larger
+velocity (as expected), the part of the profile between the
+attractor and the close end is not affected; the decrease in
+the spread is due to the other side of the profile. There-
+fore, as velocity is increased, more and more material is
+localized near the close end, while the chance of escap-
+ing through this end does not diminish - resulting in the
+overall decrease of escape time.
+We also study the effect of varying X in Fig. 9. Here
+the results conform to the intuitive expectation that a de-
+crease in asymmetry will lead to a decrease in the mag-
+nitude of the dip (with no dip at all in a completely
+symmetric geometry). An attractor placed much closer
+to the left end than the right one, for example, has two
+effects. First, it lowers the MFPT overall, since there is
+less distance to travel during the escape. Second, pre-
+venting particles from wandering too far to the right (by
+increasing a, and thus the reset rate) causes the particles
+to congregate closer to the left end in the more asymmet-
+ric situation, leading to a lower MFPT.
+One-layer limit
+Dynamics of probability density
+The analytical approach in the one-layer limit is out-
+lined in the Methods section, with details in Appendix
+B. These predictions are verified by simulations (see Ap-
+pendix D). Here we present results of analytical calcula-
+tions.
+In Fig. 10 we show several snapshots in the evolution of
+the probability density profiles for a specific placement of
+!!
+"(!)
+𝑣 = 1
+!!
+"(!)
+𝑣 = 20
+FIG. 10: X = 0.85, x0 = 0.35. The distributions are shown
+for t = 1.3×10−5, t = 1.3×10−4, t = 1.3×10−3, t = 1.3×10−2,
+t = 1.3 × 10−1. Top: v = 20, bottom: v = 1. For v = 1, the
+distributions never reach an asymptotic form that is centered
+on x0 = X.
+the attractor and specific initial condition, for two values
+of the advective velocity. Following a δ-function initial
+condition, there is a quick diffusive spread. While this
+spread is happening, the center of the distribution is also
+advected towards the attractor. Note that in the v = 1
+case, the average position of particles reaches 1/2. On
+the other hand, for the case of stronger at v = 20, the
+center of the distribution reaches the attractor at x = X.
+At v = 20 we begin to see the emergence of large-
+time asymptotic profile centered on the attractor.
+At
+large times, the distribution reaches a stationary limit-
+ing form. As this profile develops, diffusive spread of the
+density profile is followed by a contraction, as particles
+congregate around the attractor and σx decreases. At
+t ≈ 0.06 the width stops evolving, and the cusp-shaped
+profile is established in the vicinity of the attractor. After
+that, the probability to remain in the domain continues
+to decrease (the area under the curve will continue to
+decrease), although the shape of the profile remains sta-
+tionary. We will call this limiting profile the large-time
+distribution or the limiting distribution. The width of
+this cusp-shaped limiting distribution decreases with in-
+
+80
+60
+40
+20
+0.2
+0.4
+0.6
+0.8
+1.080
+60
+40
+20
+0.2
+0.4
+0.6
+0.8
+1.010
+creasing v. At lower v, the width also saturates to a con-
+stant value at large times, and the limiting distribution
+also emerges, but it is not centered on the attractor.
+Thus, the picture is this: the attractor captures some
+particles and pulls them in to its vicinity at larger v,
+whereas at lower v, most of the particles escape be-
+fore this happens. The decay rate also decreases - as v
+grows ever larger, the large-time limiting profile localized
+around the attractor will decay ever slower, its rate of de-
+cay decreasing exponentially with v (this is for sufficiently
+large v, i.e. it is an asymptotic scaling). In this large v
+regime, the profile that develops after an initial rapid re-
+laxation may be called quasistationary - as it decays on a
+time scale much smaller than all other time scales in the
+problem. This is the regime of rare events, and we now
+discuss the scaling of MFPT and escape probabilities in
+this limiting regime.
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+0.2
+0.4
+0.6
+0.8
+pr
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+�
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+0.2
+0.4
+0.6
+0.8
+pr
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+�
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+�r
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+0.2
+0.4
+0.6
+0.8
+pr
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+10
+20
+30
+40
+50
+�r
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+10
+20
+30
+40
+50
+�
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+0.2
+0.4
+0.6
+0.8
+pr
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+0.1
+0.2
+0.3
+0.4
+�r
+(𝑎)
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+50000
+100000
+150000
+200000
+250000
+�r
+0.2
+0.4
+0.6
+0.8
+1.0
+x0
+50000
+100000
+150000
+200000
+250000
+�
+𝑝
+𝜏
+𝜏
+(𝑏)
+(𝑐)
+(𝑑)
+Right
+Left
+𝑝
+𝜏
+𝜏
+𝑝
+𝜏
+𝜏
+𝑝
+𝜏
+𝜏
+𝑥!
+𝑥!
+𝑥!
+𝑥!
+𝑥!
+𝑥!
+𝑥!
+𝑥!
+𝑥!
+𝑥!
+𝑥!
+𝑥!
+FIG. 11: Escape probability and MFPT through both ends
+versus the location x0 of the IC. The attractor is located at
+X = 0.51. (a) v = 5, (b) v = 10, (c) v = 20, (d) v = 40. The
+aberrations at the edge are numerical artifacts.
+Scaling of MFPT in the rare event limit
+In this regime, various functions of x0 - such as the
+escape probability and escape time - develop character-
+istic distinctions between a boundary layer and interior
+
+C11
+regions. This is shown in Fig. 11. As v increases, the
+MFPT to exit increases, and eventually this time be-
+comes much larger than all the other characteristic time
+scales of the problem. In this large v regime, escape be-
+comes a rare event. Starting from an initial condition x0,
+a particle will, with overwhelming probability drift to-
+wards the fixed point, and fluctuate around it for a time
+that scales exponentially with v as stated above. There-
+fore, the initial condition will be forgotten. This effect
+is manifested in Fig. 11 by distinct plateaus, that show
+the absence of dependence on x0. We show the compar-
+ison between such analytical predictions and simulation
+results of the one-layer regime in Appendix D.
+Escape rates in these plateaus will follow the usual
+Arrhenius scaling 1/τ ∼ e−∆Ueff /D in physical units.
+The effective barrier to escape to the left will be vX =
+∆Ul and the effective barrier to escape to the right will
+be v(1 − X) = ∆Ur. A small difference between X and
+(1−X) will be exponentially amplified by large v. Thus,
+for 0.5 < X < 1, the dominant factor will be v(1 − X),
+and therefore, τ ∼ ev(1−X)/D, or in dimensionless units,
+simply
+τ ∼ ev(1−X).
+(14)
+A more detailed analysis [27] predicts the prefactor as
+well, so the asymptotic expression (i.e. in the rare event
+regime) is given by τ = 4v−2ev(1−X).
+One comment regarding MFPT results is in order. We
+notice that the overall MFPT τ in Fig. 11 is ≈ 2 times
+smaller than the a = b → ∞ limit in Fig. 4 (see v = 10
+and v = 20 graphs). While a small difference is due to
+slightly different X (0.51 in Fig. 11 vs. 0.5 in Fig. 4),
+the main reason for this difference is that in the two-
+layer problem, the advection and diffusion take turns,
+while they take place simulataneously in the two-layer
+model. Thus, all timescales are slowed down by exactly
+a factor of two in the two-layer model than its truly one-
+layer equivalent.
+In other words, to make the proper
+comparison, we must multiply the one layer result by 2
+to match the a = b → ∞ limit of the two-layer model.
+Small asymmetry leads to a large bias in the exit location
+One prominent feature of Fig. 11 is the amplification in
+the asymmetry in results (for example pl and pr - proba-
+bilities to escape through the left and right ends respec-
+tively) due to a small asymmetry in the placement of
+the attractor. Note that pr = ae−∆Ur and pl = ae−∆Ul,
+where a is some constant.
+We can find this constant
+from the fact that pr + pl = 1 (a particle definitely
+exists through one of the two ends eventually). Thus,
+a =
+�
+e−∆Ur + e−∆Ul�−1, altogether giving
+pr − pl = tanh [(X − 1/2)v]
+(15)
+We overlay this prediction on top of ∆p obtained from
+the analytic results (depicted in Fig. 11) in Fig. 12 (a).
+!
+∆"
+!
+∆"
+(a)
+(b)
+FIG. 12: (a) ∆p vs. v. Top (blue): X = 0.55, bottom (or-
+ange): X = 0.51. Dots - full theory, solid curves - Eqn. (15).
+(b) ∆p vs. X, given by Eqn. (15). Top (orange): v = 20,
+bottom (blue): v = 10.
+Conclusion
+In this paper, we looked at a one-dimensional model of
+intracellular transport via a combination of advection on
+microtubules and diffusion in the cytoplasm. This one-
+dimensional model was motivated by a scenario involv-
+ing an attractor in the interior of the cell - for example,
+MTOC. There are other situations where attractors may
+arise. Consider, the β cell example from the Introduc-
+tion. Here motors transport insulin granules along MTs.
+Due to orientational disorder [29], several MTs can meet
+with ends of the same polarity facing each other, forming
+an aster-like morphological trap (or attractor) for mo-
+tors that would all congregate at this junction [11]. It
+is meaningful to talk about the domain of attraction of
+such a trap in the following sense. A molecular motor
+that attaches to a MT anywhere within this domain will
+be taken towards the attractor, while a motor that at-
+taches to a mirotubule outside of the domain has a non-
+zero probability to be taken away from the trap. When
+placed inside such a domain - where advective motion
+along microtubules tends to only attract particles - they
+can nevertheless escape the domain of attraction of the
+attractor by desorbing from MTs and diffusing within
+the cytoplasm until they end up outside of the domain.
+
+1.0
+0.8
+0.6
+0.4
+0.2
+10
+20
+30
+40
+50
+601.0
+0.8
+0.6
+0.4
+0.2
+0.6
+0.7
+0.8
+0.9
+1.012
+Naturally, a question about the time to be stuck in the
+vicinity of the attractor arises - along with the question
+of how formation of such traps affects the functioning
+of the cell and the overall transport of insulin granules
+across it.
+Using our one-dimensional model, We calculated es-
+cape probability through each end, pl(x0) and pr(x0),
+and overall p(x0). We also calculated the mean first pas-
+sage time (MFPT) to escape the domain through each
+end, τl(x0) and τr(x0), and overall τ(x0). The initial lo-
+cation inside the cell is determined by the organelles pro-
+ducing the cargo. The other parameters in the problem
+were the dimensionless location of the attractor toward
+which the advective motion is directed, and the dimen-
+sionless advective velocity v.
+In situations like these, when there is either orienta-
+tional or polarity disorder, we can think of cells as being
+divided into domains.
+We made several predictions. When the attractor is
+placed symmetrically and a and b are finite, there is a
+crossover between τ ∼ 0.1 - diffusive timescale to τ that
+grows exponentially in v. The range of a = b over which
+this crossover happens is wide - a couple of orders of
+magnitude, but it corresponds to the values of a and b
+actually found in cells. This served as our first example
+of “fine-tuning” that allows cells to achieve the biggest
+change in the functionality with the smallest change in
+parameter.
+For a = b significantly below the crossover, a particle
+that was released into the diffusive layer has a chance to
+escape the domain purely diffusively without ever visiting
+the AL. For a = b around the crossover value, the proba-
+bility of this goes to zero - every particle will be advected
+towards the attractor for at least some of the time. For
+a = b significantly above the crossover, the transport en-
+ters the effective one-layer regime and exhibits rare event
+physics.
+Asymmetric placement of the attractor gives rise to an
+interesting phenomenon of an optimal coupling. Thus,
+we found that it is possible to minimize the residence
+time in the domain by increasing the coupling, because
+that will lower the diffusive spread, and bring particles
+close to one end of the domain.
+We discussed the effective one-layer regime that re-
+sults at sufficiently large couplings. We also discussed
+rare event physics that happens at large dimensionless
+advective velocities. In such a rare event regime, a por-
+tion of particles will be localized in the vicinity of the
+attractor for a time exponentially long in v.
+We pro-
+vide an explicit formula formula for the overall MFPT
+- including not only the exponential part, but also the
+prefactor, which scales as v−2.
+The idea of exponential sensitivity, and phenomena
+such as strong amplification of the preferred exit end
+due to a slight asymmetry is tantalizing. Extrapolating
+this finding to two dimensions suggests that in complex,
+crowded environments that allow for multiple trap-like
+morphologies (for example, asters), the distribution of
+cargo around the cell will be non-homogeneous. This re-
+mains to be verified in the future, by extending our model
+two two dimensions.
+Our work is complementary to prior theoretical models
+of transport that involves a combination of diffusion and
+advection along microtubules [30] and [31], as neither of
+these sources are focusing on questions of residence time
+or the role of asymmetry.
+To continue our current work, we would like to study
+models with reflecting-reflecting or absorbing-reflecting
+boundary conditions, or models in which the source is
+on one end and the target is on the other. Such mod-
+els would be better suited for transport of cargo in cilia
+[4], transport between the plasma membrane and Golgi
+apparatus [5], [6], or between Endoplasmic Reticulum
+and Golgi [7], [3], transport of viruses towards replication
+sites [8], [9], and other intracellular transport situations
+[3], [10].
+This
+work
+was
+supported
+by
+the
+National
+Sci-
+ence Foundation (NSF-DMS-1616926 to AG) and NSF-
+CREST: Center for Cellular and Bio-molecular Ma-
+chines at UC Merced (NSF-HRD-1547848 and 2112675
+to AG). AG and NS also acknowledge partial sup-
+port from the NSF Center for Engineering Mechanobi-
+ology grant CMMI-154857 and computing time on the
+Multi-Environment Computer for Exploration and Dis-
+covery (MERCED) cluster at UC Merced (NSF-ACI-
+1429783).
+NS acknowledges Graduate Student Oppor-
+tunity Program Fellowship from the University of Cal-
+ifornia, Merced.
+BR acknowledges the support of the
+William and Linda Cal Poly Frost fund for undergradu-
+ate research.
+
+13
+A: Details of the two-layer calculations
+We start with the full one-dimensional, two-layer model in dimensionless form (primes have been omitted for clarity):
+∂ρ
+∂t = − ∂
+∂x (v(x)ρ) + aθ − bρ
+(16)
+∂θ
+∂t = −aθ + bρ + ∂2θ
+∂x2
+(17)
+Here a and b are respectively the rates of adsorption to and desorption from microtubules, v is the dimensionless
+velocity profile, ρ is the density of particles on microtubules, and θ is the density of particles diffusing in the cytoplasm.
+We seek modal solutions (or eigensolutions) of the form
+�
+ρ
+θ
+�
+=
+�
+R(x)
+Θ(x)
+�
+e−σt.
+(18)
+The vector
+�
+R(x)
+Θ(x)
+�
+is an eigenvector of the operator (see Eq. (11) of the text) that represents minus the right hand
+side of Eqs. (16)-(17), and σ is an eigenvalue of this operator.
+However, due to the mass accumulation at the attractor, we must also include a δ-function term to accommodate
+for this mathematically. The mass will not accumulate at the junction point due to the diffusive term that acts on the
+diffusive layer density. Note also that the δ- function in the advective layer acts like a point source for the diffusive
+layer. When we study a simple diffusive problem with a δ-function source plus absorbing boundaries, and seek a
+steady-state (time independent) solution, the density profile does not acquire a δ-function response - the diffusion
+acts infinitely quickly to dissipate such a singularity. With this in mind, we must augment the above formula to
+�
+ρ
+θ
+�
+=
+�
+R(x)
+Θ(x)
+�
+e−σt +
+�
+1
+0
+� �
+M0e−σt�
+δ(x − X).
+(19)
+Substituting this back to Eqs. (16)-(17), and setting Q = dΘ
+dx , we get
+d
+dx
+�
+�
+R
+Q
+Θ
+�
+� =
+�
+�
+(− b
+v + σ
+v ) 0
+a
+v
+−b
+0 (a − σ)
+0
+1
+0
+�
+�
+�
+�
+R
+Q
+Θ
+�
+�
+(20)
+for 0 ≤ x < X (call it Region-I) and
+d
+dx
+�
+�
+R
+Q
+Θ
+�
+� =
+�
+�
+( b
+v − σ
+v ) 0
+− a
+v
+−b
+0 (a − σ)
+0
+1
+0
+�
+�
+�
+�
+R
+Q
+Θ
+�
+� ,
+(21)
+for X < x ≤ 1 (call it Region-II). The solutions, will take the form:
+�
+�
+RI
+QI
+ΘI
+�
+� = A
+�
+�
+v1
+R
+v1
+Q
+v1
+Θ
+�
+� eλ1x + B
+�
+�
+v2
+R
+v1
+Q
+v2
+Θ
+�
+� eλ2x + C
+�
+�
+v3
+R
+v3
+Q
+v3
+Θ
+�
+� eλ3x
+(22)
+in Region-I and
+�
+�
+RII
+QII
+ΘII
+�
+� = D
+�
+�
+w1
+R
+w1
+Q
+w1
+Θ
+�
+� eµ1x + E
+�
+�
+w2
+R
+w1
+Q
+w2
+Θ
+�
+� eµ2x + F
+�
+�
+w3
+R
+w3
+Q
+w3
+Θ
+�
+� eµ3x,
+(23)
+in Region-II. The ⃗vs and λs are eigenvectors and eigenvalues of the matrix in Eq. (20), while ⃗ws and µs are eigenvectors
+and eigenvalues of the matrix in Eq. (21). The λs satisfy the equation
+−λ3 +
+�σ − b
+v
+�
+λ2 + (a − σ)λ + σ2 − σ(a + b)
+v
+= 0,
+(24)
+
+14
+and the µs satisfy the equation
+−µ3 −
+�σ − b
+v
+�
+µ2 + (a − σ)µ − σ2 − σ(a + b)
+v
+= 0.
+(25)
+The eigenvectors have the structure
+⃗v =
+�
+�
+−λ2+a−σ
+b
+λ
+1
+�
+� ,
+(26)
+and
+⃗w =
+�
+�
+−µ2+a−σ
+b
+µ
+1
+�
+� .
+(27)
+The functions on both sides of the attractor are different, and they need to be stitched correctly. The stitching
+is determined by the boundary conditions, so we now discuss these. The boundary conditions will determine the
+eigenvalues σn. We note that there are seven unknowns: coefficients A - F (see Eqs. (22)-(23)), and the mass growth
+rate M0 (see Eq. (19), so we need seven constraints (or conditions).
+First, there are absorbing boundary conditions at each end, which require that R(x = 0) = Θ(x = 0) = 0 and
+R(x = 1) = Θ(x = 1) = 0. The additional three conditions come from the location of stitching, i.e. the attractor
+location at x = X. The diffusive layer density must be continuous to avoid infinite currents. Thus, ΘI(X) = ΘII(X).
+The remaining two boundary conditions come from mass conservation. To extract these, we integrate Eqs. (16)-(17)
+through the junction point, i.e. from x − ϵ to x + ϵ for arbitrarily small ϵ. Performing this on Eq. (16) gives
+−σM0 = −bM0 − (vIIRII(X) − vIRI(X)) = −bM0 + v (RII(X) + RI(X)) .
+(28)
+Note that the temporal terms would not be absent if the δ-function component of ρ was not proportional to e−σt.
+This equation says that the rate of growth of the advective layer mass at x = X (i.e. of the strength of the δ-function)
+is driven by the inflow from this layer, and outflow into the diffusive layer. Performing the integration on Eq. (5)
+gives
+bM0 = dΘI
+dx
+����
+x=X
+− dΘII
+dx
+����
+x=X
+.
+(29)
+This equation says that any difference in the outflow rates (i.e.
+different slopes of the diffusive layer density) is
+balanced by the inflow from the advective layer.
+We now implement these boundary conditions algebraically. We have:
+1. Absorbing boundary condition at x = 0 in the advective layer: RI(x = 0) = 0
+A
+�−λ2
+1 + a − σ
+b
+�
++ B
+�−λ2
+2 + a − σ
+b
+�
++ C
+�−λ2
+3 + a − σ
+b
+�
+= 0
+(30)
+2. Absorbing boundary condition at x = 0 in the diffusive layer: ΘI(x = 0) = 0
+A + B + C = 0
+(31)
+3. Absorbing boundary condition at x = 1 in the advective layer: RII(x = 1) = 0
+D
+�−µ2
+1 + a − σ
+b
+�
+eµ1 + E
+�−µ2
+2 + a − σ
+b
+�
+eµ2 + F
+�−µ2
+3 + a − σ
+b
+�
+eµ3 = 0
+(32)
+4. Absorbing boundary condition at x = 1 in the diffusive layer: ΘII(x = 1) = 0
+Deµ1 + Eeµ2 + Feµ3 = 0
+(33)
+5. Continuity at x = X in the diffusive layer (to prevent infinite diffusive currents): ΘI(x = X) = ΘII(x = X)
+Aeλ1X + Beλ2X + Ceλ3X = Deµ1X + Eeµ2X + Feµ3X
+(34)
+
+15
+6. Mass conserving boundary condition in advective layer: RII(x = X) + RI(x = X) = b−σ
+v M0
+D
+�−µ2
+1 + a − σ
+b
+�
+eµ1X + E
+�−µ2
+2 + a − σ
+b
+�
+eµ2X + F
+�−µ2
+3 + a − σ
+b
+�
+eµ3X
++ A
+�−λ2
+1 + a − σ
+b
+�
+eλ1X + B
+�−λ2
+2 + a − σ
+b
+�
+eλ2X + C
+�−λ2
+3 + a − σ
+b
+�
+eλ3X = b − σ
+v
+M0
+(35)
+7. Mass conserving boundary condition in diffusive layer:
+dΘI
+dx
+��
+x=X − dΘII
+dx
+��
+x=X = bM0
+Aλ1eλ1X + Bλ2eλ2X + Cλ3eλ3X − Dµ1eµ1X − Eµ2eµ2X − Fµ3eµ3X = bM0.
+(36)
+We can write all these seven equations in the compact matrix form:
+�
+�
+�
+�
+�
+�
+�
+�
+−λ2
+1+a−σ
+b
+−λ2
+2+a−σ
+b
+−λ2
+3+a−σ
+b
+0
+0
+0
+0
+1
+1
+1
+0
+0
+0
+0
+0
+0
+0
+−µ2
+1+a−σ
+b
+eµ1
+−µ2
+2+a−σ
+b
+eµ2
+−µ2
+3+a−σ
+b
+eµ3
+0
+0
+0
+0
+eµ1
+eµ2
+eµ3
+0
+eλ1X
+eλ2X
+eλ3X
+−eµ1X
+−eµ2X
+−eµ3X
+0
+−λ2
+1+a−σ
+b
+eλ1X
+−λ2
+2+a−σ
+b
+eλ2X
+−λ2
+3+a−σ
+b
+eλ3X
+−µ2
+1+a−σ
+b
+eµ1X
+−µ2
+2+a−σ
+b
+eµ2X
+−µ2
+3+a−σ
+b
+eµ3X
+σ−b
+v
+λ1eλ1X
+λ2eλ2X
+λ3eλ3X
+−µ1eµ1X
+−µ2eµ2X
+−µ3eµ3X
+−b
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+A
+B
+C
+D
+E
+F
+M0
+�
+�
+�
+�
+�
+=
+�
+�
+�
+�
+�
+0
+0
+0
+0
+0
+0
+0
+�
+�
+�
+�
+�
+.
+(37)
+Because of the structure of this equation, we see that (i) the determinant must be non-zero for a non-trivial solution
+and (ii) the nontrivial solution is non-unique - it has at least one degree of freedom. For instance, we are free to
+choose one of the variables, or we are free to choose the normalization. Making use of this freedom, we chose to set
+M0 = 1. These equations were then used to solve for the remaining coefficients A, B, C, D, E, and F.
+Thus, calling the matrix involved in Eq. (37), M, Det(M) = 0 should provide an algebraic equation for σ.
+Expanding determinant in terms of minors, we have
+0 = bDet (m77) +
+�σ − b
+v
+�
+Det (m67)
+(38)
+where the minor mij is a matrix obtained by removing ith row and jthe column from M.
+Once M0 is chosen, the coefficients (A, ..., F) must be unique. This means that both Det (m77) and Det (m67) must
+both be non-zero. If Det (m77) is non-zero, then the solution (A, ..., F) obtained from the first six equations can be
+found with the inverse of m77, and is unique. This implies that Det (m67) must also be non-zero (otherwise, the
+solution (A, ..., F) obtained from the first five and the seventh equation is non-unique, leading to a contradiction).
+Thus, the kind of a zero of Det(M) that we want is one in which Det (m77) and Det (m67) are both non-zero.
+Therefore, we’re interested in the zeros of the following quantity:
+Det′ = b +
+�σ − b
+v
+� Det (m67)
+Det (m77).
+(39)
+It is the zeros of this determinant that gives us σ in terms of (a, b, v, X).
+We were primarily interested in the lowest (ground state) eigenvalue σ1, and the inverse 1/σ1 that serves as a
+characteristic measure of the escape time[33]. Because the set of eigenfunctions and eigenvalues turned out to be
+finite, they are of limited value in being able to construct a solution that fits the δ-function initial condition, and
+thereby to properly compute MFPT.
+
+16
+B: One-layer theory
+We now discuss the computation of the eigenfunctions p(x). The subscript n will be dropped to lighten the notation.
+Recall that 0 < x < X is Region-I, and that X < x < 1 is Region-II. The eignfunctions satisfy
+σp = −v dp
+dx + d2p
+dx2
+(40)
+in Region-I, and
+σp = v dp
+dx + d2p
+dx2
+(41)
+in Region-II. The solution in Region-I is pI = aIeλ+x + bIeλ−x, where the λs satisfy
+λ± = v ±
+√
+v2 + 4σ
+2
+.
+(42)
+The solution in Region-II is pII = aIIeµ+x + bIIeµ−x, where the µs satisfy
+µ± = −v ±
+√
+v2 + 4σ
+2
+.
+(43)
+The coefficients a and b will be fixed with the following four boundary conditions (BCs).
+The first two are the
+absorbing BCs at the ends, pI(0) = pII(1) = 0. The third boundary condition is the continuity of the solution
+pI(X) = pII(X). A discontinuous solution is unphysical due to the diffusion term. In a one-layer theory, there will
+not be an accumulation of mass at the trap, i.e. there will be no term like δ(x − X). Any such density would be
+immediately smoothed out by the action of the diffusion. Note that in the full, two-layer theory, such term existed
+only in the advective layer, but not in the diffusive layer. In the absence of a δ-function-like accumulation of mass at
+x = X, the currents across x = X will be continuous. This gives us the fourth boundary condition that enforces the
+continuity of currents at the junction: vpI(X) − dpI
+dx
+���
+x=X = −vpII(X) − dpII
+dx
+���
+x=X.
+Applying these four boundary conditions leads to four equations:
+aI + bI
+= 0
+(44)
+aIIeµ+ + bIIeµ−
+= 0
+(45)
+aIeλ+X + bIeλ−X
+= aIIeµ+X + bIIeµ−X
+(46)
+v �
+aIeλ+X + bIeλ−X�
+− �
+λ+aIeλ+X + λ−bIeλ−X�
+= −v �
+aIIeµ+X + bIIeµ−X�
+− �
+µ+aIIeµ+X + µ−bIIeµ−X�
+(47)
+Substituting the first two into the last two gives
+aI
+�
+eλ+X − eλ−X�
+= aII
+�
+eµ+X − eµ+−µ−eµ−X�
+vaI
+�
+eλ+X − eλ−X�
+− aI
+�
+λ+eλ+X − λ−eλ−X�
+= −vaII
+�
+eµ+X − eµ+−µ−eµ−X�
+− aII
+�
+µ+eµ+X − µ−eµ+−µ−eµ−X�
+Using the first of these, and substituting into the second we obtain
+v �
+eλ+X − eλ−X�
+−�
+λ+eλ+X − λ−eλ−X�
+−�
+−v �
+eµ+X − eµ+−µ−eµ−X�
+− �
+µ+eµ+X − µ−eµ+−µ−eµ−X���
+eλ+X − eλ−X
+eµ+X − eµ+−µ−eµ−X
+�
+= 0,
+(48)
+where λs and µs are given by Eqs. (42) and (43) respectively. Eq. (48) is an equation for eigenvalues σ as a function
+of v and X. Moreover,
+pI =
+�
+eλ+x − eλ−x�
+,
+(49)
+and
+pII =
+�
+eλ+X − eλ−X
+eµ+X − eµ+−µ−eµ−X
+� �
+eµ+x − eµ+−µ−eµ−x�
+(50)
+The modes given this way are not normalized; they will be normalized below. We will see below that eigenvalues turn
+out to be real.
+
+17
+The coefficients cn are determined as usual by the initial condition, P(x, t = 0) = �
+n cnpn(x). Because the operator
+O is non-Hermitian, eigenfunctions are generally non-orthogonal, i.e.
+� 1
+0 p∗
+n(x)pm(x) dx ̸= 0, so we can’t compute cm
+with the help of an inner product
+� 1
+0 P(x, 0)pm(x) dx. However, eigenfunctions of the adjoint operator O† have the
+property that they are either orthogonal to the eigenfunctions of O, or otherwise have eigenvalues that are complex
+conjugates of each other.
+Therefore, in order to be able to express initial conditions, we need to compute a set of eigenfunctions and eigenvalues
+of O†. Even after this, there is no guarantee that we will be able to express any initial condition, because there’s also
+no guarantee of completeness, due to operators being non-Hermitian.
+The adjoint of O is given by
+O† = v(x) d
+dx + d2
+dx2 .
+(51)
+To find the eigenfunctions of this operator, it helps to look back at the original equation with operator O. We note
+that both Eqs. (40)-(41) can be written compactly as one equation
+σp = d
+dx
+�dU
+dx p + d2p
+dx2
+�
+,
+(52)
+where the potential (in analogy with physics) U is given by
+U(x) =
+�
+v(X − x) x ≤ X,
+v(x − X) x ≥ X,
+(53)
+or, more compactly, v(x) = − dU
+dx .
+Now, let p = q(x)e−U(x) - we can always do this. Substituting this ansatz we find that q(x) obeys
+σ1q = −dU
+dx
+dq
+dx + d2q
+dx2
+= v(x) dq
+dx + d2q
+dx2 .
+(54)
+That is, q = p(x)eU(x) is the eigenfunction of the adjoint operator that we were seeking! Moreover, it has the same
+eigenvalue as the operator O. The modes given this way are not normalized; they will be normalized below.
+For operators with a finite dimensional eigenspace, eigenvalues of an adjoint operator O† are complex conjugates
+of the eigenvalues of the operator O. In such cases, equality of the two sets of eigenvalues implies that they are real.
+In our case the eigenspace is not guaranteed to be finite (in fact, we hope that it isn’t, if there is any chance at
+completeness). However, our numerical investigation revealed that eigenvalues σ are always real (and negative).
+Next, we give an example of the result of several hundred low-lying eigenvalues. The first example is for X = 0.85
+and v = 1. We observe an interesting feature that eigenvalues appear in groups. The second example is for X = 0.6
+20
+40
+60
+80
+100Mode
+-100000
+-80000
+-60000
+-40000
+-20000
+σ
+FIG. 13: Lowest 100 eigenvalues for X = 0.85 and v = 1.
+and v = 1. We notice that the size of groups has changed. There is no obvious relation between X the the size of
+
+18
+20
+40
+60
+80
+100Mode
+-150000
+-100000
+-50000
+σ
+FIG. 14: Lowest 100 eigenvalues for X = 0.6 and v = 1.
+groups - for instance, for X = 0.55 the groups are again increased in size.
+We verified numerically the orthogonality of several eigenfunctions belonging to different eigenvalues, and found it
+to hold true. Eigenfunctions were also normalized by multiplying by the following factors:
+ap =
+1
+�� 1
+0 p∗n(x)pn(x)
+,
+Aq =
+1
+�� 1
+0 q∗n(x)qn(x)
+,
+where ps and qs are given by Eqs. (49)-(50) and q = p(x)eU(x). The resultant modes came out to be either purely
+real or purely imaginary. In this latter case, they can be made real by multiplying by −i.
+The following are examples of eigenfunctions.
+The discontinuity in the slope of ps - but not of qs - is clearly visible
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Im[p1]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+x
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Im[q1]
+0.2
+0.4
+0.6
+0.8
+1.0
+x
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+1.5
+Im[p20]
+0.2
+0.4
+0.6
+0.8
+1.0
+x
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+1.5
+Im[q20]
+Mode 1
+Mode 20
+𝑞!
+𝑝!
+𝑞"#
+𝑝"#
+FIG. 15: The first and the twentieth modes for X = 0.85, v = 1.
+
+19
+0.2
+0.4
+0.6
+0.8
+1.0
+x
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Re[p1]
+0.2
+0.4
+0.6
+0.8
+1.0
+x
+0.5
+1.0
+1.5
+Re[q1]
+0.2
+0.4
+0.6
+0.8
+1.0
+x
+-3
+-2
+-1
+1
+2
+3
+Im[p20]
+0.2
+0.4
+0.6
+0.8
+1.0
+x
+-3
+-2
+-1
+1
+2
+3
+4
+Im[q20]
+Mode 1
+Mode 20
+𝑞!
+𝑝!
+𝑞"#
+𝑝"#
+FIG. 16: The first and the twentieth modes for X = 0.85, v = 10.
+in the first mode. We can understand this by substituting the form p(x) = q(x)e−U(x) into the fourth boundary
+condition on p (i.e. vpI(X) − dpI
+dx
+���
+x=X = −vpII(X) − dpII
+dx
+���
+x=X), and find that dq
+dx is continuous across the junction,
+i.e.
+qI
+dx
+��
+x=X = dqII
+dx
+���
+x=X. The other three boundary conditions for q are the same as for p.
+With all this information, we conclude that the set of functions {qn} is then sufficient for us to be able to find the
+coefficients cn in the series P(x, t) = �
+n cnpn(x)eσnt - as long as there is completeness. The coefficients are given by
+cn =
+� 1
+0 P(x, t = 0)q∗
+n(x) dx
+� 1
+0 q∗n(x)pn(x) dx
+(55)
+Completeness is not guaranteed, but unlike the two-layer case, we found that the method works, provided enough
+modes are used. We will not discuss convergence properies of the series here.
+In relation to the mean first passage time problem, we are interested in the δ-function initial condition, P(x, t =
+0) = δ(x − x0), in which case the coefficients are given by
+cn =
+q∗
+n(x0)
+� 1
+0 q∗n(x)pn(x) dx
+.
+(56)
+
+20
+C: Trajectory examples
+Fig. 4 and the subsequent discussion in our main text discussed several regimes of MFPT, depending on the value
+of a = b, for symmetric trap placement. We now show trajectories in each of those regimes.
+First, we show trajectories in the plateau regime that precedes the second crossover. This takes place for a roughly
+in the range [10−2, 10]. This is shown in Fig. 17.
+𝑎 = 0.1
+𝑎 = 1
+𝑎 = 10
+Time
+Time
+FIG. 17: Nine trajectories at lower as. All particles are placed initially at x0 = 0.5 on the DL. Here X = 0.5 and v = 20. The
+right panels show a smaller window of time.
+We can clearly see that as a increases, thee probability of switching into the AL increases. Once a particle switches
+to the AL, it will move towards the attractor.
+As a increases further, the likelihood of the advective motion towards the attractor all in one ride on the AL
+decreases. Instead, a typical particle will experience episodes of a little bit of advective motion, followed by a little
+bit of diffusive motion, and so on - see Fig. 18. This happens in the second crossover regime that begins for a ≈ 10
+and continues for several decades.
+
+..21
+Time
+Time
+𝑎 = 100
+𝑎 = 1000
+FIG. 18: Nine trajectories at intermediate as. All particles are placed initially at x0 = 0.5. Here X = 0.5, v = 20. The right
+panels show a smaller window of time.
+For a even larger - the system enters the second plateau, when any further increase in a does not increase MFPT.
+This means that the system behaves in accordance to the one-layer model [34]. The the episodes of diffusion and
+advection become even shorter. Trajectories in such a regime are shown in Fig. 19, for progressively narrower windows
+of time, from left to right.
+
+22
+𝑎 = 𝑏 = 10!
+𝑎 = 𝑏 = 10"
+𝑎 = 𝑏 = 10#
+Time (m.u.)
+Time (m.u.)
+Time (m.u.)
+Time (m.u.)
+FIG. 19: Trajectories for a between 104 to 106 in powers of 10. Here again X = 0.5 and v = 20. Leftmost column has 10
+trajectories, while the other columns show one trajectory for progressively narrower windows of time, from left to right. In
+these right three columns, the red color indicates advective portions of trajectories, while grey are diffusive portions.
+
+0.70
+1:
+0.65
+0.60
+0.55
+X 0.50
+0.45
+0.40
+0.35-
+0.30
+20
+40
+60
+80
+100 0
+2
+4
+6
+8
+10 0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+Time (m. u.)
+Time (m. u.)
+Time (m. u.)0.70
+0.65
+0.60
+0.55
+X 0.50111
+0.45
+0.40
+0.35-
+0.30
+20
+40
+60
+80
+100 0
+2
+8
+10 0.0
+0.2
+0.4
+0.6
+0
+6
+0.8
+1.0
+Time (m. u.)
+Time (m. u.)
+Time (m. u.)0.70
+0.65
+0.60-
+0.55
+X 0.50
+0.45
+0.40
+0.35-
+0.30
+0
+20
+40
+60
+80
+100 0
+2
+6
+8
+10 0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Time (m. u.)
+Time (m. u.)
+Time (m. u.)0.70
+0.65
+0.60
+0.55
+X 0.50
+0.45
+0.40
+0.35
+0.30
+0
+2
+4
+6
+8
+10
+Time (m. u.)0.70
+0.65
+0.60
+0.55
+X 0.50
+0.45
+0.40
+0.35
+0.30
+0
+2
+4
+6
+8
+10
+Time (m. u.)0.70
+0.65
+0.60
+0.55
+X 0.50
+0.45
+0.40
+0.35
+0.30
+0
+2
+4
+6
+8
+10
+Time (m. u.)23
+D: Theory and simulation comparison - one-layer limit
+In this section we show the comparison between the one-layer analytical predictions of pl, pr, τl, τr, and τ with
+results of simulations of the two-layer model.
+𝑥
+𝑥
+𝑥
+𝑝
+𝜏
+𝜏
+Right
+Left
+FIG. 20: Comparison between analytical quantities (open circles) and simulation results (filled circles - Monte Carlo simulation
+as described in the main paper, filled triangles - forward flux sampling algorithm [32]). Left column: probabilities to escape
+through the left end (blue) pl and right end (orange) pr. Middle column: escape time conditioned on the left exit (blue) τl and
+right exit (orange) τr. Right column: net MFPT τ. The growing discrepancy between simulation and analytical results is due
+to the diffusive approximation of the latter; the details will be discussed in the coming publication [27]. Here X = 0.5. Top
+row: v = 0.1, middle row v = 5, bottom row v = 20.
+
+.24
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+[33] This is especially true in the rare event regime that devel-
+ops at sufficiently large v - when σ1 should be separated
+from the rest of σs by a gap that grows exponentially in
+v - while there is no such gap between the rest of the
+eigenvalues.
+[34] This is not what makes escape events rare. The signa-
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+tial growth of MFPT with v.
+

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@@ -0,0 +1,1139 @@
+First passage time statistics of non-Markovian random walker: Onsager’s
+regression hypothesis approach
+Yuta Sakamoto and Takahiro Sakaue∗
+Department of Physical Sciences, Aoyama Gakuin University,
+5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Japan
+First passage time plays a fundamental role in dynamical characterization of stochastic processes.
+Crucially, our current understanding on the problem is almost entirely relies on the theoretical
+formulations, which assume the processes under consideration are Markovian, despite abundant non-
+Markovian dynamics found in complex systems. Here we introduce a simple and physically appealing
+analytical framework to calculate the first passage time statistics of non-Markovian walkers grounded
+in a fundamental principle of nonequilibrium statistical physics that connects the fluctuations in
+stochastic system to the macroscopic law of relaxation. Pinpointing a crucial role of the memory
+in the first passage time statistics, our approach not only allows us to confirm the non-trivial
+scaling conjectures for fractional Brownian motion, but also provides a formula of the first passage
+time distribution in the entire time scale, and establish the quantitative description of the position
+probability distribution of non-Markovian walkers in the presence of absorbing boundary.
+How long does it take for a random walker to reach
+a destination? Such a question on the first passage
+time (FPT) is relevant to a broad range of situa-
+tions in science, technology and every-day life applica-
+tions as encountered, for instance, in diffusion-limited
+reactions [1–3], barrier crossing [4–7], target search
+processes [8, 9], cyclization of DNA molecule [10–
+13], price fluctuation in market [2] and spread of dis-
+eases [14]. Today, the concept of the FPT and its im-
+portance in the study of stochastic processes are well
+recognized, and theoretical methods for its computa-
+tion are standardized [1, 2]. However, most of them
+are devised for Markovian random walkers, whose de-
+cision making does not depend on its past history, thus
+not applicable to non-Markovian walkers despite their
+ubiquitousness.
+Indeed, a growing body of evidence suggests that
+the non-Markovian dynamics is found quite gener-
+ally in rheologically complex matters typically, but
+not exclusively, with viscoelastic properties. Classi-
+cal examples are found in the diffusion of interact-
+ing particles in narrow channels [15] and the motion
+of tagged monomers in long polymer chain [16, 17].
+Other notable representatives include colloidal parti-
+cles in polymer solutions [18] or nematic solvents [19],
+lipids molecules and cholesterols in cellular mem-
+brane [20], proteins in crowded media [21], and chro-
+mosome loci [22] as well as membraneless organelles
+in living cells [23]. Such systems commonly exhibit a
+slow dynamics in the form of sub-diffusion MSD(t) ∼
+tα characterized by the anomalous exponent α < 1,
+where MSD(t) stands for the mean-square displace-
+ment of the observed particle during the time scale t
+. Here the physical mechanism at work is the inter-
+action of observed degree of freedom with the collec-
+tive modes with broad range of time scales underly-
+ing complex environment. Because of its importance
+in e.g.
+intracellular transport, the theoretical tools
+to describe/diagnose such anomalous diffusion phe-
+nomenology have been well developed in the last few
+decades [24].
+However, most of them rely on MSD
+and related quantities, while much less attention has
+been paid to the FPT, despite its fundamental and
+practical importance to characterize the underlying
+stochastic process. This is particularly true for sys-
+tems possessing memory, as nontrivial information on
+the history dependence of the system is encoded in
+the FPT statistics [25]. It has long been known that
+the anomalous transport properties affect the rates
+of chemical and biochemical reactions [26], and such
+reactions are initiated by the encounter of reactant
+molecules, so precisely quantified by means of the FTP
+statistics.
+Unfortunately, our current understanding on the
+FPT of non-Markovian walker lags far behind that
+of Markovian counterpart, where the difficulty is
+largely associated to the lack of appropriate theoret-
+ical foothold [25, 27, 28].
+While the Fokker-Planck
+equation and its related methods play a key role to
+analyze the time evolution of the probability distri-
+bution of the Markovian walkers, their careless ap-
+plication is problematic for walkers with memory, a
+defining property of the non-Markovian process. At
+𝑡 = 𝜏
+(a)
+(b)
+𝑡 = 𝜏
+𝜏
+FIG. 1. Regression hypothesis applied to non-Markovian
+walkers.
+(a) Example trajectory of fBM with α = 0.5
+starting from the initial position x = x0. Before (after)
+the first hitting on absorbing boundary at x = 0, the
+trajectory is drawn by solid (dotted) curve.
+First pas-
+sage event can be viewed as a large fluctuation to create
+a non-equilibrium state at t = τ. (b) After the first pas-
+sage (t > τ), the process follows, on average, the macro-
+scopic relaxation law, for sub-diffusive fBM, represented
+by the harmonic restoring force, whose spring constant
+gets smaller algebraically in longer time scales.
+arXiv:2301.13466v1  [cond-mat.stat-mech]  31 Jan 2023
+
+Absorbing wall
+Time
+0
+0
+o
+PositionAbsorbing wall
+Potential
+0
+o
+Position2
+present, available results are quite limited with no-
+table examples being the perturbative and scaling ar-
+guments to estimate the asymptotic exponents charac-
+terizing the distribution of FPT and related quantities
+in unbounded domain [25, 29–31], some approxima-
+tion schemes to calculate the mean FPT of polymer
+looping process [3, 10–13], and more recent analytical
+treatment to compute the mean FPT in confined do-
+mains [28]. However, neither of the full distribution of
+FPT or position distribution of non-Markovian walk-
+ers in the presence of boundary are available, making
+the computation of these quantities in non-Markovian
+processes fundamental challenge.
+In this Letter, we provide a simple and physically
+appealing method to calculate the FPT statistics of
+non-Markovian walkers by identifying the moment of
+first passage (t = τ) as an initial condition for the re-
+laxation process afterwards (t > τ), see Fig. 1. Our
+argument is thus rooted in a non-Markovian exten-
+sion of the regression hypothesis of Onsager, a corner
+stone for the development in the nonequilibrium sta-
+tistical physics [32]. We obtain an exact integral equa-
+tion for the FPT distribution, the analysis of which
+yields, in addition to its asymptotic decay exponent,
+full functional form in leading order over entire time
+scales, and also the walker’s probability distribution
+function.
+Importantly, our formalism allows one to
+unveil how and why the textbook standard “method
+of image” [2, 33] breaks down by pinpointing the role
+of memory built up during the first passage process.
+Here we focus on the sub-diffusive fractional Brownian
+motion [34] (fBM with α < 1), an important class of
+non-Markovian walkers found in widespread complex
+systems including living cells and nuclei [20–23].
+FIG. 2. Illustration of the method of image. For Marko-
+vian walkers (α
+=
+1), Q(x, t; 1) can be constructed
+by the method of image.
+Integrating Eqs. (2) over
+the entire space (including negative domain), one finds
+S(t; 1) = 1 −
+� ∞
+−∞ Q(x, t; 1)dx, where the surviving proba-
+bility S(t; 1) =
+� ∞
+0
+P+(x, t; 1)dx is denoted by the hatched
+area. Equivalent to the above relation is
+� ∞
+0
+Q(x, t; 1)dx =
+(1−S(t; 1))/2 thanks to the reversal symmetry of Q(x, t; 1)
+with respect to x = 0, producing a factor 1/2. The same
+relation is obtained by integrating Eq. (3) over the positive
+x domain with ⟨x(t)⟩FPT=τ = 0.
+Generalized Langevin equation and power-law mem-
+ory kernel
+– As a paradigm, consider a random
+walker in one dimensional half space with an absorb-
+ing boundary at origin. A walker is initially positioned
+at x = x0(> 0) at t = 0, and evolves according to the
+following generalized Langevin equation:
+dx(t)
+dt
+=
+� t
+0
+µ(t − t′)f(t′)dt′ + η(t)
+(1)
+where
+f(t)
+and
+η(t)
+are,
+respectively,
+a
+time-
+dependent external force and the noise acting on the
+walker [17].
+The latter is assumed to be Gaussian
+with zero mean and its auto-correlation is related to
+the mobility kernel via the fluctuation-dissipation re-
+lation ⟨η(t)η(t′)⟩ = Tµ(|t − t′|) with T being the
+noise strength. The memory effect is encoded in µ(t),
+for which we assume for large t the power-law de-
+cay µ(t) ≃ −T −1Dαtα−2 (0 < α < 1) in addition
+to instantaneous response µ(t) = 2γ−1δ(t) at short
+time, where γ is a bare friction coefficient. Finally,
+we require on physical ground
+� ∞
+0
+µ(t)dt = 0 such
+that Eq. (1) describes the sub-diffusive fBM with the
+MSD exponent α.
+This sum rule is a consequence
+of the relaxation nature of the sub-diffusive fBM,
+which is caused by the visco-elastic effect [17]. For
+a free walker (f = 0) in free space (no boundrary),
+its position probability distribution P(x, t; x0) is sim-
+ply given by N(x, x0, 2Dαtα), where N(x, A, B) =
+(2πB)−1/2e(x−A)2/2B denotes Gaussian distribution
+of x with the average A and the variance B.
+Process after first-passage – We now set a stage by
+introducing an absorbing boundary at the origin x = 0
+such that the walker performs fBM in half space x > 0
+with the same initial condition as before. Using the
+free space propagator P(x, t; x0), the walker’s position
+probability P+(x, t; x0) is now represented as
+P+(x, t; x0) = P(x, t; x0) − Q(x, t; x0)
+(2)
+where Q(x, t; x0) is the position distribution of dead
+walker, who touched the absorbing boundary by this
+moment.
+Note that while one usually looks at the
+walker’s behavior in physical domain (x ≥ 0) up to the
+absorption (t ≤ τ) in the context of FPT, Eq. (2) holds
+in entire space and time domains in a sprit similar to
+[28]; the absorbing boundary at x = 0 necessitates
+P(x, t; x0) = Q(x, t; x0) for x ≤ 0. Using the FPT
+distribution F(τ; x0), Q(x, t; x0) is represented as
+Q(x, t; x0) =
+� t
+0
+F(τ; x0) P(x, t; x0|FPT = τ)dτ (3)
+where P(x, t; x0|FPT = τ) is the conditional proba-
+bility of the walker’s position at time t after its first
+passage at time τ. Being the Gaussian process, one
+expects the form
+P(x, t; x0|FPT = τ) = N(x, ⟨x(t)⟩FPT=τ, 2Dα(t − τ)α)
+.
+(4)
+In the absence of memory effect, ⟨x(t)⟩FPT=τ = 0 ir-
+respective of the starting position x0. Then, by not-
+ing
+� t
+0 F(t′; x0)dt′ = 1 − S(t; x0), integrating Eq. (2)
+over half space leads to a classical result of the
+survival probability S(t; x0) ≡
+� ∞
+0
+P+(x, t; x0)dx =
+erf(x0/√4D1t) for Markovian case, see Fig. 2.
+Al-
+though not applicable to non-Markovian walker, the
+
+0.5
+Q(c, t; 1)
+t=1
+P(c, t; 1)
+0.4
+P(c,t; - 1)
+0.3
+0.2
+0.1
+0.0
+-3
+-2
+-1
+0
+1
+2
+33
+above calculation highlights ⟨x(t)⟩FPT=τ, which gen-
+erally depends on x0, as a central quantity to account
+for the memory effect in the first passage statistics.
+History-dependent relaxation: regression hypothesis
+view – A key idea to quantify ⟨x(t)⟩FPT=τ comes from
+the fundamental connection between fluctuation and
+response in nonequilibrium statistical physics. In his
+seminal paper, Onsager pointed out that the decay of
+mesoscopic fluctuations follow, on average, the macro-
+scopic law of relaxation [32]. Applying this so-called
+regression hypothesis to our problem, we view the pro-
+cess after the first passage t > τ as a relaxation pro-
+cess, whose “initial” condition x(τ) = 0 can be pre-
+pared either naturally (by fluctuation) or artificially
+(by external force), see Fig. 1. In the latter scenario,
+we take the sub-ensemble of walkers whose FPT is τ,
+and describe their average time evolution using Eq. (1)
+with the constant force f(t) = f0 for t < τ. This leads
+to
+⟨ ˙x(t)⟩FPT=τ = f0
+� t
+0
+µ(t′)dt′
+(t < τ)
+(5)
+then, identifying ⟨ ˙x(τ)⟩FPT=τ ≃ −x0/τ, we find
+f0 ≃ −Tx0
+Dα
+τ −α.
+(6)
+Now the desired non-equilibrium state is prepared at
+t = τ, at which we switch off the force.
+The sub-
+sequent relaxation is described, again using Eq. (1),
+by
+⟨ ˙x(t)⟩FPT=τ = f0
+� t
+t−τ
+µ(t′)dt′,
+(t > τ)
+(7)
+whose
+integral
+with
+respect
+to
+time
+leads
+to
+⟨x(t)⟩FPT=τ, where a numerical coefficient implicit in
+Eq. (6) is fixed by requiring ⟨x(t)⟩FPT=τ → x0 for
+t/τ ≫ 1 as a consequence of the sum rule. Collecting
+all together, our analytical formulation is summarized
+as the following integral equation [35]:
+1 − erf
+�
+1
+√
+2tα
+�
+=
+� t
+0
+F(τ; 1) [1 − erf(h(t, τ))] dτ
+(8)
+with the memory function
+h(t, τ) =
+1
+�
+2(t − τ)α
+�
+1 +
+� t
+τ − 1
+�α
+−
+� t
+τ
+�α�
+.(9)
+From here onwards, we measure the length and the
+time in unit of x0 and τx0 = (x2
+0/2Dα)1/α, respec-
+tively, which are the sole characteristic length and
+time scales in the problem, making the initial posi-
+tion x0 = 1 upon rescaling.
+First passage time distribution – We now determine
+the leading order solution of Eq. (8) in the form
+F(τ; 1) = Cα exp
+�
+−
+� 1
+2τ
+�ω�
+τ −(1+p)
+(10)
+where Cα is a normalization constant. This function,
+a generalization of the Markovian result [2] ω = 1,
+(a)
+(b)
+FIG. 3.
+FPT distribution of non-Markovian walk-
+ers.
+(a) FPT distribution F(τ; 1) for sub-diffusive fBM
+(α = 0.8, 0.5). Inset shows the double logarithmic plot of
+large τ regime, where the asymptotic slope p+1 = 2−α/2
+is clearly visible. The data for α = 0.8 is shifted downward
+(×10−2) for visual clarity. Both in main panel and inset,
+symbols represent simulation results and the curves corre-
+spond to the analytical formula (10) with p = 1−α/2 and
+ω given by Eq. (11). The error bars represent 95 % CI.
+(b) Exponent ω as a function of α, which characterizes the
+early time regime in FPT distribution. Blue solid circles
+are obtained by fitting the numerical simulation data for
+several α values (two of them shown in Fig. 2(a)) with the
+formula (10). Fitting these data with Eq. (11) fixes the
+parameter c1 = 0.1.
+p = 1/2, exhibits a peak at τ = τ ∗ = (1/2)(ω/(1 +
+p))1/ω and develops a power-law tail F(τ; 1) ∼ τ −(1+p)
+at τ ≫ τ ∗. With this in mind, we plug the ansatz
+(10) into Eq. (8) and perform the asymptotic analysis,
+which yields p = 1 − α/2 in agreement with previous
+scaling argument [25, 29]. In addition, our formulation
+allows us to obtain the exponent ω, which satisfies the
+relation
+(2 − α)2ω(2 + α)α
+(2ω)α
+=
+�3
+2
+�ω
+cω(α−1)
+1
+(11)
+with a numerical constant c1 of order unity [35].
+In Fig. 3 , we compare our analytical formula for
+F(τ; 1) with the results obtained from numerical sim-
+ulation [35].
+As shown, the agreement is excellent,
+encompassing the short time singularity to the peak,
+and the eventual long time power-law tail, which are
+characterized by the exponents ω and p, respectively.
+The peak position τ ∗ is rather sensitive to the value
+of ω. This is particularly true for small ω, which is
+the case for the small α, shifting the peak position τ ∗
+vanishingly small in the limit α → 0.
+Probability distributions of dead and survived walk-
+ers – We are now in a position to take a close look
+at Q(x, t; 1) that is the distribution of walkers af-
+ter their first passage.
+From Eqs. (3) and (4), we
+immediately find the memory effect in the form of
+restoring force represented by nonzero ⟨x(t)⟩FPT=τ
+breaks the reversal symmetry with respect to x = 0,
+i.e., Q(x, t; 1) ̸= Q(−x, t; 1) that clearly manifests the
+breakdown of the image method (Figs. 2, 4) [35]. The
+value of ⟨x(t)⟩FPT=τ corresponds to the peak position
+of P(x, t; 1|FPT = τ), which is zero initially (t = τ),
+and slowly evolves with time towards x = x0. Such
+a distribution P(x, t; 1|FPT = τ) characterizes the
+
+4
+O
+α= 0.5
+0
+0
+α= 0.8
+3
+F(T)
+2
+1
+0
+0
+0.2
+0.0
+0.4
+0.6
+0.8
+1.0
+T1.0
+Eq. (11)
+0.8
+w=α
+0.6
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0101
+10-1,
+10-3
+10-5
+10-7
+10-1
+100
+101
+102
+103
+1044
+FIG. 4.
+Probability distribution Q(x, t; 1) of the position of absorbed sub-diffusive walkers. Plots of Q(x, t; 1) for sub-
+diffusive fBM (a)-(c) with α = 0.8 and (d)-(f) with α = 0.5 at early, middle and late times (t = 0.2, 1, 10, respectively).
+Analytical prediction (green solid curve) is obtained using Eqs. (3), (4) and (10), which quantitatively reproduces the
+numerical simulation results (red circles). The error bar evaluated as 95 % CI is smaller than the size of symbol. Blue
+dashed curve represent the free space propagator P(x, t; 1). The asymmetry in Q(x, t; 1) grows with the memory effect,
+which is stronger for smaller α.
+(a)
+~
+(b)
+~
+FIG. 5. Probability distribution P+(x, t; 1) of the position
+of survived sub-diffusive walkers.
+Plots of the normal-
+ized position probability ˜P+(x, t; 1) ≡ P+(x, t; 1)/S(x; 1)
+for sub-diffusive fBM with (a) α = 0.8 and (b) α = 0.5
+at early, middle and late times (t = 0.2, 1, 10, respec-
+tively). Analytical prediction (dashed curve) is obtained
+using Eq. (2), which reproduces the numerical simulation
+results (symbols). Error bars represent 95 % CI.
+subensemble of walkers with fixed FPT, whose super-
+imposition with the weight F(τ; 1) results in Q(x, t; 1),
+see Eq. (3). As Fig. 4 shows, our analytical predic-
+tion of Q(x, t; 1) quantitatively captures the results
+obtained by numerical simulations.
+In Fig. 5, we plot the normalized position prob-
+ability ˜P+(x, t; 1) ≡ P+(x, t; 1)/S(x; 1) of the sur-
+vival walker from Eq. (2).
+Again, our prediction
+captures all the salient features seen in numerical
+simulations.
+One notable feature here is that the
+slope (∂ ˜P+(x, t; 1)/∂x)x→0 at the boundary is van-
+ishingly small [36]. Such an anomalous behavior of
+˜P+(x, t; 1) ∼ xδ close to the boundary with non-trivial
+exponent δ can be quantified from our expression for
+Q(x, t; 1) as follows. Note first that in long time limit
+t ≫ 1 ( ⇔
+x2
+0/Dαtα ≪ 1 in original unit), the
+asymptotic behavior of ˜P+(x, t; 1) is obtained by tak-
+ing x0 → 0 limit [30].
+For the walker absorbed at
+time τ, its characteristic travel distance during the
+subsequent time interval s = t − τ is evaluated as
+∆x(s) ∼ sα/2. This indicates that, for a given loca-
+tion x, the walker only starts substantially contribut-
+ing to Q(x, t; 1) after the time t(x) = x2/α. From Eq.
+(3), we thus find
+Q(x, t; 1) ∼
+� t−τ ∗
+t(x)
+(t − s)−(2−α/2) s−α/2 ds
+∼ t−α/2 �
+1 − t−(2−α)x(2−α)/α�
+(12)
+The first term cancels the free space propaga-
+tor
+P(x, t; 1)
+∼
+t−α/2,
+leaving
+P+(x, t; 1)
+∼
+t−(2−α/2)x(2−α)/α,
+or
+equivalently,
+˜P+(x, t; 1)
+∼
+t−1x(2−α)/α. The predicted exponent δ = (2 − α)/α
+agrees with that obtained from heuristic scaling argu-
+ment [30].
+For the Markovian case α = 1, the slope at the
+boundary is finite (δ = 1), which multiplied by dif-
+fusion coefficient is the outgoing flux. The peculiar
+nature of the flux for α ̸= 1 case implies the break-
+down of the Fick’s law, and makes the implementation
+of a reflective boundary non-trivial. This rephrases a
+fact that there is no diffusion (more generally Fokker-
+
+(f)
+Q(c,t= 10; 1)
+0.6
+P(c,t; 1)
+α=0.5
+0.5
+Q(αc,t; 1)
+0.4
+0.3
+0.2
+0.1
+0.0
+.4
+-2
+0
+2
+4
+6(d)
+Q(α,t=0.2; 1)
+0.6
+P(c, t; 1)
+α=0.5
+0.5
+Q(αc,t; 1)
+0.4
+0.3
+0.2
+0.1
+0.0
+-2
+0
+2
+4
+6(e)
+Q(α,t=1; 1)
+0.6
+P(c,t; 1)
+α=0.5
+0.5
+Q(c,t; 1)
+0.4
+0.3
+0.2
+0.1
+0.0
+0
+2
+-2
+4
+6(c)
+Q(α,t= 10; 1)
+0.5
+P(c,t; 1)
+α=0.8
+0.4
+Q(c,t; 1)
+0.3
+0.2
+0.1
+0.0
+-2
+0
+2
+4
+4
+6
+8(a)
+Q(α,t=0.2; 1)
+0.5
+P(c, t; 1)
+α = 0.8
+0.4
+Q(αc,t; 1)
+0.3
+0.2
+0.1
+0.0
+-6
+-2
+0
+2
+4
+6
+8(b)
+Q(α,t=1; 1)
+0.5
+P(c,t; 1)
+α=0.8
+0.4
+Q(αc,t; 1)
+0.3
+0.2
+0.1
+0.0
+-4
+-6
+-2
+0
+2
+4
+6
+8P+(α,t; 1)
+1.0
+t=0.2
+0.8
+t=1
+t=10
+0.6
+α= 0.8
+0.4
+0.2
+0.0
+0
+2
+3
+4
+5
+1
+6
+7
+8P+(α,t; 1)
+1.0
+t=0.2
+0.8
+t=1
+t=10
+0.6
+α= 0.5
+0.4
+0.2
+0.0
+0
+2
+3
+4
+5
+6
+7
+8
+75
+Planck) equation for non-Markovian walkers in the
+ordinary sense.
+In conclusion, we have provided a natural frame-
+work with which the first passage process of non-
+Markovian walkers can be analyzed. It is very sim-
+ple, yet has a quantitative predictability as we have
+demonstrated here for the system with persistent
+memory, i.e., sub-diffusive fBM. We expect that the
+proposed method with suitable extension and general-
+ization will find versatile applicability to explore rich
+FPT problems in non-Markovian processes.
+Acknowledgements
+We thank E. Carlon for fruitful discussion.
+This
+work is supported by JSPS KAKANHI (Grants No.
+JP18H05529 and JP21H05759).
+∗ corresponding author, sakaue@phys.aoyama.ac.jp
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+Supplementary Material
+Yuta Sakamoto and Takahiro Sakaue∗
+Department of Physical Sciences, Aoyama Gakuin University,
+5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Japan
+1
+arXiv:2301.13466v1  [cond-mat.stat-mech]  31 Jan 2023
+
+DERIVATION OF INTEGRAL EQUATION
+We start with Eq. (2) in the main text;
+P+(x, t; 1) = P(x, t; 1) − Q(x, t; 1)
+(1)
+Here the walker’s initial position x0 > 0 is a sole length scale in the problem, and we measure
+the length in unit of x0. Similarly, we introduce the unit of time τx0 = (x2
+0/2Dα)1/α, which
+corresponds to the time scale for a walker to diffuse over the length scale x0. Note the rescaled
+initial position x0 = 1, and
+P(x, t; 1) =
+1
+√
+2πtαe− (x−1)2
+2tα
+(2)
+Q(x, t; 1) =
+� t
+0
+F(τ; 1) P(x, t; 1|FPT = τ)dτ
+=
+� t
+0
+F(τ; 1)
+1
+�
+2π(t − τ)αe− {x−⟨x(t)⟩FPT=τ }2
+2(t−τ)α
+dτ
+(3)
+where
+⟨x(t)⟩FPT=τ = 1 +
+� t
+τ − 1
+�α
+−
+� t
+τ
+�α
+(t ≥ τ)
+(4)
+is the average trajectory of the walkers after the first-passage at t = τ, which is calculated by
+applying the regression hypothesis idea of Onsager as explained in the main text.
+The integral of Eq. (1) over the half space (x ≥ 0) leads to
+S(t; 1) = 1
+2
+�
+erf
+�
+1
+√
+2tα
+�
++ 1
+�
+− 1
+2
+� t
+0
+F(τ; 1) erf
+�
+⟨x(t)⟩FPT=τ
+�
+2(t − τ)α
+�
+dτ
+(5)
+where S(t; 1) is the survival probability. Noting the relation S(t; 1) = 1 −
+� t
+0 F(τ; 1)dτ, the
+above equation is transformed to
+1 − erf
+�
+1
+√
+2tα
+�
+=
+� t
+0
+F(τ; 1) [1 − erf(h(t, τ))] dτ
+(6)
+with the memory function h(t, τ) = ⟨x(t)⟩FPT=τ
+√
+2(t−τ)α , which is an exact integral equation to determine
+F(τ, 1) (Eq. (8) in the main text).
+ANALYSIS OF INTEGRAL EQUATION
+To analyze the integral equation (6), we first rewrite the memory function as
+h(t, τ) = t−α/2
+√
+2 g(u)
+(7)
+2
+
+with
+g(u) = (1 − u)−α/2(1 − u−α) + (1 − u)α/2u−α
+(8)
+where u ≡ τ/t. The error function in the integrand is expanded as
+erf(h(t, τ)) = erf
+�t−α/2
+√
+2
+�
++
+�
+2
+πt−α/2(g(u) − 1) + O(t−3α/2)
+(9)
+Neglecting higher order terms O(t−3α/2), Eq. (6) becomes
+S(t; 1)
+�
+1 − erf
+�t−α/2
+√
+2
+��
+≃
+�
+2
+π t1−α/2
+� 1
+0
+F(τ(u); 1) {1 − g(u)} du
+(10)
+Motivated by the known analytical solution
+F(τ; 1) = C1 exp
+�
+−
+� 1
+2τ
+��
+τ −3/2
+(11)
+for the Markovian case (α = 1), where C1 is a normalization constant, we seek for the solution
+in the form
+F(τ; 1) = Cα exp
+�
+−
+� 1
+2τ
+�ω�
+τ −(1+p)
+= Cαt−(1+p) exp
+�
+−
+� 1
+2tu
+�ω�
+u−(1+p)
+(12)
+Substituting the above ansatz in Eq. (10), we obtain
+S(t; 1)
+�
+1 − erf
+�t−α/2
+√
+2
+��
+≃
+�
+2
+πCα t−(p+α/2)
+� 1
+0
+e−(
+1
+2tu)
+ω �
+αu−(α+p)(1 + O(u)) − α
+2 u−p(1 + O(u))
+�
+du
+(13)
+To evaluate the above integral, we note the following:
+� 1
+0
+e−(
+1
+2tu)
+ω
+u−κdu ≃
+� 1
+u∗ u−κdu
+(14)
+where u∗ = c1t−1(ω/κ)1/ω with c1 being a numerical constant of order unity.
+Then, at leading order in 1/t, Eq. (13) becomes
+S(t; 1) ≃
+�
+2
+π Cαt−(1−α/2)
+α
+α + p − 1
+�
+c1
+�
+ω
+α + p
+�1/ω�1−α−p
+(15)
+which is asymptotically correct at large t. Calculating −dS(t; 1)/dt and comparing it with the
+assumed form of F(t; 1), we find the persistence exponent
+p = 1 − α
+2
+(16)
+3
+
+in agreement with earlier scaling argument [1]. In addition, by comparing two expressions of
+prefactor, we find a relation between ω and α;
+(2 − α)
+�2 + α
+2ω
+�α/(2ω)
+c−α/2
+1
+= c2
+(17)
+where we introduce another numerical constant c2 of order unity to make the evaluated relation
+equality. Since we know ω = 1 for the Markovian limit α = 1, one of the numerical constants
+can be eliminated through
+c2 =
+�3
+2
+�1/2
+c−1/2
+1
+(18)
+This leads to Eq. (11) in main text with one fitting parameter c1, which should be determined
+through the comparison with numerical simulation data. As discussed in the main text, we
+found c1 = 0.1 describes the simulation results well. The resultant dependence of ω on α is
+shown in Fig. 3(b) in the main text. Apparently, the relation is close to ω = α, but the value of
+ω is slightly larger than α in a systematic way. We note that, while irrelevant to the long time
+asymptotic power-law behavior, the short time behavior is highly sensitive to this ω exponent.
+For example, we show in Fig. S1 the short time part of the FPT distribution F(τ) for the case
+of α = 0.4 and 0.5, where our formula for ω(α), but not ω = α, provides satisfactory fittings.
+FIG. 1:
+Short time part of FPT distribution of non-Markovian walkers. Plot of F(τ) for
+the case (a) α = 0.4 and (b) α = 0.5. The best fit values are ω = 0.45 for α = 0.4 and ω = 0.544 for
+α = 0.5 , which are included in the plot of Fig. 3(b) in the main text.
+FAILURE OF THE METHOD OF IMAGE
+The effect of the persistent memory in fBM becomes stronger with the departure from the
+Markovian limit α = 1. This is seen, for instance, in the spatial profile of Q(x, t; 1) shown in
+Fig. 4 in the main text, where the degree of the asymmetry Q(x, t; 1) ̸= Q(−x, t; 1), a hallmark
+of the memory effect, clearly shows up in α = 0.5 case, but less evident in α = 0.8 case. In
+4
+
+6
+w= 0.5
+5
+w = 0.544
+4
+α = 0.5
+F(T)
+3
+2
+1
+xxxxxxxxxxxxx
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T25
+w= 0.4
+20
+w = 0.45
+α=0.4
+F(T)
+15
+10
+5
+0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Tsuch a situation, one may expect that the method of image, a standard method used in the
+Markovian system, might provide an acceptable approximate solution. In Fig. S2, we show the
+probability of the survival walkers P+(x, t; 1) for α = 0.8, 0.5 cases, where the comparison is
+made for our solution and that constructed by the method of image. Clearly, the method of
+image fails to capture the profile even qualitatively. In contrast, our method is capable of a
+quantitative description.
+FIG. 2:
+Failure of the method of image. Plot of P+(x, t; 1) for (a) α = 0.8 and (b) α = 0.5. Solid
+curves are obtained from our theory, which quantitatively describe the numerical simulation result
+(symbols). In contrast, the method of image provide qualitatively wrong profiles (dashed curves).
+NUMERICAL SIMULATION
+To simulate fBM trajectories {x0, x1, · · · , xN} of length N, we numerically integrated the
+discretized version of Eq. (1) in main text with f = 0.
+The Gaussian variables ηi, called
+fractional Gaussian noise, have temporal correlation, whose long time part is characterized by
+the power-law memory as described after Eq. (1) in main text. To generate the fractional
+Gaussian noise, we employed the Davies and Harte algorithm [2], and generated m samples
+of length N for each α. From these simulations, we calculated the standard deviation of the
+walker’s displacement ∆xN ≡
+�
+⟨(xN − x0)2⟩ after N steps. To analyze the FPT statistics, we
+placed the hypothetical absorbing wall at x = x0 − ˜c ∆xN such that the initial separation from
+the walker to the boundary is ˜c ∆x. We then reanalyzed each of m trajectories to find its first
+arrival at the wall, and constructed the FPT distribution and the walkers’ distribution after
+the FPT. We adopted N = 105, m = 105 and ˜c = 1 except for the FPT distribution data for
+long time regime (Fig. 2 (a) inset), where we adopted N = 106 and m = 104 and ˜c = 0.5.
+∗ corresponding author, sakaue@phys.aoyama.ac.jp
+5
+
+P+(α,t=1;1)
+0.4
+α= 0.5
+0.3
+0.2
+0.1
+0.0
+0
+2
+3
+4P+(α,t=l; 1)
+0.4
+α= 0.8
+0.3
+0.2
+0.1
+0.0
+0
+2
+3[1] J. Krug, H. Kallabis, S. N. Majumdar, S. J. Cornell, A. J. Bray, and C. Sire, Phys. Rev. E 56,
+2702 (1997), URL https://link.aps.org/doi/10.1103/PhysRevE.56.2702.
+[2] R. Davies and D. Harte, Biometrika 74, 95 (1987).
+6
+

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