# On-Demand Redundancy Grouping: Selectable Soft-Error Tolerance for a
Multicore Cluster ††thanks: We acknowledge support by the EU H2020 Fractal
project funded by ECSEL-JU grant agreement #877056.

Michael Rogenmoser1, Nils Wistoff1, Pirmin Vogel2, Frank Gürkaynak1, Luca
Benini13  1ETH Zurich, Zurich, Switzerland  2lowRISC C.I.C., Cambridge, United
Kingdom  3University of Bologna, Bologna, Italy

###### Abstract

With the shrinking of technology nodes and the use of parallel processor
clusters in hostile and critical environments, such as space, run-time faults
caused by radiation are a serious cross-cutting concern, also impacting
architectural design. This paper introduces an architectural approach to run-
time configurable soft-error tolerance at the core level, augmenting a six-
core open-source RISC-V cluster with a novel On-Demand Redundancy Grouping
(ODRG) scheme. ODRG allows the cluster to operate either as two fault-tolerant
cores, or six individual cores for high-performance, with limited overhead to
switch between these modes during run-time. The ODRG unit adds less than 11%
of a core’s area for a three-core group, or a total of 1% of the cluster area,
and shows negligible timing increase, which compares favorably to a commercial
state-of-the-art implementation, and is 2.5$\times$ faster in fault recovery
re-synchronization. Furthermore, when redundancy is not necessary, the ODRG
approach allows the redundant cores to be used for independent computation,
allowing up to 2.96$\times$ increase in performance for selected applications.

###### Index Terms:

Reliability, Adaptive Fault Tolerance, RISC-V, Space Vehicle Computers

##  I Introduction

In hostile environments such as space, radiation is ever-present and can have
adverse effects on integrated circuits. Particle strikes due to radiation can
cause soft errors in combinational and sequential elements, where a resulting
Single Event Upset (SEU) in a state-saving element. Traditionally, SEUs have
impacted sequential elements such as memories and flip-flops and resulted in
bit-flips in stored information. With the advent of smaller technology nodes
and higher clock frequencies, transient errors that initially affect
combinational logic and subsequently end up altering the state within
sequential elements started to become a major concern [1].

In recent years, advances in computing power have enabled new applications,
such as machine learning. In space, these new algorithms can be leveraged, for
example to pre-process satellite images [2], however adaptation is more
difficult, as most processors for space lack the required computing power.
When investigating the different application scenarios more closely, however,
certain patterns appear: While some processing requires high levels of
redundancy to ensure correct operation, other tasks may not require the full
fault tolerance. As a more concrete example, a satellite will require
reliability and correctness when processing commands and telemetry, however
not for the imprecise processing of noisy data or running a machine-learning
algorithm over an image, which could greatly benefit from increased processing
power [3].

In this work, we tackle this variability in application requirements by
leveraging a run-time architectural configuration approach. We introduce _On-
Demand Redundancy Grouping (ODRG)_ , an augmentation to a shared memory multi-
core cluster architecture, which allows RISC-V cores to be grouped on-demand,
enabling Triple Modular Redundancy (TMR) at the core level. ODRG allows a six-
core cluster to switch between operating as six individual cores for higher
processing power and operating as two cores backed by TMR for critical tasks
requiring fault tolerance.

##  II Related Work

As reliable processing is an important requirement for critical missions in
space, several commercial processors that focus specifically on radiation
tolerance have been developed. These radiation-hardened processors, such as
the RAD line of processors [4] implementing the PowerPC Instruction Set
Architecture (ISA) or the LEON line of processors [5] implementing the SPARC
ISA, are designed using radiation-hardened processes, leveraging
characteristics of the fabrication and cell design that increase tolerance to
SEUs, as well as ECC and modular redundancy for critical sequential elements
in the design. Architectural redundancy mechanisms, such as ODRG presented in
this work, can leverage standard commercial technologies in contrast to the
specialized space-grade technologies, both reducing cost and improving
performance for processors in space.

In recent years, the open-source RISC-V ISA has gained significantly in
popularity, leading to its consideration for processors for space [2, 1, 6].
The continuing development of the RISC-V ISA can allow processors for space to
leverage developments in efficiency and improvements in design, with its
modularity targeting specialized application.

In [7], ARM presents a complete implementation to add fault tolerance within a
core with their Triple Core Lock-Step (TCLS) architecture, a redundancy
mechanism for their Cortex-R5 processor cores. To protect the execution of a
single processor core, this core is triplicated, adding a TCLS Assist Unit as
a wrapper around the three cores. With this TMR approach at the core level,
all cores are connected to identical inputs, and the outputs are passed into a
majority voter. Operating in lock-step, the outputs of the individual cores
will be identical unless a fault corrupts the internal state of one of the
cores. In case an error occurs within one of the cores, the voter will detect
the fault if it reaches the core’s interface and the TCLS Assist Unit will
correct it.

##  III Architecture

In this work, we focus on a class of emerging computing architectures that
consist of a powerful general-purpose host processor complemented by one or
more accelerators based on multi-core clusters as seen in Figure 1. These
clusters are designed to accelerate parallel tasks dispatched by a host
processor. In a fault tolerance scenario, we will assume the host processor
and peripherals are completely fault tolerant, protected by other means.

We focus on implementing a configurable redundancy mechanism in the multi-core
cluster, accelerating computation tasks for critical applications requiring
fault tolerance, as well as non-critical tasks that benefit from increased
parallelism. As a baseline for this work, we use the open-source cluster from
the Parallel Ultra-Low Power (PULP) platform [8].

Figure 1: Overall architecture of an SoC with a general purpose host system
supported by multi-core accelerators for data-parallel applications

###  III-A Parallel Ultra-Low Power (PULP) cluster

The PULP cluster contains multiple 32-bit RISC-V cores. In this work the
cluster was configured to include a total of six “Ibex” RISC-V cores [9],
sharing an instruction cache. The cores’ data ports are demultiplexed,
allowing each core to primarily interact with a tightly coupled data memory
(TCDM) with single-cycle access latency, here configured with
$64\text{\,}\mathrm{KiB}$ of memory grouped into 16 word-interleaved SRAM
banks, connected through a logarithmic interconnect implemented as a crossbar
with round-robin priority. The cores also have direct access to an event unit
to manage synchronization across the parallel cores, as well as a DMA and
various other peripherals and elements connected to a peripheral interconnect.
The SoC, connected to the cluster through two AXI interfaces, encompasses a
capable host processing system with a larger L2 memory, as well as main memory
controllers and I/O.

###  III-B On-Demand Redundancy Grouping (ODRG)

The PULP cluster offers a unique advantage when designing a reliable
architecture: Multiple cores are readily available, and a programming
environment that supports the systems in various configurations has been
developed independently. While usually the cluster cores are used in parallel
to process different data more quickly, they can also be used to process
identical data, which can subsequently be compared to detect errors in
computing. ODRG leverages the multiple cores and adds architectural elements
to directly allow three cores to operate on the same input, comparing the
output using a majority voter. The ODRG architecture, shown in Figure 2,
groups together three cores within the cluster to one ODRG unit and can
operate in the following two modes:

The performance mode enables all cores to operate independently for maximum
parallel performance. In this configuration, each core processes its own
inputs and outputs, individually connecting to the instruction cache and the
data interconnect. Each core is assigned its own hart-ID and has its own
synchronization port within the event unit.

Figure 2: Block diagram of ODRG, wrapping 3 cores with a configurable voter

The soft-error tolerant mode binds groups of three cores together into lock-
step operation. In this configuration, all three cores receive identical
inputs. The outputs of all cores are compared and voted upon to detect and
correct any errors that may be present within the cores. The three cores
operate as a single core, using only one interface to the rest of the system,
which is otherwise occupied only by the first core in the group. The other two
interfaces, typically occupied by the two cores now offering redundancy, are
not used.

The majority voter in the ODRG is implemented individually for each bit,
featuring additional detector circuitry comparing the output with the signals
prior to voting to detect if a core has a fault. The number of faults per core
are logged within the ODRG unit, allowing for software to distinguish between
random faults, or a systematic issue within one core.

The two operational modes of an ODRG unit are implemented as states in a
Finite State Machine (FSM).To facilitate re-synchronization once an error is
detected, the soft-error tolerant mode in the FSM is split into three parts,
one for normal operation, and one each to unload the internal state of the
core and reload it back into the core.

In soft-error tolerant mode, once a mismatch is detected, the ODRG unit will
trigger a re-synchronization sequence, as a mismatch indicates differing
internal states of the cores. This is done by switching the FSM into the
unload state and signaling an interrupt to all cores in the ODRG group, which
starts the re-synchronization software routine. This routine copies all
internal registers onto the stack in memory, including relevant CSRs (such as
the MEPC register). The stack pointer is saved to a register in the ODRG unit,
signalling the unload is complete and giving all cores a reference to reload
from. For this entire unload process, the majority voter is active, which
allows any internal errors within one core to be corrected.

In reload, the cores load the voted and stored state from memory back into the
internal registers. As the stack pointer is saved at a known location, the
state to reload is unique. Once the reload process is complete, the core
returns to the PC stored prior to the interrupt, which due to synchronization
of the MEPC register is identical for all three cores.

Within the PULP cluster, the ODRG unit exposes its configuration registers to
the peripheral interconnect, allowing configuration from the host. These
configuration registers are used to group and ungroup the cores, allowing the
switch between soft-error tolerant and performance modes. Additional registers
allow for the re-synchronization to be delayed, as well as store the stack
pointer during re-synchronization. Finally, mismatches are counted for each of
the cores and can be accessed through the same connection.

Figure 3: Block diagram of the PULP cluster with included ODRG units

In the implemented configuration shown in Figure 3, the six-core PULP cluster
has two ODRG units, grouping three cores each.

##  IV Evaluation

###  IV-A Setup

To verify proper functionality and evaluate the performance impact ODRG has on
the system, we simulate it at RTL level using Siemens/Mentor Questasim
v2019.3. For these investigations, the cluster is integrated next to an Ibex
host processor, additional memory, and peripherals within a host System-on-
Chip (SoC). As our investigation concerns only the cluster, the Ibex host
processor emulates a more capable host processing system and is responsible
for configuring the ODRG unit and subsequently starting the cluster with the
corresponding task. We use regression tests, which run highly parallel,
performance-critical mathematical functions used in such as matrix-matrix
multiplication and convolutions.

###  IV-B Performance

The main advantage of our approach is the ability to trade-off performance
with fault tolerance by activating ODRG only when needed. Table I shows the
performance improvement for three test applications that run in the ODRG soft-
error tolerant mode, as well as independently in performance mode, with the
corresponding parallelization across two or six cores. In a typical
application scenario, the cores in the cluster will be used to accelerate
parallelizable tasks to support the host system. For the highly parallelizable
tasks simulated, we observe speedups between 2.63$\times$ and 2.96$\times$,
which is close to the theoretical maximum of 3.00, showing an almost linear
trade-off between redundancy and performance for tasks well suited to
parallelization across six cores.

TABLE I: Performance of PULP cluster with ODRG in different modes Test Application | Cycle count | Speedup
---|---|---
|  soft-error tolerant | performance |
16-bit 2D Convolution | 100’938 | 36’379 | 2.77$\times$
32-bit 24x24 MatMul | 57’489 | 19’432 | 2.96$\times$
32-bit 32x32 MatMul | 135’908 | 51’668 | 2.63$\times$

While enabling re-configuring ODRG between soft-error tolerant and performance
modes requires a reboot of the cluster, setting the mode requires very few
cycles, as only a few memory-mapped registers are written. Rebooting and re-
configuring the cluster takes less than 40’000 cycles, which equates to
approximately $160\text{\,}\mathrm{\SIUnitSymbolMicro s}$ at
$250\text{\,}\mathrm{MHz}$.Therefore, switching between modes 100$\times$ per
second would require less than 2% overhead. Relating to specific applications,
non-critical tasks which require more than 60’000 cycles in soft-error
tolerant mode can benefit from a speedup if 3$\times$ performance improvement
is assumed. As typical applications, such as a machine learning algorithm,
require many functions to execute in series, this overhead becomes negligible.

###  IV-C Fault Injection

To evaluate the redundancy functions of ODRG, a limited fault injection
analysis is performed in simulation. Using the force command within
Siemens/Mentor Questasim, registers and signals are forced to an inverse value
for a single cycle, simulating an SEU.

Fault injection is performed both directly at a core interface, forcing a
signal exiting a core and entering the voter to the opposite value, as well as
within a core’s internal registers. As the interface is where errors are
detected, a re-synchronization is immediately triggered. Injecting faults
within a core’s register file allows for observing how an internal corrupted
state is handled. Once an error propagates to the core interface, a mismatch
is detected and the re-synchronization is successfully able to correct the
faulty register. As analyzed in more detail in [10], not all faults within a
core cause a noticeable failure of the system, as many of the errors are
masked.

Triggering faults during the execution of a matrix multiplication show that
the re-synchronization routine requires around 700 cycles to load the core
state to memory through the voter, and back into the cores, roughly
$2.8\text{\,}\mathrm{\SIUnitSymbolMicro s}$ at $250\text{\,}\mathrm{MHz}$.
There is some variability of $\pm 20$ cycles depending on the time of
injection, due to the second grouped core continuing operation normally,
leading to limited memory contention.

This compares favorably to the 1800-2400 cycles ARM reports are required for
TCLS re-synchronization [7]. Looking more closely, ARM stores 113 registers of
the Cortex-R5 core, while only 41 registers are stored using Ibex. In
environments such as space, while SEUs occur far more frequently than on
Earth’s surface, they are still considered to be rare. On the other hand,
environments such as particle accelerators [11] observe far higher rates of
SEUs, leading to a more stringent requirement on the re-synchronization time.
In these contexts, the 2.5$\times$ re-synchronization speedup of ODRG with
respect to TCLS may become a significant performance booster.

###  IV-D Area Overhead

To investigate the additional area required to support ODRG, we use Synopsys
DC 2019.12 to synthesize the PULP cluster in GF $22\text{\,}\mathrm{nm}$
technology at $250\text{\,}\mathrm{MHz}$ and worst-case conditions (SS,
$0.72\text{\,}\mathrm{V}$, $125\text{\,}\mathrm{\SIUnitSymbolCelsius}$). A
single ODRG adds around $6.0\text{\,}\mathrm{k}\mathrm{G}\mathrm{E}$, and the
total overhead of a cluster with two ODRG units and additional circuitry is
$25.6\text{\,}\mathrm{k}\mathrm{G}\mathrm{E}$. Compared to the base
configuration with six Ibex cores and $64\text{\,}\mathrm{KiB}$ ECC protected
memory, ODRG adds less than 1% to the total cluster area of
$2.16\text{\,}\mathrm{M}\mathrm{G}\mathrm{E}$. An ODRG wrapping unit requires
less than 11% the area of a single Ibex core, which occupies
$60.5\text{\,}\mathrm{k}\mathrm{G}\mathrm{E}$ in the implemented
configuration.

The cluster both with and without ODRG was synthesized at a maximum clock
frequency of $415\text{\,}\mathrm{MHz}$ in worst-case conditions, with no
discernible impact on the critical path.

This compares favorably to ARM’s TCLS implementation, where they claim the
TCLS unit adds around 4% to the presented system, or around 14% area of a
Cortex R5 core [7]. Furthermore, TCLS’ clock frequency is approximately 10%
lower than a reference implementation, experiencing a significant hit due to
the additional cores and voters required. Comparatively, ODRG adds less than
11% of the Ibex core, with a negligible impact on timing.

##  V Discussion

Safety is a crucial element required for a variety of processing tasks,
especially in critical environments, but comes at a significant cost. With
many implementations requiring >3$\times$ the area to protect calculations
with efficient correction, such as ARM TCLS [7], this cost can seem
unreasonable for tasks and applications not requiring this level of
redundancy.

In contrast to typical redundant architectures, ODRG leverages the additional
cores included for redundancy to increase performance of non-redundant
applications. Through simple re-configuration, these cores can operate in
parallel, speeding up non-critical calculations and reducing the overall power
requirements of the application. This can be especially valuable when using
the same architecture in both ordinary environments as well as hostile
environments, such as space, or when operating on already noisy data and
executing functions that can tolerate minor deviations in these critical
environments.

The ODRG approach is not specific to the core in use and no modifications were
made to the Ibex RISC-V cores used in this work. The ODRG method can be easily
adapted to different processing cores and cluster configurations.

##  VI Conclusion

In this work, we present On-Demand Redundancy Grouping (ODRG), a hardware
architecture that enables the dynamic configuration of a compute cluster to
run its cores either independently for high-performance, or redundantly for
high reliability. We implement ODRG on a six-core multi-core cluster based on
the openly available PULP system [8] and show that its performance mode can
speed up parallel tasks by up to 2.96$\times$, while reliably protecting
against single event upsets (SEUs) in soft-error tolerant mode. We furthermore
show that ODRG adds a minimal hardware overhead of less than
$1\text{\,}\mathrm{\char 37\relax}$ to the cluster and has little to no impact
on frequency.

We find that ODRG is a flexible and low-cost solution for computing systems
that are exposed to highly increased radiation levels, for instance when
deployed in space missions.

## References

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# Crowdsourcing Semantic Label Propagation in Relation Classification

Anca Dumitrache
Vrije Universiteit Amsterdam
IBM CAS Benelux
<EMAIL_ADDRESS>
&Lora Aroyo
Vrije Universiteit Amsterdam
<EMAIL_ADDRESS>
&Chris Welty
Google Research, New York
<EMAIL_ADDRESS>

###### Abstract

Distant supervision is a popular method for performing relation extraction
from text that is known to produce noisy labels. Most progress in relation
extraction and classification has been made with crowdsourced corrections to
distant-supervised labels, and there is evidence that indicates still more
would be better. In this paper, we explore the problem of propagating human
annotation signals gathered for open-domain relation classification through
the CrowdTruth methodology for crowdsourcing, that captures ambiguity in
annotations by measuring inter-annotator disagreement. Our approach propagates
annotations to sentences that are similar in a low dimensional embedding
space, expanding the number of labels by two orders of magnitude. Our
experiments show significant improvement in a sentence-level multi-class
relation classifier.

##  1 Introduction

Distant supervision (DS) Mintz et al. (2009) is a popular method for
performing relation extraction from text. It is based on the assumption that,
when a knowledge-base contains a relation between a pair of terms, then any
sentence that contains that pair is likely to express the relation. This
approach can generate false positives, as not every mention of a term pair in
a sentence means a relation is also expressed Feng et al. (2017).

Recent results Angeli et al. (2014); Liu et al. (2016) have shown strong
evidence that the community needs more annotated data to improve the quality
of DS data. This work explores the possibility of automatically expanding
smaller human-annotated datasets to DS scale. Sterckx et al. (2016) proposed a
method to correct labels of sentence dependency paths by using expert
annotators, and then propagating the corrected labels to a corpus of DS
sentences by calculating the similarity between the labeled and unlabeled
sentences in the embedding space of their dependency paths.

In this paper, we adapt and simplify semantic label propagation to propagate
labels without computing dependency paths, and using the crowd instead of
experts, which is more scalable. Our simplified algorithm propagates
crowdsourced annotations from a small sample of sentences to a large DS
corpus. To evaluate our approach, we perform an experiment in open domain
relation classification in the English-language, using a corpus of sentences
Dumitrache et al. (2017) whose labels have been collected using the CrowdTruth
method Aroyo and Welty (2014).

##  2 Related Work

Figure 1: Fragment of the crowdsourcing task template.

There exist several efforts to correct DS with the help of crowdsourcing.
Angeli et al. (2014) present an active learning approach to select the most
useful sentences that need human re-labeling using a query by committee. Zhang
et al. (2012) show that labeled data has a statistically significant, but
relatively low impact on improving the quality of DS training data, while
increasing the size of the DS corpus has a more significant impact. In
contrast, Liu et al. (2016) prove that a corpus of labeled sentences from a
pool of highly qualified workers can significantly improve DS quality. All of
these methods employ large annotated corpora of 10,000 to 20,000 sentences. In
our experiment, we show that a comparatively smaller corpus of 2,050 sentences
is enough to correct DS errors through semantic label propagation.

Levy et al. (2017) have shown that a small crowdsourced dataset of questions
about relations can be exploited to perform zero-shot learning for relation
extraction. Pershina et al. (2014) use a small dataset of hand-labeled data to
generate relation-specific guidelines that are used as additional features in
the relation extraction. The label propagation method was introduced by
Xiaojin and Zoubin (2002), while Chen et al. (2006) first applied it to
correct DS, by calculating similarity between labeled and unlabeled examples
an extensive list of features, including part-of-speech tags and target entity
types. In contrast, our approach calculates similarity between examples in the
word2vec Mikolov et al. (2013) feature space, which it then uses to correct
the labels of training sentences. This makes it easy to reuse by the state-of-
the-art in both relation classification and relation extraction –
convolutional Ji et al. (2017) and recurrent neural network methods Zhou et
al. (2016) that do not use extensive feature sets. To evaluate our approach,
we used a simple convolutional neural network to perform relation
classification in sentences Nguyen and Grishman (2015).

##  3 Experimental Setup

###  3.1 Annotated Data

The labeled data used in our experiments consists of 4,100 sentences: 2,050
sentences from the CrowdTruth corpus Dumitrache et al. (2017), which we have
augmented by another 2,050 sentences picked at random from the corpus of
Angeli et al. (2014). The resulting corpus contains sentences for 16 popular
relations from the open domain, as shown in in Figure 1,111The
$alternate\\_names$ relation appears twice in the list, once referring to
alternate names of persons, and the other referring to organizations. as well
as candidate term pairs and DS seed relations for each sentence. As some
relations are more general than others, the relation frequency in the corpus
is slightly unequal – e.g. $places\\_of\\_residence$ is more likely to be in a
sentence when $place\\_of\\_birth$ and $place\\_of\\_death$ occur, but not the
opposite.

The crowdsourcing task (Figure 1) was designed in our previous work Dumitrache
et al. (2017). We asked workers to read the given sentence where the candidate
term pair is highlighted, and then pick between the 16 relations or none of
the above, if none of the presented relations apply. The task was multiple
choice and run on the Figure Eight222https://www.figure-eight.com/ and Amazon
Mechanical Turk333https://www.mturk.com/ crowdsourcing platforms. Each
sentence was judged by 15 workers, and each worker was paid $0.05 per
sentence.

Crowdsourcing annotations are aggregated usually by measuring the consensus of
the workers (e.g. using majority vote). This is based on the assumption that a
single right annotation exists for each example. In the problem of relation
classification, the notion of a single truth is reflected in the fact that a
majority of proposed solutions treat relations as mutually exclusive, and the
objective of the classification task is usually to find the best relation for
a given sentence and term pair. In contrast, the CrowdTruth methodology
proposes that crowd annotations are inherently diverse Aroyo and Welty (2015),
due to a variety of factors such as the ambiguity that is inherent in natural
language. We use a comparatively large number of workers per sentences (15) in
order to collect inter-annotator disagreement, which results in a more fine-
grained ground truth that separates between clear and ambiguous expressions of
relations. This is achieved by labeling examples with the inter-annotator
agreement on a continuous scale, as opposed to using binary labels.

To aggregate the results of the crowd, we use CrowdTruth
metrics444https://github.com/CrowdTruth/CrowdTruth-core Dumitrache et al.
(2018) to capture and interpret inter-annotator disagreement as quality
metrics for the workers, sentences, and relations in the corpus. The
annotations of one worker over one sentence are encoded as a binary worker
vector with 17 components, one for each relation and including $none$. The
quality metrics for the workers, sentences and relations, are based on average
cosine similarity over the worker vectors – e.g. the quality of a worker $w$
is given by the average cosine similarity between the worker vector of $w$ and
the vectors of all other workers that annotated the same sentences. These
metrics are mutually dependent (e.g. the sentence quality is weighted by the
relation quality and worker quality), the intuition being that low quality
workers should not count as much in determining sentence quality, and
ambiguous sentences should have less of an impact in determining worker
quality, etc.

We reused these scores in our experiment, focusing on the sentence-relation
score ($srs$), representing the degree to which a relation is expressed in the
sentence. It is the ratio of workers that picked the relation to all the
workers that read the sentence, weighted by the worker and relation quality. A
higher $srs$ should indicate that the relation is more clearly expressed in a
sentence.

###  3.2 Propagating Annotations

Inspired by the semantic label propagation method Sterckx et al. (2016), we
propagate the vectors of $srs$ scores on each crowd annotated sentence to a
much larger set of distant supervised (DS) sentences (see datasets description
in Section 3.3), scaling the vectors linearly by the distance in low
dimensional word2vec vector space Mikolov et al. (2013). One of the reasons we
chose the CrowdTruth set for this experiment is that the annotation vectors
give us a score _for each relation_ to propagate to the DS sentences, which
have only one binary label.

Similarly to Sultan et al. (2015), we calculate the vector representation of a
sentence as the average over its word vectors, and like Sterckx et al. (2016)
we get the similarity between sentences using cosine similarity. Additionally,
we restrict the sentence representation to only contain the words between the
term pair, in order to reduce the vector space to the one that is most likely
to express the relations. For each sentence $s$ in the DS dataset, we find the
sentence $l^{\prime}$ from the crowd annotated set that is most similar to
$s$: $l^{\prime}=\operatorname*{arg\,max}\limits_{l\in Crowd}cos\\_sim(l,s).$
The score for relation $r$ of sentence $s$ is calculated as the weighted
average between the $srs(l^{\prime},r)$ and the original DS annotation,
weighted by the cosine similarity to $s$ ($cos\\_sim(s,s)=1$ for the DS term,
and $cos\\_sim(s,l^{\prime})$ for the $srs$ term):

$DS^{*}(s,r)=\dfrac{DS(s,r)+cos\\_sim(s,l^{\prime})\cdot
srs(l^{\prime},r)}{1+cos\\_sim(s,l^{\prime})}$ (1)

where $DS(s,r)\in\\{0,1\\}$ is the original DS annotation for the relation $r$
on sentence $s$.

###  3.3 Training the Model

The crowdsourced data is split evenly into a dev and a test set of 2,050
sentences each chosen at random. In addition, we used a training set of
235,000 sentences annotated by DS from freebase relations, used in Riedel et
al. (2013).

The relation classification model employed is based on Nguyen and Grishman
(2015), who implement a convolutional neural network with four main layers: an
embedding layer for the words in the sentence and the position of the
candidate term pair in the sentence, a convolutional layer with a sliding
window of variable length of 2 to 5 words that recognizes n-grams, a pooling
layer that determines the most relevant features, and a softmax layer to
perform classification.

We have adapted this model to be both multi-class and multi-label – we use a
sigmoid cross-entropy loss function instead of softmax cross-entropy, and the
final layer is normalized with the sigmoid function instead of softmax – in
order to make it possible for more than one relation to hold between two terms
in one sentence. The loss function is computed using continuous labels instead
of binary positive/negative labels, in order to accommodate the use of the
$srs$ in training. The features of the model are the word2vec embeddings of
the words in the sentences, together with the position embeddings of the two
terms that express the relation. The word embeddings are initialized with
300-dimensional word2vec vectors pre-trained on the Google News
corpus555https://code.google.com/archive/p/word2vec/. Both the position and
word embeddings are nonstatic and become optimized during training of the
model. The model is trained for 25,000 iterations, after the point of
stabilization for the train loss. The values of the other hyper-parameters are
the same as those reported by Nguyen and Grishman (2015). The model was
implemented in Tensorflow Abadi et al. (2016), and trained in a distributed
manner on the DAS-5 cluster Bal et al. (2016).

Figure 2: Precision / Recall curve, calculated for each sentence-relation
pair.

For our experiment, we split the crowd data into a dev and a test set of equal
size, and compared the performance of the model on the held-out test set when
trained by the following datasets:

  1. 1.

DS: The 235,000 sentences annotated by DS.

  2. 2.

DS + CT: The 2,050 crowd dev annotated sentences added directly to the DS
dataset.

  3. 3.

DS + W2V CT: The DS∗ dataset (Eq. 1), with relation scores propagated over the
2,050 crowd dev sentences.

##  4 Results and Discussion

To evaluate the performance of the models, we calculate the micro precision
and recall (Figure 2), as well as the cosine similarity per sentence with the
test set (Figure 3). In order to calculate the precision and recall, a
threshold of 0.5 was set in the $srs$, and each sentence-relation pair was
labeled either as positive or negative. However, for calculating the cosine
similarity, the $srs$ was used without change, in order to better reflect the
degree of agreement the crowd had over annotating each example. We observe
that DS + W2V CT, with a precision/recall $AUC=0.512$, significantly
outperforms DS (P/R $AUC=0.294$). DS + CT (P/R $AUC=0.372$) also does slightly
better than DS, but not enough to compete with the semantic label propagation
method. The cosine similarity result (Figure 3) shows that DS + W2V CT also
produces model predictions that are closer to the different agreement levels
of the crowd. Take advantage of the agreement scores in the CrowdTruth corpus,
the cosine similarity evaluation allows us to assess relation confidence
scores on a continuous scale. The crowdsourcing results and model predictions
are available online.666https://github.com/CrowdTruth/Open-Domain-Relation-
Extraction

Figure 3: Distribution of sentence-level cosine similarity with test set
values.

One reason for which the semantic label propagation method works better than
simply adding the correctly labeled sentences to the train set is the high
rate of incorrectly labeled examples in the DS training data. Figure 4 shows
that some relations, such as $origin$ and $places\\_of\\_residence$, have a
ratio of over 0.8 false positive sentences, meaning that a vast majority of
training examples are incorrectly labeled. The success of the DS + W2V CT
comes in part because the method relabels all sentences in DS. Adding
correctly labeled sentences to the train set would require a significantly
larger corpus in order to correct the high false positive rate, but semantic
label propagation only requires a small corpus (two orders of magnitude
smaller than the train set) to achieve significant improvements.

Figure 4: DS false positive ratio in combined crowd dev and test sets.

##  5 Conclusion and Future Work

This paper explores the problem of propagating human annotation signals in
distant supervision data for open-domain relation classification. Our approach
propagates human annotations to sentences that are similar in a low
dimensional embedding space, using a small crowdsourced dataset of 2,050
sentences to correct training data labeled with distant supervision. We
present experimental results from training a relation classifier, where our
method shows significant improvement over the DS baseline, as well as just
adding the labeled examples to the train set.

Unlike Sterckx et al. (2016) who employ experts to label the dependency path
representation of sentences, our method uses the general crowd to annotate the
actual sentence text, and is thus easier to scale and not dependent on methods
for extracting dependency paths, so it can be more easily adapted to other
languages and domains. Also, since the semantic label propagation is applied
to the data before training is completed, this method can easily be reused to
correct train data for any model, regardless of the features used in learning.
In our future work, we plan to use this method to correct training data for
state-of-the-art models in relation classification, but also relation
extraction and knowledge-base population.

We also plan to explore different ways of collecting and aggregating data from
the crowd. CrowdTruth Dumitrache et al. (2017) proposes capturing ambiguity
through inter-annotator disagreement, which necessitates multiple annotators
per sentence, while Liu et al. (2016) propose increasing the number of labeled
examples added to the training set by using one high quality worker per
sentence. We will compare the two methods to determine whether quality or
quantity of data are more useful for semantic label propagation. To achieve
this, we will investigate whether disagreement-based metrics such as sentence
and relation quality can also be propagated through the training data.

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# Splitting probabilities for dynamics in corrugated channels: passive VS
active Brownian motion

Paolo Malgaretti<EMAIL_ADDRESS>Helmholtz Institute Erlangen-
Nürnberg for Renewable Energy (IEK-11), Forschungszentrum Jülich, Cauerstr. 1,
91058 Erlangen, Germany  Tatiana Nizkaia  Helmholtz Institute Erlangen-
Nürnberg for Renewable Energy (IEK-11), Forschungszentrum Jülich, Cauerstr. 1,
91058 Erlangen, Germany  Gleb Oshanin  Sorbonne Université, CNRS, Laboratoire
de Physique Théorique de la Matière Condensée (UMR CNRS 7600), 4 place
Jussieu, 75252 Paris Cedex 05, France

###### Abstract

In many practically important problems which rely on particles’ transport in
realistic corrugated channels, one is interested to know the probability that
either of the extremities, (e.g., the one containing a chemically active site,
or connected to a broader channel), is reached before the other one. In
mathematical literature, the latter are called the ”splitting” probabilities
(SPs). Here, within the Fick-Jacobs approach, we study analytically the SPs as
functions of system’s parameters for dynamics in three-dimensional corrugated
channels, confronting standard diffusion and active Brownian motion. Our
analysis reveals some similarities in the behavior and also some markedly
different features, which can be seen as fingerprints of the activity of
particles.

Transport of particles in narrow corrugated channels is an important area of
research which has attracted a great deal of attention within the recent
several decades (see e.g., Ref. Malgaretti et al. (2019a) for a review). In
part, such an interest is due to the relevance to various realistic physical,
biophysical and chemical systems, as well applications in nanotechnology and
nanomedicine, e.g., for manufacturing of artificial molecular nanofilters. To
name just a few examples, we mention transport in porins Nestorovich et al.
(2002); Bajaj et al. (2017), in nuclear pores Rout et al. (2003); Kabachinski
and Schwartz (2015); Stawicki and Steffen (2017), in microtubules Welte (2004)
and dendritic spines Nimchinsky et al. (2002), transport of microswimmers in
capillaries Berg and Turner (1990); Malgaretti and Stark (2017), translocation
of polymers in pores Zandi et al. (2003); Sakaue (2016); Palyulin et al.
(2014) and their sequencing in nanopore-based devices seq (2018), as well as
in microfluidics Stone et al. (2004); Mukhopadhyay and Granick (2001).

The problem of random transport in corrugated channels is clearly also a
challenge for the theoretical analysis - it is too complicated to be solved
analytically in full detail and one therefore seeks approximate approaches
that are justified in particular limits. Most of the available analytical
descriptions rely on the so-called Fick-Jacobs approach Merkel (1935); Zwanzig
(1992a) and its subsequent generalizations (see, e.g., Reguera and Rubi
(2001a); Malgaretti et al. (2013); Mangeat et al. (2017a, b); Mangeat et al.
(2018)). In essence, this approach amounts to a reduction of the original
multidimensional problem to a one-dimensional diffusion in presence of some
potential, which mimics in an effective way a spatial variation of the
confining boundaries. In some cases, this approximation is physically
meaningful and provides an insight into the behavior of important
characteristic properties, e.g., currents across the channel, the mean first-
passage times to some positions and quantifying fluctuations of the first-
passage times Malgaretti and Oshanin (2019). In other systems, in which, e.g.,
diffusion in the direction perpendicular to the main axis of the channel is
important Valov et al. (2020), other approaches are to be developed.

Figure 1: Top: A colloidal particle in a simple varying-section channel with
fore-aft symmetry. Bottom: the effective piece-wise linear potential $A(x)$
within the Fick-Jacobs approach and the corresponding overall barrier $\Delta
A$.

In many important situations one is interested in understanding the behavior
of the properties which characterize a kind of a ”broken symmetry” in
otherwise symmetric dynamics : in particular, of the probability that a
particle injected at some position within the channel reaches first its
prescribed extremity without having ever reached the opposite one. This
particle can be a tracer within a channel that is attached to a broader
pathway to which all the channels are connected, or it can be a chemically
active molecule which needs to react with a target site placed at either of
the extremities. In mathematical and physical literature (see, e.g., Redner
(2001); Bénichou et al. (2015)) such probabilities - the so-called splitting
probabilities - have been analyzed in details in various settings, with and
without an external potential (see, e.g. Oshanin and Redner (2009)), providing
an important complementary insight into the dynamical behavior.

In the present paper, we study analytically the behavior of splitting
probabilities (SPs) as functions of system’s parameters for transport in
narrow corrugated channels, in terms of a suitably generalized Fick-Jacobs
approach. In regard to the dynamics, we confront two different transport
mechanisms - standard Brownian motion and active Brownian motion, capitalizing
for the latter case on the theoretical framework developed in recent Kalinay
and Slanina (2021); Bauer and Nadler (2006); Berezhkovkii and Hummer (2002);
Zilman (2009); Marconi et al. (2015); Bénichou et al. (2016, 2018). For
passive diffusion we obtain exact expressions for the SPs for channels of an
arbitrary periodic shape. For the active case for which the dynamic equations
have a much more cumbersome form Kalinay and Slanina (2021); Bauer and Nadler
(2006); Berezhkovkii and Hummer (2002); Zilman (2009); Marconi et al. (2015);
Bénichou et al. (2016, 2018), we resort to a numerical analysis. Our
theoretical findings demonstrate that the SPs are quite sensitive to both the
geometry of the channel and the activity of the particles. In particular, for
active particles the SPs exhibit a spectacular non-monotonous dependence on
the amplitude of the corrugation of the channel when the magnitude of the
entropic force emerging due to a confinement becomes comparable to the
propulsive force. This effect is absent for a passive Brownian motion.

Passive particles. Consider a particle that starts at $\mathbf{r}_{0}$ and
undergoes a passive Brownian motion within an axially-symmetric three-
dimensional channel with impermeable periodically-corrugated boundaries. It is
convenient to use the cylindric coordinates $(r,x)$, where the $x$-axis
coincides with the main axis of the channel, while $r$ is the radial
coordinate. A local thickness of the channel at point $x$ is defined by $h(x)$
and hence, $r\leq h(x)$. In view of the symmetry, the particle’s position
probability density function $\rho(\mathbf{r},t)$ and therefore all other
properties derived from it are independent of the polar angle. We focus on the
SPs - the probabilities that the particle first reaches either of the
extremities $x=\pm L/2$ of the channel (see Fig. 1) without ever hitting the
other one.

We first write down the advection-diffusion equation that governs the time
evolution of the particle’s position probability density function
$\rho(\mathbf{r},t)$ :

$\displaystyle\dot{\rho}(\mathbf{r},t)=\nabla\cdot\left[D\nabla\rho(\mathbf{r},t)+D\beta\rho(\mathbf{r},t)\nabla
W(\mathbf{r})\right]$ (1)

where $\mathbf{r}$ is the position of a particle, $D$ is the diffusion
coefficient, $\beta^{-1}=k_{B}T$ is the inverse thermal energy, $k_{B}$ is the
Boltzmann constant, $T$ \- the absolute temperature and $W(\mathbf{r})$ is the
particle-wall interaction potential,

$\displaystyle W(\mathbf{r})=\begin{cases}\phi(\mathbf{r}),&r<h(x),\\\
\infty,&\text{otherwise.}\end{cases}$ (2)

If a local thickness of the channel is a slowly varying function of the
$x$-coordinate, such that $\partial_{x}h(x)\ll 1$, it is possible to write
down the probability density function in the following approximate factorized
form (see, e.g., Malgaretti et al. (2016a)),

$\displaystyle\rho(\mathbf{r},t)=p(x,t)\dfrac{e^{-\beta
W(\mathbf{r})}}{e^{-\beta A(x)}}\,,$ (3)

where

$\displaystyle A(x)=-\dfrac{1}{\beta}\ln\left[\frac{1}{\pi
h_{0}^{2}}\int_{0}^{\infty}e^{-\beta W(x,r)}rdr\right]$ (4)

is the local free energy and $h_{0}$ is the mean cross-section of the channel.
Upon integrating over the radial coordinate, we cast Eq. (3) into the form:

$\displaystyle\dot{p}(x,t)=\partial_{x}\left[D\partial_{x}p(x,t)+D\beta
p(x,t)\partial_{x}A(x)\right]$ (5)

Such a reduction of the original three-dimensional problem to a one-
dimensional diffusion in presence of an effective potential (which, in fact,
is the local free energy defined in Eq. (4)) is called the Fick-Jacobs
approach Zwanzig (1992b); Reguera and Rubi (2001b); Malgaretti et al. (2013)
and its range of applicability is well-understood Reguera et al. (2006);
Berezhkovskii et al. (2007); Burada et al. (2007); Berezhkovskii et al.
(2015); Kalinay and Percus (2005a); Kalinay and Percus (2005b); Kalinay and
Percus (2006); Martens et al. (2011a); Pineda et al. (2012); García-Chung et
al. (2015). This approach has provided an insight into the behavior of quite
diverse confined systems, including colloidal particles Martens et al.
(2011b); Berezhkovskii et al. (2015), flow of charged fluids Martens et al.
(2013); Malgaretti et al. (2014); Chinappi and Malgaretti (2018); Malgaretti
et al. (2019b); Kalinay (2020), of polymers Bianco and Malgaretti (2016);
Malgaretti and Oshanin (2019); Carusela et al. (2021), of rigid rods
Malgaretti and Harting (2021), systems with chemical reactions Ledesma-Durán
et al. (2016), and pattern-forming ones Chacón-Acosta et al. (2020).

We quantify next the SP $E_{+}$ \- the probability that the particle first
reaches $x=L/2$ without ever touching $x=-L/2$. This SP has the form (see e.g.
Oshanin and Redner (2009))

$\displaystyle E_{+}=\frac{\tau_{-}}{\tau_{+}+\tau_{-}}\,,\quad
E_{-}=\frac{\tau_{+}}{\tau_{+}+\tau_{-}}\,,$ (6)

where $\tau\pm=\rho_{0}L/|J_{\pm}|$ and $|J\pm|$ are the magnitudes of the
steady-state currents from $x_{0}$ to the extremities $x=L/2$ and $x=-L/2$,
respectively. Note that the SP $E_{-}$ (i.e., the probability that the
particle first reaches $x=-L/2$ without ever touching $x=L/2$) is simply
defined by $E_{-}=1-E_{+}$.

Solving Eq. (5), we determine the steady-state currents $J\pm$ (see appendix)
and hence, the functions $\tau_{\pm}$ to get

$\displaystyle\tau_{-}=\tau_{0}\int_{-L/2}^{x_{0}}e^{\beta
A(x)}dx\,,\quad\tau_{+}=\tau_{0}\int_{x_{0}}^{L/2}e^{\beta A(x)}dx\,,$ (7)

with $\tau_{0}=(L/D)e^{-\beta A(L)}$. Expressions (7) totally define the SPs
$E_{\pm}$. They are fairly general and hold for arbitrary $A(x)$, i.e.,
confining boundaries of arbitrary (sufficiently smooth) shapes. In the trivial
case $A(x)\equiv 0$, we find from Eq. (7) that the functions
$\tau_{\mp}=L\left(L/2\pm x_{0}\right)/D$ and hence, recover the well-known
result Redner (2001)

$\displaystyle E_{+}=\frac{1}{2}+\frac{x_{0}}{L}\,,\quad
E_{-}=\frac{1}{2}-\frac{x_{0}}{L}\,,\quad-\frac{L}{2}\leq
x_{0}\leq\frac{L}{2}$ (8)

We will use Eq. (8) in what follows as a point of reference \- all departures
from a simple linear behavior are indicative of the effects of the confining
boundaries.

In order to get an idea of the dependence of the SPs on the overall barrier
$\Delta A$ (see Fig. 1), consider a simple form of the free energy :

$\displaystyle\beta A(x)=\begin{cases}\beta\frac{2\Delta
A}{L}\left(x+\frac{L}{2}\right),&-\frac{L}{2}<x\leq 0,\\\ \beta\frac{2\Delta
A}{L}\left(x-\frac{L}{2}\right),&0<x<\frac{L}{2}\end{cases}$ (9)

We note parenthetically that such a simple piece-wise linear form has provided
qualitatively reliable predictions on the behavior of the mean first-passage
times through a finite channel in case of ions in a charged confinement
Malgaretti et al. (2016a).

Figure 2: $E_{\pm}$ as functions of the staring point $x_{0}/L$. Panel (a):
$\beta\Delta A=1.0$ (blue), $3.0$ (cyan) and $10.0$ (red). Panel (b):
$\beta\Delta A=-1.0$ (blue), $-3.0$ (cyan) and $-10.0$ (red). Solid lines
stand for $E_{+}$ whereas dashed lines for $E_{-}$.

Fig.2 displays the SPs $E_{\pm}$ as functions of the starting point $x_{0}$ in
the case of a fore-aft symmetric channel. We observe that for moderate values
of the barrier, $\beta\Delta A=\pm 1$, i.e., for a mild corrugation the SPs
exhibit an almost linear dependence on $x_{0}$ (see blue curves in Fig. 2).
For larger values of $\beta|\Delta A|$, the corrugation of the channel starts
to play a major role and entails an essential departure from the linear
dependence. We see that upon an increase of $\beta\Delta A$ to larger positive
values, $E_{+}$ and $E_{-}$ attain an $S$-shaped form which becomes
progressively more steep in the vicinity of $x=0$ the larger $\beta\Delta A$
is. Recall that for $\beta\Delta A>0$ the potential has a maximum at $x=0$
meaning that the channel has a bottleneck at this position. In consequence,
when $x_{0}$ even only slightly exceeds $0$, it becomes much more probable for
a particle to reach the right extremity because the bottleneck does not permit
to reach the left one. Conversely, for $\beta\Delta A<0$ the channel is widest
at $x=0$. In this case, if a particle starts in a broad part of the channel,
it first diffuses there for a long time effectively ”forgetting” about its
actual starting point. Moreover, since in this case the channel narrows close
to the extremities, there emerge effective entropic barriers (see, e.g.,
Grebenkov and Oshanin (2017)) which the particle has to overcome in order to
reach any of the extremities. As a consequence, a passage to the extremity may
necessitate repeated unsuccessful attempts to overpass the entropic barrier,
which attempts are interspersed with the excursions in the broad part of the
channel. A combined effect of these two factors results in a very weak
dependence of $E_{\pm}$ on $x_{0}$, which behavior we indeed observe in the
panel (b) of Fig. 2.

Figure 3: $E_{\pm}$ as function of the entropic barrier $\beta\Delta A$ and
$x_{0}/L=0.1$ (blue), $0.2$ (cyan) and $0.4$ (red). Solid lines stand for
$E_{+}$ whereas dashed lines - for $E_{-}$.

Next, we examine the dependence of $E_{\pm}$ on the magnitude of the barrier
$\beta\Delta A$ for a few values of the starting position $x_{0}$. Figure 3
shows that for $\beta\Delta A\gg 1$ (i.e., for a strong entropic repulsion
from the bottleneck at $x=0$) the SPs are either (almost) equal to zero or to
unity, meaning that the particle most likely reaches first the closest
extremity and never gets to the opposite one. In contrast, for $\beta\Delta
A\ll-1$ (i.e., for an entropic repulsion from the extremities) the barrier to
overcome becomes very high and a particle has to undertake many attempts to
cross the barrier before it actually does it. As a consequence, for large
negative $\beta\Delta A$ the SPs $E_{\pm}\simeq 0.5$.

It is important to emphasize that within the Fick-Jacobs approach, many
characteristic properties of a particle diffusing in a channel, such as
charge, elastic moduli, deformability, are effectively encoded in the free
energy barrier $\Delta A$Malgaretti et al. (2016b); Bianco and Malgaretti
(2016); Malgaretti et al. (2019b); Malgaretti and Oshanin (2019). For example,
for uncharged particles which are much smaller than the channel bottleneck
(i.e., point-like particles) we have $\beta\Delta A\lesssim 3$ where
$\beta\Delta A=3$ implies that the maximal cross-section of the channel is
$\sim 30$ times the radius of the bottleneck. In contrast, for charged ions it
is feasible to have $\beta\Delta A\simeq 10$ when electrostatic potential at
the walls is $\beta e\zeta\simeq 10$ Malgaretti et al. (2019b). Finally, for
deformable objects, like polymers, one may have a very large effective barrier
$\beta\Delta A\simeq 100$ Bianco and Malgaretti (2016); Marenda et al. (2017);
Malgaretti and Oshanin (2019).

Active particles. The case of particles that propel themselves through the
channels, e.g., of ”active” colloids, is most challenging, because a local
violations of the equilibrium may lead to quite a different scenario as
compared to the case of a passive Brownian motion. To set-up the scene,
consider first a simple situation in which the non-interacting particles move
with a constant velocity either to the left or to the right in a one-
dimensional system and interchange the sign of the velocity at random, at a
constant rate $\alpha$. In such a model the time evolution of the densities
$\rho_{\downarrow}(x,t)$ and $\rho_{\uparrow}(x,t)$ of active particles moving
to the left or to the right, respectively, is described by Malgaretti and
Harting (2021):

$\displaystyle\dot{\rho}_{\uparrow}$ $\displaystyle=$ $\displaystyle
D\partial_{x}\left[\partial_{x}\rho_{\uparrow}+\beta\rho_{\uparrow}\partial_{x}A(x)+\beta\rho_{\uparrow}F_{act}\right]-\alpha\rho_{\uparrow}+\alpha\rho_{\downarrow}$
$\displaystyle\dot{\rho}_{\downarrow}$ $\displaystyle=$ $\displaystyle
D\partial_{x}\left[\partial_{x}\rho_{\downarrow}+\beta\rho_{\downarrow}\partial_{x}A(x)-\beta\rho_{\downarrow}F_{act}\right]+\alpha\rho_{\uparrow}-\alpha\rho_{\downarrow}$

where $F_{act}$ is the propulsive force. These equations are to be solved
subject to the boundary conditions imposed at the extremities :
$\rho_{\uparrow,\downarrow}(x=x_{0})=\rho_{0}/2$ and
$\rho_{\uparrow,\downarrow}(x=\pm\frac{L}{2})=0.$ A straightforward analysis
Malgaretti and Harting (2021) shows that the dynamical behavior is
characterized by two dimensionless parameters: the Péclet number
$\text{Pe}=\beta F_{act}R$ and the reduced hopping rate $\Gamma=\alpha
R^{2}/D$. In particular, for a Janus swimmer Malgaretti and Harting (2021) the
hopping between two states stems from a rotational diffusion of the particle
and $\Gamma_{*}=3/4$. For other types of swimmers (e.g., bacteria), the
hopping rates can be much smaller. Rewriting the equations in dimensionless
form (but keeping the same notations), and setting the length scale to $L$, we
have that the particle probability densities
$\rho=\rho_{0}^{-1}(\rho_{\uparrow}+\rho_{\downarrow})$ and
$\phi=\rho_{0}^{-1}(\rho_{\uparrow}-\rho_{\downarrow})$ obey, in the steady
state,

$\begin{split}0&=\partial_{x}[\partial_{x}\rho+\beta\Delta
A\rho\partial_{x}a(x)+\dfrac{\mathrm{Pe}L}{R}\phi],\\\
0&=\partial_{x}\left[\partial_{x}\phi+\beta\Delta
A\phi\partial_{x}a(x)+\dfrac{\mathrm{Pe}L}{R}\rho\right]-2\dfrac{L^{2}}{R^{2}}\Gamma\phi,\end{split}$
(10)

while the boundary conditions take the form

$\rho(x_{0})=\rho_{0},\rho(\pm 1/2)=0,\,\phi(x_{0})=0,\phi(\pm 1/2)=0.$ (11)

Here $a(x)$ is a piecewise-linear function such that $a(0)=1$ and $a(\pm
1/2)=0$.

Note that the reduced channel length $L/R$ appears in the equations only in a
combination with $\mathrm{Pe}$ and $\Gamma$. This means that the SPs for
various channel lengths can be obtained by taking the solution at fixed $L/R$
and changing Pe and $\Gamma$ accordingly. In the following we use $L/R$=10.

Consider first a channel with a constant cross-section ($\Delta A=0$) - the
simplest case for which, however, the SPs have not been determined as yet.
Upon some straightforward algebra (see the Suppl. Mat., Eqs.(S30) to (S42)) it
is possible to derive closed-form expressions for the currents $J_{\pm}$ and
hence, for the SPs which we depict in Fig. 4.

Figure 4: $E_{+}$ (solid curves) and $E_{-}$ (dot-dashed curves) as functions
of $x_{0}/L$. Panel (a): Fixed $\Gamma=0.1$, $L/R=10$ and varying Péclet
number $\text{Pe}=0$ (blue), $1$ (cyan) and $5$ (red). Panel (b): Fixed
$\text{Pe}=3$ and $\Gamma=0.01$ (blue), $0.1$ (cyan) and $1$ (red).

We infer from Fig. 4(a) that upon an increase of Pe the SPs $E_{\pm}$ become
(almost) independent of the starting point, except when the latter appears
close to the extremities. This resembles the behavior which we observed for a
passive particle in a channel with $\Delta A<0$ for which the bottlenecks
(entropic barriers) are at the extremities and the largest cross-section is at
$x=0$. Here, the origin of such a behavior is somewhat different : For low
values of $\Gamma$ the particle does not often change the direction of its
motion and travels towards the extremities of the channel ballistically.
Consequently, the larger the propulsive force (and hence, Pe) is, the less
sensitive are $E_{\pm}$ to the starting point. In turn, in panel (b) we plot
$E_{\pm}$ as functions of $x_{0}/L$ with fixed Pe and three different values
of $\Gamma$. We realize that upon an increase of $\Gamma$ the
$x_{0}$-dependence of the SPs approaches the linear dependence in Eq. (8)
specific to a passive Brownian motion in one-dimensional systems. This is, of
course, not counter-intuitive - the larger $\Gamma$ is, the more often the
particle changes the direction of its motion and the dynamics becomes
diffusive.

To get an additional insight into the behavior of active particles in channels
with a constant cross-section we plot in Fig. 5 the SPs $E_{\pm}$ as functions
of $\Gamma$ and Pe for fixed $x_{0}/L=0.35$. Clearly, since the starting point
is close to the right extremity of the channel, one expects that
$E_{+}>E_{-}$. We observe that $E_{+}$ ($E_{-}$) is a monotonically decreasing
(increasing) function of Pe. While for small Pe (for which the particle’s
dynamics is a passive Brownian motion) $E_{+}$ ($E_{-}$) is rather large
(small) ($E_{+}\approx 0.85$ and $E_{-}\approx 0.15$), upon an increase of Pe
dynamics becomes ballistic and both $E_{+}$ and $E_{-}$ tend to the same
universal value $1/2$, which is rather counter-intuitive. Conversely, $E_{+}$
($E_{-}$) is a monotonically increasing (decreasing) function of the rate
$\Gamma$. Interestingly enough, in the small-$\Gamma$ limit the values of
$E_{+}$ ($E_{-}$) are markedly different for small and large values of Pe :
for $\text{Pe}=1$ the SP $E_{+}$ ($E_{-}$) is noticeably higher (lower) than
$1/2$ (in fact, $E_{-}\approx 0.4$ and $E_{+}\approx 0.6$), while for
$\text{Pe}=5$ and $10$ we have $E_{-}\approx E_{+}\approx 1/2$. In the limit
$\Gamma\to\infty$ the dynamics becomes diffusive and we recover the low Péclet
number behavior depicted in panel (a).

Figure 5: $E_{+}$ (solid curves) and $E_{-}$ (dot-dashed curves) for
$x_{0}/L=0.35$ in a channel with $L/R=10$. Horizontal dashed line defines the
level $E_{+}=E_{-}=1/2$. Panel (a): The SPs are plotted as functions of the
Péclet number for $\Gamma=0.001$ (blue), $0.1$ (cyan) and $1$ (red). Panel
(b): The SPs are plotted as functions of $\Gamma$ for $\text{Pe}=1$ (blue),
$5$ (cyan) and $10$ (red).

Lastly, we consider the most difficult case - the behavior of the SPs for
dynamics of active particles in a channel with a varying cross-section,
encoded in the effective potential $A(x)$. In this case, Eqs.(Splitting
probabilities for dynamics in corrugated channels: passive VS active Brownian
motion) are too complicated to be solved analytically and we resort to a
numerical analysis of these equations, which is done by using the standard
scipy library in Python (see Suppl. Mat.). Our findings for the SP $E_{-}$ are
summarized in Figs. 6 and 7.

Figure 6: Panel (a): $E_{-}$ as a function of the initial position for
$\beta\Delta A/\mathrm{Pe}=3$ (blue), $10$ (cyan) and $20$ (red) and fixed
$L=10R$. Solid lines indicate the respective behavior for passive particles,
circles - for active particles with $\mathrm{Pe}=1$ and $\Gamma=0.1$. Panel
(b): The same for $\beta\Delta A/\text{Pe}=-3$ (blue), $-10$ (cyan) and $-20$
(red). The dashed curve depicts the analytical solution for the active
particles in a channel with a constant cross-section.

Fig. 6 displays the dependence of $E_{-}$ (recall that $E_{+}=1-E_{-}$) on the
initial position $x_{0}$ for fixed $\text{Pe}=0.1$, $\Gamma=3/4$ (Janus
colloid case) and varying $\Delta A$. In this figure, circles present the
results obtained numerically for active swimmers, while solid curves - an
analytical solution for passive particles. Fig. 6(a) demonstrates that for
positive $\Delta A$ (a bottleneck in the center), the behavior of active
swimmers is very different from that of passive particles, and depends
strongly on the values of both $\Delta A$ and Pe. For sufficiently large
values of the parameter $\beta\Delta A/\text{Pe}$ (red circles), which limit
is realized either for large values of the barrier $\beta\Delta A$ or for
small Pe, the SP $E_{-}$ for the active particles in channels with a varying
cross-section exhibits a characteristic $S$-shaped form with a very steep
dependence on $x_{0}$ close to the center of the channel. This implies, that
once $x_{0}$ only slightly exceeds (or is less than) $0$, the particle is
(almost) certain to reach the closest extremity without ever reaching the
other one. Numerically, the value of $E_{-}$ appears to be very close to the
corresponding result for passive particles, which is, of course, not a
counter-intuitive behavior. In contrast, for small $\beta\Delta A/\mathrm{Pe}$
(blue circles), i.e., either for large values of Pe or for small values of the
entropic barrier, $E_{-}$ appears to be very close to our analytical
prediction obtained for active swimmers (dashed curve) moving in a constant
cross-section channels which also physically quite plausible. Since the
behavior in these limiting cases is very different, in general, there is a
strong dependence of $E_{-}$ on $\beta\Delta A/\mathrm{Pe}$ for the
intermediate values of the system’s parameters. We can therefore expect that
particles with different activities can behave very differently in such a
channel, especially if they start in the vicinity of the bottleneck. For
negative values of $\Delta A$ (entropic repulsion from the extremities), which
case is presented in Fig. 7(b), $E_{-}$ depends weakly on the starting point,
which resembles the behavior observed earlier for passive particles.

Figure 7: Splitting probabilities $E_{\pm}$ for $x_{0}=0.1L$ (panel (a)) and
$x_{0}=0.4L$ (panel (b)) as function of $\beta\Delta A$ for active particles
with $\Gamma=0.1$ and $\text{Pe}=1$ (blue), $3$ (cyan) and $5$ (red) in a
channel with $L=10R$. The black curves depict the behavior in the passive
case, i.e., for $\text{Pe}=0$.

Further on, to highlight the difference between the passive and the active
cases, in Fig. 7 we plot $E_{\pm}$ as functions of the barrier $\Delta A$ in
situations when a particle (passive or active) starts either close to the
middle of the channel, at $x_{0}=0.1L$, or close to the right extremity of the
channel, $x_{0}=0.4L$. We observe that in the active particles case the
behavior of the SPs is indeed very different from that of a passive one,
especially when the starting point is close to either of the extremities.
While in the situation when the starting point is close to the middle of the
channel (i.e., for $x_{0}=0.1L$) all curves look very similar with the only
difference that for $\text{Pe}>0$ they become progressively (with an increase
of Pe) more shifted to the larger values of the barrier $\beta\Delta A$, in
case when $x_{0}=0.4L$ a remarkable non-monotonous behavior as function of
$\beta\Delta A$ emerges for active particles, meaning that at some corrugation
profiles the active particles more readily reach the right extremity.
Interestingly enough, the position of the local maximum (minimum) of $E_{+}$
($E_{-}$) corresponds to $-\beta\Delta A\simeq\text{Pe}/2$, i.e., the entropic
force compensates the propulsive one. For passive particles $E_{\pm}$ are
monotonously increasing functions of $\beta\Delta A$.

##  I Conclusion

To conclude, we discussed here the behavior of the splitting probabilities as
functions of system’s parameters for dynamics in three-dimensional axially-
symmetric channels with varying cross-sections. In a standard notation, the
splitting probability is the probability that either of the extremities is
reached before the opposite one. In regard to the dynamical behavior, we
focused on two models of random transport - standard Brownian motion and
active Brownian motion.

Our analytical approach was based on a suitably generalized Fick-Jacobs
approximation, which reduces an original three-dimensional model to a one-
dimensional system with a spatially-varying effective potential defined as the
local free energy. For standard diffusion, the latter model is exactly
solvable and we derive explicit expressions for the splitting probabilities in
arbitrarily shaped channels. For active Brownian motion the dynamical
equations are more complicated and we find an analytical solution for constant
cross-sections only. For more general case of a spatially-varying cross-
section we resort to a numerical analysis.

Our analysis reveals some similarities in the behavior of passive and active
Brownian motions and also some distinctly different features, which can be
seen as fingerprints of the activity of particles. A more detailed discussion
of the behavior in channels with a more complicated geometry and more
elaborate analytical analysis will be presented elsewhere.

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## Appendix

##  II Active particles

Here we address the problem of the splitting probability of active colloids
confined to move in 1D. Indeed, the colloids can be in two possible states:
moving left or moving right. Accordingly, the dynamics is controlled by the
following equations:

$\displaystyle\dot{\rho}_{\uparrow}(x,t)=D\partial_{x}\left[\partial_{x}\rho_{\uparrow}(x,t)+\beta\rho_{\uparrow}(x,t)\partial_{x}W(x)+\beta\rho_{\uparrow}(x,t)F_{act}\right]-\alpha\rho_{\uparrow}(x,t)+\alpha\rho_{\downarrow}(x,t)$
(S1)
$\displaystyle\dot{\rho}_{\downarrow}(x,t)=D\partial_{x}\left[\partial_{x}\rho_{\downarrow}(x,t)+\beta\rho_{\downarrow}(x,t)\partial_{x}W(x)-\beta\rho_{\downarrow}(x,t)F_{act}\right]+\alpha\rho_{\uparrow}(x,t)-\alpha\rho_{\downarrow}(x,t)$
(S2)

where $F_{act}$ accounts for the active motion and with boundary conditions

$\displaystyle\rho_{\uparrow}(x=x_{0})$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}}{2}$ (S3)
$\displaystyle\rho_{\downarrow}(x=x_{0})$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}}{2}$ (S4)
$\displaystyle\rho_{\uparrow}(x=\pm\frac{L}{2})$ $\displaystyle=$
$\displaystyle 0$ (S5) $\displaystyle\rho_{\downarrow}(x=\pm\frac{L}{2})$
$\displaystyle=$ $\displaystyle 0$ (S6)

In order to fulfill the above mentioned boundary conditions we split the
problem into the left problem and the right problem. Using

$\displaystyle\rho(x,t)$
$\displaystyle=\rho_{\uparrow}(x,t)+\rho_{\downarrow}(x,t)$ (S7)
$\displaystyle\phi(x,t)$
$\displaystyle=\rho_{\uparrow}(x,t)-\rho_{\downarrow}(x,t)$ (S8)

we get

$\displaystyle\dot{\rho}(x,t)$
$\displaystyle=D\partial_{x}\left[\partial_{x}\rho(x,t)+\beta\rho(x,t)\partial_{x}W(x)+\beta\phi(x,t)F_{act}\right]$
(S9) $\displaystyle\dot{\phi}(x,t)$
$\displaystyle=D\partial_{x}\left[\partial_{x}\phi(x,t)+\beta\phi(x,t)\partial_{x}W(x)+\beta\rho(x,t)F_{act}\right]-2\alpha\phi(x)$
(S10)

At steady state we get

$\displaystyle\partial_{x}\rho(x,t)+\beta\rho(x,t)\partial_{x}W(x)+\beta\phi(x,t)F_{act}$
$\displaystyle=-J/D$ (S11) $\displaystyle
D\partial_{x}\left[\partial_{x}\phi(x,t)+\beta\phi(x,t)\partial_{x}W(x)+\beta\rho(x,t)F_{act}\right]$
$\displaystyle=2\alpha\phi(x)$ (S12)

In the case in which $\partial_{x}W(x)=0$ we get (see also EPL 134 (2),
20002):

$\displaystyle\partial_{x}\rho(x,t)$
$\displaystyle=-J/D-\beta\phi(x,t)F_{act}$ (S13)
$\displaystyle\partial_{x}^{2}\phi(x,t)$ $\displaystyle=\beta
F_{act}J/D+\left(\left(\beta F_{act}\right)^{2}+2\alpha/D\right)\phi(x)$ (S14)

that should be solved with the boundary conditions

$\displaystyle\rho(x=x_{0})=\rho_{0},$ $\displaystyle\rho(x=\pm\frac{L}{2})=0$
(S15) $\displaystyle\phi(x=x_{0})=0,$ $\displaystyle\phi(x=\pm\frac{L}{2})=0$
(S16)

The general solution of $\phi$ reads

$\phi(x)=Ae^{kx}+Be^{-kx}-\frac{\beta F_{act}J}{Dk^{2}}$ (S17)

with

$k=\sqrt{\left(\beta
F_{act}\right)^{2}+2\alpha/D}=\frac{\sqrt{\text{Pe}^{2}+2\Gamma}}{R}$ (S18)

where we introduced the Péclet number and the dimensionless hopping rate

$\displaystyle\text{Pe}=\beta
F_{act}R=\frac{F_{act}}{\gamma}\frac{R}{D}=\frac{v_{0}R}{D}\,,\quad\Gamma=\frac{\alpha
R^{2}}{D}$ (S19)

and we used the Stokes-Einstein relations $\beta D=1/\gamma$ with $\gamma$ the
friction coefficient of the particle.

### Solution of the left problem

Here we have to solve

$\displaystyle\partial_{x}\rho_{-}(x,t)$
$\displaystyle=-J_{-}/D-\beta\phi(x,t)F_{act}$ (S20)
$\displaystyle\partial_{x}^{2}\phi_{-}(x,t)$ $\displaystyle=\beta
F_{act}J_{-}/D+\left(\left(\beta F_{act}\right)^{2}+2\alpha/D\right)\phi(x)$
(S21)

with the boundary conditions

$\displaystyle\rho_{-}(x=x_{0})=\rho_{0},$
$\displaystyle\rho_{-}(x=-\frac{L}{2})=0$ (S22)
$\displaystyle\phi_{-}(x=x_{0})=0,$ $\displaystyle\phi_{-}(x=-\frac{L}{2})=0$
(S23)

Hence we have

$\displaystyle A_{-}$ $\displaystyle=e^{-kx_{0}}\left[\frac{\beta
F_{act}J_{-}}{Dk^{2}}-B_{-}e^{-kx_{0}}\right]$ (S24) $\displaystyle B_{-}$
$\displaystyle=\frac{\beta
F_{act}J_{-}}{Dk^{2}}\dfrac{\left[1-e^{-k(x_{0}+L/2)}\right]}{e^{kL/2}-e^{-k(2Y+L/2)}}$
(S25)

from which we have

$\displaystyle A_{-}$ $\displaystyle=\frac{\beta
F_{act}J_{-}}{Dk^{2}}\left[e^{-kx_{0}}-\dfrac{1-e^{-k(x_{0}+L/2)}}{e^{kL/2}-e^{-k(2Y+L/2)}}e^{-2kY}\right]=\frac{\beta
F_{act}J_{-}}{Dk^{2}}\mathcal{A}_{-}$ (S26) $\displaystyle B_{-}$
$\displaystyle=\frac{\beta
F_{act}J_{-}}{Dk^{2}}\left[\dfrac{1-e^{-k(x_{0}+L/2)}}{e^{kL/2}-e^{-k(2Y+L/2)}}\right]=\frac{\beta
F_{act}J_{-}}{Dk^{2}}\mathcal{B}_{-}$ (S27)

The general solution for $\rho$ reads

$\rho_{-}(x)=-\frac{J_{-}}{D}x-\beta F_{act}\int_{-L/2}^{x}\phi(z)dz+\Pi_{-}$
(S28)

Substituting the formulas for $A_{-}$ and $B_{-}$ into the equation for $\rho$
and imposing the boundary conditions we get

$\displaystyle\Pi_{-}$ $\displaystyle=-\frac{J_{-}}{D}\frac{L}{2}$ (S29)
$\displaystyle\frac{J_{-}}{D}$
$\displaystyle=\rho_{0}\left[\left(\frac{\beta^{2}F_{act}^{2}}{k^{2}}-1\right)\left(x_{0}+\frac{L}{2}\right)-\frac{\beta^{2}F_{act}^{2}}{k^{2}}\left(\frac{\mathcal{A}_{L}}{k}\left(e^{kx_{0}}-e^{-kL/2}\right)+\frac{\mathcal{B}_{L}}{k}\left(e^{kL/2}-e^{-kx_{0}}\right)\right)\right]^{-1}$
(S30)

where we used

$\int_{-L/2}^{x_{0}}\phi(x)dx=\frac{1}{k}\left[\mathcal{A}_{-}\left(e^{kx_{0}}-e^{-kL/2}\right)+\mathcal{B}_{-}\left(e^{kL/2}-e^{-kx_{0}}\right)\right]-\frac{\beta
F_{act}J}{Dk^{2}}\left(x_{0}+\frac{L}{2}\right)$ (S31)

### Solution of the right problem

Here we have to solve

$\displaystyle\partial_{x}\rho_{+}(x,t)$
$\displaystyle=-\frac{J_{+}}{D}-\beta\phi(x,t)F_{act}$ (S32)
$\displaystyle\partial_{x}^{2}\phi_{+}(x,t)$ $\displaystyle=\frac{\beta
F_{act}J_{+}}{D}+\left(\left(\beta F_{act}\right)^{2}+2\alpha/D\right)\phi(x)$
(S33)

with the boundary conditions

$\displaystyle\rho_{+}(x=x_{0})=\rho_{0},$
$\displaystyle\rho_{+}(x=\frac{L}{2})=0,$ (S34)
$\displaystyle\phi_{+}(x=x_{0})=0,$ $\displaystyle\phi_{+}(x=\frac{L}{2})=0.$
(S35)

Hence we have

$\displaystyle A_{+}$ $\displaystyle=e^{-kx_{0}}\left[\frac{\beta
F_{act}J_{+}}{Dk^{2}}-B_{+}e^{-kx_{0}}\right]$ (S36) $\displaystyle B_{+}$
$\displaystyle=\frac{\beta
F_{act}J_{+}}{Dk^{2}}\dfrac{\left[1-e^{-k(x_{0}-L/2)}\right]}{e^{-kL/2}-e^{-k(2Y-L/2)}}$
(S37)

from which we have

$\displaystyle A_{+}$ $\displaystyle=\frac{\beta
F_{act}J_{+}}{Dk^{2}}\left[e^{-kx_{0}}-\dfrac{1-e^{-k(x_{0}-L/2)}}{e^{-kL/2}-e^{-k(2Y-L/2)}}e^{-2kY}\right]=\frac{\beta
F_{act}J_{+}}{Dk^{2}}\mathcal{A}_{+}$ (S38) $\displaystyle B_{+}$
$\displaystyle=\frac{\beta
F_{act}J_{+}}{Dk^{2}}\left[\dfrac{1-e^{-k(x_{0}-L/2)}}{e^{-kL/2}-e^{-k(2Y-L/2)}}\right]=\frac{\beta
F_{act}J_{+}}{Dk^{2}}\mathcal{B}_{+}$ (S39)

The general solution for $\rho$ reads

$\rho_{+}(x)=-\frac{J_{+}}{D}x+\beta
F_{act}\int_{x}^{L/2}\phi_{+}(z)dz+\Pi_{+}$ (S40)

Substituting the formulas for $A_{-}$ and $B_{-}$ into the equation for $\rho$
and imposing the boundary conditions we get

$\displaystyle\Pi_{+}$ $\displaystyle=\frac{J_{+}}{D}\frac{L}{2}$ (S41)
$\displaystyle\frac{J_{+}}{D}$
$\displaystyle=\rho_{0}\left[\left(\frac{\beta^{2}F_{act}^{2}}{k^{2}}-1\right)\left(x_{0}-\frac{L}{2}\right)+\frac{\beta^{2}F_{act}^{2}}{k^{2}}\left(\frac{\mathcal{A}_{+}}{k}\left(e^{kL/2}-e^{kx_{0}}\right)+\frac{\mathcal{B}_{+}}{k}\left(e^{-kx_{0}}-e^{-kL/2}\right)\right)\right]^{-1}$
(S42)

where we used

$\int_{x_{0}}^{L/2}\phi(x)dx=-\frac{1}{k}\left[\mathcal{A}_{+}\left(e^{kx_{0}}-e^{kL/2}\right)+\mathcal{B}_{+}\left(e^{-kL/2}-e^{-kx_{0}}\right)\right]+\frac{\beta
F_{act}J}{Dk^{2}}\left(x_{0}-\frac{L}{2}\right)$ (S43)

From $J_{-}$ and $J_{+}$ it is straightforward to define $\tau_{\pm}$

$\displaystyle\tau_{-}=\frac{\rho_{0}L}{|J_{-}|}$ (S44)
$\displaystyle\tau_{+}=\frac{\rho_{0}L}{|J_{+}|}$ (S45)

and hence the splitting probabilities (see Eq. in the main text).

###  II.1 Numerical solution

For arbitrary $A(x)$ Eqs. (S1),(S2) can be solved numerically. To do so, we
rewrite them in form of a system of first-order differential equations:

$\mathbf{Y}^{\prime}=\mathbf{M}(x)\mathbf{Y},$

where $\mathbf{Y}=(\rho,\rho^{\prime},\phi,\phi^{\prime})$ and

$\mathbf{M}(x)=\left(\begin{array}[]{llll}0&1&0&0\\\
-A^{\prime\prime}(x)&-A^{\prime}(x)&-\dfrac{\mathrm{Pe}L}{R}&0\\\ 0&0&0&1\\\
0&-\dfrac{\mathrm{Pe}L}{R}&(-2L^{2}/R^{2})\Gamma&-A^{\prime}(x)\end{array}\right).$
(S46)

With boundary conditions for the left and the right problem:

$Y_{1}(\pm L/2)=0,\quad Y_{3}(\pm L/2)=0\quad Y_{1}(x_{0})=1,\quad
Y_{3}(x_{0})=0.$

This system has been solved numerically using the standard Python library
`scipy`. All calculations have been performed on a grid with $N_{p}=500$
nodes. The numerical solution showed good agreement with analytical solution
for the case of passive particles in condfining potential and active particles
in a flat channel (see Fig. S1).

Figure S1:  Numerical (symbols) and analytical (solid curves) solution for
$E_{-}$ as function of $x_{0}$ for (a) passive particles and varying
$\beta\Delta A=-10,-5,5,10$ and (b) active particles with $L=10R$,
$\mathrm{Pe}=0.5$ and $\Gamma=0.1$.