††thanks: Corresponding author††thanks<EMAIL_ADDRESS>
# Resetting mediated navigation of active Brownian searcher in a homogeneous
topography

Gourab K. Sar  Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 B. T. Road, Kolkata 700108, India  Arnob Ray  Physics and
Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road,
Kolkata 700108, India  Dibakar Ghosh  Physics and Applied Mathematics Unit,
Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
Chittaranjan Hens  Center for Computational Natural Sciences and
Bioinformatics, International Institute of Information Technology, Gachibowli,
Hyderabad 500 032, India  Arnab Pal  The Institute of Mathematical Sciences,
IV Cross Road, CIT Campus, Taramani, Chennai 600 113, Tamil Nadu, India.  Homi
Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai
400094, India.

###### Abstract

Designing navigation strategies for search time optimization remains of
interest in various interdisciplinary branches in science. In here, we focus
on microscopic self-propelled searchers namely active Brownian walkers in
noisy and confined environment which are mediated by one such autonomous
strategy namely resetting. As such, resetting stops the motion and compels the
walkers to restart from the initial configuration intermittently according to
an external timer that do not require control by the walkers. In particular,
the resetting coordinates are either quenched (fixed) or annealed
(fluctuating) over the entire topography. Although the strategy relies upon
simple rules, it shows a significant ramification on the search time
statistics in contrast to the original search. We show that the resetting
driven protocols mitigate the performance of these active searchers based,
robustly, on the inherent search time fluctuations. Notably, for the annealed
condition, resetting is always found to expedite the search process. These
features, as well as their applicability to more general optimization problems
starting from queuing systems, computer science to living systems, make
resetting based strategies universally promising.

††preprint: AIP/123-QED

##  I Introduction

Active particles operate far from thermodynamic equilibrium. These
microswimmers propel themselves with directed motion and thus drive themselves
out of equilibrium Toner, Tu, and Ramaswamy (2005); Ramaswamy (2010);
Romanczuk  _et al._ (2012). In other words, a constant consumption and
dissipation of energy results in non-equilibrium activity – broken detailed
balance being a hallmark property of such motion Gnesotto  _et al._ (2018).
Several living entities such as E. coli bacteria are known to perform such
active motion Berg (2008, 2018). In addition, engineered materials such as the
Janus particles also work as active autonomous agents to locate and deliver
microscopic cargos Jiang, Yoshinaga, and Sano (2010). Moreover, active systems
can also provide cues to form spatial organizations, self assemblies through
their non-linear interaction, collective motion Martín-Gómez  _et al._ (2018),
structural organization and pattern formation Ramaswamy (2010); Gompper  _et
al._ (2020); Martínez-Calvo, Trenado-Yuste, and Datta (2021); Mijalkov and
Volpe (2013); Bechinger  _et al._ (2016); Volpe  _et al._ (2011). No wonder
why active systems have become a focal point of recent interests in physics,
chemistry and biology.

There has been quite a growing interest in active search processes which have
important technological and medical applications, such as cargo transport
Ross, Ali, and Warshaw (2008), targeted drug delivery Ghosh  _et al._ (2020)
or identification of infectious protein along a long DNA at the microscopic
level Bénichou  _et al._ (2011). Other potential applications of active search
processes span in biology, computer science, ecology, soft matter and many
other fields Martínez-Calvo, Trenado-Yuste, and Datta (2021); Bechinger  _et
al._ (2016); Wang  _et al._ (2017); Malakar  _et al._ (2018); Angelani, Di
Leonardo, and Paoluzzi (2014). With the theoretical and experimental advances
side by side on engineering microswimmers, it is only natural to ask how
efficient these microscopic devices are to deliver molecular cargo at the
desired locations. However, there is a cost in such designing which we assign
to be the mean search time to a given target(s) starting from some specific
location. Hence, the problem boils down to finding the most efficient route
towards a desired target resulting in the lowest search time. Nonetheless,
there is no generic pattern on adapting a universal search strategy for all,
but drawing insights from various class of search processes (from micro to
macro), maybe a class of optimal strategies could be idealized.

Over the years, many realistic models have been proposed to describe and
analyze how a searcher spans through the search space looking for targets
Bénichou  _et al._ (2011); Condamin  _et al._ (2007); Bénichou  _et al._
(2005, 2010); Bray, Majumdar, and Schehr (2013). These include Brownian
motion, and long range searches such as Lévy flights and Lévy walks. In
particular, Lévy statistics, among others models, have been successfully used
to describe the emergence of optimal search strategies in natural systems at
different length scales, from molecular entities to foraging animals to human
motion patterns Viswanathan  _et al._ (2011); Rhee  _et al._ (2011);
Zaburdaev, Denisov, and Klafter (2015). The active search can well be
described within the Lévy strategies since they combine random jumps when the
searchers are non-reactive followed by local diffusive jumps when the
searchers explore for targets Volpe and Volpe (2017). This is also called
intermittent search strategy Bénichou  _et al._ (2011). One notable result in
this regard is that the most efficient strategy is a Brownian search (which is
a limiting case) while Lévy strategy seems to be optimal for sparse targets
when resources are plentiful Lomholt  _et al._ (2008); Palyulin, Chechkin, and
Metzler (2014).

A similar kind of intermittent strategy namely resetting has been found to be
extremely efficient in optimizing search in many complex processes Evans and
Majumdar (2011a, b); Evans, Majumdar, and Schehr (2020); Gupta and Jayannavar
(2022); Kusmierz  _et al._ (2014); Reuveni (2016); Pal and Reuveni (2017);
Pal, Eliazar, and Reuveni (2019); De Bruyne, Randon-Furling, and Redner
(2020); Pal, Kundu, and Evans (2016); Chechkin and Sokolov (2018); Pal and
Prasad (2019a); Bhat, De Bacco, and Redner (2016); Pal and Prasad (2019b);
Bressloff (2020); Tal-Friedman  _et al._ (2020); Montanari and Zecchina
(2002); Luby, Sinclair, and Zuckerman (1993); Reuveni, Urbakh, and Klafter
(2014); Ray (2020); Ray  _et al._ (2021); Roldán  _et al._ (2016); Bonomo,
Pal, and Reuveni (2022). As such, the searcher looks for the targets utilizing
its own active motion and if successful, the search is complete. However, the
motion can often be interrupted (i.e., stopped at random times) and the
searcher is expected to return to its initial location and restart/renew the
motion. Thus resetting mixes between the exploration and return phases. Such
motion is also prevalent in nature since foraging animals return to their den
to rest, bees return to their hive and drones return to their docking station
to recharge or refuel Pal, Kuśmierz, and Reuveni (2020). Considerable progress
has been made to understand the effect of variety of ways in which resetting
mechanisms and underlying dynamics combine Campos and Méndez (2015); Kuśmierz
and Gudowska-Nowak (2015); Singh, Metzler, and Sandev (2020); Pal (2015); Ray,
Mondal, and Reuveni (2019); Huang and Chen (2021); González, Riascos, and
Boyer (2021). To this end, the universal framework of first passage under
restart and its applicability to diffusive processes under resetting is
noteworthy Reuveni (2016); Pal and Reuveni (2017); Pal, Eliazar, and Reuveni
(2019); Bonomo and Pal (2021).

In this paper, our central aim is to understand the effects of resetting based
protocols in active search processes. Furthermore, we assume that the active
micro-swimmers conduct their search in a confining topography in the presence
of target. We explore the possibility of designing different resetting
protocols – one where the searcher is compelled to return to a fixed location
(quenched condition) and the other one where the searcher is reset to a random
location (annealed condition). As will be shown here, both the geometry of the
arena and choice of the resetting protocols play a crucial role in determining
the searcher’s motion and in effect to the search efficiency. Our study
reveals that resetting can indeed reduce the search time compared to that of
the original search process. Secondly, we observe that resetting mechanism
works in advantage when the fluctuations in the search time for the underlying
active process is large – a robust feature that is observed quite ubiquitously
(see also Pal, Kostinski, and Reuveni (2022)). We distill this core idea for
the randomized/annealed initial conditions where resetting driven search
always works efficiently.

We start by looking into the navigation of active Brownian particles (ABP) or
micro-swimmers in a two dimensional finite domain in the presence of target
(Sec. II). We first analyze the search properties of this minimal model in
Sec. III — fluctuations and densities of the first passage time are computed.
Next, we introduce the autonomous resetting strategies and study the motion of
ABPs in Sec. IV. In particular, we design two different resetting protocols
namely annealed and quenched that correlate with the active motion. Using
extensive Brownian dynamics simulations, we demonstrate how the new search
protocols enhance performances compared to that obtained from the original
search process. Lastly in Sec. V, we provide analytical support and underpin
the core principle that lies beneath the robustness of resetting-mediated
strategies. We conclude with future perspectives on this useful intermittent
search strategy.

##  II Model Description

The motion of an ABP in a two-dimensional topography can be modeled by
stochastic Langevin equations namely Romanczuk  _et al._ (2012); Volpe, Gigan,
and Volpe (2014); Basu  _et al._ (2018)

$\displaystyle\dot{x}(t)$ $\displaystyle=v_{0}\cos(\theta(t)),$ (1)
$\displaystyle\dot{y}(t)$ $\displaystyle=v_{0}\sin(\theta(t)),$ (2)
$\displaystyle\dot{\theta}(t)$ $\displaystyle=\sqrt{2D_{R}}\eta(t),$ (3)

where $\eta(t)$ represents white noise with $\langle\eta(t)\rangle=0$,
$\langle\eta(t)\eta(t^{\prime})\rangle=\delta(t-t^{\prime})$ and $v_{0}$
represents non-zero constant speed of the particle. Along with the position
coordinates $(x,y)$, the particle undergoes a rotational Brownian motion in
its orientation $\theta(t)$, controlled by the rotational diffusion constant
$D_{R}$. The "active" nature of the particle is signified by the coupling
between position and orientation of the particle. Such models for ABP-s have
been used in a broad spectrum of examples including collective motion such as
swarming Topaz and Bertozzi (2004); Strefler, Erdmann, and Schimansky-Geier
(2008), flocking Toner, Tu, and Ramaswamy (2005), and clustering Slowman,
Evans, and Blythe (2016); Palacci  _et al._ (2013). Other examples include E.
coli bacteria tracing the nutrients Viswanathan  _et al._ (2011), or sperms
going towards the egg Friedrich and Jülicher (2008). We refer to these
excellent reviews for an overview of active transport in complex environments
Bechinger  _et al._ (2016); Martínez-Calvo, Trenado-Yuste, and Datta (2021);
Ramaswamy (2010).

In this article, our focus is oriented towards the active search process.
Finding optimal strategy for active particles has been a long standing problem
since it is crucial to adopt the right search protocol in complex and crowded
environments with scarce resources Bénichou  _et al._ (2011); Volpe and Volpe
(2017). Interestingly, the topography of the searching process is a
determining factor for finding the search time since boundaries, barriers, and
obstacles play a crucial role in determining the searcher’s motion Bechinger
_et al._ (2016); Volpe and Volpe (2017). In contrast to a Lévy walk searcher
as considered in Volpe and Volpe (2017), we study motion of an ABP that
searches for a target within a homogeneous topography in a confined region in
two-dimensional space (see Fig. 1(a)).

Figure 1: Typical trajectory of an active Brownian searcher in a confined
geometry. Panel (a): The ABP starts its search process from the source $\rm O$
(black dot) and the process is completed once the target $\rm S$ (red dot) is
found. Here, we portray two such independent paths (indicated by blue and
orange curves) traversed by the ABP. Panel (b): When the ABP encounters the
boundary on its way, it gets reflected inside the bounded region (for details,
see Sec. II.1). At the time of reflection, the orientation (blue arrow) of the
particle changes and gets aligned to the direction of reflection.

###  II.1 The topography and boundary conditions

We assume that the ABP is confined in a two-dimensional (2D) square region
$\mathcal{B}=\\{(x,y)\in\mathbb{R}^{2}:(x,y)\in[-a,a]\times[-a,a]\\}$. The
targets ($\rm S$) are assumed to be inside this region and not at the
boundaries. Furthermore, the boundaries are considered to be reflective so
that there is no loss of probability. In Fig. 1(a), we have shown two
independent search trials (marked by blue and orange curves) performed by the
ABP starting from the source $\rm O$. The reflecting boundary condition can be
understood in the following way (similar to Volpe, Gigan, and Volpe (2014)).
Let us imagine that the searcher is at point $\rm P$ at some time instant $t$
inside the boundary and at the next time step $t+\Delta t$, it leaps over the
boundary and arrives at a point $\rm Q$ outside the boundary. In such case, we
construct a mirror image $\rm Q^{\prime}$ of $\rm Q$ across that specified
boundary so that it is inside the region. Finally, we update the motion of the
ABP by assigning its location to $\rm Q^{\prime}$ at time $t+\Delta t$. In
parallel, we also assume that the orientation of the particle has changed
along the direction of reflection, which is shown by the blue arrow in Fig. 1
(b).

###  II.2 Target locations

As mentioned earlier, the topography is considered to be a square with
diagonal $5\sqrt{2}$. We furthermore consider one target at a time and measure
the search time. The target is located at different coordinates $\rm A(1,1)$,
$\rm B(-2,2)$, $\rm C(-3,-3)$, and $\rm D(4,-4)$ respectively in each such
realization. This is to distribute the target over the four quadrants in the
2D plane placed at different distances from the origin. However, due to the
homogeneous topography, the search time remains same along the circle centered
at the origin and spanned by the target.

###  II.3 Initial conditions

In this set-up, we have considered two types of initial conditions. Denoting
$x_{i}\equiv x(0),~{}y_{i}\equiv y(0)$ and $\theta_{i}\equiv\theta(0)$ as the
initial conditions, we consider the following.

  * •

IC1 (quenched initial condition): We consider fixed position coordinates
$x_{i}=0,y_{i}=0$, and random initial orientation $\theta_{i}\in U[0,2\pi]$,
where $U[m,n]$ is a uniform random variable between $m$ and $n$.

  * •

IC2 (annealed initial condition): We consider random position coordinates
chosen uniformly from the bounded region $\mathcal{B}$ and the orientation at
the initial time is also chosen randomly as before.

Figure 2: First passage time (FPT) densities of the reset free search process
and the dependence of MFPT ($\langle T\rangle$) on the distance of the target
from the origin. Panel (a): First passage time densities for the targets $\rm
A(1,1)$ (blue), and $\rm D(4,-4)$ (red). Here, we have considered the quenched
initial condition where the ABP always starts its search from the origin. The
respective mean and fluctuation values are shown in the inset which indicate
that $\rm CV>1$ for target $\rm A$, and $\rm CV<1$ for target $\rm D$ (see
Table 1 for details). Panel (b): FPT densities for the same targets but for
the annealed initial condition when the ABP starts its search from random
positions inside the bounded region. In this case $\rm CV$ is always greater
than 1 (see inset and Table 1). Panel (c): $\rm MFPT$ ($\langle T\rangle$) as
a function of distance from the origin ($d$) under IC1. $\langle T\rangle$
increases when the distance of the target from origin increases. A circle of
radius $d$ centered at the origin is drawn for each of the distances. Ten
points on the circle’s circumference are chosen which are equally spaced and
considered as targets. Then the average of all realizations ($10$ $\times$
$10^{6}$) is used to calculate $\langle T\rangle$.

##  III First passage time statistics of ABP

Before studying the effects of resetting on the first passage statistics, it
is first instructive to delve deeper into these properties without resetting.
Thus, starting from a source, the ABP diffuses inside the bounded region
$\mathcal{B}$ until it encounters the target for the first time, thus marking
the termination of the search process. The time required for the search
completion is random and is often known as the first passage time. Let, ${\rm
S}\equiv(x_{\rm S},y_{\rm S})$ be the position vector of the target and
$X(t)\equiv(x(t),y(t))$ be the position vector of the ABP at time $t$. The
target is said to be encountered by the ABP when $||X(t)-\rm S||<\epsilon$,
for some predefined threshold $\epsilon$ ($>0$), which is sufficiently small.
We denote this first passage time ($\rm FPT$) by $T$, which is defined as

$T=\inf\\{t>0\;:\;||X(t)-\rm S||<\epsilon\\}.$ (4)

A wide array of applications in physics, chemistry, biology and other
interdisciplinary fields have turned first passage time processes into a long-
standing focal point of scientific interest. Fundamental questions include
computation of first passage time density, its moments and designing various
efficient strategies to make it optimal. We refer to these extensive reviews
Bénichou  _et al._ (2011); Bray, Majumdar, and Schehr (2013); Redner (2001)
for a detailed overview of the subject.

In what follows we use $f_{Z}(t)$, $\langle Z^{n}\rangle$, and $\sigma(Z)$ to
denote, respectively, the probability density function, moment of $n$-th
order, and standard deviation of a non-negative random variable $Z$ (e.g., the
first passage time $T$ etc.). The corresponding coefficient of variation ($\rm
CV$) for the first passage time is defined as

$\mbox{CV}=\dfrac{\sigma(T)}{\langle T\rangle}.$ (5)

Coefficient of variation is a widely used measure of statistical dispersion
that infers how some random data points are dispersed around their mean. In
other words, it tells us how broad (large fluctuations) or narrow (small
fluctuations) a distribution is around their mean.

As our study involves two different initial conditions, we investigate these
two scenarios for the searching mechanism of the ABP in the following. In both
cases, we carry out a comprehensive study for the mean first passage time
(MFPT) and fluctuations of FPTs and support the results qualitatively. The
results for the FPTs (by performing $10^{6}$ trials) for two different initial
conditions are listed in Table 1.

Table 1: Numerically calculated values of mean first passage time ($\langle T\rangle$), standard deviation ($\sigma$), and coefficient of variation ($\rm CV$) for four targets $\rm A$, $\rm B$, $\rm C$, and $\rm D$. Here we set $D_{R}=1$, $v_{0}=1$. Target | IC1 | IC2
---|---|---
|  $\langle T\rangle$ |  $\sigma$ |  $\rm CV$ |  $\langle T\rangle$ |  $\sigma$ |  $\rm CV$
$\rm A(1,1)$ |  495.75 |  518.48 |  1.0458 |  517.91 |  518.13 |  1.0004
$\rm B(-2,2)$ |  530.79 |  532.82 |  1.0038 |  533.45 |  535.43 |  1.0037
$\rm C(-3,-3)$ |  568.25 |  566.57 |  0.997 |  561.99 |  566.16 |  1.0074
$\rm D(4,-4)$ |  635.07 |  627.42 |  0.9879 |  623.73 |  628.95 |  1.0084

###  III.1 Searching with IC1 (quenched initial condition)

We first consider the case when the ABP starts from the quenched initial
conditions (IC1) and evolves until it encounters the target. While both
$\langle T\rangle$ and $\sigma$ steadily grow as we increase $d$, the distance
between the target and the origin, we observe a reduction in the fluctuations
$\rm CV$ (see Table 1). The probability density functions $f_{T}(t)$ for the
first passage time are numerically computed and illustrated for two targets
$\rm A$ (blue) and $\rm D$ (red) [Fig. 2 (a)]. Since the search process is
taking place in a finite domain, the first passage time densities, at large
time, pertain to exponential tails in contrast to the algebraic tails which
usually appear for the search in the unbounded domain Redner (2001); Bray,
Majumdar, and Schehr (2013). We refer to Fig. 3 where we have compared the
simulation results for the first passage times against the theoretical form
$f_{T}(t)=1/\langle T\rangle e^{-t/\langle T\rangle}$ at large times. Owing to
the exponential tails, all the moments of the first passage time remain
finite. For brevity, the respective values of $\langle T\rangle$ and $\sigma$
are provided in the Table. It should be noted that the $CV=\sigma/\langle
T\rangle$ may seem exactly unity owing to the exponential distribution, but
that will be an incorrect assessment since it will leave out the short and
intermediate time contributions which are not exponential.

Clearly, the ABP takes longer time to find the targets that are far away from
the source (here $\rm D$) than the ones those are closer to the source ($\rm
A$), i.e., $\langle T\rangle_{\rm D}>\langle T\rangle_{\rm A}$ [see Fig. 2
(c)]. However, the increasing trend of fluctuations are comparatively reduced
for a long-range target, and CV is dropped below 1 [Table 1]. The average of
the $\langle T\rangle$, is plotted against the distance $d$ shown in Fig. 2
(c). The graphic also shows that when the distance between the target and the
fixed source (origin) grows, the ABP takes longer to reach the target.

Figure 3: Nature of the underlying first passage time densities. Statistical
distributions for the underlying first passage time process for individual
target show exponential tails in the asymptotic limit. This observation is
found to be robust for both the initial conditions. Blue solid line is the
theoretical fit for target $\rm A$ under IC1 where blue stars represent
numerical data of first passage times. Similar behavior is observed for target
$\rm D$ under IC2 (in red) with a different slope (determined by $\langle
T\rangle$).

###  III.2 Searching with IC2 (annealed initial condition)

In this subsection, we focus on the motion of the ABP under the annealed
random initial conditions. The numerical procedure remains the same and we
list the results for $\langle T\rangle$, $\sigma$, and $\rm{CV}$ in Table 1.
Similar to the previous case, we find that both $\langle T\rangle$ and
$\sigma$ increase with the distance. Figure. 2(b) showcases the first passage
time probability density functions for the targets $\rm A$ (blue) and $\rm D$
(red). The nature of the distribution remains essentially similar to that
shown in Fig. 2(a). As before the first passage time density for the resetting
free process has exponential tails at large times as can be seen from Fig. 3.
However, in this case, we find that ${\rm CV}$ is always larger than $1$,
i.e., the fluctuations $\sigma$ tend to exceed the mean $\langle T\rangle$.

Figure 4: Typical trajectory of an ABP subject to stochastic resetting
mechanism. Generically, motion of the ABP is governed by the Eqs. (1)-(3)
until the ABP is stopped and the motion is reset. Between two resetting
events, the stochastic motion of the ABP is depicted by blue trajectories and
the reset jumps are indicated by brown lines. Panel (a): Resetting protocol I:
After each reset, the particle is reset to the origin $\rm O$. Panel (b):
Resetting protocol II: In this protocol, the reset position is completely
random and is chosen uniformly from the domain under consideration. In both
cases, the target $\rm S$ is placed in the same location. The resetting times
for both protocols are drawn from an exponential distribution (Eq. (6)).

Figure 5: Spatial probability distribution of the ABP projected to $(x,y)$
plane under resetting protocol I. Evidently, the span (excursion) of the phase
space trajectory depends on the mean resetting time $\langle R\rangle$. Here,
we have taken three choices for the mean resetting time: (a) $\langle
R\rangle=1.0$, (b) $\langle R\rangle=5.0$, and (c) $\langle R\rangle=10.0$. In
the colorbar, black represents zero probability density and yellow represents
maximum probability. Blue, magenta, yellow, and red dots indicate the location
of the targets $\rm A(1,1)$, $\rm B(-2,2)$, $\rm C(-3,-3)$, and $\rm D(4,-4)$,
respectively. These targets are juxtaposed externally and do not really hinder
the motion of the ABP. Panel (a): For $\langle R\rangle=1.0$, the spatial
density is non-zero only within a small radius centered at the origin, within
which only A (blue dot) exists. This essentially means that A can be tracked
by the ABP with higher probability while there is almost negligible
probability to track the targets C and D. Panel (b): The intensity of spatial
density is more distributed when $\langle R\rangle=5.0$. Since, resetting
occurs less frequently, the particle can diffuse more and the spatial
distribution is more wider. Naturally, in this case, it is easier to track the
targets $\rm A$ (blue dot) and $\rm B$ (magenta dot) who are in more probable
zone of the spatial density. Panel (c): When $\langle R\rangle$ is even
further increased to 10.0, the particle can spread even further. The spatial
density is maximum at the origin in all the cases as the ABP is reset every
time to the origin. However, due to more excursion, all the targets become
accessible to the ABP. In the numerical simulation, the probability density is
calculated at the time step $10^{2}$ units. We took $10^{5}$ realizations of
the process (also see Fig. 13 for the spread of typical phase space points in
one realization) to perform the ensemble averaging.

##  IV Resetting mediated First passage time of active Brownian particle

In this section, we turn our attention to the first passage time statistics of
the ABP which is further subjected to stochastic resetting. As mentioned
earlier, resetting brings back the ABP to its initial condition after a random
time $R$ drawn from a distribution $f_{R}(t)$. While this distribution could
be arbitrary, in here, we assume that the resetting time distribution is
exponential such that

$f_{R}(t)=\langle R\rangle^{-1}e^{-t/\langle R\rangle},$ (6)

where, $\langle R\rangle$ is the mean of this distribution. Simply put,
resetting occurs with a rate $\langle R\rangle^{-1}$. Since we do not
distinguish between the resetting and initial locations, it allows us to
design two different resetting protocols. In what follows, we investigate the
effect of resetting on the mean first passage or completion time, denoted by
$\langle T_{R}\rangle$. For brevity, the resetting mechanisms are illustrated
schematically in Fig. 4 and discussed further in the following subsections.
While passing, we also note that there have been some recent works on active
systems such as ABP (like here), and RTP under stochastic resetting Evans and
Majumdar (2018); Kumar, Sadekar, and Basu (2020); Santra, Basu, and
Sabhapandit (2020); Abdoli and Sharma (2021, 2021); however these works deal
with relatively simple geometry & target configurations, fixed resetting
conditions and external fields which in turn renders many exact results. In
stark contrast, our work is motivated by realistic topography Volpe, Gigan,
and Volpe (2014); Volpe and Volpe (2017) and an arbitrary choice of targets –
although it limits the tractability of exact results, it showcases a realistic
scenario and furthermore, the effect of environmental complexities and initial
conditions is explored in details.

###  IV.1 Designing resetting protocol I – quenched initial conditions

The first resetting protocol is associated with IC1, i.e., the quenched or
fixed initial conditions. Here resetting is performed on the particle in a way
which brings it back to the same configuration (IC1) instantaneously. Fig. 4
(a) depicts this scenario where the brown lines indicate the occurrence of
resetting throughout the journey. The time interval between two successive
resetting follows Eq. (6). The FPTs of the specified target(s) depend on the
rate at which resetting occurs.

Before getting into the first passage properties, we first investigate the
spatial properties for the active Brownian motion under resetting (i.e., by
scanning the phase space coordinates of the ABP up to a given time – see Fig.
13 and then performing an ensemble average) in the absence of target(s). To be
precise, we observe the spatial distribution of the ABP under resetting in the
$(x,y)$ plane ( Fig. 5). We plot the spatial distributions for three
alternative values of $\langle R\rangle$, as illustrated in Fig. 5. When
$\langle R\rangle=1.0$, the ABP is frequently reset to the origin and the
particle’s trajectory is constrained to a tiny circular region (yellow area)
around the origin for the majority of the time, as seen in Fig. 5 (a). As we
increase $\langle R\rangle$, resetting occurs less frequently which allows the
ABP to cover a larger area from the source (which is the origin in this case).
Consequently, the accessible region is broader than before which is clear from
Fig. 5 (b)-(c) for $\langle R\rangle=5.0$ and $\langle R\rangle=10.0$,
respectively.

Figure 6: Resetting protocol I (when the ABP is always reset to the origin).
Plot of $\langle T_{R}\rangle$ as a function of the resetting rate $\langle
R\rangle^{-1}$. Small $\langle R\rangle^{-1}$ corresponds to the almost no-
resetting case (i.e., the limit $\langle T_{R}\rangle\approx\langle T\rangle$)
while large $\langle R\rangle^{-1}$ implies frequent resetting. Panel (a):
optimality in $\langle T_{R}\rangle$ is observed for the targets $\rm A(1,1)$
and $\rm B(-2,2)$. For target $\rm A$, the optimal value of $\langle
T_{R}\rangle$ is $192.8$ ($<495.75$) which is achieved for $\langle
R\rangle^{-1}\approx 0.5$. For target $\rm B$, the optimality is achieved at
$\langle R\rangle^{-1}\approx 0.1$ and $\langle T_{R}\rangle$ is $483.5$
($<530.79$). Panel (b): Resetting is not beneficial for the other two targets
$\rm C$ and $\rm D$ as $\langle T_{R}\rangle$ is larger than $\langle
T\rangle$ for all values of $\langle R\rangle^{-1}$. We perform numerical
simulations by taking $10^{6}$ realizations for each value of $\langle
R\rangle^{-1}$.

We now turn our attention to the mean search time in the presence of
resetting. We summarize our results in Figure 6 where we demonstrate the
variation of $\langle T_{R}\rangle$ as a function of the resetting rate
$\langle R\rangle^{-1}$. For relatively small values of $\langle
R\rangle^{-1}$, resetting is quite infrequent and thus $\langle
T_{R}\rangle\approx\langle T\rangle$ (i.e., the mean first passage time of the
free ABP). For target $\rm A$ (blue), $\langle T_{R}\rangle$ starts decreasing
with the increment of $\langle R\rangle^{-1}$ until it reaches a minimum at
$\langle R\rangle^{-1}\approx 0.5$ (i.e., $\langle R\rangle\approx 2.0$) (see
Fig. 6(a)). This can also be perceived from the spatial distribution in Fig. 5
(a) where the target $\rm A$ (blue dot) lies very close to the high spatial
density zone for $\langle R\rangle=1.0$ and thus it can be accessed easily by
the ABP. From this optimal, $\langle T_{R}\rangle$ increases with $\langle
R\rangle^{-1}$. For the target $\rm B$ (magenta), the optimality occurs at
$\langle R\rangle^{-1}\approx 0.1$. However, optimal $\langle T_{R}\rangle$
for the target $\rm B$ is higher than that of $\rm A$, as the spatial
intensity slowly decreases throughout the space if $\langle R\rangle$ is
gradually increased (i.e., $\langle R\rangle^{-1}$ is decreased). A systematic
reduction of $\langle T_{R}\rangle$ for targets which are far away from the
origin is not possible (see yellow and red curves in Fig. 6(b)). So, it
appears that resetting is advantageous for reduction of the mean first passage
time if the targets are located at a specific region of $\mathcal{B}$
(particularly, close to the origin in this case). In what follows, we will
demonstrate how one can improvise on this strategy to get hold of the targets
that are far away from the origin as well making the search process more
efficient in general.

###  IV.2 Designing resetting protocol II – annealed initial conditions

One way of overcoming the limitation of protocol I might be to design a
resetting mechanism which would treat all the points inside $\mathcal{B}$ in
the same footing, rather than being biased to any particular region. This
indication leads us to work with annealed initial conditions (IC2) where the
initial position and orientation of the ABP are chosen at random. Now, a new
resetting protocol can be implemented which resets the ABP’s location to any
randomly chosen point inside $\mathcal{B}$. In this way, the statistical
identical behavior of the initial and resetting locations is maintained. In
Fig. 4 (b), this resetting mechanism is portrayed where the brown lines
represent reset paths and the brown crosses indicate the points where the
particle is reset. In contrast to the resetting protocol I, here resetting may
send the particle very close to the target, independent of the target’s
location. This causes a significant reduction to the first passage times
($T_{R}$) and eventually to their mean ($\langle T_{R}\rangle$).

Figure 7: Spatial probability distribution of the ABP projected to the phase
space under resetting protocol II. Here the particle is reset to random
positions (uniformly) inside the region. We have chosen $\langle R\rangle=5.0$
for this spatial probability density plot. However, the density plot shows
same qualitative behavior for any other values of $\langle R\rangle$. Since
the particle is reset to random positions, it naturally spans the whole region
and the spatial density is almost uniform as evident from the figure. Hence,
the targets $\rm A$, $\rm B$, $\rm C$, and $\rm D$ are almost equally
accessible by the ABP. Note that, as before, the targets are superimposed
externally (in other words, the simulation is done for the target free
process) and they do not really affect the motion. In the numerical
simulation, the probability density is calculated at the time step $10^{2}$
units. We took $10^{5}$ realizations of the process (also see Fig. 14 for the
spread of typical phase space points in one realization) to perform the
ensemble averaging.

Figure 8: Resetting protocol II (when the ABP is reset to random positions).
Plot of $\langle T_{R}\rangle$ as a function of $\langle R\rangle^{-1}$ when
the particle is reset to random positions under resetting protocol II. When
$\langle R\rangle^{-1}$ is large, the particle is reset frequently to random
positions and it soon spans the whole space in relatively small time. As a
result, $\langle T_{R}\rangle$ tends to decrease as we increase $\langle
R\rangle^{-1}$. The mean first passage time plots for all the targets show the
same qualitative behavior when $\langle R\rangle^{-1}\gtrapprox 10^{0}$
(frequent resetting).

As before, to gain a better insight, we first look into the spatial
distribution of the ABP (see Fig. 7) in the above-mentioned configuration but
excluding the targets. It is easy to see that the ABP covers the whole region
$\mathcal{B}$ uniformly for $\langle R\rangle=5.0$ (also see Fig. 14 where we
have shown all the phase space points up to a given time in one realization).
The nature of this distribution remains invariant under the change of $\langle
R\rangle$, i.e., for any values of $\langle R\rangle$, the entire territory is
almost uniformly accessed by the ABP. Since the target(s) are any point(s)
within this region, the search time under this condition also remains
invariant. This emerges as a notable feature for this strategy.

Keeping this wisdom in mind, we now proceed to study the variation of $\langle
T_{R}\rangle$ as a function of the resetting rate $\langle R\rangle^{-1}$. At
large $\langle R\rangle^{-1}$, resetting occurs frequently which allows the
ABP to travel from one random location to another very rapidly. This increases
the probability of finding the target as well as reduces the time for doing
so. Consequently, the value of $\langle T_{R}\rangle$ is remarkably smaller
for large $\langle R\rangle^{-1}$ compared to resetting protocol I. This is
demonstrated in the $\langle T_{R}\rangle$ versus $\langle R\rangle^{-1}$
plots in Fig. 8. As predicted before, the mean search time for all the targets
more or less have the same qualitative behavior and for large $\langle
R\rangle^{-1}$, the $\langle T_{R}\rangle$ values are almost identical. In
particular, we note that $\langle T_{R}\rangle$ (for all the targets) is
reduced by almost three fold in comparison to the resetting free process (see
the Table 1 for comparison). Interestingly, implementation of this protocol
reveals that there is an optimal rate at $\langle R\rangle^{-1}\to\infty$ for
which the search time is the lowest for all the targets. This is in accordance
with our general prediction made in the previous paragraph. Such instances can
also be found in Besga  _et al._ (2020).

##  V Theoretical developments

So far, most of our analysis have been computational in nature owing to the
intractability of active Brownian particles in a confined topography in the
presence of targets. In this section, we develop an invasive method borrowing
ideas from the general framework of first passage under restart. Although
exact expressions can not be obtained for such model systems, this semi-
analytic approach turns out to be a useful method for computing the first
passage time statistics that is further corroborated with direct numerical
simulations à la Langevin. In what follows, we first explain the effects of
resetting and why indeed resetting expedites the active search process. We
then provide a semi-analytical estimation of the mean first search time under
resetting and compare with numerical simulations.

###  V.1 Large search time fluctuations of an ABP and effect of resetting

What determines the average search time of an ABP in such scenario? The answer
to this question clearly depends on the diffusivity, initial configuration,
overall topography, and coordinates of the target location. Naturally, these
parameters can affect the search process causing in large stochastic
fluctuations in the search time making it an inefficient process. It is thus
crucial to design strategies that could mitigate fluctuations induced search
delays. The theory of first passage time under restart teaches us that a
simple resetting mechanism can reverse the deleterious effects of large
fluctuations in arbitrary search processes Pal and Reuveni (2017); Pal,
Kostinski, and Reuveni (2022), thus making them more efficient. This property
has since been demonstrated in numerous simple diffusive and non-diffusive
processes Evans, Majumdar, and Schehr (2020). In here, we investigate this
property for the system of an ABP in the aforementioned geometry.

To simplify the problem, let us assume that the ABP starts from a source (say,
$\rm O$) and searches for a single target (say, $\rm S$). In the schematic
Fig. 9, the source ($\rm O$) and the target ($\rm S$) are represented by black
and red dots, respectively. To proceed further, we will first analyze the
search process without resetting mechanism and then compare with the resetting
induced process. This should be done from the reference frame of an external
observer who randomly arrives at some point along the trajectory and either
inspects (which is the case in the absence of resetting) or resets (naturally
in the presence of resetting) the ongoing process. For the former case, recall
that the time taken by the ABP to find the target from the source is denoted
by $T$ – the mean being $\langle T\rangle$. However, the external observer may
arrive at a point (say maroon dot point $\rm P$ in Fig. 9 which is different
than the source point $\rm O$) while the process has already started. Although
the observer will not interrupt the process, the time remaining for the
process completion would be a different random variable (denoted by $T_{\rm
residual}$) pertaining to a different distribution. The trajectory colored in
cyan in Fig. 9 corresponds to this path, called as the residue path and the
waiting time for this observer is often known as the residual time, with the
following form for its mean Pal, Kostinski, and Reuveni, 2022

$\langle T_{\rm residual}\rangle=\dfrac{\langle T^{2}\rangle}{2\langle
T\rangle}.$

Figure 9: A trajectory based analysis for the inspection and restart
mechanism. The ABP begins its search from $\rm O$ (black dot) for the target
$\rm S$ (red dot) along the blue-colored trajectory. At a random point in
space, say $\rm P$ (maroon dot), motion of the ABP is either inspected or
restarted. In the former case, the motion of the ABP remains uninterrupted and
we just inspect the remaining/residual path (cyan trajectory) and its
corresponding traverse time ($T_{\rm residual}$) till it reaches the target
$\rm S$. In the second case, the ABP is reset to the source point $\rm O$ from
the point $\rm P$ from which it restarts the search process (shown by the red
dashed trajectory). In the case of almost no-resetting afterwards, this time
should be $T$ (see the main text). Resetting will expedite the search process
iff, on average, $\langle T\rangle<\langle T_{\rm residual}\rangle$.

In the second scenario, the observer arrives at this arbitrary point $\rm P$
and resets the particle to the source $\rm O$ from the point $\rm P$
(indicated by black line) and enforces to restart the search. The path
traversed by the ABP after being reset is represented by a red dashed
trajectory in Fig. 9. Following restart, the average time taken by the ABP to
reach the target from the source again would be simply $\langle T\rangle$
since it can be considered as a new search trial. Furthermore, this assumes
the limit to an infinitesimal resetting rate so that hardly one resetting
event can be experienced. Comparing the two scenarios explained above, one can
assert that resetting expedites the search time if the average completion time
following restart is less than the mean residual time, i.e.,

$\displaystyle\langle T\rangle<\langle T_{\rm residual}\rangle$
$\displaystyle=\dfrac{\langle T^{2}\rangle}{2\langle T\rangle},$ (7)

which, after simplification, yields

${\rm CV}>1,$ (8)

where recall that $\rm CV=\frac{\sigma}{\langle T\rangle}$ is the coefficient
of variation for the search time in the absence of resetting. $\rm CV$ is a
measure of statistical dispersion that quantifies how broad a distribution is.
Thus the above result signifies that resetting is helpful in the reduction of
the mean search time only when the uninterrupted search process has large
fluctuations (namely $\rm CV>1$) Pal and Reuveni (2017); Pal, Kostinski, and
Reuveni (2022).

The set-up of the ABP can be used as a testing ground for this criterion. For
the quenched initial condition IC1 when the ABP always starts its search from
a fixed location (Sec. III.1), we find that the criterion is satisfied for the
targets A and B while it does not hold for the targets C and D. For given
$D_{R}=1$, Table 1 summarizes the values of $\rm CV$ collected numerically
from the data set of first passage time for individual targets $\rm A$, $\rm
B$, $\rm C$, and $\rm D$. This simply states that resetting is useful for the
targets that are close to the initial locations where the underlying search
times have $\rm CV>1$ (see Table 1 and Fig. 6(a)). The diffusion mediated
search gradually becomes more inefficient as the ABP looks for targets that
are far away. This results in larger mean and fluctuations but somewhat
bounding the $\rm CV$ below 1 (see Table 1). Fig. 6( b) corroborates with this
observation as resetting hinders (disobeying Eq. (8)) the completion in such
cases.

This behavior can also be understood qualitatively from the spatial
probability density plots in Fig. 5. The plot essentially shows the
accessibility of the targets for different values of resetting rate. It is
clear that the targets $\rm A$ and $\rm B$ always belong to high probability
region, and thus can be detected easily by the ABP. On the other hand, the
detection probability for the targets $\rm C$ and $\rm D$ remains quite low
for a finite range of $\langle R\rangle$. This is rather evident since
resetting occurs persistently to the origin and thus the local region around
the origin becomes more accessible by the ABP. This reasoning confers with the
numerical results as we have discussed above. Note, however, that for random
resetting locations, this argument does not hold as we will show below.

In the case of annealed initial condition, the ABP starts its search from a
random point inside the region $\mathcal{B}$. This point is statistically
drawn uniformly within the range of the area. Following reset, another random
point is chosen uniformly and the particle is teleported back there from where
it restarts its search. In this case, we find that the criterion $\rm CV>1$ is
quite robust for all the targets. Table 1 summarizes the values of $\rm CV$ of
the underlying first passage time for the targets $\rm A$, $\rm B$, $\rm C$,
and $\rm D$ which clearly shows that the randomness in the initial condition
makes the search more costly resulting in large fluctuations, i.e., $\rm
CV>1$. Naturally, these conditions are ideal for resetting to work and we find
an overall reduction in completion time (see Fig. 8).

As before, this behavior can again be underpinned by taking the route to the
heatmaps for the spatial probability density as shown in Fig. 7. Since the
initial as well resetting locations are fully randomized over the entire
geometry, there is no as such priority for the target locations. In other
words, all the targets are equally accessible to the ABP as the spatial
probability is spanned uniformly. Thus, resetting works uniformly irrespective
of the target locations.

The discussion so far has been focused on a fixed rotational diffusion
constant ($D_{R}$). However, by suitable modulation of $D_{R}$, fluctuations
can be even enhanced further thus resulting in larger coefficient of variation
($\rm CV$). We discuss these aspects in details in Appendix A.

Figure 10: Comparison between the semi-analytical and numerical simulations of
$\langle T_{R}\rangle$ for resetting protocol I and II. In all the cases, red
markers indicate numerical values of $\langle T_{R}\rangle$ calculated from
stochastic simulations. On the other hand, blue crosses mark theoretical
values of $\langle T_{R}\rangle$ obtained from using Eq. (10) and the method
mentioned in the main text. Numerically each point is a mean of $10^{6}$
realizations. Panels (a)-(d): $\langle T_{R}\rangle$ values with varying
$\langle R\rangle$ under resetting protocol I. Panels (e)-(h): the same under
resetting protocol II. Panels (a) and (e) are for the target $\rm A$, (b) and
(f) are for target $\rm B$, (c) and (g) are for target $\rm C$, (d) and (h)
are for target $\rm D$. We observe an excellent agreement between the theory
and numerical simulations.

###  V.2 Mean search time of ABP under resetting

We now turn our attention to the estimation of the mean search time in the
presence of resetting. Recently, a formalism namely “First passage under
restart” was proposed to compute the mean search time of an arbitrary first
passage process which is further subjected to generic restarts. Power of this
approach should be appreciated since it does not depend on the specific nature
of the underlying dynamics. For brevity, we will briefly sketch out this
method fitted appropriately to our context.

We begin by reminding ourselves that the random search time in the absence of
restart is denoted by $T$. Thus, if the ABP starts from some initial
conditions, and finds the target in a random time $T$ without any
interruptions, the process ends. Suppose now that the motion of ABP is stopped
and compelled to restart from the initial configuration after some random time
$R$. In such case, the ABP renews its search. The process is repeated until
the ABP finds the target. Letting $T_{R}$ be the random completion time of
this compounded search process (and accounting for the above-mentioned
possibilities), one can write

$T_{R}=\begin{cases}T,&\text{if $T$ < $R$}\\\ R+T_{R}^{\prime},&\text{if
$R\leq T$}\end{cases}$ (9)

where $T_{R}^{\prime}$ is a random variable which is independent and identical
to $T_{R}$. Using Eq. (9) and taking expectations we can find Pal and Reuveni
(2017)

$\langle T_{R}\rangle=\frac{\langle\min(T,R)\rangle}{{\rm Pr}(T<R)},$ (10)

where $\min(T,R)$ is a new random variable and ${\rm Pr}(T<R)$ is the
probability that $T<R$. Notably, both these quantities can be computed from
the distributions of $T$ and $R$. Although there exists formal expressions for
these quantities, exact results can be obtained only for simplified cases. The
model for ABP does not provide analytically tractable close expressions for
first passage time distribution $f_{T}(t)$, thus we will evaluate Eq. ((10))
numerically from the data series obtained for $T$ (already in disposal from
Sec. III) and $R$ (externally employed) respectively. If the data series
contains $X$ number of inputs for $T$ and $R$, one can then compute
$\min(T,R)$ and mark yes/no for $T<R$ (/otherwise) for each entry
$i=1,2,\cdots X$. Finally, the ensemble average is done over these entries
(see Github link sar  for more details).

Following the semi-analytical procedure stated above, we calculate the mean
search time of the ABP under resetting using Eq. (10). First passage times for
the target are collected by taking $10^{6}$ trials for both kinds of sources.
The mean first search times are plotted in Fig. 10 for different values of
$\langle R\rangle$. Recall that these quantities were also calculated using
direct Langevin simulations in Sec IV. Figure 10 demonstrates a comparison
between the results obtained from pure numerical simulations and that obtained
via the semi-analytical approach. In the upper panel of Fig. 10, we present
the results for mean search time under resetting protocol I. While the red
circles represent search times that are the numerically obtained, the blue
markers (crosses) represent the same obtained semi-analytically. Similarly,
the lower panel of Fig. 10 showcases a comparison between the numerical and
semi-analytical results for search times under resetting protocol II. In both
cases, we find the results to be in excellent agreement with each other. This
study confers the power of this invasive approach where one can make progress
with a hand-to-hand intuitive numerical analysis along with the theory.

##  VI Discussion and future perspectives

Study of first passage processes has always been a fascinating research topic
in many interdisciplinary fields. Any dynamical process that starts from a
well defined initial configuration and looks for designated or sparse search
locations can broadly be classified as FPT process. In statistical physics, a
myriad of basic questions and properties have been investigated for passive
searchers such as simple diffusing particle. However, searchers in real life
are “active” driven in non-equilibrium conditions. These searchers usually
transport cargos or other important constituents from one place to other.
Naturally, active transport is a ubiquitous phenomena that depends non-
trivially on the environmental complexities such as geometric constraints,
mechanical cues, chemical gradients, and hydrodynamic fluid flow Bechinger
_et al._ (2016); Martínez-Calvo, Trenado-Yuste, and Datta (2021). Designing
various transport and search strategies is an important challenge for the
events to take place in a timely manner. Our work proposes one such strategy
for active search. Here, we design an intermittent resetting based search
strategy for an active system which consists of a single active Brownian
particle in a confinement in the presence of targets that require detection
for the process completion.

Our work shows that a suitable implementation of resetting can result in a
significant reduction of the mean search time. This observation has been found
quite robust under various resetting protocols: namely quenched (fixed)
condition when the ABP is always returned back to the same place, and annealed
(fluctuating/random) condition when the ABP is reset back to random positions.
Our study unravels that the randomized resetting protocols are more efficient
than the quenched resetting protocol in expediting generic search processes.
Using a semi-analytical approach, we have checked our numerical results with
the analytical results which show an excellent agreement. In particular, we
examine a criterion that puts a bound on the relative fluctuations of the
first passage time for the resetting free process. The criterion asserts that
resetting works in favour if the underlying search process has significant
large fluctuations around the average first passage time. Thus, resetting
based search offers a generic advantage when the underlying active search
process faces uncertain conditions.

Search process for living active systems takes place in a complex crowded
environment. Our effort has been to capture the effect of simple confinement
in a simple resetting mediated search process. However, we posit that these
ideas could be extremely relevant to sub-cellular environment where there can
be more obstacles such as macro-molecules which can hinder the search process.
Also there can be chemical reactions between the constituents, thus one needs
to gauge the interaction appropriately to the microscopic dynamics Volpe and
Volpe (2017); van Roon  _et al._ (2022). More generally, it would be an
outstanding challenge to design and explore resetting mediated strategies in
complex systems.

## Acknowledgements

C.H. is supported by DST-INSPIRE-Faculty grant (Grant No. IFA17-PH193). A.P.
acknowledges support from the Department of Atomic Energy, India.

## Data availability

The data that supports the findings of this study is available within the
article. The MATLAB codes that have been used to do the simulations are openly
available in the GitHub repository sar, .

##  Appendix A Effect of rotational diffusion constant ($D_{R}$) in the first
passage time fluctuations

In the main text, we have presented results mostly for a fixed rotational
diffusion constant ($D_{R}$). In this appendix section, we relax this
condition and investigate the effects of $D_{R}$ on the various first passage
properties. In particular, we modulate $D_{R}$ and first study the variations
of mean, fluctuations of the underlying first passage times. We simulate the
underlying ABP dynamics (in the absence of resetting) and collect the first
passage times from which we compute these observables for each $D_{R}$. Our
first observation is that mean search time increases as a function of the
distance between the target and the origin for any value of $D_{R}$
(irrespective of the initial conditions). The mean first passage time for a
particular target also increases when the value of $D_{R}$ is increased.

Next, we would like to study the fluctuations as a function of $D_{R}$. To
this end, we plot $\rm CV$ curves for different targets $\rm A$, $\rm B$, $\rm
C$, and $\rm D$ (blue, magenta, yellow, and red, respectively) as a function
of $D_{R}$ in Fig. 11. When the initial condition is fixed to the origin
(IC1), we plot the variation of $\rm CV$ for all the targets by increasing the
value of $D_{R}$ in Fig. 11(a). For the targets $\rm A$ and $\rm B$, the value
of $\rm CV$ is greater than $1$ for almost all values of $D_{R}$. Contrarily,
the value of $\rm CV$ is always less than $1$ for $\rm C$ as well as for $\rm
D$. In Fig. 11(b), we present the variation of $\rm CV$ with increasing
$D_{R}$ when the initial condition is chosen at random in every trial (IC2).
It is observed that for $D_{R}\geq 1$, $\rm CV$ of all the targets is greater
than $1$, i.e., resetting will certainly work if $D_{R}\geq 1$. For $D_{R}<1$,
the $\rm CV$ values are approximately close to 1. We accumulate the first
passage time statistics in Table 2 for three different values of $D_{R}$.

Figure 11: Modulation of the coefficient of variation ($\rm CV$) as a function of diffusion constant ($D_{R}$). $\rm CV$ of the underlying (without resetting) first passage times as a function of rotational diffusion constant ($D_{R}$) is shown here for the four different targets. The separatrix $\rm CV=1$ line is represented by the black dashed line. Panel (a): With origin as the source, the $\rm CV$ of the first passage times of targets $\rm A$ (blue) and $\rm B$ (magenta) increases from 1 as $D_{R}$ increases. However, the $\rm CV$ of the first passage times for targets $\rm C$ (yellow) and $\rm D$ (red) is always less than the $\rm CV=1$ line. Panel (b): For random initial conditions, the $\rm CV$ values for all four targets are found to be greater than unity. Table 2: Numerically calculated values of mean first passage time ($\langle T\rangle$), standard deviation ($\sigma$), and coefficient of variation ($\rm CV$) for four targets $\rm A$, $\rm B$, $\rm C$, and $\rm D$. Here we take three different values of $D_{R}$ (viz. $D_{R}=0.5$, $D_{R}=2.0$, and $D_{R}=4.0$). $D_{R}$ | Target | IC1 | IC2
---|---|---|---
|  |  $\langle T\rangle$ |  $\sigma$ |  $\rm CV$ |  $\langle T\rangle$ |  $\sigma$ |  $\rm CV$
| $\rm A$ |  487.62 |  499.04 |  1.0234 |  498.33 |  499.28 |  1.0019
| $\rm B$ |  506.00 |  506.53 |  1.0010 |  506.98 |  508.55 |  1.0031
0.5 | $\rm C$ |  522.90 |  518.60 |  0.9917 |  521.44 |  522.74 |  1.0025
| $\rm D$ |  555.52 |  550.09 |  0.9902 |  549.92 |  551.05 |  1.0021
| $\rm A$ |  543.98 |  585.93 |  1.0771 |  585.76 |  587.73 |  1.0034
| $\rm B$ |  609.22 |  619.05 |  1.0161 |  614.69 |  619.86 |  1.0084
2.0 | $\rm C$ |  683.82 |  679.34 |  0.9934 |  673.50 |  680.56 |  1.0105
| $\rm D$ |  816.56 |  806.05 |  0.9871 |  792.83 |  801.81 |  1.0113
| $\rm A$ |  676.35 |  760.70 |  1.1247 |  761.34 |  770.35 |  1.0118
| $\rm B$ |  807.56 |  829.63 |  1.0273 |  820.17 |  836.95 |  1.0204
4.0 | $\rm C$ |  959.91 |  958.43 |  0.9984 |  935.39 |  956.28 |  1.0223
| $\rm D$ |  1212.90 |  1199.10 |  0.9886 |  1178.90 |  1203.30 |  1.0207

Figure 12: Effect of rotational diffusion constant ($D_{R}$) on $\langle
T_{R}\rangle$. The top row contains plots designed for resetting protocol I,
while the bottom row contains plots designed for resetting protocol II. For
both the resetting protocols, we plot $\langle T_{R}\rangle$ as a function of
resetting rate $\langle R^{-1}\rangle$. Three different $D_{R}$ values (0.5,
2.0, and 4.0) are selected. $D_{R}$ for the left hand panel ((a) and (d)) is
0.5, for middle panel ((b) and (e)) is 2.0, for the right-hand panel ((c) and
(f)) is 4.0. For all three $D_{R}$ values, $\langle T_{R}\rangle$ for targets
$\rm A$ (blue curve) and $\rm B$ (magenta curve) decrease under resetting
protocol I. However, for the targets $\rm C$ (yellow curve) and $\rm D$ (red
curve), $\langle T_{R}\rangle$ always increases regardless of $D_{R}$ and
$\langle R^{-1}\rangle$ (a direct ramification of $\rm CV<1$ – also see Table
2). For protocol II (bottom row), $\langle T_{R}\rangle$ is reduced as the
resetting rate $\langle R\rangle^{-1}$ is gradually increased (again, a direct
ramification of the criterion $\rm CV>1$ – also see Table 2). This qualitative
behaviour does not change as we increase $D_{R}$ further.

We now turn our attention to the first passage time statistics in the presence
of resetting. The discussions from above (namely $\rm CV$ values of the
underlying first passage times) provide us useful cues to predict the behavior
of mean search time $\langle T_{R}\rangle$. The results are summarized in Fig.
12 where we show the effect of resetting on the mean search times for
different $D_{R}$. In the upper panel of Fig. 12, we present the results for
resetting protocol I. It can be seen that $\langle T_{R}\rangle$ for target
$\rm A$ decreases significantly for all three $D_{R}$ values which is clearly
in accordance with the $\rm CV$ criterion (Eq. (8)) for IC1 (also see Table
2). The average search time is only marginally reduced for target $\rm B$. For
other two targets $\rm C$ and $\rm D$, MFPT always increases which should not
be surprising since $\rm CV$ always falls below 1 in this case. Now we move to
resetting protocol II (or IC2) which is illustrated in the lower panel of Fig.
12. Here resetting successfully achieves to expedite the search process
reducing $\langle T_{R}\rangle$ drastically for all $D_{R}$. This observation
is indeed consistent with the numerical values obtained for $\rm CV$ (see
Table 2) which is always above unity for any values of $D_{R}$. An enhancement
of $D_{R}$ results in large first passage time fluctuations which essentially
add to the inherent fluctuations present in the system due to the randomness
in the resetting/initial conditions (for IC2). These two sources of randomness
essentially play a key role in the reduction of the overall search time under
reset.

##  Appendix B Spatial spread of ABP under resetting

In this section, we delineate the phase space of the ABP under both the
resetting protocols. Essentially, here, we have just taken a single snapshot
of the ABP’s position $(x,y)$ at a given time step up to $10^{5}$ time units
(see Figs. 13 and 14). In both cases, each dot represents the position of ABP
in the 2D plane and these points are colored according to the observation time
scale. Since we have not imposed any target here, the particle keeps evolving
and the distribution is expected to reach a stationary state. Taking such data
set for many realizations, we have performed an ensemble average which
resulted in the probability distribution function. As can be seen from Fig.
13, the ABP remains close to the origin since it is reset to the origin. This
indicates that the stationary state will be peaked around the origin with
highest probability as shown in Fig. 5. Similarly, for the randomized
resetting condition, the positions are spread all over the phase space as it
is shown in Fig. 14. This renders a probability distribution function which is
uniform over the topography as is evident from Fig. 7.

Figure 13: Spatial spread of the ABP in the phase space spanned over $(x,y)$
plane under resetting protocol I. In the numerical simulation, at each
microscopic time step we have measured position $(x,y)$ of the ABP and have
marked the same as a single point on the phase space. Total time step taken is
$10^{5}$ units. This clearly indicates that the particle is most probable to
be found near the origin for all resetting times, but the spread keeps growing
as we increase the mean resetting time. Resetting times were drawn from the
distribution $1/\langle R\rangle e^{-t/\langle R\rangle}$ with the following
values for $\langle R\rangle$: $1.0$ (panel a), $5.0$ (panel b), and $10.0$
(panel c).

Figure 14: Spatial spread of the ABP in the phase space under resetting
protocol II. In the numerical simulation, at each microscopic time step we
have measured position $(x,y)$ of the ABP and have marked the same as a single
point on the phase space. Total time step taken is $10^{5}$ units. Looking at
the homogeneous spread of the phase space points, it is evident that the
particle equally likely to be found anywhere in the phase space. Resetting
times were drawn from the distribution $1/\langle R\rangle e^{-t/\langle
R\rangle}$ with $\langle R\rangle=5.0$.

## References

  * Toner, Tu, and Ramaswamy (2005) J. Toner, Y. Tu, and S. Ramaswamy, “Hydrodynamics and phases of flocks,” Annals of Physics 318, 170–244 (2005).
  * Ramaswamy (2010) S. Ramaswamy, “The mechanics and statistics of active matter,” Annual Review of Condensed Matter Physics 1, 323–345 (2010).
  * Romanczuk  _et al._ (2012) P. Romanczuk, M. Bär, W. Ebeling, B. Lindner, and L. Schimansky-Geier, “Active brownian particles,” The European Physical Journal Special Topics 202, 1–162 (2012).
  * Gnesotto  _et al._ (2018) F. S. Gnesotto, F. Mura, J. Gladrow, and C. P. Broedersz, “Broken detailed balance and non-equilibrium dynamics in living systems: a review,” Reports on Progress in Physics 81, 066601 (2018).
  * Berg (2008) H. C. Berg,  _E. coli in Motion_ (Springer Science & Business Media, 2008).
  * Berg (2018) H. C. Berg,  _Random walks in biology_ (Princeton University Press, 2018).
  * Jiang, Yoshinaga, and Sano (2010) H.-R. Jiang, N. Yoshinaga, and M. Sano, “Active motion of a janus particle by self-thermophoresis in a defocused laser beam,” Physical Review Letters 105, 268302 (2010).
  * Martín-Gómez  _et al._ (2018) A. Martín-Gómez, D. Levis, A. Díaz-Guilera, and I. Pagonabarraga, “Collective motion of active brownian particles with polar alignment,” Soft matter 14, 2610–2618 (2018).
  * Gompper  _et al._ (2020) G. Gompper, R. G. Winkler, T. Speck, A. Solon, C. Nardini, F. Peruani, H. Löwen, R. Golestanian, U. B. Kaupp, L. Alvarez, _et al._ , “The 2020 motile active matter roadmap,” Journal of Physics: Condensed Matter 32, 193001 (2020).
  * Martínez-Calvo, Trenado-Yuste, and Datta (2021) A. Martínez-Calvo, C. Trenado-Yuste, and S. S. Datta, “Active transport in complex environments,” arXiv preprint arXiv:2108.07011 (2021).
  * Mijalkov and Volpe (2013) M. Mijalkov and G. Volpe, “Sorting of chiral microswimmers,” Soft Matter 9, 6376–6381 (2013).
  * Bechinger  _et al._ (2016) C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, “Active particles in complex and crowded environments,” Reviews of Modern Physics 88, 045006 (2016).
  * Volpe  _et al._ (2011) G. Volpe, I. Buttinoni, D. Vogt, H.-J. Kümmerer, and C. Bechinger, “Microswimmers in patterned environments,” Soft Matter 7, 8810–8815 (2011).
  * Ross, Ali, and Warshaw (2008) J. L. Ross, M. Y. Ali, and D. M. Warshaw, “Cargo transport: molecular motors navigate a complex cytoskeleton,” Current opinion in cell biology 20, 41–47 (2008).
  * Ghosh  _et al._ (2020) A. Ghosh, W. Xu, N. Gupta, and D. H. Gracias, “Active matter therapeutics,” Nano Today 31, 100836 (2020).
  * Bénichou  _et al._ (2011) O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, “Intermittent search strategies,” Reviews of Modern Physics 83, 81 (2011).
  * Wang  _et al._ (2017) J. Wang, D. Zhang, B. Xia, and W. Yu, “Spatial heterogeneity can facilitate the target search of self-propelled particles,” Soft Matter 13, 758–764 (2017).
  * Malakar  _et al._ (2018) K. Malakar, V. Jemseena, A. Kundu, K. V. Kumar, S. Sabhapandit, S. N. Majumdar, S. Redner, and A. Dhar, “Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension,” Journal of Statistical Mechanics: Theory and Experiment 2018, 043215 (2018).
  * Angelani, Di Leonardo, and Paoluzzi (2014) L. Angelani, R. Di Leonardo, and M. Paoluzzi, “First-passage time of run-and-tumble particles,” The European Physical Journal E 37, 1–6 (2014).
  * Condamin  _et al._ (2007) S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, “First-passage times in complex scale-invariant media,” Nature 450, 77–80 (2007).
  * Bénichou  _et al._ (2005) O. Bénichou, M. Coppey, M. Moreau, P. Suet, and R. Voituriez, “Optimal search strategies for hidden targets,” Physical Review Letters 94, 198101 (2005).
  * Bénichou  _et al._ (2010) O. Bénichou, C. Chevalier, J. Klafter, B. Meyer, and R. Voituriez, “Geometry-controlled kinetics,” Nature Chemistry 2, 472–477 (2010).
  * Bray, Majumdar, and Schehr (2013) A. J. Bray, S. N. Majumdar, and G. Schehr, “Persistence and first-passage properties in nonequilibrium systems,” Advances in Physics 62, 225–361 (2013).
  * Viswanathan  _et al._ (2011) G. M. Viswanathan, M. G. Da Luz, E. P. Raposo, and H. E. Stanley,  _The physics of foraging: an introduction to random searches and biological encounters_ (Cambridge University Press, 2011).
  * Rhee  _et al._ (2011) I. Rhee, M. Shin, S. Hong, K. Lee, S. J. Kim, and S. Chong, “On the levy-walk nature of human mobility,” IEEE/ACM Transactions on Networking 19, 630–643 (2011).
  * Zaburdaev, Denisov, and Klafter (2015) V. Zaburdaev, S. Denisov, and J. Klafter, “Lévy walks,” Reviews of Modern Physics 87, 483 (2015).
  * Volpe and Volpe (2017) G. Volpe and G. Volpe, “The topography of the environment alters the optimal search strategy for active particles,” Proceedings of the National Academy of Sciences 114, 11350–11355 (2017).
  * Lomholt  _et al._ (2008) M. A. Lomholt, K. Tal, R. Metzler, and K. Joseph, “Lévy strategies in intermittent search processes are advantageous,” Proceedings of the National Academy of Sciences 105, 11055–11059 (2008).
  * Palyulin, Chechkin, and Metzler (2014) V. V. Palyulin, A. V. Chechkin, and R. Metzler, “Lévy flights do not always optimize random blind search for sparse targets,” Proceedings of the National Academy of Sciences 111, 2931–2936 (2014).
  * Evans and Majumdar (2011a) M. R. Evans and S. N. Majumdar, “Diffusion with stochastic resetting,” Physical Review Letters 106, 160601 (2011a).
  * Evans and Majumdar (2011b) M. R. Evans and S. N. Majumdar, “Diffusion with optimal resetting,” Journal of Physics A: Mathematical and Theoretical 44, 435001 (2011b).
  * Evans, Majumdar, and Schehr (2020) M. R. Evans, S. N. Majumdar, and G. Schehr, “Stochastic resetting and applications,” Journal of Physics A: Mathematical and Theoretical 53, 193001 (2020).
  * Gupta and Jayannavar (2022) S. Gupta and A. M. Jayannavar, “Stochastic resetting: A (very) brief review,” Frontiers in Physics , 130 (2022).
  * Kusmierz  _et al._ (2014) L. Kusmierz, S. N. Majumdar, S. Sabhapandit, and G. Schehr, “First order transition for the optimal search time of lévy flights with resetting,” Physical Review Letters 113, 220602 (2014).
  * Reuveni (2016) S. Reuveni, “Optimal stochastic restart renders fluctuations in first passage times universal,” Physical Review Letters 116, 170601 (2016).
  * Pal and Reuveni (2017) A. Pal and S. Reuveni, “First passage under restart,” Physical Review Letters 118, 030603 (2017).
  * Pal, Eliazar, and Reuveni (2019) A. Pal, I. Eliazar, and S. Reuveni, “First passage under restart with branching,” Physical Review Letters 122, 020602 (2019).
  * De Bruyne, Randon-Furling, and Redner (2020) B. De Bruyne, J. Randon-Furling, and S. Redner, “Optimization in first-passage resetting,” Physical Review Letters 125, 050602 (2020).
  * Pal, Kundu, and Evans (2016) A. Pal, A. Kundu, and M. R. Evans, “Diffusion under time-dependent resetting,” Journal of Physics A: Mathematical and Theoretical 49, 225001 (2016).
  * Chechkin and Sokolov (2018) A. Chechkin and I. Sokolov, “Random search with resetting: a unified renewal approach,” Physical Review Letters 121, 050601 (2018).
  * Pal and Prasad (2019a) A. Pal and V. V. Prasad, “First passage under stochastic resetting in an interval,” Physical Review E 99, 032123 (2019a).
  * Bhat, De Bacco, and Redner (2016) U. Bhat, C. De Bacco, and S. Redner, “Stochastic search with poisson and deterministic resetting,” Journal of Statistical Mechanics: Theory and Experiment 2016, 083401 (2016).
  * Pal and Prasad (2019b) A. Pal and V. V. Prasad, “Landau-like expansion for phase transitions in stochastic resetting,” Physical Review Research 1, 032001 (2019b).
  * Bressloff (2020) P. C. Bressloff, “Directed intermittent search with stochastic resetting,” Journal of Physics A: Mathematical and Theoretical 53, 105001 (2020).
  * Tal-Friedman  _et al._ (2020) O. Tal-Friedman, A. Pal, A. Sekhon, S. Reuveni, and Y. Roichman, “Experimental realization of diffusion with stochastic resetting,” The Journal of Physical Chemistry Letters 11, 7350–7355 (2020).
  * Montanari and Zecchina (2002) A. Montanari and R. Zecchina, “Optimizing searches via rare events,” Physical Review Letters 88, 178701 (2002).
  * Luby, Sinclair, and Zuckerman (1993) M. Luby, A. Sinclair, and D. Zuckerman, “Optimal speedup of las vegas algorithms,” Information Processing Letters 47, 173–180 (1993).
  * Reuveni, Urbakh, and Klafter (2014) S. Reuveni, M. Urbakh, and J. Klafter, “Role of substrate unbinding in michaelis–menten enzymatic reactions,” Proceedings of the National Academy of Sciences 111, 4391–4396 (2014).
  * Ray (2020) S. Ray, “Space-dependent diffusion with stochastic resetting: A first-passage study,” The Journal of Chemical Physics 153, 234904 (2020).
  * Ray  _et al._ (2021) A. Ray, A. Pal, D. Ghosh, S. K. Dana, and C. Hens, “Mitigating long transient time in deterministic systems by resetting,” Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 011103 (2021).
  * Roldán  _et al._ (2016) É. Roldán, A. Lisica, D. Sánchez-Taltavull, and S. W. Grill, “Stochastic resetting in backtrack recovery by rna polymerases,” Physical Review E 93, 062411 (2016).
  * Bonomo, Pal, and Reuveni (2022) O. L. Bonomo, A. Pal, and S. Reuveni, “Mitigating long queues and waiting times with service resetting,” PNAS Nexus 1, pgac070 (2022).
  * Pal, Kuśmierz, and Reuveni (2020) A. Pal, Ł. Kuśmierz, and S. Reuveni, “Search with home returns provides advantage under high uncertainty,” Physical Review Research 2, 043174 (2020).
  * Campos and Méndez (2015) D. Campos and V. Méndez, “Phase transitions in optimal search times: How random walkers should combine resetting and flight scales,” Physical Review E 92, 062115 (2015).
  * Kuśmierz and Gudowska-Nowak (2015) Ł. Kuśmierz and E. Gudowska-Nowak, “Optimal first-arrival times in lévy flights with resetting,” Physical Review E 92, 052127 (2015).
  * Singh, Metzler, and Sandev (2020) R. Singh, R. Metzler, and T. Sandev, “Resetting dynamics in a confining potential,” Journal of Physics A: Mathematical and Theoretical 53, 505003 (2020).
  * Pal (2015) A. Pal, “Diffusion in a potential landscape with stochastic resetting,” Physical Review E 91, 012113 (2015).
  * Ray, Mondal, and Reuveni (2019) S. Ray, D. Mondal, and S. Reuveni, “Péclet number governs transition to acceleratory restart in drift-diffusion,” Journal of Physics A: Mathematical and Theoretical 52, 255002 (2019).
  * Huang and Chen (2021) F. Huang and H. Chen, “Random walks on complex networks with first-passage resetting,” Physical Review E 103, 062132 (2021).
  * González, Riascos, and Boyer (2021) F. H. González, A. P. Riascos, and D. Boyer, “Diffusive transport on networks with stochastic resetting to multiple nodes,” Physical Review E 103, 062126 (2021).
  * Bonomo and Pal (2021) O. L. Bonomo and A. Pal, “First passage under restart for discrete space and time: Application to one-dimensional confined lattice random walks,” Physical Review E 103, 052129 (2021).
  * Pal, Kostinski, and Reuveni (2022) A. Pal, S. Kostinski, and S. Reuveni, “The inspection paradox in stochastic resetting,” Journal of Physics A: Mathematical and Theoretical 55, 021001 (2022).
  * Volpe, Gigan, and Volpe (2014) G. Volpe, S. Gigan, and G. Volpe, “Simulation of the active brownian motion of a microswimmer,” American Journal of Physics 82, 659–664 (2014).
  * Basu  _et al._ (2018) U. Basu, S. N. Majumdar, A. Rosso, and G. Schehr, “Active brownian motion in two dimensions,” Physical Review E 98, 062121 (2018).
  * Topaz and Bertozzi (2004) C. M. Topaz and A. L. Bertozzi, “Swarming patterns in a two-dimensional kinematic model for biological groups,” SIAM Journal on Applied Mathematics 65, 152–174 (2004).
  * Strefler, Erdmann, and Schimansky-Geier (2008) J. Strefler, U. Erdmann, and L. Schimansky-Geier, “Swarming in three dimensions,” Physical Review E 78, 031927 (2008).
  * Slowman, Evans, and Blythe (2016) A. Slowman, M. Evans, and R. Blythe, “Jamming and attraction of interacting run-and-tumble random walkers,” Physical Review Letters 116, 218101 (2016).
  * Palacci  _et al._ (2013) J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, “Living crystals of light-activated colloidal surfers,” Science 339, 936–940 (2013).
  * Friedrich and Jülicher (2008) B. Friedrich and F. Jülicher, “The stochastic dance of circling sperm cells: sperm chemotaxis in the plane,” New Journal of Physics 10, 123025 (2008).
  * Redner (2001) S. Redner,  _A guide to first-passage processes_ (Cambridge University Press, 2001).
  * Evans and Majumdar (2018) M. R. Evans and S. N. Majumdar, “Run and tumble particle under resetting: a renewal approach,” Journal of Physics A: Mathematical and Theoretical 51, 475003 (2018).
  * Kumar, Sadekar, and Basu (2020) V. Kumar, O. Sadekar, and U. Basu, “Active brownian motion in two dimensions under stochastic resetting,” Physical Review E 102, 052129 (2020).
  * Santra, Basu, and Sabhapandit (2020) I. Santra, U. Basu, and S. Sabhapandit, “Run-and-tumble particles in two dimensions under stochastic resetting conditions,” Journal of Statistical Mechanics: Theory and Experiment 2020, 113206 (2020).
  * Abdoli and Sharma (2021) I. Abdoli and A. Sharma, “Stochastic resetting of active brownian particles with lorentz force,” Soft Matter 17, 1307–1316 (2021).
  * Besga  _et al._ (2020) B. Besga, A. Bovon, A. Petrosyan, S. N. Majumdar, and S. Ciliberto, “Optimal mean first-passage time for a brownian searcher subjected to resetting: experimental and theoretical results,” Physical Review Research 2, 032029 (2020).
  * (76) https://github.com/gourab-sar/ABP-under-reset.
  * van Roon  _et al._ (2022) D. M. van Roon, G. Volpe, M. M. T. da Gama, and N. A. Araújo, “The role of disorder in the motion of chiral swimmers in the presence of obstacles,” arXiv preprint arXiv:2205.13509 (2022).