††thanks: Corresponding author

# Channel-facilitated transport under resetting dynamics

Suvam Pal<EMAIL_ADDRESS>Physics and Applied Mathematics Unit, Indian
Statistical Institute, 203 B.T. Road Kolkata, India  Denis Boyer
<EMAIL_ADDRESS>Instituto de Física, Universidad Nacional Autónoma de
México, Ciudad de México C.P. 04510 México.  Leonardo Dagdug<EMAIL_ADDRESS>Physics Department, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael
Atlixco 186, Ciudad de México, 09340, México.  Arnab Pal<EMAIL_ADDRESS>The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113,
India & Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India

###### Abstract

The transport of particles through channels holds immense significance in
physics, chemistry, and biological sciences. For instance, the motion of
solutes through biological channels is facilitated by specialized proteins
that create water-filled channels and valuable insights can be obtained by
studying the transition paths of particles through a channel and gathering
statistics on their lifetimes within the channel or their exit probabilities.
In a similar vein, we consider a one-dimensional model of channel-facilitated
transport where a diffusive particle is subject to attractive interactions
with the walls within a limited region of the channel. We study the statistics
of conditional and unconditional escape times, in the presence of resetting –
an intermittent dynamics that brings the particle back to its initial
coordinate randomly. We determine analytically the physical conditions under
which such resetting mechanism can become beneficial for faster escape of the
particles from the channel thus enhancing the transport. Our theory has been
verified with the aid of Brownian dynamics simulations for various interaction
strengths and extent. The overall results presented herein highlight the scope
of resetting-based strategies to be universally promising for complex
transport processes of single or long molecules through biological membranes.

††preprint: 123-QED

##  I Introduction

The subject of diffusion in constrained geometries remains an enduring subject
across diverse scales, encompassing biological and technological areas [1].
For instance, proteins responsible for forming water-filled channels inside
membranes are pivotal in aiding the transport of metabolites and other solutes
across cell membranes [2, 3]. In addition to practical applications,
fundamental aspects of channel-regulated transport have been the focus of many
theoretical explorations. See [1] which provides an exhaustive list of these
problems.

Diffusion processes in high dimensional confined geometries and their
effective reduction to one-dimensional problems have been studied over nearly
two centuries since the work of Fick, much later revisited by Jacobs and
Zwanzig [4, 5, 6, 7]. Such a reduced description underlies a rapid
equilibration of the faster degrees of freedom of the particles across the
confinements, which is the key assumption behind the Fick-Jacobs formalism.
Zwanzig studied unbiased diffusive motion in quasi-one-dimensional systems
with varying geometric constraints. Depending on the constraints, he
calculated an effective diffusivity [6], paving the way for further
developments over the past decades [8]. It has been possible to identify key
quantities controlling transport processes thanks to extensive studies on
tubes and channels with simple geometries [9, 10, 11, 12, 13, 14, 15]. Despite
numerous years of research, new phenomena linked to channel-assisted
metabolite transport keep emerging, and transport in confinement remains of
much topical interest.

Consider a particle that enters a channel of an arbitrary shape from one end.
The particle can either exit on the same side of the membrane where it entered
or it can pass through and exit from the other end [16]. Essential observables
to study transport properties of such particles are return and translocation
probabilities, the lifetime of the particle inside the channel which can be
computed from the escape time statistics of the particle from the channel. A
myriad of research has been devoted to capturing the statistics of these key
quantities over the years [17, 18, 19]. Furthermore, single-particle
experiments have allowed for the direct observation of transition or first
passage or exit trajectories from the channel [20, 21, 22]. The tracking of
such trajectories allow us to get further insights on the time when a molecule
crosses an activation barrier or two molecules undergo a chemical reaction
[17, 18, 21]. The average residence time of a diffusive particle inside a
channel has been studied theoretically in different geometries [23, 24, 25]
and remains the focal point of research in channel-facilitated problems.

Generically, first-passage processes play a pivotal role in diverse fields of
chemistry, physics, and biology [26, 27, 28, 29, 30]. What is the time
required to initiate a chemical reaction? Or how much time does it take for a
stock market order to be fulfilled? First passage times are relevant to a
number of phenomena and the expectations such as the mean first passage time
can quantify the duration of a specific task. Conditional first passage
observables, such as splitting probabilities and conditional first passage
time densities, also play a key role for a deeper understanding of the
physical problem when there are multiple pathways to complete a task.

Recently, the optimization of first passage processes via different protocols
has attracted considerable attention and in particular, resetting mediated
dynamics has been an extremely effective protocol to reduce the completion
time of a complex search process [31, 32, 33, 34, 35]. Resetting usually works
by interrupting an ongoing process at a certain rate and bringing it back to a
pre-determined state, thus by confining the errand far away trajectories. This
confining effect renders anomalous non-equilibrium behavior [36, 37, 38, 39,
40, 41] but then also improves the speed-up in search processes by curtailing
trajectories which take longer time to complete the task [33, 42, 43, 44, 32,
45, 46, 31, 47, 48, 49, 50, 51, 52, 35]. The most remarkable paradigm is
perhaps the diffusive transport for which the mean first passage time is
usually diverging in a non-confining domain, however, it has been shown that
resetting can make this time optimally finite [31]. Following this work, the
efficiency of resetting processes has been studied rigorously in different
research avenues and is found to have significant speed-up in random search
processes spanning from physics, chemistry, biology, economics, computer
science and many more [53, 54, 55, 56, 57, 58, 59, 60, 61, 62]. The impact of
resetting on first passage properties in simple one-dimensional confinement
has also been extensively investigated in stochastic systems, featuring
combinations of absorbing or reactive boundaries [43, 63, 64, 65]. Recent
experiments on resetting have also showcased these features and moreover, they
have posed many interesting and new research directions [66, 67, 68, 69, 70,
71, 72].

However, the effect of resetting in a diffusion mediated channel transport is
the subject matter of current study and it was only recently Jain et al
addressed the transport of diffusing particles under resetting inside a three-
dimensional conical channel for the first time [73]. There, it was shown that
the interplay between resetting rate and the effective potential created by
the position dependent radius of the channel can give rise to a plethora of
rich behavior for the optimal transport. Motivated by this work, in this
study, we investigate the transport of resetting Brownian particles inside a
membrane whose geometry can be visualized as a cylindrical channel with
attractive interactions between the channel and the particle [16]. The motion
of the particles can then be described by the Fick-Jacobs formalism [16, 1].
Using this, we examine the dependence of first passage statistics on the
particle/channel interaction strength and on the resetting rate. We also
compute the escape probabilities and times of escape conditioned on the
boundary. It is observed that resetting can indeed expedite transport of
particles through membrane channel – we find the corresponding phase space
spanned by the parameters such as the potential strength and initial
configuration of the particles where resetting can be beneficial, otherwise
not. We also study in detail the behavior of the resetting rate whence it
makes the transport optimal or the most efficient. We perform extensive
Brownian dynamics simulations that corroborate with our theoretical
observations.

Figure 1: Channel-facilitated transport under resetting: Schematic of
diffusing trajectories in a channel containing a potential of width $-U_{0}$.
The potential is spanned for a distance $b-a$. The entire channel is confined
within two boundaries, left at $x=0$ and right at $x=l$, with a total distance
of $l$. The trajectories initiate from $x_{0}$ and are also reset to $x_{0}$
intermittently. The paths (in black) represent unconditional trajectories that
escape from any of the boundaries. The trajectories in white and red represent
the paths conditioned on the exit to the left or right boundary, respectively.
Each trajectory gives rise to different escape time statistics which we study
here.

The rest of the paper is organized in the following way. In Sec. II, we
discuss the model and the solution of the Smoluchowski equation in the
presence of a piece-wise constant potential. We derive matching conditions for
the probability density function and the survival probability at the potential
discontinuities. Also, we discuss the moments of the first passage time
density related to the survival probability. In Sec. III, we expose the
formalism of the dynamics in the presence of resetting. In Subsection. III.1,
we study the variations of the mean first passage time as a function of the
resetting rate and initial location of the particle. The investigation of the
effect of resetting is discussed in Subsection. III.2, followed by the optimal
resetting rate is discussed in Subsection. III.3. In Sec. IV we elaborate on
the conditional mean first passage times and the splitting probabilities for
various initial locations and also portray how these quantities change
qualitatively as the resetting rate is varied. We conclude in Sec. V.

##  II Model details, formalism and quantities of interest

###  II.1 Fick-Jacobs equation

Let us consider the diffusive motion of a particle inside a cylindrical
channel of length $l$ (see Fig. 1). Our analysis is based on the Fick-Jacobs &
Zwanzig formalism where we approximate the 3D motion of a particle inside the
channel as a one-dimensional diffusion along the channel axis [6, 1]. The
applicability of this approach to three-dimensional cylinders crucially
depends on the assumption of faster equilibration in the transverse degrees of
freedom than in the longitudinal direction (i.e. along the $x$-axis) [6]. The
model also assumes an interaction of the particle with the walls of the
channel, in terms of a potential $U(x)$ (see Fig. 1). The fundamental
observable in this problem is the density of the position $x$ at time $t$ for
a particle starting at $x_{0}$, which can be denoted as $p_{0}(x,t|x_{0},0)$
[for simplicity, we have used $p_{0}(x,t)$]. The evolution of the probability
density function is given by the Smoluchowski equation [1]

$\dfrac{\partial}{\partial t}p_{0}\left(x,t\right)=D\dfrac{\partial}{\partial
x}\left[e^{-\beta U(x)}\dfrac{\partial}{\partial x}\left(e^{\beta
U(x)}p_{0}\left(x,t\right)\right)\right],$ (1)

with absorbing boundary conditions at the two ends of the interval

$\displaystyle p_{0}\left(0,t\right)=0~{},~{}~{}~{}p_{0}\left(l,t\right)=0.$
(2)

The initial condition is $p_{0}(x,0)=\delta\left(x-x_{0}\right)$, where
$\beta=\dfrac{1}{k_{B}T}$ is the inverse temperature and is related to the
diffusion constant by the fluctuation-dissipation relation namely
$\beta=\mu/D$ where $\mu$ is the mobility of the particle. The form of the
effective potential that the particle experiences in the channel can be
considered as a potential well as described in [16] and is given below (also
see Fig. 1)

$U(x)=\begin{cases}0&0\leq x\leq a\\\ -U_{0}&a<x<b\\\ 0&b\leq x\leq
l,\end{cases}$ (3)

with $U_{0}>0$ and $0<a<b<l$. The nature of the potential suggests that Eq.
(1) can be solved in three different regions and then complemented with
suitable matching conditions at the edges and boundary conditions at the
boundaries. The governing equations in each region read

$\displaystyle\dfrac{\partial}{\partial
t}p_{0}^{L}(x,t)=D\dfrac{\partial^{2}}{\partial
x^{2}}p_{0}^{L}(x,t)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0<x<a,$
$\displaystyle\dfrac{\partial}{\partial
t}p_{0}^{M}(x,t)=D\dfrac{\partial^{2}}{\partial
x^{2}}p_{0}^{M}(x,t)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a<x<b,$ (4)
$\displaystyle\dfrac{\partial}{\partial
t}p_{0}^{R}(x,t)=D\dfrac{\partial^{2}}{\partial
x^{2}}p_{0}^{R}(x,t)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}b<x<l,$

where $p_{0}^{L,M,R}(x,t)$ stands for the solutions in the left, middle, and
right segments of the channel, respectively.

It is important to note that the potential has sharp jumps at $x=a$ and $x=b$
and thus the matching conditions have to be supplemented appropriately. We
note that the fluxes are continuous across the jump coordinates, however the
probabilities are discontinuous at these points due to the jump in potential.
These are known as the imperfect boundary conditions in the literature [74,
75, 76, 77] and can be written for our set-up in the following way

$\begin{cases}\partial_{x}p_{0}^{M}\left(x=a,t\right)=\partial_{x}p_{0}^{L}\left(x=a,t\right),\\\
\partial_{x}p_{0}^{R}\left(x=b,t\right)=\partial_{x}p_{0}^{M}\left(x=b,t\right).\end{cases}$
(5)

and

$\begin{cases}p_{0}^{L}\left(x=a,t\right)=e^{-\beta
U_{0}}p_{0}^{M}\left(x=a,t\right),\\\ p_{0}^{R}\left(x=b,t\right)=e^{-\beta
U_{0}}p_{0}^{M}\left(x=b,t\right).\end{cases}$ (6)

Using the boundary conditions (2) and the matching conditions (5) and (6), the
probability density functions can be solved exactly. For brevity, we have
relegated these details to the Appendix A and we will recall the solutions
wherever needed in context.

Figure 2: Schematic representation of trajectories for different initial
locations: $x_{0}=0.5$ (blue), $x_{0}=2$ (orange), and $x_{0}=4.5$ (green) in
the absence of resetting. The shaded area represents regions with non-zero
potential strength, while it is zero in the rest of the channel.

###  II.2 Survival probability and first passage

Since we are interested into the lifetime of the particles inside the channel,
a useful quantity to measure the same is the mean first passage time. However,
to compute the mean first passage time one needs to have the knowledge of the
first passage time density which can be estimated from the survival
probability. In this section, we provide a brief outline of this formalism.
Formally, the survival probability $Q_{0}(t|x_{0})$ is defined as the
probability that the particle is inside a prescribed domain without getting
absorbed upto time $t$ given that it had started from an initial position
$x_{0}$. Thus, $Q_{0}(t|x_{0})=\int_{0}^{l}p_{0}(x,t|x_{0},0)~{}dx$. Although
the survival probability can be computed from the probability density
function, there is an alternative route via backward path formalism that
allows one to directly write a governing equation for the survival probability
without an explicit knowledge of the probability density function. This is
given by the backward Smoluchowski equation where the initial location $x_{0}$
is considered as a variable

$\frac{\partial}{\partial t}Q_{0}\left(t|x_{0}\right)=De^{\beta
U(x_{0})}\frac{\partial}{\partial x_{0}}\left[e^{-\beta
U(x_{0})}\frac{\partial}{\partial x_{0}}Q_{0}\left(t|x_{0}\right)\right].$ (7)

The above equation is supplemented with the boundary conditions
$Q_{0}\left(t|x_{0}=0\right)=0$, $Q_{0}\left(t|x_{0}=l\right)=0$ and an
initial condition $Q_{0}\left(0|x_{0}\right)=1$ for $0<x_{0}<l$. As mentioned
before, due to the nature of the potential, we can solve the survival
probability region-wise and then complement with the appropriate matching
conditions at the jump coordinates. Translating Eq. (7) into the Laplace
space, we can write

$\displaystyle
s\widetilde{Q}_{0}^{I}\left(s|x_{0}\right)-1=D\frac{\partial^{2}}{\partial
x_{0}^{2}}\widetilde{Q}_{0}^{I}\left(s|x_{0}\right)~{}~{}~{}~{}0<x_{0}<a,$ (8)
$\displaystyle
s\widetilde{Q}_{0}^{II}\left(s|x_{0}\right)-1=D\frac{\partial^{2}}{\partial
x_{0}^{2}}\widetilde{Q}_{0}^{II}\left(s|x_{0}\right)~{}~{}~{}~{}a<x_{0}<b,$
(9) $\displaystyle
s\widetilde{Q}_{0}^{III}\left(s|x_{0}\right)-1=D\frac{\partial^{2}}{\partial
x_{0}^{2}}\widetilde{Q}_{0}^{III}\left(s|x_{0}\right)~{}~{}~{}~{}b<x_{0}<l,$
(10)

where
$\widetilde{Q}_{0}^{I,II,III}(s|x_{0})=\int_{0}^{\infty}dt\,e^{-st}Q_{0}^{I,II,III}(t|x_{0})$
denote the solutions in the left, middle and right intervals, respectively in
the Laplace space. The matching conditions for the survival probabilities at
the jump coordinates read (detailed derivation of these matching conditions
are provided in Appendix-B)

$\begin{cases}\partial_{x_{0}}\widetilde{Q}_{0}^{I}\left(s|x_{0}=a\right)=e^{\mu
U_{0}/D}\partial_{x_{0}}\widetilde{Q}_{0}^{II}\left(s|x_{0}=a\right),\\\
\partial_{x_{0}}\widetilde{Q}_{0}^{III}\left(s|x_{0}=b\right)=e^{\mu
U_{0}/D}\partial_{x_{0}}\widetilde{Q}_{0}^{II}\left(s|x_{0}=b\right),\end{cases}$
(11)

and

$\begin{cases}\widetilde{Q}_{0}^{I}\left(s|x_{0}=a\right)=\widetilde{Q}_{0}^{II}\left(s|x_{0}=a\right),\\\
\widetilde{Q}_{0}^{III}\left(s|x_{0}=b\right)=\widetilde{Q}_{0}^{II}\left(s|x_{0}=b\right).\end{cases}$
(12)

Solving Eqs. (8)-(10) and imposing the boundary conditions at $x=0$ and $x=l$,
the quantities $Q_{0}^{I}$, $Q_{0}^{II}$ and $Q_{0}^{III}$ take the form

$\displaystyle\widetilde{Q}_{0}^{I}\left(s|x_{0}\right)=\frac{1}{s}(1-e^{-\alpha_{0}x_{0}})\left[1+A\,s\left(e^{\alpha_{0}x_{0}}+1\right)\right],$
$\displaystyle\widetilde{Q}_{0}^{II}\left(s|x_{0}\right)=\frac{1}{s}+C_{1}e^{\alpha_{0}x_{0}}+C_{2}e^{-\alpha_{0}x_{0}},$
(13)
$\displaystyle\widetilde{Q}_{0}^{III}\left(s|x_{0}\right)=\frac{1}{s}[1-e^{-\alpha_{0}(x_{0}-l)}]\left[1+B\,s\left(e^{\alpha_{0}x_{0}}+e^{\alpha_{0}l}\right)\right],$

where $\alpha_{0}=\sqrt{s/D}$. The constants $A,B,C_{1}$ and $C_{2}$ are
obtained using the four matching conditions in Eqs. (11) and (12). Details are
provided in Appendix-C. Using Eqs. (II.2), one can now write the complete
solution for the survival probability

$\widetilde{Q}_{0}\left(s|x_{0}\right)=\Theta\left(a-x_{0}\right)\widetilde{Q}^{I}_{0}\left(s|x_{0}\right)+\Theta\left(x_{0}-a\right)\Theta\left(b-x_{0}\right)\widetilde{Q}^{II}_{0}\left(s|x_{0}\right)+\Theta\left(x_{0}-b\right)\widetilde{Q}^{III}_{0}\left(s|x_{0}\right).$
(14)

The first passage time density $f_{0}(t)$ is related to the rate of change of
the survival probability so that $f_{0}(t)=-dQ_{0}(t)/dt$. Subsequently, one
can compute all the moments from the FPT density directly

$\langle T^{n}\rangle=\int_{0}^{\infty}t^{n}~{}f_{0}(t)~{}dt,$ (15)

In particular, the mean first passage time can be obtained by setting $n=1$
and after some simplification, one finds

$\displaystyle\langle
T\rangle=\int_{0}^{\infty}t~{}f_{0}(t)~{}dt=\widetilde{Q}_{0}(s\rightarrow
0|x_{0}),$ (16)

where $\widetilde{Q}_{0}\left(s|x_{0}\right)$ is given by Eq. (14). Thus, from
the knowledge of the survival function in the Laplace space, one can easily
compute the mean lifetime of a particle inside a given geometry.

##  III First passage properties under Resetting dynamics

So far, we have focused on the diffusive transport of the particle through the
channel. In what follows, we will introduce the resetting mechanism to the
original dynamics and study its effects on the first passage statistics.
Resetting is a simple mechanism by which motion of a particle is
intermittently stopped and it is reset back to its initial coordinate.
Following this action, the particle restarts its motion and continues to
diffuse till the next resetting occurs. The overall search process ends when
the particle escapes from one of the exit points of the channel. We assume
that resetting occurs at random times and that they are distributed according
to $f_{R}(t)=re^{-rt}$, where $r$ is the rate at which resetting occurs. Thus
the mean waiting time between two resetting events is given by $1/r$.

###  III.1 Survival probability under resetting and unconditional MFPT

We start by defining the survival probability $Q_{r}\left(t|x_{0}\right)$
under the resetting dynamics. Formally, it can be interpreted as the
probability that the particle remains inside the channel upto time $t$ in the
presence of repeated resetting to the initial position $x_{0}$. Using the
renewal techniques [32, 43], one can write

$Q_{r}\left(t|x_{0}\right)=e^{-rt}Q_{0}\left(t|x_{0}\right)+r\int_{0}^{t}d\tau
e^{-r\tau}Q_{r}\left(t-\tau|x_{0}\right)Q_{0}\left(\tau|x_{0}\right),$ (17)

where the first term on the right-hand side (rhs) of Eq. (17) implies that the
particle survives until time $t$ and that no resetting event has occurred,
whereas the second term of the rhs assumes possibility of one or multiple
resetting events in the interval $[0,t]$. Here, we have used the fact that the
probability of not having any resets upto time $t$ is given by $e^{-rt}$ and
the probability of not exiting upto time $t$ without resetting is given by
$Q_{0}(t|x_{0})$. Taking Laplace transformation on the both sides of Eq. (17)
and solving for $\widetilde{Q}_{r}$, we obtain

$\widetilde{Q}_{r}\left(s|x_{0}\right)=\dfrac{\widetilde{Q}_{0}\left(r+s|x_{0}\right)}{1-r\widetilde{Q}_{0}\left(r+s|x_{0}\right)},$
(18)

which connects the survival probabilities for the processes with and without
resetting. Utilizing Eq. (14) into Eq. (18), we find

$\widetilde{Q}_{r}\left(s|x_{0}\right)=\dfrac{\Theta\left(a-x_{0}\right)\widetilde{Q}^{I}_{0}\left(r+s|x_{0}\right)+\Theta\left(x_{0}-a\right)\Theta\left(b-x_{0}\right)\widetilde{Q}^{II}_{0}\left(r+s|x_{0}\right)+\Theta\left(x_{0}-b\right)\widetilde{Q}^{III}_{0}\left(r+s|x_{0}\right)}{1-r\left(\Theta\left(a-x_{0}\right)\widetilde{Q}^{I}_{0}\left(r+s|x_{0}\right)+\Theta\left(x_{0}-a\right)\Theta\left(b-x_{0}\right)\widetilde{Q}^{II}_{0}\left(r+s|x_{0}\right)+\Theta\left(x_{0}-b\right)\widetilde{Q}^{III}_{0}\left(r+s|x_{0}\right)\right)},$
(19)

where the region-wise survival probabilities
$\widetilde{Q}^{I}_{0},~{}\widetilde{Q}^{II}_{0}$ and
$\widetilde{Q}^{III}_{0}$ are given by Eqs. (II.2). The mean first passage
time (MFPT) from the channel, denoted by $\langle
T_{r}\left(x_{0}\right)\rangle$, in the presence of resetting can be obtained
as

$\langle T_{r}\left(x_{0}\right)\rangle=\widetilde{Q}_{r}\left(s\rightarrow
0|x_{0}\right).$ (20)

Figure 3: Unconditional mean first-passage times $\langle T_{r}\rangle$ vs.
the initial position $x_{0}$ for a fixed value of the resetting rate $r=1$ and
several potential strengths. The solid lines depict the theoretical results
derived from Eq. (20), whereas the markers indicate the values obtained from
numerical simulations. We have set $D=1,~{}\mu=0.5$, $l=5$. The potential sink
is comprised in between the two vertical lines, at $a=1$ and $b=4$. The cusps
in this figure emanate from the matching conditions Eqs. (11) and (12), which
demonstrate that although the survival propagator remains continuous across
the potential boundaries, their first derivatives are discontinuous due to the
inclusion of a pre-factor $e^{\mu U_{0}/D}$. This results in the cusps at the
potential boundaries/edges.

In Fig. 3, we showcase the behavior of $\langle
T_{r}\left(x_{0}\right)\rangle$ as a function of the initial position $x_{0}$
for a fixed resetting rate. We note that the MFPT increases with the potential
depth $U_{0}$, but this effect is milder if $x_{0}$ is close to one of the
boundary. We find a good agreement between theoretical and numerical results.
The simulation method has been outlined in Appendix D. In Fig. 4, we
illustrate the behavior of $\langle T_{r}\left(x_{0}\right)\rangle$ with
respect to the resetting rate for three initial positions. We observe that
resetting can both accelerate or prolong the escape time of the particle from
the channel. For instance, when the particle starts close to the left or right
boundary, i.e., $x_{0}=0.9$ (orange line) or $x_{0}=4.5$ (green line), we find
that resetting reduces $\langle T_{r}\left(x_{0}\right)\rangle$ and in fact,
it even attains a minima for some resetting rate. For an initial position of
$x_{0}=2$, i.e., when the particle starts inside the potential sink, $\langle
T_{r}(x_{0})\rangle$ turns out to increase monotonously with $r$, hence
resetting delays the search time or increases the lifetime of the particle.
Thus, depending on the initial location and potential shape, the resetting
strategy is found to have adverse effect on the particle exit from the
channel. It is therefore natural to ask whether there is a general principle
that governs the effect of resetting. We elaborate on this in the next
subsection.

Figure 4: In this figure, we represent the unconditional MFPT as a function of
$r$ for three initial positions $x_{0}=0.9,~{}2.0$, and $4.5$ as mentioned in
Eq. (20). Solid lines correspond to the theoretical estimation from Eq. (20)
and the symbols represent data points from the numerical simulations. Here, we
have set: $U_{0}=1.5,~{}a=1,~{}b=4$, $D=1,~{}\mu=0.5$, and $l=5$.

###  III.2 $CV$ criterion

It is now well understood that if the resetting free first passage statistics
have large fluctuations in search time, then resetting is guaranteed to
expedite the search [33, 35, 78]. This is quantified by the so-called
$CV$-condition

$CV>1,$ (21)

where $CV$ is the coefficient of variation and its squared value is defined as
$CV^{2}=\left[\langle T_{0}(x_{0})^{2}\rangle-\langle
T_{0}(x_{0})\rangle^{2}\right]/\langle T_{0}(x_{0})\rangle^{2}$ – a dispersion
ratio between the standard deviation and the mean of the first passage time in
the absence of resetting. In Fig. 5, for a given potential configuration, we
have plotted $CV$ as a function of the initial condition $x_{0}$ alongside the
phase boundary $CV=1$ which is represented by the dashed horizontal line. This
line separates between the resetting -beneficial and -detrimental regions
spanned by $x_{0}$. It is seen that resetting is useful near the edges while
it remains detrimental through out the bulk of the channel. The insets show
the behaviors of the MFPT with $r$ for three different initial conditions
corroborating with our observations. Extending this observation, we now plot
$CV$ (Fig. 6(a)) as a function of $x_{0}$ for a symmetric potential well
($a=1$, $b=4$ and $l=5$) for different potential depth $U_{0}$ alongside the
phase boundary $CV=1$. It can be noticed that as $U_{0}$ increases, the range
of values of $x_{0}$ for which resetting is beneficial widens slightly. This
implies that in the presence of an attractive potential, resetting represents
a useful mechanism which can facilitate the escape of a particle from the
channel.

Figure 5: Coefficient of variation $CV$ in terms of the initial position
$x_{0}$, for an symmetric potential with spread $a=1$, $b=4$ and depth
$U_{0}=1.5$ in a channel of length $l=5$. The horizontal dashed line
corresponds to $CV=1$ and it separates the region above and below $CV>1$ as
mentioned in Eq. (21). The mean first passage time $\langle T_{r}\rangle$ as a
function of the resetting rate is also shown for three different $x_{0}$ with
different $CV$. For this figure, we use the same set of parameters as
mentioned in Fig. 4.

(a)

(b)

Figure 6: Panel (a): Variation of $CV$ in the phase space spanned by the
initial position $x_{0}$ and potential depth $U_{0}$. The horizontal plane
indicates $CV=1$, as mentioned in Eq. (21) and here, the orange lines indicate
the intersection of two surfaces. The other parameters are $a=1$, $b=4$ and
$l=5$. Panel (b): Variation of the optimal resetting rate $r^{*}$, as defined
in Eq. (22) with respect to $x_{0}$ and $U_{0}$. The color bar indicates the
value of $r^{*}$ across that phase space. The inset in panel (b) depicts the
variation of the optimal resetting rate $r^{*}$ in terms of the initial
position $x_{0}$ for different $U_{0}$. It is seen that $r^{*}$ continuously
undergoes a transition from a non-zero value to zero thus marking a ‘resetting
transition’ as explained in the main text.

###  III.3 Optimal resetting rate and resetting transition

The figures (4) and (5) clearly indicate that for the case whence resetting
expedites the search, a unique minima emerges in the mean first passage time.
In other words, this states that there is an optimal resetting rate which can
minimize the mean time. Formally defining, the optimal rate is given by

$r^{*}=\underset{r\geq 0}{\arg\min}\langle T_{r}(x_{0})\rangle,$ (22)

where the other parameters are kept fixed. Using the result (19) with the
identity (20), the minimization can be performed to evaluate $r^{*}$
numerically. Generically, the optimal rate will depend crucially on the
initial position and the potential depth. Naturally, the question arises: what
determines the existence of a finite optimal resetting rate? It was shown only
recently that $CV>1$ condition is a sufficient condition for the existence of
a finite $r^{*}>0$. In other words, $CV>1$ condition guarantees the existence
of atleast one minima and thus a finite optimal resetting rate. In Fig. 6(b),
we have shown a contour plot spanned in $(x_{0},U_{0})$-plane for a symmetric
channel superimposed with the optimal resetting rate in color code. It is seen
that resetting is helpful for a range of initial positions $x_{0}$ close to
the edges and this range can be further increased by making the potential
depth larger. As shown by the inset of Fig. 6(b), at a fixed $U_{0}$, the
optimal rate $r^{*}$ vanishes at a critical length $x_{0}^{c}$ inside the
channel which is seen to shift as we vary $U_{0}$. The transition of $r^{*}>0$
to $r^{*}=0$ as we vary $U_{0}$ is a reminiscent of continuous phase
transition and the scaling can be found to be
$r^{*}\sim|x_{0}-x_{0}^{c}|^{\alpha}$, with $\alpha=1$ as shown from a general
theory of resetting transition [43, 63]. Thus, the phase diagram in Fig. 6a
provides access to the parameters manipulating which we can explore the
effects of resetting.

##  IV Conditional exit time statistics from the channel

The shape of the channel clearly suggests that the particle can escape from
either of the boundaries. So far, we have looked into unconstrained exits
without having any preference to the boundary. However, constrained exits are
also important for practical purposes as one maybe interested into the
transport through one or a set of specific pores or boundaries. These
conditional exits require more detailed information on the nature of
trajectories and how they propagate through the channel. In particular, these
trajectories intricately depend on the shape of the potential, distance of the
exit points from the initial condition and also the resetting rate. In this
section we aim to look into the statistics of these conditional exit time
statistics and furthermore detail out the effects of resetting.

Figure 7: Theoretical and numerical corroboration of conditional MFPT (top)
and splitting probabilities (bottom) as a function of the resetting rate. The
three columns correspond to three different initial locations, from left to
right respectively: $x_{0}=0.9$, $x_{0}=2.5$ and $x_{0}=4.1$, while the rest
of the parameters are set fixed $a=1,~{}b=4,~{}U_{0}=1.5,~{}D=1,~{}\mu=0.5$,
$l=5$. Solid lines describe the theoretical results (using Eqs. (27),(28) &
(29)) whereas the markers indicate simulation results. We observe that the
conditional mean first passage times can be minimized with the resetting for
the asymmetric initial conditions as depicted in panels (a) and (c), whereas
resetting turns out to be detrimental for the symmetric initial condition –
panel (b). From panels (d) and (f) it is observed that the resetting becomes
useful in escaping from the left and right end respectively, with its maximum
efficiency. In other words, resetting can be tuned to enhance/depreciate the
escape probabilities in asymmetric conditions. However, resetting has no
impact on the escape probabilities when the particle starts from the center of
the channel as depicted in panel (e).

Unlike the unconditional exits, here one needs to track the trajectories
exiting from respective boundaries. To quantify these, one would need the
probability density of the particle under resetting dynamics which we define
as $p_{r}(x,t|x_{0},0)$. The renewal structure of the resetting dynamics
allows us to write an equation for the probability density in the following
way [31, 32]

$\displaystyle p_{r}\left(x,t|x_{0},0\right)$
$\displaystyle=e^{-rt}p_{0}\left(x,t|x_{0},0\right)+r\int_{0}^{t}d\tau
e^{-r\tau}p_{0}\left(x,\tau|x_{0},0\right)Q_{r}\left(t-\tau|x_{0}\right),$
(23)

where the first term in Eq. (23) corresponds to those trajectories that did
not undergo a single resetting event and the second term accumulates the
contributions from those trajectories which undergo multiple resetting events
until time $t$ without exiting the channel. The probability of resetting is
indicated by the factor $r$ while the condition of not exiting the channel
upto time $t$ in the presence of resetting is captured by the survival
probability $Q_{r}(t|x_{0})$ that was introduced in the previous section.
Taking Laplace transformation on both sides of Eq. (23) and combining with Eq.
(18), we find

$\widetilde{p}_{r}(x,s|x_{0},0)=\dfrac{\widetilde{p}_{0}(x,r+s|x_{0},0)}{1-r\widetilde{Q}_{0}(r+s|x_{0})},$
(24)

which connects the probability density functions for the process with and
without resetting respectively. To track a specific set of trajectories
exiting through a specific boundary, we need to compute the current or
particle flux through each of the boundaries. Within our set-up, we denote the
particle flux as $J_{r}^{\pm}(x_{0},t)$ where $\pm$ indicates the right or
left boundary and they are defined as $J_{r}^{\pm}(x_{0},t)=\mp
D\left.\dfrac{\partial p_{r}(x,t)}{\partial x}\right|_{x=l\ {\rm or}\ 0}$. The
exit or splitting probabilities that the particle escapes through a specific
boundary is easy to compute once the flux is known namely

$\displaystyle\epsilon^{\pm}_{r}(x_{0})=\int_{0}^{\infty}dt~{}J^{\pm}_{r}(x_{0},t),$
(25)

where $\epsilon^{+}_{r}(x_{0})$ and $\epsilon^{-}_{r}(x_{0})$ are the
probabilities that a particle starting at $x_{0}$ exits to the right and to
left respectively, with the normalization
$\epsilon^{+}_{r}(x_{0})+\epsilon^{-}_{r}(x_{0})=1$. The conditional exit time
through a specific boundary can be estimated by measuring the escape time of
the particles through that boundary suitably normalized by the particle flux
through that same exit point. Formally, they can be written as

$\langle
T^{\pm}_{r}(x_{0})\rangle=\dfrac{\int_{0}^{\infty}dt~{}t~{}J^{\pm}_{r}(x_{0},t)}{\int_{0}^{\infty}dt~{}J^{\pm}_{r}(x_{0},t)}.$
(26)

Although the time dependent quantities such as $J^{\pm}_{r}(x_{0},t)$ are
difficult to compute directly, in Laplace space they take simpler form and
furthermore can be connected to the probability density via Eq. (24). This
simple procedure allows us to compute the observables exactly. For instance,
the exit probabilities can be rewritten as

$\displaystyle\epsilon^{+}_{r}(x_{0})$
$\displaystyle=\widetilde{J}_{r}^{+}(x_{0},s\rightarrow
0)=-D\dfrac{\partial_{x}\widetilde{p}_{0}(x,r|x_{0},0)|_{x=l}}{1-r\widetilde{Q}_{0}(r|x_{0})},$
(27) $\displaystyle\epsilon^{-}_{r}(x_{0})$
$\displaystyle=\widetilde{J}_{r}^{-}(x_{0},s\rightarrow
0)=D\dfrac{\partial_{x}\widetilde{p}_{0}(x,r|x_{0},0)|_{x=0}}{1-r\widetilde{Q}_{0}(r|x_{0})},$
(28)

while the mean escape times can be reformulated in the following way from Eq.
(26)

$\langle
T^{\pm}_{r}(x_{0})\rangle=\dfrac{-\left.\dfrac{\partial\widetilde{J}^{\pm}_{r}(x_{0},s)}{\partial
s}\right|_{s\rightarrow 0}}{\widetilde{J}^{\pm}_{r}(x_{0},s\rightarrow
0)}=-\left.\dfrac{\partial}{\partial
s}\ln\left[\widetilde{J}^{\pm}(x_{0},s)\right]\right|_{s=0},$ (29)

where one can again use the relation between the current and the underlying
probability density function in the Laplace space. Thus, the exit
probabilities as a function of resetting rate are obtained by combining Eqs.
(II.2), (14) with (27)-(28), where $\widetilde{p}_{0}(x,s)$ are given in
Appendix A and the conditional MFPTs follow similarly from Eq. (29). In Fig.
7, we plot these results as a function of the resetting rate. We consider the
effect of the initial position $x_{0}$, for a symmetric potential that
partially covers a channel of length $l=5$ over a distance $b-a=3$.

To gather statistics for conditional times, we need to track two different
sets of trajectories linked with each boundary for a given initial condition.
For the symmetric landscape of potential, the favorable escapes depend
crucially on the initial condition $x_{0}$. If the particles start at the
proximity to the left boundary, then the escape probability of the particle
from the left channel is higher than the other end. The converse effect can be
obtained for those $x_{0}$, closer to the right end. Effectively, one may
expect that $\epsilon^{-}_{0}$ decreases to $0$ from $1$ as $x_{0}$ varies
from $0\rightarrow l$, whereas $\epsilon^{+}_{0}$ increases conversely for the
same variation of $x_{0}$. When the particle starts its journey from
$x_{0}=l/2$ the escape events are identical i.e.,
$\epsilon^{-}_{0}(x_{0}=l/2)=\epsilon^{+}_{0}(x_{0}=l/2)=\dfrac{1}{2}$.
Conditional escape times should also be prioritized accordingly.

Additionally, imposing resetting on the dynamics causes optimization in
conditional mean escape times for some initial conditions. From Figs. 7(a) &
(c), we observe that resetting minimizes the mean conditional escape times for
asymmetric initial conditions $x_{0}=0.9~{}\&~{}4.1$. Moreover, the
trajectories associated with the long excursions can be discarded with
intermittent returns in the presence of resetting; as a consequence, the
biasing of escapes becomes stronger (see Figs. 7(d) & (f)). For higher
resetting rates, these probabilities saturate to maximum value which is unity.
In addition, we observe in Fig. 7(b), that resetting prolongs the escapes for
the symmetric initial condition $x_{0}=2.5=l/2$, and the escape probabilities
can not be optimized further. To understand better, we further analyze the
behavior of the rate of escape probabilities i.e., $d\epsilon_{\pm}/dr$ in the
limit of $r\rightarrow 0$. As the initial condition is varied from the left
side to the right side of the channel, we observe a systematic optimal
behavior from the escape trajectories barring the symmetric initial conditions
(see Fig. 8). For all $x_{0}$, like in the case of a free particle diffusing
in an interval ($U_{0}=0$) [43], resetting tends to favour the escape through
one end at the expense of the other, as shown by the fact that
$\epsilon^{+}_{r}(x_{0})\gg\epsilon^{-}_{r}(x_{0})$ or
$\epsilon^{+}_{r}(x_{0})\ll\epsilon^{-}_{r}(x_{0})$ at large $r$.
Nevertheless, when the particle starts at the bottom of the potential
($x_{0}=2$), the difference between the splitting probabilities becomes less
pronounced. In summary, we conclude that the initial condition plays a crucial
role in the resetting mediated transport statistics.

Figure 8: The rate of change in the escape probabilities associated with the
left, right boundary as a function of $x_{0}$, which are represented with the
blue and orange solid lines respectively. For instance, when the particle
starts from $x_{0}<l/2$, the slope is always positive for $\epsilon_{r}^{-}$
(blue curve) as we expect the particles to escape from the left boundary more
frequently. Eventually, $\epsilon_{r}^{-}$ becomes unity as we increase the
resetting rate (see also Fig. 7d). As we switch the initial conditions to
$x_{0}>l/2$, the particle has more probability to escape from the right
boundary and furthermore, with resetting $\epsilon_{r}^{+}$ is enhanced to
unity (see Fig. 7f). This is evident from the positive value of the slope for
$\epsilon_{r}^{+}$ for $x_{0}>l/2$ (orange curve). The slope vanishes exactly
at $x_{0}=l/2$ which is the midpoint of the channel and thus the escape is
symmetric with probabilities one-half (also see Fig. 7e). Here, the parameters
were set at: $a=1,~{}b=4,~{}U_{0}=1.5,~{}D=1,~{}\mu=0.5,~{}l=5$.

##  V conclusion

In summary, we have investigated the escape statistics of a Brownian particle
confined in a channel and subject to a binding potential in the presence of
resetting dynamics. Attractive interactions tend to hold the particle for a
longer time inside the channel thus prolonging the escape time. Thus,
designing navigational strategies which can mitigate such delays are of much
need. Here, we show that the resetting dynamics can eliminate looping as well
as wandering trajectories inside the channel and can render the transport more
efficient. In effect, resetting reduces the mean and the fluctuations of the
escape times drastically and we have underpinned the physical scenarios behind
such effects. Moreover, we have sketched out a phase diagram in the parameter
space spanned by the initial position and the potential strength that provides
us a priori knowledge on the modulation of the parameters that could be useful
for harnessing the escape from the channel.

We approached the problem using the Fick-Jacobs formalism that allows one to
reduce the three-dimensional diffusion problem to one dimension with an
effective diffusivity containing the details of the channel geometry. After
studying in detail the unconditional exit time and its modulation with respect
to the resetting rate, we looked into the optimal behavior of the escape times
and how they can be quantified in practice.

Overall, this study indicates that resetting can accelerate the crossing or
escape time of particles, a characteristic shared by numerous other stochastic
processes that involve resetting. However, this effect is not random; it has
been consistently shown that resetting is advantageous when stochastic
fluctuations in the completion time of a random process are significant,
specifically when the coefficient of variation in the absence of resetting
($CV$) exceeds unity [35, 78]. We have explicitly verified this criterion for
various potential wells. The effect of resetting has been further analyzed by
observing the behavior of the optimal resetting rate that minimizes the mean
escape time. In addition, we have reported a continuous transition for the
optimal resetting rate when the initial/resetting position is varied.
Extending this formalism, we computed the escape time statistics conditioned
on the escape nature of the particles through different boundaries. The
conditional escape times are found to display nontrivial behavior as the
resetting rate is varied. It is noticed that resetting can induce a faster
escape for the trajectories which start very close to an undesired boundary
and escape through the further boundary. Looping transient trajectories inside
the potential well, which usually cause large fluctuations in such cases, get
curtailed by resetting hence a speed-up in the overall first passage time is
observed.

The problem of matter transport through confined geometries containing narrow
openings, corners and bottlenecks has recently led to many theoretical and
experimental research fronts. Recent single particle experiments have also
shown interesting observations namely identifying the structure of a channel
or generically the number of intermediate configurations from the first
passage or escape time statistics of various living and non-living agents
[79]. It would be worthwhile to look into this inference problem within our
set-up eg. whether one can infer the number of bottlenecks from the first
passage time statistics from a channel. It is evident that resetting frequency
can be used as a probe to mitigate particle clogging inside the channel which
can otherwise hinder the transport – it will be interesting to investigate
such effects in soft-matter experiments who deal with colloidal particles and
resetting [66, 69, 67, 68].

##  VI Acknowledgements

AP gratefully acknowledges the DST-SERB Start-up Research Grant Number
SRG/2022/000080 and the Department of Atomic Energy, India for research
funding (via the project ‘Modeling of Soft Materials’). This study was
partially supported by CONACyT under the grant Frontiers Science No. 51476.
The authors thank the Raman Research Institute, Bangalore for the hospitality
during the conference ‘Frontiers in Statistical Physics’ where some initial
discussion took place. The authors also thank the Higgs Center for Theoretical
Physics, Edinburgh for hospitality during the workshop ‘New Vistas in
Stochastic resetting’ where the project concluded.

##  Appendix A Probability density functions without resetting

To utilize the renewal relation in Eq. (23), it is essential to know the
propagator of the underlying reset-free process in Laplace space. In this
section, we solve the forward Smoluchowski equation for the probability
density function, $\widetilde{p}_{0}\left(x,s|x_{0},0\right)$ with absorbing
boundaries at two ends of the channel. Taking Laplace transformation of Eq.
(1), we get

$\displaystyle s\widetilde{p}_{0}\left(x,s\right)-$
$\displaystyle\delta\left(x-x_{0}\right)=D\dfrac{\partial}{\partial
x}\left[e^{-\beta U(x)}\dfrac{\partial}{\partial x}\left(e^{\beta
U(x)}\widetilde{p}_{0}\left(x,s\right)\right)\right].$ (30)

As mentioned earlier, the shape of the potential allows us to solve Eq. (30)
in a piece-wise manner in three different regions. Regime wise i.e., based on
the final position, we denoted these corresponding propagators as
$\widetilde{p}_{0}^{L,M,R}(x,s|x_{0},s)$ in the main text. However, the
particle can propagate to any one of these regimes at the final time $t$
starting either from the same regime or elsewhere at time $0$. Thus, it is
meaningful also to split these probability densities based on the exact
initial condition. The notation for the break-up in the initial condition is
denoted by $I,II,III$. For instance, $\widetilde{p}_{0}^{L,I}(x,s|x_{0},0)$
denotes the probability density function of the particle to be found in the
domain $0<x<a$ given that it had started in the interval $0<x_{0}<a$ at time
zero. Likewise, other propagators can also be written based on the final
position and the starting point. In what follows, we solve Eq. (30) in detail
based on these splittings.

###  A.1 Interval $0<x_{0}<a$

We first consider that the initial position $x_{0}$ of the particle lies to
the left part of the channel (referred to by the index $I$ in the expression
below), but the final position $x$ can be in any of the three regions. Thus,
Eq. (30) can be modified as

$\begin{cases}s\widetilde{p}_{0}^{L,I}\left(x,s\right)-\delta\left(x-x_{0}\right)=D\dfrac{\partial^{2}}{\partial
x^{2}}\widetilde{p}_{0}^{L,I}\left(x,s\right)~{}~{}~{}~{}~{}0<x<a,\\\ \\\
s\widetilde{p}_{0}^{M,I}\left(x,s\right)=D\dfrac{\partial^{2}}{\partial
x^{2}}\widetilde{p}_{0}^{M,I}\left(x,s\right)~{}~{}~{}~{}~{}a<x<b,\\\ \\\
s\widetilde{p}_{0}^{R,I}\left(x,s\right)=D\dfrac{\partial^{2}}{\partial
x^{2}}\widetilde{p}_{0}^{R,I}\left(x,s\right)~{}~{}~{}~{}~{}b<x<l,\\\
\end{cases}$ (31)

where we denote $\widetilde{p}_{0}^{L,I},\widetilde{p}_{0}^{M,I}$ and
$\widetilde{p}_{0}^{R,I}$ as solutions of the probability density function in
the Laplace domain, and the suffixes denote the left, middle, and right
solutions respectively, for $x_{0}$ belonging to the left part. Using the
absorbing boundary conditions, one can solve Eqs. (31) and the corresponding
solutions read

$\begin{cases}\widetilde{p}_{0}^{L,I}\left(x,s|x_{0},0\right)=\dfrac{1}{2\alpha_{0}D}e^{-\alpha(x+x_{0})}\left[2\alpha_{0}DA^{L,I}\left(e^{2\alpha
x}-1\right)e^{\alpha x_{0}}+\Theta(x-x_{0})\left(e^{2\alpha x_{0}}-e^{2\alpha
x}\right)\right],\\\ \\\
\widetilde{p}_{0}^{M,I}\left(x,s|x_{0},0\right)=A^{M1,I}e^{\alpha_{0}x}+A^{M2,I}e^{-\alpha_{0}x},\\\
\\\
\widetilde{p}_{0}^{R,I}\left(x,s|x_{0},0\right)=A^{R,I}e^{-\alpha_{0}x}\left(e^{2\alpha_{0}x}-e^{2\alpha_{0}l}\right),\end{cases}$
(32)

where while solving the first equation in Eq. 32 we have also used the
following matching conditions due to the Dirac delta function: (i) continuity
of the probabilities, (ii) discontinuity in the derivative of the
probabilities across the delta function.

One can derive the relations among the constants $A^{L,I}$, $A^{M1,I}$,
$A^{M2,,I}$ and $A^{R,I}$ in Eqs. (32) by applying the matching conditions
(5)-(6):

$\begin{cases}2\sinh{\left[a\alpha_{0}\right]}A^{L,I}=\dfrac{1}{\alpha_{0}D}\sinh{\left[(a-x_{0})\alpha_{0}\right]}+e^{-a\alpha_{0}-U_{0}\beta}\left(A^{M1,I}e^{2a\alpha_{0}}+A^{M2,I}\right),\\\
\\\
A^{M1,I}e^{2b\alpha_{0}}+A^{M2,I}=A^{R,I}e^{U_{0}\beta}\left(e^{2b\alpha_{0}}-e^{2l\alpha_{0}}\right),\\\
\\\
\alpha_{0}De^{-a\alpha_{0}}\left[A^{L,I}+e^{2a\alpha_{0}}\left(A^{L,I}-A^{M1,I}\right)+A^{M2,I}\right]=\cosh{\left[(a-x_{0})\alpha_{0}\right]},\\\
\\\
\alpha_{0}e^{b\alpha_{0}}A^{M1,I}=\alpha_{0}e^{-b\alpha_{0}}\left[A^{M2,I}+A^{R,I}\left(e^{2b\alpha_{0}}+e^{2l\alpha_{0}}\right)\right].\end{cases}$
(33)

These coefficients can be obtained by using Mathematica. One can gather all
the solutions given in Eqs. (32) in one equation as

$\widetilde{p}_{0}^{I}\left(x,s|x_{0},0\right)=\Theta\left(a-x\right)\widetilde{p}_{0}^{L,I}\left(x,s|x_{0},0\right)+\Theta\left(x-a\right)\Theta\left(b-x\right)\widetilde{p}_{0}^{M,I}\left(x,s|x_{0},0\right)+\Theta\left(x-b\right)\widetilde{p}_{0}^{R,I}\left(x,s|x_{0},0\right),$
(34)

###  A.2 Interval $a<x_{0}<b$

Similarly, we rewrite Eq. (30) when the initial position lies in the middle
part of the potential (referred to by the index $II$ below),

$\begin{cases}s\widetilde{p}_{0}^{L,II}\left(x,s\right)=D\dfrac{\partial^{2}}{\partial
x^{2}}\widetilde{p}_{0}^{L,II}\left(x,s\right)~{}~{}~{}~{}~{}0<x<a,\\\ \\\
s\widetilde{p}_{0}^{M,II}\left(x,s\right)-\delta\left(x-x_{0}\right)=D\dfrac{\partial^{2}}{\partial
x^{2}}\widetilde{p}_{0}^{M,II}\left(x,s\right)~{}~{}~{}~{}~{}a<x<b,\\\ \\\
s\widetilde{p}_{0}^{R,II}\left(x,s\right)=D\dfrac{\partial^{2}}{\partial
x^{2}}\widetilde{p}_{0}^{R,II}\left(x,s\right)~{}~{}~{}~{}~{}b<x<l.\\\
\end{cases}$ (35)

From the boundary conditions, the solutions of Eqs. (35) read

$\begin{cases}\widetilde{p}_{0}^{L,II}\left(x,s|x_{0},0\right)=2A^{L,II}\sinh{\left[\alpha_{0}x\right]},\\\
\\\
\widetilde{p}_{0}^{M,II}\left(x,s|x_{0},0\right)=A^{M1,II}e^{\alpha_{0}x}+A^{M2,II}e^{-\alpha_{0}x}-\dfrac{1}{\alpha_{0}D}\sinh{\left[(x-x_{0})\alpha_{0}\right]}\Theta\left(x-x_{0}\right),\\\
\\\
\widetilde{p}_{0}^{R,II}\left(x,s|z,0\right)=-2A^{R,II}e^{L\alpha_{0}}\sinh{\left[(l-x)\alpha_{0}\right]}.\end{cases}$
(36)

Also, the relations among the constants are derived as in Sec. A.1 from the
matching conditions (5)-(6),

$\begin{cases}2\sinh{\left[a\alpha_{0}\right]}A^{L,II}=e^{-\beta
U_{0}}\left(e^{a\alpha_{0}}A^{M1,II}+e^{-a\alpha_{0}}A^{M2,II}\right),\\\ \\\
-2A^{R,II}e^{L\alpha_{0}}\sinh{\left[(l-b)\alpha_{0}\right]}=e^{-\beta
U_{0}}\left(A^{M1,II}e^{b\alpha_{0}}+A^{M2,II}e^{-b\alpha_{0}}-\dfrac{1}{\alpha_{0}D}\sinh{\left[(b-x_{0})\alpha_{0}\right]}\right),\\\
\\\
2A^{L,II}\cosh{\left[a\alpha_{0}\right]}+A^{M2,II}e^{-a\alpha_{0}}=A^{M1,II}e^{a\alpha_{0}},\\\
\\\
e^{b\alpha_{0}}\alpha_{0}A^{M1,II}=\dfrac{1}{D}\cosh{\left[(b-x_{0})\alpha_{0}\right]}+\alpha_{0}e^{-b\alpha_{0}}\left[A^{R,II}\left(e^{2l\alpha_{0}}+e^{2b\alpha_{0}}\right)+A^{M2,II}\right].\end{cases}$
(37)

After solving this system of equations with Mathematica, we get the following
form of the probability density function in Laplace space

$\widetilde{p}_{0}^{II}\left(x,s|x_{0},0\right)=\Theta\left(a-x\right)\widetilde{p}_{0}^{L,II}\left(x,s|x_{0},0\right)+\Theta\left(x-a\right)\Theta\left(b-x\right)\widetilde{p}_{0}^{M,II}\left(x,s|x_{0},0\right)+\Theta\left(x-b\right)\widetilde{p}_{0}^{R,II}\left(x,s|x_{0},0\right).$
(38)

###  A.3 Interval $b<x_{0}<l$

When the initial condition lies in the last, rightmost segment (referred to as
by the index $III$), Eq. (30) is written as

$\begin{cases}s\widetilde{p}_{0}^{L,III}\left(x,s\right)=D\dfrac{\partial^{2}}{\partial
x^{2}}\widetilde{p}_{0}^{L,III}\left(x,s\right)~{}~{}~{}~{}~{}0<x<a,\\\ \\\
s\widetilde{p}_{0}^{M,III}\left(x,s\right)=D\dfrac{\partial^{2}}{\partial
x^{2}}\widetilde{p}_{0}^{M,III}\left(x,s\right)~{}~{}~{}~{}~{}a<x<b,\\\ \\\
s\widetilde{p}_{0}^{R,III}\left(x,s\right)-\delta\left(x-x_{0}\right)=D\dfrac{\partial^{2}}{\partial
x^{2}}\widetilde{p}_{0}^{R,III}\left(x,s\right)~{}~{}~{}~{}~{}b<x<l.\\\
\end{cases}$ (39)

From the boundary conditions, we get the following form for the solutions of
Eq. (39)

$\begin{cases}\widetilde{p}_{0}^{L,III}\left(x,s|x_{0},0\right)=2A^{L,III}\sinh{\left[\alpha_{0}x\right]},\\\
\\\
\widetilde{p}_{0}^{M,III}\left(x,s|x_{0},0\right)=A^{M1,III}e^{\alpha_{0}x}+A^{M2,III}e^{-\alpha_{0}x},\\\
\\\ \widetilde{p}_{0}^{R,III}\left(x,s|x_{0},0\right)=\\\ \\\
\dfrac{1}{2D\alpha_{0}}e^{-(x+x_{0})\alpha_{0}}\left[e^{2l\alpha_{0}}-e^{2x_{0}\alpha_{0}}+\left(e^{2x_{0}\alpha_{0}}-e^{2b\alpha_{0}}\right)\Theta\left(x-x_{0}\right)\right]\\\
\\\
~{}~{}~{}~{}~{}~{}~{}-A^{R,III}e^{x_{0}\alpha_{0}}\left(e^{2l\alpha_{0}}-e^{2x\alpha_{0}}\right).\end{cases}$
(40)

Using the matching conditions (5)-(6), the constants $A^{L,III}$,
$A^{M1,III}$, $A^{M2,III}$ and $A^{R,III}$ satisfy the relations

$\begin{cases}2\sinh{\left[a\alpha_{0}\right]}A^{L,III}=e^{-\beta
U_{0}}\left(e^{a\alpha_{0}}A^{M1,III}+e^{-a\alpha_{0}}A^{M2,III}\right),\\\
\\\
\dfrac{e^{-(b+x_{0})\alpha_{0}}}{2D\alpha_{0}}\left[e^{2l\alpha_{0}}-e^{2x_{0}\alpha_{0}}+2\alpha_{0}De^{x_{0}\alpha_{0}}\left(e^{2b\alpha_{0}}-e^{2l\alpha_{0}}\right)A^{R,III}\right]\\\
\\\ ~{}~{}~{}~{}~{}=e^{-\beta
U_{0}}\left(A^{M1,III}e^{b\alpha_{0}}+A^{M2,III}e^{-b\alpha_{0}}\right),\\\
\\\
2\cosh{\left[a\alpha_{0}\right]}A^{L,III}+e^{-a\alpha_{0}}A^{M2,III}=e^{a\alpha_{0}}A^{M1,III},\\\
\\\
\alpha_{0}\left(e^{b\alpha_{0}}A^{M1,III}-e^{-b\alpha_{0}}A^{M2,III}\right)=2\alpha_{0}e^{b\alpha_{0}}A^{R,III}\\\
\\\
+\dfrac{1}{2D}e^{-(b+x_{0})\alpha_{0}}\left[e^{2x_{0}\alpha_{0}}-e^{2l\alpha_{0}}-2De^{x_{0}\alpha_{0}}\left(e^{2b\alpha_{0}}-e^{2l\alpha_{0}}\right)\alpha_{0}A^{R,III}\right].\end{cases}$
(41)

After solving this system with Mathematica, one gathers the solutions in a
single form as

$\widetilde{p}_{0}^{III}\left(x,s|x_{0},0\right)=\Theta\left(a-x\right)\widetilde{p}_{0}^{L,III}\left(x,s|x_{0},0\right)+\Theta\left(x-a\right)\Theta\left(b-x\right)\widetilde{p}_{0}^{M,III}\left(x,s|x_{0},0\right)+\Theta\left(x-b\right)\widetilde{p}_{0}^{R,III}\left(x,s|x_{0},0\right).$
(42)

Combining all the probability density functions from Eqs. (34),(38) & (42) in
one equation, one can write a full solution for the probability density
function in the following way

$\widetilde{p}_{0}\left(x,s|x_{0},0\right)=\Theta\left(a-x_{0}\right)\widetilde{p}_{0}^{I}\left(x,s|x_{0},0\right)+\Theta\left(x_{0}-a\right)\Theta\left(b-x_{0}\right)\widetilde{p}_{0}^{II}\left(x,s|x_{0},0\right)+\Theta\left(x_{0}-b\right)\widetilde{p}_{0}^{III}\left(x,s|x_{0},0\right).$
(43)

##  Appendix B Derivation of the matching conditions in Eqs. (11) and (12)

The survival probability $Q_{0}(t|x_{0})$ of the Brownian particle without
resetting inside the $1d$ channel satisfies the backward Fokker-Planck
equation

$\dfrac{\partial Q_{0}}{\partial t}=D\dfrac{\partial^{2}Q_{0}}{\partial
x_{0}^{2}}-\mu U^{\prime}(x_{0})\dfrac{\partial Q_{0}}{\partial x_{0}},$ (44)

where $D$ is the diffusion constant, $\mu$ the mobility and $U(x)$ the
external potential. This equation can be rewritten as

$\displaystyle\dfrac{\partial Q_{0}}{\partial t}=De^{\beta
U(x_{0})}\dfrac{\partial}{\partial x_{0}}\left(e^{-\beta
U(x_{0})}\dfrac{\partial Q_{0}}{\partial x_{0}}\right),$ (45)

where $\beta=\mu/D=(k_{B}T)^{-1}$. Using the Laplace transform of $Q$,

$\widetilde{Q}_{0}(s|x_{0})=\int_{0}^{\infty}dt~{}e^{-st}Q_{0}(t|x_{0}),$ (46)

the transform of Eq. (45) gives

$-1+s\widetilde{Q}_{0}=De^{\beta U(x_{0})}\dfrac{\partial}{\partial
x_{0}}\left(e^{-\beta U(x_{0})}\dfrac{\partial\widetilde{Q}_{0}}{\partial
x_{0}}\right),$ (47)

where the initial condition $Q_{0}(t=0|x_{0})=1$ for any position $x_{0}$
other than the absorbing boundaries has been applied.

To determine a first matching condition at $x=a$ given the discontinuous
potential in Eq. (3), let us multiply Eq. (47) by $e^{-\beta U(x_{0})}$ and
integrate in a small interval $\left[a-\epsilon,a+\epsilon\right]$ around $a$,

$\int_{a-\epsilon}^{a+\epsilon}dx_{0}~{}e^{-\beta
U(x_{0})}\left[-1+s\widetilde{Q}_{0}(s|x_{0})\right]=D\left[e^{-\beta
U(x_{0})}\dfrac{\partial\widetilde{Q}_{0}}{\partial
x_{0}}\right]_{a-\epsilon}^{a+\epsilon}.$ (48)

The LHS of Eq. (48) tends to $0$ as $\epsilon\rightarrow 0$, because
$\widetilde{Q}_{0}(s|x_{0})$ does not have any singularity (since $0\leq
Q_{0}(t|x_{0})\leq 1$ then $0\leq\widetilde{Q}_{0}(s|x_{0})\leq 1/s$). One
deduces the boundary condition

$\dfrac{\partial\widetilde{Q}_{0}(s|a^{+})}{\partial x_{0}}=e^{-\beta
U_{0}}\dfrac{\partial\widetilde{Q}_{0}(s|a^{-})}{\partial x_{0}},$ (49)

and similarly,

$\dfrac{\partial\widetilde{Q}_{0}(s|b^{+})}{\partial x_{0}}=e^{\beta
U_{0}}\dfrac{\partial\widetilde{Q}_{0}(s|b^{-})}{\partial x_{0}}.$ (50)

Since Eqs. (49) and (50) are valid for any $s$, they also hold in the time
domain.

To derive the other matching condition at $x_{0}=a$, we take the Laplace
transform of Eq. (44) and use the identity $U^{\prime}(x)=-U_{0}\delta(x-a)$
for the discontinuous potential in the left part of the channel. We thus have

$-1+s\widetilde{Q}_{0}=D\dfrac{\partial^{2}\widetilde{Q}_{0}}{\partial
x_{0}^{2}}+\mu U_{0}\delta(x_{0}-a)\dfrac{\partial\widetilde{Q}_{0}}{\partial
x_{0}}.$ (51)

To avoid to deal with the discontinuity of $\partial\tilde{Q}_{0}/\partial
x_{0}$ at $a$ during integration, let us multiply the above equation by
$x_{0}-a$. Making the change of variable $u=x_{0}-a$,

$u\left[-1+s\widetilde{Q}_{0}(s|u)\right]=Du\dfrac{\partial^{2}\widetilde{Q}_{0}(s|u)}{\partial
u^{2}}+\mu uU_{0}\delta(u)\dfrac{\partial\widetilde{Q}_{0}(s|u)}{\partial u}.$
(52)

We again integrate over $u$ in a small interval $[-\epsilon,\epsilon]$. The
first term of the rhs of Eq. (52) can be integrated by parts and the second
one is identically $0$. Hence,

$\int_{-\epsilon}^{\epsilon}du~{}u~{}\left[-1+s\widetilde{Q}_{0}(s|u)\right]=D\epsilon\left(\dfrac{\partial\widetilde{Q}_{0}(s|\epsilon)}{\partial
u}+\dfrac{\partial\widetilde{Q}_{0}(s|-\epsilon)}{\partial
u}\right)-D\left[\widetilde{Q}_{0}(s|\epsilon)-\widetilde{Q}_{0}(s|-\epsilon)\right].$
(53)

We see that the lhs and the first term of the rhs of Eq. (53) tend to $0$ as
$\epsilon\rightarrow 0$, therefore we conclude that $\widetilde{Q}_{0}$ is
continuous at $x_{0}=a$,

$\widetilde{Q}_{0}(s|a^{+})=\widetilde{Q}_{0}(s|a^{-}).$ (54)

Similarly near $x_{0}=b$,

$\widetilde{Q}_{0}(s|b^{+})=\widetilde{Q}_{0}(s|b^{-}).$ (55)

These relations clearly hold in the time domain as well. The matching
conditions derived here are crucial to compute the survival probability
through the entire channel.

##  Appendix C Relations among the coefficients of Eq. (II.2) for
$\widetilde{Q}_{0}(s|x_{0})$

Here, we provide the necessary relations to evaluate the coefficients
$\\{A,B,C_{1},C_{2}\\}$ in Eq. (II.2). Using the matching conditions (11)-(12)
we get the following relations

$\begin{cases}\dfrac{e^{-\alpha_{0}a}}{s}+2A\cosh{\alpha_{0}a}=e^{\mu
U_{0}/D}\left(C_{1}e^{\alpha_{0}a}-C_{2}e^{-\alpha_{0}a}\right),\\\
e^{\alpha_{0}l}+s~{}B\left(e^{2\alpha_{0}b}+e^{2\alpha_{0}l}\right)=s~{}e^{\mu
U_{0}/D}\left(e^{\alpha_{0}b}C_{1}-C_{2}\right),\\\
s~{}C_{2}=s~{}(A-C_{1})e^{2\alpha_{0}a}-A~{}s-1,\\\
s~{}C_{2}=s~{}(B-C_{1})e^{2\alpha_{0}b}-e^{\alpha_{0}l}-s~{}Be^{2\alpha_{0}l}.\end{cases}$
(56)

These coefficients are obtained explicitly with Mathematica. Since their
expressions are rather long, we do not reproduce them here.

##  Appendix D Details of the numerical simulations and implementation of the
boundary conditions

Figure 9: Schematical representation of the form of the potential in terms of
two linear potentials.

In this section, we elaborate on the simulation method of our model. The
presence of discontinuities in the potential implies delta functions in the
force. Here, we replace the discontinuities of the box-shaped potential given
in Eq. (3) by two linear functions with a large slope,

$U(x)=\begin{cases}0&0<x<a-\delta\\\
-U_{0}\dfrac{x-(a-\delta)}{2\delta}&a-\delta\leq x\leq a+\delta\\\
-U_{0}&a+\delta<x<b-\delta\\\ U_{0}\dfrac{x-(b+\delta)}{2\delta}&b-\delta\leq
x\leq b+\delta\\\ 0&b+\delta<x<l,\end{cases}$ (57)

where $\delta$ is the standard deviation of the free Brownian displacement
between two consecutive times $t$ and $t+\Delta t$. Hence,
$\delta=\sqrt{2D~{}\Delta t}$ with $\Delta t$ the numerical time step of the
simulation. We run the Langevin dynamics with the force field given by
$f(x)=-\dfrac{d}{dx}U(x)$. Starting from $x_{0}$ at $t=0$, we have updated the
particle position in the following way:

  1. 1.

Call a random number $\lambda$ from the uniform distribution in $(0,1)$ and
check the resetting condition: if $\lambda<r\Delta t$ then bring the particle
back to the initial position $x_{0}$, otherwise perform step 2.

  2. 2.

Update the position as

$x(t+\Delta t)=x(t)+\mu f(x(t))\Delta t+\sqrt{2D~{}\Delta t}~{}\xi(t+\Delta
t),$ (58)

where $\xi(t+\Delta t)$ is a normally distributed random number with zero mean
and unit variance. As usual, $\mu$ is the mobility of the particle and $D$ is
the diffusion coefficient.

  3. 3.

Depending on the updated and previous positions of the particle, we invoke the
force term. The force acts on the particle when
$x(t)\in(a-\delta,a+\delta)\cup(b-\delta,b+\delta)$. When the particle crosses
the point $x=a$, it might also happen that $|x(t+\Delta t)-x(t)|>2\delta$ with
$|x(t)-a|>\delta$, i.e., the particle did not feel the force term. To overcome
this difficulty, we have added the constant force term in Eq. (58) despite of
having $|x(t)-a|>\delta$, when the Brownian term of Eq. (58) alone makes the
particle cross the force field near the position $x=a$.

  4. 4.

To track the first passage time, we impose the boundary conditions: i.e.,
$x(t+\Delta t)<0\bigcup x(t+\Delta t)>l$. If one of these conditions is
satisfied, we terminate the simulation, make a note of the corresponding time
step number $n$, multiply with the microscopic time step $\Delta t$ and the
total time $n\Delta t$ contributes to the unconditional first passage time.
Over many realizations, we collect these random times and gather the first
passage time statistics.

## References

  * Dagdug  _et al._ [2024] L. Dagdug, J. Peña, and I. Pompa-García,  _Diffusion Under Confinement_ (Springer, Cham, 2024).
  * Rostovtseva  _et al._ [2008] T. K. Rostovtseva, K. L. Sheldon, E. Hassanzadeh, C. Monge, V. Saks, S. M. Bezrukov, and D. L. Sackett, Tubulin binding blocks mitochondrial voltage-dependent anion channel and regulates respiration, Proceedings of the National Academy of Sciences 105, 18746 (2008).
  * Hoogerheide  _et al._ [2017] D. P. Hoogerheide, P. A. Gurnev, T. K. Rostovtseva, and S. M. Bezrukov, Mechanism of $\alpha$-synuclein translocation through a vdac nanopore revealed by energy landscape modeling of escape time distributions, Nanoscale 9, 183 (2017).
  * Fick [1855] A. Fick, Ueber diffusion, Annalen der Physik 170, 59 (1855).
  * Jacobs [1967] M. Jacobs, Diffusion processes; springer, Berlin Heidelberg New York (1967).
  * Zwanzig [1992] R. Zwanzig, Diffusion past an entropy barrier, The Journal of Physical Chemistry 96, 3926 (1992).
  * Berezhkovskii and Szabo [2011] A. Berezhkovskii and A. Szabo, Time scale separation leads to position-dependent diffusion along a slow coordinate, The Journal of Chemical Physics 135 (2011).
  * Dagdug and Pineda [2012] L. Dagdug and I. Pineda, Projection of two-dimensional diffusion in a curved midline and narrow varying width channel onto the longitudinal dimension, The Journal of Chemical Physics 137 (2012).
  * Rubi [2019] J. M. Rubi, Entropic diffusion in confined soft-matter and biological systems, Europhysics Letters 127, 10001 (2019).
  * Haul [1993] R. Haul, J. kärger, dm ruthven: Diffusion in zeolites and other microporous solids, j. wiley & sons inc, new york 1992. isbn 0-471-50907-8. 605 seiten, preis:£ 117 (1993).
  * Han and Craighead [2000] J. Han and H. G. Craighead, Separation of long dna molecules in a microfabricated entropic trap array, Science 288, 1026 (2000).
  * Gershow and Golovchenko [2007] M. Gershow and J. A. Golovchenko, Recapturing and trapping single molecules with a solid-state nanopore, Nature Nanotechnology 2, 775 (2007).
  * Dagdug and Berezhkovskii [2009] L. Dagdug and A. M. Berezhkovskii, Drift and diffusion in periodic potentials: Upstream and downstream step times are distributed identically, The Journal of Chemical Physics 131 (2009).
  * Skvortsov  _et al._ [2018] A. T. Skvortsov, A. M. Berezhkovskii, and L. Dagdug, Trapping of diffusing particles by spiky absorbers, The Journal of Chemical Physics 148 (2018).
  * Verdel  _et al._ [2016] R. Verdel, L. Dagdug, A. M. Berezhkovskii, and S. M. Bezrukov, Unbiased diffusion in two-dimensional channels with corrugated walls, The Journal of Chemical Physics 144 (2016).
  * Berezhkovskii  _et al._ [2003] A. M. Berezhkovskii, M. A. Pustovoit, and S. M. Bezrukov, Channel-facilitated membrane transport: Average lifetimes in the channel, The Journal of chemical physics 119, 3943 (2003).
  * Dagdug  _et al._ [2003] L. Dagdug, A. Berezhkovskii, S. M. Bezrukov, and G. H. Weiss, Diffusion-controlled reactions with a binding site hidden in a channel, The Journal of Chemical Physics 118, 2367 (2003).
  * Berezhkovskii  _et al._ [2006] A. M. Berezhkovskii, G. Hummer, and S. M. Bezrukov, Identity of distributions of direct uphill and downhill translocation times for particles traversing membrane channels, Physical Review Letters 97, 020601 (2006).
  * Berezhkovskii  _et al._ [2017] A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, Mean direct-transit and looping times as functions of the potential shape, The Journal of Physical Chemistry B 121, 5455 (2017).
  * Chung  _et al._ [2012] H. S. Chung, K. McHale, J. M. Louis, and W. A. Eaton, Single-molecule fluorescence experiments determine protein folding transition path times, Science 335, 981 (2012).
  * Berezhkovskii  _et al._ [2020] A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, Peculiarities of the mean transition path time dependence on the barrier height in entropy potentials, The Journal of Physical Chemistry B 124, 2305 (2020).
  * Ferrer and Gomez-Solano [2024] B. R. Ferrer and J. R. Gomez-Solano, Experimental measurement of mean transition path velocities of colloidal particles surmounting energy barriers, arXiv preprint arXiv:2404.14699 (2024).
  * Reguera  _et al._ [2006] D. Reguera, G. Schmid, P. S. Burada, J. Rubi, P. Reimann, and P. Hänggi, Entropic transport: Kinetics, scaling, and control mechanisms, Physical Review Letters 96, 130603 (2006).
  * Burada  _et al._ [2007] P. S. Burada, G. Schmid, D. Reguera, J. Rubi, and P. Hänggi, Biased diffusion in confined media: Test of the fick-jacobs approximation and validity criteria, Physical Review E 75, 051111 (2007).
  * Burada  _et al._ [2009] P. S. Burada, P. Hänggi, F. Marchesoni, G. Schmid, and P. Talkner, Diffusion in confined geometries, ChemPhysChem 10, 45 (2009).
  * Redner [2001] S. Redner,  _A guide to first-passage processes_ (Cambridge University Press, Cambridge, 2001).
  * Bray  _et al._ [2013] A. J. Bray, S. N. Majumdar, and G. Schehr, Persistence and first-passage properties in nonequilibrium systems, Advances in Physics 62, 225 (2013).
  * 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).
  * Zhang and Dudko [2016] Y. Zhang and O. K. Dudko, First-passage processes in the genome, Annual review of biophysics 45, 117 (2016).
  * Iyer-Biswas and Zilman [2016] S. Iyer-Biswas and A. Zilman, First-passage processes in cellular biology, Advances in Chemical Physics 160, 261 (2016).
  * Evans and Majumdar [2011] M. R. Evans and S. N. Majumdar, Diffusion with stochastic resetting, Physical review letters 106, 160601 (2011).
  * Evans  _et al._ [2020] M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, Journal of Physics A: Mathematical and Theoretical 53, 193001 (2020).
  * Pal and Reuveni [2017] A. Pal and S. Reuveni, First passage under restart, Physical Review Letters 118, 030603 (2017).
  * Reuveni  _et al._ [2014] S. Reuveni, M. Urbakh, and J. Klafter, The role of substrate unbinding in michaelis-menten enzymatic reactions, Biophysical Journal 106, 677a (2014).
  * Pal  _et al._ [2023a] A. Pal, V. Stojkoski, and T. Sandev, Random resetting in search problems, arXiv preprint arXiv:2310.12057 (2023a).
  * Evans  _et al._ [2013] M. R. Evans, S. N. Majumdar, and K. Mallick, Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics, Journal of Physics A: Mathematical and Theoretical 46, 185001 (2013).
  * Gupta  _et al._ [2014] S. Gupta, S. N. Majumdar, and G. Schehr, Fluctuating interfaces subject to stochastic resetting, Physical review letters 112, 220601 (2014).
  * Majumdar  _et al._ [2015] S. N. Majumdar, S. Sabhapandit, and G. Schehr, Dynamical transition in the temporal relaxation of stochastic processes under resetting, Physical Review E 91, 052131 (2015).
  * Pal [2015] A. Pal, Diffusion in a potential landscape with stochastic resetting, Physical Review E 91, 012113 (2015).
  * Méndez and Campos [2016] V. Méndez and D. Campos, Characterization of stationary states in random walks with stochastic resetting, Physical Review E 93, 022106 (2016).
  * Eule and Metzger [2016] S. Eule and J. J. Metzger, Non-equilibrium steady states of stochastic processes with intermittent resetting, New Journal of Physics 18, 033006 (2016).
  * 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).
  * Pal and Prasad [2019a] A. Pal and V. Prasad, First passage under stochastic resetting in an interval, Physical Review E 99, 032123 (2019a).
  * Pal and Prasad [2019b] A. Pal and V. Prasad, Landau-like expansion for phase transitions in stochastic resetting, Physical Review Research 1, 032001 (2019b).
  * 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  _et al._ [2023b] P. S. Pal, A. Pal, H. Park, and J. S. Lee, Thermodynamic trade-off relation for first passage time in resetting processes, Physical Review E 108, 044117 (2023b).
  * 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 (2021).
  * Ghosh  _et al._ [2023] P. K. Ghosh, S. Nayak, J. Liu, Y. Li, and F. Marchesoni, Autonomous ratcheting by stochastic resetting, The Journal of Chemical Physics 159 (2023).
  * Ray and Reuveni [2020] S. Ray and S. Reuveni, Diffusion with resetting in a logarithmic potential, The Journal of chemical physics 152 (2020).
  * Ray [2020] S. Ray, Space-dependent diffusion with stochastic resetting: A first-passage study, The Journal of Chemical Physics 153 (2020).
  * Ray and Reuveni [2021] S. Ray and S. Reuveni, Resetting transition is governed by an interplay between thermal and potential energy, The Journal of Chemical Physics 154 (2021).
  * Kumar and Pal [2023] A. Kumar and A. Pal, Universal framework for record ages under restart, Physical Review Letters 130, 157101 (2023).
  * Biswas  _et al._ [2023] A. Biswas, A. Pal, D. Mondal, and S. Ray, Rate enhancement of gated drift-diffusion process by optimal resetting, The Journal of Chemical Physics 159 (2023).
  * Sar  _et al._ [2023] G. K. Sar, A. Ray, D. Ghosh, C. Hens, and A. Pal, Resetting-mediated navigation of an active brownian searcher in a homogeneous topography, Soft Matter 19, 4502 (2023).
  * Domazetoski  _et al._ [2020] V. Domazetoski, A. Masó-Puigdellosas, T. Sandev, V. Méndez, A. Iomin, and L. Kocarev, Stochastic resetting on comblike structures, Physical Review Research 2, 033027 (2020).
  * Singh  _et al._ [2020] R. Singh, R. Metzler, and T. Sandev, Resetting dynamics in a confining potential, Journal of Physics A: Mathematical and Theoretical 53, 505003 (2020).
  * Jolakoski  _et al._ [2023] P. Jolakoski, A. Pal, T. Sandev, L. Kocarev, R. Metzler, and V. Stojkoski, A first passage under resetting approach to income dynamics, Chaos, Solitons & Fractals 175, 113921 (2023).
  * Julián-Salgado  _et al._ [2024] P. Julián-Salgado, L. Dagdug, and D. Boyer, Diffusion with two resetting points, Physical Review E 109, 024134 (2024).
  * Mercado-Vásquez and Boyer [2021] G. Mercado-Vásquez and D. Boyer, Search of stochastically gated targets with diffusive particles under resetting, Journal of Physics A: Mathematical and Theoretical 54, 444002 (2021).
  * Huang and Chen [2021] F. Huang and H. Chen, Random walks on complex networks with first-passage resetting, Physical Review E 103, 062132 (2021).
  * Blumer  _et al._ [2022] O. Blumer, S. Reuveni, and B. Hirshberg, Stochastic resetting for enhanced sampling, The journal of physical chemistry letters 13, 11230 (2022).
  * Blumer  _et al._ [2024] O. Blumer, S. Reuveni, and B. Hirshberg, Combining stochastic resetting with metadynamics to speed-up molecular dynamics simulations, Nature Communications 15, 240 (2024).
  * Ahmad  _et al._ [2019] S. Ahmad, I. Nayak, A. Bansal, A. Nandi, and D. Das, First passage of a particle in a potential under stochastic resetting: A vanishing transition of optimal resetting rate, Phys. Rev. E 99, 022130 (2019).
  * Ahmad  _et al._ [2022] S. Ahmad, K. Rijal, and D. Das, First passage in the presence of stochastic resetting and a potential barrier, Phys. Rev. E 105, 044134 (2022).
  * Capała and Dybiec [2023] K. Capała and B. Dybiec, Optimization of escape kinetics by reflecting and resetting, Chaos: An Interdisciplinary Journal of Nonlinear Science 33 (2023).
  * 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 (2020).
  * 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).
  * Faisant  _et al._ [2021] F. Faisant, B. Besga, A. Petrosyan, S. Ciliberto, and S. N. Majumdar, Optimal mean first-passage time of a brownian searcher with resetting in one and two dimensions: experiments, theory and numerical tests, Journal of Statistical Mechanics: Theory and Experiment 2021, 113203 (2021).
  * Goerlich  _et al._ [2023] R. Goerlich, M. Li, L. B. Pires, P.-A. Hervieux, G. Manfredi, and C. Genet, Experimental test of landauer’s principle for stochastic resetting, arXiv preprint arXiv:2306.09503 (2023).
  * Altshuler  _et al._ [2024] A. Altshuler, O. L. Bonomo, N. Gorohovsky, S. Marchini, E. Rosen, O. Tal-Friedman, S. Reuveni, and Y. Roichman, Environmental memory facilitates search with home returns, Physical Review Research 6, 023255 (2024).
  * Paramanick  _et al._ [2024a] S. Paramanick, A. Pal, H. Soni, and N. Kumar, Programming tunable active dynamics in a self-propelled robot, The European Physical Journal E 47, 34 (2024a).
  * Paramanick  _et al._ [2024b] S. Paramanick, A. Biswas, H. Soni, A. Pal, and N. Kumar, Statistical basis of the optimal homing time in a foraging robot: Learning universal characteristics of homing paths, PRX Life (2024b).
  * Jain  _et al._ [2023] S. Jain, D. Boyer, A. Pal, and L. Dagdug, Fick–jacobs description and first passage dynamics for diffusion in a channel under stochastic resetting, The Journal of Chemical Physics 158 (2023).
  * Carr and Turner [2016] E. Carr and I. Turner, A semi-analytical solution for multilayer diffusion in a composite medium consisting of a large number of layers, Applied Mathematical Modelling 40, 7034 (2016).
  * Farago [2021] O. Farago, Thermodynamics of a brownian particle in a nonconfining potential, Physical Review E 104, 014105 (2021).
  * Sheils [2017] N. E. Sheils, Multilayer diffusion in a composite medium with imperfect contact, Applied Mathematical Modelling 46, 450 (2017).
  * Korabel and Barkai [2011] N. Korabel and E. Barkai, Boundary conditions of normal and anomalous diffusion from thermal equilibrium, Physical Review E 83, 051113 (2011).
  * Pal  _et al._ [2022] A. Pal, S. Kostinski, and S. Reuveni, The inspection paradox in stochastic resetting, Journal of Physics A: Mathematical and Theoretical 55, 021001 (2022).
  * Thorneywork  _et al._ [2020] A. L. Thorneywork, J. Gladrow, Y. Qing, M. Rico-Pasto, F. Ritort, H. Bayley, A. B. Kolomeisky, and U. F. Keyser, Direct detection of molecular intermediates from first-passage times, Science advances 6, eaaz4642 (2020).