††thanks: Corresponding author

# Fick-Jacobs description and first passage dynamics for diffusion in a
channel under stochastic resetting

Siddharth Jain<EMAIL_ADDRESS>Harish-Chandra Research Institute, HBNI,
Chhatnag Road, Jhunsi, Allahabad (Prayagraj), UP 211019, India  Denis Boyer
<EMAIL_ADDRESS>Instituto de Física, Universidad Nacional Autónoma de
México, Ciudad de México C.P. 04510 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  Leonardo Dagdug<EMAIL_ADDRESS>Physics
Department, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco
186, Ciudad de México, 09340, México.

###### Abstract

Transport of particles through channels is of paramount importance in physics,
chemistry and surface science due to its broad real world applications. Much
insights can be gained by observing the transition paths of a particle through
a channel and collecting statistics on the lifetimes in the channel or the
escape probabilities from the channel. In this paper, we consider the
diffusive transport through a narrow conical channel of a Brownian particle
subject to intermittent dynamics, namely, stochastic resetting. As such,
resetting brings the particle back to a desired location from where it resumes
its diffusive phase. To this end, we extend the Fick-Jacobs theory of channel-
facilitated diffusive transport to resetting-induced transport. Exact
expressions for the conditional mean first passage times, escape probabilities
and the total average lifetime in the channel are obtained, and their
behaviour as a function of the resetting rate are highlighted. It is shown
that resetting can expedite the transport through the channel – rigorous
constraints for such conditions are then illustrated. Furthermore, we observe
that a carefully chosen resetting rate can render the average lifetime of the
particle inside the channel minimal. Interestingly, the optimal rate undergoes
continuous and discontinuous transitions as some relevant system parameters
are varied. The validity of our one-dimensional analysis and the corresponding
theoretical predictions are supported by three-dimensional Brownian dynamics
simulations. We thus believe that resetting can be useful to facilitate
particle transport across biological membranes – a phenomena that can
spearhead further theoretical and experimental studies.

††preprint: APS/123-QED

##  I Introduction

The problem of particle transport through confined geometries containing
narrow openings and bottlenecks has recently led to many theoretical and
experimental research fronts. In his original work dating back to 1855, Adolf
Fick was the first scientist to treat diffusion into a channel of varying
cross-sections as a one-dimensional system [1]. Later, the seminal works by
Jacobs and Zwanzig triggered renewed research on this topic. The so-called
Fick-Jacobs approach consists of eliminating transverse stochastic degrees of
freedom by assuming fast equilibration in such directions [2, 3, 4]. As
Zwanzig pointed out, the key quantity that characterizes the unbiased
diffusive transport of point-like Brownian particles in quasi-one-dimensional
systems with periodically varying geometric constraints is the effective
diffusivity [3]. As a result, enormous theoretical efforts have been made to
obtain consistent position-dependent diffusion coefficients that rely upon the
exact nature of confinement [3, 2, 4, 5, 7, 9, 10, 11, 12, 6, 15, 8, 13, 14].

Diffusion of particles, molecules, or even living microorganisms in confined
geometries such as tubes and channels plays a key role across various scales
in natural and technological processes [16, 17, 18, 19, 20, 21, 22, 23, 24,
25]. It is now well understood that membrane transport of metabolites and
other solutes is facilitated by membrane proteins that form water filled
channels [26, 27]. The basic mechanism of channel-regulated transport has thus
been a focal point of both theoretical and practical research. However,
despite many years of studies, new phenomena related to channel-facilitated
metabolite transport continue to be found and a comprehensive understanding is
still to be developed.

A diffusing particle inside a channel can either escape from both sides of the
membrane, or it may be allowed to escape through one side while the other side
simply reflects it back. Key quantities in these processes are the first
passage probabilities and average escape times from the channel (or lifetimes)
conditioned to the exit side of the membrane, as well as the overall average
lifetime of the particle in the channel. These quantities are ubiquitous in
theoretical studies on channel-facilitated transport [28, 29]. Remarkably,
these observables have also taken center stage since recent advances in
single-particle experiments enable direct observation of transition or first
passage paths, marking the events when a molecule diffuses across an
activation barrier or reacts to form products [30]. Fluorescence and force
spectroscopy methods render observations which hold treasure trove of
information on mechanisms such as molecular dynamics, structures of chemical
networks or biological membrane channels [31, 32]. Moreover, single molecule
experiments have unraveled the salient roles played by the channel shape, the
dynamics of biomolecular folding, binding or catalyzing, as well as by the
interactions between the channel and the molecule – all of which pave the way
to our understanding of membrane transport of metabolites or ions. Naturally,
the study of first passage distributions is crucial to understand how the
biomolecules fulfil their function, e.g., fold, bind, or react inside a
membrane channel.

Indeed, first-passage time processes have overarching applications in physics,
chemistry, biology, computer science, ecology and other cross-disciplinary
fields [33]. From macroscopic search processes where a group of searchers or
agents look for some resources or animals forage to find prey or wild
resources [30] to microscopic search where a regulatory protein has to find a
target sequence on the DNA, or chemical and biochemical reactions [34, 35],
trafficking receptors on biological membranes [36] – these are all canonical
examples of stochastic first passage processes. Processes such as the
spreading of sexually transmitted diseases in a human social network or of
viruses across the world wide web [37] are also controlled by first encounter
events [38]. Recently, first passage processes which are further subject to
resetting have gained immense interest due to their myriad of applications in
statistical physics [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51],
chemical process [52], biology [53, 54, 55, 56], queuing theory [57], and
computer science [58, 59, 60, 61]. In these problems, a stochastic variable of
interest (e.g, the position of a diffusing particle) is brought back from time
to time to a specific position, from which it restarts anew. In Michaelis-
Menten enzyme-catalyzed reactions, unbinding can be viewed as a resetting
event [52]. Furthermore, the sudden falls in stock market price, the sharp
layoff of individual jobs due to post-pandemic recession, or the massive
extinction of population due to catastrophes are also canonical signatures of
resetting events [62, 63]. A hallmark property of resetting is its ability to
speed-up complex search processes, motivating a flurry of theoretical [39, 40,
41, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76] and experimental works
[77, 78].

In this work, we attempt to study channel-facilitated first passage transport
in the presence of resetting. First passage properties under resetting have
been studied extensively for diffusive and active systems in simple one-
dimensional intervals with two absorbing/reactive boundaries [70, 71, 72, 79],
or one reflective-one absorbing boundary [55, 79, 80]. However, to the best of
our knowledge, diffusion in $2D$ or $3D$ channels of varying width in the
presence of resetting has not been explored so far. While the lifetime
statistics of reset-diffusing particles inside the channel is of interest from
the perspective of transport, another interesting question that naturally
emerges is whether such transport can be made optimal through resetting.
Furthermore, while it has been shown that diffusion-controlled reactions of
solutes with a reaction site hidden within a protein membrane channel or a
protein cavity can be slowed down by many orders of magnitude [29], resetting
or unbinding events can expedite on the contrary the turnover rate of
enzymatic catalysis process [52]. Thus, the combination of these two phenomena
– resetting and geometric confinement – can be a reasonable attempt to delve
deeper into the understanding of Michaelis-Menten when the reaction site is
hidden in a protein membrane channel or protein cavity.

As mentioned before, particle transport inside a channel can well be captured
by the Fick-Jacobs (FJ) formalism. In here, we extend this formalism for
diffusing particles under resetting. General expressions for the escape
probabilities and the average life times for the particles escaping through
each boundary are computed. These lifetimes are conditional because they are
calculated for the two subset of all possible realizations of the escape
process. The total average lifetime of the particle in the channel is then
obtained as a weighted sum of these conditional lifetimes where the escape
probabilities are used. To illustrate some qualitative features of the general
theory, we study a special case in which the particle diffuses through a three
dimensional conical channel (see Fig. 1). Moreover, we consider that the
particle is stochastically reset back to a pre-defined location at a given
rate. We find that resetting can expedite or slow down the escape of the
particle from the channel. Utilizing the resetting criterion when the
resetting and starting positions coincide, we find the exact conditions
conferred by the system parameters such as the initial position, channel
length and channel slope that determine when resetting is going to be
beneficial. In this case, we obtain the optimal condition when the escape time
can be minimized. Interestingly, we find that the optimal resetting rate can
undergo continuous or discontinuous transitions with varying the initial
position as the control parameter. Furthermore, an intriguing optimization is
also observed for the conditional times. Our theoretical results are
corroborated with Brownian dynamics simulations in 3D channels.

The structure of the paper is as follows. We describe the model setup and
review the Fick-Jacobs formalism in Section (II.1). We then introduce the
resetting dynamics and sketch out the steps leading to the modified Fick-
Jacobs formalism in Section (II.2). Therein, we provide general expressions
and solutions for the observables of interest. In Section (III), we discuss
the first passage statistics of the particle in a three dimensional cone with
two absorbing boundaries at the end. We discuss the effects of resetting,
including emergence of an optimal resetting rate. The following section (IV)
is dedicated to the study of first passage statistics in a three dimensional
cone with one absorbing boundary and one reflecting boundary at the other end.
Some of the results have been moved to the Appendix for brevity. We summarize
our paper and put concluding remarks in Section (V).

Figure 1: Schematic representation of a three-dimensional conical tube of
variable radius $R(x)$ and length $L$ with a constant radius variation rate,
$|dR(x)/dx|=const=\lambda$. In this case, the effective diffusivity is
position-independent and a function of $\lambda$ through the relation
$D_{\lambda}=D_{0}/\sqrt{1+\lambda^{2}}$. Panels (a) and (b) show an expanding
and narrowing tube, respectively. The particle starts from the initial
position at $x_{0}$ and is reset to a fixed position $x_{r}$ at a rate $r$,
from where it resumes diffusion until the next reset. The channel ends at
$x=0$ and $x=L$ (shown in yellow) could be either reflecting or absorbing.

##  II Model details and formalism

###  II.1 Setup and Fick-Jacobs equation

We consider diffusion of a particle inside a three dimensional conical tube of
variable radius $R(x)$ and length $L$. The radius of the tube linearly grows
in space (see Fig. 1). When diffusion occurs in quasi-one-dimensional
structures, one can map particle motion onto an effective one-dimensional (1D)
description in terms of diffusion along the channel axis. The key point of the
derivation is the assumption of equilibration in the transverse direction.
When the system geometry is smoothly varying along the $x$-coordinate measured
along the tube/channel axis, the simplest one-dimensional description is given
by the Fick-Jacobs equation [2]. In terms of the probability distribution
$c(x,t)$, the FJ equation reads

$\frac{\partial{c}(x,t)}{\partial t}=D_{0}\frac{\partial}{\partial
x}\left\\{{A}(x)\frac{\partial}{\partial
x}\left[\frac{{c}(x,t)}{{A}(x)}\right]\right\\},$ (1)

where $D_{0}$ is the bare diffusion constant, $A(x)$ is the cross-section area
$\pi R^{2}(x)$ of a tube of radius $R(x)$, or the width $w(x)$ of a two-
dimensional channel.

It has been shown that a more accurate reduction to the 1D description results
in a position-dependent diffusivity $D(x)$. Then the probability density in
the channel satisfies a modified Fick-Jacobs equation which was first derived
by Zwanzig [3]

$\frac{\partial{c}(x,t)}{\partial t}=\frac{\partial}{\partial
x}\left\\{D(x){A}(x)\frac{\partial}{\partial
x}\left[\frac{{c}(x,t)}{{A}(x)}\right]\right\\}.$ (2)

Confinement in higher dimensions gives rise to an effective entropic potential
in the reduced dimension. In fact, Eq. 2 is formally equivalent to the
Smoluchowski equation

$\frac{\partial{c}(x,t)}{\partial t}=\frac{\partial}{\partial
x}\left\\{D(x)e^{-\beta U(x)}\frac{\partial}{\partial x}\left[e^{\beta
U(x)}c(x,t)\right]\right\\},$ (3)

where the entropy potential is given by -$\beta U(x)=\ln(w(x)/w(x_{0}))$ for a
2D channel and -$\beta U(x)=\ln(R^{2}(x)/R^{2}(x_{0}))$ for a 3D tube. Here,
$\beta=1/(k_{B}T)$ is the inverse temperature with $k_{B}$ the Boltzmann
constant.

There are several interpretations for the position dependent effective
diffusion coefficient. For example, in the case of a narrow 2D channel of
varying width, it was shown by Reguera and Rubi [7] that the following
relation holds

$D(x)\approx D_{RR}^{2D}(x)=\frac{D_{0}}{\left[1+\frac{1}{4}w^{\prime
2}(x)\right]^{\eta}},$ (4)

where $w^{\prime}(x)=dw(x)/dx$ and $D_{0}$ the bare diffusion coefficient. The
index $\eta$ takes the value $1/3$ for the 2D set-up. Parallel derivations
were given by Kalinay and Percus [8], Martens et al. [10] and García-Chung and
co-workers [11]. In a similar manner, the following spatial dependencies were
obtained for the 3D tube geometry by Reguera and Rubi [7] and later by Kalinay
and Percus [8]

$D(x)\approx
D_{RR}^{3D}(x)=D_{KP}^{3D}(x)=\frac{1}{\sqrt{1+R^{\prime}(x)^{2}}}D_{0},$ (5)

where $\eta=1/2$ was assumed for 3D set-up. Here we consider 3D conical tube
so that the entropic potential satisfies $e^{-\beta
U(x)}=R^{2}(x)/R^{2}(x_{0})$, as mentioned before. Furthermore, we will
consider a linear variation of the tube radius. In particular, the tube radius
$R(x)$ increases in the axial direction with a constant rate $\lambda$ so that
$R(x)=a+\lambda x$, where the $x$-coordinate is measured along the tube axis,
and $a$ is the tube radius at $x=0$. It then follows from Eq. (5) that the
effective diffusivity is position-independent, hence the Reguera-Rubi formula
reduces to

$\displaystyle D_{\lambda}=\frac{D_{0}}{\sqrt{1+\lambda^{2}}},$ (6)

which we will use in this study unless otherwise stated.

###  II.2 Fick-Jacobs equation with resetting and escape statistics

We now consider the same point-like Brownian particle diffusing in a spatially
constrained asymmetric 3D tube, but it is now intermittently reset to the
position $x_{r}$ at a constant rate $r$. Let us denote the position
distribution for this particle as $c_{r}(x,t|x_{0},0)$, where suffix $r$
indicates resetting. This position density is not normalized to unity but to
the survival probability $Q_{r}(x_{0},x_{r},t)$ since the particles will
eventually escape through the boundaries. $Q_{r}(x_{0},x_{r},t)$ is defined as
the probability that the particle has stayed in-between the boundary
coordinates up to time $t$, having started from $x_{0}$ and with resetting at
$x_{r}$,

$\displaystyle Q_{r}(x_{0},x_{r},t)=\int_{0}^{L}dx~{}c_{r}(x,t|x_{0},0).$ (7)

A similar relation holds also for the resetting-free process, i.e.,
$Q(x_{0},t)=\int_{0}^{L}~{}dx~{}c(x,t|x_{0},0)$.

After each resetting event, the particle returns to the same location $x_{r}$
and restarts its motion, keeping no memory of previous resetting events. Thus
the process is renewed after each resetting and we can write a time dependent
equation for the position density $c_{r}(x,t|x_{0},0)$ using a renewal
formalism [70]

$\displaystyle c_{r}(x,t|x_{0},0)$ $\displaystyle=e^{-rt}c(x,t|x_{0},0)$
$\displaystyle+r\int_{0}^{t}~{}d\tau~{}e^{-r(t-\tau)}~{}c(x,t|x_{r},\tau)Q_{r}(x_{0},x_{r},\tau)~{},$
(8)

where the first term indicates that there was no resetting event up to a time
$t$. On the other hand, the second term captures a physical scenario when the
particle undergoes multiple resetting to $x_{r}$ not being absorbed by the
boundaries up to time $\tau$, followed by a resetting at time $\tau$, and a
last excursion without resetting (starting from $x_{r}$) of duration $t-\tau$.
This equation clearly states that the concentration in the presence of
resetting depends on the concentration of the underlying resetting-free
process and on the survival probability of the reset process. We can then
write an effective Fick-Jacobs equation in the presence of resetting, which
reads

$\displaystyle\frac{\partial{c_{r}}(x,t)}{\partial t}=$
$\displaystyle\frac{\partial}{\partial x}\left\\{D(x)e^{-\beta
U(x)}\frac{\partial}{\partial x}\left[e^{\beta
U(x)}c_{r}(x,t)\right]\right\\}$ $\displaystyle-
rc_{r}(x,t|x_{0})+r\delta(x-x_{r})Q_{r}(x_{0},x_{r},t).$ (9)

The effective FJ equation has the usual diffusive and drift terms in addition
to new two terms. While the first one represents the removal/outflux of the
particle from its current position $x$ at rate $r$, the second one indicates a
source term only at the resetting position where the probability is being
accumulated. This is however conditioned on the particle’s survival – hence
the appearance of the survival probability $Q_{r}$ in the last term.

We need not solve these resetting equations explicitly. To obtain the survival
probability in the presence of resetting, we notice that one can similarly
write a renewal equation for this quantity [39, 45],

$\displaystyle Q_{r}(x_{0},x_{r},t)$ $\displaystyle=e^{-rt}Q(x_{0},t)$
$\displaystyle+\int_{0}^{t}~{}d\tau~{}re^{-r\tau}~{}Q(x_{0},\tau)~{}Q_{r}(x_{r},x_{r},t-\tau),$
(10)

where $Q(x_{0},t)$ denotes the survival probability of the resetting-free
process. Taking Laplace transform on both sides of Eq. (10) we arrive at the
following relation

$\displaystyle q_{r}(x_{0},x_{r},s)$
$\displaystyle=\frac{q(x_{0},s+r)}{1-rq(x_{r},s+r)},$ (11)

where $q_{r}(x_{0},x_{r},s)=\int_{0}^{\infty}dt\ e^{-st}Q_{r}(x_{0},x_{r},t)$
and $q(x_{0},s)=\int_{0}^{\infty}dt\ e^{-st}Q(x_{0},t)$. Similarly, taking the
Laplace transform on both sides of Eq. (8) and using Eq. (11), we obtain

$\displaystyle\tilde{c_{r}}(x,s|x_{0},0)$
$\displaystyle=\tilde{c}(x,s+r|x_{0},0)$
$\displaystyle+\frac{rq(x_{0},s+r)\tilde{c}(x,s+r|x_{r},0)}{1-rq(x_{r},s+r)}~{},$
(12)

which is a crucial relation relating the concentration in the presence of
resetting only in terms of the observables of the underlying process.

Since we are interested in the escape properties of the particle from the
channel, we need to examine the first passage time statistics. Generically,
this encodes the information for the lifetime of the particle in confinement,
and the first escape is marked with suitable boundary conditions. The first
passage time density of any process is related to the survival probability
through the general relation [33]

$\displaystyle f_{r}(t|x_{0},x_{r})=-\frac{\partial
Q_{r}(x_{0},x_{r},t)}{\partial t},$ (13)

which in the Laplace domain translates to
$\tilde{f_{r}}(s|x_{0},x_{r})=1-sq_{r}(x_{0},x_{r},s)$. In the following we
will consider the case where the resetting position is the starting position,
or $x_{r}=x_{0}$, unless otherwise stated. Similar to the survival
probability, the first passage time densities with and without resetting are
also related in the Laplace domain. From Eq. (11),

$\displaystyle\tilde{f}_{r}(s|x_{0})=\frac{(s+r)\tilde{f}(s+r|x_{0})}{s+r\tilde{f}(s+r|x_{0})}.$
(14)

The mean first passage time (MFPT) is simply given by

$\displaystyle\langle
T_{r}(x_{0})\rangle=\int_{0}^{\infty}dt~{}tf_{r}(t|x_{0},x_{r})=q_{r}(x_{0},x_{0},s\to
0).$ (15)

Recall that within the current setup, the particle starting from $x_{0}$, can
escape through any of the boundaries. This is captured by the unconditional
MFPT above. In other words, it measures the average lifetime of the particle
inside the channel. However, one can also look for the transit time i.e., the
first passage times that estimate the time when the particle escapes through
one designated boundary. These are conditional times and have markedly
different statistics than the MFPT.

The conditional mean transit times (in the following, ‘+’ implies observables
related to the right boundary at $L$ and ‘-’ implies the same for the left
boundary at $0$) can be computed from the current/flux conditioned on
respective boundaries, i.e.,

$\displaystyle\langle\tau_{r}(x_{0})\rangle^{\pm}=\frac{\int_{0}^{\infty}dt~{}t~{}J_{r}^{\pm}(x_{0},t)}{\int_{0}^{\infty}dt~{}J_{r}^{\pm}(x_{0},t)},$
(16)

where $J_{r}^{+}(x_{0},t)$ $[J_{r}^{-}(x_{0},t)]$ is the probability flux to
the right(L) [left(0)] boundary respectively. The terms in the denominator
correspond to the exit/splitting probabilities through the boundary $L$ and
$0$ respectively

$\displaystyle\epsilon_{r}^{\pm}(x_{0})=\int_{0}^{\infty}~{}dt~{}J_{r}^{\pm}(x_{0},t)=j_{r}^{\pm}(x_{0},s=0),$
(17)

where
$j_{r}^{\pm}(x_{0},s)=\int_{0}^{\infty}dt~{}e^{-st}~{}J_{r}^{\pm}(x_{0},t)$
are the Laplace transform of currents. Note that
$\epsilon^{+}+\epsilon^{-}=1$, as the particle eventually escapes the channel.
Eqs. (16) can be rewritten as

$\displaystyle\langle\tau_{r}(x_{0})\rangle^{\pm}=\frac{-\frac{\partial
j_{r}^{\pm}(x_{0},s)}{\partial s}|_{s\to 0}}{\epsilon_{r}^{\pm}(x_{0})}~{}.$
(18)

Note that one should always have $\langle
T_{r}(x_{0})\rangle=\epsilon_{r}^{+}\langle\tau_{r}\rangle^{+}+\epsilon_{r}^{-}\langle\tau_{r}\rangle^{-}$
for both underlying and reset process. The conditional first passage densities
for the corresponding transit times are also related to the currents with
proper normalisation [70]

$\displaystyle
f_{r}^{\pm}(t|x_{0})=\frac{J_{r}^{\pm}(x_{0},t)}{\epsilon_{r}^{\pm}(x_{0})}.$
(19)

While the currents $j_{r}^{\pm}(x_{0},s)$ can be computed from the knowledge
of the conditional propagator $\tilde{c_{r}}(x,s|x_{0},0)$, we take a
different route based on the renewal approach suitable for resetting
processes. Technically, we use the renewal relations for the currents and
hence the conditional first passage densities for the resetting process. The
usefulness of these relations should be appreciated since they connect
observables of the process to those of the underlying process. Thus, the task
significantly reduces to the computation of the currents and (un)conditional
densities for the underlying process. Then one can use the renewal relation to
obtain the desired results. A renewal relation for the current can be directly
obtained from Eq. (12) (also see [70] for detailed derivation)

$\displaystyle j_{r}^{\pm}(x_{0},s)=\frac{j^{\pm}(x_{0},s)}{1-rq(x_{0},s+r)},$
(20)

where $j^{\pm}(x_{0},s)$ is the current for the underlying process in Laplace
space. Using the above and Eq. (19), we find a similar renewal relation that
connects the conditional first passage time densities for the reset process to
the same for the underlying process

$\displaystyle\tilde{f}_{r}^{\pm}(s|x_{0})=\frac{\tilde{f}^{\pm}(s+r|x_{0})}{\tilde{f}^{\pm}(r|x_{0})}\frac{(s+r)\tilde{f}(r|x_{0})}{s+r\tilde{f}(s+r|x_{0})},$
(21)

which will prove to be a useful relation later in our study to compute the
moments $\langle\tau_{r}(x_{0})\rangle^{\pm}$ for the conditional times.

###  II.3 General solution of FJ equation

To make use of the renewal relations, one first needs to obtain the complete
statistics for the underlying process. In this section, we provide solutions
for the propagator which will serve as a backbone for computing currents,
survival and first passage time density for the underlying reset free
processes. To this end, let us first recall the Smoluchowski equation from Eq.
(3)

$\displaystyle\frac{\partial{c}(x,t)}{\partial t}=$
$\displaystyle\frac{\partial}{\partial x}\left\\{D(x)e^{-\beta
U(x)}\frac{\partial}{\partial x}\left[e^{\beta U(x)}c(x,t)\right]\right\\},$
(22)

and the modified Fick-Jacobs equation for the 3D cone (see Fig.1)

$\displaystyle\frac{\partial{c}(x,t)}{\partial t}=$ $\displaystyle
D_{\lambda}\frac{\partial}{\partial x}\left\\{R^{2}(x)\frac{\partial}{\partial
x}\left[\frac{{c}(x,t)}{R^{2}(x)}\right]\right\\}.$ (23)

To proceed further, we first scale the probability density by the radius of
the channel, namely,

$\displaystyle c(x,t)=R(x)g(x,t),$ (24)

and define the following Laplace transforms

$\displaystyle\tilde{c}(x,s|x_{0},0)$
$\displaystyle=\int_{0}^{\infty}c(x,t|x_{0},0)~{}e^{-st}dt,$ (25)
$\displaystyle G(x,s)$ $\displaystyle=\int_{0}^{\infty}g(x,t)~{}e^{-st}dt,$
(26)

so that Eq. (2) is Laplace transformed as

$\displaystyle sR(x)G(x,s)-\delta(x-x_{0})=\frac{\partial}{\partial
x}\left\\{D_{\lambda}R^{2}(x)\frac{\partial}{\partial
x}\left[\frac{G(x,s)}{R(x)}\right]\right\\}.$ (27)

We consider the conical case $R(x)=\lambda x+a$, where $\lambda\geq 0$ for an
expanding channel and $\lambda\leq 0$ for a shrinking one. Eq. (27) becomes

$\displaystyle\frac{-\delta(x-x_{0})}{\lambda
x_{0}+a}=D_{\lambda}\frac{\partial^{2}G}{\partial x^{2}}-sG.$ (28)

The general solution to the above equation on each side of $x_{0}$ is given by

$\displaystyle
G(x,s)=c_{1}e^{-\sqrt{s/D_{\lambda}}x}+c_{2}e^{\sqrt{s/D_{\lambda}}x},\quad\quad
x\in[0,x_{0})$ (29) $\displaystyle
G(x,s)=c_{3}e^{-\sqrt{s/D_{\lambda}}x}+c_{4}e^{\sqrt{s/D_{\lambda}}x},\quad\quad
x\in(x_{0},L]$ (30)

resulting in

$\displaystyle\tilde{c}(x,s)=(\lambda
x+a)[c_{1}e^{-\sqrt{s/D_{\lambda}}x}+c_{2}e^{\sqrt{s/D_{\lambda}}x}],\quad
x\in[0,x_{0})$ (31) $\displaystyle\tilde{c}(x,s)=(\lambda
x+a)[c_{3}e^{-\sqrt{s/D_{\lambda}}x}+c_{4}e^{\sqrt{s/D_{\lambda}}x}],\quad
x\in(x_{0},L].$ (32)

To find the four constants we need to impose one condition at each boundary
which we will discuss in the following. The additional two conditions are
obtained from matching the propagator, namely, $(i)$ $\tilde{c}(x,s)$ is
continuous at $x=x_{0}$, and $(ii)$ the derivative
$\partial_{x}\tilde{c}(x,s)$ is discontinuous at $x=x_{0}$, as seen from
integrating Eq. (28) over a narrow interval around $x_{0}$,

$\left.\frac{\partial G}{\partial x}\right|_{x_{0}^{+}}-\left.\frac{\partial
G}{\partial x}\right|_{x_{0}^{-}}=-\frac{1}{D_{\lambda}(\lambda x_{0}+a)}.$
(33)

##  III First passage observables in a three dimensional cone with two
absorbing boundaries

We first consider the conical tube with two absorbing points at $x=0,L$. As
stated before, it has a radius which varies spatially with a rate $\lambda$
and the effective diffusive constant is $D_{\lambda}$ given by Eq. (6). By
symmetry, it is sufficient to consider $\lambda\geq 0$. The boundary
conditions read

$\displaystyle c(0,t)=0~{},~{}c(L,t)=0,$ (34)

since a particle will be immediately absorbed if it reaches one of the
boundaries. The particle starts from an initial position $x_{0}$ within the
cone and it is reset to the same point at a rate $r$. Using the matching and
boundary conditions, we solve Eqs. (31) and (32) to obtain the exact
expressions for the constants

$\displaystyle c_{1}=\frac{-\sinh(u(L-x_{0}))}{2uD_{\lambda}(\lambda
x_{0}+a)\sinh(uL)},$ (35) $\displaystyle
c_{2}=\frac{\sinh(u(L-x_{0}))}{2uD_{\lambda}(\lambda x_{0}+a)\sinh(uL)},$ (36)
$\displaystyle c_{3}=\frac{e^{uL}\sinh(ux_{0})}{2uD_{\lambda}(\lambda
x_{0}+a)\sinh(uL)},$ (37) $\displaystyle
c_{4}=\frac{-e^{-uL}\sinh(ux_{0})}{2uD_{\lambda}(\lambda x_{0}+a)\sinh(uL)},$
(38)

where $u=\sqrt{s/D_{\lambda}}$.

###  III.1 Currents through the boundaries

The probability flux/current entering the left (right) absorbing boundary at
$x=0$ ($L$, resp.) for the resetting free process can be found by observing at
the modified FJ equation in Eq. (23) which can be written in the form of a
continuity equation $\frac{\partial c}{\partial t}=-\frac{\partial J}{\partial
x}$, where one identifies $J$ to be the probability current. Then, the
probability flux through the right boundary can be written as

$\displaystyle J^{+}(x_{0},t)=-D_{\lambda}R^{2}(x)\frac{\partial}{\partial
x}\left[\frac{c(x,t)}{R^{2}(x)}\right]{\bigg{|}_{x=L}}$
$\displaystyle=-D_{\lambda}\frac{\partial c}{\partial x}\bigg{|}_{x=L},$ (39)

where we have used $c(x,t)=0$ at the boundary $L$. A similar analysis holds
for the left boundary, where the probability flux/current is given by

$\displaystyle J^{-}(x_{0},t)=D_{\lambda}R^{2}(x)\frac{\partial}{\partial
x}\left[\frac{{c}(x,t)}{R^{2}(x)}\right]\bigg{|}_{x=0}=D_{\lambda}\frac{\partial
c}{\partial x}\bigg{|}_{x=0},$ (40)

where again we have used the boundary condition at $x=0$. The integral of the
fluxes in time from zero to infinity gives the splitting probability
$\epsilon^{\pm}(x_{0})=\int_{0}^{\infty}~{}dt~{}J^{\pm}(x_{0},t)$. Similarly,
the conditional transit time densities read
$f^{\pm}(t|x_{0})=J^{\pm}(x_{0},t)/\epsilon^{\pm}(x_{0})$ [see Eq. (19)].
These expressions will be used in the next section.

Figure 2: Variation of MFPTs as a function of $x_{0}$ without resetting (left)
and with resetting (right) in the absorbing-absorbing case predicted by Eq.
(53) (solid lines). The escape times obtained from Brownian dynamics
simulations are shown with open symbols ($a=0.1$, $D_{\lambda}=0.01$, $L=0.5$,
$\lambda$=0.2).

###  III.2 Unconditional and conditional first passage time for the underlying
process

Since a particle can get absorbed at either of the two boundaries, the total
flux will be the sum of these fluxes given by Eqs. (39) and (40). Thus, the
unconditional first passage density from either boundary is given by

$\displaystyle f(t|x_{0})=J_{+}(x_{0},t)+J_{-}(x_{0},t).$ (41)

Laplace transforming the above leads to

$\displaystyle\tilde{f}(s|x_{0})=j^{+}(x_{0},s)+j^{-}(x_{0},s),$ (42)

where each term can be computed easily from the propagator $\tilde{c}(x,s)$
and using Eqs. (39) and (40). One gets

$\displaystyle j^{+}(x_{0},s)$
$\displaystyle=-D_{\lambda}\frac{\partial\tilde{c}}{\partial
x}\bigg{|}_{x=L}=-D_{\lambda}[R(x)\frac{\partial G}{\partial
x}+G\lambda]\bigg{|}_{x=L}$ $\displaystyle=(\lambda
L+a)\frac{\sinh(ux_{0})}{(\lambda x_{0}+a)\sinh(uL)},$ (43)

and

$\displaystyle j^{-}(x_{0},s)$
$\displaystyle=D_{\lambda}\frac{\partial\tilde{c}}{\partial
x}\bigg{|}_{x=0}=D_{\lambda}[R(x)\frac{\partial G}{\partial
x}+G\lambda]\bigg{|}_{x=0}$ $\displaystyle=a\frac{\sinh(u(L-x_{0})))}{(\lambda
x_{0}+a)\sinh(uL)},$ (44)

while going from the first to the second line we have used
$\tilde{c}(x,s)=R(x)G(x,s)=(a+\lambda x)G(x,s)$. The unconditional first
passage density is then given in Laplace space by

$\displaystyle\tilde{f}(s|x_{0})=\frac{(\lambda
L+a)\sinh(ux_{0})+a\sinh(u(L-x_{0}))}{(\lambda x_{0}+a)\sinh(uL)}.$ (45)

The moments of the first passage time distribution can be obtained from the
small $s$ expansion of the quantity
$\tilde{f}(s|x_{0})\simeq\int_{0}^{\infty}dtf(t|x_{0})(1-st+\frac{s^{2}}{2}t^{2}+...)$.
In terms of the variable $u=\sqrt{s/D_{\lambda}}$, one has:

$\tilde{f}(s|x_{0})=1-D_{\lambda}\langle T(x_{0})\rangle
u^{2}+\frac{D_{\lambda}^{2}\langle T^{2}(x_{0})\rangle}{2}u^{4}+...$ (46)

By expanding Eq. (45) up to 4th order in $u$, one can obtain the mean first
passage time and the second moment for the underlying process, respectively,
as

$\displaystyle\langle
T(x_{0})\rangle=\frac{x_{0}(L-x_{0})(3a+\lambda(L+x_{0}))}{6D_{\lambda}(\lambda
x_{0}+a)},$ (47)

and

$\displaystyle\langle
T^{2}(x_{0})\rangle=\frac{x_{0}(L-x_{0})\left[15a(L^{2}+Lx_{0}-x^{2}_{0})+\lambda(x_{0}+L)(7L^{2}-3x_{0}^{2})\right]}{180D_{\lambda}^{2}(\lambda
x_{0}+a)}.$ (48)

Note that the moments can also be computed directly from Eq. (45) by using the
formula $\langle
T^{n}\rangle=(-1)^{n}\frac{d^{n}}{ds^{n}}\tilde{f}(s|x_{0})\bigg{|}_{s\to 0}$.
The conditional first passage time or exit time densities can be computed from
the conditional threshold currents (see Eq. (19))

$\displaystyle\tilde{f}^{+}(s|x_{0})$
$\displaystyle=\frac{L\sinh(ux_{0})}{x_{0}\sinh(uL)},$
$\displaystyle\tilde{f}^{-}(s|x_{0})$
$\displaystyle=\frac{L\sinh(u(L-x_{0}))}{(L-x_{0})\sinh(uL)}.$ (49)

From these densities and using Eq. (18), we obtain the mean conditional exit
times

$\displaystyle\langle\tau(x_{0})\rangle^{+}$
$\displaystyle=\frac{L^{2}-x_{0}^{2}}{6D_{\lambda}},$
$\displaystyle\langle\tau(x_{0})\rangle^{-}$
$\displaystyle=\frac{2Lx_{0}-x_{0}^{2}}{6D_{\lambda}}.$ (50)

The corresponding splitting probabilities are given by

$\displaystyle\epsilon^{+}(x_{0})$
$\displaystyle=j^{+}(x_{0},s=0)=\frac{x_{0}(\lambda L+a)}{L(\lambda
x_{0}+a)},$ $\displaystyle\epsilon^{-}(x_{0})$
$\displaystyle=j^{-}(x_{0},s=0)=\frac{a(L-x_{0})}{L(\lambda x_{0}+a)},$ (51)

which were also known for $x_{0}=0$ [6].

###  III.3 Unconditional first passage time for reset process

We now turn our attention to the resetting induced transport process. As
mentioned before, the renewal relations become crucial here. We start with the
unconditional lifetime statistics of the particle inside the channel. In this
case, the renewal relation is given by Eq. (14) where we substitute expression
for the underlying first passage time density from Eq. (45) to find

$\displaystyle\tilde{f}_{r}(s|x_{0})=\frac{(s+r)[(\lambda L+a)\sinh(\beta
x_{0})+a\sinh(\beta(L-x_{0})]}{s(\lambda x_{0}+a)\sinh(\beta L)+r[(\lambda
L+a)\sinh(\beta x_{0})+a\sinh(\beta(L-x_{0})]},$ (52)

where $\beta=\sqrt{(s+r)/D_{\lambda}}$. The mean first passage time under
resetting is deduced by using the general relation (15), or $\langle
T_{r}(x_{0})\rangle=\lim_{s\rightarrow 0}s^{-1}[1-\tilde{f}_{r}(s|x_{0})]$,
which gives

$\displaystyle\langle T_{r}(x_{0})\rangle=\frac{(\lambda x_{0}+a)\sinh(\alpha
L)-a\sinh(\alpha(L-x_{0}))-(\lambda L+a)\sinh(\alpha
x_{0})}{r[a\sinh(\alpha(L-x_{0}))+(\lambda L+a)\sinh(\alpha x_{0})]},$ (53)

where $\alpha=\sqrt{r/D_{\lambda}}$. In Fig. 2 (right), we have plotted
$\langle T_{r}(x_{0})\rangle$ as a function of the initial/resetting
coordinate, showing a larger lifetime near the center of the channel and a
faster escape near the boundaries than in the corresponding case of the
underlying process (Fig. 2 (left)). When we vary the resetting rate $r$, non-
trivial effects can be observed. In Fig. 3 (left), resetting is seen to lower
the mean first-passage time at small $r$, i.e., it expedites the escape of the
particle from the channel rendering a lower average lifetime. Nevertheless, in
an other example shown in Fig. 3, $\langle T_{r}(x_{0})\rangle$ increases
monotonically with the rate $r$. While each variant above carries intriguing
effects, it is nonetheless not evident how the parameters, e.g., the initial
position or the shape of the channel, set the criterion for resetting either
to prolong or expedite the completion of the process. To characterize this
transition, we do a detailed analysis of the resetting criterion in the next
subsection.

Figure 3: MFPTs predicted by Eq. (53) (solid lines) are compared with the
values obtained by three-dimensional Brownian dynamics simulations (open
symbols) as a function of the resetting rate for different starting positions,
when resetting is useful (left) and when it is not (right). Absorbing-
absorbing case with a expanding tube with $a=0.1$, $D_{\lambda}=0.01$,
$L=0.5$, $\lambda$=0.2. The equality $CV=1$ for the reset-free process is
achieved at $x_{0}$=0.1178 and $x_{0}$=0.3472.

###  III.4 When is resetting useful?

Resetting can often become beneficial to reduce the mean first passage time of
a process, i.e., $\langle T_{r}(x_{0})\rangle<\langle T(x_{0})\rangle$. Doing
an expansion with respect to $r$ teaches us that resetting speeds up the mean
first passage time when the coefficient of variation $CV$ of the underlying
process (with $r=0$) is larger than unity, where $CV$ is the ratio between the
standard deviation of the first passage time, $\sigma(T)$, and its mean
$\langle T\rangle$ [69, 81]. The criterion can also be recast as $\langle
T^{2}(x_{0})\rangle>2\langle T(x_{0})\rangle^{2}$. From Eqs. (47) and (48),
this implies

$\displaystyle(\lambda
v+\tilde{a})\left[\lambda(1+v)(7-3v^{2})+15\tilde{a}(1+v-v^{2})\right]\geq$
$\displaystyle 10v(1-v)\left[\lambda(1+v)+3\tilde{a}\right]^{2},$ (54)

where $\tilde{a}=a/L$ and $v=x_{0}/L$. One notices that the criterion is
fulfilled in any channel as soon as the particle starts sufficiently close to
one of the absorbing boundaries ($v\sim 0$ or $v\sim 1$) but not when it
starts out in the middle. For starting positions near the center of the tube,
increasing the reset rate increases the MFPT since any trajectory on both
sides is going to take particle closer to the boundary and reset hinders it.
But for starting positions closer to the boundaries, resetting decreases the
lifetime because there are now many more possible trajectories that are taking
the particle away from the boundary and resetting helps eliminate them.
Setting $\lambda=0$, one recovers the case of the one-dimensional Brownian
particle in an interval. In this case, Eq. (III.4) reduces to

$5v^{2}-5v+1\geq 0,$ (55)

a relation which was previously derived in [70]. Putting in the parameters
values of the system, we can find for which values of $x_{0}$ resetting is
useful. For all the starting positions satisfying the criterion $CV>1$, there
exists a finite optimal resetting rate (minimum of the curve) as seen in Fig.
3 (left), while those with $CV<1$ have a minimum lifetime at $r=0$, as shown
by Fig. 3 (right).

###  III.5 Optimal resetting rate

Figure 4: Left: Optimal restart rate $r^{*}$ vs. the starting position
$x_{0}$, for various channel slopes $\lambda$ in the absorbing-absorbing case
(the narrow boundary is on the left). The other parameters are
$D_{\lambda}=1$, $a=0.1$, $L=1$. Right: Variations of the MFPT with $r$ when
$x_{0}$ lies in the vicinity of the discontinuous transition, for
$\lambda=0.3$.

It is instructive to compute the optimal resetting rate, which is defined as
the rate at which the MFPT reaches its minimum,

$\displaystyle r^{*}=\underset{r\geq 0}{\arg\min}\langle T_{r}(x_{0})\rangle,$
(56)

the other parameters being fixed. The minimization of Eq. (53) can be carried
out numerically and $r^{*}$ is displayed in Figure 4 (left) as a function of
$x_{0}$, for several channel slopes $\lambda$. In the central part of the
channel, $r^{*}=0$ as expected. As $x_{0}$ increases and approaches the wide
boundary (to the right), a continuous transition to $r^{*}>0$ takes place at
$x_{0}=x_{0}^{c}$. The latter position precisely corresponds to the point
where $CV=1$. This transition holds for $\lambda=0$ (cylindrical channel) and
$\lambda>0$. A noticeable effect of the slope is to extend the range of values
of $x_{0}$ where resetting is beneficial, compared to the cylindrical
geometry.

The situation is qualitatively different when $x_{0}$ decreases towards the
narrow absorbing boundary (to the left). Interestingly, the optimal restart
rate $r^{*}$ undergoes a discontinuous transition at a position
$x_{0}^{c\prime}$ in this case, see Figure 4 (left). The origin of this abrupt
jump, which is absent in cylinders but occurs as soon as $\lambda>0$, can be
understood by representing the variations of $\langle T_{r}(x_{0})\rangle$
with respect to $r$, as in the example of Figure 4 (right). For $x_{0}$
slightly above $x_{0}^{c\prime}=0.207...$, the optimal parameter is $r^{*}=0$
but the MFPT has developed a local, “metastable” minimum at a finite rate
$r_{local}$. As $x_{0}$ further decreases, the local minimum moves downward
with respect to $\langle T_{r^{*}=0}(x_{0})\rangle$ and becomes the absolute
minimum after some point, where $\langle
T_{r=0}(x_{0}^{c\prime})\rangle=\langle
T_{r=r_{local}}(x_{0}^{c\prime})\rangle$. In contrast with the wide side, near
the narrow boundary the extent of the region where restart is favoured shrinks
as the slope $\lambda$ increases.

With a discontinuous transition, the criterion $CV=1$ cannot be used to obtain
the exact value of $x_{0}^{c\prime}$. The condition $CV<1$ (a positive slope
for the MFPT at $r=0$) is not sufficient to guarantee that $r^{*}=0$ in
general, as a deeper minimum might exist for some other value of $r>0$. This
is clearly the case for $\lambda=0.20$ in Figure 4 (right). Nevertheless,
since the regime with two local minima exists for a tiny range of values of
$x_{0}$, the criterion $CV=1$ provides a reasonable approximation for locating
the discontinuous jump a priori.

Discontinuous transitions for $r^{*}$ are relatively less common in resetting
processes [67, 71]. Analytical estimates can be made within a mean field
theoretical treatment of resetting transition [71]. The joint observation of a
continuous and a discontinuous transition is less common, but have actually
been reported independently in different resetting problems related to the
present one [71, 79]. Recently, diffusion in the presence of a roughly linear
potential under resetting in an interval with absorbing-absorbing boundaries
was studied in [79]. If the potential is inclined, say, toward the boundary at
$L$, the transition in $r^{*}$ as $x_{0}$ approaches this boundary is
continuous, whereas it is discontinuous when $x_{0}$ approaches the other
boundary, very much like in Figure 4 (left). That problem is qualitatively
similar to ours, since, in our case, there actually exists an effective
logarithmic potential created by the varying radius of the channel and which
tends to push the particle toward the wide side located at $L$.

Figure 5: Conditional MFPT predicted by Eq. (59) (left) and Eq. (60) (right)
as a function of the resetting rate for different starting positions:
absorbing-absorbing case for an expanding tube of parameters values $a=0.1$,
$D_{\lambda}=0.01$, $L=0.5$, $\lambda$=0.2.

###  III.6 Conditional first passage or transition path times under resetting

In this section, we study the conditional first passage time statistics and
their behavior as a function of the resetting rate. Recalling the set of
relations (16)-(21) from Section IIB, and substituting the results for the
unconditional and conditional first passage time densities for the underlying
process, we obtain

$\displaystyle\tilde{f}^{+}_{r}(s|x_{0})$
$\displaystyle=\frac{(s+r)\sinh(\beta x_{0})[(\lambda L+a)\sinh(\alpha
x_{0})+a\sinh(\alpha(L-x_{0}))]}{\sinh(\alpha x_{0})[s(\lambda
x_{0}+a)\sinh(\beta L)+r((\lambda L+a)\sinh(\beta
x_{0})+a\sinh(\beta(L-x_{0}))]},$ (57)
$\displaystyle\tilde{f}^{-}_{r}(s|x_{0})$
$\displaystyle=\frac{(s+r)\sinh(\beta(L-x_{0}))[(\lambda L+a)\sinh(\alpha
x_{0})+a\sinh(\alpha(L-x_{0}))]}{\sinh(\alpha(L-x_{0}))[s(\lambda
x_{0}+a)\sinh(\beta L)+r((\lambda L+a)\sinh(\beta
x_{0})+a\sinh(\beta(L-x_{0}))]},$ (58)

with $\alpha=\sqrt{r/D_{\lambda}}$, $\beta=\sqrt{(s+r)/D_{\lambda}}$, and
where we have made use of Eqs. (21), (45) and (49). The conditional mean first
passage times are directly obtained from there. Skipping details, we find

$\displaystyle\langle\tau_{r}(x_{0})\rangle^{+}=$
$\displaystyle\frac{-2(\lambda L+a)\sinh(\alpha x_{0})+a\alpha
L\cosh(\alpha(L-x_{0}))+2(\lambda x_{0}+a)\sinh(\alpha
L)-2a\sinh(\alpha(L-x_{0}))-a\alpha x_{0}cosech(\alpha x_{0})\sinh(\alpha
L)}{2r[(\lambda L+a)\sinh(\alpha x_{0})+a\sinh(\alpha(L-x_{0}))]},$ (59)

and

$\displaystyle\langle\tau_{r}(x_{0})\rangle^{-}=$
$\displaystyle\frac{-2a\sinh(\alpha(L-x_{0}))+(\lambda L+a)\alpha
x_{0}\cosh(\alpha x_{0})+2(\lambda x_{0}+a)\sinh(\alpha L)-(\lambda
L+a)(2+\alpha(L-x_{0})\coth(\alpha(L-x_{0})))\sinh(\alpha
x_{0})}{2r[a\sinh(\alpha(L-x_{0}))+(\lambda L+a)\sinh(\alpha x_{0})]}.$ (60)

As expected, these expressions reduce to Eqs. (50) in the $r\to 0$ limiting
case. The conditional times are plotted as a function of the resetting rate in
Fig. 5, showing an excellent agreement with simulations.

Similar to the unconditional mean-first passage time, the conditional times
also display both monotonic and non-monotonic behavior as a function of the
resetting rate (see Fig. 6 for a wider range of the parameter $r$). The
introduction of resetting can either increase or decrease the conditional
times at small $r$, as shown in Fig. 5, however for large $r$, they always
increase. Furthermore, the conditional mean times can also be optimized with
the introduction of resetting in some cases that are quite counter-intuitive,
e.g., when particle starts and resets at the opposite end of the desired
escape side. See for instance the curve corresponding to $x_{0}=0.1$ in Fig. 5
(left). Even though the particle starts very close to the left boundary,
resetting can expedite its escape from the right boundary. Similarly, take a
look at the plot corresponding to $x_{0}=0.4$ in Fig. 5 (right). Resetting can
expedite the escape from the left boundary even for those trajectories which
start and reset at a point very close to the right boundary.

This non-intuitive behavior can be explained qualitatively in the following
way. Statistically, all the trajectories set out randomly to find the targets.
Some of these realizations spend more time going to the opposite direction of
the desired boundary (where we expect them to be absorbed) without being
trapped by the other absorbing end meanwhile. Naturally, such trajectories
render huge fluctuations in the first passage time. In addition, expansion of
the tube also creates more space for exploration, contributing to these
fluctuations as well. In such cases, resetting even to a close point to the
non-desired end can help to mitigate such large fluctuations (and avoid
looping or transient excursions) – a signature characteristic also can be seen
for the unconditional time. Eventually, the particles translocate through the
tube and finally reach the desired end after a considerable time. While this
is not a typical behavior, nonetheless it happens strongly induced by
resetting. This counter-intuitive observation, which does not arise in the
reset-free process, is one noteworthy result of our work.

Figure 6: Same as Figure 5, over a wider interval of values of $r$ capturing
both monotonic and non-monotonic behavior.

##  IV First passage observables in a three dimensional Cone with One
absorbing and One reflective boundary

Figure 7: The MFPT predicted by Eqs. (67) and (72) for expanding tubes
(continuous lines) are compared with the values obtained by three-dimensional
Brownian dynamics simulations (open symbols) as function of the starting
position, without resetting (left) and with resetting (right), for the
reflecting-absorbing case. The parameters are $a=0.1$, $D_{\lambda}=0.01$,
$L=0.5$, $\lambda$=0.2.

In this section, we consider a 3D narrow-to-wide expanding cone which has a
reflective boundary at one end ($x=0$) and an absorbing boundary at the other
($x=L$). Later, we will also consider the wide-to-narrow geometry, a case
which is physically different since the absorbing boundary switches from the
wide side to the narrow side. For direct comparison, it is convenient to set
the parameter $a$ that denotes the narrow diameter in both geometries. The
wide-to-narrow quantities can therefore be deduced by making the
transformations $\lambda\rightarrow-\lambda$ and $a\rightarrow\lambda L+a$
into the narrow-to-wide expressions (with $\lambda\geq 0$ in all cases), as
discussed in Appendix A.

The particle starts from $x_{0}$, but it can escape only from one side and
thus the conditional times are identical to the unconditional times. The
absorbing boundary condition remains the same i.e., $c(L,t)=0$, while the
reflecting boundary condition imposes the probability current to vanish at
$x=0$ such that $J(0,t)=0$ where
$J(x,t)=-D_{\lambda}R^{2}(x)\frac{\partial}{\partial
x}\left[\frac{{c}(x,t)}{R^{2}(x)}\right]$. Recalling $c(x,t)=R(x)g(x,t)$ from
Eq. (24), the reflecting boundary condition simplifies to

$\displaystyle\frac{\partial g}{\partial x}\bigg{|}_{x=0}=(\lambda
g/a)\bigg{|}_{x=0}.$ (61)

As before, the next step is to compute the concentration $c(x,t)$ for which we
need to evaluate the unknown constants of the solutions (31) and (32). To this
end, we use the Laplace transform of the above-mentioned boundary conditions
and obtain

$\displaystyle c_{1}=\frac{-(\lambda-
ua)\sinh(u(L-x_{0}))}{2uD_{\lambda}(\lambda
x_{0}+a)[\lambda\sinh(uL)+ua\cosh(uL)]},$ (62) $\displaystyle
c_{2}=\frac{(\lambda+ua)\sinh(u(L-x_{0}))}{2uD_{\lambda}(\lambda
x_{0}+a)[\lambda\sinh(uL)+ua\cosh(uL)]},$ (63) $\displaystyle
c_{3}=\frac{e^{uL}[\lambda\sinh(ux_{0})+ua\cosh(ux_{0})]}{2uD_{\lambda}(\lambda
x_{0}+a)[\lambda\sinh(uL)+ua\cosh(uL)]},$ (64) $\displaystyle
c_{4}=\frac{-e^{-uL}[\lambda\sinh(ux_{0})+ua\cosh(ux_{0})]}{2uD_{\lambda}(\lambda
x_{0}+a)[\lambda\sinh(uL)+ua\cosh(uL)]},$ (65)

where $u=\sqrt{s/D_{\lambda}}$. It is worth noting that this result allows us
to take any value of $x_{0}$ within the channel, which generalizes the one
obtained in [6] where the result is restricted to $x_{0}=0$.

###  IV.1 Current and first passage times of underlying process

Since the particle can escape only through the right boundary $L$, the total
flux is given from using Eq. (39),

$\displaystyle j(s|x_{0})=\frac{(\lambda L+a)}{(\lambda
x_{0}+a)}\frac{[\lambda\sinh(ux_{0})+ua\cosh(ux_{0})]}{[\lambda\sinh(uL)+ua\cosh(uL)]},$
(66)

which is equal to the first passage time density in Laplace space. The first
and the second moment of the underlying process are then easy to compute by
expanding the above expression in powers of $u^{2}$. One obtains,

$\displaystyle\langle T(x_{0})\rangle=\frac{L^{2}(\lambda
L+3a)}{6D_{\lambda}(\lambda L+a)}-\frac{x_{0}^{2}(\lambda
x_{0}+3a)}{6D_{\lambda}(\lambda x_{0}+a)},$ (67)

and

$\displaystyle\langle T^{2}(x_{0})\rangle$
$\displaystyle=\frac{x_{0}^{4}(\lambda x_{0}+5a)}{60D^{2}(\lambda
x_{0}+a)}-\frac{L^{4}(\lambda L+5a)}{60D^{2}(\lambda L+a)}$
$\displaystyle+\frac{L^{4}(\lambda L+3a)^{2}}{18D^{2}(\lambda
L+a)^{2}}-\frac{L^{2}x_{0}^{2}(\lambda L+3a)(\lambda
x_{0}+3a)}{18D^{2}(\lambda L+a)(\lambda x_{0}+a)}.$ (68)

When $x_{0}$=0, the first passage density and MFPT are in agreement with
previously known results [6]:

$\displaystyle\tilde{f}(s|x_{0}=0)$ $\displaystyle=\frac{u(\lambda
L+a)}{\lambda\sinh(uL)+ua\cosh(uL)},$ (69) $\displaystyle\langle
T(x_{0}=0)\rangle$ $\displaystyle=\frac{L^{2}(3a+\lambda
L)}{6D_{\lambda}(a+\lambda L)}.$ (70)

In Fig. 7 (left), we compare our analytical prediction for the first moment of
the first-passage time, given by Eq. (67) with the results obtained from
three-dimensional Brownian dynamics simulations. Again, one can see excellent
agreement between the theoretical predictions and simulation results.

###  IV.2 First passage time statistics under resetting

To obtain the exit time statistics under resetting, we again use the renewal
relations obtained in Section II.2. Here, the first passage time density and
the mean escape time have the following forms, respectively:

$\displaystyle\tilde{f}_{r}(s|x_{0})=\frac{(s+r)(\lambda
L+a)[\lambda\sinh(\beta x_{0})+\beta a\cosh(\beta x_{0})]}{r(\lambda
L+a)[\lambda\sinh(\beta x_{0})+\beta a\cosh(\beta x_{0})]+s(\lambda
x_{0}+a)[\lambda\sinh(\beta L)+\beta a\cosh(\beta L)]},$ (71)

and

$\displaystyle\langle T_{r}(x_{0})\rangle=\frac{(\lambda
x_{0}+a)[\lambda\sinh(\alpha L)+\alpha a\cosh(\alpha L)]-(\lambda
L+a)[\lambda\sinh(\alpha x_{0})+\alpha a\cosh(\alpha x_{0})]}{r(\lambda
L+a)[\lambda\sinh(\alpha x_{0})+\alpha a\cosh(\alpha x_{0})]},$ (72)

where $\beta=\sqrt{(s+r)/D_{\lambda}}$ and $\alpha=\sqrt{r/D_{\lambda}}$. We
have verified the theoretical expression for the mean escape time given by Eq.
(72) against numerical simulation in Fig. 7 right, finding an excellent
agreement. In Fig. 8, we represent $\langle T_{r}\rangle$ as a function of the
resetting rate. As in the two-absorbing boundary case, resetting can lower the
mean escape time if the particle starts close enough to the absorbing boundary
(top panel), while from far away positions (bottom panel), resetting only
prolongs the escape time. This transition can again be understood by looking
at the $CV>1$ criterion, which we study next.

Figure 8: MFPT predicted by Eq. (72) (continuous lines) is compared with the
values obtained by three-dimensional Brownian dynamics simulations (open
symbols) as function of reset rate for starting positions when resetting is
useful (top) vs. when resetting is not useful (bottom): reflecting-absorbing
case with expanding tube for parameter values a=0,1, $D_{\lambda}=0.01$,
L=0.5, $\lambda$=0.2.

###  IV.3 When is resetting useful?

As mentioned before, the introduction of resetting is able to reduce the mean
escape time if the relative fluctuations ($CV$) of the escape time for the
underlying process are greater than unity. Recasting the criterion into the
form $\langle T^{2}(x_{0})\rangle>2\langle T(x_{0})\rangle^{2}$ and using Eqs.
(67) and (IV.1), we find

$\displaystyle\frac{6}{5}v^{4}\left(\frac{\lambda v+5\tilde{a}}{\lambda
v+\tilde{a}}\right)$ $\displaystyle\geq 4\left[v^{2}\frac{\lambda
v+3\tilde{a}}{\lambda
v+\tilde{a}}-\frac{\lambda+3\tilde{a}}{2(\lambda+\tilde{a})}\right]^{2}$
$\displaystyle+\frac{\lambda^{2}+6\tilde{a}\lambda-15\tilde{a}^{2}}{5(\lambda+\tilde{a})^{2}},$
(73)

with $v=x_{0}/L$. For $\lambda=0$ , this reduces to: $5v^{4}-6v^{2}+1\leq 0$.
For $v=0$, the criterion (IV.3) is never satisfied as $\langle
T^{2}(x_{0}=0)\rangle-2\langle
T(x_{0}=0)\rangle^{2}=-\frac{L^{4}}{60D_{\lambda}^{2}}\frac{\lambda+5\tilde{a}}{\lambda+\tilde{a}}<0$.
In this case, since the initial/resetting location is at the origin, where the
reflecting boundary is placed, if one resets to this boundary, reset cannot
expedite the completion as every trajectory starting here is a favourable
trajectory for the particle to reach the absorbing boundary. There exists a
finite optimal resetting rate for all starting positions satisfying the
criterion, as shown in Fig. 8 (top panel).

###  IV.4 Optimal resetting rate

Variation of the optimal resetting rate with the initial position is displayed
in Figure 9, where the effects of the channel geometry can also be
appreciated. At $\lambda=0$, one recovers a continuous transition to non-
vanishing values of $r^{*}$, which occurs at
$x_{0}^{c}/L=1/\sqrt{5}=0.4472...$. A well-known result for the 1D interval
with a single absorbing boundary [79, 80].

Figure 9:  Optimal restart rate $r^{*}$ vs. the starting position $x_{0}$, for
various channel slopes $\lambda$ in the absorbing-reflective case. The
absorbing boundary is on the right ($x_{0}=1$). The solid lines correspond to
the N$\rightarrow$W geometry and the dashed ones to the W$\rightarrow$N case.
$D_{\lambda}=1$, $L=1$, and the narrow diameter is $a=0.1$.

In the narrow-to-wide geometry (solid lines), a finite $\lambda$ results in a
shift of $x_{0}^{c}$ toward the absorbing boundary. This can be understood by
the fact that, once again, the expanding channel generates an effective
potential which biases freely diffusing Brownian particles toward the
absorbing boundary. Owing to such attraction, one expects lower fluctuations
of the first passage time. The larger the channel slope, the stronger the
effective potential, and resetting becomes less needed. This finding agrees
qualitatively with the picture that emerges in the study of diffusion under
resetting on the semi-infinite line with a constant drift toward a target
[82]: in that problem, as the drift velocity increases at fixed $x_{0}$, the
optimal resetting rate decreases, until it vanishes at a critical velocity.
Similar findings were reported by using other attractive potentials [83].

Conversely, in the wide-to-narrow configuration (dashed lines of Fig. 9), the
effective potential pushes the particle in the direction opposite to the
absorbing boundary. In this case, restart is beneficial over a larger range of
values of $x_{0}$. It is worth mentioning that the transition remains
continuous: no discontinuous jumps occur, unlike in Figure 4 (left). This
highlights the fact that the number of absorbing boundaries has quite non-
trivial consequences on the optimization of resetting processes. Finally, as
$x_{0}\rightarrow 0$, all the cases converge toward the optimal rate for a
$1d$ interval ($\lambda=0$): when the particle starts very close to the target
boundary, the channel geometry becomes irrelevant in the optimization process.
This feature is also shared by the absorbing-absorbing case (see Fig. 4-left).

##  V Conclusions

We have studied the transport of diffusing particles between the left and
right ends of a 3D expanding or narrowing conical tube/channel. Such
transitions are ubiquitous in nature and have important applications in
various biochemical processes such as isomerization reactions, enzymatic
catalysis, barrier crossing or channel facilitated transport of solutes. In
addition to the diffusion, the particles also experience an intermittent
transient dynamics, namely, resetting which puts them back to some preferred
locations. Usually, in the reset-free process, a particle can escape from both
boundaries due to the entropic potential which can also render a preference in
the escape direction. In the resetting process, wandering trajectories causing
large fluctuations in the escape time are eliminated, causing a significant
reduction in the overall escape time.

We approached the problem using an effective one-dimensional description of
the particle diffusion along the tube axis in terms of the modified Fick-
Jacobs equation. There, the three dimensional effects are replaced by the so-
called entropic potential that essentially plays a crucial role in the
particle dynamics. As such, the diffusion now occurs with an effective
diffusivity entering into the modified Fick-Jacobs equation, which is smaller
than the particle diffusivity in a cylindrical tube. This is one way how the
varying tube geometry manifests itself in the direct transits. The presence of
resetting creates an influx of probabilities to the resetting coordinate by
withdrawing probabilities from the rest of the free space – this necessitates
further modification of the Fick-Jacobs formalism for the resetting process.
Using this, we presented a comprehensive study of escape properties. In
particular, we (re)derived renewal relations for the (un)conditional escape
times which become crucial in understanding the behavior of such observables
in the presence of resetting. In particular, we computed the average lifetime
that the particle spends inside the conical channel before it exits through
one of the end points.

We find that the resetting can expedite the crossing/escape time of the
particle – a hallmark feature that is shared by many other stochastic
processes subject to resetting. This feature, however, is not arbitrary and it
has been repeatedly demonstrated that resetting works in favour when
stochastic fluctuations in the completion time of a random process are large,
i.e., $CV>1$. We explicitly verify this criterion for both the expanding and
narrowing conical tubes. The effects of resetting are further analyzed by
observing the behavior of optimal resetting rate $r^{*}$ that minimizes the
mean escape time. Interestingly, the optimal rate, as a function of the
initial/resetting coordinate, shows both continuous and discontinuous
transitions. Each of these cases are thoroughly analyzed. Finally, we provide
exact analytical results for the conditional escape times and the
corresponding probabilities for the reset induced diffusive dynamics. The
conditional escape times are found to display non-trivial 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, which
usually cause large fluctuations in such cases, get curtailed by resetting
hence a speed-up in the overall first passage time is observed.

Recent single-molecule experiments, including pulling studies on proteins and
nucleic acid folding, and single-molecule fluorescence spectroscopy have
raised a number of questions that stimulated theoretical and computational
investigation on barrier-crossing dynamics [5, 6, 28, 29, 84, 85, 86, 87]. A
model of chemical reactions considers transitions of a Brownian particle
between two deep wells of a one-dimensional double-well potential, separated
by a high barrier. The fine structure of these trajectories has been analyzed
in order to gain new insights into escape dynamics. Any trajectory of a
diffusing particle making a transition between two end points of an interval
in one dimension will have two components: the transition path segment – part
of the trajectory that leaves the starting point for the last time and goes to
the end without returning to the starting point, and the looping segment where
a number of loops that start and end at the same starting point are completed
[6]. We believe that the latter may be visualized as a non-instantaneous
resetting process – a class of process which assumes that resetting is a
physical process, and thus reset/return to a preferred location will indeed
take time [64, 88, 89, 90, 91]. Bridging this gap can be useful to analyze,
for example, the capture and escape of densely negatively charged C-terminal
tails of cytosolic proteins in nanopores of $\beta$ -barrel channels, as well
as artificial constructs assembled from albumin molecules and covalently
attached negatively charged peptides mimicking C-termini of cytosolic proteins
[26] – potential research avenues that need to be explored in future.

##  VI Acknowledgements

AP gratefully acknowledges the DST-SERB Start-up Research Grant Number
SRG/2022/000080 and DAE, India for research funding. This study was partially
supported by CONACyT under the grant Frontiers Science No. 51476.

##  Appendix A Wide-to-narrow tube

In the main text, we were working with narrow-to-wide conical tube whose
radius varied as $R(x)=\lambda x+a$ with $\lambda\geq 0$. Here we give the
results corresponding to wide-to-narrow tubes, i.e., when the radius varies as
$R(x)=\lambda(L-x)+a$ with $\lambda\geq 0$, in the case of one reflective and
one absorbing boundary. In other words, the reflective boundary at $x=0$ is
now wide, whereas the absorbing boundary at $x=L$ is narrow. The full
expressions for this case can be obtained by substituting $\lambda$ by
$-\lambda$ and $a$ by $a+\lambda L$ in the expressions of Section IV.

We do not repeat the entire methodology here, but just summarize the main
results of our interest. For instance, the first passage density of underlying
process is given by

$\displaystyle\tilde{f}(s|x_{0})=\frac{a}{(\lambda(L-x_{0})+a)}\frac{[-\lambda\sinh(ux_{0})+u(\lambda
L+a)\cosh(ux_{0})]}{[-\lambda\sinh(uL)+u(\lambda L+a)\cosh(uL)]}.$ (74)

The first two moments of the first passage time distribution are given by

$\displaystyle\langle T(x_{0})\rangle=\frac{L^{2}(2\lambda
L+3a)}{6D_{\lambda}a}-\frac{x_{0}^{2}[\lambda(3L-x_{0})+3a]}{6D_{\lambda}[\lambda(L-x_{0})+a]},$
(75)

and

$\displaystyle\langle
T^{2}(x_{0})\rangle=\frac{x_{0}^{4}[\lambda(5L-x_{0})+5a]}{60D_{\lambda}^{2}[\lambda(L-x_{0})+a]}-\frac{L^{4}(4\lambda
L+5a)}{60D_{\lambda}^{2}a}+\frac{L^{4}(2\lambda
L+3a)^{2}}{18D_{\lambda}^{2}a^{2}}-\frac{L^{2}x_{0}^{2}(2\lambda
L+3a)[\lambda(3L-x_{0})+3a]}{18D^{2}a[\lambda(L-x_{0})+a]}.$ (76)

For all values of the starting position and $\lambda$ as small as $0.2$, the
MFPT is significantly larger than in the narrow-to-wide configuration, as seen
in Fig. 10 in comparison to Fig. 7. In the expanding case the trajectories are
subject to an effective entropic force in the direction of the absorbing wall.

Figure 10: Variation of MFPTs as a function of starting position $x_{0}$
without resetting (left) and with resetting (right) in the reflecting-
absorbing case with a narrowing tube of parameters $a=0.1$,
$D_{\lambda}=0.01$, $L=0.5$, $\lambda=0.2$.

The first passage density under resetting is given in the Laplace domain by:

$\displaystyle\tilde{f}_{r}(s|x_{0})=\frac{a(s+r)[-\lambda\sinh(\beta
x_{0})+\beta(\lambda L+a)\cosh(\beta x_{0})]}{ra[-\lambda\sinh(\beta
x_{0})+\beta(\lambda L+a)\cosh(\beta
x_{0})]+s(\lambda(L-x_{0})+a)[-\lambda\sinh(\beta L)+\beta(\lambda
L+a)\cosh(\beta L)]},$ (77)

from which we obtain the MFPT with resetting as

$\displaystyle\langle
T_{r}(x_{0})\rangle=\frac{(\lambda(L-x_{0})+a)[-\lambda\sinh(\alpha
L)+\alpha(\lambda L+a)\cosh(\alpha L)]-a[-\lambda\sinh(\alpha
x_{0})+\alpha(\lambda L+a)\cosh(\alpha x_{0})]}{ra[-\lambda\sinh(\alpha
x_{0})+\alpha(\lambda L+a)\cosh(\alpha x_{0})]}.$ (78)

Figure 11: Variation of the MFPT as a function of the resetting rate for
starting positions such that resetting is not useful (left) and useful
(right), in the reflecting-absorbing case with a narrowing tube. The
parameters are $a=0.1$, $D_{\lambda}=0.01$, $L=0.5$, $\lambda$=0.2. The
criterion $CV=1$ gives a critical initial position $x_{0}^{c}$=0.2067.

We see that for $v=x_{0}/L=0$, the criteria favouring resetting is never
satisfied since
$(\frac{2\lambda+3\overline{a}}{\overline{a}})^{2}\leq\frac{-3(4\lambda+\overline{a})}{10\overline{a}}$.
In the example of Figure 11 (left), resetting increases the MFPT of all the
initial conditions that are too close to the reflective wall
($x_{0}<0.2067...$), whereas an optimal $r^{*}>0$ exists for the initial
conditions that are further away (right). When $\lambda=0$, the criteria
$CV>1$ reduces to $-25v^{4}+70v^{2}-31\geq 0.$

##  Appendix B Numerical simulations

When running simulations, we considered an overdamped point-like Brownian
particle diffusing into the 3D conical tube, where the particle dynamics can
be described by means of the Langevin equation, namely,

$\frac{d\vec{r}}{dt}=\;\vec{\xi}(t),$ (79)

where $\vec{r}=(x,y,z)$. Consequently, the Brownian particle is subject to a
Gaussian noise $\vec{\xi}(t)$ of zero mean and uncorrelated in time i.e.,
$\langle\vec{\xi}(t)\rangle=\langle(\xi_{x}(t),\xi_{y}(t),\xi_{z}(t))\rangle=0$
and the auto-correlation functions are
$\langle\xi_{i}(t),\xi_{j}(t^{\prime})\rangle=2D_{0}\delta_{ij}\delta(t-t^{\prime})$
where $i,j=x,y,z$. When running simulations we took the time step $\Delta
t=10^{-7}$ and the effective diffusivity $D_{\lambda}=0.01$, so that
$\sqrt{2D_{0}\Delta t}\ll 1$, and $k_{B}T$ is set equal to 1. Using
$D_{\lambda}=D_{0}/\sqrt{1+\lambda^{2}}$ and $\lambda=0.2$, we have a factor
of $\sqrt{1.04}=1.0198$ between them. Stochastic averages were obtained as
ensemble averages over $5.0\times 10^{4}$ trajectories. Our simulation results
have shown that the reduction of axial diffusion for three-dimensional conical
tubes with resetting, to the effective one-dimensional description in terms of
the modified FJ equation, are in good agreement with a relative error of less
than 3%.

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