# Local resetting with geometric confinement

Asaf Miron

###### Abstract

“Local resetting” was recently introduced to describe stochastic resetting in
interacting systems where particles independently try to reset to a common
“origin”. Our understanding of such systems, where the resetting process is
itself affected by interactions, is still very limited. One ubiquitous
constraint that is often imposed on the dynamics of interacting particles is
geometric confinement, e.g. restricting rigid spherical particles to a channel
so narrow that overtaking becomes difficult. We here explore the interplay
between local resetting and geometric confinement in a system consisting of
two species of diffusive particles: “bath” particles, and “tracers” which
undergo local resetting. Mean-field analysis and numerical simulations show
that the resetting tracers, whose stationary density profile exhibits a
typical “tent-like” shape, imprint this shape onto the bath density profile.
Upon varying the ratio of the degree of geometric confinement over particle
diffusivity, the system is found to transition between two states. In one
tracers expel bath particles away from the origin, while in the other they
ensnare them instead. Between these two states, we find a special case where
the mean field approximation becomes exact.

Department of Physics of Complex Systems, Weizmann Institute of Science,
Rehovot 7610001, Israel

##  1 Introduction

Stochastic resetting is a non-equilibrium process that is typically used to
model dynamical observables that are “instantaneously” 111i.e. much faster
than any other relevant time-scale in the problem. returned to their initial
state some random time after having started their temporal evolution [1, 2, 3,
4, 5, 6, 7, 8, 9]. With applications ranging from search and optimization
algorithms [10, 11, 12, 13], through chemical reactions [14, 15, 16], to
processes inside biological cells [17, 18, 19], the study of stochastic
resetting is attracting broad interest from a diverse scientific audience.

Stochastic resetting has already been vigorously explored under a variety of
settings. These efforts have been greatly aided by the “renewal” framework, an
analytical tool that allows deriving exact results in situations where
stochastic resetting acts “globally” \- when stochastic resetting is either
applied to a single degree of freedom, or when it is simultaneously applied to
multiple degrees of freedom [20, 21, 22, 23, 24]. Perhaps due to its
effectiveness and applicability to a wide range of systems, far less attention
has been directed towards scenarios where stochastic resetting acts “locally”
and the resetting process is itself affected by interactions, i.e. when
different interacting degrees of freedom reset independently of one-another.
Yet in real complex systems whose dynamics are modeled by stochastic
resetting, there is no a-priori reason to assume interactions could not
dramatically affect resetting. Exploring the interplay between locally-acting
stochastic resetting and various forms of interaction is thus a natural step
forward, and a research direction that should be pursued to clarify the nature
of stochastic resetting and its consequences.

The “local resetting” process has recently been introduced in an attempt to
fill-in this gap [25]. Conditioning the success of a resetting attempt made by
one degree of freedom, on the states of all other degrees of freedom has
already been applied [25, 26] to study local resetting in a canonical model
for interacting systems, the simple exclusion model [27]. Since the renewal
approach is limited to global resetting, where all degrees of freedom undergo
resetting simultaneously, local resetting is inherently out of its reach.
Instead the mean field (MF) approximation was utilized as a means of dealing
with the correlations generated by introducing interactions into the resetting
process, and interesting observations immediately followed: in the
thermodynamic limit and for a fixed particle density, it was shown that the
stationary density profile becomes entirely independent of the resetting rate,
and that the profile’s shape scales with system size [25]. It was also found
that introducing a system-size dependence into the resetting rate has a non-
trivial effect on the stationary profile’s shape [25, 26]. Another striking
observation, which has yet to be explained, is the existence of parameter
regimes where density profiles computed using the MF approximation yield an
unexpectedly good fit to direct numerical simulations [25, 26]. Local
resetting evidently opens an exciting door to the exploration of the interplay
between stochastic resetting and various types of interactions.

One ubiquitous setting which is known to generate non-trivial dynamics in
interacting systems is geometric confinement. The prototypical setup for
probing the consequence of geometric constraints on the dynamics of
interacting systems, is that of diffusive spherical particles with repulsive
interactions that are confined to a narrow channel. Geometric confinement and
repulsive interactions then typically generate strong spatial and temporal
correlations that make it difficult for particles to overtake one-another. The
resulting plethora of phenomena, which have been investigated in numerous
studies over the past several decades, include tracer sub-diffusion, negative
absolute and differential mobility, and the vanishing of the velocity of a
driven probe [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46]. Following the recent introduction of local resetting, where
interactions enter the resetting process, one may wonder: “how would local
resetting affect interacting systems whose dynamics are restricted by
geometric confinement?”.

Here we study the interplay between stochastic resetting and the geometrically
constrained dynamics of interacting diffusive particles. Envision a narrow
channel with rigid walls occupied by two species of diffusive spherical
particles that interact via short-ranged repulsion. Particles of both species
are completely identical except for two properties: 1) each species may have a
different mass, and 2) the particles of one of the species undergo local
resetting, which here means that an attempt to reset to a target location in
the channel is successful only if some fixed volume surrounding the target is
vacant of any other particle. Consider particles whose radius is not much
smaller than one quarter of the channel’s width, where interactions and
geometric confinement conspire to reduce the rate at which neighboring
particles overtake one another. Analytically tackling this interacting non-
equilibrium system is a very challenging task with current methods and tools.
Yet its main ingredients, the interplay between local resetting and geometric
constraints, can be studied using a toy model that is more susceptible to
analytical treatment. Extending the efforts of [25, 26], we model the narrow
channel setup via the stochastic dynamics of hopping particles on a $1D$ ring
lattice of $L$ sites. We refer to particles of the species undergoing local
resetting as “tracer” particles, while particles of the other species are
called “bath” particles. Hard-core exclusion interactions [47, 48, 49], by
which particles are excluded from entering an occupied site, replace the
channel’s short-ranged repulsive interactions. The different diffusive
dynamics of each particle species in the channel, which follow from their mass
difference, are modeled by the different rates with which particles attempt to
hop into adjacent neighboring lattice sites, setting the tracer hopping rate
to $1$ and the bath particle hopping rate to $D$. Tracers also undergo local
resetting, independently attempting to reset to the “origin” subject to
exclusion. Finally, the geometric confinement imposed by the narrow channel’s
walls is modeled by an “overtaking rate” $\varepsilon$, at which a lattice
particle attempts to exchange places with a neighboring particle at an
adjacent site.

After formulating equations for the evolution of the mean bath and tracer
particle density profiles, we apply the MF approximation and compute the
profiles in the stationary limit $t\rightarrow\infty$. As in the single
species case [25], the stationary tracer density profile
$\tau\left(x=\frac{\ell}{L}\right)$ satisfies the same (stationary) evolution
equation as a single diffusive particle undergoing stochastic resetting.
$\tau\left(x\right)$ correspondingly exhibits the typical “tent-like” shape
[1], with a cusp at the origin $x=0$. Yet the main result of this work
concerns the behavior of the stationary bath density profile
$\rho\left(x\right)$, whose behavior changes dramatically as the ratio
$\frac{\varepsilon}{D}$ crosses unity. For $\frac{\varepsilon}{D}<1$ bath
particles manage to escape the dense region surrounding the origin, their
density profile resembling an inverted tent. But for $\frac{\varepsilon}{D}>1$
bath particles diffuse too slowly to escape and a macroscopic number of them
are ensnared near the origin, resulting in a bath density profile whose shape
is similar to $\tau\left(x\right)$. When the overtaking and bath hopping rates
satisfy $\frac{\varepsilon}{D}=1$, the bath profile becomes flat and equal to
$\overline{\rho}$. These three regimes are predicted by MF theory and are
validated by direct numerical simulations of the lattice model’s dynamics.
While the MF approximation becomes exact at $\varepsilon=D=1$ the agreement
between the MF and simulated profiles persists for $\varepsilon\approx
D\approx 1$. We hope that the intriguing results that emerge through the
interplay of geometric confinement and local resetting will fuel experimental
and/or numerical investigations of the narrow channel setup and, more
generally, spark interest in how different types of interactions effect the
local resetting process.

The paper is organized as follows: Section 2 introduces the $1D$ lattice model
used to probe local resetting and geometric confinement in the narrow-channel
setup. Our main results, which consist of analytical MF calculations and
numerical simulations for the lattice model, are presented in Sec. 3. Section
4 presents the derivation of the evolution equations for the mean bath and
tracer density profiles, which contain correlations due to local resetting and
geometric confinement. In Sec. 5 we apply the MF approximation to the bath and
tracer density profile equations, solve them, and obtain the stationary
profiles. The paper is concluded in Sec. 6, where we discuss outstanding open
questions and future prospects.

##  2 The model

We aim to construct a lattice model to describe the dynamics of two species of
diffusive spherical particles, locally resetting tracers of mass $m_{t}$ and
bath particles of mass $m_{b}$, that are confined to a narrow periodic channel
and subject to short-ranged repulsive interactions (right panel of Fig. 1). To
this end, consider a $1D$ ring lattice of sites labeled $\ell=0,...,L-1$
occupied by $N$ bath particles and $M$ tracer particles, whose respective mean
densities $\overline{\rho}=N/L$ and $\bar{\tau}=M/L$ satisfy
$\overline{\rho}+\overline{\tau}\leq 1$. The system evolves in continuous time
with the following stochastic dynamics: a bath particle at site $\ell$
attempts hopping to adjacent neighboring sites $\ell+1$ and $\ell-1$ with rate
$D$ to each side. Tracer particles similarly attempt hopping to neighboring
sites, but with rate $1$ to each side. In addition to hopping, each tracer
also tries to reset its position to the “origin” site $\ell=0$ with rate $r$,
independently of all other tracers, a process termed “local resetting” in
[25]. Two mechanisms mediate interactions between the particles. The first is
hard-core “exclusion” [47, 48, 49], by which hopping and resetting attempts
that would lead a particle into an occupied site are rejected.
Correspondingly, each lattice site can be occupied by one particle at most.
The second mechanisms is “exchange”, where a particle at site $\ell$ attempts
with rate $\varepsilon$ to exchange sites with a neighboring particle at sites
$\ell+1$ and $\ell-1$. The attempt is rejected if the target site is vacant.
This mechanism has been previously used to model geometric constraints in
similar settings, including the narrow channel setup discussed above [41, 50,
44, 45, 46]. Note that since particles of the same species are
indistinguishable, considering exchanges between two particles of the same
species is immaterial. Figure 1 provides schematic illustrations of the
lattice dynamics (left panel) and the narrow channel setup (right panel).

Figure 1: Illustration of the lattice dynamics and the narrow channel setup.
Bath particles are depicted in blue and resetting tracers in red. Left:
Diagram detailing the $1D$ lattice dynamics. Bath and tracer particles attempt
hopping to adjacent sites with rates $D$ and $1$ respectively, exchanges
between neighboring particles are attempted with rate $\varepsilon$, and
tracers attempt resetting to the origin site $\ell=0$ with rate $r$. Right:
Schematic illustration of the narrow channel setup with periodic boundaries,
showing a tracer in the midst of a resetting event.

##  3 Main results

The mean-field (MF) analysis and numerical simulation results presented below
demonstrate the non-trivial interplay between local resetting and geometric
confinement. In the thermodynamic limit the stationary bath and tracer density
profiles, $\rho_{\ell}$ and $\tau_{\ell}$ respectively, are not functions of
$\ell$ and $L$ separately but are instead found to depend on the scaling
variable $x=\ell/L\in\left[0,1\right)$, i.e. as $L\rightarrow\infty$ we find
$\rho_{\ell}\rightarrow\rho\left(x\right)$ and
$\tau_{\ell}\rightarrow\tau\left(x\right)$. The tracer density profile
$\tau\left(x\right)$ is derived using the MF approximation in Eq. (27) and
features a tent-like shape with a positive curvature
$\tau^{\prime\prime}\left(x\right)>0$ and a cusp at the origin. This can be
seen on the right column of Fig. 2, which compares the profiles obtained from
MF and numerical simulations for different values of $\varepsilon$ and $D$. We
find that, through the combined effect of local resetting and geometric
confinement, the tent-like shape of $\tau\left(x\right)$ is imprinted on the
stationary density profile of the non-resetting bath particles
$\rho\left(x\right)$. However, unlike the tracer profile $\tau\left(x\right)$
which retains its shape and curvature for any $\varepsilon,D\neq 0$, the bath
profile $\rho\left(x\right)$ transitions between “repelled” and “trapped”
states as the value of $\varepsilon/D$ crosses unity (see Eqs. (23) and (27)).
For $\varepsilon/D<1$ bath particles are strongly repelled from the dense
region near the origin, so that $\rho\left(x\right)$ features a negative
curvature $\rho^{\prime\prime}\left(x\right)<0$ and the shape of an inverted
tent, as shown in the top left row of Fig. 2. For $\varepsilon/D>1$ a finite
fraction of bath particles remain trapped in the dense region surrounding the
origin. The left panel in the middle row of Fig. 2 shows that, in this case,
$\rho\left(x\right)$ has a positive curvature
$\rho^{\prime\prime}\left(x\right)>0$ and a shape similar to that of
$\tau\left(x\right)$. At $\varepsilon/D=1$ the bath density profile becomes
completely flat, with its value equal to the mean bath density
$\overline{\rho}$ (see left panel at the bottom row of Fig. 2). The insets
appearing throughout the right column of Fig. 2 plot the difference between
the MF and simulated tracer profiles, which is non-zero but still too small to
see at scale.

Figure 2: Stationary bath and tracer density profiles for
$\overline{\rho}=1/4$, $\overline{\tau}=1/8$, and $r=1$, with a fixed value of
$\varepsilon/D$ shown in each row. Notice that the horizontal axis was taken
to be $x-1/2$, which simply amounts to shifting the “origin” from $x=0$ to
$x=1/2$, to highlight the behavior near the origin. Solid black lines denote
the MF profiles, computed in the thermodynamic limit $L\rightarrow\infty$. In
this limit Eqs. (23), (27), and (29), show that $\varepsilon$ and $D$ only
enter the MF profiles via their ratio $\varepsilon/D$. Dashed, dot-dashed, and
dotted lines denote the simulated profiles, obtained for $L=128$ and different
values of $\varepsilon$ and $D$.

The different states of $\rho\left(x\right)$ are most easily understood upon
fixing the value of the exchange rate $\varepsilon$ to unity, thus setting it
equal to the tracer hopping rate. This limit corresponds to a channel that is
wide enough for the tracers’ motion to be free of geometric constraints over
their diffusive time-scale. We start by considering the repelled state
$\varepsilon/D<1$, corresponding to $D>1$. In this state tracers attempt
hopping and exchanges with equal rates, but bath particles hop at a higher
rate. Imagine a bath particle trapped in the dense region surrounding the
origin. Clearly, it has a finite probability to escape the dense region via
exchanges with tracers. Once it succeeds in doing so, reaching a region where
there is a high density on one side and a low density on the other, the bath
particle is more likely to distance itself by hopping away from the origin,
rather than to remain stuck therein. The opposite happens in the trapped state
$\varepsilon/D>1$ (i.e. $D<1$), where the lower hopping rate implies a bath
particle has a higher chance to remain trapped in the dense region near the
origin. For $\varepsilon/D=1$ (i.e. $D=1$) the bath particles experience a
completely homogeneous environment, since hopping into a vacant site and
exchanging sites with a tracer occur with the same rate.

This leads us to the last main result of this study, which is that the MF
stationary density profiles becomes exact for $\varepsilon=D=1$. When
substituting $\varepsilon=D=1$ into Eqs. (4) for $\rho\left(x\right)$ and
$\tau\left(x\right)$, the correlations due to the overtaking process vanish
from both equations. The equation for $\rho\left(x\right)$ becomes effectively
decoupled from $\tau\left(x\right)$, and is easily solved to give
$\overline{\rho}$, but the remaining equation for $\tau\left(x\right)$ still
contains correlations due to the exchange process. For $\varepsilon=D=1$ the
MF equation for $\tau\left(x\right)$ is nearly identical to the one studied in
[25] for locally resetting particles with exclusion. There, for a fixed
particle density and large $L$, the remarkable agreement between the simulated
and MF profiles was noted. The same system was later studied numerically in
[26], where the MF approximation was suggested to become exact in the limit
$L\rightarrow\infty$. Following a numerical investigating the system we here
report that for $\varepsilon=D=1$ the MF approximation does indeed seem to
become exact in the thermodynamic limit $L\rightarrow\infty$, despite the
existence of correlations coming from the local resetting process. Figures 3
and 4 demonstrate this claim by showing data collapse of the difference
between the simulated and MF profiles
$\delta\rho\left(x\right)\coloneqq\rho^{sim}\left(x\right)-\rho\left(x\right)$
and
$\delta\tau\left(x\right)\coloneqq\tau^{sim}\left(x\right)-\tau\left(x\right)$
for different system sizes $L$, with each plot showing data for different
values of the parameters $\varepsilon$ and $D$. Data for $\varepsilon/D=1$ is
shown in Fig. 3 for $\varepsilon=D=0.1$, for $\varepsilon=D=1.5$, and for the
exact limit $\varepsilon=D=1$. For all parameter sets we find
$\delta\rho\left(x\right)\approx 0$, up to structure-less noise due to finite
sampling. For $\varepsilon=D=1$ we observe that
$L^{2}\times\delta\tau\left(x\right)$ exhibits a convincing data collapse,
which suggests that
$\delta\tau\left(x\right)\sim\mathcal{O}\left(L^{-2}\right)$ and is consistent
with the claim that corrections to MF decay as $L\rightarrow\infty$. Data
collapse of $\delta\tau\left(x\right)$ for $\varepsilon=D=10$ and for
$\varepsilon=D=0.1$ shows that the correction to MF remain finite as
$L\rightarrow\infty$. We complement this picture in Fig. 4, which shows data
collapse for $\varepsilon=1$ and $D=2$ (i.e. $\varepsilon/D<1$), and for
$\varepsilon=1.5$ and $D=1$ (i.e. $\varepsilon/D>1$). Here the resulting
differences $\delta\rho\left(x\right)$ and $\delta\tau\left(x\right)$ are
small but also remain finite as $L$ increases. Notice that, as in Fig. 2, the
horizontal axis in Figs. 3 and 4 was set to $x-1/2$ to ensure the behavior
near the origin is easily and clearly visible.

A final comment about the model parameters is in order. In an effort to
isolate the consequence of changing only a subset of the model’s parameters at
a time, the figures described above all correspond to the mean bath density
$\overline{\rho}=1/8$, mean tracer density $\overline{\tau}=1/4$, and
resetting rate $r=1$. Nevertheless, the validity of the entailing results has
been confirmed for a variety of mean densities and values of $r$.

Figure 3: Data collapse of the difference between the simulated and MF
profiles for different system sizes $L=64,128,$ and $256$ for the parameters
$\overline{\rho}=1/4$, $\overline{\tau}=1/8$, $r=1$, and $\varepsilon/D=1$.
Data for the bath and tracers particles is presented on the left and right
columns respectively. The first row shows the data collapse for
$\varepsilon=D=1$, with the tracer density difference scaled by $L^{2}$, i.e.
$L^{2}\delta\tau\left(x\right)$. The convincing collapse for $\varepsilon=D=1$
implies that the MF and simulated tracer density profiles become identical as
$L\rightarrow\infty$. For $\varepsilon=D=10$ and $\varepsilon=D=0.1$, shown in
the second and third rows, the tracer density difference is not scaled with
system size $L$ and the corrections to the MF profiles remain finite. Notice
that the horizontal axis is $x-1/2$, which simply amounts to shifting the
“origin” from $x=0$ to $x=1/2$.

##  4 Evolution equations for the mean density profiles

We proceed by formulating the equations governing the evolution of the mean
bath and tracer density profiles. Let $\tilde{\rho}_{\ell}\left(t\right)$
denote the instantaneous occupation of site $\ell$ by a bath particle at time
$t$, assuming the value $1$ if a bath particle is present and $0$ otherwise,
and let $\tilde{\tau}_{\ell}\left(t\right)$ denote the same for tracers.
Considering the configurations affecting the occupation of a bath particle at
site $\ell$ and averaging over the system’s stochastic dynamics, we obtain an
evolution equation for the mean bath density profile
$\left\langle\tilde{\rho}_{\ell}\left(t\right)\right\rangle$

$\partial_{t}\left\langle\tilde{\rho}_{\ell}\right\rangle=D\left\langle\left(1-\tilde{\rho}_{\ell}-\tilde{\tau}_{\ell}\right)\left(\tilde{\rho}_{\ell+1}+\tilde{\rho}_{\ell-1}\right)\right\rangle-D\left\langle\tilde{\rho}_{\ell}\left[\left(1-\tilde{\rho}_{\ell+1}-\tilde{\tau}_{\ell+1}\right)+\left(1-\tilde{\rho}_{\ell-1}-\tilde{\tau}_{\ell-1}\right)\right]\right\rangle$
$+\varepsilon\left\langle\tilde{\tau}_{\ell}\left(\tilde{\rho}_{\ell+1}+\tilde{\rho}_{\ell-1}\right)\right\rangle-\varepsilon\left\langle\tilde{\rho}_{\ell}\left(\tilde{\tau}_{\ell+1}+\tilde{\tau}_{\ell-1}\right)\right\rangle.$
(1)

The first term on the right-hand side describes the bath density gain at site
$\ell$ due to configurations where a bath particle at site $\ell\pm 1$
attempts hopping to site $\ell$ with rate $D$. Notice the “exclusion factor”
$\left(1-\tilde{\rho}_{\ell}\left(t\right)-\tilde{\tau}_{\ell}\left(t\right)\right)$
weighing this term, which vanishes if site $\ell$ is already occupied by a
bath or tracer particle, and is $1$ otherwise. The second term describes the
loss of bath density due to the bath particle attempting to leave site $\ell$
by hopping into an adjacent site. The third and fourth terms analogously
describe gain and loss terms associated with exchanges between bath and tracer
particles, which are attempted with rate $\varepsilon$. The equation for the
mean tracer density is

$\partial_{t}\left\langle\tilde{\tau}_{\ell}\right\rangle=\left\langle\left(\tilde{\tau}_{\ell+1}+\tilde{\tau}_{\ell-1}\right)\left(1-\tilde{\rho}_{\ell}-\tilde{\tau}_{\ell}\right)\right\rangle-\left\langle\tilde{\tau}_{\ell}\left[\left(1-\tilde{\rho}_{\ell+1}-\tilde{\tau}_{\ell+1}\right)+\left(1-\tilde{\rho}_{\ell-1}-\tilde{\tau}_{\ell-1}\right)\right]\right\rangle$
$+\varepsilon\left\langle\tilde{\rho}_{\ell}\left(\tilde{\tau}_{\ell+1}+\tilde{\tau}_{\ell-1}\right)\right\rangle-\varepsilon\left\langle\tilde{\tau}_{\ell}\left(\tilde{\rho}_{\ell+1}+\tilde{\rho}_{\ell-1}\right)\right\rangle+\tilde{\chi}_{\ell},$
(2)

and its construction is analogous to that of Eq. (1). The two equations have
nearly the same structure, with the exception of the last term
$\tilde{\chi}_{\ell}$ on the right-hand side of Eq. (2). This term, which
describes the local resetting of tracers

$\tilde{\chi}_{\ell}=\begin{cases}\sum_{n=1}^{L-1}r\left\langle\left(1-\tilde{\rho}_{0}-\tilde{\tau}_{0}\right)\tilde{\tau}_{n}\right\rangle&\ell=0\\\
-r\left\langle\left(1-\tilde{\rho}_{0}-\tilde{\tau}_{0}\right)\tilde{\tau}_{\ell}\right\rangle&\ell\neq
0\end{cases},$ (3)

behaves differently at site $\ell=0$ versus sites $\ell\neq 0$:
$\tilde{\chi}_{\ell=0}$ is a gain term at site $\ell=0$ that accounts for the
attempts made by tracers, located anywhere besides the origin, to reset their
position to the origin with rate $r$, whereas $\tilde{\chi}_{\ell\neq 0}$
describes the loss of tracer density at sites $\ell\neq 0$ due to resetting
attempts with rate $r$. The exclusion factor
$\left(1-\tilde{\rho}_{0}-\tilde{\tau}_{0}\right)$ in $\tilde{\chi}_{\ell}$
ensures that resetting attempts are rejected if the origin is occupied.

Figure 4: Data collapse of the difference between the simulated and MF
profiles for different system sizes $L=128,256,$ and $512$. The parameters
$\overline{\rho}=1/4$, $\overline{\tau}=1/8$, and $r=1$ are used throughout
with $\varepsilon/D\neq 1$. The first row shows the collapse for
$\varepsilon=1$ and $D=2$, while the second row is for $\varepsilon=1.5$ and
$D=1$. The collapse implies that the MF and simulated density profiles remain
different as $L\rightarrow\infty$. Notice that the horizontal axis is $x-1/2$,
which simply amounts to shifting the “origin” from $x=0$ to $x=1/2$.

Defining the lattice Laplacian $\nabla_{\ell}^{2}$ and gradient
$\nabla_{\ell}$ operators, which act on a lattice variable $u_{\ell}$ as
$\nabla_{\ell}^{2}u_{\ell}=u_{\ell+1}-2u_{\ell}+u_{\ell-1}$ and
$\nabla_{\ell}u_{\ell}=u_{\ell}-u_{\ell-1}$, and respectively denoting the
average bath and tracer density profiles by
$\rho_{\ell}=\left\langle\tilde{\rho}_{\ell}\right\rangle$ and
$\tau_{\ell}=\left\langle\tilde{\tau}_{\ell}\right\rangle$ simplifies Eqs. (1)
and (2) to

$\frac{1}{D}\partial_{t}\rho_{\ell}=\nabla_{\ell}^{2}\rho_{\ell}+\left(1-\frac{\varepsilon}{D}\right)\nabla_{\ell}\left(\left\langle\tilde{\rho}_{\ell}\tilde{\tau}_{\ell+1}\right\rangle-\left\langle\tilde{\tau}_{\ell}\tilde{\rho}_{\ell+1}\right\rangle\right),$
(4)

and

$\partial_{t}\tau_{\ell}=\nabla_{\ell}^{2}\tau_{\ell}-\left(1-\varepsilon\right)\nabla_{\ell}\left(\left\langle\tilde{\rho}_{\ell}\tilde{\tau}_{\ell+1}\right\rangle-\left\langle\tilde{\tau}_{\ell}\tilde{\rho}_{\ell+1}\right\rangle\right)+\tilde{\chi}_{\ell}.$
(5)

It is now apparent that the mean density profiles $\rho_{\ell}$ and
$\tau_{\ell}$ are affected by correlations that separate into two
contributions: “exchange correlations”, which appear in the second term on the
right-hand sides of both equations and are generated by the exchange
interactions, and “local resetting” correlations, which only enter Eq. (5) via
$\tilde{\chi}_{\ell}$ and are generated by the local resetting process.
Particle conservation places an additional constraint on $\rho_{\ell}$ and
$\tau_{\ell}$, requiring that

$\sum_{\ell=0}^{L-1}\rho_{\ell}\left(t\right)=N\text{ and
}\sum_{\ell=0}^{L-1}\tau_{\ell}\left(t\right)=M.$ (6)

Notice that the exchange correlations vanish from both equations for
$\varepsilon=D=1$. In this case at sites $\ell\neq 0$ (in the “bulk” of the
system) Eqs. (4) and (5) reduce to

$\partial_{t}\rho_{\ell}=\nabla_{\ell}^{2}\rho_{\ell},$ (7)

and

$\partial_{t}\tau_{\ell}=\nabla_{\ell}^{2}\tau_{\ell}-r\left\langle\left(1-\tilde{\rho}_{0}-\tilde{\tau}_{0}\right)\tilde{\tau}_{\ell}\right\rangle,$
(8)

where the last term in the $\tau_{\ell}$ equation comes from Eq. (3) for
$\ell\neq 0$. With the effect of correlations eliminated from Eq. (7) for
$\rho_{\ell}$, we can now solve it independently of Eq. (8) for $\tau_{\ell}$.
Imposing periodic boundary conditions and bath particle number conservation
$\sum_{\ell=0}^{L-1}\rho_{\ell}=N$ yields the stationary solution
$\overline{\rho}$. Equation (8) for $\tau_{\ell}$ is nearly identical to the
one considered in [25], which studied identical particles undergoing diffusion
and local resetting. For a finite particle density and large $L$, the
stationary density profile obtained there using the MF approximation was found
to be remarkably similar to the empirical profile obtained via numerical
simulations of the model’s dynamics, a result which was later validated in
[26]. Together with the support provided by the numerical evidence presented
in Sec. 3, when local resetting is the only source for correlations, as in
Eqs. (8) and (7), the MF approximation seems to become exact in the
thermodynamic limit. Applying the MF approximation to the “full” Eqs. (7) and
(8), which contain general overtaking rates and general bath and tracer
diffusion rates, is correspondingly expected to provide an increasingly
accurate description of the model as $\varepsilon,D\rightarrow 1$.

##  5 Stationary mean-field profiles for general $\varepsilon$ and $D$

Equations (4) and (5) for the average density profiles $\rho_{\ell}$ and
$\tau_{\ell}$ involve correlations between the instantaneous particle
occupations. These pair of equations are actually the first members of an
extensive hierarchy of partial difference equations for all moments of the
full $N$ bath particle and $M$ tracer particle distribution, which one would
have to formulate and solve222One may apply the procedure used in deriving
Eqs. (1) and (2) for $\left\langle\tilde{\rho}_{\ell}\right\rangle$ and
$\left\langle\tilde{\tau}_{\ell}\right\rangle$ to formulate equations for the
two-point correlations sitting at the next level of the hierarchy, as well as
any higher order correlations. to rigorously account for the effect of
interactions between the particles on the density profiles. Unfortunately,
this hierarchy quickly becomes intractable and there are only a handful of
models for which exact methods are known to apply [51, 52]. To make progress
we thus resort to the MF approximation, which essentially sets all cumulants
to zero, so that $n$-point correlations factorize into products (e.g.
$2$-point functions become
$\left\langle\tilde{u}_{n}\tilde{v}_{m}\right\rangle\rightarrow\left\langle\tilde{u}_{n}\right\rangle\left\langle\tilde{v}_{\ell}\right\rangle$).
Applying the MF approximation to Eqs. (4) and (5) yields

$\frac{1}{D}\partial_{t}\rho_{\ell}=\nabla_{\ell}^{2}\rho_{\ell}+\left(1-\frac{\varepsilon}{D}\right)\nabla_{\ell}\left(\rho_{\ell}\tau_{\ell+1}-\tau_{\ell}\rho_{\ell+1}\right),$
(9)

and

$\partial_{t}\tau_{\ell}=\nabla_{\ell}^{2}\tau_{\ell}-\left(1-\varepsilon\right)\nabla_{\ell}\left(\rho_{\ell}\tau_{\ell+1}-\tau_{\ell}\rho_{\ell+1}\right)+\chi_{\ell},$
(10)

where $\tilde{\chi}_{\ell}$ in Eq. (5) is replaced by

$\chi_{\ell}=\begin{cases}\sum_{n=1}^{L-1}r\left(1-\rho_{0}-\tau_{0}\right)\tau_{n}&\ell=0\\\
-r\left(1-\rho_{0}-\tau_{0}\right)\tau_{\ell}&\ell\neq 0\end{cases}.$ (11)

Despite the apparent crudeness of uncontrollably factorizing correlations, as
shown in Sec. 3, comparing the MF solutions to numerical simulation results
suggest that MF effectively captures the salient features arising from the
interplay between geometrically constrained diffusion and local resetting (at
least) in the stationary limit $t\rightarrow\infty$. That said, we shall
henceforth focus our attention on the model’s stationary density profiles.

We seek solutions for the stationary density profiles of the form
$\rho_{\ell}\cong\rho\left(x\right)$ and $\tau_{\ell}\cong\tau\left(x\right)$,
where $x=\ell/L\in\left[0,1\right)$ denotes the macroscopic distance and
$\cong$ is used to denoted asymptotic equality in the thermodynamic limit
$L\rightarrow\infty$. If such solutions exist for all values of the problem’s
parameters and satisfy the boundary conditions, they are the unique solutions
of the problem. We shall thus self-consistently assume this scaling. Expanding
out terms of the form

$u_{\ell\pm
1}=u\left(x\right)+\frac{1}{L}\partial_{x}u\left(x\right)+\frac{1}{2L^{2}}\partial_{x}^{2}u\left(x\right)+\mathcal{O}\left(L^{-3}\right),$
(12)

while keeping the leading order behavior in large $L$, yields the stationary
bath density equation

$0=\partial_{x}^{2}\rho+\left(1-\frac{\varepsilon}{D}\right)\left(\rho\partial_{x}^{2}\tau-\tau\partial_{x}^{2}\rho\right),$
(13)

and the stationary tracer density equation

$0=\partial_{x}^{2}\tau-\left(1-\varepsilon\right)\left(\rho\partial_{x}^{2}\tau-\tau\partial_{x}^{2}\rho\right)-rL^{2}\left(1-\rho_{0}-\tau_{0}\right)\tau.$
(14)

It is interesting to notice that Eq. (14) can only be affected by local
resetting, diffusion, and exchange processes if
$\left(1-\rho_{0}-\tau_{0}\right)\propto L^{-2}$ as $L\rightarrow\infty$, as
was previously noted and discussed in [25]. Otherwise, the only contribution
to $\tau$ would come from local resetting. For now we keep the densities at
the origin $\rho_{0}$ and $\tau_{0}$ as parameters whose value will ultimately
be set by demanding self-consistency with the stationary MF profiles at $x=0$.
This approach allows us to restrict our analysis to the “bulk” of the system,
i.e. sites $\ell\neq 0$, explaining why the more general $\chi_{\ell}$ is
replaced by $\chi_{\ell\neq 0}$ in the last term of Eq. (14).

###  5.1 Relating $\rho$ and $\tau$

Our first step towards the solution of Eqs. (13) and (14) involves relating
the stationary density profile $\rho\left(x\right)$ to the stationary tracer
density profile $\tau\left(x\right)$. Rewriting the bath density Eq. (13) as

$-\rho\left(1-\frac{\varepsilon}{D}\right)\partial_{x}^{2}\tau=\left[1-\left(1-\frac{\varepsilon}{D}\right)\tau\right]\partial_{x}^{2}\rho,$
(15)

and performing integration-by-parts yields

$-\left(1-\frac{\varepsilon}{D}\right)\left[\rho\left(y\right)\partial_{y}\tau\right]\biggl{|}_{y=0}^{x}+\left(1-\frac{\varepsilon}{D}\right)\int_{0}^{x}dy\partial_{y}\rho\partial_{y}\tau$
$=\left[\left(1-\left(1-\frac{\varepsilon}{D}\right)\tau\left(y\right)\right)\partial_{y}\rho\right]\biggl{|}_{y=0}^{x}+\left(1-\frac{\varepsilon}{D}\right)\int_{0}^{x}dy\partial_{y}\tau\partial_{y}\rho.$
(16)

The integrals in Eq. (16) cancel out, leaving

$C_{0}-\left(1-\frac{\varepsilon}{D}\right)\rho\partial_{x}\tau=\left(1-\left(1-\frac{\varepsilon}{D}\right)\tau\right)\partial_{x}\rho,$
(17)

where $C_{0}$ collects the boundary terms and is given by

$C_{0}=\left(1-\frac{\varepsilon}{D}\right)\rho\left(0\right)\partial_{y}\tau\biggl{|}_{y=0}+\left(1-\left(1-\frac{\varepsilon}{D}\right)\tau\left(0\right)\right)\partial_{y}\rho\biggl{|}_{y=0}.$
(18)

We proceed by guessing that $C_{0}=0$ and will show a-posteriori that this
guess is correct. If $C_{0}=0$ Eq. (17) becomes

$-\frac{\left(1-\frac{\varepsilon}{D}\right)\partial_{x}\tau}{1-\left(1-\frac{\varepsilon}{D}\right)\tau}=\frac{\partial_{x}\rho}{\rho},$
(19)

whose solution is

$\rho\left(x\right)=C_{1}\left(1-\left(1-\frac{\varepsilon}{D}\right)\tau\left(x\right)\right),$
(20)

where $C_{1}$ is an integration constant. To set its value we use the
conservation of the number of bath and tracer particles in Eq. (6),
transforming the discrete conservation laws to integral form as

$\int_{0}^{1}dx\rho\left(x\right)=\overline{\rho}\text{ and
}\int_{0}^{1}dx\tau\left(x\right)=\overline{\tau}.$ (21)

Integrating Eq. (20) over the interval $\left[0,1\right]$ and using Eqs. (21)
sets the integration constant to

$C_{1}=\frac{\overline{\rho}}{1-\left(1-\frac{\varepsilon}{D}\right)\overline{\tau}}.$
(22)

We have thus derived the relation between the bath density profile
$\rho\left(x\right)$ and the tracer density profile $\tau\left(x\right)$

$\rho\left(x\right)=\overline{\rho}\frac{1-\left(1-\frac{\varepsilon}{D}\right)\tau\left(x\right)}{1-\left(1-\frac{\varepsilon}{D}\right)\overline{\tau}}.$
(23)

Verifying self-consistency of the assumption $C_{0}=0$, for $C_{0}$ given in
Eq. (18), is now immediate. Note that the denominator in Eq. (23) only
vanishes if $\frac{\varepsilon}{D}=1-\frac{1}{\overline{\tau}}$, which cannot
happen since $0<\overline{\tau}<1$ so the right-hand side is always negative
while both $\varepsilon$ and $D$ are positive.

###  5.2 Tracer density profile

We next use the relation between $\rho\left(x\right)$ and $\tau\left(x\right)$
in Eq. (23) to solve Eq. (14) for $\tau\left(x\right)$ which becomes

$0=\partial_{x}^{2}\tau-\alpha^{2}\tau,$ (24)

where the parameter $\alpha^{2}$ is defined as

$\alpha^{2}\coloneqq
L^{2}r\frac{1-\left(1-\frac{\varepsilon}{D}\right)\overline{\tau}}{1-\left(1-\frac{\varepsilon}{D}\right)\overline{\tau}-\left(1-\varepsilon\right)\overline{\rho}}\left(1-\tau_{0}-\rho_{0}\right).$
(25)

The solution to Eq. (24) is simply

$\tau\left(x\right)=A_{-}e^{-\alpha x}+A_{+}e^{\alpha x},$ (26)

with the constants $A_{\pm}$ set using symmetry about the origin, which
dictates $\tau\left(x\right)=\tau\left(1-x\right)$, and tracer number
conservation $\int_{0}^{1}dx\tau\left(x\right)=\overline{\tau}$ of Eq. (21).
The former yields $A_{-}=A_{+}e^{\alpha}$ and the latter
$A_{+}=\frac{\alpha\overline{\tau}e^{-\frac{\alpha}{2}}}{4\sinh\left[\frac{\alpha}{2}\right]}$,
so that we arrive at

$\tau\left(x\right)=\frac{\alpha\overline{\tau}}{2}\frac{\cosh\left[\alpha\left(\frac{1}{2}-x\right)\right]}{\sinh\left[\frac{\alpha}{2}\right]}.$
(27)

The stationary tracer density profile $\tau\left(x\right)$ has the same form
as that found in [25], which studied resetting particles with exclusion. The
only difference will appear when determining the value of $\alpha$, which here
also depends on the bath density at the origin $\rho_{0}$. The tracer density
profile $\tau\left(x\right)$ exhibits the characteristic tent-like shape, that
was first obtained for the stationary distribution of a single resetting
Brownian particle [1].

Since the bath density profile $\rho\left(x\right)$ is related to the tracer
density profile $\tau\left(x\right)$ via Eq. (23), it may appear that our
solution is complete. However, at this point, $\alpha$ still remains an
unknown parameter. An equation for $\alpha$ is derived by demanding that
$\tau\left(x\right)$ at $x=0$ be consistent with the tracer density at the
origin

$\tau_{0}=1-\frac{\overline{\rho}}{\frac{D}{\varepsilon}+\left(1-\frac{D}{\varepsilon}\right)\left(\overline{\rho}+\overline{\tau}\right)}+\mathcal{O}\left(L^{-2}\right).$
(28)

Equation (28) is obtained by first using Eq. (23) to get
$\rho_{0}=\overline{\rho}\frac{1-\left(1-\frac{\varepsilon}{D}\right)\tau_{0}}{1-\left(1-\frac{\varepsilon}{D}\right)\overline{\tau}}$,
then solving Eq. (25) for $\tau_{0}$ to leading order in $L$, and finally
demanding agreement with $\tau\left(x\right)$ of Eq. (27) at $x=0$. The result
is a transcendental equation for $\alpha$

$\frac{\alpha\overline{\tau}}{2\tanh\left[\frac{\alpha}{2}\right]}=1-\frac{\overline{\rho}}{\frac{D}{\varepsilon}+\left(1-\frac{D}{\varepsilon}\right)\left(\overline{\rho}+\overline{\tau}\right)},$
(29)

which we must solve numerically to compute the profiles. The denominator on
the right-hand side of Eq. (29) vanishes if
$\frac{\varepsilon}{D}=1-\frac{1}{\overline{\rho}+\overline{\tau}}$ but, since
$\frac{\varepsilon}{D}>0$ and $0<\overline{\rho}+\overline{\tau}<1$, this
equality cannot be satisfied as $1-\frac{1}{\overline{\rho}+\overline{\tau}}$
is negative.

The set of Eqs. (23), (27), and (29) constitute the stationary MF solutions to
the bath and tracer density profiles. An inspection of Eq. (23) reveals that
the ratio $\frac{\varepsilon}{D}$, which describes the competition between the
degree of geometric constraints and the bath diffusivity, plays a crucial role
in shaping the bath density profile $\rho\left(x\right)$. Specifically, it
determines the sign of $\rho\left(x\right)$’s curvature. Taking a second
derivative of the relation between $\rho\left(x\right)$ and
$\tau\left(x\right)$ in Eq. (23) gives

$\frac{\partial_{x}^{2}\rho}{\partial_{x}^{2}\tau}=\frac{\overline{\rho}\left(\frac{\varepsilon}{D}-1\right)}{1-\left(1-\frac{\varepsilon}{D}\right)\overline{\tau}}.$
(30)

Local resetting causes the tracer density to be highest at the origin, while
the tracers’ exchange and diffusive dynamics ensure that it falls off rapidly
(faster than linear) with distance from the origin. This leads to a positive
curvature for $\tau\left(x\right)$. As such, the curvature of
$\rho\left(x\right)$ is set by the right-hand side of Eq. (30). We immediately
notice the two limits $\frac{\varepsilon}{D}\rightarrow 0$ and
$\frac{\varepsilon}{D}\rightarrow\infty$, respectively corresponding to
$\partial_{x}^{2}\rho=-\frac{\overline{\rho}}{1-\overline{\tau}}\partial_{x}^{2}\tau<0$
and
$\partial_{x}^{2}\rho=\frac{\overline{\rho}}{\overline{\tau}}\partial_{x}^{2}\tau>0$.
A closer look shows that the right-hand side of Eq. (30) is negative for
$\frac{\varepsilon}{D}<1$ and positive for $\frac{\varepsilon}{D}>1$, with
$\rho\left(x\right)$’s curvature changing sign at $\frac{\varepsilon}{D}=1$.
This result can be understood as follows: When $D>\varepsilon$ the bath
particles manage to diffuse away from the high tracer-density region
surrounding the origin. Since the high-density region decays strongly, once
outside its reach bath particles quickly spread out and inhabit the remaining
regions, implying a negative curvature for $\rho\left(x\right)$. But when
$D<\varepsilon$ the bath particles are too slow and become trapped inside the
high-density region. The bath density thus remains slightly higher near the
origin and slowly falls-off as the tracer density diminishes, indicative of a
positive curvature for $\rho\left(x\right)$. The transition in
$\rho\left(x\right)$’s curvature is also reflected in the effective MF
potential $V\left(x\right)$ that the bath particles experience due to the
tracer profile (see Fig. 5). To find $V\left(x\right)$ we compare
$\rho\left(x\right)$ to the equilibrium Boltzmann distribution
$\overline{\rho}\frac{e^{-\beta V\left(x\right)}}{Z}$, where
$Z=\int_{0}^{1}dxe^{-\beta V\left(x\right)}$ is the partition function. We
find

$Z=\left(1-\left(1-\frac{\varepsilon}{D}\right)\overline{\tau}\right)^{-1},$
(31)

and the potential $V\left(x\right)$

$V\left(x\right)=-\frac{\log\left[1-\left(1-\frac{\varepsilon}{D}\right)\tau\left(x\right)\right]}{\beta}.$
(32)

Notice that the bath particles feel no external potential when
$\varepsilon/D=1$, which is consistent with their homogeneous profile. For
$\varepsilon/D\neq 1$ we can find the shape of $V\left(x\right)$ near the
origin by substituting $\tau\left(x\right)$ of Eq. (27) into $V\left(x\right)$
and expanding around $x=0$. This gives

$V\left(x\right)=V_{0}-\frac{\alpha^{2}\overline{\tau}\left(1-\frac{\varepsilon}{D}\right)}{\alpha\overline{\tau}\left(1-\frac{\varepsilon}{D}\right)\coth\left[\frac{\alpha}{2}\right]-2}\left|x\right|+\mathcal{O}\left(x^{2}\right),$
(33)

with
$V_{0}=\log\left[1-\frac{\alpha}{2}\coth\left[\frac{\alpha}{2}\right]\left(1-\frac{\varepsilon}{D}\right)\overline{\tau}\right]$,
which is a linear potential whose slope can be shown to change sign at
$\frac{\varepsilon}{D}=1$.

Figure 5: Comparison of the MF effective potential $V\left(x\right)$ appearing
in Eq. (32) for $\varepsilon/D=1.5$ ($\varepsilon=1.5$ and $D=1$) and for
$\varepsilon/D=0.5$ ($\varepsilon=1$ and $D=2$), with model parameters
$\overline{\rho}=1/4$, $\overline{\tau}=1/8$, and $r=1$. These values of
$\varepsilon$ and $D$ correspond to those considered in Fig. 4, showing the
resetting tracers behave as an attractive potential for $\varepsilon/D=1.5$
and a repulsive potential for $\varepsilon/D=0.5$.

We conclude this section with two additional points that are worth mentioning.
As found in [25], here too the resetting rate $r$ never enters the profiles in
the limit $L\rightarrow\infty$. This can be heuristically understood from the
fact that, in this limit, there is an infinite number of tracers in the
system. Correspondingly, there is an infinite number of tracers attempting to
reset their positions to the origin at any given moment. As long as the
resetting attempt rate $r$ is finite (and independent of $L$), its precise
value is irrelevant. Another interesting point is that, although Eqs. (13) and
(14) depend on $\varepsilon$ and $D$ separately, the MF profiles depend only
on the ratio $\frac{\varepsilon}{D}$ in the thermodynamic limit. It can be
shown that $r$, $\varepsilon$, and $D$ do individually enter the higher order,
finite-size corrections to the asymptotic profiles.

##  6 Conclusion and discussion

This work studies the interplay of geometric constraints and “local
resetting”, where particles attempt stochastic resetting independently of one-
another, thus introducing interactions into the resetting process. The
dynamics associated with the classical setup of diffusive spherical particles
with short-ranged repulsive interactions confined to a narrow channel is
extended to include local resetting, and then its main ingredients are encoded
into the dynamics of hopping particles on a $1D$ lattice ring. The lattice
contains two particles species, locally resetting “tracers” of mean density
$\overline{\tau}$ and non-resetting “bath” particles of mean density
$\overline{\rho}$. Their continuous time evolution dynamics mimic those of the
narrow channel setup: hard-core exclusion replaces the short-ranged
interactions, the different masses are modeled by setting the bath and tracer
diffusion rates to $1$ and $D$ respectively, and the channel’s width is
replaced by a finite exchange rate $\varepsilon$. Evolution equations for the
mean bath and tracer density profiles are derived, and then solved in the
stationary limit using the mean-field (MF) approximation in the thermodynamic
limit $L\rightarrow\infty$. The MF tracer density profile exhibits the typical
tent-like shape with a cusp at the origin, as found for a single resetting
diffusive particle [1]. Yet the interplay of local resetting and geometric
confinement manifests most dramatically in the MF bath density profile
$\rho\left(x\right)$, which transitions between “repelled” and “trapped”
states as the value of $\varepsilon/D$ crosses $1$. For $\varepsilon/D<1$ bath
particles are strongly repelled from the origin, with $\rho\left(x\right)$
taking the shape of an inverted tent with a negative curvature
$\rho^{\prime\prime}\left(x\right)<0$. For $\varepsilon/D>1$ the bath particle
are too slow to escape the dense origin and remain trapped. The shape of
$\rho\left(x\right)$ is then similar to that of $\tau\left(x\right)$ and
$\rho^{\prime\prime}\left(x\right)>0$. For $\varepsilon/D=1$ the bath
particles experience a homogeneous environment, since hopping into a vacant
site and exchanging with a tracer happen with the same rate, and we obtain
$\rho\left(x\right)=\overline{\rho}$ . While the MF approximation successfully
predicts the existence of both states and the transition between them, there
is no a-priori reason for it to be exact in any regime. However, numerical
investigations of the model provide strong evidence to suggest that the MF
approximation does in fact become exact for $\varepsilon=D=1$ as
$L\rightarrow\infty$. This result, whose underlying origins remain unclear,
joins the observations of [25, 26] where the MF approximation was found to
stand in unexpectedly good agreement with numerical simulation results of
models exhibiting local resetting.

Looking forward, many interesting open questions still remain unsolved and
demand attention. First and foremost, why does the MF approximation seem to
work well in models featuring local resetting? Is there any way to adapt exact
methods like the matrix product ansatz [51, 52], which have been used to
derive exact solutions for various lattice models with exclusion interactions,
to treat local resetting? Another exciting direction is to explore the
temporal evolution in the presence of local resetting. While this is
interesting for local resetting in general, as dynamical phase transitions
were found in the temporal evolution of the density profile of a single
resetting Brownian particle [9], it is even more interesting in the context of
geometric confinement and tracer sub-diffusion [28, 29, 30]. In addition, both
experimental and theoretical research following this line of work would of
greatly benefit from pursuing a quantitative relation between the parameters
describing the narrow channel setup, and the corresponding parameters of the
lattice model.

##  7 Acknowledgments

I wish to thank David Mukamel, Oren Raz, and Shlomi Reuveni for critically
reading this manuscript and for their very helpful suggestions and remarks.
This work was supported by a research grant from the Center of Scientific
Excellence at the Weizmann Institute of Science. But, most crucially, it has
been made possible by the grace of DBM and the mercy of GM, to whom I am
indebted to for their unconditional support.

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