# Exact fluctuation and long-range correlations in a single-file model under
resetting

Saikat Santra1 and Prashant Singh2<EMAIL_ADDRESS><EMAIL_ADDRESS>1 International Centre for Theoretical Sciences, Tata
Institute of Fundamental Research, Bengaluru 560089, India 2 Niels Bohr
International Academy, Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, 2100 Copenhagen, Denmark

###### Abstract

Resetting is a renewal mechanism in which a process is intermittently repeated
after a random or fixed time. This simple act of stop and repeat profoundly
influences the behaviour of a system as exemplified by the emergence of non-
equilibrium properties and expedition of search processes. Herein, we explore
the ramifications of stochastic resetting in the context of a single-file
system called random average process (RAP) in one dimension. In particular, we
focus on the dynamics of tracer particles and analytically compute the
variance, equal time correlation, autocorrelation and unequal time correlation
between the positions of different tracer particles. Our study unveils that
resetting gives rise to rather different behaviours depending on whether the
particles move symmetrically or asymmetrically. For the asymmetric case, the
system for instance exhibits a long-range correlation which is not seen in
absence of the resetting. Similarly, in contrast to the reset-free RAP, the
variance shows distinct scalings for symmetric and asymmetric cases. While for
the symmetric case, it decays (towards its steady value) as $\sim
e^{-rt}/\sqrt{t}$, we find $\sim te^{-rt}$ decay for the asymmetric case ($r$
being the resetting rate). Finally, we examine the autocorrelation and unequal
time correlation in the steady state and demonstrate that they obey
interesting scaling forms at late times. All our analytical results are
substantiated by extensive numerical simulations.

##  I INTRODUCTION

Deciphering the behaviour of complex systems consisting of many interacting
units is a fundamental problem often encountered in statistical physics  [1,
2]. A classic example of an interacting particle system in non-equilibrium
statistical mechanics is the single-file system, in which particles in a one-
dimensional line move alongside each other, strictly obeying the constraint of
non-overtaking, wherein one particle cannot pass another  [3, 4, 5]. Due to
this non-overtaking constraint (also referred to as single-file constraint),
the dynamics of different particles become strongly correlated [6, 7]. For
example, in a collection of diffusing particles in one dimension with single-
file constraint, the mobility of a tracer particle is drastically reduced and
as result, the mean-squared displacement grows sub-diffusively as
$\sim\sqrt{t}$ at late times, instead of the linear growth for a freely
diffusing particle  [8, 9, 10, 11]. The coefficient of this sub-diffusive
growth, in turn, depends on the particle number density, the precise
interaction among the particles and also on the statistical properties of the
initial state of the system  [9, 12, 13, 14, 15]. In fact based on
hydrodynamic approach, a recent work showed that this sub-diffusive scaling
holds true only for short-range interactions and changes to an interaction-
dependent exponent for long-range interactions [16]. Beyond diffusion, such
slowing down effects have also been studied for Hamiltonian systems [17, 18,
19] as well as for other stochastic systems like randomly accelerated process
[20] and active processes [21, 22, 23, 24]. In this paper, we set out to study
the tracer dynamics for a single-file model in presence of a renewal mechanism
called resetting that has garnered significant attention in the last decade
[25].

Stochastic resetting is a simple and natural mechanism in which a dynamical
process is intermittently interrupted after some random time, after which it
again starts anew. A quintessential example of this phenomenon is the
resetting Brownian motion, which was first studied in  [25]. In this model, a
particle undergoing free diffusion is returned to its starting position at a
certain rate $r$, after which it recommences diffusion until the next
resetting event. As a result of this, the particle experiences an effective
confinement around its initial position. However, it is important to note that
this confinement arises solely from the dynamics of the system and does not
stem from any physical potential. Indeed, as time progresses, the system
eventually reaches a non-equilibrium steady state, which is characterised by
the presence of a non-zero probability current. Another notable aspect of this
model is that, unlike in free diffusion, the particle exhibits a finite mean
first-passage time. Remarkably, this mean time depends non-monotonically on
the resetting rate $r$ which indicates its optimisation for an optimal rate
$r^{*}$ [26]. Beyond the standard Brownian motion, resetting has also been
explored within a broader spectrum of other stochastic processes  [27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] as well as in cross-
disciplinary fields such as search theory  [42, 43], computer science  [44,
45] and in chemical and biological processes  [46, 47, 48, 49]. Furthermore,
rigorous studies have been made to comprehend non-Poissonian strategies  [50,
29, 51] and the implications of resetting in quantum settings  [52, 53, 54,
55, 56]. On the experimental side, resetting was recently realised in
experiments involving single particle in optical traps  [57, 58, 59, 59]. We
refer to  [60, 61, 62, 63, 41] and references therein for recent reviews on
the subject.

While most of the aforementioned studies primarily focused into single
particle dynamics, there has also been a substantial surge of interest in
understanding the effects of resetting for interacting particles. Examples
include exclusion processes  [64, 65, 66, 67], Ising model  [68], fluctuating
interfaces  [69, 70], and predator-prey models  [71, 72, 73], among numerous
others [see  [62] and references therein for a review on stochastic resetting
in interacting systems]. These studies investigated the scenario where
multiple particles are reset simultaneously after a random duration (global
resetting). This is contrary to some other studies where particles reset
independently of the other particles (local resetting)  [74, 75, 76]. In a
recent work involving independently diffusing particles, but undergoing global
resetting at a rate $r$, the authors showed that the simultaneous resetting
induces a strong long-range correlation in the system  [77]. Despite this
correlation, the model still possesses analytical solvability based on the
renewal formula and many results on the joint probability distribution of the
positions and extremal statistics were derived. However, to the best of our
knowledge, the implications of stochastic resetting on the dynamics of tracer
particles in an interacting multi-particle system still remains unexplored. A
natural question that particularly arises - Does one still get a resetting
induced long-range correlation in such interacting scenarios? In this paper,
we present an example of a single-file model (called random average process)
[78, 79] where these questions can be thoroughly addressed through exact
analytic computations.

Our system consists of a collection of particles moving in an infinite line
and distributed with density $\rho$. We denote the position of $i$-th particle
at time $t$ by $x_{i}(t)$ where $i\in\mathbb{Z}$ and $x_{i}(t)\in\mathbb{R}$.
Initially, the particles are located at a fixed distance $a=\frac{1}{\rho}$
apart:

$\displaystyle x_{i}(0)=ia=i/\rho,~{}~{}~{}~{}~{}\text{for all
}i\in\mathbb{Z}.$ (1)

For simplicity, we take $\rho=1$ without any loss of generality. Starting from
this configuration, each particle performs the random average process
interspersed by resetting events, during which the entire system is (globally)
reset to the configuration in Eq. (1). This means that at any small time
interval $[t,t+\Delta t]$, two events can occur: (i) either particles globally
reset their positions to $x_{i}(0)$ in Eq. (1) with probability $r\Delta t$
(ii) or they perform the random average motion with the remaining probability
$(1-r\Delta t)$. In the case of latter event, the $i$-th particle can jump to
its right with probability $p\Delta t$ and to its left with probability
$q\Delta t$. With remaining probability $\left[1-(p+q)\Delta t\right]$, the
position $x_{i}(t)$ does not change. The successful jump, either to the left
or to the right, is by a random fraction $\eta_{i}$ of the space available
between the particle and its neighbour. This means that the jump to the right
takes place by an amount $\eta_{i}\left[x_{i+1}(t)-x_{i}(t)\right]$ while to
the left, the particle jumps by the amount
$\eta_{i}\left[x_{i-1}(t)-x_{i}(t)\right]$. Here $\eta\in[0,1)$ is a random
variable drawn from the distribution $R(\eta)$. Notice that throughout time
evolution, a particle can never overtake its neighbouring particles and
maintains its initial order.

The overall update rule for the position can then be written as

$\displaystyle x_{i}(t+\Delta
t)=\begin{cases}&x_{i}(0),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{w.p.
}r\Delta t,\\\ &x_{i}(t)+\Gamma_{i}(t),~{}~{}~{}~{}\text{w.p. }(1-r\Delta
t),\end{cases}$ (2)

where ‘w.p.’ stands for ‘with probability’ and $\Gamma_{i}(t)$ for the
increment which is given by

$\displaystyle\Gamma_{i}(t)=\begin{cases}&\eta_{i}\left[x_{i+1}(t)-x_{i}(t)\right],~{}~{}\text{w.p.
}p\Delta t,\\\ &\eta_{i}\left[x_{i-1}(t)-x_{i}(t)\right],~{}~{}\text{w.p.
}q\Delta t,\\\
&~{}~{}~{}~{}0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{w.p.
}\left[1-(p+q)\Delta t\right].\end{cases}$ (3)

Given this model, our aim in this paper is two fold: (i) Firstly, we aim to
study the dynamics of tracer particles and explore analytically the effect of
stochastic resetting on the single-file model. For this purpose, we
investigate the variance and different two-point correlation functions for the
positions of tracer particles. In particular, the role of resetting on the
autocorrelation involving two different times has so far been studied only for
few cases such as the drift-diffusion  [38] and the fractional Brownian motion
[39]. It also played a crucial role in income dynamics modelled using
geometric Brownian motion [80]. However, all of these studies dealt with the
single particle dynamics. Here, our aim is to investigate this with multiple
particles and understand how fluctuations corresponding to one particle at a
given time affect the position of some other particle at some later time. (ii)
Secondly, our work will also shed light on determining whether the long-range
correlation, stemming from the simultaneous resetting of non-interacting
particles [77], persists even when the system involves interacting particles
in its underlying dynamics.

The remainder of our paper is organized as follows: Section II briefly recalls
the main results of the reset-free random average process. In Section III, we
illustrate the effect of resetting on the mean position of a tracer particle.
These results will then be used to calculate the variance in Section IV.1 and
equal time correlation in Section IV.2. We devote Sections V.1 and V.2 to
compute respectively, the autocorrelation and the unequal time correlation in
the steady state. Finally, we conclude in Section VI

##  II Random average process without resetting

Before we delve into discussing the consequences of resetting on tracer
particles, it is instructive to review some known results for the random
average process (henceforth RAP) in absence of resetting $(r=0)$. Studied
originally by Fontes and Ferrari as a generalisation of the smoothing process
and the voter model  [81], the RAP has appeared in several physical problems
like force fluctuations in bead packs  [82], in mass transport models  [83,
84], in models of wealth distribution and traffic  [85] and generalized
Hammersley process  [86]. As an interacting multi-particle system, this model
is particularly interesting since the particles do not overtake each other and
maintain their initial order throughout the time evolution as indicated by Eq.
(2). Therefore, every particle performs the single-file motion. Due to this
non-overtaking constraint, the motion of different particles get strongly
correlated [78, 79, 87, 88, 89, 90]. For instance, various studies based on
both microscopic calculations as well as hydrodynamic approaches have shown
that the variance and the correlation of the displacement variable $z_{i}(t)$
(with $z_{i}(t)=x_{i}(t)-x_{i}(0)$) at late time is given by  [79]

$\displaystyle\langle z_{i}(t)\rangle=\mu_{1}(p-q)t,$ (4)
$\displaystyle\langle z_{i}^{2}(t)\rangle-\langle
z_{i}(t)\rangle^{2}\simeq\zeta\sqrt{t},$ (5) $\displaystyle\langle
z_{0}(t)z_{i}(t)\rangle-\langle z_{i}(t)\rangle\langle z_{0}(t)\rangle$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\simeq\zeta\sqrt{t}~{}f\left(\frac{|i|}{2\sqrt{\mu_{1}(p+q)t}}\right),$
(6)

where the pre-factor
$\zeta=\mu_{2}\sqrt{\mu_{1}(p+q)}/\left[\sqrt{\pi}(\mu_{1}-\mu_{2})\right]$
depends on first two moments $\mu_{1}$ and $\mu_{2}$ of the jump distribution
$R(\eta)$ and the scaling function $f(y)$ in Eq. (6) is given by

$\displaystyle f(y)=e^{-y^{2}}-\sqrt{\pi}y~{}\text{Erfc}(y).$ (7)

Due to the translational symmetry in the model, both mean and variance in Eqs.
(4) and (5) do not depend on the particle index $i$. Also the mean vanishes
for the symmetric case $p=q$ since particles do not experience any drive under
this condition. Meanwhile the sub-diffusive scaling of the variance at large
times in Eq. (5) and the scaling function $f(y)$ in Eq. (7) are hallmark
properties of many single-file systems that possess diffusive hydrodynamics at
the macroscopic scales  [90, 9].

It turns out that the variance and the correlation for single-file systems
depend crucially on the statistical properties of the initial state of the
system. For RAP, results in Eqs. (4)-(7) hold true only for the quenched
initial condition where the initial positions are fixed to Eq. (1) for all
realisations  [79]. On the other hand, for annealed initial condition where
initial positions are drawn from the steady-state, the initial positions
themselves vary from realisation to realisation. Under this circumstance, the
variance $l_{0}(t)$ at late time grows as

$\displaystyle l_{0}(t)$
$\displaystyle\simeq\zeta\sqrt{2t},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}p=q,$ (8)
$\displaystyle\simeq\frac{\mu_{1}\mu_{2}(p-q)}{(\mu_{1}-\mu_{2})}t,~{}~{}~{}~{}~{}\text{for
}p>q.$ (9)

As indicated, the temporal scaling of $l_{0}(t)$ depends sensitively on
whether particles experience drive or not. This is contrary to the quenched
case in Eq. (5) where we obtain same scaling for both cases. Furthermore, in
the case of symmetric RAP, the ratio of the variances for two cases is found
to be $\sqrt{2}$ which is also observed in the context of other single-file
models  [9, 13]. In what follows, we will investigate these quantities for
non-zero $r$ and illustrate how resetting modifies them.

##  III Average position

For $p\neq q$, we saw in Eq. (4) that the particles experience a net drive
which gives rise to a non-zero value of mean that grows linearly with time.
Let us investigate what happens to this average in presence of resetting. Here
again, it turns out convenient to work in terms of the displacement variable
$z_{i}(t)=x_{i}(t)-x_{i}(0)$ and rewrite the update rules in Eq. (2) as

$\displaystyle z_{i}(t+\Delta
t)=\begin{cases}&0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{with
probability }r\Delta t,\\\
&z_{i}(t)+\Gamma_{i}(t),~{}~{}~{}~{}~{}~{}~{}\text{with probability
}(1-r\Delta t),\end{cases}$ (10)

where the increment $\Gamma_{i}(t)$ can be written as

$\displaystyle\Gamma_{i}(t)=\begin{cases}&\eta_{i}\left[z_{i+1}(t)-z_{i}(t)+1\right],~{}~{}~{}\text{with
probability }p\Delta t,\\\
&\eta_{i}\left[z_{i-1}(t)-z_{i}(t)-1\right],~{}~{}~{}\text{with probability
}q\Delta t,\\\
&0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{with
probability }1-(p+q)\Delta t.\end{cases}$ (11)

Denoting the mean as $h_{i}(t)=\langle z_{i}(t)\rangle$, we can write its
evolution for a small time interval $[t,t+\Delta t]$ as

$\displaystyle h_{i}(t+\Delta t)$ $\displaystyle\simeq(1-r\Delta
t)\left[h_{i}(t)+\langle\Gamma_{i}(t)\rangle\right].$ (12)

Plugging $\langle\Gamma_{i}(t)\rangle$ from Eq. (11) and taking $\Delta t\to
0$ limit, we obtain the following differential equation for $h_{i}(t)$:

$\displaystyle\frac{dh_{i}(t)}{dt}$
$\displaystyle=\mu_{1}\left[ph_{i+1}(t)+qh_{i-1}(t)-(p+q)h_{i}(t)\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-rh_{i}(t)+\mu_{1}(p-q).$ (13)

Recall that
$\mu_{m}=\langle\eta^{m}\rangle=\int_{0}^{1}d\eta~{}\eta^{m}R(\eta)$
represents the $m$-th moment of the jump distribution $R(\eta)$. One needs to
solve this equation with the initial condition $h_{i}(0)=0$. Since it is a
linear equation in $h_{i}(t)$, we proceed to solve it by taking the Fourier
transformation with respect to the index $i$. Defining the Fourier
transformation as

$\displaystyle\bar{h}(k,t)=\sum_{i=-\infty}^{\infty}h_{i}(t)~{}e^{jik},$ (14)

and the inverse Fourier transform as

$\displaystyle
h_{i}(t)=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk~{}\bar{h}(k,t)~{}e^{-jik},$ (15)

with $j^{2}=-1$, we recast Eq. (13) in terms of the Fourier variable as

$\displaystyle\frac{\partial\bar{h}(k,t)}{\partial
t}=-\alpha(k)~{}\bar{h}(k,t)+2\pi\mu_{1}(p-q)\delta(k),$ (16)

with $\alpha(k)=r+\mu_{1}(p+q)-\mu_{1}pe^{jk}-\mu_{1}qe^{-jk}$. Finally
solving Eq. (16) and performing the inverse Fourier transformation, we get

$\displaystyle h_{i}(t)=h(t)=\frac{\mu_{1}(p-q)}{r}\left(1-e^{-rt}\right).$
(17)

Notice that the final expression turns out to be independent of the index $i$,
since we have assumed translational invariance of our infinite system. Also,
the mean expectedly vanishes for the symmetric $p=q$ case. Meanwhile for
$r=0$, our result reduces to Eq. (4) where mean grows linearly with time.
However, for any non-zero $r$, it approaches a steady value at large times.
This steady value decays as $\sim 1/r$ with respect to the resetting rate.
Physically, a larger resetting rate confines the tracer particle to move in
the vicinity of its initial position which gives rise to the smaller mean. In
Figure 1, we have plotted the mean $h_{i}(t)$ and compared it with the same
obtained from numerical simulations. We observe an excellent match between
them. In the following sections, we will use this expression of mean to
compute different correlation functions for the tracer particles.

Figure 1: Comparison of the mean $h_{i}(t)=\langle z_{i}(t)\rangle$ in Eq.
(17) with the numerical simulation for $p=1.0,~{}q=0.1,~{}r=1$ and jump
distribution $R(\eta)=1$. Simulation is conducted with $N=101$ particles.
Figure 2: Comparison of $\langle z_{0}^{2}(t)\rangle_{c}$ in Eq. (33) with the
numerical simulation for two sets of parameters : $p=1.0,~{}q=0.1$ in panel
(a) and $p=0.5,~{}q=0.5$ in panel (b). For both panels, we have performed
simulation with $N=101$ particles, resetting rate $r=1$ and jump distribution
$R(\eta)=1$.

##  IV Variance and Equal time correlation

In this section, we look at the variance and equal time correlations of the
positions of two tagged particles when the entire system is reset to the
configuration in Eq. (1) with rate $r$. Let us denote this correlation by
$C_{i}(t)=\langle z_{0}(t)z_{i}(t)\rangle$. Recall that for free RAP, the
correlation function satisfies a scaling behaviour in
$\big{|}i\big{|}/\sqrt{t}$ with the associated scaling function given in Eq.
(7). Here, our aim is to illustrate how this scaling behaviour gets modified
in presence of resetting. To this aim, we start by deriving the time evolution
differential equation for $C_{i}(t)$. In a small time interval $[t,t+\Delta
t]$, the correlation $C_{i}\left(t+\Delta t\right)$ for $i\neq 0$ changes by
an amount

$\displaystyle C_{i}\left(t+\Delta t\right)$ $\displaystyle\simeq
C_{i}\left(t\right)-r\Delta t~{}C_{i}\left(t\right)+2\mu_{1}(p-q)\Delta
t~{}h(t)$ $\displaystyle+\mu_{1}(p+q)\Delta
t\left[C_{i+1}(t)+C_{i-1}(t)-2C_{i}(t)\right].$

On the other hand, applying the same procedure for $i=0$ gives

$\displaystyle C_{0}\left(t+\Delta t\right)$ $\displaystyle\simeq
C_{0}\left(t\right)-r\Delta t~{}C_{0}\left(t\right)+2\mu_{1}(p-q)\Delta
t~{}h(t)$ $\displaystyle+\mu_{2}(p+q)\Delta
t~{}\left[1+2\\{C_{0}(t)-C_{1}(t)\\}\right]$ $\displaystyle+\mu_{1}(p+q)\Delta
t\left[2C_{1}(t)-2C_{i}(t)\right]$

Combining both contributions for $i=0$ and $i\neq 0$ and taking the limit
$\Delta t\to 0$, we obtain

$\displaystyle\frac{dC_{i}(t)}{dt}=$
$\displaystyle\mu_{1}(p+q)\left[C_{i+1}(t)+C_{i-1}(t)-2C_{i}(t)\right]$
$\displaystyle+\delta_{i,0}~{}\mu_{2}(p+q)\left[1+2\\{C_{0}(t)-C_{1}(t)\\}\right]$
$\displaystyle~{}~{}~{}~{}~{}+2\mu_{1}(p-q)h(t)-rC_{i}(t).$ (18)

Fortunately, this equation involves only mean and two-point correlation
function and does not involve higher order correlation functions. This closure
property enables us to obtain exact solution for this equation. For this, let
us take the Laplace transformation with respect to $t~{}(\to s)$ as

$\displaystyle\hat{C}_{i}(s)=\int_{0}^{\infty}dt~{}e^{-st}~{}C_{i}(t),$ (19)

and rewrite Eq. (18) in terms of $\hat{C}_{i}(s)$ as

$\displaystyle s\hat{C}_{i}(s)=$
$\displaystyle~{}~{}\mu_{1}(p+q)\left[\hat{C}_{i+1}(s)+\hat{C}_{i-1}(s)-2\hat{C}_{i}(s)\right]$
$\displaystyle+\delta_{i,0}~{}\mu_{2}(p+q)\left[\frac{1}{s}+2\\{\hat{C}_{0}(s)-\hat{C}_{1}(s)\\}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-r\hat{C}_{i}(s)+2\mu_{1}(p-q)\hat{h}(s).$
(20)

While writing this equation, we have taken the initial condition $C_{i}(0)=0$
and introduced the notation $\hat{h}(s)$ to denote the Laplace transformation
of the mean $h(t)$ which from Eq. (17) turns out to be

$\displaystyle\hat{h}(s)=\frac{\mu_{1}(p-q)}{r}\left(\frac{1}{s}-\frac{1}{s+r}\right).$
(21)

We now proceed to solve Eq. (20). First notice that, due to the single-file
constraint, we get coupling of different $i$ in Eq. (20). To decouple them, we
take the Fourier transformation

$\displaystyle\mathcal{Z}(k,s)=\sum_{i=-\infty}^{\infty}\hat{C}_{i}(s)~{}e^{jik},$
(22)

and insert this in Eq. (20) to yield

$\displaystyle\mathcal{Z}(k,s)=$
$\displaystyle\frac{\mu_{2}(p+q)\left[1+2s\\{\hat{C}_{0}(s)-\hat{C}_{1}(s)\\}\right]}{s\left[s+r+2\mu_{1}(p+q)(1-\cos
k)\right]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{4\pi\mu_{1}(p-q)\hat{h}(s)}{(s+r)}~{}\delta(k).$
(23)

Everything on the right hand side is known except two functions, namely
$\hat{C}_{0}(s)$ and $\hat{C}_{1}(s)$. One of them can be expressed in terms
of the other by putting $i=0$ in Eq. (20). This results in the relation

$\displaystyle\hat{C}_{1}(s)$
$\displaystyle=\frac{\left[2(p+q)(\mu_{1}-\mu_{2})+s+r\right]\hat{C}_{0}(s)}{2(\mu_{1}-\mu_{2})(p+q)}$
$\displaystyle~{}~{}~{}~{}-\frac{\mu_{2}(p+q)/s+2\mu_{1}(p-q)\hat{h}(s)}{2(\mu_{1}-\mu_{2})(p+q)},$
(24)

which we substitute in Eq. (23) to yield

$\displaystyle\mathcal{Z}(k,s)=$
$\displaystyle\frac{\mu_{2}}{s}\frac{\mu_{1}(p+q)-s(s+r)\hat{C}_{0}(s)+2\mu_{1}s(p-q)\hat{h}(s)}{(\mu_{1}-\mu_{2})\left[s+r+2\mu_{1}(p+q)(1-\cos
k)\right]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{4\pi\mu_{1}(p-q)\hat{h}(s)}{(s+r)}~{}\delta(k).$
(25)

We now have only one unknown $\hat{C}_{0}(s)$. However, this can be computed
self-consistently as shown later. Finally inverting $\mathcal{Z}(k,s)$ in $k$
gives the exact correlation as

$\displaystyle\hat{C}_{i}(s)=\frac{2\mu_{1}(p-q)\hat{h}(s)}{s+r}+\hat{\mathcal{G}}_{i}(s),$
(26)

where the function $\hat{\mathcal{G}}_{i}(s)$ is defined as

$\displaystyle\hat{\mathcal{G}}_{i}(s)$
$\displaystyle=\frac{\mu_{2}\left[\mu_{1}(p+q)-s(s+r)~{}\hat{\mathcal{G}}_{0}(s)\right]}{s(\mu_{1}-\mu_{2})}\times\mathcal{V}_{i}(s+r),$
(27) $\displaystyle\mathcal{V}_{i}(s)$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk~{}\frac{e^{-jik}}{s+2\mu_{1}^{\prime}(1-\cos
k)},$
$\displaystyle=\frac{1}{\sqrt{s^{2}+4\mu_{1}^{\prime}s}}\left[\frac{s+2\mu_{1}^{\prime}-\sqrt{s^{2}+4\mu_{1}^{\prime}s}}{2\mu_{1}^{\prime}}\right]^{|i|}$
(28)

with $\mu_{1}^{\prime}=\mu_{1}(p+q)$. To summarise, we have calculated the
exact two-point correlation functions in Eq. (26) in terms of the Laplace
variable $s$. The idea now is to perform the inversion and obtain them in the
time domain. For simplicity, we carry out this inversion separately for $i=0$
and $i\neq 0$ cases below.

Figure 3: Comparison of the relaxation behaviour of the variance $\langle
z_{0}^{2}(t)\rangle_{c}$ for the symmetric $(p=q)$ and asymmetric $(p\neq q)$
cases. For illustration, we have plotted $d(t)=\langle
z_{0}^{2}(t\to\infty)\rangle_{c}-\langle z_{0}^{2}(t)\rangle_{c}$ in Eq. (36)
and compared it with the numerical simulation for two sets of parameters :
$p=1.0~{},q=0.1$ in panel (a) and $p=q=0.5$ in panel (b). For both panels, we
have performed simulation with $N=101$ particles, resetting rate $r=1$ and
jump distribution $R(\eta)=1$.

###  IV.1 Variance of $z_{0}(t)$

We first look at the variance for which we put $i=0$ in Eq. (26). Looking at
this expression, it is clear that one needs to invert the Laplace
transformation $\hat{\mathcal{G}}_{0}(s)$. We rewrite its expression from Eq.
(27) as

$\displaystyle\hat{\mathcal{G}}_{0}(s)=\frac{\mu_{1}\mu_{2}(p+q)(\mu_{1}-\mu_{2})^{-1}(s\sqrt{s+r})^{-1}}{\left[\sqrt{(s+r)+4\mu_{1}(p+q)}+\frac{\mu_{2}}{\mu_{1}-\mu_{2}}\sqrt{s+r}\right]}.$
(29)

Fortunately, one can perform this inversion exactly and we show in Appendix A
that

$\displaystyle\mathcal{G}_{0}(t)=$
$\displaystyle\frac{\mu_{2}(\mu_{1}-\mu_{2})(p+q)}{\sqrt{r}(\mu_{1}-2\mu_{2})}$
$\displaystyle\times\int_{0}^{t}dT~{}e^{-rT}~{}\text{Erf}\left(\sqrt{r(t-T)}\right)~{}Z(T),$
(30)

where $\mathcal{G}_{0}(t)$ stands for the Laplace transformation of
$\hat{\mathcal{G}}_{0}(s)$ and the function $Z(t)$ is defined as

$\displaystyle Z(t)=$ $\displaystyle\ \frac{1}{\sqrt{\pi
t}}\left[e^{-4\mu_{1}t(p+q)}-\frac{\mu_{2}}{\mu_{1}-\mu_{2}}\right]$
$\displaystyle-\frac{\mu_{2}}{\mu_{2}-\mu_{2}}~{}\sqrt{-B_{1}}~{}e^{-B_{1}t}~{}\text{Erf}(\sqrt{-B_{1}~{}t})$
(31)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{-B_{2}}~{}e^{-B_{1}t}~{}\text{Erf}(\sqrt{-B_{2}~{}t}).$

Constants $B_{1}$ and $B_{2}$ depend on the model parameters as
$B_{1}=4(p+q)(\mu_{1}-\mu_{2})^{2}/(\mu_{1}-2\mu_{2})$ and
$B_{2}=B_{1}-4\mu_{1}(p+q)$. Finally, using Eq. (30) in Eq. (26), we obtain

$\displaystyle
C_{0}(t)=\mathcal{G}_{0}(t)+\frac{2\mu_{1}^{2}(p-q)^{2}}{r^{2}}\left[1-e^{-rt}\left(rt+1\right)\right],$
(32)

from which the variance of $z_{0}(t)$ turns out to be

$\displaystyle\langle z_{0}^{2}(t)\rangle_{c}$ $\displaystyle=C_{0}(t)-\langle
z_{0}(t)\rangle^{2},$
$\displaystyle=\mathcal{G}_{0}(t)+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}\left[1-e^{-rt}\left(2rt+e^{-rt}\right)\right].$
(33)

This represents the exact variance of the position of a tagged particle in RAP
with resetting. Figure 2 illustrates the comparison of our analytical results
with the numerical simulations for both symmetric and asymmetric cases. An
excellent match is seen in both cases. For any non-zero $r$, we anticipate the
expression in Eq. (33) to attain a stationary value at large times. To show
this, one can, in principle, directly put $t\to\infty$ in Eq. (30) and carry
out the integration. However, it turns out more convenient to put $s\to 0$
limit in $\hat{\mathcal{G}}_{0}(s)$ and use the formula
$\mathcal{G}_{0}\left(t\to\infty\right)=\lim_{s\to
0}\left[s\hat{\mathcal{G}}_{0}(s)\right]$. With this procedure, the stationary
value of the variance turns out to be

$\displaystyle\langle z_{0}^{2}(t\to\infty)\rangle_{c}=$
$\displaystyle\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}+\frac{\mu_{1}\mu_{2}(p+q)}{\sqrt{r}(\mu_{1}-\mu_{2})}$
$\displaystyle\times\left[\sqrt{r+4\mu_{1}(p+q)}+\frac{\mu_{2}\sqrt{r}}{\mu_{1}-\mu_{2}}\right]^{-1}.$
(34)

Recall that for $r=0$, the tracer particle does not attain any stationary
value. This indicates that the stationary value of the variance should diverge
as $r\to 0$. From the exact expression in Eq. (34), we find that $\langle
z_{0}^{2}(t\to\infty)\rangle_{c}$ diverges differently depending on whether
the particles move symmetrically $(p=q)$ or asymmetrically $(p\neq q)$. For
$p=q$, the stationary value diverges as $\sim r^{-1/2}$ as $r\to 0$ whereas
for $p\neq q$, it diverges as $\sim r^{-2}$. Contrarily for large $r$,
$\langle z_{0}^{2}(t\to\infty)\rangle_{c}$ decays as $\sim 1/r$ for both
cases.

Figure 4: Comparison of the equal time correlation $\langle
z_{0}(t)z_{i}(t)\rangle_{c}$ in the steady state $(t\to\infty)$ in Eq. (44)
for for two sets of parameters : $p=1.0,~{}q=0.1$ in panel (a) and
$p=0.5,~{}q=0.5$ in panel (b). In both panels, simulation has been carried out
with $N=101$ particles, resetting rate $r=1$ and jump distribution
$R(\eta)=1$.

After analysing the stationary value of the variance, let us look at its time-
dependent form in Eq. (33). While this is an exact expression, it has rather a
complicated form. To gain some insights in it, we will analyse it for large
$t$ and study the relaxation properties. For continuity of presentation, we
have shown this calculation in Appendix A and quote only the final result
here. We find $d(t)=\langle z_{0}^{2}(t\to\infty)\rangle_{c}-\langle
z_{0}^{2}(t)\rangle_{c}$ is given by

$\displaystyle d(t)$
$\displaystyle\simeq\frac{\mu_{2}\sqrt{\mu_{1}p}}{\sqrt{2\pi}r(\mu_{1}-\mu_{2})}\frac{e^{-rt}}{\sqrt{t}},~{}~{}~{}~{}\text{for
}p=q,$ (35)
$\displaystyle\simeq\frac{2\mu_{1}^{2}(p-q)^{2}}{r^{2}}rte^{-rt},~{}~{}~{}~{}~{}~{}~{}\text{for
}p\neq q.$ (36)

Interestingly, we find different relaxation behaviours depending on whether
$p=q$ or $p\neq q$. While for the symmetric case, the variance relaxes as
$\sim e^{-rt}/\sqrt{t}$ to its stationary value, we obtain $\sim te^{-rt}$
relaxation for the asymmetric case. Note that such difference between the
symmetric and the asymmetric variances is not seen for free RAP $(r=0)$ and we
obtain the same sub-diffusive scaling for both cases as shown in Eq. (5).
Indeed, writing the time evolution equation for the variance $\langle
z_{0}^{2}(t)\rangle_{c}$ for the free RAP, one can show that it can be made
independent of $p$ and $q$ by suitably scaling $t\to(p+q)t$ [79]. Therefore,
we get same temporal scaling of the variance for both symmetric and the
asymmetric cases. However, in presence of resetting, we get an additional time
scale in the problem $\left(\sim 1/r\right)$ and the time evolution equation
cannot be rendered independent of $p$ and $q$. This gives rise to different
behaviours for two cases. This key difference is one of the consequences of
the resetting. In Figure 3, we have compared the relaxation properties with
the numerical simulations for $p=q$ case (right panel) and $p\neq q$ case
(left panel). We observe an excellent agreement between our theory and
numerics for both cases.

###  IV.2 Correlation $C_{i}(t)=\langle z_{0}(t)z_{i}(t)\rangle$ for $i\neq 0$

After analysing the variance of the position of a tagged particle, let us now
look at the position correlation for two different tagged particles. For free
RAP, we saw in Eq. (6) that the two-point correlation function satisfies non-
trivial scaling behaviour in $\sim|i|/\sqrt{t}$ with $i$ being the separation
between two particles. The associated scaling function is given in Eq. (7). In
this section, we study the correlation function in presence of resetting and
calculate $C_{i}(t)=\langle z_{0}(t)z_{i}(t)\rangle$ for $i\neq 0$. Its
expression in the Laplace domain is given in Eq. (26) from which it is clear
that we have to perform the inverse Laplace transformation of
$\hat{\mathcal{G}}_{i}(s)$. Rewriting its expression from Eq. (27)

$\displaystyle\hat{\mathcal{G}}_{i}(s)=\hat{\mathcal{G}}_{0}(s)\times\hat{w}_{i}(s+r),$
(37) with
$\displaystyle\hat{w}_{i}(s)=\left[\frac{s+2\mu_{1}^{\prime}-\sqrt{s^{2}+4\mu_{1}^{\prime}s}}{2\mu_{1}^{\prime}}\right]^{|i|},$
(38)

where we have defined $\mu_{1}^{\prime}=\mu_{1}(p+q)$. In order to perform the
inverse Laplace transformation of $\hat{\mathcal{G}}_{i}(s)$, we have to carry
out two inversions: one is for $\hat{\mathcal{G}}_{0}(s)$ and the other is for
$\hat{w}_{i}(s)$. Now for $\hat{\mathcal{G}}_{0}(s)$ , we have already
computed this inversion in Eq. (30). On the other hand, for $\hat{w}_{i}(s)$,
its inversion in the time domain (denoted by $w_{i}(t)$) is given by [91]

$\displaystyle
w_{i}(t)=\frac{|i|}{t}~{}e^{-2\mu_{1}^{\prime}t}~{}I_{|i|}\left(2\mu_{1}^{\prime}t\right),~{}~{}~{}\text{for
}i\neq 0.$ (39)

Finally using the convolution form of $\hat{\mathcal{G}}_{i}(s)$ in Eq. (37)
gives

$\displaystyle\mathcal{G}_{i}(t)=\int_{0}^{t}dT~{}e^{-rT}~{}w_{i}(T)~{}\mathcal{G}_{0}(t-T).$
(40)

inserting which in Eq. (26), we obtain

$\displaystyle C_{i}(t)=$
$\displaystyle\int_{0}^{t}dT~{}e^{-rt}~{}w_{i}(T)~{}\mathcal{G}_{0}(t-T)$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{2\mu_{1}^{2}(p-q)^{2}}{r^{2}}\left[1-e^{-rt}\left(rt+1\right)\right].$
(41)

Remember that $C_{i}(t)=\langle z_{0}(t)z_{i}(t)\rangle$ and in order to
obtain the connected-correlation, we subtract the mean contributions as
follows

$\displaystyle\langle z_{0}(t)z_{i}(t)\rangle_{c}$ $\displaystyle=\langle
z_{0}(t)z_{i}(t)\rangle-\langle z_{0}(t)\rangle\langle z_{i}(t)\rangle,$
$\displaystyle=\int_{0}^{t}dT~{}e^{-rt}~{}w_{i}(T)~{}\mathcal{G}_{0}(t-T)$
(42)
$\displaystyle~{}~{}~{}~{}~{}+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}\left[1-e^{-rt}\left(2rt+e^{-rt}\right)\right].$

It is worth mentioning that even for free RAP, only asymptotic results for
$\langle z_{0}(t)z_{i}(t)\rangle_{c}$ are known  [79]. Our analysis here
provides an exact expression for the correlation valid for all times and not
just at large times. However, at large times, one can simplify this expression
In fact, putting $t\to\infty$ in Eq. (40), one can see that

$\displaystyle\mathcal{G}_{i}(t\to\infty)=\mathcal{G}_{0}\left(t\to\infty\right)~{}\hat{w}_{i}(r),$
(43)

where the steady value $\mathcal{G}_{0}\left(t\to\infty\right)$ is given in
Eq. (71) and $\hat{w}_{i}(r)$ is given in Eq. (38). Using these expressions,
we find that the correlation between two tracer particles in the steady state
is given by

$\displaystyle\langle z_{0}(t)z_{i}(t)\rangle_{c}~{}\stackrel{{\scriptstyle
t\to\infty}}{{=}}~{}\mathcal{G}_{0}^{\text{st}}~{}\text{Exp}\left(-\frac{|i|}{\xi_{p}}\right)+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}},$
(44)

where $\mathcal{G}_{0}^{\text{st}}=\mathcal{G}_{0}(t\to\infty)$ is defined in
Eq. (71) and the decay length $\xi_{p}$ is defined as

$\displaystyle\xi_{p}=|\log\left(r+2\mu_{1}^{\prime}-\sqrt{r^{2}+4\mu_{1}^{\prime}r}\right)-\log(2\mu_{1}^{\prime})|^{-1}.$
(45)

Interestingly for $p\neq q$, we find that the correlation decays to a non-zero
constant value as $\big{|}i\big{|}\to\infty$. Contrarily, it decays to zero
for $p=q$. For reset-free RAP, this correlation decays to zero both for the
symmetric and the asymmetric cases. Our study unravels that resetting affects
these two cases in different manners and manifestly gives rise to long-range
correlations only for the $p\neq q$ case. Recently, resetting induced long-
range correlation was also found in independently diffusing particles but
subjected to simultaneous resetting at a rate $r$ [77]. Here, we have extended
this result for interacting single-file systems. Physically, the long-range
correlation can be understood as follows: Consider two particles with
positions $x_{0}(t)$ and $x_{l}(t)$ with $l\gg 1$. Both these particles
experience an effective attraction around their initial positions due to the
resetting event. However, this attraction has a purely dynamical
interpretation and does not arise due to any physical potential. Furthermore,
for $p>q$, all particles that lie between $x_{0}(t)$ and $x_{l}(t)$ experience
a net drift towards $x_{l}(t)$. As a result of this combined effect of
attraction and drift, the positions $x_{0}(t)$ and $x_{l}(t)$ of two particles
become strongly correlated. Figure 4 shows the comparison of our analytical
results with the same obtained using numerical simulation. Indeed, even in
simulations, we find that the correlation $C_{i}(t)$ does not decay to zero
for the asymmetric case.

##  V Unequal time correlations

So far, we have presented rigorous results on the variance and the equal time
correlation and demonstrated how resetting modifies these quantities. Our
analysis showed that contrary to the reset-free RAP model, these quantities,
in presence of resetting, behave differently depending on the presence or
absence of drive in the dynamics. Continuing on this, we now look at the
autocorrelation and the unequal time position correlations for two tracer
particles. Let us denote this correlation by $S_{i}(t_{0},t_{0}+t)=\langle
z_{0}(t_{0})z_{i}(t_{0}+t)\rangle$. As done before, we again consider a small
time interval $[t_{0}+t,t_{0}+t+dt]$ and follow the update rules in Eqs. (10)
and (11) to write down the total change in $S_{i}(t_{0},t_{0}+t)$ within this
interval. This results in the following differential equation:

Figure 5: Autocorrelation $\mathcal{L}_{0}(t)$ has been compared with the
numerical simulations for $p=1.0,q=0.1$ in panel (a) and $p=0.5,q=0.5$ in
panel (b). We have plotted both the exact expression in Eq. (52) as well as
the asymptotic (large-$t$) expression in Eq. (54). Simulation has been
performed with $N=101$ particles, resetting rate $r=1$ and jump distribution
$R(\eta)=1$.

$\displaystyle\frac{dS_{i}(t_{0},t_{0}+t)}{dt}=$ $\displaystyle-
rS_{i}+\mu_{1}(p-q)h(t_{0})$ (46)
$\displaystyle+\mu_{1}\Big{[}pS_{i+1}+qS_{i-1}-(p+q)S_{i}\Big{]}.$

Solving this equation by taking the joint Fourier-Laplace transformations as

$\displaystyle\tilde{S}(k,t_{0},t_{0}+t)=\sum_{i=-\infty}^{\infty}e^{jik}~{}S_{i}(t_{0},t_{0}+t),$
(47)
$\displaystyle\mathbb{S}(k,s,t)=\int_{0}^{\infty}dt_{0}~{}e^{-st_{0}}~{}\tilde{S}(k,t_{0},t_{0}+t),$
(48)

and inserting them in Eq. (46), we obtain

$\displaystyle\frac{d\mathbb{S}}{dt}=-\alpha(k)\mathbb{S}+2\pi\mu_{1}(p-q)\hat{h}(s)\delta(k),$
(49)

where $\alpha(k)=r+\mu_{1}(p+q)-\mu_{1}pe^{jk}-\mu_{1}qe^{-jk}$. Since, we are
interested in computing the correlations in the steady state $(\text{i.e.
}t_{0}\to\infty)$, we will analyse Eq. (49) in the small-$s$ limit. In
Appendix B, we have explicitly carried out this analysis and obtained the
correlation $S_{i}\left(t_{0},t_{0}+t\right)$ measured from the steady-state
$(t_{0}\to\infty)$ as

$\displaystyle S_{i}\left(t_{0},t_{0}+t\right)$
$\displaystyle\simeq~{}\frac{\mathcal{B}r}{2\pi}\int_{-\pi}^{\pi}dk~{}\frac{e^{-jik-\alpha(k)t}}{r+2\mu_{1}^{\prime}(1-\cos
k)}$ $\displaystyle~{}~{}~{}+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}(1+e^{-rt})$
(50) $\displaystyle\text{with }~{}~{}\mathcal{B}$
$\displaystyle=\sqrt{\frac{r+4\mu_{1}^{\prime}}{r}}~{}\mathcal{G}_{0}^{\text{st}},$
(51)

where again we have used the notation $\mu_{1}^{\prime}=\mu_{1}(p+q)$ and
$\mathcal{G}_{0}^{\text{st}}=\mathcal{G}_{0}(t_{0}\to\infty)$ is defined in
Eq. (71). Subtracting the mean contribution from this correlation, we obtain
the connected-correlation $\mathcal{L}_{i}(t)$ as

$\displaystyle\mathcal{L}_{i}(t)$
$\displaystyle=\Big{[}S_{i}\left(t_{0},t_{0}+t\right)-\langle
z_{0}(t_{0})\rangle~{}\langle z_{i}(t_{0}+t)\rangle\Big{]}_{t_{0}\to\infty},$
$\displaystyle=\frac{\mathcal{B}r}{2\pi}\int_{-\pi}^{\pi}dk~{}\frac{e^{-jik-\alpha(k)t}}{r+2\mu_{1}^{\prime}(1-\cos
k)}+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}e^{-rt}.$ (52)

For $t=0$, one can perform the integration over $k$ and the result matches
with the equal time correlation in Eq. (44) in the steady state. On the other
hand, for non-zero $t$, performing this integration turns out to be difficult.
However, for large $t$, we could carry out the integration rigorously and
obtain some asymptotic results. Below we discuss this first for the
autocorrelation $(i=0)$ and then for general $i$.

Figure 6: Unequal time correlation $\mathcal{L}_{i}(t)$ in Eq. (52) is
compared with the numerical simulation for asymmetric ($p>q$) RAP in the left
panel and symmetric ($p=q$) RAP in the right panel for $t=1$. Parameters
chosen are $r=1$, $R(\eta)=1$ and number of particles $N=101$ for simulation.

###  V.1 Autocorrelation $\mathcal{L}_{0}(t)$

To get large $t$-behaviour of $\mathcal{L}_{0}(t)$, we first notice that one
gets exponentially decaying terms like
$\sim\exp\left[-2\mu_{1}(p+q)t\sin^{2}(k/2)\right]$ inside the integration in
Eq. (52). For large $t$, such integrations will be dominated by smaller values
of $k$. Therefore, performing small-$k$ approximation in Eq. (52), we get

$\displaystyle\mathcal{L}_{0}(t)\simeq$
$\displaystyle\frac{\mathcal{B}e^{-rt}}{2\pi}\int_{-\pi}^{\pi}dk~{}e^{jk\mu_{1}(p-q)t-\mu_{1}t(p+q)k^{2}/2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}e^{-rt}.$
(53)

Next, we change the variable $u=k\sqrt{\mu_{1}(p+q)t}$ in this equation and
carry out the integration for large $t$ to obtain

$\displaystyle\mathcal{L}_{0}(t)\simeq\frac{\mathcal{B}~{}e^{-\left[r+\frac{\mu_{1}(p-q)^{2}}{2(p+q)}\right]t}}{\sqrt{2\pi\mu_{1}(p+q)t}}+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}e^{-rt}.$
(54)

Using this expression, we again find that the large time decay of
$\mathcal{L}_{0}(t)$ depends on whether the particles experience a drift or
not. While for the symmetric case, the autocorrelation in the steady state
decays as $\sim e^{-rt}/\sqrt{t}$ at late times, we observe an exponential
decay $\sim e^{-rt}$ for the asymmetric $(p\neq q)$ case. This is contrary to
the case of simple resetting Brownian motion, where autocorrelation in the
steady state decays exponentially as $e^{-rt}$ at all times [38, 39]. However,
for interacting particles, $\mathcal{L}_{0}(t)$ has a more complicated form
and picks up exponential decay (or otherwise) only at large times. In Figure
5, we have compared this late time decay with the numerical simulations. We
observe that while the simulation data deviates from Eq. (54) at small times,
the agreement becomes better at larger times.

###  V.2 Unequal time correlation $\mathcal{L}_{i}(t)$

We now analyse Eq. (52) for general $i$. For this case also, we can perform
small-$k$ approximation in Eq. (52) for larger values of $t$ since the
integral has exponentially decaying terms like
$\sim\exp\left[-2\mu_{1}(p+q)t\sin^{2}(k/2)\right]$. We then obtain

$\displaystyle\mathcal{L}_{i}(t)\simeq$
$\displaystyle\frac{\mathcal{B}e^{-rt}}{2\pi}\int_{-\pi}^{\pi}dk~{}e^{-jik+jk\mu_{1}(p-q)t-\mu_{1}t(p+q)k^{2}/2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}e^{-rt}.$
(55)

To perform the integration over $k$, we change the variable
$u=k\sqrt{\mu_{1}(p+q)t}$ and plug it in this equation. Finally, we find that
the $\mathcal{L}_{i}(t)$ satisfies the following scaling relation

$\displaystyle\mathcal{L}_{i}(t)-\mathcal{L}_{|i|\to\infty}(t)\simeq\frac{\mathcal{B}~{}e^{-rt}}{\sqrt{2\pi\mu_{1}(p+q)t}}~{}\mathcal{M}\left(\frac{i-\mu_{1}(p-q)t}{\sqrt{2\mu_{1}(p+q)t}}\right),$
(56)

where the scaling function $\mathcal{M}(y)$ and
$\mathcal{L}_{|i|\to\infty}(t)$ are given by

$\displaystyle\mathcal{M}(y)=e^{-y^{2}},~{}\text{and
}~{}~{}\mathcal{L}_{|i|\to\infty}(t)=\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}e^{-rt}.$
(57)

Note that this scaling behaviour is entirely an outcome of the resetting
dynamics and does not appear for the reset-free RAP [79]. Looking at Eq. (56),
once again we see that for the asymmetric RAP, $\mathcal{L}_{i}(t)$ takes a
non-zero value as $|i|\to\infty$ indicating a long-range correlation between
two particles. However, this value decays exponentially with time and as
$t\to\infty$, this long-range correlation vanishes. As discussed in the case
of equal time correlation, the appearance of long-range correlation turns out
to be an interplay of the effective attraction experienced by the particles
around their resetting sites and a net drift due to asymmetric rates.

In Figure 6, we have compared the exact expression of $\mathcal{L}_{i}(t)$ in
Eq. (52) with the numerical simulations for $p\neq q$ in the left panel and
$p=q$ in the right panel. For both cases, we see an excellent agreement
between theory and numerics. However, demonstrating the scaling behaviour of
$\mathcal{L}_{i}(t)$ in Eq. (56) in simulation turns out to be difficult. It
turns out that one needs to go to very large values of $t$ in order to observe
this scaling relation. For instance, in Figure 7, we see that this scaling
behaviour becomes valid at around $t=200$. However, the value of
$\mathcal{L}_{i}(t)$ at such large times is very small due to the presence
$\sim e^{-rt}$ term in Eq. (56). Measuring such small values in simulation is
difficult. Therefore, to validate this scaling relation, we have plotted the
exact $\mathcal{L}_{i}(t)$ in Figure 7 for different values of $t$ by
numerically performing the integration over $k$. At large $t$, we recover the
scaling function $\mathcal{M}(y)$ [see Figure 7].

Figure 7: Scaling function $\mathcal{M}(y)$ in Eq. (56) for the unequal time
correlation $\mathcal{L}_{i}(t)$ is compared with the exact expression of
$\mathcal{L}_{i}(t)$ in Eq. (52) for different values of $t$ and
$p=1.0,~{}q=0.1,~{}r=1$ and jump distribution $R(\eta)=1$.

##  VI Conclusion

In conclusion, we have studied the motion of tracer particles in a one
dimensional single-file model called random average process which is subjected
to stochastic resetting. The resetting mechanism, characterized by a constant
rate $r$ causes the entire system being reinstated to the configuration given
in Eq. (1). Utilizing an exact microscopic analysis, we calculated key
statistical quantities such as variance, equal-time correlation,
autocorrelation, and unequal time correlation for the positions of tracer
particles. Through these calculations, we demonstrated how resetting modifies
the system and gives rise to properties which are otherwise not observed in
absence of the resetting

We first looked at the variance $\langle z_{0}^{2}(t)\rangle_{c}$ whose exact
expression is given in Eq. (33). At large times, it expectedly attains a
stationary value given in Eq. (34). To gain some physical insights, we further
explored the relaxation behaviour of $\langle z_{0}^{2}(t)\rangle_{c}$ as it
approaches its stationary value. Interestingly, this relaxation process turns
out to crucially depend on whether the particles move symmetrically $(p=q)$ or
asymmetrically $(p\neq q)$ on either side. While for the symmetric case, the
variance relaxes as $\sim e^{-rt}/\sqrt{t}$ to its stationary value, we obtain
$\sim te^{-rt}$ relaxation for the asymmetric case. Note that such difference
between the symmetric and the asymmetric variances is not seen for free RAP
$(r=0)$ and we obtain the same sub-diffusive scaling for both cases as shown
in Eq. (5). Resetting introduces an additional time scale in the model which
leads to different behaviours for two cases. This key difference is one of the
consequences of the resetting.

We next turned our attention to the equal-time position correlation for two
different tracer particles. Focussing on the steady state, our study revealed
that resetting induces a long-range correlation only for the asymmetric case.
On the other hand, for the symmetric case, we obtained correlations that decay
exponentially with the distance in Eq. (44). In a recent work involving
independently diffusing particles, but undergoing global resetting at a rate
$r$, the authors showed that the simultaneous resetting induces a strong long-
range correlation in the system  [77]. Our work generalises these results in
the interacting single-file set-up and shows that simultaneous resetting
induces a long-range correlation only when particles experience a bias $(p\neq
q)$.

Finally, we investigated the autocorrelation and the unequal time position
correlation in the steady state. For the autocorrelation $\mathcal{L}_{0}(t)$,
once again, we find that the large-$t$ decay is different for $p=q$ and $p\neq
q$ cases. Specifically, for the symmetric case $\mathcal{L}_{0}(t)$ exhibited
a decay of $\sim e^{-rt}/\sqrt{t}$, while for the asymmetric case, the decay
followed $\mathcal{L}_{0}(t)\sim e^{-rt}$. Conversely, the unequal time
position correlation $\mathcal{L}_{i}(t)$ exhibits a scaling behaviour in
terms of the variable $y=\big{(}i-\mu_{1}(p-q)t\big{)}/\sqrt{t}$. The
associated scaling function $\mathcal{M}(y)$ is written in Eq. (57). We
emphasize that this scaling behaviour is entirely an outcome of the resetting
dynamics and does not appear for the reset-free RAP [79].

Studying analytically an interacting multi-particle system is difficult
because of the correlation between different particles. Here, we presented a
specific single-file model for which exact microscopic computations can be
carried out. Our work pointed at a crucial difference in the tracer dynamics
for symmetric RAP and asymmetric RAP, both subjected to resetting at a rate
$r$. For future direction, it would be interesting to explore an intermediate
case where only some of the particles experience bias while all others move
symmetrically [87, 88] and see if one still gets a resetting induced long-
range correlation. Also, our paper focused on one specific model of single-
file motion called random average process. It remains a promising direction to
study effects of resetting on other single-file models like single-file
diffusion  [8, 9, 10, 11], in active particles [92, 93, 94, 21, 22, 23, 24]
and also in experiments  [57, 58, 59, 59]. Finally, we have only looked at
different two-point correlation functions in our paper. Obtaining higher
moments and the distribution function for the position of a tracer particle
still remains an open problem even for the reset-free RAP.

###### Acknowledgements.

We thank Arnab Pal and R. K. Singh for their useful comments on the paper. SS
acknowledges the support of the Department of Atomic Energy, Government of
India, under Project No. 19P1112$\&$D. PS acknowledges the support of Novo
Nordisk Foundation under the grant number NNF21OC0071284.

##  Appendix A Derivation of the variance $\langle z_{0}^{2}(t)\rangle_{c}$ in
Eq. (33)

In this appendix, we provide a detailed derivation of the variance $\langle
z_{0}^{2}(t)\rangle_{c}=\langle z_{0}^{2}(t)\rangle-\langle
z_{0}(t)\rangle^{2}$ in Eq. (33). The starting point is to find
$C_{0}(t)=\langle z_{0}^{2}(t)\rangle$ for which we need the Laplace transform
$\hat{C}_{0}(s)$ in Eq. (26). Rewriting this expression here as

$\displaystyle\hat{C}_{0}(s)=\frac{2\mu_{1}^{2}(p-q)^{2}}{s(s+r)^{2}}+\hat{\mathcal{G}}_{0}(s),$
(58)

with $\hat{\mathcal{G}}_{0}(s)$ defined as

$\displaystyle\hat{\mathcal{G}}_{0}(s)=\frac{\mu_{2}(\mu_{1}-\mu_{2})(p+q)}{(\mu_{1}-2\mu_{2})}\hat{H}(s)\hat{Z}(s+r),~{}~{}~{}\text{with
}\hat{Z}(s)=\frac{\sqrt{s+4\mu_{1}(p+q)}-\frac{\mu_{2}}{\mu_{1}-\mu_{2}}\sqrt{s}}{s+\frac{4(p+q)(\mu_{1}-\mu_{2})^{2}}{\mu_{1}-2\mu_{2}}},\text{
and }\hat{H}(s)=\frac{1}{s\sqrt{s+r}}.$ (59)

To evaluate first term in the right hand side of Eq. (58), we use the relation

$\displaystyle\int_{0}^{\infty}dt~{}e^{-st}\left[\frac{1-e^{-rt}(rt+1)}{r^{2}}\right]=\frac{1}{s(s+r)^{2}}.$
(60)

On the other hand, for second term, we use the convolution structure of
$\hat{\mathcal{G}}_{0}(s)$ which gives $\mathcal{G}_{0}(t)$ as

$\displaystyle\mathcal{G}_{0}(t)=$
$\displaystyle\frac{\mu_{2}(\mu_{1}-\mu_{2})(p+q)}{(\mu_{1}-2\mu_{2})}\int_{0}^{t}dT~{}e^{-rT}~{}H(t-T)~{}Z(T).$
(61)

Here $H(t)$ and $Z(t)$ are inverse Laplace transforms of $\hat{H}(s)$ and
$\hat{Z}(s)$ respectively. Using Mathematica, we find
$H(t)=\text{Erf}\left(\sqrt{rt}\right)/\sqrt{r}$ and $Z(t)$ as

$\displaystyle Z(t)=$ $\displaystyle\ \frac{1}{\sqrt{\pi
t}}\left[e^{-4\mu_{1}t(p+q)}-\frac{\mu_{2}}{\mu_{1}-\mu_{2}}\right]-\frac{\mu_{2}}{\mu_{2}-\mu_{2}}~{}\sqrt{-B_{1}}~{}e^{-B_{1}t}~{}\text{Erf}(\sqrt{-B_{1}~{}t})+\sqrt{-B_{2}}~{}e^{-B_{1}t}~{}\text{Erf}(\sqrt{-B_{2}~{}t}),$
(62)

with $B_{1}=4(p+q)(\mu_{1}-\mu_{2})^{2}/(\mu_{1}-2\mu_{2})$ and
$B_{2}=B_{1}-4\mu_{1}(p+q)$. Substituting these expressions in Eq. (61) yields

$\displaystyle\mathcal{G}_{0}(t)=$
$\displaystyle\frac{\mu_{2}(\mu_{1}-\mu_{2})(p+q)}{\sqrt{r}(\mu_{1}-2\mu_{2})}\int_{0}^{t}dT~{}e^{-rT}~{}\text{Erf}\left(\sqrt{r(t-T)}\right)~{}Z(T).$
(63)

We now have both terms from the right hand side of Eq. (58). Performing the
inverse Laplace transformation, we get

$\displaystyle
C_{0}(t)=\mathcal{G}_{0}(t)+\frac{2\mu_{1}^{2}(p-q)^{2}}{r^{2}}\left[1-e^{-rt}\left(rt+1\right)\right],$
(64)

from which the variance of $z_{0}(t)$ turns out to be

$\displaystyle\langle z_{0}^{2}(t)\rangle_{c}$ $\displaystyle=C_{0}(t)-\langle
z_{0}(t)\rangle^{2},$
$\displaystyle=\mathcal{G}_{0}(t)+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}\left[1-e^{-rt}\left(2rt+e^{-rt}\right)\right],$
(65)

with $\mathcal{G}_{0}(t)$ given in Eq. (63). This expression of $\langle
z_{0}^{2}(t)\rangle_{c}$ has been quoted in Eq. (33) in the main text.

Let us now analyse this expression to extract the asymptotic behaviour of the
variance. To perform the integration in Eq. (63), we substitute the error
function for large $t$ as

$\displaystyle\text{Erf}\left(\sqrt{r(t-T)}\right)\simeq
1-\frac{e^{-r(t-T)}}{\sqrt{\pi r(t-T)}}$ (66)

and plug it in Eq. (63) as

$\displaystyle\mathcal{G}_{0}(t)\simeq$
$\displaystyle\frac{\mu_{2}(\mu_{1}-\mu_{2})(p+q)}{\sqrt{r}(\mu_{1}-2\mu_{2})}\left[\int_{0}^{t}dT~{}e^{-rT}Z(T)-\frac{e^{-rt}}{\sqrt{\pi
r}}~{}\int_{0}^{t}dT~{}\frac{Z(T)}{\sqrt{t-T}}\right].$

For $t\to\infty$, first term inside the bracket $[..]$ simply becomes the
Laplace transform $\hat{Z}(r)$ in Eq. (59). Therefore, one can recast
$\mathcal{G}_{0}(t)$ as

$\displaystyle\mathcal{G}_{0}(t)\simeq$
$\displaystyle\frac{\mu_{2}(\mu_{1}-\mu_{2})(p+q)}{\sqrt{r}(\mu_{1}-2\mu_{2})}\left[\hat{Z}(r)-\frac{e^{-rt}}{\sqrt{\pi
r}}~{}\int_{0}^{t}dT~{}\frac{Z(T)}{\sqrt{t-T}}\right].$ (67)

Next, we have to evaluate the integral over $T$ for larger values of $t$.
Defining $J(t)=\int_{0}^{t}dT~{}\frac{Z(T)}{\sqrt{t-T}}$, we take its Laplace
transform with respect to $t~{}(\to s)$ as

$\displaystyle\hat{J}(s)=\sqrt{\frac{\pi}{s}}~{}\hat{Z}(s).$ (68)

We take small $s$ limit of this equation which corresponds to its large $t$
limit in the time domain

$\displaystyle\hat{J}(s\to
0)\simeq\frac{(\mu_{1}-2\mu_{2})\sqrt{\mu_{1}}}{\sqrt{4(p+q)}(\mu_{1}-\mu_{2})^{2}}\sqrt{\frac{\pi}{s}}.$
(69)

Now performing the inverse Laplace transformation, we obtain

$\displaystyle
J(t)\simeq\frac{(\mu_{1}-2\mu_{2})\sqrt{\mu_{1}}}{\sqrt{4(p+q)t}(\mu_{1}-\mu_{2})^{2}},~{}~{}~{}~{}\text{as
}t\to\infty$ (70)

Using this in Eq. (67), we obtain

$\displaystyle\mathcal{G}_{0}(t)\simeq$
$\displaystyle\frac{\mu_{1}\mu_{2}(p+q)}{\sqrt{r}(\mu_{1}-\mu_{2})}\left[\sqrt{r+4\mu_{1}(p+q)}+\frac{\mu_{2}\sqrt{r}}{\mu_{1}-\mu_{2}}\right]^{-1}-\frac{\mu_{1}\sqrt{\mu_{1}(p+q)}}{2r(\mu_{1}-\mu_{2})}~{}\frac{e^{-rt}}{\sqrt{\pi
t}},$ (71)

plugging which in Eq. (65) gives the relaxation behaviour of $\langle
z_{0}^{2}(t)\rangle_{c}$ in Eqs. (35) and (36).

##  Appendix B Unequal time correlation $S_{i}\left(t_{0},t_{0}+t\right)$ in
the steady state $(t_{0}\to\infty)$

This appendix provides a derivation of the unequal time correlation
$S_{i}\left(t_{0},t_{0}+t\right)$ in Eq. (50). Let us first denote by
$\mathbb{S}(k,s,t)$ the joint Fourier-Laplace transform of
$S_{i}\left(t_{0},t_{0}+t\right)$ [see Eq. (48)]. We showed in Eq. (49) that
$\mathbb{S}(k,s,t)$ satisfies the differential equation

$\displaystyle\frac{d\mathbb{S}(k,s,t)}{dt}=-\alpha(k)\mathbb{S}(k,s,t)+2\pi\mu_{1}(p-q)\hat{h}(s)\delta(k),$
(72)

where $\alpha(k)=r+\mu_{1}(p+q)-\mu_{1}pe^{jk}-\mu_{1}qe^{-jk}$. Solving this
equation with the initial condition $\mathbb{S}(k,s,t=0)=\mathcal{Z}(k,s)$ in
Eq. (25) gives

$\displaystyle\mathbb{S}(k,s,t)=\mathcal{Z}(k,s)~{}e^{-\alpha(k)t}+2\pi\hat{h}(s)h(t).$
(73)

Since we are interested in finding the correlation at the steady state (i.e.
$t_{0}\to\infty$), we analyse Eq. (73) in the small-$s$ limit. For
$\hat{h}(s)$, we use Eq. (21) to obtain $\hat{h}(s\to
0)\simeq\mu_{1}(p-q)/rs$. On the other hand, using Eq. (25), we find
$\mathcal{Z}(k,s)$ as

$\displaystyle\mathcal{Z}(k,s\to
0)\simeq\frac{\mathcal{B}r}{s\left[r+2\mu_{1}(p+q)(1-\cos
k)\right]}+\frac{4\pi\mu_{1}^{2}(p-q)^{2}}{r^{2}s}~{}\delta(k),$ (74)

where $\mathcal{B}$ is a constant that depends on model parameters and is
given in Eq. (51). Finally plugging Eq. (74) in $\mathbb{S}(k,s,t)$ in Eq.
(73) and performing the inverse Fourier transformation, we find

$\displaystyle S_{i}\left(t_{0},t_{0}+t\right)\Big{|}_{t_{0}\to\infty}$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk~{}e^{-jik}~{}s~{}\mathbb{S}(k,s,t)\Big{|}_{s\to
0},$ (75)
$\displaystyle=\frac{\mathcal{B}r}{2\pi}\int_{-\pi}^{\pi}dk~{}\frac{e^{-jik-\alpha(k)t}}{r+2\mu_{1}(p+q)(1-\cos
k)}+\frac{\mu_{1}^{2}(p-q)^{2}}{r^{2}}(1+e^{-rt}).$ (76)

This result has been quoted in Eq. (50) in the main text.

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