# Current fluctuations in finite-sized one-dimensional
non-interacting passive and active systems

Arup Biswas<EMAIL_ADDRESS>The Institute of Mathematical Sciences, CIT
Campus, Taramani, Chennai 600113, India & Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai 400094, India  Stephy Jose
<EMAIL_ADDRESS>Tata Institute of Fundamental Research, Hyderabad
500046, India  Arnab Pal<EMAIL_ADDRESS>The Institute of Mathematical
Sciences, CIT Campus, Taramani, Chennai 600113, India & Homi Bhabha National
Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
Kabir Ramola<EMAIL_ADDRESS>Tata Institute of Fundamental Research,
Hyderabad 500046, India

###### Abstract

We investigate the problem of effusion of particles initially confined in a
finite one-dimensional box of size $L$. We study both passive as well active
scenarios, involving non-interacting diffusive particles and run-and-tumble
particles, respectively. We derive analytic results for the fluctuations in
the number of particles exiting the boundaries of the finite confining box.
The statistical properties of this quantity crucially depend on how the system
is prepared initially. Two common types of averages employed to understand the
impact of initial conditions in stochastic systems are annealed and quenched
averages. It is well known that for an infinitely extended system, these
different initial conditions produce quantitatively different fluctuations,
even in the infinite time limit. We demonstrate explicitly that in finite
systems, annealed and quenched fluctuations become equal beyond a system-size
dependent timescale, $t\sim L^{2}$. For diffusing particles, the fluctuations
exhibit a $\sqrt{t}$ growth at short times and decay as $1/\sqrt{t}$ for time
scales, $t\gg L^{2}/D$, where $D$ is the diffusion constant. Meanwhile, for
run-and-tumble particles, the fluctuations grow linearly at short times and
then decay as $1/\sqrt{t}$ for time scales, $t\gg L^{2}/D_{\text{eff}}$, where
$D_{\text{eff}}$ represents the effective diffusive constant for run-and-
tumble particles. To study the effect of confinement in detail, we also
analyze two different setups (i) with one reflecting boundary and (ii) with
both boundaries open.

###### pacs:

Valid PACS appear here

##  I Introduction

The study of the effect of initial conditions on the transport properties of
stochastic systems has attracted considerable interest in the past years Le
Doussal and Machta (1989); Van Beijeren (1991); Derrida  _et al._ (2004);
Derrida and Gerschenfeld (2009a); Ferrari (2010); Lizana  _et al._ (2010);
Krapivsky and Meerson (2012); Bodineau and Derrida (2004); Leibovich and
Barkai (2013); Krapivsky  _et al._ (2014); Banerjee  _et al._ (2022); Mallick
_et al._ (2022); Dandekar  _et al._ (2023); Dean  _et al._ (2023). Notably,
these studies have revealed that the distributions of quantities such as the
tracer particle displacement or the integrated current across a region are
different depending on the initial condition involving the positions of
particles Mallick  _et al._ (2022); Dandekar  _et al._ (2023); Dean  _et al._
(2023); Di Bello  _et al._ (2023). Two ensembles of initial conditions that
are commonly used to study this effect are (i) annealed setting, which allows
for random fluctuations in the initial condition, and (ii) quenched setting,
where the initial condition is deterministic Derrida and Gerschenfeld (2009b);
Cividini and Kundu (2017); Banerjee  _et al._ (2022); Krapivsky  _et al._
(2014). To gain an initial understanding of the relevance of initial
conditions, imagine a set of particles initially confined in a one-dimensional
channel, free to diffuse. Several intriguing questions arise: Does a static
disorder in the initial arrangement of particles influence the dynamic
behavior of the system? Furthermore, does this effect persist over large
times, particularly when the channel length is finite? What happens if there
is an asymmetry in the boundary conditions of this confining channel?

Previous studies have extensively examined the problem of effusion using model
systems that are infinitely extended. One such model considers a semi-infinite
confining channel bounded between $x\in(-\infty,0]$, where the fluctuations in
the number of particles crossing the origin $x=0$ up to time $t$ are
investigated Banerjee  _et al._ (2020); Dean  _et al._ (2023). For the case of
diffusive particles, the annealed setting exhibits larger fluctuations by a
factor of $\sqrt{2}$ as compared to the quenched setting Leibovich and Barkai
(2013); Derrida and Gerschenfeld (2009a); Krapivsky and Meerson (2012); Dean
_et al._ (2023); Banerjee  _et al._ (2020). Even for non-interacting active
particles, the annealed setting exhibits larger fluctuations by a factor of
$\sqrt{2}$ at large times, as the dynamics effectively becomes diffusive
Banerjee  _et al._ (2020); Jose  _et al._ (2023a, b).

Figure 1: Schematic representation of the trajectories of $N=8$ non-
interacting particles, initially confined in a box bounded between $[-L,0]$.
We are interested in the number of particles present outside this box at time
$t$, denoted as $Q(t)$. The green trajectories indicate those contributing to
a non-zero $Q(t)$, while the red ones do not contribute. We examine two
setups: (a) In the left panel, a reflecting wall is present at $x=-L$,
allowing particles to escape only through the origin $x=0$. (b) In the right
panel, both sides are open, enabling particles to exit through either
boundary. In the illustrated figure, $Q(t)=3$ for the former case and $Q(t)=5$
for the latter.

In this paper, we focus on the dynamic properties of the particle flux $Q$
across the boundaries of a finite confining box bounded between $x\in[-L,0]$.
We specifically investigate the interplay between the initial conditions and
the system geometry on the fluctuations of $Q$. We consider two setups: (i)
with a reflecting boundary condition at $-L$, and (ii) with both the
boundaries at $-L$ and $0$ open. Interestingly, we demonstrate that in both
these cases, annealed and quenched fluctuations converge and become equal at a
timescale determined by the system size $L$ and the parameters of the model
studied. For diffusing particles, the fluctuations exhibit a $\sqrt{t}$ growth
at short times and decay as $1/\sqrt{t}$ for time scales $t\gg L^{2}/D$, where
$D$ represents the diffusion constant. Meanwhile, for run-and-tumble particles
Malakar  _et al._ (2018); Evans and Majumdar (2018); Mori  _et al._ (2020a,
b); Singh and Kundu (2019); Angelani  _et al._ (2014); Martens  _et al._
(2012); Mori  _et al._ (2021); Jose  _et al._ (2022); Jose (2022); Mallikarjun
and Pal (2023), the fluctuations grow linearly at short times and then decay
as $1/\sqrt{t}$ for time scales $t\gg L^{2}/D_{\text{eff}}$, where
$D_{\text{eff}}$ denotes the effective diffusive constant for run-and-tumble
particles. For diffusive systems, the ratio of the fluctuations due to
annealed and quenched initial conditions changes from a value of $\sqrt{2}$
(which is equal to the ratio observed in an infinite system) at short times to
$1$ at large times for both the geometries; for active particles, it changes
from the value of $2$ (infinite system) at short times to $1$ at large times
in a similar vein. Intriguingly, we also show that the boundary conditions of
the confining box play a crucial role in determining the dynamic behavior of
$Q$. The setup with two open boundaries displays larger fluctuations by a
factor of $2$ at short times and smaller fluctuations by a factor of $1/2$ at
large times as compared to the setup with one open boundary for both passive
and active cases.

The paper is organized as follows. In Sec. II, we introduce the models that we
use to study the fluctuations in the particle flux $Q$. In Sec. III and Sec.
IV, we present exact analytical results for the fluctuations in both diffusive
and active systems. We present the conclusions from the study in Sec. V.
Finally, we present details related to some of the calculations in Appendices
A and B.

##  II The formalism

In this section, we generalize the formalism developed in Banerjee  _et al._
(2020) to study the fluctuations in the current of particles across the
boundaries of a finite-sized box. We consider $N$ non-interacting particles
initially distributed with a uniform density $\rho=N/L$ in a finite one-
dimensional box bounded between $[-L,0]$. These particles evolve over time
following their underlying dynamics such as diffusion or run-and-tumble
motion. The quantity of interest is the number of particles exiting the
boundaries of the box up to time $t$, equivalent to the number of particles
present outside the box at time $t$ (see Fig. 1 for a schematic
representation). We denote this quantity as $Q(t)$, representing the flux or
integrated current through the boundaries of the box up to time $t$. The
number of particles present outside the box can be expressed using an
indicator function $\mathcal{I}(t)$, defined as

${\cal I}_{i}(t)=\begin{cases}1,&\text{if the $i{\text{th}}$ particle is
outside $[-L,0]$ at $t$},\\\ 0,&\text{otherwise}.\end{cases}$ (1)

The current $Q(t)$ is then given as

$\displaystyle Q(t)=\sum_{i=1}^{N}\mathcal{I}_{i}(t).$ (2)

We are primarily interested in the statistical properties of the random
variable $Q$. Generally, two sources of randomness are associated with the
measurement of $Q$; the randomness in the initial positions of the particles
and the randomness due to the inherent stochasticity of the underlying
dynamics of the particles. There are two distinct methods for averaging over
these sources of randomness: (i) the annealed average - which corresponds to
simultaneous averaging over all initial conditions and noise history (ii) the
quenched average - where one first averages over noise history for a fixed
initial realization, followed by averaging over all possible initial
realizations. The formal definitions of these averages are provided in the
subsequent sections.

Let us denote by $\\{x_{i}\\}$ a distinct set of initial positions of the
particles. For the fixed initial positions $\\{x_{i}\\}$, the probability
distribution of $Q$ is given as

$\displaystyle
P(Q,t,\\{x_{i}\\})=\left\langle\delta\left(Q-\sum_{i=1}^{N}\mathcal{I}_{i}(t)\right)\right\rangle_{\\{x_{i}\\}}.$
(3)

The angular bracket $\langle...\rangle_{\\{x_{i}\\}}$ in the above expression
denotes an average over all trajectories of the particles for a fixed initial
condition $\\{x_{i}\\}$. Moving forward it will be convenient to work with the
moment-generating function of $Q$ defined as

$\displaystyle\sum_{Q=0}^{\infty}e^{-pQ}P(Q,t,\\{x_{i}\\})$
$\displaystyle=\langle e^{-pQ}\rangle_{\\{x_{i}\\}}$
$\displaystyle=\left\langle\text{exp}\left(-p\sum_{i=1}^{N}\mathcal{I}_{i}(t)\right)\right\rangle_{\\{x_{i}\\}}.$
(4)

We next use the identity
$e^{-p\mathcal{I}_{i}(t)}=1-(1-e^{-p})\mathcal{I}_{i}(t)$ and the independent
nature of the dynamics of the particles to obtain

$\displaystyle\langle
e^{-pQ}\rangle_{\\{x_{i}\\}}=\prod_{i=1}^{N}\left[1-(1-e^{-p})\langle\mathcal{I}_{i}(t)\rangle_{\\{x_{i}\\}}\right].$
(5)

Here $\langle\mathcal{I}_{i}(t)\rangle_{\\{x_{i}\\}}$ represents the
probability that the $i$th particle is present outside the region $x\in[-L,0]$
at time $t$. Depending on the underlying dynamics and the geometry of the
system under consideration, this quantity will be different. We study two
different cases where (i) there is a reflecting boundary at $x=-L$ (see Fig. 1
(a)) and (ii) when both the boundaries at $x=0$ and $x=-L$ are open (see Fig.
1 (b)). In the first case, particles exit only through the boundary at $x=0$,
however, in the latter case, they exit either through $x=0$ or $x=-L$.
Denoting the expectation $\langle\mathcal{I}_{i}(t)\rangle_{\\{x_{i}\\}}$ by
$U(x_{i},t)$, we obtain

$\displaystyle U(x_{i},t)=\int_{0}^{\infty}G(x,t|x_{i})dx,$ (6)

for the case with a reflecting boundary and

$\displaystyle
U(x_{i},t)=\int_{-\infty}^{-L}G(x,t|x_{i})dx+\int_{0}^{\infty}G(x,t|x_{i})dx,$
(7)

when both boundaries are open. Here, $G(x,t|x_{i})$ is the Green’s function
defined as the probability density to find a particle at a position $x$ at
time $t$ starting from the position $x_{i}$ at time $t=0$. From Eq. (5), we
obtain the expression for the generating function of $Q$ as

$\displaystyle\langle
e^{-pQ}\rangle_{\\{x_{i}\\}}=\prod_{i=1}^{N}\left[1-(1-e^{-p})U(x_{i},t)\right],$
(8)

where the expressions for the function $U$ for the settings with reflecting
boundary and open boundaries are given in Eqs. (6) and (7) respectively. The
average over the initial conditions $\\{x_{i}\\}$ can now be done in two ways,
as discussed below.

###  II.1 Annealed setting

Let us denote by the symbol $\overline{(...)}$ as an average over the initial
conditions on the positions of the particles. Performing an average over the
initial positions in Eq. (8), we obtain

$\displaystyle\overline{\langle
e^{-pQ}\rangle_{\\{x_{i}\\}}}=\prod_{i=1}^{N}\left[1-(1-e^{-p})\overline{U(x_{i},t)}\right].$
(9)

Since the position of each particle is distributed independently according to
a uniform distribution in the interval $x_{i}\in[-L,0]$, this expectation can
be further simplified to

$\displaystyle\overline{\langle e^{-pQ}\rangle_{\\{x_{i}\\}}}$
$\displaystyle=\prod_{i=1}^{N}\left[1-(1-e^{-p})\frac{1}{L}\int_{-L}^{0}U(x_{i},t)dx_{i}\right]$
$\displaystyle=\left[1-(1-e^{-p})\frac{1}{L}\int_{-L}^{0}U(z,t)dz\right]^{N},$
(10)

where we have assigned a general variable $z\equiv x_{i}$ as the motion of the
particles is independent. Defining $P_{\text{an}}(Q,t)$ as the probability
distribution for $Q$ in the annealed setting, we have

$\displaystyle\sum_{Q=0}^{\infty}e^{-pQ}P_{\text{\text{an}}}(Q,t)=\overline{\langle
e^{-pQ}\rangle_{\\{x_{i}\\}}}.$ (11)

For finite $N,~{}L$, a small $p$ expansion of Eq. (10) yields the expressions
for the first few moments from which we can obtain the expressions for the
mean $\mu_{\text{an}}(L,t)$ and the variance $\sigma^{2}_{\text{an}}(L,t)$ of
$Q$ as

$\displaystyle\mu_{\text{an}}(L,t)$ $\displaystyle=\overline{\langle
Q\rangle}=\langle Q\rangle_{\text{an}}~{},$ (12)
$\displaystyle\sigma^{2}_{\text{an}}(L,t)$ $\displaystyle=\overline{\langle
Q^{2}\rangle}-{\overline{\langle Q\rangle}^{2}}$ $\displaystyle=\langle
Q^{2}\rangle_{\text{an}}-{\langle Q\rangle_{\text{an}}}^{2}$
$\displaystyle=\mu_{\text{an}}(L,t)-\frac{1}{\rho L}\mu_{\text{an}}^{2}(L,t).$
(13)

In the above expression, we have replaced $N$ by $\rho L$. The quantity
$\mu_{\text{an}}(L,t)$ can be computed as

$\mu_{\text{an}}(L,t)=\rho\int_{-L}^{0}U(z,t)dz,$ (14)

where the expression for $U(z,t)$ is given in Eq. (6) and Eq. (7) for the
cases with one and two open boundaries respectively. So far, the majority of
studies on the dynamic behavior of $Q$ have focused on infinite systems
($L\to\infty$). For an infinitely extended system with non-interacting
particles, the mean and the variance are the same in the annealed setting.
However, as we observe from Eqs. (12) and (II.1), they are not identical when
the system size is finite and furthermore, there is a $L$ dependent correction
term in the variance. In the limit $L\to\infty$ one recovers the known result,
$\mu_{\text{an}}(L\to\infty,t)=\sigma^{2}_{\text{an}}(L\to\infty,t)$ Banerjee
_et al._ (2020).

###  II.2 Quenched setting

In the quenched setting, we first perform an average over the trajectories for
a fixed initial condition and then average over the initial conditions of the
system with a mean density $\rho$. The generating function for $Q$ in the
quenched setting can be mathematically computed as

$\displaystyle\sum_{Q=0}^{\infty}P_{\text{qu}}(Q,t)e^{-pQ}=\exp[\overline{\text{ln}\langle
e^{-pQ}\rangle_{\\{x_{i}\\}}}].$ (15)

Taking a logarithm of both sides of Eq. (8), we obtain

$\displaystyle\text{ln}\langle
e^{-pQ}\rangle_{\\{x_{i}\\}}=\sum_{i=1}^{N}\ln[1-(1-e^{-p})U(x_{i},t)].$ (16)

Next performing an average over the initial positions in the above equation
yield

$\displaystyle\overline{\text{ln}\langle e^{-pQ}\rangle_{\\{x_{i}\\}}}$
$\displaystyle=\sum_{i=1}^{N}\frac{1}{L}\int_{-L}^{0}\ln[1-(1-e^{-p})U(x_{i},t)]dx_{i}$
$\displaystyle=\frac{N}{L}\int_{-L}^{0}\ln[1-(1-e^{-p})U(z,t)]dz$
$\displaystyle=\rho\int_{-L}^{0}\ln[1-(1-e^{-p})U(z,t)]dz$
$\displaystyle=I(p,t),$ (17)

where

$\displaystyle I(p,t)=\rho\int_{-L}^{0}\ln[1-(1-e^{-p})U(z,t)]dz.$ (18)

Finally, we obtain the expression for the generating function for the
distribution of $Q$ in the quenched setting as

$\displaystyle\sum_{Q=0}^{\infty}P_{\text{qu}}(Q,t)e^{-pQ}=\exp[I(p,t)].$ (19)

Performing a small $p$ expansion and collecting the terms at first and second
orders of $p$, we obtain the expression for the mean $\mu_{\text{qu}}(L,t)$
and the variance $\sigma^{2}_{\text{qu}}(L,t)$ of $Q$ in the quenched setting
as

$\displaystyle\mu_{\text{qu}}(L,t)$ $\displaystyle=\overline{\langle
Q\rangle}=\langle Q\rangle_{\text{qu}}=\mu_{\text{an}}(L,t),$ (20)
$\displaystyle\sigma^{2}_{\text{qu}}(L,t)$ $\displaystyle=\overline{\langle
Q^{2}\rangle-\langle Q\rangle^{2}}$ $\displaystyle=\langle
Q^{2}\rangle_{\text{qu}}-{\langle Q\rangle_{\text{qu}}}^{2}$
$\displaystyle=\mu_{\text{qu}}(L,t)-\rho\int_{-L}^{0}dz~{}U^{2}(z,t),$ (21)

where $\mu_{\text{qu}}(L,t)=\mu_{\text{an}}(L,t)$ is given by Eq. (14). The
mean in the annealed and quenched settings are the same even when the system
size is finite. However, the higher-order cumulants are different.

In what follows, we study two specific examples of a system of diffusive
particles and active run-and-tumble particles.

| $t\rightarrow 0$ | $t\rightarrow\infty$ |
---|---|---|---
Diffusion reflecting boundary | $\frac{\rho\sqrt{Dt}}{\sqrt{\pi}}$ | $\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{Dt}}$ | $\sigma^{\text{diff}}_{\text{an}}(L,t)^{2}$
$\frac{1}{\sqrt{2}}\frac{\rho\sqrt{Dt}}{\sqrt{\pi}}$ | $\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{Dt}}$ | $\sigma^{\text{diff}}_{\text{qu}}(L,t)^{2}$
Diffusion finite interval | $2\frac{\rho\sqrt{Dt}}{\sqrt{\pi}}$ | $\frac{1}{2}\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{Dt}}$ | $\sigma^{\text{diff}}_{\text{an}}(L,t)^{2}$
$\sqrt{2}\frac{\rho\sqrt{Dt}}{\sqrt{\pi}}$ | $\frac{1}{2}\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{Dt}}$ | $\sigma^{\text{diff}}_{\text{qu}}(L,t)^{2}$
Table 1: Asymptotic behavior of current fluctuations for diffusive motion in
the annealed and quenched settings.

##  III Diffusive particles

In this section, we consider a set of diffusive particles initially confined
in a finite one-dimensional box bounded between $[-L,0]$. We also consider two
distinct set-ups; one in the presence of a reflecting boundary and the other
with both boundaries open. We summarize the asymptotic behaviors of current
fluctuations for both these cases in Table 1.

We first focus on the case with a reflecting wall at $x=-L$.

Figure 2: Behavior of the (a) mean and the (b) variance of current through the
origin in the presence of a reflecting wall at $x=-L$ for $N$ diffusive
Brownian particles. The mean in the annealed and quenched settings are the
same. The variance in the quenched setting (solid curves) differs from the
annealed (dashed curves) by a factor of $\sqrt{2}$ at times $t\ll L^{2}/D$,
however, they become equal at times $t\gg L^{2}/D$. The parameter values used
are $\rho=1,D=1$. The stars and circles represent the results obtained through
numerical simulations of the microscopic model for the quenched and annealed
settings respectively.

###  III.1 One reflecting wall

In this section, we study the scenario where the boundary at $x=-L$ is a
reflecting wall. The Green’s function for a single diffusive Brownian particle
in this case can be derived as Feller (1991)

$\displaystyle G(x,t|x_{i})=\frac{1}{\sqrt{4\pi
Dt}}\left(e^{-\frac{(2L+x+x_{i})^{2}}{4Dt}}+e^{-\frac{(x-x_{i})^{2}}{4Dt}}\right),$
(22)

which can be substituted in Eq. (6) to obtain

$\displaystyle
U(x_{i},t)=\frac{1}{2}\left(1+\text{erf}\left(\frac{x_{i}}{2\sqrt{Dt}}\right)+\text{erfc}\left(\frac{2L+x_{i}}{2\sqrt{Dt}}\right)\right),$
(23)

where erf($z$) and erfc($z$) are the error function and complementary error
function, respectively. Having obtained the expression for $U(x_{i},t)$, we
can now compute the expressions for the mean and the variance of $Q$ for both
annealed and quenched settings as detailed below.

####  III.1.1 Annealed setting

Substituting Eq. (23) in Eq. (14), we obtain the expression for the mean of
$Q$ for a system of diffusing particles in the annealed setting as

$\displaystyle\mu^{\text{diff}}_{\text{an}}(L,t)$
$\displaystyle=\underbrace{\frac{\rho\sqrt{Dt}}{\sqrt{\pi}}}_{\text{infinite
size limit}}+\underbrace{\rho
L\left(\text{erfc}\left(\frac{L}{\sqrt{Dt}}\right)-\frac{\sqrt{Dt}}{\sqrt{\pi}L}e^{-\frac{L^{2}}{Dt}}\right)}_{\text{finite
size correction}}.$ (24)

The first term in the mean does not have any explicit dependence on the system
size $L$. This is the result one expects in the case of an infinite system
size limit ($L\to\infty$). The second term in the parentheses contains the
finite size corrections which vanishes in the limit $L\to\infty$. The
expression for the variance follows from Eq. (II.1) as

$\displaystyle{\sigma^{\text{diff}}_{\text{an}}(L,t)}^{2}$
$\displaystyle=\mu^{\text{diff}}_{\text{an}}(L,t)-\frac{1}{\rho
L}{\mu^{\text{diff}}_{\text{an}}(L,t)}^{2},$ (25)

with $\mu^{\text{diff}}_{\text{an}}(L,t)$ given by Eq. (24). Since the exact
expression for the variance is quite lengthy, we do not provide it here. Fig.
2 shows the behavior of the mean and the variance as a function of time
obtained from Eqs. (24) and (25) for different system sizes keeping the
density $\rho=1$ fixed.

Both the mean and variance increase monotonically with time for $t\ll
L^{2}/D$. Taking this limit in Eqs. (24) and (25), we obtain

$\displaystyle\mu^{\text{diff}}_{\text{an}}(L,t\ll
L^{2}/D)\approx\frac{\rho\sqrt{Dt}}{\sqrt{\pi}},$ (26)
$\displaystyle{\sigma^{\text{diff}}_{\text{an}}(L,t\ll
L^{2}/D)}^{2}\approx\frac{\rho\sqrt{Dt}}{\sqrt{\pi}}.$ (27)

At very short time scales, the mean and the variance in the annealed setting
are the same and also correspond to the infinite system results. However, at
larger time scales $t\gg L^{2}/D$, the mean saturates to the value $N$ and the
variance goes to zero as

$\displaystyle\mu^{\text{diff}}_{\text{an}}(L,t\gg L^{2}/D)\approx
N-\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{Dt}},$ (28)
$\displaystyle{\sigma^{\text{diff}}_{\text{an}}(L,t\gg
L^{2}/D)}^{2}\approx\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{Dt}}.$ (29)

Figure 3: The ratio of the variance of $Q$ in the annealed and quenched
settings plotted against rescaled time $t/(L^{2}/D)$ for two different set-
ups: (a) in the presence of a reflecting wall (the solid line) and (b) when
both sides are open (the dashed line). The variance in the annealed setting is
given by Eq. (II.1) and the variance in the quenched setting has been
numerically evaluated using Eq. (21), for $L=10,10^{2},10^{3}$ and $D=1$. When
$t\ll L^{2}/D$, the ratio is $\sqrt{2}$, and when $t\gg L^{2}/D,$ the
variances in the annealed and quenched settings become exactly equal and the
ratio becomes unity.

In deriving the asymptotic time limits we have used the following properties
of the complementary error function

$\displaystyle\begin{array}[]{l}\text{erfc}(z)\approx\left\\{\begin{array}[]{lll}1-\frac{2z}{\sqrt{\pi}},&\text{when
}z\to 0,\\\ \frac{e^{-z^{2}}}{\sqrt{\pi}z},&\text{when
}z\to\infty.\end{array}\right.\end{array}$ (33)

At very short times, particles that are close to the boundary at $x=0$ can
only get out of the region $[-L,0]$. Meanwhile, particles that are situated
near the boundary at $x=-L$ do not get sufficient time to escape through the
origin. In effect, the finite size of the system does not come into the
picture at very short times. Consequently, the results obtained match with
those obtained for the case of an infinite system. Conversely, as time
progresses, particles in the bulk or near the reflecting wall at $x=-L$ have
sufficient time to exit the box through the origin. It is expected that
eventually, all $N$ particles will exit this region, resulting in a mean
current of $N$. At these large time scales, with all particles leaving the
box, the variance tends to approach zero.

Figure 4: Behavior of the (a) mean and the (b) variance of $Q$ as a function
of time when both the boundaries at $x=0,-L$ are open for a diffusive system.
The mean in the annealed and quenched settings are the same. The variance in
the quenched setting (solid curves) differs from the annealed (dashed curves)
by a factor of $\sqrt{2}$ at times $t\ll L^{2}/D$, however, they become equal
at times $t\gg L^{2}/D$. The parameter values used are $\rho=1,D=1$. The stars
and circles represent the results obtained through numerical simulations of
the microscopic model for the quenched and annealed settings respectively.

####  III.1.2 Quenched setting

From Eq. (20) we see that the mean in the quenched setting is the same as the
annealed setting. Therefore we obtain

$\displaystyle\mu^{\text{diff}}_{\text{qu}}(L,t)=\mu^{\text{diff}}_{\text{an}}(L,t),$
(34)

with the limiting behaviors given in Eq. (26) and Eq. (28).

Calculating the variance in the quenched setting is challenging because the
integral in Eq. (21) cannot be explicitly computed. However, we can determine
the asymptotic behaviors of the variance of $Q$ using simple arguments. At
short times ($t\ll L^{2}/D$), the system does not experience the effects of
finite size and the results obtained are similar to those obtained for
infinite systems (as also seen for the annealed case). We thus take the limit
$L\to\infty$ in Eq. (23) to obtain

$\displaystyle U(x_{i},t)$ $\displaystyle\xrightarrow[L\rightarrow\infty]{}$
$\displaystyle\frac{1}{2}\left(1+\text{erf}\left(\frac{x_{i}}{2\sqrt{Dt}}\right)\right).$
(35)

Using this result, the integration in Eq. (21) can be easily performed to
obtain

$\displaystyle{\sigma^{\text{diff}}_{\text{qu}}(L,t\ll
L^{2}/D)}^{2}\approx\frac{\rho\sqrt{Dt}}{\sqrt{2\pi}}.$ (36)

Similarly, at very large times $t\gg L^{2}/D$, one can take the limit $L\to 0$
to obtain

$\displaystyle U(x_{i},t)$ $\displaystyle\xrightarrow[L\rightarrow 0]{}$
$\displaystyle 1-\frac{Le^{-\frac{x_{i}^{2}}{4Dt}}}{\sqrt{\pi}\sqrt{Dt}}.$
(37)

We next compute the integral in Eq. (21) in this limit yielding

$\displaystyle{\sigma^{\text{diff}}_{\text{qu}}(L,t\gg
L^{2}/D)}^{2}\approx\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{Dt}}.$ (38)

Note that, at time scales where finite size effects are not present ($t\ll
L^{2}/D$), the variance for the quenched setting given in Eq. (36) is
suppressed by a factor of $\sqrt{2}$ compared to the annealed setting provided
in Eq. (27). However at time scales $t\gg L^{2}/D$, the finite size effects
are dominant and the variance in the quenched and annealed settings become
exactly equal to each other. Fig. 2 shows the behavior of the mean and the
variance as a function of time for both annealed and quenched settings. The
mean is given by Eq. (24) for both annealed and quenched settings. The
variance is given by Eq. (II.1) in the annealed setting and by Eq. (21) in the
quenched setting. Fig. 3 displays the plot of the ratio of the variance in the
annealed and quenched settings as a function of the rescaled time
$t/(L^{2}/D)$ for different system sizes $L=10,10^{2}$ and $10^{3}$. All the
curves for different system sizes collapse into a single curve. At time scales
$t\ll L^{2}/D$, finite size effects can be neglected and the ratio is close to
$\sqrt{2}$. However at large time scales $t\gg L^{2}/D$, the finite size
effects become prominent. Consequently, the annealed and the quenched averages
become the same, and the ratio becomes one.

###  III.2 Finite size interval

We next focus on the case where there is no reflecting wall in the system so
that particles can escape through either of the boundaries at $x=0$ or $x=-L$.
The diffusion propagator in this case is given by

$\displaystyle G(x,t|x_{i})=\frac{1}{\sqrt{4\pi
Dt}}e^{-\frac{(x-x_{i})^{2}}{4Dt}}.$ (39)

Substituting this expression in Eq. (7), we obtain

$\displaystyle
U(x_{i},t)=\frac{1}{2}\left(1+\text{erf}\left(\frac{x_{i}}{2\sqrt{Dt}}\right)+\text{erfc}\left(\frac{L+x_{i}}{2\sqrt{Dt}}\right)\right).$
(40)

We next focus on annealed and quenched settings separately.

####  III.2.1 Annealed setting

We substitute Eq. (40) in the expression for mean provided in Eq. (14). This
yields the exact expression for the mean in the annealed setting as

$\displaystyle\mu^{\text{diff}}_{\text{an}}(L,t)$
$\displaystyle=\underbrace{\frac{2\rho\sqrt{Dt}}{\sqrt{\pi}}}_{\text{infinite
size limit}}+\underbrace{\rho
L\left(\text{erfc}\left(\frac{L}{2\sqrt{Dt}}\right)-\frac{2\sqrt{Dt}}{\sqrt{\pi}L}e^{-\frac{L^{2}}{4Dt}}\right)}_{\text{finite
size correction}}.$ (41)

In the asymptotic limit, we obtain the simplified expressions

$\displaystyle\begin{array}[]{l}\mu^{\text{diff}}_{\text{an}}(L,t)\approx\left\\{\begin{array}[]{lll}\frac{2\rho\sqrt{Dt}}{\sqrt{\pi}},&t\ll
L^{2}/D,\\\ N-\frac{\rho L^{2}}{2\sqrt{\pi}\sqrt{Dt}},&t\gg
L^{2}/D.\end{array}\right.\end{array}$ (45)

The expression for the variance of $Q$ can now be exactly computed using Eq.
(41) and Eq. (25). Since this expression is quite long, we do not quote it
here. Fig. 4 shows the behavior of the mean and the variance as a function of
time obtained from Eqs. (41) and (25) for different system sizes keeping the
density $\rho=1$ fixed. In the asymptotic limits, we obtain the simple
expressions,

$\displaystyle\begin{array}[]{l}\sigma^{\text{diff}}_{\text{an}}(L,t)^{2}\approx\left\\{\begin{array}[]{lll}\frac{2\rho\sqrt{Dt}}{\sqrt{\pi}},&t\ll
L^{2}/D,\\\ \frac{\rho L^{2}}{2\sqrt{\pi}\sqrt{Dt}},&t\gg
L^{2}/D.\end{array}\right.\end{array}$ (49)

At short times, the variance is larger by a factor of $2$ as compared to the
case with a single reflecting boundary in the annealed setting. However, at
large times, the variance is lesser by a factor of $2$ as compared to the
previous case.

####  III.2.2 Quenched setting

The expression for the mean in the quenched setting is the same as the
annealed setting and is given in Eq. (41). It is difficult to compute the
exact closed-form expression for the variance in the quenched setting using
the expression for $U(x_{i},t)$ provided in Eq. (40). Nevertheless, it is
possible to perform a careful asymptotic analysis in the Laplace space
(details given Appendix A) which yields

$\displaystyle\begin{array}[]{l}\sigma^{\text{diff}}_{\text{qu}}(L,t)^{2}\approx\left\\{\begin{array}[]{ll}\frac{\sqrt{2}\rho\sqrt{Dt}}{\sqrt{\pi}},&t\ll\frac{L^{2}}{D},\\\
\frac{\rho
L^{2}}{2\sqrt{\pi}\sqrt{Dt}},&t\gg\frac{L^{2}}{D}.\end{array}\right.\end{array}$
(53)

Similar to the case with a reflecting wall, the variance in the annealed and
quenched settings are distinct at short times ($t\ll L^{2}/D$) and become
exactly equal to each other at times $t\gg L^{2}/D$. Fig. 4 shows the behavior
of the mean and the variance as a function of time for both annealed and
quenched settings. The mean is given by Eq. (41) for both annealed and
quenched settings. The variance is given by Eq. (II.1) in the annealed setting
and by Eq. (21) in the quenched setting. In Fig. 3, we display a plot of the
exact ratio of the annealed to quenched variance. As before, even in the
quenched setting, the variance of $Q$ is larger by a factor of $2$ compared to
the case with a single reflecting boundary at short times. However, at large
times, the variance is lower by a factor of $2$. This demonstrates how
boundary conditions can influence the transport properties of stochastic
systems over time.

##  IV Run-and-tumble particles

| $t\rightarrow 0$ | $t\rightarrow\infty$ |
---|---|---|---
RTP reflecting boundary | $\frac{1}{2}\rho vt$ | $\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{D_{\text{eff}}t}}$ | $\sigma^{\text{rtp}}_{\text{an}}(L,t)^{2}$
$\frac{1}{4}\rho vt$ | $\ \frac{\rho L^{2}}{\sqrt{\pi}\sqrt{D_{\text{eff}}t}}$ | $\sigma^{\text{rtp}}_{\text{qu}}(L,t)^{2}$
RTP finite interval | $\rho vt$ | $\frac{1}{2}\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{D_{\text{eff}}t}}$ | $\sigma^{\text{rtp}}_{\text{an}}(L,t)^{2}$
$\frac{1}{2}\rho vt$ | $\frac{1}{2}\frac{\rho L^{2}}{\sqrt{\pi}\sqrt{D_{\text{eff}}t}}$ | $\sigma^{\text{rtp}}_{\text{qu}}(L,t)^{2}$
Table 2: Asymptotic behavior of current fluctuations for run and tumble
particle motion in the annealed and quenched settings. Figure 5: Behavior of
the (a) mean and the (b) variance in both annealed and quenched settings for
RTP in the presence of a reflecting wall at $x=-L$. The mean in the annealed
and quenched settings are the same. The variance in the quenched setting
(solid curves) differs from the annealed (dashed curves) by a factor of ${2}$
at times $t\ll L^{2}/D_{\text{eff}}$, however, they become equal at times
$t\gg L^{2}/D_{\text{eff}}$. The parameter values used are
$v=\sqrt{0.2},\gamma=0.1$ to have $D_{\text{eff}}=v^{2}/2\gamma=1$. The
circles with crosses represent the results obtained through numerical
simulations of the microscopic model and the dashed curves represent the
results obtained through numerical Laplace inversion of Eq. (58) along with
Eq. (II.1) for the annealed setting. The variance in quenched setting (solid
curves) has been obtained through microscopic simulations.

In this section, we focus on a system of non-interacting run-and-tumble
particles in one dimension. We analyze the statistics of the integrated
current $Q$ in a one-dimensional system of non-interacting run-and-tumble
particles (RTPs). The dynamics of an RTP consist of run and tumble phases.
During the run phase, the particle moves with a constant velocity $v$ and
during the tumble phase, the particle instantaneously changes its direction of
velocity. The Langevin equation governing the motion of an RTP can be written
as

$\displaystyle\frac{dx}{dt}=v\sigma(t),$ (54)

where $\sigma(t)=\pm 1$ is a dichotomous noise and it switches between the two
values after a random time $\tau$ which is distributed according to an
exponential distribution $p(\tau)=\gamma e^{-\gamma\tau}$. The asymptotic
behaviors of current fluctuations for non-interacting RTPs in various settings
are summarized in Table 2.

As for the diffusive case, we first focus on the situation with a reflecting
wall at $x=-L$.

###  IV.1 One reflecting wall

The boundary conditions for an RTP in the presence of a reflecting wall have
to be defined carefully. After a reflection from the wall, each particle has
two possibilities for its orientation, (i) it continues to move in the same
direction i.e. towards the wall or (ii) it changes the orientation after
reflection and starts moving away from the wall. In this paper, we consider
the latter case where velocity is reversed after each reflection. This
prevents the accumulation of particles near the wall Angelani (2023). The
Green’s function for RTP has a simple form in the Laplace space. The Laplace
transform of a function $f(t)$ is defined as
$\widetilde{f}(s)=\int_{0}^{t}dtf(t)e^{-st}$. The propagator of RTP can be
computed using the image method as Feller (1991)

$\displaystyle\widetilde{G}(x,s|x_{i})$
$\displaystyle=\frac{\sqrt{s(s+2\gamma)}}{2vs}\left(e^{-\frac{\sqrt{s(s+2\gamma)}}{v}|x-x_{i}|}+e^{-\frac{\sqrt{s(s+2\gamma)}}{v}|x+2L+x_{i}|}\right).$
(55)

Using Eq. (6), we next compute the Laplace transform of $U(x_{i},t)$ as,

$\displaystyle\widetilde{U}(x_{i},s)=\frac{e^{\frac{x_{i}\sqrt{s(2\gamma+s)}}{v}}}{2s}+\frac{e^{-\frac{\left(2L+x_{i}\right)\sqrt{s(2\gamma+s)}}{v}}}{2s}.$
(56)

In the subsequent sections, we focus on the annealed and quenched settings
separately.

####  IV.1.1 Annealed setting

We take a Laplace transform of the expression for the mean provided in Eq.
(14) to obtain

$\displaystyle\widetilde{\mu}_{\text{an}}(L,s)=\rho\int_{-L}^{0}\widetilde{U}(z,s)dz.$
(57)

Substituting Eq. (56) in the above equation, we obtain the expression for the
mean of $Q$ in Laplace space as

$\displaystyle{\widetilde{\mu}}^{\text{rtp}}_{\text{an}}(L,s)=\underbrace{\frac{\rho
v}{2s\sqrt{s(2\gamma+s)}}}_{\text{infinite size limit}}-\underbrace{\frac{\rho
ve^{-\frac{2L\sqrt{s(2\gamma+s)}}{v}}}{2s\sqrt{s(2\gamma+s)}}}_{\text{finite
size correction}}.$ (58)

The first term in the above expression represents the infinite size
($L\to\infty$) limit while the second term is a finite size correction. Since
the exact inversion of the above expression is difficult, we focus on the
asymptotic behaviors taking different limits of $s$ as explained below.

For RTPs, there are two important time scales, (i) one timescale is associated
with the mean run time $t=1/\gamma$ between consecutive tumbles and (ii) the
other timescale is associated with the finite size of the system
$t=L^{2}/D_{\text{eff}}$ where $D_{\text{eff}}=v^{2}/2\gamma$ is the effective
diffusion constant for an RTP in one dimension. At very large times, $t\gg
1/\gamma$ the statistical properties of an RTP become similar to that of a
Brownian particle with an effective diffusion constant $D_{\text{eff}}$. In
this paper, we consider the case where $L^{2}/D_{\text{eff}}\gg 1/\gamma$.
Thus the limit $s\to\infty$ corresponds to timescales $t\ll
L^{2}/D_{\text{eff}}$. In this limit, we observe that the second term in Eq.
(58) is exponentially suppressed as compared to the first term and we obtain

$\displaystyle{\widetilde{\mu}}^{\text{rtp}}_{\text{an}}(L,s)$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$ $\displaystyle\frac{\rho
v}{2s\sqrt{s(2\gamma+s)}},$ (59)

which upon inversion yields

${\mu}^{\text{rtp}}_{\text{an}}(L,t\ll L^{2}/D_{\text{eff}})=\frac{\rho
vt}{2}e^{-\gamma t}(I_{0}(t\gamma)+I_{1}(t\gamma)),$ (60)

where $I_{0}(z)$ and $I_{1}(z)$ are the modified Bessel functions of the first
kind. The asymptotic behaviors of the modified Bessel function of the first
kind (and order $\nu$) are given as

$\displaystyle\begin{array}[]{l}I_{\nu}(z)\approx\left\\{\begin{array}[]{lll}z^{\nu}\left(\frac{2^{-\nu}}{\Gamma(\nu+1)}+\frac{2^{-\nu-2}z^{2}}{(\nu+1)\Gamma(\nu+1)}\right),&\text{when
}z\to 0,\\\ \frac{e^{z}}{\sqrt{2\pi}\sqrt{z}},&\text{when
}z\to\infty.\end{array}\right.\end{array}$ (64)

Substituting these expressions in Eq. (60), one obtains the limiting behaviors
of the mean of $Q$ as

$\displaystyle\begin{array}[]{l}{\mu}^{\text{rtp}}_{\text{an}}(L,t)\approx\left\\{\begin{array}[]{lll}\frac{1}{2}\rho
vt,&t\ll 1/\gamma,\\\
\frac{\rho\sqrt{D_{\text{eff}}t}}{\sqrt{\pi}},&\frac{1}{\gamma}\ll
t\ll\frac{L^{2}}{D_{\text{eff}}}.\end{array}\right.\end{array}$ (68)

To obtain the large time ($t\gg L^{2}/D_{\text{eff}}$) behavior, we take the
$s\to 0$ limit of the expression provided in Eq. (58) yielding

$\displaystyle{\widetilde{\mu}}^{\text{rtp}}_{\text{an}}(L,s)$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$
$\displaystyle\frac{N}{s}-\frac{\rho L^{2}}{\sqrt{D_{\text{eff}}}\sqrt{s}},$
(69)

which upon inversion yields

${\mu}^{\text{rtp}}_{\text{an}}(L,t\gg L^{2}/D_{\text{eff}})=N-\frac{\rho
L^{2}}{\sqrt{\pi}\sqrt{D_{\text{eff}}t}}.$ (70)

The asymptotic behavior of the variance can be found by substituting the
asymptotic expressions for the mean provided in Eqs. (60) and (70) directly in
Eq. (II.1). This yields

$\displaystyle\begin{array}[]{l}\sigma^{\text{rtp}}_{\text{an}}(L,t)^{2}\approx\left\\{\begin{array}[]{lll}\frac{1}{2}\rho
vt,&t\ll 1/\gamma,\\\
\frac{\rho\sqrt{D_{\text{eff}}t}}{\sqrt{\pi}},&\frac{1}{\gamma}\ll
t\ll\frac{L^{2}}{D_{\text{eff}}},\\\ \frac{\rho
L^{2}}{\sqrt{\pi}\sqrt{D_{\text{eff}}t}}&t\gg
L^{2}/D_{\text{eff}}.\end{array}\right.\end{array}$ (75)

At time scales $t\gg 1/\gamma$, the mean and the variance behave similar to
that of the diffusive case as in Eq. (26)-(29) with the diffusion constant $D$
replaced by $D_{\text{eff}}$. Fig. 5 shows the behavior of the mean and the
variance as a function of time obtained through numerical inversion of Eq.
(58) and using these results in Eq. (II.1).

Figure 6: Ratio of the variances of $Q$ in the annealed and quenched settings
for RTPs in the presence of a reflecting wall. For small timescales ($t\ll
1/\gamma$), the ratio is $2$. With time it starts decreasing and at
intermediate time scales $1/\gamma\ll t\ll L^{2}/D_{\text{eff}}$, the ratio
saturates to the value $\sqrt{2}$. This saturation is more evident for system
size $L=10^{3}$ as the intermediate region is broad here. At time scale $t\gg
L^{2}/D_{\text{eff}}$ all the curves merge and eventually saturate to unity.
While computing the ratio, the numerator has been obtained through numerical
Laplace inversion of Eq. (58) along with Eq. (II.1) while the denominator has
been estimated using microscopic simulations.

####  IV.1.2 Quenched setting

Similar to the case of diffusion, the mean in the quenched setting is the same
as that in the annealed setting. That is,

$\displaystyle\mu^{\text{rtp}}_{\text{qu}}(L,t)=\mu^{\text{rtp}}_{\text{an}}(L,t).$
(76)

The exact asymptotic behaviors of the mean are provided in Eqs. (60) and (70).
The asymptotic limits for the variance can be computed using similar arguments
we applied for the diffusive case. At very short times $t\ll
L^{2}/D_{\text{eff}}$, we take the limit $L\to\infty$ in Eq. (56) and follow a
similar calculation as we did for the diffusive case to obtain

$\displaystyle\sigma^{\text{rtp}}_{\text{qu}}(L,t\ll
L^{2}/D_{\text{eff}})^{2}=\frac{\rho v}{8}te^{-2\gamma t}\big{[}(4+\pi
L_{0}(2\gamma t))I_{1}(2\gamma t)$ $\displaystyle\hskip 85.35826pt+(2-\pi
L_{1}(2\gamma t))I_{0}(2\gamma t)\big{]},$ (77)

where $L_{0}(z),L_{1}(z)$ are the modified Struve functions. Further depending
on the time scale $1/\gamma$, we obtain the limiting behaviors for the
variance of $Q$

$\displaystyle\begin{array}[]{l}\sigma^{\text{rtp}}_{\text{qu}}(L,t)^{2}\approx\left\\{\begin{array}[]{lll}\frac{1}{4}\rho
vt,&t\ll 1/\gamma,\\\
\frac{\rho\sqrt{D_{\text{eff}}t}}{\sqrt{2\pi}},&\frac{1}{\gamma}\ll
t\ll\frac{L^{2}}{D_{\text{eff}}}.\end{array}\right.\end{array}$ (81)

Here, we have used the following asymptotic behaviors of the Struve functions

$\displaystyle\begin{array}[]{l}L_{\nu}(z)\approx\left\\{\begin{array}[]{lll}z^{\nu}\left(\frac{2^{-\nu}z}{\sqrt{\pi}\Gamma\left(\nu+\frac{3}{2}\right)}+\frac{2^{-\nu-1}z^{3}}{3\sqrt{\pi}\Gamma\left(\nu+\frac{5}{2}\right)}\right),&\text{when
}z\to 0,\\\ \frac{e^{z}}{\sqrt{2\pi}\sqrt{z}},&\text{when
}z\to\infty.\end{array}\right.\end{array}$ (85)

The large-time asymptotic behavior of the variance can be computed by taking
the limit $L\to\infty$ in Eq. (56) and performing a similar calculation as for
the diffusive case, or it can be derived directly from the fact that at this
timescale, the statistical properties of an RTP is similar to that of a
Brownian particle with a modified diffusion constant $D=D_{\text{eff}}$. We
thus obtain

$\displaystyle\sigma^{\text{rtp}}_{\text{qu}}(t\gg
L^{2}/D_{\text{eff}})^{2}=\frac{\rho
L^{2}}{\sqrt{\pi}\sqrt{D_{\text{eff}}t}},$ (86)

Figure 7: Behavior of the (a) mean and the (b) variance in both annealed and
quenched settings for run-and-tumble particles when both sides at $x=0,-L$ are
open. The mean in the annealed and quenched settings are the same. The
variance in the quenched setting (solid curves) differs from the annealed
(dashed curves) by a factor of ${2}$ at times $t\ll L^{2}/D_{\text{eff}}$,
however, they become equal at times $t\gg L^{2}/D_{\text{eff}}$. The parameter
values used are $v=\sqrt{0.2},\gamma=0.1$ so that
$D_{\text{eff}}=v^{2}/2\gamma=1$. The stars represent the results obtained
through numerical simulations of the microscopic model and the dashed curves
represent the results obtained through numerical Laplace inversion of Eq. (89)
along with Eq. (II.1) for the annealed setting. The variance in quenched
setting (solid curves) has been entirely obtained through microscopic
simulations.

which is the same as the large time behavior of the variance in the annealed
setting given in Eq. (75). Fig. 5 displays the behavior of the mean and the
variance as a function of time for both annealed and quenched settings. The
mean is given by numerical inversion of Eq. (58) for both annealed and
quenched settings. The variance is given by Eq. (II.1) in the annealed setting
and by Eq. (21) in the quenched setting. Fig. 6 displays the plot of the ratio
of the variance in the annealed and quenched settings as a function of the
rescaled time $t/(L^{2}/D_{\text{eff}})$ for different system sizes
$L=10,10^{2}$ and $10^{3}$. Unlike the diffusive case, the curves do not
collapse into a single curve as RTPs have different timescales involved in
addition to the diffusion timescale. At time scales $t\ll 1/\gamma$, finite
size effects can be neglected and the ratio is close to ${2}$. We see that at
intermediate time scales $1/\gamma\ll t\ll L^{2}/D_{\text{eff}}$, the ratio
saturates close to the value $\sqrt{2}$. However at large time scales $t\gg
L^{2}/D_{\text{eff}}$, the finite size effects become prominent. Consequently,
the annealed and the quenched averages become the same, and the ratio becomes
one.

###  IV.2 Finite size interval

We next focus on the case where the particles can escape either through the
boundary at $x=0$ or $x=-L$. The propagator for an RTP in the Laplace space is
given as Banerjee  _et al._ (2020)

$\displaystyle\widetilde{G}(x,s|x_{i})=\frac{\sqrt{s(s+2\gamma)}}{2vs}e^{-\frac{\sqrt{s(s+2\gamma)}}{v}|x-x_{i}|}.$
(87)

Substituting this expression in Eq. (7), we obtain

$\displaystyle\widetilde{U}(x_{i},s)=\frac{e^{\frac{x_{i}\sqrt{s(2\gamma+s)}}{v}}}{s}+\frac{e^{\frac{-(L+x_{i})\sqrt{s(2\gamma+s)}}{v}}}{s}.$
(88)

We next focus on the cases of annealed and quenched averages separately.

####  IV.2.1 Annealed setting

We first focus on the annealed setting where the positions of the particles
are allowed to fluctuate initially. Substituting Eq. (88) in Eq. (57) we
obtain the expression for the mean in Laplace space as

$\displaystyle{\widetilde{\mu}}^{\text{rtp}}_{\text{an}}(L,s)=\underbrace{\frac{\rho
v}{s\sqrt{s(2\gamma+s)}}}_{\text{infinite size limit}}-\underbrace{\frac{\rho
ve^{-\frac{L\sqrt{s(2\gamma+s)}}{v}}}{s\sqrt{s(2\gamma+s)}}}_{\text{finite
size correction}}.$ (89)

Using this expression, the asymptotic behaviors of the mean and the variance
in real-time can be computed as before. For the mean, we obtain

$\displaystyle\begin{array}[]{l}\mu^{\text{rtp}}_{\text{an}}(L,t)\approx\left\\{\begin{array}[]{lll}\rho
vt,&t\ll 1/\gamma,\\\
\frac{2\rho\sqrt{D_{\text{eff}}t}}{\sqrt{\pi}},&\frac{1}{\gamma}\ll
t\ll\frac{L^{2}}{D_{\text{eff}}},\\\ N-\frac{\rho
L^{2}}{2\sqrt{\pi}\sqrt{D_{\text{eff}}t}}&t\gg
L^{2}/D_{\text{eff}}.\end{array}\right.\end{array}$ (94)

and for the variance, we obtain

$\displaystyle\begin{array}[]{l}\sigma^{\text{rtp}}_{\text{an}}(L,t)^{2}\approx\left\\{\begin{array}[]{lll}\rho
vt,&t\ll 1/\gamma,\\\
\frac{2\rho\sqrt{D_{\text{eff}}t}}{\sqrt{\pi}},&\frac{1}{\gamma}\ll
t\ll\frac{L^{2}}{D_{\text{eff}}},\\\ \frac{\rho
L^{2}}{2\sqrt{\pi}\sqrt{D_{\text{eff}}t}}&t\gg
L^{2}/D_{\text{eff}}.\end{array}\right.\end{array}$ (99)

Figure 8: Ratio of the variance of $Q$ in the annealed and quenched settings
for RTPs in a finite-sized interval with open boundaries. The variance in the
annealed setting has been obtained through numerical Laplace inversion of Eq.
(89) along with Eq. (II.1). The variance in the quenched setting has been
obtained through numerical simulations of the microscopic model. Unlike the
Brownian case, the curves for different system sizes $L$ do not collapse into
a single curve at short and intermediate times. However, when $t\gg
L^{2}/D_{\text{eff}}$, all the curves merge and saturate to the value $1$.

####  IV.2.2 Quenched setting

The mean in the quenched setting is the same as the mean in the annealed
setting and is given in Eq. (89). Since it is difficult to find the exact
expression of the variance in the quenched setting in closed form, we focus on
the asymptotic behaviors. A careful asymptotic analysis in the Laplace space
(details given in Appendix B) yields

$\displaystyle\begin{array}[]{l}\sigma^{\text{rtp}}_{\text{qu}}(L,t)^{2}\approx\left\\{\begin{array}[]{ll}\frac{1}{2}{\rho
vt},&t\ll\frac{1}{\gamma},\\\ \frac{\rho L^{2}}{2\sqrt{\pi
D_{\text{eff}}t}},&t\gg\frac{L^{2}}{D_{\text{eff}}}.\end{array}\right.\end{array}$
(103)

Fig. 7 displays the behavior of the mean and the variance as a function of
time for both annealed and quenched settings. The mean is given by numerical
inversion of Eq. (89) for both annealed and quenched settings. The variance is
given by Eq. (II.1) in the annealed setting and by Eq. (21) in the quenched
setting. In Fig. 8, we display a plot of the ratio of the variance in the
annealed and quenched settings as a function of the rescaled time
$t/(L^{2}/D_{\text{eff}})$ for different system sizes $L=10,10^{2}$ and
$10^{3}$. Similar to the set-up with a reflecting wall, the curves do not
collapse into a single curve as RTPs have different timescales involved in
addition to the diffusion timescale. At time scales $t\ll 1/\gamma$, the ratio
is close to ${2}$. At intermediate time scales $1/\gamma\ll t\ll
L^{2}/D_{\text{eff}}$, the ratio is close to the value $\sqrt{2}$. However at
large time scales $t\gg L^{2}/D_{\text{eff}}$, the finite size effects become
prominent and the ratio saturates to $1$. Compared to the reflecting case, the
variance is larger by a factor of $2$ at times $t\ll L^{2}/D_{\text{eff}}$.
However, at large times, the variance gets suppressed by a factor of $2$ as
compared to the reflecting case. This is exactly the same behavior we observed
for the system of diffusing particles. This demonstrates how boundary
conditions can influence the transport properties of stochastic systems over
time. A more intricate understanding of these various factors would require a
detailed study of current fluctuations in different system geometries across
various spatial dimensions.

##  V Discussion and outlook

In this paper, we have studied the fluctuations in the number of particles
exiting the boundaries of a finite-sized one-dimensional box. We investigated
specific examples of passive as well as active systems; namely non-interacting
diffusive and run-and-tumble particles respectively. We demonstrated how
various initial conditions, system geometry, and boundary conditions affect
the transport properties of these systems over time.

For the system of diffusive particles, we showed that the ratio of
fluctuations in the annealed and quenched settings changes from a value of
$\sqrt{2}$ at short times ($t\ll{L^{2}}/{D}$) to $1$ at large times
($t\gg{L^{2}}/{D}$). While for run-and-tumble particles, this ratio changes
from a value of $2$ at short times ($t\ll 1/\gamma$) to $1$ at large times
($t\gg{L^{2}}/{D_{\text{eff}}}$) through an intermediate saturation regime
where the ratio takes up the value $\sqrt{2}$. This intermediate saturation
regime corresponds to the time scale ${1}/{\gamma}\ll
t\ll{L^{2}}/{D_{\text{eff}}}$ at which the dynamics of run-and-tumble
particles becomes effectively diffusive. The timescale at which the ratio
saturates to $1$ is the diffusive timescale which goes as $t\approx L^{2}/D$
for diffusive systems and $t\approx L^{2}/D_{\text{eff}}$ for active systems,
where $D_{\text{eff}}$ is the effective diffusion constant for run-and-tumble
particles in one dimension.

Interestingly, we demonstrated that the boundary conditions also play a
crucial role in determining the dynamic behavior of current fluctuations. The
setup with two open boundaries displays larger fluctuations by a factor of $2$
at short times compared to the setup with only one open boundary. However, the
former setup exhibits lesser fluctuations by the same factor of $2$ at large
times. This can be qualitatively understood as follows: At short times, the
particles in the setup with two open boundaries have two escape routes,
thereby increasing the fluctuations by a factor of $2$. However, at large
times, the fluctuations are predominantly determined by single-particle
events, and the probability that an unbiased single particle escapes through
one of the boundaries is $1/2$. Consequently, this reduces the fluctuations by
a factor of $2$.

Our exact analytical results reveal how slight variations in the initial
conditions and system geometry can affect the dynamic behavior of current
fluctuations in stochastic systems. Our study is a first step towards
understanding the effusion of particles through finite-sized regions across
different spatial dimensions, which can be investigated using similar methods
discussed in this paper. The study of particle effusion has applications in
designing membranes and porous materials, where controlled diffusion or
leakage plays a pivotal role, as well as in the transportation of ions or
molecules across cellular membranes Kärger (2008); Post (1991); Marbach
(2021). Naturally, a careful analytical analysis of the problem of effusion
through different confining volumes in higher dimensions will help to
understand the underlying factors governing current fluctuations.

It would be intriguing to investigate whether a universal behavior of current
fluctuations exists, one that depends on the system’s geometry, determined by
factors such as the number of reflecting boundaries and available escape
routes. Testing the results of this paper using coarse-grained field theories
such as macroscopic fluctuation theory (MFT) Bertini  _et al._ (2005, 2006,
2007, 2009); Derrida (2007); Agranov  _et al._ (2023); Jose  _et al._ (2023c)
is also a worthwhile future investigation. Finally, it would also be
interesting to extend the computations presented in this paper to interacting
systems such as the symmetric simple exclusion process (SSEP) Levitt (1973);
Arratia (1983); Derrida  _et al._ (2004); Krapivsky  _et al._ (2015); Dandekar
and Mallick (2022) and the ABC model Clincy  _et al._ (2003); Bodineau and
Derrida (2011).

##  VI Acknowledgement

The numerical calculations reported in this work were carried out on the
Nandadevi cluster, which is maintained and supported by The Institute of
Mathematical Science’s High-Performance Computing Center. KR acknowledges
funding through intramural funds at TIFR Hyderabad and the SERB-MATRICS grant
MTR/2022/000966. AP gratefully acknowledges research support from the
Department of Science and Technology, India, SERB Start-up Research Grant
Number SRG/2022/000080, and the Department of Atomic Energy, India.

## Appendices

##  Appendix A  Current fluctuations for Brownian particles confined in a
finite interval - quenched setting

For a Brownian particle confined in a finite interval, the function
$U(x_{i},t)$ can be computed as in Eq. (40). In Laplace space, this expression
becomes

$\widetilde{U}(x_{i},s)=\frac{e^{-\sqrt{\frac{s}{D}}\left(L+x_{i}\right)}\left(1+e^{\sqrt{\frac{s}{D}}\left(L+2x_{i}\right)}\right)}{2s}.$
(104)

The variance in the quenched setting can be computed by taking a Laplace
transform of the expression in Eq. (21). This yields

$\displaystyle\widetilde{\sigma}_{\text{qu}}(L,s)^{2}=\widetilde{\mu}_{\text{qu}}(L,s)-\rho\int_{0}^{L}dz~{}\mathcal{L}(s)[U^{2}(z,t)].$
(105)

The expression for $\widetilde{\mu}_{\text{qu}}^{\text{diff}}(L,s)$ for
Brownian motion can be computed as

$\widetilde{\mu}_{\text{qu}}^{\text{diff}}(L,s)=\rho\int_{0}^{L}\widetilde{U}(z,s)dz=\rho\left(1-e^{-L\sqrt{\frac{s}{D}}}\right)\sqrt{\frac{D}{s^{3}}}.$
(106)

The integral in the second term of Eq. (105) can be computed using the
identity

$\displaystyle\int_{0}^{L}dz~{}\mathcal{L}(s)[U^{2}(z,t)]$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk\int_{0}^{L}dz~{}\widetilde{U}(z,s/2-ik)\widetilde{U}(z,s/2+ik).$
(107)

We provide a short derivation of this identity below.

$\displaystyle\int_{0}^{L}dz~{}\mathcal{L}(s)[U^{2}(z,t)]$
$\displaystyle=\int_{0}^{L}dz~{}\int_{0}^{\infty}dte^{-st/2}U(z,t)\times$
$\displaystyle\hskip
102.43008pt\int_{0}^{\infty}dt^{\prime}\delta(t-t^{\prime})e^{-st^{\prime}/2}U(z,t^{\prime})$
$\displaystyle=\int_{0}^{L}dz\int_{0}^{\infty}dte^{-st/2}U(z,t)\times$
$\displaystyle\hskip
39.83368pt\int_{0}^{\infty}dt^{\prime}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}dke^{ik(t-t^{\prime})}\right)e^{-st^{\prime}/2}U(z,t^{\prime})$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk\int_{0}^{L}dz\int_{0}^{\infty}dte^{-st/2}e^{ikt}U(z,t)\times$
$\displaystyle\hskip
113.81102pt\int_{0}^{\infty}dt^{\prime}e^{-st^{\prime}/2}e^{-ikt^{\prime}}U(z,t^{\prime})$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk\int_{0}^{L}dz~{}\widetilde{U}(z,s/2-ik)\widetilde{U}(z,s/2+ik).$
(108)

The integral over $z$ in Eq. (107) can be done explicitly. Since the resultant
expression is quite long, we do not quote it here. However, this expression
admits scaling forms in the limits, $s\xrightarrow{}0$ and
$s\xrightarrow{}\infty$. Let us denote

$\displaystyle\widetilde{F}(k,s)=\int_{0}^{L}dz~{}\widetilde{U}(z,s/2-ik)\widetilde{U}(z,s/2+ik).$
(109)

Using the substitution $u=k/s$, we obtain the following scaling forms for the
function $\widetilde{F}(k,s)$,

$\displaystyle\widetilde{F}(k,s)$ $\displaystyle\xrightarrow[s\rightarrow
0]{}$ $\displaystyle C_{1}(s)G_{1}(u),$ (110)
$\displaystyle\widetilde{F}(k,s)$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$ $\displaystyle
C_{2}(s)G_{2}(u),$ (111)

where

$C_{1}(s)=\frac{L}{s^{2}}-\frac{L^{2}}{\sqrt{D}s^{3/2}},$ (112)

and

$C_{2}(s)=\frac{\sqrt{D}}{s^{5/2}}.$ (113)

The expression in Eq. (107) can now be written as

$\displaystyle\int_{0}^{L}dz~{}\mathcal{L}(s)[U^{2}(z,t)]$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}dk~{}\widetilde{F}(k,s)$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$
$\displaystyle\frac{s~{}C_{1}(s)}{2\pi}\int_{-\infty}^{\infty}du~{}G_{1}(u).$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$
$\displaystyle\frac{s~{}C_{2}(s)}{2\pi}\int_{-\infty}^{\infty}du~{}G_{2}(u).$

It can be shown that the values of the integrals
$\int_{-\infty}^{\infty}du~{}G_{1}(u)$ and
$\int_{-\infty}^{\infty}du~{}G_{2}(u)$ appearing in the above expressions are
exactly equal to $2\pi$ and $\pi(2-\sqrt{2})$ respectively. Finally, we obtain

$\displaystyle\int_{0}^{L}dz~{}\mathcal{L}(s)[U^{2}(z,t)]$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$ $\displaystyle s~{}C_{1}(s)$
(115) $\displaystyle=$ $\displaystyle\frac{L}{s}-\frac{L^{2}}{\sqrt{sD}},$

and

$\displaystyle\int_{0}^{L}dz~{}\mathcal{L}(s)[U^{2}(z,t)]$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$
$\displaystyle{s}\frac{\sqrt{2}-1}{2}~{}C_{2}(s)$ (116) $\displaystyle=$
$\displaystyle\frac{2-\sqrt{2}}{2}\frac{\sqrt{D}}{s^{3/2}}.$

Using Eq. (106), it can also be shown that

$\displaystyle\widetilde{\mu}_{\text{qu}}^{\text{diff}}(L,s)$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$
$\displaystyle\rho\left(\frac{L}{s}-\frac{L^{2}}{2\sqrt{sD}}\right),$ (117)
$\displaystyle\widetilde{\mu}_{\text{qu}}^{\text{diff}}(L,s)$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$
$\displaystyle\frac{\rho\sqrt{D}}{s^{3/2}}.$ (118)

Combining results from Eq. (115)-(117) in Eq. (105), we obtain

$\displaystyle\widetilde{\sigma}_{\text{qu}}^{\text{diff}}(L,s)^{2}$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$
$\displaystyle\rho\left(\frac{L}{s}-\frac{L^{2}}{2\sqrt{sD}}\right)-\rho\left(\frac{L}{s}-\frac{L^{2}}{\sqrt{sD}}\right)$
(119) $\displaystyle=$ $\displaystyle\frac{\rho L^{2}}{2\sqrt{sD}},$

and

$\displaystyle\widetilde{\sigma}_{\text{qu}}^{\text{diff}}(L,s)^{2}$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$
$\displaystyle\rho\frac{\sqrt{D}}{s^{3/2}}-\rho\frac{2-\sqrt{2}}{2}\frac{\sqrt{D}}{s^{3/2}}$
(120) $\displaystyle=$ $\displaystyle\frac{\rho\sqrt{D}}{\sqrt{2}s^{3/2}},$

which on inversion yield

$\displaystyle\sigma^{\text{diff}}_{\text{qu}}(L,t)^{2}$
$\displaystyle\xrightarrow[t\rightarrow\infty]{}$ $\displaystyle\frac{\rho
L^{2}}{2\sqrt{\pi Dt}},$ (121)

and

$\displaystyle\sigma^{\text{diff}}_{\text{qu}}(L,t)^{2}$
$\displaystyle\xrightarrow[t\rightarrow 0]{}$
$\displaystyle\sqrt{2}\frac{\rho\sqrt{Dt}}{\sqrt{\pi}}.$ (122)

The expression in Eq. (121) is exactly equal to the large time asymptotic
expression for the variance in the annealed setting we obtained previously in
Eq. (49).

##  Appendix B  Current fluctuations for run and tumble particles confined in
a finite interval - quenched setting

For a run-and-tumble particle confined in a finite interval, the function
$\widetilde{U}(x_{i},s)$ can be computed as in Eq. (88). Similar to the case
for Brownian motion, the variance in the quenched setting can be computed
using the expression in Eq. (105). The exact expression for the mean in
Laplace space $\widetilde{\mu}_{\text{qu}}^{\text{rtp}}(L,s)$, is given in Eq.
(89). As for the Brownian case, the integral in the second term in Eq. (105)
can be computed using the identity given in Eq. (107). After performing the
integral over $z$, we obtain scaling forms of the resultant expression in the
asymptotic limits, $s\xrightarrow{}0$ and $s\xrightarrow{}\infty$. Using the
substitution $u=k/s$, we obtain the following scaling forms for the function
$\widetilde{F}(k,s)$ defined in Eq. (109),

$\displaystyle\widetilde{F}(k,s)$ $\displaystyle\xrightarrow[s\rightarrow
0]{}$ $\displaystyle C_{1}(s)G_{1}(u),$ (123)
$\displaystyle\widetilde{F}(k,s)$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$ $\displaystyle
C_{2}(s)G_{2}(u),$ (124)

where

$C_{1}(s)=\frac{L}{s^{2}}-\frac{L^{2}}{\sqrt{D_{\text{eff}}}s^{3/2}},$ (125)

and

$C_{2}(s)=\frac{v}{s^{3}}.$ (126)

Thus we obtain

$\displaystyle\int_{0}^{L}dz~{}\mathcal{L}(s)[U^{2}(z,t)]$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}dk~{}\widetilde{F}(k,s)$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$
$\displaystyle\frac{s~{}C_{1}(s)}{2\pi}\int_{-\infty}^{\infty}du~{}G_{1}(u).$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$
$\displaystyle\frac{s~{}C_{2}(s)}{2\pi}\int_{-\infty}^{\infty}du~{}G_{2}(u).$

It can be shown that the values of the integrals
$\int_{-\infty}^{\infty}du~{}G_{1}(u)$ and
$\int_{-\infty}^{\infty}du~{}G_{2}(u)$ appearing in the above expressions are
exactly equal to $2\pi$ and $\pi$ respectively. Finally, we obtain

$\displaystyle\int_{0}^{L}dz~{}\mathcal{L}(s)[U^{2}(z,t)]$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$ $\displaystyle s~{}C_{1}(s)$
(128) $\displaystyle=$
$\displaystyle\frac{L}{s}-\frac{L^{2}}{\sqrt{sD_{\text{eff}}}},$

and

$\displaystyle\int_{0}^{L}dz~{}\mathcal{L}(s)[U^{2}(z,t)]$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$
$\displaystyle\frac{s}{4}~{}C_{2}(s)$ (129) $\displaystyle=$
$\displaystyle\frac{v}{2s^{2}}.$

Using Eq. (89), it can also be shown that

$\displaystyle\widetilde{\mu}_{\text{qu}}^{\text{rtp}}(L,s)$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$
$\displaystyle\rho\left(\frac{L}{s}-\frac{L^{2}}{2\sqrt{sD_{\text{eff}}}}\right),$
(130) $\displaystyle\widetilde{\mu}_{\text{qu}}^{\text{rtp}}(L,s)$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$ $\displaystyle\frac{\rho
v}{s^{2}}.$ (131)

Combining results from Eq. (128)-(130) in Eq. (105), we obtain

$\displaystyle\widetilde{\sigma}_{\text{qu}}^{\text{rtp}}(L,s)^{2}$
$\displaystyle\xrightarrow[s\rightarrow 0]{}$
$\displaystyle\rho\left(\frac{L}{s}-\frac{L^{2}}{2\sqrt{sD_{\text{eff}}}}\right)-\rho\left(\frac{L}{s}-\frac{L^{2}}{\sqrt{sD_{\text{eff}}}}\right)$
(132) $\displaystyle=$ $\displaystyle\frac{\rho
L^{2}}{2\sqrt{sD_{\text{eff}}}},$

and

$\displaystyle\widetilde{\sigma}_{\text{qu}}^{\text{rtp}}(L,s)^{2}$
$\displaystyle\xrightarrow[s\rightarrow\infty]{}$
$\displaystyle\rho\frac{v}{s^{2}}-\rho\frac{v}{2s^{2}}$ (133) $\displaystyle=$
$\displaystyle\frac{\rho v}{2s^{2}},$

which on inversion yield

$\displaystyle\sigma_{\text{qu}}^{\text{rtp}}(L,t)^{2}$
$\displaystyle\xrightarrow[t\rightarrow\infty]{}$ $\displaystyle\frac{\rho
L^{2}}{2\sqrt{\pi D_{\text{eff}}t}},$ (134)

and

$\displaystyle\sigma^{\text{rtp}}_{\text{qu}}(L,t)^{2}$
$\displaystyle\xrightarrow[t\rightarrow 0]{}$ $\displaystyle\frac{\rho
vt}{2}.$ (135)

The expression in Eq. (134) is exactly equal to the large time asymptotic
expression for the variance in the annealed setting we obtained previously in
Eq. (99).

## References

  * Le Doussal and Machta (1989) P. Le Doussal and J. Machta, Physical Review B 40, 9427 (1989).
  * Van Beijeren (1991) H. Van Beijeren, Journal of Statistical Physics 63, 47 (1991).
  * Derrida  _et al._ (2004) B. Derrida, B. Douçot, and P.-E. Roche, Journal of Statistical Physics 115, 717 (2004).
  * Derrida and Gerschenfeld (2009a) B. Derrida and A. Gerschenfeld, Journal of Statistical Physics 137, 978 (2009a).
  * Ferrari (2010) P. L. Ferrari, Journal of Statistical Mechanics: Theory and Experiment 2010, P10016 (2010).
  * Lizana  _et al._ (2010) L. Lizana, T. Ambjörnsson, A. Taloni, E. Barkai, and M. A. Lomholt, Physical Review E 81, 051118 (2010).
  * Krapivsky and Meerson (2012) P. Krapivsky and B. Meerson, Physical Review E 86, 031106 (2012).
  * Bodineau and Derrida (2004) T. Bodineau and B. Derrida, Phys. Rev. Lett. 92, 180601 (2004).
  * Leibovich and Barkai (2013) N. Leibovich and E. Barkai, Physical Review E 88, 032107 (2013).
  * Krapivsky  _et al._ (2014) P. Krapivsky, K. Mallick, and T. Sadhu, Physical Review Letters 113, 078101 (2014).
  * Banerjee  _et al._ (2022) T. Banerjee, R. L. Jack, and M. E. Cates, Physical Review E 106, L062101 (2022).
  * Mallick  _et al._ (2022) K. Mallick, H. Moriya, and T. Sasamoto, Physical Review Letters 129, 040601 (2022).
  * Dandekar  _et al._ (2023) R. Dandekar, P. Krapivsky, and K. Mallick, Physical Review E 107, 044129 (2023).
  * Dean  _et al._ (2023) D. S. Dean, S. N. Majumdar, and G. Schehr, Journal of Statistical Mechanics: Theory and Experiment 2023, 063208 (2023).
  * Di Bello  _et al._ (2023) C. Di Bello, A. K. Hartmann, S. N. Majumdar, F. Mori, A. Rosso, and G. Schehr, Physical Review E 108, 014112 (2023).
  * Derrida and Gerschenfeld (2009b) B. Derrida and A. Gerschenfeld, Journal of Statistical Physics 136, 1 (2009b).
  * Cividini and Kundu (2017) J. Cividini and A. Kundu, Journal of Statistical Mechanics: Theory and Experiment 2017, 083203 (2017).
  * Banerjee  _et al._ (2020) T. Banerjee, S. N. Majumdar, A. Rosso, and G. Schehr, Physical Review E 101, 052101 (2020).
  * Jose  _et al._ (2023a) S. Jose, A. Rosso, and K. Ramola, Phys. Rev. E 108, L052601 (2023a).
  * Jose  _et al._ (2023b) S. Jose, A. Rosso, and K. Ramola, arXiv preprint arXiv:2310.16811 (2023b), 10.48550/arXiv.2310.16811.
  * Malakar  _et al._ (2018) K. Malakar, V. Jemseena, A. Kundu, K. V. Kumar, S. Sabhapandit, S. N. Majumdar, S. Redner, and A. Dhar, Journal of Statistical Mechanics: Theory and Experiment 2018, 043215 (2018).
  * Evans and Majumdar (2018) M. R. Evans and S. N. Majumdar, Journal of Physics A: Mathematical and Theoretical 51, 475003 (2018).
  * Mori  _et al._ (2020a) F. Mori, P. Le Doussal, S. N. Majumdar, and G. Schehr, Physical Review Letters 124, 090603 (2020a).
  * Mori  _et al._ (2020b) F. Mori, P. Le Doussal, S. N. Majumdar, and G. Schehr, Physical Review E 102, 042133 (2020b).
  * Singh and Kundu (2019) P. Singh and A. Kundu, Journal of Statistical Mechanics: Theory and Experiment 2019, 083205 (2019).
  * Angelani  _et al._ (2014) L. Angelani, R. Di Leonardo, and M. Paoluzzi, The European Physical Journal E 37, 1 (2014).
  * Martens  _et al._ (2012) K. Martens, L. Angelani, R. Di Leonardo, and L. Bocquet, The European Physical Journal E 35, 1 (2012).
  * Mori  _et al._ (2021) F. Mori, P. Le Doussal, S. N. Majumdar, and G. Schehr, Physical Review E 103, 062134 (2021).
  * Jose  _et al._ (2022) S. Jose, D. Mandal, M. Barma, and K. Ramola, Physical Review E 105, 064103 (2022).
  * Jose (2022) S. Jose, Journal of Statistical Mechanics: Theory and Experiment 2022, 113208 (2022).
  * Mallikarjun and Pal (2023) R. Mallikarjun and A. Pal, Physica A: Statistical Mechanics and its Applications 622, 128821 (2023).
  * Feller (1991) W. Feller,  _An introduction to probability theory and its applications, Volume 2_ , Vol. 81 (John Wiley & Sons, 1991).
  * Angelani (2023) L. Angelani, Journal of Physics A: Mathematical and Theoretical 56, 455003 (2023).
  * Kärger (2008) J. Kärger, Adsorption and diffusion , 329 (2008).
  * Post (1991) M. F. Post, in  _Studies in Surface Science and Catalysis_, Vol. 58 (Elsevier, 1991) pp. 391–443.
  * Marbach (2021) S. Marbach, The Journal of Chemical Physics 154 (2021), 10.1063/5.0047380.
  * Bertini  _et al._ (2005) L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Physical Review Letters 94, 030601 (2005).
  * Bertini  _et al._ (2006) L. Bertini, A. D. Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Journal of statistical physics 123, 237 (2006).
  * Bertini  _et al._ (2007) L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Journal of Statistical Mechanics: Theory and Experiment 2007, P07014 (2007).
  * Bertini  _et al._ (2009) L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Journal of Statistical Physics 135, 857 (2009).
  * Derrida (2007) B. Derrida, Journal of Statistical Mechanics: Theory and Experiment 2007, P07023 (2007).
  * Agranov  _et al._ (2023) T. Agranov, S. Ro, Y. Kafri, and V. Lecomte, SciPost Physics 14, 045 (2023).
  * Jose  _et al._ (2023c) S. Jose, R. Dandekar, and K. Ramola, Journal of Statistical Mechanics: Theory and Experiment 2023, 083208 (2023c).
  * Levitt (1973) D. G. Levitt, Physical Review A 8, 3050 (1973).
  * Arratia (1983) R. Arratia, The Annals of Probability 11, 362 (1983).
  * Krapivsky  _et al._ (2015) P. L. Krapivsky, K. Mallick, and T. Sadhu, Journal of Statistical Physics 160, 885 (2015).
  * Dandekar and Mallick (2022) R. Dandekar and K. Mallick, Journal of Physics A: Mathematical and Theoretical (2022), 10.1088/1751-8121/ac9766.
  * Clincy  _et al._ (2003) M. Clincy, B. Derrida, and M. Evans, Physical Review E 67, 066115 (2003).
  * Bodineau and Derrida (2011) T. Bodineau and B. Derrida, Journal of Statistical Physics 145, 745 (2011).