# Exact fluctuating hydrodynamics of active lattice gases –
Typical fluctuations

Tal Agranov  Department of Physics, Technion-Israel Institute of Technology,
Haifa, 3200003, Israel.  Department of Applied Mathematics and Theoretical
Physcis, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United
Kingdom.  Sunghan Ro  Department of Physics, Technion-Israel Institute of
Technology, Haifa, 3200003, Israel.  Yariv Kafri  Department of Physics,
Technion-Israel Institute of Technology, Haifa, 3200003, Israel.  Vivien
Lecomte  Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France.

###### Abstract

We extend recent results on the exact hydrodynamics of a system of diffusive
active particles displaying a motility-induced phase separation to account for
typical fluctuations of the dynamical fields. By calculating correlation
functions exactly in the homogeneous phase, we find that two macroscopic
length scales develop in the system. The first is related to the persistence
length of the particles and the other to the collective behavior of the
particles. The latter diverges as the critical point is approached. Our
results show that the critical behavior of the model in one dimension belongs
to the universality class of a mean-field Ising model, both for static and
dynamic properties, when the thermodynamic limit is taken in a specified
manner. The results are compared to the critical behavior exhibited by the ABC
model. In particular, we find that in contrast to the ABC model the density
large deviation function, at its Gaussian approximation, does not contain
algebraically decaying interactions but is of a finite, macroscopic, extent
which is dictated by the diverging correlation length.

###### Contents

  1. I Introduction
  2. II The model and its known hydrodynamics
  3. III Fluctuating hydrodynamics of active lattice gas
    1. III.1 Complementing the hydrodynamic equations with a small Gaussian noise
    2. III.2 Two-point correlation functions in the homogeneous phase
    3. III.3 Self-consistency Ginzburg type criterion
    4. III.4 Free-energy for the density profile
  4. IV Derivation of the fluctuating hydrodynamics
    1. IV.1 Conservative dynamics and the ABC model
    2. IV.2 Accounting for tumbles
    3. IV.3 Superposition
  5. V Derivation of the two-point correlation functions
    1. V.1 Static Correlation Functions
    2. V.2 Dynamic Correlation Functions
    3. V.3 Correlation functions for finite $\ell$
  6. VI Discussion

##  I Introduction

In recent years there has been a lot of interest in the statistical mechanics
of active systems. These consist of particles that self-propel by consuming
energy. They have attracted much attention both due to their relevance to
biological and synthetic materials, and due to the host of novel non-
equilibrium behaviors that arise from the interplay between the interactions
of the agents and the non-equilibrium drive that acts on the single-particle
level Ramaswamy (2010); Thompson  _et al._ (2011); Romanczuk  _et al._ (2012);
Marchetti (2013); Takatori and Brady (2015); Ramaswamy (2017); Needleman and
Dogic (2017); Loi  _et al._ (2008); Solon  _et al._ (2015a); Fodor (2016).
Most theoretical studies of active matter involve either phenomenological
approaches based on field theoretical models constructed using symmetry
arguments, or derivations which start from microscopic dynamics and involve
mean-field approximations and gradient expansions, see for example Wittkowski
_et al._ (2014); Nardini  _et al._ (2017); Solon  _et al._ (2018a, b); Tjhung
_et al._ (2018); Peshkov  _et al._ (2012); Bertin  _et al._ (2013, 2015).
While these have been extremely successful in predicting and explaining a
large variety of interesting behaviors, there is a clear lack of exactly
solvable models in the field. Technically, it is the presence of interactions
and the non-equilibrium drive that make active systems very difficult to study
using exact methods. Indeed, even the full dynamical problem of two active
particles on a lattice interacting via hard-core interactions is not trivial
Mallmin  _et al._ (2019).

In parallel to the advancements in active matter, there have been major
developments in our understanding of the fluctuating hydrodynamics of
diffusive non-equilibrium systems. This started with the identification of the
hydrodynamic equations which capture typical fluctuations by Spohn for a class
of lattice models (see Spohn (1991) for a review) and the discovery of the
generic long-range nature of correlations when driven out of equilibrium, say
by coupling the system to boundaries at different densities Spohn (1983). More
recently a formalism, known as the Macroscopic Fluctuation Theory (MFT), was
put forward. The MFT utilizes fluctuating hydrodynamics equations, whose exact
form can be derived in several cases starting from a microscopic description
Derrida (2007, 2011); Bertini  _et al._ (2015). It enables one to study the
probability density and dynamics of large non-equilibrium fluctuations in
spatially extended systems of diffusive interacting particles (see Bertini
_et al._ (2015) for a review) in the limit of a large number of degrees of
freedom. In this limit, the probability distributions take an exponential form
where the exponent (the ‘quasi-potential’) is the direct analog of the
equilibrium free energy. Using MFT, it was understood that for many non-
equilibrium systems the probability density is highly non-trivial. For
example, the probability distribution of the density field is generically a
non-local functional of the field and the probability distributions of various
quantities are often singular Bertini  _et al._ (2015); Bodineau and Lagouge
(2010); Shpielberg and Akkermans (2016); Baek  _et al._ (2017); Jack  _et al._
(2015); Lecomte  _et al._ (2012); Agranov  _et al._ (2019); Baek and Kafri
(2015). To date, an analogous set of results for active matter systems does
not exist.

A major progress in this direction has recently been achieved in Erignoux
(2018); Kourbane-Houssene  _et al._ (2018), where the noiseless coarse-grained
hydrodynamics of two active matter models, defined at the microscopic level,
were derived. Here we focus on one of the models which on a microscopic scale,
and in one dimension, is defined on a lattice where each site can be either
empty or occupied by a right or a left moving particle. Right (left) moving
particles perform a biased diffusion with a bias to the right (left) and a
right moving particle can tumble into a left moving one and vice versa (see
Fig. 1 for an illustration). The model bears many similarities with the well-
studied diffusive lattice gases Kipnis and Landim (1999); Bertini  _et al._
(2007); Liggett (2012). In particular, the motion of the particle is biased
with a rate that scales as $L^{-1}$ with $L$ the system size (i.e. the number
of lattice sites), and the tumbling rates scale as $L^{-2}$. This choice
ensures a non-trivial diffusive scaling in the hydrodynamic limit (see details
below). It is important to stress that this scaling of the rates places the
model in a different class than active systems where the rates are not scaled
with the system size. In particular, in contrast to the latter models where
the only slow degree of freedom is the density, here both the polarization
(defined as the density difference between right and left moves) and the
density are slow degrees of freedom. Building on recent developments in the
derivation of hydrodynamic equations in non-equilibrium systems Erignoux
(2018), the authors present the exact diffusive hydrodynamic description of
the model Erignoux (2018); Kourbane-Houssene  _et al._ (2018). Interestingly,
the model displays a motility-induced phase separation (MIPS), one of the most
studied collective behavior in active matter. This corresponds to the ability
of active systems to phase separate even when the interaction between the
particles is purely repulsive Tailleur and Cates (2008); Fily and Marchetti
(2012); Buttinoni  _et al._ (2013); Cates and Tailleur (2013); Stenhammar  _et
al._ (2013); Redner  _et al._ (2013); Cates and Tailleur (2015); Solon  _et
al._ (2015b); Redner  _et al._ (2016); Whitelam  _et al._ (2018); Tjhung  _et
al._ (2018); Paliwal  _et al._ (2018); Geyer  _et al._ (2019); Solon  _et al._
(2018a, b). Using the hydrodynamic equations, the exact phase diagram was
derived using the method presented in Solon  _et al._ (2018a, b).

In this article, we build upon this work and complement the deterministic
hydrodynamic equations with noise terms at the Gaussian level which captures
typical fluctuations. This allows us to obtain two-point correlation functions
exactly to leading order in $1/L$, and in regions of the phase diagram where
the system is not phase separated. We find that in these regions of the phase
diagram the model exhibits two length scales each scaling with the system
size. The first corresponds to the length scale associated with the typical
distance traveled until a tumble. The second describes the emergence of
collective behavior in the system and diverges as the critical point is
approached. Importantly, we characterize the critical behavior exhibited by
the model exactly and we find that when the thermodynamic limit is taken in a
specified way it is in the Ising mean-field universality class. While, as we
explain below, we are not able to study the model exactly at the critical
point, we find that the dynamics exhibit critical slowing down close to the
critical point on scales smaller than the correlation length, with a mean-
field model B exponent $z=4$.

It is interesting to place these results in the context of recent studies on
the universality of MIPS. This has been a topic of debate due to conflicting
numerical and theoretical results. Studies based on a phenomenological theory
for active systems, known as active model B+ Wittkowski  _et al._ (2014);
Nardini  _et al._ (2017); Tjhung  _et al._ (2018) suggest that MIPS belongs to
the Ising universality class Caballero  _et al._ (2018) (see also Partridge
and Lee (2019)). In contrast, several numerical studies present contradictory
findings with some numerics suggesting an Ising universality Partridge and Lee
(2019); Maggi  _et al._ (2021), while others Siebert  _et al._ (2018);
Dittrich  _et al._ (2021) a different one. Note that as stressed above these
models, while similar, belong to a distinct class. In particular, the standard
active models do not phase separate in one-dimension Thompson  _et al._
(2011).

Our results also allow us to consider the structure of the quasi-potential for
the probability of profiles, at the Gaussian level. It was previously shown
that lattice models which phase separate have a quasi-potential with non-local
interactions. Here, in contrast, we find that the quasi-potential has a local
macroscopic structure whose range grows as the critical point is approached.
This, in turn, allows the system to supports a phase separation state in one
dimension.

The outline of the paper is the following: in Sec. II, we define the model and
recall for completeness its (noiseless) hydrodynamics, which was derived in
Ref. Kourbane-Houssene  _et al._ (2018). In Sec. III, we describe our results,
a set of fluctuating hydrodynamic equations with Gaussian noise, and the
analytically computed two-point correlation functions for the different
fields. We check the self-consistency of the calculation using a Ginzburg
criterion and compare the results to numerical simulations of the lattice
model. The quasi-potential (or free energy) is then computed at quadratic
level and its local structure is discussed. In Sec. IV, we explain how to
derive the Gaussian fluctuating hydrodynamic equations. The derivation of the
two-point correlation function is presented in Sec. V. We first derive the
correlation functions in the limit of a small persistence length of the active
particles, and discuss the dynamic scaling exponent in Sec. V.2. In Sec. V.3,
we generalize the results to any persistence length, which allows us to
consider finite-size effects. In addition, it allows us to compare the
observed transition to the one in the ABC model Evans  _et al._ (1998a, b) –
another diffusive model which is known to exhibit phase separation in one
dimension. Finally, in Sec. VI, we conclude.

##  II The model and its known hydrodynamics

Following Kourbane-Houssene  _et al._ (2018), we consider a one-dimensional
lattice model constituted of $L$ sites where each site $i$ can either be
occupied by a $+$ particle ($\sigma_{i}^{+}=1$ and $\sigma_{i}^{-}=0$), a $-$
particle ($\sigma_{i}^{+}=0$ and $\sigma_{i}^{-}=1$), or be empty
($\sigma_{i}^{+}=0$ and $\sigma_{i}^{-}=0$). The dynamics of the model is
defined through (see Fig. 1):

  1. (i)

A pair of neighboring sites exchange their state with rate $D$.

  2. (ii)

A $+$ ($-$) particle jumps to the right (left) neighboring site with rate
$\lambda/L$, provided the target site is empty.

  3. (iii)

A $+$ ($-$) particle tumbles into a $-$ ($+$) particle with rate
$\gamma/L^{2}$.

The scaling of the rates with $L$ ensures that, in the hydrodynamic limit
($L\gg 1$), all processes occur on diffusive time scales. Indeed, the time it
takes for particles to traverse a macroscopic system of size $L$ either via
diffusive motion, or on account of self-propulsion, scales as $L^{2}$ which is
also the time scale for tumbling events. .

Figure 1: Schematic representation of the microscopic dynamics described in
the text.

Using the diffusive scaling $t\rightarrow t/L^{2}$ and $x=i/L$ Kourbane-
Houssene  _et al._ (2018), so that $0\leq x\leq 1$, it is natural to define
the coarse-grained fields $\rho_{+}(x,t)$ and $\rho_{-}(x,t)$ in the large $L$
limit through

$\rho_{\pm}(x,t)=\frac{1}{2L^{\delta}}\sum_{|i-Lx|<L^{\delta}}\sigma_{i}^{\pm}\>,$
(1)

where the exponent $\delta$ can take any value in the range $0<\delta<1$. The
resulting hydrodynamic equations of the model were shown in Ref. Kourbane-
Houssene  _et al._ (2018) to be

$\displaystyle\partial_{t}\rho_{+}$ $\displaystyle=$ $\displaystyle
D\partial^{2}_{x}\rho_{+}-\lambda\partial_{x}\left[\rho_{+}(1-\rho)\right]+\gamma(\rho_{-}-\rho_{+})\;,$
(2) $\displaystyle\partial_{t}\rho_{-}$ $\displaystyle=$ $\displaystyle
D\partial^{2}_{x}\rho_{-}+\lambda\partial_{x}\left[\rho_{-}(1-\rho)\right]+\gamma(\rho_{+}-\rho_{-})\>,$
(3)

where $\rho(x,t)=\rho_{+}(x,t)+\rho_{-}(x,t)$ is the total density of
particles.

For large enough drift $\lambda$ and mean density $\rho_{0}$, the hydrodynamic
equations allow for non-homogeneous solutions. Their structure is controlled
by the persistence length $\ell\equiv\sqrt{D/\gamma}$ and defined as the
typical distance traveled by a particle before it tumbles. When this scale is
very small compared to the macroscopic system size ($\ell\ll 1$), the steady
state consists of high- and low-density coexisting phases, separated by a
narrow domain wall of width $\ell$. This phenomenon corresponds to MIPS. The
binodal curve has been calculated in Kourbane-Houssene  _et al._ (2018) in the
small $\ell$ limit, and is shown in Fig. 2 for completeness. The figure also
shows the spinodal line, beyond which the homogeneous state becomes linearly
unstable. The spinodal line is given by
$2-\text{Pe}^{2}(1-\rho_{0})(2\rho_{0}-1)=0$, where the Péclet number is
$\text{Pe}=\lambda/\sqrt{\gamma D}$. The critical point of the model
$(\rho_{c},\text{Pe}_{c})=(3/4,4)$ is located at the point where binodal and
spinodal lines meet.

The hydrodynamic equations presented above neglect the noise terms which on
general grounds should scale as $L^{-1/2}$. The aim of this paper is to derive
these at the Gaussian level and use them to compute two-point correlation
functions. Of particular interest is their behavior near the critical point of
the model. In what follows, before turning to the derivation of the different
results, we summarize our main findings and contrast the behavior of the
system with previous studies on other systems that phase separate in one
dimension. For clarity, we first focus on the regime $\ell\ll 1$ (the case of
finite $\ell$ is discussed in Sec. V.3).

Figure 2: The phase diagram in the total mean-density $\rho_{0}$ and
$\text{Pe}=\lambda/\sqrt{\gamma D}$ plane. The binodal curve is marked as a
black solid line, while the blue dashed line indicates the spinodal line
$2-\text{Pe}^{2}(1-\rho_{0})(2\rho_{0}-1)=0$. Both lines meet at the critical
point $(\rho_{0}=3/4,\text{Pe}=4)$ which is marked by a red dot. The
stationary configurations are phase-separated in the gray region, and
otherwise homogeneous in the colored region. The color code, given by the bar
in the legend, indicates the value of the correlation length (12) derived
using the fluctuating hydrodynamics. The binodal is obtained using the methods
described in Ref. Solon  _et al._ (2018a, b); Kourbane-Houssene  _et al._
(2018) in the limit of small $\ell$.

##  III Fluctuating hydrodynamics of active lattice gas

###  III.1 Complementing the hydrodynamic equations with a small Gaussian
noise

While the hydrodynamic equations are exact in the $L\to\infty$ limit, any
finite value of $L$ will lead to a fluctuating noisy dynamics. Our first
result complements the hydrodynamic equations of Kourbane-Houssene  _et al._
(2018) with Gaussian noise terms which account for the typical fluctuations
$\delta\rho\sim L^{-1/2}$ around the most probable profile predicted by the
deterministic equations (2)-(3). Adopting a representation in terms of the
total particle density and polarization fields $\rho=\rho_{+}+\rho_{-}$,
$m=\rho_{+}-\rho_{-}$ we find that the fluctuating hydrodynamical equations
take the form:

$\displaystyle\partial_{t}\rho$ $\displaystyle=$ $\displaystyle
D\partial^{2}_{x}\rho-\lambda\partial_{x}[m(1-\rho)]+\frac{1}{\sqrt{L}}\sqrt{D}\,\partial_{x}\eta_{\rho}$
(4) $\displaystyle\partial_{t}m$ $\displaystyle=$ $\displaystyle
D\partial^{2}_{x}m-\lambda\partial_{x}[\rho(1-\rho)]-2\gamma
m+\frac{1}{\sqrt{L}}\Big{(}\sqrt{D}\,\partial_{x}\eta_{m}+2\sqrt{\gamma}\,\eta_{K}\Big{)}\>.$
(5)

Here, $\eta_{\rho},\eta_{m}$, and $\eta_{K}$ are Gaussian white noises of zero
mean and correlations given by

$\displaystyle\langle\eta_{\rho}(x,t)\eta_{\rho}(x^{\prime},t^{\prime})\rangle$
$\displaystyle=$ $\displaystyle
2\rho(1-\rho)\,\delta(x-x^{\prime})\delta(t-t^{\prime})\quad;\quad\langle\eta_{\rho}(x,t)\eta_{m}(x^{\prime},t^{\prime})\rangle=2m(1-\rho)\,\delta(x-x^{\prime})\delta(t-t^{\prime}),$
(6) $\displaystyle\langle\eta_{m}(x,t)\eta_{m}(x^{\prime},t^{\prime})\rangle$
$\displaystyle=$ $\displaystyle
2(\rho-m^{2})\delta(x-x^{\prime})\delta(t-t^{\prime})\quad;\quad\langle\eta_{K}(x,t)\eta_{K}(x^{\prime},t^{\prime})\rangle=\rho\,\delta(x-x^{\prime})\delta(t-t^{\prime}),$
(7) $\displaystyle\langle\eta_{m}(x,t)\eta_{K}(x^{\prime},t^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\langle\eta_{\rho}(x,t)\eta_{K}(x^{\prime},t^{\prime})\rangle=0\;.$
(8)

The angular brackets denote an average over histories of the microscopic
dynamics described in Sec. II. The noise terms are multiplicative, and as is
usual in fluctuating hydrodynamics equations, it is straightforward to check
that the scaling with $L^{-1/2}$ implies that the choice of time
discretization, say Itō or Stratonovich, is not important. As we describe in
Sec. IV, the equations are obtained through a superposition of known results
for the ABC lattice gas Evans  _et al._ (1998a, b) and a reaction process
accounting for the tumbling. It is straightforward to check that by rescaling
$x$ by $\ell$ and $t$ with the timescale $1/\gamma$ (after the diffusive
rescaling) the noiseless hydrodynamic part is entirely controlled by the
Péclet number $\text{Pe}={\lambda}/{\sqrt{\gamma D}}$ Kourbane-Houssene  _et
al._ (2018). The noise amplitudes in Eqs. (4)-(5) then become proportional to
$1/\sqrt{\ell L}$, indicating that the noise term keeps track of the
lengthscale $\ell$ and the system size $L$.

###  III.2 Two-point correlation functions in the homogeneous phase

At large $L$, the hydrodynamic equations augmented with the Gaussian noise
terms Eqs. (4)-(8) are enough to give the exact two-point functions of the
model Derrida (2007); Krapivsky and Meerson (2012) at leading order in
$L^{-1}$. As detailed in Sec. V, we find to this order that when the system is
homogeneous, namely outside the binodal, the static connected two-point
correlation functions are given by:

$\displaystyle
C_{2}^{\rho}\equiv\langle\delta\rho(x)\delta\rho(x^{\prime})\rangle=\frac{\rho_{0}(1-\rho_{0})}{L}\left[\delta(x-x^{\prime})-1\right]+\frac{\text{Pe}^{2}\xi^{2}\rho_{0}^{2}(1-\rho_{0})^{2}}{2\ell
L(1-\xi^{2})}\\!\left[\\!\left(e^{-\frac{1}{\ell}|x-x^{\prime}|}-\ell\right)+\frac{1-2\xi^{2}}{\xi}\left(e^{-\frac{1}{\ell}\frac{|x-x^{\prime}|}{\xi}}-\ell\xi\right)\\!\right]~{}$
(9) $\displaystyle C^{m}_{2}\equiv\langle\delta m(x)\delta
m(x^{\prime})\rangle=\frac{\rho_{0}}{L}\left[\delta(x-x^{\prime})-1\right]+\frac{\text{Pe}^{2}\xi^{2}\rho_{0}^{2}(1-\rho_{0})(1-2\rho_{0})}{2\ell
L(1-\xi^{2})}\left(e^{-\frac{1}{\ell}|x-x^{\prime}|}-\frac{1}{\xi}e^{-\frac{1}{\ell}\frac{|x-x^{\prime}|}{\xi}}\right),$
(10) $\displaystyle C^{\rho,m}_{2}\equiv\left<\delta\rho(x)\delta
m(x^{\prime})\right>=\text{sign}(x-x^{\prime})\frac{\text{Pe}\,\xi^{2}\rho_{0}^{2}(1-\rho_{0})}{2\ell
L(1-\xi^{2})}\left(e^{-\frac{1}{\ell}|x-x^{\prime}|}+\frac{1-2\xi^{2}}{\xi^{2}}e^{-\frac{1}{\ell}\frac{|x-x^{\prime}|}{\xi}}\right),$
(11)

where $\rho_{0}$ is the mean particle density,
$\delta\rho(x)=\rho(x)-\rho_{0}$, $\delta m(x)=m(x)$ with the mean-
magnetization zero, and the (dimensionless) correlation length $\xi$ is given
by

$\xi=\frac{1}{\sqrt{2-\text{Pe}^{2}(1-\rho_{0})(2\rho_{0}-1)}}\>.$ (12)

The expressions are in excellent agreement with the Monte Carlo simulation
results of the underlying microscopic dynamics of the lattice model, as
illustrated in Fig. 3(b) and 3(c) 111For $\xi=1$, the expressions (9)-(11)
seems singular, due to the vanishing denominator $\sim 1-\xi$. Note however
that the numerator vanishes at the same order in this limit, and the
continuation in $\xi=1$ is analytical. E.g.,
$\frac{1}{1-\xi^{2}}\Big{(}e^{-\frac{1}{\ell}|x-x^{\prime}|}+\frac{1-2\xi^{2}}{\xi}e^{-\frac{1}{\ell}\frac{|x-x^{\prime}|}{\xi}}\Big{)}\xrightarrow[\xi=1]{}\frac{3+\frac{1}{\ell}\,|x-x^{\prime}|}{2}e^{-\frac{1}{\ell}|x-x^{\prime}|}\;.$
(13) . The results (9)-(11) are valid for $\ell\ll 1$ and we give the
expression for arbitrary $\ell$ in Sec. V.3. Several comments are in order.
First, note that by setting the self-propulsion to zero by taking
$\text{Pe}=0$, we are left only with local correlations as expected – in this
limit the model becomes identical to the simple symmetric exclusion model
(SSEP) on a ring (see Derrida (2011) for a review). Second, the integral of
these expressions over each of the coordinates $x,x^{\prime}$ separately,
vanishes to leading order in $\ell$, as it should, due to total particle
number conservation. Finally, when $\ell=0$, so that the persistence length is
microscopic, one is left with local contributions only, encoded in the delta
function.

When $\text{Pe}\neq 0$, the model develops large-scale correlations of the
order of the system size (recall that length scales are rescaled by the system
size $L$). Indeed, in the hydrodynamic limit, any finite correlation length
that does not scale with the system size will vanish and only manifest itself
through the Dirac delta function term in the correlation function. It is well
known that long-range correlations can appear in lattice gas systems when
driven out of equilibrium, for example, by coupling the system to reservoirs
of different densities at their boundaries Spohn (1983). The large-scale
correlations that we observe here are, in contrast, a consequence of the
active dynamics of the particles.

Interestingly there are two distinct length scales appearing in the
correlation functions. One, $\ell$, is independent of the density and stems
from the persistence length of the particles. Recall, that here this length
scale, which also controls the domain wall width in the phase-separated state,
is assumed to be very small. Thus, the terms in the two-point functions
associated with this scale are narrowly localized at $x=x^{\prime}$. The
second one, $\xi\ell$, is related to the collective behavior leading to MIPS.
In particular, $\xi$ diverges upon approaching the critical MIPS point
$(\rho_{0}=3/4,\text{Pe}=4)$ with a singular behavior of the form

$\xi(\rho_{0}=3/4,\text{Pe})=\frac{1}{\sqrt{4-\text{Pe}}}+\mathcal{O}\big{[}(4-\text{Pe})^{1/2}\big{]},$
(14)

corresponding to a mean-field like critical exponent $1/2$. Thus, when
approaching the critical point, the length scale $\xi\ell$ approaches the
scale of the system, and the associated terms in the expressions of the
correlation functions span the entire system. As detailed below, at this stage
our small $\ell$ approximation fails, and one has to resort to the finite
$\ell$ expressions presented in Sec. V.3. Note also that the amplitude of the
two-point correlation function (9) diverges upon approaching the critical
point as $\sim\xi$. Both these properties are consistent with a behavior
expected from an equilibrium scalar field theory in one spatial dimension
where only Gaussian fluctuations are taken into account. Indeed, here, the
hydrodynamic scaling and the smallness of the noise ensure that, within our
approach, only Gaussian fluctuations appear at leading order in $1/L$, see
also the discussion in Sec. III.3.

It is also interesting to compare the behavior to that of the ABC model which
also presents a singular correlation function when it phase separates (as
computed in Ref. Bodineau  _et al._ (2008)). In contrast to what we find, this
singularity is not related to a divergent length scale. This is related to the
fact that the quasi-potential of the ABC model is always long-ranged Bodineau
_et al._ (2008), as seen e.g. at equal densities Evans  _et al._ (1998a, b).
We return to the latter point below and a detailed comparison between the
critical behavior in the present model and the phase transition in the ABC
model is presented in Sec. V.3 for periodic systems.

Figure 3: Two-point correlation functions of (a) type-specific density fields,
(b) $\rho$ and $m$, and (c) cross-correlation between $\rho$ and $m$ are
presented. Symbols show values obtained from simulation and lines correspond
to analytic results obtained by using Eqs. (9)-(11). The inset of (b)
illustrates the non-monotonous behavior of the correlation function
$C_{2}^{m}$. The parameters used in the numerics are: $L=10^{3}$,
$\rho_{0}=0.25$, $\gamma=10^{4}$, $\mathrm{Pe}=2$, and $D=1$.

While all the correlation functions presented in Eqs. (9)-(11) depend on the
two length scales discussed above, their behavior is, as expected, distinct.
Both $C_{2}^{\rho}$ and $C_{2}^{m}$ are even and continuous, but
$C_{2}^{\rho}$ is monotonic and positive while $C_{2}^{m}$ is non-monotonic
and changes sign as its argument is varied, see the inset of Fig. 3(b). Its
extrema are located at $|x|=x^{*}\equiv
2\ell\xi\log\xi/(\xi-1)\sim\ell\log\xi$. On the other hand, $C_{2}^{\rho,m}$
is an odd function and is discontinuous at the origin [see Fig. 3(c)]. These
unusual properties can be best understood by rewriting Eq. (11) in terms of
the $+,-$ two-point functions

$C_{2}^{\rho,m}=-\langle\rho_{+}(x)\rho_{-}(x^{\prime})\rangle+\langle\rho_{-}(x)\rho_{+}(x^{\prime})\rangle.$
(15)

With this decomposition, consider a fluctuation localized at position $x$ with
an enhanced $+$ particle density. It is quite clear that it will be positively
correlated with an enhanced $-$ particle density fluctuation to the right of
$x$. The reason being that the $-$ particles to the right of $x$ would be
impeded by the surge in $+$ particles to their left, causing them to
accumulate. However, the effect on the particles to the left of $x$ is exactly
the opposite. There, the self-propulsion of the $-$ particles is directed away
from the local $+$ particles surge. These features are manifested by the
correlation functions between the densities of particles of either positive
and negative particles as shown in Fig. 3(a).

Finally, we note that the Gaussian fluctuating hydrodynamic equations also
allow us to determine the dynamical two-point functions. The results, most
conveniently written in Fourier space, are given by:

$\displaystyle\tilde{C}_{2}^{\rho}$ $\displaystyle=$
$\displaystyle\langle\delta\rho(k,\omega)\delta\rho(k^{\prime},\omega^{\prime})\rangle=\frac{2\rho_{0}(1-\rho_{0})}{\ell
LM(k,\omega)}\left[k^{2}\omega^{2}+\left\\{k^{2}+2+\frac{2\xi^{2}-1}{\xi^{2}(2\rho_{0}-1)}\right\\}k^{2}(k^{2}+2)\right]\delta(k+k^{\prime})\delta(\omega+\omega^{\prime}),$
(16) $\displaystyle\tilde{C}_{2}^{m}$ $\displaystyle=$
$\displaystyle\langle\delta m(k,\omega)\delta
m(k^{\prime},\omega^{\prime})\rangle=\frac{2\rho_{0}}{\ell
LM(k,\omega)}\left[({\xi^{-2}}-2)(1-2\rho_{0})k^{4}+(k^{4}+\omega^{2})(k^{2}+2)\right]\delta(k+k^{\prime})\delta(\omega+\omega^{\prime}),$
$\displaystyle\tilde{C}_{2}^{\rho,m}$ $\displaystyle=$
$\displaystyle\langle\delta\rho(k,\omega)\delta
m(k^{\prime},\omega^{\prime})\rangle=\frac{2ik\text{Pe}\rho_{0}(1-\rho_{0})}{\ell
LM(k,\omega)}\left[2ik^{2}\omega(1-\rho_{0})-2\rho_{0}k^{2}(k^{2}+2)\right]\delta(k+k^{\prime})\delta(\omega+\omega^{\prime}),$

where
$M(k,\omega)=[k^{2}(k^{2}+\xi^{-2})-\omega^{2}]^{2}+4(k^{2}+1)^{2}\omega^{2}$.
Most notably, by examining scaling properties of the correlation function we
find that the dynamical scaling exponent for our model is given by $z=4$ on
scales smaller than the correlation length and $z=2$ on larger scales, in
agreement with a mean-field model B and consistent with the mean-field
divergence of the static correlation function. Numerical results supporting
this are presented below.

###  III.3 Self-consistency Ginzburg type criterion

As detailed below in Sec. V the correlation functions are obtained by
linearizing the noisy hydrodynamic equations about a homogeneous profile. The
divergence of the amplitude of the fluctuations exhibited by the two-point
function (9) as criticality is approached indicates that the assumption of
small fluctuations may not be self-consistent near the critical point. To
check the this we employ a Ginzburg criterion and check if:

${\langle\delta\phi(\xi)^{2}\rangle}\ll 1,$ (17)

Here the average density fluctuation inside a correlation length is
$\delta\phi(\xi)\equiv({\xi}\ell)^{-1}\int_{\xi\ell}dx~{}\delta\rho(x)$, and
we assume that the average density is of the order of 1. We note the result of
analysis would be same by checking whether
$\langle[\delta\rho(x)-\delta\rho(0)]^{2}\rangle\ll 1$ holds for any value of
$x$. In the small $\ell$ limit, and using the Eqs. (16) the condition reads

$\frac{\mathrm{Pe}^{2}\xi}{\ell}{\rho_{0}^{2}(1-\rho_{0})^{2}}\ll L\,.$ (18)

Using the asymptotics (14) and omitting $\mathcal{O}\left(1\right)$ pre-
factors, we find that the above condition can be rewritten as

$\frac{1}{L\ell}\ll\sqrt{\text{Pe}_{c}-\text{Pe}}\>.$ (19)

The Ginzburg criterion implies that for any finite $\xi$, or
$\text{Pe}_{c}-\text{Pe}$, a large enough $L$ can be chosen so that it is
satisfied. This specifies how the thermodynamic limit has to be taken so that
the calculations of the correlation functions, around the homogeneous
profiles, are valid. Note that exactly at the critical point the calculation
is not self-consistent. While beyond the scope of this manuscript, it is
interesting to ask if this is due to a breakdown of the macroscopic
hydrodynamic description used here to describe the system or due to the
contribution of non-linearities to the fluctuations.

###  III.4 Free-energy for the density profile

The steady state probability distribution of observing density and
polarization profiles $\rho(x)$ and $m(x)$ takes a large-deviation form at
large $L$ as $e^{-L\mathcal{F}[\rho(x),m(x)]}$ where
$\mathcal{F}[\rho(x),m(x)]$ is an analog of a free-energy density. Our
linearized theory detailed in Sec. V allow us to probe the large-deviation
function $\mathcal{F}$ up to Gaussian order. In particular, the deviations
from the most probable profiles, outside the binodal curve, takes the form

$\mathcal{F}[\rho(x),m(x)]=\frac{1}{2}\int
dk\left[\begin{array}[]{c}\delta\tilde{\rho}(k)\\\
\delta\tilde{m}(k)\end{array}\right]\mathbb{C}^{-1}_{k}\left[\begin{array}[]{c}\delta\tilde{\rho}(-k)\\\
\delta\tilde{m}(-k)\end{array}\right]+o\big{[}(\delta\rho,\delta
m)^{2}\big{]}\;,$ (20)

in Fourier space, where $\tilde{f}$ denotes the Fourier transform of $f$,
$\mathbb{C}_{k}$ is a Hermitian matrix with diagonal elements given by (38)
and (39), and the off-diagonal elements given by (40) and its complex
conjugate 222In (20) and (21) we adopt the re-scaling used in Sec. V. Also,
note that the zeroth mode $k=0$ should be excluded from the integration above
due to total particle number conservation. Similarly, the free energy
$\mathcal{F}[\rho(x)]$ corresponding to total density fluctuations is obtained
by integrating over the magnetization fluctuations:

$\mathcal{F}[\rho(x)]=\frac{1}{2}\int
dk\frac{|\delta\tilde{\rho}(k)|^{2}}{C_{2}^{\rho}(k)}+\mathcal{O}(\delta\rho^{4})$
(21)

with $C_{2}^{\rho}(k)$ given by (38).

The long-wavelength physics is captured by the small-$k$ expansion, where we
find that $\mathcal{F}[\rho(x)]$ takes the form of the standard Landau theory
near the critical point

$\mathcal{F}[\rho(x)]\simeq\left(2\pi\right)^{2}\int_{0}^{1}dx\left\\{\frac{\alpha}{\xi^{2}}\delta\rho^{2}+\beta\ell^{2}\Big{(}\frac{d\delta\rho}{dx}\Big{)}^{2}\right\\},$
(22)

with

$\alpha=\frac{1}{2\rho_{0}\left(1-\rho_{0}\right)\left[2+\text{Pe}^{2}\left(1-\rho_{0}\right)\right]}\quad;\quad\beta=\frac{\text{Pe}^{2}\left(2+\xi^{-2}\right)}{2\left[2+\text{Pe}^{2}\left(1-\rho_{0}\right)\right]^{2}}.$
(23)

As expected, the “mass” term in Eq. (22) vanishes as the critical point is
approached and $\xi\rightarrow\infty$. Namely, the form of the large deviation
function is that of a Gaussian field theory. Note, however, that if a standard
Landau-Ginzburg free energy is rescaled so that $x\to x/L$ the coefficient of
$(\frac{d\delta\rho}{dx})^{2}$ would scale as $1/L$ and vanishes in the large
$L$ limit since the cost of a domain wall is finite. In the model we study, in
contrast, the cost of a domain wall scales with $L$ for any non-vanishing
$\ell$.

It is interesting to compare the result to those obtained previously for the
ABC model which also exhibits phase separation in one dimension. There the
large deviation function has been computed at equal densities Clincy  _et al._
(2003), close to the equal density point Clincy  _et al._ (2003), and for
arbitrary densities but at Gaussian order Bodineau  _et al._ (2008). In
striking contrast, the large deviation function of the ABC model does not
display similar criticality associated with a diverging correlation length
(see also Sec. V.3 for a comparison of the divergent features of the
correlation functions between the ABC model and the model studied here).

Next, we detail how the results described above are obtained.

##  IV Derivation of the fluctuating hydrodynamics

As stated above, the hydrodynamics (2)-(3) describes the noiseless dynamics
reached by the system when $L\to\infty$. However, at large but finite $L$, the
actual dynamics displays fluctuations around the deterministic hydrodynamic
equation. The full fluctuating dynamics can be written as

$\displaystyle\partial_{t}\rho_{\pm}(x,t)=-\partial_{x}J_{\pm}(x,t)\pm
K(x,t).$ (24)

Here $J_{\pm}(x,t)$ are the conservative particles fluxes, and $K(x,t)$ is the
total reaction rate contributed from tumbles where $+$ particles transform
into $-$ particles and vice versa. Both terms are stochastic fields which
fluctuate around their deterministic expressions

$\displaystyle\langle{J}_{\pm}(x,t)\rangle=D\partial^{2}_{x}\rho_{\pm}-\lambda\partial_{x}\left[\rho_{\pm}(1-\rho)\right]\quad;\quad\langle{K(x,t)}\rangle=\gamma\left(\rho_{-}-\rho_{+}\right).$
(25)

When $L$ is large, their typical fluctuations are small, and by central limit
arguments are given by Gaussian noise terms 333The account of large
fluctuation will be presented in a future publication for (2021).

$\displaystyle
J_{\pm}(x,t)=\langle{J}_{\pm}(x,t)\rangle+\sqrt{\frac{D}{L}}\eta_{\pm}(x,t)\quad;\quad
K(x,t)=\langle{K}(x,t)\rangle+\sqrt{\frac{\gamma}{L}}\eta_{K}(x,t)$ (26)

where the noise terms $\eta_{\pm}(x,t),\eta_{K}(x,t)$ are uncorrelated over
macroscopic time and length scales so that the covariance matrix is expected
to be proportional to $\delta(x-x^{\prime})\delta(t-t^{\prime})$. In addition,
they are expected to both have amplitudes which are functions of the local
densities and possibly be correlated with one another. To this end, we now
prescribe the covariance matrix associated with the noise terms. This is
achieved by building on a relation between the model and two other known
lattice gas models: one specifies the conservative noise $\eta_{\pm}(x,t)$,
while the other to the non-conserving noise $\eta_{K}(x,t)$.

###  IV.1 Conservative dynamics and the ABC model

To obtain the conservative noise, we set the tumbling rate of the model to
zero. In this case, the diffusive part of the model maps to a driven ABC model
Clincy  _et al._ (2003). In fact, to compute the covariance of the flux noise
terms, it is sufficient to consider the non-driven case where $\lambda=0$ (as
generic in MFT, in our weak-drive limit, the noise is independent of the drive
Bertini  _et al._ (2015)). In this limit, the model coincides with an ABC
model with symmetric dynamics, first presented in Evans  _et al._ (1998a)
(corresponding to $q=1$ in their notations). Within the model, each site can
be occupied by either an $A,B$ or $C$ particle. The dynamics are such that
chiral pairs $AB,BC,CA$ exchange with rate $q$ (in arbitrary units) into
$BA,CB,AC$ while the reverse exchanges occur with rate $q$. The model is known
to exhibit phase separation for any finite $q<1$ Evans  _et al._ (1998a, b)
and when the rate $q$ is scaled with the system size as $q=e^{-\beta/L}$ the
phase transition occurs at a finite value of $\beta=\beta_{c}$ Clincy  _et
al._ (2003). The mapping follows by identifying $+$, $-$ and vacant states
with $A$, $B$, and $C$ particle types respectively. The fluctuating
hydrodynamics for this model was studied in Bodineau  _et al._ (2008);
Bodineau and Derrida (2011). The derivation there was based on yet another
mapping, with each of the three species $A$, $B$ and $C$ behaving as a SSEP, a
gas composed of particles interacting only through excluded volume
interactions Derrida (2011). The corresponding flux noise terms have a
covariance:

$\displaystyle\langle\eta_{\pm}(x,t)\eta_{\pm}(x^{\prime},t^{\prime})\rangle=2\rho_{\pm}(1-\rho_{\pm})\,\delta(x-x^{\prime})\delta(t-t^{\prime})\
\ ;\ \
\langle\eta_{+}(x,t)\eta_{-}(x^{\prime},t^{\prime})\rangle=-2\rho_{+}\rho_{-}\delta(x-x^{\prime})\delta(t-t^{\prime}).$
(27)

Transforming to the density and polarization fields $\rho$ and $m$, this
yields the conservative part of the fluctuating hydrodynamics presented in
Eqs. (4)-(7). Note that in (27) the same-species diagonal variance terms
coincide with those of the SSEP, as one might anticipate. At zero bias
$\lambda=0$, the different species are neutral to one another, i.e., the
dynamics restricted to that of only one of the species coincides with that of
freely evolving gas of SSEP particles. Nevertheless, the cross-correlation
between the fluxes of different species is non-vanishing and negative. This
stems from the swapping of adjacent particles of different signs. Indeed, such
moves contribute to oppositely oriented fluxes of the two particle types which
leads to negative correlations. With this we now turn to discuss the non-
conserving part of the dynamics.

###  IV.2 Accounting for tumbles

The reaction dynamics corresponding to tumbles can be accounted for within the
fluctuating hydrodynamics of lattice gas models using several equivalent
methods such as the Doi–Peliti formalism Doi (1976a, b); Peliti (1985), a
large deviation formalism for reactive lattice gas De Masi  _et al._ (1986);
Jona-Lasinio  _et al._ (1993); Lefèvre and Biroli (2007); Tailleur  _et al._
(2008); Bodineau and Lagouge (2010), or for typical fluctuations using a van
Kampen expansion Gardiner  _et al._ (1985). Here, where the focus is on
typical fluctuations, we use a simpler approach.

Specifically, we note that the coarse grained scaling allows us to treat the
total tumble term $K(x,t)\equiv K_{-}(x,t)-K_{+}(x,t)$ as the difference of
two uncorrelated Poisson processes $K_{+}(x,t)$ and $K_{-}(x,t)$ which specify
the tumbling rates of $+$ and $-$ particles, respectively. According to Eq.
(24), ${\cal K}(x,t)\equiv\int_{x}^{x+\Delta
x}L\mathrm{d}x^{\prime}\int_{t}^{t+\Delta
t}\mathrm{d}t^{\prime}~{}K\left(x^{\prime},t^{\prime}\right)$ specifies the
total number of particles that tumbled into the $+$ state subtracted by the
number of particles that tumbled out of this state, inside a mesoscopic
compartment of length $\Delta x$ during a time interval $\Delta t$. Since the
densities evolve over macroscopic time and length scales, they can be
approximated as constant inside $\Delta x$ during the time interval $\Delta t$
if we take $\Delta x$ such that $L\Delta x\sim L^{\delta}\gg 1$ with
$1<\delta<0$. Hence, the random variable ${\cal K}(x,t)$ is simply the
difference of two independent Poisson processes with rates
$\gamma\rho_{-}(x,t)L\Delta x$ and $\gamma\rho_{+}(x,t)L\Delta x$ evolving
over a time $\Delta t$.

Next, using the fact that the Poisson distribution can be well-approximated by
a Gaussian distribution in the large $L$ limit, we can specify the
fluctuations by calculating the first two cumulants of ${\cal K}(x,t)$. These
are easily found to be

$\displaystyle\langle{\cal K}(x,t)\rangle$ $\displaystyle=$
$\displaystyle\gamma\left[\rho_{-}(x,t)-\rho_{+}(x,t)\right]L\Delta x\Delta t$
(28) $\displaystyle\left\langle{\cal K}^{2}(x,t)\right\rangle$
$\displaystyle=$ $\displaystyle\langle{\cal
K}(x,t)\rangle^{2}+\gamma\left[\rho_{-}(x,t)+\rho_{+}(x,t)\right]L\Delta
x\Delta t~{}\;,$ (29)

which then implies that

$K\left(x,t\right)=\gamma\left[\rho_{-}(x,t)-\rho_{+}(x,t)\right]+\sqrt{\frac{\gamma}{L}\left[\rho_{-}(x,t)+\rho_{+}(x,t)\right]}\,{\eta}(x,t)\>,$
(30)

where $\eta\left(x,t\right)$ is a zero mean and unit variance white Gaussian
field. Defining $\eta_{K}(x,t)\equiv\sqrt{\rho}\eta(x,t)$ with the variance

$\langle\eta_{K}(x,t)\eta_{K}(x^{\prime},t^{\prime})\rangle=\rho\,\delta(x-x^{\prime})\delta(t-t^{\prime}),$
(31)

leads to the announced result (7).

Finally, we remark that this derivation also elucidates the validity of the
Gaussian approximation. To see this, we evaluate the third cumulant of
$\delta{\cal K}(x,t)$ which reads

$\langle\delta{\cal
K}^{3}(x,t)\rangle=\gamma\left[\rho_{+}(x,t)-\rho_{-}(x,t)\right]L\Delta
x\Delta t~{},$ (32)

and is not accounted for by a Gaussian random noise. To incorporate this into
$K(x,t)$ of Eq. (25), we need to add a term of order $L^{-2/3}$, which is sub-
leading when compared to the Gaussian contribution which is of order
$L^{-1/2}$. Note, that to account for large but rare fluctuations of $K(x,t)$,
which go beyond the Gaussian approximation we present here and are described
by large deviations, one needs to account for the non-Gaussian contributions
to the statistics of $K$ for (2021).

###  IV.3 Superposition

To combine both the results presented above we consider a coarse-grained
description of the biased diffusion dynamics given by (27) in the presence of
the additional tumbling reactions. The former relies on the dynamics obeying
“local equilibrium” described by a local product measure parameterized by the
local particle densities Spohn (1991). This local equilibrium property is in
fact necessary for establishing the more basic deterministic description
(2)-(3) and was proven to hold in Erignoux (2018). That is, as one might
expect, the slow rate at which particles undergo tumbling reactions do not
violate the local equilibrium property that holds on mesoscopic scales.

Thus, we see that the two dynamics can be superimposed. Since tumbles occur
independent of the particles diffusion the noise terms of the two dynamics are
uncorrelated. Under the appropriate change of variables to $\rho$ and $m$
variables, one arrives at Eqs. (4)-(7).

##  V Derivation of the two-point correlation functions

###  V.1 Static Correlation Functions

In this section we derive the expressions for the different correlation
functions shown above. Our main focus is the case $\ell\ll 1$ where two
distinct phases coexist in the system within the binodal. However, in
subsection V.3 we also consider the case when $\ell$ is not small. While in
this case one cannot identify two distinct phases, the stability of the
homogeneous phase can be studied and the correlation functions in it can be
evaluated.

In the hydrodynamic limit $L\gg 1$, the two-point functions are found by
solving for small fluctuations $\rho=\rho_{0}+\delta\rho/\sqrt{L}+\dots$ and
$m=\delta m/\sqrt{L}+\dots$, around the steady-state homogeneous profiles. For
convenience we rescale the variables so that $x\rightarrow\ell\,x$,
$t\rightarrow\gamma\,t$. Then $-1/\ell<x<1/\ell$ and for small $\ell$ we can
use the continuum Fourier transforms

$\delta\tilde{m}\left(k,t\right)=\frac{1}{2\pi}\int dx~{}e^{-ikx}\delta
m\left(x,t\right)\quad;\quad\delta\tilde{\rho}\left(k,t\right)=\frac{1}{2\pi}\int
dx~{}e^{-ikx}\delta\rho\left(x,t\right)$ (33)

in Eqs. (4) and (5) with $k$ continuous. We then arrive at the linear system
of equations

$\partial_{t}\left(\begin{array}[]{c}\delta\tilde{\rho}(k,t)\\\
\delta\tilde{m}(k,t)\end{array}\right)=-\mathbb{M}_{k}\left(\begin{array}[]{c}\delta\tilde{\rho}(k,t)\\\
\delta\tilde{m}(k,t)\end{array}\right)+\mathbb{R}(k,t)~{}.$ (34)

Here, the matrix $\mathbb{M}_{k}$ and the random noise $\mathbb{R}(k,t)$ are

$\mathbb{M}_{k}=\left(\begin{array}[]{cc}k^{2}&iCk\\\
iEk&k^{2}+2\end{array}\right)\quad;\quad\mathbb{R}(k,t)=\left(\begin{array}[]{c}Aik\,\tilde{\eta}_{1}(k,t)\\\
\sqrt{k^{2}+2}B\,\tilde{\eta}_{2}(k,t)\end{array}\right)~{}.$ (35)

with $A=\sqrt{2\rho_{0}\left(1-\rho_{0}\right)/(\ell L)}$,
$B=\sqrt{2\rho_{0}/(\ell L)}$, $C=\text{Pe}\left(1-\rho_{0}\right)$ and
$E=\text{Pe}\left(1-2\rho_{0}\right)$, and where $\tilde{\eta}_{1,2}$ are
Gaussian white noises with variance

$\left<\tilde{\eta}_{i}(k,t_{1})\tilde{\eta}_{j}(k^{\prime},t_{2})\right>=\delta_{i,j}\delta(k+k^{\prime})\delta(t_{1}-t_{2}).$
(36)

The steady-state correlations can now be readily calculated using the solution
of Eq. (34)

$\left(\begin{array}[]{c}\delta\tilde{\rho}(k,t)\\\
\delta\tilde{m}(k,t)\end{array}\right)=\int_{-\infty}^{t}\mathrm{d}t^{\prime}~{}e^{-\mathbb{M}_{k}(t-t^{\prime})}\mathbb{R}(k,t^{\prime}).$
(37)

Note that the zeroth mode $k=0$ remains identically zero at all times due to
conservation of the total number of particles in the system. This implies that
the equal time correlation function vanishes at $k=k^{\prime}=0$. Accounting
for this using a delta function whose magnitude is proportional to the finite
inter-modes spacing $l$ gives

$\displaystyle C_{2}^{\rho}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\left\\{\frac{A^{2}\left[4+(6+CE)k^{2}+2k^{4}\right]+B^{2}C^{2}(2+k^{2})}{4(1+k^{2})(2+CE+k^{2})}-\frac{2A^{2}+B^{2}C^{2}}{2\left(2+CE\right)}l\delta\left(k\right)\right\\}\delta(k+k^{\prime}),$
(38) $\displaystyle C_{2}^{m}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\left[\frac{A^{2}E^{2}k^{2}+B^{2}(2+k^{2})(2+CE+2k^{2})}{4(1+k^{2})(2+CE+k^{2})}-\frac{B^{2}}{2}l\delta\left(k\right)\right]\delta(k+k^{\prime}),$
(39) $\displaystyle C_{2}^{\rho,m}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\frac{ik(2+k^{2})(A^{2}E-B^{2}C)}{4(1+k^{2})(2+CD+k^{2})}\delta(k+k^{\prime})\,.$
(40)

Performing the inverse Fourier transform on Eqs. (38)-(40), one arrives at
Eqs. (9)-(11). Notice that the finite $\ell$ correction introduced at the
origin $k=k^{\prime}=0$ via the additional delta functions, results in a
constant offset of the real space two point functions Eqs. (9) and (10). This
$\mathcal{O}\left(1\right)$ offset ensures that the integral over each of the
real space coordinates vanishes, to leading order in $\ell$, as it should, due
to total particle number conservation.

###  V.2 Dynamic Correlation Functions

For dynamic correlation functions, it is useful to use the Fourier transforms

$\delta\tilde{m}(k,\omega)=\frac{1}{(2\pi)^{2}}\int dxdt\>e^{-i(kx-\omega
t)}\delta
m(x,t)\quad;\quad\delta\tilde{\rho}(k,\omega)=\frac{1}{(2\pi)^{2}}\int
dxdt\>e^{-i(kx-\omega t)}\delta\rho(x,t).$ (41)

Inserting these into Eqs. (4) and (5), one obtains at leading order in $L$:

$\left(\begin{array}[]{c}\delta\tilde{\rho}\\\
\delta\tilde{m}\end{array}\right)=\mathbb{M}_{k,\omega}^{-1}\mathbb{R}(k,\omega)~{}.$
(42)

with

$\displaystyle\mathbb{M}_{k,\omega}=\left(\begin{array}[]{cc}-i\omega+k^{2}&iCk\\\
iEk&-i\omega+k^{2}+2\end{array}\right)\quad;\quad\mathbb{R}(k,\omega)=\left(\begin{array}[]{c}Aik\,\tilde{\eta}_{1}(k,\omega)\\\
\sqrt{k^{2}+2}B\,\tilde{\eta}_{2}(k,\omega)\end{array}\right).$ (47)

Here, the random noises satisfy

$\left<\tilde{\eta}_{i}(k,\omega)\tilde{\eta}_{j}(k^{\prime},\omega^{\prime})\right>=\delta_{ij}\delta(k+k^{\prime})\delta(\omega+\omega^{\prime})\,.$
(48)

Solving for $\delta\tilde{\rho}$ and $\delta\tilde{m}$ and averaging over
noise realizations one finds the two-point correlation functions Eq. (16). As
expected these expressions agree with Eqs. (9)-(11) after performing the
inverse Fourier transform and taking the equal-time limit. By rescaling $x\to
bx$ and $t\to b^{z}t$ and using the dynamic correlation given in Eq. (16) and
eigenvalues of the linear equation matrix given in Eq. (35), it is
straightforward to confirm that indeed the dynamic scaling exponent is $z=4$
on scales smaller than the correlation and $z=2$ on scales larger than the
correlation length.

To check this numerically it is more convenient to directly study the
eigenvalues of $\mathbb{M}_{k}$. As Eq. (34) indicates, the eigenvalues
determine the time scale for the relaxation of different modes. According to
Eq. (35), the eigenvalues are given as

$\lambda_{\pm}=(k^{2}+1)\left[1\pm\sqrt{1-\frac{k^{2}(k^{2}+\xi^{-2})}{(k^{2}+1)^{2}}}\right].$
(49)

Here, we are interested in the long-wavelength dynamics governed by small-$k$
modes where

$\lambda_{+}=2+\left(2-\frac{1}{2\xi^{2}}\right)k^{2}-\frac{1}{8}\left(2-\frac{1}{\xi^{2}}\right)^{2}k^{4}+{\cal
O}(k^{6}),\quad\quad\lambda_{-}=\frac{1}{2\xi^{2}}k^{2}+\frac{1}{8}\left(2-\frac{1}{\xi^{2}}\right)^{2}k^{4}+{\cal
O}(k^{6})~{}.$ (50)

Note that $\lambda_{+}\rightarrow 2$ and $\lambda_{-}\rightarrow 0$ as
$k\rightarrow 0$, so that the long-time dynamics of the fields are governed by
$\lambda_{-}$. In Fig. (4), we plot the relaxation dynamics of the principal
mode when the system is initially prepared in a square-wave form with
$\rho(x)=1$ for $0\leq x<\rho_{0}$ and $\rho(x)=0$ for $\rho_{0}<x<1$, and
$m(x)=0$. The simulation results, shown in symbols, display good agreement
with the analytical relaxation $e^{-\lambda_{-}t}$ with $\lambda_{-}$ from Eq.
(50).

As the critical point is approached, the correlation length diverges and
$\lambda_{-}$ becomes proportional to $k^{4}$ as shown by Eq. (50), indicating
the critical slowing down addressed with the correlation function above.
Unfortunately, a direct numerical confirmation of the critical slowing down
turned out to be beyond the reach of our numerics. Nearby the critical point,
the Ginzburg criterion Eq. (18) requires $L\gg\xi$ for the results to hold
while hydrodynamic time scale, required for the simulations, diverges as
$L^{2}$. This makes it difficult to capture the long-time behavior of large
systems in numerics using our numerical capabilities.

Figure 4: Relaxation dynamics of principal modes of the density fields
starting from the initial configuration given in the square-wave form.
Simulation results in symbols are compared with
$\exp[\lambda_{-}(k_{1},\xi)t]$ shown in solid lines. Parameters: $L=2^{11}$,
$\gamma=3\times 10^{2}$, and $\rho_{0}=1/4$.

###  V.3 Correlation functions for finite $\ell$

In problems presenting a phase coexistence or a discontinuous phase
transition, finite systems can present different features than those in the
thermodynamic limit. It is thus of interest to study how finite-size effects
manifest themselves in the system studied here. As we have shown, the finite
system results also allow us to compare the critical behavior that we report
here to that of the ABC model, which was studied in a finite system in
Bodineau  _et al._ (2008).

To do this, it is instructive to first discuss the deterministic steady state
of the system at finite $\ell$, as described by the noiseless hydrodynamics
(2)-(3). At finite $\ell$ the system no longer displays MIPS. Nevertheless, at
large enough Péclet numbers, the homogeneous state becomes linearly unstable
along a (modified) ‘spinodal’ line Kourbane-Houssene  _et al._ (2018) which is
found by analyzing the eigenvalues of the linearized dynamics Eq. (49).

Noting that at finite $\ell$ the discrete nature of the Fourier modes,
$k=2\pi\ell n$ with $n$ a non-zero integer, has to be accounted for (the
zeroth mode $k=0$ is stationary due to total particle number conservation) the
first mode to become unstable is $k_{1}=2\pi\ell$. This happens when
$\lambda_{-}$ changes sign at $k_{1}^{2}+\xi^{-2}<0$ (where $\xi$ in Eq.(12)
is purely imaginary), which sets the instability condition

$\text{Pe}^{2}>\frac{2+\left(2\pi\ell\right)^{2}}{(1-\rho_{0})(2\rho_{0}-1)}.$
(51)

Note that at finite $\ell$, the instability sets in at larger Péclet values
when compared to the small $\ell$ limit. In particular, there is no
instability at $\text{Pe}=4$ and density $\rho=3/4$. However, as in the small
$\ell$ limit, the homogeneous solution can become metastable before the
‘spinodal’ curve (51) is reached. While at vanishing $\ell$ this happens along
the binodal curve depicted in Fig. 2 Kourbane-Houssene  _et al._ (2018), at
finite $\ell$ at the moment there is no prescription for finding this line in
a systematic manner. Therefore, in what follows, we are able to calculate the
correlations about the flat state which may be, depending on the region in the
phase diagram and outside the binodal, stable or meta-stable. Interestingly,
even when the homogeneous state is metastable, for large $L$, it is long-
lived. Since the fluctuations which drive the system out of the metastable
state scale as $L^{-1/2}$, the mean time for the system to relax to the
globally stable state is expected to grow exponentially with $L$.

The derivation proceeds along the lines of Sec. V where the continuous Fourier
transform is replaced by a Fourier series for the fields
$f(x)=\delta\rho(x),\delta m(x)$ in a system of finite length $1/\ell$:

$\displaystyle\tilde{f}_{n}=\ell\int_{0}^{1/\ell}\mathrm{d}x~{}f(x)e^{2\pi
in\ell x}\quad;\quad f(x)=\sum_{n=-\infty}^{\infty}\tilde{f}_{n}e^{-{2\pi
in\ell x}}~{}.$ (52)

Figure 5:  Two-point correlation functions displayed throughout the whole
range $x-x^{\prime}\in[0:1)$. The average density is fixed at $\rho_{0}=3/4$
while $\mathrm{Pe}$ goes from 3 to 5.5. The simulation results are shown in
symbols and compared to Eq. (55) shown in lines. Parameters: $L=2\times
10^{3}$, $\gamma=10^{1}$, and $D=1$. Linear instability is expected at
$\mathrm{Pe}=6.90$ according to Eq. (51).

With Eqs. (4) and (5), these give a discrete version of Eq. (34)

$\partial_{t}\left(\begin{array}[]{c}\delta\tilde{\rho}_{n}(t)\\\
\delta\tilde{m}_{n}(t)\end{array}\right)=-\mathbb{M}_{n}\left(\begin{array}[]{c}\delta\tilde{\rho}_{n}(t)\\\
\delta\tilde{m}_{n}(t)\end{array}\right)+\mathbb{R}_{n}(t)~{}\;,$ (53)

where the matrix $\mathbb{M}_{n}$ and the noise $\mathbb{R}_{n}(t)$ are given
by

$\mathbb{M}_{n}=\left(\begin{array}[]{cc}k_{n}^{2}&iCk_{n}\\\
iEk_{n}&k_{n}^{2}+2\end{array}\right)\quad;\quad\mathbb{R}_{n}(t)=\left(\begin{array}[]{c}Aik_{n}\,\tilde{\eta}_{1,n}(t)\\\
\sqrt{k_{n}^{2}+2}B\,\tilde{\eta}_{2,n}(t)\end{array}\right)~{}.$ (54)

with $k_{n}=2\pi\ell n$ and
$\left<\tilde{\eta}_{i,n_{1}}\left(t_{1}\right)\tilde{\eta}_{j,n_{2}}\left(t_{2}\right)\right>=\ell\delta_{i,j}\delta_{n_{1},-n_{2}}\delta\left(t_{1}-t_{2}\right)$.
Solving the coupled equations and performing the inverse Fourier transform one
finds,

$\displaystyle\\!\\!\\!\\!C_{2}^{\rho}(x,x^{\prime})=\frac{\rho_{0}(1-\rho_{0})}{L}\left[\delta(x-x^{\prime})-1\right]+\frac{1}{L}\frac{\text{Pe}^{2}\xi^{2}\rho_{0}^{2}(1-\rho_{0})^{2}}{2\ell(\xi^{2}-1)}\left[f(|x-x^{\prime}|)+g_{\xi}(|x-x^{\prime}|)\right]$
(55)

where we defined

$f(|x-x^{\prime}|)\equiv-\left[\frac{\cosh\left(\frac{|x-x^{\prime}|}{\ell}-\frac{1}{2\ell}\right)}{2\sinh\left(\frac{1}{2\ell}\right)}-\ell\right],\quad\text{and}$
(56)

$\displaystyle g_{\xi}(|x-x^{\prime}|)\equiv$
$\displaystyle\frac{2\xi^{2}-1}{\xi}\left[\frac{\cosh\left(\frac{|x-x^{\prime}|}{\ell\xi}-\frac{1}{2\ell\xi}\right)}{2\sinh\left(\frac{1}{2\ell\xi}\right)}-\ell\xi\right]$
$0<\xi^{-2}$ (57) $\displaystyle g_{\xi}(|x-x^{\prime}|)\equiv$
$\displaystyle\frac{1}{\ell}\left(\frac{1}{6}+|x-x^{\prime}|^{2}-|x-x^{\prime}|\right)$
$\xi^{-1}=0$ (58) $\displaystyle g_{\xi}(|x-x^{\prime}|)\equiv$
$\displaystyle\frac{2\xi^{2}-1}{|\xi|}\left[\frac{\cos\left(\frac{|x-x^{\prime}|}{\ell|\xi|}-\frac{1}{2\ell|\xi|}\right)}{2\sin\left(\frac{1}{2\ell|\xi|}\right)}-\ell|\xi|\right]$
$-k_{1}^{2}<\xi^{-2}<0$ . (59)

Here $|\xi|$ is the modulus of $\xi$, and $k_{1}=2\pi\ell$. Note that Eq. (59)
is the analytic continuation of Eq. (57) obtained from the $0<\xi^{-2}$ regime
to the $\xi^{-2}<0$ one where $\xi$ becomes purely imaginary. Also, notice
that the integral of Eq. (55) over each of the coordinates $x,x^{\prime}$
separately, vanishes due to total particle number conservation. We compare the
expressions for finite $\ell$ correlation to simulation results of the
microscopic dynamics, all of which show excellent agreement as presented in
Fig. 5.

Eq. (55) can now be used to describes the finite $\ell$ rounding of the
critical behavior predicted by Eq. (9). Consider first the limit where we take
$\ell\to 0$ before approaching the critical point
$\left(\rho_{c}=3/4,\text{Pe}=4\right)$. In this limit, we recover the
infinite system result since Eq. (55) converges to Eq. (9) and the region for
Eq. (59) shrinks to zero since $k_{1}\to 0$. Here once $\xi$ becomes of the
order of $1/\ell$, the small $\ell$ approximation breaks down and one must
resort to the finite $\ell$ expression Eq. (55) where no singularity is
present at $\text{Pe}=4$ as discussed above. For $\text{Pe}<4$ where
$\xi^{-2}>0$ expression Eq. (57) holds. At $\text{Pe}=4$ and $\rho_{0}=3/4$
the correlation function takes the form of the parabola Eq. (58) in contrast
to the $\ell\to 0$ where there is no valid analytical expression. Then for
$\text{Pe}>4$ such that $-k_{1}^{2}<\xi^{-2}<0$ expression Eq. (59) holds.
These results are illustrated in Fig. 5.

Note that beyond $\text{Pe}=4$ for finite $\ell$ the correlation length spans
the whole system size. To see this it is useful to evaluate the span of the
two-point function using the first moment

$\langle x\rangle\equiv\int_{0}^{1}dx~{}\tilde{C}_{2}^{\rho}\left(x\right)x,$
(60)

with

$\tilde{C}_{2}^{\rho}\left(x\right)\equiv\frac{C_{2}^{\rho}\left(x\right)-\min_{x}C_{2}^{\rho}\left(x\right)}{\int_{0}^{1}dx\left[C_{2}^{\rho}\left(x\right)-\min_{x}C_{2}^{\rho}\left(x\right)\right]}$
(61)

a normalized and shifted two point function so that serves as a proper
distribution function. This measure grows from a small value which is of order
$\ell$ for $\text{Pe}<4$ to the order of the system size at $\text{Pe}=4$ and
remains at the order of the system size up to the linear instability point.
This is illustrated in Fig. 6 and compared to the $\ell\to 0$ case where the
span $\langle x\rangle$ diverges at $\text{Pe}=4$.

Figure 6:  The first moment (60) at $\rho_{0}=\frac{3}{4}$ and $\ell=0.02$ as
a function of the Péclet number. The dashed line corresponds to the vanishing
$\ell$ expression (9). The first moment grows from the value $\ell\ll 1$ at
$\text{Pe}=0$ until it spans the system and becomes
$\mathcal{O}\left(1\right)$ around $\text{Pe}=4$ and up to the instability
point at $\text{Pe}\approx 4.015$ [obtained from (51)]. The two insets show
the two point function (55) well before $\text{Pe}=4$ (upper inset), and
beyond that point (lower inset).

Finally, the results above also allow one to compare the phase transition to
that found in the version of the ABC model studied in Clincy  _et al._ (2003).
In the ABC model, each site can be occupied by either an A, B, or C particle
and the exchange dynamics lead to phase separation. For average densities
$r_{A,B,C}$ of the three species, and denoting
$\Delta=1-2(r_{A}^{2}+r_{B}^{2}+r_{C}^{2})$, it was shown that a continuous
transition into a phase separated state occurs at $\beta=\beta_{c}\equiv
2\pi/\sqrt{\Delta}$ Clincy  _et al._ (2003). Moreover, in the homogeneous
phase the connected two-point correlation function is Bodineau  _et al._
(2008):

$\langle\rho_{A}(x)\rho_{A}(x^{\prime})\rangle_{c}=\frac{1}{L}r_{A}(1-r_{A})\big{(}\delta(x-x^{\prime})-1\big{)}-\frac{3}{L}r_{A}^{2}r_{B}r_{C}\bigg{(}\frac{\cos\frac{\beta\sqrt{\Delta}(1-2x-2x^{\prime})}{2}}{\sqrt{\Delta}\sin\frac{\beta\sqrt{\Delta}}{2}}-\frac{2}{\beta\Delta}\bigg{)}\>.$
(62)

Note that at the critical point the amplitude of the non-local contribution
(62) to the correlation function diverges due to its denominator, in the same
way as is observed in Eq. (59). However, this phase transition is not
associated with a divergent correlation length, in contrast to the critical
features of the active model studied in this paper. Indeed, in the ABC model,
Eq. (62) shows that the correlation length spans the entire system also away
from criticality, and does not grow as $\beta\to\beta_{c}$, while in our case
$\xi\ell$ does become formally divergent as $\text{Pe}\to\text{Pe}_{{c}}$
(i.e. tends to the full system size). In comparison to the model we study
here, in the ABC model the domain wall width, which is known to behave as
$1/|\ln q|\propto L/\beta$ is always of the order of the system size. In our
case, the domain wall width is controlled by $\ell$ which can be held at any
value as the critical point is approached. In particular, in the $\ell\to 0$
limit the domain wall vanishes. Then our system fall into the more
conventional phase separation picture. In contrast, in the ABC model the
domain wall is not an independent parameter and takes a large value at the
critical point. In sum, in comparison to the ABC model, the model presents
more standard features of critical phenomena

##  VI Discussion

In this paper, we build upon a recent progress in deriving the exact noiseless
hydrodynamics of active systems Erignoux (2018); Kourbane-Houssene  _et al._
(2018). We complement the noiseless description by accounting for fluctuations
at the Gaussian level as given in (4)-(8). This description enabled us to
compute the two-point correlation functions in the homogeneous phase, and to
characterize exactly the critical behavior exhibited by the model. Such
treatment has thus far been missing, and provides a first conclusive
analytical evidence that, at least for the model studied here, MIPS belongs to
the mean field Ising universality class when the infinite system size is taken
properly. Our findings add to a number of numerical works Partridge and Lee
(2019); Caballero  _et al._ (2018), and phenomenological field theoretical
studies Partridge and Lee (2019); Maggi  _et al._ (2021) which argue the same,
even if the models studied previous are distinct in that the rates do not
scale with the system size. In addition, we considered the scaling of dynamic
correlation functions and relaxation eigenvalues of long-wavelength
excitations near the critical point and found that they exhibit critical
slowing down with a mean-field model B exponent $z=4$ (although, as we
discussed in Sec. V.2, such exponent was out of our numerical reach). It would
be interesting to study correlation in the non-homogeneous phase where the
length scale $\ell$ should have a role.

It is interesting to place the results in the context of general results on
non-equilibrium systems which phase separate in one dimension (see for example
Arndt  _et al._ (1999); Lahiri and Ramaswamy (1997); Lahiri  _et al._ (2000);
Kafri  _et al._ (2002, 2003)). In those works non-diffusive systems were
studied and a condition for the existence of phase separation, based on the
dynamics of domain walls, was derived Kafri  _et al._ (2002). Here the class
of systems is rather different in that it is diffusive and is more in line
with the diffusive ABC model introduced in Clincy  _et al._ (2003). While in
the latter phase separation was a result of a chiral structure it would be
interesting to understand how phase separation occurs in the active model
studied here. In particular, it is not clear what features of the domain wall
dynamics indicate whether a domain is small or large so that system can
eventually phase separate.

While the manuscript derives the fluctuations at the Gaussian level, in a
forthcoming publication for (2021) we will present the complete account of
large fluctuations beyond the Gaussian approximation within an adapted
Macroscopic Fluctuation Theory.

Acknowledgments – We are grateful to A. Frishman and J. Tailleur for
discussions and a careful reading of the manuscript. YK and SR are supported
by an ISF grant and the National Research Foundation of Korea
(2019R1A6A3A03033761).. VL acknowledges financial support from the ERC
Starting Grant No. 680275 MALIG, the ANR-18-CE30-0028-01 Grant LABS and the
ANR-15-CE40-0020-03 Grant LSD.

## References

  * Ramaswamy (2010) S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  * Thompson  _et al._ (2011) A. G. Thompson, J. Tailleur, M. E. Cates, and R. A. Blythe, Journal of Statistical Mechanics: Theory and Experiment 2011, P02029 (2011).
  * Romanczuk  _et al._ (2012) P. Romanczuk, M. Baer, W. Ebeling, B. Lindner, and L. Schimansky-Geier, Eur. Phys. J.-Spec. Top. 202, 1 (2012).
  * Marchetti (2013) M. C. Marchetti, Rev. Mod. Phys. 85, 1143 (2013).
  * Takatori and Brady (2015) S. C. Takatori and J. F. Brady, Phys. Rev. E 91, 032117 (2015).
  * Ramaswamy (2017) S. Ramaswamy, J. Stat. Mech. 2017, 054002 (2017).
  * Needleman and Dogic (2017) D. Needleman and Z. Dogic, Nature Reviews Materials 2, 1 (2017).
  * Loi  _et al._ (2008) D. Loi, S. Mossa, and L. F. Cugliandolo, Phys. Rev. E 77, 051111 (2008).
  * Solon  _et al._ (2015a) A. P. Solon, Y. Fily, A. Baskaran, M. E. Cates, Y. Kafri, M. Kardar, and J. Tailleur, Nature Physics 11, 673 (2015a).
  * Fodor (2016) E. Fodor, Phys. Rev. Lett. 117 (2016), 10.1103/PhysRevLett.117.038103.
  * Wittkowski  _et al._ (2014) R. Wittkowski, A. Tiribocchi, J. Stenhammar, R. J. Allen, D. Marenduzzo, and M. E. Cates, Nature communications 5, 1 (2014).
  * Nardini  _et al._ (2017) C. Nardini, É. Fodor, E. Tjhung, F. Van Wijland, J. Tailleur, and M. E. Cates, Physical Review X 7, 021007 (2017).
  * Solon  _et al._ (2018a) A. P. Solon, J. Stenhammar, M. E. Cates, Y. Kafri, and J. Tailleur, New J. Phys. 20, 075001 (2018a).
  * Solon  _et al._ (2018b) A. P. Solon, J. Stenhammar, M. E. Cates, Y. Kafri, and J. Tailleur, Phys. Rev. E 97, 020602 (2018b).
  * Tjhung  _et al._ (2018) E. Tjhung, C. Nardini, and M. E. Cates, Physical Review X 8, 031080 (2018).
  * Peshkov  _et al._ (2012) A. Peshkov, I. S. Aranson, E. Bertin, H. Chaté, and F. Ginelli, Physical review letters 109, 268701 (2012).
  * Bertin  _et al._ (2013) E. Bertin, H. Chaté, F. Ginelli, S. Mishra, A. Peshkov, and S. Ramaswamy, New journal of physics 15, 085032 (2013).
  * Bertin  _et al._ (2015) E. Bertin, A. Baskaran, H. Chaté, and M. C. Marchetti, Physical Review E 92, 042141 (2015).
  * Mallmin  _et al._ (2019) E. Mallmin, R. A. Blythe, and M. R. Evans, Journal of Statistical Mechanics: Theory and Experiment 2019, 013204 (2019).
  * Spohn (1991) H. Spohn,  _Large Scale Dynamics of Interacting Particles_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 1991).
  * Spohn (1983) H. Spohn, Journal of Physics A: Mathematical and General 16, 4275 (1983).
  * Derrida (2007) B. Derrida, J. Stat. Mech. 2007, P07023 (2007).
  * Derrida (2011) B. Derrida, J. Stat. Mech. 2011, P01030 (2011).
  * Bertini  _et al._ (2015) L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Rev. Mod. Phys. 87, 593 (2015).
  * Bodineau and Lagouge (2010) T. Bodineau and M. Lagouge, Journal of Statistical Physics 139, 201 (2010).
  * Shpielberg and Akkermans (2016) O. Shpielberg and E. Akkermans, Phys. Rev. Lett. 116, 240603 (2016).
  * Baek  _et al._ (2017) Y. Baek, Y. Kafri, and V. Lecomte, Phys. Rev. Lett. 118, 030604 (2017).
  * Jack  _et al._ (2015) R. L. Jack, I. R. Thompson, and P. Sollich, Phys. Rev. Lett. 114, 060601 (2015).
  * Lecomte  _et al._ (2012) V. Lecomte, J. P. Garrahan, and F. van Wijland, Journal of Physics A: Mathematical and Theoretical 45, 175001 (2012).
  * Agranov  _et al._ (2019) T. Agranov, P. L. Krapivsky, and B. Meerson, Phys. Rev. E 99, 052102 (2019).
  * Baek and Kafri (2015) Y. Baek and Y. Kafri, Journal of Statistical Mechanics: Theory and Experiment 2015, P08026 (2015).
  * Erignoux (2018) C. Erignoux, arXiv:1608.04937 [math-ph] (2018).
  * Kourbane-Houssene  _et al._ (2018) M. Kourbane-Houssene, C. Erignoux, T. Bodineau, and J. Tailleur, Physical Review Letters 120, 268003 (2018).
  * Kipnis and Landim (1999) C. Kipnis and C. Landim,  _Scaling limits of interacting particle systems_ , Grundlehren der mathematischen Wissenschaften No. 320 (Springer, New York, 1999).
  * Bertini  _et al._ (2007) L. Bertini, A. D. Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, J. Stat. Mech. 2007, P07014 (2007).
  * Liggett (2012) T. M. Liggett, “Interacting Particle Systems,” (2012).
  * Tailleur and Cates (2008) J. Tailleur and M. E. Cates, Phys. Rev. Lett. 100, 218103 (2008).
  * Fily and Marchetti (2012) Y. Fily and M. C. Marchetti, Physical Review Letters 108, 235702 (2012).
  * Buttinoni  _et al._ (2013) I. Buttinoni, J. Bialké, F. Kümmel, H. Löwen, C. Bechinger, and T. Speck, Physical Review Letters 110, 238301 (2013).
  * Cates and Tailleur (2013) M. E. Cates and J. Tailleur, EPL (Europhysics Letters) 101, 20010 (2013).
  * Stenhammar  _et al._ (2013) J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Marenduzzo, and M. E. Cates, Physical Review Letters 111, 145702 (2013).
  * Redner  _et al._ (2013) G. S. Redner, M. F. Hagan, and A. Baskaran, Physical Review Letters 110, 055701 (2013).
  * Cates and Tailleur (2015) M. E. Cates and J. Tailleur, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
  * Solon  _et al._ (2015b) A. P. Solon, M. E. Cates, and J. Tailleur, The European Physical Journal Special Topics 224, 1231 (2015b).
  * Redner  _et al._ (2016) G. S. Redner, C. G. Wagner, A. Baskaran, and M. F. Hagan, Physical Review Letters 117, 148002 (2016).
  * Whitelam  _et al._ (2018) S. Whitelam, K. Klymko, and D. Mandal, The Journal of Chemical Physics 148, 154902 (2018).
  * Paliwal  _et al._ (2018) S. Paliwal, J. Rodenburg, R. van Roij, and M. Dijkstra, New Journal of Physics 20, 015003 (2018).
  * Geyer  _et al._ (2019) D. Geyer, D. Martin, J. Tailleur, and D. Bartolo, Physical Review X 9, 031043 (2019).
  * Caballero  _et al._ (2018) F. Caballero, C. Nardini, and M. E. Cates, Journal of Statistical Mechanics: Theory and Experiment 2018, 123208 (2018).
  * Partridge and Lee (2019) B. Partridge and C. F. Lee, Phys. Rev. Lett. 123, 068002 (2019).
  * Maggi  _et al._ (2021) C. Maggi, M. Paoluzzi, A. Crisanti, E. Zaccarelli, and N. Gnan, Soft Matter 17, 3807 (2021).
  * Siebert  _et al._ (2018) J. T. Siebert, F. Dittrich, F. Schmid, K. Binder, T. Speck, and P. Virnau, Phys. Rev. E 98, 030601 (2018).
  * Dittrich  _et al._ (2021) F. Dittrich, T. Speck, and P. Virnau, “Critical behaviour in active lattice models of motility-induced phase separation,” (2021), arXiv:2010.08387 [cond-mat.stat-mech] .
  * Evans  _et al._ (1998a) M. Evans, Y. Kafri, H. Koduvely, and D. Mukamel, Physical review letters 80, 425 (1998a).
  * Evans  _et al._ (1998b) M. Evans, Y. Kafri, H. Koduvely, and D. Mukamel, Physical Review E 58, 2764 (1998b).
  * Krapivsky and Meerson (2012) P. L. Krapivsky and B. Meerson, Phys. Rev. E 86, 031106 (2012).
  * Note (1) For $\xi=1$, the expressions (9)-(11) seems singular, due to the vanishing denominator $\sim 1-\xi$. Note however that the numerator vanishes at the same order in this limit, and the continuation in $\xi=1$ is analytical. E.g.,  $\frac{1}{1-\xi^{2}}\Big{(}e^{-\frac{1}{\ell}|x-x^{\prime}|}+\frac{1-2\xi^{2}}{\xi}e^{-\frac{1}{\ell}\frac{|x-x^{\prime}|}{\xi}}\Big{)}\xrightarrow[\xi=1]{}\frac{3+\frac{1}{\ell}\,|x-x^{\prime}|}{2}e^{-\frac{1}{\ell}|x-x^{\prime}|}\;.$ (63) .
  * Bodineau  _et al._ (2008) T. Bodineau, B. Derrida, V. Lecomte, and F. van Wijland, J Stat Phys 133, 1013 (2008).
  * Note (2) In (20) and (21) we adopt the re-scaling used in Sec. V. Also, note that the zeroth mode $k=0$ should be excluded from the integration above due to total particle number conservation.
  * Clincy  _et al._ (2003) M. Clincy, B. Derrida, and M. R. Evans, Physical Review E 67, 066115 (2003).
  * Note (3) The account of large fluctuation will be presented in a future publication for (2021).
  * Bodineau and Derrida (2011) T. Bodineau and B. Derrida, J Stat Phys 145, 745 (2011).
  * Doi (1976a) M. Doi, J. Phys. A: Math. Gen. 9, 1465 (1976a).
  * Doi (1976b) M. Doi, J. Phys. A: Math. Gen. 9, 1479 (1976b).
  * Peliti (1985) L. Peliti, Journal de Physique 46, 15 (1985).
  * De Masi  _et al._ (1986) A. De Masi, P. A. Ferrari, and J. L. Lebowitz, J Stat Phys 44, 589 (1986).
  * Jona-Lasinio  _et al._ (1993) G. Jona-Lasinio, C. Landim, and M. E. Vares, Probab. Th. Rel. Fields 97, 339 (1993).
  * Lefèvre and Biroli (2007) A. Lefèvre and G. Biroli, Journal of Statistical Mechanics 2007, P07024 (2007).
  * Tailleur  _et al._ (2008) J. Tailleur, J. Kurchan, and V. Lecomte, J. Phys. A: Math. Theor. 41, 505001 (2008).
  * Gardiner  _et al._ (1985) C. W. Gardiner _et al._ ,  _Handbook of stochastic methods_ , Vol. 3 (springer Berlin, 1985).
  * for (2021) In preparation. (2021).
  * Arndt  _et al._ (1999) P. F. Arndt, T. Heinzel, and V. Rittenberg, Journal of Statistical Physics 97, 1 (1999).
  * Lahiri and Ramaswamy (1997) R. Lahiri and S. Ramaswamy, Physical Review Letters 79, 1150 (1997).
  * Lahiri  _et al._ (2000) R. Lahiri, M. Barma, and S. Ramaswamy, Physical Review E 61, 1648 (2000).
  * Kafri  _et al._ (2002) Y. Kafri, E. Levine, D. Mukamel, G. M. Schütz, and J. Török, Physical Review Letters 89, 035702 (2002).
  * Kafri  _et al._ (2003) Y. Kafri, E. Levine, D. Mukamel, G. M. Schütz, and R. D. Willmann, Physical Review E 68, 035101 (2003).