# Macroscopic Fluctuation Theory and Current Fluctuations in Active Lattice
Gases

Tal Agranov  Department of Applied Mathematics and Theoretical Physics,
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 study the current large deviations for a lattice model of interacting
active particles displaying a motility-induced phase separation (MIPS). To do
this, we first derive the exact fluctuating hydrodynamics of the model in the
large system limit. On top of the usual Gaussian noise terms the theory also
presents Poissonian noise terms, that we fully account for. We find a
dynamical phase transition between flat density profiles and sharply phase-
separated traveling waves, and we derive the associated phase diagram together
with the large deviation function for all phases, including the one displaying
MIPS. We show how the results can be obtained using methods similar to those
of equilibrium phase separation, in spite of the nonequilibrium nature of the
problem.

##  I Introduction

The modeling of active matter has attracted a lot of attention in recent years
Loi  _et al._ (2008); Ramaswamy (2010); Thompson  _et al._ (2011); Romanczuk
_et al._ (2012); Marchetti  _et al._ (2013); Soto and Golestanian (2014);
Solon  _et al._ (2015); Takatori and Brady (2015); Slowman  _et al._ (2016);
Fodor  _et al._ (2016); Ramaswamy (2017); Needleman and Dogic (2017); Martin
_et al._ (2021); Solon  _et al._ (2022); O’Byrne  _et al._ (2022). This is due
to the many novel phenomena exhibited by active systems that are not seen in
equilibrium. The macroscopic description of active systems is based, in many
cases, on noiseless phenomenological field theories, mean-field approximations
or gradient expansions Toner and Tu (1995, 1998); 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). Noise is typically
added in as Gaussian contributions, with limited control. Understanding the
nature of noise in a more rigorous way is becoming important with the growing
interest in fluctuations in active systems Cagnetta  _et al._ (2017); GrandPre
and Limmer (2018); Whitelam  _et al._ (2018); Tociu  _et al._ (2019);
Gradenigo and Majumdar (2019); Nemoto  _et al._ (2019); Cagnetta and Mallmin
(2020); Chiarantoni  _et al._ (2020); Fodor  _et al._ (2020); Banerjee  _et
al._ (2020); Yan  _et al._ (2022); GrandPre  _et al._ (2021); Keta  _et al._
(2021). Recently, an active lattice gas model with an exact hydrodynamic
description has been introduced in Refs. Kourbane-Houssene  _et al._ (2018);
Erignoux (2021). While the model presents a diffusive scaling, it reproduces
characteristic features of scalar active systems, namely, motility-induced
phase separation (MIPS) Tailleur and Cates (2008); Cates and Tailleur (2015).
Interestingly, this phase transition already appears in one dimension.

In a previous work Agranov  _et al._ (2021), we derived the fluctuating
hydrodynamics of this model in the regime of small typical Gaussian
fluctuations. This enabled us to find the static and dynamical correlation
functions of the model in the homogeneous phase. In particular, close to the
critical point, we showed that the scaling exponents belong to the Ising mean-
field universality class.

It is also of interest to understand large fluctuations in such models beyond
the typical regime which is captured by small Gaussian fluctuations. These are
captured by large-deviation theory and have been studied extensively in non-
equilibrium systems (see for instance Refs. Lebowitz and Spohn (1999); Derrida
(2007); Touchette (2009); Jona-Lasinio (2010)). Much work has focused on
models which admit a diffusive scaling between space and time coordinates when
taking the large system size limit Spohn (1983, 1991). In this limit the noise
is small, and in many cases both typical and atypical fluctuations are
Gaussian. Then the fluctuations are fully described by the Macroscopic
Fluctuation Theory (MFT) (see Bertini  _et al._ (2015a) for a review). This
framework allows one to successfully describe the distribution of several
observables of interest, such as the density profile or the time-integrated
current flowing through the system Derrida  _et al._ (2004); Bertini  _et al._
(2006); Derrida (2007); Bodineau and Derrida (2004a). One of the most
interesting features exhibited by the fluctuations are dynamical phase
transitions, which occur when the large-deviation function is singular Derrida
and Lebowitz (1998); Bodineau and Derrida (2005); Bertini  _et al._ (2005,
2006); Appert-Rolland  _et al._ (2008); Bodineau  _et al._ (2008); Lecomte
_et al._ (2012); Baek  _et al._ (2017, 2018)

However, systems in which particles not only diffuse by jumps but also undergo
reactions evade a Gaussian MFT description, as is was found in equilibrium
Jona-Lasinio  _et al._ (1993) or driven Bodineau and Lagouge (2010) models.
Although the scalings in such systems are still diffusive, one has to take
into account the Poissonian nature of the noise if one wants to fully describe
their fluctuations. The active model we are interested in falls in that class:
it consists of left- and right-moving particles which jump but also tumble.
The latter is represented by a reaction where one type of particle transforms
into the other.

In this paper, we derive a non-Gaussian MFT that fully encompasses the
Poissonian noise, extending our previous work on small Gaussian fluctuations
Agranov  _et al._ (2021). We then apply this extended MFT framework to study
the distribution of the time-integrated current flowing through the system.
These were studied previously analytically only for non interacting particles
Banerjee  _et al._ (2020) or numerically in the interacting case GrandPre and
Limmer (2018).

We find that current fluctuations in the active lattice gas model exhibits a
dynamical phase transition (DPT) and we explain how it can be derived using
techniques very similar to those used in equilibrium phase transitions. The
DPT occurs between a homogeneous profile and a sharply phase separated
travelling wave. We also study in detail how this transition connects to the
zero current phase diagram which exhibits MIPS. To do this we account for
finite system lengths.

The structure of the paper is as follows: the model and its hydrodynamics are
described in Sec. II. The fluctuating hydrodynamics (beyond the Gaussian
regime) is derived in Sec. III. In Sec. IV, we obtain the corresponding MFT of
the active lattice gas and use it to study the current large deviations. The
dynamical phase transition presented by the system is studied in Sec. V. Sec.
VI establishes the relation between the MIPS and the DPT. We conclude in Sec.
VII. Technical steps of our derivations are gathered in appendices.

##  II The active lattice gas model and its hydrodynamics

The active lattice gas model, introduced in Ref. Kourbane-Houssene  _et al._
(2018), is defined on a one-dimensional periodic lattice with $L$ sites. Each
site $i$ can be in either one of three states: occupied by a $+$ particle,
occupied by a $-$ particle, or empty. The dynamics is defined through the
following rates (see Fig. 1):

  1. (i)

A pair of neighboring sites exchange their states (if different) with rate
$D_{0}$.

  2. (ii)

A $+$ ($-$) particle hops using self-propulsion to the right (left)
neighboring site with rate $\lambda/L$, provided that 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 travel across $L$ sites, either through diffusive
motion or using self-propulsion, scales as $L^{2}$ which is also the time
scale for tumbling events.

Figure 1: Schematic representation of the microscopic dynamics of the active
lattice gas model. Particles self-propel with rate $\lambda/L$ to vacant sites
and tumble at rate $\gamma/L^{2}$. They also perform diffusive swap with an
occupied or vacant neighbor at equal rate $D_{0}$.

The hydrodynamic equations of the model are obtained by defining the coarse-
grained particle density fields

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

with $\sigma_{i}^{+}=1$ ($\sigma_{i}^{-}=1$) if site $i$ is occupied by a $+$
($-$) particle and $\sigma_{i}^{+}=0$ ($\sigma_{i}^{-}=0$) otherwise. The
exponent $0<\delta<1$ defines a mesoscopic length, $L^{\delta}$, that scales
sub-linearly with the system size. In what follows it is useful to use the
rescalings $t\rightarrow\gamma t/L^{2}$ and $x=\ell_{s}i/L$, where
$\ell_{s}^{-1}\equiv\sqrt{D_{0}/\gamma}$ is the typical distance traveled by a
particle using only diffusive steps until it tumbles. Note that
$x\in[0,\ell_{s}]$ so that $\ell_{s}$ plays the role of system length for the
macroscopic coordinate $x$. In Refs. Kourbane-Houssene  _et al._ (2018);
Agranov  _et al._ (2021) it was shown that the density fields defined in Eq.
(1) obey the fluctuating hydrodynamic equations

$\displaystyle\partial_{t}\rho_{+}=-\partial_{x}J_{+}-K\;,\qquad\partial_{t}\rho_{-}=-\partial_{x}J_{-}+K.$
(2)

Here $J_{+}$ and $J_{-}$ are conservative density fluxes that arise due to the
diffusion and self-propulsion. Their deterministic components are given by the
expressions

$\bar{J}_{+}=-\partial_{x}\rho_{+}+\text{Pe}\rho_{+}(1-\rho)\;,\qquad\bar{J}_{-}=-\partial_{x}\rho_{-}-\text{Pe}\rho_{-}(1-\rho)$
(3)

where $\rho(x,t)=\rho_{+}(x,t)+\rho_{-}(x,t)$ is the total density of
particles and $\operatorname{Pe}=\lambda/\sqrt{\gamma D_{0}}$ is the Péclet
number, which compares the persistence length $\lambda/\gamma$ to the
diffusive one $\ell_{s}^{-1}$. The non-conservative term $K$ arises from
tumbling. It is the local rate at which $+$ particles tumble into $-$, minus
the rate of the opposite reaction. Its deterministic component is given by

$\bar{K}=\rho_{+}-\rho_{-}.$ (4)

For our purposes, it is more convenient to use the density and polarization
fields,

$\displaystyle\rho(x,t)=\rho_{+}+\rho_{-}\;,\qquad m(x,t)=\rho_{+}-\rho_{-}$
(5)

respectively, which follow the dynamics

$\displaystyle\partial_{t}\rho$ $\displaystyle=$
$\displaystyle-\partial_{x}{J}_{\rho}$ $\displaystyle\partial_{t}m$
$\displaystyle=$ $\displaystyle-\partial_{x}{J}_{m}-2{K}\>,$ (6)

where the density and polarization fluxes are defined as $J_{\rho}\equiv
J_{+}+J_{-}$ and $J_{m}\equiv J_{+}-J_{-}$. Correspondingly, the deterministic
components of the conservative fluxes and tumbling rate in these variables are
given by

$\displaystyle\bar{J}_{\rho}=-\partial_{x}\rho+\operatorname{Pe}m\left(1-\rho\right)\;,\qquad\bar{J}_{m}=-\partial_{x}m+\operatorname{Pe}\rho\left(1-\rho\right)\;,\qquad\bar{K}=m\>.$
(7)

In the $L\to\infty$ limit, the fluctuations are suppressed and Eq. (6) reduces
to its deterministic hydrodynamics form given by $J_{\rho}$, $J_{m}$, and $K$
replaced by their average values given in Eq. (7). In this limit, the
hydrodynamics predicts that the system relaxes to a steady state given by the
stationary solutions of Eq. (6).

In Ref. Kourbane-Houssene  _et al._ (2018), it was shown that for large enough
$\operatorname{Pe}$ and mean density
$\rho_{0}=\ell_{s}^{-1}\int_{0}^{\ell_{s}}\rho\left(x,t\right)dx$, the
equations allow for non-homogeneous stationary solutions corresponding to
motility-induced phase-separation (MIPS). These solutions consist of
coexisting high- and low-density phases separated by sharp domain walls, with
the ratio between the system size and domain wall width controlled by
$\ell_{s}$. In the $\ell_{s}\gg 1$ asymptotics, the densities in the phase-
separated state are independent of the mean density $\rho_{0}$. They can be
found using an effective common tangent construction, see refs. Solon  _et
al._ (2018a, b). The resulting phase diagram, obtained in Ref. Kourbane-
Houssene  _et al._ (2018), is shown in Fig. 2. It shares similarities with
equilibrium phase separation, despite the very different origin of the two
phenomena Cates and Tailleur (2015). The figure also shows the spinodal line,
determined by the equation
$2-\operatorname{Pe}^{2}(1-\rho_{0})(2\rho_{0}-1)=0$, beyond which the
homogeneous state becomes linearly unstable. The contact point of the MIPS
line and the spinodal line defines the MIPS critical point
$(\rho_{c},\operatorname{Pe}_{c})=(3/4,4)$.

Figure 2: The phase diagram of the model. The binodal is denoted by a solid
black line and the spinodal by a dashed line. They meet at
$\left(\rho_{0},\operatorname{Pe}\right)=\left(3/4,4\right)$ which marks the
critical point (red circle). In a MIPS state, the high- and low-density
coexisting phases lay on the binodal.

##  III Fluctuating Hydrodynamics

For systems with finite $L$, the fluctuations cannot be neglected anymore and
the full statistics of ${J}_{\rho}$, ${J}_{m}$, and ${K}$ has to be taken into
account.

On general grounds the fluctuations scale as $L^{-1/2}$ at large $L$. In Ref.
Agranov  _et al._ (2021), we exploited an exact mapping to the well studied
ABC lattice gas model Evans  _et al._ (1998a, b); Bodineau and Derrida (2011)
to derive an expression for typical small Gaussian fluctuations. Using these
we were able to characterize the critical behavior of the model exactly,
finding that it belongs to the mean-field Ising universality class both for
static and dynamic properties.

In this paper, we extend the above results to account for arbitrary large
macroscopic fluctuations, beyond the Gaussian small-fluctuations regime. As we
show, the fluctuations of the conserved fluxes $J_{\rho}$ around
${\bar{J}}_{\rho}$ and $J_{m}$ around ${\bar{J}}_{m}$ are described by
Gaussian noise terms even for large fluctuations. This is the standard case in
related lattice gases which are in local equilibrium Bertini  _et al._
(2015b). The reason is that the fluxes are averaged over fast diffusive
exchanges with a rate $D_{0}$ that does not scale with $L$ Lefèvre and Biroli
(2007). In contrast, the large fluctuations of $K$ around ${\bar{K}}$ are not
Gaussian. They come from slow local tumbling events which follow Poisson
statistics. The technical steps of the derivation of these results are
presented in Appendix A.

The joint statistics of fluxes and tumbling rate fluctuations provide a
complete statistical description of macroscopic fluctuations in the active
lattice gas model, which is the first main result of this paper. It is given
in terms of the probability path measure, $P$, of observing a history of the
fields $\rho(x,t)$, $m(x,t)$, $J_{\rho,m}(x,t)$, $K(x,t)$ which, at large $L$,
follows the large-deviation form

$\displaystyle-\ln P\left[\rho,m,J_{\rho,m},K\right]$ $\displaystyle\simeq$
$\displaystyle L\ell_{s}^{-1}\hat{\mathcal{S}}_{\cal
L}\left[\rho,m,J_{\rho,m},K\right]$ $\displaystyle\hat{\mathcal{S}}_{\cal
L}\left[\rho,m,J_{\rho,m},K\right]$ $\displaystyle=$
$\displaystyle\int_{0}^{T}dt\int_{0}^{\ell_{s}}dx\left(\mathcal{L}_{J}+\mathcal{L}_{K}\right).$
(8)

Here $\mathcal{L}_{J}$ accounts for the statistics of the fluxes and is given
by the quadratic form

$\mathcal{L}_{J}=\frac{1}{2}\begin{bmatrix}J_{\rho}-\bar{J}_{\rho}\\\
J_{m}-\bar{J}_{m}\end{bmatrix}^{\mathrm{T}}\mathbf{C}^{-1}\begin{bmatrix}J_{\rho}-\bar{J}_{\rho}\\\
J_{m}-\bar{J}_{m}\end{bmatrix},$ (9)

with the correlation matrix $\mathbf{C}$ given by

$\mathbf{C}=\begin{bmatrix}\sigma_{\rho}&\sigma_{\rho,m}\\\
\sigma_{\rho,m}&\sigma_{m}\end{bmatrix},$ (10)

with entries

$\displaystyle\sigma_{\rho}$ $\displaystyle=$ $\displaystyle
2\rho\left(1-\rho\right)\quad;\quad\sigma_{m}=2\left(\rho-m^{2}\right)\quad;\quad\sigma_{\rho,m}=2m\left(1-\rho\right).$

The $\mathcal{L}_{K}$ term in the action, accounting for the Poisson tumble
statistics is given by:

$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle\rho-\sqrt{K^{2}+\left(\rho^{2}-m^{2}\right)}+K\ln\left[\frac{\sqrt{K^{2}+\left(\rho^{2}-m^{2}\right)}+K}{\left(\rho+m\right)}\right].$
(11)

The action above is presented in its Lagrangian form. As usual in MFT, the
fields $\rho(x,t)$, $m(x,t)$ and the currents $J_{\rho,m}(x,t)$, $K(x,t)$ are
imposed, in the path measure, to satisfy conservation equations given by Eqs.
(6). We now turn to use these expressions to derive the LDF for the total
integrated current flowing in the system using the macroscopic fluctuation
theory.

##  IV Macroscopic Fluctuation Theory

The total integrated current flowing through the system in a time interval $T$
is given by

$Q=\int_{0}^{\ell_{s}}dx\int_{0}^{T}dt\,J_{\rho}\>.$ (12)

At long times $T\gg 1$ and for large number of sites $L\gg 1$, the
distribution of $Q$ takes the large-deviation form

$-\ln P(Q)\simeq LTI(q),$ (13)

where

$q=\frac{Q}{T\ell_{s}},$ (14)

is a rescaled integrated current. As the system is reflection symmetric we
have $I(q)=I(-q)$. For simplicity we will consider from now on only $q>0$. To
find the rate function $I(q)$, we first determine the scaled cumulant
generating function (SCGF)

$\psi(\Lambda)=\frac{1}{LT}\ln\langle e^{\Lambda LTq}\rangle.$ (15)

Similar to equilibrium thermodynamics, following Varadhan’s lemma Touchette
(2009), the rate function $I$ is related to the SCGF by a Legendre–Fenchel
transform,

$I(q)=\sup_{\Lambda}\left[\Lambda q-\Psi(\Lambda)\right].$ (16)

The SCGF can be expressed via a path-integral formulation :

$e^{LT\Psi(\Lambda)}=\int{\cal D}\rho{\cal D}m{\cal D}J_{\rho}{\cal
D}J_{m}{\cal D}K\,e^{L\ell_{s}^{-1}\left(\Lambda Q-\hat{{\cal S}}_{\cal
L}\right)}\,\delta(\dot{\rho}+\partial_{x}J_{\rho})\,\delta(\dot{m}+\partial_{x}J_{m}+2K)\,.$
(17)

The delta functions ensure that the dynamical equations (6) are satisfied at
each point of space and time. In the large-$L$ limit, $\Psi(\Lambda)$ can be
evaluated using saddle-point asymptotics:
$\Psi(\Lambda)=-\frac{1}{\ell_{s}T}\min_{\rho,m,\hat{p}_{\rho},p_{m}}{\cal
S}_{\mathrm{tot}}$. Using standard techniques this translates to minimizing an
action given by

$\displaystyle{\cal S}_{\mathrm{tot}}$ $\displaystyle=$
$\displaystyle\int_{0}^{\ell_{s}}dx\int_{0}^{T}dt\,\left\\{\dot{\rho}\hat{p}_{\rho}+\dot{m}p_{m}-\frac{1}{2}\left[\begin{array}[]{c}\partial_{x}\hat{p}_{\rho}+\Lambda\\\
\partial_{x}p_{m}\end{array}\right]^{T}\mathbf{C}\left[\begin{array}[]{c}\partial_{x}\hat{p}_{\rho}+\Lambda\\\
\partial_{x}p_{m}\end{array}\right]-\bar{J}_{\rho}(\partial_{x}\hat{p}_{\rho}+\Lambda)-\bar{J}_{m}\partial_{x}p_{m}\right\\}$
(23)
$\displaystyle+\int_{0}^{\ell_{s}}dx\int_{0}^{T}dt\,\left(-2\rho\sinh^{2}p_{m}+m\sinh
2p_{m}\right),$

that we derive in Appendix B. Here $\hat{p}_{\rho}$ and $p_{m}$ are auxiliary
fields, periodic in the spatial direction, introduced by writing the delta-
function constraints in (17) using a Fourier representation. Introducing
${p}_{\rho}\equiv\hat{p}_{\rho}+\Lambda x$ the action takes the form

${\cal
S}_{\mathrm{tot}}=\int_{0}^{\ell_{s}}dx\int_{0}^{T}dt\,\Big{\\{}\dot{\rho}{p}_{\rho}+\dot{m}p_{m}-{\cal
H}[\rho,m,{p}_{\rho},p_{m}]-\Lambda x\dot{\rho}\Big{\\}}\>,$ (24)

with the Hamiltonian density

${\cal
H}[\rho,m,{p}_{\rho},p_{m}]=\frac{1}{2}\left[\begin{array}[]{c}\partial_{x}{p}_{\rho}\\\
\partial_{x}p_{m}\end{array}\right]^{T}\mathbf{C}\left[\begin{array}[]{c}\partial_{x}{p}_{\rho}\\\
\partial_{x}p_{m}\end{array}\right]+\bar{J}_{\rho}\partial_{x}{p}_{\rho}+\bar{J}_{m}\partial_{x}p_{m}+2\rho\sinh^{2}p_{m}-m\sinh
2p_{m}\leavevmode\nobreak\ .$ (25)

Note that $p_{\rho}$ has a jump discontinuity of size $\Lambda\ell_{s}$ at
$x=\ell_{s}$ on the ring geometry studied here:
$p_{\rho}(\ell_{s})=p_{\rho}(0)+\Lambda\ell_{s}$.

The optimal trajectories which minimize the action are solutions of the
Hamiltonian MFT equations

$\displaystyle\partial_{t}\rho$ $\displaystyle=$
$\displaystyle\frac{\delta{\cal H}}{\delta
p_{\rho}}=-\partial_{x}\left[\sigma_{\rho}\partial_{x}p_{\rho}+\sigma_{\rho,m}\partial_{x}p_{m}+\bar{J}_{\rho}\right]$
(26) $\displaystyle\partial_{t}p_{\rho}$ $\displaystyle=$
$\displaystyle-\frac{\delta{\cal
H}}{\delta\rho}=-\partial_{x}^{2}p_{\rho}-(1-2\rho)(\partial_{x}p_{\rho})^{2}+2m\partial_{x}p_{\rho}\partial_{x}p_{m}-(\partial_{x}p_{m})^{2}+\operatorname{Pe}m\partial_{x}p_{\rho}-\operatorname{Pe}(1-2\rho)\partial_{x}p_{m}-2\sinh^{2}p_{m}$
$\displaystyle\partial_{t}m$ $\displaystyle=$ $\displaystyle\frac{\delta{\cal
H}}{\delta
p_{m}}=-\partial_{x}\left[\bar{J}_{m}+\sigma_{m}\partial_{x}p_{m}+\sigma_{\rho,m}\partial_{x}p_{\rho}\right]+2(\rho\sinh
2p_{m}-m\cosh 2p_{m})$ $\displaystyle\partial_{t}p_{m}$ $\displaystyle=$
$\displaystyle-\frac{\delta{\cal H}}{\delta
m}=-\partial_{x}^{2}p_{m}+2m(\partial_{x}p_{m})^{2}-2(1-\rho)\partial_{x}p_{\rho}\partial_{x}p_{m}-\operatorname{Pe}(1-\rho)\partial_{x}p_{\rho}+\sinh
2p_{m}\leavevmode\nobreak\ .$

Their solutions inserted in the action Eq. (24) give $\Psi(\Lambda)=-{\cal
S}_{\mathrm{tot}}/(\ell_{s}T$).

Note that in the limit of $T\gg 1$ the initial and final boundary conditions
on the different fields do not play an important role, except for fixing the
total particles mass $\rho_{0}$ (however, for an exception see Baek  _et al._
(2019)).

With the above formalism we now turn to solve the MFT equations. As a starting
point we will consider a system in the homogeneous phase, i.e., away from
MIPS. We comment about the extension to the MIPS state in the discussion Sec.
VII.

###  IV.1 Constant-profile solutions

The simplest solutions of the MFT equations obey the additivity principle
Bodineau and Derrida (2004b) so that $\rho$ and $m$ are time independent,
except for short time intervals around $t=0$ and $t=T$, and the same holds for
$\partial_{x}p_{\rho}$ and $\partial_{x}p_{m}$. The structure of the MFT
equations implies that $p_{m}$ is time independent but that $p_{\rho}$ has a
contribution growing linearly in time as $vt$ (see Agranov  _et al._ (2016)
for a similar problem). Assuming, in addition, translational invariance we
find

$\rho=\rho_{0}\quad;\quad v=\Lambda\quad;\quad
m=\frac{\rho_{0}C\Lambda}{\sqrt{1+(C\Lambda)^{2}}}\quad;\quad
p_{m}=\frac{1}{2}\operatorname{arcsinh}\left(C\Lambda\right),$ (27)

with $C=\operatorname{Pe}\left(1-\rho_{0}\right)$. The resulting SCGF given by
Eq. (17), that corresponds to the homogeneous solution $\Psi_{H}(\Lambda)$ is
then

$\Psi_{H}(\Lambda)=\rho_{0}(1-\rho_{0})\Lambda^{2}-\rho_{0}\left({1-\sqrt{1+C^{2}\Lambda^{2}}}\right).$
(28)

Performing the Legendre–Fenchel transform (16) we find

$\displaystyle
I_{H}(\rho_{0},q)=q\Lambda-\Psi(\Lambda)=\rho_{0}(1-\rho_{0})\Lambda^{2}+\rho_{0}\left(1-\frac{1}{\sqrt{1+C^{2}\Lambda^{2}}}\right),$
(29)

with $\Lambda(\rho_{0},q)$ given by the inverse of

$\displaystyle q$ $\displaystyle=$
$\displaystyle\partial_{\Lambda}\psi(\Lambda)=2\rho_{0}(1-\rho_{0})\Lambda+\frac{\rho_{0}C^{2}\Lambda}{\sqrt{1+C^{2}\Lambda^{2}}}\>.$
(30)

Note that for $\operatorname{Pe}=0$ we obtain

$I_{H}(q,\rho_{0},\operatorname{Pe}=0)=\frac{q^{2}}{4\rho_{0}\left(1-\rho_{0}\right)}\>,$
(31)

which coincides with the integrated current rate function of the simple
symmetric exclusion process Derrida (2007) 111In fact, to make the comparison
with the SSEP complete, one has to reinstate the variable rescaling
$x\rightarrow x/\ell_{s}$ and $t\rightarrow t/\gamma$. This also results in a
rescaling of the fluxes that enter in the un-scaled version of (6) and also
enter in the definition (12). Taking all of these into account, one verifies
the coincidence of the integrated current statistics of the two models .
Indeed, in this case the self-propulsion of both types of particles is set to
zero and the two models coincide.

As we now show, a non-zero self-propulsion ($\operatorname{Pe}\neq 0$) induces
a dynamical phase transition. Indeed, the space-time constant solution (27)
loses linear stability in a parameter regime that we identify. Linear
instability can signal the onset of a second order dynamical phase transition,
see, e.g. Baek  _et al._ (2017). Nevertheless, as we find, the phase diagram
here involves a first order transition.

As we now show, the full phase diagram, including first order transitions, can
be derived by establishing an analogy with equilibrium phase separating
systems with conserved order parameters.

##  V Dynamical phase transitions

We now turn to show that the current large deviation function $I$ exhibits
dynamical phase transitions as a function of $q,\rho_{0},\operatorname{Pe}$.
The analysis in this section focuses on the $\ell_{s}\to\infty$ asymptotics,
where we recall that $\ell_{s}=\sqrt{\gamma/D_{0}}$ plays the role of the
system length for the macroscopic coordinate $x\in[0,\ell_{s}]$. In the
$\ell_{s}\to\infty$ limit, the system exhibits MIPS with sharp domain walls
whose width becomes independent of $\ell_{s}$. The finite-$\ell_{s}$ effects
are studied in Sec. VI.

As we show, the signature of the DPT is the emergence of space- and time-
dependent optimal profiles. To obtain these, one has to address the non-linear
spatial and temporal minimization problem specified by Eqs. (26). As we now
argue, the analysis in the $\ell_{s}\to\infty$ limit simplifies considerably.
In this case, gradient terms in the action functional Eq. (23) can be
neglected to leading order. This can be seen by noting that long-wavelength
spatial modulations contribute as $O(\ell_{s}^{-1})$ to the action via
gradient terms, which is much smaller than the $O(\ell_{s})$ coming from the
contribution of the bulk terms. Moreover, sharp domain walls with finite
extension also have a subextensive $O(1)$ contribution to the action. This
negligible cost of gradient terms in the $\ell_{s}\to\infty$ limit is
reminiscent of the negligible interface tension terms in the free-energy
functional of equilibrium phase-separating systems. In addition, as we show in
Appendix C, optimal histories are either flat or given by sharply phase-
separated profiles that travel at a constant velocity $V$. As such, the time
derivative terms present in Eq. (23) can also be neglected; for the flat
profile, the derivative terms are zero, and for the phase-separated profile,
the contribution to the time derivative terms only arises from the localized
boundaries which scale as $O(1)$ and can therefore be ignored.

All in all, the above implies that ideas very similar to those used in
equilibrium phase separating systems can be used to analyze the phase diagram.
In particular, optimal solutions can be found by minimizing a “free energy”
functional for these fields, which is controlled by a bulk term given by the
homogeneous rate function $I_{H}$ (29)

$I\simeq\ell_{s}^{-1}\min_{\rho,J_{\rho}}\int_{0}^{T}dt\int_{0}^{\ell_{s}}dx\>I_{H}\big{(}\rho(x,t),J_{\rho}(x,t)\big{)}\>,$
(32)

subject to the constraints of the total mass conservation and the conditioning
on the space-time averaged current (12)

$\rho_{0}=\frac{1}{\ell_{s}}{\int_{0}^{\ell_{s}}dx\>\rho(x,t)}\>,\qquad
q=\frac{1}{\ell_{s}}{\int_{0}^{\ell_{s}}dx\>J_{\rho}(x,t)}\>.$ (33)

Both $\rho$ and $J_{\rho}$ are spatially conserved by virtue of the integral
constraints (33). The explicit derivation of the minimization problem (32) and
(33) is presented in Appendix C. The terms omitted from the expression (32)
are surface tension terms which, as explained above, have a sub-leading
contribution.

In analogy with equilibrium phase separation, whenever the bulk rate function
$I_{H}$ (29) becomes locally non-convex, the homogeneous solution (27) must
lose linear stability. In the next section we derive the associated spinodal,
that is shown to coincide with an explicit linear stability analysis of the
full action (23) against spatial and temporal variations. Next, we turn to
study the analogue of the binodal and the related first-order transition which
corresponds to the loss of global convexity of $I_{H}$ (occurring when $I_{H}$
differs from its convex hull). The rate function $I$ (32) is then given by the
convex hull of $I_{H}$.

###  V.1 Linear stability

The function $I_{H}(\rho_{0},q)$ (29) is a convex function of $q$. Therefore,
the linear instabilities can be identified by checking when the determinant of
the Hessian becomes negative,

$\frac{\partial^{2}I_{H}}{\partial\rho_{0}^{2}}\frac{\partial^{2}I_{H}}{\partial
q^{2}}-\left(\frac{\partial^{2}I_{H}}{\partial\rho_{0}\partial
q}\right)^{2}<0\>.$ (34)

which can be expressed using the parametric representation, Eq. (29), and the
differentiation rules

$\displaystyle\left.\frac{\partial}{\partial\rho_{0}}\right|_{q}$
$\displaystyle=$
$\displaystyle\left.\frac{\partial}{\partial\rho_{0}}\right|_{\Lambda}+\left.\frac{\partial\Lambda}{\partial\rho_{0}}\right|_{q}\left.\frac{\partial}{\partial\Lambda}\right|_{\rho_{0}}=\left.\frac{\partial}{\partial\rho_{0}}\right|_{\Lambda}-\frac{\left.\frac{\partial
q}{\partial\rho_{0}}\right|_{\Lambda}}{\left.\frac{\partial
q}{\partial\Lambda}\right|_{\rho_{0}}}\left.\frac{\partial}{\partial{\Lambda}}\right|_{\rho_{0}}$
$\displaystyle\left.\frac{\partial}{\partial q}\right|_{\rho_{0}}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial\Lambda}{\partial
q}\right|_{\rho_{0}}\left.\frac{\partial}{\partial\Lambda}\right|_{\rho_{0}}=\frac{1}{\left.\frac{\partial
q}{\partial\Lambda}\right|_{\rho_{0}}}\left.\frac{\partial}{\partial\Lambda}\right|_{\rho_{0}}\>.$
(35)

Employing these, we find that the region where the Hessian determinant
vanishes is given by the solution of $h_{0}=0$ with

$\displaystyle h_{0}$ $\displaystyle=$
$\displaystyle\Lambda^{2}\left\\{BC^{2}\operatorname{Pe}(4C-B\operatorname{Pe})+2A(1+C^{2}\Lambda^{2})\left[4(1+C^{2}\Lambda^{2})^{2}+\sqrt{1+C^{2}\Lambda^{2}}(4C\operatorname{Pe}-B\operatorname{Pe}^{2}+4C^{3}\operatorname{Pe}\Lambda^{2})\right]\right\\}$
(36) $\displaystyle+$
$\displaystyle\Lambda^{2}4BC^{2}\left[C^{3}\Lambda^{2}\operatorname{Pe}+(1+C^{2}\Lambda^{2})^{3/2}\right]$

where $\Lambda(q,\rho_{0},\operatorname{Pe})$ is given by Eq. (30),
$A=2\rho_{0}\left(1-\rho_{0}\right)$, $B=2\rho_{0}$, and
$C=\operatorname{Pe}\left(1-\rho_{0}\right)$. A non-trivial solution to (36),
with $\Lambda\neq 0$, emerges only for
$\operatorname{Pe}>\operatorname{Pe}^{*}=\sqrt{2}$. The region of linear-
instability grows with increasing $\operatorname{Pe}$ spreading from the point
$(\rho_{0}=1,q=0)$ of the phase space, as shown in Figs. 3 and 4. The saddle-
point solutions are linearly unstable in the purple region.

---
Figure 3:  The emergence of local non-convexity of $I_{H}$ when increasing
$\operatorname{Pe}>\operatorname{Pe}^{*}=\sqrt{2}$. The boundary of this
region, marked by a purple line is given by the implicit relation $h_{0}=0$
with $h_{0}$ given in (36) together with (30).

A complementary derivation of this result is obtained by computing the second
variation of the action functional Eq. (24) with respect to space and time
dependent field variations at finite $\ell_{s}$ and then taking the limit
$\ell_{s}\to\infty$. This calculation is given in Appendix D where we show
explicitly that Eq. (36) is recovered (thus fully justifying the equilibrium-
like analysis we put forward in this Section). The methodology used also
allows us to explore the dynamical phase transitions for finite $\ell_{s}$,
see Sec. VI.

As in equilibrium phase separation, the linear stability analysis alone does
not allow one to obtain the optimal profile. The latter is determined by a
global stability analysis. As we show below this allows us to identify a
binodal for the transition. As in equilibrium, the linear stability analysis
then identifies the spinodal which plays a role in determining the transient
nucleation towards the final binodal decomposition, see e.g. Baek  _et al._
(2018) for related analysis. The study of such phenomenon is beyond the scope
of the current study and would be an interesting subject for future
investigations.

Figure 4:  (A) The rate function $I_{H}(\rho_{0},q)$ showing non-convexity and
a pinch singularity at $(\rho_{0}=1,q=0)$. (B) The difference between the rate
function $I_{H}$ and its convex hull $I$. The orange line is the boundary of
the region of global non-convexity, given by (38). (C) The phase diagram for
local and global non-convexity for $I_{H}$. Local non-convexity is denoted by
a dashed purple curve, given by the implicit relation $h_{0}=0$ with $h_{0}$
given in (36) together with (30). Global non-convexity is marked by the orange
curve and is given by the implicit relation (38). At large $\ell_{s}$ these
regions define local and global instability regions of the constant solution
respectively (27). (Lower panels) Emergence of non convexity in $I_{H}$ at
increasing $\operatorname{Pe}>\sqrt{2}$. An example of a cross section for
$I_{H}$, marked by a green line in panel (C) and given by $q=(1-\rho_{0})/2$.
The dashed blue line is the convex hull construction. The orange and purple
points corresponds to the intersection with the local and global convexity
curves that are shown in panel (C).

.

###  V.2 Global stability

In Fig. 4 (A) and (B) we plot $I_{H}$ and the difference between $I_{H}$ and
its convex hull as a function of $q$ and $\rho_{0}$ for
$\operatorname{Pe}=3.5$. One can identify a curved surface with a concave
region, which is reminiscent of a first-order phase transition in equilibrium
systems. In the large-$\ell_{s}$ limit in which the domain wall contributions
are negligible, one can then obtain the rate function by constructing the
convex hull of $I_{H}$.

Interestingly, numerically we find that the convex hull always consists of tie
lines between $(\rho_{0}=1,q=0)$ and other points on the $I_{H}$ surface,
e.g., the points on the orange line in Fig. 4(C). This is supported by the
fact that $I_{H}$ has a singularity at $(\rho_{0}=1,q=0)$. Indeed, $I_{H}$
presents global minima along the entire line $q=0$, and diverges as
approaching $\rho_{0}=1$. This results in a pinch-point singularity $I_{H}\sim
q^{2}/(1-\rho_{0})$. The convex hull then needs to be constructed by drawing
tie lines that pass through $(\rho_{0}=1,q=0)$ and extend beyond the linearly
unstable region. Denoting the density and the current in the low and high
density phases by $(\rho_{l},q_{l})$ and $(\rho_{h},q_{h})$ respectively, then
when coexistence occurs, the high density phase always satisfies $\rho_{h}=1$
and $q_{h}=0$.

With this in mind, we are left with finding the location of the curve
$\left(\rho_{l},q_{l}\right)$ in the $(\rho_{0},q)$ plane which defines the
location of the low density phase. As explained above, it is found by
demanding that a tie line emanating from the point $(\rho_{h}=1,q_{h}=0)$ is
tangent to $I_{H}$ at the low density point $(\rho_{l},q_{l})$:

$\left[I_{H}+\left(1-\rho\right)\frac{\partial
I_{H}}{\partial\rho}-q\frac{\partial I_{H}}{\partial
q}\right]_{\left(\rho_{l},q_{l}\right)}=0\>.$ (37)

Using the differentiation rules (35) one then arrives at the relation:

$h_{1}(\Lambda,\rho_{0};\operatorname{Pe})\equiv
1-\sqrt{1+C^{2}\Lambda^{2}}+\Lambda^{2}(1-\rho_{0})^{2}(\sqrt{1+C^{2}\Lambda^{2}}+\operatorname{Pe}C)=0\>,$
(38)

which together with the relation (30) defines the “binodal” curve for
dynamical phase separation in an implicit form. The result for
$\operatorname{Pe}=3.5$ is shown in Fig. 4(C). Interestingly, the result
implies that at large $\ell_{s}$ this transition starts at a vanishingly small
value of $q$. The sub-leading interface tension cost in (32) shifts the
critical value away from zero. As we detail in Sec.VI, its scaling is given by
$q=O(\ell_{s}^{-1/2})$.

Note that for a given average density $\rho_{0}$, and total current $q$, the
optimal configuration is phase-separated with the density and the current in
each phase given by the geometric construction in the
$\left(\rho_{0},q\right)$ plane that is shown in Fig. 5. See also the lower
panels of Fig. 4. The point $\left(\rho_{l},q_{l}\right)$ is found by
identifying a straight line between $\left(\rho_{h}=1,q_{h}=0\right)$ and the
binodal line (38) which passes through the point of interest
$\left(\rho_{0},q\right)$. This implies, for example, that for a given density
$\rho_{0}$, upon increasing the value of the current $q$ towards
$q_{l}(\rho_{0})$, the portion of the low density phase $\rho_{l}$ becomes
larger, until it spans the entire system at $q=q_{l}(\rho_{0})$.

Finally, we now show that when coexistence occurs, it is in the form of a
traveling wave. To see this, we consider the interface between the high
density phase with $(\rho_{0}=1,q=0)$ and the low density one with
$(\rho_{l},q_{l})$. A balance of fluxes implies that the domain wall between
the two phases must propagate with a velocity

$V=-\frac{J_{l}}{1-\rho_{l}}=-\frac{q}{1-\rho_{0}},$ (39)

where $J_{l}$ is the current in the low density phase. The second equality
arises using $J_{l}f_{l}=q$ with $f_{l}$ the fraction of the low density phase
so that $f_{l}\rho_{l}+(1-f_{l})=\rho_{0}$. A snapshot of this dynamics is
illustrated in Fig. 5(B): A band ofa high-density phase$\rho_{h}=1$ propagates
through the system in a direction opposite to the current in the low-density
phase, as seen from the expression (39) of its velocity.

Figure 5: (A) The phase diagram for current fluctuations $q$ at
$\operatorname{Pe}=3.5$. The orange curve marks the boundary between the
traveling band phase and a homogeneous phase at $\ell_{s}\rightarrow\infty$.
It is given in a parametric form by Eqs. (38) and (30). The dashed line
denotes the convex hull construction between the high density phase which is
always at close packing $\left(\rho_{0}=1,q=0\right)$, and the low density
phase. It is given by the intersection of this line with the orange curve
denoting the edge of the traveling band phase. (B) An instance of the
traveling band solution that corresponds to the geometrical construction
illustrated in panel (A). As discussed in Sec. VI, the sharp interface is
smoothed at finite $\ell_{s}$.

This sums up our findings for the rate function (13). For points in parameter
space that are inside the homogeneous phase, see Fig. 5, it is given by
$I=I_{H}$, with the homogeneous rate function $I_{H}$ given by (29). For
points inside the traveling band phase it is given by

$I=\frac{1-\rho_{0}}{1-\rho_{l}}I_{H}(\rho_{l},q_{l})$ (40)

with $\rho_{l}(\rho_{0},q),q_{l}(\rho_{0},q)$ given by the geometric
construction involving the “binodal” (38). It has a jump in its second
derivative at the point of intersection with the curve given by Eq. (38).
These results are shown in Fig. 6.
Notice that finite-$\ell_{s}$ corrections leave the immediate neighborhood of
$q=0$ protected from the dynamical phase transition. These corrections will be
discussed in detail in the next section.

---
Figure 6:  The rate function $I(q)$ of Eq. (13) at $\rho_{0}=0.9$ and
$\ell_{s}^{-1}\to 0$. Panel (B) is a closeup of the region marked in panel (A)
by a dashed rectangle. The continuous black line represents the rate function
$I(q)$ while the dashed purple curve is the sub-optimal rate function that
corresponds to a homogeneous profile (29). $I$ is constructed from the convex
hull of $I_{H}$ in the parameter space $(\rho_{0},q)$, see main text. The rate
function has a jump in its second derivative at the orange point where these
two rate functions are tangent. This point corresponds to the boundary of the
traveling band phase denoted by the orange curve in Fig. 5 (A).
---
Figure 7: The phase diagrams of the DPT and of MIPS. The MIPS binodal is shown
as a black solid line and the corresponding spinodal as a dashed black line.
The binodal for dynamical phase transition at $q=O(\ell_{s}^{-1/2})$, which is
given by Eq. (41), is plotted in orange. The dashed purple line is the
corresponding spinodal, Eq. (42). Their extension in the MIPS phase is denoted
by dotted lines.

##  VI Finite-$\ell_{s}$ corrections and relation to the MIPS criticality

Since both MIPS, which occurs at $q=0$, and the travelling wave phase that we
studied above exhibit phase separation, it is interesting to compare their
phase diagrams and see how the transitions relate to each other. As we now
show, this requires one to go beyond the analysis considered above and study
the sub-leading finite-$\ell_{s}$ corrections. We also comment on the values
of the cumulants in this limit at the end of the Section.

###  VI.1 A primer: comparing MIPS and the DPT in the limit $\ell_{s}^{-1}\to
0$

Fig. 7 shows the phase diagrams of the two transitions in the limit of
$\ell_{s}^{-1}\to 0$. The binodal for the DPT is plotted in orange in the
$q\to 0$ limit, obtained from Eq. (38) using a small-$\Lambda$ expansion. This
leads to the expression

$\rho_{l,1}=\frac{1}{2}+\frac{1}{\operatorname{Pe}^{2}}+O(q^{2})\,,$ (41)

for the low density phase while the high density phase has a density
$\rho_{h}=1$. A similar small-$\Lambda$ expansion of Eq. (36) shows that the
spinodal at $q\to 0$, plotted as a purple dashed line on Fig. 7, is given by

$\rho_{l,2}=\frac{2}{3}\left(1+\frac{1}{\operatorname{Pe}^{2}}\right)+O(q^{2})\,.$
(42)

A naive comparison with the phase diagram of MIPS (whose spinodal and binodal
are shown as dashed and solid black lines respectively in Fig. 7) seems to
suggest that the two transitions are not related. As we now show, this is in
fact not the case. The link between MIPS and the DPT is revealed by studying
finite-$\ell_{s}$ corrections to the rate function.

###  VI.2 Relating the DPT at small $q$ to the MIPS at $q=0$ through
finite-$\ell_{s}$ corrections

We start by summarizing the results before turning to their derivation. To do
this, we consider finite-$\ell_{s}$ corrections in the different regions of
the phase diagram Fig. 7.

First, we note that outside the binodal (denoted by an orange line) there is
no transition. Thus, as $q$ is increased from $0$, the system remains
homogeneous and there are no finite-$\ell_{s}$ corrections. Inside the orange
binodal in the $\ell_{s}^{-1}\to 0$ limit, the system transitions to a sharply
separated traveling wave at a finite $q$ (with two bulk densities dictated by
the binodal). Accounting for a finite $\ell_{s}$ shows that such a state
emerges only when $q$ becomes of the order $q_{\text{s}}\sim\ell_{s}^{-1/2}$.
This is a result of the finite cost of the domain walls in the system. In
other words, at finite $\ell_{s}$, there is a boundary layer
$q\sim\ell_{s}^{-1/2}$ separating the $q=0$ behavior (which display MIPS) from
the DPT. In fact, by considering more closely the boundary layer we find two
regions:

Region I – inside the spinodal ($\rho_{l,2}<\rho_{0}<1$ and
$\operatorname{Pe}>\sqrt{2}$) [which encloses the MIPS critical point
$(\rho_{0}=3/4,\operatorname{Pe}=4)$].
In this region in the $\ell_{s}^{-1}\to 0$ limit the homogeneous state is
linearly unstable for any $q>0$. This is not longer true for large but finite
$\ell_{s}$, where we find that the linear instability occurs at a finite
threshold $q_{c,2}$, that scales as $q_{c,2}\sim\ell_{s}^{-1}$. When
$q>q_{c,2}$, the systems exhibits a smooth spatial modulation which becomes
more pronounced until a sharply-phase-separated state emerges at
$q\sim\ell_{s}^{-1/2}$ (and the bulk densities of the profile are given to
leading order by the orange binodal). Hence, the system first undergoes a
linear instability into a smoothly modulated state and then crosses over to a
sharply separated state over a range which scales as $O(\ell^{-1/2})$.

Region II – between the binodal and the spinodal
($\rho_{l,1}<\rho_{0}<\rho_{l,2}$ and $\operatorname{Pe}>\sqrt{2}$). Here the
system is linearly stable for any value of $\ell_{s}$. Then one has a
discontinuous transition at $q_{c,1}\sim\ell_{s}^{-1}$ into a traveling wave
solution whose domain walls become sharper with increasing $q$ as in region I.

Interestingly, at the critical point of the MIPS transition
$(\rho_{0}=3/4,\operatorname{Pe}=4)$ both the DPT and the MIPS are initiated
by a linear instability. This implies that the critical current $q_{c,2}$ of
the DPT vanishes at this point.

In sum, the MIPS transition, and the dynamical phase transition are separated
by a finite layer at $q>0$ everywhere in phase space, except at the MIPS
critical point where the two transitions merge. This is presented in Fig. 8
and Fig. 9. We now provide a derivation of these results.

---
Figure 8:  A comparison of the phase diagrams of the DPT in the infinite
system-size limit ($\ell_{s}^{-1}=0$) and at finite $\ell_{s}$, at
$\operatorname{Pe}=3.9$ . The region marked by a dashed box in panel (A) is
shown in panel (B). As in Fig. 4 (C), the regions of linear and global
instability for $\ell_{s}^{-1}\to 0$ are shown in purple and orange. The
numerically obtained region of linear instability for a small but finite value
$\ell_{s}^{-1}=0.02$ is delimited by a black curve. The dashed blue line in
panel (B) is the analytical expression (43) which is valid at first order for
small $\ell_{s}^{-1}$, and is in good agreement with the previous numerical
results. These curves illustrate the boundary layer described in the main
text.

Figure 9:  The critical current $q_{c,2}$, given by Eq. (43), for three
$\operatorname{Pe}$ values. Panel (A) shows how the critical current (43)
vanish at the MIPS critical density $\rho_{0}=3/4$ upon approaching the MIPS
critical point $\operatorname{Pe}=4$. (B) The same as (A) but presented in the
$(\rho_{0},\operatorname{Pe},q)$ space. The critical current threshold, Eq.
(43), vanishes at the MIPS critical point which is marked by the red dot. The
black and dashed black lines are the MIPS binodal and spinodal respectively.
The orange and purple curves are the boundaries of the globally and linearly
unstable regions at $\ell_{s}^{-1}\to 0$ which are given by Eqs. (41) and
(42), respectively.

####  VI.2.1 The scalings of $q_{\text{s}}$ and $q_{c,1}$

To evaluate the scalings of these current values, we study the effect of the
sub-leading surface tension terms that were omitted in Eq. (32). Consider
first a sharply phase separated state. Then, as explained at the beginning of
Sec.V, the contribution of interface terms to the rate function (32),
evaluated over a sharply phase separated solution, scales as
$O(\ell_{s}^{-1})$. Comparing this to the gain in the bulk term in the
expression (32), which scale as $q^{2}$ for small $q$, we find that a sharply
separated state only emerges when $q$ becomes of the order of
$q_{\text{s}}\sim\ell_{s}^{-1/2}$.

Next, consider the transition into a smoothly modulated traveling wave state
with a finite amplitude. In this case, the interface cost scales as
$O(\ell_{s}^{-2})$. Correspondingly, the threshold value scales as
$q_{c,1}\sim\ell_{s}^{-1}$. In the large $\ell_{s}$ limit, we therefore expect
that the transition into a smoothly modulated profiles precedes the emergence
of traveling wave solutions everywhere in phase space.

####  VI.2.2 Critical current $q_{c,2}$ in region I

The finite critical value $q_{c,2}$ can be obtained exactly for any
$\ell_{s}$, from a stability analysis of the action (24) to small space-and-
time dependent fluctuations. As detailed in Appendix E, one finds

$q_{c,2}=\pi\ell_{s}^{-1}\frac{\sqrt{\left(2+CD\right)^{2}\left(A+\rho_{0}C^{2}\right)}}{\sqrt{\operatorname{Pe}^{2}\left(3\rho_{0}-2\right)-2}}+O(\ell_{s}^{-2}),$
(43)

with
$A(\rho_{0})=2\rho_{0}(1-\rho_{0}),C(\rho_{0},\operatorname{Pe})=\operatorname{Pe}(1-\rho_{0})$
and $D(\rho_{0},\operatorname{Pe})=\operatorname{Pe}(1-2\rho_{0})$. The
expression under the square root in the denominator is positive in the
linearly unstable region $\rho_{0}>\rho_{l,2}$, see Eq. (42). Importantly, the
scaling $q_{c,2}\sim\ell_{s}^{-1}$ guarantees that linear instability occurs
before the transition into the sharply separated state in this region (which
occurs at $q\sim\ell_{s}^{-1/2}$).

These results are presented in Fig. 8 and Fig. 9. Notice that, as argued
above, $q_{c,2}$ vanishes at the MIPS critical point
$(\rho_{0}=2/3,\operatorname{Pe}=4)$, given by $2+CD=0$. Indeed, from Eq.
(43), $q_{c,2}$ is proportional to $|2+CD|$; however, $q_{c,2}>0$ everywhere
inside the region enclosed by the MIPS spinodal $2+CD<0$. The reason is that
the linear instability into MIPS (at $q=0$) is distinct from the transition
into a traveling wave (at $q>0$). They only coincide at the MIPS critical
point. Interestingly, the distinction between the transitions can be seen in
the spectrum of the linearized problem. The MIPS linear instability describes
an exponential growth of spatial modulations towards a state which is phase
separated and stationary. Therefore, the corresponding excitation frequency
$\omega$ that describe this linear instability is imaginary. In contrast, the
linear instability associated with the dynamical phase transition is
characterized by the emergence of traveling waves. Therefore, its associated
excitation frequency $\omega$ is real.

It is only at the MIPS critical point that the excitation frequencies of both
instabilities vanish and the two transitions merge.

###  VI.3 Remarks on the current cumulants and finite-$\ell_{s}$ effects

As usual in large-deviation theory, the rate function $I(q)$ is related to the
cumulants of the current through derivatives of $I$ at $q=0$ Touchette (2009).
Notice that, in the $\ell_{s}\to\infty$ limit, the curvature at $q=0$ of the
rate function $I$ differs from that of $I_{H}$, the rate function for a
homogeneous profile (shown in Fig. 6), since $I$ is the convex envelope of
$I_{H}$. However, at large but finite $\ell_{s}$, as we have seen in the
previous paragraph, the rate function exhibits a small boundary layer around
$q=0$, in which it is equal to $I_{H}$. This means that the cumulants of the
current are in fact obtained from the set of derivatives of $I_{H}(q)$ at
$q=0$. The presence of a DPT in the rate function of the current is in some
cases associated to an anomalous scaling of the finite-size corrections to the
rate function (see e.g. Appert-Rolland  _et al._ (2008) for the cumulants of
the current in the weakly asymmetric simple exclusion process (WASEP)). This
is an interesting open question which remains to be addressed for this
problem.

In fact we can draw a parallel between the large deviations of the current in
our problem and in the (asymmetric) exclusion processes: in the periodic
WASEP, a DPT for the current large deviation is observed between a homogeneous
phase and an heterogeneous-profile phase, with smooth interfaces Bodineau and
Derrida (2005). As the asymmetry is increased, the interfaces become sharper
and a DPT point occurs closer to the average current, making the current
distribution more singular. In the very large asymmetry limit, one recovers
the TASEP (totally asymmetric simple exclusion process). When doing so, the
dynamical scaling switches from diffusive to KPZ Derrida and Lebowitz (1998)
and the nature of the fluctuations change considerably. In our case, in the
presence of phase separation, the domains are separated by sharp walls only in
the large-$\ell_{s}$ limit, and the location of the DPT becomes closer to the
average current which occurs at $q=0$ in the same limit – the situation
therefore is quite similar, except that the dynamical scaling remains
diffusive all along the limiting process. This is also what allows one to
safely determine the current cumulants in this limit, although the system
presents sharp walls.

##  VII Summary and discussion

In this work we have derived the exact fluctuating hydrodynamics for the
active lattice gas model of Ref. Kourbane-Houssene  _et al._ (2018). This is
the first such derivation for a system of interacting active particles. We
note here that the active lattice gas model differs from standard active
matter models. In contrast to more standard models, the tumbling rate is
rescaled by $L^{-2}$, see Sec. II. As a result, the polarization enters as a
slow field in the coarse-grained hydrodynamics. Still, this model shares many
similarities with the more standard active matter models, most notably MIPS.

Building on the fluctuating hydrodynamics, we have extended the classical MFT
framework to the active lattice gas model and employed it to study the
integrated current fluctuations. We provide a full mapping of the current
fluctuations phase diagram where, notably, we identify a dynamical phase
transition. The MFT problem could be tackled analytically despite the fact
that it is significantly more involved compared to the standard Gaussian noise
case. Here we find Poissonian noise and the hydrodynamics involve two scalar
fields rather than one. Nevertheless, the analysis can be carried out
analytically using $\ell_{s}^{-1}$ as a small parameter.

For the unbiased system, this small parameter controls the ratio of the domain
wall width in the MIPS phase to the system size. As we have shown, the same
small parameter sets a vanishing probability cost for gradient terms in the
MFT action. This enables us to establish an analogy between the MFT action and
the free energy of equilibrium phase separating systems. A non-convexity of
the bulk term of the ‘free energy’ can then be associated with a dynamical
phase transition into a traveling wave phase separated state. We find that
this transition occurs at vanishingly small $O(\ell_{s}^{-1})$ current
fluctuations. Interestingly, the dynamical phase transition of the biased
system and the MIPS of the unbiased system are shown to coincide at the MIPS
critical point where both of these are initiated through a linear instability.
This link is exposed by accounting for finite-$\ell_{s}$ corrections to the
MFT problem.

Our results were derived assuming that the unbiased system is homogeneous.
Nevertheless, we anticipate that to leading order at small $\ell_{s}^{-1}$,
the results will hold also in the MIPS phase. In particular, the minimization
determined by the procedure described above which allows for the biased system
to be phase separated.

It is interesting to use the formalism developed above to study other large-
deviation functions. In particular, there has been recent interest in entropy
production rates in active systems (even at the absence of any external drive
either at the bulk or at the boundaries). The study of entropy production for
this active lattice gas is the subject of a future publication Agranov  _et
al._ .

From a broader perspective, it could be interesting to investigate other
models presenting a diffusive scaling but non-Gaussian fluctuations coming
from processes other than the specific case of tumbling events we have studied
here (for instance from chemical reactions or non-Gaussian sources of noise).

###### Acknowledgements.

TA is funded by Blavatnik Postdoctoral Fellowship Program. VL is supported by
the ANR-18-CE30-0028- 01 Grant LABS and by the EverEvol CNRS MITI project. SR
and YK are supported by an Israel Science Foundation grant (2038/21) and an
NSF-BSF grant (2016624)

##  Appendix A Deriving the large deviation function

###  A.1 Contributions from conserved fluxes

Flux fluctuations are given by the Gaussian noise terms
$J_{\rho}-\bar{J}_{\rho}=\eta_{\rho}/\sqrt{L\ell_{s}^{-1}}$ and
$J_{m}-\bar{J}_{m}=\eta_{m}/\sqrt{L\ell_{s}^{-1}}$ 222Notice the factor
$\ell_{s}^{-1}$ in the noise amplitude, as compared to Ref. Agranov  _et al._
(2021). It follows from the rescaling of space and time that we employ
throughout this work, see the definition bellow Eq. (1) with the covariances

$\displaystyle\langle\eta_{\rho}(x,t)\eta_{\rho}(x^{\prime},t^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\sigma_{\rho}\,\delta(x-x^{\prime})\delta(t-t^{\prime})\quad;\quad\langle\eta_{m}(x,t)\eta_{m}(x^{\prime},t^{\prime})\rangle=\sigma_{m}\,\delta(x-x^{\prime})\delta(t-t^{\prime})$
$\displaystyle\langle\eta_{\rho}(x,t)\eta_{m}(x^{\prime},t^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\sigma_{\rho,m}\,\delta(x-x^{\prime})\delta(t-t^{\prime})$

given by

$\displaystyle\sigma_{\rho}$ $\displaystyle=$ $\displaystyle
2\rho\left(1-\rho\right)\quad;\quad\sigma_{m}=2\left(\rho-m^{2}\right)\quad;\quad\sigma_{\rho,m}=2m\left(1-\rho\right).$

These have been derived in Ref. Agranov  _et al._ (2021) using a mapping to
the ABC model Bodineau and Derrida (2011); Evans  _et al._ (1998a, b).
Importantly, as shown in Ref. Bodineau and Derrida (2011), these also capture
non-typical fluctuations of $O\left(1\right)$.

###  A.2 Contributions from tumbling events

The Gaussian approximation to the tumbling probability was derived in Ref.
Agranov  _et al._ (2021). Here we go beyond this approximation by accounting
for the underlying Poisson process.

To account for the probability of tumbles we note that these are transmutation
reactions between two particle species. Such processes were treated in Ref.
Jona-Lasinio  _et al._ (1993); Lefèvre and Biroli (2007); Bodineau and Lagouge
(2010). One finds that the probability $P(\mathcal{K}_{+})$ to observe
${\mathcal{K}_{+}}$ flips of $+$ particles into $-$ particles in a space
interval $\Delta x$ and time interval $\Delta t$ follows a large deviation
principle

$-\ln P\big{(}\mathcal{K}_{+}=K_{+}L\ell_{s}^{-1}\Delta t\Delta\big{)}\simeq
L\ell_{s}^{-1}\Delta x\Delta t\psi_{\rho_{+}}(K_{+}),$ (45)

with the expected large-deviation function of a Poisson process:

$\psi_{\rho_{+}}\big{(}K_{+}\big{)}=K_{+}\log\left(\frac{K_{+}}{\rho_{+}}\right)-K_{+}+\rho_{+}.$
(46)

Here $\rho_{+}$ is the density of $+$ particles in the mesoscopic interval
$[x,x+\Delta x]$. Similarly, the probability $P(\mathcal{K}_{-})$ to observe
${\mathcal{K}_{-}}$ flips of $-$ particles into $+$ particles in a space
interval $\Delta x$ and time interval $\Delta t$ follows a similar large
deviation principle with

$\psi_{\rho_{-}}\left(K_{-}\right)=K_{-}\log\left(\frac{K_{-}}{\rho_{-}}\right)-K_{-}+\rho_{-}.$
(47)

Here $\rho_{-}$ is the density of $+$ particles in the mesoscopic interval
$[x,x+\Delta x]$.

The total change in the number of $-$ particles in a mesoscopic interval, due
to tumbling reactions is then given by the difference
$\mathcal{K}=\mathcal{K}_{+}-\mathcal{K}_{-}$. Following the contraction
principle Touchette (2009), the probability of this variable is described by
the large deviation function

$-\ln P\left(\mathcal{K}=KL\ell_{s}^{-1}\Delta t\Delta x\right)\simeq
L\ell_{s}^{-1}\Delta x\Delta t\mathcal{L}_{K}\big{(}K\big{)},$ (48)

where $\mathcal{L}_{K}$ is found by minimizing the combined probability cost
of the previous two processes under the constraint of a given total tumbling
rate

$\mathcal{L}_{K}=\inf_{K_{+}}\left[\psi_{\rho_{+}}\left(K_{+}\right)+\psi_{\rho_{-}}\left(K_{+}-K\right)\right]=\rho-\sqrt{K^{2}+\left(\rho^{2}-m^{2}\right)}+K\ln\left[\frac{\sqrt{K^{2}+\left(\rho^{2}-m^{2}\right)}+K}{\left(\rho+m\right)}\right].$
(49)

###  A.3 Joint large deviation function

The above results can be combined into a single large deviation function as
the conserved dynamics of the fluxes are uncorrelated with the tumbling
dynamics. The microscopic rates for tumble events are much slower than the
hopping dynamics. In particular, this allows one, despite the presence of
tumbling events, to use a local equilibrium conditions for the hopping rates.
In sum, this means that the total action is written as a sum of the (Gaussian)
flux action and the tumbling action $\mathcal{L}_{K}$, as announced in Eq.
(8).

Notice that, as expected, the quadratic expansion
$\mathcal{L}_{K}(K)\simeq\left(K-m\right)^{2}/2\rho$ close to the minimum the
joint large-deviation function exactly matches the Gaussian noise term derived
in Ref. Agranov  _et al._ (2021).

##  Appendix B Deriving the MFT equations

In this Appendix, we derive the action (23) starting from the path integral
(17). To do so we use the Martin–Siggia–Rose–Janssen–de-Dominicis formalism
Bertini  _et al._ (2015a); Martin  _et al._ (1973); Janssen (1976); Dominicis
(1976). The generating function then takes the form

$e^{LT\Psi(\Lambda)}=\int{\cal D}\rho{\cal D}m{\cal D}J_{\rho}{\cal
D}J_{m}{\cal D}K{\cal D}p_{\rho}{\cal
D}p_{m}\,e^{-L\ell_{s}^{-1}\left\\{\hat{{\cal S}}_{\cal
L}-\int_{0}^{\ell_{s}}dx\int_{0}^{T}dt\left[\Lambda
J_{\rho}-p_{\rho}(\dot{\rho}+\partial_{x}J_{\rho})-p_{m}(\dot{m}+\partial_{x}J_{m}+2K)\right]\right\\}}.$
(50)

with two auxiliary response fields $p_{\rho}(x,t)$ and $p_{m}(x,t)$ arising
from representing the $\delta$ functions in Eq. (17) using a Fourier
transform. To obtain the MFT equations we then use a saddle-point evaluation
at large $L$. Minimizing with respect to $K$ one arrives at the optimal value

$K=m\cosh 2p_{m}-\rho\sinh 2p_{m}\>.$ (51)

Next, minimizing with respect to the flux fields $J_{\rho}$ and $J_{m}$ yields

$\begin{bmatrix}J_{\rho}-\bar{J}_{\rho}\\\
J_{m}-\bar{J}_{m}.\end{bmatrix}=\mathbf{C}\begin{bmatrix}\partial_{x}p_{\rho}+\Lambda\\\
\partial_{x}p_{m}\end{bmatrix}.$ (52)

Finally, using the expressions (51) and (52), one arrives at the announced
action (23).

Notice that since the fields $\rho,m,J_{\rho},J_{m},K$ are continuous, they
obey periodic boundary conditions on the ring geometry as do the fields
$p_{\rho}$ and $p_{m}$. As stated in Sec. IV, the boundary conditions in time
for the optimal fields become irrelevant in the large-$T$ limit.

##  Appendix C Establishing an equilibrium analogue

In this Appendix we show how the rate function $I(q)$ of Eq. (13) can be
found, to leading order at large $\ell_{s}$, by the minimization problem (32)
subjected to the constraints (33). It is more convenient to start from the
Lagrangian formulation (8), rather then the Hamiltonian one (24). From Eqs.
(8) and (13), we have

$\displaystyle
I(q)=\frac{1}{\ell_{s}T}\min_{\rho,m,J_{\rho},J_{m},K}\int_{0}^{\ell_{s}}dx\int_{0}^{T}dt\left(\mathcal{L}_{J}+\mathcal{L}_{K}\right),$
(53)

subject to the integrated current constraint (12), and the dynamical
constraints (6). In contrast to the Hamiltonian formulation (24), these
constrains are not built into the Lagrangian minimization and have to be
enforced explicitly.

At large times, if the additivity principle is verified, optimal solutions are
time independent and we have shown in Sec. IV.1 that this leads to homogeneous
density and polarization optimal profiles, with a corresponding rate function
$I_{H}$ given by Eq. (29). When the additivity principle is broken, it happens
in general with optimal profiles taking the form of traveling waves that
propagate at a constant velocity. This is the form that we assume now.
Furthermore, in the large-$\ell_{s}$ asymptotics, the width of the walls is
$O(1)$: it does not scale with the system size $\ell_{s}$. As discussed in the
main text, the contribution of the domain walls to the action is then sub-
extensive. We now explain how, for such profiles, the rate function $I(q)$ can
still be obtained from the homogeneous one $I_{H}$ with the adequate
constraints – in a picture analogous to what happens in equilibrium phase
separation.

The minimization in (53) becomes time-independent for the optimal traveling
profiles $\left\\{\rho,m,J_{\rho},J_{m},K\right\\}$ (which depend only on
$x$):

$\displaystyle
I(q)=\frac{1}{\ell_{s}}\min_{\left\\{\rho,m,J_{\rho},J_{m},K\right\\}}\int_{0}^{\ell_{s}}dx\left(\mathcal{L}_{J}+\mathcal{L}_{K}\right).$
(54)

The time dependency is also eliminated from the current constraint (12)

$q=\frac{1}{\ell_{s}}{\int_{0}^{\ell_{s}}dx\,J_{\rho}(x)}\>.$ (55)

We next consider the dynamical constraints (6). The first one implies that the
total mass is conserved, which we now assume implicitly:

$\displaystyle\rho_{0}=\frac{1}{\ell_{s}}{\int_{0}^{\ell_{s}}dx\,\rho(x)}\>.$
(56)

Also, it implies that the traveling wave solution moves at a constant speed

$V=\frac{J_{h}-J_{l}}{\rho_{h}-\rho_{l}},$ (57)

where $J_{h,l}$ and $\rho_{h,l}$ are the high and low density values for the
flux and total density. The second constraint in Eq. (6),
$\partial_{t}m=-\partial_{x}J_{m}-2K$, implies that optimal tumbling rate
vanishes in the bulk phases

$K=0.$ (58)

We are thus left with the four field minimization

$\displaystyle I(q)$ $\displaystyle=$
$\displaystyle\frac{1}{\ell_{s}}\min_{\left\\{\rho,m,J_{\rho},J_{m}\right\\}}\int_{0}^{\ell_{s}}dx\>\mathcal{L},$
(59) $\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\left\\{\frac{1}{2}\begin{bmatrix}J_{\rho}-\operatorname{Pe}m(1-\rho)\\\
J_{m}-\operatorname{Pe}\rho(1-\rho)\end{bmatrix}^{\mathrm{T}}\mathbf{C}^{-1}\begin{bmatrix}J_{\rho}-\operatorname{Pe}m(1-\rho)\\\
J_{m}-\operatorname{Pe}\rho(1-\rho)\end{bmatrix}+\rho-\sqrt{\rho^{2}-m^{2}}\right\\}$
(60)

subject to the integrated current constraint (55), which can be imposed via a
scalar Lagrange multiplier $\Lambda$.

In (60) we have omitted negligible gradient terms, but one has to be cautious
because this implies that the Euler–Lagrange equations become algebraic and
only admit constant value solutions. To retain possible non-homogeneous
optimal solutions one introduces a space-dependent Lagrange multiplier
$\tilde{\Lambda}(x)$ enforcing $J_{\rho}(x)$ to be equal to a profile
$\tilde{J}_{\rho}(x)$. One arrives at

$\displaystyle I(q)$ $\displaystyle=$
$\displaystyle\frac{1}{\ell_{s}}\min_{\left\\{\rho,m,J_{\rho},J_{m},\tilde{\Lambda},\tilde{J}_{\rho},\Lambda\right\\}}\left\\{\int_{0}^{\ell_{s}}dx\Big{[}\mathcal{L}+\tilde{\Lambda}(x)\big{(}\tilde{J}_{\rho}(x)-J_{\rho}(x)\big{)}+\Lambda\big{(}\tilde{J}_{\rho}(x)-q\big{)}\Big{]}\right\\},$
(61)

where again $\rho,m,J_{\rho},J_{m},\tilde{\Lambda},\tilde{J}_{\rho}$ depend on
$x$ while $\Lambda$ is constant. Omitting the dependencies on $x$ for
simplicity and minimizing with respect to the flux fields we find

$\begin{bmatrix}J_{\rho}-\operatorname{Pe}m(1-\rho)\\\
J_{m}-\operatorname{Pe}\rho(1-\rho)\end{bmatrix}=\mathbf{C}\begin{bmatrix}\tilde{\Lambda}\\\
0\end{bmatrix}.$ (62)

Replacing these into (61) and minimizing with respect to $m$ then gives

$m=\frac{\rho\operatorname{Pe}(1-\rho)\tilde{\Lambda}}{\sqrt{1+(\operatorname{Pe}(1-\rho)\tilde{\Lambda})^{2}}}\>,$
(63)

which is the same expression as was obtained for the homogeneous solutions in
Eq. (27), but with $\Lambda$ replaced by space-dependent field
$\tilde{\Lambda}$. Minimizing with respect to $\tilde{\Lambda}$ yields the
equation

$\displaystyle\tilde{J}_{\rho}=2\rho(1-\rho)\tilde{\Lambda}+\frac{\rho\operatorname{Pe}^{2}(1-\rho)^{2}\tilde{\Lambda}}{\sqrt{1+\operatorname{Pe}^{2}(1-\rho)^{2}\tilde{\Lambda}^{2}}},$
(64)

which is again, a space-dependent version of Eq. (30). Last, minimizing with
respect to $\Lambda$ merely gives the constraint

$q=\frac{1}{\ell_{s}}{\int_{0}^{\ell_{s}}dx\,\tilde{J}_{\rho}}\>.$ (65)

Gathering the previous results into Eq. (60), we arrive at

$\displaystyle
I=\frac{1}{\ell_{s}}\min_{\rho(x),\tilde{J}_{\rho}(x)}\int_{0}^{\ell_{s}}dx\bigg{\\{}\rho(1-\rho)\tilde{\Lambda}^{2}+\rho\left(1-\big{[}1+\operatorname{Pe}^{2}(1-\rho)^{2}\tilde{\Lambda}^{2}\big{]}^{-1/2}\right)\bigg{\\}},$
(66)

where $\tilde{\Lambda}$ is given implicitly by the algebraic relation (64),
and $\tilde{J}_{\rho}$ satisfies (65). Comparing Eqs. (66) and (64) with (29)
and (30), we conclude that

$\displaystyle
I=\frac{1}{\ell_{s}}\min_{\rho(x),\tilde{J}_{\rho}(x)}\int_{0}^{\ell_{s}}dx\,I_{H}\big{[}\rho(x),\tilde{J}_{\rho}(x)\big{]},$
(67)

subject to the constraint (65) (and the total mass constraint (56) which was
assumed all along). Changing the dummy variable $\tilde{J}_{\rho}(x)$ to
$J_{\rho}(x)$ we finally arrive at Eqs. (32) and (33) which concludes our
proof.

##  Appendix D Action second variation analyses and its relation to local
concavity of $I_{H}$

We expand the action to second order in path variations around the homogeneous
solutions (27). To do so, we define the variation fields $\delta\rho$, $\delta
m$, $\delta p_{\rho}$, and $\delta p_{m}$ through 333Note that the conjugate
momentum variations are imaginary since the field is imaginary. It only takes
a real value at the saddle point, see e.g. Ref. Baek  _et al._ (2018) and also
Appendix B.

$\rho=\rho_{0}+\delta\rho\quad;\quad p_{\rho}=\Lambda x+i\delta
p_{\rho}\quad;\quad m=\frac{\rho_{0}C\Lambda}{\sqrt{1+(C\Lambda)^{2}}}+\delta
m\quad;\quad
p_{m}=\frac{1}{2}\operatorname{arcsinh}\left(C\Lambda\right)+i\delta p_{m}$
(68)

and the Fourier transforms

$\delta f(x,t)=\sum_{n,m}\;\delta f^{n,m}e^{i(k_{n}x+\omega_{m}t)}\quad;\quad
k_{n}=2\pi n/\ell_{s}\quad;\quad\omega_{m}=2\pi m/T$ (69)

where $\delta X$ is the variation of the field $X$ and $n$ and $m$ are
integers. After expansion, the quadratic variation term of the action Eq.
(24), which we denote by $\delta S_{2}$, takes the form

$\displaystyle\delta
S_{2}=\ell_{s}T\sum_{n,m}V_{-n,-m}^{T}\mathbf{B}(n,m)V_{n,m},$ (70)

with the matrix

$\displaystyle\mathbf{B}(n,m;\Lambda,\rho_{0},\operatorname{Pe})=$
$\displaystyle\begin{pmatrix}\Lambda^{2}&\frac{\omega_{m}}{2}+\frac{Dk_{n}\Lambda}{\operatorname{Pe}}-\frac{k_{n}BC\Lambda\operatorname{Pe}}{4\sqrt{1+(C\Lambda)^{2}}}+i\frac{k_{n}^{2}}{2}&\frac{\operatorname{Pe}\Lambda}{2}&\frac{k_{n}}{2}\left(D-\frac{BC\Lambda^{2}}{\sqrt{1+(C\Lambda)^{2}}}\right)-iC\Lambda\\\
-\frac{\omega_{m}}{2}-\frac{Dk_{n}\Lambda}{\operatorname{Pe}}+\frac{k_{n}BC\Lambda\operatorname{Pe}}{4\sqrt{1+(C\Lambda)^{2}}}+i\frac{k_{n}^{2}}{2}&\frac{Ak_{n}^{2}}{2}&-\frac{k_{n}C}{2}&\frac{BC^{2}k_{n}^{2}\Lambda}{2\operatorname{Pe}\sqrt{1+(C\Lambda)^{2}}}\\\
\frac{\operatorname{Pe}\Lambda}{2}&\frac{k_{n}C}{2}&0&\frac{\omega_{m}}{2}+\frac{k_{n}C\Lambda}{\operatorname{Pe}}+i\frac{k_{n}^{2}}{2}+i\sqrt{1+(C\Lambda)^{2}}\\\
-\frac{k_{n}}{2}\left(D-\frac{BC\Lambda^{2}}{\sqrt{1+(C\Lambda)^{2}}}\right)-iC\Lambda&\frac{BC^{2}k_{n}^{2}\Lambda}{2\operatorname{Pe}\sqrt{1+(C\Lambda)^{2}}}&-\frac{\omega_{m}}{2}-\frac{k_{n}C\Lambda}{\operatorname{Pe}}+i\frac{k_{n}^{2}}{2}+i\sqrt{1+(C\Lambda)^{2}}&k_{n}^{2}\left(\frac{B}{2}-\frac{B^{2}(C\Lambda)^{2}}{4\left(1+(C\Lambda)^{2}\right)}\right)+\frac{B}{\sqrt{1+(C\Lambda)^{2}}}\end{pmatrix}$

where $A=2\rho_{0}\left(1-\rho_{0}\right)$, $B=2\rho_{0}$,
$C=\operatorname{Pe}\left(1-\rho_{0}\right)$,
$D=\operatorname{Pe}\left(1-2\rho_{0}\right)$, and

$V_{n,m}=\begin{bmatrix}\delta\rho^{n,m}\\\ \delta p_{\rho}^{n,m}\\\ \delta
m^{n,m}\\\ \delta p_{m}^{n,m}\end{bmatrix}.$ (72)

The homogeneous solution becomes unstable when one of the eigenvalues of the
matrix ${\mathbf{B}}$ becomes negative for some mode $n,m$. To find the region
in parameter space $\left(\rho_{0},\Lambda\right)$ where this happens we first
note that by direct inspection of the eigenvalues of $\mathbf{B}$, and similar
to related problems Baek  _et al._ (2017, 2018), the most unstable mode is
$n=1$. This is expect as the larger values of $n$ have a larger cost in the
action due to spatial modulations.

For this mode we find that for any unstable point in the parameter space
$(\rho_{0},\Lambda)$, there exists a set of unstable frequencies $\omega_{m}$.
In the large $T$ limit, the frequency can be treated as a continuous variable
$\omega$ and the set of unstable frequencies become an interval. This interval
shrinks to a point at the boundary of the unstable region. The boundary of the
unstable region can then be identified by solving the equations

$\text{Det}[\mathbf{B}]\left(\omega;n=1,\Lambda,\rho_{0},\operatorname{Pe},\ell_{s}\right)=0\leavevmode\nobreak\
\quad;\quad\partial_{\omega}\text{Det}[\mathbf{B}]\left(\omega;n=1,\Lambda,\rho_{0},\operatorname{Pe},\ell_{s}\right)=0\leavevmode\nobreak\
.$ (73)

The first equation ensures that there are eigenvalues whose value is zero,
while the second equation ensures that there is only one such eigenvalue 444We
find numerically that for each point in parameter space there is only a single
interval of frequencies where the determinant drops bellow zero.. The two
equations (73) can be cast into a single implicit algebraic relation between
$\Lambda$ and $\rho_{0}$ for given values of $\operatorname{Pe}$ and
$\ell_{s}$ which we write as:

$h_{2}(\Lambda,\rho_{0};\operatorname{Pe},\ell_{s})=0\,.$ (74)

Using Eq. (30) this relation then defines a curve in the parameter space
$\left(\rho_{0},q\right)$ which encloses the linearly unstable region. This
curve is plotted as a black line in Fig. 8 for $\ell_{s}^{-1}=0.02$ and
$\mathrm{Pe}=3.9$. The figure also shows, as a purple line, the limiting curve
that is approached as $\ell_{s}\rightarrow\infty$.

As stated in the main text, the curve defined Eq. (74), which describes the
local instability, must coincides with the curve Eq. (36), which describes
local non-convexity. That is

$\lim_{\ell_{s}\to\infty}h_{2}(\Lambda,\rho_{0};\operatorname{Pe},\ell_{s})=h_{0}(\Lambda,\rho_{0};\operatorname{Pe})\>.$
(75)

This can be shown explicitly by expanding $h_{2}$ at large $\ell_{s}$,
assuming a travelling wave form for the unstable mode, and taking the limit
$\ell_{s}\to\infty$. Since the derivation is rather lengthy but
straightforward, we omit it from the text.

##  Appendix E Deriving the critical threshold $q_{c,2}$ Eq. (43)

The curve $q_{c,2}$, given by Eq. (43) serves as a finite-$\ell_{s}$
correction to the limit (75). To account for it we must retain the next order
expansion to the determinant keeping terms of $O(\ell_{s}^{-4})$. In addition,
being interested only the region near $q=0$, we use the scaling
$\Lambda=O(\ell_{s}^{-1})$. This also implies the same scaling for the wave
velocity $V=\omega k_{1}=O(\ell_{s}^{-1})$. Using these and expanding up to
$O(\ell_{s}^{-4})$ we now arrive at the explicit expression

$\Lambda^{2}=(2\pi)^{2}\ell_{s}^{-2}\frac{(2+CD)^{2}}{(2A+BC^{2})(B\text{Pe}^{2}-4-4\text{Pe}C)}\,,$
(76)

with $A,B,C$ and $D$ defined in Eq. (D). Finally, expanding Eq. (30), we find

$\Lambda(q)=\frac{2q}{2A+BC^{2}}+O(q^{3})\>,$ (77)

which, used into Eq. (76), yields Eq. (43).

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  * Note (3) Note that the conjugate momentum variations are imaginary since the field is imaginary. It only takes a real value at the saddle point, see e.g. Ref. Baek  _et al._ (2018) and also Appendix B.
  * Note (4) We find numerically that for each point in parameter space there is only a single interval of frequencies where the determinant drops bellow zero.