[2]Raphaël Chetrite

1]Faculty of Mathematics, National Research University Higher School of
Economics, 6 Usacheva, Moscow, 119048 , Russia

2]Institut de Physique de Nice (INPHYNI), Université Côte d’Azur, CNRS, 17 rue
Julien Lauprêtre, Nice, 06200, France

# Macroscopic Fluctuation Theory for Ginzburg-Landau dynamics with long range
interactions

Cédric Bernardin<EMAIL_ADDRESS>Raphael.CHETRITE@univ-
cotedazur.fr [  [

###### Abstract

Focusing on a famous class of interacting diffusion processes called Ginzburg-
Landau (GL) dynamics, we extend the Macroscopic Fluctuations Theory (MFT) to
these systems in the case where the interactions are long-range, and
consequently, the macroscopic effective equations are described by non-linear
fractional diffusion equations.

###### Contents

  1. 1 Introduction
    1. 1.1 Macroscopic Fluctuation Theory (MFT): Overwiew
    2. 1.2 Objectives
    3. 1.3 GL dynamics: Set-up
    4. 1.4 GL dynamics: Motivations and bibliography
    5. 1.5 GL dynamics: Set-up with long-range interactions
    6. 1.6 Plan of the paper
  2. 2 Long-range GL dynamics
    1. 2.1 Long-range GL dynamics with free boundary conditions
    2. 2.2 GL thermodynamical quantities
    3. 2.3 Long-range GL dynamics in contact with two baths
  3. 3 Hydrodynamic and hydrostatic limits
    1. 3.1 Hydrodynamic limit
    2. 3.2 Hydrostatic limit and properties of the NESS
    3. 3.3 Hydrodynamic limit for a time inhomogeneous perturbed dynamics
    4. 3.4 Einstein relation
  4. 4 Large deviations
    1. 4.1 Dynamical large deviations
    2. 4.2 Static large deviations in the NESS : non-equilibrium free energy or quasi-potential
  5. 5 Diffusive limit
    1. 5.1 Diffusive limit of the hydrodynamics and hydrostatics
    2. 5.2 Diffusive limit of the large deviations functionals
  6. 6 Particular case
    1. 6.1 Additive noise
    2. 6.2 Brownian energy model (BEM)
      1. 6.2.1 BEM-GL transformation
      2. 6.2.2 Thermodynamical properties
      3. 6.2.3 Nearest neighbour case
  7. 7 Comparison with the standard diffusive case
    1. 7.1 GL dynamics with nearest neighbour interactions
    2. 7.2 GL dynamics with a mix between long and short range interactions
  8. A Proof of the form (2.6) of the Bulk Markovian generator.
  9. B Conserved quantity of the GL dynamics and ergodicity
  10. C Few properties of the hydrodynamic and hydrostatic equations
    1. C.1 Weak solutions for the hydrodynamic and hydrostatic equations
    2. C.2 Convergence of the hydrodynamic and hydrostatic equations as $\gamma\to 2^{-}$
      1. C.2.1 Choice of the normalisation
      2. C.2.2 Convergence of some integral-differential operator to a second order differential operator
  11. D Derivation of the hydrodynamic limit for the perturbed dynamics
  12. E Markovian dynamics for the energy field associated to the Brownian Energy Model

##  1 Introduction

###  1.1 Macroscopic Fluctuation Theory (MFT): Overwiew

Macroscopic Fluctuation Theory (MFT) is the cornerstone of modern non-
equilibrium statistical physics [30, 101, 53] that describes far from
equilibrium processes and improves on Onsager’s theory, in which fluctuations
are modeled by Gaussian processes. It can be seen as an infinite-dimensional
version of the Freidlin-Wentzel theory [66]. One of the main interest of MFT
is that it provides, in the context of interacting particle systems driven by
external forces, a definition of a non-equilibrium free energy for the non-
equilibrium stationary state (NESS) as the solution of a dynamical variational
problem. When the system is at equilibrium, the variational problem becomes
trivial 111Of course, Gibbs equilibrium measures may have a very complicated
form so that the variational problem becomes difficult to solve. in the sense
that one recovers the usual equilibrium free energy given in the Gibbs
formalism.

Based on large deviations theory [60, 126, 62, 56, 52, 51, 122], MFT has been
developed for interacting particle systems, mainly lattice gas, whose typical
behaviour is given by a diffusion equation (e.g., Simple Symmetric Exclusion
Process [25]), a viscous conservation law (e.g., Weakly Asymmetric Exclusion
Process [34, 33, 28]) or a conservation law (e.g., Asymmetric Simple Exclusion
Process [128, 103, 8]). MFT has been extended to other lattice systems (e.g.
[29]), to some reaction diffusion processes like Glauber–Kawasaki dynamics
[88, 35] but also to systems with more than one conservation law [13, 22, 44,
125] or to systems with ballistic transport [61]. Without being exhaustive we
mention other applications of the MFT like [97, 102]. It is also expected that
it will be applicable to other fields like geophysics [89] or turbulence
[114].

###  1.2 Objectives

The aim of this paper is to study the extension of the MFT framework for a
certain class of lattice field models, known in the literature as Ginzburg-
Landau (GL) dynamics. A convenient picture is to see them as stochastic
fluctuating interfaces, see Figure 1. The novelty of the current paper w.r.t.
the existing literature is that long-range interactions are considered. The
hydrodynamic equation is then governed by a fractional diffusion equation
[129] (see Eq. (3.1)) instead of a usual diffusion equation. Our main interest
is to develop MFT for super-diffusive lattice field models and in particular
to obtain some informations on the NESS of these open systems in the
thermodynamic limit.

###  1.3 GL dynamics: Set-up

To be more precize and in order to motivate the choice to consider GL
dynamics, let us first define them in a very general context, ignoring for the
moment some external effects. We are interested in the transport properties of
a conservative lattice field model
$\varphi_{t}:=\\{\varphi_{t}(x)\in{\mathcal{M}}\;;\;x\in\Lambda\\}\in{\mathcal{M}}^{\Lambda}$,
where $\mathcal{M}$ is an arbitrary Riemannian sub-manifold of
$\mathbb{R}^{N}$ and $\Lambda$ is a $D$-dimensional lattice. Depending on the
physical context, the variable $\varphi_{t}(x)$ represents the value at time
$t$ of, e.g. a continuous scalar spin (${\mathcal{M}}=\mathbb{R}$), a charge
(${\mathcal{M}}=\mathbb{R}$), an energy ($\mathcal{M}=\mathbb{R}^{+}$), a
continuous N- spin model ($\mathcal{M}=S^{N-1}(1)$) etc.

A (lattice) $\varphi$-conservation law is locally expressed by an equation in
the form

$\partial_{t}\varphi_{t}(x)+\sum_{y\in\Lambda}J_{t}(x,y)=0,\quad x\in\Lambda\
,$ (1.1)

where $J_{t}(x,y)$ is the instantaneous rate of the $\varphi$-current between
site $x$ and site $y$ at time $t$, positively counted from $x$ to $y$, and
such that $J_{t}(x,y)=-J_{t}(y,x)$.

The previous equation express that the change in time of $\varphi_{t}(x)$ is
only due to exchanges between two arbitrary sites (there is no local creation
or annihilation) so that, thanks to the antisymmetry of $J_{t}$, the following
$\varphi$-conservation law holds:

$\partial_{t}\mathcal{V}_{\Lambda}(\varphi_{t})=0\quad\text{where}\quad{\mathcal{V}}_{\Lambda}(\varphi)=\sum_{x\in\Lambda}\varphi(x)\
.$ (1.2)

Note that the number of conserved quantities is the same as the dimension of
$\mathcal{M}$.

A particular sub-class of $\varphi$-conservation law concerns (stochastic)
lattice fields model where Eq. (1.1) is a finite dimensional stochastic
differential equation, i.e. whose the instantaneous rates of currents satisfy

$J_{t}(x,y)=-F_{\varphi_{t}}(x,y)-\sqrt{2\Gamma_{\varphi_{t}}(x,y)}\
\dot{\zeta}_{t}(x,y),$ (1.3)

where the deterministic vector field $F_{\varphi_{t}}(x,y)$ is a drift term,
and $\dot{\zeta}_{t}(x,y)$ is a $N$-dimensional standard white Gaussian noise
and the matricial field $\Gamma_{\varphi_{t}}(x,y)>0$ is the
diffusivity222Ito’s convention is assumed everywhere in this article. To
insure the antisymmetry of the current $J_{t}$, we have to impose that for any
$x,y\in\Lambda$,

$F_{\varphi_{t}}(x,y)=-F_{\varphi_{t}}(y,x),\quad\dot{\zeta}_{t}(x,y)=-\dot{\zeta}_{t}(y,x),\quad\Gamma_{\varphi_{t}}(x,y)=\Gamma_{\varphi_{t}}(y,x)\
.$

We also require that the white noises
$\\{\dot{\zeta}_{t}(x,y)\;;\;x,y\in\Lambda\\}$ are independent apart from the
antisymmetry constraint333Hence there are in fact only
$|\Lambda|(|\Lambda|-1)/2$ independent $N$-dimensional white noises..
Injecting Eq. (1.3) in Eq. (1.1) we obtain then a so-called diffusion
$\varphi$-conservation law.

######  Remark 1.1.

Note that the main physical restrictions of this set-up are:

  1. 1.

The noise $\dot{\zeta}_{t}(x,y)$ associated to a link $\\{x,y\\}$ is a
$N$-dimensional white Gaussian noise. The physical origin of the noise is not
fully justified. Relaxing the Gaussian and white hypothesis does not seem to
be very considered until now in the probabilistic literature, despite that
conservative lattice gas are peculiar case of $\varphi$-conservation law like
in Eq. (1.1) with white Poissonnian noise.

  2. 2.

The instantaneous current $J_{t}(x,y)$ is a function of the conservative field
$\varphi_{t}:=\\{\varphi_{t}(x)\in{\mathcal{M}}\;;\;x\in\Lambda\\}$. There
exist in the literature (see e.g. [5, 100]) plenty of $\varphi$-conservation
law like Eq. (1.1) where this is not the case and which therefore do not lead
to a closed equation for the conservative field $\varphi$.

Anyway, these two assumptions permit to ensure that the $\varphi$-conservation
law Eq. (1.1) defines a Markovian process.

Now, given a positive functional $\mathcal{E}_{\Lambda}(\varphi)$ called the
GL Hamiltonian, our aim is to define a diffusion $\varphi$-conservation law of
the previous form which is moreover invariant w.r.t. the Gibbs measure
$d{\tilde{\mu}}^{\Lambda}_{0}(\varphi)\propto
e^{-\mathcal{E}_{\Lambda}(\varphi)}d\varphi$. We call such a dynamics a
conservative GL dynamics. If $\Gamma_{\varphi}(x,y)$ is constant, say equal to
one, it is sufficient to take the drift term associated to the gradient flow
associated to $\mathcal{E}_{\Lambda}$ and respecting the
$\varphi$-conservation law, i.e.
$F_{\varphi}(x,y):=-\partial_{\varphi(x)}\mathcal{E}_{\Lambda}+\partial_{\varphi(y)}\mathcal{E}_{\Lambda}$.
However, as soon as $\Gamma_{\varphi}$ is not constant, the previous choice
does not respect the fluctuation-dissipation relation and
${\tilde{\mu}}_{0}^{\Lambda}$ is no longer invariant. A possible choice to
restore this discrepancy is to take the drift term $F_{\varphi}$ in the
form444The fact that this choice implies that the Gibbs probability measure
$d{\tilde{\mu}}^{\Lambda}_{0}(\varphi)\propto
e^{-\mathcal{E}_{\Lambda}(\varphi)}d\varphi$ is an invariant measure of the
$\varphi$-conservation law, will be proven in a peculiar set-up in the Section
2. (see [113, Section 3.2.3.2] for more physical motivations in a slightly
different context)

$\begin{split}F_{\varphi}(x,y)&=-\Omega_{\varphi}(x,y)\
\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\
\mathcal{E}_{\Lambda}(\varphi)\ +\
\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\
\Omega_{\varphi}(x,y)\ ,\end{split}$ (1.4)

where for all $x,y\in\Lambda$, $\Omega_{\varphi}(x,y)=\Omega_{\varphi}(y,x)$
is a matricial field called mobility, and which satisfies the Einstein
relation

$\Omega_{\varphi}(x,y)+\Omega^{\dagger}_{\varphi}(x,y)=2\
\Gamma_{\varphi}(x,y)\ .$ (1.5)

The anti-symmetric part of $\Gamma_{\varphi}$ can be of interest to model
Hamiltonian (underdamped) effects.

Observe that since the dynamics is a $\varphi$-conservation law, we have more
invariant Gibbs measures apart from
$d{\tilde{\mu}}_{0}^{\Lambda}(\varphi)\propto
e^{-\mathcal{E}_{\Lambda}(\varphi)}d\varphi$: for any chemical potential
$\lambda\in\mathbb{R}$555Depending on the growth properties of $U$ we have
sometimes to restrict the domain of the admissible $\lambda$’s in order to
have well defined Gibbs measures. We will always assume that the dynamics is
well defined and that the canonical Gibbs probability measures make sense, at
least for a non-empty range of chemical potentials $\lambda$. Then
$\mathbb{R}$ will have to be replaced by this domain., the (canonical) Gibbs
probability measure

$d\tilde{\mu}^{\Lambda}_{\lambda}(\varphi)\propto\exp\left(-\mathcal{E}_{\Lambda}(\varphi)-\lambda\mathcal{V}_{\Lambda}(\varphi)\right)\
d\varphi$ (1.6)

is also invariant for the GL dynamics666Hence the GL dynamics is not ergodic.
But under suitable generic assumptions, one can prove it is, when the dynamics
is restricted to a $\varphi$-invariant manifold
$\left\\{\varphi\in{\mathcal{M}}^{\Lambda}\;;\;\mathcal{V}_{\Lambda}(\varphi)=C\right\\}$,
$C\in\mathbb{R}$ fixed. . In fact, if $\Gamma_{\varphi}=\Omega_{\varphi}$,
then the canonical Gibbs measures $\tilde{\mu}^{\Lambda}_{\lambda}$ are
reversible (i.e. satisfy the detailed balance condition).

External interactions (boundary effects, external forces …) can be
incorporated to the free dynamics in several ways depending on the physical
context of interest. We discuss few of them, without being exhaustive. The
free dynamics corresponds to the situation discussed above: the system is
isolated from the exterior universe. If $\Lambda$ is the discrete
$D$-dimensional torus (we say that we have periodic boundary conditions), we
are are roughly in a similar physical situation. In both cases the canonical
Gibbs measures $\tilde{\mu}^{\Lambda}_{\lambda}$ are invariant for the
dynamics.

A first very different typical physical situation consists to add some
$\varphi$-baths on the sites of the boundary $\partial\Lambda$ of $\Lambda$.
The action of a bath acting on $x\in\partial\Lambda$ is to fix the
distribution of $\varphi(x)$ equal to the $\varphi(x)$-marginal of
$\tilde{\mu}^{\Lambda}_{\lambda(x)}$, where $\lambda(x)$ is an arbitrary fixed
chemical potential depending on $x$. For example, we can model such a bath by
a Langevin dynamics, acting on $\varphi(x)$ only, and making the
$\varphi(x)$-marginal of $\tilde{\mu}^{\Lambda}_{\lambda(x)}$ reversible
w.r.t. the Langevin dynamics. In this case, if the $\lambda(x)$ are non
constant, none canonical Gibbs measure is invariant for the full dynamics777If
all the $\lambda(x)=\lambda$ are equal to a fixed value, under suitable
generic conditions, we expect then that the unique invariant measure of the
dynamics is $\tilde{\mu}^{\Lambda}_{\lambda}$.. We will be concerned with this
last situation in the present article.

A second interesting situation (not discussed in the present paper) can result
from the presence of some external forces acting in the bulk and not breaking
the $\varphi$-conservation law. For example, transversal forces $F^{\perp}$
satisfying the antisymmetry property
$F^{\perp}_{\varphi}(x,y)=-F^{\perp}_{\varphi}(y,x)$ and the transversal
condition

$\sum_{x,y\in\Lambda}\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\
\left(e^{-\mathcal{E}(\varphi)}\ F^{\perp}_{\varphi}(x,y)\right)=0$ (1.7)

do not break the invariance of the canonical Gibbs measures. More generally,
an external antisymmetric force
$F^{ext}_{\varphi}(x,y)=-F^{ext}_{\varphi}(y,x)$ will still give rise to a
diffusion $\varphi$-conservation law but will in general break the invariance
of the canonical Gibbs measures888We will investigate in fact a particular
case of this situation in Section 3.3. [70] .

$\varphi_{t}(x)$$x$$0$$1$$2$$3$$4$$5$$6$$n$$\varphi_{t}(5)$ Figure 1: The GL
dynamics on the lattice $\Lambda_{n}=\\{1,\ldots,n\\}$ can be seen as a
fluctuating interface conserving the algebraic volume (represented in gray)
$\mathcal{V}_{n}(\varphi_{t}):={\mathcal{V}}_{\Lambda_{n}}(\varphi_{t})=\sum_{x\in\Lambda_{n}}\varphi_{t}(x)$
between the interface and the $x$-axis.

###  1.4 GL dynamics: Motivations and bibliography

The discrete GL dynamics described in the previous paragraph can be physically
motivated in many ways.

First: One goal [119, Part II, Section 7.3] is to makes sense, via a
discretisation, of some continuous ill-defined continuous GL dynamics, see
e.g. [42, Chap. 8.6] or [85]. In fact, eight (labeled by the letters A,B,C,
…,H) other ill-defined stochastic partial differential equations have been
introduced in physics to model the dynamics of phase transitions. For example,
the model A concerns a scalar continuous field, and is more or less an ad-hoc
coarse grained version of Glauber Ising spin dynamics, hence without any
conservation law. On the other hand, the model B is typically an ad-hoc scalar
coarse grained version of Kawasaki spin dynamics model, and as one
conservation law, and discretised version of it is is a case of our GL
dynamics. The other models in this list concern vectorial continuous field
will typically contain also non dissipative Hamiltonian forces. Discretised
versions of theses models did not seem to have been considered until now in
the physical mathematics literature.

Second: They also appear as effective equations for the transport of energy in
chains of oscillators with conservative bulk noise in a suitable weak coupling
limit [100] or in weakly coupled Hamiltonian dynamics [58]. Even if [100, 58]
are only concerned with short-range dynamics, understanding of energy
transport problems in chains of oscillators with long-range interactions is
the subject of recent several studies, e.g. [57, 10, 104, 131, 121, 1].

Third: GL dynamics are standard models of interacting particle systems studied
by probabilists since more than three decades and which has a rich history,
see e.g. [117, 67, 68, 70, 72, 71, 83, 59, 112, 134, 127, 132, 43, 94, 110,
32, 80, 3] and references therein. Translated in our previous formalism, all
these studies are only concerned with the case $\mathcal{M}=\mathbb{R}$ or
$\mathcal{M}=\mathbb{R}^{+}$ [100, 58] (but in any case $N=1$), so that
$\Omega_{\varphi}(x,y)=\Gamma_{\varphi}(x,y)$ (see Eq. (1.5)) and with
periodic boundary conditions (except999The BEP model in [80] is a GL dynamics
but it is not mentioned in the paper. [80]), so that
$\Lambda:=\mathbb{T}_{n}$, the discrete torus of length $n$ (which will go to
infinity is the scaling limit). In these works, the GL Hamiltonian
$\mathcal{E}_{\mathbb{T}_{n}}(\varphi)$ is always supposed to be in the
translation-invariance form

$\mathcal{E}_{\mathbb{T}_{n}}(\varphi)=\sum_{x\in\mathbb{T}_{n}}{\mathcal{U}}(\tau_{x}\varphi)$
(1.8)

where ${\mathcal{U}}$ is a function of $\varphi$ and the shift $\tau_{x}$ is
defined via $(\tau_{x}\varphi)(z)=\varphi(z+x)$. Most of the previous papers
consider the case where ${\mathcal{U}}(\varphi)=U(\varphi_{0})$ for some given
potential function $U$ (it will be our case also). Consistently, it is usually
assumed that $\Gamma_{\varphi}(x,y)=\Gamma_{\tau_{x}\varphi}(0,y-x)$ and
similarly for $\Omega_{\varphi}$. Moreover, as far as we know, [110, 12, 80]
are the only papers in the previous list to consider the case
$\Gamma_{\varphi}$ non constant. However [80] considers a reversible gradient
system while [110] considers reversible models which are non-gradient ([127],
[119, Part II, Section 2.4] [91, Chapter 7], [111], [4]). This means that the
current in [110] cannot be written as the discrete gradient of a local
function. This adds very serious difficulties and [110] has to rely on the
famous Varadhan’s non-gradient method. On the other hand [12] considers a GL
dynamics with a transverse drift and the model is non reversible101010Like in
[80] the author of [12] did not notice that the dynamics introduced is a GL
dynamics.. But, in fact, the most important remark for us is that in all these
papers, only short-range interactions are considered in the sense that there
exist $K,L\geq 1$ and $M\geq 0$ such that111111Usually, the authors take for
convenience $K=1$, and apart from [112], $M=0$.

$\begin{cases}&\Gamma_{\varphi}(x,y)=0\ ,\quad\text{if}\quad|x-y|>K\ ,\\\
&\Gamma_{\varphi}(x,y)\quad\text{depends only of $\varphi(z)$ for $|z-x|\leq
L$ }\ ,\quad\text{if}\quad|x-y|\leq K\ ,\\\ &\mathcal{U}\quad\text{ is local,
i.e. it depends only of $\varphi(z)$ for $|z|\leq M$ \ . }\end{cases}$ (1.9)

In these works are studied hydrodynamic limits [67, 68, 70, 72, 71, 83, 112,
127, 132, 32, 80], equilibrium and non-equilibrium fluctuations (i.e. Central
Limit Theorems) [117, 134, 43] and dynamical large deviations [59, 94, 110,
3]. All of them are for isolated dynamics with periodic boundary conditions.
The hydrodynamic limit obtained takes the form of a non-linear diffusion
equations with periodic boundary conditions, apart from [80] for which the
diffusion equation is linear, with inhomogeneous boundary conditions, and [32]
where is derived a ‘generalized’ Cahn-Hillliard equation121212The assumptions
done in the paper unfortunately rule out the the obtention of the classical
Cahn-Hilliard equation with quartic double well potential.. The dynamical
large deviations results for the empirical density associated to the field
$\varphi$ appeared first in the seminal paper [59] for a gradient Ginzburg-
Landau dynamics defined on the one-dimensional discrete torus $\mathbb{T}_{n}$
of length $n$ going to infinity (with time rescaled diffusively by $n^{2}$).
The result has been generalized in [94] where the authors consider directly
the dynamics in infinite volume, i.e. $\mathbb{T}_{n}$ is replaced by
$\tfrac{1}{n}\mathbb{Z}$. A different approach to derive the result of [59] is
provided in [3]. In [110], Quastel derived a result similar to [59] for
$\Gamma_{\varphi}$ non constant and, as mentioned above, he has therefore to
rely on the complex non-gradient methods and does not obtain an explicit form
for the diffusion coefficient in the non-linear hydrodynamic equations.
However, in all theses paper, the form of the dynamical large deviation
function is the one considered in MFT for diffusive systems (see Eq. (7.3)).

###  1.5 GL dynamics: Set-up with long-range interactions

The aim of this work is to study GL dynamics with long-range interactions when
the lattice dimension is $D=1$ and the space dimension is $N=1$ (in fact we
will restrict even our study to $\mathcal{M}=\mathbb{R}$ or
$\mathcal{M}=\mathbb{R}^{+}$). There are several ways to introduce long-range
effects in the GL dynamics since it suffices to break (at least) one of the
conditions appearing in Eq. (1.9) by taking $K,L$ or $M$ infinite. We consider
only the case where we break one of the three lines of Eq. (1.9), while
maintaining the other ones valid. Reading the existing literature, we think
that the definition of long-range effects is very far to be uniform in the
physical or mathematical literature. Hence, to be precize, for us, by
definition, a long-range effect occurs as soon as, in its definition, the
model investigated incorporates interactions which are not of finite-range.
Observe that with this definition, long-range models can in fact behave like
finite-range models at the macroscopic level and, more surprisingly, some
short-range models can behave macroscopically like long-range models [87, 16].
We describe below some possibilities while in the current paper we will
discuss only one of them, i.e. the first one.

  1. 1)

Break of the validity of first line of Eq. (1.9). This is the kind of long-
range effects considered in this paper. We consider $\Gamma_{\varphi}(x,y)$ in
the non-local form

$\Gamma_{\varphi}(x,y)=K(y-x)\beta(\varphi(x),\varphi(y))$

where $\beta$ is a smooth positive function on $\mathbb{R}^{2}$ and $K$ is a
long-range coupling given by

$K(z)=\frac{\sc 1_{z\neq 0}}{|z|^{1+\gamma}}$ (1.10)

where $\gamma>0$. Moreover, we assume that the GL Hamiltonian
$\mathcal{E}_{\Lambda}$ is in the form (1.8) but with a local function
$\mathcal{U}$, i.e. $\mathcal{U}(\varphi)$ depends only on the variables
$\varphi_{z}$ for $|z|\leq M$, $M$ fixed. For example

$\mathcal{E}_{\Lambda}(\varphi)=\sum_{x}{\mathcal{U}}(\tau_{x}\varphi)=\sum_{x}U(\varphi(x))$
(1.11)

where $U$ is a given nice131313Recall that we will always assume that the
dynamics are well defined and that the Gibbs canonical measures make sense, at
least for some non-empty range of chemical potentials $\lambda$. potential
function, i.e. $\mathcal{U}(\varphi)=U(\varphi(0))$.

For lattice gas, this kind of long-range interactions have been investigated
in [86, 21, 116, 18, 19, 84, 14] and is restricted to hydrodynamic limits
which are linear (apart from [86, 116]).

  2. 2)

Break of the validity of the second line of Eq. (1.9). While this possibility
would make sense we are note aware of any result in this direction (by
breaking moreover or not the first line of Eq. (1.9)).

  3. 3)

Break of the validity of the third line of Eq. (1.9). In this case we maintain
thus a finite range $\Gamma$ in the sense of the two first lines of the
definition (1.9), but introduce long-range interactions in the GL Hamiltonian
$\mathcal{E}$ by assuming the function $\mathcal{U}$ is non local. Typically
(the reader will generalise easily) we have still Eq. (1.8), but now
$\mathcal{U}$ is in the form

$\mathcal{U}\left(\varphi\right)=U\left(\varphi(0)\right)+\sum_{z\in\Lambda}K(z)\
V\left(\varphi(0),\varphi(z)\right)$

where $U$ is the one-site potential, $V$ is the interaction potential and $K$
the coupling function. As usual we assume $U$ and $V$ are so that the dynamics
are well defined and that the Gibbs canonical measures make sense, at least
for some non-empty range of chemical potentials. In particular we do not
consider singular potentials and we assume that $U\geq 0$ grows sufficiently
fast to infinity in order to control also the growth of $V$.

There are then several possibilities to introduce long-range effects (in $K$,
in $V$ or in both):

    1. a)

For the first possibility we take $K$ a function like in Eq. (1.10) and $V$
smooth, e.g. a polynomial function. A quite similar case to this has been
considered in [133]. There Yau considers a GL Hamiltonian in the Kac’s form

$\mathcal{E}_{\mathbb{T}_{n}}(\varphi)=\sum_{x\in\mathbb{T}_{n}}U(\varphi(x))\
+\
\sum_{x,y\in\mathbb{T}_{n}}n^{-a}J\left(\tfrac{y-x}{n^{a}}\right)\varphi(x)\varphi(y)$

with $0<a<1$ and $J$ a non-negative function with compact support such that
$\int_{\mathbb{R}}du\ J(u)=1$. We recall that $\mathbb{T}_{n}$ is the discrete
torus of length $n$ and that in the scaling limit, $n$ goes to infinity.
Hence, here, $K(z)=n^{-a}J(z/n^{a})$ is long-range and
$V(\phi,\phi^{\prime})=\phi\phi^{\prime}$ is short-range. The hydrodynamic
equation is still given by a nonlinear diffusion equation like in the short-
range case since the ‘long-range’ effects in [133] are in fact sufficiently
weak. This kind of long-range effects have been lengthly studied in the
context of lattice gas (without being exhaustive, we refer the reader to [74,
98, 76, 2, 77, 81, 49, 106]).

    2. b)

We take $K$ with finite range but $V$ with long-range support. While it could
make sense we are not aware of any result in this direction.

    3. c)

A third possibility is to take $K$ long-range and $V$ long-range. We can for
example consider $\mathcal{U}(\varphi)$ in the form

${\mathcal{U}}(\varphi)=U(\varphi(0))\ +\ \sum_{z}V(\varphi(0),\varphi(z))\ ,$

where $V:\mathbb{R}\to\mathbb{R}$ is long-range, i.e.
$V(\phi,\phi^{\prime})=1/{|\phi-\phi^{\prime}|^{1+\gamma}}$, $\gamma>0$. It is
two times long-range in the sense that $K(z):=1$ does not have even any decay
as $|z|\to\infty$. Apparently this case has not be investigated in the
literature for GL dynamics. Of course, this possibility can not provide long-
range effects for lattice gas since the occupation variables
$\varphi(0),\varphi(z)$ take values in $\\{0,1\\}$.

As mentioned above, in this paper, we will be only concerned with the
situation 1) with $\gamma<2$. If $\gamma>2$ we expect to have a diffusive
behaviour, but the treatment of this case would require, apart from specific
cases, to develop non-gradient tools like in the short range case [110]. This
is out of the scope of the present paper (see Remark 3.4).

######  Remark 1.2.

To conclude this bibliographical interlude, let us mention there exists also
an important literature for interacting Brownian particles [119, Part II,
Section 7.2] with long range interactions (also called sometimes mean-field
models) [50, 46, 115, 47, 73, 48, 38, 39, 37, 9] and an exponentially
increasing interest for related models motivated by active matter [130, 6,
108, 7, 41]. Of course, a system of interacting Brownian particles (or models
in active matter) can be seen as lattice field models if one labels the
particles and defines $\varphi_{t}(x)$ as the position141414For models in
active matter, it would be also necessary to incorporate extra variables like
velocity or angles. of the particle with label $x$ at time $t$. However they
do not provide a $\varphi$-conservation law. The conserved quantity of
interest in these models is the particle number so that the interpretation as
lattice models is not really interesting151515This is different in the context
of lattice gas where typically there, for $x$ in $\Lambda$ and $\eta$ a given
configuration, $\eta(x)\in\\{0,1\\}$ denotes the number of particles on site
$x$. A (conservative) lattice gas is an $\eta$-conservation law so that the
analogy with GL dynamics is clear when we replace $\eta$ by $\varphi$.. In
[95, 64] the authors study properties of the stationary state of a one
dimensional Lieb-Liniger delta Bose gas. In [45] the authors developed MFT for
a system composed of Brownian motions on the line interacting through the long
range Riesz potential. The authors then obtained there the long time behaviour
of the variance of the fluctuations of the integrated current and of the
position of a tagged particle. Connected to these models exists also some
literature motivated by active matter. In particular, let us mention [123,
124] where is studied a Dyson Brownian motions model in which run and tumble
particles interact via a logarithmic repulsive potential in the presence of a
harmonic well. The hydrodynamic limit of the systems there is then governed by
a nonlinear fractional diffusion equation161616In [45], if $s>1$, the
effective dynamics is in fact short-range and the hydrodynamics is given by a
standard diffusion equation. but with a quadratic flux. The obtention of a
quadratic flux will be a very particular case for us. Let us also mention that
in these models particles evolve on the infinite line and that the system is
not subject to boundary thermal forces (without external forces in [95, 45,
64] and with external ones in [123, 124]).

###  1.6 Plan of the paper

After having introduced the model in Section 2 we derive the hydrodynamic and
hydrostatic limits of the model in Section 3 (see Eq. (3.1) and Eq. (3.12)).
The hydrodynamic limit is derived in a subdiffusive time scale, i.e. shorter
than the usual diffusive time. This is done for the original model as well as
for a perturbed dynamics (see Section 3.3) used later to establish a large
deviation principle. In Section 4.1 we derive the (dynamical) large deviations
functional of the empirical density of the volume in the previous subdiffusive
time scale: the probability to observe an atypical macroscopic profile, i.e.
different from the one provided by the solution of the hydrodynamic equation,
is exponentially small in the system size, and we identify the corresponding
rate (see Eq. (4.9)). In Section 4.2 we identify the non-equilibrium free
energy (or quasi-potential) of the NESS as the solution of a stationary
Hamilton-Jacobi equation (see Eq. (4.14)). In Section 5 we study our results
in the diffusive limit $\gamma\to 2^{-}$ and argue the existence of a $0$-th
order phase transition. Section 6 considers special cases. In Section 7 we
investigate the differences and similarities between the diffusive case
(nearest neighbour interactions) and the superdiffusive case ( heavy tails
long range interactions). The paper is completed by several appendices. In a
companion paper [15] we will extend the results of this paper to a more
general set-up including in particular lattice gas with long-range
interactions, and revisit the arguments in the framework of fluctuating
hydrodynamics.

##  2 Long-range GL dynamics

Let $\gamma<2$ and let $K:\mathbb{R}\to(0,\infty]$ be the function defined by

$K(u)=\frac{1}{|u|^{1+\gamma}}\;\;\text{if}\;\;u\neq 0,\quad K(0)=0\ .$ (2.1)

We restrict our general set-up of the introduction to the case where the
lattice $\Lambda$, now denoted $\Lambda_{n}=\\{1,\ldots,n\\}$, $n\geq 1$, is
of dimension $D=1$ and the $\varphi$’s-space $\mathcal{M}$ is equal to to
$\mathbb{R}$ or $\mathbb{R}^{+}$ (hence $N=1$). We also assume the GL
Hamiltonian is without interactions, i.e. given by Eq. (1.11) with $U$ a
smooth potential

$\mathcal{E}(\varphi):=\mathcal{E}_{\Lambda_{n}}(\varphi)=\sum_{x\in\Lambda_{n}}U(\varphi(x))\
.$ (2.2)

Finally, we assume also the choice

$\Gamma_{\varphi}(x,y):=K(y-x)\beta(\varphi(x),\varphi(y))\ ,$ (2.3)

in Eq. (1.4), with $\beta(\varphi(x),\varphi(y))$ is a symmetric positive
function. By Eq. (1.4) and the one-dimensional version of Eq. (1.5), the drift
term $F_{\varphi}$ is then

$\begin{split}&F_{\varphi}(x,y):=K(y-x)\alpha(\varphi(x),\varphi(y))\\\
&=-K(y-x)\beta\left(\varphi(x),\varphi(y)\right)\left(U^{\prime}\left(\varphi(x)\right)-U^{\prime}\left(\varphi(y)\right)\right)+K(y-x)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\beta\left(\varphi(x),\varphi(y)\right)\\\
&=K(y-x)e^{\mathcal{E}(\varphi)}(\partial_{\varphi(x)}-\partial_{\varphi(y)})\left(e^{-\mathcal{E}(\varphi)}\beta(\varphi(x),\varphi(y))\right)\end{split}$
(2.4)

which is an antisymmetric function in $\varphi(x),\varphi(y)$.

This choice is mainly motivated by the fact there are the long-range version
of the models appearing in [127, 110, 100, 58].

###  2.1 Long-range GL dynamics with free boundary conditions

Before to introduce the GL dynamics with long-range interactions in contact
with baths, let us consider the closed dynamics with free boundary conditions.
A typical configuration of a Ginzburg-Landau dynamics is denoted by
$\varphi=\\{\varphi(x)\in\mathbb{R}\;;\;x\in\Lambda_{n}\\}$ and the
configuration space is thus $\mathbb{R}^{\Lambda_{n}}$.

The equations of motion defined through Eqs. (1.1), (1.3), (2.3) and (2.4),
are given for any $x\in\Lambda_{n}$ by the Ito’s stochastic differential
equation

$\partial_{t}\varphi_{t}(x)=\sum_{y\in\Lambda_{n}}K(y-x)\alpha(\varphi_{t}(x),\varphi_{t}(y))+\sqrt{2}\sum_{y\in\Lambda_{n}}\sqrt{K(y-x)\beta(\varphi_{t}(x),\varphi_{t}(y))}\
\dot{\zeta}_{t}(x,y)\ .$ (2.5)

Here, we recall that $\\{\dot{\zeta}_{t}(x,y)\;;\;x<y\\}$ are independent
standard white noises and $\dot{\zeta}(y,x)=-\dot{\zeta}(x,y)$ (by convention
$\dot{\zeta}(x,x)=0$).

The Gibbs probability measure171717To lighten the notations the dependence in
$\Lambda_{n}$ is omitted. $d{\tilde{\mu}}_{0}(\varphi)\propto
e^{-\mathcal{E}(\varphi)}d\varphi$ is a reversible, hence invariant, measure
for this dynamics. This comes from the form of the Markovian generator
$\mathcal{G}_{b}^{n}$ of the dynamics (2.5) (see Appendix A for a proof)

$\begin{split}\mathcal{G}_{b}^{n}&=\frac{1}{2}\sum_{x,y\in\Lambda_{n}}K(y-x)\
e^{\mathcal{E}(\varphi)}(\partial_{\varphi(y)}-\partial_{\varphi(x)})\left[e^{-\mathcal{E}(\varphi)}\
\beta(\varphi(x),\varphi(y))\
(\partial_{\varphi(y)}-\partial_{\varphi(x)})\right]\ .\end{split}$ (2.6)

Moreover, since the volume
$\mathcal{V}(\varphi):=\mathcal{V}_{\Lambda_{n}}(\varphi)$ is a conserved
quantity of the dynamics, the Gibbs probability measures
${\tilde{\mu}}_{\lambda}:={\tilde{\mu}}_{\lambda}^{\Lambda_{n}}$ (see Eq.
(1.6)) parameterised by $\lambda\in\mathbb{R}$181818Depending on the growth
properties of $U$ we have sometimes to restrict the domain of the admissible
$\lambda$’s in order to have well defined Gibbs measures. and given by

${\tilde{\mu}}_{\lambda}(d\varphi)={\mathcal{Z}}_{\lambda}^{-1}\exp\left(-\mathcal{E}(\varphi)-\lambda\mathcal{V}(\varphi)\right)d\varphi\
,$ (2.7)

forms the family of extremal191919This is a consequence of Appendix B.
product202020The fact that the invariant measure is produced by i.i.d. random
variables does not mean that there are no interactions in the dynamics: the
stochastic differential equations appearing in Eq. (2.5) igive rise to an
interacting diffusion process, even when $\beta=1$.  invariant probability
measures of the dynamics. Here $\mathcal{Z}_{\lambda}=(Z(\lambda))^{n}$, where

$Z(\lambda)=\int d\phi\exp\left(-U\left(\phi\right)-\lambda\phi\right)$

is the partition function. In fact, Eq. (2.23) below shows that the dynamics
satisfies detailed balance condition with respect to $\tilde{\mu}_{\lambda}$.

Since the quantity of interest is the volume it is more convenient to perform
a change of parameterization in the equilibrium measures. We denote by
$\nu_{\Phi}$ the probability measure on $\mathbb{R}$ such that

$\nu_{\Phi}(d\phi)=Z^{-1}(\lambda)\ \exp\left(-U(\phi)-\lambda\phi\right)\
d\phi$ (2.8)

with $Z(\lambda)$ the normalisation constant and $\lambda:=\lambda(\Phi)$
chosen such that $\langle\phi\rangle_{\nu_{\Phi}}=\Phi$. Then the product
probability measures

$\mu_{\Phi}:={\tilde{\mu}}_{\lambda(\Phi)}$ (2.9)

obtained in this way are the canonical equilibrium measures of the dynamics.

######  Remark 2.1.

In this one-dimensional set-up, the $\varphi$-conservation law given by Eq.
(1.1) or Eq. (2.5) can be rewritten in the form of the discrete continuity
equation

$\partial_{t}\varphi_{t}(x)\ +\ J_{t}(x)-J_{t}(x-1)=0\ .$

where $J_{t}(x)$ is the instantaneous local rate of $\varphi$-current at $x$,
expressed in terms of the instantaneous rate of $\varphi$-currents

$J_{t}(x)=\sum_{y\leq x}\sum_{z\in\Lambda_{n}}J_{t}(y,z)\ .$ (2.10)

Then, with the relations provided in Eqs. (1.3), (2.3) and (2.4), we obtain

$J_{t}(x)=-\sum_{y\leq
x}\sum_{z\in\Lambda_{n}}\left\\{K(z-y)\alpha(\varphi_{t}(y),\varphi_{t}(z))+\sqrt{K(z-y)\beta(\varphi_{t}(y),\varphi_{t}(z))}\
\dot{\zeta}_{t}(y,z)\right\\}.$ (2.11)

######  Remark 2.2.

The conserved quantity

$\mathcal{V}(\varphi_{t}):=\sum_{x\in\Lambda_{n}}\varphi_{t}(x)$ (2.12)

defined in Eq. (1.2) is sometimes called in the following the ‘volume’, and it
is in fact the only conserved quantity (see Appendix B for a proof). Moreover,
if we restrict the dynamics to the sub-manifold (hyperplane)
$E_{C}=\\{\varphi\in{\mathbb{R}}^{\Lambda_{n}}\;;\;\mathcal{V}(\varphi)=C\\}$,
where $C$ is an arbitrary constant, then the dynamics is ergodic and its
unique invariant measure is the uniform measure on $E_{C}$ (see Appendix B for
a proof). In other words the microcanonical measures of the dynamics with free
boundaries are given by the uniform probability measures on
$(E_{C})_{C\in\mathbb{R}}$.

######  Remark 2.3.

In the GL dynamics derived in [100, 58], in the context of transport of energy
in chains of oscillators, we have that

$\beta(\phi,\phi^{\prime})=\phi\phi^{\prime}\Theta(\phi,\phi^{\prime})$

where $\Theta$ is a smooth positive function. It is then easy to check that in
this case, if we start the dynamics with an initial condition $\varphi_{0}$
such that $\varphi_{0}(x)>0$ for any $x$, then at any time $t\geq 0$,
$\varphi_{t}(x)>0$ for any $x$. This is consistent with the fact that the
variable $\varphi_{t}(x)$ is the value of the energy (which is positive) in
the effective dynamics derived by a suitable scaling limit in [100, 58].

###  2.2 GL thermodynamical quantities

Since the Gibbs equilibrium measures are product, see Eq. (2.7), the
corresponding thermodynamic relations are reduced to the study of the ones for
the marginal $\nu_{\Phi}$. By introducing the (concave212121It follows from
Hölder’s inequality.) equilibrium free energy

$F(\lambda)=-\log Z(\lambda)$ (2.13)

and the thermodynamical (microcanonical) convex222222The supremum over a
family of affine functions is convex. entropy function

$S(\Phi)=\sup_{\lambda}\left\\{F(\lambda)-\lambda\Phi\right\\}$ (2.14)

we get that, if $F(\lambda)$ is differentiable, the usual Legendre duality
relations hold

$\lambda(\Phi)=-S^{\prime}(\Phi),\quad\Phi=F^{\prime}(\lambda(\Phi))\ .$
(2.15)

Moreover, if $F^{\prime\prime}$ exists and is fintie, we also have the static
fluctuation dissipation theorem [92, 93, 96]:

$-F^{\prime\prime}(\lambda(\Phi))=\left\langle\phi^{2}\right\rangle_{\nu_{\Phi}}-(\left\langle\phi\right\rangle_{\nu_{\Phi}})^{2}=\cfrac{1}{S^{\prime\prime}(\Phi)}\geq
0\ ,$ (2.16)

where the last equality follows from (2.15). The last inequality shows then
that the entropy function $S(\Phi)$ is strictly convex, i.e.

$S^{\prime\prime}(\Phi)>0\ .$ (2.17)

######  Remark 2.4.

For the reader not very familiar with the physicist’s jargon, let us point out
that the equilibrium free energy $F$ and the entropy $S$ are just standard
probabilistic objects appearing in the large deviations theory. More
precisely, we have the relations

$F\left(\lambda\right)=F(0)-\Lambda_{\widetilde{\mu}_{0}}(-\lambda)\ ,\quad
S(\Phi)=F(0)+I_{\widetilde{\mu}_{0}}(\Phi)\ ,$

where $\Lambda_{\widetilde{\mu}_{0}}$ (resp. $I_{\widetilde{\mu}_{0}}$ ) is
the scaled cumulant generating function (resp. the large deviation rate
function) of the conserved quantity
$\mathcal{V}_{{\Lambda_{n}}}(\varphi):=\sum_{x\in\Lambda_{n}}\varphi(x)$ under
the product probability measure
${\tilde{\mu}}^{\Lambda_{n}}_{0}(d\varphi)\propto\exp\left(-\mathcal{E}_{\Lambda_{n}}(\varphi)\right)d\varphi$,
defined formally as [60, 126, 62, 56, 52, 51, 122]

$\begin{split}\Lambda_{\widetilde{\mu}_{0}}(\lambda)&:=\lim_{n\rightarrow\infty}n^{-1}\log\left\langle
e^{\lambda\mathcal{V}_{\Lambda_{n}}(\varphi)}\right\rangle_{\widetilde{\mu}_{0}}\
,\\\
I_{\widetilde{\mu}_{0}}(\Phi)&:=-\lim_{n\rightarrow\infty}n^{-1}\log\left\langle\delta\left(\mathcal{V}_{\Lambda_{n}}(\varphi)-n\Phi\right)\right\rangle_{\widetilde{\mu}_{0}}\
.\end{split}$ (2.18)

With this in mind, the relation of Eq.(2.14) is a particular case of Ellis-
Gärtner Theorem [60, 126, 62, 56, 52, 51, 122], valid as soon as $F$ exists,
modulo some more technical assumptions apart from differentiability. We must
also point that the entropy in the physical literature is most of the time
defined as $-S$, and is then concave. We did not choose this convention here
in order to avoid many negative signs in several formula.

###  2.3 Long-range GL dynamics in contact with two baths

At the boundaries we introduce some baths, which will break the detailed
balance condition in the steady state, and which will be responsible for a non
vanishing macroscopic volume flux. Fix two constants $\Phi_{\ell}$ and
$\Phi_{r}$. Then the equations of motion for the particle $1$ and particle $n$
are now

$\begin{split}\partial_{t}\varphi_{t}(1)&=-(\lambda(\Phi_{\ell})+U^{\prime}(\varphi_{t}(1)))+\sqrt{2}\dot{\zeta}_{t}(0,1)\\\
&\ +\
\sum_{y\in\Lambda_{n}}K(y-1)\alpha(\varphi_{t}(1),\varphi_{t}(y))+\sqrt{2}\
\sum_{y\in\Lambda_{n}}{\sqrt{K(y-1)\beta(\varphi_{t}(1),\varphi_{t}(y))}}\
\dot{\zeta}_{t}{(x,y)}\ ,\\\
\partial_{t}\varphi_{t}(n)&=-(\lambda(\Phi_{r})+U^{\prime}(\varphi_{t}(n)))+\sqrt{2}\
\dot{\zeta}_{t}{(n,n+1)}\\\ &\ +\
\sum_{y\in\Lambda_{n}}K(y-n)\alpha(\varphi_{t}(n),\varphi_{t}(y))\
+\sqrt{2}\sum_{y\in\Lambda_{n}}\sqrt{K(y-n)\beta(\varphi_{t}(n),\varphi_{t}(y))}\
\dot{\zeta}_{t}{(x,y)}\ ,\end{split}$ (2.19)

where $\dot{\zeta}{(0,1)},\dot{\zeta}{(n,n+1)}$ are two new independent
standard white noises. The two quantities
$\lambda(\Phi_{\ell}),\lambda(\Phi_{r})$ have been defined previously by Eq.
(2.15).

Let us comment about the choice of the form of the baths. Consider for example
the bath on the left. We want to define a dynamics at the left boundary (site
$1$) such that the invariant measure of this dynamics satisfies the detailed
balance condition with respect to $\nu_{\Phi_{\ell}}$ (recall Eq. (2.8)),
since we want to fix the value of the volume at the left boundary to be
$\Phi_{\ell}$. The dynamics introduced above is thus the most natural choice.

In the presence of the baths, the volume $\mathcal{V}(\varphi_{t})$ defining
in Eq. (1.2) is no longer conserved but, if $\Phi:=\Phi_{\ell}=\Phi_{r}$, then
the systems is still at equilibrium: it satisfies the detailed balance
condition with respect to $\mu_{\Phi}={\tilde{\mu}}_{\lambda(\Phi)}$.

The Markovian generator of the open dynamics is now given by

$\mathcal{G}^{n}=\mathcal{G}_{\ell}+\mathcal{G}_{r}+\mathcal{G}^{n}_{b}\ ,$

where $\mathcal{G}_{b}^{n}$ is the generator of the dynamics with free
boundary conditions given in Eq. (2.6) and

$\mathcal{G}_{\ell}=-(\lambda(\Phi_{\ell})+U^{\prime}(\varphi(1)))\partial_{\varphi(1)}+\partial^{2}_{\varphi(1)},\quad\mathcal{G}_{r}=-(\lambda(\Phi_{r})+U^{\prime}(\varphi(n)))\partial_{\varphi(n)}+\partial^{2}_{\varphi(n)}$
(2.20)

are the generators of the baths.

######  Remark 2.5.

In other words, the Fokker-Planck equation of the GL dynamics described above
takes the form

$\partial_{t}P_{t}(\varphi)={\mathcal{L}}^{n}P_{t}(\varphi)$

where $\mathcal{L}^{n}$ is given by

$\mathcal{L}^{n}=\mathcal{L}_{\ell}+\mathcal{L}_{r}+\mathcal{L}^{n}_{b}\ .$

The bulk part is given by

$\begin{split}\mathcal{L}_{b}^{n}&=\frac{1}{2}\sum_{x,y\in\Lambda_{n}}K(y-x)\
(\partial_{\varphi(y)}-\partial_{\varphi(x)})\left[e^{-\mathcal{E}(\varphi)}\
\beta(\varphi(x),\varphi(y))\ (\partial_{\varphi(y)}-\partial_{\varphi(x)})\
e^{\mathcal{E}(\varphi)}\right]\\\ \end{split}$ (2.21)

and the boundary parts by

$\mathcal{L}_{\ell}=\partial^{2}_{\varphi(1)}+\partial_{\varphi(1)}(U^{\prime}(\varphi(1))+\lambda(\Phi_{\ell})),\quad\mathcal{L}_{r}=\partial^{2}_{\varphi(n)}+\partial_{\varphi(n-1)}(U^{\prime}(\varphi(n))+\lambda(\Phi_{r}))\
.$ (2.22)

In particular, with this expression, it is easy to prove that the dynamics
with free boundary conditions satisfies the detailed balance condition with
respect to $\tilde{\mu}_{\lambda}$ for any $\lambda$ since for any function
$f$, we have

$\widetilde{\mu}_{\lambda}(\varphi)\mathcal{G}_{b}^{n}\left(\frac{f(\varphi)}{\widetilde{\mu}_{\lambda}(\varphi)}\right)=\mathcal{L}_{b}^{n}\left(f(\varphi)\right)\
.$ (2.23)

##  3 Hydrodynamic and hydrostatic limits

###  3.1 Hydrodynamic limit

In this section, we will prove that the hydrodynamic limit is given by the
nonlinear fractional232323The word fractional is justified by the fact the
operator $\mathcal{A}$ can be seen as a generalisation of the classical linear
fractional Laplacian [129]. diffusion equation with Dirichlet boundary
condition

$\begin{cases}&\partial_{t}\Phi_{t}(u)-{\mathcal{A}}(\Phi_{t})\ (u)=0\ ,\\\
&\Phi_{t}(0)=\Phi_{\ell},\quad\Phi_{t}(1)=\Phi_{r}\ ,\\\
&\Phi_{t}|_{t=0}=\Phi_{0}\ ,\end{cases}$ (3.1)

where the nonlinear operator $\mathcal{A}$ is defined by

$\begin{split}{\mathcal{A}}(\Phi_{t})\ (u)&:=\int_{0}^{1}dv\ K(v-u)\
A(\Phi_{t}(u),\Phi_{t}(v))\ ,\end{split}$ (3.2)

with

$A(\Phi,\Phi^{\prime})=\int
d\nu_{\Phi}(\phi)d\nu_{\Phi^{\prime}}(\phi^{\prime})\
\alpha(\phi,\phi^{\prime})\ .$ (3.3)

######  Remark 3.1.

Note that this equation depends on the ‘microscopic’ quantities $K,U,\beta$,
present in the microscopic SDE defined by Eq. (2.5), but also on the
‘emergent’ quantity $S(\Phi)$ (which appears in the measure $\nu_{\Phi}$ given
by Eq. (2.8)).

######  Remark 3.2.

We can rewrite

$\begin{split}{\mathcal{A}}(\Phi_{t})\ (u)&:=\partial_{u}\left(\int_{0}^{u}dv\
\int_{0}^{1}dw\ K(w-v)\
A\left(\Phi_{t}(v),\Phi_{t}(w)\right)\right)\end{split}$ (3.4)

and the term

$J(\Phi(u))=-\int_{0}^{u}dv\ \int_{0}^{1}dw\ K(w-v)\
A\left(\Phi(v),\Phi(w)\right)=-\int_{0}^{u}dw\ \mathcal{A}(\Phi)(w)$ (3.5)

is then the macroscopic instantaneous current associated to the nonlinear
fractional diffusion appearing in Eq. (3.1). As mentioned in the Remark 1.2 of
the Introduction, this current is usually nonlinear and even not necessarily
quadratic.

###### Proof.

Since $\gamma<2$, we have to look at the dynamics in a subdiffusive time
scale, i.e. a time scale shorter than the diffusive one. Therefore we define
the empirical volume profile at time $tn^{\gamma}$ by

$\pi^{n}_{tn^{\gamma}}(u)=\cfrac{1}{n}\sum_{x\in\Lambda_{n}}\varphi_{tn^{\gamma}}(x)\delta(u-\tfrac{x}{n}),\quad
u\in[0,1]\ .$ (3.6)

For any smooth function $G:[0,1]\to\mathbb{R}$ we denote

$\langle\pi_{tn^{\gamma}}^{n},G\rangle=\int_{0}^{1}du\ G(u)\
\pi_{tn^{\gamma}}^{n}(u)=\cfrac{1}{n}\sum_{x\in\Lambda_{n}}\varphi_{tn^{\gamma}}(x)\
G(\tfrac{x}{n})\ .$

We assume that we start from some initial condition associated to a
macroscopic profile $\Phi_{0}:[0,1]\to\mathbb{R}$, i.e.

$\lim_{n\to\infty}\langle\pi_{0}^{n},G\rangle=\int_{0}^{1}du\
G(u)\,\Phi_{0}(u)\ .$ (3.7)

We expect that at any time $t>0$, we have

$\lim_{n\to\infty}\langle\pi_{tn^{\gamma}}^{n},G\rangle=\int_{0}^{1}du\
G(u)\,\Phi_{t}(u)$

where $\\{\Phi_{t}\;;\;t\geq 0\\}$ is the solution of a suitable PDE with
initial condition $\Phi_{0}$.

By using the equations of motions and Eq. (2.1), and the homogeneity of $K$
(i.e. $K(z)=n^{-1-\gamma}K(z/n)$), we have that

$\begin{split}&\langle\pi_{tn^{\gamma}}^{n},G\rangle-\langle\pi_{0}^{n},G\rangle\\\
=&\int_{0}^{t}ds\
\left\\{\frac{1}{n^{2}}\sum_{x,y\in\Lambda_{n}}K\Big{(}\tfrac{y-x}{n}\Big{)}\alpha(\varphi_{sn^{\gamma}}(x),\varphi_{sn^{\gamma}}(y))\
G\Big{(}\tfrac{x}{n}\Big{)}\right.\\\
&\left.\quad-n^{\gamma-1}\left[(\lambda(\Phi_{\ell})+U^{\prime}(\varphi_{sn^{\gamma}}(1)))G\Big{(}\tfrac{1}{n}\Big{)}-(\lambda(\Phi_{r})+U^{\prime}(\varphi_{sn^{\gamma}}(n))G(1)\right]\right\\}\\\
&+\mathcal{M}_{t}^{n}(G)\end{split}$ (3.8)

where $\mathcal{M}^{n}(G)$ is an explicit stochastic noise (martingale) such
that, by a direct computation, satisfies

$\lim_{n\to\infty}\left\langle\left[\mathcal{M}_{t}^{n}(G)\right]^{2}\right\rangle=0\
.$

Here $\langle\cdot\rangle$ denotes the average with respect to the initial
condition and the diverse white noises involved in the dynamics.

Assume that $G(0)=G(1)=0$. Then, since $G$ is smooth, $G(1/n)$ and $G(1-1/n)$
are of order $1/n$ and since $\gamma<2$, the second term on the right-hand
side of Eq. (3.8) vanishes.

To manage the first term on the right-hand side of Eq. (3.8) we rely on the
propagation of local equilibrium assumption [119, Section 2.3, Section 3.1]:

Propagation of local equilibrium (see Figure 2): Let $\varepsilon\ll 1$. Split
$\Lambda_{n}=\cup_{j=1}^{K_{\varepsilon}}\Lambda_{2\varepsilon n}(x_{j})$ into
$K_{\varepsilon}=(2\varepsilon)^{-1}$ consecutive disjoint boxes of size
$2\varepsilon n$, centred around the points $x_{j}$, namely
$\Lambda_{2\varepsilon
n}(x_{j})=\\{x\in\Lambda_{n}\;;\;|x-x_{j}|\leq\varepsilon n\\}$. Then at time
$tn^{\gamma}$, in the double limit first in $n\to\infty$ and then in
$\varepsilon\to 0$, the distribution of
$\\{\varphi_{tn^{\gamma}}(z)\;;\;z\in\Lambda_{n}\\}$ is very close to242424If
$\Lambda^{\prime}\subset\Lambda_{n}$, $\mu_{\Phi}\Big{|}_{\Lambda^{\prime}}$
denotes the marginal of $\mu_{\Phi}$ defined by Eq. (2.9) when restricted to
the box $\Lambda^{\prime}$.

$\mu_{t,\varepsilon,n}^{\rm{loc}}:=\prod_{j=1}^{K_{\varepsilon}}\
\mu_{\Phi_{t}(x_{j}/n)}\Big{|}_{\Lambda_{2\varepsilon n}(x_{j})}\ ,$

where $\Phi_{t}$ is the macroscopic profile we are looking for in the
hydrodynamics limit252525Of course we have to assume that this property is
satisfied for $t=0$, which explains the term ‘propagation’.. We recall that
the Gibbs equilibrium measure $\mu_{\Phi}$ has been defined by Eq. (2.9).

######  Remark 3.3.

The propagation of local equilibrium assumption is a very strong assumption
well accepted in physics but whose proof is one of the main problem in the
mathematical theory of hydrodynamic limits [119, 91]. In the case of short-
range interacting particle systems in contact with reservoirs, a general
theory has been developed in [63] for gradient stochastic lattice gas with
bounded spins, following the seminal work [83]. In the case of systems with
long-range interactions, but which do not encapsulate the models under
investigation, we refer the interested reader to [86, 116] (in these papers
there are no reservoirs and the systems considered are lattice gas). For
systems with long range interactions in contact with reservoirs, the fact that
the hydrodynamic profile satisfies the suitable boundary conditions is a non-
trivial mathematical problem considered in details in [19, 14].

$\Lambda_{n}=\\{1,\ldots,n\\}$$2\varepsilon n$$2\varepsilon n$$2\varepsilon
n$$2\varepsilon n$$2\varepsilon n$$2\varepsilon n$$2\varepsilon
n$$2\varepsilon n$$2\varepsilon n$$2\varepsilon
n$$x_{1}$$x_{2}$$x_{j}$$\Lambda_{2\varepsilon
n}(x_{j})$$x_{K_{\varepsilon}}$$\Phi_{t}(u)$$\mu_{\Phi_{t}\big{(}\tfrac{x_{1}}{n}\big{)}}$$\mu_{\Phi_{t}\big{(}\tfrac{x_{2}}{n}\big{)}}$$\mu_{\Phi_{t}\big{(}\tfrac{x_{j}}{n}\big{)}}$
Figure 2: Propagation of local equilibrium. On top is represented the
microscopic space $\Lambda_{n}$ while in the middle is represented the
macroscopic space $[0,1]$ on which is living the macroscopic hydrodynamic
profile $\Phi_{t}(u)$. On top appears the decomposition of the box
$\Lambda_{n}$ into disjoint successive boxes $\Lambda_{2\varepsilon n}(x_{j})$
of length $2\varepsilon n$ and centred around $x_{j}$. These boxes are
microscopically large in the thermodynamic limit $n\to\infty$ but
macroscopically small since $(2\varepsilon n)/n=2\varepsilon\to 0$. The Gibbs
local equilibrium assumption appears at the bottom of the picture: the
distribution of $\\{\varphi_{tn^{\gamma}}(z)\;;\;z\in\Lambda_{2\varepsilon
n}(x_{j})\\}$ is given by the marginal (w.r.t. the box involved) of
$\mu_{\Phi_{t}(x_{j}/n)}$.

######  Remark 3.4.

If $\gamma>2$, the correct time scale to observe a non-trivial time evolution
of the macroscopic volume profile is the diffusive one. This has been
established in [19] in the context of exclusion process with long jumps. In
[14], a complete picture is provided for almost all complete values of
$\gamma$ (the cases $\gamma\in\\{1,2\\}$ being pathological, they are not
considered). However, extending these results to the GL dynamics considered
here would require a very important work (for a generic case this would ask to
develop non-gradient methods with long-range interactions in the presence of
thermal baths).

For $x\neq y$ denote the local function
$\alpha_{x,y}(\varphi)=\alpha(\varphi(x),\varphi(y))$ and let

$A(\Phi,\Phi^{\prime})=\int
d\nu_{\Phi}(\phi)d\nu_{\Phi^{\prime}}(\phi^{\prime})\
\alpha(\phi,\phi^{\prime})$ (3.9)

be the expectation of $\alpha_{x,y}$ w.r.t. the product probability measure
$d\nu_{\Phi}(\varphi(x))\otimes d\nu_{\Phi^{\prime}}(\varphi(y))$. Then, with
the notations and by the propagation of local equilibrium property above, we
claim then that the term appearing in Eq. (3.8) satisfies

$\begin{split}&\lim_{n\to\infty}\int_{0}^{t}ds\
\left\\{\frac{1}{n^{2}}\sum_{x,y\in\Lambda_{n}}K\Big{(}\tfrac{y-x}{n}\Big{)}\alpha_{x,y}(\varphi_{sn^{\gamma}})\
G\Big{(}\tfrac{x}{n}\Big{)}\right\\}\\\ &=\lim_{\varepsilon\to
0}\lim_{n\to\infty}\int_{0}^{t}ds\
\left\\{\frac{1}{n^{2}}\sum_{x,y\in\Lambda_{n}}K\Big{(}\tfrac{y-x}{n}\Big{)}\
\left\langle\mu^{\rm{loc}}_{s,\varepsilon,n},\alpha_{x,y}\right\rangle\
G\Big{(}\tfrac{x}{n}\Big{)}\right\\}\\\ &=\int_{0}^{t}ds\
\left\\{\int_{[0,1]^{2}}dudv\ K(v-u)\ A(\Phi_{s}(u),\Phi_{s}(v))G(u)\right\\}\
.\end{split}$ (3.10)

To justify the second equality in the previous display, we split the sum as

$\begin{split}&\frac{1}{n^{2}}\sum_{x,y\in\Lambda_{n}}K\Big{(}\tfrac{y-x}{n}\Big{)}\
\left\langle\mu^{\rm{loc}}_{s,\varepsilon,n},\alpha_{x,y}\right\rangle\
G\Big{(}\tfrac{x}{n}\Big{)}\\\
&=\frac{1}{n^{2}}\sum_{i,j=1}^{K_{\varepsilon}}\sum_{x\in\Lambda_{2\varepsilon
n}(i)}\sum_{y\in\Lambda_{2\varepsilon n}(j)}K\Big{(}\tfrac{y-x}{n}\Big{)}\
\left\langle\mu^{\rm{loc}}_{s,\varepsilon,n},\alpha_{x,y}\right\rangle\
G\Big{(}\tfrac{x}{n}\Big{)}\\\ &=\frac{1}{n^{2}}\sum_{i\neq
j=1}^{K_{\varepsilon}}\sum_{x\in\Lambda_{2\varepsilon
n}(i)}\sum_{y\in\Lambda_{2\varepsilon n}(j)}K\Big{(}\tfrac{y-x}{n}\Big{)}\
\left\langle\mu^{\rm{loc}}_{s,\varepsilon,n},\alpha_{x,y}\right\rangle\
G\Big{(}\tfrac{x}{n}\Big{)}\\\
&+\frac{1}{n^{2}}\sum_{i=1}^{K_{\varepsilon}}\sum_{x\neq
y\in\Lambda_{2\varepsilon n}(i)}K\Big{(}\tfrac{y-x}{n}\Big{)}\
\left\langle\mu^{\rm{loc}}_{s,\varepsilon,n},\alpha_{x,y}\right\rangle\
G\Big{(}\tfrac{x}{n}\Big{)}\ .\end{split}$ (3.11)

To deal with the first sum in the last equality of Eq. (3.11) we observe that
for $i\neq j$ and $x\in\Lambda_{2\varepsilon n}(i)$,
$y\in\Lambda_{2\varepsilon n}(j)$,

$\begin{split}&\left\langle\mu^{\rm{loc}}_{s,\varepsilon,n},\alpha_{x,y}\right\rangle=\left\langle\mu_{\Phi_{t}(x_{j}/n)}\big{|}_{\Lambda_{2\varepsilon
n}(x_{i})}\otimes\mu_{\Phi_{t}(x_{j}/n)}\big{|}_{\Lambda_{2\varepsilon
n}(x_{j})}\ ,\ \alpha_{x,y}(\varphi)\right\rangle\ =\
A\left(\Phi_{t}(x_{i}/n),\Phi_{t}(x_{j}/n)\right)\ ,\end{split}$

where the last equality follows from a trivial computation, remembering that
$\mu_{\Phi}$ is a product measure with marginals given by $\nu_{\Phi}$. We
also argue that in the first sum in the last equality of Eq. (3.11) we can
replace $K\Big{(}\tfrac{y-x}{n}\Big{)}G\Big{(}\tfrac{x}{n}\Big{)}$ by
$K\Big{(}\tfrac{x_{j}-x_{i}}{n}\Big{)}G\Big{(}\tfrac{x_{i}}{n}\Big{)}$ since
the difference between the two previous terms will vanish as262626In fact,
this would be trivial if $K$ was a regular function but we have to be careful
here because $K$ is singular at the origin. We prefer to omit this technical
problem to not increase the length of the paper. $\varepsilon$ will go to
zero. After this last replacement we recognise a discrete Riemann sum
converging, as $n$ goes to infinity and then $\varepsilon$ to $0$, to

$\int_{[0,1]^{2}}dudv\ K(v-u)\ A(\Phi_{s}(u),\Phi_{s}(v))G(u)\ .$

Now, for the second sum in the last equality of Eq. (3.11), we observe that

$\frac{1}{n^{2}}\sum_{i=1}^{K_{\varepsilon}}\sum_{x\neq
y\in\Lambda_{2\varepsilon n}(i)}K\Big{(}\tfrac{y-x}{n}\Big{)}\
\left\langle\mu_{\Phi_{s}(x_{i}/n)}\ ,\ \alpha_{x,y}(\varphi)\right\rangle\
G\Big{(}\tfrac{x}{n}\Big{)}=0\ ,$

since for any fixed $\Phi$, we have, thanks to the antisyammetry of
$\alpha_{x,y}=-\alpha_{y,x}$, that

$\left\langle\mu_{\Phi}\ ,\ \alpha_{x,y}\right\rangle=0\ .$

This concludes the justification of the second equality in (3.10). It follows
that the hydrodynamic limit of the system is given by the solution of Eq.
(3.1).

∎

######  Remark 3.5.

The attentive reader will notice that in Eq. (3.2), the kernel $K$ is singular
and that it is not clear that the integral makes sense (in fact it does not
make sense!). The precise notion of solution to Eq. (3.1) has to be
interpreted in a weak (distributional) sense (see Appendix C).

###  3.2 Hydrostatic limit and properties of the NESS

Taking the large time limit in the hydrodynamic equation we have that the
stationary profile $\Phi_{ss}(u)$ in the NESS is the solution to

$\begin{cases}&{\mathcal{A}}(\Phi_{ss})\ (u)=0\ ,\\\
&\Phi_{ss}(0)=\Phi_{\ell},\quad\Phi_{ss}(1)=\Phi_{r}\ .\end{cases}$ (3.12)

######  Remark 3.6.

Here also the precise notion of solution to Eq. (3.12) has to be interpreted
in a weak (distributional) sense (see Appendix C).

######  Remark 3.7.

It is also possible to show that in the NESS the the instantaneous local rate
of $\varphi$-current at $x$ defined by Eq. (2.10), scales as $n^{1-\gamma}$
(for a standard diffusive system it would scale as $n^{-1}$). Indeed, the
instantaneous microscopic current at $x\in\\{2,\ldots,n-2\\}$ was introduced
in Eq. (2.11) and given by

$J_{t}(x)=-\sum_{y\leq
x}\sum_{z\in\Lambda_{n}}\left\\{K(z-y)\alpha(\varphi_{t}(y),\varphi_{t}(z))+\sqrt{K(z-y)\beta(\varphi_{t}(y),\varphi_{t}(z))}\
\dot{\zeta}_{t}(y,z)\right\\}.$ (3.13)

At the boundaries the definition of the current has to be modified to take
into account the exchange with the baths. By using the local equilibrium
property, we deduce that the current satisfies, for any $u\in[0,1]$,

$\lim_{n\to\infty}n^{\gamma-1}\left\langle
J_{t}([nu])\right\rangle_{ss}=\int_{0}^{u}dv\ \int_{0}^{1}dw\ K(w-v)\
A\left(\Phi_{ss}(v),\Phi_{ss}(w)\right)\ .$

Of course, the quantity on the right hand side is falsely dependent on $u$,
since taking its derivative in $u$, we get

$\int_{0}^{1}dw\ K(w-u)\ A\left(\Phi_{ss}(u),\Phi_{ss}(w)\right)$

which is equal to $0$ since the profile $\Phi_{ss}$ satisfies Eq. (3.12).

###  3.3 Hydrodynamic limit for a time inhomogeneous perturbed dynamics

Consider a macroscopic time-dependent scalar field $H_{t}(u)$, $u\in[0,1]$,
and consider the perturbed dynamics obtained by addingto the stochastic
differential equation Eq. (2.5) a drift term in the form

$d_{t}(x,\varphi)=-\sum_{y\in\Lambda_{n}}K(y-x)\beta(\varphi(x),\varphi(y))\left(H_{t}\big{(}\tfrac{y}{n}\big{)}-H_{t}\big{(}\tfrac{x}{n})\big{)}\right)\
.$ (3.14)

By proceeding like in a previous section (see Appendix D for details) we
deduce that the hydrodynamic equations of the perturbed system are given by

$\begin{cases}&\partial_{t}\Phi_{t}(u)-\mathcal{A}(\Phi_{t})\ (u)\ =\
-\mathcal{B}_{\Phi_{t}}H_{t}\ (u)\ ,\\\
&\Phi_{t}(0)=\Phi_{\ell},\quad\Phi_{t}(1)=\Phi_{r}\ ,\\\
&\Phi_{t}|_{t=0}=\Phi_{0}\ ,\end{cases}$ (3.15)

with the linear mobility operator $\mathcal{B}_{\Phi_{t}}$ given by

${\mathcal{B}}_{\Phi_{t}}H_{t}\ (u):=\int_{0}^{1}dv\
K(v-u)(H_{t}(v)-H_{t}(u))B(\Phi_{t}(u),\Phi_{t}(v))$ (3.16)

with

$B(\Phi,\Phi^{\prime})=\int
d\nu_{\Phi}(\phi)d\nu_{\Phi^{\prime}}(\phi^{\prime})\
\beta(\phi,\phi^{\prime})>0\ .$ (3.17)

######  Remark 3.8.

The bulk part of the Markovian generator, given previously by Eq. (2.6), has
now, for the perturbed dynamics, the form

$\mathcal{G}_{b,t}^{n,d}:=\mathcal{G}_{b}^{n}-\frac{1}{2}\sum_{x\in\Lambda_{n}}\sum_{y\in\Lambda_{n}}K(y-x)\beta\left(\varphi\left(x\right),\varphi\left(y\right)\right)\left(H_{t}\left(\tfrac{y}{n}\right)-H_{t}\left(\tfrac{x}{n}\right)\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\
.$ (3.18)

Therefore
$\left[\mathcal{G}_{b,t}^{n,d}\right]^{\dagger}\left(e^{-\mathcal{E}\left(\varphi\right)}\right)\neq
0$: the Gibbs probability measure ${\tilde{\mu}}_{0}(d\varphi)\propto
e^{-\mathcal{E}(\varphi)}d\varphi$ is no longer invariant in the bulk for this
perturbed dynamics. Anyway, the drift $d_{t}(x,\varphi)$ defined in Eq. (3.14)
conserves the ‘gradient structure’ given in Eq. (1.4) of the GL stochastic
differential equation, i.e. the total drift of the perturbed stochastic
differential equation takes the form

$F_{\varphi,t}(x,y):=\exp\left(\mathcal{E}\left(\varphi\right)+\mathcal{E}_{t}^{ext}\left(\varphi\right)\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\left[K(y-x)\beta\left(\varphi\left(x\right),\varphi\left(y\right)\right)\exp\left(-\mathcal{E}\left(\varphi\right)-\mathcal{E}_{t}^{ext}\left(\varphi\right)\right)\right]\
,$

with the perturbed Hamiltonian

$\mathcal{E}_{t}^{ext}\left(\varphi\right)=-\sum_{x\in\Lambda_{n}}\varphi(x)\
H_{t}\left(\tfrac{x}{n}\right)\ .$

This implies that we still have
$\left[\mathcal{G}_{b,t}^{n,d}\right]^{\dagger}\left(e^{-\mathcal{E}\left(\varphi\right)-\mathcal{E}_{t}^{ext}\left(\varphi\right)}\right)\neq
0$: the Gibbs probability measure proportional to
$e^{-\mathcal{E}\left(\varphi\right)-\mathcal{E}_{t}^{ext}\left(\varphi\right)}d\varphi$
is an accompanying measure (see [113] for physical implication of this type of
measure.)

######  Remark 3.9.

The precise notion of solution to Eq. (3.15) has again to be interpreted in a
weak (distributional) sense (see Appendix C).

###  3.4 Einstein relation

Observe first that by Eq. (1.4), Eq. (1.5) (or Eq.(2.4) in this set-up), which
are kind of microscopic Einstein relations, we have the following hydrodynamic
level Einstein relation

$A(\Phi,\Phi^{\prime})=\left(S^{\prime}(\Phi^{\prime})-S^{\prime}(\Phi)\right)B(\Phi,\Phi^{\prime})$
(3.19)

where $S$ is the thermodynamic entropy defined in Eq. (2.14). Thanks to the
relation Eq. (3.2) and Eq. (3.16), this implies also a second form of
hydrodynamic level Einstein relation:

$\mathcal{A}(\Phi)=\mathcal{B}_{\Phi}\Big{(}S^{\prime}(\Phi)\Big{)}\ .$ (3.20)

Hence the hydrodynamic equation given by Eq. (3.1) becomes

$\begin{cases}&\partial_{t}\Phi_{t}(u)-\mathcal{B}_{\Phi_{t}(u)}\Big{(}S^{\prime}(\Phi_{t}(u))\Big{)}=0\
,\\\ &\Phi_{t}(0)=\Phi_{\ell},\quad\Phi_{t}(1)=\Phi_{r}\ ,\\\
&\Phi_{t}|_{t=0}=\Phi_{0}\ .\end{cases}$ (3.21)

###### Proof of (3.19).

The definition (3.9) of $A$ and Eq. (2.4) give

$\begin{split}A\left(\Phi,\Phi^{\prime}\right)&=\int d\nu_{\Phi}(\phi)\
d\nu_{\Phi^{\prime}}(\phi^{\prime})\
\left[\left(U^{\prime}(\phi^{\prime})-U^{\prime}(\phi)\right)\beta\left(\phi,\phi^{\prime}\right)+\partial_{\phi}\beta\
(\phi,\phi^{\prime})-\partial_{\phi^{\prime}}\beta\
(\phi,\phi^{\prime})\right]\\\ &=\int d\phi\ d\phi^{\prime}\
e^{-\left(U(\phi)+\lambda(\Phi)\phi\right)+F\left(\lambda(\Phi)\right)}\
e^{-(U(\phi^{\prime})+\lambda(\Phi^{\prime})\phi^{\prime})+F(\lambda(\Phi^{\prime}))}\\\
&\quad\quad\quad\quad\quad\times\left[\left(U^{\prime}(\phi^{\prime})-U^{\prime}(\phi)\right)\beta(\phi,\phi^{\prime})+\partial_{\phi}\beta\
(\phi,\phi^{\prime})-\partial_{\phi^{\prime}}\beta\
(\phi,\phi^{\prime})\right]\end{split}$ (3.22)

where in the second equality we used the definition (2.8). We have then

$\begin{split}A\left(\Phi,\Phi^{\prime}\right)&=\int d\phi
d\phi^{\prime}\left(\frac{d}{d\phi}+\lambda({\Phi})\right)\left[e^{-\left(U(\phi)+\lambda(\Phi)\phi\right)+F\left(\lambda(\Phi)\right)}\
e^{-(U(\phi^{\prime})+\lambda(\Phi^{\prime})\phi^{\prime})+F(\lambda(\Phi^{\prime}))}\beta(\phi,\phi^{\prime})\right]\\\
&-\int d\phi
d\phi^{\prime}\left(\frac{d}{d\phi^{\prime}}+\lambda({\Phi^{\prime}})\right)\left[e^{-\left(U(\phi)+\lambda(\Phi)\phi\right)+F\left(\lambda(\Phi)\right)}\
e^{-(U(\phi^{\prime})+\lambda(\Phi^{\prime})\phi^{\prime})+F(\lambda(\Phi^{\prime}))}\beta(\phi,\phi^{\prime})\right]\end{split}$
(3.23)

and then

$A\left(\Phi,\Phi^{\prime}\right)=\left[\lambda({\Phi})-\lambda({\Phi^{\prime}})\right]\int
d\nu_{\Phi}(\phi)d\nu_{\Phi^{\prime}}(\phi^{\prime})\beta\left(\phi,\phi^{\prime}\right)$

By the duality relation (2.15) and the definition (3.17), we get finally the
relation (3.19).

∎

##  4 Large deviations

###  4.1 Dynamical large deviations

Fixing some horizon time $T>0$ and starting from an initial configuration
associated to a macroscopic profile $\Phi_{0}$ (in the sense of Eq. (3.7)), we
want now to estimate the probability, as $n$ goes to infinity, to observe an
atypical profile $\Phi$, i.e.

$\mathbb{P}\left(\pi_{tn^{\gamma}}^{n}\approx\Phi_{t}\;;\;0\leq t\leq
T\right)\approx\exp(-nI_{[0,T]}(\Phi))$ (4.1)

where
$\pi^{n}_{tn^{\gamma}}(u)=\tfrac{1}{n}\sum_{x\in\Lambda_{n}}\varphi_{tn^{\gamma}}(x)\delta(u-\tfrac{x}{n})$
is the empirical volume profile at time $tn^{\gamma}$ and where
$I_{[0,T]}(\Phi)$ is the so-called dynamical large deviations function. This
probability will be therefore exponentially small in $n$, with however
$I_{[0,T]}(\Phi)$ vanishing if and only if $\Phi$ is solution of the
hydrodynamic equation (3.1).

We prove in this section Eq. (4.1) and that

$I_{[0,T]}(\Phi)=\int_{0}^{T}dt\ \mathbb{L}(\Phi_{t},\partial_{t}\Phi_{t})\ ,$
(4.2)

where the Lagrangian $\mathbb{L}$ takes the form

${\mathbb{L}}(\Phi,p)=\dfrac{1}{4}\|p-\mathcal{A}(\Phi)\|_{-1,{\mathcal{B}_{\Phi}}}^{2},$
(4.3)

with the weighted fractional Sobolev norm272727While the first equality is
only formal, the second one is well defined. defined by

$\begin{split}\|\Psi\|^{2}_{-1,\mathcal{B}_{\Phi}}&:=\left\langle\Psi\ ,\
(-\mathcal{B}_{\Phi})^{-1}\Psi\right\rangle\ =\
\sup_{p}\left\\{\langle\Psi,p\rangle-\tfrac{1}{4}\langle
p,(-\mathcal{B}_{\Phi})p\rangle\right\\}\ .\end{split}$ (4.4)

######  Remark 4.1.

Even if we did not give the details here, it can be shown by similar arguments
that the dynamical rate function ${R}_{[0,T]}(\Phi,\mathcal{J})$ corresponding
to the probability to observe an atypical volume profile $\Phi$ and an
atypical $\varphi$-current $\mathcal{J}$ (satisfying the continuity equation
$\partial_{t}\Phi_{t}(u)+\partial_{u}\mathcal{J}_{t}(u)=0$) is given by the
quadratic form

$\begin{split}R_{[0,T]}(\Phi,\mathcal{J})&=\frac{1}{4}\
\int_{0}^{T}dt\left\|\mathcal{J}_{t}-J(\Phi_{t})\right\|^{2}_{-1,\mathcal{B}_{\Phi_{t}}}\
\end{split}$

where $J(\Phi_{t})$ is the typical current given in Eq. (3.5)

$J(\Phi_{t}(u))=-\int_{0}^{u}dw\ \mathcal{A}(\Phi_{t})(w)\ .$

###### Proof of (4.2).

In the hydrodynamic time scale $tn^{\gamma}$, the ratio between the paths
measure of the perturbed process $\mathbb{Q}^{n}_{H}$ (introduced in the
previous section) and the paths measure of the unperturbed process
$\mathbb{Q}^{n}_{0}$ satisfies

$\begin{split}&\cfrac{\mathbb{Q}^{n}_{0}(\\{\pi_{t}\;;\;0\leq t\leq
T\\})}{\mathbb{Q}^{n}_{H}(\\{\pi_{t}\;;\;0\leq t\leq T\\})}\\\
&=\exp\left\\{-\frac{n}{2}\left[\langle\pi_{T},H_{T}\rangle-\langle\pi_{0},H_{0}\rangle-\int_{0}^{T}\langle\pi_{t},\partial_{t}H_{t}\rangle\
dt-\int_{0}^{T}n^{\gamma}\mathcal{G}^{n}\left(\langle\pi_{t},H_{t}\rangle\right)\
dt\right.\right.\\\
&\left.\left.\quad\quad\quad\quad\quad\quad-\cfrac{1}{4}\int_{0}^{T}n^{\gamma+1}\Gamma^{n}\left(\langle\pi_{t},H_{t}\rangle,\langle\pi_{t},H_{t}\rangle\right)\
dt\right]\right\\}\ .\\\ \end{split}$ (4.5)

Here $\Gamma^{n}$ is the bilinear ‘carré du champ’ operator associated to
$\mathcal{G}^{n}$ and is given, for two test functions $f,g$, by

$\begin{split}\Gamma^{n}(f,g)&=\mathcal{G}^{n}(fg)-f\mathcal{G}^{n}g-g\mathcal{G}^{n}f\\\
&=\sum_{x,y\in\Lambda_{n}}K(y-x)\ \beta(\varphi(x),\varphi(y))\
\left[\left(\partial_{\varphi(y)}f-\partial_{\varphi(x)}f\right)\right]\left[\left(\partial_{\varphi(y)}g-\partial_{\varphi(x)}g\right)\right]\\\
&+2\ \partial_{\varphi(1)}f\ \partial_{\varphi(1)}g+2\
\partial_{\varphi(n-1)}f\ \partial_{\varphi(n-1)}g\ .\end{split}$ (4.6)

In particular we have that

$\begin{split}&\Gamma^{n}\left(\langle\pi_{t},H_{t}\rangle,\langle\pi_{t},H_{t}\rangle\right)\\\
=&\frac{1}{n^{2}}\sum_{x,y\in\Lambda_{n}}K(y-x)\ \beta(\varphi(x),\varphi(y))\
\left[H_{t}\Big{(}\tfrac{y}{n}\Big{)}-H_{t}\Big{(}\tfrac{x}{n}\Big{)}\right]^{2}+\cfrac{2}{n^{2}}\
\left(H_{t}(1/n)\right)^{2}+\cfrac{2}{n^{2}}\left(H_{t}(1)\right)^{2}\
.\end{split}$

The atypical profile $\Phi$ being fixed we choose the scalar field $H_{t}(u)$
such that it satisfies the linear fractional Poissson equation with
homogeneous Dirichlet boundary conditions:

$\begin{split}&-\ {\mathcal{B}}_{\Phi_{t}}H_{t}\ (u)=\
\partial_{t}\Phi_{t}(u)-\mathcal{A}(\Phi_{t})(u)\ ,\quad H_{t}(0)=H_{t}(1)=0\
.\end{split}$ (4.7)

Again, Eq. (4.7) has to be interpreted in a weak sense (i.e. multiplying
everything in this equation by a smooth test function with compact support,
integrating in space, and using symmetry arguments to write well defined
integrals) because Eq. (3.16) does not make sense as a classical integral.

Thanks to this particular choice, we observe that as $n$ goes to infinity,

$\mathbb{Q}^{n}_{H}(\\{\pi_{t}\approx\Phi_{t}\;;\;0\leq t\leq T\\})\approx 1\
.$

It follows that

$\begin{split}&\mathbb{P}\left(\pi_{tn^{\gamma}}^{n}\approx\Phi_{t}\;;\;0\leq
t\leq T\right)\\\ &=\mathbb{Q}^{n}_{0}(\\{\pi_{t}\approx\Phi_{t}\;;\;0\leq
t\leq
T\\})=\mathbb{E}^{n}_{H}\left[{\mathbb{1}}_{\\{\pi_{t}\approx\Phi_{t}\;;\;0\leq
t\leq T\\}}\ \frac{{\mathbb{Q}}^{n}_{0}(\\{\pi_{t}\;;\;0\leq t\leq
T\\})}{{\mathbb{Q}}^{n}_{H}(\\{\pi_{t}\;;\;0\leq t\leq T\\})}\right]\\\
&=\mathbb{E}^{n}_{H}\left[\frac{{\mathbb{Q}}^{n}_{0}(\\{\pi_{t}\;;\;0\leq
t\leq T\\})}{{\mathbb{Q}}^{n}_{H}(\\{\pi_{t}\;;\;0\leq t\leq T\\})}\right]\
.\end{split}$

We deduce, by using that for the perturbed system, the hydrodynamic profile is
$\Phi$, and the propagation of local equilibrium assumption, that as $n$ goes
to infinity

$\mathbb{P}\left(\pi_{tn^{\gamma}}^{n}\approx\Phi_{t}\;;\;0\leq t\leq
T\right)\approx\exp(-nI_{[0,T]}(\Phi))$

where the dynamical large deviations function is

$\begin{split}I_{[0,T]}(\Phi)&=\frac{1}{2}\int_{0}^{T}dt\
\left\\{\langle\partial_{t}\Phi_{t}-{\mathcal{A}}(\Phi_{t})\ ,\ H_{t}\rangle\
+\tfrac{1}{2}\langle{\mathcal{B}}_{\Phi_{t}}H_{t},H_{t}\rangle\right\\}\\\
\end{split}$

if $\Phi$ satisfies $\Phi_{t}(0)=\Phi_{\ell}$, $\Phi_{t}(1)=\Phi_{r}$ for any
$t\in[0,T]$ and $+\infty$ otherwise (because the probability to observe a
profile which does not satisfy the Dirichlet boundary conditions will be
superexponentially small in $n$).

Let us now give an equivalent expression of the dynamical large deviation
functional. By Eq. (4.7), we have that

$\langle\partial_{t}\Phi_{t}-{\mathcal{A}}(\Phi_{t})\ ,\
H_{t}\rangle=-\langle{\mathcal{B}}_{\Phi_{t}}H_{t},H_{t}\rangle$ (4.8)

so that

$\begin{split}I_{[0,T]}(\Phi)&=\frac{1}{4}\
\int_{0}^{T}dt\langle-{\mathcal{B}}_{\Phi_{t}}H_{t},H_{t}\rangle\\\
&=\frac{1}{8}\int_{[0,1]^{2}}dudv\ \ K(v-u)(H_{t}(v)-H_{t}(u))^{2}\
B(\Phi_{t}(u),\Phi_{t}(v))\ .\end{split}$ (4.9)

The last equality shows that for any scalar field $\Phi$,
$-\mathcal{B}_{\Phi}$ is a non-negative linear operator whose kernel is
constituted of constant functions. Hence restricted on the set of functions
$H$ satisfying homogeneous Dirichlet boundary conditions $H(0)=H(1)=0$, it is
invertible. It is then easy to check that (use Eq. (4.8) and Eq. (4.4))

$\begin{split}I_{[0,T]}(\Phi)&=\frac{1}{4}\
\int_{0}^{T}dt\left\|\partial_{t}\Phi_{t}-\mathcal{A}(\Phi_{t})\right\|^{2}_{-1,\mathcal{B}_{\Phi_{t}}}\
.\end{split}$ (4.10)

∎

###  4.2 Static large deviations in the NESS : non-equilibrium free energy or
quasi-potential

The MFT claims that the non-equilibrium free energy, i.e. the large deviations
function of the empirical volume density in the NESS, can be recovered as the
solution (called the quasi-potential) of a variational problem involving the
dynamical large deviation function $I_{[0,T]}$ introduced above. If we believe
this principle is still correct we obtain hence an indirect way to compute the
non-equilibrium free energy of the NESS since we have an explicit expression
for $I_{[0,T]}$.

Classical arguments in analytic mechanics imply that the quasi-potential $V$,
defined on profiles $\Phi(u)$ such that $\Phi(0)=\Phi_{\ell},\Phi(1)=\Phi_{r}$
by

$V(\Phi)=\inf_{\begin{subarray}{c}\pi(-\infty)=\Phi_{ss}\\\
\pi(0)=\Phi\end{subarray}}\
I_{(-\infty,0]}(\pi)=\inf_{\begin{subarray}{c}\pi(-\infty)=\Phi_{ss}\\\
\pi(0)=\Phi\end{subarray}}\
\int_{-\infty}^{0}\mathbb{L}(\pi_{t},\partial_{t}\pi_{t})\ dt\ ,$ (4.11)

solves the stationary Hamilton-Jacobi equation

$\mathbb{H}\left(\Phi,\tfrac{\delta V}{\delta\Phi}\right)=0,\quad\tfrac{\delta
V}{\delta\Phi}(0)=\tfrac{\delta V}{\delta\Phi}(1)=0\ $ (4.12)

where the Hamiltonian $\mathbb{H}$ is defined by

$\mathbb{H}(\Phi,p)=\sup_{\xi}\left\\{\langle\xi,p\rangle-\mathbb{L}(\Phi,\xi)\right\\}=\langle{\mathcal{A}}(\Phi),p\rangle+\langle
p,(-\mathcal{B}_{\Phi})p\rangle\ .$ (4.13)

Observe that by using the Einstein relation (3.20), it can be rewritten as

$\left\langle\tfrac{\delta
V}{\delta\Phi},\mathcal{B}_{\Phi}\left(-S^{\prime}(\Phi)+\tfrac{\delta
V}{\delta\Phi}\right)\right\rangle=0,\quad\tfrac{\delta
V}{\delta\Phi}(0)=\tfrac{\delta V}{\delta\Phi}(1)=0\ .$ (4.14)

######  Remark 4.2.

Here, we simplify considerably the mathematical difficulties associated to the
problem by avoiding to specify correctly the topological setting associated to
this infinite-dimensional Hamilton-Jacobi equation, see [28] for an
interesting discussion. In fact, with a complete mathematical rigour, there is
not uniqueness of the solution of the stationary Hamilton-Jacobi equation but
it can be shown that the non-equilibrium free energy is bigger than any
stationary Hamilton-Jacobi equation vanishing for $\Phi=\Phi_{ss}$, the volume
profile in the NESS (to see it it is is sufficient to follow mutatis mutandis
the proof of Lemma 6.1 in [26]).

######  Remark 4.3.

MFT claims that $V$ is the non-equilibrium free energy of the NESS [30,
Section IV]. While the theory is very general, it is usually very difficult to
compute $V$ explicitly, even numerically. We observe that $V$ depends only on
the two macroscopic parameters $A$ and $B$ appearing respectively in Eq. (3.3)
and Eq. (3.17), which are related by the Einstein relation (3.19). On the
other hand $V$ does not depend on the microscopic parameters of the model.

######  Remark 4.4.

In Section 6, we will consider special cases where we can obtain the explicit
maximal solution of the Hamilton-Jacobi equation (4.14).

######  Remark 4.5.

In the literature (Eq. (4.12) in [30] in a diffusive set-up or [45] in a long-
range situation), an important role is played by the so called ‘MFT’s
deterministic equations’, which are the canonical equations associated to the
Hamiltonian $\mathbb{H}$. This provides a theoretical way to solve the
Hamilton-Jacobi equation by characteristics. In our set-up, theses equation
associated to Eq.(4.13) are given by

$\begin{cases}&\left(\partial_{t}\Phi_{t}\right)(u)=\cfrac{\delta\mathbb{H}\left[\Phi_{t},p_{t}\right]}{\delta
p_{t}}\
(u)=\mathcal{A}\left[\Phi_{t}\right](u)-2\mathcal{B}_{\Phi_{t}}\left[p_{t}\right](u)\
,\\\ &\\\
&\left(\partial_{t}p_{t}\right)(u)=-\cfrac{\delta\mathbb{H}\left[\Phi_{t},p_{t}\right]}{\delta\Phi_{t}}\
(u)=-\left\langle\mathcal{A}_{\Phi_{t}}^{\ell}\left[\delta_{u}\right],p_{t}\right\rangle-\left\langle
p_{t},\left(-\mathcal{B}_{\Phi_{t},u}^{\ell}\right)\left[p_{t}\right]\right\rangle\
,\end{cases}$

with $p_{t}(0)=p_{t}(1)=0$ and the linearised operators defined by

$\begin{cases}&\mathcal{A}_{\Phi}^{\ell}\left[h\right]:=\lim_{\varepsilon\rightarrow
0}\frac{\mathcal{A}\left[\Phi+\varepsilon
h\right]-\mathcal{A}\left[\Phi\right]}{\varepsilon}\ ,\\\ &\\\
&\mathcal{B}_{\Phi,u}^{\ell}:=\lim_{\varepsilon\rightarrow
0}\frac{\mathcal{B}_{\Phi+\varepsilon\delta_{u}}-\mathcal{B}_{\Phi}}{\varepsilon}\
.\end{cases}$

These MFT’s deterministic equations are non linear non local equations, which
seem a priori impossible to solve in general.

######  Remark 4.6.

Like in the diffusive case [24, Section 2.4], the quasi-potential $V$ is a
Lyapounov function for the hydrodynamic equations. Indeed, by using the
Hamilton-Jacobi equation (4.12) satisfied by $V$, we see that if $\Phi_{t}(u)$
is solution of Eq. (3.1) then

$\begin{split}\frac{d}{dt}V(\Phi_{t})&=\left\langle\tfrac{\delta
V}{\delta\Phi}(\Phi_{t}),\partial_{t}\Phi_{t}\right\rangle=\left\langle\tfrac{\delta
V}{\delta\Phi}(\Phi_{t}),\mathcal{A}(\Phi_{t})\right\rangle\\\
&=-\left\langle\tfrac{\delta V}{\delta\Phi}(\Phi_{t})\ ,\
\left(-{\mathcal{B}}_{\Phi_{t}}\right)\tfrac{\delta
V}{\delta\Phi}(\Phi_{t})\right\rangle\ \leq\ 0\ ,\end{split}$

the last inequality coming from the fact that the linear operator
$-{\mathcal{B}}_{\Phi}$ is a non-negative operator (this follows from Eqs.
(3.16) and (3.17)).

This can be seen as an extension of the second law for non-equilibrium
systems.

######  Remark 4.7.

At equilibrium, i.e. if $\Phi_{\ell}=\Phi_{r}={\bar{\Phi}}$, the stationary
profile $\Phi_{ss}$ is constant equal to $\bar{\Phi}$ and the quasi-potential,
maximal solution of Eq. (4.12), is given by

$\begin{split}V(\Phi)&=\int_{0}^{1}du\
\left\\{S(\Phi(u))-S(\bar{\Phi})-S^{\prime}(\bar{\Phi})(\Phi(u)-\bar{\Phi})\right\\}\
.\end{split}$ (4.15)

On one hand, this formula can be directly obtained by a contraction principle
from Sanov’s Theorem since we know that in this case the stationary state is
given by the product probability measure $\mu_{\bar{\Phi}}$.

On the other hand, it can be shown that $V$ defined above solves the Hamilton-
Jacobi equation (4.12). This can be proved by observing first that

$\tfrac{\delta
V}{\delta\Phi}(u)=S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\bar{\Phi}\right)\
,$

satisfies the boundary conditions $\tfrac{\delta
V}{\delta\Phi}(0)=\tfrac{\delta V}{\delta\Phi}(1)=0$, and moreover that

$\displaystyle\left\langle\mathcal{A}\left(\Phi\right),\frac{\delta
V}{\delta\Phi}\right\rangle+\left\langle\frac{\delta
V}{\delta\Phi},\left(-{\mathcal{B}}_{\Phi}\right)\frac{\delta
V}{\delta\Phi}\right\rangle$ $\displaystyle=\int_{0}^{1}du\ \int_{0}^{1}dv\
K(v-u)\
\left(S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\bar{\Phi}\right)\right)\
A(\Phi_{t}(u),\Phi_{t}(v))$ $\displaystyle-\int_{0}^{1}du\ \int_{0}^{1}dv\
K(v-u)\left(S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\bar{\Phi}\right)\right)\
B(\Phi_{t}(u),\Phi_{t}(v))\left(S^{\prime}\left(\Phi(v)\right)-S^{\prime}\left(\bar{\Phi}\right)-\left(S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\bar{\Phi}\right)\right)\right)$
$\displaystyle=\int_{0}^{1}du\ \int_{0}^{1}dv\ K(v-u)\
\left(S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\bar{\Phi}\right)\right)\
A(\Phi_{t}(u),\Phi_{t}(v))$ $\displaystyle-\int_{0}^{1}du\ \int_{0}^{1}dv\
K(v-u)\left(S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\bar{\Phi}\right)\right)\
B(\Phi_{t}(u),\Phi_{t}(v))\left(S^{\prime}\left(\Phi(v)\right)-S^{\prime}\left(\Phi(u)\right)\right)$
$\displaystyle=0$

where the last equality comes from the Einstein relation Eq. (3.19). This is
not sufficient to show that $V$ is really the quasi-potential but it can be
shown it is the maximal solution of the Hamilton-Jacobi equation (see Remark
4.2).

##  5 Diffusive limit

###  5.1 Diffusive limit of the hydrodynamics and hydrostatics

As already mentioned in Remark 3.4, if $\gamma>2$ (resp. $\gamma=2$), we
expect that the correct time scale to observe a non-trivial evolution of the
macroscopic volume profile, is the diffusive one, i.e. $tn^{2}$ (resp.
$tn^{2}/\log n$). Then we expect the dynamics will behave in fact as if it was
short-range, and the hydrodynamic limits will be given by a diffusion equation
(see [84, 14] for rigorous proofs in the case of the exclusion process with
long jumps and [45] for convincing arguments in the context of Riesz
gas282828In [45] the parameter $s$ there corresponds to $\gamma/2$ for the
current paper.).

However we observe that as $\gamma\to 2^{-}$, the renormalised operator
$\tfrac{2-\gamma}{2}{\mathcal{A}}$ converges to a second order differential
operator (see Appendix C). The reason to choose the prefactor
$\tfrac{2-\gamma}{2}$ is explained in Appendix C.2.1. This shows that the
hydrodynamics (3.1) (resp. hydrostatics (3.12)) converge as $\gamma\to 2^{-}$
to the solution of the parabolic equation

$\partial_{t}\Phi_{t}=\partial_{u}\left(D(\Phi)\ \partial_{u}\Phi\right),\quad
D(\Phi)=S^{{}^{\prime\prime}}(\Phi)B(\Phi,\Phi)>0\ ,$ (5.1)

(resp. of the elliptic equation $0=\partial_{u}\left(D(\Phi_{ss})\
\partial_{u}\Phi_{ss}\right)$) with Dirichlet boundary conditions provided by
$\Phi_{\ell}$ on the left and $\Phi_{r}$ on the right. The coefficient $B$ is
defined by Eq. (3.17) and the thermodynamic (convex) entropy $S$ in Eq. (2.14)
(with Eq.(2.17)).

Finally, If $\gamma$ is formally infinite, then $K$ is formally nearest
neighbour, i.e. $K(z)={\sc 1}_{|z|=1}$. Then the hydrodynamics is also in a
similar diffusive form, but with a non explicit diffusion coefficient
$D_{\infty}$ defined by the Green-Kubo formula, see Eqs. (1.13)-(1.15) in
[110]292929In [110], to have the same notations as us, one has to replace
there $h$ by $S$, $\alpha$ by $\beta/2$.. More exactly, we have that

$D_{\infty}(\Phi)=S^{\prime\prime}(\Phi)\
\inf_{g}\left\langle\beta(\varphi(1),\varphi(2))\
\left\\{1-\left[\partial_{\varphi(1)}-\partial_{\varphi(2)}\right]\zeta_{g}\right\\}^{2}\right\rangle_{\mu_{\Phi}}\
.$ (5.2)

Here, $\mu_{\Phi}$ is define in Eq. (2.9), the infimum is taken over local
smooth functions $g:\mathbb{R}^{\mathbb{Z}}\to\mathbb{R}$, i.e. which depends
only on a finite number of coordinates, and $\zeta_{g}$ is defined by

$\zeta_{g}(\varphi)=\sum_{y}g(\tau_{y}\varphi)$ (5.3)

where the shift $\tau_{y}$ is defined via $(\tau_{y}\varphi)(z)=\varphi(z+y)$.
By choosing $g=0$ in the previous variational formula we deduce, by a direct
computation involving Eq. (2.9), that

$D_{\infty}(\Phi)\leq D(\Phi)\ .$ (5.4)

This inequality is consistent with the fact that the GL dynamics is less
diffusive for $\gamma>2$ than for $\gamma<2$.

######  Remark 5.1.

We conjecture that for any $\gamma>2$, the hydrodynamic limit is given by a
diffusion equation with the diffusion coefficient $D_{\infty}$ obtained in
[110] 303030Of course this requires to renormalise correctly $K$ so that the
variance of $K$ is the same as for the nearest-neighbour case. . We also
conjecture that the inequality (5.4) is usually strict313131If $\beta$ is
constant then the inequality is an equality [83].. If these conjectures are
correct, this means we would have generically a $0$-th order phase transition
for the diffusion coefficient at the critical value $\gamma=2$.

###  5.2 Diffusive limit of the large deviations functionals

Recall from Eq. (4.3) that the rate functional for the dynamical large
deviations of the empirical density is given, for a profile $\Phi$ satisfying
the Dirichlet boundary conditions $\Phi_{t}(0)=\Phi_{\ell}$,
$\Phi_{t}(1)=\Phi_{r}$, by

$\begin{split}I_{[0,T]}(\Phi)&=\frac{1}{4}\
\int_{0}^{T}dt\left\|\partial_{t}\Phi_{t}-\mathcal{A}(\Phi_{t})\right\|^{2}_{-1,\mathcal{B}_{\Phi_{t}}}\
.\end{split}$ (5.5)

In the Hamiltonian formulation, we can also write it as

$\begin{split}I_{[0,T]}(\Phi)&=\sup_{p}\ \int_{0}^{T}dt\ \left\\{\left\langle
p_{t}\ ,\
\partial_{t}\Phi_{t}\right\rangle-\mathbb{H}(\Phi_{t},p_{t})\right\\}\end{split}$

where the supremum is taken over smooth functions
$p:[0,T]\times[0,1]\to\mathbb{R}$ and the Hamiltonian $\mathbb{H}$ is defined
by Eq. (4.13). Let $\tilde{\Phi}$ be the renormalised profile

${\tilde{\Phi}}_{t}=\Phi_{\tfrac{2t}{2-\gamma}}\ .$

We can then rewrite $I_{[0,T]}(\Phi)$ as

$\begin{split}I_{[0,T]}(\Phi)={\tilde{I}}_{\left[0,{\tiny{\tfrac{2T}{2-\gamma}}}\right]}(\tilde{\Phi})&=\
\sup_{p}\left\\{\ \int_{0}^{\tfrac{2T}{2-\gamma}}du\ \left\\{\left\langle
p_{u}\ ,\
\partial_{u}\tilde{\Phi}_{u}-\tfrac{2-\gamma}{2}\mathcal{A}(\tilde{\Phi}_{u})\right\rangle-\left\langle
p_{u}\ ,\
-\tfrac{2-\gamma}{2}\mathcal{B}_{\tilde{\Phi}_{u}}\,p_{u}\right\rangle\right\\}\right\\}\
.\end{split}$

By using the results of Appendix C we conclude that the rate functional
${\tilde{I}}_{\left[0,T\right]}(\tilde{\Phi})$ converges323232In fact, we
should use $\Gamma$-convergence to be totally correct. to

$\ \sup_{p}\left\\{\ \int_{0}^{T}du\ \left\\{\left\langle p_{u}\ ,\
\partial_{u}\tilde{\Phi}_{u}-\partial_{u}\left(D(\tilde{\Phi}_{u})\
\partial_{u}\tilde{\Phi}_{u}\right)\right\rangle-\left\langle p_{u}\ ,\
\frac{D(\tilde{\Phi}_{u})}{S^{\prime\prime}(\tilde{\Phi}_{u})}\,p_{u}\right\rangle\right\\}\right\\}$
(5.6)

where $D$ is defined by Eq. (5.1). This coincides with the rate functional of
a GL diffusive dynamics having diffusion coefficient $D$ and entropy function
$S$ [30]. Since $D$ usually does not coincide with $D_{\infty}$, it is not the
rate functional associated to the GL dynamics with nearest neighbour
interactions.

######  Remark 5.2.

Using the definition (4.11) of the quasi potential, we also deduce that the
quasi-potential converges as $\gamma\to 2^{-}$ to the quasi-potential of a
short-range GL dynamics.

##  6 Particular case

###  6.1 Additive noise

In this section we consider the GL dynamics (2.5) and (2.19) where $\beta$ is
constant, say $\beta=1$ to simplify, so that by Eq. (2.4), we have

$\alpha(\varphi(x),\varphi(y))=U^{\prime}(\varphi(y))-U^{\prime}(\varphi(x))\
.$

Moreover, thanks to the form $\nu_{\Phi}$ defined by Eq. (2.8), we have
$\langle U^{\prime}(\phi)\rangle_{\nu_{\Phi}}=S^{\prime}(\Phi)$, and then Eqs.
(3.3), (3.17) become

$A\left(\Phi,\Phi^{\prime}\right)=S^{\prime}\left(\Phi^{\prime}\right)-S^{\prime}\left(\Phi\right)\
,\quad B\left(\Phi,\Phi^{\prime}\right)=1\ .$

This imply that the operators appearing in Eqs. (3.2), (3.16) take the
particular form

$\begin{split}&\mathcal{A}(\Phi)(u)=\int_{0}^{1}dv\
K(v-u)\left[S^{\prime}(\Phi(v))-S^{\prime}(\Phi(u))\right]\ ,\\\
&(\mathcal{B}_{\Phi}H)\ (u)=\int_{0}^{1}dv\ K(v-u)\left[H(v)-H(u)\right]\
.\end{split}$

It is then not difficult to see that

$\begin{split}V(\Phi)&=\int_{0}^{1}du\
\left(\int_{\Phi_{ss}(u)}^{\Phi(u)}\left[S^{\prime}(v)-S^{\prime}(\Phi_{ss}(u))\right]\
dv\right)\\\ &=\int_{0}^{1}du\
\left\\{S(\Phi(u))-S(\Phi_{ss}(u))-S^{\prime}(\Phi_{ss}(u))(\Phi(u)-\Phi_{ss}(u))\right\\}\,\end{split}$
(6.1)

solves the Hamilton-Jacobie equation (4.12). This can be proved by observing
first that

$\frac{\delta
V}{\delta\Phi}(u)=S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\Phi_{ss}(u)\right)\
,$

and then that

$\displaystyle\left\langle\mathcal{A}\left(\Phi\right),\frac{\delta
V}{\delta\Phi}\right\rangle+\left\langle\frac{\delta
V}{\delta\Phi},\left(-{\mathcal{B}}_{\Phi}\right)\frac{\delta
V}{\delta\Phi}\right\rangle$ $\displaystyle=\int_{0}^{1}du\ \int_{0}^{1}dv\
K(v-u)\left(S^{\prime}\left(\Phi(v)\right)-S^{\prime}\left(\Phi(u)\right)\right)\left(S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\Phi_{ss}(u)\right)\right)$
$\displaystyle-\int_{0}^{1}du\ \int_{0}^{1}dv\
K(v-u)\left(S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\Phi_{ss}(u)\right)\right)\left(S^{\prime}\left(\Phi(v)\right)-S^{\prime}\left(\Phi_{ss}(v)\right)-\left(S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\Phi_{ss}(u)\right)\right)\right)$
$\displaystyle=\int_{0}^{1}du\
\left(S^{\prime}\left(\Phi(u)\right)-S^{\prime}\left(\Phi_{ss}(u)\right)\right)\int_{0}^{1}dv\
K(v-u)\left(S^{\prime}\left(\Phi_{ss}(v)\right)-S^{\prime}\left(\Phi_{ss}(u)\right)\right)$
$\displaystyle=0$

where the last equality come the fact that the non-equilibrium stationary
profile $\Phi_{ss}$ satisfies

$0=\mathcal{A}(\Phi_{ss})(u)=\int_{0}^{1}dv\
K(v-u)\left[S^{\prime}(\Phi_{ss}(v))-S^{\prime}(\Phi_{ss}(u))\right],\quad\Phi_{ss}(0)=\Phi_{\ell},\quad\Phi_{ss}(1)=\Phi_{r}\
.$ (6.2)

######  Remark 6.1.

This quasi-potential $V$ is then a local functional so that the NESS should
not have long-range correlations. We expect that if $\beta$ is not constant,
then the quasi-potential is non-local and that long-range correlations are
present. We will study more precisely this point in the forthcoming paper
[15].

######  Remark 6.2.

In the particular $U(\phi)=\phi^{2}$, the NESS is product and the non-
equilibrium free energy can be computed directly [17]. This can be proved
similarly as in [20] where is considered the NESS of the boundary driven
symmetric long-range zero-range process.

###  6.2 Brownian energy model (BEM)

In this section we consider a lattice conservative model which is not a GL
dynamics but which becomes a GL dynamics after a simple change of variable.
The BEM appeared for example in [23, 11, 13, 12, 79, 80] in order to derive
the Fourier’s law and it has remarkable duality properties. Given the sub
lattice $\Lambda_{n}=\\{1,\ldots,n\\}$ of $\mathbb{Z}$, the Brownian energy
model is the Markov process
$\omega_{t}:=\left\\{\omega_{t}(x)\in\mathbb{R}\;;\;x\in\Lambda_{n}\right\\}$
with state space $\mathbb{R}^{{\Lambda}_{n}}$ whose generator is given by

$\mathbb{G}_{b}^{n}=\frac{1}{2}\sum_{x,y\in\Lambda_{n}}K(y-x)\left(\omega(y)\partial_{\omega(x)}-\omega(x)\partial_{\omega(y)}\right)^{2},$
(6.3)

where the kernel $K:\mathbb{Z}\to(0,\infty)$ is chosen to be symmetric, i.e.
$K(z)=K(-z)$.

On one hand, it is straightforward to check that for any temperature $T>0$,
the Gibbs measure
$\exp\left[-\frac{1}{2T}\sum_{x\in\Lambda_{n}}\left(\omega(x)\right)^{2}\right]\
d\omega$ is invariant (and in fact reversible) for the BEM, i.e.

$\left[\mathbb{G}_{b}^{n}\right]^{\dagger}\left(\exp\left[-\frac{1}{2T}\sum_{x\in\Lambda_{n}}\left(\omega(x)\right)^{2}\right]\right)=0\
.$ (6.4)

On the other hand, the BEM is not a $\omega$-conservation law as defined in
the sense of Eq. (1.1) because
$\mathbb{G}_{b}^{n}\left[\omega(z)\right]=-\sum_{x\in\Lambda_{n}}K(x,z)\omega(z)$
and then

$\mathbb{G}_{b}^{n}\left[\sum_{z\in\Lambda}\omega(z)\right]=-\sum_{x\in\Lambda,z\in\Lambda}K(x,z)\omega(z)$

is not identically zero, i.e. for any configuration $\omega$. Hence
$\frac{d}{dt}\mathbb{E}\left[\sum_{z\in\Lambda}\omega_{t}(z)\right]$ is not
identically zero for any initial configuration.

####  6.2.1 BEM-GL transformation

The Brownian Energy model can be transformed into a GL model. Indeed, the
energy field
$\left\\{\varphi_{t}(x):=[\omega_{t}(x)]^{2}\in\mathbb{R}^{+}\;;\;x\in\Lambda\right\\}$
is also a Markovian process (thanks to Dynkin’s criterium) but now with
generator (see Appendix E)

${\mathcal{G}}_{b}^{n}=\sum_{x,y\in\Lambda_{n}}\left\\{K(y-x)\left(\varphi(y)-\varphi(x)\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)+2K(y-x)\varphi(y)\varphi(x)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)^{2}\right\\}\
.$ (6.5)

This dynamics is then a peculiar case of general GL dynamics Eq. (1.3) with
$\mathcal{M}=\mathbb{R}^{+}$ and

$\begin{cases}F_{\varphi}(x,y)=2K(y-x)\left(\varphi(y)-\varphi(x)\right)\ ,\\\
\varGamma_{\varphi}\left(x,y\right)=4K(y-x)\varphi(y)\varphi(x)\ .\end{cases}$
(6.6)

More precisely, it is a particular case of Eq. (1.4) by choosing the GL
Hamiltonian $\mathcal{E}\left(\varphi\right)$ and the mobility field
$\Omega_{\varphi}$ as

$\begin{cases}\mathcal{E}\left(\varphi\right)=\frac{1}{2}\sum_{x\in\Lambda_{n}}\ln\left(\varphi(x)\right)\
,\\\
\varOmega_{\varphi}\left(x,y\right)=\varGamma_{\varphi}\left(x,y\right)=4K(y-x)\varphi(y)\varphi(x)\
.\end{cases}$ (6.7)

This follows from

$\displaystyle-\varOmega_{\varphi}\left(x,y\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\mathcal{E}\left(\varphi\right)+\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\left[\varOmega_{\varphi}\left(x,y\right)\right]$
$\displaystyle=-4K(y-x)\varphi(y)\varphi(x)\
\frac{1}{2}\left(\frac{1}{\varphi(x)}-\frac{1}{\varphi(y)}\right)+4K(y-x)\left(\varphi(y)-\varphi(x)\right)$
$\displaystyle=2K(y-x)\left(\varphi(y)-\varphi(x)\right)\ .$

Therefore it corresponds then to take in Eq. (2.4) and Eq. (2.2)

$\beta(\phi,\phi^{\prime})=4\phi\phi^{\prime},\quad
U(\phi)=\tfrac{1}{2}\log\phi\ .$

######  Remark 6.3.

It is interesting to observe that this very particular GL dynamics also
appears as an effective limit of an harmonic chain perturbed by an energy
conserving noise in the weak coupling limit [100].

######  Remark 6.4.

It is also possible to add to the dynamics defined by Eq. (6.3) a transversal
force (similarly to Eq. (1.7) for GL dynamics):

$\mathbb{G}_{b}^{n}=\frac{1}{2}\sum_{x\in\Lambda_{n}}\sum_{y\in\Lambda_{n}}\left\\{K(y-x)\left(\omega(y)\partial_{\omega(x)}-\omega(x)\partial_{\omega(y)}\right)^{2}+a(y-x)\omega(x)\omega(y)\left(\omega(y)\partial_{\omega(x)}-\omega(x)\partial_{\omega(y)}\right)\right\\}\
.$

Here $a:\mathbb{Z}\to\mathbb{R}$ is an arbitrary function333333This model has
been studied in the nearest neighbour case [12].. It is then a direct
calculation to see that the Gibbs probability measure proportional to
$\exp\left[-\frac{1}{2T}\sum_{x\in\Lambda_{n}}\left(\omega(x)\right)^{2}\right]$
is still invariant (like in Eq. (6.4)) but no longer reversible. The energy
field
$\left\\{\varphi_{t}(x):=[\omega_{t}(x)]^{2}\in\mathbb{R}^{+}\;;\;x\in\Lambda\right\\}$
is still a peculiar choice of the general GL dynamics defined through Eq.
(1.4) where in the first line of Eq. (6.6) we have to add the transverse drift

$F_{\varphi}^{\bot}(x,y)=\left(a(y-x)-a(x-y)\right)\varphi(x)\varphi(y)$

which satisfies Eq. (1.7). The thermodynamics properties defined in the next
section are then still the same.

####  6.2.2 Thermodynamical properties

We observe that the free energy is given, for any $\lambda>0$, by

$F(\lambda)=-\log Z(\lambda)=-\tfrac{1}{2}\log\pi+\tfrac{1}{2}\log\lambda\ .$

This follows from the simple computation

$\begin{split}\int_{0}^{\infty}dx\ e^{-\tfrac{1}{2}\log x-\lambda
x}&=\int_{0}^{\infty}dx\ x^{-1/2}e^{-\lambda x}=\sqrt{2}\int_{0}^{\infty}dy\
e^{-\tfrac{\lambda y^{2}}{2}}=\frac{1}{\sqrt{2}}\int_{-\infty}^{\infty}dy\
e^{-\tfrac{\lambda y^{2}}{2}}=\sqrt{\cfrac{\pi}{\lambda}}\ .\end{split}$

Consequently, recalling Eq. (2.14), we get that the entropy satisfies for any
$\Phi>0$

$\begin{split}S(\Phi)=\sup_{\lambda}\left\\{F(\lambda)-\lambda\Phi\right\\}=-\frac{1+\log
2+\log\pi}{2}\ -\ \frac{1}{2}\log\Phi\ ,\quad\lambda(\Phi)=\frac{1}{2\Phi}\
.\end{split}$

We have also that

$\sigma(\Phi)=\left\langle\phi^{2}\right\rangle_{\nu_{\Phi}}-(\left\langle\phi\right\rangle_{\nu_{\Phi}})^{2}=\cfrac{1}{S^{\prime\prime}(\Phi)}=2\Phi^{2}\
.$ (6.8)

####  6.2.3 Nearest neighbour case

In the nearest neighbour case $K(z)={\sc 1}_{|z|=1}$ one easily check that,
for $x\in\Lambda_{n}=\\{1,\ldots,n\\}$,

${\mathcal{G}}_{b}^{n}\left[\varphi(x)\right]=2\left[\varphi(x-1)+\varphi(x+1)-2\varphi(x)\right]\
.$

Hence, in the diffusive time scale $tn^{2}$, the hydrodynamic equations are
given by

$\begin{cases}&\partial_{t}\Phi_{t}=2\partial_{u}^{2}\Phi_{t}\ ,\\\
&\Phi_{t}(0)=\Phi_{\ell},\quad\Phi_{t}(1)=\Phi_{r}\ ,\\\
&\Phi_{t}|_{t=0}=\Phi_{0}\ .\end{cases}$ (6.9)

Hence, by Eqs. (6.8) and (6.9), with the notations of [26, Section 4], we have
that thermodynamical quantities are given by

$D(\Phi)=2,\quad\sigma(\Phi)=2\Phi^{2},\quad\chi(\Phi)=2\sigma(\Phi)=4\Phi^{2}\
.$

The coefficients above are, up to irrelevant constants, the same as for the
Kipnis-Marchioro-Presutti (KMP) model. In [29], the authors were able to
compute explicitly the quasi-potential. The same computations can therefore be
reproduced for the GL dynamics derived from the BEM.

######  Remark 6.5.

In the long-range case, we have not been able to reproduce the computations of
[29]. The reason is that the method of [29] relies crucially on differential
calculus.

##  7 Comparison with the standard diffusive case

###  7.1 GL dynamics with nearest neighbour interactions

In order to illustrate the differences and similarities with the diffusive
case, consider a GL dynamics with nearest neighbour interactions, i.e. take
$\Lambda_{n}=\\{1,\ldots,n\\}$ and Eqs. (2.4), (2.19) with (1.1).

The hydrodynamics for the empirical volume profile (3.6) is given by the
continuity equation343434Strictly speaking these references prove this
hydrodynamic equations only with free boundary conditions. [127, 110] (compare
with Eq. (3.1))

$\begin{cases}\partial_{t}\Phi_{t}(u)+\partial_{u}\left[J(\Phi_{t}(u))\right]=0\
,\\\ \Phi_{t}(0)=\Phi_{\ell}\ ,\quad\Phi_{t}(1)=\Phi_{r}\ ,\\\
\Phi_{t}|_{t=0}\Phi_{0}\end{cases}$ (7.1)

where the macroscopic instantaneous current associated to the density profile
$\Phi$ is given by (compare with Eq. (3.5))

$J(\Phi)=-D_{\infty}(\Phi)\,\partial_{u}\Phi\ .$ (7.2)

Here $D_{\infty}(\Phi)$ is the diffusion coefficient defined by Eq. (5.2). The
values of $\Phi$ at the boundaries are fixed by some thermostats:
$\Phi_{\ell}$ on the left and $\Phi_{r}$ on the right

The dynamical large deviations function takes then the form [59, 110, 94, 30,
3] (to compare with Eq. (4.10))

$\begin{split}I_{[0,T]}(\Phi)&=\frac{1}{4}\
\int_{0}^{T}dt\left\|\partial_{t}\Phi_{t}+\partial_{u}\left[J(\Phi_{t}(u))\right]\right\|^{2}_{-1,{\mathcal{B}}_{\Phi_{t}}}\
.\end{split}$ (7.3)

Here we introduced the weighted (diffusive) Sobolev norm (compare with Eq.
(4.4))

$\|\Psi\|^{2}_{-1,\mathcal{B}_{\Phi}}=\left\langle\Psi\ ,\
(-{\mathcal{B}}_{\Phi})^{-1}\Psi\right\rangle\ $

but now with the mobility operator (compare with Eq. (3.16))

$\mathcal{B}_{\Phi}=\partial_{u}\ \chi(\Phi)\ \partial_{u}\ ,$ (7.4)

and $\chi$ is the mobility function of the model. The latter is related to the
diffusion coefficient by the associated hydrodynamic level Einstein relation
[110]

$\chi(\Phi)=\frac{D_{\infty}(\Phi)}{S^{\prime\prime}(\Phi)}\ =\\\
inf_{g}\left\langle\beta(\varphi(1),\varphi(2))\
\left\\{1-\left[\partial_{\varphi(1)}-\partial_{\varphi(2)}\right]\zeta_{g}\right\\}^{2}\right\rangle_{\mu_{\Phi}}\
,$ (7.5)

where we use Eq.(5.2) in the second equality.

The previous Einstein relation can be equivalently stated with the macroscopic
instantaneous current Eq. (7.2) as

$-\partial_{u}\left[J(\Phi)\right]={\mathcal{B}}_{\Phi}(S^{\prime}(\Phi))$

which has the advantage to be also the same as for GL dynamics with long-range
interactions, see Eq. (3.20) with Eq. (3.5), modulo the definition of the
mobility operator $\mathcal{B}_{\Phi}$.

This discussion is resumed in the tabular below.

######  Remark 7.1.

To compute the non-equilibrium free energy $V(\Phi)$, in the nearest-neighbour
case, we have to solve a variational formula, like in Eq. (4.11), but
involving now the Lagrangian appearing in Eq. (7.3). The quasi-potential is
then the maximal353535See Remark (4.2) for more explanations. solution of the
Hamilton-Jacobi equation (4.14), but with the mobility operator
$\mathcal{B}_{\Phi}$ defined by Eq. (7.4). This is usually an inextricable
problem which has been solved only in very specific nearest-neighbour cases
[26, Section 4]: Exclusion process, Zero-range-process, (gradient363636In our
context it corresponds to $\beta=1$.-)Ginzburg-Landau process, Kipnis-
Marchioro-Presutti (KMP) process. The fact that this variational problem has
been solved for the Zero-Range-process and the (gradient)-Ginzburg-Landau
process is due to the fact that in this case the NESS is explicitly product.
In the Exclusion process case, the ‘tour de force’ was obtained in [55] by
Derrida and coauthors thanks to the explicit representation of the NESS as a
product of matrices [54]. The solution for the KMP process was provided by
analogy with the exclusion process [31] (even if the representation of the
NESS is not in a known matrix product form like for the Exclusion process).

Table 1: This table resumes the discussion of Section 7

.

$\begin{array}[]{|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\text{\bf{Bulk generator dynamics in
$\Lambda_{n}=\\{1,\ldots,n\\}$:}}\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\mathcal{L}_{b}^{n}=\frac{1}{2}\sum_{x,y\in\Lambda_{n}}K(y-x)\
(\partial_{\varphi(y)}-\partial_{\varphi(x)})\left[e^{-\mathcal{E}(\varphi)}\
\beta(\varphi(x),\varphi(y))\ (\partial_{\varphi(y)}-\partial_{\varphi(x)})\
e^{\mathcal{E}(\varphi)}\right]\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr&&\\\
&\text{\bf{Diffusive case}}&\text{\bf{Super diffusive case}}\\\ &&\\\
\hline\cr&&\\\ &\text{GL with nearest neighbour interactions}&\text{GL with
long-range interactions}\\\ &&\\\ &K(z)={\sc 1}_{|z|=1}&K(z)={\sc 1}_{z\neq
0}|z|^{-(1+\gamma)},\quad\gamma\in(1,2)\\\ &&\\\
\hline\cr\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\text{\bf{Hydrodynamic
equations:}}\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\
\partial_{t}\Phi_{t}(u)+\partial_{u}\left[J(\Phi_{t}(u))\right]=0,\quad\Phi_{t}(0)=\Phi_{\ell}\
,\quad\Phi_{t}(1)=\Phi_{r},\quad\Phi_{t}|_{t=0}\Phi_{0}\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr&&\\\
\text{Currents}&J(\Phi(u))=-D_{\infty}(\Phi(u))\,\partial_{u}\Phi(u)&J(\Phi(u))=-\int_{0}^{u}dv\
\int_{0}^{1}dw\ K(w-v)\ A\left(\Phi(v),\Phi(w)\right)\\\ &&\\\ &D_{\infty}\
\text{defined in Eq.}\ \eqref{eq:D_infty}&A\ \text{defined in Eq.}\
\eqref{eq:A}\\\ &&\\\ \hline\cr\text{Mobility
operators}&\begin{split}&[\mathcal{B}_{\Phi}H](u)=\left[\partial_{u}\,\chi(\Phi)\,\partial_{u}\,H\right](u)\\\
&{\text{with \
}}\chi(\Phi)=\cfrac{D_{\infty}(\Phi)}{S^{\prime\prime}(\Phi)}\end{split}&\begin{split}&\\\
&[\mathcal{B}_{\Phi}H](u)\\\ &=\int_{0}^{1}dv\ K(v-u)\
B(\Phi(u),\Phi(v))[H(v)-H(u)]\\\ &{\text{with }}\
B(\Phi,\Phi^{\prime})=\tfrac{A(\Phi,\Phi^{\prime})}{S^{\prime}(\Phi^{\prime})-S^{\prime}(\Phi)}\end{split}\\\
&&\\\ \hline\cr\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\text{\bf{Einstein
relation:}}\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil-\partial_{u}J(\Phi)={\mathcal{B}}_{\Phi}(S^{\prime}(\Phi))\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\text{{\bf{Dynamical Large Deviations
functional:}}}\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil I_{[0,T]}(\Phi)=\frac{1}{4}\
\int_{0}^{T}dt\left\|\partial_{t}\Phi_{t}+\partial_{u}J(\Phi_{t})\right\|^{2}_{-1,{\mathcal{B}}_{\Phi_{t}}},\quad\quad\|\Psi\|^{2}_{-1,\chi(\Phi)}=\left\langle\Psi\
,\
(-\mathcal{B}_{\Phi})^{-1}\Psi\right\rangle\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\text{\bf{Hamilton-Jacobi equation for the quasi-
potential $V$ \eqref{eq:HJ-equation-
einstein}:}}\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\left\langle\tfrac{\delta
V}{\delta\Phi},\mathcal{B}_{\Phi}\left(-S^{\prime}(\Phi)+\tfrac{\delta
V}{\delta\Phi}\right)\right\rangle=0,\quad\tfrac{\delta
V}{\delta\Phi}(0)=\tfrac{\delta
V}{\delta\Phi}(1)=0\hfil\lx@intercol\vrule\lx@intercol\\\
\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

###  7.2 GL dynamics with a mix between long and short range interactions

To conclude this comparison with the diffusive case, let us observe that we
could also consider a GL dynamics where the kernel $K$ incorporates at the
same time super-diffusive and diffusive effects. This was investigated in [38]
for interacting Brownian motions and in [76, 77]) for lattice gas models.

Since the diffusive effects appear only at the diffusive time scale $tn^{2}$,
much larger that the sub-diffusive time scale $tn^{\gamma}$, we have to
enhance the diffusive effects to see them appearing in the sub-diffusive time
scale. We can therefore consider for example a kernel $K$ in the form

$K(z)=\ an^{2-\gamma}{\sc 1}_{|z|=1}+\frac{{\sc 1}_{z\neq 0}}{|z|^{1+\gamma}}$
(7.6)

where $a$ is a positive constant. In this case we expect that the macroscopic
current will take the form

$J(\Phi(u))=-a\,D_{\infty}(\Phi(u))\,\partial_{u}\Phi(u)-\int_{0}^{u}dw\
\mathcal{A}(\Phi)(w)\ .$ (7.7)

with the mobility operator $\mathcal{B}_{\Phi}$ acting on functions
$H:[0,1]\to\mathbb{R}$ as

$\big{[}{\mathcal{B}}_{\Phi}H\big{]}(u)=a\
\big{[}\partial_{u}\big{(}\chi(\Phi(u))\
\partial_{u}H\big{)}\big{]}(u)+\int_{0}^{1}dv\ K(v-u)\
B(\Phi(u),\Phi(v))\big{[}H(v)-H(u)\big{]}\ $

where $B$ is defined by Eq. (3.17) and $\chi$ by Eq. (7.5). The dynamical
large deviations function will take the form

$I_{[0,T]}(\Phi)=\frac{1}{4}\
\int_{0}^{T}dt\left\|\partial_{t}\Phi_{t}+\partial_{u}J(\Phi_{t})\right\|^{2}_{-1,{\mathcal{B}}_{\Phi_{t}}},\quad\|\Psi\|^{2}_{-1,\chi(\Phi)}=\left\langle\Psi\
,\ (-\mathcal{B}_{\Phi})^{-1}\Psi\right\rangle$ (7.8)

and the quasi-potential will be again the maximal solution of the Hamilton-
Jacobi equation (4.14).

######  Remark 7.2.

Thanks to the additive structure of (7.6), both for the hydrodynamic equation
(7.7) and for the Lagrangian expression (7.8) of the large deviation function,
the structure is just additive between short-range and long-range terms. This
is not longer true at the level of the Hamiltonian formulation (4.13) of the
large deviations function , or then, for the non-equilibrium free energy
(4.11).

Acknowledgements The authors are grateful to J. Barré, A. Dhar, M. Jara, P.
Gonçalves, A. Kundu and K. Mallick for useful discussions. This work was
supported by the projects RETENU ANR-20-CE40-0005-01, LSD ANR-15-CE40-0020-01
of the French National Research Agency (ANR).

##  Appendix A Proof of the form (2.6) of the Bulk Markovian generator.

Due to the antisymmetry $\dot{\zeta}_{t}(x,y)=-\dot{\zeta}_{t}(y,x)$, we have

$\left\langle d\zeta_{t}(x,y)\
d\zeta_{t}(z,w)\right\rangle=dt\left(\delta_{x,z}\delta_{y,w}-\delta_{x,w}\delta_{y,z}\right)$

and then the Markov generator corresponding to the stochastic differential
equation in Eq. (2.5) is given by

$\begin{split}&\mathcal{G}_{b}^{(n)}\\\
&=\sum_{\left(x,y\right)\in\Lambda_{n}}K(y-x)\left(\alpha\left(\varphi(x),\varphi(y)\right)\partial_{\varphi(x)}+\beta\left(\varphi(x),\varphi(y)\right)\partial^{2}_{\varphi(x)}-\beta\left(\varphi(x),\varphi(y)\right)\partial^{2}_{\varphi(x),\varphi(y)}\right)\\\
&=\sum_{\left(x,y\right)\in\Lambda_{n}}K(y-x)\left(\alpha\left(\varphi(x),\varphi(y)\right)\partial_{\varphi(x)}+\beta\left(\varphi(x),\varphi(y)\right)\partial_{\varphi(x)}\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\right)\\\
&=\sum_{\left(x,y\right)\in\Lambda_{n}}K(y-x)\left(\alpha\left(\varphi(x),\varphi(y)\right)\partial_{\varphi(x)}+\frac{1}{2}\beta\left(\varphi(x),\varphi(y)\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)^{2}\right)\end{split}$
(A.1)

where in the last equality we used the symmetry of $K$ and $\beta$, i.e.
$K(-z)=K(z)$, $\beta(\varphi(x),\varphi(y))=\beta(\varphi(y),\varphi(x))$.
Finally, by substituting the relation (2.4) in the previous display, we get

$\begin{split}\mathcal{G}_{b}^{(n)}&=\sum_{\left(x,y\right)\in\Lambda_{n}}K(y-x)\left\\{\left(e^{\mathcal{E}(\varphi)}\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\left[e^{-\mathcal{E}(\varphi)}\beta\left(\varphi(x),\varphi(y)\right)\right]\right)\partial_{\varphi(x)}\right.\\\
&\left.\qquad\quad+\frac{1}{2}\beta\left(\varphi(x),\varphi(y)\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)^{2}\right\\}\\\
&=\sum_{\left(x,y\right)\in\Lambda_{n}}K(y-x)\left\\{\frac{1}{2}\left(e^{\mathcal{E}(\varphi)}\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\left[e^{-\mathcal{E}(\varphi)}\beta\left(\varphi(x),\varphi(y)\right)\right]\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)\right.\\\
&\left.\qquad\quad+\frac{1}{2}\beta\left(\varphi(x),\varphi(y)\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)^{2}\right\\}\end{split}$
(A.2)

where in the last equality, we used again the symmetry of $K$ and $\beta$.
Finally, we obtain the relation (2.6).

##  Appendix B Conserved quantity of the GL dynamics and ergodicity

Consider the GL dynamics with free boundary conditions on
$\Lambda_{n}=\\{1,\ldots,n\\}$. It’s generator is denoted by (2.6)
$\mathcal{G}_{b}^{n}$.

Let us first show that the volume is the unique conserved quantity. A simple
computation based on integration by parts shows that if
$f:{\mathbb{R}}^{\Lambda_{n}}\to f(\varphi)$ is an arbitrary function then

$\begin{split}\int_{{\mathbb{R}}^{\Lambda_{n}}}&\left(\mathcal{G}_{b}^{n}f\right)(\varphi)\
f(\varphi)\ d\varphi\\\
&=-\frac{1}{2}\sum_{x,y\in\Lambda_{n}}K(y-x)\int_{{\mathbb{R}}^{\Lambda_{n}}}e^{-\mathcal{E}(\varphi)}\beta(\varphi(x),\varphi(y))\left[(\partial_{\varphi(y)}-\partial_{\varphi(x)})f\right]^{2}(\varphi)d\varphi\
.\end{split}$ (B.1)

If $f(\varphi)$ is a conserved quantity of the Ginzburg-Landau dynamics with
free boundary conditions then $\mathcal{G}_{b}^{n}(f)=0$ so that for any
$x,y\in\Lambda_{n}$, $\partial_{\varphi(y)}f=\partial_{\varphi(x)}f$, i.e
there exists a constant $K$ such that for any $x\in\Lambda_{n}$,
$\partial_{\varphi(x)}f=K$ . Let $C$ be an arbitrary constant, define the flat
sub-manifold of ${\mathbb{R}}^{\Lambda_{n}}$ defined by
$E_{C}=\\{\varphi\in{\mathbb{R}}^{\Lambda_{n}}\;;\;{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathcal{V}}(\varphi)=}\sum_{x\in\Lambda_{n}}\varphi(x)=C\\}$.
Consider the restriction of $f$ to $E_{C}$ and observe that on $E_{C}$ we
have373737Here the $d$ on the left-hand side denotes the total differential of
$f$ and $d\varphi(x)$ the elements of the tangent space at $\varphi$ obtained
as the differential of the coordinate function $\varphi\in
E_{C}\to\varphi(x)$.

$df(\varphi)=\sum_{x\in\Lambda_{n}}\partial_{\varphi(x)}f\ (\varphi)\
d\varphi(x)=K\sum_{x\in\Lambda_{n}}d\varphi(x)=0$ (B.2)

where the last equality results from the definition of $E_{C}$. Hence there
exists a function $g:{\mathbb{R}}^{\Lambda_{n}}\to\mathbb{R}$ such that for
any configuration $\varphi$, $f(\varphi)=g({\mathcal{V}}(\varphi))$. This
proves that the volume is the only conserved quantity of the dynamics with
free boundary conditions.

Let now $C\in\mathbb{R}$ and consider the GL dynamics restricted to the
hyperplane $E_{C}$ defined above. If $f:E_{C}\to\mathbb{R}$ satisfied
$\mathcal{G}_{b}^{n}f=0$ then the previous argument shows that $f$ is a
function of the volume $\mathcal{V}(\varphi)$ which is equal to $C$ on
$E_{C}$. Hence $f$ is constant and the dynamics restricted to $E_{C}$ is
ergodic.

##  Appendix C Few properties of the hydrodynamic and hydrostatic equations

###  C.1 Weak solutions for the hydrodynamic and hydrostatic equations

Here we explain how to interpret in a rigorous sense the formal hydrodynamic
equation (3.1) and the hydrostatic equation (3.12). Such interpretation can be
adapted also for the hydrodynamic equations (3.15) of the perturbed dynamics.
This adaptation is left to the reader. We recall that $B$ is defined in Eq.
(3.17) and the relation between $A$ and $B$ is provided in Eq. (3.19).
Moreover we assume the condition $B(\Phi,\Phi^{\prime})\geq c>0$ for any
$\Phi,\Phi^{\prime}$.

We say that a function $\Phi$ is a solution of the hydrodynamic equation (3.1)
on $[0,T]$ if it bounded and satisfies the three following conditions for any
$t\in[0,T]$:

  1. 1.

We have that

$\int_{0}^{t}ds\ \int_{[0,1]^{2}}dudv\ B(\Phi_{s}(u),\Phi_{s}(v))\ K(v-u)\
\left[S^{\prime}(\Phi_{s}(v))-S^{\prime}(\Phi_{s}(u))\right]^{2}\ <\ \infty\
.$

  2. 2.

For any continuous function $G(u)$ with compact support included into $(0,1)$
we have that383838Observe that in Eq. (3.10), thanks to the antisymmetry of
$\alpha$, we could have equivalently written the limit as
$\frac{1}{2}\int_{0}^{t}ds\ \int_{[0,1]^{2}}dudv\
K(v-u)A(\Phi_{s}(u),\Phi_{s}(v))(G(u)-G(v))$.

$\begin{split}&\int_{0}^{1}du\ G(u)\Phi_{t}(u)-\int_{0}^{1}du\
G(u)\Phi_{0}(u)\\\ &=-\frac{1}{2}\int_{0}^{t}ds\ \left\\{\int_{[0,1]^{2}}dudv\
K(v-u)B(\Phi_{s}(u),\Phi_{s}(v))\left[S^{\prime}(\Phi_{s}(v))-S^{\prime}(\Phi_{s}(u))\right]\left[G(v)-G(u)\right]\right\\}\
.\end{split}$

  3. 3.

$\Phi_{t}(0)=\Phi_{\ell},\quad\Phi_{t}(1)=\Phi_{r}$ .

Using Cauchy-Schwarz inequality and the item $1$, we see that the term in item
$2$ makes sense. We also observe that item $1$ and the assumption on $B$
implies that for any time $t$, $S^{\prime}(\Phi_{t})$ belongs to the factional
Sobolev space $W^{\gamma/2,2}$ so that it is $(\gamma-1)/2$-Hölder and hence
continuous. It follows that $\Phi_{t}$ is continuous and item $3$ makes sense.
We expect that $\Phi_{t}$ is smooth on $(0,1)$ but not differentiable at the
boundaries.

We say that $\Phi_{ss}$ is a solution the hydrostatic equation (3.12) if it
bounded and satisfies the three following conditions:

  1. 1.

We have that ($B$ is defined in Eq. (3.17) and the relation between $A$ and
$B$ is provided in Eq. (3.19))

$\int_{[0,1]^{2}}dudv\ B(\Phi_{ss}(u),\Phi_{ss}(v))\ K(v-u)\
\left[S^{\prime}(\Phi_{ss}(v))-S^{\prime}(\Phi_{ss}(u))\right]^{2}\ <\ \infty\
.$

  2. 2.

For any continuous function $G(u)$ with compact support included into $(0,1)$
we have that

$\begin{split}\int_{[0,1]^{2}}dudv\ B(\Phi_{ss}(u),\Phi_{ss}(v))\ K(v-u)\
\left[S^{\prime}(\Phi_{ss}(v))-S^{\prime}(\Phi_{ss}(u))\right]\
\left[G(v)-G(u)\right]=0\ .\end{split}$

  3. 3.

$\Phi_{ss}(0)=\Phi_{\ell},\quad\Phi_{ss}(1)=\Phi_{r}$ .

###  C.2 Convergence of the hydrodynamic and hydrostatic equations as
$\gamma\to 2^{-}$

####  C.2.1 Choice of the normalisation

Let $G,H:[0,1]\to\mathbb{R}$ be smooth functions. We have that

$\begin{split}&\lim_{\gamma\to 2^{-}}\cfrac{2-\gamma}{2}\
\int_{[0,1]^{2}}dudv\ K(v-u)\left[G(v)-G(u)\right]\ \left[H(v)-H(u)\right]\\\
&=\int_{[0,1]}du\ G^{\prime}(u)\ H^{\prime}(u)\ .\end{split}$

A detailed proof is given in the next subsection (take $B=1$ and
$S^{\prime}(\Phi)=\Phi$ there). Roughly, we expand

$\begin{split}G(v)-G(u)&=G^{\prime}(u)\ (v-u)+o((v-u)^{2})\ ,\\\
H(v)-H(u)&=H^{\prime}(u)\ (v-u)+o((v-u)^{2})\ .\end{split}$

Hence, since $\gamma<2$, we have, at first order, to control, for $u\in(0,1)$,

$\int_{0}^{1}du\ G^{\prime}(u)\ H^{\prime}(u)\ \int_{0}^{1}dv\ |v-u\
|^{1-\gamma}\ .$

Observe that

$\begin{split}\int_{0}^{1}dv\ |v-u\ |^{1-\gamma}&=\int_{0}^{u}\ dz\
z^{1-\gamma}+\int_{0}^{1-u}dz\ z^{1-\gamma}=\frac{1}{2-\gamma}\
\left\\{u^{2-\gamma}+(1-u)^{2-\gamma}\right\\}\end{split}$

so that

$\lim_{\gamma\to 2^{-}}\ \frac{2-\gamma}{2}\int_{0}^{1}du\ G^{\prime}(u)\
H^{\prime}(u)\ \int_{0}^{1}dv\ |v-u\ |^{1-\gamma}=\int_{0}^{1}du\
G^{\prime}(u)\ H^{\prime}(u)\ .$

####  C.2.2 Convergence of some integral-differential operator to a second
order differential operator

Here we show that if $G:[0,1]\to\mathbb{R}$ is a smooth function with compact
support included in $(0,1)$ and $\Phi:[0,1]\to\mathbb{R}$ is a smooth function
on $(0,1)$, then

$\begin{split}&\lim_{\gamma\to 2^{-}}\cfrac{2-\gamma}{2}\
\int_{[0,1]^{2}}dudv\
K(v-u)B(\Phi(u),\Phi(v))\left[S^{\prime}(\Phi(v))-S^{\prime}(\Phi(u))\right]\left[G(v)-G(u)\right]\\\
&=\int_{[0,1]}du\ B(\Phi(u),\Phi(u))\ S^{{}^{\prime\prime}}(\Phi(u))\
\Phi^{\prime}(u)\ G^{\prime}(u)\ ,\end{split}$

which proves that, after a suitable renormalisation, the hydrodynamic (resp.
hydrostatic) equation (3.1) (resp. (3.12)) converges as $\gamma\to 2^{-}$ to
the solution of the parabolic equation

$\partial_{t}\Phi_{t}=\partial_{u}\left(D(\Phi)\ \partial_{u}\Phi\right),\quad
D(\Phi)=\ S^{{}^{\prime\prime}}(\Phi)B(\Phi,\Phi)>0\ ,$

(resp. of the elliptic equation $0=\partial_{u}\left(D(\Phi_{ss})\
\partial_{u}\Phi_{ss}\right)$) with Dirichlet boundary conditions provided by
$\Phi_{\ell}$ on the left and $\Phi_{r}$ on the right. To see the previous
convergence we write

$\begin{split}&\int_{[0,1]^{2}}dudv\
K(v-u)B(\Phi(u),\Phi(v))\left[S^{\prime}(\Phi(v))-S^{\prime}(\Phi(u))\right]\left[G(v)-G(u)\right]\\\
&=\int_{|v-u|\leq\varepsilon_{\gamma}}dudv\
K(v-u)B(\Phi(u),\Phi(v))\left[S^{\prime}(\Phi(v))-S^{\prime}(\Phi(u))\right]\left[G(v)-G(u)\right]\\\
&+\int_{|v-u|\geq\varepsilon_{\gamma}}dudv\
K(v-u)B(\Phi(u),\Phi(v))\left[S^{\prime}(\Phi(v))-S^{\prime}(\Phi(u))\right]\left[G(v)-G(u)\right]\
,\end{split}$

where $\varepsilon_{\gamma}\to 0$ as $\gamma\to 2^{-}$ such that

$\lim_{\gamma\to 2^{-}}\varepsilon_{\gamma}^{4-\gamma}=\lim_{\gamma\to
2^{-}}\varepsilon_{\gamma}^{3-\gamma}=0\quad\text{but}\quad\lim_{\gamma\to
2^{-}}\varepsilon_{\gamma}^{2-\gamma}=1\ .$

For example we can take
$\varepsilon_{\gamma}=\exp\left(\tfrac{\log(1-\sqrt{2-\gamma})}{2-\gamma}\right)$.

In the first integral we can replace the integrand
$[S^{\prime}(\Phi(v))-S^{\prime}(\Phi(u))]\ [G(v)-G(u)]$ by
$\tfrac{d}{dv}[S^{\prime}(\Phi(v))]|_{v=u}\ G^{\prime}(u)\ (v-u)^{2}$ and the
error term resulting in the integral is (by Taylor expansion) of order
$\int_{|v-u|\leq\varepsilon_{\gamma}}dudv\
|v-u|^{3-\gamma}\lesssim\varepsilon_{\gamma}^{4-\gamma}$. The second integral
can be bounded by a constant times $\int_{|v-u|\geq\varepsilon_{\gamma}}dudv\
|v-u|^{1-\gamma}=\int_{1\geq|z|\geq\varepsilon_{\gamma}}dz\
|z|^{1-\gamma}\left(\int_{0}^{1}du\ {\sc 1}_{0<u+z<1}\right)\leq
2\tfrac{1-\varepsilon_{\gamma}^{2-\gamma}}{2-\gamma}$. We have to multiply
these error terms by $2-\gamma$ and then they disappear. Hence as $\gamma\to
2^{-}$, we are left with the convergence of

$(2-\gamma)\ \int_{|v-u|\leq\varepsilon_{\gamma}}dudv\ B(\Phi(u),\Phi(v))\
\frac{d}{du}[S^{\prime}(\Phi(u))]\ G^{\prime}(u)\ K(v-u)\ (v-u)^{2}$

which can be replaced by

$(2-\gamma)\ \int_{|v-u|\leq\varepsilon_{\gamma}}dudv\ B(\Phi(u),\Phi(u))\
\frac{d}{du}[S^{\prime}(\Phi(u))]\ G^{\prime}(u)\ |v-u|^{1-\gamma}$

with a cost at most of order $(2-\gamma)\varepsilon_{\gamma}^{3-\gamma}$. The
last integral is equal to

$\begin{split}&\int_{0}^{1}du\ B(\Phi(u),\Phi(u))\
\frac{d}{du}[S^{\prime}(\Phi(u))]\ G^{\prime}(u)\
\left(\int_{|z|\leq\varepsilon_{\gamma}}dz\ |z|^{1-\gamma}{\sc
1}_{0<u+z<1}\right)\\\ &=\int_{0}^{1}du\ B(\Phi(u),\Phi(u))\
\frac{d}{du}[S^{\prime}(\Phi(u))]\ G^{\prime}(u)\
\left(\int_{\sup(-u,-\varepsilon_{\gamma})}^{\inf(1-u),\varepsilon_{\gamma})}dz\
|z|^{1-\gamma}\right)\ .\end{split}$

For every $u\in(0,1)$ we have that

$\lim_{\gamma\to 2^{-}}(2-\gamma)\
\int_{\sup(-u,-\varepsilon_{\gamma})}^{\inf((1-u),\varepsilon_{\gamma})}dz\
|z|^{1-\gamma}=\lim_{\gamma\to 2^{-}}(2-\gamma)\
\int_{-\varepsilon_{\gamma}}^{\varepsilon_{\gamma}}dz\
|z|^{1-\gamma}=2\lim_{\gamma\to 2^{-}}\varepsilon_{\gamma}^{2-\gamma}=2\ ,$

the first equality resulting from the fact that for $\gamma$ sufficiently
large, $-u<-\varepsilon_{\gamma}<\varepsilon_{\gamma}<1-u$. By using the
dominated convergence theorem, the proof is complete.

##  Appendix D Derivation of the hydrodynamic limit for the perturbed dynamics

Our aim is to derive Eq. (3.15) for the system where we add on each site
$x\in{1,\ldots,n}$ the drift given by (3.14)

$d_{t}(x,\varphi)=-\sum_{y\in\Lambda_{n}}K(y-x)\beta(\varphi(x),\varphi(y))\left(H_{t}\big{(}\tfrac{y}{n}\big{)}-H_{t}\big{(}\tfrac{x}{n})\big{)}\right),$
(D.1)

so that the bulk Markovian generator of the dynamics is modified according to
(3.18) while $\mathcal{G}_{\ell}$ and $\mathcal{G}_{r}$ remain the same
(2.20). The evolution of the empirical density tested against a test function
$G$ is the same as Eq. (3.8) except that in the integral appearing on the
right hand side of this equation we have the extra term

$\begin{split}&n^{\gamma-1}\sum_{x\in\Lambda_{n}}d_{s}(x,\varphi_{sn^{\gamma}})\
G\left(\tfrac{x}{n}\right)\\\
&=-\cfrac{1}{n^{2}}\sum_{x,y\in\Lambda_{n}}K\Big{(}\tfrac{y}{n}-\tfrac{x}{n}\Big{)}\beta(\varphi_{sn^{\gamma}}(x),\varphi_{sn^{\gamma}}(y))\left(H_{s}\big{(}\tfrac{y}{n}\big{)}-H_{s}\big{(}\tfrac{x}{n})\big{)}\right)\
G\left(\tfrac{x}{n}\right)\ .\end{split}$ (D.2)

Arguing like for the derivation of Eq. (3.10) by using the local equilibrium
assumption, we have that

$\begin{split}&\lim_{n\to\infty}\int_{0}^{t}ds\
\left\\{\frac{1}{n^{2}}\sum_{x,y\in\Lambda_{n}}K\Big{(}\tfrac{y-x}{n}\Big{)}\beta(\varphi_{sn^{\gamma}}(x),\varphi_{sn^{\gamma}}(y))\left(H_{s}\big{(}\tfrac{y}{n}\big{)}-H_{s}\big{(}\tfrac{x}{n})\big{)}\right)\
G\Big{(}\tfrac{x}{n}\Big{)}\right\\}\\\ &=\int_{0}^{t}ds\
\left\\{\int_{[0,1]^{2}}dudv\ K(v-u)B(\Phi_{s}(u),\Phi_{s}(v))\
\left[H_{s}(v)-H_{s}(u)\right]G(u)\right\\}\ .\end{split}$ (D.3)

Hence we get that the hydrodynamic limit is given by Eq. (3.15).

##  Appendix E Markovian dynamics for the energy field associated to the
Brownian Energy Model

Let us prove the form (6.5) of the generator after the change of variables
$\varphi=\omega^{2}$. This is proved by letting the Markovian generator (6.3)
acting on an arbitrary function in the form
$f(\omega):=g(\omega^{2})=g(\varphi)$, then

$\begin{split}&\left(\mathbb{G}_{b}^{n}f\right)(\omega)=\frac{1}{2}\sum_{x,y\in\Lambda_{n}}K(y-x)\left(\omega(y)\partial_{\omega(x)}-\omega(x)\partial_{\omega(y)}\right)^{2}g(\omega^{2})\\\
&=\sum_{x,y\in\Lambda_{n}}K(y-x)\left(\omega(y)\partial_{\omega(x)}-\omega(x)\partial_{\omega(y)}\right)\left(\omega(y)\omega(x)\left(\partial_{\varphi(x)}g\right)(\omega^{2})-\omega(x)\omega(y)\left(\partial_{\varphi(y)}g\right)(\omega^{2})\right)\
.\end{split}$ (E.1)

Observe now that

$\begin{split}&\left(\omega(y)\partial_{\omega(x)}-\omega(x)\partial_{\omega(y)}\right)\left(\omega(y)\omega(x)\left(\partial_{\varphi(x)}g\right)(\omega^{2})-\omega(x)\omega(y)\left(\partial_{\varphi(y)}g\right)(\omega^{2})\right)\\\
&=\left(\omega(y)\right)^{2}\left(\partial_{\varphi(x)}g\right)(\omega^{2})+2\left(\omega(y)\right)^{2}\left(\omega(x)\right)^{2}\left(\partial^{2}_{\varphi(x)}g\right)(\omega^{2})\\\
&-\left(\omega(y)\right)^{2}\left(\partial_{\varphi(y)}g\right)(\omega^{2})-2\left(\omega(y)\right)^{2}\left(\omega(x)\right)^{2}\left(\partial^{2}_{\varphi(x),\varphi(y)}g\right)(\omega^{2})\\\
&-\left(\omega(x)\right)^{2}\left(\partial_{\varphi(x)}g\right)(\omega^{2})-2\left(\omega(y)\right)^{2}\left(\omega(x)\right)^{2}\left(\partial^{2}_{\varphi(x),\varphi(y)}g\right)(\omega^{2})\\\
&+\left(\omega(x)\right)^{2}\left(\partial_{\varphi(y)}g\right)(\omega^{2})+2\left(\omega(y)\right)^{2}\left(\omega(x)\right)^{2}\left(\partial^{2}_{\varphi(y)}g\right)(\omega^{2})\\\
&=\left[\varphi(y)\left(\partial_{\varphi(x)}g\right)+2\varphi(y)\varphi(x)\left(\partial^{2}_{\varphi(x)}g\right)-\varphi(y)\left(\partial_{\varphi(y)}g\right)-2\varphi(y)\varphi(x)\left(\partial^{2}_{\varphi(x),\varphi(y)}g\right)\right](\varphi)\\\
&+\left[-\varphi(x)\left(\partial_{\varphi(x)}g\right)-2\varphi(x)\varphi(y)\left(\partial^{2}_{\varphi(x),\varphi(y)}g\right)+\varphi(x)\left(\partial_{\varphi(y)}g\right)+2\varphi(x)\varphi(y)\left(\partial^{2}_{\varphi(y)}g\right)\right](\varphi)\end{split}$
(E.2)

so that

$\begin{split}&\left(\omega(y)\partial_{\omega(x)}-\omega(x)\partial_{\omega(y)}\right)\left(\omega(y)\omega(x)\left(\partial_{\varphi(x)}g\right)(\omega^{2})-\omega(x)\omega(y)\left(\partial_{\varphi(y)}g\right)(\omega^{2})\right)\\\
&=\left[\left(\varphi(y)-\varphi(x)\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)+2\varphi(y)\varphi(x)\partial^{2}_{\varphi(x)}\right.\\\
&\left.\quad\quad-4\varphi(y)\varphi(x)\partial^{2}_{\varphi(x),\varphi(y)}+2\varphi(x)\varphi(y)\partial^{2}_{\varphi(y)}\right]g(\varphi)\\\
&=\left[\left(\varphi(y)-\varphi(x)\right)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)+2\varphi(y)\varphi(x)\left(\partial_{\varphi(x)}-\partial_{\varphi(y)}\right)^{2}\right]g(\varphi)\end{split}.$
(E.3)

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