∎

11institutetext:  O. Feliachi 22institutetext: Institut Denis Poisson,
Université d’Orléans, CNRS, Université de Tours, France
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique,
Lyon, France
22email<EMAIL_ADDRESS>33institutetext: F. Bouchet
44institutetext: Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS,
Laboratoire de Physique, Lyon, France
44email<EMAIL_ADDRESS>
# Dynamical large deviations for homogeneous systems with long range
interactions and the Balescu–Guernsey–Lenard equation

Ouassim Feliachi and Freddy Bouchet

(Received: date / Accepted: date)

###### Abstract

We establish a large deviation principle for time dependent trajectories
(paths) of the empirical density of $N$ particles with long range
interactions, for homogeneous systems. This result extends the classical
kinetic theory that leads to the Balescu–Guernsey–Lenard kinetic equation, by
the explicit computation of the probability of typical and large fluctuations.
The large deviation principle for the paths of the empirical density is
obtained through explicit computations of a large deviation Hamiltonian. This
Hamiltonian encodes all the cumulants for the fluctuations of the empirical
density, after time averaging of the fast fluctuations. It satisfies a time
reversal symmetry, related to the detailed balance for the stochastic process
of the empirical density. This explains in a very simple way the increase of
the macrostate entropy for the most probable states, while the stochastic
process is time reversible, and describes the complete stochastic process at
the level of large deviations.

###### Keywords:

Plasma Balescu–Guernsey–Lenard equation Large deviation theory Macroscopic
fluctuation theory Widom theorem

††journal: Journal of Statistical Physics

##  1 Introduction

We consider the Hamiltonian dynamics of particles that interact through a
mean-field potential. The dynamics reads

$\begin{cases}{\displaystyle\frac{\text{d}\mathbf{r}_{n}}{\text{d}t}}&=\mathbf{v}_{n}\\\
\\\
{\displaystyle\frac{\text{d}\mathbf{v}_{n}}{\text{d}t}}&{\displaystyle=-\frac{1}{N}\sum_{m\neq
n}\frac{\text{d}}{\text{d}\mathbf{r}_{n}}W\left(\mathbf{r}_{n}-\mathbf{r}_{m}\right)}\end{cases}$
(1)

where $\left\\{\mathbf{r}_{n}\right\\}_{1\leq n\leq N}$ are the positions and
$\left\\{\mathbf{v}_{n}\right\\}_{1\leq n\leq N}$ the velocities. This set-up
is relevant for plasmas in the weak coupling regime Nicholson_1991 , self-
gravitating systems Padmanabhan_1990 ; Chavanis_2006IJMPB_Revue_Auto_Gravitant
, and many particle systems with long range interactions
Bouchet_Gupta_Mukamel_PRL_2009 . It also shares many theoretical analogies
with two-dimensional and geostrophic turbulence, through the point vortex
model Chavanis_2001PhRvE_64_PointsVortex ;
Kiessling_Lebowitz_1997_PointVortex_Inequivalence_LMathPhys , or
stochastically forced models of two-dimensional and geostrophic turbulence
Bouchet_Nardini_Tangarife_2013_Kinetic_JStatPhys . The kinetic theory of
systems with mean-field potentials (or long range interactions) is a classical
piece of theoretical physics. The relaxation to equilibrium of the empirical
density

$g_{N}(\mathbf{r},\mathbf{v},t)=\frac{1}{N}\sum_{n=1}^{N}\delta(\mathbf{r}-\mathbf{r}_{n}(t))\delta(\mathbf{v}-\mathbf{v}_{n}(t)),$

is described by the Balescu–Guernsey–Lenard kinetic equation in the limit of a
large number of particles. This result has been formally derived by Balescu,
Guernsey and Lenard balescu1960irreversible ; balescu1963kinetic . In the
context of plasma physics where we consider $N$ charged particles submitted to
Coulomb interactions, we refer to Nicholson Nicholson_1991  for a derivation
using the BBGKY hierarchy, or to Lifshitz and Pitaevskii
Lifshitz_Pitaevskii_1981_Physical_Kinetics  who follow the Klimontovich
approach.

In this paper we extend this classical kinetic theory by describing the
statistics of the large deviations for time dependent trajectories (paths) of
the empirical density. For simplicity, we restrict our analysis to paths of
the empirical density which remain close to homogenous distributions. We
consider the projection of the empirical density on homogeneous distributions:
$f_{N}\left(\mathbf{v},t\right)=N^{-1}L^{-3}\int\text{d}\mathbf{r}\,g_{N}\left(\mathbf{r},\mathbf{v},t\right)=N^{-1}L^{-3}\sum_{n=1}^{N}\delta(\mathbf{v}-\mathbf{v}_{n}(t))$,
where $L^{3}$ is the volume of the system. The natural evolution of $f_{N}$
occurs on time scales of order $N$ (except in dimension $d=1$
Yamaguchi_Barre_Bouchet_DR:2004_PhysicaA ). After time rescaling $\tau=t/N$,
we study the probability of
$f_{N}^{s}\left(\mathbf{v},\tau\right)=f_{N}\left(\mathbf{v},N\tau\right)$ (by
abuse of notation and for convenience, we still denote $f_{N}^{s}=f_{N}$). We
justify that the probability that a path
$\left\\{f_{N}(\tau)\right\\}_{0\leq\tau\leq T}$ remains in the neighborhood
of a prescribed path $\left\\{f(\tau)\right\\}_{0\leq t\leq T}$ satisfies the
large deviation principle

$\mathbf{P}\left(\left\\{f_{N}(\tau)\right\\}_{0\leq\tau\leq
T}=\left\\{f(\tau)\right\\}_{0\leq\tau\leq
T}\right)\underset{N\rightarrow\infty}{\asymp}\text{e}^{-NL^{3}\int_{0}^{T}\text{d}\tau\,\text{Sup}_{p}\left\\{\int\text{d}\mathbf{v}\,\dot{f}p-H[f,p]\right\\}}\text{e}^{-NI_{0}\left[f^{0}\right]},$
(2)

where $\dot{f}$ is the time derivative of $f$, $p$ is a function over the
velocity space and is called the conjugated momentum of $\dot{f}$, the
Hamiltonian $H$ is a functional of $f$ and $p$ that characterizes the
dynamical fluctuations, $I_{0}$ is a large deviation rate function for the
initial conditions of $f_{N}$, and where the symbol
$\underset{N\rightarrow\infty}{\asymp}$ means a large deviation principle
(roughly speaking, the log of the left hand side is equivalent to the log of
the right hand side, see classical textbooks Dembo_Zeitouni_1993_Book  for a
more precise definition). We note that $H$ is not the Hamiltonian of the
microscopic dynamics but it rather defines a statistical field theory that
quantifies the probabilities of paths of the empirical density.

The main result of this paper is the first computation of an explicit
expression for $H$ and the study of its symmetry properties. The explicit
expression for $H$ is

$H\left[f,p\right]=-\frac{1}{4\pi
L^{3}}\sum_{\mathbf{k}}\int\text{d}\omega\,\log\left\\{1-\mathcal{J}\left[f,p\right]\left(\mathbf{k},\omega\right)\right\\},$
(3)

with

$\mathcal{J}\left[f,p\right]\left(\mathbf{k},\omega\right)=4\pi\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\frac{\partial
p}{\partial\mathbf{v}_{1}}\cdot\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)\cdot\left\\{\frac{\partial
f}{\partial\mathbf{v}_{2}}f(\mathbf{v}_{1})-f(\mathbf{v}_{2})\frac{\partial
f}{\partial\mathbf{v}_{1}}\right\\}\\\
+4\pi\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\left\\{\frac{\partial
p}{\partial\mathbf{v}_{1}}\frac{\partial
p}{\partial\mathbf{v}_{1}}-\frac{\partial
p}{\partial\mathbf{v}_{1}}\frac{\partial
p}{\partial\mathbf{v}_{2}}\right\\}:\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)f(\mathbf{v}_{1})f(\mathbf{v}_{2}),$
(4)

where the
$\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)$ is a
symmetric tensor which can be expressed through the Fourier transform of the
interaction potential and the dielectric function.

Equations (3-4) clearly show that the Hamiltonian is not quadratic in the
conjugated momentum $p$. This shows that the fluctuations that lead to large
deviations are not locally Gaussian, by contrast with many other cases, for
instance when diffusive limits are involved as in the case of macroscopic
fluctuation theory bertini2015macroscopic , or for plasma fluctuations at
scales much smaller than the Debye length 2021arXiv210104455F . It is striking
that it is possible to get explicit formulas (3-4) for the large deviation
Hamiltonian, for which cumulants of all order are relevant and non trivial.
The four key theoretical ideas and technical tools we use are: making the
connection with large deviation theory for slow-fast systems and identifying
the statistics of the fast motion, expressing the Hamiltonian as a functional
determinant on a space of functions that depend both on time and velocity,
using the Szegö–Widom theorem to reduce this functional determinant to a
simpler one on a space of functions that depend on velocity only, and finally
computing explicitly those determinants on the space of functions that depend
on velocity only.

This article is the last of a series of three aimed at establishing dynamical
large deviation principles related to the main classical kinetic theories,
starting from Hamiltonian dynamics. The first one bouchet2020boltzmann  dealt
with the large deviations for dilute gases (associated with the Boltzmann
equation), the second one dealt with large deviations for plasma fluctuations
at scales much smaller then the Debye length 2021arXiv210104455F  (associated
with the Landau equation), and this one deals with the case of systems with
mean-field interactions (associated with the Balescu–Guernsey–Lenard
equation). The key point of these three works is to establish large deviation
principles for particle systems with Hamiltonian dynamics. At first sight it
might seem surprising to obtain a stochastic process for an effective kinetic
description, starting from a deterministic dynamics. However it is well known
that, after taking the limit with an infinite time scale separation between
the slow and fast degrees of freedom, the effective dynamics of a slow-fast
dynamical system, with chaotic fast degrees of freedom, is stochastic. At the
level of large deviations, for deterministic dynamical systems, mathematicians
have proven theorems that establish large deviation principles for the
effective stochastic process of the slow variable, from natural hypotheses
kifer1992averaging ; kifer2004averaging . This behavior can also be
illustrated numerically, for instance coupling a slow dynamics with a fast
chaotic Lorenz model dynamics Melbourne_2011 . As far as we know, our three
works (bouchet2020boltzmann ; 2021arXiv210104455F  and this one) and
bodineau2020fluctuation  establish the first large deviation principles, in
kinetic theory that do not start from stochastic dynamics, like for instance
in macroscopic fluctuation theory bertini2015macroscopic . While in
bodineau2020fluctuation  the result is proven for dilute gases in the
Boltzmann–Grad limit for times of order of one collision times, our
derivations are not mathematical proofs. All the steps of our derivation are
however exact computations, once natural hypothesis are made, in the spirit of
the most precise classical works by theoretical physicists in kinetic theory,
and the result is expected to be valid for times much large that the kinetic
times.

The large deviation results of the present paper (3-4) have some direct
relations with the large deviations for plasma fluctuations at a scale much
smaller than the Debye length, considered in 2021arXiv210104455F . While the
result in the present paper is much more general, we explain in section 5.3
that for plasma fluctuations for scales much smaller than the Debye length,
the locally Gaussian large deviation Hamiltonian of 2021arXiv210104455F  can
be recovered. We also prove in this paper a conjecture made in
2021arXiv210104455F  about the structure of the cumulant expansion. We stress
that the results of 2021arXiv210104455F  are not just a sub-case of the
results in the present paper. There, we were also considering questions of a
different nature: the relation with the large deviation Hamiltonian for dilute
gases, on one hand, and the relation with large deviations for effective mean-
field diffusions, on the other hand.

We stress that our large deviation principle for paths immediately implies a
gradient flow structure for the Balescu–Guernsey–Lenard operator, adapting to
this specific case the general connection between path large deviation and
gradient flows first discussed in mielke2014relation  and simply explained in
section 5 of bouchet2020boltzmann . As far as we know, no gradient flow
structure was known before for the Balescu–Guernsey–Lenard operator. As the
large deviations are not locally Gaussian, this gradient structure is not a
standard one, but a generalized one mielke2014relation  (please see section 5
of bouchet2020boltzmann  for a simple definition standard of non standard
gradient flow structure).

In parallel to our results, many mathematical results have recently been
obtained for the kinetic theory of plasma and systems with long range
interactions. The derivation of the Vlasov equation from the $N$ particle
dynamics had been first proved by Neunzert neunzert1977vlasov , Braun and Hepp
Braun_Hepp_CommMathPhys_1977  and Dobrushin dobrushin1979vlasov , for
interactions through a smooth potential. This question is still under study
for interaction potentials with singularities, for instance with the Coulomb
interaction (see for instance Hauray_Jabin_ARMA_2007 ). Kiessling’s review
kiessling2008microscopic  provides a recent report on the mathematical
justification of the Vlasov equation from the microscopic dynamics of
interacting particles. The stability of stationary states of the Vlasov
equation, for describing the dynamics of the empirical density over time
scales that diverge with $N$, but which are much smaller than the kinetic
time, has been proven in Caglioti_Rousset_2007_JStatPhys_QSS . The description
of Gaussian fluctuations of the potential, for dynamics close to the Vlasov
equilibrium, has been established by Braun and Hepp for smooth interaction
potentials, or in the book Spohn_1991 . More recent works
lancellotti2009fluctuations ; lancellotti2010vlasov ; lancellotti2016time ;
paul2019size ; velazquez2018two ; duerinckx2021size  discuss the Gaussian
process of the fluctuations of the potential close to a Vlasov solution. A
recent proof has been proposed for the the validity of the
Balescu–Guernsey–Lenard equation up to time scales of order $N^{r}$ with $r<1$
duerinckx2021lenard .

We also stress that the mathematics community is also very active in studying
large deviation principles related to kinetic theories. Rezakhanlou pioneered
large deviation principle results, related to the Boltzmann equations
rezakhanlou1998large , in the case of stochastic toy models of collisions. The
first mathematical result which deals with Hamiltonian dynamics, more
specifically hard sphere dynamics, has been obtained by Bodineau, Gallagher,
Saint-Raymond, and Simonella bodineau2020fluctuation ; bodineau2020statistical
. They basically prove the validity of the large deviation Hamiltonian for
times of order one collision time, in the spirit of Lanford’s proof for the
Boltzmann equation. Large deviation principles have also been obtained for
Kac’s models heydecker2021large ; basile2021large . As far as we know, no
mathematical results exists for the large deviations of plasma or systems with
mean-field interactions.

In section 2 we define the Hamiltonian dynamics, as well as the classical
kinetic equation that describes the relaxation to equilibrium. In section 4,
we establish a large deviation principle for the empirical density using the
slow-fast decomposition of the quasilinear dynamics. In section 4, we provide
an explicit computation of the large deviation Hamiltonian. In section 5, we
check that this Hamiltonian is fully compatible with the conservation laws of
the system, as well as its time-reversal symmetry, and that it is consistent
with statistics in the microcanonical ensemble. We discuss perspectives in
section 6.

##  2 Dynamics of particles with long range interactions

In this section we set up the definitions, and present classical results about
the kinetic theory of the dynamics of $N$ particles with long range
interactions, in the limit of large $N$. In section 2.1, we define the
Hamiltonian dynamics. In section 2.2, we introduce the Vlasov equation that
describes the evolution of the empirical density on timescales of order one.
In section 2.3, we introduce the Balescu–Guernsey–Lenard equation that
describes the long time relaxation of the empirical density, from Vlasov
stationary solutions to the Maxwell-Boltzmann equilibrium distribution, and
some of its important physical properties.

###  2.1 Hamiltonian dynamics of $N$ particles with long range interactions

We consider $N$ particles with positions
$\left\\{\mathbf{r}_{n}\right\\}_{1\leq n\leq N}$ and velocities
$\left\\{\mathbf{v}_{n}\right\\}_{1\leq n\leq N}$ governed by a Hamiltonian
dynamics

$\begin{cases}{\displaystyle\frac{\text{d}\mathbf{r}_{n}}{\text{d}t}}&=\mathbf{v}_{n}\\\
\\\
{\displaystyle\frac{\text{d}\mathbf{v}_{n}}{\text{d}t}}&{\displaystyle=-\frac{1}{N}\sum_{m\neq
n}\frac{\text{d}}{\text{d}\mathbf{r}_{n}}W\left(\mathbf{r}_{n}-\mathbf{r}_{m}\right)}\end{cases}$
(5)

where the interaction potential $W(\mathbf{r})$ is an even function of
$\mathbf{r}$. In the following, we consider that $\mathbf{r}_{n}$ belongs to a
3-dimensional torus of size $L^{3}$, and $\mathbf{v}_{n}\in\mathbb{R}^{3}$. We
stress that our results are actually valid for any space dimension $d>1$. We
assume that the potential $W$ is a long range potential: the decay of $W$ is
slow enough, so that the interaction is dominated by the collective effects of
the $N$ particle rather than by local effects. In an infinite space this
condition would be met if the potential would be non-integrable, for instance
in dimension $d$ it would decay asymptotically like a power law
$1/\left|\mathbf{r}\right|^{d}$ or more slowly. This condition is met in many
physical systems, for instance self-gravitating systems or weak interacting
plasma (with a large plasma parameter). For any finite $L$, the condition that
the potential decays more slowly than $1/r^{d}$ is a sufficient condition for
the potential to be long range.

We call $\mu-$space the $\left(\mathbf{r},\mathbf{v}\right)$ space. The
$\mu-$space is of dimension $6$. Let us define $g_{N}$ the $\mu-$space
empirical density for the positions and velocities of the $N$ particles

$g_{N}(\mathbf{r},\mathbf{v},t)=\frac{1}{N}\sum_{n=1}^{N}\delta(\mathbf{r}-\mathbf{r}_{n}(t))\delta(\mathbf{v}-\mathbf{v}_{n}(t)).$

In the following, we will study the stochastic process of the asymptotic
dynamics of $g_{N}$, as the number of particles $N$ goes to infinity.

###  2.2 The Vlasov equation

From equation (5), one immediately obtains the Klimontovich equation

$\frac{\partial g_{N}}{\partial t}+\mathbf{v}\cdot\frac{\partial
g_{N}}{\partial\mathbf{r}}-\frac{\partial
V\left[g_{N}\right]}{\partial\mathbf{r}}\cdot\frac{\partial
g_{N}}{\partial\mathbf{v}}=0,$ (6)

where
$V[g_{N}](\mathbf{r},t)=\int\text{d}\mathbf{v}^{\prime}\text{d}\mathbf{r}^{\prime}W(\mathbf{r}-\mathbf{r}^{\prime})g_{N}(\mathbf{r}^{\prime},\mathbf{v}^{\prime},t)$.
This is an exact equation for the evolution of $g_{N}$, if $W$ is regular
enough. For the Coulomb interaction, the formal equation (6) has to be
interpreted carefully. In the following, we do not discuss the divergences
that might occur related to small scale interactions. At a mathematic level,
this would be equivalent to considering a potential which is regularized at
small scales, and smooth. The Klimontovich equation (6) contains all the
information about the trajectories of the $N$ particles. We would like to
build a kinetic theory, that describes the stochastic process for $g_{N}$ at a
mesoscopic level.

An important first result is that the sequence $\\{g_{N}\text{\\}}$ obeys a
law of large numbers when $N\rightarrow+\infty$. More precisely, if we assume
that there is a set of initial conditions $\\{g_{N}^{0}\text{\\}}$ such that
$\lim_{N\rightarrow+\infty}g_{N}^{0}\left(\mathbf{r},\mathbf{v}\right)=g^{0}\left(\mathbf{r},\mathbf{v}\right)$,
then over a finite time interval $t\in\left[0,T\right]$,
$\lim_{N\rightarrow+\infty}g_{N}\left(\mathbf{r},\mathbf{v},t\right)=g\left(\mathbf{r},\mathbf{v},t\right)$
where $g$ solves the Vlasov equation

$\frac{\partial g}{\partial t}+\mathbf{v}\cdot\frac{\partial
g}{\partial\mathbf{r}}-\frac{\partial
V\left[g\right]}{\partial\mathbf{r}}\cdot\frac{\partial
g}{\partial\mathbf{v}}=0\,\,\,\text{with}\,\,\,g\left(\mathbf{r},\mathbf{v},t=0\right)=g^{0}\left(\mathbf{r},\mathbf{v}\right).$
(7)

While the solution of the Klimontonvich equation is a distribution that
carries the whole information about the positions and velocities of all the
particles, the Vlasov equation describes the evolution of a continuous
mesoscopic density for the same evolution. As the Klimontovich and the Vlasov
equations are formally the same, this law of large numbers is actually a
stability result for the Vlasov equation in a space of distributions. Such a
result has first been proven for smooth potentials by Braun and Hepp
Braun_Hepp_CommMathPhys_1977  and Neunzert neunzert1977vlasov .

The Vlasov equation has infinitely many Casimir conserved quantities. As a
consequence, it has an infinite number of stable stationary states
Yamaguchi_Barre_Bouchet_DR:2004_PhysicaA . Any homogeneous distribution
$g\left(\text{$\mathbf{r}$},\mathbf{v}\right)=f(\mathbf{v})$ is a stationary
solution of the Vlasov equation. In the following, we will consider dynamics
close to any homogeneous $f$ which is a linearly stable stationary solution of
the Vlasov equation. This linear stability can be assessed by studying the
dielectric susceptibility $\varepsilon[f](\mathbf{k},\omega)$ Nicholson_1991 ;
Lifshitz_Pitaevskii_1981_Physical_Kinetics , defined by

$\varepsilon[f]\left(\mathbf{k},\omega\right)=1-\hat{W}\left(\mathbf{k}\right)\int\text{d}\mathbf{v}\frac{\mathbf{k}.\frac{\partial
f}{\partial\mathbf{v}}}{\mathbf{k}.\mathbf{v}-\omega-i\eta},$ (8)

where $\hat{W}\left(\mathbf{k}\right)$ is the $\mathbf{k}$-th Fourier
component of the interaction potential:
$\hat{W}\left(\mathbf{k}\right)=\int\text{d}\mathbf{r}\,\exp\left(-i\mathbf{k}.\mathbf{r}\right)W\left(\mathbf{r}\right)$.
Equation (8) and every other equations involving $\pm i\eta$ have to be
understood as the limit as $\eta$ goes to zero with $\eta$ positive. The
dielectric susceptibility function $\varepsilon$ plays the role of a
dispersion relation in the linearized dynamics, and a solution $f$ is stable
if $\varepsilon[f]$ has no zeros except for $\omega$ on the real line. We note
that
$\varepsilon\left[f\right]\left(-\mathbf{k},-\omega\right)=\varepsilon^{*}\left[f\right]\left(\mathbf{k},\omega\right)$.
Another important property of the dielectric susceptibility is
$\varepsilon\left[I\left[f\right]\right]\left(\mathbf{k},-\omega\right)=\varepsilon^{*}\left[f\right]\left(\mathbf{k},\omega\right)$,
where $I\left[f\left(\mathbf{v}\right)\right]=f\left(-\mathbf{v}\right)$. This
last property, associated to the time-reversal symmetry of the Hamiltonian
dynamics, will be used in section 5.2. In this section we have discussed the
linear stability of stationary solutions of the Vlasov equation while
Yamaguchi_Barre_Bouchet_DR:2004_PhysicaA  defines different other notions of
stability.

From the point of view of dynamical systems, those homogeneous solutions might
be attractors of the Vlasov equation, with some sort of asymptotic stability.
At a linear level, this convergence for some of the observables, for instance
the potential, is called Landau damping Nicholson_1991 ;
Lifshitz_Pitaevskii_1981_Physical_Kinetics . Such a stability might also be
true for the full dynamics. Indeed some non-linear Landau damping results have
recently been proven Mouhot_Villani:2009 .

In the following we will study the dynamics of $g_{N}$, when its initial
condition is close to a homogeneous stable state $f(\mathbf{v})$. On time
scales of order one, the distribution is stable and remains close to $f$
according to the Vlasov equation. However a slow evolution occurs on a
timescale $\tau$ of order $N$, in spaces of dimension $d>1$. For this reason,
such $f$ are called quasi-stationary states
Yamaguchi_Barre_Bouchet_DR:2004_PhysicaA . In the following section, we
explain that this slow evolution is described by the Balescu–Guernsey–Lenard
equation for most initial conditions, or as a law of large numbers.

As a conclusion, the Balescu–Guernsey–Lenard equation appears as a mesoscopic
description of the solution of the Klimontovich equation, for homogeneous
solutions, which is valid up to time scales of order $N$, while the Vlasov
equation is valid only up to time scales of order one. The
Balescu–Guernsey–Lenard equation is a crucial correction to the Vlasov
equation close to homogeneous solutions. Indeed homogeneous solution have no
evolution theourgh the Vlasov equation as they are stationary, while they have
an evolution of order one over times scales of order $N$ through the
Balescu–Guernsey–Lenard equation.

###  2.3 The Balescu–Guernsey–Lenard equation

With the rescaling of time $\tau=t/N$, we expect a law of large numbers in the
sense that “for almost all initial conditions” the empirical density $g_{N}$
converges to $f$, with $f$ that evolves according to the
Balescu–Guernsey–Lenard equation

$\frac{\partial
f}{\partial\tau}=\frac{\partial}{\partial\mathbf{v}}.\int\text{d}\mathbf{v}_{2}\,\mathbf{B}\left[f\right](\mathbf{v},\mathbf{v}_{2})\left(-\frac{\partial
f}{\partial\mathbf{v}_{2}}f(\mathbf{v})+f(\mathbf{v}_{2})\frac{\partial
f}{\partial\mathbf{v}}\right),$ (9)

with

$\mathbf{B}\left[f\right](\mathbf{v}_{1},\mathbf{v}_{2})=\frac{\pi}{L^{3}}\int_{-\infty}^{+\infty}\text{d}\omega\,\sum_{\mathbf{k}\in\left(2\pi/L\right)\mathbb{Z}^{3}}\frac{\hat{W}\left(\mathbf{k}\right)^{2}\mathbf{k}\mathbf{k}}{\left|\varepsilon[f]\left(\mathbf{k},\omega\right)\right|^{2}}\delta\left(\omega-\mathbf{k}.\mathbf{v}_{1}\right)\delta\left(\omega-\mathbf{k}.\mathbf{v}_{2}\right),$
(10)

where $\mathbf{k}\mathbf{k}$ denotes the tensor product
$\mathbf{k}\otimes\mathbf{k}$. The tensor $\mathbf{B}$ is called the collision
kernel of the Balescu–Guernsey–Lenard equation (by analogy with the Boltzmann
equation).

A recent proof has been proposed for the the validity of the
Balescu–Guernsey–Lenard equation up to time scales $t$ of order $N^{r}$ with
$r<1$ duerinckx2021lenard . We know no mathematical proof of such a result for
time scales $t$ of order $N$ ($\tau$ of order one). In the theoretical physics
literature, this equation is derived as an exact consequence of the dynamics
once natural hypotheses are made. Two classes of derivations are known, either
the BBGKY hierarchy detailed in Nicholson_1991  or the Klimontovich approach
presented for instance in Lifshitz_Pitaevskii_1981_Physical_Kinetics . The
Klimontovich derivation is the more straightforward from a technical point of
view. We now recall the main steps of the Klimontovich derivation, that will
be useful later.

In the following we will consider statistical averages over measures of
initial conditions for the $N$ particle initial conditions
$\left\\{\mathbf{r}_{n}^{0},\mathbf{v}_{n}^{0}\right\\}$. We denote
$\mathbb{E}_{S}$ the average with respect to this measure of initial
conditions. As an example the measure of initial conditions could be the
product measure
$\prod_{n=1}^{N}g^{0}\left(\mathbf{r}_{n}^{0},\mathbf{v}_{n}^{0}\right)\mbox{d}\mathbf{r}_{n}\mbox{d}\mathbf{v}_{n}$.
But we might consider other measures of initial conditions. We assume that for
the statistical ensemble of initial conditions, the law of large numbers
$\lim_{N\rightarrow\infty}g_{N}^{0}\left(\mathbf{r},\mathbf{v}\right)=g^{0}\left(\mathbf{r},\mathbf{v}\right)$
is valid at the initial time. This is true for instance for the product
measure. In the following, for simplicity, we restrict the discussion to cases
when the initial conditions are statistically homogenous:
$g^{0}\left(\mathbf{r},\mathbf{v}\right)=P^{0}(\mathbf{v})$. In the following,
we define $f$ as the statistical average of $g_{N}$ over the initial
conditions
$f(\mathbf{v},t)=\mathbb{E}_{S}\left(g_{N}(\mathbf{r},\mathbf{v},t)\right)$.

We define the fluctuations $\delta g_{N}$ by
$g_{N}(\mathbf{r},\mathbf{v},t)=f(\mathbf{v})+\delta g_{N}/\sqrt{N}$. The
scaling $1/\sqrt{N}$ is natural when we see the Vlasov equation (7) as a law
of large numbers for the empirical density. For the potential we obtain
$V\left[g_{N}\right]=V\left[\delta g_{N}\right]/\sqrt{N}$, as $f$ is
homogeneous. If we introduce this decomposition in the Klimontovich equation
(6), we obtain

$\displaystyle\frac{\partial f}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\mathbb{E}_{S}\left(\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right)$ (11) $\displaystyle\frac{\partial\delta
g_{N}}{\partial t}+\mathbf{v}.\frac{\partial\delta
g_{N}}{\partial\mathbf{r}}-\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial f}{\partial\mathbf{v}}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{N}}\left[\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}-\mathbb{E}_{S}\left(\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right)\right].$ (12)

In the first equation, the right hand side of the equation
$\frac{1}{N}\mathbb{E}_{S}\left(\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right)$ is called the averaged non linear term and
is responsible for the long term evolution of the distribution $f$. The right
hand side of the second equation $\frac{1}{\sqrt{N}}\left[\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}-\right.$ $\left.\mathbb{E}_{S}\left(\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right)\right]$ describes the fluctuations of the
non-linear term. For stable distributions $f,$ and on timescales much smaller
than $\sqrt{N}$, we can neglect this term, following Klimontovich and
classical textbooks Lifshitz_Pitaevskii_1981_Physical_Kinetics . Please see
Caglioti_Rousset_2007_JStatPhys_QSS  for a mathematical proof of a sufficient
condition of stability on time scales of order $N^{\alpha}$, for some
$\alpha<1$. Neglecting the terms much smaller than $\sqrt{N}$ closes the
hierarchy of the correlation functions. The Bogoliubov approximation then
amounts to using the time scale separation between the evolution of $f$ and
$\delta g_{N}$. Then for fixed $f$, the equation for $\delta g_{N}$ (12) is
linear when $f$ is fixed. One computes the correlation function
$\mathbb{E}_{S}\left(\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right)$ resulting from (12) with fixed $f$, and
argues that this two point correlation function converges to a stationary
quantity on time scales much smaller than $\sqrt{N}$. Using this quasi-
stationary correlation function $\mathbb{E}_{S}\left(\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right)$, one can compute the right hand side of
(11) as a function of $f$.

After time rescaling $\tau=t/N$, we define
$g_{N}^{s}\left(\mathbf{r},\mathbf{v},\tau\right)=g_{N}\left(\mathbf{r},\mathbf{v},N\tau\right)$.
By abuse of notation and for convenience, we still denote
$g_{N}^{s}(\tau)=g_{N}(\tau)$. The closed equation for $g_{N}(\tau)$, which is
obtained from (11) is the Balescu–Guernsey–Lenard equation (9). We do not
reproduce these lengthy and classical computations that can be found in plasma
physics textbooks, for instance in Chapter 51 of
Lifshitz_Pitaevskii_1981_Physical_Kinetics . A natural conjecture is that we
have a law of large numbers
$\lim_{N\rightarrow\infty}g_{N}\left(\mathbf{r},\mathbf{v},\tau\right)=f(\mathbf{v},\tau)$,
where $f$ solves the Balescu–Guernsey–Lenard equation (9), and valid for any
finite time $\tau$.

#### Symmetries and conservation properties.

The Balescu–Guernsey–Lenard equation (9) has several important physical
properties:

  1. 1.

It conserves the mass $M[f]$, momentum $\mathbf{P}[f]$ and total kinetic
energy $E[f]$ defined by

$M[f]=\int\text{d}\mathbf{v}\,f\left(\mathbf{v}\right),\,\,\,\mathbf{P}[f]=\int\text{d}\mathbf{v}\,\textbf{v}f\left(\mathbf{v}\right)\,\,\,\text{and}\,\,\,E[f]=\int\text{d}\mathbf{v\,}\frac{\textbf{v}^{2}}{2}f\left(\mathbf{v}\right).$
(13)

  2. 2.

It increases monotonically the entropy $S[f]$ defined by

$S[f]=-k_{B}\int\text{d}\mathbf{v}\,f\left(\mathbf{v}\right)\log
f\left(\mathbf{v}\right),$

where $k_{B}$ is the Boltzmann constant.

  3. 3.

It converges towards the Boltzmann distribution for the corresponding energy

$f_{B}\left(\mathbf{v}\right)=\frac{\beta{}^{3/2}}{\left(2\pi\right)^{3/2}}\exp\left(-\beta\frac{\mathbf{v}^{2}}{2}\right).$

##  3 Derivation of the large deviation principle from the quasi-linear
dynamics

In this section, we derive a large deviation principle for the empirical
density of $N$ particles with long range interactions, directly from the
dynamics (5).

In section 3.1, we introduce the quasi-linear dynamics of the empirical
density of $N$ long range interacting particles, for which the law of large
numbers is the Balescu–Guernsey–Lenard kinetic theory. In section 3.1, we
explain that this quasi-linear dynamics for the empirical density can be seen
as a slow-fast system, for which we can define the path large deviation
functional for the slow variable. In section 3.2, we characterize the
stochastic process for the quasi-linear dynamics of the fluctuations of the
empirical density as a stationary Gaussian process.

###  3.1 The Klimontovich approach, quasilinear and slow-fast dynamics

We begin by equations which are similar to (11-12), but by contrast to the
discussion of the previous section, we will not compute just the average for
the effect of fluctuations on the evolution of $f_{N}$, but all the cumulants
after time averaging.

We consider the empirical density

$g_{N}\left(\mathbf{r},\mathbf{v},t\right)=\frac{1}{N}\sum_{n=1}^{N}\delta\left(\mathbf{v}-\mathbf{v}_{n}\left(t\right)\right)\delta\left(\mathbf{r}-\mathbf{r}_{n}\left(t\right)\right),$

of $N$ particles which interact through a long range pair potential according
to the dynamics (5). From these equations of motion, we can deduce the
Klimontovich equation

$\frac{\partial g_{N}}{\partial t}+\mathbf{v}\cdot\frac{\partial
g_{N}}{\partial\mathbf{r}}-\frac{\partial
V\left[g_{N}\right]}{\partial\mathbf{r}}\cdot\frac{\partial
g_{N}}{\partial\mathbf{v}}=0.$ (14)

We consider the decomposition

$g_{N}\left(\mathbf{r},\mathbf{v},t\right)=f_{N}\left(\mathbf{v}\right)+\frac{1}{\sqrt{N}}\delta
g_{N}\left(\mathbf{r},\mathbf{v},t\right),$

where
$f_{N}\left(\mathbf{v},t\right)=\frac{1}{L^{3}}\int\text{d}\mathbf{r}\,g_{N}\left(\mathbf{r},\mathbf{v},t\right)$
is the projection of $g_{N}$ on homogeneous distributions (distributions that
depend on velocity only) and $\delta g_{N}$ describes the inhomogeneous
fluctuations of the empirical density $g_{N}$. Alternately, we can understand
$f_{N}$ as the empirical density of the $N$ particles in the velocity space:
$f_{N}\left(\mathbf{r},\mathbf{v},t\right)=N^{-1}L^{-3}\sum_{n=1}^{N}\delta\left(\mathbf{v}-\mathbf{v}_{n}\left(t\right)\right)$.
From the Klimontovich equation (14), we straightforwardly write

$\displaystyle\frac{\partial f_{N}}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{1}{NL^{3}}\int\text{d}\mathbf{r}\,\left(\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right),$ (15) $\displaystyle\frac{\partial\delta
g_{N}}{\partial t}$ $\displaystyle=$
$\displaystyle-\mathbf{v}.\frac{\partial\delta
g_{N}}{\partial\mathbf{r}}+\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial f_{N}}{\partial\mathbf{v}}$
(17) $\displaystyle+\frac{1}{\sqrt{N}}\left[\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}-\frac{1}{L^{3}}\int\text{d}\mathbf{r}\,\left(\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right)\right].$

Just like in the previous section, we will consider statistical averages over
a probability measure for the initial conditions
$\left\\{\mathbf{r}_{n}^{0},\mathbf{v}_{n}^{0}\right\\}$ of the $N$ particles.
As the microscopic dynamics is deterministic, the only source of randomness is
the ensemble of initial conditions. We assume that this ensemble of initial
conditions is sampled from a spatially homogeneous measure and that the set of
corresponding $g_{N}$ is concentrated close to homogeneous distributions in
the $(\mathbf{r},\mathbf{v})$ space. Moreover we assume that the large
deviation principle

$\mathbf{P}\left(f_{N}\left(\tau=0\right)=f^{0}\right)\underset{N\rightarrow\infty}{\asymp}\text{e}^{-NI_{0}\left[f^{0}\right]},$
(18)

holds, where $I_{0}$ is a large deviation rate function for $f_{N}(\tau=0)$
the initial conditions of $f_{N}$. As an example, the measure of initial
conditions $\left\\{\mathbf{r}_{n}^{0},\mathbf{v}_{n}^{0}\right\\}$ could be
the homogeneous product measure
$\prod_{n=1}^{N}P^{0}\left(\mathbf{v}_{n}^{0}\right)\mbox{d}\mathbf{v}_{n}\mbox{d}\mathbf{r}_{n}/L^{3}$,
with $\int\mbox{d}\mathbf{v}P^{0}\left(\mathbf{v}\right)=1$. Then $I_{0}$
would then be the Kullback–Leibler divergence of $f^{0}$ with respect to
$P^{0}$. But we might consider other ensembles of initial conditions.

We now assume the validity of the quasi-linear approximation, which amounts to
neglecting terms of order $N^{-1/2}$ in the evolution equation for $\delta
g_{N}$. We also change the timescale $\tau=t/N$ and obtain the quasilinear
dynamics

$\displaystyle\frac{\partial f_{N}}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{L^{3}}\int\text{d}\mathbf{r}\,\left(\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right),$ (19) $\displaystyle\frac{\partial\delta
g_{N}}{\partial\tau}$ $\displaystyle=N$
$\displaystyle\left\\{-\mathbf{v}.\frac{\partial\delta
g_{N}}{\partial\mathbf{r}}+\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial
f_{N}}{\partial\mathbf{v}}\right\\}.$ (20)

When $N$ goes to infinity, we observe that the equation for $\delta g_{N}$ is
a fast process, with timescales for $\tau$ of order $1/N$, while the equation
for $f_{N}$ is a slow one with timescales for $\tau$ of order $1$. For such
slow-fast dynamics, it is natural to consider $f_{N}$ fixed (frozen) in
equation (20) on time scales for $\tau$ of order $1/N.$ For fixed $f_{N}$ the
dynamics for $\delta g_{N}$ is linear and can be solved. Computing then the
average of the term $\int\text{d}\mathbf{r}\,\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}$, for the asymptotic process for $\delta g_{N}$ for
fixed $f_{N}$ leads to the Guernsey–Lenard–Balescu equation, as explained in
section 3. Those computation can be found in classical textbooks
Lifshitz_Pitaevskii_1981_Physical_Kinetics .

In the following we want to go beyond these classical computations, by
estimating not just the average of the left hand side in (15),
$\int\text{d}\mathbf{r}\,\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}$, but all the cumulants of the time averages
$\int_{0}^{\Delta T}\int\text{d}\mathbf{r}\,\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}$ in order to describe the large deviations for the
process $f_{N}$. For slow-fast dynamics, the theory for the large deviations
of the effective evolution of the slow variable is a classical one both in
theoretical physics (see for instance Bouchet_Grafke_Tangarife_Vanden-
Eijnden_2015_largedeviations ) and mathematics. In the mathematics literature,
it is for instance treated for diffusions Freidlin_Wentzel_1984_book ;
Veretennikov , or chaotic deterministic systems kifer1992averaging ;
kifer2004averaging . The result for the path large deviations for the slow
dynamics is explained in section 2.2.2 of 2021arXiv210104455F  (see equations
(20-21)). After rescaling time $\tau=t/N$, we then have

$\mathbf{P}\left(\left\\{f_{N}\left(\mathbf{v},\tau\right)\right\\}_{0\leq\tau\leq
T}=\left\\{f\left(\mathbf{v},\tau\right)\right\\}_{0\leq\tau\leq
T}\right)\underset{N\rightarrow\infty}{\asymp}\text{e}^{-NL^{3}\text{Sup}_{p}\int_{0}^{T}\text{d}\tau\left\\{\int\text{d}\mathbf{v}\,\dot{f}p-H[f,p]\right\\}}\text{e}^{-NI_{0}\left[f_{0}\right]},$
(21)

where $I_{0}$ is a large deviation rate function for the initial conditions of
$f_{N}$, see equation (18), and with

$H\left[f,p\right]=\underset{T\rightarrow\infty}{\lim}\frac{1}{TL^{3}}\log\mathbb{E}_{f}\left[\text{exp$\left(\int_{0}^{T}\text{d}t\,\int\text{d}\mathbf{v}\,p\left(\mathbf{v}\right)\int\text{d}\mathbf{r}^{\prime}\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}^{\prime}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right)$}\right]$ (22)

and where $\mathbb{E}_{f}$ denotes the expectation on the process for $\delta
g_{N},$ evolving according to

$\frac{\partial\delta g_{N}}{\partial t}=-\mathbf{v}.\frac{\partial\delta
g_{N}}{\partial\mathbf{r}}+\frac{\partial V\left[\delta
g_{N}\right]}{\partial\mathbf{r}}.\frac{\partial f}{\partial\mathbf{v}}.$ (23)

In this equation, $f_{N}=f$ is fixed and time independent. We note that the
classical mathematical results to justify (22) would require to prove mixing
properties for the fast process, and stability of the invariant measure, that
nobody has proven yet for (23).

We note that to obtain equation (22) from equation (21) of 2021arXiv210104455F
, we have considered $f_{N}$ as a function of the $\mu$-space. Then the
conjugated momentum $p\left(\mathbf{r},\mathbf{v}\right)$ should also be a
function of the $\mu$-space and the scalar product be the one of the
$\mu$-space. However, recognizing that for homogeneous $f$, $p$ should also be
homogeneous ($p\left(\mathbf{r},\mathbf{v}\right)=p\left(\mathbf{v}\right)$),
and performing trivial integration over $\mathbf{r}$ leads to (22). The
$L^{3}$ factor in the large deviation principle (21) also comes from a trivial
integration over $\mathbf{r}$ of $\dot{f}p$. In the definition of $H$, in (22)
we have divided the scaled cumulant generating function by $L^{3}$ for
convenience, such that the action in (21) appears as a natural action for
homogenous distributions.

The goal of the following sections and the contribution of this work is to
obtain an explicit expression for (22).

###  3.2 The quasi-stationary Gaussian process for $\delta g_{N}$

In order to compute (22), we need to estimate averages over the stochastic
process which corresponds to generic sets of initial condition for $\delta
g_{N}$, and where $\delta g_{N}$ satisfies equation (23). We first note that
for fixed $f$, equation (23) is linear. If the set of initial conditions for
$\delta g_{N}(t=0)$ is a Gaussian random variable, then the stochastic process
$\left\\{\delta g_{N}(t)\right\\}_{t\geq 0}$ will be a Gaussian process.
Several recent mathematical works lancellotti2009fluctuations ;
lancellotti2010vlasov ; lancellotti2016time ; paul2019size ; velazquez2018two
; duerinckx2021size  discuss some properties of the Gaussian process of the
fluctuations close to a Vlasov solution. For instance lancellotti2016time
proves that, when starting from sets of Gaussian initial conditions of the
form of relevant central limit theorems, at long times, the stochastic process
converges to a statistically stationary Gaussian process. The fact that for
generic sets of initial conditions, the stochastic process of the fluctuations
$\delta g_{N}$ converges to a stochastically stationary process, which is
independent of the initial conditions, has long been understood by physicists.
This is for instance explained in §51 of
Lifshitz_Pitaevskii_1981_Physical_Kinetics , where the asymptotic stationary
process is precisely characterized. This striking convergence result is
related to the Landau damping and the fact that we deal with particle systems.
The work Bouchet:2004_PRE_StochasticProcess  derives another characterization
of this stationary process, based on an integral equation, and illustrates
numerically the convergence. In the following we will thus consider averages
in equation (22) as averages over this stationary Gaussian process. Such
stationary averages are denoted $\mathbb{E}_{S}$.

We do not reproduce the classical and lengthy computations of the correlation
functions of this stationary process, but just report the formulas which can
be found for instance in §51 of Lifshitz_Pitaevskii_1981_Physical_Kinetics .
The potential autocorrelation function are homogeneous because of the space
translation symmetry. Then

$\mathbb{E}_{S}\left(V\left[\delta
g_{N}\right]\left(\mathbf{r}_{1},t_{1}\right)V\left[\delta
g_{N}\right]\left(\mathbf{r}_{2},t_{2}\right)\right)=\mathcal{C}_{VV}\left(\mathbf{r}_{1}-\mathbf{r}_{2},t_{1}-t_{2}\right),$

We define $\tilde{\varphi}$ the space-time Fourier transform of a function
$\varphi$ as

$\tilde{\varphi}\left(\mathbf{k},\omega\right)=\int_{\left[0,L\right]^{3}}\text{d}\mathbf{r}\int_{-\infty}^{\infty}\text{d}t\,\text{e}^{-i\left(\mathbf{k}.\mathbf{r}-\omega
t\right)}\varphi\left(\mathbf{r},t\right),$ (24)

following the same convention as in Lifshitz_Pitaevskii_1981_Physical_Kinetics
. According to equation (51.20), §51 of
Lifshitz_Pitaevskii_1981_Physical_Kinetics , with the identification $V=e\phi$
and $\hat{W}(k)=4\pi e^{2}/k^{2}$, the space-time Fourier transform of the
autocorrelation function of the potential then reads

$\widetilde{\mathcal{C}_{VV}}\left(\mathbf{k},\omega\right)=2\pi\left[\int\text{d}\mathbf{v}^{\prime}\,f\left(\mathbf{v}^{\prime}\right)\delta\left(\omega-\mathbf{k}.\mathbf{v}^{\prime}\right)\right]\frac{\hat{W}\mathbf{\left(k\right)}^{2}}{\left|\varepsilon\left[f\right]\left(\mathbf{k},\omega\right)\right|^{2}}.$
(25)

Similarly the time stationary correlation function between the potential and
distribution fluctuation is space-time homogeneous

$\mathbb{E}_{S}\left(V\left[\delta
g_{N}\right]\left(\mathbf{r}_{1},t_{1}\right)\delta
g_{N}\left(\mathbf{r}_{2},\text{$\mathbf{v}$},t_{2}\right)\right)=\mathcal{C}_{VG}\left(\mathbf{r}_{1}-\mathbf{r}_{2},t_{1}-t_{2},\mathbf{v}\right).$

According to equation (51.21) of Lifshitz_Pitaevskii_1981_Physical_Kinetics ,
its space-time Fourier transform reads

$\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}\right)=-\frac{\mathbf{k}}{\omega-\mathbf{k}.\mathbf{v}-i\eta}.\frac{\partial
f}{\partial\mathbf{v}}\left(\mathbf{v}\right)\widetilde{\mathcal{C}_{VV}}\left(\mathbf{k},\omega\right)+2\pi\frac{\hat{W}\left(\mathbf{k}\right)}{\varepsilon\left[f\right]\left(\mathbf{k},\omega\right)}f\left(\mathbf{v}\right)\delta\left(\omega-\mathbf{k}.\mathbf{v}\right).$
(26)

We also define the autocorrelation function of the distribution fluctuations

$\mathbb{E}_{S}\left(\delta
g_{N}\left(\mathbf{r}_{1},\mathbf{v}_{1},t_{1}\right)\delta
g_{N}\left(\mathbf{r}_{2},\mathbf{v}_{2},t_{2}\right)\right)=\mathcal{C}_{GG}\left(\mathbf{r}_{1}-\mathbf{r}_{2},t_{1}-t_{2},\mathbf{v}_{1},\mathbf{v}_{2}\right).$

According to equation (51.23) of Lifshitz_Pitaevskii_1981_Physical_Kinetics ,
its space-time Fourier transform reads

$\displaystyle\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)$
$\displaystyle=$ $\displaystyle
2\pi\delta\left(\mathbf{v}_{1}-\mathbf{v}_{2}\right)f\left(\mathbf{v}_{1}\right)\delta\left(\omega-\mathbf{k}.\mathbf{v}_{1}\right)$
$\displaystyle+$
$\displaystyle\frac{\widetilde{\mathcal{C}_{VV}}\left(\mathbf{k},\omega\right)}{\left(\omega-\mathbf{k}.\mathbf{v}_{1}+i\eta\right)\left(\omega-\mathbf{k}.\mathbf{v}_{2}-i\eta\right)}\mathbf{k}.\frac{\partial
f}{\partial\mathbf{v}}\left(\mathbf{v}_{1}\right)\mathbf{k}.\frac{\partial
f}{\partial\mathbf{v}}\left(\mathbf{v}_{2}\right)$ $\displaystyle-$
$\displaystyle 2\pi\hat{W}\mathbf{\left(k\right)}\mathbf{k}.\frac{\partial
f}{\partial\mathbf{v}}\left(\mathbf{v}_{1}\right)\frac{f(\mathbf{v}_{2})\delta\left(\omega-\mathbf{k}.\mathbf{v}_{2}\right)}{\varepsilon\left(\mathbf{k},\omega\right)\left(\omega-\mathbf{k}.\mathbf{v}_{1}+i\eta\right)}$
$\displaystyle-$ $\displaystyle
2\pi\hat{W}\mathbf{\left(k\right)}\mathbf{k}.\frac{\partial
f}{\partial\mathbf{v}}\left(\mathbf{v}_{2}\right)\frac{f(\mathbf{v}_{1})\delta\left(\omega-\mathbf{k}.\mathbf{v}_{1}\right)}{\varepsilon^{*}\left(\mathbf{k},\omega\right)\left(\omega-\mathbf{k}.\mathbf{v}_{2}-i\eta\right)}.$

We note that the order in the correlation functions for $V$ and $g_{N}$
matters. We have

$\mathbb{E}_{S}\left(\delta
g_{N}\left(\mathbf{r}_{1},\text{$\mathbf{v}$},t_{1}\right)V\left[\delta
g_{N}\right]\left(\mathbf{r}_{2},t_{2}\right)\right)=\mathcal{C}_{GV}\left(\mathbf{r}_{1}-\mathbf{r}_{2},t_{1}-t_{2},\mathbf{v}\right),$

with

$\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}\right)=\widetilde{\mathcal{C}_{GV}}\left(-\mathbf{k},-\omega,\mathbf{v}\right)=\widetilde{\mathcal{C}_{GV}}^{*}\left(\mathbf{k},\omega,\mathbf{v}\right).$

We also note the symmetry property for $\widetilde{\mathcal{C}_{GG}}$:
$\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)=\widetilde{\mathcal{C}_{GG}}\left(-\mathbf{k},-\omega,\mathbf{v}_{2},\mathbf{v}_{1}\right)$.
It is a consequence of the symmetry
$\mathcal{C}_{GG}\left(\mathbf{r},t,\mathbf{v}_{1},\mathbf{v}_{2}\right)=\mathcal{C}_{GG}\left(-\mathbf{r},-t,\mathbf{v}_{2},\mathbf{v}_{1}\right)$.
Moreover, since $C_{GG}$ is real, we have
$\widetilde{\mathcal{C}_{GG}}\left(-\mathbf{k},-\omega,\mathbf{v}_{2},\mathbf{v}_{1}\right)=\widetilde{\mathcal{C}_{GG}}^{*}\left(\mathbf{k},\omega,\mathbf{v}_{2},\mathbf{v}_{1}\right)$.
We thus have the symmetry

$\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)=\widetilde{\mathcal{C}_{GG}}^{*}\left(\mathbf{k},\omega,\mathbf{v}_{2},\mathbf{v}_{1}\right).$
(28)

We note that as a mere consequence of the definition of $V\left[\delta
g_{N}\right]$, we have the following relations between the two-point
correlation functions

$\displaystyle\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}_{1}\right)=\hat{W}\mathbf{\left(k\right)}\int\text{d}\mathbf{v}_{2}\,\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right),$
(29)
$\displaystyle\widetilde{\mathcal{C}_{VV}}\left(\mathbf{k},\omega\right)=\left(\hat{W}\mathbf{\left(k\right)}\right)^{2}\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right).$
(30)

##  4 Computation of the large deviation Hamiltonian

In this section, we obtain an explicit formula for the large deviation
functional of the empirical density of $N$ particles with long range
interactions, starting from equation (22). We noticed in section 3.2 that the
fluctuations of the homegeneous part empirical density are described by the
average of a quadratic form over a Gaussian stationary process. In section
4.1, we explain how this makes the computation of the Hamiltonian (22)
equivalent to the computation of a functional determinant. In section 4.2,
this functional determinant is explicitly computed, using the Szegö–Widom
theorem and an explicit computation of determinants in the space of observable
over velocity distributions.

###  4.1 The large deviation Hamiltonian as a functional Gaussian integral

Within the quasi-linear approximation, the fluctuations of the empirical
density $\delta g_{N}$ describe a stationary Gaussian process over functions
of the $\mu$-space. The goal of this subsection is to show that the
computation of the large deviation Hamiltonian is equivalent to the
computation of a Gaussian functional integral of the fluctuations of the
empirical density $\delta g_{N}$.

We consider $\mathcal{H}_{\mathbf{v}}$, the Hilbert space of complex functions
over the velocity space, with $\left\langle.,.\right\rangle$, the Hermitian
product: $\left\langle
a,b\right\rangle=\int\text{d}\mathbf{v}\,a^{*}\left(\mathbf{v}\right)b\left(\mathbf{v}\right)$.
We can conveniently express the argument of the exponential in the formula
(22) for the large deviation Hamiltonian using a spatial Fourier decomposition
of the fluctuations of the empirical density
$\delta\hat{g}\left(\mathbf{k},\mathbf{v},t\right)=\int_{\left[0,L\right]^{3}}\text{d}\mathbf{r}\,\text{e}^{-i\mathbf{k}.\mathbf{r}}\delta
g_{N}\left(\mathbf{r},\mathbf{v},t\right)$. Using this Fourier decomposition,
the definition of the potential $V\left[\delta g_{N}\right]$ and partial
integration with respect to the velocity integral, we obtain

$\int\text{d}\mathbf{v}\,p\left(\mathbf{v}\right)\int\text{d}\mathbf{r}^{\prime}\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}^{\prime}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}=\frac{1}{2}\sum_{\mathbf{k}\in\left(2\pi/L\right)\mathbb{Z}^{3}}\langle\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right),\mathbf{M}\left(\mathbf{k}\right)\left[\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)\right]\rangle,$
(31)

where we define the Hermitian operator $\mathbf{M}\left(\mathbf{k}\right)$
acting on $\varphi\in\mathcal{H}_{\mathbf{v}}$ as

$\mathbf{M}\left(\mathbf{k}\right)\left[\varphi\right]\left(\mathbf{v}_{1}\right)=\int\text{d}\mathbf{v}_{2}\,M\left(\mathbf{k};\mathbf{v}_{1},\mathbf{v}_{2}\right)\varphi\left(\mathbf{v}_{2}\right),$
(32)

with the kernel $M$ defined by

$M\left(\mathbf{k};\mathbf{v}_{1},\mathbf{v}_{2}\right)=\frac{i}{L^{6}}\hat{W}\left(\mathbf{k}\right)\mathbf{k}.\left\\{-\frac{\partial
p}{\partial\mathbf{v}}\left(\mathbf{v}_{1}\right)+\frac{\partial
p}{\partial\mathbf{v}}\left(\mathbf{v}_{2}\right)\right\\}.$ (33)

There is a factor $1/2$ on the r.h.s. of (31) because we chose to symmetrize
the expression, such that
$M\left(\mathbf{k};\mathbf{v}_{1},\mathbf{v}_{2}\right)=M\left(\mathbf{k};\mathbf{v}_{2},\mathbf{v}_{1}\right)^{*}$.
$\mathbf{M}\left(\mathbf{k}\right)$ is then an Hermitian operator.

The goal of the following of this subsection is to express the sum on the
r.h.s. of (31) as a sum of independent terms to make the computation of (22)
easier. Since for every $\mathbf{k}\in\left(2\pi/L\right)\mathbb{Z}^{3}$,
$\mathbf{M}\left(\mathbf{k}\right)$ is an Hermitian operator,
$\mathbf{M}\left(\mathbf{k}\right)^{*}=\mathbf{M}\left(-\mathbf{k}\right)$,
and

$\delta\hat{g}_{N}^{*}\left(\mathbf{k},\cdot,t\right)=\delta\hat{g}_{N}\left(\mathbf{-k},\cdot,t\right),$

we have the following relation

$\langle\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right),\mathbf{M}\left(\mathbf{k}\right)\left[\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)\right]\rangle=\langle\delta\hat{g}_{N}\left(-\mathbf{k},\cdot,t\right),\mathbf{M}\left(\mathbf{-k}\right)\left[\delta\hat{g}_{N}\left(\mathbf{-k},\cdot,t\right)\right]\rangle.$
(34)

This implies that on the r.h.s. of (31), the contribution of an index
$\mathbf{k}\in\left(2\pi/L\right)\mathbb{Z}^{3}$ will be equal to the
contribution of its negative $-\mathbf{k}$.

Because the stochastic process $\delta g_{N}(\mathbf{r},\cdot,t)$, for the
fluctuations of the distribution function is spatially homogeneous, the
stochastic process $\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)$ is
statistically mutually independent with every other
$\delta\hat{g}_{N}\left(\mathbf{k}^{\prime},\cdot,t\right)$ as long as
$\mathbf{k}^{\prime}\neq-\mathbf{k}$. Because
$\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)$ is not statistically
independent from $\delta\hat{g}_{N}\left(-\mathbf{k},\cdot,t\right)$, it is
useful to treat them together. We define
$\mathbb{Z}_{\pm}^{3}=\mathbb{Z}^{3}/Z_{2}$ the quotient of group
$\mathbb{Z}^{3}$ with $Z_{2}$, the cyclic group of order 2. In other words,
$\mathbb{Z}_{\pm}^{3}$ is the set of triplets of integers where we identify a
triplet $(a,b,c)\in\mathbb{Z}^{3}$ with its negative $(-a,-b,-c)$. Then, using
(34), the sum over $\mathbf{k}\in\left(2\pi/L\right)\mathbb{Z}^{3}$ can be
rewritten as

$\frac{1}{2}\sum_{\mathbf{k}\in\left(2\pi/L\right)\mathbb{Z}^{3}}\left\langle\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right),\mathbf{M}\left(\mathbf{k}\right)\left[\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)\right]\right\rangle=\sum_{\mathbf{k}\in\left(2\pi/L\right)\mathbb{Z}_{\pm}^{3}}\left\langle\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right),\mathbf{M}\left(\mathbf{k}\right)\left[\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)\right]\right\rangle.$
(35)

As a consequence, the r.h.s of (35) is a sum of statistically independent
terms. We can then use the fact that the expected value of a product of
independent random variables is the product of their expected values, as well
as equations (31) and (35) to obtain

$\mathbb{E}\left[\text{exp$\left(\int_{0}^{T}\text{d}t\,\int\text{d}\mathbf{v}\,p\left(\mathbf{v}\right)\int\text{d}\mathbf{r}^{\prime}\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}^{\prime}}.\frac{\partial\delta
g_{N}}{\partial\mathbf{v}}\right)$}\right]=\prod_{\mathbf{k}\in\left(2\pi/L\right)\mathbb{Z}_{\pm}^{3}}\mathbb{E}\left[\exp\left(\int_{0}^{T}\text{d}t\,\langle\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right),\mathbf{M}\left(\mathbf{k}\right)\left[\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)\right]\rangle\right)\right].$
(36)

We can then go back to (22) using (34) and (36) to express the large deviation
Hamiltonian as a sum over the wavevectors

$H\left[f,p\right]=\sum_{\mathbf{k}\in\left(2\pi/L\right)\mathbb{Z}^{3}}\hat{H}\left[f,p\right]\left(\mathbf{k}\right),$
(37)

where

$\hat{H}\left[f,p\right]\left(\mathbf{k}\right)=\underset{T\rightarrow\infty}{\lim}\frac{1}{2TL^{3}}\log\mathbb{E}\left[\text{exp$\left(\int_{0}^{T}\text{d}t\,\left\langle\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right),\mathbf{M}\left(\mathbf{k}\right)\left[\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)\right]\right\rangle\right)$}\right].$
(38)

###  4.2 Application of the Szegö–Widom theorem

The computation of (37-38) requires to estimate large time large deviations of
a quadratic functional of a Gaussian stochastic process. More precisely, the
Gaussian process involved in (38) is the stochastic process of the
$\mathbf{k}$-th Fourier mode of the fluctuations of the empirical density
$\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)$, and the quadratic
functional is defined by the Hermitian operator
$\mathbf{M}\left(\mathbf{k}\right)$ (32). Since
$\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)$ is a Gaussian process, it
is possible to compute (38) via functional determinants. Thanks to the
Szegö–Widom theorem, it is possible to evaluate the asymptotics of this
Fredholm determinant in terms of much simpler determinants of an operator on
$\mathcal{H}_{\mathbf{v}}$. This program was first implemented in Bouchet-
Roger-Zaboronski , with a nice application to a model inspired by 2D and
geophysical turbulence. In appendix A we explain the details of this program,
easily adapting Bouchet-Roger-Zaboronski , for Gaussian processes with complex
variables. The result (58) of the appendix A, adapted to the case where the
Hilbert space is $\mathcal{H}_{\mathbf{v}}$, reads

$\log\mathbb{E}\left[\text{exp$\left(\int_{0}^{T}\text{d}t\left\langle\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right),\mathbf{M}\left(\mathbf{k}\right)\left[\delta\hat{g}_{N}\left(\mathbf{k},\cdot,t\right)\right]\right\rangle\right)$}\right]\underset{T\rightarrow\infty}{\sim}-\frac{T}{2\pi}\int\text{d}\omega\,\log\underset{\mathcal{H}_{\mathbf{v}}}{\det}\left(u_{\mathbf{k},\omega}\right),$
(39)

where, for any $\mathbf{k}$ and $\omega$, and
$\varphi\in\mathcal{H}_{\mathbf{v}}$,
$u_{\mathbf{k,\omega}}\left[\varphi\right]$ is defined by

$\displaystyle
u_{\mathbf{k},\omega}\left[\varphi\right]\left(\mathbf{v}_{1}\right)$
$\displaystyle=$
$\displaystyle\varphi\left(\mathbf{v}_{1}\right)+\int\text{d}\mathbf{v}_{2}\text{d}\mathbf{v}_{3}\,M\left(\mathbf{k};\mathbf{v}_{1},\mathbf{v}_{2}\right)\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{2},\mathbf{v}_{3}\right)\varphi\left(\mathbf{v}_{3}\right).$

$u_{\mathbf{k},\omega}$ is a linear operator of $\mathcal{H}_{\mathbf{v}}$.
The subscript $\mathcal{H}_{\mathbf{v}}$ in (39) indicates that the
determinant is a determinant of an operator over $\mathcal{H}_{\mathbf{v}}$.

Then, combining equations (38) and (39) yields

$\hat{H}\left[f,p\right]\left(\mathbf{k}\right)=-\frac{1}{4\pi
L^{3}}\int\text{d}\omega\,\log\underset{\mathcal{H}_{\mathbf{v}}}{\det}\left(u_{\mathbf{k},\omega}\right).$
(40)

Our next task is to obtain an explicit formula for
$\hat{H}\left[f,p\right]\left(\mathbf{k}\right)$ and thus for the full large
deviation Hamiltonian $H$ (37) is to compute
$\underset{\mathcal{H}_{\mathbf{v}}}{\det}\left(u_{\mathbf{k},\omega}\right)$.
This determinant can be easily computed once we realize that the range of
$u_{\mathbf{k},\omega}-\text{Id}$ is two-dimensional. The explicit computation
is performed in appendix B. The result reads

$\underset{\mathcal{H}_{\mathbf{v}}}{\det}\left(u_{\mathbf{k},\omega}\right)=1-\mathcal{J}\left[f,p\right]\left(\mathbf{k},\omega\right),$
(41)

with

$\mathcal{J}\left[f,p\right]\left(\mathbf{k},\omega\right)=-2\int\text{d}\mathbf{v}_{1}\,\mathbf{k}.\frac{\partial
p}{\partial\mathbf{v}_{1}}\Im\left(\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}_{1}\right)\right)\\\
-\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\mathbf{k}.\frac{\partial
p}{\partial\mathbf{v}_{1}}\mathbf{k}.\frac{\partial
p}{\partial\mathbf{v}_{2}}\left\\{\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}_{1}\right)\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}_{2}\right)^{*}\right.\\\
-\left.\widetilde{\mathcal{C}_{VV}}\left(\mathbf{k},\omega\right)\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)\right\\}.$
(42)

Using the expressions of the two-point correlation functions (25-3.2), we
obtain that

$\mathcal{J}\left[f,p\right]=\mathcal{L}\left[f,p\right]+Q\left[f,p,p\right],$
(43)

where $\mathcal{L}$ depends linearly on $p$ and $Q$ depends on $p$ as a
quadratic form. We have

$\mathcal{L}\left[f,p\right]\left(\mathbf{k},\omega\right)=4\pi\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right):\frac{\partial
p}{\partial\mathbf{v}_{1}}\left\\{\frac{\partial
f}{\partial\mathbf{v}_{2}}f(\mathbf{v}_{1})-f(\mathbf{v}_{2})\frac{\partial
f}{\partial\mathbf{v}_{1}}\right\\}$ (44)

and

$Q\left[f,p,q\right]\left(\mathbf{k},\omega\right)=2\pi\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right):\left\\{\frac{\partial
p}{\partial\mathbf{v}_{1}}\frac{\partial q}{\partial\mathbf{v}_{1}}\right.\\\
+\left.\frac{\partial p}{\partial\mathbf{v}_{2}}\frac{\partial
q}{\partial\mathbf{v}_{2}}-\frac{\partial
p}{\partial\mathbf{v}_{1}}\frac{\partial
q}{\partial\mathbf{v}_{2}}-\frac{\partial
p}{\partial\mathbf{v}_{2}}\frac{\partial
q}{\partial\mathbf{v}_{1}}\right\\}f(\mathbf{v}_{1})f(\mathbf{v}_{2}),$ (45)

with

$\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)=\pi\frac{\mathbf{k}\mathbf{k}\hat{W}(\mathbf{k})^{2}}{\left|\varepsilon\left[f\right]\left(\mathbf{k},\omega\right)\right|^{2}}\delta\left(\omega-\mathbf{k}.\mathbf{v}_{1}\right)\delta\left(\omega-\mathbf{k}.\mathbf{v}_{2}\right).$
(46)

We note that the tensor $\mathbf{A}$ is related to the tensor $\mathbf{B}$ of
the Balescu–Guernsey–Lenard equation (9):

$\mathbf{B}\left[f\right]\left(\mathbf{v}_{1},\mathbf{v}_{2}\right)=\frac{1}{L^{3}}\sum_{\mathbf{k}}\int\text{d}\omega\,\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right),$

and that it shares all of its properties: it is symmetric as a tensor, it is
symmetric in its velocities argument

$\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)=\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)$

(momentum conservation), and we have

$\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right).\left(\mathbf{v}_{1}-\mathbf{v}_{2}\right)=0$

(energy conservation). These properties are related to the conservation laws
of the physical system, as we will see in section 5.1. Using
$\varepsilon\left[f\right]\left(-\mathbf{k},-\omega\right)=\varepsilon^{*}\left[f\right]\left(\mathbf{k},\omega\right)$,
we also have

$\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)=\mathbf{A}\left[f\right]\left(-\mathbf{k},-\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right).$

$\mathbf{A}$ also has a symmetry property related to the time reversal
symmetry. Recalling that
$I\left[f\right](\mathbf{v})=f\left(-\mathbf{v}\right)$ is the velocity
inversion involution, we recall that
$\varepsilon\left[I\left[f\right]\right]\left(\mathbf{k},-\omega\right)=\varepsilon^{*}\left[f\right]\left(\mathbf{k},\omega\right)$
and as a consequence

$\mathbf{A}\left[I\left[f\right]\right]\left(\mathbf{k},-\omega,-\mathbf{v}_{1},-\mathbf{v}_{2}\right)=\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right).$

We will discuss more deeply this property in section 5.2.

Using equations (37), (40) and (41) we obtain an explicit formula for the
large deviation Hamiltonian

$H\left[f,p\right]=-\frac{1}{4\pi
L^{3}}\sum_{\mathbf{k}}\int\text{d}\omega\,\log\left\\{1-\mathcal{J}\left[f,p\right]\left(\mathbf{k},\omega\right)\right\\},$
(47)

where $\mathcal{J}\left[f,p\right]\left(\mathbf{k},\omega\right)$ is defined
in equations ((43)-45).

As a conclusion, in this section, we have established the path large deviation
principle

$\mathbf{P}\left(\left\\{f_{N}(\tau)\right\\}_{0\leq\tau\leq
T}=\left\\{f(\tau)\right\\}_{0\leq\tau\leq
T}\right)\underset{N\rightarrow\infty}{\asymp}\text{e}^{-NL^{3}\int_{0}^{T}\text{d}\tau\,\text{Sup}_{p}\left\\{\int\text{d}\mathbf{v}\,\dot{f}p-H[f,p]\right\\}}\text{e}^{-NI_{0}\left[f\left(\tau=0\right)\right]},$

where $H$ is given by (47) and where $\tau=t/N$.

Density-current formulation of the large deviation principle. We define the
current as

$\mathbf{j}_{N}\left(\mathbf{v},t\right)=-\frac{1}{NL^{3}}\int\text{d}\mathbf{r}\,\left(\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}}\delta g_{N}\right).$

In appendix D, we prove that the large deviation principle (4.2) is equivalent
to a empirical density-current formulation:

$\mathbf{P}\left(\left\\{f_{N}(\tau),\mathbf{j}_{N}\left(\tau\right)\right\\}_{0\leq
t\leq T}=\left\\{f(\tau),\mathbf{j}\left(\tau\right)\right\\}_{0\leq t\leq
T}\right)\underset{N\rightarrow\infty}{\asymp}\text{e}^{-N\mathcal{A}\left[f,\mathbf{j}\right]}\text{e}^{-NI_{0}\left[f\left(\tau=0\right)\right]},$

where $\mathbf{j}_{N}\left(\tau\right)$ should be interpreted as a time-
averaged current after time rescaling, with

$\mathcal{A}\left[f,\mathbf{j}\right]=\begin{cases}\begin{array}[]{ll}L^{3}\int_{0}^{T}\text{d}\tau\,\tilde{L}\left[f,\mathbf{j}\right]&\text{if\,\,\,}\text{$\dot{f}$+$\frac{\partial}{\partial\mathbf{v}}\cdot$$\mathbf{j}$=0,}\\\
+\infty&\text{otherwise}.\end{array}\end{cases}$

and where
$\tilde{L}\left[f,\mathbf{j}\right]=\underset{\mathbf{E}}{\text{Sup}}\left\\{\int\text{d}\mathbf{v}\,\mathbf{j}\cdot\mathbf{E}-\tilde{H}[f,\mathbf{E}]\right\\},$
and $\tilde{H}$ is defined by $H\left[f,p\right]=\tilde{H}\left[f,\partial
p/\partial\mathbf{v}\right]$.

##  5 Properties of the large deviation Hamiltonian

In this section we check that the large deviation Hamiltonian (47) satisfies
all the expected symmetry properties. In section 5.1, we check that the
Hamiltonian (47) is consistent with the mass, momentum and energy conservation
laws. In section 5.2, we show that the Hamiltonian (47) has a time-reversal
symmetry, and has the negative of the entropy, with conservation law
constraints and up to constants, as a quasipotential.

###  5.1 Conservation laws

It is a classical exercise to prove that any conservation law is equivalent to
a symmetry property of the large deviation Hamiltonian, see for instance
section 7.3.2 of bouchet2020boltzmann . From section 7.3.2 of
bouchet2020boltzmann , we know that a functional $C[f]$ is a conserved
quantity of the large deviation principle (21) if and only if for any $f$ and
$p$

$\int\text{d}\mathbf{v}\,\frac{\delta H}{\delta
p\left(\mathbf{v}\right)}\frac{\delta C}{\delta f\left(\mathbf{v}\right)}=0,$
(48)

or equivalently, if for any $f$, $p$ and $\alpha\in\mathbb{R}$:

$H[f,p]=H\left[f,p+\alpha\frac{\delta C}{\delta f}\right].$ (49)

We will need the expression of the functional derivative of the Hamiltonian
$H$ with respect to its conjugate momentum $p$ throughout this section. It
reads

$\frac{\delta H}{\delta p\left(\mathbf{v}\right)}[f,p]=\frac{1}{4\pi
L^{3}}\sum_{\mathbf{k}}\int\text{d}\omega\,\frac{\frac{\delta\mathcal{J}}{\delta
p\left(\mathbf{v}\right)}[f,p]\left(\mathbf{k},\omega\right)}{1-\mathcal{J}\left[f,p\right]\left(\mathbf{k},\omega\right)},$
(50)

with

$\frac{\delta\mathcal{J}}{\delta
p\left(\mathbf{v}\right)}[f,p]\left(\mathbf{k},\omega\right)=-4\pi\int\text{d}\mathbf{v}_{2}\,\frac{\partial}{\partial\mathbf{v}}\left\\{\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v},\mathbf{v}_{2}\right)\left[\frac{\partial
f}{\partial\mathbf{v}_{2}}f(\mathbf{v})-\frac{\partial
f}{\partial\mathbf{v}}f(\mathbf{v}_{2})+2f(\mathbf{v})f(\mathbf{v}_{2})\left(\frac{\partial
p}{\partial\mathbf{v}}-\frac{\partial
p}{\partial\mathbf{v}_{2}}\right)\right]\right\\}.$ (51)

#### Mass conservation.

The conservation of the total mass $M\left[f\right]=\int\text{d}\mathbf{v}\,f$
is immediately visible from equation (49) as $H$ only depends on the
derivative of the conjugated momentum $p$.

#### Momentum conservation.

We define the total momentum
$\mathbf{P}\left[f\right]=\int\text{d}\mathbf{v}\,\mathbf{v}f.$ It follows
that $\frac{\delta\mathbf{P}}{\delta f\left(\mathbf{v}\right)}=\mathbf{v}$.
Using equation (51) and partial integration, the relation

$\int\text{d}\mathbf{v}_{1}\frac{\delta\mathcal{J}}{\delta
p\left(\mathbf{v}_{1}\right)}[f,p]\left(\mathbf{k},\omega\right)\frac{\delta\mathbf{P}}{\delta
f\left(\mathbf{v}_{1}\right)}=0$

is a direct consequence of the symmetry
$\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)=\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{2},\mathbf{v}_{1}\right).$
Then, using the relation (50) between the functional derivatives of
$\mathcal{J}$ and $H$, we obtain

$\int\text{d}\mathbf{v}_{1}\frac{\delta H}{\delta
p\left(\mathbf{v}_{1}\right)}[f,p]\left(\mathbf{k},\omega\right)\frac{\delta\mathbf{P}}{\delta
f\left(\mathbf{v}_{1}\right)}=0.$

We have thus checked that the large deviation principle conserves momentum.

The conservation of mass and momentum should have been expected as momentum
and mass conservations were already granted from the expression of the
Hamiltonian (22), as a direct consequence of mass and momentum conservations
for $f_{N}$ that can be deduced from either equation (15) or equation 19) .

#### Energy conservation.

We define the total kinetic energy
$E\left[f\right]=\int\text{d}\mathbf{v}\,\frac{\mathbf{v}^{2}}{2}f.$ It
follows that $\frac{\delta E}{\delta
f\left(\mathbf{v}\right)}=\mathbf{v}^{2}/2$. Using equation (51) and partial
integration, one can check that the relation

$\int\text{d}\mathbf{v}_{1}\frac{\delta\mathcal{J}}{\delta
p\left(\mathbf{v}_{1}\right)}[f,p]\left(\mathbf{k},\omega\right)\frac{\delta
E}{\delta f\left(\mathbf{v}_{1}\right)}=0$

is a direct consequence of the following symmetries of the tensor:

$\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right).\left(\mathbf{v}_{1}-\mathbf{v}_{2}\right)=0,$

and

$\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)=\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{2},\mathbf{v}_{1}\right).$

Then, using the relation (50) between the functional derivatives of
$\mathcal{J}$ and $H$, we obtain

$\int\text{d}\mathbf{v}_{1}\frac{\delta H}{\delta
p\left(\mathbf{v}_{1}\right)}\left(\mathbf{k},\omega\right)\frac{\delta
E}{\delta f\left(\mathbf{v}_{1}\right)}=0.$

From the result (48) we deduce that the large deviation principle conserves
the kinetic energy.

The conservation of the kinetic energy is not a trivial consequence of
equation (15) or equation (19). Indeed, from equation (15) or equation (19),
at any time some energy can be exchanged between the kinetic part
$\int\text{d}\mathbf{v}\,\frac{\mathbf{v}^{2}}{2}f_{N}$ and the potential part
related to $\delta g_{N}$. However
$\int_{0}^{T}\text{d}t\int\text{d}\mathbf{v}\,\frac{\mathbf{v}^{2}}{2}\frac{\partial
f_{N}}{\partial t}$ is equal to the negative of the variations of the
potential energy. Then, over any time $T$, these variations should remain
bounded, for the system to stay close to the set of homogenous solutions. As a
consequence, in accordance with our hypothesis of spatial homogeneity,
$\underset{T\rightarrow\infty}{\lim}\frac{1}{T}\int_{0}^{T}\text{d}t\int\text{d}\mathbf{v}\,\frac{\mathbf{v}^{2}}{2}\frac{\partial
f_{N}}{\partial t}=0$. This is the reason why we should have expected the
conservation of kinetic energy by the large deviation principle. The
conservation of kinetic energy by the large deviation principle, which is a
conservation for the slow effective dynamics for the empirical density, should
thus be interpreted as a conservation for time averages for the fast process.
If the system became inhomogeneous, this conservation could be broken.

###  5.2 Time-reversal symmetry, quasipotential, and entropy

For the Hamiltonian dynamics (5), we consider the microcanonical measure with
fixed energy $E$ and momentum fixed and equal to zero, and denote
$\mathbb{E}_{m}$ averages with respect to the microcanonical measure. We
expect the stationary probability to observe $f_{N}=f$, to satisfy a large
deviation principle

$\mathbb{E}_{m}\left[\delta\left(f_{N}-f\right)\right]\underset{N\rightarrow\infty}{\asymp}\exp\left\\{-NU\left[f\right]\right\\},$
(52)

where this large deviation principle defines the quasipotential $U$.

From classical equilibrium statistical mechanics considerations, for this
system with long range interactions, it is easy to justify that the
quasipotential is

$U\left[f\right]=\begin{cases}\begin{array}[]{ll}-\frac{S\left[f\right]}{k_{B}}+\frac{S_{m}(E)}{k_{B}}&\text{if\,\,\,}\int\text{d}\mathbf{v}\,f=1,\,\,\,\int\text{d}\mathbf{v}\,\mathbf{v}f=0,\,\,\,\text{and}\,\,\,\int\text{d}\mathbf{v}\,\frac{\mathbf{v}^{2}}{2}=E;\\\
+\infty&\text{otherwise},\end{array}\end{cases}$ (53)

where

$S\left[f\right]=-k_{B}\int\text{d}\mathbf{v}\,f\log f$

is the entropy of the macrostate $f$ and

$S_{m}(E)=-k_{B}\inf_{f}\left\\{\int f\log
f\left|\int\text{d}\mathbf{v}\,f=1,\,\,\,\int\text{d}\mathbf{v}\,\mathbf{v}f=0,\,\,\,\text{and}\,\,\,\int\text{d}\mathbf{v}\,\frac{\mathbf{v}^{2}}{2}=E\right.\right\\}.$

is the equilibrium entropy. We have
$S_{m}(E)=k_{B}\left[3\log(E)/2+3\log(4\pi)/2+3/2\right]$.

It is also classically known that the Hamiltonian dynamics (5) is time-
reversible: the dynamics is symmetric by the change of variable
$(t,\mathbf{r}_{n},\mathbf{v}_{n})\rightarrow(-t,\mathbf{r}_{n},-\mathbf{v}_{n})$.
This is equivalent to say that if
$\left\\{\mathbf{r}_{n}(t),\mathbf{v}_{n}(t)\right\\}_{t\in\left[0,T\right]}$
is a solution of the Hamiltonian dynamics, then
$\left\\{\mathbf{r}_{n}(T-t),-\mathbf{v}_{n}(T-t)\right\\}_{t\in\left[0,T\right]}$
is also a solution. In order to take into account the change of sign for the
velocity, we define the linear operator on the set of function of the velocity
$I\left[f\right]\left(\mathbf{v}\right)=f\left(-\mathbf{v}\right)$. We note
that $I$ is an involution: $I^{2}=\text{Id}$. From the time reversal symmetry
for the Hamiltonian dynamical system, it is straightforward to conclude that
the stochastic process for the empirical density $f_{N}$ should verify a
generalized detailed balance symmetry. This symmetry writes

$\mathbf{P}_{T}\left(f_{N}(T)=f_{2}\left|f_{N}(0)=f_{1}\right.\right)\mathbf{P}_{m}\left(f_{N}=f_{1}\right)=\mathbf{P}_{T}\left(f_{N}(T)=I\left[f_{2}\right]\left|f_{N}(0)=I\left[f_{2}\right]\right.\right)\mathbf{P}_{m}\left(f_{N}=I\left[f_{2}\right]\right),$
(54)

where $\mathbf{P}_{m}$ is the stationary measure with respect to the
microcanonical measure, $\mathbf{P}_{T}$ are the transition probabilities for
the microcanonical measure. The term “generalized” means that the symmetry
holds using the involution $I$. It is a classical exercise, see for instance
section 7.3.2 of bouchet2020boltzmann , to prove that the detailed balance
condition (54) implies a detailed balance symmetry at the level of the
Hamiltonian: for any $f$ and $p$,

$H\left[I\left[f\right],-I\left[p\right]\right]=H\left[f,p+\frac{\delta
U}{\delta f}\right].$ (55)

From the relation (53) between the quasipotential $U$ and the entropy $S$,
using the conservation law symmetries of the large deviation Hamiltonian (49)
we can conclude that the generalized detailed balance symmetry (55) is
equivalent to the symmetry: for any $f$ and $p$,

$H\left[I\left[f\right],-I\left[p\right]\right]=H\left[f,p-\frac{1}{k_{B}}\frac{\delta
S}{\delta f}\right].$ (56)

One may directly check this symmetry, from (47), using the time reversal
symmetry for $A$:

$\mathbf{A}\left[I\left[f\right]\right]\left(\mathbf{k},-\omega,-\mathbf{v}_{1},-\mathbf{v}_{2}\right)=\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right).$

It is however simpler to first note that for spatially homogeneous systems,
which is the case in this paper, one has the further symmetry :

$H\left[I\left[f\right],I\left[p\right]\right]=H\left[f,p\right].$

This symmetry can be checked starting from (47) and (43), using

$\mathbf{A}\left[I\left[f\right]\right]\left(\mathbf{k},-\omega,-\mathbf{v}_{1},-\mathbf{v}_{2}\right)=\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right),$

to conclude that

$\mathcal{J}\left[I\left[f\right],I\left[p\right]\right]\left(\mathbf{k},\omega\right)=\mathcal{J}\left[f,p\right]\left(\mathbf{k},-\omega\right).$

With this remark, we can conclude that the generalized detailed balance
condition is equivalent to: for any $f$ and $p$,

$H\left[f,-p\right]=H\left[f,p-\frac{1}{k_{B}}\frac{\delta S}{\delta
f}\right].$ (57)

This last condition is a detailed balance condition at the level of large
deviations (see for instance section 7.3.2 of bouchet2020boltzmann ). In order
to check directly (57), one can start from (47) and (43), and see that this
follows from $\mathcal{J}\left[f,p-k_{B}^{-1}\delta S/\delta
f\right]-\mathcal{J}\left[f,-p\right]=0$. One can see that this last equality
is equivalent to the relation: for any $f$ and $p$,
$\mathcal{L}\left[f,p\right]=Q\left[f,p,k_{B}^{-1}\delta S/\delta f\right],$
using (43) and that $\mathcal{L}$ is linear and $\mathcal{Q}$ quadratic with
respect to $p$. Using (44) and (45) and $\partial/\partial\mathbf{v}(\delta
S/\delta f)/k_{B}=1/f\partial f/\partial\mathbf{v}$, this is easily verified
using
$\text{$\mathbf{A}$}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)=\text{$\mathbf{A}\left[f\right]\left(\mathbf{k},\omega,\mathbf{v}_{2},\mathbf{v}_{1}\right)$}$.

As a final remark, we note that the quasipotential and the entropy are
solutions to the stationary Hamilton-Jacobi equation

$H\left[f,\frac{\delta U}{\delta
f}\right]=H\left[f,-\frac{1}{k_{B}}\frac{\delta S}{\delta f}\right]=0.$

Those are direct consequences of any of the detailed balance symmetries: (57),
(55) or (56).

In this section we have explained that $U$ (53) is the quasipotential. We have
argued that the large deviation Hamiltonian satisfies the generalized detailed
balance symmetry (56) as a consequence of the microscopic time reversibility,
and checked directly this relation from the explicit Hamiltonian equations. We
have moreover justified that that the large deviation Hamiltonian satisfies
the detailed balance symmetry (57). This proves that $U$ satisfies the
stationary Hamilton-Jacobi equation.

###  5.3 A remark on Gaussian approximations of the large deviation principle
and cumulant expansions

In a recent paper 2021arXiv210104455F  dedicated to the path large deviations
for homogeneous plasma, for fluctuations on scales $k\lambda_{D}\gg 1$ where
$\lambda_{D}$ is the Debye length and $k$ a wavenumber, we obtained a large
deviation Hamiltonian which is quadratic in $p$ featuring locally Gaussian
distribution of the large deviations. We have obtained this quadratic in $p$
Hamiltonian, either from the large deviation Hamiltonian associated with the
Boltzmann equation, or from an expansion using $k\lambda_{D}\gg 1$ of the
expression (22). The technical approach in 2021arXiv210104455F  is different.
In 2021arXiv210104455F  we used a tedious cumulant expansion. This paper
rather uses the Szegö–Widom theorem, a very efficient approach.

The present paper considers any interaction, and not just the Coulomb
interaction case, but it is clear that the result in 2021arXiv210104455F
should be recovered from the results of the present paper. In appendix C, we
show that the large deviation Hamiltonian of this paper (47), is fully
consistent with our previous perturbative result for $k\lambda_{D}\gg 1$. We
also justify an assumption we used in 2021arXiv210104455F  about the cumulant
series expansion of (22) from the formula (47) for the large deviation
Hamiltonian associated with the Balescu–Guernsey–Lenard equation.

##  6 Perspectives

The main result of this paper is the derivation of a large deviation principle
(2), for the velocity empirical density, for the Hamiltonian dynamics of $N$
particles which interact through mean-field interactions. We have obtained an
explicit formula for the large deviation Hamiltonian (3-4) and we have checked
all its symmetry properties. This result opens many mathematical and
theoretical questions, as well as interesting applications.

This large deviation result relies on natural assumptions. Some of these
assumptions are also required to establish the Balescu–Guernsey–Lenard kinetic
equation, but the hypotheses made to obtain the large deviation principle seem
stronger. The first assumption is the validity of the quasilinear
approximation: we neglected non linear terms of order $1/\sqrt{N}$ in the
equation for the fluctuations of the empirical density. This amounts to
neglecting possible effects of large deviations of the fluctuations, and
describing the fluctuation process at a Gaussian level only. The second
assumption is the convergence of the process of fluctuations to a stationary
Gaussian process and more specifically the convergence of the large time
asymptotics for the large deviation estimates over this process. A proof would
also require the study of the mixing properties for this Gaussian processes.
The mixing properties are critical to justify the Markov behavior described by
the slow-fast large deviation principle. While the proofs of these assumptions
are beyond the scope of this paper, they open very interesting questions for
both theoretical physicists and mathematicians.

Systems with long range interactions are important for many phenomena.
However, more elaborate models than the one we used in this paper could be
more appropriate to describe physical situations where rare event are
important for applications. Of special interest, would be the derivation of
large deviation principles for inhomogeneous systems with long range
interactions, for instance self-gravitating systems. This is an exciting
application, that would open the way to the study of the rare destabilization
of globular clusters or galaxies, or the formation of inhomogeneous structures
of smaller scales in self gravitating systems. Another very interesting
generalization would be for the dynamics of $N$ point-vortices for two-
dimensional hydrodynamics. Another generalization should also consider
dynamics of particles driven by stochastic forces, which generically lead to
irreversible stochastic processes. For those systems, explicit results for the
large deviation theory would be extremely useful for explaining non-
equilibrium phase transitions in two dimensional Bouchet_Simonnet_2008  and
geostrophic turbulence BOUCHET:2019:C , or in systems with long range
interactions Nardini_Gupta_Ruffo_Dauxois_Bouchet_2012_kinetic ;
NardiniGuptaBouchet-2012-JSMTE .

##  Appendix A Long time large deviations for quadratic observables of
Gaussian processes, functional determinants and the Szegö–Widom theorem for
Fredholm determinants

In this appendix, we explain how we can use the Szegö–Widom theorem in order
to evaluate the large time asymptotics of Fredholm determinants that appears
when computing the cumulant generating function of a quadratic observable of a
Gaussian process. We follow the ideas in Bouchet-Roger-Zaboronski , adapting
the discussion for the case of Gaussian processes with complex variables.

Let $Y_{t}$ be a stationary $\mathbb{C}^{n}$-valued Gaussian process with
correlation matrix $C\left(t\right)=\mathbb{E}\left(Y_{t}\otimes
Y_{0}^{*}\right)$ and with a zero relation matrix
$R(t)=\mathbb{E}\left(Y_{t}\otimes Y_{0}\right)=0$, let
$M\in\mathcal{M}_{n}\left(\mathbb{C}\right)$ be a $n\times n$ Hermitian
matrix. The aim of this appendix is to prove that

$\log\mathbb{E}\exp\left(\int_{0}^{T}\text{d}t\,Y_{t}^{*\intercal}MY_{t}\right)\underset{T\rightarrow\infty}{\sim}-\frac{T}{2\pi}\int\text{d}\omega\,\log\det\left(I_{n}-M\tilde{C}\left(\omega\right)\right),$
(58)

where $\tilde{C}\left(\omega\right)=\int_{\mathbb{R}}\text{e}^{i\omega
t}C\left(t\right)\text{d}t$ is the Fourier transform of the correlation matrix
$C\left(t\right)$ and $I_{n}$ is the $n\times n$ identity matrix. We note that
the determinant of the r.h.s. of (58) is a real number. Indeed, as $Y_{t}$ is
a stationary process, $\tilde{C}\left(\omega\right)$ and $M$ are Hermitian
matrices, then the determinant is the determinant of a Hermitian operator and
is a real number.

For pedagogical reasons, in this appendix the result (58) is stated for a
process $Y_{t}$ that takes values in a finite-dimensional space. However with
adapted hypotheses, this result can be generalized when $Y_{t}$ is a
stationary $\mathcal{H}$-valued Gaussian process, where $\mathcal{H}$ is a
Hilbert space, and where $M$ is a Hermitian operator on $\mathcal{H}$.

In section A.1, we state the Szegö–Widom theorem. In section A.2, we explain
that the left hand side of (58) is the log of the determinant of a Gaussian
integral, that this quantity can be expressed as a functional determinant for
linear operators on $L^{2}\left(\left[0,T\right],\mathbb{C}^{n}\right)$, and
that Szegö–Widom theorem reduces it to the computation of frequency integrals
of determinants of operators on the space $\mathbb{C}^{n}$, as expressed by
(58).

###  A.1 The Szegö–Widom theorem

We first define integral operators on
$L^{2}\left(\left[0,T\right],\mathbb{C}^{n}\right)$. We considers maps
$\varphi:\left[0,T\right]\rightarrow\mathbb{C}^{n}$ and
$K:\mathbb{R}\rightarrow\mathcal{M}_{n}\left(\mathbb{C}\right)$, where
$\mathcal{M}_{n}\left(\mathbb{C}\right)$ is the set of $n\times n$ complex
matrices. We define the integral operator $\mathbf{K}_{T}$ by

$\mathbf{K}_{T}\varphi\left(t\right)=\int_{0}^{T}K\left(t-s\right)\varphi\left(s\right)\text{d}s,$
(59)

$\mathbf{K}_{T}$ is a linear operator of
$L^{2}\left(\left[0,T\right],\mathbb{C}^{n}\right)$. $K$ is called the kernel
of the operator $\mathbf{K}_{T}$.

The Szegö–Widom theorem allows to compute large $T$ asymptotics of the
logarithm of the Fredholm determinant of the integral operator
$\text{Id}+\mathbf{K}_{T}$. The result is

$\log\det_{\left[0,T\right]}\left(\text{Id}+\mathbf{K}_{T}\right)\underset{T\rightarrow\infty}{\sim}\frac{T}{2\pi}\int\text{d}\omega\,\log\det\left(I_{n}+\int_{\mathbb{R}}\text{e}^{i\omega
t}K\left(t\right)\text{d}t\right),$ (60)

where $I_{n}$ is the $n\times n$ identity matrix. Whereas the determinant on
the l.h.s. of this expression, denoted by the subscript $\left[0,T\right]$ is
a Fredholm determinant, the determinant on the r.h.s. is a matrix determinant
which can be more easily computed. Further details about this theorem and its
possible applications can be found in Bouchet-Roger-Zaboronski .

###  A.2 Expectation of functionals of Gaussian processes

Let $Y_{t}$ be a $\mathbb{C}^{n}$-valued stationary Gaussian process with
correlation matrix

$C\left(t\right)=\mathbb{E}\left(Y_{t}\otimes Y_{0}^{*}\right),$

and with zero relation matrix

$R(t)=\mathbb{E}\left(Y_{t}\otimes Y_{0}\right)=0.$

We will compute the large time asymptotics of

$\mathcal{U}\left(T\right)=\log\mathbb{E}\exp\left(\int_{0}^{T}\text{d}t\,Y_{t}^{*\intercal}\mathbf{M}_{T}Y_{t}\right),$

where $\mathbf{M}_{T}$ is an integral operator on
$L^{2}\left(\left[0,T\right],\mathbb{C}^{n}\right)$ whose integral kernel is
given by $M\left(t\right)$ (see the definition (59)). We assume that for all
times $t$, $M\left(t\right)$ is a $n\times n$ Hermitian matrix. As $Y_{t}$ is
a Gaussian process we can compute the expectation as a Gaussian integral. It
is straightforward to check that

$\mathbb{E}\exp\left(\int_{0}^{T}\text{d}t\,Y_{t}^{*\intercal}\mathbf{M}_{T}Y_{t}\right)=\det_{\left[0,T\right]}\left(\text{Id}-(\mathbf{M}\mathbf{C})_{T}\right)^{-1},$

where $(\mathbf{MC})_{T}$ is the integral operator whose kernel is $(M\star
C)(t)$ the convolution product on $[0,T]$ of the kernels $M(t)$ and $C(t)$.

Then, we can deduce the following expression for $u$

$\mathcal{U}\left(T\right)=-\log\det_{\left[0,T\right]}\left(\text{Id}-(\mathbf{M}\mathbf{C})_{T}\right),$

where the determinant is the Fredholm determinant of the integral operator
$\text{Id}-(\mathbf{M}\mathbf{C})_{T}$. Generally, it is not obvious how to
compute this kind of Fredholm determinant. Fortunately, we can use the
Szegö–Widom theorem to obtain an expression for large $T$ asymptotics as a
finite-dimensional determinant. Using the result (60) from section A.1, we get

$\mathcal{U}\left(T\right)\underset{T\rightarrow\infty}{\sim}-\frac{T}{2\pi}\int\text{d}\omega\,\log\det\left(I_{n}-\int_{\mathbb{R}}\text{e}^{-i\omega
t}\left(M\star C\right)\left(t\right)\text{d}t\right).$

In the special case when $\mathbf{M}_{T}$ is a diagonal integral operator,
i.e. when its kernel is $M(t)=M\delta(t)$, we can write

$\mathcal{U}\left(T\right)\underset{T\rightarrow\infty}{\sim}-\frac{T}{2\pi}\int\text{d}\omega\,\log\det\left(I_{n}-M\int_{\mathbb{R}}\text{e}^{-i\omega
t}C\left(t\right)\text{d}t\right),$

which is the result (58). In these expressions, the determinant to be computed
on the r.h.s. is the determinant of a $n\times n$ matrix.

##  Appendix B Computation of the determinant of the operator
$u_{\mathbf{k},\omega}$

In this appendix, we compute the determinant of the operator
$u_{\mathbf{k},\omega}$, encountered in section 4.2, and defined by

$\displaystyle
u_{\mathbf{k},\omega}\left[\varphi\right]\left(\mathbf{v}_{1}\right)$
$\displaystyle=$
$\displaystyle\varphi\left(\mathbf{v}_{1}\right)-\int\text{d}\mathbf{v}_{2}\text{d}\mathbf{v}_{3}\,M\left(\mathbf{k},\mathbf{v}_{1},\mathbf{v}_{2}\right)\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{2},\mathbf{v}_{3}\right)\varphi\left(\mathbf{v}_{3}\right),$

for any $\varphi\in\mathcal{H}_{\mathbf{v}}$, $\mathcal{H}_{\mathbf{v}}$ being
the Hilbert space of complex functions over the velocity space. Using equation
(33), we can simplify this expression

$u_{\mathbf{k},\omega}\left[\varphi\right]\left(\mathbf{v}_{1}\right)=\varphi\left(\mathbf{v}_{1}\right)-i\hat{W}\left(\mathbf{k}\right)\mathbf{k}\cdot\int\text{d}\mathbf{v}_{2}\text{d}\mathbf{v}_{3}\,\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{2},\mathbf{v}_{3}\right)\left\\{\frac{\partial
p}{\partial\mathbf{v}}\left(\mathbf{v}_{2}\right)-\frac{\partial
p}{\partial\mathbf{v}}\left(\mathbf{v}_{1}\right)\right\\}\varphi\left(\mathbf{v}_{3}\right).$
(61)

We note that the operator $u_{\mathbf{k},\omega}$ has the form

$u_{\mathbf{k},\omega}:\varphi\longmapsto\varphi-\left\langle
w,\mathbf{Q}\varphi\right\rangle v-\left\langle
v,\mathbf{Q}\varphi\right\rangle w,$ (62)

where $\mathbf{Q}$ is a Hermitian operator over $\mathcal{H_{\mathbf{v}}}$,
$w$ and $v$ are complex functions over the velocity space, and
$\left\langle.,.\right\rangle$ denotes the Hermitian product: $\left\langle
a,b\right\rangle=\int\text{d}\mathbf{v}\,a^{*}\left(\mathbf{v}\right)b\left(\mathbf{v}\right)$.
The connection is made between formulas (61) and (62) by setting
$v\left(\mathbf{v}\right)=-i\mathbf{k}.\frac{\partial p}{\partial\mathbf{v}}$
, $w\left(\mathbf{v}\right)=\hat{W}\left(\mathbf{k}\right)$ and
$\mathbf{Q}\left[\phi\right]\left(\mathbf{v}_{1}\right)=\int\text{d}\mathbf{v}_{2}\,\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)\phi\left(\mathbf{v}_{2}\right)$.
Using (28), we see that $Q$ is a Hermitian operator. We note that, whenever
$\frac{\partial p}{\partial\mathbf{v}}$ is not a constant in the velocity
space, $v$ and $w$ are linearly independent.

Formula (62) shows that $u_{\mathbf{k},\omega}-\text{Id}$ is a rank two linear
operator. Then $\det_{\mathcal{H}_{\mathbf{v}}}u_{\mathbf{k},\omega}$ is the
determinant of the operator $u_{\mathbf{k},\omega}$ restricted to
span$\left(v,w\right)$:

$\det_{\mathcal{H}_{\mathbf{v}}}u_{\mathbf{k},\omega}=\begin{vmatrix}1-\left\langle
w,\mathbf{Q}v\right\rangle&-\left\langle w,\mathbf{Q}w\right\rangle\\\
-\left\langle v,\mathbf{Q}v\right\rangle&1-\left\langle
v,\mathbf{Q}w\right\rangle\end{vmatrix}.$

Then

$\det_{\mathcal{H}_{\mathbf{v}}}u_{\mathbf{k},\omega}=1-2\Re\left[\left\langle
v,\mathbf{Q}w\right\rangle\right]+\left\langle
v,\mathbf{Q}w\right\rangle\left\langle
v,\mathbf{Q}w\right\rangle^{*}-\left\langle
w,\mathbf{Q}w\right\rangle\left\langle v,\mathbf{Q}v\right\rangle.$

where we have used $\left\langle
w,\mathbf{Q}v\right\rangle=\left\langle\mathbf{Q}w,v\right\rangle=\left\langle
v,\mathbf{Q}w\right\rangle^{*}$, as $\mathbf{Q}$ is an Hermitian operator.

We can explicitly compute the determinant of (61). We have

$\left\langle
v,\mathbf{Q}v\right\rangle=\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\mathbf{k}.\frac{\partial
p}{\partial\mathbf{v}_{1}}\mathbf{k}.\frac{\partial
p}{\partial\mathbf{v}_{2}}\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right),$
$\left\langle
v,\mathbf{Q}w\right\rangle=i\int\text{d}\mathbf{v}_{1}\,\mathbf{k}.\frac{\partial
p}{\partial\mathbf{v}_{1}}\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}_{1}\right)^{*},$

and

$\left\langle
w,\mathbf{Q}w\right\rangle=\widetilde{\mathcal{C}_{VV}}\left(\mathbf{k},\omega\right),$

where $\widetilde{\mathcal{C}_{VG}}$, $\widetilde{\mathcal{C}_{VV}}$ and
$\widetilde{\mathcal{C}_{GG}}$ are the two-point correlations functions of the
quasi-linear problem computed in section 3.2, and we have used (29-30).

We conclude that

$\underset{\mathcal{H}_{\mathbf{v}}}{\det}\left(u_{\mathbf{k},\omega}\right)=1+2\int\text{d}\mathbf{v}_{1}\,\mathbf{k}.\frac{\partial
p}{\partial\mathbf{v}_{1}}\Im\left(\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}_{1}\right)\right)\\\
+\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\mathbf{k}.\frac{\partial
p}{\partial\mathbf{v}_{1}}\mathbf{k}.\frac{\partial
p}{\partial\mathbf{v}_{2}}\left\\{\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}_{1}\right)\widetilde{\mathcal{C}_{VG}}\left(\mathbf{k},\omega,\mathbf{v}_{2}\right)^{*}-\widetilde{\mathcal{C}_{VV}}\left(\mathbf{k},\omega\right)\widetilde{\mathcal{C}_{GG}}\left(\mathbf{k},\omega,\mathbf{v}_{1},\mathbf{v}_{2}\right)\right\\}.$

##  Appendix C Consistency and validation of the cumulant series expansion

In this appendix, we expand $H$ from the formula (47) in powers of $p$. This
amounts at a cumulant expansion for the statistics of the fluctuations. We use
this expansion to prove a conjecture made in the paper 2021arXiv210104455F ,
and to recover the Gaussian (order two) truncation we computed in
2021arXiv210104455F .

We expand the logarithm in formula (47) to obtain

$H\left[f,p\right]=\frac{1}{4\pi
L^{3}}\sum_{\mathbf{k}}\int\text{d}\omega\,\sum_{n=1}^{+\infty}\frac{1}{n}\left(\mathcal{J}\left[f,p\right]\left(\mathbf{k},\omega\right)\right)^{n}=\sum_{n=1}^{+\infty}H^{(n)}\left[f,p\right].$
(63)

The second equality defines $H^{(n)}$ as being the terms homogeneous of order
$n$ in $p$ in this expansion. It is the $n$-th cumulant.

We also define $\mathbf{B}^{(m)}$ as

$\mathbf{B}^{(m)}\left(\mathbf{v}_{1},\ldots,\mathbf{v}_{2m}\right)=\frac{\left(2\pi\right)^{2m}}{4\pi
mL^{3}}\sum_{\mathbf{k}}\int_{\Gamma}\text{d}\omega\,\frac{W\left(\mathbf{k}\right)^{2m}}{\left|\varepsilon\left(\mathbf{k},\omega\right)\right|^{2m}}\mathbf{k}^{\otimes
2m}\prod_{i=1}^{2m}\delta\left(\omega-\mathbf{k}.\mathbf{v}_{i}\right).$

$\mathbf{B}^{(m)}$ is a rank $2m$ tensor. $l^{(k)}$ and $q^{(k)}$ are defined
by the relations. We have
$\mathcal{J}\left[f,p\right]=\mathcal{L}\left[f,p\right]+Q\left[f,p,p\right],$
where $\mathcal{L}$ and $Q$ are defined in equations (44) and (45). We will
need to compute $\left(\mathcal{L}\left[f,p\right]\right)^{k}$, which is
$\mathcal{L}\left[f,p\right]$ to the power $k$. We define $l^{(k)}$ and
$q^{(k)}$ by

$\left(\mathcal{L}\left[f,p\right]\right)^{k}=\int\text{d}\mathbf{v}_{1}\cdots\text{d}\mathbf{v}_{2k}\,l^{(k)}\left[f,p\right]\prod_{j=1}^{k}\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{2j-1},\mathbf{v}_{2j}\right),$

and

$\left(Q\left[f,p,p\right]\right)^{k}=\int\text{d}\mathbf{v}_{1}\cdots\text{d}\mathbf{v}_{2k}\,q^{(k)}\left[f,p,p\right]\prod_{j=1}^{k}\mathbf{A}\left(\mathbf{k},\omega,\mathbf{v}_{2j-1},\mathbf{v}_{2j}\right).$

$l^{(k)}$ and $q^{(k)}$ are both tensors of order $2k$. $l^{(k)}$ depends on
$p$ as a homogeneous function of order $k$. $q^{(k)}$ depends on $p$ as a
homogeneous function of order $2k$.

In the expansion of $\left(\mathcal{J}\left[f,p\right]\right)^{n}$ using
$\mathcal{J}\left[f,p\right]=\mathcal{L}\left[f,p\right]+Q\left[f,p,p\right],$
we see that for all $m\in\left[n/2,n\right]\cap\mathbb{N}$,
$\mathcal{L}^{2m-n}\left[f,p\right]Q^{n-m}\left[f,p,p\right]$ is homogeneous
of order $n$ in $p$. Using this remark, from equation (63) we obtain

$H^{(n)}\left[f,p\right]=\sum_{m\in\left[n/2,n\right]\cap\mathbb{N}}\int\text{d}\mathbf{v}_{1}\cdots\text{d}\mathbf{v}_{2m}\,\left(\begin{array}[]{c}m\\\
2m-n\end{array}\right)\frac{1}{\left(2\pi\right)^{2m}}\mathbf{B}^{(m)}\left(\mathbf{v}_{1},\ldots,\mathbf{v}_{2m}\right):l^{(2m-n)}\left[f,p\right]q^{(n-m)}\left[f,p,p\right],$
(64)

where the symbol $":"$ means a contraction of a tensor of order $2m$ with
another tensor of order $2m$.

This result ensures that as soon as $n>2,$ $H^{(n)}$ only includes terms
proportional to the tensors $\mathbf{B}^{(m)}$ with $m\geq n/2\geq 2.$ In
2021arXiv210104455F , we used a conjecture on $H^{(n)}$ to justify the
Gaussian truncation of the cumulant series expansion of (22). The conjecture
was that only the two first cumulants $H^{(1)}$ and $H^{(2)}$ do involve the
tensor $\mathbf{B}=\mathbf{B}^{(1)}$, whereas all the other cumulants
$H^{(n)}$ for $n>2$ only involve tensors $\mathbf{B}^{(m)}$ with $m\geq 2$.
Expansion thus (64) justifies this conjecture.

In 2021arXiv210104455F , from a cumulant series expansion, we obtained that

$H\left[f,p\right]=H_{\text{quad}}\left[f,p\right]+O\left(p^{2}\right),$

where

$H_{\text{quad}}\left[f,p\right]=\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\mathbf{B}\left(\mathbf{v}_{1},\mathbf{v}_{2}\right):\frac{\partial
p}{\partial\mathbf{v}_{1}}\left\\{\frac{\partial
f}{\partial\mathbf{v}_{2}}f(\mathbf{v}_{1})-f(\mathbf{v}_{2})\frac{\partial
f}{\partial\mathbf{v}_{1}}\right\\}\\\
+\int\text{d}\mathbf{v}_{1}\text{d}\mathbf{v}_{2}\,\mathbf{B}\left(\mathbf{v}_{1},\mathbf{v}_{2}\right):\left\\{\frac{\partial
p}{\partial\mathbf{v}_{1}}\frac{\partial
p}{\partial\mathbf{v}_{1}}-\frac{\partial
p}{\partial\mathbf{v}_{1}}\frac{\partial
p}{\partial\mathbf{v}_{2}}\right\\}f(\mathbf{v}_{1})f(\mathbf{v}_{2})\\\
+\int\text{d$\mathbf{v}_{1}$}\text{d}\mathbf{v}_{2}\text{d$\mathbf{v}_{3}$}\text{d}\mathbf{v}_{4}\,\mathbf{B}^{(2)}\left(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4}\right):\frac{\partial
p}{\partial\mathbf{v}_{1}}\frac{\partial
p}{\partial\mathbf{v}_{2}}\left\\{f(\mathbf{v}_{1})f(\mathbf{v}_{2})\frac{\partial
f}{\partial\mathbf{v}_{3}}\frac{\partial f}{\partial\mathbf{v}_{4}}\right.\\\
\left.-2f(\mathbf{v}_{1})\frac{\partial
f}{\partial\mathbf{v}_{2}}f(\mathbf{v}_{3})\frac{\partial
f}{\partial\mathbf{v}_{4}}+\frac{\partial
f}{\partial\mathbf{v}_{1}}\frac{\partial
f}{\partial\mathbf{v}_{2}}f(\mathbf{v}_{3})f(\mathbf{v}_{4})\right\\},$ (65)

and where $O\left(p^{2}\right)$ designates terms that are of order more than
two in the conjugate momentum $p$, and the symbol $":"$ means a contraction of
two tensors of order 2 and 4, for the two first lines and the third line,
respectively. We see that $H^{(1)}+H^{(2)}=H_{\text{quad}}.$ We conclude that
(47) is consistent with the quadratic approximation of the large deviation
Hamiltonian we obtained in 2021arXiv210104455F .

##  Appendix D Current formulation of the large deviation principle

Because the particle number is conserved, it is clear that the dynamics of the
empirical density has a conservative form $\frac{\partial f_{N}}{\partial
t}+\frac{\partial}{\partial\mathbf{v}}\cdot\mathbf{j}_{N}=0$. For the
microscopic dynamics (before time averaging), this is a consequence of
equations (15) or (19) with

$\mathbf{j}_{N}\left(\mathbf{v},t\right)=-\frac{1}{NL^{3}}\int\text{d}\mathbf{r}\,\left(\frac{\partial
V\left[\delta g_{N}\right]}{\partial\mathbf{r}}\delta g_{N}\right).$

After time averaging, we could have obtained the path large deviations by
studying the large deviations of the time averaged current. Alternatively, we
can rephrase our large deviation principle as a large deviation principle for
the current, through a change of variable. This is the subject of this
appendix.

The conservative nature of the dynamics is visible because the large deviation
Hamiltonian $H$ (47) does depend on the conjugate momentum $p$ only through
its gradient $\partial p/\partial\mathbf{v}$. We define $\tilde{H}$ as
$\tilde{H}\left[f,\partial p/\partial\mathbf{v}\right]=H\left[f,p\right].$ We
start from the definition of the large deviation Lagrangian

$L\left[f,\dot{f}\right]=\text{Sup}_{p}\left\\{\int\text{d}\mathbf{v}\,\dot{f}p-H[f,p]\right\\}.$

Writing $\dot{f}$ as the divergence of a current
$\dot{f}+\partial/\partial\mathbf{v}\cdot\mathbf{j}=0$, we have

$L\left[f,\dot{f}\right]=\underset{\left\\{\mathbf{j}\mid\dot{f}+\frac{\partial}{\partial\mathbf{v}}\cdot\mathbf{j}=0\right\\}}{\text{Sup}}\text{Sup}_{p}\left\\{-\int\text{d}\mathbf{v}\,p\frac{\partial}{\partial\mathbf{v}}\cdot\mathbf{j}-H[f,p]\right\\}.$

Using $H\left[f,p\right]=\tilde{H}\left[f,\partial
p/\partial\mathbf{v}\right]$, and integrating by part, we have

$L\left[f,\dot{f}\right]=\underset{\left\\{\mathbf{j}\mid\dot{f}+\frac{\partial}{\partial\mathbf{v}}\cdot\mathbf{j}=0\right\\}}{\text{Sup}}\tilde{L}\left[f,\mathbf{j}\right]$

with

$\tilde{L}\left[f,\mathbf{j}\right]=\underset{\mathbf{E}}{\text{Sup}}\left\\{\int\text{d}\mathbf{v}\,\mathbf{j}\cdot\mathbf{E}-\tilde{H}[f,\mathbf{E}]\right\\}.$

where $\mathbf{E}$ designates the conjugate quantity of the current
$\mathbf{j}$.

We thus have the large deviation principle

$\mathbf{P}\left(\left\\{f_{N}(\tau)\right\\}_{0\leq\tau\leq
T}=\left\\{f(\tau)\right\\}_{0\leq\tau\leq
T}\right)\underset{N\rightarrow\infty}{\asymp}\text{e}^{-NL^{3}\underset{\left\\{\mathbf{j}\mid\dot{f}+\frac{\partial}{\partial\mathbf{v}}\cdot\mathbf{j}=0\right\\}}{\text{Sup}}\int_{0}^{T}\text{d}\tau\,\tilde{L}\left[f,\mathbf{j}\right]}\text{e}^{-NI_{0}\left[f\left(\tau=0\right)\right]}.$
(66)

We note that we can also write a large deviation principle for the joint
probability of the empirical density and the time averaged current
$\mathbf{j}_{N}\left(\tau\right)$

$\mathbf{P}\left(\left\\{f_{N}(\tau),\mathbf{j}_{N}\left(\tau\right)\right\\}_{0\leq
t\leq T}=\left\\{f(\tau),\mathbf{j}\left(\tau\right)\right\\}_{0\leq t\leq
T}\right)\underset{N\rightarrow\infty}{\asymp}\text{e}^{-N\mathcal{A}\left[f,\mathbf{j}\right]}\text{e}^{-NI_{0}\left[f\left(\tau=0\right)\right]},$

with

$\mathcal{A}\left[f,\mathbf{j}\right]=\begin{cases}\begin{array}[]{ll}L^{3}\int_{0}^{T}\text{d}\tau\,\tilde{L}\left[f,\mathbf{j}\right]&\text{if\,\,\,}\text{$\dot{f}$+$\frac{\partial}{\partial\mathbf{v}}\cdot$$\mathbf{j}$=0,}\\\
+\infty&\text{otherwise}.\end{array}\end{cases}$

###### Acknowledgements.

We warmly thank Oleg Zaboronski for first teaching us the Szegö–Widom theorem,
used in this paper. We thank Julien Barré, Charles-Edouard Bréhier, Jules
Guioth, and Julien Reygner for useful comments on our manuscript. The research
leading to this work was supported by a sub-agreement from the Johns Hopkins
University with funds provided by Grant No. 663054 from Simons Foundation. Its
contents are solely the responsibility of the authors and do not necessarily
represent the official views of Simons Foundation or the Johns Hopkins
University. We thank the two anonymous reviewers for their useful
recommendations, which helped us to improve our first version of our
manuscript.

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