# Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time
propagation of chaos for weakly interacting diffusions

Matías G<EMAIL_ADDRESS>[ Rishabh S.
<EMAIL_ADDRESS>[ Grigorios A.
<EMAIL_ADDRESS>[ Scott A.
<EMAIL_ADDRESS>[ Department of Mathematics, The University of
Texas at Austin  Max-Planck-Institut für Mathematik in den Naturwissenschaften
Department of Mathematics, Imperial College London  Academy of Mathematics and
Systems Sciences, Chinese Academy of Sciences

60K35,

82B26,

39B62,

Interacting particle systems,

log Sobolev inequalities,

phase transitions,

propagation of chaos,

###### keywords:

[class=MSC2020]

###### keywords:

, , and

Abstract In this article, we study the mean field limit of weakly interacting
diffusions for confining and interaction potentials that are not necessarily
convex. We explore the relationship between the large $N$ limit of the
constant in the logarithmic Sobolev inequality (LSI) for the $N$-particle
system and the presence or absence of phase transitions for the mean field
limit. The non-degeneracy of the LSI constant is shown to have far reaching
consequences, especially in the context of uniform-in-time propagation of
chaos and the behaviour of equilibrium fluctuations. Our results extend
previous results related to unbounded spin systems and recent results on
propagation of chaos using novel coupling methods. As incidentals, we provide
concise and, to our knowledge, new proofs of a generalised form of Talagrand’s
inequality and of quantitative propagation of chaos by employing techniques
from the theory of gradient flows, specifically the Riemannian calculus on the
space of probability measures.

##  1 Introduction

Interacting particle systems have attracted a lot of attention in recent years
since they appear in diverse areas ranging from plasma physics and galactic
dynamics to machine learning and optimization. For systems of identical (or
exchangeable) particles in which the pair-wise interactions scale like the
inverse of the number of particles, it is possible to pass to the mean field
limit and obtain a coarse-grained description of the system via a nonlinear
nonlocal PDE that governs the evolution of the one-particle density. In this
paper, we consider systems of weakly interacting diffusions driven by pair-
wise interactions, confinement and independent Brownian motions (see (2.1) ).
In this case the mean field PDE is the so-called McKean–Vlasov equation.

A natural problem that one would like to address is how to obtain sharp
quantitative estimates on the rate at which the empirical measure of the
particle system converges to the mean field limit, as the number of particles
$N$ goes to infinity. When considering arbitrarily long time scales, this
problem is intimately connected to the rate of convergence to steady states as
time $t$ goes to infinity. For the study of such quantitative results, a
crucial role is played by the Poincaré (PI) and logarithmic Sobolev (LSI)
inequalities. Our focus in this paper is to elucidate the connection between
the validity of the LSI for the $N$-particle Gibbs measure uniformly in the
number of particles $N$ and the properties of the mean field limit. We
establish connections with uniform-in-time propagation of chaos,
(non-)uniqueness of steady states of the mean field equation, exponential
convergence to equilibrium, and the behaviour of equilibrium fluctuations. We
show the effect that the presence of multiple steady states for the mean field
equation, which correspond to invariant measures of the associated nonlinear
McKean SDE, has on the various quantitative and qualitative features of the
underlying interacting particle system.

In the equilibrium statistical mechanics of lattice systems, the non-
uniqueness of the infinite-volume grand canonical Gibbs measure is referred to
as a phase transition [30]. The presence of such a phase transition can then
be detected with the help of some order parameter, for example the
thermodynamic pressure for the nearest-neighbour Ising model, or the average
magnetisation of the ensemble for the mean-field Curie–Weiss model. Based on
the behaviour of these quantities (or rather on the behaviour of the infinite-
volume partition function), one can then characterise the phase transitions as
either continuous (second-order) or discontinuous (first-order).

The situation in our setting is more complicated but closely mirrors the one
for lattice systems. Indeed, the system we consider can be thought of as spin
system with mean field interaction and the “spins” taking values in some
uncountable state space $\Omega$. The difference between the system we
consider and the Ising and Curie–Weiss model lies in the fact that, except in
a small number of specific examples, it is extremely hard to specify an order-
parameter or understand the exact behaviour of the infinite-volume partition
function. An additional important difference is that for our system we will be
more interested in the non-uniqueness of critical points of the free energy,
which correspond to important changes in the features of the system, as
opposed to non-uniqueness of its minimisers. In either case, the similarity
with spin systems is instructive enough that it will serve the reader well to
remember this analogy as we discuss the notion of phase transition we work
with.

A detailed characterisation of phase transitions for McKean-Vlasov PDEs on the
torus without a confining potential was given in [11]. In particular, the
presence of phase transitions for this setting as it relates to non-uniqueness
of minimisers of the free energy functional was discussed in detail. However,
in this paper, we are more interested in characterising these phase
transitions as they relate to non-uniqueness of critical points of the free
energy.

For convex confining and interaction potentials (when the state space is
Euclidean), the system does not undergo phase transitions. In fact, uniform-
in-time propagation of chaos and uniqueness of the steady state for the mean
field PDE have been established, for e.g. in [44]. Moreover, in [13] under a
uniform convexity assumption of the potentials, the authors show exponentially
fast relaxation to the unique steady state of the mean field system. Our focus
in this paper is to deal with non-convex potentials which may exhibit phase
transitions and thus could not be expected to always (for all temperatures)
exhibit uniform-in-time propagation of chaos. In terms of the LSI, we show
that the existence of a non-minimising steady state implies the (quantitative)
degeneracy of the constant in the LSI of the $N$-particle Gibbs measure in the
limit as $N\rightarrow+\infty$. On the other hand, we show that the non-
degeneracy of the LSI, implies uniform-in-time propagation of chaos and
Gaussianity of the fluctuations around the mean field limit at equilibrium.
Furthermore, we conjecture that the limit of the LSI constant for the
$N$-particle system can be uniquely characterised in terms of the dissipation
inequality for the mean field system.

The relation between the non-degeneracy of the constant in the PI or the LSI,
the absence of phase transitions, and the exponentially fast decay of
correlations has been studied extensively for unbounded spin systems [63].
Conversely, in these works the equivalence between the slow decay of
correlations and the fact that the constant in the LSI becomes degenerate at
the phase transition has been established. Uniform estimates on the constant
in the LSI beyond the convex case have been established recently [32], under
the Lipschitzian spectral gap condition for the single particle. We remark
that this assumption is reminiscent to the assumptions on the conditional
measures for the two-scale LSI [49, 31, 41]. We utilize the latter approach to
show the non-degeneracy of the LSI for our $N$-particle Gibbs measure in the
high temperature/weak interaction regime.

In the probability literature, the study of the LSI in the context of linear
Fokker–Planck equations goes back to the classical $\Gamma_{2}$ functional
introduced by Bakry and Emery [3]. More recently, contractivity for
interacting particle systems has been studied in the context of entropic
interpolation and Schrödinger bridges [51, 29, 2, 28]. These techniques yield
proofs of both the Talagrand [16] and Sobolev [22] inequality, under lower
curvature conditions on the underlying manifold. We also mention the novel
coupling techniques introduced by Eberle in [24] that produce contractivity
estimates in a tailored transportation cost distance. This approach was then
later used to prove uniform-in-time propagation of chaos estimates under a
smallness assumptions on the interaction potential [23].

Our approach in this paper exploits the fact that both the $N$-particle system
(or rather its Fokker–Planck equation) and its mean field limit are gradient
flows of a particular energy functional with respect to the $2$-Wasserstein
distance. We can use this structural feature of the system to study the limit
of all the relevant quantities as $N\to\infty$. This approach was pioneered by
Hauray and Mischler in [35], and later used by the authors in [10, 20] to
study both propagation of chaos and periodic homogenization for the
interacting particle system. The advantage of this approach is that we can
often make minimal assumptions on the regularity of the confining and
interaction potentials: we will essentially assume that they are both only
semi-convex, which is natural for 2-Wasserstein gradient flows (see [1]). We
refer to [54, 38, 9] for the reader interested in propagation of chaos results
with more singular potentials.

### Organization of the paper.

Section 2 sets up the problem we are interested in studying along with our
notation and main assumptions. Section 3 contains the statements of all our
main results. Section 4 connects our results to the phenomenon of phase
transitions and discusses possible properties that could capture the radical
change of behavior in our system in the presence multiple steady states of the
mean field equation. Section 5 contains some technical results on the
convergence of the relevant quantities and functionals as $N\to\infty$ which
play an important role in the proofs of our main theorems. Sections 6, 7, 8,
9, 10, 11 and 12 contain the proofs of Theorems 3.2, 3.3, 3.4, 3.6, 3.7, 3.14
and 3.11, respectively.

##  2 Set up, assumptions, and notation

We consider $\\{X_{t}^{i}\\}_{i=1,\dots,N}\subset\mathbb{R}^{d}$, the
positions of $N$ indistinguishable interacting particles at time $t\geq 0$,
satisfying the following system of SDEs:

$\begin{cases}\displaystyle dX_{t}^{i}=-\nabla
V(X_{t}^{i})\mathop{}\\!\mathup{d}t-\frac{1}{N}\sum_{j=1}^{N}\nabla_{1}W(X_{t}^{i},X_{t}^{j})\mathop{}\\!\mathup{d}t+\sqrt{2\beta^{-1}}dB_{t}^{i}\\\
\displaystyle\mathrm{Law}(X_{0}^{1},\dots,X_{0}^{N})=\rho_{\mathrm{\mathrm{in}}}^{\otimes
N}\in\mathcal{P}_{2,\mathrm{\mathrm{sym}}}((\mathbb{R}^{d})^{N}),\end{cases}$
(2.1)

where $V:\mathbb{R}^{d}\to\mathbb{R}$,
$W:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$, $\beta^{-1}>0$ is the
inverse temperature, $B_{t}^{i},i=1,\dots,N$ are independent $d$-dimensional
Brownian motions, and the initial position of the particles is i.i.d with law
$\rho_{\mathrm{\mathrm{in}}}$. The chaoticity assumption on the initial data
is not necessary but it greatly simplifies the exposition. Similarly, the
state space $\mathbb{R}^{d}$ can be replaced by the periodic domain
$\mathbb{T}^{d}$ or any convex set $\Omega\subset\mathbb{R}^{d}$ with normal
reflecting boundary conditions, see, for example, [56]. We denote the space of
symmetric Borel probability measures on $\Omega^{N}$ with finite second moment
by $\mathcal{P}_{2,\mathrm{\mathrm{sym}}}(\Omega^{N})$, i.e. probability
measures which are invariant under the relabeling of variables (or probability
measures that arise as laws of exchangeable random variables). Throughout the
paper, we will always work with probability measures that have finite second
moment; to avoid burdensome notation, we forego the subscript $2$ from now on
and simply write $\mathcal{P}_{\mathrm{\mathrm{sym}}}(\Omega^{N})$.

To ensure well-posedness of the evolutionary flows and coercivity, we make the
following assumptions.

######  Assumption 2.1.

The confining potential $V$ is lower semicontinuous, bounded below,
$K_{V}$-convex for some $K_{V}\in\mathbb{R}$ and there exists $R_{0}>0$ and
$\delta>0$, such that $V(x)\geq|x|^{\delta}$ for $|x|>R_{0}$.

######  Assumption 2.2.

The interaction potential $W$ is lower semicontinuous, $K_{W}$-convex for some
$K_{W}\in\mathbb{R}$, bounded below, symmetric $W(x,y)=W(y,x)$, vanishes along
the diagonal $W(x,x)=0$, and there exists $C$ such that

$|\nabla_{1}W(x,y)|\leq C(1+|W(x,y)|+V(x)+V(y))$ (2.2)

######  Remark 2.3.

The $K$-Convexity assumptions on the potentials is short hand for global lower
bounds on their Hessians

$D^{2}V\geq K_{V}I^{d\times d}\qquad\mbox{and}\qquad D^{2}W\geq
K_{W}I^{2d\times 2d}$

with $K_{V},$ $K_{W}\in\mathbb{R}$. Unlike a convexity assumption on the
potentials, i.e. $K$-convexity with $K=0$ (see [44] for results in the convex
case), these assumptions are weak enough to include models that exhibit phase
transitions with respect to changes in the system’s temperature, $\beta^{-1}$,
for example, the double well potential $V(x)=(1-|x|^{2})^{2}$ with quadratic
interactions $W(x,y)=|x-y|^{2}$, also known as the Desai–Zwanzig model (for
more details see [18].

######  Remark 2.4.

The more technical bound (2.2) replaces the more classical doubling condition
[1, Section 10.4.42] which is used to characterise the minimal sub-
differential of the interaction energy [1, Theorem 10.4.11].

###  2.1 The Fokker–Planck equation.

It is well known that the curve
$\rho^{N}:[0,\infty)\to\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$ which describes
the evolution of the law of the process
$(X_{t}^{1},...,X_{t}^{N})\in\Omega^{N}$ satisfies the following linear
Fokker–Planck equation

$\begin{cases}\partial_{t}\rho^{N}=\beta^{-1}\Delta\rho^{N}+\nabla\cdot(\rho^{N}\;\nabla
H_{N})&\mbox{in $(0,\infty)\times\Omega^{N}$},\\\
(\nabla\rho^{N}+\rho^{N}\nabla H_{N})\cdot\vec{n}_{\Omega^{N}}=0&\mbox{on
$(0,\infty)\times\partial\Omega^{N}$}\\\
\rho^{N}(0)=\rho_{\mathrm{in}}^{\otimes N},\end{cases}$ (2.3)

where the Hamiltonian $H_{N}$ is given by

$H_{N}(x)=\sum_{i=1}^{N}V(x_{i})+\frac{1}{2N}\sum_{j=1}^{N}\sum_{i=1}^{N}W(x_{i},x_{j}),$

$\rho_{\mathrm{in}}\in\mathcal{P}(\Omega)$, and $\vec{n}_{\Omega^{N}}$ is the
unit normal to $\Omega^{N}$.

###  2.2 de Finetti/Hewitt–Savage.

To take the limit as $N\to\infty$, we will crucially use the exchangeability
of the underlying particle system whose law is governed by (2.3). This implies
that the joint law $\rho^{N}$ is symmetric for all times, that is

$\rho^{N}(t)\in\mathcal{P}_{\mathrm{sym}}(\Omega)\quad\textrm{ for all }t\geq
0.$

The main idea is that we can characterise the limit $N\to\infty$ of
$\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$ as
$\mathcal{P}(\mathcal{P}(\Omega))$, which denotes the Borel probability
measures with bounded second moment defined over the metric space
$(\mathcal{P}(\Omega),d_{2})$, where $d_{2}$ is the 2-Wasserstein distance.

Following de Finetti [19] and Hewitt–Savage [37], we know that any tight
sequence $(\rho^{N})_{N\in\mathbb{N}}$, with
$\rho^{N}\in\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$, i.e. with tight (in
$\mathcal{P}(\Omega)$) $l$th marginals for all $l\in\mathbb{N}$, has a limit
$P_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))$ along a subsequence which we
do not relabel such that

$\rho^{N}\rightharpoonup P_{\infty},$

where weak convergence is given by duality with cylindrical test functions.
That is, for any $l\in\mathbb{N}$ and $\varphi\in C_{c}(\Omega^{l})$, we have

$\lim_{N\to\infty}\int_{\Omega^{l}}\varphi(y)\;\mathop{}\\!\mathup{d}\rho^{N}_{l}(y)=\int_{\mathcal{P}(\Omega)}\left(\int_{\Omega^{l}}\varphi(y)\;\mathop{}\\!\mathup{d}\rho^{\otimes
l}(y)\right)\;\mathop{}\\!\mathup{d}P_{\infty}(\rho),$ (2.4)

where

$\rho^{N}_{l}\in\mathcal{P}_{\mathrm{sym}}(\Omega^{l})\qquad\mbox{is the
$l$-th marginal of $\rho^{N}$.}$

In essence, this means that in the limit $N\to\infty$ of symmetric probability
measures can be characterised as convex combinations of chaotic measures. For
more details, we refer the reader to [35, 10, 52]. In the sequel, we will use
the notation $\rho^{N}\rightharpoonup
P_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))$ to denote this notion of weak
convergence for any sequence $(\rho^{N})_{N\in\mathbb{N}}$ with
$\rho^{N}\in\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$.

Moreover, a metric version of this result can be obtained by considering the
appropriately scaled 2-Wasserstein distance, i.e.

$\overline{d}_{2}:=\frac{1}{\sqrt{N}}d_{2},$ (2.5)

where $d_{2}$ is the classical 2-Wasserstein on $\mathcal{P}(\Omega^{N})$.
More specifically, Hauray–Mischler [35] showed that under the topology of
(2.4)

$\lim_{N\to\infty}\overline{d}_{2}=\mathfrak{D}_{2},$ (2.6)

where $\mathfrak{D}_{2}$ is the 2-Wasserstein distance defined on the
probabilities with second moment bounded over the metric space
$(\mathcal{P}_{2}(\mathbb{R}^{d}),d_{2})$. For more details, see Theorem 5.3.
This metric on $\mathcal{P}(\mathcal{P}(\Omega))$ is closely related to the
convergence discussed above as will be seen in Proposition 5.5.

$\beta<\beta_{c}$$\beta=\beta_{c}$$\beta>\beta_{c}$

Figure 1: A rough schematic showing two possible kinds of phase transition:
The upper diagram shows a typical continuous phase transition. In this
setting, the unique critical point (shown in blue) loses its local stability
through a local (pitchfork) bifurcation which gives rise to new locally stable
critical points. The lower diagram shows a typical discontinuous phase
transition. In this setting, the unique critical point retains its local
stability but new critical points arise in the free energy landscape through a
saddle node bifurcation.

###  2.3 The mean field limit and nonlinear behaviour

Within this formalism, the limit $N\to\infty$ of the equation (2.3) can be
written as

$\rho^{N}(t)\rightharpoonup\delta_{\rho(t)}\in\mathcal{P}(\mathcal{P}(\Omega))\qquad\mbox{for
all $t\geq 0$,}$ (2.7)

where $\delta_{\rho(t)}$ denotes the delta measure concentrated on $\rho(t)$,
the unique solution of the nonlinear McKean equation

$\begin{cases}\partial_{t}\rho=\beta^{-1}\Delta\rho+\nabla\cdot\left(\rho\left(\nabla
V+\nabla W\star\rho\right)\right)&\mbox{on $(0,\infty)\times\Omega$}\\\
(\nabla\rho+\rho(\nabla V+\nabla W\star\rho)\cdot\vec{n}_{\Omega}=0&\mbox{on
$(0,\infty)\times\partial\Omega$}\\\ \rho(0)=\rho_{\mathrm{in}},\end{cases}$
(2.8)

with

$W\star\rho(x):=\int_{\mathbb{R}^{d}}W(x,y)\;\mathop{}\\!\mathup{d}\rho(y)\,.$
(2.9)

One of the most salient differences between the particle dynamics (2.3) and
the mean field dynamics (2.8) is that, whereas the Fokker-Planck equation
governing the evolution of the $N$-particle system is linear, the mean field
PDE (2.8) is nonlinear. As is well known, a consequence of this is that for
non-convex confining/interaction potentials the mean field dynamics might have
more than one stationary state, in contrast to the particle dynamics. Indeed,
the unique steady state of the $N$-particle Fokker–Planck equation (2.3) is
given by the Gibbs measure

$M_{N}:=\frac{e^{-\beta
H_{N}(x)}}{Z_{N}}\mathop{}\\!\mathup{d}x\qquad\mbox{with}\qquad
Z_{N}:=\int_{\Omega^{N}}e^{-\beta H_{N}(y)}\;\mathop{}\\!\mathup{d}y.$ (2.10)

On the other hand, the mean field limit can admit more than a single steady
state, with the full characterisation being the set of solutions to the self-
consistency equation

$\beta^{-1}\log\rho_{*}+W\star\rho_{*}+V=C_{*}\qquad\mbox{on $\Omega$ for some
$C_{*}\in\mathbb{R}$,}$ (2.11)

This is discussed in detail in Proposition 4.1. See also [11, 18] and the
references therein.

###  2.4 Phase transitions

The uniqueness/non-uniqueness of steady states of the mean field system (2.8)
depends both the temperature of the system (or, equivalently, the strength of
the interaction) and on the convexity properties of the confining and
interaction potentials $V$ and $W$. At sufficiently high temperatures, the
diffusion is strong enough that the expected escape time of particles from
local minima of the potentials is bounded uniformly in the number of
particles. Indeed, a perturbation argument shows that, at sufficiently high
temperatures, the self-consistency equation (2.11) has a unique solution, see
Proposition 4.2. When we cool the system and the potentials are non-convex,
particles can get trapped for arbitrarily long time scales in local minima and
condense [4]. In statistical physics terms, the system changes from a gaseous
state to a liquid or solid state. In the mean field limit, this change of
behavior can be characterised by the local or global instability of the
minimisers of the mean field energy (see (2.15)), see for instance [11, 15]
and Fig. 1 for a loose picture of the change in the mean field energy
landscape as the temperature of the system is varied. For more details on the
possible definitions of a phase transition, see Section 4.

###  2.5 Qualitative long time behavior

Due to the linearity of the $N$-particle system, we know that (2.3) admits a
unique steady state (given by the Gibbs measure $M_{N}$). If the potentials
prevent mass from escaping to infinity, by the La-Salle’s principle for
graident flows [12, Theorem 2.13], we know that independent of the initial
condition

$\lim_{t\to\infty}\rho^{N}(t)=M_{N}\,,$ (2.12)

in the sense of weak convergence of probability measures. Furthermore, entropy
methods and the LSI provide us with exponentially fast convergence to the
steady state [45]. On the other hand, if the McKean–Vlasov equation admits
multiple steady states, then the limit $t\to\infty$ for the mean field
dynamics will depend on the choice of initial condition. In this case, the
limiting dynamics (2.8) do not approximate the particle dynamics (2.1) for
arbitrarily long times. In this paper we provide evidence that phase
transitions constitute the natural obstruction to obtaining uniform-in-time
propagation of chaos estimates. See Theorem 3.7 for sufficient conditions for
the mean field approximation to be valid uniformly in time.

###  2.6 $\Gamma$-convergence

Taking advantage of the 2-Wasserstein gradient flow structure of (2.3) and
(2.8) (see [1, 10, 53]), the main tool that we will use to obtain a
quantitative understanding of the limit $N\to\infty$ is $\Gamma$-convergence
with respect to the topology introduced by de Finetti/Hewitt-Savage-type
convergence (2.4). To illustrate this technique, we use it to characterise the
limit $N\to\infty$ of the Gibbs measure $M_{N}$. Following the pioneering work
of Messer and Spohn [47], we notice that $M_{N}$ is the unique minimiser over
probability measures of the energy per unit particle

$E^{N}[\rho^{N}]=\frac{1}{N}\left(\beta^{-1}\int_{\Omega^{N}}\rho^{N}\log\rho^{N}\;\mathop{}\\!\mathup{d}x+\int_{\Omega^{N}}H_{N}\rho^{N}\;\mathop{}\\!\mathup{d}x\right).$
(2.13)

Taking $N\to\infty$, we can characterise the thermodynamic limit of the energy
$E^{N}$ as
$E^{N}\to^{\Gamma}E^{\infty}:\mathcal{P}(\mathcal{P}(\Omega))\to\mathbb{R}\cup\\{+\infty\\}$,
where

$\displaystyle
E^{\infty}[P]:=\int_{\mathcal{P}(\Omega)}E^{MF}[\rho]\mathop{}\\!\mathup{d}P(\rho),$
(2.14)

with the mean field energy
$E^{MF}:\mathcal{P}(\Omega)\to\mathbb{R}\cup\\{+\infty\\}$ given by

$E^{MF}[\rho]:=\beta^{-1}\int_{\Omega}\rho\log(\rho)\;\mathop{}\\!\mathup{d}x+\frac{1}{2}\int_{\Omega^{2}}W(x,y)\;\mathop{}\\!\mathup{d}\rho(x)\;\mathop{}\\!\mathup{d}\rho(y)+\int_{\Omega}V(x)\;\mathop{}\\!\mathup{d}\rho(x)\,,$
(2.15)

see [35] or Theorem 5.6 for a more modern proof. Using the fact that $M_{N}$
is the minimiser of (2.13), we know that any accumulation point of the
sequence $M_{N}$ (in the sense of de Finetti/Hewitt–Savage)
$P_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))$ needs to be a minimiser of
(2.14). Hence, we may conclude that the minimal energy converges

$\lim_{N\to\infty}\left(-\frac{1}{N}\log
Z_{N}\right)=\lim_{N}E^{N}[M_{N}]=E^{\infty}[P_{\infty}]=\inf_{P\in\mathcal{P}(\mathcal{P}(\mathbb{R}^{d}))}E^{\infty}[P]=\inf_{\rho\in\mathcal{P}(\mathbb{R}^{d})}E^{MF}[\rho],$

where we have used that (2.14) is a potential energy, which also implies that
$P_{\infty}$ needs to be supported on the minimisers of $E^{MF}$. This
convergence and related results are discussed in further detail in Section 5.

Under our previous hypothesis, we have the following standard result.

######  Theorem A.

Under Assumptions 2.1 and 2.2, $E^{MF}$ is bounded below and has at least one
minimiser.

The lower bound follows from Jensen’s inequality, while the existence of a
minimiser follows from the direct method of calculus of variations. Under the
extra assumption that the $E^{MF}$ admits a unique minimiser
$\rho_{\beta}\in\mathcal{P}(\mathbb{R}^{d})$, we have

$M_{N}\rightharpoonup\delta_{\rho_{\beta}}\in\mathcal{P}(\mathcal{P}(\Omega))\,.$

The perspective of the proofs in this paper is that the evolution of the
$N$-particle law (2.3) and the mean field limit (2.8) are respectively the
gradient flows of $E^{N}$ and $E^{MF}$ with respect to the scaled
2-Wasserstein distance $\bar{d}_{2}$ and the $2$-Wasserstein distance on
$\mathcal{P}(\Omega)$. In fact, the $\lambda$-convexity assumption on the
potentials and the doubling condition (2.2) can be used to obtain uniqueness
of the gradient flow solutions. The next fundamental result is essentially a
restatement of [1, Theorem 11.2.8], for the compact case see [53].

######  Theorem B.

If Assumptions 2.1 and 2.2 hold, then for any
$\rho_{\mathrm{in}}\in\mathcal{P}(\Omega)$ there exists unique distributional
solutions $\rho^{N}\in C([0,\infty);\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$
and $\rho\in C([0,\infty);\mathcal{P}(\Omega)$ to (2.3) and (2.8),
respectively, which are the gradient flows of $E^{N}$ (2.13) and $E^{MF}$
(2.15) with respect to the scaled 2-Wasserstein distance $\bar{d}_{2}$ and the
$2$-Wasserstein distance $d_{2}$ on $\mathcal{P}(\Omega)$, respectively.

We remark here that in [10] an alternative proof of propagation of chaos is
provided which employs the $\Gamma$-convergence result and the convergence of
the gradient flow structures.

##  3 Main results

To quantify the convergence as $t\to\infty$ in (2.12), we can apply the
standard relative entropy estimate [60]. More specifically, we consider the
Lyapunov functional given by the scaled relative entropy of $\rho^{N}(t)$ with
respect to the equilibrium measure $M_{N}$:

$E^{N}[\rho^{N}(t)]-E^{N}[M_{N}]=\overline{\mathcal{E}}(\rho^{N}(t)|M_{N}):=\frac{1}{N}\int_{\Omega^{N}}\log\left(\frac{\rho^{N}(t)}{M_{N}}\right)\rho^{N}(t)\;\mathop{}\\!\mathup{d}x,$
(3.1)

where we use the notation
$\overline{\mathcal{E}}:\mathcal{P}(\Omega)\times\mathcal{P}(\Omega)\to[0,\infty]$
to denote the scaled relative entropy. Taking a time derivative and using the
PDE (2.3) we obtain the scaled relative Fisher information
$\overline{\mathcal{I}}:\mathcal{P}(\Omega)\times\mathcal{P}(\Omega)\to[0,\infty]$:

$\displaystyle\frac{\mathop{}\\!\mathup{d}missing}{\mathop{}\\!\mathup{d}t}\overline{\mathcal{E}}(\rho^{N}(t)|M_{N})=-\beta^{-1}\frac{1}{N}\int_{\Omega^{N}}\left|\nabla\log\left(\frac{\rho^{N}(t)}{M_{N}}\right)\right|^{2}\rho^{N}(t)\;\mathop{}\\!\mathup{d}x=:-\beta^{-1}\overline{\mathcal{I}}(\rho^{N}(t)|M_{N}).$
(3.2)

The convergence (2.12) in relative entropy is exponential whenever we can show
that the $N$-particle log Sobolev constant is bounded away from zero:

$0<\lambda_{\operatorname{LS}}^{N}:=\inf_{\rho^{N}\in\mathcal{P}(\Omega^{N})\setminus\\{M_{N}\\}}\frac{\beta^{-1}\overline{\mathcal{I}}(\rho^{N}|M_{N})}{\overline{\mathcal{E}}(\rho^{N}|M_{N})}.$
(3.3)

Following the classical work of Bakry–Emery [3, 40, 36], we can find mild
conditions for the positivity of the log Sobolev constant whenever the domain
$\Omega$ is $\mathbb{R}^{d}$.

######  Theorem C.

Under Assumptions 2.1 and 2.2, if there exists $R>1$ and $\lambda>0$ such that

$D^{2}H^{N}(x)\geq\lambda I^{Nd\times Nd}\qquad\mbox{for every $|x|>R$,}$
(3.4)

then we have that $\lambda_{\operatorname{LS}}^{N}>0$.

######  Remark 3.1.

The convexity condition (3.4) in the far field can arise from either the
convexity of the interaction or the confining potential. We expect that the
sharp condition for the Gibbs measure $M_{N}$ to satisfy
$\lambda_{\operatorname{LS}}^{N}>0$ uniformly in $N$ is related to the
behavior of the mean field limit dissipation inequality (3.6). We will discuss
this in more detail in 1.

For the mean field limit, we can perform a similar analysis with the relative
mean field energy. More specifically, given $\rho(t)$ the solution to (2.8) we
can differentiate to obtain the dissipation

$\frac{\mathop{}\\!\mathup{d}missing}{\mathop{}\\!\mathup{d}t}E^{MF}[\rho(t)]-\inf
E^{MF}=-\int_{\Omega}|\beta^{-1}\nabla\log\rho(t)+\nabla W\star\rho(t)+\nabla
V|^{2}\rho(t)\;\mathop{}\\!\mathup{d}x=:-D(\rho(t)).$ (3.5)

Hence, we obtain exponential decay of the mean field energy to its minimum
value, as long as the so-called infinite colume log Sobolev constant, given by

$0<\lambda_{\operatorname{LS}}^{\infty}:=\inf_{\begin{subarray}{c}\rho\in\mathcal{P}(\Omega)\\\
\rho\notin\mathcal{K}\end{subarray}}\frac{D(\rho)}{E^{MF}[\rho(t)]-\inf
E^{MF}},$ (3.6)

is positive, where

$\mathcal{K}=\\{\rho\in\mathcal{P}(\Omega)\;:\;E^{MF}[\rho]=\inf E^{MF}\\}.$
(3.7)

In both cases, when the log Sobolev constant is positive we can show that the
relative energy behaves quadratically with respect to the 2-Wasserstein
distance. This is essentially the content of Talagrand’s inequality [58]. One
of our contributions in this paper is a new proof of a generalised version of
this inequality using gradient flow techniques.

######  Theorem 3.2.

Under Assumptions 2.1 and 2.2,

$E^{N}[\rho^{N}]-E^{N}[M_{N}]\geq\frac{\lambda^{N}_{\operatorname{LS}}}{2}\overline{d}^{2}_{2}(\rho^{N},M_{N})\qquad\mbox{and}\qquad
E^{MF}[\rho]-\inf
E^{MF}\geq\frac{\lambda^{\infty}_{\operatorname{LS}}}{2}d^{2}_{2}(\rho,\mathcal{K}),$
(3.8)

where $\mathcal{K}$ is the set of minimisers of $E^{MF}$ and
$d^{2}_{2}(\rho,\mathcal{K})=\inf_{\mu\in\mathcal{K}}d_{2}^{2}(\rho,\mu).$

An optimal transport-based proof of inequality (3.8) for $E^{N}$ and $E^{MF}$
can be found in [61, Theorem 22.17]. We also refer to [16] for a proof using
entropic interpolation. In Section 6, we provide a different more intuitive
proof of (3.8) for general energies
$E:\mathcal{P}(\Omega)\to\mathbb{R}\cup\\{+\infty\\}$. Our strategy is to use
the associated gradient flow structure and what is sometimes referred to as
Otto calculus [48], a formal Riemannian calculus on
$(\mathcal{P}(\Omega),d_{2})$. We should note that one of the main differences
in Talagrand’s inequality for the $N$-particle energy and for the mean field
energy is that the set of minimisers $\mathcal{K}$ does not need to be a
single point in the mean field case.

Having established the need for understanding the behavior of the log Sobolev
constant, our first result relates the limit of the particle system log
Sobolev constant (3.3) with the mean field or infinite volume log Sobolev
constant(3.6):

######  Theorem 3.3.

Under Assumptions 2.1 and 2.2, we have

$\limsup_{N\to\infty}\lambda_{\operatorname{LS}}^{N}\leq\lambda_{\operatorname{LS}}^{\infty}.$
(3.9)

Moreover, if the mean field energy $E^{MF}$ (2.15) admits a critical point
that is not a minimiser, then $\lambda_{\operatorname{LS}}^{\infty}=0$, and
there exists $C>0$ such that

$\lambda_{\operatorname{LS}}^{N}\leq\frac{C}{N}.$ (3.10)

Our result complements similar results that have been obtained for unbounded
spin systems [63] and the references therein. On the other hand, when
$\lambda^{\infty}_{\operatorname{LS}}>0$, we can show that the regularized log
Sobolev constant

$\lambda_{\operatorname{LS}}^{N,\varepsilon}:=\inf_{\rho^{N}:\;\overline{\mathcal{E}}(\rho^{N}|M_{N})>\varepsilon}\frac{\beta^{-1}\overline{\mathcal{I}}(\rho^{N}|M_{N})}{\overline{\mathcal{E}}(\rho^{N}|M_{N})}$
(3.11)

does not degenerate. More specifically,

$\lim_{N\to\infty}\lambda_{\operatorname{LS}}^{N,\varepsilon}\geq\lambda^{\infty}_{\operatorname{LS}}>0.$
(3.12)

This result implies that relaxation to neighborhoods of the stationary state
of the particle dynamics $\eqref{eq:forwardK}$ happens exponentially fast,
uniformly in $N$.

######  Theorem 3.4.

Under Assumptions 2.1 and 2.2, assume that
$\lambda^{\infty}_{\operatorname{LS}}>0$, and that
$\rho_{\mathrm{\mathrm{in}}}$ in (2.1) has finite energy and bounded higher
order moments,

$E^{MF}[\rho_{\mathrm{\mathrm{in}}}]<\infty\qquad\mbox{and}\qquad\int_{\Omega}|x|^{2+\delta}\;\mathop{}\\!\mathup{d}\rho_{\mathrm{\mathrm{in}}}<\infty,\qquad\mbox{for
some $\delta>0$.}$ (3.13)

Then, for every $\varepsilon>0$, there exists $N_{0}\in\mathbb{N}$, such that
for every $N>N_{0}$ we have

$\overline{\mathcal{E}}(\rho^{N}(t)|M_{N})\leq\max\left\\{\varepsilon,e^{-\frac{1}{2}\lambda_{\operatorname{LS}}^{\infty}t}\;\overline{\mathcal{E}}(\rho^{\otimes
N}_{\mathrm{in}}|M_{N})\right\\}.$

######  Remark 3.5.

For chaotic measures, we can take the limit $N\to\infty$ of the relative
entropy obtaining

$\lim_{N\to\infty}\overline{\mathcal{E}}(\rho^{\otimes
N}_{\mathrm{in}}|M_{N})=E^{MF}[\rho_{\mathrm{in}}]-\inf E^{MF}\,.$

This is discussed in Theorem 5.6.

Unfortunately, we are not able to fully characterise the limit of
$\lambda^{N}_{\operatorname{LS}}$ in terms of the mean field limit. Despite
this, Theorem 3.3 and Theorem 3.4, provide us with evidence which is
convincing enough to make the following conjecture.

######  Conjecture 1.

Under Assumptions 2.1 and 2.2, we have the equality

$\lim_{N\to\infty}\lambda^{N}_{\operatorname{LS}}=\lambda^{\infty}_{\operatorname{LS}}.$

The results of our paper provide us with a strong indication that the absence
of phase transitions (loosely defined to mean that the mean field limit has a
unique stationary state), the non-degeneracy of the infinite volume log
Sobolev constant, and the validity uniform-in-time propagation of chaos are
all equivalent.

###  3.1 Consequences of the non-degeneracy of the log Sobolev inequality

Bearing 1 in mind, we now explore the implications of the non-degeneracy of
the LSI constant in the limit $N\rightarrow+\infty$. We begin by noticing that
if the log Sobolev constant does not degenerate in $N$, then the invariant
Gibbs measure $M_{N}$ of the $N$-particle system is well approximated by the
unique minimiser of the mean field energy.

######  Theorem 3.6.

Under Assumptions 2.1 and 2.2, assume that
$\limsup_{N\to\infty}\lambda^{N}_{\operatorname{LS}}>0$. Then, there exists a
unique steady state $\rho_{\beta}$ to (2.8). Moreover, there exists $C>0$,
such that

$\overline{d}^{2}_{2}(\rho_{\beta}^{\otimes
N},M_{N})\leq\frac{2}{\lambda_{\operatorname{LS}}^{N}}\overline{\mathcal{E}}(\rho_{\beta}^{\otimes
N}|M_{N})\leq\frac{2}{(\lambda_{\operatorname{LS}}^{N})^{2}}\overline{\mathcal{I}}(\rho_{\beta}^{\otimes
N}|M_{N})\leq\frac{C}{N}.$ (3.14)

Interpolating the previous result with more standard propagation of chaos
estimates that depend on the convexity constant of the potentials, we obtain
the following uniform-in-time propagation of chaos result.

######  Theorem 3.7.

Under Assumptions 2.1 and 2.2, let $\rho^{N}$ and $\rho$ denote the unique
solutions to the particle (2.3) and mean field (2.8) dynamics given by B.
Assume that $\rho_{\mathrm{in}}$ has finite energy

$E^{MF}[\rho_{\mathrm{in}}]<\infty\,,$

that the gradient of the square of the interaction potential is uniformly
integrable

$\sup_{t\in[0,\infty]}\int_{\Omega}|\nabla_{1}W|^{2}\star\rho(t)\rho(t)\;\mathop{}\\!\mathup{d}x<\infty\,,$
(3.15)

and that
$\liminf_{N\to\infty}\lambda^{N}_{\operatorname{LS}}=:\lambda^{\infty}>0$.
Then,

$\overline{d}_{2}(\rho^{N}(t),\rho^{\otimes
N}(t))\leq\frac{C}{N^{\theta}}\qquad\mbox{for all $t>0$,}$ (3.16)

where

$\theta=\begin{cases}1/2&\mbox{if $K_{V}+K_{W}(1-1/N)>0$}\\\
\frac{1}{2}\frac{\lambda^{\infty}}{\lambda^{\infty}-2(K_{V}+K_{W}(1-1/N))}&\mbox{if
$K_{V}+K_{W}(1-1/N)<0$}\end{cases}$

with $K_{V}$ and $K_{W}$ the convexity constants of $V$ and $W$ in Assumptions
2.1 and 2.2. In the case where $K_{V}+K_{W}(1-1/N)=0$, we can pick any
$\theta<1/2$.

######  Remark 3.8.

The integrability assumption (3.15) is trivially true when $W$ is uniformly
Lipschitz. Also, this assumption is satisfied when the potentials are
attractive enough in the far field, i.e. outside a ball of radius $R$, that we
can obtain uniform exponential bounds for the tail behavior of the mean field
solution.

######  Remark 3.9.

We note that for convex potentials $K_{V},$ $K_{W}\geq 0$ the uniform
propagation of chaos with $\theta=1/2$ has already been shown in [44]. The
main difference with this work is that our approach utilizes the convexity of
the entropy along the 2-Wasserstein distance to obtain a contraction estimate.
This approach can be easily extended to manifolds with Ricci curvature bounded
from below [43], where we need to consider the sign of

$K_{\mathrm{Ric}}+K_{V}+K_{W}(1-1/N)$

with $K_{\mathrm{Ric}}$ the lower bound on the Ricci curvature of the
underlying Riemannian manifold.

######  Remark 3.10.

We do not expect $\theta$ in the above theorem to be sharp for
$K_{V}+K_{W}(1-1/N)<0$. At sufficiently high temperatures and
$K_{V}+K_{W}(1-1/N)<0$, a comparable result for the 1-Wasserstein distance is
shown by coupling methods in [25] with $\theta=1/2$.

We can also use the non-degeneracy of the LSI constant to identify the
fluctuations at equilibrium.

######  Theorem 3.11.

Let $\Omega=\mathbb{T}^{d}$ and assume that
$\liminf_{N\to\infty}\lambda^{N}_{\operatorname{LS}}>0$, then the fluctuations
process

$\eta^{N}(t)=\sqrt{N}\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}}-\rho_{\beta}\right)\,,$
(3.17)

where $(X_{t}^{1},...,X_{t}^{N})$ are solutions to (2.1) with initial law
given by the invariant Gibbs measure $M_{N}$, satisfies

$\sup_{N\in\mathbb{N},t\in[0,T]}\mathbb{E}\left[\|\eta^{N}(t)\|_{H^{-s}(\mathbb{T}^{d})}^{2}\right]<\infty,$
(3.18)

for any $T>0,\,s>d/2+1$.

Moreover, assume that $V$, $W$ are smooth and that the linearised operator
(2.8) around $\rho_{\beta}$

$\displaystyle\mathcal{L}_{\rho_{\beta}}\eta=\beta^{-1}\Delta\eta+\nabla\cdot(\rho_{\beta}\nabla
W\star\eta)+\nabla\cdot(\eta\nabla W\star\rho_{\beta})+\nabla\cdot(\nabla
V\eta)\,,$ (3.19)

satisfies, for all $\phi\in\textrm{C}^{\infty}(\mathbb{T}^{d})$, the following
coercivity inequality

$\langle-\mathcal{L}_{\rho_{\beta}}\phi,\phi\rangle_{L^{2}(\mathbb{T}^{d})}\geq
c\|\nabla\phi\|^{2}_{L^{2}(\mathbb{T}^{d})}$ (3.20)

for some $c>0$.

Then, for any $m>d/2+3$, $\eta^{N}$ converges in law, as a
$C([0,T];H^{-m}(\mathbb{T}^{d}))$-valued random variable, to the unique
stationary solution $\eta^{\infty}$ of the following linear SPDE

$\partial_{t}\eta^{\infty}=\mathcal{L}_{\rho_{\beta}}\eta^{\infty}+\nabla\cdot(\sqrt{\rho_{\beta}}\xi)$
(3.21)

where $\xi$ is space-time white noise.

######  Remark 3.12.

We mention the result of Fernandez and Meleard [26] (see also [59, 55]) which
characterises the fluctuations of the particle dynamics with respect to the
mean field limit for finite time horizons, under a stronger closeness
assumption for the initial data. We also mention the recent preprint [62]
which studies fluctuations for singular potentials.

To our knowledge, the first available results for fluctuations at equilibrium
is due to Dawson [18] in which he shows that, for the specific case of
Desai–Zwanzig model, at the phase transition temperature, equilibrium
fluctuations are non-Gaussian. It is an interesting open problem if this
behavior is universal for any system which undergoes a (continuous) phase
transition. The different notions of phase transition will be discussed in
Section 4.

######  Remark 3.13.

For The smoothness assumptions on $V$ and $W$ are used to have a well-defined
semigroup associated to the linearised operator $\mathcal{L}_{\rho_{\beta}}$
which regularizes instantaneously arbitrary initial data in
$H^{-m}(\mathbb{T}^{d})$. Up to technical results, this can be quantified by
requiring $V$ and $W\in W^{m+\epsilon,\infty}(\mathbb{T}^{d})$ for any
$\epsilon>0$. For fluctuation results with singular potentials we refer the
reader to the recent preprint [62].

The unique invariant measure of the SPDE (3.21)
$\mathcal{G}\in\mathcal{P}(H^{-m}(\mathbb{T}^{d}))$ is a centred Gaussian
measure with covariance operator $Q_{\mathcal{G}}$ given by

$Q_{\mathcal{G}}(\varphi,\psi):=\lim_{t\to\infty}\int_{0}^{t}\int_{\mathbb{T}^{d}}\nabla
e^{t\mathcal{L}_{\rho_{\beta}}^{*}}\varphi\cdot\nabla
e^{t\mathcal{L}_{\rho_{\beta}}^{*}}\psi\,\rho_{\beta}\,\mathop{}\\!\mathup{d}x\mathop{}\\!\mathup{d}t\,,$
(3.22)

for any mean-zero $\varphi,\;\psi\in C^{\infty}(\mathbb{T}^{d})$ and where
$\mathcal{L}_{\rho_{\beta}}^{*}$ denotes the flat $L^{2}$-adjoint.

In the specific case that $V\equiv 0$ and $W(x,y)=W(x-y)$, we can obtain a
more explicit characterisation of $\mathcal{G}$. Indeed, since
$\liminf_{N\to\infty}\lambda_{\operatorname{LS}}^{N}>0$, we know from
Proposition 4.2 (and the discussion following it) that $E^{MF}$ has a unique
critical point which is given by
$\rho_{\infty}(\mathop{}\\!\mathup{d}x)=\mathop{}\\!\mathup{d}x$. We can then
write down an explicit representation of the action of the semigroup
$e^{t\mathcal{L}^{*}_{\mathop{}\\!\mathup{d}x}}$ in Fourier space as follows

$\hat{\varphi}_{t}(k)=e^{-4\pi^{2}\lvert
k\rvert^{2}(\beta^{-1}+\hat{W}(k))t}\hat{\varphi}(k)\quad
k\in\mathbb{Z}^{d},k\neq 0\,,$ (3.23)

where $\varphi_{t}=e^{t\mathcal{L}^{*}_{\mathop{}\\!\mathup{d}x}}\varphi$ for
some mean-zero $\varphi\in C^{\infty}(\mathbb{T}^{d})$. This leaves us with
the formula

$Q_{\mathcal{G}}(\varphi,\psi)=\sum_{k\in\mathbb{Z}^{d},k\neq
0}\frac{\hat{\varphi}(k)\hat{\psi}(k)}{8\pi^{2}(\beta^{-1}+\hat{W}(k))}\,,$
(3.24)

where we have used the fact that the coercivity inequality (3.20) is
equivalent to the fact that $\beta^{-1}+\hat{W}(k)>0$ for all
$k\in\mathbb{Z}^{d},k\neq 0$, which is also equivalent to the condition
$\beta<\beta_{\sharp}$, see Proposition 4.3.

Another way of rewriting this is that $\mathcal{G}$ is the unique centred
Gaussian measure with Cameron–Martin space given by the closure of all smooth
mean-zero functions $\varphi$ under the norm

$\lVert\varphi\rVert_{\mathcal{H}_{\mathcal{G}}}^{2}=8\pi^{2}\left(\beta^{-1}\int_{\mathbb{T}^{d}}\varphi^{2}\mathop{}\\!\mathup{d}x+\int_{\mathbb{T}^{d}}(W*\varphi)\varphi\mathop{}\\!\mathup{d}x\right)\,.$
(3.25)

Since, $\beta^{-1}+\hat{W}(k)>0$ for all $k\in\mathbb{Z}^{d},k\neq 0$, the
above norm is equivalent to the standard $L^{2}(\mathbb{T}^{d})$ norm. The
above norm is also the same, up to a multiplicative constant, as the norm
introduced in Property A.

One can further use the structure of the covariance operator $Q_{\mathcal{G}}$
to read off that $\mathcal{G}$ is supported on
$H^{-\frac{d}{2}-}(\mathbb{T}^{d})$ distributions. Thus, the limiting
equilibrium fluctuations have the regularity of spatial white noise, which is
not surprising considering the fact that their Cameron–Martin space is “
_basically_ ” $L^{2}(\mathbb{T}^{d})$.

###  3.2 Non-degeneracy of the LSI constant in specific cases

Putting aside for the time being the validity of 1, we show that the LSI
constant $\lambda_{\operatorname{LS}}^{N}$ does not degenerate in the high
temperature regime when $\Omega$ is compact, or when the confinement $V$
satisfies an LSI inequality and the interaction strength is small enough.

######  Theorem 3.14.

Assume that there exists a constant $C>0$ such that

$\displaystyle\lVert W\rVert_{{L}^{\infty}(\Omega^{2})},\,\lVert
D^{2}_{xy}W\rVert_{{L}^{\infty}(\Omega^{2})}<C\,.$ (3.26)

We then have the following two scenarios:

  1. (a)

Compact case: Assume $\Omega$ is compact and its normalised Lebesgue measure
$\mathop{}\\!\mathup{d}x$ satisfies a log Sobolev inequality. Then, there
exists a $0<\beta_{\operatorname{LS}}=\beta_{\operatorname{LS}}(C)$ such that
for all $\beta<\beta_{\operatorname{LS}}$, we have

$\liminf_{N\to\infty}\lambda^{N}_{\operatorname{LS}}>0\,.$ (3.27)

  2. (b)

weak interaction case: Assume $\Omega=\mathbb{R}^{d}$ and that the one-
particle measure $Z_{V}^{-1}e^{-V}\mathop{}\\!\mathup{d}x$ satisfies a log
Sobolev inequality with constant $\lambda^{V}_{\operatorname{LS}}>0$. Then
there exists an
$\epsilon_{\operatorname{LS}}=\epsilon_{\operatorname{LS}}(C,\lambda^{V}_{\operatorname{LS}},\beta)>0$,
such that for any $0\leq\epsilon<\epsilon_{\operatorname{LS}}$ we have

$\liminf_{N\to\infty}\lambda^{\epsilon,N}_{\operatorname{LS}}>0\,,$ (3.28)

where $\lambda^{\epsilon,N}_{\operatorname{LS}}$ is the log Sobolev constant
of the Gibbs measure $\tilde{M}_{N}=Z_{N}^{-1}e^{-\beta
H^{\epsilon}_{N}}\mathop{}\\!\mathup{d}x$ with

$H^{\epsilon}_{N}(x)=\sum_{i=1}^{N}V(x_{i})+\frac{\epsilon}{2N}\sum_{j=1}^{N}\sum_{i=1}^{N}W(x_{i},x_{j})\,.$
(3.29)

The above result relies crucially on the two-scale approach to log Sobolev
inequalities introduced in [49], which is based on the analysis of the
marginal and conditional measures of $M_{N}$. For the convenience of the
reader, we describe the main result of [49] in Theorem 11.2. In addition to
the above high temperature result, it is also possible to obtain a sharper
result in certain specific scenarios. For instance, in the case in which
$\Omega=\mathbb{T}$, $W(x,y)=-\cos(2\pi(x-y))$, and $V\equiv 0$. The
corresponding system is referred to by many names: the noisy Kuramoto model,
the mean field classical $XY$ model, the mean field $O(2)$ model or the
Brownian mean field model [14]. It is known that this system exhibits a phase
transition (of type A, B, and C, see Definition 4.6). Due to the particularly
simple nature of the model, it is possible to show that the $N$ particle log
Sobolev constant $\lambda^{N}_{\operatorname{LS}}$ is asymptotically non-
degenerate all the way up to the critical inverse temperature $\beta_{c}=2$.
We state without proof the following result due to Bauerschmidt and Bodineau
[5], see also [6].

######  Theorem D ([5, Theorem 1]).

Consider the Gibbs measure $M_{N}$ of the mean field $O(2)$ model and denote
by $\lambda_{\operatorname{LS}}^{N}$ its log Sobolev constant. Then, for all
$0<\beta<\beta_{c}=2$, we have that

$\liminf_{N\to\infty}\lambda^{N}_{\operatorname{LS}}>0\,.$ (3.30)

######  Remark 3.15.

An essentially similar argument as in [5, Theorem 1], can be used to show that
the system with $\Omega=\mathbb{T}$, $W(x,y)=-\cos(2\pi k(x-y))$, and $V\equiv
0$ for some $k\in\mathbb{N}$ has a uniform LSI all the way up to the critical
inverse temperature $\beta_{c}$ which coincides with $\beta_{\sharp}$ from
Proposition 4.3.

##  4 Phase transitions

We start our discussion by stating and proving the following result which
provides us with a particularly useful characterisation of steady states.

######  Proposition 4.1.

Under Assumptions 2.1 and 2.2 the following statements are equivalent:

  1. 1.

$\rho\in\mathcal{P}(\Omega)$ is a critical point of the mean field free energy
$E^{MF}$, that is to say

$\lvert\partial E^{MF}\rvert(\rho)=D(\rho)=0.$

  2. 2.

$\rho\in\mathcal{P}(\Omega)$ is a steady state of the mean field equation
(2.8), i.e. it is distributional weak solution of the PDE

$\beta^{-1}\Delta\rho+\nabla\cdot(\rho(\nabla W\star\rho+\nabla V))=0\,\quad
x\in\Omega\,.$

  3. 3.

$\rho$ solves the self-consistency equation:

$\displaystyle\rho-\frac{1}{Z_{\beta}}e^{-\beta(W\star\rho+V)}=0\,,\qquad
Z_{\beta}=\int_{\Omega}e^{-\beta(W\star\rho+V)}\mathop{}\\!\mathup{d}x\,.$
(4.1)

Furthermore, for all $\beta>0$, $E^{MF}$ has at least one global minimiser
which is a critical point, and any critical point $\rho\in\mathcal{P}(\Omega)$
of $E^{MF}$ is Lipschitz, strictly positive, and has moments of all orders.

###### Proof.

The proof of the equivalence of the three characterisations follows from
similar arguments to [11, Proposition 2.4] and so we omit the proof. The fact
that $E^{MF}$ has a critical point follows from A.

Now, using (4.1), we know that any critical point $\rho\in\mathcal{P}(\Omega)$
is of the form

$\displaystyle\frac{1}{Z_{\beta}}e^{-\beta(W\star\rho+V)}\,,$ (4.2)

which is Lipschitz and strictly positive. We now use Eq. 2.2 to assert that,
since $W$ is bounded below,

$\displaystyle\rho\leq Z_{\beta}^{-1}e^{\beta C}e^{-\beta V}\,.$ (4.3)

Thus, by Assumption 2.1, any critical point $\rho\in\mathcal{P}(\Omega)$ of
$E^{MF}$ has moments of all orders. ∎

Furthermore, we have the following result regarding uniqueness of critical
points.

######  Proposition 4.2.

Under the assumptions of Theorem 3.14 there exists a unique critical point
$\rho_{\beta}\in\mathcal{P}(\Omega)$ of the mean field free energy $E^{MF}$.

###### Proof.

The proof of this result follows by combining the results of Theorems 3.6 and
3.14. We know from Theorem 3.14 that the logarithmic Sobolev inequality holds
uniformly for $M_{N}$. We can then use Theorem 3.6 to argue that for $\beta$
sufficiently small, the mean field free energy $E^{MF}$ must have a unique
critical point $\rho_{\beta}$. ∎

When we discuss non-uniqueness of critical points, we will often restrict
ourselves to the case in which $V=0,W(x,y)=W(x-y)$, and
$\Omega=\mathbb{T}^{d}$. This case lends itself particularly well to analysis
as the Lebesgue measure
$\rho_{\infty}(\mathop{}\\!\mathup{d}x)=\mathop{}\\!\mathup{d}x$, what we
shall hereafter refer to as the “ _flat state_ ”, is always a critical point
of $E^{MF}$ for all $\beta>0$. We shall see later that in this setting it is
possible to provide relatively clean examples of the different types of phase
transitions that we consider in this paper. For an example of what a typical
phase transition looks like, see Fig. 1 for a schematic of the free energy
landscape in the vicinity of a phase transition.

We now introduce and motivate a list of properties that may serve as a proxy
for the absence of well-defined order parameter. We discuss how these
properties relate to each other, specifically in the “ _flat case_ ”. We start
by looking at the linearisation of the mean field dynamics (2.8):

######  Property A.

Fix $\beta>0$. We say that the system (2.8) satisfies _Property A_ if $E^{MF}$
has a unique critical point $\rho_{\beta}$ and the linearised operator
associated to right hand side of (2.8) has a spectral gap under the
interaction weighted inner product. More specifically, we have that

$\displaystyle\inf_{\eta\in
C^{\infty}_{0}(\Omega),\int_{\Omega}\eta\mathop{}\\!\mathup{d}x=0}\frac{\langle-\mathcal{L}_{\rho_{\beta}}\eta,\eta\rangle_{W,\rho_{\beta}}}{\lVert\eta\rVert_{W,\rho_{\beta}}^{2}}>0\,,$
(A)

where

$\displaystyle\mathcal{L}_{\rho_{\beta}}\eta=\beta^{-1}\Delta\eta+\nabla\cdot(\nabla
V\eta)+\nabla\cdot(\rho_{\beta}\nabla W\star\eta)+\nabla\cdot(\eta\nabla
W\star\rho_{\beta})\,,$ (4.4)

and

$\displaystyle\langle\eta,\nu\rangle_{W,\rho_{\beta}}=\frac{\beta^{-1}}{2}\int_{\Omega}\eta\nu\rho_{\beta}^{-1}\mathop{}\\!\mathup{d}x+\frac{1}{2}\int_{\Omega}(W\star\eta)\nu\mathop{}\\!\mathup{d}x$
(4.5)

with
$\lVert\eta\rVert_{W,\rho_{\beta}}^{2}=\langle\eta,\eta\rangle_{W,\rho_{\beta}}$.

The above property captures essentially the local stability of the critical
point $\rho_{\beta}$. Loss of the local stability of $\rho_{\beta}$ is a
precursor to a phase transition. In the setting of a continuous phase
transition (see the upper half of Fig. 1), one expects this property to fail
exactly at the critical temperature $\beta=\beta_{c}$. The weighted inner
product arises naturally through the linearisation of the mean field log
Sobolev constant, or more specifically through the linearisation of the mean
field energy $E^{MF}$, see (4.7). We note that the bilinear form
$\langle\cdot,\cdot\rangle_{W,\rho_{\beta}}$ is positive semi-definite when
$\rho_{\beta}$ is the unique critical point. For the classical $XY$ model (see
the discussion before Eq. 3.30), it is known that the system exhibits a
continuous phase transition [11, Proposition 6.1]. In this case, the non-local
inner product above it is related to the inner product that was introduced in
[8] to understand the spectral gap ahead of the phase transition. In the
absence of the interaction term, it reduces to the standard weighted inner
product that symmetrises the Fokker-Planck operator, see [50]. In the periodic
translation invariant case $\Omega=\mathbb{T}^{d}$, $W(x,y)=W(x-y)$, and
$V\equiv 0$, when $\rho_{\infty}=\mathop{}\\!\mathup{d}x$ is the unique
critical point of $E^{MF}$, the weighted inner product
$\langle\cdot,\cdot\rangle_{W,\rho_{\infty}}$ is equivalent to the standard
$L^{2}$ inner product. Thus, in this situation, checking Property A is
satisfied is equivalent to checking that $\mathcal{L}_{\rho_{\infty}}$ has a
spectral gap in the standard $L^{2}$ inner product. This relationship will be
made clearer in Proposition 4.3. Before discussing Property A in more detail,
we introduce the next property which measures the local degeneracy of the
self-consistency equation (4.1).

######  Property B.

Fix $\beta>0$. We denote by $T:{L}^{2}(\Omega)\to{L}^{2}(\Omega)$ the map
associated to the self-consistency equation (4.1):

$\displaystyle
T(\rho)=\rho-\frac{1}{Z_{\beta}}e^{-\beta(W\star\rho+V)}=0\,,\qquad
Z_{\beta}=\int_{\Omega}e^{-\beta(W\star\rho+V)}\mathop{}\\!\mathup{d}x\,.$ (B)

We say that the system (2.8) satisfies _Property B_ if $E^{MF}$ has a unique
critical point $\rho_{\beta}$ and the Fréchet derivative $D_{\rho}T$ of $T$ at
$\rho_{\beta}$ is non-degenerate, that is to say it has a trivial kernel
consisting only of constant functions.

Under additional conditions, see [11], the above property being violated
implies the presence of a local bifurcation around the minimiser
$\rho_{\beta}$. A simple condition that implies the presence of a local
bifurcation is when the algebraic multiplicity of the $0$ eigenvalue of
$D_{\rho}T$ at $\rho_{\beta}$ is odd. Local bifurcations also arise, if the
map $T$ can be rewritten as a so-called potential operator and its Frechét
derivative $D_{\rho}T$ has a non-zero crossing number, see [39]. The above
property exactly captures the second-order degeneracy of the mean field free
energy around the unique critical point $\rho_{\beta}$. Indeed, formally
expanding $E^{MF}[\rho]$ about $\rho_{\beta}$ we obtain

$\displaystyle E^{MF}[\rho]=$ $\displaystyle
E^{MF}[\rho_{\beta}]+\int_{\Omega}\beta^{-1}\frac{\eta^{2}}{2}\rho_{\beta}^{-1}+\frac{1}{2}W\star\eta\eta\mathop{}\\!\mathup{d}x+O(\eta^{3})\,$
(4.6) $\displaystyle=$ $\displaystyle
E^{MF}[\rho_{\beta}]+\beta^{-1}\int_{\Omega}\left(D_{\rho}T[\rho_{\beta}]\eta\right)\eta\rho_{\beta}^{-1}\mathop{}\\!\mathup{d}x+O(\eta^{3})\,,$
(4.7)

with $\eta=\rho-\rho_{\beta}$. Thus, if Property B is satisfied the mean field
free energy is non-degenerate at second order near the critical point
$\rho_{\beta}$.

The third and final property considers the validity of the Dissipation
Inequality.

######  Property C.

Fix $\beta>0$. We say that the system (2.8) satisfies _Property C_ if the
infinite-volume log Sobolev constant is positive. More precisely, if we have
that

$\displaystyle\lambda^{\infty}_{\operatorname{LS}}:=\inf_{\rho\in\mathcal{P}(\Omega),\rho\not\in\mathcal{K}}\frac{D(\rho)}{E^{MF}[\rho]-\min_{\mathcal{P}(\Omega)}E^{MF}}>0\,.$
(C)

The third and final property captures global aspects of the free energy
landscape. Indeed, one would expect it to be violated in both the situations
described in Fig. 1. For the case of the discontinuous phase transition, the
lower half of Fig. 1, Property C would be violated because of the presence of
a non-minimising critical point of $E^{MF}$ which is represented by the red
circles. Thus, the numerator of (C) vanishes while the denominator is strictly
positive. As an explicit example of a system which exhibits such a phase
transition, one can consider $\Omega=\mathbb{T}$, with the bi-chromatic
interaction potential $W(x,y)=-\cos(2\pi(x-y))-\cos(4\pi(x-y))$, and $V\equiv
0$ (cf. [11, Theorem 5.11]). On the other hand, in the case of a continuous
phase transition, the upper half of Fig. 1, the fact that Property C should be
violated is more subtle. To observe this, we linearise the right hand side of
(C) about the unique minimiser $\rho_{\beta}$ at $\beta=\beta_{c}$. For the
dissipation, we have that

$\displaystyle D(\rho)=$ $\displaystyle
2\int_{\Omega}\left\lvert\beta^{-1}\nabla\frac{\eta}{\rho_{\beta}}+\nabla
W\star\eta\right\rvert^{2}\mathop{}\\!\mathup{d}\rho_{\beta}+O(\eta^{3})\,,$
(4.8)

with $\eta=\rho-\rho_{\beta}$. Combining the above expression with (4.7), we
obtain to leading order

$\displaystyle\frac{D(\rho)}{E^{MF}[\rho]-E^{MF}[\rho_{\beta}]}\approx\frac{4\int_{\Omega}\left\lvert\beta^{-1}\nabla\frac{\eta}{\rho_{\beta}}+\nabla
W\star\eta\right\rvert^{2}\mathop{}\\!\mathup{d}\rho_{\beta}}{\int_{\Omega}\beta^{-1}\eta^{2}+W\star\eta\eta\rho_{\beta}\mathop{}\\!\mathup{d}\rho_{\beta}^{-1}}\,.$
(4.9)

Moreover, we notice that

$\displaystyle(-\mathcal{L}_{\rho_{\beta}}\eta,\eta)_{W,\rho_{\beta}}=$
$\displaystyle\frac{1}{2}\int_{\Omega}\left\lvert\beta^{-1}\nabla\frac{\eta}{\rho_{\beta}}+\nabla
W\star\eta\right\rvert^{2}\mathop{}\\!\mathup{d}\rho_{\beta}\,.$ (4.10)

Using (4.9), it follows that to leading order

$\displaystyle\frac{D(\rho)}{E^{MF}[\rho]-E^{MF}[\rho_{\beta}]}\approx
4\frac{\langle-\mathcal{L}_{\rho_{\beta}}\eta,\eta\rangle^{2}_{W,\rho_{\beta}}}{\lVert\eta\rVert_{W,\rho_{\beta}}^{2}}\,.$
(4.11)

Hence, in the setting of a continuous phase transition the infinite volume log
Sobolev constant captures the spectral gap of $\mathcal{L}_{\rho_{\beta}}$,
and thus also captures the loss of local stability of $\rho_{\beta}$.

We now present the following result which characterises how the various
properties we have discussed relate to each other in the periodic spatially
homogeneous case.

######  Proposition 4.3.

Assume $\Omega=\mathbb{T}^{d}$, $V\equiv 0,$ $W(x,y)=W(x-y)$, and denote

$\beta_{\sharp}:=\frac{1}{\min(0,-\min_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}\hat{W}(k))}\in(0,\infty].$

Then, for $\beta<\beta_{\sharp}$ the linearised operator
$\mathcal{L}_{\mathop{}\\!\mathup{d}x}$ satisfies the spectral gap property
(10.4), and the kernel of the Fréchet derivative of the self-consistency
equation (10.5) is non degenerate at the “ _flat state_ ”. For
$\beta\geq\beta_{\sharp}$, Properties A and B are violated.

Furthermore, for all $\beta\neq\beta_{\sharp}$, Property C implies Property A
and Property B.

######  Remark 4.4.

We emphasize that Proposition 4.3 does not ensure that Properties A and B are
satisfied for $\beta<\beta_{\sharp}$. In fact, for the bi-chromatic potential
case $W(x,y)=-\cos(2\pi(x-y))-\cos(4\pi(x-y))$, we know that the “ _flat
state_ ” is not the unique steady state for some $\beta<\beta_{\sharp}$
violating the uniqueness requirement of Properties A and B, see [11, Theorem
5.11].

###### Proof.

We will first argue that Properties A and B are equivalent. It follows from
the fact that
$\rho_{\infty}(\mathop{}\\!\mathup{d}x)\equiv\mathop{}\\!\mathup{d}x$ is
always a critical point of $E^{MF}$ (in the flat case) that if $E^{MF}$ has a
unique critical point, it must be $\rho_{\infty}$. For any mean-free $\eta\in
C_{0}^{\infty}(\Omega)$, we have from the previous calculation, that

$\displaystyle\frac{\langle-\mathcal{L}_{\rho_{\infty}}\eta,\eta\rangle_{W,\rho_{\beta}}}{\lVert\eta\rVert_{W,\rho_{\beta}}^{2}}=$
$\displaystyle\frac{1}{2}\frac{\int_{\Omega}\left\lvert\beta^{-1}\nabla\eta+\nabla
W*\eta\right\rvert^{2}\mathop{}\\!\mathup{d}x}{\int_{\Omega}(\beta^{-1}\eta+W*\eta)\eta\mathop{}\\!\mathup{d}x}\,,$
(4.12)

where we have used the fact that $\rho_{\beta}$ is the Lebesgue measure on
$\mathbb{T}^{d}$. It is easy to check that the above expression is strictly
positive if and only if
$\beta<\frac{1}{\min(0,-\min_{k\in\mathbb{Z}^{d}\setminus\\{0\\}}\hat{W}(k))}=\beta_{\sharp}$.
Similarly, computing the Fréchet derivative of $T$ at $\rho_{\infty}$, we
obtain

$\displaystyle D_{\rho}T[\rho_{\infty}]\eta=\eta+\beta W*\eta\,.$ (4.13)

Again, the above linear operator has a trivial kernel if and only if
$\beta<\beta_{\sharp}$. It follows that Properties A and B are equivalent.

We will now show that Property C implies Property B for
$\beta\neq\beta_{\sharp}$. We know that Property B is satisfied if and only if
$E^{MF}$ has unique critical point and $\beta<\beta_{\sharp}$. Assume it is
violated. Then, either $E^{MF}$ has more than one critical point or
$\beta\geq\beta_{\sharp}$ (or both).

Consider the case in which $\beta<\beta_{\sharp}$ but $E^{MF}$ has more than
one critical point one of which is $\rho_{\infty}$. Furthermore, we know from
[33, Lemma 5.6] that $\rho_{\infty}$ is a strict local minimum. If at least
one of the other critical points is not a minimiser then clearly Property C is
violated since the numerator can be chosen to be zero while the denominator
remains positive. We will now argue that this is the case. Assume it is not,
that is all other critical points are minimisers. Then, we can apply the
mountain pass theorem in $\mathcal{P}(\Omega)$ (using the fact that
$\rho_{\infty}$ is a strict local minimum) [33, Theorem 1.1] to construct a
new non-minimising critical point thus obtaining a contradiction.

Consider now the case in which $\beta>\beta_{\sharp}$. In this situation, we
note from [11, Proposition 5.3] that $\rho_{\infty}$ is a non-minimising
critical point of $E^{MF}$. Thus, Property C is again violated.

∎

######  Remark 4.5.

We note here that one would expect the Properties A, B and C to be equivalent
to each other. However, we unable to show that this holds true in general.

Given the above result, we are now finally in a position to present our
definition of a phase transition in the “ _flat case_ ”.

######  Definition 4.6 (Phase transition).

Assume $V\equiv 0,W(x,y)=W(x-y),$ and $\Omega=\mathbb{T}^{d}$. Then, we say
that the system (2.8) exhibits a phase transition of type A (resp. type B,
type C) if there exists a $0<\beta_{c}<\infty$ such that for all
$0<\beta<\beta_{c}$ Property A (resp. type B, type C) is satisfied, and for
$\beta_{c}<\beta$ Property A (resp. type B, type C) is violated.

Except for the Brownian mean field model see Eq. 3.30, we cannot ensure that
the existence of a phase transition in the sense of Definition 4.6. Combining
Theorem 3.14 with analysis in [15, 11], we can show the following result.

######  Theorem 4.7.

Assume $V\equiv 0,\;W(x,y)=W(x-y),$ and $\Omega=\mathbb{T}^{d}$. If $W$ is
$\mathbf{H}$-stable, that is to say $\hat{W}(k)\geq 0$ for all
$k\in\mathbb{Z}^{d}\setminus\\{0\\}$, then the system (2.8) satisfies
Properties A and B for all $\beta\in(0,\infty)$.

If there exists $k\in\mathbb{Z}^{d}\setminus\\{0\\}$ such that $\hat{W}(k)<0$,
then there exists $\beta_{1}<\beta_{\sharp}$ such that Properties A, B and C
are satisfied for all $\beta<\beta_{1}$, and Properties A, B and C are
violated for all $\beta>\beta_{\sharp}$.

###### Proof.

If $W$ is $\mathbf{H}$-stable, it follows from linear convexity
$\rho_{\infty}=\mathop{}\\!\mathup{d}x$ is the unique critical point of
$E^{MF}$ for all $0<\beta<\infty$, hence Properties A and B cannot be
violated, see [11, Proposition 5.8]. On the other hand, if
$\beta_{\sharp}<\infty$, we know from Proposition 4.3 that Properties A, B and
C are violated for all $\beta>\beta_{\sharp}$.

By Theorem 3.14 and Theorem 3.3, we know that there exists
$\beta_{\operatorname{LS}}$ such that $0<\beta<\beta_{\operatorname{LS}}$ then
Property C is satisfied. Finally, by Proposition 4.3 we have that Properties A
and B are also satisfied. ∎

##  5 The limit
$\mathcal{P}_{\mathrm{sym}}(\Omega^{N})\to\mathcal{P}(\mathcal{P}(\Omega))$

In this section, we recall some useful results that allows us the characterise
the various relevant objects in the limit as $N\to\infty$. We follow the
approach taken by Hauray and Mischler in [35]. We start by defining the
empirical measure.

######  Definition 5.1.

Given some $\rho^{N}\in\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$, the empirical
measure is the $\mathcal{P}(\Omega)$-valued random variable which is given by

$\mu^{(N)}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}\qquad\mbox{where
$(x_{1},...,x_{N})$ is distributed according $\rho^{N}$.}$

We denote the law of the empirical measure $\mu^{(N)}$ on
$\mathcal{P}(\Omega)$ by

$\hat{\rho}^{N}\in\mathcal{P}(\mathcal{P}(\Omega))\,.$

The topology we consider throughout most of the manuscript is the one induced
by the scaled 2-Wasserstein distance.

######  Definition 5.2.

Given $\rho_{1}^{N},\,\rho_{2}^{N}\in\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$,
the scaled 2-Wassertein distance between them is given by

$\overline{d}_{2}(\rho_{1}^{N},\rho_{2}^{N})=\inf_{\begin{subarray}{c}X\sim\rho_{1}^{N}\\\
Y\sim\rho_{2}^{N}\end{subarray}}\left(\frac{1}{N}\mathbb{E}[|X-Y|^{2}]\right)^{1/2},$

where $X,\,Y$ are $\Omega^{N}$-valued random variables.

Similarly, given $P_{1},\,P_{2}\in\mathcal{P}(\mathcal{P}(\Omega))$, the
2-Wasserstein distance between them is given by

$\mathfrak{D}_{2}(P_{1},P_{2})=\inf_{\begin{subarray}{c}\mathcal{X}\sim
P_{1}\\\ \mathcal{Y}\sim
P_{2}\end{subarray}}\left(\mathbb{E}[d_{2}^{2}(\mathcal{X},\mathcal{Y})]\right)^{1/2}$

where $\mathcal{X},\,\mathcal{Y}$ are $\mathcal{P}(\Omega)$-valued random
variables.

A fundamental result we need to understand the convergence
$\mathcal{P}_{\mathrm{sym}}(\Omega^{N})\to\mathcal{P}(\mathcal{P}(\Omega))$ is
the fact that the mapping $\rho^{N}\mapsto\hat{\rho}^{N}$ is an isometry for
the appropriately scaled 2-Wasserstein distance defined in Definition 5.2.
Indeed, we have the following result.

######  Theorem 5.3 ( [35, Proposition 2.14]).

For each
$\mu^{N},\,\nu^{N}\in\mathcal{P}_{\mathrm{\mathrm{sym}}}(\Omega^{N})$, we have

$\overline{d}_{2}(\mu^{N},\nu^{N})=\mathfrak{D}_{2}(\hat{\mu}^{N},\hat{\nu}^{N})\,.$

###### Proof.

We present only a sketch of the proof of this result. The first inequality
$\overline{d}_{2}(\mu^{N},\nu^{N})\geq\mathfrak{D}_{2}(\hat{\mu}^{N},\hat{\nu}^{N})$
follows from the mapping in Definition 5.1 $X\mapsto\mathcal{X}$, i.e. given
$X=(X_{1},...,X_{N})$ an $\Omega^{N}$-valued random variable with law
$\mu^{N}$, we define

$\mathcal{X}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}\,,$

is a $\mathcal{P}(\Omega)$-valued random variable with law $\hat{\mu}^{N}$.
The converse inequality follows from taking the inverse mapping and exploiting
the symmetry of $\mu^{N}$ and $\nu^{N}$. ∎

Using this result, we can provide a metric notion of the de
Finetti/Hewitt–Savage convergence.

######  Definition 5.4.

We say that the sequence $(\rho^{N})_{N\in\mathbb{N}}$ such that
$\rho^{N}\in\mathcal{P}_{\mathrm{\mathrm{\mathrm{sym}}}}(\Omega^{N})$
converges in the 2-Wasserstein distance to
$P_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))$ if

$\lim_{N\to\infty}\mathfrak{D}_{2}(\hat{\rho}^{N},P_{\infty})=0,$

where $\hat{\rho}^{N}$ is the law of the empirical measure (as
$\mathcal{P}(\Omega)$-valued random variable) associated to $\rho^{N}$.

We now have the following result which connects the above convergence to the
one introduced in (2.4).

######  Proposition 5.5.

For any sequence $(\rho^{N})_{N\in\mathbb{N}}$ with
$\rho^{N}\in\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$ and

$\sup_{N\in\mathbb{N}}\int_{\Omega}|x|^{\gamma}\;\mathop{}\\!\mathup{d}\rho_{1}^{N}(x)<\infty$
(5.1)

for some $\gamma>2$, the metric convergence in Definition 5.4 is equivalent to
the original marginal convergence of (2.4).

###### Proof.

We first notice that by the results of Diaconis and Freedman [21], for a fixed
$l\in\mathbb{N}$, the marginal $\rho^{N}_{l}$ coincides in the limit as
$N\to\infty$ with the $l$th product of the empirical measure. More
specifically, for any $\varphi\in C^{\infty}_{c}(\Omega^{l})$ we have that

$\left|\int_{\Omega^{l}}\varphi\;\mathop{}\\!\mathup{d}\rho^{N}_{l}-\int_{\mathcal{P}(\Omega)}\left(\int_{\Omega^{l}}\varphi\;\mathop{}\\!\mathup{d}\mu^{\otimes
l}\right)\;\mathop{}\\!\mathup{d}\hat{\rho}^{N}(\mu)\right|\leq
l^{2}\frac{\|\varphi\|_{L^{\infty}(\Omega^{l})}}{N}\,.$ (5.2)

Furthermore, (5.1) implies that $\hat{\rho}^{N}$ is tight in
$\mathcal{P}(\Omega)$. Indeed, we have

$\displaystyle\int_{\mathcal{P}(\Omega)}d_{2}^{2\gamma}(\rho,\delta_{0})\;\mathop{}\\!\mathup{d}\hat{\rho}^{N}(\rho)=\int_{\Omega}|x|^{\gamma}\;\mathop{}\\!\mathup{d}\rho^{N}_{1}\,.$
(5.3)

Now if $\hat{\rho}^{N}$ converges in the sense of Definition 5.4 to
$P_{\infty}$ we know that it must converge when tested against every
$C_{b}(\mathcal{P}(\Omega))$. Since cylindrical test functions are a subset of
$C_{b}(\Omega)$, it is clear from (5.2) that $\rho^{N}$ must also converge to
$P_{\infty}$ in the sense of (2.4).

On the other hand if $\rho^{N}$ converges to $P_{\infty}$ in the sense of
(2.4) we know that, under (5.1), the sequence $\hat{\rho}^{N}$ is relatively
compact in $(\mathcal{P}(\mathcal{P}(\Omega)),\mathfrak{D}_{2})$. By the
Stone–Weierstrass theorem cylindrical functions are dense in
$C_{b}(\mathcal{P}(\Omega))$ (see [52, Theorem 2.1 ]), which tells us that the
limits in (5.2) and Definition 5.4 coincide. ∎

Next, we discuss the limit of the associated free energy and dissipation
functionals.

######  Theorem 5.6.

Under Assumptions 2.1 and 2.2, consider the sequence
$(\rho^{N})_{N\in\mathbb{N}}$ with
$\rho^{N}\in\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$ such that $\rho^{N}\to
P_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))$ in the sense of Definition 5.4.
Then,

$\liminf_{N\to\infty}E^{N}[\rho^{N}]\geq
E^{\infty}[P_{\infty}]\qquad\mbox{and}\qquad\liminf_{N\to\infty}\overline{\mathcal{I}}(\rho^{N}|M_{N})\geq\int_{\mathcal{P}(\Omega)}D(\rho)\;dP_{\infty}(\rho)$
(5.4)

where $E^{N}$, $E^{\infty}$, $\overline{\mathcal{I}}$, and $D(\rho)$ are
defined by (2.13), (2.14), (3.2), and (3.5), respectively. Moreover, for any
$P_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))$ the sequence

$\rho^{N}_{*}=\int_{\mathcal{P}(\Omega)}\rho^{\otimes
N}\;\mathop{}\\!\mathup{d}P_{\infty}(\rho)\in\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$

attains the limits the limits, i.e.

$\lim_{N\to\infty}E^{N}[\rho_{*}^{N}]=E^{\infty}[P_{\infty}]\qquad\mbox{and}\qquad\lim_{N\to\infty}\overline{\mathcal{I}}(\rho_{*}^{N}|M_{N})=\int_{\mathcal{P}(\Omega)}D(\rho)\;dP_{\infty}(\rho)$
(5.5)

###### Proof.

The convergence of the relative entropy functional follows directly from the
classical arguments in [47]. A more modern proof can be found in [35]. The
convergence of the relative Fisher information with respect to the Lebesgue
measure is covered in [35]. Our case is slightly more involved due to the
minimal regularity assumptions on the potentials. Formally, expanding the
square we obtain after integrating by parts

$\displaystyle\overline{\mathcal{I}}(\rho^{N},M_{N})=$
$\displaystyle\frac{1}{N}\int_{\Omega^{N}}|\nabla\log\rho^{N}-\nabla
H^{N}|^{2}\mathop{}\\!\mathup{d}\rho^{N}$ (5.6) $\displaystyle=$
$\displaystyle\int_{\Omega^{N}}(|\nabla\log\rho^{N}|^{2}+2\Delta H^{N}+|\nabla
H^{N}|^{2})\mathop{}\\!\mathup{d}\rho^{N}\,.$ (5.7)

The first term of the above expression is exactly the relative Fisher
information with respect to the Lebesgue measure and so is already covered in
[35]. Under stronger regularity assumptions on the potentials, the second and
third term fall within the type of functionals already considered in the
classical [47]. The main obstruction to conclude under Assumptions 2.1 and 2.2
is that $\Delta V$ and $\Delta W$ are merely signed measures bounded below, so
we need to adapt the proofs of [35] to a lower regularity setting. We will
circumvent the regularity problem, by appealing again and again to convexity.
More specifically, by Assumptions 2.1 and 2.2, we know that

$\overline{\mathcal{I}}^{1/2}(\rho^{N}|M_{N})=\sup_{\nu^{N}\in\mathcal{P}_{\mathrm{sym}}(\Omega^{N})}\left(\frac{E^{N}[\rho^{N}]-E^{N}[\nu^{N}]}{\overline{d}_{2}(\rho^{N},\nu^{N})}-\frac{K_{V}+K_{W}}{2}\overline{d}_{2}(\rho^{N},\nu^{N})\right)_{+}\,,$
(5.8)

see [1, Theorem 2.4.9]. We pick $\nu^{N}$ to be the recovery sequence for a
generic $Q_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))$, using the lower
semicontinuous convergence of $E^{N}$, and the isometry Theorem 5.3, to obtain

$\liminf_{N\to\infty}\overline{\mathcal{I}}^{1/2}(\rho^{N}|M_{N})\geq\sup_{Q_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))}\left(\frac{E^{\infty}[P_{\infty}]-E^{\infty}[Q_{\infty}]}{\mathfrak{D}_{2}(P_{\infty},Q_{\infty})}-\frac{K_{V}+K_{W}}{2}\mathfrak{D}_{2}(P_{\infty},Q_{\infty})\right)_{+}.$

To obtain equality, we consider
$S_{t}:\mathcal{P}(\Omega)\to\mathcal{P}(\Omega)$ the solution operator
associated to (2.8) which is well-defined by B and consider the curve
$Q_{\infty}=S_{t}\\#P_{\infty}$, which coincides with the unique gradient flow
of $E^{\infty}$ with respect to $\mathfrak{D}_{2}$ (see [10, Lemma 19]).
Taking $t\to 0^{+}$, we obtain

$\displaystyle\lim_{t\to 0^{+}}$
$\displaystyle\left(\frac{E^{\infty}[P_{\infty}]-E^{\infty}[S_{t}\\#P_{\infty}]}{\mathfrak{D}_{2}(P_{\infty},S_{t}\\#P_{\infty})}-\frac{K_{V}+K_{W}}{2}\mathfrak{D}_{2}(P_{\infty},S_{t}\\#P_{\infty})\right)_{+}$
(5.9) $\displaystyle=$
$\displaystyle\left(\int_{\mathcal{P}(\Omega)}D(\rho)\;\mathop{}\\!\mathup{d}P_{\infty}(\rho)\right)^{1/2}\,,$
(5.10)

obtaining the desired lower semicontinuity result.

$\liminf_{N\to\infty}\mathcal{I}(\rho^{N}|M_{N})\geq\int_{\mathcal{P}(\Omega)}D(\rho)\;\mathop{}\\!\mathup{d}P_{\infty}(\rho)$
(5.11)

Now we focus on showing that given $P\in\mathcal{P}(\mathcal{P}(\Omega))$, the
sequence

$\rho_{*}^{N}=\int_{\mathcal{P}(\Omega)}\rho^{\otimes
N}\;\mathop{}\\!\mathup{d}P(\rho)$ (5.12)

attains the limit (5.5). For convenience, we consider the auxiliary functional
$\mathcal{I}^{\prime}:\mathcal{P}(\mathcal{P}(\Omega))\to\mathbb{R}\cup\\{\infty\\}$,
which is given by

$\mathcal{I}^{\prime}[P]=\lim_{N\to\infty}\overline{\mathcal{I}}(\rho_{*}^{N}|M_{N}),$

where $\rho_{*}^{N}$ is given by (5.12). Next, we show that
$\mathcal{I}^{\prime}$ is well defined and that it is weakly lower
semicontinuous.

To show this, we regularize $\rho_{*}^{N}$ by using the $N$-particle dynamics
(2.3). Using the convexity in Assumptions 2.1 and 2.2, we obtain that

$e^{t(\lambda_{V}+\lambda_{W})}\overline{\mathcal{I}}(S^{N}_{t}\rho_{*}^{N}|M_{N})\leq\overline{\mathcal{I}}(\rho_{*}^{N}|M_{N})\,,$

where $S^{N}_{t}:\mathcal{P}(\Omega^{N})\to\mathcal{P}(\Omega^{N})$ is the
solution operator associated to the Fokker–Planck equation (2.3). Using the
weak lower semicontinuity of $\overline{\mathcal{I}}$ from the expression
(5.8), we obtain

$\lim_{t\to
0^{+}}\overline{\mathcal{I}}(S^{N}_{t}\rho_{*}^{N}|M_{N})=\overline{\mathcal{I}}(\rho_{*}^{N}|M_{N})\,.$
(5.13)

Using the fact that $S_{t}^{N}$ is immediately regularizing, we can use the
formal expression (5.7) freely for the measure $S^{N}_{t}\rho_{*}^{N}$. By
using the sub-additivity of the Fisher information, we can notice that for any
$3<j<N$

$\overline{\mathcal{I}}((S^{N}_{t}\rho_{*}^{N})_{j}|M_{j})\leq\overline{\mathcal{I}}(S^{N}_{t}\rho_{*}^{N}|M_{N})\,,$
(5.14)

where $(S^{N}_{t}\rho_{*}^{N})_{j}$ is the $j$th marginal of
$S^{N}_{t}\rho_{*}^{N}$. Taking $t\to 0^{+}$, using (5.13) and the weak lower
semicontinuity from (5.8), we obtain for any $3<j<N$

$\overline{\mathcal{I}}(\rho^{j}_{*}|M_{j})\leq\overline{\mathcal{I}}(\rho_{*}^{N}|M_{N})\,,$

which implies that we can characterise

$\mathcal{I}^{\prime}[P]=\sup_{N\geq
3}\overline{\mathcal{I}}(\rho_{*}^{N}|M_{N}).$

By the weak lower semicontinuity of $\overline{\mathcal{I}}$, we obtain the
weak lower semicontinuity of $\mathcal{I}^{\prime}$.

Using (5.14) and the propagation of chaos result of [10], we obtain that

$e^{t(\lambda_{V}+\lambda_{W})}\overline{\mathcal{I}}(\rho_{t,*}^{j}|M_{j})\leq
e^{t(\lambda_{V}+\lambda_{W})}\liminf_{N\to\infty}\overline{\mathcal{I}}((S^{N}_{t}\rho_{*}^{N})_{j}|M_{j})\leq\mathcal{I}^{\prime}[P]\,,$
(5.15)

where

$\rho_{t,*}^{j}=\int_{\mathcal{P}(\Omega)}\rho^{\otimes
j}\;\mathop{}\\!\mathup{d}(S_{t}\\#P)(\rho)=\int_{\mathcal{P}(\Omega)}(S_{t}\rho)^{\otimes
j}\;\mathop{}\\!\mathup{d}P(\rho)$

with $S_{t}$ the solution operator to (2.8). Taking $j\to\infty$ in (5.15), we
obtain the inequality

$e^{t(\lambda_{V}+\lambda_{W})}\int_{\mathcal{P}(\Omega)}D(S_{t}\\#\rho)\;\mathop{}\\!\mathup{d}P(\rho)=e^{t(\lambda_{V}+\lambda_{W})}\mathcal{I}^{\prime}[S_{t}\\#P]\leq\mathcal{I}^{\prime}[P]\,,$

where the equality in the left hand side follows from using the expression
(5.7) and the result for the standard Fisher information in [35]. Taking $t\to
0^{+}$ and using the lower semicontinuity of $\mathcal{I}^{\prime}$ we have
showed above, we obtain the desired conclusion (5.5)

$\mathcal{I}^{\prime}[P]=\int_{\mathcal{P}(\Omega)}D(\rho)\;\mathop{}\\!\mathup{d}P(\rho).$

∎

Standard arguments also yield the convergence of the minimal energy.

######  Corollary 5.7.

Any accumulation point $P_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))$ of the
sequence $(M_{N})_{N\in\mathbb{N}}$ satisfies

$\mathrm{supp}(P_{\infty})\subset\\{\rho\;:\;E^{MF}[\rho]=\min_{\mathcal{P}(\Omega)}E^{MF}\\}$

and

$E^{N}[M_{N}]=-\frac{1}{N}\log(Z_{N})\stackrel{{\scriptstyle
N\to\infty}}{{\to}}\min_{\mathcal{P}(\Omega)}E^{MF}\,.$ (5.16)

We remark that the convergence (5.16) coincides with the standard definition
of the thermodynamic limit from statistical mechanics (see [36, Ch. 3]).

######  Theorem 5.8 (HWI inequality).

Let $V$ and $W$ be $K$-convex. Given two arbitrary probability measures
$\mu^{N},\,\nu^{N}\in\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$, it holds

$\overline{\mathcal{E}}(\mu^{N}|M_{N})\leq\overline{\mathcal{E}}(\nu^{N}|M_{N})+\overline{d}_{2}(\mu^{N},\nu^{N})\sqrt{\overline{\mathcal{I}}(\mu^{N}|M_{N})}-\frac{K}{2}\overline{d}^{2}_{2}(\mu^{N},\nu^{N})\,.$
(5.17)

###### Proof.

The proof of this result can be found in [61, Theorem 30.22]. We just present
the main idea of the proof for the reader’s convenience. We consider
$\gamma:[0,1]\to\mathcal{P}_{\mathrm{sym}}(\Omega^{N})$ the 2-Wasserstein
geodesic between $\mu^{N}$ and $\nu^{N}$. Under the hypothesis of $K$
convexity of the potentials, we obtain that
$t\to\overline{\mathcal{E}}(\gamma(t)|M_{N})$ is $K$-convex, which implies the
desired inequality (5.17). ∎

######  Corollary 5.9.

Assume that Assumptions 2.1 and 2.2 hold true and that the sequence
$(\rho^{N})_{N\in\mathbb{N}}$ with
$\rho^{N}\in\mathcal{P}_{\mathrm{\mathrm{sym}}}(\Omega^{N})$ such that
$\rho^{N}$ converges to $P_{\infty}\in\mathcal{P}(\mathcal{P}(\Omega))$ in the
sense of Definition 5.4 and

$\sup_{N\in\mathbb{N}}\overline{\mathcal{I}}(\rho^{N}|M_{N})<\infty\,,$ (5.18)

then

$\lim_{N\to\infty}\overline{\mathcal{E}}(\rho^{N}|M_{N})=E^{\infty}[P_{\infty}]-\min_{\mathcal{P}(\Omega)}E^{MF}.$
(5.19)

###### Proof.

Given $P_{\infty}$, we consider the recovery sequence
$(\rho^{N}_{*})_{N\in\mathbb{N}}$ from Theorem 5.6. By the HWI inequality
(5.17), we have

$\overline{\mathcal{E}}(\rho^{N}|M_{N})\leq\overline{\mathcal{E}}(\rho_{*}^{N}|M_{N})+\overline{d}_{2}(\rho^{N},\rho_{*}^{N})\sqrt{\overline{\mathcal{I}}(\rho^{N}|M_{N})}-\frac{K}{2}\overline{d}^{2}_{2}(\rho^{N},\rho_{*}^{N})\,.$
(5.20)

Using that both sequences converge to $P_{\infty}$ and that
$(\rho^{N}_{*})_{N\in\mathbb{N}}$ is a recovery sequence, we obtain

$\limsup_{N\to\infty}\left(E^{N}[\rho^{N}]-E^{N}[M_{N}]\right)=\limsup_{N\to\infty}\overline{\mathcal{E}}(\rho^{N}|M_{N})\leq
E^{\infty}[P_{\infty}]-\min_{\mathcal{P}(\Omega)}E^{MF}\,.$ (5.21)

The reverse inequality follows from Theorem 5.6. ∎

######  Remark 5.10.

A version of this result without the potentials $V$ and $W$ can be found in
[35].

##  6 Proof of Theorem 3.2

###### Proof of Theorem 3.2.

We prove the result only for $E^{MF}$, as the proof for $E^{N}$ and even more
general energies $E:\mathcal{P}(\Omega)\to[0,\infty]$ is analogous, see Remark
6.1. We consider $\rho:[0,\infty)\to\mathcal{P}(\Omega)$, the unique
2-Wasserstein gradient flow of $E^{MF}$ with initial condition
$\rho_{0}\in\mathcal{P}(\Omega)$, see B.

We notice that if $\lambda^{\infty}_{\operatorname{LS}}=0$, then there is
nothing left to prove. Hence, we can assume without loss of generality that
$\lambda^{\infty}_{\operatorname{LS}}>0$, which implies (cf. Properties C and
4.1) that $E^{MF}$ does not admit non-minimising steady state. Using the
version of LaSalle’s invariance principle for gradient flows proved in [12,
Theorem 4.11], we know that $\rho(t)$ accumulates on the set of steady states
of $E^{MF}$, as $t\to\infty$. Using the fact that all steady states are
minimisers, we can find a sequence of times $t_{n}\to\infty$ such that

$\rho(t_{n})\rightharpoonup\rho_{\infty}$ (6.1)

for some $\rho_{\infty}\in\mathcal{K}$. Differentiating $E^{MF}[\rho(t)]$ with
respect to time and using that $\lambda^{\infty}_{\operatorname{LS}}>0$ we
obtain

$\displaystyle\frac{\mathop{}\\!\mathup{d}}{\mathop{}\\!\mathup{d}t}\left(E^{MF}[\rho(t)]-\inf_{\mathcal{P}(\Omega)}E^{MF}\right)$
$\displaystyle=-\int_{\Omega}\left\lvert\beta^{-1}\nabla\log\rho(t)+\nabla
V+\nabla W\star\rho(t)\right\rvert^{2}\;\mathop{}\\!\mathup{d}\rho(t)$ (6.2)
$\displaystyle\leq$
$\displaystyle-\sqrt{\lambda_{\operatorname{LS}}^{\infty}}\left(E^{MF}[\rho(t)]-\inf_{\mathcal{P}(\Omega)}E^{MF}\right)^{1/2}$
(6.3)
$\displaystyle\times\left(\int_{\Omega}\left\lvert\beta^{-1}\nabla\log\rho(t)+\nabla
V+\nabla
W\star\rho(t)\right\rvert^{2}\;\mathop{}\\!\mathup{d}\rho(t)\right)^{1/2}.$
(6.4)

Integrating from $0$ to $t_{n}$ for some $n\in\mathbb{N}$, we obtain

$\displaystyle\left(E^{MF}[\rho(t_{n})]-\inf_{\mathcal{P}(\Omega)}E^{MF}\right)^{1/2}-\left(E^{MF}[\rho_{0}]-\inf_{\mathcal{P}(\Omega)}E^{MF}\right)^{1/2}$
(6.5) $\displaystyle\leq$
$\displaystyle-\left(\frac{\lambda_{\operatorname{LS}}^{\infty}}{2}\right)^{1/2}\int_{0}^{t_{n}}\left(\frac{1}{2}\int_{\Omega}|\beta^{-1}\nabla\log\rho(s)+\nabla
V+\nabla
W\star\rho(s)|^{2}\;\mathop{}\\!\mathup{d}\rho(s)\right)^{1/2}\;\mathop{}\\!\mathup{d}s$
(6.6) $\displaystyle\leq$
$\displaystyle-\left(\frac{\lambda_{\operatorname{LS}}^{\infty}}{2}\right)^{1/2}d_{2}(\rho_{0},\rho(t_{n})),$
(6.7)

where the last inequality follows from the Benamou–Brenier [7] formulation of
the 2-Wasserstein distance. Taking $t_{n}\to\infty$, applying (6.1), and
rearranging, we end up with

$\displaystyle\left(E^{MF}[\rho_{0}]-\inf_{\mathcal{P}(\Omega)}E^{MF}\right)^{1/2}\geq\left(\frac{\lambda_{\operatorname{LS}}^{\infty}}{2}\right)^{1/2}d_{2}(\rho_{0},\rho_{\infty})\geq\left(\frac{\lambda_{\operatorname{LS}}^{\infty}}{2}\right)^{1/2}d_{2}(\rho_{0},\mathcal{K}).$
(6.8)

The desired inequality (3.8) follows by squaring both sides. ∎

######  Remark 6.1.

This proof can easily be adapted to general
$E:\mathcal{P}(\Omega)\to(0,\infty]$, that are regular enough to admit
gradient flow solutions from arbitrary initial data, and that sub-level sets
are weakly compact to ensure convergence of the gradient flows to steady
states [12, Theorem 2.12].

##  7 Proof of Theorem 3.3

###### Proof of Theorem 3.3.

For the proof of the first part of the theorem, we consider some
$\rho\in\mathcal{P}(\Omega)$ such that
$E^{MF}[\rho]>\inf_{\mathcal{P}(\Omega)}E^{MF}>-\infty$, see A. By Theorem 5.6
and Corollary 5.9, we can find a recovery sequence $P^{N}=\rho^{\otimes N}$,
such that

$\lim_{N\to\infty}\overline{\mathcal{E}}(P^{N}|M_{N})=E^{MF}[\rho]-\inf_{\mathcal{P}(\Omega)}E^{MF}\qquad\mbox{and}\qquad\lim_{N\to\infty}\overline{\mathcal{I}}(P^{N}|M_{N})=D(\rho).$
(7.1)

Using the definition of the log Sobolev constant and passing to the limit as
$N\to\infty$, we observe

$\limsup_{N\to\infty}\lambda^{N}_{\operatorname{LS}}\leq\limsup_{N\to\infty}\frac{\overline{\mathcal{I}}(P^{N}|M_{N})}{\overline{\mathcal{E}}(P^{N}|M_{N})}=\frac{D(\rho)}{E^{MF}[\rho]-\inf
E^{MF}}.$

Taking the infimum over all $\rho\notin\mathcal{K}$, we obtain

$\limsup_{N\to\infty}\lambda^{N}_{\operatorname{LS}}\leq\lambda^{\infty}_{\operatorname{LS}}=\inf_{\rho\notin\mathcal{K}}\frac{D(\rho)}{E^{MF}[\rho]-\inf_{\mathcal{P}(\Omega)}E^{MF}}.$

This completes the proof of the first half of the theorem.

Next, we consider the case when there exists $\rho_{*}\in\mathcal{P}(\Omega)$
a non-minimising steady state, that is to say

$D(\rho_{*})=0\qquad\mbox{and}\qquad
E^{MF}[\rho_{*}]>\inf_{\mathcal{P}(\Omega)}E^{MF}.$

Consider now the sequence $\rho_{*}^{\otimes
N}\in\mathcal{P}_{\mathrm{\mathrm{sym}}}(\Omega^{N})$. We have

$\displaystyle\lambda_{N}^{\operatorname{LS}}\leq$
$\displaystyle\frac{\overline{\mathcal{I}}(\rho_{*}^{\otimes
N}|M_{N})}{\overline{\mathcal{E}}(\rho_{*}^{\otimes N}|M_{N})}$ (7.2)

By Theorem 5.6 and Corollary 5.9, we can pass to the limit in the denominator
to obtain

$\lim_{N\to\infty}\overline{\mathcal{E}}(\rho_{*}^{\otimes
N}|M_{N})=E^{MF}[\rho_{*}]-\inf_{\mathcal{P}(\Omega)}E^{MF}>0.$

Thus, we have that for $N$ large enough

$\lambda_{N}^{\operatorname{LS}}\leq\frac{2}{E^{MF}[\rho_{*}]-\inf_{\mathcal{P}(\Omega)}E^{MF}}\overline{\mathcal{I}}(\rho_{*}^{\otimes
N}|M_{N})\,.$ (7.3)

The desired bound

$\lambda_{N}^{\operatorname{LS}}\leq\frac{C}{N}$

follows from the decay estimate for the Fisher information proved in Lemma
7.1. ∎

######  Lemma 7.1.

Under Assumptions 2.1 and 2.2, assume that $\rho_{*}$ is a critical point of
the mean field free energy $E^{MF}$, then we have the following bound

$\displaystyle\mathcal{I}(\rho_{*}^{\otimes N}|M_{N})$ $\displaystyle=$
$\displaystyle\left(1-\frac{1}{N}\right)\int_{\Omega}\left(|\nabla_{1}W|^{2}\star\rho_{*}(x_{1})-\left\lvert\nabla_{1}W\star\rho_{*}(x_{1})\right\rvert^{2}\right)\rho_{*}(x_{1})\;\mathop{}\\!\mathup{d}x_{1}$
$\displaystyle+\frac{1}{N}\int\left\lvert\nabla_{1}W\star\rho_{*}(x_{1})\right\rvert^{2}\rho_{*}(x_{1})\mathop{}\\!\mathup{d}x_{1}$
$\displaystyle\leq C\,.$

###### Proof of Lemma 7.1.

We start by expanding $\mathcal{I}(\rho_{*}^{\otimes N}|M_{N})$ as follows

$\displaystyle\mathcal{I}(\rho_{*}^{\otimes N}|M_{N})=$
$\displaystyle\int_{\Omega^{N}}\left|\nabla\left(\log\rho_{*}^{\otimes
N}+\beta\sum_{i}V(x_{i})+\frac{\beta}{2N}\sum_{i,j}W(x_{i},x_{j})\right)\right|^{2}\rho_{*}^{\otimes
N}\;\mathop{}\\!\mathup{d}x$ (7.4) $\displaystyle=$
$\displaystyle\sum_{i}\int_{\Omega^{N}}\left|\frac{\nabla\rho_{*}(x_{i})}{\rho_{*}(x_{i})}+\beta\nabla
V(x_{i})+\frac{\beta}{N}\sum_{j}\nabla_{1}W(x_{i},x_{j})\right|^{2}\rho_{*}^{\otimes
N}\;\mathop{}\\!\mathup{d}x$ (7.5) $\displaystyle=$ $\displaystyle
N\int_{\Omega^{N}}\left|-\beta\nabla_{1}W\star\rho_{*}(x_{1})+\frac{\beta}{N}\sum_{j}\nabla_{1}W(x_{1},x_{j})\right|^{2}\rho_{*}^{\otimes
N}\;\mathop{}\\!\mathup{d}x$ (7.6) $\displaystyle=$ $\displaystyle
N\beta^{2}\int_{\Omega^{N}}\left(\left\lvert\nabla_{1}W\star\rho_{*}(x_{1})\right\rvert^{2}-\frac{2}{N}\nabla_{1}W\star\rho_{*}(x_{1})\cdot\sum_{j}\nabla_{1}W(x_{1},x_{j})\right.$
(7.7)
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad+\frac{1}{N^{2}}\sum_{j,k}\nabla_{1}W(x_{1},x_{j})\cdot\nabla_{1}W(x_{1},x_{k})\rho_{*}^{\otimes
N}\right)\;\mathop{}\\!\mathup{d}x,$ (7.8)

where we have used symmetry of the particle system to write the integrand in
terms of the variable $x_{1}$ and that $\rho_{*}$ is critical point of the
mean field energy $E^{MF}$ which itself implies that

$\beta^{-1}\log\rho_{*}+W\star\rho_{*}+V=C.$

Next, proceeding term by term, we notice the following cancellation (for
simplicity, we assume that $W(x,x)=0$, otherwise we can change $V$ by an
additive constant such that this holds):

$\displaystyle\int_{\Omega^{N}}\left\lvert\nabla_{1}W\star\rho_{*}(x_{1})\right\rvert^{2}\rho_{*}^{\otimes
N}\;\mathop{}\\!\mathup{d}x$ $\displaystyle=$
$\displaystyle\int_{\Omega}\left\lvert\nabla_{1}W\star\rho_{*}(x_{1})\right\rvert^{2}\rho_{*}(x_{1})\;\mathop{}\\!\mathup{d}x_{1},$
$\displaystyle-\int_{\Omega^{N}}\frac{2}{N}\nabla_{1}W\star\rho_{*}(x_{1})\cdot\sum_{j}\nabla_{1}W(x_{1},x_{j})\rho_{*}^{\otimes
N}\;\mathop{}\\!\mathup{d}x$ $\displaystyle=$
$\displaystyle-\left(2-\frac{2}{N}\right)\int_{\Omega}\left\lvert\nabla_{1}W\star\rho_{*}(x_{1})\right\rvert^{2}\rho_{*}(x_{1})\;\mathop{}\\!\mathup{d}x_{1},$

and

$\displaystyle\int_{\Omega^{N}}\frac{1}{N^{2}}\sum_{j,k}\nabla_{1}W(x_{1},x_{j})\cdot\nabla_{1}W(x_{1},x_{k})\rho_{*}^{\otimes
N}\;\mathop{}\\!\mathup{d}x=$
$\displaystyle\frac{(N-1)(N-2)}{N^{2}}\int_{\Omega}\left\lvert\nabla_{1}W\star\rho_{*}(x_{1})\right\rvert^{2}\rho_{*}(x_{1})\;\mathop{}\\!\mathup{d}x_{1}$
(7.9)
$\displaystyle+\frac{N-1}{N^{2}}\int\left\lvert\nabla_{1}W\right\rvert^{2}\star\rho_{*}(x_{1})\rho_{*}(x_{1})\mathop{}\\!\mathup{d}x_{1}\,.$
(7.10)

Putting the previous identities together, we obtain

$\displaystyle\mathcal{I}(\rho_{\beta}^{\otimes N}|M_{N})=$
$\displaystyle\left(1-\frac{1}{N}\right)\int_{\Omega}\left(\left\lvert\nabla_{1}W\right\rvert^{2}\star\rho_{*}(x_{1})-\left\lvert\nabla_{1}W\star\rho_{*}(x_{1})\right\rvert^{2}\right)\rho_{*}(x_{1})\;\mathop{}\\!\mathup{d}x_{1}$
(7.11)
$\displaystyle+\frac{1}{N}\int\left\lvert\nabla_{1}W\star\rho_{*}(x_{1})\right\rvert^{2}\rho_{*}(x_{1})\mathop{}\\!\mathup{d}x_{1}\,.$
(7.12)

Finally, the second inequality in the statement follows from applying Jensen’s
inequality to obtain

$\left\lvert\nabla_{1}W\right\rvert^{2}\star\rho_{*}(x)\geq\left\lvert\nabla_{1}W\star\rho_{*}(x)\right\rvert^{2}\qquad\mbox{for
every $x\in\Omega$.}$

The final bound follows from the assumption that $W$ grows at most
polynomially (cf. Eq. 2.2) and the fact that all steady states have finite
moments of every order by Proposition 4.1. ∎

##  8 Proof of Theorem 3.4

###### Proof of Theorem 3.4.

We prove the statement of the theorem by contradiction. To this end, we assume
that there exists a sequence $N_{i}\to\infty$ and a sequence of times
$t_{i}>0$

$0<t_{i}<\frac{2}{\lambda^{\infty}_{\operatorname{LS}}}\log((E^{N_{i}}[\rho_{\mathrm{in}}^{\otimes
N_{i}}]-E^{N_{i}}[M_{N_{i}}])/\varepsilon)$ (8.1)

such that

$\overline{\mathcal{E}}(\rho^{N_{i}}(t_{i})|M_{N_{i}})>\max(\varepsilon,e^{-\frac{\lambda_{\operatorname{LS}}^{\infty}t_{i}}{2}}(E^{N_{i}}[\rho_{\mathrm{in}}^{\otimes
N_{i}}]-E^{N_{i}}[M_{N_{i}}]).$

Using the finite energy assumption on the initial condition (3.13), and
Assumptions 2.1 and 2.2 which imply that $E^{N}$ is uniformly bounded below,
we have that

$\sup_{i}t_{i}\leq\sup_{i}\frac{2}{\lambda^{\infty}_{LSI}}\log((E^{N_{i}}[\rho_{\mathrm{in}}^{\otimes
N_{i}}]-E^{N_{i}}[M_{N_{i}}])/\varepsilon)=:T(\varepsilon)<\infty.$ (8.2)

By the relative entropy dissipation estimate (3.2), we obtain

$\overline{\mathcal{E}}(\rho^{N_{i}}(t_{i})|M_{N_{i}})>e^{-\frac{\lambda_{\operatorname{LS}}^{\infty}t_{i}}{2}}(E^{N_{i}}[\rho_{\mathrm{in}}^{\otimes
N_{i}}]-E^{N_{i}}[M_{N_{i}}])$

which implies that there exists at least one $s_{i}\in(0,t_{i})$ such that

$\frac{\beta^{-1}\overline{\mathcal{I}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})}{\overline{\mathcal{E}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})}<\frac{\lambda_{\operatorname{LS}}^{\infty}}{2}.$

The contradiction will arise if we can show that

$\liminf_{i\to\infty}\frac{\beta^{-1}\overline{\mathcal{I}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})}{\overline{\mathcal{E}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})}\geq\lambda^{\infty}_{\operatorname{LS}}=\inf_{\rho\notin\mathcal{K}}\frac{D(\rho)}{E^{MF}[\rho]-\inf_{\mathcal{P}(\Omega)}E^{MF}}.$

First, we consider the case when

$\liminf_{i\to\infty}\overline{\mathcal{I}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})=\infty\,.$

By using the monotonicity of the relative entropy (3.2) and the finite energy
assumption in the statement of the theorem, we obtain

$\overline{\mathcal{E}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})\leq\overline{\mathcal{E}}(\rho_{\mathrm{in}}^{\otimes
N_{i}}|M_{N_{i}})=E^{MF}[\rho_{\mathrm{in}}]-E^{N}[M_{N}]<\infty.$

Hence, we can conclude

$\liminf_{i\to\infty}\frac{\beta^{-1}\overline{\mathcal{I}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})}{\overline{\mathcal{E}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})}=\infty>\lambda^{\infty}_{\operatorname{LS}}.$

Therefore, we can assume that up to a subsequence, which we do not relabel,

$\sup_{i}\overline{\mathcal{I}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})<\infty.$

By the bounded higher moment hypothesis in (3.13), the uniform boundedness of
$s_{i}$ (8.2), and the propagation of moments along the flow for $K$-convex
potentials [10], we have that $\rho^{N_{i}}(s_{i})$ has uniformly bounded
moments of order $2+\delta$. By Proposition 5.5, we obtain that there exists
$P_{*}\in\mathcal{P}(\mathcal{P}(\mathbb{R}^{d}))$, such that up to a (not
relabeled) subsequence, $\rho^{N_{i}}(s_{i})\to P_{*}$ in the sense of
Definition 5.4. By the HWI inequality Theorem 5.8 and Corollary 5.9, we obtain
the strong convergence in the relative entropy term, which yields

$\varepsilon\leq\lim_{i\to\infty}\overline{\mathcal{E}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})=\int_{\mathcal{P}(\mathbb{R}^{d})}E^{MF}[\rho]-\inf
E^{MF}\;\mathop{}\\!\mathup{d}P_{*}(\rho).$

Combining this limit with the $\liminf$-inequality of Theorem 5.6 for the
dissipation we obtain that

$\liminf_{i\to\infty}\frac{\beta^{-1}\overline{\mathcal{I}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})}{\overline{\mathcal{E}}(\rho^{N_{i}}(s_{i})|M_{N_{i}})}\geq\frac{\int_{\mathcal{P}(\mathbb{R}^{d})}D(\rho)\;\mathop{}\\!\mathup{d}P_{*}(\rho)}{\int_{\mathcal{P}(\mathbb{R}^{d})}E^{MF}[\rho]-\inf
E^{MF}\;\mathop{}\\!\mathup{d}P_{*}(\rho)}\geq\lambda^{\infty}_{\operatorname{LS}},$

where the last inequality follows from the point-wise inequality

$D(\rho)\geq\lambda^{\infty}_{\operatorname{LS}}(E^{MF}[\rho]-\inf E^{MF})\,.$

This is the desired contradiction and the result now follows. ∎

##  9 Proof of Theorem 3.6

###### Proof of Theorem 3.6.

The uniqueness of the minimiser in the limit follows from applying the
Talagrand inequality (3.8), which states that the energy grows quadratically
around the Gibbs measure $M_{N}$. We take $\rho_{1}$ and $\rho_{2}$, two
minimisers of $E^{MF}$, and show that they must coincide. By the triangle and
Talangrand inequality, we have

$\displaystyle
d^{2}_{2}(\rho_{1},\rho_{2})=\overline{d}^{2}_{2}(\rho_{1}^{\otimes
N},\rho_{2}^{\otimes N})$ $\displaystyle\leq
2\overline{d}^{2}_{2}(\rho_{1}^{\otimes
N},M_{N})+2\overline{d}^{2}_{2}(M_{N},\rho_{2}^{\otimes N})$ (9.1)
$\displaystyle=\frac{4}{\lambda^{N}_{\operatorname{LS}}}(E^{MF}[\rho_{1}]+E^{MF}[\rho_{2}]-2E^{N}[M_{N}]).$
(9.2)

The fact that $\rho_{1}$ is equal to $\rho_{2}$ follows from taking the limit
in the previous inequality, using the hypothesis that
$\limsup_{N\to\infty}\lambda^{N}_{\operatorname{LS}}>0$, and Corollary 5.9 to
obtain

$\lim_{N\to\infty}E^{N}[M_{N}]=E^{MF}[\rho_{1}]=E^{MF}[\rho_{2}]=\inf_{\mathcal{P}(\Omega)}E^{MF}.$

The quantitative convergence of $M_{N}$ to $\rho_{\beta}$, follows from the
bound in Lemma 9.1. ∎

######  Lemma 9.1.

Under Assumptions 2.1 and 2.2, assume that
$\liminf_{N\to\infty}\lambda_{\operatorname{LS}}^{N}=:\lambda^{\infty}>0$.
Consider $M_{N}$ the $N$ particle Gibbs measure and $\rho_{\beta}$ the unique
minimiser of the mean field energy. Then, for $N$ large enough,

$\overline{d}_{2}(M_{N},\rho_{\beta}^{\otimes
N})\leq\frac{2}{\lambda^{\infty}}\frac{\displaystyle\int_{\Omega}|\nabla_{1}W|^{2}\star\rho_{\beta}\rho_{\beta}\;\mathop{}\\!\mathup{d}x}{\sqrt{N}}\leq\frac{C}{\sqrt{N}}.$

###### Proof of Lemma 9.1.

Using the Talagrand and log Sobolev inequality, we have for $N$ large enough

$\overline{d}^{2}_{2}(M_{N},\rho_{\beta}^{\otimes
N})\leq\frac{2}{\lambda_{\operatorname{LS}}^{N}}\frac{\mathcal{E}(\rho_{\beta}^{\otimes
N}|M_{N})}{N}\leq\frac{2}{(\lambda_{\operatorname{LS}}^{N})^{2}}\frac{\mathcal{I}(\rho_{\beta}^{\otimes
N}|M_{N})}{N}\leq\frac{4}{(\lambda^{\infty})^{2}}\frac{\mathcal{I}(\rho_{\beta}^{\otimes
N}|M_{N})}{N}.$

The result now follows from bounding the Fisher information using Lemma 7.1. ∎

##  10 Proof of Theorem 3.7

We start by revisiting the classical propagation of chaos results [57, 44] by
using a convexity approach based on the 2-Wasserstein distance.

######  Theorem 10.1.

Under Assumptions 2.1 and 2.2, if $K_{V}+K_{W}(1-1/N)\neq 0$, then

$\overline{d}_{2}(\rho^{N}(t),\rho^{\otimes
N}(t))\leq\frac{1-e^{-\frac{K_{V}+K_{W}(1-1/N)}{2}t}}{K_{V}+K_{W}(1-1/N)}\left(\frac{\sup_{s\in[0,t]}\left(\int_{\Omega}|\nabla_{1}W|^{2}\star\rho(s)\rho(s)\;\mathop{}\\!\mathup{d}x\right)^{1/2}}{N^{1/2}}\right),$
(10.1)

else if $K_{V}+K_{W}(1-1/N)=0$, then

$\overline{d}_{2}(\rho^{N}(t),\rho^{\otimes
N}(t))\leq\frac{t}{2}\left(\frac{\sup_{s\in[0,t]}\left(\int_{\Omega}|\nabla_{1}W|^{2}\star\rho(s)\rho(s)\;\mathop{}\\!\mathup{d}x\right)^{1/2}}{N^{1/2}}\right),$
(10.2)

###### Proof of Theorem 10.1.

Using [1, Theorem 8.4.7], we differentiate the 2-Wasserstein distance between
$\rho^{N}$ and $\rho^{\otimes N}$ along their respective flows (2.3) and
(2.8), to obtain

$\displaystyle\frac{\mathop{}\\!\mathup{d}}{\mathop{}\\!\mathup{d}t}\overline{d}_{2}^{2}(\rho^{N},\rho^{\otimes
N})$
$\displaystyle=-\frac{1}{N}\int_{\Omega^{N}\times\Omega^{N}}(x-y)\cdot\bigg{(}\nabla_{x}\left(\beta^{-1}\log\rho^{N}(x)+H_{N}(x)\right)$
(10.3)
$\displaystyle\qquad-\nabla_{y}\left(\sum_{i=1}^{N}\beta^{-1}\log\rho(y_{i})+V(y_{i})+W\star\rho(y_{i})\right)\bigg{)}\;\mathop{}\\!\mathup{d}\Pi(x,y)$
(10.4)

where $\Pi\in\mathcal{P}(\Omega^{N}\times\Omega^{N})$ denotes the optimal
transport plan between $\rho^{N}$ and $\rho^{\otimes N}$. The convexity along
2-Wasserstein geodesics of the entropy functional as discussed in [46] implies
that (cf. [1, Section 10.1.1])

$\int_{\Omega^{N}\times\Omega^{N}}(x-y)\cdot(\beta^{-1}\nabla_{x}\log\rho^{N}(x)-\beta^{-1}\nabla_{y}\log\rho^{\otimes
N}(y))\;\mathop{}\\!\mathup{d}\Pi(x,y)\geq 0,$ (10.5)

where we have used that $\nabla\log\rho^{N}$ and $\nabla\log\rho^{\otimes N}$
are in the sub-differential of the entropy at $\rho^{N}$ and $\rho^{\otimes
N}$, respectively (cf. [1, Theorem 10.4.6]).

Applying inequality (10.5) to (10.4), we obtain

$\displaystyle\frac{\mathop{}\\!\mathup{d}}{\mathop{}\\!\mathup{d}t}\overline{d}_{2}^{2}(\rho^{N},\rho^{\otimes
N})$
$\displaystyle\leq-\frac{1}{N}\int_{\Omega^{N}\times\Omega^{N}}(x-y)\cdot\bigg{(}\nabla_{x}H_{N}(x)-\nabla_{y}H_{N}(y)$
(10.6)
$\displaystyle\qquad\qquad+\nabla_{y}\left(\frac{1}{2N}\sum_{i,j=1}^{N}W(y_{i},y_{j})-\sum_{i=1}^{N}W\star\rho(y_{i})\right)\bigg{)}\;\mathop{}\\!\mathup{d}\Pi(x,y)$
(10.7)
$\displaystyle\leq-(K_{V}+K_{W}(1-1/N))\underbrace{\frac{1}{N}\int_{\Omega^{N}\times\Omega^{N}}|x-y|^{2}\;\mathop{}\\!\mathup{d}\Pi(x,y)}_{=\overline{d}_{2}^{2}(\rho^{N},\rho^{\otimes
N})}$ (10.8)
$\displaystyle\qquad-\underbrace{\frac{1}{N}\int_{\Omega^{N}\times\Omega^{N}}(x-y)\cdot\nabla_{y}\bigg{(}\frac{1}{2N}\sum_{i=1}^{N}\sum_{j=1}^{N}W(y_{i},y_{j})-\sum_{i=1}^{N}W\star\rho(y_{i})\bigg{)}\;\mathop{}\\!\mathup{d}\Pi(x,y)}_{\mathcal{R}},$
(10.9)

where the last inequality follows from the convexity hypothesis on the
potentials (cf. Assumptions 2.1 and 2.2). To estimate the second term
$\mathcal{R}$, we employ Cauchy-Schwarz inequality and use the fact that $\Pi$
is the optimal transference plan to obtain

$\displaystyle\mathcal{R}\leq\overline{d}_{2}(\rho^{N},\rho^{\otimes
N})\left(\underbrace{\frac{1}{N}\int_{\Omega^{N}}\left|\nabla\left(\frac{1}{2N}\sum_{i=1}^{N}\sum_{j=1}^{N}W(y_{i},y_{j})-\sum_{i=1}^{N}W\star\rho(y_{i})\right)\right|^{2}\;\mathop{}\\!\mathup{d}\rho^{\otimes
N}}_{I}\right)^{1/2}$ (10.10)

Expanding the square, using the symmetry of the underlying system and the fact
that $W(y,y)=0$, we obtain

$\displaystyle I$
$\displaystyle=\int_{\Omega^{N}}\left|\frac{1}{N}\sum_{j=2}^{N}\nabla_{1}W(y_{1},y_{j})-\nabla_{1}W\star\rho(y_{1})\right|^{2}\;\mathop{}\\!\mathup{d}\rho^{\otimes
N}$ (10.11)
$\displaystyle=\int_{\Omega^{N}}\underbrace{\frac{1}{N^{2}}\sum_{j=2}^{N}\sum_{k=2}^{N}\nabla_{1}W(y_{1},y_{j})\nabla_{1}W(y_{1},y_{k})}_{A}-\underbrace{\frac{2}{N}\sum_{j=2}^{N}\nabla_{1}W(y_{1},y_{j})\nabla_{1}W\star\rho(y_{1})}_{B}$
(10.12)
$\displaystyle\qquad\qquad+(\nabla_{1}W\star\rho(y_{1}))^{2}\;\mathop{}\\!\mathup{d}\rho^{\otimes
N}.$ (10.13)

Going term by term, we have

$A=\frac{(N-1)(N-2)}{N^{2}}\int_{\Omega}(\nabla_{1}W\star\rho(y_{1}))^{2}\;\mathop{}\\!\mathup{d}\rho(y_{1})+\frac{(N-1)}{N^{2}}\int_{\Omega}|\nabla_{1}W|^{2}\star\rho(y_{1})\;\mathop{}\\!\mathup{d}\rho(y_{1}),$
$B=-2\frac{N-1}{N}\int_{\Omega}(\nabla_{1}W\star\rho(y_{1}))^{2}\;\mathop{}\\!\mathup{d}\rho(y_{1}).$

Using these identities, we are left with

$\displaystyle I$
$\displaystyle=\frac{1}{N}\left(1-\frac{1}{N}\right)\int_{\Omega}|\nabla_{1}W|^{2}\star\rho(y_{1})-(\nabla_{1}W\star\rho(y_{1}))^{2}\;\mathop{}\\!\mathup{d}\rho(y_{1})+\frac{1}{N^{2}}\int_{\Omega}(\nabla_{1}W\star\rho(y_{1}))^{2}\;\mathop{}\\!\mathup{d}\rho(y_{1})$
(10.14)
$\displaystyle\leq\frac{1}{N}\int_{\Omega}|\nabla_{1}W|^{2}\star\rho(y_{1})\;\mathop{}\\!\mathup{d}\rho(y_{1}),$
(10.15)

where the last inequality follows from Jensen’s inequality. Replacing the
previous equation in (10.10), combined with (10.9), we obtain

$\displaystyle\frac{\mathop{}\\!\mathup{d}}{\mathop{}\\!\mathup{d}t}\overline{d}_{2}^{2}(\rho^{N},\rho^{\otimes
N})\leq$
$\displaystyle-(K_{V}+K_{W}(1-1/N))\overline{d}_{2}^{2}(\rho^{N},\rho^{\otimes
N})$ (10.16) $\displaystyle+\overline{d}_{2}(\rho^{N},\rho^{\otimes
N})\frac{\left(\int_{\Omega}|\nabla_{1}W|^{2}\star\rho\,\rho\;\mathop{}\\!\mathup{d}x\right)^{1/2}}{N^{1/2}}.$
(10.17)

The estimates (10.1) and (10.2) now follow from Grönwall’s inequality. ∎

We are now ready to prove Theorem 3.7.

###### Proof of Theorem 3.7.

Using the uniform integrability assumption on the gradients of $W$ (3.15), the
estimates of Theorem 10.1 simplify to

$\overline{d}_{2}(\rho^{N}(t),\rho^{\otimes
N}(t))\leq\frac{C}{N^{1/2}}\begin{cases}\frac{1-e^{t(K_{V}+K_{W}(1-1/N))/2}}{K_{V}+K_{W}(1-1/N)}&\mbox{if
$K_{V}+K_{W}(1-1/N)\geq 0$}\\\ \frac{t}{2}&\mbox{if $K_{V}+K_{W}(1-1/N)=0$}\\\
\frac{e^{t|K_{V}+K_{W}(1-1/N)|/2}-1}{|K_{V}+K_{W}(1-1/N)|}&\mbox{if
$K_{V}+K_{W}(1-1/N)\leq 0$}\end{cases}$ (10.18)

Next, we derive a competing estimate by employing the triangle inequality and
the long time behavior of the flows. More specifically,

$\displaystyle\overline{d}_{2}(\rho^{N}(t),\rho^{\otimes N}(t))\leq$
$\displaystyle\;\overline{d}_{2}(\rho^{N}(t),M_{N})+\overline{d}_{2}(M_{N},\rho_{\beta}^{\otimes
N})+\overline{d}_{2}(\rho_{\beta}^{\otimes N},\rho^{\otimes N}(t))$ (10.19)
$\displaystyle=$
$\displaystyle\;\overline{d}_{2}(\rho^{N}(t),M_{N})+\overline{d}_{2}(M_{N},\rho_{\beta}^{\otimes
N})+d_{2}(\rho_{\beta},\rho(t))\,.$ (10.20)

For the first term, we use the Talagrand inequality (3.8)

$\overline{d}_{2}(\rho^{N}(t),M_{N})\leq\left(\frac{2}{\lambda_{\operatorname{LS}}^{N}}\right)^{1/2}\overline{\mathcal{E}}^{1/2}(\rho^{N}(t)|M_{N}).$

By the log Sobolev inequality we obtain exponential contraction of the
relative entropy

$\overline{d}_{2}(\rho^{N}(t),M_{N})\leq
e^{-\frac{\lambda_{\operatorname{LS}}^{N}}{2}t}\left(\frac{2}{\lambda_{\operatorname{LS}}^{N}}\right)^{1/2}\overline{\mathcal{E}}^{1/2}(\rho_{\mathrm{in}}^{\otimes
N}(t)|M_{N})\leq Ce^{-\frac{\lambda^{\infty}}{4}t},$ (10.21)

where in the last equality we have used the hypothesis $\rho_{\mathrm{in}}$
has finite energy and that the log Sobolev constant does not degenerate.

For the second term, we use Lemma 9.1 to obtain

$\overline{d}_{2}(M_{N},\rho_{\beta}^{\otimes N})\leq\frac{C}{\sqrt{N}}.$
(10.22)

For the third term, we use the limiting Talgrand inequality and the limiting
log Sobolev inequality to obtain the exponential contraction estimate

$\displaystyle d_{2}(\rho(t),\rho_{\beta})$
$\displaystyle\leq\left(\frac{2}{\lambda_{\infty}}\right)^{1/2}(E^{MF}[\rho(t)]-E^{MF}[\rho_{\beta}])^{1/2}$
$\displaystyle\leq\left(\frac{2}{\lambda_{\infty}}\right)^{1/2}e^{-\frac{\lambda_{\infty}}{2}t}(E^{MF}[\rho_{\mathrm{in}}]-E^{MF}[\rho_{\beta}])^{1/2}$
$\displaystyle\leq Ce^{-\frac{\lambda_{\infty}}{2}t}.$ (10.23)

Combining (10.20) with (10.21), (10.22) and (10.23) we obtain the estimate

$\overline{d}_{2}(\rho^{N}(t),\rho^{\otimes N}(t))\leq
C\left(e^{-\frac{\lambda_{\infty}}{4}t}+\frac{1}{\sqrt{N}}\right).$ (10.24)

The result now follows from interpolating the estimates (10.18) and (10.24).
In the case, $K_{V}+K_{W}(1-1/N)>0$ the desired estimate follows directly from
(10.18). For $K_{-}:=K_{V}+K_{W}(1-1/N)<0$, we consider the distinguished time
scale $T_{N}:=\log N^{\gamma}$, for some $\gamma>0$ to be chosen in terms of
$K_{-}$. Applying (10.24), we obtain

$\displaystyle\overline{d}_{2}(\rho^{N}(t),\rho^{\otimes N}(t))\leq
C\left(N^{-\gamma\frac{\lambda_{\infty}}{4}}+N^{-\frac{1}{2}}\right)\qquad\mbox{for
$t>T_{N}$}.$ (10.25)

For $t<T_{N}$, we apply (10.18)1 to obtain

$\displaystyle\overline{d}_{2}(\rho^{N}(t),\rho^{\otimes N}(t))\leq
CN^{-\frac{1+\gamma K_{-}}{2}}\,.$ (10.26)

Choosing

$\gamma=\left(\frac{\lambda_{\infty}}{2}-K_{-}\right)^{-1},$

we obtain that for every $t\in(0,\infty)$

$\overline{d}_{2}(\rho^{N}(t),\rho^{\otimes N}(t))\leq\frac{C}{N^{\theta}}$

is satisfied with

$\theta=\frac{1}{2}\frac{\lambda_{\infty}}{\lambda_{\infty}-2K_{-}}.$

∎

##  11 Proof of Theorem 3.14

As mentioned earlier, our proof of Theorem 3.14 will rely on the two-scale
approach to log Sobolev inequalities introduced in [49] and discussed further
in [31], see also [41]. Before we introduce the main result of [49], we
introduce some preliminary notions.

######  Definition 11.1 (Conditional measures).

Given a probability measure $\mu_{N}\in\mathcal{P}(\Omega^{N})$ we define the
conditional measure $\mu_{N,i},i\in{1,\dots,N}$ as the family of measures
indexed by $x_{j},\,j\neq i$ such that for all $\varphi\in C_{b}(\Omega^{N})$

$\displaystyle\int_{\Omega^{N}}\varphi\mathop{}\\!\mathup{d}\mu_{N}=\int_{\Omega^{N-1}}\int_{\Omega}\varphi\mathop{}\\!\mathup{d}\mu_{N,i}\mathop{}\\!\mathup{d}\mu_{N\setminus\\{i\\}}\,,$
(11.1)

where $\mu_{N\setminus\\{i\\}}$ is the marginal of $\mu_{N}$ obtained by
integrating out $x_{i}\in\Omega$.

We can now the state the result of interest.

######  Theorem 11.2 ([49, Theorem 1]).

Let $\Omega$ be a smooth, connected, and complete Riemannian manifold and
assume that the measure $\mu_{N}$ has a Gibbs structure, that is to say

$\displaystyle\mu_{N}(\mathop{}\\!\mathup{d}x)=Z_{N}^{-1}e^{-\beta
H_{N}}\mathop{}\\!\mathup{d}x\,,$ (11.2)

where $\mathop{}\\!\mathup{d}x$ is the Riemannian volume measure and
$H_{N}:\Omega^{N}\to\mathbb{R}$ is some smooth Hamiltonian. Assume there
exists some constants $\kappa_{ij}$, such that for all $i\neq j$

$\displaystyle\lVert
D^{2}_{x_{i}x_{j}}H_{N}\rVert\leq\kappa_{ij}\qquad\mbox{for all
$x\in\Omega^{N}$},$ (11.3)

where $\lVert\cdot\rVert$ is the operator norm of $D^{2}_{x_{i}x_{j}}H_{N}$.
Furthermore, assume that the conditional measures $\mu_{N,i}$ satisfy a log
Sobolev inequality with uniform constant $\lambda_{\operatorname{LS}}^{N,i}$
for all $\hat{x}\in\Omega^{N-1}$. Consider the matrix $A\in\mathbb{R}^{N\times
N}$ with entries $A_{ii}=\lambda^{N,i}_{\operatorname{LS}}$ and
$A_{ij}=-\beta\kappa_{ij}$; if

$\displaystyle A\geq CI^{N\times N}\,,$ (11.4)

the measure $\mu_{N}$ satisfies a log Sobolev inequality with constant $C$.

Relying on Theorem 11.2, we present now the proof of Theorem 3.14.

###### Proof of Theorem 3.14.

We note first that from the Definition 11.1, the conditional measure $M_{N,i}$
of $M_{N}$ can be expressed as

$\displaystyle M_{N,i}(\mathop{}\\!\mathup{d}x_{i})=$ $\displaystyle
M_{N}(\mathop{}\\!\mathup{d}x_{i}|x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{N})=\frac{M_{N}}{(M_{N})_{N\setminus\\{i\\}}}\,,$
(11.5)

where $(M_{N})_{N\setminus\\{i\\}}$ is the marginal of $M_{N}$ obtained by
integrating out $x_{i}$. We are thus left with

$\displaystyle M_{N,i}=$
$\displaystyle\frac{\exp\left(-\beta(V(x_{i})+\frac{1}{N}\sum_{j=1}^{N}W(x_{i},x_{j}))-\beta(\sum_{j\neq
i,j=1}^{N}V(x_{j})+\frac{1}{2N}\sum_{j,k=1,j,k\neq
i}W(x_{j},x_{k}))\right)}{\int_{\Omega}\exp\left(-\beta
H_{N}\right)\mathop{}\\!\mathup{d}x_{i}}$ (11.6) $\displaystyle=$
$\displaystyle
Z_{N,i}^{-1}\exp\left(-\beta(V(x_{i})+\frac{1}{N}\sum_{j=1}^{N}W(x_{i},x_{j}))\right)\,,$
(11.7)

where

$\displaystyle
Z_{N,i}=\int_{\Omega}\exp\left(-\beta(V(x_{i})+\frac{1}{N}\sum_{j=1}^{N}W(x_{i},x_{j}))\right)\mathop{}\\!\mathup{d}x_{i}\,.$
(11.8)

We now assert that the conditional measure $M_{N,i}$ satisfies a log Sobolev
inequality. We first treat the case in which $\Omega$ is compact, that is to
say Theorem 3.14 3.27. By the Holley–Stroock perturbation Theorem [3,
Proposition 5.1.6], we have that

$\displaystyle\lambda^{N,i}_{\operatorname{LS}}\geq e^{-2\beta(\lVert
W\rVert_{{L}^{\infty}(\Omega^{2})}+\lVert
V\rVert_{{L}^{\infty}(\Omega^{)}})}\lambda_{\operatorname{LS}}^{\Omega}\,,$
(11.9)

for all $x_{j},j\neq i,j=1,\dots,N$ and where
$\lambda_{\operatorname{LS}}^{\Omega}$ is the optimal log Sobolev constant of
the Lebesgue measure on $\Omega$.

Note that because of the exchangeability of the underlying particle system, we
have that
$\lambda^{N,i}_{\operatorname{LS}}=\lambda^{N,j}_{\operatorname{LS}}$ for all
$i,j=1,\dots,N$.

Note now that

$\displaystyle
D^{2}_{x_{i}x_{j}}H_{N}(x_{1},\dots,x_{N})=\frac{1}{N}D^{2}_{x_{i}x_{j}}W(x_{i},x_{j})\,.$
(11.10)

Using the hypothesis that $W\in W^{2,\infty}(\Omega^{2})$, we can bound

$\displaystyle\lVert
D^{2}_{x_{i}x_{j}}H_{N}\rVert_{{L}^{\infty}(\Omega^{N})}\leq\frac{1}{N}\lVert
D^{2}_{xy}W\rVert_{{L}^{\infty}(\Omega^{2})}=:\kappa_{ij}\,,$ (11.11)

for all $i,j=1,\dots,N$. We will show that the matrix $A\in\mathbb{R}^{N\times
N}$ from Theorem 11.2 is positive definite, by showing that it is diagonally
dominant. In fact, for $\beta$ sufficiently small we have that

$\displaystyle A_{ii}-\sum_{j\neq i}|A_{ij}|\geq e^{-2\beta(\lVert
W\rVert_{{L}^{\infty}(\Omega^{2})}+\lVert
V\rVert_{{L}^{\infty}(\Omega^{)}})}\lambda_{\operatorname{LS}}^{\Omega}-\beta\frac{N-1}{N}\lVert
D^{2}_{xy}W\rVert_{{L}^{\infty}(\Omega^{2})}>c>0\,,$ (11.12)

holds true for all $N$ with the constant $c$ independent of $N$. Applying
Theorem 11.2, Theorem 3.14 3.27 now follows.

For the proof of Theorem 3.14 3.29 we can apply essentially the same
perturbative argument as before but now around the measure
$Z_{V}^{-1}e^{-V}\mathop{}\\!\mathup{d}x$. For $\epsilon$ sufficiently small,
we obtain that the analogous bound

$\displaystyle A_{ii}-\sum_{j\neq i}|A_{ij}|\geq e^{-2\epsilon\beta\lVert
W\rVert_{{L}^{\infty}(\Omega^{2})}}\lambda_{\operatorname{LS}}^{V}-\epsilon\beta\frac{N-1}{N}\lVert
D^{2}_{xy}W\rVert_{{L}^{\infty}(\Omega^{2})}>c>0\,$ (11.13)

holds true for all $N$ with the constant $c$ independent of $N$. ∎

##  12 Proof of Theorem 3.11

To simplify the computations in this section, we will take the following
definition of the negative Sobolev norm $H^{-s}_{0}(\mathbb{T}^{d})$ of mean-
zero distributions is given by

$\|h\|_{H^{-s}(\mathbb{T}^{d})}^{2}:=\sum_{j\in\mathbb{N}}|\langle
h,\phi_{j}\rangle|^{2},$ (12.1)

where $(\phi_{j})_{j\in\mathbb{N}}$ is a given a smooth orthonormal basis for
the Sobolev space $H^{s}_{0}(\mathbb{T}^{d})$.

We remark that Theorem 3.11 can also be proved when $\Omega=\mathbb{R}^{d}$,
under appropriate assumptions on the confining and interaction potentials. In
particular, we can construct an appropriate orthonormal basis using the
eigenfunctions of the linearised McKean-Vlasov operator with a weighted inner-
product. Conditions on the growth of the confining potential so that the
Sobolev embedding theorems needed in the proof in the appropriate weighted
spaces are given in [42].

######  Lemma 12.1 (Law of large numbers).

Assume that $\liminf_{N\to\infty}\lambda^{N}_{\operatorname{LS}}>0$. Let
$\mu^{(N)}$ be the empirical measure associated to the $N$-particle Gibbs
measure $M_{N}\in\mathcal{P}_{\mathrm{sym}}((\mathbb{T}^{d})^{N})$. Then, for
any $s>\frac{d+2}{2}$, there exists $C>0$ such that

$\mathbb{E}\left[\|\mu^{(N)}-\rho_{\beta}\|_{H^{-s}(\mathbb{T}^{d})}^{2}\right]\leq\frac{C}{N}\,,$

where $\rho_{\beta}\in\mathcal{P}(\mathbb{T}^{d})$ is the unique critical
point of the mean field energy $E^{MF}$.

######  Remark 12.2.

The solutions to the linear SPDE (12.10) are supported in $H^{-s}$ with
$s>d/2$, see Lemma 12.4. Hence, the Law of large numbers does not hold for any
$H^{-s}$ with $s<d/2$.

###### Proof.

We consider $\rho_{\beta}^{(N)}$ the empirical measure associated to
$\rho_{\beta}^{\otimes N}\in\mathcal{P}_{\mathrm{sym}}((\mathbb{T}^{d})^{N})$,
i.e. $\rho_{\beta}^{(N)}$ the probability-measure valued random variable
defined as

$\rho_{\beta}^{(N)}=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}\qquad\mbox{such
that $(X_{1},...,X_{N})$ are distributed according to $\rho_{\beta}^{\otimes
N}$}\,.$

By the triangle inequality we have that

$\mathbb{E}\left[\|\mu^{(N)}-\rho_{\beta}\|^{2}_{H^{-s}(\mathbb{T}^{d})}\right]\leq
2\left(\mathbb{E}\left[\|\mu^{(N)}-\rho_{\beta}^{(N)}\|^{2}_{H^{-s}(\mathbb{T}^{d})}\right]+\mathbb{E}\left[\|\rho_{\beta}^{(N)}-\rho_{\beta}\|^{2}_{H^{-s}(\mathbb{T}^{d})}\right]\right)$
(12.2)

We start by controlling the first term
$\mathbb{E}\left[\|\mu^{(N)}-\rho_{\beta}^{(N)}\|^{2}_{H^{-s}(\mathbb{T}^{d})}\right]$.
We consider the optimal coupling between $\mu^{(N)}$ and $\rho_{\beta}^{(N)}$
such that

$\mathbb{E}\left[d_{2}^{2}(\mu^{(N)},\rho_{\beta}^{(N)})\right]=\mathfrak{D}_{2}^{2}\left(\hat{\mu}^{N},\hat{\rho}_{\beta}^{N}\right)\,,$

where
$\hat{\mu}^{N}=\mathrm{Law}\left(\mu^{(N)}\right)\in\mathcal{P}(\mathcal{P}(\mathbb{T}^{d}))$,
$\hat{\rho}_{\beta}^{N}=\mathrm{Law}\left(\rho_{\beta}^{(N)}\right)\in\mathcal{P}(\mathcal{P}(\mathbb{T}^{d}))$,
and $\mathfrak{D}_{2}$ is the 2-Wassertein distance defined on the space of
probability measures of the metric space
$(\mathcal{P}(\mathbb{T}^{d}),d_{2})$. Then by the isometry from Theorem 5.3,
we obtain

$\mathfrak{D}_{2}^{2}\left(\hat{\mu}^{N},\hat{\rho}_{\beta}^{N}\right)=\overline{d}_{2}^{2}(M_{N},\rho_{\beta}^{\otimes
N})\leq\frac{C}{N},$

where we have used Lemma 9.1 and the fact that
$\liminf_{N\to\infty}\lambda^{N}_{\operatorname{LS}}>0$ for the last
inequality. Next, using the compactness of $\mathbb{T}^{d}$ and the fact that
$s>d/2+1$, we have the pointwise bound

$d_{2}^{2}(\mu^{(N)},\rho_{\beta}^{(N)})\geq
c_{1}\|\mu^{(N)}-\rho_{\beta}^{(N)}\|_{W^{-1,\infty}(\mathbb{T}^{d})}^{2}\geq
c_{2}\|\mu^{(N)}-\rho_{\beta}^{(N)}\|^{2}_{H^{-s}(\mathbb{T}^{d})}\,,$

for some constants $c_{1},c_{2}>0$ independent of
$\mu^{(N)},\rho_{\beta}^{(N)}$. Combining the previous three expressions, we
obtain the first desired bound

$\mathbb{E}\left[\|\mu^{(N)}-\rho_{\beta}^{(N)}\|^{2}_{H^{-s}(\mathbb{T}^{d})}\right]\leq\frac{C}{N}.$
(12.3)

Next, we bound
$\mathbb{E}\left[\|\rho_{\beta}^{(N)}-\rho_{\beta}\|^{2}_{H^{-s}(\mathbb{T}^{d})}\right]$.
We take $\\{\phi_{j}\\}_{j=0}^{\infty}$ to be the orthonormal basis of
$H^{s}_{0}(\mathbb{T}^{d})$ from (12.1). Expanding the square, we obtain

$\displaystyle\mathbb{E}\left[\|\rho_{\beta}^{(N)}-\rho_{\beta}\|^{2}_{H^{-s}(\mathbb{T}^{d})}\right]=$
$\displaystyle\mathbb{E}\left[\sum_{j=1}^{\infty}\left\lvert\langle\rho_{\beta}^{(N)}-\rho_{\beta},\phi_{j}\rangle\right\rvert^{2}\right]$
(12.4) $\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\mathbb{E}\left[\left|\frac{1}{N}\sum_{i=1}^{N}\phi_{j}(X_{i})-\int\phi_{j}\rho_{\beta}\;\mathop{}\\!\mathup{d}x\right|^{2}\right]$
(12.5) $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{j=1}^{\infty}\int\phi_{j}^{2}\rho_{\beta}\;\mathop{}\\!\mathup{d}x-\frac{1}{N}\left(\int\phi_{j}\rho_{\beta}\;\mathop{}\\!\mathup{d}x\right)^{2}$
(12.6) $\displaystyle\leq$
$\displaystyle\frac{\|\rho_{\beta}\|_{L^{\infty}(\mathbb{T}^{d})}}{N}\sum_{j=1}^{\infty}\|\phi_{j}\|_{{L}^{2}(\mathbb{T}^{d})}^{2}.$
(12.7)

Noticing that $s>d/2$ implies that the embedding
$H^{s}_{0}(\mathbb{T}^{d})\to{L}^{2}(\mathbb{T}^{d})$ is Hilbert-Schmidt, we
can use the boundedness of $\rho_{\beta}$ to obtain

$\mathbb{E}\left[\|\rho_{\beta}^{(N)}-\rho_{\beta}\|^{2}_{H^{-s}(\mathbb{T}^{d})}\right]\leq\frac{C}{N}\,.$
(12.8)

Combining (12.2) with (12.3) and (12.8), we obtain the desired conclusion. ∎

######  Remark 12.3.

An alternative proof of inequality (12.8) can be found in [35, Theorem 5.1].

We now consider the implications of having uniform control of the log Sobolev
constant the fluctuations of the stationary solutions to (2.1), i.e. solutions
of (2.1) started at stationarity. To this end, we consider the empirical
measure process $t\mapsto\mu^{(N)}(t)$ defined by

$\mu^{(N)}(t):=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}},$

where $X_{t}=(X_{t}^{1},\dots,X_{t}^{N})$ is the solution to (2.1) started
from the unique invariant Gibbs measure $M_{N}$. Our goal is to analyze the
corresponding fluctuation process $t\mapsto\eta^{N}(t)$ defined by

$\eta^{N}(t):=\sqrt{N}(\mu^{(N)}(t)-\rho_{\beta}),$

As a direct consequence of the estimates from the previous lemma and using the
fact that $\liminf_{N\to\infty}\lambda^{N}_{LS}>0$, we have the uniform bound

$\sup_{N,t}\mathbb{E}\left[\|\eta^{N}(t)\|_{H^{-s}(\mathbb{T}^{d})}^{2}\right]<\infty\qquad\mbox{for
any $s>d/2+1$.}$ (12.9)

In the sequel, we will use the above estimate together with the classical
martingale method, c.f. [17, Chapter 8], to establish convergence in law as
$N\to\infty$ of $\eta^{N}$ to the stationary solution $\eta$ of the following
linear SPDE

$\partial_{t}\eta=\mathcal{L}_{\rho_{\beta}}\eta+\nabla\cdot(\sqrt{\rho_{\beta}}\xi),$
(12.10)

where $\xi$ is a mean-zero space-time white noise on
$\mathbb{R}_{+}\times\mathbb{T}^{d}$ and $\mathcal{L}_{\rho_{\beta}}$ is the
linearisation of the McKean-Vlasov operator (2.8) around the unique invariant
measure $\rho_{\beta}$ defined by

$\mathcal{L}_{\rho_{\beta}}\psi=\beta^{-1}\Delta\psi+\nabla\cdot(\nabla
W\star\psi\,\rho_{\beta})+\nabla\cdot(\nabla
W\star\rho_{\beta}\,\psi)+\nabla\cdot(\nabla V\,\psi).$ (12.11)

The SPDE (12.10) can be solved using classical methods, the three typical
notions of solution being the mild, weak, and martingale formulations. As is
typical, the martingale formulation is the most convenient for identifying the
limiting law of a tight subsequence of $\eta^{N}$, while the mild formulation
provides a clearer picture of the uniqueness in law and hence the convergence
in law of the full sequence.

Denote $U$ as the closed mean-zero subspace of
$L^{2}(\mathbb{T}^{d};\mathbb{T}^{d})$ and $H:=H^{-s}(\mathbb{T}^{d})$ for
$s>\frac{d}{2}$. Let $(\mathcal{X},\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},{\mathbb{P}})$ be a stochastic basis, i.e a complete filtered probability
space with a right continuous filtration endowed with an
$\mathcal{F}_{t}$-adapted $U$-valued cylindrical Wiener process W and let
$\eta_{0}$ be an $H$-valued $\mathcal{F}_{0}$-measurable random variable
independent of W. The mild formulation of (12.10) with initial condition
$\eta_{0}$ involves stochastic integration in Hilbert spaces, c.f. [17,
Chapter 4], which we quickly review for our specific case below. Let
$\mathbf{L}^{2}_{0}(U;H)$ denote the Hilbert–Schmidt operators from $U$ to $H$
equipped with the standard Hilbert–Schmidt norm
$\lVert\cdot\rVert_{\mathbf{L}_{2}^{0}(U;H)}$. Given $T>0$ and $\Phi\in
L^{2}([0,T];\mathbf{L}^{2}_{0}(U;H))$, the stochastic integral $[0,T]\ni
t\mapsto\int_{0}^{t}\Phi(s)d\textbf{W}(s)$ is well defined as a continuous
$\mathcal{F}_{t}$-martingale with trajectories in $C([0,T];H)$. We remind the
reader that the relevance of $\mathbf{L}^{2}_{0}(U;H)$ in this context is that
the Itô isometry takes the following form

$\mathbb{E}\left[\left\|\int_{0}^{t}\Phi(s)d\textbf{W}(s)\right\|_{H}^{2}\right]=\int_{0}^{t}\|\Phi(s)\|_{\mathbf{L}^{2}_{0}(U;H)}^{2}\mathop{}\\!\mathup{d}s.$

The mild solution to $t\mapsto\eta(t)\in H$ to (12.10) with initial condition
$\eta_{0}$ is then given by the following stochastic convolution:

$\eta^{\infty}(t):=e^{t\mathcal{L}_{\rho_{\beta}}}\eta_{0}+\int_{0}^{t}e^{(t-s)\mathcal{L}_{\rho_{\beta}}}\nabla\cdot\big{(}\sqrt{\rho_{\beta}}d\textbf{W}(s)\big{)}.$
(12.12)

######  Lemma 12.4.

The mild solution (12.12) is well-defined as a stochastic process with
trajectories in $C([0,T];H)$.

###### Proof.

In light of our remarks in the preceding paragraph, it suffices to show that
$s\mapsto\Phi(s)$ defined by $U\ni u\mapsto
e^{(t-s)\mathcal{L}_{\rho_{\beta}}}\nabla\cdot(\sqrt{\rho_{\beta}}u)\in H$
belongs to $L^{2}([0,T];\mathbf{L}^{2}_{0}(U;H))$. Given an orthornormal basis
$\\{e_{k}\\}_{k=1}^{\infty}$ of $U$, using the definition of the
Hilbert–Schmidt norm and integrating by parts leads to

$\displaystyle\int_{0}^{T}\|\Phi(s)\|_{\mathbf{L}^{2}_{0}(U;H)}^{2}\mathop{}\\!\mathup{d}s$
$\displaystyle=\sum_{k=1}^{\infty}\int_{0}^{T}\|e^{(t-s)\mathcal{L}_{\rho_{\beta}}}\nabla\cdot\big{(}\sqrt{\rho_{\beta}}e_{k}\big{)}\|_{H}^{2}\mathop{}\\!\mathup{d}s$
$\displaystyle=\sum_{j,k=1}^{\infty}\int_{0}^{T}\langle
e_{k},\sqrt{\rho_{\beta}}\nabla
e^{(t-s)\mathcal{L}_{\rho_{\beta}}^{*}}\phi_{j}\rangle_{U}^{2}\mathop{}\\!\mathup{d}s$
$\displaystyle=\sum_{j=1}^{\infty}\int_{0}^{T}\|\sqrt{\rho_{\beta}}\nabla
e^{(t-s)\mathcal{L}_{\rho_{\beta}}^{*}}\phi_{j}\|_{U}^{2}\mathop{}\\!\mathup{d}s,$
(12.13)

where in the last step we used Parseval’s identity in $U$. Finally, we note
that the coercivity hypothesis for $\mathcal{L}_{\rho_{\beta}}$ (see Lemma
12.8) implies that

$\int_{0}^{t}\|\nabla
e^{(t-s)\mathcal{L}_{\rho_{\beta}}^{*}}\phi_{j}\|_{U}^{2}\mathop{}\\!\mathup{d}s\leq
C\|\phi_{j}\|_{L^{2}(\mathbb{T}^{d})}^{2}.$

Combining this with (12.13) and using the fact that $\rho_{\beta}\in
L^{\infty}(\mathbb{T}^{d})$, yields

$\int_{0}^{T}\|\Phi(s)\|_{\mathbf{L}^{2}_{0}(U;H)}^{2}\mathop{}\\!\mathup{d}s\leq
C\|\rho_{\beta}\|_{L^{\infty}(\mathbb{T}^{d})}\sum_{j=1}^{\infty}\|\phi_{j}\|_{L^{2}(\mathbb{T}^{d})}^{2}.$
(12.14)

Since $s>\frac{d}{2}$, the embedding of $H^{s}_{0}(\mathbb{T}^{d})$ into
$L^{2}(\mathbb{T}^{d})$ is Hilbert–Schmidt. Thus, the above series converges,
completing the proof of the lemma. ∎

######  Remark 12.5.

We note that the representation (12.12) immediately implies that solutions to
(12.10) are unique in law, i.e. (12.10) satisfies weak uniqueness. That is to
say, given two different stochastic bases
$(\mathcal{X},\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},{\mathbb{P}},\textbf{W})$ and
$(\tilde{\mathcal{X}},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_{t})_{t\geq
0},\tilde{{\mathbb{P}}},\widetilde{\textbf{W}})$ defining solutions $\eta$ and
$\tilde{\eta}$ to (12.10) on their respective probability spaces through the
formula (12.12), the laws of $\eta$ and $\tilde{\eta}$ agree on $C([0,T];H)$
for any $T>0$, as long as $\eta_{0}$ and $\tilde{\eta}_{0}$ are equal in law
on $H$.

For our purposes, it is easier to work with the martingale formulation of
(12.10), which is in turn motivated by the weak formulation of (12.10). Hence,
we note in passing that the mild solution (12.12) has the property that for
each $t\in[0,T]$ the following equality holds in $H^{-s}(\mathbb{T}^{d})$, for
$s>\frac{d+2}{2}$,

$\eta^{\infty}(t)=\eta_{0}+\int_{0}^{t}\mathcal{L}_{\rho_{\beta}}\eta^{\infty}(s)\;\mathop{}\\!\mathup{d}s+\int_{0}^{t}\nabla\cdot(\sqrt{\rho_{\beta}}d\textbf{W}_{s}).$
(12.15)

######  Lemma 12.6.

Let $(\mathcal{X},\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}})$ be a
filtered probability space. Assume that $\eta$ is a continuous-time
$\mathcal{F}_{t}$-adapted $H$-valued stochastic process and define $t\mapsto
M(t)$ by

$M(t):=\eta(t)-\eta_{0}-\int_{0}^{t}\mathcal{L}_{\rho_{\beta}}\eta(s)\;\mathop{}\\!\mathup{d}s.$
(12.16)

Assume that the following two conditions hold:

  * •

For all $\varphi\in C^{\infty}(\mathbb{T}^{d})$ it holds that

$t\mapsto\langle M(t),\varphi\rangle$ (12.17)

is an $\mathcal{F}_{t}$-martingale.

  * •

For all $\varphi,\psi\in C^{\infty}(\mathbb{T}^{d})$ it holds that

$t\mapsto\langle M(t),\varphi\rangle\langle
M(t),\psi\rangle-t\int_{\mathbb{T}^{d}}\nabla\phi\cdot\nabla\psi\rho_{\beta}\;\mathop{}\\!\mathup{d}x$
(12.18)

is an $\mathcal{F}_{t}$ martingale.

Then, $\eta$ is equal in law on $C([0,T];H)$ to the mild solution (12.12).

###### Proof.

By [17, Theorem 8.2], it follows that on a suitable extension of the
probability space, $\eta$ is a weak solution in the sense of (12.15). By [17,
Theorem 5.4], the weak and mild solutions coincide on that probability space,
so equality in law follows from Remark 12.5. ∎

Using the above lemma, we are now finally in a position to prove the
convergence of the fluctuations.

######  Theorem 12.7.

Assume that $\liminf_{N\to\infty}\lambda^{N}_{LS}>0$ and that $V$ and $W$ are
smooth. Then, for any $m>d/2+3$ the fluctuation process $\eta^{N}$ converges
in law on $C([0,T];H^{-m}(\mathbb{T}^{d}))$ to the unique stationary mild
solution $\eta^{\infty}$ of the SPDE (12.10).

###### Proof.

The proof has four steps. In Step 1, we apply Itô’s formula to show that
$\eta^{N}$ satisfies (12.19), an approximate version of the weak formulation
(12.15). In Step 2, we combine Step 1 with the uniform bound (12.9) to show
that the laws of $(\eta^{N})_{N\in\mathbb{N}}$ on
$C([0,T];H^{-m}(\mathbb{T}^{d}))$ are uniformly tight for $m$ large enough. In
Step 3, we pass to the limit in the martingale problem and verify the
assumptions of Lemma (12.6) to identify the limit along any tight subsequence.
In Step 4, we conclude the uniqueness of the limit and hence the proof of the
theorem.

Step 1\. In this step, we show that for all $\varphi\in
C^{\infty}(\mathbb{T}^{d})$ it holds

$\mathop{}\\!\mathup{d}\langle\eta^{N},\varphi\rangle=\langle\eta^{N},\mathcal{L}_{\rho_{\beta}}^{*}\varphi\rangle\mathop{}\\!\mathup{d}t+\langle
R_{N},\varphi\rangle\mathop{}\\!\mathup{d}t+\sqrt{2}(\beta
N)^{-\frac{1}{2}}\sum_{i=1}^{N}\nabla\varphi(X_{t}^{i})\cdot dB_{t}^{i}\,,$
(12.19)

with $X_{t}^{i}$ the solution to (2.1) and where $R_{N}$ is defined as

$R_{N}:=N^{-\frac{1}{2}}\nabla\cdot(\eta^{N}\nabla_{1}W\star\eta^{N})\,.$
(12.20)

Indeed, Itô’s formula gives

$\displaystyle\mathop{}\\!\mathup{d}\varphi(X_{t}^{i})=$
$\displaystyle\left(\beta^{-1}\Delta-\nabla
V\cdot\nabla\right)\varphi(X_{t}^{i})\mathop{}\\!\mathup{d}t-\frac{1}{N}\sum_{j=1}^{N}\nabla_{1}W(X_{t}^{i},X_{t}^{j})\cdot\nabla\varphi(X_{t}^{i})\mathop{}\\!\mathup{d}t$
(12.21) $\displaystyle+\sqrt{2}\beta^{-1}\nabla\varphi(X_{t}^{i})\cdot
dB_{t}^{i}\,.$ (12.22)

We now sum over $i=1,\dots,N$, divide by $N$, and use the identity

$\displaystyle\frac{1}{N^{2}}\sum_{i,j=1}^{N}\nabla_{1}W(X_{t}^{i},X_{t}^{j})\cdot\nabla\varphi(X_{t}^{i})=$
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\nabla_{1}W*\mu^{(N)}(X_{t}^{i})\cdot\nabla\varphi(X_{t}^{i})$
(12.23) $\displaystyle=$
$\displaystyle\langle\mu^{(N)},\nabla_{1}W\star\mu^{(N)}\cdot\nabla\varphi\rangle\,,$
(12.24)

to obtain

$\mathop{}\\!\mathup{d}\langle\mu^{(N)},\varphi\rangle=\langle\mu^{(N)},\beta^{-1}\Delta\varphi-(\nabla
V+\nabla_{1}W\star\mu^{(N)})\cdot\nabla\varphi\rangle\mathop{}\\!\mathup{d}t+\sqrt{2}\beta^{-\frac{1}{2}}\frac{1}{N}\sum_{i=1}^{N}\nabla\varphi(X^{i}_{t})\cdot
dB_{t}^{i}$

Next, we insert the identity
$\mu^{(N)}=\rho_{\beta}+N^{-\frac{1}{2}}\eta^{N}$, to deduce

$\displaystyle\mathop{}\\!\mathup{d}\langle\eta^{N},\varphi\rangle=$
$\displaystyle
N^{1/2}\underbrace{\langle\rho_{\beta},\beta^{-1}\Delta\varphi-(\nabla
V+\nabla_{1}W\star\rho_{\beta})\cdot\nabla\varphi\rangle}_{=0}\mathop{}\\!\mathup{d}t$
(12.25)
$\displaystyle+\underbrace{\langle\eta^{N},\beta^{-1}\Delta\varphi-(\nabla
V+\nabla_{1}W\star\rho_{\beta})\cdot\nabla\varphi\rangle+\langle\rho_{\beta},\nabla_{1}W\star\eta^{N}\cdot\nabla\varphi\rangle}_{=\langle\eta^{N},\mathcal{L}_{\rho_{\beta}}^{*}\varphi\rangle}\mathop{}\\!\mathup{d}t$
(12.26)
$\displaystyle+\underbrace{N^{-1/2}\langle\eta^{N},\nabla_{1}W\star\eta^{N}\cdot\nabla\varphi\rangle}_{\langle
R^{N},\varphi\rangle}+\frac{\sqrt{2}}{\beta
N^{\frac{1}{2}}}\sum_{i=1}^{N}\nabla\varphi(X^{i}_{t})\cdot dB_{t}^{i}$

The first identity follows from the fact that $\rho_{\beta}$ is a steady state
and the second follows from integration by parts and Fubini’s theorem (using
the symmetry of $W$).

Step 2\. In this step, we will show that the laws of