MODULATED FREE ENERGY AND MEAN FIELD LIMIT

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MODULATED FREE ENERGY AND MEAN FIELD LIMIT

Didier Bresch, Pierre-Emmanuel Jabin, Zhenfu Wang

To cite this version:

Didier Bresch, Pierre-Emmanuel Jabin, Zhenfu Wang. MODULATED FREE ENERGY AND MEAN FIELD LIMIT. 2019. �hal-02397471�

MODULATED FREE ENERGY AND MEAN FIELD LIMIT

DIDIER BRESCH, PIERRE-EMMANUEL JABIN, AND ZHENFU WANG

Abstract. This is the document corresponding to the talk the first author gave at IH ´ES for the Laurent Schwartz seminar on November 19, 2019. It concerns our recent introduction of a modulated free energy in mean-field theory inBrJaWa

[4]. This physical object may be seen as a combination of the modulated potential energy introduced by S. Serfaty [See Proc. Int. Cong. Math. (2018)] and of the relative entropy introduced in mean field limit theory by P.–E. Jabin, Z. Wang [See Inventiones 2018]. It allows to obtain, for the first time, a convergence rate in the mean field limit for Riesz and Coulomb repulsive kernels in the presence of viscosity using the estimates inDu

[8] andSe1

[20]. The main objective in this paper is to explain how it is possible to cover more general repulsive kernels through a Fourier transform approach as announced inBrJaWa

[4] first in the caseσN →0 when N →+∞and then if σ >0 is fixed. Then we end the paper with comments on the particle approximation of the Patlak-Keller-Segel system which is associated to an attractive kernel and refer to [C.R. Acad Science Paris 357, Issue 9, (2019), 708–720]

by the authors for more details.

1. Introduction

This paper concerns interaction particles system and quantitative estimates in mean field limit theory in the spirit of [20],^Se1 [8] and^Du [15].^JaWa Namely we consider N particles, identical and interacting two by two through a kernel K. For simplicity, we consider periodic boundary conditions and we assume the position of the i-th particle X_i(t)∈Π^d evolves through as follows

dXi = 1 N

X

j6=i

K(Xi−Xj)dt+√ 2σdWi

forN independent Brownian motionsW_iwith a gradient flow hypothesisK=−∇V where V ∈L¹(Π^d) will be discussed later-on: For possible vanishing viscosity with respect toN namelyσ_N →0 whenN →+∞we consider singular repulsive kernels with some pointwise and Fourier controls (including Riesz and Coulomb Kernels) and for σ > 0 we consider more general singular repulsive kernels with Fourier control and then we conclude with some comments on the Patlak-Keller-Segel attractive kernel. The main objective in this document is to give some details in the repulsive case to complete the note written by the authors [C.R. Acad Sciences357, Issue 9, (2019), 708–720] which focused on the attractive Patlak-Keller-Segel kernel. A full paper is still in progress to propose a single complete document, see ^BrJaWa1[5]. Readers interested by some reviews on mean field limit for stochastic particle systems are referred to^JaWa1[16],^Sa[18] and on mean field limit for deterministic particle systems are referred to ^Go[11],[14],^{Ja Se1}[20].

Date: December 6, 2019.

2010Mathematics Subject Classification. 35Q35, 76N10.

1

As usually, we introduce ρ_N the joint law of the process (X₁,· · · , X_N) which satisfies the Liouville equation

∂_tρ_N+

N

X

i=1

div_xi



ρ_N 1 N

N

X

j6=i

K(x_i−x_j)



=σ

N

X

i=1

∆_xiρ_N.

The main objective is to obtain a rate of convergence with respect to the number of particles N from ρN to ¯ρN = ¯ρ^⊗N = Π^N_i=1ρ¯with

∂_tρ¯+ div( ¯ρu) =σ∆ ¯ρ, u=−∇V ?_xρ¯ where ¯ρ is a 1-particle distribution namely ¯ρ ≥0 and R

¯

ρ = 1. More precisely, the main objective is to prove (for σ > 0 fixed) that there exists constants C_{T ,¯ρ,k} > 0 and θ > 0 such that

kρ_N,k−Π^k_i=1ρ(t, x¯ _i)k_L1(Π^kd)≤C_{T ,ρ,k¯} N^−θ where ρ_N,k is the marginal of the system at a fixed rank k,

ρ_N,k(t, x1, . . . , x_k) = Z

T^(N−k)d

ρN(t, x1, . . . , xN)dx_k+1. . . xN.

under assumptions of global existence of entropy-weak solutionsρN of the Liouville equa- tion and global existence of classical solution ¯ρof the limiting system. Due to the Csisz´ar- Kullback-Pinsker Inequality, it suffices to control the quantity

1 k

Z

Π^kd

ρN,klogρ_N,k

¯ ρ^⊗k and therefore

1 N

Z

Π^{N d}

ρ_Nlogρ_N

¯ ρN

due to the inequality 1 k

Z

Π^kd

ρN,klogρ_N,k

¯

ρ^⊗k ≤ 1 N

Z

Π^{N d}

ρNlogρN

¯ ρ_N.

In the caseσN →0 whenN →+∞, the information will be related to a rescaled modulated potential energy similar to the one introduced by S. Serfaty.

As explained in[19] (and commented by F. Golse during the talk) the keys of the paper^Se by S. Serfaty is to introduce a clever truncation of the kernel at a length-scaleri depending on the point i and equal to a minimal distance from x_i to its nearest neighbors. Using such truncation it is possible to prove that the truncated energy can be controlled by the full energy and conversely. The next crucial result is that, even though positivity is lost, when using the ri as truncation parameters she can still control for each time slice the expression by the modulated energy itself, up to a small error , and provided the limiting solution is regular enough. This, which is the most difficult part of her proof, uses two ingredients: the first is to re-express as a single integral in terms of a stress-energy tensor, and the second is to show that the expression is in fact close to the same expression with truncated fields.

In this document, we will explain how to construct an appropriate regularized kernel (Lemmalemmatruncate

4.1) which in some sense play the role of the appropriate truncation in[20]. In the^Se1 case σ_N →0, it will be used to control the contribution to the potential energy for close particles switching between the kernel and its regularization. It will also be used coupled

with a Fourier transform property to get rid of the re-expression as a single integral in terms of a stress-energy tensor. In the case σ >0 fixed the Fourier transform hypothesis may be relaxed playing with the classical convexity inequality (^ineg6.5) and large deviation type estimates.

The paper is divided in seven sections. Section 1 is the present introduction. Section 2 is dedicated to the modulated free energy (2.1) (with^freeenergyGN andGρ¯N defined respectively by (^GN2.2) and (2.3)) introduced by the authors in^{G BrJaWa1}[5] which allows to make the link between^Se1[20]

and[15]. The analysis is based on the important inequality (^JaWa 2.6) with (^{INEG IN}2.7) with Gronwall Lemma and some appropriate estimates. Section 3 concerns the assumptions (3.1)–(^{hyp01 hyp06}3.7) on the kernels and the main results Theorem ^Conv3.1 and Theorem3.2. We focus on repulsive^Convgene kernels and provide Theorems first in the case σN → 0 when N → +∞ and secondly when σ is fixed. We provide simple comments for the attractive kernel corresponding to the Patlak-Keller-Segel system and refer to ^BrJaWa[4] for more details. Section 4 is dedicated to an important regularization lemma which will helpful to switch between the kernel and an appropriate regularized one. Section 5 and Section 6 concern the proofs of different controls from below and above coupled with inequality (2.6) with (^INEG 2.7) firstly when^IN σN →0 when N →+∞ and secondly whenσ is fixed. Section 7 concludes the proofs of Theorem 3.1 and TheoremConv ^Convgene3.2 from Gronwall arguments. In Section 8, we provide comments on the interesting attractive case corresponding to the particle approximation of the Patlak- Keller-Segel system. We end the paper by Section 9 with some comments and open problems.

2. Physics provides the right mathematical Object

As firstly introduced by the authors in [4], keeping advantage of the idea to introduce^BrJaWa appropriate weights from[3], we define the following modulated free energy^BrJa

Eρ_N G_N| ρ¯_N

G_ρ¯N

= 1 N

Z

Π^dN

ρ_N(t, X^N) log

ρ_N(t, X^N) G_N(X^N)

Gρ¯_N(t, X^N)

¯

ρ_N(t, X^N)

dX^N. (2.1) freeenergy where the Gibbs equilibrium G_N of the system and G_ρ¯N the corresponding distribution

where the exact field is replaced by the mean field limit according to the law ¯ρ are given by

G_N(t, X^N) = exp



− 1 2N σ

X

i6=j

V(x_i−x_j)



 (2.2) GN

and

Gρ¯N(t, X^N) = exp −1 σ

N

X

i=1

V ?ρ(x¯ i) + N 2σ

Z

Π^d

V ?ρ¯ρ¯

!

(2.3) G It may be written

Eρ_N G_N| ρ¯_N

G_ρ¯N

=H_N(ρ_N|¯ρ_N) +K_N(G_N|G_ρ¯N) where

H_N(ρN|¯ρN) = 1 N

Z

Π^dN

ρN(t, X^N) log

ρN(t, X^N)

¯

ρ_N(t, X^N)

dX^N (2.4) relativeentropy

is exactly the relative entropy introduced by Jabin-Wang and K_N(GN|G_ρ¯N) =−1

N Z

Π^dN

ρN(t, X^N) log

GN(X^N) Gρ¯N(t, X^N)

dX^N (2.5) modulatedenergy is the expectation of the modulated potential energy introduced by S. Serfaty multiplied

by 1/σ. It is then possible to show that the free energy has the right algebraic structure for any V even. More precisely, using smoothing properties and definition of global entropy- weak solution of the Liouville system and global classical solution of the limit system, we get the inequality

E_N ρ_N

G_N | ρ¯_N G_ρ¯N

(t) + σ N

Z t 0

Z

Π^{d N}

dρ_N

∇logρ_N

¯

ρ_N − ∇log G_N G_ρ¯N

2

≤E_N ρ_N

GN

| ρ¯_N Gρ¯N

(0) +I_N

(2.6) INEG

where I_N =−1

2 Z t

0

Z

Π^dN

Z

Π^2d∩{x6=y}

∇V(x−y)·

∇log ρ¯

G_ρ¯(x)− ∇log ρ¯ G_ρ¯(y)

(dµN −d¯ρ)^⊗2dρN,

(2.7) IN

where µ_N =P_N

i=1δ(x−x_i)/N is the empirical measure. It is important to note that the right-hand side is exactly the expectation of the quantity obtained by S. Serfaty with the modulated potential energy when σ= 0 and to remark that the parasite terms involving divKin the work by P.–E. Jabin and Z. Wang have disappeared. In order to use Gronwall Lemma, the previous expression leaves two main points in the proof namely the existence θ >0 and C >0 such that we have

I) An upper bound of IN given by (2.7).^IN If the viscosityσ >0 is fixed namely I_N ≤

Z t 0

E_N ρ_N

GN

| ρ¯_N Gρ¯N

(s)ds+ C N^θ and if σN →0 whenN →+∞

σNIN ≤CσN

Z t 0

K_N ρN

G_N | ρ¯N

G_ρ¯N

(s)ds+ C N^θ

II) A control from below on modulated quantities. For σ > 0 fixed, the modulated free energy E_N is almost positive or more specifically that for some constant

E_N ρ_N

G_N| ρ¯_N G_ρ¯N

(t)≥ 1

CH_N(ρ_N|¯ρ_N)(t)− C N^θ

and, for σ_N → 0 when N → +∞, the rescaled modulated potential energy is almost positive namely

σ_NK_N ≥ − C N^θ.

Remark 2.1. Combining the relative entropy with a modulated energy has bee successively used for various limit in kinetic theory such as quasi neutral limit (see[12],^{Kw PuSa}[17]) or Vlasov- Maxwell-Boltzmann to incompressible viscous EMHD (see[1]).^ArSa

Remark 2.2. Note the presence of modulated Fisher type information D= σ

N Z t

0

Z

Π^{d N}

dρ_N

∇logρ_N

¯ ρN

− ∇log G_N Gρ¯N

2

(2.8) diffusive in the inequality (2.7).^IN

3. Assumptions and main results.

We will split the discussions in three parts. The first part concerns the case with viscosityσ →0 when N →+∞. We show how to consider more general singular kernels than in^Se[19],[8] extending their methods. The second part concerns a fixed viscosity^Du σ >0 using the modulated free energy where we indicate kernels that may be considered. In the last part, we give comments regarding the Patlak-Keller-Segel system which concerns an attractive kernels and viscosity 2dσ > λwhereλmeasures the intensity of the kernel.

I) Repulsive Kernels. In this part, the first assumptions onV are

V(−x) =V(x), V ∈L^p(Π^d) for some p >1 (3.1) hyp01 with following Fourier sign

Vˆ(ξ)≥0 for allξ ∈R^d (3.2) hyp03

and the following pointwise controls for all x ∈T^d: There exists constants k and C > 0 such that

|∇V(x)| ≤ C

|x|^k, |∇²V(x)| ≤ C

|x|^k0 for somek, k⁰>1/2 (3.3) hyp001 and

|∇V(x)| ≤CV(x)

|x| . (3.4) hyp02

We also assume that

|x|→0lim V(x) = +∞, V(x)≤C V(y) for all|y| ≤2|x| (3.5) hyp04 and

|∇_ξVˆ(ξ)| ≤ C 1 +|ξ|

Vˆ(ξ) +f(σ) 1 1 +|ξ|^d−α

with 0< α < dfor all ξ ∈R^d (3.6) hyp05 where

f(σ) = 0 ifσ →0 andf(σ) = 1 if σ is fixed. (3.7) hyp06 Remark 3.1. Remark that Riesz and Coulomb kernels perturbed to get periodic kernels

satisfy hypothesis (3.1)–(^{hyp01 hyp06}3.7). Note that constraintsk, k⁰ >1/2 are chosen for simplicity in the argument: the results cover any kand k⁰.

I-1) Case σN →0 and repulsive kernels. The convergence rate theorem reads as follows Conv Theorem 3.1. AssumeK =−∇V withV satisfying (^hyp013.1)–(^hyp063.7)withf(σ) = 0. Consider

¯

ρ a smooth enough solution with inf ¯ρ >0. There exists constants C >0 and a function η(N) with η(N) →0 as N → ∞ s.t. for ρ¯_N = Π^N_i=1ρ(t, x¯ _i), and for the joint law ρ_N on Π^dN of any entropy-weak solution to the SDE system,

σ_NK_N(t)≤e^Cρ¯kKkt (σ_NK_N(t= 0) +σ_NH_N(t= 0) +η(N)),

Hence if σ_NK⁰_N+σ_NH⁰_N ≤η(N), for any fixed marginal ρ_N,k W₁(ρ_N,k, Π^k_i=1ρ(t, x¯ _i))≤C_{T ,¯ρ,k}η(N), where W1 is the Wasserstein distance.

I-2) Case σ > 0 fixed with respect to N and repulsive kernels. The convergence rate theorem reads as follows

Convgene Theorem 3.2. AssumeK =−∇V withV satisfying (^hyp013.1)–(^hyp063.7)withf(σ) = 1. Consider

¯

ρ a smooth enough solution with inf ¯ρ >0. There exists constants C > 0 and θ >0 s.t.

for ρ¯N = Π^N_i=1ρ(t, x¯ i), and for the joint law ρN on Π^dN of any entropy-weak solution to the SDE system,

HN(t) +|K_N(t)| ≤e^Cρ¯kKkt

HN(t= 0) +|K_N(t= 0)|+ C N^θ

, Hence if H_N⁰ +|K⁰_N| ≤C N^−θ, for any fixed marginal ρ_N,k

kρ_N,k−Π^k_i=1ρ(t, x¯ _i)k_L1(Π^{k d})≤C_{T ,¯ρ,k}N^−θ.

Remark 3.2. In the case σ > 0 fixed, it is possible to enlarge the class of kernels for instance assuming only V ≥ −C withC >0 and choosing α= 0 in (3.6). To allow^hyp05 α= 0 requires a novel large deviation inequality, similar in spirit to Prop. LargeDeviation

6.1 but for singular attractive potentials. More precisely, we can use the following proposition for which the proof is more complex and we refer to our upcoming article[5] for it. There exists^BrJaWa1 η₀ >0, θ >0 s.t. if G(x)≤C log_|x|¹ +C withC >0 andη ≤η0 then

Z

Π^{d N}

Z

{x6=y}∩{|x−y|≤η}

G(x−y) (dµN−d¯ρ)^⊗2dρN ≤CH_N + C N^θ.

Note that such control is also central in the proof for the attractive Patlak-Keller-Segel kernel, see[4] for a sketch.^BrJaWa

II)An Attractive case. This part concerns the first quantitative estimate related to par- ticle approximation of the Patlak-Keller-Segel system. Namely we get the same conclusion than in Theorem ^Convgene3.2 for

V =λlog|x|+perturbation

if 0≤λ <2dσwith the perturbation being regular kernel to getV periodic. We refer the readers to [4] for the explanation of the steps in the Proof. Some comments are given at^BrJaWa the end of the present paper.

4. An important regularization lemma

In our approach an important Lemma will be the construction of an appropriate regu- larized kernel using hypothesis (3.1)–(^hyp01 3.5). More precisely we prove the following Lemma^hyp04

Lemma 4.1. Let V satisfying (^hyp013.1)–(^hyp023.4). Then there exists a smooth approximation V_ε of V and a function ofη(ε) with η(ε)→0 as ε→0 such that

Vˆ_ε≥0,

kV_ε−Vk_L1(Π^d)≤η(ε), kI|x|≥δ(V −V_ε)k_L1 ≤C ε

δ^k, kI|x|≥δ(∇V − ∇V_ε)k_L1 ≤C ε δ^k0, V_ε(x)≤V(x) +εfor all x.

lemmatruncate

This step asks for appropriate regularization which in some sense depend on the total number of particles N: It uses the pointwise properties of the kernels (3.3)–(^hyp0013.5) which^hyp04 includes a doubling variable property.

Proof. Some preliminary controls. Consider a smooth kernel K¹ with compact support in B(0,1) and s.t. Kˆ¹ ≥ 0. Observe that for |x| ≥ 2δ and since |∇V(y)| ≤ _|x|^C_k for

|y| ≥ |x|/2,

|K_δ¹? V(x)−V(x)| ≤ Z

|z|≤1

K¹(z)|V(x−δ z)−V(x)|dz ≤C δ

|x|^k. Therefore

K_δ¹? V(x)≤V(x) +C δ^1/2, ∀ |x| ≥δ^1/2k. (4.1) |x|geqdelta On the other hand, we also have that K_δ¹ ? V(x) ≤ C δ^−dkVk_L1. Since V(x) → ∞ as

|x| →0, we also have that for some increasing function f(δ)

K_δ¹? V(x)≤V(x), ∀ |x| ≤f(δ). (4.2) |x|leqdelta As k is chosen such that k > 1/2 then δ^1/2k ≥2δ, so we need to be more precise where

f(δ)≤ |x| ≤δ^1/2k.

Construction of an appropriate regularized kernel W_ε.

Case |x| ≥ 2δ. First of all we notice that by the doubling property, we have directly if

|x| ≥2δ that|x|/2≤ |x−δ z| ≤2|x|for|z| ≤1 and thus

K_δ¹? V(x)≤C V(x), ∀ |x| ≥2δ. (4.3) |x|sim2delta Case |x| ≤2δ. For|x| ≤2δ, the doubling property on its own only gives that

K_δ¹? V(x)≤C K_δ¹? V(0)≤C δ^−d Z

B(0,δ)

V(y)dy. (4.4) K*V(0)

We now define a sequenceδn→0 s.t.

Z

B(0,δn)

V(y)dy≤C Z

δn/2≤|y|≤δn

V(y)dy. (4.5) defdelta_n

The existence of such a sequence is straightforward to show by contradiction, as otherwise, we would have for some ¯δ and all δ≤δ¯that

Z

B(0,δ/2)

V(y)dy≥(C−1) Z

δ/2≤|y|≤δ

V(y)dy.

By induction, this would imply that Z

B(0,¯δ/2^k)

V(y)dy≥(1−1/C)^k−1(C−1) Z

¯δ/2≤|y|≤δ¯

V(y)dy. (4.6) contrad

Using theL^p bound onV, we have Z

B(0,¯δ/2^k)

V(y)dy≤C⁰2^{−kp/(p−1)}

this provides a contradiction with (4.6) if^contradC−1 is too large. Note by the way that if we assume some explicit rate on V then we can have explicit bound on how large δn can be w.r.t. δ_n+1.

If δ=δn, then (^defdelta_n4.5) and (^K*V(0)4.4) together implies that K_δ¹? V(x)≤C δ^−d_n

Z

δn/2≤|y|≤δ_n

V(y)dy.

This is where we use (3.4) which implies that if^hyp02 |x| ≤ 2δn and δn/2 ≤ |y| ≤ δn then V(y)≤C V(x). Hence eventually forδ =δn, we obtain the counterpart to (|x|sim2delta

4.3) and find

K_δn? V(x)≤C V(x), ∀x. (4.7) roughgeneral

Now for every ε, we are going to choose M of the parameters δn, withM the integer part of εlarge, and define first

W_ε= 1 M

M

X

i=1

K_δni ? V.

We start by taking δn1 ≤ε²/C and we then define theni recursively s.t.

δ_ni+1 ≤min(f(δ_ni), δ_n^2k_i).

We of course have automatically that ˆWε ≥0. Moreover since maxδni →0 as ε→0, the standard approximation by convolution shows that kV −W_εk_L1 → 0. By using (^hyp0013.3), we also directly have that

kI|x|≥δ(V −W_ε)k_L1 ≤Cmaxδ_ni

δ^k ≤C ε δ^k, and similarly

kI|x|≥δ(∇V − ∇W_ε)k_L1 ≤C ε δ^k0.

It only remains to compareW_εand V. For this consider anyx, if |x| ≥δ_n^1/2k₁ then (|x|geqdelta

4.1) directly implies that Wε(x) ≤ V(x) +ε. If |x| ≤ f(δnM) then (|x|leqdelta

4.2) also directly implies that W_ε(x)≤V(x).

This only leaves the case where |x|is somewhere between δnM and δn1. In that case, there exists is.t. δ_ni+1 ≤ |x| ≤δ_ni. By the definition of theδ_nj, one has that|x| ≤f(δ_nj) ifj < i and |x| ≥δn^1/2kj ifj > i+ 1. Using again (|x|geqdelta

4.1) and (|x|leqdelta

4.2), we then have K_δnj ? V(x)≤V(x) +ε ifj > i+ 1, K_δnj ? V(x)≤V(x) ifj < i.

Using (roughgeneral

4.7) for j=iand j=i+ 1, we get

Wε(x)≤(1 + 2C/M)V(x) +ε= (1 + 2C ε)V(x) +ε.

Definition of V and conclusion. This leads to the final definition Vε = Wε/(1 + 2C ε) which indeed satisfiesVε≤V+ε, ˆVε≥0 and stillkV_ε−Vk_L1 →0 together with the other

convergences since obviously kV_ε−W_εk_L1 →0.

5. Control of terms for Repulsive Kernels with vanishing viscosity σ_N →0 As a corollary of Lemma lemmatruncate

4.1, we obtain a straightforward control on the contribution to the potential energy from close particles

Lemma 5.1. Under the assumptions of Lemma lemmatruncate

4.1, there exists a function η(N) with η(N)→0 as N → ∞, one has that

σ_NK_N(t)≥ −η(N).

Furthermore there exists η(δ) withη(δ)→0 as η→0, such that for any δ E 1

N² X

i6=j

V(xi−xj)I|x_i−x_j|≤δ

≤4σNK_N(t) +η(N) +η(δ).

KNgeq0

Proof. We start by noticing that σ_NK_N(t) = 1

2 Z

|x−y|≥δ

V(x−y) (µ_N−ρ)¯^⊗2+1 2

Z

|x−y|<δ, x6=y

V(x−y) (µ_N −ρ)¯ ^⊗2. By using the regularity of ¯ρand the L^p integrability of V, we may bound from below the second term in the right-hand side by

Z

|x−y|<δ, x6=y

V(x−y) (µ_N −ρ)¯^⊗2 ≥ 1 N²

X

i6=j

V(x_i−x_j)I|x_i−x_j|≤δ−C δ^α, for some positive exponent α. Using Lemma lemmatruncate

4.1 and more precisely the inequality V_ε ≤ V +ε, we obtain that

Z

|x−y|<δ, x6=y

V(x−y) (µ_N −ρ)¯^⊗2≥ 1 N²

X

i6=j

V_ε(x_i−x_j)I|x_i−x_j|≤δ−C δ^α−ε.

Observe now that by the second point in Lemma lemmatruncate

4.1, we have that Z

|x−y|≥δ

V(x−y) (µ_N −ρ)¯^⊗2 ≥ Z

|x−y|≥δ

V_ε(x−y) (µ_N −ρ)¯^⊗2−C ε

δ^k. (5.1) intermed0 Therefore by summing, we obtain that

Z

x6=y

V(x−y) (µN −ρ)¯^⊗2 ≥ Z

x6=y

Vε(x−y) (µN −ρ)¯^⊗2−C δ^α−ε−C ε δ^k. We may simply add the diagonal to find

Z

x6=y

V(x−y) (µ_N−ρ)¯^⊗2 ≥ Z

V_ε(x−y) (µ_N−ρ)¯^⊗2−CkV_εk_L^∞

N −C δ^α−ε−C ε

δ^k. (5.2) intermed1 Since ˆV_ε≥0, this yields

Z

x6=y

V(x−y) (µ_N −ρ)¯^⊗2 ≥ −CkV_εk_L^∞

N −C δ^α−ε−C ε δ^k, and conclude the first point by optimizing in εand δ.

To prove the second point we first remind that 1

N² X

i6=j

V(xi−xj)I|x_i−x_j|≤δ≤ Z

|x−y|<δ, x6=y

V(x−y) (µN −ρ)¯^⊗2+C δ^α, and thus

1 N²

X

i6=j

V(x_i−x_j)I|x_i−x_j|≤δ ≤ Z

x6=y

V(x−y) (µ_N −ρ)¯^⊗2+C δ^α

− Z

|x−y|≥δ, x6=y

V(x−y) (µN −ρ)¯ ^⊗2. Using again (5.1), we get^intermed0

1 N²

X

i6=j

V(x_i−x_j)I|xi−xj|≤δ ≤ Z

x6=y

V(x−y) (µ_N −ρ)¯^⊗2+C δ^α

− Z

|x−y|≥δ, x6=y

Vε(x−y) (µN −ρ)¯ ^⊗2+C ε δ^k.

Remark that since we are on the torus, I|x|≥δ has L¹ norm less than 1. Therefore the Fourier transform of V_εI|x|≥δ is dominated by ˆV_ε and

− Z

|x−y|≥δ, x6=y

Vε(x−y) (µ_N −ρ)¯^⊗2 ≤ − Z

Vε(x−y) (µ_N −ρ)¯^⊗2. Appealing to (^intermed15.2), we hence finally get

1 N²

X

i6=j

V(x_i−x_j)I|x_i−x_j|≤δ≤2 Z

x6=y

V(x−y) (µ_N −ρ)¯^⊗2 +C δ^α+C ε

δ^k +ε+CkV_εk_L^∞ N ,

which concludes the second point, again by optimizing inε.

We need to control terms from above like IN =−

Z

Π^dN

dρN

Z

Π^2d∩{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµN −d¯ρ)^⊗2,

for ψ regular enough in terms of the potential energy. We use Fourier transform for the regularized kernel that not use explicit formula of the kernel as in[19],^Se [8]. This procedure^Du allows to treat more general kernels because it is not based on the reformulation of the energy in terms of potential or extension representation (for the fractional laplacian) by Caffarelli-Silvestre as in ^Se[19],[20],^Se1 [8]. More precisely, we prove that^Du

Lemma 5.2. Let ψ ∈W^1,∞(Π^d) and if V satisfies (3.1)–(^hyp01 3.2), then for any measure^hyp03 ν, we have that

− Z

∇V(x−y)(ψ(x)−ψ(y))ν^⊗2≤C Z

|ˆν(ξ)|²Vˆ(ξ)dξ.

lemfourier

Remark 5.1. Remark that Riesz and Coulomb Kernel satisfy Hypothesis (3.1)–(^{hyp01 hyp05}3.6).

Proof. This is where we need (^hyp053.6) which gives studying|ξ| ∼ |ζ|

|Vˆ(ξ)−Vˆ(ζ)| ≤ C|ξ−ζ|

1 +|ζ| Vˆ(ζ)

using Gronwall Lemma. Forψ regular enough, we use the following calculation

− Z

∇V(x−y)(ψ(x)−ψ(y))ν^⊗2=−Re Z

i(ξVˆ(ξ)−ζVˆ(ζ)) ˆψ(ξ−ζ)ˆν(ξ)ˆν(ζ)dξdζ

=−Re Z

i

ξ( ˆV(ξ)−Vˆ(ζ)) + (ξ−ζ) ˆV(ζ)

ψ(ξˆ −ζ)ˆν(ξ)ˆν(ζ)dξdζ

≤C Z

|ξ−ζ||ψ(ξˆ −ζ)|

qVˆ(ξ)

qVˆ(ζ)|ˆν(ζ)||ˆν(ξ)|dξdζ and therefore, using some regularity on ψ, by Cauchy-Schwartz

− Z

∇V(x−y)(ψ(x)−ψ(y))ν^⊗2≤C Z

|ˆν(ζ)|²Vˆ(ζ)dζ.

Of course we cannot directly use Lemma6.7 with^fourier V onI_N as

Z

V(x−y) (dµN −d¯ρ)^⊗2

will in general be infinite as the diagonal is not removed. But we can now easily combine Lemma ^lemfourier5.2 with Lemma^KNgeq05.1 to obtain

controlINvanishing Corollary 5.1. Assume thatψ∈W^s,∞and thatV satisfies (3.1)–(^hyp01 3.6). Then there exists^hyp05 a function η(N) with η(N)→0 as N → ∞ and such that

IN =− Z

Π^dN

dρN

Z

Π^2d∩{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµN−d¯ρ)^⊗2 ≤C σNK_N(t) +η(N).

Proof. The basic strategy is again to splitI_N into two parts IN =−

Z

Π^dN

dρN

Z

{|x−y|≤δ}∩{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµN −d¯ρ)^⊗2

− Z

Π^dN

dρ_N Z

{|x−y|>δ}

∇V(x−y)·(ψ(x)−ψ(y))(dµ_N −d¯ρ)^⊗2.

For the second term in the right-hand side, we want to replace∇V by∇V_ε. We simply use the second point of Lemma lemmatruncate

4.1 again (similarly to the obtention of (^intermed05.1) in Lemma ^KNgeq05.1) to get that

− Z

{|x−y|>δ}

∇V(x−y)·(ψ(x)−ψ(y))(dµ_N −d¯ρ)^⊗2

≤ − Z

{|x−y|>δ}

∇V_ε(x−y)·(ψ(x)−ψ(y))(dµN −d¯ρ)^⊗2+C ε δ^k0.

(5.3) intermed0’

For the first term in the right-hand side, we first use the regularity of ¯ρ together with the L^p bound onV from (^hyp013.1) to deduce that

− Z

{|x−y|≤δ}∩{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµ_N −d¯ρ)^⊗2

≤ − Z

{|x−y|≤δ}∩{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))dµ^⊗2_N +C δ^α,

for some α > 0. We know use the pointwise bound on ∇V from (3.4) and the Lipschitz^hyp02 bound onψ to obtain that

− Z

{|x−y|≤δ}∩{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµ_N−d¯ρ)^⊗2

≤C Z

{|x−y|≤δ}∩{x6=y}

V(x−y)dµ^⊗2_N +C δ^α. Using the second point in Lemma5.1, we hence have that^KNgeq0

− Z

Π^dN

dρ_N Z

{|x−y|≤δ}∩{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµ_N −d¯ρ)^⊗2

≤C σ_NK_N+η(N) +η(δ).

(5.4) intermed1’

By the construction of V_ε the same estimate applies

− Z

Π^dN

dρN

Z

{|x−y|≤δ}∩{x6=y}

∇V_ε(x−y)·(ψ(x)−ψ(y))(dµN −d¯ρ)^⊗2

≤C σ_NK_N+η(N) +η(δ).

(5.5) intermed2’

We may combine (5.5) with (^{intermed2’intermed0’}5.3) to obtain that

− Z

Π^dN

dρN

Z

{|x−y|>δ}∩{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµN−d¯ρ)^⊗2

≤ − Z

Π^dN

dρ_N Z

{x6=y}

∇V_ε(x−y)·(ψ(x)−ψ(y))(dµ_N −d¯ρ)^⊗2 +C ε

δ^k0 +C σNK_N +η(N) +η(δ).

Together with (^intermed1’5.4), this finally concludes that

− Z

Π^dN

dρ_N Z

{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµ_N −d¯ρ)^⊗2

≤ − Z

Π^dN

dρN

Z

∇V_ε(x−y)·(ψ(x)−ψ(y))(dµN −d¯ρ)^⊗2 +C ε

δ^k0 +C σ_NK_N+η(N) +η(δ).

(5.6) intermed3’

We know apply Lemma ^lemfourier5.2 onV_ε which by construction still satisfies (^hyp013.1)-(3.6) and this^hyp05 yields

− Z

Π^dN

dρ_N Z

{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµ_N −d¯ρ)^⊗2

≤C Z

Π^dN

dρN

Z

Vε(x−y) (dµN −d¯ρ)^⊗2 +C ε

δ^k0 +C σ_NK_N +η(N) +η(δ).

We may now remove the diagonal

− Z

Π^dN

dρN

Z

{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµN −d¯ρ)^⊗2

≤C Z

Π^dN

dρ_N Z

{x6=y}

V_ε(x−y) (dµ_N−d¯ρ)^⊗2+CkV_εk_L^∞ N +C ε

δ^k0 +C σNK_N+η(N) +η(δ), and using the third point in Lemma lemmatruncate

4.1,

− Z

Π^dN

dρN

Z

{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµN −d¯ρ)^⊗2

≤C Z

Π^dN

dρ_N Z

{x6=y}

V(x−y) (dµ_N−d¯ρ)^⊗2+C ε+C kV_εk_L^∞ N +C ε

δ^k0 +C σ_NK_N +η(N) +η(δ).

The conclusion follows by optimizing in εandδ.

6. Control of terms for Repulsive Kernels with fixed viscosity σ >0 6.1. A large deviation result.

LargeDeviation Proposition 6.1. There exists δ₀ > 0 and some exponent θ > 0, s.t. for any ρ¯∈ L^∞∩ P(Π^d) with log ¯ρ ∈ W^1,∞, for any W ∈ L^p(Π^2d) for some p > 1 with W(−x) = W(x), W ≥0, |∇W(x)| ≤ _|x|^C_k and finallykWk_L1 ≤δ0 then

1 N log

Z

Π^dN

e^−N

R

{x6=y}W(x−y) (dµ_N−d¯ρ)^⊗2

¯

ρ^⊗NdX^N ≤ C N^θ. Proof. We denote

F(µN) = Z

{x6=y}

W(x−y) (dµN −d¯ρ)^⊗2, and introduce a simple truncation Wε of W by

W_ε(x) =W(x)I|x|≥ε. We define as well

Fε(µ) = Z

Π^2d∩{x6=y}

Wε(x−y) (dµ−d¯ρ)^⊗2.

We may expand F to find F(µN) =

Z

Π^2d∩{x6=y}

W(x−y) (dµN −d¯ρ)^⊗2

= 1 N²

X

i6=j

W(xi−xj)−2 1 N

X

i

W ?ρ(x¯ i) +

Z

Π^2d∩{x6=y}

W(x−y) ¯ρ(x) ¯ρ(y)dx dy.

Now note that by theL^p integrability onW and theL^∞ integrability on ¯ρ,

|W_ε?ρ(x)¯ −W ?ρ(x)| ≤ kW¯ _ε−Wk_L1kρk¯ _L^∞ ≤ kW I|z|≤εk_L1kρk¯ _L^∞

≤C ε^1/2p∗kρk¯ _L^∞.

Further note that since W ≥0, we have that W_ε(x)≤W(x) so this directly implies that

−F(µ_N)≤ −F_ε(µ_N) +Ckρk¯ _L^∞ε^1/2p∗. We are hence led to bounding

Z_N,ε= 1 N log

Z

Π^dN

e^{−N RΠ2dWε(x−y) (dµN−d¯ρ)⊗}

2

¯

ρ^⊗NdX^N, since

Z_N = 1 N log

Z

Π^dN

e^−N

R

{x6=y}W(x−y) (dµ_N−d¯ρ)^⊗2

¯

ρ^⊗NdX^N ≤Z_N,ε+C ε^1/2p∗. (6.1) boundZN We now rely on a quantitative variant of a classical large deviation result

Theorem 6.1. Assume that log ¯ρ ∈W^1,∞ and that L is a standard convolution kernel.

Then there exists a constant C depending only on d, L, s.t. for any F : P(Π^d) → R, continuous on continuous functions, one has that

1 N log

Z

Π^dN

e^{−N F(Lδ?µN)}ρ¯_NdX^N ≤I(F)

+ C

N^1/(d+1)δ^d/(d+1) (logN+|logδ|+klog ¯ρk_L^∞) +C δklog ¯ρk_W^1,∞, where

I(F) = max

µ∈P(Π^d)

−

F(µ) + Z

Π^d

µ logµ

¯ ρ dxi

. (6.2) largedeviationfunct

variantlargedeviation

Proof. The proof of Theoremvariantlargedeviation

6.1 relies on classical arguments and we refer to our upcoming article for more details. Since|∇W| ≤C/|x|^k, the potential W_ε is smooth and hence

−F_ε(µ_N)≤ −F_ε(L_δ? µ_N) +C δ ε^k. Using Theorem variantlargedeviation

6.1, we thus have that Z_N,ε ≤I(F_ε) +C δ

ε^k + C

N^1/(d+1)δ^d/(d+1)(logN+|logδ|+klog ¯ρk_L^∞) +C δklog ¯ρk_W^1,∞.

(6.3) boundZNeps The last step of the proof is hence to estimate I(F_ε) for which we appeal to

Lemma 6.1. For anyρ¯∈L^∞, there exists a truncationδ0 s.t. for anyW˜ withkW˜k_L1 ≤δ0

and W(x)≥0 then, defining FW˜(µ) =

Z

{x6=y}

W˜(x−y) (µ(dx)−ρ(x)¯ dx) (µ(dy)−ρ(y)¯ dy), one then has that I(F_W˜) = 0.

I(F)log

Assuming that Lemma ^I(F)log6.1 holds, we may apply it to ˜W =Wε which impliesI(Fε) = 0 and finally combining (^boundZN6.1) and (^boundZNeps6.3)

ZN ≤C ε^1/2p∗+C δ

ε^k + C

N^1/(d+1)δ^d/(d+1) (logN +|logδ|+klog ¯ρk_L^∞) +C δklog ¯ρk_W^1,∞.

We may immediately conclude by optimizing in εand δ.

Sketch of the proof of Lemma 6.1.^I(F)logSince ˜W ≥0, I(FW˜) is coercive and by considering a maximizing sequence, we may find a maximum ¯µ which is bounded in L logL. Such a maximum must satisfy that

1 + logµ¯

¯

ρ + 2 ˜W ?(¯µ−ρ) =¯ κ, where the constant κ is chosen so thatR

¯

µ= 1. This may be rewritten as

¯ µ= ρ¯

M e^{−2 ˜W ?(¯µ−¯ρ)}, M = Z

¯

ρ e^{−2 ˜W ?(¯µ−¯ρ)}dx.

Let us denoteu=−W ?˜ (¯µ−ρ) and to emphasize the dependence on¯ uin M M =M_u=

Z

¯

ρ e^2u(x)dx.

We observe thatu is a solution to

u=−W ?˜ ρ¯ e^2u(x) M_u −1

!!

, (6.4) eulerlagrange

which is in fact a sort of non-linear elliptic equation. It is straightforward to show that the unique solution to (eulerlagrange

6.4) isu= 0 provided that kW˜k_L1 is small enough.

6.2. Control on EN. The second ingredient to bound EN from below is the following classical convexity inequality

Z

Π^dN

ψ(X^N)dρN ≤ 1 α

1 N

Z

dρN logρN

¯ ρ_N + 1

α 1 N log

Z

Π^dN

e^{αN ψ(XN)}d¯ρN. (6.5) ineg The proper control ofE_N however requires truncating interactions after some distance so

that we define K^η_N = 1

2σ Z

Π^{d N}

Z

{x6=y}

V(x−y)χ(|x−y|/η) (dµ_N −d¯ρ)^⊗2ρNdX^N,

where χ is some smooth non-negative function with χ=1 on [0,1] and χ = 0 on [2,∞), together with ˆχ ≥0. Finally we define E_N^η =H_N +K^η_N. Combined with Prop. LargeDeviation

6.1 this inequality lets us bound from below E_N^η as per

boundEN Proposition 6.2. Assume that V satisfies (^hyp013.1)-(3.4)^hyp02and (^hyp063.7) then there exists η0 >0 and θ >0 so that for any η≤η₀

E_N^η ≥ − C N^θ. Moreover for any δ≤η with η≤η0

E



 1 N²

X

i6=j

V(xi−xj)I|x_i−x_j|≤δ



≤E^η_N+ C

N^θ +C δ^θ. Proof. For the first point, using (^ineg6.5) onK^η_N, we find that

−K_N^η ≤H_N + 1 N log

Z

Π^dN

e^−2Nσ

R

{x6=y}V(x−y)χ(|x−y|/η) (dµ_N−d¯ρ)^⊗2

d¯ρ_N. We now apply Prop. LargeDeviation

6.1 to W = ₂¹_σV(x)χ(|x|/η). From (^hyp013.1) and (^hyp0013.3), W trivially satisfies all assumptions of Prop. LargeDeviation

6.1 with the exception of kWk_L1 ≤ δ₀. For this, we remark that

kV(x)χ(|x|/η)k_L1 ≤ Z

|x|≤2η

V(x)dx≤CkVk_Lpη^1/p∗,

so thatkWk_L1 ≤δ₀ is ensured by choosingη≤η₀ withη₀ small enough. Therefore Prop.

LargeDeviation

6.1 implies that

−K^η_N ≤HN + C N^θ. For the second point, observe first that

1 N²

X

i6=j

V(x_i−x_j)I|x_i−x_j|≤δ≤ 1 N²

X

i6=j

V(x_i−x_j)χ(|x_i−x_j|/δ)

≤ Z

{x6=y}

V(x−y)χ(|x−y|/δ) (dµ_N −d¯ρN)^⊗2+C δ^θ. Therefore

E



 1 2σ N²

X

i6=j

V(x_i−x_j)I|xi−xj|≤δ



−E_N^η

≤ −H_N− 1 2σ

Z

{x6=y}

V(x−y) (χ(|x−y|/η)−χ(|x−y|/δ)) (dµ_N−d¯ρ_N)^⊗2+C δ^θ. Using (^ineg6.5)

E



 1 2σ N²

X

i6=j

V(xi−xj)I|x_i−x_j|≤δ



−E_N^η

≤ 1 N log

Z

Π^{d N}

e⁻

N 2σ

R

{x6=y}V(x−y) (χ(|x−y|/η)−χ(|x−y|/δ)) (dµ_N−d¯ρN)^⊗2

d¯ρN +C δ^θ.

Of courseχ(|x−y|/η)−χ(|x−y|/δ)≥0 ifδ≤η so that we may again apply Prop. LargeDeviation

6.1 toW(x) =V(x) (χ(|x|/η)−χ(|x|/δ)) and find as claimed

E



 1 2σ N²

X

i6=j

V(xi−xj)I|x_i−x_j|≤δ



−E_N^η ≤C δ^θ+ C N^θ.

6.3. Control of the right-hand side. As for the caseσ→0, the goal is to control

I_N =− Z

Π^{d N}

Z

{x6=y}

∇V(x−y)·(ψ(x)−ψ(y)) (dµ_N −d¯ρ)^⊗2dρ_N.

The Fourier assumption (3.6) for^hyp05 σ >0 is more general that assumption (^hyp053.6) in the case σ = 0 because we can use Inequality (^ineg6.5) and Theorem LargeDeviation

6.1 to control the new term in the Fourier procedure. More precisely, we have

Lemma 6.2. Assume thatψ∈W^k,+∞withklarge enough and thatV satisfies (3.1)-(^hyp01hyp023.4) and

|Vˆ(ξ)−Vˆ(ζ)| ≤C|ξ−ζ|

1 +|ζ|

Vˆ(ζ) + χ(ζ) 1 +|ζ|^d−α

(6.6) diffhatV with 0< α < dand χ∈L^∞_ζ . Then for any measure ν, we have that

− Z

∇V(x−y)(ψ(x)−ψ(y))ν^⊗2≤C Z

|ˆν(ξ)|²( ˆV(ξ) +χ(ξ)/(1 +|ξ|^d−α)dξ. (6.7) fourier lemfourier2

Proof. The proof follows the same lines as for Lemma5.2.^lemfourier Note that the extra term in the right-hand side of inequality (^fourier6.7) is controlled by the relative entropy because going back to the physical space, we will get a term written as

Z

x6=y

G(x−y)(µ_N −ρ)¯^⊗2(dxdy)

withcontrolINvanishingG(x) ∼ |x|^−α ∈ L^p for some p >1. We can now prove the equivalent of Corollary 5.1.

controlINnonvanishing Corollary 6.1. Assume that ψ∈W^s,∞ and that V satisfies all assumptions (^hyp013.1)–(^hyp063.7).

Then there exists η0 andθ >0 s.t. if η≤η0

IN =− Z

Π^dN

dρN

Z

Π^2d∩{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµN −d¯ρ)^⊗2 ≤C E_N^η(t) + C N^θ. Proof. The proof closely follows the one for Corollary (controlINvanishing

5.1). More precisely all arguments up to obtaining (5.6) are identical and hence we have still^intermed3’

I_N =− Z

Π^dN

dρ_N Z

{x6=y}

∇V(x−y)·(ψ(x)−ψ(y))(dµ_N−d¯ρ)^⊗2

≤ − Z

Π^dN

dρN

Z

{x6=y}

∇V_ε(x−y)·(ψ(x)−ψ(y))(dµN−d¯ρ)^⊗2 +C ε

δ^k0 +CK_N + C

N^θ +C η^θ.