Statistical mechanics of the <span class="mathjax-formula">$N$</span>-point vortex system with random intensities on a bounded domain

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Statistical mechanics of the N -point vortex system with random intensities on a bounded domain

Mécanique statistique des systèmes de N vortex ponctuels à intensités aléatoires sur un domaine borné

Cassio Neri

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, Brazil Received 5 April 2003; accepted 29 April 2003

Available online 16 December 2003

Abstract

The system of N point vortices on a bounded domainΩ is considered under the hypothesis that vortex intensities are independent and identically distributed random variables with respect to a lawP supported on a bounded subset ofR. It is shown that, in the limitN→ +∞, the 1-vortex distribution onΩis a minimizer of the free energy functional (a combination of entropy and energy functionals) and is associated to (some) solutions of the following non-linear Poisson Equation (called Mean Field Equation):



−u(x)=

e^−βru(y)dyP (dr) ₋₁

r₁e^−βru(x)P (dr), ∀x∈Ω,

u(x)=0, ∀x∈∂Ω.

(1)

2003 Elsevier SAS. All rights reserved.

Résumé

Le systéme deN vortex ponctuels sur un domaineΩ borné est considéré sous l’hypothèse que les intensités des vortex sont des variables aléatoires indépendentes et identiquement distribuées selon la loiP de support inclus dans un borné deR. On montre que, à la limite quandN→ +∞, la distribution d’un seul vortex dansΩest un minimiseur de la fonctionnelle d’énergie libre (qui est une composition de l’entropie et de l’énergie) et qui est associée à (certaines) solutions de l’équation de Poisson non linéaire (dite de Champ Moyen) (1).

2003 Elsevier SAS. All rights reserved.

Keywords: Statistical mechanics;N-point vortex system; Onsager theory; Limits of Gibbs’s measures; Entropy; Mean field equation

E-mail address: [email protected] (C. Neri).

1 Supported by CNPq grant number 200491/97-0, Brazil.

0294-1449/$ – see front matter 2003 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2003.05.002

1. Introduction

Systems ofN-point vortices in a smooth, bounded, connected open domainΩ⊂R²of Lebesgue measure|Ω| have been studied with some success since Onsager [15]. This paper studies such systems assuming the vortex intensities to be independent and identically distributed random variables with respect to a lawP. We assume that P is supported on a bounded subset ofR(for simplicity we shall considerP supported on[−1,1]).

We follow the approach of Cagliotti et al. [2] who handled the case in which all vortices have intensity equal to 1. This situation corresponds to the lawP being the Dirac measure concentrated at 1 and, hence, our results will generalize some of those in [2].

The first motivation for considering random intensities is to provide a mathematical explanation to certain results which are well known to physicists. An example is the work of Joyce and Montgomery [5] on the statistical mechanics of the so-called neutral systems, which are defined by an equal number of vortices of intensity 1 and−1.

This situation is similar to a particular case in our approach (also called neutral) in which each vortex has intensity either 1 or−1 with probabilities 1/2 in each case.

In Section 2 we introduce the canonical Gibbs measure µ^N associated with the system of N vortices and temperatureN/β. The phase space of the Hamiltonian system is, essentially,Ω^N and thus it has finite measure.

In Section 3 we shall show that the problem is well posed forβ in (−8π,8π ). Forβ <0 we mean states of negative temperatures as predicted by Onsager and considered by Joyce and Montgomery [5,6,12]. The length of this interval is connected to the support ofP.

As it happens in all statistical theories, we expect the results should have more physical meaning whenN is large. So we letN go to infinity and look for cluster points of(µ^N)N>1in the following way: we introduceµ^N_k (k∈N), the marginal distribution ofkvortices induced byµ^N. Then we look for weak cluster points (inL^p) ofµ^N_k asN→ +∞withkfixed. Since for eachkwe have a cluster point ofµ^N_k , we find sequencesµ_∗=(µ_k)_k∈N, called weak cluster points of(µ^N)N>1, which we characterize by variational arguments. This is the subject of Section 5.

We remark thatµ^N minimizes the free-energy functionalF^N (which is a composition of two terms: entropy and energy). We define the limit functionalF^∗for which weak cluster points are minima.

AsN increases, we expect vortex positions to become independent of one another. This means the following:

the field created by vortices converges to a mean field; thek-vortex distributionsµ^N_k “factorize” and, at the limit N→ +∞, this distribution behaves like a product ofkcopies of 1-vortex distributions (this factorization property is called “propagation of chaos”.)

Our main tool is the Hewitt–Savage theorem [7]. Using this result, we rewrite the limit problem to show that weak cluster points are averages of product distributions, i.e.µ_k=

ρ^⊗kξ(dρ), whereξ is a probability measure on the space of 1-vortex distributions. Whenβ >0, ξ is a Dirac measure and thus we have the propagation of chaos. On the other hand, for negative temperatures, the propagation depends on several factors (P,β and the geometry of the domain). In any case, we can show that in the support ofξwe have only 1-vortex distributions that are minima of a certain functionalF.

The mean fields are nothing but the Newton potentials associated with minima of F. These potentials are solutions of Eq. (1). This is a semi-linear Poisson equation (with an exponential non-linearity) called Mean Field Equation (MFE for short). In the neutral case, up to a constant, we find the same as that found in [1,4–6,9–13].

This constant is a nonlocal and nonlinear term of MFE (this makes numerical implementation more difficult).

The MFE is studied in Sections 6, 8, 9 and 10. We introduce another functionalGwhich acts on the potentials and for which the MFE is the associated Euler–Lagrange equation. We show existence of minimizers forGusing the sharp form of Moser–Trundiger inequality [14]. We show thatG preserves the minimizers of F, i.e. each potential associated with a minimizer ofF is a minimizer ofGand conversely. We find a relationship between the minimal values of these two functionals.

We are interested in the uniqueness of solutions for the MFE, as this is directly related to the propagation of chaos. If uniqueness holds, then we have a unique minimizer ofF and henceξ is a Dirac measure (concentrated at this point). As we have mentioned before, there is propagation of chaos in a positive temperature state, and this

is indeed a consequence of the uniqueness of solutions for the MFE, which arises from the strict convexity of the functionalG.

2. Notation

We introduce some notation which will be used in the sequel. SetΩ=Ω× [−1,1].X=(x˜₁, . . . ,x˜_N)denotes an arbitrary point inΩ^N, wherex˜i =(xi, ri)(xi∈Ω andri∈ [−1,1]). Allri’s are random variables identically distributed with respect to a Borelian probability measureP on[−1,1]. OnΩwe consider the product measure Lebesgue×P. Bya.e.we mean almost everywhere with respect to Lebesgue,P, or Lebesgue×P measures without precising which one we are considering.

For all 1kN andX∈Ω^N we set X=(x₁, . . . , x_N)and defineX_k =(x˜₁, . . . ,x˜_k)andX^N−k =(x˜_k+1, . . . ,x˜N)(Xk andX^N−k are analogous defined).

For the purpose of integration we set dx˜_i=dx_iP (dr_i), dX=dx˜₁. . .dx˜_N, and dX=dx₁. . .dx_N. In an obvious way we define dX_k, dX^N−k, dX_k, and dX^N−k.

The Hamiltonian of theN-point vortex system is given by

H^N(X)=1 2

N

i=j

rirjV (xi, xj),

whereV is the Green function of the Poisson equation inΩwith homogeneous Dirichlet boundary conditions. For simplicity, we setH=H².

As it is classical,V is given by V (x1, x2)= − 1

2π log|x1−x2| +γ (x1, x2), (2)

whereγ is its regular part, which is bounded from above. We know thatV is positive and

Ω

V (x1, x2)dx2C, ∀x1∈Ω,

for some constantC(depending onΩ).

The associated Gibbs measure, with inverse temperatureβ⁻¹, is defined by µ^N(X)= 1

Z(N, β)e^−NβHN(X), where

Z(N, β)=

Ω^N

e^−NβHN(X)dX (3)

is the partition function.

FinallyC, with or without indices, denotes several positive constants and 1_Adenotes the characteristic function of a setA.

3. Bounds for the partition function

First of all, we should find the range ofβfor whichµ^Nmakes sense, i.e. for which the integral in (3) converges.

The following proposition yields this result and also provides some bounds for the partition function. These bounds will be useful when we shall letN go to+∞.

Proposition 1. Letβ∈(−8π,8π ). There existC1andC2=C2(β)such that C₁^NZ(N, β)C₂^N, ∀N2.

Moreover,C2is uniform on compact subsets of(−8π,8π ).

Proof. Let us fixα∈(0,8π )such that[−α, α]is a compact interval which containsβ. By Hölder’s inequality and the boundness ofγ from above we have

Z(N, β)

Ω^N

N i=1

e

|β| 2N

N j=1, j=i

V (xi,xj)

dX

N i=1

Ω^N

e

α 2

N j=1, j=i

V (xi,xj)

dX ¹

N

Ω

Ω^N−1

N j=2

e^C

|x₁−x_j|^α/4π dX^N−1dx₁C^N−1

Ω Ω

1

|x₁−x₂|^α/4π dx_{2 N−1}

dx₁.

The innermost integral is finite sinceα/4π <2. It follows thatZ(N, β)C^N−1|Ω|C₂^N (C₂depends onαbut not onβ).

The lower bound is a consequence of Jensen’s inequality, the positivity ofV and (2). Indeed, Z(N, β)|Ω|^Nexp

− β 2N|Ω|^N

N

i=j

Ω^N

rirjV (xi, xj)dX

|Ω|^Nexp

−4π(N−1)

|Ω|²

Ω

Ω

V (x1, x2)dx1dx2

|Ω|^Ne^−C(N−1)C^N₁.

This completes the proof. 2

From now onβwill be fixed in(−8π,8π ).

4. Existence of weak cluster points for Gibbs measures

The elements of the Gibbs sequence(µ^N)_N>1 are functions defined on different domains. They are points in different functional spaces. This leads to a problem when looking for limits of this sequence. To overcome this problem we introduce the family of correlation functions(ρ_k)_1kNof a functionρ∈L¹(Ω^N), defined by

ρ_k(X_k)=

Ω^N−k

ρ(X)dX^N−k.

In the probability jargon, whenρis a probability density onΩ^N,ρkis the marginal density ofXk.

Now, for eachk∈N,(µ^N_k)N>k is a sequence onL¹(Ω^k)and thus we can look for its cluster points. Before findingL^pestimates for these sequences we find point-wise ones. First we have the following lemma.

Lemma 2. There existsC (depending only onΩ but not onβ)such that Z

k,βk

N

C^N−kZ(N, β), ∀k2, ∀N > k.

Proof. It is easy to see that Z

k+1,β(k+1) N

=

Ω^k

e^−NβHk(Xk)f (X_k)dX_k, (4)

where

f (X_k)=

Ω

e⁻

β N

k i=1

rirk+1V (xi,xk+1)

dx˜_k+1.

By Jensen’s inequality, the positivity ofV and (2) we have f (X_k)|Ω|exp

− β

|Ω|N

k

i=1

Ω

r_ir_k+1V (x_i, x_k+1)dx˜_k+1

|Ω|exp

− 8π

|Ω|N

k

i=1

Ω

V (x_i, x_k+1)dx_k+1

|Ω|exp

−8π kC N

|Ω|e^{−8π C}=C,

which, with Eq. (4), yields Z

k,βk

N

CZ

k+1,β(k+1) N

. By induction we finish the proof. 2

The next proposition yields point-wise estimates for(µ^N_k)N>k. Proposition 3. There existsC=C(β)such that

µ^N_k(X_k)C^ke^−NβHk(Xk), ∀k2, ∀N large enough.

Proof. LetN₀, N∈Nandr, p, p ∈Rbe such that

• r >1 withβr∈(−8π,8π );

• N₀=min{N∈N|N >2kandN/(N−2k) < r}andNN₀;

• p=N/(N−2k)andp =N/2k.

It is easy to see that

µ^N_k(Xk)= 1

Z(N, β)e^−NβHk(Xk)

Ω^N−k

e

−N^βH^N−k(X^N−k)−N^β

k i=1

N j=k+1

H (x˜i,x˜j)

dX^N−k.

By Hölder’s inequality the last integral is bounded above by

Ω^N−k

e^{−βpNHN−k(XN−k)}dX^N−k 1/p

Ω^N−k

e

−^βpN

k i=1

N j=k+1

H (x˜i,x˜j)

dX^N−k 1/p

.

Since|β|pk/2π N= |β|/4π <2, by an analogous argument as in the proof of Proposition 1, we show that the second term is bounded above byC^k (C=C(β)). We easily see that the first term is equal toZ(N−k, βp(N− k)/N )^1/p. To finish the proof, we shall show that there existsC=C(β)such that

Z(N−k, βp(N−k)/N )^1/p Z(N, β) C^k. By Lemma 2 there existsC >1 such that

Z(N−k, βp(N−k)/N )^1/p

Z(N, β) C^kZ(N, βp)^1/p Z(N, β) . Again, by Hölder’s inequality, we have

Z(N, βp)^1/pZ(N, βr)^θ/rZ(N, β)^1−θ, i.e.

Z(N, βp)^1/p

Z(N, β) Z(N, βr)^θ/rZ(N, β)^−θ,

whereθ∈(0,1)is such that 1/p=θ /r+(1−θ )which yieldsθ=2kr/(N (r−1)).

Sinceβr∈(−8π,8π ), Proposition 1 providesC1=C1(β)andC2=C2(β)such thatZ(N, βr)^θ/rC₂^Nθ/r C^kandZ(N, β)^−θC₁^−NθC^k. 2

Finally, we have theL^pestimates:

Corollary 4. Letp∈ [1,+∞). We haveµ^N_k ∈L^p(Ω^k)for allk∈NandN large enough. Moreover, there exists C=C(β, p)such that

µ^N_k L^pC^k ∀k∈N, ∀N large enough.

Hence, if p >1, then there exist µk ∈L^p(Ω^k)and a subsequence (µ^N_k^j)k∈N such thatµ^N_k^j µk weakly in L^p(Ω^k).

Proof. From Proposition 3, for allk2 andN large enough, we have

Ω^k

µ^N_k (X_k)^pdX_kC^kpZ

k,βkp N

.

But (βkp/N )_Nkp is in a compact subset of (−8π,8π ) so, by Proposition 1, there exists C such that Z(k, βkp/N )C^k. It follows thatµ^N_kL^pC^kC^k/pC^kforN large enough. 2

The set of indices(N_j)_j∈Ndepends onkandpbut, by a diagonal process, we can take the same one for allk andp. It holds even forp=1 sinceΩ^kis of finite measure. In the sequel we shall always noteµ_∗=(µk)_k∈Nas a weak cluster point of(µ^N)N>1, that is,

µ^N_k^j µk weakly inL^p(Ω^k),∀k∈N, ∀p∈ [1,+∞).

The sequence of indices will be always noted by(Nj)_j∈N.

5. Looking for cluster points: a variational way

We wish to find the weak cluster points of(µ^N)N>1. We shall see that eachµ^N is a solution of a variational problem and thus one can ask if the cluster points of(µ^N)N>1are also solutions of some limit problem (whenN goes to+∞). This is the goal of this section.

We setD(F^N)= {ρ∈L¹(Ω^N)|ρlogρ∈L¹(Ω^N)}. Forρ∈D(F^N)we define the following functionals:

S^N(ρ)=

Ω^N

ρ(X)logρ(X)dX (entropy),

E^N(ρ)= 1 N

Ω^N

H^N(X)ρ(X) dX (energy), F^N(ρ)=S^N(ρ)+βE^N(ρ) (free energy).

Remark thatS^Nis convex andE^Nis linear thusF^Nis convex.

By the inequality srrlogr+1

ee^s, ∀r0,∀s∈R, (5)

applied tor=ρands=H^N/N, it follows thatE^N(ρ)∈Rprovidedρ∈D(F^N). HenceF^Nis well defined from D(F^N)toR.

Remark 5. Wheneverρis symmetric a simpler expression forE^Nholds:

E^N(ρ)=N−1 2

Ω

Ω

H (x˜1,x˜2)ρ2(x˜1,x˜2)dx˜1dx˜2.

An important property of entropy is sub-additivity given by the following proposition.

Proposition 6. Let 1k < Nandρ∈D(F^N)symmetric such thatρL¹=1. We have S^k(ρ_k)+S^N−k(ρ_N−k)S^N(ρ).

Proof. It is clear that

S^k(ρ_k)+S^N−k(ρ_N−k)−S^N(ρ)=

Ω^N

ρ(X)log

ρ_k(X_k)ρ_N−k(X^N−k) ρ(X)

dX.

By Jensen’s inequality the last integral is bounded above by log

Ω^N

ρ_k(X_k)ρ_N−k(X^N−k)dX

,

which is null sinceρ_L¹=1. The proposition follows. 2 Lemma 7. For allC >0 the set

MC=

ρ∈D(F^N)

Ω^N

ρ(X)logρ(X)₊ dXC

is weakly compact onL¹(Ω^N).

Proof. We remark thatMC is convex since D(F^N)and t→ [tlogt]⁺ are convex. By Fatou’s lemma, MC is strongly closed inL¹(Ω^N)and thus, by convexity, it is weakly closed.

We have to show that every sequence(ρn)_n∈N inMC has a subsequence weakly convergent inL¹(Ω^N). For M >1 andn∈Nwe have

{ρnM}

ρ_n(X)logMdX

Ω^N

ρ_n(X)logρ_n(X)₊

dXC.

Hence sup

n∈N

{ρ_nM}

ρn(X) dX C

logM →0 asM→ +∞. The result follows from Dunford–Pettis’s theorem. 2

Here we have our first variational problem:

Theorem 8. We have thatµ^Nis the unique solution of min

F^N(ρ)|ρ∈D(F^N), ρ_L¹ =1 .

Proof. We split the proof into two steps: in the first step we shall show that the problem has a solutionµ, and in˜ the second step we shall prove thatµ˜ =µ^N.

Step 1: Letρ ∈D(F^N)andt 1 such that βt∈(−8π,8π ). From inequality (5), applied to r=ρ/t and s= −βtH^N/N, it follows that

ρlogρ+ β

NH^Nρ

1−1 t

ρlogρ+1

tρlogt−1

ee^−βtNHN. (6)

In particular, fort=1, one has ρlogρ+ β

NH^Nρ+1

ee^−NβHN0.

By Fatou’s lemma it follows thatF^N is an l.s.c. functional in the strong topology ofL¹(Ω^N). Hence, by convexity, F^N is also l.s.c. in the weak topology ofL¹(Ω^N).

Let(ρn)_n∈N⊂D(F^N)be a minimizing sequence of the problem. Takingt >1 in (6) and integrating onΩ^Nwe obtain

CF^N(ρ_n)

1−1 t Ω^N

ρ_n(X)logρ_n(X)dX+1

t logt−1

eZ(N, βt).

Hence,

Ω^N

ρ_n(X)logρ_n(X)dXC, ∀n∈N.

Since the mappingt∈ [0,+∞)→tlogtis bounded from below, we have shown that there existsC >0 such that (ρ_n)_n∈N is in a setM_C as in the Lemma 7 and thus it has a subsequence weakly convergent toµ˜ ∈M_C. Hence,

˜

µ∈D(F^N)and ˜µL¹=1. By the lower semi-continuity ofF^Nin the weak topology ofL¹(Ω^N), it follows that

˜

µis a solution of the problem.

Step 2: Formallyµ^N is the unique solution of the Euler–Lagrange equation associated to the problem and thus we should haveµ˜ =µ^N. But ifµ˜ vanishes on a set of positive measure, then the derivative ofS^N will have a

singularity atµ. Therefore, a rigorous proof of this result is not so straightforward. To avoid this problem we set,˜ forδ >0,

Λδ=X∈Ω^N| ˜µ(X) > δ

and Uδ=

ϕ∈L^∞(Ω^N)| ϕL^∞< δ/2 . We define the following functionals

J_δ:U_δ→R and G_δ:U_δ→R, ϕ→F^N(µ˜+1Λ_δϕ), ϕ→

Ω^N

1Λ_δ(X)ϕ(X)dX.

We easily see that both functionals are well defined. Clearlyϕ=0 is a minimizer ofJδrestricted to the constraint Gδ(ϕ)=0. Now, one can easily (and rigorously) derivate the associated Euler–Lagrange equation to show that there existsC=C(δ)such thatµ˜ =Ce^−NβHN almost everywhere inΛδ. ButΛδ⊂Λ_δ wheneverδ < δ. Hence the constantC does not depend onδandµ˜ is given by the previous expression on the subsetΛofΩ^N whereµ˜ does not vanish. From ˜µ_L¹=1 we obtain

C=

Λ

e^−NβHN(X)dX ₋1

. It is easy to show that

e^−FN(µ)˜ =

Λ

e^−NβHN(X)dX and e^−FN(µN)=

Ω^N

e^−NβHN(X) dX.

SinceF^N(µ)˜ F^N(µ^N)we should haveΛ=Ω^N, and thusµ˜ =µ^N. 2

Remark 9. In the last proof we have shown thatF^N is l.s.c. in the weak topology ofL¹(Ω^N). Hence, sinceΩ^Nis of finite measure,F^Nis l.s.c. in the weak topology ofL^p(Ω^N)for allp∈ [1,+∞). The same result holds forS^N (sinceS^N=F^Nwhenβ=0).

We define the setD(F^∗)of allρ_∗=(ρ_k)_k∈N∈_+∞

k=1D(F^k)which verifies, for allk∈N, (i) ρ_kL¹=1;

(ii) ρkis symmetric;

(iii) ρk(Xk)=

Ωρk+1(Xk+1)dx˜k+1;

(iv) ρ_kL∞(Ω^k)C^kfor some constantC=C(ρ_∗).

Forρ_∗∈D(F^∗)we define the functionals S^∗(ρ_∗)= lim

k→+∞

1 k

Ω^k

ρ_k(X_k)logρ_k(X_k)dX_k= lim

k→+∞

1 kS^k(ρ_k), E^∗(ρ_∗)=1

2

Ω

Ω

H (x˜1,x˜2)ρ2(x˜1,x˜2)dx˜1dx˜2, F^∗(ρ_∗)=S^∗(ρ_∗)+βE^∗(ρ_∗).

Forρ_∗∈D(F^∗)we have, again by (5), thatE^∗(ρ_∗)∈R. The property (iii) implies, by induction, that ρk(Xk)=

Ω^N−k

ρN(X)dX^N−k, ∀k∈N, ∀N > k.

Hence, by Proposition 6, we have that(S^k(ρk))_k∈Nis a sub-additivity sequence and thus the limit definingS^∗exists but it is possibly infinite. However, by the property (iv), we have the following bounds

1 k

Ω^k

ρ_k(X_k)logρ_k(X_k)dX_k1 k

Ω^k

ρ_k(X_k)logC^kdX_k=logC.

We conclude thatS^∗(ρ_∗)∈Rand thusF^∗is well defined fromD(F^∗)toR.

Proposition 10. Letρ_∗∈D(F^∗)andµ_∗be a weak cluster point of(µ^N)_N>1. We have the following convergences:

(i) N⁻¹E^N(ρ_N)→E^∗(ρ_∗)asN→ +∞; (ii) N⁻¹S^N(ρ_N)→S^∗(ρ_∗)asN→ +∞; (iii) N⁻¹F^N(ρN)→F^∗(ρ_∗)asN→ +∞; (iv) N_j⁻¹E^Nj(µ^Nj)→E^∗(µ_∗)asj → +∞;

(v) N_j⁻¹F^Nj(µ^Nj)→F^∗(µ_∗)asj → +∞; (vi) N_j⁻¹S^Nj(µ^Nj)→S^∗(µ_∗)asj→ +∞;

(vii) k⁻¹S^k(µ_k)lim inf_j→+∞k⁻¹S^k(µ^N_k^j)lim sup_j→+∞k⁻¹S^k(µ^N_k^j)S^∗(µ_∗).

Proof. (i) follows trivially from symmetry ofρ_kand Remark 5 while (ii) is just the definition ofS^∗. Clearly, (iii) is consequence of (i) and (ii).

(iv) follows from symmetry ofµ^Nj and from the weak convergenceµ^N₂^j µ₂ inL²(Ω²) (notice that by Proposition 1 we haveH∈L²(Ω²)).

Let us show (v). Letk∈Nbe fixed. For everyj∈N, large enough, we can find two integersmj andnj such thatN_j=m_jk+n_jand 0< n_j k. By sub-additivity ofS^Nwe have

m_j Nj

S^k µ^N_k^j

+ 1 Nj

S^nj µ^N_nj^j

1

Nj

S^Nj µ^Nj

.

The functiont∈ [0,+∞)→tlogtis bounded from below and so is the entropy. Hence, mj

N_jS^k µ^N_k^j

− C N_j 1

N_jS^Nj µ^Nj

. (7)

By addingβN_j⁻¹E^Nj(µ^Nj)to both sides of last inequality and by minimality ofF^N(µ^N)it follows that m_j

N_jS^k µ^N_k^j

+ β N_jE^Nj

µ^Nj

− C N_j 1

N_jF^Nj µ^Nj

1

N_jF^Nj(µN_j).

Taking limits asj→ +∞, from (iii), (iv), weak lower semi-continuity ofS^k(see Remark 9) and sincem_j/N_j→ 1/kit follows that

1

kS^k(µ_k)+βE^∗(µ_∗)lim inf

j→+∞

1 N_jF^Nj

µ^Nj

lim sup

j→+∞

1 N_jF^Nj

µ^Nj

F^∗(µ_∗).

Finally, we letk→ +∞and we use (ii) to conclude.

(vi) follows from (iv) and (v).

It remains to show (vii). The first inequality is just the lower semi-continuity ofS^k. The second one is trivial.

The last one follows from (vi) and by taking limits, asj→ +∞, in (7). 2

Theorem 11. Letµ_∗be a weak cluster point of(µ^N)_N>1. We have thatµ_∗is a solution of min

F^∗(ρ_∗)|ρ_∗∈D(F^∗) .

Proof. Forρ_∗∈D(F^∗), by minimality of F^Nj(µ^Nj), we haveF^Nj(µ^Nj)F^Nj(ρN_j). So the theorem follows from Proposition 10 (iii) and (v). 2

We recall that the Gibbs measure gives the distribution of vortices on the phase space. So, it is physically reasonable that the more vortices we have the more their positions will be independent from one another. So, at limitN → +∞, we can think that the distribution ofkvorticesµ_k will be induced by a distributionµof one- vortice, that is,µ_k(X_k)=µ(x˜₁) . . . µ(x˜_k). This factorization property is usually called “propagation of chaos”.

This does not always holds. In fact,µk is not a product measure but an average of product measures (in this case we are talking about “partial propagation of chaos”). Partial propagation holds not only for cluster points of (µ^N_k)N>kbut for allρ_∗∈D(F^∗). The property ofρ_∗which assures this result is the symmetry of allρk’s.

ByP(Ω) we denote the space of all Borelian probabilities onΩendowed with the weak topology. We denote Q(Ω) the set of all Borelian probabilitiesνonP(Ω) which are supported on a bounded set ofL^∞(Ω).

Theorem 12. The application which maps eachν∈Q(Ω) toρ_∗∈D(F^∗)defined by ρ_k(X_k)=

P(Ω)

ρ(x˜₁) . . . ρ(x˜_k)ν(dρ), ∀k∈N,

which is equivalent to

Ω^k

f (X_k)ρ_k(X_k)dX_k=

P(Ω)

Ω^k

f (X_k)ρ(x˜₁) . . . ρ(x˜_k)dX_kν(dρ), ∀f ∈L¹(Ω^k), (8)

is onto.

Proof. Letρ_∗∈D(F^∗). We can viewρ_∗as the probability onΩ^Nfor whichρkis the marginal density ofΩ^k(for allk∈N). By the Hewitt–Savage theorem (see Theorem 7.4 of [7]) there exists a unique Borelian probabilityνon P(Ω) such that

Ω^k

f (X_k)ρ_k(X_k)dX_k=

P(Ω)

Ω^k

f (X_k)ρ(dx˜₁) . . . ρ(dx˜_k)ν(dρ). (9)

Takingf (X_k)=g(x˜₁) . . . g(x˜_k)(g∈L¹(Ω)) and recalling that ρ_kL^∞C^kwe find

P(Ω) Ω

g(x˜₁)ρ(dx˜₁) k

ν(dρ)C^kg^k_L1, ∀g∈L¹(Ω), ∀k∈N.

Hence,

Ω

g(x˜₁)ρ(dx˜₁)

CgL¹, ν-almost allρ∈P(Ω).

This means thatνis supported inside the ball ofL^∞(Ω) of radiusCcentered at the origin. It follows thatν∈Q(Ω) and Eq. (9) becomes (8). 2

Using (8) tof=H we obtain E^∗(ρ_∗)=

P(Ω)

E(ρ)ν(dρ), whereE(ρ)=1 2

Ω

Ω

H (x˜1,x˜2)ρ(x˜1)ρ(x˜2)dx˜1dx˜2.

In [16] the author shows that S^∗(ρ_∗)=

P(Ω)

S(ρ)ν(dρ), whereS(ρ)=

Ω

ρ(x˜1)logρ(x˜1)dx˜1.

Hence, settingF =S+βE, we find F^∗(ρ_∗)=

P(Ω)

F (ρ)ν(dρ). (10)

The functionalsE,SandF are well defined fromD(F )=D(F¹)toR.

Let µ_∗∈D(F^∗)be a weak cluster point of (µ^N)N>1 and consider the corresponding ξ ∈Q(Ω) given by Theorem 12, i.e.µ_∗andξ are related by

µ_k(X_k)=

P(Ω)

µ(x˜₁) . . . µ(x˜_k)ξ(dµ), ∀k∈N. (11)

By using Eq. (10) one can rewrite the claim of Theorem 11 to find thatξ is a solution of min

P(Ω)

F (ρ)ν(dρ)

ν∈Q(Ω)

.

Thus we obtain easily the following theorem.

Theorem 13. The functionalF isξ-almost all constant on suppξ and equal to its minimum value. In other words, ξ-almost allµ∈suppξ is a solution of

min

F (ρ)|ρ∈P(Ω) ∩L^∞(Ω) .

We finish this section precising what happens ifF has a unique minimizer.

Proposition 14. Ifµis the unique minimizer ofFonP(Ω) ∩L^∞(Ω), then (µ^N_k )N>kconverges strongly inL^p(Ω^k) toµ^⊗k, for allk∈Nand for allp∈ [1,+∞).

Proof. Letk∈Nandp∈ [1,+∞). Take a weak cluster pointµ_∗ of(µ^N)N>1 andξ such that (11) holds. By Theorem 13,µ is the unique point in the support ofξ, i.e.ξ is a Dirac measure supported atµ. Consequently, µk(Xk)=µ(x˜1) . . . µ(x˜k)and

1

kS^k(µ_k)=1 k

Ω^k

µ_k(X_k)logµ_k(X_k)dX_k=

Ω

µ(x˜₁)logµ(x˜₁)dx˜₁=S(µ).

We know also that S^∗(µ_∗)=

P(Ω)

S(ρ)ξ(dρ)=S(µ).

From Proposition 10 (vii) we obtainS^k(µ^N_k^j)→S^k(µk). Sinceµk∈L^∞(Ω^k),µ^N_k^j µk inL^p(Ω^k)for all p∈ [1,+∞)andt→tlogt is a strictly convex function, we conclude thatµ^N_k^j →µ_k strongly in L^p(Ω^k). We have shown that every weak cluster point of(µ^N)N>1is a strong one and unique. The result follows. 2