Faculty of Pharmaceutical Sciences B.Pharm. - Jodhpur National

Equilibrium Statistical Theory forNearly Parallel Vortex Filaments

PIERRE-LOUIS LIONSCeremade

Université Paris-Dauphine

AND

ANDREW MAJDACourant Institute

Abstract

The first mathematically rigorous equilibrium statistical theory for three-dimen-sional vortex filaments is developed here in the context of the simplified asymp-totic equations for nearly parallel vortex filaments, which have been derived re-cently by Klein, Majda, and Damodaran. These simplified equations arise froma systematic asymptotic expansion of the Navier-Stokes equation and involvethe motion of families of curves, representing the vortex filaments, under lin-earized self-induction and mutual potential vortex interaction. We consider herethe equilibrium statistical mechanics of arbitrarily large numbers of nearly par-allel filaments with equal circulations. First, the equilibrium Gibbs ensembleis written down exactly through function space integrals; then a suitably scaledmean field statistical theory is developed in the limit of infinitely many interact-ing filaments. The mean field equations involve a novel Hartree-like problemwith a two-body logarithmic interaction potential and an inverse temperaturegiven by the normalized length of the filaments. We analyze the mean field prob-lem and show various equivalent variational formulations of it. The mean fieldstatistical theory for nearly parallel vortex filaments is compared and contrastedwith the well-known mean field statistical theory for two-dimensional point vor-tices. The main ideas are first introduced through heuristic reasoning and thenare confirmed by a mathematically rigorous analysis. A potential application ofthis statistical theory to rapidly rotating convection in geophysical flows is alsodiscussed briefly.c© 2000 John Wiley & Sons, Inc.

Contents

1. Introduction 772. Gibbs Ensembles for Nearly Parallel Filaments and the Broken Path

Models 813. Heuristic Derivation of Mean Field Theory 894. Rigorous Mean Field Theory for the Broken Path Models 925. Rigorous Mean Field Theory for Vortex Filaments 946. Alternative Formulations for the Mean Field Equations 1197. The Current and Some Scaling Limits 1238. Concluding Discussion and Future Directions 137

Communications on Pure and Applied Mathematics, Vol. LIII, 0076–0142 (2000)c© 2000 John Wiley & Sons, Inc. CCC 0010–3640/00/000076-67

EQUILIBRIUM STATISTICAL THEORY 77

Appendix. Remarks on the KMD Equations 137Bibliography 141

1 Introduction

Over the last fifteen years, Chorin [4, 5, 6, 7] has proposed several novel heuris-tic models for fully developed turbulence based on the equilibrium statistical me-chanics of collections of three-dimensional vortex filaments. Chorin’s pioneeringwork has emphasized both the similarities and differences between statistical the-ories for heuristic models for ensembles of three-dimensional vortex filaments andthe earlier two-dimensional statistical theories for point vortices (Onsager [21],Joyce and Montgomery [10], and Montgomery and Joyce [20]).

Here, we develop the first mathematically rigorous equilibrium statistical the-ory for three-dimensional vortex filaments in the context of a model involving sim-plified asymptotic equations for nearly parallel vortex filaments. These equationshave been derived recently by Klein, Majda, and Damodaran [13] through system-atic asymptotic expansion of the Navier-Stokes equations where the nearly parallelvortex filaments are represented by families of curves that move through linearizedself-stretch and mutual induction as leading-order asymptotic approximations ofthe Biot-Savart integral.

Each vortex filament is concentrated near a curve that is nearly parallel to thex3-axis. Thus, each vortex filament is described by a functionXi(σ,t) ∈ R

2 whereσ ∈ R

1 parametrizes the asymptotic center curve of the filament. The family ofnearly parallel vortex filamentsXj(σ,t)N

j≥1 evolves according to the 2N coupledsystem of equations

Γ j∂Xj

∂t= J

[α jΓ2

j∂2

∂σ2Xj +12

N

∑k6= j

Γ jΓkXj −Xk

|Xj −Xk|2]

(1.1)

for all 1≤ j ≤N, where the parameterΓ j denotes the circulation of thej th filament,α j is the vortex core structure,N is the number of filaments, andJ = (0 −1

1 0).The simplified asymptotic equations in (1.1) are derived in a formal asymptotic

limit from the Navier-Stokes equations under the conditions that

1. the wavelength of the nearly parallel filament perturbations is much longerthan the separation distance between filaments,

2. the separation distance is much larger than the core thickness of each fila-ment, and

3. the Reynolds number is very large.

The technical aspects of the derivation of (1.1) as well as more details beyondthe discussion below are given in the work of Klein, Majda, and Damodaran [13]while a more leisurely treatment can be found in Majda [18] or Majda and Bertozzi

78 P.-L. LIONS AND A. MAJDA

[19]. The term in (1.1) involving∂2Xj/∂σ2 arises from the linearized self-inductionof the individual filaments. The contribution of the terms

J

(12

N

∑k6= j

Γ jΓkXj −Xk

|Xj −Xk|2)

is the velocity induced at a given vortex filament for a fixed value ofσ by theother vortex filaments; this contribution is the same one that occurs for the motionof point vortices in the plane (Lamb [14] and Chorin and Marsden [8]). In fact,special exact solutions of (1.1) without anyσ-dependence coincide with solutionsof the two-dimensional point vortex equations. In this sense, the equations in (1.1)generalize the physics of two-dimensional point vortex dynamics by allowing forthe purely three-dimensional effect of self-induction. Numerical solutions of (1.1)for pairs of filaments show a remarkable, genuinely three-dimensional behaviorthat agrees qualitatively with many aspects of solutions of the complete Navier-Stokes equations.

From the mathematical viewpoint, (1.1) can be recast as a system of nonlinearSchrödinger equations by settingϕ j = X1

j + iX2j , so that (1.1) becomes

−iΓ j∂ϕ j

∂t= α jΓ2

j∂2ϕ j

∂σ2 +12

N

∑k6= j

Γ jΓkϕ j −ϕk

|ϕ j −ϕk|2 .(1.2)

Because of the singularity of the nonlinear term, this evolution problem is not wellunderstood: The existence and uniqueness of regular solutions are not known, andshould depend on the parametersΓ jN

j=1 and their respective signs. In an appendixwe collect a few mathematical observations on that system while some accessibleopen problems for (1.1) and (1.2) are discussed elsewhere (Majda [18] and Majdaand Bertozzi [19]).

In this paper, we develop the equilibrium statistical mechanics for solutionsof (1.1) in a suitable scaled limit as the number of filamentsN gets arbitrarilylarge as the model for the equilibrium statistical mechanics of nearly parallel vortexfilaments. Here we assume that each filament is periodic inσ, i.e.,

Xj(σ+L) = Xj(σ) , 1≤ j ≤ N ,(1.3)

for someL> 0. We also assume that all filaments have the same circulation,Γ j =Γ, and the same core structure,α j = α > 0; thus, without loss of generality, wemay assume thatΓ> 0 and by a trivial scaling that we have

Γ j = 1, α j = α , for 1≤ j ≤ N ,(1.4)

for the solutions of (1.1) considered here. The assumption of identical signs for allthe circulationsΓ j , i.e., corotating filaments, is a genuine physical restriction sincemore complex dynamical phenomena occur for solutions of (1.1) with both positiveand negative circulations (Klein, Majda, and Damodaran [13], Majda [18], andMajda and Bertozzi [19]). Designing an equilibrium statistical mechanics model


in that case is an interesting open problem to which we hope to return in a futurepublication.

Theories for equilibrium statistical mechanics are based on the conserved quan-tities for the Hamiltonian system in (1.1). With the special assumptions in (1.3) and(1.4), these conserved quantities are given by the Hamiltonian,

H =N

∑j=1

12α

∫ L

0

∣∣∣∣∂Xj

∂σ

∣∣∣∣2 dσ− 12

N

∑j 6=k

∫ N

0log|Xj(σ)−Xk(σ)|dσ(1.5)

as well as the center of vorticityM, the mean angular momentumI , and a quan-tity that we denote byC and call the mean current by analogy with quantum me-chanics. The currentC has an indefinite character much like the helicity in three-dimensional flows. These additional conserved quantities are given explicitly by

M =N

∑j=1

L∫

0

Xj(σ)dσ ,

I =N

∑j=1

L∫

0

|Xj(σ)|2dσ ,

C =N

∑j=1

L∫

0

JXj(σ) · ∂Xj

∂σ(σ)dσ .

(1.6)

In Section 2 we introduce the Gibbs measures defined through the conservedquantities in (1.5) and (1.6). These Gibbs measures naturally involve suitable func-tion space integrals with respect to Wiener measure (more precisely, some kind of“discounted conditional Wiener measure”). We also introduce a natural discreteapproximation of these Gibbs measures through a broken path discretization inσ. This broken path discretization has several conceptual advantages: First, forextremely coarse broken paths with only a single segment, we recover the Gibbsmeasures for the statistical theory for point vortices in the plane; second, in theother extreme limit of infinitely fine discretization, we recover the Gibbs measuresof the continuum problem associated with (1.1). In this fashion, we can compareand contrast the equilibrium statistical theories for two-dimensional point vorticesand three-dimensional nearly parallel vortex filaments as well as build intermedi-ate theories involving many fixed broken paths. These intermediate theories alsosuggest the manner in which we can adapt and generalize the rigorous statisticalmechanical arguments for two-dimensional point vortex systems (Caglioti, Lions,Marchioro, and Pulvirenti [2, 3], Kiessling [12], Lions [17]) to the present situationinvolving the statistical mechanics of nearly parallel vortex filaments.

In Section 3 we give a heuristic discussion of mean field theory for the statisticalmechanics of nearly parallel vortex filaments. This theory requires the specificscaling relation in (3.3) below for the nondimensional form of the Hamiltonian


in (2.2) and (2.3). The tacit assumption of such mean field theories is that theempirical distribution of the filament curves

1N

(N

∑i=1

δXi(σ)

)converges toρ(x) asN → ∞ for eachσ(1.7)

whereρ(x) is a probability density onR2 independent ofσ. In fact, we shall showin Section 5 that this property holds with probability 1 under the Gibbs measure,and we shall also determine the limit of the empirical law ofXj(σ). Withoutloss of generality, we have setL = 1 (see Section 3). In addition, in the casewhere we only use the conserved quantitiesH andI , we give a heuristic derivationin Section 3 that the probability densityρ(x) is determined through the Green’sfunction p(x,y,t), x∈ R

2, y∈ R2, of the following PDE:

∂p∂t

− 12β

∆p+aβ

[(− 1

2πlog|x|

)∗ρ]

p+µ|x|2p = 0 in R2× (0,1)

p|t=0 = δy(x) ,(1.8)

whereβ is the inverse temperature,µ is the chemical potential forI , and the con-stanta in (1.8) is determined by the mean field scaling limit described in (3.3) fromSection 3. The probability densityρ is recovered fromp by the formula

ρ(x) = p(x,x,1)

∫

R2

p(x,x,1)

−1

.(1.9)

In Section 4 we give a sketch of the rigorous a priori derivation of the meanfield limit for the broken path approximations following ideas from Caglioti et al.[2] and Lions [17] for two-dimensional point vortices with positive temperatures.Section 5 contains the main mathematical results in this paper, namely, rigorous apriori proof of mean field behavior for any positive inverse temperatureβ > 0 forthe statistical mechanics of nearly parallel vortex filaments as described in (1.7),(1.8), and (1.9) above and motivated heuristically in Section 3. The techniquesthat we utilize here in the proof are similar to those of Angelescu, Pulvirenti, andTeta [1] in their study of the classical limit for a quantum Coulomb system inR

3

although our limiting mean field theory is completely different and we have othertechnical difficulties associated with logarithmic interactions. We also prove thatthe mean field limit for the broken path approximations converges to the continuummean field limit equation in (1.8) and (1.9).

In Section 6 we present several alternative characterizations of the mean fieldlimit problem in (1.8) and (1.9). One of these involves a Hartree-like problem witha two-body logarithmic interaction potential and an inverse temperature given bythe normalized length of the filaments. We also utilize these alternative variationalcharacterizations to compare the mean field statistical theory for nearly parallelvortex filaments with the mean field theory for point vortices in the plane. UntilSection 7 we do not use the conserved quantityC in order to keep the presentation


as simple as possible. In Section 7 we show how the use of the currentC rigorouslyleads to a modified mean field theory. We also discuss other scaling limits such asthe case of infinite-length vortex filaments. Finally, in Section 8 we briefly discussseveral possible directions for future work.

2 Gibbs Ensembles for Nearly Parallel Filaments and the BrokenPath Models

2.1 The Continuous Path Models

Here we discuss the definition of Gibbs measures forN-vortex filaments. For anappropriate range of parameters, we would like to define Gibbs measures formallygiven by

µN =1Z

exp(−βH −λλλ ·M−µI −vC )dX1 · · ·dXN(2.1)

on the path space ofN filaments where the HamiltonianH and the other conservedquantitiesM, I , andC are given in (1.4) and (1.5), respectively.

We begin by rewriting the HamiltonianH in convenient nondimensional units.With the notationX = (X1, . . . ,XN) ∈ R

2N, we nondimensionalize the amplitude ofthe curves byA and the period interval by the dilation factorλ; i.e., we changevariables byσ′ = λσ where bothA andλ−1 have the units (length). We introducethe nondimensional variableX′(σ′) = X(σ′/λ)/A into the Hamiltonian. By mul-tiplying the Hamiltonian by a constant, ignoring additive constants, and droppingthe prime in notation, we obtain the nondimensional Hamiltonian

H (X(σ)) =12

∫ λL

0

N

∑j=1

∣∣∣∣∂Xj

∂σ

∣∣∣∣2 dσ+12

a∫ λL

0

N

∑j 6=k

− log|Xj(σ)−Xk(σ)|dσ .(2.2)

With the natural choice forλ andλ= L−1, the nondimensional factor ¯a is given by

a =L2

αA2 ,(2.3)

and the nondimensional Hamiltonian has the form

H (X(σ)) =12

∫ 1

0

N

∑j=1

∣∣∣∣∂Xj

∂σ

∣∣∣∣2 dσ+12

a∫ 1

0

N

∑j 6=k

− log|Xj(σ)−Xk(σ)|dσ .(2.4)

With these preliminaries, we build the Gibbs measures in (2.1). For pedagog-ical purposes, we begin with the special case of (2.1) withµ,v = 0 and a = 0for the Hamiltonian in (2.4); in this special case the Gibbs measure is simplythe Wiener measure on(R2)N with diffusion constant 1/β conditioned on pe-riodic paths, which we denote byνβ. In fact, νβ may be written asνβX,X dX,whereνβX,X is the usual conditional Wiener measure conditioned on paths suchthatω(0) = ω(1) = X ∈ R

2 (recall thatνβX,X is not a probability measure, since∫dνβX,X = (2πβt)−1). In particular,νβ is not a bounded measure(

∫dνβ = +∞!)


on the Banach spaceΩN endowed with the usual norm(maxi,t∈[0,1] |ωi(t)|). LetΩN = (ω1, . . . ,ωN) denote periodic continuous paths withω j ∈ C([0,1];R2) andω j(0) = ω j(1) for all 1≤ j ≤ N. The rigorous way to defineνβX,X is to write downits marginals explicitly through its action on arbitrary bounded continuous func-tions of the typeF = F(Ω(t1), . . . ,Ω(tm)) with m≥ 0, 0< t1 < t2 < · · · < tm ≤ 1,and we assume, for instance, thatF has compact support on(R2)m. Thus we have

∫F dνβ =

∫

R2N

dX∫

R2n

dX1 · · ·∫

R2N

dXmF(X1, . . . ,Xm)(2.5)

· pβ0(X,X1,t1)pβ0(X1,X2,t2− t1) · · ·

pβ0(Xm−1,Xm,tm− tm−1) · pβ0(Xm,X,1− tm)

=∫

R2Nm

dX1 · · ·dXmpβ0(Xm,X1,1+ t1− tm)

· pβ0(X1,X2,t2− t1) · · ·

pβ0(Xm−1,Xm,tm− tm−1)F(X1, . . . ,Xm)

wherepβ0(X,Y,t) = ∏Nj=1 pβ0(Xj ,Yj ,t) with pβ0(x,y,t), the Gaussian kernel onR2,

pβ0(x,y,t) =(

2πtβ

)−1

exp

(− β|x−y|2

2t

),(2.6)

(see, for example, Ginibre [9], Lebowitz, Rose, and Speer [15], Simon [23], orAngelescu et al. [1]). With this definition ofdνβX,X as background, we next turn tothe definition of the Gibbs ensemble in (2.1) withβ > 0,λ 6= 0, andµ> 0, but herewe requirev = 0.

In this general case, the Gibbs measureµN is given in a straightforward fashionas

µN = (Z(N))−1exp

−

∫ 1

0dσ

[βa2

N

∑j 6=k

− log|ω j(σ)−ωk(σ)|]

(2.7)

+N

∑j=1

λλλ ·ω j(σ)+µN

∑j=1

|ω j(σ)|2

dνβX,X(Ω)dX


with

Z(N) =∫

R2N

dXEβX,X

[exp

−

∫ 1

0dσ

[βa2

N

∑j 6=k

− log|ω j(σ)−ωk(σ)|(2.8)

+N

∑j=1

(λλλ ·ω j(σ)+µ|ω j(σ)|2)]]

.

In (2.8),EβX,X denotes the expected value with respect todνβX,X. At this stage, oneneeds to explain whyZ(N) < ∞ and thus justify thatµN is well-defined by (2.7).Indeed, one can clearly boundZ(N) for some positive constantC = C(N)

Z(N) ≤C∫

R2N

dX EβX,X exp

−µ

2

∫ 1

0dσ

N

∑j=1

|ω j(σ)|2

≤C∫

R2N

dX EβX,X

∫ 1

0dσ exp

(−µ

N

∑j=1

|ω j(σ)|2)

= C∫ 1

0dσ

∫∫

R2×R2

pβ(X,Y,σ)e−µ2 |Y|2 pβ(Y,X,1−σ)dX dY

N

= C

∫

R2

(2π)−1βe−µ2 |Y|2dY

N

= C

(4π2 µ

β

)−N

.

As in the definition in (2.5) forνβX,X, the marginal distributions for the GibbsmeasureµN can be written down via the Green’s function of a PDE. The way tosee this is to observe that the potentialV(X) defined by

V(X) =βa2

N

∑j 6=k

− log|Xj −Xk|+N

∑j=1

(λλλ ·Xj +µ|Xj |2

)(2.9)

satisfies the hypotheses for the Feynman-Kac formula with respect todvβX,X pro-videdµ satisfiesµ > 0 (see Simon [23, chap. 2]). Thus, from (2.7) and (2.9), foran arbitrary, bounded, continuous function onR

2Nm, F(Ω(t1), . . . ,Ω(tm)), and anypartition with 0≤ t1 < t2 < · · ·< tm ≤ 1 with m≥ 1, we have

∫F dµN = (Z(N))−1

∫

R2Nm

dX1 · · ·dXmF(X1, . . . ,Xm)

(2.10)

· p(Xm,X1,1+ t1− tm)p(X1,X2,t2− t1) · · · p(Xm−1,Xm,tm− tm−1)


and

Z(N) =∫

R2N

dX p(X,X,1) .(2.11)

Moreover, from the Feynman-Kac formula and (2.7),p(X,Y,t) is the Green’sfunction for the PDE,

∂p∂t

− 12β

N

∑j=1

∆Xj p−(βa2

N

∑i 6= j

log|Xi −Xj |)

p+N

∑j=1

(λλλ ·Xj +µ|Xj |2)p = 0

in R2N × (0,1) ,

p|t=0 = δY(X) onR2N .

(2.12)

Of course, as is well-known,p(X,Y,t) is a positive kernel, symmetric in(X,Y),and, by classical results on parabolic equations,p is C∞ in (X,Y,t) for t > 0 andaway from the sets(X,Y) ∈ (R2N)2 : ∃i 6= j Xi = Xj or Yi = Yj with p> 0 fort > 0; ∂p/∂t, Dα

X,Y p∈ Lq(R2N ×R2N × (δ,1)) for |α| ≤ 2, and for all 1≤ q< ∞,

δ > 0.Finally, using the maximum principle, one may check the bound onR

2N ×R

2N × (0,1),

0< p(X,Y,t) ≤ eC(N)t(µβ)N/2(sinh(bt))−N

exp

−(µβ)1/2

[12

cotanh(bt)(|X|2+ |Y|2)− 2sinh(bt)

X ·Y]

with b = (µ/β)1/2. It is worth remarking that the special case with ¯a = 0, µ >0, is the parabolic quantum oscillator and can be solved explicitly by Mehler’sformula (Simon [23]) in terms of appropriate Gaussians. The situation withλλλ 6=0 can be reduced to the situation withλλλ = 0 by elementary transformations sowithout loss of generality, we assumeλλλ= 0 in the following section. The explicitformula for the parabolic oscillator kernel, combined with the trivial comparisonpotential∑N

i=1 log|Xi −Xj | ≤ µ2βa|X|2 + C (β,N), leads to the explicit upper bound

on p(X,Y,t) stated above. On the other hand, in order to include the conservedquantity given by the currentC from (1.6), we need to utilize the Ito calculus.We will not do this here in the continuum setting for simplicity in exposition; untilSection 7, we will always assumev= 0. However, we will retain an approximationto C in the broken path models discussed in the next section.

We have seen above two equivalent ways of defining the Gibbs measureµN.We shall also justify (and recover the equivalence of) these definitions in the nextsection by letting the broken paths “converge” to continuous paths. This asymp-totic approach yields the derivation of a third way of definingµN, which is also aconsequence of the Feynman-Kac formula. Indeed, we see that we have

dµN = hN dµN0 ,(2.13)


whereµN0 is the probability measure onΩN defined below in (2.16), (2.17), and

hN =1

Z′(N)exp

−

∫ 1

0dσ

[βa2

N

∑j 6=k

− log|ω j(σ)−ωk(σ)|(2.14)

+N

∑j=1

λ ·ω j(σ)+µ

2

N

∑j=1

|ω j(σ)|2]

with

Z′(N) = EN0

[exp

−

∫ 1

0dσ

[βα

2

N

∑j 6=k

− log|ω j(σ)−ωk(σ)|(2.15)

+N

∑j=1

λ ·ω j(σ)+µ

2

N

∑j=1

|ω j(σ)|2]]

.

HereEN0 denotes the expectation with respect toµN

0 .The probability measureµN

0 corresponds to the special case whena = λ =0 above; i.e.,µN

0 is the law of the “quantum oscillator” process, which can beequivalently defined by

1R(N)

exp

−µ

2

∫ 1

0dσ

N

∑j=1

|ω j(σ)|2

dXνβX,X

or by

EN0 [F(Ω(t1), . . . ,Ω(tm))](2.16)

=∫

R2Nm

dX1 · · ·dXmq(Xm,X1,1+ t1− tm)q(X1,X2,t2− t1)

· · ·q(Xm−1,Xm,tm− tm−1)F(X1, . . . ,Xm)

for any bounded continuous functionF onR2m, whereq(X,Y,t) is given by

q(X,Y,t) = π−N(µβ)N/2(sinh(bt))−N(2.17)

exp

−(µβ)1/2

[12

coth(bt)(|X|2+ |Y|2)− 1sinh(bt)

X ·Y]

with b = (µ/β)1/2. In particular, we have

Z(N) = Z′(N)R(N),R(N) = (coshb−1)N.(2.18)

Under the lawµN0 , Ω(t) is obviously a Gaussian process andω1(t), . . . ,ωN(t) are

independent. These formulas guarantee thatZ′(N) is finite, and thushN is boundedon ΩN. Notice finally thatµN andµN

0 are both symmetric probability measures onΩN.

We conclude this section with an important observation on the invariance of theabove Gibbs measuresµN andµN

0 by time shifts. Extending periodically the paths


ω j(1 ≤ j ≤ N) in Ω to [0,∞), we denote byθt the shift byt, namely,θtΩ(s) =Ω(s+ t). Then, (2.5) and (2.10) immediately yield

EN[F(θtΩ)] = EN[F(Ω)],EN0 [F(θtΩ)] = EN[F(Ω)](2.19)

for any, say, bounded, measurable random variableF on ΩN, where we denote byEN the expectation with respect toµN. Of course, the densityhN satisfies the sameinvariance property, namely,hN(θtΩ) = hN(Ω) for all t ≥ 0.

2.2 The Broken Path Models

In these models, we replace the continuous paths utilized in definingµN in (2.1)by discrete curves. Thus, we consider periodic broken chains

xσj , 1≤ j ≤ N , 0≤ σ ≤ M , Mδ = 1,

with the periodicity conditionxMj = x0

j . To denote the individual broken path fil-

aments, we utilize the notationXj = (x0j , . . . ,x

M−1j ) ∈ R

2M for 1 ≤ j ≤ N. In thebroken path models, we simply approximateH in (2.4) and the conserved quanti-ties in (1.6) by straightforward discretizations,

H δ =M−1

∑σ=0

N

∑j=1

12δ

|xσ+1j −xσj |−

a2

M−1

∑σ=0

N

∑j 6=k

δ log|xσj −xσk | ,

I δ =M−1

∑σ=0

N

∑j=1

δ|xσj |2 ,

C δ =M−1

∑σ=0

N

∑j=1

xσj ·J(xσ+1j −xσj ) .

(2.20)

The Gibbs measuresµN,δ for the broken path approximation are absolutely contin-uous with respect to Lebesgue measure on(R2M)N with density given by

µN,δ(X1, . . . ,XN) = Z−1exp−βH δ(X1, . . . ,XN)−µI δ−vC δ)(2.21)

and

Z =∫

(R2M)N

exp(−βH δ−µI δ−vC δ)dX1 · · ·dXN.(2.22)

We remark that in the special case of the coarsest broken path model withM = 1,H δ reduces to the point vortex Hamiltonian for two-dimensional flows,I δ becomesthe moment of inertia, andC δ vanishes identically. Also, asM ↑ ∞, the Gibbs mea-sures for the broken path approximation formally converge to the Gibbs measuresfor continuous paths, which we discussed earlier in this section. We shall comeback to that point at the end of this section.

With a nonzero currentC δ, the Gibbs measures in (2.21) and (2.22) are notwell-defined unless the Lagrange multiplierv satisfies certain restrictions given


the values ofβ, µ, andM. To see this, we consider the quadratic terms in theexponential in (2.21) given by

βH δ∣∣∣a=0

+µI δ +vC δ ≡N

∑j=1

B(Xj)(2.23)

with B(X), the quadratic form on a periodic broken path, given by

B(X) =β

2

M−1

∑σ=1

1δ|xσ+1−xσ|2 + δµ

M−1

∑σ=1

|xσ|2(2.24)

+vM−1

∑σ=1

xσ ·J(xσ+1−xσ) , δ = M−1 .

In standard fashion for discrete periodic problems, this quadratic form is diagonal-ized by 2M orthonormal eigenvectors with the form

eee±l =(

1,e2πilM ,e

2πi2lM , . . . ,e

2πi(M−1)lM

)e±l for 0≤ l ≤ M−1,(2.25)

with e±l ∈ C2, the orthonormal eigenvectors of an appropriate 2×2 Hermitian ma-

trix with eigenvalues

λ±l = βM

(1−cos

(2π

lM

))+M−1µ±vsin

(2πlM

),(2.26)

l = 0,1,. . . ,M−1.

Thus,B(X) is positive definite if and only ifλ±l > 0, l = 0,1,. . . ,M −1, and weimmediately have the following:

PROPOSITION2.1 Given fixedβ, µ, and M withβ > 0, µ > 0, and M any positiveinteger, the Gibbs measures in(2.20)are well-defined only forv that satisfies theconditionsλ±l > 0, l = 0,1,. . . ,M−1, with λ±l the explicit numbers in(2.26)and,in particular, forv2< 2βµ. Under these conditions, there are constants C1,C2> 0so that B(X) in (2.24)satisfies

C2|X|2 ≤ B(X) ≤C1|X|2 .(2.27)

For fixedβ > 0, µ> 0, the numbersλ±l are always positive for all sufficiently largeM ≥ M0. If eitherλ+

l or λ−l for some l satisfiesλ±l < 0, then the Gibbs measurecannot be defined for this value ofv.

We now sketch a proof of the fact that, in the case whenv = 0, the GibbsmeasuresµN,δ “converge” asM goes to+∞, i.e., asδ tends to zero to the prob-ability measuresµN defined in the preceding section. More precisely, we de-fine (or extend) a probability measure onΩN from µN,δ, which is concentratedon piecewise linear curves, by setting for any bounded continuous functionF =F(Ω(t1), . . . ,Ω(tm)) with m≥ 0, 0< t1 < t2 < · · ·< tm ≤ 1,∫

F dµN,δ =∫

F(X(t1), . . . ,X(tm))µN,δ(X1, . . . ,XN)dX1 · · ·dXN(2.28)


where

Xj(ti) = xσij +(ti −σi)

(xσi+1

j −xσij

δ

)(1≤ i ≤ m,1≤ j ≤ N) with σi = [ti/δ]. We keep the same notation,µN,δ, for thisnatural extension, and we claim that, asM = 1

δ goes to+∞, µN,δ converges weaklyto µN. A complete proof of this fact is somewhat tedious and is certainly notneeded here. However, it is worth explaining the main idea of the proof, namely,the use of a Trotter product formula. In order to do so, we only consider the simplecase whenF = F(Ω(0)) andF is, say, smooth with compact support. In fact, thisproof immediately adapts to the case whenF = F(Ω(t1)) and then to the case whenF = ∏m

i=1 Fi(Ω(ti)−Ω(ti−1)), and the general case follows by linearity and density.Next, if F = F(Ω(0)), takingβ = 1 in order to simplify notation, we have∫

F(Ω(0))dµN,δ

=∫

F(x01, . . . ,x0

N)µN,δ(X1, . . . ,XN)dX1 · · ·dXN

=1Zδ

∫∫F(y0)pδ

(1,y0,yM−1)ce−

12δ |yM−1−y0|2

2πδe−δV(y0) dy0dyM−1

where

yσ = (xσ1 , . . . ,xσN) , σ = 0,. . . ,M−1,

V(y) =a2

N

∑j 6=k

log|xj −xk|−µ|y|2 ,

Zδ =∫∫

pδ(1,y0,yM−1)e−

12δ (yM−1−y0)2

2πδe−δV(y0) dy0dyM−1 ,

and

pδ(1,y0,yM−1) =∫ (M−2

∏i=0

e−1

2δ |yi+1−yi |2

2πδe−δV(yi+1)

)e−δV(yn−1) dy1 . . . dyM−1 .

As a consequence of the Trotter formula, the kernelpδ(1,y0,yM−1) is easilyseen to converge top(1,y0,yM−1), at least formally, wherep is the Green’s functionof

∂p∂t

−∆p+V p= 0 in R2N × (0,1) .

Therefore,Zδ converges toZ =∫

p(1,y0,y0)dy, and∫

F(Ω(0))dµN,δ−→ 1Z

∫F(y0)p(1,y0,y0)dy0 ,


which proves our claim. All the above can be justified, but we choose not to do sohere since the precise argument is not needed in this paper (and quite tedious!).

Finally, we mention that this type of argument also allows one to check therepresentations ofµN mentioned in the preceding section, namely, (2.7)–(2.8) and(2.13)–(2.15).

3 Heuristic Derivation of Mean Field Theory

We begin the discussion by motivating the scaling regime for mean field the-ory utilized in this paper for the Gibbs ensembles from Section 2 in the limit asN → ∞. The main scaling assumption for mean field statistical theories for pointvortices in the plane (see Caglioti et al. [2, 3] and Lions [17]) involves the scalingexp(− β

NHN(X)) in the Gibbs ensembles withX = (x1, . . . ,xN) ∈ R2N and

1N

HN(X) =1N

N

∑j 6=k

− ln|xj −xk| .(3.1)

The second heuristic idea in the mean field theory for point vortices in the plane isthat the empirical measure1N ∑N

i=1δxi converges weakly to a probability measure,ρ(y), asN → ∞ so that

1N

N

∑k6= j

− ln |xj −xk| ∼= −∫

R2

ln(|xj −y|)ρ(y)dy.(3.2)

In other words, the velocity potential induced on an individual vortex by the othervortices is insensitive to the detailed locations of these vortices asN → ∞ andinstead can be computed by the mean velocity potential defined by the probabilitydensityρ(y) ∈ R

2; i.e., fluctuations are arbitrarily small asN increases.In the mean field theory for nearly parallel filaments developed in this paper, we

scale the logarithmic contributions to the Hamiltonian in (2.4) in a similar fashionas in the two-dimensional theory described above asN → ∞. Thus, we assume thatthe nondimensional factor ¯a defined in (2.3) has the form

a =a

2πN(3.3)

with some prescribed constanta> 0. Next, we present a heuristic derivation ofmean field theory for the continuous Gibbs ensembles from Section 2. Rigorousa priori proofs of mean field limiting behavior with the scaling in (3.3) are givenin Sections 4 and 5 below for the broken path and continuum Gibbs measures,respectively.

Recall from (2.10)–(2.12) that, with the scaling in (3.3), the marginal distribu-tions of the Gibbs measure are defined in (2.10) and (2.11) through the Green’s


function for the PDE,∂p∂t

− 12β

N

∑i=1

∆Xi p−(βa

2πN

N

∑i 6= j

log|Xi −Xj |)

p+µN

∑j=1

|Xj |2p = 0

in R2N × (0,1) ,


(3.4)

In the heuristic derivation, we assume that (3.2) is satisfied with a densityρ(y) thatis independent oft, i.e., translation invariant. This assumption merely reflects thetime translation invariance of the Gibbs measures shown at the end of Section 2.1.By replacing the logarithmic sums in (3.4) by the convolution appearing in (3.2),we obtain heuristically that asN → ∞

p(X,Y,t) ∼=N

∏i=1

p(Xi ,Yi ,t)(3.5)

wherep(x,y,t) for x∈ R2, y∈ R

2, satisfies

∂p∂t

− 12β

∆xp− (βa)2π

(log|x| ∗ρ)p+µ|x|2p = 0 in R2× (0,1) ,(3.6)

p|t=0 = δy(x) onR2 .

To complete this formal derivation of mean field theory, we need to determinethe densityρ(x). According to (3.2),ρ is the limiting single-point probability dis-tribution of the filament curves; in general, this distribution is determined by settingm= 1 in (2.11) so that

∫F dµN =

∫

R2N

p(X,X,1)

−1 ∫

R2N

F(X)p(X,X,1)(3.7)

for any bounded continuous functionF(X). The formal factorized approximationin (3.5) combined with (3.7) yields the identity

ρ(x) = p(x,x,1)

∫

R2

p(x,x,1)

−1

.(3.8)

This completes the heuristic derivation since the equations in (3.6) and (3.8) con-stitute the Green’s function formulation of the mean field approximation. Otherequivalent formulations of this mean field approximation are presented in Sec-tion 6.

It is also possible to give a heuristic derivation of the mean field probabil-ity measure on paths (i.e., filaments). Indeed, we consider the marginals ofµN,namely,

µN,k =∫

dµN(. . . ,ωk+1, . . . ,ωN)(3.9)


or equivalently

µN,k = hN,k ·µk0(3.10)

wherehN,k =∫

hNdµ0(ωk+1) · · ·dµ0(ωN). Notice thatµN,k andhN,k are symmetricin (ω1, . . . ,ωk) and are invariant by time shifts. Then, if we believe that a meanfield theory is relevant asN goes to+∞, as we will in fact prove in the subsequentsections,µN,k should factorize asymptotically for eachk≥ 1 fixed

µN,k−→N

k⊗j=1

µ(3.11)

whereµ is the mean field law of a single filament. In order to determineµ, we goback to (2.7) (for instance), recalling that ¯a= 1/2πN, and we integrate with respectto ωk+1, . . . ,ωN. Then, using the rule1N ∑N

i=1δωi(σ)≈Nρ(x)dx for eachδ ∈ [0,1], we

deduce at least formally

µ=1Z

exp

−∫ 1

0dσ

−aβ2π

∫

R2

log|ω(σ)−x|ρ(x)dx+µ|ω(σ)|2(3.12)

dνβx,x dx,

Z =∫

exp

−∫ 1

0dσ

−aβ2π

∫

R2

log|ω(σ)−x|ρ(x)dx+µ|ω(σ)|2(3.13)

dνβx,x(ω)dx,

or equivalently

µ= h·µ0 ,(3.14)

h =1Z′ exp

−∫ 1

0dσ

aβ2π

∫

R2

log|ω(σ)−x|ρ(x)dx+µ

2|ω(σ)|2

,

Z′ = E0

exp

−∫ 1

0dσ

−aβ2π

∫

R2


2|ω(σ)|2

.

(3.15)

In particular, we should expect for any bounded, measurableF = F(ω1, . . . ,ωk) onΩk ∫

F dµN−→N

∫F(ω1, . . . ,ωk)dµ(ω1) · · · dµ(ωk) .(3.16)

Clearly, the above heuristic derivation gives no insight into a rigorous a prioriproof. Completely different considerations are needed that involve the characteri-zation of the limiting mean field problem as the unique minimizer of an appropriatefree energy functional. This rigorous procedure is carried out in Section 4 for the


broken path models following Caglioti et al. [2] and for the continuum filamentmodels in Section 5.

Finally, we mention some heuristic motivation for the mean field scaling in(3.3) for the Hamiltonian in (2.4) in terms of vortex dynamics. With the scalingin (3.3), the dynamic equations for interacting nearly parallel filaments from (1.1)have the form

∂Xj

∂t= J

[∂2

∂σ2Xj +a

2πN

N

∑k6= j

Xj −Xk

|Xj −Xk|2].(3.17)

Thus, forN → ∞, the scaling in (3.3) for statistical behavior for the Hamiltoniancorresponds to the circumstances where the linearized self-induction of each indi-vidual filament is much stronger than the potential vortex interaction of individualfilaments. For the actual vortex dynamics of nearly parallel vortex filaments, otheradditional nonlinear corrections to the self-induction of individual filaments mightbe needed (Majda [18]), but the model in (3.9) probably still retains a number ofsignificant features.

4 Rigorous Mean Field Theory for the Broken Path Models

Here we sketch a rigorous proof of a priori convergence to a suitable meanfield limit for the Gibbs measures in (2.21) and (2.22) for the broken path modelswith the mean field scaling from (3.3) providedβ andµ satisfyβ ≥ 0, µ > 0,while the multiplier for the discrete currentv necessarily satisfies the restrictions inProposition 2.1. We will not give details of the proofs since they closely mimic thearguments of Caglioti et al. [2] for mean field behavior of statistical point vorticesin R

2 in their simplest situation with positive temperatureβ > 0.Thus, setting ¯a= a/2πN in (2.20), we introduce the correlation functions asso-

ciated with the Gibbs measuresµN,δ in (2.15) and (2.16). The correlation functionsρN,δ

k (X1, . . . ,Xk) are probability densities defined by

ρN,δk (X1, . . . ,Xk) =

∫µN,δ dXk+1 · · ·dXN for 1≤ k≤ N−1.(4.1)

These probability densities are symmetric in(X1, . . . ,Xk) as a consequence of thesymmetry ofµN,δ with respect to the broken paths(X1, . . . ,XN). We have the fol-lowing:

THEOREM 4.1 Assumea = a/πN in (2.20) with β > 0, µ > 0, and v satisfyingthe conditions of Proposition2.1 in the Gibbs measures in(2.21)and (2.22) forthe broken path models with fixed M≥ 1. Then, for any k≥ 1, the correlationfunctionsρN,δ

k converge in Lp((R2M)k) for all 1≤ p<∞ to ∏kj=1ρ

δ(Xj) as N→ ∞.The probability densityρδ(X) onR

2M is translation invariant so that

ρδ(X) = ρδ(TkX)(4.2)


where for a given periodic broken path,

X =(x0,x1, . . . ,xM−1) , TkX =

(xk, . . . ,xM+k−1),

with the convention that xσ+mM = xσ for all 0 ≤ σ ≤ M −1, m≥ 1. The densityρδ(X) is the unique solution of the following mean field equation:

ρδ(X) = Z−1exp

−B(X)− δ

M−1

∑σ=0

V(xσ)

onR

2M ,

V(x) = −aβ2π

∫

R2

log|x−y|ρδ1(y)dy onR2 ,

ρδ1(y) =∫

R2(M−1)

ρδ(x0,y,x2, . . . ,xM−1)dx0dx2 · · ·dxM−1

(4.3)

with

ρδ ∈ L∞(R2M) , ρδ logρδ ∈ L1(R2M) , ρδ|X|2 ∈ L1(R2M) ,

and B(X,β,µ,v) given in(2.18). In fact,ρδ is smooth and rapidly decreasing andis the unique minimum of the following strictly convex(free energy) functional

Fδ(ρ) =1β

∫

R2M

[ρ logρ+B(X)ρ]dX

− a4π

∫∫

R2M×R2M

ρ(X)ρ(Y)M−1

∑σ=0

log|xσ−yσ|dX dY.

(4.4)

For the special case withM = 1, the equations in (4.3) reduce to the familiarmean field equations for point vortices inR2 in the positive temperature regime(Caglioti et al. [3]). In Section 5, we establish that forv = 0, ρδ1 converges toρ asδ → 0 whereρ(x) is the probability density in (3.8) arising from the meanfield theory for continuous-path vortex filaments described heuristically in (3.6)and (3.8).

Under the restrictions on the multiplierv in Proposition 2.1 and with the nota-tion from (2.18), the density for the Gibbs measures in (2.15) with ¯a = a(πN)−1

has the form

µN,δ(X1, . . . ,XN) = Z−1exp

(βa

2πN

M−1

∑σ=0

N

∑k6= j

ln |xσk −xσj |)

N

∏j=1

e−B(Xj )(4.5)

where

e−C1|Xj |2 ≤ e−B(Xj ) ≤ e−C2|Xj |2 .(4.6)

With the structure in (4.5) and (4.6), simple modifications of the estimates in Sec-tion 3 of Caglioti et al. [2] and identical to those needed in Section 6 of that paper


yield the uniform bounds on the correlations

ρN,δj (X1, . . . ,Xj) ≤Cj for all N .(4.7)

With (4.7) and the subadditivity and strict convexity of entropy, we can copy theargument in section 4 of Caglioti et al. [2] (also see section 4 of Lions [17]) withonly minor changes to conclude the theorem provided the free energy functionaldefined in (4.4) has a unique solution. Proposition 2.1 guarantees that the integrandis strictly convex so a unique solution exists.

The calculation for the minimizerρδ(X) for the free energy in (4.4) yields

ρδ(X) = Z−1exp

−B(X)−

M−1

∑σ=0

δVσ(xσ)

onR

2M ,

Vσ(x) = −aβ2π

∫

R2

log|x−y|ρδσ(y)dy

ρδσ(y) =∫

R2(M−1)

ρδ(x0, . . . ,xσ−1,y,xσ+1, . . . ,xM−1)

dx0 · · ·dxσ−1dxσ+1 · · ·dxM−1 for all 0≤ σ ≤ M−1.

(4.8)

The free energy functional in (4.4) is translation invariant, i.e.,F(ρ(X)) =F(ρ(Tk(X)), and since the minimizer is unique, we deduce (4.2). The equationin (4.2) and the last one in (4.8) together imply thatρσ(y) = ρ1(y) for all σ with0≤ σ ≤ M−1, and (4.8) reduces to the mean field equation stated in (4.3).

Recall that the derivation for the heuristic mean field theory for the continuumfilament model in Section 3 tacitly assumed that the one-point density is translationinvariant; here we have deduced this property forρ in an a priori fashion for thebroken path models. We will do this in a similar manner for the continuum modelsin Section 5.

5 Rigorous Mean Field Theory for Vortex Filaments

For each fixedk ≥ 1, we consider forN ≥ k the probability measuresµN,k andtheir densities with respect to the probability measuresµk

0 andhN,k, given by (3.9)and (3.10), respectively. We also assume thatλ = v = 0 (see Section 7 for theextension to the case whenλ andv do not vanish). Of course, we set ¯a = a/2πNfor the reasons explained in Sections 3 and 4 above. These assumptions are madethroughout this section and will not be repeated.

5.1 Main ResultsTHEOREM 5.1 For each k≥ 1, µN,k converges weakly(in the sense of probabilitymeasures onΩk), as N goes to+∞, to some product measure

⊗kj=1µ. Further-

more, hN,k is bounded onΩk uniformly in N and converges, as N goes to+∞,


in Lp(Ωk,µk0) for all 1 ≤ p< ∞ to ∏k

j=1h(ωi) for some h that is continuous andbounded onΩ. In addition, we have

dµ= hdµ0

h = 12 exp

−∫ 1

0dσ

−aβ2π

∫

R2


2|ω(σ)|2

Z′ = E0 exp

−∫ 1

0dσ

−aβ2π

∫

R2


2|ω(σ)|2]

(5.1)

dµ=12

exp

−∫ 1

0dσ

−aβ2π

∫

R2

log|ω(σ)−x|ρ(x)dx

+µ|ω(σ)|2]

dνβx,x dx

Z =∫

R2

dx∫

dνβx,x(ω)

exp

−∫ 1

0dσ

−aβ2π

∫

R2

log|ω(σ)−x|ρ(x)dx+µ|ω(σ)|2 ,

(5.2)

whereρ is the probability measure onR2 defined by∫

R2

ϕ(x)ρ(x)dx= E[ϕ(ω(σ))] = E0[ϕ(ω(σ))h(ω)](5.3)

for anyσ ∈ [0,1] and for anyϕ that is bounded and measurable onR2, where E

denotes the expectation with respect toµ. In other words,ρ(x) is the density of thelaw ofω(σ) underµ for eachσ ∈ [0,1].

The densityρ is smooth, radially symmetric, and rapidly decreasing and is theunique solution, say, in L1∩L∞(R2) of

ρ(x) =p(1,x,x)∫

R2

p(1,x,x)dx(5.4)

where p(t,x,y) is the Green’s function of∂p∂t

− 12β

∆p− aβ2π

(log|x| ∗ρ)p+µ|x|2p = 0 in R2× (0,1) ,

p|t=0 = δy(x) in R2 .

(5.5)


Remark5.2. For any bounded, measurableF on (R2)m(m≥ 1) and for any 0≤t1 < t2 < · · ·< tm = 1, we obviously have

E[F(ω(t1), . . . ,ω(tm))] =

[ ∫

R2m

dy1 · · ·dymp(t2− t1,y1,y2) · · ·

p(tm− tm−1,ym−1,ym)p(1− tm+ t1,ym,y1)

F(y1, . . . ,ym)

] ∫

R2

p(1,x,x)dx

−1

.

(5.6)

In other words, the joint law of(ω(t1), . . . ,ω(tm)) underµ admits a density withrespect to the Lebesgue measure onR

2m that is given by

q = q(t2− t1, . . . ,tm,tm−1,1− (tm− t1);y1, . . . ,ym)

= (p(t2− t1,y1,y2) . . .

p(tm− tm−1,ym−1,ym)p(1− tm+ t1,ym,y1))

∫

R2

p(1,x,x)dx

−1

.

(5.7)

Then, the above result yields for any bounded, measurableF onR2mk and for each

fixedk≥ 1 ∫dµNF(Ωk(t1), . . . ,Ωk(tm))

=∫

dµN,kF(Ωk(t1), . . . ,Ωk(tm))

−→N

∫dµ(ω1) · · ·dµ(ωk)F(Ωk(t1), . . . ,Ωk(tm))

=∫

R2mk

F(Y1k , . . . ,Ym

k )q1 · · ·qk dY

whereY jk = (yj

1, . . . ,yjk) for 1≤ j ≤ m, qi = q(y1

i , . . . ,ymi ) for 1≤ i ≤ k. Notice that

the convergence is indeed valid for any bounded, measurableF as a consequenceof the strong convergence of the densitieshN,k.

The proof of the above result is given in Sections 5.2 and 5.3. We first drawsome consequences of it.

COROLLARY 5.3 For any m≥ 1, 0≤ t1 ≤ ·· · ≤ tm≤ 1, the empirical law1N(δω1 +

· · ·+δωN) underµN weakly converges(in the sense of probability measures onR2m)

to q(s1, . . . ,sm;y1, . . . ,ym)dy where we denote by

s1 = t2− t1 , . . . , sm−1 = tm− tm−1 , sm = 1− (tm− t1) ,


and ωi = (ωi(t1), . . . ,ωi(tm))

for 1≤ i ≤ N. More precisely, we have for any p∈ [1,+∞)∥∥∥∥∥∥ 1N

N

∑j=1

ϕ(ω j)−∫

R2m

ϕqdy

∥∥∥∥∥∥Lp(ΩN,µN)

−→N

0(5.8)

for anyϕ ∈ Lp(R2m)+L∞(R2m).

PROOF OFCOROLLARY 5.3: We begin with the simple case whenp = 2. Set-ting ϕ= ϕ− ∫

R2mϕqdy, we then have∥∥∥∥∥∥ 1N

N

∑j=1

ϕ(ω j)−∫

R2m

ϕqdy

∥∥∥∥∥∥2

L2

=1

N2

N

∑i 6= j

∫dµNϕ(ωi)ϕ(ω j)+

1N2

N

∑i=1

∫dµNϕ(ωi)

=N(N−1)

N2

∫dµ2

0ϕ(ω1)ϕ(ω2)hN/2 +1N

∫dµ1

0ϕ2(ω1)hN,1

using the symmetries ofµN. Then this converges, asN goes to+∞, to

∫dµ2

0ϕ(ω1)ϕ(ω2)h(ω1)h(ω2) =(∫

dµ0ϕ(ω)h(ω))2

in view of Theorem 5.1, provided we check thatϕ(ω) ∈ L2(Ω,µ0). Then we finishthe proof easily forp = 2, since we have

∫dµ0ϕ(ω)h(ω) =

∫dµ ϕ(ω(t1), . . . ,ω(tm))−

∫

R2m

ϕqdy= 0.

Finally, ϕ(ω) ∈ L2(Ω,µ0) sinceϕ ∈ L2 +L∞(R2m), in view of the explicit densityof (ω(t1), . . . ,ω(tm)) underµ0 exhibited in (2.16) and (2.17) of Section 2 (whichbelongs to the Schwartz classS of rapidly decreasing smooth functions—it is aGaussian).

For a general exponentp, we writeϕ = ϕ1 +ϕ2 with ϕ1 ∈ Lp, ϕ ∈ L∞, andwe decomposeϕ1 into ϕ11(|ϕ|<R) +ϕ1(|ϕ|≥R). Then, for eachR∈ (0,∞), ψ =ϕ2 +ϕ11(|ϕ|<R) ∈ L∞ +Lp∩L∞ ⊂ (L2 +L∞)∩L∞. Hence, by the preceding proof

1N

N

∑i=1

ψ(ωi)−∫

R2m

ψqdy−→N

0 in L2(ΩN,µN)


and thus inLp if p≤ 2; it also converges to 0 inLp if p> 2 since it is obviouslybounded inL∞ by 2‖ψ‖L∞ . We conclude observing that we have∫

R2m

ϕ1q1(|ϕ1|≥R) dy→ 0 asR→ +∞

(sinceq∈ L1∩L∞(R2m)) and∥∥∥∥∥ 1N

N

∑i=1

ϕ1(ωi)1(|ϕ1(ωi)|≥R)

∥∥∥∥∥Lp(ΩN,µN)

≤ ‖ϕ1(ω1)1(|ϕ1(ω1)|≥R)‖Lp(ΩN,µN)

= ‖|ϕ1(ω1)|p1(|ϕ1(ω1)|≥R)hN,1‖1/p

L1(Ω,µ0)

≤C‖|ϕ1(ω1)|1(|ϕ1(ω1)|≥R)‖Lp(Ω,µ0)

≤C‖|ϕ1|1(|ϕ1|≥R)‖Lp(R2m) → 0 asR→ +∞ ,

whereC denotes various positive constants independent ofN.

As mentioned above, the proof of Theorem 5.1 is given in the next two sub-sections. In Section 5.2, we present the heart of the matter, leaving aside the ver-ification of some technical (but crucial) facts that are proved in Section 5.3. Inparticular, as in the case of point vortices (see Caglioti et al. [3] or Lions [17]),the proof relies upon a variational characterization ofµN and more precisely ofhN,which yields asymptotically the following variational characterization ofh (alsoproved in the next subsections):

THEOREM 5.4 The mean field density h is a continuous, bounded function onΩand is the unique minimum of the following free energy functional:

minF( f ) : f ≥ 0, f ∈ L∞(Ω,µ0), E0( f ) = 1(5.9)

where

F( f ) =1β

E0( f log f )+µ

2βE0(V0 f )+

a2

E2(V(ω−ω′) f (ω) f (ω′))(5.10)

and we denote by

V0(ω) =∫ 1

0|ω(0)|2dσ , V(ω−ω′) = − 1

2π

∫ 1

0log|ω(σ)−ω′(σ)|dσ .

Remark5.5. It is possible to extend the minimization class in (5.9) to thosef ∈L1

+(Ω0,µ0) with E( f | log f |)<∞. It is also possible to obtain a variational charac-

terization of the densityg of µ with respect todxdνβx,x = dµ. It is given by

min

F ( f ) : f ≥ 0, f ∈ L1∩L∞(Ω,µ),∫

Ω

f dµ= 1

(5.11)


with

F ( f ) =1β

∫

Ω

f log f dµ(ω)+µ

β

∫

Ω

V0(ω) f dµ(ω)

+a2

∫∫

Ω×Ω

V(ω−ω′) f (ω) f (ω′)dµ(ω)dµ(ω′) .(5.12)

In fact, this formulation may be deduced from the preceding one (and is equivalentto the preceding one). We prefer to work withµ0 instead of ¯µ, sinceµ0(Ω) = 1while µ(Ω) = +∞!

Remark5.6. We note that the entropy functionalE0( f log f ) is nothing but therelative entropy of ˆµ= f ·µ0 with respect toµ0, namely,

E0( f log f ) =∫

Ω

f log f dµ=∫

Ω

log

(dµdµ0

)dµ .

Remark5.7. Let us check immediately thatF is well-defined and in fact finite onL1

+ ∩L∞ and thatF is indeed strictly convex. The first term is obviously finite andstrictly convex since(t 7→ t logt) is strictly convex on[0,∞) and

f −1≤ f log f ≤ ‖ f‖L∞ log[max(‖ f‖L∞ ,1)] .

In particular, ifE0( f ) = 1, this term is obviously nonnegative.The second term is linear inf and clearly nonnegative. In addition, we have

0≤ V f ≤ V‖ f‖L∞ and

E0(V) = E0(|ω(0)|2) =

∫R2

|x|2q(x,x,1)dx

∫R2

q(x,x,1)dx< ∞

by the time shift invariance and the explicit representation ofq, (2.17).The third term is obviously quadratic inf , and we claim it is both finite and

nonnegative for eachf ∈ L1+∩L∞. This suffices to complete the proof of the above

claim onF—let us remark in passing that it also shows the nonnegativity ofF. Wefirst observe that we have

logt ≤ t −1 for all t ≥ 0,(5.13)

hence,

V(ω−ω′) f (ω) f (ω′) ≤ ‖ f‖2L∞

∫ 1

0|ω(σ)|+ |ω′(σ)|dσ ∈ L1(Ω)

while we have

V(ω−ω′) f (ω) f (ω′) ≥−∫ 1

0| log|ω(σ)−ω′(σ)|1(|ω(σ)−ω′(σ)|≤1) dσ‖ f‖2

L∞


and by the time shift invariance ofµ0

E2∫ 1

0| log|ω(σ)−ω′(σ)||1(|ω(σ)−ω′(σ)|≤1) dσ

= E20| log|ω(0)−ω′(0)||1(|ω(0)−ω′(0)|≤1)

=

∫∫

R2×R2

| log|x−y||1(|x−y|≤1)q(x,x,1)q(y,y,1)dxdy

∫

R2

q(x,x,1)dx

−2

<+∞ ,

since

q(x,x,1) =√µβ

πsinh

(√µ

β

)exp

−√µβ

cosh(√

µβ

)−1

sinh(√

µβ

) |x|2 .

Finally, in order to prove the nonnegativity of this quadratic term, it clearly sufficesto check it whenf = F(ω(t1), . . . ,ω(tm)) wherem≥ 1, 0≤ t1 < t2 < · · ·< tm ≤ 1,F ∈C∞

0 (Rm). We observe that for eachσ ∈ [0,1]

∫∫

Ω×Ω

(− 1

2πlog|ω(σ)−ω′(σ)|

)F(ω(t1), . . . ,ω(tm))F(ω′(t1), . . . ,ω′(tm))dµ0(ω)dµ0(ω′)

=∫∫

R2×R2

(− 1

2πlog|x−y|

)G(x)G(y)dxdy

for some smooth and rapidly decreasingG, in view of (2.16)–(2.17), and this ex-pression is nonnegative, as is well-known.

5.2 The Heart of the Matter

We begin by stating various facts, whose proofs will be given in Section 5.3.

PROPOSITION5.8 The density hN is the unique minimum of the following strictlyconvex functional:

FN = minFN( f ) : f ≥ 0, f ∈ L∞(ΩN,µN0 ), EN

0 ( f ) = 1(5.14)


where

FN( f ) =1β

EN0 ( f log f )+

µ

2βEN

0

((N

∑i=1

V0(ωi)

)f

)

+aN

EN0

((12

N

∑i 6= j

V(ωi −ω j ·))

f

).

(5.15)

Remark5.9. The same argument as in Remark 5.7 (Section 5.1) shows that eachterm in Fn is well-defined (and finite) onL1

+ ∩ L∞ and nonnegative on the mini-mization class.

PROPOSITION5.10 There exists a positive constant C0 such that for all N≥ k≥ 1

0≤ hN,k ≤Ck0 on ΩN .(5.16)

We can now give the proofs of Theorems 5.1 and 5.4, which we divide intoseveral steps.

Step 1: Any Weak Limit Point Is a Minimum of a Free Energy Function

In view of the uniform bound (5.16), we may extract by a diagonal procedure asubsequence, still denoted byN in order to simplify notation, such that

hN,k → hk weaklyL∞∗for some bounded, measurablehk ≥0 such thatEk

0(hk)= 1. SincehN,k is symmetric

in (ω1, . . . ,ωk), so ishk. Furthermore,∫

hN,k+1dµ0(ωk+1) = hN,k ;

hence, we have ∫hk+1dµ0(ωk+1) = hk .

Applying the classical Hewitt-Savage theorem, we deduce that there exists a prob-ability measure ¯π on f ∈ L1

+(Ω,µ0) : E0( f ) = 1 supported on the ball (inL∞) f ∈ P : ‖ f‖L∞ ≤C0 in view of the bound (5.16) such that we have for allk≥ 1

hk =∫ k

∏i=1

f (ωi)dπ( f ) a.s. inΩk .(5.17)

We then denote byP the set of all probability measuresπ on P supported in anarbitrary ball ofL∞.

We now claim thatπ is a minimum of the following free energy functional:

F = min

F(π) : π ∈ P

(5.18)


whereF is given by

F(π) =1β

E0

(∫f log f dπ( f )

)+µ

2βE0

(V0

∫f dπ( f )

)+

a2

E20

(V(ω−ω′)

∫∫f (ω) f (ω′)dπ( f )

).

(5.19)

Once more, the assumption made upon the support ofπ allows us to check, as inRemark 5.7 (Section 5.1), that each term inF is finite and nonnegative onP .

In order to prove our claim, we first observe that we obviously have

1N

EN0

((N

∑i=1

V0(ωi)

)hN

)= E0(V0hN,1)

−→N

E0(V0h1) = E0

(V0

∫f dπ( f )

)(5.20)

1N2EN

0

(N

∑i 6= j

V(ωi −ω j)hN

)

=(

N−1N

)E2

0(V(ω−ω′)hN,2(ω,ω′))−→N

E20(V(ω−ω′)h2)

= E0(V0(ω−ω′)∫

f (ω) f (ω′)dπ( f ))

(5.21)

where, for instance, we use the observations made in Remark 5.7 (Section 5.1)to check that one can pass to the limit asN goes to+∞ despite the growth andsingularities ofV0 andV. In conclusion, we have shown for someεN−→

N0 that

1N

FN =1N

FN(hN) =1

NβEN

0 (hN loghN)+µ

2βE0

(V0

∫f dπ( f )

)+

a2

E0(V0(ω−ω′)∫

f (ω) f (ω′)dπ( f ))+εN .

(5.22)

Similarly, denoting byf N =∫

∏Ni=1 f (ωi)dπ( f ) for anyπ ∈ P , we have for some

δN−→N

0

F(π) =1β

E0


)+µ

2βEN

0

((N

∑i=1

V0(ωi) f N

))

+a

2NEN

0

(N

∑i 6= j

V(ωi −ω j) f N

)+ δN .

(5.23)

We then recall some classical facts on the entropy (see, for instance, Ruelle[22]):

EN0 ( f N log f N) ≥ Em

0 ( f N,m log f N,m)+EN−m0 ( f N,N−m log f N,N−m)(5.24)


for each symmetric probability densityf N on ΩN, and for all 1≤ m≤ N−1,

limN

1N

EN0 ( f N log f N) = E0


)(5.25)

for eachπ ∈ P where f N =∫

∏Ni=1 f (ωi)dπ( f ). We briefly sketch a proof of these

facts for the sake of completeness (and also because of the slightly particular settingwe use). (5.24) follows readily from the convexity inequality

f N log

(f N

f N,mgN,m

)+ f N,mgN,m− f N ≥ 0

wheregN,m = f N,N−m(ωm+1, . . . ,ωN). Then, (5.24) implies that we have for anyN ≥ k≥ 1

1N

EN0 ( f N log f N) ≥ 1

kEk

0( f N,k log f N,k)+1N

Er0( f N,r log f N,r)(5.26)

with r = N−[Nk

]k. In particular, if f N =

∫∏N

i=1 f (ωi)dπ( f ) for someπ ∈P , f N,k =f k for all 1≤ k ≤ N and thus1

NEN0 ( f N log f N) converges, in view of (5.26), asN

goes to+∞. The limit obviously coincides withE0( f log f ) whenπ is concentratedon f. Therefore, (5.25) follows upon proving that this limit is linear inπ. Thisis obvious if we use the following inequality valid for alla,b≥ 0:

0≤ 12

aloga+12

blogb− a+b2

loga+b

2≤ log2

2|a−b| .

Indeed, we then deduce

1N

EN0

((f N1 + f N

2

2

)log

(f N1 + f N

2

2

))+

log22N

EN0 | f N

1 − f N2 |

≥ 12

(1N

EN0 ( f N

1 log f N1 ))

+12

(1N

EN0 ( f N

2 log f N2 ))

≥ 1N

EN0

((f N1 + f N

2

2

)log

(f N1 + f N

2

2

)),

and we finish the proof sinceEN0 (| f N

1 − f N2 |) ≤ EN

0 ( f N1 )+EN

0 ( f N2 ) = 2.

Collecting now (5.22), (5.23), and (5.24)–(5.26), we deduce on the one handthat for eachπ

F(π) = limN

1N

FN( f N) ≥ 1N

FN =1N

FN(hN) ;


henceF≥ limN

1NFN. On the other hand, we have for eachk≥ 1

limN

1N

FN = limN

1N

FN(hN)

≥ 1β

Ek0(h

k loghk)+µ

2βE0

(V0

∫f dπ( f )

)+

a2

E0

(V(ω−ω′)

∫f (ω) f (ω′)dπ( f )

);

hence, lettingk go to+∞,

limN

1N

FN ≥ F(π) ≥ F.

And we have shown that1NFN = 1NFN(hN) converges, asN goes to+∞, to

F(π) = F. Finally, we have also shown that we have

1N

EN0

(hN loghN)−→

NE0


).(5.27)

Step 2: Strong Convergence to the Unique Minimum

We first show that ¯π is concentrated on the unique minimum ofF. Indeed, weobserve that we have

F = F(π) =∫

F( f )π(d f) ≥ F,

while if fn is a minimizing sequence of F, thenF(δ fn) = F( fn) converges by defi-nition to F. Hence,F = F andF( f ) = F π-a.s. This shows that Fadmits at least aminimumh∈ L∞(‖h‖L∞ ≤C0), and sinceF is strictly convex onP (see Remark 5.7in Section 5.1), ¯π= δh whereπ is the unique minimum ofF overP. In other words,we have shown at this stage that, for allk ≥ 1, hN,k converges weakly (inL∞(Ωk)weak∗) to hk = ∏k

i=1h(ωi). Furthermore, this convergence, by the uniqueness ofthe limit, is in fact true for the whole sequence and not only for any particularsubsequence we extracted in Step 1.

In addition, in view of (5.26) and (5.27),

1k

Ek0

(hk loghk)= E0(hlogh) = lim

N

1N

EN0

(hN loghN)

≥ limN

1k

Ek0(h

N,k loghN,k) .

This, combined with the strict convexity of the entropy, yields, as is well-known,the strong convergence inL1(Ωk) of hN,k to hk, and thus inLp(Ωk) for all 1≤ p<∞in view of the uniform bound (5.16).

We also observe that, sinceF( f (θt ·)) = F( f ) for all t ≥ 0, the uniqueness ofthe minimum implies thath(θtω) = h(ω) a.s. inΩ for all t ≥ 0, i.e., the translation


invariance that we were expecting in view of the phenomenon already observed forthe broken path models in the preceding section.

Another consequence of these facts concerns the law ofω(σ) underµ, which,by the invariance ofµ0 andh, is independent ofσ ∈ [0,1]. Indeed, lettingϕ ∈L∞(R2), we have

E[ϕ(ω(0)] = E0[ϕ(ω(0))h]and thus

|E[ϕ(0))]| ≤C0

∫

Ω

|ϕ(ω(0)|dµ0(ω) ≤C∫

R2

|ϕ(x)|q(x,x,1)dx

≤C∫

R2

|ϕ(x)|e−δ|x|2 dx

for some positive constantsC andδ independent ofϕ.This bound shows that the law ofω(σ) for all σ ∈ [0,1] underµ admits a density

ρ with respect to the Lebesgue measure onR2 and that we have

0≤ ρ≤Ce−δ|x|2

onR2.(5.28)

In other words, not only is this density bounded onR2, but the rapid decay stated

in Theorem 5.1 is established.

Step 3: Conclusion of the Proofs of Theorems 5.1 and 5.4

We begin with a lemma that is nothing but the justification of the formal Euler-Lagrange equation satisfied by the minimumh of F overP. Once more, becausethe verification is purely technical, we will not address it until the next subsection.

LEMMA 5.11 The minimum h of F over P satisfies the following Euler-Lagrangeequation:

h =1Z′ exp

−∫ 1

0dσ

µ2|ω(σ)|2

−aβ2π

∫

Ω

log|ω(σ)−ω′(σ)|h(ω′)dµ0(ω′)

Z′ = E0

exp

−∫ 1

0dσ

µ2|ω(σ)|2

−aβ2σ

∫

Ω


(5.29)


At this stage, in order to complete the proofs of Theorems 5.1 and 5.4, thereonly remains to prove (5.1), the PDE characterization ofρ as the unique solutionof (5.4)–(5.5) and its smoothness, and we shall do so in that order. Indeed, (5.2)is clearly equivalent to (5.1) in view of the definition ofµ0, while the radial sym-metry follows from the uniqueness of the solution of (5.4)–(5.5) together with theelementary rotational invariance of that system of equations.

In order to prove (5.1), we first remark that, by the invariance ofh andµ0, theexpression

− 12π

∫

Ω

log|x−ω′(σ)|h(ω′)dµ0(ω′)

is independent ofσ ∈ [0,1] for all x ∈ R2 and thus is a function ofx only. We

denote byΨ(x) this potential, and we observe that we have

Ψ(x) = − 12π

E(log|x−ω′(0)|) =(− 1

2πlog|x|

)∗ρ(5.30)

by the definition ofρ. We observe that the bound in (5.28) immediately shows thatΨ is radial, nonincreasing,C1 on R

2 (in fact, W2,ploc (R2), DΨ ∈ W1,p(R2) for all

p∈ [2,∞], D2Ψ ∈ Lp(R2) for all p∈ (1,∞]—even forp = +∞ sinceρ is radial—and−∆Ψ = ρ onR

2) with Ψ(x) = − 12π log|x|+O(1/|x|) as|x| goes to+∞. Thus,

the representation (5.1) ofh is shown, as well as the continuity ofh overΩ.Next, the PDE characterization ofρ, namely, (5.4)–(5.5), is now immediate

since the total potentialaβΨ +µ|x|2 is smooth onR2(C1,1) and grows at infinity.It is indeed a simple consequence of the Trotter formula (see, for instance, Simon[23] or the argument sketched in Section 2.2 above). It will also be a consequenceof another variational argument that we present below in Section 5.3. Finally, thesmoothness ofρ follows from regularity results for parabolic equations: Indeed,sinceaβΨ +µ|x|2 ∈ C1,α

loc for all α ∈ (0,1), Schauder estimates easily yield thatp(1,x,y) ∈ C3,k(R2 ×R

2) and thusρ ∈ C3,α(R2) for all α ∈ (0,1). Hence,Ψ ∈C5,α(R2) and we may bootstrap the regularity exponents, showing thus thatρ ∈C∞(R2). One can also check easily that all derivatives ofρ have at least someGaussian decay asρ has, as shown by (5.28).

5.3 Some Technical Facts

We begin with the proofs of the formally obvious Proposition 5.8 and Lem-ma 5.11. Indeed, if we ignore the lack of differentiability of the function,t 7→t logt at t = 0, then these two results are immediate consequences of the Euler-Lagrange equations associated with the convex variational problems (5.14) and(5.10), respectively.

PROOF OFPROPOSITION5.8: We first observe that loghN is easily seen to be-long toLp(ΩN) for all 1≤ p<∞ by arguments similar to ones made several times


above. This allows one to write the following convexity inequality valid for allf ∈ L∞(ΩN):

f log f ≥ hN loghN +(loghN +1)( f −hN)

= hN loghN +

[µV0 +

aβ2N ∑

i 6= j

V(ωi −ω j)

]( f −hN)+ f −hN .

Hence, taking the expectation with respect toµ0, we conclude

FN( f ) ≥ FN(hN)+EN0 ( f −hN) = FN(hN)

of EN0 ( f ) = 1.

PROOF OFLEMMA 5.11: In order to circumvent the possible vanishing of theminimumh, we follow an argument introduced in Caglioti et al. [3] (see also Lions[17]). We consider, forδ > 0 (small enough), the eventBδ = h≥ δ, and we setB= limδ↓0+ ↑Bδ. Of course,µ0(B)> 0 sinceE0(h) = 1. Then,h is still a minimumof δ over the set f ∈ L∞(Ω,µ0),E0( f ) = 1, f ≥ 0, f = h a.s. onBc

δ. Sinceh doesnot vanish onBδ, we may now write the Euler-Lagrange equation associated to thatrestricted minimization problem, and we find

h =1Z′δ

exp

−∫ 1

0dσ

µ2|ω(σ)|2− aβ

2π

∫

Ω


a.s. onBδ

where

Z′δ = E0

exp

−∫ 1

0dσ

µ2|ω(σ)|2

− aβ2π

∫

Ω


Bδ

E0[h1Bδ]−1 .

Lettingδ go to 0+, we easily deduce

h =1

Z′ exp

−∫ 1

0dσ

σ2|ω(σ)|2− aβ

2π

∫

Ω


a.s. onB

(5.31)


where

Z = E0

exp

−∫ 1

0dσ

µ2|ω(σ)|2

−aβ2π

∫

Ω


1B

sinceE0[h1Bδ

] → E0[h1B] = E0[h] = 1 asδ goes to 0+.We conclude by proving by contradiction thatµ0(A) = 0 whereA= BC. Indeed,

if µ0(A) = a> 0, we may consider, forδ > 0, h = (h+ δ1A)(1+ aδ)−1, and wecheck easily, using the boundedness ofh, that we have

F(h) ≤ F(h)+Cδ+1β

∫hlogh dµ0(ω)− 1

β

∫hloghdµ0(ω)

≤ F(h)+Cδ+1β

∫h

1+aδlog

h1+aδ

dµ0(ω)

− 1β

∫hloghdµ0(ω)+

1β

aδ1+aδ

logδ

1+aδ

≤ F(h)+Cδ+aβδ logδ < F(h) for δ small enough,

whereC denotes various positive constants independent ofδ > 0. The contradictioncompletes the proof of Lemma 5.11.

We now complete the proofs of Theorems 5.1 and 5.4 by proving the bounds,(5.16), which played a crucial role in the analysis performed above and in thepreceding subsection (Section 5.2).

PROOF OFPROPOSITION5.10: The bounds (5.16) may be obtained by a ratherstraightforward chain of estimates that involve the following quantities defined for1≤ l ≤ N, µ > 0,

Z′(N, l ,µ) =∫

Ωl

exp

−[

aβ2N

l

∑i 6= j

V(ωi −ω j)+µ

2

l

∑i=1

V0(ωi)

]

dµ0(ω1) · · ·dµ0(ωl ) ,

so that, in particular,Z′(N) = Z′(N,N,µ). Finally, we denote byC various positiveconstants independent ofN andk.

Step 1: 0≤ hN,k ≤ Z′(N,N−k,µ(1− kN ))Z′(N,N,µ)−1Ck

By definition, we have

hN,k =∫

hNdµ0(ωk+1) · · ·dµ0(ωN)


=1

Z′(N)

[exp

−[

aβ2N

k

∑i 6= j

V(ωi −ω j)+µ

2

k

∑i=1

V0(ω)

]]∫

ΩN−k

exp

(−aβ

N

k

∑i=1

N

∑j=k+1

V(ωi −ω j)

)

·[

exp

−[

aβ2N

N

∑i 6= j≥k+1

V(ωi −ω j)+µ

2

N

∑i=k+1

V0(ωi)

]]dµ0(ωk+1) · · ·dµ0(ωN) .

We then use (5.13) in order to estimate

− aβ2N

k

∑i 6= j

V(ωi −ω j)− µ

2

k

∑i=1

V0(ωi)

≤ aβk2N

k

∑i=1

∫ 1

0dσ|ωi(σ)|− µ

2

k

∑i=1

V0(ωi)

≤Ck− µ

4

k

∑i=1

V0(ωi)− aβN

k

∑i=1

N

∑j=k+1

V(ωi −ω j)− µ

4

k

∑i=1

V0(ωi)

≤ aβk

∑i=1

∫ 1

0dσ|ωi(σ)|− µ

4

k

∑i=1

V0(ωi)+aβkN

N

∑j=k+1

∫ 1

0dσ|ω j(σ)|dσ

≤Ck+µk

4N

N

∑j=k+1

V0(ω j)+CkN

(N−k)

≤Ck+µk

4N

N

∑j=k+1

V0(ω j) .

Collecting these estimates, we deduce

hN,k ≤ Ck

Z′(N)

∫

ΩN−k

exp

−[

aβ2N

N

∑i 6= j≥k+1

V(ωi −ω j)+µ

2

(1− k

2N

) N

∑i=k+1

V0(ωi)

]

dµ0(ωk+1) · · ·dµ0(ωN) ;

hence we have obtained the desired estimate

0≤ hN,k ≤CkZ′(

N,N−k,µ

(1− k

2N

))Z′(N,N,µ)−1 .(5.32)

Step 2:Z′(N,N−k,µ) ≤ CkZ′(N,N,µ)

It suffices to show the inequality

Z′(N, l ,µ) ≤CZ′(N, l +1,µ)(5.33)


for all 1 ≤ l ≤ N−1 for some constant that only depends on a lower bound onµ.We write

Z′(N, l +1,µ)

=∫

exp

−[

aβ2N

l+1

∑i 6= j

V(ωi −ω j)+µ

2

l+1

∑i=1

V0(ωi)

]dµ0(ω1) · · ·dµ0(ωl+1)

=∫

exp

−aβ

2N

l

∑i 6= j

V(ωi −ω j)+µ

2

l

∑i=1

V0(ωi)

dµ0(ω1) · · ·dµ0(ωl )

∫dµ0(ω)

·exp

−[

aβN

l

∑i=1

V(ωi −ω)+µ

2V0(ω)

]

≥ Z′(N, l ,µ) inf(ω1,...,ωl )

∫dµ0(ω)exp

−[

aβN

l

∑i=1

V(ωi −ω)+µ

2V0(ω)

]

≥ Z′(N, l ,µ)exp

[inf

(ω1,...,ωl )

∫dµ0(ω)

−aβ

N

l

∑i=1

V(ωi −ω)− µ

2V0(ω)

]using Jensen’s inequality. Then we write

inf(ω1,...,ωl )

∫dµ0(ω)

−aβ

N

l

∑i=1

V(ωi −ω)− µ

2V0(ω)

≥−C+∫ 1

0dσ inf

(ω1,...,ωl )

∫dµ0(ω)

aβN

l

∑i=1

log|ωi(ω)−ω(σ)|

≥−C+∫ 1

0dσ inf

(x1,...,xl )∈R2l

∫dµ0(ω)

aβN

l

∑i=1

log|xi −ω(σ)|

≥ −C+aβN

inf(x1,...,xl )∈R2l

∫dµ0(ω)

l

∑i=1

log|xi −ω(0)|

using the invariance ofµ0 by time shifts. Then we remark that we have

1N

inf(x1,...,xl )∈R2l

∫dµ0(ω)

l

∑i=1

log|xi −ω(0)|

≥ lN

infx∈R2

∫dµ0(ω) log|x−ω(0)|

=lN

infx∈R2

∫

R2

q(y,y,1) log|x−y|dy

R(1)−1 ≥−C.

Therefore, (5.33) holds and we deduce

Z′(

N,N−k,µ

(1− k

2N

))≤CkZ′

(N,N,µ

(1− k

2N

)).(5.34)


Combining (5.32) and (5.34), we have shown the following bound onhN,k:

0≤ hN,k ≤CkZ′(

N,N,µ

(1− k

2N

))Z′(N,N,µ)−1 .(5.35)

Step 3: Conclusion

Using Hölder’s inequality withp = (1−k/2N)−1(1+k/2N), we obtain

Z′(

N,N,µ

(1− k

2N

))≤

∫dµN

0 exp

−[

aβ2N

N

∑i 6= j

V(ωi −ω j)+µ

2

N

∑i=1

V0(ωi)

]

·exp

−[

aβ2N

(p−1)N

∑i 6= j

V(ωi −ω j)+µ

2k

2N

N

∑i=1

V0(ωi)

].

Next, we observe that we have, in view of (5.13),

−[

aβ2N

(p−1)N

∑i 6= j

V(ωi −ω j)+µk4N

N

∑i=1

V0(ωi)

]

≤ aβ2

(p−1)N

∑i=1

∫ 1

0dσ|ωi(σ)|dσ− µk

4N

N

∑i=1

∫ 1

0dσ|ωi(σ)|2

≤ aβkN

N

∑i=1

∫ 1

0dσ|ωi(σ)|dσ− µk

4N

N

∑i=1

∫ 1

0dσ|ωi(σ)|2

≤Ck.

Therefore, we deduce

Z′(

N,N,µ

(1− k

2N

))≤CkZ′(N,N,µ) ,

and the proof of Proposition 5.10 is complete.

5.4 Direct Derivation of a Hartree-like Variational Problem for ρ

As explained in Section 2, the Gibbs measureµN is completely determined bythe Green’s functionpN of the linear parabolic second-order PDE (2.12). Recallthat we choose ¯a = a/2πN andλ= 0 so thatpN solves

∂pN

∂t− 1

2β∆X pN −

(βa

4πN

N

∑i 6= j

log|Xi −Xj |)

pN +N

∑j=1

µ|Xj |2µ= 0

in R2N× (0,1) ,

pN∣∣∣l=0

= δY(X) onR2N .

(5.36)


Furthermore, the law ofΩ(σ) = (ω1(σ), . . . ,ωN(σ)) underµN is given by

ρN(X) =pN(X,X,1)∫

R2N

pN(z,z,1)dz.

In this section, we first observe that

ρN(X,Y) =pN(X,Y,1)∫

R2N

pN(z,z,1)dz,

viewed as a kernel, is determined by a variational problem. Then we use thisvariational problem to analyze the behavior ofρN asN goes to+∞. This will leadto a Hartree-like variational problem forp(x,y,1)(

∫R2 p(x,x,1)dx)−1, namely, the

mean field kernel introduced in Theorem 5.1 (see equations (5.4) and (5.5)) whosediagonal,p(x,x,1), is nothing but the lawρ(x) of ω(σ) (for eachσ ∈ [0,1]) underthe mean field measureµ.

Of course,ρN is the kernel of an operatorρN on L2(R2N), which is self-adjoint

and nonnegative sinceρN = e−1

2β ∆X+βUNfor some potential

UN = − a4πN

N

∑i 6= j

log|Xi −Xj |+N

∑j=1

µ

β|Xj |2

and which has a finite trace since we have

Tr ρN =∫

R2N

ρN(X,X)dX = 1.(5.37)

We denote byK1(R2N) the closed convex set of such self-adjoint, nonnegative op-erators with trace equal to 1. We then introduce the following free energy, definedfor all K ∈ K1(R2N) by

FN(K) =1β

Tr(K logK)+Tr(UN ·K)+Tr(H0K)(5.38)

whereH0 =− 12β2 ∆X. Of course, we have to make the meaning of Tr(H0K) precise,

which can be defined by several equivalent formulations such as

Tr(H0K) =1

2β2

∫

R2N

∇x ·∇yk(X,Y)∣∣Y=X dX(5.39)

or

Tr(H0K) =1

2β2

∫

R2N

∑i

λi |Vϕi (X)|2dX(5.40)

wherek(X,Y) is the kernel associated toK, andλ1 ≥ λ2 ≥ ·· · are the eigenvaluesof K, while (ϕ1)i is the orthonormal basis of eigenfunctions ofK corresponding to(λi)i . Obviously, Tr(H0K) is linear inK, nonnegative, and possibly infinite onK1.


We first observe that ˆρN is determined by a variational problem.

PROPOSITION5.12 The operatorρN is the unique minimum of the convex func-tional FN over K1.

PROOF: We only sketch the proof, since this is a classical fact in quantummechanics. One possible proof consists in observing that the minimization ofFN

overK1 is equivalent to the following minimization problem:

min

1β ∑

i≥1

λi logλi + ∑i≥1

λi

∫

R2n

12β2 |Vϕi |2 +UN|ϕi|2dX : 0≤ λi ,∀i ≥ 1;

∑i≥1

λi = 1,(ϕi)i≥1 is an orthonormal basis ofL2

,

(5.41)

formulation, which amounts to writing anyK in K1 as∑i≥1λiϕi(X)ϕi(Y). Remark-ing thatUN is a potential that is bounded from below, grows at infinity, and belongsto Lp

loc(R2N) (for all 1≤ p<∞), one can easily deduce that up to permutations and

orthogonal transforms (in case of multiple eigenvalues), the Euler-Lagrange equa-tions are equivalent to requiring that(ϕi)i≥1 is an orthonormal basis of eigenvaluesof − 1

2β∆X +UN corresponding to eigenvaluesΛ1<Λ2 ≤ Λ3 · · · ≤ Λn−→n

+∞, andthat

λi =1

Z(N)e−Λi , Z(N) = ∑

i≥1

e−Λi .(5.42)

Therefore, the minimum is given by

1Z(N) ∑

i≥1

e−Λiϕi(x)ϕi(y) ,

which is nothing but the kernel representation ofe−1

2β ∆X+UNwith

Z(N) = ∑i≥1

e−Λi = Tr(

e−1

2β ∆X+UN)

=∫

R2N

pN(X,X,1)dX .

We may then consider the reduced operators onL2(R2k) for 1≤ j ≤ N, definedby the following kernels:

ρN, j(X,Y) =1

Z(N)

∫

R2(N− j)

ρN(X,zj+1, . . . ,zN;Y,zj+1, . . . ,zN)dzj+1 · · ·dzN .(5.43)

Let us observe that the law of(ω1(σ), . . . ,ω j(σ)) (for eachσ ∈ [0,1]) underµN

admits the densityρN, j(X,X) with respect to Lebesgue measureR2 j . In addition,

ρN, j ∈ K1(R2 j). And we have the following:


THEOREM 5.13 (i) There exist some positive constants C,δ > 0 independent of1≤ k≤ N such that we have

0≤ ρN,k(X,Y) ≤Cke−δ2 (|X|2+|Y|2) .(5.44)

(ii) As N goes to+∞, ρN,k converges in Lp(R2k×R2k) (for all 1 ≤ p< ∞), for

each k≥ 1, to ∏kj=1ρ(xj ,yj) whereρ is the kernel of the unique minimum of

the following strictly convex free energy functional:

minF(K) : K ∈ K1(R2)(5.45)

where

F(K) =1β

Tr(K logK)+Tr((H0 +V0)K)+a2

Tr(V1,2K ⊗K)(5.46)

where V0(x) = µ|x|2, V1,2 = − 12π log|x1−x2|, so that

Tr(V1,2(K⊗K)) = − 12π

∫∫

R2×R2

log|x−y|k(x,x)k(y,y)dxdy.

(iii) The minimumρ of F over K1 is given by

ρ(x,y) =p(x,y,1)∫

R2

p(x,x,1)dx

where p is the Green’s function of(5.5)andρ(x,x) ≡ ρ(x) on R2 whereρ is

the density determined in Theorem5.1.

Remark5.14. The mean field minimization problem (5.45)–(5.46) is nothing buta temperature-dependent Hartree model for bosons interacting with a logarithmicpotential in an external potential given byV0 (and ~

2

m = 1β2 · · · ). We refer the inter-

ested reader to Lions [16] for more mathematical details on temperature-dependentHartree or Hartree-Fock equations.

PROOF OFTHEOREM 5.13: It is possible to make a self-contained proof thatdoes not rely on any of the facts proved in the preceding section, but we shall notdo so here in order to restrict the length of this paper.

We begin with the proof of (5.44). SinceρN,k is the kernel of a nonnegativeself-adjoint operator, we have

|ρN,k(X,Y)| ≤ ρN,k(X,X)1/2ρN,k(Y,Y)1/2 for all X,Y ∈ R2k .

Next, we have for allϕ ∈Cb(R2k)∫

R2k

ϕρN,k(X,X)dX = EN[ϕ(ω1(0), . . . ,ωk(0))] = EN0 [ϕ(ω1(0), . . . ,ωk(0))hN]

= Ek0[ϕ(ω1(0), . . . ,ωk(0))hN,k] .


Hence, thanks to Proposition 5.10,∣∣∣∣∣∣∫

R2k

ϕρN,k dX

∣∣∣∣∣∣≤Ck0Ek

0[|ϕ(ω1(0), . . . ,ωk(0))|]

≤Ck∫

R2k

|ϕ(x1, . . . ,xk)|k

∏j=1

q(xj ,xj ,1)dx1 · · · dxk

≤Ck∫

R2k

|ϕ(x1, . . . ,xk)|e−δ|x|2 dx

and (5.44) follows.We next prove the convergence ofρN,k to ∏k

j=1ρ(xj ,yj) whereρ is the kerneldescribed in part (iii) of the above result. In order to do so, we first remark thatρN

is given by

ρN(X,Y) =1

Z(N)

∫dνβX,Y(ω)

exp

−[

aβ2N

N

∑i 6= j

V(ωi −ω j)+µN

∑i=1

V0(ωi)

].

(5.47)

Therefore, we have for all 1≤ k≤ N,

ρN,k(X,Y) =Z(N−k)

Z(n)

∫dνβX,Y(ω1, . . . ,ωk)e

−µk∑

i=1V0(ωi)

∫dµN−k

ωk+1,...,ωN)e− aβ

N

k∑

i=1

N∑

j≥k+1V(ωi−ω j )

,

(5.48)

whereµN−k is defined likeµN−k was, replacing

1N−k

n−k

∑i 6= j

V(ωi −ω j) by1N

N−k

∑i 6= j

V(ωi −ω j) .

Adapting easily the proof of Theorem 5.1 and of Corollary 5.3, we deduce that,in view of (5.44),ρN,k converges pointwise (and thus inLp) for all 1≤ p< ∞ toρk(X,Y) given by

ρk(X,Y) =1Zk

∫dνβx1,y1

(ω1) · · ·dνβxk,yk(ωk)e

−µk∑

i=1V0(ωi)

e−aβ

k∑

i=1Ψ(ωi)

(5.49)

with

Zk =∫

R2k

dX∫

dνβX,X(ω)e−µ

k∑

i=1V0(ωi)

e−aβ

k∑

i=1Ψ(ωi)

,


and

Ψ =(− 1

2πlog|x|

)∗ρ.

Hence,ρk(X,Y) = 1Zk ∏k

j=1 p(xj ,yj ,1), and we have proven our claims.Finally, the variational formulation (5.45)–(5.46) is derived in a similar fashion

to the proof of Proposition 5.12, yielding the following equivalent representationto a minimum:

k(x,y) =1Z ∑

i≥1

e−Λiϕi(x)ϕi(y)

where(ϕi)i≥1 are the eigenfunctions of(− 1β∆ + µ|x|2 + aΨ), that is, k(x,y) =

1Z p(x,y,1).

5.5 Convergence of Mean Field Densities for Broken Path Models

In this section, we briefly describe why the mean field measure on broken pathsρδ determined in Theorem 4.1 converges to the mean field measure on continuouspathsµ determined in Theorem 5.1, while the (invariant) densityρδ1 converges tothe densityρ asδ = 1

M goes to+∞. We shall not state a result, even though thestatements are easily deduced from the considerations that follow. And, we shallnot provide all the details of the proofs since the topic covered in this section ismore a consistency check than a real necessity for the statistical theories developedin this paper.

We claim thatρδ converges, asδ goes to 0, toµ in the sense made precise inSection 2.2 (i.e., asµN,δ converges toµN), and thatρδ1 converges toρ in Lp(R2) forall 1≤ p<∞ (for instance, the convergence is, in fact, stronger). In order to provethese claims, we need to introduce some notation. First of all, we denote by

µδ0 =1Rδ

exp

− β

2δ

M−1

∑σ=0

|xσ+1−xσ|2− δµ

2

M−1

∑σ=0

|xσ|2(

2πδ

β

)−M

,(5.50)

where

hδ =∫

R2M

exp

− β

2δ

M−1

∑σ=0

|xσ+1−xσ|2− δµ

2

M−1

∑σ=0

|xσ|2(

2πδ

β

)−M

(5.51)

dx0 · · ·dxM−1 ,

hδ =1Z′δ

exp

−aβδ

M−1

∑σ=0

Ψδ1(x

σ)− δµ

2

M−1

∑σ=0

|xσ|2

,(5.52)

where

Z′δ =

∫

R2n

hδdµδ0 ,(5.53)


and

Ψδ1 = − 1

2πlog|x| ∗ρδ1 .(5.54)

(Recall thatρδ1(x) =∫R2(n−1) ρδ(. . . ,xσ−1,x,xσ+1, . . .)dxσ, for all 0 ≤ σ ≤ M − 1,

wheredxσ0 denotes the integration with respect to allxσ butxσ0.)We begin with a few straightforward observations. First of all,µδ0 converges, as

δ goes to 0, toµ0 (in the sense made precise in Section 2.2 above). Once more, thisis a more or less standard consequence of Trotter’s formula. Next, we check, as wedid in Section 5, thathδ is the unique minimum of the following convex variationalproblem:

Fδ = min

Fδ(h) : h≥ 0, h∈ L∞(R2M),

∫hdµδ0 = 1

(5.55)

where

Fδ(h) =1β

∫hloghdµδ0 +

µ

2β

∫ M−1

∑σ=0

δV0(xσ)hdµδ0

+a2

∫∫ M−1

∑σ=0

δV(xσ−yσ)h(x)h(y)dµδ0(x)dµδ0(y) .

(5.56)

As noticed above, each of the three terms definingFδ is nonnegative on theminimization class defined in (5.55). Choosing, for instance,h≡ 1, so that

Fδ(1)−→δ

µ

2βE0(V0)+

a2

E20(V(ω−ω′)),

we deduce the following a priori bounds:∫

hδ loghδ dµδ0 ≤C(5.57)

∫

R2

|x|2ρ21(x)dx=

M−1

∑σ=0

δ

∫|xσ|2hdµδ0 ≤C(5.58)

(5.59)∫∫

R2×R2

V(x−y)ρδ1(x)ρδ1(y)dxdy=

∫∫ M−1

∑σ=0

δV(xσ−yσ)h(x)h(y)dµδ0(x)dµδ0(y) ≤C,

whereC denotes, here and below, various positive constants independent ofδ.These bounds allow us to obtain some bound onΨδ. Indeed, on one hand, we


have for allx∈ R2

Ψδ(x) = − 12π

∫

R2

log|x−y|ρδ1(y)dy≥− 12π

∫

R2

|x−y|ρδ1(y)dy

≥− |x|2π

− 12π

∫

R2

|y|ρδ1(y)dy≥−C(|x|+1) ,

in view of (5.58), and, on the other hand, we have

Ψδ(x) = − 12π

∫hδ log|x−x0|dµδ0

≤ 12π

∫hδ(

log1

|x−x0|)

1|x−x0|≤1dµδ0

≤C∫

hδ loghδ dµδ0 +C∫

1|x−x0|ν dµδ0 ≤C,

for someν > 0 small enough(0< ν < 2), where we used the estimate (5.57). Thisbound allows us to obtain the following estimate onhδ:

0≤ hδ ≤C in R2M ,(5.60)

and we deduce for someα > 0 independent ofδ,

0≤ ρδ1 ≤Ce−α|x|2

onR2 .(5.61)

Indeed, we have for anyϕ ∈C∞0 (R2), ϕ≥ 0,∫

R2

ϕ(x)ρδ1(x)dx=∫

R2M

ϕ(x0)ρδ(x0, . . . ,xM)dx0 · · ·dxM−1

=∫

R2M

ϕ(x0)hδµδ0 dx0 · · ·dxM−1

≤C∫

R2

ϕ(x0)(∫

µδ0 dx1 · · ·dxM−1)

dx0

≤C∫

R2

ϕ(x0)e−α|x0|2 dx0 ,

with some straightforward computation of Gaussian integrals that we skip.

Once these crucial bounds are obtained, several proofs are possible. First of all,extracting a subsequence if necessary, we may assume thatρδ1 converges weakly inLp (for all 1≤ p< ∞) to someρ1, which satisfies (5.61). Because of (5.61),Ψδ

1

converges inW2,ploc to Ψ1 = 1

2π log|x| ∗ρ1, VΨδ1 converges inLq(R2) to VΨ1 for all

1≤ q≤ ∞, andΨδ1 converges toΨ1 in C1,α(R2) for all 0≤ α < 1. Then one may

simply use Trotter’s formula to complete the proofs of our claims by showing thatρ1 = ρ (andΨ1 = Ψ) becauseρ1 satisfies (5.4) and (5.5).


Another possible argument consists in passing to the limit in the variationalproblem (5.55) that “goes” to the variational problem introduced in Theorem 5.4(Section 5.1 above). Indeed, by a single approximation procedure, we may checkthat

limsupδ

Fδ ≤ F.

On the other hand, denoting byh the weak limit ofhδ (or of a subsequence),

Fδ = Fδ(hδ) =1β

∫hδ loghδ dµδ0 +

µ

2β

∫V0(x)ρδ1dx

+a2

∫∫V(x−y)ρδ1(x)ρ

δ1(y)dxdy.

Hence, lettingδ go to 0+, we deduce

liminfδ

Fδ ≥ 1β

F0(hlogh)+µ

2β

∫V0(x)ρ1dx

+a2

∫∫V(x−y)ρ1(x)ρ1(y)dxdy

= F(h) ≥ F,

sinceh− 1Z′ exp(−µ

2V0(ω)−aβΨ1(ω)) where

Z′ = E0

(exp(−µ

2V0(ω)−aβΨ1(ω)

)).

Therefore, Fδ converges to F, and thush = h and∫

hδ loghδdµδ0 converges toF0(hlogh). We then deduce from (5.52) the strong convergence ofhδ to h (extend-ing hδ to a “continuous path” function as we did in Section 2.2), and thus the strongconvergence ofρδ1 to ρ.

6 Alternative Formulations for the Mean Field Equations

We have seen in the previous sections two variational formulations of the meanfield problems. The first one, in Theorem 5.4, yields the mean field measure onpathsµ or, more precisely, the mean field (Radon-Nykodym) densityh with respectto µ0. The second one, in Theorem 5.13, yields a direct variational determinationof the invariant density onR2, ρ. We present in this section one more variationalformulation that allows one to determine directly the potentialΨ created byρ,namely, 1

2π log|x|∗ρ. This formulation is, in a sense we do not wish to make precisehere, a dual convex problem to the “formulation inρ” introduced in Theorem 5.13.It is also the analogue of a formulation introduced in Caglioti et al. [3], at leastin the case of two-dimensional point vortices in a bounded region (with no-slipboundary conditions). In the case of the whole planeR

2 (for two-dimensionalpoint vortices), the logarithmic divergence ofΨ at infinity makes the adaptation ofthis formulation rather delicate. This difficulty is circumvented in Lions [17]. Weshall follow the same approach to take care of the fact that∇Ψ /∈ L2(R2) while


introducing a new variational problem associated to the mean field limit for three-dimensional vortex filaments.

In order to keep the ideas clear, we first present formally the variational formu-lation, ignoring the lack of integrability of|∇Ψ|2. Afterwards, we detail the neces-sary mathematical (simple) machinery that allows us to formulate this variationalproblem rigorously. Thus we wish to emphasize the fact that the functional we aregoing to write now, strictly speaking, does not make sense! With this convention,we may now introduce

G(φ) =12

∫

R2

|∇φ|2dx+1

aβlog

Tr[e−(− 1

2β ∆+µ|x|2+aβφ)] .(6.1)

We claim that, at least formally,Ψ is the “unique minimum ofG.” In order toconvince ourselves that this is indeed the case, we only need to explain thatΨ isa solution of the Euler-Lagrange equation associated to (6.1) and thatG is convex,i.e., thatG2(φ) = logTr[e−(H+aβφ)] is a convex functional ofφ, where we denoteby H = − 1

2β∆ +µ|x|2. These verifications, in turn, depend upon the computation

of a directional derivative ofG2 of Ψ in some directionφ (whereφ ∈C∞0 (R2), for

instance). Identifyingφ with the multiplication operator (byφ) whose kernel isgiven byφ(y)δ0(x−y), we easily check that we have, asε goes to 0,

1ε(G(Ψ+εφ)−G(Ψ)) →−(Tr

e−(H+aβΨ))−1

aβTr(e−(H+aβΨ)φ

)= −aβ

(Tr

e−(H+aβΨ))−1∫

R2

p(x,x,1)φ(x)dx.

Hence, the Euler-Lagrange equation associated to (6.1) is nothing but

−∆Ψ =p(x,x,1)∫

R2

p(z,z,1)dzonR

2 ,(6.2)

i.e., precisely the equation we expected (see (5.4) and (5.5)).

Next, we prove thatG2 is convex (and, in fact, strictly convex modulo the ad-dition of constants) forφ ∈ L∞

loc(R2), φ(x)/ log(1+ |x|) ∈ L∞(R2). Let φ1,φ2 ∈

L∞loc(R

2), φ1/ log(1+ |x|), φ2/ log(1+ |x|)∈ L∞(R2), φ1 6≡ φ2 up to a constant, andlet θ ∈ (0,1). We denote byp1, p2, andp the Green’s function associated to, re-spectively,H +aβφ1, H +aβφ2, andH +aβφ whereφ= θφ1 +(1−θ)φ2, and weclaim that

p(x,y,t)< (p1(x,y,t))θ(p2(x,y,t))1−θ = p onR2×R

2× [0,1] .


If this inequality holds, we deduce immediately

G2(φ) = log∫

R2

p(x,x,1)dx< log∫

R2

p(x,x,1)dx

≤ log

∫

R2

p1(x,x,1)dx

θ ∫

R2

p2(x,x,1)dx

1−θ= θG2(φ1)+(1−θ)G2(φ2) .

Finally, the above inequality follows from the (strong) maximum principle and thefollowing computation:

∂p∂t

− 12β

∆p+µ|x|2p

= θ

(p2

p1

)1−θ(∂p∂r

− 12β

∆p1 +µ|x|2p1

)+(1−θ)

(p1

p2

)θ·(

∂p2

∂t− 1

2β∆p2 + p|x|2p2

)+θ(1−θ)

2β

[|∇p1|2 p1−θ

2

p2−θ1

+ |∇p2|2 pθ1p1+θ

2

−2∇p1 ·∇p2

p1−θ1 pθ2

]

≥−θ(

p2

p1

)1−θ(aβφ1p1)− (1−θ)

(p1

p2

)θ(aβφ2p2) = −aβφp

and the equality holds if and only if∇ logp1 = ∇ logp2; hence the strict inequalityis shown unlessp1 ≡ p2ect andφ1 ≡ φ2− c

aβ .Having thus “checked formally” the above variational formulation, we may

now turn to make it precise and mathematically rigorous. In order to do so, weintroduce (for instance)φ0(x) = − 1

2π logmax(|x|,1) so that we have

−∆φ0 =1

2πδS1 onR

2 ,

and we easily check thatΨ − φ0 decays at infinity like 1/|x| while ∇(Ψ− φ0)decays like 1/|x|2, and thus∇(Ψ−φ0) ∈ L2(R2). We next define a “corrected”functionalG by

G(φ) =12

∫

R2

|∇(φ−φ0)|2dx+1

aβlog[Tre−(H+aβφ)](6.3)

on the spaceφ ∈ H1loc(R

2), φ−φ0 ∈ L∞(R2), ∇(φ−φ0) ∈ L2(R2)—in fact, witha little more work, we may even get rid of the constraintφ− φ0 ∈ L∞(R2). Adifficulty remains, however: We need to normalizeφ sinceG is not bounded from


below, sinceG(φ+C) = G(φ)−C for all C ∈ R! Thus we choose the followingnormalization: ∫

S1

φds= 0,(6.4)

which makes sense sinceφ ∈ H1loc(R

2). In conclusion, we define the followingminimization class:

M =

φ ∈ H1loc(R

2), φ−φ0 ∈ L∞(R2), ∇(φ−φ0) ∈ L2(R2),∫

S1

φds= 0

.(6.5)

We have the following:

THEOREM 6.1 The normalized potential(Ψ− ∫-S1Ψds) is the unique minimum of

the strictly convex functionalG over the setM .

Remark6.2. In other words, the functionalG over the setM allows one to identify(Ψ− ∫

-S1Ψds) and thus the mean field density(ρ = −∆Ψ) = −∆(Ψ− ∫-S1Ψds).

Therefore, sinceΨ = − 12π log|a| ∗ρ, it allows us to identify the full potentialΨ as

well.

PROOF OFTHEOREM 6.1: First of all, the argument made above shows thatG2 is convex, and even strictly convex, onM , while the first term in the definitionof G, namely,

∫R2 |∇(φ−φ0)|2dx, is obviously strictly convex onM .

Next, if we fix ψ ∈ M , we may adapt the formal argument made above anddeduce that the Euler-Lagrange equation atψ reads, denoting by ¯p the Green’sfunction associated toH +aβΨ,

∫

R2

∇(Ψ−φ0) ·∇φdx=∫

R2

p(x,x,1)∫R2

p(z,z,1)dz

φ(x)dx(6.6)

for all φ ∈ L∞(R2) such that∇φ ∈ L2(R2) and∫

S1φds= 0. Next, we observethat

∫R2 ∇φ0 ·∇φdx =

∫-S1φ ds= 0, at least if

∫|x|≥1 |∇φ|/|x|dx< ∞. Hence, the

preceding Euler-Lagrange equation implies, in particular,

∫

R2

∇Ψ ·∇φdx=∫

R2

p(x,x,1)∫R2

p(z,z,1)dz

φ(x)dx

for all φ ∈ L∞(R2) such that∇φ ∈ L2(R2),∫

S1φds= 0, and∫|x|≥1 |∇φ|/|x|dx<∞.

This equation is obviously satisfied byΨ = Ψ − ∫-S1Ψds. Therefore, (6.6)

holds withΨ = Ψ− ∫-S1Ψds, providedφ ∈ L∞(R2), ∇φ ∈ L2(R2),

∫S1φds= 0,

and∫|x|≥1 |∇φ(x)|/|x|dx< ∞.


At this stage, there only remains to show, by a truncation argument, that thisequation holds, in fact, for allφ ∈ L∞(R2) such that∇φ ∈ L2(R2) and

∫S1φds= 0.

In order to do so, we considerφn = φζ(x/a) for n≥ 1, whereζ ∈C∞0 (R2), 0≤ ζ ≤

1, onR2, ζ ≡ 1 onB1, andζ ≡ 0 for |x| ≥ 2. We only need to show that∫

R2

∇(Ψ−φ0) ·∇φndx−→n

∫

R2

∇(Ψ−φ0) ·∇φdx.

This is immediate since we have, denoting byC various positive constants inde-pendent ofn≥ 1,∣∣∣∣∣∣

∫

R2

∇(Ψ−φ0) ·∇ζnφdx

∣∣∣∣∣∣≤ Cn

∫

R2

11+ |x|2 1(n≤|x|≤2n) dx≤ C

nlog

1+4n2

1+n2 ≤ Cn.

7 The Current and Some Scaling Limits

In this section, we consider and study various problems related to what we didabove. First of all, in Section 7.1, we go back to the issue of Gibbs measuresinvolving the conserved quantityC defined in (1.6) that we called the current andwe explain how to adapt everything we did before to that general case. Next, inSection 7.2, we consider infinite-length filaments. Finally, in Section 7.3, we studyvarious asymptotic limits for the mean field equations.

7.1 CurrentsHere we consider the Gibbs measuresµN defined formally by (2.1) using the

same normalization (see (2.3)) as in Section 2 and the same scaling (see (3.3)) as inSection 3. Also, as in Section 2, we restrict the parametersβ, µ, andv by requiring

v2 < 2βµ ,(7.1)

which allows us to define the measureµN properly. The precise mathematical def-inition of µN is exactly the same as in Section 2 provided we replace the equation(2.12) by

∂p∂t

− 12β

N

∑j=1

∆xj p−(βa2

N

∑i 6= j

log|Xi −Xj |)

p

+N

∑j=1

(λ ·Xj +µ|Xj |2)p− v

β(JXj ,∇xj p)

= 0 in R

2N × (0,1) ,


(7.2)

We may also defineµN by its densityhN with respect to a “background” proba-bility measure onN independent periodic paths inRN denoted byµN

0 so that (2.13)remains true. The measureµN

0 is defined as in Section 2 replacing the Gaussian


kernelq (see (2.17)) by the Green’s function (which is still a Gaussian kernel thatcan be computed) of the same equation as (7.2) with ¯a = 0 andµ replaced byµ′whereµ′ is chosen in(0,µ) in such a way thatv2 < 2βµ′ (and thus replacingµ/2by µ−µ′ in the definition ofhN).

Exactly as in Section 7.2, one can recover the Gibbs measureµN from the cor-responding measure on broken pathsµN,δ. The heart of the matter is the follow-ing easy computation, which also sheds some light on the new term appearing inequation (7.2), namely,vβ (Jx,∇xp). Indeed, we consider, forδ ∈ (0,1) and for

(x0,x1) ∈ (R2)2, the quantity

G(δ) =(

2πδβ

)−1

exp

−β (x1−x0)2

2h−v(Jx0,x1−x0)−λ ·x0δ−µ|x0|2δ

,

(7.3)

and we claim that we haveG(δ)− δx0(x1)

δ

12β

∆δx0(x1)− (λ ·x0 +µ|x0|2)δx0(x1)

+v

β(Jx0,∇δx0(x1))

(7.4)

(in the sense of distributions) asδ goes to 0+. Indeed, we have for anyϕ ∈C∞0 (R2)

∫

R2

G(δ)ϕ(x1)−ϕ(x0)δ

dx1 =∫

R2

dze−

|z|22

2πϕ

(x0 +

√δ

βz

)exp

−[v

√δ

β(Jx0,z)+λ ·x0δ+µ|x0|2δ

]−ϕ(x0)

,

and, by a trivial expansion, we deduce∫

R2

G(δ)ϕ(x1)−ϕ(x0)δ

dx1

−→δ

∫

R2

dze−

|z|22

2π

1

2β

2

∑i, j=1

∂i jϕ(x0)zizj − v

β(Jx0,z) · (z,∇ϕ(x0))

−λ ·x0ϕ(x0)−µ(x0)2ϕ(x0)

=1

2β∆ϕ(x0)− v

β(Jx0,∇ϕ(x0))(λ ·x0 +µ|x0|2)ϕ(x0) ,

and our claim is shown.Having thus defined the Gibbs measuresµN, we may now consider the limit as

N goes to+∞ (choosing ¯a = a/2πN). Then we claim that Theorems 5.1 and 5.4


together with Corollary 5.3 hold, with a few modifications, namely, we skip therepresentation (5.2) and equation (5.5) is replaced by

∂p∂t

− 12β

∆p− aβ2π

(log|x| ∗ρ)p+(µ|x|2+λx)p− σ

β(Jx,∇xp) = 0

onR2× (0,1) ,

p|t=0 = δy(x) onR2 .

(7.5)

In all terms involving the Radon-Nykodym densities,µ/2 is to be replaced byµ−µ′. The proofs made in Sections 5.2 and 5.3 can then be copied mutatis mutandis.

However, the introduction of the currentC leads to a Green’s functionp(x,y,t)that is not symmetric in(x,y) (the time reversal symmetry is broken), and this isprobably why we are not aware of any variational formulation for the densityρ, thepotentialΨ = − 1

2π log|x| ∗ρ, or the kernel

ρ(x,y) =p(x,y,1)∫

R2 p(z,z,1)dz

analogous to the variational formulations that were developed in Section 5.4 andSection 6 in the case whenv = 0.

7.2 Infinite-Length Filaments

Here we consider another variant where we allow an infinite length for thevortex filaments, or, in other words, we wish to letL go to+∞. Since our originalformulation of the Gibbs measures used a scaling argument leading to a normalizedlengthL = 1, we have to go back to the definition ofµN, leaving out explicitly thedependence upon the lengthL of the filaments, i.e.,

µN =1Z

exp(−βH −µI)dX1 · · ·dXn(7.6)

whereβ,µ > 0 and we take, in order to simplify notation,λ= v = 0 (even thougheverything we do below can be generalized toλ 6= 0, |v|<√

2βµ), and

H (X) =12

∫ L

0

N

∑j=1

∣∣∣∣∂Xj

∂σ

∣∣∣∣2 dσ+12

a∫ L

0

N

∑j 6=k

− log|Xj(σ)−Xk(σ)|dσ ,(7.7)

I(X) =∫ L

0

N

∑j=1

|Xj(σ)|2dσ .(7.8)

Then everything we did above applies, provided, of course, that we replaceeverywhere the final time 1 byL. In particular,µN is defined precisely by itsaction upon any bounded continuous function onR

2Nm F = F(Ω(t1), . . . ,Ω(tm))


(m≥ 0, 0≤ t1 < t2 < · · ·< tm ≤ 1), namely,

∫F dµN =

1Z

∫

(R2N)m

dX F(X1, . . . ,Xm)p(X1,X2,t2− t1)

· · · p(Xm−1,Xm,tm− tm−1)p(Xm,X1,L− (tm− t1))

Z =∫

R2N

p(X,X,L)dX ,

(7.9)

wherep is the Green’s function of the PDE (2.12) (withλ = 0), and the law ofΩ(t), for anyt ∈ [0,L], admits a density, with respect to the Lebesgue measure onR

2N, given by

ρN(X) =p(X,X,L)

Z.(7.10)

We next wish to sendL to+∞, and we denote byµNL andρN

L the above quantitiesto recall the dependence uponL. In order to understand the asymptotics inL, oneneeds to introduce the complete set of normalized eigenfunctionsφN

k (1 ≤ k) inL2(R2N) of the Schrödinger operator

− 12β

∆− βa2

N

∑i 6= j

log|Xi −Xj |+µN

∑j=1

|Xj |2 ,(7.11)

and we denote byλNk (1≤ k) the corresponding eigenvalues withλN

1 ≤ λN2 ≤ ·· · ≤

λNk −→k + ∞. Let us also recall thatλN

1 is simple (i.e.,λN1 < λN

2 ) and that we maychooseφN

1 to be positive onR2N. With this notation, we have for allX,Y ∈ R2N,

t > 0,

pN(X,Y,t) = ∑k≥1

e−λNk tφN

k (x)φNk (Y) .(7.12)

In particular, one may easily check that we have for anyt ≥ 0

pN(X,Y,L− t) = e−λN1 Leλ

N1 tφN

1 (x)φN1 (y)

+o(e−λ

N1 L) ,

ZL = e−λN1 L +o

(e−λ

N1 L) ,

(7.13)

where the remainder termo(e−λN1 t) is small inL1∩L∞(R2N ×R

2N).This allows us to deduce thatµN

L converges weakly (in the sense of probabilitymeasures) to the probability measure onC([0,∞);R2)N defined by, for all boundedcontinuous functions onR2NmF = F(Ω(t1), . . . ,Ω(tm)) (m≥ 1, 0≤ t1< t2< · · ·<


tm<+∞), the following expression:∫F dµN =

∫

R2Nm

dX F(X1, . . . ,Xm)p(X1,X2,t2− t1)

· · · p(Xm−1,Xm,tm− tm−1)eλN1 (tm−t1)φN

1 (X1)φN1 (Xm) .

(7.14)

In addition,ρN1 converges, asL goes to+∞, toρN(X) = (φN

1 (X))2 in L1∩L∞(R2N)(for instance), andρN is the density of the law ofΩ(t) (under the probability mea-sureµN) for all t ≥ 0.

We may now turn to the limit asN goes to+∞ under the scaling ¯a = a/2πN.The analysis of the behavior ofµN is somewhat intricate, and this is why we onlyconsider the behavior of the law ofΩ(t) (∀t ≥ 0), i.e., the behavior ofρN(X) asNgoes to+∞. Before we do so, we first argue formally in order to guess the rightanswer by commuting limits, that is, first lettingN go to +∞, in which case werecover the mean field problems studied and justified in the preceding sections, andthen lettingL go to+∞. The mean field lawµ = µL is defined (see Theorem 5.1)by

∫F dµL =

1ZL

∫

R2m

dxF(x1, . . . ,xm)pL(x1,x2,t2− t1)(7.15)

· · · pL(xm−1,xm,tm− tm−1)pL(xm,x1,L− (tm− t1))

for anyF = F(ω(t1), . . . ,ω(tm)), F bounded and continuous onR2m, m≥ 1, 0≤t1 < t2 < · · · ≤ tm ≤ L, wherep is the Green’s function of

∂pL

∂t− 1

2β∆pL +

(−aβ

2πlog|x| ∗ρL

)pL +µ|x|2pL = 0 in R

2× (0,L)

pL|t=0 = δy(x) in R2 ,

(7.16)

and

ρL(x) = pL(x,x,L)/ZL ,(7.17)

ZL =∫

R2

pL(z,z,L)dz.(7.18)

Let us recall thatρL is the density of the mean field law ofω(t) for all t ∈ [0,L].We now letL go to +∞ and argue formally, although a rigorous argument,

which we skip for the sake of brevity, is possible. We thus assume thatρL con-verges, asL goes to+∞, to some probability densityρ on R

2. Then, denoting byφ1 the first eigenfunction, which we take to be normalized inL2(R2) and positive,of the Schrödinger operator[− 1

2β∆ +(−aβ2π log|x| ∗ρ)+µ|x|2] and byλ1 the cor-

responding (simple) eigenvalue, we deduce by a similar argument to the one made


above thatµL “converges” to a probability measure onC([0,∞);R2) defined by∫F dµ=

∫

R2m

dxF(x1, . . . ,xm)p(x1,x2,t2− t1)(7.19)

· · · p(xm−1,xm,tm− tm−1)φ1(xm)φ1(x1)eλ1(tm−t1)

for anyF = F(ω(t1), . . . ,ω(tm)), F bounded and continuous onR2m, m≥ 1, 0≤t1 < t2 < · · ·< tm, wherep is the Green’s function of

∂p∂t

− 12β

∆p+(−aβ

2πlog|x| ∗ρ

)p+µ|x|2p = 0 in R

2× (0,∞)

p|t=0 = δy(x) in R2

(7.20)

and

ρ(x) = (φ1(x))2 onR2 .(7.21)

Of course,ρ is the density of the law ofω(t) for all t ≥ 0 under the probabilitymeasureµ.

In other words, the limit measureµ is entirely determined by the probabilitydensityρ= φ2

1 onR2 that solves the following Hartree equation:

− 12β

∆φ1 +(−aβ

2πlog|x| ∗φ2

1

)φ1 +µ|x|2φ1 = λ1φ1 in R

2

φ1 > 0 onR2 ,

∫

R2

φ21dx= 1.

(7.22)

We expectρ to be the minimum of the following strictly convex functional:

Λ = min

Λ(ρ) : ρ ∈ L1(R2), ρ|x|2 ∈ L1(R2), ρ≥ 0 onR2,(7.23)

∫

R2

ρdx= 1,√ρ ∈ H1(R2)

,

with

Λ(ρ) =∫

R2

12β

|∇√ρ|2 +µ|x|2ρdx− aβ

4π

∫∫

R2×R2

log|x−y|ρ(x)ρ(y)dxdy.(7.24)

As is well-known, the first term of the energylike functionalΛ, namely, 12βS1(ρ),

whereS1(ρ) =∫R2 |∇√

ρ|2dx, is convex inρ. S1 is the so-called Fisher informa-tion functional in information theory, Linnik functional in kinetic theory, and thevon Weiszäcker correction for kinetic energy in density-dependent quantum mod-els.


Having thus determined formally the mean field limit, we now turn to a rigorousproof of the “convergence,” asN goes to+∞, of ρN = (φN

1 )2 to ρ = (φ1)2 (or“products ofρ”). More precisely, we introduce, for each 1≤ k≤N, ρN,k the densityof the law of(ω1(t), . . . ,ωk(t)) (∀t ≥ 0)

ρN,k(x1, . . . ,xk) =∫

R2(N−k)

ρN(x1, . . . ,xk,xk+1, . . . ,xN)dxk+1 · · ·dxN.(7.25)

We may state our main result on the mean field limit for infinite-length vortexfilaments.

THEOREM 7.1 For each k≥ 1, ρN,k converges in L1 ∩ Lk

k−1 (R2k), as N goes to+∞, to ρk = ∏k

i=1ρ(xi) and√ρN,k converges in H1 to

√ρk, whereρ = (φ1)2 is

the unique minimum of the Hartree variational problem(7.23)–(7.24), andφ1 issmooth, decays rapidly at infinity, and solves(7.22). Furthermore,λN

1 /N −→ Λ asN goes to+∞.

PROOF OFTHEOREM 7.1: We first recall thatφN1 is the unique minimum (up

to a change of sign) of

λN1 = min

∫

R2N

12β

|∇φ|2 +

(µ

N

∑j=1

|xj |2− aβ4πN

N

∑i 6= j

log|xi −xj |)φ2dx :

|x|φ ∈ L2(R2N), φ ∈ H1(R2N),∫

R2n

φ2dx= 1

;

thereforeρN = (φN1 )2 is the unique minimum of the following convex problem:

λN1 = min

12β

S1(ρ)+∫

R2N

(µ

N

∑j=1

|xj |2− aβ4πN

N

∑i 6= j

log|xi −xj |)ρdx :

r ∈ L1(RN), ρ≥ 0 onRN,

√ρ ∈ H1(R2N), ρ|x|2 ∈ L1(R2N),

∫

R2N

ρdx= 1

.(7.26)

In the course of proving Theorem 7.1, we shall need some properties of thefunctionalS1 that we isolate in the next lemma, whose proof is postponed until theconclusion of the proof of Theorem 7.1.

LEMMA 7.2 Letρ≥ 0∈ L1(Rn).

(i) Let k∈ 1,. . . ,n. We denote by

ρk =∫ρ(x1, . . . ,xk,xk+1, . . . ,xn)dxj+1 · · ·dxn


and by

ρn−k =∫ρ(x1, . . . ,xk,xk+1, . . . ,xn)dx1 · · ·dxk .

Then we have

S1(ρ) ≥ S1(ρk)+S1(ρn−k) .(7.27)

(ii) We denote byρt = ρ∗((2πt)−n/2e−|x|2/2t) for t > 0. Then, we have, assuming(for instance) thatρ|x|δ ∈ L1(Rn) for someδ > 0,

2S1(ρ) = supt>0

1tS0(ρ)−S0(ρt)(7.28)

where S0(ρ) =∫Rn ρ logρdx.

We first show thatλN1 /N is bounded and thatS1(ρN,k) and

∫R2k ρN,k(x)|x|2dxare

bounded for eachk ≥ 1. First of all, letρ be an element of the minimizing classdefined in (7.23); then, introducingρ(x1, . . . ,xN) = ∏N

i=1ρ(xi), we have, in view of(7.26),

λN1 ≤ NΛ(ρ)+

aβ2πN

∫

R2

(log|x| ∗ρ)ρdx;

hence

limN

λN1

N≤ Λ(ρ)

or

limN

λN1

N≤ Λ .(7.29)

On the other hand, we have

λN1 ≥ min

12β

∫

R2N

|∇φ|2dx+∫

R2N

(µ

N

∑j=1

|xj |2− aβ4πN

N

∑i 6= j

|xi −xj |)

dx :

|x|φ ∈ L2(R2N), φ ∈ H1(R2N),∫

R2N

φ2dx= 1

≥−CN

sinceµ∑Nj=1 |xj |2− aβ

4πN ∑Ni 6= j |xi − xj | ≥ −CN on R

2N, where, here and below, wedenote byC various positive constants independent ofN.

The preceding argument also yields immediately

1N

∫

R2N

N

∑j=1

|xj |2ρN dx=∫

R2

|x|2ρN,1dx≤C,(7.30)

1N

∫

R2N

|∇√ρN|2dx≤C.(7.31)


Indeed, we just need to observe thatµ2 ∑N

j=1 |xj |2 − aβ4πN ∑N

i 6= j |xi − xj | ≥ −CN onR

2N; hence

12β

∫

R2N

|∇√ρN|2dx+

µ

2

∫

R2N

N

∑j=1

|xj |2ρN dx≤ λN1 +CN≤CN.

Our claim then follows since we have for eachk≥ 1∫

R2k

|x|2ρN,k dx= k∫

R2

|x|2ρN,1dx≤Ck

while (7.27) implies

CN≥ S1(ρN) ≥[

Nk

]S1(ρN,k) .

At this stage, we may now letN go to+∞. We first observe that, by a standarddiagonal procedure, we may extract a subsequence, still denoted byρN, to simplifynotation such that, by Sobolev imbeddings,

ρN,kNρk weakly inL1∩L

kk−1(R

2k)and∇

√ρk ∈ L2(R2k), ρk|x|2 ∈ L1(R2k),

∫R2k ρk dx= 1, ρk is symmetric in(x1, . . . ,

xk), ρk =∫R2 ρk+1(x1, . . . ,xk,xk+1)dxk+1, and for eachk≥ 1

1k

S1(ρk) ≤ limN

1k

S1(ρN,k) ≤ limN

1N

S1(ρN) .(7.32)

In addition, we have, in view of the above bounds and convergences,

1NλN

1 − 12β

1N

S1(ρN) = µ∫

R2

|x|2ρ1dx− aβ2π

∫∫

R2×R2

log|x−y|ρ2(x,y)dxdy.(7.33)

Finally, using the Hewitt-Savage theorem again, we obtain a probability measureπ on the set of probability measures onR

2 such that

ρk =∫ k

∏j=1

ρ(xj)dπ(ρ) for all k≥ 1.

Next we have, in view of (7.32) and of the convexity ofS,

1k

S1(ρk) ≤∫

S1(ρ)dπ(ρ) ≤ +∞ for all k≥ 1.

On the other hand, we claim that

limk

1k

S1(ρk) =∫

S1(ρ)dπ(ρ) .

This is indeed a straightforward consequence of part (ii) of Lemma 7.2, observingthat we have for allt > 0

1k

S1(ρk) ≥ 12t

1k

S0(ρk)−S0(ρk

t )

;


hence

limk

1k

S1(ρk) ≥ 12t

∫S0(ρ)dπ(ρ)− 1

2t

∫S0(ρt)dπ(ρ)

=∫

12tS0(ρ)−S0(ρt)dπ(ρ) ,

and we easily conclude the proof.The above arguments show, in particular, thatS1(ρ)+

∫R2 ρ|x|2dx< ∞ π-a.s.,

and we deduce from (7.33)

limN

λN1

N≥

∫Λ(ρ)dπ(ρ) .(7.34)

Comparing (7.29) and (7.34), we deduce that limNλN1 /N = Λ and thatΛ(ρ) = Λ π-

a.s. SinceΛ is strictly convex, there is a unique minimumρ andπ = δρ. Thereforewe haveρk = ∏k

j=1ρ(xj) and 1NS1(ρN)−→N S1(ρ) = 1

kS1(ρk). Furthermore, wededuce from (7.32) that

∇√ρN,k−→

N∇√ρk

strongly inL2(R2k). We then easily conclude the proof of the convergence part ofTheorem 7.1.

The smoothness and the decay ofφ1 =√ρ follows immediately from ellip-

tic regularity after writing the Euler-Lagrange equation of (7.23)–(7.24) recast interms of

√ρ, namely,

Λ = min

∫

R2

12β

|∇φ|2 +µ|x|2φ2dx

+aβ2

∫∫

R2×R2

(− 1

2πlog|x−y|

)φ2(x)φ2(y)dxdy:

φ ∈ H2(R2), φ|x| ∈ L2(R2),∫

R2

|φ|2dx= 1

.Indeed, the corresponding Euler-Lagrange equation is nothing but (7.22), and weconclude the proof.

PROOF OFLEMMA 7.2: (i) We begin with the case whenρ is smooth, decaysrapidly at infinity, and is strictly positive onRn. SinceS1 is convex and positivelyhomogeneous of degree 1, we have

S1(ρ) ≥ S′1(ρk ·ρn−k) ·ρ

=∫

Rn

∇(√

ρk

√ρn−k

)·(

1√ρk

√ρn−k

)dx


=∫

Rn

∇√ρk ·∇

(ρ√ρk

)dx+

∫

Rn

∇√ρn−k ·∇

(ρ√ρn−k

)dx

=∫

Rk

∣∣∇√ρk∣∣2dx+

∫

Rn−k

∣∣∣∇√ρn−k∣∣∣2dx= S1(ρk)+S1

(ρn−k) .

For a generalρ such that∇√ρ ∈ L2(Rn), we approximateϕ =

√ρ in H1(Rn)

byϕε, which is smooth, strictly positive, and rapidly decreasing at infinity, and wesetρε = ϕ2

ε. Thus,ρε−→ερ in L1(Rn), and(ρε)k and(ρε)n−k converge inL1(Rk)

andL1(Rn−k), respectively. Therefore, we have

S1(ρ) = limε

S1(ρε) ≥ limε

S1((ρε)k)+ limε

S1((ρε)n−k)

≥ S1(ρk)+S1(ρn−k)

sinceS1 is convex. The inequality (7.27) is shown in full generality.

(ii) We first observe thatρt logρt ∈ L1(Rn) and thatρ(logρ)− ∈ L1(Rn) so thatS0(ρ)−S0(ρt) makes sense inR∪+∞. Indeed, on one hand, we have a.e. onR

n

ρ(logρ+ |x|δ)+e−|x|δ −ρ≥ 0;

henceρ(logρ)− ∈ L1(Rn). On the other hand,ρt ∈ L∞(Rn), and it is easily checkedthatρt |x|δ ∈ L1(Rn). Hence, as before,

ρt(logρt)− ∈ L1(Rn) while ρt(logρt)+ ≤ (log‖ρt‖L∞(Rn))+ρt ∈ L1(R2) .

Next, we consider first the case whenρ is smooth, rapidly decreasing at infin-ity, and strictly positive onRn. We then compute (this is, by the way, a classicalcomputation in kinetic theory)

ddt

S0(ρt) =∫

Rn

(logρt +1)∂ρt

∂tdx=

∫

Rn

(logρt)12

∆ρt dx

= −12

∫

Rn

|∇ρt |2ρ−1t dx= −2S1(ρt) ,

ddt

S1(ρt) =14

∫

R2

∇ρt ·∇(∆ρt)ρt

− 12|∇ρt |2ρ2

t∆ρt dx

= −14

∫

R2

|D2ρt |2ρt

dx+12

∫

R2

D2ρt(∇ρt ,∇ρt)

ρ2t

dx− 14

∫

R2

|∇ρt |4ρ3

tdx

= −14

∫

R2

1ρt

∣∣∣∣D2ρt − ∇ρt ⊗∇ρt

ρt

∣∣∣∣2 dx≤ 0.


Hence,

supt>0

1tS0(ρ)−S0(ρt) = lim

t→0+

1tS0(ρ)−S0(ρt) = 2S1(ρ) .

Finally, for a generalρ such thatS1(ρ) < ∞, we may adopt the approximationargument sketched above in the proof of part (i) and obtain for allt > 0

2S1(ρ) ≥ 1tS0(ρ)−S0(ρt).

We also deduce from the above proof thatS0(ρt) is a convex, decreasing functionof t; hence we have for allh> 0

limt→0+

S0(ρ)−S0(ρt)t

≥− ddt

S0(ρh) = 2S1(ρh) ,

and we conclude upon lettingh go to 0+.

7.3 Asymptotic Limits

In this section, we investigate briefly some asymptotic limits involving the var-ious parameters (L, β, a, µ) of the mean field problem. In order to clearly see therole of the various parameters, we recall the variational problem (6.3) and (6.5),which allows the determination of the mean field potentialΨ and the mean fielddensity. We rewrite it in the case of filaments of lengthL, i.e.,

min

G(φ) : φ ∈ H1loc(R

2), φ−φ0 ∈ L∞(R2),(7.35)

∇(φ−φ0) ∈ L2(R2),∫

S1

− φds= 0

,

with

G(φ) =12

∫

R2

|∇(φ−φ0)|2dx+1

aβLlog[Tr(e−L(− 1

2β ∆+µ|x|2+aβφ))] .(7.36)

These expressions show, in particular, that only the normalized parametersL/β,Lµ, andLaβ matter.

We concentrate now upon the limits whenβ goes to 0+ (“infinite temperature”)or whenβ goes to+∞. First of all, if β goes to 0+, µβ goes to ˜µ > 0 andaβ2 goesto a> 0; keepingL fixed, we immediately see that this amounts to sending (for a“new” β = 1) the normalized lengthL/β to +∞ while keeping the other parametersµ anda essentially equal (or converging) to ˜µ anda, respectively. In other words,this limit is precisely the one we investigated in Section 7.2.

Next, if we letβ go to+∞ while keeping the other parametersL,µ,a> 0 fixed,we see that this amounts to sending the normalized lengthL/β to 0 while takingthe other parametersµ anda equal (or equivalent) to ˜µ/L and a/L, respectively.


In other words, this limit is equivalent to sending the length of the filaments to 0.Observe also that the limitβ going to+∞ is nothing but a classical limit for theHartree equation in terms of quantum mechanics problems.

Thus, we now investigate this distinguished limit, namely,L goes to 0+, β > 0fixed,µ= µ/L, anda = a/L with µ anda> 0 fixed. At least intuitively, we expectto recover at the limit the two-dimensional mean field theory for point vortices, andthis is precisely what we state in the following result where we denote byρL andψL the mean field expressions for a length equal toL by GL, the functional givenby (7.36), and byGL, the minimum given by (7.35).

THEOREM 7.3 As L goes to0+, ρL converges in Lp(R2) for 1 ≤ p< ∞ to ρ, ΨL

converges toΨ = (− 12π log|x|)∗ρ in C1,α

loc (R2) for 0<α< 1, ∇(ΨL−φ0) convergesto ∇(Ψ−φ0) in W1,p(R2) for 2≤ p<∞ whereΨ−∫

-S1Ψ ds is the unique minimumof

G = min

G(φ) : φ−φ0 ∈ L∞(R2), ∇(φ−φ0) ∈ L2(R2),

∫–S1φds= 0

(7.37)

where

G(φ) =12

∫

R2

|∇(φ−φ0)|2dx+1

aβlog

∫

R2

e−µ|x|2−aβφdx.(7.38)

In addition, we have

GL −1

aβlog

β

2πL−→ G as L goes to0+ .(7.39)

We sketch only one possible proof, since it is in fact possible to copy the argu-ment developed in Angelescu, Pulvirenti, and Teta [1] for three-dimensional quan-tum particles with a Coulomb (repulsive) interaction. One possible proof is verysimilar to the one presented in Section 5.5 for the continuous limit of broken pathmean field theories. Indeed, denoting byµL the mean field probability measure oncontinuous paths inR2, periodic with periodL, so thatµL = hLµ

0L and

hL =1Z′

Lexp

(−

∫–0

L µ

2|ω(σ)|2+ aβΨL(ω(σ))dσ

),

Z′L = E0

L

exp

(−

∫–0

L µ

2|ω(σ)|2+ aβΨL(ω(σ))dσ

).


We recall thathL is the unique minimum inL∞(ΩL) of

min

1β

E0L( f log f )+

µ

2βE0

L

(f

(∫–0

L|ω(σ)|2dσ

))

+a2

E0L

(f (ω) f (ω′)

(− 1

2π

∫–0

L log|ω(σ)−ω′(σ)|dσ))

:

f ≥ 0, E0L( f ) = 1

.

This allows us to prove the following bounds uniformly inL ∈ (0,1]:

E0L(hL loghL) ≤C,

E0L

(hL

∫–0

L|ω(σ)|2dx

)=

∫

R2

ρL|x|2dx≤C,

E0L

(hL(ω)hL(ω′)

(− 1

2π

∫–0

L log|ω(σ)−ω′(σ)|dσ))

=

∫∫

R2×R2

ρL(x)ρL(y)(− 1

2πlog|x−y|

)dxdy≤C.

Then these bounds yieldL∞ bounds onρL and thus (ρL being radially symmetric)W1,∞ bounds on∇ΨL andL∞ bounds onΨL −φ0.

These bounds are sufficient to prove that we have

2πLβ

Tr

eL

2β ∆−µ|x|2−aβΨL−

∫

R2

e−µ|x|2−aβΨL dx−→

L0,

by using, for instance, the Feynman-Kac representation of the trace (or making an

analytical proof where we approximate the Green’s function at timeL of L2β∆−

µ|x|2− aβΨL by β2πLe

−β|x−y|22L e−µ|y|2e−aβΨL(y) . . . ). Finally, we recall the semiclas-

sical inequality (see, for instance, B. Simon [23] and the references therein) valid

for anyφ such thatφ ∈ L∞loc(R

2), aβφ+ µ|x|2 is bounded from below,

Tr

e1

2β ∆−µ|x|2−aβφ≤ β

2πL

∫

R2

exp(µ|x|2− aβφ)dx.

This allows us to complete the proof of the convergence ofΨL and of the be-havior of GL. The rest of the proof is then straightforward, going back to therepresentation ofµL andρL.


8 Concluding Discussion and Future Directions

In this paper, we introduced Gibbs measures for nearly parallel filaments withidentical circulations, and then we rigorously derived a novel mean field theory asthe number of filaments tends to infinity. Here we would like to mention brieflysome related theoretical problems as well as a potential application.

Perhaps the most interesting and difficult theoretical issue for equilibrium statis-tical mechanics involves the behavior of the continuum Gibbs measures asN → ∞without the mean field scaling ¯a = a(πN)−1, which was utilized in Sections 3through 7 of this paper. It is very interesting to link even the equilibrium Gibbsmeasures with a finite number of filaments to actual dynamic behavior of thefilament equations. WithXj(σ,t) = (Xj,1,Xj,2) and the complex notationφ j =Xj,1+ iXj,2, the equations in (1.1) for identical filaments are equivalent to unusualcoupled nonlinear Schrödinger equations

1i

∂φ j

∂t= α

∂2φ j

∂σ2 +12 ∑

k6= j

φ j −φk

|φ j −φk|2 for 1≤ j ≤ N, φ j(σ+L,t) = φ j(σ,t) .

(8.1)

There is some computational and theoretical work connecting dynamics with equi-librium statistical mechanics of a single focusing nonlinear Schrödinger equation(Lebowitz, Rose, and Speer [15]) that might provide interesting background.

Finally, we briefly mention a potential physical application of the mean fieldtheory. Recently Julien et al. [11] presented numerical simulations of rapidly rotat-ing convection at high Rayleigh numbers. They observed that with small convec-tive Rossby numbers, the interior flow is dominated by the interaction of cyclonic(all rotating in the same sense), nearly parallel vortex filaments with most of theheat transport in the filament cores (Werne, private communications; Julien et al.[11]). Thus, in the interior region the turbulent flow is dominated by nearly parallelfilaments with very similar circulations. This is essentially the same regime as inthis paper, where we have developed the equilibrium statistical mechanics of nearlyparallel filaments. Do the equations for mean field theory developed here predicta large-scale interior flow resembling the simulations? Are other nonlinear correc-tions to self-induction, as discussed in the last paragraph of Section 3, needed foran accurate statistical prediction?

Appendix: Remarks on the KMD Equations

In this appendix, we make a few brief remarks on the coupled Schrödingerequations (8.1) to which we add some initial conditions

φ j(σ,0) = φ0j (σ)(A.1)


whereφ0j ∈ H1(0,L), φ0

j is periodic with periodL for 1≤ j ≤ N, and

∑j 6=k

∫ L

0

∣∣log|φ0j −φ0

k|∣∣dσ < ∞ .(A.2)

This system of coupled, nonlinear Schrödinger equations is not understood mathe-matically because of the singularity of the coupling nonlinear term.

We sketch here an argument that shows that there exists a global “very weak”solution of (8.1) together with the initial condition (A.1) such thatφ j ∈ L∞(0,∞;H1(0,L))∩C([0,∞);H1(0,L)) for all s∈ [0,1) (∀1 ≤ j ≤ N). We postpone theprecise definition of such solutions, since it is plausible that one could obtain amore satisfactory notion anyway. But we wish to emphasize that we only knowthat, for allt ≥ 0,

measσ ∈ (0,L) : ∃ j 6= k, φ j(σ,t) = φk(σ,t) = 0,

since we obtain, in fact, an estimate on

supt≥0

∑j 6=k

∫ L

0

∣∣log|φ j −φk|∣∣dσ ,

and, in particular, we are not aware of any bound on1|φ j−φk| for j 6= k.

One simple way to construct solutions is to smooth out the singularity by re-placing

φ j −φk

|φ j −φk|2 byφ j −φk

δ2 + |φ j −φk|2 for δ ∈ (0,1) .

Then the resulting system is trivial to solve (with the initial conditions (A.1)), andwe obtain a unique solution(φδj ) j ∈C([0,∞);H1), periodic inσ with periodL, ofthat regularized system. In addition, we have the following conservation laws:

ddt

∫ L

0

N

∑j=1

|φδj |2dσ = 0(A.3)

ddt

α2

∫ L

0

N

∑j=1

∣∣∣∣∣∂φδj∂σ

∣∣∣∣∣2

dσ− 18

∫ L

0

N

∑j 6=k

log(δ2 + |φδj −φδk|2)dσ= 0,(A.4)

from which we easily deduce bounds, uniform inδ, onφδ in C([0,∞);H1) and

N

∑j 6=k

12

log(δ2 + |φδj −φδ2|2) in C([0,∞);L1) .

From this point on, everything we say and do is really up to the extraction ofsubsequences. In particular, we may assume thatφδ converges weakly (weak-∗) toφ in L∞(0,∞;H1). We next proceed to show thatφδ converges toφ in C([0,T];L2)(and thus inC([0,T];Hs) for all s∈ [0,1)) for all T ∈ (0,∞). In order to do so,we recall thatφδ is bounded, uniformly inδ, in L∞((0,T)× (0,L)) (and thusφ ∈


L∞((0,T)× (0,L))) by the trivial one-dimensional Sobolev imbeddings. Then wedenote byφδj = ϕδj + iψδj , and we consider arbitraryC2 functionsF(ϕ1,ψ1; . . . ;ϕN,ψN) onR

2N such that we have for all 1≤ j ≤ N

F ′(ϕ j ,ψ j ) = 0 if (ϕk,ψk) = (ϕ j ,ψ j) for somek 6= j .(A.5)

Then a straightforward computation shows that we have (skipping the indexδ inorder to simplify notation)

∂∂t

F = αN

∑j=1

∂∂σ

(F ′ψ j

∂pj

∂σ−F ′

ϕ j

∂ψ j

∂σ

)+α

N

∑j,k

(F ′′ϕ jϕk

−F ′′ψ jψk

)∂ϕ j

∂σ∂ψk

∂σ

+αN

∑j,k

F ′′ϕ jψk

(∂ψ j

∂σ∂ψk

∂σ− ∂ϕ j

∂σ∂ϕk

∂σ

)+

12 ∑

k6= j

[F ′ψ j

(ϕ j −ϕk)−F ′ϕ j

(ψ j −ψk)] · (δ2 + |φ j −φk|2)−1 .

(A.6)

Then, in view of the bounds onφ j and condition (A.5), we deduce that∂F/∂t isbounded inL∞(0,∞;H−1).

We may then choose

F = Fεj = φ jβ

N∏k<`

|φ`−φk|2

ε

where 1≤ j ≤ N, ε∈ (0,1), β ∈C∞([0,∞)), β(t) = t if 0 ≤ t ≤ 1

2, β(t) = 1 if t ≥ 2,and 0≤ β′(t) ≤ 1 for all t ≥ 0. Obviously, condition (A.5) holds, and thus∂Fεj /∂tis bounded inL∞(0,∞;H−1) while Fεj is bounded inL∞(0,∞;H1). ThereforeFεj isrelatively compact inC([0,T);L2) asδ goes to 0 for eachε > 0 fixed.

Next we observe that we haveFεj = φ j if ∏Nk<` |φk−φ`|2 ≥ ε, and thus if|φk−

φ`|2 ≥ εν for all k 6= `, whereν = (N(N−1)2 )−1. Hence, we have for allδ,δ′ ∈ (0,1)

supt∈[0,T]

‖φδj (t)−φδ′

j (t)‖L2

≤ supt∈[0,T]

‖Fε,δj (t)−Fε,δ′

j (t)‖L2

+C supt∈[0,T]

measσ ∈ (0,L) : |φδk −φδ` |2 < εν for somek 6= `1/2

+C supt∈[0,T]

measσ ∈ (0,L) : |φδ′k −φδ′` |2 < εν for somek 6= `1/2 ,

≤ supt∈[0,T]

‖Fε,δj (t)−Fε,δ′

j ‖L2 +C

| log(δ2 +εν)| +C

| log(δ′2 +εν)| ,


in view of the bound on| log(|φδj −φδk|2 + δ2)| in L∞(0,∞;L1) for all j 6= k. Thisbound immediately yields the convergence ofφδj to φ j in C([0,T];L2) for all T ∈(0,∞).

Next, we claim thatφ j ∈C([0,∞)×L2). The argument above shows thatF(φ)∈C([0,∞)×L2) for any F satisfying (A.5). This is not enough, however, to proveour claim. In addition, we need to make some further observations. First of all, weremark that we have for any subsetI ⊂ 1,. . . ,N

1i

∂∂t

(∑j∈I

φδj

)= α

∂2

∂σ2

(∑j∈I

φδj

)+ ∑

j∈I∑k6= jk∈I

φδj −φδkδ2 + |φδδ−φδk|2

.(A.7)

In particular, we have, ifI = 1,. . . ,N,

1i

∂∂t

(N

∑j=1

φ j

)= α

∂2

∂σ2

(∞

∑j=1

φ j

),(A.8)

and thus∑Nj=1φ j ∈C([0,∞)× [0,L]).

WhenN = 2, this suffices to finish the proof, since

φ1 +φ2 and (φ1−φ2)|φ1−φ2|2 = φ1|φ1−φ2|2−φ2|φ1−φ2|2

are both continuous. Let us remark indeed thatF1 = φ1|φ1−φ2|2 andF2 = φ2|φ1−φ2|2 both satisfy (A.5). For a generalN, the proof requires some tedious combi-nations that we leave to the reader once we observe that we obtain from (A.7) abound on

∂∂t

(∑j∈I

φδj

)N

∏j=1

∏k/∈Ik6= j

|φδj −φδk|2 in L∞(0,T;H−1) .

Once the continuity is shown, we obtain for all 1≤ j ≤ N the existence of anopen setOj in [0,∞)×R such that meas(t,σ)(Oc

j) = 0 and∀k 6= j, |φ j −φk|> 0 onOj . Indeed, we have, from the convergence ofφδ to φ, for all j ∈ 1,. . . ,N, thefollowing estimate:

supt≥0

∑k6= j

∫ L

0

∣∣log|φ j −φk|∣∣dσ < ∞ .

In particular, (8.1) holds onOj , and we denote byO =⋂N

j=1Oj in order thatmeas(t,σ)(Oc) = 0.

Finally, we claim that we can pass to the limit, asδ goes to 0+, in (A.6) andrecover a formulation of the equation on(0,∞)× [0,L], at least whenFφ jφk = 0 ifφ` = φ j or φ` = φk for some` = k, ` 6= j. Let us also mention, by the way, thatit is possible to recover other “nonlinear equalities” based upon (A.7) by a similar


argument. We just sketch the argument for the above passage to the limit. First ofall, one deduces easily from the div-curl lemma that for allj andk,

∂ϕ j

∂σ∂ψk

∂σand

∂ϕ j

∂σ∂ϕk

∂σ− ∂ψ j

∂σ∂ψk

∂σweakly pass to the limit onOj ∩Ok. Sinceφ is continuous andF ′′

φ jφkvanishes on

(Oj ∩Ok)c, we may then pass to the limit in (A.6).

Acknowledgment. Andrew Majda is partially supported by NSF Grant DMS-9625795, ARO Grant DAAG55-98-1-0129, and ONR Grant N00014-96-0043.

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PIERRE-LOUIS LIONS ANDREW MAJDA

University of Paris IX Courant InstituteCeremade 251 Mercer StreetPlace de Lattre de Tassigny New York, NY 10012-118575775 Paris, FRANCEE-mail: [email protected]

Received August 1998.

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