Equilibrium Statistical Theory for
Nearly Parallel Vortex Filaments
PIERRE-LOUIS LIONS
Ceremade
Université Paris-Dauphine
AND
ANDREW MAJDA
Courant Institute
Abstract
The first mathematically rigorous equilibrium statistical theory for three-dimensional vortex filaments is developed here in the context of the simplified asymptotic equations for nearly parallel vortex filaments, which have been derived recently by Klein, Majda, and Damodaran. These simplified equations arise from
a systematic asymptotic expansion of the Navier-Stokes equation and involve
the motion of families of curves, representing the vortex filaments, under linearized self-induction and mutual potential vortex interaction. We consider here
the equilibrium statistical mechanics of arbitrarily large numbers of nearly parallel filaments with equal circulations. First, the equilibrium Gibbs ensemble
is written down exactly through function space integrals; then a suitably scaled
mean field statistical theory is developed in the limit of infinitely many interacting filaments. The mean field equations involve a novel Hartree-like problem
with a two-body logarithmic interaction potential and an inverse temperature
given by the normalized length of the filaments. We analyze the mean field problem and show various equivalent variational formulations of it. The mean field
statistical theory for nearly parallel vortex filaments is compared and contrasted
with the well-known mean field statistical theory for two-dimensional point vortices. The main ideas are first introduced through heuristic reasoning and then
are confirmed by a mathematically rigorous analysis. A potential application of
this statistical theory to rapidly rotating convection in geophysical flows is also
discussed briefly. c 2000 John Wiley & Sons, Inc.
Contents
1. Introduction
2. Gibbs Ensembles for Nearly Parallel Filaments and the Broken Path
Models
3. Heuristic Derivation of Mean Field Theory
4. Rigorous Mean Field Theory for the Broken Path Models
5. Rigorous Mean Field Theory for Vortex Filaments
6. Alternative Formulations for the Mean Field Equations
7. The Current and Some Scaling Limits
8. Concluding Discussion and Future Directions
77
81
89
92
94
119
123
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
Appendix. Remarks on the KMD Equations
Bibliography
77
137
141
1 Introduction
Over the last fifteen years, Chorin [4, 5, 6, 7] has proposed several novel heuristic models for fully developed turbulence based on the equilibrium statistical mechanics of collections of three-dimensional vortex filaments. Chorin’s pioneering
work has emphasized both the similarities and differences between statistical theories for heuristic models for ensembles of three-dimensional vortex filaments and
the 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 theory for three-dimensional vortex filaments in the context of a model involving simplified asymptotic equations for nearly parallel vortex filaments. These equations
have been derived recently by Klein, Majda, and Damodaran [13] through systematic asymptotic expansion of the Navier-Stokes equations where the nearly parallel
vortex filaments are represented by families of curves that move through linearized
self-stretch and mutual induction as leading-order asymptotic approximations of
the Biot-Savart integral.
Each vortex filament is concentrated near a curve that is nearly parallel to the
x3 -axis. Thus, each vortex filament is described by a function Xi (σ,t) ∈ R2 where
σ ∈ R1 parametrizes the asymptotic center curve of the filament. The family of
nearly parallel vortex filaments {X j (σ,t)}Nj≥1 evolves according to the 2N coupled
system of equations
#
"
2
∂X j
X j − Xk
1 N
2 ∂
(1.1)
= J α j Γ j 2 X j + ∑ Γ j Γk
Γj
∂t
∂σ
2 k6= j
|X j − Xk |2
for all 1 ≤ j ≤ N, where the parameter Γ j denotes the circulation of the jth filament,
α j is the vortex core structure, N is the number of filaments, and J = ( 01 −10 ).
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 longer
than the separation distance between filaments,
2. the separation distance is much larger than the core thickness of each filament, and
3. the Reynolds number is very large.
The technical aspects of the derivation of (1.1) as well as more details beyond
the 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 ∂2 X j /∂σ 2 arises from the linearized self-induction
of the individual filaments. The contribution of the terms
!
X j − Xk
1 N
J
Γ j Γk
2 k6∑
|X j − Xk |2
=j
is the velocity induced at a given vortex filament for a fixed value of σ by the
other vortex filaments; this contribution is the same one that occurs for the motion
of point vortices in the plane (Lamb [14] and Chorin and Marsden [8]). In fact,
special exact solutions of (1.1) without any σ-dependence coincide with solutions
of 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 for
the purely three-dimensional effect of self-induction. Numerical solutions of (1.1)
for pairs of filaments show a remarkable, genuinely three-dimensional behavior
that agrees qualitatively with many aspects of solutions of the complete NavierStokes equations.
From the mathematical viewpoint, (1.1) can be recast as a system of nonlinear
Schrödinger equations by setting ϕ j = X j1 + iX j2 , so that (1.1) becomes
(1.2)
−iΓ j
∂ϕ j
∂2 ϕ j 1 N
ϕ j − ϕk
+ ∑ Γ j Γk
.
= α j Γ2j
2
∂t
∂σ
2 k6= j
|ϕ j − ϕk |2
Because of the singularity of the nonlinear term, this evolution problem is not well
understood: The existence and uniqueness of regular solutions are not known, and
should depend on the parameters {Γ j }Nj=1 and their respective signs. In an appendix
we collect a few mathematical observations on that system while some accessible
open problems for (1.1) and (1.2) are discussed elsewhere (Majda [18] and Majda
and Bertozzi [19]).
In this paper, we develop the equilibrium statistical mechanics for solutions
of (1.1) in a suitable scaled limit as the number of filaments N gets arbitrarily
large as the model for the equilibrium statistical mechanics of nearly parallel vortex
filaments. Here we assume that each filament is periodic in σ, i.e.,
(1.3)
X j (σ + L) = X j (σ) ,
1≤ j ≤N,
for some L > 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, we
may assume that Γ > 0 and by a trivial scaling that we have
(1.4)
Γj = 1,
αj = α,
for 1 ≤ j ≤ N ,
for the solutions of (1.1) considered here. The assumption of identical signs for all
the circulations Γ j , i.e., corotating filaments, is a genuine physical restriction since
more complex dynamical phenomena occur for solutions of (1.1) with both positive
and negative circulations (Klein, Majda, and Damodaran [13], Majda [18], and
Majda and Bertozzi [19]). Designing an equilibrium statistical mechanics model
EQUILIBRIUM STATISTICAL THEORY
79
in that case is an interesting open problem to which we hope to return in a future
publication.
Theories for equilibrium statistical mechanics are based on the conserved quantities 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,
N
(1.5)
1
H=∑ α
j=1 2
Z L
∂X j 2
0
∂σ
1
dσ −
2
N
∑
Z N
j6=k 0
log |X j (σ) − Xk (σ)|dσ
as well as the center of vorticity M, the mean angular momentum I, and a quantity that we denote by C and call the mean current by analogy with quantum mechanics. The current C has an indefinite character much like the helicity in threedimensional flows. These additional conserved quantities are given explicitly by

L

N Z



M = ∑ X j (σ)dσ ,




j=1

0




L

N Z
(1.6)
I = ∑ |X j (σ)|2 dσ ,


j=1

0




L

N Z

∂X


C = ∑ JX j (σ) · j (σ)dσ .



∂σ
j=1
0
In Section 2 we introduce the Gibbs measures defined through the conserved
quantities in (1.5) and (1.6). These Gibbs measures naturally involve suitable function space integrals with respect to Wiener measure (more precisely, some kind of
“discounted conditional Wiener measure”). We also introduce a natural discrete
approximation of these Gibbs measures through a broken path discretization in
σ. This broken path discretization has several conceptual advantages: First, for
extremely coarse broken paths with only a single segment, we recover the Gibbs
measures for the statistical theory for point vortices in the plane; second, in the
other extreme limit of infinitely fine discretization, we recover the Gibbs measures
of the continuum problem associated with (1.1). In this fashion, we can compare
and contrast the equilibrium statistical theories for two-dimensional point vortices
and three-dimensional nearly parallel vortex filaments as well as build intermediate theories involving many fixed broken paths. These intermediate theories also
suggest the manner in which we can adapt and generalize the rigorous statistical
mechanical arguments for two-dimensional point vortex systems (Caglioti, Lions,
Marchioro, and Pulvirenti [2, 3], Kiessling [12], Lions [17]) to the present situation
involving the statistical mechanics of nearly parallel vortex filaments.
In Section 3 we give a heuristic discussion of mean field theory for the statistical
mechanics of nearly parallel vortex filaments. This theory requires the specific
scaling relation in (3.3) below for the nondimensional form of the Hamiltonian
80
P.-L. LIONS AND A. MAJDA
in (2.2) and (2.3). The tacit assumption of such mean field theories is that the
empirical distribution of the filament curves
!
1 N
(1.7)
∑ δXi (σ) converges to ρ(x) as N → ∞ for each σ
N i=1
where ρ(x) is a probability density on R2 independent of σ. In fact, we shall show
in Section 5 that this property holds with probability 1 under the Gibbs measure,
and we shall also determine the limit of the empirical law of {X j (σ)}. Without
loss of generality, we have set L = 1 (see Section 3). In addition, in the case
where we only use the conserved quantities H and I, we give a heuristic derivation
in Section 3 that the probability density ρ(x) is determined through the Green’s
function p(x, y,t), x ∈ R2 , y ∈ R2 , of the following PDE:

 ∂p − 1 ∆p + aβ − 1 log |x| ∗ ρ p + µ|x|2 p = 0 in R2 × (0, 1)
∂t 2β
2π
(1.8)

p|t=0 = δy (x) ,
where β is the inverse temperature, µ is the chemical potential for I, and the constant a in (1.8) is determined by the mean field scaling limit described in (3.3) from
Section 3. The probability density ρ is recovered from p by the formula

−1
(1.9)
ρ(x) = p(x, x, 1) 
Z
p(x, x, 1)
.
R2
In Section 4 we give a sketch of the rigorous a priori derivation of the mean
field 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 a
priori proof of mean field behavior for any positive inverse temperature β > 0 for
the 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 techniques
that we utilize here in the proof are similar to those of Angelescu, Pulvirenti, and
Teta [1] in their study of the classical limit for a quantum Coulomb system in R3
although our limiting mean field theory is completely different and we have other
technical difficulties associated with logarithmic interactions. We also prove that
the mean field limit for the broken path approximations converges to the continuum
mean field limit equation in (1.8) and (1.9).
In Section 6 we present several alternative characterizations of the mean field
limit problem in (1.8) and (1.9). One of these involves a Hartree-like problem with
a two-body logarithmic interaction potential and an inverse temperature given by
the normalized length of the filaments. We also utilize these alternative variational
characterizations to compare the mean field statistical theory for nearly parallel
vortex filaments with the mean field theory for point vortices in the plane. Until
Section 7 we do not use the conserved quantity C in order to keep the presentation
EQUILIBRIUM STATISTICAL THEORY
81
as simple as possible. In Section 7 we show how the use of the current C rigorously
leads to a modified mean field theory. We also discuss other scaling limits such as
the case of infinite-length vortex filaments. Finally, in Section 8 we briefly discuss
several possible directions for future work.
2 Gibbs Ensembles for Nearly Parallel Filaments and the Broken
Path Models
2.1 The Continuous Path Models
Here we discuss the definition of Gibbs measures for N-vortex filaments. For an
appropriate range of parameters, we would like to define Gibbs measures formally
given by
1
λ · M − µI − v C )dX1 · · · dXN
µN = exp(−β H −λ
(2.1)
Z
on the path space of N filaments where the Hamiltonian H and the other conserved
quantities M, I, and C are given in (1.4) and (1.5), respectively.
We begin by rewriting the Hamiltonian H in convenient nondimensional units.
With the notation X = (X1 , . . . , XN ) ∈ R2N , we nondimensionalize the amplitude of
the curves by A and the period interval by the dilation factor λ; i.e., we change
variables by σ 0 = λσ where both A and λ−1 have the units (length). We introduce
the nondimensional variable X 0 (σ 0 ) = X(σ 0 /λ)/A into the Hamiltonian. By multiplying the Hamiltonian by a constant, ignoring additive constants, and dropping
the prime in notation, we obtain the nondimensional Hamiltonian
(2.2) H (X(σ)) =
1
2
Z λL N
0
∑
j=1
∂X j
∂σ
2
1
dσ + ā
2
Z λL N
0
∑ − log |X j (σ) − Xk (σ)|dσ .
j6=k
With the natural choice for λ and λ = L−1 , the nondimensional factor ā is given by
L2
,
αA2
and the nondimensional Hamiltonian has the form
ā =
(2.3)
(2.4)
H (X(σ)) =
1
2
Z 1 N
∑
0 j=1
∂X j
∂σ
2
1
dσ + ā
2
Z 1 N
∑ − log |X j (σ) − Xk (σ)|dσ .
0 j6=k
With these preliminaries, we build the Gibbs measures in (2.1). For pedagogical purposes, we begin with the special case of (2.1) with µ, v = 0 and ā = 0
for the Hamiltonian in (2.4); in this special case the Gibbs measure is simply
the Wiener measure on (R2 )N with diffusion constant 1/β conditioned on peβ
riodic paths, which we denote by ν β . In fact, ν β may be written as νX,X
dX,
β
where νX,X
is the usual conditional Wiener measure conditioned on paths such
β
that ω(0) = ω(1) = X ∈ R2 (recall that νX,X
is not a probability measure, since
R
R
β
dνX,X
= (2πβt)−1 ). In particular, ν β is not a bounded measure ( dν β = +∞!)
82
P.-L. LIONS AND A. MAJDA
on the Banach space ΩN endowed with the usual norm (maxi,t∈[0,1] |ωi (t)|). Let
ΩN = (ω1 , . . . , ωN ) denote periodic continuous paths with ω j ∈ C([0, 1]; R2 ) and
β
ω j (0) = ω j (1) for all 1 ≤ j ≤ N. The rigorous way to define νX,X
is to write down
its marginals explicitly through its action on arbitrary bounded continuous functions of the type F = F(Ω(t1 ), . . . , Ω(tm )) with m ≥ 0, 0 < t1 < t2 < · · · < tm ≤ 1,
and we assume, for instance, that F has compact support on (R2 )m . Thus we have
Z
(2.5)
F dν β =
Z
Z
dX1 · · ·
dX
R2N
R2n
Z
dXm F(X1 , . . . , Xm )
R2N
· pβ0 (X, X1 ,t1 )pβ0 (X1 , X2 ,t2 − t1 ) · · ·
pβ0 (Xm−1 , Xm ,tm − tm−1 ) · pβ0 (Xm , X, 1 − tm )
=
Z
dX1 · · · dXm pβ0 (Xm , X1 , 1 + t1 − tm )
R2Nm
· pβ0 (X1 , X2 ,t2 − t1 ) · · ·
pβ0 (Xm−1 , Xm ,tm − tm−1 )F(X1 , . . . , Xm )
where pβ0 (X,Y ,t) = ∏Nj=1 p̃β0 (X j ,Y j ,t) with p̃β0 (x, y,t), the Gaussian kernel on R2 ,
(2.6)
p̃β0 (x, y,t) =
2πt
β
−1
β|x − y|2
exp −
,
2t
(see, for example, Ginibre [9], Lebowitz, Rose, and Speer [15], Simon [23], or
β
Angelescu et al. [1]). With this definition of dνX,X
as background, we next turn to
the definition of the Gibbs ensemble in (2.1) with β > 0, λ 6= 0, and µ > 0, but here
we require v = 0.
In this general case, the Gibbs measure µN is given in a straightforward fashion
as
(2.7)
( Z
"
1
β ā
µN = (Z(N))−1 exp − dσ
2
0
+
#
N
∑ − log |ω j (σ) − ωk (σ)|
j6=k
N
N
j=1
j=1
∑ λ · ω j (σ) + µ ∑ |ω j (σ)|2
)
β
(Ω)dX
dνX,X
EQUILIBRIUM STATISTICAL THEORY
with
Z(N) =
(2.8)
"
Z
β
dXEX,X
R2N
"
( Z
1
β ā
dσ
exp −
2
0
83
N
∑ − log |ω j (σ) − ωk (σ)|
j6=k
N
#)#
λ · ω j (σ) + µ|ω j (σ)|2 )
+ ∑ (λ
.
j=1
β
β
In (2.8), EX,X
denotes the expected value with respect to dνX,X
. At this stage, one
N
needs to explain why Z(N) < ∞ and thus justify that µ is well-defined by (2.7).
Indeed, one can clearly bound Z(N) for some positive constant C = C(N)
(
)
Z
Z 1
N
µ
β
Z(N) ≤ C
dX EX,X
exp −
dσ ∑ |ω j (σ)|2
2 0
j=1
R2N
!
Z
Z
β
dX EX,X
≤C
R2N
=C
Z 1


=C
0
N
dσ exp −µ ∑ |ω j (σ)|2
j=1
N
ZZ
dσ 
0
1
µ
pβ (X,Y , σ)e− 2 |Y | pβ (Y , X, 1 − σ)dX dY 
R2 ×R2
Z
(2π)−1 βe
R2
− µ2 |Y |2
2
N
µ −N
dY  = C 4π 2
.
β
β
As in the definition in (2.5) for νX,X
, the marginal distributions for the Gibbs
N
measure µ can be written down via the Green’s function of a PDE. The way to
see this is to observe that the potential V (X) defined by
V (X) =
(2.9)
β ā
2
N
N
j6=k
j=1
∑ − log |X j − Xk | + ∑
λ · X j + µ|X j |2
β
satisfies the hypotheses for the Feynman-Kac formula with respect to dvX,X
provided µ satisfies µ > 0 (see Simon [23, chap. 2]). Thus, from (2.7) and (2.9), for
an arbitrary, bounded, continuous function on R2Nm , F(Ω(t1 ), . . . , Ω(tm )), and any
partition with 0 ≤ t1 < t2 < · · · < tm ≤ 1 with m ≥ 1, we have
(2.10)
Z
F dµN = (Z(N))−1
Z
dX1 · · · dXm F(X1 , . . . , Xm )
R2Nm
· p(Xm , X1 , 1 + t1 − tm )p(X1 , X2 ,t2 − t1 ) · · · p(Xm−1 , Xm ,tm − tm−1 )
84
P.-L. LIONS AND A. MAJDA
and
(2.11)
Z(N) =
Z
dX p(X, X, 1) .
R2N
Moreover, from the Feynman-Kac formula and (2.7), p(X,Y ,t) is the Green’s
function for the PDE,
!
N
β ā N
∂p
1 N
λ · X j + µ|X j |2 )p = 0
∆X j p −
log |Xi − X j | p + ∑ (λ
−
∑
∑
∂t 2β j=1
2 i6= j
j=1
(2.12)
2N
in R × (0, 1) ,
p|t=0 = δY (X) on R2N .
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 and
away from the sets {(X,Y ) ∈ (R2N )2 : ∃i 6= j Xi = X j or Yi = Y j } with p > 0 for
t > 0; ∂p/∂t, DαX,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 on R2N ×
2N
R × (0, 1),
0 < p(X,Y ,t) ≤ eC(N)t (µβ)N/2 (sinh(bt))−N
2
1/2 1
2
2
exp −(µβ)
cotanh(bt)(|X| + |Y | ) −
X ·Y
2
sinh(bt)
with b = (µ/β)1/2 . It is worth remarking that the special case with ā = 0, µ >
0, is the parabolic quantum oscillator and can be solved explicitly by Mehler’s
formula (Simon [23]) in terms of appropriate Gaussians. The situation with λ 6=
0 can be reduced to the situation with λ = 0 by elementary transformations so
without loss of generality, we assume λ = 0 in the following section. The explicit
formula for the parabolic oscillator kernel, combined with the trivial comparison
potential ∑Ni=1 log |Xi − X j | ≤ 2βµā |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 conserved
quantity given by the current C from (1.6), we need to utilize the Ito calculus.
We will not do this here in the continuum setting for simplicity in exposition; until
Section 7, we will always assume v = 0. However, we will retain an approximation
to C in the broken path models discussed in the next section.
We have seen above two equivalent ways of defining the Gibbs measure µN .
We shall also justify (and recover the equivalence of) these definitions in the next
section by letting the broken paths “converge” to continuous paths. This asymptotic approach yields the derivation of a third way of defining µN , which is also a
consequence of the Feynman-Kac formula. Indeed, we see that we have
(2.13)
dµN = hN dµN0 ,
EQUILIBRIUM STATISTICAL THEORY
85
where µN0 is the probability measure on ΩN defined below in (2.16), (2.17), and
"
( Z
1
β ā N
1
N
h = 0
(2.14)
− log |ω j (σ) − ωk (σ)|
exp − dσ
Z (N)
2 ∑
0
j6=k
#)
N
µ N
+ ∑ λ · ω j (σ) + ∑ |ω j (σ)|2
2 j=1
j=1
with
(2.15)
"
Z 0 (N) = E0N
"
( Z
1
β ᾱ
dσ
exp −
2
0
N
∑ − log |ω j (σ) − ωk (σ)|
j6=k
µ
+ ∑ λ · ω j (σ) +
2
j=1
N
N
∑ |ω j (σ)|
#)#
2
.
j=1
Here E0N denotes the expectation with respect to µN0 .
The probability measure µN0 corresponds to the special case when a = λ =
0 above; i.e., µN0 is the law of the “quantum oscillator” process, which can be
equivalently defined by
)
(
Z
N
1
µ 1
β
2
dσ ∑ |ω j (σ)| dX νX,X
exp −
R(N)
2 0
j=1
or by
(2.16)
E0N [F(Ω(t1 ), . . . , Ω(tm ))]
=
Z
R2Nm
dX1 · · · dXm q(Xm , X1 , 1 + t1 − tm )q(X1 , X2 ,t2 − t1 )
· · · q(Xm−1 , Xm ,tm − tm−1 )F(X1 , . . . , Xm )
for any bounded continuous function F on R2m , where q(X,Y ,t) is given by
(2.17) q(X,Y ,t) = π −N (µβ)N/2 (sinh(bt))−N
1
1/2 1
2
2
coth(bt)(|X| + |Y | ) −
X ·Y
exp −(µβ)
2
sinh(bt)
with b = (µ/β)1/2 . In particular, we have
(2.18)
Z(N) = Z 0 (N)R(N), R(N) = (cosh b − 1)N .
Under the law µN0 , Ω(t) is obviously a Gaussian process and ω1 (t), . . ., ωN (t) are
independent. These formulas guarantee that Z 0 (N) is finite, and thus hN is bounded
on ΩN . Notice finally that µN and µN0 are both symmetric probability measures on
ΩN .
We conclude this section with an important observation on the invariance of the
above Gibbs measures µN and µN0 by time shifts. Extending periodically the paths
86
P.-L. LIONS AND A. MAJDA
ω j (1 ≤ j ≤ N) in Ω to [0, ∞), we denote by θt the shift by t, namely, θt Ω(s) =
Ω(s + t). Then, (2.5) and (2.10) immediately yield
(2.19)
E N [F(θt Ω)] = E N [F(Ω)], E0N [F(θt Ω)] = E N [F(Ω)]
for any, say, bounded, measurable random variable F on ΩN , where we denote by
E N the expectation with respect to µN . Of course, the density hN satisfies the same
invariance property, namely, hN (θt Ω) = hN (Ω) for all t ≥ 0.
2.2 The Broken Path Models
In these models, we replace the continuous paths utilized in defining µN in (2.1)
by discrete curves. Thus, we consider periodic broken chains
xσj ,
1 ≤ j ≤ N , 0 ≤ σ ≤ M , Mδ = 1 ,
0
with the periodicity condition xM
j = x j . To denote the individual broken path filaments, we utilize the notation X j = (x0j , . . . , xM−1
) ∈ R2M for 1 ≤ j ≤ N. In the
j
broken path models, we simply approximate H in (2.4) and the conserved quantities in (1.6) by straightforward discretizations,

M−1 N
1 σ+1
ā M−1 N

δ
σ

H
=
−
x
|
−
|x

∑ ∑ 2δ j
∑ ∑ δ log |xσj − xkσ | ,
j


2

σ=0 j=1
σ=0 j6=k



M−1 N
I δ = ∑ ∑ δ|xσj |2 ,
(2.20)


σ=0 j=1


M−1 N



δ

C
=
− xσj ) .

∑ ∑ xσj · J(xσ+1
j
σ=0 j=1
The Gibbs measures µN,δ for the broken path approximation are absolutely continuous with respect to Lebesgue measure on (R2M )N with density given by
(2.21)
µN,δ (X1 , . . . , XN ) = Z −1 exp{−β H δ (X1 , . . . , XN ) − µI δ − v C δ )}
and
(2.22)
Z=
Z
exp(−β H δ − µI δ − v C δ )dX1 · · · dXN .
(R2M )N
We remark that in the special case of the coarsest broken path model with M = 1,
H δ reduces to the point vortex Hamiltonian for two-dimensional flows, I δ becomes
the moment of inertia, and C δ vanishes identically. Also, as M ↑ ∞, the Gibbs measures for the broken path approximation formally converge to the Gibbs measures
for continuous paths, which we discussed earlier in this section. We shall come
back to that point at the end of this section.
With a nonzero current C δ , the Gibbs measures in (2.21) and (2.22) are not
well-defined unless the Lagrange multiplier v satisfies certain restrictions given
EQUILIBRIUM STATISTICAL THEORY
87
the values of β, µ, and M. To see this, we consider the quadratic terms in the
exponential in (2.21) given by
βH δ
(2.23)
ā=0
+ µI δ + v C δ ≡
N
∑ B(X j )
j=1
with B(X), the quadratic form on a periodic broken path, given by
B(X) =
(2.24)
M−1
β M−1 1 σ+1
− xσ |2 + δµ ∑ |xσ |2
|x
∑
2 σ=1 δ
σ=1
+v
M−1
∑ xσ · J(xσ+1 − xσ ) ,
σ=1
δ = M −1 .
In standard fashion for discrete periodic problems, this quadratic form is diagonalized by 2M orthonormal eigenvectors with the form
2πil 2πi2l
2πi(M−1)l
M ,e M ,...,e
M
(2.25)
=
1,
e
for 0 ≤ l ≤ M − 1 ,
e±
e±
l
l
2
with e±
l ∈ C , the orthonormal eigenvectors of an appropriate 2 × 2 Hermitian matrix with eigenvalues
2πl
l
±
−1
λl = βM 1 − cos 2π
(2.26)
+ M µ ± v sin
,
M
M
l = 0, 1, . . . , M − 1 .
Thus, B(X) is positive definite if and only if λ±
l > 0, l = 0, 1, . . . , M − 1, and we
immediately have the following:
P ROPOSITION 2.1 Given fixed β, µ, and M with β > 0, µ > 0, and M any positive
integer, the Gibbs measures in (2.20) are well-defined only for v that satisfies the
±
conditions λ±
l > 0, l = 0, 1, . . . , M − 1, with λl the explicit numbers in (2.26) and,
2
in particular, for v < 2βµ. Under these conditions, there are constants C1 ,C2 > 0
so that B(X) in (2.24) satisfies
C2 |X|2 ≤ B(X) ≤ C1 |X|2 .
(2.27)
For fixed β > 0, µ > 0, the numbers λ±
l are always positive for all sufficiently large
−
M ≥ M0 . If either λ+
or
λ
for
some
l satisfies λ±
l
l
l < 0, then the Gibbs measure
cannot be defined for this value of v.
We now sketch a proof of the fact that, in the case when v = 0, the Gibbs
measures µN,δ “converge” as M goes to +∞, i.e., as δ tends to zero to the probability measures µN defined in the preceding section. More precisely, we define (or extend) a probability measure on ΩN from µN,δ , which is concentrated
on piecewise linear curves, by setting for any bounded continuous function F =
F(Ω(t1 ), . . . , Ω(tm )) with m ≥ 0, 0 < t1 < t2 < · · · < tm ≤ 1,
Z
(2.28)
F dµN,δ =
Z
F(X(t1 ), . . . , X(tm ))µN,δ (X1 , . . . , XN )dX1 · · · dXN
88
P.-L. LIONS AND A. MAJDA
where
X j (ti ) =
xσj i
xσj i +1 − xσj i
+ (ti − σi )
!
δ
(1 ≤ i ≤ m, 1 ≤ j ≤ N) with σi = [ti /δ]. We keep the same notation, µN,δ , for this
natural extension, and we claim that, as M = δ1 goes to +∞, µN,δ converges weakly
to µN . A complete proof of this fact is somewhat tedious and is certainly not
needed 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 simple
case when F = F(Ω(0)) and F is, say, smooth with compact support. In fact, this
proof immediately adapts to the case when F = F(Ω(t1 )) and then to the case when
F = ∏m
i=1 Fi (Ω(ti ) − Ω(ti−1 )), and the general case follows by linearity and density.
Next, if F = F(Ω(0)), taking β = 1 in order to simplify notation, we have
Z
F(Ω(0))dµN,δ
=
Z
1
=
Zδ
F(x10 , . . . , xN0 ) µN,δ (X1 , . . . , XN )dX1 · · · dXN
ZZ
e− 2δ |yM−1 −y0 |2 −δV (y0 ) 0 M−1
c
dy dy
e
2πδ
1
F(y )pδ 1, y , y
0
0
M−1
where
yσ = (x1σ , . . . , xNσ ) ,
ā
2
V (y) =
Zδ =
σ = 0, . . . , M − 1 ,
N
∑ log |x j − xk | − µ|y|2 ,
j6=k
ZZ
e− 2δ (yM−1 −y0 )2 −δV (y0 ) 0 M−1
dy dy
,
e
2πδ
1
pδ 1, y0 , yM−1
and
pδ (1, y0 , yM−1 ) =
Z
1 i+1
M−2 − 2δ
|y −yi |2
∏
i=0
e
2πδ
!
e
−δV (yi+1 )
e−δV (y
n−1 )
dy1 . . . dyM−1 .
As a consequence of the Trotter formula, the kernel pδ (1, y0 , yM−1 ) is easily
seen to converge to p(1, y0 , yM−1 ), at least formally, where p is the Green’s function
of
∂p
− ∆p +V p = 0 in R2N × (0, 1) .
∂t
R
Therefore, Zδ converges to Z = p(1, y0 , y0 )dy, and
Z
F(Ω(0))dµN,δ −→
1
Z
Z
F(y0 )p(1, y0 , y0 )dy0 ,
EQUILIBRIUM STATISTICAL THEORY
89
which proves our claim. All the above can be justified, but we choose not to do so
here 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 the
representations of µN 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 theory utilized in this paper for the Gibbs ensembles from Section 2 in the limit as
N → ∞. The main scaling assumption for mean field statistical theories for point
vortices in the plane (see Caglioti et al. [2, 3] and Lions [17]) involves the scaling
exp(− Nβ HN (X)) in the Gibbs ensembles with X = (x1 , . . . , xN ) ∈ R2N and
(3.1)
1
1
HN (X) =
N
N
N
∑ − ln|x j − xk | .
j6=k
The second heuristic idea in the mean field theory for point vortices in the plane is
that the empirical measure N1 ∑Ni=1 δxi converges weakly to a probability measure,
ρ(y), as N → ∞ so that
(3.2)
1 N
− ln |x j − xk | ∼
=−
N k6∑
=j
Z
ln(|x j − y|)ρ(y)dy .
R2
In other words, the velocity potential induced on an individual vortex by the other
vortices is insensitive to the detailed locations of these vortices as N → ∞ and
instead can be computed by the mean velocity potential defined by the probability
density ρ(y) ∈ R2 ; i.e., fluctuations are arbitrarily small as N 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 fashion
as in the two-dimensional theory described above as N → ∞. Thus, we assume that
the nondimensional factor ā defined in (2.3) has the form
(3.3)
ā =
a
2πN
with some prescribed constant a > 0. Next, we present a heuristic derivation of
mean field theory for the continuous Gibbs ensembles from Section 2. Rigorous
a priori proofs of mean field limiting behavior with the scaling in (3.3) are given
in 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 distributions of the Gibbs measure are defined in (2.10) and (2.11) through the Green’s
90
P.-L. LIONS AND A. MAJDA
function for the PDE,
!

N
N
N
∂p
βa
1



∆
p
−
log
|X
−
X
|
p
+
µ
−
X
i
j
∑ i
∑ |X j |2 p = 0

 ∂t 2β i=1
2πN i6∑
j=1
=j
(3.4)
2N

in R × (0, 1) ,




p|t=0 = δY (X) on R2N .
In the heuristic derivation, we assume that (3.2) is satisfied with a density ρ(y) that
is independent of t, i.e., translation invariant. This assumption merely reflects the
time 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 as N → ∞
N
p(X,Y ,t) ∼
= ∏ p(Xi ,Yi ,t)
(3.5)
i=1
where p(x, y,t) for x ∈
(3.6)
R2 ,
y∈
R2 ,
satisfies
∂p
(βa)
1
− ∆x p −
(log |x| ∗ ρ)p + µ|x|2 p = 0
∂t 2β
2π
p|t=0 = δy (x) on R2 .
in R2 × (0, 1) ,
To complete this formal derivation of mean field theory, we need to determine
the density ρ(x). According to (3.2), ρ is the limiting single-point probability distribution of the filament curves; in general, this distribution is determined by setting
m = 1 in (2.11) so that

−1
Z
(3.7)
F dµ = 
Z
p(X, X, 1)
N
R2N
Z
F(X)p(X, X, 1)
R2N
for any bounded continuous function F(X). The formal factorized approximation
in (3.5) combined with (3.7) yields the identity

−1
(3.8)
ρ(x) = p(x, x, 1) 
Z
p(x, x, 1)
.
R2
This completes the heuristic derivation since the equations in (3.6) and (3.8) constitute the Green’s function formulation of the mean field approximation. Other
equivalent formulations of this mean field approximation are presented in Section 6.
It is also possible to give a heuristic derivation of the mean field probability measure on paths (i.e., filaments). Indeed, we consider the marginals of µN ,
namely,
(3.9)
µN,k =
Z
dµN (. . . , ω k+1 , . . . , ω N )
EQUILIBRIUM STATISTICAL THEORY
91
or equivalently
(3.10)
µN,k = hN,k · µk0
R
where hN,k = hN dµ0 (ω k+1 ) · · · dµ0 (ω N ). Notice that µN,k and hN,k are symmetric
in (ω 1 , . . . , ω k ) and are invariant by time shifts. Then, if we believe that a mean
field theory is relevant as N goes to +∞, as we will in fact prove in the subsequent
sections, µN,k should factorize asymptotically for each k ≥ 1 fixed
µN,k −→
(3.11)
N
k
O
µ
j=1
where µ is the mean field law of a single filament. In order to determine µ, we go
back to (2.7) (for instance), recalling that ā = 1/2πN, and we integrate with respect
to ωk+1 , . . . , ωN . Then, using the rule N1 ∑Ni=1 δωi (σ) ≈ ρ(x)dx for each δ ∈ [0, 1], we
N
deduce at least formally



Z 1
Z


1
aβ
µ = exp − dσ −
(3.12)
log |ω(σ) − x|ρ(x)dx + µ|ω(σ)|2


Z
2π
0
R2
(3.13)
β
dνx,x
dx ,



Z
Z

 Z 1
aβ
2

log |ω(σ) − x|ρ(x)dx + µ|ω(σ)|
Z = exp − dσ −


2π
0
R2
β
(ω)dx ,
dνx,x
or equivalently
(3.14) µ = h · µ0 ,



Z 1
Z


aβ
µ
1
dσ 
log |ω(σ) − x|ρ(x)dx + |ω(σ)|2  ,
h = 0 exp −

 0
Z
2π
2
R2
(3.15)



Z
 Z 1

aβ
µ
Z 0 = E0 exp −
dσ −
log |ω(σ) − x|ρ(x)dx + |ω(σ)|2   .
 0

2π
2

R2
In particular, we should expect for any bounded, measurable F = F(ω1 , . . . , ωk ) on
Ωk
Z
(3.16)
F dµN −→
N
Z
F(ω1 , . . . , ωk )dµ(ω1 ) · · · dµ(ωk ) .
Clearly, the above heuristic derivation gives no insight into a rigorous a priori
proof. Completely different considerations are needed that involve the characterization of the limiting mean field problem as the unique minimizer of an appropriate
free energy functional. This rigorous procedure is carried out in Section 4 for the
92
P.-L. LIONS AND A. MAJDA
broken path models following Caglioti et al. [2] and for the continuum filament
models 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 scaling
in (3.3), the dynamic equations for interacting nearly parallel filaments from (1.1)
have the form
#
"
∂X j
∂2
a N X j − Xk
Xj +
(3.17)
.
=J
2
∂t
∂σ 2
2πN k6∑
= j |X j − Xk |
Thus, for N → ∞, the scaling in (3.3) for statistical behavior for the Hamiltonian
corresponds to the circumstances where the linearized self-induction of each individual filament is much stronger than the potential vortex interaction of individual
filaments. For the actual vortex dynamics of nearly parallel vortex filaments, other
additional nonlinear corrections to the self-induction of individual filaments might
be needed (Majda [18]), but the model in (3.9) probably still retains a number of
significant features.
4 Rigorous Mean Field Theory for the Broken Path Models
Here we sketch a rigorous proof of a priori convergence to a suitable mean
field limit for the Gibbs measures in (2.21) and (2.22) for the broken path models
with the mean field scaling from (3.3) provided β and µ satisfy β ≥ 0, µ > 0,
while the multiplier for the discrete current v necessarily satisfies the restrictions in
Proposition 2.1. We will not give details of the proofs since they closely mimic the
arguments of Caglioti et al. [2] for mean field behavior of statistical point vortices
in R2 in their simplest situation with positive temperature β > 0.
Thus, setting ā = a/2πN in (2.20), we introduce the correlation functions associated with the Gibbs measures µN,δ in (2.15) and (2.16). The correlation functions
ρkN,δ (X1 , . . . , Xk ) are probability densities defined by
(4.1)
ρkN,δ (X1 , . . . , Xk )
=
Z
µN,δ dXk+1 · · · dXN
for 1 ≤ k ≤ N − 1 .
These probability densities are symmetric in (X1 , . . . , Xk ) as a consequence of the
symmetry of µN,δ with respect to the broken paths (X1 , . . . , XN ). We have the following:
T HEOREM 4.1 Assume ā = a/πN in (2.20) with β > 0, µ > 0, and v satisfying
the conditions of Proposition 2.1 in the Gibbs measures in (2.21) and (2.22) for
the broken path models with fixed M ≥ 1. Then, for any k ≥ 1, the correlation
functions ρkN,δ converge in L p ((R2M )k ) for all 1 ≤ p < ∞ to ∏kj=1 ρδ (X j ) as N → ∞.
The probability density ρδ (X) on R2M is translation invariant so that
(4.2)
ρδ (X) = ρδ (Tk X)
EQUILIBRIUM STATISTICAL THEORY
93
where for a given periodic broken path,
X = x0 , x1 , . . . , xM−1 , Tk X = xk , . . . , xM+k−1 ,
with the convention that xσ+mM = xσ for all 0 ≤ σ ≤ M − 1, m ≥ 1. The density
ρδ (X) is the unique solution of the following mean field equation:

(
)
M−1



ρδ (X) = Z −1 exp −B(X) − δ ∑ V (xσ )
on R2M ,




σ=0


Z


aβ
V (x) = −
log |x − y|ρδ1 (y)dy on R2 ,
(4.3)
2π


R2


Z




ρδ (y) =
ρδ (x0 , y, x2 , . . . , xM−1 )dx0 dx2 · · · dxM−1


 1
R2(M−1)
with
ρδ ∈ L∞ (R2M ) ,
ρδ log ρδ ∈ 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 and
is the unique minimum of the following strictly convex (free energy) functional
F δ (ρ) =
1
β
(4.4)
−
Z
[ρ log ρ + B(X)ρ]dX
R2M
a
4π
ZZ
ρ(X)ρ(Y )
R2M ×R2M
M−1
∑ log |xσ − yσ |dX dY .
σ=0
For the special case with M = 1, the equations in (4.3) reduce to the familiar
mean field equations for point vortices in R2 in the positive temperature regime
(Caglioti et al. [3]). In Section 5, we establish that for v = 0, ρδ1 converges to
ρ as δ → 0 where ρ(x) is the probability density in (3.8) arising from the mean
field theory for continuous-path vortex filaments described heuristically in (3.6)
and (3.8).
Under the restrictions on the multiplier v in Proposition 2.1 and with the notation from (2.18), the density for the Gibbs measures in (2.15) with ā = a(πN)−1
has the form
!
N
βa M−1 N
N,δ
−1
σ
σ
µ (X1 , . . . , XN ) = Z exp
(4.5)
ln
|x
−
x
|
∑ ∑ k j ∏ e−B(X j )
2πN σ=0
j=1
k6= j
where
(4.6)
e−C1 |X j | ≤ e−B(X j ) ≤ e−C2 |X j | .
2
2
With the structure in (4.5) and (4.6), simple modifications of the estimates in Section 3 of Caglioti et al. [2] and identical to those needed in Section 6 of that paper
94
P.-L. LIONS AND A. MAJDA
yield the uniform bounds on the correlations
j
ρN,δ
j (X1 , . . . , X j ) ≤ C
(4.7)
for all N .
With (4.7) and the subadditivity and strict convexity of entropy, we can copy the
argument in section 4 of Caglioti et al. [2] (also see section 4 of Lions [17]) with
only minor changes to conclude the theorem provided the free energy functional
defined in (4.4) has a unique solution. Proposition 2.1 guarantees that the integrand
is strictly convex so a unique solution exists.
The calculation for the minimizer ρδ (X) for the free energy in (4.4) yields
(
)

M−1



ρδ (X) = Z −1 exp −B(X) − ∑ δVσ (xσ )
on R2M ,



σ=0



Z


aβ


log |x − y|ρδσ (y)dy
Vσ (x) = −


2π

R2
(4.8)
Z



ρδσ (y) =
ρδ x0 , . . . , xσ−1 , y, xσ+1 , . . . , xM−1





R2(M−1)






dx0 · · · dxσ−1 dxσ+1 · · · dxM−1 for all 0 ≤ σ ≤ M − 1 .


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 equation
in (4.2) and the last one in (4.8) together imply that ρσ (y) = ρ1 (y) for all σ with
0 ≤ 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 continuum
filament model in Section 3 tacitly assumed that the one-point density is translation
invariant; here we have deduced this property for ρ in an a priori fashion for the
broken path models. We will do this in a similar manner for the continuum models
in Section 5.
5 Rigorous Mean Field Theory for Vortex Filaments
For each fixed k ≥ 1, we consider for N ≥ k the probability measures µN,k and
their densities with respect to the probability measures µk0 and hN,k , given by (3.9)
and (3.10), respectively. We also assume that λ = v = 0 (see Section 7 for the
extension to the case when λ and v do not vanish). Of course, we set ā = a/2πN
for the reasons explained in Sections 3 and 4 above. These assumptions are made
throughout this section and will not be repeated.
5.1 Main Results
T HEOREM 5.1 For each k ≥ 1, µN,k converges weakly (in the sense of probability
N
measures on Ωk ), as N goes to +∞, to some product measure kj=1 µ. Furthermore, hN,k is bounded on Ωk uniformly in N and converges, as N goes to +∞,
EQUILIBRIUM STATISTICAL THEORY
95
in L p (Ωk , µk0 ) for all 1 ≤ p < ∞ to ∏kj=1 h(ωi ) for some h that is continuous and
bounded on Ω. In addition, we have
(5.1)

dµ = h dµ

0





Z
Z



1

aβ
µ


h = 12 exp −
dσ −
log |ω(σ) − x|ρ(x)dx + |ω(σ)|2 

 0
2π
2
R2






Z 1
Z



aβ
µ

2 

0 = E exp −


Z
dσ
−
log
|ω(σ)
−
x|ρ(x)dx
+
]
|ω(σ)|
0


 0

2π
2
R2



Z
 Z 1


1
aβ



dµ
=
dσ
−
log |ω(σ) − x|ρ(x)dx
exp
−



2
2π

0


R2
#)





2
β

+ µ|ω(σ)|
dx
dνx,x


(5.2)
Z
Z

β


(ω)
Z = dx dνx,x




R2






Z

 Z 1


aβ

2



exp
−
,
dσ
−
log
|ω(σ)
−
x|ρ(x)dx
+
µ|ω(σ)|




2π
0
R2
where ρ is the probability measure on R2 defined by
Z
(5.3)
ϕ(x)ρ(x)dx = E[ϕ(ω(σ))] = E0 [ϕ(ω(σ))h(ω)]
R2
for any σ ∈ [0, 1] and for any ϕ that is bounded and measurable on R2 , where E
denotes the expectation with respect to µ. In other words, ρ(x) is the density of the
law of ω(σ) under µ for each σ ∈ [0, 1].
The density ρ is smooth, radially symmetric, and rapidly decreasing and is the
unique solution, say, in L1 ∩ L∞ (R2 ) of
(5.4)
ρ(x) = R
p(1, x, x)
p(1, x, x)dx
R2
where p(t, x, y) is the Green’s function of

 ∂p − 1 ∆p − aβ (log |x| ∗ ρ)p + µ|x|2 p = 0
∂t 2β
2π
(5.5)

p|t=0 = δy (x) in R2 .
in R2 × (0, 1) ,
96
P.-L. LIONS AND A. MAJDA
Remark 5.2. For any bounded, measurable F on (R2 )m (m ≥ 1) and for any 0 ≤
t1 < t2 < · · · < tm = 1, we obviously have
" Z
E[F(ω(t1 ), . . . , ω(tm ))] =
dy1 · · · dym p(t2 − t1 , y1 , y2 ) · · ·
R2m
p(tm − tm−1 , ym−1 , ym )p(1 − tm + t1 , ym , y1 )
(5.6)
−1
# Z
F(y1 , . . . , ym )  p(1, x, x)dx .
R2
In other words, the joint law of (ω(t1 ), . . . , ω(tm )) under µ admits a density with
respect to the Lebesgue measure on R2m that is given by
q = q(t2 − t1 , . . . ,tm ,tm−1 , 1 − (tm − t1 ); y1 , . . . , ym )
(5.7)
= (p(t2 − t1 , y1 , y2 ) . . .

p(tm − tm−1 , ym−1 , ym )p(1 − tm + t1 , ym , y1 )) 
Z
−1
p(1, x, x)dx
.
R2
Then, the above result yields for any bounded, measurable F on R2mk and for each
fixed k ≥ 1
Z
dµN F(Ωk (t1 ), . . . , Ωk (tm ))
=
Z
dµN,k F(Ωk (t1 ), . . . , Ωk (tm ))
−→
Z
N
=
dµ(ω1 ) · · · dµ(ωk )F(Ωk (t1 ), . . . , Ωk (tm ))
Z
F(Yk1 , . . . ,Ykm )q1 · · · qk dY
R2mk
j
where Yk
j
j
(y1 , . . . , yk )
=
for 1 ≤ j ≤ m, qi = q(y1i , . . . , ym
i ) for 1 ≤ i ≤ k. Notice that
the convergence is indeed valid for any bounded, measurable F as a consequence
of the strong convergence of the densities hN,k .
The proof of the above result is given in Sections 5.2 and 5.3. We first draw
some consequences of it.
C OROLLARY 5.3 For any m ≥ 1, 0 ≤ t1 ≤ · · · ≤ tm ≤ 1, the empirical law N1 (δω1 +
· · ·+δωN ) under µN weakly converges (in the sense of probability measures on R2m )
to q(s1 , . . . , sm ; y1 , . . . , ym )dy where we denote by
s1 = t 2 − t 1 ,
...
,
sm−1 = tm − tm−1 ,
sm = 1 − (tm − t1 ) ,
EQUILIBRIUM STATISTICAL THEORY
97
and ωi = (ωi (t1 ), . . . , ωi (tm ))
for 1 ≤ i ≤ N. More precisely, we have for any p ∈ [1, +∞)
1
N
(5.8)
N
∑ ϕ(ω j ) −
j=1
Z
ϕq dy
R2m
−→ 0
N
L p (ΩN ,µN )
for any ϕ ∈ L p (R2m ) + L∞ (R2m ).
P ROOF OFR C OROLLARY 5.3: We begin with the simple case when p = 2. Setting ϕ
e = ϕ − R2m ϕq dy, we then have
1
N
N
∑ ϕ(ω j ) −
j=1
=
N
1
N 2 i6∑
=j
Z
2
Z
ϕq dy
R2m
L2
dµN ϕ(ω
e i )ϕ(ω
e j) +
N(N − 1)
=
N2
Z
1 N
∑
N 2 i=1
Z
dµ20 ϕ(ω
e 1 )ϕ(ω
e 2 )hN/2 +
dµN ϕ(ω
e i)
1
N
Z
dµ10 ϕ
e2 (ω1 )hN,1
using the symmetries of µN . Then this converges, as N goes to +∞, to
Z
2
Z
2
dµ0 ϕ(ω
e 1 )ϕ(ω
e 2 )h(ω1 )h(ω2 ) =
dµ0 ϕ(ω)h(ω)
e
in view of Theorem 5.1, provided we check that ϕ(ω)
e
∈ L2 (Ω, µ0 ). Then we finish
the proof easily for p = 2, since we have
Z
dµ0 ϕ(ω)h(ω)
e
=
Z
dµ ϕ(ω(t1 ), . . . , ω(tm )) −
Z
ϕq dy = 0 .
R2m
Finally, ϕ(ω)
e
∈ L2 (Ω, µ0 ) since ϕ
e ∈ L2 + L∞ (R2m ), in view of the explicit density
of (ω(t1 ), . . . , ω(tm )) under µ0 exhibited in (2.16) and (2.17) of Section 2 (which
belongs to the Schwartz class S of rapidly decreasing smooth functions—it is a
Gaussian).
For a general exponent p, we write ϕ = ϕ1 + ϕ2 with ϕ1 ∈ L p , ϕ ∈ L∞ , and
we decompose ϕ1 into ϕ1 1(|ϕ|<R) + ϕ1(|ϕ|≥R) . Then, for each R ∈ (0, ∞), ψ =
ϕ2 + ϕ1 1(|ϕ|<R) ∈ L∞ + L p ∩ L∞ ⊂ (L2 + L∞ ) ∩ L∞ . Hence, by the preceding proof
1 N
∑ ψ(ωi ) −
N i=1
Z
R2m
ψq dy −→ 0 in L2 ΩN , µN
N
98
P.-L. LIONS AND A. MAJDA
and thus in L p if p ≤ 2; it also converges to 0 in L p if p > 2 since it is obviously
bounded in L∞ by 2kψkL∞ . We conclude observing that we have
Z
ϕ1 q1(|ϕ1 |≥R) dy → 0 as R → +∞
R2m
(since q ∈ L1 ∩ L∞ (R2m )) and
1 N
∑ ϕ1(ωi )1(|ϕ1 (ωi )|≥R)
N i=1
≤ kϕ1 (ω1 )1(|ϕ1 (ω1 )|≥R) kL p (ΩN ,µN )
L p (ΩN ,µN )
1/p
= k|ϕ1 (ω1 )| p 1(|ϕ1 (ω1 )|≥R) hN,1 kL1 (Ω,µ0 )
≤ Ck|ϕ1 (ω1 )|1(|ϕ1 (ω1 )|≥R) kL p (Ω,µ0 )
≤ Ck|ϕ1 |1(|ϕ1 |≥R) kL p (R2m ) → 0
as R → +∞ ,
where C denotes various positive constants independent of N.
As mentioned above, the proof of Theorem 5.1 is given in the next two subsections. In Section 5.2, we present the heart of the matter, leaving aside the verification of some technical (but crucial) facts that are proved in Section 5.3. In
particular, as in the case of point vortices (see Caglioti et al. [3] or Lions [17]),
the proof relies upon a variational characterization of µN and more precisely of hN ,
which yields asymptotically the following variational characterization of h (also
proved in the next subsections):
T HEOREM 5.4 The mean field density h is a continuous, bounded function on Ω
and is the unique minimum of the following free energy functional:
min{F( f ) : f ≥ 0, f ∈ L∞ (Ω, µ0 ), E0 ( f ) = 1}
(5.9)
where
(5.10)
F( f ) =
1
µ
a
E0 ( f log f ) + E0 (Vb0 f ) + E 2 (Vb (ω − ω 0 ) f (ω) f (ω 0 ))
β
2β
2
and we denote by
Vb0 (ω) =
Z 1
0
|ω(0)| dσ ,
2
1
Vb (ω − ω 0 ) = −
2π
Z 1
0
log |ω(σ) − ω 0 (σ)|dσ .
Remark 5.5. It is possible to extend the minimization class in (5.9) to those f ∈
1 (Ω , µ ) with E( f | log f |) < ∞. It is also possible to obtain a variational characL+
0 0
β
terization of the density g of µ with respect to dx dνx,x
= d µ̄. It is given by


Z


(5.11)
min F ( f ) : f ≥ 0, f ∈ L1 ∩ L∞ (Ω, µ̄), f d µ̄ = 1


Ω
EQUILIBRIUM STATISTICAL THEORY
with
F (f) =
(5.12)
1
β
Z
f log f d µ̄(ω) +
Ω
a
+
2
ZZ
µ
β
Z
99
Vb0 (ω) f d µ̄(ω)
Ω
Vb (ω − ω 0 ) f (ω) f (ω 0 )d µ̄(ω)d µ̄(ω 0 ) .
Ω×Ω
In fact, this formulation may be deduced from the preceding one (and is equivalent
to the preceding one). We prefer to work with µ0 instead of µ̄, since µ0 (Ω) = 1
while µ̄(Ω) = +∞!
Remark 5.6. We note that the entropy functional E0 ( f log f ) is nothing but the
relative entropy of µ̂ = f · µ0 with respect to µ0 , namely,
Z
Z
d µ̂
E0 ( f log f ) = f log f d µ̄ = log
d µ̂ .
dµ0
Ω
Ω
Remark 5.7. Let us check immediately that F is well-defined and in fact finite on
1 ∩ L∞ and that F is indeed strictly convex. The first term is obviously finite and
L+
strictly convex since (t 7→ t logt) is strictly convex on [0, ∞) and
f − 1 ≤ f log f ≤ k f kL∞ log[max(k f kL∞ , 1)] .
In particular, if E0 ( f ) = 1, this term is obviously nonnegative.
The second term is linear in f and clearly nonnegative. In addition, we have
0 ≤ Vb f ≤ Vb k f kL∞ and
R
E0 (Vb ) = E0 (|ω(0)| ) =
2
R2
|x|2 q(x, x, 1)dx
R
q(x, x, 1)dx
<∞
R2
by the time shift invariance and the explicit representation of q, (2.17).
The third term is obviously quadratic in f , and we claim it is both finite and
1 ∩ L∞ . This suffices to complete the proof of the above
nonnegative for each f ∈ L+
claim on F—let us remark in passing that it also shows the nonnegativity of F. We
first observe that we have
logt ≤ t − 1 for all t ≥ 0 ,
(5.13)
hence,
Vb (ω − ω 0 ) f (ω) f (ω 0 ) ≤ k f k2L∞
while we have
Vb (ω − ω 0 ) f (ω) f (ω 0 ) ≥ −
Z 1
0
Z 1
0
|ω(σ)| + |ω 0(σ)|dσ ∈ L1 (Ω)
| log |ω(σ) − ω 0 (σ)|1(|ω(σ)−ω0 (σ)|≤1) dσk f k2L∞
100
P.-L. LIONS AND A. MAJDA
and by the time shift invariance of µ0
E
2
Z 1
0
| log |ω(σ) − ω 0 (σ)||1(|ω(σ)−ω0 (σ)|≤1) dσ
= E02 | log |ω(0) − ω 0 (0)||1(|ω(0)−ω0 (0)|≤1)

ZZ
=


| log |x − y||1(|x−y|≤1) q(x, x, 1)q(y, y, 1)dx dy
R2 ×R2
Z

−2
q(x, x, 1)dx
< +∞ ,
R2
since


q µ


r 
 p cosh
β −1
µβ
µ
2
q |x|
sinh
exp − µβ
.
µ


π
β


sinh
√
q(x, x, 1) =
β
Finally, in order to prove the nonnegativity of this quadratic term, it clearly suffices
to check it when f = F(ω(t1 ), . . . , ω(tm )) where m ≥ 1, 0 ≤ t1 < t2 < · · · < tm ≤ 1,
F ∈ C0∞ (Rm ). We observe that for each σ ∈ [0, 1]
ZZ Ω×Ω
−
1
log |ω(σ) − ω 0 (σ)|
2π
F(ω(t1 ), . . . , ω(tm ))F(ω 0 (t1 ), . . . , ω 0 (tm ))dµ0 (ω)dµ0 (ω 0 )
ZZ 1
=
− log |x − y| G(x)G(y)dx dy
2π
R2 ×R2
for some smooth and rapidly decreasing G, in view of (2.16)–(2.17), and this expression 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.
P ROPOSITION 5.8 The density hN is the unique minimum of the following strictly
convex functional:
(5.14)
FN = min{F N ( f ) : f ≥ 0, f ∈ L∞ (ΩN , µN0 ), E0N ( f ) = 1}
EQUILIBRIUM STATISTICAL THEORY
101
where
1
µ
F ( f ) = E0N ( f log f ) + E0N
β
2β
N
(5.15)
∑ Vb0 (ωi )
f
i=1
1 N b
V (ωi − ω j ·)
2 i6∑
=j
a
+ E0N
N
! !
N
! !
f
.
Remark 5.9. The same argument as in Remark 5.7 (Section 5.1) shows that each
1 ∩ L∞ and nonnegative on the miniterm in F n is well-defined (and finite) on L+
mization class.
P ROPOSITION 5.10 There exists a positive constant C0 such that for all N ≥ k ≥ 1
0 ≤ hN,k ≤ C0k
(5.16)
on ΩN .
We can now give the proofs of Theorems 5.1 and 5.4, which we divide into
several 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 a
subsequence, still denoted by N in order to simplify notation, such that
hN,k → hk
weakly L∞ ∗
for some bounded, measurable hk ≥ 0 such that E0k (hk ) = 1. Since hN,k is symmetric
in (ω1 , . . . , ωk ), so is hk . Furthermore,
Z
hence, we have
hN,k+1 dµ0 (ω k+1 ) = hN,k ;
Z
hk+1 dµ0 (ω k+1 ) = hk .
Applying the classical Hewitt-Savage theorem, we deduce that there exists a prob1 (Ω, µ ) : E ( f ) = 1} supported on the ball (in L∞ )
ability measure π̄ on { f ∈ L+
0
0
{ f ∈ P : k f kL∞ ≤ C0 } in view of the bound (5.16) such that we have for all k ≥ 1
(5.17)
h =
k
Z
k
∏ f (ωi ) d π̄( f )
a.s. in Ωk .
i=1
We then denote by P the set of all probability measures π on P supported in an
arbitrary ball of L∞ .
We now claim that π is a minimum of the following free energy functional:
b
b
F = min F(π)
:π∈P
(5.18)
102
P.-L. LIONS AND A. MAJDA
where Fb is given by
Z
Z
1
µ
b
b
F(π) = E0
f log f dπ( f ) + E0 V0 f dπ( f )
β
2β
(5.19)
ZZ
a 2 b
0
0
+ E0 V (ω − ω )
f (ω) f (ω )dπ( f ) .
2
Once more, the assumption made upon the support of π allows us to check, as in
Remark 5.7 (Section 5.1), that each term in Fb is finite and nonnegative on P .
In order to prove our claim, we first observe that we obviously have
! !
N
1 N
E
∑ Vb0 (ωi) hN = E0 (Vb0 hN,1)
N 0
i=1
(5.20)
Z
−→ E0 (Vb0 h1 ) = E0 Vb0 f d π̄( f )
N
(5.21)
!
1 N N b
N
V (ωi − ω j )h
E
N 2 0 i6∑
=j
N −1
=
E02 (Vb (ω − ω 0 )hN,2 (ω, ω 0 )) −→ E02 (Vb (ω − ω 0 )h2 )
N
N
= E0 (Vb0 (ω − ω 0 )
Z
f (ω) f (ω 0 )d π̄( f ))
where, for instance, we use the observations made in Remark 5.7 (Section 5.1)
to check that one can pass to the limit as N goes to +∞ despite the growth and
singularities of Vb0 and Vb . In conclusion, we have shown for some εN −→ 0 that
N
Z
1 N
1 N N
1 N N
µ
N
b
F = F (h ) =
E (h log h ) + E0 V0 f d π̄( f )
N
N
Nβ 0
2β
(5.22)
Z
a
+ E0 (Vb0 (ω − ω 0 ) f (ω) f (ω 0 )d π̄( f )) + εN .
2
R N
N
Similarly, denoting by f = ∏i=1 f (ωi )dπ( f ) for any π ∈ P , we have for some
δN −→ 0
N
!!
Z
N
1
µ
N
N
b
F(π)
= E0
f log f dπ( f ) + E0
∑ Vb0 (ωi ) f
β
2β
i=1
!
(5.23)
a N N b
N
+
V (ωi − ω j ) f
+ δN .
E
2N 0 i6∑
=j
We then recall some classical facts on the entropy (see, for instance, Ruelle
[22]):
(5.24)
E0N ( f N log f N ) ≥ E0m ( f N,m log f N,m ) + E0N−m ( f N,N−m log f N,N−m )
EQUILIBRIUM STATISTICAL THEORY
103
for each symmetric probability density f N on ΩN , and for all 1 ≤ m ≤ N − 1,
1
lim E0N ( f N log f N ) = E0
N N
(5.25)
Z
f log f dπ( f )
R
for each π ∈ 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 setting
we use). (5.24) follows readily from the convexity inequality
f N log
fN
N,m
f gN,m
+ f N,m gN,m − f N ≥ 0
where gN,m = f N,N−m (ωm+1 , . . . , ωN ). Then, (5.24) implies that we have for any
N≥k≥1
(5.26)
1 N N
1
1
E0 ( f log f N ) ≥ E0k ( f N,k log f N,k ) + E0r ( f N,r log f N,r )
N
k
N
k
R
with r = N − Nk . In particular, if f N = ∏Ni=1 f (ωi )dπ( f ) for some π ∈ P , f N,k =
f k for all 1 ≤ k ≤ N and thus N1 E0N ( f N log f N ) converges, in view of (5.26), as N
goes to +∞. The limit obviously coincides with E0 ( f log f ) when π is concentrated
on { f }. Therefore, (5.25) follows upon proving that this limit is linear in π. This
is obvious if we use the following inequality valid for all a, b ≥ 0:
0≤
1
1
a+b
a + b log 2
a log a + b log b −
log
≤
|a − b| .
2
2
2
2
2
Indeed, we then deduce
N
f1N + f2N
f1 + f2N
log 2 N N
log
+
E | f − f2N |
2
2
2N 0 1
1 1 N N
1 1 N N
N
N
≥
E ( f log f1 ) +
E ( f log f2 )
2 N 0 1
2 N 0 2
N
N
f1 + f2N
f1 + f2N
1 N
≥ E0
log
,
N
2
2
1 N
E
N 0
and we finish the proof since E0N (| f1N − f2N |) ≤ E0N ( f1N ) + E0N ( f2N ) = 2.
Collecting now (5.22), (5.23), and (5.24)–(5.26), we deduce on the one hand
that for each π
1
1
1
b
F(π)
= lim F N ( f N ) ≥ FN = F N (hN ) ;
N N
N
N
104
P.-L. LIONS AND A. MAJDA
hence b
F ≥ lim N1 FN . On the other hand, we have for each k ≥ 1
N
lim
N
1 N
1
F = lim F N (hN )
N
N N
Z
1 k k
µ
k
b
≥ E0 (h log h ) + E0 V0 f d π̄( f )
β
2β
Z
a
0
0
+ E0 Vb (ω − ω ) f (ω) f (ω )d π̄( f ) ;
2
hence, letting k go to +∞,
lim
N
1 N b
F.
F ≥ F(π̄) ≥ b
N
And we have shown that
= N1 F N (hN ) converges, as N goes to +∞, to
b π̄) = b
F(
F. Finally, we have also shown that we have
Z
1 N N
N
(5.27)
f log f d π̄( f ) .
E h log h −→ E0
N
N 0
1 N
NF
Step 2: Strong Convergence to the Unique Minimum
We first show that π̄ is concentrated on the unique minimum of F. Indeed, we
observe that we have
b
b π̄) =
F = F(
Z
F( f )π(d f ) ≥ F ,
b fn ) = F( fn ) converges by defiwhile if fn is a minimizing sequence of F, then F(δ
b
nition to F. Hence, F = F and F( f ) = F π-a.s. This shows that F admits at least a
minimum h ∈ L∞ (khkL∞ ≤ C0 ), and since F is strictly convex on P (see Remark 5.7
in Section 5.1), π̄ = δh where π̄ is the unique minimum of F over P. In other words,
we have shown at this stage that, for all k ≥ 1, hN,k converges weakly (in L∞ (Ωk )
weak ∗) to hk = ∏ki=1 h(ωi ). Furthermore, this convergence, by the uniqueness of
the limit, is in fact true for the whole sequence and not only for any particular
subsequence we extracted in Step 1.
In addition, in view of (5.26) and (5.27),
1 k k
1
E0 h log hk = E0 (h log h) = lim E0N hN log hN
N N
k
1
≥ lim E0k (hN,k log hN,k ) .
N k
This, combined with the strict convexity of the entropy, yields, as is well-known,
the strong convergence in L1 (Ωk ) of hN,k to hk , and thus in L p (Ωk ) for all 1 ≤ p < ∞
in view of the uniform bound (5.16).
We also observe that, since F( f (θt ·)) = F( f ) for all t ≥ 0, the uniqueness of
the minimum implies that h(θt ω) = h(ω) a.s. in Ω for all t ≥ 0, i.e., the translation
EQUILIBRIUM STATISTICAL THEORY
105
invariance that we were expecting in view of the phenomenon already observed for
the broken path models in the preceding section.
Another consequence of these facts concerns the law of ω(σ) under µ, which,
by the invariance of µ0 and h, is independent of σ ∈ [0, 1]. Indeed, letting ϕ ∈
L∞ (R2 ), we have
E[ϕ(ω(0)] = E0 [ϕ(ω(0))h]
and thus
|E[ϕ(0))]| ≤ C0
Z
|ϕ(ω(0)|dµ0 (ω) ≤ C
Ω
Z
|ϕ(x)|q(x, x, 1)dx
R2
≤C
Z
|ϕ(x)|e−δ|x| dx
2
R2
for some positive constants C and δ independent of ϕ.
This bound shows that the law of ω(σ) for all σ ∈ [0, 1] under µ admits a density
ρ with respect to the Lebesgue measure on R2 and that we have
(5.28)
0 ≤ ρ ≤ Ce−δ|x|
2
on R2 .
In other words, not only is this density bounded on R2 , 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 EulerLagrange equation satisfied by the minimum h of F over P. Once more, because
the verification is purely technical, we will not address it until the next subsection.
L EMMA 5.11 The minimum h of F over P satisfies the following Euler-Lagrange
equation:
(5.29)



Z 1



1
µ


h = 0 exp −
dσ  |ω(σ)|2




Z
2
0








Z


aβ

0
0
0 


−
log
|ω(σ)
−
ω
(σ)|h(ω
)dµ
(ω
)
0



2π

Ω





 Z 1



0
 µ |ω(σ)|2

exp
−
Z
=
E
dσ
0




2

0







Z



aβ

0
0
0 

−
log
|ω(σ)
−
ω
(σ)|h(ω
)dµ
(ω
)

0



2σ
Ω
106
P.-L. LIONS AND A. MAJDA
At this stage, in order to complete the proofs of Theorems 5.1 and 5.4, there
only remains to prove (5.1), the PDE characterization of ρ as the unique solution
of (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 of µ0 , while the radial symmetry follows from the uniqueness of the solution of (5.4)–(5.5) together with the
elementary rotational invariance of that system of equations.
In order to prove (5.1), we first remark that, by the invariance of h and µ0 , the
expression
Z
1
log |x − ω 0 (σ)|h(ω 0 )dµ0 (ω 0 )
−
2π
Ω
is independent of σ ∈ [0, 1] for all x ∈ R2 and thus is a function of x only. We
denote by Ψ(x) this potential, and we observe that we have
1
1
0
(5.30)
Ψ(x) = − E(log |x − ω (0)|) = − log |x| ∗ ρ
2π
2π
by the definition of ρ. We observe that the bound in (5.28) immediately shows that
2,p
Ψ is radial, nonincreasing, C1 on R2 (in fact, Wloc
(R2 ), DΨ ∈ W 1,p (R2 ) for all
p ∈ [2, ∞], D2 Ψ ∈ L p (R2 ) for all p ∈ (1, ∞]—even for p = +∞ since ρ is radial—
1
and −∆Ψ = ρ on R2 ) with Ψ(x) = − 2π
log |x| + O(1/|x|) as |x| goes to +∞. Thus,
the representation (5.1) of h is shown, as well as the continuity of h over Ω.
Next, the PDE characterization of ρ, namely, (5.4)–(5.5), is now immediate
since the total potential aβΨ + µ|x|2 is smooth on R2 (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 consequence
of another variational argument that we present below in Section 5.3. Finally, the
smoothness of ρ follows from regularity results for parabolic equations: Indeed,
1,α
for all α ∈ (0, 1), Schauder estimates easily yield that
since aβΨ + µ|x|2 ∈ Cloc
3,k
2
2
p(1, x, y) ∈ C (R × R ) and thus ρ ∈ C3,α (R2 ) for all α ∈ (0, 1). Hence, Ψ ∈
C5,α (R2 ) and we may bootstrap the regularity exponents, showing thus that ρ ∈
C∞ (R2 ). One can also check easily that all derivatives of ρ have at least some
Gaussian decay as ρ has, as shown by (5.28).
5.3 Some Technical Facts
We begin with the proofs of the formally obvious Proposition 5.8 and Lemma 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 EulerLagrange equations associated with the convex variational problems (5.14) and
(5.10), respectively.
P ROOF OF P ROPOSITION 5.8: We first observe that log hN is easily seen to belong to L p (ΩN ) for all 1 ≤ p < ∞ by arguments similar to ones made several times
EQUILIBRIUM STATISTICAL THEORY
107
above. This allows one to write the following convexity inequality valid for all
f ∈ L∞ (ΩN ):
f log f ≥ hN log hN + (log hN + 1)( f − hN )
"
#
aβ
Vb (ωi − ω j ) ( f − hN ) + f − hN .
= hN log hN + µVb0 +
2N i6∑
=j
Hence, taking the expectation with respect to µ0 , we conclude
F N ( f ) ≥ F N (hN ) + E0N ( f − hN ) = F N (hN )
of E0N ( f ) = 1.
P ROOF OF L EMMA 5.11: In order to circumvent the possible vanishing of the
minimum h, we follow an argument introduced in Caglioti et al. [3] (see also Lions
[17]). We consider, for δ > 0 (small enough), the event Bδ = {h ≥ δ}, and we set
B = limδ↓0+ ↑ Bδ . Of course, µ0 (B) > 0 since E0 (h) = 1. Then, h is still a minimum
of δ over the set { f ∈ L∞ (Ω, µ0 ), E0 ( f ) = 1, f ≥ 0, f = h a.s. on Bcδ }. Since h does
not vanish on Bδ , we may now write the Euler-Lagrange equation associated to that
restricted minimization problem, and we find



Z 1
Z


µ
1
aβ
h = 0 exp − dσ  |ω(σ)|2 −
log |ω(σ) − ω 0 (σ)|h(ω 0 )dµ0 (ω 0 )


Zδ
2
2π
0
Ω
a.s. on Bδ
where


 Z 1
µ
Zδ0 = E0  exp − dσ  |ω(σ)|2

2
0

−
aβ
2π
Z
Ω
 

E0 [h1Bδ ]−1 .
log |ω(σ) − ω 0 (σ)|h(ω 0 )dµ0 (ω 0 )

Bδ
Letting δ go to 0+ , we easily deduce
(5.31)



Z
 Z 1

1
σ
aβ
h = exp −
dσ  |ω(σ)|2 −
log |ω(σ) − ω 0 (σ)|h(ω 0 )dµ0 (ω 0 )
 0

2
2π
Ze0
Ω
a.s. on B
108
P.-L. LIONS AND A. MAJDA
where


 Z 1
µ
Ze = E0 exp − dσ  |ω(σ)|2

2
0

−
aβ
2π
Z
Ω
 

log |ω(σ) − ω 0 (σ)|h(ω 0 )dµ0 (ω 0 ) 1B

since E0 [h1Bδ ] → E0 [h1B ] = E0 [h] = 1 as δ goes to 0+ .
We conclude by proving by contradiction that µ0 (A) = 0 where A = BC . Indeed,
if µ0 (A) = a > 0, we may consider, for δ > 0, h̃ = (h + δ1A )(1 + aδ)−1 , and we
check easily, using the boundedness of h, that we have
Z
Z
1
1
h̃ log h̃ dµ0 (ω) −
F(h̃) ≤ F(h) +Cδ +
h log h dµ0 (ω)
β
β
Z
h
1
h
≤ F(h) +Cδ +
log
dµ0 (ω)
β 1 + aδ
1 + aδ
Z
1
1 aδ
δ
−
h log h dµ0 (ω) +
log
β
β 1 + aδ
1 + aδ
a
≤ F(h) +Cδ + δ log δ < F(h) for δ small enough,
β
where C denotes various positive constants independent of δ > 0. The contradiction
completes 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 the
preceding subsection (Section 5.2).
P ROOF OF P ROPOSITION 5.10: The bounds (5.16) may be obtained by a rather
straightforward chain of estimates that involve the following quantities defined for
1 ≤ l ≤ N, µ > 0,
( "
#)
Z
aβ l b
µ l b
0
Z (N, l, µ) = exp −
V (ωi − ω j ) + ∑ V0 (ωi )
2N i6∑
2 i=1
=j
Ωl
dµ0 (ω1 ) · · · dµ0 (ωl ) ,
so that, in particular, Z 0 (N) = Z 0 (N, N, µ). Finally, we denote by C various positive
constants independent of N and k.
Step 1: 0 ≤ hN,k ≤ Z0 (N, N − k, µ(1 − Nk ))Z0 (N, N, µ)−1 Ck
By definition, we have
hN,k =
Z
hN dµ0 (ωk+1 ) · · · dµ0 (ωN )
EQUILIBRIUM STATISTICAL THEORY
109
#)#
"
( "
aβ k b
1
µ k b
V (ωi − ω j ) + ∑ V0 (ω)
= 0
exp −
Z (N)
2N i6∑
2 i=1
=j
!
Z
aβ k N
exp − ∑ ∑ Vb (ωi − ω j )
N i=1 j=k+1
ΩN−k
"
( "
#)#
N
N
aβ
µ
· exp −
∑ Vb (ωi − ω j ) + 2 ∑ Vb0 (ωi )
2N i6= j≥k+1
i=k+1
dµ0 (ωk+1 ) · · · dµ0 (ωN ) .
We then use (5.13) in order to estimate
−
aβ k b
µ k
V (ωi − ω j ) − ∑ Vb0 (ωi )
∑
2N i6= j
2 i=1
≤
aβk k
∑
2N i=1
≤ Ck −
Z 1
0
dσ|ωi (σ)| −
µ k b
∑ V0 (ωi )
2 i=1
µ k b
aβ k N b
µ k
V0 (ωi ) −
V (ωi − ω j ) − ∑ Vb0 (ωi )
∑
∑
∑
4 i=1
N i=1 j=k+1
4 i=1
k
≤ aβ ∑
Z 1
i=1 0
dσ|ωi (σ)| −
≤ Ck + µ
k
4N
≤ Ck + µ
k
4N
µ k b
k
V0 (ωi ) + aβ
∑
4 i=1
N
N
∑
Z 1
j=k+1 0
dσ|ω j (σ)|dσ
N
k
Vb0 (ω j ) +C (N − k)
N
j=k+1
∑
N
∑
Vb0 (ω j ) .
j=k+1
Collecting these estimates, we deduce
( "
#)
N
Z
N
k
aβ
C
µ
k
hN,k ≤ 0
exp −
∑ Vb (ωi − ω j ) + 2 1 − 2N ∑ Vb0 (ωi )
Z (N)
2N i6= j≥k+1
i=k+1
ΩN−k
dµ0 (ωk+1 ) · · · dµ0 (ωN ) ;
hence we have obtained the desired estimate
k
(5.32)
Z 0 (N, N, µ)−1 .
0 ≤ hN,k ≤ Ck Z 0 N, N − k, µ 1 −
2N
Step 2: Z0 (N, N − k, µ) ≤ Ck Z0 (N, N, µ)
It suffices to show the inequality
(5.33)
Z 0 (N, l, µ) ≤ CZ 0 (N, l + 1, µ)
110
P.-L. LIONS AND A. MAJDA
for all 1 ≤ l ≤ N −1 for some constant that only depends on a lower bound on µ.
We write
Z 0 (N, l + 1, µ)
( "
Z
#)
aβ l+1 b
µ l+1 b
V (ωi − ω j ) + ∑ V0 (ωi ) dµ0 (ω1 ) · · · dµ0 (ωl+1 )
= exp −
2N i6∑
2 i=1
=j
(
)
Z
Z
aβ l b
µ l b
= exp −
V
(ω
V
−
ω
)
+
(ω
)
dµ
(ω
)
·
·
·
dµ
(ω
)
dµ0 (ω)
i
j
0 1
0 l
∑ 0 i
2N i6∑
2 i=1
=j
( "
#)
aβ l b
µb
· exp −
∑ V (ωi − ω) + 2 V0 (ω)
N i=1
( "
#)
Z
l
aβ
µ
≥ Z 0 (N, l, µ) inf
dµ0 (ω) exp −
∑ Vb (ωi − ω) + 2 Vb0 (ω)
N i=1
(ω1 ,...,ωl )
"
(
)#
Z
aβ l b
µb
0
≥ Z (N, l, µ) exp
inf
dµ0 (ω) − ∑ V (ωi − ω) − V0 (ω)
N i=1
2
(ω1 ,...,ωl )
using Jensen’s inequality. Then we write
(
)
Z
aβ l b
µb
inf
dµ0 (ω) −
∑ V (ωi − ω) − 2 V0 (ω)
N i=1
(ω1 ,...,ωl )
)
(Z
Z 1
aβ l
≥ −C +
dσ inf
dµ0 (ω) ∑ log |ωi (ω) − ω(σ)|
N i=1
(ω1 ,...,ωl )
0
≥ −C +
Z 1
Z
dσ
0
inf
(x1 ,...,xl )∈R2l
aβ
≥ −C +
inf
N (x1 ,...,xl )∈R2l
Z
dµ0 (ω)
aβ l
∑ log |xi − ω(σ)|
N i=1
l
dµ0 (ω) ∑ log |xi − ω(0)|
i=1
using the invariance of µ0 by time shifts. Then we remark that we have
1
inf
N (x1 ,...,xl )∈R2l
≥
=
l
inf
N x∈R2
Z
Z

l
inf 
N x∈R2
l
dµ0 (ω) ∑ log |xi − ω(0)|
i=1
dµ0 (ω) log |x − ω(0)|
Z

q(y, y, 1) log |x − y|dy R(1)−1 ≥ −C .
R2
Therefore, (5.33) holds and we deduce
k
k
0
k 0
Z N, N − k, µ 1 −
(5.34)
≤ C Z N, N, µ 1 −
.
2N
2N
EQUILIBRIUM STATISTICAL THEORY
111
Combining (5.32) and (5.34), we have shown the following bound on hN,k :
k
N,k
k 0
0 ≤ h ≤ C Z N, N, µ 1 −
(5.35)
Z 0 (N, N, µ)−1 .
2N
Step 3: Conclusion
Using Hölder’s inequality with p = (1 − k/2N)−1 (1 + k/2N), we obtain
k
0
Z N, N, µ 1 −
2N
( "
#)
Z
aβ N b
µ N b
N
V (ωi − ω j ) + ∑ V0 (ωi )
≤ dµ0 exp −
2N i6∑
2 i=1
=j
( "
#)
N
N
aβ
µ
k
· exp −
(p − 1) ∑ Vb (ωi − ω j ) +
∑ Vb0 (ωi) .
2N
2
2N
i=1
i6= j
Next, we observe that we have, in view of (5.13),
"
#
N
N
aβ
µk
−
(p − 1) ∑ Vb (ωi − ω j ) +
∑ Vb0 (ωi)
2N
4N
i=1
i6= j
N
aβ
≤
(p − 1) ∑
2
i=1
k N
≤ aβ ∑
N i=1
Z 1
0
Z 1
0
µk N
dσ|ωi (σ)|dσ −
∑
4N i=1
µk N
dσ|ωi (σ)|dσ −
∑
4N i=1
Z 1
0
Z 1
0
dσ|ωi (σ)|2
dσ|ωi (σ)|2
≤ Ck .
Therefore, we deduce
Z
0
k
≤ Ck Z 0 (N, N, µ) ,
N, N, µ 1 −
2N
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 measure µN is completely determined by
the Green’s function pN of the linear parabolic second-order PDE (2.12). Recall
that we choose ā = a/2πN and λ = 0 so that pN solves
!

N
∂pN
βa N
1

N
N


p
−
log
|X
−
X
|
p
+
µ|X j |2 µ = 0
−
∆
X
i
j
∑
∑


∂t
2β
4πN

j=1
i6= j
(5.36)
2N
in R × (0, 1) ,





 pN
= δY (X) on R2N .
l=0
112
P.-L. LIONS AND A. MAJDA
Furthermore, the law of Ω(σ) = (ω1 (σ), . . . , ωN (σ)) under µN is given by
ρN (X) = R
pN (X, X, 1)
.
pN (z, z, 1)dz
R2N
In this section, we first observe that
ρN (X,Y ) = R
pN (X,Y , 1)
,
pN (z, z, 1)dz
R2N
viewed as a kernel, is determined by a variational problem. Then we use this
variational problem to analyze the behavior of ρN as
R N goes to +∞. This will lead
to a Hartree-like variational problem for p(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)) whose
diagonal, p(x, x, 1), is nothing but the law ρ(x) of ω(σ) (for each σ ∈ [0, 1]) under
the mean field measure µ.
N on L2 (R2N ), which is self-adjoint
Of course, ρN is the kernel of an operator ρc
1
N
N = e− 2β ∆X +βU for some potential
and nonnegative since ρc
UN = −
N
µ
a N
log |Xi − X j | + ∑ |X j |2
∑
4πN i6= j
β
j=1
and which has a finite trace since we have
(5.37)
N =
Tr ρc
Z
ρN (X, X)dX = 1 .
R2N
We denote by K1 (R2N ) the closed convex set of such self-adjoint, nonnegative operators with trace equal to 1. We then introduce the following free energy, defined
for all K ∈ K1 (R2N ) by
1
F N (K) = Tr(K log K) + Tr(U N · K) + Tr(H0 K)
(5.38)
β
where H0 = − 2β1 2 ∆X . Of course, we have to make the meaning of Tr(H0 K) precise,
which can be defined by several equivalent formulations such as
Z
1
Tr(H0 K) = 2
(5.39)
∇x · ∇y k(X,Y ) Y =X dX
2β
R2N
or
(5.40)
Tr(H0 K) =
1
2β 2
Z
∑ λi |V ϕ (X)|2 dX
i
R2N
i
where k(X,Y ) is the kernel associated to K, and λ1 ≥ λ2 ≥ · · · are the eigenvalues
of K, while (ϕ1 )i is the orthonormal basis of eigenfunctions of K corresponding to
(λi )i . Obviously, Tr(H0 K) is linear in K, nonnegative, and possibly infinite on K1 .
EQUILIBRIUM STATISTICAL THEORY
113
We first observe that ρ̂N is determined by a variational problem.
P ROPOSITION 5.12 The operator ρ̂N is the unique minimum of the convex functional F N over K1 .
P ROOF : We only sketch the proof, since this is a classical fact in quantum
mechanics. One possible proof consists in observing that the minimization of F N
over K1 is equivalent to the following minimization problem:

Z
1
1
min
λ
log
λ
+
λ
|V ϕi |2 +U N |ϕi |2 dX : 0 ≤ λi , ∀i ≥ 1;
i
i
i
∑
∑
2
 β i≥1
2β
i≥1
R2n
(5.41)
)
∑ λi = 1, (ϕi )i≥1 is an orthonormal basis of L2
,
i≥1
formulation, which amounts to writing any K in K1 as ∑i≥1 λi ϕi (X)ϕi (Y ). Remarking that U N is a potential that is bounded from below, grows at infinity, and belongs
p
(R2N ) (for all 1 ≤ p < ∞), one can easily deduce that up to permutations and
to Lloc
orthogonal transforms (in case of multiple eigenvalues), the Euler-Lagrange equations are equivalent to requiring that (ϕi )i≥1 is an orthonormal basis of eigenvalues
1
∆X +U N corresponding to eigenvalues Λ1 < Λ2 ≤ Λ3 · · · ≤ Λn −→ + ∞, and
of − 2β
n
that
1 −Λi
λi =
(5.42)
Z(N) = ∑ e−Λi .
e ,
Z(N)
i≥1
Therefore, the minimum is given by
1
∑ e−Λi ϕi (x)ϕi(y) ,
Z(N) i≥1
which is nothing but the kernel representation of e− 2β ∆X +U with
1
Z
N
Z(N) = ∑ e−Λi = Tr e− 2β ∆X +U =
pN (X, X, 1)dX .
1
i≥1
N
R2N
We may then consider the reduced operators on L2 (R2k ) for 1 ≤ j ≤ N, defined
by the following kernels:
(5.43)
ρN, j (X,Y ) =
Z
1
ρN (X, z j+1 , . . . , zN ;Y , z j+1 , . . . , zN )dz j+1 · · · dzN .
Z(N)
R2(N− j)
Let us observe that the law of (ω1 (σ), . . . , ω j (σ)) (for each σ ∈ [0, 1]) under µN
admits the density ρN, j (X, X) with respect to Lebesgue measure R2 j . In addition,
ρN, j ∈ K1 (R2 j ). And we have the following:
114
P.-L. LIONS AND A. MAJDA
T HEOREM 5.13 (i) There exist some positive constants C, δ > 0 independent of
1 ≤ k ≤ N such that we have
δ
0 ≤ ρN,k (X,Y ) ≤ Ck e− 2 (|X|
(5.44)
2 +|Y |2 )
.
(ii) As N goes to +∞, ρN,k converges in L p (R2k × R2k ) (for all 1 ≤ p < ∞), for
each k ≥ 1, to ∏kj=1 ρ(x j , y j ) where ρ is the kernel of the unique minimum of
the following strictly convex free energy functional:
min{F(K) : K ∈ K1 (R2 )}
(5.45)
where
(5.46)
F(K) =
1
a
Tr(K log K) + Tr((H0 +V0 )K) + Tr(V1,2 K ⊗ K)
β
2
1
log |x1 − x2 |, so that
where V0 (x) = µ|x|2 , V1,2 = − 2π
1
Tr(V1,2 (K ⊗ K)) = −
2π
ZZ
log |x − y|k(x, x)k(y, y)dx dy .
R2 ×R2
(iii) The minimum ρ̂ of F over K1 is given by
ρ(x, y) = R
p(x, y, 1)
p(x, x, 1)dx
R2
where p is the Green’s function of (5.5) and ρ(x, x) ≡ ρ(x) on R2 where ρ is
the density determined in Theorem 5.1.
Remark 5.14. The mean field minimization problem (5.45)–(5.46) is nothing but
a temperature-dependent Hartree model for bosons interacting with a logarithmic
2
potential in an external potential given by V0 (and ~m = β12 · · · ). We refer the interested reader to Lions [16] for more mathematical details on temperature-dependent
Hartree or Hartree-Fock equations.
P ROOF OF T HEOREM 5.13: It is possible to make a self-contained proof that
does not rely on any of the facts proved in the preceding section, but we shall not
do so here in order to restrict the length of this paper.
We begin with the proof of (5.44). Since ρN,k is the kernel of a nonnegative
self-adjoint operator, we have
|ρN,k (X,Y )| ≤ ρN,k (X, X)1/2 ρN,k (Y ,Y )1/2
for all X,Y ∈ R2k .
Next, we have for all ϕ ∈ Cb (R2k )
Z
ϕρN,k (X, X)dX = E N [ϕ(ω1 (0), . . . , ωk (0))] = E0N [ϕ(ω1 (0), . . . , ωk (0))hN ]
R2k
= E0k [ϕ(ω1 (0), . . . , ωk (0))hN,k ] .
EQUILIBRIUM STATISTICAL THEORY
115
Hence, thanks to Proposition 5.10,
Z
ϕρN,k dX ≤ C0k E0k [|ϕ(ω1 (0), . . . , ωk (0))|]
R2k
≤C
Z
k
k
|ϕ(x1 , . . . , xk )| ∏ q(x j , x j , 1)dx1 · · · dxk
j=1
R2k
≤ Ck
Z
|ϕ(x1 , . . . , xk )|e−δ|x| dx
2
R2k
and (5.44) follows.
We next prove the convergence of ρN,k to ∏kj=1 ρ(x j , y j ) where ρ is the kernel
described in part (iii) of the above result. In order to do so, we first remark that ρN
is given by
Z
1
β
ρ (X,Y ) =
dνX,Y
(ω)
Z(N)
( "
#)
N
aβ N b
exp −
V (ωi − ω j ) + µ ∑ Vb0 (ωi )
.
2N i6∑
i=1
=j
N
(5.47)
Therefore, we have for all 1 ≤ k ≤ N,
e − k) Z
−µ ∑ Vb0 (ωi )
Z(N
β
(X,Y ) =
dνX,Y
(ω1 , . . . , ωk )e i=1
Z(n)
k
ρ
(5.48)
N,k
Z
k
d µ̃N−k
ωk+1 ,...,ωN ) e
− aβ
N ∑
N
∑ Vb (ωi −ω j )
i=1 j≥k+1
,
where µ̃N−k is defined like µN−k was, replacing
1 n−k b
V (ωi − ω j ) by
N − k i6∑
=j
1 N−k b
V (ωi − ω j ) .
N i6∑
=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 in L p ) for all 1 ≤ p < ∞ to
ρk (X,Y ) given by
(5.49)
1
ρ (X,Y ) =
Zk
Z
k
k
dνxβ1 ,y1 (ω1 ) · · · dνxβk ,yk (ωk )e
k
b i)
−µ ∑ Vb0 (ωi ) −aβ ∑ Ψ(ω
i=1
e
with
Zk =
Z
Z
dX
R2k
k
k
b i)
−µ ∑ Vb0 (ωi ) −aβ ∑ Ψ(ω
β
dνX,X
(ω)e i=1
e i=1
,
i=1
116
P.-L. LIONS AND A. MAJDA
and
1
Ψ = − log |x| ∗ ρ .
2π
Hence, ρk (X,Y ) = Z1k ∏kj=1 p(x j , y j , 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 representation
to a minimum:
1
k(x, y) = ∑ e−Λi ϕi (x)ϕi (y)
Z i≥1
where (ϕi )i≥1 are the eigenfunctions of (− β1 ∆ + µ|x|2 + aΨ), that is, k(x, y) =
1
Z 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 paths
ρδ determined in Theorem 4.1 converges to the mean field measure on continuous
paths µ determined in Theorem 5.1, while the (invariant) density ρδ1 converges to
the density ρ as δ = M1 goes to +∞. We shall not state a result, even though the
statements are easily deduced from the considerations that follow. And, we shall
not provide all the details of the proofs since the topic covered in this section is
more a consistency check than a real necessity for the statistical theories developed
in this paper.
We claim that ρδ converges, as δ goes to 0, to µ in the sense made precise in
Section 2.2 (i.e., as µN,δ converges to µN ), and that ρδ1 converges to ρ in L p (R2 ) for
all 1 ≤ p < ∞ (for instance, the convergence is, in fact, stronger). In order to prove
these claims, we need to introduce some notation. First of all, we denote by
(
)
M−1
M−1
1
β
δµ
δ −M
δ
σ+1
σ 2
σ 2
µ0 =
(5.50)
exp −
,
∑ |x − x | − 2 ∑ |x | 2π β
Rδ
2δ σ=0
σ=0
where
(5.51)
δ
h =
Z
R2M
(
β M−1 σ+1
δµ M−1 σ 2
exp −
|x
− x σ |2 −
∑
∑ |x |
2δ σ=0
2 σ=0
dx0 · · · dxM−1 ,
(5.52)
(
)
M−1
1
δµ M−1 σ 2
δ σ
h = 0 exp −aβδ ∑ Ψ1 (x ) −
∑ |x | ,
Zδ
2 σ=0
σ=0
δ
where
(5.53)
Zδ0 =
Z
R2n
hδ dµδ0 ,
)
δ
2π
β
−M
EQUILIBRIUM STATISTICAL THEORY
117
and
Ψδ1 = −
(5.54)
1
log |x| ∗ ρδ1 .
2π
R
(Recall that ρδ1 (x) = R2(n−1) ρδ (. . . , xσ−1 , x, xσ+1 , . . . )d x̃σ , for all 0 ≤ σ ≤ M − 1,
where d x̃σ0 denotes the integration with respect to all xσ but xσ0 .)
We begin with a few straightforward observations. First of all, µδ0 converges, as
δ goes to 0, to µ0 (in the sense made precise in Section 2.2 above). Once more, this
is a more or less standard consequence of Trotter’s formula. Next, we check, as we
did in Section 5, that hδ is the unique minimum of the following convex variational
problem:
Z
δ
δ
∞
2M
δ
F = min F (h) : h ≥ 0, h ∈ L (R ), h dµ0 = 1
(5.55)
where
F δ (h) =
(5.56)
1
β
+
Z
h log h dµδ0 +
µ
2β
Z M−1
∑ δV0 (xσ )h dµδ0
σ=0
ZZ M−1
a
∑ δV (xσ − yσ )h(x)h(y)dµδ0 (x)dµδ0 (y) .
2
σ=0
As noticed above, each of the three terms defining F δ is nonnegative on the
minimization class defined in (5.55). Choosing, for instance, h ≡ 1, so that
F δ (1) −→
δ
µ
a
E0 (Vb0 ) + E02 (Vb (ω − ω 0 )),
2β
2
we deduce the following a priori bounds:
Z
(5.57)
Z
(5.58)
R2
ZZ
(5.59)
R2 ×R2
hδ log hδ dµδ0 ≤ C
|x|2 ρ21 (x)dx
=
M−1
∑δ
σ=0
Z
|xσ |2 h dµδ0 ≤ C
V (x − y)ρδ1 (x)ρδ1 (y)dx dy =
ZZ M−1
∑ δV (xσ − yσ )h(x)h(y)dµδ0 (x)dµδ0 (y) ≤ C ,
σ=0
where C denotes, here and below, various positive constants independent of δ.
These bounds allow us to obtain some bound on Ψδ . Indeed, on one hand, we
118
P.-L. LIONS AND A. MAJDA
have for all x ∈ R2
1
Ψ (x) = −
2π
δ
≥−
Z
log |x − y|ρδ1 (y)dy
R2
|x|
1
−
2π 2π
Z
1
≥−
2π
Z
|x − y|ρδ1 (y)dy
R2
|y|ρδ1 (y)dy ≥ −C(|x| + 1) ,
R2
in view of (5.58), and, on the other hand, we have
Z
1
δ
hδ log |x − x0 |dµδ0
Ψ (x) = −
2π Z
1
1
δ
≤
h log
dµδ
1
2π
|x − x0 | |x−x0 |≤1 0
Z
Z
1
≤ C hδ log hδ dµδ0 +C
dµδ ≤ C ,
|x − x0 |ν 0
for some ν > 0 small enough (0 < ν < 2), where we used the estimate (5.57). This
bound allows us to obtain the following estimate on hδ :
0 ≤ hδ ≤ C
(5.60)
in R2M ,
and we deduce for some α > 0 independent of δ,
0 ≤ ρδ1 ≤ Ce−α|x|
2
(5.61)
on R2 .
Indeed, we have for any ϕ ∈ C0∞ (R2 ), ϕ ≥ 0,
Z
ϕ(x)ρδ1 (x)dx =
R2
Z
ϕ(x0 )ρδ (x0 , . . . , xM )dx0 · · · dxM−1
R2M
=
Z
ϕ(x0 )hδ µδ0 dx0 · · · dxM−1
R2M
≤C
Z
ϕ(x )
0
Z
µδ0 dx1 · · · dxM−1
dx0
R2
≤C
Z
ϕ(x0 )e−α|x | dx0 ,
0 2
R2
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 that ρδ1 converges weakly in
L p (for all 1 ≤ p < ∞) to some ρ1 , which satisfies (5.61). Because of (5.61), Ψδ1
2,p
1
converges in Wloc
to Ψ1 = 2π
log |x| ∗ ρ1 , V Ψδ1 converges in Lq (R2 ) to V Ψ1 for all
δ
1 ≤ q ≤ ∞, and Ψ1 converges to Ψ1 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 that
ρ1 = ρ (and Ψ1 = Ψ) because ρ1 satisfies (5.4) and (5.5).
EQUILIBRIUM STATISTICAL THEORY
119
Another possible argument consists in passing to the limit in the variational
problem (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 check
that
lim sup Fδ ≤ F .
δ
On the other hand, denoting by h̃ the weak limit of hδ (or of a subsequence),
Z
Z
1
µ
Fδ = F δ (hδ ) =
hδ log hδ dµδ0 +
V0 (x)ρδ1 dx
β
2β
ZZ
a
+
V (x − y)ρδ1 (x)ρδ1 (y)dx dy .
2
Hence, letting δ go to 0+ , we deduce
Z
1
µ
δ
lim inf F ≥ F0 (h̃ log h̃) +
V0 (x)ρ1 dx
δ
β
2β
ZZ
a
V (x − y)ρ1 (x)ρ1 (y)dx dy
+
2
= F(h̃) ≥ F ,
b 1 (ω)) where
since h̃ − Z10 exp(− µ2 Vb0 (ω) − aβ Ψ
µ
b 1 (ω) .
Z 0 = E0 exp − Vb0 (ω) − aβ Ψ
2
R
Therefore, Fδ converges to F, and thus h̃ = h and hδ log hδ dµδ0 converges to
F0 (h̃ log h̃). We then deduce from (5.52) the strong convergence of hδ to h̃ (extending hδ to a “continuous path” function as we did in Section 2.2), and thus the strong
convergence of ρδ1 to ρ.
6 Alternative Formulations for the Mean Field Equations
We have seen in the previous sections two variational formulations of the mean
field problems. The first one, in Theorem 5.4, yields the mean field measure on
paths µ or, more precisely, the mean field (Radon-Nykodym) density h with respect
to µ0 . The second one, in Theorem 5.13, yields a direct variational determination
of the invariant density on R2 , ρ. We present in this section one more variational
formulation that allows one to determine directly the potential Ψ created by ρ,
1
log |x|∗ρ. This formulation is, in a sense we do not wish to make precise
namely, 2π
here, a dual convex problem to the “formulation in ρ” introduced in Theorem 5.13.
It is also the analogue of a formulation introduced in Caglioti et al. [3], at least
in the case of two-dimensional point vortices in a bounded region (with no-slip
boundary conditions). In the case of the whole plane R2 (for two-dimensional
point vortices), the logarithmic divergence of Ψ at infinity makes the adaptation of
this formulation rather delicate. This difficulty is circumvented in Lions [17]. We
shall follow the same approach to take care of the fact that ∇Ψ ∈
/ L2 (R2 ) while
120
P.-L. LIONS AND A. MAJDA
introducing a new variational problem associated to the mean field limit for threedimensional vortex filaments.
In order to keep the ideas clear, we first present formally the variational formulation, ignoring the lack of integrability of |∇Ψ|2 . Afterwards, we detail the necessary mathematical (simple) machinery that allows us to formulate this variational
problem rigorously. Thus we wish to emphasize the fact that the functional we are
going to write now, strictly speaking, does not make sense! With this convention,
we may now introduce
(6.1)
G(φ) =
1
2
Z
|∇φ|2 dx +
R2
1
2
1
log Tr e−(− 2β ∆+µ|x| +aβφ) .
aβ
We claim that, at least formally, Ψ is the “unique minimum of G.” In order to
convince ourselves that this is indeed the case, we only need to explain that Ψ is
a solution of the Euler-Lagrange equation associated to (6.1) and that G is convex,
i.e., that G2 (φ) = log{Tr[e−(H+aβφ) ]} is a convex functional of φ, where we denote
1
by H = − 2β
∆ + µ|x|2 . These verifications, in turn, depend upon the computation
of a directional derivative of G2 of Ψ in some direction φ (where φ ∈ C0∞ (R2 ), for
instance). Identifying φ with the multiplication operator (by φ) whose kernel is
given by φ(y)δ0 (x − y), we easily check that we have, as ε goes to 0,
−1
1
aβ Tr e−(H+aβΨ) φ
(G(Ψ + εφ) − G(Ψ)) → − Tr e−(H+aβΨ)
ε
−(H+aβΨ) −1 Z
= −aβ Tr e
p(x, x, 1)φ(x)dx .
R2
Hence, the Euler-Lagrange equation associated to (6.1) is nothing but
(6.2)
−∆Ψ = R
p(x, x, 1)
p(z, z, 1)dz
on R2 ,
R2
i.e., precisely the equation we expected (see (5.4) and (5.5)).
Next, we prove that G2 is convex (and, in fact, strictly convex modulo the ad∞ (R2 ), φ(x)/ log(1 + |x|) ∈ L∞ (R2 ). Let φ , φ ∈
dition of constants) for φ ∈ Lloc
1 2
∞ (R2 ), φ / log(1 + |x|), φ / log(1 + |x|) ∈ L∞ (R2 ), φ 6≡ φ up to a constant, and
Lloc
1
2
1
2
let θ ∈ (0, 1). We denote by p1 , p2 , and p the Green’s function associated to, respectively, H + aβφ1 , H + aβφ2 , and H + aβφ where φ = θφ1 + (1 − θ)φ2 , and we
claim that
p(x, y,t) < (p1 (x, y,t))θ (p2 (x, y,t))1−θ = p̄
on R2 × R2 × [0, 1] .
EQUILIBRIUM STATISTICAL THEORY
121
If this inequality holds, we deduce immediately
G2 (φ) = log
Z
p(x, x, 1)dx < log
R2
Z
p̄(x, x, 1)dx
R

θ 
1−θ 


Z

 Z




≤ log
p1 (x, x, 1)dx
p2 (x, x, 1)dx



 2
2
2
R
R
= θG2 (φ1 ) + (1 − θ)G2 (φ2 ) .
Finally, the above inequality follows from the (strong) maximum principle and the
following computation:
∂ p̄
1
− ∆ p̄ + µ|x|2 p̄
∂t 2β
1−θ p2
∂p
1
=θ
− ∆p1 + µ|x|2 p1
p1
∂r 2β
θ p1
∂p2
1
2
+ (1 − θ)
·
− ∆p2 + p|x| p2
p2
∂t
2β
#
"
1−θ
θ
p
θ(1 − θ)
p
·
∇p
∇p
1
2
1
+
− 2 1−θ θ
|∇p1 |2 22−θ + |∇p2 |2 1+θ
2β
p2
p1
p1 p2
1−θ
θ
p2
p1
≥ −θ
(aβφ1 p1 ) − (1 − θ)
(aβφ2 p2 ) = −aβφ p̄
p1
p2
and the equality holds if and only if ∇ log p1 = ∇ log p2 ; hence the strict inequality
c
.
is shown unless p1 ≡ p2 ect and φ1 ≡ φ2 − 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, we
1
introduce (for instance) φ0 (x) = − 2π
log{max(|x|, 1)} so that we have
1
δ 1 on R2 ,
2π S
and we easily check that Ψ − 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”
e by
functional G
Z
1
1
e
G(φ)
=
(6.3)
|∇(φ − φ0 )|2 dx +
log[Tr{e−(H+aβφ) }]
2
aβ
−∆φ0 =
R2
1 (R2 ), φ − φ ∈ L∞ (R2 ), ∇(φ − φ ) ∈ L2 (R2 )}—in fact, with
on the space {φ ∈ Hloc
0
0
a little more work, we may even get rid of the constraint φ − φ0 ∈ L∞ (R2 ). A
e is not bounded from
difficulty remains, however: We need to normalize φ since G
122
P.-L. LIONS AND A. MAJDA
e + C) = G(φ)
e
below, since G(φ
− C for all C ∈ R! Thus we choose the following
normalization:
Z
(6.4)
φ ds = 0 ,
S1
1 (R2 ). In conclusion, we define the following
which makes sense since φ ∈ Hloc
minimization class:
(6.5)


Z


1
M = φ ∈ Hloc
(R2 ), φ − φ0 ∈ L∞ (R2 ), ∇(φ − φ0 ) ∈ L2 (R2 ), φ ds = 0 .


S1
We have the following:
R
T HEOREM 6.1 The normalized potential (Ψ − -S1 Ψ ds) is the unique minimum of
e over the set M .
the strictly convex functional G
e
Remark
R 6.2. In other words, the functional G over the set M allows one toR identify
(Ψ − S1 Ψ ds) and thus the mean field density (ρ = −∆Ψ) = −∆(Ψ − -S1 Ψ ds).
1
Therefore, since Ψ = − 2π
log |a| ∗ ρ, it allows us to identify the full potential Ψ as
well.
P ROOF OF T HEOREM 6.1: First of all, the argument made above shows that
G2 is convex, and even strictly convex, on M , while the first term in the definition
R
e namely, 2 |∇(φ − φ0 )|2 dx, is obviously strictly convex on M .
of G,
R
Next, if we fix ψ ∈ M , we may adapt the formal argument made above and
deduce that the Euler-Lagrange equation at ψ reads, denoting by p̄ the Green’s
function associated to H + aβΨ,


Z
(6.6)
∇(Ψ − φ0 ) · ∇φ dx =
R2
Z
R2
 R p̄(x, x, 1) 

 φ(x)dx
p̄(z, z, 1)dz
R2
R
∞
2
2
2
for all
R φ ∈ L (R ) such
R that ∇φ ∈ L (R ) and
R S1 φ ds = 0. Next, we observe
that R2 ∇φ0 · ∇φ dx = -S1 φ ds = 0, at least if |x|≥1 |∇φ|/|x| dx < ∞. Hence, the
preceding Euler-Lagrange equation implies, in particular,


Z
R2
∇Ψ · ∇φ dx =
Z
R2
 R p̄(x, x, 1) 

 φ(x)dx
p̄(z, z, 1)dz
R2
for all φ ∈ L∞ (R2 ) such that ∇φ ∈ L2 (R2 ),
R
R
and |x|≥1 |∇φ|/|x| dx < ∞.
R
(6.6)
This equation is obviously
satisfied by Ψ = Ψ − -S1 Ψ ds. Therefore,
R
R
holdsR with Ψ = Ψ − -S1 Ψ ds, provided φ ∈ L∞ (R2 ), ∇φ ∈ L2 (R2 ), S1 φ ds = 0,
and |x|≥1 |∇φ(x)|/|x| dx < ∞.
S1 φ ds = 0,
EQUILIBRIUM STATISTICAL THEORY
123
At this stage, there only remains to show, by a truncation argument,
R that this
∞
2
2
2
equation holds, in fact, for all φ ∈ L (R ) such that ∇φ ∈ L (R ) and S1 φ ds = 0.
In order to do so, we consider φn = φζ(x/a) for n ≥ 1, where ζ ∈ C0∞ (R2 ), 0 ≤ ζ ≤
1, on R2 , ζ ≡ 1 on B1 , and ζ ≡ 0 for |x| ≥ 2. We only need to show that
Z
∇(Ψ − φ0 ) · ∇φn dx −→
Z
n
R2
∇(Ψ − φ0 ) · ∇φ dx.
R2
This is immediate since we have, denoting by C various positive constants independent of n ≥ 1,
Z
∇(Ψ − φ0 ) · ∇ζn φ dx ≤
R2
C
n
Z
R2
1
C
1 + 4n2 C
1
dx
≤
≤ .
log
1 + |x|2 (n≤|x|≤2n)
n
1 + n2
n
7 The Current and Some Scaling Limits
In this section, we consider and study various problems related to what we did
above. First of all, in Section 7.1, we go back to the issue of Gibbs measures
involving the conserved quantity C defined in (1.6) that we called the current and
we explain how to adapt everything we did before to that general case. Next, in
Section 7.2, we consider infinite-length filaments. Finally, in Section 7.3, we study
various asymptotic limits for the mean field equations.
7.1 Currents
Here we consider the Gibbs measures µN defined formally by (2.1) using the
same normalization (see (2.3)) as in Section 2 and the same scaling (see (3.3)) as in
Section 3. Also, as in Section 2, we restrict the parameters β, µ, and v by requiring
(7.1)
v 2 < 2βµ ,
which allows us to define the measure µN properly. The precise mathematical definition of µN is exactly the same as in Section 2 provided we replace the equation
(2.12) by
!

N
N
∂p
β
ā
1


 −
∑ ∆x j p − 2 ∑ log |Xi − X j | p


∂t 2β j=1

i6= j



N v
(7.2)
2
+ ∑ (λ · X j + µ|X j | )p − (JX j , ∇x j p) = 0 in R2N × (0, 1) ,


β

j=1






p|t=0 = δY (X) on R2N .
We may also define µN by its density hN with respect to a “background” probability measure on N independent periodic paths in RN denoted by µN0 so that (2.13)
remains true. The measure µN0 is defined as in Section 2 replacing the Gaussian
124
P.-L. LIONS AND A. MAJDA
kernel q (see (2.17)) by the Green’s function (which is still a Gaussian kernel that
can be computed) of the same equation as (7.2) with ā = 0 and µ replaced by µ0
where µ0 is chosen in (0, µ) in such a way that v 2 < 2βµ0 (and thus replacing µ/2
by µ − µ0 in the definition of hN ).
Exactly as in Section 7.2, one can recover the Gibbs measure µN from the corresponding measure on broken paths µN,δ . The heart of the matter is the following easy computation, which also sheds some light on the new term appearing in
equation (7.2), namely, βv (Jx, ∇x p). Indeed, we consider, for δ ∈ (0, 1) and for
(x0 , x1 ) ∈ (R2 )2 , the quantity
(7.3)
G(δ) =
2πδ
β
−1
(x1 − x0 )2
exp − β
− v(Jx0 , x1 − x0 ) − λ · x0 δ − µ|x0 |2 δ ,
2h
and we claim that we have
G(δ) − δx0 (x1 )
1
*
∆δx0 (x1 ) − (λ · x0 + µ|x0 |2 )δx0 (x1 )
δ
2β
(7.4)
v
+ (Jx0 , ∇δx0 (x1 ))
β
(in the sense of distributions) as δ goes to 0+ . Indeed, we have for any ϕ ∈ C0∞ (R2 )
Z
R2
Z
|z|2
G(δ)ϕ(x1 ) − ϕ(x0 )
e− 2
dx1 = dz
δ
2π
R2
s !
#)
)
( " s
(
δ
δ
− ϕ(x0 ) ,
z exp − v
(Jx0 , z) + λ · x0 δ + µ|x0 |2 δ
ϕ x0 +
β
β
and, by a trivial expansion, we deduce
Z
R2
G(δ)ϕ(x1 ) − ϕ(x0 )
dx1
δ
−→
δ
Z
R2
|z|2
e− 2
dz
2π
(
1 2
v
∂i j ϕ(x0 )zi z j − (Jx0 , z) · (z, ∇ϕ(x0 ))
∑
2β i, j=1
β
)
− λ · x0 ϕ(x0 ) − µ(x0 )2 ϕ(x0 )
1
v
∆ϕ(x0 ) − (Jx0 , ∇ϕ(x0 ))(λ · x0 + µ|x0 |2 )ϕ(x0 ) ,
2β
β
and our claim is shown.
Having thus defined the Gibbs measures µN , we may now consider the limit as
N goes to +∞ (choosing ā = a/2πN). Then we claim that Theorems 5.1 and 5.4
=
EQUILIBRIUM STATISTICAL THEORY
125
together with Corollary 5.3 hold, with a few modifications, namely, we skip the
representation (5.2) and equation (5.5) is replaced by

σ
1
aβ
∂p
2


 ∂t − 2β ∆p − 2π (log |x| ∗ ρ)p + (µ|x| + λx)p − β (Jx, ∇x p) = 0

(7.5)
on R2 × (0, 1) ,




p|t=0 = δy (x) on R2 .
In all terms involving the Radon-Nykodym densities, µ/2 is to be replaced by µ −
µ0 . The proofs made in Sections 5.2 and 5.3 can then be copied mutatis mutandis.
However, the introduction of the current C leads to a Green’s function p(x, y,t)
that is not symmetric in (x, y) (the time reversal symmetry is broken), and this is
probably why we are not aware of any variational formulation for the density ρ, the
1
log |x| ∗ ρ, or the kernel
potential Ψ = − 2π
p(x, y, 1)
R2 p(z, z, 1)dz
ρ(x, y) = R
analogous to the variational formulations that were developed in Section 5.4 and
Section 6 in the case when v = 0.
7.2 Infinite-Length Filaments
Here we consider another variant where we allow an infinite length for the
vortex filaments, or, in other words, we wish to let L go to +∞. Since our original
formulation of the Gibbs measures used a scaling argument leading to a normalized
length L = 1, we have to go back to the definition of µN , leaving out explicitly the
dependence upon the length L of the filaments, i.e.,
µN =
(7.6)
1
exp(−β H − µI)dX1 · · · dXn
Z
where β, µ > 0 and we take, in order to simplify notation, λ√= v = 0 (even though
everything we do below can be generalized to λ 6= 0, |v| < 2βµ), and
(7.7)
(7.8)
H (X) =
I(X) =
1
2
Z L N
∑
0 j=1
Z L N
∂X j
∂σ
2
1
dσ + ā
2
Z L N
∑ − log |X j (σ) − Xk (σ)|dσ ,
0 j6=k
∑ |X j (σ)|2 dσ .
0 j=1
Then everything we did above applies, provided, of course, that we replace
everywhere the final time 1 by L. In particular, µN is defined precisely by its
action upon any bounded continuous function on R2Nm F = F(Ω(t1 ), . . . , Ω(tm ))
126
P.-L. LIONS AND A. MAJDA
(m ≥ 0, 0 ≤ t1 < t2 < · · · < tm ≤ 1), namely,
Z
Z
1
N


dX F(X1 , . . . , Xm )p(X1 , X2 ,t2 − t1 )
 F dµ = Z



2N
m

(R )




· · · p(Xm−1 , Xm ,tm − tm−1 )p(Xm , X1 , L − (tm − t1 ))
(7.9)




Z





Z=
p(X, X, L)dX ,


R2N
where p is the Green’s function of the PDE (2.12) (with λ = 0), and the law of
Ω(t), for any t ∈ [0, L], admits a density, with respect to the Lebesgue measure on
R2N , given by
ρN (X) =
(7.10)
p(X, X, L)
.
Z
We next wish to send L to +∞, and we denote by µNL and ρNL the above quantities
to recall the dependence upon L. In order to understand the asymptotics in L, one
needs to introduce the complete set of normalized eigenfunctions φNk (1 ≤ k) in
L2 (R2N ) of the Schrödinger operator
(7.11)
−
N
1
β ā N
log |Xi − X j | + µ ∑ |X j |2 ,
∆−
∑
2β
2 i6= j
j=1
and we denote by λNk (1 ≤ k) the corresponding eigenvalues with λN1 ≤ λN2 ≤ · · · ≤
λNk −→ + ∞. Let us also recall that λN1 is simple (i.e., λN1 < λN2 ) and that we may
k
choose φN1 to be positive on R2N . With this notation, we have for all X,Y ∈ R2N ,
t > 0,
(7.12)
pN (X,Y ,t) =
∑ e−λ t φNk (x)φNk (Y ) .
N
k
k≥1
In particular, one may easily check that we have for any t ≥ 0
(7.13)
N N
N pN (X,Y , L − t) = e−λ1 L eλ1 t φN1 (x)φN1 (y) + o e−λ1 L ,
N
N ZL = e−λ1 L + o e−λ1 L ,
where the remainder term o(e−λ1 t ) is small in L1 ∩ L∞ (R2N × R2N ).
This allows us to deduce that µNL converges weakly (in the sense of probability
measures) to the probability measure on C([0, ∞); R2 )N defined by, for all bounded
continuous functions on R2Nm F = F(Ω(t1 ), . . . , Ω(tm )) (m ≥ 1, 0 ≤ t1 < t2 < · · · <
N
EQUILIBRIUM STATISTICAL THEORY
127
tm < +∞), the following expression:
Z
F dµN =
Z
dX F(X1 , . . . , Xm )p(X1 , X2 ,t2 − t1 )
R2Nm
(7.14)
· · · p(Xm−1 , Xm ,tm − tm−1 )eλ1 (tm −t1 ) φN1 (X1 )φN1 (Xm ) .
N
In addition, ρN1 converges, as L goes to +∞, to ρN (X) = (φN1 (X))2 in L1 ∩ L∞ (R2N )
(for instance), and ρN is the density of the law of Ω(t) (under the probability measure µN ) for all t ≥ 0.
We may now turn to the limit as N goes to +∞ under the scaling ā = a/2πN.
The analysis of the behavior of µN is somewhat intricate, and this is why we only
consider the behavior of the law of Ω(t) (∀t ≥ 0), i.e., the behavior of ρN (X) as N
goes to +∞. Before we do so, we first argue formally in order to guess the right
answer by commuting limits, that is, first letting N go to +∞, in which case we
recover the mean field problems studied and justified in the preceding sections, and
then letting L go to +∞. The mean field law µ = µL is defined (see Theorem 5.1)
by
Z
(7.15)
F dµL =
1
ZL
Z
dx F(x1 , . . . , xm )pL (x1 , x2 ,t2 − t1 )
R2m
· · · pL (xm−1 , xm ,tm − tm−1 )pL (xm , x1 , L − (tm − t1 ))
for any F = F(ω(t1 ), . . . , ω(tm )), F bounded and continuous on R2m , m ≥ 1, 0 ≤
t1 < t2 < · · · ≤ tm ≤ L, where p is the Green’s function of


 ∂pL − 1 ∆pL + − aβ log |x| ∗ ρL pL + µ|x|2 pL = 0 in R2 × (0, L)
∂t
2β
2π
(7.16)

 p | = δ (x) in R2 ,
L t=0
y
and
(7.17)
(7.18)
ρL (x) = pL (x, x, L)/ZL ,
ZL =
Z
pL (z, z, L)dz .
R2
Let us recall that ρL is the density of the mean field law of ω(t) for all t ∈ [0, L].
We now let L go to +∞ and argue formally, although a rigorous argument,
which we skip for the sake of brevity, is possible. We thus assume that ρL converges, as L goes to +∞, to some probability density ρ on R2 . Then, denoting by
φ1 the first eigenfunction, which we take to be normalized in L2 (R2 ) and positive,
1
2
∆ + (− aβ
of the Schrödinger operator [− 2β
2π log |x| ∗ ρ) + µ|x| ] and by λ1 the corresponding (simple) eigenvalue, we deduce by a similar argument to the one made
128
P.-L. LIONS AND A. MAJDA
above that µL “converges” to a probability measure on C([0, ∞); R2 ) defined by
Z
(7.19)
F dµ =
Z
dx F(x1 , . . . , xm )p(x1 , x2 ,t2 − t1 )
R2m
· · · p(xm−1 , xm ,tm − tm−1 )φ1 (xm )φ1 (x1 )eλ1 (tm −t1 )
for any F = F(ω(t1 ), . . . , ω(tm )), F bounded and continuous on R2m , m ≥ 1, 0 ≤
t1 < t2 < · · · < tm , where p is the Green’s function of


 ∂p − 1 ∆p + − aβ log |x| ∗ ρ p + µ|x|2 p = 0 in R2 × (0, ∞)
∂t 2β
2π
(7.20)

 p| = δ (x) in R2
t=0
y
and
ρ(x) = (φ1 (x))2
(7.21)
on R2 .
Of course, ρ is the density of the law of ω(t) for all t ≥ 0 under the probability
measure µ.
In other words, the limit measure µ is entirely determined by the probability
density ρ = φ21 on R2 that solves the following Hartree equation:

1
aβ

2
 − ∆φ1 + − log |x| ∗ φ1 φ1 + µ|x|2 φ1 = λ1 φ1 in R2

 2β
2π
Z
(7.22)
2


φ
>
0
on
R
,
φ21 dx = 1 .
1


R2
We expect ρ to be the minimum of the following strictly convex functional:


Λ = min Λ(ρ) : ρ ∈ L1 (R2 ), ρ|x|2 ∈ L1 (R2 ), ρ ≥ 0 on R2 ,
(7.23)


Z

√
ρ dx = 1, ρ ∈ H 1 (R2 ) ,

R2
with
(7.24)
Λ(ρ) =
Z
R2
1 √ 2
aβ
|∇ ρ| + µ|x|2 ρ dx −
2β
4π
ZZ
log |x − y|ρ(x)ρ(y)dx dy .
R2 ×R2
1
As is well-known, the first term of the energylike functional Λ, namely, 2β
S1 (ρ),
R
√ 2
where S1 (ρ) = R2 |∇ ρ| dx, is convex in ρ. S1 is the so-called Fisher information functional in information theory, Linnik functional in kinetic theory, and the
von Weiszäcker correction for kinetic energy in density-dependent quantum models.
EQUILIBRIUM STATISTICAL THEORY
129
Having thus determined formally the mean field limit, we now turn to a rigorous
proof of the “convergence,” as N goes to +∞, of ρN = (φN1 )2 to ρ = (φ1 )2 (or
“products of ρ”). More precisely, we introduce, for each 1 ≤ k ≤ N, ρN,k the density
of the law of (ω1 (t), . . ., ωk (t)) (∀t ≥ 0)
(7.25)
ρN,k (x1 , . . . , xk ) =
Z
ρN (x1 , . . . , xk , xk+1 , . . . , xN )dxk+1 · · · dxN .
R2(N−k)
We may state our main result on the mean field limit for infinite-length vortex
filaments.
k
T HEOREM 7.1 For each k ≥p1, ρN,k converges in L1 ∩p
L k−1 (R2k ), as N goes to
+∞, to ρk = ∏ki=1 ρ(xi ) and ρN,k converges in H 1 to ρk , where ρ = (φ1 )2 is
the unique minimum of the Hartree variational problem (7.23)–(7.24), and φ1 is
smooth, decays rapidly at infinity, and solves (7.22). Furthermore, λN1 /N −→ Λ as
N goes to +∞.
P ROOF OF T HEOREM 7.1: We first recall that φN1 is the unique minimum (up
to a change of sign) of

!
 Z 1
N
aβ N
N
2
2
λ1 = min
|∇φ| + µ ∑ |x j | −
∑ log |xi − x j | φ2 dx :

2β
4πN
j=1
i6= j
R2N

Z

2
2N
1
2N
2
|x|φ ∈ L (R ), φ ∈ H (R ), φ dx = 1 ;

R2n
therefore ρN = (φN1 )2 is the unique minimum of the following convex problem:

!
Z
 1
N
N
aβ
λN1 = min
log |xi − x j | ρ dx :
µ ∑ |x j |2 −
S (ρ) +
 2β 1
4πN i6∑
j=1
=j
R2N
(7.26)
r ∈ L1 (RN ), ρ ≥ 0 on RN ,

Z

ρ dx = 1 .

√
ρ ∈ H 1 (R2N ), ρ|x|2 ∈ L1 (R2N ),
R2N
In the course of proving Theorem 7.1, we shall need some properties of the
functional S1 that we isolate in the next lemma, whose proof is postponed until the
conclusion of the proof of Theorem 7.1.
L EMMA 7.2 Let ρ ≥ 0 ∈ L1 (Rn ).
(i) Let k ∈ {1, . . . , n}. We denote by
ρk =
Z
ρ(x1 , . . . , xk , xk+1 , . . . , xn )dx j+1 · · · dxn
130
P.-L. LIONS AND A. MAJDA
and by
ρn−k =
Z
ρ(x1 , . . . , xk , xk+1 , . . . , xn )dx1 · · · dxk .
Then we have
S1 (ρ) ≥ S1 (ρk ) + S1 (ρn−k ) .
(7.27)
(ii) We denote by ρt = ρ ∗ ((2πt)−n/2 e−|x| /2t ) for t > 0. Then, we have, assuming
(for instance) that ρ|x|δ ∈ L1 (Rn ) for some δ > 0,
1
(7.28)
2S1 (ρ) = sup {S0 (ρ) − S0 (ρt )}
t>0 t
2
where S0 (ρ) =
R
Rn ρ log ρ dx.
R
We first show that λN1 /N is bounded and that S1 (ρN,k ) and R2k ρN,k (x)|x|2 dx are
bounded for each k ≥ 1. First of all, let ρ be an element of the minimizing class
defined in (7.23); then, introducing ρ(x1 , . . . , xN ) = ∏Ni=1 ρ(xi ), we have, in view of
(7.26),
Z
aβ
(log |x| ∗ ρ)ρ dx ;
λN1 ≤ NΛ(ρ) +
2πN
R2
hence
lim
N
or
(7.29)
λN1
≤ Λ(ρ)
N
lim
N
λN1
≤ Λ.
N
On the other hand, we have

Z
 1 Z
λN1 ≥ min
|∇φ|2 dx +
 2β
!
aβ N
|xi − x j | dx :
µ ∑ |x j | −
4πN i6∑
j=1
=
j
R2N
R2N

Z

|x|φ ∈ L2 (R2N ), φ ∈ H 1 (R2N ),
φ2 dx = 1 ≥ −CN

N
2
R2N
aβ
4πN
since µ ∑Nj=1 |x j |2 −
∑Ni6= j |xi − x j | ≥ −CN on R2N , where, here and below, we
denote by C various positive constants independent of N.
The preceding argument also yields immediately
(7.30)
(7.31)
1
N
1
N
Z
R2N
Z
R2N
N
∑ |x j |2ρN dx =
j=1
Z
R2
p
|∇ ρN |2 dx ≤ C .
|x|2 ρN,1 dx ≤ C ,
EQUILIBRIUM STATISTICAL THEORY
Indeed, we just need to observe that
R2N ; hence
Z
Z
p
1
µ
2
N
|∇ ρ | dx +
2β
2
R2N
R2N
µ
2
131
aβ
∑Nj=1 |x j |2 − 4πN ∑Ni6= j |xi − x j | ≥ −CN on
N
∑ |x j |2ρN dx ≤ λN1 +CN ≤ CN .
j=1
Our claim then follows since we have for each k ≥ 1
Z
|x| ρ
2 N,k
dx = k
Z
|x|2 ρN,1 dx ≤ Ck
R2
R2k
while (7.27) implies
N
S1 ρN,k .
k
At this stage, we may now let N go to +∞. We first observe that, by a standard
diagonal procedure, we may extract a subsequence, still denoted by ρN , to simplify
notation such that, by Sobolev imbeddings,
k
ρN,k * ρk weakly in L1 ∩ L k−1 R2k
N
p
R
and ∇ ρRk ∈ 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 each k ≥ 1
CN ≥ S1 (ρN ) ≥
(7.32)
1
1
1
S1 (ρk ) ≤ lim S1 (ρN,k ) ≤ lim S1 (ρN ) .
N N
k
N k
In addition, we have, in view of the above bounds and convergences,
Z
ZZ
1 N
1 1
aβ
(7.33)
log |x − y|ρ2 (x, y)dx dy .
λ1 −
S1 (ρN ) = µ |x|2 ρ1 dx −
N
2β N
2π
R2
R2 ×R2
Finally, using the Hewitt-Savage theorem again, we obtain a probability measure
π on the set of probability measures on R2 such that
ρk =
Z
k
∏ ρ(x j )dπ(ρ)
for all k ≥ 1 .
j=1
Next we have, in view of (7.32) and of the convexity of S,
Z
1
k
S1 (ρ ) ≤ S1 (ρ)dπ(ρ) ≤ +∞ for all k ≥ 1 .
k
On the other hand, we claim that
Z
1
k
lim S1 (ρ ) = S1 (ρ)dπ(ρ) .
k k
This is indeed a straightforward consequence of part (ii) of Lemma 7.2, observing
that we have for all t > 0
1
1 1
S0 (ρk ) − S0 (ρtk ) ;
S1 (ρk ) ≥
k
2t k
132
P.-L. LIONS AND A. MAJDA
hence
1
1
lim S1 (ρk ) ≥
2t
k k
Z
Z
1
S0 (ρ)dπ(ρ) −
2t
Z
S0 (ρt )dπ(ρ)
1
{S0 (ρ) − S0 (ρt )}dπ(ρ) ,
2t
and we easily conclude the proof.
R
The above arguments show, in particular, that S1 (ρ) + R2 ρ|x|2 dx < ∞ π-a.s.,
and we deduce from (7.33)
=
λN
lim 1 ≥
N N
(7.34)
Z
Λ(ρ)dπ(ρ) .
Comparing (7.29) and (7.34), we deduce that limN λN1 /N = Λ and that Λ(ρ) = Λ πa.s. Since Λ is strictly convex, there is a unique minimum ρ and π = δρ . Therefore
we have ρk = ∏kj=1 ρ(x j ) and N1 S1 (ρN ) −→N S1 (ρ) = 1k S1 (ρk ). Furthermore, we
deduce from (7.32) that
p
p
∇
ρN,k −→ ∇
N
ρk
strongly in L2 (R2k ). We then easily conclude the proof of the convergence part of
Theorem 7.1.
√
The smoothness and the decay of φ1 = ρ follows immediately from elliptic regularity after writing the Euler-Lagrange equation of (7.23)–(7.24) recast in
√
terms of ρ, namely,

Z 1
Λ = min
|∇φ|2 + µ|x|2 φ2 dx
 2β
R2
1
− log |x − y| φ2 (x)φ2 (y)dx dy :
2π
R2 ×R2

Z

φ ∈ H 2 (R2 ), φ|x| ∈ L2 (R2 ), |φ|2 dx = 1 .

aβ
+
2
ZZ R2
Indeed, the corresponding Euler-Lagrange equation is nothing but (7.22), and we
conclude the proof.
P ROOF OF L EMMA 7.2: (i) We begin with the case when ρ is smooth, decays
rapidly at infinity, and is strictly positive on Rn . Since S1 is convex and positively
homogeneous of degree 1, we have
S1 (ρ) ≥ S10 (ρk · ρn−k ) · ρ
Z √ p
= ∇ ρk ρn−k ·
Rn
1
√ p n−k
ρk ρ
!
dx
EQUILIBRIUM STATISTICAL THEORY
133
Z p
ρ
ρ
√
n−k
= ∇ ρk · ∇ √
dx + ∇ ρ · ∇ √
dx
ρk
ρn−k
n
n
R
R
Z
Z
p
2
√ 2
=
∇ ρn−k dx = S1 (ρk ) + S1 ρn−k .
∇ ρk dx +
Z
Rk
Rn−k
√
√
For a general ρ such that ∇ ρ ∈ L2 (Rn ), we approximate ϕ = ρ in H 1 (Rn )
by ϕε , which is smooth, strictly positive, and rapidly decreasing at infinity, and we
set ρε = ϕ2ε . Thus, ρε −→ ρ in L1 (Rn ), and (ρε )k and (ρε )n−k converge in L1 (Rk )
ε
and L1 (Rn−k ), respectively. Therefore, we have
S1 (ρ) = lim S1 (ρε ) ≥ lim S1 ((ρε )k ) + lim S1 ((ρε )n−k )
ε
ε
ε
≥ S1 (ρk ) + S1 (ρ
n−k
)
since S1 is convex. The inequality (7.27) is shown in full generality.
(ii) We first observe that ρt log ρt ∈ L1 (Rn ) and that ρ(log ρ)− ∈ L1 (Rn ) so that
S0 (ρ) − S0 (ρt ) makes sense in R ∪ {+∞}. Indeed, on one hand, we have a.e. on Rn
δ
ρ(log ρ + |x|δ ) + e−|x| − ρ ≥ 0 ;
hence ρ(log ρ)− ∈ L1 (Rn ). On the other hand, ρt ∈ L∞ (Rn ), and it is easily checked
that ρt |x|δ ∈ L1 (Rn ). Hence, as before,
ρt (log ρt )− ∈ L1 (Rn ) while ρt (log ρt )+ ≤ (log kρt kL∞ (Rn ) )+ ρt ∈ L1 (R2 ) .
Next, we consider first the case when ρ is smooth, rapidly decreasing at infinity, and strictly positive on Rn . We then compute (this is, by the way, a classical
computation in kinetic theory)
d
S0 (ρt ) =
dt
Z
(log ρt + 1)
Rn
1
=−
2
d
1
S1 (ρt ) =
dt
4
Z
Z
=−
1
4
1
4
Z
Rn
1
(log ρt ) ∆ρt dx
2
|∇ρt |2 ρt−1 dx = −2S1 (ρt ) ,
Rn
R2
=−
∂ρt
dx =
∂t
∇ρt · ∇(∆ρt ) 1 |∇ρt |2
−
∆ρt dx
ρt
2 ρt2
Z
R2
Z
R2
|D2 ρt |2
1
dx +
ρt
2
Z
D2 ρt
R2
1 2
∇ρt ⊗ ∇ρt
D ρt −
ρt
ρt
(∇ρt , ∇ρt )
1
dx −
2
4
ρt
2
dx ≤ 0 .
Z
R2
|∇ρt |4
dx
ρt3
134
P.-L. LIONS AND A. MAJDA
Hence,
1
1
sup {S0 (ρ) − S0 (ρt )} = lim {S0 (ρ) − S0 (ρt )} = 2S1 (ρ) .
t→0+ t
t>0 t
Finally, for a general ρ such that S1 (ρ) < ∞, we may adopt the approximation
argument sketched above in the proof of part (i) and obtain for all t > 0
1
2S1 (ρ) ≥ {S0 (ρ) − S0 (ρt )}.
t
We also deduce from the above proof that S0 (ρt ) is a convex, decreasing function
of t; hence we have for all h > 0
d
S0 (ρ) − S0 (ρt )
lim
≥ − S0 (ρh ) = 2S1 (ρh ) ,
t→0+
t
dt
and we conclude upon letting h go to 0+ .
7.3 Asymptotic Limits
In this section, we investigate briefly some asymptotic limits involving the various parameters (L, β, a, µ) of the mean field problem. In order to clearly see the
role of the various parameters, we recall the variational problem (6.3) and (6.5),
which allows the determination of the mean field potential Ψ and the mean field
density. We rewrite it in the case of filaments of length L, i.e.,


1
e
: φ ∈ Hloc
(7.35)
(R2 ), φ − φ0 ∈ L∞ (R2 ),
min G(φ)


Z

2
2
∇(φ − φ0 ) ∈ L (R ), − φ ds = 0 ,

S1
with
(7.36)
1
e
G(φ)
=
2
Z
R2
|∇(φ − φ0 )|2 dx +
1
2
1
log Tr e−L(− 2β ∆+µ|x| +aβφ) .
aβL
These expressions show, in particular, that only the normalized parameters L/β,
Lµ, and Laβ matter.
We concentrate now upon the limits when β goes to 0+ (“infinite temperature”)
or when β goes to +∞. First of all, if β goes to 0+ , µβ goes to µ̃ > 0 and aβ 2 goes
to ã > 0; keeping L fixed, we immediately see that this amounts to sending (for a
“new” β = 1) the normalized length L/β to +∞ while keeping the other parameters
µ and a essentially equal (or converging) to µ̃ and ã, respectively. In other words,
this limit is precisely the one we investigated in Section 7.2.
Next, if we let β go to +∞ while keeping the other parameters L, µ, a > 0 fixed,
we see that this amounts to sending the normalized length L/β to 0 while taking
the other parameters µ and a equal (or equivalent) to µ̃/L and ã/L, respectively.
EQUILIBRIUM STATISTICAL THEORY
135
In other words, this limit is equivalent to sending the length of the filaments to 0.
Observe also that the limit β going to +∞ is nothing but a classical limit for the
Hartree equation in terms of quantum mechanics problems.
Thus, we now investigate this distinguished limit, namely, L goes to 0+ , β > 0
fixed, µ = µ̃/L, and a = ã/L with µ̃ and ã > 0 fixed. At least intuitively, we expect
to recover at the limit the two-dimensional mean field theory for point vortices, and
this is precisely what we state in the following result where we denote by ρL and
eL , the functional given
ψL the mean field expressions for a length equal to L by G
e
by (7.36), and by GL , the minimum given by (7.35).
T HEOREM 7.3 As L goes to 0+ , ρL converges in L p (R2 ) for 1 ≤ p < ∞ to ρ, ΨL
1,α
1
converges to Ψ = (− 2π
log |x|)∗ρ in Cloc
(R2 ) for 0 < α < 1, ∇(ΨL −φ0 ) converges
R
to ∇(Ψ− φ0 ) in W 1,p (R2 ) for 2 ≤ p < ∞ where Ψ− -S1 Ψ ds is the unique minimum
of
Z
∞
2
2
2
e
e
(7.37) G = min G(φ) : φ − φ0 ∈ L (R ), ∇(φ − φ0 ) ∈ L (R ), – φ ds = 0
S1
where
(7.38)
1
e
G(φ)
=
2
Z
R2
|∇(φ − φ0 )|2 dx +
1
log
ãβ
Z
e−µ̃|x|
2 −ãβφ
dx .
R2
In addition, we have
(7.39)
eL − 1 log β −→ G
e as L goes to 0+ .
G
ãβ
2πL
We sketch only one possible proof, since it is in fact possible to copy the argument developed in Angelescu, Pulvirenti, and Teta [1] for three-dimensional quantum particles with a Coulomb (repulsive) interaction. One possible proof is very
similar to the one presented in Section 5.5 for the continuous limit of broken path
mean field theories. Indeed, denoting by µL the mean field probability measure on
continuous paths in R2 , periodic with period L, so that µL = hL µ0L and
Z
µ̃
1
L
2
hL = 0 exp − – |ω(σ)| + ãβΨL (ω(σ))dσ ,
ZL 0Z 2
µ̃
0
0
2
L
.
ZL = EL exp − – |ω(σ)| + ãβΨL (ω(σ))dσ
0 2
136
P.-L. LIONS AND A. MAJDA
We recall that hL is the unique minimum in L∞ (ΩL ) of
(
!!
Z
1 0
µ̃ 0
L
2
min
E ( f log f ) + EL f – |ω(σ)| dσ
β L
2β
0
Z
a
1 L
– log |ω(σ) − ω 0 (σ)|dσ
+ EL0 f (ω) f (ω 0 ) −
2
2π 0
)
!!
:
f ≥ 0, EL0 ( f ) = 1 .
This allows us to prove the following bounds uniformly in L ∈ (0, 1]:
EL0 (hL log hL ) ≤ C ,
Z
Z
0
2
L
EL hL – |ω(σ)| dx = ρL |x|2 dx ≤ C ,
0
R2
Z
1 L
0
0
0
–
EL hL (ω)hL (ω ) −
log |ω(σ) − ω (σ)|dσ
=
2π 0
ZZ
1
ρL (x)ρL (y) − log |x − y| dx dy ≤ C .
2π
R2 ×R2
Then these bounds yield L∞ bounds on ρL and thus (ρL being radially symmetric)
W 1,∞ bounds on ∇ΨL and L∞ bounds on ΨL − φ0 .
These bounds are sufficient to prove that we have
Z
L n 2βL ∆−µ̃|x|2 −ãβΨL o
2
2π Tr e
− e−µ̃|x| −ãβΨL dx −→ 0 ,
L
β
R2
by using, for instance, the Feynman-Kac representation of the trace (or making an
analytical proof where we approximate the Green’s function at time L of
µ̃|x|2 − ãβΨL by
β
2πL e
−β|x−y|2
2L
L
2β ∆ −
e−µ̃|y| e−ãβΨL (y) . . . ). Finally, we recall the semiclas2
sical inequality (see, for instance, B. Simon [23] and the references therein) valid
∞ (R2 ), ãβφ + µ̃|x|2 is bounded from below,
for any φ such that φ ∈ Lloc
Z
1 ∆−µ̃|x|2 −ãβφ
β
2β
≤
Tr e
exp(µ̃|x|2 − ãβφ)dx .
2πL
R2
This allows us to complete the proof of the convergence of ΨL and of the beeL . The rest of the proof is then straightforward, going back to the
havior of G
representation of µL and ρL .
EQUILIBRIUM STATISTICAL THEORY
137
8 Concluding Discussion and Future Directions
In this paper, we introduced Gibbs measures for nearly parallel filaments with
identical circulations, and then we rigorously derived a novel mean field theory as
the number of filaments tends to infinity. Here we would like to mention briefly
some related theoretical problems as well as a potential application.
Perhaps the most interesting and difficult theoretical issue for equilibrium statistical mechanics involves the behavior of the continuum Gibbs measures as N → ∞
without the mean field scaling ā = a(πN)−1 , which was utilized in Sections 3
through 7 of this paper. It is very interesting to link even the equilibrium Gibbs
measures with a finite number of filaments to actual dynamic behavior of the
filament equations. With X j (σ,t) = (X j,1 , X j,2 ) and the complex notation φ j =
X j,1 + iX j,2 , the equations in (1.1) for identical filaments are equivalent to unusual
coupled nonlinear Schrödinger equations
(8.1)
∂2 φ j 1
φ j − φk
1 ∂φ j
=α 2 + ∑
i ∂t
∂σ
2 k6= j |φ j − φk |2
for 1 ≤ j ≤ N, φ j (σ + L,t) = φ j (σ,t) .
There is some computational and theoretical work connecting dynamics with equilibrium 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 field
theory. Recently Julien et al. [11] presented numerical simulations of rapidly rotating convection at high Rayleigh numbers. They observed that with small convective 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 the
heat transport in the filament cores (Werne, private communications; Julien et al.
[11]). Thus, in the interior region the turbulent flow is dominated by nearly parallel
filaments with very similar circulations. This is essentially the same regime as in
this paper, where we have developed the equilibrium statistical mechanics of nearly
parallel filaments. Do the equations for mean field theory developed here predict
a large-scale interior flow resembling the simulations? Are other nonlinear corrections to self-induction, as discussed in the last paragraph of Section 3, needed for
an accurate statistical prediction?
Appendix: Remarks on the KMD Equations
In this appendix, we make a few brief remarks on the coupled Schrödinger
equations (8.1) to which we add some initial conditions
(A.1)
φ j (σ, 0) = φ0j (σ)
138
P.-L. LIONS AND A. MAJDA
where φ0j ∈ H 1 (0, L), φ0j is periodic with period L for 1 ≤ j ≤ N, and
∑
(A.2)
Z L
j6=k 0
log |φ0j − φ0k | dσ < ∞ .
This system of coupled, nonlinear Schrödinger equations is not understood mathematically 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 that φ j ∈ L∞ (0, ∞;
H 1 (0, L)) ∩ C([0, ∞); H 1(0, L)) for all s ∈ [0, 1) (∀1 ≤ j ≤ N). We postpone the
precise definition of such solutions, since it is plausible that one could obtain a
more satisfactory notion anyway. But we wish to emphasize that we only know
that, for all t ≥ 0,
meas{σ ∈ (0, L) : ∃ j 6= k, φ j (σ,t) = φk (σ,t)} = 0 ,
since we obtain, in fact, an estimate on
sup ∑
Z L
t≥0 j6=k 0
log |φ j − φk | dσ ,
1
for j 6= k.
and, in particular, we are not aware of any bound on |φ j −φ
k|
One simple way to construct solutions is to smooth out the singularity by replacing
φ j − φk
φ j − φk
by
for δ ∈ (0, 1) .
2
2
|φ j − φk |
δ + |φ j − φk |2
Then the resulting system is trivial to solve (with the initial conditions (A.1)), and
we obtain a unique solution (φδj ) j ∈ C([0, ∞); H 1), periodic in σ with period L, of
that regularized system. In addition, we have the following conservation laws:
(A.3)
(A.4)
d
dt
Z L N
∑ |φδj |2 dσ = 0
0 j=1

Z
d  α L N ∂φδj
∑ ∂σ
dt 2 0 j=1
2
1
dσ −
8

Z L N
∑ log(δ2 + |φδj − φδk |2)dσ = 0 ,
0 j6=k
from which we easily deduce bounds, uniform in δ, on φδ in C([0, ∞); H 1) and
N
1
∑ 2 log(δ2 + |φδj − φδ2 |2)
in C([0, ∞); L1 ) .
j6=k
From this point on, everything we say and do is really up to the extraction of
subsequences. In particular, we may assume that φδ converges weakly (weak-∗) to
φ in L∞ (0, ∞; H 1 ). We next proceed to show that φδ converges to φ in C([0, T ]; L2 )
(and thus in C([0, T ]; H s ) for all s ∈ [0, 1)) for all T ∈ (0, ∞). In order to do so,
we recall that φδ is bounded, uniformly in δ, in L∞ ((0, T ) × (0, L)) (and thus φ ∈
EQUILIBRIUM STATISTICAL THEORY
139
L∞ ((0, T ) × (0, L))) by the trivial one-dimensional Sobolev imbeddings. Then we
denote by φδj = ϕδj + iψ δj , and we consider arbitrary C2 functions F(ϕ1 , ψ1 ; . . . ; ϕN ,
ψN ) on R2N such that we have for all 1 ≤ j ≤ N
0
F(ϕ
=0
j ,ψ j )
(A.5)
if (ϕk , ψk ) = (ϕ j , ψ j ) for some k 6= j .
Then a straightforward computation shows that we have (skipping the index δ in
order to simplify notation)
N
N
∂ϕ j ∂ψk
∂
∂
0 ∂p j
0 ∂ψ j
F =α∑
− Fϕ j
Fψ j
+ α ∑(Fϕ00j ϕk − Fψ00j ψk )
∂t
∂σ
∂σ
∂σ ∂σ
j=1 ∂σ
j,k
N
∂ψ j ∂ψk ∂ϕ j ∂ϕk
(A.6)
+ α ∑ Fϕ00j ψk
−
∂σ ∂σ
∂σ ∂σ
j,k
+
1
[Fψ0 j (ϕ j − ϕk ) − Fϕ0 j (ψ j − ψk )] · (δ 2 + |φ j − φk |2 )−1 .
2 k6∑
=j
Then, in view of the bounds on φ j and condition (A.5), we deduce that ∂F/∂t is
bounded in L∞ (0, ∞; H −1 ).
We may then choose
 N

2
∏ |φ` − φk | 
 k<`

F = Fjε = φ j β 


ε
where 1 ≤ j ≤ N, ε ∈ (0, 1), β ∈ C∞ ([0, ∞)), β(t) = t if 0 ≤ t ≤ 12 , β(t) = 1 if t ≥ 2,
and 0 ≤ β 0 (t) ≤ 1 for all t ≥ 0. Obviously, condition (A.5) holds, and thus ∂Fjε /∂t
is bounded in L∞ (0, ∞; H −1 ) while Fjε is bounded in L∞ (0, ∞; H 1 ). Therefore Fjε is
relatively compact in C([0, T ); L2 ) as δ goes to 0 for each ε > 0 fixed.
Next we observe that we have Fjε = φ j if ∏Nk<` |φk − φ` |2 ≥ ε, and thus if |φk −
)−1 . Hence, we have for all δ, δ 0 ∈ (0, 1)
φ` |2 ≥ εν for all k 6= `, where ν = ( N(N−1)
2
0
sup kφδj (t) − φδj (t)kL2
t∈[0,T ]
0
≤ sup kFjε,δ (t) − Fjε,δ (t)kL2
t∈[0,T ]
+C sup {meas{σ ∈ (0, L) : |φδk − φδ` |2 < εν for some k 6= `}1/2
t∈[0,T ]
0
0
+C sup {meas{σ ∈ (0, L) : |φδk − φδ` |2 < εν for some k 6= `}1/2 ,
t∈[0,T ]
0
≤ sup kFjε,δ (t) − Fjε,δ kL2 +
t∈[0,T ]
C
C
+
,
2
ν
| log(δ + ε )| | log(δ 0 2 + εν )|
140
P.-L. LIONS AND A. MAJDA
in view of the bound on | log(|φδj − φδk |2 + δ 2 )| in L∞ (0, ∞; L1 ) for all j 6= k. This
bound immediately yields the convergence of φδj to φ j in C([0, T ]; L2 ) for all T ∈
(0, ∞).
Next, we claim that φ j ∈ C([0, ∞)×L2). The argument above shows that F(φ) ∈
C([0, ∞) × L2) for any F satisfying (A.5). This is not enough, however, to prove
our claim. In addition, we need to make some further observations. First of all, we
remark that we have for any subset I ⊂ {1, . . . , N}
!
!
2
φδj − φδk
1∂
∂
(A.7)
φδj = α 2 ∑ φδj + ∑ ∑ 2
.
∑
δ
δ 2
i ∂t j∈I
∂σ
j∈I
j∈I k6= j δ + |φδ − φk |
k∈I
In particular, we have, if I = {1, . . . , N},
!
N
1∂
∂2
φj = α 2
(A.8)
∑
i ∂t j=1
∂σ
∞
∑ φj
!
,
j=1
and thus ∑Nj=1 φ j ∈ C([0, ∞) × [0, L]).
When N = 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 that F1 = φ1 |φ1 − φ2 |2 and F2 = φ2 |φ1 −
φ2 |2 both satisfy (A.5). For a general N, the proof requires some tedious combinations that we leave to the reader once we observe that we obtain from (A.7) a
bound on



!



N
∂
δ
δ
δ 2 
|φ
−
φ
|
 ∑ φj
∏ ∏ j k  in L∞ (0, T ; H −1) .

∂t
 j=1 k∈I

j∈I
/
k6= j
Once the continuity is shown, we obtain for all 1 ≤ j ≤ N the existence of an
open set O j in [0, ∞) × R such that meas(t,σ) (Ocj ) = 0 and ∀k 6= j, |φ j − φk | > 0 on
O j . Indeed, we have, from the convergence of φδ to φ, for all j ∈ {1, . . . , N}, the
following estimate:
sup ∑
Z L
t≥0 k6= j 0
log |φ j − φk | dσ < ∞ .
T
In particular, (8.1) holds on O j , and we denote by O = Nj=1 O j in order that
meas(t,σ) (Oc ) = 0.
Finally, we claim that we can pass to the limit, as δ goes to 0+ , in (A.6) and
recover a formulation of the equation on (0, ∞) × [0, L], at least when Fφ j φk = 0 if
φ` = φ j or φ` = φk for some ` = k, ` 6= j. Let us also mention, by the way, that
it is possible to recover other “nonlinear equalities” based upon (A.7) by a similar
EQUILIBRIUM STATISTICAL THEORY
141
argument. We just sketch the argument for the above passage to the limit. First of
all, one deduces easily from the div-curl lemma that for all j and k,
∂ϕ j ∂ψk
∂σ ∂σ
and
∂ϕ j ∂ϕk ∂ψ j ∂ψk
−
∂σ ∂σ
∂σ ∂σ
weakly pass to the limit on O j ∩ Ok . Since φ is continuous and Fφ00j φk vanishes on
(O j ∩ Ok )c , we may then pass to the limit in (A.6).
Acknowledgment. Andrew Majda is partially supported by NSF Grant DMS9625795, ARO Grant DAAG55-98-1-0129, and ONR Grant N00014-96-0043.
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P IERRE -L OUIS L IONS
University of Paris IX
Ceremade
Place de Lattre de Tassigny
75775 Paris, FRANCE
E-mail: lions@dmi.ens.fr
Received August 1998.
A NDREW M AJDA
Courant Institute
251 Mercer Street
New York, NY 10012-1185