- Email: [email protected]
About the stationary states of vortex systems
Ann. Inst. Henri Poincark, Prohabilite’s et Statistiques
Vol. 35, no 2, 1999, p. 205-237
About the stationary states of vortex systems Thierry BODINEAU
and Alice GUIONNET
URA 743, CNRS, Bat. 425, Universitk de Paris Sud. 91405 Orsay, France e-mail:[email protected]
ABSTRACT. - We investigate the precise behaviour of a gas of vortices approximating the vorticity of an incompressible, inviscid, two dimensional fluid as proposed by Onsager [ 161. For such mean field interacting particles system with positive vortices, the convergence of the empirical measure was proven in [3]. We improve this result by showing, for more general values of the vortices, that a large deviation principle holds. We also prove a central limit theorem for neutral gases. 0 Elsevier, Paris Key words: Statistical mechanics of two-dimensional systems, Large deviations, Central limit theorem.
Euler equations, Interacting particle
R&uM!+. - Nous Ctudions le comportement asymptotique d’un gaz de tourbillons decrivant un fluide incompressible bidimensionnel. Ce systeme est modelise par une interaction de type champ moyen. La convergence faible des mesures empiriques associees a ete prouvee darts [3]. Nous Ctendons ce resultat et prouvons un principe de grandes deviations pour des valeurs quelconques de l’intensite des tourbillons. Dans le cas d’un gaz neutre, nous Ctablissons aussi le theoreme de la limite centrale. 0 Elsevier. Paris
Mathematics Subject of Classification : 82 C 22, 60 F IO, 60 F 05. 60 E IS M. Degli Annales de I’lnsrirur
Hem+ Poincar6 Probabilites et Statistiquer Vol. 35/99/02/O Elsevier. Paris
0246-0203
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AND
A. GUIONNET
1. INTRODUCTION Recently, methods of approximations by particles systems have been widely studied. For an incompressible, inviscid, two dimensional fluid a natural approximating scheme follows the point vortex method. The basic idea of the point vortex method is to approximate the vorticity of a fluid by a “gas of vortices” which is represented by a linear combination of Dirac measures C RiSzc. The investigation of the 2 dimensional turbulence by this method has been initiated by Onsager [ 161. Onsager notices that the gas of vortices exhibits 3 different regimes. When the inverse of the temperature 0 is positive and large the vortices are mostly close to the boundary whereas they will be more or less uniformly distributed for smaller but positive /3. But also, Onsager argued that there is no reason to consider only positive temperatures: when the energy of the system is increased the vortices of the same sign are forced to be close to each other. This can be interpreted as a negative temperature state. This tendency to create local clusters of the same sign has been observed in numerical experiments by Joyce and Montgomery [ 121. Since then, many attempts have been made to understand this phenomenon. In the standard thermodynamic limit, Friihlich and Ruelle [lo] showed that this negative temperature regime does not exists. Nevertheless, it was then argued by Caglioti, Lions, Marchioro and Pulvirenti [3] and also by Eyink and Spohn [8] that the mean field scaling is relevant for the study of this negative temperature phase. In [3], it was proven that the weak limit of the Gibbs measures associated to the N-vortex systems converges towards some measure concentrated on particular stationary solution of the 2-D Euler equation. They also computed the behavior of these solutions as p converges to the critical temperature -8~. Viewing the vortex method as a way to approximate these solutions, it is natural to wonder what is the speed of this convergence. Our goal is to precise it by proving large deviations and central limit results. Also, we will investigate more precisely the role of the signs of the vortices. We follow the discretization procedure described in [ 151. The vorticity field in a bounded domain A is approximated by a linear combination of N Dirac measures concentrated in points zi of A with intensity R;. Then, the N-vortex system in A is described by the Hamiltonian
Ann&s
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where X = (xi,. . . , z,) and VA is the Green function of the Laplacian in A with Dirichlet boundary conditions. More precisely ‘dx, y E A,
vA(x,y)
= -&log/l
- Yl +Y*(z:Y).
where yA is symmetric and harmonic in each variable. Moreover. K(x) = Y&,4. In the following, we will assume that the intensities are bounded. Without loss of generality we can assume that they are bounded by 1. In [3], Caglioti et al. consider all the vorticities equal to +1 and in [lo], Frohlich and Ruelle consider the neutral case, i.e. the sum of the intensities equal 0. Here, we wish to consider general { - 1,l) valued intensities. First, we shall assume that the Ri’s have a fixed ratio of -1 and 1. We will refer to this setting as quenched. On the other hand we wish to consider as well the case where the intensities are randomly distributed, we will assume that they are independent and identically distributed (i.i.d.) with Bernoulli law QN = Q@-V. We denote by
the product of Lebesgue measures on A N. In the following, C will be either A or 52 = A @ { - 1, 1). We denote by M(C) the space of measures on C and by MT(C) the space of probability measures on C. We define M.
= {v E M:(R)
: rTT2o v = (2)
the set of probability measures with intensities marginal Q. Let [j be the inverse of the temperature. For a given sequence (Ri)iEfa of intensities, we introduce the canonical quenched Gibbs measures on M;t(A”) by d$NW
1 = p&q
exp
-!+(X,
R)}dP”(X),
where -;H;(X,
R)
>> We consider as well the averaged Gibbs measures on MF(RN) dvg,N(X,R) Vol. 35. no 2.1999
= 1 ZN(P> exp C
208
T. BODINEAU
where ZN(P) = PN 8
AND
A. GUIONNET
QN exp
(
+lf(X,
R)
1
. >>
To state our large deviation principles we need to introduce the following energy functional
=ssRR’V*(x, vvEM:(fq,&(I/) 9)dv(z, R)dv(y, R’). Define
Vv EMT(R), ;Fp(v) =H(vJP @ Q) +;f(v): (1) where H(vIP ~3 Q) is the relative entropy of v with respect to the product measure P @ Q. In section 2, we will see that Fo is lower semi-continuous. To state in the simplest and more complete way our result, we shall restrict ourselves in this introduction to the case where A is a disk. Then, we have the following quenched large deviation result THEOREM 1.1. - For any ,B in ] - 8~, CO[, if (l/N) a measure Q, the law of the empirical measure
Ci 6~, converges to
(2) under vFN satisjies a large deviation principle with ratefinction
a4
=
$v)
-inTp
ifv f MQ, otherwise.
1 Moreover, the following
averaged large deviation principle holds
THEOREM 1.2. - For any p in ] - 87r, OO[, the law of the empirical measure PN under vp,N obeys a large deviation principle with action functional
The same results hold for any compact set A provided p E (-8x, 8~); for larger values of ,B, our controls on the partition function are not good enough to ensure the exponential tightness in general. For such p’s, we need to restrict ourselves to the case of the disk. The range of temperature ,Ll E] - 87r, cc[ is optimal in the case where all the intensities Ri = 1 (see Ann&s
de I’hstirur
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[3]). On the other hand, for /3 E] - 8n, 87r[, our proof mainly depends on the uniform bound on the intensities so that the generalization of our results to any [-1, l] valued R; is straightforward. In section 2, we prove that for p 2 0 or negative and sufficiently small, !& and 8, admit a unique minimizer. Therefore, Theorems 1 .l and 1.2 imply the almost sure convergence of the empirical measure. In the last section, we investigate the fluctuations of the empirical measure. This problem turned out to be difficult because of the logarithmic singularity of the interaction. This is why we shall restrict ourselves to the case where A is a disk, p is positive and the gas neutral. In this case, the empirical measure converges towards V* = P @3Q. To describe the fluctuations of the empirical measure around v*, let us first introduce the operator E in L2(v*) with kernel l&(x, y)RR’ and I the identity in L2(v*). In the statement of the following central limit theorem, C is a subset of L2(u*) described in section 4.1. Then THEOREM 1.3. - If A is a circular disk and ,8 is positive, I) [fj card(i : Ri = +l} - card{i : R; = -l}] = o(Na), forany,@nction .f E cc,
f(z;, converges in law under viN covariance
R;) -
s
R)
towards a centered Gaussian variable with
2) If Q = +(6+1 + S-I), for any function
f(zi,
f(z, R)dv*(z.
Ri) -
.I
f E Cc.
f(x. R)dv*(q
R)
converges in law under Vp,N towards a centered Gaussian variable with covariance
One can check that Z is a non negative operator so that, at positive temperature, I + DE is always non degenerate. Vol. 35, no 2-1999.
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As a conclusion, we wish to stress that the main difficulty in this paper is to deal with the logarithmic singularity of the interaction. In particular, at high but positive inverse temperature, we have to restrict to the case where A is a disk (since we wish to consider signed intensities) in order to control the partition function. Once we consider the fluctuations of the empirical measure, the problem becomes even deeper and the hypotheses more numerous. To overcome the problem of the logarithmic singularity of the interaction, we therefore had to develop new techniques which should be useful for other models where the interaction is singular.
2. STUDY
OF THE RATE FUNCTION
In this section, we study the quenched (resp. averaged) rate function 6, (resp. G,) which is equal, up to a normalizing constant, to 30 defined by (1). We will show that GQ is a good rate function and study its minima. The generalization of these results to G, is straightforward and stated in the last subsection. 2.1. 6, is a good rate function
if ,B is larger than -8~
Our purpose here is to show that, in the range of temperature ,L?> -%r, 6, is a good rate function or in other words that the sets KM = {.v E MQ : 3&9
< M}
are compact subsets of MQ for any real number M. Since R is compact, MQ is compact so that this is equivalent to prove that the sets KM are closed, that is that 30 is lower semi-continuous. The logarithmic singularity of the energy will be controlled by the relative entropy thanks to entropic inequalities. The first step of this study is to show that for any /3 > -87r, there exists a finite constant Mp so that KM
C {u
: ff(+‘@
&)
5
MO>,
(3)
this means that 3~ is bounded from below in terms of the entropy. By definition of the Green function VA, if pf is the heat kernel in A with Dirichlet boundary conditions, the energy functional is given by
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Therefore, E is non negative which yields, if p 2 0, q3(4
2 H(4P
@ Q).
(5)
To establish such a bound in the negative temperature regime, we use the definition of the relative entropy H. Indeed, by the definition of the relative entropy and monotone convergence theorem, we have, for any rl > 0 and y E A,
But the exponent in the second term diverges as (1/27r) log ]z - y]-I. Hence, if 7 < 47r, the last term is uniformly bounded independently of y E A. Therefore, we get that, for q < 4n, there exists a finite constant C(v) = sup, log J’exp{rllVA(z, y)lVJ’(z) so that
As a consequence, for any q < 47r,
(8) where 1 + * is positive if p E] - &r, O[ and v E] q, 47r[. Therefore (5) and (8) gives (3) for p > -8~. We are now going to show that the sets of bounded entropy. This that the KM’s are closed. As a consequence of (3), any respect to P @ Q. Let us denote v with respect to P I t--1,11
the energy functional E is continuous on is enough, according to (3), to conclude v in KAY is absolutely continuous with p8, the density of the first marginal of
u(dz, dR) = pl,(z)P(ds).
Observe as well that, for any continuous function C$which vanishes in a neighborhood of {(z, y) E A2 : x = y}, the truncated energy
E&u):= Vol. 35. no 2-1999.
II
4(x:,Y)RW*(z, Y)
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is bounded continuous in MQ. Thus, in order to prove that E is continuous, it is enough to show that
vanishes when E goes to zero uniformly on {H 2 M} for a given M. In view of the singularity of VA, this is also equivalent to prove this property for
J
log (2 - yl(-ldv(z,
R)dv(y, R’).
bYI
We can apply the result proved in [4] (Proposition 2.1) which yields, for any positive t
J l%l~ LJ 1% JIZ-yl
Iz-yl
(z - y/(-l log jz - y(-1P(d2++iy)
Ix--Yl
+
(9)
(py(~)pv(Y>>py(~>P(dz)p,(y)P(dy).
The first term goes to zero with E. For the second term we notice that
I H(~,J’IP)~~P~({Y
: 1~ - y/) i cl),
Moreover, by property of the relative entropy, we know that
so that we deduce
J 1%
(PY(~.))PV(Z)P(drC)p,(Yl)P(dY)
Ix-YIP
But, by convexity H(P~,PIP)
1 s log(l/e)
HbJVW
+ %d’JJ’)).
of the relative entropy L
J
W(VR[P)~Q(R) Annuks
de I’lnstitur
= H(v(P Hem-~ Poincd
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so that we deduce from (9) that there exists 6, going to zero with F so that I.1 /3.-!//
log 15 - yl-‘pv(~)P(d~)pv(Y)P(dy)
I &Cl + H(4P
@ Q)12.
This term goes uniformly to zero with E on {H 5 111) so that E is continuous on this set. The proof is complete with (3). 0 2.2. Existence and uniqueness of the minimum
of 4,
We first tackle the positive temperature regime where the rate function satisfies the following PROPERTY2.1. - If ,0 > 0,
I) G4 is strictly convex. 2) 4, achieves its minimal value at a unique probability which is dejined by the nonlinear equation
nu*(x,R) 1 dP@Q
~‘I,(:7q/)R’rlu*(y
where
Z&Y*)
= /ixp{
-+‘R/
~~\(z.y)R’dv*(:~,
measure on MQ
R’)
R’)}dPc,,.
Proof. - Since the entropy H is strictly convex and the energy & is convex (see (4)) it follows that 8, is strictly convex if [? is non negative. Hence, it admits a unique minimizer v*. Moreover, since Gn achieves its minimum value, it is not hard to check that the minima are described, on MQ, by the nonlinear equation
&(x1.R)=& exp{ -/JR I’T/:1 (x, y)R2u*(y. I?)}.(10) Z,(u*)
= I’~xp{-NRS~~(~,~)R’d~,*(p.R’)}iiP(~r).
q
Of course, in the negative temperature case, the convexity of the rate function is not clear. We can nevertheless prove PROPERTY2.2. - There exists a negative temperature /3”, -8~ 2 /Jo < 0 so that, ,for any p E],&, 0[, the functionnal Gn achieves its minimum value at a unique probability measure u* so that
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Proof. - The proof now follows a fixed point argument. Namely, let us assume that there is two minima v and v’. As before, both of them satisfy the non linear equation (10). We are going to show that, if
then
is null. According to (lo), it implies that v = V’ and therefore gives the uniqueness of the minima. To prove the existence (which is already known in view of our construction ), we could also apply a fixed point argument based on the estimation of D(v, Y’). We leave it to the reader. We begin our argument by finding a bound on ] ]U,, (loo uniform on the minima of Gq and which will be crucial later on. Indeed, using (6) and (S), we find that, for any r] E] $!, 4~[, for any ?/ E A
Notice that, since the energy E is non negative, inf .FD is a non decreasing function of ,B and is therefore non positive for ,0 5 0. Hence,
IlQJcu L / Iww)I
d4GR) 6 &C(‘r).
(11)
We shall choose in the following q = 718 = max{ (2/3)],B], 1) so that is uniformly bounded for 0 > -6~. c(P) = &C(Q) To estimate D(v, v’), note that (10) shows that for any z in A
IUv(~)- U&:)1
Therefore, it is not hard to check that for all z E A, jUv(x) - Uvf(x)l I IPI (/
IV~ld(v + u’))e2’(j)1810(I/.V1).
5 (2c(P)e2c(P)IPI)IPJD(v, v’). Annales
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Taking the supremum over the X’S, we conclude that D(v, 21’) 5 (2c(P)e2’(P)‘P1)IPlo(v,
dj.
Since (2c(P)e”“@)~@l)IPI goes to zero with IpI, we find a positive ,& so that it is smaller than one for all /? E (-PO, 0). In that range of temperature, we deduce that D(v, v’) = 0 so that v = v’. 0 2.3. Properties
of 4,
It is not hard to generalize the preceeding to the average setting and see that LEMMA 2.3. - G, is a good rate function. There exists a positive ,& so thatfor /3 E (-,/? 0, oo), 6, admits a unique minimizer described by the non linear equation
dv*(z:R) dP@Q
1 =---Zlx(v*j
exp
3. LARGE
V’I(.X, :y)R’dv*(y,
R’)
DEVIATIONS
3.1. Existence of the Gibbs measures First we need to compute the range of temperature for which the Gibbs measures are defined. PROPOSITION 3.1. [3] - Let A be any compact set in Iw2. For any p in ] - 87r, 8n[ and any sequence of intensities with values in { - 1, l}, there is a constant C such that the following holds for all N sufficiently large
This lemma is similar to the one proven by Caglioti et al. (see Lemma 2.1 [3]). If A is a disk, a more accurate statement holds for any positive temperature PROPOSITION 3.2. - Zf A is a disk, for any ,b’ in 10, co[ and any sequence
of intensities with values fl, there is a finite real number p such that the following holds for all N sufjiciently large z,“(P)
L NP.
(13)
Note that the previous results imply that the same bounds are also valid for the averaged partition function. The proof of this sharp upper bound Vol. 35, no 2.1999.
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relies on the very specific form of VA when A is a disk. For more general domains, we do not know if such a bound holds and even if one can get any exponential bounds for /YJ> 87r and general { -1, +1}-intensities. Proof. - In order to study the case of positive (and eventually large) temperatures, we shall restrict ourselves to the case where A is a disk. To simplify the notations, we will assume that A is centered at the origin and with radius one so that VA has the specific form V*(x, y) = &log
I~-W%
IX-7J(+log I+!.?I-log
F-Y/I>
(14)
where jj is the reflection of y at the circle dh (see Lemma 3.3 of Frijhlich [9]). Note that, if z = yr + iya is the complex representation of y, rYjhas complex coordinate (l/z). In the general case, we do not know how to control the singularity of ye near the boundary of A. To prove (13), let us first remark that by using Holder’s inequality, we have
LJA’”
X
l
J
[s
dP(x) exp{ -
(15)
AN
The last term in the above r.h.s. is clearly bounded uniformly us therefore focus on the second term
in N. Let
%m=1AN dPN(X) exp( -h lcgcN xi)}. - - R&VA(xi, P
Here, we follow which reads c
Frijhlich
R;RjVA(x;,xj)
I
[9] who used a representation for the energy
= -;
c
ii;&
1% 1% - zjl
l
l
;
c
log
ljii
-
ifi+NI
(16)
where 2 is the vector in RzN so that
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iTi = +Ri
if l
-%a> 5
UN(X)
[J AN
X
C
exp{
iiiiij
log
]Xi - iijl}
2!
l
f 1
[I I to check that the expectation ?z (x
-
(17)
2
i(dP(z)
It is not difficult in the last term of the r.h.s. of (17) is finite. Let us now focus on the first term. If we denote I+ (resp. I-) the indices for which ii = +l (resp. Ri = -l), we have l
To use this representation, we adapt the argument used by Deutsch and Lavaud [7] (see also Frijhlich [9]). Indeed, let us recall the following formula from [7] valid for any complex numbers (z+, Yi)l
-
Zj)
(
rI
l
(
Yi -
Yj)
=
n
(.G
-
yj )det
lsi;j
l
=CE(O) n (.G - Yj),(19) D l
where the sum is over all the permutation o of {I! .., 7~) and E(O) is the signature of (T. Therefore, (18) and (15) shows that, since ]I- ) = J+ ) = N, if We denote (zi(X))i
dPN(X)
exp{
C
EiRj
= J
ljii
AN
dPN(X)det C ziCx>
c Vol. 35, no 2-1999.
log
-
Xjl}
l
dJ?X) CT J AN
1 - YjCx) I J-J
l
149
-
ycT(i)(x)I-‘.
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The last term in the above r.h.s. is bounded independently of the permutation 0. Therefore, we have shown that for some constant C
This completes the proof of Proposition 3.2 with (15). 3.2. A large deviations
principle
To derive the large deviations principle we have to control the singularities of the Hamiltonian. We define the new functional vu E MyyR),
2(u) =
R R’VA(x, y) d/(x, R)dv(y,
ss
R’).
X#Y
We note that for any probability measure v with finite entropy i(v) = E(v). Therefore, the main point in the proof of theorem 1.1 is to show that the energy 2 is quasi-continuous i.e. for any probability measure u in M:(a) with finite entropy, any S positive and for all R;‘s, lim lim sup b log PN (p” E {B(u, E) n {I&)
E-0
- QN)I
> 5))) = -00,
N-ace
(20) where fiN was defined in (2) and B(v, E) is the ball of radius E around v for the distance d defined by
where C”(A) is the set of continuous functions on the compact A bounded by 1. Before going on, we explain briefly how to recover the large deviations principle from the quasi-continuity (20). Since the space M:(R) is compact, it is enough to prove a weak large deviations principle. We fix u a probability measure with finite entropy. We first compute the denominator of uzN(bN E B(u,E)) and we will deal with Z:(p) in a second step. As it has been noticed in the previous section, the term $ CL1 W(Q) does not contribute in the limit, so that we omit it in the computations. PN (l~p~~+,~)l
exp(-DNi(P”))) B(V,E) l~qjP)-~(v)l>6 + exp(-N@(v) Annales
de l’lnstirut
e4-MD~GN))) + N@)PN Henri
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In order to get rid of the first term of the RHS we use Holder inequality with a coefficient Q: > 1 such that ap belongs to ] - 8x, oc[ PN ( lB(w)
l@($ye(v)j>s
exd-NM@))) I-$ < Z~(&PN
(1 B(V,i) l,r(f-.~r)-C(v),>n >
Propositions 2.1 and 2.2 and inequality (20) tells us that the above quantity vanishes exponentially fast lim limsup L log PN (l~(,,,) E’O N-m iv
l,~~~)-g(~)l>6
exp(-NB&(,i”)))
= --oc.
It remains to control the last term of the RHS. Well known large deviations results imply lirn lirn sup $
c-4
log
NL+m
PN ( lB(V,Ej exp(--Ni@(fiN)))
I -3-,,3(1/).
(21)
To prove the lower bound we note that p*
I
~B(u,E)
=p(-“&fiN)))
>P"
1 ( B(v.E) 11i(,“)-&)l<6
exp( -N/3i(fi,N))).
thus P” I l~(~,~) ev(-NBi(P))) L exp(-N@(v)
- NPS) P” (lob.;)
l~~~fil~)~:;(v~~~h).
By using inequality (20), we derive the lower bound h; _
limmzf $ log PN (l~(~,~j exp(-N@(P”))) I
2 -F/j(V).
(22)
Finally, we will check that (23)
Since M f (0) is compact, we cover it with a finite number of open balls of radius E and we get from (21)
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The reverse inequality follows immediately from (22). Combining previous results, we have proven a weak large deviation principle
the
(24) According to Theorem 4.1.11 of [5] the large deviation principle follows from (24). Similarly, Theorem 1.2 can be derived from
This can be proved in the same way as (20) so that in the following we will focus on the proof of (20). In fact, the quasi-continuity property does not depend on the intensities R;. Indeed, let us denote N N- -P
;
and
pr,
vx E A,
P&d
=
J’f--1,11
i=l
Y(X, dR),
and define an error energy Vm E M:(A),
Then (20) can be deduced from the following
Lemma
LEMMA 3.3. - For any measure m in M:(A) with finite entropy with respect to the Lebesgue measure and for any 6 positive, there is a function f~ with values in [0, W[ such that
liio lip s:p $ log PN (pN E { B( m,c) n {EIM(IL~) Furthermore
> ST}) 5 -fh(J4.
f~ satis$es li~nr f6(M)
= cc.
We postpone the proof of Lemma 3.3 and we derive (20). Annales
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In fact the potential VA is singular on {X = y} because of the logarithmic term but also near the frontier of A because of the logarithmic divergence of yI\. First we control the singularity on the diagonal. Let us be given S > 0 and a probability measure v in M ;‘( 0) with finite entropy. We introduce the functional Vu’ E M;(R): E(d)
dv’(q
=
R)dv’(y,
R’).
For any finite M the functional EI, defined on M f (62) by
Ei,(u’) = E(v’) - E&f(v’) = ll
RR’irlf
(M,iog~)du’(:r.R)dV’(!,,R’)
is continuous. Therefore, there exists a constant EATsmall enough such that
YE L edGN
E B(v, 4,
IE’ILf(v ) - E’hf (/IiN) 5 6; 4’
(25)
Since v has a finite entropy, we know that E(v) is finite (cf (7)). By dominated convergence Theorem, there is a constant M large enough such that
Combining
(E(v) - E’Al (.!I))1 5 5 4 (25) and (26) we get for all E 5 &nl
(26)
PN(fi” E {B(v) n W(iN) - EWl > v})
As EM does not depend on the sign of the vortices, we obtain an upper bound which depends only on pN
pN(GNE {B(V) n {EdfiN) > 6))) I pN(pNE {B(P~ n PM(P~)> 6))). The measure v has finite entropy, so that pv satisfies also the same property; this enables us to apply lemma 2.3. Therefore for any M large Vol. 35, Ilo 2.1999.
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enough there exists a constant EM such for all E less than EM the following holds
Letting E tends to 0 and M go to infinity, we derive the quasi-continuity of the logarithmic part of the interaction liio lilim
f log PN (pN E { B(m, &) fl {EM
> 6)))
= --co.
(27)
It remains to control the interaction term which depends on 7~ Iknli~&logP”(B(~,~)n where the functional E’(d)
=
{(E’(bN)
-E’(V)/
> S}) = -cc,
(28)
V’ --f E’(.v’) is defined by
JSA A
RRh(x,
Y) d~‘(x, RW’(y,
R’)
Noticing that 7~ is harmonic in the interior of A and has a logarithmic singularity at the boundary of A, there is some constant C such that
Therefore we can derive (28) by the same arguments as the ones used to control the logarithmic singularity. Combining (27) and (28), we complete Theorem 1.1. 3.3. Proof of Lemma 3.3 Let m be a probability measure with finite entropy. To control the singularity of the logarithm we introduce a coarse graining procedure : we partition the compact set A into cubes {Q;}il~ with side length exp( -1M). For any E positive, we define the set AE(m) by AC(m) =
c
p E M:(A)
) Vi I K,
. Is Q%
For any E’ sufficiently small B(m, E’) is included in A,(m), Chebyshev’s inequality, we have for any positive T PN (B(m, E’) n {EM
so that by
> 6)) I exp(-NWPN Annales
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Any empirical measure in A,(m) is associated to configurations that the number ‘n; of particles in the cube Q; satisfies lni - N~~m(dz)l<
NE
and
gni
= N.
such
(29)
Therefore, summing over all the K-uplets {nl, . . . . ‘II,} which satisfy (29), we get P”
(qm,
E’)
n
{E&P)
>
S})
N! exp(-NST) -< c n1!. . . njy! IL1,...,7LK exp(TNEdd#r)).
J’QIX...XQK
(30)
The number of K-uplets (7~1, . . . , no} is less than exp( K log N) so that we have just to compute the upper bound of the RHS for a given uplet (7Ll). . . , no} which satisfies (29). By using Stirling formula, we get that
Noticing that x log z > --e-l
1 -1, we get
where 1Qi 1 denotes the area of the cube Q;. Finally, we derive the upper bound N! n1!.
< exp NJRI - 5 ni .7Lk! i=l
log
I&i1 + No(N)
(31)
Let us now consider the last term in the RHS of (30). By definition of E~J, we have EM(m)
= 2
5
J,. J,
i=l
j=l
1
L#,
SUP
(o,l%
(j--&j)
-
M)
dm(z)dm(y).
3
Thus in the above sum, only the terms where Q; and Qj have a common side (including the case Qi = Qj) contribute. Let us denote by Q; the Vol. 35, no 2.1999
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T. BODINEAU
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union of the cubes which have a face in common with Qi and by 71i the number of particles in Qi. We are going to show that for any T > 0 there exist a finite constant M(T) and a positive constant E(T) such that for any M > M(T) and E < E(T) the following uniform bound holds
L (21Qil>“1,(32) where the supremum is taken over all the configurations y = { yr , . . . , y;, } in 0,. Combining (30), (31) and (32), we get that for any T there is A4 large enough such that liio liz sup $ log PN (B( m,~) II {EM(,u~)
> S}) 5 -ST + log2 + [RI.
This completes Lemma 2.2. It remains to prove (32). First we derive a preliminary
estimate
LEMMA 3.4. - For any measure m in M t(R) with finite entropy with respect to the Lebesgue measure the following holds
Proof. - For any cube IQi 1, we get from Jensen inequality
applied to
2 + zlogz
log hs&~ dx 4x1 IJ’
dx m(x) log m(x)
-
Q%
Noticing that sn dx m(x) and sA dx m(z) log m(x) are finite we deduce that the RHS goes uniformly to 0 as M grows. This completes the 0 Lemma. Ann&s
de I’lnstitut
Henri
Poincare!
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et Statistiques
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SYSTEMS
By using Holder inequality, we split (32) into two terms. The first term contains the interaction energy of the particles in Qi
J’
dxl...
dxnz exp
QZ
2T 3
n’ c
(34)
log
l#k=l
the second one bounds the interaction energy between Q; and Q;
s
sup dxl . . . dx:,% exp Y&X Q.
.
(35)
The next step is to estimate (34). From Holder inequality we get dxl . . . dx,, exp
2T
F
/ &I
c
z.#k
log 2Tni
dxp exp( 7
A straightforward
computation
log
gives for any cr in [O. 2[
;;; k, dxexp(ah&)
5 Cl+~‘~)I&il~-~‘~,
(36)
where c’ is a constant. According to (33), we know that when h4 goes to infinity and E tends to 0, 3 goes to 0 uniformly. Hence for M sufficiently large and E small enough, we get
By definition of ni (see (29)) and Lemma 2.4, we check that for M sufficiently large and E sufficiently small
Therefore,
we get
J’
dxl . . . dx:,%exp
5 exp(n; log. IQil)P.
Qt
(37) Vol. 35, no 2-1999
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T. BODINEAU
AND
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_ Let us now consider (35). We recall that 6; is the number of vortices in Qi. Since each cell interacts only with its nearest neighbors, the number 6; is of the same order as ni. By using again Lemma 2.4, if A4 is sufficiently large and E sufficiently small such that
we have sup
YE& s Qi
dxl . .
By applying Holder inequality sup YE%
s
dxi . ..dx.% exp
we obtain
2T
-
Qt
c
N
1.J
this leads to
.I
2T
sup dxl . ..dx.% exp y Qt
- c N l,j
(38)
Finally, combining (37) and (38), we complete the Lemma.
4. CENTRAL
LIMIT
THEOREM
In this last section, we study the fluctuations of the empirical measure N -N k’-
---
;
c i=l
6~i&
around V* (dx, dR), the limiting law of the empirical measures. Of course, the problems due to the logarithmic singularity of the potential become Annales
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VORTEX SYSTEMS
even more difficult than for the study of the large deviations. This is the reason why the strategy followed in [2] seems to fail. We propose here to follow an approach developed in [I l] for strongly interacting particles. This method allows to study fluctuations as soon as the empirical measure converges. Its advantage is that it can easily deal with a logarithmic singularity of the interaction. Its weakness is that it describes the fluctuations of < f, $v - L/* > only for f in a subset of L2(~*) even if V* is non degenerate. Also, this method requires a good control on the partition function that we only got in Proposition 3.2 for neutral gases at positive temperature on the disk. However, we believe that central limit theorems should hold for more general choices of the intensities, small negative temperatures and more general compact domains. The proof is performed in two steps : first we apply our strategy to get a biaised central limit theorem where the fluctuations are shifted by a remaining term. Secondly, we show that this remaining term goes to zero in probability via controls on the partition function to obtain the standard central limit result. 4.1. A biaised central limit Theorem Let S be the operator in L2(v*) with kernel V1i(:c,y)RR’ and I be the identity in L2(v*). We are going to prove biaised fluctuations for test functions f in a subset L of L2(v*) where L =
f E L2(V*)lf(x,
where
R) = (I+@)
(divtk))
, k E C’(A): k&l = 0 }
V*(dZ, dR) xR(zc) = dP(z)dQ(R) ’
Then, LEMMA 4.1. - For any function f in Is, there exists a random variable RN(~) so that 1) Under vgN and vp,N,
fbi, R)- 1 f(y,R’W*b,R’))+
RN(f)
converges in law to a centered Gaussian variable with covariance
g(f) = f(I -I-/3z)-“fdv*. .I Vol. 35, no 2-1999.
228
T. BODINEAU
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2) Quenched convergence of RN : There exists a finite constant C, so that, ij for Q > (l/2) and for N large enough, we have
LN-“, then, for any positive E we have N
> 6) 5 exp{-Cffi(t
@u&(lR~(f)l
- CfN-“)}.
i=l
3) Averaged convergence of the rest : There exists a finite constant Cf so that for any positive t we have N
QBN @ .4,(lR~(f)l
> t) I exp{-CffiE).
i=l
Let us also notice that RR’V,d(cN
- y*)d(fiN
- v*)
where
%%P) =/w{-BNl,,
RR’V,d(fiN
- V*)d(bN - v”)
The same formula holds for vp,N with the partition function .&F(p) =
J
.Z$(,f3)dQ@N(dR).
Thus, we deduce from Lemma 4.1 that LEMMA
4.2. - I) If; for a in a neighborhood l$m;p
-& Ann&s
log z;(a) de l’lnstirut
of /3,
= 0,
Henn’ Poincark
Probabilitks
et Statistiques
VORTEX
then, for any function
229
SYSTEMS
f in L, N f(G,
R)
-
J
f(Yl
wq
--CC &,
converges in law under vP”,~ to a centered Gaussian variable with covariance
o(f)= f(I + pq-‘fdv”. .I’ 2) i” for (Y in a neighborhood
then for any function
of p,
f in C, N f(%
-4 Ai=,
R)
-
f(Y>
wdv*
.i
)
converges in law under vp,N to a centered Gaussian variable with covariance G(f) =
f(I+ ps:)-‘.fdv*. s Proof. - Indeed, this assumption allows us, by Holder inequality, to compare viN (resp. vp,N) with @E, v& (resp. v*> and to conclude, according to Lemma 4.1.2 (resp. Lemma 4.1.3), that Z$N( /RN(~) 1 > t) oes to zero. Therefore, Lemma 4.2 is a direct tresP. %N(IRN(f)i > E)) g consequence of Lemma 4.1. q We will see in the next subsection that the assumption of Lemma 4.2 are fulfilled in the setting of Proposition 4.2. Proof of Lemma 4.1. - Let us first notice that, according to our large deviations principle, the term in the density of v/& containing IV,, is converging almost surely. Therefore, we can easily approximate it by its averaged value and neglect it. In the following, we will assume that this term disappears. We want to prove fluctuations in the scale (l/v%) as a consequence of the sensitivity to perturbations in the scale (l/v%). To this end, let us consider a smooth function k = (ICI, k,) and the change of variables 5; + gy; where
Q/i) xi = &(Yi) = yyi+ VT Vol. 35, no 2-1999.
230
T. BODINEAU
AND
A. GUIONNET
which is possible as soon as (IlVk((,/fi) < 1. Doing this change of variables in the partition function, we find that, if k is null at the boundary of A then q& is a bijection of A,
(40) where, if & is the derivative with respect to the ith variable (do not forget x = (xl, x2) in d = 2),
Furthermore, expanding the first term in the exponent, it is not hard to see that, since if k is continuously differentiable,
h(4 - h(y) h(x) - b(y) ,div(k),J(k) ’ - Y2 N xl-Y1 > > x2
are bounded continuous, there exists a function EN going to zero when N goes to infinity such that , R ) -H,N(K
R)
1 cNR&Dh(xi, xJk(xi); k(q)] =2Nq ;zj>l 1 + 4Nz
5 i#jZ
RiRj~(2)VA(xi,xj)[k(x;);
k(xj)12
1
+
EN
(41)
where all the terms in the expansion are bounded continuous. Here, we have denoted DV(x, y)[w, w] = lii and DC21 = 00. DV(x,
Y)[V,
WI
=
$qx
+ EU,y + ew) - I+:, y))
Notice that a,‘Jqx,
Y)Vl
+
wx:,
+ Ann&s
Y)V2
@yx:, de l’lnstitut
Y)W Henri
+
@V(x,
Poincark
Y>W2 - F’robabilitks
(42) et Statistiques
VORTEX
231
SYSTEMS
where $ denotes the derivation with respect to the jth coordinate in the ith variable. Therefore, (40) gives
x exp +q(X, 1
R)
CuqX),
(43)
1
where, we denote
(44)
(45) In other words, (40) reads exp{-EN}
=
J
exp{-A:(k)
- A~(k~)}dz$~(X,
R).
(46)
Let us interpret (46) in terms of central limit Theorem. We define f(z, R) = PR
R’DV~(y,z)[k(y);
k(z)]dv*(y,
R’) - div k(z),
(47)
J where Y* is the limiting law of the empirical measures. Recalling that BV*(y, z)[k(y); k(z)] is bounded continuous and denoting rf the bounded centered continuous function
V((G 3
(YY,R’)) = (P/4
/ R&~V&, G&R,
Vol. 35, no 2-1999.
-
d~*&2>
~2)[@,);
R1&,,~7
&2)1
-
d~*)(~2r
R2)>
232
T. BODINEAU
AND
A. GUIONNET
we find
+ dz
P ( J +Jf(x, Rw* tx,R) > -2
RR’Dvgx,
y)[k(x);
k(y)]dv*(x,
R)dv*(y,
R’)
(48)
We prove in the Appendix, Lemma 5.2 (i) that the last term in (48) is null. The second term in the r.h.s of (48) corresponds to the remainder and we let RN(~)
= +
2
c vth z#Li
Ri), txj, Rj>).
It is well known (see [1] for instance ), that, since r-f is bounded, for S small enough, S;P
J
e~p{SfiR~(f)}d(~*)‘~
< o;).
Chebyshev’s the quenched v* by the h The price is
inequality shows that RN(~) satisfies Lemma 4.1.3). To get analogue Lemma 4.1.2) of this result, one needs to replace CL1 Sn, z&, in rf. Once this is done, the same result holds. of order ]]r,]lmd(& CE, Sn,, Q). Thus, we find also that RN(~) verifies Lemma 4.1.2. Hence, the main contribution in (48) is given by the first term which is on the scale of the central limit theorem and describes the fluctuations of < f,jP - v* >. Let us consider the second term Ap in our expansion and denote
F(x, y, R, R’) = +RW2)l/n(z, #C(X); k(y)12+ $c(k)(~)~. If Ic is continuously differentiable, it is not hard to see that F(z, 9, R, R’) is bounded continuous. Moreover, h;(k)
=
J
F(x, y, R, R’)dfiN(x, Annales
de I’hstitut
Henri
R)djIiN(y, Poincart!
R’).
- Probabilitks
et Statistiques
VORTEX
233
SYSTEMS
Thus, the second term in (46) is governed by the law of large numbers and we almost surely have lim h;(k)
=
N-+02
s
F(z, y, R, R/)&*(x,
R)dv*(y,
R’).
(4%
As a consequence of our large deviations result, we have therefore proved that
- RN(~) d+v(X, 4 > = exp
F(x, y, R, R’)dv*(x,
R)dv*(y,
R’)
>
.
Extending our computation to ak for real numbers QI, our result shows that the moment generating functions of f(z;, Ri) - / f(z, R)du’hR))
+ RN(~)
converges in law to a centered Gaussian converge so that X,(f) variable with covariance 2 s F(z, y, R, R’)dv*(x, R)dv*(y, R’). Moreover, according to Lemma 5.1 in the appendix, the relation between k and f reads
f(z, R) = -(I + PS)( “tk’)
(2, R).
Finally, we prove in Lemma 5.2 (ii) in the appendix that o(f)
= 2 =
J’
.I
F(z, y, R, R’)dv*(z,
f(I+
R)dv*(y,
R’),
,E)-‘fdv*.
which achieves the proof of Lemma 4.1. 4.2. Control on the remaining
cl terms; the neutral case
Let us assume that the medium is neutral, that is that there are roughly as many positive vortices as negative vortices. In this case, it is not difficult to see that u*(dz, dR) = P(ds) @ Q(dR) Vol. 35, no 2-1999.
234
T. BODINEAU
AND
A. GUIONNET
is a minimum of Sg. Therefore, at least when the temperature is not too negative, the positive and negative vortices are both distributed uniformly over A and 2: = Zg . To check the hypothesis of Lemma 4.2 and complete the proof of Theorem 1.3, we shall rely on Proposition 3.2 and therefore restrict to the positive temperature and disk setting. We are going to see that LEMMA 4.3. - Let A be a disk. If/l Ri = -1}1 5 o(N:), then
l$JsZP I -j$
2
0
and Jcard{i : R; = +l}
logz~(p)(
- card{i :
= 0
and 6~~) (l/2)(6+1
d Y$ c
+ 6-i))
= u(N=+).
z About the averaged setting, we can prove the following LEMMA
4.4. - Let
A be a disk. rf/3 2 0 and Q = (1/2)(~7+~ limm:p I -&
logZ&?)I
+
S-,)
then,
= 0.
These two Lemmas and Lemma 4.2 complete the proof of Theorem 1.3. Proof. - The proof of the above Lemmas follows from proposition 3.2 and Jensen’s inequality. Indeed, proposition 3.2 shows that when A is a disk and /? is positive, the partition function grows at most polynomialy in N so that the upper bound of Lemma 4.2 is proven. To control the lower bound, let us apply Jensen’s inequality. For the quenched setting we get
If Cc1 Ri = o(Nz), the above r.h.s. is clearly of order o(o). The second assertion of Lemma 4.3 is clear. For the average setting, Jensen’s inequality immediatly shows that the free energy is non negative and proposition 3.2 gives the upper bound since it is uniform on the intensities. 0 Annales
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Poincark
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235
VORTEX SYSTEMS
5. APPENDIX 5.1. - If f is deJined by (47), then :
LEMMA
f(z, R) = -(I + pq(z)[divyy Proof. - Indeed, the following algebra due to integration by parts formula holds : f(z, R) = PR
J
R’DV*(y,
(S
= PR
J
R’D~V~(y,z)[k(y)Jdv*(y,
R’DA(y,
s - divk(z)
=
+ @) (kdiv(h&))
R’) - divk(x)
R’) - D(~ogbt)(~)[Wl
~[WY)]~V*(Y,
R’V~(y,z)div(kAa,(y,
= -(I
R’) - divk(z)
k(z)]dv*(y,
R’VA(Y, ~W*(Y> R’) b9[~(41 >
= /?RD + ,!?R
z)[k(y);
- &div(A&)(x)
R’))dydQ(R)
(x)
where we have denoted at the second line D and (Di)i=l,2 operators such that
DW(+u] = %W(+I D;V(z,
for any differentiable LEMMA
y)[u] = afV(x,
the differential
+ &W(x)w
Y)ZLI + @V(q
y)uz
functions W on A and V on R2.
0
5.2.
=-f /- RR’Dhb, Y)[&); Vol. 35, no 2-1999.
Vy)ld~*(a:,
RP*(y,
R’).
236
T. BODINEAU
AND
A. GUIONNE’I
(ii)
o(f) = 2
I
F(z,y,R,R')dv*(2,R)dv*(y,R')
= f(I + /3q-lfdv*. J
Proof. - Again, we could apply integration by parts formula to get our result. A short cut (but equivalent way) is to do a perturbation x + X+&(CIZZ) in the partition function of the limit law v*. Expending in E and writing that the second term in the expansion is null yields (i). Writing down that the third term is null gives
+ PRR’D,2K+, Y)[q412)d~*(R’, YP*(R,x) / (L@)(z) = ,b’ RR’D~~~(x, y)[k(z)]‘dv*(y, R’) - div (k)(z) 2d~*(R, > I( J From the definition g(f)
= p
J
Moreover, I
of D and of f, we get,
RR’DlD2v&,
+ J (f(Q)
x).
Y)p+$,
- AJ
fqY)ld~*(RJ)d~*(R’,
RR’D2h(?
Y)
Y)[wY)ld~*(fc Y> 2rlv*(R,4 >
by integration by parts, one sees that
?/)[@4l[~(Y/)l~~*(z,fw*(Y> w
RR’aD2V(T
div(kXRt)
I I 1 j E ;v;:;
(YP*(T
;;;:;y;;;;.,.
Rw*(Y,
R’)
(2, R).
and
/~~~~~v~(z,y)[k(y)]h*(R',y)
= -E(divj\:k))(c).
Thus, according to Lemma 5.1,
a(f) = iiJ5(div~~k))(5)div~~~)(~)~~*(~,~) +J(-(I+~~)(~"~~))(~)+"~(~"~~~))(z))~~~*~R,~~ =J(div~"k))(~~~~+~z)(div~k))(z)~~*(~,5) =
I
f(I + pz)-lfdv*. Annales de l’lnsritut
Henri
Poincar6
Probabilith
et Statistiques
237
VORTEX SYSTEMS ACKNOWLEDGEMENTS
We are grateful to G. Ben Arous, who told us about the problem treated in this paper, for many constructive discussions and advices. T. B. acknowledges A. TrouvC for enlightening discussions.
REFERENCES ]I] M. A. ARCONES,E. GINE, Limit Theorems for U-processes. Ann. Probub., Vol. 21, 1993. pp. 1494-1542. [2] G. BEN AROUS, M. BRUNAUD,MCthode de Laplace : Etude variationnelle des fluctuations de diffusions de type “champ moyen”, Srochastic.s , Vol. 31-32, 1990, pp. 79-144 131 E. CAGLIOTI,P. L. LYONS,C. MARCHIORO,M. PULVIRENTI,A special class of stationary flows for two dimensional Euler equations : A statistical mechanics description, Commun. Math. Phys., Vol. 143, 1991, pp. 501-525. [4] E. CACLIOTI, P. L. LIONS, C. MARCHIORO,M. PULVIRENTI,A special class of stationary flows for two dimensional Euler equations : A statistical mechanics description. Part 2 Commun. Math. Phys., Vol. 174, 1995, pp. 229-260. [S] A. DEMBO,0. ZEITOUNI.Large deviations techniques and Applications. Jones and Bartlett. 1993. 161 J.-D. DEUSCHEL,D. W. STROOCK,Larges deviations, Academic Press, 1989. [7] C. DEUSTCH,M. LAVAUD,Equilibrium properties of a two dimensional Coulomb gas, Ph~s. Rev. A, Vol. 9, 6, 1973, pp. 2598-2616. [R] G. L. EYINK, H. SPOHN,Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence, J. Stat.Phvs.. Vol. 70, Nov. 3/4, 1993. pp. 833-886. [9] J. FR~HLICH, Classical and quantum statistical mechanics in one and two dimensions. two component Yukawa and Coulomb systems. Commun. M&h. Phys.. Vol. 47, 1976, pp. 233-268. [IO] J. FK~HLICH,D. RUELLE, Statistical mechanics of vortices in an inviscid two-dimensional fluid, Commun. M&h. Phys., Vol. 87, 1982, pp. I-36. [I I] A. GUIONNET,Fluctuations for strongly interacting random variables and Wigner’s law. to appear
in Prob.
Th. Rel. Fields,
1996.
[ 121 G. JOYCE,D. MONTGOMERY,Negative temperature states for the two dimensional guidingcentre plasma, J. Plasma Phys., Vol. 10, part 1, 1973, pp. 107-I 2 1. [ 131 M. KIESSLING,Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math., Vol. 46, 1, 1993, pp. 27-56. [ 141 M. KIESSLING, J. LEBOWITZ, The micro-canonical point vortex ensemble : beyond equivalence, Left. Math. Phys., Vol. 42, 1, 1997, pp. 43-58. [ 151 C. MARCHIORO,M. PULVIRENTI,Mathematical theory of incompressible nonviscous fluids. Springer, Applied Math. sciences, Vol. 96, 1995. [ 161 L. ONSAGER,Statistical hydrodynamics, Supplemen?o al Nuovo Cimento, Vol. 6, 1949. pp. 279-287. [l7] S. SCHOCHET,The point vortex method for periodic weak solutions of the 2-D Eulet equations, Comm. Pure Appl. Math.. Vol. 49, 9, 1996, pp. 91 l-965. (Munuscript
Vol. 35, IlO z-1999
received Junuury 12, 1998; Revised \~rsion June 25, 1998.)
Vol. 35, no 2, 1999, p. 205-237
About the stationary states of vortex systems Thierry BODINEAU
and Alice GUIONNET
URA 743, CNRS, Bat. 425, Universitk de Paris Sud. 91405 Orsay, France e-mail:[email protected]
ABSTRACT. - We investigate the precise behaviour of a gas of vortices approximating the vorticity of an incompressible, inviscid, two dimensional fluid as proposed by Onsager [ 161. For such mean field interacting particles system with positive vortices, the convergence of the empirical measure was proven in [3]. We improve this result by showing, for more general values of the vortices, that a large deviation principle holds. We also prove a central limit theorem for neutral gases. 0 Elsevier, Paris Key words: Statistical mechanics of two-dimensional systems, Large deviations, Central limit theorem.
Euler equations, Interacting particle
R&uM!+. - Nous Ctudions le comportement asymptotique d’un gaz de tourbillons decrivant un fluide incompressible bidimensionnel. Ce systeme est modelise par une interaction de type champ moyen. La convergence faible des mesures empiriques associees a ete prouvee darts [3]. Nous Ctendons ce resultat et prouvons un principe de grandes deviations pour des valeurs quelconques de l’intensite des tourbillons. Dans le cas d’un gaz neutre, nous Ctablissons aussi le theoreme de la limite centrale. 0 Elsevier. Paris
Mathematics Subject of Classification : 82 C 22, 60 F IO, 60 F 05. 60 E IS M. Degli Annales de I’lnsrirur
Hem+ Poincar6 Probabilites et Statistiquer Vol. 35/99/02/O Elsevier. Paris
0246-0203
206
T. BODINEAU
AND
A. GUIONNET
1. INTRODUCTION Recently, methods of approximations by particles systems have been widely studied. For an incompressible, inviscid, two dimensional fluid a natural approximating scheme follows the point vortex method. The basic idea of the point vortex method is to approximate the vorticity of a fluid by a “gas of vortices” which is represented by a linear combination of Dirac measures C RiSzc. The investigation of the 2 dimensional turbulence by this method has been initiated by Onsager [ 161. Onsager notices that the gas of vortices exhibits 3 different regimes. When the inverse of the temperature 0 is positive and large the vortices are mostly close to the boundary whereas they will be more or less uniformly distributed for smaller but positive /3. But also, Onsager argued that there is no reason to consider only positive temperatures: when the energy of the system is increased the vortices of the same sign are forced to be close to each other. This can be interpreted as a negative temperature state. This tendency to create local clusters of the same sign has been observed in numerical experiments by Joyce and Montgomery [ 121. Since then, many attempts have been made to understand this phenomenon. In the standard thermodynamic limit, Friihlich and Ruelle [lo] showed that this negative temperature regime does not exists. Nevertheless, it was then argued by Caglioti, Lions, Marchioro and Pulvirenti [3] and also by Eyink and Spohn [8] that the mean field scaling is relevant for the study of this negative temperature phase. In [3], it was proven that the weak limit of the Gibbs measures associated to the N-vortex systems converges towards some measure concentrated on particular stationary solution of the 2-D Euler equation. They also computed the behavior of these solutions as p converges to the critical temperature -8~. Viewing the vortex method as a way to approximate these solutions, it is natural to wonder what is the speed of this convergence. Our goal is to precise it by proving large deviations and central limit results. Also, we will investigate more precisely the role of the signs of the vortices. We follow the discretization procedure described in [ 151. The vorticity field in a bounded domain A is approximated by a linear combination of N Dirac measures concentrated in points zi of A with intensity R;. Then, the N-vortex system in A is described by the Hamiltonian
Ann&s
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where X = (xi,. . . , z,) and VA is the Green function of the Laplacian in A with Dirichlet boundary conditions. More precisely ‘dx, y E A,
vA(x,y)
= -&log/l
- Yl +Y*(z:Y).
where yA is symmetric and harmonic in each variable. Moreover. K(x) = Y&,4. In the following, we will assume that the intensities are bounded. Without loss of generality we can assume that they are bounded by 1. In [3], Caglioti et al. consider all the vorticities equal to +1 and in [lo], Frohlich and Ruelle consider the neutral case, i.e. the sum of the intensities equal 0. Here, we wish to consider general { - 1,l) valued intensities. First, we shall assume that the Ri’s have a fixed ratio of -1 and 1. We will refer to this setting as quenched. On the other hand we wish to consider as well the case where the intensities are randomly distributed, we will assume that they are independent and identically distributed (i.i.d.) with Bernoulli law QN = Q@-V. We denote by
the product of Lebesgue measures on A N. In the following, C will be either A or 52 = A @ { - 1, 1). We denote by M(C) the space of measures on C and by MT(C) the space of probability measures on C. We define M.
= {v E M:(R)
: rTT2o v = (2)
the set of probability measures with intensities marginal Q. Let [j be the inverse of the temperature. For a given sequence (Ri)iEfa of intensities, we introduce the canonical quenched Gibbs measures on M;t(A”) by d$NW
1 = p&q
exp
-!+(X,
R)}dP”(X),
where -;H;(X,
R)
>> We consider as well the averaged Gibbs measures on MF(RN) dvg,N(X,R) Vol. 35. no 2.1999
= 1 ZN(P> exp C
208
T. BODINEAU
where ZN(P) = PN 8
AND
A. GUIONNET
QN exp
(
+lf(X,
R)
1
. >>
To state our large deviation principles we need to introduce the following energy functional
=ssRR’V*(x, vvEM:(fq,&(I/) 9)dv(z, R)dv(y, R’). Define
Vv EMT(R), ;Fp(v) =H(vJP @ Q) +;f(v): (1) where H(vIP ~3 Q) is the relative entropy of v with respect to the product measure P @ Q. In section 2, we will see that Fo is lower semi-continuous. To state in the simplest and more complete way our result, we shall restrict ourselves in this introduction to the case where A is a disk. Then, we have the following quenched large deviation result THEOREM 1.1. - For any ,B in ] - 8~, CO[, if (l/N) a measure Q, the law of the empirical measure
Ci 6~, converges to
(2) under vFN satisjies a large deviation principle with ratefinction
a4
=
$v)
-inTp
ifv f MQ, otherwise.
1 Moreover, the following
averaged large deviation principle holds
THEOREM 1.2. - For any p in ] - 87r, OO[, the law of the empirical measure PN under vp,N obeys a large deviation principle with action functional
The same results hold for any compact set A provided p E (-8x, 8~); for larger values of ,B, our controls on the partition function are not good enough to ensure the exponential tightness in general. For such p’s, we need to restrict ourselves to the case of the disk. The range of temperature ,Ll E] - 87r, cc[ is optimal in the case where all the intensities Ri = 1 (see Ann&s
de I’hstirur
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[3]). On the other hand, for /3 E] - 8n, 87r[, our proof mainly depends on the uniform bound on the intensities so that the generalization of our results to any [-1, l] valued R; is straightforward. In section 2, we prove that for p 2 0 or negative and sufficiently small, !& and 8, admit a unique minimizer. Therefore, Theorems 1 .l and 1.2 imply the almost sure convergence of the empirical measure. In the last section, we investigate the fluctuations of the empirical measure. This problem turned out to be difficult because of the logarithmic singularity of the interaction. This is why we shall restrict ourselves to the case where A is a disk, p is positive and the gas neutral. In this case, the empirical measure converges towards V* = P @3Q. To describe the fluctuations of the empirical measure around v*, let us first introduce the operator E in L2(v*) with kernel l&(x, y)RR’ and I the identity in L2(v*). In the statement of the following central limit theorem, C is a subset of L2(u*) described in section 4.1. Then THEOREM 1.3. - If A is a circular disk and ,8 is positive, I) [fj card(i : Ri = +l} - card{i : R; = -l}] = o(Na), forany,@nction .f E cc,
f(z;, converges in law under viN covariance
R;) -
s
R)
towards a centered Gaussian variable with
2) If Q = +(6+1 + S-I), for any function
f(zi,
f(z, R)dv*(z.
Ri) -
.I
f E Cc.
f(x. R)dv*(q
R)
converges in law under Vp,N towards a centered Gaussian variable with covariance
One can check that Z is a non negative operator so that, at positive temperature, I + DE is always non degenerate. Vol. 35, no 2-1999.
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As a conclusion, we wish to stress that the main difficulty in this paper is to deal with the logarithmic singularity of the interaction. In particular, at high but positive inverse temperature, we have to restrict to the case where A is a disk (since we wish to consider signed intensities) in order to control the partition function. Once we consider the fluctuations of the empirical measure, the problem becomes even deeper and the hypotheses more numerous. To overcome the problem of the logarithmic singularity of the interaction, we therefore had to develop new techniques which should be useful for other models where the interaction is singular.
2. STUDY
OF THE RATE FUNCTION
In this section, we study the quenched (resp. averaged) rate function 6, (resp. G,) which is equal, up to a normalizing constant, to 30 defined by (1). We will show that GQ is a good rate function and study its minima. The generalization of these results to G, is straightforward and stated in the last subsection. 2.1. 6, is a good rate function
if ,B is larger than -8~
Our purpose here is to show that, in the range of temperature ,L?> -%r, 6, is a good rate function or in other words that the sets KM = {.v E MQ : 3&9
< M}
are compact subsets of MQ for any real number M. Since R is compact, MQ is compact so that this is equivalent to prove that the sets KM are closed, that is that 30 is lower semi-continuous. The logarithmic singularity of the energy will be controlled by the relative entropy thanks to entropic inequalities. The first step of this study is to show that for any /3 > -87r, there exists a finite constant Mp so that KM
C {u
: ff(+‘@
&)
5
MO>,
(3)
this means that 3~ is bounded from below in terms of the entropy. By definition of the Green function VA, if pf is the heat kernel in A with Dirichlet boundary conditions, the energy functional is given by
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Therefore, E is non negative which yields, if p 2 0, q3(4
2 H(4P
@ Q).
(5)
To establish such a bound in the negative temperature regime, we use the definition of the relative entropy H. Indeed, by the definition of the relative entropy and monotone convergence theorem, we have, for any rl > 0 and y E A,
But the exponent in the second term diverges as (1/27r) log ]z - y]-I. Hence, if 7 < 47r, the last term is uniformly bounded independently of y E A. Therefore, we get that, for q < 4n, there exists a finite constant C(v) = sup, log J’exp{rllVA(z, y)lVJ’(z) so that
As a consequence, for any q < 47r,
(8) where 1 + * is positive if p E] - &r, O[ and v E] q, 47r[. Therefore (5) and (8) gives (3) for p > -8~. We are now going to show that the sets of bounded entropy. This that the KM’s are closed. As a consequence of (3), any respect to P @ Q. Let us denote v with respect to P I t--1,11
the energy functional E is continuous on is enough, according to (3), to conclude v in KAY is absolutely continuous with p8, the density of the first marginal of
u(dz, dR) = pl,(z)P(ds).
Observe as well that, for any continuous function C$which vanishes in a neighborhood of {(z, y) E A2 : x = y}, the truncated energy
E&u):= Vol. 35. no 2-1999.
II
4(x:,Y)RW*(z, Y)
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is bounded continuous in MQ. Thus, in order to prove that E is continuous, it is enough to show that
vanishes when E goes to zero uniformly on {H 2 M} for a given M. In view of the singularity of VA, this is also equivalent to prove this property for
J
log (2 - yl(-ldv(z,
R)dv(y, R’).
bYI
We can apply the result proved in [4] (Proposition 2.1) which yields, for any positive t
J l%l~ LJ 1% JIZ-yl
Iz-yl
(z - y/(-l log jz - y(-1P(d2++iy)
Ix--Yl
+
(9)
(py(~)pv(Y>>py(~>P(dz)p,(y)P(dy).
The first term goes to zero with E. For the second term we notice that
I H(~,J’IP)~~P~({Y
: 1~ - y/) i cl),
Moreover, by property of the relative entropy, we know that
so that we deduce
J 1%
(PY(~.))PV(Z)P(drC)p,(Yl)P(dY)
Ix-YIP
But, by convexity H(P~,PIP)
1 s log(l/e)
HbJVW
+ %d’JJ’)).
of the relative entropy L
J
W(VR[P)~Q(R) Annuks
de I’lnstitur
= H(v(P Hem-~ Poincd
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so that we deduce from (9) that there exists 6, going to zero with F so that I.1 /3.-!//
log 15 - yl-‘pv(~)P(d~)pv(Y)P(dy)
I &Cl + H(4P
@ Q)12.
This term goes uniformly to zero with E on {H 5 111) so that E is continuous on this set. The proof is complete with (3). 0 2.2. Existence and uniqueness of the minimum
of 4,
We first tackle the positive temperature regime where the rate function satisfies the following PROPERTY2.1. - If ,0 > 0,
I) G4 is strictly convex. 2) 4, achieves its minimal value at a unique probability which is dejined by the nonlinear equation
nu*(x,R) 1 dP@Q
~‘I,(:7q/)R’rlu*(y
where
Z&Y*)
= /ixp{
-+‘R/
~~\(z.y)R’dv*(:~,
measure on MQ
R’)
R’)}dPc,,.
Proof. - Since the entropy H is strictly convex and the energy & is convex (see (4)) it follows that 8, is strictly convex if [? is non negative. Hence, it admits a unique minimizer v*. Moreover, since Gn achieves its minimum value, it is not hard to check that the minima are described, on MQ, by the nonlinear equation
&(x1.R)=& exp{ -/JR I’T/:1 (x, y)R2u*(y. I?)}.(10) Z,(u*)
= I’~xp{-NRS~~(~,~)R’d~,*(p.R’)}iiP(~r).
q
Of course, in the negative temperature case, the convexity of the rate function is not clear. We can nevertheless prove PROPERTY2.2. - There exists a negative temperature /3”, -8~ 2 /Jo < 0 so that, ,for any p E],&, 0[, the functionnal Gn achieves its minimum value at a unique probability measure u* so that
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Proof. - The proof now follows a fixed point argument. Namely, let us assume that there is two minima v and v’. As before, both of them satisfy the non linear equation (10). We are going to show that, if
then
is null. According to (lo), it implies that v = V’ and therefore gives the uniqueness of the minima. To prove the existence (which is already known in view of our construction ), we could also apply a fixed point argument based on the estimation of D(v, Y’). We leave it to the reader. We begin our argument by finding a bound on ] ]U,, (loo uniform on the minima of Gq and which will be crucial later on. Indeed, using (6) and (S), we find that, for any r] E] $!, 4~[, for any ?/ E A
Notice that, since the energy E is non negative, inf .FD is a non decreasing function of ,B and is therefore non positive for ,0 5 0. Hence,
IlQJcu L / Iww)I
d4GR) 6 &C(‘r).
(11)
We shall choose in the following q = 718 = max{ (2/3)],B], 1) so that is uniformly bounded for 0 > -6~. c(P) = &C(Q) To estimate D(v, v’), note that (10) shows that for any z in A
IUv(~)- U&:)1
Therefore, it is not hard to check that for all z E A, jUv(x) - Uvf(x)l I IPI (/
IV~ld(v + u’))e2’(j)1810(I/.V1).
5 (2c(P)e2c(P)IPI)IPJD(v, v’). Annales
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Taking the supremum over the X’S, we conclude that D(v, 21’) 5 (2c(P)e2’(P)‘P1)IPlo(v,
dj.
Since (2c(P)e”“@)~@l)IPI goes to zero with IpI, we find a positive ,& so that it is smaller than one for all /? E (-PO, 0). In that range of temperature, we deduce that D(v, v’) = 0 so that v = v’. 0 2.3. Properties
of 4,
It is not hard to generalize the preceeding to the average setting and see that LEMMA 2.3. - G, is a good rate function. There exists a positive ,& so thatfor /3 E (-,/? 0, oo), 6, admits a unique minimizer described by the non linear equation
dv*(z:R) dP@Q
1 =---Zlx(v*j
exp
3. LARGE
V’I(.X, :y)R’dv*(y,
R’)
DEVIATIONS
3.1. Existence of the Gibbs measures First we need to compute the range of temperature for which the Gibbs measures are defined. PROPOSITION 3.1. [3] - Let A be any compact set in Iw2. For any p in ] - 87r, 8n[ and any sequence of intensities with values in { - 1, l}, there is a constant C such that the following holds for all N sufficiently large
This lemma is similar to the one proven by Caglioti et al. (see Lemma 2.1 [3]). If A is a disk, a more accurate statement holds for any positive temperature PROPOSITION 3.2. - Zf A is a disk, for any ,b’ in 10, co[ and any sequence
of intensities with values fl, there is a finite real number p such that the following holds for all N sufjiciently large z,“(P)
L NP.
(13)
Note that the previous results imply that the same bounds are also valid for the averaged partition function. The proof of this sharp upper bound Vol. 35, no 2.1999.
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relies on the very specific form of VA when A is a disk. For more general domains, we do not know if such a bound holds and even if one can get any exponential bounds for /YJ> 87r and general { -1, +1}-intensities. Proof. - In order to study the case of positive (and eventually large) temperatures, we shall restrict ourselves to the case where A is a disk. To simplify the notations, we will assume that A is centered at the origin and with radius one so that VA has the specific form V*(x, y) = &log
I~-W%
IX-7J(+log I+!.?I-log
F-Y/I>
(14)
where jj is the reflection of y at the circle dh (see Lemma 3.3 of Frijhlich [9]). Note that, if z = yr + iya is the complex representation of y, rYjhas complex coordinate (l/z). In the general case, we do not know how to control the singularity of ye near the boundary of A. To prove (13), let us first remark that by using Holder’s inequality, we have
LJA’”
X
l
J
[s
dP(x) exp{ -
(15)
AN
The last term in the above r.h.s. is clearly bounded uniformly us therefore focus on the second term
in N. Let
%m=1AN dPN(X) exp( -h lcgcN xi)}. - - R&VA(xi, P
Here, we follow which reads c
Frijhlich
R;RjVA(x;,xj)
I
[9] who used a representation for the energy
= -;
c
ii;&
1% 1% - zjl
l
l
;
c
log
ljii
-
ifi+NI
(16)
where 2 is the vector in RzN so that
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and
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iTi = +Ri
if l
-%a> 5
UN(X)
[J AN
X
C
exp{
iiiiij
log
]Xi - iijl}
2!
l
f 1
[I I to check that the expectation ?z (x
-
(17)
2
i(dP(z)
It is not difficult in the last term of the r.h.s. of (17) is finite. Let us now focus on the first term. If we denote I+ (resp. I-) the indices for which ii = +l (resp. Ri = -l), we have l
To use this representation, we adapt the argument used by Deutsch and Lavaud [7] (see also Frijhlich [9]). Indeed, let us recall the following formula from [7] valid for any complex numbers (z+, Yi)l
-
Zj)
(
rI
l
(
Yi -
Yj)
=
n
(.G
-
yj )det
lsi;j
l
=CE(O) n (.G - Yj),(19) D l
where the sum is over all the permutation o of {I! .., 7~) and E(O) is the signature of (T. Therefore, (18) and (15) shows that, since ]I- ) = J+ ) = N, if We denote (zi(X))i
dPN(X)
exp{
C
EiRj
= J
ljii
AN
dPN(X)det C ziCx>
c Vol. 35, no 2-1999.
log
-
Xjl}
l
dJ?X) CT J AN
1 - YjCx) I J-J
l
149
-
ycT(i)(x)I-‘.
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The last term in the above r.h.s. is bounded independently of the permutation 0. Therefore, we have shown that for some constant C
This completes the proof of Proposition 3.2 with (15). 3.2. A large deviations
principle
To derive the large deviations principle we have to control the singularities of the Hamiltonian. We define the new functional vu E MyyR),
2(u) =
R R’VA(x, y) d/(x, R)dv(y,
ss
R’).
X#Y
We note that for any probability measure v with finite entropy i(v) = E(v). Therefore, the main point in the proof of theorem 1.1 is to show that the energy 2 is quasi-continuous i.e. for any probability measure u in M:(a) with finite entropy, any S positive and for all R;‘s, lim lim sup b log PN (p” E {B(u, E) n {I&)
E-0
- QN)I
> 5))) = -00,
N-ace
(20) where fiN was defined in (2) and B(v, E) is the ball of radius E around v for the distance d defined by
where C”(A) is the set of continuous functions on the compact A bounded by 1. Before going on, we explain briefly how to recover the large deviations principle from the quasi-continuity (20). Since the space M:(R) is compact, it is enough to prove a weak large deviations principle. We fix u a probability measure with finite entropy. We first compute the denominator of uzN(bN E B(u,E)) and we will deal with Z:(p) in a second step. As it has been noticed in the previous section, the term $ CL1 W(Q) does not contribute in the limit, so that we omit it in the computations. PN (l~p~~+,~)l
exp(-DNi(P”))) B(V,E) l~qjP)-~(v)l>6 + exp(-N@(v) Annales
de l’lnstirut
e4-MD~GN))) + N@)PN Henri
Poincar6
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In order to get rid of the first term of the RHS we use Holder inequality with a coefficient Q: > 1 such that ap belongs to ] - 8x, oc[ PN ( lB(w)
l@($ye(v)j>s
exd-NM@))) I-$ < Z~(&PN
(1 B(V,i) l,r(f-.~r)-C(v),>n >
Propositions 2.1 and 2.2 and inequality (20) tells us that the above quantity vanishes exponentially fast lim limsup L log PN (l~(,,,) E’O N-m iv
l,~~~)-g(~)l>6
exp(-NB&(,i”)))
= --oc.
It remains to control the last term of the RHS. Well known large deviations results imply lirn lirn sup $
c-4
log
NL+m
PN ( lB(V,Ej exp(--Ni@(fiN)))
I -3-,,3(1/).
(21)
To prove the lower bound we note that p*
I
~B(u,E)
=p(-“&fiN)))
>P"
1 ( B(v.E) 11i(,“)-&)l<6
exp( -N/3i(fi,N))).
thus P” I l~(~,~) ev(-NBi(P))) L exp(-N@(v)
- NPS) P” (lob.;)
l~~~fil~)~:;(v~~~h).
By using inequality (20), we derive the lower bound h; _
limmzf $ log PN (l~(~,~j exp(-N@(P”))) I
2 -F/j(V).
(22)
Finally, we will check that (23)
Since M f (0) is compact, we cover it with a finite number of open balls of radius E and we get from (21)
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The reverse inequality follows immediately from (22). Combining previous results, we have proven a weak large deviation principle
the
(24) According to Theorem 4.1.11 of [5] the large deviation principle follows from (24). Similarly, Theorem 1.2 can be derived from
This can be proved in the same way as (20) so that in the following we will focus on the proof of (20). In fact, the quasi-continuity property does not depend on the intensities R;. Indeed, let us denote N N- -P
;
and
pr,
vx E A,
P&d
=
J’f--1,11
i=l
Y(X, dR),
and define an error energy Vm E M:(A),
Then (20) can be deduced from the following
Lemma
LEMMA 3.3. - For any measure m in M:(A) with finite entropy with respect to the Lebesgue measure and for any 6 positive, there is a function f~ with values in [0, W[ such that
liio lip s:p $ log PN (pN E { B( m,c) n {EIM(IL~) Furthermore
> ST}) 5 -fh(J4.
f~ satis$es li~nr f6(M)
= cc.
We postpone the proof of Lemma 3.3 and we derive (20). Annales
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In fact the potential VA is singular on {X = y} because of the logarithmic term but also near the frontier of A because of the logarithmic divergence of yI\. First we control the singularity on the diagonal. Let us be given S > 0 and a probability measure v in M ;‘( 0) with finite entropy. We introduce the functional Vu’ E M;(R): E(d)
dv’(q
=
R)dv’(y,
R’).
For any finite M the functional EI, defined on M f (62) by
Ei,(u’) = E(v’) - E&f(v’) = ll
RR’irlf
(M,iog~)du’(:r.R)dV’(!,,R’)
is continuous. Therefore, there exists a constant EATsmall enough such that
YE L edGN
E B(v, 4,
IE’ILf(v ) - E’hf (/IiN) 5 6; 4’
(25)
Since v has a finite entropy, we know that E(v) is finite (cf (7)). By dominated convergence Theorem, there is a constant M large enough such that
Combining
(E(v) - E’Al (.!I))1 5 5 4 (25) and (26) we get for all E 5 &nl
(26)
PN(fi” E {B(v) n W(iN) - EWl > v})
As EM does not depend on the sign of the vortices, we obtain an upper bound which depends only on pN
pN(GNE {B(V) n {EdfiN) > 6))) I pN(pNE {B(P~ n PM(P~)> 6))). The measure v has finite entropy, so that pv satisfies also the same property; this enables us to apply lemma 2.3. Therefore for any M large Vol. 35, Ilo 2.1999.
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enough there exists a constant EM such for all E less than EM the following holds
Letting E tends to 0 and M go to infinity, we derive the quasi-continuity of the logarithmic part of the interaction liio lilim
f log PN (pN E { B(m, &) fl {EM
> 6)))
= --co.
(27)
It remains to control the interaction term which depends on 7~ Iknli~&logP”(B(~,~)n where the functional E’(d)
=
{(E’(bN)
-E’(V)/
> S}) = -cc,
(28)
V’ --f E’(.v’) is defined by
JSA A
RRh(x,
Y) d~‘(x, RW’(y,
R’)
Noticing that 7~ is harmonic in the interior of A and has a logarithmic singularity at the boundary of A, there is some constant C such that
Therefore we can derive (28) by the same arguments as the ones used to control the logarithmic singularity. Combining (27) and (28), we complete Theorem 1.1. 3.3. Proof of Lemma 3.3 Let m be a probability measure with finite entropy. To control the singularity of the logarithm we introduce a coarse graining procedure : we partition the compact set A into cubes {Q;}il~ with side length exp( -1M). For any E positive, we define the set AE(m) by AC(m) =
c
p E M:(A)
) Vi I K,
. Is Q%
For any E’ sufficiently small B(m, E’) is included in A,(m), Chebyshev’s inequality, we have for any positive T PN (B(m, E’) n {EM
so that by
> 6)) I exp(-NWPN Annales
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Any empirical measure in A,(m) is associated to configurations that the number ‘n; of particles in the cube Q; satisfies lni - N~~m(dz)l<
NE
and
gni
= N.
such
(29)
Therefore, summing over all the K-uplets {nl, . . . . ‘II,} which satisfy (29), we get P”
(qm,
E’)
n
{E&P)
>
S})
N! exp(-NST) -< c n1!. . . njy! IL1,...,7LK exp(TNEdd#r)).
J’QIX...XQK
(30)
The number of K-uplets (7~1, . . . , no} is less than exp( K log N) so that we have just to compute the upper bound of the RHS for a given uplet (7Ll). . . , no} which satisfies (29). By using Stirling formula, we get that
Noticing that x log z > --e-l
1 -1, we get
where 1Qi 1 denotes the area of the cube Q;. Finally, we derive the upper bound N! n1!.
< exp NJRI - 5 ni .7Lk! i=l
log
I&i1 + No(N)
(31)
Let us now consider the last term in the RHS of (30). By definition of E~J, we have EM(m)
= 2
5
J,. J,
i=l
j=l
1
L#,
SUP
(o,l%
(j--&j)
-
M)
dm(z)dm(y).
3
Thus in the above sum, only the terms where Q; and Qj have a common side (including the case Qi = Qj) contribute. Let us denote by Q; the Vol. 35, no 2.1999
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T. BODINEAU
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union of the cubes which have a face in common with Qi and by 71i the number of particles in Qi. We are going to show that for any T > 0 there exist a finite constant M(T) and a positive constant E(T) such that for any M > M(T) and E < E(T) the following uniform bound holds
L (21Qil>“1,(32) where the supremum is taken over all the configurations y = { yr , . . . , y;, } in 0,. Combining (30), (31) and (32), we get that for any T there is A4 large enough such that liio liz sup $ log PN (B( m,~) II {EM(,u~)
> S}) 5 -ST + log2 + [RI.
This completes Lemma 2.2. It remains to prove (32). First we derive a preliminary
estimate
LEMMA 3.4. - For any measure m in M t(R) with finite entropy with respect to the Lebesgue measure the following holds
Proof. - For any cube IQi 1, we get from Jensen inequality
applied to
2 + zlogz
log hs&~ dx 4x1 IJ’
dx m(x) log m(x)
-
Q%
Noticing that sn dx m(x) and sA dx m(z) log m(x) are finite we deduce that the RHS goes uniformly to 0 as M grows. This completes the 0 Lemma. Ann&s
de I’lnstitut
Henri
Poincare!
- Pmbabilit&
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SYSTEMS
By using Holder inequality, we split (32) into two terms. The first term contains the interaction energy of the particles in Qi
J’
dxl...
dxnz exp
QZ
2T 3
n’ c
(34)
log
l#k=l
the second one bounds the interaction energy between Q; and Q;
s
sup dxl . . . dx:,% exp Y&X Q.
.
(35)
The next step is to estimate (34). From Holder inequality we get dxl . . . dx,, exp
2T
F
/ &I
c
z.#k
log 2Tni
dxp exp( 7
A straightforward
computation
log
gives for any cr in [O. 2[
;;; k, dxexp(ah&)
5 Cl+~‘~)I&il~-~‘~,
(36)
where c’ is a constant. According to (33), we know that when h4 goes to infinity and E tends to 0, 3 goes to 0 uniformly. Hence for M sufficiently large and E small enough, we get
By definition of ni (see (29)) and Lemma 2.4, we check that for M sufficiently large and E sufficiently small
Therefore,
we get
J’
dxl . . . dx:,%exp
5 exp(n; log. IQil)P.
Qt
(37) Vol. 35, no 2-1999
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T. BODINEAU
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_ Let us now consider (35). We recall that 6; is the number of vortices in Qi. Since each cell interacts only with its nearest neighbors, the number 6; is of the same order as ni. By using again Lemma 2.4, if A4 is sufficiently large and E sufficiently small such that
we have sup
YE& s Qi
dxl . .
By applying Holder inequality sup YE%
s
dxi . ..dx.% exp
we obtain
2T
-
Qt
c
N
1.J
this leads to
.I
2T
sup dxl . ..dx.% exp y Qt
- c N l,j
(38)
Finally, combining (37) and (38), we complete the Lemma.
4. CENTRAL
LIMIT
THEOREM
In this last section, we study the fluctuations of the empirical measure N -N k’-
---
;
c i=l
6~i&
around V* (dx, dR), the limiting law of the empirical measures. Of course, the problems due to the logarithmic singularity of the potential become Annales
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VORTEX SYSTEMS
even more difficult than for the study of the large deviations. This is the reason why the strategy followed in [2] seems to fail. We propose here to follow an approach developed in [I l] for strongly interacting particles. This method allows to study fluctuations as soon as the empirical measure converges. Its advantage is that it can easily deal with a logarithmic singularity of the interaction. Its weakness is that it describes the fluctuations of < f, $v - L/* > only for f in a subset of L2(~*) even if V* is non degenerate. Also, this method requires a good control on the partition function that we only got in Proposition 3.2 for neutral gases at positive temperature on the disk. However, we believe that central limit theorems should hold for more general choices of the intensities, small negative temperatures and more general compact domains. The proof is performed in two steps : first we apply our strategy to get a biaised central limit theorem where the fluctuations are shifted by a remaining term. Secondly, we show that this remaining term goes to zero in probability via controls on the partition function to obtain the standard central limit result. 4.1. A biaised central limit Theorem Let S be the operator in L2(v*) with kernel V1i(:c,y)RR’ and I be the identity in L2(v*). We are going to prove biaised fluctuations for test functions f in a subset L of L2(v*) where L =
f E L2(V*)lf(x,
where
R) = (I+@)
(divtk))
, k E C’(A): k&l = 0 }
V*(dZ, dR) xR(zc) = dP(z)dQ(R) ’
Then, LEMMA 4.1. - For any function f in Is, there exists a random variable RN(~) so that 1) Under vgN and vp,N,
fbi, R)- 1 f(y,R’W*b,R’))+
RN(f)
converges in law to a centered Gaussian variable with covariance
g(f) = f(I -I-/3z)-“fdv*. .I Vol. 35, no 2-1999.
228
T. BODINEAU
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2) Quenched convergence of RN : There exists a finite constant C, so that, ij for Q > (l/2) and for N large enough, we have
LN-“, then, for any positive E we have N
> 6) 5 exp{-Cffi(t
@u&(lR~(f)l
- CfN-“)}.
i=l
3) Averaged convergence of the rest : There exists a finite constant Cf so that for any positive t we have N
QBN @ .4,(lR~(f)l
> t) I exp{-CffiE).
i=l
Let us also notice that RR’V,d(cN
- y*)d(fiN
- v*)
where
%%P) =/w{-BNl,,
RR’V,d(fiN
- V*)d(bN - v”)
The same formula holds for vp,N with the partition function .&F(p) =
J
.Z$(,f3)dQ@N(dR).
Thus, we deduce from Lemma 4.1 that LEMMA
4.2. - I) If; for a in a neighborhood l$m;p
-& Ann&s
log z;(a) de l’lnstirut
of /3,
= 0,
Henn’ Poincark
Probabilitks
et Statistiques
VORTEX
then, for any function
229
SYSTEMS
f in L, N f(G,
R)
-
J
f(Yl
wq
--CC &,
converges in law under vP”,~ to a centered Gaussian variable with covariance
o(f)= f(I + pq-‘fdv”. .I’ 2) i” for (Y in a neighborhood
then for any function
of p,
f in C, N f(%
-4 Ai=,
R)
-
f(Y>
wdv*
.i
)
converges in law under vp,N to a centered Gaussian variable with covariance G(f) =
f(I+ ps:)-‘.fdv*. s Proof. - Indeed, this assumption allows us, by Holder inequality, to compare viN (resp. vp,N) with @E, v& (resp. v*> and to conclude, according to Lemma 4.1.2 (resp. Lemma 4.1.3), that Z$N( /RN(~) 1 > t) oes to zero. Therefore, Lemma 4.2 is a direct tresP. %N(IRN(f)i > E)) g consequence of Lemma 4.1. q We will see in the next subsection that the assumption of Lemma 4.2 are fulfilled in the setting of Proposition 4.2. Proof of Lemma 4.1. - Let us first notice that, according to our large deviations principle, the term in the density of v/& containing IV,, is converging almost surely. Therefore, we can easily approximate it by its averaged value and neglect it. In the following, we will assume that this term disappears. We want to prove fluctuations in the scale (l/v%) as a consequence of the sensitivity to perturbations in the scale (l/v%). To this end, let us consider a smooth function k = (ICI, k,) and the change of variables 5; + gy; where
Q/i) xi = &(Yi) = yyi+ VT Vol. 35, no 2-1999.
230
T. BODINEAU
AND
A. GUIONNET
which is possible as soon as (IlVk((,/fi) < 1. Doing this change of variables in the partition function, we find that, if k is null at the boundary of A then q& is a bijection of A,
(40) where, if & is the derivative with respect to the ith variable (do not forget x = (xl, x2) in d = 2),
Furthermore, expanding the first term in the exponent, it is not hard to see that, since if k is continuously differentiable,
h(4 - h(y) h(x) - b(y) ,div(k),J(k) ’ - Y2 N xl-Y1 > > x2
are bounded continuous, there exists a function EN going to zero when N goes to infinity such that , R ) -H,N(K
R)
1 cNR&Dh(xi, xJk(xi); k(q)] =2Nq ;zj>l 1 + 4Nz
5 i#jZ
RiRj~(2)VA(xi,xj)[k(x;);
k(xj)12
1
+
EN
(41)
where all the terms in the expansion are bounded continuous. Here, we have denoted DV(x, y)[w, w] = lii and DC21 = 00. DV(x,
Y)[V,
WI
=
$qx
+ EU,y + ew) - I+:, y))
Notice that a,‘Jqx,
Y)Vl
+
wx:,
+ Ann&s
Y)V2
@yx:, de l’lnstitut
Y)W Henri
+
@V(x,
Poincark
Y>W2 - F’robabilitks
(42) et Statistiques
VORTEX
231
SYSTEMS
where $ denotes the derivation with respect to the jth coordinate in the ith variable. Therefore, (40) gives
x exp +q(X, 1
R)
CuqX),
(43)
1
where, we denote
(44)
(45) In other words, (40) reads exp{-EN}
=
J
exp{-A:(k)
- A~(k~)}dz$~(X,
R).
(46)
Let us interpret (46) in terms of central limit Theorem. We define f(z, R) = PR
R’DV~(y,z)[k(y);
k(z)]dv*(y,
R’) - div k(z),
(47)
J where Y* is the limiting law of the empirical measures. Recalling that BV*(y, z)[k(y); k(z)] is bounded continuous and denoting rf the bounded centered continuous function
V((G 3
(YY,R’)) = (P/4
/ R&~V&, G&R,
Vol. 35, no 2-1999.
-
d~*&2>
~2)[@,);
R1&,,~7
&2)1
-
d~*)(~2r
R2)>
232
T. BODINEAU
AND
A. GUIONNET
we find
+ dz
P ( J +Jf(x, Rw* tx,R) > -2
RR’Dvgx,
y)[k(x);
k(y)]dv*(x,
R)dv*(y,
R’)
(48)
We prove in the Appendix, Lemma 5.2 (i) that the last term in (48) is null. The second term in the r.h.s of (48) corresponds to the remainder and we let RN(~)
= +
2
c vth z#Li
Ri), txj, Rj>).
It is well known (see [1] for instance ), that, since r-f is bounded, for S small enough, S;P
J
e~p{SfiR~(f)}d(~*)‘~
< o;).
Chebyshev’s the quenched v* by the h The price is
inequality shows that RN(~) satisfies Lemma 4.1.3). To get analogue Lemma 4.1.2) of this result, one needs to replace CL1 Sn, z&, in rf. Once this is done, the same result holds. of order ]]r,]lmd(& CE, Sn,, Q). Thus, we find also that RN(~) verifies Lemma 4.1.2. Hence, the main contribution in (48) is given by the first term which is on the scale of the central limit theorem and describes the fluctuations of < f,jP - v* >. Let us consider the second term Ap in our expansion and denote
F(x, y, R, R’) = +RW2)l/n(z, #C(X); k(y)12+ $c(k)(~)~. If Ic is continuously differentiable, it is not hard to see that F(z, 9, R, R’) is bounded continuous. Moreover, h;(k)
=
J
F(x, y, R, R’)dfiN(x, Annales
de I’hstitut
Henri
R)djIiN(y, Poincart!
R’).
- Probabilitks
et Statistiques
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233
SYSTEMS
Thus, the second term in (46) is governed by the law of large numbers and we almost surely have lim h;(k)
=
N-+02
s
F(z, y, R, R/)&*(x,
R)dv*(y,
R’).
(4%
As a consequence of our large deviations result, we have therefore proved that
- RN(~) d+v(X, 4 > = exp
F(x, y, R, R’)dv*(x,
R)dv*(y,
R’)
>
.
Extending our computation to ak for real numbers QI, our result shows that the moment generating functions of f(z;, Ri) - / f(z, R)du’hR))
+ RN(~)
converges in law to a centered Gaussian converge so that X,(f) variable with covariance 2 s F(z, y, R, R’)dv*(x, R)dv*(y, R’). Moreover, according to Lemma 5.1 in the appendix, the relation between k and f reads
f(z, R) = -(I + PS)( “tk’)
(2, R).
Finally, we prove in Lemma 5.2 (ii) in the appendix that o(f)
= 2 =
J’
.I
F(z, y, R, R’)dv*(z,
f(I+
R)dv*(y,
R’),
,E)-‘fdv*.
which achieves the proof of Lemma 4.1. 4.2. Control on the remaining
cl terms; the neutral case
Let us assume that the medium is neutral, that is that there are roughly as many positive vortices as negative vortices. In this case, it is not difficult to see that u*(dz, dR) = P(ds) @ Q(dR) Vol. 35, no 2-1999.
234
T. BODINEAU
AND
A. GUIONNET
is a minimum of Sg. Therefore, at least when the temperature is not too negative, the positive and negative vortices are both distributed uniformly over A and 2: = Zg . To check the hypothesis of Lemma 4.2 and complete the proof of Theorem 1.3, we shall rely on Proposition 3.2 and therefore restrict to the positive temperature and disk setting. We are going to see that LEMMA 4.3. - Let A be a disk. If/l Ri = -1}1 5 o(N:), then
l$JsZP I -j$
2
0
and Jcard{i : R; = +l}
logz~(p)(
- card{i :
= 0
and 6~~) (l/2)(6+1
d Y$ c
+ 6-i))
= u(N=+).
z About the averaged setting, we can prove the following LEMMA
4.4. - Let
A be a disk. rf/3 2 0 and Q = (1/2)(~7+~ limm:p I -&
logZ&?)I
+
S-,)
then,
= 0.
These two Lemmas and Lemma 4.2 complete the proof of Theorem 1.3. Proof. - The proof of the above Lemmas follows from proposition 3.2 and Jensen’s inequality. Indeed, proposition 3.2 shows that when A is a disk and /? is positive, the partition function grows at most polynomialy in N so that the upper bound of Lemma 4.2 is proven. To control the lower bound, let us apply Jensen’s inequality. For the quenched setting we get
If Cc1 Ri = o(Nz), the above r.h.s. is clearly of order o(o). The second assertion of Lemma 4.3 is clear. For the average setting, Jensen’s inequality immediatly shows that the free energy is non negative and proposition 3.2 gives the upper bound since it is uniform on the intensities. 0 Annales
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VORTEX SYSTEMS
5. APPENDIX 5.1. - If f is deJined by (47), then :
LEMMA
f(z, R) = -(I + pq(z)[divyy Proof. - Indeed, the following algebra due to integration by parts formula holds : f(z, R) = PR
J
R’DV*(y,
(S
= PR
J
R’D~V~(y,z)[k(y)Jdv*(y,
R’DA(y,
s - divk(z)
=
+ @) (kdiv(h&))
R’) - divk(x)
R’) - D(~ogbt)(~)[Wl
~[WY)]~V*(Y,
R’V~(y,z)div(kAa,(y,
= -(I
R’) - divk(z)
k(z)]dv*(y,
R’VA(Y, ~W*(Y> R’) b9[~(41 >
= /?RD + ,!?R
z)[k(y);
- &div(A&)(x)
R’))dydQ(R)
(x)
where we have denoted at the second line D and (Di)i=l,2 operators such that
DW(+u] = %W(+I D;V(z,
for any differentiable LEMMA
y)[u] = afV(x,
the differential
+ &W(x)w
Y)ZLI + @V(q
y)uz
functions W on A and V on R2.
0
5.2.
=-f /- RR’Dhb, Y)[&); Vol. 35, no 2-1999.
Vy)ld~*(a:,
RP*(y,
R’).
236
T. BODINEAU
AND
A. GUIONNE’I
(ii)
o(f) = 2
I
F(z,y,R,R')dv*(2,R)dv*(y,R')
= f(I + /3q-lfdv*. J
Proof. - Again, we could apply integration by parts formula to get our result. A short cut (but equivalent way) is to do a perturbation x + X+&(CIZZ) in the partition function of the limit law v*. Expending in E and writing that the second term in the expansion is null yields (i). Writing down that the third term is null gives
+ PRR’D,2K+, Y)[q412)d~*(R’, YP*(R,x) / (L@)(z) = ,b’ RR’D~~~(x, y)[k(z)]‘dv*(y, R’) - div (k)(z) 2d~*(R, > I( J From the definition g(f)
= p
J
Moreover, I
of D and of f, we get,
RR’DlD2v&,
+ J (f(Q)
x).
Y)p+$,
- AJ
fqY)ld~*(RJ)d~*(R’,
RR’D2h(?
Y)
Y)[wY)ld~*(fc Y> 2rlv*(R,4 >
by integration by parts, one sees that
?/)[@4l[~(Y/)l~~*(z,fw*(Y> w
RR’aD2V(T
div(kXRt)
I I 1 j E ;v;:;
(YP*(T
;;;:;y;;;;.,.
Rw*(Y,
R’)
(2, R).
and
/~~~~~v~(z,y)[k(y)]h*(R',y)
= -E(divj\:k))(c).
Thus, according to Lemma 5.1,
a(f) = iiJ5(div~~k))(5)div~~~)(~)~~*(~,~) +J(-(I+~~)(~"~~))(~)+"~(~"~~~))(z))~~~*~R,~~ =J(div~"k))(~~~~+~z)(div~k))(z)~~*(~,5) =
I
f(I + pz)-lfdv*. Annales de l’lnsritut
Henri
Poincar6
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et Statistiques
237
VORTEX SYSTEMS ACKNOWLEDGEMENTS
We are grateful to G. Ben Arous, who told us about the problem treated in this paper, for many constructive discussions and advices. T. B. acknowledges A. TrouvC for enlightening discussions.
REFERENCES ]I] M. A. ARCONES,E. GINE, Limit Theorems for U-processes. Ann. Probub., Vol. 21, 1993. pp. 1494-1542. [2] G. BEN AROUS, M. BRUNAUD,MCthode de Laplace : Etude variationnelle des fluctuations de diffusions de type “champ moyen”, Srochastic.s , Vol. 31-32, 1990, pp. 79-144 131 E. CAGLIOTI,P. L. LYONS,C. MARCHIORO,M. PULVIRENTI,A special class of stationary flows for two dimensional Euler equations : A statistical mechanics description, Commun. Math. Phys., Vol. 143, 1991, pp. 501-525. [4] E. CACLIOTI, P. L. LIONS, C. MARCHIORO,M. PULVIRENTI,A special class of stationary flows for two dimensional Euler equations : A statistical mechanics description. Part 2 Commun. Math. Phys., Vol. 174, 1995, pp. 229-260. [S] A. DEMBO,0. ZEITOUNI.Large deviations techniques and Applications. Jones and Bartlett. 1993. 161 J.-D. DEUSCHEL,D. W. STROOCK,Larges deviations, Academic Press, 1989. [7] C. DEUSTCH,M. LAVAUD,Equilibrium properties of a two dimensional Coulomb gas, Ph~s. Rev. A, Vol. 9, 6, 1973, pp. 2598-2616. [R] G. L. EYINK, H. SPOHN,Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence, J. Stat.Phvs.. Vol. 70, Nov. 3/4, 1993. pp. 833-886. [9] J. FR~HLICH, Classical and quantum statistical mechanics in one and two dimensions. two component Yukawa and Coulomb systems. Commun. M&h. Phys.. Vol. 47, 1976, pp. 233-268. [IO] J. FK~HLICH,D. RUELLE, Statistical mechanics of vortices in an inviscid two-dimensional fluid, Commun. M&h. Phys., Vol. 87, 1982, pp. I-36. [I I] A. GUIONNET,Fluctuations for strongly interacting random variables and Wigner’s law. to appear
in Prob.
Th. Rel. Fields,
1996.
[ 121 G. JOYCE,D. MONTGOMERY,Negative temperature states for the two dimensional guidingcentre plasma, J. Plasma Phys., Vol. 10, part 1, 1973, pp. 107-I 2 1. [ 131 M. KIESSLING,Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math., Vol. 46, 1, 1993, pp. 27-56. [ 141 M. KIESSLING, J. LEBOWITZ, The micro-canonical point vortex ensemble : beyond equivalence, Left. Math. Phys., Vol. 42, 1, 1997, pp. 43-58. [ 151 C. MARCHIORO,M. PULVIRENTI,Mathematical theory of incompressible nonviscous fluids. Springer, Applied Math. sciences, Vol. 96, 1995. [ 161 L. ONSAGER,Statistical hydrodynamics, Supplemen?o al Nuovo Cimento, Vol. 6, 1949. pp. 279-287. [l7] S. SCHOCHET,The point vortex method for periodic weak solutions of the 2-D Eulet equations, Comm. Pure Appl. Math.. Vol. 49, 9, 1996, pp. 91 l-965. (Munuscript
Vol. 35, IlO z-1999
received Junuury 12, 1998; Revised \~rsion June 25, 1998.)