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Transport Theory and Statistical Physics, 34: 523-535, 2005Copyright © Taylor & Francis Group, LLCISSN 0041-1450 print/1532-2424 onlineDOl: 10.1080/00411450600878516

Q Taylor& Francis~ Taylor&FrancisGroup

On the Vlasov Limit for Systems ofNonlinearly Coupled Oscillators

without Noise

Carlo LancellottiDepartment of Mathematics, City University of New York-CSI,

Staten Island, New York

Abstract: We discuss the application to systems of coupled nonlinear oscillators ofmethods originally developed for the Vlasov-Poisson equations of plasma kinetictheory. These methods lead to a simple but rigorous derivation of the infinite-oscillatorlimit for the well-known Kuramoto model, and also to a compact proof of globalexistence and uniqueness of solutions to the resulting Kuramoto-Sakaguchiequations. Vlasov-limit techniques can also be adapted to prove a central limittheorem for statistical ensembles of systems of oscillators. However, we emphasizethat the noiseless Kuramoto model itself is strictly deterministic, and that theKuramoto-Sakaguchi equations can be formulated without any reference to a stochasticdistribution of oscillator frequencies.

Keywords: Kuvamoto model, mean field limit, coupled oscillators, kinetic theory

1. THE KURAMOTO-SAKAGUCHI MODEL

Large systems of coupled nonlinear oscillators constitute an important familyof mathematical models and have found many applications in a broad range offields, from mathematical biology to electronics. Of particular interest are themodels that exhibit spontaneous self-synchronization, i.e., the ability todevelop synchronized oscillatory patterns-via nonlinear coupling-starting

Received 1 March 2003, Accepted 23 September 2004Address correspondence to Carlo Lancellotti, Department of Mathematics, City

University of New York-CSI, 2800 Victory Blvd., Staten Island, NY 10314, USA.E-mail: carlo@math.csi.cuny.edu

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524 c.Lancellotti

(1)

from generic, nonsynchronous initial conditions. The oldest and best-knownmodel that exhibits self-synchronization is due to Kuramoto (1975). IffJi(t) E [0, 21T] denotes the phase of the ith oscillator at a time t, andco, E ~ is its natural frequency, the Kuramoto model is governed by Nordinary differential equations (ODEs)

. K NfJi == Wi +-Lsin(8j - 8d i == 1, ... ,N,

N j=l

where K ::: 0 is the coupling strength and the frequencies are usually assumedto be randomly distributed according to a symmetric, one-humped densityfunction g(w). Since each oscillator is coupled in the same way to allothers, this represents a mean-field model for the set of oscillators, and it isnatural to ask whether it possesses a meaningful Vlasov limit. Sakaguchi(1988) proposed that as N -+ 00 the individual oscillators should bereplaced by a continuum of oscillators distributed in 8 over [0, 21T],according to a density function g( (J, W, t). For each wEIR, this functionmust satisfy the normalization condition f 61Tg( fJ, w, t) dfJ == 1 and thecontinuity equation

(2)

where the velocity v( fJ, w, t) is given by the continuous version ofEquation (1):

vee, UJ, t) = w + K 1:'T fIR sin(6' - e){?(6', w', t)g(w') dw ' de I. (3)

The combination of Equations (2) and (3) is usually called the Kuramoto­Sakaguchi equation, and has received much attention in the literature(Strogatz and Mirollo 1991; Bonilla et al. 1992; Crawford 1994; Crawfordand Davies 1999; Balmforth and Sassi 2000; Strogatz 2000; Acebron et al.2005). However, its derivation from the discrete Kuramoto model seems tohave been left at the formal level (e.g., see Appendix B in Crawford andDavies 1999), and we have found no notice in the literature of the fact thatthe derivation of Equations (2) and (3) from Equation (1) can be made fullyrigorous simply by applying the elegant Vlasov-limit theory of Neunzert(Neunzert 1978, 1984; Spohn 1991). We should mention that the stochasticversion of the Kuramoto model (obtained by adding a random noise term inEquation (1» has received a rigorous treatment (Dai Pra and den Hollander1996) as an example of a McKean - Vlasov stochastic process (McKean1966). In fact, McKean-Vlasov results also apply to the Kuramoto modelwithout noise but with an ensemble of initial conditions (see Section 3).Neunzert's method, on the other hand, leads to the Kuramoto-Sakaguchiequations within a completely deterministic framework. His approach,which was originally developed in view of the well-known Vlasov-Poisson

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Systems of Nonlinearly Coupled Oscillators 525

equations for plasmas and gravitating systems (see also Braun and Hepp 1977;Dobrushin 1979), is remarkably general, provided that the equations underconsideration satisfy certain boundedness and Lipschitz-continuity assump­tions. Whereas these assumptions lead to some unwelcome restrictions inthe many-particle case (because they are incompatible with the non-mollifiedCoulombINewton interparticle potential), they are fulfilled naturally in themany-oscillator case, where the coupling is usually modeled by well-behavedfunctions like the sine in the Kuramoto equations.

We remark that from a technical point of view the application ofthe Vlasov-limit methods to the Kuramoto model is quite straightforward.Nevertheless, from the perspective of the Kuramoto theory itself it is anelegant and rigorous way to settle a few mathematical questions, besides theN -+ 00 limit itself:

1. By showing that Neunzert's general theory applies to this problem, wealso prove global existence and uniqueness of either weak or classicalsolutions to the Kuramoto-Sakaguchi equation (depending on the smooth­ness of the initial condition). To our knowledge, existence and uniquenessresults have been established (Lavrentiev and Spigler 2000) only for thedissipative version of the Kuramoto-Sakaguchi equation, which isobtained by adding a diffusive term Da2~/arl on the right-hand side inEquation (2). The existence and uniqueness proof in Lavrentiev andSpigler (2000) relies heavily on the parabolic nature of the equationand does not appear to extend easily to the nondissipative case (D == 0).On the other hand, in our approach existence and uniqueness resultsfor the "noiseless" Kuramoto-Sakaguchi equation come essentially forfree. It may be worth recalling that this model is believed to possesssingular solutions (Balmforth and Sassi 2000); our global existenceresults imply that such solutions cannot arise in finite time from smoothinitial conditions.

2. Also, a careful discussion of the N -+ 00 limit helps clarify the role of thefrequency distribution g(w) in the Kuramoto model. Clearly, g(w) doesnot play any role in the dynamics of any individual set on N oscillatorswith assigned frequencies, since the Kuramoto model without noiseis strictly deterministic. Of course, one can consider a statisticalensemble of initial conditions (including an ensemble of independentand identically distributed [i.i.d] random frequencies Wi), but this is notrelevant to either the evolution of a given system of oscillators or the deri­vation of the Kuramoto-Sakaguchi equation, which is also a determinis­tic evolution equation for a density in phase space (as opposed to anensemble probability density). The function g(w) does arise in theN -+ 00 limit, but only a posteriori as the time-independent marginal ofthe oscillator density p(O, W, t) :::::: g(w){?((}, W, t). In fact, we will obtainthe Kuramoto-Sakaguchi equation as an equation for p (Equation (6)),in which g does not appear at all and is just "buried" in the choice of

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526 c. Lancellotti

the initial condition p( (}, co, 0). This clarification may be helpful tonewcomers to the literature, where g(w) is often introduced as a constitu­tive part of the Kuramoto model. This gives the impression that the theoryhas an intrinsically stochastic character, which it does not.

3. Having carried out the mean-field limit for the (deterministic) Kuramotomodel, we are then able to study statistical ensembles and transfer someprobabilistic results that have been obtained in the Vlasov case (law oflarge numbers, propagation of chaos, central limit theorem, etc.). In par­ticular, by adapting to the Kuramoto case the Braun-Hepp central limittheorem (Braun and Hepp 1977), we establish O(N- I/2) behavior forthe fluctuations for the Kuramoto model, a question that has beenbrought up in the physics literature (see, e.g., Strogatz 2000; Acebronet a1. 2005) and that was already answered in a more generalframework by Dai Pra and den Hollander (1996).

The article is naturally divided in two parts. In the first part we presentthe Vlasov limit for the Kuramoto model in the purely deterministic setup,and we obtain existence and uniqueness results for the Kuramoto-Sakaguchiequation. In the second part we move to the probabilistic context and wepresent results on the evolution of statistical ensembles of systems ofoscillators.

2. DETERMINISTIC VLASOV LIMIT

Because the oscillators' frequencies do not change in time, the Kuramoto­Sakaguchi equation has traditionally been written in terms of the functionf2( (), w, t), understood as a density with respect to the phase variable () E[0, 21T] for each fixed wand t. Then, the density in to has to be assigned bya separate function g(w). Here, both for the sake of symmetry and in viewof more general models where the oscillators' frequencies are time­dependent (Acebron and Spigler 1998), we will consider the function p(f),to, t), the density with respect to both variables «(), w) on the slab [0,27T] x ~ with the normalization J67T JIRp(O, to, t)dw dO~ 1. Note that, inprobabilistic language, f2 «(}, lV, t) is the conditional density for () given co,since e«(), to, t) == pCB, co, t)/g(w) and J67T {!((), to, t)dfJ == 1, which impliesthat g(w) is the (time-independent) marginal

J27r

g(w) = 0 p(B, w, t)dB.

Multiplying Equation (2) by g(w) gives the continuity equation for P

ap aat + ao (vp) == o.

(4)

(5)

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Systems of Nonlinearly Coupled Oscillators 527

We will consider a more general form of the equation for vee, w, t), Equation(3), namely,

v(O, to, t) = w + K J:7T

J~f(O' - O)p(O', w', t) dw' do' (6)

wherefis a given 21T-periodic odd function that models the coupling betweenpairs of oscillators; clearly, Equation (3) is recovered for f(x) == sin x. Fromnow on, f will be assumed to be bounded and Lipschitz continuous over [0,217]. To emphasize that we are working in the O-w slab, we write the general­ized version of the Kuramoto ODEs in terms of the pairs (O/t), Wi(t)),i == 1, ... , N:

(7a)

(7b)

The Wi(t), of course, are just going to be constant functions of t. The general­ized equations, Equations (5)-(6) and Equations (7), will still be called,respectively, the Kuramoto-Sakaguchi and Kuramoto equations.

It order to compare their solutions, we reformulate both problems in termsof probability measures on B(G), the Borel rr-algebra of subsets of G == [0,21T] x ~. For finite N this is done by associating with each configuration[eJ, ... , ON, WI, ... , WN] of the oscillators the discrete (empirical) measure

(8)

where Pi(t) == (Oi(t), Wi(t)) is the state of the ith oscillator at time t and 0 is theusual Dirac measure in G. In the continuous case we simply interpret the dis­tribution function p(O, to, t) in the Kuramoto-Sakaguchi equation as thedensity (with respect to the Lebesgue measure on G) of a probabilitymeasure J.Lt such that

ILt(M) = Lp(O, to, t)dOdw (9)

for every Borel set M E B(G). Then, the mean-field limit will be carried outwithin the set M of all probability measures on B(G), endowed with thebounded Lipschitz metric

d(J-L, v) == sup If c!Jd/L - J c!Jdvl¢EL G G

(10)

where L == {c!J: G ~ [0,1], 1q,(P) - cP(Q) I :s liP - Qlll (11·11 denotes the usualEuclidean norm in 1R2

) . As is well known, this metric generates the weakw

convergence in M, which will be denoted by the symbol ~.

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528 c. Lancellotti

Remark 1. We recall that the use of probability measures in the Vlasov limitis just a matter of normalization of the initial densities, and does not have anyreal probabilistic meaning. In fact, the limit itself does not require anystatistical assumptions on the initial conditions for finite N's, which onlyhave to well approximate the smooth initial condition for the Kuramoto­Sakaguchi equation as N ---+ 00. In this sense, assuming a statisticaldistribution of initial data for the Kuramoto model is (in the words ofH. Spohn for the Vlasov case; Spohn 1991, p. 81) "some sort of luxury,"which will be briefly discussed in the next section. Here, as anticipated, wewill construct a purely deterministic theory, in which g(w) is just themarginal of p( fJ, W, t) in Equation (4).

Just like in the Vlasov-Poisson context, both the Kuramoto andKuramoto-Sakaguchi equations will be obtained as special cases ofNeunzert's abstract continuity equation in M

(11)

Here J.t represents a M-valued function of t E [0, T], and the subscripts areused to denote the values of JL at different times; Tt,s[JL] is some appropriate"flow" on G, i.e., a family of Borel-measurable bijective mappings on Gthat satisfy (a) Ts,s = Id and (b) Tt,s 0 Ts,r :::: Tt,r; Tt,s "moves" the point P attime s to the point Tt,sP at time t. The nonlinearity of the problem iscontained in the functional dependence of Tt,sLu] on the measure JL itself,assumed to be in the space eM of all weakly continuous time-dependentmeasures (meaning that f t/JdJLt must be a continuous function of t for anyt/J E Cb(G), the continuous and bounded functions on G). The flow Tt,s[JL]will be taken to be differentiable and to be determined by J.L through avelocity field V[/L] , in the sense that pet) == Tt,sQ is a solution to the initialvalue problem

dPdt == V[J.L] (P, t), pes) == Q. (12)

In this framework! Neunzert (1978, 1984) proved the following generalresults.

Theorem 1. Let the velocity field V[JLJ(P, t) satisfy the following hypotheses:

(a). V[ JL](P, t) is a continuous function on G x [0, T] and satisfies the(uniform) Lipschitz condition

II V[J.L](P, t) - V[JL](Q, t)1! ::: LIIP - QII (13)

for all t E [0, T], P,Q E G, and J.L E CM.

INeunzert's proof applies to any domain G in [Rn.

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Systems of Nonlinearly Coupled Oscillators 529

(b). For each fixed P and t, V [J.L](P, t) as afunction on eM also is Lipschitzcontinuous, i.e.,

II V[J-L](P, t) - V[v](P, t)1I ::; Md(J-L, v). (14)

Then, if Tt,s[J-L] is the flow generated by V [J-L],1. For every fLo E M Equation (11) has a unique solution J-L E eM'2. If fLO is absolutely continuous with density Po E L1(G), then J-L is

also absolutely continuous, and its density p is a weak solution to

ap + div(pV) = 0 pep, 0) = Po(P). (15)at3. If !-La, Va E M and if fL, v E eM are the corresponding solutions to

Equation (11), then

(16)

where Land M are the constants that appear in Equations (13)and (14).

Remark 2.

1. The first hypothesis ensures that Equation (12) has a unique solution on[0, T], i.e., V[/L] generates Tt,s[/L] uniquely.

2. Weak solvability is defined here (Neunzert 1978) in terms of the space oftest functions that are continuously differentiable and compactlysupported in G x [0, T]. See Definition 1 below for the specificKuramoto-Sakaguchi case.

In order to show that, for the appropriate choices of the initial measures,both the Kuramoto and Kuramoto-Sakaguchi equations are .special cases ofEquation (11), we introduce the velocity field

V[JL] (e, to, t) = ( w + K fo!((j ~ e)JLtCd(j, dw') ). (17)

We proceed as follows: first, we prove that Neunzert' s general theorem appliesto this case. Then, we show that for this velocity field Equation (11) yields(unique) solutions to either the Kuramoto or Kuramoto-Sakaguchiequations, depending on the choice of the initial condition, and that theexpected limit behavior occurs as N ~ 00.

Lemma 1. When f is bounded and Lipschitz continuous on [0, 217'], thevelocity field V[J.L]( 8, W, t) in Equation (17) satisfies the hypotheses ofTheorem 1.

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530 c. Lancellotti

Proof (a) Since J-L is weakly continuous and! is Lipschitz continuous, V[J,L]is continuous in G x [0, T], and for each given !L,

II V[J.L] ( fh, WI, t) - V[JL] (O2 , W2, t)ll

~ IWI - Wzi + KIL[f(8' - (1) - f(d - 02)] JLt(dd,duf)!

~ IWI - Wzi +K 10 41 01 -021JLtCdd, duf)

== IWI - W21 + KLfl 0l - 02 1 :s LII(wl, Or) - (W2' ( 2) /1 (18)

where Lf is the Lipschitz constant for f and L == ,J2 max(l, KLj ) .

(b) For (8, w) and t fixed

IIV[JL](O, w, t) - V[v](O, w, 011 = KI Lf(O' - O)JLld(/,dw/)

- Lf(d - O)vt(d(/, dW'+ (19)

This expression is formally similar to the difference that appears in thedefinition of the metric d(J.L,v), Equation (10), except that f({}' - 8) (as afunction of (J', with fJ fixed) is not necessarily in the set L. Like in theVlasov-Poisson theory (Neunzert 1978), this difficulty is easily overcomeby noticing that even if f may not be in L, the modified function(2BLf )"«+ B) will always be in L since

J( (J' - 0) + B 1-----<-

2BLf - Lf IJ ((/ - ( 1) + B _ J(ff - (2) + BI < ~

2BLf 2BLf - 2B(20)

(one can always choose B :::!' Lf ::: 1). Thus, we rewrite Equation (19) in theform

2KBL.IJ Jeff - fJ) +B (dff dul) - JJ(ff - 8) +B v (df! dw') I'1 G 2BLf JLt' G 2BLf t, ,

(21)

which is indeed S Md(j.L,v), with M == 2KBLf . o

From the properties of the velocity field V[J-L] it follows that all ofNeunzert's results apply to the Kuramoto-Sakaguchi equation. In order toconsider a larger class of initial conditions we make precise what we meanby weak solutions.

Definition 1. The density function p( (), co, t) is a weak solution to theKuramoto-Sakaguchi equation if it is weakly continuous in t and for all

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Systems of Nonlinearly Coupled Oscillators 531

cP E C~(G x [O,T}) ic' with support in K x [0, T), K compact in G) theequation

is satisfied.

Now we are ready for the following.

Theorem 2. When f is bounded and Lipschitz continuous the initial valueproblem for the Kuramoto-Sakaguchi equation has a unique weak solutionin L leG).

Proof This follows directly from Neunzert's results 1 and 2 in Theorem 1,since for V[,u] given by Equation (17) and for an absolutely continuous J.Lt withdensity p

div(pV) = 880

[(W+ K Li(e' - O)p(O, w, t)dOdW)P] = 88e(Vp) (23)

Thus, in this case Equation (15) is given precisely by the Kuramoto­Sakaguchi equation. D

Remark 3. If p(8, W, 0) andf(O) are sufficiently smooth (say, C1) then the

same argument leads to existence and uniqueness of a classical solution.

The last theorem is based on the fact that the Kuramoto-Sakaguchiequation has been shown to be a special case of Neunzert's abstract continuityequation, Equation (l l), A similar identification can be carried out in the finiteN case, via the following.

Lemma 2. If Pi(t) == (Oi(t), Wi(t», i == 1, ... ,N is the solution to the(generalized) Kuramoto initial value problem, Equations (7), then

1 Np,}N) = NL 8(Pit)) (24)

j=l

is the solution to Equation (11) with T[J.L(N)} determined by V[ ,urN)} accordingto Equations (12) and (17), and with initial density

N

J4N) = ~L 8(Pj(O)) PiO) == (0;°), w;o»). (25)j=l

Proof Since f is globally Lipschitz continuous, Equations (7a) and (7b) havea unique solution for t E [0, T], and the associated measure /L/N) is weakly

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532 c. Lancellotti

continuous in t. Then, for each fixed i:::: 1, ... ,N, Pi(t) == (Olt), Wi(t)) solvesEquation (12) with Pi(O) = (0/°), w/O», since

By definition then Pi(t) = Tt,oLu(N)]Pi(O), which implies o(Pj(t)) 0 Tt,o[JL(N)] ::::o(Pj(O»), which implies JL/N) 0 Tt,o[JL(N)] = JLbN>, i.e., Equation (11). D

Having thus identified solutions to the discrete Kuramoto equation withmeasures that solve Equation (11), we are ready to conclude that theKuramoto-Sakaguchi equation is the correct limit as N ---+ 00.

Theorem 3. Let /LO(N) E M be a se~uence ofdiscrete measures of the [ormin Equation (25) such that JL~N) ---+ /La, where J.Lo E M is absolutelycontinuous. Let J..L(N) and JL be the (unique) solutions to Equation (11)associated, respectively, with the initial conditions J..LO(N) and J..Lo. ThenJL(N)~./-L as N ~ 00, for all t E [0, T].

Proof This follows immediately from Neunzert's estimate, Equation (16).0

Remark 4.

1. Of course, the density associated with JL is the unique solution to theKuramoto-Sakaguchi equation from Theorem 2.

2. Note that it is always possible to find a sequence /LtN)~ /La becausediscrete measures are dense in M.

3. PROPAGATION OF CHAOS AND FLUCTUATIONS

Even if the N -+ 00 limit for solutions to the Kuramoto model has been carriedout in terms of probability measures on G, all the development so far has beencompletely deterministic. One can also consider a genuinely probabilisticsetup by assigning a statistical superposition of N-oscillator initial conditions(in particular, of oscillator frequencies) and studying the time evolution of anensemble probability density on the N-oscillator phase space GN

. Like in theVlasov case, this is not necessary in order to justify the kinetic (Kurarnoto­Sakaguchi) equation, and does not require any fundamentally new ideas,

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Systems of Nonlinearly Coupled Oscillators 533

because the Kuramoto dynamics itself remains deterministic and the evolutionof the probability density on GN is simply induced by the flow Tt,s acting oneach pure state. Moreover, the development is very similar to the correspond­ing theory for the Vlasov equation. Thus, we only state the main results andpoint to the relevant references in the Vlasov literature for the proofs.

Let n == GN be the space of G-valued seqt:ences '1T == (Ph' .. , Pn ...),

Pi == (Oi, Wi)' endowed with the rr-algebra B generated by the finite­dimensional sets. We denote by '1TN ==- (PJ, ... ,PN) the projection of '1T onGN and we consider the evolution of symmetric product measures//.LO(N) == n f /Lo, J.Lo E M. The complete sequence ILbN)(d'1TN) , N ==- 1,2, ... , defines an infinite product measure fio(d'1T) on (0, B), and for eachfinite N the measure JLO(N) at time zero leads to the measure at time t

".CN) _ ,,(N) 0 T CN)[II .(N)]rt - rO O,t r: (27)

where Tt,/N) is the flow on GN induced by the flow Tt,s[/L(N)] on G (i.e., T~~maps each component Pi of '1TN E GN to Tt,s[J.L(N)]pj, IL/N) being theempirical measure in Equation (24).

Theorem 4. Let Cb ( G) be the continuous and bounded real functions on G.

1. Law of large numbers: For all g E Cb(G) and t > 0,

1 N Jlim - L g(Oi(t), Wi(t» == g(O,w)J.LtCdOdw)N-HXJN i=i G

(28)

with probability 1 in (n, 8, flo) (i.e., J.L~N) ~ ILt a.s. with respect to fLo).Here ILt is the solution to the Kuramoto-Sakaguchi equation for measures(Equation (11) together with Equations (12) and (17), with initialcondition Mo.

2. Propagation of chaos: For all n == 1, 2, ... and gt, ... , S» E Cb(G),

where IL~N,n\d7Tn) denotes the n-th marginal of 1L~N)(d'1TN).

3. Central limit theorem: Consider the fluctuation field

2Por initial probabilities on n that do not factorize, the N -+ 00 limit will lead to aKuramoto-Sakaguchi hierarchy analogous to the well-known Vlasov hierarchy (e.g.,see Spohn 1981).

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534

and assume that f( (J) has three bounded derivatives. Then

c. Lancellotti

(31)

where g(g, t) is a Gaussian random variable.

Proof The law of large numbers is proved along the lines of the Vlasovproof due to Braun and Hepp (1977, Theorem 3.3).3 For a somewhatstreamlined argument see also Golse (2003). Propagation of chaos follows,as discussed in Gottlieb (1998). We remark that propagation of chaos forthe Kuramoto model is a special case of a general result due to McKean(1966). The proof of the central limit theorem is identical to the originalVlasov argument by Braun and Hepp (1977); see also the discussion ofVlasov fluctuations in Chapter 7 of Spohn (1991). 0

Of course, whenever J-Lo is absolutely continuous with density poe f), W), ILlis absolutely continuous with density p( (J, to, t) and p is a solution to theKuramoto-Sakaguchi equation (weak or classical, depending on the smooth­ness of Po). As is well known, pointwise canvergence of the characteristicfunctionals in Equation (31) means that gN ~ g. By adapting Braun andHepp's proof to our problem it is easy to see that the Gaussian randomvariable g(g, t) satisfies dgt == L(t) gt dt, where L(t) is the linearizedKuramoto-Sakaguchi operator"

(L(t)cjJ)(£J, w) = _ a~:)

- K :£J [PC £J, w, t) Ie.r (£J - O')cjJ(8', w')dO' dwl (32)

Here v is the velocity determined by p via Equation (6) and g(g, 0) is theGaussian random variable associated with the central limit theorem att == 0; see Spohn (1991), and Braun and Hepp (1977) for further details.

ACKNOWLEDGMENTS

This work was supported by NSF grant DMS-0318532 and by PSC-CUNYgrant 60074-33-34. I would like to thank Michael Kiessling for helpful discus­sions and Jack Doming for his comments.

3Actually, in this theorem Braun and Hepp also proved propagation of chaos, butonly for "purely atomic initial data": see Gottlieb (1998).

4For absolutely continuous J..to. The extension to any J..to E M is straightforward.

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