Skrypnik, W. I.

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DIAS-STP-87-56INTEGRABLE SOLUTIONS OF THE HIERARCHYOF THE BBGKY-TYPE FOR BROWNIAN PARTICLESIN THE MEAN-FIELD LIMITW.I. SkrypnikDublin Institute for Advanced Studies10 Burlington Road, Dublin 4, IrelandIntroductionLet us consider the gradient hierarchy of theBBGKY-type for correlation functionsdm d .X= (x1,...,x) € ll , x € 1P , of a nonequ.l.braumsystem of diffusing particles, interacting via apair, integrable smooth potential (x). Let us assumealso that the following asymptotic relation holdsPt(2Cm) = m P(Xm)i € 0where the functions P(Xm) have a limit when E 0. Then10n leave of absence from the Institute of mathematics ofthe Ukrainian Academy of Sciences, 252004, Kiev 4, RepinStreet 3, USSR1the hierarchy Is written as follows(1)=‘ P(Xm) ++(x—x1) P(Xm+i) dx1 }where ,9 is the inverse temperature and EU(X) is thepotential energyU(X) =1 i < jm= { + h 2 }3 j ax.ax. 8(x.)3 3 3In the mean—field limit ( e’o ) the consideredhierarchy is transformed into a hierarchy of the Vlasovtype(2)t P(Xm) { p‘P(Xm) ++ jdj) (Xj_Xm+i) P(Xm+i) dXm+l }In this paper we propose a justification of themean—field limit in a class of integrable correlation2functions satisfying eq.(1) in a weak sense. We introduce asemigroup in a Banach space of sequences ofsymmetric itegrable functions and prove that the sequensePt(X)=)(X), pE Esatisfies eq.(1) in a weak sense.We show that, if theE 1sequence exp ( U(Xm) m1, belongs to thenP(Xm) converges weakly toP(Xm) = ( it p° )(X)i m1,where rc as defined as a mapif (t) = / exp{ (t) } < 1We also prove that the sequence p is a weak solution ofeq (2)1.The norm an the Banach space is defined as followsmax I’PflILl(dnniThe mean-field limit in a mechanical and a specialrandom mechanical systems was studied earlier, respectivelyan [2,3)( see also [4,5]) The mean—field limit for eq (1)an a class of bounded correlation functions as investigatedin [6]. It is found in [7] that there is a space in which asolution of eq.(2) exists and is unique.3I Main theoremLet Pt(X;Y) be a fundamental solution of then-particle Smoluchowski equation ( eq.(1) without thetintegral term an the r.s. ). The operators P 1 to,(pt p)(X ) = f pt() ;Y )w(Y )dYn 2dn € n n n ndefines a contraction strongly continuous semigroup inL1(2). Let be a diogonal operator in given by(Pw)(X) = (P)(X)It is evident that is a contraction stronglycontinuous semigroup an ll Let us define an operator fdwhich is bounded in II)fdW )(X)=X,X)dXThen m(1 1) = exp{ e’fd } exp{is a strongly continuous semigroup in L. Its structurecoincide with a structure of an evolution operator of theBBGKY-hierarchy [8).Lemma 1.1The sequence = icpE, pE€ L ,is a weak solution ofeq.(1), that is4(1.2)‘dm m>h(X) dX ==dmP’P(Xm) 2 h(X)-(—h(X)) L E P(2m) U(X) ++ d() (x._Xm+i) p (X1) dx1 } d XLemma 1.2Let P(dx) be the Wiener measure on = andmx (dXm) = 1P (dx) . Let* be an operation ofm j=1 jmultiplication, defined on sequencesX E dn tp(ø,ø) = 1by(*)(XX){s}EE(1,..n){n\s) = (1,. ,.,n)\{s}Let (ip)’ be the inverse element to p with respect to *andD(Xm;m)P )(X;X) = P(Xm,Xn;imiXn)5I I=E( (exp{ } exp{ — U } )(X ;Xt finm mdX P ‘(dXdX)X,X m nn+m m nIf the potential is a positive-definite function from3dC (JR ) and(x) Vc(x)l IA(x)j ,o A =then the following uniform in € bound holds(1.4)I —I — —Iess sup exp{- U(X ,X ;X ,X ) } II (X ,X jX ,X )all(x,x’,x,x’) m m n € m m n nm+nn!( .,‘ 2 exp { i(t)There exists a measurable function U(XX) such that, iffl (X ,X X ,X ) =m m n nwhere the n—th component of the sequence exp{ — U }equals exp{ - tJ(X;X) ,then the cluster expansion forc holdsE(1.3)t E:(rr p )(X =t m L T dnno JRxd— —Ix fl (X ,X IX ,X ) p(X (t),X(t1))€ m mn n mLemma 1.3(t) = ( 4>° x(t))2 + (O) +4 t (-A)(O)and the functions flc(XmiXmlXnXn) converge a.e. to functions6a)aicnQ)4.)3-’•rl00U)xII4-’VCt,>1(,)rir-1•0U)m+-(1)1.)4.)r1X0--IIIVu—a)E--•oE-1a)-I14.4+-CZ2a)0r-t-.PE•r-110-rg-1CtSU)W0—c-I‘4’E“E-cO(-••rlVEE‘(1E-1‘i-i(I)iii+(1—n-—EV4.)0Q03-’Xa)a)0::)--L*P---04-’4j0H4-’rj-‘4-4-•---->1-r-i144E‘4.-I0U)IU)-l-4‘.--r.j0U)(ISa)4-’4.)4—i-Cl)(D(O(IS(IS4-’0(1)—0$(0(0Cl)-‘r-IQ)-inQ.U)E-4-’0Cl)‘Cs‘Ciri(IS-a)00‘Cl0)Eriu0)0a)(IiII4-’tD—(I)4-ri-0(1)-r-j—10(1)4-’(‘4(0(00(1)‘-40-if)-4)4‘-10)(1)o0(IS0-10S14—i4J‘4-i2 Cluster expansaonLet us prove the equality (1.3). We start fromresumming in (1.1)(2.1)= F dX n! (l)fl_SD Pt Dn!‘dn n s!(n—s)! X X{\}no U? s—awhere ( Dp ) (X) = ( X,X )Let * be the operation of multiplication defined onsequences of kernels { K(X;Y) K(ø,ø) = 1 ( seeLemma 1.2 ) . e0 = ( 1,0,0,...) is the unit with therespect to * Define(K)’ = (l)’ (Ke)*. .*(K_e) , (K)’ K = e0non<K> = —i--- dX F K(X ;Y ) (Y ) dYn!j dn n nno II? U?Then (1.2) is rewritten in the following fashion(2.2)( 7 PE)()= ‘dme1*D(Xm;Ym)P E’D e dY11? mwhere ( e÷)(Xn;Yn) = (±l)n6(X_Y),(l)(P)(X)EnPE(X)As a result we derive the cluster expansion(2.3)T p )(X)= jdm(pt)1*D(Xm;Ym)P >E lDYmpE: dYm(2.3) follows from the law of conservation of probabilityfor the SmoJ.uchowski equation and the equality8in the following way<(P)_l*D(Xm;ym)PE>W = <(Pe)**(pt_e)*Dpt>no= <(e1-e) * * (e1- ) *D(X;y) Pp_fl;O= < (e+l) 1*D Y) > = < e *D Y) >It is well known that, if a function s(X) satisfiesthe Smoluchowski equation, the functionexp {-3 U(X) }satisfies the heat equation with the potential V(X)V(X)=U(X) - e ( U(X) )2 }Applying the Feynman-Kac formula we derive the followingrepresentation for the kernel(2.4) P(Xn;Yn) = exp { - ( U(X) - U(Y) ) }x—1-tp - - —‘x J’ PX(dXn) exp { I V(X(t))dt } ö(Xn(tP ) - Y)(2.2) and (2.4) yield (1.3) with- -- 1U(X;X) = E{ U(X)_U(X(t )) } + S f V(X(r))dt9C)4-’U)•rlr1-rl4ISCl)0ocXC)a)‘ci0x*.1U)4-’4-)II*r0Cl)-‘ci*.4‘ci—IJ-a)SWW£q-(CI-4-’0S*(14).r-j*‘ciU)oClirj4)X*4-’a)‘0‘CI.0•raicli-2(E+SU)H-CliU).+-F-I-Cl44-,4—’a•,—.._c)**(-G:Dx*x*E-i((DI-‘t-—I—criC)Vrla—-—,oU)(14c::4)1---1(CI4JC__0‘—I-,-a)o‘-I•*--II4-’U)G)I,oCD‘ci‘-I:>I.’4)9-iC%1‘-I-o:44>1‘I’I—-1-—‘-I-9.o04-’IIU)_o4-’‘-i—--—-—4ci)4)U)U)C(14II‘rj—a)(1)x4-’—‘TI‘ciU)SC’)•-(1)—*4-)IIoU)4-’r4JS(1.1cI)rjU)(i(CIV•1f-iU)oU)U)flr-1Ia)>a)r-jU)U)4)0o(Y)II£..C1(C’)9-i—4-’= j—*tp_1 *(xjx ) = f f ( (Vq)(x(t)), dx (r)The sequence fl (X IX ) m1, n1 satisfies the standardrn nrelation [9](3 3) = exp {-P )ljx KE(xjiX{s}) E m(j)IX{s}{n\s}){s}E(1, n)Em = exp { —e Ut(Xm) } fl(ølX) = 0 ,m(j)=(1,. .m)\jK(xX) = 1(exp { -(xx) }- 1)(3.3) has the limit for € -* 0(3.4) 11omn =tjl fom(j)IX{s}X{n\s}){s)E(1,..n)Proposition 1.3 C The main boundIf the conditions of Lemma 1.3 are satisfied then thefollowing uniform in all variabJ.es,except t, bound(3 5){ SdX)[ fP(dXt) II (X IX >1 ] } n!(/ exp{L.(t)})m11ProofTo derive (3.5) we have to symmetrize (3.3), takinginto consideration that the potential+ x(t))is stable ( see also C6,9,1OJ). Now we prove (3.5) byinduction for = 0Let us integrate (3.4) by P(dX) and apply theSchwartz inequalityr — -P(dX, j1E{s}d1*x r P(dX ) II CX . ,X X ){n\s} o m(j) (s} {n\s}r —J P(dX 1 2lE{s)xl P(dX )E P(dX )fl (X . ,X x )IiJ<> (s} L <n\s> {n\s) o m(j) {s) {n\s} jwhere <s> is the number of elements in the sequence {s).Let us square the obtained inequality, split the sum over{s) into two sums (over {s1} and {s2}) and integrate byP(dX) the resulting expression, utilizing the Schwartzinequality Assume that (3 5) holds for all <s> m+n-1 Inthis casej’P(dX) [ S miXnH P(dX) ] (n! _--- K0)2 xx ( exp { v(t) ))2(m+nl)12K = ess sup P(dx) [5,0 alI(x,x) QS 11d dMaking use of the Schwartz and generalized Helderinequalities we obtain the following bound for K50K ess sup I P(dX) r P(dx*)It(xjxl)I2]SICall(x,x) L Sd dess sup I P(dx) P(dx*Ht(xlxl) 2s ]all(x,x) 1=1dIt as well known that—1*2s 2s’ r 2P(dx ((V) (x(t) ,dx (t)) = (, ); I () (x(t) )dt]-IQWith the help of the inequalities1(a5+b) (a+b) , (a+b)5 25(a+b) , 3rnn &we haveK50 (s!) {72 ( I1 + 21V10t’ )} (s!)(°x0t )5where= ess supcP(x) ,x(t) = 72(1+j5t),=(x( )°)XEXNow it is not difficult to prove the proposition for > o,using the symmetrized (3 3)[6) and the above arguements Asa resultx(t) = exp { p0(1+4j t1)} x(t)13Now we return to the Theorem 1.1. At first we considerthe simplest case= exp { - 2 U )Since the algebraic structure of the functions fl(XmXn)are known , they converge to the functions flo(XmXn) a.e.From the Lebesque theorem at follows thatS (7c PE)(Xm) h(Xm) dXm (rr P°Hx) h(X) dxdn o dnIt means that the r.s. of (1.2) converges to the r.s. of(1.5) . Since the r.s. of (1.5) is a continuous function oft ( the series in (1.3) converges uniformely on a finitetime interval ) the l.s. of (1.2) converges to the leftside of (1.5) . It is clear that in a general casepC=exp{_EU} (p0pE)From Lemmas 1.1 , 1.2 it follows that the remaining termTcexp{ - 3 U } p converges to zero in if(t) =/ exp { v(t) } < 1The theorem Is proved4. Proof of Lemma 1.1Let us put= exp{ J’ d } E , 4( X) = 0 , n >NThen14PN(Xm) = (itt p)(X) (exp{ fd }P HX) ==dmdX (P )(X,X)It can be shown that satisfies (1) in a classicalsense. Now let us differentiate it ,taking intoconsideration that ( IJEN)(Xn) satisfies theSmoJ.uchowski equation= { ++ Edfl8)cJm+ltmnnThe derivatives in the inner variables(X) disappearedsincer )(x)dx = 0 , f , €: L1(IR)and E L’fl C2[iO]. As a result the last term is equalN-rn-idj (XjXm+i) [ dn J(XmPXn) dX] dXm+ir a= J (— ) (X.-X) P. N(Xm+1) dXm+i m N.jHence the sequence { P,N(2m) m<N satisfies (1) in aweak sense. Let N *x .Then converges to in thetopology of 114 . By the Lebesque theorem the r.s. of (1.2)for t,N converges to the r.s. of (1.2), Since is astrongly continuous semigroup the same is true for the15corresponding left sides .The proof is completeREMARK . Our theorem does not allow the canonicalcorrelation functions to converge in the mean—field limitsince their limit satisfies the compatibility conditiondtnn = pt(X1)and E ll only if 1But there is a possibility to improve our bounds in such away that (t) goes to 0 when either t or goes to 0ACKNOWLEDGEMENT .The author expresses his sincere grati—tude to professor John Lewis for invitation, kindness,andhospitality He thanks also for help Margaret Matthews,Michel Vandyck and Guido RaggioREFERENCES1.Streltsova E.A. Ukrainian Math.Journ. ,1959,11, 1, 83—92.2.BraunW.., Hepp K. CommMath.Phys.,1977,56, 2, 101-113.3 Skorohod A V Stochastic equations for complicatedsystems , Moscow, Nauka, 1983 (in Russian)4.Dobrushin R.L., Sinai Ya.G., Suhov U.M. in Itogi naukii tehniki, series “Modern problems of mathematics”, v.2,233—284; Moscow, VINITI,1985 (in Russian)5 Spohn H , Rev Mod Phys ,1980, 53, 3, 569—6156.Skrypnik W.I. ,Teor.Mat.Fiz. ,1986, 69, .1, 128—141;1988(toappear)7.Chueshov I.D., Teor.Mat.Fiz.,1986, 67, 2, 304—308.8 .Petrina D .Ya, , Gerasimenko V.1., Malyshev P .V. Mathematical foundations of Classical Statistical Mechanics , Kiev,Naukova Dumka, 1985 (in Russian )169.Ruelle D. ,StatIstical Mechanics.Rigorous results.Benjamin1970.10.Dynki.n E.B. Markov processes, Moscow,’M, 1963 (inRussian)17

Integrable Solutions of the Hierarchy of the BBGKY-type for Brownian Particles in the Mean-field Limit

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Integrable Solutions of the Hierarchy of the BBGKY-type for Brownian Particles in the Mean-field Limit

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