The von Neumann Hierarchy for Correlation Operators of Quantum Many-Particle Systems

Transcription

1 The von Neumann Herarchy for Correlaton Operators of Quantum Many-Partcle Systems V I Gerasmenko V O Shtyk Insttute of Mathematcs of NAS of Ukrane arxv: v2 [math-ph] 5 Jun 2008 Bogolyubov Insttute for Theoretcal Physcs of NAS of Ukrane E-mal: gerasym@math.kev.ua, vshtyk@btp.kev.ua The Cauchy problem for the von Neumann herarchy of nonlnear equatons s nvestgated. One descrbes the evoluton of all possble states of quantum many-partcle systems by the correlaton operators. A soluton of such nonlnear equatons s constructed n the form of an expanson over partcle clusters whose evoluton s descrbed by the correspondng order cumulant sem-nvarant of evoluton operators for the von Neumann equatons. For the ntal data from the space of sequences of trace class operators the exstence of a strong and a weak soluton of the Cauchy problem s proved. We dscuss the relatonshps of ths soluton both wth the s- partcle statstcal operators, whch are solutons of the BBGKY herarchy, and wth the s-partcle correlaton operators of quantum systems. Keywords: von Neumann herarchy; BBGKY herarchy; quantum knetc equatons; cumulant sem-nvarant; cluster expanson; correlaton operator; statstcal operator densty matrx; quantum many-partcle system.

2 The von Neumann Herarchy for Correlaton Operators 2 Contents Introducton 3 2 Evoluton equatons of quantum many-partcle systems 4 2. Quantum systems of partcles The von Neumann equaton Dervaton of von Neumann herarchy for correlaton operators Cluster expansons of evoluton operator of von Neumann equaton 8 3. Cluster expansons Propertes of cumulants Intal-value problem for the von Neumann herarchy 0 4. The formula of a soluton Chaos property Propertes of a group of nonlnear operators The unqueness and exstence theorem BBGKY herarchy 8 5. Nonequlbrum grand canoncal ensemble On a soluton of the BBGKY herarchy Correlaton operators of nfnte-partcle systems References 24

3 The von Neumann Herarchy for Correlaton Operators 3 Introducton In the paper we consder the von Neumann herarchy for correlaton operators that descrbes the evoluton of quantum correlatons of many-partcle systems. The necessty to descrbe the evoluton of correlatons arses n many problems of modern statstcal mechancs. Among them we refer to such fundamental problems that are challengng for mathematcs, n partcular, the rgorous dervaton of quantum knetc equatons [2, 4 8, 5, 9, 2, 27, 36], for example, the knetc equatons descrbng Bose gases n the condensate phase [, 6 8, 35], and the descrpton of nonequlbrum quantum correlatons n ultracold Ferm and Bose gases [28]and references theren. In the paper we ntroduce the herarchy of equatons for correlaton operators the von Neumann herarchy that descrbes the quantum correlatons from the mcroscopc pont of vew and shows how such dynamcs s orgnated n the dynamcs of an nfnte-partcle system the BBGKY herarchy [0, 26, 33] and nonequlbrum fluctuatons of macroscopc characterstcs of such systems. The am of the work s to formulate the evoluton equatons descrbng correlatons n quantum many-partcle systems wth the general type of an nteracton potental and construct a soluton of the correspondng Cauchy problem, then usng the constructed soluton to establsh ts relatonshps wth the solutons of herarches of the evoluton equatons of nfntely many quantum partcles. We outlne the structure of the paper and the man results. In secton 2 we ntroduce prelmnary facts about the dynamcs of quantum systems of non-fxed number of partcles and deduce the von Neumann herarchy for correlaton operators whch gves the alternatve descrpton of the evoluton of states of the many-partcle systems. In secton 3 we defne the cumulants sem-nvarants of evoluton operators of the von Neumann equaton and nvestgate some of ther typcal propertes. It turned out that the concept of cumulants of evoluton operators s a bass of the expansons for the solutons of varous evoluton equatons of quantum systems of partcles, n partcular, the von Neumann herarchy for correlaton operators and the BBGKY herarchy for nfntepartcle systems. In secton 4 the soluton of the ntal-value problem for the von Neumann herarchy s constructed and proved that the soluton generates a group of nonlnear operators of the class C 0 n the space of trace class operators. In ths space we state an exstence and unqueness theorem of a strong and a weak solutons for such a Cauchy problem. In the last secton 5 we dscuss the relatonshps of a soluton of the von Neumann herarchy for correlaton operators both wth the s-partcle statstcal operators, whch are solutons of the BBGKY herarchy and wth the s-partcle correlaton operators of quantum systems. We also state the general structure of the BBGKY herarchy generator of nfnte-partcle quantum systems.

4 The von Neumann Herarchy for Correlaton Operators 4 2 Evoluton equatons of quantum many-partcle systems 2. Quantum systems of partcles We consder a quantum system of a non-fxed.e. arbtrary but fnte number of dentcal spnless partcles wth unt mass m = n the space R ν, ν n the termnology of statstcal mechancs t s known as a nonequlbrum grand canoncal ensemble []. The Hamltonan of such a system H = H n s a self-adjont operator wth doman DH = {ψ = ψ n F H ψ n DH n H n, H n ψ n 2 < } F H, n where F H = H n s the Fock space over the Hlbert space H H 0 = C. As- sume H = L 2 R ν coordnate representaton then an element ψ F H = L 2 R νn s a sequence of functons ψ = ψ 0, ψ q,..., ψ n q,...,q n,... such that ψ 2 = ψ n= dq...dq n ψ n q,...,q n 2 < +. On the subspace of nfntely dfferentable functons wth compact supports ψ n L 2 0 Rνn L 2 R νn n-partcle Hamltonan H n acts accordng to the formula H 0 = 0 H n ψ n = 2 2 n n n q ψ n + Φ k q,...,q k ψ n, = k= <...< k = where Φ k s a k-body nteracton potental satsfyng Kato condtons [30] and h = 2π s a Planck constant. An arbtrary observable of the many-partcle system A = A 0, A,..., A n,... s a sequence of self-adjont operators A n defned on the Fock space F H. We wll consder the observables of the system as the elements of the space LF H of sequences of bounded operators wth an operator norm [3,33]. The contnuous lnear postve functonal on the space of observables s defned by the formula A = e a D 0 n! Tr,...,nA n D n, 2 where Tr,...,n s the partal trace over,...,n partcles, e a D 0 = Tr n!,...,nd n s a normalzng factor grand canoncal partton functon, for whch the notaton from secton 5 s used. The sequence D = I, D,...,D n,... s an nfnte sequence of selfadjont postve densty operators D n I s a unt operator defned on the Fock space F H. Ths sequence descrbes the state of a quantum system of non-fxed number of partcles. The densty operators D n, n also called statstcal operators whose kernels are known as densty matrces [9], defned on the n-partcle Hlbert space H n = H n = L 2 R νn, we wll denote by D n,...,n. For a system of dentcal partcles descrbed by Maxwell- Boltzmann statstcs, one has D n,...,n = D n,..., n f {,..., n } {,..., n}. We consder the states of a system that belong to the space L F H = L H n of sequences f = I, f,...,f n,... of trace class operators f n = f n,...,n L H n,

5 The von Neumann Herarchy for Correlaton Operators 5 satsfyng the above-mentoned symmetry condton, equpped wth the trace norm f L F H = f n L H n = Tr,...,n f n,...,n. We wll denote by L 0 the everywhere dense set n L F H of fnte sequences of degenerate operators [30] wth nfntely dfferentable kernels wth compact supports. Note that the space L F H contans sequences of operators more general than those determnng the states of systems of partcles. For the D L F H and A LF H functonal 2 exsts. One s nterpreted as an average value expectaton value of the observable A n the state wth densty operator D nonequlbrum grand canoncal ensemble for the Maxwell-Boltzmann statstcs. We remark that n the case of a system of a fxed number N of partcles nonequlbrum canoncal ensemble the observables and states are the one-component sequences, respectvely, A N = 0,...,0, A N, 0,..., D N = 0,..., 0, D N, 0,.... Therefore, the formula for an average value reduces to A N = Tr,...,N D N Tr,...,N A N D N. 2.2 The von Neumann equaton The evoluton of all possble states Dt = I, D t,,..., D n t,,..., n,... s descrbed by the ntal-value problem for a sequence of the von Neumann equatons the quantum Louvlle equatons [0,3,33] d Dt = NDt, dt 3 Dt t=0 = D0, 4 where for f L 0F H DN L F H the von Neumann operator s defned by Nf n = [ ] fn, H n := fn H n H n f n. 5 In the space of sequences of trace-class operators L F H for an abstract ntal-value problem 3 4 the followng theorem s true. Theorem [24,33]. The soluton of ntal-value problem 3 4 s determned by the formula where U t = U n t and Dt = U td0u t, 6 U n t = e thn, U n t = e thn, 7 U 0 t = I s a unt operator. For D0 L 0 F H L F H t s a strong classcal soluton and for arbtrary D0 L F H t s a weak generalzed soluton.

6 The von Neumann Herarchy for Correlaton Operators 6 Note that the nature of notatons 7 used for untary groups e ± thn s related to the correspondence prncple between quantum and classcal systems for the later the evoluton operator for the Louvlle equaton for the densty of the dstrbuton functon s defned n analogous terms. It s a consequence of the exstence of two approaches to the descrpton of the evoluton of systems based on the descrpton of the evoluton n framework of observables or states. The evoluton operator generated by soluton 6 for f L F H we wll denote by U tfu t := G tf. 8 The followng propertes of the group G t follow from the propertes of groups 7. In the space L F H the mappng 8 : t G tf s an sometrc strongly contnuous group whch preserves postvty and self-adjontness of operators. For f L 0 F H D N n the sense of the norm convergence L F H there exsts a lmt [33] by whch the nfntesmal generator: N = N n of the group of evoluton operator 8 s determned lm t t 0 G tf f = Hf fh := Nf, 9 where H = H n s the Hamltonan and the operator: Hf fh s defned n the doman DH F H. It should be emphaszed that the densty operator belongng to the space L F H descrbes only fntely many-partcle systems,.e. systems for whch the average number of partcles n the system s fnte. Indeed, for the observable of the number of partcles N = 0, I, 2I,..., ni,... functonal 2 has a form N t = e a D0 0 and for any sequence Dt L F H we get n! Tr,...,n+D n+ t,,...,n +, 0 N t D0 L F H <. 2.3 Dervaton of von Neumann herarchy for correlaton operators Let us represent the state Dt of a quantum system n the form of cluster expansons [0,34] over the new operators gt = 0, g t,,...,g n t,,...,n,... D t, = g t,, D 2 t,, 2 = g 2 t,, 2 + g t, g t, 2, D n t, Y = P: Y = S X X P g X t, X, n, where the followng notatons are used: Y,..., n, Y = n s a number of partcles of the set Y, s a sum over all possble parttons P of the set Y nto P nonempty P mutually dsjont subsets X.

7 The von Neumann Herarchy for Correlaton Operators 7 It s evdently that, n terms of the sequences of operators gt, the state of the system s descrbed n an equvalent way. The operators g n t, n are nterpreted as correlaton operators of a system of partcles. The evoluton of correlaton operators s descrbed by the ntal-value problem for the von Neumann herarchy for correlaton operators d dt g nt, Y = N n Y g n t, Y + P: Y = S X, P > N nt X,...,X P X P g X t, X, 2 g n t, Y t=0 = g n 0, Y, n. 3 The von Neumann operator N n Y = N n for the system of partcles wth Hamltonan s defned by formula 5, N nt X,...,X P = Z X, Z... Z P X P, Z P N P P Z r r= nt where s a sum over all subsets Z j X j and for k =,..., n Z j X j Z,...,Z P, 4 N k nt, 2,..., k = [, Φ k, 2,..., k ]. 5 The smplest examples of von Neumann herarchy 2 are gven by d dt g t, = N g t,, d dt g 2t,, 2 = N 2, 2g 2 t,, 2 N 2 nt, 2g t, g t, 2, d dt g 3t,, 2, 3 = N 3, 2, 3g 3 t,, 2, N 2 nt + N 2 nt + N 2 nt, 2 N 2 nt, 2 N 2 nt, 3 N 2 nt N 3 nt, 2, 3g t, g t, 2g t, 3., 3 N 3 nt, 2, 3 g t, g 2 t, 2, 3 + 2, 3 N 3 nt, 2, 3 g t, 2g 2 t,, 3 + 2, 3 N 3 nt, 2, 3 g t, 3g 2 t,, 2 + We note that, n the case of a two-body nteracton potental k = 2, the von Neumann herarchy 2 s smplfed. For example, the expresson for the operator g 3 t does not contan terms wth the operator N 3 nt. In ths case the nonlnear terms n herarchy 2 have the form 2 N nt, 2 g X t, X g X2 t, X X 2 P: Y =X S X2 X For classcal systems herarchy 2 wth nonlnear terms 6 s an equvalent form of the correspondng nonlnear Louvlle herarchy [37] formulated by Green [29].

8 The von Neumann Herarchy for Correlaton Operators 8 The von Neumann herarchy 2 can be formally derved from the sequence of lnear von Neumann equatons 3 provded that the state of a system s descrbed n terms of correlaton operators defned by cluster expansons. We remark that, n the case of a system of partcles satsfyng Ferm or Bose statstcs, cluster expansons and herarchy 2 have another structure. The analyss of these cases wll be gven n a separate paper. 3 Cluster expansons of evoluton operator of von Neumann equaton 3. Cluster expansons Let us expand the group G t of evoluton operator 8 as follows cluster expansons: G n t, Y = P:Y = S X X P A X t, X, n = Y 0, 7 where s the sum over all possble parttons of the set Y,..., n nto P nonempty P mutually dsjont subsets X Y. The operators A n t, Y we refer to as the nth-order cumulant sem-nvarant of evoluton operators 8. The smplest examples of cluster expansons 7 have the form G t, = A t,, G 2 t,, 2 = A 2 t,, 2 + A t, A t, 2, G 3 t,, 2, 3 = A 3 t,, 2, 3 + A 2 t,, 2A t, 3 + A 2 t,, 3 A t, 2 +A 2 t, 2, 3 A t, + A t, A t, 2A t, 3. Solvng prevous relatons, one obtans A t, = G t,, A 2 t,, 2 = G 2 t,, 2 G t, G t, 2, A 3 t,, 2, 3 = G 3 t,, 2, 3 G t, 3G 2 t,, 2 G t, 2G 2 t,, 3 G t, G 2 t, 2, 3 + 2!G t, G t, 2G t, 3. In general case the followng lemma s true. Lemma. A soluton of recurrence relatons 7 s determned by the expanson A n t, Y = P P! G X t, X, 8 X X P P:Y = S n = Y, where s the sum over all possble parttons of the set Y nto P nonempty mutually P dsjont subsets X Y.

9 The von Neumann Herarchy for Correlaton Operators 9 Proof. Let us consder the lnear space of sequences f = f 0, f,...,f n,..., n,... of operators f n L H n f 0 s an operator that multples a functon by an arbtrary number. We ntroduce n ths lnear space the tensor -product [20,34] f h Y Y = Z Yf Z Z h Y \Z Y \Z, 9 where h = 0, h,..., h n,..., n,... s a sequence of operators h n L H n and s the sum over all subsets Z of the set Y,...,n. Accordng to defnton 9 for the sequence At = 0, A t,, A 2 t,, 2,......,A n t,,...,n,... the followng equalty s true P:Y = S X X P A X t, X = Exp At Y, n = Y, n where Exp s defned as the -exponental mappng,.e. Exp f = + n= Z Y n! } f {{ f, 20 } f f 0, f,...,f n,... and I, 0, 0,... s the unt sequence. As a result, we can represent recurrence relatons 7 n the form + G t = Exp At, where G t = 0, G t,,...,g n t,,...,n,... and the elements of the sequence G t + G t = I, G t,,...,g n t,,...,n,... are the evoluton operators 8, G tf n Y = U tfu t n Y = U n t, Y f n Y Un t, Y for Y =,...,n. Smlarly, defnng the mappng Ln on the sequences h 0, h,...,h n,... as the mappng nverse to Exp,.e. one obtans P:Y = S X P P! n n Ln + h = n } h {{ h}, 2 n= n X P Hence, relaton 8 can be rewrtten as G X t, X = Ln + G t Y, n At = Ln + G t, n = Y. and, therefore, expresson 8 s a soluton of recurrence relatons 7. For systems of classcal partcles cumulants 8 were ntroduced n [22].

10 The von Neumann Herarchy for Correlaton Operators Propertes of cumulants We wll now deal wth the propertes of cumulants 8. As was proved n Lemma the cumulants A n t, n of evoluton operators 8 of the von Neumann equatons are solutons of recurrence relatons 7,.e. cluster expansons of the group of evoluton operators 8, smlar to. For the quantum system of non-nteractng partcles for n 2 we have: A n t = 0. Indeed, for a non-nteractng Maxwell-Boltzmann gas we have: G n t,,...,n = n = G t,, then A n t, Y = P: Y = S X P P! where the followng equalty s used: P: Y = S X P P! = X X P j = = G t, j = n k sn, kk! k= n G t, = 0. = n k sn, kk! = δ n,, 22 k= sn, k s the Strlng numbers of the second knd and δ n, s a Kroneker symbol. In the general case a generator of the nth-order cumulant, n 2, s an operator N n nt defned by an n-body nteracton potental 5. Accordng to equalty 22 for the nth-order cumulant, n 2, n the sense of pont-by-pont convergence of the space L H n we have lm t 0 t A nt, Y g n Y = P P! N Zk Z k g n Y = Z k Z k P = N n nt g Y n Y, 23 P: Y = S k where the operator N n nt s gven by 5. For n = the generator of the frst-order cumulant s gven by lm A t, Y I g n Y = N n Y g n Y, t 0 t where Y =,...,n and N n Y s defned by formula 9. 4 Intal-value problem for the von Neumann herarchy 4. The formula of a soluton We consder two approaches to the constructon of a soluton of the von Neumann herarchy 2. Snce herarchy 2 has the structure of recurrence equatons we deduce

11 The von Neumann Herarchy for Correlaton Operators that the soluton can be constructed by successve ntegraton of the nhomogeneous von Neumann equatons. Indeed, for solutons of the frst two equatons we have g t, = G t, g 0,, g 2 t,, 2 = G 2 t,, 2g 2 0,, t 0 dt G 2 t + t,, 2 N 2 nt, 2 G t, G t, 2g 0, g 0, Then let us consder the second term on the rght hand sde of 24 t 0 dt G 2 t + t,, 2 N 2 nt, 2 G t, G t, 2 = = G 2 t,, 2 t 0 d dt G 2 t,, 2G t, G t, 2 = dt = G 2 t,, 2 G t, G t, The operator G 2 t,, 2 G t, G t, 2 := A 2 t,, 2 n 25 s the second-order cumulant of evoluton operators 8. Formula 25 s an analog of the Duhamel formula, whch holds rgorously, for example, for bounded nteracton potental [3]. Makng use of the transformatons smlar to 25, for n > 2 a soluton of equatons 2, constructed by the teratons s presented by expressons 28. The formula for a soluton of the von Neumann herarchy 2 3 can also be formally derved from the soluton Dt = I, D t,,..., D n t,,..., n,... of von Neumann equatons 6 Theorem on the bases of cluster expansons. Indeed, a soluton of recurrence equatons s defned by the expressons g n t, Y = P P! D X t, X, n, 26 X X P P: Y = S where s a sum over all possble parttonspof the set Y,...,n nto P nonempty P mutually dsjont subsets X. If we substtute soluton 6 n expressons 26 and takng nto account cluster expansons for t = 0, we derve g n t, Y = P P! G X t, X g Z k 0, Z k. 27 X X P Z k P P: Y = S P : X = S k Z k As a result of the regroupng n expresson 27 the tems wth smlar products of ntal operators X P g X 0, X, one obtans g n t, Y = A P t, Y P g X 0, X, n, 28 X X P P: Y = S where Y =,...,n, Y P X,...,X P s a set whose elements are P subsets X Y of partton P : Y = X and s a sum over all possble parttons P of the set Y X P: Y = S

12 The von Neumann Herarchy for Correlaton Operators 2 nto P nonempty mutually dsjont subsets. Evoluton operators A P t for every P n expresson 28 are defned by A P t, Y P := P P! G Zk t, Z k. 29 Z k Z k P P : Y P= S k For P 2 the P th-order cumulants A P t of evoluton operators 8 of the von Neumann equatons [25] have smlar structure, n contrast to the frst-order cumulant. For example, for P = for P = 2 A t,... n = G n t,,...,n, A 2 t,... X, X +... Y = = G Y t,,...,n G X t,,..., X G X2 t, X +,..., n, where {,..., Y } {,..., n} and the followng notatons are used: the symbol... X, X +... Y denote that the sets {,..., X } and { X +,..., Y } are the connected subsets clusters, respectvely, of X and X 2 partcles of a partton of the set Y =,..., X, X +,..., Y nto two elements. The smplest examples for soluton 28 are gven by g t, = A t, g 0,, g 2 t,, 2 = A t, 2g 2 0,, 2 + A 2 t,, 2g 0, g 0, 2, g 3 t,, 2, 3 = A t, 2 3g 3 0,, 2, 3 + A 2 t, 2 3, g 0, g 2 0, 2, 3 + +A 2 t, 3, 2g 0, 2g 2 0,, 3 + A 2 t, 2, 3g 0, 3g 2 0,, 2 + +A 3 t,, 2, 3g 0, g 0, 2g 0, 3. We remark that at the ntal nstant t = 0 soluton 28 satsfes ntal condton 3. Indeed, for P 2 accordng to defntons 7 U n ± 0 = I s a unt operator and n vew of equalty 22, we have A P 0, Y P = P P!I = 0. Z k P : Y P= S k 4.2 Chaos property Let us consder the structure of soluton 28 for one physcally motvated example of the ntal data; that s to say, f the ntal data for Cauchy problem 2 3 satsfy the chaos property statstcally ndependent partcles [],.e. the sequence of correlaton operators s the followng one-component sequence g0 = 0, g 0,, 0, 0, In fact, n terms of the sequence D0 ths condton means that D0 =, D 0,, D 0, D 0, 2,....

13 The von Neumann Herarchy for Correlaton Operators 3 Makng use of relaton 26 we obtan ntal condton 30 for correlaton operators. A more general property, namely, decay of correlatons for the classcal system of hard spheres, was consdered n [23]. For ntal data 30 the formula for soluton 28 of the ntal-value problem 2 3 s smplfed and s reduced to the followng formula: g n t,,...,n = A n t,,...,n n g 0,, n. 3 It s clear from 3 that, f at the ntal nstant there are no correlatons n the system, the correlatons generated by the dynamcs of a system are completely governed by the cumulants of evoluton operators 8. In the case of ntal data 30 soluton 3 of the Cauchy problem for the von Neumann herarchy 2 3 may be rewrtten n another form. For n =, we have = g t, = A t, g 0,. Then, wthn the context of the defnton of the frst-order cumulant, A t, and the nverse to t evoluton operator A t, we express the correlaton operators g n t, n 2 n terms of the one-partcle correlaton operator g t, makng use of formula 3. Fnally, for n 2 formula 3 s gven by g n t,,..., n = Ânt,,..., n n g t,, n 2, where Ânt,,...,n s the nth-order cumulant of the scatterng operators Ĝ t,..., n := G n t,,...,n = n G t, k, n. The generator of the scatterng operator Ĝt,...,n s determned by the operator: n where N k nt acts accordng to 5. n k=2 <...< k = k= N k nt,..., k, 4.3 Propertes of a group of nonlnear operators On L H n soluton 28 generates a group of nonlnear operators of the von Neumann herarchy. The propertes of ths group are descrbed n the followng theorem. Theorem 2. For g n L H n, n, the mappng t A t g Y A P t, Y P g X X 32 n X X P P: Y = S

14 The von Neumann Herarchy for Correlaton Operators 4 s a group of nonlnear operators of class C 0. In the subspace L 0H n L H n the nfntesmal generator N nl of group 32 s defned by the operator N nl g Y := N n ny g n Y + N nt X,...,X P g X X, 33 P: Y = S X, P > where the notatons are smlar to those n 4. Proof. Mappng 32 s defned for g n L H n, n and the followng nequalty holds: X P A t g n L H n n!e 2n+ c n, 34 where c := max P: Y = S g X X X L. H X Indeed, nasmuch as for g n L H n the equalty holds [24] we have A t g n L H n P: Y = S X P : Y S P= k Tr,...,n G n tg n = g n L H n, P c P P: Y = S X k= Z k P! X P s P, kk! g X L H X P c P P: Y = S X k= k P n!e 2n+ c n, where s P, k are the Strlng numbers of the second knd. That s, A t g n L H n for arbtrary t R and n. We can now formulate the group property of the one-parametrc famly of nonlnear operators A t whch are defned by 32,.e. A t At2 g = A t2 At g = A t +t 2 g. Indeed, for g n L H n, n and for any t, t 2 R, accordng to 28 and 8, we have At At2 g Y = A n P t, Y P A P t 2, {X } P g X X PP : X = S Zl Z l = Z l Z l P l X P P : X = S l Z l = P: Y = S P: Y = S X P : Y S P= j P P! Q j R k P Q j P G Qj t, Q j P P P : {X } P = S! G Rk t 2, R k R k k Z l P g Zl Z l, where {X } P Z,...,Z P s a set whose elements are P subsets Z l X of the partton P : X = l Z l. Havng collected the tems at dentcal products of the ntal

15 The von Neumann Herarchy for Correlaton Operators 5 data g n 0, n, and takng nto account the group property of the evoluton operators t, n 7, we obtan U ± n At At2 g n Y = X P Smlarly, we establsh P: Y = S X P : Y S P= k g X X = P: Y = S P P! Q k X A P t + t 2, Y P A t2 At g = A t +t 2 g. X P Q k P G Qk t t 2, Q k g X X = A t +t 2 g Y. n The strong contnuty property of the group A t g0 over the parameter t R s a consequence of the strong contnuty of group 8 of the von Neumann equatons [3]. Indeed, accordng to dentty 22 the followng equalty holds P: Y = S X P : Y S P= k P P! Z k Therefore, for g n L 0H n L H n, n, we have lm t 0 P: Y = S X P : Y S P= k P: Y = S X P : Y S P= k P P! Z k X P Z k P G Zk t, Z k P! lm G Zk t 0 t, Z k Z k Z k P g X X = g n Y. X P X P g X X X P g X X g n Y L H n g X X L H n. In vew of the the fact that group G n t 8 Theorem s a strong contnuous,.e. n the sense of the norm convergence L H n there exsts the lmt lmg n tg n g n = 0, t 0 whch mples that, for mutually dsjont subsets X Y, the followng equalty s also vald: lm G Zk t 0 t, Z k g n g n = 0. Z k P For g n L 0 H n L H n, we fnally have lm A t g g t 0 n n L H n = 0. We wll now construct the generator N nl of group 32. Takng nto account that for g n L 0 H n DN nl n equalty 9 holds, let us dfferentate the group A t g n ψ n for

16 The von Neumann Herarchy for Correlaton Operators 6 all ψ n DH n H n. Accordng to equalty 23 for the P th-order cumulant, P 2, we obtan lm t 0 t A P t, Y P g n ψ n = = Z X, Z... P : Y P= S k Z P X P, Z P P P! Z k N P P Z r r= nt Z k P N Zk Z k g n ψ n = Z,...,Z P g n ψ n = N nt Y P g n ψ n, where Y P X,...,X P s a set whose elements are P subsets X Y of partton P : Y = X and the operator N n nt s gven by formula 5. Therefore, for group 32 we derve At lm g t 0 t g n n ψ n = lm A P t, Y P t 0 t P: Y = S X = lm A t, Y g n g n ψn + lm t 0 t t 0 P: Y = S X, P > = N n g n Y ψn + P: Y = S X, P > X P g X X g n Y ψ n = t A P t, Y P X P N nt Y P g X X ψ n = X P g X X ψ n. 35 Then n vew of equalty 35 and the proof of the theorem 2 for g n L 0 H n DNn nl L H n, n, n the sense of the norm convergence n L H n, we fnally have lm At g t 0 t g n n N nl g n = 0, L H n where N nl s gven by formula The unqueness and exstence theorem For abstract ntal-value problem 2 3 n the space L H n of trace class operators the followng theorem holds. Theorem 3. The soluton of ntal-value problem 2 3 for the von Neumann herarchy 2 s determned by formula 28. For g n 0 L 0 H n L H n t s a strong classcal soluton and for arbtrary ntal data g n 0 L H n t s a weak generalzed soluton. Proof. Accordng to theorem 2 for ntal data g n 0 L 0 H n L H n, n, sequence 28 s a strong soluton of ntal-value problem 2-3.

17 The von Neumann Herarchy for Correlaton Operators 7 Let us show that n the general case g0 L F H expansons 28 gve a weak soluton of the ntal-value problem for the von Neumann herarchy 2. Consder the functonal ϕn, g n t := Tr,...,n ϕ n g n t, 36 where ϕ n L 0 H n are degenerate bounded operators wth nfntely tmes dfferentable kernels wth compact supports. The operator g n t s defned by 28 for the arbtrary ntal data g k 0 L H k, k =,..., n. Accordng to estmate 34 for g n L H n and ϕ n L 0 H n functonal 36 exsts. We transform functonal 36 as follows ϕn, g n t = ϕ n, A P t, Y P g X 0, X = X P = P: Y = S X P : Y S P= k P:Y = S X P P! Z k Z k P G Zk t; Z k ϕ n, X P g X 0, X,37 where the group of operators G n t s adjont to the group G n t n the sense of functonal 36. For g n 0 L H n and ϕ n L 0 H n wthn the context of the theorem 2 we have lm t 0 t G ntϕ n ϕ n, g n 0 N n ϕ n, g n 0 = 0, and, therefore, t holds that lm G Zk t 0 t; Z k ϕ n ϕ n, t Z k P X P = N Zl Z l ϕ n, G Zk t; Z k Z l P Z k P g X 0, X = X P g X 0, X. Then, for representaton 37 of functonal 36, one obtans d ϕn, g n t = N n ϕ n, g n t + P P! dt X, Z k = P: Y = S P > N Zl Z l ϕ n, G Zk t; Z k Z l P Z k P X P N n ϕ n, g n t + P: Y = S X, P > P : Y P= S k g X 0, X = N nt X,...,X P ϕ n, X P g X t, X where the operator N nt X,...,X P s defned by 4 and 5. For functonal 36 we fnally have d ϕn, g n t = dt Nn = ϕ n Y, gn t, Y + P: Y = S X, P >, N nt X,...,X P ϕ n, g X t, X. 38 X P

18 The von Neumann Herarchy for Correlaton Operators 8 Equalty 38 means that for arbtrary ntal data g n 0 L H n, n, the weak soluton of the ntal-value problem of the von Neumann herarchy 2 3 s determned by formula BBGKY herarchy 5. Nonequlbrum grand canoncal ensemble As we have seen above the two equvalent approaches to the the descrpton of the state evoluton of quantum many-partcle systems were formulated, namely, both on the bass of von Neumann equatons 3 for the statstcal operators Dt and of the von Neumann herarchy 2 for the correlaton operators gt. For the system of a fnte average number of partcles there exsts another possblty to descrbe the evoluton of states, namely, by sequences of s-partcle statstcal operators that satsfy the BBGKY herarchy [0]. Tradtonally such a herarchy s deduced on the bass of solutons of the von Neumann equatons the nonequlbrum grand canoncal ensemble [, 25, 33] or the canoncal ensemble [2,0,5] n the space of sequences of trace class operators. The sequence Ft = I, F t,,...,f s t,,...,s,... of s-partcle statstcal operators F s t,,...,s, s can be defned n the framework of the sequence of operators Dt = I, D t,,..., D n t,,..., n,... the operator D n t beng regarded as densty operator 6 of the n-partcle system by expressons [33] F s t,,..., s = e a D0 0 where e a D0 0 = + n= n! Tr s+,...,s+nd s+n t,,...,s + n, s, 39 Tr n!,...,nd n 0,,..., n s a partton functon see defnton of functonal 2. For D0 L F H seres 39 converges. If we descrbe the states of a quantum system of partcles n the framework of correlaton operators gt the s-partcle statstcal operators, that are solutons of the BBGKY herarchy, are defned by the expanson F s t,,...,s = n! Tr s+,...,s+ng +n t,... s, s +,...,s + n, s, 40 where g +n t,... s, s +,..., s + n, n 0, are correlaton operators 28 that satsfy the von Neumann herarchy 2 for a system consstng both from partcles and the cluster of s partcles... s s a notaton as to formula 29. Expanson 40 can be derved from 39 as a result of the followng representaton for rght hand sde of expanson 39 F s t,,..., s = = n! Tr s+,...,s+n P:{... s,s+,...,s+n}= S X P P! X P D X t, X, s, 4

19 The von Neumann Herarchy for Correlaton Operators 9 where s a sum over all possble parttons P of the set {... s, s +,..., s + n} P nto P nonempty mutually dsjont subsets X. If Dt L F H seres 4 converges. We remark that expanson 39 can be defned for more general class of operators then from the space L F H. To prove representaton 4 we wll frst ntroduce some necessary facts. We wll use the notatons of lemma. Accordng to the defnton of -product 9 for the sequence f = f 0, f, f 2, 2,..., f n,...,n,... of elements f n L H n the mappng Exp s defned by seres 20. The mappng Ln that nverse to Exp s defned by seres 2. For f = f 0, f,...,f n,..., f n L H n we defne the mappng d : f d f by d f n,...,n := f n+,...,n, n +, n 0, and for arbtrary set Y =,...,s we defne the lnear mappng d...d s : f d...d s f by d...d s f n := f s+n,...,s + n. 42 We note that for sequences f = f 0, f,...,f n,..., f n L H n and f 2 = f 2 0, f2,...,f2 n,..., f2 n L H n the followng dentty holds [34] d f f 2 = d f f 2 + f d f 2. Further, snce Y P = X,..., X P then Y =... s cluster of s partcles s one element Y = of the partton P P =, we ntroduce the mappng d Y : f d Y f, as follows d Y f n,..., n = f n+ Y,,...,n,, n Then for arbtrary f = f 0, f,...,f n,..., f n L H n accordng to defnton 20 of the mappng Exp the followng equalty holds d Exp f = d f Exp f. 44 For f n L H n an analog of the annhlaton operator s defned and, therefore e a f s,...,s = af s,...,s := Tr s+ f s+,..., s, s +, 45 n! Tr s+,...,s+nf s+n,...,s, s +,...,s + n, s 0. Usng prevous defntons and 9 for sequences f = f 0, f,...,f n,...,f n L H n and f 2 = f 2 0, f2,...,f2 n,..., f2 n L H n we have [34] e a f f 2 0 = e a f 0 e a f From equalty 46 we deduce that expressons 39 and 40 can, respectvely, be rewrtten as F s t, Y = e a D0 0 e a d Y Dt 0 and F s t, Y = e a d Y gt 0, where Y =,...,s. Hence, n vew of equaltes, 2 to derve expressons 40 we have to prove the followng lemma.

20 The von Neumann Herarchy for Correlaton Operators 20 Lemma 2. For f = f 0, f,...,f n,... and f n L H n the followng dentty holds e a Exp f 0 e a d Y Exp f = 0 ea d Y f Proof. Indeed, usng equaltes 43, 44 and 46, we obtan e a Exp f 0 e a d Y Exp f = 0 ea Exp f 0 e a Exp f d Y f = 0 = e a Exp f 0 ea Exp f 0 e a d Y f = e a d 0 Y f On a soluton of the BBGKY herarchy In ths subsecton we wll turn to the soluton of the ntal-value problem of the BBGKY herarchy. In the space L αf H = α n L H n, where α > s a real number, we consder the followng ntal-value problem for the BBGKY herarchy for quantum systems of partcles obeyng Maxwell-Boltzmann statstcs d dt F st = N s F s t + + n! Tr s+,...,s+n n= Z Y, Z N Z +n nt Z, s +,...,s + n F s+n t, 48 F s t t=0 = F s 0, s, 49 where Y =,...,s and the operator N n nt s defned on L α,0f H L αf H by formula 5. The equaton from herarchy 48 for s = has the followng transparent form: d dt F t = N F t + n= n! Tr 2,...,n+ N n+ nt, 2,..., n + F n+ t. In the space L α F H there s an equvalent representaton for the generator of the BBGKY herarchy: e a Ne a = N + [...[N, a],..., a] n! }{{} n= n tmes that follows from 39. For a two-body nteracton potental one s reduced to the form [24]: N + [ N, a ]. We remark that n terms of the s-partcle densty matrx margnal dstrbuton F s t, q,...,q s ; q,...,q s that are kernels of the s-partcle densty operators F s t, for a two-body nteracton potental see for k = 2 the evoluton operator 48 takes the canoncal form of the quantum BBGKY herarchy [0] s t F st; q,...,q s ; q,...,q s = 2 q 2 q + = s + Φq q j Φq q j F s t; q,...,q s ; q,...,q s + + <j= s dq s+ Φq q s+ Φq q s+ F s+ t; q,...q s, q s+ ; q,...,q s, q s+. =

21 The von Neumann Herarchy for Correlaton Operators 2 For the soluton of the BBGKY herarchy the followng theorem s true [25]. Theorem 4. If F0 L α F H and α > e, then for t R there exsts a unque soluton of ntal-value problem gven by F s t,,..., s = = n! Tr s+,...,s+na +n t, Y, s +,...,s + nf s+n 0,,..., s + n, s, 50 where for n 0 A +n t, Y, s +,...,s + n = P:{Y,X\Y }= S X P P! X P G X t, X s the + nth-order cumulant of operators 8, X \ Y = {s +,..., s + n} and Y = {... s}. For ntal data F0 L α,0 L α F H t s a strong soluton and for arbtrary ntal data from the space L α F H t s a weak soluton. The condton α > e guarantees the convergence of seres 50 and means that the average number of partcles 0 s fnte: N < α/e. Ths fact follows f to renormalze a sequence 50 n such a way: Fs t = N s F s t. For arbtrary F0 L α F H the average number of partcles expectaton value 0 expressed n terms of s-partcle operators n state 50 s fnte, n fact, N t = Tr F t, 5 N t c α F0 L α F H <, where c α = e 2 e α s a constant. We emphasze the dfference between fnte and nfnte systems wth non-fxed number of partcles. To descrbe an nfnte partcle system we have to construct a soluton of ntal-value problem n more general spaces than L αf H, for example, n the space of sequences of bounded operators to whch the equlbrum states belong [20,34]. We remark that the formula for soluton 50 can be drectly derved from soluton 28 of von Neumann herarchy 2 [26]. In papers [2, 27, 32, 35] a soluton of ntalvalue problem s represented as the perturbaton teraton seres, whch for a two-body nteracton potental has the form F s t = t 0 dt... t n 0 dt n Tr s+,...,s+n G s t+t... G s+n t n + t n s+n n= s = N 2 nt, s+ G s+ t +t 2 N 2 nt n, s + n G s+n t n F s+n 0. 52

22 The von Neumann Herarchy for Correlaton Operators 22 In the space L αf H expanson 48 s equvalent to ths teraton seres. Ths follows from the valdty of analogs of the Duhamel formula for cumulants of evoluton operators of the von Neumann equatons. For the second-order cumulant an analog of the Duhamel formula has the form 25 and n the general case the followng formula takes place A +n t, Y, s +,...,s + n = =... t 0 dt... t n 0 dt n k I G k t + t, k k 2 I 2 G k2 t + t 2, k 2 k n I n G kn t n +t n, k n n<j n I n l n n 2 <j 2 I 2 l 2 2 <j I l m j N 2 nt l, m m 2 j 2 N 2 nt l 2, m 2... m n j n N 2 nt l n, m n G s+n t n,,..., s+n, where I {Y, s +,...,s + n}, I n { n j n } I n \ { n, n }. We remark that for classcal systems of partcles the frst few terms of the cumulant expanson 50 were consdered n [2]. In [25] we dscuss other possble representatons of a soluton of the BBGKY herarchy n the space L α F H. 5.3 Correlaton operators of nfnte-partcle systems Correlaton operators 28 may be employed to drectly calculate the macroscopc values of a system, n partcular, fluctuatons characterzed by the average values of the square devatons of observables from ts average values. For example, for an addtve-type observable a = a 0, a,..., n a,... from the formula for expectaton value 2 we = derve the formula for fluctuatons the dsperson of an addtve-type observable [0] a a t 2 t = = Tr a 2 a 2 t F t, + Tr,2 a a 2 F 2 t,, 2 F t, F t, 2. Therefore, the dsperson of the addtve-type observable s defned not drectly through the solutons of the BBGKY herarchy but by the followng correlaton operators: F 2 t,, 2 F t, F t, 2 = G 2 t,, 2 or n the general case by the s-partcle correlaton operators G s t,,..., s := P P! F X t, X. 53 X X P P: {,...,s}= S The s-partcle correlaton operators G s t, s can be expressed n terms of correlaton operators 28 by the formula G s t,,..., s = n! Tr s+,...,s+ng s+n t,,..., s + n, s, 54 where g s+n t,,..., s + n s a soluton of the von Neumann herarchy 2. To derve expresson 54 we state the followng lemma.

23 The von Neumann Herarchy for Correlaton Operators 23 Lemma 3. Let f = f 0, f,..., f n and f n L H n, then the equalty holds e a Exp f 0 e a Exp f = Exp e a f. Proof. Indeed, usng equalty 46 and the equalty d...d s Exp f = Exp f d...d X f... d X P +... d X P f, P:Y = S X that follows from 42, 44, one obtans here Y =,..., s e a Exp f 0 ea Exp f Y Y = e a Exp f 0 ea d... d s Exp f 0 = = e a Exp f 0 e a Exp f d...d X f...d X P P:Y = S +...d X P f = X 0 = e a Exp f 0 e a Exp f 0 e a d...d X f... d X P P:Y = S +...d X P f = X 0 = e a d...d X f 0 = Exp e a f Y. Y P: Y = S X X P We now derve representaton 54 for s-partcle correlaton operators. Usng equalty n terms of the mappng Exp 20, representaton 39 for s-partcle statstcal operators can be rewrtten n the form Ft = e a Exp g0 0 ea Exp gt. In terms of the mappng Exp formula 53 has the form Ft = Exp Gt. Further, accordng to prevous formula and Lemma 3 we have and, therefore, we fnally derve Exp Gt = e a Exp g 0 ea Exp gt = Exp e a gt, Gt = e a gt 55 or n component-wse form 54. For chaos ntal data 30 obeyng Maxwell-Boltzmann statstcs accordng to 55 we have G 0 = g 0. Usng soluton 3 of the von Neumann herarchy 2 the expanson for a soluton of the ntal-value problem for s-partcle correlaton operators can be represented as follows G s t,,...,s = s+n n! Tr s+,...,s+na s+n t,,..., s + n G 0,, s. It should be noted that sequence 55 s a soluton of the nonlnear BBGKY herarchy for s-partcle correlaton operators [29] whch descrbes the dynamcs of correlatons of nfnte-partcle systems. =

24 The von Neumann Herarchy for Correlaton Operators 24 Acknowledgement Ths work was partally supported by the WTZ grant No M/24 UA 04/2007 and by the Specal programm of the PAD of NAS of Ukrane. References [] Adam R, Bardos C, Golse F, Teta A 2004 Towards a rgorous dervaton of the cubc nonlnear Schrödnger equaton n dmenson one Asymptot. Anal [2] Arnold A 2008 Mathematcal propertes of quantum evoluton equatons to appear n Lecture Notes n Mathematcs, Berln: Sprnger [3] Banasak J, Arlott L 2006 Perturbatons of Postve Semgroups wth Applcatons Berln: Sprnger p 438 [4] Bardos C, Ducomet B, Golse F, Gottleb A, Mauser N 2007 The TDHF approxmaton for Hamltonans wth m-partcle nteracton potentals Commun. n Math. Sc. 5-9 [5] Bardos C, Golse F, Gottleb A and Mauser N 2003 Mean feld dynamcs of fermons and the tme-dependent Hartree-Fock equaton J. Math. Pures et Appl [6] Benedetto D, Castella F, Esposto R and Pulvrent M 2007 A short revew on the dervaton of the nonlnear quantum Boltzmann equatons Commun. Math. Sc [7] Benedetto D, Castella F, Esposto R and Pulvrent M 2004 Some consderaton on dervaton nonlnear quantum Boltzmann equaton J. Stat. Phys. 6/ [8] Benedetto D, Castella F, Esposto R, Pulvrent M 2006 Some consderatons on the dervaton of the nonlnear quantum Boltzmann equaton II: the low densty regme J. Stat. Phys [9] Berezn F A, Shoubn M A 99 Schrödnger Equaton Amsterdam: Kluwer p 576 [0] Bogolyubov N N 949 Lectures on Quantum Statstcs. Kyv: Rad. shkola p 227 n Ukranan [] Cercgnan C, Gerasmenko V I, Petrna D Ya 997 Many-Partcle Dynamcs and Knetc Equatons Dordrecht: Kluwer Acad. Publ. p 252 [2] Cohen E G D 968 The knetc theory of dense gases n Fundamental Problems n Statstcal Mechancs II ed. E. G. D. Cohen. Amsterdam: North-Holland [3] Dautray R and Lons J L 992 Mathematcal Analyss and Numercal Methods for Scence and Technology vol 5 Berln: Sprnger-Verlag p 562 [4] Elgart A, Schlen B Mean feld dynamcs of boson stars 2007 Comm. Pure Appled Math

25 The von Neumann Herarchy for Correlaton Operators 25 [5] Erdös L, Salmhofer M and Yau H-T 2004 On quantum Boltzmann equaton J. Stat. Phys [6] Erdös L, Schlen B, Yau H-T 2007 Rgorous dervaton of the Gross-Ptaevsk equaton Phys. Rev. Lett [7] Erdös L, Schlen B, Yau H-T 2006 Dervaton of the Gross-Ptaevsk herarchy for the dynamcs of a Bose-Ensten condensate Comm. Pure Appled Math [8] Erdös L, Schlen B, Yau H-T 2007 Dervaton of the cubc nonlnear Schrödnger equaton from quantum dynamcs of many-body systems Invent Math [9] Fröhlch J, Lenzmann E 2005 Nonlnear Equatons descrbng the Mean-Feld Lmt of Bose Gases Workshop: Math. Methods n Quantum Mechancs, Bressanone p 25 [20] Genbre J 97 Some applcatons of functonal ntegratons n statstcal mechancs n Statstcal Mechancs and Quantum Feld Theory eds. S. de Wtt and R. Stord - N.Y.: Gordon and Breach [2] Gerasmenko V I, Petrna D Ya 997 On generalzed knetc equaton Reports of NAS of Ukrane n Ukranan [22] Gerasmenko V I and Ryabukha T V 2002 Cumulant representaton of solutons of the BBGKY herarchy of equatons Ukranan Math. J [23] Gerasmenko V I, Shtyk V O 2008 Bogolyubov decay of correlatons prncple for nfnte system of hard spheres to appear n Reports of NAS of Ukrane, 3 n Ukranan [24] Gerasmenko V I and Shtyk V O 2007 The Cauchy problem for the nonlnear von Neumann herarchy Proceedngs of Insttute of Mathematcs of NASU n Ukranan [25] Gerasmenko V I and Shtyk V O 2006 Intal-value problem for the Bogolyubov herarchy for quantum systems of partcles Ukranan Math. J [26] Gerasmenko V I, Shtyk V O 2006 Exstence crteron of cumulant representaton for soluton of ntal value problem of quantum BBGKY herarchy Reports of NAS of Ukrane n Ukranan [27] Golse F 2003 The mean-feld lmt for the dynamcs of large partcle systems Journees equatons aux dervees partelles 9 p 47 [28] Gottleb A D and Mauser N J 2005 New measure of electron correlaton Phys. Rev. Let [29] Green M S 956 Boltzmann equaton from the statstcal mechancal pont of vew J. Chem. Phys [30] Kato T 995 Perturbaton Theory for Lnear Operators Berln: Sprnger-Verlag p 69

26 The von Neumann Herarchy for Correlaton Operators 26 [3] Markowch P A, Rnghofer C and Schmeser C 990 Semconductor Equatons Berln: Sprnger p 248 [32] Petrna D Ya 972 On solutons of Bogolyubov knetc equatons. Quantum statstcs Theor. and Math. Phys [33] Petrna D Ya 995 Mathematcal Foundatons of Quantum Statstcal Mechancs. Contnuous Systems Amsterdam: Kluwer p 624 [34] Ruelle D 999 Statstcal Mechancs. Rgorous Results Sngapur: World Sc. Publ. Co. p 34 [35] Schlen B 2006 Dervaton of the Gross-Ptaevsk herarchy Mathematcal Physcs of Quantum Mechancs, Lecture Notes n Physcs 690 SprngerProceedngs of QMath 9 [36] Spohn H 2007 Knetc equatons for quantum many-partcle systems arxv: v [37] Shtyk V O 2007 On the solutons of the nonlnear Louvlle herarchy J. Phys. A: Math. Theor