On Evolution Equations for Marginal Correlation Operators
aa r X i v : . [ m a t h - ph ] A ug On Evolution Equations for MarginalCorrelation Operators
V.I. Gerasimenko ∗ and D.O. Polishchuk ∗∗ ∗ Institute of Mathematics of NAS of Ukraine,3, Tereshchenkivs’ka Str.,01601 Kyiv-4, Ukraine ∗∗ Taras Shevchenko National University of Kyiv,Department of Mechanics and Mathematics,2, Academician Glushkov Av.,03187 Kyiv, Ukraine
Abstract.
This paper is devoted to the problem of the description of nonequi-librium correlations in quantum many-particle systems. The nonlinear quantumBBGKY hierarchy for marginal correlation operators is rigorously derived from thevon Neumann hierarchy for correlation operators that give an alternative approachto the description of states in comparison with the density operators. A nonpertur-bative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchyfor marginal correlation operators is constructed.
Key words: nonlinear quantum BBGKY hierarchy; von Neumann hierarchy; cor-relation operator; density matrix; quantum many-particle system. E-mail: [email protected] E-mail: [email protected] volution of marginal correlation operators 2
Contents
The importance of the mathematical description of correlations in numerous problems of themodern statistical mechanics is well-known. Among them in particular, we refer to such fun-damental problems as the problem of quantum measurements and of a description of collectivebehavior of interacting particles by quantum kinetic equations [1–10]. Owing to the intrinsiccomplexity and richness of these problems, primarily it is necessary to develop an adequatemathematical theory of underlying evolution equations.The goal of this paper is to derive rigorously the evolution equations for marginal correlationoperators that give an equivalent approach to the description of the evolution of states incomparison with marginal density operators governed by the quantum BBGKY hierarchy andto construct a solution of the corresponding Cauchy problem.We briefly outline the results and structure of the paper. In introductory section 2 we setforth an approach to the description of the evolution of states of quantum many-particle systemswithin the framework of correlation operators governed by the von Neumann hierarchy [11, 12].In section 3 we introduce the notion of marginal correlation operators. To justify this notion,we discuss in detail the motivation of the description of states within the framework of marginalcorrelation operators or in other words, the origin of the microscopic description of correlationsin quantum many-particle systems. Then we rigorously derive the nonlinear quantum BBGKYhierarchy for marginal correlation operators from the von Neumann hierarchy for correlationvolution of marginal correlation operators 3operators. The nonlinear quantum BBGKY hierarchy gives an alternative method of the de-scription of the evolution of states of infinitely many particles in comparison with the quantumBBGKY hierarchy for the marginal density operators [13, 14]. In section 4 we construct thenonperturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy.The nonperturbative solution is determined in the form of expansion over particle clusters,which evolution is governed by the corresponding-order cumulant of the nonlinear group ofoperators generated by the solution of the von Neumann hierarchy. The existence theorem forinitial data from the space of trace-class operators is proved. We also give some comments onthe mean field (self-consistent field) asymptotic behavior of the constructed solution. Finally,in section 5 we conclude with some observations and perspectives for future research in thelight of the results we present here.
We consider a quantum system of a non-fixed, i.e. arbitrary but finite, number of identical(spinless) particles with unit mass m = 1 in the space R ν , ν ≥
1, that obey the Maxwell-Boltzmann statistics. Let F H = L ∞ n =0 H n be the Fock space over the Hilbert space H , wherethe n -particle Hilbert space H n ≡ H ⊗ n is a tensor product of n Hilbert spaces H and weadopt the usual convention that H = C . The Hamiltonian H n of the n -particle system is aself-adjoint operator with the domain D ( H n ) ⊂ H n H n = n X i =1 K ( i ) + n X i
1. The evolution of all possible states isvolution of marginal correlation operators 4determined by the initial-value problem of the von Neumann hierarchy [11, 12] ddt g s ( t, Y ) = N ( Y | g ( t )) , (1) g s ( t, Y ) (cid:12)(cid:12) t =0 = g s (0 , Y ) , s ≥ , (2)where the following notations are used: N ( Y | g ( t )) . = −N s ( Y ) g s ( t, Y ) + (3)+ X P: Y = X S X X i ∈ X X i ∈ X ( −N int ( i , i )) g | X | ( t, X ) g | X | ( t, X ) , P P: Y = X S X is the sum over all possible partitions P of the set Y ≡ (1 , . . . , s ) into twononempty mutually disjoint subsets X ⊂ Y and X ⊂ Y , the operator ( −N s ) defined on L ( H s ) by the formula ( −N s ( Y )) f s . = − i ~ ( H s f s − f s H s ) , (4)is the generator of the von Neumann equation [15] and the operator ( −N int ) is defined by( −N int ( i , i )) f s . = − i ~ (Φ( i , i ) f s − f s Φ( i , i )) . (5)Hereafter we use the following notations: ( { X } , . . . , { X | P | } ) is a set, elements of whichare | P | mutually disjoint subsets X i ⊂ Y ≡ (1 , . . . , s ) of the partition P : Y = ∪ | P | i =1 X i ,i.e. | ( { X } , . . . , { X | P | } ) | = | P | . In view of these notations we state that ( { Y } ) is the setconsisting of one element Y = (1 , . . . , s ) of the partition P ( | P | = 1) and | ( { Y } ) | = 1. Weintroduce the declasterization mapping θ : ( { X } , . . . , { X | P | } ) → Y , by the following formula: θ ( { X } , . . . , { X | P | } ) = Y . For example, let X ≡ (1 , . . . , s + n ), then for the set ( { Y } , X \ Y ) itholds: θ ( { Y } , X \ Y ) = X .On the space L ( H n ) we also introduce the mapping: R ∋ t → G n ( − t ) f n , which is generatedby the solution of the von Neumann equation of n particles [15, 16] G n ( − t ) f n . = e − i ~ tH n f n e i ~ tH n . (6)This mapping is an isometric strongly continuous group that preserves positivity and self-adjointness of operators [15]. On L ( H n ) ⊂ D ( −N n ) the infinitesimal generator of group (6) isdetermined by operator (4).A solution of the Cauchy problem (1)-(2) is given by the following expansion [11, 12], [17] g s ( t, Y ) = G ( t ; Y | g (0)) . = (7) . = X P: Y = S i X i A | P | ( t, { X } , . . . , { X | P | } ) Y X i ⊂ P g | X i | (0 , X i ) , s ≥ , where P P: Y = S i X i is the sum over all possible partitions P of the set Y ≡ (1 , . . . , s ) into | P | nonempty mutually disjoint subsets X i ⊂ Y , the evolution operator A | P | ( t ) is the | P | th -ordervolution of marginal correlation operators 5cumulant of groups of operators (6) defined by the formula A | P | ( t, { X } , . . . , { X | P | } ) . = (8) . = X P ′ : ( { X } ,..., { X | P | } )= S k Z k ( − | P ′ |− ( | P ′ | − Y Z k ⊂ P ′ G | θ ( Z k ) | ( − t, θ ( Z k )) . Here P P ′ : ( { X } ,..., { X | P | } )= S k Z k is the sum over all possible partitions P ′ of the set ( { X } , . . . , { X | P | } ) into | P ′ | nonempty mutually disjoint subsets Z k ⊂ ( { X } , . . . , { X | P | } ). For operators(7) the estimate holds (cid:13)(cid:13) g s ( t ) (cid:13)(cid:13) L ( H s ) ≤ s ! e s c s , (9)where c ≡ max P: Y = S i X i ( k g | X | (0) k L ( H | X | ) , . . . , k g | X | P | | (0) k L ( H | X | P || ) ).If g n (0) ∈ L ( H n ) ⊂ L ( H n ) , n ≥
1, expansion (7) is a strong (classical) solution of theCauchy problem (1)-(2) and for arbitrary initial data g n (0) ∈ L ( H n ) , n ≥
1, it is a weak(generalized) solution [11, 12].In case of the absence of correlations between particles at initial time, i.e. initial datasatisfying a chaos condition, the sequence of correlation operators has the form g (0) = (0 , g (0 , , , . . . ) . (10)The corresponding solution of the initial-value problem of the von Neumann hierarchy is givenby the expansion g s ( t, Y ) = A s ( t, Y ) s Y i =1 g (0 , i ) , (11)where A s ( t ) is the sth -order cumulant defined by A s ( t, Y ) = X P: Y = S i X i ( − | P |− ( | P | − Y X i ⊂ P G | X i | ( − t, X i ) . For operators (11) estimate (9) takes the corresponding form: (cid:13)(cid:13) g s ( t ) (cid:13)(cid:13) L ( H s ) ≤ s ! e s (cid:13)(cid:13) g (0) (cid:13)(cid:13) s L ( H ) . We note that correlations created in evolutionary process of a system are described byformula (11) and determined by the corresponding-order cumulant of the groups of operators(6) of the von Neumann equations.
The evolution of states of infinite-particle quantum systems is traditionally described by themarginal (or s -particle) density operators governed by the quantum BBGKY hierarchy [13, 14].In this section we introduce the marginal correlation operators that give an equivalent approachto the description of the evolution of such states and describe the nonequilibrium correlationsin quantum systems. We also rigorously derive the nonlinear quantum BBGKY hierarchy formarginal correlation operators from the von Neumann hierarchy (1) for correlation operators.volution of marginal correlation operators 6 In the capacity of an example of a mean-value functional of observables [12] we consider thedefinition of the mean-value functional of the additive-type observable A (1) = (0 , a (1) , . . . , P ni =1 a ( i ) , . . . ) h A (1) i ( t ) = ∞ X n =0 n ! Tr ,..., n a (1) g n ( t, , . . . , n ) , (12)where the operators g n ( t ) , n ≥
0, are determined by expansions (7), and the functional ofthe dispersion for this type of observables h ( A (1) − h A (1) i ) i ( t ) = (13)= ∞ X n =0 n ! Tr ,..., n ( a (1) − h A (1) i ( t )) g n ( t, , . . . , n ) ++ ∞ X n =0 n ! Tr ,..., n a (1) a (2) g n ( t, , . . . , n ) . For A (1) ∈ L ( F H ) and g ∈ L ( F H ) functionals (12),(13) exists.Following to formula (13), we introduce the marginal correlation operators by the series G s ( t, , . . . , s ) . = ∞ X n =0 n ! Tr s +1 ,...,s + n g s + n ( t, , . . . , s + n ) , s ≥ , (14)where the sequence g s + n ( t, , . . . , s + n ) , n ≥
0, is a solution of the Cauchy problem of the vonNeumann hierarchy (1). According to estimate (9), series (14) exists and the estimate holds: (cid:13)(cid:13) G s ( t ) (cid:13)(cid:13) L ( H s ) ≤ s !(2 e ) s c s P ∞ n =0 (2 e ) n c n . Thus, macroscopic characteristics of fluctuations ofobservables are determined by marginal correlation operators (14) on the microscopic level h ( A (1) − h A (1) i ) i ( t ) = Tr ( a (1) − h A (1) i ( t )) G ( t,
1) + Tr , a (1) a (2) G ( t, , . Traditionally marginal correlation operators are introduced by means of the cluster expan-sions of the marginal density operators F s ( t ) , s ≥
1, governed by the quantum BBGKY hier-archy [13] F s ( t, Y ) = X P : Y = S i X i Y X i ⊂ P G | X i | ( t, X i ) , s ≥ , (15)where P P: Y = S i X i is the sum over all possible partitions P of the set Y ≡ (1 , . . . , s ) into | P | nonempty mutually disjoint subsets X i ⊂ Y . Hereupon solutions of cluster expansions (15) G s ( t, Y ) = X P : Y = S i X i ( − | P |− ( | P | − Y X i ⊂ P F | X i | ( t, X i ) , s ≥ , (16)are interpreted as the operators that describe correlations of many-particle systems. Thus,marginal correlation operators (16) are cumulants (semi-invariants) of the marginal densityoperators.volution of marginal correlation operators 7As follows from formula (12) and its generalization [12] the marginal density operators F s ( t )are defined in terms of the correlation operators of clusters of particles g ( s ) ( t ) = ( g ( t, { Y } ) , . . . ,g n ( t, { Y } , s + 1 , . . . , s + n ) , . . . ) by the expansion F s ( t, Y ) . = ∞ X n =0 n ! Tr s +1 ,...,s + n g n ( t, { Y } , s + 1 , . . . , s + n ) , s ≥ , (17)where the sequence g n ( t, { Y } , s + 1 , . . . , s + n ) , n ≥
0, is a solution of the Cauchy problem ofthe von Neumann hierarchy for correlation operators of particle clusters [12], namely g n ( t, { Y } , X \ Y ) = G ( t ; { Y } , X \ Y | g (0)) . = (18) . = X P: ( { Y } , X \ Y )= S i X i A | P | (cid:0) t, { θ ( X ) } , . . . , { θ ( X | P | ) } (cid:1) Y X i ⊂ P g | X i | (0 , X i ) , s ≥ , n ≥ , where A | P | ( t ) is the | P | th -order cumulant defined by formula (8). According to estimate (9),series (17) exists and the estimate holds: (cid:13)(cid:13) F s ( t ) (cid:13)(cid:13) L ( H s ) ≤ e c P ∞ n =0 e n c n . We note that everyterm of marginal correlation operator expansion (14) is determined by the ( s + n )-particlecorrelation operator (7) as contrasted to marginal density operator expansion (17) which isdefined by the (1 + n )-particle correlation operator (18).The correlation operators of particle clusters g ( s ) ( t ) = ( g ( t, { Y } ) , . . . , g n ( t, { Y } , X \ Y ) ,. . . ) ∈ L ( ⊕ ∞ n =0 H s + n ) can be expressed in terms of correlation operators of particles (7) g n ( t, { Y } , X \ Y ) = (19)= X P:( { Y } , X \ Y )= S i X i ( − | P |− ( | P | − Y X i ⊂ P X P ′ : θ ( X i )= S ji Z ji Y Z ji ⊂ P ′ g | Z ji | ( t, Z j i ) . In particular case n = 0, i.e. the correlation operator of a cluster of | Y | particles, these relationstake the form g ( t, { Y } ) = X P: Y = S i X i Y X i ⊂ P g | X i | ( t, X i ) . By the way we observe on that cluster expansions (15) follow from definitions (14) and (17) inconsequence of relations (19) between correlation operators of particle clusters and correlationoperators of particles.The marginal ( s -particle) density operators (17) are determined by the Cauchy problem ofthe quantum BBGKY hierarchy [13] ddt F s ( t, Y ) = −N s ( Y ) F s ( t, Y ) + X i ∈ Y Tr s +1 ( −N int ( i, s + 1)) F s +1 ( t ) , (20) F s ( t ) | t =0 = F s (0) , s ≥ . (21)We remind that usually the marginal density operators F s ( t ) , s ≥
1, are defined by the well-known formula of the nonequilibrium grand canonical ensemble [18, 19] in terms of the densityvolution of marginal correlation operators 8operators D = ( I, D ( t ) , . . . , D n ( t ) , . . . ) governed by the von Neumann equations (the quantumLiouville equation) F s ( t, Y ) = ( I, D ( t )) − ∞ X n =0 n ! Tr s +1 ,...,s + n D s + n ( t, X ) , where ( I, D ( t )) = P ∞ n =0 1 n ! Tr ,...,n D n ( t ) is a normalizing factor, I is the identity operator and Y ≡ (1 , . . . , s ), X ≡ (1 , . . . , s + n ). Thus, along with the definition within the framework ofthe non-equilibrium grand canonical ensemble the marginal density operators can be definedwithin the framework of dynamics of correlations that allows to give the rigorous meaning ofthe states for more general classes of operators than the trace-class operators.If F (0) ∈ L α ( F H ) = L ∞ n =0 α n L α ( H n ) and α > e , then for t ∈ R a unique solution ofthe Cauchy problem (20)-(21) of the quantum BBGKY hierarchy exists and is given by theexpansion [11], [20] F s ( t, Y ) = ∞ X n =0 n ! Tr s +1 ,...,s + n A n ( t, { Y } , X \ Y ) F s + n (0 , X ) , s ≥ , (22)where the (1 + n ) th -order cumulant A n ( t ) of groups of operators (6) is defined by A n ( t, { Y } , X \ Y ) = X P :( { Y } , X \ Y )= S i X i ( − | P |− ( | P | − Y X i ⊂ P G | θ ( X i ) | ( − t, θ ( X i )) , (23) P P is the sum over all possible partitions P of the set ( { Y } , X \ Y ) into | P | nonempty mutuallydisjoint subsets X i ⊂ ( { Y } , X \ Y ).Formally, the evolution equations for marginal correlation operators are derived from thequantum BBGKY hierarchy for marginal density operators (20) on basis of expression (16).Then the evolution of all possible states of quantum many-particle systems obeying the Maxwell-Boltzmann statistics with the Hamiltonian (1) can be described within the framework ofmarginal correlation operators governed by the nonlinear quantum BBGKY hierarchy ddt G s ( t, Y ) = N ( Y | G ( t )) + Tr s +1 X i ∈ Y ( −N int ( i, s + 1)) (cid:0) G s +1 ( t, Y, s + 1) + (24)+ X P : (
Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X G | X | ( t, X ) G | X | ( t, X ) (cid:1) ,G s ( t, Y ) (cid:12)(cid:12) t =0 = G s (0 , Y ) , s ≥ . (25)In equation (24) the operator N ( Y | G ( t )) is generator of the von Neumann hierarchy (1)defined by formula (3), i.e. N ( Y | G ( t )) . = ( −N s ( Y )) G s ( t, Y ) ++ X P: Y = X S X X i ∈ X X i ∈ X ( −N int ( i , i )) G | X | ( t, X ) G | X | ( t, X ) , volution of marginal correlation operators 9where the operators ( −N s ) and ( −N int ) are defined by (4) and (5) respectively, P P: Y = X S X is the sum over all possible partitions P of the set Y ≡ (1 , . . . , s ) into two nonempty mutuallydisjoint subsets X ⊂ Y and X ⊂ Y , and P P : (
Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X is the sum over all possiblepartitions of the set ( Y, s + 1) into two mutually disjoint subsets X and X such that ith particle belongs to the subset X and ( s + 1) th particle belongs to X . As far as we knowhierarchy (24) was introduced by M.M. Bogolyubov [13] and in the papers of J. Yvon [21] andM.S. Green [22] for systems of classical particles.Another method of the justification of evolution equations for marginal correlation operatorsconsists in their derivation from the von Neumann hierarchy for correlation operators (1) onthe basis of definition (14). In this section we establish that marginal correlation operators (14) are governed by the non-linear quantum BBGKY hierarchy (24). With this aim we differentiate by time variable themarginal correlation operators defined by series (14) in the sense of the pointwise convergenceon the space L ( H s ) ddt G s ( t, Y ) = ∞ X n =0 n ! Tr s +1 ,...,s + n (cid:0) ( −N s + n ( X )) g s + n ( t, X ) ++ X P: X = X ∪ X X i ∈ X X i ∈ X ( −N int ( i , i )) g | X | ( t, X ) g | X | ( t, X ) (cid:1) , (26)where we use the notations: X ≡ (1 , . . . , s + n ), Y ≡ (1 , . . . , s ), P P: Y = X S X is the sum overall possible partitions P of the set Y ≡ (1 , . . . , s ) into two nonempty mutually disjoint subsets X ⊂ Y and X ⊂ Y , and operators ( −N s + n ) and ( −N int ) are defined by formulas (4) and (5)respectively. Taking into account the equality −N s + n ( X ) = −N s ( Y ) − N n ( X \ Y ) + X i ∈ Y X i ∈ X \ Y ( −N int ( i , i )) , and the identity Tr s +1 ,...,s + n ( −N n ( X \ Y )) g s + n ( t ) = 0 , and according to the symmetry property of the correlation operators g s + n ( t, X ), for the firstterm in the right-hand side of identity (26) in terms of definition (14) we obtain ∞ X n =0 n ! Tr s +1 ,...,s + n ( −N s + n ( X )) g s + n ( t, , . . . , s + n ) = (27)= −N s ( Y ) G s ( t, Y ) + ∞ X n =1 n ! Tr s +1 ,...,s + n X i ∈ Y X i ∈ X \ Y ( −N int ( i , i )) g s + n ( t, X ) == −N s ( Y ) G s ( t, Y ) + ∞ X n =0 n ! Tr s +1 ,...,s + n +1 X i ∈ Y ( −N int ( i , s + 1)) g s + n +1 ( t, X, s + n + 1) =volution of marginal correlation operators 10= −N s ( Y ) G s ( t, Y ) + Tr s +1 X i ∈ Y ( −N int ( i , s + 1)) G s +1 ( t, , . . . , s + 1) . Let us consider successively the following four parts of the second term of the right-hand sideof identity (26) X P: X = X ∪ X X i ∈ X X i ∈ X ( −N int ( i , i )) = X P: X = X ∪ X (cid:0) X i ∈ Y ∩ X X i ∈ Y ∩ X + (28)+ X i ∈ X \ Y ∩ X X i ∈ Y ∩ X + X i ∈ Y ∩ X X i ∈ X \ Y ∩ X + X i ∈ X \ Y ∩ X X i ∈ X \ Y ∩ X (cid:1) ( −N int ( i , i )) . Taking into account that the equality is true X P : X = X ∪ X ,Y ∩ X = ∅ , Y ∩ X = ∅ g | X | ( t, X ) g | X | ( t, X ) == X P : Y = Y S Y ,Y = ∅ , Y = ∅ X Z ⊂ X \ Y g | Y | + | Z | ( t, Y , Z ) g | Y | + | ( X \ Y ) \ Z | ( t, Y , ( X \ Y ) \ Z ) , and the validity of the following equality (according to the symmetry property of operators g n ( t )) Tr s +1 ,...,s + n X Z ⊂ X \ Y g | Y | + | Z | ( t, Y , Z ) g | Y | + | ( X \ Y ) \ Z | ( t, Y , ( X \ Y ) \ Z ) == Tr s +1 ,...,s + n n X k =0 n ! k !( n − k )! g | Y | + n − k ( t, Y , s + 1 , . . . , s + n − k ) ×× g | Y | + k ( t, Y , s + n − k + 1 , . . . , s + n ) , for the first part of equality (28) of the second term of identity (26) it holdsTr s +1 ,...,s + n X P: X = X ∪ X X i ∈ Y ∩ X X i ∈ Y ∩ X ( −N int ( i , i )) g | X | ( t, X ) g | X | ( t, X ) == Tr s +1 ,...,s + n X P: Y = Y S Y X i ∈ Y X i ∈ Y ( −N int ( i , i )) n X k =0 n ! k !( n − k )! ×× g | Y | + n − k ( t, Y , s + 1 , . . . , s + n − k ) g | Y | + k ( t, Y , s + n − k + 1 , . . . , s + n ) . Then in terms of definition (14) the last expression takes the form ∞ X n =0 n ! Tr s +1 ,...,s + n X P: Y = Y S Y X i ∈ Y X i ∈ Y ( −N int ( i , i )) n X k =0 n ! k !( n − k )! ×× g | Y | + n − k ( t, Y , s + 1 , . . . , s + n − k ) g | Y | + k ( t, Y , s + n − k + 1 , . . . , s + n ) == X P: Y = Y S Y X i ∈ Y X i ∈ Y ( −N int ( i , i )) G | Y | ( t, Y ) G | Y | ( t, Y ) . volution of marginal correlation operators 11Hence for the first part of equality (28) of the second term of the right-hand side of identity(26) we have ∞ X n =0 n ! Tr s +1 ,...,s + n X P: X = X ∪ X X i ∈ Y ∩ X X i ∈ Y ∩ X ( −N int ( i , i )) g | X | ( t, X ) g | X | ( t, X ) = (29)= X P: Y = Y S Y X i ∈ Y X i ∈ Y ( −N int ( i , i )) G | Y | ( t, Y ) G | Y | ( t, Y ) . Similarly the second and third parts of equality (28) of the second term of the right-hand sideof identity (26) are expressed in terms of definition (14) in the form ∞ X n =0 n ! Tr s +1 ,...,s + n X P: X = X ∪ X (cid:0) X i ∈ X \ Y ∩ X X i ∈ Y ∩ X + (30)+ X i ∈ Y ∩ X X i ∈ X \ Y ∩ X (cid:1) ( −N int ( i , i )) g | X | ( t, X ) g | X | ( t, X ) == X i ∈ Y Tr s +1 ( −N int ( i, s + 1)) X P : (
Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X G | X | ( t, X ) G | X | ( t, X ) , where P P : (
Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X is the sum over all possible partitions of the set ( Y, s + 1) intotwo mutually disjoint subsets X and X such that ith particle belongs to the subset X and( s + 1) th particle belongs to X .Then taking into account the following identity for the fourth part (28) of the second termof (26) ∞ X n =0 n ! Tr s +1 ,...,s + n X P: X = X ∪ X X i ∈ X \ Y ∩ X X i ∈ X \ Y ∩ X ( −N int ( i , i )) g | X | ( t, X ) g | X | ( t, X ) = 0 , and identities (29),(30), for the second term of the right-hand side of (26) it holds ∞ X n =0 n ! Tr s +1 ,...,s + n X P: X = X ∪ X X i ∈ X X i ∈ X ( −N int ( i , i )) g | X | ( t, X ) g | X | ( t, X ) = (31)= X P: Y = Y S Y X i ∈ Y X i ∈ Y ( −N int ( i , i )) G | Y | ( t, Y ) G | Y | ( t, Y ) ++ X i ∈ Y Tr s +1 ( −N int ( i, s + 1)) X P : (
Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X G | X | ( t, X ) G | X | ( t, X ) . In consequence of identities (27) and (31) we finally derive ddt G s ( t, Y ) = −N s ( Y ) G s ( t, Y ) + X P: Y = Y S Y X i ∈ Y X i ∈ Y ( −N int ( i , i ) G | Y | ( t, Y ) G | Y | ( t, Y )+volution of marginal correlation operators 12+ X i ∈ Y Tr s +1 ( −N int ( i, s + 1))( G s +1 ( t ) + X P : (
Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X G | X | ( t, X ) G | X | ( t, X )) , where we use notations accepted above in (26),(30). The constructed identity for the marginalcorrelation operators defined by expansion (14) we treat as the hierarchy of evolution equations,which governs the marginal correlation operators of quantum many-particle systems.We also formulate the nonlinear quantum BBGKY hierarchy in case of many-particle systemsobeying quantum statistics with the Hamiltonian H n = n X i =1 K ( i ) + n X k =1 n X i <... X Z ⊂ X ,Z = ∅ . . . X Z | P | ⊂ X | P | ,Z | P | = ∅ (cid:0) − N ( | P | P r =1 | Z r | )int ( Z , . . . , Z | P | ) (cid:1) S ± s Y X i ⊂ P G | X i | ( t, X i ) ++ ∞ X n =1 n X k =1 s X j <... To construct a nonperturbative solution of the Cauchy problem (24)-(25) of the nonlinearquantum BBGKY hierarchy we first consider its structure for physically motivated example ofinitial data, namely, initial data satisfying a chaos property G s ( t, Y ) | t =0 = G (0 , δ s, , s ≥ , (34)where δ s, is a Kronecker symbol. Chaos property (34) means the absence of state correlationsin a system at the initial time.According to definition (14) and solution expansion (11), in the case under consideration thefollowing relation between the marginal correlation operators and correlation operators is true G (0 , i ) = g (0 , i ) . (35)Taking into account the form (11) of a solution of the initial-value problem of the von Neumannhierarchy (1) in case of initial data (10), for expansion (14) we obtain G s ( t, Y ) = ∞ X n =0 n ! Tr s +1 ,...,s + n A s + n ( t, , . . . , s + n ) s + n Y i =1 g (0 , i ) , (36)where A s + n ( t ) is ( s + n ) th -order cumulant (8). In consequence of relation (35) we finally derive G s ( t, Y ) = ∞ X n =0 n ! Tr s +1 ,...,s + n A s + n ( t, , . . . , s + n ) s + n Y i =1 G (0 , i ) , s ≥ . (37)If k G (0) k L ( H ) ≤ (2 e ) − , series (37) converges, since for cumulants (8) the estimate holds [11] k A n ( t ) f k L ( H n ) ≤ n ! e n k f k L ( H n ) . volution of marginal correlation operators 14From the structure of series (37) it is clear that in case of absence of correlations at initial instantin a system the correlations generated by the dynamics of quantum many-particle systems arecompletely governed by cumulants (8) of groups of operators (6).Thus, the cumulant structure of solution (7) of the von Neumann hierarchy (1) inducesthe cumulant structure of solution expansion (37) of the initial-value problem of the quantumnonlinear BBGKY hierarchy for marginal correlation operators.The evolution equations which satisfy expression (37) are derived similarly to the derivationof hierarchy (24) given in section 3.2 on the base of definition (36).We note, that in case of initial data (10) solution (11) of the Cauchy problem (1)-(2) of thevon Neumann hierarchy may be rewritten in another representation. For n = 1, we have g ( t, 1) = A ( t, g (0 , . Then, within the context of the definition of the first-order cumulant, A ( − t ), and the dualgroup of operators A ( t ), we express the correlation operators g s ( t ) , s ≥ 2, in terms of theone-particle correlation operator g ( t ) using formula (11). Hence for s ≥ g s ( t, Y | g ( t )) = b A s ( t, Y ) s Y i =1 g ( t, i ) , s ≥ , where b A s ( t, Y ) is sth -order cumulant (8) of the scattering operators b G s ( t, Y ) . = G s ( − t, Y ) s Y i =1 G ( t, i ) , s ≥ . (38)The generator of the scattering operator b G t ( Y ) is determined by the operator ddt b G s ( t, Y ) | t =0 = s X k =2 s X i <...
2, are represented by the expansions G s (cid:0) t, Y | G ( t ) (cid:1) = ∞ X n =0 n ! Tr s +1 ,...,s + n V n (cid:0) t, θ ( { Y } ) , s + 1 , . . . , s + n (cid:1) s + n Y i =1 G ( t, i ) , (39)where the operator G ( t, i ) is given by (36) for s = 1, and we use the notion of the declasteriza-tion mapping defined in section 2 [20]. In expansion (39) the (1 + n ) th -order evolution operator V n ( t ) is defined by the formula [9]volution of marginal correlation operators 15 V n ( t, θ ( { Y } , X \ Y ) . = n ! n X k =0 ( − k n X n =1 . . . n − n − ... − n k − X n k =1 n − n − . . . − n k )! ×× b A s + n − n − ... − n k ( t, { Y } , s + 1 , . . . , s + n − n − . . . − n k ) ×× k Y j =1 X D j : Z j = S l j X l j , | D j | ≤ s + n − n − · · · − n j | D j | ! s + n − n − ... − n j X i = ... = i | D j | =1 Y X lj ⊂ D j | X l j | ! b A | X lj | ( t, i l j , X l j ) , where P D j : Z j = S lj X lj is the sum over all possible dissections D j of the linearly ordered set Z j ≡ ( s + n − n − . . . − n j + 1 , . . . , s + n − n − . . . − n j − ) on no more than s + n − n − . . . − n j linearly ordered subsets, and the operator b A n ( t ) is the (1 + n )-order cumulant (8) of thescattering operators (38). For example, the lower orders evolution operators V n (cid:0) t, θ ( { Y } ) , s +1 , . . . , s + n (cid:1) , n ≥ 0, have the form V ( t, θ ( { Y } )) = b A s ( t, θ ( { Y } ) , V ( t, θ ( { Y } ) , s + 1) = b A s +1 ( t, θ ( { Y } ) , s + 1) − b A s ( t, θ ( { Y } )) s X i =1 b A ( t, i, s + 1) , and in case of s = 2, it holds V ( t, θ ( { , } )) = b G ( t, , − I. We point out also that in case of chaos initial data solution expansion (22) of the quantumBBGKY hierarchy (20) for marginal density operators differs from solution expansion (37) ofthe nonlinear quantum BBGKY hierarchy (24) for marginal correlation operators only by theorder of the cumulants of the groups of operators of the von Neumann equations [12], [20] F s ( t, Y ) = ∞ X n =0 n ! Tr s +1 ,...,s + n A n ( t, { Y } , X \ Y ) s + n Y i =1 F (0 , i ) , s ≥ , (40)where A n ( t ) is the (1 + n ) th -order cumulant (23). Series (40) converges under the condition: k F (0) k L ( H ) ≤ e − . The direct method of the construction of a solution of the nonlinear quantum BBGKY hi-erarchy (24) in the form of nonperturbative expansion consists in its derivation on the basisof expansions (16) from nonperturbative solution (22) of initial-value problem of the quan-tum BBGKY hierarchy (20)-(21). Following stated above approach, we derive a formula fora solution of the quantum nonlinear BBGKY hierarchy for marginal correlation operators incase of general initial data on the basis of definition (14) and nonperturbative solution (7) ofinitial-value problem of the von Neumann hierarchy (1)-(2). With this aim on f n ∈ L ( H n ) weintroduce an analogue of the annihilation operator( a f ) s (1 , . . . , s ) . = Tr s +1 f s +1 (1 , . . . , s, s + 1) , s ≥ , (41)volution of marginal correlation operators 16and, therefore we have( e ± a f ) s (1 , . . . , s ) = ∞ X n =0 ( ± n n ! Tr s +1 ,...,s + n f s + n (1 , . . . , s + n ) . According to definition (14) of the marginal correlation operators, i.e. G ( t ) = e a g ( t ) , where the sequence g ( t ) is a solution of the von Neumann hierarchy for correlation operatorsdefined by group (7), i.e. g ( t ) = G ( t | g (0)), and to the equality: g (0) = e − a G (0), we finallyderive G ( t ) = e a G ( t | e − a G (0)) . (42)To set down formula (42) in componentwise form we observe, that the following equalityholds Y X i ⊂ P ( e − a G (0)) | X i | ( X i ) = ∞ X k =0 ( − k k ! Tr s + n +1 ,...,s + n + k k X k =0 k ! k !( k − k )! . . . (43) . . . k | P |− X k | P |− =0 k | P |− ! k | P |− !( k | P |− − k | P |− )! G | X | + k − k (0 , X , s + n + 1 , . . . , s + n + k − k ) . . .. . . G | X | P | | + k | P |− (0 , X | P | , s + n + k − k | P |− + 1 , . . . , s + n + k ) . Then according to formulas (42) and (7), for s ≥ G s ( t, Y ) = ∞ X n =0 n ! Tr s +1 ,...,s + n X P: (1 ,...,s + n )= S i X i A | P | (cid:0) t, { X } , . . . , { X | P | } (cid:1) Y X i ⊂ P ( e − a G (0)) | X i | ( X i ) , where A | P | ( t ) is | P | th -order cumulant (8), and as a result of the validity of equality (43) forsequence (42) we obtain G s ( t, , . . . , s ) == ∞ X n =0 n ! Tr s +1 ,...,s + n n X k =0 ( − k n ! k !( n − k )! X P: (1 ,...,s + n − k )= S i X i A | P | (cid:0) t, { X } , . . . , { X | P | } (cid:1) ×× k X k =0 k ! k !( k − k )! . . . k | P |− X k | P |− =0 k | P |− ! k | P |− !( k | P |− − k | P |− )! G | X | + k − k (0 , X ,s + n − k + 1 , . . . , s + n − k ) . . . G | X | P | | + k | P |− (0 , X | P | , s + n − k | P |− + 1 , . . . , s + n ) . Consequently the solution expansion of the nonlinear quantum BBGKY hierarchy has thefollowing structure G s ( t, Y ) = ∞ X n =0 n ! Tr s +1 ,...,s + n U n ( t ; { Y } , s + 1 , . . . , s + n | G (0)) , s ≥ , (44)volution of marginal correlation operators 17where we introduce the notion of the (1 + n ) th -order reduced cumulant U n ( t ) of nonlineargroups of operators (7) U n ( t ; { Y } , s + 1 , . . . , s + n | G (0)) . = (45) . = n X k =0 ( − k n ! k !( n − k )! X P: ( θ ( { ,...,s } ) ,s +1 ,...,s + n − k )= S i X i A | P | (cid:0) t, { X } , . . . , { X | P | } (cid:1) ×× k X k =0 k ! k !( k − k )! . . . k | P |− X k | P |− =0 k | P |− ! k | P |− !( k | P |− − k | P |− )! G | X | + k − k (0 , X ,s + n − k + 1 , . . . , s + n − k ) . . . G | X | P | | + k | P |− (0 , X | P | , s + n − k | P |− + 1 , . . . , s + n ) . We give simplest examples of reduced nonlinear cumulants (45): U ( t ; { Y } | G (0)) = G ( t ; Y | G (0)) == X P: (1 ,...,s )= S i X i A | P | (cid:0) t, { X } , . . . , { X | P | } (cid:1) Y X i ⊂ P G | X i | (0 , X i ) ,U ( t ; { Y } , s + 1 | G (0)) = X P: ( Y,s +1)= S i X i A | P | (cid:0) t, { X } , . . . , { X | P | } (cid:1) Y X i ⊂ P G | X i | (0 , X i ) −− X P: (1 ,...,s )= S i X i A | P | (cid:0) t, { X } , . . . , { X | P | } (cid:1) | P | X j =1 Y X i ⊂ P ,X i = X j G | X i | (0 , X i ) G | X j | +1 (0 , X j , s + 1) . We remark that in case of solution expansion (22) of the quantum BBGKY hierarchy, ananalog of reduced cumulant (45) is the reduced cumulant of groups of operators (6) defined byformula [14] U n ( t ; { Y } , s + 1 , . . . , s + n ) . = n X k =0 ( − k n ! k !( n − k )! G s + n − k ( − t ) . We indicate some properties of reduced nonlinear cumulants (45) of groups of operators (7).According to formula (44) and properties of cumulants (8), namely A n (0) = Iδ n, , the followingequality holds U n (0; { Y } , s + 1 , . . . , s + n | G (0)) == n X k =0 ( − k n ! k !( n − k )! A (cid:0) , { , . . . , s + n − k } (cid:1) G s + n (0 , , . . . , s + n ) == G s + n (0 , , . . . , s + n ) δ n, , and hence the marginal correlation operators determined by series (44) satisfy initial data (25).volution of marginal correlation operators 18In case of n = 0 for f ∈ L ( F H ) in the sense of the norm convergence of the space L ( H s )the infinitesimal generator of first-order reduced cumulant (45) coincides with generator (3) ofthe von Neumann hierarchy (1)lim t → t ( U ( t ; { Y } | f ) − f s ( Y )) = N ( Y | f ) , s ≥ , where the operator N ( Y | f ) is defined by formula (3). In case of n = 1 for second-orderreduced cumulant (45) in the same sense we obtain the following equalityTr s +1 lim t → t U ( t ; { Y } , s + 1 | f ) = X i ∈ Y Tr s +1 ( −N int ( i, s + 1)) (cid:0) f s +1 ( t, Y, s + 1) ++ X P : ( Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X f | X | ( t, X ) f | X | ( t, X ) (cid:1) , where notations are used as above for hierarchy (24), and for n ≥ s +1 ,...,s + n lim t → t U n ( t ; { Y } , s + 1 , . . . , s + n | f ) = 0 . In case of initial data satisfying a chaos property, i.e. G (1) (0) ≡ (0 , G (0 , , , . . . ), for the(1 + n ) th -order reduced cumulant we have U n ( t ; { Y } , s + 1 , . . . , s + n | G (1) (0)) = A s + n (cid:0) t, , . . . , s + n (cid:1) s + n Y i =1 G (0 , i ) , i.e. the only summand that gives contribution to the result is the one with k = 0 and | P | = s + n ,since otherwise there is at least one operator G s (0) with s ≥ n ) th -order reduced cumulant (45) the following inequality holds (cid:13)(cid:13) U n ( t ; { Y } , s + 1 , . . . , s + n | f ) (cid:13)(cid:13) L ( H s + n ) ≤ n ! s !(2 e ) s + n c s + n , (46)where c ≡ max P : (1 , . . . , s + n − k ) = S i X i max k, k , . . . , k | P |− ∈∈ ( s + n − k + 1 , . . . , s + n ) (cid:0) k f | X | + k − k k L ( H | X | + k − k ) , . . .. . . , k f | X | P | | + k | P |− k L ( H | X | P || + k | P |− ) (cid:1) .To prove this inequality we first remark that for cumulant (8) the following estimate holds k A | P | ( t, { X } , . . . , { X | P | } ) f n k L ( H n ) ≤ | P | ! e | P | k f n k L ( H n ) . (47)Indeed, we have k A | P | ( t, { X } , . . . , { X | P | } ) f n k L ( H n ) ≤≤ X P ′ : ( { X } ,..., { X | P | } )= S k Z k ( | P ′ | − k Y Z k ⊂ P ′ G | θ ( Z k ) | ( − t, θ ( Z k )) f n k L ( H n ) == k f n k L ( H n ) | P | X l =1 s( | P | , l )( l − , volution of marginal correlation operators 19where s( | P | , l ) are the Stirling numbers of second kind and we use the isometric property of thegroups G n ( − t ) , n ≥ 1. Estimate (47) holds as a consequence of the inequality | P | X l =1 s( | P | , l )( l − ≤ | P | ! e | P | . Then owing to estimate (47), for the (1 + n ) th -order reduced cumulant (45) we have (cid:13)(cid:13) U n ( t ; { Y } , s + 1 , . . . , s + n | f ) (cid:13)(cid:13) L ( H s + n ) ≤≤ n X k =0 n ! k !( n − k )! X P: (1 ,...,s + n − k )= S i X i | P | ! e | P | k X k =0 k ! k !( k − k )! . . . k | P |− X k | P |− =0 k | P |− ! k | P |− !( k | P |− − k | P |− )! k f | X | + k − k k L ( H | X | + k − k ) . . . k f | X | P | | + k | P |− k L ( H | X | P || + k | P |− ) ≤≤ n X k =0 n !( n − k )! X P: (1 ,...,s + n − k )= S i X i | P | ! e | P |− c | P | . As result of using of the definition of the Stirling numbers of second kind s( s + n − k, l ) andthe inequalities n X k =0 n !( n − k )! X P: (1 ,...,s + n − k )= S i X i | P | ! e | P |− = n X k =0 n !( n − k )! s + n − k X l =1 s( s + n − k, l ) l ! e l − ≤≤ n X k =0 n !( s + n − k )!( n − k )! e s + n − k ) ≤ n ! s !(2 e ) s + n , we obtain estimate (46).Thus, according to estimate (46), for initial data from the space L ( H n ) series (44) convergesunder the condition: c ≡ max n ≥ (cid:13)(cid:13) G n (0) (cid:13)(cid:13) L ( H n ) < (2 e ) − , and the following inequality holds (cid:13)(cid:13) G s ( t ) (cid:13)(cid:13) L ( H s ) ≤ s !(2 e c ) s ∞ X n =0 (2 e ) n c n . (48)A solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginalcorrelation operators (24) is determined by the following one-parametric mapping R ∋ t → U ( t | f ) = e a G ( t | e − a f ) , (49)which is defined on the space L ( F H ) owing to estimate (48), and has the group property U (cid:0) t | U ( t | f ) (cid:1) = U (cid:0) t | U ( t | f ) (cid:1) = U (cid:0) t + t | f (cid:1) . Indeed, according to definition (41) and taking into consideration the group property of themapping G ( t | · ), we obtain U (cid:0) t + t | f (cid:1) = e a G ( t + t | e − a f ) = e a G ( t | G ( t | e − a f ) == e a G ( t | e − a e a G ( t | e − a f ) = e a G ( t | e − a U (cid:0) t | f (cid:1) ) = U (cid:0) t | U ( t | f ) (cid:1) . volution of marginal correlation operators 20To construct the generator of the strong continuous group U ( t ; Y | · ) we differentiate it inthe sense of the norm convergence on the space L ( H s ) ddt U ( t ; Y | f ) | t =0 = ddt ( e a G ( t | e − a f )) s ( Y ) | t =0 == ∞ X n =0 n ! Tr s +1 ,...,s + n N ( X | G ( t | e − a f )) | t =0 = ( e a N ( · | e − a f )) s ( Y ) , where N ( · | f ) is a generator of the von Neumann hierarchy (1) defined by formula (3) on thesubspaces L ( H s ) ⊂ L ( H s ) , s ≥ 1, or in the componentwise form( e a N ( · | e − a f )) s ( Y ) = N ( Y | f ) + Tr s +1 X i ∈ Y ( −N int ( i, s + 1)) (cid:0) f s +1 ( Y, s + 1) + (50)+ X P : ( Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X f | X | ( X ) f | X | ( X ) (cid:1) , where we use notations as above for formula (24), and transformations similar to equalities (27)and (31) have been applied.Indeed, to set down a generator of mapping (49) in componentwise form we observe thataccording to definitions (41) and (3), the following equality holds( e a N ( · | e − a f )) s ( Y ) = ∞ X n =0 n ! Tr s +1 ,...,s + n (cid:0) − N s + n ( X )( e − a f ) s + n ( X ) ++ X P: X = X S X X i ∈ X X i ∈ X ( −N int ( i , i ))( e − a f ) | X | ( X )( e − a f ) | X | ( X ) (cid:1) . Then in view of formulas (41) and (43) we have( e − a f ) | X | ( X )( e − a f ) | X | ( X ) = ∞ X k =0 ( − k k ! Tr s + n +1 ,...,s + n + k k X k =0 k ! k !( k − k )! f | X | + k − k ( X ,s + n + 1 , . . . , s + n + k − k ) f | X | + k ( X , s + n + k − k + 1 , . . . , s + n + k ) , and as a result we obtain( e a N ( · | e − a f )) s (1 , . . . , s ) == ∞ X n =0 n ! Tr s +1 ,...,s + n n X k =0 ( − k n ! k !( n − k )! (cid:0) − N s + n − k (1 , . . . , s + n − k ) f s + n ( X ) ++ X P: (1 ,...,s + n − k )= X S X X i ∈ X X i ∈ X ( −N int ( i , i )) k X k =0 k ! k !( k − k )! ×× f | X | + k − k ( X , s + n − k + 1 , . . . , s + n − k ) f | X | + k ( X , s + n − k + 1 , . . . , s + n ) (cid:1) . Therefore the first term of this series is generator (3) of the von Neumann hierarchy I ≡ ( −N s ) f s ( Y ) + X P: Y = X S X X i ∈ X X i ∈ X ( −N int ( i , i )) f | X | ( X ) f | X | ( X ) , volution of marginal correlation operators 21and, as stated above, it coincides with the generator of the first order cumulant (45). For thesecond term of series (50) we have I ≡ Tr s +1 (cid:0) ( −N s +1 ( Y, s + 1)) f s +1 − ( −N s ( Y )) f s +1 ++ X P: ( Y,s +1)= X S X X i ∈ X X i ∈ X ( −N int ( i , i )) f | X | ( X ) f | X | ( X ) −− X P: Y = X S X X i ∈ X X i ∈ X ( −N int ( i , i ))( f | X | +1 ( X , s + 1) f | X | ( X ) ++ f | X | ( X ) f | X | +1 ( X , s + 1)) (cid:1) . Taking into account equality (28) in case of the set ( Y, s + 1), it holds X P: ( Y,s +1)= X S X X i ∈ X X i ∈ X ( −N int ( i , i )) f | X | ( X ) f | X | ( X ) == X P: ( Y,s +1)= X S X X i ∈ X T Y X i ∈ X T Y ( −N int ( i , i )) f | X | ( X ) f | X | ( X ) ++ X P : ( Y, s + 1) = X S X ,s + 1 ∈ X X i ∈ X T Y ( −N int ( i , s + 1)) f | X | ( X ) f | X | ( X ) == X P: Y = Y S Y X i ∈ Y X i ∈ Y ( −N int ( i , i ))( f | Y | +1 ( Y , s + 1) f | Y | ( Y ) ++ f | Y | ( Y ) f | Y | +1 ( Y , s + 1)) + X i ∈ Y ( −N int ( i, s + 1)) X P : ( Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X f | X | ( X ) f | X | ( X ) , where P P : ( Y, s + 1) = X S X ,s + 1 ∈ X is the sum over all possible partitions of the set ( Y, s + 1) intotwo mutually disjoint subsets X and X such that ( s + 1) th particle index belongs to set X .As a result we obtain I = Tr s +1 (cid:0) X i ∈ Y ( −N int ( i, s + 1)) f s +1 ++ X i ∈ Y ( −N int ( i, s + 1)) X P : ( Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X f | X | ( X ) f | X | ( X ) (cid:1) , i.e. this term coincides with the generator of second-order cumulant (45).In case of a two-body interaction potential other terms of series (50) are identically equal tozero. This statement is a consequence of the structure of expansion (50) and of the fact thatits third term equals zero. Indeed as a result of regrouping terms in the expression of the thirdvolution of marginal correlation operators 22term we obtain I = 12! Tr s +1 ,s +2 (cid:0) ( −N s +2 ( Y, s + 1 , s + 2) − −N s +1 ( Y, s + 1)) + ( −N s ( Y ))) f s +2 ++ X P: ( Y,s +1 ,s +2)= X S X X i ∈ X T Y X i ∈ X T Y ( −N int ( i , i )) f | X | ( X ) f | X | ( X ) −− X P: ( Y,s +1)= X S X X i ∈ X T Y X i ∈ X T Y ( −N int ( i , i ))( f | X | +1 ( X , s + 1) f | X | ( X ) ++ f | X | ( X ) f | X | +1 ( X , s + 1)) ++ X P: Y = X S X X i ∈ X T Y X i ∈ X T Y ( −N int ( i , i ))( f | X | +2 ( X , s + 1 , s + 2) f | X | ( X ) ++ f | X | ( X ) f | X | +2 ( X , s + 1 , s + 2) + 2! f | X | +1 ( X , s + 1) f | X | +1 ( X , s + 2)) (cid:1) = 0 . Thus, we conclude the validity of formula (50) in case of a two-body interaction potentialwhich describes the structure of the infinitesimal generator of mapping (49) in the general case. For an abstract initial-value problem of hierarchy (24) in the space L ( F H ) the following theoremis true. Theorem 1. If max n ≥ (cid:13)(cid:13) G n (0) (cid:13)(cid:13) L ( H n ) < (2 e ) − , then in case of bounded interaction potentialsfor t ∈ R a solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy (24)-(25) is determined by expansion (44). If G n (0) ∈ L ( H n ) ⊂ L ( H n ) , it is a strong (classical)solution and for arbitrary initial data G n (0) ∈ L ( H n ) it is a weak (generalized) solution.Proof. It will be recalled that according to estimate (48), series (44) converges under the con-dition: max n ≥ (cid:13)(cid:13) G n (0) (cid:13)(cid:13) L ( H n ) < (2 e ) − . To prove that a strong solution of the nonlinearBBGKY hierarchy (24) is given by expansion (44) we first differentiate it over time variablein the sense of a pointwise convergence on the space L ( H n ), i.e. for every function from thedomain ψ s ∈ D ( H s ) ⊂ H s . Taking into account the group property of mapping (49) generatedby expansion (44) and properties of reduced nonlinear cumulants (45) of groups of operators(7), for G n (0) ∈ L ( H n ) ⊂ L ( H n ) , n ≥ 1, we obtainlim ∆ t → t (cid:0) ∞ X n =0 n ! Tr s +1 ,...,s + n ( U n (∆ t ; { Y } , X \ Y | G ( t )) − G s ( t, Y ) (cid:1) ψ s = (51)= N ( Y | G ( t )) ψ s + Tr s +1 X i ∈ Y ( −N int ( i, s + 1)) (cid:0) G s +1 ( t, Y, s + 1) ++ X P : ( Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X G | X | ( t, X ) G | X | ( t, X ) (cid:1) ψ s , where expansion (44) is denoted by the symbol G s ( t, Y ). Since G n (0) ∈ L ( H n ) ⊂ L ( H n ) , n ≥ 1, then using equality (51), in the sense of the norm convergence in L ( H n ) we finally establishvolution of marginal correlation operators 23the validity of the equalitylim ∆ t → Tr ,...,s (cid:12)(cid:12) t (cid:0) ∞ X n =0 n ! Tr s +1 ,...,s + n U n ( t + ∆ t ; { Y } , X \ Y | G (0)) −− ∞ X n =0 n ! Tr s +1 ,...,s + n U n ( t ; { Y } , X \ Y | G (0)) (cid:1) −− (cid:16) N ( Y | G ( t )) + Tr s +1 X i ∈ Y ( −N int ( i, s + 1)) (cid:0) G s +1 ( t, Y, s + 1) ++ X P : ( Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X G | X | ( t, X ) G | X | ( t, X ) (cid:1)(cid:17)(cid:12)(cid:12) = 0 , which means that a strong solution of the nonlinear BBGKY hierarchy (24) is given by expan-sion (44) in case of initial data from the subspaces L ( H n ) ⊂ L ( H n ) , n ≥ G n (0) ∈ L ( H n ) , n ≥ f, G ( t )) . = ∞ X s =0 s ! Tr ,...,s f s ( Y ) G s ( t, Y ) , (52)where f = (0 , f , . . . , f n , . . . ) ∈ L ( F H ) is the finite sequence of degenerate bounded operatorswith infinitely times differentiable kernels with compact supports. For G n (0) ∈ L ( H n ) and f n ∈ L ( H n ) functional (52) exists.We transform functional (52) to the following form( f, G ( t )) = ( f, e a G ( t | e − a G (0))) = ( e a + f, G ( t | e − a G (0))) , (53)where the operator a is defined by (41) and on f s ∈ L ( H s ) the operator a + is defined by theformula (an analog of the creation operator)( a + f ) s ( Y ) . = s X j =1 f s − ( Y \ ( j )) . To differentiate obtained functional (53) with respect to the time variable we use the corre-sponding result [12] of the differentiation of group (7) of the von Neumann hierarchy (1). As aresult we derive that ddt ( f, G ( t )) = ∞ X s =0 s ! Tr ,...,s (cid:0) N s ( Y )( e a + f ) s ( Y ) G ( t, Y | e − a G (0)) ++ X P: Y = X S X X i ∈ X X i ∈ X N int ( i , i )( e a + f ) s ( Y ) G ( t, X | e − a G (0)) G ( t, X | e − a G (0)) (cid:1) . volution of marginal correlation operators 24Taking into account the structure of expansion (44) of the nonlinear quantum BBGKY hierarchysolution, for f s ∈ L ( H s ) , s ≥ 1, the following equality holds ddt ( f, G ( t )) = ∞ X s =0 s ! Tr ,...,s (cid:16)(cid:0) N s ( Y ) f s ( Y ) + s X i, j = 1 i = j N int ( i, j ) f s − ( Y \ ( j )) (cid:1) G s ( t, Y ) ++ X P: Y = X S X X i ∈ X X i ∈ X X i ∈ X N int ( i , i ) f s ( Y ) G | X | ( t, X ) G | X | ( t, X ) ++ s X i, j = 1 i = j N int ( i, j ) f s − ( Y \ ( j )) X P : Y = X S X i ∈ X ; j ∈ X G | X | ( t, X ) G | X | ( t, X ) (cid:1)(cid:17) . This equation means that in case of arbitrary initial data G n (0) ∈ L ( H n ) , n ≥ 1, a weaksolution of the initial-value problem (24)-(25) is given by expansion (44). We give comments on the mean field asymptotic behavior [23] of constructed solution (44).Let us suppose the existence of the mean field limit of initial state in the following senselim ǫ → (cid:13)(cid:13) ǫ s G n (0) − g n (0) (cid:13)(cid:13) L ( H n ) = 0 , n ≥ . (54)Then there exists the mean field limit g s ( t, , . . . , s ) , s ≥ 1, of marginal correlation operators(44) lim ǫ → (cid:13)(cid:13) ǫ s G s ( t ) − g s ( t ) (cid:13)(cid:13) L ( H s ) = 0 , s ≥ , which is governed by the nonlinear Vlasov quantum hierarchy ddt g s ( t, Y ) = X i ∈ Y ( −N ( i )) g s ( t, Y ) + Tr s +1 X i ∈ Y ( −N int ( i, s + 1)) × (55) × (cid:0) g s +1 ( t, Y, s + 1) + X P : ( Y, s + 1) = X S X ,i ∈ X ; s + 1 ∈ X g | X | ( t, X ) g | X | ( t, X ) (cid:1) , s ≥ , where notations similar to hierarhy (24) are used.If initial data satisfies chaos property, then we establishlim ǫ → (cid:13)(cid:13) ǫ s G s ( t ) (cid:13)(cid:13) L ( H s ) = 0 , s ≥ , (56)since solution expansions (37) for marginal correlation operators are defined by the ( s + n ) th -order cumulants as contrasted to solution expansions (22) for marginal density operators de-fined by the (1 + n ) th -order cumulants and in the consequence of the following formula on anasymptotic perturbation of cumulants of groups of operators [24]lim ǫ → (cid:13)(cid:13) ǫ n A s + n ( t, , . . . , s + n ) f s + n (cid:13)(cid:13) L ( H s + n ) = 0 . volution of marginal correlation operators 25In case of s = 1 provided that (54) we havelim ǫ → (cid:13)(cid:13) ǫG ( t ) − g ( t ) (cid:13)(cid:13) L ( H ) = 0 , where for finite time interval the limit one-particle marginal correlation operator g ( t, 1) is givenby the norm convergent on the space L ( H ) series g ( t, 1) = (57)= ∞ X n =0 t Z dt . . . t n − Z dt n Tr ,...,n +1 G ( − t + t , −N int (1 , Y j =1 G ( − t + t , j ) . . .. . . n Y i n =1 G ( − t n + t n , i n ) n X k n =1 ( −N int ( k n , n + 1)) n +1 Y j n =1 G ( − t n , j n ) n +1 Y i =1 g (0 , i ) , which obviously coincides with iteration series of the Vlasov quantum kinetic equation [10]. Forbounded interaction potential (1) series (57) is norm convergent on the space L ( H ) under thecondition: t < t ≡ (cid:0) k Φ k L ( H ) k g (0) k L ( H ) (cid:1) − .In view of the validity of limit (56) from the Vlasov nonlinear quantum hierarchy (55) we alsoconclude that limit one-particle marginal correlation operator (57) is governed by the Cauchyproblem of the Vlasov quantum kinetic equation ddt g ( t, 1) = −N (1) g ( t, 1) + Tr ( −N int (1 , g ( t, g ( t, , (58)and consequently for pure states we derive the Hartree equation.Thus, the nonlinear Vlasov quantum hierarchy (55) describes the evolution of initial corre-lations. In the paper the origin of the microscopic description of non-equilibrium correlations of quantummany-particle systems obeying the Maxwell-Boltzmann statistics has been considered. Thenonlinear quantum BBGKY hierarchy (24) for marginal correlation operators was introduced.It gives an alternative approach to the description of the state evolution of quantum infinite-particle systems in comparison with quantum BBGKY hierarchy for marginal density operators[13, 14]. The evolution of both finitely and infinitely many quantum particles is described byinitial-value problem of the nonlinear quantum BBGKY hierarchy (24) and in case of finitelymany particles the nonlinear quantum BBGKY hierarchy is equivalent to the von Neumannhierarchy (1).A nonperturbative solution of the nonlinear quantum BBGKY hierarchy is constructed inthe form of expansion (44) over particle clusters which evolution is governed by corresponding-order cumulant (45) of the nonlinear groups of operators generated by solution (7) of the vonNeumann hierarchy (1). We established that in case of absence of correlations at initial time thecorrelations generated by the dynamics of quantum many-particle systems (37) are completelydetermined by cumulants (8) of groups of operators (6).volution of marginal correlation operators 26Thus, the cumulant structure of solution (7) of the von Neumann hierarchy (1) induces thecumulant structure of solution expansion (44) of initial-value problem of the nonlinear quantumBBGKY hierarchy (24).We emphasize that intensional Banach spaces for the description of states of infinite-particlesystems, which are suitable for the description of the kinetic evolution or equilibrium states, aredifferent from the exploit spaces [14], [19]. Therefore marginal correlation operators from thespace of trace-class operators describe finitely many quantum particles. In order to describethe evolution of infinitely many particles we have to construct solutions for initial data frommore general Banach spaces than the space of sequences of trace-class operators. For example,it can be the space of sequences of bounded translation invariant operators which contains themarginal density operators of equilibrium states [25]. In that case every term of the solutionexpansion of the nonlinear quantum BBGKY hierarchy (44) contains the divergent traces, whichcan be renormalized due to the cumulant structure of solution expansion (45).The mean field asymptotic behavior of constructed solution (44) is governed by the nonlinearVlasov quantum hierarchy (55). In such approximation this hierarchy describes the evolutionof initial correlations and in case of its absence the nonlinear Vlasov hierarchy (55) is equivalentto the Vlasov quantum kinetic equation (58).Following to the paper [12] the obtained results can be also generalized on many-particlesystems obeying the Fermi-Dirac and Bose-Einstein statistics (32). References [1] A. Arnold, Mathematical properties of quantum evolution equations . Lecture Notes in Math. , (2008), 45-110.[2] C. Bardos, B. Ducomet, F. Golse, A.D. Gottlieb and N.J. Mauser, The TDHF approx-imation for Hamiltonians with m-particle interaction potentials . Commun. Math. Sci. ,(2007), 1-9.[3] L. Erd¨os, B. Schlein and H.-T. Yau, Derivation of the cubic nonlinear Schr¨odinger equationfrom quantum dynamics of many-body systems . Invent. Math. , (3), (2007), 515-614.[4] L. Erd¨os, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii Equation for theDynamics of Bose-Einstein Condensate . Ann. of Math., , (2010), 291-370.[5] J. Fr¨ohlich, S. Graffi and S. Schwarz, Mean-field and classical limit of many-bodySchr¨odinger dynamics for bosons . Commun. Math. Phys. , (2007), 681-697.[6] A. Michelangeli, Role of scaling limits in the rigorous analysis of Bose-Einstein condensa-tion . J. Math. Phys. , (2007), 102102.[7] F. Pezzotti and M. Pulvirenti, Mean-field limit and semiclassical expansion of quantumparticle system . Ann. Henri Poincar´e. , (2009), 145-187.[8] L. Saint-Raymond, Kinetic models for superfluids: a review of mathematical results . C. R.Physique, , (2004), 6575.volution of marginal correlation operators 27[9] V.I. Gerasimenko and Zh.A. Tsvir, A description of the evolution of quantum states bymeans of the kinetic equation . J. Phys. A: Math. Theor. , (48), (2010), 485203.[10] V.I. Gerasimenko, Heisenberg picture of quantum kinetic evolution in mean-field limit ,Kinet. Relat. Models, , (1), (2011), 385-399.[11] V.I. Gerasimenko and V.O. Shtyk, Evolution of correlations of quantum many-particlesystems . J. Stat. Mech. Theory Exp. , (2008), P03007, 24p.[12] V.I. Gerasimenko and D.O. Polishchuk, Dynamics of correlations of Bose and Fermi par-ticles . Math. Meth. Appl. Sci. , (1), (2011), 76-93.[13] M.M. Bogolyubov, Lectures on Quantum Statistics. Problems of Statistical Mechanics ofQuantum Systems . Kyiv, 1949 (in Ukrainian).[14] D.Ya. Petrina, Mathematical Foundations of Quantum Statistical Mechanics. ContinuousSystems . Kluwer, 1995.[15] R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science andTechnology . , Springer-Verlag, 1992.[16] O. Bratelli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics . ,Springer, 1997.[17] V.I. Gerasimenko, Groups of operators for evolution equations of quantum many-particlesystems . Oper. Theory Adv. Appl. , (2009), 341-355.[18] V.I. Gerasimenko and D.Ya. Petrina, A mathematical description of the evolution of thestates of infinite systems of classical statistical mechanics . Russ. Math. Surv., , (5),(1983), 3-58.[19] C. Cercignani, V.I. Gerasimenko and D.Ya. Petrina, Many-Particle Dynamics and KineticEquations . Kluwer, 1997.[20] D.O. Polishchuk, BBGKY hierarchy and dynamics of correlations . Ukrainian J. Phys. ,(5), (2010), 593-598.[21] J. Yvon, La theorie statistique des fluides et l’equation d’etat . Actualites Scientifiques etIndustrielles, , (203). Paris: Hermann, 1935.[22] M.S. Green, Boltzmann equation from the statistical mechanical point of view . J. Chem.Phys. , (5), (1956), 836-855.[23] H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits . Rev. ModernPhys. , (1980), 569-615.[24] T. Kato, Perturbation Theory for Linear Operators . Springer-Verlag, 1995.[25] J. Ginibre, Some applications of functional integrations in statistical mechanics . (in