Mean Field Asymptotic Behavior of Quantum Particles with Initial Correlations
aa r X i v : . [ m a t h - ph ] J u l Mean Field Asymptotic Behavior ofQuantum Particles with Initial Correlations
V.I. Gerasimenko ∗ ∗ Institute of Mathematics of NAS of Ukraine,3, Tereshchenkivs’ka Str.,01601, Kyiv-4, Ukraine
Abstract.
In the paper we consider the problem of the rigorous description of thekinetic evolution in the presence of initial correlations of quantum large particle sys-tems. One of the developed approaches consists in the description of the evolutionof quantum many-particle systems within the framework of marginal observables inmean field scaling limit. Another method based on the possibility to describe theevolution of states within the framework of a one-particle marginal density opera-tor governed by the generalized quantum kinetic equation in case of initial statesspecified by a one-particle marginal density operator and correlation operators.
Key words: quantum kinetic equation; quantum Vlasov equation; dual quantumVlasov hierarchy; mean field scaling limit; correlation operator. E-mail: [email protected] uantum kinetic equations with correlations 2
Contents
As is known the collective behavior of quantum many-particle systems can be effectively describedwithin the framework of a one-particle marginal density operator governed by the kinetic equationin a suitable scaling limit of underlying dynamics. At present the considerable advances in therigorous derivation of the quantum kinetic equations in the mean (self-consistent) field scalinglimit is observed [1]- [6]. In particular, the nonlinear Schr¨odinger equation [3]- [10] and the Gross–Pitaevskii equation [7]- [15] was justified.The conventional approach to this problem is based on the consideration of an asymptotic be-havior of a solution of the quantum BBGKY hierarchy for marginal density operators constructedwithin the framework of the theory of perturbations in case of initial data specified by one-particlemarginal density operators without correlations, i.e. such that satisfy a chaos condition [16], [17].In paper [18] it was developed more general method of the derivation of the quantum kineticequations. By means of a non-perturbative solution of the quantum BBGKY hierarchy constructedin [19] it was established that, if initial data is completely specified by a one-particle marginaldensity operator, then all possible states of many-particle systems at arbitrary moment of time canuantum kinetic equations with correlations 3be described within the framework of a one-particle density operator governed by the generalizedquantum kinetic equation (see also [20]). Then the actual quantum kinetic equations can bederived from the generalized quantum kinetic equation in appropriate scaling limits, for example,in a mean field limit [21].Another approach to the description of the many-particle evolution is given within the frame-work of marginal observables governed by the dual quantum BBGKY hierarchy [22]. In paper [23]a rigorous formalism for the description of the kinetic evolution of observables of quantum particlesin a mean field scaling limit was developed.In this paper we consider the problem of the rigorous description of the kinetic evolution inthe presence of initial correlations of quantum particles. Such initial states are typical for thecondensed states of quantum gases in contrast to the gaseous state. For example, the equilibriumstate of the Bose condensate satisfies the weakening of correlation condition specified by corre-lations of the condensed state [24]. One more example is the influence of initial correlations onultrafast relaxation processes in plasmas [25], [26].Thus, our goal consists in the rigorous derivation of the quantum kinetic equations in thepresence of initial correlations of quantum large particle systems.We outline the structure of the paper. In section 2, we establish the mean field asymptoticbehavior of marginal observables governed by the dual quantum BBGKY hierarchy. The limitdynamics is described by the set of recurrence evolution equations, namely by the dual quan-tum Vlasov hierarchy. Furthermore, the links of the dual quantum Vlasov hierarchy for the limitmarginal observables and the quantum Vlasov-type kinetic equation with initial correlations areestablished. In section 3, we consider the relationships of dynamics described by marginal ob-servables and within the framework of a one-particle marginal density operator governed by thegeneralized quantum kinetic equation in the presence of initial correlations. In section 4, wedevelop one more approach to the description of the quantum kinetic evolution with initial cor-relations in the mean field limit. We prove that a solution of the generalized quantum kineticequation with initial correlations is governed by the quantum Vlasov-type equation with initialcorrelations. The property of the propagation of initial correlations is also established. Finally, insection 5, we conclude with some perspectives for future research.
The kinetic evolution of many-particle systems can be described within the framework of ob-servables. We consider this problem on an example of the mean field asymptotic behavior ofa non-perturbative solution of the dual quantum BBGKY hierarchy for marginal observables.Moreover, we establish the links of the dual quantum Vlasov hierarchy for the limit marginalobservables with the quantum Vlasov-type kinetic equation in the presence of initial correlations.
We shall consider a quantum system of a non-fixed (i.e. arbitrary but finite) number of identical(spinless) particles obeying Maxwell–Boltzmann statistics in the space R . We will use units where h = 2 π ~ = 1 is a Planck constant, and m = 1 is the mass of particles.uantum kinetic equations with correlations 4Let the space H be a one-particle Hilbert space, then the n -particle space H n = H ⊗ n is a tensorproduct of n Hilbert spaces H . We adopt the usual convention that H ⊗ = C . The Fock spaceover the Hilbert space H we denote by F H = L ∞ n =0 H n .The Hamiltonian H n of a system of n particles is a self-adjoint operator with the domain D ( H n ) ⊂ H n H n = n X i =1 K ( i ) + ǫ n X i K ( i ) acts on functions ψ n , that belong to the subspace L ( R n ) ⊂ D ( H n ) ⊂ L ( R n ) of infinitely differentiable functionswith compact supports, according to the formula: K ( i ) ψ n = − ∆ q i ψ n . Correspondingly, we have:Φ( i , i ) ψ n = Φ( q i , q i ) ψ n , and we assume that the function Φ( q i , q i ) is symmetric with respectto permutations of its arguments, translation-invariant and bounded function.Let a sequence g = ( g , g , . . . , g n , . . . ) be an infinite sequence of self-adjoint bounded operators g n defined on the Fock space F H . An operator g n defined on the n -particle Hilbert space H n = H ⊗ n will be also denoted by the symbol g n (1 , . . . , n ). Let the space L ( F H ) be the space of sequences g = ( g , g , . . . , g n , . . . ) of bounded operators g n defined on the Hilbert space H n that satisfysymmetry condition: g n (1 , . . . , n ) = g n ( i , . . . , i n ), for arbitrary ( i , . . . , i n ) ∈ (1 , . . . , n ), equippedwith the operator norm k . k L ( H n ) . We will also consider a more general space L γ ( F H ) with thenorm (cid:13)(cid:13) g (cid:13)(cid:13) L γ ( F H ) . = max n ≥ γ n n ! (cid:13)(cid:13) g n (cid:13)(cid:13) L ( H n ) , where 0 < γ <
1. We denote by L γ , ( F H ) ⊂ L γ ( F H ) the everywhere dense set in the space L γ ( F H )of finite sequences of degenerate operators with infinitely differentiable kernels with compactsupports.For g n ∈ L ( H n ) it is defined the one-parameter mapping R ∋ t
7→ G n ( t ) g n . = e itH n g n e − itH n , (2)where the Hamilton operator H n has the structure (1). On the space L ( H n ) one-parametermapping (2) is an isometric ∗ -weak continuous group of operators. The infinitesimal generator N n of this group of operators is a closed operator for the ∗ -weak topology, and on its domain ofthe definition D ( N n ) ⊂ L ( H n ) it is defined in the sense of the ∗ -weak convergence of the space L ( H n ) by the operator w ∗ − lim t → t (cid:0) G n ( t ) g n − g n (cid:1) = − i ( g n H n − H n g n ) . = N n g n , (3)where H n is the Hamiltonian (1) and the operator N n g n defined on the domain D ( H n ) ⊂ H n hasthe structure N n = n X j =1 N ( j ) + ǫ n X j 0) cumulant of groups of operators (2) as follows [19] A n ( t, { Y \ X } , X ) . = X P: ( { Y \ X } , X )= S i X i ( − | P |− ( | P | − Y X i ⊂ P G | θ ( X i ) | ( t, θ ( X i )) , (6)where the symbol P P means the sum over all possible partitions P of the set ( { Y \ X } , j , . . . , j n )into | P | nonempty mutually disjoint subsets X i ⊂ ( { Y \ X } , X ), and θ ( · ) is the declusterizationmapping defined as follows: θ ( { Y \ X } , X ) = Y . For example, A ( t, { Y } ) = G s ( t, Y ) , A ( t, { Y \ ( j ) } , j ) = G s ( t, Y ) − G s − ( t, Y \ ( j )) G ( t, j ) . In terms of observables the evolution of quantum many-particle systems is described by thesequence B ( t ) = ( B , B ( t, , . . . , B s ( t, , . . . , s ) , . . . ) of marginal observables (or s -particle ob-servables) B s ( t, , . . . , s ) , s ≥ , determined by the following expansions [22]: B s ( t, Y ) = s X n =0 n ! s X j = ... = j n =1 A n ( t, { Y \ X } , X ) B ,ǫs − n ( Y \ X ) , s ≥ , (7)where B (0) = ( B , B ,ǫ (1) , . . . , B ,ǫs (1 , . . . , s ) , . . . ) ∈ L γ ( F H ) is a sequence of initial marginal ob-servables, and the generating operator A n ( t ) of expansion (7) is the (1 + n ) th -order cumulant ofgroups of operators (2) defined by expansion (6). The simplest examples of marginal observables(7) are given by the expressions: B ( t, 1) = A ( t, B ,ǫ (1) ,B ( t, , 2) = A ( t, { , } ) B ,ǫ (1 , 2) + A ( t, , B ,ǫ (1) + B ,ǫ (2)) . If γ < e − , for the sequence of operators (7) the following estimate is true (cid:13)(cid:13) B ( t ) (cid:13)(cid:13) L γ ( F H ) ≤ e (1 − γe ) − (cid:13)(cid:13) B (0) (cid:13)(cid:13) L γ ( F H ) . uantum kinetic equations with correlations 6We note that a sequence of marginal observables (7) is the non-perturbative solution of recur-rence evolution equations known as the dual quantum BBGKY hierarchy [22]: ∂∂t B s ( t, Y ) = (cid:0) s X j =1 N ( j ) + s X j Theorem 1. Let for B ,ǫn ∈ L ( H n ) , n ≥ , in the sense of the ∗ -weak convergence on the space L ( H s ) it holds: w ∗ − lim ǫ → ( ǫ − n B ,ǫn − b n ) = 0 , then for arbitrary finite time interval there existsthe mean field limit of marginal observables (7): w ∗ − lim ǫ → ( ǫ − s B s ( t ) − b s ( t )) = 0 , s ≥ , that aredetermined by the following expansions: b s ( t, Y ) = s − X n =0 t Z dt . . . t n − Z dt n Y l ∈ Y G ( t − t , l ) s X i = j =1 N int ( i , j ) Y l ∈ Y \ ( j ) G ( t − t , l ) . . . (8) Y l n ∈ Y \ ( j ,...,j n − ) G ( t n − − t n , l n ) s X i n = j n = 1 ,i n , j n = ( j , . . . , j n − ) N int ( i n , j n ) × Y l n +1 ∈ Y \ ( j ,...,j n ) G ( t n , l n +1 ) b s − n ( Y \ ( j , . . . , j n )) , where the operator N int ( i , j ) is defined on g n ∈ L ( H n ) by formula (5) . The proof of Theorem 1 is based on formulas for cumulants of asymptotically perturbed groupsof operators (2).For arbitrary finite time interval the asymptotically perturbed group of operators (2) has thefollowing scaling limit in the sense of the ∗ -weak convergence on the space L ( H s ):w ∗ − lim ǫ → (cid:0) G s ( t, Y ) − s Y j =1 G ( t, j ) (cid:1) g s = 0 . (9)uantum kinetic equations with correlations 7Taking into account analogs of the Duhamel equations for cumulants of asymptotically perturbedgroups of operators, in view of formula (9) we havew ∗ − lim ǫ → (cid:16) ǫ − n n ! A n (cid:0) t, { Y \ X } , j , . . . , j n (cid:1) −− t Z dt . . . t n − Z dt n Y l ∈ Y G ( t − t , l ) s X i = j =1 N int ( i , j ) Y l ∈ Y \ ( j ) G ( t − t , l ) . . . Y l n ∈ Y \ ( j ,...,j n − ) G ( t n − − t n , l n ) s X i n = j n = 1 ,i n , j n = ( j , . . . , j n − ) N int ( i n , j n ) Y l n +1 ∈ Y \ ( j ,...,j n ) G ( t n , l n +1 ) (cid:17) g s − n = 0 , where we used notations accepted in formula (8) and g s − n ≡ g s − n ((1 , . . . , s ) \ ( j , . . . , j n )) , n ≥ b ∈ L γ ( F H ), then the sequence b ( t ) = ( b , b ( t ) , . . . , b s ( t ) , . . . ) of limit marginal observables(8) is a generalized global solution of the Cauchy problem of the dual quantum Vlasov hierarchy ∂∂t b s ( t, Y ) = s X j =1 N ( j ) b s ( t, Y ) + s X j = j =1 N int ( j , j ) b s − ( t, Y \ ( j )) , (10) b s ( t ) | t =0 = b s , s ≥ , (11)where the infinitesimal generator N ( j ) of the group of operators G ( t, j ) of j particle is definedon g n ∈ L ( H n ) by formular (4).It should be noted that equations set (10) has the structure of recurrence evolution equations.We give several examples of the evolution equations of the dual quantum Vlasov hierarchy (10)in terms of operator kernels of the limit marginal observables i ∂∂t b ( t, q ; q ′ ) = − 12 ( − ∆ q + ∆ q ′ ) b ( t, q ; q ′ ) ,i ∂∂t b ( t, q , q ; q ′ , q ′ ) = − X i =1 ( − ∆ q i + ∆ q ′ i ) b ( t, q , q ; q ′ , q ′ ) ++ (cid:0) Φ( q ′ − q ′ ) − Φ( q − q ) (cid:1)(cid:0) b ( t, q ; q ′ ) + b ( t, q ; q ′ ) (cid:1) . We consider the mean field limit of a particular case of marginal observables, namely theadditive-type marginal observables B (1) (0) = (0 , B ,ǫ (1) , , . . . ). We remark that the k -ary marginalobservables are represented by the sequence B ( k ) (0) = (cid:0) , . . . , , B ,ǫk (1 , . . . , k ) , , . . . (cid:1) . In case ofadditive-type marginal observables expansions (7) take the following form: B (1) s ( t, Y ) = A s ( t ) s X j =1 B ,ǫ ( j ) , s ≥ , (12)where the operator A s ( t ) is s -order cumulant (6) of groups of operators (2).uantum kinetic equations with correlations 8 Corollary 1. If for the additive-type marginal observable B ,ǫ ∈ L ( H ) it holds w ∗ − lim ǫ → ( ǫ − B ,ǫ − b ) = 0 , then, according to statement of Theorem 1, for additive-type marginal observable (12) wehave: w ∗ − lim ǫ → ( ǫ − s B (1) ,ǫs ( t ) − b (1) s ( t )) = 0 , s ≥ , where the limit additive-type marginal observ-able b (1) s ( t ) is determined by a special case of expansion (8) b (1) s ( t, Y ) = t Z dt . . . t s − Z dt s − Y l ∈ Y G ( t − t , l ) s X i = j =1 N int ( i , j ) (13) × Y l ∈ Y \ ( j ) G ( t − t , l ) . . . Y l s − ∈ Y \ ( j ,...,j s − ) G ( t s − − t s − , l s − ) × s X i s − = j s − = 1 ,i s − , j s − = ( j , . . . , j s − ) N int ( i s − , j s − ) Y l s ∈ Y \ ( j ,...,j s − ) G ( t s − , l s ) b ( Y \ ( j , . . . , j s − )) . We make several examples of expansions (13) for the limit additive-type marginal observables b (1)1 ( t, 1) = G ( t, b (1) ,b (1)2 ( t, , 2) = t Z dt Y i =1 G ( t − t , i ) N int (1 , X j =1 G ( t , j ) b ( j ) . Thus, for arbitrary initial states in the mean field scaling limit the kinetic evolution of quantummany-particle systems is described in terms of limit marginal observables (8) governed by the dualquantum Vlasov hierarchy (10). Furthermore, the relationships between the evolution of observables and the kinetic evolution ofstates described in terms of a one-particle marginal density operator are discussed.We shall consider initial states of a quantum many-particle system specified by the one-particle(marginal) density operator F ,ǫ ∈ L ( H ) in the presence of correlations, i.e. initial state specifiedby the following sequence of marginal density operators F c = (cid:0) , F ,ǫ (1) , g ǫ Y i =1 F ,ǫ ( i ) , . . . , g ǫn n Y i =1 F ,ǫ ( i ) , . . . (cid:1) , (14)where the bounded operators g ǫn ≡ g ǫn (1 , . . . , n ) ∈ L ( H n ) , n ≥ 2, are specified the initial correla-tions. We remark that such assumption about initial states is intrinsic for the kinetic descriptionof a gas. On the other hand, initial data (14) is typical for the condensed states of quantum gases,for example, the equilibrium state of the Bose condensate satisfies the weakening of correlationcondition with the correlations which characterize the condensed state [24].We assume that for the initial one-particle (marginal) density operator F ,ǫ ∈ L ( H ) exists themean field limit lim ǫ → (cid:13)(cid:13) ǫ F ,ǫ − f (cid:13)(cid:13) L ( H ) = 0 , and it holds: lim ǫ → (cid:13)(cid:13) g ǫn − g n (cid:13)(cid:13) L ( H n ) = 0 , then inuantum kinetic equations with correlations 9the mean field limit initial state is specified by the following sequence of limit operators f c = (cid:0) , f (1) , g Y i =1 f ( i ) , . . . , g n n Y i =1 f ( i ) , . . . (cid:1) . (15)We note that in case of initial states specified by sequence (15) the average values (meanvalues) of limit marginal observables (8) are determined by the following positive continuouslinear functional [20] (cid:0) b ( t ) , f c (cid:1) . = ∞ X n =0 n ! Tr ,...,n b n ( t, , ..., n ) g n (1 , ..., n ) n Y i =1 f ( i ) . (16)For b ( t ) ∈ L γ ( F H ) and f ∈ L ( H ), functional (16) exists under the condition that k f k L ( H ) < γ .We consider relationships of the constructed mean field asymptotic behavior of marginal ob-servables with the quantum Vlasov-type kinetic equation in case of initial states (15).For the limit additive-type marginal observables (13) the following equality is true (cid:0) b (1) ( t ) , f c (cid:1) = ∞ X s =0 s ! Tr ,...,s b (1) s ( t, , . . . , s ) g s (1 , . . . , s ) s Y i =1 f ( i ) == Tr b (1) f ( t, , where the operator b (1) s ( t ) is determined by expansion (13) and the one-particle (marginal) densityoperator f ( t, 1) is represented by the series expansion f ( t, 1) = ∞ 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 ) . . . (17) × 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 ) g n (1 , . . . , n + 1) n +1 Y i =1 f ( i ) . In series (17) the operator N ∗ int ( j , j ) f n = −N int ( j , j ) f n is an adjoint operator to operator (3)and the group G ∗ ( t, i ) = G ( − t, i ) is dual to group (2) in the sense of functional (16). For boundedinteraction potentials series (17) is norm convergent on the space L ( H ) under the condition that t < t ≡ (cid:0) k Φ k L ( H ) k f k L ( H ) (cid:1) − .The operator f ( t ) represented by series (17) is a solution of the Cauchy problem of the quantumVlasov-type kinetic equation with initial correlations: ∂∂t f ( t, 1) = N ∗ (1) f ( t, 1) + (18)+Tr N ∗ int (1 , Y i =1 G ∗ ( t, i ) g (1 , Y i =1 ( G ∗ ) − ( t, i ) f ( t, f ( t, ,f ( t ) | t =0 = f , (19)uantum kinetic equations with correlations 10where the operator N ∗ (1) = −N (1) is an adjoint operator to operator (4) in the sense of functional(16) and the group ( G ∗ ) − ( t ) = G ∗ ( − t ) = G ( t ) is inverse to the group ( G ∗ )( t ). This fact is provedsimilarly as in case of a solution of the quantum BBGKY hierarchy represented by the iterationseries [20] (see also [27], [28]).Thus, in case of initial states specified by one-particle (marginal) density operator (15) weestablish that the dual quantum Vlasov hierarchy (10) for additive-type marginal observablesdescribes the evolution of quantum large particle system just as the quantum Vlasov-type kineticequation with initial correlations (18). The property of the propagation of initial correlations is a consequence of the validity of thefollowing equality for the mean value functionals of the limit k -ary marginal observables in caseof k ≥ (cid:0) b ( k ) ( t ) , f c (cid:1) = ∞ X s =0 s ! Tr ,...,s b ( k ) s ( t, , . . . , s ) g s (1 , . . . , s ) s Y j =1 f ( j ) = (20)= 1 k ! Tr ,...,k b k (1 , . . . , k ) k Y i =1 G ∗ ( t, i ) g k (1 , . . . , k ) k Y i =1 G ∗ ( − t, i ) k Y j =1 f ( t, j ) , k ≥ , where the limit one-particle (marginal) density operator f ( t, j ) is represented by series expansion(17) and therefore it is governed by the Cauchy problem of the quantum Vlasov-type kineticequation with initial correlations (18),(19).This fact is proved similarly to the proof of a property on the propagation of initial chaos in amean field scaling limit [23].Thus, in case of the limit k -ary marginal observables a solution of the dual quantum Vlasovhierarchy (10) is equivalent to a property of the propagation of initial correlations for the k -particlemarginal density operator in the sense of equality (20) or in other words the mean field scalingdynamics does not create correlations.We remark that the general approaches to the description of the evolution of states of quantummany-particle systems within the framework of correlation operators and marginal correlationoperators were given in papers [29], [30] and [31], respectively (see also review [20]). We consider the relationships of dynamics of quantum many-particle systems described in termsof marginal observables and dynamics described within the framework of a one-particle (marginal)density operator governed by the quantum kinetic equation in the presence of initial correlationsin the general case, i.e. without any approximations like scaling limits as above in Section 2. Ifinitial states is completely specified by a one-particle (marginal) density operator, using a non-perturbative solution of the dual quantum BBGKY hierarchy we prove that all possible states atarbitrary moment of time can be described within the framework of a one-particle density operatorgoverned by the generalized quantum kinetic equation with initial correlations.uantum kinetic equations with correlations 11 In case of initial states specified by sequence (14) the average values (mean values) of marginalobservables (7) are defined by the positive continuous linear functional on the space L ( F H ) (cid:0) B ( t ) , F c (cid:1) . = ∞ X s =0 s ! Tr ,...,s B s ( t, , . . . , s ) g ǫs (1 , . . . , s ) s Y i =1 F ,ǫ ( i ) . (21)For F ,ε ∈ L ( H ) and B ,ǫs ∈ L ( H s ) series (21) exists under the condition that k F ,ε k L ( H ) < e − .For mean value functional (21) the following representation holds (cid:0) B ( t ) , F c (cid:1) = (cid:0) B (0) , F ( t | F ( t )) (cid:1) , (22)where B (0) = ( B , B ,ǫ (1) , . . . , B ,ǫs (1 , . . . , s ) , . . . ) ∈ L γ ( F H ) is a sequence of initial marginal ob-servables, and F ( t | F ( t )) = (1 , F ( t ) , F ( t | F ( t )) , . . . , F s ( t | F ( t )) , . . . ) is a sequence of explic-itly defined marginal functionals F s ( t | F ( t )) , s ≥ 2, with respect to the following one-particle(marginal) density operator F ( t, 1) = ∞ X n =0 n ! Tr ,..., n A ∗ n ( t ) g ǫn +1 (1 , . . . , n + 1) n +1 Y i =1 F ,ǫ ( i ) . (23)The generating operator A ∗ n ( t ) ≡ A ∗ n ( t, , . . . , n + 1) of series expansion (23) is the (1 + n ) th -order cumulant of groups of operators G ∗ n ( t ) , n ≥ , dual to groups (2) in the sense of functional(21), namely A ∗ n ( t, , . . . , n + 1) . = X P: (1 ,...,n +1)= S i X i ( − | P |− ( | P | − Y X i ⊂ P G ∗| X i | ( t, X i ) , where the symbol P P means the sum over all possible partitions P of the set (1 , . . . , n + 1) into | P | nonempty mutually disjoint subsets X i ⊂ (1 , . . . , n + 1).The marginal functionals of the state F s ( t | F ( t )) , s ≥ 2, are represented by the followingseries expansions: F s (cid:0) t, Y | F ( t ) (cid:1) . = ∞ X n =0 n ! Tr s +1 ,...,s + n G n (cid:0) t, { Y } , X \ Y (cid:1) s + n Y i =1 F ( t, i ) , (24)where we denote: Y ≡ (1 , . . . , s ) , X \ Y ≡ ( s + 1 , . . . , s + n ), and the (1 + n ) th -order generatingoperator G n ( t ) , n ≥ 0, of this series is determined by the following expansion G n ( t, { 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 )! (25) × ˘ A 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 | ! ˘ A | X lj | ( t, i l j , X l j ) . uantum kinetic equations with correlations 12In formula (25) we denote by P D j : Z j = S lj X lj the sum over all possible dissections of the linearlyordered 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 we introduced the (1 + n ) th -order scatteringcumulants ˘ A n ( t, { Y } , X \ Y ) . = A ∗ n ( t, { Y } , X \ Y ) g ǫs + n ( θ ( { Y } ) , X \ Y ) s + n Y i =1 ( A ∗ ) − ( t, i ) , where the operator g ǫs + n ( θ ( { Y } ) , X \ Y ) is specified initial correlations (15), the operator ( A ∗ ) − ( t )is inverse to the operator A ∗ ( t ) and it is used notations accepted above. We give examples of thescattering cumulants G ( t, { Y } ) = ˘ A ( t, { Y } ) . = A ∗ ( t, { Y } ) g ǫs ( θ ( { Y } )) s Y i =1 ( A ∗ ) − ( t, i ) , G ( t, { Y } , s + 1) = A ∗ ( t, { Y } , s + 1) g ǫs +1 ( θ ( { Y } ) , s + 1) s +1 Y i =1 ( A ∗ ) − ( t, i ) −− A ∗ ( t, { Y } ) g ǫs ( θ ( { Y } )) s Y i =1 ( A ∗ ) − ( t, i ) s X i =1 A ∗ ( t, i, s + 1) g ǫ ( i, s + 1)( A ∗ ) − ( t, i )( A ∗ ) − ( t, s + 1) . If k F ( t ) k L ( H ) < e − (3 s +2) , then for arbitrary t ∈ R series expansion (22) converges in the norm ofthe space L ( H s ) [20].We emphasize that marginal functionals of the state (24) characterize the correlations generatedby dynamics of quantum many-particle systems in the presence of initial correlations. We prove the validity of equality (22) for mean value functional (21).In a particular case of initial data specified by the additive-type marginal observables, i.e. B (1) (0) = (0 , B ,ǫ (1) , , . . . ), equality (22) takes the form (cid:0) B (1) ( t ) , F c (cid:1) = Tr B ,ǫ (1) F ( t, , (26)where the one-particle (marginal) density operator F ( t ) is determined by series expansion (23).The validity of this equality is a result of the direct transformation of the generating operators ofexpansions (12) to adjoint operators in the sense of the functional (21).In case of initial data specified by the s -ary marginal observables i.e. B ( s ) (0) = (0 , . . . , ,B ,ǫs (1 , . . . , s ) , , . . . ) , s ≥ 2, equality (22) takes the following form: (cid:0) B ( s ) ( t ) , F c (cid:1) = 1 s ! Tr ,...,s B ,ǫs (1 , . . . , s ) F s (cid:0) t, , . . . , s | F ( t ) (cid:1) , (27)where the marginal functional of the state F s ( t | F ( t )) is represented by series expansion (24).The proof of equality (27) is based on the application of cluster expansions to generatingoperators (6) of expansions (7) which is dual to the kinetic cluster expansions introduced inpaper [18]. Then the adjoint series expansion can be expressed in terms of one-particle (marginal)density operator (23) in the form of the functional from the right-hand side of equality (27).In case of the general type of marginal observables the validity of equality (22) is proven inmuch the same way as the validity of particular equalities (26) and (27).uantum kinetic equations with correlations 13 As a result of the differentiation over the time variable of operator represented by series (23) inthe sense of the norm convergence of the space L ( H ), then the application of the kinetic clusterexpansions [18], [32] to the generating operators of obtained series expansion, for the one-particle(marginal) density operator we derive the following identity ∂∂t F ( t, 1) = N ∗ (1) F ( t, 1) + ǫ Tr N ∗ int (1 , F ( t, , | F ( t )) , (28)where the operators N ∗ (1) = −N (1) and N ∗ int (1 , 2) = −N int (1 , 2) are adjoint operators in the senseof functional (16) to operators (4) and (5), respectively, and the collision integral is determinedby series expansion (24) for the marginal functional of the state in case of s = 2. This identity wetreat as the non-Markovian quantum kinetic equation. We refer to this evolution equation as thegeneralized quantum kinetic equation with initial correlations.We emphasize that the coefficients in an expansion of the collision integral of kinetic equation(28) are determined by the operators specified initial correlations (14). We remark also thatin case of a system of particles with a n -body interaction potential the collision integral of thecorresponding quantum kinetic equation is determined by the marginal functional of the state (24)in case of s = n [20].For the generalized quantum kinetic equation with initial correlations (28) on the space L ( H )the following statement is true.If k F ,ǫ k L ( H ) < ( e (1 + e )) − , the global in time solution of initial-value problem of kineticequation (28) is determined by series expansion (23). For initial data F ,ǫ ∈ L ( H ) it is a strong(classical) solution and for an arbitrary initial data it is a weak (generalized) solution.We note that for initial data (14) specified by a one-particle (marginal) density operator, theevolution of states described within the framework of a one-particle (marginal) density operatorgoverned by the generalized quantum kinetic equation with initial correlations (28) is dual to thedual quantum BBGKY hierarchy for additive-type marginal observables with respect to bilinearform (21), and it is completely equivalent to the description of states in terms of marginal densityoperators governed by the quantum BBGKY hierarchy.Thus, the evolution of quantum many-particle systems described in terms of marginal observ-ables can be also described within the framework of a one-particle (marginal) density operatorgoverned by the generalized quantum kinetic equation with initial correlations (28). We construct a mean field asymptotics of a solution of the generalized quantum kinetic equationwith initial correlations (28). This asymptotics is governed by the quantum Vlasov-type kineticequation with initial correlations (18) derived above from the dual quantum Vlasov hierarchy(10) for the limit marginal observables. Moreover, a mean field asymptotic behavior of marginalfunctionals of the state (24) describes the propagation in time of initial correlations like establishedproperty (20).uantum kinetic equations with correlations 14 For solution (23) of the generalized quantum kinetic equation with initial correlations (28) thefollowing mean field limit theorem is true [32]. Theorem 2. If for the initial one-particle density operator F ,ǫ ∈ L ( H ) exists the followinglimit: lim ǫ → k ǫ F ,ǫ − f k L ( H ) = 0 and lim ǫ → (cid:13)(cid:13) g ǫn − g n (cid:13)(cid:13) L ( H n ) = 0 , then for t ∈ ( − t , t ) , where t ≡ (cid:0) k Φ k L ( H ) k f k L ( H ) (cid:1) − , there exists the mean field limit of solution (23) of the Cauchyproblem of the generalized quantum kinetic equation with initial correlations (28) lim ǫ → (cid:13)(cid:13) ǫ F ( t ) − f ( t ) (cid:13)(cid:13) L ( H ) = 0 , (29) where the operator f ( t ) is represented by series expansion (17) and it is a solution of the Cauchyproblem of the quantum Vlasov-type kinetic equation with initial correlations (18) , (19) . The proof of this theorem is based on formulas of asymptotically perturbed cumulants of groupsof operators G ∗ n ( t ) , n ≥ , adjoint to groups (2) in the sense of functional (21). Indeed, in a meanfield limit for generating evolution operators (25) of series expansion (24) the following equalitiesare valid: lim ǫ → (cid:13)(cid:13) ǫ n G n ( t, { Y } , X \ Y ) f s + n (cid:13)(cid:13) L ( H s + n ) = 0 , n ≥ , (30)and in case of the first-order generating evolution operator we havelim ǫ → (cid:13)(cid:13)(cid:0) G ( t, { Y } ) − s Y j =1 G ∗ ( t, j ) g s (1 , . . . , s ) s Y j =1 G ∗ ( − t, j ) (cid:1) f s (cid:13)(cid:13) L ( H s ) = 0 , (31)respectively.In view that under the condition t < t ≡ (2 k Φ k L ( H ) k ǫ F ,ǫ k L ( H ) ) − , for a bounded interactionpotential the series for the operator ǫ F ( t ) is norm convergent, then for t < t the remainder ofsolution series (23) can be made arbitrary small for sufficient large n = n independently of ǫ .Then, using stated above asymptotic formulas, for each integer n every term of this series convergesterm by term to the limit operator f ( t ) which is represented by series (17).As stated above the mean field scaling limit (17) of solution (23) of the generalized quantumkinetic equation in the presence of initial correlations is governed by the quantum Vlasov-typekinetic equation with initial correlations (18).Thus, we derived the quantum Vlasov-type kinetic equation with initial correlations (18) fromthe generalized quantum kinetic equation (28) in the mean field scaling limit. It is the same asthe kinetic equation derived from the dual quantum Vlasov hierarchy (10) for the mean field limitmarginal observables. As we noted above in Section 3 in case of initial data (14) the evolution of all possible correlationsof quantum many-particle systems is described by marginal functionals of the state (24).uantum kinetic equations with correlations 15Since solution (23) of initial-value problem of the generalized quantum kinetic equation withinitial correlations (28) converges to solution (17) of initial-value problem of the quantum Vlasov-type kinetic equation with initial correlations (18) as (29), and equalities (30) and (31) hold,then for a mean field asymptotic behavior of marginal functionals of the state (24) the followingequalities are true:lim ǫ → (cid:13)(cid:13) ǫ s F s ( t, , . . . , s | F ( t )) − s Y j =1 G ∗ ( t, j ) g s (1 , . . . , s ) s Y j =1 G ∗ ( − t, j ) s Y k =1 f ( t, k ) (cid:13)(cid:13) L ( H s ) = 0 ,s ≥ . These equalities describe the propagation of initial correlations in time in the mean field scalingapproximation. In the paper the concept of quantum kinetic equations in case of the kinetic evolution, involvingcorrelations of particle states at initial time, for instance, correlations characterizing the condensedstates, was considered. Two approaches were developed with a view to this purpose. One approachbased on the description of the evolution of quantum many-particle systems within the frameworkof marginal observables. Another method consists in the possibility in case of initial states specifiedby a one-particle marginal density operator and correlation operators to describe the evolutionof states within the framework of a one-particle (marginal) density operator governed by thegeneralized quantum kinetic equation with initial correlations.In case of pure states the quantum Vlasov-type kinetic equation with initial correlations (18)can be reduced to the Gross–Pitaevskii-type kinetic equation. Indeed, in this case the one-particledensity operator f ( t ) = | ψ t ih ψ t | is a one-dimensional projector onto a unit vector | ψ t i ∈ H andits kernel has the following form: f ( t, q, q ′ ) = ψ ( t, q ) ψ ∗ ( t, q ′ ). Then, if we consider quantumparticles, interacting by the potential which kernel Φ( q ) = δ ( q ) is the Dirac measure, from kineticequation (18) we derive the Gross–Pitaevskii-type kinetic equation i ∂∂t ψ ( t, q ) = − 12 ∆ q ψ ( t, q ) + Z dq ′ dq ′′ g ( t, q, q ; q ′ , q ′′ ) ψ ( t, q ′′ ) ψ ∗ ( t, q ) ψ ( t, q ) , where the coupling ratio g ( t, q, q ; q ′ , q ′′ ) of the collision integral is the kernel of the scattering lengthoperator Q i =1 G ∗ ( t, i ) g (1 , Q i =1 G ∗ ( − t, i ). If we consider a system of quantum particleswithout initial correlations, then this kinetic equation is the cubic nonlinear Schr¨odinger equation.This paper deals with a quantum system of a non-fixed (i.e. arbitrary but finite) number ofidentical (spinless) particles obeying Maxwell–Boltzmann statistics. 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