Quantum Kinetic Evolution of Marginal Observables

We develop a rigorous formalism for the description of the evolution of observables of quantum systems of particles in the mean-field scaling limit. The corresponding asymptotics of a solution of the initial-value problem of the dual quantum BBGKY hierarchy is constructed. Moreover, links of the evolution of marginal observables and the evolution of quantum states described in terms of a one-particle marginal density operator are established. Such approach gives the alternative description of the kinetic evolution of quantum many-particle systems to generally accepted approach on basis of kinetic equations.

It is well known that a description of quantum many-particle systems is formulated in terms of two sets of objects: observables and states. The functional of the mean value of observables defines a duality between observables and states and as a consequence there exist two approaches to the description of evolution. Usually the evolution of many-particle systems is described in the framework of the evolution of states by the quantum BBGKY hierarchy for marginal density operators. An equivalent approach of the description of the evolution of quantum systems is given in terms of marginal observables by the dual quantum BBGKY hierarchy (the Heisenberg picture of evolution) [10,11].
The conventional philosophy of the description of kinetic evolution consists in the following. The evolution of states can be effectively described by a one-particle marginal density operator governed by the kinetic equation as a result of some approximations [12] or in suitable scaling limits [13]. In the paper we develop an approach to the description of kinetic evolution of quantum many-particle systems in framework of the evolution of marginal observables. For this purpose we investigate the mean-field asymptotics of a solution of the initial-value problem of the quantum dual BBGKY hierarchy. In addition links between the evolution of observables and the kinetic evolution of states described in terms of a one-particle marginal density operator are discussed in the general case.
We now outline the structure of the paper and the main results. In Section 2 we introduce some preliminary definitions and construct a solution of the Cauchy problem of the dual quantum BBGKY hierarchy for marginal observables as an expansion over particle clusters which evolution is governed by the corresponding-order cumulant (semi-invariant) of the groups of operators of the Heisenberg equations of finitely many particles. In Section 3 the main result is proved, namely, the mean-field asymptotic behavior of stated above solution of the dual quantum BBGKY hierarchy is established. The constructed asymptotics is governed by the recurrence evolution equations set (the dual quantum Vlasov hierarchy). In Section 4 typical properties of the dual kinetic dynamics are formulated, in particular the relation of the dual quantum Vlasov hierarchy with the nonlinear Schrödinger equation is considered. Moreover, the relation of the dual quantum BBGKY hierarchy and the generalized quantum kinetic equation is established. Finally we conclude with some observations and perspectives for future research.
2 The evolution of observables of quantum many-particle systems In order to construct the asymptotic form of the marginal (s-particle) observables of quantum many-particle systems we describe the evolution by means of the dual quantum BBGKY hierarchy. We introduce such a hierarchy of evolution equations and formulate necessary properties of a solution of the Cauchy problem of this hierarchy.

Preliminary facts
We 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 ν , ν ≥ 1. We will use units where h = 2π = 1 is a Planck constant, and m = 1 is the mass of particles. The Hamiltonian H = ∞ n=0 H n of such system is a self-adjoint operator with the domain D(H) = {ψ = ⊕ψ n ∈ F H | ψ n ∈ D(H n ) ∈ H n , n H n ψ n 2 < ∞} ⊂ F H , where F H = ∞ n=0 H ⊗n is the Fock space over the Hilbert space H. We adopt the usual conven- . . dq n |ψ n (q 1 , . . . , q n )| 2 < +∞. On the subspace of infinitely differentiable functions with compact supports ψ n ∈ L 2 0 (R νn ) ⊂ L 2 (R νn ) the Hamiltonian H n of n ≥ 1 particles acts according to the formula where K(i)ψ n = − 1 2 ∆ q i ψ n is the operator of the kinetic energy, Φ(i, j)ψ n = Φ(|q i − q j |)ψ n is the operator of a two-body interaction potential Φ and ǫ > 0 is a scaling parameter. Hereinafter we shall consider the bounded interaction potentials.
Let a sequence g = (g 0 , g 1 , . . . , 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 denoted by g n (1, . . . , n). Let the space L(F H ) be the space of sequences g = (g 0 , g 1 , . . . , g n , . . .) of bounded operators g n defined on the Hilbert space H n that satisfy symmetry condition: g n (1, . . . , n) = g n (i 1 , . . . , i n ), for arbitrary (i 1 , . . . , i n ) ∈ (1, . . . , n), equipped with the operator norm . L(Hn) [15]. We will also consider a more general space where 0 < γ < 1. We denote by L γ ,0 (F H ) ⊂ L γ (F H ) the everywhere dense set in of finite sequences of degenerate operators with infinitely differentiable kernels with compact supports. Observables of finitely many quantum particles are sequences of self-adjoint operators from the space L γ (F H ). The case of unbounded operators of observables can be reduced to the case under consideration [15].
. . , f n , . . .) of trace class operators f n = f n (1, . . . , n) ∈ L 1 (H n ), satisfying the mentioned above symmetry condition, equipped with the trace norm Tr 1,...,n |f n (1, . . . , n)|, where Tr 1,...,n is the partial trace over 1, . . . , n particles. The everywhere dense set of finite sequences of degenerate operators with infinitely differentiable kernels with compact supports in the space L 1 (F H ) we denote by L 1 0 (F H ). The sequences of operators f n ∈ L 1 (H n ), n ≥ 1, which kernels are known as density matrices defined on the n-particle Hilbert space H n = L 2 (R νn ), describe the states of a quantum system of non-fixed number of particles.
The space L(F H ) is dual to the space L 1 (F H ) with respect to the bilinear form g|f . = ∞ n=0 1 n! Tr 1,...,n g n f n , where g n ∈ L(H n ) and f n ∈ L 1 (H n ). The mean value of observables are given by the positive continuous linear functional (2) on the space of observables L γ (F H ).

The dual quantum BBGKY hierarchy
The evolution of marginal observables is described by the initial-value problem of the dual quantum BBGKY hierarchy where on L 0 (H n ) ⊂ L(H n ) the operators N 0 and N int are consequently defined by formulas where ·, · is a commutator of operators. We refer to equations (3) as the dual quantum BBGKY hierarchy since the canonical quantum BBGKY hierarchy [16][17][18] for the marginal density operators is the dual hierarchy of evolution equations with respect to bilinear form (2) to evolution equations (3). In case of the space H = L 2 (R ν ), evolution equations (3) for kernels of the operators G s (t), s ≥ 1, are given by To construct a solution of abstract initial-value problem (3) we formulate some necessary facts. For g n ∈ L(H n ) it is defined the one-parameter mapping On the space L(H n ) one-parameter mapping (6) is an isometric * -weak continuous group of operators, i.e. it is a C * 0 -group. The infinitesimal generator N n of this group of operators is a closed operator for the * -weak topology and on its domain of the 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 [19] w * − lim where H n is the Hamiltonian (1) and the operator: Let us introduce some abridged notations: Y ≡ (1, . . . , s), X ≡ (j 1 , . . . , j n ) ⊂ Y , and {Y \X} is the set consisting of one element X = (j 1 , . . . , j n ), i.e. the set (j 1 , . . . , j n ) is a connected subset of the partition P such that |P| = 1 (|P| denotes the number of partitions of a set).
We define the n-order cumulant of group of operators (6) as follows [20,21] A 1+n t, {Y \X}, X . = where P is the sum over all possible partitions P of the set ({Y \X}, j 1 , . . . , j n ) into |P| nonempty mutually disjoint subsets X i ⊂ ({Y \X}, X).
Let us indicate some properties of cumulants (8). If n = 1, for g 1 ∈ L 0 (H) ⊂ L(H) in the sense of the * -weak convergence of the space L(H) the generator of the first-order cumulant is given by the operator In the case n = 2 for and for n > 2 as a consequence of the fact that we consider a system of particles interacting by a two-body potential, it holds On the space L γ (F H ) for abstract initial-value problem (3)-(4) the following statement holds [10].
The solution where the (1 + n)-order cumulant A 1+n t, {Y \X}, X is determined by formula (8) and the estimate holds For it is a classical solution and for arbitrary initial data G(0) ∈ L γ (F H ) it is a generalized solution.
We note that expansion (9) can be represented in the form of the perturbation (iteration) series as a result of applying of analogs of the Duhamel equation to cumulants (8) of groups of operators (6) from solution expansion (9) [18].
As stated above (see formula (2)) the mean value of the marginal observable According to estimate (10), functional (11) exists under the condition that γ < e −1 .

The dual quantum Vlasov hierarchy
We consider the problem of the rigorous description of the quantum kinetic evolution on the basis of many-particle dynamics of observables by way of example of the mean-field asymptotic behavior of stated above solution of the dual quantum BBGKY hierarchy.

A mean-field limit of the dual quantum BBGKY hierarchy
Consider the mean-field scaling limit of a solution of initial-value problem (3)-(4) of the dual quantum BBGKY hierarchy.
then for arbitrary finite time interval there exists the mean-field limit of solution (9) of the Cauchy problem (3)-(4) of the dual quantum BBGKY hierarchy in the sense of the * -weak convergence of the space L(H s ) and it is defined by the expansion where the following notation of the group of operators (6) of noninteracting particles is used Before to prove this statement we give some comments. If g(0) ∈ L(F H ), the sequence g(t) = (g 0 , g 1 (t), . . . , g s (t), . . .) of limit marginal observables (14) is a generalized global solution of the initial-value problem of the dual quantum Vlasov hierarchy This fact is proved similar to the case of an iteration series of the dual quantum BBGKY hierarchy [10]. It should be noted that equations set (15) has the structure of recurrence evolution equations. We make a few examples of the dual quantum Vlasov hierarchy (15) in terms of operator kernels of the limit marginal observables Let us consider a particular case of observables, namely the mean-field limit of the additivetype marginal observables 2 : G (1) (0) = (0, G 1 (0, 1), 0, . . .). In this case solution (9) of the dual BBGKY hierarchy (3) has the form where A s (t) is the s-order cumulant (8) of groups of operators (6). If for the additive-type observables G (1) (0) condition (12) is satisfied, i.e. it holds 1 (0) = 0, then according to statement (13), we have where the limit operator g (1) s (t) is defined by the expression 1 (0, Y \(j 1 , . . . , j s−1 )), s ≥ 1, as a special case of expansion (14). We give examples of expressions (18) g 1 (t, 1) = G 1 (t, 1) g 1 (0, j).
where the operator N int (i, j) is defined by formula (5) and Y ≡ (1, . . . , s). Then for (1+n)-order cumulant of groups of operators (6) the analog of the Duhamel equation holds where accepted above notations are used, G s−n (t n ) ≡ G s−n (t n , Y \(j 1 , . . . , j n )) and we take into consideration the identity: For arbitrary finite time interval * -weak continuous group of operators (6) has the following scaling limit in the sense of the * -weak convergence of the space L(H s ) [19] w Taking into account assumption (12) and an analog of the Duhamel equation (19), in view of formula (20) of an asymptotic perturbation of group (6) for the n term of expansion (9) we have where it is used the following notations: X ≡ (j 1 , . . . , j n ) and G 0 As a result of equality (21) we establish the validity of statement (13) for solution (9) of the dual quantum BBGKY hierarchy (3).
To construct the evolution equations which satisfy limit expression (14) we differentiate over the time variable expansion (14) in the sense of pointwise convergence of the space L(H s ). In view equality (7) it holds . . . , j n+1 )).
According to definition (14), the second summand in the right-hand side of this equality is expressed in terms of the limit marginal observable g s−1 (t, Y \j 1 ) and consequently, we get the dual quantum Vlasov hierarchy (15).

Some properties of the dual kinetic dynamics
The links of constructed mean-field asymptotic behavior of marginal observables with the nonlinear Schrödinger equation are considered. Furthermore the relation between the evolution of observables and the generalized description of the kinetic evolution of states in terms of a one-particle marginal density operator is discussed.

The propagation of a chaos
Hereinafter we shall consider initial data satisfying the factorization property or a "chaos" property [17], which means the lack of correlations at initial time. For a system of identical particles, obeying the Maxwell-Boltzmann statistics, we have The assumption about initial data is intrinsic for the kinetic description of a gas, because in this case all possible states are characterized only by a one-particle marginal density operator. Let lim then the limit of the initial state F (c) satisfies a chaos property too If g(t) ∈ L γ (F H ) and f 0 1 ∈ L 1 (H), then under the condition f 0 1 L 1 (H) < γ, there exists the mean-field limit of mean value functional (11) which is determined by the expansion In consequence of the following equality for the limit additive-type marginal observables (18) (it is proved below in more general case) where operator g (1) s (t) is given by (18) and f 1 (t, 1) is the solution of the initial-value problem of the Vlasov quantum kinetic equation we establish that hierarchy (15) for additive-type marginal observables and initial state (23) describe the evolution of quantum many-particle systems as by the Vlasov quantum kinetic equation (27). Indeed for bounded interaction potential (1) series (26) is norm convergent on the space L 1 (H) under the condition and hence the functional in right-hand side of equality (25) exists. Taking into account the validity for f n ∈ L 1 0 (H n ) in the sense of the norm convergence of the equality where the operator (−N n ) is defined by formulas (5), we differentiate over the time variable expression (26) in the sense of pointwise convergence of the space L 1 (H) Using the product formula for the one-particle marginal density operator f 1 (t, i) defined by expansion (26) for initial data (24) where the group property of one-parameter mapping (6) is applied, we express the second summand in the right-hand side of equality (29) in terms of 2 i=1 f 1 (t, i) and consequently, we get equation (27).
Correspondingly, a chaos property in the Heisenberg picture of evolution of quantum manyparticle systems is fulfil. It follows from the equality for the limit k-ary marginal observables, i.e. g (k) (0) = (0, . . . , g where the limit one-particle marginal density operator f 1 (t, i) is defined by expansion (26) and therefore it is governed by the Cauchy problem (27)-(28). Really, taking into account the validity of the following equality for the expression g . . .
and product formula (30) for f 1 (t, i) defined by series (26), we finally get equality (31). Thus, in the mean-field scaling limit an equivalent approach to the description of the kinetic evolution of quantum many-particle systems in terms of the Cauchy problem (27)-(28) of the Vlasov kinetic equation is given by the Cauchy problem (15)-(16) of the dual quantum Vlasov hierarchy for the additive-type marginal observables. In case of the k-ary marginal observables a solution of the dual quantum Vlasov hierarchy (15) is equivalent in the sense of equality (31) to preserving of a chaos property for k-particle marginal density operators.

The dual mean-field dynamics and the nonlinear Schrödinger equation
If the initial state is a pure state, i.e. f 0 1 = |ψ 0 ψ 0 |, the Heisenberg picture of evolution of quantum many-particle systems described by the dual quantum Vlasov hierarchy (15) is an equivalent to the Schrödinger picture of evolution governed by the Hartree equation.

On the generalized quantum kinetic equation
We consider the relations of the evolution of observables and the evolution of quantum states described in terms of a one-particle marginal density operator in the general case. In case of initial states specified by a one-particle marginal density operator, the dual BBGKY hierarchy describes the dual picture of evolution to the picture of the evolution of states governed by the generalized quantum kinetic equation and an infinite sequence of explicitly defined functionals of the solution of such evolution equation. In fact, the following equality is true where the initial state F c is defined by (22) and F (t | F 1 (t)) = (F 1 (t), F 2 (t | F 1 (t)), . . . , F s (t; 1, . . . , s | F 1 (t)) is a sequence of marginal functionals of the state. The functionals F s t, 1, . . . , s | F 1 (t) , s ≥ 2, are represented by the expansions over products of the one-particle density operator F 1 (t) 1 n! Tr s+1,...,s+n V 1+n t, {Y }, s + 1, . . . , s + n where the (n + 1)-order evolution operator V 1+n (t), n ≥ 0, are defined as follows and D j :Z j = l j X l j is the sum over all possible dissections D j of the linearly ordered set Z j ≡ (s + n−n 1 −. . . −n j + 1, . . . , s + n−n 1 −. . . −n j−1 ) on no more than s + n−n 1 −. . . −n j linearly ordered subsets [22]. In expression (34) it is denoted by A 1+n (t) the (1 + n)-order cumulant of the groups of scattering operators where P is the sum over all possible partitions of the set ({Y }, s + 1, . . . , s + n) into |P| nonempty mutually disjoint subsets X i ⊂ ({Y }, s + 1, . . . , s + n) and the group of operators G |X i | (−t) is adjoint to the group G |X i | (t) in the sense of functional (11).
The one-particle density operator F 1 (t) is determined by the following series where the cumulants A 1+n (−t), n ≥ 0, are defined by the formula similar to (8) A 1+n (−t, 1, . . . , n + 1) where P is the sum over all possible partitions of the set (1, . . . , n + 1) into |P| nonempty mutually disjoint subsets X i ⊂ (1, . . . , n + 1). The one-particle density operator (36) is a solution of the following initial-value problem where the evolution operator V 1+n (t) is defined by formula (34). We refer to evolution equation (37) as the generalized quantum kinetic equation. For systems of classical particles such equation was formulated in [17] and for quantum many-particle systems in [22]. To verify equality (32) we transform functional G(t)|F c as follows where the (1 + n)-order cumulant A 1+n (−t, {Y }, s + 1, . . . , s + n) is defined by (35). For F 0 1 ∈ L 1 (H) and G s (0) ∈ L(H s ) obtained functional (39) exists under the condition F 0 1 L 1 (H) < e −1 . Then we expand the cumulants A 1+n (−t) over the new evolution operators V 1+n (t), n ≥ 0, into the kinetic cluster expansion [22] A 1+n (−t, {Y }, s + 1, . . . , s + n) = n n 1 =0 where D:Z= l X l , |D|≤s+n−n 1 is the sum over all possible dissections D of the linearly ordered set Z ≡ (s + n−n 1 + 1, . . . , s + n) on no more than s + n−n 1 linearly ordered subsets. Representing series over the summation index n and the sum over the summation index n 1 in functional (39) as the two-fold series and identifying the series over the summation index n 1 with the products of one-particle density operators similar to formula (30) ∞ n 1 =0 Tr s+n+1,...,s+n+n 1 we transform functional (39) to the form in terms of marginal functionals of the state (33). Thus, equality (32) holds.
Thus, in case of initial data (22) which is completely characterized by the one-particle marginal density operator F 0 1 , solution (9) of the Cauchy problem (3)-(4) of the dual quantum BBGKY hierarchy for marginal observables and a solution of the Cauchy problem of the generalized kinetic equation (37)-(38) together with marginal functionals of the state (33) give two equivalent approaches to the description of the evolution of quantum many-particle systems.

Conclusions
We develop an approach of the description of kinetic evolution of quantum many-particle systems in terms of the evolution of marginal observables. One of the advantage of such approach is the possibility to construct the kinetic equations in scaling limits if there are correlations of particle states at initial time [17], for instance, correlations characterizing the condensate states [12].
In the case of quantum systems of particles obeying Fermi or Bose statistics [23] quantum kinetic equations have the different structure from formulated above. The analysis of these cases will be given in a separate paper.
Finally we point out the relation of the generalized quantum kinetic equation (37) and the specific quantum kinetic equations. The last can be derived from the generalized quantum kinetic equation in the appropriate scaling limits [13] or as a result of certain approximations. For example, in the mean-field limit we derive the quantum Vlasov kinetic equation [22]. Observing that in the kinetic (macroscopic) scale of the variation of variables [24] the groups of operators (6) of finitely many particles depend on microscopic time variable ε −1 t, where ε ≥ 0 is a scale parameter, the dimensionless marginal functionals of the state are represented in the form: F s ε −1 t, Y | F 1 (t) . Then in the limit ε → 0 the first two terms of the dimensionless marginal functional expansions (33) (−N int (i 1 , s + 1)) G s+1 (ε −1 t, Y, s + 1) coincide with corresponding terms constructed by the perturbation method with the use of the weakening of correlation condition by Bogolyubov [17] Thus, in the kinetic scale the collision integral of the generalized kinetic equation (37) takes the form of Bogolyubov's collision integral [17], [25] which enables to control correlations of infinite-particle systems. In a space homogeneous case the collision integral of the first approximation has a more general form than the quantum Boltzmann collision integral.