## Abstract

We review new approaches to the description of the evolution of states of large quantum particle systems by means of the marginal correlation operators. Using the definition of marginal correlation operators within the framework of dynamics of correlations governed by the von Neumann hierarchy, we establish that a sequence of such operators is governed by the nonlinear quantum BBGKY hierarchy. The constructed nonperturbative solution of the Cauchy problem to this hierarchy of nonlinear evolution equations describes the processes of the creation and the propagation of correlations in large quantum particle systems. Furthermore, we consider the problem of the rigorous description of collective behavior of quantum many-particle systems by means of a one-particle (marginal) correlation operator that is a solution of the generalized quantum kinetic equation with initial correlations, in particular, correlations characterizing the condensed states of systems.

### Keywords

- von Neumann hierarchy
- nonlinear quantum BBGKY hierarchy
- quantum kinetic equation
- correlation of states
- scaling limit
- 2000 Mathematics Subject Classification: 35Q40; 47D06

## 1. Introduction

In this chapter, we consider mathematical problems concerning the description of processes of a creation and a propagation of correlations in quantum many-particle systems, namely, correlations in quantum systems both finitely and infinitely many particles and the description of correlations by means of the state of typical particle of large quantum particle system.

As known, the marginal correlation operators give an equivalent approach to the description of the evolution of states of quantum systems of many particles in comparison with marginal density operators [1]. The physical interpretation of marginal correlation operators is that the macroscopic characteristics of fluctuations of mean values of observables are determined by them on the microscopic level [1, 2].

Traditionally marginal correlation operators are introduced by means of the cluster expansions of the marginal density operators [2, 3, 4]. In articles [5, 6] an approach based on the definition of the marginal correlation operators within the framework of dynamics of correlations governed by the von Neumann hierarchy was developed. As a result of which, it is established that the marginal correlation operators are governed by the hierarchy of nonlinear evolution equations, known as the quantum nonlinear BBGKY (Bogoliubov-Born-Green-Kirkwood-Yvon) hierarchy, and its solution is represented in the form of series, the generating operator of every term of which are the corresponding-order cumulant of groups of nonlinear operators of the von Neumann hierarchy for correlation operators [7].

In the chapter, we also consider the problem of the rigorous description of the evolution of correlations in quantum many-particle systems by means of a one-particle (marginal) density operator that is a solution of the generalized quantum kinetic equation with initial correlations [8]. We remark that initial states specified by correlations are typical for the condensed states of many-particle systems in contrast to their gaseous state [1].

We note that in modern researches, the conventional approach to the problem of the rigorous derivation of kinetic equations lies in the construction of various scaling limits of a solution of equations, describing the evolution of the state of many-particle systems [9], in particular, a mean field limit of a perturbative solution of the BBGKY hierarchy for a sequence of marginal density operators [10, 11, 12, 13, 14, 15, 16, 17].

## 2. Dynamics of quantum correlations

As known [1, 2], quantum systems of fixed number of particles are described in terms of 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 the evolution of quantum systems, namely, in terms of observables that are governed by the Heisenberg equation and in terms of states governed by the von Neumann equation for the density operator, respectively. An equivalent approach to the description of states of quantum systems is given by means of operators determined by the cluster expansions of the density operator which are interpreted as correlation operators. In this section we consider fundamental equations describing the evolution of correlations of quantum systems with a finite number of particles.

### 2.1 Preliminaries

We denote by

where

On the space of trace class operators

where the following units are used:

On its domain of the definition, the infinitesimal generator

that has the following structure:

Let the symbol

where the generating operator

and

Below we adduce the examples of mapping expansions (3):

For

the following estimate is true:

where

### 2.2 The von Neumann hierarchy for correlation operators

The evolution of all possible states of a quantum system of non-fixed, i.e., arbitrary but finite, number of identical particles, obeying the Maxwell-Boltzmann statistics, can be described by means of the sequence

where

We remark that correlation operators can be introduced by means of the cluster expansions [2] of the density operators (the kernel of a density operator is known as a density matrix) governed by a sequence of the von Neumann equations, and hence, they describe the evolution of states by an equivalent method in comparison with the density operators. For quantum systems of fixed number of particles, the state is described by finite sequence of correlation operators governed by a corresponding system of the von Neumann equations (6).

A solution (nonperturbative solution) of the Cauchy problem of the von Neumann hierarchy for correlation operators (6) and (7) is represented by group of nonlinear operators (3)

where a sequence of initial correlation operators (7) is denoted by

We remark, if at initial time there are no correlations between particles, i.e., in the case of initial states, satisfying a chaos condition [2], a sequence of initial correlation operators takes the form

where the operator

and we used notations accepted in formula (3).

We remark also that nonperturbative solution (8) of the Cauchy problem of the von Neumann hierarchy (6) and (7) can be transformed to the perturbation (iteration) expansion as a result of the application of analogs of the Duhamel equation to cumulants (4) of groups of operators (1).

The following statement is true [6]. In the case of bounded interaction potentials for

The stated above results can be extended to quantum systems of bosons and fermions like in paper [6].

## 3. The evolution of correlations in large quantum particle systems

An equivalent approach in describing the states of quantum systems of many particles consists in describing states by means of marginal density operators governed by the BBGKI hierarchy or by means of operators determined by their cluster expansions, which are interpreted as marginal correlation operators [1]. On the microscopic scale, the macroscopic characteristics of fluctuations of observables are directly determined by the marginal correlation operators. Such approach allows us to describe the evolution of correlations in quantum systems both with finite and infinite number of particles.

### 3.1 The hierarchy of evolution equations for marginal correlation operators

Traditionally marginal correlation operators are determined by means of the cluster expansions of the marginal density operators [2, 3, 4]. We introduce the marginal correlation operators in the framework of the solution of the Cauchy problem for the von Neumann hierarchy (6) and (7) by the following series expansions:

According to estimate (5), series (10) exists and the following estimate holds:

We remark that the macroscopic characteristics of fluctuations of observables are directly determined by marginal correlation operators (10), for example, the functional of the dispersion of the additive-type observables, i.e.,

where

Then the evolution of all possible states of large quantum particle systems, obeying the Maxwell-Boltzmann statistics, can be described by means of the sequence

where

If

where the generating operator

and composition of mappings (3) of the corresponding noninteracting groups of particles we denote by

Below we adduce the examples of expansions (14). The first-order cumulant of the groups of nonlinear operators (3) is the same group of nonlinear operators, i.e.,

In the case of

where the operator

is the third-order cumulant (9) of groups of operators (1).

In the case of initial data specified by the sequence of marginal correlation operators

i.e., initial states satisfying a chaos property [9], according to definition (14), marginal correlation operators (13) are represented by the following series expansions:

where the generating operator

We note that within the framework of the description of states by means of marginal density operators defined by cluster expansions over marginal correlation operators

initial states described like to sequence (15) is specified by the sequence

where the generating operator

One of the possible methods to derive series expansion (13) for the marginal correlation operators lies in the substitution of the cluster expansions of groups of nonlinear operators (3) over cumulants (14) and the sequence of initial correlation operators

into the definition of marginal correlation operators (10). Indeed, developing the generating operators of series (13) as the following cluster expansions:

according to definition (17), we derive expressions (13). The solutions of recursive relations (18) are represented by expansions (14).

We remark that on the space

as examples, we adduce the simplest examples of reduced cumulants (19):

We note also that a nonperturbative solution of the nonlinear quantum BBGKY hierarchy (13) or in the form of series expansions with generating operators (19) can be transformed to the perturbation (iteration) series as a result of the application of analogs of the Duhamel equation to cumulants (4) of groups of operators (1).

The following statement is true [7]. If

### 3.2 A mean field asymptotic behavior of marginal correlation operators

Now we deal with a scaling asymptotic behavior of the constructed marginal correlation operators in a mean field limit in the case of initial state satisfied condition (15).

Let us observe that if

As a result of this for the

We assume the existence of a mean field limit for initial marginal correlation operator (or a one-particle density operator) in the following sense:

Then, taking into account equality (20), and since the

If for the initial marginal correlation operator equality (21) holds, then in the case of

where for arbitrary finite time interval, the limit one-particle marginal correlation operator

In series expansion (23), the operator

As a result of differentiation in the sense of the norm convergence of the space

Then for pure states we derive the Hartree equation [2], indeed, in terms of the kernel

where the function

We note that in the case of pure states, kinetic equation (24) can be reduced to the nonlinear Schrödinger equation [12] or to the Gross-Pitaevskii kinetic equation [13].

## 4. The description of processes of a creation and a propagation of correlations by means of kinetic equations

In this section we consider mathematical problems concerning the description of processes of creation and propagation of correlations within framework of the state of typical particle of quantum systems of many particles; in other words, an approach to the description of evolution of correlations by means of quantum kinetic equations is developing.

### 4.1 Marginal correlation functionals of the state

Further we shall consider the case of initial states specified by a one-particle marginal density operator with correlations, namely, initial states specified by the following sequence of marginal correlation operators:

where the operators

For initial states specified in terms of a one-particle density operator and correlation operators (26), the evolution of states given within the framework of the sequence

In the case under consideration, the marginal correlation functionals

where the generating operator

where the

In formula (29) the sum of all possible dissections [18] of the linearly ordered set

where the operator

A method of the construction of marginal correlation functionals (28) is based on the application of kinetic cluster expansions [2] to the generating operators of series (13). If

We emphasize that marginal correlation functionals (28) describe all the possible correlations generated by dynamics of large quantum particle system with initial correlations by means of a one-particle density operator.

### 4.2 The generalized quantum kinetic equation with initial correlations

Now we establish the evolution equation for one-particle (marginal) density operator (27). As a result of the differentiation over time variable of the operator represented by series expansion (27) in the sense of the norm convergence of the space

where the second part of the collision integral in equality (30) is determined in terms of the marginal correlation functional represented by series expansions (28) in the case of

We emphasize that the coefficients in an expansion of the collision integral of the non-Markovian kinetic equation (30) are determined by the operators specified initial correlations (26).

On the space

The proof of this existence statement is similar to the proof in the case of the generalized quantum kinetic equation given in [18].

### 4.3 On a propagation of initial correlations in a mean field limit

Further we establish the mean field asymptotic behavior of constructed marginal correlation functionals (28) in the case of initial states specified by the one-particle density operator with correlations (26).

We assume the existence of a mean field limit of an initial one-particle density operator in sense (21) and for initial correlation operators as follows:

Then in consequence of the validity of equalities (20) for one-particle density operator (27), the following statement is true [8]. If conditions (21) and (31) hold, then for series expansion (27) the equality holds:

where for finite time interval, the limit one-particle density operator

In series expansion (32), the operator

For marginal correlation functionals (28), the following statement is true [8]. Under conditions (21) and (31) on initial state (26), there exists a mean field limit of marginal correlation functionals (28) in the following sense:

where the limit marginal correlation functionals

and, respectively, the limit one-particle density operator

The proof of these statements is based on the validity of equality (20) for cumulants of asymptotically perturbed groups of operators (1) and the explicit structure of the generating operators of series expansions (28) of marginal correlation functionals and of series expansion (27).

We remark that limit marginal correlation functionals (32) and (33) are a solution of the Cauchy problem of the quantum Vlasov hierarchy of nonlinear evolution equations [6], which describes a mean field asymptotic behavior of marginal correlation operators in the case of arbitrary initial states, namely,

where we used notations similar to accepted above.

It should be noted that limit marginal correlation functionals (33) describe the process of the evolution of correlations of large quantum particle systems by means of a one-particle density operator in a mean field approximation.

Similar to the derivation of kinetic equation (30), we establish that the one-particle density operator represented by series expansion (32) is a solution of the Cauchy problem of the Vlasov-type quantum kinetic equation with initial correlations:

and consequently, for pure states we derive the Hartree-type equation with initial correlations. We point out that Eq. (34) is the non-Markovian quantum kinetic equation.

Thus, we established that a mean field behavior of processes of the creation of correlations and the propagation of initial correlations in large quantum particle systems are governed by kinetic equation (34).

Moreover, in the case under consideration, the processes of the creation of correlations generated by dynamics of many-particle systems and the propagation of initial correlations are described by the constructed marginal functionals of the state (28) governed by the non-Markovian generalized kinetic equation with initial correlations (26).

## 5. Conclusion

In this chapter the process of a creation and a propagation of correlations in quantum many-particle systems has been described by means of the Cauchy problem of the quantum BBGKY hierarchy of nonlinear equations (11) and (12). A nonperturbative solution for a sequence of marginal correlation operators is represented in the form of series (13) the generating operator of every term of which are corresponding-order cumulant (14) of groups of nonlinear operators (3). In the case of initial state specified by a sequence of the marginal correlation operators that satisfy chaos property (15), the correlations generated by dynamics of large quantum particle system (16) are completely determined by the corresponding-order cumulants (4) of groups of operators (1). The obtained results can be extended to large quantum systems of bosons and fermions like in paper [6].

In the case of initial state satisfied condition (15), a mean field asymptotic behavior of the processes of a creation and a propagation of correlations was described. It was directly proven the property called the propagation of initial chaos (22), which underlies in mathematical derivation of effective evolution equations of systems of infinitely many particles [16].

The problem of the rigorous description of collective behavior of quantum many-particle systems by means of a one-particle (marginal) correlation operator that is a solution of the generalized quantum kinetic equation [18] with initial correlations [19], for instance, the initial correlations, characterizing the condensed states [1], or initial correlations that influence on ultrafast relaxation processes in plasmas [4] has been also considered.

In particular, such an approach to the derivation of the Vlasov-type quantum kinetic equation with initial correlations (34) from underlying dynamics governed by the generalized quantum kinetic equation with initial correlations (30) enables to construct the higher-order corrections to the mean field evolution of large quantum systems of particle.

We note that in paper [20] other approach to the description of the propagation of initial correlations of large quantum particle systems in a mean field limit was developed, namely, the process of the propagation of initial correlations was described within the framework of the evolution of marginal observables governed by the dual BBGKY hierarchy [2, 21].