Evolution of correlations of quantum many-particle systems

Abstract

The Cauchy problem for the von Neumann hierarchy of nonlinear equations is investigated. One describes the evolution of all possible states of quantum many-particle systems by the correlation operators. A solution of such nonlinear equations is constructed in the form of an expansion over particle clusters whose evolution is described by the corresponding order cumulant (semi-invariant) of evolution operators for the von Neumann equations. For the initial data from the space of sequences of trace class operators the existence of a strong and a weak solution of the Cauchy problem is proved. We discuss the relationships of this solution both with the sparticle statistical operators, which are solutions of the BBGKY hierarchy, and with the s-particle correlation operators of quantum systems.

Groups of Operators for Evolution Equations of Quantum Many-particle Systems

“…The concept of cumulants (1.9) of groups (1.1) of the von Neumann equations forms the basis of group expansions for quantum evolution equations, namely, the von Neumann hierarchy for correlation operators [23], as well as the BBGKY hierarchy for s-particle density operators [24] and the dual BBGKY hierarchy [26]. In the case of quantum systems of particles obeying Fermi or Bose statistics groups (2.1),(3.2) and (4.2) have different structures.…”

“…The operator A n (t) we refer to as the nthorder cumulant (semi-invariant) of evolution operators (1.1). Some properties of cumulants (1.9) considered in [23]. The generator of the 1st-order cumulant is given by operator…”

“…As is well known, there are various possibilities to describe the evolution of quantum manyparticle systems [1]. The sequence of the von Neumann equations for density operators [2], [11], the von Neumann hierarchy for correlation operators [23], the BBGKY hierarchy for marginal density operators [11] and the dual BBGKY hierarchy for marginal observables [1] give the equivalent approaches for the description of the evolution of finitely many particles. Papers [23]- [27] constructed the one-parametric families of operators that define solutions of the Cauchy problem for these evolution equations.…”

“…The sequence of the von Neumann equations for density operators [2], [11], the von Neumann hierarchy for correlation operators [23], the BBGKY hierarchy for marginal density operators [11] and the dual BBGKY hierarchy for marginal observables [1] give the equivalent approaches for the description of the evolution of finitely many particles. Papers [23]- [27] constructed the one-parametric families of operators that define solutions of the Cauchy problem for these evolution equations. It was established that the concept of cumulants (semi-invariants) of groups of operators for the von Neumann equations forms the basis of the one-parametric families of operators of various evolution equations of quantum systems of particles, in particular, the BBGKY hierarchy for infinitely many particles [23].…”

A nonperturbative solution of the nonlinear BBGKY hierarchy for marginal correlation operators

“…We describe states of a system by means of sequences g.t/ D .g 0 , g 1 .t, 1/, : : : , g n .t, 1, : : : , n/, : : :/ 2 L 1 .F H / of the correlation operators g n .t/, n 1. The evolution of all possible states is determined by the initial-value problem of the von Neumann hierarchy [14,15]…”