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Evolution of Correlations of Many-Particle Quantum Systems in Condensed States

V. I. Gerasimenko

Published Dec 24, 2018• 0 citations
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Abstract

We review some new approaches to the description of the evolution of states of many-particle quantum systems by means of the correlation operators. Using the denition 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 many-particle quantum systems. Moreover, we consider the problem of the rigorous description of collective behavior of many-particle quantum 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.

DOI
10.20944/preprints201812.0286.v1

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The paper deals with the problem of the rigorous description of the evolution of states of large particle quantum systems by means of correlation operators. A nonperturbative solution of the Cauchy problem of the hierarchy of nonlinear evolution equations for a sequence of marginal correlation operators is constructed. For initial states specified in terms of a one-particle density operator and correlation operators we also develop an approach to the description of the processes of the creation and the propagation of correlations within the framework of a one-particle density operator. Moreover, the mean field asymptotic behavior of constructed marginal correlation operators is established.
We consider the origin of the microscopic description of correlations in quantum systems. For correlation operators that give the alternative description of the quantum state evolution of many-particle systems, we deduce the von Neumann hierarchy of nonlinear equations and discuss the relationships of its solution both with the solution of the BBGKY hierarchy and with the s -particle correlation operators. A solution of the von Neumann hierarchy 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 of the von Neumann equations for the density operators. 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 origin of the microscopic description of correlations in quantum many-particle systems obeying Fermi–Dirac and Bose–Einstein statistics. For correlation operators that give the alternative description of the quantum state evolution of Bose and Fermi particles, we deduce the von Neumann hierarchy of nonlinear equations and construct the solution of its initial-value problem in the corresponding spaces of sequences of trace class operators. The links of constructed solution both with the solution of the quantum BBGKY hierarchy and with the nonlinear BBGKY hierarchy for marginal correlation operators are discussed. The solutions of the Cauchy problems of these hierarchies are constructed, in particular for initial data satisfying a chaos property. Copyright © 2010 John Wiley & Sons, Ltd.
We consider the problem of the rigorous description of nonequilibrium quantum correlations. Within the framework of an alternative approach to the description of the evolution of states of finitely many particles in terms of correlation operators governed by the von Neumann hierarchy, we derive the nonlinear quantum Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy for marginal correlation operators that is adopted to the description of quantum infinite‐particle systems as well. A nonperturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed. Copyright © 2013 John Wiley & Sons, Ltd.
In the paper we discuss possible approaches to the problem of the rigorous derivation of quantum kinetic equations from underlying many-particle dynamics. For the description of a many-particle evolution we construct solutions of the Cauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitable Banach spaces. In the framework of the conventional approach to the description of kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchy solution is constructed. We develop also alternative approaches. One method is based on the construction of the solution asymptotics of the initial-value problem of the quantum dual BBGKY hierarchy. One more approach is based on the generalized quantum kinetic equation that is a consequence of the equivalence of the Cauchy problems of such evolution equation and the BBGKY hierarchy with initial data determined by the one-particle density operator.