Quantum Kinetic Equations and Evolution of Many-Particle Systems

TL;DR

This paper explores rigorous derivations of quantum kinetic equations from many-particle quantum dynamics, constructing solutions to BBGKY hierarchies and proposing alternative approaches including dual hierarchies and generalized kinetic equations.

Contribution

It introduces new methods for deriving quantum kinetic equations, including solutions to BBGKY hierarchies and alternative approaches based on dual hierarchies and generalized equations.

Findings

Constructed solutions to quantum BBGKY hierarchy in Banach spaces

Developed alternative approaches using dual hierarchies

Established equivalence between kinetic equations and BBGKY hierarchy

Abstract

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.