From BBGKY Hierarchy to Non-Markovian Evolution Equations

TL;DR

This paper derives a generalized non-Markovian evolution equation for microscopic phase densities from the BBGKY hierarchy, providing a microscopic foundation for non-Markovian kinetic descriptions.

Contribution

It introduces a sequence of marginal microscopic phase densities and derives a non-Markovian evolution equation, extending the Fokker-Planck collision integral.

Findings

Derivation of a generalized evolution equation for microscopic phase density

Introduction of a non-Markovian Fokker-Planck collision integral

Formulation of the BBGKY hierarchy for microscopic distributions

Abstract

The problem of description of the evolution of the microscopic phase density and its generalizations is discussed. With this purpose, the sequence of marginal microscopic phase densities is introduced, and the appropriate BBGKY hierarchy for these microscopic distributions and their average values is formulated. The microscopic derivation of the generalized evolution equation for average value of the microscopic phase density is given and the non-Markovian generalization of the Fokker-Planck collision integral is deduced.