On Microscopic Origin of the Fokker - Planck Kinetic Evolution of Hard Spheres

TL;DR

This paper rigorously derives a generalized Fokker-Planck kinetic equation describing the evolution of a trace hard sphere in an environment of finitely many hard spheres, providing a detailed mathematical framework.

Contribution

It introduces a rigorous derivation of the Fokker-Planck kinetic equation for a trace hard sphere interacting with finitely many spheres, including explicit functionals of the distribution.

Findings

Derivation of the generalized Fokker-Planck equation for hard spheres

Explicit functionals describing the evolution of the trace sphere

Mathematical framework for kinetic evolution in finite environments

Abstract

The rigorous approach to the description of the kinetic evolution of a many-particle system composed of a trace hard sphere and an environment of finitely many hard spheres is developed. We prove that the evolution of states of a trace hard sphere in an environment can be described within the framework of the marginal distribution function governed by the generalized Fokker -- Planck kinetic equation and an infinite sequence of the explicitly defined functionals of this function.