Derivation of the particle dynamics from kinetic equations

TL;DR

This paper explores the paradox of time-reversible microscopic solutions within the time-irreversible Boltzmann-Enskog equation, analyzing solution classes and proposing a novel approach to derive microdynamics from kinetic equations.

Contribution

It demonstrates how the reversibility of solutions depends on their class and introduces approximate microscopic solutions, offering a new perspective on deriving microdynamics from kinetic equations.

Findings

Reversible solutions are delta-function sums.

Irreversible solutions are continuously differentiable.

Approximate microscopic solutions are reversible on bounded intervals.

Abstract

We consider the microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov. The fact that the time-irreversible kinetic equation has time-reversible microscopic solutions is rather surprising. We analyze this paradox and show that the reversibility or irreversibility property of the Boltzmann-Enskog equation depends on the considered class of solutions. If the considered solutions have the form of sums of delta-functions, then the equation is reversible. If the considered solutions belong to the class of continuously differentiable functions, then the equation is irreversible. Also, we construct the so called approximate microscopic solutions. These solutions are continuously differentiable and they are reversible on bounded time intervals. This analysis suggests a way to reconcile the time-irreversible kinetic equations with the time-reversible particle dynamics. Usually one tries to derive the kinetic equations from the particle dynamics. On the contrary, we postulate the Boltzmann-Enskog equation or another kinetic equation and treat their microscopic solutions as the particle dynamics. So, instead of the derivation of the kinetic equations from the microdynamics we suggest a kind of derivation of the microdynamics from the kinetic equations.