# Derivation of the particle dynamics from kinetic equations

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## Abstract

The microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov are considered. 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, the so called approximate microscopic solutions are constructed. These solutions are continuous and they are reversible on bounded time intervals. This analysis suggests a way to reconcile the time-irreversible kinetic equations with the timereversible 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.

## Key words

Boltzmann-Enskog equation Boltzmann equation kinetic equations microscopic solutions reversibility paradox## Preview

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