Microscopic solutions of the Boltzmann-Enskog equation in the series representation

TL;DR

This paper provides a rigorous mathematical framework for microscopic solutions of the Boltzmann-Enskog equation for hard sphere gases, using a series representation to clarify their meaning.

Contribution

It introduces a mathematically rigorous approach to define microscopic solutions via a series representation, resolving previous ambiguities.

Findings

Established a formal series representation for microscopic solutions.

Provided a rigorous mathematical interpretation of empirical measure solutions.

Clarified the connection between microscopic solutions and the Boltzmann-Enskog equation.

Abstract

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation.