Microscopic and soliton-like solutions of the Boltzmann-Enskog and generalized Enskog equations for elastic and inelastic hard spheres

TL;DR

This paper rigorously defines microscopic and soliton-like solutions of the Boltzmann-Enskog and generalized Enskog equations for hard spheres, connecting them to the BBGKY hierarchy and multisoliton solutions.

Contribution

It introduces a regularization of delta-functions to give a rigorous meaning to microscopic solutions and links the Boltzmann-Enskog equation to the BBGKY hierarchy, also showing soliton-like solutions for the generalized Enskog equation.

Findings

Regularized delta-functions provide a rigorous framework for microscopic solutions.

Collision integral matches the BBGKY hierarchy for hard spheres.

Exact smooth soliton-like solutions exist for the generalized Enskog equation.

Abstract

N.N. Bogolyubov discovered that the Boltzmann-Enskog kinetic equation has microscopic solutions. They have the form of sums of delta-functions and correspond to trajectories of individual hard spheres. But the rigorous sense of the product of the delta-functions in the collision integral was not discussed. Here we give a rigorous sense to these solutions by introduction of a special regularization of the delta-functions. The crucial observation is that the collision integral of the Boltzmann-Enskog equation coincides with that of the first equation of the BBGKY hierarchy for hard spheres if the special regularization to the delta-functions is applied. This allows to reduce the nonlinear Boltzmann-Enskog equation to the BBGKY hierarchy of linear equations in this particular case. Also we show that similar functions are exact smooth solutions for the recently proposed generalized Enskog equation. They can be referred to as "particle-like" or "soliton-like" solutions and are analogues of multisoliton solutions of the Korteweg-de Vries equation.