On the validity of the Boltzmann equation for short range potentials

TL;DR

This paper proves that a classical particle system with short-range interactions behaves according to the Boltzmann equation in the low-density limit, extending previous results to smooth potentials and providing explicit convergence rates for repulsive cases.

Contribution

It extends Lanford's and King’s results to smooth, short-range potentials, demonstrating the validity of the Boltzmann equation in this broader context and quantifying convergence rates for repulsive interactions.

Findings

System behaves as predicted by Boltzmann equation at low density

Extension of previous results to smooth, short-range potentials

Explicit convergence rate for repulsive potentials

Abstract

We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low--density (Boltzmann--Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King (unpublished), appeared after the well known result of Lanford for hard spheres, and of a recent paper by Gallagher et al (arXiv: 1208.5753v1). Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence.