THE LINEAR BOLTZMANN EQUATION FOR LONG-RANGE FORCES: A DERIVATION FROM PARTICLE SYSTEMS

Abstract

In this paper we consider a particle moving in a random distribution of obstacles. Each obstacle generates an inverse power law potential εs/|x|s, where ε is a small parameter and s>2. Such a rescaled potential is truncated at distance εγ-1, where γ∈] 0, 1[ is suitably large. We also assume that the scatterer density is diverging as ε-d+1, where d is the dimension of the physical space. We prove that, as ε→0 (the Boltzmann–Grad limit), the probability density of a test particle converges to a solution of the linear (uncutoff) Boltzmann equation with the cross-section relative to the potential V(x)=|x|-s.