Boltzmann collision operator for the infinite range potential: A limit problem

Abstract

The conventional Boltzmann collision operator for the infinite range inverse power law model was derived by Maxwell by adopting a collision kernel which is a limit of that for the finite range model by ignoring the glancing angles. Since the interpretation of collision operator for the infinite range potential through limit process to the one with finite range potential is natural in regard to the derivation of the Boltzmann equation. It is the purpose of this paper to clarify the physical meaning of the conventional collision operator for the infinite range inverse power law model through the study of the limiting process of the collision operator as the cutoff radius tends to infinity. We first estimate the extent in which the glancing angles can be ignored in the limiting process. Furthermore we prove that taking limit to collision operator with finite range potential directly will lead to the conventional one with algebraic convergence rate.