Small-parameter expansions of linear Boltzmann collision operators

Abstract

An important problem in the study of irreversible processes is the approximation of the linear Boltzmann operator by the Fokker-Planck operator or, when this is insufficient, by still further terms. The first attempt at a solution of this problem was the Kramers-Moyal expansion. Van Kampen and Siegel have reformulated this expansion with the aim of systematizing terms according to order of magnitude. Very few studies have been made of this problem in the multi-dimensional case, particularly collision processes. The present paper derives a new and greatly simplified formula for the coefficients of the Kramers-Moyal series in collision processes. This is applied to Siegel's order-of-magnitude rearrangement (“CD expansion”), which is found to have the required property at least in the case of power-law molecular interactions. Finally, we present a new stochastic model, related to the theory of collision processes, for the sake of its mathematical interest and simplicity.