Interaction integrals in the kinetic theory of gases

Abstract

Interaction integrals V^ℓ v arise naturally in so-called moment theories of solution of Boltzmanns equation in which the velocity distribution function is represented by an expansion in Burnett functions. They are essentially matrix elements of scattering cross sections with respect to Sonine polynomials. In this paper, we discuss firstly the way in which the V^ℓ v arise in the test particle problem when representing the linear Boltzmann collision operator in the Burnett function basis with two free parameters, temperature and drift velocity respectively. Secondly, properties of V^ℓ v are discussed, recurrence relationships derived and both analytic and numerical evaluation is carried out for both elastic and inelastic processes, described by various model and realistic cross sections. Finally, the significance of these results for transport coefficient calculation is briefly discussed.