Momentengleichungen mit Boltzmann- und Fokker-Planck-Stoßintegralen

Abstract

There are discussed some questions on the calculation of the collision integrals for the moment equations of the kinetic theory, using once the Boltzmann collision term, on the other hand the Fokker-Planck form of this term, which is obtained by expanding the Boltzmann term in a Taylor series. We than show that both forms give the same results for the collision integrals in the moment equation of the mth order, if the Taylor expansion was carried out to the mth order too. This equivalence does not depend on the law of interaction between the colliding particles. For strong deviations from equilibrium the interaction therefore determines the convergence of the Fokker-Planck expansion and, with the equivalence derived in section 3 of this paper, the number of essential moments. Especially for the two extreme cases of Coulomb potential and rigid spheres we get the following consequences: In the first case (long range forces) the number of essential terms in the Fokker-Planck equation is two. Therefore the first two moment equations take account of the principal properties of the Boltzmann collision term in the kinetic equation. But for short range forces and strong deviations from equilibrium higher moment equations must not be neglected in the approximative solution of the Boltzmann equation. The integration in velocity space, needed fot this moment method, is easily carried out with the integral superposition of Weitzsch for the distribution function. This was done by Sucky, using the whole BOLTZMANN term, for every interaction potential U ∼ r−p, p > 0. In the case of p = 1, however, the calculation employing the Fokker-Planck: term becomes much simpler, since the Taylor series breakes off after the second term, and one may use the Rosenbluth-McDonald-Judd formula.