The cumulative measure of a force: A unified kinetic theory for rigid-sphere and inverse-square force law interactions

Abstract

By introducing a cutoff on the cumulative measure of a force, a unified kinetic theory is developed for both rigid-sphere and inverse-square force laws. The difference between the two kinds of interactions is characterized by a parameter, γ, which is 1 for rigid-sphere interactions and -3 for inverse-square force law interactions. The quantities governed by γ include the specific reaction rates, kernels, collision frequencies, arbitrarily high orders of transition moments, arbitrarily high orders of Fokker-Planck expansion (also called Kramers-Moyal expansion) coefficients, and arbitrarily high orders of energy exchange rates. The cutoff constants are shown to be incomplete gamma functions of different orders. The widely used cutoff constant in plasma physics (usually known as Coulomb logarithm) is found to be exactly the zeroth order of the incomplete gamma function. The well known Arrhenius reaction rate formula comes from the first order of the incomplete gamma functions, while the negative first order can be used for fitting the fusion reaction rate between deuterium and tritium.