A non-linear Boltzmann equation with analytic solutions

Abstract

An analytic solution is obtained for a certain non-linear one-dimensional Boltzmann equation describing the temporal relaxation to equilibrium of a system of particles, and its general features are elucidated. A solution is also found for the corresponding linearised problem and for two BGK models, one with the correct energy-dependent and the other with a mean energy-independent relaxation time. The accuracy of the linearised equation is superior to that of the models, even for large displacements from equilibrium, and the gain in accuracy of the energy-dependent BGK model over the energy-independent one may well be offset by the additional computational work involved in using the former. A calculation of the successive time derivatives of the entropy, based on the exact non-linear equation, show that these alternate in sign, at least up to the tenth derivative.