Homogeneous kinetic equations for probabilistic linear collisions in multiple space dimensions

Abstract

We analyze the convergence to equilibrium in a family of Kac-like kinetic equations in multiple space dimensions. These equations describe the change of the velocity distribution in a spatially homogeneous gas due to binary collisions between the particles. We consider a general linear mechanism for the exchange of the particles' momenta, with interaction coefficients that are random matrices with a distribution that is {independent} of the velocities of the colliding particles. Applying a synthesis of probabilistic methods and Fourier analysis, we are able to identify sufficient conditions for the existence and uniqueness of a stationary state, we characterize this stationary state as a mixture of Gaussian distributions, and we prove equilibration of transient solutions under minimal hypotheses on the initial conditions. In particular, we are able to classify the high-energy tails of the stationary distribution, which might be of Pareto type. We also discuss several examples to which our theory applies, among them models with a non-symmetric stationary state.