The Boltzmann equation for driven systems of inelastic soft spheres

Abstract

We study a generic class of inelastic soft sphere models with a binary collision rate $g^\nu$ that depends on the relative velocity $g$. This includes previously studied inelastic hard spheres ($\nu=1$) and inelastic Maxwell molecules ($\nu=0$). We develop a new asymptotic method for analyzing large deviations from Gaussian behavior for the velocity distribution function $f(c)$. The framework is that of the spatially uniform nonlinear Boltzmann equation and special emphasis is put on the situation where the system is driven by white noise. Depending on the value of exponent $\nu$, three different situations are reported. For $\nu<-2$, the non-equilibrium steady state is a repelling fixed point of the dynamics. For $\nu>-2$, it becomes an attractive fixed point, with velocity distributions $f(c)$ having stretched exponential behavior at large $c$. The corresponding dominant behavior of $f(c)$ is computed together with sub-leading corrections. In the marginally stable case $\nu=-2$, the high energy tail of $f(c)$ is of power law type and the associated exponents are calculated. Our analytical predictions are confronted with Monte Carlo simulations, with a remarkably good agreement.