Application of Simple Models to the Study of Nonequilibrium Behaviour of Inelastic Gases

Abstract

The dynamics of cooling inelastic gases and the evolution of their velocity fields is addressed by studying a series of simplified models. We first discuss a model of inelastic hard particles, which shows an uncorrelated transient phase (homogeneous cooling state or Haff regime) followed by the emergence of structures in the velocity and density field. Motivated by the linear stability analysis of the Haff regime, which predicts the appearance of density clusters only after the formation of structures in the velocity field, we focus our attention on the velocities of the gas particles. We study the so-called inelastic Maxwell model (IMM), first in a version with infinite connectivity which is the analog of mean-field spin systems. Secondly, we embed the model onto a lattice (in one dimension and two dimension), in order to observe spatial correlations. The mean-field IMM has the advantage that it lends itself to analytical treatment: in one dimension we find an exact asymptotic scaling solution for the probability density function (p.d.f.) of velocities. On the other hand, the lattice version displays typical spatial features of an inelastic gas, e.g., the transient Haff regime followed by the coarsening of structures in the velocity field (shocks in one dimension, vortices and shocks in two dimension), the so-called “return to the Gaussian” phenomenon of the velocity p.d.f. observed in MD simulations, etc. We show that the growth of structures in the lattice model is similar to that of domains in a diffusive field, but presents a short-scale disorder (“internal noise”) which is induced by the randomizing effect of collisions. In the lattice model, we can also establish the presence or absence of a mesoscopic scale which is required for a hydrodynamics description of the field evolution.