The rich behavior of the Boltzmann equation for dissipative gases

¹ Theoretische Fysica, Universiteit Utrecht - Postbus 80.195 3508 TD Utrecht, The Netherlands
² Theoretical Biological Physics, UC San Diego - La Jolla, CA 92093, USA
³ LPTMS (UMR CNRS 8626), Université Paris-Sud - 91405 Orsay, France
⁴ LPT (UMR CNRS 8627), Université Paris-Sud - 91405 Orsay, France

Received: 23 May 2006
Accepted: 3 August 2006

Abstract

Within the framework of the homogeneous nonlinear Boltzmann equation, we present a new analytic method, without the intrinsic limitations of existing methods, for obtaining asymptotic solutions. This method permits the extension of existing results for Maxwell molecules and hard spheres to large classes of particle interactions, from very hard spheres to softer than Maxwell molecules, as well as to more general forcing mechanisms, beyond free cooling and white-noise driving. By combining this method with numerical solutions, obtained from the Direct Simulation Monte Carlo (DSMC) method, we study a broad class of models relevant for the dynamics of dissipative fluids, including granular gases. We establish a criterion connecting the stability of the non-equilibrium steady state to an exponentially bound form for the velocity distribution F, which varies depending on the forcing mechanism. Power laws arise in marginal stability cases, of which several new cases are reported. Our results provide a minimal framework for interpreting large classes of experiments on driven granular gases.