Asymptotic Solutions of the Nonlinear Boltzmann Equation for Dissipative Systems

Abstract

Analytic solutions F(v,t) of the nonlinear Boltzmann equation in d dimensions are studied for a new class of dissipative models, called inelastic soft spheres, interacting through pseudo-power law repulsions, and characterized by a collision frequency ∝ g ^ν, where g is the relative speed of the colliding particles. These models embed inelastic hard spheres (ν = 1) and inelastic Maxwell models (ν = 0). The systems are either freely cooling without energy input or driven by thermostats, e.g. white noise, and approach stable nonequilibrium steady states, or marginally stable homogeneous cooling states, where the data, v ^d₀ (t)F(v,t) plotted versus c=v/v ₀(t), collapse on a scaling or similarity solution f(c), where v ₀(t) is the r.m.s. velocity. The dissipative interactions generate overpopulated high energy tails, described generically by stretched Gaussians, f(c) ∼ exp[−βc ^b] with 0 < b < 2, where b = ν with ν > 0 in free cooling, and $b = 1 + \tfrac{1}{2}v$ with ν ≥ 0 when driven by white noise. Power law tails, f(c) ∼ 1/c ^a+d, are only found in marginal cases, where the exponent a is the root of a transcendental equation. The stability threshold depend on the type of thermostat, and is for the case of free cooling located at ν = 0.

Moreover we analyze an inelastic BGK-type kinetic equation with an energy dependent collision frequency coupled to a thermostat, that captures all qualitative properties of the velocity distribution function in Maxwell models, as predicted by the full nonlinear Boltzmann equation, but fails for harder interactions with ν > 0. This review was completed early February 2003 and covers most of the relevant literature available at that time.