Exact steady-state solution of the Boltzmann equation: a driven one-dimensional inelastic Maxwell gas

The exact nonequilibrium steady-state solution of the nonlinear Boltzmann equation for a driven inelastic Maxwell model was obtained by Ben-Naim and Krapivsky [Phys. Rev. E 61, R5 (2000)] in the form of an infinite product for the Fourier transform of distribution function f(c). In this paper we have inverted the Fourier transform to express f(c) in the form of an infinite series of exponentially decaying terms. The dominant high-energy tail is exponential, f(c) approximately A0 exp(-a|c|), where a identical with 2/square root[1-alpha(2)] and amplitude A0 is given in terms of a converging sum. This is explicitly shown in the totally inelastic limit (alpha-->0) and in the quasielastic limit (alpha-->1). In the latter case, the distribution is dominated by a Maxwellian for a very wide range of velocities, but a crossover from a Maxwellian to an exponential high-energy tail exists for velocities |c-c(0)| approximately 1/square root[q] around a crossover velocity c(0) approximately ln q(-1)/square root[q], where q identical with (1-alpha)/2<<1. In this crossover region the distribution function is extremely small, ln f(c(0)) approximately q(-1) ln q.