Exact steady state solution of the Boltzmann equation: A driven 1-D 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 the 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, $f(c)\simeq A_0\exp (-a\mathrm{\mid }c\mathrm{\mid })$, where $a\equiv 2/\sqrt{1-\alpha^2}$andtheamplitude$A_0$ is given in terms of a converging sum. This is explicitly shown in the totally inelastic limit ($\alpha \to 0$) and in the quasi-elastic limit ($\alpha \to 1$). 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 $\mathrm{\mid }c-c_0\mathrm{\mid }\sim 1\mathrm{/}\sqrt{q}$ around a crossover velocity $c_0\simeq \ln q^{-1}/\sqrt{q}$,where$q\equiv (1-\alpha)/2\ll 1$. In this crossover region the distribution function is extremely small, $\ln f(c_0)\simeq q^{-1}\ln q$.