Exact steady state solution of the Boltzmann equation: A driven 1-D inelastic Maxwell gas

TL;DR

This paper derives an explicit series form of the steady state velocity distribution for a driven inelastic Maxwell gas, revealing an exponential high-energy tail and detailed crossover behavior near elastic limits.

Contribution

It provides an explicit series solution for the steady state distribution of an inelastic Maxwell model, including high-energy tail analysis and crossover behavior near elastic and inelastic limits.

Findings

High-energy tail is exponential, $f(c) o A_0 e^{-a|c|}$.

Distribution approaches Maxwellian in the quasi-elastic limit.

Crossover from Maxwellian to exponential tail occurs at specific velocity scales.

Abstract

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, $f(c)\simeq A_0\exp(-a|c|)$, where $a\equiv 2/\sqrt{1-\alpha^2}$ and the amplitude $A_0$ 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 $|c-c_0|\sim 1/\sqrt{q}$ 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, $\ln f(c_0)\simeq q^{-1}\ln q$.