Nonequilibrium Stationary Solutions of Thermostated Boltzmann Equation in a Field

TL;DR

This paper analyzes the stationary solutions of a thermostated Boltzmann equation under an external field, revealing velocity distributions that vanish beyond a cutoff and exhibit specific asymptotic behaviors, with explicit solutions in one dimension.

Contribution

It derives the form of nonequilibrium stationary velocity distributions for a Boltzmann system with a Gaussian thermostat under an external field, including explicit solutions in one dimension.

Findings

Velocity distribution vanishes for |v| > v_max(E) with v_max(E) ~ 1/|E| as E -> 0.

Distribution approximates exp(-c|v|^3) for fixed v in the small E limit.

Explicit distribution solutions are obtained in one-dimensional cases.

Abstract

We consider a system of particles subjected to a uniform external force E and undergoing random collisions with "virtual" fixed obstacles, as in the Drude model of conductivity. The system is maintained in a nonequilibrium stationary state by a Gaussian thermostat. In a suitable limit the system is described by a self consistent Boltzmann equation for the one particle distribution function f. We find that after a long time f(v,t) approaches a stationary velocity distribution f(v) which vanishes for large speeds, i.e. f(v)=0 for |v|>vmax(E), with vmax(E)~1/|E| as |E| -> 0. In that limit f(v)~exp(-c|v|^3) for fixed v, where c depends on mean free path of the particle. f(v) is computed explicitly in one dimension.