Nonequilibrium stationary state of a current-carrying thermostated system

¹ School of Mathematics, Georgia Institute of Technology - Atlanta, GA 30332, USA
² Department of Mathematics, University of Alabama at Birmingham - Birmingham, AL 35294, USA
³ Departments of Mathematics and Physics, Rutgers University - Piscataway, NJ 08854, USA

Received: 17 January 2013
Accepted: 12 March 2013

Abstract

We find an explicit expression for the long time evolution and stationary speed distribution of N point particles in 2D moving under the action of a weak external field E, and undergoing elastic collisions with either a fixed periodic array of convex scatterers, or with virtual random scatterers. The total kinetic energy of the N-particles is kept fixed by a Gaussian thermostat which induces an interaction between the particles. We show analytically and numerically that for weak fields this distribution is universal, i.e., independent of the position or shape of the obstacles, as far as they form a dispersing billiard with finite horizon, or the nature of the stochastic scattering. Our results are nonperturbative. They exploit the existence of two time scales; the velocity directions become uniformized in times of order unity while the speeds change only on a time scale of O(|E|⁻²).