Approach to a stationary state in an external field

TL;DR

This paper analyzes the relaxation process of a particle in a thermal bath under an external field, providing analytical solutions and demonstrating the exponential relaxation of velocity distribution and the validity of the Green-Kubo formula for diffusion.

Contribution

It offers analytical solutions for specific models and confirms the Green-Kubo formula's applicability in stationary non-equilibrium states.

Findings

Velocity distribution relaxes exponentially to stationary form

Hydrodynamic diffusive mode governs position distribution

Green-Kubo formula accurately predicts diffusion coefficient

Abstract

We study relaxation towards a stationary out of equilibrium state by analizing a one-dimensional stochastic process followed by a particle accelerated by an external field and propagating through a thermal bath. The effect of collisions is described within Botlzmann's kinetic theory. We present analytical solutions for the Maxwell gas and for the very hard particle model. The exponentially fast relaxation of the velocity distribution toward the stationary form is demonstrated. In the reference frame moving with constant drift velocity the hydrodynamic diffusive mode is shown to govern the distribution in the position space. We show that the exact value of the diffusion coefficient for any value of the field is correctly predicted by Green-Kubo autocorrelation formula generalized to the stationary state.