# Non-isothermal inertial Brownian motion

## Abstract

The stationary distribution of a dilute gas of Brownian particles in an inhomogeneous thermal bath and in the presence of a force field is considered. Our aim is to understand the dynamics of the Brownian particles in cases when inertial effects are non-negligible and in conditions of zero mass current. We restrict the analysis to the one-dimensional case. Beyond the requirement that there is no mass transport, the temperature and force fields are arbitrary. The statistical description of this processes is governed by the Kramers equation, whose solution we find as an expansion in powers of the inverse of the friction coefficient. The resulting expressions for the position and velocity distribution, heat flux, etc., are tested against numerical simulations of the corresponding Langevin equation. We show that the appropriate interpretation of the Langevin equations, as inertia becomes less important (overdamped limit), is the Itô interpretation. For sufficiently large friction, the heat flux is linear in the temperature gradient, and the system can be analyzed using the tools of irreversible thermodynamics. We show that the entropy production is never negative, vanishing only at thermodynamic equilibrium. © 1998 Published by Elsevier Science E.V.