Non equilibrium density profiles in Lorentz tubes with thermostated boundaries

TL;DR

This paper analyzes non-equilibrium density profiles in Lorentz tubes with thermostated boundaries, demonstrating linear density in finite tubes and constant density in semi-infinite tubes, with convergence described by the heat equation.

Contribution

It provides new results on Lorentz particles, including convergence to the Brownian meander and local limit theorems, advancing understanding of non-equilibrium states.

Findings

Density profiles are constant in semi-infinite tubes.

Density profiles are linear in finite tubes.

Convergence to equilibrium follows the heat equation.

Abstract

We consider a long Lorentz tube with absorbing boundaries. Particles are injected to the tube from the left end. We compute the equilibrium density profiles in two cases: the semi-infinite tube (in which case the density is constant) and a long finite tube (in which case the density is linear). In the latter case, we also show that convergence to equilibrium is well described by the heat equation. In order to prove these results, we obtain new results for the Lorentz particle which are of independent interest. First, we show that a particle conditioned not to hit the boundary for a long time converges to the Brownian meander. Second, we prove several local limit theorems for particles having a prescribed behavior in the past.