Nonequilibrium Density Profiles in Lorentz Tubes with Thermostated Boundaries

We consider a long Lorentz tube with absorbing boundaries. Particles are injected into 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 that 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. © 2016 Wiley Periodicals, Inc.