  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.
