Abstract. 
We consider a two dimensional Lorentz tube of finite horizon and of
length L. Starting from time -T, non-interacting particles are
injected from both ends to the tube with different rates and both ends
are assumed to be absorbing. In the case of infinite T, we observe a
linear density profile of particles for large L. If T is of order L^2,
then the density profile converges to the solution of the heat
equation with Dirichlet boundary conditions. In order to prove the
above results, we establish the convergence to the Brownian meander of
a Lorentz particle conditioned on not retuning to its starting cell.
This is a joint work in progress with D. Dolgopyat.