ABSTRACT:

We consider the 2D periodic Lorentz gas in the Boltzmann-Grad limit and propose a kinetic equation to describe it. In the case of a random (Poisson) distribution of obstacles the limit is described by a linear Boltzmann equation for the phase space density. In the periodic case the collision times are not exponentially distributed and the impact parameters of successive collisions are not indipendent. The limiting behavior is described by a kinetic equation for a density distribution in an extended phase space which includes position, velocity and two additional variables: the time to the next (or last) collision and the impact parameter of the last collision. The results presented here have been obtained in a collaboration with Francois Golse.