Typical Recurrence for the Ehrenfest Wind-tree Model

We show that the typical wind-tree model, in the sense of Baire, is recurrent and has a dense set of periodic orbits. The recurrence result also holds for the Lorentz gas : the typical Lorentz gas, in the sense of Baire, is recurrent. These Lorentz gases need not be of finite horizon! In 1912 Paul and Tatiana Ehrenfest proposed the wind-tree model of diffusion in order to study the statistical interpretation of the second law of thermodynamics and the applicability of the Boltzmann equation [EhEh]. In the Ehrenfest wind-tree model, a point particle (the “wind”) moves freely on the plane and collides with the usual law of geometric optics with randomly placed fixed square scatterers (the “trees”). The notion of “randomness” was not made precise, in fact it would have been impossible to do so before Kolmogorov laid the foundations of probability theory in the 1930s. We will call the subset of the plane obtained by removing the obstacles the billiard table, and the the motion of the point the billiard flow. From the mathematical rigorous point of view, there have been two results on recurrence for wind-tree models, both on a periodic version where the scatterers are identical rectangular obstacles located periodically along a square lattice on the plane, one obstacle centered at each lattice point. Hardy and Weber [HaWe] proved recurrence and abnormal diffusion of the billiard flow for special dimensions of the obstacles and for very special directions, using results on skew products above rotations. More recently Hubert, Lelièvre, and Troubetzkoy have studied the general full occupancy periodic case [HuLeTr]. They proved that, if the lengths of the sides of the rectangles belong to a certain dense Gδ subset E ′, then the dynamics is recurrent and they gave a lower bound on the diffusion rate. The recurrence was proven by analysis of the case when the lengths of the sides are rational and