DIFFUSION-APPROXIMATION AND HYPERBOLIC AUTOMORPHISMS OF THE TORUS

In this article a diffusion equation is obtained as a limit of a rever sible kinetic equation scaled appropriately. This limiting diffusion i s produced by the collisions of the particles with the boundary. Indee d, these particles follow a reversible reflection law having convenien t mixing properties. This model, based on ''Arnold's cat map'', can be handled with Fourier series instead of the symbolic dynamics associat ed to a Markov partition. As a consequence, optimal convergence result s can be obtained by elementary means and illustrate the apparition of irreversibility in macroscopic limits.