Billiards Systems and the Transport Equation
Consider a system of like particles that do not interact between themselves but collide with a periodic array of strictly convex obstacles. This type of dynamical system is referred to as a dispersive billiards systems in this work. The most genuine example of such systems is the so-called Lorentz gas model of hard spheres7: the interaction between the particles and the obstacles is the pure (Descartes) reflection law. In large time, due to multiple collisions and the strict convexity of the obstacles, the distribution of the particles tends to forget its velocity dependence7. Bunimovich and Sinai7 have proved that the density of particles approaches in some sense the solution of a diffusion (heat) equation for large time and on a large space scale. Their proof relies on symbolic dynamics via the construction of a Markov partition8 , 9. The role of the strict convexity of the scatterers is reminiscent of the ergodic properties of the geodesic flow on compact manifolds with negative curvature: see Arnold 1 (although the explanation provided there is not fully rigorous). However, the method used by Bunimovich-Sinai7 does not provide the value of the diffusion coefficient, except through the use of the ubiquitous Kubo series. See also Gaspard-Nicolis11 for an interesting expression of the diffusion coefficient. None of these expressions clearly provides the dependence of the diffusion coefficient in terms of the geometry of the array of scatterers. Numerical experiments have been conducted by Machta-Zwanzig, as well as some asymptotics in the case where the distance between neighboring scatterers goes to zero (see Machtal4).
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