Narrow Results

For two (or more) interacting classical particles the existing few results (for diffusion or energy transfer, for instance) assume that the mass of one of them - as compared to the other mass - becomes negligible in the limit (cf. [ChD 09]). Here two models are presented for energy transfer in systems with two or more identical hard disks. The first one, a stochastic paradigm for two Lorentz disks, suggests that the joint diffusive limit of two disks is the mixture of independent pairs of Wiener processes (cf. [P-GySz 09]). In the second one it is shown that in a quasi-1D mechanical chain of localized hard disks - in the scaling of [GG 08] - the limit for the energies of the disks is a n. n. interacting Markov process (cf. [SzT 09]). This latter result should open the way toward a rigorous derivation of Fourier law of heat conduction for a deterministic particle system.

First we briefly review a recent breakthrough in the verification of the Boltzmann-Sinai ergodic hypothesis formulated for systems of elastic hard balls known to be isomorphic to semi-dispersing billiards. A basic tool behind these results: the Fundamental Theorem for Semi-Dispersing Billiards is, however, also applicable to establishing the ergodicity of planar billiards with convex boundary components. As an application, we elucidate the idea of the proof of ergodicity of Wojtkowski's planar billiards with convex-scattering boundary components generalizing Bunimovich's famous stadium and, moreover, survey related results and questions.