Kinetic limits of the HPP cellular automaton

Abstract

We study the Boltzmann-Grad limit in various versions of the two-dimensional HPP cellular automaton. In the completely deterministic case we prove convergence to an evolution that is not of kinetic type, a well-known phenomenon after Uchyiama's paper on the Broadwell gas, whereas the limiting equation becomes of kinetic type in the model with random collisions. The main part of the paper concerns the case where the collisions are deterministic and the randomness comes from inserting, between any two successive HPP updatings,ɛ ^-ν stirring updatings, ν<1 being any fixed positive number andɛ a parameter which tends to 0. The initial measure is a product measure with average occupation numbers of the order ofɛ (low-density limit) and varying on distances of the order of ɛ⁻¹. The limit asɛ → 0 of the system evolved for times of the order ofɛ ^-1-ν corresponds to the Boltzmann-Grad limit. We prove propagation of chaos and that the renormalized average occupation numbers (i.e., divided byɛ) converge to the solution of the Broadwell equation. Convergence is proven at all times for which the solution of the Broadwell equation is bounded.