Kac’s Mean-Field Model and the Boltzmann Equation

We consider a stochastic many-particle system introduced by Kac as a mean field approximation to the spatially homogeneous Boltzmann equation. A first question one can ask is to prove a hydrodynamic limit for the many-particle system as the number of particles $N\to\infty$, which justifies Boltzmann’s assumption of molecular chaos, and to quantify the rate of convergence. We will address this problem in the cases where the dynamics are governed by a range of collision kernels, which represent either a) localised interactions (hard spheres), or b) a class of long-range repulsive potentials (hard potentials).

The available results and required techniques depend strongly on the collision kernel in question. For the case a), many previous results are available and we use a `top-down’ approach in the spirit of Mischler and Mouhot, using the stability of the Boltzmann equation to obtain estimates with good time dependence. In the case b), we introduce a coupling of the kind first introduced by Tanaka, and show that the coupling is stable for a Wasserstein-type optimal transportation problem with a well-chosen cost function; this leads to a law of large numbers and stability for the Boltzmann equation.