A few ideas about quantitative convergence of collison models to the mean field limit ( draft )

We consider a stochastic N -particle model for the spatially homogeneous Boltzmann evolution and show its convergence to the associated Boltzmann equation when N −→ ∞. More precisely, for any time T > 0 we bound over the distance between the empirical measure of the particle system and the measure given by Boltzmann evolution. That distance is computed in some homogeneous Sobolev space. The control we get is Gaussian, i.e. we prove that the distance is bigger than xN−1/2 with a probability of type e−x 2 at most. The two ingredients needed are first a control of fluctuations due to the discrete nature of collisions, secondly a kind of Lipschitz continuity for the Boltzmann collision kernel. The latter condition, in our present setting, is only satisfied for Maxwellian models. We also have to control the initial situation of the particle evolution, which we do by a kind of Chernoff inequality for the i.i.d. case. Numerical applications tend to show that our results are useful in practice.