Some Ideas About Quantitative Convergence of Collision Models to Their Mean Field Limit

We consider a stochastic N-particle model for the spatially homogeneous Boltzmann evolution and prove its convergence to the associated Boltzmann equation when N -> infinity, with non-asymptotic estimates: for any time T > 0, we bound the distance between the empirical measure of the particle system and the measure given by the Boltzmann evolution in a relevant Hilbert space. The control got is Gaussian, i.e. we prove that the distance is bigger than xN(-1/2) with a probability of type O(e(-x2)). The two main ingredients are a control of fluctuations due to the discrete nature of collisions and a kind of Lipschitz continuity for the Boltzmann collision kernel. We study more extensively the case where our Hilbert space is the homogeneous negative Sobolev space (H) over dot(-s). Then we are only able to give bounds for Maxwellian models; however, numerical computations tend to show that our results are useful in practice.