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

Abstract

We consider a stochastic N-particle model for the spatially homogeneous Boltzmann evolution and prove its convergence to the associated Boltzmann equation when N⟶∞, 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 $O(e^{-x^{2}})$. 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 $\smash {\dot {H}}^{-s}$. Then we are only able to give bounds for Maxwellian models; however, numerical computations tend to show that our results are useful in practice.