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

Rémi Peyre (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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 \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.

Domaines: Probabilités [math.PR]

Rémi Peyre. Some Ideas About Quantitative Convergence of Collision Models to Their Mean Field Limit. Journal of Statistical Physics, 2009, 136, pp.1105-1130. ⟨10.1007/s10955-009-9820-3⟩. ⟨hal-01282584⟩