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 \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.
Domains
Probability [math.PR]
Origin : Files produced by the author(s)
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