Microscopic derivation of Vlasov equation with compactly supported pair potentials

Abstract

We present a probabilistic proof of the mean-field limit and propagation of chaos of a N-particle system in three dimensions with compactly supported pair potentials of the form $N^{3\beta -1}\varphi (N^{\beta }x)$ for $\beta \in \left[0,\frac{1}{7}\right)$ and $\varphi \in L^{\infty }({\mathbb{R}}^3)\cap L^1({\mathbb{R}}^3)$. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Dirac-Benney system with delta-like interactions. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Dirac-Benney equation from the microscopic $N$-particle dynamics with a strong short range force.