Microscopic derivation of Vlasov–Dirac–Benney equation with short-range 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 pair potentials of the form N3β−1ϕ(Nβx) for β∈[0,17] and ϕ∈ L∞(R3)∩L1(R3) . 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.