Microscopic derivation of Vlasov–Dirac–Benney equation with short-range pair potentials

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 <?CDATA N3β-1 φ(Nβx)?> N 3 β − 1 ϕ ( N β x ) for <?CDATA β∈[0,1/7]?> β ∈ [ 0 , 1 7 ] and <?CDATA φ∈~L∞(R³)∩ L¹(R³)?> ϕ ∈ L ∞ ( R 3 ) ∩ L 1 ( 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.