Kinetic limit of N-body description of wave-particle self-consistent interaction

A system of N particles ξ^N = (X_I , V_I ,..., X_N , V_N) interacting sell-consistently with one wave Z = A exp(iφ) is considered. Given initial data (Z^(N)(0), ξ^N(0)), it evolves according to Hamiltonian dynamics to (Z^(N)(t), ξ^N(t)). In the limit N → ∞, this generates a Vlasov-like kinetic equation for the distribution function f(x, v, t), abbreviated as f(t), coupled to the envelope equation for Z: initial data (Z^(∞)(0), f(0)) evolve to (Z^(∞)(t), f(t)). The solution (Z, f) exists and is unique for any initial data with finite energy. Moreover, for any time T > 0, given a sequence of initial data with N particles distributed so that the particle distribution f^N(0) → f(0) weakly and with Z^(N)(0) → Z(0) as N → ∞, the states generated by the Hamiltonian dynamics at all times 0 ≤ t ≤ T are such that (Z^(N)(t), f^N(t)) converges weakly to (Z^(∞)(t), f(t)).