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

A system of N particles xi(N) = (x(1), v(1),..., x(N), v(N)) interacting self-consistently with one wave Z=A exp(i phi) is considered. Given initial data (Z((N))(0), xi(N)(0)), it evolves according to Hamiltonian dynamics to (Z((N))(t), xi(N)(t)). In the limit N --> infinity, 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((infinity))(0), f(0)) evolve to (Z((infinity))(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 --> infinity, the states generated by the Hamiltonian dynamics at all times 0 less than or equal to t less than or equal to T are such that (Z((N))(t), f(N)(t)) converges weakly to (Z((infinity))(t), f(t)).