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

A system of N particles ξ = (x1, v1, ..., xN , vN ) interacting selfconsistently with M waves Zn = An exp(iφn) is considered. Given initial data (Z (0), ξ (0)), it evolves according to hamiltonian dynamics to (Z (t), ξ (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 envelope equations for the Zn: 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(0) → f(0) weakly and with Z (0)→ Z(0) as N →∞, the states generated by the hamiltonian dynamics at all times 0 ≤ t ≤ T are such that (Z (t), f (t)) converges weakly to (Z(t), f(t)).