Kinetic theory and large-N limit for wave-particle self-consistent interaction

A system of N particles xi(N) = (x(1), v(1), x(N), v(N)) interacting self-consistently with M waves Z = (A(j)e(-10j)) is considered in d-dimensional space. Given initial data [Z(0), xi(N)(0)], it evolves according to Hamiltonian dynamics to [Z(N)(t), xi(N)(t)]. Assuming that the particles interact only with the waves and conversely, in a mean-held way, we obtain an upper bound for the largest Liapunov exponent which depends on the total energy per particle but not on N. In the limit N --> infinity, this dynamics generates a Vlasov-like kinetic system for distribution functions f(t) = [f(sigma)(x, v, t)] for all species sigma, coupled to envelope equations for Z(j). 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, the kinetic limit N --> infinity commutes with time evolution for all times 0 less than or equal to t less than or equal to T.