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

A system of N particles ξ ^N = (x ₁, v ₁, ..., x _N, v _N) interacting self-consistently with M waves Z = (A _je- ^iθ) is considered in d-dimensional space. Given initial data [Z(0), ξ ^N(0)], it evolves according to Hamiltonian dynamics to [Z ^N(t), ξ ^N(t)]. Assuming that the particles interact only with the waves and conversely, in a mean-field 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 → ∞, this dynamics generates a Vlasov-like kinetic system for distribution functions f(t) ≡ [f _σ(x, v, t)] for all species σ, 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 → ∞ commutes with time evolution for all times 0 ≤ t ≤ T.