Kinetic Limit of N-Body Description of Wave–Particle Self-Consistent Interaction

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A system of N particles ξN=(x1, v1, …, xN, vN) interacting self-consistently with one wave Z = A exp(i φ) is considered. Given initial data (Z(N)(0), ξN(0)), it evolves according to Hamiltonian dynamics to (Z(N)(t), ξN(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 fN(0) → f(0) weakly and with Z(N)(0) → Z(0) as N → ∞, the states generated by the Hamiltonian dynamics at all times 0 ≤ tT are such that (Z(N)(t), fN(t)) converges weakly to (Z(∞)(t), f(t)).

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