Nonlinear theory of cyclotron resonant wave-particle interactions: Analytical results beyond the quasilinear approximation.

Cyclotron resonant wave-particle interactions are studied in the context of Hamiltonian theory with utilization of Lie transform techniques. The canonical perturbation method for single particle motion is used for providing results for the collective particle behavior under interaction with wave fields of either localized or periodic profiles. Analytical expressions for the calculation of phase-averaged quantities of physical interest as well as the diffusion equation are derived. In the lowest order of perturbation, the method reformulates in a rigorous and unifying context the derivation of well-known results, namely Madey's theorem and quasilinear diffusion equation. Proceeding to higher order the method provides results consisting of fourth-order accurate analytical expressions for the calculation of phase-averaged quantities as well as the derivation of a fourth-order accurate diffusion equation, with higher-order derivatives, which is the extension of the well-known Fokker-Planck equation beyond the quasilinear approximation. Higher-order terms are related to the effect of nonlinear resonant coupling between different spectral components of the waves, on the evolution of the particle distribution function.