In this paper, we address the motion of charged particles subjected to a discrete spectrum of electrostatic waves. We focus on situations when transport dominates, leading to significant variations in particle velocity. Nonetheless, these velocity changes remain finite due to the presence of Kolmogorov-Arnold-Moser (KAM) tori bounding phase space. We analyze the conditions under which transport can be modeled as a diffusion process and evaluate the relevance of the so-called quasilinear value of the diffusion coefficient. We distinguish between traditional quasilinear diffusion, when wave–particle interaction is perturbative, and the so-called chaotic regime of diffusion, when the particle motion looks erratic. In the perturbative regime, we demonstrate both numerically and theoretically that diffusion occurs only when wave–particle interaction is local in phase velocity; that is, when wave contributions from phase velocities far from the particles' instantaneous velocities are negligible. Conversely, numerical results indicate that chaotic diffusion can occur even when wave–particle interaction is not local. However, without locality, the diffusion coefficient is not the quasilinear one. Furthermore, in regimes when quasilinear diffusion is applicable, we introduce a simple analytical expression for the time evolution of the velocity distribution function that accounts for phase space boundaries.
Didier Bénisti (Sun,) studied this question.