Phase space transport, quasilinear diffusion and locality in phase velocity

TL;DR

This work analyzes velocity-space transport of particles in a fixed spectrum of electrostatic waves, clarifying when a diffusion (Fokker-Planck) description is valid and when chaos dominates. It develops a locality criterion in phase velocity, derives a discrete-spectrum quasilinear diffusion coefficient $D_{QL}$, and shows that uniform transport emerges under specific spectral scalings, with the Hamiltonian $H_p$ serving as a benchmark. The authors confirm, via extensive test-particle numerics, that locality yields universal transport properties and that chaotic diffusion remains of the same order as $D_{QL}$ but can depend on the number of modes, especially when locality fails. They also propose a simple, boundary-aware analytical form for the time evolution of the velocity distribution in bounded chaotic transport, validated against simulations, and discuss implications for plasma contexts where wave spectra are discrete and evolving.

Abstract

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 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.

Figures (17)

• Figure 1: Phase portrait of the pendulum. The blue dashed line is an untrapped orbit, the red dashed-dotted line is a trapped orbit and the black solid line is the separatrix.
• Figure 2: Poincaré sections for the Hamiltonian $H_p$ when $N=21$ and $s=0.53$. Each color corresponds to one of the initial condition 1-7, as defined by Eqs. (\ref{['pp1']})-(\ref{['pp7']}). In the vicinity of each Poincaré section, and using the same color, is indicated the corresponding initial condition.
• Figure 3: Same as Fig. \ref{['p1']} for $s=4$.
• Figure 4: Time dependence of $\langle\Delta v^2(t)\rangle/2 D_{\rm{QL}} t=\langle\Delta v^2(t)\rangle/\pi A^2t$ when $\gamma=-1$, $A=1$ and $v_0=50$. The blue solid line is for $N=142$, the green dashed line for $N=448$, the red dashed line for $N=1415$ and the black dashed-dotted line for $N=14143$. These values for $N$ have been chosen so that the maximum phase velocity be, respectively, 2, 20, 200 and $2\times10^4$ times $v_0$.
• Figure 5: The black solid line shows the velocity distribution function, $f(v)$, found numerically when $\gamma=-1$, $v_0=50$, $N=142$, and at time $t=50$, by repeating the simulations with $10^7$ different phase realizations. $f(v)$ is normalized so that its mean is zero and its variance is unity. The red dashed line shows the Maxwellian with same mean and same variance.