When can the Fokker–Planck equation describe anomalous or chaotic transport? Intuitive aspects

Abstract

The Fokker-Planck equation (FPE) is a quite flexible tool for experimental and numerical data analysis. However, advective-diffusive effects may compete with a curvature pinch or the impact of a localized source to determine profiles of transported quantities. Depending on the statistics of interest and/or on the radial extension of the transport process, the same dynamical system may be found diffusive or dominated by its Lévy flights. The Kubo number plays a leading role in determining whether diffusive transport is in the quasilinear regime. It would be advisable to systematically compute it from experimental or numerical data, in order to clarify the issue of its value in magnetic fusion turbulence.

Basic microscopic plasma physics from N-body mechanics

“…where the last expression makes cosh(β σ t) = 1, since β σ goes to zero in the continuous limit, and uses a Fourier transform identity in this limit. Equations (58) and (59) yields a contribution to the wave…”

“…Nevertheless, the diffusive picture for the chaos due to waves needed to be substantiated. It was shown to be right, provided adequate averages are performed on the dynamics; however, this picture is wrong if one averages only over the initial positions of particles with the same initial velocity [12] (see also section 6.2 of [44], and [57,58]).…”

From thermonuclear fusion to Hamiltonian chaos

“…These uncorrelated random steps yield to numerically measure the diffusion coefficient by following the dynamics either of a single particle for a series of random outcomes of the wave phases, or of many particles for a single typical outcome of the phases. 25 See also section 6.2 of [Elskens 2003], and [Escande 2007,Escande 2008. 26 The idea of locality was already present in the resonance broadening concept introduced in 1966 by Dupree [Dupree 1966].…”

“…The change of topography forces particles to jump to a nearby trough or peak. The successive jumps produce a random walk whose order of magnitude of the corresponding diffusion coefficient can be easily computed [Ottaviani 1992,Vlad 2004,Escande 2007,Escande 2008. In a series of works, Vlad and coworkers clarified the issue of diffusion with trajectory trapping.…”

Contributions of plasma physics to chaos and nonlinear dynamics

“…In reality, when the diffusive picture is correct, there is a pinch or dynamic friction part on top of the diffusive part, and the correct model is the Fokker-Planck equation (Escande & Sattin 2007, Escande & Sattin 2008. For the advection of particles in drift waves or in 2-dimensional turbulence, the direction of this pinch part depends on K (Vlad, M. & Benkadda 2006).…”