Diffusion limit of the Vlasov equation in the weak turbulent regime

TL;DR

This paper analyzes the asymptotic behavior of charged particles in a weak turbulent plasma, showing that the Vlasov equation converges to a velocity-space diffusion equation, with implications for plasma physics theories.

Contribution

It establishes the diffusion limit of the Vlasov equation in the weak turbulent regime and relates it to existing plasma diffusion theories, highlighting the role of electric field autocorrelation.

Findings

Vlasov equation converges to a velocity-space diffusion equation

Diffusion matrix relates to resonance broadening and quasilinear theories

Homogenization occurs in the self-consistent deterministic case

Abstract

In this paper we study the Hamiltonian dynamics of charged particles subject to a non-self-consistent stochastic electric field, when the plasma is in the so-called weak turbulent regime. We show that the asymptotic limit of the Vlasov equation is a diffusion equation in the velocity space, but homogeneous in the physical space. We obtain a diffusion matrix, quadratic with respect to the electric field, which can be related to the diffusion matrix of the resonance broadening theory and of the quasilinear theory, depending on whether the typical autocorrelation time of particles is finite or not. In the self-consistent deterministic case, we show that the asymptotic distribution function is homogenized in the space variables, while the electric field converges weakly to zero. We also show that the lack of compactness in time for the electric field is necessary to obtain a genuine diffusion limit. By contrast, the time compactness property leads to a "cheap" version of the Landau damping: the electric field converges strongly to zero, implying the vanishing of the diffusion matrix, while the distribution function relaxes, in a weak topology, towards a spatially homogeneous stationary solution of the Vlasov-Poisson system.