Diffusion limit of the Vlasov equation in the weak turbulent regime
Abstract
In this paper, we study the Hamiltonian dynamics of charged particles subject to a nonselfconsistent stochastic electric field when the plasma is in the socalled 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 selfconsistent 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, toward a spatially homogeneous stationary solution of the VlasovPoisson system.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 October 2021
 DOI:
 10.1063/5.0022130
 arXiv:
 arXiv:2110.05914
 Bibcode:
 2021JMP....62j1505B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics
 EPrint:
 J. Math. Phys. 62, 101505 (2021)