Stochastic magnetic hydrodynamic hierarchy in a strong external magnetic field

S. V. Bogomolov ; N. B. Esikova

Matematičeskoe modelirovanie, Tome 31 (2019) no. 8, pp. 120-142

Résumé

Based on a stochastic microscopic collisional model of the motion of charged particles in a strong external magnetic field, a hierarchy of equations of magnetic hydrodynamics is constructed. The transition to increasingly rough approximations occurs in accordance with a decrease in the dimensioning parameter, similar to the Knudsen number in gas dynamics. The result is stochastic and nonrandom macroscopic equations that differ from the magnetic analog of the Navier–Stokes system of equations as well as from the systems of magnetic quasi-hydrodynamics. The main feature of this derivation is a more accurate velocity averaging due to the analytical solution of stochastic differential equations with respect to the Wiener measure, in the form of which the intermediate meso model is presented in the phase space. This approach differs significantly from the traditional one, which uses not the random process itself, but its distribution function. Emphasis is placed on clarity of assumptions when moving from one level of detail to another, and not on numerical experiments that contain additional approximation errors.

Détail

Keywords: Boltzmann equation, Navier–Stokes equation, random processes, stochastic differential equations with respect to Poisson and Wiener measures, particle method.
Mots-clés : Kolmogorov–Fokker–Planck equation, Lorentz force, magnetohydrodynamic and quasi-gasdynamic equations