CARTESIAN TENSOR EXPANSION OF THE FOKKER–PLANCK EQUATION

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Abstract
The Fokker–Planck equation is expanded via Cartesian tensors. The results, valid for arbitrary mass ratio, reduce to known expressions for the f0, (isotropic) and f1 (directional) parts of the distribution. The same analysis also gives the relation for the f2 (tensor) part of the distribution. Nonlinear and recoil terms are also investigated. The f0 relation yields relaxation times and energy equipartition times for arbitrary velocity distributions. The ion recoil term is important for low-frequency waves in the electron f1 equation but is negligible in the electron f2 equation. The electron–electron contributions to the f1 and f2 collisional terms are of the same order as the electron–ion contributions. Except for momentum transfer calculations, the ion–ion collisional terms dominate over the ion–electron terms in the ion collision equations referring to the directional (F1) and tensor (F2) parts of the ion distribution.