Describing local and non-local electron heating by the Fokker–Planck equation

Abstract

The common description of kinetic effects in low-pressure plasmas is based on the Boltzmann equation. This applies especially to the description of Ohmic (collisional) and non-local (stochastic/collisionless) electron heating, where the Boltzmann equation is the starting point for the derivation of the corresponding heating operator. Here, it is shown, that an alternative and fully equivalent approach for describing the interaction between electrons and electric fields can be based on the Fokker–Planck equation in combination with the corresponding Langevin equation. Although, ultimately the final expressions are the same in both cases, the procedures are entirely different. While the Fokker–Planck/Langevin approach provides physical insights in a very natural way, the linearized Boltzmann equation allows straightforward calculation but requires some effort to interpret the mathematical structure in terms of physics. The Fokker–Planck equation for the present problem is derived, with particular emphasis on the consistent treatment of velocity-dependent elastic collision frequencies. The concept is tested for a simple case by comparing it with results from an ergodic Monte-Carlo simulation. Finally, the concept is applied to the problem of combined Ohmic and stochastic heating in inductively coupled plasmas. The heating operator is first analyzed for an exponential model field profile. Self-consistent field profiles are determined subsequently. In this context, a generalization of the plasma dispersion function is introduced, which allows for arbitrary forms of the distribution function and velocity dependence of the elastic collision frequency. Combined with the Fokker–Planck heating operator, a fully self-consistent description of the plasma and the fields is realized. Finally, a concept for integrating the operator in a standard local Boltzmann solver and using the local solver for determination of the global electron velocity distribution function in a low-pressure plasma is provided.