Fokker-Planck-Boltzmann Model for Low-Pressure Plasmas

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 stochastic (collisionless) electron heating, where the Boltzmann equation is the starting point for the derivation of the corresponding heating operator. Here, it will be shown that an alternative and fully equivalent approach for describing the interaction between electrons and fields can be based on the Fokker-Planck equation in combination with the corresponding Langevin equation. While the final results are the same in both cases, the procedure is entirely different. The Fokker-Planck approach provides physical insights in a very natural way, which in contrast requires some effort to extract when starting with the Boltzmann equation. The alternative concept is introduced and applied to the combined Ohmic and stochastic heating in inductively coupled plasmas. Further, a generalization of the plasma dispersion function is introduced, for arbitrary forms of the distribution function and for velocity-dependent elastic collision frequencies. Combined with the Fokker-Planck heating operator, a fully self-consistent description of the plasma and the fields is realized. Finally, an outlook on how to integrate the operator in a standard local Boltzmann solver will be provided.