Appendix A - Heuristic derivation of the kinetic equation

Summary

Introduction

The basic model describing MHD and transport theory in a plasma is the kinetic-Maxwell equations, which consist of a set of coupled electromagnetic and kinetic equations. The electromagnetic behavior is governed by the full Maxwell’s equations (i.e., displacement current and Poisson’s equation are included). In the kinetic model each species is described by a distribution function f_α(r, v, t), which satisfies a 6-D plus t integro-differential equation including the effect of collisions. The equations are very general and very, very difficult to solve. They accurately describe behavior ranging from the fast ω_cα and ω_pα time scales, down to the slower MHD time scale and the even slower transport time scale.

Since the ideal MHD model is based on the kinetic-Maxwell equations, the first step in the theoretical development of MHD is a derivation of the kinetic equation. A simple heuristic derivation is presented below.