Plasma Kinetic Theory: Collective Equation in Phase Space

Abstract

The motion of a large number of charged particles in plasma could be determined completely if the initial conditions are known since an individual particle follows Newton’s equations of motion. The flow of the probability distribution function of the system in 6N phase space consisting of the position and momentum of N particles shows incompressibility (Liouville theorem). This property leads to an important theorem of the isolated dynamical system “Poincare’s recurrence theorem,” which guarantees that the system will return to be arbitrarily close to the initial state. The kinetics equation represented by the Boltzmann equation is derived from reversible mechanics equations, but is often irreversible. In the Boltzmann equation, a statistical assumption “Stosszahl Ansatz” leads to a collision term exhibiting the arrow of time. Thus, there is a fundamental difference between the reversible dynamical equation and the kinetic equation.

In the kinetic equation for high temperature plasma, a strange phenomenon (called Landau damping) occurs where the oscillating electric field damps with time even when collisions are negligible through the mechanismof “phase mixing” in the velocity space, since the operator of the kinetic equation ν ∙ ∂f / ∂ x has a continuous spectrum. In this chapter, the basics of plasma kinetic equations including Coulomb collisions, the drift kinetic equation, and the gyro kinetic equations are introduced based on the orbit theories described in Chapter 4.