Fokker-planck equation for a plasma with a constant magnetic field

Abstract

Starting from the Liouville equation, a chain of equations is obtained by integrating out the co-ordinates of all but one, two-etc. particles. One particle is singled out. All other particles are considered to be initially in thermal equilibrium. For the time evolution of the distribution function of the `singled out' particle an equation is obtained whose asymptotic form is of the usual Fokker-Planck type. It is characterized by a frictional drag force that slows the particle down and a fluctuation tensor that speeds it up and produces diffusion in velocity space. The objective of the calculation is to determine these quantities for a plasma consisting of electrons and protons in a constant external magnetic field. The chain of equations contains two dimensionless parameters λ = ωc/ωp and g = l/n LD3. A solution for the s-body correlation function is obtained in the form fs = fs(0)(λ) + gfs(1)(λ) + etc. fs(0) and fs(1) have been determined to all orders of λ. The frictional drag consists of a part due to collisions and a part due to plasma wave emission. When λ=1 the modification of the collisional part due to the magnetic field is negligible. There is a significant change in the properties of plasma waves of wavelength greater than the Larmor radius which modifies the force due to plasma wave emission. When λ=1 the force due to plasma wave emission disappears. The collisional force is altered to the extent that the maximum impact parameter is sometimes the Larmor radius instead of the Debye length, or something in between. A more interesting modification obtains for the particular case of a slow ion moving perpendicular to the field. It is due to repeated collisions with fast electrons. This collisional force is of a qualitatively different form, but the quantitative modification is not large. In addition to the drag force anti-parallel to the velocity of the particle, there is a collisional force anti-parallel to the Lorentz force. This force arises because the particle and its `shield cloud' are spiralling about field lines. The force on the particle is equal and opposite to the centripetal force acting on the `shield cloud.' It is much smaller than the Lorentz force. The main result is that to the lowest order in g, the frictional drag and fluctuation tensor are slowly varying functions of λ.