13 Fokker-Planck theory of collisions
13.1 Introduction
Logically, this chapter ought to be located at the beginning of Chapter 2, just after phase-
space concepts had been introduced. The reason this chapter is not located there is that
the theory in this chapter is too involved to be at the beginning of the book and would have
delayed the introduction of important other topics that do not need the detail of thischapter.
The discussion of collisions in Chapter 1 was very approximate. Collisions were shown
to scale as an inverse power of temperature, but this was based on a “one size fits all”
analysis since it was assumed that the collision frequency of both slow andfast particles
were nominally the same as that of a particle going at the thermal velocity. Because the
collision frequency scales asv−^3 , it is dubious to assume that the collision rates of both
super-thermal and subthermal particles can be reasonably represented by a single collision
frequency. A more careful averaging over velocities is clearly warranted. This is provided
by a Fokker-Planck analysis due to Rosenbluth, Macdonald and Judd (1957) but ifall
this analysis provided was more accuracy for collision calculations, itwould not be worth
the considerable effort except possibly for exceptional situations wherehigh accuracy is
important. However, it turns out that more than increased accuracy results from this theory,
because some new and important phenomena become evident from the analysis.
We begin the analysis by reviewing collisions between two particles. Suppose a test
particle having massmTand chargeqTcollides with a field particle having massmFand
chargeqF.Since the electric field associated with a chargeqisE=ˆrq/ 4 πε 0 r^2 wherer
is the distance from the charge, the equations of motion for the respective test and field
particles are
mT ̈rT =
qTqF
4 πε 0 |rT−rF|^3(rT−rF) (13.1a)mF ̈rF =qTqF
4 πε 0 |rT−rF|^3(rF−rT). (13.1b)The center of mass vector is defined to beR=
mTrT+mFrF
mT+mF