Fundamentals of Plasma Physics

34 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD

particles atx.Thus we see that

∫

f(x,v)dv=n(x);the transition from a phase-space
description (i.e.,x,vare dependent variables) to a normal space description (i.e.,xis a
dependent variable) involves “integrating out” the velocity dependence to obtain a quantity
(e.g., density) depending only on position. Since the number of particles is finite, and since
fis a positive quantity, we see thatfmust vanish asv→∞.
Another way of viewingf is to consider it as the probability that a randomly selected
particle at positionxhas the velocityv.Using this point of view, we see that averaging over
the velocities of all particles at∫ xgives the mean velocityu(x)determined byn(x)u(x) =
vf(x,v)dv.Similarly, multiplyingfbyv^2 and integrating over velocity will give an
expression for the mean energy of all the particles. This procedure of multiplyingf by
various powers ofvand then integrating over velocity is calledtaking moments of the
distribution function.
It is straightforward to generalize this “moment-taking” to three dimensional problems
simply by taking integrals over three-dimensional velocity space. Thus, in three dimensions
the density becomes

n(x) =

∫

f(x,v)dv (2.10)

and the mean velocity becomes

u(x) =

∫

vf(x,v)dv

n(x)

. (2.11)

x

v

initially fast particle

moving to right

initially slow particle

moving to right

apparentannihilation

sudden change in v

due to collision

apparent creation

Figure 2.4: Detailed view of collisions causing ‘jumps’ in phase space

2.3.1 Treatment of collisions in the Vlasov equation

It was shown in Sec. 1.8 that the cumulative effect of grazing collisions dominates the