13.2 Statistical argument for the development of the Fokker-Planck equation 385invoking Eq.(13.13) this can be recast as
∂f
∂t=
−
∂
∂v·
(∫
∆vF(v,∆v)d∆v
∆tf(v,t))
+
1
2
∂
∂v∂
∂v:
(∫
∆v∆vF(v,∆v)d∆v
∆t
f(v,t))
. (13.17)
By defining
〈
∆v
∆t
〉
=
∫
∆vF(v,∆v)d∆v
∆t(13.18)
and 〈
∆v∆v
∆t
〉
=
∫
∆v∆vF(v,∆v)d∆v
∆t(13.19)
the standard form of the Fokker-Planck equation is obtained,
∂f
∂t=−
∂
∂v·
(〈
∆v
∆t〉
f(v,t))
+
1
2
∂
∂v∂
∂v:
(〈
∆v∆v
∆t〉
f(v,t))
. (13.20)
The first term gives the slowing down of a beam and is called the frictional term while the
second term gives the spreading out of a beam and is called the diffusive term.
The goal now is to compute〈∆v/∆t〉and〈∆v∆v/∆t〉. To do this, it is necessary
to consider all possible ways there can be a change in velocity∆vin time∆tand then
average all values of∆vand∆v∆vweighted according to their respective probability of
occurrence. We will first evaluate〈∆v/∆t〉and then later use the same method to evaluate
〈∆v∆v/∆t〉.The problem will first be solved in the center of mass frame and so involves
a particle with reduced massμ, speedvrel=|vT−vF|, and impact parameterbcolliding
with a stationary scattering center at the origin. This was discussed inChapter 1 and the
geometry was sketched in Fig.1.3. The deflection angleθfor small angle scattering was
shown to be
θ=
qTqF
2 πε 0 bμv^2 rel
. (13.21)
The averaging procedure is done in two stages:- The effect of collisions in time∆tis calculated. The functional dependence of∆von
the scattering angleθis determined and then used to calculate the weighted average
∆vfor all possible impact parameters and all possible azimuthal angles of incidence.
This result is then used to calculate the weighted average change in target particle
velocity in the lab frame. - An averaging is then performed over all possible field particle velocities weighted
according to their probability, i.e., weighted according to the field particle distribution
functionfF(vF).
Energy is conserved in the center of mass frame and since the energy before and after
the collision is entirely composed of kinetic energy, the magnitude ofvrel^2 must be the same
before and after the collision. The collision thus has the effect of causing a simple rotation
of the relative velocity vector by the angleθ.Letzbe the direction of the relative velocity
before the collision and letˆbbe the direction of the impact parameter. Ifvrel 1 ,vrel 2 are