13.2 Statistical argument for the development of the Fokker-Planck equation 387
This result was for a particular value ofvFand summing over all possiblevFweighted
according to their probability, gives
∆vT|allb,allφ,allvF=−∆t
qT^2 q^2 FlnΛ
4 πε^20 μmT
∫
vrel
v^3 rel
fF(vF)dvF. (13.28)
The integrand can be expressed in a simpler form by considering that
∂r
∂r
=
(
ˆx
∂
∂x
+ˆy
∂
∂y
+ˆz
∂
∂z
)√
x^2 +y^2 +z^2 =
r
r
(13.29)
and also∂/∂vT=∂/∂vrel.Thus
∂vrel
∂vT
=
vrel
vrel
(13.30)
and
∂
∂vT
1
vrel
=−
1
v^2 rel
vrel
vrel
. (13.31)
Equation (13.28) can therefore be written as
∆vT|allb,allφ,allvF=∆t
qT^2 q^2 FlnΛ
4 πε^20 μmT
∂
∂vT
∫
1
vrel
fF(vF)dvF. (13.32)
The∂/∂vThas been factored out of the integral becausevTis independent ofvF, the
variable of integration. SincevFis a dummy variable, it can be replaced byv′and since
vTis the velocity of the given incident particle, the subscriptTcan be dropped. After
making these re-arrangements we obtain the desired result,
〈
∆v
∆t
〉
=
q^2 TqF^2 lnΛ
4 πε^20 μmT
∂
∂v
∫
fF(v′)
|v−v′|
dv′. (13.33)
The averaging procedure for∆v∆vis performed in a similar manner, starting by noting
that
∆v∆v=
(
θcosφ,θsinφ,−θ^2 / 2
)(
θcosφ,θsinφ,−θ^2 / 2
)
(13.34)
and so Eq.(13.25) is replaced by
∆vrel∆vrel|allb,allφ = vrel∆tfF(vF)dvF
∫ 2 π
0
dφ
∫λD
bπ/ 2
bdb×
v^2 rel
(
θcosφ,θsinφ,−θ^2 / 2
)(
θcosφ,θsinφ,−θ^2 / 2
)
.
(13.35)
The terms linear insinφorcosφvanish upon performing theφintegration. Also, theˆzˆz
term scales asθ^4 and so may be neglected relative to theˆxxˆandyˆˆyterms which scale as