Fundamentals of Plasma Physics

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