Fundamentals of Plasma Physics

13.2 Statistical argument for the development of the Fokker-Planck equation 391

which is the ratio of the beam velocity to the thermal velocity of the ionsor electrons. The
slowing down relationship can then be expressed as

∂u

∂t

=

nee^2 lnΛ

4 πε^20

q^2 T

m^2 T

[

Z

(

1+

mT

mi

)

mi

2 κTi

d

dξi

(

erf(ξi)

ξi

)

+

(

1+

mT

me

)

me

2 κTe

d

dξe

(

erf(ξe)

ξe

)]

. (13.62)

0 1.25 2.5 3.75 5

0.4

0.3

0.2

0.1

−ddx

erf
x

x

x

lim x

 1 −ddx

erfx

x 

4 x

3 

lim x 1 −ddx

e rfx

x 

1

x^2

Figure 13.1: Plot of−(x−^1 erfx)′v.x.

Figure 13.1 plots−(erf(x)/x)′as a function ofx;this quantity has a maximum when
x= 0. 9 indicating that the friction increases with respect toxwhenxis less than 0. 9 but
decreases with respect toxwhenxexceeds this value. Examination of this figure suggests
that the function has a linear dependence forxwell to the left of the maximum and varies
as some inverse power ofxwell to the right of the maximum.
This conjecture is verified by noting that Eq.(13.56) has the limiting values

lim

x<< 1

erf(x) ≃

2

√

π

∫x

0

(1−w^2 )dw=

2

√

π

(

x−

x^3

3

)

lim

x>> 1

erf(x) ≃ 1 (13.63)

and so

lim

x<< 1

(

−

d

dx

erfx

x

)

=

4 x

3

√

π

(13.64a)

lim

x>> 1

(

−

d

dx

erfx

x

)

=

1

x^2

(13.64b)

which is consistent with the figure.
The existence of these two different asymptotic limits according to the ratio of the test
particle velocity (i.e., beam velocity) to the thermal velocity indicates three distinct regimes
can occur, namely: