Environmental Engineering Reference
In-Depth Information
In the second-order expansion over the small parameter
Δ
v
we have
Z
Δ
v
d
v
2
W
1
2
f
1
@
v
1
α
@
v
1
@
f
1
@
v
1
α
@
f
2
@
v
2
@
I
ee
(
f
)
D
d
2
Δ
α
Δ
f
2
Δ
α
Δ
2
f
2
@
v
2
α
@
v
2
1
2
Δ
α
Δ
@
C
f
1
Z
d
@
Δ
v
d
v
2
1
@
v
1
α
@
W
W
@
v
2
α
f
1
@
v
1
@
f
2
@
v
2
@
2
Δ
α
Δ
f
2
f
1
,
whereweusethenotation
f
1
f
(
v
1
),
f
2
f
(
v
2
), and
Δ
α
Δ
v
α
;indices
α
and
denote components of the indicated vectors, and summation is made over
two repeated indices. Integrating over parts, one can evaluate some terms of this
expression. We have
Z
Z
1
2
f
1
@
v
1
α
@
f
2
@
v
2
C
@
1
2
Δ
α
Δ
@
W
@
v
2
α
f
1
@
v
1
@
d
Δ
v
d
v
2
W
Δ
α
Δ
d
Δ
v
d
v
2
f
2
Z
d
1
2
f
1
@
v
1
α
@
@
@
v
2
D
Δ
v
d
v
2
W
Δ
α
Δ
(
Wf
2
)
D
0,
Z
Z
2
f
2
@
v
2
α
@
v
2
C
1
2
@
1
2
Δ
α
Δ
@
W
@
v
2
α
f
2
@
v
2
@
d
Δ
v
d
v
2
W
Δ
α
Δ
f
1
d
Δ
v
d
v
2
W
f
1
W
@
D
@
W
@
v
2
α
f
2
@
v
2
@
@
@
v
2
α
f
2
@
v
2
f
1
f
1
D
0,
since the distribution function is zero at
v
2
!˙1
. After elimination of these
terms we obtain
Z
1
2
I
ee
(
f
)
D
d
Δ
v
d
v
2
Δ
α
Δ
W
2
f
1
@
v
1
α
@
v
1
@
W
@
f
1
@
v
1
α
@
v
2
C
@
@
f
2
W
@
v
1
α
f
1
@
v
1
@
@
W
@
v
2
α
f
1
@
f
2
@
v
2
f
2
f
2
D
@
j
@
v
1
,
where the flux in electron-velocity space is
Z
d
v
2
f
1
@
f
2
D
α
Z
@
v
2
@
f
2
f
1
@
v
1
1
2
j
D
,
D
α
D
Δ
α
Δ
Wd
Δ
v
.
This symmetric form of the electron-electron collision integral is called the Lan-
dau collision integral after evaluation of the diffusion coefficient in a space of elec-
tron velocities. For evaluation of tensor
D
α
we use (2.34) for variation of the elec-
tron velocity after electron-electron collision with scattering into a small angle:
2
e
2
α
Δ
α
D
,
2
gm
e
