Landau (Fokker-Planck) kinetic equation

PHYS 7500 Plasma Transport Theory #2 c○Jeong-Young Ji 5
Landau collision operator
Introducing the Landau tensor
and its derivative
U = u2 I − uu
u 3 ,
∂
∂v a
· U = − 2u
u 3 ,
we can write Eqs. (21) and (24), respectively, as
and
̂∆u ≈ −4π α2 ln Λ
m 2 u = γ abm a 1 ∂
ab u4 2m 2 · U, (25)
ab
u ∂v a
̂∆u∆u ≈
4π α2 ln Λ U
m 2 ab
u = γ abm a U
m 2 ab
u , (26)
where
γ ab = 4πα2 ln Λ ab
m a
= q2 a q2 b ln Λ ab
4πɛ 2 0 m .
a
Let’s write Eq. (18) term by term (t i denotes the ith term and t ij denotes the
jth term in t i )
t 1 = γ ∫ (
ab ∂ 1 ∂f a
dv b · U · f b + 1 )
∂f a
f b = t 11 + t 12 ,
2 ∂v a m a ∂v a m b ∂v a
t 2 = γ ∫ (
ab ∂
dv b · U · − 1 ∂f b
f a − m )
a ∂f b
2 ∂v a m b ∂v b m 2 f a = t 21 + t 22 ,
b
∂v b
t 3 = γ ∫
ab 1 ∂ ∂f a
dv b U : f b ,
2 m a ∂v a ∂v a
t 4 = − γ ∫
ab 2
2 m b
= γ [ ∫
ab 1
−
2 m b
= t 41 + t 42 ,
dv b U : ∂f a
∂v a
∂f b
dv b U : ∂f a
∂v a
∂f b
∂v b
−
( 1 IBP →)
∂v b 2
∫
dv b
( ∂
∂v a
· U ) · ∂f a
∂v a
f b
]
and
t 5 = γ ab
2
= γ ab
2
m a
m 2 b
f a
∫
∫
m a
m 2 f a
b
dv b U :
∂
∂v b
∂f b
∂v b
(IBP →)
dv b
( ∂
∂v a
· U ) · ∂f b
∂v b
.