406 Chapter 14. Wave-particle nonlinearities
14.2.2Conservation properties of the quasilinear diffusion equation
Conservation of particles
Conservation of particles occurs automatically because Eq.(14.35) has theform of a
derivative in velocity space. Thus, the zeroth moment of Eq.(14.35) issimply
∂
∂t
∫
dvf 0 =
∫
dv
∂
∂v
(
DQL
∂f 0
∂v
)
=
[
DQL
∂f 0
∂v
]v=∞
v=−∞
=0 (14.43)
and so the quasi-linear diffusion equation conserves the densityn=
∫
dvf 0.
Conservation of momentum
Examination of momentum conservation requires taking the first moment of Eq.(14.35),
∂
∂t
(nmu)=m
∫
dvv
∂
∂v
(
DQL
∂f 0
∂v
)
=−m
∫
dvDQL
∂f 0
∂v
. (14.44)
Using Eq.(14.36) this becomes
∂
∂t
(nmu) = −m
∫
dv
2ie^2
ε 0 m^2
∫
dk
E(k,t)
ω−kv
∂f 0
∂v
= −2iω^2 p
∫
dkE(k,t)
∫
dv
1
ω−kv
∂f ̄ 0
∂v
. (14.45)
However, the linear dispersion relation Eq.(14.41) shows that
ω^2 p
∫
dv
1
ω−kv
∂f ̄ 0
∂v
=−k (14.46)
and so
∂
∂t
(nmu)=2i
∫
dkE(k,t)k=0 (14.47)
which vanishes because the integrand is an odd function ofk.Thus, the constraint provided
by the linear dispersion relation shows that the quasi-linear velocity diffusion equation also
conserves momentum.
Conservation of energy
Consideration of energy conservation starts out in a similar manner but leads to some
interesting, non-trivial results. The mean particle kinetic energy is defined to be
WP=
∫
dv
mv^2
2
f 0. (14.48)
The time evolution ofWPis obtained by taking the second moment of the quasi-linear
diffusion equation, Eq.(14.35),
∂WP
∂t
=
∫
dv
mv^2
2
∂
∂v
DQL
∂f 0
∂v
= −
∫
dvmvDQL
∂f 0
∂v