404 Chapter 14. Wave-particle nonlinearities
and
ω(−k)+ω(k)=2iωi(k) (14.28)
in which case Eq.(14.19) reduces to
〈E 1 f 1 〉=
i
2 πLe
m∫
dk∣∣
∣E ̃ 1 (k)∣∣
∣
2
e^2 ωi(k)t
ω−kv∂f 0
∂v. (14.29)
Substitution of〈E 1 f 1 〉into Eq.(14.10) gives the time evolution of the equilibrium distrib-
ution function,
∂f 0
∂t=
i
2 πLe^2
m^2∂
∂v∫
dk∣
∣
∣E ̃ 1 (k)∣
∣
∣
2
e^2 ωi(k)t
ω−kv∂f 0
∂v. (14.30)
The quantity∣
∣
∣E ̃ 1 (k)∣
∣
∣
2
has a physical meaning that is made evident by considering thevolume average of the electric field energy
〈WE〉 =
〈
ε 0 E 12
2〉
=
ε 0
2 L∫
dx(
1
2 π∫
dkE ̃ 1 (k)eikx−iω(k)t)(
1
2 π∫
dk′E ̃ 1 (k′)eik′x−iω(k′)t)
=
ε 0
4 πL∫
dk∫
dk′E ̃ 1 (k)E ̃ 1 (k′)e−i[ω(k)+ω(k′)]t 1
2 π∫
dxei(k+k′)x
.(14.31)However, using Eq.(14.14) this can be written as
〈WE〉 =
ε 0
4 πL∫
dkE ̃ 1 (k)E ̃ 1 (−k)e−i[ω(k)+ω(−k)]t=
∫
dkE(k,t) (14.32)where
E(k,t)=
ε 0
4 πL∣
∣
∣E ̃ 1 (k)∣
∣
∣
2
e^2 ωi(k)t (14.33)is the time-dependent electric field energy density associated withthe wavevectork.A
possible initial condition for the distribution function and wave energy spectrum is shown
in Fig.14.1(a);the wave energy spectrum is shown in the insert and is finite in the range
kmin<k<kmaxand the distribution function is monotonically decreasing so as to cause
Landau damping of the waves.
Combination of Eqs.(14.30) and (14.33) shows that the evolution of the equilibrium
distribution function is given by
∂f 0
∂t=
2i
ε 0e^2
m^2∂
∂v∫
dkE(k,t)
ω−kv∂f 0
∂v. (14.34)
This can be summarized as a velocity space-diffusion equation
∂f 0
∂t=
∂
∂v(
DQL
∂f 0
∂v