402 Chapter 14. Wave-particle nonlinearities
On applying this postulate, Eq.(14.10) reduces to thequasi-linear velocity-space diffusion
equation
∂f 0
∂t
=
e
m
∂
∂v
〈E 1 f 1 〉. (14.11)
This implies that even thoughf 0 is of orderǫ^0 ,the time derivative off 0 is of orderǫ^2 so
thatf 0 is a very slowly changing equilibrium.
The nonlinear term〈E 1 f 1 〉can be explicitly calculated by first expressing the perturba-
tions as a sum (i.e., integral) over spatial Fourier modes, for example the first order electric
field can be expressed as
E 1 (x,t)=
1
2 π
∫
dkE ̃ 1 (k,t)eikx. (14.12)
This allows the product on the right hand side of Eq.(14.11) to be evaluated as
〈E 1 f 1 〉 =
1
L
∫
dx
[(
1
2 π
∫
dkE ̃ 1 (k,t)eikx
)(
1
2 π
∫
dk′f ̃ 1 (k′,v,t)eik
′x
)]
=
1
2 πL
∫
dk
∫
dkE ̃ 1 (k,t)f ̃ 1 (k′,v,t)
1
2 π
∫
dxei(k+k
′)x
(14.13)
where the order of integration has been changed in the second line. Then, invoking the
representation of the Dirac delta function
δ(k)=
1
2 π
∫
dxeikx, (14.14)
Eq.(14.13) reduces to
〈E 1 f 1 〉=
1
2 πL
∫
dkE ̃ 1 (−k,t)f ̃ 1 (k,v,t). (14.15)
The linear perturbationsE 1 andf 1 are governed by the system of linear equations discussed
in Section 4.5. This means that associated with each wavevectorkthere is a complex
frequencyω(k)which is determined by the linear dispersion relationD(ω(k),k)=0.This
gives the explicit time dependence of the modes,
E ̃ 1 (k,t) = E ̃ 1 (k)e−iω(k)t (14.16a)
f ̃ 1 (k,v,t) = f ̃ 1 (k,v)e−iω(k)t. (14.16b)
Furthermore, Eq.(14.8) provides a relationship betweenE ̃ 1 (k)andf ̃ 1 (k,v)since the spa-
tial Fourier transform of this equation is
−iωf 1 +ikvf 1 −
e
m
E 1
∂f 0
∂v
=0 (14.17)
which leads to the familiar linear relationship
f ̃ 1 (k,v)=ie
m
E ̃ 1 (k)∂f^0 /∂v
ω−kv
. (14.18)
Inserting Eqs.(14.18) and (14.16a) into Eq.(14.15) shows that
〈E 1 f 1 〉=
i
2 πL
e
m
∫
dkE ̃ 1 (−k)E ̃ 1 (k)
∂f 0 /∂v
ω−kv
e−i[ω(−k)+ω(k)]t. (14.19)