Fundamentals of Plasma Physics

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)