Fundamentals of Plasma Physics

14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 403

This expression can be further evaluated by invoking the parity properties ofE ̃ 1 (k)and
ω(k).These parity properties are established by writing

E 1 (x,t)=

1

2 π

∫

dkE ̃ 1 (k)eikx−iω(k)t (14.20)

and noting that the left hand side is real because it describes a physical quantity. Taking the
complex conjugate of Eq.(14.20) gives

E 1 (x,t)=

1

2 π

∫

dkE ̃ 1 ∗(k)e−ikx+iω

∗(k)t

. (14.21)

The parity properties are determined by defining a temporary new integration variablek′=
−kand noting that

∫∞

−∞dkcorresponds to

∫∞

−∞dk

′sincedk′=−dkand the klimits

(−∞,∞)correspond to thek′limits(∞,−∞). Thus Eq.(14.21) can be recast as

E 1 (x,t) =

1

2 π

∫

dk′E ̃ 1 ∗(−k′)eik

′x+iω∗(−k′)t

=

1

2 π

∫

dkE ̃ 1 ∗(−k)eikx+iω

∗(−k)t

(14.22)

where the primes have now been removed in the second line. Since Eq.(14.20)and (14.22)
have the same left hand sides, the right hand sides must also be the same. BecauseE ̃ 1 (k)
andω(k)are arbitrary, they must individually satisfy the respective parity conditions

E ̃ 1 (k) = E ̃∗ 1 (−k)

ω(k) = −ω∗(−k) (14.23)

or equivalently

E ̃ 1 (−k) = E ̃ 1 ∗(k)

ω(−k) = −ω∗(k). (14.24)

If the complex frequency is written in terms of explicit real and imaginary partsω(k) =
ωr(k)+iωi(k)then the frequency parity condition becomes

ωr(−k)+iωi(−k) = −[ωr(k)+iωi(k)]∗

= −ωr(k)+iωi(k) (14.25)

from which it can be concluded that

ωr(−k) = −ωr(k)

ωi(−k) = ωi(k). (14.26)

The real part ofωis therefore an odd function ofkwhereas the imaginary part is an even
function ofk.
Application of the parity conditions gives

E ̃ 1 (−k)E ̃ 1 (k)=E ̃ 1 ∗(k)E ̃ 1 (k)=

∣

∣

∣E ̃ 1 (k)

∣

∣

∣

2

(14.27)