Dispersion Relations in the Generalized Boltzmann Kinetic Theory Neglecting the Integral Collision Term

Appendix 4

Dispersion Relations in the Generalized Boltzmann Kinetic Theory Neglecting the Integral Collision Term We are concerned with developing (within the GBE framework) the dispersion relation for plasma in the absence of a magnetic field. We make the same assumptions used in developing this relation within the BE model, namely: (a) the integral collision term is neglected; (b) the evolution of electrons and ions in a self-consistent electric field corresponds to a one-dimensional, unsteady model; (c) distribution functions for ions fi and electrons fe deviate only slightly from their respective equilibrium values f0i and f0e: f i ¼ f 0i ðuÞ + df i ðx, u, tÞ,

(A.4.1)

f e ¼ f 0e ðuÞ + df e ðx, u, tÞ;

(A.4.2)

(d) we consider a wave mode corresponding to a certain wave number k and a complex frequency o, so that the solution of the GBE can be written in the form: df i ¼ hdf i ieiðkxotÞ ,

(A.4.3)

df e ¼ hdf e ieiðkxotÞ ;

(A.4.4)

(e) the quadratic terms in the GBE, determining the deviation from the equilibrium DFs, are neglected, and (f) the self-consistent forces Fi and Fe are small: Fi ¼  Fe ¼

e @c , mi @x

e @c , me @x

(A.4.5) (A.4.6)

where e is the absolute electron charge, mi are the ion masses, me the electron mass, and finally c ¼ hcieiðkxotÞ :

(A.4.7)

Under these assumptions, the GBE is written as follows (we seek the solution for the ion plasma component, to be specific):
2 2 @f i @f i @f i @ fi @2f i @2f i 2@ fi +u + Fi  ti + u + 2u + 2F i @t @x @u @t2 @t@x @x2  @t@u (A.4.8) 2 @Fi @f i @f i @Fi @f i @ f @2f i + Fi +u + F2i 2i + 2uFi ¼ 0: + @t @u @x @x @u @u @u@x

559

560

Appendix 4 Dispersion Relations Neglecting the Integral Collision Term

Using the assumptions listed above, we find the relations @f i @f @f e @c @f 0i ¼ iodf i , u i ¼ ikudf i , Fi i ¼  , @t @x @u mi @x @u 2 @2f i @2f i 2 2@ fi ¼ 2oukdf ¼ o df , 2u , u ¼ u2 k2 df i , i i @t2 @t@x @x2 @2f i @Fi @f i e @f @f ¼ 0, ¼  okc 0i , Fi i ¼ 0, 2Fi @u@t @t @u @u @x mi 2 2 @f @Fi e 2 @f 0i @ f @ fi ¼ k uc , F2i 2i ¼ 0, 2uFi ¼ 0, u i @u @x mi @u @u @u@x

(A.4.9)

which when substituted into Eq. (A.4.8) yield iðku  oÞhdf i i  i


 e @f ek @f 0i ¼ 0, khci 0i  ðku  oÞti ðku  oÞhdf i i + hci @u mi mi @u

giving the ion density fluctuation

(A.4.10)

ð e @f 0i =@u du hdni i ¼  hcik mi o  ku

(A.4.11)

ð e @f 0e =@u du: hcik me o  ku

(A.4.12)

and the electron density fluctuation hdne i ¼

Equations (A.4.11) and (A.4.12) are identical to their BE analogues. Substituting Eqs. (A.4.11) and (A.4.12) into the Poisson equation e0 k2 c ¼ eðdni  dne Þ we arrive at the classical dispersion relation (see, for instance, Ref. [144])
ð  ð e2 1 + 1 @f 0e =@u 1 + 1 @f 0i =@u 1¼ du + du : e0 k me 1 o  ku mi 1 o  ku

(A.4.13)

(A.4.14)

Although Eqs. (A.4.11) and (A.4.12) are a consequence of the general statement that in the absence of the integral collision term the relation Df a ¼0 Dt

(A.4.15)

(the Vlasov equation) is the solution of the equation

  Df a D Df a  ¼ 0: ta Dt Dt Dt

(A.4.16)

The above argument shows that the GBE can produce correct and expected results, when treated perturbatively.