Kinetic theory of slightly nonideal plasmas
Citation for published version (APA):Brouwer, H. H. (1987). Kinetic theory of slightly nonideal plasmas. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR258680
DOI:
10.6100/IR258680
Document status and date: Published: 01/01/1987
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KINETIC THEORY OF
SLIGHTLY NONIDEAL PLASMAS
(1.) j ( j
(tu.i
5.)
=
_i_
{d~},
Zfd'l
Z
jd~
Z
d
3
Pi
!/
.~t
(z.
71)
3s
t
}J fris /Tlt
Ca
J.
1-
f.G ._,
J l/
J _....",.,__...,.."__-...,.__. ___ ~..-,.,KINETIC THEORY OF
SLIGHTLY NONIDEAL PLASMAS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE TECHNISCHE UNIVERSITEIT EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF.DR. F.N. HOOGE, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
VRIJDAG 13 MAART TE 16.00 UUR
DOOR
HARM HENDRIK BROUWER
GEBOREN TE GRONINGENDIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOREN: PROF.DR.IR. P.P.J.M. SCHRAM
EN
CONTENTS page I INTRODUCTION I I III IV V VI BASIC E':!UATIONS
1. Microscopie and averaged density functions 2. Multiple time scale formalism
3. Fourier transformation
THE LANDAU PERTURBATION SCHEME
9 9 10 13 15 I, Introduetion 15 ,2. Initial conditions 16
3. The Landau kinetic equation 17
4. The first order correction to the Landau equation 23
5. Discnssion 25
THE LENARD-BALESCU'PERTURBATION SCHEME
I. Introduetion
2. Initial conditions
3. The Lenard-Balescu equation
4. Gorreetiens to the Lenard-Balescu Collision integral
5. Discussion
SOME FURTHER REMARKS ON THE PERTURBATION SCHEMES I. Gomparisen of the Landau and Lenard-Balescu
28 28 29 31 35 38 40 perturbation schemes 40
2. The Boltzmann scheme 43
3. Construction of a convergent cellision integral 43
THE ELECTRICAL CONDUCTIVITY AT LOW FREQUENCIES
J, Introduetion
2. Outline of the metbod 3. Calculation of cr1B 4. Calculation of crc 45 45 46 48 51
VII THE ELECTRICAL CONDUCTIVITY AT HIGH FREQUENCIES 60
1 . Introduetion 60
2. Zeroth order conductivity 60
3. Second order conductivity 63
4. The conductivity in case of a homogeneaus electric
field 68
5. Comparison with Kubo's formalism 75
6. Discussion 79
VIII THE DISPERSION RELATION
I' The conneetion between the conductivity and the
dispersion relation
2. The dispersion relation in zeroth order
3. The dispersion re lation in secoud order
4. Discussion
APPENDICES
A Calculation of the field-density moments in third order
of the Landau scheme
B Some features of the calculation of the correction to
the Lenard-Balescu collision integral
c
Expansion of the Boltzmann integral into powers ofthe interaction parameter
D The second order equilibrium density-density moments
in the Lenard-Balescu scheme
E The correction to the high frequency conductivity
of a hydragenie plasma LIST OF SYMBOLS 82 82 85 86 89 91 95 102 105 107 I I 4
REFERENCES 119
SUMMARY 122
SAMENVATTING 125
DANKWOORD 128
I INTRODUCTION
The first kinetic equation for plasmas was derived by Landau in 1936 [LAN 36]. This equation is still widely used,
The Landau cellision integral gives the rate of change of the ene-partiele distribution functions as caused by those two-particle collisions for which the deflection angle is small,
It can be derived by solving the first two equations of the B.B.G,K.Y, hierarchy in the weak coupling limit,
Another derivation consists of the substitution of the Rutherford cross-section into the Boltzmann integral, which incorporates all two-particle collisions, and the subsequent approximation with respect to small deflection angles,
The latter derivation clearly shows the origin of the shortcomings connected with the Landau equation,
The Boltzmann integral is based on the assumption that variations in the distribution functions are mainly caused by isolated two-particle collisions.
It is clear that this assumption is not justified for plasmas, because of the long range of the Coulomb potential.
In a plasma each partiele interacts simultaneously with a large number of ether particles.
This fact and the approximation with respect to the small deflection angles causes a twofold logarithmic divergence in the Landau integral.
These divergencies are usually dealt with by the introduetion of two
cut-off lengths, which amounts to the neglect of both distant and close collisions. This procedure leads to the introduetion of the well-known Coulomb logarithm.
The argument of this Coulomb logarithm is inversely proportional to the plasma parameter, which is therefore defined here as the ratio of the lower to the upper cut-off length.
The nature of these lengths will be discussed later.
As a function of density and temperature the plasma parameter is given by:
where q and n are the electron charge and density respectively, kB
e e
is the Boltzmann constant and T the temperature.
Results obtained from the Landau equation are accurate only when the Coulomb logarithm is very large, i.e. when the plasma parameter is extremely small. This is the case for plasmas with a very high temperature and a very low charged partiele density. These plasmas are called ideal plasmas.
Many plasmas, however, such as those in M.H.D.-generators or are discharges, do not satisfy these conditions.
This has caused a vivid interest in both the experimental and theoretical investigations of nonideal plasmas.
In the field of kinetic theory, an important improvement with respect to the Landau equation was found in 1960.
The incorporation of dynamical screening, which is one of the main aspects of the collective behaviour of the plasma particles, has led to the so-called Lenard-Balescu equation [LEN 60, BAL 60].
Lenard obtained the electron-electron Collision integral from a partial solution of an integral equation for the asymptotic binary correlation function, which was derived by Bogoliubov [BOG 62] from the first two equations of the B.B.G.K.Y. hierarchy in the weak coupling
approximation, including the effects caused by the long range of the Coulomb potential.
This integral equation was treated in a similar way by Balescu, but he derived it on basis of a triple expansion in density, time and coupling strength, which was partially summed by means of a diagram technique.
Goropared to the Landau equation, only one of the two divergencies is present in the Lenard-Balescu equation, namely the one connected with close collisions.
Both the kinetic equations mentioned:pertain to the classical theory of plasmas .,
Later on a vast amount of work has been done on the quantum-mechanical description of nonideal plasmas.
functions technique and the metbod of second quantization [KAD 62].
A recent review of this metbod and tbe results obtained by it, has been publisbed by Kraeft et al.
[KRA 86].
It is, however, not always necessary to invoke quantummechanics. If the distance of closest approach is much larger than the partiele sizes and the thermal, de Broglie wavelength for electrons, given by
À
e
h
(I. 2)
wbere me is the electron mass, one may assume that the particles can be described as classical point charges.
The distance of closest approach corresponding to electrens having thermal energy is given by the Landau length, defined by
2
qe
À -
-;---"';---:;;-1 4'11' e0 kBT (I. 3)
It is this length which is used as the lower cut-off for the Landau and the Lenard-Balescu callision integrals.
Adopting it as the characteristic distance of approach for all types of close collisions between charged particles and taking the Bohr radius a
0 = 4'11' e0 h
2/(meq!) as a measure for the partiele sizes, we find the following conditions:
2 ÀL qe (me )~ >> I ~ 4'11' e:o h ~T (1.4a) 4 qe me 2 >> 1 2 .. e:B ao (41r e: 0h) ~T (1.4b)
If e:B' which is called the Born parameter, satisfies condition (1.4a), then obviously condition (1.4b) is also satisfied.
Note that, except for very high values of the plasma parameter, condition (1.4a) also implies the non-degeneracy of the electrons:
n
e ( 1.5)
simplifications.
In the first place we consider a plasma that is homogeneous, i.e. the model can be applied if the characteristic lengtbs of temperature and density variations are much larger than the microscopie lengtbs which are characteristic for the interactions and are determined by the plasma temperature and density. The largest of these is the electron Debye length, defined by:
À
=
De (1.6)
In chapter IV a slightly more generai Debye length is introduced. It is this length which serves as the upper cut-off for the Landau Collision integral.
Secondly, the plasma considered is fully ionized, i.e. the only potential occurring in the theory is the Coulomb potential.
This restrietion in principle corresponds to an ionization degree a of unity, where a is defined by
n.
l. a = - - -n.+n
l. a
(I. 7)
n. and n being the densities of ions and neutral atoms respectively.
l. a
Due to the long range of the Coulomb potential, however, collisions between charged particles may dominate the collisions invalving neutrals at much lower degrees of ionization already.
The average cellision frequency for electron-electron collisions is given by
v ee
=
n e ee Q (k_Tfm --B e)i
(1.8)where the average elastic callision cross-sectien Q can be derived ee
from the Landau equation. This results into:
4
!
2 -1For the electron-atom callision frequency we may write:
\)
ea (1. I O)
where Qea is an average elastic callision cross-section. Both Q and Q pertain to momentum transfer.
ee ea
Futhermore Qea depends on temperature only, whereas Qee depends also weakly on the density.
With T
~
2 x 104 K, we have for argon:(1.11)
For e ~ 0.01 we then find:
p
(l. 12)
We consider a plasma to be fully ionized if a~ O.I.
The formalism presented in this thesis is partly based on that of Klimontovich, which involves the use of the microscopie density
functions.
These describe the position in phase space of all particles. In fact the present work covers some of the topics which have also been treated in Klimontovich' monographof 1982 [KLI 82].
Three of the major diffèrences between the methods preseneed in the latter and this thesis are the following:
a. Starting with the Klimontovich equation for the microscopie densities one has to take statistica! averages sooner or later in order to derive macroscopie properties.
In this thesis this is done in a later stage of the calculations than in Klimontovich' work.
In this way a straightforward formalism is constructed with a remarkable mathematica! simplicity.
binary collision or polarization approximation, the formalism in the present work consists of a systematic expansion into powers of a small parameter.
c, Klimontovich introduces nonideality only by considering the retardation effect, i.e. the change of the one-particle distribution function during the interaction process.
In this thesis, a plasma is considered to be ideal when some dimensionless parameter connected with the interactions, e.g. the plasma parameter, is very small.
The effect of (slight) nonideality on a quantity can then be investigated by expansion of that quantity into powers of this parameter.
Of course, the differences b. and c. are not entirely independent, as with the definition of a nonideality parameter, we have a perturbation parameter at our disposal at the same time.
The perturbation schemes, as presented in chapters III and IV, are the Landau and Lenard-Balescu schemes, which have been introduced by Lambert [LAM 83, 85] •.
These names have been chosen because, in lowest significant order, the schemes lead to the Landau and Lenard-Balescu callision integrals respectively.
The. schemes which are described in this thesis are essentially the same as Larobert's schemes; but in contrast to his work they are provided here with a firm basis. This is done by invoking the multiple time scale formalism, which bas been described in detail by Sandri
[SAN 63).
In chapter V the relation between the two schemes and their relation to the Boltzmann kinetic theory for gases is discussed, as well as the possibility to construct a convergent cellision integral. Another convergent collieion integral, derived by Lifshitz and
Pitaewskii [LIF 83], and, for comparison, the Lenard-Balescu integral, are used in chapter VI to calculate the a.c. conductivity, for
frequencies much lower than the plasma frequency. This chapter is basedon an artiele that bas been accepted for publication {BRO 87]. The necessity of the restrietion to low frequencies is caused by the
limited validity of the kinetic equations. In chapter VII is shown how the Lenard-Balescu perturbation scheme can be used to calculate the high frequency conductivity.
The results of chapter VII are used in chapter VIII to discuss the dispersion relation for plasma waves.
Since we use the nonideality parameter as a perturbation parameter, it is clear that the formalism does not yield results which are valid for all degrees of nonideality.
Most results presented in this thesis are valid for small values of the plasma parameter E p only. This is still consideiably better than the lowest order results, e.g. as obtained from the Landau equation, which are valid only for large values of lln epi·
Tagether with the restrictions corresponding to the classical model and full ionization we now have three conditions which have to be satisfied.
In the density-temperature plane, drawn in fig. 1.1, we have shaded the region where:
metals -1025r---m ~ c: semi conductors
Fig. 1.1 Density-temperature plane with real plasmas
and region of validity for the theory presented in this thesis.
The degree of ionization a is calcul*ted according to the Saha equation: n . n a e I -U. /kBT - e J. h3 (1.6) ni I =
-where for the ionization potential Ui we have taken the value 15.8 eV corresponding to the first ionization step of argon. Argon is a gas which is frequently used in are discharge experiments.
As can be seen from the figure, there exist laboratory plasmas for which the model presented in this thesis should give reliable results, although quantum effects may not be entirely negligible.
II BASIC EQUATIONS
1. Microscopie and averaged density functions
We consider a fully ionized plasma consisting of N particles of each
s
species s.
A partiele of species s is characterized by a mass m and an electric
s
charge q
5•
The specific microscopie density is defined by n (t,r,v)
=
Ea Es
~(r - r (t)) o(V - V (t)) (2. I)
s - - - - a - - a .
where (r ,v ), aEs is the position in phase space of a partiele of
-a -a.
species s.
This density function satisfies the Klimontovich equation [KLI 82],
which reads, in the electrostatic approximation
a
qs["t +
y_.'ïJ
+ -m!
(t,r).V] n (t,r,v) = 0o - V S - - (2.2)
S
where E is the total microscopie electric field.
When there is no external field, it is related to the microscopie density functions by
(2.3)
The statistical average of a microscopie function A of a N - partiele system is defined by
<A>(t) Jdr A(r) n(r,t) (2.4)
where r is a vector repreaenting the position of the system in the 6N- dimensional r - space and D(r,t) is the ensemble density function. In eq. (2.4)'A is considered to depend only implicitly on time, i.e. through the orbit of the system in r - space: r (ro,t) where
r
0 = r(t
=
O).We can however also consider A to be explicitly dependent on time and
In that case we write
(2.5)
where
o
0<r
0)= D(r
0,o).
In eq. (2.1) we actually should have written
!a(r
0 ,t) and!a(r
0 ,t), sowe will use eq. (2.5) rather than eq. (2.4).
In the sequel we use the notation <A(t)> instead of <A>(t). The averaged density functions are defined by
f (t,r,v)
=
<n (t,r,v)>s - - s
-and the fluctuations by
ön (t,r,v) = n (t,r,v) - f (t,r,v)
5 - - s. - - s
-(2.6)
(2. 7)
Averaged qqantities will be considered only in the thermodynamic limit, i.e. N5- ,
v-,
n5 = N5 /V is finite., where V is the volume of the
system.
For homogeneous systems, as considered in chapters III - VI, we have
Jd3v f8(t,v)
=
n- s (2.8)
In the sequel arguments will be omitted when there is no cause for confusion.
2. Multiple time scale formalism
In chapters III and IV the Klimontov~ch equation, eq. (2.2), is solved iteratively within the multiple time scale formalism.
An extensive description of this formalism has been given by Sandri
(SAN 63].
In this section a simple example is worked out, showing the most important features of the formalism.
Consider the equation df + e:f
with E << I, and the initial condition
f(O) = A (2.10)
Trying to solve eq. (2.9) by means of a perturbation expansion in E, i.e. we write
(2. IJ)
we find, by successively solving eq. (2.9) for each order in E
A (2.12)
-At (2 .13)
(2. 14)
etc ••
However, in an expansion like eq. (2,11) physicists want the second term to be much smaller than the first one, which is clearly not the case for t ~ 1/E.
This is what we call a secular divergence or secularity. To apply the multiple time scale formalism, we define a function of N+l variables
f(•
0 , • 1, - 'N), for which we postulate the following differential equation(2.15)
and the initial condition
I<•o- o, •
1=
o, - - -
'N=
o)
=A (2,16) If f satisfies the condition-f(t,Et, - - N
E t)
=
f(t) (2. 17)problem for f(t) to a N - dimensional time space. The space of solutions is also extended, which gives us the possibility to demand that all secularities vanish.
The removal of secularities has a central position in this method. Writing
2-f = f
0 + e:f 1 + e f2 + - - - (2 .IS)
with the initial condition
A;
o,
i > (2.19)we find in zeroth order
,,
(2.20)which gives
(2. 21)
In first order one finds:
ac
a"f,
- - + - - +c"'O
i1T1 a-ro (2.22)
of' which the solution is
(2.23)
Here we have a secularity that we remove by demanding
(2.24)
Satisfying the initial conditions we find
Choosing N = I, we have from eqs, (2,21) and (2,17)
f(t) = Ae-et (2.26)
Of course, in most applications of this metbod one will not find the exact solution, as we did bere.
3. Fourier transformation
In the following chapters one - sided Fourier transformations are used frequently.
For a function a(t) this is defined by
. def. . +
a(w)
=
lim J~dt e1(w+ie)t a(t) /~dt e100 t a(t)e+O o o (2.27)
An alternative notation for w+ is w + iO.
Of course, the inverse transformation returns the function a(t) for positive values of t only:
+w -iwtÀ
h(t) a(t)
=
2
~ -~~ dwe a (w) (2.28)where h(t) is Heavisides step function:
h(t)
o.
t < 0 h(t) I, t > 0 (2.29)As most calculations are performed for t > 0, the function h(t) will be written only when necessary.
If lim a(t) exists one may use the final value theorem t~
lim a(t) • lim - iwa(w)
t~ ~o
Spatial Fourier transformation will also be applied.
(2.30)
This will simply be indicated by writing a k variable instead of ~·
e.g.
A
The relation between E and n8 can be obtained straightforwardly from the Fourier transformed Poisson equation.
If one uses eq. (2.3), however, a factore-ar should be introduced in the right hand side.
Taking the limit ~~o afterwards one finds
III THE LANDAU PERTURBATION SCHEME
1. Introduetion
In this chapter a perturbation scheme for homogeneaus plasmas is
presented, that gives as one of its results the Landau kinetic equation. The perturbation parameter is not ehosen in advanee, but is suggested by the choice of the eharacteristic quantities that are needed to write the Klimontovich equation [eq.(2.2) ] in a dimensionless form.
In this ehapter these quantities are taken to be: the thermal veloeities vT , the avera~e interpartiele the eharaeteristie field ~trength ~ (the charges sameorder of magnitude), eorav
Next we define the dimensionless quantities:
3 3 r' I v• I
n•
nsravvT = - - r = - - V s s r -s VT -av VT 2 s t• s---
s t E' = e:orav E rav qWe now find for the Klimontovich equation
With
m
s
distanee r and av are all of the
(3.1)
(3.2)
(3.3)
where kB is the Boltzmann constant and Ts the speeific temperature, we assume that, for plasmas not too far from equilibrium, the third term of eq. (3,2) is of the order of the interactionparameter e:
1, defined
by
(3.4)
For ~I << l this implies a restrietion to a hot dilute, i.e. nearly ideal, plasma.
In the sequel of this chapter, €I will serve as the perturbation parameter.
As we will use the Klimontovich equation itself, and not its dimensionless form,
like
will not oeeur explieitly; but in expansions
n = n(O) + n(l)
s s s + - - - (3.5)
~t ~s assumed that n(i) · f d d t (i-I)
~ ~
5 ~s o or er ~I eompare o ns .
2. Initial eonditions
As we are ultimately only interested in averaged quantities, it is suffieient to impose initial eonditions on the eorrelation funetions, Usually these functions are defined in terms of the (averaged) multiple partiele density funetions, but they ean also be defined in
the quantities we have defined so far by
g2(I,2)
=
<on(l)on(2)> - ö(I-2) f(2)3 g3(I,2,3) +
i
L o(i-j) g2(j,k)=
i,j ,k=l <on(l)on(2)on(3)> - o(I-2)o(2-3) f(3) 4 g4(t,2,3,4) + E {}o(i-j)g3(j,k,l) + !ö(i-j)f(j)g2(k,l) i,j,k,l=l + t/8 ö(i-j)ö(k-l)g2(j,l) + t/6 o(i-j)ö(j-k)g2(k,l)}=
4 <ón(l)ón(2)ön(3)ón(4)> - 1/8 E ó(ij)ö(kl)f(j)f(l) -i,j,k,l=1 ö(l-2)ö(2-3)ö(3-4) f(4) etc •• (3.6a) (3.6b) (3.6e)In these equations the partiele variables have been abbreviated to a single number, e.g.
ö(l-2)
and in the summations all indices should be taken differently. The initial conditions are taken in tbe spirit of Bogoliubovs
condition of complete weakening of the initia! correlations [BOG 62] ,
i > 2 (3. 7)
If there exist initia! correlations, and they satisfy certain conditions, their influence will decay fast so that the kinetic behaviour will not be impeded.
The conditions they have to satisfy have been called by Sandri [SAN 63] the "absence of parallel motions".
A more detailed investigation of these conditions can be found in [SCH 64].
After extension to multiple time scales wedemand that eq, (3,7) is satisfied on the entire r
0 axis, ~.e. :
i > 2 (3.8)
This is different from eq. (2.16), but it givesfue correct result on the "physical line" "n
=
e:nt.Furthermore, by making this choice we avoid complications involving the nonphysical functions gi(o,-r1, - - TN) [FRI 66].
In the sequel it will turn out that eq. (3.8) leaves enough freedom to remove the secularities up to the highest order considered in this thesis.
It is however possible that it cannot be maintained in still higher orders.
3. The Landau kinetic equation
In this section we will solve the Klimontovich equation, making use of the multiple time scale formalism, up to second order in e
1•
Applying this method, we choose N
=
2 (see section 11,2), e.g.:Zeroth order theory
The Klimontovich equation in lowest order reads
[ ~ + v.V]n(O) c 0
o"to - s (3.10)
Averaging this equation, and recalling the homogeneity of the plasma,
we find _a_ f(o) • o a-r
0 s
Subtraction. of eq. (3.11) from eq. (3.10) gives
(3. 11)
(3.12) Applying Fourier transformation with respect to
'o
and r (one-sided with respect to -r0) we obtain6n~O)
<•o
=
o,
'1•'2·~·~)
Ml~O)
(w,'P'z•
~.~)= _;:,_ _ __: ___
_;... ___
_
i(~.~- w+)
(3.13)
6n!0) are the fluctuations of a plasma without any interaction, i.e. a plasma where all partiele orbits are straight lines.
First order theory In first order we have
Let us first consider the average of the fourth term.
Using Fourier transforms, eqs. (2.32), (3.13) and the initia! conditions, which can be written in this case as
(3.14)
we find
(3. 16)
If the integration volume in the right hand side of eq, (3.16) is restricted to k < k • the integral vanishes for any value of kmax•
max
as the integrand is antisymmetrie in ~·
This may be used as an argument to put this expression equal to zero, There exists however a more physical justification. The right hand side of eq. (3.16) corresponds toa "self-field", which does not exist in the classical electrastatic approximation,
The self-field of a partiele ~ would follow from the replacement of
in eq. (2.3) by
qNo(~ + ~
1
- r ) o(v - v )~ -a - -a
In the expression
(3.17)
the self-field term is equal to
- 1- f d3k 3 k d3r ei~. <!.1-~) E f d VI qa f o(rl-r )ó(vl-v ) (2'11') 3 ~ES i k2 e I - -a - -a 0 k o(r-r )o(v-v )= - 1- f d3k - - - - qs f (:'!} (3. 18) - -a - -a (2'11')3 i k2 eo s
So, < E(O) n(O) > is equal to the zeroth order of this and is excluded
- s
from the calculation.
In the sequel we will sametimes invoke the antisymmetry argument for practical reasons, keeping in mind the physical argument just ~xpounded.
We thus find for the average of eq. (3.14): __ a __ f(l) +_a_ f(O) ~ 0
oTO s (hl s (3. 19)
As
f~O)
is independent of TO and we demand thatf~l)
remains finitefor TO + ~ we must have:
(3.20)
For the first order fluctuations we find:
(3. 21)
Second order theory
The Klimontovich equation in secoud order reads:
[--ê-- + v.V]n(2) + [--l!-+ qs E(O) .V ]n(l) + [ l! + E(l) .V ]n(O) ". 0 •
liTO - s l!T
1 m6 - v s a1:2 m5 - v s (3.22) Averaging this equation we obtain:
(3.23) Remaval of the secularity leads to:
(3.24)
where
Applying the final value theorem [eq. (2.30)1 and inverse Fourier transformation we find:
>
=
(3.26)
Using eqs. (2.32), (3.13) and (3.21) and integrating over TO we obtain:
o
3 3 ei(11+12>·E 3 11lim - -8 fdwl/d k 1Jdw2Jd k2 + E fd v1
qt-.-2
-R+O (2~) Q -w
1-w2 t ~k
1
&03 (O) -(o) -<o>
Jdw
3Jd k3 <önt (T
0
=o,~1
,y1
)!
(w3,!_3)öns (w2-w3,!_2-!_3,y)>]} • The first two terms between curly brackets, tagether with theircounterparts from lim < i<t)ön(O)> , give no contribution.
- s
To--The first one, < ön(O)(T =O)ön(I)(T =0) > +
t 0 s 0
(3.27)
< ön(l)(T =O)ön(O)(T =0) > gives the right hand side of eq. (3.16)
t 0 s 0 •
with f(O) replaced by f(l).
s s
The second term, containing:
< ön(O)(T =0) __
a
__
o (O)(T =0) > + < on(O)(T•O) __a
__
ón(O)(T =0) >t 0 <lT I s s <lT 1 t 0
is proportional to
~
f(O) • which is zero. <1T SThe integrand of the
l
3-integral can be shown to be antisymmetrie and therefore the corresponding term does not contribute either.In the remaining part we apply once more eqs. (2.32), (3.13) and
(3.15) after which we perferm a contour integration intheupper w
2 and the lower w
1 half-plane.
Finally we obtain, dropping the subscript of k1:
2 -i 3 3 qsqt - - 3 f d k r.J d v 1 -k-4m;;....;e:'--2 ~ (21T) t s 0 ft(O) (~l) k. V f(O) (v) V S -~· (~-~~ )+iO (3.28) Using the Plemelj formula
1Tiö(x) (3.29)
we can replace (~.(~-~
1
)+i0)-l by -1Tiö(1,(~-~1
)), as the principalvalue part' is antisymmetrie in~·
Along the same lines we derive
Inserting eqs. (3.28) and 3.30) into the right hand side of eq. (3;25) we obtain
(3.30)
(3.31)
We have dropped the T
1- variable in , eq. (3.31) is independentof T
1,
J(2), as the right hand side of
! s
(3.32)
2
Going back to the "physical line". i.e. taking •o=t, ,1=e:t,
•z=e:
t, and recalling that the small parameter does not explicitly occur. we may replace t2 by t.
The kinetic equation, obtained by using eqs. (3.32) and (3.24), i.e.:
(3.33) is the well-known Landau equation [LAN 36].
4. The first order correction to the Landau equation
Averaging the third order Klimontovich equation and removing the secularity we find:
• (3.34)
where
J~
3)
- :svv.
lim [ < !(O)on!
2) > + < !(O)on!
1) > + < !(Z)on!
0)>]s t
0-As we can express ons (Z) in lower order quantities by means of eq. (3.22), the right hand side of eq. (3.35) can be calculated straightforwardly, see appendix A.
The result is
(3.3S)
J(3) + J(3) + J(3)
s,l s,2 s,3 (3.36)
k.V k.V - vl - v2 [ mtmp
~· (!_
1
~)+i0
1·
(Y[!_)+iO + + k.\l k,V I - V - VI msmt ~· (!.-.!2)-iO1·
<!.t-y2)+i0 f ( s O) (• '2'-v) f t (O) ( ,.2•!.1 ) f p (O) ( ,.2'-2 v ) (3.37) (3.38)where each differential operator acts on everything following it, and J( 3) is equal to J( 2) with f(O) f(O) replaced by J(O) f(l) + f(l) f(O)
s,3 s s t s t s t •
J!
3) is also independent of ,.1 and thus we obtain from eq, (3.34):
(3.39)
Replacing T
2 by t again, we obtain
_a_
f(l) = /3) (t,y>at
s s (3.40)Summing eqs. (3.33) and (3.41) and dropping the superscripts on the density functions, we see that
of
(t,v) (3) (3)s - = J(2) (t,y) + J (t,y) + J (t,y)
ot
s s,l s,2is the kinetic equation for fs(t,!_) up to third order in the interaction parameter.
Note that, in this notation, J(3) is contained in J(Z).
s,3 s
It can be shown that
(3.42)
s s
i.e. particles, momentum and kinetic energy are conserved.
Kinetic energy is not conserved in fourth order, as the rate of change of field energy, proportional to ~<EL> , is not zero in that order
T2
[KLI 82].
If Maxwell distributions with the same temperature are substituted in the right hand side of eq. (3.42) one finds that it vanishes.
Thus, as one may expect, the Maxwell distributions are equilibrium distributions for slightly non-ideal plasmas toa,
5. Discussion
In the preceding sections we have seen that the combination of the Klimontovich theory for microscopie density functions [KLI 82] and Sandri's multiple time scale formalism [SAN 63] leads to a straight-forward and unambiguous expansion of correlation functions, and thus of the callision integral, in powers of some small parameter. In the present chapter this small parameter is the interaction parameter.
In lewest order the callision integral is found to be the Landau integral.
The calculation of the next term offers na difficulties and is in full agreement with an expression found by Sandri [SAN 63}.
This term can be seen as a correction for non-ideality.
A difference between our methad and that of Sandri is, that he imposed an extra initial condition, namely
f(i) (t
s 0) - 0 • i > 2 (3.43)
This resulted in [see eq. (3.20)]
and a correction to the kinetic equation that looks like
(3.45) In the first place, there is no physical argument to impose eq. (3.44), and secondly there is no physical interpretation of the time scale ,
3, in contrast to ,0 and ,
2, ,0 being the time scale on which partiele
interactions take place, and <
2 being the time scale on which the
system relaxes towards equilibrium. The time scale 1:
1 does not allow for ,such a physical interpretation. However, there is no need for such ari interpretation since all relevant quantities are independent of 1:
1, see eqs. (3,20), (3.32) and (3.40), The importance of using an expansion like the one presented here, can be seen by comparing our results with those of Klimontovich [KLI 82].
He assumes that non-ideality is caused by a time retardation effect, i.e. the variation of the distribution during the interaction process. The incorporation of this effect leads to
2 2
:s
qt 2~
• Vv Reol"
d<e-2e:-r-i~.
(,Y."".Y,J )-rk mse:O
e:+O (3.46)
If the distributions are expanded in 1:, one retains in zeroth order
the Landau collision integral. In first order, writing
f (t-t) - f (t) .. -rJ(Z) (3.47)
one obtains an expression, that is of order e:14 •
In other words, there are non-ideality effects which are more important than the time retardation, as they cause a correction of order e:13 insteadof e:14 •
What these other effects are will become clearer in chapter V, where , the conneetion between the Landau, Lenard-Balescu and Boltzmann
It should be noted here that the usefulness of corrections like eq, (3,41) is questionable.
As the integration over wave-number space in eq. (3.31) is divergent for both small and large values of k, one has to introduce cut-off values. leading to the well-known Coulomb logarithm, which is linear
in ln (e: 1-1).
The resulting integral is insensitive to the precise values of the cut-offs, if the parameter e:
1 is small enough.
This situation is not altered by adding a finite number of terms like
J~
3)
to the collision integral.
IV THE LENARD-BALESCU PERTURBATION SCHEME
I. Introduetion
In the previous chapter we have determined the perturbation parameter by defining a set of characteristic quantities and writing the Klimontovich equation in a dimensionless form.
These characteristic quantities however can be defined in more than one way and the choice made in chapter III is not necessarily the best one.
Indeed, the averaged interpartiele distance r av is not the characteristic length for the interaction forces and correlation functions.
Thus the interaction range should be a better choice than r av •
The interaction range of the bare Coulomb potential is infinite, but, due to screening, an effective potential exists with a finite range. In thermal equilibrium the collective behaviour of the charged particles, surrounding a test partiele at rest, is such that its effective potential is
where C is a constant and ÀD is the Debye length, defined by
In this chapter the Debye length is chosen as the characteristic length in the plasma.
We now adopt as the new set of dimensionless quantities:
n'
s n s n T - v I 3 r' = -À -1 r -s v'=-
1 V VT s D s VT (4. 1) (4.2) EO t' s .. __ s t E'=--E
(4.3) ÀD qÀDn-where the subscripts of q and n have been dropped as they are of the same order of magnitude for all partiele species.
The dimensionless Klimontovich equation now reads
2 2
q lD n
+ v' •
-s
v•
+ e:o~T!'.
] n' s 0 (4.4)The factor in front of!' is of order unity, so that this equation, in contrast to eq. (3.2), does not suggest a perturbation parameter. In the next section it is shown thát this parameter is provided by the initial conditions.
2. Initial conditions
The initial conditions are taken to be the same as in chapter III, see eq. (3,8).
When we substitute eq. (3.8) into eqs. (3.6) and write the resulting equations in a dimensionless form according to eq. (3.1), noparameter appears.
Doing the same using eq. (4.3) however, we find for the two-particle density moments
" ' (0 ' ') " ' (0 ' ' == - -1- ö "( '
')~(
' ')f' ( ')<un , E.l , ~I un , E.2, ~
2
)> 3 u r -r u v -v v ,s 1 s 2
"n
n s 1 s 2 -1 -2 -1 -2 s 1 -1 (4.5)The dimensionless number appearing in the right hand side is of the order of the plasma parameter, which is defined bere by
where
). = De
is the electron Debye length.
(4.6)
(4. 7)
Note that e: is proportional to the inverse of the number of particles
p 3/2
in a Debye sphere and to e:1 •
densities and the right hand side is proportional to ~ , suggests that p
we should take ~~ as the new perturbation parameter,
p
We thus wri te n
=
n(O) + n(l)s s s + • • (4.8)
and assume that 11(i) I n(i-1) is of the order
Ie. •
s s p
Assuming that the correlation function gy' defined by eq, (3,6a), is at least of first order in
~~p•
i.e.g~
0 = 0, as correlations are caused by interactions, it follows tbat(4.9)
Taking s 1 = s2,
Ej
=.Ez,
~~ = ~2
we easily see tb at the zero tb order fluctuations ön(O (t,r,v) themselves vanish.s
-The physical picture bebind this result is that the lowest order quap.tities ,correspond to e:p = 0, i.e. the limit of an infinite number of particles in the Debye sphere so that fluctuations in the partiele density can be neglected.
These considerations result in the following lowest order initial conditions:
t .. 0 (4. JOa)
t 0 (4.10b)
t
=
0 I (4.10c)where the abbreviated nota ti on of section III. 2 has been used for the partiele variables and in each term of the summatien over i,j,k and 1 all labels should be taken differently.
Eqs. (4.10a)- (4.10c) are given bere as an example; corresponding higher order equations are also used in the sequel of this chapter.
3, The Lenard-Balescu equation
This section is analogous to section 111.3, but the results are different because of the different role played by the perturbation parameter.
Zeroth order theory
As there are no fluctuations (and no electric field) in zeroth order, the Klimontovich equation reduces to
0 (4. 11)
First order theory
In first order we obtain for the Klimontovich equation
0 (4. 12)
Averaging this equation, we find
0 ( 4. 13)
Thus we find bere the same result as in the Landau scheme and the removal of the secularity again gives
0 (4. 14)
Fourier transforming eq. (4.12) we obtain
(4.15)
Using eq. (2.32) we can solve eq. (4.15) for Ê(l) and we find k Ê(l) (oo k) - 1 - - - i: q ! - •- - e:(oo,~) .k2 s l.
e:o
s (I) 3 ön s (T0=o,
-k, -v) d V + i<!~: .• !_-IJ) ) (4. 16)where the (Vlasov) dielectric function E(w,~) is defined for Imw2 0 by k.V f(O)(v)
f d 3 - v s -V +
~.y-w
( 4. 17)
For Imw < 0 the analytic continuatien should be used.
The name Vlasov is connected with this function because for Vlasov plasmas, where the interactions between individual particles are neglected, the dispersion relation for plasma waves is given by E(w,~)
=
0.Substitution of eq. (4.16) into eq. (4.15) gives:
with the "Vlasov kemel"
Second order theory In second order we find:
which gives, after averaging:
(I) önt
'"o•O,
~·.!1) i (~·.!(-w +) k. V f(O) (v) - V S -k.v-w+ 0Removal of the secularity is obtained by the condition
where ( 4. 18) (4.19) (4.20) (4.21) (4 .22)
qs = - - V •
m s v (4. 23)
Applying the final value theorem (eq. (2.30)), eqs. (4.16) and (4.18) we obtain:
lim < E(l)
(To•E)
ön!l)(To•E•~)
> =To""""
lim~
J dw 1 J d 3 k 1 J dw2 J d 3k 2eiQ~.J
n-+-0 (2'11") ~I (4 .24)We now assume that the distribution functions satisfy the Penrose criterion [PEN 60, SCH 64], so that, for k
~
0,e-
1(w,~)
is analytic for Imw ~ 0. This corresponds to a Vlasov plasma where only damped oscillations exist.For k
=
0, one can see by inspeetion of eq, (4.17) that e(w,~) has two zero's on the real w-axis: w=
+ w , where w is the plasma frequency- p p
defined by
(4.25)
For k ~ 0 we can perform a contour integration in the upper w1 half-plane.
Using eqs. (4.10a) and (4.19) we find for the right hand side of eq. (4. 24):
-lim-0- J n-+-0 (2'11")4 k>k .
llll.n
where k . can be taken arbitrarily small. mln
Let us denote the zerosof e(w,~), i.e. the analytic continuatien of the right hand side of eq. (4.17), in the lower w half-plane by
~.(k) and let us assume for simplicity that they are all simple zeros. 1
-Their residues are denoted by Ri(~).
We then find by contour integration in the upper w2 half~plane:
1: - - - - " - - _ _;:_ _ _ _ _ :..._ _ _ _ .:.._...;..._
i e:(w~i.-~> ~._!+w~i ~·.!:t-vi ~·.!:t+rl-vi
R.
1 (4.27)
Substituting this into the second term of eq. (4.26) only the first term of the right hand side of eq. (4.27) contributes.
The last term of eq. (4.27) corresponds to damped plasma waves and therefore does not contribute to the asymtotic limit.
The first term of eq. (4.26) can be evaluated by means of a contour integration in the lower w
2 half-plane.
We obtain:
where the limit k • + 0 bas been taken. mn
(4.28)
Using the definition (4.19) of w
8t aqd the fact that e(-~·.!:
1
,-~) isthe complex conjugate of &~·.!:
1
,~), we can write this as2 2. qsqt · 1
4 2
----"---::-2 k. V k mseO je:~·.!:•~>l - v [1r cS(~.(~-~1
)) ~.( m V -.!_V ) f(O)(v) f(O)(v )] s v mt v 1s
-
t -1 (4.29)It is clear that in this scheme too we have:
= 0 (4.30)
so that the kinetic equation obtained is
(4.31)
This is called the Lenard-Balescu equation (LEN 60, BAL 60]. The difference between the Landau and the Lenard-Balescu callision integrale, i.e.
J!
2) andr!
2). is the factorle~·~·~>l-
2in
the latter. The Landau callision integral may be interpreted as repreaenting the collisions of two particles interacting through the bare Coulomb potential, whereas in the Lenard-Balescu callision integral the potential is modified by the collective behaviour of the surrounding particles.This collective behaviour has the effect of dynamical screening and causes the Lenard-Balescu integral to converge for small values of the wave number k, in contrast to the Landau integral.
From eq. (4.20) we also derive an expression for the secend order density fluctuations
(4.32)
4. Corrections to the Lenard-Balescu callision integral
kinetic equation we have to go to third and higher order in the expansion in Ie .
p
Third order theory
The averaged Klimontovich equation in third order reads
Removing the secularity results into
lim
<
_a_ f(Z)) +_a_ f(l) "' I(3) i1T I sa-r
2 s s.. o.._
where
r<3) -...,.V • qs lim [<!(I) dn(Z)> + < E(Z) ön (I)>
s m V s s
s
•o--(4.33)
(4.34)
(4. 35)
The asymptotic moments on the right hand side of eq. (4.35) contain two terms invalving expressions like < ön(l) --3- ön(l) > , see
s
a-r
1 teq. (4.32).
In the Landau scheme similar terms could be summed into an expression proportional to
~
f(O) "' 0.oT 8
Unfortunately this1is not possible in the Lenard-Balescu scheme, as these terms have different operators in front of them.
Therefore we make the assumption that, as there are no correlations at T
0 "' 0, we may write
(4.36)
(I)
a
(I)< ons <•o=O,_EI' .!1>a,
1 ont <•o=O,_E2' .!2) > +
(I)
a
(I)< ont (•o=O,_E2' ,!2)ClTI ons (•o=O,_El' .!1) > =
(4.37)
Thus we have
(4. 38)
The remaining part of 1(3) can be calculated straightforwardly and one
s obtains: I : : -2 k. V Ie:(~·.!·~) I - V + ) } ] (4.39) e: <-~·.!·-~) ~· <.!2-.!)+io
The same expression is found if in eqs. (4.17) and (4.29) f(O) is replaced by f(O) + f(l) and subsequently the right hand side of eq. (4.29) is linearized in f(I).
Fourth order theory
The first significant correction to the kinetic equation is given by
()f (2) lim _ s _ d'[
'o-!<>0
2 where (4.40)> +
(4. 41)
The right hand side of eq. (4.41) can be calculated along the familiar lines, which results in an expression'of great length that is not presented here.
The most important features of the calculation, among which the proof that the limit TO~ can indeed be taken, are shown in appendix B. In analogy with the Landau scheme we can write
+ 1(4) s,2 where 1(4)
2 is bilinear in f(l) and linear in lim
(4.42)
Th is f (O)
s
s,
part can be obtained from an expansion TO~ of eq, (4.29) with replaced by f(O) + f(l) + f(Z).
S S. S
The kinetic equation up to second ord~r in the plasma parameter can be written as
af
(t,v)s
-at
Is(Z) (t,.:y) +r<
s, 4) I (t,_y) 5. Discussion(4 .43)
As already mentioned the main difference between the Landau and Lenard-Balescu iteration schemes is that in the latter dynamical screening is incorporated.
This screening modifies the effective potential at large distances and I
the divergence at small wavenumbers i.n the Landau integral is not found in the Lenard-Balescu integral.
The situation at small distauces however is the same in both schemes and collisions with small impact parameter are not properly
incorporated in either cellision integral, which results in the same divergence at large wavenumbers.
Therefore one also has to use a cut-dff value in the Lenard-Balescu integral, which is only insensitive to this value if e is small.
p
chapter.
Therefore the treatment of slight nonideality given here is not complete.
One possibility to construct a convergent collision integral is discussed in the next chapter.
V SOME FURTHER REMARKS ON THE PERTURBATION SCHEMES
I. Comparison of the Landau and Lenard-Balescu perturbation schemes In the previous chapters it is shown that there are (at least) two parameters which can serve as expansion parameters in the kinetic theory of slightly nonideal plasmas: the plasma parameter e: and the
p interaction parameter e:I,
If the distinction between these parameters would be the only difference between the Lenard-Balescu and Landau scheme, the conneetion between these schemes would be simple, since e: is
. 1 3/2 p
proport~ona to e:I •
The difference in the expansion parameters, however, is not the only consequence of the different set-ups of the two schemes.
Their relation is more complicated because of the different scalings of the position vectors.
Let us define the exact collision i~tegral K
8;
(5. I)
where the subscript A refers to the asymptotic limit TO~ resulting from correlationfree initial conditions.
Taking dynamical screening into account we expand this in the plasma parameter and write
K
=
s
'f
K (i)
i=l s
with K (i+l) / K (i) • (!) (e: }.
s s p
From chapter IV it is clear that we can make the following identification: K (1) s I (2) s K (2) s I (4) s,l (5 .2) (5.3)
where the expansion of the distribution functions is left out of the considerations.
The integrals K (i) can be expanded once more, this time with
s
To this end we regard the electric charges as small and collect the terms that are of the samepower in q2•
We write this as
K (i)
s
'f
K (i,j) j=l s
with K (i,j+l)
IK
(i,j)=
@(E ).(5.4)
s s (" 1) I 2+2i
The lowest order term, K 1' , is of the order of q see the expressions for I ( 2} an: I (4}.
s s
In order to see how these terms are related with the expressions found in chapter III, we write
(5.5)
Substituting this into the right hand side of eq. (4.29} we find:
K (1 ,I)
s
J (2J s
and we also must have
K (2 ,1} s J (3} s,2 K (1,2) s J (3) s,J (5.6} (5.7)
Assuming that in the Lenard-Balescu scheme only the even order contributions are significant, we can identify them with the rows of the "matrix'' K s (i,j) •
A similar statement is true for the Landau scheme and the diagonals from the lower left to the upper right:
J (i) s i-1 E K (j,i-j) j=l s (5.8)
This is shown scliematically in fig. 5,1,
K(l,l) K (1, 2) K (1, 3) K( I ,4) s / s / s s
I
/ / / / / / /I
/I
/ /I
/I
/I
/ K(2,l) K(2,2) K(2,3) s / s / sI
/I
/I
I
/I
/I
I
/I
/I /
K(3,1) K(3,2)I /
s s Landau Lenard-Baleseu - - - BoltzmannFig. 5.1 Schematical representation of the iteration schemes.
2. The Boltzmann scheme
Going from left to right in the matrix K (i,j) we improve the
s
description of the interactions, i.e. more and more particles are incorporated.
Therefore we make the conjecture that the first column at the left represents all two partiele interactions, the second column all three partiele interactions and so forth.
This is in agreement with the expressions which have been calculated explicitly in the preceding chapters. So we might think of a third scheme where the expansion parameter is proportional to the density and of which the subsequent contributions to the collision integral can be identified with the columns of K (i,j)•
s
Such a scheme is not very appropriate for plasmas as it does not describe collective properties, but it would describe low density gases very well.
Therefore the expressions E K (i,l)
• s and
~
Ks (i,Z} should be equal tothe Roltzmann and Choh-Uhlênbeck L
fUHL
63] collision integralsrespectively.
In analogy with the other two schemes this third scheme should be called the Boltzmann scheme. It is shown in appendix C that it is iudeed possible to make an expansion of the Boltzmann integral of which the first two te:rms are / 2) and J(3
2).
s s.
From the theory of dense gases it is well known that the contributton to the collision integral corresponding to four partiele interactions is divergent for all (short range} interaction potentials {DOR 67]. This divergence is due to th.e secular growth in time of correlation functions in isolated systems of more than three particles.
As we have shown in appendix B that Is (4) is convergent, we conclude that either at least one of the terms K (i,3
1,
i~
2; is divergent,s
in th.e sense that the limit Ë K (i', 3) is divergent.
1:
0-- does not exist1 or that the sum
. l s
1.= This subject is not investigated further in this thesis.
3. Construction of a convergent collision integral
in the Lenard-Balescu and Landau collision integrals, are divergent. The same is true for the Boltzmann integral, when applied to a plasma. In contrast to the Lenard-Balescu integral, this divergence occurs for small wavenumbers.
As both divergences are present in the Landau integral one can construct a convergent cellision integral by summing the Lenard-Balescu and Boltzmann integrals and subtracting from this the Landau integral.
By inspeetion of fig. (5.1) we see that this amounts to summing the elements of the first row and first column of K (i,j), after which
s
the element that is counted twice is subtracted.
This method was first presented by Kibara and Aono [KIH 63).
Other derivations of convergent collision integrals can be found in [GOU 67, LIF 83, MON 77],
The convergent cellision integrals imply corrections of the order ep' if the order of magnitude of the Lenard-Balescu integral with the
wavenumber cut-off k s (e ÀD )-1 is indicated bye ln(e -1}.
, max p e p p
Therefore these integrals constitute the main correction in the limit
of very small nonideality.
The corrections mentioned in cliapters III and IV are important for slightly larger degrees of nonideality.
VI THE ELECTRICAL CONDUCTIVITY AT LOW FREQUENCIES
I. Introduetion
In the large numher of papers that have appeared on plasma transport coefficients, much attention bas been ~aid to the d.c. electrical conductivity, descrihing the response of a plasma to the presence of a static external electric field.
Both classica! [cf. SPI 53, ODE 83] and quantummechanical [cf. RÖP 81,
KRA 83, HÖH 84] models have been applied to partly or fully ionized plasmas.
In the case of a classical, fully ionized plasma the Landau equation, eq. (3.33), often forms the basis for the calculation of the
conductivity. In that case the contribution of the electron-ion collisions to the conductivity, to zeroth order in the electron-ion mass ratio (Lorentz plasma), can be found straightforwardly [LIF 83]. The incorporation of the electron-electron collisions however is less simple. This problem bas been solved numerically by Spitzer and Härm
[SPI 53].
Using the same kinetic equation one can also easily derive an
expression for the a.c. conductivity, for frequencies much higher than the average collision frequencies but much lower than the electron plasma frequency [LIF 83].
The latter restrietion is caused by the fact that tbe kinetic equation is not valid for very high frequencies.
Tbe former validates the Lorentz model, i.e. the neglect of electron-electron collision.
The use of the Landau equation however, involves the use of the cut-offs mentioned in section III.5, which means that the results obtained from it are valid only for very high values of ln (&~1). The metbod outlined in section 2 of this chapter is applied to the Lenard-Balescu equation, eq. (4.31), which, however, suffers from the same restriction, and to the kinetic equation given by Lifshitz and Pitaewskii [LIF 83] invalving a completely convergent callision
integral.
These kinetic equations are solved approximately by means of the Galerkin metbod
[KAN
58], wbich is in this application equivalent tothe Chapman-Enskog metbod [CHA 61).
2. Outline of the metbod
The general form of a kinetic equation in lowest order for a homogeneaus plasma can be written as
a
q-;;-t fs + -..! E V fs
a mS c::.t' V E t J t(f ,ft) s s
see expressions (3.31) and (4.29), where we have dropped the superscripts on the distributions.
Jst are binary cellision operators a~d ~ is the electric field which is taken to be of the form
total averaged
(6. 1)
(6.2)
a ,
where Et is small enough to allow for linearization of the problem. Note that for a homogeneaus plasma ! t is equal to the external electric field, as there is no polarization field.
For simplicity we assume that the plasma consists of electrens (e) and one kind of ions (i). Moreover we use the fact that the ion mass mi is much larger than the electron mass me.
To calculate the electric current we have to solve eq. (6,1), with s = e, for fe'
In general this cannot be done exactly, so that we have to use some approximation scheme. Therefore we apply the Galerkin metbod [KAN 58) to the equation for fe.
We write f as e f (v}
=
~(v}
(I + '(v)) e - e - -where 4>(!} is lines.r in E~ ~(v) s-and
~
are Maxwell2 -msv ( 2~T } (6. 3) distributions: (6.4)
temperature T which is assumed to be the same for electrans and ions. Now ~(y) is approximated by
(6 .5)
The coefficients a are determined by multiplication of the linearized n
form of eq. (6.1) with v2ny, n = 0,1 - - - N, and integration over velocity space. We then obtain the following set of equations:
N
E a [iw (2(m+n) + 3)!! + E Cs ] + (2n+3):!
=
0m=O m s=e,i nm ·
where (2p-l)!! = ( 2p)! and zPp!
with the Kronecker-delta öes and
h (v)
n
-2n = V V
The electric current density is given by
2 N
3 qe me n 3 2n+2 _M
1.•
=
qe f d V V f (v) =3k T
!t
E a ( - k T ) f d V V r-(y)- e - B n=O n B e
which leads to the following expression for the conductivity cr:
N 1:
n=O
a (2n+3)!!
n
where wpe is the electron plasma frequency, defined by
2
2
=
qene wpe &OmeWe apply this metbod for N=l and find
(6.6)
(6. 8)
(6. 9)
(6.10)
