Relativistic Boltzmann theory for a plasma

Physica 89A (1977) 225-244 (~) North-Holland Publishing Co.

RELATIVISTIC BOLTZMANN THEORY FOR A PLASMA II. RECIPROCAL RELATIONS H. van ERKELENS and W.A. van LEEUWEN Institute of Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands

Received 23 May 1977

The: linear or phenomenological laws such as Ohm's law, Fourier's law and Fick's law are derived for a relativistic plasma in an electromagnetic field. It is shown that the choice of a reference frame as proposed by Landau and Lifshitz entails-in contrast to, for instance, the choice of Eckart-the validity of Onsager's reciprocity relations.

I. Introduction T h e p u r p o s e of the p r e s e n t p a p e r , w h i c h c o n t i n u e s a p r e c e d i n g o n e * , is to s t u d y the p h e n o m e n a of t h e r m a l c o n d u c t i o n , electrical c o n d u c t i o n , diffusion a n d t h e r m a l diffusion in a relativistic p l a s m a , as well as to find o u t w h e t h e r a n d w h e n the O n s a g e r or r e c i p r o c a l r e l a t i o n s b e t w e e n t r a n s p o r t coefficients are valid. In o r d e r to c a r r y o u t this p r o g r a m , we first split up the l i n e a r i z e d t r a n s p o r t e q u a t i o n s o b t a i n e d in p a p e r I into a set of n e w a n d s i m p l e r e q u a t i o n s ( s e c t i o n 2). T h e n , b y m e a n s of a n e l e g a n t m a n o e u v r e w h i c h we b o r r o w f r o m the w e l l - k n o w n t e x t b o o k of C h a p m a n a n d Cowling2), we r e c o m b i n e these n e w e q u a t i o n s into a s m a l l e r set of e q u a t i o n s ( s e c t i o n 3). A f t e r these p r e p a r a t o r y m a n i p u l a t i o n s h a v e b e e n c a r r i e d out, we m a y o b t a i n the linear laws d e s c r i b i n g diffusion, t h e r m a l diffusion a n d t h e r m a l c o n d u c t i o n as well as e x p r e s s i o n s for the t r a n s p o r t coefficients figuring in these laws ( s e c t i o n 4). In s e c t i o n 5 the O n s a g e r r e l a t i o n s are d i s c u s s e d . It is s h o w n t h a t - i n c o n t r a s t to the s i t u a t i o n e n c o u n t e r e d in the n e u t r a l case or in the n o n r e l a t i v i s t i c t r e a t m e n t of a p l a s m a - t h e r e exist no r e c i p r o c i t y r e l a t i o n s i n a n a r b i t r a r y s y s t e m of r e f e r e n c e , b u t that t h e s e r e l a t i o n s hold true o n l y with r e s p e c t to two p a r t i c u l a r s y s t e m of r e f e r e n c e viz. (i) the local L o r e n t z - f r a m e in w h i c h the total net electrical c u r r e n t v a n i s h e s , a n d (ii) the local L o r e n t z f r a m e in w h i c h there is no total n e t e n e r g y flux. S e c t i o n 6, the final s e c t i o n of this p a p e r , is d e v o t e d to electrical c o n d u c t i o n . *The preceding paper ~) of this series will be referred to as paper I. Furthermore, (1 S • m. . . . . n) refers to the formulae m. . . . . n of the sth section of that paper. 225

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We obtain a generalized form of O h m ' s law and expressions for the electrical and the electro-thermal conductivities. A m o n g the latter occurs the so-called Hall coefficient, characteristic for the current which is transverse to both the electric and the magnetic field. A typically relativistic effect found here is the o c c u r r e n c e of the thermal diffusion coefficients in the expressions for the electrical conductivities. The article closes with an appendix in which some integral relations that have been used in the main text are explained.

2. The linearized transport equations As a basis for the description of an N - c o m p o n e n t plasma we choose the linearized relativistic transport equations 1) p ~ouf{ °~- c - 2 q i ( U U E ~"_ U"E~')pi,,,gf[m/apff N

= c ~qiB"~pi~a(f[%,oi)/ap~-fl°~,

~#(~),

i = I .....

N.

(2.1)

j--I

The linearized collision operators ~ j occurring in these equations are defined in the same way as in the case of a system of neutral particles, i.e.,

where W~i and W~} are the transition probabilities for the collision p~ + Pi-* p~+ p} and its inverse p}+ p } + p~ + p,, respectively. The relation of the function ¢i(x,p~), to be solved f r o m (2.1), to the distribution function fi(x,p~) describing the ith c o m p o n e n t of the system is f i ( x , Pi)

=

f/m[1 + ¢i(x, Pi)],

(2.3)

where f!0~ is the zeroth-order or Jtittner distribution function given by f~0~= ,~ 3 exp {[milxi(x) - p ~ U a ( x ) ] / k R T ( x ) } ,

(2.4)

with /x~(x) the local t h e r m o d y n a m i c potential of c o m p o n e n t i of the mixture [see (I 2.13) and (I 4.3)], and T ( x ) the local kinetic temperature. Furthermore, we recall that p;' is the e n e r g y - m o m e n t u m f o u r - v e c t o r of a particle of c o m p o n e n t i, q~ its charge and m~ its mass, while we abbreviated d~o~ := d3pi/p °. The symbol x stands for the s p a c e - t i m e point (ct, x), with c the speed of light, while a~ represents the corresponding s p a c e - t i m e derivative (c ~ O[Ot, a/cgx). The quantity ~ is an arbitrary constant with the dimensions of Planck's constant and kB is Boltzmann's constant. The four-vector U " ( x ) is the local h y d r o d y n a m i c velocity, normalized according to UuU~ = c 2. The metric tensor has diagonal elements (+1, - 1 , - 1 , -1). The local h y d r o d y n a m i c rest f r a m e is defined by the requirement that in this f r a m e U u equals (c, 0, 0, 0). Finally, we defined E ~ ( x ) := - c ~F~U,,,

B~(x)

:= A ~ F ~ , A *~,

(2.5)

RELATIVISTIC BOLTZMANN THEORY FOR A PLASMA, II

227

where F " ~ ( x ) is the local electromagnetic field tensor and A ~" := g~'~-c - 2 U ~ ' U ~. In the local h y d r o d y n a m i c frame the nonvanishing c o m p o n e n t s of (2.5) are the electric and magnetic field as felt by a particle at x; the electromagnetic field is treated as a given function of time and space. In order to cast the transport equations in a more manageable form, we first rewrite the streaming term. Using the explicit form (2.4) for f}0) and carrying out the differentiations we find for the l.h.s, of (2.1): p~Off[ °) - c-2q~(U~'E ~ _ U.E~.)p,.af}o)/apl • = - (kBT) Jf[°){p,a[m~h, 1

-

p;~Ua]Xq~

~v

-

mipt~Xi~

A

+ (pt'p~)(O~U~) + ~A p,~p~O~U - (phc)-lp{U~p~'B~,~J~},

(2.6)

where h~ is the specific enthalpy of c o m p o n e n t i. The " t h e r m o d y n a m i c f o r c e s " X~ and X7 are defined as [see (I 4.28)]

x ; --- r - l o ~ r -

( (oh)-'\Zl~O"p + E~

(2.7)

j=l

and Xi ~ := Ta~(l~i/T) + ( q i / m i ) E ~ + hiX~,

(2.8)

where h := p-t yqp~hi is the total specific enthalpy of the system, p~ is the mass density of c o m p o n e n t i, p = Eip; is the total rest mass density and p is the ]hydrostatic pressure. The effect of pointed brackets around a tensor t,~ is given by (t -~5 : = (½A ~zl "~ + ~A ~'AA " -- ~A ~ a ~ ) t ~ .

(2.9)

In the derivation of (2.6), use has furthermore been made of the equation of motion (I 3.42) (oh/c2)DU ~ = A"

OKp + EKi~=] n i q i -

c-~B,,J~ ,

(2.10)

where D := U~'O~, and J ~ = AU~Q~ with Q~ the total charge four-flow and ni = pi[mi the particle number density of c o m p o n e n t i. In the local hydrodynamic rest frame D is the time derivative while J~ equals (0, Jo) with Jo an electric current density. Instead of the quantity c3~(t~/T) we shall henceforth use (c3d.L~)r, the derivative at constant temperature. If we choose ~ to be a function of the variables of state p, T and p~(i = 1, 2 . . . . . N ) , (ad~i)T is defined by

Hence, in the case of uniform pressure, (0dxg)r is proportional to 0~oi, i.e. to derivatives of quantities characterizing the composition of the system. From the thermodynamic relation •,

T 2'

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H. V A N

ERKELENS

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W.A. VAN

LEEUWEN

with hi the partial specific enthalpy of c o m p o n e n t i, one finds for (2.11), ( ~ ) _ (OdXi)v O~ T

h~ T20~T.

(2.13)

The thermodynamic force X~~ may thus be written Xs" = (0"~s)r - P ~(h~/h)zl20"P + (qi/mi) - (hi~h) ~ (qj/mj)c i E ~, ]=!

(2.14)

where cj := O~[P is the concentration of c o m p o n e n t ]. The most important difference between the driving force X2 (2.7) and its nonrelativistic counterpart is the occurrence of two terms that are important in the relativistic regime only (in the nonrelativistic regime they tend to zero since the specific enthalpy h then tends to c2). The first of these, which is proportional to the gradient of pressure, is the well-known Eckart term4), the second, which is characteristic for a plasma because it is proportional to the charges qj, has first been obtained by Balescu, Brenig and Paiva-Veretennicoff 5) in a thermodynamic treatment. In the t h e r m o d y n a m i c force Xg" no such extra terms occur (in the nonrelativistic limit both the partial and total specific enthalpy tend to c2). We may split up the thermodynamic forces (2.7) and (2.14) according to X qv = A uvX qIX + c 2 U " T - I D T , l,

ix

(2.15)

v

X i ~= AixXi + c 2U (Dlzi)T,

(2.16)

using E"U~ = 0 as follows with (2.5). The last terms on the r.h.s, of these equations contain derivatives which may be eliminated with the help of the zeroth-order conservation laws. Because of the fact that the only formula which in this stage of the calculation differs from the corresponding one of the neutral case is equation (2.10), which has already been used to eliminate the term D U , , we can take over some results of the neutral case 6) T ~DT = (1 - y)O~U ~,

( D ~ i ) r = - (ykBT/mi)O~U ~,

(2.17)

where y is the quotient of the specific heats at constant pressure (c,) and constant volume (c,): cf, = (Oh/OT),,

c, = (Oe/OT),,

(2.18)

with h = e + pp ~ the specific enthalpy and e the specific energy of the system. Inserting (2.15) and (2.16) combined with (2.17) into (2.6) we obtain (mihi -p,~ UA )piixA ""Xq~ - mip~uA '~"Xi,, + (p~p]')(OuU~) - Qicg,U" j::l

d N

+ c ~qikBTl)i,B"~aq~/c)p7 + k , T ~

5t~ii(q~), j

i

(2.19)

RELATIVISTIC

BOLTZMANN

THEORY

FOR

A PLASMA,

Il

229

where =

Qi

- ~ (I m i c ) 2 - [mihi(l - y) + y k B T ] c 2p~U~ + c - 2 ( 43 - y)(P~U~) 2.

(2.20)

From (2.14) ensues N

N

~, ciA.~Xi~ = ~, CiA"~(a~l~i)r -- p 'A.~a~p, i-I

(2.21)

i=1

or, with (2.13) and (I 4.26,27), N

ciA ~ X f = 0. i

(2.22)

I

Hence, the t h e r m o d y n a m i c forces A~Xf are not independent. The latter relation may alternatively be written N /t

u

A~Xi = - ~

I ~

v

v

(ci - 6ii)zl~(Xj - XN),

(2.23)

i-1

where 8ij is the K r o n e c k e r delta: 8~ = 1 and 6 ~ j - 0 ( i ¢ ]). Substituting (2.23) into (2.19) we arrive at the final expression for the linearized transport equations: N-I

(mihi - p{U~,)Pi~.A~'~X,~ + mlpi,. ~

(ci - a,i)A"~(Xj~ - XN~)

i-I

+ (pl'pY)(a~,U~) - Q~a~U ~ =

(oh)-~P~U.prB~,

qi

P~'Cifl °~ da, j

j=l N

+ c-~qikBTpl'B~,~Oq~/Opi~ + kBT~_, .~#(qO,

i = 1,2 .....

N.

(2.24)

j=t

The, solutions ~0~ of these equations must obey the conditions of fit (I 3.45,46) mic

f p?U,q~J} °~ d¢o~ = O,

i = 1,2 .....

u , ) g'ili dooi = 0

N,

(2.25) (2.26)

i=l

and the L a n d a u - L i f s h i t z condition c~,

p{U,A""&,wd}°~do)~ = 0,

(2.27)

i=1

which results f r o m J ] = 0 (I 5.2) with the help of (I2.6), (I 3.26) and the condition of fit (2.26). We note that in the first E n s k o g a p p r o x i m a t i o n the electric field occurs in the transport equation (2.24) only through its a p p e a r a n c e in the t h e r m o d y n a m i c forces (2.7) and (2.14), while the magnetic field is found only next to the collision terms: in the present picture of a plasma the magnetic field lines a p p e a r as scattering obstacles by which the particles are deflected. Each of

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H. VAN ERKELENS AND W.A. VAN LEEUWEN

the o p e r a t i o n s which are applied on q~ in the three terms of the r.h.s, of (2.24) is linear and isotropic. This leads us <7) to look for a solution q~i of the f o r m N

1

q~i(X, Pi) = Ait`A t ` ~ q v + E A iit,At`,,(Xj 1,-- X~) i t + C~'(Ot`U,)+ Dia~U ~ + '¢~h....

(2.28)

w h e r e the f u n c t i o n ~hom (X, p~) is the general solution of the h o m o g e n e o u s equation 0 = (ph)

I

A

t`

N

f

j=l

3

Pi U, pi B~,,,~ qi

P[q~if~°)d°°i N

+ c ~qikBTp~Bt`,,oq~i/Op~,, + k s T ~

j-I

,LPsi(q~).

(2.29)

Multiplying this e q u a t i o n by f[mq~Jp°kBT, integrating with r e s p e c t to p~, and s u m m i n g o v e r i = ! , 2 . . . . . N we find 0 = c ' ~ q~

fI°)q~pff'B~,~Oq~s/Opiv do2~ +

f~°)q~ifij(q~ ) d~o~,

(2.30)

i=1 j = l

w h e r e the L a n d a u - L i f s h i t z condition (2.27) has also been used. By partial integration we m a y then p r o v e ~_ q~

f}°)th~pffBt`,,Oq~JOpi,, dw~ = -

q~

f~ q~sPsBt`~OOi/Opi~ dw~,

(2.31)

i=l

f o r arbitrary f u n c t i o n s ~ ( x , pi) and ~bs(x, Pi). H e n c e , the first term of (2.30) v a n i s h e s and we have ~ f ~ i ~ j ( ~ ) f l °) d~oi = O.

i=1 i=1

(2.32)

C o n s e q u e n t l y 6 : ) , ~#~homis a s u m m a t i o n a l invariant i.e. ~iho~ = al +/3t`py.

(2.33)

w h e r e ai and /~t` are i n d e p e n d e n t of p~. Since the t h e r m o d y n a m i c f o r c e s are i n d e p e n d e n t , not only ~ , but also ~ihom must satisfy the conditions of fit as well as the L a n d a u - L i f s h i t z condition. Inserting (2.33) into (2.25) and (2.26) we find 8) oei = 0,

/~t`Ut` = 0.

(2.34)

Substitution of ~h,,,, = /Bt`p~ into (2.27) yields with (2.34) /~t` = 0.

(2.35)

Hence, ~ o m = 0,

i = 1,2 . . . . .

N.

(2.36)

N o w , substituting (2.28) with (2.36) into (2.24) and equating the coefficients of

RELATIVISTIC BOLTZMANN THEORY FOR A PLASMA, 1I

231

the (independent) t h e r m o d y n a m i c f o r c e s we obtain the integral e q u a t i o n s

(mihi - p ~ U,~)A ~'Pi~, = (P h)-lp ~ U,~p~'B,~,A'~" ~_, qk f P ~Ak,f~ °) doJk k=l

N

+ c-lqikBTp'~B,~t~A"~OAidOpit~ + k~T ~ ~ik(A"~A~), k=l

(2.37) a

mi(c i - 6i~)A~'~p~, = (oh) ~p{U~pi B ~ A

~zv N

f

k=l

J

~ qk

p~Aik,ftk °) dtOk N

+ c lqikBTp']B,~oA~'~OAi~,/Opi,i

+kBT~_~ ~'wik(A~'"A~), k=l

(2.38)

(p~p~) = (ph) •-1 PiA U~pia B ~ k ~N qk f~ P~C~ftk°) dtok N

+ c ~qikaTp?B~OC~/Opi~ + k a T ~ - Q~ = (ph)-'p~U~pf'B~ ~ qk

~,k(C~),

(2.39)

PfDff~ °' dwk

k=l N

+ c-lqikaTp~B~ODi/Opi~ + kBT ~ ~ k ( D ) ,

(2.40)

k=l

i = l, 2 . . . . . N ; j = l, 2 . . . . . N - l, with Q~ given b y (2.20). T h e first two of these e q u a t i o n s are related to the p h e n o m e n a that f o r m the s u b j e c t of the p r e s e n t article, i.e. diffusion, thermal diffusion, heat c o n d u c t i o n and electrical c o n d u c t i o n . T h e remaining two e q u a t i o n s are related to viscous p h e n o m e n a and will be d i s c u s s e d in the next paper~).

3. The integral equations related to conduction and diffusion In o r d e r to p r e p a r e the solution of the integral e q u a t i o n s (2.37) and (2.38) we ,decompose the f o u r - v e c t o r s A f A ~ and AfA~ ~ (which are p e r p e n d i c u l a r to U ~') with r e s p e c t to the f o u r - v e c t o r s p e r p e n d i c u l a r to U ~ that can be c o n s t r u c t e d f r o m p f , U ~ and B~L First, define

~

:= ½ e ~ B ~ ,

(3.1)

w h e r e E~ is the L e v i - C i v i t a p e r m u t a t i o n s y m b o l in f o u r d i m e n s i o n s : E~ is +1 ,or - 1 d e p e n d i n g on w h e t h e r (/zvKA) is an e v e n or an odd p e r m u t a t i o n of (0123), and v a n i s h e s if two or m o r e indices are equal. In the local r e s t f r a m e , w h e r e U ~ = (c, 0, 0, 0) we h a v e B k~ = B k l = ~klmBm,

~ok = -- Bok = Bk,

B °~ = O, ~oo = ~kl = 0,

(3.2) (3.3)

232

H. VAN E R K E L E N S AND W.A. VAN L E E U W E N

w h e r e E~j,, is the L e v i - C i v i t a s y m b o l in three d i m e n s i o n s (e~23 = + 1) and B = (BI, B~, B3) is the magnetic field in the local restframe. A useful relation b e t w e e n the magnetic field tensor B ~" and its dual /~"~ is

B"~I3A~ - B"~B~,, = ~g"~B*~B~,,

(3.4)

w h e r e g"" is the metric tensor. F r o m (3.2) we find B := (B~+ B ~ + B3) '12 = (½BU"B,,) "2,

(3.5)

for the magnitude of the magnetic field in the local restframe. Using the notation i n t r o d u c e d in (3.5) we find with (3.4)

OuKI~K~A~p[" - BUKB~,p,

=

B2A2ps ",

(3.6)

w h e r e use has been made of the identity

B

KA

u

A,~vpi = B

KV

Pi~,

(3.7)

w h i c h is a c o n s e q u e n c e of B~*U, = 0. T h e relation (3.6) limits the n u m b e r of different f o u r - v e c t o r s p e r p e n d i c u l a r to U " that can be c o n s t r u c t e d f r o m p¢, U " and B"" using also g"" and e"~*. As an example, c o n s i d e r the f o u r - v e c t o r B"~B~,B*Kp~, which, b e c a u s e of the relation B**/?A, = 0

(3.8)

and (3.6) equals ( - B Z B " " p ~ ) . It turns out that each c o m b i n a t i o n that is p e r p e n d i c u l a r to U " m a y be written as a linear c o m b i n a t i o n of the three four-vectors

A"~p~,,

B"~ps~,

B~'AB~A~,pf.

(3.9)

Hence, the f o l l o w i n g decompositions can be made:

A"~A;~ = A!~A"~p;~ - B 'AI2~B~p;, + B - 2 A I 3 ) ] ~ ' ~ B , , . A ~p;~,

(3.10)

A'~"Ai~ = A{t')A"~p;~ - B ~Ait2)BU~p~ + B 2Ai'3)U""U,,,~AX"pi,,.

(3.1 1)

T h e f a c t o r s B J and B 2 have been inserted for dimensional r e a s o n s ; the subtraction signs b e f o r e the f u n c t i o n s A (2~ have been c h o s e n to omit a s u b t r a c t i o n sign in eq. (4.8). The scalar f u n c t i o n s A}"~ and A{~"~ (m = 1,2, 3) are functions of the scalars which can be c o n s t r u c t e d f r o m p~', B "", U ", g"~ and ~"~"*. B e c a u s e of the relations AU~'pi~pi~, = m { c 2 -- ( p ~ U , / c )

2,

(3.12)

and

B""B.~ A ;~Pi~Pi. -- B " K B , , ~ P [ P i . = B 2A ""Pi.Pi~,

(3.13)

the n u m b e r of different scalars is limited. I n d e p e n d e n t scalars are

and it turns out that e a c h scalar c o m b i n a t i o n m a y be written as a linear

RELATIVISTIC BOLTZMANN THEORY FOR A PLASMA, II

233

combination of this limited number of scalars. Hence, we have

a ~ ' ~ = A(k")(A~'~p,,p,,, B, B~'",B,,aAX*p,~,p,,),

m = 1,2, 3.

(3.15)

The treatment of the integral equations (2.37) and (2.38) being very similar, we confine our attention for a m o m e n t to the first of the two only. Using the decomposition (3.10) and the equations (A.5) and (A.7) of the appendix, the first term on the r.h.s, may be rewritten as /3 (0) dtok Pia B,,t~A /~v ~ a qk f pgAkffk k=! N

f ~

~

A(2)#(O)

= ½B 'B~AB; ~_, q k j Pk~ek.'~k Sk dw*A~"Pi. k=l

+ ~B 2B~"B2 ~_. qk

pkopk~A~!!f (°)~ dcokB"~p~

k=l

' 2 ' " ° ' d t oJkk B " ' 1 3 . c A ~ p i - ½B-3B~*B~ ~, qk f ~ A IJk3Pkl,rXk

~.

(3.16)

k=l

The second term may be rewritten according to

PiaB~,~A~'~'OAi,,/OPiO = - - ~RA(:)A~" ) t ~ Pie'*-" -t- u - ! az~ ( 2i )ut ' ~ '~',~x~ ~ a ~ Piv, (3.17) .-iP i v - - ~a i( 1 J-P a relation which may be verified using the ancillary relations

Pia B~oOAi(m) ~SPit3 = 0,

m = 1,2, 3,

(3.18)

whic, h follow with (3.15). Inserting the expressions (3.10), (3.16) and (3.17) into the integral equation (2.37) we obtain terms proportional to the three linearly independent four-vectors (3.9). Consequently, the equation thus obtained may be split up into three s e p a r a t e - but c o u p l e d - integral equations for the scalar functions a t " A! 2~ and A~3)

(mihi - p ~ U~ )ll ~Pi~ = ½(ph) !B -~

qk

uaa~t-~3PkPkZ'~tk Jk dtokp

k=l N

- c !qik~TBA!2!A~'"pi~ + k~T~_~ ~ig(A(~)A"~p~),

(3.19)

k=l

0 = - ½ ( p h ) - ! B -2

q~

L~a,~V BPkPkZ"ak Jk dtokpiU,~B

Pi~,

k=l N

+c

I

qik~TAi (I) B

~v

Pi~ + k ~ T B -~ ~ ~it(A(2)B"~P~),

(3.20)

k=l

0 = -~(ph)

B

qt k=l

B,,~B~p~

0a(2)¢~o)

U~B

BK~Z~ Ply

N

+ c 'qikBTB-'A}2'B~'"B,,c-~e~pi~ + k B T B -2 E "~'ik(A°)B~'"B,¢"~e~p~)" (3.21) k=l

These equations may be recombined into two new and uncoupled integral

234

H. VAN E R K E L E N S AND W.A. VAN L E E U W E N

equations, one of which is complex. Contracting (3.19) as well as (3.21) with /~,~ and adding the result we find N

( m i h i - p ~ Ux)A•Vpiv = kB

T~

~ik(,~(1)A ~ p v ) ,

(3.22)

k-1

where we have used the connexion

which follows from (3.6) and (3.8). F u r t h e r m o r e , we introduced the function

MI 1~= A!" + AI 3).

(3.24)

Next, we combine (3.19) and (3.20) into a single equation. Adding (3.20) multiplied by iB to (3.19) contracted with Bn, we obtain N

f~

/~X ~ a ~,8 ,¢/(2)/(0)

( m , h , - p;?UA)A"~pi~ = - ½i(oh) -IB '~, qk j~,~,-, ~lJkldk'3~k Jk dtokpXiUxA~'~Pi~ k=l

N

+ ic lqikaTBM}2)A""pi~ + kaTk~_~ 5Cik(M(2)A"~p~),

(3.25)

where M~"-~= A~')+ iA~2'.

(3.26)

The (complex) function M(~2~together with the function M},~ determine the three functions (3.15) and vice versa. The first equation, eq. (3.22), is identical to the corresponding equation of the neutral case6'7). We now note that, since the linearized collision operators 2g~k are isotropic, the last term of (3.25) will in general be proportional to some linear combination of the three four-vectors (3.9). Since the remaining terms of (3.25) are proportional to the vector AU"p~ only, it is permissible to look for solutions of the form ,~I 2 ) = ,~}2)(A"Vpi.Pi~.,

B),

(3.27)

which, when inserted into the collision term of (3.25) will yield a function that is proportional to AU~p~, only. We stress that it is not immediately evident that the possibility of a solution which does not depend upon all of the scalars (3.14)- as one would a priori e x p e c t - i s excluded. In a following article it will turn out, however, that a solution of type (3.27) can indeed be constructed. Using that M~~) does not depend upon B""BK~AX~piv,pi~ the first term on the r.h.s, may be simplified. We then get f A

c: fl,..4(2)¢(0)

,av

k=l N

+ iBc-tqikBTsd}2)A~'~pi. + kBT Z ~'~ik(S~(2)At~VPv)" k I

(3.28)

In the neutral case (qk = 0, k = 1.2 . . . . . N ) this equation has the same form as

RELATIVISTIC BOLTZMANN THEORY FOR A PLASMA, II

235

(3.22;) and we have M}1)= M}2) implying A~2)= A!3)= 0. The functions M}t) (l = 1,2) must furthermore obey the conditions of fit (2.25) and (2.26) and the L a n d a u - L i f s h i t z conditions (2.27), i.e.

m,cfpt 'ra'''.-,,,..~,~,"d(t)"r(°)dtoiJi= 0 ,

1= 1 , 2 ; i = 1,2, . . . , N,

c~_, f(ptU~)2A~'Vpi~}"f}°'dwi=O,

(3.29)

l = 1,2,

(3.30)

i=1

N~=lfp;?U~,A"'pi,..~"~pi,~sd~')f~°)do)i=O,

/ = 1,2,

(3.31)

where (2.28), (2.36), (3.10), (3.24) and (3.26), the fact that the thermodynamic forces are independent, and the fact that the three four-vectors (3.9) are independent have been used. We note that an integral of the type f puG do), with G an arbitrary function of za"'po.p,, [or, equivalently, p"U,~, see (3.12)] is a four-vector and, consequently, proportional to U , the only four-vector available. Hence, since A"~U~ vanishes, it follows that the first two of these conditions are trivially fulfilled, so that only the third one, the L a n d a u - L i f s h i t z condition, is a real subsidiary condition. As stated above, the treatment of the integral equation (2.38) is analogous. Defining

M['"

= A i m + A{(3),

M/~2~=

(3.32)

A{~1~+ iA{~2~,

(3.33)

we find N

Y~ ,~ik(di(I)At~Vp~),

m i ( c i -- 6ii)Am'pi~, = k B T

(3.34)

k=l

mi(c~ - 6ii)A"~piv = ~iB(ph)

~

qk

k=l

AaopkP~dt,:

J

fk

dOOkPi U ~ a

Pi~,

N

+iBc JqikB rgd[ (2)A ~'~Pi.+ ka r ~. ,~ik(S~ j(2)A t~vp~), k-I

i=l ..... N;j=

1. . . . . N - I ,

(3.35)

l = 1 , 2 , ; j = 1,2 . . . . . N - I .

(3.36)

with the L a n d a u - L i f s h i t z conditions

i=1

~,i,-,~,-. ~,i~-~ ~'ie~i

ji

dwi=0,

The integral equations (3.22) and (3.28) for the functions sO}I~ (l = 1,2; i = 1,2 . . . . . . N ) obeying the conditions (3.31), as well as the integral equations (3.34) and (3.35) for the functions M[m ( l = 1 , 2 ; i=1,2 ..... N; j= 1,2 . . . . . . N - 1) satisfying the conditions (3.36), are the basis for the description of the p h e n o m e n a related to diffusion and (heat) conduction. These will be considered in some detail in the next sections.

236

H. VAN

ERKELENS

AND W.A. VAN

LEEUWEN

4. Diffusion and conduction

We shall now investigate the relation between the gradients that exist in the system and the various resulting irreversible flows. The transition from the functions A to the functions s¢ carried out in the preceding section enables us here to give concise expressions for the transport coefficients. The diffusion flows I," (i = 1,2 . . . . . N), eq. (I 3.33) and the heat flow I~, eq. (I 3.34) may be written N

=_

,~ ~A..Mm

(4.1)

j=l I r~T(I)A'~" -- hA"VM

(4.2)

(I)

by making use of (I 3.45) and the defining expression of the projector A "~ (I 2.5). The first order corrections M! 1)" and M (l~ to the partial and total mass flows, and the first order correction T m"~ to the e n e r g y - m o m e n t u m tensor occurring in these expressions are defined in eqs. (I 3.24,26): M,•m" := mic

pffqgif} °) do)i,

Tm"" := c ~,

M ~'"

z

~ M, .c'" ,

pl~pf¢if} °) dtoi.

(4.3) (4.4)

i=1

We thus have I ff --

h k I ff = -- c

-

( mkhk

-- p k xU x )

A ~,vP k v q ~ k f kCO) d o ) k ,

(4.5)

21f

k=l

(

°) do)

(4.6)

k=l J

Define =

X 2 "= A ~ X , - X I . = - B 2 B . . B ~ , X ~, X t := B IBu~XX.

(4.7)

The three quantities X~, X2 and X~ form an orthogonal, right-handed triad of space-like four-vectors orthogonal to U ". In the local rest frame X II is the c o m p o n e n t of X along B, X ± the c o m p o n e n t of X perpendicular to B in the plane through X and B, while X ' is a vector perpendicular to both X IIand X ~. We have the relations A ~ ~"X, = pi-"t"4~'lvlet,~ A , + A ,. ( I ),y. ,± + A~2)X~]

(4.8)

where ~Q¢ stands for ~QCior sCj, A for A; or Ai and X for Xq or X;, as may be verified with (3.10) or (3.11) and (4.7), using also (3.24) or (3.32), With this

RELATIVISTIC BOLTZMANN THEORY FOR A PI_,ASMA, II

237

e x p r e s s i o n and (2.36) we m a y rewrite ~Ok (2.28) as N-I

II p f f A k(])Xq,~ ± ~a(2)vt ~Ok = pff.S~ ko)Xq~-I+ [Jk...--Xk Aqu-I'- pff Z

"~k'ctiO)tvllt'aiU__ X I~,)

i=1

N-1

N-I

i=l

i=1

+ Cku~(O~U ") + DkO~,U".

(4.9)

Inserting this e x p r e s s i o n into (4.5) we find a generalized f o r m of F o u r i e r ' s law o f heat c o n d u c t i o n : N

Jr~- Z •

,t t, hilff = i,iiyil~ + l q,±q X °±u + l,~qXq "qq~=q

l=l

N-I

N-I

+ Z

4,(xI

/=l

+

./=l

N-I

lqi{Xj ±

w h e r e the thermal c o n d u c t i o n coefficients lq~ are defined as

- X k ~`) + Y~ Iq,{X ' i" /=l

coefficients

I~,

-

Xk~'),

(4.10)

and the t h e r m o - d i f f u s i o n

t~:= -- k c ~k=l f

(m,h~

-- p k~U x ) A ~ v P ~ P k ~ ' ~ k (,,f k,o) dtok,

(4.11)

I'q~ : = -- ~C ~a

(mkh~-- Pkl, Ux)A~,,,P~pkAk ~ (l)fk(o) dtOk,

(4.12)

l'~ := - ~c ~ f (mkhk -- p g~'U x ) A t ~ P f f p k~A k (2,fk(o, dtok,

(4.13)

k=l

1

A

l,~i:=-~cY~

k=l

lqi3 _". - - l c ~ . ,

1.7i t .--" --

~.

f

k=l

.I

(mkhk--pkUDA.~P~pkS~k

v i(1)fk(0) do)k,

(4.14)

( m k h k - - P k ~Air/ , X ) ~x A. v p k p/~k Z a~Ai(1)4¢(0) k jg dtok,

(4.15)

~cl ~ f .(mkhk -- p k'~U A ) A , ~ P ~ p k,,A k i(2)fk(0) dtOk, i = 1 . . . . . N - I . k=l

(4.16) In deriving (4.11)-(4.16) we h a v e used that the last two terms of (4.9) give no c o n t r i b u t i o n to the linear law (4.5). F o r the first of these (i.e. the one with C~'~ as factor) this follows with the help of (A.4), (A.I 1) and (A.12) and the fact that a n u m b e r of integrals is odd in at least one of the c o m p o n e n t s of the f o u r - m o m e n t u m ; the vanishing o f the c o n t r i b u t i o n of the o t h e r term (the one with Dk) m a y be verified by m e a n s of a r e a s o n i n g identical to the one t h r o u g h which it was f o u n d in the p r e c e d i n g section that the conditions of fit (3.29) and (3.30) were o b e y e d . F u r t h e r m o r e , the f a c t o r ~ in (4.11)-(4.16) also results f r o m the use of (A.4), (A.I1) and (A.12). With the help of (3.26) or (3.33) the

238

H. VAN ERKELENS AND W.A. VAN LEEUWEN

e x p r e s s i o n s (4.12) a n d (4.13) o r (4.15) a n d (4.16) m a y be c o m b i n e d to: .,, i ~,( l'q~ + llqq = -- ~C ~ k=l

[q~iq- llqi : --Tc ~

k=l

.I

,

J

~

v

(mkhk - p k U ~ ) A ~ , ~ p k P k ~ k

(mkhk -- Pk U,~)Z.l,,~pkpk.~k

(2) (0~

f~ dwk,

(4.17)

.tk dwk.

(4.18)

S i m i l a r l y we o b t a i n , b y s u b s t i t u t i n g (4.9) into (4.6), a g e n e r a l i z e d f o r m of F i c k ' s law of diffusion: : "ia''u -- liqXq

+ li,iX,1

N-I

N

+ Z t!l(x?-

1

N

+Z

i.1

I

+Z

j-I

x',.,.),

j-I

i = 1. . . . . w h e r e the t h e r m a l d i f f u s i o n coefficients defined by l!~ = - - ~(: ~.j ( m k ( C i k=l J

li~• +

- t

1 I,~_N f

lliq : -

3c

- - :~C

mk(Ci

,il-}-i,~j=

- - ~C ' ~ f

(4.19)

a n d t h e d i f f u s i o n coefficients l~i are

-- n ° i k l ,Za~ t ~ l ' P kuP k ' Y~..4(,,.c,o, ' ~ k 3 k dOgk,

(4.20)

(2)fk(0)

dOgk,

(4.21)

- - °~i k )XZ A' i ~ l "_ktP~k ' ~ tJ,,~/j(1)¢(O) 'k Jk

dtok,

(4.22)

dtok.

(4.23)

m k ( c "i _

k=l

liq

N - 1,

mk(ci - -

8 i k ) A u v p f f P k S ~ ,,k

~° i k "~A ' z ~ k°at "v'4/J(2)¢(O) l l " ~ v P~ t kt-' k Jk

B e c a u s e of the c o n n e x i o n N

~] I~~ = 0,

(4.24)

i I

[see (I 3.14)] w e n e e d n o t c o n s i d e r the N t h d i f f u s i o n flow. All of the t r a n s p o r t c o e f f i c i e n t s r e l a t e d to c o n d u c t i o n a n d d i f f u s i o n h a v e n o w b e e n w r i t t e n as integral e x p r e s s i o n s c o n t a i n i n g the f u n c t i o n s ~/, the s o l u t i o n s o f t h e integral e q u a t i o n s (3.23), (3.28), (3.34) a n d (3.35). In the n e x t s e c t i o n w e s t u d y u n d e r w h a t c o n d i t i o n s the O n s a g e r or r e c i p r o c i t y r e l a t i o n s b e t w e e n the t r a n s p o r t coefficients h o l d true in a p l a s m a .

5. The

Onsager

or reciprocity

relations

T h e t r a n s p o r t c o e f f i c i e n t s f o u n d a b o v e m a y be r e w r i t t e n in a c o n c i s e f o r m with the h e l p of t w o t y p e s o f " b r a c k e t s " , w h i c h , f o r a r b i t r a r y t e n s o r f u n c t i o n s

RELATIVISTIC BOLTZMANN THEORY FOR A PLASMA, II Fk and

Gk

239

of the f o u r - m o m e n t u m Pk, are defined by

[F, G] :=

cn -2 ~ ~_a f "~Pik(F)Gkftk°) do)k,

(5.1)

i=l k=l

{F, G} := ~ qk

(5.2)

fkGkf~k °' do)k,

k=l

with qk the charge of a particle of c o m p o n e n t k (k = 1, 2 . . . . . N ) and n = ~;ini the total particle n u m b e r density. The brackets (5.1) and (5.2) are s y m m e t r i c [F, GI = [G, F],

(5.3)

{F, G} = {G, F},

(5.4)

while the first of these brackets is positive semi-definite in the sense that, for arbitrary functions Fk [F, F] I> 0.

(5.5)

The p r o p e r t y (5.3) is essentially based upon the time reversal invariance 6) of the iinteraction, the equality (5.4) is trivial, while the p r o p e r t y (5.5) m a y be proved by means of an e l e m e n t a r y symmetrization procedure6). Inserting now (3.22) and (3.28) into (4.11) and (4.17) respectively, we obtain l,o~ = _ ~n2kBT[sgtl)A~,~p~, ~(l)p~],

(5.6)

1',~ + il'oto = - ~nZkBT[..rg(2)A ~'~p~, ~g(2)pu ] - ½iBkB T{.~g(2)A"~p~,s~2)p~,}.

(5.7)

and

where also the L a n d a u - L i f s h i t z condition (3.31) has been employed. In the same way we find for the remaining transport coefficients the expressions l~i = - ~ n Z k B T [ s g m A " ~ p .

lq~-t- ilq, =

(5.8)

sgimp,~],

- ~n2ksT[s~l(2)AU~p~, M~(2)p.] - ~iBksT{s~(Z'A""p., .~U(2'p.},

i!lq = _ ~n2kBr[sci(J)AU~p~, .~mpu],

(5.10)

Ii~ + il[q = - ~n2kB T[sg"2~A"~p. sf~2'p.] - ~iBkB T { ~ " 2 ' A " " p . ,

d2~p.},

(5. I 1) (5.12)

1!I = _ In2kB T[..~l,,) A ~,~p~. .fgimp. ],

l#± + i/~= - gn~2kBT ~" ~"2)A - I t , . s~i(2)P.]

(5.9)

-

~iBkBT{sg"Z~A"~p...5:~i(2)p,}.

(5.13)

Since the brackets are symmetric, one has d~ d liq-lqi,

ld=

d lji,

i=1,2 ..... N-1;j=I,2

..... N-I,

(5.14)

where I d stands for III, i • or I t. These relations, the so called Onsager or reciprocity relations, are essentially based both on time reversal invariance and the e m p l o y m e n t of the L a n d a u - L i f s h i t z formalism. If, however, no use would have been made of the L a n d a u - L i f s h i t z condition, the expressions for l ± and l' would have been more complicated.

240

H. VAN E R K E I , E N S AND W.A. VAN L E E U W E N

Instead of, for instance, the e x p r e s s i o n (5.13) for the nonparallel coefficients of diffusion we w o u l d have f o u n d

1;} + illi =

~n2k,T[,~,2,A,,.p,,,

~iBk,T{,N;,2,A~p,,, sgi,2~p~,} N {iB(ph) l[~ql f Aa,pff[)ff~Q~ff2)do)lJf~lf ~z ,~,~//(21((()) -LI=I PkAITA t~a~#~PkPk~k 3k doJk]. ,~i,2,p~ ] _

(5.15) Since the last t e r m - in c o n t r a s t to the first t w o - i s not invariant with r e s p e c t to the interchange of the indices i and j, the reciprocity relations are not fulfilled. T h e s y m m e t r y is restored w h e n

I=1

qt

zl~plpl~l

11 ~o/ = 0

(i)

or

~ f p~ UaAuvp~'p[~f,~2)f~°' d~ok

k=l

0,

(ii)

since then the n o n - s y m m e t r i c a l term of (5.15) vanishes. As follows f r o m (I 5.4) with the help of (I 3.24), (3.33) and (4.9), condition (i) is fulfilled if the t h e o r y is f o r m u l a t e d with r e s p e c t to a local r e f e r e n c e f r a m e c o - m o v i n g with the local total electric current, i.e. in a L o r e n t z s y s t e m in which the local total electric c u r r e n t vanishes (see I, section 5). Condition (ii) is implied b y the L a n d a u Lifshitz conditions (3.36), which are satisfied if the t h e o r y is f o r m u l a t e d with respect to a s y s t e m c o - m o v i n g with the local e n e r g y - c e n t r e (see I, section 5). Since similar r e m a r k s can be m a d e with r e s p e c t to the other t r a n s p o r t coefficients, we m a y state in c o n c l u s i o n that the r e c i p r o c i t y relations hold true (i) with respect to a r e f e r e n c e s y s t e m 9) in which there is no electrical c u r r e n t and (ii) with r e s p e c t to a r e f e r e n c e s y s t e m in w h i c h there is no e n e r g y flux, and that O n s a g e r relations are not fulfilled in o t h e r f r a m e s of r e f e r e n c e , such as, for instance, that used by Eckart4). F o r r e a s o n s explained in paper I, we c h o o s e for the s e c o n d possibility, w h i c h c o r r e s p o n d s to a choice for the L a n d a u - L i f s h i t z formalism ~0). In the next and final section of this paper we give s o m e alternative f o r m s of the linear laws f o u n d a b o v e , paying particular attention to electrical c o n d u c tion. 6. O h m ' s law

One s o m e t i m e s i n t r o d u c e s thermal diffusion ratios k~ a c c o r d i n g to N-I E j=l

d d -liiki

liq,d

i = 1,2 . . . . .

N - 1,

(6.1)

w h e r e the index d indicates 11, 3_ or t. In o r d e r to m a k e use of these ratios we

RELATIVISTIC

BOLTZMANN

THEORY

FOR

A PLASMA,

II

241

split up each of the N - l linearly independent diffusion flows I~ ( i = 1 . . . . . N - 1) into three f o u r - v e c t o r s in the following w a y N-!

Lag := "'q''qldyd'`+ E "',"tg¢ Vd"., -- X~"),

d = [1, 3_, t.

(6.2)

i=1

One easily verifies with the help of the Onsager reciprocity relations that N-I

N-I N 1

N 1

i=J

i=l

j=]

j=l

(6.3)

where (6.1) and (6.2) have been used. C o m p a r i n g this equation with (4.10) we can conclude that N

I~- ~, h,I¢i-I

N-1

N-I

~ , k!lI] I ~ -

~ k[I/-'`- ~ k[I/"

N-1

i=l

i=l

i=l

y l l t , 4± ±'` + 14qX t = Ill .qq..q - looX. qt~ ,

(6.4)

where N IN

lqd "--lqo. _ ,d E i-I

1

E Idkdkd ,ii.*i "-i,

d = I[, _L, t.

(6.5)

j-I

As follows f r o m (6.4), the coefficients of thermal conduction Iq% are those accessible for direct m e a s u r e m e n t when the diffusion flows vanish. The electric current density I(~ or charge-flux f o u r - v e c t o r is defined by

N

)

Q'` =: (k~_j nkqk ( M ' ` / p ) + ' ~ .

(6.6)

It is equal the total electric current Q'` minus the total net charge carried a w a y with the total mass flow. We note that the splitting of the total current Q'` into a conductive part I ~ and a c o n v e c t i v e part (~,krtkqk)(M'`/p) is to some extent arbitrary. We might equally well have written [cf. I 3.20]: N =

in which case we could consider J ~ as the conduction current and (~,knkqk)U'` as the convection current. The division actually made here is the one indicated in (6.6). Eliminating M'` f r o m (6.6) with the help of (I 3.14) we find N

I~ = ~

(qJmi)U',

(6.8)

i=l

an expression which shows that the electric current density represents the total net charge carried by the diffusing particles. Define new t h e r m o d y n a m i c

242

H. VAN ERKELENS AND W.A. VAN LEEUWEN

f o r c e s Y~ and Y/~ (j = 1,2 . . . . .

N - 1) a c c o r d i n g to

Y2 : : TX2 = a " T - T(ph)-~(A,a'p+ 2 nkqkE~),

(6.9)

k=l

m• = E" +

oh

~':'=1nkqk (X/*- X~)

~

ph

x [(a"ttj-

k:l nkqk

hj - hN A,~aKp].

a%*~)r

(6.10)

o1~

In terms of these new t h e r m o d y n a m i c f o r c e s we have, with (4.24),

)

I~= ~

qi mNqNI¢" N

:X

I N

I

N

I

X X ~ J Y f " + X E 'p~Y~", d

i-l

)-I

d

i

(6.11)

I

w h e r e £a indicates s u m m a t i o n o v e r the directions [[, _L and t, and _ _

5[( q,

_

mN/L\m,

q:,/do.=(q~m~

mN)-h'-"-~-hN'~-," nkq,] ,,,, qN

qN) 1 d

7 1'°'

d

d = I1,±, t.

(6.12)

The linear law (6.11) is a generalized version of O h m ' s law. It gets a particularly simple f o r m if the t e m p e r a t u r e , pressure and c o m p o s i t i o n are u n i f o r m , i.e. w h e n O~T, O"p and (O"l-ti)r vanish. Then (6.11) r e d u c e s to

I¢) = ~ craEa",

(6.13)

d

w h e r e the electrical conductivities (r a are given by

o'"

=

~

O-ij

"=

ntqkq~a,

--

d = II, ± , t.

(6.14)

k=l

T h e o c c u r r e n c e of the coefficients of thermal diffusion in the s e c o n d term of (6.14) is a typically relativistic effect: it vanishes in the non-relativistic limit, since in this limit the specific e n t h a l p y h tends to c 2. The c u r r e n t in the direction p e r p e n d i c u l a r to both the electric and magnetic field, o-'E '~, is k n o w n as the Hall current, the c o r r e s p o n d i n g electrical c o n d u c t i v i t y o-' is t e r m e d the Hall c o n d u c t i v i t y .

Acknowledgement This investigation is part of the r e s e a r c h p r o g r a m of the "Stichting v o o r f u n d a m e n t e e l o n d e r z o e k der materie (FOM)", w h i c h is financially s u p p o r t e d

RELATIVISTIC BOLTZMANN THEORY FOR A PLASMA, II

243

by the "Nederlandse organisatie voor zuiver-wetenschappelijk onderzoek (ZWO)".

Appendix

Some integral theorems Let A be any function of the scalars (3.14), i.e.

A=A(A~p~p~,

B,

B~KB~A*~p~p~),

(A.I)

where B .• = tgl/~ ~ o , ~l~/x~'~ , j 1/2 is the strength of the magnetic field in the local restframe. Since the only symmetric tensors of the second rank which can be formed from U"./~"~, g"~ and ~.K~ are

U . U ~,

A ~,

B"B. ~

(A.2)

or linear combinations thereof, we must have f

~

"K

p,,pBAf (°~do) = Fi U,,U~ + F:A,,~ + F3B,..B ~,

(A.3)

where F~, F2 and F3 are scalar functions. Multiplying this equation by A"~A ~° and B"~A ~a we obtain

A"~A ~

p,p~Af t°~doJ = F2 A"~ + FaB ~B ,

(A.4)

B ~ A ~t~f p~p¢Af(O) do) = F2B "~,

(A.5)

respectively, where we have used the properties B"@~K = 0 and B""A2 = B "~. Next, multiplying (A.5) by B,~ we find

B"B.

p~pt3Af (°) do) = F2B B , ,

(A.6)

F2 = -½B z f B,,~,B,,t3p~pt3Af(O)do),

(A.7)

so that

where we used that B ~ = - B ~". S e t t i n g / x = u in ( A . 4 ) we get

f A~p~pBAf~°) do) = 3Fz + 2B2F3, where we employed B = t~,_,,,o

(A.8)

) . Hence,

F3 = ½B 2 f A,,t~p,~pt~Af(O)do) - 3B 2 F 2. If the function A is assumed not to depend upon B

A = A(A"~p~,p~, B),

(A.9)

B~,A p,p~, i.e. if (A.10)

244

H. VAN ERKELENS AND W.A. VAN LEEUWEN

then F~=0.

(A.Ii)

From (A.9) it then follows that

F2 =

f

A"t3p,~p~Af (°~ d~o.

(A. 12)

The results (A.4), (A.5), (A.7), (A.11) and (A.12) are used in the main text.

References I) H. van Erkelens and W.A. van Leeuwen, Physica 89A (1977) 113 (paper I); Paper I11, Physica, to be published. 2) S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge Univ. Press, London, 1970), p. 358. A modernized version of the theory of ionized gases can be found in ref. 3. 3) J.H. Ferziger and H.G. Kaper, Mathematical Theory of Transport Processes in Gases (North-Holland Publishing Company, Amsterdam~ 1972), p. 423. 4) C. Eckart, Phys. Rev. 58 (1940) 919. 5) R. Balescu, L. Brenig and 1. Paiva-Veretennicoff, I- hysica 81A (1975) 1. 6) See e.g.S.R, de Groot, C,G. van Weert, W.Th. Hermens and W.A. van Leeuwen, Physica 40 (1969) 581. 7) W. Israel, J. Math. Phys. 4 (1963) 1163. 8) This may be proved with the help of formulae (145) of W.A. van Leeuwen, P.H. Polak and S.R. de Groot, Physica 63 (1973) 65. 9) J.L. Anderson, Physica 85A (1976) 287 and Variational Principles for Transport Coefficients of a Relativistic Multi-Component Plasma (manuscript). 10) L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1959), p. 499.