On relativistic kinetic gas theory

Physica 51 (1971) 32-49 © North-Holland Publishing Co.

ON R E L A T I V I S T I C K I N E T I C GAS T H E O R Y vI. THE VISCOSITY FOR A GAS OF MAXWELLIAN MOLECULES W. A. van LEEUWEN and S. R. de GROOT [nstituut voor theoretische [ysica, Universiteit van Amsterdam, Amsterdam, Nederland Received 13 July 1970

Synopsis The tensor pseudo-eigenvalue equation of the linearized collision operator of a system of Maxwellian particles is determined. With tile help of this equation the first relativistic correction to the viscosity is calculated. 1. Introduction. In the non-relativistic t h e o r y the scalar, vector, tensor (of r a n k two), etc. eigenvalue problems of the collision o p e r a t o r for Maxwellian molecules can be solved simultaneously, b u t in the relativistic t h e o r y the situation is more involved. E a c h case has to be dealt with separately. The scalar and v e c t o r eigenvalue problems h a v e been solved b y Israel1). In the following sections we shall be concerned with the tensor eigenvalue problem. In section 2 we give the m a t h e m a t i c a l a p p r o a c h (which is of the same t y p e as in the scalar and v e c t o r case) t h a t will enable us to solve the tensor eigenvalue problem in sections 3, 4 and 5. T h e result is given in eq. (80). I n the last section the viscosity is calculated up to the first relativistic correction, which is of the order e -2 [c/. eq. (90)J. The connexion between the viscosity *1, the heat c o n d u c t i v i t y coefficient 2 and the specific heat Cv, which is k n o w n to exist in the non-relativistic kinetic t h e o r y of a Maxwellian gas, does not hold in the relativistic t h e o r y [c/. eq. (92)~. The thermal-diffusion coefficient and the v o l u m e viscosity, quantities which vanish in the non-relativistic t h e o r y for Maxwell particles, are of the order c-2 and e -4, respectively, in the relativistic theory. 2. A special choice o/ the re/erence /rame," the trans/ormation to new variables 1). The tensor integral e q u a t i o n (II.39) becomes for a simple gas £# [C(p) (PuP~)] = - - (1/kT) (PuPv).

(1)

In order to o b t a i n the solution C(p), we solve the eigenvalue problem of the o p e r a t o r ~ . The action of the o p e r a t o r £0 on a function z(P) reads 32

RELATIVISTIC

KINETIC

GAS THEORY.

VI

33

explicitly ~[z(P)] ------IJS/o(Pl){z(p) + z(Pl) - z(P') - z(pl)} × W ( p , P l lP', Pl) dc91 d o ' de)i

(2)

[c/. (II.6) t and (IV.26)]. For a gas of Maxwellian molecules the collision rate W occurring in (2) has the form which follows from the insertion of (IV. 14) into (IV. 12) or (IV.21). The equilibrium distribution function, /o, is given b y (I.89). The main part of this paper will be devoted to the determination of the tensor pseudo-eigenvalue equation of the operator ~ for Maxwellian particles. Having obtained this equation we can solve (1) and compute the viscosity (II.81) in the same way as we have calculated the coefficients of heat conduction, diffusion and thermal diffusion using the vector pseudoeigenvalue equations (IV.27) and (IV.28). In the same way that we obtained (IV.23), we find here with the help of (I.89) and (IV. 33, 34) for (2) the expression r~ 2r¢

~Ez(q)] = m4c3o~ S J J {exp(zn,q•)}{z(q) 0 -

-

+ z(ql)

0

z(q') -- z(ql)}(1 + g2/4m2c2)-~/1(O) sin 0 d0 dq0 (dql/q°).

(3)

We have introduced the normalized four-vectors q. ~ p./rnc ; nu =-- Utdc ;

qaq. = _ 1,

{4)

nt~nl, = -- 1.

{5)

The polar angles 0 and 90 occurring in (3) characterize the direction in the centre of momentum system of the relative momentum three-vector after the collision, i.e., of g'. The scattering angle O occurring in (3) as the argument of the function F, is given b y cos O --= g ' g ' / l g l ! g ' l = g~,g'/g2.

(6)

The last step in (6) is a consequence of the fact that g, is purely spacelike in the centre of momentum system. If the 3-axis of the centre of momentum system is chosen parallel to the incident relative momentum g of the colliding particles we have O = 0. Let us introduce the trial function z(q) =
(7)

where a~ is a fixed but arbitrary, spacelike unit four-vector perpendicular to the bulk velocity Uv: aua~ = 1,

aaU~ = 0.

(8)

t R e f e r e n c e s t o o n e of t h e p r e c e d i n g articles 2) will b e r e f e r r e d t o b y R o m a n n u m e r als, while t h e A r a b i c n u m e r a l s i n d i c a t e t h e f o r m u l a i n q u e s t i o n .

34

W.A. van LEEUWEN AND S. R. de GROOT

Since the expression ,£f[Z] with 9~ given by (7) is a scalar we m a y calculate . f [Z] in the special Lorentz frame A which is a t t a c h e d to the particle with energy m o m e n t u m p . : pu = mc(1, O, O, 0);

qa -- (1, 0, 0, 0);

(9)

a n d which is oriented such t h a t the 3-axis is parallel to the local threevelocity, or (ha = Uu/c): nu -~ (cosh Z, sinh ze);

e = (0, 0, 1).

(10)

In the frame A the normalized e n e r g y - m o m e n t u m four-vector q~ = p ~ / m c of the other particle is set equal to (11)

q~ = (cosh Z1, sinh Zlel) ; el = (sin 01 cos 91, sin 01 sin ~01, cos 01), so t h a t in the Lorentz frame A we have gt, = p~ _ pu = 2 m c sinh ½Xl (sinh 19Cl, cosh {Xlel),

(12)

g = 2 m c sinh ½Zl,

(13)

(1 + g2/4m2c2)~ = cosh ½Zl,

(14)

/~ = /5'u = (cosh 1gl, sinh ½Zlel).

(15)

The last expression is found from (IV.18) with the help of (9), (11) and (14). Now we fix the orientation of the reference frame A in three-space completely b y the requirement t h a t the spacelike four-vector a" has no c o m p o n e n t along the 2-axis of A : a" = (sinh Xa, cosh zaea) ;

ea = (sin Oa, O, cos Oa).

(16)

Since au is perpendicular to n. (10) we have the relation cos Oa = tgh )Ca cotgh ;~.

(17)

Now we shall construct a special centre of m o m e n t u m system Xu, Y., Z" to which we can relate g. [c/, (24)]. Also, we shall give the connexion between this system and quantities defined in A [c/. (23)], In this way g~ is connected with quantities in A. Let i., ]'~ and k. be the four-vectors t h a t are associated with three m u t u a l l y perpendicular three-vectors in the centre of m o m e n t u m system. Then one has t

i~'i~, = i . k . = ].&, = / 5 . i . = / S a i . =/Saku : 0.

(18)

As a first axis we m a y choose ku = ga, because Saga ~- O.

(19)

RELATIVISTIC

KINETIC

GAS THEORY.

VI

35

The 3-axis of this centre of momentum system is thus chosen parallel to the incident relative momentum. The second four-vector m a y be obtained with the help of Schmidt's orthogonalization procedure: ~, = n , + (n'~.) p , - g-2(n.g.) g,.

(20)

The last four-vector which is perpendicular to i5#,/'# and k# m a y be found b y calculating the vector product of these four-vectors:

i" = ~,'~°p,n~g,.

(21)

Here, d ~"" is the Levi-Civita permutation symbol. A cartesian coordinate system in the centre of momentum frame can thus be characterized b y the three spacelike unit four-vectors X~, Ya and Zv: x . = i"/i,

Y . = kv/k,

Zv = g~,/g,

(22)

(1.2 = i"i., k~ = kl'k,, g2 = gvg.).

In the special Lorentz frame A we find with (10), (12), (13) and (15) from (19)-(22) X~ ----- (0, sin -1 01e -- cotg 01el), Yv ---- (0, sin 01e A el),

(23)

Z~ -----(sinh ½Xl, cosh ½Zlel). In the centre of momentum system the angle between the relative threemomenta g and g' before and after the collision is, b y definition, O. The axis Z~ of the coordinate system (22) is parallel with the relative momentum g~ before collision. Let # be, in this system, the angle between the X~ axis and the projection of the relative three-momentum after collision g'~ on the plane of the axes X~ and Y~. Then we have g'~/g = sin 8 cos q~xv + sin 8 sin q~Yv + cos OZv.

(24)

The angles 0 and ~0 occurring in (3) were the polar angles in the centre of momentum system. In the special centre of momentum system with cartesian axis Xv, Yv, Z~ (22) one has 9 = q5 + constant,

0 = O.

(25)

Inserting (7) into (3) we find, with (25), S°[x(q)3 -= macaa ~ (1 + I1 - - 1' - -

Ii) F(O) sin O dO,

(26)

0

where 2n

I = (aaat)(qt*q'){exp(~no, q~')} S S exp(znaq~) cosh-1½Zl dq~(dql/q°), 0

(27)

36

W.A.

van LEEUWEN

A N D S. R. de G R O O T

27~

I1 =

c~ /~ v I exp{(z + 2) n eql} cosh --i ½21 dq)(dql/q°),

(28)

0 2=

I exp(znc~q~ + 2naq'a) cosh-1 ½gl dO(dql/q°),

(29)

o 2r~

Ii = S exp(zn~q? + •naqi a) cosh -1 821 dq~(dql/q°l), (30) 0

and where (14) also has been used. In order to carry out these four-fold integrations (27)-(30) we introduce the new variables t x = sinh ½Zl sin 01 cos ¢, y = sinh ½21 sin 01 sin qs,

(31)

z = sinh ½Zl cos 01. The connexion between the corresponding differential elements is found by calculating the Jacobian of this transformation. The result is dx dy dz = ½sinh 2 ½21 cosh ½Z1 sin 01 d21 d01 d # .

(32)

Calculation of the Jacobian of the t r a n s f o r m a t i o n (11) gives the connexion

[c/. (1.56)] dql/q ° == sinh2 %1 sin 01 d21 d01 d~l.

(33)

W i t h (32) and (33) we find cosh-1 ½21 dq)(dql/q °) = 8 d~l dx dy dz.

(34)

Next we introduce the new variable of integration z = (1 + x 2 + y2).~ sinh ½t.

(35)

This gives cosh-1 ½21 dq~(dql/q °) = 4(1 + x 2 + y2).t cosh ½t dcp1 dt dx dy.

(36)

In the following two sections we shall calculate the integrals (27)-(30) using the new variables t, x and y.

3. The calculation o/the integral I'. With the help of (10), (11), (31) and (35) we find

n~q~ = --(1 -: x 2 + y2) cosh(t -- 2) -- (x2 - / y 2 ) cosh 2.

(37)

F r o m the law of conservation of energy m o m e n t u m (IV.7) and the defit T h e v a r i a b l e z as it is i n t r o d u c e d in eq. (31) will o c c u r o n l y in t h e eqs. (32), (34) a n d (35). I n all o t h e r f o r m u l a e z = m c 2 / k T Iv. (IV.34)].

RELATIVISTIC KINETIC GAS THEORY. VI

37

nition of the relative m o m e n t u m (IV.6), we obtain t

t

q, = ½(q, + q l , - - g J m c ) ,

(38a)

qia = ½(q. + q l . + g'~/mc).

(38b)

W i t h (9)-(11), (13), (23), (24), (31), (35) and (38a), we find nc,q '~ = --(1 + x 2 + y2) sin 2 ½0 cosh(t -- Z) -- (x2 + y2) sin 2 ½0 cosh z

(39)

-- x sin 0 sinh X -- c°s2 ½0 cosh Z. Using the abbreviations z' = z + 4 s i n

2½0;

4'=~cos

2½0,

(40)

we can write the argument of the exponential of the integrand of (29) in the form zn~q~ + ~naq'~ -- --z'(1 + x 2 + y2) cosh(t -- ;~) z,(x 2 + y2) cosh ;~ -- 4x sin 0 sinh z -- 4' cosh ;~.

_

(41)

Note that the argument of the exponential is independent of 91- W h e n we insert (36) and (41) into (29), we find I ' = 4{exp(--4' cosh 2)} ×

T --co

_

~ --co

co

2r~

I

f exp{--z'(1 + x 2 + y 2 ) c o s h ( t - ) ~ )

--oo

0

z,(x 2 + y2) cosh g -- ~x sin 0 sinh X} × (1 + x 2 + y2)½ cosh it dgl dt dx dy.

(42)

' ' After having expressed the scalar product in terms of x, y, t and 91 with the help of (9)-(I 1), (13), (16), (17), (24), (31), (35) and (38a), we carry out the integration over 91. The terms containing either sin 91 or cos 91 vanish, since the integration is carried out over a whole period. The terms which do not contain 91 give a factor 2~, while the terms containing either cos 2 91 or sin 2 91 yield a factor r~. In this w a y we arrive after a lengthy calculation at

I'=

8~ e x p ( - - 4 ' cosh 2) 7 --co

_

? --oo

~ exp{--z'(1 + x 2 + - y 2 ) c o s h ( t - Z) --co

z,(x 2 + y2) cosh Z -- 4x sin 0 sinh )~} P ( x , y, t)

× (1 + x 2 + y2)~ cosh ½t dt d x dy,

(43)

P ( x , y, t) =~ (sinh 2 Za -- ½ sinh 2 ;~) Q(x, y, t).

(44)

where

38

W . A . van L E E U W E N AND S. R. de GROOT

The factor Q(x, y, t) in (44) is given b y

Q(x, y, t) - cos 4 ½0 + sin 4 ½0(x 2 + y2)2 + cotgh e X sine Oxe + sinh-e Z sin4 ½0(1 + x 2 + y2)2 sinhe(t _ X) + 2 sin e ½0 cos 2 ½0(x 2 + yz) + 2 cotgh Z cose ½0 sin Ox -- 2 sinh -1Z sin2 1 0 c°s2 ½0(1 q- x u + ye) sinh(t -- Z) + 2 cotgh Z sin2 ½0 sin Ox(x e + yZ) - - 2 sinh-1Z sin4 ½0(x2 + Y2)( 1 + xe + Y~) sinh(t -- Z) -

-

2 sinh -e Z cosh Z sin2 ½0 sin Ox(1 + x 2 q- y.Z) s i n h ( / - - Z)

-- sinh-2 Z sin4 ½0( xe q- y2) (1 q- x 2 q- y2) (1 + cosh t) -- ½sinh-2 X sin2 0(1 + x 2 + y'~) cosh ') ½t + ½ sinh-2 Z sin2 0(1 + x e) -

-

sinh-2 Z sine 1 0 sin Ox(1 -- x 2 + ye) sinh t.

(45)

L e t us introduce the abbreviations b -- z'(1 + x 2 + y2), if = cosh ½g, K =- cos 10,

(46) .9°

sinh ½Z,

i ~ = sin ~0.

(47)

(48)

T h e expression (43) m a y then be r e w r i t t e n I ' -- 8n(z')-~ e x p { - - z ' -- 2'(1 + 29o2)} ×

II

dx dy exp{--2z'ife(x 2 + ye) _ 42Kagor~x}

× eb bt I dt e x p { - - b cosh(t -- Z)} cosh ½tQ(x, y, t),

(49)

where also (44) and the relation

-- sinh2 Za -- ½ sinhe Z,

(50)

which follows from (9), (10) and (16), h a v e been used. W i t h the abbreviations (46)-(48), the expression for Q(x, y, t) (45) becomes Q(x, y, t) = K4 @ o-4(x 2 ~- y2)2 _@ o.eK2(1 _~_ 49o2 ~_ 4,~g4) ,~g-2if-2x2 + l(z') -e a49O-2~-eb2 sinhe( t -- Z) + 2a2K2( x2 + y2)

+ 2aKa(1 + 250 e) 9o-l~f-lx -- (z') -1 o'2K2~90-lif-lb sinh(t -- X) + 2aaK(1 + 29o2) . ~ - l i f - l x ( x 2 + ye)

RELATIVISTIC KINETIC GAS THEORY. VI _

_

-

-

39

(Z')-I a4~-l(~-l(x2 + y2) b sinh(t -- Z) (z')-I aaK(1 + 25¢2) 5a-2~-'~xb sinh(t -- •)

__ ¼(g')--i {74~-2(~--2(X2 _~ y2) b(1 + cosh t) - - ¼(Z') -1 0"2K2,9°-2(~-2b(1 -~- c o s h

t)

+ 10"2/<2,.~-2~--2

(51)

+ ½a2K25g-2(6~-2X2 -- l(z')-i aaK,Sa-2c6~-2xb sinh t. It is convenient to introduce the operators St and Sxy by St[g(t)] =- (27:) -'~ b~ e ~ cg-1 T dt exp{--b cosh(t -- Z)} cosh(½t) g(t)

(52)

-co

Sxy[h(x, y)] -- 2=-lz-2(z') 8 ~6 exp(_2~2a2K25¢2/z')

×

~

~ dx dy exp{--2z'~e(x 2 + y2) _ 42aK2f~} h(x, y).

(53)

-co -co

In the appendix it will be shown t h a t Still = 1, St[b sinh(t -- Z)] : ½5p%~-1, St[ b2sinh2( t - Z ) ] = z '

1 -}-~-z,+X 2+y2

St[b(l + c o s h t ) ] : 2 z '

~2
St[b sinh t] = 2z'

APW +

~ z'

)

,

3 } 4z' + ( x 2 + y ' a ) ~ 2 ' 5" 4 z ' ~ + (x~ + y'>) ~

and Sxy[l] :

<~4, z'

( z 'z

z'

S~u[x2 + y2] = ½7 ~ + k ~ 1

Sxv[(x 2+y2)2] = ~2z Z 2

+ \~

+ 2

.7,'

~ K2

2 --z + 1] ~ - y,2~2,

z'

z2 Z

2 -F

z

z' ] a 2

Z2 ~ K4

4--z + 6 - 4-Z + 7 ~ ] ~ - ~4, Z'2 / &" Z2 (~

5 ~'z

}

'

(54)

40

W.A.

l

van LEEUWEN

Z'

( Z '2

S~v[x2]- 4 z2 w +

Sxv[X(X2 +

y2)] =

A N D S. R. d e G R O O T

Z'

\ ~ - - 2 - -z + i ]~ ~a2 s ~ w , z

z2

(z z' z'" ~ ~3 + ~z - 3 + 3--z + --z2 / --~3 9°~"

(55)

The factor in St (52) has been chosen such t h a t St[l] -~ 1. The factor in the integral operator Sxy (53) is such t h a t if we write I ' (49) with the help of St and Sxv:

I' =/'Sxy[St{Q(x, y, t)}],

(56)

the f a c t o r / ' is equal to ]' -- (2r:)~ z2(z ') -~ ~ - 5 exp{--(z + 2) -- 22'zSe2/z'}(a~a~}(q"q~).

(57)

We have chosen to write /' separately for later convenience. Taking together the 102 terms resulting from the 15 integrals of (56), we find with the help of (51), (54) and (55) the result I'

=

1'[½•2(3K2

--

~2) + ~z-l~2(K2 + ~z-la2)

@ K2{3K2~q~2 -r- z-lo'2( ~5#2 - - ~)}(z/z') dr- K4oG°2(~ 2 - - ½)(Z/Z')2].

(58)

Note t h a t all terms with z' in the n u m e r a t o r have vanished.

4. The calculation o/I~, I1 and I. Now t h a t we have obtained I', we m a y calculate the remaining integrals I i , I1 and I relatively easily since we are already in possession of the m a t h e m a t i c a l material needed. First we note t h a t we m a y obtain I i (30) from I' (29) by replacing q~ by qiu, or, in view of (38a, b), by replacing g~ by --g~. This can also be achieved b y replacing O and ~b in (24) with n -- O a n d r: + ~b, respectively. This replacement a m o u n t s to the interchange of a and K in (48), so t h a t we m a y also obtain I i from I' by interchanging a and K in (58). In this way we find I i -- 11[½~2(3~ 2 -- ~2) + ~z-lK2(~2 + _~z-lK2) ~ - O'2{3(12~ 2 -4- z - - l t ~ 2 ( ~ 2 - -

7)}(Z/Zl) -~- Cr4oG°2(,.9a2

--

1)(Z/Zl)2],

(59)

where /i = (2~)~ z2(zl) -~ ~ - 5 exp{--(z + ~) -- 221z~2/Z'l}(a,a,)
(6o)

zi=z+2cos

(61)

210;

21=2sin2½0

RELATIVISTIC KINETIC GAS THEORY. VI

41

[c/. (40) and (57)j. The c o m p u t a t i o n of I1 (28) proceeds analogously to that of I'. We find

I1 = / 1 T x y [ T t { R ( x , y, t)}~,

(62)

TtEg(t)] =-- (2~)-t d½ eaCK-1 ~o dt exp{--d cosh(t -- %)} cosh(½t) g(t),

(63)

--oo

d --= (z + 2)(1 + x 2 + y2),

(64)

Tzv[h(x, y)J =-- 2~-lz-2(z + 2) z c~6 ×

II

dx dy exp{--2(z + 2) ff"{x 2 + y2)} h(x, y),

(65)

R(x, y, t) =~ }(z + 1) -1 5:-2
_

6a-lC~-l(z + 2)-1 (x 2 + y2) d sinh(t -- Z)

-- ~(z -f- 2) -1 Sf-zcK-Z(x2 -t- y2) d(1 + cosh t), /1 =---(2~)I z2(z + 2)-½~ -5 exp{--(z + 2)}(a/,a,>(qUq,>.

(66) (67)

Comparison of (46), (52) with (53) and (63), (64) with (65) shows t h a t expressions for T¢[...3 and Txv[...J are obtained from those for St[...J (54) and Sxu[...J (55) by setting K = 0 and replacing of z' by z + 4. Using these expressions for Tt[...] and T~v[...3, we find from (62) and (66), that I1 =- 3/1/16z 2.

(68)

Making use of the first expression of (54) we can evaluate immediately (27). This gives I ----- (27:)t z-~Cd-1 exp(--z -- 2 cosh %)(aua~)(quqv).

(69)

We shall make use of the integrals 1 (69), 11 (68), I' (58) and I i (59) to solve the eigenvalue problem for tensor eigenfunctions in the following section.

5. The tensor pseudo-eigenvalue equation. Let us introduce the new parameters s -= 2/(~ + z),

s' = K2s,

s l = ~2s.

(70)

The generating function of the Laguerre polynomials is a) WP(z, s) = (1

--

s -(p+I))

e -'s/(1-s).

(71)

The connexion with the Laguerre polynomials is given by z) (n l) -1 {~n~p~(r, s)/~sn},=o = L~(r).

(72)

Insertion of (58), (59), (68) and (69) into (26) followed by multiplication of

42

W . A . van LEEUWEN AND S. R. de GROOT

b o t h sides of (26) with mZc2eZ/(1 - - s) ~ and use of (IV.30, 34), (4), (9), (10) and (71), yields 5e [~o~(7-,s) (aua~'/
(73)

0

where T(7-, s) -- (1 + z-17- + ~z-27-2) W~(7-, s) + ~ z - 2 + ( - ~ K 4 + ½K~ - Iz-1~27- + Iz-17- _ ~z-'~2 + ~z-1~4 _ ~ z - 2 ~ 4 _ l z-2~2~ _ Iz-27-2) ~ ( ~ , s') + ~(z-17- + ½z-27-2 _ ]z-la27- + Iz-2~27- -- ~z-laz) ~(7-, s')

+ ~4(z-17- _ z-27-2) ~0~(7-,s') + ( _ ~.~4 + 1~2 _ ~z-1~2~- + ~z-17- _ ~z-1,~2 + ~z-l,,4 _ ~ z - 2 ~ 4 _ ~z-2K27- _ ¼z-27-2) ~(7-, sl) + ~2(z-1~ + ~z-27-2 _ ~z-1~27- + ~z-2~27- _ ~ z - l ~ ) ~ ( ~ , si) + ¼K4(z-17- -- z-27-2) ~(7-, s~).

(74)

Now we note t h a t we can eliminate
(75) we arrive, after having differentiated (73) and (74) n times with respect to s, at the expression [L~(7-)
× [{I(3en+2 -- en+l) -- ~ z - l ( 7 e n -- 19en+l + 12en+2) + ~z-2(en

- - 2en+l + en+2 -- (~onen) -- ¼z-17-(en - - 5en+~)

@ Iz-27-(en -- en+l) @ ¼z-27-2en} L~(r)

+ {~Z-l(7 ---

-

1Z-27-2(en

4z-17-)(en --

--

2en+l -]- en+2) --

-lz-17-(en

--

4en+l -~- 3en+2)

en+l)} L ;n(7-)

¼z-17-(l -- z - l r ) ( e n -- 2en+l + en+2) L~(7-)I.

(76)

Using the relations 3) L~ +1(7") =

~ L~m(*), m-O

(77)

R E L A T I V I S T I C K I N E T I C GAS T H E O R Y . VI

43

• L~+I(T) = (~" -- n) L~(T) + (n + P) ~n-l~rP 17~,,

(78)

~'LPn+I(T) ---~ (n -~- p -~- 1) LnP(T) -- (n -~- 1) Ln~+m('r),

(79)

for the associated Laguerre polynomials, we can eliminate the terms containing • or r 2 as a factor, as is shown in section 2 of the appendix. We then obtain the tensor pseudo-eigenvalue equation: ~[L~(T)] = (1 + ½Z--IT)-~ '~ + ¼Z-1 [(n + ~)(en -- Sen+l)Ln_I(T i × [1(3en+2 - - en+l) L ~(~) )

3- {(2n -- 1) en+l + 3(2n + 5) en+2} L~(T) + (n + l)(e.+l

5e.+2) Ln+I(*)~ + al,z-Z[(2n + 3)(2n + 5) enL~_2(, ) -

-

-- (21, + 5){4nen + 2(2n + 5) en+a}L~_m(, ) + {(--3~0n + 4n 2 + 4n + 3) en + 2(8n 2 + 28n + 1 l) en+l

+ 2(2n2 + 12n + 19) en+2}L~('r) -- 4(n + 1){(2n + 1) en+l + (2n + 7) en+2} Ln+I(T ~ ) + 4(n + 1)(n + 2) en+zL~+ 2(~')1]
(80)

This equation is used to determine the viscosity.

6. The viscosity/or a gas o] Maxwellian molecules. H a v i n g obtained the tensor pseudo-eigenvalue equation (80), we are in a position to solve the integral equation (1), which reads [C I = G ,

(81)

where C is in the function to be determined, and where

G :

-- 1/kT.

(82)

One m a y verify that G satisfies the solubility conditions. Since the set of associated Laguerre polynomials L~ is complete, we can t r y oo

C :

~, , Y~ cn L ,At},

(83)

n=0

and write oo

(1 + ½,-17)0 C =

Z g.L2(*). n=0

(84)

F r o m (82) and (84) we find, up to order z-i: go = -- (1 + ~ z -1)/kT,

g . -- 0

(n > 2).

gl = ---~z-1/kT,

(8s)

44

W.A. van LEEUWEN AND S. R. de GROOT

If now (85) and (84) are inserted into the pseudo-eigenvalue equation (80) we find, by identification of the coefficients of Ln(T), the recursion relations ~(3en+2 - - en+l) c,,

+ ¼z-l[(n @ ~)(en+l -- Sen+2) Cn+X + {(2n -- 1) en+x + 3 ( 2 n + 5) en+2} Cn + n ( e n - - 5en+j c n - l l

-~- l~6Z-2[(2n -~- 7)(2n + 9) en+2Cn+2 - - (2n q- 7){4(n + 1) en+l + 2(2n + 7) en+2} Cn+l @ {(--3b0n @ 4n 2 -~- 4n + 3) en + 2(8n 2 + 28n + 11) en+l + 2(2n 2 + 12n + 19) en+2}cn --

4 n { 2 n e n + (2n + 5) en+l} Cn-1

-p 4(n -- 1) nenCn-2] = gn

(n :

O, 1 , 2 , . . . ) .

(86)

The first of these recursion relations (n = 0) contains the three u n k n o w n coefficients co, Cl and c2. To obtain one particular set of coefficients Cn which satisfy (86) we proceed in the same m a n n e r as in section 5 of article IV of this series. First we determine the non-relativistic solution by neglecting in (86) the terms of order z -1 and z -2. This gives for the non-relativistic a p p r o x i m a t i o n gn to the coefficients Cn: Cn :

2gn/(3en+2 - - en+l)

(n : 0, 1,2 . . . . ),

(87)

where gn is the non-relativistic limit of gn. The first approximation to the coefficients Cn can now be found b y inserting these values into the terms of (86) which are of order z -1 and z -~ and by solving the resulting set of equations. If this procedure is continued we find a solution of (86) as a power series in z -1. We carry out the first step of this programme by substitution of (87) and (85) into the terms of ( 8 6 ) w h i c h are of order z -1, while we neglect the terms of order z-~. Solving the resulting set of equations we find, up to order z -1 (z - - m c Z / k T ) : co--

3kTe2

1 + c -~

8

rn

'

1

Cl = - - ~c -2 m ( 3 e a - - e2) ' Cn

=

0

(88)

(n > 2).

The coefficient of viscosity (11.81) can now be calculated. W i t h the help of (83)-(85), (88), (IV.43, 53) and (V.28) we obtain, up to order z - l : ---- (2z:)~ m ~ ( k T ) ~ ~x e - z cogo, where we have also used the fact t h a t = §(A~vpapv) 2.

(89)

RELATIVISTIC KINETIC GAS THEORY. VI

45

Insertion of the values (85) and (88) into (89) results finally in

kT(

45

r/-- 37~z2 1 + c - 2 - 4

kT)

m

,

(90)

up to the order c-2. The viscosity is seen to differ from its non-relativistic limit b y an additional term of order c-2. Remark I. In the non-relativistic theory of a Maxwell gas, the conductionviscosity r a t i o / , given by

/ ~ 2/cv~1

(91)

is known 4j to be ~. In the relativistic theory, however, from (91) and (II.21) with (III. 18), (V.43, 90) and (90) we find the viscosity ratio to be /=~

1 - - c -2 17 2

m"

(92)

Remark 2. In the non-relativistic kinetic theory, the volume viscosity is a vanishing quantity, while in the relativistic theory it is, in general, a quantity of order c-2 [c]. (III.74)]. The volume viscosity of a relativistic Maxwell gas has been calculated b y Israeli). The result is a quantity of order c-4: ~v = c-4 5(kT)a/12rcm2z2.

(93)

The thermal-diffusion coefficient is, in general, a non-vanishing quantity in the kinetic gas theory, but vanishes for a gas of Maxwellian molecules in the non-relativistic theory. In the relativistic theory we find [c[. (V.66, 88)1 a quantity of order c-2: ,

s

D T = c-2 - 16~

k2T

2 --

pm

1)

--

w2

--

(94)

The difference in dimensions having been noted, the order of magnitude of the thermal-diffusion coefficient is still a factor c2 bigger than the order of magnitude of the volume viscosity.

APPENDIX

1. Calculation o[ integrals appearing in sections 3 and 4. Let us consider the integral a =-- T exp{--b cosh(t-- Z)}cosh ½t dt. --co

(A.1)

46

W.A. van LEEUWEN AND S. R. de GROOT

After the substitution t' = t -- Z, we m a y obtain for (A. 1) : co

a = 2 cosh ½X ~ e x p ( - - b cosh ½t') cosh ½t' dt'.

(A.2)

0

W i t h the substitution u = sinh ½t' the integral (A.2) takes the form of a Poisson integral. Thus we find a = (2~)~ b-~ e -b cosh ½Z.

(A.3)

Differentiation of (A.1) and (A.3) with respect to Z yields ~° exp{--b cosh(t -- Z)} b sinh t -- ;g cosh ½t dt = la,9~W-1.

(A.4)

--oo

Similarly we obtain oo

S exp{--b cosh(t -- Z)} b2 sinh2( t -- ;g) cosh ½t dt -- a(b + }),

(A.5)

--oo oo

exp{-- b cosh(t--z) } b(1 + cosh t) cosh {t dt = a{2(b -t- 1) cg2_ ~_}, (A.6) --co oo

exp{--b cosh(t -- 9~)}b sinh t cosb ½t dt = a{2(b -¢- 1) 5 ~

-- ½~9~-1}.

--oo

(A.7) F r o m (A.3)-(A.7) with (46) and (47) we find the expressions (54) for the integrals St[...]. Let us also consider the integral T --oo

T dx dy exp{--p(x 2 + y~) -- qx} = r@-1 exp(q~/4p).

(A.8)

--Co

We introduce the abbreviation r =-- ~p-a exp(q2/4p).

(A.9)

After differentiation of (A.8) with respect to p and/or q we obtain the following integrals: SS dx dy e x p { - - p ( x 2 + y2) _ qx}(x 2 + y2) = r(p + ¼q2),

(A.10)

SS dx dy exp{--p(x 2 + y2) _ qx}(x 2 + y2)2 = r(2 + q2/p + q4/16p), (A.11) SS dx dy e x p { - - p ( x 2 + y2) _ qx} x = --½rpq,

(A. 12)

SS dx dy exp{--p(x 2 + y2) _ qx} x 2 = r(½p + ¼q2),

(A.13)

SS dx dy e x p { - - p ( x 2 -t- y2) _ qx} x(x 2 + y2) = --r(q + qa/81b).

(A. 14)

The expressions tor the integrals Sxv[...], eq. (55), are found b y choosing in

(A.8)-(A.14) p = 2z'~2,

q = 4~a~9°~,

and elimination of ~ with the help of (40).

(A. 15)

RELATIVISTIC KINETIC GAS THEORY. VI

47

2. The elimination o/the/actors r /rom the expression (76). W i t h (77) and (79), the s u m of the two terms of (76) which contain z-lLnl and z - l r L nl as a factor, m a y be written ¼z-l(n + 1)(en -- 2en+l + en+~) L~+I -=- C1.

(A.16)

Addition of (78) multiplied with an a r b i t r a r y constant k and of (79) multiplied with (1 -- k) yields the connexion rL~+I(z) : k-rL~(r) + k{(n + p) L~_t(r ) -- nL~(z)} + (1 -- k){(n + p + 1) L~(,) -- (n + 1) L~+x(T)}.

(A.17)

We m u l t i p l y this relation with r/4z 2, choose k : ½ and p : {. Then using also (79) we find t h a t the term of (76) which contains z-2~'2L~ as a factor, m a y be rewritten

lz-2~'(en -- 2en+l q- en+2)/-Tr'z z q- rL~} t2~n q- 9_f{ 2--n -- Ln+l z + C2,

(A.18)

where

C2 ~ ~z-2n(en -- 2en+l + en+2){(n + 6) Ln-1 -- nL~

-- (n +

~)

L~+ 1 + (n + 2) L~+ 2}.

(A. 19)

W i t h the help of (A.16), (A.18) and the abbreviation Ca for the first three terms of (76): Ca --= {½(3en+2 -- en+l) -- ~z-l(7en -- 19en+l + 12en+2)

+ ~z-U(en -- 2en+l + en+2 -- 6onen)} L~(T),

(A.20)

we can now rewrite (76) to read

.W[ L ~(pap~)] = (1 + ½z-it) -a [{--l(en -- 5en+l) z - i t .~ + ½(en -- en+l) z-2r + lenz-2r2} L~ + ½(en -- 4en+l + 3en+2) z-l'rL~ + 1Ag(en -- 2en+l q- en+2) z-2~'(7L~n_x q- L~ -- 2L]n+l) --

~(3en -- 2en+l -- en+2) z-2r2L~ + C1 + C2 + Ca] (PuPv).

(A.21)

F r o m (A. 17) with p = ~ and

k = (en -- 5en+l)/(2en -- 8en+l + 6en+9.), we m a y rewrite the two terms of (A.21) which contain z - i t as a factor:

lz-l[(en -- 5en+l){(n + 5) Lln_l -- nLn) q- ( e n - - 3 e n + l +6en+2){(n + ~ ) L :n -

(n 4- 1)L~+I} ] ~ C 4 .

(A.22)

48

W.A. van LEEUWEN AND S. R. de GROOT

Now we choose in (A.17)" p = ~ and k = 2en/(3en - - 2en+l -- en+2)

and multiply the equation then obtained with r. The terms of (A.21) which contain z-2r 2 as a factor m a y now be eliminated. Then we find with (A.21) and (A.22) that

i ~ [ L ,~(P~,P,)I - - (1 + ½z-lr)-t[z-2"r[--¼(n + ~) enLn_ 1 + {l(en -- en+l + ½ n e n ) -- ~(en - - 2en+l -- en+2)(n + z)} L~

+ ~(n + 1)(en -- 2en+l -- en+2) L~+I~ + ~ z - 2 r ( e n - - 2en+l + en+2)(7Ln_ ~ 1 + L~~ -- 2Ln+l)

+ C1 + C2 + Ca + C4] .

(A.23)

In (A.17) we now set p ---- ~ and replace n by n -- 1. Furthermore we make the choice k =

2en(2n +

5)/7(en - 2en+l + en+2)

for the arbitrary constant in (A.17). With the help of the resulting expression we find for z2 times the sum of those two terms of (A.23) which contain a Laguerre polynomial with a lower index (n -- 1):

g_~

--¼(n + ~ ) enr ~-1 + 13-~(en - - 2en+l + en+2) "rL~_1

= -~{(2n + 3)(2n + 5)L~_2 + (en - 8hen - 14en+l + 7en+2) × (n + {) L ~ _ 1 + (3en + 4nen + 14en+l -- 7en+2) nL2}.

(A.24)

Now the factor r has been eliminated from the left-hand side of (A.24). Choosing suitable values for k, the other terms of (A.23) containing a Laguerre polynomial with a lower index n or n + 1, m a y be similarly rewritten. Thus, with the help of the explicit expressions for C1 (A.16), C2 (A. 19), C3 (A.20) and C4 (A.22), we obtain the tensor pseudo-eigenvalue equation (80). A c k n o w l e d g e m e n t s . The authors wish to t h a n k P. H. Polak, who was willing to check some of the more involved calculations, and Dr. L. P. Staunton, who carefully read the manuscript. This investigation is part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)", which is financially supported by the "Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)".

R E L A T I V I S T I C K I N E T I C GAS T H E O R Y . VI

49

REFERENCES 1) Israel, W., J. math. Phys. 4 (1963) 1163. 2) de Groot, S. R., van Weert, C. G., Hermens, W. Th. and van Leeuwen, W. A., Physica 40 (1968) 257; 40 (1969) 581; 42 (1969) 309; van Leeuwen, W. A. and de Groot, S. R., Physica 51 (1971) 1 ; 51 (1971) 16. 3) Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products, Academic Press (New York, 1965) p. 1037, 1038. 4) Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, University Press (Cambridge, 1970) p. 174.