Derivation of the equations of relativistic hydrodynamics from the relativistic transport equation

Volume 5, number 2

DERIVATION

PHYSICS

OF THE FROM THE

LETTERS

15 June 1963

HYDRODYNAMICS EQUATIONS OF RELATIVISTIC EQUATION RELATIVISTIC TRANSPORT N. A. CHERNIKOV

Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna Received 22 May 1963

The. Boltzmann t r a n s p o r t equation taking into account the theory of relativity and gravitation has been derived and investigated in a number of p a p e r s 1). A s y s t e m a t i c consideration of this p r o b lem was started by the author 2-4). Bogoliubov 5) has shown how it is possible to come in the nonrelativistic case to the t r a n s p o r t equation starting f r o m the m e c h a n i c s of the particle s y s t e m and has established by this the connection between the t r a n s p o r t equation and mechanics. In view of the latter the formulation and the investigation of the relativistic t r a n s p o r t equation is of i n t e r e s t in connection with the development of relativistic mechanics. In the p r e s e n t paper the relativistic hydrodynamic equations of dissipative p r o c e s s e s o c c u r r i n g in the gas are derived f r o m the r e l a t i vistic kinetic equation. These equations were known so far only in a phenomenological aspect 6). Here we establish in the relativistic case a hitherto unknown dependence of the heat conductivity and v i s cosity coefficients on the differential c r o s s section for the gas particle interaction and on the local gas t e m p e r a t u r e . The problem is solved by the method of m o m e n t s developed in the n o n - r e l a t i v i s t i c case by Grad 7). In o r d e r to extend the method of m o ments to the relativistic case, a relativistic generalisation of the H e r m i t e - C h e b y s h e v polynomials was needed. Some of the f i r s t such polynomials are stated in the p r e s e n t paper. It is supposed that all the gas p a r t i c l e s are of the same kind and the collisions of p a r t i c l e s a r e elastic. We denote the m o m e n t s of the gas distribution function A(x,p) and the collision integral I(x,p) as follows:

A~I ''.an(x ) = J pal . . . pan A(x,p) dP ,

(1)

/ ~ l ' " a n ( x ) = J'p al . . . p an l(x,p) rip.

(2)

F r o m the equation of t r a n s f e r 2) for the function (~(x),p) n the equation of m o m e n t s follows: V~ A ~ 1 " " " an : / ~ 1 " " ~n

(3)

The f i r s t moment AS(x) coincides with the v e c t o r of

the gas particle flux. The second moment AaB(x) coincides with the energy momentum t e n s o r of the gas. In ref. 2) it was proved that the zeroth m o ment I(x) and the f i r s t m o m e n t I[3(x) are equal to z e r o and, consequently, V~xA s = 0

,

V~A ~fl=0.

(4)

Owing to the equality (p,p) = m2c 2 the m o m e n t s of A(x,p) and I(x,p) a s , in general, of any function B(x, p) p o s s e s s an important p r o p e r t y

gl~ B#V~l"" an = m2 c2 B~I'" "an

(5)

In p a r t i c u l a r , if the r e s t m a s s m of a gas particle is equal to z e r o then the t r a c e of any m o m e n t is equal to z e r o too. The quadratic f o r m Aa~(x)~C~ fl of ~ is positive definite. Hence, there is one and only one eigenv e c t o r of the energy momentum t e n s o r (AaB~~:=.u~) lying inside the light cone. It d e t e r m i n e s the velocity u(x) of the gas m a s s at the point x:

A~u~

=~(0) u ~ ,

(u,u) = 1 ,

Uo > 0 .

(6)

The eigenvalue ~(0) is positive, the other eigenvalues #(1), ~(2), ~(3) a r e negative. The s c a l a r ~(0)c -1 is the density of the gas m a s s at the point x. The s c a l a r p(x) = ~c [,~(0) - m 2c 2A (x)] = - ~c [t~(t) + ~(~.) + ~(3)]

(7)

is the mean hydrostatic p r e s s u r e of the gas at the point x. We p a s s to the relativistic hydrodynamics without dissipative p r o c e s s e s , putting

A(x, p) .~ ~

e -(x(x),p) ,

(8)

and substituting in (4) the m o m e n t s of the distribution functions (8). To take into account the dissipative p r o c e s s e s o c c u r r i n g in the gas it is n e c e s s a r y to use a m o r e exact expression for the distribution function:

A(x,p)~e-(kJ~)!

a~Ha a ~ H ~ + b~H~ } , {a++ - ~/(k,k)

2(k,k)

5 J(k,k)

115

Volume 5, number 2

PHYSICS

where H,y H~ s

ks ~ {.u ,

H s s = h~u hSu ~# ~u - ~o h s s ,10~/

=[5~o_ hup ~p~ ~p] k s ~

~

, ~s=j~(~pS

LETTERS

15 June 1963

w h e r e 0 is the local t e m p e r a t u r e and Kn(7 ) is the c y l i n d r i c a l function. F o r the t h i r d m o m e n t we write approximately Aa~ = a ¢~. We have

(hu~ h~v - ~ bus h~u) V~ a ~ U v

,

(15)

h s s = ua, u s - g s S ,

xs =~

u~' •

(11)

The t e n s o r s (10) a r e the r e l a t i v i s t i c g e n e r a l i s a t i o n of the H e r m i t e - C h e b y s h e v p o l y n o m i a l s . It m a y be a s s u m e d that aSu s = 0, bSu~ = 0, aS~u s = 0, a~S = a/~. We put t h a t the v e c t o r u(x) in (9) - (10) r e p r e sents the v e l o c i t y of the gas m a s s . T h i s m e a n s that it m u s t be the e i g e n v e c t o r of the s e c o n d m o m e n t of the function (9). H e n c e , it follows a ~ = bs. In addition, we put that the m e a n h y d r o s t a t i c p r e s s u r e is d e t e r m i n e d by the l o c a l - e q u i l i b r i u m p a r t (8) of the sum (9). F r o m h e r e we find hc~sa a s = O. T h u s , in the a p p r o x i m a t i o n c o n s i d e r e d the gas d i s t r i b u t i o n function is given by t h i r t e e n c o m p o n e n t s : four c o m p o n e n t s of the v e c t o r X(x), one s c a l a r function a(x), t h r e e independent c o m p o n e n t s of the v e c t o r a~(x) and five independent c o m p o n e n t s of the t e n s o r a~t~(x). To d e t e r m i n e t h e s e c o m p o n e n t s t h i r t e e n e q u a tions of m o m e n t s a r e needed. We m u s t c o n s e r v e five equations (4) which m e a n the c o n s e r v a t i o n of n u m b e r of p a r t i c l e s , m o m e n t u m and e n e r g y . To e s t a b l i s h eight additional equations, we c o n s i d e r the t e n s o r T~ u = ~ A ~ u - I~ u. By v i r t u e of (5) and the f i r s t equation of (4) its t r a c e equals z e r o . T h u s , the t e n s o r T~ ~ has nine independent c o m p o nents. It is n a t u r a l to imit the equation u~uuT~ u = 0 and not to take it into a c c o u n t a s all h i g h e r equations of m o m e n t s . T h e r e r e m a i n five + t h r e e i n d e pendent equations:

[hs~ hSu - -} h s s hm~]

: a

m5c 5

K2(7)

v . "S +

hs~ uuV~a ¢~u~

v.

-

hss v . . " } ,

__m5c5 K2(7) K2(7) Cp 74 K3(7) k h'~ V# a (16)

In deriving (16) use has been made of the second equation of (4), cp is the heat capacity of the gas per particle for the constant pressure. The q u a d r a t i c t e r m a(x)a(x) in the c o l l i s i o n int e g r a l v a n i s h e s . Neglecting the t e r m not containing a(x) we find /~S = a D [ a s k s + a s ks] - a S a s s ,

(17)

where B = ~ {40 b(0) - 21 0b'(0) + 302b"(0)} ,

(18) D:V~32 {20 b(O) -

oo b(0) = f

7 Ob'(e) + S2b"(O)}

K3(27(o)) S(p) (p2+ m2c2) [27(P)]3 dp,

o

(19)

),(p) = ( ~ , k ~ , / p

2 +

m2c 2 ,

S(p) is e x p r e s s e d in t e r m s of the differential c r o s s s e c t i o n in the c e n t r e of m a s s s y s t e m , a ~ = h((p,q), cos e) sin 0 dO d~ a s follows:

S((P'q>)

= 21r,1" h ( ( p , q ) , c o s O ) s i n 3 O d O ,

"d(P+q, P+qY

(20)

o


[ V ~ A ~ p ~ - Ip ~ ] = 0 ,

(12)

Substituting (15), (16) and (17) in (12) we find:

hs/z u u [V~ A c~/zv - l#V] = 0 . T h e s e equations t o g e t h e r with eqs. (14) g e n e r a l i s e the t h i r t e e n equations of G r a d to the r e l a t i v i s t i c c a s e . We shall c o n s i d e r t h e s e equations in just the s a m e w a y a s it h a s been done in the n o n - r e l a t i v i s t i c c a s e in the w o r k by S o m m e r f e l d 8) in o r d e r to c o m e s h o r t l y to the r e l a t i v i s t i c h y d r o d y n a m i c s equation s. We denote:

¢ a l . . . o%

1 f pS 1

=4-~

.

.

pan e-(X ' p) dP . .

(13)

The m o m e n t s of the function (9) a r e

A(x) = a @ ,

7 = mcJ(X,X) = m c 2 / k O , 116

m5c5 K3(7 ) {h~ V~ u~ + lz~ Vp z,s 72 B

as :

hsS %, . u } ,

m6c6 (K2(7)~ 2 cp - K3(7) ~- Iz~ V# In a - 75 D \K3(7)/

(14)

(21) (22)

The e x p r e s s i o n s (21), (22) should be substituted in (14) and the obtained A s and A s S should be obeyed eqs. (4). As a r e s u l t we a r e led to the r e l a t i v i s t i c h y d r o d y n a m i c equations of the d i s s i p a t i v e p r o c e s s e s o c c u r r i n g in the c a s e . The h e a t conductivity coefficient x is equal to m 10c 11 x =K2(7 ) K3(7 ) Cp . 75D

m4c4 AS(x) = a ~)s + ~ 2 - - K2(7 ) a s

A s S ( x ) = a CsS + m6c6 K3(7 ) a s s ~

ass -

(23)

The v i s c o s i t y coefficient ~ is equal to n =

roll c l l 75B

K3(7 ) K-(~) .

(24)

Volume 5, number 2

PHYSICS

T h e s e c o n d v i s c o s i t y c o e f f i c i e n t ~ i s e q u a l to z e r o . In the n o n - r e l a t i v i s t i c l i m i t x/~ = 15k/4m i n d e p e n d e n t l y on t h e f o r m of t h e d i f f e r e n t i a l c r o s s s e c t i o n . F o r the c o n s t a n t d i f f e r e n t i a l c r o s s s e c t i o n h ( ( p , q ) , c o s O ) = a / 4 ~ and f o r m = 0, we h a v e

3ck x-

2(7 '

6 ~-SgJ(~.,X)'

x 5c 2 ~ - 4S "

(25)

In c o n c l u s i o n t h e a u t h o r e x p r e s s e s h i s g r a t i t u d e to N. N. B o g o l i u b o v f o r the i n t e r e s t in the w o r k and the valuable discussions. 1) N.A.Chernikov, Dokl. Akad. Nauk SSSR 112 (1957) 1030; 114 (1957) 530; 133 (1960) 84, 333; 144 (1962) 89, 314, 544; Nauchnye Doklady vysshel Shkoly, Fiz.-Mat. Nauki no. 1 (1959) 168.

HFS

COUPLING

OF

THE

LETTERS

15 June 1963

2) N.A. Chernikov, The relativistic gas in the gravitational field, Acta Phys. Pol. 23, no. 5 (1963), in course of publication; l>reprint JINR P-1028 (1952). 3) N.A. Chernikov, P r e p r i n t JINR P-1159 (1962). 4) N.A. Chernikcv,, A microscopic basis for relativistic hydrodynamics, l>reprint JINR (1963), in course of publication. 5) N.N. Bogoliubov, J. Expfl. Theoret. Phys. (USSR) 16 (1946) 691; also in: Studies in Statistical mechanics, ed. J.De Boer and G. E. Uhlenbeck (North-Holland Publishing Company, Amsterdam, 1961). 6) L.D. Landau anc~ E. M. Lifshitz, Electrodynamics of continuous media (Oxford, 1960). A. Z. Petrov, Einstein space (Moscow, 1961). 7) H. Grad, Commun. pure appl. Math. 2 (1949) 331. 8) A. Sommerfeld, Thermodyn~mik und Statistik (Wiesbaden, 1952).

123

keV

STATE

OF

Gd 1 5 4

P . B. T R E A C Y

Research School of Physical Sciences, Australian National University, Canberra Received 17 May 1963

R e c e n t e x p e r i m e n t s 1 , 2 ) on the c o u p l i n g of v - e m i t t i n g s t a t e s of n u c l e i in p a r a m a g n e t i c i o n s with e x t e r n a l m a g n e t i c f i e l d s , h a v e r a i s e d t h e q u e s t i o n a s to how s u c h t i m e - d e p e n d e n t i n t e r a c t i o n s a r e to be a n a l y s e d so a s to g i v e v a l u e s of H F S c o u p l i n g s , and h e n c e n u c l e a r m a g n e t i c m o m e n t s , of t h e s e s t a t e s . T h e u s u a l t h e o r y 3) of t i m e d e p e n d e n t e x t r a - n u c l e a r i n t e r a c t i o n s i s b a s e d on the a s s u m p t i o n t h a t t h e r e e x i s t s a s h o r t - l i v e d c o r r e l a t i o n t i m e r c such t h a t ~ r c << 1, w h e r e o~ i s the n u c l e a r p r e c e s s i o n f r e q u e n c y c a u s e d b y t h e i n t e r a c t i o n . W i t h v a l u e s of r c n e a r 10 -10 s e c , t h i s c o n d i t i o n i s not u s u a l l y f u l f i l l e d with e l e c t r o n s p i n p r e c e s s i o n in f i e l d s of o r d e r 1000 O e , f o r w h i c h ,,, ~ 2 × 1010 r a d i a n s / s e c . It i s the p u r p o s e of t h i s l e t t e r to show t h a t a t h e o r y 4) d e v e l o p e d f o r j u s t such a s i t u a t i o n c a n b e a p p l i e d to the e x p e r i m e n t s of r e f . 1). S t i e n i n g and D e u t s c h 1) h a v e m a d e e x t e n s i v e s t u d i e s of V-V a n g u l a r c o r r e l a t i o n s i n v o l v i n g Gd 154 (123 k e V , l i f e t i m e ~ = 1.7 × 10 - 9 s e c , s p i n I = 2) f o r m e d f r o m /3-decay of Eu154 in an a q u e o u s s o l u tion. P a r a m a g n e t i c r e s o n a n c e e x p e r i m e n t s 5) s u g g e s t t h a t t h e Gd154 n u c l e u s r e s i d e s in an 8S~ s t a t e of the Gd 3+ ion, with a s p h e r i c a l h a l f - f i l l e d 4f s h e l l , but with i n t e r a c t i o n f r o m c o n f i g u r a t i o n s s u c h a s 4 f 7 5 s 6 s m i x e d to 4f75s 2, and r e s u l t a n t w e a k h y p e r f i n e i n t e r a c t i o n . In t h i s e n v i r o n m e n t the c o r r e l a t i o n

b e t w e e n t h e 123 k e V ~ - r a y and a p r e c e d i n g 7 - r a y i s r e l a t i v e l y u n d i s t u r b e d , i . e . , t h e c o e f f i c i e n t G 2 in the c o r r e l a t i o n W(O)= 1 + G 2 A 2 P 2 ( c ° s e ) a p p r o a c h e s u n i t y (G 2 ~ 0.8). The t i m e - d e c a y of the a t t e n u a t i o n c o e f f i c i e n t G2(t ) i s r e l a t i v e l y s l o w ( r e f . 1) fig. 5a) and f u r t h e r m o r e the a t t e n u a t i o n c a n be almost completely removed by a sufficiently s t r o n g f i e l d a p p l i e d parallel to t h e a x i s of one c o u n t e r . ( H e r e we s h a l l n e g l e c t the s m a l l r e s i d u a l e f f e c t , a m o u n t i n g to a b o u t 3%, a t t r i b u t e d to e l e c t r i c q u a d r u p o l e i n t e r a c t i o n s . ) T h e t h e o r y of t h e s e e f f e c t s w i l l now b e d i s c u s s e d in t e r m s of the t r e a t m e n t of r e f . 4). In r e f . 4) an e x a c t c a l c u l a t i o n w a s m a d e of the f o r m e x p e c t e d f o r G2(t ) f o r e l e c t r o n i c s t a t e s of t o t a l a n g u l a r m o m e n t u m J = ½ and ~. In the l i m i t w h e r e t h e e l e c t r o n (ion) i n t e r a c t i o n with t h e a p p l i e d f i e l d ((-pj/J). H) i s l a r g e c o m p a r e d to the h y p e r f i n e c o u p l i n g 6) 2;friAr = (I + ½)a, one f i n d s

G2(H,t)= 1 - - ~A (1 - c o s o~t) ,

(1)

where A :

2J~J+l)( I+½)a2 (-#j/j)2

'

(2)

117