Transport properties of a neutrino-antineutrino mixture

Physica 93A (1978) 485-492 © North-Holland Publishing Co.

T R A N S P O R T PROPERTIES OF A N E U T R I N O - - A N T I N E U T R I N O MIXTURE

W.P.H. de BOER and S.R. de GROOT Institute of Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

Received 10 May 1978

The transport properties - heat conduction, diffusion, thermal diffusion and viscous flow - of a neutrino-antineutrino mixture are studied. It turns out that in a temperature field the more abundant component will diffuse against the temperature gradient.

1. Introduction T h e t r a n s p o r t p r o p e r t i e s o f a n e u t r i n o - a n t i n e u t r i n o m i x t u r e will b e disc u s s e d o n t h e b a s i s of r e l a t i v i s t i c t r a n s p o r t e q u a t i o n s a n d a f o u r - f e r m i o n H a m i l t o n i a n f o r t h e w e a k i n t e r a c t i o n . E x p r e s s i o n s a r e d e r i v e d f o r t h e first a p p r o x i m a t i o n to t h e s h e a r v i s c o s i t y , t h e h e a t c o n d u c t i v i t y , t h e d i f f u s i o n coefficient a n d the t h e r m a l d i f f u s i o n coefficient. It is s h o w n t h a t t h r o u g h t h e r m a l d i f f u s i o n t h e m o r e a b u n d a n t c o m p o n e n t t e n d s to c o n c e n t r a t e in t h e relatively colder parts of the system.

2. The equations. Linear theory The neutrino-antineutrino system can be described by means of relativistic t r a n s p o r t equations~): 2

p~'OM"(i)l~x ., p ) = ~ E('°(x, p ) , j=t

(i = 1, 2)

(1)

w h e r e f(o is t h e d i s t r i b u t i o n f u n c t i o n o f t h e n e u t r i n o s (i = 1) o r t h e a n t i n e u t r i n o s (i = 2). It d e p e n d s u p o n t h e t i m e - s p a c e c o o r d i n a t e s x ~ (with 0~ = 485

486

W.P.H. DE BOER AND S.R. DE GROOT

0/0x ~) and the e n e r g y - m o m e n t u m p~'. The collision terms in (1) are E(ii)(x, p ) = f

d3pl

d3p ' d3p; --~--y-¢ p~--~w(")(p ', p~[p, p,)

× Lf(°(x, p')fo')(x, p ;) - f")(x, p)f(J)(x, p 01,

(i, j = 1, 2)

(2)

(P " Pl)(P' " P[)8(4)(P + P, - P ' - P~),

( i = 1,2)

with transition rates of the f o r m

w(ii)(P ', Pi [ P, P,) = ~

4G 2

(3)

w"J>(p', p i I p, p ,) = ~

8G 2

(p • p i)

×(P~'P')St4)(p+P~-P'-Pl),

(i,j=l,2;i#j)

(4)

which follows f r o m the F e r m i - t y p e 2) or effective W e i n b e r g - S a l a m 3) Hamiltonian density H = (4~/2)-1G~/~(1 + Vs)v15~/~(1 + Vs)~',

(5)

involving the w e a k interaction constant G, the neutrino and antineutrino field operator u and Dirac matrices y" and ys. The theory m a y be linearized by writing

f(i) = f~)(1 + ~p(i)), [~) = a -3 exp ¢](i.ti - p~'Uj,),

(i = 1, 2)

(6)

with a a constant, [3 = ( k B T ) -~ the reciprocal temperature, /~i = fl-i log(8~r)-~(aClc)3ni the specific Gibbs function (where ni is the particle density of species i) and U ~ the h y d r o d y n a m i c velocity. F r o m (1-2) one then obtains the linear equations

(r - 4)p+'Xqt, + ( 8 , , - x,)[Jpt'Xlt~ -- ~](p~tp ")Xttv 2

= - n ( 8 ~ r f l c ) - ' ~ xi{I"~)[q~ (i)] + i(°)[q~°)]},

(i = 1, 2)

(7)

with r := t i p . U, the density n = nj + n2 and particle fractions x~ = nJn of species i. Pointed brackets indicate that the tensor enclosed should be s y m m e t r i z e d and made traceless. F u r t h e r m o r e the " t h e r m o d y n a m i c f o r c e s "

X~ = _(V~_+

fl Xh-n: W'p~,

X ~ = ~1v ~

x, log~,

x~=(wu~>

(8)

occur, with p = n B -~ the pressure and the operator V ~ = A"~Ov with A +'~= g " V - c - 2 U ~ ' U ~. The metric tensor is g+" = d i a g ( 1 , - 1 , - 1 , - 1 ) . The collision

TRANSPORT PROPERTIES OF A NEUTRINO-ANTINEUTRINO MIXTURE

487

expressions are defined as I"~)[~b] = (flc)4 f "-~l d3pl -d3p' d3p[ W@(p ', p ; I p, p O e-"[ ~b(p ) - ~b(p') ], - ~ "-ff~-~ ~"d3pl d3p ' d3p;

i(#)[~b] = (~c)4 j

~ lo ~

.

p ;o w t g ' ( p ' , p ; [ p, p l ) e - " [ l ~ ( p l ) - ~b(p [)l.

(9) (lO)

Both are functions of the energy-momentum p".

3. The solution

The solution of (7) is linear in the thermodynamic forces (8):

~(i) = 87r~cn-l[_Bi(.r)A~p~Xq~ _ ~Di(.r)A~p~Xi ~ + ~C'O')(p*'p ~)X~,v].

(11)

The parameters occurring here will be written as power series

s =0

bsr ,

DiO ") =

s =0

Ci(r) =

ds'r,

s =0

Cs'r.

(12)

Insertion of (11) and (12) into the equations (7) leads to equations for the coefficients, for instance for b is

-4

r(r+

& 2. x,(b

s=0 j=l

^ij j sb's+b.,bs),

(r=O,I .... )

.3)

with collision integrals

d3p za~p 7 rr(ij)r bri~ = (tic) 4 f --~-e • tP u,~.s - ,l -~--

(14)

~

and / ~ depending in the same way upon /*0. Insertion of the collision expressions (9) and (10)-with the transition rates (3) and (4) substituted- and introduction of total and relative momenta according to

P'=pt'+p~,

P'U=p''+p~,

q'*=p~'-P*t,

q'~=P''-P~

(15)

has as a result that in the expressions (14) for bi,~ and the similar/;~, integrals appear of the form:

d3p d3pl d3p , d3p; J (j, k, 1, m, n) := (~c )4 f --~- --~- --~- --~Tf u~t*" _ p,) e-ae . U x(~Pc):~(~IP • U)k(~q • U)~(~lq '. U)"(~2c2q • q')L

(16)

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W.P.H. DE BOER AND S.R. DE GROOT

T h e y involve invariant combinations of P " , q~' and q~'. Upon evaluation (v. the appendix) we find J ( j , k, I, m, n) = 2(27r)3(- I)"(2j + k + l + m + 2n + 3)!

~+p~-~0

(-1)P

× = 2p+l+m+3 t,m ~

K(l, m, n) =

i = max(0, ( n - m ) / 2 )

(j +n)K(l,m,n), p

C ( m , n, i) l + n - 2i + 1'

n!m! C ( m , n, i) = (n - 2i)!(m + n + 1)!!(m - n + 2i)!!(2i)!!'

(17)

(18)

(19)

With the help of these expressions the collision integrals (14) may be calculated and subsequently the coefficients bi~ may be solved from the equations (13). A similar procedure can be followed for the coefficients dis and c~. In the linear theory the energy flow I~ and the diffusion flow I~' are linear expressions in ~"), and thus, via formula (ll), in the thermodynamic forces (8). In this way we get the linear laws I ~ = IqqX~ + l q l X ~ ,

(20)

I~ = llqX~ + INX~,

(21)

with transport coefficients lqq for heat conduction, lql for the diffusion-thermoeffect (Dufour effect), lLq for thermal diffusion and IL~ for diffusion. They depend upon the coefficients b~s and d~s as: 41r

C

s ( s + 3)!(xlb Is+ x2b2),

(22)

c = s ( s + 3)!(xldJ~ + x2d~),

(23)

3 ~ s=0

lqqlql = ~

4~r

llq=_.~_ c ~ (s + 3)!xlx2(b I, - b,), 2

(24)

s=0

IN =

BC =

(s + 3).xLx2(d, v 1 - d 2~),

(25)

where, incidentally, lq~ and l~q should be equal as an example of the relativistic Onsager reciprocity relations. The symmetric, traceless part of the pressure tensor becomes in the linear theory /P'~ = 2 n ( W U ~) with the shear viscosity

(26)

TRANSPORT PROPERTIES OF A NEUTRINO-ANTINEUTRINO MIXTURE 47r 1 ~ - - -/3c ,=o (s + 5).(xlcs */ = -15 v 1+ x2c2).

489 (27)

The trace of the pressure tensor, and thus the volume viscosity, vanishes as always for massless particles.

4. The results

The equation (13) may be solved in successive approximations. In the first one of these one restricts oneself to terms with r as well as s equal to 0 and 1 only. Explicit evaluation of (22-25) then yields for the transport flows (20-21) with (8) (deleting V"p) inserted:

3~h4c 5 I112xtx2) Y ~ - + 2 4 ( x t - xe)V~xl], I~ = 80(97 + 12x~x2)kaTG2[(97 +

(28)

3,h'c' [ , V'T ] I~ = 80(97 + 12xlx:)(kBT)2G 2 24(xl - x2)xlx2 T + (95 - 12xlx2)V"xl . (29) A remarkable feature is that the cross-coefficient, proportional to lqt = llq, which describes the Dufour effect and thermal diffusion, vanishes if the number of neutrinos is equal to the number of antineutrinos (x~ = x2 = ½). Otherwise (that is for x~ ¢ x2) temperature gradients will enhance the concentration of the more abundant component in the relatively colder parts of the system, as is shown by the form of (29). The first approximation to the law (26--27) for viscous flow turns out to be 3 119+ 184x~x2 h4c 3 H "~ = ~ ~r 119 + 52xlx~ kBTG"~(V~U~)"

(30)

It is interesting to compare the values of the heat conductivity and the shear viscosity coefficients as they appear in the linear laws (28) and (30) for the case of equal amounts of neutrinos and antineutrinos (xl = x2 = ½) with the corresponding values for the pure neutrino gas (xl = 1, xz = 0). One may check then that both coefficients mentioned are enhanced by a factor -~. The transport coefficients for the pure neutrino system are in accordance with results obtained previously4). A succinct version of the present work has already been publishedS).

490

W.P.H. DE BOER AND S.R. DE GROOT

Appendix The p r o o f that evaluation of the integral J(j, k, l, m, n) as defined in (16), yields the result (17-19) runs along the following lines. Beforehand we note that the integral

d3p 8~4)(p f -d3P -p-rc-~f

-

n')q"q'~.

. .

q'~'~,

(A.1)

involving total and relative m o m e n t a , as introduced in (15), has the value 6)

2~'O(P)(a + l) J(-l)~/zPaA~)(a even),

0(a odd).

(A.2)

H e r e P is the length of the f o u r - v e c t o r P~' and A(,a) the symmetric tensor A~a' := .( -a - - - -1)l T 1t paws ~ A~''2A~'3"*"" A~°-'"°(a even)

(A.3)

f o r m e d out of products of p r o j e c t o r s Ae with c o m p o n e n t s

A ~ = g,,~ _ p 2p,,p~.

(A.4)

The sum in (A.3) has to be extended over all different w a y s in which the indices/~1,/z2 . . . . . /za can be divided into pairs. To evaluate (16) we start by integrating over p ' and p~, that is by evaluating [- d3p, ,~3 ,o u pl 8(4)(p J--P~--~I - P ' ) ( f l q " u)'~(~2c2q " q')~"

(A.5)

We e m p l o y the l e m m a (A.1-2), with a = m + n even, and obtain thus

2~'O(P)(a + 1)-'(-1)a12P~A~) ! (flU)m(~2c2q)L

(A.6)

The inner product of A(p~ with m vectors 13U ~" and n vectors fl2c2q~ occurring here is A(pa) i ( [ ~ U ) m ( [ ~ 2 c 2 q ) n = (~[~c) 2n

In/2] ~ D(m, i =max(O,(n-m)/2)

n, i)(q • A p • q)i

x ( q • A , • flU)~-2i(flU • Ap • f l U ) (m-n)/2+i,

(A.7)

where D ( m , n, i) is the n u m b e r of different ways to divide the set of indices /z~. . . . . p.,,, v~. . . . . l,~ into i pairs of indices (Izo,/x~), n - 2i pairs (/z~, v~) and ½(m-n)+i pairs (v~, v,), divided by the total n u m b e r (m + n - 1)![ of pairs:

n!m! D ( m , n, i) = (n - 2i)!(m + n - 1)[!(rn - n + 2i)!!(2i)!!"

(A.8)

TRANSPORT PROPERTIES OF A NEUTRINO-ANTINEUTRINO MIXTURE

491

With (A.7) and (A.8) inserted into (A.6) we find for the integral (A.5): In/2]

2 zrO(P)(-1)n(tPc) 2"

~

C(m, n, i)

/=max(0, (n-m)/2)

x (tq " U)n-2i[(tP • U) 2 - (tPc)2] tm-È)12+i

(A.9)

with coefficients

C(m, n, i) := (m + n + 1)-lD(m, n, i).

(A.10)

To obtain (A.9) we made use of the identity q • Ap • q = _ p 2

(A.11)

valid for massless particles. If one inserts (A.8) into (A.10) one finds the formula (19) for C(m, n, i). As the next step in the calculation of (16) we p e r f o r m the integration over p and p~, which involves an integral of the type

f --~ d3p ~d3pl F(P ~)(tq " U) b

(A. 12)

with F(P ~) a scalar function of P~, and where the p o w e r b is equal to (l + n - 2i). With the l e m m a - similar to (A.1-2) -

f d3p d3pl F(p~)~t~,q~2 " " q ~ -~---~1 2~r(d + 1)-1 f d4PO(P)F(PV)(-1)djEpdA~)(d even), 0(d odd),

(A.13)

the integral (A.12) b e c o m e s 2~r(d + 1)-~ f d4po(P)F(P ~)[(tP • U) 2 - ( t P c ) 2]~2.

(A. 14)

With the help of this result we obtain for (16)

J(j, k, l, m, n) = (2~')2(-1)nK(l, m, n)(tc)4 f d4po(P) e-~P "U(tPc)2~+n) X (tiP. u)k[(flp. U) 2 - (flpc)2] (t+m)/2

(A.15)

with K(l, m, n) as introduced in (18). The remaining integral is calculated by writing the vector P~ in polar coordinates: P~ = P ( c h X, I sh X).

(A.16)

H e r e I is the unit three-vector. The integration element is then d4P = d P d x d r i P 3 sh 2 X-

(A.17)

492

W.P.H. DE BOER AND S.R. DE GROOT

The integral in (A.15), multiplied by (/3c)4, becomes, if we write y for ¢]Pc: 41r ( dy dxy 2o+n~+k+t+m+3e - y c h x

chkx sh t+"+zXI

(A.18)

0

The value of this integral is

4 ¢ f ( 2 j + k + l + m + 2 n + 3 ) ! J+p=~=~02 p + (-1)P l+m+3

( j +pn ) "

(A.19)

With this result the integral (A.15) becomes equal to the form (17) to be proved.

Acknowledgements The authors wish to express their gratitude to Mr. L.J. van den Horn for his assistance in checking the results. This investigation is part of the research programme of the "Stichting voor fundamenteel onderzoek der materie (FOM)", which is financially supported by the "Nederlandse organisatie voor zuiver-wetenschappelijk onderzoek (Z.W.O.)".

References 1) W.P.H. de Boer and Ch.G. van Weert, Physica $$A (1976) 566, 86A (1977) 67. Th.J. Siskens and Ch.G. van Weert, Physica 86A (1977) 80. 2) S.R. de Groot, A.J. Kox and W.A. van Leeuwen, Physica 84A (1976) 613. 3) Th.J. Siskens and Ch.G. van Weert, Physica 89A (1977) 163. 4) W.A. van Leeuwen and S.R. de Groot, Lett. Nuovo Cimento 6 (1973) 470. W.A. van Leeuwen, P.H. Meltzer and S.R. de Groot, C.R. Acad. Sc. Paris 279B (1974) 45. S.R. de Groot, W.A. van Leeuwen and P.H. Meltzer, Nuovo Cimento 25A (1975) 229. 5) W.P.H. de Boer and S.R. de Groot, C.R. Acad. Sc. Paris B (1978). 6) Th.J. Siskens and Ch.G. van Weert, Physica 91A (1978) 303.