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A relativistic relaxation-time model for the Boltzmann equation
Physica 74 (1974) 466-488
0
North-Holland
Publishing Co.
A RELATIVISTIC RELAXATION-TIME MODEL FOR THE BOLTZMANN EQUATION* J. L. ANDERSON Department
and H. R. WITTING
of Physics, Stevens Institute of Technology,
Hoboken,
New Jersey
07030,
Received 20 November
USA
1973
Synopsis
A relaxation-time model is proposed for the collisions term of the relativistic Boltzmann equation for a single-component gas. The resultant equation is solved as a power series in the relaxation time and this solution is used to calculate the transport coefficients for the gas. These coefficients are then compared with those obtained using the relativistic Grad moment method and Marle’s version of the relativistic relaxation-time model. The results indicate that the proposed relaxation-time model gives values for the transport coefficients which functionally agree with the values obtained using the relativistic Grad moment method in the asymptotic limits.
1. Introduction. In recent years interest in the Boltzmann equation and its applications to problems in plasma physics, astrophysics, and cosmology has increased tremendously. In the areas of relativistic astrophysics and cosmology this interest has led to the formulation of the relativistic Boltzmann equation’)
and C(f) represents the phase-space density of collisions. To solve the relativistic Boltzmann equation for nonequilibrium processes approximation methods had to be developed, just as in the classical case. Studies by Chereniko?), Marle3), Stewart4), and Anderson5) led to the formulation of the relativistic Grad moment method which allows one to formally solve eq. (1) for nonequilibrium process. This method suffers, however, from the practical problem of one’s inability to evaluate the resulting integrals (even numerically) unless he restricts himself to expansions about the Maxwell-Boltzmann distribution. This same problem exists with the classical Grad moment method and can be overcome by use of the BGK (Bhatnagar, Gross, and Krook) relaxation-time * This research was supported
by National Research Foundation 466
Grant GP-3462 I
RELATIVISTIC
RELAXATION-TIME
MODEL
467
modeF). The BGK model allows one to effectively handle distributions other than the Maxwell-Boltzmann and is relatively much easier to deal with than the Grad moment method, albeit one must pay the price of less accuracy. To handle distributions other than Maxwell-Boltzmann in the relativistic case it seems natural to develop a relativistic BGK model. Marle formulated such a model in his 1969 paper with C(f) = --m U -
f0)lr ,
(2)
were r is the collision time, m the mass,f, the equilibrium distribution andf the distribution itself. In the classical limit Marle’s formulation of the relativistic BGK model gives the same results for the transport coefficients as the classical BGK model. However, in the extreme relativistic limit the results for the transport coefficients with Marle’s formulation differ functionally from the results one calculates with the relativistic Grad moment method. In this paper we propose a relativistic BGK model with
ccf>= - %P’ (f
- fo)/t >
(3)
where U, is the mean four-velocity of the gas. Calculations with this formulation will give the same classical results as Marle’s formulation since u,p’ --t m in the classical limit. However, in the extreme relativistic limit our formulation gives results for the transport coefficients which are functionally the same as those calculated z;iu the relativistic Grad moment method. The explicit purpose of this paper is to calculate the transport coefficients with our proposed relativistic BGK model and compare the results with those obtained from the Marle formulation of the relativistic BGK model and from the relativistic Grad moment method. The results of these calculations are shown in figs. 1, 2 and 3.
Fig. 1. Plot of shear viscosity as a function of y as calculated by the Grad moment method (qo), the Marle relaxation-time model (qhl) and our relaxation-time model (7). Fig. 2. Plot of heat conduction as a function of y as calculated by the Grad moment method (A,), the Marle relaxation-time model (&) and our relaxation-time model (3.).
468
J. L. ANDERSON
AND H. R. WITTING
Fig. 3. Plot of bulk viscosity as a function of 7 as calculated by the Grad moment method the Marie relaxation-time model (CM) and our relaxation-time model (5).
2. Relativistic BGK model and the transport coef$cients.
The relativistic Boltzmann equation with our proposed relativistic BGK model is = -Z.&p* L.f
L(f)
fo
-
z
In Minkowski space-time this equation reduces to
p’$
= _U,P’f_ 7.
so that, in the local rest frame of the gas [u’ = (1, 0, 0, 0)] and when there is no spatial variation in f, we have
po - af =
_o
a.9
f-.fo
.
T
The PO’Scancel and we get the solution f(t)
=fo + [f(t
= 0) -So] eet”.
Thus t is seen to play the role of a relaxation time. In Marle’s formulation of the relativistic BGK model on the other hand the above assumptions lead to the equation
af
PO j-g
=
_-)11
f-fo. z
RELATIVISTIC
The
RELAXATION-TIME
MODEL
469
p” does not cancel now and we get a solution f(t) =
fo + [f(t
= 0) -
fo] e-mt’por.
If we assume, as Marle does, that z depends only on the macroscopic properties of the gas, then we see that Marle’s model leads to an energy-dependent relaxation time. We also see that Marle’s equation cannot be applied directly to the case of a photon gas (m = 0) while our equation is seen to be the relativistic version of the relaxation-time approximation of the equation of radiative transfer. It was these facts that led us to our formulation of the relativistic BGK model. In this paper we wish to determine approximate expressions for the transport coefficients of a gas whose distribution function obeys eq. (4). These coefficients can be obtained from knowledge of the energy-momentum tensor Tab and the particle-number flux vector N*, given respectively by
Tab= sp”p*fn,
(5)
N* = J-pbf?d,
(6)
and
where f is a solution of eq. (4) and z is the four-dimensional momentum-space volume element. We require that Tab and N* be conserved in our gas
T"** = 0,
(7)
hfb = 0,
(8)
as indeed they must in any gas satisfying the full relativistic Boltzmann equation. The consequence of requiring these conservation equations is determined by use of the lemma’)
(9) where t represents the product of n pa’s.Using this lemma and eq. (4) we get
@(f)n
= (_,-pbfn>;, = Nbb= -Ub(Nb - N;)/t,
j-P*L(f >?‘t = (j&‘ff..) ;
b
=
T""b
=
--u, (Tab - T;*)/t ,
(10)
(11)
where Noband T,O*are respectively the first and second moments of the equilibrium distribution f. (see appendix A). To insure that the conservation equations are
470
J.L. ANDERSON
satisfied
AND H.R. WITTING
the conditions
ubNb = u bNb0,
(12)
ubTab = ubTGb,
(13)
must therefore be imposed. These conditions are the ones used by Landau and Lifschitz in their phenomenological treatment of transport processes in a relativistic gas to fix the local rest frame of the gas and to define the local temperature and density. Imposing the Landau matching condition insures that the conservation equations (7) and (8) will be satisfied, but also means that we will obtain expressions for the particle-number flux vector (first moment off) and the energy-momentum tensor (second moment off) in a Landau form of decompositions): N” = nLui - Al, Tub = (,uL + pL)
(14)
&A; - pLgub - Ph.
(15)
In identifying the transport coefficients it is somewhat more convenient expressions for N” and Tab in terms of the Eckart decomposition9): N” = n,uE,
The relationship between the two representations where it is shown that to first order in deviations equilibrium values n
to have
(16)
is established in appendix B of Tab and N” from their local
=
nL,
(18)
iuE
=
iuL>
(19)
PE
=
PL>
(20)
E
Having thus obtained expressions for the energy-momentum tensor and the particle-number flux vector we can identify the phenomenological transport coefficients with the elements of the Eckart decomposition. Eckart9), as well as other authors, has shown that the relationship between his form of the decomposition
RELATIVISTIC
RELAXATION-TIME
MODEL
of the energy-momentum tensor and the phenomenological is given by the following:
471
transport coefficients
1) Coefficient of shear viscosity 7 tab = -2?j@,
(22)
where Oab ~
@;b)
_ 3(jhab _
CC”&
(23)
and oabUb = # (I = 0
e = u;;,,
9
h”b = g’b -
gUb,
(24)
tia= u;bub.
2) Coefficient of heat conduction 1 qU= -lhab[:-(b);J,
where
y=g.
(25)
3) Coefficient of bulk viscosity 5 PK - PE =
-ye,
(26)
where pE is the thermodynamic pressure and pK is the kinetic pressure. This difference in pressure is calculated in terms of the energy-momentum tensors as pK - pE = -4 (T: - T&j. 3. Evaluation of the$rst and second moments. There are a number of methods available to us for constructing approximate solutions of eq. (4). We might, for example, use a relativistic version of the Grad moment method as developed by Marle, Anderson and Stewart. However, the form of eq. (4) suggests an alternate procedure, namely an expansion of the distribution function in a power series in t in which all but the first few terms are assumed to be sufficiently small that they can be set equal to zero. Such an approximation is widely used in the theory of radiative transfer where it goes by the name of the optically thick approximation and has been used by Marle to find approximate solutions to his relaxation-time equation. Let us then substitute the expansion f =fo + tf1 + Z”fi + ***,
(27)
412
J.L. ANDERSON
AND H.R. WITTING
into eq. (4) and equate to zero the coefficients of the various powers oft. In this way we get a set of equations which allows us to calculatef, in terms Of fi1:
afi-1
+ rycpbpC -
axa
‘”
afi-1
- UrP’f I >
=
apa
For our present calculation we will expand about the Maxwell-Boltzmann bution7) f.
= ew%P~,
distri-
(2%
in Minkowski space-time and keep terms to first order in T. The distribution function is, therefore, f = fo + Zfl 2
(30)
where fi
=
-(l/urpr)pd
b
;d -
(b’b)
;dP=lfO*
The first and second moments offare first moment N” is
(31)
now calculated using eqs. (5) and (6). The
(32) where Ng = n,u”
(33)
and N,“=
s
p4pd[ _a
;d + (bc)
;dpClfOn
%P’
(34)
RELATIVISTIC
RELAXATION-TIME
MODEL
473
Likewise the second moment can be written as Tab = Tlb + tTib,
(35)
where Tib = (p,, + PO) U”Ub- Poi?‘”
(36)
and
Tfb=
--LX
;*
s
s
(37)
Since u,p’ is a scalar we can define three tensors Tad = J (PaPdlurPr)fon,
(38)
Sac* = J (PaPcPdl%P’)fo~,
(39)
QnbCd= f (PaPbPcPdlurPr)fo~ 9
(40)
which depend only on the equilibrium distribution and must, therefore, have decompositions of the form T’* = TIu’ud - T&*,
(41)
s”” = S~U”U=U* - s, (u”g’d + u=g”*+ ud$c),
(42)
Q abe*=
Q1&&U*
+ gwu=
-
Q2(g”bucud + gacubud
+ g%v
+ gb*U’Uf + g%“ub)
+ 93 (gbgcd + g”‘g”” + g”*g”‘).
(43)
To determine the coefficients of these decompositions we equate their scalar contractions with suitable combinations of u, and g,b to the corresponding contractions of the integrals and solve the resulting equations for the coefficients. Doing this we find that T1 = 3 [4no - 4xm3 en (Kl - Kil)],
(44)
T, = 3 [no - 45cm3 ea (ICI - K,,)],
(45)
S, = 2~, - m2A,,
(46)
474
J. L. ANDERSON
AND H. R. WITTING
(47)
Q1 = 3 [16S, - 48S2 - 12m2n,, f 4ii e.5tns (K, - K,,)],
(48)
Q2 = & [6S, - 18S2 - 7~7~~2,+ 4~ ea ms
(49)
(K,
-
K,,)],
Q3 = & [S, - 3S, - 2m2n,, + 4x e” FH~(KI - K,,)], where K, is the x dt ‘O), and moment of the ing expressions
(50)
nth-order modified Bessel function of the second kind, K,,, = J‘8K,(t) S1 and S, are the coefficients of the decomposition of the third equilibrium distribution (see appendix A). We now have the followfor Nf and Tfb
N,” = -LX ;dTBd + @u,) ;d Sacd,
(51)
Tfb = -T ;dSabd+ (/h,) ;d Qabcd.
(52)
To express N” in a form compatable with the Landau decomposition use of the definitions
we make
(53) (54) (55) (56) in eq. (51) to get N,” = [@(S, - 2S2) - T,& - S@]
zP - S,p ;’ - ,&ti”
Rewriting this in terms of the dimensionless parameter y =
m@,
+ Tza “.
(57)
where (58) (59)
we get N; =
C
- T,a - -?- S28 m
+s,yz_A_ r m
0 Y
1
U”
;’
-
m
S2ti” + T20c ;a.
(60)
RELATIVISTIC RELAXATION-TIME
By making the approximation [see eq. (C.7), appendix C] &;a
=
y2- (2S2
mPo
mpo
+ PO + PO g PO
475
Ttbi b % 0 we obtain the following expression for 01;a
ys,
,& -
[
MODEL
-
0L + 0_!_ .
S,)
&J + POfj p PO
Y
y2
1
;a
s2
“PO
(61)
Y
Substituting this into eq. (60) we obtain
-T2p(1-3+-f&
+
(3s,
-
s,)
,3s2-sl,(~j-~s2ep
0-!- - L s,e - 1 Y
no&
m
24’.
(62)
By eqs. (A-23) and (C.8) the second to the last term in the above equations is zero and in appendix D we prove that the last term is zero. We therefore arrive at an expression for the first moment
~~=~~+~~(~_s2)hob[~_(t>;~,
(63)
which has the Landau form of the decomposition. To express Tab in a form compatible with the Landau decomposition we first perform the differentiation in the coefficient of Qabedin eq. (52) then use eq. (23) to obtain T;b = --oc;dsabd +
B;dt& + ,%,d + f-
hcd + /%i&
pbcd.
(64)
> Using the decompositions in eqs. (42) and (43) we obtain T,Ob= [b (Qi-
50,+ 29,)- k:(S, - 2s~)
-
2/9’3(02 +
3Qd1uaub
- [b (Qz - Qd - kS2 - $Qdel fb - 2 (Q2 - Q3) (@‘” + ,6 ;@ - /@“) ub) + 2s201 ;@zF - 2S2&‘ub + 2~Q3u”OL,
(6%
J. L. ANDERSON AND H. R. WITTING
476
upon inserting the quantity [2&Szuaub + 2 (Q2 - Q3) /haub - 2bS2z.hb - 2 (Q2 - Q3) ~u‘k”]. In appendix D it is shown by use of the approximation
(66)
7,‘; b z 0 that
(Q, - Q3) (/3ti” + /3 ;= - /3’zP)+ S2&zP - .Szn ;’ = 0.
(67)
We can now write the expression for the second moment as Tab = {duo+ po + z [B (91 - 592 + 203) - dr(S, - 2%) - 2F’ (92 + SQ,>l> uaub - {PO + r Lb (92 - 93) - 6
- $Q,P’l) gab + ~$QN’~.
(68)
This expression can be further simplified by using the matching condition ubPb = UbGb t0 obtain the rehtiOn P
(9, - 69, + 303) - k (S, - 3%) - PC202
- Qd = 0,
(69)
which, when subtracted from the coefficient of the U”U~term in Tab, gives an expression which takes the form Tab =
+ 2t 2 Q,db, m
(70)
upon use of eqs. (58) and (59). This expression has the desired Landau form of the decomposition for the second moment. 4. Transport coef$cients. From the expressions for the first and second moments derived in the last section [eqs. (63) and (70)] wecandetermine the transport coefficients using the Eckart form of the decompositions and their relation to the Landau form as given in section 2, with the result shear viscosity
7 = t (y/m) QJ,
(71)
RELATIVISTIC RELAXATION-TIME
heat conduction
I = z IuO+ PO --(y y2 -
MODEL
477
%),
(72)
n0
bulk viscosity
c = -4 e
(92
-
93)
$
1
.+ ctS,+ ;Q,y 8 . m
0
(73)
Putting the above expressions for the transport coefficient explicitly in terms of K,, K,, ,a, and the enthalpy h 3 KS/K2 we get (74)
il=
T4xe”m4h
c=
-t4xeam4
Y
> Y2 1
h s+Kil-Kl [(
-35,
(75)
K2 y2h’ + yh KS 1 -y2 y2h’ + 1 y y2h’ + 1
C
(76)
We would like to compare these results for the transport coefficients with those arrived at via the relativistic Grad moment method’). Calculation of the transport coefficients with the relativistic Grad moment method gives
qc =
$ $ K:(Y) 4K4 (2~)
+ 2K3 (2~)
---- K2 (2~)
Y2
Y
Kl(2r)
Y
Y2
-’ >
, (77)
_ K4(y)
2
+
’ 2
4K4 (2~) -
+K3 (2~) + 2K3 (2y) --Y
cc= _$$(3r_4) -
(35-r
K,(y) - $ 1
x [K3 (2~) -
Kl (2y)l-’
,
- $)K+Kd,,
K3(y) -
>
-
(78)
- $j
-
(79)
478
J.L. ANDERSON AND H.R. WITTING
where J’is the ratio of specific heats and is given in terms of the enthalpy by
r = yvz’/(y%
+ 1).
(80)
We would also like to compare our results in the relativistic and extremerelativistic regions with the transport coefficients calculated uia Marie’s relativistic BGK mode13). The expressions for the transport coefficients calculated via Marle’s formulation are ?jM =
(81)
z4xe"m4 (K3/y2),
A, = z4xeam4 [K4 - (K,/y) CM = t4xe”m4
-
(Kz/K,)],
[y’hf+ l(
Y
K4(F--
(82)
y
KS (r - 1) - K”(12F
yh’ + h K4(FyV2 + 1 (
+1.
-
l)-
1) - K”(9C y
Y
Qlr--)
- 13) + 5
y2
)
(9r - IO)
10)
>I.
> (83)
In order to compare the result of the various methods we will express Z, the collision time, in terms of the cross section cr. Using the results of appendix E we get r = [Y2/U + Y2P (~/ml
(u>>,
(84)
where y in the above equation represents (1 - v*/c*)-~. We wish to compare the results in the classical limit, the extreme-relativistic limit, and in the relativistic region. In the classical limit y 9 1 (y = mc*/kT) and we use the asymptotic expansion K,(y)=
($)Le+(l+++
‘“-2~~~2-9’
+ (P - 1) (P - 9) (P - 25) + ... > 3!83y3 >
(85)
where p = 4n2 for K,,(y). An asymptotic expansion for K,, is obtained from the asymptotic expansion of K,(y) using K,, =
7K,,(z) dz.
Y
(86)
RELATIVISTIC
The integrals encountered the result
K,,(y)=
RELAXATION-TIME
MODEL
can be performed via the Whittaker functionls)
479 with
(a)‘emy(l--&+J&-~ +
903105
163451925 +
98304~~ -
3932160~~
(87)
Inserting the various expansions into the expressions for the transport coefficients we get the following results in the classical limit, y B 1 11= 0.25Om
2. ’
1+ 3
0 O(Y
+ .. . ,
8Y
(88)
(89)
(90)
1
7123 +
(91)
- =G 3
~=0.207-!?
X v2
& = 0.273 $
2. ’ (1 + e-e). 0Y
In the extreme-relativistic K(Y)
= 4 (n -
(92)
O(Y
(93)
limit y < 1 and we use
I)! (2/Y)
and
K,, --) 0.
(94)
Inserting the various expressions into the expressions for the transport coefficients we get the following results in the extreme-relativistic limit, y 6 1 7 = 0.800 (m/a) (l/y) (1 - 0.16y2 +
l j,
?jG = 0.800 (m/U) (l/y) (1 + y/50 + ***),
(95) (96)
480
J.L. ANDERSON AND H.R. WITTING Q,, =
(4m/qJ) (1 - 0.08~~ + ..a),
(97)
1 = 1.33 (m/o) (1 - 2.08~~ + a..),
(98)
A, = 1.33 (m/a) (1 + y/18 + ..a),
(99)
1, = (4m/q) (1 - 0.08y2 + **a),
(100)
c = 0.0556 (m/c) y3 (1 - 0.08~~ + a..),
(101)
to = 0.091 (m/o) y3 (1 + ;ty’ + *a*),
(102)
&, = 0.296 (m/a) y2 (1 + 0.42~’ + ..a).
(103)
The values of the coefficients in the relativistic regime were evaluated numerically and the results are presented in figs. 1, 2 and 3. The actual transport coefficients are not plotted, but rather the o/m coefficient. The results shown in figs. I,2 and 3 confirm our formulation of the relativistic BGK model and can be used as an indicator of the accuracy one might expect in various regions when the model is applied to distributions other than the Maxwell-Boltzmann distribution.
APPENDIX A
The moments of a distribution function fare
44
= jfn,
N”(x) = j”PYk
defined as follows:
T"*(x)
= fp”pbfn, (A-1)
P’(X)
= Jp”pbp”fn,
@*‘d(X) = Jp=p*p=pdfn;
where YZ= 2H (p) 6 (pop” - m’) (-g)*
H(P) =
d4p,
1 P>O, 0
p
Choosing as a representation p” = mchx,
of the momenta
p’ = m sh x sin 6 cos 4,
(A.21 p2 =mshxsinesin4,
p3 = mshxcost3,
RELATIVISTIC
RELAXATION-TIME
MODEL
481
we find
n = U/PO) 6 (PO- mchx)dp”m3sh2xchxdxsin6d8dr$.
(A.3)
Iffis a locally dynamically symmetric distribution the various moments have the following decompositions : Ao
=
Ao,
gac= S~zl’UbU~Qo
abed =
Gb= (PO+ PO) u’ub-
= n,u”,
N:
s, (ZPgbC+
PO$~,
28-p + ZJCgOb) ,
QluaubuCud - Q2 (fbucud
+ g“=ubud + gUdubuc
+ gb=Uaud+ g%Pu= + g%=d) + Q3 (fbgcd
+ g”‘g’“’ + gOdgbc).
(A-4)
To determine the coefficients in the above decomposition we perform the following contractions u,N: = no,
(A.5)
W&ib = PO,
64.6)
7% = rue - 3~0,
(A.7)
U,,UbU,S;bc
u&s;; =
=
S,
s,
-
-
3s,,
G4.8)
6S2,
U&,U,UdQ;bCd = QI
(A.9
-
6Q2
+
3Q3,
(A. 10)
U&QtbZ = Q, - 9Q2 + 6Q3,
(A. 11)
Q oat
(A.12)
al?
-
Q, - 12Qz + 24Q3.
The above equation can be solved for the coefficients of the decomposition upon evaluating the various moments for a particular locally dynamically symmetric distribution. In this paper we are concerned with the Maxwell-Boltzmann distribution f.
= ea-8%o”*
(A.13)
482
J. L. ANDERSON
AND H. R. WITTING
Performing the required integrations and solving for the coefficients one finds A0 = 4xm2 ea Kl Jy ,
(A.14)
no = 41cm3 ea
(A.15)
PO =
K,ly,
4xm4ea [(K31y)
- (K,/Y~)I >
(A. 16)
p. = 4xm4 ea K21y2,
(A.17)
S, = 4xm5 ea K4/y,
(A.18)
S, = 4xm5 ex K3/y2,
(A.19)
Q, = 4xm6 ea KJy,
(A.20)
Q2 = 4xm6 e” K4/y2,
(A.21)
Q3 = 4xm6 en K3/y3,
(A.22)
where K, is the nth-order modified Bessel function of the second kind and y =mc2/ kT. Many useful expressions can be obtained from the above coefficients, among them 1u0 + PO =
n0
=
(A.23)
(y/m>S2,
(A.24)
(y/m)PO. APPENDIX
The decomposition
B
of the first two moments offin
the Eckart form are
N” = n&,
(B.1)
Tab = (,uE + pE) I.&; - pEgab + 292 z&’ - tab,
(B.2)
while the decompositions N” = n&
in the Landau form are
- ;I:,
Tab = (,uL + pL) u;z&’ - pLgab - Pb.
(B-3) (B.4)
RELATIVISTIC
RELAXATION-TIME
MODEL
483
The difference between the two representations lies in the choice of a four-velocity. In the Eckart form one defines the four-velocity in terms of the mass-flux density whereas in the Landau form one defines the four-velocity in terms of the energyflux density. When there is a component of flux (e.g. heat flux) other than mass the two definitions differ, but can be related by
where LY“represents the components of flux other than mass. This component is nonzero only when a nonequilibrium situation exists and reduces to zero when equilibrium is reached. Since we are interested in distributions only slightly off equilibrium, LY’will be a small term of first-order deviation from equilibrium. In our calculations we will be interested only in first-order corrections and will drop higher-order corrections such as c&~. Using eq. (B.5) we obtain
U&
= (24; + oc”)UEn= 1.
(B.7)
We can now determine the relationship between the two forms. Starting with the Landau form of the decomposition for the first moment N” = nLu; - 1: =
nL
(24: + 0~“) - Af, = nLuz + nLa” - it,
P.8)
by requiring
we obtain Na = nLuE,
(B.lO)
which is the Eckart form with nE = nL.
(B.ll)
Now looking at the Landau form for the second moment
Tab= (,uL -t- pL) u$; - pLgab - T=~
=(,,+,,@+~)(u:+$)
-PPLP--b
J.L. ANDERSON AND H.R. WITTING
484
z.& = @L + PL) && + -+-
nL
nL
= (/JL + PL)
z& -
UbT EL
pLg”b +
_
pL8ab
_
tab
>
2‘“;: pL)@4;) -
tab,
(B.12)
which is the Eckart form. Taking the scalar contractions of the two representations we get (B.13)
T:: = /.~r.- 3p, T::
=
PE
-
3~9,I
*PL
=
(B. 14)
PE*
Comparing eq. (B.12) with the Eckart form we obtain (B.15) APPENDIX C
Expressions for oiand (l/y)* are obtained by simultaneously solving the equations derived from the approximation Tib; b NN0 and A$ ; II z 0. From the first condition Tib; b = I(,‘& + PO)’ + c/b + PO) 01 u” + t,% + PO) ~2”- PO:=> but (,%I +Pd;b
=
(4nm4
ea
; b
&/Y)
= 4i7m4ea
=
4xm4 ea
- 2 m
and
=
4xm4 ea
m
(2& - S,)
L 0Y
;b
cc.11
RELATIVISTIC
RELAXATION-TIME
MODEL
485
where use has been made of quantities defined in appendix A and the relations
dK.(Y) _ -6 (19 - Kn+l(l/), dY
(C.4)
Y
Kn+l(r) = K - I(Y) + (WY) K.(Y)-
(C.5)
We, therefore, get
1
- 2 (2Sz - S,) l_ m 0Y
TGb;b = +
+ (/Al) + PO) 8
u”
cU0+ PO)9 - PO& ;’ - (y”/m) S2(l/y) ;‘,
(C.6)
so a;a =
(2S2 - S,)
_L 0
+po+po
p
Y”S2
--
PO
-1
. +
Cl0
Y ;a
+
po
PO
I3
1 24”
.
(C.7)
mPo 0 Y
Contracting the above expression with U, we obtain the first equation
(l-25 >
Y2 (3& -S,) mpo
&:=-_
mpo
1 + Po +po ( Y >’ PO
8.
(C.8)
From the second condition Nl ;a x 0 N”0;a
--
tie+ noe,
(C-9)
but noGa
=
(4xm3 e"K2/y)
;a
=
4xm3 ea
a
1 ;cr
+
-
Kzia+ K,
Y ;a-(yK3-G)
01 =noci _i
,
L
01Y
;a
(C.10)
Y ;’
tie =
47cm3ea
K2
-&
Y
(C.11)
J.L. ANDERSON
486 We, therefore,
iv-0 ;D
H.R.
WITTING
get 0 = n,k - (y”/m)
=
AND
(l/l’)’
/lo
(C.12)
+ n$,
for the second equation. Using the definition into the form (yh jl +
of enthalpy,
h E K,/rCr,, eqs. (C.8) and (C. 12) can be put
1) k + y2 (y + 3h) (l/Y)’ + $0
y2 (h - l/y)(I/y)*
Solving these equations
(C.13)
= 0,
(C. 14)
+ e = 0.
first for (l/v)’
we get
butll)
SO
tw = WV)MY2h
+ 111.
(C.15)
For k we get d: =
-(jm
+ yh)/(y2h’ + 1) 0.
(C.16)
APPENDIX
D
1. ProofoJ’(Q2 - Q3) (/W + /3 ;a - $zP) + S,c?u” we get (Q2 - Q3) = S,. By contracting Tib we get
T,“, = ,uo - 3p, = m2Ao, Using the above expressions
(92 -
p. = 3C1uo- mZAo) = S2.
and eqs. (C.7)
(58) and (59) we get
Q,) (/‘&” + /? Go- @u”)+ s,k’u”
+ p&P
- -ys2 &”
m
Szb ;’ = 0. From
+
f
(2s
m
2
s#
-
-
s
1
)
”
L .p 0Y
section
3
(D.1)
RELATIVISTIC
-
RELAXATION-TIME
(PO+ PO)fh” - (PO+ PO)ZP+ f
s2
MODEL
487
;”
J_
0
_~o,(l-~)~+~(3s2-s~( e-p=0 (D-2) by eq. (C.8) of appendix C. 2. Proofof(y/m) (3SZ - S,) (l/y)* - S, (r/m) 13- no& = 0. From section 3 we S, = ,uo. Using this we get get 3SZ 1 * J- (3s 2 - S1) - s,h m m 0Y 2
= --ok+~po
m
0
1 .
- n,ci
-~oe+noe-s2-?i-e=0,
Y
m
(D.3)
by eqs. (C.12), (A.24) and (D.l).
APPENDIX
E
The relation between the collision time r and the cross section u is
where (0) is the average velocity and
(p/m)(1 + p"/m')+
W)
and f. normalized is f. = (1/4xm2) (y/K,) eWychx. Performing the above calculation we get
= WG)
WY + W2) fP -
(P-3)
M41~
where E,(y) is the exponential integral E,(y) = y (eaYS/S) dS.
(E.4)
1
The relative velocity of two particles is12) V= [ur +(1/--
l)(Y,.u/~2)Y-~yY][l/y(l
-U1*U)l,
(E.5)
488
J. L. ANDERSON
AND H. R. WITTING
where y is used to signify (1 - vz/c’)-’ in the remainder of this appendix. In calculating the average relative velocity the term u1 . v -+ 0 sincef, is an equilibrium distribution so
v + (VI -
ye/y
7
v = (V’ v)+ =
[(VI + y’v’)/yqf.
Since we are interested in a gas of similar particles close to equilibrium (~7~) w (0) and we get
(V> = (v>.u
+ Y2)lY213.
(Jw
We therefore have
-r = W% )lY'/(l
+ Y’P.
(E. 7)
In the classical limit y + 1 and t -+ l/2* an, (v). In the extreme-relativistic z + l/on,
w3)
limit y -+ co
.
(E-9) REFERENCES
1) Ehlers, J., in Proceedings Varenna Summer School on Relativistic Astrophysics 1969, Academic Press (New York, 1971). 2) Chernikov, N.A., Acta Physica Polonica 23 (1963) 629; Physics Letters 5 (1963) 115. 3) Marle, C., Ann. Inst. H. Poincare, 10 (1969) 67, 127. 4) Stewart, J.M., MNRAS 145 (1969) 347. 5) Anderson, J.L., in Proceedings of the Midwest Conference on Relativity, Plenum Press (1970). 6) Bhatnagar, D., Gross, E. and Krook, M., Phys. Rev. 94 (1954) 511. 7) Stewart, J. M., Non-Equilibrium Relativistic Kinetic Theory, Lecture Notes in Physics Vol. 10, Springer-Verlag (Berlin, 1971). 8) Landau, L.D. and Lifschitz, E.M., Fluid Mechanics, Pergamon Press (New York, 1959) p. 505. 9) Eckart, C., Phys. Rev. 58 (1940) 919. 10) Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover (New York, 1965). 11) Israel, W., J. math. Phys. 4 (1963) 1163. 12) Anderson, J.L., Principles of Relativity Physics, ch. 6, Academic Press (New York, 1967).
0
North-Holland
Publishing Co.
A RELATIVISTIC RELAXATION-TIME MODEL FOR THE BOLTZMANN EQUATION* J. L. ANDERSON Department
and H. R. WITTING
of Physics, Stevens Institute of Technology,
Hoboken,
New Jersey
07030,
Received 20 November
USA
1973
Synopsis
A relaxation-time model is proposed for the collisions term of the relativistic Boltzmann equation for a single-component gas. The resultant equation is solved as a power series in the relaxation time and this solution is used to calculate the transport coefficients for the gas. These coefficients are then compared with those obtained using the relativistic Grad moment method and Marle’s version of the relativistic relaxation-time model. The results indicate that the proposed relaxation-time model gives values for the transport coefficients which functionally agree with the values obtained using the relativistic Grad moment method in the asymptotic limits.
1. Introduction. In recent years interest in the Boltzmann equation and its applications to problems in plasma physics, astrophysics, and cosmology has increased tremendously. In the areas of relativistic astrophysics and cosmology this interest has led to the formulation of the relativistic Boltzmann equation’)
and C(f) represents the phase-space density of collisions. To solve the relativistic Boltzmann equation for nonequilibrium processes approximation methods had to be developed, just as in the classical case. Studies by Chereniko?), Marle3), Stewart4), and Anderson5) led to the formulation of the relativistic Grad moment method which allows one to formally solve eq. (1) for nonequilibrium process. This method suffers, however, from the practical problem of one’s inability to evaluate the resulting integrals (even numerically) unless he restricts himself to expansions about the Maxwell-Boltzmann distribution. This same problem exists with the classical Grad moment method and can be overcome by use of the BGK (Bhatnagar, Gross, and Krook) relaxation-time * This research was supported
by National Research Foundation 466
Grant GP-3462 I
RELATIVISTIC
RELAXATION-TIME
MODEL
467
modeF). The BGK model allows one to effectively handle distributions other than the Maxwell-Boltzmann and is relatively much easier to deal with than the Grad moment method, albeit one must pay the price of less accuracy. To handle distributions other than Maxwell-Boltzmann in the relativistic case it seems natural to develop a relativistic BGK model. Marle formulated such a model in his 1969 paper with C(f) = --m U -
f0)lr ,
(2)
were r is the collision time, m the mass,f, the equilibrium distribution andf the distribution itself. In the classical limit Marle’s formulation of the relativistic BGK model gives the same results for the transport coefficients as the classical BGK model. However, in the extreme relativistic limit the results for the transport coefficients with Marle’s formulation differ functionally from the results one calculates with the relativistic Grad moment method. In this paper we propose a relativistic BGK model with
ccf>= - %P’ (f
- fo)/t >
(3)
where U, is the mean four-velocity of the gas. Calculations with this formulation will give the same classical results as Marle’s formulation since u,p’ --t m in the classical limit. However, in the extreme relativistic limit our formulation gives results for the transport coefficients which are functionally the same as those calculated z;iu the relativistic Grad moment method. The explicit purpose of this paper is to calculate the transport coefficients with our proposed relativistic BGK model and compare the results with those obtained from the Marle formulation of the relativistic BGK model and from the relativistic Grad moment method. The results of these calculations are shown in figs. 1, 2 and 3.
Fig. 1. Plot of shear viscosity as a function of y as calculated by the Grad moment method (qo), the Marle relaxation-time model (qhl) and our relaxation-time model (7). Fig. 2. Plot of heat conduction as a function of y as calculated by the Grad moment method (A,), the Marle relaxation-time model (&) and our relaxation-time model (3.).
468
J. L. ANDERSON
AND H. R. WITTING
Fig. 3. Plot of bulk viscosity as a function of 7 as calculated by the Grad moment method the Marie relaxation-time model (CM) and our relaxation-time model (5).
2. Relativistic BGK model and the transport coef$cients.
The relativistic Boltzmann equation with our proposed relativistic BGK model is = -Z.&p* L.f
L(f)
fo
-
z
In Minkowski space-time this equation reduces to
p’$
= _U,P’f_ 7.
so that, in the local rest frame of the gas [u’ = (1, 0, 0, 0)] and when there is no spatial variation in f, we have
po - af =
_o
a.9
f-.fo
.
T
The PO’Scancel and we get the solution f(t)
=fo + [f(t
= 0) -So] eet”.
Thus t is seen to play the role of a relaxation time. In Marle’s formulation of the relativistic BGK model on the other hand the above assumptions lead to the equation
af
PO j-g
=
_-)11
f-fo. z
RELATIVISTIC
The
RELAXATION-TIME
MODEL
469
p” does not cancel now and we get a solution f(t) =
fo + [f(t
= 0) -
fo] e-mt’por.
If we assume, as Marle does, that z depends only on the macroscopic properties of the gas, then we see that Marle’s model leads to an energy-dependent relaxation time. We also see that Marle’s equation cannot be applied directly to the case of a photon gas (m = 0) while our equation is seen to be the relativistic version of the relaxation-time approximation of the equation of radiative transfer. It was these facts that led us to our formulation of the relativistic BGK model. In this paper we wish to determine approximate expressions for the transport coefficients of a gas whose distribution function obeys eq. (4). These coefficients can be obtained from knowledge of the energy-momentum tensor Tab and the particle-number flux vector N*, given respectively by
Tab= sp”p*fn,
(5)
N* = J-pbf?d,
(6)
and
where f is a solution of eq. (4) and z is the four-dimensional momentum-space volume element. We require that Tab and N* be conserved in our gas
T"** = 0,
(7)
hfb = 0,
(8)
as indeed they must in any gas satisfying the full relativistic Boltzmann equation. The consequence of requiring these conservation equations is determined by use of the lemma’)
(9) where t represents the product of n pa’s.Using this lemma and eq. (4) we get
@(f)n
= (_,-pbfn>;, = Nbb= -Ub(Nb - N;)/t,
j-P*L(f >?‘t = (j&‘ff..) ;
b
=
T""b
=
--u, (Tab - T;*)/t ,
(10)
(11)
where Noband T,O*are respectively the first and second moments of the equilibrium distribution f. (see appendix A). To insure that the conservation equations are
470
J.L. ANDERSON
satisfied
AND H.R. WITTING
the conditions
ubNb = u bNb0,
(12)
ubTab = ubTGb,
(13)
must therefore be imposed. These conditions are the ones used by Landau and Lifschitz in their phenomenological treatment of transport processes in a relativistic gas to fix the local rest frame of the gas and to define the local temperature and density. Imposing the Landau matching condition insures that the conservation equations (7) and (8) will be satisfied, but also means that we will obtain expressions for the particle-number flux vector (first moment off) and the energy-momentum tensor (second moment off) in a Landau form of decompositions): N” = nLui - Al, Tub = (,uL + pL)
(14)
&A; - pLgub - Ph.
(15)
In identifying the transport coefficients it is somewhat more convenient expressions for N” and Tab in terms of the Eckart decomposition9): N” = n,uE,
The relationship between the two representations where it is shown that to first order in deviations equilibrium values n
to have
(16)
is established in appendix B of Tab and N” from their local
=
nL,
(18)
iuE
=
iuL>
(19)
PE
=
PL>
(20)
E
Having thus obtained expressions for the energy-momentum tensor and the particle-number flux vector we can identify the phenomenological transport coefficients with the elements of the Eckart decomposition. Eckart9), as well as other authors, has shown that the relationship between his form of the decomposition
RELATIVISTIC
RELAXATION-TIME
MODEL
of the energy-momentum tensor and the phenomenological is given by the following:
471
transport coefficients
1) Coefficient of shear viscosity 7 tab = -2?j@,
(22)
where Oab ~
@;b)
_ 3(jhab _
CC”&
(23)
and oabUb = # (I = 0
e = u;;,,
9
h”b = g’b -
gUb,
(24)
tia= u;bub.
2) Coefficient of heat conduction 1 qU= -lhab[:-(b);J,
where
y=g.
(25)
3) Coefficient of bulk viscosity 5 PK - PE =
-ye,
(26)
where pE is the thermodynamic pressure and pK is the kinetic pressure. This difference in pressure is calculated in terms of the energy-momentum tensors as pK - pE = -4 (T: - T&j. 3. Evaluation of the$rst and second moments. There are a number of methods available to us for constructing approximate solutions of eq. (4). We might, for example, use a relativistic version of the Grad moment method as developed by Marle, Anderson and Stewart. However, the form of eq. (4) suggests an alternate procedure, namely an expansion of the distribution function in a power series in t in which all but the first few terms are assumed to be sufficiently small that they can be set equal to zero. Such an approximation is widely used in the theory of radiative transfer where it goes by the name of the optically thick approximation and has been used by Marle to find approximate solutions to his relaxation-time equation. Let us then substitute the expansion f =fo + tf1 + Z”fi + ***,
(27)
412
J.L. ANDERSON
AND H.R. WITTING
into eq. (4) and equate to zero the coefficients of the various powers oft. In this way we get a set of equations which allows us to calculatef, in terms Of fi1:
afi-1
+ rycpbpC -
axa
‘”
afi-1
- UrP’f I >
=
apa
For our present calculation we will expand about the Maxwell-Boltzmann bution7) f.
= ew%P~,
distri-
(2%
in Minkowski space-time and keep terms to first order in T. The distribution function is, therefore, f = fo + Zfl 2
(30)
where fi
=
-(l/urpr)pd
b
;d -
(b’b)
;dP=lfO*
The first and second moments offare first moment N” is
(31)
now calculated using eqs. (5) and (6). The
(32) where Ng = n,u”
(33)
and N,“=
s
p4pd[ _a
;d + (bc)
;dpClfOn
%P’
(34)
RELATIVISTIC
RELAXATION-TIME
MODEL
473
Likewise the second moment can be written as Tab = Tlb + tTib,
(35)
where Tib = (p,, + PO) U”Ub- Poi?‘”
(36)
and
Tfb=
--LX
;*
s
s
(37)
Since u,p’ is a scalar we can define three tensors Tad = J (PaPdlurPr)fon,
(38)
Sac* = J (PaPcPdl%P’)fo~,
(39)
QnbCd= f (PaPbPcPdlurPr)fo~ 9
(40)
which depend only on the equilibrium distribution and must, therefore, have decompositions of the form T’* = TIu’ud - T&*,
(41)
s”” = S~U”U=U* - s, (u”g’d + u=g”*+ ud$c),
(42)
Q abe*=
Q1&&U*
+ gwu=
-
Q2(g”bucud + gacubud
+ g%v
+ gb*U’Uf + g%“ub)
+ 93 (gbgcd + g”‘g”” + g”*g”‘).
(43)
To determine the coefficients of these decompositions we equate their scalar contractions with suitable combinations of u, and g,b to the corresponding contractions of the integrals and solve the resulting equations for the coefficients. Doing this we find that T1 = 3 [4no - 4xm3 en (Kl - Kil)],
(44)
T, = 3 [no - 45cm3 ea (ICI - K,,)],
(45)
S, = 2~, - m2A,,
(46)
474
J. L. ANDERSON
AND H. R. WITTING
(47)
Q1 = 3 [16S, - 48S2 - 12m2n,, f 4ii e.5tns (K, - K,,)],
(48)
Q2 = & [6S, - 18S2 - 7~7~~2,+ 4~ ea ms
(49)
(K,
-
K,,)],
Q3 = & [S, - 3S, - 2m2n,, + 4x e” FH~(KI - K,,)], where K, is the x dt ‘O), and moment of the ing expressions
(50)
nth-order modified Bessel function of the second kind, K,,, = J‘8K,(t) S1 and S, are the coefficients of the decomposition of the third equilibrium distribution (see appendix A). We now have the followfor Nf and Tfb
N,” = -LX ;dTBd + @u,) ;d Sacd,
(51)
Tfb = -T ;dSabd+ (/h,) ;d Qabcd.
(52)
To express N” in a form compatable with the Landau decomposition use of the definitions
we make
(53) (54) (55) (56) in eq. (51) to get N,” = [@(S, - 2S2) - T,& - S@]
zP - S,p ;’ - ,&ti”
Rewriting this in terms of the dimensionless parameter y =
m@,
+ Tza “.
(57)
where (58) (59)
we get N; =
C
- T,a - -?- S28 m
+s,yz_A_ r m
0 Y
1
U”
;’
-
m
S2ti” + T20c ;a.
(60)
RELATIVISTIC RELAXATION-TIME
By making the approximation [see eq. (C.7), appendix C] &;a
=
y2- (2S2
mPo
mpo
+ PO + PO g PO
475
Ttbi b % 0 we obtain the following expression for 01;a
ys,
,& -
[
MODEL
-
0L + 0_!_ .
S,)
&J + POfj p PO
Y
y2
1
;a
s2
“PO
(61)
Y
Substituting this into eq. (60) we obtain
-T2p(1-3+-f&
+
(3s,
-
s,)
,3s2-sl,(~j-~s2ep
0-!- - L s,e - 1 Y
no&
m
24’.
(62)
By eqs. (A-23) and (C.8) the second to the last term in the above equations is zero and in appendix D we prove that the last term is zero. We therefore arrive at an expression for the first moment
~~=~~+~~(~_s2)hob[~_(t>;~,
(63)
which has the Landau form of the decomposition. To express Tab in a form compatible with the Landau decomposition we first perform the differentiation in the coefficient of Qabedin eq. (52) then use eq. (23) to obtain T;b = --oc;dsabd +
B;dt& + ,%,d + f-
hcd + /%i&
pbcd.
(64)
> Using the decompositions in eqs. (42) and (43) we obtain T,Ob= [b (Qi-
50,+ 29,)- k:(S, - 2s~)
-
2/9’3(02 +
3Qd1uaub
- [b (Qz - Qd - kS2 - $Qdel fb - 2 (Q2 - Q3) (@‘” + ,6 ;@ - /@“) ub) + 2s201 ;@zF - 2S2&‘ub + 2~Q3u”OL,
(6%
J. L. ANDERSON AND H. R. WITTING
476
upon inserting the quantity [2&Szuaub + 2 (Q2 - Q3) /haub - 2bS2z.hb - 2 (Q2 - Q3) ~u‘k”]. In appendix D it is shown by use of the approximation
(66)
7,‘; b z 0 that
(Q, - Q3) (/3ti” + /3 ;= - /3’zP)+ S2&zP - .Szn ;’ = 0.
(67)
We can now write the expression for the second moment as Tab = {duo+ po + z [B (91 - 592 + 203) - dr(S, - 2%) - 2F’ (92 + SQ,>l> uaub - {PO + r Lb (92 - 93) - 6
- $Q,P’l) gab + ~$QN’~.
(68)
This expression can be further simplified by using the matching condition ubPb = UbGb t0 obtain the rehtiOn P
(9, - 69, + 303) - k (S, - 3%) - PC202
- Qd = 0,
(69)
which, when subtracted from the coefficient of the U”U~term in Tab, gives an expression which takes the form Tab =
+ 2t 2 Q,db, m
(70)
upon use of eqs. (58) and (59). This expression has the desired Landau form of the decomposition for the second moment. 4. Transport coef$cients. From the expressions for the first and second moments derived in the last section [eqs. (63) and (70)] wecandetermine the transport coefficients using the Eckart form of the decompositions and their relation to the Landau form as given in section 2, with the result shear viscosity
7 = t (y/m) QJ,
(71)
RELATIVISTIC RELAXATION-TIME
heat conduction
I = z IuO+ PO --(y y2 -
MODEL
477
%),
(72)
n0
bulk viscosity
c = -4 e
(92
-
93)
$
1
.+ ctS,+ ;Q,y 8 . m
0
(73)
Putting the above expressions for the transport coefficient explicitly in terms of K,, K,, ,a, and the enthalpy h 3 KS/K2 we get (74)
il=
T4xe”m4h
c=
-t4xeam4
Y
> Y2 1
h s+Kil-Kl [(
-35,
(75)
K2 y2h’ + yh KS 1 -y2 y2h’ + 1 y y2h’ + 1
C
(76)
We would like to compare these results for the transport coefficients with those arrived at via the relativistic Grad moment method’). Calculation of the transport coefficients with the relativistic Grad moment method gives
qc =
$ $ K:(Y) 4K4 (2~)
+ 2K3 (2~)
---- K2 (2~)
Y2
Y
Kl(2r)
Y
Y2
-’ >
, (77)
_ K4(y)
2
+
’ 2
4K4 (2~) -
+K3 (2~) + 2K3 (2y) --Y
cc= _$$(3r_4) -
(35-r
K,(y) - $ 1
x [K3 (2~) -
Kl (2y)l-’
,
- $)K+Kd,,
K3(y) -
>
-
(78)
- $j
-
(79)
478
J.L. ANDERSON AND H.R. WITTING
where J’is the ratio of specific heats and is given in terms of the enthalpy by
r = yvz’/(y%
+ 1).
(80)
We would also like to compare our results in the relativistic and extremerelativistic regions with the transport coefficients calculated uia Marie’s relativistic BGK mode13). The expressions for the transport coefficients calculated via Marle’s formulation are ?jM =
(81)
z4xe"m4 (K3/y2),
A, = z4xeam4 [K4 - (K,/y) CM = t4xe”m4
-
(Kz/K,)],
[y’hf+ l(
Y
K4(F--
(82)
y
KS (r - 1) - K”(12F
yh’ + h K4(FyV2 + 1 (
+1.
-
l)-
1) - K”(9C y
Y
Qlr--)
- 13) + 5
y2
)
(9r - IO)
10)
>I.
> (83)
In order to compare the result of the various methods we will express Z, the collision time, in terms of the cross section cr. Using the results of appendix E we get r = [Y2/U + Y2P (~/ml
(u>>,
(84)
where y in the above equation represents (1 - v*/c*)-~. We wish to compare the results in the classical limit, the extreme-relativistic limit, and in the relativistic region. In the classical limit y 9 1 (y = mc*/kT) and we use the asymptotic expansion K,(y)=
($)Le+(l+++
‘“-2~~~2-9’
+ (P - 1) (P - 9) (P - 25) + ... > 3!83y3 >
(85)
where p = 4n2 for K,,(y). An asymptotic expansion for K,, is obtained from the asymptotic expansion of K,(y) using K,, =
7K,,(z) dz.
Y
(86)
RELATIVISTIC
The integrals encountered the result
K,,(y)=
RELAXATION-TIME
MODEL
can be performed via the Whittaker functionls)
479 with
(a)‘emy(l--&+J&-~ +
903105
163451925 +
98304~~ -
3932160~~
(87)
Inserting the various expansions into the expressions for the transport coefficients we get the following results in the classical limit, y B 1 11= 0.25Om
2. ’
1+ 3
0 O(Y
+ .. . ,
8Y
(88)
(89)
(90)
1
7123 +
(91)
- =G 3
~=0.207-!?
X v2
& = 0.273 $
2. ’ (1 + e-e). 0Y
In the extreme-relativistic K(Y)
= 4 (n -
(92)
O(Y
(93)
limit y < 1 and we use
I)! (2/Y)
and
K,, --) 0.
(94)
Inserting the various expressions into the expressions for the transport coefficients we get the following results in the extreme-relativistic limit, y 6 1 7 = 0.800 (m/a) (l/y) (1 - 0.16y2 +
l j,
?jG = 0.800 (m/U) (l/y) (1 + y/50 + ***),
(95) (96)
480
J.L. ANDERSON AND H.R. WITTING Q,, =
(4m/qJ) (1 - 0.08~~ + ..a),
(97)
1 = 1.33 (m/o) (1 - 2.08~~ + a..),
(98)
A, = 1.33 (m/a) (1 + y/18 + ..a),
(99)
1, = (4m/q) (1 - 0.08y2 + **a),
(100)
c = 0.0556 (m/c) y3 (1 - 0.08~~ + a..),
(101)
to = 0.091 (m/o) y3 (1 + ;ty’ + *a*),
(102)
&, = 0.296 (m/a) y2 (1 + 0.42~’ + ..a).
(103)
The values of the coefficients in the relativistic regime were evaluated numerically and the results are presented in figs. 1, 2 and 3. The actual transport coefficients are not plotted, but rather the o/m coefficient. The results shown in figs. I,2 and 3 confirm our formulation of the relativistic BGK model and can be used as an indicator of the accuracy one might expect in various regions when the model is applied to distributions other than the Maxwell-Boltzmann distribution.
APPENDIX A
The moments of a distribution function fare
44
= jfn,
N”(x) = j”PYk
defined as follows:
T"*(x)
= fp”pbfn, (A-1)
P’(X)
= Jp”pbp”fn,
@*‘d(X) = Jp=p*p=pdfn;
where YZ= 2H (p) 6 (pop” - m’) (-g)*
H(P) =
d4p,
1 P>O, 0
p
Choosing as a representation p” = mchx,
of the momenta
p’ = m sh x sin 6 cos 4,
(A.21 p2 =mshxsinesin4,
p3 = mshxcost3,
RELATIVISTIC
RELAXATION-TIME
MODEL
481
we find
n = U/PO) 6 (PO- mchx)dp”m3sh2xchxdxsin6d8dr$.
(A.3)
Iffis a locally dynamically symmetric distribution the various moments have the following decompositions : Ao
=
Ao,
gac= S~zl’UbU~Qo
abed =
Gb= (PO+ PO) u’ub-
= n,u”,
N:
s, (ZPgbC+
PO$~,
28-p + ZJCgOb) ,
QluaubuCud - Q2 (fbucud
+ g“=ubud + gUdubuc
+ gb=Uaud+ g%Pu= + g%=d) + Q3 (fbgcd
+ g”‘g’“’ + gOdgbc).
(A-4)
To determine the coefficients in the above decomposition we perform the following contractions u,N: = no,
(A.5)
W&ib = PO,
64.6)
7% = rue - 3~0,
(A.7)
U,,UbU,S;bc
u&s;; =
=
S,
s,
-
-
3s,,
G4.8)
6S2,
U&,U,UdQ;bCd = QI
(A.9
-
6Q2
+
3Q3,
(A. 10)
U&QtbZ = Q, - 9Q2 + 6Q3,
(A. 11)
Q oat
(A.12)
al?
-
Q, - 12Qz + 24Q3.
The above equation can be solved for the coefficients of the decomposition upon evaluating the various moments for a particular locally dynamically symmetric distribution. In this paper we are concerned with the Maxwell-Boltzmann distribution f.
= ea-8%o”*
(A.13)
482
J. L. ANDERSON
AND H. R. WITTING
Performing the required integrations and solving for the coefficients one finds A0 = 4xm2 ea Kl Jy ,
(A.14)
no = 41cm3 ea
(A.15)
PO =
K,ly,
4xm4ea [(K31y)
- (K,/Y~)I >
(A. 16)
p. = 4xm4 ea K21y2,
(A.17)
S, = 4xm5 ea K4/y,
(A.18)
S, = 4xm5 ex K3/y2,
(A.19)
Q, = 4xm6 ea KJy,
(A.20)
Q2 = 4xm6 e” K4/y2,
(A.21)
Q3 = 4xm6 en K3/y3,
(A.22)
where K, is the nth-order modified Bessel function of the second kind and y =mc2/ kT. Many useful expressions can be obtained from the above coefficients, among them 1u0 + PO =
n0
=
(A.23)
(y/m>S2,
(A.24)
(y/m)PO. APPENDIX
The decomposition
B
of the first two moments offin
the Eckart form are
N” = n&,
(B.1)
Tab = (,uE + pE) I.&; - pEgab + 292 z&’ - tab,
(B.2)
while the decompositions N” = n&
in the Landau form are
- ;I:,
Tab = (,uL + pL) u;z&’ - pLgab - Pb.
(B-3) (B.4)
RELATIVISTIC
RELAXATION-TIME
MODEL
483
The difference between the two representations lies in the choice of a four-velocity. In the Eckart form one defines the four-velocity in terms of the mass-flux density whereas in the Landau form one defines the four-velocity in terms of the energyflux density. When there is a component of flux (e.g. heat flux) other than mass the two definitions differ, but can be related by
where LY“represents the components of flux other than mass. This component is nonzero only when a nonequilibrium situation exists and reduces to zero when equilibrium is reached. Since we are interested in distributions only slightly off equilibrium, LY’will be a small term of first-order deviation from equilibrium. In our calculations we will be interested only in first-order corrections and will drop higher-order corrections such as c&~. Using eq. (B.5) we obtain
U&
= (24; + oc”)UEn= 1.
(B.7)
We can now determine the relationship between the two forms. Starting with the Landau form of the decomposition for the first moment N” = nLu; - 1: =
nL
(24: + 0~“) - Af, = nLuz + nLa” - it,
P.8)
by requiring
we obtain Na = nLuE,
(B.lO)
which is the Eckart form with nE = nL.
(B.ll)
Now looking at the Landau form for the second moment
Tab= (,uL -t- pL) u$; - pLgab - T=~
=(,,+,,@+~)(u:+$)
-PPLP--b
J.L. ANDERSON AND H.R. WITTING
484
z.& = @L + PL) && + -+-
nL
nL
= (/JL + PL)
z& -
UbT EL
pLg”b +
_
pL8ab
_
tab
>
2‘“;: pL)@4;) -
tab,
(B.12)
which is the Eckart form. Taking the scalar contractions of the two representations we get (B.13)
T:: = /.~r.- 3p, T::
=
PE
-
3~9,I
*PL
=
(B. 14)
PE*
Comparing eq. (B.12) with the Eckart form we obtain (B.15) APPENDIX C
Expressions for oiand (l/y)* are obtained by simultaneously solving the equations derived from the approximation Tib; b NN0 and A$ ; II z 0. From the first condition Tib; b = I(,‘& + PO)’ + c/b + PO) 01 u” + t,% + PO) ~2”- PO:=> but (,%I +Pd;b
=
(4nm4
ea
; b
&/Y)
= 4i7m4ea
=
4xm4 ea
- 2 m
and
=
4xm4 ea
m
(2& - S,)
L 0Y
;b
cc.11
RELATIVISTIC
RELAXATION-TIME
MODEL
485
where use has been made of quantities defined in appendix A and the relations
dK.(Y) _ -6 (19 - Kn+l(l/), dY
(C.4)
Y
Kn+l(r) = K - I(Y) + (WY) K.(Y)-
(C.5)
We, therefore, get
1
- 2 (2Sz - S,) l_ m 0Y
TGb;b = +
+ (/Al) + PO) 8
u”
cU0+ PO)9 - PO& ;’ - (y”/m) S2(l/y) ;‘,
(C.6)
so a;a =
(2S2 - S,)
_L 0
+po+po
p
Y”S2
--
PO
-1
. +
Cl0
Y ;a
+
po
PO
I3
1 24”
.
(C.7)
mPo 0 Y
Contracting the above expression with U, we obtain the first equation
(l-25 >
Y2 (3& -S,) mpo
&:=-_
mpo
1 + Po +po ( Y >’ PO
8.
(C.8)
From the second condition Nl ;a x 0 N”0;a
--
tie+ noe,
(C-9)
but noGa
=
(4xm3 e"K2/y)
;a
=
4xm3 ea
a
1 ;cr
+
-
Kzia+ K,
Y ;a-(yK3-G)
01 =noci _i
,
L
01Y
;a
(C.10)
Y ;’
tie =
47cm3ea
K2
-&
Y
(C.11)
J.L. ANDERSON
486 We, therefore,
iv-0 ;D
H.R.
WITTING
get 0 = n,k - (y”/m)
=
AND
(l/l’)’
/lo
(C.12)
+ n$,
for the second equation. Using the definition into the form (yh jl +
of enthalpy,
h E K,/rCr,, eqs. (C.8) and (C. 12) can be put
1) k + y2 (y + 3h) (l/Y)’ + $0
y2 (h - l/y)(I/y)*
Solving these equations
(C.13)
= 0,
(C. 14)
+ e = 0.
first for (l/v)’
we get
butll)
SO
tw = WV)MY2h
+ 111.
(C.15)
For k we get d: =
-(jm
+ yh)/(y2h’ + 1) 0.
(C.16)
APPENDIX
D
1. ProofoJ’(Q2 - Q3) (/W + /3 ;a - $zP) + S,c?u” we get (Q2 - Q3) = S,. By contracting Tib we get
T,“, = ,uo - 3p, = m2Ao, Using the above expressions
(92 -
p. = 3C1uo- mZAo) = S2.
and eqs. (C.7)
(58) and (59) we get
Q,) (/‘&” + /? Go- @u”)+ s,k’u”
+ p&P
- -ys2 &”
m
Szb ;’ = 0. From
+
f
(2s
m
2
s#
-
-
s
1
)
”
L .p 0Y
section
3
(D.1)
RELATIVISTIC
-
RELAXATION-TIME
(PO+ PO)fh” - (PO+ PO)ZP+ f
s2
MODEL
487
;”
J_
0
_~o,(l-~)~+~(3s2-s~( e-p=0 (D-2) by eq. (C.8) of appendix C. 2. Proofof(y/m) (3SZ - S,) (l/y)* - S, (r/m) 13- no& = 0. From section 3 we S, = ,uo. Using this we get get 3SZ 1 * J- (3s 2 - S1) - s,h m m 0Y 2
= --ok+~po
m
0
1 .
- n,ci
-~oe+noe-s2-?i-e=0,
Y
m
(D.3)
by eqs. (C.12), (A.24) and (D.l).
APPENDIX
E
The relation between the collision time r and the cross section u is
where (0) is the average velocity and
(p/m)(1 + p"/m')+
W)
and f. normalized is f. = (1/4xm2) (y/K,) eWychx. Performing the above calculation we get
= WG)
WY + W2) fP -
(P-3)
M41~
where E,(y) is the exponential integral E,(y) = y (eaYS/S) dS.
(E.4)
1
The relative velocity of two particles is12) V= [ur +(1/--
l)(Y,.u/~2)Y-~yY][l/y(l
-U1*U)l,
(E.5)
488
J. L. ANDERSON
AND H. R. WITTING
where y is used to signify (1 - vz/c’)-’ in the remainder of this appendix. In calculating the average relative velocity the term u1 . v -+ 0 sincef, is an equilibrium distribution so
v + (VI -
ye/y
7
v = (V’ v)+ =
[(VI + y’v’)/yqf.
Since we are interested in a gas of similar particles close to equilibrium (~7~) w (0) and we get
(V> = (v>.u
+ Y2)lY213.
(Jw
We therefore have
-r = W% )lY'/(l
+ Y’P.
(E. 7)
In the classical limit y + 1 and t -+ l/2* an, (v). In the extreme-relativistic z + l/on,
w3)
limit y -+ co
(E-9) REFERENCES
1) Ehlers, J., in Proceedings Varenna Summer School on Relativistic Astrophysics 1969, Academic Press (New York, 1971). 2) Chernikov, N.A., Acta Physica Polonica 23 (1963) 629; Physics Letters 5 (1963) 115. 3) Marle, C., Ann. Inst. H. Poincare, 10 (1969) 67, 127. 4) Stewart, J.M., MNRAS 145 (1969) 347. 5) Anderson, J.L., in Proceedings of the Midwest Conference on Relativity, Plenum Press (1970). 6) Bhatnagar, D., Gross, E. and Krook, M., Phys. Rev. 94 (1954) 511. 7) Stewart, J. M., Non-Equilibrium Relativistic Kinetic Theory, Lecture Notes in Physics Vol. 10, Springer-Verlag (Berlin, 1971). 8) Landau, L.D. and Lifschitz, E.M., Fluid Mechanics, Pergamon Press (New York, 1959) p. 505. 9) Eckart, C., Phys. Rev. 58 (1940) 919. 10) Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover (New York, 1965). 11) Israel, W., J. math. Phys. 4 (1963) 1163. 12) Anderson, J.L., Principles of Relativity Physics, ch. 6, Academic Press (New York, 1967).