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Relativistic thermodynamics of gases
OF PHYSICS
169, 191-219 (1986)
Relativistic
Thermodynamics I-SHIH
Institute
de Matemritica.
Universidade
of Gases
LIU Federal
do Rio de Janeiro,
Brazil
I. MCJLLER Hermann-Fiitiinger-Institut,
Technische
Universitiit,
Berlin,
West Germany
AND
T. RUGGERI Dipartimento
di Matematica,
Universitd
di Bologna,
Bologna,
Italy
Received June 10, 1985 Relativistic thermodynamics of degenerate gases is presented here as a field theory of the 14 lields of particle density - particle flux, and stress-energy -momentum. The field equations are based on the conservation laws of particle numbers, and energymomentum and on a balance of fluxes. The necessary constitutive equations are strongly restricted by the principle of relativity, entropy principle, requirement of hyperbolicity. It turns out that the resulting field equations contain only viscosity, bulk viscosity and heat conductivity as unknown functions. All other constitutive coeflicients may be calculated from the equilibrium equations of state that are known from statistical arguments. The paper offers a more systematic version of relativistic thermodynamics of gases than the earlier papers by Miiller and Israel. At the same time the present version contains less unknown functions than those earlier papers. All speeds of propagation are linite. The relation between the present theory and the classical one formulated by Eckart is described. ‘c 1986 Academc Press. Inc.
1. INTRODUCTION Relativistic extended thermodynamics is a theory with the principal objective of determining the 14 fields of the particle number-particle flux vector V” and of the stress-energy-momentum tensor PP. The necessary field equations are the conservation laws of particle number and of energy-momentum and a balance equation for the fluxes. Constitutive equations for the third order flux tensor and for the flux 191 0003-4916/86 $7.50 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved
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LIU,
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production tensor relate these quantities to the 14 basic fields. The principle of relativity, the entropy principle and the requirement of hyperbolicity are used to restrict the generality of the constitutive functions. We assume symmetry of the flux tensor and of the flux production and thereby we limit the applicability of the theory to ideal gases which, however, may well be degenerate. It turns out that in a linear theory the knowledge of the thermal and caloric equations of state in equilibrium suffices to determine the field equations specifically except for one function of a single variable and three negative-valued functions of two variables each. The latter functions are closely related to the bulk viscosity, the shear viscosity and the heat conductivity and are therefore amenable for experimental determination in principle. The results are summarized in Chapter 7, Sections 7.2 and 7.3. The main motif in the formulation of extended thermodynamics and, in particular, relativistic extended thermodynamics is the derivation of a hyperbolic set of field equations so that disturbances propagate at a finite speed. The field equations presented in Chapter 7 satisfy that criterion. A natural question to ask of extended thermodynamics is how to get back to the ordinary thermodynamic theory that was first developed by Eckart [ 11, in whose theory we have only 5 fields, namely the particle number-particle flux vector and the energy density (or temperature). That question is addressed and answered in Chapter 7, Section 7.5. Also there have been earlier versions of relativistic extended thermodynamics, formulated by Miiller [2] and Israel [3]. These were essentially reformulations of non-relativistic extended thermodynamics in a form that is invariant under spacetime transformations. In Chapter 7, Section 7.6 we compare the present theory with those earlier ones and we conclude that the present theory provides stronger restrictions on the field equations. These results are made possible by the use of Lagrange multipliers in the exploitation of the entropy principle, as proposed by Liu [4], and by choosing these multipliers as variables, an idea that was conceived by Ruggeri [ 51. While the present version of relativistic extended thermodynamics is conceptually very simple and the results are very restrictive, the detailed calculations are extremely involved. Indeed, the Lagrange multipliers must be used as variables, they must then be identified in physical terms and eventually we must return to the suggestive variables Va and T”fl. In a previous draft of this paper we have tried to lead the reader through all twists of the argument, but that attempt met the resistance of the reviewer. Therefore now we just indicate the procedure, cite certain intermediate results, skip many technicalities and remit a large part of the arguments to the appendix. We invite the interested reader to get into contact with one of the authors who will be happy to supply all intermediate calculations. The notation in the paper is the standard notation used in many books and papers. Latin indices indicate spatial coordinates and components, while Greek ones denote spa=-time components. Angular brackets like in fcaB> denote the symmetric traceless part of a tensor.
RELATIVISTIC
2. 2.1.
THERMODYNAMIC
PROCESSES
THERMODYNAMICS AND
THE PRINCIPLES
The Objective of Thermodynamics
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OF GASES OF A CONSTITUTIVE
THEORY
and the Field Equations
The objective of extended thermodynamics mination of the 14 fields.
of relativistic
fluids is the deter-
Va(xa) - particle number - particle flux vector, YB(xs) - stress - energy - momentum
tensor.
(2.1)
FP is assumed symmetric so that it has 10 independent components. For the determination of the 14 fields (1) one needs field equations and these are formed by the conservation laws of particle number and energy-momentum, viz.
v*,, = 0, T@,= 0>
(2.2), (2.2),
and by the balance law of fluxes A”8’.
.i
=I”P
(2.2),
AxPYis the completely symmetric tensor of fluxes and I@ is its production density. The symmetry of AaBYand of Pp is suggested by the relativistic kinetic theory of gases (see [6], [7], [S], [9]). In that theory d is a third moment of the distribution function and ,I is its collision production. The kinetic theory also shows that I*, = 0 and A@, = m2c2V
(2.3)
holds, so that the trace of the tensor equation (2.2), reduces to the conservation law of particle numbers. The formulation of non-relativistic extended thermodynamics of real gases by Kremer [lo] has made it clear that symmetry assumptions like the one on AMP? limit the applicability of the theory to ideal gases. We do not consider this limitation as a serious one, because in all conceivable applications of relativistic thermodynamics the matter is in a state that is well described by ideal gas laws. We accept the conditions (2.3) also in the phenomenological theory and conclude that among the 10 equations (2.2), there are only 9 independent ones. Thus the set of equations (2.2) is a set of 14 independent equations. However, the equations (2.2) cannot serve as field equations for the thermodynamic fields (2.1). Because the additional fields AaBy and I@ have appeared. In this situation we follow the usual procedure of thermodynamics and formulate constitutive equations for the flux tensor and its production. Such equations relate the
194
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MCLLER,
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values of 4 and ,I to the fields y and T in a materially extended thermodynamics we write
dependent manner, and in
(2.4) The functions aapy and pp are called constitutive functions and, if they were known, the equations (2.2) and (2.4) would form an explicit set of field equations. Each solution of these field equations is called a thermodynamic process. 2.2 Restrictive
Principles
In reality the constitutive functions are not known and therefore thermodynamicists are usually working in the field of the constitutive theory. In that theory restrictions on the form of the constitutive functions are extracted from universal physical principles. The most important such principles are the entropy principle, the principle of relativity and the requirement of hyperbolicity. The entropy principle states that the entropy-entropy flux vector ha is a constitutive quantity which obeys the inequality h”,, 3 0 for all thermodynamic
In particular form
in extended thermodynamics
processes.
the constitutive
h” = h”( V, F”).
(2.5)
equation for h” has the (2.6)
The principle of relativity states that the field equations have the same form in all frames, and this statement implies that the constitutive functions be invariant under a change of frame i” = P(xp). When some constitutive quantity, if referred to two different frames, is denoted by C and ?, the principle of relativity reads
Note that the constitutive function c is the same one in both frames! The requirement of hyperbolicity ensures that Cauchy problems of our field equations are well-posed and all wave speeds are finite. Therfore we require that our set of field equations be symmetric hyperbolic in the sense of Friedrichs & Lax [ 111. The formal statement of that requirement will follow in Chapter 6.
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3. MORE SUGGESTIVE VARIABLES AND EQUILIBRIUM 3.1. Particle Density and Velocity We introduce
so that the particle number-particle
flux vector may be written in the form VZ=nU”.
(3.2)
The advantage of this decomposition of v” is easy interpretation: Indeed, n is the number density of molecules in the Lorentz frame that is locally and momentarily at rest in the fluid. u” is the 4-velocity of the fluid and we have U, U” = cz so that u” has only three independent components. 3.2. Pressure, Energy, Stress Deviator and Heat Flux
The metric tensor gxB is defined such that in a Lorentz frame it assumes the form
By use of U, and of the projector
h,,=lc2U,U,-g,,j, we may identify four separate parts of the stress viz.
t<@> = h;h{-; h@h w’ TP’ ( f’=-h 1 flu TJ’” 3 q” = -ha, U,, Tp”
energy -
momentum
tensor,
stress deviator,
~ pressure, (3.5) - heat flux energy density
The names indicated in (5) are motivated by the observation that t<@), P, qa and e in the rest Lorentz frame are the deviator of the stress tensor, the pressure, the heat flux and the energy density respectively. The components I<*), tcm> and q” vanish in that frame.
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The definitions (3.5) imply an identity which gives a suggestive decomposition 7@ into parts, namely m=t
of
(3.6)
and F8 in Equilibrium
It is obvious that, instead of v”, Fp we may now use n, u*, tcxp), P, q’ and e as variables. In particular I@ may be considered to be a function of these variables. Equilibrium is delined as a process in which the production densities I”” vanish. We write I”” 1E = 0
(3.7)
and thus, by (2.3) we have nine conditions of equilibrium. Eight of those conditions can be satisfied by setting tca8) and q’ equal to zero. The remaining condition obviously must give an equilibrium relation between the remaining non-zero variables P, n and e. We write this relation as PI E= ~(6 e). It is useful to rewrite the stressenergy-momentum
(3.8) tensor by introducing
P=p+17,
(3.9)
where IZ, the non-equilibrium part of the pressure, may be called the dynamic pressure. Thus we obtain from (6) (3.10)
r”=t(“B’+(p+n)h^“f~(U”q”+U”y’)+;fjeli”l:B, in which form the three quantities t (OIB>,Z7 and qp, which vanish in equilibrium, displayed explicitly. In particular in equilibrium we have
are
(3.11) 4. PRINCIPLE OF RELATIVITY
AND LINEAR REPRESENTATIONS RESPECT TO II, q”, t(i‘v).
WITH
In (2.7) we have already generally stated the requirement which the principle of relativity imposes upon the constitutive functions. With the more suggestive variables introduced in Chapter 3 we write this condition in the form C= C(n, e, II, Up, qfl, t<““>)
and
k=
C(n,
e, IZ,
ii@, Go, Fcpy’).
(4.1)
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In the present case we have the vectorial constitutive quantity ha and two tensorial quantities Z@ and AaBy of order two and three respectively. For these quantities the two equations (4.1) combine to give
(4.2)
One expresses the conditions (2) by saying that h”, PP, AaPr are isotropic functions with respect to arbitrary transformations k’ = P(xp). There are explicit representation theorems for such functions (see [ 123 or [ 131). At this moment we are not interested in the general solution of the functional equations (4.2) but rather we shall write the part of the representation theorem for I@ and AXBy that is linear in Z7, q’ and t<@). These read I@ = B; ng”p -$ Aapy=(C~+C;17)
B;17U”UP
+ B3tcaS> + B,(qaUP + q8U”),
(4.3),
UaVU~+~(nm2-C~--C~IZ)(grflU~+g~~U”fg~”U~)
+ C,( g@qy + gPYq” + gy*qfi) - f C,( UYPq + Cs(PD)UY+
+ UflU’qX + UYulq~)
tQY)zP + t(yW”).
(4.3 12
Even in a linear theory as characterized by the linear relations (4.3),,, we shall be able to calculate the coefficients of linear and quadratic terms of the entropyentropy flux vector. Therefore we shall write down the general representation of h” up to second order. It reads h’=(ns+A;Z7+A~2Z72+AfqPql,+A~t~BY)t~,~~~) + (A; + A;Z7) q’+ A; t
U
(4.3 )3
The trace conditions (2.3) have been observed in writing down (4.3),,, and I@ in (4.3), conforms to the requirement that ,I vanish in equilibrium. All coefficients B; through A!j’ may be functions of n and e and it is our purpose in this paper to determine these functions. The tool in the determinition of the coefficients of the equations (4.3) is the entropy principle whose exploitation is indicated in Chapter 5 and carried out in some detail in the Appendix.
198
LIU, MijLLER, 5. ENTROPY
5.1. Entropy Inequality
ANDRUGGERI PRINCIPLE
and Lagrange Multipliers
The entropy principle has been stated in Section 2.2, see the inequality (2.5) where ha is a constitutive quantity. The key to the evaluation of the entropy principle is the statement (2.5) by which the entropy inequality need not hold for all fields V’(Y), F’“(x”) but only for thermodynamic processes, i.e., for those fields that satisfy the field equations. Liu [4] has shown that we can take care of this constraint by the use of Lagrange multipliers. He showed that the new inequality h”,, + [I”,,
+ A, TP”,, + C,,(A”““,, - Ik) > 0
(5.1)
holds for all fields v”, Yp. The quantities 5, A, and C,, are called Lagrange multipliers by Liu and they may be functions of VP, TP’, dependent on material. In other words <, A, and C,, are constitutive quantities. C,{, is symmetric and traceless which reflects the fact that, by (2.3), only the traceless part of equation (2.2)3 belongs to the field equations. 5.2. A Change of Variables In preparation for a change of variables we rewrite the inequality following equivalent form (h” + t I’” + A, Tfi’ + Z, A+),,
- V’r,, - T’%,,
(5.1) in the
- A”%Z’,:,,, - C,,Z”’ 2 0. (5.2)
In the sequel we shall abbreviate the first bracket in (5.2) by h’” h’“=h”+~V”+ApT~l+C~Aysi’.
(5.3)
Obviously h’” is a constitutive quantity since it is composed of variables and other constitutive quantities. Now we make a transformation of variables from
(v”,m to (5,A,,C/l,) and a corresponding
shift of constitutive
quantities
(5.4)
from
(ABY”, IBy, h”(or h”), 5, A,, C,,) to (A@‘, ZPy,h’“(or h”), VP, TV”) Inserting h’“({, A,, C,,) into (5.2) and performing h’” we obtain the inequality
This inequality
(5.5)
the indicated differentiation
must hold for all fields 5, A,, C,, and in particular
for arbitrary
of
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199
values of the derivatives r,,, /i,,, , C,,, which occur linearly on its left hand side. It follows that we must have
lest the entropy inequality be violated. We conclude from (5.7) that V, Yp, A( as functions of <, A,, C,, can be obtained by differentiation of a single vector h’“. This is the constraint that the entropy principle imposes upon our constitutive functions. The exploitation of this constraint is described in the appendix, the results are summarized in Chapter 7, see Sections 7.2 and 7.3. 5.3. Equilibrium
Properties
We proceed to exploit the residual inequality (5.7),. Equilibrium is a process in which all productions ZaB vanish. Therefore we see thr$ the left hand side of the residual inequality, whichhis the entropy production 0, assumes its minimum in equilibrium, namely zero. (T may be considered as a function ofh5, /i,, C,, since I”” is a function of these variables. Alternatively we may consider 0 to be a function of r, A,, IaD, and C,, to depend on these variables as well. If we do that we have
where the index E refers to equilibrium. have of necessity
Since i is minimal
in equilibrium
we must
and - ~~&~,~ In particular,
= ~~($+$$)~~I
the Lagrange multiplier
6. In order to state the requirements into a more compact form. Let
F”.4
-negative
595/169/l-14
(5.9)
C vanishes in equilibrium.
HYPERBOLICITY
of hyperbolicity
formally we put our problem
1vu,T”,,Aa
I”, stand for the set (0, 0, Z@,} x,4
semidelinite
{l, A,, qT,>,
(6.1)
m, MUELLER,AND RUGGERI
200
where A runs from 1 through 14. Thus the balance laws (2.2) and the restrictions (5.7) assume the forms
p Combination
-I
A,or -
A
Andy =dh’” A ax;
of these two relations puts the balance laws into the symmetric form a2hfr
(6.3)
-XB.z=IA,
ax, ax,
or, with X,,, = -$XB U, - Xg,php, : 1
a2hra U,B,-&hp,X,,,=I,, f ax, ax, A B
(6.4)
where XB = X,,, UP is the material time derivative of X,. For this symmetric system of equations to be hyperbolic we require that (see [ 111 and [5]) I& Alternatively,
(ix11- positive definite.
by use of (6.2),, this requirement
reads
u, 6F, 6X, > 0 or, more explicitly,
(6.5 1
(6.6)
by (6.1) U,SV’S< + U,67%%4,
for arbitrary variations. Some consequences of the condition tion A.5.
+ U,6AaPYGZ,, > 0
(7) are derived in the appendix,
(6.7) see Sec-
7. RESULTS AND COMPARISON WITH EARLIER THEORIES 7.1. State Functions in Equilibrium In this Chapter we shall summarize the results which the entropy principle imposes upon the coefficients E; through A: in the constitutive relations (4.3). In the Appendix it is indicated how the results are obtained. It turns out that these coefficients are strongly restricted in their dependence on c( and T once the equilibrium state functions n(a, T), e(a, T), p(cr, T), CT(c+ T) and S(U, T) are known. -crT is the specific Gibbs free energy or chemical potential and T is the absolute temperature. The equilibrium state functions can in principle be measured but of
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OF GASES
course in a relativistic degenerate gas this is practically impossible. The natural alternative is to calculate the state functions in statistical mechanics as follows: The quantities introduced in Chapter 2 as variables and constitutive quantities, nameley V/*, Y8, AEBYand h” have counterparts in statistical mechanics. They are defined there as moments of the distribution functionf(x’, p”) V”=c
I
p’f dP
T@=c s p"pfifdP (7.1)
dP
pz is the four momentum of the particles so that we have p”p, = m2cZ and dP = &/pa dp’ dp2 dp3 is the invariant element of momentum space. k is the Boltzmann constant and y is equal to w/h3 where h is Planck’s constant and w is equal to (2s + 1) for particles with spin s. h/2x The upper and lower sign in (7.1 )4
refers to Bosons and Fermions respectively. Jiittner [14], [15] has shown that the equilibrium ideal gas has the form
distribution
function of an
(7.2)
The equilibrium form of the equations (7.1) is obtained by introducing f 1E from (7.2) into their right hand sides and V’, T*flI E, AaBY1E and h” 1E from (3.2), (3.11), (4.3), and (4.3)3 into their left hand sides. We thus get c
s
p'f
l,dP=nU
c I papppf ~,=ph@+-$UTJP (7.3) c
s
p~pBpYf1.dP=C~LI”UBU.+~(nm2-C~)(g~~U~+g”~I/^+gv~~R),
202
LIU,
MtiLLER,
AND
RUGGERI
This set of equations permits the calculation we obtain n = 47cym3c3Jz.,(tl,
of the equilibrium
state functions and
y)
e = 47~yrn~c~J~.~(~x,y) p=4v~J4.0(a,
Y)
C:=47VmSc3(Jz.1(a,
(7.4)
Y)+ 2J4.1(& Y))
1 e+p T n
s=a+--.
The expressions J,,” are defined by
J,da, Y)= lam
sinh” p cash” p
exp(i+ycoshp)il It will also be useful to have a separate notation
(7.5),
dp’ for aJ,,,,/&
and we introduce
y stands for mc2/kT and this parameter determines how much relativity affects the gas. The functions J,,,n and Z,,,, satisfy the following recurrence relations
= Jm.,+2- J,.,, kyz,+2,,=nJ,,~-,-(n+m+1)J,,,+, Jm +
2,n
ZJ
=kZ,,,,+, ay m-n
(7.6)
& W+lJm,,)=bm+‘L,,. Note that the equations (7.4),,2 may in principle be used to calculate a and T as functions of n and e. When this is done we may calculate p, Cy and s from (7.4)3,4,5 as functions of n and e. However, since the analytic solution of (7.4)1,2 is impossible, we shall have to be content with results expressed in terms of a and T. This is not different from the non-relativistic theory of degenerate gases. Of course, there are special cases in which either a or T or both can be calculated as functions of n and e. These are certain limiting cases of no degeneracy or complete degeneracy or the non-relativistic and the ultra relativistic case. An
RELATIVISTIC
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203
investigation of most of these special cases is to follow in future papers: here we shall merely give the general results and we shall only quote results for the nondegenerate case, where a is an explicit function of n and T (see Section 7.3). 7.2. Summary of General Results The main result of this paper consists in the observation that the knowledge of the equilibrium state functions (7.4) may be used to make the representations (4.3), and (4.3), very explicit. Thus the coefficients C;, C3 and C, of non-equilibrium terms in AzP7 are given bjl
=---
6m NY c2 D;
m N, 5D,
(7.71,
(7.7L
(7.713
where A is an arbitrary function of the single variable a and where the second members of (7.7)1,2 introduce abbreviations N;, D;, N, and D, for the determinants in those equations. Also the coefficients A; through A! of non-equilibrium terms in h” are given by A;=0
(7.8),
(7.8 )z Aq=L 3 ’ 4ny2mSc9
D3I4,1--3Z4.0
D:
(7.8)3
(7.814
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LIU,
MijLLER,
AND
RUGGERI
(7.8)s
(78)
A;=
Ai=-
.
l-3N, 4nyn
13 D3Z6,,=~m5c7
D,
6
(7.8)7
’
A; and A! are given in two alternative forms which will both be needed later. Thus all coefficients of the linear non-equilibrium terms of AmByare determined to within a single function of cc The same is true for the coefficients of linear and quadratic non-equilibrium terms of h”, except that there A: and Ai are completely explicit, not containing A(a). Moreover the requirements of non-negative entropy production and of hyperbolicity imply the following restrictions on the coefficients B;, B3, B, and A:*, A:, Ai respectively. These read
By
I62.1 12.2 n I
2.2
II,3
>O,+O,
6.0
0;
dP ,>O,;T>O,
Bqd
A;‘
-140 20 D3
(7.9) ~ I
Af
The conditions (7.9), ensure the positivity of bulk viscosity, shear viscosity and heat conductivity (see (7.22) below). Actually, since I,,, is negative we may conclude that B, is non-positive and similar conditions on By and B, could be obtained, if we were to investigate the signs of (I,,, I,,, - Z,.,‘)/D; and of Z4,0/D3, which are explicit functions. The conditions (7.9)2 ensure that the entropy Uahr(n, e, 17, q@(,t(““)) has its maximum in equilibrium. Among (7.9)2 the first two inequalities are the classical “stability conditions” on compressibility and specific heat, while the last three inequalities, together with (7.9),, ensure positive relaxation times for l7, qp and T<“‘>. The whole set of inequalities (7.9) ensures finite wave speeds. The derivation of all these results is indicated in the appendix. 7.3. Summary of Results for Non-Degenerate Gases As indicated before, we wish to list the results of the previous section for the limiting case of a non-degenerate gas. Formally this limit is taken by letting CIgo to infinity so that the Jiittner distribution tends to flE=.W
aIke - (U.lkTh
.
(7.10)
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205
In this case the integrals (7.3) over the distribution function are easily expressed in terms of modified Bessel functions of the second kind, viz. (7.11)
K,(y) = jam cash npe-ycosh P dp.
The functions J,,, that appeared in the results of Sections 7.1 and 7.2 always have an even integer as first index and therefore they can be expressed in terms of Jon by use of the recurrence formulae (7.6),. In the non-degenerate limit these functions assume the form JO,n(~, ~)=e-“‘~
IO: coshnpe-YcoShP dp,
(7.12)
which may be reduced to the functions K,(y) by use of trigonometric identities. In this limit the functions I,,, are simply equal to - l/k J,,,,. Thus the state functions (7.4) in the non-degenerate case can be derived as
p=nmc’!.,
(7.13) Y
s=k
ln(4~~m3c3)+ln~+yG
where G stands for K3/K2. The values of CT, C3 and C5 for the non-degenerate by taking the non-degenerate limit and we obtain
I
,
gas can be derived from (7)
206
LIU,
MtiLLER,
m3c3 kA 15 ny
l+;G-G’---? c3= -IIf Y
AND
RUGGERI
7
l+;G-G2 (7.14)
Similarly the coefficients A; through At of equation assume the values
(7.8) in a non-degenerate
A;=O,
gas
(7.15h 1-$+;G-G2
Af=-&;
(7.1%
G+!?G2-2G3’ Y
m3c3 kA G-!!G2+(;33ny6 Y 2
7
(7.1513
l+tC_G2 A; =--- -k y2 1 4m2c4 n G2
(7.1514
A;=;,
(7.151,
A”= --3k y 2 m2c4 n
c7.15)6
l+fG-G2--A;=---&;
m3c3 kA 15 ny (7.15),
>
.
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7.4. Field Equations The balance equations (2.2) with the representations (3.2), (3.10), (4.3)1,2 and the expressions (7.7) furnish a set of field equations for the variables n, e, U”, Z7, t<“fl) and qa which contains only one unknown function of one variable, viz. A, and three negative valued functions of n, e. viz. B;, B, and B,. There is a strong suspicion that the function A is equal to zero, because that is the case in the kinetic theory of gases as was shown by Dreyer [16]. In order to clear the way for the following comparison of the present theory with earlier ones we rewrite the balance equations (2.2) by splitting them into spatial and temporal parts and by making right hand sides explicit by the insertion of the representation (4.3), for I”@. We have V”,, = 0
(7.161,
h,, T@, = 0
(7.161,
u, T”fi,, = 0
(7.161,
h,,k,- f h/A,
7.5. Transition Eckart [ 1]
to Ordinary
The thermodynamic
,i
(7.16),
h,, U, A@, = - B4c2qr
(7.16),
U, U,A”Pv,v = - 3B;c211,
(7.161,
A@“.=B
Thermodynamics,
3
t
Comparison
theory of the 14 independent
with the Theory of
fields
n, e, U”, II, t<@>, qa that has been developed here is called “Extended from the older theory which we call “Ordinary theory has 5 independent fields, viz. n, e, U
Thermodynamics” Thermodynamics”.
(7.17) to set it off That older (7.18)
and it was formulated by Eckart. The field equations of extended thermodynamics have been written down in the equations (7.16). Ordinary thermodynamics has only the first live of those equations, viz. (7.16)1,2,J and it must formulate constitutive equations for t<‘“), q’ and A’ which relate these quantities to the fields (7.18). Liu & Mtiller in a nonrelativistic paper [ 171 have shown that such constitutive equations for t(‘“), qP and
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LIU,
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AND
RUGGERI
IZ can be derived from the equations (7.16)4,5,6 of extended thermodynamics by an iterative procedure that is akin to the Maxwellian iteration used in the kinetic theory of gases. The iterative scheme proceeds as follows: Into the left hand side of the equations (7.16)4,5,6 we insert the zeroth iterates of t<‘“), qp and 17 which we take to be their equilibrium values t (rv>j E = 0, qpl E = 0 and 171E = 0. By (4.3)2 the corresponding value of AmBYis given by (7.19)
A’BY~,=C~U”U~Uy+~(nm2-C~)(gZ”LiY+~~yUa+~yaU~).
Insertion of this expression into (7.16)4,5,6 p reduces the first iterates which we denote by t’$), 4;) and 17(l) These are used to calculate a new value for Aapy from (4.3),. When that new value is introduced into (7.16)4 5 6 we obtain second iterates t(Z) c2)’ n(2) etc. In the kinetic theory one can argue that the results of the
q1r iterative procedure furnish good approximate expressions and we shall proceed on that assumption. Actually we shall be content here with only the first step in that procedure and therefore we calculate Aa@ )4y and insert it into (7.16)4,5,6. In this calculation in order to simplify the resulting expressions we use the equations (7.16)1,2,3 in the form (7.20) v’,, = 0, h,,, T*’ 1 E$ = 0, u, T*p 1 E,fl = O, as is appropriate
t&
for the first iterative step. Thus we obtain
=g
(nm2- Cy)[h;hp,y,,)]. 3
2
(U=C qP 6TB,
f T,a - 4 VU,,, c
)I
(7.21)
In obtaining (7.21)2, we have used the relation (A. 11) from the Appendix. Among these first iterates the equations (7.21),,2 are easily recognized as the relativistic versions of the laws of Navier-Stokes and Fourier respectively that were first derived by Eckart. This interpretation identifies the coefficients of the square brackets as viscosity p and heat conductivity K. We must recall that in relativity the expression h;( l/T T,, - l/c2 tY’Uor,y) plays the role of the “thermodynamic force” for heat conduction. The equation (7.21), relates the dynamic pressure to the divergence of
RELATIVISTIC
THERMODYNAMICS
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209
velocity and the coefficient of ,U6,a must be interpreted as the bulk viscosity v of the gas. If we introduce the state functions (4) we obtain more specific forms for p, K and v viz. /l=471yu=471y--
m6c7
I
15TB3 6*o -m6c7 D3
3T=&
4.0
+ -;nkTG 3
+ k nmk(y + 5G - yG2) 4
(7.22) nm2c2
+3j@
l-f-+&-G= Y2 Y
The second members of these equations refer to the non-degenerate limit a + co. Note that the relations (7.22) offer a possibility to determine the unknown functions BT(n, T), B,(n, T) and B,(n, T) from measurements of viscosity, heat conductivity and bulk viscosity. We conclude that extended thermodynamics contains Eckart’s ordinary thermodynamics as an approximation. 7.6. Comparison with older Theories of Extended Thermodynamics, notably the Theories of Miiller [2] and Israel [3]
Miiller’s paper was written when linear irreversible thermodynamics’ was the only systematic thermodynamic theory of non-equilibrium and it was within this theory that he derived his results. We change the notation of Miiller’s equations ([2] 6.28) through ([2] 6.30) so as to make it conform to the notation of the present paper and write these equations in the form
qr = KTh”P II=
LT T ,a--tUYJ,, c2
+ kjm + MD,, - Nt,,P>.,
(7.23)
-v[U~,~ + Gil+ Mqq,].
k stands for U”R*,. Israel’s equations ([3] 14) are identical to these except for notation. Comparison of (7.21) and (7.23) shows that Miller’s and Israel’s results differ from Eckart’s by the terms with the coefficients A, B, C and M, N. Nothing specific can be said about these 5 coefficients from thermodynamic arguments even if we wish the equations (7.23) to apply to an ideal gas. In particular, there is no relation between the five coefficients A through N and the thermal and caloric equations of state of a degenerate gas. It is tempting to assume that the 9 equations (7.23) of the old theories corres’ Sometimes called TIP for thermodynamics or irreversible processes.
210
LIU,
MtiLLER,
AND
RUGGERI
pond to the 9 independent equations (2.2)3 or (7.16)4,5,6 of the present theory when the representation (4.3)2 is introduced. And indeed that assumption is valid, even though only approximately. Insertion of (4.3), into the equations (7.16)4,5,6 gives after some calculation
c2(nm2 -
t
Cy) h;ht
I
(7.24)
where the brackets ( ) and ( } in (7.24)2,3 are equal to the corresponding brackets in (7.21),,,. The equations (7.24) are approximate, because in their derivation all terms of the general form flUp,,, nn,,, 17e,,, qa Up,,, qorn,y, qae,?, tcap> Up,,, t<,,g,,, tcor8>e,, have been ignored. Such terms would have been ignored by Miiller and Israel, if in fact they had appeared in their work, because they would have considered them as non-linear. Comparison of (7.23) and (7.24) reveals strong similarities between the two sets of equations. There is one difference though which makes the equations (7.24) of the present theory superior to the equations (7.23) of the old theories. Indeed, in (7.24) all coefficients in the square brackets are explicitly known to within one function of a single variable, see (7.7). Thus, while the old theories contain the eight unknown functions ~1,rc, v, A, B, C, M and N of two variables each, the present theory contains only three unknown functions of two variables, viz. B;, B,, B, plus one unknown function of a single variable. In addition, of course, the new theory contains a number of explicit non-linear terms which had formerly never appeared. Another difference between the sets (7.23) and (7.24) is only apparent. The factors of qcm,sj and - t<,~>,8 in (7.23),,2 and the factors of n,, and qb,a in (7.23),,, are the same ones while this is not obvious for the corresponding factors in (7.24). However, the alternative forms of A; and A! given in (7.8),,, guarantee the same relation between those pairs of factors as in (7.23).
RELATIVISTIC
Inspection
THERMODYNAMICS
211
OF GASES
of the equations (7.24) shows that the three coefficients
71 5c3-~~m2-~)~~=4~~2m6~~
--
3
B.4
c2 c; --= 6 B;
D,
A4
BJ,.,
’
2m6c7
-4ny 7
(7.25)
0; e12.1
z2,2
z2,2
z2,3
I A;*
B;
are to be considered as relaxation times for t<“‘), q” and 17 respectively. Note that the inequalities (7.9) ensure the positivity of these relaxation times. The second members in (7.25) are derived by use of (7.7) and (7.8). 7.7. The Non-Relativistic Limit for a Non-Degenerate Gas
The non-relativistic theory of extended thermodynamics with 14 variables has been formulated by Kremer C7.183. In that theory the variables are given by nonrelativistic moments of up to fourth order. It is therefore not obvious how the present theory, which uses only first and second order moments, is related to the non-relativistic one. Dreyer & Weiss C7.193 have investigated this question and they have found complete agreement between the weakly relativistic limit y B 1 of the present theory and Kremer’s non-relativistic theory. In the process of comparing the two theories Dreyer & Weiss were able to determine the orders of magnitude of the functions B;, B, and B, that remain unknown in the present paper. They grouped all quantities according to their “relativistic order of magnitude” as O(y), O(l), 0(1/y) etc. and they found that in case of the non-degenerate gas we have’ (i) B;isofO(l) (ii) B, is of O( 1)
(iv) CT tends to -6--
m2 1
(7.26)
kTy
z Kremer’s theory contains two arbitrary constants which we have taken to be zero in (26), because they vanish in the kinetic theory of gases.
212
LIU,
MijLLER,
AND
RUGGERI
From (i) it might seem that the bulk viscosity v in (7.22) were of O(l), but this is not so. In fact, on the contrary, the factor of l/B; in (7.22) is of @(l/y*) so that v vanishes in the non-relativistic case as is to be expected for an ideal gas. The viscosity and the heat conductivity are both of U(1) as can be seen from (ii), (iii) and (7.22). APPENDIX
A.l. Evaluation of Entropy Principle, The vector h’* that determines V”, of the Lagrange multipliers 5, /i, corresponding constitutive function transformations i.e. we must have
The general solution of this functional
Symmetry Conditions and Trace Condition
Y@ and A@? according to (5.7), is a function and ,Z’,,. By the principle of relativity the must be isotropic with respect to arbitrary
equation is given by the representation
A=0
where ,4cA)= (ZA),Bns has been defined with Z$ = gaS and where the coefficients yA may be Rnctions of the scalars <, GA=A”AkA)(A=O,
1, 2, 3), Qi= (Zi+‘)OLa (i= 1, 2, 3).
We insert (A.2) into the equations (5.7),,,,, and apply the requirement that 7@ and AaBy be symmetric as well as the requirement (2.3)2 on the trace of 4. These requirements severely limit the generality of the coefficient functions ya in (A.2). In fact, if we are interested to have & to within third order terms in g the coefficients ?A must be calculated so as to contain C in up to (3 - A)th order and these read
Y*=r*+ac,G
ar,
1 a*r, ar3 ar2 YI=rl+~G’+Z~G:+dC,G,+~r,Q, ar,
1 a*r,
ar,
i
(A.31
i
Yo=ro+ac,G’+Z~G:+acoG,+4r2Q,
1 a*r, p-C:+ +zaG:,
a*r, -G,G,+~G,+t~G,Q~+~r,e*, aG; 0
0
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THERMODYNAMICS
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213
where the rA are functions of 5 and GO which can be shown to satisfy the differential equations.
&
(G;T,)
=f
(m*c*G;r,)
0
&(G~r2)=$z2c2G;r,)
(A.41
0
j$ (GV,)=-$(m*c*G;r,) 0
In conclusion we may say that the constitutive functions for ha, v*, T”fl and AaBy are now specific except for the unknown function r,(<, Go) and functions of < which emerge as constants in the integration of (A.4). A.2. Determination
of rA
The evaluation of the entropy principle was much simplified by the choice of the Lagrange multipliers 5, ,4’ and ,PB as variables. However, since these multipliers have no a priori significance, we should much prefer to state our results in terms of the original variables V”, T”fl of Chapter 2 or, even better, in terms of the suggestive variables n, e, Z7, qp, t(““> of Chapter 3. The first step in that transformation is effected by equating
(A.51
to the equilibrium
expressions implied
by (5.7),,2,j and (A.2), (A.3), viz.
are /ialE.
are hOLIE=- tag+2GOdco
>
214
LIU,
It follows by some calculation
MijLLER,
AND
RUGGERI
that we have
which identifies &,/c as the reciprocal of the absolute temperature T. Moreover, we obtain an interpretation of the equilibrium values of the Lagrange multipliers t 1E, /i a 1E in terms of IX and T, namely
(A.81 The second part of (A.7) follows from (7.4),. We conclude that g IE is closely related to the Gibbs free energy g = l/n(e +p) - Ts and to the variable a which occurs in the equilibrium state functions (7.4),. Finally the comparison of (A.5) and (A.6) gives
adGo at
q/Gko
= cn,
=o
ar, -- c3 co @)= JG,3 I7
e
z-o= -p
(A.9)
=x’
r,= -2 %
nm2 - Cy)
3&
of which the latter two equations imply an integrability
condition
for Cy, viz.
‘(e+p).
(A.ll)
The key to the further treatment, and indeed to the whole procedure of this paper is the observation that the knowledge of n(a, T), p(a, T) and Cy(a, T) from equilibrium statistical mechanics (see (7.4),,,,d) allows us to calculate r. and f1 from (A.9)3 and (A.10)2. The remaining functions r2 and r3 may then be calculated integrating (A.4). Frequent use of the recurrence relations (7.6) finally provides the solutions To(a, T) = 47ty q
m4c5J4,0
Tl(a, T) = 47cy 2
m6c7z6&,
RELATIVISTIC
THERMODYNAMICS
215
OF GASES
+pw+w 2 y6
-
y*
(A.12)
The functions A(a), B(a) emerge as constants of integration. A.3. Constitutive Relation for Aa@ as a Linear Function of II, qa and t<@) Insertion of rA from (A.12) into the formulae (A.3) for ya and insertion of those into the representation (A.2) gives h’” and by differentiation we obtain AaPY according to (5.7). However, there is a drawback, because the AmBythus calculated is still a function of the Lagrange multipliers 5, /i, and Z,,. To be sure, in the previous section we have identified the equilibrium values of the Lagrange multipliers and we obtained
So now we have to calculate non-equilibrium values of the Lagrange multipliers. We shall be forced to limit the attention to linear departures from equilibrium and write
(A.13)
The quantities x, Aa, 1, a(@), ca and c are considered so small that products may be neglected. A’, cra and a(@) are normal to V and o(OLB>is trace-less. We write (5.7),., with (A.2) and obtain with (3.2) and (3.10) and by dropping non-linear terms in C:
+ 22
(nYPyn, 0
595/169/l-15
+ A!rYAy).
(A.14)
216
LIU,
MtiLLER,
AND
RUGGERI
On the right hand side we introduce (A.13), dropping all non-linear terms in 1, A”, i, o
4ny 27 zz -- 3 ‘=G4m”c7
0; 6 I,,,
I,,, I ’
1 12,1 I,,2 D; 6 I2,2 12,,I ’
(A.15)
In the next step we calculate a linear constitutive relation for A,,, as follows. We write the linear part of (5.7), with the coefficients yA inserted from Section A.l. Thus we obtain
+
+g= (A,A&JS
+ Ll,n,c,,n~ + flyAaCpsA~).
(A.16)
0
We insert (A.13) with (A.15) and, again dropping all non-linear terms, we arrive at an explicit expression for A,,, which contains but one unknown function of a single variable, viz. A(a). Comparison of that expression for A.,, with the representation (4.3), identifies the coefficients C;, C3 and Cg. The results have been listed in Chapter 7, see (7.7).
RELATIVISTIC
THERMODYNAMICS
OF GASES
217
A.4 The Entropy-Entropy
Flux Vector and the Entropy Production The equations ( 5.7)1,2,3 may be combined in the form dh’” = V= d[ + T@ d/i, + A@ dC,
By use of (5.3) this relation entropy-entropy flux vector
(A.17)
may be converted into a differential
dh*= -
form for h”, the (A.18)
We replace Vu and 7’@ by their decompositions (3.2) and (3.10) and insert the representations (4.3)* and (4.3), for AmByand h”. Moreover we introduce the linear expressions (A.1 3) with (A.1 5) for r, /i, and L’,,. In this manner we obtain a differential form in dZ7, dq” and dtCru> on both sides of (A.18). Comparison of coefficients then identifies the coefficients A; through Ai in (4.3),. The results have been listed in Chapter 7, see (7.8). Thus we have obtained explicit expressions for the entropy density (A.19) and the entropy flux (A.20)
Note that the entropy density contains non-equilibrim contributions which are quadratic in U, q” and t<@). The coefficients A;‘, AT and Ai of these terms will be shown to be negative by the hyperbolic argument of Section A.5 below, thus ensuring that the entropy assumes its maximum in equilibrium. Note also that the entropy flux is not just equal to the heat flux divided by temperature, rather it contains two quadratic contributions. The residual inequality (5.7), can be written more explicitly by use of the linear representation (4.3), and of the decompositions (A.13),. We obtain for the entropy production ;=
12B;on-2B,o”q,-B,~(“P>t(,B>
20
(A.21)
or by use of (A.15)
n2
+ - 3 B4 z4,0 Tg--4%ol+y--mc 4zyD3
15 B, 1 tt
218
LIU, MULLER, ANLI RUGGERI
We conclude that there are three “mechanisms” for entropy production. One is related to the dynamic pressure, one to heat flux and one to shear stress. For the entropy production to be non-negative the coefficients By, B,, and B, must satisfy the inequalities that were listed in (7.9), . Inspection of (A.22) confirms this statement. AS. Some Consequences of Hyperbolicity In the requirement (6.7) of hyperbolicity we replace I/*, Fp and A’/? by their representations (3.2), (3.10) and (4.3),. Also we introduce the linear expressions (13) with (15) for 5, /1, and C,,. In this manner the left hand side of the inequality (6.7) becomes a quadratic form in the variations an, 6T, 6U”, SIT, 6qP, &(P”). Since the inequality must hold for arbitrary values of these variations, the coefficients of (&I)~, (6T)2, @If)‘, 6q’dq,Jt<“‘> 6t<,,> must be positive. This requirement leads to the conditions that have been listed in (7.9),, if we make use of the explicit form (7.8) for the coefficients A$, A? and A:. However, the conditions (7.9), do not exhaust all consequences of the requirement of hyperbolicity. Indeed, some of the remaining consequences impose bounds on the values of rc, qa and t<@) in terms of values of n and T. As long as we stay close to equilibrium, these constraints will be satisfied but they cannot hold far from equilibrium. This situation is not different from the non-relativistic case investigated in [ 171.
ACKNOWLEDGMENT One of the authors (I-S.L.) gratefully acknowledges the support of the Humboldt Foundation.
REFERENCES 1. C. ECKART, The Thermodynamics of Irreversible Processes III: Relativistic Theory of the Simple Fluid. Phys. Rev. 58 (1940). 2. I. MOLLER, Zur Ausbreitungsgeschwindigkeit von Storungen in kontinuierlichen Medien. Dissertation TH Aachen (1966). 3. W. ISRAEL, Nonstationary Irreversible Thermodynamics: A Causal Relativistic Theory. Annuls of Physics 100 (1976). 4. I-SHIH LIU, Method of Lagrange Multipliers for Exploitation of Lagrange Multipliers. Arch. Rational Mech. Anal. 46 (1972). 5. T. RUGGERI, Symmetric hyperbolic system of conservative equations for a viscous-heat conducting fluid. Acta Mech. 47 (1983). 6. N. A. CHERNIKOV, The Relativistic Gas in the Gravitational Field. Acta Phys. Polonica 23 (1963). 7. N. A. CHERNIKOV, Equilibrium Distribution of the Relativistic Gas. Actn Phys. Polonica 26 (1964). 8. N. A. CHERNIKOV, Microscopic Foundation of Relativistic Hydrodynamics. Acra Phys. Polonica 27 (1964). 9. C. MARLE, Sur l’l?tablissement des Equations de I’Hydrodynamique des Fluides Relativistes Dissipatives. Ann. Inst. Henri PoincarC 10 (1969).
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10. G. M. KREMER, Zur erweiterten Thermodynamik idealer und dichter Gase. Disserration TU Berlin (1985). 11. K. 0. FRIEDRICHS AND P. D. LAX, Systems of Conservation Equations with a Convex Extension. Proc. Nat. Acad. Sci. 68 (1971). 12. G. F. SMITH, On Isotropic Integrity Bases. Arch. Rarional Mech. Anal. 18 (1965). 13. C. C. WANG, A New Representation Theorem for Isotropic Functions: Answer to Professor G. F. Smith’s Criticism of my Paper on Representations for Isotropic Functions. Part I: Scalar Valued Isotropic Functions. Arch. Rational Mech. Anal. 36 (1970); Part II: Vector Valued Isotropic, Symmetric Tensor Valued Isotropic, Skew-Symmetric Tensor Valued Isotropic Functions, ibid. 14. F. J~TTNER, Das Maxwell’sche Gesetz der Geschwindigkeitsverteilung in der Relativitatstheorie. Annalen der Physik 34 ( 1911). 15. F. J~~TTNER, Die relativistische Quantentheorie des idealen Gases. Zeitschriff 1: Physik 47 (1928), 542.
16. W. DREYER, Statistical Mechanics of a Relativistic Gas in Non-Equilibrium (in preparation). 17. I-SHIH Ltu AND I. MILLER, Extended Thermodynamics of Classical and Degenerate Ideal Gases. Arch. Rational Anal. 83 (1983). 18. G. M. KREMER, Extended Thermodynamics of a Gas with 14 Moments (in preparation). 19. W. DREYER AND W. WEISS, The Classical Limit of Relativistic Extended Thermodynamics (in preparation).