Relativistic Boltzmann equation and relativistic irreversible thermodynamics

(1)

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Relativistic Boltzmann equation and relativistic irreversible thermodynamics

Kefei Mao, Byung Eu

To cite this version:

Kefei Mao, Byung Eu. Relativistic Boltzmann equation and relativistic irreversible thermodynamics.

Journal de Physique I, EDP Sciences, 1993, 3 (8), pp.1757-1776. �10.1051/jp1:1993214�. �jpa-00246830�

(2)

Classification

Physics

Abstracts

05.20D 05.70L 04.90 51.10

Relativistic Boltzmann equation and relativistic irreversible

thermodynamics

~~

Kefei Mao

(')

and

Byung

Chan Eu (~.

~)

(~

Department

of

Physics

and Centre for the

Physics

of Materials, McGill

University,

Montreal, Quebec H3A 2K6, Canada

(2)

Department

of

Chemistry,

McGill

University,

Montreal, Quebec H3A 2K6, Canada (Receii,ed 23 September J992, revised 29 March J993,

accepted

9

April

J993)

Abstract. The covariant Boltzmann

equation

for _a relativistic gas mixture is used _to formulate _a theory of relativistic irreversible

thermodynamics.

The modified moment method is

applied

to derive various evolution

equations

for

macroscopic

variables from the covariant Boltzmann equation. The method

rigorously yields

the entropy differential which is _not _an _exact differential if the system ^is away from

equilibrium.

Therefore, _an extended Gibbs relation does not hold valid for the entropy

density

in _contrast _to the usual surmise taken in extended irreversible

thermodynamics.

However, _an extended Gibbs relation-like

equation

holds for the

compensation

differential _as has been shown _to be the _case for nonrelativistic gas mixtures in _a recent work. The entropy ^balance

equation

is cast into _an

equivalent

form in _terms of _a _new function called the Boltzmann function.

The

equation

is _seen _to be _a local expression of the H theorem.

Macroscopic

evolution

equations

(I.e.,

generalized hydrodynamic equations)

are

presented

for various

macroscopic

variables.

Together

with the equivalent form for the entropy ^balance

equation,

^these

macroscopic

evolution

equations

form _a mathematical structure for _a

theory

of irreversible processes in relativistic monatomic gases.

1. Introduction.

The relativistic Boltzmann

equation [1-6]

_was used _to

study

irreversible processes in

relativistic gases

by

_some authors who

employed

either the

Chapman-Enskog

method

[7]

_or the

Maxwell-Grad moment method

[8]

_to solve the kinetic

equation. Especially,

these methods

were used _as _a _means _to obtain _or

justify

_a

macroscopic theory [9-12]

of irreversible processes

in the relativistic framework

[4-6].

Since the aforementioned methods do _not afford _an exact

solution for the

problem,

_one

justifies,

within the

validity

of the

approximation used,

_a

theoretical structure of irreversible

thermodynamics

_one has in mind.

However,

the

justifica-

tion is _not without _an uneasy

feeling

since

thermodynamics

makes

rigorous

demands _on

theories of

macroscopic

irreversible processes

occurring

at all

degrees

^of removal from

equilibrium

^but

approximate

kinetic

theory

^results can

hardly

be relied on for _a

sufficiently

general

and

rigorous theory, especially,

if the

system

^{is far} from

equilibrium.

When _one aims

(3)

1758 JOURNAL DE PHYSIQUE I N° 8

to formulate _a

theory

of irreversible processes, the

question

of the nature of entropy

unavoidably arises,

but relativistic kinetic

theory

has not _as yet

yielded

_a

satisfactory

_answer _to

the

question.

The range of

validity

of the

existing approximate

relativistic

therrnodynamic

theories

depends

on the _answer _to this

question.

Recently,

the modified _moment method

[13]

for nonrelativistic Boltzmann

equations

has been shown _to

yield

a

rigorous

_answer _to the

question

of whether the entropy differential for irreversible processes is _exact _or _not. The _answer _turns _out _io be that it is not e,<act

if

the system is away

fi.om equilibrium.

In this paper we aim _to show that the same answer remains valid

even for relativistic gases of finite _masses that _can be described

by

_a relativistic Boltzmann

equation.

2. Relativistic kinetic

equation

and balance

equations.

We _assume _an _r component ^mixture ^of

nonequilibrium

relativistic monatomic gases of finite

masses. The _case of massless

particles

is excluded in this

theory

and will be discussed

elsewhere

[14].

It is well established that for such _a mixture the

singlet

distribution function

f,

_{(.<, p,}

obeys

the covariant Boltzmann

equation [1-6]

Pf ?»f, (x,

_Pi

)

,

=

I

C

~f,, fj (2. II

,

where _x

= (ct, r

),

_p, ₌

(cp$,

_p, with

pi

=

(p)

₊

m) c~)'~~,

and the collision

integral

is

given by

the formula

C

~f,, fj

=

G,j ld~Pj ^d3p,*

^d3p~*

^w,~

^{~p, p~}

^p,*

^p~*

^j

x

lf,* (x, P,* fj* _{(x, Pj*} f, (x,

_Pi

) fj

(x,

Pj Ii

,

(2.

2

j

with d~fi~ =

d~p~/p),

_d~fi~* =

d~p~/p),

^etc. ^and _W~~~p, p~

p,*

_p~*

denoting

the _rate of transition from the initial _state

~p,, pj

^to

~p,*,

_p~* as a result of _a collision between

particles

I and

j.

The

factor G,~ =

3,~/2

^insures ^that ^the ^final state is _not counted twice. The asterisk denotes the

post-collision

value. Note that the

subscripts

I and

j play

a dual role of

labelling

a

species

and _a

particle

of that

species.

The transition _rate

W,~~p, p~(p,p~)

^is a scalar under Lorentz transformation and

obeys

the

microscopic reversibility [6].

The distribution function is normalized _to the number

density

of the

species

in

question

:

n,(x)

=

d3p, f, (x, p~) (2.3)

The

particle

four-flow

Nf

and the covariant energy-momentum tensor of

species

I _are

defined, respectively, by

the statistical mechanical formulas

Nf

(x

j

=

(pi _f~

_(x, _p,

j

,

(2.4)

and

Tf

^"

(x)

=

(Pf

P,"

f, (x,

_P,

)) (2.5)

Here the

angular

brackets _are the abbreviation of the

integral l'' )

" ~

d~P>~pl

~ ~

d~»..

(4)

Then, the total number four-vector N" and total

energy-momentum

tensor T"" _are

given,

,

respectively, by

the _sum of the

species

_components _over all

species

_: N"

=

jj Nf

and

, =i

~Mv

~j

^~Mv

,

i =I

The entropy four-flow may be also defined in covariant forrn

s~

(x

=

f _St

(.<

j

=

f k~ (pi _{tin f~ (x,}

_p,

_j

i

j f,

(x, _p,

j (2.6j

for the whole system ^of ^the ^mixture ⁱⁿ

question.

We will later _return _to this

important quantity

to discuss its consequences for irreversible

thermodynamics.

Following

Eckart

[15],

_we define the

hydrodynamic velocity by

U~

=

cN~/(N

"

N~

)'~~

(2.7)

This may be put into the form

U"

= p ^' N"

(2. 8)

where _p is the

hydrodynamic

number

density

defined

by

p =

f

p, =

f _c~~Nf

U~

₌

c~~N" U~. (2.9)

,

These formulas

imply

that

p(x)

=

n(x)(I u~/c~)'~~

and U"

=

(c, u)(I u~/c~)~

^'~~

Various relevant

macroscopic

variables such _as energy

density,

heat

flux,

_etc., _can be

defined in covariant form with the

help

of the

hydrodynamic four-velocity just

defined. The

scalar internal energy

density

_E~ of

species

I is

given

in _terms of the internal energy

df, per

particle

_or in _terrns of the energy-momentum tensor

by E,

= p,

8,

= c~ ^~

U~ Tf" U~ (2. lo)

,

The total internal energy

density

will be denoted

by

E

=

£ E,.

With the definition of the

, i

projector

A"~

=

g""

₊

c~~

_U"

U", g"~ being

the metric tensor, the heat flux

Qf

and diffusion flux

Jf

are defined

by

Qf

=

U~ T)" AS, (2. I)

Jf =Nf-c,N", ^(2.12)

where c, is the number fraction defined

by

c, =

p,/p

^(2.131

We will find it _more convenient to use a new heat flux defined

by

Q,'"

=

Qf h, Jf (2.14)

where _h~ is the

enthalpy

_per

particle

of

species

I

h,

=

8,

₊ _{p, v,} and h

=

£

_c,

(8,

₊ _p, _{v, )}

=

8 ₊ pv

(2.15)

1=1

(5)

1760 JOURNAL DE

PHYSIQUE

I N° 8

Here p, is the

hydrostatic

_pressure of

species

I, _v,

= p/ I, and v ₌ p ^l The heat fluxes of

,

different

species

_sum _up to the total heat flux

Q~' Q~

=

£ Qf.

The stress tensor

=1

Pf"

is defined _as

Pf"

=

AS T[~ A). (2.16)

This may be further

decomposed

into the traceless

symmetric

part

1I,"",

the _excess _trace part A, ^and ^the

hydrostatic

_pressure _p, of

species

I _as follows _:

Pf"

₌ _p, A"" ₊ A, ^A~"

+1I,~" (2.17)

The traceless part ^is ^related to the viscous

phenomena

while the _excess trace part is associated with the dilatation

(or compression)

of the gas. The

hydrostatic

_pressure is defined in _terrns of

equilibrium (or

local

equilibrium)

energy~momentum tensor

T$"

p, =

A~~ T$" (2.18)

Then,

it is easy ^to see that

A, ₌

A~~ Pf"

_p, ₌

_A~~ Tf"

_p,,

(2.19)

1I,""

₌ P

f" A~~ T$~

A""

=

AS T)~ A) A~~ T)~

A~"

(2.20)

3 3

The definitions made for the energy

density,

heat

flux,

and stress tensor are

equivalent

to the

decomposition

of the energy-momentum tensor as follows _:

Tf"

= c~ ^~

E,

U" U" ₊ c~

~(Qf

U" ₊ U"

Qi)

₊

Pf"

=

c~~ E,

U" U" ₊ p, A~" ₊

c~~(Qf

_U" ₊ _U"

_Q,")

₊ _A, A""

+1I,"" (2.21)

The entropy

density S(x)

and the entropy 5P _per

particle

_are scalars defined

by

_:

s(x)

=

psP(x)

= c- ² s»

u~ (2.22)

This is the

quantity

_we _are

going

to deal with in the

theory

of irreversible processes that will be constructed _on the basis of the covariant Boltzmann

equation.

The

particle

number and energy-momentum conservation laws _are

easily

derived from the

covariant Boltzmann

equation

^since ^the

particle

number and energy-momentum are the

invariants of the Boltzmann collision _terra. The

particle

number balance

equation

is

3~N~ (x)

=

0

(2.23)

while the energy-momentum balance

equation

is

3~T~"(x)

= 0

(2.24)

With the definitions of the operators ^D = U~

3~

^and ^V~ =

A~ "

3~,

^{it is} easy to obtain from

(2.23)

_or

(2.24)

the various fluid

dynamic equations

_as follows _:

Equation of continuity

:

Dp

= p

V~U~ (2.25)

(6)

Density fraction

balance

equation

For systems without _a chemical reaction

pDc,

=

3~Jf _(2.26)

Momentum balance

equation

c~ ^~

hpDU"

₌

V~p

₊

AS 3~fi""

₊

c~~(Af _DQ" Q" V~U" Q" V~U") (2.27)

where

~~~

"

f ~~'

~"" ₊

Hi")

= AA"~ ₊ n«v

'

(2.28)

Energy

balance

equation

_:

pD£

=

V~Q"

_p

V~U~

+

fi"" V~U~

+ 2

c~~ Q~ DU~ (2.29)

These

equations

and the flux evolution

equations presented

later,

together

with the entropy

balance

equation,

form _a set of evolution

equations

in the

theory

of irreversible processes

presented

in this work.

3. Flux evolution

equations.

The various balance

equations

derived earlier contain

macroscopic

variables such _as the _stress tensor, heat flux and diffusion

fluxes,

etc. which

require

their _own evolution

equations.

We

can derive them from the kinetic

equation.

To this end, we first define _a _tensor

~/"~""...

^~" _as the average of its molecular

expression pi h)"'""

~,~"~ ~"~"

=

lpi h)~~~"..~ f,(x, P,)) (3.I)

where

h)"~~"...~

is the n-th element of _a set of molecular moments in _terms of which the distribution function may be

expanded.

The

leading

moments relevant _to

physical applications

of the

theory developed

here _are _:

h)~~⁼ )

(U> P)

_)~^'

A~ ~ Pf PI

- P,/P,

,

_(3.2b)

h)~~

"

= -

~(U>

)

)~ (AS _p[ _pi _U~

etc.

where

~i "

At 4~ (P$ P~ fe, _gp

v

[AS 4~ (h, Pi PI

_~P~ U

> g

~~

(3.2e)

These _moments tend _to the nonrelativistic moments

appearing

in the nonrelativistic kinetic

theory [13]. They

_are constructed

by

_means of the Schmidt

orthogonalization

method such that

they,

when

averaged, give

rise _to

physically

relevant variables. We also _note that a; - I _as u/c

- 0.

We then define

macroscopic

moments

di,~~~~"

_~, which _are _tensors of various

ranks, by taking

contraction of

~,~"~~"...

^~" with

U~.

W,~"'~"

= c~ ^~

U~ ~,l"~~"

^~"

_(3.3)

(7)

j762 JOURNAL DE

PHYSIQUE

I N° 8

It is then _easy to

show, by using

the definitions

of1I,~~

etc. that

~ ~

~j,,~v

_~/~»>.

#,12'

= A, W>~~~" _"

Q/"

_~°>^~~

~'

'

~~~ ~~ ~~

Thus, _we _are

treating,~,~"

' ~~ ~"

as a component ^of « four-tensor _» and

macroscopic

_moments di,~" ^~"~ are their contractions with the

hydrodynamic four-velocity

in the _sense similar to

the scalars such _as the

density,

_entropy

density,

_etc. This way,

they

are put on the

equal

conceptual footing

to the conserved variables and the entropy

density

to which

they

_are related

as will be shown later. The evolution

equation

for

~,~"'"~

^~" is

easily

derived from the covariant Boltzmann

equation

?«4r,~"~

^"~ ^~" ₌

lpi ?«hl"

^~"~

_f,

_{(,<, Pi}

))

+ A,~~~ ^"~

(3.5)

where

~(al»v.

^f

j~ jj~(«)»v

^f

~~ ~ jj

_~~

_~j

, , >, j

j=1

With the definition of

d)~'"~

as

di,~~'"~...~

_per

particle,

Wj~~~"~.~=pd)"J"~ _(3.7)

and

by making

_use of U"

V~

=

0,

_we obtain

pDdi°~»~ ~=zi~~»~ _~+A,~~'»~. _(3.81

where the kinematic term

Z)"~"~

is

given by Z)"~"~

"

(Pi 3«h)"~"~ f>(.t, Pill

₊

_?T(4$ ~)"~"~' ~") (3.9)

This statistical mechanical formula may be worked _out _more

explicitly,

but it is _not necessary to do _so at this

point

since _our immediate interest does _not lie in the relativistic

theory

of

transport _processes. See table I for their

explicit

forrnulas. We

emphasize

that the evolution

equations (3.8)

are

coupled

to the momentum, energy ^and concentration balance

equation

introduced earlier. As in the nonrelativistic

theory

this _set of

equations

is open and therefore must be

suitably

closed

by

means of _a closure relation. When the set is thus closed, it

provides

relativistic

generalized hydrodynamic equations consisting

of

(2.25)-(2.29)

and

(3.8)

with the

number of _moments limited _to _a finite value

by

_a suitable closure. The _moment evolution

equation (3.8)

can be

easily

shown to tend to the nonrelativistic counterpart ⁱⁿ ^the ^limit ^of

c _- ~x~. The nonrelativistic limit of

(3.8)

is

quite

clear from table I for

Z)"~

in which the

relativistic effect terms are

easily

discernable. The _manner in which the moments

(di,~"')

_are defined in this work is

significantly

different from the _one in the conventional

moment method

[5]

and _as _a consequence their evolution

equations

are

accordingly

different.

4. The H theorem,

temperature

and _pressure.

4.I THE H THEOREM. The entropy ^balance

equation

_can be derived from the kinetic

equation. By covariantly differentiating (2.6),

_we obtain

~

~s»

= ~r

~~,

(x ) (4, )

(8)

Table I. Kinematic Terms

Z)~~.

Definitions

[VU]~~~""

=

(V"U"

+

V"U")

A""

V~U~, [A.

B

]~~J""

=

(A~" B$

₊ A""

Bf )

A""A"~

B~~.

2 3

Kinematic Terms

Z)~~""

=

V~ ~,~~J"""

2[1I~

VU]~~'"~

₊ _{2 p~}

[VU]~~J"~

₊ 2 A,

[VU]~~~"~

2 _C~^~

IQ, DU]

~~~""

+

If, _{PI PI} P) _~Pl U~)~ V«U~ (Af

U~ ₊ U"

Ail j ^~~'

^~' ^~' ^~'

^{~' ~E}

^~~

^~fT

^~~

^~"~

^~ ^~

^~"~

~ ^~ ~> ~> ~' ~' ~l

~f' ~F~ ~W~l ~~ ~T

~

~fT

~

~~

^~ ^~ ^~ ^~ ~~ ^~ ^~~

c~ ^~

(P f~

U~ ₊

P)~

U~

DU~

+ ~p, + A,

)(U~

DU~ ₊ U~ DU"

Ii

,

Z)~~ =

V~

~,~~~~ +

~1I,"~[VU](~)

_p, D In ~p, v~~~) +

~

A,

V~U~ _V~ (J, p,/p,

+

j

^C~ ^~

^Qf Du~ j

^c2

^{f, pi pj} p( p) _~pj _u~)-

²

v~u> A~t

,

~

zl~~»

=

v~

~r,~~>~» +

H,»~Duv

_~p, ₊ _A, Du»

Jn Dh,

Q,>"

vvu»

+ a,

J,"

U"

V~h,

+ ~p, + A, ^U"

V~U~

U"

HI" V~U~

+ c~ _a,

f, p," pf ~p[ U~

)~ ^'

V~h,

c~ ^~ U"

Qi'" DU~

+ C~

If, _P," _Pi PI _{~Pl U~}

_)~^~ _a,

_h,

V~U> +

(a, h,

_n,

E,

DU"

+ U~

NT h, ?«a,

C~

If, _Pi _P," _PI _~Pl _U~

_)~

_{hi ?«a,}

+ C.~

If,

_P,"

Pi PI _(U~ PI

)~^~

h,

V~U>

,

z)~~~

=

V,'

^~"

Jj V~u"

_p, Du~ _c~ ^~ u"

Jf Du~

C~

If, Pi PI PI _{~Pl u~}

_j- ²

_v~u>

where

"ent (X "

iB I I (i~ f>

(X, _p,I C ~f>,

fj11 ^(4.

21

j

By following

the well known

procedure [6]

based on the symmetry

properties

of the transition

probability W~p,

_p~

p,*

_{p~* ),} it is easy to show that the entropy

production «~~~(x)

is

positive

~ent(.t)

~ °

(4.3)

with the

equality applying only

at

equilibrium.

This is the content of the H theorem. It is _a

statistical mechanical

representation

of the second law of

thermodynamics.

The entropy

balance

equation

for the entropy

density

introduced in

(2.22)

_can be obtained from

(4.I ) by using (2.23)

pD5~

"

?MJf

+ "ent

(Xl (4.4j

(9)

where

Jf

₌ S" SAN "

=

S"

p5PU" (4.5)

Equation (4.4)

_may be _recast into the form

pD5P

=

V~Jf

+

c~~ Jf DU~

+

«~~~(x). (4.6)

Various balance

equations

and flux evolution

equations previously

introduced _are

subject

to the entropy ^balance

equation

in the _sense that the entropy

production «~~~(x)

must remain

positive

so that the second law of

thermodynamics

is satisfied. As it stands, the entropy balance

equation provides

_no clue _as _to how this

requirement

of the second law is fulfilled.

This

point

is

usually

clarified

by using

an

approximate

(e.g.,

the

Chapman-Enskog solutions),

but since the

therrnodynamic

laws must be satisfied

rigorously,

any conclusion drawn in that respect on the basis of _an

approximate

solution to the kinetic

equation

cannot be relied _on if the system ^is ^removed ^from

equilibrium by

_an

arbitrary degree.

Wi therefore _must

employ

_a

rigorous approach.

The modified moment method

[13]

meets this

stringent billing,

and _we will _use it in this work.

As _a further

preparation

for this method, we note that the

equilibrium

solution _to the

covariant Boltzmann

equation

is

given by

the well known Jiittner function

[6]

fe> "

~Xpl~ Pelj'f

_UeM _P

Ill (4.7)

where

p)

is the normalization factor which _tums _out _to be the chemical

potential,

U~ ^is ^the

velocity

_at

equilibrium,

and

fl~

tums out to be

proportional

_to the inverse temperature :

fl~

_m

I/k~

_T~

(4.8)

This parameter

p~

is deterrnined

by comparing

the

thermodynamic

_entropy, _pressure, and

internal energy with the statistical mechanical entropy

S~,

pressure and intemal energy

calculated with the

equilibrium

solution

(4.7)

of the kinetic

equation by following

the basic strategy ^of ^the ^Gibbs ^ensemble

theory.

Since the chemical

potential

is defined

by

p)/T~

=

(35~/3p,) (4.9)

in

thermodynamics,

the

statistically

derived forrnula for S~ indicates that the norrnalization factor

pi

is identified with the chemical

potential.

The ideal gas

equation

of _state is

similarly

identified

by

p=

pk~T~, (v

=

p~'), _(4,10)

This forrnula is also

easily

derived from

(2.18)

when the statistical forrnula for p is

computed

with the

equilibrium

(4.7).

Therefore, the

thermodynamically

defined pressure

through

the

equation

of state

(4,10)

is consistent with the statistical mechanical

definition of p, The situation thus is _seen _to be the _same _as in the nonrelativistic

theory

in that the

kinetically

and the

thermodynamically

defined pressure are identical,

4.2 TEMPERATURE IN RELATIVISTIC KINETIC THEORY. In the nonrelativistic kinetic

theory

the temperature ^of ^the ^fluid ^is ^defined ^such ^that ^the

following

relation holds true between the intemal _energy and the

temperature [7]

£

=

~k~T. _(4.ll)

(10)

The

temperature

defined in this _manner is called the kinetic temperature ^and ^coincides with the

therrnodynamic temperature

defined

through

the

thermodynamic

relation

T~ ^'

=

(35~/3E) (4.12)

However,

in the relativistic

theory

the situation is altered

significantly

since if the kinetic temperature ^is defined

by (4, II),

then the temperature so defined does not agree with the

thermodynamic

temperature ^defined

by

means of

(4.12), Nevertheless, (4.12)

is consistent with

(4.10).

One _recovers the coincidence between

(4.ll)

and

(4.12)

in the limit of u/c

-

0, provided

that the

mc~

_term is

neglected

_; see reference

[6]

for the

expression

for 8.

However,

it would be

preferable

to have the

thermodynamic

temperature ^coincide ^with the kinetic

temperature

ⁱⁿ ^the relativistic

theory

_as

well,

since the kinetic

theory

should be _a

molecular

representation

of

thermodynamics underlying macroscopic

_processes. Here _we

define the kinetic temperature for _a system ^of

particles

of finite _masses such that it

precisely

coincides with the

thermodynamic

temperature as follows

pkB

_Te

"

I (2

C~)~

~Pf PI

_UeM

Uev ~)

_C~ _{fe> (X,} _P,

)) (4.13)

This definition

yields

a correct nonrelativistic limit. It is still rooted

basically

in the idea that the temperature is _a _measure of kinetic energy. Since the _rest _mass should not be _a factor in the measure, it is subtracted out in the definition, Since the

integrand

in

(4.13)

is

scalar,

the

integral

_may be evaluated in the local _rest frame where

U~~

=

(c,

0, 0,

0).

Then,

fe> " eXp

j- pe(Cp) /lj)j (4.14)

Now,

by following

the well-known

procedure

in the Gibbs ensemble

theory

we can

identify p~ by p~

₌

I/k~

_T~ as in

(4.8).

This also proves the coincidence of the kinetic temperature ^with the

thermodynamic

temperature, ^if ^the ^kinetic temperature ^is ^defined as stated in

(4.13).

Equation (4.13)

is another way of

expressing

Tolman's

equipartition

^theorem

[16]. Therefore,

the definition of temperature

by (4.13)

is _seen to be based _on the

equipartition

law. In other

words,

the statistical mechanical temperature may be defined

by

_means of the

equipartition

theorem of

Tolman,

which

equally

holds

for

the classical and relativistic theories. This

interpretation

of the definition of temperature

provides

_us with _a universal _way

of defining

temperature

for

both classical and relativistic theories _: the

equipariition

law

of

Tolman.

The local _rest frame version of the definition

(4.13)

of kinetic temperature ⁱⁿ

relativity

was

suggested by

ter Haar and

Wergeland [17],

and it _was later taken up

by

Menon and

Agrawal [18],

but _as

pointed

out

by Landsberg [19],

their definition is

preceded by

Tolman

[16].

On the other

hand,

in the

Chapman-Enskog theory [6]

the kinetic

temperature

^is ^defined ^such ^that ^the

equation

of _state in

therrnodynamics

and in kinetic

theory

^coincide ^with ^each other. Therefore, there appear to be _two different modes of definition for temperature, but, in

fact, they

are

identical _at least in the _case of ideal gases. We show it below. Since there holds the relation

Pi P) _{Ue» Uev} m)

C~

=

pf pi Ue» Uev Pi

_P>»

U( Uev

_" C~

Pi P) 4v»

,

(4.15)

substitution of

(4.15)

into

(4.13) yields

~

pk~

_T~

=

A~~ Tf"

3 2 ^~ which

implies

the

equation

of _state.

(11)

1766 JOURNAL DE

PHYSIQUE

I N° 8

5. Modified moment method.

With the

equilibrium

uniquely

determined

by

the H theorem and the parameters ^therein ^identified ^with ^the

thermodynamic

temperature ^and ^chemical

potential,

it is

now

possible

to

develop

_a method _to determine the

nonequilibrium

distribution function such that it is consistent with the H theorem. In the nonrelativistic

theory

such determination of a

nonequilibrium

distribution function is made

possible by

the modified moment method

[13].

We

apply

the _same method to the covariant Boltzmann

equation.

To

begin with,

the

following

consideration is

helpful

for

adopting

_a

general

strategy.

Since the

nonequilibrium

distribution function must reduce _to the

equilibrium

distribution function _as the system ^tends ^toward

equilibrium,

it is natural _to construct the former _on the latter. Therefore, we look for the

nonequilibrium

distribution function in the form

f>

~

fi(Pe,

_p>e,

_US

₎

_Af>

where

f)(p~,

_p,~,

Uf)

is the

equilibrium

already

determined and

Af,

is the

nonequilibrium

correction. This correction _term may be

expressed

in a

variety

^of

ways. Since the system is in

nonequilibrium,

it is

preferable

_to relax the

equilibrium

parameters

p~,

_p,~,

US appearing

in

f)

and make them

space-time dependent, namely,

p~

_~ p ; p,~ ~ p ;

US

_~ U"

(M.C.)

The

equilibrium

distribution function _so relaxed and made

space-time dependent

becomes the local

equilibrium

distribution function. The latter is the

building

block of the solution methods

in the kinetic

theory

of gases. These

correspondences

mentioned above _can be made

mathematically precise by

the

matching

conditions used in kinetic

theory.

Since the main aim of this paper is in

deriving

the entropy differential and

developing

the basic

equations

for transport processes, it is not necessary to delve into the

question

of

matching

conditions for temperature,

density, velocity

mentioned earlier. Such

question, however,

will have _to be dealt with when the

theory

of irreversible

thermodynamics

is further

developed

in

depth.

Being space-time dependent, p,

_p,, and U" _are not

equilibrium quantities. They

_are, in

fact, statistically

defined in _terms of

nonequilibrium

f,.

In

particular,

for

p

=

I/k~

T

3

PkB

T

=

£

C~^~

~Pf PI u~ u~ m)

c4

j f, (x,

_p,

)) (4.13

_>)

, i

This definition is the

nonequilibrium analog

of

(4.13) holding

for systems ^of

particles

of finite

mass.

Now, the

nonequilibrium

distribution function is taken with the

following

ansatz which contains

f~,

_as a part ôf ît ând is written in _an

exponential

form

f,

= exP

I-

P

~f u~

+

H)' _p,11 _(5.1)

where

H)~~

_~

i ~)~~

_fh~"~~

~f

"~~

(5.2)

£ xia)

_~

~(a)

«»j

(12)

and p, is the normalization factor

given by

the formula

exp(- pp,

₌ pi ^' c- ^~

p) _u>

_exp _p

pi u~

₊

£ X)()

_t

h)"~

^~"

l. ^(5.3j

« ~

The parameter p ₌

I/k~

T has the

meaning

_as fixed

by (4.13')

and elaborated earlier. The

tensors

X)j)

t ^are ^as

yet

undetermined functions of

macroscopic

variables, but will be

determined such that any

approximation

to them does _not violate the H theorem and

consequently

the

resulting

formalism for irreversible processes remains consistent with the second law of

thermodynamics.

To _construct such _a

formalism,

_we _assume that

X)"~

are somehow known and then

proceed

to construct the formalism.

(In

this _sense the attitude taken here is similar _to that

by

the Gibbs

ensemble

theory

where the

partition

function is assumed _to be known and with it

thermodynamic

functions and their relations _are

sought

after.

By making

sure that the entropy

production

remains

positive

in the

approximation

to be made _to the unknowns X,l~~ ⁱⁿ ^the

future,

the

consistency

of the formalism with the second law of

thermodynamics

is

assuredly preserved.

For _n

= I the unknown is _a second rank tensor since

h)'

is _a second rank covariant

tensor for _n

= 2 it is _a scalar _; while for _cy

=

3 and 4 the unknowns are vectors. We will

dispense

with the first mode of

expression

in

(5.2)

and use the

symbol

O whenever convenient.

The _sum _over

cy in

(5.2)

in

principle

_runs _over _any number of terms, but it must be such that

f,

is normalizable.

Only

this restriction is necessary in

formulating

the

theory

^since the distribution function is

explicitly

used

only

to define

macroscopic

observables in the moment

method, and _once that is done, ^its role in the kinetic

theory

is

basically

_over except ^for the entropy

production

which _must be

computed

with

f,

in

(5.I)

such that the restriction

just

mentioned is met. The

exponential

form for

f,

will be called the

nonequilibrium

canonical forrn.

Substitution of the

nonequilibrium

canonical form for

f,

into

(4.2)

for the entropy

production yields

the formula

"ent ~

~

i I l~)~)

_f

A/~~~

^~~

« >

1

~-

j~ j~ ^~(a

^j ^~ ^~ ^(a)

^{(5 ~j}

, j

i=laml

for which the collisional invariants of the Boltzmann collision

integral

_are

exploited.

As in the

case of its nonrelativistic version

[13],

the entropy

production

in this form suggests ^that energy

dissipation

arises

owing

to the

dissipative

evolution of nonconserved

macroscopic

variables

di,~"'"~

_As it stands in

(5.4),

_«~~~ is

easily

_seen to remain

positive

no matter what

approximation

is made to the unknown

Xl"

~, thanks to the H theorem and the

nonequilibrium

canonical forrn for

f,.

Since it is _not

possible

_to determine

X)"~ exactly,

^this property ^of

«~~~ ^to remain

positive

is very useful _to have in

formulating

a

thermodynamically

consistent

theory

^of irreversible processes. This is made

possible by

the

nonequilibrium

canonical form for the distribution function.

The entropy ^balance

equation (4.4),

_as it

stands,

does not reveal how the second law works to conform irreversible processes ^to it. With the

nonequilibrium

^canonical

form,

the situation is

greatly improved

and _we _are _now able to calculate the entropy differential

accompanying

irreversible processes in _a way consistent with the second law.

On substitution of the

nonequilibrium

canonical forrn into the statistical mechanical formula

(2.6)

and the entropy flux

(4.5)

and _use of the

decomposition (2.21)

for

Tf~,

_we obtain the

(13)

entropy ^flux four-vector in the forrn

, ,

Jf

₌ ^T~ ^'

£ (Qf

_P,

Jf )

₊ T~

£ £ XIII,

v

n;~"

~' "~

(5.5j

,=1 ,=1«~i

where

n,(«

^v «~

= ~r,(«> ^v ^«P

dim

~ « N»

= ~f,~" ^" ^"~

At ~~.~l

Now,

by substituting (5.4)

and

(5.5)

into the entropy ^balance

equation (4.4)

and

eliminating

the

dissipation

_term _A,~" and the

divergences

of

Qf

and

Jf

^with the

help

of the evolution

equation

for

di)~~

^and ^the balance

equations,

_we obtain DSP in the form

DSP

= T~

lD£

⁺

^pDv f

p,

Dc,

+

f _£

X)~~

3

Ddi)~

_c- ^~

0~ _DU~

₊ R

(5.7)

, =1 _, =1 _« _~

where

R

=

R~ R~, (5.8)

Rd

"

(PT)~ £ (- 1I,"" V~U~ A,

A""

V~U~

+

Qf _3~

In T ₊

Jf

T

3~jZ, (5.9a)

,

Rc

₌ _p ^'

£ £ _1?~ (k)a~

O

n,(a~~)

₊

ii"

_O

z)a'j

,

(5.9b)

, « ~ i

X)~~

=

X)~ ^J/T jZ,

=

p,/T

v ₌ p ^'

0"

=

Q"/p (5. lo)

Note that

-A""V~U~ =V"U~.

As in the nonrelativistic

theory [13],

we define the

compensation

differential

D V/

=

T- ^'

lD8

⁺

^pDv ^jj

^p,

^Dc,

⁺

f _jj X)~'3 Dd)"~

_c- ^~

0" _DU~ _(5.

_II

, , am1

which differs from the classical formula

only by

the appearance of the

velocity

derivative term that is

O(c~~)

and thus vanishes in the nonrelativistic limit. It is

interesting

to _see that in

relativistic

theory

the fluid

velocity

_appears

explicitly

_as _a Gibbs variable in _contrast _to the nonrelativistic

theory.

Since the term R does not vanish if the system ^is away from

equilibrium owing

to the fact

that the fluxes do not vanish in _a

nonequilibrium condition,

the entropy differential

DSP is _not

equal

to the

compensation

differential D V/ and hence does _not become _a Pfaffian form.

Equation (5.7)

_presages the

invalidity

of the extended Gibbs relation for DSP unless R

=

0

identically.

To _answer this

question

_more

definitely,

we examine R in detail.

The

nonequilibrium

contribution R may be _cast into _a _more transparent and useful form. To this end, the

following

^relations are useful. The

nonequilibrium

canonical form for

f,

_may be _recast into the fomi

£ Xl"

8

h)"'

=

AA,

kB ⁱⁿ

~f,/fi), _{(5. '2)}

« ~

where

AR,

₌

ji, p), p)

=

p)/T.

(14)

By using

the kinetic

equation

and forrnula

(3.9),

it is

possible

to show

, ,

I I Xl"

₃

Z)"

=

£ £ If, Pf _?~ (i)"~

_O

h)" _~))

₊

_d~ (Jr I)"~

₃ _~,~"~

_")]

, «~ , a~

£ £

,

If, Pi (?~ X)"~)

O

h)"J)

+

(d~ I)"~)

_O

_At

_4r,1"~

_"j _(5.131

,=1«~i

Since from the kinetic

equation

£ If, Pi _?~

ln

~f,/f)))

=

o

(5.14)

,

and in view of

(2.27)

and the

decomposition

3~

=

c~~

_{U~ D}

+ V"

If, Pf (?~ Xl"

_~l 8

h)"~)

⁺

(?~ X)"~)

³

^AS ~,~"'"

~(~p~i _)3~> ~~v(~M~' )3~'

=

(c-

²

u~

D ₊ v

~

j k)"

_{O ~,~~} "

(v~ I)a~)

_{O ~,~~} "

= ~- ² ~ ~ ^(a)M z

Dg(«)

=

w,("

o

Dk)"'

,

(5,15)

there follows the relation

,

Rc

= P ^'

£ INS ?» AA,

4~,~"~ 3

DX)"~]

, i

= p

f _Jf

V~ Ajz, f _jj _di)"

O

Dk)"

_{c, D}

_Ajz, (5.16)

,=1 >=1 «»1

The Gibbs-Duhem relation follows from the statistical mechanical definition of

p)

if the local

equilibrium

Jfittner function

f)

= exp (-

ppf U~

+

p J1))

is used. It may be written in the form

f

c,

Dji)

=

8D

(1/T)

₊ vD

~p/T) (5.17)

=1

Here,

the

equilibrium

energy

8~

is matched with the

nonequilibrium

_energy ⁸ ^and ^the

equilibrium density

_n~ matched with the

density

_n to obtain

(5.17).

Since

Jf U~

= o and thus

, ,

jjJ[3~ji,= jjJ)V~ji,,

,=i >=i

with

(5.16)

for

R~

^and

(5.17),

R _can be _recast into _a _more transparent ^form

R

=

f _£ _ail"~

3

Dk)"~

_c,

jZ~j

+ 8D

(1/T)

+ vD

~p/T) &~ (5.18)

, =1 « ~

where

&~

₌

(pT)~

^'

£ (- 1If" V~U~

+ A,

V~U~

+

Qf _V~

In T ₊

Jf

T

V~jZ)). (5.19)

(15)

1770 JOURNAL DE

PHYSIQUE

I N° 8

Note that this

quantity

is _a _measure of energy

dissipation

due _to irreversible _processes in the system.

By defining

the Boltzmann function fl3

,

fl3 = 5P T~ ^' 8 ₊ pv

£

_c, _p,

£ £ d)"~

₃

_X)~~ _(5.20)

>=i >=i«~i

and

combining (5.7)

with

(5.18),

we

finally

obtain _a differential

equation

for fl3 which is

equivalent

to the entropy balance

equation (4.4)

Dfl3

=

&~ c~~

_T~ ^'

0" _DU~ (5.21)

This

equation

differs from the classical

analog [13] by

the last _term which vanishes in the nonrelativistic limit. Since

by

the momentum balance

equation

c~ ^~

0~ _DU~

= h~

l~~ lv~p

+

A~~ 3~fi

^"" + c~

~(A~~ DQ" Q~ V~U" Q" V~U~)] (5.22)

we may write

(5.21)

in the form

Dfl3

=

£~ (5.23)

where

£~

=

(pT)~ f [-1I~"" V~U~ A,

A""

V~U~

+

Qf(V~

In T ₊ h~

V~p)

+

Jf

T

V~jZ)]

, =1

T~ ^'

11" lA~, ?~P""

₊

c-~(A~~ DQ" Q~ V~u" Q" V~u~ )j (5.24j

The

dissipation

term

£~

contains _terms

quadratic

in fluxes such _as

Q"

and P ^"~ which _are _one order

higher

than the terms in

&~. They

_are relativistic effects.

The

dissipation

term

£~

does not vanish if there is _an irreversible process present ⁱⁿ the system, ^and

consequently

Dfl3 _# 0 except at

equilibrium.

This _means that DSP is _not _even _a

Pfaffian differential and therefore _not _an _exact differential in the space of variables

8,

_v, _c,,

di)~', (lwiwr,

_cy ml ). It is not

equal

to the

compensation

differential D V/ away from

equilibrium.

That

is,

the extended Gibbs relation

for

D3P does not e.<ist

if

the

system ^is aM>ay

from equilibrium,

in contradiction with the

premise

made in the relativistic version

ofextended

irreversible

thermodynamics.

In _a _recent work

[13, 20]

it is shown that the extended Gibbs relation for DSP does _not exist in the nonrelativistic

theory

^either. ^This ^is a

rigorous

result based _on the covariant Boltzmann

equation

and does not

require

an

explicit

solution for the unknowns _X~~" for its

validity

_as

long

_as the latter exist. There, however, exists

a differential form for the

compensation

function V/ that looks like _an extended Gibbs relation.

It may ^be used to construct _a

thermodynamic theory

^of irreversible processes in _a way

parallel

to

equilibrium thermodynamics.

The

theory

will

provide

various relations between

nonequilib-

rium

thermodynamic quantities

that _can be related _to

experimental

observables. In any case, the present ^result on DSP calls for revision of the

existing

notions of

nonequilibrium

_entropy

and entropy ^fluxes ^and ^of ^the

applications

thereof in extended irreversible

thermodynamics [21], including

its relativistic version.

6. Cumulant

expansion

for the

dissipation

terms

A)"'

The

dissipation

_terrns _A,~"~ defined in

(3.6)

must be calculated in terms of

macroscopic

variables before the evolution

equations

for di,~" are solved for

specific

flow

problems.

^Since

(16)

A,~"~ are

intimately

related _to the entropy

production

and the latter must remain

positive regardless

of the

approximation

made _to

A,~"',

it is best _to calculate the

dissipation

terms

through

the entropy

production

that is

manifestly positive.

As in the nonrelativistic kinetic

theory,

the aim is

nicely

achieved if _a cumulant

expansion

is

employed.

To this end, we first write the distribution function in the form

f,

=

f)

_exP

_(- _w,) _(6.ij

where

f$

_is the local Jiittner

(equilibrium distribution)

function

fj

= exp

i- p ~pt U~

p

iii

,

(6.2a) exp(- p p,~)

=

p/

^'

(c~

^~

^p) U~ exp(- ppf U~ ))

,

(6.2b)

w, =

P (Hl~'- _Ap1) _(6.31

We also define the abbreviations

'J

~' ~ ~J

'

~'J ~'~

_~

~J~ ^'

and the

following

reduced variables

gr,"

₌

cpf

p

,

tf"

=

U"/c.

(6.5)

We also scale the distribution function with the factor

p/(mk~

_T)~~~ where _m is the _mean _rest

mass of the mixture _; that

is,

fi

=

lP/(mkB

T)~/~

ff (6.61

Therefore,

ff

is dimensionless. The transition _rate W,~ is scaled

by A~/mkBT

where A is _a

parameter

^of ^dimension

length

we may take it with the _mean free

path

_or with the

interaction range or the size parameter. Thus, we write

W,~ = W,~

(A ~/mk~ T) (6.7)

so that

Wj~ is dimensionless. It is then convenient to define the parameter

g =

(mc~/k~ T)~/cA

^~ _p^~

(6.8)

which has _a dimension of volume _x time. If this parameter ^is

multiplied

to the collision

integral

for the entropy

production,

it is nondimensionalized _as follows

"ent(X)

" ~B

~ent~~ (6.9)

where the nondimensional reduced entropy

production

_&~~~ is

given by

the

integral

&~~~ =

l f f

~<~~

y~~

i iexp

(-

y~~

i

_exp

(-

_x,~₁₁

,

(6. 101

where

(. )

=

G,~ d~it,

_d~fi~

d~t,*

_d~it~*

ff f(

_W,~

,

(6.

II