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On the domain of validity of Gibbs' entropy law
V o l u m e 26A, n u m b e r 12
PHYSICS
LETTERS
ON T H E D O M A I N O F V A L I D I T Y
6 M a y 1968
OF GIBBS'
ENTROPY
LAW
M. G. VELARDE and J. WALLENBORN
Facult~ des Sciences, Universit~ Libre de Bruxeltes, Bruxelles, Belgique Received 5 April 1968
The microscopic definition of entropy as introduced by Prigogine, Henin and George, in terms of a quasiparticle description is used to derive the entropy balance equation for a moderately strong coupled classical non-uniform gas. Both entropy flow and entropy production are found to agree with the expressions postulated by non-equilibrium theory on the basis of the Gibbs entropy law.
Non-equiUbrium thermodynamics has been used widely both as a tool for phenomenolngical considerations [1] and as the starting point for the response formalism of the theory of transport processes [2]. However the statistical foundation of non-equilibrium thermodynamics was till now only for dilute s y s tems (behaving as a perfect gas at equilibrium) [3]. The difficulty is to introduce a microscopic definition of entropy which can be used as a starting point to obtain the expressions for entropy flow and entropy production and which would take into account correlations. This is now possible through the quasi-particle description introduced by Prigogine et al. [4]. In this representation the thermodynamic entropy appears as the Boltzmann-type entropy constructed in t e r m s of quasi.particle distribution function only [5,6]. We shall consider here the case of a moderately strong coupled classical non-uniform gas but we believe that this proof may be extended to higher orders of concentration (or coupling constant). In the linear domain of i r r e v e r s i b l e p r o c e s s e s , the transformed kinetic equation is [7] ~
i ~-~'~¢+ i ~~ n 17n" flN+~n
nj~[VJV(Ixn-Xj[)]" ( a ; n - ~ j )af l
N
- l ~ ° 2 f ~ = (0~°2+0~°3+0~°4)~ ,
(1)
where only t e r m s expressing the first strong coupling corrections (beyond the perfect gas limit) have been retained. The subscript and the superscript on ~o (transformed collision operator) indicates the order in the coupling parameter and the order in the gradient, respectively; p is the momentum and v the bare particle velocity; V is the interaction potential and Vn - a/ax n . We have set: 1
7f-- x- sf, =
N
x+A
(3)
where A is a microscopic bare particle quantity. To this order, the dressing operator X is:
x= l-¼8na S s(xi- xj) f dlV?
Oi - aoja ) i. (vi_ vj) l. (~_~__o "'
w) o
.
(4)
We define the local entropy density:
ps = -
/ (dx)N(dp)N6(x-
7
log
N + C.
(5)
It is shown that this definition yields, at equilibrium, the c o r r e c t expression for the thermodynamic entropy (including correlations). With the aid of kinetic equation (1), we easily obtain the usual entropy balance [1]: 584
Volume 26A, number 12
PHYSIC S L E TTER S
a
TiPs =
- V"
Js )
(psu+
6 May 1968
+ o',
(6)
where
"Is
=
-k
/
(dx)N(dp)N O( x- xa)(~ - u)'flNlOg .~/"
n,.1
(7)
(8)
+ log ~N(i 0~p2 +i l(p2 +i 0~P3 +i 0(p4)~N I . In o r d e r to calculate a, we now solve the kinetic equation (1) following the Chapman-Enskog method with the usual subsidiary conditions on summational invariants [8]. The solubility conditions r e d u c e h e r e to the balance equation f o r m a s s , m o m e n t u m and total energy. Introducing the solution of eq. (1) (up to 2nd o r d e r in the deviation f r o m local equilibrium) in the e x p r e s s i o n of a eq. (8), we finally obtain the Gibbs law c o r r e s p o n d i n g to heat conduction and viscous flow: J o" = - ~--~-"
I T T - ~1 ( P - p l )
: V u >/0
(9)
where J is the total energy flow, P the p r e s s u r e t e n s o r and p the local equilibrium p r e s s u r e . The details of calculations will be published elsewhere [9]. Both the entropy flow [7] and the entropy porduction [9] coincide with the e x p r e s s i o n s postulated in n o n - e q u i l i b r i u m t h e r m o d y n a m i c s on the basis of the Gibbs law. Invbrsely this observation adds a supplementary a r g u m e n t in favor of m i c r o s c o p i c definition of entropy as introduced by Prigogine, Henin and G e o r g e and i l l u s t r a t e s the usefulness of the q u a s i - p a r t i c l e concept in the descriptions of linear t r a n s p o r t p r o p e r t i e s . We want to e x p r e s s our gratitute to P r o f e s s o r P r i g o g i n e for his constant i n t e r e s t in this work. We also thank Dr. G. Nicolis f o r stimulating discussions. This r e s e a r c h has been s p o n s o r e d in part, by the Air F o r c e Office of Scientific R e s e a r c h through the European Office of A e r o s p a c e R e s e a r c h , OAR, United States Air F o r c e under Grant AF EOAR 67-25. One of us (M. G.V.) is grateful to the Fundaci6n "Juan March" (Spain) for a fellowship f r o m October 1965 to August 1966.
References 1. S . R . De Groot and P. Mazur. Non-equilibrium thermodynamics (North-Holland, Amsterdam, 1963). 2. R.Kubo, J. Phys. Soc. (Japan) 12 (1957) 570. 3. I. Prigogine, Physica 15 (1949) 272. 4. I. Prigogine, F.Henin and C.George, Proc. Nat. Acad. Sci. 59 (1968) 7. 5. I. Prigogine, F. Henin and C. George, Physica 32 (1966) 1873. 6. M.G. Velarde, to be published. 7. J. Wallenborn. to be published. 8. S. Chapman and T.G. Cowling, The mathematical theory of non-uniform gases (Cambridge University Press, rep~ 1964). 9. J. Wallenborn and M. G. Velarde, ~11 be submitted for publication to J. Chem. Phys. * * * * *
585
PHYSICS
LETTERS
ON T H E D O M A I N O F V A L I D I T Y
6 M a y 1968
OF GIBBS'
ENTROPY
LAW
M. G. VELARDE and J. WALLENBORN
Facult~ des Sciences, Universit~ Libre de Bruxeltes, Bruxelles, Belgique Received 5 April 1968
The microscopic definition of entropy as introduced by Prigogine, Henin and George, in terms of a quasiparticle description is used to derive the entropy balance equation for a moderately strong coupled classical non-uniform gas. Both entropy flow and entropy production are found to agree with the expressions postulated by non-equilibrium theory on the basis of the Gibbs entropy law.
Non-equiUbrium thermodynamics has been used widely both as a tool for phenomenolngical considerations [1] and as the starting point for the response formalism of the theory of transport processes [2]. However the statistical foundation of non-equilibrium thermodynamics was till now only for dilute s y s tems (behaving as a perfect gas at equilibrium) [3]. The difficulty is to introduce a microscopic definition of entropy which can be used as a starting point to obtain the expressions for entropy flow and entropy production and which would take into account correlations. This is now possible through the quasi-particle description introduced by Prigogine et al. [4]. In this representation the thermodynamic entropy appears as the Boltzmann-type entropy constructed in t e r m s of quasi.particle distribution function only [5,6]. We shall consider here the case of a moderately strong coupled classical non-uniform gas but we believe that this proof may be extended to higher orders of concentration (or coupling constant). In the linear domain of i r r e v e r s i b l e p r o c e s s e s , the transformed kinetic equation is [7] ~
i ~-~'~¢+ i ~~ n 17n" flN+~n
nj~[VJV(Ixn-Xj[)]" ( a ; n - ~ j )af l
N
- l ~ ° 2 f ~ = (0~°2+0~°3+0~°4)~ ,
(1)
where only t e r m s expressing the first strong coupling corrections (beyond the perfect gas limit) have been retained. The subscript and the superscript on ~o (transformed collision operator) indicates the order in the coupling parameter and the order in the gradient, respectively; p is the momentum and v the bare particle velocity; V is the interaction potential and Vn - a/ax n . We have set: 1
7f-- x- sf, =
N
x+A
(3)
where A is a microscopic bare particle quantity. To this order, the dressing operator X is:
x= l-¼8na S s(xi- xj) f dlV?
Oi - aoja ) i. (vi_ vj) l. (~_~__o "'
w) o
.
(4)
We define the local entropy density:
ps = -
/ (dx)N(dp)N6(x-
7
log
N + C.
(5)
It is shown that this definition yields, at equilibrium, the c o r r e c t expression for the thermodynamic entropy (including correlations). With the aid of kinetic equation (1), we easily obtain the usual entropy balance [1]: 584
Volume 26A, number 12
PHYSIC S L E TTER S
a
TiPs =
- V"
Js )
(psu+
6 May 1968
+ o',
(6)
where
"Is
=
-k
/
(dx)N(dp)N O( x- xa)(~ - u)'flNlOg .~/"
n,.1
(7)
(8)
+ log ~N(i 0~p2 +i l(p2 +i 0~P3 +i 0(p4)~N I . In o r d e r to calculate a, we now solve the kinetic equation (1) following the Chapman-Enskog method with the usual subsidiary conditions on summational invariants [8]. The solubility conditions r e d u c e h e r e to the balance equation f o r m a s s , m o m e n t u m and total energy. Introducing the solution of eq. (1) (up to 2nd o r d e r in the deviation f r o m local equilibrium) in the e x p r e s s i o n of a eq. (8), we finally obtain the Gibbs law c o r r e s p o n d i n g to heat conduction and viscous flow: J o" = - ~--~-"
I T T - ~1 ( P - p l )
: V u >/0
(9)
where J is the total energy flow, P the p r e s s u r e t e n s o r and p the local equilibrium p r e s s u r e . The details of calculations will be published elsewhere [9]. Both the entropy flow [7] and the entropy porduction [9] coincide with the e x p r e s s i o n s postulated in n o n - e q u i l i b r i u m t h e r m o d y n a m i c s on the basis of the Gibbs law. Invbrsely this observation adds a supplementary a r g u m e n t in favor of m i c r o s c o p i c definition of entropy as introduced by Prigogine, Henin and G e o r g e and i l l u s t r a t e s the usefulness of the q u a s i - p a r t i c l e concept in the descriptions of linear t r a n s p o r t p r o p e r t i e s . We want to e x p r e s s our gratitute to P r o f e s s o r P r i g o g i n e for his constant i n t e r e s t in this work. We also thank Dr. G. Nicolis f o r stimulating discussions. This r e s e a r c h has been s p o n s o r e d in part, by the Air F o r c e Office of Scientific R e s e a r c h through the European Office of A e r o s p a c e R e s e a r c h , OAR, United States Air F o r c e under Grant AF EOAR 67-25. One of us (M. G.V.) is grateful to the Fundaci6n "Juan March" (Spain) for a fellowship f r o m October 1965 to August 1966.
References 1. S . R . De Groot and P. Mazur. Non-equilibrium thermodynamics (North-Holland, Amsterdam, 1963). 2. R.Kubo, J. Phys. Soc. (Japan) 12 (1957) 570. 3. I. Prigogine, Physica 15 (1949) 272. 4. I. Prigogine, F.Henin and C.George, Proc. Nat. Acad. Sci. 59 (1968) 7. 5. I. Prigogine, F. Henin and C. George, Physica 32 (1966) 1873. 6. M.G. Velarde, to be published. 7. J. Wallenborn. to be published. 8. S. Chapman and T.G. Cowling, The mathematical theory of non-uniform gases (Cambridge University Press, rep~ 1964). 9. J. Wallenborn and M. G. Velarde, ~11 be submitted for publication to J. Chem. Phys. * * * * *
585