- Email: [email protected]
Fluctuations and non-equilibrium thermodynamics
Physica A 261 (1998) 451– 457
Fluctuations and non-equilibrium thermodynamics P. Mazur Instituut Lorentz, Rijksuniversiteit, Leiden, P.O. Box 9504, 2300 RA Leiden, The Netherlands Received 22 June 1998
Abstract A closed macroscopic system with uctuating local properties is treated as a thermodynamic system with internal degrees of freedom. Gibb’s entropy postulate is used to de ne the systems’ entropy as a functional of the probability density in internal coordinate space. It is shown that application of the scheme of thermodynamics of irreversible processes then leads directly to the theory of uctuations as Markov processes described by a multivariate Fokker–Planck equation. In this perspective uctuation theory may be said to have become integrated into non-equilibrium c 1998 Elsevier Science B.V. All rights reserved. thermodynamics.
1. Introduction Non-equilibrium thermodynamics provides a formalism to set up systematically, following a small number of rules, the linear phenomenological equations characterising irreversible processes [1]. One of the important advantages of this formalism is that it yields automatically the proper choice of dissipative currents and thermodynamic forces, for which the linear phenomenological laws describing irreversible processes have a coecient scheme satisfying Onsager’s reciprocal relations [2]. Does this formalism, and its rules, also yield the equations describing the dynamics of thermodynamic uctuations? The answer to this question would seem to be: no. True, there is a connection between the theory of irreversible processes and the theory of spontaneous uctuations of thermodynamic variables. Indeed uctuation theory was used (‘brought in’ as Onsager and Machlup [3] say in their paper on uctuations and irreversible processes) to derive the reciprocal relations. But the basic equations of thermodynamic uctuation theory were not obtained from the theory of non-equilibrium thermodynamics. Some reservation must be made regarding this last assertion. When Prigogine and Mazur [4] introduced the notion of internal degrees of freedom into non-equilibrium thermodynamics to extend its range of applicability, Fokker–Planck like di erential equations resulted to describe the irreversible processes in internal coordinate space. c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 3 5 3 - 7
452
P. Mazur / Physica A 261 (1998) 451– 457
Examples were Brownian motion in velocity space and orientation of dipoles. In both cases the theory deals with densities which are distribution functions in the internal coordinate space. Thus implicitly uctuations in this space are also accounted for. Similar treatments were used later to discuss Brownian motion problems in a nonequilibrium heat bath [5]. In this paper we reformulate the theory of non-equilibrium thermodynamics for systems with internal degrees of freedom in such a way that the theory of thermodynamic
uctuations as Markov processes follows straightforwardly when applying the former theory’s conventional rules. In their standard form these rules are: (1) the statement of Gibbs’ equation in terms of the relevant thermodynamic variables; (2) the statement of the conservation laws for these variables; (3) the establishment of an entropy balance equation and calculation of the entropy production as a sum of products of dissipative uxes and thermodynamic forces; (4) the establishment of linear phenomenological relations between the uxes and forces, and (5) the use of these relations in the conservation equations to determine the di erential equations governing the dynamics of the thermodynamic variables. Section 2 deals with point (1) considering the uctuations of thermodynamic variables as internal degrees of freedom and using in addition Gibbs’ entropy postulate to de ne entropy as a functional of the probability density in internal co-ordinate space. In Section 3 we discuss appropriately points (2) – (4). In Section 4 we substitute the phenomenological relations found in Section 3 which are relations for the components of the probability ux, into the continuity equation for the probability density and obtain a multivariate Fokker–Planck equation, describing therefore uctuations as Markov processes. A brief discussion of these results, which integrate thermodynamic uctuation theory as well as the derivation of reciprocal relations into the formalism of non-equilibrium thermodynamics, is given in Section 5. 2. A material system and its uctuations as a thermodynamic system with internal degrees of freedom Consider a closed adiabatically insulated system. From a macroscopic point of view its state will be described by a set of variables A1 ; A2 ; : : : ; An , corresponding to extensive properties, masses, energies, etc., of small subsystems. These subsystems while small compared to the size of the global system are large enough, as far as the number of their constituent particles is concerned, to make a thermodynamic description applicable. We introduce the probability density P(A; ) that the system is at time in a state de ned by a point A = (A1 ; A2 ; : : : ; An ) in an n dimensional cartesian phase space. This probability density satis es the normalisation condition. Z Z (1) P(A1 ; A2 ; : : : ; An ) dA1 dA2 · · · dAn = P(A; t) dA = 1 :
P. Mazur / Physica A 261 (1998) 451– 457
453
At equilibrium, that is in a suciently aged system (or equivalently in the limit as t tends to in nity) the distribution function P(A; t) becomes the equilibrium distribution function P0 (A) for which the state variables Ai have mean values hAi i Z (2) hAi i = Ai P0 (A) dA : The state for which the variables Ai have values A0i ≡ hAi i
(i = 1; 2; : : : ; n)
(3)
may be referred to as the equilibrium state in particular when this state is also the state of maximum probability around which the equilibrium distribution function P0 (A; t) is suciently sharply peaked. Moreover for this equilibrium state thermodynamic equilibrium conditions between various subsystems should be satis ed. We now de ne thermodynamic uctuations = ( 1 ; 2 ; : : : ; n ) i = Ai − A0i
(4)
obeying the distribution functions P( ; t) and P0 ( ) de ned above. The uctuations i may be considered as n internal degrees of freedom of the system in the sense of non-equilibrium thermodynamics referred to above. The system as a whole can then be viewed as a thermodynamic system of (in nitely) many components with mass densities MP( ; t), and of total constant mass M and total constant energy E. Taking this point of view the total entropy S satis es a Gibbs equation Z ( ; t) @P( ; t) dS = −M d ; (5) dt T0 @t where ( ) is the chemical potential per unit of mass of the chemical component with mass density MP( ; t) while T0 is the equilibrium temperature of the closed thermodynamically insulated system. The chemical potential ( ) can be expressed in terms of the distribution function P( ; t) by making use of Gibbs’ entropy postulate which states that the system’s entropy with respect to its equilibrium value S0 (E; M ) is given by Z P( ; t) d + S0 (E; M ) (6) S(t) = −k P( ; t) ln P0 ( ) with k the Boltzmann constant. Di erentiating expression (6) with respect to time and using Eq. (1), we obtain Z P( ; t) @P( ; t) dS = −k d ln : (7) dt P0 ( ) @t Comparing Eqs. (5) and (7) we therefore have ( ; t) = 0 +
kT0 P( ; t) ln ; M P0 ( )
(8)
where 0 , the equilibrium value of ( ; t) is independent of state , and is the chemical potential per unit of mass of the system as a whole.
454
P. Mazur / Physica A 261 (1998) 451– 457
In the subsequent sections we shall now apply the standard formalism (or scheme) of non-equilibrium thermodynamics to Eqs. (5) and (8) (or equivalently Eq. (7)) in order to obtain the di erential equation describing the evolution of the probability distribution function of the uctuations, as well as the irreversible processes connected with this evolution.
3. Conservation law, entropy production and phenomenological relation We interpreted the uctuating system as consisting of in nitely many components of mass density MP( ; t). The total mass of the system is conserved according to Eq. (1). In the n dimensional -space the probability uid will obey a continuity equation X @ @P( ; t) P( ; t)vi ( ; t) ; =− @t @ i i
(9)
where vi ( ; t) represents the velocity of the uid along the i axis. Eq. (9) is so to say the local form in -space of mass, or rather probability, conservation. We can now substitute Eq. (9) into the Gibbs equation (7) and obtain after partial integration in -space (since P( ; t) → 0 suciently rapid as → ∞) for the rate of change of entropy, that is for the irreversible entropy production of the closed system Z dS = d P( ; t)( ; t)¿0 (10) dt with ( ; t) = −
X
vi ( ; t)k
i
@ P( ; t) ln @ i P0 ( )
(11)
the local entropy source strength in -space (entropy per unit of time). The total entropy production is according to Eqs. (10) and (11) a sum of products of velocities of the probability uid and their conjugate thermodynamic forces, the respective gradients in -space of the thermodynamic potential (8). Following the scheme of non-equilibrium thermodynamics [1] linear relations may be established between these quantities. These phenomenological relations have the form vi ( ) = −
X j
Lij k
@ P( ; t) ; ln @ j P0 ( )
(12)
determining the velocities in the continuity equation (9). The phenomenological (Onsager) coecients Lij in these relations, which obey the reciprocal relations, may in principle depend on . Introducing the thermodynamic variables conjugate to the variables Xj ( ) = −k
@ ln P0 ( ) : @ j
(13)
P. Mazur / Physica A 261 (1998) 451– 457
The phenomenological relations (12) may also be written in the form X @ Lij Xj + k ln P( ; t) : vi ( ) = − @ j j
455
(14)
Notice that, as a consequence of the fact that inequality (10) is valid for an arbitrary choice of the probability distribution, it follow that the simpler inequality ( ; t)¿0
(15)
must hold and therefore that the second law is valid locally in -space. This necessarily leads to the phenomenological laws of the form (12). 4. The Fokker–Planck equation for the thermodynamic uctuations If we substitute the phenomenological relations (12) into the conservation law (9) we obtain the following di erential equation for the irreversible evolution of the probability distribution function @P @P X @ = : (16) Lij PXj + k @t @ i @ j ij De ning the two functions Hij ( ) = k(Lij + Lji ) ; X @Lij Lij Xj − k ; Gi ( ) = − @ j j
(17) (18)
Eq. (16) can be written in the usual form of the multivariate non-linear Fokker–Planck equation [6] X @ 1 X @2 @P =− Gi P + Hij P : @t @ i 2 i; j @ i @ j i
(19)
This signi es that one obtains from the scheme of non-equilibrium thermodynamics for a system with uctuations treated as internal degrees of freedom Markov processes for the description of the dynamics of these quantities. If in particular Lij is constant and the equilibrium distribution function Gaussian, so that Xi is linear in (cf. Eq. (13)), Eq. (16) or (19) reduces to the linear multivariate Fokker–Planck equation of which the solution is a multidimensional Gaussian Markov process [7]. Thus the theory of thermodynamic uctuations as Gaussian Markov processes can be integrated into (follows from) the formalism of non-equilibrium thermodynamics. Finally we observe that if the Fokker–Planck equation is solved with the initial condition P( ; 0) = i ( i − i0 )
(20)
456
P. Mazur / Physica A 261 (1998) 451– 457
the solution P( ; t) is the conditional d.f. (P( 0 | ; t) P( ; t) = P( 0 | ; t) : Let us then calculate for the rst moment of the uctuating variable i , Z 0 i = i P( ; t) d ;
(21)
(22)
its behaviour in time. If we multiply both terms of Eq. (19) by i and integrate over we obtain the ordinary di erential equation X d i 0 Lij X j 0 : =− dt j
(23)
This is Onsager’s regression hypothesis from which the reciprocal relations [2] follow in a straightforward way applying microscopic reversibility, or detailed balance, a property which the distribution function P( 0 | t) must satisfy. What is to be stressed in the present context is that it is not necessary here to bring in uctuation theory to prove the reciprocity theorem, which then becomes a law for macroscopic non-equilibrium thermodynamics. Rather it is so, as found above, that thermodynamic uctuation theory has itself become part of thermodynamics of irreversible processes.
5. Discussion It has been shown above that the concept of internal degrees of freedom, introduced originally to allow for the description of a wider class of irreversible processes with the methods of non-equilibrium thermodynamics, can be used in such a way that the theory of thermodynamic uctuations as Gaussian Markov processes is obtained in a simple manner. These processes are described equivalently by a linear multivariate Fokker– Planck equation, as found above by the methods of non-equilibrium thermodynamics, or by a set of coupled linear Langevin equations as in Onsager and Machlup’s theory. That theory has therefore indirectly also been recovered by thermodynamic non-equilibrium methods. As a corollary this included, as we have seen, the derivation of reciprocal relations. We note as we did previously in a similar case that the extension of thermodynamic methods given above goes beyond the traditional macroscopic domain. Indeed it introduces statistical notions such as the distribution function in the space of uctuating thermodynamic variables and Gibbs’ entropy postulate relating the system’s entropy to this function. However it is an extension limited (and therefore still legitimate) to a mesoscopic domain comprising uctuations and dealing with events taking place on a time scale which is extremely slow compared to molecular or microscopic times.
P. Mazur / Physica A 261 (1998) 451– 457
457
Acknowledgements The author wishes to express his gratitude to Joel Keizer and Dick Bedeaux for reading the manuscript and for their pertinent remarks and criticism, and to Miguel Rubi for suggesting that the concept of internal degrees of freedom might be of wider use in non-equilibrium thermodynamics. References [1] S.R. de Groot, P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962; Dover, New York, 1984. [2] L. Onsager, Phys. Rev. 37 (1931) 405; 38 (1931) 2265. [3] L. Onsager, S. Machlup, Phys. Rev. 91 (1953) 1505; S. Machlup, L. Onsager, Phys. Rev. 91 (1953) 1512. [4] I. Prigogine, P. Mazur, Physica 19 (1953) 241. [5] A. Perez Madrid, J.M. Rubi, P. Mazur, Physica A 212 (1994) 231; J.M. Rubi, P. Mazur, Physica A 250 (1998) 253. [6] N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Ch. VIII, North-Holland, Amsterdam, 1992, p. 6. [7] M.C. Wang, G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323.
Fluctuations and non-equilibrium thermodynamics P. Mazur Instituut Lorentz, Rijksuniversiteit, Leiden, P.O. Box 9504, 2300 RA Leiden, The Netherlands Received 22 June 1998
Abstract A closed macroscopic system with uctuating local properties is treated as a thermodynamic system with internal degrees of freedom. Gibb’s entropy postulate is used to de ne the systems’ entropy as a functional of the probability density in internal coordinate space. It is shown that application of the scheme of thermodynamics of irreversible processes then leads directly to the theory of uctuations as Markov processes described by a multivariate Fokker–Planck equation. In this perspective uctuation theory may be said to have become integrated into non-equilibrium c 1998 Elsevier Science B.V. All rights reserved. thermodynamics.
1. Introduction Non-equilibrium thermodynamics provides a formalism to set up systematically, following a small number of rules, the linear phenomenological equations characterising irreversible processes [1]. One of the important advantages of this formalism is that it yields automatically the proper choice of dissipative currents and thermodynamic forces, for which the linear phenomenological laws describing irreversible processes have a coecient scheme satisfying Onsager’s reciprocal relations [2]. Does this formalism, and its rules, also yield the equations describing the dynamics of thermodynamic uctuations? The answer to this question would seem to be: no. True, there is a connection between the theory of irreversible processes and the theory of spontaneous uctuations of thermodynamic variables. Indeed uctuation theory was used (‘brought in’ as Onsager and Machlup [3] say in their paper on uctuations and irreversible processes) to derive the reciprocal relations. But the basic equations of thermodynamic uctuation theory were not obtained from the theory of non-equilibrium thermodynamics. Some reservation must be made regarding this last assertion. When Prigogine and Mazur [4] introduced the notion of internal degrees of freedom into non-equilibrium thermodynamics to extend its range of applicability, Fokker–Planck like di erential equations resulted to describe the irreversible processes in internal coordinate space. c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 3 5 3 - 7
452
P. Mazur / Physica A 261 (1998) 451– 457
Examples were Brownian motion in velocity space and orientation of dipoles. In both cases the theory deals with densities which are distribution functions in the internal coordinate space. Thus implicitly uctuations in this space are also accounted for. Similar treatments were used later to discuss Brownian motion problems in a nonequilibrium heat bath [5]. In this paper we reformulate the theory of non-equilibrium thermodynamics for systems with internal degrees of freedom in such a way that the theory of thermodynamic
uctuations as Markov processes follows straightforwardly when applying the former theory’s conventional rules. In their standard form these rules are: (1) the statement of Gibbs’ equation in terms of the relevant thermodynamic variables; (2) the statement of the conservation laws for these variables; (3) the establishment of an entropy balance equation and calculation of the entropy production as a sum of products of dissipative uxes and thermodynamic forces; (4) the establishment of linear phenomenological relations between the uxes and forces, and (5) the use of these relations in the conservation equations to determine the di erential equations governing the dynamics of the thermodynamic variables. Section 2 deals with point (1) considering the uctuations of thermodynamic variables as internal degrees of freedom and using in addition Gibbs’ entropy postulate to de ne entropy as a functional of the probability density in internal co-ordinate space. In Section 3 we discuss appropriately points (2) – (4). In Section 4 we substitute the phenomenological relations found in Section 3 which are relations for the components of the probability ux, into the continuity equation for the probability density and obtain a multivariate Fokker–Planck equation, describing therefore uctuations as Markov processes. A brief discussion of these results, which integrate thermodynamic uctuation theory as well as the derivation of reciprocal relations into the formalism of non-equilibrium thermodynamics, is given in Section 5. 2. A material system and its uctuations as a thermodynamic system with internal degrees of freedom Consider a closed adiabatically insulated system. From a macroscopic point of view its state will be described by a set of variables A1 ; A2 ; : : : ; An , corresponding to extensive properties, masses, energies, etc., of small subsystems. These subsystems while small compared to the size of the global system are large enough, as far as the number of their constituent particles is concerned, to make a thermodynamic description applicable. We introduce the probability density P(A; ) that the system is at time in a state de ned by a point A = (A1 ; A2 ; : : : ; An ) in an n dimensional cartesian phase space. This probability density satis es the normalisation condition. Z Z (1) P(A1 ; A2 ; : : : ; An ) dA1 dA2 · · · dAn = P(A; t) dA = 1 :
P. Mazur / Physica A 261 (1998) 451– 457
453
At equilibrium, that is in a suciently aged system (or equivalently in the limit as t tends to in nity) the distribution function P(A; t) becomes the equilibrium distribution function P0 (A) for which the state variables Ai have mean values hAi i Z (2) hAi i = Ai P0 (A) dA : The state for which the variables Ai have values A0i ≡ hAi i
(i = 1; 2; : : : ; n)
(3)
may be referred to as the equilibrium state in particular when this state is also the state of maximum probability around which the equilibrium distribution function P0 (A; t) is suciently sharply peaked. Moreover for this equilibrium state thermodynamic equilibrium conditions between various subsystems should be satis ed. We now de ne thermodynamic uctuations = ( 1 ; 2 ; : : : ; n ) i = Ai − A0i
(4)
obeying the distribution functions P( ; t) and P0 ( ) de ned above. The uctuations i may be considered as n internal degrees of freedom of the system in the sense of non-equilibrium thermodynamics referred to above. The system as a whole can then be viewed as a thermodynamic system of (in nitely) many components with mass densities MP( ; t), and of total constant mass M and total constant energy E. Taking this point of view the total entropy S satis es a Gibbs equation Z ( ; t) @P( ; t) dS = −M d ; (5) dt T0 @t where ( ) is the chemical potential per unit of mass of the chemical component with mass density MP( ; t) while T0 is the equilibrium temperature of the closed thermodynamically insulated system. The chemical potential ( ) can be expressed in terms of the distribution function P( ; t) by making use of Gibbs’ entropy postulate which states that the system’s entropy with respect to its equilibrium value S0 (E; M ) is given by Z P( ; t) d + S0 (E; M ) (6) S(t) = −k P( ; t) ln P0 ( ) with k the Boltzmann constant. Di erentiating expression (6) with respect to time and using Eq. (1), we obtain Z P( ; t) @P( ; t) dS = −k d ln : (7) dt P0 ( ) @t Comparing Eqs. (5) and (7) we therefore have ( ; t) = 0 +
kT0 P( ; t) ln ; M P0 ( )
(8)
where 0 , the equilibrium value of ( ; t) is independent of state , and is the chemical potential per unit of mass of the system as a whole.
454
P. Mazur / Physica A 261 (1998) 451– 457
In the subsequent sections we shall now apply the standard formalism (or scheme) of non-equilibrium thermodynamics to Eqs. (5) and (8) (or equivalently Eq. (7)) in order to obtain the di erential equation describing the evolution of the probability distribution function of the uctuations, as well as the irreversible processes connected with this evolution.
3. Conservation law, entropy production and phenomenological relation We interpreted the uctuating system as consisting of in nitely many components of mass density MP( ; t). The total mass of the system is conserved according to Eq. (1). In the n dimensional -space the probability uid will obey a continuity equation X @ @P( ; t) P( ; t)vi ( ; t) ; =− @t @ i i
(9)
where vi ( ; t) represents the velocity of the uid along the i axis. Eq. (9) is so to say the local form in -space of mass, or rather probability, conservation. We can now substitute Eq. (9) into the Gibbs equation (7) and obtain after partial integration in -space (since P( ; t) → 0 suciently rapid as → ∞) for the rate of change of entropy, that is for the irreversible entropy production of the closed system Z dS = d P( ; t)( ; t)¿0 (10) dt with ( ; t) = −
X
vi ( ; t)k
i
@ P( ; t) ln @ i P0 ( )
(11)
the local entropy source strength in -space (entropy per unit of time). The total entropy production is according to Eqs. (10) and (11) a sum of products of velocities of the probability uid and their conjugate thermodynamic forces, the respective gradients in -space of the thermodynamic potential (8). Following the scheme of non-equilibrium thermodynamics [1] linear relations may be established between these quantities. These phenomenological relations have the form vi ( ) = −
X j
Lij k
@ P( ; t) ; ln @ j P0 ( )
(12)
determining the velocities in the continuity equation (9). The phenomenological (Onsager) coecients Lij in these relations, which obey the reciprocal relations, may in principle depend on . Introducing the thermodynamic variables conjugate to the variables Xj ( ) = −k
@ ln P0 ( ) : @ j
(13)
P. Mazur / Physica A 261 (1998) 451– 457
The phenomenological relations (12) may also be written in the form X @ Lij Xj + k ln P( ; t) : vi ( ) = − @ j j
455
(14)
Notice that, as a consequence of the fact that inequality (10) is valid for an arbitrary choice of the probability distribution, it follow that the simpler inequality ( ; t)¿0
(15)
must hold and therefore that the second law is valid locally in -space. This necessarily leads to the phenomenological laws of the form (12). 4. The Fokker–Planck equation for the thermodynamic uctuations If we substitute the phenomenological relations (12) into the conservation law (9) we obtain the following di erential equation for the irreversible evolution of the probability distribution function @P @P X @ = : (16) Lij PXj + k @t @ i @ j ij De ning the two functions Hij ( ) = k(Lij + Lji ) ; X @Lij Lij Xj − k ; Gi ( ) = − @ j j
(17) (18)
Eq. (16) can be written in the usual form of the multivariate non-linear Fokker–Planck equation [6] X @ 1 X @2 @P =− Gi P + Hij P : @t @ i 2 i; j @ i @ j i
(19)
This signi es that one obtains from the scheme of non-equilibrium thermodynamics for a system with uctuations treated as internal degrees of freedom Markov processes for the description of the dynamics of these quantities. If in particular Lij is constant and the equilibrium distribution function Gaussian, so that Xi is linear in (cf. Eq. (13)), Eq. (16) or (19) reduces to the linear multivariate Fokker–Planck equation of which the solution is a multidimensional Gaussian Markov process [7]. Thus the theory of thermodynamic uctuations as Gaussian Markov processes can be integrated into (follows from) the formalism of non-equilibrium thermodynamics. Finally we observe that if the Fokker–Planck equation is solved with the initial condition P( ; 0) = i ( i − i0 )
(20)
456
P. Mazur / Physica A 261 (1998) 451– 457
the solution P( ; t) is the conditional d.f. (P( 0 | ; t) P( ; t) = P( 0 | ; t) : Let us then calculate for the rst moment of the uctuating variable i , Z 0 i = i P( ; t) d ;
(21)
(22)
its behaviour in time. If we multiply both terms of Eq. (19) by i and integrate over we obtain the ordinary di erential equation X d i 0 Lij X j 0 : =− dt j
(23)
This is Onsager’s regression hypothesis from which the reciprocal relations [2] follow in a straightforward way applying microscopic reversibility, or detailed balance, a property which the distribution function P( 0 | t) must satisfy. What is to be stressed in the present context is that it is not necessary here to bring in uctuation theory to prove the reciprocity theorem, which then becomes a law for macroscopic non-equilibrium thermodynamics. Rather it is so, as found above, that thermodynamic uctuation theory has itself become part of thermodynamics of irreversible processes.
5. Discussion It has been shown above that the concept of internal degrees of freedom, introduced originally to allow for the description of a wider class of irreversible processes with the methods of non-equilibrium thermodynamics, can be used in such a way that the theory of thermodynamic uctuations as Gaussian Markov processes is obtained in a simple manner. These processes are described equivalently by a linear multivariate Fokker– Planck equation, as found above by the methods of non-equilibrium thermodynamics, or by a set of coupled linear Langevin equations as in Onsager and Machlup’s theory. That theory has therefore indirectly also been recovered by thermodynamic non-equilibrium methods. As a corollary this included, as we have seen, the derivation of reciprocal relations. We note as we did previously in a similar case that the extension of thermodynamic methods given above goes beyond the traditional macroscopic domain. Indeed it introduces statistical notions such as the distribution function in the space of uctuating thermodynamic variables and Gibbs’ entropy postulate relating the system’s entropy to this function. However it is an extension limited (and therefore still legitimate) to a mesoscopic domain comprising uctuations and dealing with events taking place on a time scale which is extremely slow compared to molecular or microscopic times.
P. Mazur / Physica A 261 (1998) 451– 457
457
Acknowledgements The author wishes to express his gratitude to Joel Keizer and Dick Bedeaux for reading the manuscript and for their pertinent remarks and criticism, and to Miguel Rubi for suggesting that the concept of internal degrees of freedom might be of wider use in non-equilibrium thermodynamics. References [1] S.R. de Groot, P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962; Dover, New York, 1984. [2] L. Onsager, Phys. Rev. 37 (1931) 405; 38 (1931) 2265. [3] L. Onsager, S. Machlup, Phys. Rev. 91 (1953) 1505; S. Machlup, L. Onsager, Phys. Rev. 91 (1953) 1512. [4] I. Prigogine, P. Mazur, Physica 19 (1953) 241. [5] A. Perez Madrid, J.M. Rubi, P. Mazur, Physica A 212 (1994) 231; J.M. Rubi, P. Mazur, Physica A 250 (1998) 253. [6] N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Ch. VIII, North-Holland, Amsterdam, 1992, p. 6. [7] M.C. Wang, G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323.