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Reciprocity relations and higher-order fluxes
Volume 153, number 2,3
PHYSICS LETTERS A
25 February 1991
Reciprocity relations and higher-order fluxes Maria Ferrer Departament de Fisica (FIsica EstadIstica), UniversitatAutànoma de Barcelona, 08193 Bellaterra, Catalonia, Spain Received 20 July 1990; revised manuscript received 3 December 1990; accepted for publication 1 7 December 1990 Communicated by A.R. Bishop
Reciprocity relations between coefficients linking the evolution equations of higher-order hydrodynamic fluxes in a heat conducting fluid at rest are derived from a thermodynamic point of view. Such relations have some subtle differences with respect to the usual Onsager—Casimir relations. Our paper provides a simple illustration of the breaking of the reciprocity relations in nonequilibrium steady states.
Onsager—Casimir reciprocity relations [1,2] are one of the basic ingredients of classical nonequilibrium thermodynamics [3—5].They are very important because they reduce in a nontrivial way the number of independent coefficients appearing in the thermodynamic evolution equations. These relations may be formulated in the following way. Let a, be a set of thermodynamic variables or, rather, their deviations from their respective equilibrium values, If the evolution equations of the a, may be written as —L~,t9S/9a~~ (1) with S the entropy and L0 constant phenomenological coefficients and an upper dot the material time derivative, then the L~satisfy the following reciprocity relations, =
L~= e,L11,
(2)
~,
where = + 1 or 1 according to the parity of a, with respect to the time reversal operation. In the usual literature, the a, are named the thermodynamic fluxes and ÔS/ôa1 the thermodynamic forces, In this paper, another kind of relations is presented, according to which if the evolution equations for the a, are written as ~,
a, = ~ then, 76
—
(3)
~1~= —~1~1~
(4)
The main difference of (4) with the Onsager— Casimir relations (2) is the change of sign. Furthermore, (4) may be obtained from purely thermodynamic arguments, whereas (2) may only be derived from a microscopic point of view. The Onsager—Casimir relations are derived from fluctuation theory from the hypothesis of microscopic reversibility [3—51. Such a hypothesis is valid in equilibrium but not in nonequilibrium steady states. Thus, one could ask for deviations from reciprocity relations in nonequilibrium steady states [6]. Another purpose of this Letter is to provide a simple example of the breaking of reciprocity relations for a heat-conducting system (for instance, a monatomic gas at rest) when a description including as independent variables not only the usual conserved ones (density, linear momentum, energy) but also the heat flux and higher-order fluxes [7,8]. We follow here the macroscopic approach known as extended irreversible thermodynamics (EIT) [9—141. Our approach to higher-order hydrodynamics thus differs from some previous approaches [15,16] in the choice of independent variables. The authors of refs. [15,16] have referred to an approximate closed dynamics for the local conserved densities and have studied the breakdown of Onsager’s relations for fluxes beyond the Navier—Stokes order in gradients, due to the finite time required for the relaxation of
0375-9601/91/s 03.50 © 1991
—
Elsevier Science Publishers B.V. (North-Holland)
Volume 153, number 2,3
PHYSICS LETTERS A
nonhydrodynamical variables. Violations of Onsager symmetry in the course of a contraction from a fully microscopic to a macroscopic description have been found by Geigenmuller et al. [161 in a two-step contraction of fast variables. After the first contraction, the system is described by a relatively large number of macroscopic variables, some of them much faster than the other ones. They assume that the relaxation matrix describing the approach to equilibrium in the first contracted level obeys the Onsager—Casimir relations and they are able to show that after elimination of the fast variables the relaxation matrix of the remaining contracted set ofvanables shows small deviations from Onsager—Casimir symmetry of second order in the ratio of time scales. We address a different problem and we take as basic variables of our description not only the conserved ones but also the fast higher-order fluxes and we do not contract the description. This makes that our equations are first-order instead of higher-order differential equations as in refs. [15,16]. Furthermore, the inclusion of the fast variables (higher-order fluxes) in the formalism means that our analysis should be valid not only for long times but also for short times ofthe order ofthe relaxation times ofthe fluxes (i.e. before the “aging time” assumed in Onsager’s theory). We assume, for simplicity, that the system (a rigid conductor or a gas at rest) is able to transport heat. We denote as q1 the heat flux, as q2 the flux of the heat flux and so on, in such a way that q~is an nth ordertensor which is the flux ofq~_1.From the point of view of kinetic theory of monatomic ideal gases, the q,, could be interpreted as the reduced nth order moments ofthe distribution function with respect to the velocity [7,8]. Following the usual development of EIT we write for the entropy differential the generalized Gibbs equation [9,14] ds=T’du+T’pdv—>~~q~odq~,
(5)
where s and u are the entropy and the internal energy per unit mass, T and p the absolute temperature and thermodynamic pressure, and 0 denotes the total contraction of the respective tensors. The coefficients a~are related to the relaxation times of the q~ as is well known [9,14] and as will be seen below.
25 February 1991
The entropy flux is also generalized to have the form [9,14] Js=Tlq+ ~fl~q~÷1oq~, (6) with 0 meaning the contraction of q~+ with q~to give a vector. Of course, more complicated nonlinear couplings could be added to (5) and (6), but our purpose is to provide the simplest possible illustration, so that we do not include them. When the time derivative of s is obtained from (5) and when one takes into account the mass and energy balance equations pi’= V v, (7) = V•q+p(V. v) (8) with v the specific volume per unit mass and v the baricentric velocity, one obtains
~
—
ps=
—
,
T’Vq1
—
~ ~
(9)
The entropy production is easily obtained from (6) and (9) through the usual definition ps+V.Js=a= ~
~
(10)
Our aim is to obtain evolution equations for q~compatible with the positivity ofthe entropy production a as expressed in (10). The simplest possibility is to assume a linear relation of the form ~ +V(/i~_1q~_1)+fl~V•q~+1 q,,, (11) with hz,, ~ 0. The resulting evolution equations may be written as = ~,,
tnà,,,, =
—
(AVT)ö1~ —q~+
+~V•q~~1.
-~-
V(fl~_1q~_~)
(12)
The definition of q~~1 as the flux of q~allows us to identify fl~/ji,,= r~/pwith ‘r~the relaxation time of q~identified as x,, = ~ in (11) and (12). From this set of equations one may obtain a generalized frequency- and wavevector-dependent thermal conductivity in the form of the continued-fraction ex—
77
Volume 153, number 2,3
PHYSICS LETTERS A
pansion [9,14], but here we are interested in a different topic. The equation for the heat flux obtained from (12) (n= 1) takes the form ~-V’q2. (13) p When q2=0, this equation reduces to the Maxwell— Cattaneo equation used in hyperbolic heat transport [9, 17,181, which in turn reduces to the classical Fourier’s law when r~ = 0 or in steady state situations. We write now the equations for q,, and q,,~1 in terms ofthe derivatives of the entropy. According to (5) and (12) we have p q~ = V0S/aq~_1+ ôS/8q~ = —AVT—q1
—
—
+
i-V. [(p/a~~1)t9S/9q~+1]
,
P
1 V’ [(p/a~÷2)ôS/0q~~2]
—
.
steady states characterized by VT~0 unless p/an does not depend on T. One could consider (16) in still another form. One could define the coefficients Mnn+i as the one relating with VôS/ôq~÷1 and M~÷1~ the one relating ~ and VôS/ôq~,and ~ andL~÷1,~ the ones relating ~n with aS/aq~+1 and ± with ôS/äq~.In this form, (16) yields, on the one hand, ~,,
~,,
~
=
= M~+
,,,.
(14)
If we assume that p/an depend only on T we have 1 p ~1n= V8S/8q~_1+ ôS/ôq~ —
(17)
an+I
This is relation (4). On the other hand, (18) implying a breaking of the Onsager relations (1) out of equilibrium. Another example may be helpful in clarifying this issue. We consider an incompressible fluid. The linear momentum and energy balance equations are L’nn±i_ran±i(VT), L’~+~=0,
(19)
pv=—VP+pF,
~S/öqn~
+
25 February 1991
pü—V.q—P:V, with P the pressure tensor, q the heat flux and V(20) the symmetric part of the velocity gradient. These equations must be complemented with constitutive equations for q and the viscous pressure tensor P—pU. In contrast with the classical theory EIT ~V,
}~V,
considers q and as independent variables for which evolution equations are needed. These evolution equations turn out to be ~V
+
~ln±I
V- aS/a~+1+ an+I (VT)• (aS/aq~+1) —~—V8S/ôq~+ ~ a~±1
~ (21) the symmetric part of the gradient of q.
r2P=_(P+2~~I)+2,~T(Vq), +
~~(85/8qn÷2)+an~2(VT)’~/aqn±2, (15)
with a~=(1 /p)ö(p/a~)/ôT. Thus, the coefficients Lnn+i relating to 8S/t3q~+1 and L~+i,n linking ~n±l to ôS/ôq~are ~,,
1
(16)
It is thus seen that 1=L~~ i,n are satisfied in equilibrium, when VT= 0, but not in nonequilibrium 78
(Vq)s
The generalized Gibbs equation is now ds=T’ du+T’pdv— ~-~qdq ~L..pv:dp~ 2i1T
Lnn+1V+an±1VT,
Ln±i,n~~V.
with
whereas the entropy flux is
.P=T~q+flP°’q.
(22)
(23)
We will only consider the linear case when/i is constant and we concentrate our attention on relations
Volume 153, number 2,3
PHYSICS LETTERS A
(4). According to (23) we may write (21) and (22) as
4= ~2—8S/8q— 2fJA~1TS~8~,8pv_ -~-VT, 3 2T 2~ 2fl)~T T~~2 VoS/aq+ öS/8P~ V. 1112
—
11
(24) Thus we obtain here Mqv=
2 ~1 T3 ‘~ II 12
“
=M~
(25)
as stated in (4). It is worthwhile to mention that these relations are supported from the kinetic theory of gases [7], where the coefficient /1 turns out to be 13= —2/5pT. Note that in a steady state eqs. (21) lead to q= —2VT—2flAi~T2VV+NL, PV2~V2fl)~7’V(VT)+NL
(26)
with NL standing for nonlinear terms. These relations have been obtained from the Boltzmann equalion by means of several methods, as Chapman— Enskog developments (7) or projection-operator techniques (19). Recall that in the framework ofthe local-equilibrium theory T2VT and T ‘V are the respective forces conjugated to q and It is thus seen that the sign in the relation connecting the cross coefficients in (26) is not the usual Onsager—Casimir sign but it has the opposite sign. Indeed, P’ has even parity whereas q has odd parity, so that a minus sign instead of a plus sign should to expected. Several comments are in order. In spite of their similar appearance, the reciprocity relations (1) and (4) between the respective L 0 and M,~coefficients defined by (3), there are some obvious differences, One of them is the sign. Note, indeed, that q~and q~±1 have opposite parity with respect to time reversal (from the point of view of the kinetic theory of monatomic gases, they are related to the moments with respect to the molecular velocity). Thus, according to (2) an antireciprocity relation should be expected, rather than a reciprocity relation. Relation (4) could be seen as a special case ofgeneralized Onsager—Casimir relations relating cit, with ôS/ôa~,but with L,~V= E1~L~, ( V). This relation could be ob—
—
-
~V,
—
25 February 1991
tamed from the usual fluctuations arguments: in a nonequilibrium situation, the microscopic reversibility would not be valid unless we change externally the sign of V. The situation is analogous to the one encountered in the ofbeexternal magnetic fields, which must be presence reversed to able to apply microscopic reversibility arguments. This is also similar to the problem studied by Dufty and Rubi [61 concerning Onsager’s relation under a velocity gradient: the externally imposed velocity gradient must be reversed in order to be able to apply microscopic reversibility. Another difference between the relations defined by L and by M is that the latter ones are closely related to the extra part of the entropy flux, and not with the entropy production (the only dissipative terms in (12) are those connecting~~ with q~).Thus, they are different from those of the Onsager—Casimir theory, where the entropy flux is not taken into account or it is supposed to keep its classical form. Furthermore, EIT allows one to obtain the reciprocity relations in M from a thermodynamic point of view, as is seen in detail in ref. [9], section 3.4, whereas the Onsager—Casimir relations rely on a microscopic underlying hypothesis. The usual thermodynamic theory is based on the local-equilibrium hypothesis. Such a local equilibrium is not obtained until the fast variables have decayed to their local-equilibrium value. The use of the generalized Gibbs equation of EIT includes the effects of the relaxation of the fluxes, and thus it is expected to be valid at shorter times than the classical theory, on the time scale settled by the relaxation times of the fluxes. Since in the classical theory only the classical variables are considered, Onsager relations are usually sought for in terms of only these variables and their respective conjugate forces, but not in the more general set including the fast variables. As conclusions of the present Letter we underline two points: (1) Some reciprocity relations in the evolution equations for the fluxes arise from the couplings induced by the entropy flux. Such relations have some differences with the usual Onsager— Casimir relations, as they imply a further change of sign: for instance, they imply reciprocity instead of antireciprocity when they connect variables of different parity. (2) Furthermore, our equations pro79
Volume 153, number 2,3
PHYSICS LETTERS A
25 February 1991
vide a simple example of the breaking of some reciprocity relations which are fulfilled in equilibrium but not in nonequilibrium steady states.
[511. Gyarmati, Nonequilibrium thermodynamics (Springer, Berlin, 1970). [61 J.W. Dufty and J.M. Rubi, Phys. Rev. A 36 (1987) 222. [7]H. Grad, Handbuch der Physik, Vol. 12. Principles of the
I thank Professors D. Jou and J. Casas-Va.zquez for stimulating discussions and interest in our work. This work has been partially financed by the Dirección General de Investigación CientIfica y TecnolOgica of the Spanish Ministry of EducaciOn y Ciencia.
kinetic theory of gases, ed. W. Flugge (Springer, Berlin, 1958). [8] L. Waldmann and H. Vestner, Physica A 80 (1975) 523. [9] D. iou, J. Casas-Vazquez and G. Lebon, Rep. Progr. Phys. 51(1988)1105. [1011. Muller, Thermodynamics London,344. 1985). [11] L.S. Garcia-Cohn, Rev. Mex.(Pitman, FIs. 34 (1988)
References [1] L. Onsager, Phys. Rev. 37 (1931) 405; 38 (1931) 2265. [2] H.B. Casimir, Rev. Mod. Phys. 17 (1945) 843. [3] S.R. de Groot and P. Mazur, Nonequilibrium thermodynamics (North-Holland, Amsterdam, 1962). [4] P. Glansdorff and 1. Prigogine, Thermodynamic theory of structure, stability and fluctuations (Wiley—Interscience, NewYork, 1971).
80
[12] R.E. Netleton, Phys. Fluids 2 (1959) 256; 3(1960)216. [13] B.C. Eu, J. Chem. Phys. 73 (1980) 2958. [14] C. Perez-Garcia and D. Jou, J. Phys. A 19 (1986) 1272. [15]J.A.McLennan,Phys.Rev.A 10(1974)1272. [161 U. Geigenmuller, U.M. Titulaer and B.U. Felderhof, Physica A 119 (1983) 53. [171 D.D. Joseph and L. Preziosi, Rev. Mod. Phys. 61 (1989) 41; 62 (1990) 375. [18] B. Vick and MN. Ozisik, ASME Trans. C, J. Heat Transfer 105 (1983) 902. [l9]M.H.Ernst,Am.J.Phys.38 (1970) 908.
PHYSICS LETTERS A
25 February 1991
Reciprocity relations and higher-order fluxes Maria Ferrer Departament de Fisica (FIsica EstadIstica), UniversitatAutànoma de Barcelona, 08193 Bellaterra, Catalonia, Spain Received 20 July 1990; revised manuscript received 3 December 1990; accepted for publication 1 7 December 1990 Communicated by A.R. Bishop
Reciprocity relations between coefficients linking the evolution equations of higher-order hydrodynamic fluxes in a heat conducting fluid at rest are derived from a thermodynamic point of view. Such relations have some subtle differences with respect to the usual Onsager—Casimir relations. Our paper provides a simple illustration of the breaking of the reciprocity relations in nonequilibrium steady states.
Onsager—Casimir reciprocity relations [1,2] are one of the basic ingredients of classical nonequilibrium thermodynamics [3—5].They are very important because they reduce in a nontrivial way the number of independent coefficients appearing in the thermodynamic evolution equations. These relations may be formulated in the following way. Let a, be a set of thermodynamic variables or, rather, their deviations from their respective equilibrium values, If the evolution equations of the a, may be written as —L~,t9S/9a~~ (1) with S the entropy and L0 constant phenomenological coefficients and an upper dot the material time derivative, then the L~satisfy the following reciprocity relations, =
L~= e,L11,
(2)
~,
where = + 1 or 1 according to the parity of a, with respect to the time reversal operation. In the usual literature, the a, are named the thermodynamic fluxes and ÔS/ôa1 the thermodynamic forces, In this paper, another kind of relations is presented, according to which if the evolution equations for the a, are written as ~,
a, = ~ then, 76
—
(3)
~1~= —~1~1~
(4)
The main difference of (4) with the Onsager— Casimir relations (2) is the change of sign. Furthermore, (4) may be obtained from purely thermodynamic arguments, whereas (2) may only be derived from a microscopic point of view. The Onsager—Casimir relations are derived from fluctuation theory from the hypothesis of microscopic reversibility [3—51. Such a hypothesis is valid in equilibrium but not in nonequilibrium steady states. Thus, one could ask for deviations from reciprocity relations in nonequilibrium steady states [6]. Another purpose of this Letter is to provide a simple example of the breaking of reciprocity relations for a heat-conducting system (for instance, a monatomic gas at rest) when a description including as independent variables not only the usual conserved ones (density, linear momentum, energy) but also the heat flux and higher-order fluxes [7,8]. We follow here the macroscopic approach known as extended irreversible thermodynamics (EIT) [9—141. Our approach to higher-order hydrodynamics thus differs from some previous approaches [15,16] in the choice of independent variables. The authors of refs. [15,16] have referred to an approximate closed dynamics for the local conserved densities and have studied the breakdown of Onsager’s relations for fluxes beyond the Navier—Stokes order in gradients, due to the finite time required for the relaxation of
0375-9601/91/s 03.50 © 1991
—
Elsevier Science Publishers B.V. (North-Holland)
Volume 153, number 2,3
PHYSICS LETTERS A
nonhydrodynamical variables. Violations of Onsager symmetry in the course of a contraction from a fully microscopic to a macroscopic description have been found by Geigenmuller et al. [161 in a two-step contraction of fast variables. After the first contraction, the system is described by a relatively large number of macroscopic variables, some of them much faster than the other ones. They assume that the relaxation matrix describing the approach to equilibrium in the first contracted level obeys the Onsager—Casimir relations and they are able to show that after elimination of the fast variables the relaxation matrix of the remaining contracted set ofvanables shows small deviations from Onsager—Casimir symmetry of second order in the ratio of time scales. We address a different problem and we take as basic variables of our description not only the conserved ones but also the fast higher-order fluxes and we do not contract the description. This makes that our equations are first-order instead of higher-order differential equations as in refs. [15,16]. Furthermore, the inclusion of the fast variables (higher-order fluxes) in the formalism means that our analysis should be valid not only for long times but also for short times ofthe order ofthe relaxation times ofthe fluxes (i.e. before the “aging time” assumed in Onsager’s theory). We assume, for simplicity, that the system (a rigid conductor or a gas at rest) is able to transport heat. We denote as q1 the heat flux, as q2 the flux of the heat flux and so on, in such a way that q~is an nth ordertensor which is the flux ofq~_1.From the point of view of kinetic theory of monatomic ideal gases, the q,, could be interpreted as the reduced nth order moments ofthe distribution function with respect to the velocity [7,8]. Following the usual development of EIT we write for the entropy differential the generalized Gibbs equation [9,14] ds=T’du+T’pdv—>~~q~odq~,
(5)
where s and u are the entropy and the internal energy per unit mass, T and p the absolute temperature and thermodynamic pressure, and 0 denotes the total contraction of the respective tensors. The coefficients a~are related to the relaxation times of the q~ as is well known [9,14] and as will be seen below.
25 February 1991
The entropy flux is also generalized to have the form [9,14] Js=Tlq+ ~fl~q~÷1oq~, (6) with 0 meaning the contraction of q~+ with q~to give a vector. Of course, more complicated nonlinear couplings could be added to (5) and (6), but our purpose is to provide the simplest possible illustration, so that we do not include them. When the time derivative of s is obtained from (5) and when one takes into account the mass and energy balance equations pi’= V v, (7) = V•q+p(V. v) (8) with v the specific volume per unit mass and v the baricentric velocity, one obtains
~
—
ps=
—
,
T’Vq1
—
~ ~
(9)
The entropy production is easily obtained from (6) and (9) through the usual definition ps+V.Js=a= ~
~
(10)
Our aim is to obtain evolution equations for q~compatible with the positivity ofthe entropy production a as expressed in (10). The simplest possibility is to assume a linear relation of the form ~ +V(/i~_1q~_1)+fl~V•q~+1 q,,, (11) with hz,, ~ 0. The resulting evolution equations may be written as = ~,,
tnà,,,, =
—
(AVT)ö1~ —q~+
+~V•q~~1.
-~-
V(fl~_1q~_~)
(12)
The definition of q~~1 as the flux of q~allows us to identify fl~/ji,,= r~/pwith ‘r~the relaxation time of q~identified as x,, = ~ in (11) and (12). From this set of equations one may obtain a generalized frequency- and wavevector-dependent thermal conductivity in the form of the continued-fraction ex—
77
Volume 153, number 2,3
PHYSICS LETTERS A
pansion [9,14], but here we are interested in a different topic. The equation for the heat flux obtained from (12) (n= 1) takes the form ~-V’q2. (13) p When q2=0, this equation reduces to the Maxwell— Cattaneo equation used in hyperbolic heat transport [9, 17,181, which in turn reduces to the classical Fourier’s law when r~ = 0 or in steady state situations. We write now the equations for q,, and q,,~1 in terms ofthe derivatives of the entropy. According to (5) and (12) we have p q~ = V0S/aq~_1+ ôS/8q~ = —AVT—q1
—
—
+
i-V. [(p/a~~1)t9S/9q~+1]
,
P
1 V’ [(p/a~÷2)ôS/0q~~2]
—
.
steady states characterized by VT~0 unless p/an does not depend on T. One could consider (16) in still another form. One could define the coefficients Mnn+i as the one relating with VôS/ôq~÷1 and M~÷1~ the one relating ~ and VôS/ôq~,and ~ andL~÷1,~ the ones relating ~n with aS/aq~+1 and ± with ôS/äq~.In this form, (16) yields, on the one hand, ~,,
~,,
~
=
= M~+
,,,.
(14)
If we assume that p/an depend only on T we have 1 p ~1n= V8S/8q~_1+ ôS/ôq~ —
(17)
an+I
This is relation (4). On the other hand, (18) implying a breaking of the Onsager relations (1) out of equilibrium. Another example may be helpful in clarifying this issue. We consider an incompressible fluid. The linear momentum and energy balance equations are L’nn±i_ran±i(VT), L’~+~=0,
(19)
pv=—VP+pF,
~S/öqn~
+
25 February 1991
pü—V.q—P:V, with P the pressure tensor, q the heat flux and V(20) the symmetric part of the velocity gradient. These equations must be complemented with constitutive equations for q and the viscous pressure tensor P—pU. In contrast with the classical theory EIT ~V,
}~V,
considers q and as independent variables for which evolution equations are needed. These evolution equations turn out to be ~V
+
~ln±I
V- aS/a~+1+ an+I (VT)• (aS/aq~+1) —~—V8S/ôq~+ ~ a~±1
~ (21) the symmetric part of the gradient of q.
r2P=_(P+2~~I)+2,~T(Vq), +
~~(85/8qn÷2)+an~2(VT)’~/aqn±2, (15)
with a~=(1 /p)ö(p/a~)/ôT. Thus, the coefficients Lnn+i relating to 8S/t3q~+1 and L~+i,n linking ~n±l to ôS/ôq~are ~,,
1
(16)
It is thus seen that 1=L~~ i,n are satisfied in equilibrium, when VT= 0, but not in nonequilibrium 78
(Vq)s
The generalized Gibbs equation is now ds=T’ du+T’pdv— ~-~qdq ~L..pv:dp~ 2i1T
Lnn+1V+an±1VT,
Ln±i,n~~V.
with
whereas the entropy flux is
.P=T~q+flP°’q.
(22)
(23)
We will only consider the linear case when/i is constant and we concentrate our attention on relations
Volume 153, number 2,3
PHYSICS LETTERS A
(4). According to (23) we may write (21) and (22) as
4= ~2—8S/8q— 2fJA~1TS~8~,8pv_ -~-VT, 3 2T 2~ 2fl)~T T~~2 VoS/aq+ öS/8P~ V. 1112
—
11
(24) Thus we obtain here Mqv=
2 ~1 T3 ‘~ II 12
“
=M~
(25)
as stated in (4). It is worthwhile to mention that these relations are supported from the kinetic theory of gases [7], where the coefficient /1 turns out to be 13= —2/5pT. Note that in a steady state eqs. (21) lead to q= —2VT—2flAi~T2VV+NL, PV2~V2fl)~7’V(VT)+NL
(26)
with NL standing for nonlinear terms. These relations have been obtained from the Boltzmann equalion by means of several methods, as Chapman— Enskog developments (7) or projection-operator techniques (19). Recall that in the framework ofthe local-equilibrium theory T2VT and T ‘V are the respective forces conjugated to q and It is thus seen that the sign in the relation connecting the cross coefficients in (26) is not the usual Onsager—Casimir sign but it has the opposite sign. Indeed, P’ has even parity whereas q has odd parity, so that a minus sign instead of a plus sign should to expected. Several comments are in order. In spite of their similar appearance, the reciprocity relations (1) and (4) between the respective L 0 and M,~coefficients defined by (3), there are some obvious differences, One of them is the sign. Note, indeed, that q~and q~±1 have opposite parity with respect to time reversal (from the point of view of the kinetic theory of monatomic gases, they are related to the moments with respect to the molecular velocity). Thus, according to (2) an antireciprocity relation should be expected, rather than a reciprocity relation. Relation (4) could be seen as a special case ofgeneralized Onsager—Casimir relations relating cit, with ôS/ôa~,but with L,~V= E1~L~, ( V). This relation could be ob—
—
-
~V,
—
25 February 1991
tamed from the usual fluctuations arguments: in a nonequilibrium situation, the microscopic reversibility would not be valid unless we change externally the sign of V. The situation is analogous to the one encountered in the ofbeexternal magnetic fields, which must be presence reversed to able to apply microscopic reversibility arguments. This is also similar to the problem studied by Dufty and Rubi [61 concerning Onsager’s relation under a velocity gradient: the externally imposed velocity gradient must be reversed in order to be able to apply microscopic reversibility. Another difference between the relations defined by L and by M is that the latter ones are closely related to the extra part of the entropy flux, and not with the entropy production (the only dissipative terms in (12) are those connecting~~ with q~).Thus, they are different from those of the Onsager—Casimir theory, where the entropy flux is not taken into account or it is supposed to keep its classical form. Furthermore, EIT allows one to obtain the reciprocity relations in M from a thermodynamic point of view, as is seen in detail in ref. [9], section 3.4, whereas the Onsager—Casimir relations rely on a microscopic underlying hypothesis. The usual thermodynamic theory is based on the local-equilibrium hypothesis. Such a local equilibrium is not obtained until the fast variables have decayed to their local-equilibrium value. The use of the generalized Gibbs equation of EIT includes the effects of the relaxation of the fluxes, and thus it is expected to be valid at shorter times than the classical theory, on the time scale settled by the relaxation times of the fluxes. Since in the classical theory only the classical variables are considered, Onsager relations are usually sought for in terms of only these variables and their respective conjugate forces, but not in the more general set including the fast variables. As conclusions of the present Letter we underline two points: (1) Some reciprocity relations in the evolution equations for the fluxes arise from the couplings induced by the entropy flux. Such relations have some differences with the usual Onsager— Casimir relations, as they imply a further change of sign: for instance, they imply reciprocity instead of antireciprocity when they connect variables of different parity. (2) Furthermore, our equations pro79
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vide a simple example of the breaking of some reciprocity relations which are fulfilled in equilibrium but not in nonequilibrium steady states.
[511. Gyarmati, Nonequilibrium thermodynamics (Springer, Berlin, 1970). [61 J.W. Dufty and J.M. Rubi, Phys. Rev. A 36 (1987) 222. [7]H. Grad, Handbuch der Physik, Vol. 12. Principles of the
I thank Professors D. Jou and J. Casas-Va.zquez for stimulating discussions and interest in our work. This work has been partially financed by the Dirección General de Investigación CientIfica y TecnolOgica of the Spanish Ministry of EducaciOn y Ciencia.
kinetic theory of gases, ed. W. Flugge (Springer, Berlin, 1958). [8] L. Waldmann and H. Vestner, Physica A 80 (1975) 523. [9] D. iou, J. Casas-Vazquez and G. Lebon, Rep. Progr. Phys. 51(1988)1105. [1011. Muller, Thermodynamics London,344. 1985). [11] L.S. Garcia-Cohn, Rev. Mex.(Pitman, FIs. 34 (1988)
References [1] L. Onsager, Phys. Rev. 37 (1931) 405; 38 (1931) 2265. [2] H.B. Casimir, Rev. Mod. Phys. 17 (1945) 843. [3] S.R. de Groot and P. Mazur, Nonequilibrium thermodynamics (North-Holland, Amsterdam, 1962). [4] P. Glansdorff and 1. Prigogine, Thermodynamic theory of structure, stability and fluctuations (Wiley—Interscience, NewYork, 1971).
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[12] R.E. Netleton, Phys. Fluids 2 (1959) 256; 3(1960)216. [13] B.C. Eu, J. Chem. Phys. 73 (1980) 2958. [14] C. Perez-Garcia and D. Jou, J. Phys. A 19 (1986) 1272. [15]J.A.McLennan,Phys.Rev.A 10(1974)1272. [161 U. Geigenmuller, U.M. Titulaer and B.U. Felderhof, Physica A 119 (1983) 53. [171 D.D. Joseph and L. Preziosi, Rev. Mod. Phys. 61 (1989) 41; 62 (1990) 375. [18] B. Vick and MN. Ozisik, ASME Trans. C, J. Heat Transfer 105 (1983) 902. [l9]M.H.Ernst,Am.J.Phys.38 (1970) 908.