An experiment is proposed to check the physical reality of a nonequilibrium absolute temperature previously proposed from theoretical grounds in the framework of extended irreversible thermodynamics

A. Santos

B. C. Eu

D. D. Joseph

D. Jou

G. Lebon

H. Grad

I. Müller

J. Casas-Vázquez

J. Keizer

J. Meixner

L. S. García-Colín

M. Criado-Sancho

P. Glansdorff

R. E. Nettleton

S. R. de Groot

W. Muschick

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Physical Review A

Possible experiment to check the reality of a nonequilibrium temperature

Jou i Mirabent, David,

Casas Vázquez, José

American Physical Society

Diposit Digital de Documents de la UAB

PHYSICAL REVIEW A VOLUME 45, NUMBER 12 15 JUNE 1992Possible experiment to check the reality of a nonequilibrium temperatureD. Jou and J. Casas-VazquezDepartament de Fisica (Fisica Estadistica), Universitat Autonoma de Barcelona, 08193 Bellaterra, Catalonia, Spain(Received 25 November 1991)An experiment is proposed to check the physical reality of a nonequilibrium absolute temperature pre-viously proposed from theoretical grounds in the framework of extended irreversible thermodynamics.PACS number(s): 05.70.Ln, 44.10.+ iSBQwith s the entropy (or the specific entropy per unit mass),u the internal energy (or specific internal energy per unitmass), and v the volume (or specific volume per unitmass).The generalized entropy arising in extended irreversi-ble thermodynamics for a rigid heat conductor has thefollowing forms(u, q)=s, (u) —(v 2'/. T )q.q, (2)where s,z(u) is the local-equilibrium entropy, T thelocal-equilibrium absolute temperature, q the heat flux, A,Some of the most basic questions in nonequilibriumthermodynamics are those concerning the fundamentalthermodynamic concepts, namely, the definition andmeaning of entropy and of absolute temperature out ofequilibrium. Most of the nonequilibrium thermodynamictheories have assumed that these concepts do not needany reformulation out of equilibrium, and that the usualequilibrium concepts may be applied locally in nonhomo-geneous nonequilibrium situations.For some years, we have proposed a nonequilibriumabsolute temperature [1—4] stemming as the partialderivative of a nonequilibrium entropy that incorporatesas variables the dissipative fluxes, in the framework of theso-called extended irreversible thermodynamics (EIT)[5—10]. Up to now, we have analyzed such a nonequili-brium temperature from a theoretical point of view, ei-ther from a formal point of view involving dissipativeCarnot cycles [2] or, from a more specific point of view,by studying some of the possible indirect consequences ofsuch a generalized temperature in the propagation of heatwaves in nonequilibrium steady states [3,4] or in the ap-parent thermal conductivity in Poiseuille phonon flow[4].The purpose of this paper is to suggest an experimentto directly check the physical reality of such a nonequili-brium temperature. First of all, we review the introduc-tion of the generalized temperature; afterwards, we sug-gest the experiment and, finally, we evaluate the order ofmagnitude of the expected effects.We recall that the standard thermodynamic definitionof the absolute temperature isthe thermal conductivity, and ~ the relaxation time of theheat flux, defined according to the relaxational evolutionequationr+q= —A,V8BqBt (3)where the quantity 8 is a generalized absolute tempera-ture defined according to (1), i.e.,8 '=T ' —yq q, (4)with y=B(v r/2A, T )/t)u.An equation of the form (3), with T instead of 8, isknown as Maxwell-Cattaneo-Vernotte equation, and hasbeen used in several occasions to describe heat waves insolids at low temperatures [11]. For a detailed discussionof this equation and of its thermodynamic consequences,the reader is referred to [5(b)]. The appearance of 8 in-stead of T is imposed by thermodynamic requirementswhich are a direct consequence of the use of the general-ized entropy (2) instead of the local-equilibrium entropy,and which are examined at length in [3—5]. It must benoted that EIT does not pretend to be an exact theory,but rather a model which allows for formulation of anonequilibrium thermodynamic theory not based a priorion the local-equilibrium hypothesis, and which allows forexploration of the limitations of this hypothesis and fornew questions which are out of the scope of the classicaltheory.It must be pointed out that a nonequilibrium absolutetemperature is a concept that does not arise in the classi-cal local-equilibrium theory [12—13], but that is not ex-clusive of extended irreversible thermodynamics. For in-stance, different concepts of "nonequilibrium tempera-tures" have been considered in other contexts by otherauthors, such as Meixner (who sets the concept of a"dynamical temperature" [14]), Miiller (who introduces a"coldness" [15]), Muschick (who postulates a "contacttemperature" [16]), and Keizer (in the framework of hisformulation of nonequilibriurn thermodynamics based onstatistical considerations of molecular fluctuations[17,18]). To our knowledge, none of these abstract tem-peratures has had any direct experimental confirmation.As compared with these other temperatures, the non-equilibrium temperature of extended thermodynamicsand that of Keizer have the advantage of having an expli-cit expression allowing for an evaluation of the nonclassi-45 8371 1992 The American Physical Society8372 D. JOU AND J. CASAS-VAZQUEZ 45cal effects.In order to check whether the quantity 8 defined in (4)is a mathematical artifact arising from an undue exten-sion of the classical definition (1), or whether it is an actu-al physical quantity, we propose the following experi-ment. We connect by means of a plate of a good thermalconductor (such as, for instance, Cu or Ag) two thermo-dynamic systems. One of such systems is at equilibrium,whereas the other one is in a nonequilibrium steady statecharacterized by a heat flux q perpendicular to the con-nection between both systems (Fig. 1). Such a connectionis installed in such a form that the local-equilibrium tem-perature of the nonequilibrium system at the position ofthe connection is equal to the temperature of the equilib-rium system. According to (3), a heat flux q„should beobserved between both systems. Such a flux would begiven byq, =A, (8—T)/I, (5)where A, is the thermal conductivity of the metallicplate providing the connection between both systems andIt is the separation between them. In contrast, in thelocal-equilibrium theory q, should be given asq =A, ( T —T) /1 =0, because we have assumed thatboth ends of the connection are at the same local-equilibrium temperature T.According to (4), the heat flux q„should be, up to thesecond order in q,q„=(A, /l)yT q (6)The presence of such a heat flux q„as predicted in (6)would be a strong support to the physical reality of thegeneralized absolute temperature 8.Finally, we proceed to make an estimation of the orderof magnitude of the predicted nonclassical effect. For anideal gas and for a metallic conductor, the explicit ex-pressions of the generalized temperature (4) are given by8 '=T '+(2/5)(m/n k T )q q,8 '=T '+(9/vr )(mcF/n k T')q q,(7)T qgFIG. 1. Two thermodynamic systems are connected througha highly conducting metallic plate. The system on the left is atequilibrium at temperature T. The system on the right is in anonequilibrium steady state under a heat flux q„. Both ends ofthe conducting plate are at the same local-equilibrium absolutetemperature T. According to the classical theory, no heatshould flow from one system to the other one. In contrast, EITpredicts a flow q from the left to the right because the non-equilibrium thermodynamic temperature of the right-hand sidesystem, 8, is not equal to the local-equilibrium absolute temper-ature T.where m is the mass of the molecules in (7} or the elec-tron mass in (8), n is the number particle density, eF isthe Fermi energy of the metal, and k is the Boltzmannconstant (k =1.38X 10 J/K).To estimate q, assume, for instance, that the subsys-tems are composed of CO2 at 300 K and 0.1 atm. Then,m =4X 10 kg and n =2.6X 10 particles/m, and wehave for (6) q„=(A, /I)(9. 6X10 ' )q, with q expressedin W/m . Thus, for q of the order of 10 W/m, theeffective temperature difference 8—T would be of the or-der of 9.6X 10 K. We think it would be better to mea-sure q„rather than 8—T itself because of the conceptualdiSculties in the definition and measurement of tempera-ture in nonequilibrium. Note that to achieve q =102W/m one would need a temperature gradient in the ydirection of the order of 10 K/m because the thermalconductivity of the gas is of the order of 10 W/Km.Such a temperature gradient is easy to achieve. At suchconditions, and using a copper plate ( A. = 3.8 X 10W/Km) of 1 cm in length to connect both systems, onewould have q„=3.65X10 W/m, which should be per-ceptible.If the two subsystems are made of copper[n =8.45 X 10 cm, TF =Ez/k =8.12X10 K,m =9.1X10 ' Kg], the difference 8—T for a tempera-ture gradient in the y direction of the order of 10 K/mwould be of the order of 2X10 K, less than for the gasof the previous example. In this situation, the heat-fluxq due to the difference 8—T would be q, =76 W/m .It is worthwhile to point out that the entropy (2} isconfirmed from kinetic theory of gases in the 13-momentGrad's approximation [19]. More general expressions fors as a function of q may be found in more sophisticateddevelopments in kinetic theory [20]. But in kinetictheory of monatomic gases the temperature is identifiedthrough the expression u'=(3/2)kT, u' being the meankinetic energy per molecule, whereas the thermodynamicdefinition (1) is not used. Our suggested experimentcould contribute to clarify the physical significance ofboth temperatures. Thus the interest of such an experi-ment is not restricted to extended irreversible thermo-dynamics, but may be of more general interest in non-equilibrium thermodynamics and statistical mechanics.The relation between the generalized temperature ofextended thermodynamics and that of Keizer deservessome special mention. In fact, both temperatures aredifferent than the local-equilibrium absolute temperature,and both depend on the fluxes besides on the classicalvariables. The difference between both theories is foundin their respective starting points: extended thermo-dynamics is motivated by the dynamics of the fluxes con-sidered as independent variables; Keizer's theory has itsorigin in the second moments of nonequilibrium fluctua-tions. It would be of interest to explore whether boththeories lead to a same expression for the nonequilibriumtemperature in a given definite situation: it is likely to ex-pect a confluence of both theories in this point. Onemust note, however, that the experiment we are propos-ing would lead to the conclusion that a thermometer doesnot measure the local-equilibrium temperature; but the45 POSSIBLE EXPERIMENT TO CHECK THE REALITY OF A. . . 8373generalized temperature, because the heat exchange be-tween the system and the thermometer is related todifferences in the latter one rather than to differences inthe local-equilibrium temperature, according to extendedthermodynamics. This point of view is different fromthat of Keizer, who assumes [17] that the thermometerreads the local-equilibrium temperature and that theeffects of the nonequilibrium temperature would be seenthrough an analysis of nonequilibrium fluctuations.Temperature would not be the only thermodynamicquantity to contain nonequilibrium corrections: theywould also arise in the pressure or in the chemical poten-tial. Keizer has studied theoretically and experimentallythe corrections to the electromotive force in a nonequili-brium situation [21]. The experimental consequences ofthe nonequilibrium contributions to the chemical poten-tial in polymeric solutions have been studied in theframework of extended irreversible thermodynamics [22].In both analyses, the corrections turn out to be measur-able. However, the corrections have been more elusive inthe analysis of temperature. This is not surprising, as themeasurement of temperature in nonequilibrium steadystates is still an open problem. Though our experimentdoes not set a completely general nonequilibrium temper-ature, as it is restricted to situations where the heat flux isthe only dissipative quantity, it is evident that theconfirmation of the physical reality of a nonequilibriumabsolute temperature would be a very important fact innonequilibrium thermodynamics, either from a theoreti-cal point of view concerning the foundations themselvesof irreversible thermodynamics, or from an experimentalpoint of view, since intense heat fluxes may be found inactual physical systems as, for instance, in the sun andother stars, in gravitational collapse, in explosions, or inimplosions of hydrogen pellets in laser-induced fusion re-actions.This work has been supported by the Direccion Gen-eral de Investigacion Cienti'fica y Tecnologica of theSpanish Ministry of Education and Science under GrantNo. PB-90/676. We acknowledge stimulating discussionswith Professor G. Lebon, from Liege University, andProfessor J. J. Brey, from Sevilla University.[1]J. Casas-Vazquez and D. Jou, J. Phys. A 14, 1221 (1981)[2] D. Jou and J. Casas-Vazquez, J. Phys. A 20, 5371 (1988).[3]J. Casas-Vazquez and D. Jou, Acta Phys. Acad. Sci.Hung. 66, 99 (1989).[4] D. Jou and J. Casas-Vazquez, Physica A 163, 47 (1990).[5] (a) Recent Developments in Nonequilibrium Thermodynamics, edited by J. Casas-Vazquez, D. Jou, and G. Le-bon (Springer, Berlin, 1984); (b) D. Jou, J. Casas-Vazquez,and G. Lebon, Rep. Prog. Phys. 51, 1105 (1988); (c) G. Le-bon, D. Jou, and J. Casas-Vazquez, J. Phys. A 13, 275(1980).[6] L. S. Garcia-Colin, Rev. Mex. Fis. 34, 344 (1988); L. S.Garcia-Cohn, M. Lopez de Haro, R. F. Rodriguez, J.Casas-Vazquez, and D. Jou, J. Stat. Phys. 37, 465 (1984);L. S. Garcia-Colin, R. Luzzi, and A. Vasconcellos, Phys.Rev. A 43, 6622 (1991);43, 6633 (1991).[7] R. E. Nettleton, Phys. Fluids 3, 216 (1960); J. Phys. A 20,4017 (1990).[8] I. Miiller, Z. Phys. 198, 329 (1967); Thermodynamics (Pit-man, London, 1985).[9] B. C. Eu, J. Chem. Phys. 73, 2958 (1980).[10]Advances in Thermodynamics: Nonequilibrium Thermodynamics, edited by S. Sieniutycz and P. Salamon (Taylorand Francis, New York, 1992).[11]D. D. Joseph and L. Preziosi, Rev. Mod. Phys. 61, 41(1989).[12]S. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics (North-Holland, Amsterdam, 1962)~[13]P. Glansdorff and I. Prigogine, Thermodynamic Theory ofStructure, Stability and Fluctuations (Wiley, New York,1971).[14]J. Meixner, in Foundations of Continuum Thermodynamics, edited by J. J. D. Domingos, M. N. R. Nina, and J. H.Whitelaw (MacMillan, London, 1974).[15]I. Miiller, Arch. Ration. Mech. Anal. 40, 1 (1971).[16]W. Muschick, Arch. Ration. Mech. Anal. 66, 379 (1977);J. Non-Equilib. Thermodyn. 4, 277 (1979).[17]J. Keizer, J. Chem. Phys. 65, 4431 (1976); 69, 2609 (1978);82, 2751 (1985).[18]J. Keizer, Statistical Thermodynamics of NonequilibriumProcesses (Springer, New York, 1987).[19]H. Grad, in Principles of the Kinetic Theory of Gases, edit-ed by S. Flugge, Handbuch der Physik Vol. XII (Springer,Berlin, 1958).[20] A. Santos and J.J. Brey, Physica A 174, 355 (1991).[21]J. Keizer, J. Chem. Phys. 87, 4064 (1987);87, 4074 (1987).[22] M. Criado-Sancho, D. Jou, and J. Casas-Vazquez, Macro-molecules 24, 2834 (1991).

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Possible experiment to check the reality of a nonequilibrium temperature

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