Robustness of the nonequilibrium entropy related to the Maxwell-Cattaneo heat equation

The connection between the Maxwell-Cattaneo heat transport equation and a nonequilibrium entropy is examined through four different thermodynamic approaches, and it is shown that all of them lead to the same form of the nonequilibrium entropy. Furthermore, it is seen that this form is also consistent with three microscopic formalisms. This robustness underlines the consistency and relevance of the entropy.


I. INTRODUCTION
Nonequilibrium thermodynamics is an open and active field of research where a relatively high number of different approaches coexist ͓1-13͔.The differences between them have been pointed out on several occasions, but their confluences have not yet received sufficient emphasis, maybe because of the difficulties caused by the choice of different variables, different formulations of the second law, and attention to different kinds of physical systems and different problems.The aim of this paper is to emphasize that different formalisms lead to the same nonequilibrium entropy in some circumstances.This confluence of different approaches deserves to be outlined because it gives more consistency to the final common result.Since entropy outside equilibrium is a particularly debated topic, expressions for nonequilibrium entropies are especially worthy of attention.
The present analysis is focused on a concrete situation, arising in heat transport with some memory effects.The Maxwell-Cattaneo equation is a well-known extension of the Fourier heat transport equation incorporating a relaxational term, which leads to a finite velocity of propagation of heat pulses ͓1,7,8,14-18͔.It reads where is the thermal conductivity and the relaxation time of the heat flux.The speed of the heat pulses is given by v 2 = / c v , with the mass density and c v the heat capacity per unit mass at constant volume; when tends to zero, the velocity of the heat pulses diverges, whereas ͑1͒ reduces to the classical Fourier law.Despite its simplicity, Eq. ͑1͒ has stimulated much research in nonequilibrium thermodynamics, in order to obtain theoretical models in which this equation becomes compatible with the second law of thermodynamics ͓1,5,7,8,14,19͔.This is necessary because Eq. ͑1͒ is not compatible with the local-equilibrium formulation of the second law, as it leads to negative entropy production in some situations.Here, we want to stress a result that is common to all the mentioned theoretical approaches, when expressed in terms of the same variables, namely, the consistency between ͑1͒ and the following form of the entropy per unit volume: where s eq ͑u͒ is the local-equilibrium entropy per unit volume, a function of the internal energy per unit volume, u ͑here, we focus our attention on heat transfer in incompressible systems and neglect other possible classical variables of the entropy͒.This is a nonequilibrium entropy, because of the contribution of the second term on the right-hand side, which depends on the heat flux and vanishes at equilibrium.Note the close relation between this term in ͑2͒ and the relaxational term in ͑1͒: both of them vanish when tends to zero.
However, the robustness of the connection between ͑1͒ and ͑2͒ has not been appreciated enough up to now, because it is usually credited only to the theory known as extended irreversible thermodynamics ͑EIT͒ ͓7,8,14,16,19͔.Here we will show, however, that this connection has also been established in other approaches, in the linear domain, with different choices of variables and different methods of exploitation of the second law.This was not sufficiently realized up to now because the corresponding entropies were always expressed in terms of different variables and without using an explicit physical identification of the coefficient of the nonequilibrium term.To be more faithful to the original developments and to better understand the difficulties in realizing the mentioned relation, we will keep the notation of each theory rather than unifying the different notations a priori.We think that this robustness of relation ͑2͒ to different methods of derivation has much interest, because it illustrates an explicit form of a nonequilibrium entropy common to different approaches.͓1,7,14,19͔, one starts from an entropy of the form s͑u , q͒.In fact, more general forms are used, incorporating more variables, either classical, such as the volume and the composition of the system, and nonequilibrium ones, such as diffusion flux, electric current, or viscous pressure.Here, we will focus our attention on heat conduction.The heat flux is considered as an independent variable because ͑1͒ is an evolution equation for q which requires initial conditions for q to be solved.In second order in the heat flux, it is assumed that where is the temperature, s the entropy, u the internal energy per unit volume, and ␣ a coefficient which should be identified in physical terms.In fact, the temperature is assumed to be not exactly the local equilibrium temperature itself, but to contain a second-order contribution in the heat flux stated by the integrability condition of ͑3͒, namely, ‫ץ‬ −1 / ‫ץ‬q ‫␣ץ͑−=‬ / ‫ץ‬u͒q, which leads, after integration, to If we are interested only in first-order corrections we can approximate the temperature by its local equilibrium temperature value T.
To simplify the presentation and concentrate our attention on the central point, we will assume incompressible systems.The energy balance equation has the form Then the rate of change of s is The first term in the right-hand side is the divergence of the entropy flux, namely, the exchange of entropy with the surroundings, and the second term is the entropy production.Since we want to find an evolution equation for q, compatible with the positive definite character of the entropy production, we write with a positive coefficient.Comparison of ͑7͒ with ͑1͒ leads to = ͑T2 ͒ −1 and ␣ = ͑ / T 2 ͒.Introducing the latter identification in ͑3͒ we have for the entropy ds = −1 du − T 2 q • dq.͑8͒ Here we have used the notation for the nonequilibrium temperature; relation ͑4͒ defines a nonequilibrium thermodynamic temperature which has been compared with other proposals for effective nonequilibrium temperatures ͓20͔.
Integration of ͑8͒ leads to ͑2͒, and relation ͑4͒ becomes ͑9͒ Now we will compare results ͑8͒ and ͑9͒ with those obtained in other approaches.

III. INTERNAL VARIABLES
The preceding derivation of the connection between ͑1͒ and ͑2͒ is almost immediate.The choice of q as independent variable is not shared by all theories but is typical of EIT which analyzes the minimum ingredients for the thermodynamic consistency of ͑1͒.One would like to understand ͑1͒ from some structural details of the system, rather than deriving it in a direct but less microscopically suggestive way.A possibility for deriving ͑1͒ on thermodynamical grounds is to assume the existence of a vectorial internal variable characterizing some internal features of the system, to be identified later on ͓4,10͔.Such an approach yields an equation that is more general than ͑1͒, but that, when specialized to ͑1͒, yields again the entropy ͑2͒.
To see this, we write the time derivative of s͑u , ͒ as where the coefficient ␣Ј plays a similar role as ␣ in Eq. ͑3͒ of the previous section.
Taking into account the energy balance equation ͑5͒, ͑10͒ is written as As in Eq. ͑6͒, the first term represents the divergence of the entropy flux and the two last ones are the entropy production, which has a different structure than in ͑6͒.For the moment we will neglect nonlinear contributions of q to , and we approximate it by T. By following the usual procedure in classical irreversible thermodynamics ͓1,2,21͔, we will express q and ˙in terms of ١T −1 and −␣Ј in linear approximation as with L ij being phenomenological coefficients.We assume that is odd under time reversal, a feature that will be discussed below.This leads, according to the Onsager-Casimir reciprocal relations, to L 12 =−L 21 because ١T −1 is even and odd, and the Onsager-Casimir relations state that L ji = ⑀ i ⑀ j L ij , with ⑀ i and ⑀ j being +1 or −1 if the corresponding variable is even or odd with respect to time reversal.The positiveness of the entropy production requires The last term is not found in ͑1͒.An equation of the form ͑14͒, or in more explicit terms Jeffrey's equation ͓1,2͔.Such an equation does not lead to hyperbolic equations for heat transport, and it leads to an infinite velocity for heat pulses.In EIT, an equation of the form ͑15͒ is obtained by taking as the independent variable not only q but also the flux of the heat flux, as we will discuss below ͓1,7͔.
If we want to obtain ͑1͒ from ͑14͒, one must identify L 11 = 0, and it follows that L 22 ␣Ј= −1 and L 12 2 ␣Ј= T 2 / .But if L 11 = 0, then, according to ͑12͒, =−͑L 12 ␣Ј͒ −1 q.In fact, ␣Ј may be taken as 1, by normalizing the definition of , and ͑10͒ may be rewritten in the form which is the same expression as ͑8͒ because of the identification of L 12 2 in terms of and presented above and leads, under the same conditions, to ͑2͒.The importance of considering as an odd variable with respect to time reversal may be realized a posteriori.Indeed, to eliminate the last term in ͑14͒ one must take L 11 = 0. Furthermore, one needs to identify −L 12 L 21 / T 2 = Ն0; therefore, it is necessary that L 12 and L 21 have different signs, which, by virtue of the Onsager-Casimir reciprocal relations requires that has opposite parity to ١T −1 , i.e., that it is an odd variable.This is consistent with the identification =−L 12 −1 q because q is an odd variable.In different systems, the physical meaning of may be different, and also the meaning of L 12 .For instance, in ideal gases may be related to the energy flux transported by moving molecules, whereas in superfluids it may be related to the relative motion between normal and superfluid components ͓22͔.
In going from ͑10͒ to ͑16͒ we have assumed that ͑ / T 2 ͒ 1/2 is constant.However, ͑16͒ is also valid in the general case, provided T is not identified with the local equilibrium temperature.Indeed, in strict terms, combining ͑10͒, the identification of in terms of q, and the identification of L 12 as ͑T 2 / ͒ 1/2 , we should have written When the differential in the second term is developed we obtain

͑19͒
which is the same nonequilibrium thermodynamic temperature as in ͑9͒.Thus, the use of a vectorial internal variable odd with respect to time inversion allows one to obtain the Maxwell-Cattaneo relation ͑1͒ provided the entropy takes the form ͑2͒.

IV. RATIONAL EXTENDED THERMODYNAMICS
In the two previous sections, the second law has been exploited along the lines of classical irreversible thermodynamics, namely, the Onsager linear relation between fluxes and forces.One may exploit the second law along the lines of rational thermodynamics ͓1,12,13͔.In this section we do so by using the Liu technique in rational extended thermodynamics, selecting u and q as variables; we will look for an evolution equation for q obeying the restrictions of the second law ͓7,8͔.Now, the second law is expressed as with J s the entropy flux.This equation is supplemented by the energy balance equation, and the evolution equation of q, which is assumed to be of the general form where Q is the heat flux and q is the source term.Note that this form is simply the general form of the evolution equations, incorporating the divergence of a flux-i.e., the exchange with the surroundings of the system-and a source term.To have a concrete equation, q and q must be specified.This is what we will do below.
To take into account the constraints of the evolution equations ͑5͒ and ͑21͒ besides the constraints of the second law ͑20͒, in Liu's method one incorporates these dynamical constraints by means of Lagrange multipliers, as Note that, for the moment, s and J s have not been assigned an explicit form, but they will be found consistently throughout the development.This is a difference with respect to the classical theory of nonequilibrium thermodynamics, in which s is taken as the local equilibrium entropy and J s is identified a priori with T −1 q.In EIT and internal variable theory, s is not fully identified a priori, but some coefficients in the differential of the entropy must be identified in consistency with the evolution equations.Condition ͑22͒ leads to This expression must be positive for any values of u ˙, q ˙, ١u, and ١q; this implies that the coefficients of these quantities in ͑23͒ must vanish because u ˙, q, ١u, and ١q may take arbitrary positive or negative values.Thus it follows that ‫ץ‬s / ‫ץ‬u = ⌳ 0 , ‫ץ‬s / ‫ץ‬q = ⌳ 1 , and J s = ⌳ 0 q.According to these conclusions one may write From a physical point of view, these expressions are not very illustrative, unless ⌳ 0 and ⌳ 1 are identified in physical terms and related to measurable quantities.For q = 0, ͑24͒ should reduce to the classical Gibbs equation, which leads to the identification ⌳ 0 = T −1 near equilibrium; this is also consistent, when introduced in ͑25͒, with the classical form of the entropy flux.A main point is to identify ⌳ 1 when q is not zero.Furthermore, the integrability condition of ͑24͒ is which leads to One may identify ⌳ 0 ϵ −1 as a nonequilibrium temperature, as in previous sections.
The expression for ⌳ 1 will be related to the evolution equation for q, given by ͑21͒.Now, we assume Q = a͑u͒U and q =− −1 q as a particular and simplest possibility, which does not exclude other more general possibilities.In this case the residual terms in ͑23͒ may be written as Since ‫ץ‬J s / ‫ץ‬u = ‫ץ͑‬ −1 / ‫ץ‬u͒q, it follows from the first term in parentheses in ͑28͒ that This shows explicitly the connection between ⌳ 1 and the evolution equation ͑21͒.To be totally explicit, the function a͑u͒ must yet be identified.
Introducing into ͑21͒ the forms of q and q postulated above, we obtain q ˙= − ١a − −1 q. ͑30͒ Comparing ͑30͒ to ͑1͒, and writing ١a = ͑‫ץ‬a / ‫͒ץ‬ ١ , we may identify ͑‫ץ‬a / ‫͒ץ‬ = / .From here and ͑29͒ it follows that Then the Gibbs equation may be written as which has, again, the form ͑6͒ or ͑16͒, and leads to ͑2͒, under the assumption of the integrability conditions of ͑32͒.This condition is ͑26͒ and leads to the same form as ͑9͒ for the nonequilibrium temperature .
Unfortunately, in the standard presentations of rational extended thermodynamics ͓8͔, the explicit identification of ⌳ 1 is usually not given.This makes it difficult to realize that the entropy ͑2͒ is recovered in this case.

V. INTERNAL VARIABLES AND RATIONAL THERMODYNAMICS
The assumption that some internal variables must be included in the description of the system does not imply by itself which kind of interpretation and of exploitation of the second law must be used.In Sec.III, we have used the Onsager approach.Here, we will use instead the framework of rational thermodynamics.
In particular, we will follow a paper by Cimmelli and Frischmuth based on a particular interpretation of the internal variable ͓23͔.Following a model proposed by Koscinsky they assume that the heat flux is given by where ␤ is a generalized dynamical temperature which in steady state becomes equal to the temperature , according to an evolution equation of the generic form ␤ ˙= f͑,␤͒, ͑34͒ and they denote p ϵ ١␤.They assume that ␤ does not enter the constitutive equations explicitly, and that p ϵ ١␤ can be regarded simply as an additional internal variable related to the heat flux.They postulate for p the equation where and are positive quantities given by ϵ 1 / f 1 and ϵ 1 / f 1 ␤.In the classical case, = and The second law of thermodynamics is written as where g ϵ ١, is the specific entropy, and ϵ ⑀ − is the Helmholtz free energy.Cimmelli and Frischmuth postulate that q, ⑀, and are functions of , g, and p.The corresponding constitutive equations for q, ⑀, and are then examined from the perspective of their compatibility with the second law ͑37͒ by using Liu's method of Lagrange multipliers ͓8͔.Thus, they write

͑38͒
This leads, after a thorough application of the method of Lagrange multipliers, analogous to that shown in the previous section, to the necessary and sufficient conditions and the residual inequality

͑41͒
with k͑ , p͒ being a dynamical heat conductivity.This leads for the entropy inequality ͑40͒ to It follows that p = 0 and k Ͼ 0. To investigate the consequences of the hypothesis ͑41͒ they assume Then, by the restrictions ͑40͒ and ͑39͒, they find that The result ͑45͒ is valid for k / 2 = const.In general, the entropy is To make contact with our previous results it must be recalled that eq should be referred to eq ͑⑀͒.According to Cimmelli and Frischmuth ͓23͔, one has whereas in ͑46͒ eq is expressed in terms of corresponding to ⑀.Thus, in ͑46͒ we should rewrite eq ͑͒ in terms of ͑⑀ eq ͒.We have, from ͑47͒, Expanding this expression up to second order in q and recalling that ‫ץ‬ eq / ‫⑀ץ‬ =1/ , ͑48͒ yields When this is introduced into ͑46͒ we have This has the same form as the entropy ͑2͒, if one uses the symbol s instead of to denote the entropy.
Note that to have the same result as in ͑2͒ it is necessary to be especially careful about the local equilibrium reference state, making sure that it is the same as in ͑2͒.Indeed, ͑46͒ seems rather different from ͑2͒, and one realizes the close connection only after realizing that in ͑2͒ entropies at the same internal energy are compared.

VI. MICROSCOPIC DERIVATIONS OF (2)
The form ͑2͒ of the entropy, related to the form ͑1͒ for the evolution of q, is not only obtained in different macroscopic formalisms but is also supported from several microscopic theories: ͑a͒ Grad's development of kinetic theory ͓24,25͔, ͑b͒ the Chapman-Enskog development of kinetic theory ͓24,26͔, and ͑c͒ fluctuation expressions.
In kinetic theory for dilute gases, the entropy is given by the Boltzmann expression s͑r,t͒ = − k B͵ f͑r,c,t͒ln f͑r,c,t͒dc, ͑51͒ with k B the Boltzmann constant, c the peculiar molecular velocity, i.e., the velocity of the molecules with respect to the barycentric velocity of the system, and f͑r , c , t͒ the probability distribution function for the molecular velocity.At equilibrium f is given by the well-known Maxwell-Boltzmann expression, which will be denoted as f eq ͑c͒.Near equilibrium, it is usual to write where ⌽ i are suitable nonequilibrium contributions, which must satisfy some side conditions in order that ͑51͒ refers to a given density, energy, and velocity.Introduction of ͑52͒ into ͑51͒ leads, up to second order, to ͓1,8,24,25͔

͑53͒
The second term on the right-hand side of ͑53͒ indicates the nonequilibrium contribution to the entropy.Our aim is now to explore whether these nonequilibrium contributions have the same form as ͑2͒ for heat transport.We will use two different formalisms: ͑a͒ Grad's approach, where ⌽ i is expressed in terms of q, and ͑b͒ the Chapman-Enskog approach, where ⌽ i is given in terms of ١T.It will be seen that, up to second order, both formalisms lead to the entropy ͑2͒.

A. Grad's approach
In Grad's approach, the distribution function f is expanded in terms of its several moments with respect to the velocity; the zeroth-order moment is the density, the firstorder moment is the barycentric velocity, and the trace of the second-order moment is the internal energy.The heat flux is given by and therefore it is a part of the third-order moments of f.For heat transport, Grad's approach takes the form ͓24,26͔ ⌽ 1 ͑r,t͒ = ͩA͑r,t͒•c+B͑r,t͒• 1 2 mc 2 cͪ , ͑55͒ where A and B are coefficients determined in terms of the barycentric velocity v and the heat flux q.When this is done, the explicit form of ͑55͒ is ͓24,26͔ When this expression is inserted into ͑53͒ and after performing a suitable integration we find This shows that a contribution of second order in q to the entropy is to be expected on microscopic grounds.However, we want more than a qualitative corroboration.Therefore, we must examine whether ͑57͒ has the same form as ͑2͒.To do that, we need an evolution for the heat flux.In Grad's approach, this is obtained by inserting ͑55͒ into Boltzmann's equation and integrating it over the velocity.In the absence of viscous pressure contributions, this leads to ͓24,26͔ where ␥ is a function of T and p related to an integral of the collision term.This equation has the form of ͑2͒ provided one identifies From the expression relating and 1 , it is seen that 1 / =2m / 5pk B and therefore that ͑57͒ may be written as which is precisely the form ͑2͒ of the entropy.

B. Relaxation-time approximation
In nonequilibrium steady states one may approximate q by its classical expression in terms of the temperature gradient, according to Fourier's law.Then, ͑2͒ becomes This expression for the entropy is also obtained from a microscopic basis in the relaxation-time approximation in the Chapman-Enskog approach ͓25͔.In such an approach, the nonequilibrium correction to the distribution function is expressed in terms of ١T instead of in terms of q.In this case, one has When introduced into the entropy ͑53͒ and after performing the integration we find that s = s eq − 5 4 pk B 1 2 mT 2 ١ T • ١T.͑63͒ In view of the form ͑59͒ for in terms of 1 , which is also found in the Chapman-Enskog approach, this may be rewritten just as ͑61͒.

C. Fluctuation-dissipation expressions
A third microscopic proof of the internal consistency of ͑1͒ and ͑2͒ may be obtained by studying the second moment of the fluctuations of the heat flux around equilibrium.This is more general than the two previous derivations in this section, because it is not restricted to classical diluted gases, but is valid for general systems.To do that, we use the wellknown Einstein formula for the probability of fluctuations, namely, where ␦ 2 s is the second differential of the entropy.In fact, the Einstein formula is applicable to the fluctuations of extensive quantities; thus, instead of working with q itself, one should work with Vq, which is an extensive variable, as is given by where N is the number of particles.By combining ͑2͒ and ͑64͒, and writing the volume explicitly, we have From here one obtains for the second moments of the fluctuations with ␦ ij being Kronecker's delta, and the angular brackets denoting an equilibrium average.This expression is a particular case of the general Green-Kubo expression for the thermal conductivity, according to which ͓1,5͔

͑68͒
If one uses ͑1͒ to describe the decay of the fluctuations ␦q͑t͒ near equilibrium one has ␦q i ͑t͒ = ␦q i ͑0͒exp͑−t / ͒.When this is introduced into ͑68͒ and the integration is performed, expression ͑67͒ is recovered.In other words, if one writes and one uses the Einstein expression for fluctuations, one finds

͑70͒
Then, consistency with ͑68͒ in the relaxation-time approximation would lead to ␣ = / T 2 , as in expression ͑2͒.Consistency between ͑1͒ and ͑2͒ is corroborated again.

VII. COMMENTS
In this paper we have shown that seven different methods, four of them macroscopic and three others microscopic, lead to the same expression for the nonequilibrium entropy ͑2͒ related to the Maxwell-Cattaneo equation ͑1͒.Such generality of s͑u , q͒ reinforces its role as a nonequilibrium entropy.Of course, the validity of such entropy is restricted to situations where Eq. ͑1͒ is valid, but it is of interest as a particular explicit illustration of a nonequilibrium entropy.
If quartic terms in q were added to the entropy, the corresponding evolution equation for the heat flux would not be the linear Maxwell-Cattaneo equation.Probably, the quarticorder approximation to the entropy would not be universal; for instance, approaches based on kinetic theory or on maximum entropy lead to the same result for the quadratic contributions of the heat flux to the entropy, but they lead to different coefficients for the quartic term ͓7͔.In any case, the universality at the quadratic order is interesting by itself, as it covers a considerable phenomenology and it uncovers some degree of universality beyond the local-equilibrium hypothesis.This is valid not only for the heat flux, but also for other fluxes, such as, for instance, the momentum flux or viscous pressure tensor P v , in which case the quadratic nonequilibrium entropy per unit volume has the form ͓7,19͔ s͑u,P v ͒ = s eq ͑u͒ −

2T
P v :P v , ͑71͒ where is the shear viscosity and the viscoelastic relaxation time, appearing in the Maxwell viscoelasticity equation with V the symmetric part of the velocity gradient.Particular simplified models which are able to grasp some essential physics-in our case, a finite speed velocity of heat pulses-are useful as a basis for more exhaustive and detailed models.In this aspect, the entropy ͑2͒ plays a role analogous to that of the van der Waals equation in the analysis of real gases.The van der Waals equation is not an exact equation, and it does not lead to very accurate quantitative predictions, but it grasps the existence of a finite critical temperature.The entropy ͑2͒ is not an exact expression, and Eq.͑1͒ does not lead to accurate expressions for heat wave speeds, but it is able to grasp a finite value for the speed of heat pulses.To improve the quantitative agreement, higherorder approaches are needed.In the van der Waals equation, they are related to higher-order powers of the density; in extended thermodynamics, they are related to the higherorder fluxes of the energy.