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We present the state of the art on the modern mathematical methods of exploiting the entropy principle in thermomechanics of continuous media. A survey of recent results and conceptual discussions of this topic in some well-known non-equilibrium theories (Classical irreversible thermodynamics CIT, Rational thermodynamics RT, Thermodynamics of irreversible processes TIP, Extended irreversible thermodynamics EIT, Rational Extended thermodynamics RET) is also summarized.

In continuum physics the entropy principle constitutes a valuable tool in modeling material properties. In rational thermodynamics (RT) [

These authors also proposed a rigorous mathematical procedure to exploit the requirement above, currently referred to as Coleman-Noll procedure [

On the other hand, the Coleman-Noll approach requires an appropriate formulation of second law. In continuum physics [

^{(}^{s}^{)}

For the entropy flux Coleman and Noll postulated the constitutive equation

When dealing with the formulation above, one should look for an appropriate constitutive equation not only for the entropy flux but also for the entropy density, and for an appropriate definition of temperature in situations far from equilibrium. In extended irreversible thermodynamics (EIT), taking the fluxes as independent variables, the three questions mentioned above give the following answers [

in processes which are not too far from equilibrium, the extended entropy is the local equilibrium entropy plus a negative contribution proportional to the square of the fluxes; the corresponding coefficient is proportional to the relaxation time of the corresponding flux and inversely proportional to the respective transport coefficient; thus, the non-equilibrium contributions to the entropy are related to relaxational contributions to generalized transport equations;

the entropy flux is the Coleman-Noll entropy flux

the absolute temperature, given by the reciprocal of the derivative of the extended entropy with respect to the internal energy, differs from the local-equilibrium temperature and depends on the fluxes; it is related to the average kinetic energy of the particles in the plane perpendicular to the fluxes, and it is in general different from the temperatures of the other degrees of freedom, which may have different values.

Some conceptual problems arise by analyzing the different mathematical methods for the exploitation of the entropy principle. First of all, the entropy principle itself, such as formulated by Coleman and Noll, could appear as an arbitrary assumption, since nothing prevents second law of thermodynamics to restrict the processes instead of the constitutive equations. Thus, Muschik and Ehrentraut [

As a consequence of this statement, they proved that for regular processes,

It is worth noting that if one applies the classical Coleman–Noll or Liu procedures, weakly nonlocal continuum theories [

An alternative method for circumventing that problem is to generalize the classical exploitation procedures, by regarding the governing equations for the gradients entering the state space as additional constraints for the entropy inequality, [

Recently Fabrizio, Lazzari and Nibbi [

where

In [

In classical irreversible thermodynamics (CIT) [

A different perspective has been open in rational extended thermodynamics (RET) [

The formulation (

_{B}^{3} × ^{3}.

Thus, it seems to be quite natural to investigate what is the relation between the two statements above. Recent results have shown that the procedure of MEP for the closure of the moments equations for rarefied gases, complies with the recent macroscopic approach of extended thermodynamics to real and perfect gases [

Although entropy maximization in isolated systems is a well-known procedure, it becomes problematic in non equilibrium situations. For instance, entropy maximization around an equilibrium steady state, submitted to further additional constraints on fluxes, or higher-order moments (which are null at equilibrium) is not so evident nor well-known, but it is not in contradiction with maximization in equilibrium. Nonetheless, several quantum systems, such as interacting fermionic and bosonic gases, semiconductor heterostructures with confined carrier transport, require a quantum formulation of MEP in situations out of equilibrium. In achieving that task, the main difficulty is to define, in non equilibrium situations, an appropriate quantum entropy that explicitly incorporates quantum statistics [

As far as the relation between macroscopic and microscopic approaches is concerned, one should observe that in kinetic theory higher order weakly nonlocal continuum extensions can be obtained by Chapmann-Enskog expansion. However, the expansion beyond first order results in unstable equations. Thus, the family of Burnett equations cannot be much improved, contrary to serious efforts [

The examples above show that the proper formulation of the entropy principle in continuum physics is still an open problem. The aim of this paper is to provide an overview on the state of the art and on recent researches on this topic, which represents a milestone in thermodynamics of continuous media.

To achieve that task, different approaches, corresponding to the different point of view of the authors on non-equilibrium continuum theories, are presented and illustrated by some recent applications. In this way, different thermodynamic theories are analyzed.

The foundation of different theories in the last decades has stimulated several discussions into the thermodynamic community and produced many progresses, which regard both the mathematical tools of investigation and the understanding of new physical phenomena. Moreover, if we look at the different schools of thermodynamics with more attention, we discover that these are not so far as it could appear at the first sight, since several connections emerge. A further aim of this presentation is just to point out the similarities between the various approaches.

Then, in Section 2, Péter Ván illustrates the role of the entropy principle in classical irreversible thermodynamics. In Section 3, David Jou analyzes the application of the entropy principle in extended irreversible thermodynamics. In Section 4, Tommaso Ruggeri shows the consequences of the entropy principle in rational extended thermodynamics. In Section 5, Vito Antonio Cimmelli presents some classical results of rational thermodynamics and recent results regarding the mathematical analysis of the entropy inequality.

Throughout the paper, summation with respect to repeated indices will be assumed. Moreover we adopt the following notations:

_{t}

_{k}_{k}

_{t}_{i}∂_{i}

_{i}_{i}

_{ij}_{ij}

∇ ≡ (_{i}

Δ ≡ (_{ii}

∇(∇ · _{i}_{j}x_{j}

_{(}_{ij}_{)}—symmetric part of tensor _{ij}

_{〈}_{ij}_{〉} —symmetric and traceless part of tensor _{ij}

^{T}

^{n}_{1} . . . _{n}

the word “density” means the amount of a given quantity per unitary volume

the word “specific” means the amount of a given quantity per unitary mass

The problems illustrated in the present paper have been discussed during the “12

It is worth observing that, although in our analysis second law of thermodynamics plays a fundamental role, the present review is not devoted to the wide topic of the different formulations of second law, but to the more restricted one of the entropy principle,

The authors gratefully acknowledge the chairman of the conference, Prof. Gian Paolo Beretta, and the staff of JETC 2013, for the perfect organization, the friendly ambient and the opportunity of several interesting discussions on new trends in non-equilibrium thermodynamics.

In his pioneering works about reciprocal relations Lars Onsager established a uniform approach of different interactions in discrete (homogeneous) thermodynamic bodies, where the basic variables are the classical extensive thermodynamic variables [

CIT constructs evolution equations for dissipative phenomena. Instead of variational principles the thermodynamic compatibility of continuum material properties plays a key role in the methodology. Eckart established the connection between homogeneous thermodynamic bodies and classical continua with the help of the hypothesis of local equilibrium. According to this hypothesis one assumes the validity of the Gibbs relation and the corresponding equations of state for densities or specific quantities of the extensive thermodynamic variables—that is local equilibrium—and introduces the entropy inequality as a balance. There are two crucial points of this procedure:

What is internal energy? How could one separate mechanics and thermodynamics, distinguish between motion and equilibrium?

What is the form of the entropy current density?

The answers of these questions are technically simple. The internal energy is the difference between the conserved total energy and the kinetic energy. One component simple fluids are characterized by the internal energy and the mass. Their Gibbs relation, expressed with specific quantities is

Here the extensive quantities are the specific internal energy

In this simple case the entropy inequality (

In the expression above we denoted by _{i}

where _{ij}

Then the entropy balance

follows, where _{ij}_{i}_{ij}_{ij}_{i}_{j}v_{i}

Here _{F}^{2} is the Fourier heat conduction coefficient, _{v}

In case of several components and more complex materials one generalizes the previous procedure and try to obtain this quadratic form of the entropy production as a sum of product of thermodynamic fluxes and forces. The linear relationship with symmetric and positive definite coefficient matrix is assumed for less symmetric materials, too. In this respect the concepts of crystal physics proved to be instructive [

In the previous heuristic treatment one can identify several problematic points, where a deeper analysis is necessary for further developments:

The weakly nonlocal extension of the constitutive state space is a safe procedure, because the gradient is a covector, that does not depend on the motion, e.g., does not transform by a Galilean transformation [

In RET, weakly nonlocal constitutive equations are derived by truncated balance laws (see Section 4 for more details), so that they do not need to satisfy any constitutive principle.

In the following, we focus on these last questions. Exploring the conditions behind of classical theory, we show some possible generalizations and the way toward extensions.

The limitations of classical irreversible thermodynamics in its original form became more and more evident soon after the appearance of the monograph of de Groot and Mazur [

Other challenges came from the experimental observations and from the part of statistical theories. Thermodynamic compatibility of rheology and plasticity are old challenges, relativistic fluids and power law tail statistics provide new ones. Any new approaches of thermodynamics of irreversible processes (TIP) that generalize CIT should answer these challenges, when keeping the universal applicability and uniting character of the theory as a framework of dissipative gases, fluids and solids.

From this point of view the Truesdellian program of rationalization encountered various difficulties. For instance, the extension of the state space of elastic solids with gradients of the basic state variables (e.g., strain) was forbidden by mathematically rigorous analysis of the second law [

However, the mathematical methods, the Coleman-Noll and Liu procedures, are neutral, and one should scrutinize the applied conditions and background concepts also from a physical point of view. In the following we compare the heuristic methods to the more precise ones. In order to focus on the second law and simplify the treatment, we choose an abstract notation, that hides the details of the particular continua together with the aspects of material frame indifference.

The formal vector of the densities of extensive quantities _{α}_{α}_{α}

if the material is considered at rest. The entropy inequality (

where _{s}_{k}_{s}_{s}_{α}

Here
_{α}

We may recognize the quadratic flux-force structure and identify the current densities of the extensives,
_{k}_{α}

if the symmetric part of the conductivity matrix

In this procedure the constitutive quantities are the extensive fluxes
_{s}_{k}

Let us assume that the constitutive functions
_{k}_{s}

The balances _{α}

In this case the vector of highest derivatives is (_{t}z_{α}_{t}∂_{k}z_{α}

According to the second Liu equation there is local equilibrium, the entropy does not depend on the gradient of the state space. The first Liu

This is identical to

Moreover,

is a solution of _{k}_{k}z_{α}

Let us observe, that the final result of this simple derivation, the entropy production in the form of

The extension of the basic state space by additional variables or/and higher order gradients has two conceptually important consequences.

In case of higher order weak nonlocality it is necessary to introduce the gradients of the constraints (e.g., the balances) as independent new restrictions, too [

In case of internal variables (without balances as constraints) one can derive the complete evolution equations with thermodynamic methods and a weakly nonlocal extension provides also natural boundary conditions. The thermodynamic approach can substitute variational considerations. For example evolution equations of generalized continua can be obtained with the help of the entropy principle also in the ideal, nondissipative limit [

Writing about entropy principle in continuum physics we should mention the method of GENERIC (General Equation of Reversible and Irreversible Coupling) [

The evolution of the nondissipative part is generated by the functional derivative of the energy

The entropy principle in extended irreversible thermodynamics (EIT) is analogous to that in CIT, but with some formal and conceptual differences which we examine here. One starts with an entropy and an entropy flux which depend on the fluxes, besides on the classical variables. As said in the introduction, not too far from equilibrium the entropy is the local equilibrium entropy minus some expressions quadratic in the fluxes, and the entropy flux is the classical entropy flux plus a correction proportional to the product of the flux of the flux times the flux itself. In principle, these non-equilibrium contributions are not identified a priori, but depend on coefficients whose physical meaning will be interpreted later. From here, using the definition (

The thermodynamic forces are more general than their local equilibrium counterparts, and in them appear the time derivatives and the gradients of the fluxes, as a consequence of using an entropy and entropy flux which depend on the fluxes.

Since the fluxes are considered as independent variables, the constitutive equations do not aim to express the fluxes in terms of thermodynamic forces, but to express the evolution equations for the fluxes.,

As a consequence of (

Since the corresponding evolution equations are not exact equations but approximate equations, requiring that the entropy production is positive may be interpreted as giving a limit of validity to these equations; thus, whereas the restrictions of the entropy principle on exact evolution equations could establish that such equations must be valid for any process, the entropy principle as applied to approximate evolution equations may indicate that these equations are only valid for a range of processes, but not for every process. This is not in contradiction with the amendment to the second law proposed by Muschik and Ehrentraut [

The temperature appearing in the Gibbs equations,

Two other points worth of comment are the following ones:

The non-classical contributions to the entropy and entropy flux may be given a microscopic physical meaning by fluctuation theory, namely, by combining the generalized entropy with the Einstein’s equation for the probability of fluctuations to obtain the second moments of the fluctuations of the fluxes, which coincide with the usual forms of the fluctuation-dissipation theorems [

The non-classical contributions to the entropy and entropy flux may also be compared with the corresponding second-order expressions obtained from kinetic theory [

After the evolution equations for the fluxes have been formulated, one is able to give a physical interpretation to the several coefficients appearing in such equations (which are related to the coefficients of the non-equilibrium contributions to the entropy and the entropy flux) in terms of the classical transport coefficients, the relaxation times of the fluxes, the correlation lengths of the fluxes, and so on [

where

with _{eq}

The second term in _{eq}_{eq}_{rel}_{rel}

When the mean free path is negligible,

while

Absolute temperature is given by the reciprocal of the derivative of the entropy with respect to the internal energy (at constant values of the other extensive variables). When the extended entropy (

with the (temperature dependent) specific heat at constant volume
_{eq}

Relation (

In order to show the important role played by the nonlocal constitutive _{0}, and thickness _{0}, acting as a steady heat source at constant temperature _{0}, connected to a graphene circular layer, of outer radius _{g}, and the thickness _{g}), which removes the heat from that source. We analyze the steady-state situation in an isotropic layer, in which the heat removed is equal to the heat dissipated, thus allowing the temperature to remain constant.

This situation can be studied starting from the hydrodynamic-like equation

which, in steady states and when the heat flux may be neglected with respect to its spatial derivative, as it often happens in nanosystems, is a direct consequence of the Guyer-Krumhansl heat-transport

with _{0}/ (2_{g}), _{0} being the constant amount of heat per unit time irradiated by the point source.

Thus, in a generic annular region between two radial distances _{1} and _{2}, such that _{1} < _{2}, the inequality (

wherein _{1}) is the incoming entropy flux, and _{2}) is the outgoing one. Owing to

Moreover, since the conservation of the total heat flux across each circular zone implies 2_{1}_{1}) = 2_{2}_{2}), we have

From this relation it follows that when ^{2}Γ/^{2} (or in the limit case of Γ ^{2}/ (^{2}) → 0), then inequality (_{1} > _{2}. On the other hand, when _{1}, _{2} the condition ^{2}Γ/^{2} holds. Then we conclude that for radial distances from the center of the hot point larger than the mean free path, as expected, the larger the radial distance, the smaller the temperature.

When _{1}, _{2} < ^{2}Γ/^{2}, and the inequality (

which is not violated for _{1} < _{2}. Then, for radial distances from the center of the hot point smaller than the mean free path, the larger the radial distance, the bigger the temperature, so that the temperature hump arises.

Thus, the nonlocal form (

Another field in which nonlocal interactions are important is the analysis of thermoelectric effects,

_{e}_{e}_{p}

Let us focus our attention on a cylindrical nanowire with transversal radius _{e}_{p}_{e}_{p}

where

represents the heat flux in the bulk, calculated by solving the stationary Guyer-Krumhansl

_{p}_{p}

while, the figure-of-merit may be calculated to be [

As it is possible to observe, the figure-of-merit depends on _{p}_{p}_{lim} = ^{2}_{e}_{e}_{p}

Modelling non-equilibrium phenomena in which steep gradients and rapid changes occur represents a challenging task which has been tackled in the case of gas by means of two complementary approaches: the

The continuum model consists in describing the system by means of macroscopic equations (e.g., fluid-dynamic equations) obtained on the basis of conservation laws and appropriate constitutive equations. As an example, classical irreversible thermodynamics (CIT), which relies on the assumption of local thermodynamic equilibrium (LTE), has proven to be a useful and sound theory characterized by a systematic and comprehensive theoretical structure [

On the other hand, the approach based on the kinetic theory, which postulates that the state of the gas can be described by a velocity distribution function whose evolution is governed by the Boltzmann equation, is applicable to processes characterized by a

Rational extended thermodynamics is a phenomenological theory aiming at filling the gap between this two limit cases.

After the well known observation of Cattaneo in the case of a rigid heat conductor [

It was observed by Ruggeri in [

Taking into account that the most clear situation in which the NSF theory is not valid is when the gas is very rarefied, Liu and Müller [

More precisely, the kinetic theory of monatomic gases is based on the assumption that the state of the gas can be described by the distribution function

where the collision integral

Finding solution of (

where

where

with _{ll}

The balance laws obtained for the collision invariants _{i}^{2}/2 (^{2} = _{j}ξ_{j}

where _{i}_{i}_{i}

is the stress tensor, _{ij}

When we cut the infinite hierarchy (_{k}_{1}, _{k}_{1}_{k}_{2}, . . . _{k}_{1}_{k}_{2...}_{kN}

According to the continuum theory, restrictions on the constitutive equations come only from

An important example of the procedure outlined above is given by the “13-moment theory”, in which the truncated system includes only the quantities _{i}_{ll}_{i}_{1}_{i}_{2} (momentum flux), and _{lli}

The closure procedure is very complex. To achieve that task, first we need to require the Galilean invariance of the system. Hence, as a consequence of a general theorem for generic balance laws proved in [

where _{S}_{q}

The obtained closed system may be considered as the extension of the Navier-Stokes-Fourier theory for rarefied monatomic gases. In fact, by performing the Maxwellian iteration [

In the case of a large number of moments it is too difficult to proceed with a pure macroscopic theory (as the 13-moments theory) and it is necessary to recall that the

where, the densities _{j}

with

The MEP procedure states that actual distribution function _{1} are prescribed. Thus, the approximate distribution function comes out as solution of a variational problem, with constraints and yields the following solution [

where _{i}_{1}, _{i}_{1}_{i}_{1}, . . . , _{i}_{1}_{i}_{2...}_{iN}_{E}

where _{E}_{E}

Plugging (_{1}, we obtain a linear algebraic system that permits to evaluate the Lagrange multipliers

(_{E}_{2} and source terms (_{3}, the truncated system of moment

where _{S}_{max} becomes unbounded when _{max} for

For the case of rarefied monatomic gases, the well-established RET theory, endowed with MEP to achieve the closure, has proven to be very successful. Even in those cases in which the 13-moment theory is not so satisfactory (problems involving high-frequency sound waves, light scattering, or shock waves structure [

Unfortunately the RET theory, being strictly connected with the kinetic theory, suffers from the same limit as the Boltzmann equation, which is notoriously valid only for rarefied monatomic gases since the internal energy

A RET theory of dense gases and of rarefied polyatomic ones has recently been developed by Arima, Taniguchi, Ruggeri and Sugiyama which adopts the

As far as the kinetic counterpart is concerned, a crucial step towards the development of a theory of rarefied polyatomic gases is the work by Borgnakke and Larsen [

Recently Pavić, Ruggeri and Simić have proven [

More precisely, the 14 field RET theory of dense gases adopts the following 14 independent fields

whose evolution is governed by the balance equations

where _{ijk}_{ppik}_{ij}_{ppi}_{ij}_{ppi}_{ij}_{ppi}_{i}_{ii}

In the case of 14 moments it is possible to deduce using the previous hierarchy (

In the case of rarefied polyatomic gases, with the thermal and caloric equations of state

where ^{3} × ^{3} ×[0, ∞). Its rate of change is determined by the Boltzmann equation which has the same form as for monatomic gas but collision integral

where

The closure with both the method of Entropy Principle and MEP, leads to the following differential system for polyatomic rarefied gas

The entropy density and the entropy flux are expressed as

and concavity of the entropy density is ensured in equilibrium provided

The previous results are very similar to the ones of the EIT but now all the coefficients in the system (

The limit

Using the procedure of Maxwellian iteration the Navier Stokes equations emerge as limit case of (_{4,5}, while _{6} reduces to the Fourier law. These results together with similar ones obtained for Fick’s and Darcy’s law convinced one of us (T. R.) that equations that are non local in space are not real constitutive equations but approximations of balance laws equations (as the present case) and the real constitutive equations are local and, as consequence, the differential systems of mathematical physics are hyperbolic rather than parabolic [

Although the differential system of RET is very complex, it is possible to give some qualitative analysis. In fact, not only every nesting theories of RET are governed by symmetric hyperbolic systems with the property of well posedness of local (in time) Cauchy problem (see Section 5) but there can exists also global smooth solutions due to the overlap between the first 5 conservation laws and the remaining dissipative ones. In fact, for generic hyperbolic systems of balance laws (

_{2} ≤

Moreover Ruggeri and Serre have proved, in the one-dimensional case, that the constant states are stable [

Recently Lou and Ruggeri [

Let us consider a local state space
_{α}

We suppose that the evolution of the basic fields _{α}

with
_{α}_{α}_{α}

The dissipation inequality reads [

with _{k}

Thus, the set of governing equations, evaluated on the state space, reads

In

The inequality above is linear in the _{t}z_{α}_{0}_{0}), let us introduce the vector _{n}

We call _{0}_{0}) we can write the balance equations and the entropy inequality as follows

where _{0}_{0}), so that

[_{0}_{0})

Since

must be assigned. By successive differentiations with respect to space, from

ensue.

From _{0}). The remaining _{t}z_{α}_{0}) are determined by the balance _{0}_{0}). We get so

The 7

The problem now is to exploit the new dissipation inequality obtained by the inequality (

In the light of the previous result, it is worth investigating if, for arbitrary initial conditions,

We have two possibilities, each of them excluding the other one:

in (_{0}_{0}) all (local) solutions of the balance equations have to satisfy the second law,

in (_{0}_{0}) there are (local) solutions of the balance equations which satisfy the second law and others which do not,

In (_{0}_{0}) a process-direction vector

[

[

Let us suppose that in nonequilibrium second law represents a restriction on the processes, ^{1} and ^{2} an admitted process-direction vector and a forbidden one, and let us prove that, in such a case, there exists a process-direction vector

with

which is reversible.

In fact, let us evaluate ^{1} and multiply them by a scalar ^{2} and multiply them by the scalar

Moreover, since ^{1} is an admitted process-direction vector while ^{2} is a forbidden one, the entropy production densities ^{1} and ^{2} in correspondence of them will be

It is worth observing that ^{3} satisfies the system of balance laws (

On the other hand, the entropy production density corresponding to ^{3}, namely

has not definite sign, and therefore nothing prevents that ^{1} and ^{2} are such that

However, since ^{3} < 0 characterizes processes which do not exist in nature, for real ones we have ^{3} = 0, so that ^{3} is a reversible process-direction, contradicting so Axiom 1.

Moreover, the following relation, corresponding to reversible processes, holds

As a direct consequence of the admissibility of ^{1} we infer ^{2} > 0 and ^{1} – ^{2}) > 0, from which we conclude that

Due to the result above, presupposing the validity of Axiom 1, we are forced to conclude that in correspondence of nonequilibrium states, the process-direction vectors are either all admissible or all forbidden. On the other hand, since the real processes are necessarily admissible, the second hypothesis is false. As a consequence, in nonequilibrium second law cannot exclude forbidden process-direction vectors (since they do not exist) but it may only restrict the constitutive equations. This completes the proof.

In rational thermodynamics (RT) the sole balance equations are the balances of mass, linear momentum, angular momentum and energy [

where _{i}_{i j}_{i}

with the Helmholtz free energy

Alternative approaches introduce an additional rate of supply of mechanical energy, the interstitial working, engendered by long–range interactions among the molecules [

Without loss of generality we can pursue our analysis under the hypothesis that the relation (

Thus, it is easy to verify that in an arbitrary but fixed point (_{0}_{0}) the inequality above may be rewritten as

where

[^{n},^{n}

^{n}

The proof is straightforward. It is clear that the conditions (

_{0}_{0}), to ensure that the entropy inequality is fulfilled for arbitrary process-direction vectors. On the other hand, since (_{0}_{0}) is arbitrary, we can write the relations (

The technique illustrated so far is known in literature as Coleman-Noll procedure. Since the form of

[

^{n}, with

^{n}, with

It is trivial that 2 implies 1. To see that 2 and 3 are equivalent, let us observe that the inequality (

Being

Similarly, let

Observe that _{0},
_{0} and
^{n}_{0}, for any _{0} ⊂ _{0}. In fact, if we suppose that _{0} ⊄ _{0}, then there exists _{0} such that _{0}, _{0} is a linear subspace of ^{n}_{0}, which implies, for any

Since _{0}, _{0} ⊂ _{0}. That implies
_{0} ⊂ _{0.} Observe that, by definition,
^{n}_{1}, . . . _{n}_{1}, . . . _{n}^{n}

A simple but meaningful application of this procedure can be seen in Section 2, and the results are shown in

For the sake of simplicity all the results above have been proved for a system at rest, in the case that only the first-order gradients of the unknown quantities enter the state space. It is worth noting that the proofs may be given in more general situations, _{α}

Although Coleman-Noll and Liu procedures are quite different, it can be proven that for some class of materials they lead to the same results [

An interesting situation arises when the state pace is spanned by _{α}

The internal variables of state constitute an important tool in dealing with nonequilibrium processes involving complex thermodynamical systems. This is the case in the inelastic behavior of solids or in the relaxation effects in thermo-viscous fluids [

where the _{α}

Both the _{k}a_{α}_{t}a_{α}

where ^{ω}_{t}a_{α}_{k}a_{α}

In the inequality above, the time and space derivatives of _{α}

with

As first observed by Truesdell [

which is always satisfied if

It is worth noting that the vanishing of

which represents the set of ordinary differential equations we are looking for. That way, the form of the kinetic equations is determined by the constitutive equation of the specific entropy [

Since

The necessary and sufficient conditions to satisfy (

Thus, we conclude that the Onsager procedure is capable to provide less information with respect to Coleman-Noll and Liu ones. However, it is the sole method which can lead to a set of differential equations as thermodynamic restrictions, since the Coleman-Noll procedure restricts the functions _{α}_{2}.

In continuum physics the systems of governing equations often my be also put in the first-order quasi-linear form

with the unknown N-column vector _{1}, _{2}, . . . _{N}^{T}_{0} and _{i}

A wave is a moving surface ∑ represented mathematically by the equation

which defines the wave front. The unit normal

Weak waves have a continuous velocity across the front but a jump of the acceleration. Meantime, the unknown fields _{α}^{+} and ^{−} will denote the limits of _{α}_{i}u_{α}_{i}^{+} – ^{−} is the jump of

The system (_{0} ≠ 0, and the problem (_{0} is positive definite and _{i}_{i}^{T}

A remarkable consequence of the symmetry is the well-posedness of the Cauchy problem under very general conditions [^{p}^{,2} with ^{p}^{,2} in the neighborhood of the initial manifold [

A particular case of the quasi-linear first order system (

compatible with a supplementary balance law

with _{0} = −_{k}_{k}_{k}

[

_{0}

The compatibility of (

As the system (

The condition (

Supposing that the map (

where we have put

From (

and we conclude that the entropy principle implies that the vectors _{β}

Moreover, the production term

or, equivalently

where the matrices _{α}

Obviously they are all symmetric. Now we choose the components of the field _{0}, which we may do without essential loss of generality, and we assume that _{0} is a strictly convex function of

and therefore the Jacobian matrix

is positive definite and the map from _{0} and therefore

Thus, by (_{0} is positive definite. We have in this case that all matrices _{β}_{0} is also positive definite. Consequently our system (

New mathematical problems arise when dealing with higher-grade materials, namely those materials with higher-order gradients in the state space. In such a case, since the number of balance laws is smaller than that of the state variables, we have some time derivatives that appear into the entropy inequality but not in the system of balance laws. As a consequence, in the inequality (

where ^{A}^{B}^{A}^{B}

Since the components of the vector ^{B}

where

However, it is possible to regard this problem from a purely mathematical point of view, by considering a wider set of governing equations, some of them not in the balance form, by taking the gradients of the basic balance laws up to the order of the gradients entering the state space. That way, the number of differential constraints to be taken into account is always equal to the number of thermodynamic variables. Moreover, since both the Coleman-Noll and Liu procedures reduce the exploitation to an algebraic problem in an arbitrary point of the space-time (see Theorems 1–4 above), and since in a fixed point a function and its derivatives are independent quantities, the additional constraints are independent of the original ones. Thus, it is worth investigating:

what is the form of the entropy inequality once the gradients of the balance laws have been taken into account;

what is the new set of thermodynamic restrictions implied by this new, generalized, entropy inequality.

From now on, we pursue our analysis by considering the Liu procedure only. Similar conclusions may be achieved for the Coleman-Noll one [

In the classical Liu procedure, the obtained entropy inequality (Liu inequality) is linear with respect to those time derivatives which cannot be expressed through the balance laws as functions of the elements of the state space. Moreover, it is linear also with respect to the spatial gradients which are one order higher with respect to the order of the gradients entering the state space. In the extended procedure, instead, gradients which are two orders higher with respect to the order of the gradients in the state space, enter the entropy inequality [

Let us introduce in (_{0}_{0}) the vector of the highest derivatives

which, in the present case, is an element of ^{m}

and is an element of ^{n}

[_{0}_{0})

[_{0}_{0})

^{m}^{p}

[

[

^{m}, the vectors^{n}^{m}^{n}

The result above does not prevent the constitutive equations to depend on the whole set of the thermodynamic variables. However, particular solutions of

That way, the thermodynamic compatibility of higher-order nonlocal theories can be achieved without any change of first or second law of thermodynamics.

Let us consider now an initial value problem with non-regular data. In particular, let us focus our attention on initial conditions suffering jump discontinuities across a surface, which could lead to non-regular solutions. They can be written as [

where Ω_{0} is the discontinuity surface at a fixed instant _{0},
_{0} with positive normal and

Finally, let us consider the following projections of the state space

onto

On the other hand, the balance equations and the entropy inequality in an arbitrary point of
_{0}, read [

where ^{+} and ^{−} denote the process directions in an arbitrary point of
_{0}, while

[_{0}

The meaning of Axiom 2 is straightforward. Since Axiom 1 expresses the well-known experimental evidence that reversible thermodynamic transformations in

The main consequence of the axiom above, is the following.

[

We have examined different approaches to non-equilibrium thermodynamics starting from different applications of the entropy principle in different thermodynamic theories. We have pointed out that, in spite of the apparent differences, the theories present several similarities.

We reviewed the methods of exploitation of the entropy inequality and showed that they are robust from the mathematical point of view, unless in CIT, in which some methodologies are rather heuristic and not based on rigorous mathematical proofs (see [

In RT (Rational Thermodynamics) the entropy principle is characterized by the following properties: The entropy inequality in postulated in the form (

In CIT (Classical Irreversible Thermodynamics), it is characterized by the following properties: Local equilibrium, usually given in the form of a classical Gibbs relation or, equivalently, by a local entropy function. Heuristic state spaces. Entropy current density as in RT. Solution of the entropy inequality in the form of thermodynamic fluxes and forces. Balance form of evolution equations.

In TIP (Thermodynamics of Irreversible Processes):

Gibbs relation is not necessary, but possible. The state spaces are extended including weak nonlocality. Entropy current density is calculated. A solution of the entropy inequality is obtained in the form of thermodynamic fluxes and forces. Balance form of evolution equations is not required.

The entropy principle in EIT (Extended Irreversible Thermodynamics) is characterized by: Generalized Gibbs relation, with the dissipative fluxes as new variables. Generalized entropy current. Solution of the entropy inequality either in the form of thermodynamic fluxes and forces or by Coleman-Noll or Liu procedures. Balance form of evolution equations. Compatibility with the structure of kinetic theory, with transport coefficients not necessarily calculated.

The entropy principle in RET (Rational Extended Thermodynamics) has the following distinguishing properties: Rigorous compatibility with moment series expansion of kinetic theory, e.g., transport coefficients are calculated. Gibbs relation as consequence of the general mathematical structure. Balance form of evolution equations, and generalized entropy current density. Entropy inequality (beyond H-theorem) used in the closure and also in the construction of a symmetric system of governing equations.

In general, we can say that with more compatibility with kinetic theory the continuum theories loose from their universality, because they can be applied for a restricted class of materials, but gain in predictability, because the mathematical structure becomes more rigorous and material quantities can be calculated.

All the presented theories are able to reproduce important physical phenomena. Moreover, also local and nonlocal theories seem to be not so far as it could appear at a first sight. For instance, although the Guyer-Krumhansl

As we pointed out in Section 4, in RET the requirements of causality and predictability are achieved by applying the entropy principle to find the main field which symmetrizes the system of field equations.

In weakly nonlocal theories (RT, EIT, TIP) instead, finite speeds of propagation can be obtained in generalized sense, by observing that a correct estimation of the speed of propagation of the perturbations requires to compare the order of magnitude of the solution of the system of balance equations with that of the error affecting the experimental data which the theory aims to fit [

In his article Fichera also commented an observation made by Maxwell [

This idea suggests a second line of defense of Fourier theory [

Of course, the arguments above loose their validity in a relativistic context, where the experimental evidence—always confirmed until now—imposes the speed of light as upper bound for the velocity of propagation of thermomechanical disturbances. Thus, in this case the system of balance laws must be necessarily hyperbolic. A generalized constitutive principle, which is compatible with parabolic theories in the classical case but also forces the theories to be hyperbolic in the relativistic one, has been proposed in [

In RT memory effects, as those arising in heat conduction with finite speed, are often modeled through integral constitutive equations [

Some final comments are in order.

First we observe that, according to the results of kinetic theory, extended thermodynamics is not a single theory, but a family of theories, each of which is extracted by an hierarchy of infinite balance equations. Nonetheless, each theory has a finite number of equations, which is determined according to a general criterion established in [

When the boundary value of higher order fluxes cannot be determined by direct measurements, one can use the so called fluctuation principle, according to which the systems on the boundary adjust naturally to the mean values of the fluctuating boundary values [

There are subtle differences between EIT and RET, the most important one being the choice of the state space. A further difference is the starting point of the theories, since EIT assumes a modified Gibbs equation and does not require balance laws a priori, while RET starts from the balance structure dictated by kinetic theory. The first approach is more general while the second approach is well-suited for a restricted class of materials (e.g., rarefied monoatomic gases, polyatomic and dense gases). However, it provides more accurate models of these materials, determining all the material properties from the thermal and caloric equations of state. Also, it leads to a satisfactory qualitative analysis of shock wave propagation.

Some authors consider CIT with internal variables the real counterpart of EIT and RET [

A comparative analysis of the main properties of the different thermodynamic theories is resumed in the tables below. In

It is worth observing that what we presented above is only a first tentative scheme of comparison. The detailed comparison and classification of the different theories with respect to the properties mentioned above could be the subject of a future separate review.

At this point, it seems natural to investigate if there exists a more general approach to non-equilibrium thermodynamics which encompasses all the previous ones. This important problem, which still remains to be solved, is left to the young generations of thermodynamicists.

Vito Antonio Cimmelli acknowledges the financial support from the University of Basilicata, research project

David Jou acknowledges the financial support from the

Tommaso Ruggeri acknowledges the financial support from the University of Bologna Farb Project 2012

The authors declare no conflicts of interest.

In this article Péter Ván illustrated the role of the entropy principle in classical irreversible thermodynamics; David Jou analyzed the application of the entropy principle in extended irreversible thermodynamics; Tommaso Ruggeri showed the consequences of the entropy principle in rational extended thermodynamics; Vito Antonio Cimmelli presented some classical results of rational thermodynamics and recent results regarding the mathematical analysis of the entropy inequality. All 4 authors were fully involved in: substantial conception and design of the paper; drafting the article and revising it critically for important intellectual content; final approval of the version to be published.

One day before the receipt of the proofs of this article, the authors have learned of the untimely death of Prof. Witold Kosiński, from the Polish-Japanese Institute of Information Technology in Warsaw and Institute of Mechanics and Applied Computer Science at Kazimierz-Wielki University in Bydgoszcz. For all of them Witold was a dear colleague and a renowned scientist. For V. A. Cimmelli he was also a memorable teacher and a sincere friend. The authors remember him with affection and they wish to dedicate this article to his memory.

Comparison of the different thermodynamic theories: Predictability, Causality, Agreement with kinetic theories.

Predictability | Causality | Agreement with kinetic theories | |
---|---|---|---|

CIT | Yes, to be proved case by case | Yes, in the generalized sense | Not considered |

EIT | Yes, to be proved case by case | Yes, in the generalized sense | Yes |

RET | Yes, in general | Yes, in general | Yes |

RT | Yes, to be proved case by case | Yes, in the generalized sense | Not considered |

Comparison of the different thermodynamic theories: Number of field equations, Constitutive equations, Boundary conditions.

Number of field equations | Constitutive equations | Boundary conditions | |
---|---|---|---|

CIT | Finite | Weakly nonlocal | Experiments |

EIT | Finite, from an infinite hierarchy | Weakly nonlocal | Experiments or fluctuation theory |

RET | Finite, from an infinite hierarchy | Local | Experiments or fluctuation theory |

RT | Finite | Weakly nonlocal | Experiments |