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Journal of Non-Equilibrium Thermodynamics

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Volume 43, Issue 1

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Gradient Dynamics and Entropy Production Maximization

Adam Janečka
  • Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic
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/ Michal Pavelka
  • Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic
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Published Online: 2017-09-19 | DOI: https://doi.org/10.1515/jnet-2017-0005

Abstract

We compare two methods for modeling dissipative processes, namely gradient dynamics and entropy production maximization. Both methods require similar physical inputs–-how energy (or entropy) is stored and how it is dissipated. Gradient dynamics describes irreversible evolution by means of dissipation potential and entropy, it automatically satisfies Onsager reciprocal relations as well as their nonlinear generalization (Maxwell–Onsager relations), and it has statistical interpretation. Entropy production maximization is based on knowledge of free energy (or another thermodynamic potential) and entropy production. It also leads to the linear Onsager reciprocal relations and it has proven successful in thermodynamics of complex materials. Both methods are thermodynamically sound as they ensure approach to equilibrium, and we compare them and discuss their advantages and shortcomings. In particular, conditions under which the two approaches coincide and are capable of providing the same constitutive relations are identified. Besides, a commonly used but not often mentioned step in the entropy production maximization is pinpointed and the condition of incompressibility is incorporated into gradient dynamics.

Keywords: non-equilibrium thermodynamics; constitutive relations; entropyproduction maximization; dissipation potential; non-Newtonian fluids

1 Introduction

Irreversible phenomena are ubiquitous and it is a goal of non-equilibrium thermodynamics to describe evolution equations governing such processes. There are many frameworks of the non-equilibrium thermodynamics leading to countless different ways of prescribing the irreversible terms in evolution equations, see Lebon et al. [1]. We shall discuss two of them, namely the gradient dynamics (GD), employing dissipation potentials, and the entropy production maximization (EPM).

In the case of gradient dynamics the irreversible part of evolution equations is given by thermodynamic fluxes J which are gradients of a dissipation potential Ξ with respect to thermodynamic forces X1 J=ΞX.(1)

Such framework is advocated, for example, by Edelen [3]. The non-potential part can be regarded as the reversible part as in Eringen [4], p. 28, or by recent findings based on the theory of large deviations by Mielke et al. [5, 6]. A multiscale statistical optimization principle also leads to gradient dynamics, Turkington [7]. Further, gradient dynamics plays a crucial role in the GENERIC framework, Grmela and Öttinger [2, 8], where it guarantees approach to the equilibrium driven by the irreversible part of the evolution equations.2

There are several concepts that can be regarded as a maximization of entropy production. We shall use the approach developed by Rajagopal and Srinivasa [10, 11, 12], which is a generalization of the method proposed by Ziegler [13, 14, 15].3 Maximizing a prescribed entropy production ξ subject to the constraint that the entropy production is a sum of products of thermodynamic forces and fluxes4

Xξ(X)+λξ(X)JX=0,(2)

λ being a Lagrange multiplier, leads to generally implicit constitutive relations between thermodynamic fluxes and forces.5 EPM is used to describe many nonlinear irreversible phenomena in particular in non-Newtonian fluids.

A thermodynamic motivation for EPM can be found when studying the CR-thermodynamics introduced in Ref. [16], see the text below eq. (40) therein. The CR-thermodynamics is a method of natural extension and subsequent reduction recovering a constitutive relation (CR) that was present in the equations before the extension, and it is demonstrated in Section 2.5. When performing the reduction, Legendre transformation of a dissipation potential is carried out, which can be interpreted as maximization of the dissipation potential subject to constraints. When the dissipation potential is proportional to the corresponding entropy production (e.g. when it is a homogeneous function of forces or fluxes –-as discussed later in this paper), the Legendre transformation can be interpreted as maximization of entropy production subject to constraints given by the force–flux relations. In other words, for homogeneous dissipation potentials (or homogeneous entropy productions), the fast reducing evolution studied within the CR-thermodynamics can be seen as gradual EPM.

Steepest Entropy Ascent (SEA), a framework closely related to EPM, was developed by G. P. Beretta, see [17] and references therein. In this case, the entropy production (as a product of forces and fluxes) is also maximized subject to constraints. One of the constraints is a formula for the entropy production, which is interpreted as a metric on the vector space of fluxes (or forces). Therefore, SEA essentially means XJXλ˜ξ(X)=0,(3)

which is equivalent to eq. (2). SEA and EPM are thus essentially equivalent, as shown in Beretta [17], and relations between gradient dynamics (GD) and EPM presented in this paper could be interpreted as relations between GD and SEA.

The novel insight brought in this paper is: (i) Comparison of EPM and GD and identification of formulas for entropy production for which both approaches coincide. (ii) Explicit identification of a step in the procedure of EPM that is usually tacitly performed, but for which substantial physical insight is necessary. (iii) Formulation of the incompressibility condition within GENERIC.

2 Gradient dynamics

We refer to the GD as to dynamics generated by a dissipation potential Ξ (sufficiently regular function, zero at the origin and convex near the origin). Let x denote the set of state variables.6 Thermodynamic forces X are then related to thermodynamic fluxes J through eq. (1). A relation between thermodynamic forces and fluxes is called a constitutive relation.

2.1 Legendre transformation

Relation (1) can be seen as a consequence of the Legendre transformation XΞ(X)+JX=0J=ΞXX˜(J),(4)

giving the dependence X=X˜(J). The dual dissipation potential is Ξ(J)=Ξ(X˜(J))+JX˜(J).(5)

The backward Legendre transformation from Ξ(J) to Ξ(X) proceeds as

2.1 JΞ(J)+XJ=0X=ΞJJ˜(X),(6)

and Ξ(X)=Ξ(J˜(X))+XJ˜(X).(7)

Legendre transformation is the natural way for passing between Ξ(X) and Ξ(J) with relation (1) because no information is lost during the passage, see e.g. Callen’s textbook [18], where the pertinence of the Legendre transformation in equilibrium thermodynamics is explained. Note that the forward and backward Legendre transformations need the dissipation potential to be non-degenerate, see Esen and Gümral [19, 20] for the degenerate case. GD can be also motivated from the point of view of differential geometry, where the Legendre transformation is the natural transformation between functions on tangent and cotangent bundles, see [21, 22].

Equation (4) can be interpreted as a constitutive relation prescribing fluxes as functions of forces. Its Legendre transform (5) is then interpreted as an inverse constitutive relation, where forces are given in terms of fluxes. GD thus naturally leads to explicit constitutive relations and their inversions. Both types are straightforwardly converted to each other by the Legendre transform, see e.g. Janečka and Pavelka [23].

Apart from constitutive relations where fluxes are functions of forces and vice versa, fully implicit constitutive relations h(X,J)=0, h being a functional, were advocated by Rajagopal [24]. Such a fully implicit constitutive relation can be obtained by EPM when entropy production is expressed as a function of both forces and fluxes. Although GD could also yield fully implicit constitutive relation by letting the dissipation potential depend on both forces and fluxes as well, such a dependence would ruin the structure of Legendre transformations behind GD. The fully implicit constitutive relations can be thus obtained by EPM naturally while GD cannot give them without sacrificing its geometrical structure.

Legendre transformations provide GD with a clear geometrical picture. However, imposing additional constraints is not as straightforward as within EPM. We suggest a way how to formulate the incompressibility condition (a prototype of constrains) into GD in Section 2.5.

2.2 Maxwell–Onsager reciprocal relations

Onsager reciprocal relations are generalized into far-from-equilibrium regime7 by using a non-quadratic dissipation potential as shown for example in [2, 8, 25, 26]. Indeed, taking the derivative of both sides of eq. (1) with respect to the force X2, we obtain J1X2X1=J2X1X2,(8)

and change of variables then leads to equivalent relations X1J2J1=X2J1J2, J1J2X1=X2X1J2, J2J1X2=X1X2J1,

called the Maxwell–Onsager reciprocal relations (MORR), see Grmela et al. [25].

MORR can be also seen as conditions necessary for existence of a dissipation potential generating the dynamics. If they turn out not to be fulfilled, no dissipation potential can be constructed for the given set of thermodynamic forces and fluxes (and state variables). By means of MORR Onsager reciprocal relations are generalized to the fully nonlinear regime.

2.3 Identification of forces and fluxes

Dissipation potential generates irreversible evolution of a set of state variables x (x˙)irr=Ξxx=Sx,(12)

where the derivative is interpreted as a functional derivative. Conjugate state variables x are identified with derivatives of entropy S with respect to the state variables, see Grmela and Öttinger [2, 8, 27]. Another possibility, which is often preferred in the nonlinear regime, is to identify the conjugate variables with derivatives of the thermodynamic potential, see Grmela and Öttinger [2, 8] for more details. The thermodynamic potential then plays a role of Lyapunov function driving the evolution into thermodynamic equilibrium. The relation between x and x can be also interpreted as a Legendre transformation.

The dissipation potential typically depends only on gradients of the conjugate variables, and thermodynamic forces are then just a shorthand for writing that dependence X=Γ(x),

where Γ is an operator, usually Γ=. We than have Ξx=ΞX=J.(14)

For example in classical hydrodynamics, momentum density u is among the state variables (together with density and entropy density), and conjugate variables can be identified with derivatives of entropy S=drsρ,eu22ρ,

where s is the local equilibrium entropy density and e is total energy density. In the isothermal case, conjugate momentum is thus u=Su=1Tu/ρ, which is proportional to the velocity. The corresponding thermodynamic force is the gradient of conjugate momentum, i.e. the velocity gradient. That is why the strain rate tensor is to be interpreted as a thermodynamic force while the irreversible Cauchy stress as the corresponding flux (of momentum). It is thus possible to distinguish between thermodynamic forces and fluxes (as in the statistical approach to non-equilibrium thermodynamics, de Groot and Mazur [28]) and one should not interchange them. This point of view is shared with the EPM approach.

2.4 Entropy production

Within GD, entropy production is given as a product of fluxes and forces ξ=JX=XΞX=JΞJ.(16)

These relations can be seen from eqs. (12) and (14), since the total entropy of a system evolves as S˙=drSx(x˙)irr=drSxJ=drSxJ=drXJ.

Entropy production ξ is thus equal to the product of thermodynamic forces and fluxes as usually in the non-equilibrium thermodynamics, see de Groot and Mazur [28].

2.5 Incompressibility constraint

Let us now propose a way how to incorporate the incompressibility condition into classical isothermal hydrodynamics formulated by means of the GENERIC framework. In particular, we rely on recent results by Grmela [16] and Janečka and Pavelka [23], where the so-called CR-thermodynamics was developed and applied (CR stands for constitutive relation). CR-thermodynamics is a form of thermodynamic extension compatible with the GENERIC framework (see Section 1 for references), where a thermodynamic flux (or rather its conjugate) is promoted to a new state variable, whose evolution approaches a constitutive relation used before introducing the new state variable.

Standard compressible isothermal Navier–Stokes equations, Lebon et al. [1], read ρt=divρu,(18) ut=p(ρ)+divuudivPδP0,(19)

where the deviatoric and isotropic parts of the irreversible pressure stress tensor (negative of the Cauchy stress tensor) are specified as Pδ=defΞδDδandP0=defΞ0div(u),(20)

where Dδ=defD13div(u)I is the negative of the deviatoric part of the symmetric part of the velocity gradient (shear rate) D=12u+u and the conjugate variables are identified with negative derivatives of the Helmholtz-free energy F, i.e. u=Fu=u/ρ=v, which comes from the kinetic part of free energy.

The dissipation potential Ξ is chosen as Ξ(D)=drμDδ:DδΞδ(Dδ)+drνdiv(u)2Ξ0(div(u)),(21)

which recovers the standard compressible isothermal Navier–Stokes equations.

Let us now promote conjugate of the isotropic irreversible pressure P0 to an independent state variable. This conjugate isotropic stress is related to P0 by Legendre transformation with respect to an extended free energy F(ex) dependent on P0 explicitly, i.e. P0=F(ex)P0.

The extended free energy depends on the particular physical system under consideration. It can be sought by means of kinetic theory of dense gases, see Hirschfelder [29], we prefer leaving it unspecified (albeit convex).

Since the Legendre transformation between P0 and P0 is idempotent, i.e. second Legendre transformation is identity, we can write that P0=P0. Therefore, the last term in eq. (19) can be interpreted as coupling with P0. The reciprocal antisymmetric coupling satisfying Onsager–Casimir reciprocal relations affects the evolution equation of P0, which thus becomes P0t=div(u)+Ξ0P0.(23)

The antisymmetry of the coupling can be seen from8 dr()ab=dradiv(b),a,b.

Equation (23) also contains an irreversible contribution from dissipation potential conjugate to the isotropic part of Ξ0, which is given by Ξ0(P0)=Ξ0div(u)(P0)+drP0div(u)(P0).

From eq. (20) and eq. (21), one can infer the dependence div(u)(P0)=12νP0,

and the conjugate isotropic dissipation potential becomes Ξ0(P0)=dr14νP02.

Equation (28) thus finally becomes P0t=div(u)+12νP0(28)

with P0=F(ex)P0, F(ex) being the Legendre transformation of F(ex).

How does the extra evolution equation (28) affect the evolution equation of momentum density (19)? First, setting the left-hand side of eq. (28) to zero, which means that the extra variable has relaxed to a stationary state, the right- hand side of the equation reduces to the constitutive relation (20). The extra evolution equation thus recovers the constitutive relation, and by reducing the extra variable P0 the system approaches the Navier–Stokes equations (18).

Let us now formulate the incompressibility condition. The material can be regarded as incompressible if any compression would trigger extremely high dissipation. In other words, the volume viscosity coefficient ν is very high or the value of P0 is very low. Therefore, eq. (28) can be approximated in the stationary state by 0div(u)+'0',

which leaves the field P0 and thus also P0 undetermined. By combining with Navier–Stokes equations (18) we obtain the incompressible Navier–Stokes equations, ρt=ρv,(30) ut=p˜divρvvdivPδ, 0=div(v),

where the deviatoric pressure tensor Pδ is given by eq. (20) and p˜=defp+P0 is new undetermined pressure.

In summary, extending the set of state variables by the conjugate isotropic part of the pressure tensor leads to an extra evolution equation for P0. When this evolution equation approaches a stationary state, the standard constitutive relation (20) is recovered. However, for high volume viscosities or low isotropic pressure tensor fields, the extra evolution equation can be approximated by the incompressibility condition div(v)=0 and the extra variable P0 becomes undetermined. That is how the standard incompressible Navier–Stokes equations can be obtained. Moreover, by solving eq. (28) with the compressible Navier–Stokes equation simultaneously, one could reveal how the incompressible limit is approached.

3 Entropy production maximization

Let us now formulate the procedure of EPM. Here, we shall consider the entropy production to be a function of thermodynamic forces ξ=ξ(X), hence the maximization is performed with respect to the forces X while keeping the fluxes J constant, but it would be very well possible to proceed in an opposite way. The entropy production ξ(X) should be maximized while keeping the constraint ξ=JX.(33)

The maximization is carried out by means of the Lagrange multipliers Xξ(X)+λ(ξ(X)JX)=0,(34) λξ(X)+λ(ξ(X)JX)=0,

where λ is the Lagrange multiplier. Solving these two equations, it is generally possible to obtain an implicit relationship between J and X. Note that the second equation is equivalent to constraint (33).

After multiplying by X, eq. (34) implies 1+λλXξX=ξ,

which gives λ as a function of X. eq. (34) then leads to9 J=ξXξXξX,(37)

which is the general result (constitutive relation) obtained by the method of EPM. One of the main advantages of this procedure is the ease of enforcing further constrains as, e.g. incompressibility of the material. For each constraint there would be one additional equation in (34). Enforcing of such constraints is rather involved in GD, see Section 2.5.

In the particular case of k-homogeneous entropy production, we have XξX=kξ,(38)

and eq. (37) becomes J=1kξX,(39)

see Ziegler [13]. For example the quadratic entropy production ξ=ijLijXiXj(40)

is 2-homogeneous, and eq. (39) yields the standard linear force–flux relations Ji=LijXj,(41)

see de Groot and Mazur [28]. Rajagopal and Srinivasa [11] have shown that in this case, Onsager reciprocity relations are fulfilled, i.e. Lij=Lji, as otherwise eq. (41) would not be compatible with eqs. (37) and (38).

3.1 Non-uniqueness of the choice of fluxes and forces

Let us assume system with two fluxes J1 and J2 which are known functions of the forces X1 and X2. The entropy production is then given by ξ(X1,X2)=J1(X1,X2)X1+J2(X1,X2)X2.(42)

It could be tempting to identify the thermodynamic fluxes with J1 and J2, but the method of maximum entropy production leads to generally different fluxes J1 and J2, given by eq. (37). It can be shown that the relation between the known fluxes Ji and the fluxes resulting from the maximum entropy production principle Ji is J1=J1+ΔX1, J2=J2ΔX2,

where the discrepancy Δ is given by Δ=X1X2XξXJ1X1J1X2J2X2J2X1+J1X2J2X2J2X1J1X1.

The fluxes are unique if the discrepancy is equal to zero, i.e. J1(X1,X2)ξX2=J2(X1,X2)ξX1,(45)

see for example Martyushev and Seleznev [30]. For quadratic entropy production, condition (45) holds if Onsager reciprocal relations are satisfied, a result consistent with Rajagopal and Srinivasa [11].

In summary, it is advisable to determine the thermodynamic fluxes by going through the whole procedure of EPM, which leads to formula (37). On the other hand, if the fluxes are identified simply from writing down entropy production in the form ξ=J(X)X, the result can be misleading, since there are usually more ways of casting entropy production in that form.

3.2 General framework of the EPM procedure

To find the constitutive relations specifying the system by the means of the EPM, we need to determine how the system stores energy and how it produces entropy–-to this end, we need to stipulate two scalar functions. The general framework goes as follows, for more details see Málek and Pruša [31].

  • [STEP 1:] Determine the state variables x and specify the storage mechanism of the system by virtue of the fundamental thermodynamic relation in terms of one of the thermodynamic potentials (internal energy e, Helmholtz-free energy ψ, Gibbs potential G, enthalpy h, ) as a function of the state variables.

  • [STEP 2:] From the balance of internal energy e and the fundamental thermodynamic relation derive the local form of the balance of entropy–-the Clausius–Duhem inequality

    ρdηdt+divjη=JX>0,(46) where ρ is the density, the thermodynamic temperature is defined as θ=defeη, jη is the entropic flux10 and ξ=defJX>0 is the entropy production, where the dot product can be understood as a summation of different mechanisms of the entropy production. The non-negativity of the entropy production function ξ is a consequence of the second law of thermodynamics.

  • [STEP 3:] Specify the constitutive relation for the entropy production function in terms of the thermodynamic fluxes J or the thermodynamic forces X as

    ξ=ξ(J),or ξ=ξ(X).

  • [STEP 4:] Maximize the entropy production function ξ with respect to the thermodynamic fluxes J or the thermodynamic forces X.11 As a constrain of this maximization procedure, the definition of the entropy production ξ=JX arising form (46) must hold.

    • [STEP 4a:] In case that there is no coupling among individual terms in the entropy production function, i.e. it can be written as

      ξ(J)=αξα(Jα),or ξ(X)=αξα(Xα),(48)

      we can additionally require that the sought forces or fluxes would depend solely on their corresponding counterparts. Then, we can maximize the particular terms ξα separately using the Lagrange multipliers λα as

      ddJαξα(Jα)+λαξα(Jα)JαXα=0,α,

      ddλαξα(Jα)+λαξα(Jα)JαXα=0,α,

      or

      ddXαξα(Xα)+λαξα(Xα)JαXα=0,α,

      ddλαξα(Xα)+λαξα(Xα)JαXα=0,α.

      From the same procedure as described in Section 3, it follows for the particular fluxes or forces

      Jα=ξαdξαdXαXαdξαdXα,or Xα=ξαdξαdJαJαdξαdJα,

      where there is no summation over α and we truly have Jα=Jα(Xα) or Xα=Xα(Jα).

    • [STEP 4b:] When the entropy production cannot be written as (48) or without the additional requirement made inSTEP 4a, we need to maximize the entropy production as a whole, thus arriving at the relation (37) or its counterpart for the thermodynamic forces. In this case, the resulting fluxes might depend on all the other forces or the other way around.

Even though the requirement made inSTEP 4a is not necessary, it is usually tacitly considered, since it leads to the commonly used constitutive relations. On the other hand, maximization of the entropy production as a whole leads to rather complicated but possibly richer expressions, see the example in Section 4.1.2. As an example where the maximization was conducted truly as a whole, we refer to Hron et al. [33].

3.3 Summary of EPM

The method of EPM is summarized in Figure 1.

Possible routes of EPM. In many cases, the general entropy production ξ=ξ(X1,…,Xn)$\xi = \xi(X_1,\dots,X_n)$ can be split into parts each of which depends only on one force ξ=∑αξα(Xα)$\xi = \sum_\alpha \xi_\alpha(X_\alpha)$. Then there are two possible ways how to proceed. If coupling between different entropy production mechanisms is forbidden, each flux depends only on the corresponding thermodynamic force. One can then infer the constitutive relations by maximizing each part of the entropy production separately. That corresponds to the Step 4a in Section 3.2. The forces can be often separated from each other in a physically plausible way (e.g. separating heat from mechanical stress), although one should be aware of subjectivity of such a choice, see Section 3.1. If coupling between forces is allowed, i.e. the corresponding fluxes depend on all the forces in general, the entropy production should be maximized as a whole, otherwise the result would depend on the ambiguous choice of thermodynamic forces, see Section 3.1. On the other hand, if the entropy production cannot be split as in the preceding case, ξ≠∑αξα(Xα)$\xi \neq \sum_\alpha \xi_\alpha(X_\alpha)$, one needs to maximize it directly as a whole. That corresponds to Step 4b in Section 3.2 of the EPM procedure.
Figure 1:

Possible routes of EPM. In many cases, the general entropy production ξ=ξ(X1,,Xn) can be split into parts each of which depends only on one force ξ=αξα(Xα). Then there are two possible ways how to proceed. If coupling between different entropy production mechanisms is forbidden, each flux depends only on the corresponding thermodynamic force. One can then infer the constitutive relations by maximizing each part of the entropy production separately. That corresponds to the Step 4a in Section 3.2. The forces can be often separated from each other in a physically plausible way (e.g. separating heat from mechanical stress), although one should be aware of subjectivity of such a choice, see Section 3.1. If coupling between forces is allowed, i.e. the corresponding fluxes depend on all the forces in general, the entropy production should be maximized as a whole, otherwise the result would depend on the ambiguous choice of thermodynamic forces, see Section 3.1. On the other hand, if the entropy production cannot be split as in the preceding case, ξαξα(Xα), one needs to maximize it directly as a whole. That corresponds to Step 4b in Section 3.2 of the EPM procedure.

4 GD vs. EPM

Assume now that the dissipative evolution is described by GD as in Section 2, where fluxes can be expressed in terms of forces by means of eq. (1), i.e. there is an explicit dependence. Entropy production is then given by eq. (16), and eq. (37) becomes J=XΞXXΞX+ΞXXX(ΞX+ΞXXX).

If the dissipation potential is k-homogeneous, see Section A, the last equation becomes eq. (1), and GD can be regarded as EPM.

In the case of only one thermodynamic force X and on the reasonable assumption that the dissipation potential depends on this force through its norm, Ξ=ΞX, the GD also coincides with EPM. Indeed, from (16), we can compute for the quantities figuring in (37) ξ=XdΞdX=XΞXX=ΞX,dξdX=Ξ′′X+ΞXX,dξdXX=Ξ′′X2+ΞX,

where the prime () denotes the derivation with respect to the argument–-the norm X, and the dot () should be understood as a corresponding vector/tensor inner product. Then, we can substitute into (37) J=ΞXΞ′′X2+ΞXΞ′′+ΞXX=ΞXX=dΞdX,

and we have recovered relation (1).

The previous observations can be summarized in the following Propositions.

Proposition 4.1

Let the entropy production ξ be either k-homogeneous function in all the forces or let it depend on only one force. The EPM is then equivalent to the GD.

Proposition 4.2

Let the entropy production ξ be without coupling. The GD is then equivalent to the EPM done by parts (Step 4a).

4.1 Non-homogeneous entropy production

The purpose of this section is to demonstrate cases when EPM and GD are not equivalent.

4.1.1 Chemical kinetics

Consider for example chemical reactions with a dissipation potential Ξ=k1coshX121+k2coshX221,

where k1,k2 are constants. Dissipation potential of this form leads to reaction rates (fluxes) Ji=ki2sinhXi2,i=1,2.(57)

This description of chemical kinetics was shown to be compatible with the Guldberg–Waage law of mass action by Grmela [34]. From eq. (16), we see that ξ=k12sinhX12X1+k22sinhX22X2,(58)

and using (37) we obtain the reaction rates Ji=kksinhXk2Xk2klXlXlcoshXl2+2sinhXl2kiXicoshXi2+2sinhXi2,i=1,2,(59)

where there is no summation over the index i. This does not seem to be the right result due to its complexity and because eq. (57) is compatible with the well established law of mass action. But note that the entropy production done by parts (Step 4a) gives the same result as the GD as in this case the maximization is done always over only one force. This result corresponds to the case in Proposition 4.2.

In summary, EPM with the entropy production (58) leads to thermodynamic fluxes (reaction rates) (59), which are different from fluxes (57). The latter choice of fluxes was however shown to be compatible with the widely accepted law of mass action and Butler–Volmer equation, see Grmela [34] and Pavelka et al. [35], and we thus prefer them to the former choice of fluxes in the case of nonlinear chemical kinetics.

Nevertheless, EPM has been also successfully used in modeling of chemically reacting systems at least when using quadratic entropy production. For example Kannan and Rajagopal [36] employed it to derive a model of vulcanization of rubber.

4.1.2 Incompressible heat-conducting non-Newtonian fluid

As a simple but physically relevant example of non-homogeneous non-quadratic entropy production, we can consider the following formula ξ=2μ1+αDδ2r1Dδ2+κθ2,(60)

where μ, α and κ are positive constants, r is a constant, Dδ=defD13TrDI is the deviatoric part of the symmetric part of the velocity gradient and θ is the temperature. The fluxes associated with Dδ and the temperature gradient θ are the deviatoric part of the Cauchy stress Tδ and the negative heat flux q respectively, see Rajagopal and Srinivasa [11] for details. This entropy production describes heat-conducting non-Newtonian fluids.

Using the EPM principle (37), we arrive at Tδ=2μ1+αDδ2r1Dδ2+κθ24μ1+αDδ2r21+αrDδ2Dδ2+2κθ24μ1+αDδ2r21+αrDδ2Dδ,(61) q=2μ1+αDδ2r1Dδ2+κθ24μ1+αDδ2r21+αrDδ2Dδ2+2κθ22κθ,(62)

which is not particularly neat (and it would be even worse in a hypothetical situation when the viscosity μ was dependent on the temperature gradient θ).

Separate maximization of eq. (60) or using the dissipation potential12 Ξ=μαr1+αDδ2r+12κθ2,

leads to a much more luminous relations Tδ=2μ1+αDδ2r1Dδ,(64) q=κθ,(65)

where the first equation is the Carreau model, see [37], and the second equation is the well-known Fourier law of thermal conductivity.

We see that for r=1, i.e. when both terms in (60) are quadratic, constitutive relations (61) and (64) are tantamount, see Proposition 4.1, and we recover the standard model for incompressible heat-conducting Newtonian fluid.

Since there are materials with the thermal conductivity κ depending on the shear rate, see for example Lee [38], we hoped that the constitutive relation (62) given by the EPM could capture this dependence. Unfortunately, this was not the case neither for the entropy production (60) nor for entropy productions motivated by the Ostwald–de Waele power-law model [39, 40], the Sisko model [41] or the Cross model [42]. Therefore, maximization of entropy production as a whole, where the rather subjective step splitting entropy production into parts is skipped, does not seem to pay off in this concrete example. Indeed, the coupling resulting from the maximization does not provide explanation for the experimental behavior.

One should bear in mind, however, that the statement that EPM as a whole does not describe the considered particular experimental behavior, is only valid for a constant κ. Assuming κ to be shear rate dependent κ=κ(D), one can get any dependence using the EPM.

There are two possible ways to introduce the observed coupling between stress and heat flux within GD. Firstly, one can let the heat conductivity κ depend on gradient of the velocity (interpreted as gradient of momentum divided by density). Indeed, when the heat conductivity does not depend on conjugate momentum, but only on momentum and density, derivative of the dissipation potential with respect to conjugate momentum treats the conductivity as constant. Second, one could introduce explicit coupling between heat flux and stress tensor into the dissipation potential, i.e. the dissipation potential could contain terms dependent on both conjugate momentum and conjugate energy.

In summary, although there is no apparent coupling in entropy production (60), nonlinear coupling appears after maximization of the whole entropy production. Such a coupling would suggest for example rather complex dependence of effective thermal conductivity on shear rate. However, the magnitude of the coupling is not in agreement with experimental observations, so it is still a question whether such coupling is desirable or not.

4.1.3 Incompressible Navier–Stokes fluid

The classical incompressible Navier–Stokes fluid can be derived by means of EPM from the entropy production function ξ=2μD2,

where μ>0 is the viscosity of the fluid. The maximization procedure is executed under the standard constraint that the entropy production is a product of forces and fluxes ξ=T:D and under an additional constraint enforcing the incompressibility TrD=0.

Then, the maximization with respect to the symmetric part of the velocity gradient D of the Lagrange function L=ξ+λ1ξT:D+λ2TrD,

where λ1 and λ2 are the Lagrange multipliers leads to T=1+λ1λ1ξD+λ2λ1I.

Taking the scalar product with D and exploiting the incompressibility condition, we obtain 1+λ1λ1=ξξD:D=12,

since the entropy production function is 2-homogeneous. Thus we have T=pI+2μD,

where we have denoted p=defλ2λ1. Taking trace of the final constitutive relation, we see that p=13TrT is the mean normal stress.

This was just an example to show the simplicity of including additional constraints within EPM. Note that one could also take into account the incompressibility condition prior the maximization, i.e., rewrite the entropy production function as ξ=2μDδ2 and then maximize with respect to the deviatoric part of the symmetric part of the velocity gradient Dδ under only one constraint ξ=Tδ:Dδ, see Rajagopal and Srinivasa [12] for details.

Similarly as here, in GD one could take take derivative of the dissipation potential with respect to shear rate at constant divergence of velocity. The constantness of divergence of velocity would be enforced by a Lagrange multiplier. This way, however, we would lose the structure of Legendre transformations.

Maxwell–Onsager relations

When discussing compatibility of GD and EPM, a question arises whether the Maxwell–Onsager relations from Section 2.2, which are necessary for existence of a dissipation potential, are fulfilled when the evolution is obtained by EPM. Taking the derivative of the general equation for flux given by EPM, eq. (37), with respect to force X2 and requiring relation (8) to hold leads to the condition Xk2ξXkXjξXi2ξXkXiξXj=0,i,j.(71)

If this condition is fulfilled, there may be constructed a dissipation potential describing the evolution and MORR are fulfilled even in the far-from-equilibrium regime. The condition is fulfilled for example for khomogeneous entropy productions. If the condition is not fulfilled, no dissipation potential can be constructed. In particular, the condition is fulfilled for entropy production (40) when the matrix Lij is symmetric. That means that Onsager reciprocal relations are fulfilled by EPM near equilibrium, where the original Onsager’s derivation [43, 44] is formulated.

Condition (71) is not fulfilled for all entropy productions. For example entropy production ξ=X1sinhX1+X2sinhX22,

violates it, which is compatible with Rajagopal and Srinivasa [11] as it is not k-homogeneous.

Maxwell–Onsager reciprocal relations, which are fulfilled even far from equilibrium by GD, are fulfilled by EPM near equilibrium, but not necessarily far from equilibrium. Assuming that a constitutive relation is generated by EPM, MORR can be regarded as the compatibility condition necessary for constructing a dissipation potential leading to the same constitutive relation.

Conclusion

Relations between the GD and the EPM are summarized in Figure 2.

Relations between GD and EPM as a Venn diagram. The left top set contains models obtained by EPM carried out by parts, i.e. by Step 4a in Section 3.2. No coupling between the forces or fluxes is allowed in this case. Due to Proposition 4.2 all models coming from EPM by parts can be formulated as GD. In particular, if the entropy production formula is homogeneous of the same degree in all forces (or fluxes in the conjugate representation) without coupling or if there is only one force in the formula, EPM by parts is equivalent with EPM when entropy production is maximized as a whole, see Proposition 4.1, which corresponds to Step 4b in Section 3.2. EPM as a whole is equivalent to GD if the entropy production formula is homogeneous (including coupling) or if there is only one force in the formula, see Proposition 4.1. Otherwise, it can be incompatible with GD if conditions (71), which are necessary for validity of MORR, are violated. Finally, GD gives generally different results than EPM (both by parts and as a whole) if the dissipation potential (and thus also entropy production) is non-homogeneous with coupling satisfying MORR.
Figure 2

Relations between GD and EPM as a Venn diagram. The left top set contains models obtained by EPM carried out by parts, i.e. by Step 4a in Section 3.2. No coupling between the forces or fluxes is allowed in this case. Due to Proposition 4.2 all models coming from EPM by parts can be formulated as GD. In particular, if the entropy production formula is homogeneous of the same degree in all forces (or fluxes in the conjugate representation) without coupling or if there is only one force in the formula, EPM by parts is equivalent with EPM when entropy production is maximized as a whole, see Proposition 4.1, which corresponds to Step 4b in Section 3.2. EPM as a whole is equivalent to GD if the entropy production formula is homogeneous (including coupling) or if there is only one force in the formula, see Proposition 4.1. Otherwise, it can be incompatible with GD if conditions (71), which are necessary for validity of MORR, are violated. Finally, GD gives generally different results than EPM (both by parts and as a whole) if the dissipation potential (and thus also entropy production) is non-homogeneous with coupling satisfying MORR.

In Section 2, we review GD, where irreversible evolution is generated by a dissipation potential. A feature of GD is the generalization of Onsager reciprocal relations to the far-from-equilibrium (nonlinear) regime, where dissipation potential is non-quadratic, implying the Maxwell–Onsager reciprocal relations (MORR). Moreover, GD guarantees approach to thermodynamic equilibrium, which is characterized as the minimum of the dissipation potential, where all forces (or fluxes) disappear assuming that equilibrium is implied by disappearing of the forces. Another feature of GD is the geometric way for passing between explicit constitutive relations, where fluxes are functions of forces, to inverse constitutive relations, where forces are functions of fluxes. The passage is carried out simply by the Legendre transformation. Finally, GD is often implied by statistical arguments, see Section 1.

In Section 3, the method of entropy production maximization (EPM) is recalled and subsequently compared with GD in Section 4. Both methods are compatible if entropy production is a homogeneous function of thermodynamic forces or fluxes or if it depends on only one force (or flux). Otherwise the compatibility is rather rare.

When performing EPM, a step is usually tacitly made (namely Step 4a in Section 3.2) where entropy production is split into several parts, each of which is maximized separately. In the case of possible coupling between thermodynamic forces, this step can be done only with great caution, since the resulting coupling is affected by the splitting and since the splitting depends on the not always objective definition of thermodynamic forces, see Section 3.1. EPM carried out by parts thus usually does not cover coupling. Step 4a is mostly done to naturally separate different mechanisms, for example, mechanical forces and heat flux. One should always at least mention the non-trivial input into the procedure of EPM by parts when performing the splitting.

Instead of splitting the entropy production into several independent parts, it can be maximized at once. The resulting constitutive relations then contain nonlinear coupling between the thermodynamic forces even if the entropy production is written as a sum of terms each dependent on only one force. It is a question whether such coupling is desirable or not. For example, it seems to be incompatible with experimental observations in the case of cylindrical Couette flow of suspensions under high temperature gradient when considering constant thermal conductivity κ, see Section 4.1.2.

To compare GD and EPM, a condition is identified, namely eq. (71), which is necessary for validity of Maxwell–Onsager reciprocal relations for constitutive relations obtained by EPM. This condition is satisfied for homogeneous entropy productions, but it can be violated in the inhomogeneous case. Validity of Onsager relations is thus guaranteed by EPM near equilibrium, where entropy production is approximately quadratic (i.e. 2-homogeneous), but not generally far from equilibrium.

In summary, the method of EPM has been widely used and it provides a lot of insight into modeling of complex materials. GD is also a practical method for generating constitutive relations (and generally reductions onto less detailed levels of description), which is automatically compatible with thermodynamics–-second law, Onsager reciprocal relations, approach to equilibrium, Maxwell–Onsager relations. All but the last mentioned compatibility conditions are fulfilled also by EPM. Both approaches are reviewed and compared in this paper and cases are identified when they coincide and when they differ, see Fig. 2.

Let us now summarize advantages and shortcomings of the two methods.

  • Entropy production maximization (EPM):

    • Advantages: Simple imposition of constraints. Fully implicit constitutive relations.

    • Shortcomings: Non-homogeneous entropy production either leads to very complex relations that do not explain experimental observations at least in the considered cases or it has to be split into several parts that are maximized separately. Carrying out the latter option is, however, subjective as the result depends on how the total entropy production is written down in a split form.

  • Gradient dynamics (GD):

    • Advantages: Geometrical and statistical interpretation. Generalization of Onsager reciprocal relations to the far-from-equilibrium regime.

    • Shortcomings: Imposition of constraints not as straightforward as within EPM (although feasible and with additional physical interpretation, see Section 2.5) without sacrificing the structure of Legendre transformations behind GD. Impossibility of fully implicit constitutive relations without sacrificing the structure of Legendre transformations behind GD.

Acknowledgements:

We are grateful to Oğul Esen for discussing the geometric origin of gradient dynamics and to Vít Pruša for discussing the method of entropy production maximization and for assistance during the review process.

This work was supported by Czech Science Foundation, project no. 17-15498Y.

Adam Janečka acknowledges the support of Project No. LL1202 in the programme ERC-CZ, funded by the Ministry of Education, Youth and Sports of the Czech Republic, and of Project 260 449/2017 “Student research in the field of physics didactics and mathematical and computer modelling”.

A k-homogeneous functions

A function is k-homogeneous if f(αx)=αkf(x),αR.(73)

Taking derivative with respect to α at α=1, we get xf(x)=kf(x).

Taking derivative of eq. (73) with respect to x, we get αf(αx)=αkf(x),

thus f is (k1)-homogeneous.

When f is a function of several variables, k-homogeneous in each variable, the same results hold for partial derivatives and for gradients.

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Footnotes

  • 1

    The thermodynamic forces X are expressed as operators (e.g. gradient) acting on conjugate variables x, i.e., X=Γ(x) with Γ=, and conjugate variables can be identified with derivatives of entropy S (or a thermodynamic potential) with respect to the state variables, x=Sx, see Grmela and Öttinger [2]. Therefore, the knowledge of the entropy is crucial for deriving constitutive relations explicitly, see also Section 2.3. 

  • 2

    It should be noted that many authors prefer the quasilinear version of GENERIC, where the irreversible evolution is constructed by assuming generalized Green–Kubo relations and not by a dissipation potential, see Öttinger [9]. 

  • 3

    See the papers by Rajagopal and Srinivasa regarding the differences between their and Ziegler’s approaches. 

  • 4

    Other constraints as the incompressibility of the material can be easily included using additional multipliers. 

  • 5

    The maximization can be also conducted with respect to the thermodynamic fluxes. 

  • 6

    For example fields of density, momentum density and entropy density within classical hydrodynamics. 

  • 7

    I.e. beyond the validity of linear relations. 

  • 8

    Note that all boundary terms are dropped as only spatially local effects are anticipated. 

  • 9

    As well as the SEA framework mentioned in the Introduction 

  • 10

    Note that the entropy flux jη is not given uniquely, as it can be mixed with the fluxes and forces. This can be easily seen from the relation ϕΔφ=divϕφϕφ, where the term under the divergence is an additional component of the entropy flux jη. The remedy might be the time reversibility of the indistinct terms. In case they are odd with respect to time reversal, they do not produce entropy and should be moved into the entropy flux. If they are even with respect to time reversal, there is no objective way how to decide whether to put them into the flux or the entropy production. This is not a problem in GD, since the method does not rely on the entropy production itself but on the dissipation potential, from which the evolution equations are generated directly. GD does not rely on which terms one puts under divergence and which not. 

  • 11

    In case of non-linear non-equilibrium thermodynamics, the requirement to maximize the entropy production is still considered as a working assumption, see Martyushev and Seleznev [32]. 

  • 12

    The requirement for the dissipation potential to be convex yields an additional restriction r12. 

About the article

Received: 2017-01-20

Revised: 2017-07-27

Accepted: 2017-08-23

Published Online: 2017-09-19

Published in Print: 2018-01-26


Citation Information: Journal of Non-Equilibrium Thermodynamics, Volume 43, Issue 1, Pages 1–19, ISSN (Online) 1437-4358, ISSN (Print) 0340-0204, DOI: https://doi.org/10.1515/jnet-2017-0005.

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