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Like in mechanics and electrodynamics, the fundamental laws of the thermodynamics of dissipative processes can be compressed into Gyarmati’s variational principle. This variational principle both in its differential (local) and in integral (global) forms was formulated by Gyarmati in 1965. The consistent application of both the local and the global forms of Gyarmati’s principle provides all the advantages throughout explicating the theory of irreversible thermodynamics that are provided in the study of mechanics and electrodynamics by the corresponding classical variational principles, e.g., Gauss’ differential principle of least constraint or Hamilton’s integral principle.

A number of efforts have been made to formulate variational principles for dissipative processes, namely those accounted by (“classical”) irreversible thermodynamics. The results seemed, and even seem, to contradict the known result in mathematics that says the parabolic equations of heat conduction, diffusion,

A brief and rather good survey can be found on Wikipedia [

In this paper, we put Gyarmati’s variational principle into the limelight. This principle was applied to several fields of irreversible processes: first of all, by his colleagues (Verhás [

Gyarmati’s principle is based on the fact that the generalization of the dissipation functions that were introduced by Rayleigh and Onsager for special cases always exists locally in continua [

and:

The _{ik}_{ik}

The most important property of the dissipation function, Ψ(_{i}_{i}_{i}_{k}_{k}

Finally, the equality of the mixed second derivatives of Ψ with respect to the forces are equivalent to Onsager’s reciprocal relations:

Because of the above properties, the function, Ψ, is called a dissipation potential, more precisely: it is the flux potential (see

The function, Φ, has similar properties. In the strictly linear theory the function, Φ, is a homogeneous quadratic function of the currents, _{k}_{k}

Due to this relation, the function, Φ, is also a dissipation potential, more exactly: it is the force potential.

The equality of the mixed second derivatives of Φ with respect to the _{ik}

Hence, it can be seen that the necessary and sufficient condition of the existence of the dissipation potentials, Ψ and Φ, is the existence of Onsager’s reciprocal relations.

Some weighted potentials, Ψ^{G}^{G}_{s}^{2} leads to the “Fourier picture”,

The quantity, _{s}_{j}

regarded as forces, and substituted into the original form, give the linear laws in the “

The coefficients obey Onsager’s reciprocal relations. By choosing various functions for

Finally, we note another essential property of the functions, Ψ and Φ; namely, that they are invariant scalar quantities with respect to the linear transformations of the currents and forces.

Gyarmati’s variational principle of non-equilibrium thermodynamics can be derived from the properties in

of the Onsager relations corresponds to. These relations are necessary and sufficient conditions of the existence of dissipation potentials, obeying

Notice that

where, in executing the partial differentiation, the currents must be regarded variables independent of the forces and local state variables. This means that the constitutive relations given by

has a stationary point in the space of the currents. This form of the principle, which stands nearest to Onsager’s principle for small fluctuations around an equilibrium in an adiabatically closed discontinuous system, is called the flux representation of Gyarmati’s principle [

The force representation of Gyarmati’s principle is obtained by putting the relation in

During partial differentiation, the forces and the fluxes must be regarded again as independent variables. Thus, those forces correspond to a given set of currents and state variables at which the function:

has a stationary point in the space of the forces.

It is easily seen that the functions, _{J}_{X}_{i}_{J}_{X}

by which the extremum properties in

It can be said, quite generally, that if a sufficient number of the currents and forces is known, that is either every force or every current, or even one part of the currents and the other part of the forces, then the remaining variables must be chosen, so that the universal Lagrangian in

In the quasi-linear theory, the functions, Ψ and Φ, depend on the state variables through the conductivities, _{ik}_{ik}_{ik}_{ik}

Let us calculate the partial derivative of the

The partial derivatives _{ik}/∂_{ik}_{ik}/∂

Substituting this in

is obtained. Hence, it is seen that the partial derivatives of the universal Lagrangian with respect to the local state variables, at real processes, are zero. Therefore, the parameters, Γ, can also be varied independently.

This theorem is Gyarmati’s supplementary theorem [

The universal form of the local Gyarmati principle states, consistently with the supplementary theorem, that the Lagrangian _{s}

In examining the type of the extremum, instead of considering second variations, we had better use another form of the Lagrangian, which is advantageous in other respects, as well. This form is:

Executing the multiplications, the form in _{ik}_{ik}_{i}_{s} L_{is}X_{s}

The local Gyarmati principle of irreversible thermodynamics is of universal validity, yet its primary importance is that it is the ground the integral principles are built on. Before the discussion of integral principles, however, the place of the local principle in the framework of the theory should be examined. To this end, the local principle is resumed more explicitly.

The essence of the local principle is that it replaces the set of linear laws by a single scalar function. If either the function, Ψ, or the function, Φ, is known, the constitutive equations can be obtained by the variational principle. Actually, it is sufficient to know only one of the dissipation potentials, Ψ or Φ, since the matrix of the coefficients can be read from one of them, and the other potential is determined by the elements of the reciprocal matrix. This calculation can be executed via a more elegant method. Let us regard, for example, the function, Ψ, as the given one. Then, the Legendre-transformation of the function, Ψ, leads to the function,

The advantage of the method of Legendre transformation lies in the fact that its formulation and application is independent of the linear or quasi-linear character of the theory; thus, it is applicable to dissipation potentials of an entirely different character. From the fact that the dissipation potentials, Ψ and Φ, are the Legendre transforms of each other, it is also seen that the validity of Gyarmati’s supplementary theorem is not restricted to the quasi-linear case, but holds to any strictly non-linear theory, subject to the Gyarmati-Li generalized reciprocal relations (and where the higher order coefficients also depend on the variables of state). This, at the same time, means that the Lagrangian _{s}

The next question is how a dissipation potential can be constructed from the constitutive equations. The potential character of the functions, Ψ and Φ, is defined by

The function, Ψ, can be obtained from the bilinear form of the entropy production by introducing

for the entropy production. This expression can be regarded as a quasi-linear inhomogeneous first order partial differential equation. Its only solution subject to the condition Ψ(0) = 0 is the function:

A similar formula is obtained for Φ(

The knowledge of the function, Ψ or Φ, defined so, is equivalent to the knowledge of the original constitutive equations; nevertheless, to get them, it is enough to know the single function, _{s}_{i}_{s}_{i}

Dissipation potentials for non-linear cases were given first (and independently) by Verhás [

Here, for an example, we present a very simple application of the local principle to a typical non-linear case: we derive Mises’ theory of plastic flow (Verhás [

The existence of the function, Φ, is assumed.

Let us consider a homogeneous, isotropic, incompressible fluid continuum in local equilibrium and ignore heat conduction. The energy dissipation for this case reads:

where _{0} for the deviatoric part of Cauchy’s stress, and

It is well known that the direct application of the linear laws to this expression leads to Newton’s viscosity law [

During isochoric motions, the trace of

If Φ is assumed to be a continuous function of its variables (remember that Φ = 0 at ^{3} variable, too, as its value is small relative to tr ^{2}. The trace of ^{3} is strictly zero for viscometric flows. Thus, the form of the potential, Φ, for viscometric flows exactly and for nearly viscometric flows approximately is:

Since Φ is regarded as a function of

where

Thus, instead of

It is obvious that the function,

can be taken, where

is obtained, from which the constitutive relation for the deviatoric part of Cauchy’s stress:

follows. Introducing the notation:

the expression:

is obtained for the stress, which is well known form for the ideal plastic body in von Mises’ approximation.

From the above results, it is clear that Gyarmati’s local principle furnished with various approximations for the potential,

The possibility of generalizing the reciprocal relations for non-linear constitutive equations has already been mentioned earlier when describing Gyarmati’s variation principle. However, the practical value of the suggestions made there is doubtful. The reason for this is that neither the macroscopic reversibility principle proposed by Meixner [

Nevertheless, in the following, a generalization will be presented, which is proven by strict mathematics and whose validity is not restricted if we have twice continuously differentiable constitutive equations.

Start from the bilinear form of entropy production:

but let us drop our usual notation, _{i}_{i}_{i}_{i}_{i}

where _{1}, _{2}, . . . are the independent variables selected from the expression of entropy production. For brevity, let us now call them forces; Γ_{1}, Γ_{2}, . . . are other local state parameters whose specification is not necessary at present. Let us then take function:

depending on

results, which at

We note that reciprocal relations do not follow from the expression; their validity has been taken from the linear theory. Now, for brevity, we introduce coefficients:

Following from the nature of the Taylor series, derivatives should be taken at _{1}, _{2}, . . ., where

where coefficients _{ijk}

hold between them. Let us now introduce coefficients:

for which, on the one hand, Onsager’s reciprocal relations of the linear theory are valid and, on the other hand, as a consequence of

hold. By using them, the constitutive equations may be written as:

Since the last term on the right-hand side is zero,

Our results can be summarized as follows. The constitutive equations, also for non-linear cases, may be written in the form:

where coefficients, _{ik}

hold if they are valid in the linear limiting case, _{ik}

The selection of independent variables can be varied by a linear transformation of forces and currents, which is formally analogous to that applied in the linear theory. This means that in the equation with the changed independent variables, Casimir-type reciprocal relations also appear, similar to the linear theory. Of course, from the viewpoint of the validity of the generalized Onsager-Casimir reciprocal relations between coefficients _{ik}_{ik}_{ik}_{ik}_{ik}

The above generalization of Onsager’s reciprocal relations permits the writing of the Lagrange function belonging to Gyarmati’s principle provided by

where numbers _{ik}_{ik}_{ik}

The Lagrange function can be reduced to its well-known simpler form by removing the parentheses:

From this expression, dissipation potentials may be determined as:

It is very important and striking that relative to the linear theory, the only change is that dissipation potential, Ψ, is not a quadratic function of

holds also here, partial derivative:

does not give the currents, since in this non-linear theory, Ψ cannot be regarded as a potential. It should also be noted that the problem of these derivatives becomes even more complicated if

The unchanged validity of the local form of Gyarmati’s principle in non-linear cases and the existence of an absolute maximum allows the integration of the local form with respect to time and space. Therefore, we can say that the validity of the Governing Principle of Dissipative Processes whose basis is the integrated form of

After the above results were reached, a new variational technique was proposed [

Up to now, the symmetry of the conductivity matrix was presumed. Nevertheless, the complete symmetry is frequent; it does not always hold. A magnetic field or Coriolis force in the rotating frame may result in the skew-symmetric part, even in the linear approximation. Getting rid of the skew-symmetric part is easy and always possible. Colman and Truesdell showed a good method [

If Onsager’s reciprocity does not hold, the constitutive equations have the form:

with:

Introducing new variables for the fluxes by:

we find that the bilinear form of the entropy production remain unchanged;

as the last term in the right-hand side is zero.

The above train of thought is valid both for the linear and the non-linear approximation. This means that, even in the presence of a skew-symmetric part in the matrix of the conductivity (or the resistivity) coefficients, with changing the variables accounting on processes, the results of the previous subsection can be applied; nevertheless, the marvelous freedom of picking up the fluxes has been reduced.

Though the local form of Gyarmati’s principle is indispensable for the description of local constraints, an integral form of the principle is of much greater importance in practical calculations. The integral forms are obtained by the integration of the universal Lagrange density with respect to space or space and time coordinates. The universal (global) principle, obtained so, is called the “Governing Principle of Dissipative Processes” (GPDP) [

Since the universal Lagrange density is everywhere and always stationary, it is also true that:

and:

The Governing Principle of Dissipative Processes given by Gyarmati can be regarded as the most widely valid and the most widely applied integral principle of irreversible thermodynamics. From this principle, the parabolic transport equations of irreversible transport processes can be derived both in the linear and quasi-linear case, as well as in all those non-linear cases where dissipation potentials can be determined by

The application of the governing principle can be understood through the properties of the local principle. The variational principle alone does not contain sufficient information about the system; the functional takes its absolute maximum in several points of the (Γ

Hence, it follows that the variational principles in

The extraordinary importance of the formula in

The situation with the time integrated form in

The Governing Principle of Dissipative Processes, like any other integral principle of physics, contains information on the boundary conditions, too. They have to be given, so that the absolute maximum is provided,

We mention that for strictly linear problems, there are two partial forms that are also valid:

and:

The first of these is called force, and the second is called flux representation. Both representations were widely applied to the solution of several practical problems [

Here, a more or less “classical” framework of Gyarmati’s variational principle has been surveyed, but life does not stop; new fields of applications and new aspects emerge. The unification of and relating the different approaches is a permanent task. I mention only some works of Sieniutycz [

Some of them (e.g., [

For those readers who are interested in how a parabolic equation can be derived from a variational principle, a very simple example is put forward here.

Assume one-dimensional heat conduction with constant density, specific heat and heat conductivity in, e.g., a wall with constant thickness; moreover, use such units that both the heat capacity of the unit volume and the heat conductivity are one (and dimensionless). In this model, the conservation of energy is expressed by:

Here, _{q}

The entropy balance reads:

In this simple case, the so-called Fourier picture is the most convenient one. The quantity:

is a comfortable starting point to write Onsager’s linear law, which reads:

Nevertheless, substituting this into

and the Lagrangian in

The two functions looked for are correlated by the conservation of the energy in

or better, introducing a potential,

with which the conservation of the energy automatically holds. Now, the form of the variational problem in

The Euler-Lagrange equation reads:

or:

or

The usual initial and boundary conditions prescribe the temperature; now, they prescribe the value of
_{2}, consequently, the transversality conditions:

also hold. Introducing temporarily the:

auxiliary variable, the equation turns into:

with the initial (now final) and boundary conditions:

The

This work has been supported by the Hungarian National Scientific Research Funds, OTKA(62278 and K81161).

The author declares no conflict of interest.

This is based on the paper of Verhás, J. Gyarmati’s Variational Principle of Dissipative Processes. In Proceedings of the 12th Joint European Thermodynamics Conference, JETC 2013, Brescia, Italy, 1–5 July 2013; pp. 127–132.