# Thermodynamic hierarchies of evolution equations/Evolutsioonivorrandite termodunaamilised hierarhiad.

1. INTRODUCTION

Theories and material models of multiscale phenomena in space and time treat the scale changes either as a step from a micro- or mesoscopic statistical level to the phenomenological one or as a reduction of the degrees of freedom by averaging over a field variable or spatial dimension. The characteristic methodology is similar to the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of the kinetic theory [1,2]. In these approaches, the modelling of transitional effects requires detailed knowledge or drastic simplification of the material structure.

In this work, we show that a hierarchical arrangement of evolution equations is apparent in thermodynamics with internal variables where the different levels of the hierarchy are regulated by material parameters. The scale transitions are natural and dynamical.

We show three examples. First, a hierarchy of ordinary differential equations is presented in the thermodynamic rheology of solids with a single internal variable. The building block of the hierarchy is the basic constitutive equation of elasticity. This is a hierarchy between the different time scales of the evolution, a time hierarchy.

Then a hierarchy of hyperbolic partial differential equations is shown in the thermodynamic theory of generalized continua with dual internal variables. The building block of the hierarchy is the wave equation. This is a dynamic hierarchy between different time and length scales of the evolution, a space-time hierarchy.

Finally, a hierarchy of parabolic partial differential equations is shown in the thermodynamic theory of heat conduction with current multipliers. The building block of the hierarchy is the Fourier equation. Like the previous example, it is a dynamic hierarchy between different time and length scales of the evolution, a space-time hierarchy.

2. THE HIERARCHY OF RHEOLOGICAL BODIES AND THE KLUITENBERG-VERHAS MODEL

In the thermodynamic approach to rheology, an extended state space is chosen, which is spanned by the following variables: specific internal energy e, strain e, and an internal variable %. This modelling approach is well known for fluids [3-7] and was introduced recently for solids [8]. The internal variable is a second order symmetric tensor, based on our purpose to gain an extension of the mechanical aspects (the 'material law') of the initial system and to obtain corrections to the relation between stress and strain, which are both symmetric tensors.

We shift entropy by a concave non-equilibrium term depending--quadratically--on [xi] only. According to the Morse lemma, this new entropy term can be chosen as a pure square term; hence, the extended specific entropy function [??] is

[??](e,[epsilon], [xi]) = s(e, [epsilon]) - [1/2] tr([[xi].sup.2]), (1)

where tr denotes the trace of a second order tensor and we have denoted the classical specific entropy without tilde. The Gibbs relation for the extended entropy is a convenient particular thermodynamic notation for the partial derivatives, the intensive quantities:

[rho]d[??] = [[rho]/T] de - [1/2]tr([sigma]d[epsilon]) - [rho]tr([xi]d[xi]). (2)

Here [rho] is density, T is temperature, and [sigma] is thermostatic stress. We assume that the thermostatic stress is different from the stress [??] in the balance of momentum and internal energy. The difference is characterized introducing a rheological (non-equilibrium) term:

[??] = [sigma] + [??]. (3)

Consequently, the mechanical power, and correspondingly the energy balance, gets shifted as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Here je is the conductive current density of the internal energy, the heat flux. With the choice [j.sub.s] = [j.sub.e]/T, and using (2) and (4), the entropy production is found to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Here the symmetric stress, strain, and internal variable [xi] tensors are divided into spherical and deviatoric parts: [sigma] = [[simga].sup.d] + [[sigma].sup.s], [epsilon] = [[epsilon].sup.d] + [[epsilon].sup.s], and [xi] = [[xi].sup.d] + [[xi].sup.s]. Here As = (trA)I/3 is the spherical part of the symmetric tensor A, I denotes the unity tensor, and the deviatoric part is defined as [A.sup.d] = A - [A.sup.s]. In the last row of (5), vectors are present in the first term, scalars in the third and fifth terms, and symmetric traceless tensors in the second and fourth terms. In an isotropic material, these three types of quantities cannot couple to one another. Therefore, concerning the term containing vectors, we consider Fourier heat conduction, je = [lambda] [nabla](1/T). For the remaining two pairs of terms, the most general Onsagerian solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where the [l.sup.d.sub.11], [l.sup.d.sub.12],[l.sup.d.sub.21],[l.sup.d.sub.22] and [l.sup.s.sub.11], [l.sup.s.sub.12], [l.sup.s.sub.21], [l.sup.s.dub.22] material parameters are subjects of thermodynamic restrictions, due to the entropy inequality (5).

Eliminating the internal variable in the constant temperature case also leads to two independent models,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

with thermodynamics-originated inequalities for the altogether eight coefficients. The complete model is a deviatoric and a spherical K/uitenberg-Verhas body. When [E.sup.d.sub.2] = 0, the deviatoric part reduces to the standard or Poynting-Thomson body of solid rheology. Several simpler rheological bodies may be obtained by a particular choice of the parameters.

A suitable rearrangement reveals the hierarchical structure of the equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

In both the deviatoric and the spherical case, the first term is a pure elastic stress-strain relation, the second is the time derivative of a similar relation with different coefficients, and the third one with the highest derivative is an incomplete block, closing the two terms' hierarchy.

If the coefficients in the consecutive blocks are the same, we may speak about hierarchical resonance. If the closure term is zero, a hierarchical resonance need not be dissipative.

In case of specific loading conditions, the deviatoric and spherical parts are coupled but the hierarchical structure may be conserved. It is straightforward to calculate the effective rheological equation in case of uniaxial loading conditions. Denoting the uniaxial stress by 0, one obtains:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [[tau].sup.[parallel].sub.1], [[tau].sup.[parallel].sub.2], [[tau].sup.|.sub.2] and[E.sup.[parallel].sub.0], [E.sup.[parallel].sub.1], [E.sup.[parallel].sub.2], [E.sup.|.sub.3], [E.sup.[parallel].sub.4] coefficients are calculated from the spherical and deviatoric coefficients of (7) [8].

In typical experimental situations, the time scales of the different blocks are clearly separated.

3. HIERARCHY OF WAVE EQUATIONS IN THE THEORY OF DUAL INTERNAL VARIABLES

Dual internal variables extend the modelling capability of non-equilibrium thermodynamics by connecting inertial phenomena with dissipation [9]. Dual internal variables coupled to continuum mechanics lead to generalized continua [10,11]. In this case, the elimination of the internal variables results in a hierarchical structure of wave equations [10,12,13].

In what follows, we introduce in brief a one-dimensional version of the theory of weakly nonlocal dual internal variables coupled to small-strain elasticity. Therefore, the state space is given by the strain, [epsilon], and the internal variables are denoted by [phi] and [xi]. In this illustrative example, specific entropy is a quadratic function of the internal variables and their gradients:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The Gibbs relation of the weakly nonlocal theory fixes the partial derivatives of the entropy function as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Assuming the following form of the entropy current density:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

one obtains the entropy production similarly to the previous section:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Here, we have introduced a shorthand notation for the internal variable related weakly to nonlocal thermodynamic forces. The above form of the entropy current density and entropy production (dissipation inequality) can be also derived with the help of a more detailed thermodynamic analysis, as it is shown in [11]. Then a linear solution of the above inequality is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

In our simple case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

The elimination of the internal variables leads to the following constitutive relation of stress and strain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where the coefficients [[alpha].sub.1],[[alpha].sub.2],[[beta].sub.1],[[beta].sub.2],[[beta].sub.3] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are simple polynomials of the thermodynamic material parameters. The consequence of the momentum balance and the compatibility condition is the well-known relation of stress and strain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Eliminating stress from (16), one obtains the following partial differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Mixed space and time derivatives of coupled wave equations are analysed in detail and are compared to various wave propagation models in [13].

4. HIERARCHY OF FOURIER EQUATIONS AND GENERALIZED HEAT CONDUCTION WITH CURRENT MULTIPLIERS

Non-equilibrium thermodynamics with current multipliers introduces a unified constitutive theory of heat conduction where several generalizations of Fourier equation may be obtained as special cases [14]. Moreover, it is shown that the structure is compatible with the moment series expansion of kinetic theory, at least up to the third moment [15]. In this framework, the basic state space is extended by the heat flux q and also by a second order tensorial internal variable Q. We assume the usual quadratic form of the entropy function at the extended part of the state space,

[??](e, q, Q) = s(e) - [m.sub.1]/2 [q.sup.2] - [m.sub.2]/2[Q.sup.2]. (19)

Then a generalized entropy current is introduced in the following form:

[j.sub.q] = b x q + B : Q. (20)

Here, the current multipliers b and B are second and third order tensors, respectively. This form of the generalized entropy current was introduced by Nylri [16]. The multipliers b and B are to be determined as constitutive functions with the help of the entropy inequality. A short calculation results in

[SIGMA] = (b - [1/T] I): [nabla]q - ([nabla]b - [m.sub.1][??]) x q - ([nabla] x B - [m.sub.2]Q) : Q + B x: [nabla]Q [greater than or equal to] 0. (21)

Here the number of the central dots denotes one, two, and three contractions of the first, second, and third order tensors, respectively. The first and the third terms are products of second order tensors, the second term is vectorial, and the last term is a product of third order tensors. Therefore, for isotropic materials, cross effects may appear only between the first and the third terms. Hence, in a one dimensional simplification, linear relations between the thermodynamic fluxes and forces are as follows:

[m.sub.1][??] - [[partial derivative].sub.x]b = -[l.sub.1]q, (22)

[m.sub.2][??] - [[partial derivative].sub.x]B = -[k.sub.1]Q + [k.sub.12][[partial derivative].sub.x] q, (23)

b - 1/T = -[k.sub.21] Q + [k.sub.2][[partial derivative].sub.x]q, (24)

B = [n.sub.3][[partial derivative].sub.x] Q, (25)

where dx denotes the one dimensional spatial derivative and the material coefficients [m.sub.1],[m.sub.2],[l.sub.1]} [k.sub.1],[k.sub.2], [k.sub.12], [k.sub.21], [n.sub.3] are subjects to thermodynamical constraints. It is straightforward to eliminate the current multipliers and the tensorial internal variable Q. Then one obtains the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

In our case, the balance of internal energy (4) is

[rho]cT + [[partial derivative].sub.x] q = 0, (27)

where [rho] is the density and c is the specific heat. The combination of (26) and (27) may be written in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

We can observe various time and space derivatives of the Fourier equation in different forms, plus the last term with the highest derivatives. The arrangement is space-time hierarchical, like in the previous section.

4.1. Hierarchical resonance: the example of the Guyer-Krumhansl equation

The hierarchical rearrangement of an evolution equation may help in recognizing solution patterns. In this subsection, we give a simple example with the help of the Guyer-Krumhansl equation.

The Guyer-Krumhansl equation is obtained when [n.sub.3] = [m.sub.2] = [k.sub.12] = 0 in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Here, [[tau].sub.q] = [m.sub.1]/[l.sub.1], [lambda] = 1/([rho][cl.sub.1]) and a = [k.sub.2]/[l.sub.1], and these coefficients are considered constant. This is a two-level hierarchical arrangement. If [[tau].sub.q] = 0, then the first term is zero, and the hierarchy is reduced to a single Fourier equation. If a = [[tau].sub.q][lambda], then there appears the same Fourier equation in both terms. This is the case of hierarchical resonance, and the solutions of the coupled set of equations may be identical to those of the single Fourier equation [15].

The resonance may help to classify the solutions. Let us introduce adiabatic boundary at the end of a rod and heat pulse boundary conditions at the front side in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [t.sub.p] is the duration of the pulse and [q.sub.max] is the maximum of the heat flux at the boundary. Initially, the temperature is uniform and there is no heat flux q(t = 0, x) = 0, T(t = 0, x) = [T.sub.0]. Then the solutions show characteristic differences depending on whether the parameters are above or below the resonance value. This is represented in Fig. 1, where temperature and time are the following dimensionless quantities: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here L is the length of the rod and [T.sub.end] is the asymptotic value of the temperature after the equilibration.

* If a = [[tau].sub.q][lambda], then we obtain the solution of the Fourier equation. This is the solid line in Fig. 1.

* If a < [[tau].sub.q][lambda], then we obtain solutions where temperature starts to increase later than in the Fourier solution. For short rods, the heat pulse is observable. The solution is zero at the beginning and a visible heat pulse appears similarly to the solutions of the Maxwell-Cattaneo-Vernotte equation. This is the dashed line in Fig. 1.

* If a > [[tau].sub.q][lambda], then temperature starts to increase earlier than for the Fourier solution. The remnants of the heat pulse are not observable, and sometimes there is a change in the steepness of the solution, a kink. The solution is more damped than the Fourier one. This is the dashed-dotted line in Fig. 1.

[FIGURE 1 OMITTED]

5. SUMMARY

Eliminating internal variables in non-equilibrium thermodynamics results in a hierarchical structure of the evolution equation. The buiding block of the hierarchy is the evolution equation of the original theory, which was supplemented by the internal variable.

The solution of the original equation may appear at different particular values of the parameters. If this happens with more than one nonzero element of the hierarchy, we can call it hierarchical resonance.

ACKNOWLEDGEMENT

The work was supported by grants OTKA K81161 and K104260.

REFERENCES

[1.] Grmela, M., Lebon, G., and Dubois, C. Multiscale thermodynamics and mechanics of heat. Phys. Rev. E, 2011, 83, 061134.

[2.] Liboff, R. L. Kinetic Theory (Classical, Quantum, and Relativistic Descriptions). Prentice Hall, Englewood Cliffs, New Jersey, 1990.

[3.] Kluitenberg, G. A. Thermodynamical theory of elasticity and plasticity. Physica, 1962, 28, 217-232.

[4.] Verhas, J. Thermodynamics and Rheology. Akademiai Kiado and Kluwer Academic Publishers, Budapest, 1997.

[5.] Kluitenberg, G. A. and Ciancio, V. On linear dynamical equations of state for isotropic media. Physica A, 1978, 93, 273-286.

[6.] Ciancio, V. and Kluitenberg, G. A. On linear dynamical equations of state for isotropic media II: some cases of special interest. Physica A, 1979, 99, 592-600.

[7.] Maugin, G. A. and Muschik, W. Thermodynamics with internal variables. Part I. General concepts. J. Non-Equil. Thermody., 1994, 19, 217-249.

[8.] Asszonyi, Cs., Fulop, T., and Van, P. Distinguished rheological models for solids in the framework of a thermodynamical internal variable theory. Continuum Mech. Therm., 2014. doi: 10.1007/s00161-014-0392-3. arXiv:1407.0882.

[9.] Van, P., Berezovski, A., and Engelbrecht, J. Internal variables and dynamic degrees of freedom. J. Non-Equil. Thermody., 2008, 33(3), 235-254. cond-mat/0612491.

[10.] Berezovski, A., Engelbrecht, J., and Berezovski, M. Waves in microstructured solids: a unified viewpoint of modelling. Acta Mech., 2011, 220, 349-363.

[11.] Van, P., Papenfuss, C., and Berezovski, A. Thermodynamic approach to generalized continua. Continuum Mech. Therm., 2014, 25(3), 403-420. Erratum: 421-422, arXiv:1304.4977.

[12.] Berezovski, A., Engelbrecht, J., and Peets, T. Multiscale modeling of microstructured solids. Mech. Res. Commun., 2010, 37(6), 531-534.

[13.] Berezovski, A. and Engelbrecht, J. Thermoelastic waves in microstructured solids: dual internal variables approach. Journal of Coupled Systems and Multiscale Dynamics, 2013, 1(1), 112-119.

[14.] Van, P. and Fulop, T. Universality in heat conduction theory: weakly nonlocal thermodynamics. Ann. Phys., 2012, 524(8), 470-478. arXiv:1108.5589.

[15.] Kovacs, R. and Van, P. Generalized heat conduction in laser flash experiments. International Journal of Heat and Mass Transfer, 2015, 83, 613-620.2014. arXiv:1409.0313v2.

[16.] Nyiri, B. On the entropy current. J. Non-Equil. Thermody., 1991, 16, 179-186.

Peter Van (a, b,c) *, Robert Kovacs (a,b,c), and Tamas Fulop (b,c)

(a) Department of Theoretical Physics, Wigner Research Centre for Physics, Institute for Particle and Nuclear Physics, PO Box 49, 1525 Budapest, Hungary

(b) Department of Energy Engineering, Budapest University of Technology and Economics, POBox91, 1521 Budapest, Hungary

(c) Montavid Thermodynamic Research Group, Igmandi u. 26, 1112 Budapest, Hungary

Received 14 December 2014, accepted 30 March 2015, available online 28 August 2015

* Corresponding author, van.peter@wigner.mta.hu

Theories and material models of multiscale phenomena in space and time treat the scale changes either as a step from a micro- or mesoscopic statistical level to the phenomenological one or as a reduction of the degrees of freedom by averaging over a field variable or spatial dimension. The characteristic methodology is similar to the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of the kinetic theory [1,2]. In these approaches, the modelling of transitional effects requires detailed knowledge or drastic simplification of the material structure.

In this work, we show that a hierarchical arrangement of evolution equations is apparent in thermodynamics with internal variables where the different levels of the hierarchy are regulated by material parameters. The scale transitions are natural and dynamical.

We show three examples. First, a hierarchy of ordinary differential equations is presented in the thermodynamic rheology of solids with a single internal variable. The building block of the hierarchy is the basic constitutive equation of elasticity. This is a hierarchy between the different time scales of the evolution, a time hierarchy.

Then a hierarchy of hyperbolic partial differential equations is shown in the thermodynamic theory of generalized continua with dual internal variables. The building block of the hierarchy is the wave equation. This is a dynamic hierarchy between different time and length scales of the evolution, a space-time hierarchy.

Finally, a hierarchy of parabolic partial differential equations is shown in the thermodynamic theory of heat conduction with current multipliers. The building block of the hierarchy is the Fourier equation. Like the previous example, it is a dynamic hierarchy between different time and length scales of the evolution, a space-time hierarchy.

2. THE HIERARCHY OF RHEOLOGICAL BODIES AND THE KLUITENBERG-VERHAS MODEL

In the thermodynamic approach to rheology, an extended state space is chosen, which is spanned by the following variables: specific internal energy e, strain e, and an internal variable %. This modelling approach is well known for fluids [3-7] and was introduced recently for solids [8]. The internal variable is a second order symmetric tensor, based on our purpose to gain an extension of the mechanical aspects (the 'material law') of the initial system and to obtain corrections to the relation between stress and strain, which are both symmetric tensors.

We shift entropy by a concave non-equilibrium term depending--quadratically--on [xi] only. According to the Morse lemma, this new entropy term can be chosen as a pure square term; hence, the extended specific entropy function [??] is

[??](e,[epsilon], [xi]) = s(e, [epsilon]) - [1/2] tr([[xi].sup.2]), (1)

where tr denotes the trace of a second order tensor and we have denoted the classical specific entropy without tilde. The Gibbs relation for the extended entropy is a convenient particular thermodynamic notation for the partial derivatives, the intensive quantities:

[rho]d[??] = [[rho]/T] de - [1/2]tr([sigma]d[epsilon]) - [rho]tr([xi]d[xi]). (2)

Here [rho] is density, T is temperature, and [sigma] is thermostatic stress. We assume that the thermostatic stress is different from the stress [??] in the balance of momentum and internal energy. The difference is characterized introducing a rheological (non-equilibrium) term:

[??] = [sigma] + [??]. (3)

Consequently, the mechanical power, and correspondingly the energy balance, gets shifted as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Here je is the conductive current density of the internal energy, the heat flux. With the choice [j.sub.s] = [j.sub.e]/T, and using (2) and (4), the entropy production is found to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Here the symmetric stress, strain, and internal variable [xi] tensors are divided into spherical and deviatoric parts: [sigma] = [[simga].sup.d] + [[sigma].sup.s], [epsilon] = [[epsilon].sup.d] + [[epsilon].sup.s], and [xi] = [[xi].sup.d] + [[xi].sup.s]. Here As = (trA)I/3 is the spherical part of the symmetric tensor A, I denotes the unity tensor, and the deviatoric part is defined as [A.sup.d] = A - [A.sup.s]. In the last row of (5), vectors are present in the first term, scalars in the third and fifth terms, and symmetric traceless tensors in the second and fourth terms. In an isotropic material, these three types of quantities cannot couple to one another. Therefore, concerning the term containing vectors, we consider Fourier heat conduction, je = [lambda] [nabla](1/T). For the remaining two pairs of terms, the most general Onsagerian solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where the [l.sup.d.sub.11], [l.sup.d.sub.12],[l.sup.d.sub.21],[l.sup.d.sub.22] and [l.sup.s.sub.11], [l.sup.s.sub.12], [l.sup.s.sub.21], [l.sup.s.dub.22] material parameters are subjects of thermodynamic restrictions, due to the entropy inequality (5).

Eliminating the internal variable in the constant temperature case also leads to two independent models,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

with thermodynamics-originated inequalities for the altogether eight coefficients. The complete model is a deviatoric and a spherical K/uitenberg-Verhas body. When [E.sup.d.sub.2] = 0, the deviatoric part reduces to the standard or Poynting-Thomson body of solid rheology. Several simpler rheological bodies may be obtained by a particular choice of the parameters.

A suitable rearrangement reveals the hierarchical structure of the equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

In both the deviatoric and the spherical case, the first term is a pure elastic stress-strain relation, the second is the time derivative of a similar relation with different coefficients, and the third one with the highest derivative is an incomplete block, closing the two terms' hierarchy.

If the coefficients in the consecutive blocks are the same, we may speak about hierarchical resonance. If the closure term is zero, a hierarchical resonance need not be dissipative.

In case of specific loading conditions, the deviatoric and spherical parts are coupled but the hierarchical structure may be conserved. It is straightforward to calculate the effective rheological equation in case of uniaxial loading conditions. Denoting the uniaxial stress by 0, one obtains:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [[tau].sup.[parallel].sub.1], [[tau].sup.[parallel].sub.2], [[tau].sup.|.sub.2] and[E.sup.[parallel].sub.0], [E.sup.[parallel].sub.1], [E.sup.[parallel].sub.2], [E.sup.|.sub.3], [E.sup.[parallel].sub.4] coefficients are calculated from the spherical and deviatoric coefficients of (7) [8].

In typical experimental situations, the time scales of the different blocks are clearly separated.

3. HIERARCHY OF WAVE EQUATIONS IN THE THEORY OF DUAL INTERNAL VARIABLES

Dual internal variables extend the modelling capability of non-equilibrium thermodynamics by connecting inertial phenomena with dissipation [9]. Dual internal variables coupled to continuum mechanics lead to generalized continua [10,11]. In this case, the elimination of the internal variables results in a hierarchical structure of wave equations [10,12,13].

In what follows, we introduce in brief a one-dimensional version of the theory of weakly nonlocal dual internal variables coupled to small-strain elasticity. Therefore, the state space is given by the strain, [epsilon], and the internal variables are denoted by [phi] and [xi]. In this illustrative example, specific entropy is a quadratic function of the internal variables and their gradients:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The Gibbs relation of the weakly nonlocal theory fixes the partial derivatives of the entropy function as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Assuming the following form of the entropy current density:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

one obtains the entropy production similarly to the previous section:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Here, we have introduced a shorthand notation for the internal variable related weakly to nonlocal thermodynamic forces. The above form of the entropy current density and entropy production (dissipation inequality) can be also derived with the help of a more detailed thermodynamic analysis, as it is shown in [11]. Then a linear solution of the above inequality is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

In our simple case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

The elimination of the internal variables leads to the following constitutive relation of stress and strain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where the coefficients [[alpha].sub.1],[[alpha].sub.2],[[beta].sub.1],[[beta].sub.2],[[beta].sub.3] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are simple polynomials of the thermodynamic material parameters. The consequence of the momentum balance and the compatibility condition is the well-known relation of stress and strain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Eliminating stress from (16), one obtains the following partial differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Mixed space and time derivatives of coupled wave equations are analysed in detail and are compared to various wave propagation models in [13].

4. HIERARCHY OF FOURIER EQUATIONS AND GENERALIZED HEAT CONDUCTION WITH CURRENT MULTIPLIERS

Non-equilibrium thermodynamics with current multipliers introduces a unified constitutive theory of heat conduction where several generalizations of Fourier equation may be obtained as special cases [14]. Moreover, it is shown that the structure is compatible with the moment series expansion of kinetic theory, at least up to the third moment [15]. In this framework, the basic state space is extended by the heat flux q and also by a second order tensorial internal variable Q. We assume the usual quadratic form of the entropy function at the extended part of the state space,

[??](e, q, Q) = s(e) - [m.sub.1]/2 [q.sup.2] - [m.sub.2]/2[Q.sup.2]. (19)

Then a generalized entropy current is introduced in the following form:

[j.sub.q] = b x q + B : Q. (20)

Here, the current multipliers b and B are second and third order tensors, respectively. This form of the generalized entropy current was introduced by Nylri [16]. The multipliers b and B are to be determined as constitutive functions with the help of the entropy inequality. A short calculation results in

[SIGMA] = (b - [1/T] I): [nabla]q - ([nabla]b - [m.sub.1][??]) x q - ([nabla] x B - [m.sub.2]Q) : Q + B x: [nabla]Q [greater than or equal to] 0. (21)

Here the number of the central dots denotes one, two, and three contractions of the first, second, and third order tensors, respectively. The first and the third terms are products of second order tensors, the second term is vectorial, and the last term is a product of third order tensors. Therefore, for isotropic materials, cross effects may appear only between the first and the third terms. Hence, in a one dimensional simplification, linear relations between the thermodynamic fluxes and forces are as follows:

[m.sub.1][??] - [[partial derivative].sub.x]b = -[l.sub.1]q, (22)

[m.sub.2][??] - [[partial derivative].sub.x]B = -[k.sub.1]Q + [k.sub.12][[partial derivative].sub.x] q, (23)

b - 1/T = -[k.sub.21] Q + [k.sub.2][[partial derivative].sub.x]q, (24)

B = [n.sub.3][[partial derivative].sub.x] Q, (25)

where dx denotes the one dimensional spatial derivative and the material coefficients [m.sub.1],[m.sub.2],[l.sub.1]} [k.sub.1],[k.sub.2], [k.sub.12], [k.sub.21], [n.sub.3] are subjects to thermodynamical constraints. It is straightforward to eliminate the current multipliers and the tensorial internal variable Q. Then one obtains the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

In our case, the balance of internal energy (4) is

[rho]cT + [[partial derivative].sub.x] q = 0, (27)

where [rho] is the density and c is the specific heat. The combination of (26) and (27) may be written in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

We can observe various time and space derivatives of the Fourier equation in different forms, plus the last term with the highest derivatives. The arrangement is space-time hierarchical, like in the previous section.

4.1. Hierarchical resonance: the example of the Guyer-Krumhansl equation

The hierarchical rearrangement of an evolution equation may help in recognizing solution patterns. In this subsection, we give a simple example with the help of the Guyer-Krumhansl equation.

The Guyer-Krumhansl equation is obtained when [n.sub.3] = [m.sub.2] = [k.sub.12] = 0 in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Here, [[tau].sub.q] = [m.sub.1]/[l.sub.1], [lambda] = 1/([rho][cl.sub.1]) and a = [k.sub.2]/[l.sub.1], and these coefficients are considered constant. This is a two-level hierarchical arrangement. If [[tau].sub.q] = 0, then the first term is zero, and the hierarchy is reduced to a single Fourier equation. If a = [[tau].sub.q][lambda], then there appears the same Fourier equation in both terms. This is the case of hierarchical resonance, and the solutions of the coupled set of equations may be identical to those of the single Fourier equation [15].

The resonance may help to classify the solutions. Let us introduce adiabatic boundary at the end of a rod and heat pulse boundary conditions at the front side in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [t.sub.p] is the duration of the pulse and [q.sub.max] is the maximum of the heat flux at the boundary. Initially, the temperature is uniform and there is no heat flux q(t = 0, x) = 0, T(t = 0, x) = [T.sub.0]. Then the solutions show characteristic differences depending on whether the parameters are above or below the resonance value. This is represented in Fig. 1, where temperature and time are the following dimensionless quantities: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here L is the length of the rod and [T.sub.end] is the asymptotic value of the temperature after the equilibration.

* If a = [[tau].sub.q][lambda], then we obtain the solution of the Fourier equation. This is the solid line in Fig. 1.

* If a < [[tau].sub.q][lambda], then we obtain solutions where temperature starts to increase later than in the Fourier solution. For short rods, the heat pulse is observable. The solution is zero at the beginning and a visible heat pulse appears similarly to the solutions of the Maxwell-Cattaneo-Vernotte equation. This is the dashed line in Fig. 1.

* If a > [[tau].sub.q][lambda], then temperature starts to increase earlier than for the Fourier solution. The remnants of the heat pulse are not observable, and sometimes there is a change in the steepness of the solution, a kink. The solution is more damped than the Fourier one. This is the dashed-dotted line in Fig. 1.

[FIGURE 1 OMITTED]

5. SUMMARY

Eliminating internal variables in non-equilibrium thermodynamics results in a hierarchical structure of the evolution equation. The buiding block of the hierarchy is the evolution equation of the original theory, which was supplemented by the internal variable.

The solution of the original equation may appear at different particular values of the parameters. If this happens with more than one nonzero element of the hierarchy, we can call it hierarchical resonance.

ACKNOWLEDGEMENT

The work was supported by grants OTKA K81161 and K104260.

REFERENCES

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Peter Van (a, b,c) *, Robert Kovacs (a,b,c), and Tamas Fulop (b,c)

(a) Department of Theoretical Physics, Wigner Research Centre for Physics, Institute for Particle and Nuclear Physics, PO Box 49, 1525 Budapest, Hungary

(b) Department of Energy Engineering, Budapest University of Technology and Economics, POBox91, 1521 Budapest, Hungary

(c) Montavid Thermodynamic Research Group, Igmandi u. 26, 1112 Budapest, Hungary

Received 14 December 2014, accepted 30 March 2015, available online 28 August 2015

* Corresponding author, van.peter@wigner.mta.hu

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Title Annotation: | PHYSICS |
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Author: | Van, Peter; Kovacs, Robert; Fulop, Tamas |

Publication: | Proceedings of the Estonian Academy of Sciences |

Article Type: | Report |

Date: | Sep 1, 2015 |

Words: | 2943 |

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