D. FEKETE: Application of Gyarmati’s Wave Theory of Thermodynamics 161

phys. stat. sol. (b) 105, 161 (1981)

Subject classification: 8; 6

Central Research Institute for Chemistry, Hungarian Academy of Sciences, Budapest1)

A Systematic Application of Gyarmati’s Wave Theory of Thermodynamics to Thermal Waves in Solids

BY D. FEKETE

An analysis is made of the approximate character of Fourier’s heat conduction equation with respect to the theory of irreversible thermodynamics and with the various generalizations of the linear theory. The different explicit forms of the thermal wave equations in solids are deduced on the basis of the wave theory of thermodynamics developed by Gyarmati and the generalizations of the linear wave equations to quasilinear equations are treated. Finally, the derived equations are compared with the equations of the phonon dynamical theory and an analysis of the experi- mental situation is given.

Eine Analyse der Niiherungseigenschaften der Fourierschen Wkrmeleitungsgleichung in Hinsicht auf die Theorie der irreversiblen Thermodynamik und die verschiedenen Verallgemeinerungen der linearen Theorie wird durchgefiihrt. Die verschiedenen expliziten Formen der Warmewellen - gleichungen in Festkorpern werden auf der Grundlage der Gyarmatischen thermodynamischen Wellentheorie abgeleitet, und es wird die Verallgemeinerung der linearen Wellengleichungen auf quasilineare Gleichungen behandelt. Endlich werden diese Witrmewellengleichungen mit jener Gleichung verglichen, die in der dynamischen Phonon-Theorie abgeleitet wird. Eine Analyse der experimentellen Situation wird gegeben.

1. Introduction Let us consider the equation of energy conservation for solids

au at eo- + v * I = 0 ,

where eo is the constant density, u the specific internal energy, and I the heat current [l to 31. One can transform this equation into the form of the transport equation of heat conduction by using two different types of constitutive equations. One is the partial time derivative of the caloric equation of state u = u(T) , i.e.

a% aT _ - at - cw at 3

whereas the other is Fourier’s linear law I = -AgT

which is a simple example for the linear kinematical constitutive equation of the Onsager type [ 1,2]. Since, in general, the specific heat c,, and the heat conductivity A are functions of the absolute temperature T , with (l.l), (1.2), and (1.3), we can write

l) Pusztaszeri ut 59 to 67, Budapest, Hungary.

11 physioa (b) 105/1

162 D. FEKETE

or, more explicitly,

where v2 = is the Laplace operator. This equation is a quasilinear partial differential equation due to two quite different reasons [3 to 51. The first reason is the temperature dependence of cv( T ) , which is a consequence of the adequate form of the caloric equa- tion of state, i.e. this type of quasilinearity has a thermostatic origin. The second rea- son is the temperature dependence of A( T ) , i.e. this type of quasilinearity has a thermo- dynamic origin. Sharp distinction must be made between these two types of quasi- linearity in the quasilinear approach of irreversible thermodynamics [S to 101 and in the quasilinear form of the Gyarmati theory of thermodynamical waves [4]. If both c, = Zv and h = x are approximately constant, we have the ordinary linear heat trans- port equation of Fourier type

which is a special representative of Onsager’s strictly linear theory [l to 31. As Gyarmati has point,ed out the linear theory and similarly the quasilinear ap-

proach of irreversible thermodynamics may be generalized into two directions [4]. First there is a possibility to extend it to the domain of the strictly nonlinear theories, among them the oldest one is the Gyarmati-Li theory [I1 to 171. This theory leads in the special case of heat conduction in solids to the nonlinear equation [4, 61

8T eocv at - v * [4T) +I‘(T) (VTj2 + ... I v r = 0 Y (1.7)

where ?L( T ) + A’( T ) ( v T ) 2 is the heat conductivity coefficient that depends on T

The second possibility to generalize (1.5) and (1.6) is to supplement them in such a way that they can describe thermal wave phenomena with finite speed. Confining ourselves to the linear approximation sucK wave equation could be

V T , too.

where & is the heat diffusivity coefficient. This quantity is a material constant in the linear theory and is a positive and constant relaxation time very small on the time scales of most practical problems. It would be appropriate to mention that Cattaneo [18] and Vernotte [19] were the pioneers to modify the linear equation (1.6) so as to describe thermal waves (second sound) in solids with (1.8). Their attempts have been based only on an ad hoc generalization of Fourier’s linear law (1.3), by supposing t,he validity of the following type of constitutive equation:

(1.9)

However, if we neglect some unsuccessful attempts [20 to 231 it can be said that the exact and complete theory of thermodynamical wave phenomena was developed by Gyarmati [4]. He derived the total set of constitutive equations of Cattaneo-Vernotte type from the original Onsager theory. In his approach it was assumed that the local entropy depends simultaneously on the equilibrium type “a” parameters and velocity

Application of Gyarmati's Wave Theory of Thermodynamics in Solids 163

type "B" parameters as well. The fl-parameters are represented in Gyarmati's theory by the currents of the purely dissipative transport phenomena and the derived constitu- tive equations are

(1.10)

where I( are the currents and g r , the conjugated thermodynamic forces. By using (1.10) a complete set of the wave equations of thermodynamics

(1.11)

was obtained. As special examples the telegraph equations of electrodynamics, the wave equations of diffusion phenomena, and the equation of temperature wave were mentioned in the original paper [4]. The latter was given as

+ 0 1 PT-1 1 i3T-1 -- h2 at2 +sat (1.12)

which represents a form in the so-called entropy picture [4, 91. I n (1.12) & is the finite speed of the propagation of thermal wave, the relaxation time and both are con- sidered in the original work as constants.

Gyarmati's wave equations only correspond to the linear theory of thermodynamics. As a consequence of experimental situations it is necessary to extend the linear wave theory to quasilinear realm when the material coefficients (w, z, etc.) are not constant but state-dependent (temperature-dependent etc.) quantities. On the other hand, when considering experiments the form of the thermal wave equation in the entropy picture (1.12) is not the most adequate one. Anyway, it seems to be useful to generalize the linear theory and to give the most important representations of thermal waves in different pictures.

2. On the Thermodynamic Treatment of Heat Conduction in Different Pictures

Let us consider an isotropic solid body of volume V in which the conductive transport of internal energy is the only possible process. In this case we can characterize the reversible state variation of the system by the special form of the Gibbs relation,

d U = T M . (2.1)

Here T is the absolute temperature, d U and dS are the infinitesimal equilibrium varia- tions of internal energy U and entropy S, respectively. In the contemporary thermo- dynamic theory of heat conduction [l, 91 the following transformation:

r = P(T) (2.2) h

defines a temperature scale in the r-picture. Supposing that the function r is con- tinuously differentiable, and its inverse

A

T = Pl(I')

exists, we have

> o af( T) dT

d r ( T ) ,

or ___ ( 0 d T

(2.3)s

(2.4)

11 *

164 D. FEKETE

corresponding to the strictly monotonic property of .f' in the whole interval 0 < T < < 00. The most frequently used scale transformations are the so-called Fourier picture F(T) = T, the energy picture r * ( T ) = In T, and the entropy picture r * * ( T ) = T-l [l, 91.

It is well known that in local equilibrium state of solids the specific entropy s is determined only by the specific internal energy u as

s(r, t ) = S [ U ( T , Ql from which we have the specific form of the Gibbs relation

Let, u s give now the ordinary definition of specific heat at constant volume

which is only a special form of the following most general definition referring to an arbitrary r-picture :

Clearly, the relation between the ordinary specific heat c,, and the quantity cf referring to the r-picture is

(2.9)

where y is the scale fact,or defined by Parkas [9]. With the following special selection of this quantity,

dr(T) - dT Fourier picture y = ___ - .- -

dT - d T V 1 '

dr*(T) - d In T - T-l )

energy picture y * E ___ - __ - dT d T

dr**(T) dT-l entropy picture y * * = __ = - T-2 dT dT

we liave the relations C, , C: = Tc, ) c:* = - T

(2.10)

(2.11)

among the various specific heat.s, where c,* and ct* are the specific heat defined in energy and entropy picture, respect,ively.

Let us consider the partial t'ime derivative of the Gibbs relation (2.6)

(2.12)

(2.13)

and t.he partial time derivative of (2.8) au a r - = 6,' - at at *

Application of Gyarmati’s Wave Theory of Thermodynamics in Solids 165

Now, the most important special pictures are : au aT

Fourier picture - - at - at ’

- CVT , au * 31n T at - cv ~ - at

energy picture - -

8T-1 -cVTz ___ - a U aT-1 at

entropy picture - = cf* - at at

(2.14)

from which one can infer that the time derivatives of the specific internal energy aulat remain invariant under scale transformations. I n the classical thermodynamic theory of heat conduction t,he use of the first time derivatives is sufficient for the description of the heat transport phenomena. However, in case of wave phenomena the use of the second time derivatives is also necessary. Accordingly, from (2.13) one has

and the most important pictures are :

- a2u Fourier picture ~ - c, at2

a 2 u a2 In T energy picture

~ at2 = c: -.

(2.15)

(2.16)

These relations refer to the general cases when the specific heats c:, c,, cz, ct * are not constant. I n the special case when anyone of these is a constant quantity, the second term of the adequate expression vanishes. Unfortunately, the temperature dependence of specific heats is, at least in general, so complicated that the second term cannot be eliminated from none of the special pictures. Nevertheless, such a reduction can always be carried out by means of the general r-picture (2.15)) i.e. in the case of any tem- perature dependence of heat capacity such a general r-picture can be selected that the reduced expression

(2.17)

is valid. Knowing the special form of the internal energy balance equation [l, 21

au eo- + v . I = 0; v I = div I (2.18) at

and the time derivative of the Gibbs relation one has the entropy balance as

e o ~ + v - I * = o (2.19)

with I, = IT-I and 0 = I - V T - I Z 0 . (2.20)

166 D. FEKETE

Here I , is the entropy current density, a the non-negative entzopy production accord- ing to the second law. Now, it was already shown [l, 91 that it is possible to transform the original Fourier linear law (1.3) into different pictures by postulate of the invarian- ce of heat current with respect the scale transformations. Indeed, we can write

I = -2 VZ = -L* v In T = L** vT- l= Lr v r = LrXr (2.21) from which we can infer that the thermodynamic force X r = v r given in the r-pic- ture has the following special representations :

(2.22) Fourier picture X = -vT , energy picture X* = -v In T , entropy picture X** = vT- l ,

whereas between the coefficients the relations A = T-lL* = -27-2L** (2.23)

are valid. Of course, the entropy product'ion belonging to the heat conduction can also be given in different pictures as

(2.24)

Let us turn now to the construction of the heat transport equation. By using (2.13) and the last expression of (2.21), from (2.18) one has

(2.25)

which is the most general form of the Fourier equation in the generalized r-picture. Evidently, the most important part.icular representations of it are :

ar eocf at + v . L r v r = o

(2.26)

(2.27)

aT at 8 In T

Fourier picture

energypicture

eocr - - v - 2 BT = 0 ,

e0$ ___ - v - L* v In T = 0 , at

$. * L*" 7 T - l ~ 0 . (2.28) 8 T-l

ent,ropy pict,ure eOc:* - at

It frequently happens in applications that the solid system is relatively near to its equilibrium state. In t,his case, for example, we can find that A = h is a material con- stant at least for certain materials. Now, from (2.26) we have the common form

(2.29)

of the linear heat conduction equation, where x is the thermal diffusivity coefficient in the Fourier picture. If we can use the Fourier equation with A = const, we speak about t,he applicability of the classical linear theory [3].

Suppose now that experimentally we find that i! depends on temperature linearly A(T) = aT or in a quadratic way A(T) = bTz in a certain interval To < T < TI, In such cases due to the relations (2.23) we can use instead of (2.26) the equations

(2.30)

Application of Gyarmati’s Wave Theory of Thermodynamics in Solids 167

(2.31)

because now the coefficients i* or i** are constant. Of course, in a more complicated temperature dependence of A( T ) we can use a general scale transformation by which the coefficient Lr becomes a material constant. In this case from (2.25) we have

(2.32)

which is again a linear equation. Thus, we can conclude that in any case of quasi- linearity of the heat conduction equation we can reduce our problems to the solution of linear equations by a selection of the adequate r-picture [l, 91.

We should like to mention an important property of the thermal diffusivity x . In general cases the coefficients Lr, cf, 1, c,,, L*, c:, L**, c:* depend in a complicated way on temperature, and the thermal diffusivity is also a function of temperature. However, x is a picture-invariant quantity, since the relations

(2.33)

are valid. This means that if x is a constant quantity in any of the special pictures, this remains constant for all pictures. Similarly if x depends on temperature, for example in the Fourier picture as x = A/pocw = b‘T2, this temperature dependence of x remains valid for all the other pictures, too, due to the picture-invariant property of x.

3. Deduction of Constitutive Equations for Thermal Waves in Different Pictures

Following the method of Gyarmati [4] we assume that the entropy density is simul- taneously a function of the internal energy density and its conductive current density, i.e.

This assumption means that the entropy density has two different parts. One part is the equilibrium entropy density seq(u), while the second skin(I) represents the kinetic part of the local entropy. Of course, by this assumption the validity of the postulate of local equilibrium is violated because I is a typical /?-parameter in Casimir’s sense. To write the assumption of (3.1) more explicitly we can write [4]

= S ( U ; 1) = Seq(U) + Skin(z) . (3.1)

s = seq(u) + &in(I) = Se,(U) + ;m**12, (3.2)

where m** is a non-positive quantity, the negative of which may be called thermal inductivity and refers to the entropy picture. Considering the partial time derivative of the equilibrium part seq(u) given by (2.12) the partial time derivative of (3.2) will be

From this with (2.18) the balance equation of entropy

(3.3)

168 D. PEKETE

is obtained. Here the entropy production u is expressed in terms of the generalized (**-force given in the entropy picture (see [4]), i.e.

Of course, X** = 7T- l is the ordinary dissipative force, while aY**/at is the partial time derivative of the inertial force Y** determined by the kinetic part of the entropy as

= m**z . (3.6) az askin y** =I __

It is easy to determine the generalized thermodynamic forces in various pictures. Since, in the generalized r-picture we can write

thus evidently ay* 8Z - and ~ * = X * + - = - ~ l n T + m * - - X f - = -gT f rn " aY .-

at at at at

(3.8)

(3.9)

ere the generalized forces in Fourier picture and energy picture, respectively, where m = Tm* = Tam**

are the relations among the thermal inductivities of various pictures.

quasilinear approximation as Let us now write the constitutive equation in the generalized r-picture and in

aI I = L r ( r ) tr = L r ( r ) = Lr(r ) Xr+ Lr(r ) mr at . (3.10)

By introducing t,he non-negative picture-invariant quantity

instead of (3.10), we have

t = -Lr ( r ) mr = -Am = -L*m* = L * * ~ * * (3.11)

(3.12)

which is an alternative form of (3.10). It is clear from (3.12) that z is a time of relaxa- tion of the thermal inertial phenomena, furtheron that the particular pictures of constitutive equations are:

8Z I = LrXr - z - at

a i at

Fourier picture Z = L = --I. g T - z - ,

energy picture I = L* X* + = -AT v In T - z - , (3.13) ( at at

entropy picture Z = L** at

It is remarkable that the first equation was used by Cattaneo [18] and Vernotte [19] in the ad hoc construction of t,he equation of a temperature wave.

Application of Gyarmati's Wave Theory of Thermodynamics in Solids

4. Deduction of Temperature Wave Equations in Quasilinear Approximation and in Different Pictures

169

Let us now derive the temperature wave equations in different pictures on the basis of the general Gyarmati theory of the thermodynamic waves. First we formulate our problem in the generalized r-picture.

The time derivative of the balance equation of internal energy (2.18) will be a z u ar eo -@ + v * - at = 0 , (4.1)

since eo is const.ant in solids. Multiplying this equation by rand adding (2.18), we have as, au ar

@,Z -@ + e0 at at + z V - - + 7. I = 0 : Now, from this equation by means of the divergence relation

the quasilinear equation

is attained. By eliminating the first and second time derivatives of internal energy with (2.13) and (2.15), we have

(4.5) which is the most general equation of temperature waves in solids. Of course, in general it is impossible to select a generalized r-picture so that all the quantites cf, Lr, and z are simultaneously constants if the state of the solid is relatively far from the equilib- rium state. It is clear from the various special pictures which are:

Fourier picture

az -~ an(T) (VT)2 - A(T) AT - Vz(T) .at = 0 , aT

entropy picture

(4.8)

170 D. FEKETE

that at least in general the quantites cv, A, z, and similarly c,*, L*, z, etc. are not simul- taneously constant quantities for various materials. Of course, there are practical cases when we can select such a general r-picture in which the quantities cur, Lr, z are constants. If these conditions are fulfilled, from (4.5) we have

which is the linear temperature wave equation in the generalized picture. Evidently, with the more special conditions cv = 8,, A = i, z = from (4.6) we have

(4.10)

which is the more common form of the linear temperature wave equation. Of course, the constancy of c,,, I , and z is frequently not fulfilled under given experi-

mental conditions. However, sometimes the constancy of c,*, L*, and z or c$*, L**, and z is guaranteed by the experiments. Hence, sometimes the use of linear equations

a2 In T a In T e0+c: ~ + eoc: 7 - L" In 27 = 0

at2

or

(4.11)

(4.12)

is more fruitful than the use of (4.10). The last equation was first derived by Gyarmati, who preferred to use the entropy picture, since this picture is the most adequate one in theoretical works [4]. If we as usual define the speed of the temperature wave phenomenon in the r-picture by

(4.13)

we can see that similarly to x and z, w is also an objective picture-invariant quantity. This fact is evident from the expression

(4.14)

which may be written alternatively by (3.11) as

(4.16)

where rnr, m, m*, and m** are negat,ive quantities by definition. Substituting the finite speed of a thermal wave int,o (4.5), we have

1 - 1 1 - - 1 w2 = -~

eOc:mr eocvm eoc:m* eoc:*m** '

from which, in the linear case, when cf, Lr, and z are constant, the equat'ion

(4.16)

(4.17)

Application of Gyarmati’s Wave Theory of Thermodynamics in Solids 171

follows. We would like to emphasize that in (4.17) only the picture-invariant quantities z and w are present, which are now constant quantities. Of course, this condition is satisfied only in one well-selected picture. I n more general cases, when the quantities cf, Lr are not constants neither in the generalized T-picture nor in some particular picture the temperature dependence of z and w may be so complicated that none of the linear equations (4.9) to (4.12) is adequate for the description of thermal wave phenomena. I n such cases one of the quasilinear equations (4.5) to (4.8) must be used.

Now, let us suppose for a moment that the strictly nonlinear transport equation mentioned a t the very beginning in (1.7) is experimentally verifiable. It is evident that the transport equation (1.7) is a consequence of

(4.18)

the strictly nonlinear constitutive equation which is a very special example of the Gyarmati-Li nonlinear thermodynamic theory [ 1, 61. Now, assuming the validity of (4.18) together with (4.18) the wave theory of Gyarmati leads t o the constitutive equation

I = --h(T) V T - L‘(T) (VT)’vT + ... ,

(4.19) ar at I = -A(T) V T - -h’(T) (VT)’VT - z (T ) -

instead of (3.13). Hence, instead of (4.6) we have

ar at

zv.-- ( vT)2 - A(T) AT -vz(T) . -- az -- 8T at

We emphasize that this nonlinear equation is adequate only to the first higher-order approximation of the Gyarmati-Li nonlinear theory. However, we can infer from (4.20) that the experimental verification of the “wave terms” is perturbed by the last two “strictly nonlinear terms”. Hence, if the existence of the strictly nonlinear heat conduction is a real problem (?), we experimentally should try to separate it from the wave phenomena, the experimental verification of which is very difficult. Thus, we should like to fix here that the more complicated nonlinear theories of temperature waves which are proposed independently of Onsager’s theory are absolutely illusory in real physical respects [20 to 221.

5. Comparison of the Ctyarmati Theory with the Phonon Hydrodynamics Theories and Experiments

It is self-evident that the Gyarmati theory of thermodynamic waves is a direct exten- sion of the experimentally well-proved Onsager theory to the domain of wave phenome- na. This theory is based on the macroscopic assumption (3.2) that the currents contri- but,e appreciably to the local entropy. With this single assumption the Gyarmati wave theory is self-consistent. However, in general, although it remains desirable to preserve this part of thermodynamics as a self-contained branch of macroscopic discipline, it is nevertheless interesting to recast Gyarmati’s wave theory so that its basic postulate is most directly related to its ultimate microscopic foundation. Indeed, from a microscopic point of view, a temperature wave (second sound) is represented

172 D. FEKETB

by oscillations of the density of thermal excitations and can be interpreted as ordinary sound in a gas of phonons. The corresponding kinetic theory is based on the Boltz- mann-Peierls equat,ion and in this theory two main cases can be distinguished. I n the first case we assume that the momentum of quasiparticles is conserved (normal pro- cesses). For this case we have

where p = ( k T ) - l , k is the Boltzmann constant and CII the speed of second sound. Here the first formula is due to the kinetic theory of phonon hydrodynamics in solids [24], and the second is its thermodynamic alternative. We can conclude here that in the kinetic theory the entropy picture is used as a consequence of the use of ,3 = = (kT)- l variables, further that the second form above is equivalent with the special form of the linear wave equation of Gyarmati given in the entropy picture by (4.12). Of course, (5.1) refers t o an ideal situation, when the momentum of quasiparticles is conserved. This is the reason that (5.1) exhibits the archetype of the hyperbolic partial differential equation.

When momentum destroying (umklapp) processes cannot be neglected, from the kinetic theory follows (see [25], p. 145) that

where zE is the relaxation time for the resistive scattering of phonons under umklapp processes. Evidently, this equation is given in the Fourier picture and is absolutely identical to (4.10), supposing that the phenomenological z is equal to zE of the kinetic theory.

Let us consider a more general case, when the phenomenological z is a combination of the relaxation times of zI1 and zE, where z11 is characteristic of momentum con- serving normal processes. As usual we suppose that

z-1 = zE1 + & , (5.3)

then we cite here the equation

(5.4)

deduced from the theory of phonon hydrodynamics [24]. Let us translate this equa- tion into the language of thermodynamics. Evidently

since in (5.4) by the dot partial time derivatives are denoted. We wish to present here the consistency of this equation with some particular form of t,he Gyarmati quasilinear theory.

For consistency, let us start from the wave equation

Application of Gyarmati’s Wave Theory of Thermodynamics in Solids 173

which follows from (4.8) assuming that L** and z are constants. By introducing the finite speed of second sound with (4.14) we have

Now, if we suppose that in a certain domain of the state of crystals the condition zR > ZII is fulfilled (consequently z zz zII) and, at the same time, we approximate I as in the classical dissipative theory of heat conduction by I = L** DT-l, we can transform the last term of (5.7) as

Here we assumed as above that L** is constant. Considering that $* = -T%, and L** = -Tail and using (5 .8) from (5.7)) we have

which equation is nearly identical to (5.5) or (5.4) derived by the kinetic theory, since w2 = $1. For complete agreement perhaps the analysis of the zR =ZII = 22 condition or others would be more favourable, but in our opinion further analysis is not important because of the rather approximate character of the hydrodynamic equations of phonon yield (5.4).

Although it is impossible to enter into the details of the experimental works [24 t o 281 the existence of thermal waves in solids is experimentally proved. However, concerning the theoretical interpretation of the data it should be mentioned that the most serious problem seems to be the interpretation of the discrepancy between the different temperature dependences of z. Indeed, the experiments show temperature dependences of z (more exactly zII) for different materials from z - T-8 to T-5. On the other hand, the theoretical investigations all lead to the result z - T-6. Sometimes, it is explicitly emphasized [24] that the linearized form of the phonon transport equation is not adequate to describe exactly thermal wave phenomena. We do not want to criticize the theory of phonon hydrodynamics, but it is very surprising to think that linear equations were proposed to describe such a type thermal wave phenomena, for which experiments have proved long ago the temperature dependence of heat capacity, heat conductivity, and even relaxation time and speed of second sound.

Finally, we should like to mention an important representation problem. It is clear that without the conlplete quasilinear Gyarmati wave theory i t is not possible to treat the problems in different pictures. On the other hand, we learnt from the comparison of Gyarmati’s theory with the theory of phonon hydrodynamics that in the latter theory the entropy picture is used. At the same time, the measured data probably referred to quantities given in the Fourier picture. Now, we show that this very simple fact can be a source of serious discrepancies.

For illustration only, let us suppose that an experimentator in measurements in the Fourier picture found that

3 = 21T-3 and m = m, = const, (5.10) i.e. the ordinary heat conductivity depends on temperature as Ta8. Now, on the basis of the definition of z we can write this quantity in Fourier and entropy pictures a8

(5.11) By siibstituting the functions (5.10) into the expressions of (5.11), one has

(5.12)

z = -Am. = L**m**;

2(T) = -Am = L**m** = -1 1 T-3 172.1,

m = T2m**, I = --T-BL**.

174 D. FERETE : Application of Gyarmati’s Wave Theory of Thermodynamics

since z is picture invariant. Of course, this result is a trivial one. However, we suppose now that experimentators are not familiar with the picture representations of the thermodynamic theory. On the other hand, the theorist writes his equations in the entropy picture, which was made in the theory of phonon hydrodynamics. He thinks without hesitation that 3, is the heat conductivity and z is a “relaxation time” defined as

(5.13) However, z,ixed is wrong, but the difference between the temperature dependence of z given in (5.12) andtmixed is the factor T-2, which is just the problematic factor with respect t o the agreement of theory and experiment. Of course, we do not make the statement here that the inconsistency of the kinetic theories and the experiments is a consequence of the obscure representation problems presented above, but it is doubtless that the clear basis of Gyarmati’s thermodynamic theory may contribute to the clarification of situations.

The results reported here were obtained in the course of research supported by Central Besearch Institute of Chemistry of the Hungarian Academy of Sciences, Budapest. The author is deeply indebted to Prof. Dr. S. Lengyel and Dr. D. K. Bhattacharaya for their helpful criticism of a previous version of this manuscript.

tmixed(T) = -Am** = -A 1 T-3mT-2 = -AlmlT-5.

Acknowledgements

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(Received December 18, 1980)