Continuum Mech. Thermodyn. (2014) 26:447–463DOI 10.1007/s00161-013-0313-x

ORIGINAL ARTICLE

Cs. Mészáros · I. Kirschner · Á. Bálint

Relevance of the time–quasi-polynomials in the classic linearthermodynamic theory of coupled transport processes

Received: 11 March 2013 / Accepted: 10 July 2013 / Published online: 25 July 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract A general description of the basic system of ordinary differential equations of coupled transportprocesses is given within framework of a linear approximation and treated by tools of matrix analysis andgroup representation theory. It is shown that the technique of hyperdyads directly generalizes the method ofsimple dyadic decomposition of operators used earlier in the classical linear irreversible thermodynamics andleads to possible new applications of the concept of quasi-polynomials at descriptions of coupled transportprocesses.

Keywords Coupled transport processes · Non-equilibrium thermodynamics · Matrix analysis ·Group representation theory

1 Introduction

Since the non-equilibrium thermodynamics plays a decisive role [1,2] in accurate description of the genuine(usually: coupled) transport processes taking place in macroscopic dissipative continua, the refined mathemat-ical modeling of such phenomena represents a permanent objective of the contemporary mathematical physicstoo. Namely, many transport processes also show features of the anomalous diffusion [3] of non-local charac-ter and with memory effects on macroscopic level (corresponding usually to the percolative fractal characteron the mesoscopic level), and require, e.g., use of the Riemann–Liouville operators, leading to a widespreadapplication of the fractional partial derivatives with respect to time and spatial coordinates, as well [4,5]. Therelevant mathematical features have also been analyzed in detail in studies related to general problems offractional diffusion, e.g., [6–9] as well as in [10,11] where the crucial importance of applying of the theory ofLie groups in this research field has been pointed out.

The classic domains of the extended irreversible thermodynamics (EIT) have also many open fundamentalquestions even in some earlier developed areas of them. It is enough to mention the archetypal problem of

Communicated by Andreas Öchsner.

Cs. MészárosDepartment of Physics and Process Control, Faculty of Mechanical Engineering,Szent István University, Páter K.u.1. 2103 Gödöllo, Hungary

I. KirschnerInstitute of Physics, Eötvös University, 1117 Budapest, Hungary

Á. Bálint (B)Department of Chemistry and Biochemistry, Faculty of Agricultural and Environmental Sciences,Szent István University, Gödöllo, HungaryE-mail: balint.agnes@mkk.szie.hu

448 Cs. Mészáros et al.

infinitely large propagation velocity connected with the application of separate parabolic-type partial differ-ential equations (PDEs), to which the classic equations of Fourier and Fick belong, too. Since existence ofthe infinitely large propagation velocities contradicts to the basic principles of physics, this problem inducedintroduction of novel-type formalisms based on the existence of non-zero relaxation time constants, whichresulted in possibility to work with hyperbolic type PDEs instead of parabolic ones. One of the most generalvariants of these descriptions is the formalism of the wave approach of thermodynamics (WAT) elaborated byGyarmati [12] resulting in systems composed from coupled hyperbolic type PDEs.

First, we recall here a frequently used system of PDEs for mathematical modeling of drying processes,representing an archetype of coupled transport processes [13,14]:

∂M

∂t= ∇2a11 M + ∇2a12T + ∇2a13 P,

∂T

∂t= ∇2a21 M + ∇2a22T + ∇2a23 P, (1)

∂P

∂t= ∇2a31 M + ∇2a32T + ∇2a33 P,

where M is moisture content, T temperature, and P pressure. The system of PDEs describing coupled heatand mass transfer through capillary-porous bodies is usually based on application of the zone picture, i.e.,it is assumed that the whole porous bulk may be divided into thin enough layers (zones) inside which theconductivity and coupling coefficients (incorporated into matrix elements ai j , 1 ≤ i, j ≤ 3) have constantvalues.

Moreover, the system of equations (1), as well as its simplified variants, is usually written in “criterial” form,i.e., the time t is to be understood as the Fourier criterion (t → aT t

l2 , where aT denotes the heat conductivitycoefficient of the mixture and l is the characteristic length of the body being dried). The relevant systemsof coupled PDEs derived from (1) have been studied extensively in the literature of transport processes andpermanently represent a very active research object from the point of view of modeling of such phenomena[15–19]. For us, it is particularly important here that the matrix analysis methods have also been applied [14]at setting up of solution methods of the system (1), which we intend to use here in a very general form, basedon an effective use of the Jordan-type block matrices, too.

Therefore, within frame of the classical irreversible thermodynamics (CIT) [2], mathematical modelingof transport processes has been performed mainly in the linear approximation, and even this well-elaboratedbranch of thermodynamics still represents an open research area. Some simple symmetry assumptions aboutthese generalized systems (more precisely about their possible solution types) of the coupled heat and masstransfer processes through porous media led us [20,21] to possibility of introducing of Lommel-type specialfunctions previously not used in classic transport theories, and a direct, natural way for eliminating of theinfinitely large propagation velocity problem via taking into account the thermodynamic cross-effects.

Finally, we also point out in this introductory section that forms of the temperature and moisture-levelfunctions etc., are well-known and in the frequently studied one-dimensional case (for the concrete exampleof temperature function) are usually represented as [18]:

T (�r , t) ≡ T (x, y, z, t) → T (x, t) =∑

n

cnTn (x) · e−λnt , (2)

where the series expansion coefficients cn are constants, and quantity λn denotes the reciprocal value ofrelaxation time constant of the relevant nth expansion term.

2 Some earlier general applications of matrix analysis methods in the CIT

In order to summarize the most general features of the mathematical formalism of coupled transport processesthrough dissipative macroscopic continua (but still within linear approximation), we recall here briefly theresults of the theory based on classic results of the linear approximation of the CIT and refined later byKirschner et al. [22–24]. According to this theory, the basic thermodynamic equations of motion are given by

�ζ = L �X , (3)

Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory 449

and

�X = −g�ζ , (4)

where �X = �X (t) denotes the abstract vector-column of thermodynamic forces and �ζ = �ζ (t) is similarly therelevant vector-column composed from ζ—parameters (non-equilibrium deviations of extensive thermody-namic parameters from their equilibrium values), while g is the entropy matrix and L is the conductivity matrix,which beyond time-reversal symmetry must also reflect the material symmetries of the actual dissipative con-tinuum. Both matrices are of nth order quadratic type, positive definite and symmetric ones. Throughout thispaper, the symbols of type A will be used for general notation of matrices, while without operator notation(A), they will be assigned to their concrete representation corresponding to a given basis of the actual abstractvector space. Since we are studying changes in the vicinity of equilibrium states (where the entropy matrix is

composed from entirely constant elements gik = −(∂2ζs∂ζi ∂ζk

)

0= −

(∂2ζs∂ζk∂ζi

)

0= gki , 1 ≤ i, k ≤ F f e.g. [1]),

the total time derivative of (4) can be written in the form �X = −g �ζ . Then, in the most general case of thelinear approximation of the non-equilibrium thermodynamics, we arrive at the following systems of ordinarydifferential equations (ODEs) written again in matrix form

�ζ = −L g�ζ − �G ≡ − A+�ζ − �G, (5)

�X = −g L �X + �G ≡ − A �X + �G, (6)

where by the symbol “+”—as usual—the hermitian conjugation operation is denoted and �G = �G (t) representsthe abstract vector-column of the externally generated thermodynamic forces. Therefore, we obtain directlyseparated equations in the systems for both thermodynamic forces and extensive thermodynamic parameters.The solutions of these systems can immediately be given in the form

�ζ (t) = e− A+t · �ζ0 − e− A+t

t∫

0

eA+τ �G (τ )dτ, (7)

�X (t) = e− At �X0 − e− At

t∫

0

eAτ �G (τ ) dτ , (8)

where �X0 = �X (0) and �ζ0 = �ζ (0) denote the initial values of the vectors of thermodynamic forces andextensive thermodynamic parameters, respectively. It important to note [22–24] that when only one of thethermodynamic forces differs from its equilibrium value for a given system, it may induce non-equilibriumvalues for all the other forces, too. According to the basic properties of the systems of ODEs, each onefrom the thermodynamic forces may change its sign (F f − 1) times and approach its equilibrium valueasymptotically. The perturbations propagate through system via all possible types of interactions present in it[i.e., interaction types described by systems (3–4) and (5–6)]. Furthermore, since the matrix A = g L has onlypositive eigenvalues: λi > 0,

(1 ≤ i ≤ F f

), it is suitable to represent it in the below given form of the sum of

dyadic products

A =F f∑

i=1

λi �ψi ◦ �ϕ+i , (9)

where the basic vectors from{ �ψ1, . . . , �ψF f

}and

{ �ϕ1, . . . , �ϕF f

}are elements of a biorthogonal system:

�ψ+i · �ϕk = �ϕ+

k · �ψi = δik,(1 ≤ i, k ≤ F f

). Their dyadic products are of idempotent character, i.e.:

( �ψi ◦ �ϕ+i

)2 = �ψi

(�ϕ+

i�ψi

)�ϕ+

i ≡ �ψi E �ϕ+i = �ψi ◦ �ϕ+

i , 1 ≤ i ≤ F f (10)

450 Cs. Mészáros et al.

(E is the identity operator). Then, it is obvious that all matrix polynomials F(

A)

= ∑l cl Al ,can (with all

cl -s as constants) can also be represented as

F(

A)

=F f∑

i=1

F (λi ) �ψi ◦ �ϕ+i , (11)

leading directly to e− At = ∑F fi=1 e−λi t �ψi ◦ �ϕ+

i . In the case, when external generating forces are equal to zero,i.e., �G (t) = �0, we will obtain the following particular forms for the homogeneous variants of the solutions(5–6)

�ζ (t) =F f∑

i=1

ci e−λi t �φi ,

(ci = �ψ+

i · �ζ0

), (12)

and

�X (t) =F f∑

i=1

c∗i e−λi t �ψi ,

(c∗

i = �φ+i

�ζ0

). (13)

Therefore, the coefficients ci and c∗i can be calculated on the base of the general solution functions of type

(7–8) and expressions in the parentheses in (12) and (13).

3 Application of quasi-polynomials

3.1 Novel solutions of the systems of ODEs of the CIT

In this section, we give a proposal for further generalization of the above-discussed abstract formalism ofcoupled transport processes. Although this analysis will be again performed within frame of the strictly linearapproximation, it can trace out pathway for applications of some well-known and effective methods of thetheory of systems of ODEs. We start from the WAT formalism used by Gyarmati [12] in its original form.Accordingly, in the case of convection-free coupled transport processes in macroscopic dissipative continua, thesystems of differential equations related to intensive (Γ ) and specific extensive (a) thermodynamic quantitiescan be given concisely as

ρd ��dt

+(

s∇ · L∇) �� = s �σ,

ρd�adt

+(∇ · L s∇

)�a = �σ,

(14)

i.e., we arrange them into vectors, as well as the different contributions of the general entropy productionfunction into the vector �σ . For the sake of completeness and further forthcoming applications, we mention

here immediately that σ = (s)σ + (v)

σ + (a)σ + (t)

σ ≥ 0 (e.g. [2]) and that this inequality emanating from the secondlaw of thermodynamics must be fulfilled separately for each of the scalar-(s), polar-vector-(v), axial-vector-(a),and “genuine tensorial”-(t) type contributions to the entropy production function. Since within frame of CIT,all these components of σ can be separated one from each other on the base of the Curie principle, it is obviousthat group representation theory may play a more important role in the refinements of the general solutionsemanating from the systems of type (14), than it has been applied. The elements of the quadratic and symmetricmatrix s are defined by sik ≡ −gik , and in the strictly linear case, we have:

d ��dt

− κ0��� = 1

ρ0s0 �σ ,

(κ0 ≡ − 1

ρ0s0 L0

),

d�adt

− D0��a = 1

ρ0�σ, (D0 ≡ − 1

ρ0L0s0),

(15)

Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory 451

where�denotes the Laplace operator and the lower indices zero correspond to constant quantities. The matricesκ0 and D0 are related to each other [12] via similarity transformation κ0 = s0 D0s−1

0 . It is characteristic forsystems (14) and (15) that they contain total time derivatives, but the spatial derivatives are of partial character.For further purposes, it was suitable [25] to perform a Fourier transformation procedure for both systems (withrespect to the spatial coordinate) and to obtain the following abstract column vectors:

�� = �� (�r , t) → ��(�k, t

),

�a = �a (�r , t) → �a(�k, t

). (16)

As a novel result within CIT, it was possible to write down the solutions of these systems of ODEs bytaking into account the possibility of multiple eigenvalues of their characteristic equations [26–29]. Then, forthe inhomogeneous systems of ODEs �x (t) = A (t) �x (t) + �f (t) [symbolically unifying (15) and (16) intoODE systems explained by block matrices] obeying initial condition �x (t0) = �x0 and omitting the intermediatesteps at its solving [25,28] we arrive at:

�x (t; t0, �x0) =s∑

k=1

eλk (t−t0)Lk

(A)�x0 +

s∑

k=1

t∫

t0

eλk (t−τ)Lk

(A)

· �f (τ )dτ, (17)

in the case, when the minimal equation of the matrix A has only (“s” number of) single roots (Lk

(A)

denotes a

Lagrangian-type matrix-polynomial). In this case, we have a direct prescription of the formula (11) in the form

of F(

A)

= ∑sk=1 F (λk)Lk

(A)

≡ ∑sk=1

∑αkν=1 F (λk) Lk

(�ukν ◦ �vTkν

). In case, when the minimal equation

of the matrix A also has multiple solutions, we have

�x (t; t0, �x0) =s∑

k=1

eλk (t−t0)γk−1∑

υ=0

(t − t0)υ Hkυ

(A)

�x0

+s∑

k=1

t∫

t0

eλk (t−τ)γk−1∑

υ=0

(t − τ)υHkυ

(A)

· �f (τ )dτ, (18)

where Hkυ

(A)

denotes the Hermitian-type interpolation (or Lagrange–Sylvester-) matrix-polynomial

and γk is the multiplicity of the root λk in the minimal equation. In this case, we have F(

A)

=∑s

k=1∑γk−1ν=0 F (ν) (λk)Hkν

(A)

≡ ∑sk=1

∑γk−1ν=0 F (ν) (λk) Hkν

(�ukν ◦ �vTkν

), which may be considered as a

direct generalization of (11). In an earlier paper of ours [25], we demonstrated (including brief descriptionof a method allowing calculation of the necessary inverse Fourier-transformations) that in Lagrangian rep-resentation of continuum mechanics, application of (18) on the Fourier-transformed form of (15) provides anovel, effective method for general treatment of simultaneous convection–diffusion sytems in macroscopicdissipative continua.

3.2 A novel-type application of hyperdyadic decomposition of matrices

In order to justify and generalize further the above-discussed results explained by (17–18) at solving ofsystems of the type (5–6) in the case of absence of the forces of external generation, we recall here the factthat (by a suitable similarity transformation via adequate non-singular matrices) all quadratic matrices can betransformed into matrices with Jordan-type normal forms. The Jordan-type matrices are given by direct sumsof Jordan-blocks [28–31] i.e. J = J1 ⊕ · · · ⊕ Jq ⊕ · · · ⊕ Jp, where Jq = λq Eq + Nq , 1 ≤ q ≤ p, andλq denotes an eigenvalue of the matrix A, Eq is the unit matrix of order “q” corresponding to the identityoperator, while Nq is a nilpotent matrix of order uq . It is important to note here that—due to abstract vectorsubspaces of cyclic character (carrying the separate Jordan-blocks)—dimensions of submatrices on the main

452 Cs. Mészáros et al.

diagonal are strictly determined [30]. Moreover, if a similarity transformation between matrices A and J is

realized by use of the non-singular matrix T , i.e., A = T −1 J T , for a matrix function f(

A)

, we will have

f(

A)

= T −1[

f(

J1

)⊕ · · · ⊕ f

(Jq

)⊕ · · · ⊕ f

(Jp

)]T ). Particularly, it can be shown [28,32] that for

operator functions given in general form eAt , all matrix elements on the main diagonal are equal to eλt [λ isan eigenvalue emanating from the basic equation A �X (t) = λ �X (t) (in which is suitable to use the form of�X (t) = eAt �d; �d = const.)], all matrix elements below it are zeros, while all the other matrix elements aregiven as products of eλt with polynomial terms tk

k! , (0 ≤ k ≤ n − 1) of the relevant order. In the case of multipleeigenvalues, we must apply these Jordan’s canonical forms of the matrices, the so-called Jordan chains, on thebase of which (for a given eigenvalue λ with multiplicity “p”) the following basic formula is valid:

eAt �ds = eλt(

�ds + t

1! �ds−1 + · · · + t s−1

(s − 1)! �d1

), (1 ≤ s ≤ p). (19)

Then, (at least in the case of finite-dimensional abstract vector spaces applied here, too) the complete eigenvalueproblem being investigated may be decomposed and reduced to simultaneous application of several Jordanchains.

According to our assumptions on the eigenvalues of A, basis functions of the abstract vector spaces offunctions leading to representation in the form of Jordan-blocks are realized by quasi-polynomials, which ingeneral case have the form [33]:

ϕ(t) =k∑

l=1

eλl tνl−1∑

m=0

cl,m · tm, cl,m = const., (20)

where νl denote the algebraic multiplicities of different eigenvalues λ1, λ2, . . . , λl , . . . , λk in the eigenvalueproblem being investigated.

3.3 Symmetry consequences

The time-dependent part of the solution form (2) widely used in the literature may be taken as a specialcase of (20), where νl denotes the multiplicity of the eigenvalue λl . For a function t �→ ϕ (t),the expression(20) gives possibility to connect the linear thermodynamic formalism demonstrated in this paper by formulae(5–6) to other well-established descriptions of the classical irreversible thermodynamics, e.g., [1]. Namely,according to this formalism, the coupling between thermodynamic forces and fluxes of different character andtensorial order at point group symmetry transformations of their Cartesian components is restricted: From allcouplings, only those with identical tensorial order are allowed according to the Curie principle. In order tosimplify the forthcoming analyses, we also recall here the abstract group theoretical formulation of the principleand consider complex physical systems, whose subsystems are characterized by particular symmetry groups.Accordingly, if subsystems of the given complete system do not affect each other significantly, the symmetryof the whole system is given as a direct product of symmetry groups of the subsystems. If, however, thereexists a significant interaction between subsystems, the symmetry group of the whole system is intersectionbetween symmetry groups of the interaction and those of the subsystems. Often, this intersection is restricted tosymmetry groups of the subsystems only, because the whole system remains invariant in the case of coincidentaction of common symmetries.

Thus, the Curie principle makes possible a direct generalization of mathematical formalism of the classicalirreversible thermodynamics. Since the actual thermodynamic systems being examined are characterized byinvariance property with respect to some symmetry transformations (e.g., if we consider the basic symmetries ofthe macroscopic order parameter realized via magnetization- and electric polarization vectors, stress- and defor-mation tensors, electric conductivity tensors, etc.). In crystals, the relevant system is simultaneously invariant[34] to the actual crystallographic point group-, and/or to the time-reversal symmetry transformations, and therelevant results of the group representation theory may be directly applied here. Due to the product structure A =g L of the matrix A, we can formulate a result of the present investigation in the following general form: sincethe tensor L can be decomposed into irreducible blocks (according to the actual symmetry group of the given

Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory 453

dissipative system), and the matrix g can be transformed into direct sum of matrices having Jordan-type normalforms, the matrix A can be represented as a simple product of two matrices having block-diagonal forms, i.e.,

A = g L =[

J g1 ⊕ · · · ⊕ J g

q ⊕ · · · ⊕ J gp

] [L(1) ⊕ · · · ⊕ L(μ) ⊕ · · · ⊕ L(μ f )

]

≡

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

J g1 0 0 . 0

0 . . . 0

0 . J gq . 0

. . . . .

0 0 0 . J gp

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

L(1) . 0 . 0

0 . . . 0

0 . L(μ) . 0

. . . . .

0 0 0 . L(μ f )

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(21)

Moreover, the transformation procedures leading to quasi-diagonal forms may result in multiple appearanceof the same blocks. Analogous statement will be valid for the block-diagonal decomposition of the entropymatrix realized via Jordan-blocks, too. The symbolμ f indicates the last non-equivalent irreducible componentof the tensor representation of symmetry group GS of the conductivity tensor. Having collected all these basicformulae, we are now in position to generalize the functions (2) and (20). In order to ensure a perfect fitting ofdimensions of the block matrices appearing in the product decomposition (21), we emphasize here that if wefirstly apply Schur’s lemma on the reduced form of the conductivity matrix [see below Eq. (23) in this respect],this operation makes possible transformation of the entropy matrix to the required form with adequate arrange-

ment and direct sum of blocks with dimensions identical to dimensions of the relevant block matrices(

L(μ))

.

At first, since the symmetry theory plays a fundamental role in the above-proposed generalization procedure,we base the introduction of the spatial dependence of the coefficients from (20) on it. Namely, via extensioncl,m = const. → cl,m · Sl,m (�r), the functions Sl,m (�r)must obey certain spatial [≡ orthogonal group] symme-

tries in general case according to the continuous groups SO (3,R) or O (3,R)(≡ SO (3,R)⊗

({e, I

}, ·))

(if we take into account the spatial inversion operation I , too)] or some of their discrete subgroups—(e.g.,in the case of describing of the secondary order parameters like electric polarization, magnetization etc. incrystals) invariance properties. This extension means also that the replacement t �→ ϕ (t) → (�r , t) �→ ϕ (�r , t)must be used in (20).

Secondly, if we take into account the particular general forms of basis functions allowing transformationof matrices to their Jordan normal forms, the formula (20) leads directly to the proposed generalization

ϕ(�r , t) =k∑

l=1

eλl tνl−1∑

m=0

cl,m Sl,m(�r)tm, cl,m = const. (22)

of the function system (19) [i.e., the choice νl = 1 corresponding to the case of non-multiple eigenvalueswill lead back the form of (2)]. Namely, by taking into account the general method of the decomposition ofthe product of matrices and using the so-called hyperdyads [26,27,32] on the base of (21) (after multiply-

ing it from the right-hand side by a block-diagonal matrix[γ1 I1 ⊕ · · · ⊕ γμ Iμ ⊕ · · · ⊕ γμ f Iμ f

]), we may

write:[

J g1 ⊕ · · · ⊕ J g

q ⊕ · · · ⊕ J gp

] [γ1 I1 ⊕ · · · ⊕ γμ Iμ ⊕ · · · ⊕ γμ f Iμ f

] [L(1) ⊕ · · · ⊕ L(μ) ⊕ · · · ⊕ L(μ f )

]

≡

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

J g1 0 0 . 0

0 . . . 0

0 . J gq . 0

. . . . .

0 0 0 . J gp

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

γ1 I1 . 0 . 0

0 . . . 0

0 . γμ Iμ . 0

. . . . .

0 0 0 . γμ f Iμ f

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

L(1) . 0 . 0

0 . . . 0

0 . L(μ) . 0

. . . . .

0 0 0 . L(μ f )

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(23)

where the scalar matrices γμ Iμ, μ ∈ {1, . . . , μ f

}are chosen according to Schur’s lemma, e.g., [35,36] to

commute with all matrices of the adequate irreducible representation D(μ) (GS)of the group GS . Then, on the

454 Cs. Mészáros et al.

base of the relations (A.1) and (A.2) from the “Appendix,” we may realize the following decomposition of(23) into hyperdyads with coefficients γμ of the scalar matrices as linear combination coefficients:

γ1

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

J g1

0

0

.

0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(L(1) 0 0 . 0

)+ · · · + γμ

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

.

J gq

.

0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(0 . L(μ) . 0

)+ · · · + γμ f

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

0

0

.

J gp

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(0 0 0 . L

(μ f) ). (24)

This decomposition also justifies introduction of functions (22) in a direct manner, where the relevant basisfunctions are obtained as products of those belonging to abstract subspaces carrying Jordan-block decomposi-tions of the entropy matrix (related explicitly to the time variable) and decomposition into irreducible blocksof the conductivity tensor (and related therefore explicitly to spatial coordinates). It demonstrates that dyadicdecomposition (11) has a specific character, whose generalization is realized here by novel-type partitioning ofthe matrix A = g L represented by dyadic products of rows and columns of matrix blocks instead of the earlierapplications of simple rows and columns of scalars. At writing down the basis functions of type (22), whichcorrespond to decomposition (24), the scalars γμ, μ ∈ {

1, . . . , μ f}

may be incorporated into coefficientscl,m = const. It is also assumed here (on the base of the previously mentioned application of Schur’s lemma)

that dimensions of the submatrices building up hyper-columns ( J gq ) and hyper-rows (L(μ)) fit, i.e., they have

values, which allow realization of (24) type products of block matrices.As a direct continuation of this refinement of the basic CIT formalism, an additional treatment of the

time-reversal symmetries via Onsager–Casimir relations Lik = εiεk Lki(1 ≤ i, k ≤ F f

)will be taken into

account. Here, εiεk = +1 in the case of cross-effects (i) ↔ (k) , 1 ≤ i, k ≤ F f when either purely αor purely β parameters are involved, while εiεk = −1 when in the (i) ↔ (k) , 1 ≤ i, k ≤ F f cross-effects α and β parameters are simultaneously involved and the β-parameters are quantities which changetheir signs under the influence of the time-reversal symmetry operation t → (−t) [1,2,37–40]. Althoughattempts for interconnecting the quantum-mechanical treatment of time-reversal symmetry with macroscopicmanifestations are known for decades [41] and represent a still completely open research area even nowadays,this fact (i.e., the existence of α- and β-type parameters) would make possible further refinement of descriptionof the hyperdyads in (24) by taking into account the explicit block structure of the matrix A. In order to realizethis program, we would like to point out a relatively rarely emphasized crucial but classic result of the non-equilibrium thermodynamics [40] that while the entropy function S itself is a quantity of α-character, the

entropy production is of β-type, because it is defined [1,2,12,40] as σ = ∑F fi=1 Ji Xi , where the fluxes �J =

(. . . , Ji , . . .)T , as well as the relevant conjugate thermodynamic forces �X = (. . . , Xi , . . .)

T may separatelybe of both α-, or β-type, but in conjugate pairs, they always have an opposite character. Besides, and withinframe of the strictly linear approximation of the non-equilibrium thermodynamics corresponding to strictlyconstant values of the conductivity tensor matrix elements Lik (after direct use of the well-known relationships

Ji = ∑F fk=1 Lik Xk ⇒ σ = ∑F f

i=1 Ji Xi =∑F fi,k=1 Lik Xi Xk), we arrive at a contradiction [40], because

the entropy production is of α-type in this case (this fact becomes particularly obvious if we approach theequilibrium state, which itself is invariant with respect to the time-reversal symmetry operation, i.e., it is of α-type) and not ofβ-type as it should be. (The same contradiction is valid in the case of quasi-linear approximation[40], i.e., when the matrix elements of the generalized conductivity tensor are not constants, but depend onlocal equilibrium state variables.) The well-elaborated general formalism of macroscopic reversibility due toMeixner [42] also leads to the final conclusion that entropy production is a quantity of α-type. These problemssuggest immediately that the entropy production must be treated as a quantity of pseudo-scalar [36] character ingeneral case instead of simple scalar and call for a direct application of some advanced techniques of the grouprepresentation theory in strong connection with the also well-elaborated methods of tensor products withinframe of the general multilinear algebra [29,36]. In order to justify this statement more precisely, we first recalla well-known fact from continuum mechanics [43] that the stress (or: pressure) tensor may also be representedin a dyadic product form, which corresponds to the direct product representation D(1) (G) ⊗ D(1) (G) ofthe simple rotation group [36]. Since this direct product representation can be expressed via direct sum ofirreducible representations as D(1) (G)⊗ D(1) (G) = D(0) (G)⊕ D(1) (G)⊕ D(2) (G) ; G ≡ SO (3,R), theusual decomposition of matrix of the, e.g., pressure tensor (pik) , 1 ≤ i, k ≤ 3 into equilibrium part (expressed

Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory 455

by a simple diagonal matrix), antisymmetric part (composed from three independent components whose cyclicpermutation allows introducing of an equivalent axial-vector), and symmetric part having five independent

elements as(

pi j) =

((13

∑3k=1 pkk

)δi j

)+(

pi j −p ji2

)+(

pi j +p ji2 −

(13

∑3k=1 pkk

)δi j

), we can uniquely

assign the above-listed three irreducible representations to them. This simple example illustrates how we canset up the most general form of the entropy production function. Namely, if we apply adjoint representation,which can be expressed by matrix elements of the “original” representation D (g) ≡ (

Di j (g)), g ∈ G as

D (g) ≡ (D ji

(g−1

)), g ∈ G [36,44] [with respect to the representation decomposed into direct product

of its irreducible parts as D (G) = ∑μ

⊕aμD(μ) (G)—and corresponding to the block-diagonal form of theconductivity matrix in Eqs. (21), (23)] of the actual orthogonal symmetry group, we may construct an invariant,which is de facto identical with the entropy production function. Since the latter can be explained by a bilinearform of flux (Ji )-, and the relevant conjugate thermodynamic force (Xi ) components [1,2,40] as

σ =F f∑

i=1

Ji Xi ≡R∑

j=1

J j A j + pvXv + �Jq · �Xq +K−1∑

k=1

�Jk · �X ′k + �Pva · �Xa

v +0

Pvs :0

Xsv, (25)

where R denotes the number chemical reactions, K is the number of chemical components, A j ≡− 1

T

∑Kk=1 μkνk j is chemical affinity of the j th reaction, Xv ≡ − 1

T ∇ · �vis the scalar viscous force conjugated to

the viscous pressure defined by scalar product of tensors as pv ≡ 13 Pv : E, �Xk ≡ �Fk

T − ∇ (μkT

), (1 ≤ k ≤ K )

is a polar-vector of thermodynamic force of diffusion, �Xq ≡ ∇ ( 1T

)is again a polar-vector of the thermo-

dynamic force causing heat conduction,0

Xsv ≡ − 1

T

0(∇ ◦ �v)s is the tensorial viscous force coupled to the

symmetric part0

Pvs (having trace zero) of the full, second-order viscous pressure tensor, and the axial-vectoris �Xa

v ≡ − 1T (∇ × �v − 2 �ω) conjugated to the axial-vector Pvaformed by the antisymmetric part of the general

viscous pressure tensor. (A detailed analysis of contributions of different tensorial character can be found inthe well-known, classic monographs of the topic [1,2,43]). It must also be mentioned here that in more generalcases, contributions of further various system models of electromagnetic fields are included [1,40] into (25)(their simplest variant is 1

T �ε · �i, �ε ≡ �E + �v × �B with �ias electric current density, �E electric field, and �Bmagnetic induction contributing to Lorentz-force), but they can also be subjected to classification accordingto the irreducible tensor representations of the group O (3,R) extended by time-reversal symmetry operation.In order to perform symmetry analysis of the entropy production function in detail, we apply the adjointrepresentation, mentioned previously. First of all, since direct product of an irreducible representation of agiven group with its adjoint representation (realized via mutually contravariant abstract vector-space bases)contains the unit representation only once [36], it is possible to set up invariants composed from sums of scalarproducts of coordinates of two (generally: different) vectors expressed by these bases (more precisely, byconvolution defined on abstract vectors taken from direct product of a given vector space and its contravariantcounterpart, e.g., [29]). This method is of rather general character and serves as a basis for determining possiblereal equivalent of a given representation on the base of the Schur–Frobenius theorem [45], widely applied inthe representation theory of crystallographic space groups, too [34,36,46]. According to this crucial theorem(emanating from some elementary properties of group representations adjoint to D (G), e.g., [44]), there existthree types of irreducible representations, from which the so-called second-type is equivalent to its complexconjugate representation, but cannot be brought to real form. According to Wigner’s terminology [35,44], thisrepresentation is of half-integer type and is sensitive to the presence of the time-reversal symmetry. Accord-ingly, if we have a coset decomposition in the form of G = H + s H (⇒ H � G), then for an irreduciblerepresentation�(H) of the subgroup H , we may apply the so-called *-induction procedure defined via coset-representative element sas �∗

s (h) := �∗ (s−1hs), h ∈ H (* denotes the complex conjugation operation).

Then, in the case of simple orbitals �∗s (H) ∼ �(H) ⇔ �∗

s (H) = Z ·�(H) · Z−1, where the non-singularmatrix Z obeys the relationship Z · Z∗ = cZ�

(s2)

and cZ > 0 for representation of the first-, and cZ < 0 forrepresentations of the second kind. In the case of representations of the third kind, we have orbitals consistingof two rays corresponding to �∗

s (H) �= �(H) (in the sense of non-equivalence of representations). Then,for cZ > 0, we have real representations, while for all second-type representations and conjugated pairs ofthird-type representations generally explained in the form of D(μ) (G) ⊕ D(μ)∗ (G), we may derive equiv-alent real representations [47]. Therefore, even in the case of abstract vector sub-spaces relevant for β-typeparameters, we may find real representations and the general bilinear form of the entropy production function

456 Cs. Mészáros et al.

can always be represented as a real quantity. The usual formalism relevant to such representations is knownas theory of corepresentations and—particularly by use of its relatively new variants [48]—has made possiblerefined descriptions of non-collinear magnetic systems in different type condensed matter systems, e.g., [49–51]. (The thermodynamically analogous problem of cholesteric liquid crystals—despite of use of some of themost effective variational methods of non-equilibrium thermodynamics [40] has not been solved in detail).

Therefore (if we use notation D (G) for adjoint of an “initial” group representation decomposed into its irre-ducible constituents as D (G) = �⊕

μ aμD(μ) (G)), according to the reduced form of the general conductivitytensor in (21), we will have:

D (G)⊗ D (G) = �κ

⊕D(κ) (G), (26)

(the irreducible representations D(κ) (G) on the right-hand side of (26) are all non-equivalent irreducible rep-resentations)) which can uniquely be connected to the entropy production function, if we identify the particularirreducible representations D(μ) (G)with D(0) (G) , D(1) (G) , D(2) (G) , . . . etc. (the half-integer representa-tions are also allowed in the above-discussed sense of direct sums of corepresentations to be transformed to realform). At writing down the reduced form of the Kronecker product representation in (26), we supposed thatdetailed presentation of application of the Clebsch–Gordan coefficients (e.g., [44,46]) at decomposition of theproduct representation D (G)⊗ D (G) is not necessary. This fact is now also justified by previously mentioned

unique decomposition of the entropy production function as σ = ∑F fi=1 Ji Xi ≡ (s)

σ + (v)σ + (a)

σ + (t)σ ≥ 0, as well

as by the fact, that the full rotation group O (3,R) and most of the crystallographic point groups belong tosimply reducible groups [44] (The Lemma on page 152 of Hamermesh’s book about such groups is of particularimportance from this point of view, because it fits completely to the fundamental symmetry analysis result ofthe entropy production function represented here by Eqs. (23–26). Accordingly, the Kronecker product of twointeger or two half-integer representations of such groups contains only integer representations, while the Kro-necker product of an integer and half-integer representations contains only half-integer representations.). If wetake into account the time-reversal symmetry, we may apply the technique of two-valued representations [44]for accomplishing of symmetry analysis of the entropy production function in sense of the relation (26). Allthese methods may contribute to further refinements of formalizations of derivations of the Onsager–Casimirrelations representing active research areas for decades [3,52].

Finally, we may conclude that application of the Jordan-type chains (whose highest power of the timeindependent variable is de facto determined by dimension of a submatrix appearing in block-diagonal form ofthe conductivity matrix) directly eliminates the problem of contradictory character of the entropy productionfunction, because at time-reversal operation, the resulting sign of thermodynamic state-functions [obtainedby solutions of the ODE systems of type (5–6)] will be determined by summation of different power termsof the time variable, and the dominant term may have both signs depending on the odd or even value of itsexponent and/or concrete values of the power term constants multiplying it. This is also in agreement with thefrequently emphasized crucial fact of the whole extended irreversible thermodynamics, e.g., [3] [Chapter 2.,Fig. 2.1. illustrating evolution of the entropy function of the classical entropy (described by the Maxwell–Cattaneo equation) presented together with evolution of the extended entropy function] and [53] that theentropy function of a non-equilibrium isolated system during its equilibration has a non-monotonic (or evenoscillatory character) versus time, which fact contradicts the local equilibrium hypothesis.

4 A concrete application of the developed formalism

4.1 Removal of the infinitely large propagation velocity problem by a novel-type use of special functions

As a concrete illustration of the previously explained general formalism, in the present section, we would liketo call attention of resolving the infinitely large propagation velocity problem, but without explicit use of thehyperbolic formalism. This method is based on fundamental symmetry properties of conductivity coefficientsin linear approximation of the EIT and points out relevance of thermodynamic cross-effects at eliminatingof infinitely large propagation velocities even in this approximation. It is based on an earlier paper of ours[20], where the Lommel-type special functions [54,55] have been applied in this well-elaborated branchof thermodynamics. In order to demonstrate the general character and possible further applications of ourmethod, we recall here some basic results from non-equilibrium thermodynamics. Accordingly, if we considera simultaneous diffusion in a system consisting of “K” number of components, with relative concentrations

Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory 457

ck = ck (�r , t) ≡ ρk (�r ,t)ρ , (k = 1, . . . , K − 1) obeying

∑K−1k=1 ck = 1 (with ρk (�r , t) = dMk

dV , (k = 1, . . . , K )as density of the kth component, in the absence of chemical reactions and under isothermal conditions) andwithout convective motions, the following coupled system of non-linear partial differential equations is valid:

∂ck

∂t=

K−1∑

j=1

∇ · (Dkj∇c j), (1 ≤ k ≤ K − 1), (27)

[1–3], where the diffusion coefficients (not to be confused with previously applied matrix elements ofoperators of the group elements: D (g) , g ∈ G → D (g) ≡ (

Di j (g)), 1 ≤ i, j ≤ n in a given,

finite, n-dimensional vector space) can be explained by use of the conductivity coefficients Lik as Dkj =∑K−1i=1 Lki · ∂(μi −μk )

∂ ρ j, (1 ≤ k, j ≤ K − 1). (The symbols μi , as usual, denote here the chemical potentials

of the i th component in a given, dissipative multi-component continuum). In this sub-section, we use linearapproximation of the EIT and demonstrate that thermodynamic cross-effects effectively remove the problemof infinite propagation velocities and may provide a novel mathematical method for studying deep relationsbetween the parabolic- and hyperbolic formalisms of coupled transport processes. (Simultaneously, we alsoeliminate some mistakes mainly of formal character, which remained in our earlier papers [20] and [21].)Concretely, we use here linear variant of (27) at K = 3, and consider spatially one-dimensional case, i.e.,

∂c1

∂t= D11

∂2c1

∂x2 + D12∂2c2

∂x2 ,

∂c2

∂t= D21

∂2c1

∂x2 + D22∂c2

∂x2 .

(28)

It will be assumed throughout this section that we are allowed to work in the so-called energy representation[1,2], since the temperature of the multicomponent system being investigated is permanently be assumed tobe of constant value.

4.2 Symbolic calculation simulation results

The form of the temperature and moisture-level functions is well-known from the classical literature onthermodynamics and hydrodynamics [18] and in the simplified, one-dimensional case for, e.g., concentrationfunction can also be written as [the series expansion coefficients k jn are constants, and the quantities λ jn (j =1,2) denote the reciprocal values of relaxation time constants of the nth harmonics]:

c (�r , t) ≡ c (x, y, z, t) → c(x, t) =∑

n

kncn (x) · e−λnt . (29)

Substitution of temperature and moisture-level functions also represented in this series expansion form intosystem (28) gives the following equation:

∑

n

k1n

[(D11 − D21)

d2c1n(x)

dx2 + λ1n · c1n(x)

]· e−λ1nt

=∑

n

k2n

[(D22 − D12)

d2c2n(x)

dx2 + λ2n · c2n(x)

]· e−λ2nt . (30)

Since the equation system (28) is symmetric, we assume, following our earlier basic conception [20], thatspatial harmonics of the same order of different relative concentration functions have identical functionalform, i.e.:

d2c1n(x)

dx2 + λ1n

D11 − D21c1n(x) = dc2n(x)

dx2 + λ2n

D22 − D12c2n(x) ∝ ψn(x). (31)

It is obvious that homogeneous parts of (31) represent archetypal linear harmonic oscillator equation [for thesake of simplicity, we will always assume here that diagonal diffusion coefficients (i.e., those, which have thesame indices) corresponding to “direct” material flow are always significantly larger than those corresponding

458 Cs. Mészáros et al.

to thermodynamic cross-effects (i.e., non-diagonal ones)]. For the sake of simplicity, we identify here thefunctions on their right-hand sights as simplest polynomials of the same order as the order of the relevantspatial harmonic is, i.e., ψn(x) ∝ xn, n ∈ N. In general case, the functions on the right-hand side in therelations (31) must be presented as linear combinations of such elementary polynomials. In order to simplifythe forthcoming calculations, we introduce here a unifying notation system valid for both ordinary differentialequations (ODE-s) in (31):

{c1n(x), c2n(x)} → y(x),

{λ1n

D11 − D22,

λ2n

D21 − D12

}→ k. (32)

According to this simplifying notation system, the ODEs (31) have the following solution obtained directly,via application of the MAPLE 10 symbolic computer algebra system [56]:

y(x) = K1 · cos(√

kx)

+ K2 · sin(√

kx)

− x1+n

k√

x√

k(3n + n2 + 2

) ×⎡

⎢⎣− 1(

x√

k)n (n + 2) (n + 1) cos

(x√

k)

×(

x1+n√

k cos(

x√

k)

− xn sin(

x√

k)

LommelS1(n + 1

2,

1

2, x

√k)

)

+ 1(

x√

k)n (n+2)LommelS1

(n+ 3

2,

1

2, x

√k

)(x1+n

√k sin

(x√

k)

cos(

x√

k)−xn +xn cos2

(x√

k))

+ x1+n√

k

⎛

⎜⎝n

(x√

k)n (n + 2)(cos(x

√k − 1)2 LommelS1(n + 1

2,

3

2, x

√k)

⎞

⎟⎠

− cos(

x√

k)

sin(

x√

k)⎛

⎜⎝√

x√

k + (n + 1)1 + n(

x√

k)n LommelS1

(n + 3

2,

3

2, x

√k

)⎞

⎟⎠

⎤

⎥⎦ .

(33)

Therefore, this symbolic calculation gave us directly an analytical result explained by Lommel-functions[54,55], where K1 and K2 are integration constants. In the literature about Lommel-type special functions, e.g.,[57] there are two types of them, namely LommelS1 (μ, ν, z) and LommelS2 (μ, ν, z), which are solutionsof the ODE z2 · y′′ + z · y′ + (z2 − ν2

)y = zχ+1, where “z” is an independent variable. The precise difference

between these two basic types of Lommel-functions can be simply explained by special functions, namely byν-th order Bessel-functions of the first (Jν (z))—and second (Yν (z)) kind, and �-functions [56,57] in the formof:

LommelS2 (χ, ν, z) = −2(χ−1) · �(χ + ν + 1

2

)[− sin

(χ − ν) π

2

]Jν (z)

+�(χ − ν + 1

2

)cos

(χ − ν) π

2Yν (z)+ LommelS1 (χ, ν, z) . (34)

The first two terms in (33), multiplied by constants K1 and K2, respectively, give the general solution of thewell-known free harmonic oscillator problem. Earlier, the Lommel-type special functions were very effectivelyapplied, e.g., in plasma physics [58–60], while except our own earlier papers [20,21]—at least according toour knowledge—there have not been other applications of them in the classic theories of transport processes.Moreover, another solution form of the same type of the ODE (31) can be represented [20] as

y (x) = K1 cos(√

kx)

+ K2 sin(√

kx)

+ine−i

√kx�

(n + 1,−i

√kx)

+ �(

n + 1, i√

kx) (

i3n cos(√

kx)

+ i (−i)n sin(√

kx))

2 · kn2 +1

.

(35)

Since (35) also contains incomplete �-functions, which are complicated to deal with [20], we will not use(35) further, but will prefer the solution form (33). In low-order variants (n = 2,3), the graphics of the integrals

Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory 459

leading to appearance of the Lommel-function in the solution are given on the Fig. 1a, b. Therefore, anoscillatory character of the spatial parts of relative concentration functions (which de facto directly removesthe classic infinitely large propagation velocity problem emanating from general solutions of the separate linearparabolic-type PDEs relevant for the CIT [3,61]) described in this section has been obtained by taking intoaccount the thermodynamic cross-effects and without explicit use of the earlier developed WAT [12], whichis based on application of the hyperbolic type PDE-s (and their coupled systems) instead of parabolic ones.

4.3 Application of Lagrange’s method of variation of constants

The type of the linear ODE, which has to be solved here in its most general form, may be represented by:

y′′ (x)+ f1 (x) · y′ (x)+ f2(x) · y (x) = h(x), (36)

and according to the standard Lagrange’s method of variation of constants, it can be solved directly. Lety1(x), y2(x) be two linearly independent solution functions of the homogeneous part of (36). Then, as it iswell-known on the base of the classic general theory of variation of constants of ODEs (and their systems)e.g., [62], we have the following solution:

y(x) = K1 y1(x)+ K2 y2(x)+ y2(x)∫

y1(x)h(x)

W (y1, y2)dx + y1(x)

∫y2(x)h(x)

W (y1, y2)dx, (37)

where K1, K2 are integration constants and the Wronskian of the system can be expressed by

W(

sin(√

kx), cos

(√kx))

= √k. Since in our case f1 (x) ≡ 0, f2 (x) ≡ k, the integrals appearing

on the right-hand side of (37) were calculated again by MAPLE software package directly and lead to thefollowing finite expressions:

∫xn cos

(√kx)

dx = (−1)

1 + n

(− x1+n cos

(√kx)

−√kxn+2n

(√kx)−

(32 +n

)

LommelS1

(n + 1

2,

3

2,√

kx

)sin(√

kx)

+k · x (3+n)(√

kx)−

(n+ 5

2

)

LommelS1

(n + 3

2,

1

2,√

kx

)cos

(√kx)

(38a)

−√kx (n+2)

(√kx)−

(n+ 5

2

)

LommelS1

(n + 3

2,

1

2,√

kx

)sin(√

kx)),

∫xnsin

(√kx)

dx = (−1)

n + 2

(− x1+nsin

(√kx)

+√kx (n+2)

(√kx)−

(32 +n

)

LommelS1

(n + 3

2,

3

2,√

kx

)sin

(√kx)

+k · x (3+n)(√

kx)−

(n+ 5

2

)

LommelS1

(n + 1

2,

1

2,√

kx

)(n + 1) · cos

(√kx)

(38b)

−√kx (n+2)

(√kx)−

(n+ 5

2

)

LommelS1

(n + 1

2,

1

2,√

kx

)(n + 1) · sin

(√kx)),

i.e., they are represented again by use of Lommel-type special functions. In order to illustrate the equivalence ofthe solutions explained by (33) and (37–38a,38b), instead of direct and laboursome transformation calculations,we compare them here graphically. Their similarity can be recognized immediately (Figs. 2, 3 are also drawnfor n = 2, and (similarly to Fig. 1a, b) the value k = 1 is taken for the constant in (33)).

Despite of the reversed position of the graphics on the Figs. 2 and 3 compared to the graphic given onFig. 1a, it can be recognized directly that the same inflection point appears on all Figs. 1a, 2, 3 (at the centersof intervals presented on the x-axes).

Since all of the solutions explained explicitly by expressions (33), (34), and (38a–38b) show non-monotoniccharacter with clearly visible inflexion points (and differ therefore significantly from simple Gaussian-type

460 Cs. Mészáros et al.

Fig. 1 a, b Graphics of solutions of the ODE of type (32) for n = 2 and n = 3, respectively (both graphics are represented inrelative units, and all constants K1, K2 and k are assumed to be equal to one)

Fig. 2 Graphic of solution function of the ODE (32) represented by formulae (38) and (39a–b) for values K1 = +20 and K2 =−20 of the integration constants

curves relevant for solution of separate linear parabolic-type PDEs), we may state with confidence that thebasic symmetry assumption expressed by (31) leads directly to the result, that thermodynamic cross-effectsare also responsible for effective elimination of infinitely large propagation velocities directly connected tosolutions of Fourier’s equation of heat conduction or Fick’s equation of diffusion. Furthermore, the solutionresults represented here may also serve as a basis for further refined studies of oscillatory integrals [61] beingrelated to simultaneous and coupled diffusion processes (and other type of coupled transport processes, aswell) in order to give an exact proof for non-existence of singular solutions (explained by use of Dirac-typedelta-functions) of coupled transport problems.

5 Conclusions

The general formalism of the classical irreversible thermodynamics in linear approximation is connected tothe concept of quasi-polynomials used in the general theory of differential equations and matrix analysis. It

Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory 461

Fig. 3 Graphic of solution function of the ODE (32) represented by formulae (38) and (39a–b) for values K1 = +12 and K2 =−12 of the integration constants

is demonstrated that this refinement of the existing descriptions emanates directly from the Curie principleand is based on use of the group representation theory. The hyperdyadic decomposition of the product of theentropy- and conductivity matrices leads directly to the introduction of new and very general basis functions,which naturally explain the meaning of degenerate eigenvalues of the matrix A = g L by use of the productof abstract vector spaces, with suitably chosen bases corresponding to the block-diagonal forms of them. Thenewly introduced concept of symmetry properties of the series expansion coefficients cl,m · Sl,m (�r)may refinefurther the mathematical formalism of the classic, linear irreversible thermodynamics, together with moredetailed studies of some special functions on the base of the general group representation theory. It is shownthat well-known problem of contradictory character of the entropy production function with respect to the time-reversal symmetry operation may successfully be eliminated by application of Jordan chains firstly introducedinto mathematical formalism of the non-equilibrium thermodynamics in the present study by us. A furtherdirect continuation of this work may be realized by effective use of relations describing the influence of time-reversal symmetries on the general thermodynamic relations, which may lead to the revealing of additionaldetails of the block structure of the hyperdyads applied here and connected to other methods developed indetail in order to give phenomenological foundations of the Onsager–Casimir reciprocity relations. Finally,as an illustration for the general formalism developed, a concrete example has been given via simultaneousdiffusion of two, chemically non-interacting components and the solution of the relevant coupled systemof partial differential equations is presented in a novel manner, by detailed use of the Lommel-type specialfunctions.

Appendix

Products of hyperdyads

Considering the product of matrices represented by blocks obtained on the basis of their suitable equal partitioning, i.e., thenumbers of blocks fit in the relevant rows and columns, we can form immediately the relevant products of them, which in thecase of, e.g., three matrices reads

(Aik

) (Bkl

) (Cl j

)=(

n∑

l=1

n∑

k=1

Aik Bkl Cl j

), 1 ≤ i, k, l, j ≤ n, (A.1)

which in the case of, e.g., quasi-diagonal form of the block-matrix B =(

Bkl

), (1 ≤ k, l ≤ n) → B =

(Bk

), (1 ≤ k ≤ n) can

be conveniently given by

462 Cs. Mészáros et al.

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

A11 A12 . . A1n

A21 A22 . . A2n

. . . . .

. . . . .

An1 An2 . . Ann

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

B1 0 . . 0

0 B2 . . 0

. . . . .

. . . . .

0 0 . . Bn

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

C11 C12 . . C1n

C21 C22 . . C2n

. . . . .

. . . . .

Cn1 Cn2 . . Cnn

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

A11

A21

.

.

An1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

(B1

) (C11 . . . C1n

)+ · · · +

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

An1

An2

.

.

Ann

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

(Bn

) (Cn1 . . . Cnn

).

(A.2)

The procedure may be directly applied to the case of block matrices with non-quadratic block structures, too [26].

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